Coordination of Power System Controllers for Optimal ... · system controllers. In this context,...

Coordination of Power System Controllers for Optimal Damping of Electromechanical Oscillations

by

Rudy Gianto

This thesis is presented for the degree of Doctor of Philosophy of

The University of Western Australia

Energy Systems Centre School of Electrical, Electronic and Computer Engineering

2008

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ABSTRACT This thesis is devoted to the development of new approaches for control coordination

of PSSs (power system stabilisers) and FACTS (flexible alternating current

transmission system) devices for achieving and enhancing small-disturbance stability in

multi-machine power systems. The key objectives of the research reported in the thesis

are, through optimal control coordination of PSSs and/or FACTS devices, those of

maintaining satisfactory power oscillation damping and secure system operation when

the power system is subject to persisting disturbances in the form of load demand

fluctuations and switching control. Although occurring less frequently, fault

disturbances are also considered in the assessment of the control coordination

performance.

Based on the constrained optimisation method in which the eigenvalue-based objective

function is minimised to identify the optimal parameters of power system damping

controllers, the thesis first develops a procedure for designing the control coordination

of PSSs and FACTS devices controllers. The eigenvalue-eigenvector equations

associated with the selected electromechanical modes form a set of equality

constraints in the optimisation. The key advance of the procedure is that there is no

need for any special software system for eigenvalue calculations, and the use of

sparse Jacobian matrix for forming the eigenvalue-eigenvector equations leads to the

sparsity formulation which is essential for large power systems. Inequality constraints

include those for imposing bounds on the controller parameters. Constraints which

guarantee that the modes are distinct ones are derived and incorporated in the control

coordination formulation, using the property that eigenvectors associated with distinct

modes are linearly independent. The robustness of the controllers is achieved very

directly through extending the sets of equality constraints and inequality constraints in

relation to selected eigenvalues and eigenvectors associated with the state matrices of

power systems with loading conditions and/or network configurations different from that

of the base case.

On recognising that the fixed-parameter controllers, even when designed with optimal

control coordination, have an inherent limitation which precludes optimal system

damping for each and every possible system operating condition, the second part of

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the research has a focus on adaptive control techniques and their applications to power

system controllers. In this context, the thesis reports the development of a new design

procedure for online control coordination which leads to adaptive PSSs and/or

supplementary damping controllers (SDCs) of FACTS devices for enhancing the

stability of the electromechanical modes in a multi-machine power system. The

controller parameters are adaptive to the changes in system operating condition and/or

configuration. Central to the design is the use of a neural network synthesised to give

in its output layer the optimal controller parameters adaptive to system operating

condition and configuration. A novel feature of the neural adaptive controller is that of

representing the system configuration by a reduced nodal impedance matrix which is

input to the neural network. Only power network nodes with direct connections to

generators and FACTS devices are retained in the reduced nodal impedance matrix.

The system operating condition is represented in terms of the measured generator

power loadings, which are also input to the neural network. The parallel structure of the

neural adaptive controller is amenable to its implementation by a cluster of high-

performance processors for real-time applications.

The final part of the thesis develops a new controller design procedure for addressing a

key deficiency in all of the eigenvalue-based control coordination techniques currently

used in the power industry. The deficiency is that of omitting the representation of

controller output limiters in the traditional designs. The adverse consequence of the

omission is the possible system damping impairment due to controller output

saturation. With the objective of eliminating the deficiency and avoiding controller

output saturation, the thesis develops a new method by which the controller output

limits are taken into account in the eigenvalue-based control coordination design for

achieving optimal dampings of the electromechanical oscillations for specified

disturbances. The method combines the nonlinear time-domain simulations with the

constrained optimisation of the eigenvalue-based objective function. The time-domain

simulations are used to form the relations between the maximum controllers outputs

and controllers parameters, for any specified disturbances. The relations combined

with specified controller output limits lead to additional set of constraints in the design

by which the possibility of controller output saturation is prevented. The time-domain

simulations are performed independently of one another and outside the optimisation

procedure. These features lead to lower computing time requirement and the possibility

of using parallel processors for implementing the design algorithm.

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ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my supervisor, Associate Professor T.T.

Nguyen, for providing me the opportunity to undertake this research, along with his

excellent guidance, constant support and invaluable encouragement throughout my

PhD candidature at The University of Western Australia.

I would like to thank the staff at the Energy Systems Centre for their assistance and the

use of the facilities of the centre. Thanks are also extended to all of the postgraduate

students studying at the Energy Systems Centre for their friendship, support and

encouragement.

I would like to thank my wife, Emil Merry Simanungkalit, and my daughter, Regina Prita

Masaki Hutagalung, for their love, patience and understanding during all my time spent

in carrying out the work in this research.

Finally, I would like to express my special appreciation to the scholarship granted by

Government of Indonesia (Directorate General of Higher Education, The Ministry of

National Education) through the Technological and Professional Skills Development

Sector Project, and the SIRF Scholarship provided by The University of Western

Australia.

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CONTENTS

Chapter 1 INTRODUCTION 1

1.1 Background and Scope of the Research 1

1.2 Objectives 4

1.3 Outline of the Thesis 5

1.4 Contribution of the Thesis 7

Chapter 2 DAMPING CONTROLLER DESIGN: REVIEW OF NON-ADAPTIVE METHOD 10

2.1 Introduction 10

2.2 Energy/Lyapunov Function-Based Method 11

2.2.1 Method Proposed By Machowski et al. [21,22] 11

2.2.2 Method Proposed By Lo et al. [24] 13

2.2.3 Method Proposed By Noroozian et al. [26] 15

2.2.4 Method Proposed By Ghandari et al. [25] 16

2.2.5 Method Proposed By Januszewski et al. [7] 19

2.3 Control Coordination Method 23

2.3.1 Method Proposed By Pourbeik et al [18,19] 23

2.3.2 Method Proposed By Lei et al. [11] 25

2.3.3 Method Proposed By Ramirez et al. [8] 27

2.3.4 Method Proposed By Cai et al. [5] 28

2.4 Eigenvalue-Distance Minimisation Method 30

2.5 Conclusions 32

Chapter 3 DAMPING CONTROLLER DESIGN: REVIEW OF NON-ADAPTIVE METHOD (H∞ APPROACH) 33

3.1 Introduction 33

3.2 Overview of H∞ Control Theory 33

3.2.1 H∞ Norm 33

3.2.2 Controller Design 35

3.2.3 Standard H∞ Control Problem 36

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3.2.4 LMI-Based H∞ Design 39

3.2.5 LMI-Based H∞ Design with Pole-Placement 42

3.2.6 H∞ Mixed-Sensitivity Design 43

3.2.7 H∞ Loop-Shaping Design 46

3.2.7.1 Coprime Factorisation 47

3.2.7.2 Robust Stabilisation 47

3.2.7.3 Loop-Shaping Design Procedure 50

3.3 Summary of H∞ Damping Control in Power System 51

3.4 Disadvantages of H∞ Controller 54

3.5 Conclusions 55

Chapter 4 DYNAMIC MODELING: POWER SYSTEM COMPONENTS 56

4.1 Introduction 56

4.2 Synchronous Machine Model 57

4.3 Excitation and Prime-Mover Controllers 58

4.4 PSS Model 59

4.5 FACTS Device Models 60

4.5.1 SVC Model 60

4.5.2 TCSC Model 61

4.5.3 STATCOM Model 62

4.5.4 UPFC Model 64

4.6 Supplementary Damping Controller Model 67

4.7 Load Models 68

4.7.1 Static Loads 68

4.7.2 Dynamic Loads (Induction Motors) 68

4.8 Multi-Machine Equation System 70

4.9 State Equation for Multiple FACTS Devices 72

4.10 Multi-Induction-Motor Equation System 75

4.11 Linearisation of Equations 76

4.11.1 Linearisation of Machine and PSS Equations 76

4.11.2 Linearisation of SVC State Equations 80

4.11.3 Linearisation of TCSC State Equations 81

4.11.4 Linearisation of STATCOM State Equations 81

4.11.5 Linearisation of UPFC State Equations 84

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4.11.6 Linearisation of SDC State Equations 87

4.11.7 Linearisation of Induction Motor Equations 87

4.12 Conclusions 88

Chapter 5 DYNAMIC MODELING: MULTI-MACHINE POWER SYSTEM 89

5.1 Introduction 89

5.2 Network Model 89

5.3 Multi-Machine Power System with SVCs 92

5.4 Multi-Machine Power System with TCSCs 94

5.5 Multi-Machine Power System with STATCOMs 96

5.6 Multi-Machine Power System with UPFCs 99

5.7 Linearisations of Algebraic Equations 102

5.7.1 Linearised Network Model 102

5.7.2 System Installed with SVCs 103

5.7.3 System Installed with TCSCs 105

5.7.4 System Installed with STATCOMs 106

5.7.5 System Installed with UPFCs 108

5.8 Summary of Variables and Nonlinear Equations 111

5.8.1 Summary of State and Non-State Variables 111

5.8.2 Summary of State Equations 112

5.8.3 Summary of Algebraic Equations 113

5.9 Summary of Linearised Equations 114

5.9.1 Summary of Linearised State Equations 114

5.9.2 Summary of Linearised Algebraic Equations 116

5.10 System State Matrix 117

5.11 System with FACTS Devices of Different Types 118

5.12 Conclusions 118

Chapter 6 OPTIMISATION-BASED CONTROL COORDINATION: DESIGN PROCEDURE 119

6.1 Introduction 119

6.2 Optimisation-Based Control Coordination 121

6.2.1 Objective Function and Variables 121

6.2.2 Equality Constraints 122

6.2.3 Inequality Constraints 124

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6.2.4 Alternative Objective Function 124

6.2.5 Selection of Modes for Design 125

6.2.6 Robust Controller Design 125

6.2.7 Prevention Against Convergence to the Same Eigenvalues 126

6.2.7.1 Practical Approach 126

6.2.7.2 Approach Based on Linearly Independent Eigenvectors Property 128

6.2.8 Constrained Minimisation Methods 130

6.3 Sparsity Formulation 130

6.4 Advantages of the Proposed Method 131

6.4.1 Selection of Modes in the Control Coordination 131

6.4.2 Elimination of Eigenvalue Shift Approximation 131

6.4.3 Simultaneous Coordination 132

6.4.4 Preserving the Matrix Sparse Structure 132

6.5 Nonlinear Time-Domain Simulation Method 133

6.6 Conclusions 134

Chapter 7 OPTIMISATION-BASED CONTROL COORDINATION: DESIGN RESULT AND VALIDATION 136


7.2 Multi-Machine System with TCSC 136

7.2.1 Test System and Initial Investigation 136

7.2.2 Application of PSSs 138

7.2.3 Application of PSSs and TCSC 140

7.2.4 Time-Domain Simulations 143

7.3 Multi-Machine System with UPFC 147

7.3.1 Test System and Initial Investigation 147

7.3.2 Application of PSSs 148

7.3.3 Application of PSSs and UPFC 149

7.3.4 Time-Domain Simulations 152

7.4 Conclusions 156

Chapter 8 REVIEW OF ADAPTIVE DAMPING CONTROLLERS AND WAM-BASED STABILISERS 157


8.2 Self-Tuning Controller 158

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8.2.1 Overview of Self-Tuning Controller 158

8.2.1.1 RLS Parameters Identification Method 159

8.2.1.2 Kalman Filter (KF) State Estimation 161

8.2.1.3 Kalman Filter Interpretation 163

8.2.1.4 Pole-Shifting Controller Design 163

8.2.2 Application of STC in Power Oscillation Damping 166

8.3 Neural Network-Based Controller 167

8.3.1 Overview of the Neural Network Theory 167

8.3.1.1 Architecture of the FNN 167

8.3.1.2 FNN Training Algorithm 170

8.3.1.3 Sizing of FNN 171

8.3.2 Neural Network-Based Damping Controller 172

8.4 WAM-Based Stabilisers 174

8.5 Conclusions 177

Chapter 9 NEURAL ADAPTIVE CONTROLLER DESIGN PROCEDURE 178


9.2 Representing System Configuration 179

9.2.1 Concept 179

9.2.2 Forming Reduced Nodal Impedance Matrix 180

9.3 Development of Neural Network-Based Adaptive Controller 183

9.3.1 Principle of Neural Network Application 183

9.3.2 Overall Neural Adaptive Controller Structure 186

9.3.3 Training Procedure for Neural Adaptive Controller 187

9.3.4 Neural Network Testing and Sizing 188

9.4 Conclusions 190

Chapter 10 NEURAL ADAPTIVE CONTROLLER DESIGN RESULTS AND VALIDATION 191


10.2 Test System 191

10.3 Design of the Neural Adaptive Controller 193

10.3.1 Neural Network Training and Testing Data 193

10.3.2 Training, Testing and Sizing the Neural Network 195

10.4 Dynamic Performance of the System in the Study 196

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10.5 Time-Domain Simulations 198

10.6 Possible Improvements 207

10.7 Discussion on Large Power System Application 209


Chapter 11 OPTIMAL DESIGN OF PSSs AND FACTS DEVICES SDCs WITH CONTROLLER SATURATION LIMITS 212


11.2 Representation of Controller Limit Cosntraints in the Design 215

11.2.1 Basic Concept 215

11.2.2 Formulation of the Inequality 217

11.2.3 Flowchart of the Controller Design with Saturation Limits 218

11.3 Design Result and Validation 220

11.3.1 Power System Configuration 220

11.3.2 Dynamic Performance for the Design Without Considering

Saturation Limits 221

11.3.3 Design with SDC Output Limiter 222

11.3.3.1 Effects of SDC Output Saturation 222

11.3.3.2 Dynamic Performance for the Design Considering

Saturation Limits 225

11.3.4 Design with PSSs and SDC Output Limiters 228


Chapter 12 CONCLUSIONS AND FUTURE WORK 233


12.2 Future Work 235

12.2.1 Real-Time Implementation of the Adaptive Control Coordination 235

12.2.2 Implementation of WAM-Based Stabilisers 236

REFERENCES 237

APPENDICES 248

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LIST OF FIGURES

2.1 Current injection model of the UPFC 132.2 Power injection model for FACTS device 172.3 Single generator infinite bus system with a UPFC 202.4 General feedback control 303.1 Standard feedback control system 353.2 General control configuration 373.3 A conic sector 423.4 Control system for mixed-sensitivity formulation 443.5 General representation of Fig.3.4 463.6 Control system for loop-shaping formulation 493.7 Loop-shaping design procedure 504.1 PSS control block diagram 604.2 Control block diagram of SVC 614.3 Control block diagram of TCSC 624.4 STATCOM connection and vector diagram 644.5 Control block diagram of STATCOM 644.6 UPFC block diagram 654.7 Control block diagram of UPFC 664.8 SDC control block diagram 675.1 Multi-machine power system 905.2 Multi-machine power system with SVCs 925.3 Multi-machine power system with TCSCs 945.4 Multi-machine power system with STATCOMs 965.5 Multi-machine power system with UPFCs 997.1 Two-area system with a TCSC 1377.2 Transient for the system of Fig.7.1 (without PSSs and TCSC) 1437.3 Transients for the system of Fig.7.1 (with PSSs only) 1447.4 Transients for the system of Fig.7.1 (with PSSs and TCSC) 1457.5 PSS (in G1) output transients for the system of Fig.7.1 1457.6 PSS (in G3) output transients for the system of Fig.7.1 1467.7 SDC output transients for the system of Fig.7.1 1467.8 Two-area system with a UPFC 1477.9 Transient for the system of Fig.7.8 (without stabilisers) 1527.10 Transients for the system of Fig.7.8 (with PSSs only) 153

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7.11 Transients for the system of Fig.7.8 (with PSSs and UPFC) 1547.12 PSS (in G3) output transients for the system of Fig.7.8 1557.13 PSS (in G4) output transients for the system of Fig.7.8 1557.14 SDC output transients for the system of Fig.7.8 1568.1 Self-tuning controller 1588.2 Closed-loop control system 1648.3 Single-input neuron 1688.4 Multiple-input neuron(s) 1698.5 Multilayer feedforward neural network 1709.1 Input and output structure of the neural network 1849.2 Neural adaptive controller block diagram 1869.3 Flowchart for training, testing and sizing of the neural network 18910.1 Two-area system 19210.2 Relative voltage phase angle transients for case A disturbance 19910.3 Relative voltage phase angle transients for case B disturbance 20010.4 Relative speed (G2-G1) transients for case A disturbance 20010.5 Relative speed (G2-G1) transients for case B disturbance 20110.6 PSSs gain transients for case A disturbance 20210.7 SDC gain transient for case A disturbance 20210.8 PSSs gain transients for case B disturbance 20310.9 SDC gain transient for case B disturbance 20310.10 Relative voltage phase angle transients for different time delays 20410.11 SDC gain transients for different time delays 20510.12 Relative speed (G4-G1) transients for different time delays 20610.13 Relative speed (G3-G2) transients for different time delays 20611.1 Flowchart of the control coordination taking into account the saturation

limits 21911.2 Two-area 230 kV system 22011.3 System transients (effects of SDC saturation) 22411.4 System transients for new design with SDC output limiter and ideal SDC 22711.5 Inter-area mode transients and SDC outputs 23011.6 Local mode transients and PSSs outputs 231

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LIST OF TABLES

5.1 State and non-state variables for multi-machine system 111

5.2 Algebraic equations for multi-machine system 114

7.1 Participation factor magnitudes for the system of Fig.7.1 137

7.2 Eigenvalues for uncoordinated PSSs in the system of Fig.7.1 138

7.3 PSS parameters obtained from the uncoordinated design

in the system of Fig.7.1

138

7.4 Eigenvalues for coordinated PSSs in the system of Fig.7.1 139

7.5 PSSs parameters obtained from the coordinated design


139

7.6 Limiting values of controller parameters of PSSs 139

7.7 Eigenvalues for uncoordinated PSSs and TCSC with SDC 140

7.8 TCSC and SDC parameters obtained from the uncoordinated design 141

7.9 Eigenvalues for coordinated PSSs and TCSC 141

7.10 PSSs, TCSC main controller and SDC parameters obtained from

the coordinated design

142

7.11 Limiting values of controller parameters of TCSC with SDC 142


7.13 Eigenvalues for uncoordinated PSSs in the system of Fig.7.8 148

7.14 PSSs parameters obtained from the uncoordinated design


148

7.15 Eigenvalues for coordinated PSSs in the system of Fig.7.8 149

7.16 PSSs parameters of the coordinated design in the system of Fig.7.8 149

7.17 Eigenvalues for uncoordinated PSSs and UPFC 150

7.18 UPFC main controller and SDC parameters of uncoordinated design 150

7.19 Limiting values of controller parameters of UPFC and SDC 151

7.20 Eigenvalues for coordinated PSSs and UPFC 151

7.21 PSSs, UPFC and SDC parameters of coordinated design 151

8.1 Summary of the RLS algorithm 161

8.2 Summary of the KF algorithm 162

9.1 Neural adaptive controller outputs for PSS parameters 185

9.2 Neural adaptive controller outputs for SVC parameters 185

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9.3 Neural adaptive controller outputs for TCSC parameters 185

9.4 Neural adaptive controller outputs for STATCOM parameters 185

9.5 Neural adaptive controller outputs for UPFC parameters 185

9.6 Neural adaptive controller outputs for SDC parameters 186


10.2 Line(s) outages cases 194

10.3 Variations of load and power generation 194

10.4 Dynamic performances of controllers 196

10.5 Descriptions of line(s) outage cases and disturbances 199

10.6 Range of optimal controller parameter variation for different operating

conditions and system configurations

208

11.1 Electromechanical modes with optimal controller parameters

(Design without considering the controller saturation limits)

221

11.2 Optimal controller parameters


222

11.3 Description of system disturbances 223


(Output limit of SDC considered in the design)

226



226


(Output limits of PSSs and SDC considered in the design)

228


(Output limit of PSSs and SDC considered in the design)

229

xiv

LIST OF ABBREVIATIONS

AC Alternating Current

ARE Algebraic Riccati Equation

ARI Algebraic Riccati Inequality

AVR Automatic Voltage Regulator

CDI Comprehensive Damping Index

CGBP Conjugate Gradient Backpropagation

CSC Controllable Series Capacitor

DAE Differential Algebraic Equation

FACTS Flexible AC Transmission System

FDS FACTS Device Stabiliser

FNN Feedforward Neural Network

KF Kalman Filter

LMBP Levenberg-Marquardt Backpropagation

LMI Linear Matrix Inequality

LP Linear Programming

MIMO Multi Input Multi Output

MOBP Momentum Modification Backpropagation

MSE Mean Square Error

MVA Mega Volt-Ampere

PF Power Factor

PI Proportional Integral

PMU Phasor Measurement Unit

PSS Power System Stabiliser

QBT Quadrature Boosting Transformer

RHS Right Hand Side

RLS Recursive Least Square

RNN Recurrent Neural Network

SDBP Steepest Descent Backpropagation

SDC Supplementary Damping Controller

SMES Superconducting Magnetic Energy Storage

SMIB Single Machine Infinite Bus

xv

SPFC Series Power Flow Controller

SQP Sequential Quadratic Programming

STATCOM Static Compensator

STC Self-Tuning Controller

SVC Static VAr Compensator

TCSC Thyristor Controlled Series Capacitor

TEF Transient Energy Function

TF Transfer Function

ULSNN Ultra-Large-Scale Neural Network

UPFC Unified Power Flow Controller

VLBP Variable Learning-Rate Backpropagation

VSC Voltage Source Converter

WACS Wide-Area Control System

WAM Wide-Area Measurement

LIST OF PRINCIPAL SYMBOLS

SYMBOLS USED IN CHAPTER 2 x state variable

)x(ν Lyapunov/energy function

NGEN number of generators ω machine speed VEf machine emf D damping coefficient Xd, X’d synchronous and transient d-axis generator reactances T’d0 open–circuit transient time constant

qq E,E pre-fault and post-fault q-axis synchronous emfs

ff E,E pre-fault and post-fault excitation voltages

K controller gain Xad d-axis armature reaction reactance if field current Qinj,upfc reactive-power injected by UPFC ks, kp, γs UPFC control variables V, θ voltage magnitude and phase bse, bsh series and shunt admittances of UPFC coupling

transformer P active-power in transmission line

TCSCu

TCSC0

TCSC x,x,x total, fixed and controlled reactances of TCSC

SVCu

SVC0

SVC b,b,b total, fixed and controlled shunt admittances of SVC

Vshunt, Vseries SVC and compensated line voltage magnitude m1, m2 number of SVCs and TCSCs xL line reactance ITCSC, UTCSC current and voltage of TCSC kSVC, kTCSC SVC and TCSC gains Ps, Qs active- and reactive-power injection u1, u2, uq, uc control variables of FACTS devices xc reactance of CSC r, γ quantities that determine the magnitude and phase of

UPFC series voltage

xvi

xvii

Vse UPFC series voltage ICSC, VCSC current and voltage of CSC k1, k2, k3, k4 FACTS device positive gains Xa, Xb left-hand side and right-hand side reactances of

transmission line

η transformer complex transformation ratio

BBr susceptance of shunt part of UPFC

γv, βv parameters of UPFC

δ rotor angle

Pb, Qb active- and reactive-power injections at node b

Kγ, Kβ, KBBgains of UPFC controller

GPSS, GFDS PSS and FDS transfer functions nPSS, nFDS number of PSSs and FDSs

τPSS, τFDS PSS and FDS time constants

F TF between the generator reference input to its electrical power output

λ eigenvalue

p participation factor M two-times the inertia constant w weighting coefficients vector

μf mode frequency

J quadratic performance index QJ diagonal matrix containing the relative weighting factor NQ number of state variables adopted for tuning stabilisers Pe vector of transmission line and generator active-powers NLINE number of transmission lines

λdes, λact desired and actual eigenvalues

G, H plant and stabiliser transfer function matrices I identity matrix nmod number of dominant eigenvalues

ζ damping ratio

σ, ω real and imaginary parts of eigenvalue

Eeq equality constraint Ein inequality constraint

SYMBOLS USED IN CHAPTER 3 σ maximum singular value

V unitary matrix

v1 vector of the first column elements of unitary matrix V

G plant model (transfer function)

K controller model (transfer function)

r reference input

d disturbance

n measurement noise

y plant output

ym measured plant output

u controller output signal

v controller input

P generalized plant model (transfer function)

w exogenous input

z exogenous output

x state-space variable

s Laplace transform operator

A, BB1, B2B , C1, C2

D11, D12, D21, D22

variables of the realisation of generalized plant P

AK, BK, CK, DK controller variables

xcl state-space variable of the closed-loop system

Acl, Bcl, Ccl, Dcl variables of the realisation of closed-loop system

TFzw closed-loop TF from w to z

A plant state matrix

n order of the plant state matrix

k order of the controller

D,C,B,A ˆˆˆˆ new controller variables

W1, W2 weight matrices in mixed-sensitivity control system design

Mc, Nc left coprime transfer functions

Gs shaped plant model

S1, S2 pre- and post-compensators in loop-shaping control

system design

xviii

SYMBOLS USED IN CHAPTER 4 AND 5

rΨ vector of rotor flux linkage of synchronous machine

ωr rotor angular frequency of synchronous machine

δr rotor angle of synchronous machine

Vr vector of rotor voltage of synchronous machine Pm, Pe mechanical and electrical powers H synchronous machine inertia constant

ωR synchronous speed

Am, Fm, AM, FM matrices depending on synchronous machine parameters Is vector of stator current of synchronous machine Efd synchronous machine field voltage Gss, Gsr matrices of synchronous machine parameters Gm, Sm, GM, SM constant matrices depending on synchronous machine

parameters Vs vector of stator voltage of synchronous machine Pm, Zm, PM, ZM matrices depending on synchronous machine parameters

and rotor angular frequency s Laplace transform operator xe state vector for the excitation system VPSS supplementary signal from power system stabiliser (PSS)

refsV voltage reference for excitation control system

Ae, BBe, Ce, De

AeM, BBeM, CeM, DeM

matrices depending on gains and time constants of excitation system controller

xg state vector for the prime-mover controller

ωref speed reference 0mP initial mechanical power

Ag, BBg, Cg, Dg

AgM, BBgM, CgM, DgM

matrices depending on gains and time constants of prime-mover controller

xp state vector for power system stabiliser (PSS) KPSS, TPSS, TPSS1 – TPSS4 gain and time constants of PSS Ap, Cp, ApM, CpM matrices depending on gains and time constants of PSS

controller BBc SVC susceptance XSDC supplementary signal from supplementary damping

controller (SDC) xs state vector for SVC main controller

xix

KS, TS, TS1,TS2 gain and time constants of SVC main controller

As, BBs, Cs, Ds

AsM, BBsM, CsM, DsM

matrices depending on gains and time constants of SVC

main controller

Xt TCSC reactance

xt state vector for TCSC main controller

KF, KT, TF, Tt gains and time constants of TCSC main controller

At, BBt, Ct, Dt, Et

AtM, BBtM, CtM, DtM, EtM

matrices depending on gains and time constants of TCSC

main controller

PT, Pref, Qref line active-power and active-, reactive-power reference

Vdc, refdcV DC capacitor voltage and DC voltage reference

Idc DC capacitor current

Cdc capacitance of DC capacitor

VC, IC STATCOM voltage, current

VCp, VCq p and q components of STATCOM voltage

ICp, ICq p and q components of STATCOM current

VT, refTV FACTS device terminal voltage and voltage reference

xso state vector for STATCOM main controller

KC1, KC2, Tc, TC2 gains and time constants of STATCOM main controller

Aso, BBso, Cso, Dso, Eso,

Fso, Gso, Hso, Oso, Jso,

Kso, Lso, Mso, Nso

AsoM, BBsoM, CsoM, DsoM, EsoM,

FsoM, GsoM, HsoM, OsoM, JsoM,

KsoM, LsoM, MsoM, NsoM

matrices depending on the STATCOM and its controller

droop slope of the voltage-current characteristic Ish, Ise shunt and series current

Vsh, Vse shunt and series voltage

Ishp, Ishq p and q components of shunt current

Isep, Iseq p and q components of series current

Vshp, Vshq p and q components of shunt voltage

Vsep, Vseq p and q components of series voltage

k ratio between ac and dc voltages

m1, m2 pulse width modulation (PWM) ratios for shunt and series

converters

xx

Ψ1, Ψ2 pulse width modulation (PWM) phases for shunt and

series converters

Vshp0, Vshq0 p and q components of shunt voltage initial value

Vsep0, Vseq0 p and q components of series voltage initial value refq

refp I,I p and q components of series current reference

xu state vector for UPFC main controller

Ksh1, Ksh2, Kse1, Kse2

Tsh1, Tsh2, Tse1, Tse2

gains and time constants of UPFC main controller

Au, BBu, Cu, Du, Eu, Fu, Gu,

Hu, Iu, Ju, Ku, Lu, Mu, Nu

AuM, BBuM, CuM, DuM, EuM,

FuM, GuM, HuM, IuM, JuM, KuM,

LuM, MuM, NuM

matrices depending on the UPFC and its controller

xsu state vector for SDC KSDC, TSDC, TSDC1 – TSDC4 gain and time constants of SDC Asu, Csu, AsuM, CsuM matrices depending on gains and time constants of PSS

controller YL, VL, PL, QL static load admittance, voltage, active- and reactive-power

ms

ms ,IV stator voltage and current vectors of induction motor

mM

mM

mM

mM

mm

mm

mm

mm

,,,

,,,

FAZPFAZP

matrices depending on induction motor parameters and

rotor angular speed

mrω angular speed of induction motor

Te induction motor electromagnetic torque

TL induction motor load torque mM

mM

mm

mm ,,, SGSG matrices depending on induction motor parameters

Se, Sg, Sp, SC, Ssu

SeM, SgM, SpM, SCM, SsuM

MVM, Mse, LshM, LseM

selection matrices

Δ small change notation used in the linearisation process

I, V nodal current and voltage vectors

Y system admittance matrix

IN, VN, YN real forms of I, V and Y

NB, NG number of nodes and generators

ISN, VSN vectors of nodal currents and voltages at generator nodes

xxi

VLN vectors of nodal voltages at non-generator nodes

YSS, YSL, YLS, YLL, YLC

YCL, YCC, YLU, YUL, YUU

submatrices from partitioning of Y matrix

IsM, VsM d-q components of ISN and VSN

TδM d-q frame of reference transformation matrix

NS number of SVCs

NT number of TCSCs

NC number of STATCOMs

XC reactance of STATCOM coupling transformer

ICN, VCN vectors of STATCOM currents and voltages in D-Q axes

ICM, VCM vectors of STATCOM currents and voltages in p-q axes

TαM p-q frame of reference transformation matrix

NU number of UPFCs

Xsh, Xse transformer reactance of UPFC shunt and series

converters

IshN, IseN vectors of UPFC shunt and series currents in D-Q axes

IshM, IseM vectors of UPFC shunt and series currents in p-q axes

VshN, VseN vectors of UPFC shunt and series voltages in D-Q axes

VshM, VseM vectors of UPFC shunt and series voltages in p-q axes

BBCM vector of SVC susceptances

XtM vector of TCSC reactances

rMΨ vector of rotor flux linkages

ωrM vector of rotor angular frequencies

δrM vector of rotor angles

ωRM vector of synchronous speeds

MM diagonal matrix of synchronous machine inertia constants xeM state vector for the excitation controllers xgM state vector for the prime-mover controllers xpM state vector for PSSs xsM state vector for SVC main controllers

xtM state vector for TCSC main controllers

xsoM state vector for STATCOM main controllers

xuM state vector for UPFC main controllers

xsuM state vector for SDCs msM

msM,IV stator voltage and current vectors of induction motors

xxii

mrMω vector of angular speeds of induction motors

TeM vector induction motor electromagnetic torques

TLM vector of induction motor load torques

AstM, BBstM, CstM, DstM matrices depending on STATCOMs and their controllers

AucM, BBucM, CucM, DucM, EucM,

FucM, GucM, HucM, IucM

matrices depending on UPFCs and their controllers

PeM vector of synchronous machine electrical powers PTM vector of transmission line active-powers superscripts ‘0’ initial steady-state condition

SYMBOLS USED IN CHAPTER 6 AND 7 K vector of controller parameters

m number of eigenvalues

A state matrix

λ eigenvalue

λR real part of λ

λI imaginary part of λ z eigenvector associated with λ zR real part of z zI imaginary part of z AC matrix derived from A matrix λC real matrix formed from λR and λI

zC real vector formed from the eigenvector associated with λ

U unit matrix ζ damping ratio f frequency ω angular frequency L number of stabiliser gains

a positive gain of the stabiliser

w weighting coefficient assigned to a

ci, i = 1, 2,….., m scalar coefficients in the linear combination of eigenvectors

C vector of the scalar coefficients ci’s CR, CI real, and imaginary parts of C Z eigenvectors matrix

xxiii

CC vector of CR and CI

ZR, ZI real, and imaginary parts of Z ZC real matrix formed from ZR and ZI

J1, J2, J3, J4 Jacobian submatrices

x vector of state variables w vector of non-state variables

SYMBOLS USED IN CHAPTER 8 y plant output

u plant input

hi, bi plant model parameters

ε random noise

n sampling instant kq− backward shift (or delay) operator

H, B, G polynomials that are function of delay operator z nh, nb, ng order of polynomials H, B and G gi coefficients of polynomial G

ΦΘ, parameters and regression vectors

P error covariance I identity matrix

αc constant with large value K gains vector ρf forgetting factor x state vector y observation vector wx, wy process and measurement noise vectors A, C state transition and measurement matrices Q, R process and measurement noise covariance matrices G Kalman gain Gz plant transfer function Kz controller transfer function C, D polynomials that are function of delay operator z nc, nd order of polynomials C and D P pole characteristic polynomial of the closed-loop system αs pole-shifting factor p, p neural network input a, a neural network output

xxiv

w, W neural network weight b, b neural network bias f transfer function or activation function r, r argument of the transfer function R number of input elements S number of neurons t neural network target output Q number of training cases δ vectors of neural network weights (and biases) k iteration count σ search direction α learning rate g gradient M number of neural network layers s sensitivity matrix γ momentum coefficient e, e error J Jacobian matrix v error vector S Marquardt sensitivity matrix PG synchronous machine real-power PL, QL transmission line real- and reactive-power λ eigenvalue

SYMBOLS USED IN CHAPTER 9 Y system admittance matrix I, V nodal current and voltage vectors Z nodal impedance matrix Zreduced reduced nodal impedance matrix p neural network input vector a neural network output vector f transfer function or activation function m number of input elements M number of output elements t neural network target output vector Q number of training cases x vectors of neural network weights (and biases)

xxv

SYMBOLS USED IN CHAPTER 11 K controllers parameters vector K* initial vector of the controller parameters y maximum magnitude of the controller output a vector of coefficients in the linearised relationship between

y and K n number controller parameters M number of controllers ε small change of controller parameter

K controllers parameters vector after small perturbation

L number of disturbances

xxvi

1

1.1 Background and Scope of the Research Low frequencies electromechanical oscillations, with frequencies ranging from 0.1 to

2.0 Hz, are phenomena inherent in power systems with interconnected synchronous

generators [1-4]. In the context of multi-area power systems, these electromechanical

oscillations are classified into two types: local modes and inter-area modes. The local

modes of oscillations are associated with one generator swinging against other

generators in the same area, and typically have the frequencies of 0.8 to 2.0 Hz. The

impact of these oscillations are mainly localised to one area. On the other hand, the

inter-area modes of oscillations are associated with two groups of generators in

different areas swinging against each other at the frequency in the range about 0.1 to

0.8 Hz. These oscillations are observed over a large part of the system and more

complex in comparison with the local modes [2].

Following the restructuring of the power supply industry and increased trend of

interconnecting power systems for forming or expanding the electricity markets, the

phenomena of electromechanical modes of oscillations among the interconnected

synchronous generators, particularly the inter-area modes, are a growing concern, and

their damping constitutes one of the essential criteria for secure system operation [5-8].

Given the increased significance at present of the phenomena, the thesis has the focus

on, in its scope, the development and design of optimal control techniques, including

those which are adaptive, for maintaining and enhancing the stability of the

electromechanical modes in power systems.

Power system stabiliser (PSS) has been used for many years to enhance the damping

of the electromechanical oscillations [8-15]. With increasing transmission line loading

1 IINNTTRROODDUUCCTTIIOONN

CHAPTER 1

2

over long distances, the use of PSS may not be, in some cases, effective in providing

sufficient damping for inter-area oscillations [5,8]. In such cases, other effective

alternatives are needed in addition to PSS. At present, the availability of FACTS

(Flexible AC Transmission System) devices which have been developed primarily for

active- and/or reactive-power flow and voltage control function in the transmission

system has led to their use for a secondary function of enhancing the damping of

power system oscillations [16,17].

In particular, FACTS device stabilisers (FDSs), often referred to as supplementary

damping controllers (SDCs), have been proposed to augment the main control systems

for the purpose of damping the rotor modes or inter-area modes of oscillation [18-20].

However, it has been acknowledged that, to achieve an optimal performance in terms

of small-disturbance stability improvement, the coordination among power system

damping controllers is necessary [5,10,11,18,20].

There have been numerous publications reporting or proposing methods for control

coordination design of PSSs and/or FACTS device stabilisers, particularly in off-line

environment, which lead to fixed-parameter controllers, to achieve damping

improvement of power system oscillations [5-15,18-42]. Within the context of fixed-

parameter controllers, it has been acknowledged that, with the existing methods, a

large power system will lead to difficulty in the design of simultaneous control

coordination of multiple controllers, particularly when a standard

eigenvalue/eigenvector calculation software (for example, the QR method-based

software) is used.

With the above acknowledgement, the first part of the research reported in the thesis is

devoted to the development of a new procedure for identifying simultaneously the

controller parameters to achieve optimal damping of electromechanical modes. The

new procedure directly takes into account the sparsity in the system Jacobian matrix,

and thus, offers an important advantage for large power system applications. In

addition, the procedure is entirely based on constrained optimisation, and does not

require any standard eigenvalue/eigenvector calculation software. Any limitation of the

standard eigenvalue/eigenvector software, particularly in terms of power system size, is

therefore, removed.

INTRODUCTION

3

However, It is generally accepted that, with fixed-parameter controllers, optimum

damping performance cannot be achieved for each and every operating condition or

contingency. In an attempt to address this issue, many adaptive control techniques,

where the controller parameters are determined online and adaptive to the changes in

system operating conditions, have been proposed in the open literature to overcome

the disadvantage of fixed-parameter controllers design [43-60]. Notwithstanding the

extensive research, there remain key deficiencies in the previously-proposed adaptive

control techniques, which are identified and described in the following:

(i) a need for postulating a reduced-order/approximate power system model and

online model parameter identification

(ii) a lack of simultaneous coordination of multiple controllers

(iii) the difficulty in representing simultaneously a number of electromechanical modes

of interest in the online controllers coordination

(iv) the difficulty in taking into account the variation of power system configuration in the

coordination. This is due to the combinatorial nature of the topology variation.

Given the above state-of-the-art progress in relation to the adaptive control techniques,

the second part of the research reported in the thesis has a focus on the development

of a novel adaptive control technique by which power system damping controllers

parameters are tuned online, with the deficiencies identified in (i) – (iv) of the previous

techniques completely eliminated. The novel technique presented in the thesis is based

on the establishment, using the control coordination procedure developed in the first

part of the research, of the nonlinear relationship between the optimal controller

parameters and power system configuration and operating condition. The relationship

is formed off-line, and expressed in terms of a nonlinear multi-variable vector function

the output of which is the set of optimal controller parameters, and the input is the

vector of variables representing power system configuration and operating condition.

A key and novel concept used in the technique is that of representing the variation of

power system configuration, through a series of transformation based on power system

impedance characteristics, by a vector of continuous variables. The combinatorial

problem arising from system topology variation previously encountered is completely

eliminated, with the new concept. The nonlinear vector function referred to in the above

lends itself to implementation in a multi-processor environment for real-time application

CHAPTER 1

4

by which power system controllers parameters are adaptive to the prevailing system

configuration and operating condition.

Within the scope of the thesis, the final part of the research addresses a crucial aspect

which has hitherto been omitted or discounted in the design of control coordination of

power system controllers, including those with adaptive parameters. The aspect is

related to the output limits of the power system controllers and the adverse impact of

the controller output saturation on the system damping. For the first time, a procedure

is developed in the thesis by which the eigenvalue-based control coordination is

combined with time-domain simulation to achieve in an effective and efficient manner

the inclusion of controller output limit constraints in the design of power system

controllers and their coordination. The procedure developed ensures that the controller

outputs will not exceed their limits, and the system damping will not be impaired due to

controllers output saturation.

1.2 Objectives Given the scope of the research described in Section 1.1, the thesis has the following

objectives:

(a) Developing a new procedure for control coordination design of PSSs and FACTS

devices together with their SDCs in multi-machine power systems which identifies

the optimal controller parameters in a simultaneous manner. The new design

procedure will draw on the constrained optimisation in which the sparsity of the

power system Jacobian matrix is preserved. With the new procedure, state

matrices with high dimension for large power systems do not present any difficulty.

(b) Developing a novel design procedure for online tuning and control coordination of

PSSs and FACTS devices for maintaining and enhancing the stability of the

electromechanical modes in multi-machine power systems. The controller

parameters will be adaptive to the changes in system operating condition and/or

configuration. The adaptivity will be achieved by a neural network-based controller

synthesised to give the nonlinear mapping between optimal power system

controllers parameters and system configuration together with operating condition.

INTRODUCTION

5

All of the variables in the mapping are continuous ones, and the combinatorial

problem caused by system topology variation does not arise in the new procedure.

(c) Developing a new method for the control coordination of PSSs and FACTS devices

for achieving optimal dampings of the electromechanical oscillations, where the

controller output limits are taken into account in the design for any specified

disturbances. The design is applicable to multiple controllers and any number of

electromechanical modes specified together with their damping requirements.

1.3 Outline of the Thesis The thesis is organised in twelve main chapters. Starting with the background and

scope of the research, the first chapter presents the objectives, outline and

contributions of the thesis.

Chapter 2 presents and discusses an overview of the previous works published in the

area of design methods for the PSSs and/or FDSs to achieve electromechanical mode

damping improvement in a power system. Apart from the H∞ approach, the non-

adaptive design methods for power system controllers is presented and discussed. The

key disadvantages or deficiencies of the design methods are also identified in this

chapter.

Chapter 3 discusses the previously-published methods of designing H∞ controller in

damping power system oscillation. The review presented in this chapter covers two

aspects related to the robust controller design. The first aspect is the examination of a

range of linear controller design techniques, based on the H∞ control methodology. The

second aspect in the review is the applications of the H∞-based technique to power

systems damping controller designs. The key disadvantages with the application of the

H∞ controller in the damping of power system oscillations are also presented in this

chapter.

In Chapter 4 the dynamic models for individual items of power system are presented

and discussed. The equations that describe the models are, in general, nonlinear. The

CHAPTER 1

6

linearised forms of the equations valid for small-signal stability analysis are then

derived and given in this chapter.

Chapter 5 develops a composite set of state equations and algebraic equations which

describes the dynamics of the complete power system in relation to its

electromechanical responses. The composite equations set forms the basis for stability

analysis and coordination of power system damping controllers which will be developed

in the subsequent chapters.

Chapter 6 is devoted to the development of a new procedure for optimal control

coordination design of multiple PSSs and FACTS devices in a multi-machine power

system. The coordination procedure proposed draws on constrained optimisation in

which the eigenvalue-based objective function is minimised to identify the optimal

controller parameters. A key feature is that of including the eigenvalue-eigenvector

equations in the set of constraints in the optimisation. The advantages of the new

control coordination design method, over other previous methods, are also given and

discussed in this chapter.

Chapter 7 applies the new control coordination design procedure developed in Chapter

6 to representative power systems. The design procedure is applied to multi-machine

power systems in which FACTS devices are installed. Verification by both eigenvalues

calculations and time-domain simulations is carried out to confirm the effectiveness of

the procedure in providing substantial electromechanical mode damping improvement.

Chapter 8 presents an overview of the previous works published in the area of adaptive

damping controller designs. The widely-proposed methods, i.e. self-tuning controllers

and neural network-based controllers, are reviewed. An overview of WAM (wide-area

measurement)-based controllers is also given in this chapter. The comprehensive

review identifies the key disadvantages or deficiencies of the previously-proposed

adaptive schemes and the WAM-based controllers which depend on the remote signals

for their operation.

Chapter 9 develops a new adaptive control algorithm and procedure for online tuning of

the PSSs and SDCs of FACTS devices. The procedure is based on the use of a neural

network which adjusts the parameters of the controllers to achieve system stability and

INTRODUCTION

7

maintain optimal dampings as the system operating condition and/or configuration

change. The new procedure addresses the issues or difficulties encountered in the

previous adaptive schemes reviewed in Chapter 8.

Chapter 10 discusses the design results and validation of the proposed neural adaptive

controller described in Chapter 9. Application of the proposed method to power

systems having PSSs and FACTS device is investigated and discussed in this chapter

in relation to the effectiveness of the design procedure proposed in achieving the

enhancement in system oscillation damping.

Chapter 11 develops, for the first time, an effective and efficient design procedure for

the control coordination of PSSs and SDCs of FACTS devices in which the

dependence on the controller parameters of both the dampings of electromechanical

modes and controller maximum outputs are taken into account. In this way, the

controller output limiters are represented directly and effectively in the eigenvalue-

based control coordination. Case studies are also presented in this chapter to verify the

correctness of the design procedure, and quantify the improvements in terms of

damping performance over the traditional controller designs which do not represent

controller output limiters.

The overall conclusion in Chapter 12 summarises the main features and advances of

the research reported in the thesis. Future research work is also suggested and

included in the chapter.

1.4 Contributions of the Thesis The thesis has made four original contributions as described in the following:

(a) Development of a new eigenvalue-based control coordination design of multiple

PSSs and FACTS devices together with their SDCs for optimal oscillation damping

in a multi-machine power system. The coordination procedure proposed draws on

constrained optimisation in which the eigenvalue-based objective function is

minimised to identify the optimal controller parameters in a simultaneous manner.

By representing the relationship among eigenvalues, eigenvectors and system

state matrix as equality constraints in the optimisation, a key advance is made in

CHAPTER 1

8

the research by which the need for any special software system for eigenvalue

calculations does not arise. The limitation of eigenvalue calculation software

systems in terms of the size of the state matrix is, therefore, removed. In addition,

the proposed algorithm does not require separate and time-consuming eigenvalue

calculations at each iteration during the control coordination. Eigenvalues together

with optimal controller parameters are available simultaneously at the convergence

of the optimisation process.

(b) A second contribution made in the development of the control coordination design

procedure referred to in (a) is that of preserving fully the sparsity of the power

system Jacobian matrix in the optimisation process. This is achieved by using, in

the constrained optimisation, the eigenvalue-eigenvector equations without forming

explicitly the system state matrix. The coefficient matrix in the equations retains the

sparsity of the Jacobian matrix. The contribution represents a key advance and an

important feature, particularly in the context of control coordination in a large power

system.

(c) Development of a new adaptive control coordination method based on the use of a

neural network which identifies online the optimal controller parameters of PSSs

and FACTS devices together with their SDCs. Online tuning and coordination of

multiple controllers is very important for achieving system stability and maintaining

optimal dampings as the system operating condition and/or configuration change. A

particular contribution of the method is that of representing the power system

configuration and its variation by a reduced nodal impedance matrix. This allows

any variation of system configuration to be formed in terms of continuous variables

derived from the nodal impedance matrix, and input to the neural adaptive

controller.

(d) Development of a new method for control coordination design of PSSs and FACTS

devices with SDCs where the controller output limits are represented in the design.

The combination of eigenvalue-based technique with nonlinear time-domain

simulations is used for achieving the design objective. A key feature of the new

method is that the time-domain simulations which are used for forming the

controller output limit constraints are performed outside the eigenvalue-based

control coordination loop. The interface between the eigenvalue-based control

INTRODUCTION

9

coordination and the time-domain simulations for any specified disturbances is the

set of inequality constraints in an algebraic form expressed in terms of controllers

parameters, which represent the controller output limiters. The number of variables

is, therefore, not increased in the eigenvalue-based control coordination. The

design procedure is amenable to implementation by parallel computing system for

reducing the computing time required.

The thesis is supported by four publications as follows:

1. Nguyen, T.T., and Gianto, R.: ‘Application of optimization method for control co-

ordination of PSSs and FACTS devices to enhance small-disturbance stability’.

Proceedings of the IEEE PES 2005/2006 Transmission and Distribution

Conference & Exposition, Dallas-Texas, May 2006, pp. 1478-1485.

2. Nguyen, T.T., and Gianto, R.: ‘Stability improvement of electromechanical

oscillations by control co-ordination of PSSs and FACTS devices in multi-machine

systems’. Proceedings of the IEEE PES General Meeting 2007, Tampa-Florida,

June 2007, pp. 1-7.

3. Nguyen, T.T., and Gianto, R.: ‘Optimisation-based control co-ordination of PSSs

and FACTS devices for optimal oscillations damping in multimachine power

system’, IET Generation Transmission and Distribution, 2007, 1, (4), pp.564-573.

4. Nguyen, T.T., and Gianto, R.: ‘Neural networks for adaptive control coordination of

PSSs and FACTS devices in multimachine power system’, IET Generation

Transmission and Distribution, 2008, 2, (3), pp.355-372.

For reference, copies of the above four publications are given in Appendix J.

10

2.1 Introduction This chapter discusses an overview on the previous works published in the area of

design methods of the Power System Stabilisers (PSSs) and/or Flexible AC

Transmission System Device Stabilisers (FDSs) to achieve electromechanical mode

damping improvement in a power system. Off-line based-design which leads to non-

adaptive (fixed-parameter) controller design will be reviewed in Chapters 2 and 3 of this

thesis. The previously published methods of non-adaptive controller approaches can

be classified into the following categories.

In the first category, the control strategy based on Lyapunov function or energy function

[6,7,21-25] has been proposed. This method offers robust and decentralised control

structure. In parallel with the work using Lyapunov functions, active research on control

coordination has been carried out [5,8,11,18,19]. In the control coordination methods,

which belong to the second category, parameters of all damping controllers are

identified in a coordinated manner to achieve optimal damping of electromechanical

modes.

In addition to the research published regarding the above categories, there have been

publications reporting the application of the eigenvalue-distance minimisation technique

[2,26] and also there have been extensive research and numerous publications

reporting the applications of the H∞ controller design method in the power system

damping improvement. The overview of the H∞ control-based method will be presented

in Chapter 3.

2 DDAAMMPPIINNGG CCOONNTTRROOLLLLEERR DDEESSIIGGNN:: RREEVVIIEEWW OOFF NNOONN--AADDAAPPTTIIVVEE MMEETTHHOODD

DAMPING CONTROLLER DESIGN: REVIEW OF NON-ADAPTIVE METHOD

2.2 Energy/Lyapunov Function-Based Method Lyapunov method has traditionally been used in power system analysis to assess the

stability and determine the stability margins [61]. It has been shown in [6,7,21-25] that

the Lyapunov method can also be used for determining control strategies for power

system damping controllers (PSSs or FDSs). In such applications, the Lyapunov

method is employed to form the control laws for achieving the desired control objective.

For a dynamic system described by a set of nonlinear differential equations of the form

, the Lyapunov stability theorem states that the post-fault equilibrium point is

stable if there is a Lyapunov function

)x(Fx =•

x

)x(ν such that: (i) the function is positive definite

and has a minimum value at , and (ii) the function time-derivative is negative semi-

definite along the trajectory x(t), i.e. [21,22]. It is important to note that the more

negative the value of the faster the system returns to the post-equilibrium point.

x

0≤ν•

•ν

The above explanations show that, in the Lyapunov-based design method, for a control

strategy to be optimal it has to be designed to maximize the negative value of at

each instant of time. Therefore, in Lyapunov-based approach, the steps for design

procedure are usually as follows [21,22]: (i) find the Lyapunov function which is an

explicit function of the control variable, (ii) select the control laws that maximize the

negative value of . The key points of the previous published methods for damping

controller design based on energy/Lyapunov function are discussed in the following.

•ν

)x(ν

•ν

2.2.1 Method Proposed By Machowski et al. [21,22] In [21,22], an approach to the design of power system stabilisers (PSSs) based on the

application of Lyapunov’s direct method has been developed. In the design, the third-

order generator model has been used for forming the control strategies. The Lyapunov

function is expressed as the sum of the system kinetic energy, potential energy and a

term proportional to the sum of squared deviation of the transient emf for all machines.

It has been shown in [21,22] that the derivative of the Lyapunov function can be

expressed as:

11

CHAPTER 2

( )∑ +−−

∑ −ωΔ−=ν=

•

=

• GENGEN n

1iEf

2qiqi

didii0d

n

1i

2ii VEE

)'XX(1

'T1D (2.1)

where:

( )( )∑ −−−

−==

• GENn

1iqiqififi

didii0dEf EEEE

)'XX(1

'T1V (2.2)

where NGEN is the number of generators in the system, D is the damping coefficient, Δω

is the speed deviation, Xdi and X’di are the synchronous and transient d-axis generator

reactance respectively, Eq is the q-axis synchronous emf, T’d0 is the open-circuit

transient time constant and Ef is the excitation voltage. A “hat” on the top of a symbol

corresponds to the post-fault equilibrium point.

The first and the second components in (2.1) are negative semi-definite and always

contribute to the overall system damping. The third component is given by (2.2) and is

influenced by the excitation control. In order to optimally damp the oscillations, the RHS

of (2.2) must be negative maximum at any instant. This can be achieved if each of the

components of is positive maximum at any instant. Therefore, in

order to develop the relevant control strategy, the following control law was proposed

[21]:

)EE)(EE( qiqififi −−

(2.3) GENqiqiififi N,1,2,i ; )EE(K)EE( L=−=−

where K is the controller gain.

The control law defined by (2.3) can be rewritten as:

GENfadiiref,qifi N,1,2,i ; )t(iXKE)t(E L=Δ−= (2.4)

where:

(2.5) GENref,fififi

ref,fiadiref,qi N,,2 ,1i i)t(i)t(i

iXEL=

⎪⎭

⎪⎬⎫

−=Δ

=

12


13

where Xad is the d-axis armature reaction reactance and if is the field current.

In [21], it has been proposed to execute the control strategy (2.4) by using two different

control structures: (i) a hierarchical structure in which the automatic voltage regulator

(AVR) is the master controller and the PSS is the slave controller, and (ii) a traditional

structure in which the PSS constitutes a supplementary loop to the main AVR.

Although the controllers as proposed in [21,22] are shown to be robust, it has been

mentioned also in [21,23] that there is a limit in choosing the proposed controller gain

where beyond this value the damping will be reduced. This disadvantage may be due

to the simplifications of the power system model used in the derivation of the

theoretical control law [21].

2.2.2 Method Proposed By Lo et al. [24] In [24], a damping control strategy for a Flexible AC Transmission System (FACTS)

device, i.e. unified power flow controller (UPFC), has been proposed to damp the low

frequency of electromechanical oscillations. The control strategy is based on the time-

domain analysis of the system transient energy function (TEF). A UPFC current

injection model as shown in Fig.2.1 has been used in [24] to develop the time-domain

transient simulation for tuning controller parameters and demonstrating the

effectiveness of the control strategy proposed.

Fig.2.1: Current injection model of the UPFC

In order to damp the electromechanical oscillations, the value of the energy function

must be a decreasing time function. Therefore, in [24], it has been proposed to control

the UPFC in a way that the value of the TEF is decreasing with respect to time,

upfc,injjI

upfc,injiI

UPFC jjV θ∠ iiV θ∠

CHAPTER 2

namely . It has been shown in [24] that, in order to achieve the desired damping,

the UPFC must be able to adjust the related control variables so that the following

constraint is satisfied:

0≤ν•

[ ] 0Qdtd upfc,inj ≤ (2.6)

where is the reactive-power injected into the power system by the UPFC and

given by [24]:

upfc,injQ

[ ] [ ])1k(b)V(cos)V(bk)cos(VVbkQ psh2

is2

isessijjisesupfc,inj −+γ+γ+θ−= (2.7)

where ks, kp, and γs are the UPFC control variables which determine the series and

shunt converter output voltages; Vi and Vj are the voltage magnitude at nodes i and j

respectively; θij is the relative voltage phase angle between nodes i and j; bse and bsh

are the UPFC coupling transformer series and shunt admittances respectively.

Based on the control criterion stated in (2.6) and the energy function in (2.7), the

following relationships must be satisfied, in order to achieve the damping effects:

[ ] 0)(dtd)sin(VVbk ijsijjises ≤θγ+θ (2.8)

[ ] 0dt

dVV1

dtdV

V1)cos(VVbk j

j

i

isijjises ≤⎟

⎟⎠

⎞⎜⎜⎝

⎛−γ+θ (2.9)

[ ] 0dt

dVV)1k(b2bk2 iipshse

2s ≤−+ (2.10)

[ ] 0dt

dk)cos(VVbVbk2cosVb ssijjise

2isess

2ise ≤γ+θ−+γ (2.11)

[ ] 0dt

dsinVbk)sin(VVbk ss

2isessijjises ≤

γγ−γ+θ (2.12)

0dt

dkVb p2

ish ≤ (2.13)

14


Based on (2.8) – (2.13), the following three control schemes have been proposed in

[24] to achieve the damping effects:

- a PI controller with as the input signal and imaginary part of the UPFC

series converter voltage as the output signal (where P

ijP•

ij is the active-power in

the transmission line flowing from bus i to bus j)

- a PI controller with )]dt/dV)(V/1()dt/dV)(V/1[( jjii − as the input signal and real

part of the UPFC series converter voltage as the output signal

- a PI controller with as the input signal and Qdt/dVi shunt as the output signal

(Qshunt is the reactive-power injection).

In [24], a transient simulation program with the assistance of the user-defined output

environment has been used to obtain the best value of the parameters of the controller.

However, as mentioned also in [24], the best tuning of the controller can only be

achieved by using the mathematical optimisation methods. The controller tuning by

exercising the transient simulation results may not achieve the best control results.

2.2.3 Method Proposed By Noroozian et al. [6] In [6], the control strategies based on energy function methods for damping of

electromechanical oscillations have been developed. It has been shown in [6] that the

damping effect of a TCSC and an SVC can be added using the control strategies

developed for these devices. The control strategies are formed by incorporating the

controller models in an energy function, and then the time derivative of the energy

function is determined to form the control laws.

In [6], the TCSC has been modeled as a series combination of a fixed

reactance and a controlled part , whereas the SVC is modeled as a parallel

combination of a fixed shunt admittance and a controlled part . It has been

shown in [6] that, for m

TCSC0x TCSC

ux

SVC0b SVC

ub

1 SVCs and m2 TCSCs in a power system, the derivative of the

energy function can be expressed as follows:

∑ ∑−−=ν• 1 2m

1

m

1

2series

TCSCu

2shunt

SVCu V

dtdx

21V

dtdb

21 (2.14)

15

CHAPTER 2

where is the shunt admittance of the SVC; is the reactance of the TCSC;

V

SVCub TCSC

ux

shunt is the voltage magnitude across the SVC, and Vseries is the magnitude of the

voltage across the compensated line. This voltage magnitude is given by:

(2.15) TCSCLTCSC2series UxIV +=

where xL is the line reactance; ITCSC is the current through the TCSC, and UTCSC is the

voltage across TCSC.

By satisfying in (2.14), the following control laws have been suggested in [6] to

damp the electromechanical power oscillations:

0≤υ•

Control law for SVC:

⎪⎩

⎪⎨

⎧

<<

>=

SVCmax

SVCSVCmin

SVC2shuntSVC

SVCu

bbb

0k ; Vdtdkb

(2.16)

Control law for TCSC:

⎪⎩

⎪⎨

⎧

<<

>=

TCSCmax

TCSCTCSCmin

TCSC2seriesTCSC

TCSCu

xxx

0k ; Vdtdkx

(2.17)

where is the total parallel combination of and is the total series

combination of and

SVCb SVC0b SVC

ub ; TCSCx

TCSC0x TCSC

ux .

Although the proposed control strategies in [6] is robust with respect to loading

condition, fault location and network structure, each damping controller only contributes

individually to the power swing damping without any coordination with other controllers.

2.2.4 Method Proposed By Ghandhari et al. [25] Similar to the approach in [6], a control strategy for the FACTS devices based on the

energy function or Lyapunov function has also been developed in [25]. Input signals

16


and control laws for the FACTS devices, such as Unified Power Flow Controller

(UPFC), Controllable Series Capacitor (CSC) and Quadrature Boosting Transformer

(QBT), are discussed and derived in the paper.

It has been proposed in [25] to use energy (or energy-like) functions as the Lyapunov

function candidates. These Lyapunov function candidates are then used in the

feedback control design by making the Lyapunov function derivative negative when

choosing the control. Based on this theory, the control laws for the FACTS devices are

then formed to damp the electromechanical power oscillations.

In deriving the control laws, the model referred to as power injection model has been

used in [25] to represent the FACTS devices. Fig.2.2 shows the power injection model

for a FACTS device which is located between bus i and bus j. For UPFC and QBT, xs is

the effective reactance seen from the transmission line side of the series transformer,

and for CSC, it is the reactance of the line where the CSC is installed.

jjV θ∠ iiV θ∠

jxs

Psi+jQsi Psj+jQsj

Fig.2.2: Power injection model for FACTS device

It can be seen from Fig.2.2 that the nodal powers for the FACTS devices can be written

as follows [25]:

UPFC:

( )

(⎪⎩

⎪⎨⎧

θ−θ−=−=

=θ+θ=

ij2ij1jissjsisj

2is1siij2ij1jissi

sinucosuVVbQ ; PP

VbuQ ; cosusinuVVbP) (2.18)

QBT:

(2.19) ⎪⎩

⎪⎨⎧

θ=−=

θ+=θ=

ijjisqsjsisj

ijjisq2isqsiijjisqsi

sinVVbuQ ; PP

sinVVbuVbuQ ; cosVVbuP

17

CHAPTER 2

CSC:

( )

( )⎪⎩

⎪⎨⎧

θ−=−=

θ−=θ=

ijji2jscsjsisj

ijji2iscsiijjiscsi

cosVVVbuQ ; PP

cosVVVbuQ ; sinVVbuP (2.20)

In (2.18) – (2.20), bs, u1, u2, uq and uc are given by:

cL

ccq21

ss xx

xu ; sinruu ; cosru ; x1b

−=γ==γ== (2.21)

where u1, u2, uq and uc are the control variables; xL is the line reactance where the

FACTS device is installed; xc is the reactance of CSC; r and γ are the quantities that

determine the magnitude and phase of UPFC series voltage (Vse), i.e. γ= jise erVV .

It has been shown in [25] that the time derivative of the energy function can be written

as:

j

jsj

i

isiijsi V

VQVVQP

••••

−−θ−=ν (2.22)

By using (2.22), the time derivatives of the energy function for the various FACTS

devices are as follows.

Time derivative of the energy function for UPFC:

( ) ( ⎥⎦⎤

⎢⎣⎡ θ+θ−−=ν

•

ijj2ijji1isUPFC sinVdtducosVV

dtduVb ) (2.23)

Time derivative of the energy function for QBT:

( ijjiqsQBT sinVVdtdub θ−=ν

• ) (2.24)

Time derivative of the energy function for CSC:

18


[ 2CSCCSCLcsCSC VIx

dtdub

21

−−=ν•

] (2.25)

where Icsc and Vcsc are the CSC current and voltage respectively.

By making the time derivative of energy function negative, the following control

(feedback) laws are proposed in [25] to damp the electromechanical power oscillations:

Control law for the UPFC:

( ) ijj22ijji11 θsinVdtdku ; θcosVV

dtdku =−= (2.26)

Control law for the QBT:

( ijji3q θsinVVdtdku = ) (2.27)

Control law for the CSC:

[ 2csccscL4

2ij4c VIx

dtdkV

dtdku −== ] (2.28)

where k1, k2, k3 and k4 are the positive gains which are chosen individually to obtain

appropriate dampings.

Although the Lyapunov function-based method offers robust and decentralised control

structure, some issues have been identified in [25] for further research. One of them is

the inclusion of detailed dynamic models for synchronous generators and loads, and

transmission system with losses. The other is related to the effects of modeling on the

control laws.

2.2.5 Method Proposed By Januszewski et al. [7] In [7], an approach based on the use of the nonlinear system model and application of

the direct Lyapunov method to improve damping of power swings using the UPFC has

19

CHAPTER 2

been proposed. A state-variable control strategy has been derived using locally

available signals of active- and reactive-power.

In order to derive the control strategy, it has been assumed in [7] that the UPFC is

installed at node b (see Fig.2.3a). The series part of the UPFC is modeled by the series

reactance included in the reactance of the left-hand side of the transmission line Xa

(see Fig.2.3b) and by an ideal transformer in series with the line with complex

transformation ratio given by:

b

a

UU

=η (2.29)

The shunt part of the UPFC is modeled as controlled shunt susceptance Br (Fig.2.3b).

g

20

Fig.2.3: Single generator infinite bus system with a UPFC

(a) One-Line Diagram

(b) Equivalent Circuit

On discounting the network resistance, and using the simplified machine model, the

energy function is formed in [7] from the kinetic energy and potential energy of a single

Us Ub Ua Eg

η

Br

Xb Xa s b a g

●●●●

s b

●

a

(a)

(b)


generator system. It has been shown in [7] that the time derivative of the energy

function for the single generator infinite bus system in Fig.2.3 can be written as:

(2.30) ωΔδ+ωΔδ−γ+

ωΔδ−β−ωΔ−=ν

∑∑

∑

•

sinbXBcosb)BX1( sinb)BX1(D

SHCrrSHCv

rSHCv2

where γv and βv are the UPFC parameters used for forming the series converter voltage

in terms of the voltage at busbar b injected by the transformer in Fig.2.3b; δ is the

power (rotor) angle and Δω is the speed deviation. In deriving the control laws, γv, βv

and Br are chosen as the control variables [7].

In (2.30), bΣ and XSHC are defined by:

b

2a X||X

1bη+

=∑ (2.31)

b

2a

baSHC X||X

XXXη+

= (2.32)

Equation (2.30) shows that each control variable γv, βv or BBr can contribute to the power

system damping by increasing the negative value of . It is noted that the product

X

•ν

SHCBrB is very small as Xr >> XSHC (where Xr = 1/BBr). Consequently, the factor (1-XSHCBrB )

≈ 1 has no influence on the sign of the first or second component in (2.30).

In order to contribute to the negative value of , the control strategy should ensure that

all of the components in (2.30) are negative, independently of the sign of power angle δ

and speed deviation Δω. This can be achieved if the following equations are satisfied:

•ν

ωΔδ−=γ ∑γ ]cosb[Kv (2.33)

ωΔδ+=β ∑β ]sinb[Kv (2.34)

ωΔδ−= ∑ ]sinb[KB Br (2.35)

where Kγ, Kβ and KB are positive coefficients. B

21

CHAPTER 2

It can be seen that, the control strategy as described in (2.33) – (2.35) uses the state

variables δ and Δω and valid for a single machine system. It does not appear that the

control strategy has been derived for a multi-machine power system, using input state

variables (machine rotor angles and speed deviations).

As mentioned in [7], the implementation of this state-variable control in a real multi-

machine system requires the estimation of all of the state variables (rotor angles and

speed deviations of all of the generators). This is a very complicated problem which

requires reliable wide-area measurements and communication channels. However,

even in the case of a single machine system, communication channels are also

required unless the UPFC is located at the machine terminal.

Therefore, in [7], it has been proposed to use a modified control strategy which is

based on the local signals. It is shown in [7] that the control strategy given by (2.33) –

(2.35), can be replaced (with good accuracy) by the following strategy using the signals

local to the UPFC:

dt

dPK bv γ+≅γ (2.36)

dt

dQKdt

dQKK b

)Q(b

Xv β

β +=+≅β (2.37)

dt

dQKdt

dQKKB b

)Q(Bb

X

Br =−≅ (2.38)

where Pb and Qb are the active- and reactive-power injections at busbar b respectively

(see Fig.2.3a), and KX is determined by the system reactances and the ratio η of the

transformer in Fig.2.3b.

The control strategy in (2.36) – (2.38) is implemented by a supplementary damping

controller based on differentiators with small time constants for reducing noises from

differentiator operations.

The main disadvantage of the method proposed in [7] is that of the difficulty in choosing

the controller gains Kγ, Kβ and KB [23]. As mentioned in [23], this difficulty is linked to

the fact that the controller gains used have to be constrained when adopting the control

B

22


23

strategy based on the local signals. For high values of the gains, the control system

becomes oscillatory unstable [23]. Another disadvantage is that the approximate

control law in (2.36) – (2.37) has been derived from the single-machine control strategy

in (2.33) – (2.35). The effects of adopting (2.36) – (2.38) in the multi-machine case are,

in general, not known.

2.3 Control Coordination Methods There has been extensive research in the design methods for control coordination of

PSSs and/or FACTS devices in the context of small-disturbance stability enhancement

[5,8,11,18,19]. In the control coordination methods, detailed dynamic models for

generators and loads can be represented directly. Parameters of all of the

supplementary controllers are identified in a coordinated manner to achieve optimal

damping of electromechanical modes.

In general, the coordination is eigenvalue-based in the context of multiple FACTS

controllers and/or PSSs and detailed representation for the power system. The

previous published methods for control coordination of PSSs and/or FACTS device

stabilizers are reviewed in the following.

2.3.1 Method Proposed By Pourbeik et al. [18,19] In [18,19], a scheme for simultaneous coordination of PSSs and FACTS device

stabilizers (FDSs) based on linear programming and eigenvalue analysis has been

developed. Central to the scheme is the approximation by which the shifts in

eigenvalues are formed as linear functions of the changes in stabilisers gains.

In the paper [18], a two-stage method for coordinating the gains of all stabilisers has

been proposed. The first stage is the determination of the transfer functions (TFs) of

stabilisers to provide appropriate phase compensation and ensure a left-shift in the

rotor modes of oscillation. The second stage is the solution of a linear programming

problem to calculate the minimum values of stabiliser gains to satisfy specified left-

shifts in the modes of rotor oscillation. It has also been proposed that for the stabiliser

TF, kiGi(s), the transfer function Gi(s) will ensure a left-shift in the mode(s) of rotor

oscillation, while the gain ki will be responsible for the extent of the left-shift [18].

CHAPTER 2

In the first stage of the stabilisers coordination discussed in [18], the transfer functions

for PSSs and FDSs are proposed to take the following forms:

PSSi,PSSmi,PSS1i,PSS

i,PSS

iii,PSS n1,2,...,i ;

)sτ1()sτ1(1

sτ1sτ

)s(F1)s(G =

+××++=

L (2.39)

FDS

p

j,FDS2

j,FDS1

j,FDS

j,FDSj,FDS n1,2,...,j ;

sτ1sτ1

sτ1sτ

)s(Gj

=⎟⎟⎠

⎞⎜⎜⎝

⎛

+

+

+= (2.40)

where GPSS(s) and GFDS(s) are the transfer functions for PSS and FDS respectively;

F(s) is the TF between the voltage reference input of the generator to its electrical

power output; τPSS and τFDS are the washout block time constants of PSS and FDS

respectively; τPSS1 to τPSSm are the lead-lag block time constants of the PSS; τFDS1 and

τFDS2 are the lead-lag block time constants of the FDS; nPSS and nFDS are the number of

PSSs and FDSs respectively.

In the second stage of the stabilisers coordination proposed in [18], a procedure to left-

shift the rotor modes by -Δσ while minimizing the stabiliser gain incremental ΔK has

been developed. It is also proposed to approximate the shift in selected modes Δλh for

a given ΔK by:

[ ] K.φφφλFDSPSS nh,nh2h1h Δ=Δ +K (2.41)

In (2.41), are determined by the following equation: FDSPSS nh,nh2h1 φ and ,,,φφ +K

FDSPSS

n

1i ihhjhij

i

ihhj nn1,2,...,j ;

u)λ(G)λ(F

Mpφ

PSS+=∑

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

=

∗∗ hl uc (2.42)

where pih is the participation factor of the generator i in mode λh; Mi is two-times the

generator inertia constant; Fij(λh) is the TF between the reference input of device j (PSS

or FDS) and the electrical power output of generator i; cl* is the lth row of the matrix

used for forming the inputs to the controllers from the state variables; u*h is the hth right

24


eigenvector of the state matrix; uih is the ith element of u*h corresponding to the speed

of generator i.

Based on (2.42), a linear programming (LP) problem has been formulated for the

simultaneous coordination of the stabiliser gains as follows:

(2.43) { }{ }

KKK

KImKRe

:to subject

Kw :minimize

minmax

ff

nn

1jjj

FDSPSS

⎪⎩

⎪⎨

⎧

≥Δ≥Δ≥ΔΔ−≥Δ≥Δ

Δ−≤Δ

∑ Δ+

=

0μφμ

σφ

where wj is the weighting coefficients; Δμf is the change in the frequency of the mode.

The LP problem (2.43) has been solved in [18] by using the simplex algorithm. It has

been mentioned also in [18] that the weighting coefficients wj can be chosen to be unity

(i.e. all stabiliser gain increments are weighted equally), or it can be chosen in such a

way to bias the solution in favour of the most effective stabilisers.

A drawback of the method proposed in [18] is that the accuracy of the predicted shift in

an eigenvalue diminishes as the changes in stabilisers gains become large, and the

requirement of a separate procedure using frequency response for the design of

stabiliser transfer functions. Another disadvantage of the scheme in [18] is that the

method draws on the calculations of the eigenvalues of the state matrix by the QR

algorithm, which does not exploit the sparsity structure in power system Jacobian

matrices.

2.3.2 Method Proposed By Lei et al. [11] In [11], a method for optimisation and coordination of damping controls based on time-

domain approach using a postulated disturbance has been proposed. The proposed

procedure is based on the nonlinear system analysis and function optimisation.

25

CHAPTER 2

In order to satisfy all of the requirements of the simultaneous optimisation and

coordination of the parameter settings of the FDSs and PSSs, a quadratic performance

index given by:

[ ]dt ˆˆJ min0

JT∫=

∝xQx (2.44)

has been chosen as a target function in [11]. The tuning procedure starts with the pre-

selected initial values of the stabiliser parameters involved, and iteratively adjusts all of

the selected parameters simultaneously, until the target function (2.44) is minimised.

These determined parameters are the optimal settings of the stabilisers involved. A

constrained quasi-Newton algorithm has been used in [11] to solve the nonlinear

optimisation problem arising from iteratively adjusting the selected parameters

simultaneously.

In (2.44), is a vector of state variable deviations used to form the performance index,

and is defined by:

x

)0()t(ˆ xxx −= (2.45)

where is the state at time t and the initial state. Q)t(x )0(x J is an NQ×NQ diagonal

matrix given by:

[ ]QqN3q2q1qJ wwwwdiag L=Q (2.46)

where wqi is the relative weighting factor for the ith state vector in relation to its

contribution to the system performance index in (2.44) , and NQ is the number of the

state variables adopted for tuning stabilisers.

As the active-power deviation ΔPe can easily be measured and contains relevant

features of the power swings, it has been recommended in [11] to use ΔPe as a state

variable for forming a performance index. This results in a performance index of the

form:

26


[ ]dt eeJ min0

T∫ ΔΔ=∝

PP (2.47)

In (2.47), wqi is set to 1 and ΔPe is defined by:

[ ]TTN3T2T1T LINEppppe ΔΔΔΔ=Δ LP (2.48)

where ΔpTi is active-power deviation on the ith transmission line where a FACTS device

is installed, and NLINE is the number of line to be considered. In the case of a shunt

compensator such as an SVC, the power is measured on one of the main infeeders of

the station where the SVC is installed.

In order to achieve overall damping improvement, it is also proposed in [11] to include

the deviations of the generator active-power (ΔpGj) in the performance index. Thus, the

controller parameters are determined by minimising a performance index that includes

both ΔpTi and ΔpGj. By including ΔpGj in the performance index, ΔPe is, therefore,

defined by:

(2.49) [ ]TGN1GTn1T GENppppe ΔΔΔΔ=Δ LLP

where NGEN is the number of generators equipped with stabilisers.

Although the nonlinear system analysis has been used in [11], as mentioned also in

[11], the results depend on the nature of the disturbances used to excite the system,

and the controller robustness might be compromised. Another disadvantage of the

method proposed in [11] is that it does not provide the flexibility of selecting the

electromechanical modes for optimisation.

2.3.3 Method Proposed By Ramirez et al. [8] A scheme has been proposed in [8] for coordinating FACTS-based stabilisers, using

the method of closed-loop characteristic polynomial and eigenvalue assignment. The

scheme solves the problem of coordinating the stabilisers sequentially, that is, in a pre-

specified sequence, rather than simultaneously.

27

CHAPTER 2

The proposed method in [8] is based on the fact that, if is a desired eigenvalue for

the closed-loop system, then the following equation holds:

desiλ

( ) ( )[ ] 0det desi

desi =λλ+ HGI (2.50)

where I is the identity matrix, G is the matrix of plant transfer function and H is the

matrix of stabiliser transfer function.

Since the plant matrix G and the desired eigenvalue are known, the proposed

procedure is based on the solution of the stabiliser parameters H so that (2.50) can be

satisfied. In order to solve this problem, the optimisation formulation to estimate the

stabiliser parameters is proposed as follows:

desiλ

( ) ( )[ ]{ }desi

desidetmin λλ+ HGI (2.51)

Once (2.51) has been solved, the loop is closed, and the design proceeds with the next

stabiliser using the same procedure. It is to be noted that, the evaluation of matrix G for

the next stabiliser must include all of the previously determined stabilisers.

As the proposed scheme in [8] solves the problem of coordinating the stabilisers

sequentially, it has a disadvantage that the pre-specified sequence used in the

coordination may not lead to the optimal results. It has also been mentioned in [8] that

a compromise should be established among the stabilisers to avoid them penalising

each other. Another disadvantage of the method proposed in [8] is that it requires the

state matrix to be formed explicitly. This will destroy the sparsity structure of the

Jacobian matrix.

2.3.4 Method Proposed By Cai et al. [5] In [5], an optimisation-based tuning algorithm has been proposed to coordinate among

multiple controllers simultaneously. The proposed algorithm is based on the linearised

power system model and parameter constrained nonlinear optimisation technique.

28


The proposed method in [5] is to search the best parameter sets of the controllers, so

that a comprehensive damping index (CDI) can be minimised:

29

) (2.52) (∑ ζ−==

modn

1ii1CDI

where 2i

2iii / ω+σσ−=ζ is the damping ratio of the ith eigenvalue, σi and ωi are the

real and imaginary parts of the ith eigenvalue respectively and nmod is the total number

of the dominant eigenvalues. In order to minimise (2.52), the nonlinear optimisation

algorithm based on sequential quadratic programming (SQP) has been used in the

proposed method [5].

It has been proposed in [5] to formulate the parameter optimisation problem as a

nonlinear programming formulation as follows:

(2.53) ( )

⎩⎨⎧

≥

=

∑ ς===

0)(E0)(E

:to subject

-1CDI)f( :minimize

in

eq

n

1ii

mod

KK

K

where f(K) is the objective function defined in (2.52), K is a vector which consists of the

parameters of the PSSs and FDSs controllers to be tuned; Eeq(K) is the equality

constraints, and Ein(K) is the inequality constraints. The constraints in (2.53) are given

in a general form. In the proposed method [5], only the inequality constraints on the

controller parameters are applicable.

The optimisation starts with the pre-selected initial values of the controllers. Then the

nonlinear optimisation algorithm is employed to adjust the controller parameters

iteratively until the objective function (2.52) is minimised. These determined parameters

are the optimal settings of the PSSs and FDSs controllers.

On the basis of the information presented in the paper [5], it appears that the method

reported draws on the calculations of the eigenvalues of the state matrix by the QR

algorithm, which does not exploit the sparsity structure in power system Jacobian

CHAPTER 2

matrices. Therefore, it may be difficult to apply the proposed method to large power

systems.

2.4 Eigenvalue-Distance Minimisation Method In the method proposed in [2,26], the desired closed-loop eigenvalues are specified

and the controller parameters are determined such that the distance between the

actual and the desired closed-loop eigenvalues is minimised. The robustness issue in

the proposed method has been addressed by considering a range of operating

conditions and optimising over the worst case scenario [26].

Fig.2.4 describes the general feedback control set-up where, G(s) and H(s) are the

transfer functions of the system and the controller respectively. Using s-plane

polynomials, these transfer functions can be represented as follows [2,26]:

G0

1nG1n

nGn

G0

1nG1n

nGn

DsDsD

NsNsN)s(G

g

g

g

g

g

g

g

g

+++

+++= −

−

−−

K

K (2.54)

K0

1nK1n

nKn

K0

1nK1n

nKn

DsDsDNsNsN

)s(Hk

kk

k

kk

kk

+++

+++= −

−

−−

K

K (2.55)

30

Fig.2.4: General feedback control

The closed-loop transfer function (TF) for the feedback system in Fig.2.4 is, therefore,

given by:

)s(H)s(G1

)s(H)s(G)s(TF+

= (2.56)

and the characteristic polynomial of the transfer function is given by:

y ur

-

+ H(s) G(s) Σ


(2.57) P0

P1

1nnP1nn

nnPnn

P sss)s( kg

kg

kg

kgδ+δ++δ+δ=δ −+

−++

+ K

It has been shown in [2,26] that the following equation is valid:

(2.58) PP δKP =

where:

(2.59)

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=−−

−−

G0

G0

G1

G1

G1n

G1n

G0

G0

Gn

Gn

G1

G1

G0

G0

Gn

Gn

G1n

G1n

Gn

Gn

P

DNDN

DNDN00DNDNDN

00DNDN0000DN

gg

gg

gggg

gg

LMMMM

LMMMM

MMLMMMM

L

L

MMLMMMM

L

L

P

(2.60) [ ]TK0

K0

K1

K1

K1n

K1n

Kn

Kn DNDNDNDN

kkkkL−−=K

[ ]TP0

P1

P1nn

Pnn

Pkgkg

δδδδ= −++ Lδ (2.61)

It is difficult to find a vector of controller parameters K by directly solving (2.58) for a

desired characteristic polynomial δP*. On the other hand, it is easier to solve K such

that ∗− PP δKP is minimised. The solution of this minimisation problem will results in

δP close to δP*, but the closed-loop eigenvalues might not be close to the desired one.

Due to this reason, it has been proposed in [2,26] to minimise the distance between the

desired and the actual eigenvalues. Therefore, the function to be minimised is given by:

∑λ

λ−λ=

+

=λ

kg nn

1i desi

acti

desi

i

)(w)(F

KK (2.62)

where and are the desired and actual values of the idesiλ act

iλ th closed-loop eigenvalue,

and wi is the weight associated with it. The proper selection of the weights can be

found in [2].

31

CHAPTER 2

32

The robustness issue of the proposed method is addressed by extending the above

technique to include other matrices PP

P associated with other operating conditions. The

eigenvalue-distances F (K) are then evaluated for certain values of δλjP and K.

In [2,26], the design methodology has been used to design a single FACTS device

controller in a power system. The capability of the method in designing multiple

controllers is, therefore, still not known. Another disadvantage of the method in [2,26] is

that it requires approximation or simplification where the order of the power system is

significantly reduced.

2.5 Conclusion Apart from the H∞ approach, the present chapter has presented and discussed non-

adaptive design methods for power system controllers which have the main function of

damping electromechanical oscillations, including the inter-area modes of oscillations.

The focus of the review is on the non-adaptive controllers that uses the local signal.

On examining the design principles of the methods, the key disadvantages or

deficiencies have been identified and discussed in the chapter.

33

3.1 Introduction As mentioned in Chapter 2, there have been extensive research and numerous

publications reporting the applications of the H∞ controller design method in the power

system damping improvement [10,27-40]. This chapter will discuss the previously-

published methods of H∞ controller in damping power system oscillation.

This chapter is organised as the following. First, the overview of the H∞ control theory

will be discussed to introduce the terminologies used in the H∞ design framework and

explain its principle. Then, the summary will be given of the published reports of the H∞

approaches for power system damping control which have been investigated over the

last decade. The last section of this chapter will discuss the key disadvantages with the

application of the H∞ controller in the damping of power system oscillations.

3.2 Overview of H∞ Control Theory 3.2.1 H∞ Norm

A control system is robust if it is insensitive (i.e. remains stable and achieves certain

performance criteria) to the differences between the actual system and the model of

the system which was used to design the controller. These differences are referred to

as model uncertainty [62-64]. Typical sources of the difference include unmodelled

(usually high-frequency) dynamics, neglected nonlinearities in modeling, effects of

deliberate reduced-order models and changes in system operating conditions [62-64].

H∞ control theory, which was originally formulated by Zames in the early 1980s, is

aimed to obtain satisfactory performance specifications even for the “worst-case” of

uncertainty [62].

3 DDAAMMPPIINNGG CCOONNTTRROOLLLLEERR DDEESSIIGGNN:: RREEVVIIEEWW OOFF NNOONN--AADDAAPPTTIIVVEE MMEETTHHOODD ((HH∞∞ AAPPPPRROOAACCHH))

CHAPTER 3

The H∞ norm has been extensively used in H∞ control problem formulation because it is

the convenient way for representing the model uncertainty [62]. It is to be noted that, in

H∞ design framework, the uncertainty can be modeled as perturbations to the nominal

model. In H∞ controller design, the H∞ norm is minimised in order to obtain the robust

design for the controller. It will be shown that minimising this H∞ norm corresponds to

minimising the peak of the largest singular value (“worst direction, worst frequency”),

and therefore, it can be used as a measure of the worst possible performance of the

control system [62].

The H∞ norm of a system is the peak value of the transfer function magnitude over the

whole frequency range. In a multi-input-multi-output (MIMO) system, the H∞ norm is the

peak of the largest singular value and can be expressed as [2,62]:

( ))j(max)s( ωσ=ω∞

GG (3.1)

Since the singular value provides maximum gain in the principal direction, H∞ norm can

be seen as the magnitude of the transfer function in the worst direction over the entire

frequency range [2,62].

The maximum singular value σ of transfer matrix G is determined by [2,62]:

( )21

21

vGv

G =σ (3.2)

where v1 is the vector of the first column elements of unitary matrix V. The unitary

matrix V can be found by using the singular value decomposition of G, i.e.

(note that the superscript H represents the matrix complex conjugate). In (3.2),

HVUΣG =

2. is a

vector 2-norm and defined by [62,64]:

L++= 22

212

|x||x|x (3.3)

where |xi| is the magnitude of the ith element of vector x.

34

DAMPING CONTROLLER DESIGN: REVIEW OF NON-ADAPTIVE METHOD (H∞ APPROACH)

3.2.2 Controller Design This section discusses the transfer function shaping approach for controller design. In

this approach, the designer specifies the “magnitude” of some transfer function(s) as a

function of frequency, and then finds a controller which gives the desired shape(s) [62].

The transfer function shaping approach can be subdivided into two approaches as

follows:

(i) Loop-shaping approach. This is the classical approach in which the magnitude of

the open-loop transfer function is shaped. However, classical loop-shaping is

difficult to apply for complicated systems, and therefore, the Glover-McFarlane H∞

loop-shaping design is preferred instead. This will be discussed later in this chapter.

(ii) Closed-loop transfer function shaping approach. In this approach, the closed-loop

transfer functions such as S, T and KS are to be shaped in the design. Optimization

is usually used in the approach, resulting in various H∞ control problems such as

mixed-sensitivity (this will also be discussed later in this chapter). The following is

the explanation of the S, T and KS transfer functions.

Consider the standard feedback control system shown in Fig.3.1 [62,63]. In Fig.3.1, G

is the plant model, K is the controller model to be designed, r is the reference inputs

(commands, set-points), d is the disturbances, n is the measurement noise, y is the

plant outputs (these signals include the variables to be controlled), ym is the measured

y, u is the controller output signals (manipulated plant inputs), and v is the controller

inputs (i.e. the difference between the reference inputs and measured plant outputs).

n

ym

+

+

+

+

-

+ y

d

u v rK G

Σ

ΣΣ

Fig.3.1: Standard feedback control system

35

CHAPTER 3

For the control system of Fig.3.1, it can be shown that the following relationships hold

[62,63]:

TnSdTry −+= (3.4)

KSnKSdKSru −−= (3.5)

TnSdSrrye −+−=−= (3.6)

where is the sensitivity function, and is the

complementary sensitivity function. It can be seen that S is the closed-loop transfer

function from the disturbances to the outputs, while T is the closed-loop transfer

function from the reference signals to the outputs.

( 1−+= GKIS ) ( ) GKGKIT 1−+=

The objective of the robust control design is to find a controller such that the closed-

loop system is robust. As mentioned in the previous discussion, in order to achieve

this, the H∞ norm of the transfer matrix should be minimised. Similarly, for the control

system shown in Fig.3.1, in order to obtain the best performance specifications such as

disturbance rejection or noise attenuation for any r, d or n, the H∞ norm of the

corresponding transfer matrices should also be minimised.

Therefore, the controller design problem can be formulated as follows: over the set of

all stabilising controllers K’s (i.e. those K’s make the closed-loop system internally

stable), find the optimal one that minimises [62,63]:

• ∞

S ; for good disturbance rejection or tracking

• ∞

T ; for good noise attenuation, and

• ∞

KS ; for control energy reduction

3.2.3 Standard H∞ Control Problem Fig.3.2 shows a general control configuration where any particular control problem can

be manipulated into this configuration [62-64]. The standard control system in Fig.3.1

can be transformed into an equivalent form of the general structure in Fig.3.2 which is

more convenient to formulate the H∞ control problem.

36


The system of Fig.3.2 is described by:

(3.7) ⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡uw

PPPP

uw

Pvz

2221

1211

Kvu = (3.8)

where P is the generalised plant model (this will include the plant model G and the

interconnection structure between the plant and the controller), w is the exogenous

inputs (commands, disturbances and noise), z is the exogenous outputs (“error” signals

to be minimised to meet the control objectives, i.e. y – r).

w z

P

u v

K

Fig.3.2: General control configuration

In state-space approaches to H∞ control, it is common to introduce the realisation of the

generalised plant P in the form of [65]:

(3.9) ( ) [ 211

2

1

2221

1211 BBAsICC

DDDD

P −−⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡= ]

This realisation corresponds to the state-space equations:

(3.10) uDwDxCvuDwDxCz

uBwBAxx

22212

12111

21

++=

++=++=

•

37

CHAPTER 3

Assume that the realisation of the controller to be determined in Fig.3.2 is:

, and the corresponding state-space equations is of the form: ( ) K1

KKK BAsICDK −−+=

(3.11) vDxCu

vBxAx

KKK

KKKK

+=

+=•

With the generalised plant defined as (3.10) and the controller model defined as (3.11),

it can be shown that the realisation of the closed-loop system shown in Fig.3.2 in state-

space form is given by [65,66]:

(3.12) wDxCz

wBxAx

clclcl

clclclcl

+=

+=•

where:

(3.13)

[ ]21K1211cl

K122K121cl

21K

21K21cl

K2K

K22K2cl

DDDDDCDCDDCC

DBDDBB

B

ACBCBCDBA

A

+=

+=

⎥⎦

⎤⎢⎣

⎡ +=

⎥⎦

⎤⎢⎣

⎡ +=

The results in (3.13) have been obtained by assuming D22 in (3.10) equal to 0. This

assumption will incur no loss of generality and has been made only to simplify the

calculations [65].

It can also be shown that the closed-loop transfer function from w to z for the system

configuration in Fig.3.2 is given by [2,62-65]:

(3.14) cl1

clclclzw B)AI(CDTF −−+= s

The standard H∞ optimal control problem is to find all stabilising controllers which

minimize the H∞ norm of the closed-loop transfer function:

38


( ))j(max ωσ=ω∞ zwzw TFTF (3.15)

In practice, it is usually not necessary to obtain an optimal controller for the H∞

problem, and it is often simpler to design a sub-optimal one. Therefore, the H∞ sub-

optimal control problem consists of finding all stabilising controllers such that [2,62-65]:

γ<∞zwTF (3.16)

where γ is greater than the minimum value of ∞zwTF over all stabilising controllers.

The standard H∞ optimal control problem (3.16) can be solved by: (i) analytical

approach using a positive semi-definite solution to the algebraic Ricatti equations

(AREs), or (ii) numerically optimise certain performance index such that the algebraic

Ricatti inequalities (ARIs) are satisfied. Although ARIs are nonlinear, they can be

converted into linear matrix inequalities (LMIs) by using linearisation techniques [2].

The numerical approach using LMIs has a distinct advantage as additional constraints

(such as minimum damping ratio) can be included in the design in a straight forward

manner [2,67]. In order to ensure a minimum damping ratio, a method known as pole-

placement is used in the design. In the method, the poles of the closed-loop system are

placed within a certain region in the complex plane. LMI-based solution to the H∞

control problem is described in the following section.

3.2.4 LMI-Based H∞ Design By using the Bounded Real Lemma and the Schur’s formula, it can be concluded that

the H∞ constraint (3.16) is equivalent to the existence of a solution of a symmetric

matrix to the following matrix inequality [2,65]: 0X >cl

(3.17) 0

clclcl

Tcl

Tcl

Tclclclclclcl

Tcl

<⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

γ−γ−

+

IDXCDIB

CXBAXXA

39

CHAPTER 3

In the matrix inequality (3.17), Acl, BBcl, Ccl and Dcl are functions of the controller

variables AK, BKB , CK and DK, and the controller variables are functions of Xcl. This

makes the products of the terms involving Xcl in (3.17) nonlinear. The following

techniques are used to change the controller variables and convert the problem into a

linear one.

Let n be the number of the plant states (size of A) and k be the order of the controller

(with k ≤ n), and also let Xcl (of dimension (n+k) (n+k)) and its inversion ( ) be

partitioned as:

1cl−X

(3.18) ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

HTH

HHcl

HTH

HHcl VN

NSX ;

UMMR

X 1-

where SH and RH are of dimension nn × and symmetric. It can be shown that Xcl will

satisfy the identity for [2,66]: 12cl ΠΠX =

(3.19) ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡= T

H

HTH

H

N0SI

Π ; 0MIR

Π 21

Also, let the new controller variables be defined as:

(3.20)

K

2KK

K2K

2K22KK

DD

RCDMCC

DBSBNB

)RCDB(ASRCBNMANA

=

+=

+=

+++=

ˆ

ˆ

ˆ

ˆ

HTH

HH

HHHHTHH

By examining the identity , it can be shown that: 12cl1

clcl ΠΠXIXX ==− or

(3.21) HHTHH SRINM −=

Pre- and post-multiplying the inequality by respectively leads to the

following LMI problem [2,66], the solution of which is used for forming X

0X >cl 2T2 Π Π and

cl.

40


0SIIR >

⎥⎥⎦

⎤

⎢⎢⎣

⎡

H

H(3.22)

Similarly, pre- and post-multiplying the inequality (3.17) by ( ) ( )II,,ΠII,,Π 2T2 diag and diag

respectively, and carrying out appropriate change of variables according to (3.20), the

following LMI is obtained [2,66] for determining the controller variables:

(3.23) 0ΨΨΨΨ

2221

T2111 <⎥

⎦

⎤⎢⎣

⎡

where:

( )( )

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ−+++++=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+++++=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ−+

++++=

ICDDCCDDCBCCBASSAΨ

DDDDCDRCDBBSCDBAAΨ

IDDBBDDBBBCCBARAR

Ψ

2121

T2121

TT22

T22

211211121

211T

2221

T2121

2121T2

T2

T

11

)

)))

))

)))

)

)))

HH

H

H

HH

(3.24)

Therefore, the LMI-based solution to the H∞ control problem consists of the following

steps:

• Solve the LMIs (3.22) and (3.23) for DC,B,A,S,R))))

and HH

• Compute by using a full-rank factorisation of THH and NM HH

THH SRINM −=

• Based on (3.20), determine the controller variables as follows:

( )( )

( )( )( ) 1

KKK1

K

K1

K

1KK

K

MRCBDASMSBCCRNBANA

BDSBNB

MCRDCC

DD

−−

−

−

+−−−=

−=

−=

=

THHH

THH

H

THH

)

)

)

( )

)

• Determine the controller transfer function using K1

KKK B)A(sICDK −−+=

41

CHAPTER 3

3.2.5 LMI-Based H∞ Design with Pole-Placement Satisfactory closed-loop pole damping ratios can be achieved by placing the closed-

loop poles into a certain region of the open-left-half complex plane. In order to include

this requirement in the controller design, the previous H∞ control problem formulation

must be modified to find the controller such that [2]:

• γ<∞zwTF

• Poles of the close-loop system lie in the desired region in the complex plane

Regions of interest for control purposes include those having certain geometric shapes

such as: vertical/horizontal strips, disks, conic sectors, etc, or combinations of these

geometric shapes. In particular, a ‘conic sector’ with inner angle θ and apex at the

origin as shown in Fig.3.3 is the region of interest in power system applications as it

ensures a minimum damping ratio for closed-loop poles [2]. )2/(cos 1min θ=ς −

42

Fig.3.3: A conic sector

0 Real

Imag

θ


43

has been shown in [2,67] that the closed-loop system matrix Acl has all the poles

It

inside the conical region if and only if there exists a symmetric matrix XD > 0 such that:

( ) ( )( ) ( 0

sincoscossin

TclDDclDcl

TclD

TclDDcl

TclDDcl <

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+θ−θ−θ+θ

AXXAXAAXAXXAAXXA

) (3.25)

he design (with incorporating the pole-placement constraint) is feasible if (3.17) and

imilar to (3.17), the matrix inequality (3.25) is also nonlinear. As in the previous

T

(3.25) hold for some positive definite matrices Xcl, XD. However, the problem is not

jointly convex in Xcl and XD unless it is solved for the same matrix Xcl. Therefore, the H∞

problem with pole-placement can be stated as follows [2]: find 0X >cl and controller K,

such that (3.17) and (3.25) are satisfied with Xcl = XD.

S

section, it can be linearised by pre- and post- multiplying (3.25) by 2T2 and ΠΠ

respectively, and carrying out the change of variables according to (3.20) which will

result in the following LMI [2]:

( ) ( )( ) ( ) 0

sincoscossin

TT

TT

<⎥⎦

⎤⎢⎣

⎡

+θ−θ−θ+θ

ΦΦΦΦΦΦΦΦ (3.26)

here:

w

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+++

=2

2122CBASADDBACBARΦ ))

))

H

H (3.27)

he solution steps for determining controller K in the H∞ design with pole-placement

3.2.6 H∞ Mixed-Sensitivity Design en required to minimise the impact of

any disturbance d on the plant output (i.e. disturbance rejection) and to limit the size

T

constraint are similar to those described in the previous section. The only difference is

that for solving the H∞ control problem with pole-placement constraint, the LMI (3.26)

has to be solved in addition to solving the LMIs (3.22) and (3.23).

For the control system shown in Fig.3.4, it is oft

CHAPTER 3

and bandwidth of the controller (control energy reduction). To achieve these, it makes

sense to shape the closed-loop transfer functions S and KS. As referred to in Section

3.2.2, S is the transfer function between d and the output, and KS the transfer function

between d and the control signals. Therefore, the design problem can be formulated as

[2,62-64]:

44

∞

∈ ⎥⎦

⎤⎢⎣

⎡KSS

K Smin (3.28)

where S is the set of all stabilising controllers K. The design problem in (3.28) is

ferred to as the S/KS (S-over-KS) mixed-sensitivity optimisation.

Fig.3.4: Control system for mixed-sensitivity formulation

owever, it is usually not required to minimise the norm in (3.28) over all frequencies.

he disturbance d is typically a low frequency signal, thus S can be minimised over low

re

H

T

frequency signals. On the other hand, KS can be minimised at higher frequencies

where limited control action is required. In order to do this, the appropriate weighting

•x x

G

z2

z1

v

r=0

w=d

y +

- +

+ yp +

+

+

+ u 1/s Σ Cm Σ Σ ΣBm K

Am

Dm

-W2 W1


45

lem, the control system shown

Fig.3.4 can be represented in the form of general configuration. Vector r in the

(3.29)

By referring to Fig.3.5 and substituting

filters W1 and W2 can be used to emphasise the individual transfer function

minimisation at frequency ranges of interest. In practice, it is common to select W1 as

an appropriate low-pass filter for disturbance rejection and W2 as a high-pass filter to

reduce the control effort over the high frequency range.

In order to solve this S/KS mixed-sensitivity design prob

in

linearised system in Fig.3.4 which represents the changes in the input references is

zero for fixed commands. The equivalent general representation of the control system

is shown in Fig.3.5. In Fig.3.4, it can be seen that the plant has the transfer function

represented by ( ) mmmm DBAsICG 1 +−= − , and the corresponding state-space

equations of:

uDxCy

uBxAx

p mm

mm

+=

+=•

KyKvu −== into Guwy += , it can be shown

at . Also, in Fig.3.5, the disturbance d can be seen as a single

rror signal) z is defined a

uWz

z222

⎥⎦

⎢⎣

⎥⎦

⎢⎣−⎥

⎦⎢⎣

(3.30)

From (3.30), it can be concluded that the closed-loop transfer function from w to z for

e system configuration in Fig.3.5 is given by:

KSWS

2zw (3.31)

Therefore, the S/KS mixed-sensitivity problem is to find a stabilizing controller which

inimises the infinity-norm of (3.31):

th SwwGK)(Iy 1 =+= −

exogenous input w, and the exogenous output (e s [62]:

SWyWz 111 ⎤⎡=

⎤⎡=

⎤⎡= w

KSW

th

⎡

=W

TF 1 ⎥⎦

⎤⎢⎣

m

CHAPTER 3

46

∞

⎥⎦

⎤⎢⎣

⎡KSWSW

2

1 (3.32)

Fig.3.5: General representation of Fig.3.4

y solving the H∞ optimisation problem given in (3.32), the solution of the S/KS mixed-

.2.7 H∞ Loop-Shaping Design d-sensitivity design described in the previous

loop-shaping design technique which is discussed in this section does not have the

P

z2

B

sensitivity design problem which gives the controller transfer function K can be

obtained. The solution procedure is the same as described in the previous discussion.

3The difficulty in implementing the mixe

discussion is that there is no systematic procedure for selection the weights W1 and

W2. Moreover, there may exist undesirable pole-zero cancellation between the nominal

plant model and the controller [63].

A

drawbacks as those in the mixed-sensitivity design methodology. In this approach, the

uncertainty is represented by the perturbations directly on the coprime factors of the

+

+

+

-y

r = 0 v u

w = d

z1

z

G Σ Σ

K

-W2

W1


nominal plant model [63]. This method combines the characteristics of the classical

open-loop shaping and H∞ optimisation.

47

he coprime factorisation-based loop-shaping method was first introduced by

.2.7.1 Coprime Factorisation

ant G is defined as follows [2,62-64]:

(3.33)

here Nc and Mc are stable coprime transfer functions.

he left coprime factorisation can be calculated as follows. Suppose G has a state-

are in the left side o

T

McFarlane and Glover [62]. It is essentially a two stage design procedure. First, the

open-loop plant is augmented by pre- and post-compensators to give a desired shape

to the open-loop frequency response. Then the resulting shaped plant is robustly

stabilised with respect to coprime factor uncertainties by solving the H∞ optimisation

problem [62,63].

3

A left coprime factorisation of a pl

c1

c NMG −=

w

T

space realisation of m1

mmm )s( BAICDG −−+= , and let Lc be a matrix such that the

eigenvalues of mA + f the complex plane, then a left coprime

factorisation of is given by [64]:

mcCL

c1

c NMG −=

( )( ) ( )mcm

1mcmmmc

c1

mcmmc

s

s

DLBCLAICDM

LCLAICIN

+−−+=

−−+=−

−

(3.34)

.2.7.2 Robust Stabilisation

e factorisation defined by (3.33), the corresponding

3

For a plant G with left coprim

perturbed plant can be written as [2,62-64]:

( ) ( )N1

Mp ΔNΔMG ++=−

cc (3.35)

CHAPTER 3

48

here ΔM and ΔN are stable unknown transfer functions which represent the uncertainty

ig.3.6 shows the standard feedback system with the perturbations on the coprime

w

in the nominal plant model G.

F

factors of G. By referring to Fig.3.6 and substituting GuwMy += −1c into KyKvu −== ,

it can be shown that wKMKG)(Iu 1 1−−+−= . Also, the exogenous output z :

c is defined as

(3.36)

y using the identities:

(3.37)

quation (3.36) can be rewritten as follows:

(3.38)

rom (3.38), it can be concluded that the closed-loop transfer function from w to z for

(3.39)

herefore, the loop-shaping design problem can be stated as follows: find a stabilising

wKMKGI

MKKGIGIuy

zz

z2

1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−+−

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡= −−

−−

1c

1

1c

1

)(])([

B

( ) 1

11

GK)(IKKGIGI

GK)K(IKKG)(I−−

−−

+=+−

+=+1

E

wMGKIK

MGKIuy

zz

z2

1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−+

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡= −−

−−

1c

1

1c

1

)()(

F

the system configuration in Fig.3.6 is given by:

( ) ⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡−

= −− 1c

1MGKIKI

TFzw

T

controller which minimises:

( )∞

−−+⎥⎦

⎤⎢⎣

⎡−

11MGKIKI

c (3.40)


49

Fig.3.6: Control system for loop-shaping formulation

(a) Left coprime factor perturbed system

(b) General representation of the perturbed system

z1 z2 -+

+

+

-

+ y

w

u v r = 0

K Nc

Σ

ΣΣ Mc-1

ΔN ΔM

(a)

P

+

+

+

-y

r = 0 v u

w

z1

z2

z

Nc Σ Σ

K

Mc-1

(b)

CHAPTER 3

3.2.7.3 Loop-Shaping Design Procedure

Two stages in the loop-shaping design procedure can be described as follows:

(1) Loop-Shaping. In this stage, the open-loop frequency response is shaped by using

the pre- and post-compensators. This stage is carried out to specify the

performance requirements prior to robust stabilisation of the shaped plant. Suppose

S1 and S2 are the pre- and post-compensators respectively. The shaped plant is,

therefore, given by 12s GSSG = as shown in Fig.3.7. Recommendations for

choosing the appropriate pre- and post-compensators can be found in [62].

However, some trial and error is still required to select S1 and S2.

S1 G S2

50

Fig.3.7: Loop-shaping design procedure

(a) Shaped plant

(b) Compensated plant

(c) Equivalent controller

G S1 S2

K S1 S2

(a)

K

Gs

(b)

G

K∞ (c)


(2) Robust Stabilisation. In this stage, the controller (K∞) is designed by solving the

robust stabilisation problem for the shaped plant Gs, i.e. find a stabilising controller

K∞ which minimises:

( )∞

−−∞

∞+⎥

⎦

⎤⎢⎣

⎡−

1s

1s MKGI

KI

(3.41)

The solution technique to this robust stabilisation problem is the same as that in the

standard H∞ control design problem. In (3.41), can be determined by using the

coprime factorisation of G

1s−M

s such that . The final controller K is then

constructed by combining the designed controller with the compensators, i.e.

as shown in Fig.3.7.

s1

ss NMG −=

21KSSK =∞

3.3 Summary of H∞ Damping Control in Power System

H∞ approaches for power system damping control have been investigated over the last

decade [10,27-40]. The results of the investigation have also been reported in many

literatures and can be summarised as follows:

- In [10], the methodology for the design of robust damping controllers for PSSs has

been discussed. The design procedure was based on a formulation of the output

feedback control problem, which is suited for damping controller design. With this

formulation, the design problem can be expressed directly in the form of LMIs. Also,

the inclusion of a regional pole placement criterion, as the design objective, allows

the specification of a minimum damping factor for all modes of the controlled system.

It has been shown in [10] that the controller is able to provide adequate damping for

the oscillation modes of interest.

- In [27,28], the design of an H∞ controller for FACTS device for enhancing the

electromechanical mode damping has been presented. The H∞-based design

procedure has been developed in an attempt to obtain a robust damping controller for

a thyristor controlled series compensator (TCSC). In the procedure, two Riccati

51

CHAPTER 3

52

equations were used and solved in order to obtain the solution for the H∞ optimization

problem.

- The design process and a method to formulate the H∞ optimal PSS design problem in

terms of a general H∞ control design framework have been discussed in [29]. The H∞

design problem has been solved by using two algebraic Riccati equations. The H∞-

based PSS was tested by simulation on a single-machine infinite bus (SMIB) system.

Results of the testing show that the proposed H∞ PSS satisfies the design

specifications.

- Design of a robust supplementary controller for a static VAr compensator (SVC) to

improve the damping of a two-machine power system has been proposed in [30]. In

the paper, the formulation of the damping control problem has been based on the H∞

optimization. The solution to the design problem was obtained by solving the

standard mixed-sensitivity control problem.

- In [31,32], a Glover-McFarlane H∞ loop-shaping approach has been used to design a

robust control for a FACTS device and PSS respectively. In [31], the H∞ loop-shaping

was used to design a robust control for static compensator (STATCOM), series

power flow controller (SPFC), voltage source converter (VSC)-based static phase

shifter (SPS) and unified power flow controller (UPFC). The simulation has been

carried out in [31] to show the effectiveness of the proposed controllers in improving

the system damping. In [32], it has been shown that the H∞ PSS can guarantee the

stability of a set of perturbed plants with respect to the nominal system and exhibit a

good oscillation damping ability.

- In [33], a method for designing low-order controllers for damping power swings has

been proposed. The method was based on an H∞ design formulation and uses LMI

solver to obtain controller parameters. In particular, the proposed method has been

used for design of a PSS for a SMIB system and a decentralised control for a TCSC

and an SVC in a three-area system. Although the proposed method might not

guarantee to provide global convergence, the convergence to a good damping

controller design can be achieved [33].


53

- In [34], a robust design and tuning of a PSS in a SMIB system has been presented.

In the method, maintaining stability and performance over a range of uncertain plant

parameters (due to variations in generation and load patterns) was handled by

imposing an upper bound on the H∞ norm of the closed-loop transfer function. A pole

region constraint was also included in the design. The solution to these design

problems has been obtained by solving a standard LMI formulation.

- An H∞ mixed-sensitivity design of a damping device employing a UPFC has been

presented in [35]. The problem is posed in the LMI framework. The controller design

was aimed at providing adequate damping to interarea oscillations over a range of

operating conditions. The results obtained in a two-area four-machine test system

have shown to be satisfactory both in the frequency domain and through nonlinear

simulations.

- In [36,37], a design procedure for robust damping controller of superconducting

magnetic energy storage (SMES) device has been presented. The mixed-sensitivity

H∞ design based on the LMI formulation was used in the power system damping

control design. Furthermore, a regional pole placement objective was also addressed

in the design process [37].

- A H∞ damping control design based on the mixed-sensitivity formulation in an LMI

framework has been carried out in [38,39]. In [38], a power system containing a

controllable series capacitor (CSC), a static VAr compensator and a controllable

phase shifter (CPS) was considered. It has been shown in [38] that the H∞ controllers

designed for these devices can improve the damping of interarea oscillation and also

robust in the face of operating condition changes such as: varying power-flow

patterns, load characteristics and tie-line strengths. In [39], a multiple-input single-

output (MISO) H∞ controller has been designed for a TCSC to improve the damping

of the critical interarea modes. Also, in [39], the stabilising signals are obtained from

remote locations based on observability of the critical modes.

- The application of loop-shaping technique in H∞ damping control design has been

proposed in [40]. In [40], the problem of robust stabilisation of a normalised coprime

factor plant description was converted into a generalised H∞ problem. The problem

was solved using LMIs with additional pole-placement constraints. In addition to

CHAPTER 3

54

robust stabilisation of the shaped plant, a minimum damping ratio can thus be

ensured for the critical modes. The proposed method has been used to design a

supplementary damping controller for a TCSC.

3.4 Disadvantages of H∞ Controller

Although the H∞ damping controller can guarantee the stability and robust operation of

power system as described in the previous discussion, there remain some

disadvantages with the application of this controller in damping of power system

oscillation. The disadvantages are identified as follows:

- As H∞ controller is a fixed-parameter controller, it is, in general, not possible to

achieve maximum damping performance for each and every operating condition or

contingency.

- The H∞ technique does not provide simultaneous and optimal control coordination

of multiple controllers. The multiple controllers have to be designed in a sequential

manner [2,32,36,38,39], i.e., the damping controller for one device is designed and

the loop is closed before designing the next one. At each stage of this sequential

design, the plant model is to be updated to include the controllers designed in the

previous stages. In general, the sequential design will not lead to optimal control

coordination of multiple controllers in a power system.

- The application of H∞ controller requires a significant simplification of the power

system dynamic model to achieve a low plant order suitable for H∞ controller

design, and the sparsity in the power system Jacobian matrix is not taken into

account in the model order reduction process [2].

- Although it has been mentioned in [2,40] that the primary task in H∞ loop-shaping

design is to choose appropriate pre- and post-compensator, there does not appear

to be a systematic and general procedure for selecting the compensator transfer

functions. Also, there is no general procedure for shaping the open-loop plant. The

procedure is only specific to the particular application and some trial and error is

involved [2,40].


55

3.5 Conclusions

The review presented in this chapter covers two aspects related to the robust controller

design. The first aspect is the examination of a range of the linear controller design

techniques, based on the H∞ control methodology, which are applied in general to a

linear plant the uncertainty or variations of the model parameters are represented by

disturbances on the plant outputs.

The second aspect in the review is the applications of the H∞ based technique to power

systems damping controller designs. In the context of power systems, the plant model

parameters variations arise from the plant nonlinearities combined with possible

changes in plant operating conditions or topology.

Although it appears that the H∞ control methodology in its general form is suitable to

the applications, the review of many publications on this subject reveals that there are

important disadvantages in the H∞ control approach. The key reasons identified in the

review include those related to the very high dimension of the state matrix of the power

system, and for a large power system, the very high dimensions of the system

Jacobian matrices the sparsity of which will be destroyed when the explicit state matrix

is formed for H∞ control design.

As the H∞ control technique developed in the control theory applies to the design of

one controller only, the simultaneous and optimal control coordination of multiple

controllers in a power system cannot be achieved with this theory. Approximation

based on a sequential design technique is then required if the theory is used in the

case of multiple controllers.

56

4.1 Introduction In the context of electromechanical oscillations in a power system, the system dynamic

responses derive from the rotating machines (including their controllers) and FACTS

devices together with their SDCs which are interconnected by the network. On this

basis, the dynamic modeling required for the study and design related to

electromechanical oscillations has two main aspects to be considered: the first is that

of the modeling of individual items of plants, and the second the overall system model

where their interconnection is represented.

This chapter has a focus on the first aspect in which dynamic models are presented for

the following items:

• Synchronous generators

• Excitation control systems

• Governor and turbine systems

• Power system stabilisers

• FACTS devices

• Supplementary damping controllers

• Loads

In general, the differential/algebraic equations derived from the plant models are

nonlinear. In preparation for small-disturbance stability to be considered in the

subsequent development, the individual plant equations will be linearised about a

specified operating point.

4 DDYYNNAAMMIICC MMOODDEELLIINNGG:: PPOOWWEERR SSYYSSTTEEMM CCOOMMPPOONNEENNTTSS

DYNAMIC MODELING: POWER SYSTEM COMPONENTS

57

iven the frequency range encountered in electromechanical oscillations, the network

.2 Synchronous Machine Model ented by the fifth-order model in the d-

(4.1)

(4.2)

(4.3)

here Ψr , ωr , and δr are rotor flux linkage vector, rotor angular frequency and rotor

G

which has the function of interconnecting the above items of plant will be modeled in a

static form as discussed in the next chapter which combines the network with other

items of plant in forming the overall system model.

4In this thesis, the synchronous machine is repres

q axes having the rotor frame of reference [68]:

rSmrmr VIFΨAΨ ++=•

( ) M/PPω emr −=•

Rrr ωωδ −=•

w

angle respectively; [ ]Tfdr 00E=V is the rotor voltage vector; Pm and Pe are the

mechanical and elec pectively; M is calculated from R/H2Mtrical powers res ω= (H is

the machine inertia constant and ωR is the synchronous speed); Am re the

matrices depending on machine parameters (see Appendix A.1 for the expressions of

these matrices); I

and Fm a

he derivation of (4.1) is given in Appendix A.1. Whereas, (4.2) which is related to the

lectrical power Pe in (4.2) can be expressed in terms of generator current as follows:

(4.4)

is eliminated from (4.4) by using (A.13) in (4.4) to give:

S is the stator current vector, and Efd is the field voltage. The

transients in the rotor fluxes set up by the field winding, and damper windings on the d

and q axes are represented in the model.

T

machine mechanical axes, is referred to as the machine swing equation. Equation

(4.3), which relates the rotor angle to rotor angular frequency, is needed to complete

the representation in the mechanical axes.

E

rsrTsrsss

TsreP IGIIGI ω+ω=

Ir

CHAPTER 4

58

(4.5)

(4.6)

(4.6), Gm and Sm are constant matrices depending on machine parameters and

iven the low frequency encountered in electromechanical transients, and to be

rmsmeP ΨCIB +=

where:

mTsrmm

Tsrm ; SICGIB ω=ω=

In

defined by (A.18) in Appendix A.

G

consistent with the network model in a static form, the stator flux transients are

discounted. With the stator flux linkage in a non-transient form, Appendix A.1 derives

the relationship between the stator current vector I and the stator voltage vector Vs s,

which is given by the following algebraic equation:

smrms IZΨPV −= (4.7)

here Pm and Zm are the matrices depending on machine parameters and rotor angular

.3 Excitation and Prime-Mover Controllers ted in a general and

relation to the excitation system, the inputs comprise the terminal voltage magnitude,

w

frequency and defined by (A.17).

4At present, in terms of modeling, these controllers are represen

flexible structure by which any particular control block diagrams for them including

those defined by the users can be accommodated. A wide range of block diagrams for

modeling various types of excitation systems and prime-mover controllers have been

developed by the IEEE [69,70].

In

its reference value and the supplementary signal from the PSS, and the output is the

field voltage. Irrespective of the control block diagram, the first-order differential

equation set for describing the excitation system dynamics can be arranged in the

following form:

refsePSSeseeee VVV DBCxAx +++=

• (4.8)


59

where is the state vector for the excitation system; Vs is the synchronous machineex

terminal voltage; VPSS is the supplementary signal from the PSS; refV is the voltage

reference; As

Be, B

quation (4.8) provides the interaction between excitation controller and synchronous

r prime-mover (turbine and governor) system, the inputs comprise the machine

(4.9)

here is the state vector for the prime-mover controller; is the speed

ce; g g

e, Ce and De are matrices of constant values which depend on the gains

and time constants of the controller.

E

machine. The interaction is accounted for via the interface state variable for machine

field voltage (Efd) which is embedded in the state vector xe.

Fo

speed, its reference value and the initial power, and the output is the mechanical

power. The system dynamics of the prime-mover controller can be represented by the

set of first-order differential equation as follows:

0mg

refgrgggg PDBCxAx +ω+ω+=

•

refωgxw

referen 0P is the initial power; A , C , Bm Bg g

nd

quation (4.9) also provides the interaction between the prime-mover controller and

.4 PSS Model ost common stabiliser to damp out the oscillations. The PSS

ig.4.1 shows the general structure of a PSS [12] which is adopted in this thesis. The

and D are matrices of constant values

which depe on the gains and time constants of the controller.

E

synchronous machine. One of the variables in state vector xg is the mechanical power

Pm.

4PSS has been the m

function is to introduce a modulating signal through the excitation system to contribute

to rotor oscillation damping. The machine speed, terminal frequency and/or power can

be used as the input signals to PSS.

F

structure consists of a gain block, a washout, lead-lag blocks and a limiter. A washout

term/filter (i.e. with a time derivative operator) in the PSS structure is needed to

CHAPTER 4

guarantee that the PSS responds only to disturbances, and does not respond to any

steady-state condition, when speed or power is input. Here, the rotor speed is used for

the PSS input. The PSS output is added to the exciter voltage error signal and served

as a supplementary signal.

60

he state equation derived by examining the PSS transfer functions can be arranged in

(4.10)

here is the vector of state variables of the PSS; Ap and Cp are

Fig.4.1: PSS control block diagram

.5 FACTS Device Models

function of voltage/reactive-power control, an SVC can provide

Fig.4.2 is shown in a block diagram form the control system of an SVC [71,72] used

T

the following form (see Appendix A.2.1 for the derivation):

rpppp••ω+= CxAx

[ ]TPSS2P1Pp Vxx=xw

matrices the elements of which depend on the gains and time constants of the PSS

controllers and defined by (A.32).

VPSS,max

44.5.1 SVC Model In addition to the main

auxiliary control of active-power flow through a transmission line. The possibility of

controlling the transmittable power implies the potential application of this device for

damping of power system electromechanical oscillations.

In

in the present work. In Fig.4.2, Bc represents SVC susceptance and VT is the terminal

voltage where the SVC is installed. For electromechanical oscillation damping purpose,

XP2 XP1

VPSS,min

VPSS ωr

PSS

PSS

sT1sT+

4PSS

3PSS

sT1sT1

++

2PSS

1PSS

sT1sT1

++

PSSK


a supplementary signal (XSDC) derived from a separate controller is input to the main

controller as shown in Fig.4.2.

T

61

he state equations for the SVC main control system can be arranged as follows (see

Appendix A.2.2 for the derivation):

refTsSDCsTssss VXV DBCxAx +++=

• (4.11)

here is the state vector for the SVC main control system; As, Bs, Cs,

matrices th

Fig.4.2: Control block diagram of SVC

.5.2 TCSC Model ies Capacitor (TCSC) is a FACTS device that can provide fast

Fig.4.3 is shown in a block diagram form the control system of a TCSC

[ ]Tc1cs Bx=xw

and Ds are e elements of which depend on the gains and time constants of

the controllers and defined by (A.40).

VTref

Bc max.

+

4Thyristor Controlled Ser

and continuous changes of transmission line impedance, and can regulate power flow

in the line. The possibility of controlling the transmittable power also implies the

potential application of this device for the improvement of power oscillations damping

[6,73,74].

In

[41,42,73,75]. In the figure, Xt is the reactance of TCSC. The TCSC control block

diagram contains Proportional-Integral (PI) controller block and the block that

xc1

Supplementary signal (XSDC)

--

Bc

Bc min.

|VT| SK Σ 2S

1S

sT1sT1

++

SsT11

+

CHAPTER 4

represents the TCSC thyristor firing delays. The PI block is the TCSC main controller.

The power flow control is usually implemented with a slow controller which is typical for

a PI controller with a large time constant.

62

can be shown that the state equations for the TCSC main control system in Fig.4.3

(4.12)

here is the state vector for the TCSC main control system; At, Bt, Ct, Dt

are matrices th

Fig.4.3: Control block diagram of TCSC

he TCSC reactance limits shown in Fig.4.3 have a dynamic form which depends

.5.3 STATCOM Model mary voltage/reactive-power control function can also be

It

can be arranged as follows (see Appendix A.2.3 for the derivation):

reftTtTtSDCtttt PPPX EDCBxAx ++++=

••

[ ]TtPFt Xx=xw

and Et e elements of which depend on the gains and time constants of

the controllers and defined by (A.47).

Pref

Xt,max

T

nonlinearly on the current in the transmission line for which the TCSC provides the

compensation [73].

4A STATCOM which has the pri

used to improve the damping of power system oscillations [76]. The basic principle of

STATCOM is to use a voltage source inverter which generates a controllable ac

voltage source behind the transformer leakage reactance (see Fig.4.4a). The voltage

difference across the transformer reactance leads to active- and/or reactive-power

xPFXt

+PT

-

Xt,min.


+ - t

t

sT1K+

ΣF

FF sT

)sT1(K + Σ


flows to the network. The exchange of reactive-power with the network is obtained by

controlling the voltage magnitude at the STATCOM terminal, and the exchange of

active-power results from the control of the phase shift between STATCOM terminal

voltage and the network voltage VT. The exchange of active-power is used to control

the dc voltage.

63

is to be noted that the phase reference for STATCOM voltage VC is the terminal

ased on the dc voltage expression in Fig.4.4a and the transfer functions in Fig.4.5,

It

voltage VT as shown in Fig.4.4b. It can be seen in Fig.4.4b that p is the axis in phase

with VT and q is the axis perpendicular to p. In Fig.4.5 is shown in a block diagram form

the control system of the STATCOM [72,73] used in the present work.

B

the state equations for the STATCOM main controller in Fig.4.5 are formed as follows

(see Appendix A.2.4 for the derivation):

φ+=φ

++++++=

+++=

=

•

φ

•

•

φ

•

socso

soSDCsoCqsoTsorefTsodcsosoc

SDCsoCqsoTsorefTso

sodc

x

VXIVVVVx

XIVVV

VV

NM

LKJOHGF

EDCB

A

(4.13)

here:

w

φ=φ sinVV T (4.14)

(4.13), Aso, Bso, Cso, Dso, Eso, Fso, Gso, Hso, Jso, Kso, Lso, Mso, Nso and Oso are matrices

In

the elements of which depend on the STATCOM and its controllers parameters as

defined by (A.66). From (4.13), the state vector for the STATCOM main control system

is assembled as [ ]Tcdcso xVV φ=x .

The first equation in (4.13) is derived from the dc side of the capacitor which has

and the input for the droop is the reactive component of the STATCOM current.

capacitance Cdc (see Fig.4.4a), and the remaining equations are derived from the main

block diagram of the STATCOM (see Fig.4.5). The droop is also included in the model,

CHAPTER 4

64

Fig.4.4: STATCOM connection and vector diagram

(a) STATCOM connection to the network

Fig.4.5: Control block diagram of STATCOM

.5.4 UPFC Model he UPFC is a versatile FACTS controller which has a wide range of control functions

rmance [73,77]. Fig.4.6 shows the general

(b) Vector diagram

4T

for the improvement of power system perfo

xc

Limit max.

Supplementary signal(XSDC)

ICq

VTref

k

Vdc

φ

-+

--

-

+ Vdcref

|V|

Limit min.

|VT|

● ●

2C

2C2C

sT)sT1(K +

sK 1C

droop

csT11

+ Σ Σ

φ

α

VCq

VCp

VC

VT

q

p

Q

D

CqCpC jIII +=

∫= dcdcdc I)C/1(V

XC

VT = |VT|.ejα

VC = VCp + jVCq = kVdcejφ

φ

(a) (b)


65

4.7(b) are shown the dynamic models for the shunt and series

onverter controllers respectively [73,77,78]. In general, for system damping purpose,

Fig.4.6: UPFC block diagram

Based on the controller block he equations system for the

PFC main control system in Fig.4.7 can be arranged as follows (see Appendix A.2.5

(4.15)

structure of the UPFC [73,77,78]. The UPFC combines two voltage source converters

linked by a dc bus.

In Figs. 4.7(a) and

c

there is, as shown in Fig.4.7b, a supplementary signal obtained from an SDC and input

to the shunt converter controller. However, it is also possible to use an SDC in

conjunction with the series converter controller.

diagrams in Figs.4.7a and b, t

U

for the derivation):

IVdc VCV =•

shushudcudcu

refdcu

SDCuSDCuuTush XXV•

+++= EDBAV (4.16) TuT

ref

VVV

VV•

•••

+++++

+

IJIIHGF

C

seuseuTuuse VVV••

δα

•+++= INIMLKV

(4.17)

2dc2seqsepse ΨVkmjVVV ∠=+=

Vdc Series- converter

Shunt- converter

I1

Ise = Isep + jIseq

Xse VT VU

Ish = Ishp + jIshq

Xsh

1dc1shqshpsh ΨVkmjVVV = + = ∠

CHAPTER 4

66

Fig.4.7: Control block diagram of UPFC

(a) Shunt part

(b) Series part

In (4.15) – (4.17), Vsh, V Vα and Vδ are defined by:

se, Ish, Ise, CV, VI,

kVdc

Isep |VT|

|VT|

Ipref

Iseq

+

+

+

+

+

Vsep0

Vseq0

-

Ψ2

m2 +

Iqref

Qref

-

Pref

Σ

Σ

|V| Vsep

Vseq

sepV

seqVatan2Ψ

2seqV

2sepV|V|

=

+=

••

••

••

se1

se1se1

sT)sT(1K +

se2

se2se2

sT)sT(1K +

Σ

Σ

(b)

m1

Ψ1

kVdc


+

-

Ishq

-

+

+

+

+

Vshq0

Vshp0

-

|VT|

-

+

|V| Vshq

Vshp

Vdc

Vdcref

VTref

Σ

Σ

sh1sT

)1shsT1(sh1K +

sh2sT

)sh2sT(1sh2K +

shpV

shqVatan1Ψ

2shqV

2shpV|V|

=

+=

••

Σ

Σ

droop

(a)


67

⎡⎤⎡⎤⎡⎤⎡ sepshpsepshp IIV;

VV (4.18)

⎥⎦

⎢⎣

=⎥⎦

⎢⎣

=⎥⎦

⎢⎣

=⎥⎦

⎢⎣

=seq

seshq

shseq

seshq

sh I ;

I ;

V

VIIV

⎤

seTsesh

TshI

dcdcV V;

VC1C IVIV −==

(4.19)

2TT V1 V;

V1V == δα (4.20)

Also, in (4.16) and (4.17), A Bu, B Cu, Du, Eu, Fu, Gu,

matrices the elements of which depend on the UPFC and its controllers parameters

nd defined by (A.79) and (A.89). It can also be seen from (4.15) – (4.17) that the state

u, Hu, Iu, Ju, Ku, Lu, Mu and N are u

a

vector of the UPFC main control system is: [ ]TTse

Tshdcu V VVx = .

4.6 Supplementary Damping C elontroller Mod ions with FACTS devices is effected through power

. The SDC block

Fig.4.8: SDC control block diagram

Damping of power oscillat

modulation by a supplementary damping controller (SDC). Fig.4.8 shows the SDC

control block diagram used in the present work [41,42,72,75,78]

provides a modulation for power oscillation damping or small-disturbance stability

improvement control. The SDC block contains a gain, a washout, lead-lag blocks and a

limiter. The washout block is used to make the controller inactive to the input signal dc

offset. The lead-lag blocks are needed to obtain the necessary phase-lead

characteristics.

XSDC,max

XS2 XS1

XSDC,min

XSDC PT

SDC

SDC

sT1sT+

4SDC

3SDC

sT1sT1

++

2SDC

1SDC

sT1sT1

++

SDCK

CHAPTER 4

68

Many different power system quantities have been proposed or used for the input

signal to the SDC. They include voltage phase angle, frequency, line current and

active-power flow. The principal SDC function is to improve the inter-area mode

damping. As there is a strong interaction between active-power and electromechanical

oscillations, the use of active-power flow input appears to be the most common one

[41], which is also adopted in the present work.

he state equation for the s be written in compact

rm as follows (see Appendix A.2.6 for the derivation):

T upplementary damping controller can

fo

Tsusususu P••

+= CxAx (4.21)

where [ ]TSDC2s1ssu Xxx=x is the vector of state variables of the SDC; Asu and Csu

are matrices the elements of which depend on the gains and time constants of the SDC

ontrollers and defined by (A.91).

.7.1 Static Loads odeled as equivale

data for these admittances calculations are obtained from the load-flow study. Thus if a

load bus has a voltag

c

4.7 Load Models 4In this thesis, the static loads are m nt admittances [79]. The needed

certain e VL, active-power PL and reactive-power QL, then the

equivalent load admittance at that bus is given by:

2L

L2

L

LL

VQj

VPY −= (4.22)

e constant admittance form is the most popular one,

) representation of induction motors in the system dynamic model, where required,

may easily be incorporated as a particular case of the generator representation of

Although the static load model of th

other static models such as those based on constant current, constant power and

exponential functions have also been proposed and reported [80-82].

4.7.2 Dynamic Loads (Induction MotorsA


69

dix A.3, the algebraic

motor, to be solved in conjunction with those of the network system, has the form [68]:

Section 4.2. As is derived in Appen stator equation for induction

ms

mm

mr

mm

mrmr (4.24)

ms IZΨPV −= (4.23)

while the rotor flux linkage vector is calculated from:

mmmmm IFΨAΨ +=•

s

In (4.23) and (4.24), ms

ms and IV are the stator voltage and current vectors respectively;

mm

mm

mm

mm

arameters and rotor angular speed. The expressions for these vectors/matrices are

(4.23) and (4.24),

ave the same form as (4.1) and (4.7) respectively, as previously derived for a

In the mechanical axes, the equation of motion is:

(4.25)

in which is the

and ,, FAZP are the matrices the elements of which depend on the machine

p

given in Appendix A.3. Equations as applied to the induction motor,

h

generator.

( ) M/TT Lemr −=ω

•

mrω angular speed of the induction motor; Te is the electromagnetic

torque developed by the motor; TL is the motor load torque which is, frequently,

specified as a function of shaft speed, using load-torque indices, as described in [83],

and M can be determined from: R/H2M ω= .

The motor electromagnetic torque is given by:

( ) [ ]mr

mm

ms

mm

TmseT ΨSIGI += (4.26)

are constant matrices depending on machine parameters and

mm

mm and SG where

defined by (A.110).

CHAPTER 4

70

e number of individual induction motors in a power

ystem, equivalent representation based on grouping of induction motors of similar

rge scale system analysis and design

3].

Multi-Machine Equation System on (4.1) - (4.3

(4.27)

The use of dynamic load model will, therefore, increase the number of state variables.

In practice, due to a very larg

s

dynamic characteristics is often adopted in la

[8

4.8Based ), and (4.8) - (4.10), the state equations for one machine and one

PSS are:

eesmrmr xSIFΨAΨ ++=•

( )egg1 PMω r −= −

•xS (4.28)

Rrr ωωδ −=•

(4.29)

refseppeseeee VV DxSBCxAx +++=

• (4.30)

PD•

(4.31)

Sg and Sp are the selection

Pm and VPSS are obtained from the state vectors xe, xg and xp using these selection

V; xS

grgggg BCxAx +ω+ω+=

••

0mg

ref

rpppp ω+= CxAx (4.32)

In (4.27), (4.28) and (4.30), Se, matrices. The variables Vr,

matrices, i.e. gmeer P ; xSxSV == ppPSSg = .

Based on (4.27) – (4.32), the state equations for multi-machine and multi-PSS system

PxSM −= −•

(4.35)

are:

(4.33) eMeMSMMrMMrM xSIFΨAΨ ++=•

rMω (4.34) ( )eMgMgM1

M

RMrMrM ωωδ −=•

ref sMeMpMpMeMsMeMeMeMeM VDxSBVCxAx +++=•

(4.36)

+ +=•

(4.37) 0MgM

refMgMrMgMgMgMgM PDωBωCxAx +


••

+= ωCxAx (4.38) rMpMpMpMpM

71

where:

[ ] [ ]TNG,s2sT2r

T1rrM VV LL= ΨΨΨ

[ ] [ ][ ] [ ]

] [ ]

[= xx (4.39)

[ ] [[ ] [ ]T2R1RRM

TTNG,s

TT

TTNP,p

T2p

T1ppM

TTNG,g

T1g

TrefNG

ref2

ref1

refM

TNG,e2e1eeM

refNG,s

ref2s

ref1s

refsM

TNG,r2r1rrM

T0001m

0M

TNG,r

1ssMTT

NG,r

ωωω ;

;

;

VVV ; δδδ

P ;

V ;

LL

LL

LL

LL

==

=

ωωω=

==

=

=

ωIIII

xxxxxx

ωxx

Vδ

P

VΨ

in

.38) are given in (A.111); NG and NP in (4.39) are the number of

synchronous machines and PSSs respectively.

Based on (4.5), the electrical power for multi-machine system, PeM, in (4.34) is defined

(4.40)

(4.40),

T

NG,m2m2r1rrM PPωωω LL=ω

]T1ggM = xx

NG,R2s1ssM

Matrices pMpMgMgMgMgMeMpMeMeMeMgMMeMMM and ,,,,,,,,,,,,,, CADBCADSBCASMSFA

(4.33) – (4

as follows:

rMMsMMeM ΨCIBP +=

( ) ( )NG,m2m1mMNG,m2m1mM ,,,diag and ,,,diag CCCCBBBB KK ==In where Bmi and

Cmi are determined based on (4.6).

ained based on (4.7), also needed to

omplete the representation of the dynamic model of multi-machine system:

The following algebraic equation, which is obt

c

sMMrMMsM IZΨPV −= (4.41)

where:

( ) ( )NG,m2m1mMNG,m2m1mM ,,,diag ; ,,,diag ZZZZPPPP LL == (4.42)

In (4.42), Pmi and Zmi are determined based on (A.17).

CHAPTER 4

4.9 State Equation for Multiple FACTS Devices

72

and (4.15) – (4.17), the state equations for one FACTS device

and its main controller are as follows:

or SVC:

Based on (4.11) - (4.13)

F

refTssususTssss VV DxSBCxAx +++=

• (4.43)

or TCSC:

PP E+•

(4.44)

F

Ttsusutttt P DCxSBxAx +++=•

reftTt

For STATCOM:

⎪⎩φ⎪⎪

⎪⎪

⎪⎧

φ+=

++++++=

+++=

=

•

φ

•

φ

•

socso

sosususoCCsoTsorefTsodcsosoc

sususoCCsoTsorefTso

sodc

x

VVVVVx

VVV

VV

NM

LxSKISJOHGF

xSEISDCB

A

⎪

⎪⎨

• (4.45)

For UPFC:

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

+++=

+++++

++++=

=

••

δα

•

••

•••

•

seuseuTuuse

shushudcudcurefdcu

susuususuuTuTurefTush

IVdc

VVV

VVV

xVVV

V

INIMLKV

IJIIHGF

SExSDCBAV

CV

(4.46)

In (4.43) – (4.46), SC and Ssu are the selection matrices and defined by:

[ ] [ ]100 ; 10 suC == SS (4.47)

Extending (4.21) and (4.43) – (4.46), leads to the following state equations for multi-

FACTS-device and multi-SDC system:


73

For SVCs:

refTMsMsuMsuMsMTMsMsMsMsM VDxSBVCxAx +++=

•

where:

(4.48)

[ ] [

][ ] [ ]Trefrefref

1TrefTM

TTNS,su

TT

T

VVV ; LL == Vxxxx (4.49)

ces and ,,, SDBCA in (4.48) are given in (A.112), and NS in (4.49) is

(4.50)

(4.

in (4.50) are giv

NS,T2T1TTMTT

NS,sT2s

T1ssM VVV ; LL == Vxxxx

NS,T2T2su1susuM

Matri suMsMsMsMsM

the number of SVCs.

For TCSCs:

ref•

MtMTMtMTMtMsuMsuMtMtMtMtM PEPDPCxSBxAx ++++=•

where:

51) [ ] [ ]

[ ] [ ]TrefNT

ref2

ref1

refM

TNTsu,su2su1suM

TNT21TM

TTNT,t

T2t

T1ttM

PPP ;xxx

PPP ;

LL

LL

==

==

Px

Pxxxx

Matrices en in (A.113), and NT in (4.51) is the

number of TCSCs.

tMtMtMtMtM and ,,, EDBCA

For STATCOMs:

⎪⎪⎪⎪

⎩ +=

++++++=

+++

•

φ

•

•

φ

MsoMcMsoMM

MsoMsuMsuMsoMCMCMsoMTMsoMrefTMsoMdcMsoMMsoMcM

suMsuMsoMCMCMsoMTMsoMrefTMsoM

MsoMdcM

φNxMφ

VLxSKISJVOVHVGVFx

xSEISDVCV

(4.52)

⎪⎪⎨

=M BV⎪⎪⎧ =

•VAV

CHAPTER 4

74

where:

[ ]

[ ] [ ][

] [ ]TTTTTM

NC,c2c1ccMNC

=φ NC,C2C1CCMNC,21

TT21M

TM21M

T

NC,dc2dc1dcdcM

;VVV

xxx ; VVV

; VVV

IIIIV

xV

φV

LL

LL

LL

=

==

φφφ=⎥⎦

⎤⎢⎣

⎡=

φφφ

(4.53)

and ,,,,,,,,,, SNMLKJOHGFE in

.52) are given in (A.114); Vφi in (4.53) is determined based on (4.14), and NC in

(4.53) is the number of STATCOMs.

For UPFCs:

soMsoMsoM ,,, DCBA CMsoMsoMsoMsoMsoMsoMsoMsoMsoMsoMsoM

(4

⎪⎪⎪⎪

⎩

⎪⎪⎨

++++= suMsuMuMsuMsuMuMTMuMTMuMrefTMuMshM xSExSDVCVBVAV

(4.54)

⎪⎪⎧

+++=

+++++

=

••

δα

•

••

•••

•

seMuMseMuMTMMuMMuMseM

shMuMshMuMdcMuMdcMuMrefdcMuM

IMVMdcM

INIMVVLVKV

IJIIVHVGVF

VCV

where:

[ ] [ ][ ] [ ]

NU,21M

TTTTTTTT

TTNU,sh

T2sh

T1shshM

TTNU,sh

T2sh

T1shshM

VVV ;

;

αααα =

==

==

L

LL

V

IIIIVVVV

atrices CVM

nd VδM are given in (A.115); in (4.55) are obtained based on (4.19) and

(4.20) respectively, and NU in (4.55) is the number of UPFCs.

ng the state equations of the form in (4.21) for individual SDCs of

devices leads to:

(4.56)

[ ] [ ]TTNU,I2I1IIM

NU,se2se1seseMNU,se2se1seseM

VVV

;

= L

LL

V

IIIIVVVV (4.55)

uMuMuMuMuMuMuMuMuMuMuMuMuMuM ,,,,,,,,,,,,, NMLKJIHGFEDCBA ,In (4.54), m

iIi Vand V αa

Assembli the FACTS

TMsuMsuMsuMsuM••

+= PCxAx


where:

75

(4.57)

M and C in (4.56) are given in (A.116), and ND in (4.57

of SDCs.

ction-Motor Equation System state equa

mmmm•

(4.59)

[ ] ( )( ) ( )[ ] [ ]TNI,L2L1LLM

TNI,e2e1eeM

NIm,m2m1MNI,sm

2sTm

1s

mNIm,

mm2

mm1

mM

TmNI,r

m2r

m1r

mrM

mNm,

mm2

mm1

mM

TTmNI,r

Tm2r

Tm1r

TTT ; TTT

M,,M,Mdiag ;

,,,diag ;

,,,diag ;

LL

KL

KL

KL

==

=⎥⎦⎢⎣⎡=

=ωωω=

=⎥⎦⎤

TT

MIIII

FFFFω

AAAAΨΨ

m y

ω= , and NI is the number of induction motors.

hich is obtained based

dynamic model for

(4.61)

here:

[ ] [ ]TND21MTT

ND,suT

2suT

1susuM PPP ; LL == Pxxxx

Matrices suA ) is the number suM

4.10 Multi-InduBased on (4.24) and (4.25), the tions for multi-induction-motor system are:

sMMrMMrM IFΨAΨ += (4.58)

m

( )LMeM1

MmrM TTMω −= −

•

where:

( )ImrM ⎢⎣

⎡= ΨΨ ) ( ) ( ) (

( ) ( )mmmm

TTmTmsM

⎤ (4.60)

In (4.60), mimi and FA are determined based on (A.107), miM is given bm m

Riimmi /H2M

The following algebraic equation, w on (4.23), also needed to

complete the representation of the multi-induction-motor system:

msM

mM

mrM

mM

msM IZΨPV −=

w

( )( )m

NI,mm

2mm

1mmM

mNI,m

m2m

m1m

mM

,,,diag

,,,diag

ZZZZ

PPPP

L

L

=

= (4.62)

CHAPTER 4

76

In (4.62), are determined based on (A.104).

inearisation of Equations For small perturbations (from the initial steady-state operating point), the system

variables change only slightly (i.e., the variable x changes from x0 to x0+Δx where Δx is

small change in x, and superscript 0 denotes the initial condition). Under this

nonlinear s m

can be approx

ux

ansients can be expressed as:

(4.63)

, the following relationship

smrmr (4.64)

gives the linearised form of (4.27) as

rotor equation of motion can also

d as the following:

r

In (4.66), the expression for ΔPe is obtained by linearising generator active-power given

be shown to be of the form (see Appendix A.6 for the

mmi

mmi and ZP

4.11 L

a

condition, the ystem of equations which describe the syste dynamic model

imated by linear system of equations.

4.11.1 Linearisation of Machine and PSS Equations Based on the above discussion, for small perturbations, (4.27) related to rotor fl

tr

)()()( e0ees

0smr

0rmr

0r xxSIIFΨΨAΨΨ Δ++Δ++Δ+=Δ+

••

For initial steady-state condition holds:

000 xSIFΨA0Ψ ++==•

0ee

Substituting (4.64) into (4.63) and rearranging

follows:

eesmrmr xSIFΨAΨ Δ+Δ+Δ=Δ•

(4.65)

By using the similar procedure, (4.28) describing the

be linearise

( )1 PMω Δ−Δ=Δ −•

xS (4.66) egg

in (4.5) which can derivation):


77

(4.67)

in (A.121).

d prime-mover and governor are obtained by linearising (4.29) –

e following:

r3s2r1eP ωΔ+Δ+Δ=Δ KIKΨK

where K1, K2 and K3 are given

The linearised forms of the remaining machines state equations for rotor angle,

excitation controller an

(4.31) as th

(4.68) rr ωδ Δ=Δ•

ppeseeee V xSBCxAx Δ+Δ+Δ=Δ (4.69) •

(4.70)

rgggg ωΔ+Δ=Δ•

CxAx

sVΔ , which is the change o pressed in f the generator voltage, can be ex

In (4.69),

terms of sVΔ as described in the following. The magnitude of Vs is given by:

( ) 2/12q

2ds VVV += (4.71)

Equation (4.71) can be rewritten as:

2q

2d

2s VVV += (4.72)

For small perturbations, (4.72) becomes:

q0s

0qVV

d0s

0d

s VV

VV

V Δ+Δ=Δ (4.73)

be rewritten as:

In vector/matrix form, (4.73) can

s0ssV VE Δ=Δ (4.74)

where:

CHAPTER 4

78

( )T0s0

s

0s

V1 VE =

(4.75), is given by:

(4.75)

0sVIn

[ ]T0q

0d

0s VV=V (4.76)

ubstituting (4.74) into (4.69) gives:

(4.77)

tate equation is given in:

+Δ= CxAx (4.78)

Equations (4.65), (4.66), (4.68), (4.70), (4.77) and (4.78) are the linearised state

PSS only. B

equations for multi-machine and multi-PSS system are assembled and arranged as

llows:

(4.79)

eMgMgM1

MMrM xS −Δ= −

(4.81)

(4.82)

(4.83)

(4.84)

d by:

S

ppes0seeee xSBVECxAx Δ+Δ+Δ=Δ

•

By linearising (4.32), the linearised form of PSS s

rpppp••ωΔΔ

equations for one machine and one ased on these equations, the state

fo

eMeMsMMrMMrM xSIFΨAΨ Δ+Δ+Δ=Δ•

)ω PΔΔ•

(4.80) (

rMrM ωδ Δ=Δ•

pMpMeMsM0sMeMeMeMeM xSBVECxAx Δ+Δ+Δ=Δ

•

rMgMgMgMgM ωCxAx Δ+Δ=Δ•

rMpMpMpMpM••

Δ+Δ=Δ ωCxAx

In (4.82), sM0sM and VE are define


( )

[ ]TTNG,s

T2s

T1ssM

0NG,s

02s

01s

0sM ,,,diag

VVVV

EEEE

L

L

=

= (4.85)

79

where is determined based on (4.75).

Based on (4.67), in (4.80) can be obtained as:

KΨ Δ+Δ

0siE

eMPΔ

rMM3sMM1eM ωKIKP Δ+=Δ (4.86) M2rM

where:

( )(( )NG,33231M3

NG,22221M2

NG,11211M1

,,,diag,,,diag,,,diag

KKKKKKK )KKKK

L

L

L

=

=

=

(4.87), K1i, K2i and K3i are obtained based on (A.121).

nous machine algebraic eq

the following. For small perturbations, the machine voltage equation (4.7) becomes:

K (4.87)

In

The linearisation of the synchro uation (4.7) is described in

( )( )

( ) s0ssrs

1rrsrrrs

1rrsr

0rssrss

0r

r

( ) IIRLLGLLGGG

Ψ

Δ+−ωΔ+ω+ωΔ−ω−−

ΔΔ+−−

(4.88)

or initial steady-state condition:

0r

1rrsrr

1rrsr

0rs

0s ΨLGLGVV +ωΔ+ω= −−

F

( )0ssrs

1rrsr

0rss

0r

0r

1rrsr

0r

0s IRLLGGΨLGV −ω+ω−−ω= −− (4.89)

ubstituting (4.89) into (4.88) and rearranging leads to the linearised form of (4.7) as

follows:

(4.90)

S

r0ms

0mr

0ms ωΔ+Δ−Δ=Δ KIZΨPV

where:

CHAPTER 4

( )[ ]( )

LGP −ω=

ulti-machine system, (4.90) becomes:

(4.92)

here:

0srs

1rrsrss

0r

1rrsr

0m

srs1

rrsrss0r

0m

ILLGGΨLGK

RLLGGZ−−

−

−+=

+−ω−= (4.91)

1rrsr

0r

0m

For m

rM0MsM

0MrM

0MsM ωKIZΨPV Δ+Δ−Δ=Δ

w

( )( )0

NG,m0

2m0

1m0M ,,,diag ZZZZ L= (4.93)

( )0NG,m

02m

01m

0M

0NG,m

02m

01m

0M

,,,diag

,,,diag

KKKK

PPPP

L

L

=

=

mimi determined based on (4.91).

Linearisation of SVC State Equations ocedure described in the previous section, (4.4

linearised to give:

In (4.93), 000 and , KZP are mi

4.11.2 By using the same linearisation pr 3) is

80

sss CxAx +Δ=Δ sususTs V xSB Δ+Δ•

(4.94)

In (4.94), TVΔ can be expr as follows: essed in the form similar to (4.74)

T0TTV VE Δ=Δ (4.95)

where:

( )T0T0

T

0T

T

TT

V1 ;

)V(im)V(re

VEV =⎥⎦

⎤⎢⎣

⎡= (4.96)

ubstituting (4.95) into (4.94) gives:

S


81

xΔ (4.97)

valid for one SVC only. Extending (4.97) gives the linearised state

quation for multi-SVC system as follows:

(4.98)

(4.98), are defined by:

0Tssss SBVECxAx +Δ+Δ=Δ

•

sususT

Equation (4.97) is

e

suMsuMsMTM0TMsMsMsMsM xSBVECxAx Δ+Δ+Δ=Δ

•

TM0TM and VEIn

( )[ ]TT

NG,TT2T

T1TTM

0NG,T

02T

01T

0TM ,,,diag

VVVV

EEEE

L

L

=

= (4.99)

(4.96).

.11.3 Linearisation of TCSC State Equations Linearisation of (4.44) gives the linearised state equation for one TCSC as follows:

(4.100)

ate equation for multi-TCSC system can be obtained and will

have the form:

(4.101)

The expressions for ΔPTM and in (4.101), which are related to transmission line

ived in the next chapter w

presented.

Linearisation of STATCOM State Equations inearised form of the STATCOM state equation can be obtained by linearising (4.45)

where 0TiE is obtained based on

4

TtTtsusutttt

PP••

Δ+Δ+Δ+Δ=Δ DCxSBxAx

Based on (4.100), the st

TMtMTMtMsuMsuMtMtMtMtM PP••


TMP•

Δ

active-power flow, will be der hen the network model is

4.11.4L

as the following:

CHAPTER 4

82

so 02) •

Δ=Δ VVdc A (4.1φ

sususoCCsoTso VV xSEISDC Δ+Δ+Δ= Δ (4.103) •

φ

•Δ+Δ+Δ+Δ+Δ+Δ=Δ VVVVx sosususoCCsoTsodcsosoc LxSKISJOGF (4.104)

socso x NM (4.105)

4), ΔVφ can be determined by linearising (4.14) as th

•

φΔ+Δ=φΔ

In (4.102) and (4.10 e following:

( ) φΔ+φΔ+Δ+φ0

T0T

0 sinVVV (4.106)

Equation (4.

=φV

106) can be simplified by using identity:

( ) φΔφ+φΔφ=φΔ+φ sincoscossinsin 000 (4.107)

For small Δφ:

φΔ≅φΔ≅φΔ sin ; 1cos (4.108)

Substituting (4.108) into (4.107):

( ) φΔφ0ossi (4.109) +φ=φΔ+φ 00 csinn

Subsituting (4.109) into (4.106) gives:

T00 VsinV ΔφΔ+φ (4.110) 00

T00

T cosVsinVV +φΔφ+φ=φ

For initial steady-state condition:

00T

0 sinVV φ=φ

.111) into (4.110) leads to:

(4.111)

Substituting (4


83

Tφ000

T VsincosVV Δφ+φΔφ=Δ (4.112)

nd (4.104) gives:

Substituting (4.112) into (4.102) a

Tdc VV ΔΔ (4.113) 0so

00Tso sincosV φ+φΔφ=

•AA

( ) Tso0

so00

Tso

sususoCCsodcsosoc GF

VsincosV

VVx

Δ+φ+φΔφ+

Δ+Δ+Δ+Δ=Δ•

OLL

xSKISJ (4.114)

o (4.103), (4.113) and (4.114) gives: Substituting (4.95) int

sususoCCsoT0TsoV xSEISDVEC Δ+Δ+Δ=Δ

•

(4.115)

TTsoTsodc sinVV VEA Δφ=Δ (4.116) 0000 cos A+φΔφ•

( ) T0Tso

0so

0

0Tso cosV L φ+

sususoCCsodcsosoc

sin

VVx

VEOL

xSKISJGF

Δ+φ+φΔ

Δ+Δ+Δ+Δ=Δ•

(4.117)

quations (4.105) and (4.115) – (4.117) are collected together and rewritten in a more

TCOM state equation (4.45) as the

llowing:

(4.118)

where:

E

compact form to give the linearised form of the STA

fo

CstTstsustsostso IDVCxBxAx Δ+Δ+Δ+Δ=Δ•

( )⎥

⎥⎥

⎢⎢=

SKB

Cso

Cso

so

sost

⎥

⎦⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+φ

φ

=

⎥⎥⎥⎥

⎦

⎤

⎢

⎢

⎣

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

φ

0J

SDD

0EOL

ECEA

C

0SSE

NM00L0FG

A000

st0Tso

0so

0Tso

0T

0so

stsu

su

soso

0Tsososo

00Tso

c

dc

; sin

sin

;

cosφV

cosφV

x

V

⎢⎢=

⎥⎥

⎢⎢= 0000Ax

0stso ; V

⎤⎡ 00 (4.119)

CHAPTER 4

84

Equation (4.118) is for one STATCOM only. By extending (4.118), the linearised state

TCOM system is:

equation for multi-STA

CMstMTMstMsuMstMsoMstMsoM IDVCxBxAx Δ+Δ+Δ+Δ=Δ (4.120)

where:

•

[ ] ( )( ) ( )

)NC,st2st1ststM

2st1ststMNC,st2st1st

NC,st2st1ststMTT

NC,soT

2soT

1sosoM

,,,,,,diag ; ,,,diag,,,diag ;

BBDDDDAAAACCCCxxxx

L

LL

LL

==

==

(4.121)

are determined based on (4.119).

4.11.5 Linearisation of UPFC State Equations 5) gives the first linearised UPFC state equation as follo

(NC,ststM

diag BB =

In (4.121), stistististi and ,, DCBA

Linearising (4.1 ws:

( ) ( ) ( ) ( ) ⎥⎦⎤

⎢⎣⎡ Δ−Δ+Δ−Δ=Δ

•

seT0

seshT0

shseT0

seshT0

sh0dcdc

dcVC

1V IVIVVIVI (4.122)

he second linearised UPFC state equation is obtained by linearising the second T

equation of (4.46) which can be shown to be of the form:

shushudcudcususuu

susuuTuTush

VV

VV•••

••

Δ+Δ+Δ+Δ+Δ+

Δ+Δ+Δ=Δ

IJIIHGxSE

xSDCBV (4.123)

Substituting (4.95) into (4.123) gives:

(4.124)

by linearising (A.86) which

shushudcudcususuu

susuuT0TuT

0Tush

VV •••

••

Δ+Δ+Δ+Δ+Δ+

Δ+Δ+Δ=Δ

IJIIHGxSE

xSDVECVEBV

The third linearised UPFC state equation can be found

gives:


85

(4.125)

N3 and N4 are given in (A.87).

in the following steps.

Linearising (A.82) leads to:

se4se3se••

Δ+Δ=Δ YNYNV

where

The expressions for and sese

•ΔΔ YY in (4.125) are derived

( ) se2T0se

0se20

Tse V

V1 INYINY Δ+Δ−=Δ (4.126)

In (4.126), is defined by:

0seY

0se20

T1

0se

V1 INNY += (4.127)

ives:

INVN

where N1 and N2 in (4.126) and (4.127) are defined by (A.83).

Substituting (4.95) into (4.126) g

seYΔ = se2T5Δ + Δ (4.128)

here:

w

( ) 0T

0se

0se20

T5

V1 EYINN −= (4.129)

n by:

(4.130)

he expressions fo as formed in (4.128) and (4.130) respectively are

then substituted into (4.125) to give:

The time-derivative of (4.128) is give

se2T5se•••

Δ+Δ=Δ INVNY

r and sese

•ΔΔ YYT

CHAPTER 4

86

•+Δ+ NNVNN

quations (4.122), (4.124) and (4.131) are collected together and rearranged in a more

equation (4.46) as the

llowing:

suucsuuc

••

••

Δ+Δ+Δ+Δ+Δ+

+Δ+Δ+

IIIHIGIFVE

xCxB

se24se253se••

Δ+ΔΔ=Δ INNIVNNV (4.131) 3T54T

E

compact form to give the linearised form of the UPFC state

fo

Tucuucu

•

ΔΔ=Δ VDxAx (4.132)

seucseucshucshucTuc

where:

( ) ( )

( ) ( )

( )

( )

( )

( )⎥⎥⎥⎤

⎢⎢⎢⎡

=⎥⎥⎥⎤

⎢⎢⎢⎡

=⎥⎥⎥

⎢⎢⎢

−=

⎥⎥⎥⎤

⎢⎢⎢

⎣

⎡= sh

dc

uucT0

seu0dcdc

dcdc

ucuuc

V ; ;

VC1VC

; Vx00

IVHHJ0

G

⎦⎣⎦⎣

⎥⎥⎥

⎥

⎦

⎤

⎢⎢⎢

⎢

⎣

⎡ −

⎦

⎥⎥⎥⎥

⎥

⎦

⎤

⎢⎢⎢⎢

⎢

⎣

⎡

+=⎥⎤

⎢⎡

==

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢

⎢⎡ −

=

se2423

T0se0

ushu0dcdc

T0sh0

suuucsuuucT0

seu0dcdc

shu0dcdc

u

T0se0

T0sh0

uc

1

VC

VC1

; ;

; ; VCVC

11

VNNNN

V

0

0

IVH

V

FEC0

EEBD

0SE0

C0SD0

B

000

IHIHG

II0

A

(4.133)

for one UPFC only. Extending it will lead to the

equation for multi-UPFC system as follows:

(4.134)

⎢⎢

−T0

dcdcdcdc11VCVC

⎢⎢⎣

⎥⎥

⎢⎢

⎥⎥⎦⎢

⎢⎣⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡T0

dcdc

uc

54

Tuuc

53

Tuuc1

NNNN

0

Equation (4.132) is linearised state

seMucMseMucMshMucMshMucMTMucM

TMucMsuMucMsuMucMuMucMuM

•••

••

Δ+Δ+Δ+Δ+Δ+

Δ+Δ+Δ+Δ=Δ

IIIHIGIFVE

VDxCxBxAx


87

where:

[ ] ( )( ) (( ) (( ) (

( ) [ ]( ) [ ]TT

NU,seT

2seT

1seseMNU,uc2uc1ucuM

TTNU,sh

T2sh

T1shshMNU,uc2uc1ucuM

NU,uc2uc1ucuMNU,uc2uc1ucucM



NU,uc2uc1ucuMTT

NU,uT2u

T1uuM

; ,,,diag

; ,,,diag

,,,diag ; ,,,diag,,,diag ; ,,,diag

,,,diag ; ,,,diag,,,diag ;

IIIIEEEE

IIIIDDDD

IIIICCCCHHHHBBBB

GGGGAAAAFFFFxxxx

LL

LL

LL

LL

LL

LL

==

==

==

==

==

==

))

) (4.135)

In (4.135), are obtained based on (4.133).

4.11.6 Linearisation of SDC State Equations linearised form of the SDC state equation in (4.21) is:

xtending (4.136) leads to the linearised state equation for multi-SDC system as

follows:

(4.137)

here are given by (A.116).

4.11.7 Linearisation of Induction Motor Equations Based on (4.24) and (4.25), the linearised forms of the state equations for multi-

otor system are as follows (see Appendix A.7 for the derivations)

(4.138)

LMeM1

MmrM TMω Δ−Δ=Δ −

•

uciuciuciuciuciuciuciuciuci and ,,,,,,, IHGFEDCBA

The

Tsusususu P••

Δ+Δ=Δ CxAx (4.136)

E


Δ+Δ=Δ PCxAx

suMsuM and CAw

induction-m :

mrM

0mM

msM

0mM

mrM

0mM

mrM ωLIFΨAΨ Δ+Δ+Δ=Δ•

)T (4.139)(

CHAPTER 4

88

earised form of the induction

te the representation of the

mall-disturbance model of the multi-induction-motor system:

mM

msM

0mM

mrM

0mMsM KIZΨP +Δ−Δ

mimi ,, LFA mimimi and , KZP in (4.138) and (4.140) are defined in (A.125)

nd (A.132).

.12 Conclusions r has presented the dynamic models for individual items of power system.

ms

so been derived and

plete the

presentation of the power system dynamic model in relation to its electromechanical

In addtition to (4.138) and (4.139), the following is the lin

motor stator voltage equation which is needed to comple

s

m0m ωV Δ=Δ (4.140) rM

Matrices m00m0m m00m0m,mi

a

4This chapte

The equations that describe the models are, in general, nonlinear. The linearised for

of the equations valid for small-signal stability analysis have al

presented in this chapter. The developed dynamic models will be used in conjunction

with the network equations which will be formed in the next chapter to com

re

responses.

5 DDYYNNAAMMIICC MMOODDEELLIINNGG:: MMUULLTTII--MMAACCHHIINNEE PPOOWWEERR SSYYSSTTEEMM

5.1 Introduction In Chapter 4, state equations together with their linearised forms have been derived for

individual items of plant related to power generation, load demand and compensation

systems. There is the requirement in stability analysis of representing the interactions

among these items of plant which are interconnected via the power network in forming

the multi-machine power system.

The present chapter will develop a composite set of state equations and algebraic

equations which describes the dynamics of the complete power system in relation to its

electromechanical responses. The composite equations set forms the basis for stability

analysis and coordination of power system controllers, which are the focus of the

thesis.

5.2 Network Model Fig.5.1 shows an NB-node power system considered in this thesis. It is to be assumed

that NG generators are connected to the power system. The network nodal current

vector I and nodal voltage vector V for the system are related by:

YVI = (5.1)

where Y is the system admittance matrix.

All of the quantities in (5.1) are, in general, complex numbers. Separating (5.1) into real

and imaginary parts and rearranging, leads to the following equation where all of the

vector/matrix coefficients are real:

89

CHAPTER 5

NNN VYI = (5.2)

where:

( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( ) ( ) ( )[ ]TNBNB2211N

TNBNB2211N

VimVreVimVreVimVre

IimIreIimIreIimIre

MLMM

MLMM

=

=

V

I (5.3)

and:

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) (( ) ( ) ( ) ( ) ( ) ( ) ⎥

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−

−−−

−−−

=

NB,NBNB,NB2,NB2,NB1,NB1,NB

NB,NBNB,NB2,NB2,NB1,NB1,NB

NB,2NB,222222121

NB,2NB,222222121

NB,1NB,112121111

NB,1NB,112121111

N

YreYimYreYimYreYimYimYreYimYreYimYre



MLMM

MLMM

LLLLMLMLLLLMLLLL

MMMOMMMMMM

LLLLMLMLLLLMLLLL

MLMM

MLMM

LLLLMLMLLLLMLLLL

MLMM

MLMM

Y

)

(5.4)

90

Fig.5.1: Multi-machine power system

IS,NG

IS2

VS1 IS1

Network System

(NB-NG nodes)

VS,NG

VS2

GNG

G2

G1

DYNAMIC MODELING: MULTI-MACHINE POWER SYSTEM

In (5.1), the static loads of constant admittance form and fixed form of reactive power

compensation are included in the system admittance matrix. In this way, the nodal

currents in vector I are non-zero only at generator nodes. Therefore, (5.2) can be

partitioned as follows:

(5.5) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= LN

SN

LLLS

SLSS

LN

SN

V

V

YY

YY

0I

ILL

M

LLLLL

M

LL

where:

S: set of generator nodes

L: set of non-generator nodes

ISN, VSN: vectors of network nodal currents and voltages at generator nodes

respectively

VLN: vector of nodal voltages at the remaining nodes in the system

YSS, YSL, YLS and YLL: submatrices from partitioning of Y matrix

Based on (5.5), the following equations are obtained:

LNSLSNSSSN VYVYI += (5.6)

LNLLSNLS VYVY0 += (5.7)

The nodal voltage vector VSN and nodal current vector ISN in (5.6) and (5.7) are in the

network D-Q frame of reference. By transforming the variables VSN and ISN into their

corresponding d-q components (VsM and IsM) using frame of reference transformation

described in Appendix B.1, (5.6) and (5.7) become:

LNSLsMMSSsMM VYVTYIT += δδ (5.8)

LNLLsMMLS VYVTY0 += δ (5.9)

where:

( )NG,21M ,,,diag δδδδ = TTTT L (5.10)

91

CHAPTER 5

92

In (5.10), Tδi is determined based on (B.8) in Appendix B.1.

It can be seen from the discussion that the network model for multi-machine power

system is described by two sets of algebraic equations (5.8) and (5.9). It is also to be

noted that the algebraic (non-state) variables of the network model of the system are

VsM, IsM and VLN.

The basic network model needs to be augmented with the models of individual FACTS

devices which were reviewed in the previous chapter. The algebraic equations for

interfacing the network model in Section 5.2 with the FACTS devices will be developed

in Sections 5.3 – 5.6, leading to a composite system of algebraic equations which

describe the system voltage-current relationship and represent the interactions

between the network and FACTS devices.

5.3 Multi-Machine Power System with SVCs Fig.5.2 shows a multi-machine power system installed with SVCs. In order to illustrate

the mathematical formulation for modeling the power system network, it is to be

assumed that the SVCs are installed as shown in Fig.5.2. In Fig.5.2, NS is the number

of SVCs.

Fig.5.2: Multi-machine power system with SVCs

Network System (NB-NG nodes)

IS,NG

IS2

IS1

VS,NG

VS2

VS1

GNG

G2

G1

jBC1 SVC 1

x1

jBC2 SVC 2

x2 xNS

jBC,NSSVC NS


93

D-Q frame of reference and partitioned form, the network model for the multi-

(5.11)

he submatrix YFS in (5.11) has the same dimension as submatrix YLL and is given by:

ach 2 2 submatrix in YFS in (5.12) which is identified by the block with i and k

In

machine power system installed with SVCs can be described by:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= LN

SN

FSLLLS

SLSS

LN

SN

V

V

YYY

YY

0I

ILL

M

LLLLL

M

LL

T

E )L,L( ki

being the node number assignments has the following structure:

(5.12)

Lx2

Lx2

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−

−

MMMMMM

MMMMMM

LLMLLMLLMLLMLLMLLMLL

MMMMMM

MMMMMM


MMMMMM

MMMMMM


MMMMMM

MMMMMM


MMMMMM

MMMMMM


MMMMMM

MMMMMM


MMMMMM

MMMMMM

2C

2C

1C

1C

B

B

B

B

FSY

LNB

Lx1

LNG+1

LNG+1 Lx1 LNB

2i

2k

2i-1

2k-1

CHAPTER 5

94

Based on (5.11), the following equation is obtained:

( ) LNFSLLSNLS VYYVY0 ++= (5.13)

to: Transforming the variable VSN in (5.13) into VsM leads

( ) LNFSLLsMMLS VYYVTY0 += +δ

multi-machine power system

stalled with SVCs can be described by two sets of algebraic equations (5.8) and

odel of the system are VsM, IsM and VLN.

ith TCSCs is shown in Fig.5.3. Similar to the

revious discussion, in order to illustrate the mathematical formulation for modeling the

ower system network, it is assumed that the TCSCs are installed as shown in Fig.5.3.

Fig.5.3: Multi-machine power system with TCSCs

(5.14)

The above discussion shows that the network model for

in

(5.14). It is also to be noted that the algebraic (non-state) variables of the network

m

5.4 Multi-Machine Power System with TCSCs A multi-machine power system installed w

p

p

In Fig.5.3, NT is the number of TCSCs.

Network System (NB-NG+NT nodes)

IS,NG

IS2

VS1 IS1

VS,NG

VS2

GNG

G2

G1 x1 w1 Zt1=-jXt1 v1

Line TCSC 1

Zt2=-jXt2 x2 w2 v2 Line TCSC 2

Line wNT xNT Zt,NT=-jXt,NT vNT

TCSC NT


95

The network model for multi-machine power system installed with TCSCs is similar to

that for the system with SVCs, and is given as follows:

(5.15)

where submatrix in (5.15) YFT is defined by:

As each TCSC is connected in series with the transmission line, the TCSC reactance

will augment both the diagonal and off-diagonal 2 2 submatrices of the network nodal

admittance matrix for nodes which are directly connected to the TCSC.

In (5.16), Yt1 and Yt2 are defined by:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= LN

SN

FTLLLS

SLSS

LN

SN

V

V

YYY

YY

0I

ILL

M

LLLLL

M

LL

2t2t

1t1t X

1Y ; X1Y == (5.17)

LNB

Lv2

Lv1

LNG+1

LNG+1 Lx1 Lx2 LNB

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎥⎥⎥⎥

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎢⎢⎢

⎢⎢⎢

⎢⎢⎢⎢

⎣

⎡

−

−

−

−

−

MMMMMMM

MMMMMMM

LLMLMLLMLLMLLMLLMLMLL

MMMMMMM


MMM

MMM


MMMMMMM

MMMMMMM

MLLMLLMLLML

MMMMMMM

MMMMMMM


MM

MMMMMMM

LLMLL

MMMMMMM

MMMMMMM

2t2t

2t

1t1t

1t1t

2t2t

2t2t

Y

Y

YY

YY

YY

YY

⎥⎥

⎢⎢= LLMLLMLMLLFTY

⎥⎥

⎢⎢⎢

−

−

MMMMM

MMMMMMM


1t1t

1t1t

YY

YY

⎢⎢

− MMMM

MMMM 2t

Y

Y

LMLMLLMLLMLLMLLM

Lv1 Lv2

Lx1

Lx2 (5.16)

CHAPTER 5

96

Based on (5.15), the following equation is obtained:

( ) LNFTLLSNLS VYYVY0 ++= (5.18)

Transforming the variable VSN in (5.18) into VsM leads to:

( ) LNFTLLsMMLS VYYVTY0 ++= δ (5.19)

The above discussion shows that the network model for multi-machine power system

installed with TCSCs can be described by two sets of algebraic equations (5.8) and

(5.19). It is also to be noted that the algebraic (non-state) variables of the network

model of the system are VsM, IsM, and VLN.

5.5 Multi-Machine Power System with STATCOMs Fig.5.4 shows a multi-machine power system installed with STATCOMs. It is also

assumed that the STATCOMs are installed as shown in Fig.5.4. In Fig.5.4, NC is the

number of the STATCOMs.

Fig.5.4: Multi-machine power system with STATCOMs

Network System (NB-NG+NC nodes)

IS,NG

IS2

VS1 IS1

VS,NG

VS2

GNG

G2

G1

NB-NG+1

XC1

VC1

IC1 x1

STATCOM 1

NB-NG+2 VC2

XC2

IC2 x2

STATCOM 2

NB-NG+NC VC,NC

XC,NC

xNC IC,NC

STATCOM NC


In D-Q frame of reference and partitioned form, the network model for the multi-

machine power system installed with STATCOMs can be described by the following

equations:

97

⎥⎥⎤

⎢

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡ =

⎥

⎥⎥⎤

⎢

⎢⎢⎡ SNSCSLSSSN V0YYYI

LLLLMLLMLL

MM

LL (5.20)

nodes is identified by C.

In (5.20), VCN and ICN are defined by:

(5.21)

(5.22)


⎥⎥

⎢⎢

⎥⎥⎥

⎢⎢⎢=

⎥⎥

⎢⎢ =

LNLCLLLSLN VYYY0ILLLLMLLMLL

MMLL

⎥⎦⎢⎣⎥⎦⎢⎣ =⎦⎣ − CNCCCLCSCN VYY0YI MM

in which the set of STATCOM

⎥

[ ]TNC,CNC,C2C2C1C1CCN )V(im)V(re)V(im)V(re)V(im)V(re MLMM=V

[ ]TNC,CNC,C2C2C1C1CCN )(im)(re)(im)(re)(im)(re IIIIIII MLMM=

CNLCLNLLSNLS VYVYVY0 ++= (5.23)

CNCCLNCLCN VYVYI +=− (5.24)

VCN and currents ICN are expressed in the

xes. In Appendix B.1, the transformation between the

ased on which (5.23) and

(5.24) are re-expressed in terms of STATCOM voltages and currents in the p-q axes:

The vectors of STATCOM terminal voltages

network D-Q axes. However, it is preferable to work with the p-q axes for individual

STATCOM where the references are the high-voltage nodes of the STATCOM

transformers, as all of the STATCOM controller state equations in Chapter 4 have been

formed in the STATCOM p-q a

network D-Q axes and STATCOM p-q axes is derived, b

CMMLCLNLLsMMLS VTYVYVTY0 αδ ++= (5.25)

CHAPTER 5

98

CMMCCLNCLCMM VTYVYIT αα +=− (5.26)

CM and ICM in (5.25) and (5.26) are vectors of STATCOM voltages and currents in the

n matrix:

V

p-q axes, and TαM is the transformatio

[ ] [ ] ( )NC,21MTT

NC,CT

2CT

1CCMTT

NC,CT

2CT1CCM ,,,diag ; ; αααα === TV TTTIIIIVVV KLL

(5.27)

(5.27), Tαi is determined based on (B.10) in Appendix B.1; VCi and ICi are defined by:

In

[ ] [ ]TCqiCpiCiT

CqiCpiCi II ; VV == IV (5.28)

imilar to the development related to SVC and TCSC in Sections 5.3 and 5.4, the

erator nodes is transfor

xes.

S

vector of voltages VSN at the gen med to VSM in the generator d-q

a

The relationship between the STATCOM terminal voltages and STATCOM dc-side

voltages as shown in Fig.4.4a is required to complete the integration of the STATCOM

model with the network representation. From Fig.4.4a, the relationship for the ith

STATCOM is given by ijdciiCi eVkV φ= , which is directly generalised for NC STATCOMs:

dcMMCM VTV φ= (5.29)

In (5.29), TφM and VdcM are given by:

( ) [ ]TNC,dc2dc1dcdcMNC,21M VVV ; ,,,diag LL == φφφφ VTTTT (5.30)

where:

[ ]Tiiiii sinkcosk φφ=φT (5.

he above development shows that the network model for a multi-machine power

system installed with STATCOMs is described by four sets of nonlinear algebraic

31)

T


99

tate variables.

.6 Multi-Machine Power System with UPFCs

ine power system with UPFCs

equations (5.8), (5.25), (5.26) and (5.29). These equations contain both state and non-

s

5A model for a multi-machine power system installed with UPFCs is shown in Fig.5.5. In

Fig.5.5, NU is the number of UPFCs.

Fig.5.5: Multi-mach

Network System (NB-NG+2*NU nodes)

IS,NG

IS2

IS1

VS,NG

VS2

VS1

GNG

G2

G1

w1 Line

UPFC 1

v1

Xse1

Vsh1

Xsh1

Ise1

- Vse1 +

Ish1

x1

w2 x2 Line

UPFC 2

v2

Xse2

Vsh2

Xsh2

Ise2

- Vse2 +

Ish2

wNU xNU Line

UPFC NU

vNU

Xse,NU

Vsh.NU

Xsh,NU

Ise,NU

- V + se,NU

Ish,NU

CHAPTER 5

100

Similar to the previous discussion, in D-Q frame of reference and partitioned form, the

network model for the multi-machine power system installed with UPFCs is described

by:

(5.32)

In (5.32), the UPFC shunt and series transformer reactances augment the network

nodal admittance matrix. Also, in (5.32), IshN is the vector of the UPFCs shunt currents;

VseN and VshN are the vectors of UPFC series and shunt voltages, which are determined

by the UPFC controllers as discussed in Chapter 4;Yuc is the matrix that relates the

system nodal currents and the UPFC series voltages. These variables are given by:

seNuc

shN

LN

SN

UUULUS

LULLLS

SUSLSS

shN

LN

SN

V

0

Y

0

V

V

V

YY0Y

YYY

0YYY

I

0I

I

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

=

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−

=

LL

LL

LL

LL

MM

LLMLLMLL

MM

LLMLLMLL

MM

LL

LL

(5.33)

Lv,NU

Lv2

LNU

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎥⎥

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−

−

−

−

MM

MM

LLMLM

MM

LLMLM

MM

MM

LLMLM

MM

MM

MM

LLMLM

MM

MM

LLMLM

MM

MM

LLMLM

MM

LLMLM

MM

MM

M

M

LLMLL

M

LLMLL

M

M

LLMLL

M

M

M

LLMLL

M

M

LLMLL

M

M

LLMLL

M

LLMLL

M

M

NU,e

NU,e

NU,e

NU,e

1e

1e

2e

2e

2e

1e

Y

Y

Y

Y

Y

YY

YY

ucY

⎥⎥⎥

⎢⎢⎢ −

MM

MM

LLMLM

M

M

LLMLL

2e

2eY

Y

⎥⎥

− MM

LLMLM

M

LLMLL

YLv1

LNG+1

Lx2

Lx1

LNB

Lx,NU

L1 L2


101

(5.34)

6)

In (5.33), Yei is given by:

[ ]TNU,shNU,sh2sh2sh1sh1shshN )V(im)V(re)V(im)V(re)V(im)V(re MLMM=V

[ ]TNU,seNU,se2se2se1se1seseN )V(im)V(re)V(im)V(re)V(im)V(re MLMM=V (5.35)

[ ]TNU,shNU,sh2sh2sh1sh1shshN )I(im)I(re)I(im)I(re)I(im)I(re MLMM=I (5.3

seiei X/1Y = .


seNucshNLULNLLSNLS VYVYVYVY0 +++= (5.37)

shNUULNULshN VYVYI +=− (5.38)

In addition to (5.37) and (5.38), the following equation is also needed to complete the

description of the network model for the multi-machine power system installed with

UPFCs, where the UPFC series currents are related to network voltages and UPFC

series voltages (see Appendix B.2 for the derivation):

seNseMLNseseMseN VYVMYI += (5.39)

In (5.39), IseN is defined by:

] (5.40)

where YseM and Mse are derived and given by (B.18) and (B.20) respectively, using the

oltage law equation applied to the individual paths between the UPFC series

converter terminal.

ilar to the development related to STATCOMs in Section 5.5, trans

VSN in (5.37) into VsM and the variables VshN, VseN, IshN and IseN in (5.37), (5.38)

VshM, VseM, IshM and IseM), the following equations

[ TNU,seNU,se2se2se1se1seseN )I(im)I(re)I(im)I(re)I(im)I(re MLMM=I

v

Sim forming the

variable

and (5.39) into their p-q components (

are obtained:

CHAPTER 5

102

VTY α shMMLULNLLsMMLST VTYVYVY0 αδ ++= + seMMuc (5.41)

shMMUULNULshMM VTYVYIT αα +=− (5.42)

seMMVTT αα seMLNseseMseMM YVMYI = + (5.43)

braic equations (5.8) and (5.41) -

.43). Similar to other FACTS devices discussed in Sections 5.3 – 5.5, the composite

contain both state and non-s

of Algebraic Equations Linearisations of the algebraic equations for a multi-machine power system which have

vious sections and given by (5.8), (5.9), (5.14), (5.19), (5.2

.26), (5.29) and (5.41) - (5.43) will be discussed in this section.

ure for the algebraic equations is based on partial derivatives of

dividual functions of the variables which appear explicitly in the equations.

r multi-machine power system without FACTS devices, as

ebraic equa

(5.9). Although the voltage and current variables in these equations are in their linear

rm, the overall equations are nonlinear as they depend on the nonlinear functions of

sed in the transformation matrix δM

quations (5.8) and (5.9) are given in the following.

inearising (5.8), the following equation is obtained:

(5.44)

where VshM, VseM, IshM and IseM are given in (4.55).

The above discussion shows that the network model for multi-machine power system

installed with UPFCs is described by four sets of alge

(5

network and UPFC equations tate variables.

5.7 Linearisations

been derived in the pre 5),

(5

Similar to the development of the linearised state equations in Chapter 4, the

linearisation proced

in

5.7.1 Linearised Network Model The network model fo

discussed in Section 5.2, is described by two sets of alg tions (5.8) and

fo

the rotor angles which are u T . The linearisations of

e

L

LNSLrM0VMSSsM

0MSSrM

0IMsM

0M VYδDYVTYδDIT Δ+Δ+Δ=Δ+Δ δδ


where:

( )( )0

NG,I02I

01I

0IM

0NG,

02

01

0M

,,,diag

,,,diag

DDDD

TTTT

L

L

=

= δδδδ

( )0NG,V

02V

01V

0VM ,,,diag DDDD L=

(5.45)

⎢⎢⎣⎥⎦

⎢⎣ δ−δ

=

⎥⎦

⎢⎣ δδ

=δ

0qi

0ri

0ri

riri0Ii

0ri

0ri

ririi

Isincos

cossin

D

T

In (5.45), 0Vi

0Ii

0i and,, DDTδ are defined by:

⎤⎡ δ−δ 000 sincos

⎤⎡⎤⎡ δ−δ−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡

δ−δδ−δ−

=

0di

00

0qi

0di

0ri

0ri

0ri

0ri0

Vi

Icossin

VV

sincoscossinD (5.46)

⎥⎥⎦

Rearranging (5.44) leads to a more compact linearised form of (5.8) as the following:

( ) 0VYδDDYITVTY =Δ+Δ−+Δ−Δ δδ LNSLrM0IM

0VMSSsM

0MsM

0MSS (5.47)

Using the similar procedure, it can be shown that the linearised form of (5.9) is given

by:

00 0VYδDYVTY =Δ+Δ+Δδ LNLLrMVMLSsMMLS (5.48)

e the linearised network equations which are to be

sed in conjunction with the machine voltage equation given by (4.92) to complete the

c equations for the multi-machine power system.

5.7.2 System Installed with SVCs

installed with SVCs is described by two sets of algebraic equations (5.8) and (5.14).

he linearised form of (5.8) has been given in the previous section. The following is the

of (5.14):

Equations (5.47) and (5.48) describ

u

algebrai

As discussed in Section 5.3, the network model for multi-machine power system

T

linearisation

103

CHAPTER 5

104

( ) 0BVVYYδDYVTY =Δ+Δ++Δ+Δδ CM0LNLN

0FSLLrM

0VMLSsM

0MLS (5.49)

where:

[ ]TNS,C2C1CCM BBB L=B (5.50)

and:

1 2 NS

Matrix 0LNV in (5.51) comprises )NGNB(NS −× 2 1 submatrix blocks. However, a

block has nonzero elements as indicated in (5.51) only if it is associated with an SVC.

Network node numbers are used to identify the rows of the matrix ranging from NG+1

to NB. For expressing in the rea ), each node corresponds to two rows

entified by the set in Lj for node j which is

l form in (5.51

id )}NGj(2 ,1)NGj(2{ −∗−−∗ .

s it is preferable to use x for the state variable vector, (5.49) is rewritten in the

form where SVC susceptances are expressed in terms of x:

A

following

( ) 0xSVVYYδDYVTY =Δ+Δ++Δ+Δδ sMCM00LNLN

0FSLLrM

0VMLSsMMLS (5.52)

where SCM is given by (A.114).

(5.51)

⎥⎥⎥

⎥⎥

⎥⎥⎥⎥⎥

⎥⎥⎥⎥⎥⎥⎥

⎥

⎦

⎤

⎢

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎢⎢⎢⎢⎢⎢⎢

⎢

⎣

⎡

−

−

=

L

M

M

LM

M

LM

M

M

M

M

LM

M

M

MM

MM

LMLML

MM

LMLML

MM

MM

MM

LMLML

MM

MM

)V(im

)V(re

)V(re)V(im

0

0NS,x

02x

2x

01x

01x

0LNV

⎥⎥

⎢⎢

LM

M

LM

M

LM

M

L

Lx1

⎥⎥

⎢⎢

−LM

MM

LMLML

)V(im 0

Lx2

⎥⎥

Lx,NS

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢⎢⎢

M

M

MLM

M

L

M

M

M

L

M

M

M

L

LMLML

MM )V(re NS,x

LNB

LNG+1


105

The above discussion concludes that the linearised form of the network model for multi-

machine power system installed with SVCs is described by (5.47) and (5.52). These

equations together with (4.92) are the required algebraic equations for power system

installed with SVCs.

5.7.3 System Installed with TCSCs The network model for a multi-machine power system installed with TCSCs which has

been derived in Section 5.4 is described by two sets of algebraic equations (5.8) and

(5.19). Linearising (5.19) gives the following equation:

( ) ( ) 0XYYVYYδDVTY =Δ+Δ++Δ+Δδ tM0tM

0VDLN

0FTLLrM

0VMsM

0MLS (5.53)

where:

1 2 NT

[ ]TNT,t2t1ttM XXX L=X (5.55)

( )0NGNB,V

02V

01V

0VD ,,,diag −= YYYY K (5.56)

(5.54)

Lv,NT

Lx,NT

⎥⎥⎥⎥⎥⎥

⎥⎥⎥⎥

⎥⎥

⎥⎥⎥⎥

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎢⎢

⎢⎢

⎢⎢⎢⎢

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

MM

LLLLLLL

MMM

LLLLLLL

MMM

LLLLLLL

LLLLLLL

M

LLL

LLL

MMM

LLLLLLL

MMM

0NT,t

0

tM

Y

Y

LNG+1

= MM

LLLL

MMM

LLLLLLL

MMM

LLLLLLL

MMM

LLLL

0NT,t

02t

01t

0

Y

Y

Y

Lx2

Lx1

M

MMM

LLLLLLL

MMM

02t

1t

Y

Y

Lv2

Lv1

LNB

CHAPTER 5

The rows and columns identifiers for matrix 0tMY in (5.54) are similar to those adopted

in matrix 0LNV of (5.51). NG)-NB,1,2,(i 0

Vi K=Y in (5.56) is a subvector with two

elements which are the D and Q components of the nodal

ent from TCSCs is not included

currents at the ith load node

(the contribution of curr in the calculations of these

nodal currents). The set of individual subvectors is calculated from nodal voltages

odes and generator nodes as follows:

0ViY

at the load n

0LNLL

0SNLS

0VM VYVYY += (5.57)

Individual subvectors s'0ViY are obtained in 0

VMY in (5.57) and used in (5.56) to form

0VDY . Using 0

VDY instead of 0VMY achieves a consistency in the dimensions of the

vectors and matrices in the vector/matrix operations in (5.53).

Equation (5.53) can be rewritten as the following to give a more preferable linearised

form of (5.19) where the vector of TCSC reactances XtM has been expressed in terms

of TCSC state vector xtM:

( ) ( ) 0xSYYVYYδDVTY =Δ+Δ++Δ+Δδ tMCM0tM

0VDLN

0FTLLrM

0VMsM

0MLS (5.58)

Equations (5.47) and (5.58) are the linearised network equations for multi-machine

power system installed with TCSCs. Similar to the discussion in the previous sections,

these equations and the linearised machine voltage equation (4.92) are the complete

algebraic equations for a power system installed with TCSCs.

.7.4 System Installed with STATCOMs .5, the network model for a

stalled with STATCOMs is described by four sets of algebraic equations (5.8), (5.25),

ST

(5.29) are given in the following.

(5.59)

5As discussed in Section 5 multi-machine power system

in

(5.26) and (5.29). The linearisations of the ATCOM equations (5.25), (5.26) and

By linearising (5.25), the following STATCOM equation in a linear form is obtained:

0αAYVTYVYδDYVTY =Δ+Δ+Δ+Δ+Δ αδ M0VMLCCM

0MLCLNLLrM

0VMLSsM

0MLS

106


where:

107

( ) ( )0NC,V

02V

01V

0VM

0NC,

02

01

0M ,,,diag ; ,,,diag AAAATTTT LL == αααα (5.60)

(5.60) are given by:

⎦⎢⎣⎦

α 0Cqi

00000i V

sinT

iables in D-Q axes which have been adopted for forming the

etwork equations. In Appendix B.3 is derived the transformation between voltage

es and D-Q voltage components, which is used to re-express

quation (5.59) in the following:

, 0Vi

0i and ATαIn

⎥⎤

⎢⎡⎥⎤

⎢⎣

⎡

α−αα−α−

=⎥⎦

⎤⎢⎣

⎡

ααα−α

=0Cpi

ii

0i

0i0

Viii

0i

0i V

sincoscossin ;

cossincos A (5.61)

⎥

In the linearised equation (5.59), the vector of voltage phase angles at the high-voltage

nodes of the STATCOM transformers, Mα , is used. However, it is required to express

Mα in terms of voltage var

n

phase angl the linearised

e

( ) 0δDYVTYVMMAYYVTY =Δ+Δ+Δ++Δ ααδ rM0VMLSCM

0MLCLNVM

0M

0VMLCLLsM

0MLS (5.62)

where 0 and MM in (5.62) are given by (B.30) and (B.32), VMMα respectively.

(5.26) is linearised to give:

The second set of STATCOM equations as given in

M0VMCCCM

0MCCLNCLM

0IMCM

0M αAYVTYVYαAIT Δ+Δ+Δ=Δ−Δ− αα (5.63)

where:

( )0NC,I

02I

01I

0IM ,,,diag AAAA L= (5.64)

, is determined by using:

0IiAIn (5.64)

(5.65) ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡

α−αα−α−

= 0Cqi

0Cpi

0i

0i

0i

0i0

Ii II

sincoscossinA

CHAPTER 5

108

ing into the D-Q axes gives:

MαSimilar to equation (5.59), transform

( ) 0ITVTYVMMAMMAYY =Δ+Δ+Δ++ αααα CMCL0MCM

0MCCLNVM

0M

0IMVM

0M

0VMCC (5.66)

he linearised form of the third STATCOM set of equations in (5.29) is given by:

0M

T

0xSMV =Δ−Δ cMstMCM (5.67)

where:

φ

( )( )NC,st2st1ststM

0NC,

02

01

0MMφ

,,,diag,,,diag

SSSSMMM

L

L

=

= φφφ (5.68)

(5.68), are determined by using:

⎡⎤⎡ φ−φ 0001sinVkcosk 0i

0dcii

0ii0 (5.69)

(5.62), (5.66) and (5.67) describe the linearised equations for the network

ith STATCOMs. These equations are used in conjunction with the machine voltage

given by (4.92) to give the algebraic equations for a multi-machine power

ystem installed with STATCOMs.

.7.5 System Installed with UPFCs ork chine power system installed with UPFCs, as

iscussed in Section 5.6, is described by four sets of algebraic equations (5.8) and

(5.41) - (5.43). The linearisations of (5.41) - (5.43) are given in the following in relation

(5.70)

st0i and SMφIn

⎥⎦

⎢⎣

=⎥⎦

⎢⎣ φφ

=φ 1000 ;

cosVksink sti0i

0dcii

0ii

i SM ⎤

Equations

w

equation

s

5The netw model for a multi-ma

d

to UPFCs.

Linearising the first set of UPFC equations leads to:

M0VEMucseM

0MucM

0VHMLUshM

0MLU

LNLLrM0VMLSsM

0MLS

αAYVTYαAYVTYVYδDYVTY0

Δ+Δ+Δ+Δ+

Δ+Δ+Δ=

αα

δ


where:

( ) ( )0NU,VE

02VE

01VE

02VH

01VH

0VHM ,,,,diag AAAAAA L= (5.71) 0

VEM0

NU,VH diag ; ,, AAL =

are determined by:

⎦⎢⎣⎦⎢⎣

⎡

α−αα−

=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤

⎣ α−αα

seqii0i

0VEi0

shqi

0shpi

0ii

0i

Vsincossin ;

VV

sincoscos A

M respectively, and the vector of voltage

hase angles , are used in the linear equation in (5.70). To achieve a consistency in

notation by which x is used for the state variable vector, VshM and VseM are grouped and

x. In addition, the voltage phase angles,

ansformed into voltage variables expressed in D-Q axes for the network. The working

, 0VEi

0VHi and AAIn (5.71)

⎥⎤

⎢⎡⎥⎤α−

⎢⎡ −α−

= 0

0sepi

0

0i

0i

0

0i0

VHiVcossinA (5.72)

⎥

Shunt and series voltage sources, VshM and Vse

, Mαp

included in state vector , needs to be Mα

tr

in the following will lead to the final form for the UPFC first set of linearised equations.

It can be shown that the following equations hold:

uMseMseMuMshMshM ; xLVxLV == (5.73)

where:

( ) ( )NU,se2se1seseMNU,sh2sh1shshM ,,,diag ; ,,,diag SSSLSSSL LL == (5.74)

In (5.74), the selection matrices Sshi and Ssei are given by:

⎥⎦

⎢⎣ 10000

010000100 seishi (5.75)

into the D-Q axes gives the final form of

C first set of linearised equations as follows:

⎡

=⎤⎡

=0

; 00010

SS ⎥⎦

⎤⎢⎣

MαOn using (5.73) in (5.70) and transforming

the UPF

( )

( ) 0xLTYLTYδDYVMMAYMMAYYVTY

=Δ++Δ+

Δ+++Δ

αα

ααδ

uMseM0MucshM

0MLUrM

0VMLS

LNVM0

M0VEMucVM

0M

0VHMLULLsM

0MLS

(5.76)

109

CHAPTER 5

110

d set of UPFC in (5.42) is linearised to give:

(5.77)

here:

The secon equations as given

M0VHMUUshM

0MUULNULM

0IHMshM

0M αAYVTYVYαAIT Δ+Δ+Δ=Δ−Δ− αα

w

( )L,,diag 02IH

01IH

0IHM AAA = (5.78)

In (5.78), 0IHiA is determined by:

⎤⎡⎤⎡ α−α− 0shpi

0i

0i0 Icossin

⎢⎢⎣⎥⎦

⎢⎣ α−α

= 0shqi

0i

0i

IHi IsincosA (5.79)

imilar to equation (5.76), using (5.73) in (5.77) and transforming into the D-Q axes

give:

⎥⎥⎦

αS M

( ) 0xLTYITVMMAMMAYY =Δ+Δ+Δ++ αααα uMUUUL shM0MUUshM

0MLNVM

0MIHMVMMVHM (5.80)

.43) is given by:

(5.81)

here:

000

The linearised form of the third UPFC set of equations in (5

M0VEMseMseM

0MseMLNseseMM

0IEMseM

0M αAYVTYVMYαAIT Δ+Δ+Δ=Δ+Δ αα

w

( )L,,diag 2IE1IEIEM AAA = 000 (5.82)

In (5.82), iA is given by:

⎢⎣⎥⎦

⎢⎣ α−α 0

seqi0i

0i

IEi Isincos(5.83)

sing (5.73) in (5.81) and transforming into the D-Q axes lead to:

0IE

⎤

⎢⎡⎤⎡ α−α−

=0sepi

0i

0i0 IcossinA

⎥⎥⎦

MαU


( ) 0xLTYITVMMAMMAYMYseM =Δ+Δ−Δ−+ αααα uMseM0MseMseM

0MLNVM

0M

0IEMVM

0M

0VEMseMse

(5.84)

quations (5.76), (5.80) and (5.84) are the linearised network equations for multi-

pow ed with UPFCs. Similar to other FACTS devices

iscussed in Sections 5.7.2 – 5.7.4, these equations and the linearised machine

voltage equation (4.92) are the complete algebraic equations for a power system

ll of the state and algebraic equations, and also the corresponding state and non-

multi-machine system installed with FACTS devices are

summarised in this section.

Summary of State and Non-State Variables d non-state variables for the dynamic models of

multi-machine system installed with FACTS devices which have been presented in the

Table 5.1: State and non-state variables for multi-machine system

Power System State Variable Non-State Variable

E

machine er system install

d

installed with UPFCs.

5.8 Summary of Variables and Nonlinear Equations A

state variables, for the

5.8.1Table 5.1 summarises the state an the

previous discussion.

with PSSs pMgMeMrMrMrM ,,,,, xxxδωΨ LNsMsM ,, VIV with SVCs (+SDCs) suMsMpMgMeMrMrMrM ,,,,,,, xxxxxδωΨ LNsMsM ,, VIV

with TCSCs (+SDCs) suMtMpMgMeMrMrMrM ,,,,,,, xxxxxδωΨ LNsMsM ,, VIV with STATCOMs

(+SDCs) suM CMCMLNsMsM ,,,, IVVIVcMpMgMeMrMrMrM ,,,,,,, xxxxxδωΨ

with UPFCs (+SDCs)

suMuMpMgMeMrMrMrM ,,,,,,, xxxxxδωΨ seMshMLNsMsM ,,,, IIVIV

Also included in the lists of state variables and non-state variables in Table 5.1 are

those for generators and their excitation controllers together with prime mo

governors

vers and

111

CHAPTER 5

5.8.2 Summary of State Equations Based on (4.34) – (4.39), (4.48), (4.50), (4.52), (4.54) and (4.56), the following

equations are obtained and presented to summarise the state equations for the

dynamic model of a multi-machine system installed with FACTS devices.

112

tate equations for machines and PSSs:

(5.85)

(5.86)

(5.87)

(5.88)

(5.89)

(5.90)

D S devices in the power system, the set of state

equa 5.90) is augmented with those for individual FACTS devices, and

w DCs, as follows:

tate equation for TCSCs:

(5.92)

S

eMeMsMMrMMrM xSIFΨAΨ ++=•

•

( )eMgMgM1

MrM PxSMω −= −

RMrMrM ωωδ −=•

ref•

sMeMpMpMeMsMSMeMeMeMeM VDxSBVECxAx +++=

0MgM

refMgMrMgMgMgMgM PDωBωCxAx +++=

•

rMpMpMpMpM••

+= ωCxAx

epending on the types of FACT

tions in (5.85) – (

here applicable, their S

State equation for SVCs:

refTMsMsuMsuMsMLNVMTMsMsMsMsM VDxSBVMECxAx +++=

• (5.91)

S

refMtMTMtMTMtMsuMsuMtMtMtMtM PEPDPCxSBxAx ++++=

••


State equations for STATCOMs:

113

⎪⎪⎩ +=

•

MsoMM1ssoMM φNxMφ

⎪⎪

⎪⎪

+•

φ

φ++++++=•

suMsuMsoMCMCM

MsoMdcM

MsoMsuMsuMsoMCMCMsoMLNVMTMsoMrefTMsoMdcMsoMMsoMcM

xSEIS

VLxSKISJVMEOVHVGVFx

(5.93)

tate equations for UPFCs:

+++=

+++++

++++=

=

••

δα

•

••

•••

seMuMseMuMLNVMTMMuMMuMseM

shMuMshMuMdcMuMdcMuMdcMuM

suMsuMuMLNVMTMuMLNVMTMuMref

IMVMdcM

x

INIMVMEVLVKV

IJIIVHVGVF

SExSDVMECVMEBVAV

VCV

in (5.93), VIM, VαM and VδΜ in (5.94) are

etermined based on (4.6), (4.14), (4.19) and (4.20). Also, ESM in (5.88), MVM in (5.91),

in (5.91), (5.93) and (5.94) are defined in (B.67), (B.32) and

(B.49), respectively. The expressions for PTM in (5.92) and in (5.92) and (5.95) are

B.6 and given by (B.97) and (B.104) respectively. Th

in (5.94) are also derived in Appendix B.6 and given by (B.155).

5.8.3 Summary of Algebraic Equations ), (5.19), (5.25), (5.26), (5.29), (5.41) - (5.43)

andard form where the right hand side is zero, and

⎧ =•

VAV

⎪⎪⎨

++= soMLNVMTMsoMrefTMsoMM DVMECVBV

S

⎨ref

suMsuMuMTMuMshM

⎪⎪⎪

⎩

⎪⎪⎪⎧ •

(5.94)

State equations for SDCs:


+= PCxAx (5.95)

It is to be noted that PeM in (5.86), VφM

d

(5.93) and (5.94) and ETM

TM•P

derived in Appendix e expressions

for seMshMLN and ,•••IIV

Equations (4.41), (5.8), (5.9), (5.14 are

now re-arranged to achieve a st

CHAPTER 5

114

collected together in Table 5.2 to summarise the algebraic equations for the multi-

machine system models installed with FACTS devices.

Table 5.2: Algebraic equations for multi-machine system

Model Equations

Multi-Machine System

0IZΨPV =+− sMMrMMsM 0VYITVTY =+− δδ LNSLsMMsMMSS

0VYVTY =+δ LNLLsMMLS Multi-Machine System

Installed with SVCs 0IZΨPV =+− sMMrMMsM

0VYITVTY =+− δδ LNSLsMMsMMSS ( ) 0VYYVTY =++δ LNFSLLsMMLS

Multi-Machine System Installed with TCSCs


( ) 0VYYVTY =++δ LNFTLLsMMLS Multi-Machine System

Installed with STATCOMs 0IZΨPV =+− sMMrMMsM

0VYITVTY =+− δδ LNSLsMMsMMSS 0VTYVYVTY =++ αδ CMMLCLNLLsMMLS

0ITVTYVY =++ αα CMMCMMCCLNCL 0VTV =− φ dcMMCM

Multi-Machine System Installed with UPFCs


0VTYVTYVYVY =+++ ααδ seMMucshMMLULNLLsMMLST ITVY 0VTY =++ α shMMUU α shMMLNUL

0VTYITVMY =+− αα seMMseMseMMLNseseM

5.9 Summary of Linearised Equations The detailed derivation of the linearised equations from he nonlinear equations

summarised in Section 5.8 has been given in Chapter 4 and Section 5.7 in Chapter 5.

thei

t

In the following sections, are summarised both the state and non-state equations in

r linear forms.

tions

5.9.1 Summary of Linearised State EquaBased on (4.79) - (4.84), (4.98), (4.101), (4.120), (4.134) and (4.137) the following

equations are obtained and presented to summarise the linearised state equations for

the dynamic model of multi-machine system installed with FACTS devices.


115

inearised state equations for machines and PSSs:

IFΨAΨ Δ+Δ=Δ•

(5.96)

(5.97)

(5.98)

(5.99)

(5.100)

(5.101)

Linearised state equation for SVCs:

xSBVMECxA Δ+Δ+Δ

Linearised state equation for TCSCs:

(5.103)

uation for UPFCs:

inearised state equation for SDCs:

(5.106)

L

eMeMsMMrMMrM xS Δ+

( )eMgMgM1

MMrM PxSω Δ−Δ=Δ −•

rMrM ωδ Δ=Δ•

pMpMeMsM0sMeMeMeMeM xSBVECxAx Δ+Δ+Δ=Δ

•

rMgMgMgMgM ωCxAx Δ+Δ=Δ•

rMpMpMpMpM••

Δ+Δ=Δ ωCxAx

sMx =Δ•

suMsuMsMLNVM0TMsMsMsM (5.102)

TMtMTMtMsuMsuMtMtMtMtM PP••


Linearised state equation for STATCOMs:

CMstMLNVMstMsuMstMcMstMsoM IDVMCxBxAx Δ+Δ+Δ+Δ=Δ•

(5.104)

Linearised state eq

seMucMseMucMshMucMshMucMLNVMucM

LNVMucMsuMucMsuMucMuMucMuM

•••

Δ+Δ+Δ+Δ+Δ+

Δ+Δ+Δ+Δ=Δ

IIIHIGIFVME

VMDxCxBxAx (5.105)

••

L


Δ+Δ=Δ PCxAx

CHAPTER 5

116

), in (5.99) and i

in (5.103) and (5.106) are derived in Appendix B.6 and given by (B.110) and (B.111)

respectively. The formulas for •

ΔΔΔ IIV in (5.105) are also derived in

by (B.165).

quations (4.92), (5.47), (5.48), (5.52), (5.58), (5.62), (5.66), (5.67), (5.76), (5.80) and

ected together in the following to summarise the

ystem linearised algebraic equations.

inearised algebraic equations for multi-machine system:

⎧

=Δ+Δ+Δ

=Δ−+Δ+Δ−Δ

=Δ−Δ−Δ+Δ

δ

δ

0δDYVYVTY0δDDYVYITV

0ωKΨPIZV

rM0VMLSLNLLsM

0MLS

rM0IM

0VMSSLNSLsM

0MsMM

rM0MrM

0MsM

0MsM

ic equations for multi-machine system installed with SV

It is to be noted that ΔPeM in (5.97 n (5.102) are defined by 0SME 0

TME

(4.86), (B.72) and (B.60), respectively. The formulations for ΔPTM in (5.103) and TMΔP •

seMshMLN and ,••

Appendix B.6 and given

5.9.2 Summary of Linearised Algebraic Equations E

(5.84) are now rearranged and coll

s

L

⎪⎪⎨ δTY 0

SS (5.107) ( )⎪⎪⎩

Linearised algebra Cs:

( )( )⎪

⎪⎩

⎪⎪⎨

⎧ −Δ−Δ+Δ KΨPIZV rM0MsM

0MsM

=Δ+Δ+Δ++Δ

=Δ−+Δ+Δ−Δ

=Δ

δ

δδ

0xSVδDYVYYVTY0δDDYVYITVTY

0ω

sMCM0LNrM

0VMLSLN

0FSLLsM

0MLS

rM0IM

0VMSSLNSLsM

0MsM

0MSS

rM0M

(5.108)

lgebraic equations for multi-machine system installed with TCSCs:

⎧

=Δ+Δ++Δ+Δ

Δ−+Δ+

=Δ−Δ−Δ+Δ

δ 0xSYYVYYδDVTYDDYVYI

0ωKΨPIZV

tMCM0tM

0VDLN

0FTLLrM

0VMsM

0MLS

0IM

0VMSSLNSLsM

rM0MrM

0MsM

0MsM

Linearised a

⎪⎪⎨ =Δ−Δ δδ 0δTVTY rM

0MsM

0MSS (5.109) ( )

( ) ( )⎪⎪⎩


117

ti-m

Linearised algebraic equations for mul achine system installed with STATCOMs:

( )( )

⎩ =Δ−Δ

=Δ+Δ+Δ

Δ+Δ

φ

αα

αα

δδ

0xSMV0ITVTYV

VTYVM

cMstM0MCM

CM0MCM

0MCCLNVM

0MLCLNVM

0M

rMIMVMSSLNSLsMMsMMSS

with UPFCs:

)( )

( )

( )⎪

⎪

⎩

⎪⎪

⎨

=Δ+Δ−Δ−+

=Δ+

=Δ++Δ+

Δ+++Δ

=Δ−

αααα

α

αα

ααδ

0xLTYITVMMAMMAYMY0xLTY

0xLTYLTYδDYVMMAYMMAYYVTY

0δDD

uMseM0MseMseM

0MLNVM

0M

0IEMVM

0M

0VEMseMseseM

uMshM0MUU

uMseM0MucshM

0MLUrM

0VMLS

LNVM0

M0VEMucVM

0M

0VHMLULLsM

0MLS

rM0IM

0VMSS

(5.111)

5.10 System State Matrix ection 5.8 shows that the dynamic model of the power system installed with FACTS

evices is described by a set of differential-algebraic equations (DAEs) which can be

here: x is the vector of state variables; w is the vector of non-state (algebraic)

ariables; f and g are nonlinear vector functions the individual expressions of which

As shown in Section 5.9, the small-disturbance model of the power system installed

S device is obtained by linearising the differential-algebraic e

can also be written in a more compact form as follows:

(⎪⎪⎪⎪⎪

⎨

⎧

++

=Δ+++Δ

=Δ−+Δ+Δ−Δ

=Δ−Δ−Δ+Δ

αα

δ

MMAMMAYY0δDYMAYYVTY

0δDDYVYITVTY0ωKΨPIZV

0M

0IMVM

0M

0VMCCCL

rM0VMLSCM

0VMLCLLsM

0MLS

0000rM

0MrM

0MsM

0MsM

(5.110)

)⎪

Linearised algebraic equations for multi-machine system installed

(⎪⎪⎧

+Δ+Δ−Δ

=Δ−Δ−Δ+Δ

δδ YVYITVTY0ωKΨPIZV

LNSLsM0MsM

0MSS

rM0MrM

0MsM

0MsM

( )⎪⎪ Δ+Δ++ ααα ITVMMAMMAYY shM

0MLNVM

0M

0IHMVM

0M

0VHMUUUL

S

d

written in a more compact for as the following:

(5.112) )()(

wx,g0wx,fx

==

•

w

v

have been given in Section 5.8.

with FACT quations which

CHAPTER 5

118

⎜⎜⎝⎟⎟⎠

⎜⎜⎝

=⎟⎠

⎜⎝ wJJ0 3 Δ4

(5.113)

where J1, J2 , J3 , and J4 are matrices the elements of which are defined in Section 5.9,

sed on the power system initial operating condition and the parameters of the

system together with its controllers.

y reducing (5.113), the following equation is obtained:

⎟⎟⎠

⎞⎛⎞⎛⎟⎞

⎜⎛Δ

• xJJx Δ21

ba

B

xAx Δ=Δ•

(5.114)

where is the system state matrix needed for evaluation the dynamic

characteristic of the power system. The state matrix is the function of controllers (PSSs

and FACTS devices together with their SDCs) parameters.

.11 System with FACTS Devices of Different Types installed with various FACT devices have

installed in the system) can be directly

e equations as previously derived.

combinations of various FACTS devices are given in Appendix B.7.

velopment in this chapter is the composite set of

ifferential-algebraic equations in the linear form which describe the power system

31

421 JJJJA −−=

5The network equations for a power system

been discussed in the previous sections. Based on the discussion, the network

equations for multi-machine power system installed with various types of FACTS

devices (more than one type of FACTS devices

expanded, drawing on individual FACTS devic

For illustration and completeness, equation sets of power systems with some

5.12 Conclusions The main outcome of the de

d

dynamics applicable to small disturbances. Subsequent chapters in the thesis will draw

on this equation set for stability analysis and control coordination design of PSSs and

FACTS device SDCs.

119

6.1 Introduction

Within the category of fixed-parameter power system damping controllers, a

comprehensive review of the previously-published control coordination methods was

carried out and presented in Chapters 2 and 3 of the thesis. Key deficiencies and/or

disadvantages encountered in these methods including those based on LMI and H∞

methods were identified and presented in the review.

For the eigenvalue-based control coordination methods, the key issues that remain to

be addressed are:

(i) Robustness of the controllers designed. Effective and efficient techniques are

needed for obtaining robust controllers, particularly with respect to changes in

power system configurations.

(ii) Sparsity formulation. The control coordination procedure needs to take into account

the sparsity in the power system Jacobian matrix, and at the same time avoid the

separate eigenvalue calculations at each iteration in the control co-ordination. This

is an important requirement, particularly in the context of large power system.

Given the above background, the objective of the present chapter is to develop a new

eigenvalue-based control coordination design of multiple PSSs and FACTS devices

together with their SDCs, which addresses the above two issues (i) and (ii).

The coordination procedure proposed draws on constrained optimisation in which the

eigenvalue-based objective function is minimised to identify the optimal controller

parameters. A key advance is that there is no need for any special software system for

6 OOPPTTIIMMIISSAATTIIOONN--BBAASSEEDD CCOONNTTRROOLL CCOOOORRDDIINNAATTIIOONN:: DDEESSIIGGNN PPRROOCCEEDDUURREE

CHAPTER 6

120

eigenvalue calculations. In the method proposed, the nonlinear relationships amongst

eigenvalues and controller parameters are expressed as eigenvalue-eigenvector

equations associated with the electromechanical modes selected in the coordination.

These equations are included directly in the optimisation in the form of equality

constraints. Therefore, for a large power system, the method lends itself to sparsity

formulation in which the sparse Jacobian matrix is used directly in forming the

eigenvalue-eigenvector equations. Sparse optimisation technique based on the Newton

algorithm [84] then provides a fast and efficient solution method for the coordination

problem in large power systems. The algorithm does not require separate eigenvalue

calculations at each iteration during the control co-ordination.

Special constraints in addition to those representing the eigenvalue-eigenvector

equations are derived in the paper to guarantee that the modes are distinct ones in the

optimisation process. By comparison, it is quite difficult, if not infeasible, to apply

sequentially the deflating procedure [85] when several modes are to be considered

simultaneously in the design.

The present work also addresses the issue of robustness in the control coordination

design through extending the set of constraints. The additional constraints are those

related to eigenvalue-eigenvector equations and eigenvalues of the power systems

with changes in configurations and/or load demands.

Based on detailed comparisons, the advantages of the new control coordination design

method, over other previous methods, are given and discusses in the chapter.

Although the principal application is in the optimal control coordination, the procedure

developed can also be adapted for calculating selected eigenvalues and eigenvectors

associated with the electromechanical modes, for known controller parameters. The

use of the QR method which is not suitable for large power systems is avoided

altogether. The design method discussed in this chapter has also been presented in

the works published jointly by the candidate in [72,75,78].

OPTIMISATION-BASED CONTROL COORDINATION: DESIGN PROCEDURE

6.2 Optimisation-Based Control Coordination

The starting point in the development of the new control coordination procedure is that

of forming the power system state matrix A, based on the formulation presented in

Chapter 5. In terms of system damping in small-disturbance conditions, it is the

eigenvalues of the state matrix that are used for evaluating the power system dynamic

performance and designing the damping controllers. The steps in the controller

coordination design are derived and presented in the following sections.

6.2.1 Objective Function and Variables

The objective of the optimisation is to find a set of appropriate controller parameters

such that the system damping is maximised or improved, i.e., when the selected

eigenvalues (poorly-damped modes) have been moved as left most as possible in the

complex plane subject to controller parameter constraints as given in Section 6.2.3.

Therefore, the objective function to be minimised with respect to controller parameters

in the control co-ordination design is [72,75,78]:

121

][∑ λ−=λλλ=

m

i

2im21m21 )(Re),...,,,,...,,,(f

1zzzK (6.1)

where:

K = vector of controller parameters to be optimised

λi = the ith eigenvalue to be placed

zi = the eigenvector associated with the ith eigenvalue

m = number of selected eigenvalues

The eigenvalues and eigenvectors associated with them are nonlinear functions of

parameter vector K. However, closed-form expressions for the functions are, in

general, not available. A key feature of the proposed method is to express the inter-

relationship amongst parameter vector, selected eigenvalues and eigenvectors in the

form of eigenvalue-eigenvector equations which are to be satisfied during the

optimisation process. The equations form a set of equality constraints in the

optimisation, and the eigenvalues and eigenvectors are treated as variables in addition

to those representing the controller parameters.

CHAPTER 6

The variables in the objective function in (6.1) to be minimised, therefore, comprise

selected eigenvalues, eigenvectors and controller parameters. The minimisation of the

objective function is subject to equality constraints formed from the eigenvalue-

eigenvector equations and inequality constraints which represent the bounds required

on the selected eigenvalues and controller parameters.

6.2.2 Equality Constraints

If λ is an eigenvalue of matrix A and z is an eigenvector associated with λ then [86]:

0zAz =λ− (6.2)

where z is not equal to 0.

Although the state matrix A is real, some or all of its eigenvalues and eigenvectors can

be complex. It is now required to rearrange (6.2) into a real form for the purpose of

including it as a set of constraints in the optimisation in which real variables and

functions are used.

Defining:

IR

IR

jjzzzλ+λ=λ

+= (6.3)

Using (6.3) in (6.2):

0zzzzA =+λ+λ−+ )j)(j()j( IRIRIR (6.4)

Separating (6.4) into the real and imaginary parts gives:

0)(0)(

RIIRI

IIRRR

=λ+λ−

=λ−λ−

zzAzzzAz

(6.5)

Grouping (6.5) into a vector/matrix form leads to [87]:

0=λ− CCCC zzA (6.6)

122


where:

C ⎟⎟⎠

⎞⎜⎜⎝

⎛=

A00A

A (6.7)

I

RC ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

zz

z (6.8)

⎟⎟⎠

⎞⎜⎜⎝

⎛λλ

λ−λ=

UUUU

RI

IRCλ (6.9)

In (6.8), zR and zI are the real and imaginary parts of z respectively, and in (6.9), λR and

λI are the real and imaginary parts of λ respectively. If the dimension of the state matrix

A is N × N, then the dimensions of matrices AC and λC are 2N × 2N. Vector zC has 2N

elements, and U is the N × N unit matrix. The form of matrix AC as defined in (6.7) is

the same as that given in [87] except that the imaginary part of matrix A in the present

work is zero. The real-valued equation in (6.6) is equivalent to that in complex form in

(6.2).

For m selected eigenvalues, the set of equality constraints to be satisfied are:

1,2,...,m ; for i CiCiCiC ==− 0zzA λ (6.10)

Equation (6.10) is the first set of equality constraints to be satisfied in the optimisation

process. The second set comes from the eigenvector constraints. As eigenvector

associated with an eigenvalue is not unique, equation (6.10) has an infinite number of

solutions for vector zCi. In particular, 0z =Ci is also a solution which is not a valid

eigenvector. The problem is avoided by imposing a constraint on vector zCi. In the

present work, the constraint is imposed on the norm of vector zCi, i.e.:

m1,2,..., i for ; 1)k(z2/1N2

1k

2Ci ==⎟

⎠⎞

⎜⎝⎛ ∑

= (6.11)

123

CHAPTER 6

With the equality constraint (6.11) imposed on each eigenvector, the trivial and non-

valid solution zCi = 0 will be avoided.

In Section 4 is developed the modification of the eigenvalue-eigenvector equation in

(6.2) to provide a sparsity formulation for large power systems.

6.2.3 Inequality Constraints

Three sets of inequality constraints will be used in the optimisation process to impose

bounds on the eigenvalues and parameter values:

n1,2,...,j for ; KKK max,jjmin,j =≤≤ (6.12)

m1,2,...,i for ; )Re(

)Re(i,des2

i2

i

i =ζ≥+λ

λ−

ω (6.13)

m1,2,...,i for ; i,maxii,min =ω≤ω≤ω (6.14)

In (6.12), Kj is the jth element of controller parameter vector K. In (6.13), ζi,des is a

desired damping ratio of the ith mode. In (6.13) and (6.14), ωi is the angular frequency

given by the imaginary part of the ith eigenvalue.

6.2.4 Alternative Objective Function

The objective function in (6.1) is formed in terms of the real parts of selected

eigenvalues. Alternatively, the control coordination design can be based on the

minimisation of the weighted sum of stabiliser gains [18]. In this case, the objective

function is:

∑=λλλ=

L

1iiim21m21 aw),...,,,,...,,,(f zzzK (6.15)

124


In (6.15), ai is the positive gain of the ith stabiliser, which is an element of the parameter

vector K; L is number of stabiliser gains, and wi is a weighting coefficient assigned to ai.

Equality and inequality constraints as developed in Sections 6.2.2 and 6.2.3 are still

applicable when the objective function in (6.15) is minimised.

6.2.5 Selection of Modes for Design

The normal practice is to investigate the dampings of individual rotor modes prior to the

control coordination. For initialisation, the initial FACTS controllers and PSSs

parameters are set to representative values within specified lower and upper limits. The

eigenvalues and eigenvectors obtained from the initial calculations will be used for the

starting values required in the optimisation procedure. The formulation in the paper

provides this analysis facility for the investigation.

Using the results of the investigation, rotor modes which are unstable or lightly-damped

are selected for subsequent design of FACTS controllers and PSSs. Once the design

based on the selected modes has been carried out, rotor mode dampings will be

evaluated again to confirm whether all of the rotor modes have adequate damping

ratios. If one or more rotor modes do not have adequate dampings, then the control

coordination will have to be repeated, with the additional modes (unstable/lightly-

damped) to be included. The design procedure can be an iterative one (first option). An

alternative is to include all of rotor modes in the control coordination design at the

outset to avoid the possibility of iterations referred to in the first option.

6.2.6 Robust Controller Design

The method can also handle any system outage (e.g. transmission circuit outage or

generator outage) or operating conditions leading to changes in the numbers of state

variables.

In order to achieve a robust controller design, the sets of equality constraints to be

satisfied in addition to (6.10) and (6.11) are, for each contingency case:

m 1,2,..., ; for i CiCiCiC ==− 0zzA λ (6.16)

125

CHAPTER 6

and:

m1,2,..., i for ; 1)k(z2/1N2

1k

2Ci ==⎟

⎠⎞

⎜⎝⎛ ∑

= (6.17)

Also, the sets of inequality constraints for each contingency case to be satisfied in

addition to (6.16) and (6.17) are:

m1,2,...,i for ; )Re(

)Re(i,des2

i2

i

i =ζ≥+λ

−ω

λ (6.18)

and:

m1,2,...,i for ; i,maxii,min =≤≤ ωωω (6.19)

The symbol ─ above the variables and quantities in (6.16) - (6.19) identifies those for

contingency cases.

It is not necessary to assume that the pre-contingency and the post-contingency

systems would have identical modes. The selection of modes for including in

contingency cases and the initialisation process can also be based on the procedure

described in Section 6.2.5.

6.2.7 Prevention Against Convergence to the Same Eigenvalues

6.2.7.1 Practical Approach

In a single-machine infinite bus system, there is only one electromechanical mode of

oscillation. However, for the case of a multi-machine power system, there are multi-

modes of electromechanical oscillations, and depending on the number of areas in the

power system, there can be more than one inter-area mode.

The rotor mode frequencies or eigenvalues can be very close to one another. This

leads to the possibility of the optimisation converging to the same mode twice or more

times. Therefore, it is essential to augment with additional constraints the coordination

design procedure described in previous discussion to ensure that distinct modes are

126


used in the optimisation. In the following is the derivation of the additional constraints

for achieving the purpose.

If the angular frequencies of the ith and kth modes (for i ≠ k) are different, then the two

modes are necessarily distinct ones. Based on this property, and to make the provision

for the situation the two mode frequencies are very close to each other, the following

inequality constraint is proposed:

ε>ω−ω ki (6.20)

In (6.20), ε is a small positive value (for example, 10-3 rad/s) specified in the

optimisation procedure. For the optimisation solution algorithm where derivatives are

required, it is preferable to use the following constraint which is equivalent to (6.20):

( ) 22ki ε>ω−ω (6.21)

Based on (6.21), the set of inequality constraints described in the previous discussion

is now extended to include the following constraints for distinct modes:

( )⎪⎩

⎪⎨

⎧

≠=

=ε>ω−ω

ki andm1,2,..., k

m1,2,..., i for22

ki (6.22)

The additional set of constraints in (6.22) will prevent the optimisation from converging

to the same mode twice or more times.

Although angular frequencies (i.e. the imaginary parts of eigenvalues) have been used

in the constraints in (6.22), it is possible to adopt instead the real parts of eigenvalues

or combinations of both the real and imaginary parts to form the constraints for distinct

modes.

In practice, the mode frequencies, even if they are close, are different from one another

to some extent. The constraints in (6.22), therefore, will ensure that distinct modes are

used in the optimisation. However, in the unlikely event that two or more distinct modes

127

CHAPTER 6

have exactly the same frequency, a mathematical technique based on the property that

eigenvectors of distinct modes are to be linearly independent can be applied for

deriving a set of equality constraints for including in the optimisation. This will

guarantee distinct modes, even if their frequencies are the same. However, the

constraints in (6.22) are simpler to implement and fulfill the practical requirements. The

approach based on the property that eigenvectors of distinct modes are to be linearly

independent is discussed in the following.

6.2.7.2 Approach Based on Linearly Independent Eigenvectors Property

The following procedure is developed to guarantee that distinct modes are used in the

optimisation process even when their frequencies or eigenvalues are close to one

another.

The set of eigenvectors associated with distinct modes must be linearly independent.

Therefore, if:

0zzz =+++ mm2211 c.....cc (6.23)

then:

0c.....cc m21 ==== (6.24)

In (6.23) and (6.24), ci’s (for i = 1, 2,…., m) are the scalar coefficients in the linear

combination in (6.23), and zi’s are linearly independent eigenvectors.

Rewriting (6.23) in vector/matrix form:

0Z.C = (6.25)

where:

[ ]m21 z....zzZ = (6.26)

[ ]Tm21 c.....cc=C (6.27)

128


Equation (6.25) is rearranged into a real form, as required in the optimisation:

0.CZ =CC (6.28)

where:

ZZZZ

ZRI

IRC ⎟⎟

⎠

⎞⎜⎜⎝

⎛ −= (6.29)

⎟⎟⎠

⎞⎜⎜⎝

⎛=

I

RC C

CC (6.30)

In (6.29), ZR and ZI are the real and imaginary parts of Z respectively, and in (6.30), CR

and CI are the real and imaginary parts of C respectively.

The number of linear equations in (6.28) is greater than the number of coefficients in

vector CC. If the pseudo-inverse of ZC exists, then there is a unique solution for CC

which is equal to a zero vector. This is the condition for the set of eigenvectors

being linearly independent. For developing the constraint

corresponding to this condition, matrix D

{ m1,2,.....,i for i =z }C is defined in:

CTCC ZZD = (6.31)

If DC in (6.31) is non-singular, then the pseudo-inverse of ZC exists.

When DC is non-singular, there exists matrix EC that satisfies the following constraint:

UED CC = (6.32)

or:

ki if ; 0

ki if ; 1)k,j(E)j,i(Dm2

1jCC

≠=

==∑= (6.33)

129

CHAPTER 6

In (6.32), U is the unit matrix, and in (6.33), Dm2m2 × C(i, j) is an element of matrix DC,

and EC(i,j) an element of matrix EC. The set of individual constraints in (6.33) is then

included in the optimisation where EC(i, j) are the additional variables.

6.2.8 Constrained Minimisation Methods

The formulation of the optimal control coordination problem in Sections 6.2.1 – 6.2.7 is

a general one. In principle, a number of standard constrained minimisation algorithms

can be applied to solve the problem formulated. For example, the quasi-Newton

algorithm or sequential quadratic programming [88,89] is directly applicable.

6.3 Sparsity Formulation

The sparsity in the Jacobian submatrices J1 – J4 in (5.113) can be directly and in a

straightforward manner taken into account in the new formulation presented in Section

6.2. This offers an important advantage in the case of large power systems. The

modification required in sparsity formulation is described in the following. Instead of

eliminating the non-state variables to form the A matrix in (5.113), the equality

constraint based on eigenvalue-eigenvector equation in (6.2) is modified to, using

(5.113):

⎟⎟⎠

⎞⎜⎜⎝

⎛λ=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛0z

yz

JJJJ

43

21 (6.34)

In (6.34), the eigenvector z is augmented with vector y to include the non-state

variables.

The modification in (6.34) can be included in a straightforward manner in the

formulation of Section 6.2. The advantage is that of preserving the sparse structure in

the matrix coefficients which are used in the Newton algorithm [84].

130


131

6.4 Advantages of the Proposed Method

6.4.1 Selection of Modes in the Control Coordination A method for optimisation and coordination of damping controls based on time-domain

approach using a postulated disturbance was reported in [11]. However, the results

depend on the nature of disturbances used to excite the system, and the controller

robustness might be compromised [11]. The method does not provide the flexibility of

selecting the electromechanical modes for optimisation. These problems do not arise in

the new method developed in the chapter.

In general, the control coordination design is an iterative process, particularly when

location of the measurement unit yielding the feedback signal and even of the FACTS

device itself is to be determined for achieving optimal damping enhancement. The

advantage of the mode selection provided by the proposed method can be exploited in

the design. Different arrangements for SDC and/or PSS input signals and FACTS

device locations can lead to different modes to be considered. The design procedure

developed which allows modes to be selected directly is applied repeatedly for different

combinations of specified input signals and/or FACTS device locations, with the

objective of determining the optimal combination.

6.4.2 Elimination of Eigenvalue Shift Approximation In [18], a scheme for simultaneous coordination of PSSs and FACTS device stabilisers

based on linear programming and eigenanalysis was developed. Central to the scheme

is the approximation by which the shifts in eigenvalues are formed as linear functions of

the changes in stabilisers gains. A drawback is that the accuracy of the predicted shift

in an eigenvalue diminishes as the changes in stabiliser gains become large. Another

disadvantage of the scheme in [18] is the requirement of a separate procedure using

frequency response for estimating the time constants of stabiliser transfer functions.

In the new method by which all of the controller transfer function coefficients are

coordinated simultaneously and optimally, the above drawbacks or disadvantages are

eliminated.

CHAPTER 6

132

6.4.3 Simultaneous Coordination A scheme was reported in [8] for coordinating FACTS-based stabilisers, using the

method of closed-loop characteristic polynomial and eigenvalue assignment. The

scheme solves the problem of coordinating the stabilisers sequentially, i.e. in a pre-

specified sequence, rather than simultaneously. For a given power system, a pre-

specified sequence used in the coordination may not lead to the optimal results.

According to [8], a compromise should be established amongst stabilisers to avoid

them penalising each other. Methods reported in [90-92] are applicable to one FACTS

controller only. Similarly as reviewed in Chapter 3, the H∞ control-based design

[32,36,38,39] cannot achieve simultaneous and optimal control coordination of multiple

controllers.

The approach in the present work offers simultaneous and optimal coordination of

multiple controllers, without any need to specify a sequence or compromise in the

design.

6.4.4 Preserving the Matrix Sparse Structure Based on the information presented, it appears that the methods reported in [5,18]

draw on the calculations of the eigenvalues of the A matrix by the QR algorithm which

does not exploit the sparsity structure in power system Jacobian matrices.

The method based on the closed-loop characteristic polynomial in [8] requires the A

matrix to be formed explicitly. This will destroy the sparsity structure of the Jacobian

matrix.

Similarly, the requirement for the system state matrix A in its explicit form when the H∞

control technique is used in the design [32,36,38,39] will preclude the full exploitation of

the sparsity structures inherent in the power system Jacobian matrices.

As described in Section 6.3, the method developed in this chapter takes into account

fully the sparsity in the Jacobian matrix of a large power system.

There are other algorithms such as the modified-Arnoldi algorithm [93] which provide

eigenvalue calculations and take advantage of the Jacobian matrix sparsity. In


principle, these algorithms can also be applied to the methods reported in [5,18].

However, the control coordination design using these algorithms needs to calculate

separately the eigenvalues at every iteration where controller parameters are updated.

This can be time-consuming.

The control coordination design proposed in the present work uses the equality

constraints provided by the eigenvalue-eigenvector equations in the optimisation, and

avoids separate calculations of eigenvalues at each iteration. Eigenvalues together

with optimal controller parameters are available at the convergence of the optimisation

process.

6.5 Nonlinear Time-Domain Simulation Method

In order to verify the results of the control coordination design proposed, and to validate

the performance of the designed controllers under transient conditions, it is always

desirable to carry out the time-domain simulations for the system to investigate its

performance under large disturbances.

The time-domain solutions are carried out by solving simultaneously the set of

differential-algebraic equations (DAEs) which is given in (5.112). The set of differential

equations is solved by using the trapezoidal rule of integration as follows:

( ) ([ ])1n(),1n()n(),n(2t)1n()n( −−+

Δ+−= wxfwxfxx ) (6.35)

where Δt is the time step length and n is the time step counter.

Equation (6.35) is rearranged to give:

( ) 0fxwxf =−+−Δ )1n()n()n(),n(2t

p (6.36)

where:

( )1n(),1n(2t)1n()1n(p −− )Δ

+−=− wxfxf (6.37)

133

CHAPTER 6

Therefore, the solutions for x(n) and w(n) of (5.112) are found by simultaneously

solving the system of nonlinear equations as follows:

( )

( )⎪⎩

⎪⎨⎧

=

=−+−Δ

0wxg

0fxwxf

)n(),n(

)1n()n()n(),n(2t

p (6.38)

The solution steps for solving the nonlinear equations (6.38) are summarised as

follows:

1. Determining the initial operating points at time step n = 0: x(0) and w(0).

Appendix C shows the procedures needed to calculate the system initial

operating conditions for a power system installed with FACTS devices, based

on power-flow analysis.

2. Solving (6.38) simultaneously for x(n) and w(n) (n = 1, 2,…..,M), where M is the

chosen total number of solutions points. Any disturbances, including fault and

fault clearance, will be imposed at the time instants of their occurrence on the

power system for which the simulation is being carried out.

6.6 Conclusions

Through the application of constrained optimisation method, the chapter has

formulated a procedure for optimal control coordination design of multiple PSSs and

FACTS devices in a multimachine power system. The key advances made by the

control coordination design procedure developed include those described in the

following:

• sparsity formulation. The formulation exploits the sparsity in the Jacobian matrix.

This is of particular benefit in control coordination for very large power systems. A

particular feature of the formulation is that separate eigenvalue calculations are not

needed at each iteration in the constrained optimisation.

134


135

• the constraints developed for obtaining always distinct modes in the optimisation

procedure. Mode frequencies which are the same or similar to one another impose

no difficulty in the coordination process.

• approximation by which eigenvalues are linearly related to controller parameters is

not required.

• robustness in the optimal controller design. Critical contingencies and/or system

load changes can be included straightforwardly in the design.

Based on the design procedure developed and its software implementation, the control

coordination in power systems with FACTS devices will be carried out, and the design

results together with their validation using time-domain simulation will be presented in

the next chapter.

136

7.1 Introduction

Drawing on the new procedure developed in Chapter 6 together with its software

implementation, the present chapter carries out the control coordination design of

damping controllers in representative power systems. The design procedure is applied

to multi-machine power systems in which FACTS devices are installed. The power

system controllers considered in the design include PSSs and SDCs associated with

the TCSC and UPFC.

By nonlinear time-domain simulations, the effectiveness of the control coordination

results in enhancing power system oscillation damping is quantified and verified for

various power system disturbances.

7.2 Multi-Machine System with TCSC

7.2.1 Test System and Initial Investigation The initial control coordination design and study for illustrating the effectiveness and

capability of the design procedure developed in Chapter 6 is based on the test system

of Fig.7.1 for which the simultaneous coordination of PSSs and the TCSC controller for

improving the small-disturbance stability is carried out. The two-area system in Fig.7.1

is based on that in [94]. It is a 4 generator, 12 bus system with a total connected load of

2734 MW. The two areas are connected by three AC tie lines. Data for this test system

together with its initial operating condition is presented in Appendix D.2.

7 OOPPTTIIMMIISSAATTIIOONN--BBAASSEEDD CCOONNTTRROOLL CCOOOORRDDIINNAATTIIOONN:: DDEESSIIGGNN RREESSUULLTT AANNDD VVAALLIIDDAATTIIOONN

OPTIMISATION-BASED CONTROL COORDINATION: DESIGN RESULT AND VALIDATION

137

The design focuses on the improvement of the electromechanical mode damping. For

the 4-machine system, the number of the eigenvalue pairs associated with the

electromechanical modes to be optimised is three, one of which is the inter-area mode.

In the initial investigation, PSSs and FACTS device controllers are not included. The

eigenvalues results and participation factors are given in Table 7.1 for the three

electromechanical modes. The damping ratio of the inter-area mode 3 is very poor. It is

only 0.03. Stabilisation measure is, therefore, required for improving the damping of the

inter-area oscillation.

Fig.7.1: Two-area system with a TCSC

Table 7.1: Participation factor magnitudes for the system of Fig.7.1 Gen. Mode 1

(local mode) λ = -0.7337 ± j6.5606 f = 1.04 Hz; ζ = 0.11

Mode 2 (local mode)

λ = -0.7248 ± j6.8685 f = 1.09 Hz; ζ = 0.10

Mode 3 (inter-area mode)

λ = -0.1264 ± j4.6665 f = 0.74 Hz; ζ = 0.03

1 0.5068 0.0051 0.2991

2 0.0010 0.6676 0.1588

3 0.0117 0.4242 0.3759

4 0.6023 0.0224 0.1944

In Table 7.1, λ denotes eigenvalue; f mode frequency and ζ damping ratio.

G4 G2

G1 G3

N1 N5 N3

N4

N7

N11

N10 N9

N2

N8

TCSC N12

N6

CHAPTER 7

138

7.2.2 Application of PSSs This section discusses the application of PSSs for the purpose of enhancing the inter-

area oscillation damping. It is proposed to use only two PSSs, one of which is installed

in each area. The participation factors in Table 7.1 indicate that it is most effective to

install PSSs in generators G1 and G3 in relation to inter-area mode damping

enhancement. The PSS structure with rotor speed input is given in Fig.4.1.

In Table 7.2 are given the electromechanical mode eigenvalues, frequencies, and

damping ratios for the system without control coordination among the PSSs. In this

case, the controller parameters are optimised individually in a sequential manner

(uncoordinated). The controller parameters are given in Tables 7.3. There are some

damping improvements for both the local and inter-area modes when PSSs are

installed (compare the results in Table 7.1 and Table 7.2). However, without proper and

simultaneous coordination among the controllers, the improvements in dampings

offered by the PSSs may not be optimal.

Table 7.2: Eigenvalues for uncoordinated PSSs in the system of Fig.7.1

Mode Eigenvalues f (Hz) ζ

1 -1.4192 ± j6.6331 1.06 0.2092

2 -1.2888 ± j6.9192 1.10 0.1831

3 -0.3492 ± j4.6883 0.75 0.0743

Table 7.3: PSS parameters obtained from the uncoordinated design


Controller parameter PSS in G1 PSS in G3

KPSS 9.9066 pu 6.8727 pu

TPSS 1.0000 s 1.0000 s

TPSS1 0.2001 s 0.2006 s

TPSS2 0.0999 s 0.0991 s

TPSS3 0.0498 s 0.0508 s

TPSS4 0.2000 s 0.1990 s

pu on 100 MVA


139

It is now proposed to maximise the dampings offered by PSSs by a control

coordination among them. The control coordination procedure described in Chapter 6

when applied to the test system of Fig.7.1 leads to the results of Table 7.4 which show

the eigenvalues and damping ratios after the simultaneous optimisation of controller

parameters. The optimal controller parameters obtained by the control coordination

design are given in Tables 7.5. The limiting values of controller parameters used in the

design are given in Table 7.6. The desired minimum damping ratios for local and inter-

area modes in the design are 0.3 and 0.1, respectively. The damping results in Tables

7.1, 7.2 and 7.4 confirm the substantial improvement for both local modes and inter-

area mode achieved with the control coordination.

Table 7.4: Eigenvalues for coordinated PSSs in the system of Fig.7.1


1 -2.4700 ± j6.5509 1.04 0.3528

2 -2.4636 ± j6.5392 1.04 0.3526

3 -0.6097 ± j4.8551 0.77 0.1246

Table 7.5: PSSs parameters obtained from the coordinated design



KPSS 12.6503 pu 19.8027 pu

TPSS 1.1442 s 1.1932 s

TPSS1 0.2236 s 0.0547 s

TPSS2 0.1651 s 0.0584 s

TPSS3 0.0441 s 0.1387 s

TPSS4 0.1160 s 0.1648 s

pu on 100 MVA

Table 7.6: Limiting values of controller parameters of PSSs

Controller parameter Limit

KPSS 1 – 20 pu

TPSS 0.1 - 20 s

TPSS1 - TPSS4 0.01 - 10 s

pu on 100 MVA

CHAPTER 7

140

7.2.3 Application of PSSs and TCSC Table 7.4 shows that by using two properly coordinated PSSs, the inter-area mode

damping ratio of the system in Fig.7.1 can be improved to 0.1246. Although it has been

considered in [5,18] that the damping ratio greater than 0.1 is acceptable; however, the

design proposed in [37,40] has been based on a minimum damping ratio of 0.15 for

ensuring faster settling time of the inter-area oscillations.

Therefore, in the present work, in order to ensure faster settling time, it is proposed to

further improve the oscillation damping by installing an SDC in the TCSC in the

transmission line between nodes N9 and N10. The TCSC is installed for the primary

purpose of power flow controls and series compensation for the long tie line in the

system. An opportunity is then taken to equip the TCSC installed with an SDC to

provide a secondary function for damping improvement of the electromechanical

modes, particularly the inter-area mode.


damping ratios for the system where the controller parameters of PSSs and TCSC

main controller together with SDC are optimised individually in a sequential manner

(uncoordinated). In Figs.4.3 and 4.8 are shown the control structures for the TCSC

main controller and SDC respectively.

The controller parameters of PSSs obtained from the sequential design are given in

Table 7.3, whereas, the controller parameters of TCSC and its SDC are given in Table

7.8. There is a damping improvement for the inter-area mode when the TCSC with

SDC is installed (compare the damping ratio of mode 3 in Table 7.2 and Table 7.7).

However, without simultaneous optimisation of the controller parameters, the

improvements in dampings offered by PSSs and TCSC with SDC are minimal.

Table 7.7: Eigenvalues for uncoordinated PSSs and TCSC with SDC


1 -1.4184 ± j6.6338 1.06 0.2091

2 -1.2896 ± j6.9264 1.10 0.1830

3 -0.4847 ± j4.7773 0.76 0.1009


141

The results in Table 7.7 confirm the need to coordinate simultaneousy the controller

parameters of PSSs and TCSC with SDC if maximum dampings are to be achieved.

The coordination procedure described in Chapter 6 when applied to the test system of

Fig.7.1 leads to the results of Table 7.9 which show the eigenvalues after the

simultaneous optimisation of the controller parameters.

The optimal controller parameters obtained by the control coordination design are

given in Tables 7.10. The limiting values of controller parameters of PSSs, TCSC main

controller and SDC used in the design are given in the Tables 7.6 and 7.11,

respectively.

Table 7.8: TCSC and SDC parameters obtained from the uncoordinated design

Controller Controller parameter Value

KF 0.0101 pu

TF 1.1432 s

Kt 0.1413 pu TCSC Main Controller

Tt 0.0150 s

KSDC 0.0194 pu

TSDC 1.0022 s

TSDC1 0.3244 s

TSDC2 0.0176 s

TSDC3 0.3891 s

SDC

TSDC4 0.3789 s

pu on 100 MVA

Table 7.9: Eigenvalues for coordinated PSSs and TCSC


1 -3.0158 ± j6.7083 1.07 0.4100

2 -3.0424 ± j6.6870 1.06 0.4141

3 -1.1479 ± j4.8887 0.78 0.2286

CHAPTER 7

142

Table 7.10: PSSs, TCSC main controller and SDC parameters obtained from the

coordinated design


KPSS 9.2505 pu/7.2189 pu

TPSS 1.0216 s/1.6989 s

TPSS1 0.4859 s/0.0493 s

TPSS2 0.2163 s/1.0237 s

TPSS3 0.0475 s/0.3144 s

PSSs

(in G1/G3)

TPSS4 0.1096 s/0.0897 s

KF 0.0126 pu

TF 1.0119 s

Kt 0.1004 pu TCSC Main Controller

Tt 0.0217 s

KSDC 0.0258 pu

TSDC 1.0334 s

TSDC1 0.2542 s

TSDC2 0.0564 s

TSDC3 0.7106 s

SDC

TSDC4 0.0933 s

pu on 100 MVA

Table 7.11: Limiting values of controller parameters of TCSC with SDC

Controller Controller parameter Limit

KF 0.01 - 1 pu

TF 0.01 – 10 s

Kt 0.01 - 1 pu TCSC Main Controller

Tt 0.01 – 0.03 s

KSDC 0.01 – 1 pu

TSDC 0.01 - 10 s SDC

TSDC1 – TSDC4 0.01 - 10 s

pu on 100 MVA


143

7.2.4 Time-Domain Simulations Although the results of the design given in Sections 7.2.2 and 7.2.3 have been

confirmed by eigenvalue calculations, it is desirable to investigate the performance of

the designed controllers in the time-domain under a large disturbance. The disturbance

is a three-phase fault on a busbar section connected to node N8 via a bus coupler. The

fault is initiated at time t = 0.1 second with respect to the time origin in Fig.7.2, and the

fault clearing time is 0.1 second with the bus coupler tripping.

The improvement in performance is quantified by comparing the time-domain

responses in Figs.7.2 - 7.4. As the critical mode is the inter-area mode, the responses

used in the comparisons are those of the relative voltage phase angle transients

between nodes N9 and N10 of the tie line having the TCSC.

Fig.7.2: Transient for the system of Fig.7.1 (without PSSs and TCSC)

From the responses, it can be seen that, without damping controllers (PSSs and/or

FACTS device), the system oscillation is poorly damped and takes a considerable time

to reach a stable condition (see Fig. 7.2). Fig.7.3 shows the system transients with two

PSSs installed in the system. There are some improvements in oscillation settling time

with PSSs installed in the system (compare the dashed line of Fig.7.3 with Fig.7.2).

0 1 2 3 4 5 6 7 8 9 10 -0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

time, s

rela

tive

volta

ge p

hase

ang

le, r

ad

CHAPTER 7

144

time, s

However, with control coordination, the time period for damping out the oscillation is

substantially reduced (see the solid line of Fig.7.3).

Further improvement in the inter-area oscillation settling time is obtained with the PSSs

and TCSC with SDC installed and coordinated properly. The oscillation is damped

more quickly and almost disappears after about 5 – 6 seconds (see the solid line of

Fig.7.4). Fig.7.4 also confirms a good performance of the proposed control coordination

method which gives better results than the uncoordinated design (compare the solid

and dashed lines of Fig.7.4).

Fig.7.3: Transients for the system of Fig.7.1 (with PSSs only)

In Figs.7.5 - 7.7 are also shown in a graphical form the outputs of PSSs and SDC of

TCSC during the transient period following the disturbance. In the time-domain

simulations, both the SDC and PSS output limiters are represented. The SDC output

amplitude is limited to a band of 10%, whereas those of PSSs are limited to 5%.

Figs.7.5 – 7.7 show further the confirmation of the proposed control coordination

performance under large disturbance. It can be seen from Figs.7.5 – 7.7 that the

proposed controller design performs better than the uncoordinated design. During a

first few seconds of the transient period, the coordinated controllers (PSSs and SDC)

give higher outputs, and therefore, provide more contribution to the system damping

than the uncoordinated ones.

0 1 2 3 4 5 6 7 8 9 10 -0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

rela

tive

volta

ge p

hase

ang

le, r

ad

Uncoordinated Coordinated


145

Fig.7.4: Transients for the system of Fig.7.1 (with PSSs and TCSC)

Fig.7.5: PSS (in G1) output transients for the system of Fig.7.1

0 1 2 3 4 5 6 7 8 9 10 -0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

time, s

PS

S (i

n G

1) o

utpu

t, pu


time, s

0 1 2 3 4 5 6 7 8 9 10 -0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

rela

tive

volta

ge p

hase

ang

le, r

ad


CHAPTER 7

146


Fig.7.7: SDC output transients for the system of Fig.7.1

0 1 2 3 4 5 6 7 8 9 10 -0.03

-0.02

-0.01

0

0.01

0.02

0.03

time, s

PS

S (i

n G

3) o

utpu

t, pu


0 1 2 3 4 5 6 7 8 9 10 -0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

time, s

SD

C o

f TC

SC

out

put,

pu



147

7.3 Multi-Machine System with UPFC

7.3.1 Test System and Initial Investigation The initial test system adopted in the design and study of Section 7.2 is now reinforced

with additional transmission circuits as shown in Fig.7.8. In particular, two areas which

are now separated by a longer distance than that in the system of Section 7.2. Data for

this test system together with its initial operating condition is given in Appendix D.3.

In the initial investigation, PSSs and FACTS device controllers are not included. The

results of the investigation, i.e. the eigenvalues and participation factors, are given in

Table 7.12. The damping ratio of the inter-area mode 3 is very poor. It is only 0.0187.

Stabilisation measure is, therefore, required for improving the damping of the inter-area

oscillation.

Fig.7.8: Two-area system with a UPFC

Table 7.12: Participation factor magnitudes for the system of Fig.7.8

Generator Mode 1 (local mode)

λ = -0.7952 ± j7.0455 f = 1.12 Hz ; ζ = 0.1122

Mode 2 (local mode)

λ = -0.7759 ± j6.8316 f = 1.09 Hz ; ζ = 0.1129


λ = -0.0595 ± j3.1784 f = 0.51 Hz ; ζ = 0.0187

Generator 1 0.558466 0.000030 0.277824

Generator 2 0.000013 0.570775 0.218618

Generator 3 0.000001 0.548829 0.238230

Generator 4 0.559638 0.000034 0.282552

G4

G1 G3

G2

N5 N3

L15 L14

L13

L12 L11

L10

L9 L8

L7

L6

L4

L3

L5

L2

L1

N4

N7

N11

N13 N10 N9

N2

N1 N8

UPFC

N12

N6

CHAPTER 7

148

7.3.2 Application of PSSs Improvement of the inter-area oscillation damping by using PSSs is investigated in this

section. As in the system in Fig.7.1, it is also proposed here to use only two PSSs for

the damping enhancement. Based on the participation factors in the inter-area mode in

Table 7.12, the two PSSs are installed in generators G3 and G4 (one PSS for each

area). However, given the distribution of the participation factors related to mode 3 in

Table 7.12, the second option of having PSSs installed in generators G1 and G2

locations would lead to similar dynamic performance.


damping ratios for the system of Fig.7.8 installed with two PSSs. The PSS controller

parameters are optimised individually in a sequential manner (uncoordinated). The

controller parameters are given in Tables 7.14. As expected, there are damping

improvements when the PSSs are installed. However, without proper coordination

among the controllers, the improvements in dampings offered by the PSSs are not

optimal.

Table 7.13: Eigenvalues for uncoordinated PSSs in the system of Fig.7.8


1 -1.4259 ± j6.4554 1.03 0.2157

2 -1.2998 ± j6.3717 1.01 0.1999

3 -0.2327 ± j3.0923 0.49 0.0750

Table 7.14: PSSs parameters obtained from the uncoordinated design



KPSS 5.6091 pu 7.5451 pu

TPSS 1.0010 s 1.0016 s

TPSS1 0.1126 s 0.1625 s

TPSS2 0.2708 s 0.0397 s

TPSS3 0.2655 s 0.2525 s

TPSS4 0.0275 s 0.3555 s

pu on 100 MVA


149

The control coordination procedure described in Chapter 6 is now used to

simultaneously optimised the controller parameters of PSSs in the test system of

Fig.7.8. Table 7.15 shows the eigenvalues after the optimisation of the controller

parameters, and the optimal values of the controller parameters are given in Tables

7.16. The limiting values of controller parameters used in the design are given in Table

7.6. The desired minimum damping ratios for local and inter-area modes in the design

are 0.3 and 0.1, respectively. The damping results in Tables 7.12 and 7.15 confirm the

substantial improvement achieved with the control coordination.

Table 7.15: Eigenvalues for coordinated PSSs in the system of Fig.7.8


1 -2.2142 ± j6.4577 1.03 0.3243

2 -2.2214 ± j6.4666 1.03 0.3249

3 -0.3548 ± j3.0922 0.49 0.1140

Table 7.16: PSSs parameters of the coordinated design in the system of Fig.7.8


KPSS 7.3164 pu 10.5331 pu

TPSS 1.0415 s 1.0403 s

TPSS1 0.3031 s 0.1666 s

TPSS2 0.1528 s 0.1232 s

TPSS3 0.0100 s 0.0490 s

TPSS4 0.0750 s 0.0689 s

pu on 100 MVA

7.3.3 Application of PSSs and UPFC With the long-distance interconnection between the two areas of the system in Fig.7.8,

it is proposed to install a UPFC at node N13 for the primary purpose of power flow and

voltage control. An opportunity is then taken here to install an SDC associated with the

UPFC in providing a secondary function of damping improvement of the

electromechanical modes, particularly the inter-area mode. The enhancement in the

damping of the inter-area mode of oscillation will be quantified in the following study

and design, with the installation of the UPFC together with SDC.

CHAPTER 7

150

Table 7.17 shows the eigenvalue results for the system of Fig.7.8 where the damping

controllers (PSSs and UPFC together with SDC) are tuned in a sequential manner

(uncoordinated). The controller parameters of PSSs are given in Table 7.14, and the

controller parameters of UPFC and its SDC are given in Table 7.18. There is a

damping improvement for the inter-area mode when the SDC is installed (compare the

inter-area mode damping ratio in Tables 7.13 and 7.17). However, without proper

coordination, the combined use of the PSSs and UPFC with an SDC hardly provides

any further damping improvement.

Table 7.17: Eigenvalues for uncoordinated PSSs and UPFC


1 -1.4237 ± j6.4602 1.03 0.2152 2 -1.3024 ± j6.3721 1.01 0.2002 3 -0.3196 ± j2.8932 0.46 0.1098

Table 7.18: UPFC main controller and SDC parameters of uncoordinated design

Controller Controller parameter Limit

Ksh1 0.2664 pu Ksh2 0.3557 pu Tsh1 0.1002 s Tsh2 0.1003 s Kse1 0.0773 pu Kse2 0.1000 pu Tse1 0.2000 s

UPFC Main Controller

Tse2 0.2000 s KSDC 0.0100 pu TSDC 1.2001 s TSDC1 0.2000 s TSDC2 0.1001 s TSDC3 0.0500 s

SDC

TSDC4 0.2000 s pu on 100 MVA

The optimal control coordination procedure developed in Chapter 6 is now applied to

further enhance the dampings of the electromechanical modes. The limiting values of

controller parameters adopted in the design are given in Tables 7.6 and 7.19. The

desired minimum damping ratios for local and inter-area modes in the design are 0.3


151

and 0.1 respectively. The eigenvalues after simultaneous coordination are shown in

Table 7.20. The results in Table 7.20 show that the properly coordinated PSSs and

UPFC further improves substantially the dampings of the electromechanical modes in

comparison with those in Table 7.17 achieved by uncoordinated design. The optimal

controller parameters are given in Fig.7.21.

Table 7.19: Limiting values of controller parameters of UPFC and SDC


Ksh1, Ksh2 0.1 - 1 pu Tsh1, Tsh2 0.1 - 1 s Kse1, Kse2 0.01 – 0.1 pu UPFC Main Controller

Tse1, Tse2 0.2 – 1 s KSDC 0.01 – 1 pu TSDC 1 - 20 s SDC

TSDC1 – TSDC4 0.01 - 10 s pu on 100 MVA

Table 7.20: Eigenvalues for coordinated PSSs and UPFC


1 -2.4443 ± j6.7629 1.08 0.3399 2 -2.4329 ± j6.7629 1.08 0.3385 3 -0.5276 ± j3.0125 0.48 0.1725

Table 7.21: PSSs, UPFC and SDC parameters of coordinated design


KPSS 15.1378 pu/12.2798 pu TPSS 1.0014 s/1.0008 s TPSS1 0.1949 s/0.2099 s TPSS2 0.0963 s/0.0896 s TPSS3 0.0716 s/0.0626 s

PSSs (in G3/G4)

TPSS4 0.2158 s/0.1918 s Ksh1/Ksh2 1.0000 pu/0.1057 pu Tsh1/Tsh2 0.1025 pu/0.1159 s Kse1/Kse2 0.0437 pu/0.0226 pu


Tse1/Tse2 0.2012 s/0.2002 s KSDC 0.1207 pu TSDC 1.2022 s TSDC1 0.1948 s TSDC2 0.1092 s TSDC3 0.0442 s

SDC


CHAPTER 7

152

7.3.4 Time-Domain Simulations In order to validate the performance of the coordinated controllers discussed in

Sections 7.3.2 and 7.3.3, the comparisons between the coordinated and uncoordinated

damping controllers (PSSs and UPFC with SDC) under large disturbance using time-

domain simulations are carried out in this section. The performance comparisons are

also carried out for the system with and without damping controllers. The disturbance

used in the performance validation is a three-phase fault near node N11 on the line

between nodes N8 and N11. The fault is initiated at time t = 0.1 second with respect to

the time origin in Fig.7.9, and the fault clearing time is 0.1 second. Following the fault

clearance, transmission line L12 is lost.

Fig.7.9: Transient for the system of Fig.7.8 (without stabilisers)

Fig.7.9 shows a transient of relative voltage phase angle between nodes N9 and N10

for the system of Fig.7.8 without any damping controllers. From the response, it can be

seen that, without damping controllers (PSSs and/or FACTS device), the required time

for damping the system oscillation is unacceptably very long.

0 1 2 3 4 5 6 7 8 9 10-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

time, s

rela

tive

volta

ge p

hase

ang

le, r

ad


153

The dashed line in Fig.7.10 shows the system transients with two uncoordinated PSSs

installed in the system. As expected, there is an improvement in oscillation damping

time with the PSSs installed in the system. However, with the proposed control

coordination, faster oscillation damping time is obtained (see the solid line of Fig.7.10).

These results confirm the coordinated controllers performance as discussed in

Sections 7.3.2.

Fig.7.10: Transients for the system of Fig.7.8 (with PSSs only)

The solid line in Fig.7.11 which shows the system transients with the PSSs and UPFC

with SDC installed and coordinated properly also confirms a good performance of the

proposed control coordination method which gives significantly better results than the

uncoordinated design (compare the solid and dashed lines in Fig.7.11). With proper

control coordination, the combined dynamic performance of PSSs and SDC leads to

the inter-area mode oscillation being damped out in 6 – 7 s following the fault

disturbance.

0 1 2 3 4 5 6 7 8 9 10-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

time, s

rela

tive

volta

ge p

hase

ang

le, r

ad


CHAPTER 7

154

Fig.7.11: Transients for the system of Fig.7.8 (with PSSs and UPFC)

The outputs of PSSs and SDC of UPFC during the transient period following the

disturbance are shown in Figs.7.12 - 7.14. As in the system of Fig.7.1, the SDC output

amplitude is limited to a band of 10%, whereas those of PSSs are limited to 5%.

The results shown in Figs.7.12 - 7.14 also confirm the capability of the proposed

control coordination. It can be seen that, during a first few seconds of the transient

period following the disturbance, the coordinated controllers give higher outputs, and

thus provide more contribution to the system damping than the uncoordinated ones.

0 1 2 3 4 5 6 7 8 9 10 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

time, s

rela

tive

volta

ge p

hase

ang

le, r

ad



155



0 1 2 3 4 5 6 7 8 9 10 -0.01

-0.005

0

0.005

0.01

0.01

0.02

time, s

PS

S (i

n G

3) o

utpu

t, pu


0 1 2 3 4 5 6 7 8 9 10 -0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

time, s

PS

S (i

n G

4) o

utpu

t, pu


CHAPTER 7

156

Fig.7.14: SDC output transients for the system of Fig.7.8

7.4 Conclusions

The control coordination design procedure based on the constrained optimisation as

developed in Chapter 6 has been applied and verified in the present chapter, in relation

to multi-machine power systems with PSSs and FACTS devices. The dynamic

performance of the coordinated power system controllers designed with the procedure

has first been verified by eigenvalues calculations, which confirms the effectiveness of

the procedure in providing substantial electromechanical mode damping improvement.

The validation studies were then reinforced by using nonlinear time-domain simulations

in which large disturbances initiated by faults were considered.

However, there remains an issue which has not been considered or included in the

procedure of Chapter 6 or previous controller design methods. It is related to the

possibility of the SDC and/or PSS output saturation and its effect on dynamic

performance. This issue and the modified procedure for controller design taking into

account output limits will be considered in Chapter 11 of the thesis.

0 1 2 3 4 5 6 7 8 9 10 -0.03

-0.02

-0.01

0

0.01

0.02

0.03

time, s

SD

C o

utpu

t, pu


8 RREEVVIIEEWW OOFF AADDAAPPTTIIVVEE DDAAMMPPIINNGG CCOONNTTRROOLLLLEERRSS AANNDD WWAAMM--BBAASSEEDD SSTTAABBIILLIISSEERRSS

8.1 Introduction

The non-adaptive (fixed-parameter) controller designs as discussed in the previous

chapters (Chapters 2, 3 and 6) are, in general, based on one particular power system

operating condition and configuration. The key disadvantage of these designs is that

the possibility of the controllers performance deterioration under other operating

conditions or configurations. Furthermore, it is not possible to achieve maximum

damping performance for each and every operating condition or contingency when the

controller parameters are fixed.

Recently, adaptive control techniques have been proposed to overcome the

disadvantage of fixed-parameter controllers design. In this adaptive controller design,

the controller parameters are determined online and adaptive to the changing in

system operating conditions. This chapter provides an overview on the previous works

published in the area of adaptive damping controller designs. The key methods

previously proposed that will be reviewed in the chapter include: self-tuning controllers

[43-52] and neural network-based controllers [53-58].

In addition to the discussion of adaptive controller methods, an overview on WAM

(wide-area measurement) based controllers is also given in this chapter. In WAM-

based damping controller, global signals or remote feedback control signals are used

as the inputs to the controller [9,39,95-103]. The advances in WAM technologies using

phasor measurement units (PMUs) which can deliver control signal at high speed, and

the advantages of using remote signals as the controller input signals have triggered

the development of the WAM-based damping controller.

157

CHAPTER 8

158

8.2 Self-Tuning Controller

8.2.1 Overview of Self-Tuning Controller Self-tuning controller (STC) is one of the techniques of adaptive control. STC was

originally proposed by Kalman in 1958. However, because of the unavailability of high-

speed computers and inadequately developed theory, this technique was not taken up

seriously at that time. The breakthrough came with the work reported by Astrom and

Wittenmark in 1973. Since then this technique has become popular, especially due to

the advent of microprocessors, which make it feasible to implement the STC algorithms

[104].

The controller is called self-tuning, since it has the ability to adjust its own parameters

according to the system conditions to obtain satisfactory control performance. The STC

can be thought of as having two loops (see Fig.8.1): an inner loop consisting of a

conventional controller (but with varying parameters), and an outer loop consisting of a

plant model parameters identifier and a controller design with the function of adjusting

the controller parameters. The controller design block diagram in Fig.8.1 represents an

online solution to a design problem for a system with known parameters [104-106].

Fig.8.1. Self-tuning controller

In the context of control theory applied in general to self-tuning controller, the

methodologies which can be used for controller design include: linear quadratic,

minimum variance, gain-phase margin design, pole assignment (pole placement) and

pole shifting. Whereas, those for plant model parameters identification schemes

include: least-squares, recursive least-squares and Kalman filtering [104-106].

Plant Model Parameters

Controller Parameters

Plant Output Reference Input CONTROLLER PLANT

PLANT MODEL IDENTIFIER

CONTROLLER DESIGN

REVIEW OF ADAPTIVE DAMPING CONTROLLERS AND WAM-BASED STABILISERS

In [43-52], the STC has been applied to design the power system oscillation damping

controller. In [43-50], the recursive least square (RLS) method has been used to

identify the system parameters online. Whereas, in [51,52], the Kalman Filter (KF) was

used for the plant identification. Furthermore, the pole shifting algorithm has been

employed in [43-52] for controller design to determine the controller parameters. The

overview on RLS method, Kalman filter and pole shifting algorithm which have been

proposed for power systems applications are given in the following.

8.2.1.1 RLS Parameters Identification Method

In STC, the model of the system to be controlled is usually described by a linear

difference equation, and the model parameters are identified every sampling interval.

The system model in the discrete-time domain is assumed to be of the form [104,106-

108]:

(8.1)

)nn(g)2n(g)1n(g)n(

)nn(ub)2n(ub)1n(ub)n(ub

)nn(yh)2n(yh)1n(yh)n(y

gn21

bn210

hn21

g

b

h

−ε++−ε+−ε+ε+

−++−+−+=

−++−+−+

L

L

L

where y is the plant output; u is the input; hi and bi are the model parameters to be

identified; ε is a sequence of independent and equally distributed random noise, and n

is the sampling instant.

The operator notation will be used here for conveniently writing the difference equation

(8.1). Let be the backward shift (or delay) operator which is used to relate [104]: kq−

) (8.2) kn(y)n(yq k −=−

On using (8.2), (8.1) can be rewritten as:

159

) (8.3) n()q(G)n(u)q(B)n(y)q(H 111 ε+= −−−

where:

(8.4) g

g

bb

hh

nn

22

11

1

nn

22

110

1

nn

22

11

1

qgqgqg1)q(G

qbqbqbb)q(B

qhqhqh1)q(H

−−−−

−−−−

−−−−

++++=

++++=

++++=

L

L

L

CHAPTER 8

The estimation of the model parameters can be simplified by assuming [104]

which modify (8.3) to become:

1)q(G 1 =−

160

) (8.5) n()n(u)q(B)n(y)q(H 11 ε+= −−

Equation (8.5) can be expressed in terms of the various model parameters as the

following:

)n()nn(ub)2n(ub)1n(ub)n(ub

)nn(yh)2n(yh)1n(yh)n(y

bn210

hn21

b

h

ε+−++−+−+=

−++−+−+

L

L (8.6)

By introducing the parameter and regression vectors:

(8.7) [ ][ ]Tbh

Tn10n21

)nn(u)1n(u)n(u)nn(y)2n(y)1n(y)n(

bbbhhh)n(bh

−−−−−−−−=

=

LL

LL

Φ

Θ

Equation (8.6) can be rewritten in a compact form, using definitions in (8.7):

) (8.8) n()n()n( )n(y T ε+= ΘΦ

The parameter vector Θ is to be estimated from the observations of system inputs and

outputs. In adaptive controllers, the observations are obtained sequentially in real time.

It is then desirable to make the computations recursively to reduce computing time.

Computation of the least-squares estimate is arranged in such a way that the results

obtained at time n-1 can be used to get the estimates at time n.

In recursive implementations of the least-squares method, the computation is started

with known initial conditions and uses the information contained in the new data

samples (from the measurements) to update the old estimates. The RLS algorithm for

estimating model parameters hi and bi is summarised in the following (see Table 8.1).

Details of the algorithm can be found in [104,105,107,108].


Table 8.1: Summary of the RLS algorithm

Stage Description

Initialisation

1) Model parameters estimates: )0(Θ2) Error covariance: IP c)0( α=

(I is the identity matrix which has the dimension of (nh+nb+1) (nh+nb+1), and αc is a specified constant)

Online computation

For each sampling instant: n = 1, 2,…., calculate:

1) Gain vector: )n()1n()n(

)n()1n()n( Tf ΦPΦ

ΦPK−+ρ

−=

2) Prediction error: )1n(ˆ)n()n(y)n( T −−=ε ΘΦ

3) Update of model parameter estimate: )n()n()1n(ˆ)n(ˆ ε+−= ΚΘΘ

4) Update of error covariance: [ ] fT /)1n()n()n()n( ρ−−= PΦKIP

(ρf is the forgetting factor which has the value of ) 10 f ≤ρ<

8.2.1.2 Kalman Filter (KF) State Estimation

KF is essentially a set of mathematical equations that provides an efficient

computational (recursive) means to estimate the state of a process [109]. KF uses a

recursive algorithm whereby the updated estimate of the state at each time step is

computed from the previous estimate and the new input data, so only the previous

estimate requires storage [108-112]. KF provides a unifying framework for the

derivation of an important family of adaptive filters known as recursive least-squares

filters [112].

KF addresses the problem of estimating the state of a linear discrete-time dynamical

system governed by the following equation [108-112]:

)n()n( )1n( xwAxx +=+ (8.9)

)n()n( )n( ywCxy += (8.10)

where: x is the state vector to be estimated; y is the observation vector which contain a

set of observed (measured) data; wx and wy are the process and measurement noise

respectively; A is the state transition matrix, and C is the measurement matrix.

The noise vector sequences wx(n) and wy(n) in (8.9) and (8.10) are assumed to be

known and independent (of each other). It is also assumed that they are white

(uncorrelated) noise, with zero mean and covariance matrix defined by:

161

CHAPTER 8

) (8.11) n( ])n()n([ Txx Qww =E

(8.12) )n( ])n()n([ Tyy Rww =E

where is the mathematical expectation of the discrete random variable ][∗E ∗ (see

Appendix E for the details of the formulation).

Equations (8.11) and (8.12) show that the process noise covariance matrix Q and

measurement noise covariance matrix R might change with each time step, however, it

will be assumed here that they are pre-specified constants [109]. The state matrix A in

(8.9) relates the state of the system at time n+1 and n, whereas, the measurement

matrix C in (8.10) relates the state to the measurement y. For a time-varying system,

matrices A and C change with each time step.

Suppose that a measurement on a linear discrete-time dynamical system described by

(8.9) and (8.10) has been made at sampling instant n. The requirement is to use the

information contained in the new measurement y(n) to update the estimate of the

unknown state x(n). The KF recursive algorithm for state estimation is summarised in

Table 8.2. Details of the algorithm can be found in [108-112].

Table 8.2: Summary of the KF algorithm

Stage Description

Initialisation 1) Initial estimate of state: [ ])0()0(ˆ xx E=

2) Error covariance: [ ][ ]{ }T)0(ˆ)0(()0(ˆ)0(()0( xxxxP −−= E

Online

computation

For each sampling instant: n = 1, 2,…., calculate:

1) State estimate: )1n(ˆ)n(ˆ −=− xAx

2) Error covariance: QAAPP +−=− T)1n()n(

3) Kalman gain: [ ] 1TT )n()n()n(−−− += RCCPCPG

4) State estimate update: [ ])n(ˆ)n()n()n(ˆ)n(ˆ −− −+= xCyGxx

5) Error covariance update: [ ] )n()n()n( −−= PCGIP

162


8.2.1.3 Kalman Filter Interpretation

It is interesting to note that, when model parameters are time-varying, the model

described by (8.8) can be interpreted as a linear state-space model of the form

[104,105,107]:

)n()n()n( )n(y

)n()n()1n(T ε+=

μ+=+

ΘΦΘΘ

(8.13)

The above observation shows that the least-squares estimate can be interpreted as a

KF for the process described by (8.13).

It is possible to express (8.13) in the form in (8.9) and (8.10), when and x

is the plant model parameters vector, to obtain [104,105,107]:

T , ΦCIA ==

)n()n()n( )n(y

)n()n()1n(T ε+=

μ+=+

xΦxx

(8.14)

With the plant model in (8.14), the KF algorithm in Table 8.2 can be for model

parameters determination.

8.2.1.4 Pole-Shifting Controller Design

In the pole-shifting controller design, it is assumed that the pole characteristic

polynomial of the closed-loop system has the same form (i.e. the same order) as the

pole characteristic polynomial of the open-loop system, but the pole locations

determined by the roots of the characteristic polynomial are shifted by a factor αs [43-

52].

Fig.8.2 shows the control system block diagram for illustrating the pole-shifting

controller design procedure. The system to be controlled in Fig.8.2 has the transfer

function of the form:

)q(H)q(B)q(G 1

11

z −

−− = (8.15)

where H(q-1) and B(q-1) are polynomials defined by (8.4). The transfer function of the

controller is assumed to be of the form:

163

CHAPTER 8

)q(D)q(C)q(K 1

11

z −

−− = (8.16)

where C(q-1) and D(q-1) are polynomials given by:

(8.17) d

d

cc

nn

22

11

1

nn

22

110

1

qdqdqd1)q(D

qcqcqcc)q(C−−−−

−−−−

++++=

++++=

L

L

164

Fig.8.2: Closed-loop control system

With the plant output y in Fig.8.2 representing the deviation from a given operating

point, the reference r input to the closed-loop system takes the value of zero as shown.

It can be shown that the pole characteristic polynomial of the closed-loop system in

Fig.8.2 is:

(8.18) )q(C)q(B)q(D)q(H)q(P 11111 −−−−− +=

In the pole-shifting method, the pole characteristic polynomial of the closed-loop

system P(q-1) has the same form as the pole characteristic polynomial of the open-loop

system H(q-1) but the pole locations are shifted radially towards the origin of unit circle

in the z-plane by a factor αs where 10 s ≤α≤ [48,51,107]. Thus the following equation

holds:

(8.19) )q(H)q(C)q(B)q(D)q(H 1s

1111 −−−−− α=+

where:

(8.20) hh

h nn

ns

22

2s

11s

1s qhqhqh1)q(H −−−− α++α+α+=α L

v y ur = 0

-

+ Kz(q-1 Gz(q-1) ) Σ


Expanding both sides of (8.19) and comparing the coefficients with the same power of

q-i will result in the linear equation system which must be solved to obtain the controller

parameters ci and di. In order to guarantee that the solution of the linear equation

system is unique, it has been suggested in [48] that the number of the parameters nc

and nd should be and 1nh − 1nb − respectively. In partitioned vector/matrix forms, the

formulation of the linear equation system for )1nnnn (and nnn GKdcGbh −===== is

shown in the following [48,49]:

(8.21) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

2Z

1Z

2Z

1Z

4Z3Z

2Z1Z

L

L

R

R

MM

MMLL

M

LLL

M

where , ,

are given by:

GGKGGKKK nn4Z

nn3Z

nn2Z

nn1Z ,,, ×××× ℜ∈ℜ∈ℜ∈ℜ∈ MMMM GK n

2Zn

1Z , ℜ∈ℜ∈ RR

KG n2Z

n1Z and ℜ∈ℜ∈ LL

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−−−−−−

03n2n1n

012

01

2Z

4n3n2n

12

1

1Z

bbbb

0bbb00bb

;

1hhh

01hh001h0001

GGGGGG

L

MOMMM

L

L

L

MOMMM

L

L

L

MM

(8.22)

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=−

−−

−

−−

G

G

GG

GGG

G

G

GG

GG

n

3n

22nn

12n1nn

4Z

n

3n

21nn

12n1n

3Z

b000

bb00bbb0bbbb

;

h00

hh0hhhhhh

L

MOMMM

L

L

L

L

MOMM

L

L

L

MM

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−α

−α−α

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

0

00

;

)1(h

)1(h)1(h

;

c

cc

;

d

dd

2Z

nsn

2s2

s1

1Z

n

1

0

2Z

n

2

1

1Z

GGKK

MMMMLLRR

It is to be noted that in (8.21), the model parameters hi and bi are identified every

sampling instant by using one of the system identification methods described in

Sections 8.2.1.1 and 8.2.1.3.

165

CHAPTER 8

166

8.2.2 Application of STC in Power Oscillation Damping In [43-50], the STC concept has been employed to design the self-tuning PSS for

damping of power oscillation. In the design, the recursive least square method was

used to estimate the system parameters online. Based on the identified system

parameters, the pole-shifting algorithm has been incorporated in the controller design

to determine the controller parameters. A similar approach has been applied to a TCSC

damping controller in [51,52]. In [51,52], Kalman Filter (KF) has been used for

parameters identification method, and pole shifting algorithm was employed in the

controller design.

Furthermore, the self-searching and self-optimising pole shifting techniques have been

incorporated in the controller design in [43-52]. The techniques have been used in the

design with the objective to enable the modification of the pole shifting factor with

respect to control signal to avoid unsatisfactory control performances. With these

techniques, excessive pole shifting is no longer a problem. Thus, the control

constraints violation and control signal saturation which might affect the control

performances can be avoided and the control signals are kept within their limits.

Although the results presented in [43-52] indicate that the proposed self-tuning

damping controller can provide good damping under varying operating conditions and

different disturbances, some issues have been identified in the application as the

following:

- Low-order system model has to be used in the application of the method due to

computation time requirement (higher order model requires longer computation time

which is not suitable for online implementation). Also, it appears that there is no

systematic technique for determining the order of the system model (i.e. na and nb). In

[43-50], the third order system model has been selected to approximate the higher

order power system. This approximation might lead to some errors and affect the

system identification and controller design. Furthermore, there is no guarantee that

the low-order model will be sufficient to track the important system dynamics and

modes of interest.


167

- It is difficult to implement the method in [43-52] for system having multiple modes of

oscillations, and it will be more difficult if the modes of interest have to be

simultaneously considered in the application of the method.

- Coordination amongst multiple damping controllers is very important for achieving

optimal oscillation dampings in multi-machine power system. It is not clear how the

controllers coordination can be implemented in the approach proposed in [43-52].

8.3 Neural Network-Based Controller

8.3.1 Overview of the Neural Network Theory Artificial neural networks are composed of elements (which imitate the nerve cells or

neurons of the biological nervous system) operating in parallel [113-118]. The neural

network function is determined largely by the connections between the elements. The

neural network can be trained to perform a particular function by adjusting the values of

the connections (weights) between the elements [113]. The neural network is usually

implemented by using electronic components or is simulated in software on a digital

computer [116].

In terms of their architectures, the neural networks can broadly be classified into: (i) the

feedforward neural network, and (ii) the recurrent neural network [116]. In feedforward

neural network (FNN), the inputs to the neurons in each layer of the network are the

output signals from the preceding layer only. A recurrent neural network (RNN)

distinguishes itself from a FNN in that it has at least one feedback loop. In RNN, the

neurons feed their output signals back to their own inputs (self-feedback) or to the

inputs of other neurons.

The multilayer feedforward neural network or multilayer perceptron, trained by

backpropagation algorithm, is the most widely used neural network [114]. This section

will first discuss the architecture of FNN, the backpropagation algorithm and then the

sizing of the FNN.

8.3.1.1 Architecture of the FNN

As mentioned in the previous discussion, neural networks consist of elements (or

neurons) operating in parallel. A “neuron” in a neural network is sometimes referred to

CHAPTER 8

as a “unit”. A single input neuron is shown in Fig.8.3a. The scalar input p is multiplied

by the scalar weight w to form wp. Here wp is the only argument of the transfer function

(or activation function) f, which produces output a for the single input case. The neuron

in Fig.8.3b has a scalar bias b. The bias is added to the product wp and shifts the

function f by an amount b [113,114]. The bias is much like a weight, except that it has a

constant input of 1. One can choose neurons with or without biases. The bias gives the

network an extra variable, and so it might be expected that the networks with biases

would be more powerful and flexible [114].

w a w r a r

168

Fig.8.3: Single-input neuron

(a) Without bias

(b) With bias

The transfer function f in Fig.8.3 can be a linear or a nonlinear function of argument r.

Log-sigmoid, tan-sigmoid and linear transfer functions are the most commonly-used

transfer functions for the neural network [113]. Note that w is the adjustable parameter

or weighting coefficient of the connection between two neurons. The network can be

trained to achieve a particular application requirement (e.g., function approximation) by

adjusting the weighting coefficients.

Typically, a neuron has more than one input. A neuron with R inputs is shown in

Fig.8.4a. The individual inputs are weighted by the corresponding weights

. The argument r of the transfer function in Fig.8.4a is then given in

terms of the weight and input vectors as follows:

R21 p,,p,p K

R11211 w,,w,w K

br += Wp (8.23)

where:

[ ]

[ TR21

R11211

ppp

www

L

L

=

=

p

W

] (8.24)

1

f f Σ p p

b

(a) (b)


Commonly, the neural network with one neuron, even with many inputs, may not be

sufficient. Two or more of the neurons shown in Fig.8.4a can be combined to operate in

parallel to form a layer. A particular neural network could contain one or more such

layers. A one-layer network (with R input elements and S neurons) is shown in

Fig.8.4b. In this one-layer network, the input vector elements enter the network through

the weight matrix W which has the form [113,114]:

(8.25) ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

SR2S1S

R22221

R11211

www

wwwwww

L

MOMM

L

L

W

a1 r1

169

Fig.8.4: Multiple-input neuron(s)

(a) Neuron with R inputs

(b) S neurons with R inputs

A neural network can have several layers. Each layer has a weight matrix W, a bias

vector b and an output vector a. A multilayer neural network starts with an input layer

followed by one or more layers of hidden units (neurons). These hidden layers will then

be connected to the output layer. The input data will be fed to the network through the

w12

w1R pR

p2

1

b

(a) (b)

ar w11

f

p1

Σ M

1

f Σ

b p1

pR

p2

1b

a2 r2 f Σ

M

1b

aS rS f Σ

M M

CHAPTER 8

170

input units. There is not any processing in the input layer. The input nodes just simply

feed the data to be processed to the subsequent layers.

The multilayer feedforward neural networks are more powerful than single-layer neural

networks. For instance, a network of two layers, the first layer is sigmoid and the

second layer is linear (see Fig.8.5), can be trained to approximate most functions

arbitrarily well [113,114]. Most practical neural networks have just two or three layers.

Four or more layers are used rarely [114].

Fig.8.5: Multilayer feedforward neural network with one hidden layer

8.3.1.2 FNN Training Algorithm

In neural network training stage, the network parameters (weights and biases) are

adjusted to optimise the performance of the neural network. This optimisation process

consists of two steps [114]. The first step is to determine a quantitative measure of the

network performance and usually refers to as performance index. The performance

index should be small when the network performs well and large when the network

performs poorly. The second step of the optimisation process is to search the network

parameters in order to reduce the performance index.

aS

a2

a1

pR

p2

p1 f

f

f

f

f

f

f

output layer

hidden layer


Training the multilayer feedforward neural network is usually carried out using

optimisation methods by which the difference between the network response and target

output is minimised.

The network is presented with a set of pairs of input and output patterns:

{ } { } { }QQ2211 t,p,.....,t,p,t,p (8.26)

In (8.26), pi is an input vector to the network, and ti is the corresponding target output

vector, for i = 1, 2,.…., Q, where Q is the number of training cases. As each input is

applied to the network, the network output is formed, and then compared to the target.

The algorithm should adjust the network parameters which are the weights and biases

in order to minimise the mean squared error [113,114]:

∑ −−==

Q

iiT

ii ()(Q1)(F

1i)atatδ (8.27)

In (8.27), and a are the vectors of network weights and outputs respectively. δ

In Appendix F, optimisation algorithms commonly used for minimising the error function

in (8.27) are presented.

8.3.1.3 Sizing of FNN

One important aspect in designing a neural network is to determine the network size,

e.g., the number of layers and the number of units in each layer [119,120]. Usually, the

numbers of input units and output units are determined from the problem. However,

determining the number of hidden layers and the number of units in each hidden layer

is not straightforward and requires experimentation.

There are a number of methods that can be used for the network size determination.

These methods can be divided into two categories, i.e. pruning methods and

constructive methods [119,120]. Pruning methods start with a large network and

reduce the size until a solution is found. Constructive methods start with a small

network and gradually increase the size. Both methods use the training and testing

errors for adjusting the neural network size.

171

CHAPTER 8

8.3.2 Neural Network-Based Damping Controller In [53-58], neural network-based controllers have been proposed for improving the

damping of power system oscillations. The neural network was trained over a wide

range of operating conditions. Once trained, the controllers were adapted in real-time

based on the system operating conditions to maintain a good damping characteristic

under different system operating conditions. The summary of the proposed methods

reported in [53-58] are presented in the following.

a) Method Proposed in [53]

In [53], the multilayer feedforward neural network with two hidden layers (four neurons

at each layer) has been employed to adapt PSS parameters according to generator

loading conditions in SMIB system environment. The inputs to the neural network were

machine real-power (PG) and power factor (PF) which characterise machine loading

conditions. The outputs of the neural network were the desired PSS parameters.

In order to obtain the network connection weights, a set of 300 training patterns have

been compiled in the training process. The weights were computed using the method

of gradient descent with adaptive learning rate. Each training pattern contains machine

PG and PF (which serve as the inputs to the neural network), and the desired PSS

parameters (the target output signals of the neural network). These PSS parameters

have been determined using the pole-assignment method with the electromechanical

mode fixed at the locations of 5646.10j3 ±− .

b) Method Proposed in [54]

In the investigation reported in [54], the gains of PI controller for TCSC in SMIB system

environment were determined adaptively by an artificial neural network. The inputs to

the neural network include the measured real- and reactive-power (PL and QL) in the

transmission line, and the outputs of the neural network were the desired PI controller

gains.

The network structure of two hidden layers with fifteen neurons in each layer has been

used in the proposed neural network controller. The data for the network training were

generated as in the following. For every combination of PL and QL within the region of

interest ( 5.1P6.0 L ≤≤ , 0.1Q4.0 L ≤≤− ), the desired controller gains were computed.

Pole assignment method has been used for determining the controller gains where the

172


poles were assigned in the region of 0.1)Re(5.4 −≤λ≤− ; . In [54],

gradient descent method with adaptive learning rate has been used for updating the

weights in the training process.

681.10)Im( =λ

c) Method Proposed in [55]

A multi-input neural network PSS in SMIB system environment has been proposed and

investigated in [55]. The generator speed deviation and the electrical power deviation

together with their delayed responses and the delayed supplementary control signal in

the excitation system were used as the inputs of the proposed neural network. The

output of the neural network was the supplementary control signal. The neural network

structure of one hidden layer with thirty-five neurons has been employed in the

proposed controller. Data for training the neural network were generated by applying

the self-optimising pole-shifting control strategy described in [49], and the gradient

descent backpropagation method has been used to train the multilayer network.

d) Method Proposed in [56]

Similar to the method proposed in [55], in [56], a neural network PSS in SMIB system

environment has also been proposed. The neural network structure of two hidden

layers with twenty neurons in each layer was used in the investigation. Data for training

were also generated by applying the pole-shifting control strategy, and the gradient

descent method has also been used to train the multilayer network.

e) Method Proposed in [57,58]

In [57,58], a neural network-based PSS has been proposed. The generator speed

deviation or the electrical power deviation together with their delayed responses and

the delayed supplementary control signal were used as the inputs of the proposed

neural network, whereas, the output of the neural network was the supplementary

control signal.

Two hidden layers with thirty neurons in the first layer and ten neurons in the second

layer have been used in the proposed controller. The neural network-based PSS was

trained over a wide range of operating conditions where the generator powers ranging

from 0.1pu to 1.0pu and power factors ranging from 0.7 lead to 0.1 lag, also the

disturbances such as governor input variations have been used to simulate the

generator loading conditions. The self-optimising pole-shifting control strategy

173

CHAPTER 8

174

described in [49] was used to control the generator in the working conditions mentioned

above and to generate data for training the neural network. The gradient descent

backpropagation method has been used to train the multilayer network.

Although the results presented in [53-58] show that the proposed neural network-based

controllers can provide good damping under different system operating conditions,

some issues have been identified in the application as the following:

- The problem of optimal control coordination of multiple PSSs and/or SDCs of FACTS

devices has not been discussed in the method proposed in [53-58]. Although multiple

PSSs in multimachine system has been addressed in [58], but, as the design was

based on the self-tuning pole-shifting method [49], it is not clear how the optimal

control coordination of multiple controllers can be implemented in the design (see

also the discussion in Section 8.2).

- The change in power system configuration is required to be represented

systematically in online tuning of controller. This requirement has not been

considered in the design of neural network-based controllers in [53-58]. If the system

configuration has changed significantly, the proposed neural network controllers in

[53-58] will have to be retrained.

8.4 WAM-Based Stabilisers

It has previously been mentioned that to have a damping effect, a power system

damping controller uses an input signal and synthesises an output control signal based

on appropriate phase-lead compensation to add to the reference signal of the main

controller for the purpose of damping power oscillations.

The input signal to the damping controller may be local or remote to the location of the

controller. The remote feedback control signal will require additional communication

channel for the signal transmission from distant location to the controller site. As the

signal is derived from a distant location, there is a problem of some time delay for the

signal to be available to the controller. This delay can typically be in the range of 0.5 –

1.0s depending on the distance, protocol of transmission and several other factors

[39,95-98]. The introduction of the time delay has a destabilising effect and reduces the


175

effectiveness of the control system damping, and in some cases, the system

synchronism may be lost [95]. Another disadvantage of the remote signal-based

controller is that the possibility of communication channel failure, and therefore, loss of

input signal to the controller. This, in turn, will affect the damping capability of the

controller [99].

Contrary to the remote signal-based controller, the additional communication facility is

not needed for the controller with local input signal, and therefore, it does not have

disadvantages as in the remote signal-based controller. However, local control signal

may not have adequate observability for some important modes. The signal with

maximum observability for a particular mode may be derived from a remote location or

as a combination of signals from several locations [39]. Moreover, with remote signal-

based controller, multiple swing modes can be damped out by using only a small

number of controllers. This is possible because the remote signals capture more swing

modes from different locations of the power system, and therefore, contain more

information about networks dynamics [39,97].

The above advantages of using remote signals and recent advances in wide-area

measurement (WAM) technologies using phasor measurement units (PMUs) which can

deliver control signals at high speed have triggered the development of the WAM-

based damping controller. In WAM-based damping controller, global signals or remote

feedback control signals are used as the input to the controller. In [9,39,95-103], the

use of WAM-based damping controller for enhancing system dynamic performance has

been investigated. The summary of the investigation is given as follows:

- A robust damping control design for multiple swing modes using global stabilising

signals has been proposed in [39]. A multiple-input-single-output (MISO) controller

was designed for a TCSC to improve the damping of the critical interarea modes.

The stabilising signals were obtained from remote locations based on observability

of the critical modes.

- In [95], the analysis of time delay impact to wide-area power system control was

addressed by a robust supervisory power system stabiliser (SPSS). The design

was aimed at improving the damping of interarea oscillations. The control design

technique was based on H∞ gain scheduling theory.

CHAPTER 8

176

- The damping controller design taking into account a delayed arrival of feedback

signals has been investigated in [96]. A predictor-based H∞ control design strategy

was proposed for such time-delayed system. The concept was utilised to design a

WAM-based damping controller using a static VAr compensator.

- A control design procedure for handling time delays encountered in transmitting the

remote signals was proposed in [97]. A H∞ control design methodology following

unified Smith predictor (USP) approach was applied for designing a centralised

controller through a TCSC.

- In [98], the implementation of a centralised control design scheme in a real-time

laboratory-based dynamic simulator has been demonstrated. A centralised

multivariable control algorithm was designed employing remote feedback signals

considering delay in signal transmission. The transmitted signals can be used for

multiple swing modes damping using a single controller.

- The capability of the synchronised phasor measurement technology in improving

the overall stability of the Hydro-Quebec’s system through supplementary

modulation of voltage regulators has been investigated in [99]. A

decentralised/hierarchical control architecture with significant advantages in terms

of reliability and operational flexibility was used in the design approach.

- In [100], a two-level control scheme for PSS design has been proposed. The first

level control was to provide damping for the local modes using local signal as an

input to the PSS. The second level was to enhance damping of interarea modes

using global signals as additional inputs to the PSS. These global signals have also

been used to enhance damping of interarea modes using SVC.

- In [101], a robust damping H∞ controller for power system oscillations has been

proposed. Using wide-area measurements, the robust controller was a supervisory

level controller that can track system dynamics online. Based on the concept of

multiagent systems, the robust controllers were embedded into a system-intelligent

agent, which is coordinated with local agents to increase system damping.


177

- A two-level hierarchical structure has been proposed in [102] to improve the

stability of multimachine power systems. The design consists of a local controller

for each generator at the first level complemented by a multivariable central

controller at the secondary level. The secondary-level controller uses remote

signals from all of the generators. The first-level controllers, on the other hand, use

only local signals to damp local oscillations.

- An adaptive wide-area control system (WACS) has been designed in [103] to

provide damping control signals to the excitation of generators. A single

simultaneous recurrent neural network was used in the realisation of the adaptive

WACS for both identification and control of the power system. The WACS has been

implemented on a digital signal processor and its performance was evaluated on a

power system implemented on the real-time digital simulator.

- In [9], the use of multiple input signals (some of which may be remote) for the

design of PSS and TCSC controllers has been investigated. It has been shown that

using multiple input signals will allow the controller to be more effective in providing

additional damping.

8.5 Conclusions

The present chapter has presented and discussed the adaptive design methods for

power system damping controllers. The most popular and widely proposed methods

that have been reviewed include: self-tuning controllers and neural network-based

controllers. In addition, an overview on WAM (wide-area measurement) based

controllers has also been given in this chapter. On examining the design principles of

the methods, the key disadvantages or deficiencies have been identified and discussed

in the chapter.

9 NNEEUURRAALL AADDAAPPTTIIVVEE CCOONNTTRROOLLLLEERR DDEESSIIGGNN PPRROOCCEEDDUURREE

9.1 Introduction The review discussed in Chapter 8 indicates that there remain two key issues that need

to be addressed in relation to the design of adaptive PSSs and SDCs:

(i) Optimal control coordination.

It is required to achieve online control coordination of multiple PSSs and/or SDCs in

a multi-machine power system. The requirement is to maximise the damping ratio

for electromechanical modes for each and every credible system operating

condition or configuration.

(ii) Representation of power system configuration.

The optimal controller parameters depend importantly on power system

configuration. Due to load demand variation and switching control, including that in

protection operation for fault clearance, power system configuration is time-varying

during system operation. There is then a need to represent directly and

systematically the change in system configuration in online tuning and coordination

of multiple controllers.

The present chapter proposes and develops an adaptive control coordination scheme

for PSSs and SDCs that addresses the above two issues. The scheme is based on the

use of a neural network which identifies online the optimal controller parameters. The

inputs to the neural network include the active- and reactive- power of the synchronous

generators which represent the power loading on the system, and elements of the

reduced nodal impedance matrix for representing the power system configuration. It is,

therefore, not required to form and store a range of system models for subsequent

online use. The outputs of the neural network are the parameters of the PSSs and

178

NEURAL ADAPTIVE CONTROLLER DESIGN PROCEDURE

179

FACTS devices together with their SDCs which lead to optimal oscillation damping for

the prevailing system configuration and operating condition.

The use of the reduced nodal impedance matrix is a novel feature in the scheme

proposed by which any power system configuration can be represented very directly

and systematically. The matrix is formed for only power network nodes that have direct

connections to synchronous generators and FACTS devices. The reduced nodal

impedance matrix is derived very efficiently from the power system nodal admittance

matrix and sparse matrix operations. The remaining inputs to the neural network in

terms of generator powers are available from measurements. The proposed design

method discussed in this chapter has also been presented in the works published

jointly by the candidate in [121] of which the candidate is a coauthor.

9.2 Representing System Configuration 9.2.1 Concept In addition to active- and reactive-power loading on the power system, the optimal

parameters of PSSs and SDCs of FACTS devices depend importantly on system

configuration. In designing adaptive controllers, it is required to represent power

system configuration which is variable. One option is to use a set of discrete variables

to describe the power system topology. However, this option is not a practical one as it

will lead to a very large number of combinations, particularly for a large power system.

This combinatorial problem represents a key difficulty encountered in designing

adaptive controllers.

On recognising that the information on the power network configuration together with

transmission circuit parameters is embedded fully in the nodal impedance matrix

confined to the generators and controllers locations, the chapter proposes a novel

concept of using the elements of the nodal impedance matrix to represent any variation

in system configuration. As the elements of the nodal impedance matrix are continuous

variables, the combinatorial problem is completely eliminated when designing adaptive

controllers based on the new concept. The nonlinear and discrete relationship between

the system configuration and optimal controller parameters is now transformed into a

continuous form in which the optimal controller parameters are nonlinear and

CHAPTER 9

180

continuous functions of nodal impedance matrix elements which are continuous

variables themselves.

In the present work, neural networks are used for the nonlinear function representation

required in the mapping between optimal controllers parameters and nodal impedance

matrix. With this approach, the elements of the nodal impedance matrix confined to the

generators and controllers locations, referred to as the reduced nodal impedance

matrix in the subsequent development, will be formed and input to the neural network-

based adaptive controller.

9.2.2 Forming Reduced Nodal Impedance Matrix The reduced nodal impedance matrix is formed using the status data of circuit breakers

and isolators together with the power system database. The steps to achieve this

include:

(i) Forming the power system configuration from circuit-breaker and isolator status

data [122].

(ii) Forming system nodal admittance matrix. The system configuration determined in

step (i) is used in conjunction with the network branch parameters stored in the

power system database to form the system nodal admittance matrix.

(iii) Reducing the system nodal admittance matrix formed in step 2 to the nodal

impedance matrix for the power system nodes that have direct connections to

generators and SDCs. This is achieved through sparse matrix operations and LU

matrix factorisation.

(iv) Online modification of the reduced nodal impedance matrix. The LU matrix

factorisation in step 3 of the system nodal admittance matrix is performed only once

in an off-line mode for the system configuration of the base case (i.e. full system).

The results of the factorisation are then stored for subsequent use in the online

mode. A scheme based on the compensation technique reported in [123] is

adopted to form the reduced nodal impedance matrix for any contingency, using the

stored results of the base-case factorisation, and only a minimal amount of


computation which does not involve the refactorisation is required. The scheme is

suitable for online application of the adaptive controller. The detail of formulation of

the proposed scheme is shown in Appendix G.

With the present advances in LU factorisation techniques and high-speed computing

systems, it is possible that the online full refactorisation can be carried out to form the

reduced nodal impedance matrix, without using the compensation method. This also

allows load models in the form of admittance to be represented in the system nodal

admittance matrix. The scheme is described in the following for the prevailing system

operating condition and configuration.

The reduced nodal impedance matrix is derived from the nodal admittance matrix for

the entire system, Y, which is formed online. A key property is that the nodal

admittance matrix, Y, is highly sparse. Only non-zero elements of the matrix are stored

for subsequent processing.

The relationship between the nodal current vector, I, and the nodal voltage vector, V,

can be written as:

YVI = (9.1)

or equivalently:

ZIV = (9.2)

where Z is nodal impedance matrix for the entire system and has the form:

1−= YZ (9.3)

The solution for V given in (9.2) and (9.3) is expressed in a symbolic form only. In

practice, the LU sparse matrix factorisation operating on the sparse matrix Y is used in

conjunction with the forward and backward substitution procedure to solve for nodal

voltage vector V, when nodal current vector I is known or specified.

181

CHAPTER 9

In the proposed method, each column of the reduced nodal impedance matrix is

calculated by sequentially injecting 1 pu current into each of the generator node or

FACTS device node, and then forming the nodal voltages for each case using the

sparse procedure described above. In each case, the elements of the nodal voltage

vector that correspond to specified generator nodes or FACTS device nodes form a

column of the required nodal impedance matrix. The following is an example to explain

the calculation steps involved in the formation of the reduced nodal impedance matrix.

Suppose the power system has 6 nodes including 2 generator nodes (nodes 1 and 2)

and 1 FACTS device node (node 4). By sequentially injecting 1 pu currents into nodes

1, 2 and 4, the nodal current vectors specified for individual cases are:

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

001000

654321

;

000010

654321

;

000001

654321

321 III (9.4)

where subscripts 1, 2 and 3 identify the cases in the calculation. Using the sparse

procedure, the nodal voltage vectors are computed for individual cases. Suppose the

results are:

(9.5)

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

)6(V)5(V)4(V)3(V)2(V)1(V

65432

;

)6(V)5(V)4(V)3(V)2(V)1(V

2

;

)6(V)5(V)4(V)3(V)2(V)1(V

3

3

3

3

3

3

3

2

2

2

2

2

2

2

1

1

1

1

1

1

1

1

6543

1

654321

VVV

The elements (1, 2 and 4) of the voltage vectors form the columns of the reduced

impedance matrix. By assembling the columns in matrix form:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

)4(V)4(V)4(V)2(V)2(V)2(V)1(V)1(V)1(V

321

321

321

reducedZ (9.6)

182


183

where Zreduced is the reduced nodal impedance matrix required. The above calculations

for large power systems can be carried out very efficiently by using the sparse

vector/matrix operations described.

9.3 Development of Neural Network-Based Adaptive Controller

9.3.1 Principle of Neural Network Application The relationship among the optimal controller parameters and power system operating

condition including system configuration is, in general, a nonlinear one. The present

paper draws on the key property of the multilayer feedforward neural network, which is

that of nonlinear multi-variable function representation [113]. The neural network is

used for the mapping between the power system configurations and/or operating

conditions and optimal controller parameters. Drawing on the key concept described in

Section 9.2.1, the power system configuration will be represented by a set of

continuous variables derived from the reduced nodal impedance matrix.

Fig.9.1 shows the general structure of the multilayer feedforward neural network which

is adopted to represent the nonlinear relationship between the optimal controller

parameters and power system operating condition together with configuration. For

compactness, only one hidden layer is shown in the neural network of Fig.9.1.

However, the structure can be extended to include two or more hidden layers in a

straightforward manner.

There are two separate sets of nodes in the inputs layer in Fig.9.1. The first set has n

nodes the inputs to which are obtained from the real and imaginary parts of the

reduced nodal impedance matrix as formed in Section 9.2.2. These inputs represent

power system configuration. If there are Nc generator nodes and FACTS device nodes,

the number of input nodes in the first set is Nc2+Nc, when the symmetry in the nodal

impedance matrix is exploited. The second set of inputs comprises active- and

reactive-power of each and every generator. Therefore, if there are Ng generators in

the power system, there will be 2Ng input nodes in the second set. These inputs in the

second set represent power system operating condition. The total number of inputs is

Nc2+Nc+2Ng.

CHAPTER 9

184

The nodes in the output layer of the neural network structure in Fig.9.1 give the optimal

values of the parameters of PSSs and FACTS device control systems, including the

SDCs. It is possible to exclude the FACTS device main controllers from the adaptive

control coordination. However, to achieve maximum benefit in terms of damping, both

FACTS device main controller and SDC are included in the adaptive control

coordination. The structure in Fig.9.1 assumes that there are M controller parameters

to be tuned online. On this basis, the output parameters from the neural network in

Fig.9.1 are described in Tables 9.1 – 9.6, for individual power system controller types.

Individual controllers structures and parameters have been presented in Chapter 4.

Fig.9.1: Input and output structure of the neural network

p1, p2,….., pn : Real and imaginary parts of the elements of the

reduced nodal impedance matrix pn+1, pn+2,….., pm : Active- and reactive- power of generators

a1, a2,….., aM : Optimal controller parameters f : Activation function

The number of hidden layers, the number of nodes in each hidden layer and the

weighting coefficients of the connections between nodes in the structure of Fig.9.1 are

a3

a2

a1

pn

p2

p1 f

f

f

f

f

f

f

input layer output layer hidden layer

aM

a5

a4

pm

pn+2

pn+1 f

f

f

f

f

f

f


185

to be determined by neural network training, and verified by testing which will be

discussed in Sections 9.3.3 and 9.3.4.

Table 9.1: Neural adaptive controller outputs for PSS parameters

Controller Type Parameters Description

KPSS PSS gain TPSS Time constant of PSS washout block

PSS TPSS1, TPSS2, TPSS3, TPSS4 Time constants of PSS lead-lag blocks

Table 9.2: Neural adaptive controller outputs for SVC parameters


KS SVC gain TS Time constant to represent the delay

SVC Main

Controller TS1, TS2 Time constants of SVC lead-lag block

Table 9.3: Neural adaptive controller outputs for TCSC parameters


KF Gain of TCSC power flow controller KF Gain of TCSC delay block TF Time constant of TCSC power flow controller

TCSC Main

Controller Tt Time constants of TCSC delay block

Table 9.4: Neural adaptive controller outputs for STATCOM parameters


KC1, KC2 STATCOM controller gains TC2 Time constant of STATCOM PI controller

STATCOM Main

Controller Tc Time constants of STATCOM delay block

Table 9.5: Neural adaptive controller outputs for UPFC parameters


Ksh1, Ksh2 Shunt converter controller gains Tsh1, Tsh2 Shunt converter controller time constants Kse1, Kse2 Series converter controller gains

UPFC Main

Controller Tse1, Tse2 Series converter controller time constants

CHAPTER 9

186

Table 9.6: Neural adaptive controller outputs for SDC parameters


KSDC SDC gain

TSDC Time constant of SDC washout block

SDC

TSDC1, TSDC2, TSDC3, TSDC4 Time constants of SDC lead-lag blocks

9.3.2 Overall Neural Adaptive Controller Structure In Fig.9.2 is shown the overall structure of which the neural adaptive controller

described in Section 9.3.1 is a part. For online tuning of the parameters of PSSs and

FACTS device main controllers together with SDCs, the inputs required are, as shown

in Fig.9.2:

- circuit-breaker and isolator status data

- power network branch parameters

- generator active- and reactive-power

Fig.9.2: Neural adaptive controller block diagram

generator active- and reactive power

t

generator speeds and line active

powers (for PSSs and SDCs)

circuit breaker and isolator status data

power system data base (branch parameters)

PSSs and

FACTS device outputs

NEURAL ADAPTIVE CONTROLLER

optimal controller

parametersPSSs, SDCs of

FACTS DEVICES AND FACTS

DEVICE MAIN CONTROLLERS

TRAINED NEURAL

NETWORK

FORMING REDUCED NODAL

IMPEDANCE MATRIX

SPARSE MATRIX OPERATIONS

FORMING POWER SYSTEM NODAL

ADMITTANCE MATRIX

FORMING POWER

SYSTEM CONFIGURATION power system

quantities controlled by main FACTS

controllers


The response of the trained neural network gives the optimal parameters for the PSSs

and FACTS device main controllers together with SDCs. The feedback inputs to these

controllers are generator speeds and transmission line active-powers, as in the case of

fixed-parameter controllers.

9.3.3 Training Procedure for Neural Adaptive Controller The training set is generated using the optimisation-based control coordination method

discussed in Chapter 6. For a given power system, a wide range of credible operating

conditions and configurations which include those arising from contingencies is

considered in the training data generation.

For the ith training case, the pair of specified input and output vectors is { }ii t,p . Based

on the structure in Fig.9.1, the input vector pi is:

( ) Qi ; mi2i1iTi ....., 2, 1, == p,.....,p,pp (9.7)

in which Q is the total number of training cases.

The target output vector ti for the ith training case is the optimal controller parameters

vector for the power system with the operating condition and configuration specified by

the input vector pi.

The requirement in the training is to minimise the difference between the target output

vector ti and response of the neural network in Fig.9.1. For Q training cases, as

mentioned in Chapter 8, it is proposed to minimise the following mean square error

(MSE):

∑ −−==

Q

1i

T

Q1F )a(t)a(t(x) iiii (9.8)

In (9.8), ai is the neural network response which has the following form, based on the

structure in Fig.9.1:

( ) Qi ; Mi2i1iTi ....., 2, 1, == a,.....,a,aa (9.9)

187

CHAPTER 9

188

Vector x in (9.8) is the vector of weighting coefficients of the connections in the neural

network to be identified. Minimising the error function F(x) with respect to x gives the

weighting coefficient vector. In the present work, the Levenberg-Marquardt algorithm

which is a second-order method with a powerful convergence property is adopted for

minimizing F(x) in (9.8). One of the criteria for the convergence in training is that the

error function F(x) has to be less than a specified tolerance.

In addition to the training performance expressed in terms of error function F(x), the

controller parameters obtained from the trained neural network are also used for

calculating the damping ratios of the rotor modes, which are then compared with the

optimal damping ratios obtained at the stage of training data generation. The

convergence in training is confirmed when both the error function F(x) and the damping

ratio comparison satisfy the specified tolerances.

9.3.4 Neural Network Testing and Sizing In addition to forming the training data set, a separate testing data set is also required.

The procedure for testing data generation is similar to that of training where the

optimisation-based control coordination method described in Chapter 6 is used.

The testing criteria are also based on the MSE formed over the testing set, and

comparisons of the damping ratios obtained from controller parameters given in the

neural network outputs and those from the specified cases in the testing set.

The sizing of the neural network is achieved by monitoring the performance of the

neural network on the basis of both training and testing criteria as shown in the

flowchart of Fig.9.3. A poor training performance will lead to an increase in the size of

the neural network in terms of hidden layers and/or hidden nodes. The neural network

size will be increased until the training convergence criteria as described in Section

9.3.3 are satisfied. Subsequent neural network testing will then be carried out to ensure

that the trained neural network has good generalisation capability. If the testing criteria

are not met, retraining the neural network, as indicated in the flowchart of Fig.9.3, will

be required. In the retraining step, the training set is expanded with additional cases,

for enhancing the generalisation capability of the neural network.


189

In general, the final neural network with satisfactory performance is obtained through

successive iterations of the training and testing processes.

Fig.9.3: Flowchart for training, testing and sizing of the neural network

YES NO

YES NO

START WITH A SMALL NUMBER OF HIDDEN NODES

TRAINING WITH THE MOST RECENT NEURAL NETWORK CONFIGURATION

ARE TESTING CRITERIA IN TERMS OF MSE AND DAMPING

RATIOS SATISFIED?

ARE TRAINING CONVERGENCE CRITERIA IN TERMS OF MSE

AND DAMPING RATIOS SATISFIED?

CALCULATE THE TRAINING MSE AND MAXIMUM DAMPING RATIO DIFFERENCE

CALCULATE THE TESTING MSE AND MAXIMUM DAMPING RATIO

DIFFERENCE

INCREASE THE SIZE OF THE NEURAL NETWORK

INCREASE THE TRAINING DATA SET WITH

ADDITIONAL NEW CASES STOP

CHAPTER 9

190

9.4 Conclusions An adaptive control algorithm and procedure have been derived and developed for

online tuning of the PSSs and SDCs of FACTS devices. The procedure is based on the

use of a neural network which adjusts the parameters of the controllers to achieve

system stability and maintain optimal dampings as the system operating condition

and/or configuration change. A particular contribution of the method is that of

representing the power system configuration in terms of a reduced nodal impedance

matrix, which is formed using sparse matrix operations. This allows any variation of

system configuration to be included and input to the neural adaptive controller.

191

10.1 Introduction The principle and method developed in Chapter 9 will be applied for designing a neural

adaptive controller for a representative multi-machine system comprising two areas

where both local and inter-area modes exist. Damping controllers considered in the

design include PSSs and SDC of the UPFC in the power system

The neural network in the adaptive controller is trained and tested off-line with a wide

range of credible power system operating conditions and configurations. For all of the

tests considered, for assessing the performance of the trained neural network, the

controller parameters obtained from the trained neural network are verified by both

eigenvalue calculations and time-domain simulations. The results from an extensive

test study confirm that significant improvements in the power system dynamic

performance are achieved with the neural adaptive controller in comparison with the

fixed-parameter controllers. The simulation results discussed in this chapter have also

been presented in the works published jointly by the candidate in [121].

The chapter also discusses the possibility of implementing the neural adaptive

controller on a cluster of high-speed and low-cost processors which are currently

available.

10.2 Test System The system in the study is based on the two-area 13-bus power system of Fig.10.1

[94]. Data for this test system together with its initial operating condition is presented in

Appendix H.

10 NNEEUURRAALL AADDAAPPTTIIVVEE CCOONNTTRROOLLLLEERR DDEESSIIGGNN RREESSUULLTTSS AANNDD VVAALLIIDDAATTIIOONN

CHAPTER 10

192

Fig.10.1: Two-area system

Initial investigations of the dynamic performance of the system without any damping

controllers are carried out, with the results in terms of eigenvalues of the

electromechanical modes and participation factors presented in Table 10.1.

Table 10.1: Participation factor magnitudes for the system of Fig.10.1

Generator Mode 1 (local mode)

λ = -0.8056 ± j7.0626 f = 1.12 Hz ; ζ = 0.1133

Mode 2 (local mode)

λ = -0.7784 ± j6.8425 f = 1.09 Hz ; ζ = 0.1133


λ = -0.0499 ± j3.3929 f = 0.54 Hz ; ζ = 0.0147

Generator 1 0.560530 0.000017 0.289657

Generator 2 0.000011 0.570346 0.213061

Generator 3 0.000004 0.549495 0.281995

Generator 4 0.558979 0.000038 0.230038

The results confirm that the inter-area mode has poor damping, and the damping ratios

of the local modes are low. Stabilisation measure based on PSSs and FACTS device

controllers with SDCs as discussed in Chapter 6 is, therefore, proposed for improving

the damping of these electromechanical modes in the power system.

Stability analysis of the power system without any PSSs and FACTS devices indicate

that, among the four generators in Fig.10.1, participation factors of the inter-area mode

in generators G1 and G3 are greater than those in the other two generators as shown

in Table 10.1. On this basis, it is proposed to install PSSs for generators G1 and G3

G4

G1 G3

G2

N5 N3

L16 L15

L14L13

L12 L11

L10

L9 L8

L7

L6

L4

L3

L5

L2

L1

N4

N7

N11 N13

N10 N9

N2

N1 N8

UPFC

N12

N6

NEURAL ADAPTIVE CONTROLLER DESIGN RESULTS AND VALIDATION

193

only. The other two generators (generators G2 and G4) do not have PSSs. The PSSs

for generators G1 and G3 have adaptive parameters. The PSSs have the structure

described in Chapter 4 with rotor speed inputs.

A FACTS device, i.e. a UPFC with an SDC, is installed at node N13 in line L16. It is

proposed to use the line active-power as the input to the SDC which has the structure

described in Chapter 4. For this system with 4 generators, there are three swing modes

(two local modes and one inter-area mode) of low frequency oscillations. All of these

electromechanical modes are represented in the control coordination and the design of

the neural adaptive controller described in the next section.

10.3 Design of the Neural Adaptive Controller

10.3.1 Neural Network Training and Test Data

The key requirement is to design a neural controller that has the capability of

generalising with high accuracy from the training cases. This requirement is achieved

through the neural network training, testing and sizing referred to in Sections 9.3.3 and

9.3.4 based on the selection of the training and testing data sets. The neural network

training set should be representative of the cases described by credible system

contingencies and changes in system operating conditions.

The possible contingencies of the system in Fig. 10.1 for line(s) outages, load and

power generation variations are shown in Tables 10.2 and 10.3 respectively. Both

single-line outages and double-line outages are considered in the postulated

contingencies where there is no loss of any generator, and the two areas remain

connected. The input and output pairs for neural network training and testing cases are

generated from the combinations of these contingencies and operating conditions.

For the system in Fig.10.1, the number of neural network inputs, as determined on the

basis of Section 9.3.1, is 38. In the present work, the parameters of both the main

controller and SDC of the UPFC are to be tuned online to achieve the maximum benefit

in terms of damping. Therefore, 26 linear neurons are needed in the output layer (6 for

each PSS controller and 14 for the UPFC controller).

CHAPTER 10

194

Table 10.2: Line(s) outages cases

Double-Line Outages No.

Single-Line

Outages No. Lines No. Lines No. Lines 1.1 Line L5 1.11 Lines L5 and L9 1.22 Lines L7 and L12 1.33 Lines L11 and L13

1.2 Line L6 1.12 Lines L5 and L10 1.23 Lines L8 and L9 1.34 Lines L12 and L13









1.21 Lines L7 and L11 1.32 Lines L10 and L13 1.43 Lines L13 and L14

Table 10.3: Variations of load and power generation

Load Demand (pu) Power Generation (pu)

Node N9 Node N10 Slack Bus (node N1) PV Bus No.

Load PF Load PF PGEN QGEN PGEN QGEN

2.1 8 + j 2 0.97 11 + j 3 0.96

2.2 8 + j 2 0.97 12 + j 3 0.97

5.4 – 6.8 -1.0 – -0.9 4.5 – 5.0 -2.0 – -1.1

2.3 9 + j 8 0.75 11 + j 9 0.77

2.4 9 + j 8 0.75 12 + j 9 0.80

5.0 – 6.4 2.1 – 2.2 5.0 – 5.5 1.6 – 3.3

2.5 10 + j 5 0.89 13 + j 6 0.85

2.6 10 + j 5 0.89 14 + j 6 0.91

6.5 – 7.9 0.9 – 1.0 5.5 – 6.0 0.1 – 1.3

2.7 11 + j 6 0.88 13 + j 7 0.92

2.8 11 + j 6 0.88 14 + j 7 0.89

6.0 - 7.4 1.3 - 1.4 6.0 – 6.5 0.7 – 2.1

2.9 12 + j 8 0.83 15 + j 9 0.86

2.10 12 + j 8 0.83 16 + j 9 0.87

7.7 – 9.0 3.0 – 3.1 6.5 – 7.0 2.0 – 4.2

2.11 13 + j 4 0.96 15 + j 5 0.95

2.12 13 + j 4 0.96 16 + j 5 0.95

7.1 – 8.5 0.7 – 0.8 7.0 – 7.5 -0.2 – 1.1

2.13 14 + j 7 0.89 17 + j 8 0.90

2.14 14 + j 7 0.89 18 + j 8 0.91

7.5 – 8.8 2.9 – 3.1 8.0 – 8.5 2.0 – 4.0

2.15 15 + j 2 0.99 17 + j 3 0.98

2.16 15 + j 2 0.99 18 + j 3 0.99

6.9 – 8.2 0.1 – 0.2 8.5 – 9.0 -0.7 – 0.4

pu on 100 MVA

The load demands together with their power factors (PFs) at nodes N9 and N10 are

varied in the representative range between minimum and maximum values. Power-flow

solutions with the specified load demands give the range of active- and reactive- power

at generator nodes as shown in Table 10.3. It has been taken that the load demands at


195

nodes N9 and N10 follow similar patterns. However, any different patterns of load

demand variations, for example, in areas in different time zones, when they arise, can

be included in the data set without difficulty.

For each contingency, the procedure described in Section 9.2 and power flow studies

are used for forming the neural network input data in the training case. The optimal

controller parameters are also determined for each case using the method described in

Chapter 6. These optimal controller parameter values are used as the specified

network output data.

In applying the optimal control coordination described in Chapter 6 for training and test

data generation, the sum of the squares of the real parts of all of the eigenvalues of the

electromechanical modes is maximised, with the constraints that the minimum damping

ratio of the local modes is to be 0.3, and that of the inter-area mode 0.1.

The cases generated from Tables 10.2 and 10.3 are sub-divided into the training set

and test set. For the training set, line outage cases 1.1 – 1.4, 1.6 – 1.9, 1.11 – 1.20,

1.22 – 1.27, 1.29 – 1.34 and 1.36 – 1.42 together with load demand variations in cases

2.1 – 2.5, 2.7 – 2.10, and 2.12 – 2.16 are selected. The remaining cases of line

outages and load demand variations in Tables 10.2 and 10.3 are used for the test set.

10.3.2 Training, Testing and Sizing the Neural network

In the present work, the neural network is initially assumed to have one hidden layer

and the number of hidden nodes is taken to be 5. The size of the neural network is then

adjusted according to the procedure described in Section 9.3.4.

The performance goals specified in terms of the error function F(x) of 0.004 (for

training) and 0.006 (for testing) are used. The maximum differences between the

optimal damping ratio and the damping ratio calculated using neural network outputs of

0.03 (for training) and 0.05 (for testing) are also used as the performance goals.

Maximum number of epoch of 100 is specified for the network training. Several network

sizes (i.e. number of hidden neurons) are investigated to achieve the performance

goals. Based on the investigation, it is found that the network with 10 hidden neurons in

one hidden layer satisfies the convergence criteria. On this basis, the trained and

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196

tested neural network is used in the application mode, and its dynamic performance is

evaluated by simulation in the following section.

10.4 Dynamic Performance of the System in the Study Table 10.4 shows the comparison of modal response characteristics

(electromechanical mode eigenvalues, frequencies and damping ratios) between non-

adaptive and adaptive controllers of the system in Fig.10.1 for a range of contingencies

and operating conditions. For non-adaptive controller, the controller parameters derived

from the base case design are used for all of the contingency cases and load changes.

Table 10.4: Dynamic performances of controllers

Non-Adaptive Controller Adaptive Controller No

Case

Eigenvalues Freq. (Hz)

DampRatio Eigenvalues Freq.

(Hz) DampRatio

1 Base Case Load (pu) :

Node N9 : 10 + j2.0 Node N10 : 13 + j2.5

-2.3699 ± j7.0642*

-2.3448 ± j6.6088**

-0.5329 ± j3.5437***

1.12 1.05 0.56

0.3181 0.3344 0.1487

-2.3089 ± j7.3413*

-2.3925 ± j7.1246**

-0.5036 ± j3.3037***

1.17 1.13 0.53

0.3000 0.3183 0.1507

2 Load-Change Case

Load (pu) : Node N9 : 15 + j7

Node N10 : 16 + j8

-1.8169 ± j7.5759*

-1.8685 ± j7.1286**

-0.3723 ± j3.1591***

1.21 1.13 0.50

0.2332 0.2535 0.1170

-2.3145 ± j7.2747*

-2.2683 ± j7.0094**

-0.5303 ± j3.1685***

1.16 1.12 0.50

0.3032 0.3079 0.1651

3 Line L5 Out

-2.3783 ± j7.3294*

-1.4468 ± j6.6625**

-0.2630 ± j3.3211***

1.17 1.06 0.53

0.3086 0.2122 0.0789

-2.2690 ± j6.9820*

-2.0757 ± j6.6838**

-0.5093 ± j3.4126***

1.11 1.06 0.54

0.3091 0.2966 0.1476

4 Lines L7 & L11 Out

-1.7872 ± j6.8207*

-1.5108 ± j6.5671**

-0.3246 ± j3.2679***

1.09 1.05 0.52

0.2535 0.2242 0.0988

-2.0100 ± j6.6457*

-2.0145 ± j6.4854**

-0.4865 ± j3.2230***

1.06 1.03 0.51

0.2895 0.2966 0.1493

5 Line L13 Out

-2.5048 ± j7.4695*

-2.5559 ± j7.3784**

-0.1622 ± j3.7914***

1.19 1.17 0.60

0.3179 0.3273 0.0427

-2.3543 ± j7.4433*

-2.4456 ± j7.3200**

-0.5619 ± j3.3340***

1.18 1.17 0.53

0.3016 0.3169 0.1662

6 Lines L5 & L14 Out

-2.5297 ± j7.5291*

-1.5138 ± j6.6691**

-0.1575 ± j3.5010***

1.20 1.06 0.66

0.3185 0.2214 0.0449

-2.4497 ± j7.3964*

-1.9943 ± j6.5613**

-0.4610 ± j3.0515***

1.18 1.04 0.49

0.3144 0.2908 0.1494

* local mode associated with generators G1 and G4 ** local mode associated with generators G2 and G3 *** inter-area mode

The base case (referred to as case 1 in Table 10.4) is that with the full system in

Fig.10.1, and load demands at nodes N9 and N10 being 10+j2 pu and 13+j2.5 pu

respectively. The comparison in Table 10.4 for case 1 confirms that the damping ratios

for the electromechanical modes achieved by the neural adaptive controller are closely

similar to those obtained from the fixed-parameter controllers (i.e. non-adaptive)

designed with the system configuration and operating condition specified in the base


197

case. In the off-line training of the neural adaptive controller, the base case has not

been included in the training set. The comparison for case 1 can, therefore, be seen as

a neural adaptive controller testing.

In case 2 of Table 10.4, the load demands at nodes N9 and N10 increase to 15+j7 pu

and 16+j8 pu respectively while the system configuration remains as that of the base

case. With non-adaptive controllers, the damping ratios of the electromechanical

modes decrease noticeably in comparison with those in the base case. However, with

the neural adaptive controller, the damping ratios are maintained at the levels similar to

those of the base case.

Further comparisons in cases 3-6 of Table 10.4 focus on contingencies where one or

two transmission circuits are lost. The load demands are those in the base case. In

case 3 where there is an outage of transmission line L5 in Fig.10.1, there is a

substantial reduction in the inter-area mode damping in comparison with the base

case. The decreases in the local mode damping are non-uniform. The local mode

associated with generators G2 and G3 is affected severely in terms of damping, given

that these generators are electrically close to the outage location. The damping ratio of

this mode is reduced to 0.2122, compared to 0.3344 in the base case. The damping of

the local mode associated with generators G1 and G4 is hardly affected by this outage.

Its damping ratio is now 0.3086 in comparison with 0.3181 of the base case. With the

adaptive controller, the damping ratios of all of the electromechanical modes are only

marginally affected by the outage, in comparison with those in the base case, as

indicated in Table 10.4.

The response characteristics of the three electromechanical modes in case 4 where

there are double outages of transmission lines L7 and L11 are given in Table 10.4. The

modal damping ratios with non-adaptive controllers are now substantially lower than

those of the base case. In comparison, the adaptive controllers are able to restore the

damping ratios to the levels which are nearly equal to those of the base case, even

though the contingency of case 4 has not been included in the off-line training of the

adaptive controller.

The outage of transmission line L13 in case 5 of Table 10.4 affects the damping of the

inter-area mode very severely when the non-adaptive controllers are used. The

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198

damping ratio of 0.1487 in the base case is now reduced to 0.0427 in the outage case

5. However, the outage does not affect the local mode dampings to any significant

extent, relative to those in the base case. This response characteristic is consistent

with the topology of the power system in Fig.10.1 where transmission line L13 has the

primary function of interconnecting the two areas. The robustness of the adaptive

controller in this outage case is confirmed by the results of Table 10.4. The controller

parameters determined by the trained neural network are able to adapt to the new

system configuration for maintaining the modal damping ratios at the levels similar to

those in the base case.

Double outages of transmission lines L5 and L14 are then considered in case 6 of

Table 10.4. As expected, the additional outage of transmission line L14 which

interconnects the two areas affects mainly the damping of the inter-area mode when

non-adaptive controllers are used. Comparisons among the damping ratios of the inter-

area mode achieved by the non-adaptive controllers in cases 1, 3 and 6 confirm the

effect of the outage of transmission line L14 on the inter-area mode damping. With

adaptive controller parameters, the adverse effects of the outages in case 6 are largely

countered, as indicated in the damping ratios results of Table 10.4 The levels of

electromechanical mode dampings are almost the same as those in the base case.

10.5 Time-Domain Simulations In order to further validate the performance of the proposed neural-adaptive controller,

time-domain simulations are carried out for the selected contingency cases (i.e. line

L13 outage and lines L5 and L14 outage). The time-step length of 50 ms is adopted for

the simulations. The descriptions of the line(s) outage cases and the disturbances used

to initiate the transients for each case are given in Table 10.5.

In Figs.10.2 – 10.5 are shown the system transients following the disturbances. As the

focus is on the inter-area mode oscillation, relative voltage phase angle transient

between nodes N9 and N10 is used in forming the responses in Figs.10.2 and 10.3.

From the responses, it can be seen that, with non-adaptive controller, the system

oscillation is poorly damped and takes a considerable time to reach a stable condition.

With the proposed neural-adaptive controller, the system reaches steady-state

condition in 6 – 7 s subsequent to the disturbance for the contingency cases

considered (see Figs. 10.2 and 10.3).


199

Further comparison in terms of the transients in the rotor speed of generator G2

relative to that of generator G1 are given in Figs.10.4 and 10.5. The comparison

confirms the noticeable improvement in electromechanical oscillation damping when

the adaptive controller is used.

Table 10.5: Descriptions of line(s) outage cases and disturbances

Case Outage Description Disturbance Description

A

Line L13 has to be disconnected

to clear the fault.

Three-phase fault near node N13 on

line L13. The fault is initiated at time

t = 0.1 s, and the fault clearing time

is 0.1 s.

B

Line L5 is initially taken out for

maintenance then line L14 has

to be disconnected to clear the

fault.

Three-phase fault near node N13 on

line L14. The fault is initiated at time

t = 0.1 s, and the fault clearing time

is 0.1 s.

Fig.10.2: Relative voltage phase angle transients for case A disturbance

10 0 1 2 3 4 5 6 7 8 9 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

time, s

rela

tive

volta

ge p

hase

ang

le, r

ad

─── adaptive ------ non-adaptive

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200

Fig.10.3: Relative voltage phase angle transients for case B disturbance

Fig.10.4: Relative speed (G2-G1) transients for case A disturbance

0 1 2 3 4 5 6 7 8 9 10 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time, s

rela

tive

volta

ge p

hase

ang

le, r

ad


0 1 2 3 4 5 6 7 8 9 10 -1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

time, s

rela

tive

spee

d va

riatio

n, ra

d/s



201

Fig.10.5: Relative speed (G2-G1) transients for case B disturbance

In Figs.10.6 – 10.9 are also shown the variations of two controller parameters (i.e.

PSSs and SDC gains) during the transient period following the disturbance. For line

L13 outage, the variations are shown in Figs.10.6 and 10.7 respectively. Whereas, the

parameter variations in the case of lines L5 and L14 outage are shown in Figs.10.8 and

10.9 respectively.

There are rapid changes in the controller gains in the initial transient period following

fault and fault clearance, due to the transients in generator powers. To facilitate the

adaptation of the controller parameters in the initial transient period typically within the

range up to about 6 seconds, the option of keeping the inputs to the neural network

representing generator powers at the base-case values, and changing only the inputs

derived from the reduced nodal impedance matrix can be used. This option is based on

the result of the study given in Table 10.4 of Section 10.4 which confirms that the

overall damping is more substantially affected by system configuration than generator

loadings.

0 1 2 3 4 5 6 7 8 9 10 -1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

time, s

rela

tive

spee

d va

riatio

n, ra

d/s


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202

Fig.10.6: PSSs gain transients for case A disturbance

Fig.10.7: SDC gain transient for case A disturbance

0 1 2 3 4 5 6 7 8 9 10 0

2

4

6

8

10

12

14

16

18

20

time, s

PS

S g

ain,

pu

----- PSS at Gen.1 −−− PSS at Gen.3

0 1 2 3 4 5 6 7 8 9 10 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SDC

gai

n, p

u

time, s


203

Fig.10.8: PSSs gain transients for case B disturbance

Fig.10.9: SDC gain transient for case B disturbance

0 1 2 3 4 5 6 7 8 9 10 0

2

4

6

8

10

12

14

16

18

20

time, s

PS

S g

ain,

pu

------ PSS at Gen.1 ─── PSS at Gen.3

time, s

0 1 2 3 4 5 6 7 8 9 10 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SDC

gai

n, p

u

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204

In Figs.10.10 and 10.11 are shown the relative voltage phase angle transient and SDC

gain transient for disturbance case B in Table 10.5 respectively, using the option

described in the above (i.e. the option of keeping the inputs to the neural network

representing generator powers at the base-case values, and changing only the inputs

derived from the reduced nodal impedance matrix).

The damping of the transient in Fig. 10.10 is similar to that in Fig.10.3, whilst the

transient in the controller parameter in Fig.10.11 is substantially reduced in comparison

with that in Fig.10.9, which will facilitate the implementation of the adaptive controller.

In practice, there will be some time delay in the communication channel before the

inputs to the neural adaptive controller which represent the power system configuration

can be updated, following a disturbance. Studies have been carried out to quantify the

performance of the neural adaptive controller when there is the time delay.

Fig.10.10: Relative voltage phase angle transients for different time delays

0 1 2 3 4 5 6 7 8 9 10 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time, s

rela

tive

volta

ge p

hase

ang

le, r

ad

: no time delay : 1s time delay : 2s time delay


205

Fig.10.11: SDC gain transients for different time delays

In [96], a signal transmission delay of 0.75s has been proposed in the design of H∞

damping controllers using remote signals. A time delay up to 2s is, therefore,

considered in the presents work for evaluating the effect on the neural adaptive

controller performance. With signal transmission delays represented in the inputs to the

neural adaptive controller, the system transient responses for disturbance case B

described in Table 10.5 are re-evaluated and shown in Figs.10.10 – 10.13.

Time delays of 1s and 2s in relation to the updating of system topology after fault

clearance have been adopted in the study. The comparisons made of the inter-area

mode responses of Fig.10.10, and the local mode responses of Figs.10.12 and 10.13

indicate that the effect of the time delay is to reduce only slightly the electromechanical

mode dampings.

0 1 2 3 4 5 6 7 8 9 10 0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

time, s

SDC

gai

n, p

u


CHAPTER 10

206

Fig.10.12: Relative speed (G4-G1) transients for different time delays

Fig.10.13: Relative speed (G3-G2) transients for different time delays

0 1 2 3 4 5 6 7 8 9 -4

-3

-2

-1

0

1

2

3

time, s

rela

tive

spee

d va

riatio

n, ra

d/s

10


0 1 2 3 4 5 6 7 8 9 10 -1.5

-1

-0.5

0

0.5

1

1.5

2

time, s

rela

tive

spee

d va

riatio

n, ra

d/s



207

However, in relation to signal transmission delay and/or communication channel failure,

the neural adaptive controller designed in the chapter offers a key advantage in

comparison with other controller designs using remote signals [39]. When there is a

loss of communication channel or substantial time delay, the neural adaptive controller

will revert back to the fixed-parameter controller, with sub-optimal damping. In the

delay period/loss of communication channel, the PSSs and SDCs still have local input

signals (rotor speed/power), and they operate normally to give continuous non-zero

outputs which contribute to the system damping. Other controllers which depend totally

on remote input signals will not be able to function without the communication channel.

10.6 Possible Improvements Table 10.6 shows the range of optimal controller parameter variation for different

operating conditions and system configurations described in Tables 10.2 and 10.3

Results in the table show that the range of variation in the controller gains is wider than

that in the controller time constants. This indicates that the controller gains are more

sensitive to system changes than the time constants. Therefore, to simplify the

adaptive controller and its training, it is possible to adapt only the controller gains to the

prevailing system condition, and keep the controller time constants at the constant

values determined in the base case.

It is also found out from the investigation that the local modes are more affected by

PSSs, whereas, the inter-area mode is more affected by the SDC. In other words,

SDCs are more important if only the inter-area modes are to be considered. Therefore,

if the damping ratios of the local modes are high in the base case, it is possible to

include only the SDCs in the neural adaptive controller design, and to have fixed-

parameter PSSs designed in the base case.

In order to check whether a smaller number of neural network inputs can be used in the

adaptive controller, representation of the system configuration with a reduced nodal

impedance matrix of a lower dimension is investigated. In the investigation, only power

system nodes with direct connections to generators with PSSs and FACTS devices are

retained. The neural network with a smaller number of inputs is then trained and tested

using the test cases described in Section 10.3.1.

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208

Table 10.6: Range of optimal controller parameter variation for different operating

conditions and system configurations

Parameters Controller Type

Symbol Type Range

KPSS gain 4 – 20 pu

TPSS time constant 0.80 – 1.33 s

TPSS1 time constant 0.16 – 0.24 s



PSS


KSDC gain 0.1 – 1.0 pu

TSDC time constant 0.16 – 0.24 s

TSDC1 time constant 0.16 – 0.24 s



SDC


Ksh1, Ksh2 gain 0.1 – 1.0 pu

Tsh1, Tsh2 time constant 0.05 – 0.16 s

Kse1, Kse2 gain 0.01 – 0.10 pu

UPFC Main

Controller

Tse1, Tse2 time constant 0.16 – 0.24 s

pu on 100 MVA

Based on the outcome of the investigation, it is found that the neural network with a

reduced number of inputs can also provide acceptable results. Further reduction in the

number of inputs is also possible by discounting the real parts of the reduced nodal

impedance matrix elements, given that the parameters of the transmission circuits are

dominated by reactances.

By applying the above measures, the size of the neural network and its training can be

greatly simplified and kept to be minimal.


209

10.7 Discussion on Large Power System Application Drawing on the measures for improvements in Section 10.6 and the development of

ultra-large-scale neural network reported in [124], it is feasible to meet the

requirements of large power system application in terms of neural network size and

response time.

For the purpose of illustration, it is taken in the discussion that a large power system

has 100 generators and 10 FACTS devices with each generator having a PSS. The

comparisons between the adaptive neural controller requirements and the available

capability of the ultra-large-scale neural network are discussed in the following.

For the above power system, neural adaptive controller size requirements are given in

the following:

• The dimension of the reduced nodal impedance matrix is 110 110. Due to the

symmetry in the impedance matrix, only 6105 elements are required to represent the

power system configuration. The impedance matrix elements are, in general,

complex numbers. However, in a transmission system (which is the focus of the

present paper), the parameters of transmission circuits are dominated by the

reactances. This means that it is possible to discount the real parts of the nodal

impedance matrix, for the purpose of representing the system configuration.

• In addition to 6105 elements (in real numbers, following the removal of the real parts

of the nodal impedance matrix) used for representing the power system

configuration, there are 200 input values for representing generator active- and

reactive powers. Therefore, in this example of the system having 100 generators

each of which has a PSS, and 10 FACTS devices, the total number of input nodes of

the neural adaptive controller is about 6300.

• Based on the controller output parameters described in Section 9.3.1, the total

number of output nodes of the neural adaptive controller is about 750.

With the availability of cluster technology based on high-speed processors, ultra-large-

scale neural network (ULSNN) has been developed and reported in the literature. For

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210

example, in [124] a ULSNN with the following processing capability has been

developed:

• Multi-processor technology (a cluster of 196 Pentium III processors)

• 1.73 million weighting coefficients

• 9 million training patterns

• Computational speed of 163.3 GFlops/s

The cost of the above ULSNN was about 150,000 US dollars (in 2000). It is highly likely

that the cost at present is much lower, given that the cost of computer hardware is

decreasing while computing capability (in terms of memory and processing speed) is

increasing.

Recently, a cluster of 40 AMD Opteron processors has been developed at The

University of Western Australia for real-time simulation with a computational speed of

36 TeraFlops at the cost of about 100,000 Australian dollars. This processing speed far

exceeds the capability of the previous ULSNN development reported in [124].

The above comparisons confirm that the available neural network capability exceeds

the requirements of the neural adaptive controller for a large power system by a large

margin.

10.8 Conclusions The neural adaptive controller trained for a representative power system with a UPFC

has been comprehensively tested to verify its dynamic performance. Both eigenvalue

calculations and time-domain simulations are applied in the testing and verification.

Many comparative studies have been carried out to quantify the improved performance

of the adaptive controller proposed in comparison with that achieved with fixed-

parameter controllers.


211

The results confirm that the deterioration in system dampings arising from the use of

fixed-parameter controllers when system operating condition changes will be removed,

and maximum or optimal damping is regained by the proposed neural adaptive

controller.

With the state-of-the-art cluster technology, there is, in principle, no difficulty in

implementing the adaptive controller designed for real-time applications.

212

11.1 Introduction Power system controllers usually have hard (non-smooth) nonlinearities such as

saturation limits. The saturation limiters of the controllers are needed to impose

practical restrictions as a result of physical system limitation such as device or

equipment rating [59].

In the case of power system damping controllers, limitations in their outputs which can

lead to saturations will affect the oscillation damping capabilities of the controllers. The

reason is that the saturation effect will enforce the actual controller outputs (which

could have been several times higher than their limiting values) within the range of

permissible output values. The restrictions in the damping controller output will reduce

the signal needed for oscillation damping and therefore affect the overall system

dynamic behavior.

On this basis, the control coordination design of the damping controllers including

those of PSSs and FACTS devices needs to represent their output limits so that the

damping ratios as predicted in the design stage will be achieved in actual operating

conditions.

There have been numerous publications [18,40,72,78,90,91] which report the control

coordination of PSSs and FACTS device controllers for maintaining or enhancing

electromechanical oscillation damping in power systems. Although detailed plant

11 OPTIMAL DESIGN OF PSSs AND FACTS DEVICES SDCs WITH CONTROLLER SATURATION LIMITS

OPTIMAL DESIGN OF PSSs AND FACTS DEVICES SDCs WITH CONTROLLER SATURATION LIMITS

213

dynamic models have been incorporated in the control coordination reported, the

nonlinearities of the damping controllers in terms of their output limits are often

discounted or neglected in the design procedure. The research in developing design

methods which take into account controller output saturation has been very limited

[59,60].

In [11], a time-domain method has been proposed for optimisation and coordination of

the damping controllers of PSSs and FACTS devices. In principle, the time-domain

control coordination technique can represent the controller saturation limits. However,

there are a number of disadvantages of this technique which were identified in [11,78].

The results provided by the technique depend on the nature of the disturbance used to

excite the power system, and the controller robustness in relation to changes in power

system configurations is not achieved. Furthermore, the technique does not provide the

flexibility of selecting the electromechanical modes and specifying the required

damping ratios for control coordination design [78].

Similar to the approach in [11], the research reported in [59,60] applied the time-

domain method to optimise the saturation limits of the damping controller of a PSS or

series compensator in the context of a single-machine infinite bus system. Therefore,

the methods share the same disadvantages as that in [11]. In addition, the optimal

controller saturation limits obtained by the method in [59,60] may be difficult to

implement in practice as the optimal values to achieve satisfactory system dynamic

performance may be quite high and unrealisable in practice due to physical system

restrictions [59].

In all of the time-domain methods in [11,59,60], there is a need to define target values

of state variables and/or generator active-powers for forming the objective function to

be minimised in the control coordination. The use of initial condition prior to the

disturbance to form the target values has been proposed in [11]. However, the

assumption in [11] that the power system will return to its initial condition or settle to a

state close to its initial condition might not be valid, depending on the nature of the

disturbance. In [59,60], post-disturbance steady-state values are adopted for forming

the target values in the objective function. This is based on the premise that the power

system is stable and will settle to a new steady-state condition. Depending on the

CHAPTER 11

214

controller parameters and the severity of the disturbance, the premise used might not

be applicable.

Against the above background, the objective of the present chapter is to develop a new

method for control coordination design of PSSs and supplementary damping

controllers (SDCs) for FACTS devices in a multimachine power system which

achieves:

(i) the selection of any electromechanical modes for control coordination

(ii) the required damping ratios of the modes selected in (i)

(iii) the representation of output limits of PSSs and/or SDCs in the design procedure,

for any specified disturbances.

The new method draws on the eigenvalue-based optimal control coordination

developed in Chapter 6. This will address points (i) and (ii) in the above. The key

contribution of the new method is that of combining the eigenvalue-based technique

with nonlinear time-domain simulations for achieving the objective in (iii). The method is

based on the principle that, for a given disturbance, the PSSs and/or SDCs maximum

outputs are functions of the controller parameters. In general, the functions are

nonlinear, and analytical techniques to derive them in a closed form are not available.

The present work uses linear approximation to form the functions, and the coefficients

of which are determined by sensitivity analysis, drawing on the results of time-domain

simulation for specified disturbances. In the sensitivity analysis, individual controller

parameters are perturbed by small amount, and the changes in controllers maximum

outputs are evaluated by time-domain simulation. The sensitivities are used for forming

the linear functions which give the relationship between controller maximum outputs

and controller parameters. The controllers specified output limits are then imposed on

the linear functions to establish the inequality constraints which are included in the

eigenvalue-based control coordination design procedure.

In this way, the controller output limits are represented in the design. A key feature of

the new method is that the time-domain simulations which are used for forming the

controller output limit constraints are performed outside the eigenvalue-based control

coordination loop. The number of variables is, therefore, not increased in the control


215

coordination. If required, the individual time-domain simulations which are independent

of one another can be performed in parallel using a cluster of computers to speed up

the design process.

As the controller output limit constraints have been derived using linear approximation,

the results of the control coordination need to be validated with time-domain

simulations to confirm whether the actual controller outputs are within their specified

limits. In the case where the limits are exceeded, a revised set of linearised constraints

will be derived, by time-domain simulation and perturbation of the controller parameters

obtained from the most recent control coordination results. The set of revised

constraints is to be included in the next control coordination design. The procedure is

applied iteratively until the controller parameters obtained in the design lead to

controller outputs being within their specified limits, as confirmed by time-domain

simulations. Typically about 5 iterations of the procedure are required in the overall

control coordination process.

The design method is applied to a multimachine power system with PSSs and a unified

power flow controller (UPFC). The results from eigenvalues calculation and time-

domain simulation confirm the capabilities of the design procedure proposed in

avoiding controller saturations and preserving the optimal system damping.

11.2 Representation of Controller Limit Constraints in the

Design 11.2.1 Basic Concept This section discusses the additional inequality constraints to be imposed on the

controller parameters in the optimisation and the modification of the control

coordination procedure discussed in Chapter 6 to avoid controller saturations in the

design. An iterative method is used, and time-domain simulations to determine the

controller outputs are employed in the procedure.

The basic concept in deriving the additional inequality constraints is to use the linear

approximation for relating the maximum absolute value of controller outputs to a small

change of controller parameter values as follows:

CHAPTER 11

( )∗∗ −+≈ KKa tjjj yy (11.1)

In (11.1):

K : vector of parameters of all of the controllers considered in the design th : the maximum magnitude of the output of the j controller when the controller

parameter vector is K

yj

: the maximum magnitude of the output of the jth controller when the controller

parameter vector is K

∗jy

*

a and K. : vector of coefficients in the linearised relationship between yj j

If there are n controller parameters in total, vectors K, K* and aj have the following

forms:

( )n21t K,,K,K K=K (11.2)

( )∗∗∗∗ = n21t K,,K,K KK (11.3)

( ) ,M 1,2,j ; a,,a,a jn2j1jtj KK ==a (11.4)

where M is the number of controllers.

The coefficient vector a in (11.1) is calculated using the following two steps: j

1) Step 1. Perform time-domain simulations for small change (ε) of controller

parameters and calculate the maximum absolute values of the controller outputs.

Each controller parameter is perturbed by ε in turn, i.e. the initial controller

parameter vector K* iK will become (for i = 1, 2,…, n). With n individual

perturbations, the following set of parameter vectors is obtained:

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

ε+

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

ε+=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡ ε+

=

∗

∗

∗

∗

∗

∗

∗

∗

∗

n

2

1

n

n

2

1

2

n

2

1

1

K

KK

; ;

K

KK

;

K

KK

ML

MMKKK (11.5)

216


For each perturbation, one time-domain simulation for a specified disturbance will

be performed to determine the maximum magnitude the output of each and every

controller participating in the control coordination design. In addition, time-domain

simulation for the same disturbance when the parameter vector is K* will also be

carried out. Controller output limits are not imposed in these time-domain

simulations. There are, therefore (n+1) time-domain simulations for each specified

disturbance. With parallel computing systems, these individual time-domain

simulations can be carried out simultaneously to reduce the computing time in the

design.

2) Step 2. Coefficient vector aj is calculated using the results of the time-domain

simulations carried out in Step 1. If ijy is the maximum magnitude of the output of

the j

217

th controller when the parameter vector ( M,,2,1j K= ) ),n1,2,i( i K=K is used in

the time-domain simulation, then coefficient aji to be used in the linear relation in

(11.1) is given in, based on sensitivity analysis:

ε

−=

∗j

ij

jiyy

a (11.6)

thIn (11.6), is the maximum magnitude of the output of the j*

jy controller when the

parameter vector is K*. In (11.5) and (11.6), ε is chosen to be a small value (for

example, 1% of the controller parameter value).

11.2.2 Formulation of the Inequality Constraints The following is the inequality constraints to be included in the control coordination

design in addition to the constraints described in Chapter 6, drawing on the relation in

(11.1):

( ) M 2,...., 1, j ; j controller of limit output y *tj

*j =≤−+ KKa (11.7)

The constraints in (11.7) are expressed as functions of the controller parameters to be

identified in the optimisation process.

CHAPTER 11

The set of inequality constraints as described in (11.7) is for one disturbance condition.

Different disturbance conditions will lead to different controller output responses, which

have different maximum magnitudes. It is, therefore, necessary to consider several

disturbance conditions (for example, different fault locations and/or different fault

clearing times) in the control coordination design. The constraint in (11.7) extends, in a

straightforward manner, to L disturbances considered in the design:

( )( )L 2,....., 1, k and ; M 2,...., 1, j

j controller of limit output ][y *tkj

*jk

==

≤−+ KKa (11.8)

where:

( ));L1,2,k ;,M 1,2,j (

a,,a,a][ kjn

k2j

k1j

tkj

KK

K

==

=a (11.9)

In (11.9), the coefficient is given by, using (11.6): kjia

( )L 2,....., 1, k and ; M 2,....., 1, j ; n 2,...., 1, i

yya

*jk

ijkk

ji

===ε

−= (11.10)

218

In (11.10), ijky is the maximum absolute output value of the jth controller for the kth

disturbance condition when controller parameters vector ),n1,2,i( i K=K is used in the

time-domain simulation; and is the maximum absolute output value of the jth*jky

controller for the kth disturbance condition when controller parameters vector K* is used

in the time-domain simulation.

11.2.3 Flowchart of the Controller Design with Saturation Limits Fig.11.1 shows the flowchart of the control coordination design taking into account the

saturation limits of the controllers. The design starts from the control coordination

procedure described in Chapter 6 where saturation limit constraints are not considered.

This leads to the initial vector of the controller parameters denoted by K*.


219

Fig.11.1: Flowchart of the control coordination taking into account the saturation limits

YES

NO

START

Determine K* using the control coordination design described in Chapter 6

Perform time-domain simulations to determine )L1,2,....,k ;M1,2,.....,j (for s'y*

jk == (Use K* in the simulations)

Are all j controller of limit outputs'y*jk ≤

Calculate )n1,2,....,i (for s'i

=K using (11.5)

Determine the new controller parameters vector K using the control coordination

procedure of Chapter 6 augmented with inequality constraints (11.8)

Set : KK =* STOP

Perform time-domain simulations to determine )L1,2,...,k ;M1,2,...,j ;n,...,2,1i (for s'yi

jk ===

(Use s'i

K in the simulations)

Using (11.10), calculate:)L1,2,..., k ;M1,2,...,j ;n1,2,...,i (for s'ak

ji ===

CHAPTER 11

The next step is to calculate the peak values of controller outputs by performing time-

domain simulations. The maximum absolute values of the outputs of controllers

obtained from the simulations are denoted by . )L,1,2,k ;M,1,2,j (for s'y*jk KK ==

The controller output values are then compared with their corresponding limits. If

the maximum absolute values of all of the controller outputs are within their individual

limits, then the design procedure is completed, and K

s'y*jk

* is the desired vector of controller

parameters. If one or more of the limits are exceeded, the control coordination is

carried out again with the inequality constraints (11.8) to be included in the optimisation

process. The above steps are repeated until all of the controllers outputs are within

their specified limits.

11.3 Design Result and Validation 11.3.1 Power System Configuration The system in the study is based on the two-area 13-bus power system of Fig.11.2

[94]. It is proposed to use PSSs and FACTS device, i.e. a UPFC with an SDC to

improve the electromechanical oscillation damping in the power system. Data for this

test system together with its initial operating condition is presented in Appendix I. The

system configuration in Fig.11.2 in the present study is the same as that used in

Chapter 10. However, the lengths of the tie lines in the network of Fig.11.2 are shorter

that those in the previous chapter.

220

Fig.11.2: Two-area 230 kV system

G4

G1 G3

G2

N5 N3 N1 L3 L1 N8

L12 L6L5 L11

L16 L15

L14L13 L10

L9 L8

N11 N12 N13

L7

L4

N6 N7

UPFC N10 L2 N9 N4 N2


221

The UPFC is installed at node N13 in line L16. The general structure of the UPFC is

shown in Fig.4.6. The dynamical models for the shunt and series converter controllers

of the UPFC are shown in Figs.4.7a and 4.7b respectively. In addition to the UPFC

main controllers, there is an SDC with the control block diagram shown in Fig.4.8. The

input to the SDC is the active-power flow in the transmission line controlled by the

UPFC series converter.

PSSs are installed in generators G1 and G3. The locations of the PSSs are determined

by participation factors. The PSS block diagram is given in Fig. 4.1, with rotor speed as

the input. It is noted that, for this system with 4 generators, there are three

electromechanical modes (two local modes and one inter-area mode) of low frequency

oscillations.

11.3.2 Dynamic Performance for the Design Without Considering

Saturation Limits

Table 11.1 shows the modal information (electromechanical mode eigenvalues,

frequencies and damping ratios) of the system in Fig.11.2. The results in the table are

obtained using the control coordination design developed in Chapter 6 without

considering the controller saturation limits. The optimisation method described in

Chapter 6 is used to obtain the results in the table. The optimal controller parameters

determined by the control coordination design are given in Table 11.2.

Table 11.1: Electromechanical modes with optimal controller parameters


No. Mode Eigenvalue Frequency (Hz) Damping Ratio

1 Local -3.3176 ± j7.0199 1.12 0.4273

2 Local -3.3385 ± j7.0029 1.11 0.4303

3 Inter-area -0.7957 ± j3.6742 0.58 0.2116

CHAPTER 11

222

Table 11.2: Optimal controller parameters




PSSs (in G1/G3)




SDC


11.3.3 Design with SDC Output Limiter 11.3.3.1 Effects of SDC Output Saturation

In the present work, it is first proposed to analyse the effects of saturation limits of

UPFC’s SDC on the inter-area mode damping in order to investigate the saturation

effects on the system dynamic performance. The SDC is chosen as a controller for the

investigation due to the following reasons:

- The inter-area mode damping ratio is much lower than the local mode damping

ratios (see Table 11.1). It is, therefore, more vulnerable to the effects of controller

saturation limits.

- The inter-area mode is more affected by the SDC, whereas, the local modes are

more affected by the PSSs.

The investigation of the controller (i.e. SDC) saturation effects on system dynamic

performance is carried out by examining the transient responses of the system


223

following the disturbances for different controller saturation levels. The different

saturation levels are obtained by setting the controller limit to different values. Two

values of controller limit of 1.0 pu and 0.1 pu respectively are used in the present work.

The limit value of 1.0 pu is used to represent the SDC with a very high output limit

which will not be exceeded in the transient period after the disturbance. The value of

0.1 pu represents a typical SDC output limit.

The transient responses of the system following the disturbances are obtained by

performing the non-linear time-domain simulations. As the focus is on the inter-area

mode oscillation, the relative voltage phase angle transients between nodes N9 and

N10 are used in forming the responses. Two disturbances as described in Table 11.3

are considered in the investigation.

Table 11.3: Description of system disturbances

Case Disturbance Description

1 Three-phase fault near node N9 on line L13. The fault is initiated at time t =

0.1 s, and cleared after 0.20 s by disconnecting line L13.

2 Three-phase fault near node N10 on line L14. The fault is initiated at time t =

0.1 s, and cleared after 0.20 s by disconnecting line L14.

Fig.11.3 shows the system transients for the disturbances in cases 1 and 2 described

in Table 11.3. With a high output limit of 1 pu, none of the disturbances lead to the SDC

output being saturated, as indicated in Figs.11.3c and 11.3d. The inter-area mode

dampings obtained from the transient responses in Figs.11.3a and 11.3b when there is

no controller output saturation are similar to that shown in Table 11.1 which was

predicted by the eigenvalue-based design.

With the typical limit of 0.1 pu, the responses in Figs.11.3c and 11.3d confirm that the

SDC output is saturated for the time period of about 5 seconds. The high degrees of

SDC output saturation lead to the deterioration of the inter-area mode dampings as

indicated in the transient responses of Figs.11.3a and 11.3b. The damping predicted in

the design is now not achieved.

CHAPTER 11

In general, a disturbance that causes a line outage which weakens the interconnection

between the two areas will affect the inter-area mode damping severely, and SDC

output saturation will reduce significantly the effectiveness of the SDC in providing the

oscillation dampings.

224

Fig.11.3: System transients (effects of SDC saturation)

(a) Relative voltage phase angle transients for disturbance case 1

(b) Relative voltage phase angle transients for disturbance case 2

(c) SDC output transients for disturbance case 1

(d) SDC output transients for disturbance case 2

0 1 2 3 4 5 6 7 8 9 10 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

time, s

rela

tive

volta

ge p

hase

ang

le, r

ad

0 1 2 3 4 5 6 7 8 9

0.3

0.25

0.2

0.15

0.1

0.05

0

10-0.05

time, s

rela

tive

volta

ge p

hase

ang

le, r

ad

(a) (b)

0 1 2 3 4 5 6 7 8 9 10 -0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

time, s

SD

C O

utpu

t, pu

0 1 2 3 4 5 6 7 8 9

1

0.8

0.6

10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4 u ut

, pp

time, s

SD

C O

ut

(c) (d)

_______: controller with large output limit of 1 pu (no saturation)

-----------: controller with output limit of 0.1pu (saturation)


225

11.3.3.2 Dynamic Performance for the Design Considering Saturation Limits

Based on the investigation discussed in Section 11.3.3.1, it is confirmed that the

controller (SDC) saturation deteriorates the system dynamic performance. In order to

preserve a good system dynamic behavior, it is, therefore, necessary to consider the

controller (SDC) saturation limit in the control coordination design.

The inequality constraints (11.8) are now included in the new control coordination

design. The controller parameters in Table 11.2 are chosen as the starting parameter

values (K*) in the new design. The small change (ε) of 1% of the parameter value is

adopted in deriving the inequality constraints (11.8). The typical value of SDC output

limit of 0.1 pu is investigated and used in the new design.

The results (i.e. modal information) of the new control coordination designs with two

disturbances described in Table 11.3 are shown in Table 11.4. The optimal controller

parameters obtained by the new design are given in Tables 11.5. It can be seen that

the electromechanical mode damping ratios for the new design are similar to those of

the traditional design where controller saturation limits are not represented.

The system transients for the new design are shown in Fig.11.4. The disturbances in

cases 1 and 2 as described in Table 11.3 are used to initiate the transients. For each

disturbance, two time-domain simulations are performed:

(i) simulation using the SDC designed with saturation limits represented. The SDC

output limit is included in the time-domain simulation.

(ii) simulation using the SDC designed without saturation limit representation. The

time-domain simulation does not model the SDC output limiter.

The dampings of the inter-area mode estimated from the relative phase angle

transients in Figs.11.4a and 11.4b are closely similar to those predicted by the design

as given in Table 11.4. As shown in Figs.11.4c and 11.4d, the SDC outputs for the

disturbances considered do not saturate. This confirms the correctness of the new

design procedure in imposing the output limits on the controller design, which avoid the

possibility of controller output saturation and deterioration of system damping.

CHAPTER 11

226



No. Mode Eigenvalue Freq. (Hz) Damp. Ratio

1 Local -2.8188 ± j6.3039 1.00 0.4082

2 Local -2.8576 ± j6.4168 1.02 0.4068

3 Interarea -0.7253 ± j3.4487 0.55 0.2058





PSSs (in G1/G3)




SDC


The comparison between the transients obtained from the two simulations described in

(i) and (ii) are given in Fig.11.4. As shown in Figs.11.4c and 11.4d, the SDC outputs

from simulation (ii) are substantially greater than those from simulation (i). However,

the dampings as indicated in the transients in Figs.11.4a and 11.4b for the two

simulations are similar. This illustrates the usefulness and effectiveness of the new

control coordination procedure which provides optimal controller design in which

practical limiters are included. The dynamic performance is comparable to that

obtained by the ideal controllers the outputs of which are not limited.


227

Fig.11.4: System transients for new design with SDC output limiter and ideal SDC


(b) Relative voltage phase angle transients for disturbance case 2

(c) SDC output transients for disturbance case 1

(d) SDC output transients for disturbance case 2

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

time, s

rela

tive

volta

ge p

hase

ang

le, r

ad

0 1 2 3 4 5 6 7 8 9

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

10-0.05

time, s

rela

tive

volta

ge p

hase

ang

le, r

ad

(a) (b)

0 1 2 3 4 5 6 7 8 9 10-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

time, s

SD

C O

utpu

t, pu

0 1 2 3 4 5 6 7 8 9

1

0.8

0.6

10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4 u ut

, pp

time, s

SD

C O

ut

(d) (c)

_______: simulation using the SDC designed with saturation limits represented (the SDC output limit is included in the time-domain simulation)

-----------: simulation using the SDC designed without saturation limit representation (the time-domain simulation does not model the SDC output limiter)

CHAPTER 11

228

11.3.4 Design with PSSs and SDC Output Limiters The new design is now applied for the control coordination of the SDC and two PSSs

where output limits (for all of the controllers) are represented. The output limit of each

PSS is 0.03 pu, and that of the SDC is 0.1 pu. All of the electromechanical modes and

disturbance cases in Table 11.3 are considered in the control coordination.

In Table 11.6 are given the modal dampings and frequencies predicted from the design

results. The optimal controller parameters obtained by the new design are given in the

Table 11.7. The comparison between the results in Tables 11.1 and 11.6 indicate that

imposing the controller output limits on the design does not affect significantly the

damping ratios of the electromechanical modes.


(Output limits of PSSs and SDC considered in the design)

No. Mode Eigenvalue Freq. (Hz) Damp. Ratio

1 Local -2.9945 ± j6.7086 1.07 0.4076

2 Local -3.1066 ± j6.5814 1.05 0.4269

3 Inter-area -0.7447 ± j3.4845 0.55 0.2090

In Fig.11.5 are shown the transient responses related to the inter-area mode when the

disturbance is that of case 1 in Table 11.3. For comparison purpose, the transient

responses obtained by the controllers designed without saturation limits being

considered are also shown in Fig.11.5. The comparison in Fig.11.5a confirms again the

effectiveness of the new controller design in damping the inter-area mode.

The output of the SDC obtained in the new design does not saturate, as indicated in

the response of Fig.11.5b. However, the output of the SDC based on the traditional

design saturates heavily following the disturbance for about 6 seconds. This saturation

directly causes the damping deterioration as indicated in the response of Fig.11.5a.


229


(Output limit of PSSs and SDC considered in the design)



PSSs (in G1/G3)




SDC


The controllers designed using the new procedure also improve the local mode

dampings as indicated in the comparisons in Figs.11.6a and 11.6b where relative rotor

speeds transients confined to individual areas are shown. The disturbance condition for

forming the transients is that of case 1 described in Table 11.3. The local mode

dampings obtained from the controllers designed without their output limit

representation deteriorate noticeably in the comparisons shown in Figs.11.6a and

11.6b.

The transients in Figs.11.6c and 11.6d show that, with traditional controller design,

there is initial saturation in the PSSs outputs following the disturbance. With new

design procedure, the PSSs outputs are within their specified limits throughout the

transient period, as shown in the responses of Figs.11.6c and 11.6d. The

improvements of the local modes dampings are due to the combined contribution of the

SDC and PSSs designed with their respective output limits represented.

CHAPTER 11

230

Fig.11.5: Inter-area mode transients and SDC outputs


(b) SDC output transients for disturbance case 1

0 1 2 3 4 5 6 7 8 9

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

10 0

time, s

rela

tive

volta

ge p

hase

ang

le, r

ad

(a)

0 1 2 3 4 5 6 7 8 9

0.1

0.08

0.06

0.04

0.02

0

-0.02

-0.04

-0.06

-0.08

10 -0.1

time, s

SD

C O

utpu

t, pu

(b)

_______: new design with considering saturation limits

------------: design without considering saturation limits


231

Fig.11.6: Local mode transients and PSSs outputs

(a) Relative speed (G3-G2) transients for disturbance case 1

(b) Relative (G4-G1) transients for disturbance case 1

(c) PSS (at G1) output transients for disturbance case 1

(d) PSS (at G3) output transients for disturbance case 1

(a) (b)

(c) (d)

0 1 2 3 4 5 6 7 8 9 10 -3

-2

-1

0

1

2

3

time, s

,rela

tive

spee

d G

3-G

2, ra

d/s

0 1 2 3 4 5 6 7 8 9

3

2

1

0

-1

-2

10-3

time, s

,rela

tive

spee

d G

4-G

1, ra

d/s

time, s 0 1 2 3 4 5 6 7 8 9 10

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

PS

S (a

t G1)

out

put,

pu

0 1 2 3 4 5 6 7 8 9

0.03

0.02

10-0.03

-0.02

-0.01

0

0.01

u ut

, p

time, s

PS

S (a

t G3)

out

p

_______: new design with considering saturation limits

------------: design without considering saturation limits

CHAPTER 11

232

11.4 Conclusions The chapter has developed an effective and efficient design procedure for the control

coordination of PSSs and SDCs of FACTS devices in which both the dependence of

the dampings of electromechanical modes and that of controller maximum outputs on

the controller parameters are taken into account.

The relationships between the controller maximum outputs and their parameters

combined with the specified output limits form the set of inequality constraints which

complement those traditionally included in the eigenvalue-based control coordination.

The additional constraints expressed in terms of the controller parameters guarantee

that, for any specified disturbances, the outputs of the controllers designed will not

saturate, and thereby, the possibility of damping deterioration due to output saturation

is avoided. The levels of modal dampings specified in the design stage will be

maintained in the controllers operation following a disturbance.

In terms of computing time, the new design procedure is efficient as the relationships

between the controller maximum outputs and their parameters are derived from time-

domain simulations performed outside the optimisation loop which forms the optimal

controller parameters. Parallel processing has been proposed in the chapter to speed

up the design process in relation to forming the relationships if computing time is a

concern.

Case studies presented have verified the correctness of the design procedure together

with its software implementation, and quantified the improvements over the traditional

controller designs in terms of damping performance. Although the controller output limit

constraints derived have been adopted for the eigenvalue-based control coordination,

they are of general application, and can also be used in other design methods,

including the H∞ and LMI techniques [40].

233

12.1 Conclusions In the following are brought together and summarised the original contributions or

advances made in the research and presented in the body of the thesis.

The initial part of the research has made several contributions to the control

coordination design of fixed-parameter damping controllers in multi-machine power

systems. These contributions have addressed the many deficiencies identified in the

review presented in Chapters 2 and 3 of the previously-published design methods for

fixed-parameter controllers. While there are a number of existing or proposed methods

for power system controllers design in the context of small-disturbance stability, only

the group of design methods that draw on the eigenvalues of the power system state

matrix has been generally accepted and applied extensively in the power industry.

Within this group of eigenvalue-based methods, the research reported in Chapter 6 has

made two key original contributions to the control coordination design.

The first contribution is that of representing directly the nonlinear dependence of

eigenvalues associated with the electromechanical modes of interest on the controller

parameters to be identified by the eigenvalue-eigenvector equation set which is

included, as equality constraints, in the constrained optimisation-based control

coordination. The objective function used in the constrained optimisation for

maximising the damping of the electromechanical modes is formed from the

eigenvalues associated with these modes. This approach leads to a number of

important advantages over the existing methods as discussed in the following.

12 CONCLUSIONS AND FUTURE WORK

CHAPTER 12

234

Separate and time-consuming eigenvalue calculations at individual iterations in the

coordination design process are not required in the new procedure. Both eigenvalues

and controllers parameters are treated as the variables in the constrained optimisation,

and their final values are obtained simultaneously at the convergence. In addition, the

procedure developed is based on the constrained optimisation in its entirety, without

any need for a special and separate eigenvalue calculation software system.

Therefore, any possible limitations, particularly on the system state matrix dimension,

of the eigenvalues calculation software systems currently used in existing design

methods are removed.

The second contribution is that of retaining completely in the new control coordination

procedure the sparsity of the power system Jacobian matrix while achieving the

advantages derived from the first contribution. The benefit derived from the second

contribution is a significant one in the context of control coordination design for a large

power system.

The third advance made in the thesis is to propose and develop a new adaptive

controller which addresses the deficiencies of the numerous adaptive schemes

previously proposed and reviewed in Chapter 8. The adaptivity achieved with the new

controller developed in Chapter 9 allows the parameters of the power system

controllers to be tuned in real time, while there is no need to identify online any

reduced-order or approximate power system models as required in the previously-

proposed adaptive schemes.

The parameters of the power system controllers which lead to optimal dynamic

performance are identified directly from the prevailing power system configuration and

operating condition via a nonlinear multi-variable vector function implemented by a

neural network which has been trained off-line for a wide range of system operating

conditions and configurations. A key aspect of the contribution is that of transforming

the discrete form of system topology variation into a set of continuous variables derived

from the power network nodal impedance matrix. The transformation removes the

difficulty imposed by the combinatorial nature of system topology which has precluded

the representation of power system configuration variation in the previously-proposed

adaptive schemes.

CONCLUSIONS AND FUTURE WORK

235

The neural network-based adaptive controller developed in the thesis has an inherent

parallel structure which makes it suitable for multi-processor implementation based on

the state-of-the-art cluster technology.

The fourth and final contribution of the research is related to a very practical aspect in

power system controller design. In addition to the bounds on controller parameters

which are always taken into account in traditional design methods, the research

develops a new technique by which limits on the controllers outputs are also

represented in the controller design. With the new design method, the possibility of

system damping deterioration due to controller output saturation is eliminated, for any

specified disturbances.

Through a series of time-domain simulations performed outside the constrained

optimisation loop required in the controller design, the relationship expressed in an

algebraic form between controllers maximum outputs and their parameters is

established, which leads to a set of inequality constraints for representing the

controllers output limiters. The eigenvalue-based control coordination described in

Chapter 6 is then augmented with the additional set of inequality constraints, for

including in the new formulation as developed in Chapter 11 to give a comprehensive

controller design method where practical aspects of power system controllers in

relation to their parameters and output limits constraints are considered.

12.2 Future Work With the foundation provided by the new concepts and developments presented in the

thesis, further research is envisaged and outlined in the following.

12.2.1 Real-Time Implementation of the Adaptive Control

Coordination The proposal is that of implementing the neural adaptive controller on a cluster of high-

speed processors for real-time testing prior to its online application in power systems.

The testing will require real-time dynamic simulation of the power system, which allows

the interaction between the neural adaptive controller and the power system to be

CHAPTER 12

236

represented, and the dynamic performance of the adaptive controller to be quantified

closely in an online environment. With the multi-processor systems currently available,

it is feasible to achieve real-time power system dynamic simulation for use in the test.

12.2.2 Implementation of WAM-Based Stabilisers The control coordination considered in this thesis is carried out by using local feedback

control signals which are input to PSSs and SDCs associated with FACTS devices.

With the advances in WAM technologies using phasor measurement units (PMUs), it is

proposed to investigate the implementation of remote feedback control signals (or

combination of local and remote control signals) in the proposed control coordination of

multi-input controllers.

Although, as acknowledged in the literature, the WAM-based multi-input controllers

offer flexibility in control, particularly in the damping of multiple electromechanical

modes with a limited number of controllers, particularly those associated with FACTS

devices, there remain important issues to be addressed prior to practical applications in

power systems. A particular and important issue is related to the robustness of the

controllers with respect to the possible loss of remote signals due to communication

channel failure. The second issue is that of the design and implementation of multi-

input WAM-based stabilisers which are adaptive to power system configuration and

operating condition. The foundation established in Chapter 9 in the context of adaptive

single-input PSSs and SDCs associated with FACTS devices will provide a starting

point for further research in adaptive multi-input WAM-based controllers.

237

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at http://www.cs.unc.edu/~welch/kalman.

http://www.cs.unc.edu/%7Ewelch/kalman

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248

A.1. Synchronous Machine Equations Expressed in d-q rotor frame of reference, the voltages and currents of a generator are

related by:

(A.1) RIIGILV +ω+=•

r

where:

[ ] [ ]TkqkdfdqdT

fdqd IIIII ; 00EVV == IV (A.2)

It is taken here that in addition to the field circuit, the rotor has 1 damper winding on

each of the d and q axes. The field winding is on rotor d-axis. The stator voltages and

stator currents are denoted respectively by:

[ ] [ ]TqdsT

qds II ; VV == IV (A.3)

The rotor voltages and currents are denoted respectively by:

[ ]Tfdr 00E=V (A.4)

[ ]Tkqkdfdr III=I (A.5)

In (A.1), L, G and R matrices are given by:

AAPPPPEENNDDIIXX AA DDEERRIIVVAATTIIOONN OOFF SSOOMMEE EEQQUUAATTIIOONNSS FFRROOMM CCHHAAPPTTEERR 44

APPENDIX A

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−

−−

=

kq

kd

fd

a

aT

mq

md

md

q

q

kqmq

kdmdmd

mdfdmd

mqq

mdmdd

R00000R00000R00000R-00000R-

;

0LL0L00LL0

;

L00L00LL0L0LL0L

L00L00LL0L

RGL

(A.6)

The sign convention for stator currents is that of flowing into the external network. The

sign convention for rotor currents is that of flowing into the field and damper windings.

The signs of individual elements in L, G and R matrices reflect the convention of

current flow directions. The meaning of individual inductance and resistance in (A.6) is

given in Table A.1.

Table A.1: Inductance and resistance descriptions

Element Meaning

Ld and Lq Stator self inductances in the d and q axes

Lmd and Lmq Mutual inductances between stator and rotor in the d and q axes

Lfd Field circuit self inductance

Lkd and Lkq Self inductances of the damper winding in the d and q axes

Ra Stator winding resistance

Rfd Field circuit resistance

R

249

kd and R Resistances of the damper winding in the d and q axes kq

Equation (A.1) can be partitioned into stator and rotor circuit equations as the following:

(A.7)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ω+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

•

•

r

s

r

s

r

ssrss

r

r

s

rrrs

srss

r

s

00 I

I

R0

0R

I

IGG

I

I

LL

LL

V

VL

M

LLL

M

L

M

LLL

M

L

M

LLL

M

L

In (A.7), Lss, Lsr, Lrs, L , Grr ss, Gsr, Rs and R matrices are given by: r

APPENDIX A

(A.8)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎦

⎤⎢⎣

⎡−

−=

⎥⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡−

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

=

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−

−=

kq

kd

fd

ra

as

mdmd

mqsr

d

qss

kq

kdmd

mdfd

rr

mq

md

md

rs

mq

mdmdsr

q

dss

R000R000R

; R00R

0LLL00

; 0LL0

L000LL0LL

; L00L0L

L000LL

; L00L

RR

GG

LL

LL

From (A.7), the following equations can be obtained:

(A.9) ssrsrrsssrrsrssss IRIGIGILILV +ω+ω++=••

(A.10) rrrrrsrsr IRILILV ++=••

Defining stator flux linkage Ψ

250

s and rotor flux linkage Ψr as:

(A.11) ⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

r

s

rrrs

srss

r

s

II

LLLL

ΨΨ

From (A.11), the following equations can be obtained:

rsrssss ILILΨ += (A.12)

rrrsrsr ILILΨ += (A.13)

Using (A.12) and (A.13) in (A.9) gives:

( ) ( )( )[ ]ssrs1

rrsrssrr1

rrrsrss IRLLGGΨLGΨV +−ω+ω+= −−•

(A.14)

where:

APPENDIX A

(A.15) rsrssss•••

+= ILILΨ

The term which is related to stator winding electromagnetic transient can be

discounted as the electromechanical transients is the focus of investigation. On

discounting term, (A.14) becomes:

s•Ψ

s•Ψ

smrms IZΨPV −= (A.16)

where:

( )smrmmrm ; RGZSP +ω−=ω= (A.17)

( ) ( ) rs1

rrsrssm1

rrsrm ; LLGGGLGS −− −== (A.18)

Thus the generator equation is algebraic in form. The following is the derivation of rotor

circuit equations. On using (A.13) in (A.10) gives:

(A.19) ( ) ( srsr1

rrrrr ILΨLRΨV −+= −•

)

or:

(A.20) rsmrmr VIFΨAΨ ++=•

where:

( ) ( ) rs1

rrrm1

rrrm ; LLRFLRA −− =−= (A.21)

A.2. Derivation of Controllers State Equations

A.2.1 PSS Controller The transfer functions of the PSS controller shown in Fig.4.1 are given by:

251

APPENDIX A

PSS

PSSPSS

r

1p

sT1KsTx

+=

ω (A.22)

2PSS

1PSS

1p

2p

sT1sT1

xx

++

= (A.23)

4PSS

3PSS

2p

PSS

sT1sT1

xV

++

= (A.24)

By using Laplace transformation and re-arranging, (A.22) – (A.23) can be rewritten as:

rPSS1pPSS

1p KxT

1x••ω+−= (A.25)

1p2PSS

1PSS2p

2PSS1p

2PSS2p x

TTx

T1x

T1x

••+−= (A.26)

2p4PSS

3PSSPSS

4PSS2p

4PSSPSS x

TTV

T1x

T1V

••+−= (A.27)

Substituting (A.25) into (A.26) gives:

r2PSS

1PSSPSS2p

2PSS1p

2PSSPSS

1PSSPSS2p

TTKx

T1x

TTTTx

••ω+−

−= (A.28)

Similarly, substituting (A.28) into (A.27) gives:

r4PSS2PSS

3PSS1PSSPSSPSS

4PSS2p

4PSS2PSS

3PSS2PSS1p

2PSSPSS

1PSSPSS

4PSS

3PSSPSS

TTTTKV

T1x

TTTTx

TTTT

TTV

••ω+−

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=

(A.29)

Equations (A.25), (A.28) and (A.29) are collected together and arranged in

vector/matrix form as the following:

252

APPENDIX A

r

4PSS2PSS

3PSS1PSS

2PSS

1PSSPSS

PSS

PSS

2p

1p

4PSS4PSS2PSS

3PSS2PSS

2PSSPSS

1PSSPSS

4PSS

3PSS

2PSS2PSSPSS

1PSSPSS

PSS

PSS

2p

1p

TTTKT

TTK

K

V

x

x

T1

TTTT

TTTT

TT

0T

1T.TTT

00T

1

V

x

x•

•

•

•

ω

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

−−

−

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

(A.30)

or in more compact form as:

(A.31) rpppp••ω+= CxAx

where:

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

−−

−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

4PSS2PSS

3PSS1PSS

2PSS

1PSSPSS

PSS

p

4PSS4PSS2PSS

3PSS2PSS

2PSSPSS

1PSSPSS

4PSS

3PSS

2PSS2PSSPSS

1PSSPSS

PSS

p

PSS

2p

1p

p

TTTKT

TTK

K

;

T1

TTTT

TTTT

TT

0T

1T.TTT

00T

1

; Vxx

CAx

(A.32)

A.2.2 SVC Controller The transfer functions of the SVC controller shown in Fig.4.2 are given by:

S

S

SDCTrefT

1csT1

KxVV

x+

=−−

(A.33)

2S

1S

1c

C

sT1sT1

xB

++

= (A.34)

By using Laplace transformation and re-arranging, (A.33) and (A.34) can be rewritten

as:

253

APPENDIX A

SDCS

S1c

ST

S

SrefT

S

S1c x

TKx

T1V

TKV

TKx −−−=

• (A.35)

C2S

1c2S

1S1c

2SC B

T1x

TTx

T1B −+=

•• (A.36)


refT

2SS

1SST

2SS

1SSSDC

2SS

1SSC

2S1c

2SS

1SSC V

TTTKV

TTTKx

TTTKB

T1x

TTTTB +−−−

−=

• (A.37)

Equations (A.35) and (A.37) can be written in matrix form as follows:

refT

2SS

1SS

S

S

T

2SS

1SS

S

S

SDC

2SS

1SS

S

S

C

1c

2S2SS

1SS

S

C

1c V

TTTK

TK

V

TTTK

TK

x

TTTK

TK

B

x

T1

TTTT

0T1

Bx

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡•

•

(A.38)

or in compact form:

refTsSDCsTssss VXV DBCxAx +++=

• (A.39)

where:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−=⎥

⎦

⎤⎢⎣

⎡=

2SS

1SS

S

S

s

2SS

1SS

S

S

s

2SS

1SS

S

S

s

2S2SS

1SS

Ss

C

1cs

TTTK

TK

;

TTTK

TK

;

TTTK

TK

;

T1

TTTT

0T1

; Bx

DBCAx

(A.40)

A.2.3 TCSC Controller The transfer functions of the TCSC controller shown in Fig.4.3 are given by:

F

FF

Tref

PF

sT)sT1(K

PPx +

=−

(A.41)

254

APPENDIX A

t

t

SDCPF

t

sT1K

xxx

+=

− (A.42)


as:

TFTF

Fref

F

FPF PKP

TKP

TKx

••−−= (A.43)

SDCt

tt

tPF

t

tt x

TKX

T1x

TKX −−=

• (A.44)

Equations (A.43) and (A.44) are collected together in matrix form as follows:

refF

F

TF

TF

F

SDC

t

t

t

PF

tt

t

t

PF P0TK

P0K

P0TK

xTK0

X

x

T1

TK

00

Xx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+⎥

⎦

⎤⎢⎣

⎡−+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−+⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ •

•

•

(A.45)

or in compact form:

(A.46) reftTtTtSDCtttt PPPX EDCBxAx ++++=

••

where:

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−=⎥⎦

⎤⎢⎣

⎡=

0TK

; 0K

; 0TK

; TK0

; T1

TK

00 ;

Xx

F

F

tF

tF

F

t

t

tt

tt

ttt

PFt EDCBAx (A.47)

A.2.4 STATCOM Controller The first state equation for the STATCOM controller shown in Fig.4.5 is derived from

the capacitor voltage on the dc side (see Fig.4.4a). From Fig.4.4a, the following

equation can be obtained:

dcdc

dc IC1V =

• (A.48)

255

APPENDIX A

The dc current Idc in (A.48) is determined by using the following equation:

dcdcdcC IVPP == (A.49)

where PC is the active-power flow into the STATCOM. PC can be calculated by using

the relationship given by:

{ }∗= CNCNC IVreP (A.50)

In (A.50), superscript * denotes the complex conjugate. V and ICN CN are the STATCOM

voltage and current, and defined respectively by:

(A.51) )(jdc

jCCN ekVeVV α+φα ==

C

CNTCN jX

VVI −= (A.52)

Substituting (A.51) and (A.52) into (A.50) gives:

[ ])sin(kV)V(re)cos(kV)V(imX1P dcTdcTC

C α+φ−α+φ= (A.53)

The dc current Idc can be obtained by using (A.49) and (A.53) which will result in:

[ ])sin()V(re)cos()V(imXkI TT

Cdc α+φ−α+φ= (A.54)

By using the identities:

1cossin

sincoscossin)sin(sinsincoscos)cos(

cosV)V(re

sinV)V(im

22

TT

TT

=α+α

αφ+αφ=α+φαφ−αφ=α+φ

α=

α=

(A.55)

256

APPENDIX A

Equation (A.54) can be simplified to the following equation:

φ−

= sinVX

kI TC

dc (A.56)


φ−

=•

sinVXCkV T

Cdcdc (A.57)

Equation (A.57) is the first state equation for STATCOM. The remaining three

equations are derived by examining the transfer functions of the STATCOM controller

shown in Fig.4.5. These transfer functions are given by:

sK

I.droopXVVV 1C

CqSDCTrefT

=−−−

(A.58)

2C

2C2C

dc

c

sT)sT1(K

VkV

x +=

− (A.59)

Cc sT11

x +=

φ (A.60)

By using Laplace transformation and re-arranging, (A.58) - (A.60) can be rewritten as:

SDC1CCq1CT1CrefT1C xKIdroopKVKVKV −−−=

•

(A.61)

dc2C2C

dc2C

2C

2C

2Cc VKV

kKV

TKV

kTKx

•••−+−= (A.62)

φ−=φ•

Cc

C T1x

T1 (A.63)


257

APPENDIX A

φ+−

−−+−=•

sinVXCkKx

kKK

IkdroopKKV

kKKV

kKKV

TKV

kTKx

TCdc

2CSDC

2C1C

Cq2C1C

T2C1Cref

T2C1C

dc2C

2C

2C

2Cc

(A.64)

Equations (A.57), (A.61), (A.63) and (A.64) are collected together and rewritten in

compact form as the following:

φ+=φ

++++++=

+++=

=

•

φ

•

•

φ

•

socso

soSDCsoCqsoTsorefTsodcsosoc

SDCsoCqsoTsorefTso

sodc

x

VXIVVVVx

XIVVV

VV

NM

LKJOHGF

EDCB

A

(A.65)

where:

[ ] [ ] [ ] [ ]

[ ]φ=⎥⎦

⎤⎢⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=⎥⎦

⎤⎢⎣

⎡−=

⎥⎦

⎤⎢⎣

⎡−=⎥⎦

⎤⎢⎣

⎡−=⎥⎦

⎤⎢⎣

⎡=⎥⎦

⎤⎢⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡=

−=−=−==⎥⎦

⎤⎢⎣

⎡−=

φ sinV V; T1 ;

T1 ;

XCkK ;

kKK

droopkKK ;

kKK ;

kKK ;

TK ;

kTK

K ; droopK ; K ; K ; XC

k

TC

soC

soCdc

2Cso

2C1Cso

2C1Cso

2C1Cso

2C1Cso

2C

2Cso

2C

2Cso

1Cso1Cso1Cso1CsoCdc

so

NMLK

JOHGF

EDCBA

(A.66)

A.2.5 UPFC Controller Similar to the STATCOM, the first state equation for the UPFC controller shown in

Fig.4.7 is also derived from the capacitor on the dc side (see Fig.4.6). Therefore, for

UPFC, the time derivative of the dc voltage is also given by (A.48). In order to

determine the dc current I , the dc power Pdc dc has to be calculated in advance. For

UPFC, this dc power is given by:

seshdc PPP −= (A.67)

258

APPENDIX A

where P

259

sh and Pse are the shunt and series converter active-power respectively. Psh

and Pse can be calculated by using the following equations:

( ) shTshshshsh IVreP IV== ∗ (A.68)

( ) seTsesesese IVreP IV== ∗ (A.69)


(A.70) seTsesh

TshdcP IVIV −=

On using (A.49) and (A.70), the expression for dc current can be obtained as the

following:

( )seTsesh

Tsh

dcdc

dcdc V

1VPI IVIV −== (A.71)


(A.72) IVdc VCV =•

where:

seTsesh

TshI

dcdcV V;

VC1 IVIVC −=⎥

⎦

⎤⎢⎣

⎡= (A.73)

Equation (A.72) is the first state equation for UPFC. The remaining state equations are

derived by examining the transfer functions of the UPFC controller shown in Fig.4.7.

The transfer functions of the UPFC shunt converter (Fig.4.7a) are given by:

( )1sh

1sh1sh

shqSDCTrefT

0shpshp

sTsT1K

I.droopXVV

VV +=

−−−

− (A.74)

APPENDIX A

( )2sh

2sh2sh

dcrefdc

0shqshq

sTsT1K

VV

VV +=

−

− (A.75)


respectively as:

shq1shshq1sh

1sh

SDC1shSDC1sh

1shT1shT

1sh

1shrefT

1sh

1shshp

IdroopKITdroopK

XKXTKVKV

TKV

TKV

•

•••

−−

−−−−= (A.76)

dc2shdc2sh

2shrefdc

2sh

2shshq VKVTKV

TKV

••−−= (A.77)

Equations (A.76) and (A.77) are collected together and rewritten in matrix form as the

following:

shushudcudcurefdcu

SDCuSDCuTuTurefTush

VVV

XXVVV••

•••

+++++

++++=

IJIIHGF

EDCBAV (A.78)

where:

⎥⎦

⎤⎢⎣

⎡ −=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡−

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡−=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=

00droopK0

; 00TdroopK0 ;

K0

; TK0

TK

0 ;

0K

; 0TK

; 0K

; 0TK

; 0

TK

1shu1sh

1sh

u2sh

u

2sh

2shu

2sh

2shu1sh

u1sh

1sh

u1sh

u1sh

1sh

u1sh

1sh

u

JIHG

FEDCBA

(A.79)

The UPFC series converter control block diagram shown in Fig.4.7b can be

represented as the diagram shown in Fig.A.1. The transfer functions of the controller

shown in Fig.A.1 are given by:

260

APPENDIX A

2se

seqT

refseq

1se

sepT

refsep K

IV

QY

; KI

VP

Y=

−=

−

(A.80)

2se

2se

seq

0sepsep

1se

1se

sep

0seqseq

sTsT1

YVV

; sT

sT1Y

VV +=

−+=

− (A.81)

Ipref

261

Fig.A.1: UPFC series converter block diagram

Equation (A.80) can be rearranged and rewritten in compact form as follows:

se2T

1se V1 INNY += (A.82)

where:

(A.83) ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−

−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

seq

sepse

2se

1se2ref

2se

ref1se

1seq

sepse I

I ;

K00K

; QKPK ;

YY

INNY

The time-derivative of (A.82) is given by:

Yseq

Ysep

kVdc

Isep |VT|

|VT| Iseq

+

+

+

+

+

Vsep0

Vseq0

-

Ψ2

m+

Iqref

Qref

-

Pref

Σ

Σ

|V| Vsep

Vseq

sep

seq2

2seq

2sep

V

VatanΨ

VV|V|

=

+=

••

••

••

se1

se1sT

sT1+

se2

se2

sTsT1+

Σ

Σ

Kse1

Kse2

APPENDIX A

se22T

T1se

V

V ••

•+−= INNY (A.84)

By using Laplace transformation and re-arranging, (A.81) can be rewritten as:

seqseq2se

sepsepsep1se

seq YYT

1V ; YYT1V

••••+=+= (A.85)

In matrix form, (A.85) becomes:

(A.86) se4se3se••

+= YNYNV

where:

⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=⎥⎦

⎤⎢⎣

⎡=

0110

; 0

T1

T10

; VV

4

1se

2se3

seq

sepse NNV (A.87)


seuseuTuuse VVV••

δα

•+++= INIMLKV (A.88)

where:

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡===

⎥⎦

⎤⎢⎣

⎡−

−==

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−==

⎥⎦

⎤⎢⎣

⎡

−−

=−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

==

δαseq

sepse

seq

sepse2

TT

1se

2se24u

1se

1se

2se

2se

23u

ref1se

ref2se

14uref

1se

1se

ref

2se

2se

13u

II

; VV

; V

1 V; V1V

0KK0

; 0

TK

TK0

PKQK ;

PTK

QTK

IV

NNNNNM

NNLNNK

(A.89)

262

APPENDIX A

Equations (A.72), (A.78) and (A.88) are the state equations for the UPFC shown in

Fig.4.7.

A.2.6 Supplementary Damping controller By using the same procedure as described in Section A.2.1, the state equation for SDC

shown in Fig.4.8 can be written in compact form as follows:

(A.90) Tsusususu P••

+= CxAx

where:

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

−−−

−−

−

4SDC2SDC

3SDC1SDCSDC

2SDC

1SDCSDC

SDC

4SDC4SDC2SDC

SDC32SDC

2SDCSDC

1SDCSDC

4SDC

3SDC

2SDC2SDCSDC

1SDCSDC

1SDC

TT

TTKT

TKK

T

1

TT

TT

TT

TT

T

T

0T

1

T.T

TT

00T

1

susu

SDC

2s

1s

su ; ; Xxx

CAx

(A.91)

A.3. Induction Motor Equations For induction motor, the voltage and current are related by [1]:

(A.92) mmS

mmmmr

mm IVIRIGΨV ω++ω+=•

where the time derivative of flux-linkage can be expressed as: mΨ

••

= mmm ILΨ (A.93)

In (A.92), is the slip with respect to free-frame angular velocity. In

partition forms, the coefficient matrices in (A.92) and (A.93) are

given by:

mrRS ω−ω=ω

mmmm and ,, VRGL

263

APPENDIX A

(A.94)

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

mrr

mrr

mss

mss

mr

ms

m

R0000R00

00R0000R

M

M

LLMLL

M

M

M

LML

M

R0

0RR

(A.95)

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

mrr

mm

mrr

mm

mm

mss

mm

mss

mrr

mrs

msr

mss

m

L0L00L0L

L0L00L0L

M

M

LLMLL

M

M

M

LML

M

LL

LLL

(A.96)

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

00000000

0L0LL0L0

mm

mss

mm

mss

msr

mss

m

M

M

LLMLL

M

M

M

LML

M

00

GGG

(A.97)

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

000L00L0

000L00L0

mm

mm

mss

mss

mrs

mss

m

M

M

LLMLL

M

M

M

LML

M

0V

0VV

where: are the stator and rotor resistances respectively; re

the total inductance of stator and rotor windings respectively, and is the mutual

inductance between rotor and stator.

mrr

mss R and R m

rrmss L and L a

mmL

In terms of partitions of the coefficient matrices , the voltage/current

relationship for the stator is:

mmmm and ,, VRGL

(A.98) ( ) ms

mss

mrR

ms

ms

mr

msr

mr

ms

mss

mr

ms

ms IVIRIGIGΨV ω−ω++ω+ω+=

•

while for the rotor:

264

APPENDIX A

(A.99) ( ) ms

mrs

mrR

mr

mr

mr

mr IVIRΨV ω−ω++=

•

where:

(A.100) ⎥⎥⎦

⎤

⎢⎢⎣

⎡

ΨΨ

=⎥⎥⎦

⎤

⎢⎢⎣

⎡

ΨΨ

=⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡= m

qr

mdrm

rmqs

mdsm

smqr

mdrm

rmqr

mdrm

rmqs

mdsm

smqs

mdsm

s ; ; II

; VV

; II

; VV ΨΨIVIV

In (A.100), are the components of stator voltage in d and q axes

respectively; are the components of stator current in d and q axes

respectively; are the components of rotor voltage in d and q axes

respectively; are the components of rotor current in d and q axes

respectively; are the components of stator flux-linkage in d and q axes

respectively; are the components of rotor flux-linkage in d and q axes

respectively.

V and V mqs

mds

I and I mqs

mds

V and V mqr

mdr

I and I mqr

mdr

and mqs

mds ΨΨ

and mqr

mdr ΨΨ

From the flux-linkage relationships:

(A.101) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

mr

ms

mrr

mrs

msr

mss

mr

ms

I

I

LL

LL

Ψ

ΨL

M

LML

M

L

the rotor current may be expressed in the form: mrI

( ) ( )ms

mrs

mr

1mrr

mr ILΨLI −=

− (A.102)

Substituting for the rotor current in (A.98) and discounting the stator winding

electromagnetic transient gives:

(A.103) ms

mm

mr

mm

ms IZΨPV −=

265

APPENDIX A

where:

( )( ) ( ) m

ssmrR

ms

mrs

1mrr

msr

mss

mr

mm

1mrr

msr

mr

mm

VRLLGGZ

LGP

ω−ω−−⎥⎦⎤

⎢⎣⎡ −ω−=

ω=−

−

(A.104)

Similarly, substituting for the rotor current in (A.99) gives:

(A.105) ( ) ( ) ( ) ms

mrs

mrR

ms

mrs

1mrr

mr

mr

1mrr

mr

mr

mr IVILLRΨLRΨV ω−ω+−+=

−−•

For , (A.105) can be rearranged to give: 0V =mr

(A.106) ms

mm

mr

mm

mr IFΨAΨ +=•

where:

( )( ) ( ) m

rsmrR

mrs

1mrr

mr

mm

1mrr

mr

mm

VLLRF

LRA

ω−ω−=

−=−

−

(A.107)

A.4. Induction Motor Electromagnetic Torque Equations The electromagnetic torque developed by the motor is given by:

(A.108) ( ) ( ) ( ) ( )mr

msr

ms

mss

Tms

mr

ms

msr

mss

Tmr

TmseT IGIGI

I

I

00

GGII +=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎦⎤

⎢⎣⎡= L

M

LML

M

M


( ) ( )mr

mm

ms

mm

TmseT ΨSIGI += (A.109)

where:

266

APPENDIX A

( ) ( ) 1mrr

msr

mm

mrs

1mrr

msr

mss

mm ;

−−=−= LGSLLGGG (A.110)

A.5. Representation of Matrices in State Equations Matrices in

(4.33) – (4.38) are defined as follows:

pMpMgMgMgMgMeMpMeMeMeMgMMeMMM and ,,,,,,,,,,,,,, CADBCADSBCASMSFA

( ) ( )( ) (( ) ( )( ) ( )( ) (( ) ( )( ) ( )( ) ( )NP,p2p1ppMNG,e2e1eeM

NP,p2p1ppMNG,e2e1eeM

NG,g2g1ggMNG,e2e1eeM

NG,g2g1ggMNG,g2g1ggM

NG,g2g1ggMNG21M

NG,g2g1ggMNG,e2e1eeM

NG,e2e1eeMNG,m2m1mM

NG,p2p1ppMNG,m2m1mM

,,,diag ; ,,,diag

,,,diag ; ,,,diag

,,,diag ; ,,,diag

,,,diag ; ,,,diag

,,,diag ; M,,M,Mdiag


,,,diag ; ,,,diag

CCCCBBBBAAAACCCCDDDDAAAA

BBBBSSSSCCCCM

AAAASSSSDDDDFFFF

SSSSAAAA

LL

LL

LL

LL

LL

LL

LL

LL

==

==

==

==

==

==

== )

)

==

(A.111)

where S , S and S

267

ei gi pi are the selection matrices; A and Fmi mi can be found based on

(A.21); A and C are determined based on (A.32); Mpi pi i is calculated by using:

; ARii /H2M ω= ei, BBei, Cei, Dei (and Agi, BgiB , Cgi, D ) are the coefficient matrices of the ithgi

excitation controller (and prime-mover controller) which can be determined by the

examining the controllers transfer functions.

Matrices in (4.48) are defined as follows: suMsMsMsMsM and ,,, SDBCA

( ) ( )( ) (( )NS,s2s1ssM

NS,su2su1susuMNS,s2s1ssM

NS,s2s1ssMNS,s2s1ssM

,,,diag,,,diag ; ,,,diag

,,,diag ; ,,,diag

CCCCSSSSBBBB

DDDDAAAA

L

LL

LL

=

==

==

) (A.112)

where can be found based on (A.40), and Ssisisisi and ,,, DBCA sui is calculated based

on (4.47).

Matrices in (4.50) are defined as follows: tMtMtMtMtM and ,,, EDBCA

APPENDIX A

( ) ( )( ) (( )NT,t2t1ttM

NT,t2t1ttMNT,t2t1ttM

NT,t2t1ttMNT,t2t1ttM

,,,diag,,,diag ; ,,,diag

,,,diag ; ,,,diag

CCCCEEEEBBBB

DDDDAAAA

L

LL

LL

=

==

==

) (A.113)

where are determined based on (A.47). tititititi and ,,, EDBCA

CMsoMsoMsoMsoMsoMsoMsoMsoMsoMsoMsoMsoMsoMsoM and ,,,,,,,,,,,,, SNMLKJOHGFEDCBA in

(4.52) are defined as follows:

( ) ( )( ) (( ) (( ) (( ) (( ) (( ) (( ) ; ,,,diag


,,,diag ; ,,,diag,,,diag ; ,,,diag,,,diag ; ,,,diag


NC,so2so1sosoM

NC,C2C1CCMNC,so2so1sosoM

NC,so2so1sosoMNC,so2so1sosoM






HHHHSSSSGGGG

NNNNFFFFMMMMEEEE

LLLLDDDDKKKKCCCC

JJJJBBBBOOOOAAAA

L

LL

LL

LL

LL

LL

LL

LL

=

==

==

==

==

==

== ))

))

))

==

(A.114)

where are found

based on (A.66), and Ssoisoisoisoisoisoisoisoisoisoisoisoisoisoi and ,,,,,,,,,,,, NMLKJOHGFEDCBA

can be calculated based on (4.47). Ci

Matrices and VVMuMuMuMuMuMuMuMuMuMuMuMuMuMuM ,,,,,,,,,,,,,, CNMLKJIHGFEDCBA δM in

(4.54) are defined as follows:

( ) ( )( ) (( ) (( ) (( ) (( ) (( ) (( ) (

V,,V,Vdiag ; ,,,diag

,,,diag ; ,,,diag,,,diag ; ,,,diag,,,diag ; ,,,diag



NU,21MNU,V2V1VVM

NU,u2u1uuMNU,u2u1uuM







δδδδ ==

==

==

==

==

==

== )))

))))

==

LL

LL

LL

LL

LL

LL

LL

LL

VCCCCNNNNGGGG

MMMMFFFFLLLLEEEE

KKKKDDDDJJJJCCCC

IIIIBBBBHHHHAAAA

(A.115)

268

APPENDIX A

where are obtained based on (A.79);

can be found based on (A.89) and C

uiuiuiuiuiuiuiuiuiui and ,,,,,,,, JIHGFEDCBA

uiuiuiui and ,, NMLK is calculated based on (A.73). Vi

Matrices in (4.56) are defined as follows: suMsuM and CA

( ) ( )ND,su2su1susuMND,su2su1susuM ,,,diag ; ,,,diag CCCCAAAA LL == (A.116)

where are found based on (A.91). suisui and CA

A.6. Derivation of Formula ePΔ

The formulation for can be derived as the following. Based on (4.5), the following

equation is obtained:

ePΔ

( ) ( ) ( )( ) ( ) ( r

0rm

Ts

T0sr

0r

s0sm

Ts

Tr

0re

0e

PP

ΨΨSII

IIGII0s

Δ+⎥⎦⎤

⎢⎣⎡ Δ+ωΔ+ω+

Δ+⎥⎦⎤

⎢⎣⎡ Δ+ωΔ+ω=Δ+

) (A.117)

Equation (A.117) can be rewritten as:

( ) ( ) ( )( ) ( ) ( ) rm

T0s

0r

0rm

T0s

0rm

Ts

0r

0rm

T0s

0r

smT0

s0r

0sm

T0s

0sm

Ts

0r

0sm

T0s

0re

0e

PP

ΨSIΨSIΨSIΨSI

IGIIGIIGIIGI

Δω+ωΔ+Δω+ω+

Δω+ωΔ+Δω+ω=Δ+ (A.118)

For steady-state initial condition:

( ) 0rm

Ts

0r

0sm

T0s

0r

0eP ΨSIIGI Δω+ω= (A.119)

Substituting (A.119) into (A.118) and rearranging, the formulation for is obtained

as follows:

ePΔ

(A.120) r3s2r1eP ωΔ+Δ+Δ=Δ KIKΨK

where:

269

APPENDIX A

( )( ) ( ) ( )

( ) ( ) 0rm

T0s

0sm

T0s3

Tm

T0r

0rm

T0s

0r

Tm

T0s

0r2

mT0

s0r1

ΨSIIGIK

SΨGIGIK

SIK

+=

ω+ω+ω=

ω=

(A.121)

A.7. Derivation of Equations (4.138), (4.139) and (4.140) For small perturbation, (4.24) becomes:

(A.122) ( ) ( ) ( ) (( )( ) ( )m

s0m

smrsR

ms

0ms

mr

0mr

mrs

ms

0ms

mrs

1mrr

mr

mr

0mr

1mrr

mr

mr

0mr

IIVIIV

IILLRΨΨLRΨΨ

Δ+ω+Δ+ωΔ+ω−

Δ++Δ+−=Δ+−−

••

)

For initial steady-state condition, the following relationship holds:

(A.123) ( ) ( ) 0ms

mrsR

0ms

0mr

mrs

0ms

mrs

1mrr

mr

0r

1mrr

mr

0r IVIVILLRΨLR0Ψ ω+ω−+−==

−−•

Substituting (A.123) into (A.122) and rearranging gives the linearised form of (4.24) as

follows:

(A.124) mr

0mm

ms

0mm

mr

0mm

mr ωΔ+Δ+Δ=Δ•

LIFΨAΨ

where:

( )( ) ( )

0ms

mrs

0mm

mrs

0mrR

mrs

1mrr

mr

0mm

1mrr

mr

0mm

IVLVLLRF

LRA

=

ω−ω−=

−=−

−

(A.125)

By using the similar procedure, (4.25) can also be linearised to give:

(A.126) ( Le1m

r TTMω Δ−Δ=Δ −•

)

270

APPENDIX A

In (A.126), the expression for ΔTe can be obtained by linearising (4.26) as follows:

(A.127) ms

0m2

mr

0m1eT IKΨK Δ+Δ=Δ

where:

(A.128) ( )( ) ( ) ( ) ( ) ( )Tm

mT0m

rTm

mT0m

smm

T0ms

0m2

mm

T0ms

0m1

SΨGIGIK

SIK

++=

=

mm

mm and SGIn (A.128), are defined by (A.110).

The linearisation of the induction motor algebraic equation (4.23) is described in the

following. For small perturbations, the motor voltage equation (4.23) becomes:

( )( )( ) m

s0m

sms

mrs

1mrr

msr

mr

mrs

1mrr

msr

0mr

mss

mr

mss

0mr

mr

0mr

1mrr

msr

mr

1mrr

msr

0mr

ms

0ms

)()(

)()(

IIRLLGLLGGGΨΨLGLGVV

Δ+−ωΔ+ω+ωΔ−ω−−

Δ+ωΔ+ω=Δ+−−

−−

271

( )

(A.129)

For initial steady-state condition:

( ) 0ms

ms

mrs

1mrr

msr

0mr

mss

0mr

0mr

1mrr

msr

0mr

0ms )()( IRLLGGΨLGV −ω+ω−−ω= −− (A.130)

Substituting (A.130) into (A.129) and rearranging leads to the linearized form of (4.23)

as follows:

(A.131) mr

0mm

ms

0mm

mr

0mm

ms ωΔ+Δ−Δ=Δ KIZΨPV

where:

(A.132) ( ) ([ ]( ) 0m

smrs

1mrr

msr

mss

mss

0mr

1mrr

msr

0mm

mssR

0mr

ms

mrs

1mrr

msr

mss

0mr

0mm

1mrr

msr

0mr

0mm

)()(

)(

)(

ILLGVGΨLGKVRLLGGZ

LGP

−−

−

−

−++=

ω−ω++−ω−=

ω=

)

APPENDIX A

By extending (A.124) and (A.126), the state equations for multi-induction-motor system

are obtained as the following:

(A.133) mrM

0mM

msM

0mM

mrM

0mM

mrM ωLIFΨAΨ Δ+Δ+Δ=Δ•

272

) (A.134) ( LMeM1

MmrM TTMω Δ−Δ=Δ −•

where:

( )(( )0m

NI,m0m2m

0m1m

0mM

0mNI,m

0m2m

0m1m

0mM

0mNI,m

0m2m

0m1m

0mM

,,,diag

,,,diag

,,,diag

LLLL

FFFF

AAAA

K

K

K

=

=

=

) (A.135)

In (A.135), are determined based on (A.125). m0mi

0mmi

0mmi and , LFA

Similarly, extending (A.131) leads to the linearised voltage equation for multi-induction-

motor system as follows:

(A.136) mrM

0mM

msM

0mM

mrM

0mM

msM ωKIZΨPV Δ+Δ−Δ=Δ

where:

( )(( )0m

NI,m0m2m

0m1m

0mM

0mNI,m

0m2m

0m1m

0mM

0mNI,m

0m2m

0m1m

0mM

,,,diag

,,,diag

,,,diag

KKKK

ZZZZ

PPPP

L

L

L

=

=

=

) (A.137)

In (A.137), are determined based on (A.132). m0mi

0mmi

0mmi and , KZP

AAPPPPEENNDDIIXX BB DDEERRIIVVAATTIIOONN OOFF SSOOMMEE EEQQUUAATTIIOONNSS FFRROOMM CCHHAAPPTTEERR 55 AANNDD EEQQUUAATTIIOONN SSEETTSS TTOO BBEE UUSSEEDD IINN SSEECCTTIIOONN 55..1111

B.1 Frame of Reference Transformation B.1.1 d-q Frame of Reference Transformation from D-Q (network) to d-q (machine) frame of reference is carried out

as follows. By examining the vector diagram shown in Fig.B.1, the following equations

are obtained:

⎪⎩

⎪⎨⎧

δ−θ=

δ−θ=

)sin(VV

)cos(VV

Sq

Sd (B.1)

⎪⎩

⎪⎨⎧

θ=

θ=

sinVV

cosVV

SQ

SD (B.2)

On using the identities:

δθ−δθ=δ−θδθ+δθ=δ−θ

sincoscossin)sin(sinsincoscos)cos(

(B.3)

Equation (B.1) becomes:

δθ−δθ=

δθ+δθ=

sincosVcossinVV

sinsinVcoscosVV

SSq

SSd (B.4)

273

APPENDIX B

Substituting (B.2) into (B.4) gives:

δ−δ=

δ+δ=

sinVcosVVsinVcosVV

DQq

QDd (B.5)

Based on (B.5), it can be shown that the following relationships also hold:

(B.6) δ+δ=

δ−δ=

cosVsinVVsinVcosVV

qdQ

qdD

Q q

VS VQ

d Vq

θ−δ Vd

δ θ VD D

Fig.B.1: d-q frame of reference transformation

Rewriting (B.6) in matrix form:

SSN VTV δ= (B.7)

where:

(B.8) ⎥⎦

⎤⎢⎣

⎡δδδ−δ

=⎥⎦

⎤⎢⎣

⎡= δ cossin

sincos ;

VV

Q

DSN TV

274

APPENDIX B

Equation (B.7) describes the relationship between D-Q and d-q frame of references

where Tδ is the corresponding transformation matrix.

B.1.2 p-q Frame of Reference Transformation from D-Q (network) to p-q (device) frame of reference will be explained

by examining Fig.B.2. Following the similar procedure as described in Section B.1.1, it

can be shown that the transformation from D-Q to p-q frame of reference is carried out

by using the following relationship:

CCN VTV α= (B.9)

where:

(B.10) ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡ααα−α

=⎥⎦

⎤⎢⎣

⎡= α

Cq

CpC

CQ

CDCN V

V ;

cossinsincos

; VV

VTV

Q q

VC VCQ

p VCq

β−α VCp

β α VCD D

Fig.B.2: p-q frame of reference transformation

B.2 Derivation of (5.39) Fig.B.3 shows the UPFC series converter connection on the transmission line. By

examining Fig.B.3, the following equation is obtained:

275

APPENDIX B

sevxsese VVVZI +−= (B.11)

where:

sese jXZ = (

B.12)

earranging (B.12) gives:

R

seseEsese VYVYI += (B.13)

here:

w

vxEsese VVV ; Z/1Y −== (B.14)

Fig.B.3: UPFC series converter connection

eparating the real and imaginary parts of (B.13) and grouping them into vector/matrix

276

S

form leads to:

senseEsesen VYVYI += (B.15)

here:

(B.16)

quation (B.15) is valid for one UPFC only. Extension of (B.15) gives the equation for

power system having more than one UPFC (as shown in Fig.B.4) as the following:

w

⎥⎦

⎤⎢⎣

⎡−

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

0X/1X/10

; )V(im)V(re

; )V(im)V(re

; )I(im)I(re

se

sese

E

EE

se

sesen

se

sesen YVVI

E

Xse Vv Vx - Vse +

Ise

APPENDIX B

seNseMENseMseN VYVYI += (B.17)

where:

277

[ ] ( )NU,se2se1seseMTT

NU,ET2E

T1EEN ,,,diag ; YYYYVVVV LL == (B.18)

are calculated based on (B.16).

In (B.18), T V seiEi and Y

Equation (B.17) can be rewritten as the following:

LNseseMseN VMYI seNseMVY+= (B.19)

defined by:

where the selection matrix Mse is

(B.20)

Lv,NU

Lx,NU

LNU

11

11

11

11

11

11

se

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−−

−−

−−

LL

LL

LL

LL

LL

LL

LL

LL

LL

LL

MM

MLM

MM

MLM

MM

MM

MLM

MM

MM

MLM

MM

MM

MLM

MM

MLM

MM

MM

MLM

MM

MM

MLM

MM

MM

MLM

MM

MLM

MM

M

LLMLL

M

LLMLL

M

M

LLMLL

M

M

LLMLL

M

M

LLMLL

M

LLMLL

M

M

LLMLL

M

M

LLMLL

M

M

LLMLL

M

LLMLL

M

M

Lv2

Lx2

Lx1

LNB

Lv1

LNG+1

L1 L2

APPENDIX B

278

Fig.B.4: Connection of multiple UPFC series converters

B.3 Formulas for M••

ΔΔ ααα and , MM By examining the vector diagram of VT shown in Fig.B.5, the following relationship is

obtained (it is to be noted that VT is the voltage of the HV side of the FACTS device

transformer):

)V(re)V(im

cossintan

T

T=αα

=α (B.21)

Fig.B.5: Vector diagram of VT

α

Im(VT)

Re(VT)

VT

Im

Re

Xse1 Vv1 Vx1 - Vse1 +

Ise1

Xse2 Vv2 Vx2 - Vse2 +

Ise2

Xse,NU

Ise,NU

- Vse,NU + Vv,NU Vx,NU

APPENDIX B

Equation (B.21) can be rewritten as:

α=α cos)V(imsin)V(re TT (B.22)

Linearisation of (B.22) gives:

( ) ( ) ( ) ( )αΔ+αΔ+=αΔ+αΔ+ 0T

0T

0T

0T cos)V(im)V(imsin)V(re)V(re (B.23)

Using the identities:

( )( ) αΔα−α≅αΔ+α

αΔα+α≅αΔ+α000

000

sincoscos

cossinsin (B.24)

Equation (B.23) is modified to:

(B.25) )V(imcossin)V(im)V(resincos)V(re T000

TT000

T Δα+αΔα−=Δα+αΔα

By using the identities:

1sincos

sinV)V(im

cosV)V(re

0202

00T

0T

00T

0T

=α+α

α=

α=

(B.26)

Equation (B.25) becomes (after rearranging into vector/matrix form):

(B.27) T0 VM Δ=αΔ α

where:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

=α 20T

0T

20T

0T0

V

)V(re

V

)V(imM (B.28)

279

APPENDIX B

For multi-FACTS-device system, (B.27) is extended to:

(B.29) TM0

MM VMα Δ=Δ α

where:

( )0NF,

02

01

0M ,,,diag αααα = MMMM L (B.30)

In (B.30), is determined based on (B.28), and NF is the number of FACTS devices. 0iαM

In (B.29), can be represented in terms of as follows: TMV LNV

LNVMTM VMV = (B.31)

where MVM is the corresponding selection matrix and given by:

280

L1 L2 LNF

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

MMM

LLMLMLLMLL

MMM

LLMLMLLMLL

MMM

MMM

LLMLMLLMLL

MMM

MMM

LLLLLLLLLL

MMM

MMM

LLLLLLLLLL

MMM

LLLLLLLLLL

MMM

MMM

11

11

11

TVMM

LNG+1

Lx1

Based on (B.31), (B.29) can be expressed as:

(B.33) LNVM0

MM VMMα Δ=Δ α

(B.32) Lx2

Lx,NF

LNB

APPENDIX B

Time derivative of (B.22) is given by:

(B.34) αα−α=αα+α••••

sin)V(imcos)V(imcos)V(resin)V(re TTTT

or:

α+αα−α

=α

•••

sin)V(imcos)V(resin)V(recos)V(im

TT

TT (B.35)

Using the identities: TTTT V/)V(recos and V/)V(imsin =α=α , (B.35) can be rewritten

as:

(B.36) TR

••=α VM

where:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

= 2T

T2

T

TR

V

)V(re

V

)V(imM (B.37)

For multi-FACTS-device system, (B.36) is modified to:

(B.38) TMRMM••

= VMα

where:

( )NF,R2R1RRM ,,,diag MMMM K= (B.39)

On using (B.31) in (B.38), the following equation is obtained:

(B.40) LNVMRMM••

= VMMα

281

APPENDIX B

The linearised form of (B.40) is obtained by taking the time derivative of (B.33) as

follows:

(B.41) LNVM0

MM•

α

•Δ=Δ VMMα

B.4 Formulas for •

ΔΔ•

TMTMTMTM and VVVV ,,

B.4.1 Formulas for •TMTM and VV

The voltage magnitude at the HV terminal of the FACTS device transformer is given by:

( ) 2/12T

2TT )V(im)V(reV += (B.42)

Rearranging (B.42) gives:

2T

2T

2T )V(im)V(reV += (B.43)

Equation (B.43) can be further rearranged to give:

2T

2TTT )V(im)V(reVV += (B.44)

or:

T

2T

T

2T

T V)V(im

V)V(reV += (B.45)

Expressed in vector form, (B.45) becomes:

TTTV VE= (B.46)

where:

282

APPENDIX B

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

T

T

T

TT V

)V(imV

)V(reE (B.47)

For multi-FACTS device system, (B.46) is modified to:

TMTMTM VEV = (B.48)

where:

( )NF,T2T1TTM ,,,diag EEEE K= (B.49)

In (B.49), ETi is determined based on (B.47).


LNVMTMTM VMEV = (B.50)

Time derivative of (B.43) is:

⎟⎠

⎞⎜⎝

⎛ +=•••

)V(im)V(im)V(re)V(re2VV2 TTTTTT (B.51)

Rearranging (B.51) gives:

)V(imV

)V(im)V(reV

)V(reV TT

TT

T

TT

•••

+= (B.52)

Rewriting (B.52) in vector form leads to:

TTTV••

= VE (

B.53)

here ET is defined in (B.47). w

283

APPENDIX B

F

284

or multi-FACTS device system, (B.53) is modified to:

TMTMTM

••

= VEV (B.54)

here ETM is defined in (B.49).

n using (B.31) in (B.54) gives:

w

O

LNVMTMTM

••

= VMEV (B.55)

B.4.2 Formulas for

•ΔΔ TMTM and VV

For small perturbations, (B.43) becomes:

)V(imV

)V(im)V(reV

)V(reV T0T

0T

T0T

0T

T Δ+Δ=Δ (B.56)

vector/matrix form, (B.56) can be rewritten as:

In

T0TTV VE Δ=Δ (B.57)

here:

w

( )T0T0

T

0T

V1 VE = (B.58)

or multi-FACTS device, (B.57) is modified to:

F

TM0TMTM VEV Δ=Δ (B.59)

here:

w

APPENDIX B

285

( )0NF,T

02T

01T

0TM ,,,diag EEEE K= (B.60)

(B.60) is calculated based on (B.58).

n using (B.31) in (B.59) gives:

In 0TiE

O

LNVM0TMTM VMEV Δ=Δ (B.61)

ime derivative of (B.61):

T

LNVM0TMTM

••

Δ=Δ VMEV (B.62)

Formulas for

sMV , sMVΔ , sM•V and sM

•ΔV B.5

B.5.1 Formulas for sMsM and VV Δ

The voltage magnitude o onous macf the synchr hine stator terminal is given by:

( ) 2/12q

2ds VVV += (B.63)

sing the same procedure as described in Section B.4.1,

sVU can be expressed in

terms of vector Vs as follows:

sSsV VE= (B.64)

here:

w

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

s

q

s

dS V

VVVE (B.65)

or multi-machine system, (B.64) is modified to: F

APPENDIX B

286

sMSMsM VEV = (B.66)

here:

w

( )NG,S2S1SSM ,,,diag EEEE K= (B.67)

(B.67), ESi is determined based on (B.65).

or small perturbations, (B.63) becomes:

In

F

q0s

0q

d0s

0d

s VV

VV

VVV Δ+Δ=Δ (B.68)

ewriting (B.68) in vector/matrix form leads to:

R

s0ssV VE Δ=Δ (B.69)

here:

w

( )T0s0

s

0s

V1 VE = (B.70)

or multi-machine system, (B.69) is modified to:

F

sM0SMsM VEV Δ=Δ (B.71)

here:

w

( )0NG,S

02S

01S

0SM ,,,diag EEEE K= (B.72)

(B.72), is determined based on (B.70).

In 0SiE

APPENDIX B

287

Formulas for sMsM•

Δ•

VV and B.5.2

The time derivative of the synchronous machine stator terminal voltage can be

(B.73)

he formulation for the time derivative of the synchronous machine stator current in

(B.74)

olving (B.74) for gives:

(B.75)

can be eliminated from (B.75) by solving (A.13) for Ir and substitute the result into

(B.76)

here:

s•V

obtained by taking the time derivative of (A.16) as follows:

smsmrmrms•

−−+=••••

IZIZΨPΨPV

T s•I

(B.73) is described in the following. As the stator winding electromagnetic transient of

the synchronous machine is discounted, (A.15) becomes:

0ILIL =+••

rsrsss

S s•I

rsr1

sss•

−•

−= ILLI

Ir(B.75) to give:

••

= rms ΨQI

w

( ) 1rrsr

1ss

1rs

1rrsr

1ssm

−−−−− −= LLLULLLLQ (B.77)

(B.77), U is a 2 2 identity matrix.

ubstituting (A.17) and (B.76) into (B.73) gives time derivative of the synchronous

(B.78)

In

S

machine stator terminal voltage as:

rmrms•••ω+= WΨRV

APPENDIX B

where:

288

smrmmmmmm ; IGΨSWQZPR +=−= (B.79)

For multi-machine system case, the vector of the stator terminal voltages is:

(B.80)

where:

rMMrMMsM += ωWΨRV•••

( )( )NG,m2m1mM

NG,m2m1mM

,,,diag,,,diag

WWWWRRRR

K

K

=

= (B.81)

In (B.81), Rmi and Wmi are determined based on (B.79).

e formulation for is described in the following. Time derivative of (4.92) is

given by:

(B.82)

On using (B.76) in (B.82), the following equation is obtained:

(B.83)

where:

(B.84)

QM

sMΔ V•

Th

rM0MsM

0MrM

0MsM

••••Δ+Δ−Δ=Δ ωKIZΨPV

rM0MrM

0MsM Δ+Δ=Δ ωKΨRV

•••

M0M

0M

0M QZPR −=

In (B.84), is defined by:

( )NG,m2m1mM ,,,diag QQQQ K= (B.85)

APPENDIX B

289

here Qmi is determined based on (B.77).

.6 Formula for Line Active-Power Flow

B.6.1 Formulas for

w

B

TMP , TMPΔ , TM•

and TM•

P ΔP Fig.B.6 shows the diagram of transmission line for illa ustrating the determination of

active-power flow. The active-power flow will be used as an input signal to SDC of the

FACTS device. By examining Fig.B.6, the line active-power is calculated as follows:

{ }∗= vwvT IVreP (B.86)

(B.86), line current Ivw is given by:

In

( ) vshLwv

seLvw VYVVYI +−= (B.87)

here are the line series and shunt admittances,

ively.

Fig.B.6: A transmission line in power system

olving (B.86) gives:

(B.88)

here:

w SHshLLL

seL jBY and jBGY =+=

respect

S

vPTwLv

TwLv

TvLT BGGP VMVVVVV +−=

w

Transmission lineIvw

Vv Vw

P

APPENDIX B

290

(B.89)

quation (B.88) can be rewritten as follows:

(B.90)

r in a more compact form:

(B.91)

(B.92)

Based on (B.91), the formulation for is obtained as the following:

⎥⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

0110

; )V(im)V(re

; )V(im)V(re

Pw

ww

v

vv MVV

E

wTP

TvLv

TwLv

TvLT BGGP VMVVVVV +−=

o

BDTP VM=

where:

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=−=

w

v

BTP

TvL

TwL

TvLD ; BGG

V

VVMVVVM LM

TMP

BMDMVMP = TM (B.93)

where:

( )[ ]TT

ND,BT2B

T1BBM

ND,D2D1DDM ,,,diag

VVVV

MMMM

L

L

=

= (B.94)

In (B.94), MDi and VBi are determined based on (B.92), and ND is the number of SDCs.

In (B.93), VBM can be expressed in terms of VLN as follows:

LNKMBM VMV = (B.95)

nd given by:

where the MKM is the selection matrix a

APPENDIX B

291


L2 L1 LND

TMP LNKMDM VMM= (B.97)

y:

(B.98)

tion (B.98) can be rewritten as follows:

(B.99)

ore compact form:

Time-derivative of (B.88) is given b

•••

++− vPTwLvP

TwLv

TwLvwLvvLvvLT BBG VMVVMVVVV

••••−+= TTT GGGP VVVVV

Equa

•••

++−−+= vPTwLw

TP

TvLv

TwLwvLvvLvvLT BBGGGGP VMVVMVVVVVVVVV

••••TTT

or in m

(B.96)

Lw,ND

Lv,ND

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

MMM

LLLLMLMLLLLMLLLL

MMM

LLLLMLMLLLLMLLLL

MMM

MMM

LLLLMLMLLLLMLLLL

MMM

MMM

LLLLMLMLLLLMLLLL

MMM

MMM

LLLLMLMLLLLMLLLL

MMM

LLLLMLMLLLLMLLLL

MMM

MMM

LLLLMLMLLLLMLLLL

MMM

MMM

LLLLMLMLLLLMLLLL

MMM

MMM

LLLLMLMLLLLMLLLL

MMM

LLLLMLMLLLLMLLLL

MMM

11

11

11

11

11

11

TKMM

Lw2

Lv2

Lv1

LNB

Lw1

LNG+1

APPENDIX B

292

(B.100)

BLTP••

= VM

where:

[ ]TP

TvL

TvLP

TwL

TwL

TvLL BGBGG2 MVVMVVVM +−+−= M (B.101)

Based on (B.100), the formulation for can be obtained as the following:

(B.102)

TM•P

••

= VMP BMLMTM

where:

( )ND,L2L1LLM ,,,diag MMMM L= (B.103)

ed on (B.101).

(B.104)

In (B.103), MLi is calculated bas


•P LNKMLMTM

•= VMM

Linearisation of (B.88) gives:

( ) ( ) ( ) ( )( ) ( ) ⎥⎦

⎤⎢⎣⎡ Δ+Δ+

⎥⎦⎤

⎢⎣⎡ Δ+Δ−⎥⎦

⎤⎢⎣

Δ

wTP

T0vvP

T0wL

wT0

vvT0

wLvT0

vvvLT

B

G

VMVVMV

VVVVV⎡ +Δ=ΔT0GP VVV

(B.105)

Equation (B.105) can be rewritten in a more compact form as follows:

(B.106)

B0LTP VM Δ=Δ

where:

APPENDIX B

( ) ( ) ( ) ( ) ( ) ⎥⎦⎤

⎢⎣⎡ +−+−= T

PT0

vLT0

vLPT0

wLT0

wLT0

vL0L BGBGG2 MVVMVVVM M (B.107)

rmulation for Based on (B.106), the fo TMPΔ can be obtained as the following:

(B.108)

BM0LM VMP ΔΔ TM =

where:

( )0ND,L

02L

01L

0LM ,,,diag MMMM L= (B.109)

following equation is obtaine

(B.110)

ven by:

(B.111)

Equations (B.104) and (B.111) are the formulas for which are to be

sed in conjunction with the state equations for SDCs of vices, i.e. (5.95)

On using (B.95) in (B.108) the d:

LNKM0LMTM VMMP Δ=Δ

The time-derivative of (B.110) is gi

•

=Δ LNKM0LMTM

•Δ VMMP

TMTM and PP

FACTS de

able to use t

••Δ

u

and (5.106), respectively. It can be seen that, to be he formulas, the

variables LNLN and ••

Δ VV need to be expressed in terms of state and non-state

variables which will be discussed in the following.

B.6.2 Formulas for LN•V and LN

•ΔV

B.6.2.1 System with SVCs ive of (5.14) is given by: It can be shown that the time-derivat

( ) sMCMLNLNFSLLsMMLSsMMLS

•••

δ

•++++= VVYYVTYVTY0 δ xS (B.112)

293

APPENDIX B

where:

294

The expression for in (B.112) is given in the following. Based on (B.8), the

llowing equation is obtained:

(B.114)

(B.115)

(B.116)

1 2 NS

sMM VTδ•

fo

•T

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

δδ−δδ

δδ−δδ−= ••

••

δ

rrrr

rrrr

sincos

cossin

Equation (B.114) can be rewritten in a more compact form as follows:

∗δδ δ= TT r

••

where:

⎥⎦

⎤⎢⎣

⎡δ−δδ−δ−

=∗δ

rr

rr

sincoscossin

T

(B.113)

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

−

=

L

M

M

M

MLM

M

M

LM

M

LM

M

M

LM

M

M

LM

M

M

M

M

L

M

M

M

M

L

M

M

M

M

L

LMLML

MM

MM

LMLML

MM

LMLML

MM

MM

LMLML

MM

MM

LMLML

MMLM

M

M

M

M

M

L

)V(re)V(im

)V(re)V(im

)V(re)V(im

NS,x

NS,x

2x

2x

1x

1x

LNV

LNG+1 LL

Lx1

Lx2

Lx,NS

LNB

APPENDIX B

295

Multiplying both side of (B.115) by Vs, the following equation is obtained:

(B.117)

ue to is a scalar quantity, (B.117) can be modified to:

(B.118)

i-machine system, (B.119) becomes:

srs VTVT ∗δ

•δ

•δ=

D rδ•

rVSs

•δ

•δ= TVT

where:

(B.119) sVS VTT ∗δ=

For mult

rM•

δ•

(B.120) VSMsMM = δTVT

where ( )NG,VS2VS1VSVSM ,,,diag TTTT K= , and TVSi is calculated based on (B.119).

(B.120) into (B.112) and solving it for gives:

LN•VSubstituting

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+++−=

••

δ

•−

•

sMCMLNsMMLSrMVSMLS1

FSLLLN xSVVTYδTYYYV (B.121)

CM .11 ctively.

sion for is found by taking the time-derivative of (5.

In (B.121), sM•V and S are given in (B.80) and (A 4), respe

The expres 52) and solved it LN•

Δ V

for LNΔ V as follows: •

( ) ⎟⎠

⎜⎝

Δ+Δ+Δ−=Δ δ sMCM0LNrM

0VMLSsM

0MLSSLLLN xSVδDYVTYYV (B.122)

⎞⎛+•••−• 10

FY

APPENDIX B

296

is given by (B.83).

B.6.2.2 System with STATCOMs

(B.123)

(B.124)

imilar to that in (B.120), it can also be shown that the following equations hold:

(B.125)

B.126)

sM•

Δ VIn (B.122), the formula for

Time-derivative of (5.25) and (5.29) are given by:

LNLLsM CMMLCCMMLC

•

αα

••••++ VTYVTYMLSsMMLS δδ ++= VYVTYVTY0

dcMMdcMMCM•

φφ••

+= VTVTV

S

MVCMCMM•

α•

= αTVT

MMVdcMM φφ = ΦTVT (••

where:

( )( )NC,dcNC,2dc21dc1MV

NC,CNC,2C21C1VCM

V,,V,Vdiag

,,,g∗φ

∗φ

∗φφ

∗α

∗α

= TTTT

VTVT

K

Ldia ∗α= VTT

(B.127)

and are defined by:

(B.128)

ubstituting (B.40) into (B.125) gives:

(B.129)

∗αiT ∗

φiVTIn (B.127),

⎥⎦

⎤⎢⎣

⎡φφ−

=⎥⎦

⎤⎢⎣

⎡α−αα−α−

= ∗φ

∗α

ii

iiiV

ii

iii cosk

sink ;

sincoscossin

TT

S

LNVMRMVCMCMM•

α•

= VMMTVT

APPENDIX B

297

9) in (B.123) the following equation is

obtained:

(B.130)

In (B.130), and are given by the first and fourth equations of (5.93),

spectively.

he following is the derivation of the formula for . Time-derivative of (5.62) and

.67) are given by:

(B.131)

(B.133)

B.6.2.3 System with TCSCs Time-derivative of (5.19) is given by:

On using (B.120), (B124), (B.126) and (B.12

( )

⎥⎦

⎤⎟⎟⎠

⎞⎜⎜⎝

⎛++

⎢⎣

⎡++−=

•

φ

•

φα

δ−

dcMMMMVMLC

sMMLSrMVSMLS1

VMRMVCMLCLL

VTΦTTY

VTYδTYMMTYY•••

LNV

dcM•V M

•Φ

re

LN•

Δ VT

(5

( )YVTY +Δ••••

δ LLsM0MLS 0δDYVTYVMMAY =Δ+Δ+Δ+ αα rM

0VMLSCM

0MLCLNVM

0M

0VMLC

(B.132) cMstM0MCM

•

φ

•Δ=Δ xSMV

Substituting (B.132) into (B.131) and rearranging gives:

( ) ⎜⎝

+Δ+−=Δ αδ−

α0MLCsM

0MLS

1VM

0M

0VMLCLLLN TYVTYMMAYYV ⎟

⎠

⎞⎛ Δ+Δ••

φ

••

rM0VMLScMstM

0M δDYxSM

( ) 0xSYYVYY =++⎞⎛ ••••

tMCMtMVDLNFTLLδTVTY +⎟⎠

⎜⎝

+δ rMVSMsMMLS (B.134)

where:

APPENDIX B

1 2 NT

298

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

MMM

LLLLLLL

MMM

LLLLLLL

MMM

LLLLLLL

MMM

LLLLLLL

MMM

LLLLLLL

MMM

LLLLLLL

MMM

LLLLLLL

MMM

LLLLLLL

MMM

LLLLLLL

MMM

LLLLLL

MMM

NT,t

2t

1t

NT,t

2t

1t

tM

Y

Y

Y

Y

Y

Y

Y

LNG+1 L

Lx1

( )NGNB,V2V1VVD ,,,diag −= YYYY K (B.136)

vector of defined by:

(B.137)

(B.138)

The expression for can be found as the following. Time-derivative of (5.58) is as

follows:

In (B.136), Y is the ith subVi VMY

LNLLSNVM VYVY += LSY

On using (5.92), (B.97) and (B.104) in (B.134), and rearranging leads to:

( )⎟⎟

⎠

⎞+++

⎝δ

refMtMLNKMDMtMsuMsuMtMtMtMCMtMVD

rMVSMLSsMMLSKMLMtMCMtMVDFTLLLN

PEVMMCxSBxASYY

( ) +⎜⎛++−=

••−• 1 δTYVTYMMDSYYYYV

LN•

Δ V

(B.135)

Lv,NT

Lx,NT

LNB

Lv2

Lv1

Lx2

APPENDIX B

( ) 0xSYYVYYδVTY =Δ+Δ++⎟⎠

⎞⎜⎝

⎛ ΔΔ••••

δ tMCM0tM

0VDLN

0FTLLrMMsM

0MLS (B.139) D+ 0

V

On using (5.103), (B.110) and (B.111) in (B.139), and rearranging leads to:

(B.140)

B.6.2.4 System with UPFCs ime-derivatives of (5.41) - (5.43) are given by:

(B.141)

(B.143)

at in (B.129), and in (B.141) – (B.143)

pressed as:

(B.145)

(B.146)

( )

( )⎟⎟⎠

⎞Δ+Δ+Δ

Δ+⎜⎝

⎛ Δ++−=Δ••

δ−•

LNKM0LMtMsuMsuMtMtMtMCM

0tM

0VD

rM0VMLSsM

0MLS

1KM

0LMtMCM

0tM

0VD

0FTLLLN

D

VMMCxSBxASYY

δYVTYMMDSYYYYV

T

YVTYVTY ++

•

δδ

•

SMMLSSMMLS

0VTYV

VTYVTYV

=+

++•

αα

•

•

αα

••

seMMucseMMuc

shMMLUshMMLULNLL

TY+

(B.142) shMMUUshMMUULNULshMMshMM•

αα

•••

αα

•++=−− VTYVTYVYITIT

seMMseMseMMseMLNseseMseMMseMM•

αα•••

αα•

++=+ VTYVTYVMYITIT

shMM VTα

•, seMM VTα

•, shMM ITα

•

seMM ITα

•Similar to th

can be ex

LNM VM (B.144)

VMRMVHshMM•

α

•= MTVT

LNVMRMVEMseMM•

α

•= VMMTVT

LNVMRMIHMshMM•

α

•= VMMTIT

299

APPENDIX B

(B.147) LNVMRMIEMseMM•

α

•= VMMTIT

where:

300

( ) ( )( ) ( NU,seNU,2se21se1IEMNU,seNU,2se21se1VEM

NU,shNU,2sh21sh1IHMNU,shNU,2sh21sh1

,,,diag ; ,,,diag

,,,diag ; ,,,diag

ITITITTVTVTVTT

ITITITTVTVTVT∗α

∗α

∗α

∗α

∗α

∗α

∗α

∗α

∗α

∗α

∗α

∗α

==

==

LL

LL )

(B.148)

On using (5.94), (5.95), (B.120), (B.104) and (B.144) – (B.147) in (B.141) – (B.143) and

rearranging into a more compact form gives:

(B.149)

where:

(B.150)

(B.151)

VHMT

UPFCUPFCUPFC bxA =

T

shMshMLNUPFC

⎥⎦

⎤⎢⎣

⎡=•••IIVx MM

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

α−α

δα+

−

+

α+α

α+

++

α+

αα

δα+

α+

++

α+

MuMMseM

VMTMMuMMseM

VMRMIEM

VMRMVEMseMseseM

MuMMUU

KMLMsuMsuMuMMUU

VMRMIHMVMRMVHMUU

VMTMuMMUUUL

uMMucuMMLU

VMTMMuMMuc

KMLMsuMsuMuMMLU

VMRMVEMucVMRMVHMLU

VMTMuMMLU LL

UPFC

TNTY0

MEVLTY

MMT

MMTYMY

0TJTY

MMCSETY

MMTMMTY

MECTYY

NTYJTY

MEVLTY

MMCSETY

MMTYMMTY

MECTYY

A

M

M

M

M

M

M

LLLLLLLLMLLLLLLLLMLLLLLLLLLLLLLLLLL

M

M

M

M

M

M

LLLLLLLLMLLLLLLLLMLLLLLLLLLLLLLLLLL

M

M

M

M

M

M

M

M

APPENDIX B

(B.152)

( )

( )

( )

( ) ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎜⎝

⎛

⎟⎠⎞

⎜⎜⎝

⎛

=

αα

α

αα

δ

+−

+•

+++

+++−

+−+•

+++

+++α−

•−

•−

MuMseMuMMseM

shMuMdcMuMdcMuMrefdcMuM

suMsuMsuMuMsuMuMLNVMTMuMrefTMuMMUU

seMuMMuMMucshMuMdcMuMdcMuMrefdcMuM

suMsuMsuMuMsuMuMLNVMTMuMrefTMuMLU

SMMLSrMVSMLS

M

UPFC

VKIMTY

IIVHVGVF

xASESDVMEBVATY

IMVKTYIIVHVGVF

xASESDVMEBVATY

VTYδTY

b

LLLLLLLLLLLLLLLLLLLLLLLLLLLLL

LLLLLLLLLLLLLLLLLLLLLLLLLLLLL

Solution for in (B.149) is given by: UPFCx

( ) UPFCUPFCUPFC1UPFCUPFC bAbAx ==−

(B.153)

where:

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

==−

UPFC3

UPFC2

UPFC1

UPFC

UPFC33

UPFC32

UPFC31

UPFC23

UPFC22

UPFC21

UPFC13

UPFC12

UPFC11

1UPFCUPFC ; bbb

bAAAAAAAAA

AA (B.154)

Based on (B.153), the formulas for are as the following: seMshMLN and ,,•••IIV

UPFC3

UPFC33

UPFC2

UPFC32

UPFC1

UPFC31seM

UPFC3

UPFC23

UPFC2

UPFC22

UPFC1

UPFC21shM

UPFC3

UPFC13

UPFC2

UPFC12

UPFC1

UPFC11LN

bAbAbAI

bAbAbAI

bAbAbAV

++=

++=

++=

•

•

•

(B.155)

The following is the derivation of the formula for , and . Time-

derivatives of (5.76), (5.80) and (5.84) are given by:

LN•

Δ V shM•

Δ I seM•

Δ I

301

APPENDIX B

(B.156) ( )

( ) 0xLTYLTYδDY

VMMAYMMAYYVTY

=Δ++Δ+

Δ+++Δ•

αα

•

•

αα

•

δ

uMseM0MucshM

0MLUrM

0VMLS

LNVM0

M0VEMucVM

0M

0VHMLULLSM

0MLS

( ) 0xLTYITVMMAMMAYY =Δ+Δ+Δ++•

α

•

α

•

αα uMshM0MUUshM

0MLNVM

0M

0IHMVM

0M

0VHMUUUL (B.157)

( ) 0xLTYITVMMAMMAYMY =Δ+Δ−Δ−+•

α

•

α

•

αα uMseM0MseMseM

0MLNVM

0M

0IEMVM

0M

0VEMseMseseM

(B.158)

On using (5.105), (5.106) and (B.111) in (B.156) – (B.158) and rearranging into a more

compact form gives:

(B.159) UPFCUPFCUPFC cyD =

where:

(B.160) T

shMshMLNUPFC

⎥⎦

⎤⎢⎣

⎡ ΔΔΔ=•••IIVy MM

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

ααα

α

α

α

α

ααα

α

α

α

α

α

α

α

α

α

α

α

α

α

α

−

+

+

−

+

+

+

+

+

+

++

+

+

+

+

+

+

0MucMseM

0MseMucMseM

0MseM

KM0LMsuMucMseM

0MseM

VMucMseM0MseM

VM0

M0IEM

VM0

M0VEMseMseseM

ucMshM0MUUucMshM

0MUU

0M

KM0LMsuMucMshM

0MUU

VMucMshM0MUU

VM0

M0IHM

VM0

M0VHMUUUL

ucMseM0Muc

ucMshM0MLU

ucMseM0Muc

ucMshM0MLU

KM0LMsuMucMseM

0Muc

KM0LMsuMucMshM

0MLU

VMucMseM0Muc

VMucMshM0MLU

VM0

M0VEMuc

VM0

M0VHMLULL

UPFC

TILTYGLTY

MMCCLTY

MELTY

MMA

MMAYMY

ILTYGLTYT

MMCCLTY

MELTY

MMA

MMAYY

ILTY

ILTY

GLTY

GLTY

MMCCLTY

MMCCLTY

MELTY

MELTY

MMAY

MMAYY

D

M

M

M

M

M

M

M

M

LLLLLLLMLLLLLLLLMLLLLLLLLLLLLLLL

M

M

M

M

M

M

M

M

LLLLLLLMLLLLLLLLMLLLLLLLLLLLLLLL

M

M

M

M

M

M

M

M

M

M

M

M

(B.161)

302

APPENDIX B

(B.162)

( )[ ]

( )()

( )() ⎥

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

Δ+Δ+Δ+

Δ++Δ−

Δ+Δ+Δ+

Δ++Δ−

Δ+Δ+Δ+⎟⎠⎞⎜

⎝⎛ ++Δ

+−•

Δ−•

Δ−

α

α

Δ

ααδ

seMucMshMucMLNVMucM

suMsuMucMucMuMucMseM0MseM

seMucMshMucMLNVMucM

suMsuMucMucMuMucMshM0MUU

seMucMshMucMVMucMsuMsuMucMucMuMucM

seM0MucshM

0MLUrM

0VMSM

0MLS

LND

LS

UPFC

IHIFVMD

xACBxALTY

IHIFVMD

xACBxALTY

IHIFVMxACBxA

LTYLTYδDYVTY

c

LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL

LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL

Solution for in (B.159) is given by: UPFCy

( ) UPFCUPFCUPFC1UPFCUPFC cDcDy ==−

(B.163)

where:

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

==−

UPFC3

UPFC2

UPFC1

UPFC

UPFC33

UPFC32

UPFC31

UPFC23

UPFC22

UPFC21

UPFC13

UPFC12

UPFC11

1UPFCUPFC ; ccc

cDDDDDDDDD

DD (B.164)

Based on (B.163), the formulas for can be obtained as follows: seMshMLN and ,,•••

ΔΔΔ IIV

UPFC3

UPFC32

UPFC2

UPFC32

UPFC1

UPFC31seM

UPFC3

UPFC23

UPFC2

UPFC22

UPFC1

UPFC21shM

UPFC3

UPFC13

UPFC2

UPFC12

UPFC1

UPFC11LN

cDcDcDI

cDcDcDI

cDcDcDV

++=Δ

++=Δ

++=Δ

•

•

•

(B.165)

303

APPENDIX B

B.7 Summary of Algebraic Equations for Multi-Machine

System with Various FACTS Devices

B.7.1 System with SVCs and TCSCs Non-state variables: LNSMSM and , VIV

Algebraic equations:

(B.166) ( )⎪

⎩

⎪⎨

⎧

=+++

=+−

=+−

δ

δδ

0VYYYVTY0VYITVTY

0IZΨPV

LNFTFSLLSMMLS

LNSLSMMSMMSS

sMMrMMsM

B.7.2 System with SVCs and STATCOMs Non-state variables: CMCMLNSMSM and ,,, IVVIV


(B.167) ( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=−

=++

=+++

=+−

=+−

φ

αα

αδ

δδ

0VTV0ITVTYVY

0VTYVYYVTY0VYITVTY

0IZΨPV

dcMMCM

CMMCMMCCLNCL

CMMLCLNFSLLSMMLS

LNSLSMMSMMSS

sMMrMMsM

B.7.3 System with SVCs and UPFCs Non-state variables: seMshMLNSMSM and ,,, IIVIV


(B.168) ( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=+−

=++

=++++

=+−

=+−

αα

αα

ααδ

δδ

0VTYITVMY0VTYITVY

0VTYVTYVYYVTY0VYITVTY

0IZΨPV

seMMseMseMMLNseseM

shMMUUshMMLNUL

seMMucshMMLULNFSLLSMMLS

LNSLSMMSMMSS

sMMrMMsM

304

APPENDIX B

B.7.4 System with TCSCs and STATCOMs Non-state variables: CMCMLNSMSM and ,,, IVVIV


(B.169) ( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=−

=++

=+++

=+−

=+−

φ

αα

αδ

δδ

0VTV0ITVTYVY

0VTYVYYVTY0VYITVTY

0IZΨPV

dcMMCM

CMMCMMCCLNCL

CMMLCLNFTLLSMMLS

LNSLSMMSMMSS

sMMrMMsM

B.7.5 System with TCSCs and UPFCs Non-state variables: seMshMLNSMSM and ,,, IIVIV


(B.170) ( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=+−

=++

=++++

=+−

=+−

αα

αα

ααδ

δδ

0VTYITVMY0VTYITVY

0VTYVTYVYYVTY0VYITVTY

0IZΨPV

seMMseMseMMLNseseM

shMMUUshMMLNUL

seMMucshMMLULNFTLLSMMLS

LNSLSMMSMMSS

sMMrMMsM

B.7.6 System with STATCOMs and UPFCs Non-state variables: seMshMCMCMLNSMSM and , ,,,, IIIVVIV


(B.171)

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

=+−

=++

=−

=++

=++++

=+−

=+−

αα

αα

φ

αα

αααδ

δδ

0VTYITVMY0VTYITVY

0VTV0ITVTYVY

0VTYVTYVTYVYVTY0VYITVTY

0IZΨPV

seMMseMseMMLNseseM

shMMUUshMMLNUL

dcMMCM

CMMCMMCCLNCL

seMMucshMMLUCMMLCLNLLSMMLS

LNSLSMMSMMSS

sMMrMMsM

305

306

C.1 Synchronous Machine and Its Controllers For initial steady-state conditions, the stator and rotor voltage vectors of the

synchronous machine as given by (A.9) and (A.10) will become:

(C.1) 0ss

0rsr

0r

0sss

0r

0s R IIGIGV +ω+ω=

(C.2) 0rr

0r R IV =

where:

[ ] [ ][ ] [ ]T0

fd0r

T0q

0d

0s

T0fd

0r

T0q

0d

0s

00I ; VV

00E ; VV

==

==

II

VV (C.3)

Based on (C.1) and (C.2), the following equations are obtained:

(C.4) ⎪⎩

⎪⎨⎧

−−=

−=0qa

0dd

0fdmd

0q

0da

0qq

0d

IRIxIxV

IRIxV

(C.5) 0fdfd

0fd IRE =

where:

(C.6) md0rmdq

0rqd

0rd Lx ; Lx ; Lx ω=ω=ω=

AAPPPPEENNDDIIXX CC DDEETTEERRMMIINNAATTIIOONN OOFF SSYYSSTTEEMM IINNIITTIIAALL OOPPEERRAATTIINNGG CCOONNDDIITTIIOONN

APPENDIX C

In (C.6), the initial value of rotor angular frequency ( ) is determined based on (4.3)

and given by:

0rω

(C.7) R0r ω=ω

The initial steady-state value for synchronous machine terminal voltage is assumed to

be of the form:

(C.8) 0Q

0D

0s jVVV +=

where the real and imaginary parts of (i.e. ) are calculated based on

(B.6) and given by:

0sV 0

Q0D Vand V

(C.9) ⎪⎩

⎪⎨⎧

δ+δ=

δ−δ=0r

0q

0r

0d

0Q

0r

0q

0r

0d

0D

cosVsinVV

sinVcosVV

It is also to be assumed that the synchronous machine power generation has the form:

(C.10) 0e

0e

0e jQPS +=

In (C.10), it can be shown that the active- and reactive-power (i.e.

respectively) are given by:

0e

0e Q and P ,

(C.11) ⎪⎩

⎪⎨⎧

−=

+=0q

0d

0d

0q

0e

0q

0q

0d

0d

0e

IVIVQ

IVIVP

In (C.9) and (C.11), the synchronous machine voltage and power (

respectively) are determined from the load-flow study. The sets of the nonlinear

equations (C.4), (C.9) and (C.11) are solved simultaneously for to

give the initial values for machine stator voltage and current ( and , respectively),

rotor voltage and current ( and , respectively), and rotor angle .

0s

0s S and V ,

0fd

0r

0q

0d

0q

0d I and ,I,I,V,V δ

0sV 0

sI

0rV 0

rI0rδ

307

APPENDIX C

The initial value of rotor flux linkage of the synchronous machine is calculated based on

(A.13) and given by:

(C.12) 0rrr

0srs

0r ILILΨ +=

The following is the formulations of the initial conditions for synchronous machine

controllers (i.e. excitation, governor and PSS controllers).

C.1.1 PSS Controller Based on (4.32), the initial value of the state vector for PSS controller can be obtained

as the following:

(C.13)

0x =0p

Equation (C.13) shows that for initial steady-state condition, there is no signal coming

from the PSS output (i.e. ) which indicates that the controller does not respond

to the steady-state condition.

0V0PSS =

C.1.2 Excitation System Controller For initial steady-state condition, the state equation of excitation system controller as

given by (4.30) becomes:

refse

0ppe

0se

0ee VV DxSBCxA0 +++= (C.14)

Substituting (C.13) into (C.14) gives:

refse

0se

0ee VV DCxA0 ++= (C.15)

Rewriting (C.15) in partitioned form leads to:

308

APPENDIX C

refs

2e

1e0s

2e

1e

0fd

02e

01e

4e3e

2e1e

VVE ⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

D

D

C

C

x

x

AA

AA

0

0LLLLLL

M

LML

M

L (C.16)

The following equations are obtained by examining (C.16):

refs1e

0s1e

0fd2e

01e1e VVE DCAxA0 +++= (C.17)

refs2e

0s2e

0fd4e

01e3e VVE DCAxA0 +++= (C.18)

Solving (C.17) for gives: 01ex

refs1e

11e

0s1e

11e

0fd2e

11e

01e VVE DACAAAx −−− −−−= (C.19)

Substituting (C.19) into (C.18) and solving it for will lead to: refsV

0s6e

0fd5e

refs VEV AA += (C.20)

where:

( ) ( )( ) ( 1e

11e3e2e

12e1e

11e3e6e

2e11e3e4e

12e1e

11e3e5e

CAACDDAAA

AAAADDAAA−−−

−−−

−−=

−−=

) (C.21)

Substituting (C.20) into (C.19) and rearranging:

0s8e

0fd7e

01e VE AAx += (C.22)

where:

(C.23) 6e1e

11e1e

11e8e

5e1e11e2e

11e7e

ADACAAADAAAA

−−

−−

−−=

−−=

309

APPENDIX C

The initial value of the state vector for excitation controller is, therefore, given as the

following:

(C.24) ⎥⎦

⎤⎢⎣

⎡= 0

fd

01e0

e Exx

where is defined by (C.22). 01ex

C.1.3 Governor System Controller For initial steady-state condition, the state equation of governor system controller as

defined by (4.31) is modified to:

(C.25) 0mg

refg

0rg

0gg PDBCxA0 +ω+ω+=

Equation (C.25) can be rewritten in partitioned form as follows:

(C.26) 0m

2g

1gref

2g

1g0r

2g

1g

0m

02g

01g

4g3g

2g1g

PP ⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+ω⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+ω⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

D

D

B

B

C

C

x

x

AA

AA

0

0LLLLLLL

M

LML

M

L

Based on (C.26), the following equations are obtained:

(C.27) 0m1g

ref1g

0r1g

0m2g

01g1g PP DBCAxA0 +ω+ω++=

(C.28) 0m2g

ref2g

0r2g

0m4g

01g3g PP DBCAxA0 +ω+ω++=

Solving (C.27) for gives: 01gx

( ) ref1g

11g

0r1g

11g

0m1g2g

11g

01g P ω−ω−+−= −−− BACADAAx (C.29)

Substituting (C.29) into (C.28) and solving it for will lead to: refω

310

APPENDIX C

(C.30) 0r6g

0m5g

ref P ω+=ω AA

where:

( ) ( )[ ]( ) ( )1g

11g3g2g

12g1g

11g3g6g

1g2g11g3g2g4g

12g1g

11g3g5g

CAACBBAAA

DAAADABBAAA−−−

−−−

−−=

+−+−= (C.31)

Substituting (C.30) into (C.29) and rearranging:

(C.32) 0r8g

0m7g

01g P ω+= AAx

where:

( )

6g1g11g1g

11g8g

5g1g11g1g2g

11g7g

ABACAA

ABADAAA−−

−−

−−=

−+−= (C.33)

Based on the above discussion, the initial value for the state vector of the governor

system controller is, therefore, given by:

(C.34) ⎥⎦

⎤⎢⎣

⎡= 0

m

01g0

g Pxx

where is defined by (C.32). 01gx

C.2 FACTS Controllers For initial steady-state condition, the state equations of SVC, TCSC, STATCOM, UPFC

and SDC as described by (4.43) – (4.46) and (4.21) will become:

0DxSBCxA =+++ refTs

0susus

0Ts

0ss VV (C.35)

311

APPENDIX C

(C.36) 0ECxSBxA =+++ reft

0t

0susut

0tt PP

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=φ+

=φ++++++

=+++

=φ

0NM

0LxSKISJOHGF

0xSEISDCB

0A

0so

0cso

00Tso

0sususo

0CCso

0Tso

refTso

0dcso

0so

0sususo

0CCso

0Tso

refTso

00Tso

x

sinVVVVV

VV

sinV

(C.37)

( ) ( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=+

=+++++

=−

0IMK

0IIGFxSDBA

0IVIV

0seu0

Tu

0shu

0dcu

refdcu

0susuu

0Tu

refTu

0se

T0se

0sh

T0sh

V1

VVVV (C.38)

(C.39) 0x =0su

Similar to the state equation for PSS controller, (C.39) also shows that for initial steady-

state condition, the SDC gives zero output (i.e. ) which indicates that the

controller does not respond to the steady-state condition.

0V0SDC =

The following sections discuss the calculations of initial values for SVC, TCSC,

STATCOM and UPFC controllers. In the calculations, the following assumptions have

been made:

0ref0

dcrefdc

0ref0T

refT

QQ ; VV

PP ; VV

==

== (C.40)

C.2.1 SVC Controller On using (C.39) and (C.40) in (C.35), the initial value of the state vector for SVC

controller is obtained as follows:

312

APPENDIX C

(C.41) 0x =0s

C.2.2 TCSC Controller Similar to the previous section, on using (C.39) and (C.40) in (C.36), the initial value of

the state vector for TCSC controller can be obtained as the following:

(C.42) 0x =0t

C.2.3 STATCOM Controller Substituting (C.39) and (C.40) in (C.37), the following initial values for STATCOM

controller are obtained.

Initial value for state vector:

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

φ

=

00

kV

Vk1

xVV

0dc

0T

0

01s

0

0dc

0cx (C.43)

Initial value for STATCOM current vector:

(C.44)

0I =0C

In addition to the above, the initial value for STATCOM voltage vector which is derived

from (5.29) is also needed and given by:

(C.45) ⎥⎦

⎤⎢⎣

⎡=

0kV0

dc0CV

C.2.4 UPFC Controller Substituting (C.39) and (D.40) in (D.38), the following equations can be obtained:

313

APPENDIX C

0V

I

0I

=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=

=

0se

0T

0

0T

00se

0sh

V/Q

V/P (C.46)

Based on (C.46) and Fig.4.6, the following initial values for UPFC controller are also

valid:

)1m (assuming k

V

0V

01

0T0

dc

0T0

sh

==

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

V

V (C.47)

Therefore, the initial value of the state vector for UPFC controller is given by:

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

00

0V

k/VV

0T

0T

0se

0sh

0dc

0u

LLL

LLL

LL

LL

V

Vx (C.48)

In order to obtain the initial values for all of the synchronous machine and FACTS

controller in multi-machine power system, the procedures described in C.1 and C.2 can

be repeated as many times as the number of synchronous machines and FACTS

controllers in the system.

314

315

D.1. Synchronous Generator and Its Controllers Data Table D.1 shows the synchronous generator data used in the design study in Chapter

7. The synchronous machine controllers, i.e. exciter and governor system, are shown

in Figs.D.1 and D.2, respectively. The exciter is based on the IEEE Type-ST1

excitation system [69], whereas, the governor is adopted from the general model of

governor and turbine in [70]. The exciter and governor system constants are given in

Tables D.2 and D.3.

Table D.1: Machine constants

Gen. 1 Gen. 2 Gen. 3 Gen. 4

Ra (pu) 0.00028 0.00028 0.00028 0.00028

xd (pu) 0.2 0.2 0.2 0.2

xq (pu) 0.19 0.19 0.19 0.19

xmd (pu) 0.178 0.178 0.178 0.178

xmq (pu) 0.168 0.168 0.168 0.168

xkd (pu) 0.50 0.50 0.50 0.50

xkq (pu) 0.2218 0.2218 0.2218 0.2218

xfd (pu) 0.1897 0.1897 0.1897 0.1897

rkd (pu) 0.01 0.01 0.01 0.01

rkq (pu) 0.001471 0.001471 0.001471 0.001471

rfd (pu) 0.000063 0.000063 0.000063 0.000063

H (s) 63 54 54 63

AAPPPPEENNDDIIXX DD SSYYSSTTEEMM DDAATTAA UUSSEEDD IINN TTHHEE SSTTUUDDYY IINN CCHHAAPPTTEERR 77

APPENDIX D

316

Fig.D.1: Exciter control block diagram

Fig.D.2: Governor and turbine control block diagram

Table D.2: Exciter constants


KA (pu) 200 200 200 200 TA (s) 0.02 0.02 0.02 0.02

Table D.3: Turbine and governor constants


KG (pu) 10 10 10 10 TG1 (s) 0.25 0.25 0.25 0.25 TG3 (s) 0.1 0.1 0.1 0.1 TCH (s) 0.25 0.25 0.25 0.25

Tg

-

-++

- Σ Ta

Pm0ωref

1G

G

sT1K

+

3GsT1

CHsT11

+

Pm - Σωr

+

Vsref

Efd, min. Supplementary

signal (VPSS)

Efd, max

Efd -

-

|Vs|

A

A

sT1K+

Σ

APPENDIX D

D

317

.2. Data for the Power System Network Shown in Fig.7.1 its initial

Table D.4: Transmission line data for the system of Fig.7.1

Line mitance (pu)

Tables D.4 and D.5 show the data for the system of Fig.7.1 together with

steady-state operating condition.

Node Impedance (pu) Shunt Ad

1 N3 – N5 0.0010 + j0.0120 0 2 N2 – N6 0.0010 + j0.0120 0 3 N1 – N8 0.0010 + j0.0120 0 4 N4 – N7 0.0010 + j0.0120 0 5 N5 – N12 0.0025 + j0.0250 j0.150 6 N6 – N9 0.0010 + j0.0100 j0.030 7 N6 – N12 0.0013 + j0.0125 j0.075 8 N7 – N10 0.0010 + j0.0100 j0.030 9 N7 – N11 0.0013 + j0.0125 j0.075

10 N8 – N11 0.0013 + j0.0125 j0.075 11 N9 – N10 0.0074 + j0.0734 j0.990

pu 00 MV

Table D.5: Initial operating condition for the system of Fig.7.1

Generation Load

on 1 A

Node Voltage

PGEN ( (pu) PLOAD (pu (pu) pu) QGEN ) QLOAD

1 1.05 0o∠ 7.1962 0.5429 0 0 2 1 o.05 ∠ -1.95 7 1.1727 0 0 3 1.05 ∠ 12.02o 7 0.6231 0 0 4 1.05 -9.66∠ o 7 1.0830 0 0 5 1.039 ∠ 7.642 o 0 0 0 0 6 1.0329 ∠ -6.33o 0 0 0 0 7 1.0340 ∠ -14.04o 0 0 0 0 8 1.0402 -4.51∠ o 0 0 0 0 9 1.0327 ∠ -13.75o 0 0 11.59 2.12

10 1.0324 -21.59∠ o 0 0 15.75 2.88 11 1.0344 -9.27∠ o 0 0 0 0 12 1.0301 -1.67∠ o 0 0 0 0

pu 00 MVon 1 A

APPENDIX D

318

.3. Data for the Power System Network Shown in Fig.7.8 dy-

Table D.6: Transmission line data for the system of Fig.7.8

Line Node Impedance (pu) Shunt Admitance (pu)

DTables D.6 - D.8 show the data for the system of Fig.7.8 together with its initial stea

state operating condition.

1 N3 – N5 0.0010 + j0.0120 0 2 N2 – N6 0.0010 + j0.0120 0 3 N1 – N8 0.0010 + j0.0120 0 4 N4 – N7 0.0010 + j0.0120 0 5 N5 – N9 0.0025 + j0.0250 j0.1500 6 N5 – N12 0.0025 + j0.0250 j0.1500 7 N6 – N9 0.0010 + j0.0100 j0.0300 8 N6 – N12 0.0013 + j0.0125 j0.0750 9 N7 – N10 0.0010 + j0.0100 j0.0300

10 N7 – N11 0.0013 + j0.0125 j0.0750 11 N8 – N10 0.0025 + j0.0250 j0.1500 12 N8 – N11 0.0013 + j0.0125 j0.0750 13 N9 – N10 0.0444 + j0.4404 j0.4950 14 N9 – N13 0.0222 + j0.2202 j0.2475 15 N10 – N13 0.0222 + j0.2202 j0.2475

pu 00 M

Table D.7: UPFC data for the system of Fig.7.8

No. Parameter Unit

on 1 VA

1 Xsh 0.1 pu

2 Xse 0.1 pu

3 Cdc 0.1875 pu

4 k 1.1

5 dr p oo 0

APPENDIX D

319

Table D.8: Initial operating condition for the system of Fig.7.8

Generation Load Node Voltage

PGEN ( GEN (pu) PLOAD (pu LOAD (pu) pu) Q ) Q

1 1.05 0o∠ 7.6167 2.2235 0 0

2 1. o05 ∠ 20.31 7 1.7854 0 0

3 1.05 23.62∠ o 7 1.1066 0 0

4 1.05 -2.51∠ o 7 2.8447 0 0

5 1 o.0337 19.24∠ 0 0 0 0

6 1.0259 15.94∠ o 0 0 0 0

7 1.0138 -6.88∠ o 0 0 0 0

8 1.0209 -4.77∠ o 0 0 0 0

9 1.0057 ∠ 11.46o 0 0 12 2

10 0.9772 ∠ -12.97o 0 0 16 3

11 1.0181 -5.71∠ o 0 0 0 0

12 1.0333 ∠ 17.57o 0 0 0 0

13 1.0362 1.38o∠ 0 0 0 0

pu 00 MVon 1 A

320

}

E.1. Expectation If X is a discrete random variable with the possible values then the

expectation or expected value of X, denoted by E[X], is defined by [125-127]:

n21 x,,x,x K

(E.1) {∑ ===

n

1iii xXPx]X[E

In words, the expected value of X is a weighted average of the possible values of X,

each value is weighted by its probability (P).

E.2. Variance If X is a random variable with mean μ, then the variance of X, denoted by Var(X), is

defined by [125-127]:

(E.2) ])X[()XVar( 2μ−= E

An alternative formula for Var(X) can be expressed as follows [125]:

(E.3) 22 ])X[(]X[)XVar( EE −=

or, in words, the variance of X is equal to the expected value of the square of X minus

the square of the expected value of X.

AAPPPPEENNDDIIXX EE EEXXPPEECCTTAATTIIOONN,, VVAARRIIAANNCCEE AANNDD CCOOVVAARRIIAANNCCEE

APPENDIX E

E.3. Covariance The covariance of two random variables X and Y, denoted by Cov(X,Y), is defined by

[125,127]:

)]Y)(X[()Y,XCov( yx μ−μ−= E (E.4)

where and are the mean values of X and Y respectively. [X]x E=μ [Y]y E=μ

A useful expression for Cov(Y,Y) can be obtained by expanding the right side of (E.4)

which yields:

]Y[]X[]XY[)Y,XCov( EEE −= (E.5)

From the definition of covariance, it can be seen that covariance satisfies the following

property [125]:

)X,YCov()Y,XCov( = (E.6)

Another important property of covariance is that, if X and Y are independent [127]:

]Y[]X[]XY[ EEE = (E.7)

From (E.5) and (E.7) it can be concluded that, if X and Y are independent, the

covariance of the two random variables is given by:

0)Y,XCov( = (E.8)

E.4. Covariance Matrix Covariance matrix is a matrix of covariances between elements of a vector. Consider a

random vector X (where each component of the vector Xi is a random variable):

[ ]Tn21 XXX L=X (E.9)

321

APPENDIX E

Then, the covariance matrix S is the matrix where its component is given by [127]:

)]X)(X[()X,XCov( jjiijiij μ−μ−== ES (E.10)

Due to symmetry property of covariance (see (E.6)), the covariance matrix is always a

symmetric matrix (i.e. ). Also, based on (E.2) and (E.4), the covariance of any

component X

jiij SS =

i with itself is the variance of the component:

(E.11) )XVar(])X[()X,Cov(X i2

iiii =μ−= E

322

323

F.1 Basic Backpropagation Algorithm

This section discusses the algorithms to optimise the neural network performance

index, or in other words, to find the value of the network parameters that minimises the

error function. In backpropagation algorithm, the mean squared error (8.27) is

approximated by:

)qqT

qq ()()(F atatδ −−= (F.1)

where the mean squared error has been replaced by the squared error for a single

input/output pair.

The algorithms are iterative and start with some initial guess and then update the

guess according to the equation of the form [113,114]:

)0(δ

)k()k((k)1)(k σδδ α+=+ (F.2)

where is the search direction, α(k) is the learning rate which determines the

length of the step, and k is the iteration count.

)k(σ

It is to be noted that, in (F.2), different choice of search direction will lead to

different optimisation algorithms. Three different algorithms which are usually used

include: steepest descent algorithm, Newton’s method and conjugate gradient

algorithm [113,114].

)k(σ

AAPPPPEENNDDIIXX FF OOPPTTIIMMIISSAATTIIOONN AALLGGOORRIITTHHMMSS FFOORR NNEEUURRAALL NNEETTWWOORRKK TTRRAAIINNIINNGG

APPENDIX F

Basic backpropagation algorithm is based on the steepest descent method. In steepest

descent method, the weights are updated in the direction of the negative gradient

where the performance function decreases most rapidly. This will occur when the

direction vector is negative of the gradient g(k) [113,114]:

)k()k( gσ −= (F.3)

On using (F.3) in (F.2), the method of the steepest descent backpropagation algorithm

can be formulated as follows:

)k()k((k)1)(k gδδ α−=+ (F.4)

In steepest descent algorithm, there are two methods that can be used for

determination the learning rate α(k) in (F.4) [114]. One method is to use a line search

to determine the optimal step length. In this approach the performance index is

minimised with respect to α(k) at each iteration, or other words, choose α(k) to

minimise . Detail discussion of the line search can be found in

[114,117]. The other method is to use a fixed value (e.g., ), or to use

variable values (e.g., ).

))k()k()k((F gδ α−

02.0)k( =α

k/1)k( =α

In (F.3) and (F.4), the gradient g(k) is calculated using:

)k()(F)k(

δδδg

=∇= (F.5)

In multilayer neural networks, the relationship between the network weights and the

error is more complex than that in single-layer network. Therefore, in multilayer

networks, the chain rule of calculus has to be used in order to calculate the partial

derivatives described in (F.5).

The following is the description of the backpropagation algorithm for training the

multilayer feedforward neural network. The detail description and derivation of the

algorithm can be found in [114]. For simplification in developing the algorithm, the

abbreviated notation for the network has been used. Fig.F.1 shows an example of tree-

layer neural network in abbreviated notation.

324

APPENDIX F

First Layer Second Layer Third Layer Input

325

Fig.F.1: Three-layer network (abbreviated notation)

The first step of the algorithm is to propagate the input forward through the network

(from the first layer to the last layer):

(F.6) ( )⎪⎪⎩

⎪⎪⎨

⎧

=

=+=

=−

M

m1mmmm

0

M,1,2,m for ; )k()k(faa

baWapa

K

where M is the number of layers in the network.

The next step is to propagate the sensitivities backward through the network (from the

last layer to the first layer):

(F.7) ( )⎪⎩

⎪⎨⎧

==

−−=−−− ,2,1M,L for ; )k()(

))((2LTL1L1L1L

MMM

KsWrFs

atrFs

where:

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂

∂∂

∂∂

= KS

KS

K

K2

K2

K

K1

K1

KKK

K

K

r)r(f

,,r

)r(f,r

)r(fdiag)( KrF (F.8)

It is to be noted that in (F.8) is the argument of the transfer function. It is the function

of the network weights and biases and has the formulation of the form:

Kir

S3 1

a3p R 1

S1

r1

S1 1

1

S1 1

S1 R

W1

b1

Σ

a1

S2

r2

S2 1

1

S2 1

b2

Σ

a2

S2 1

S3

r3

S3 1

1

S3 1

S3 S2

W3

b3

Σ

f3

f2

S1 1

f1

W2

S2 S1

APPENDIX F

(F.9) ∑ +=−

=

−1KS

1j

Ki

1Kj

Kj,i

Ki )k(ba)k(wr

Finally the weights and biases are updated using the steepest descent rule:

(F.10) ⎪⎩

⎪⎨⎧

α−=+

α−=+ −

mmm

T1mmmm

)k()k()1k(

)()k()k()1k(

sbb

asWW

Based on the above discussion, procedure in the steepest descent backpropagation

(SDBP) algorithm can be summarised as follows:

Step.1: Set 0k = and initialise the weights and biases, i.e. . )0( and )0( mm bW

Step.2: Apply an input vector to the network.

Step.3: Propagate the input forward through the network using (F.6).

Step.4: Propagate the sensitivities backward through the network using (F.7).

Step.5: Update the weights and biases using (F.10) and set 1kk += .

Step.6: Repeat Step.2 through Step.5 with all the training vectors until the error for all

vectors in the training set is reduced to an acceptable value.

The steepest descent method as discussed previously is the simplest implementation

of backpropagation algorithm. Unfortunately, this method is often too slow, and has

poor convergence property for practical problems [113,114]. The faster training

algorithms with more powerful convergence characteristics, as will be discussed in the

next sections, fall into two main categories [113,114]. The first category uses heuristic

techniques, which has been developed from an analysis of the performance of the

standard steepest descent algorithm. The second category uses standard numerical

optimisation techniques (e.g., conjugate gradient and Levenberg-Marquardt).

F.2 Heuristic Variations of Backpropagation

This section discusses two methods for improving the speed and making the basic

backpropagation algorithm more practical [113,114]. The first heuristic modification to

the basic backpropagation is to use a momentum technique. Before applying this

326

APPENDIX F

modification to the algorithm, first, recall that the network parameter updates for SDBP

are:

(F.11) ⎪⎩

⎪⎨⎧

α−=Δ

α−=Δ −

mm

T1mmm

)k(

)()k(

sbasW

When the momentum is added to the parameter changes, the following equations for

the momentum modification to backpropagation (MOBP) can be obtained:

(F.12) ⎪⎩

⎪⎨⎧

αγ−−Δγ=+Δ

αγ−−Δγ=+Δ −

mmm

T1mmmm

)1()k()1k(

)()1()k()1k(

sbbasWW

where γ ( 10 <γ≤ ) is the momentum coefficient.

The second heuristic modification to the basic backpropagation is to use a variable

learning rate technique. With standard steepest descent algorithm, the learning rate

α(k) is held constant during the training [113]. The performance of the algorithm is

sensitive to the proper choice of the learning rate. If the learning rate is too high, the

algorithm may oscillate and become unstable. If the learning rate is too small, the

algorithm will take too long to converge. The performance of the steepest descent

algorithm can be improved if variable learning rate is used during the training process.

In the following, the procedure for varying the learning rate is developed. The learning

rate is varied according to the performance of the algorithm. The procedure will attempt

to keep the learning rate as large as possible while keeping the algorithm stable. The

rules of the variable learning rate backpropagation algorithm (VLBP) are [114]:

- If the error increases by more than some set percentage ζ (typically one to five

percent) after a weight update, then the weight update is discarded, the learning

rate is multiplied by some factor 10 <ρ< , and the momentum coefficient γ (if it is

used) is set to zero.

327

APPENDIX F

- If the error decreases after a weight update, then the weight update is accepted

and the learning rate is multiplied by some factor 1>η . If γ has been previously set

to zero, it is reset to its original value.

- If the error increases by less than ζ, then the weight update is accepted but the

learning rate and the momentum coefficient are unchanged.

F.3 Conjugate Gradient Backpropagation

The basic backpropagation algorithm adjusts the weights and biases in the steepest

descent direction (negative of the gradient). In the conjugate gradient algorithms, a

search is performed along conjugate directions, which generally produces faster

convergence than steepest descent direction [113].

It is common in conjugate gradient method to begin the search in the steepest descent

direction (negative of the gradient):

)0()0( gσ −= (F.13)

Then the next search direction is determined so that it is conjugate to previous search

directions. The procedure for determining the new search direction is to combine the

new steepest descent direction with the previous search direction [113,114,117]:

)k()1k()1k()1k( σgσ +β++−=+ (F.14)

The scalar β(k+1) in (F.14) can be calculated by using several different methods.

Different choice of method will lead to the different versions of conjugate gradient. The

most common choices are described in the following.

For the Fletcher-Reeves update, the procedure is:

)k()k(

)1k()1k()1k( T

T

gggg ++

=+β (F.15)

328

APPENDIX F

For the Polak-Ribiere update, the constant β(k+1) is computed by:

)k()k(

)1k()k()1k( T

T

gggg +Δ

=+β (F.16)

For the Hestenes-Steifel update, the formulation for β(k+1) is:

)k()k(

)1k()k()1k( T

T

gggg

Δ

+Δ=+β (F.17)

where:

)k()1k()k( ggg −+=Δ (F.18)

As a summary, the conjugate gradient backpropagation algorithm (CGBP) for neural

network training can be described as follows:

Step.1: Set 0k = , initialise the network parameters and calculate the initial search

direction σ(0) using (F.13).

)0(δ

Step.2: Perform a line search to determine the step length α(k).

Step.3: Update the parameters using (F.2).

Step.4: Calculate the new search direction using (F.14).

Step.5: Check the error. If the convergence criterion is satisfied then the iteration is

stopped; otherwise set 1kk += and go to Step.2.

F.4 Levenberg-Marquardt Backpropagation

The Levenberg-Marquardt algorithm is a variation of Newton method. It was designed

to approach the quadratic convergence property of Newton method without having to

compute the second derivatives of Hessian matrix [113,114].

The application of Levenberg-Marquardt algorithm to the multilayer neural network

training problem is discussed in the following. As mentioned before, the performance

index for multilayer neural network training is the mean squared error given by (8.27).

This mean squared error is proportional to the sum of squared error over the Q targets

in training set [114]:

329

APPENDIX F

∑=∑∑=∑=== ==

QM N

1i

2i

Q

1q

S

1j

2q,j

Q

qTq ve)(F

1qeeδ (F.19)

where SM is the number of output units in the network and eq is defined by qqq ate −= .

The Levenberg-Marquardt algorithm uses an approximation to the Hessian matrix and

employs the Newton-like update as the following [114,115,118]:

[ ] ))k(())k(()k())k(())k(()k()1k( T1T δvδJIδJδJδδ −μ+−=+ (F.20)

where I is the identity matrix; μ is some scalar (will be discussed later), and the

matrix/vector J, and v are defined by: δ

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂

∂

∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂

∂

∂

=

MMMM

LL

LL

MMMM

LL

LL

11

2,11

R,S

2,11

2,1

2,11

1,1

2,1

11

1,S1

R,S

1,S1

2,1

1,S1

1,1

1,S

11

1,21

R,S

1,21

2,1

1,21

1,1

1,2

11

1,11

R,S

1,11

2,1

1,11

1,1

1,1

be

we

we

we

b

e

w

e

w

e

w

e

be

we

we

we

be

we

we

we

))k((

1

M

1

MMM

1

1

δJ (F.21)

[ ][ ]M

S21,1

1S

11

1R,S

12,1

11,1

n21T

M11

Q

bwbbwww

)k(

LLL

L

=

δδδ=δ (F.22)

(F.23) [ ][ ]Q,S1,2,1S2,11,1

N21T

MM

Q

eeeee

vvv)k(

LL

L

=

=v

In (F.22), the size of vector is , whereas

in (F.23), ), the size of vector v is .

δ )1S(S)1S(S)1R(Sn 1MM121Q ++++++= −L

MQ SQN ×=

330

APPENDIX F

It is to be noted that, in (F.20), when the scalar μ is zero, the algorithm becomes

Newton method with approximate Hessian matrix. When the scalar μ is large, the

algorithm becomes gradient descent algorithm with a small learning rate.

The Levenberg-Marquardt algorithm starts with some small value of μ (e.g. μ = 0.01). If

the step does not give a smaller value of error, then the step is repeated with μ

multiplied by some factor (e.g. 1>κ 10=κ ). If a step does produce a smaller error,

then μ is divided by κ for the next step. The algorithm will provide a compromise

between the speed of Newton method and the guaranteed convergence of steepest

descent [113,114].

As can be seen from (F.20), the key step in the Levenberg-Marquardt algorithm is the

computation of the Jacobian matrix J. In order to determine this matrix, the partial

derivatives of the errors with respect to weights and biases have to be calculated. In

[114], the formulas for computing the elements of the Jacobian have been derived and

are given as follows:

M,1,2,m for

sbe

aswe

mh,im

i

q,k

1mq,j

mh,im

j,i

q,k

K=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=∂

∂

=∂

∂ −

(F.24)

where:

(F.25) kS)1q(h M +−=

In (F.24), the Marquardt sensitivity mh,is is an element of the total Marquardt sensitivity

matrices for each layer mS which is created by augmenting the matrices computed for

each input mqS as the following:

[ ]mQ

m2

m1

m SSSS MLMM= (F.26)

331

APPENDIX F

In (F.26), mqS is determined by:

⎪⎩

⎪⎨⎧

=

−=−−− m

qm1m

q1m1m

q

Mq

MMq

))k()((

)(

SWrFS

rFS (F.27)

where is calculated based on (F.8). )( KK rF

In summary, the Levenberg-Marquardt backpropagation algorithm (LMBP) for neural

network training can be described as follows:

Step.1: Set 0k = and 10) (e.g., 0 =κ>κ . Initialise 0.01)(0) (e.g., )0( =μμ and . )0(δ

Step.2: Compute network outputs using (F.6), the error vector using (F.23), and the

error using (F.19).

Step.3: Calculate Marquardt sensitivity matrices using (F.26) and the Jacobian matrix

using (F.24) and (F.21).

Step.4: Update the parameters using (F.20).

Step.5: Recompute the error using the new parameters. If this new error is smaller than

that computed in Step.2, then divide μ(k) by κ and go back to Step.2. If the error

is not reduced, multiply μ(k) by κ and go back to Step.4.

The above algorithm is assumed to have converged when the error calculated in

Step.2 has been reduced to some error goal.

332

333

The starting point of the compensation method is LU factorisation results for the base-

case system configuration which have been obtained off-line in the adaptive controller

design stage. The results are then stored for subsequent online application of the

compensation method, following a change in the system configuration. To illustrate the

method, the case of one transmission line being lost will be discussed in details in the

following.

In Fig.G.1 is shown a portion of one-line diagram of a power system. The objective is to

calculate the reduced impedance matrix when the line between nodes k and j is

disconnected. It is assumed that the disconnected line has series admittance Ykj and

half-shunt admittances Yks and Yjs. The nodal voltages at nodes k and j are Vk and Vj

respectively, and the nodal currents (currents to be injected sequentially as described

in the previous section) at these nodes are Ik and Ij respectively.

From Fig.G.1, the relationships of the injected currents are:

j) node (at IVY)VV(YV)m,j(

k) node (at IVY)VV(YV)m,k(

jjjskjkjmm

kkksjkkjmm

=+−+∑

=+−+∑

Y

Y

(G.1)

It is assumed that, after disconnecting the line, these injected currents are denoted as

Ik’ and Ij’ which can be calculated using:

AAPPPPEENNDDIIXX GG CCOOMMPPEENNSSAATTIIOONN MMEETTHHOODD FFOORR FFOORRMMIINNGG OONNLLIINNEE TTHHEE RREEDDUUCCEEDD NNOODDAALL IIMMPPEEDDAANNCCEE MMAATTRRIIXX

APPENDIX G

j) node (at VY)VV(YI'I

k) node (at VY)VV(YI'I

jjskjkjjj

kksjkkjkk

+−+=

+−+=

(G.2)

Ik Ij Ykj

334

Fig.G.1: Portion of transmission line showing the disconnected line

Equation (G.2) can be rewritten as follows:

j) node (at VyVyI'I

k) node (at VyVyI'I

jjjkjkjj

jkjkkkkk

++=

++=

(G.3)

where:

jskjjj

kjkj

kjjk

kskjkk

YYy

Yy

Yy

YYy

+=

−=

−=

+=

(G.4)

Based on (G.3), for N-nodes system, the injected currents into each system node are

(arranged in vector form):

Yjs Yks

Vj k j Vk

APPENDIX G

j

jj

kjk

jk

kk

N

j

k

2

1

N

j

k

2

1

V

0

y

y

00

V

0

y

y

00

I

I

I

II

I

'I

'I

II

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

+

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

+

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

M

M

M

M

M

M

M

M

M

M

M

M

(G.5)

Rewriting (G.5) in a more compact form leads to:

210 IIII ++= (G.6)

where:

[ ] tNjk21 I'I'III LLL=I (G.7)

[ ] tNjk21

0 IIIII LLL=I (G.8)

k11 VθI = (G.9)

j22 VθI = (G.10)

[ ] tjkkk

1 0yy00 LLL=θ (G.11)

[ ] tjjkj

2 0yy00 LLL=θ (G.12)

If the nodal voltage vectors V, V0, V1 and V2 are the solutions of (9.2) for the nodal

current vectors I, I0, I1 and I2 respectively, then it can be shown that:

210 VVVV ++= (G.13)

335

APPENDIX G

As mentioned in Section 9.2.2, the elements of the nodal voltage vector V that

correspond to specified generator nodes or FACTS device nodes form a column of the

required nodal impedance matrix. The nodal voltage vectors V for individual cases as

described in Section 9.2.2 have to be computed for forming the complete impedance

matrix. In order to do this, the nodal current vectors I0 and nodal voltage vectors V0 for

individual cases have determined in advance. Nodal current vectors I0’s for individual

cases can be determined by sequentially injecting 1 pu currents into each of the

generator node or FACTS device node. Each case of nodal voltage vector V0 is then

can be found by substituting each case of the nodal current vector I0 into (9.2) and

solving it.

In (G.5), Vk and Vj are unknown. The following procedure is used to determine the

voltages. If it is assumed that α1 and α2 are the solution of (9.2) for the nodal current

vectors θ1 and θ2 as defined in (G.11) and (G.12), respectively, then it can be shown

that the nodal voltage vectors V1 and V2 will have the following expressions:

k11 VαV = (G.14)

j22 VαV = (G.15)

Substituting (G.14) and (G.15) into (G.13):

j2

k10 VV ααVV ++= (G.16)

Based on (G.16), the nodal voltages at nodes k and j are:

j2kk

1k

0kk VVVV α+α+= (G.17)

j2jk

1j

0jj VVVV α+α+= (G.18)

By solving (G.17) and (G.18) for Vk and Vj and substituting them into (G.16), the nodal

voltage vector V can be computed. The elements of the voltage vectors that

correspond to specified generator nodes or FACTS device nodes will form the columns

336

APPENDIX G

337

of the reduced impedance matrix required. As mentioned in Section 9.2.2, by

assembling the columns in matrix form, the complete reduced nodal impedance matrix

is obtained.

338

Tables H.1 – H.3 show the data for the system of Fig.10.1 together with its initial

steady-state operating condition. The synchronous machine, exciter and governor

constants for the test system are given in Tables D.1 – D.3 in Appendix D.

Table H.1: Transmission line data for the system of Fig.10.1


1 N3 – N5 0.0010 + j0.0120 0

2 N2 – N6 0.0010 + j0.0120 0

3 N1 – N8 0.0010 + j0.0120 0

4 N4 – N7 0.0010 + j0.0120 0

5 N5 – N9 0.0025 + j0.0250 j0.150

6 N5 – N12 0.0025 + j0.0250 j0.150

7 N6 – N9 0.0010 + j0.0100 j0.030

8 N6 – N12 0.0013 + j0.0125 j0.075

9 N7 – N10 0.0010 + j0.0100 j0.030

10 N7 – N11 0.0013 + j0.0125 j0.075

11 N8 – N10 0.0025 + j0.0250 j0.150

12 N8 – N11 0.0013 + j0.0125 j0.075

13 N9 – N13 0.0222 + j0.2202 j0.297

14 N10 – N13 0.0222 + j0.2202 j0.297

15 N9 – N13 0.0222 + j0.2202 j0.297

16 N10 – N13 0.0222 + j0.2202 j0.297

AAPPPPEENNDDIIXX HH SSYYSSTTEEMM DDAATTAA UUSSEEDD IINN TTHHEE SSTTUUDDYY IINN CCHHAAPPTTEERR 1100

APPENDIX H

Table H.2: UPFC data for the system of Fig.10.1

No. Parameter Unit

1 Xsh 0.1 pu

2 Xse 0.1 pu

3 Cdc 0.1875 pu

4 k 1.1

5 droop 0

pu on 100 MVA

Table H.3: Initial operating condition for the system of Fig.10.1


PGEN (pu) QGEN (pu) PLOAD (pu) QLOAD (pu)

1 1.02∠0o 5.4487 1.0032 0 0

2 1.02 22.00∠ o 6 1.4076 0 0

3 1.02 24.99∠ o 6 0.8224 0 0

4 1.02 -0.92∠ o 6 1.3674 0 0

5 1.0069 21.02∠ o 0 0 0 0

6 1.0000 18.03∠ o 0 0 0 0

7 1.0004 -4.89∠ o 0 0 0 0

8 1.0048 -3.60∠ o 0 0 0 0

9 0.9823 14.51∠ o 0 0 10 2.0

10 0.9803 -10.90∠ o 0 0 13 2.5

11 1.0034 -4.18∠ o 0 0 0 0

12 1.0068 19.51∠ o 0 0 0 0

13 1.0455 2.47∠ o 0 0 0 0

pu on 100 MVA

339

340

Tables I.1 – I.3 show the data for the system of Fig.11.2 together with its initial steady-

state operating condition. The synchronous machine, exciter and governor constants

for the test system are given in Tables D.1 – D.3 in Appendix D.

Table I.1: Transmission line data for the system of Fig.11.2


1 N3 – N5 0.0010 + j0.0120 0

2 N2 – N6 0.0010 + j0.0120 0

3 N1 – N8 0.0010 + j0.0120 0

4 N4 – N7 0.0010 + j0.0120 0

5 N5 – N9 0.0025 + j0.0250 j0.150

6 N5 – N12 0.0025 + j0.0250 j0.150

7 N6 – N9 0.0010 + j0.0100 j0.030

8 N6 – N12 0.0013 + j0.0125 j0.075

9 N7 – N10 0.0010 + j0.0100 j0.030

10 N7 – N11 0.0013 + j0.0125 j0.075

11 N8 – N10 0.0025 + j0.0250 j0.150

12 N8 – N11 0.0013 + j0.0125 j0.075

13 N9 – N13 0.0037 + j0.0367 j0.099

14 N10 – N13 0.0037 + j0.0367 j0.099

15 N9 – N13 0.0037 + j0.0367 j0.099

16 N10 – N13 0.0037 + j0.0367 j0.099

AAPPPPEENNDDIIXX II SSYYSSTTEEMM DDAATTAA UUSSEEDD IINN TTHHEE SSTTUUDDYY IINN CCHHAAPPTTEERR 1111

APPENDIX I

Table I.2: UPFC data for the system of Fig.11.2

No. Parameter Unit

1 Xsh 0.1 pu

2 Xse 0.1 pu

3 Cdc 0.1875 pu

4 k 1.1

5 droop 0

pu on 100 MVA

Table I.3: Initial operating condition for the system of Fig.11.2


PGEN (pu) QGEN (pu) PLOAD (pu) QLOAD (pu)

1 1.05∠0o 6.8930 2.0584 0 0

2 1.05 5.42∠ o 7 2.2956 0 0

3 1.05 8.76∠ o 7 1.4497 0 0

4 1.05 -1.66∠ o 7 2.6830 0 0

5 1.0298 4.38∠ o 0 0 0 0

6 1.0201 1.04∠ o 0 0 0 0

7 1.0156 -6.03∠ o 0 0 0 0

8 1.0228 -4.31∠ o 0 0 0 0

9 0.9946 -5.46∠ o 0 0 11.59 2.12

10 0.9804 -10.87∠ o 0 0 15.75 2.88

11 1.0201 -5.07∠ o 0 0 0 0

12 1.0283 2.70∠ o 0 0 0 0

13 0.9894 -8.08∠ o 0 0 0 0

pu on 100 MVA

341

342

[1] Nguyen, T.T., and Gianto, R.: ‘Application of optimization method for control co-

ordination of PSSs and FACTS devices to enhance small-disturbance stability’.


Conference & Exposition, Dallas-Texas, May 2006, pp. 1478-1485.

[2] Nguyen, T.T., and Gianto, R.: ‘Stability improvement of electromechanical

oscillations by control co-ordination of PSSs and FACTS devices in multi-machine

systems’. Proceedings of the IEEE PES General Meeting 2007, Tampa-Florida,

June 2007, pp. 1-7.

[3] Nguyen, T.T., and Gianto, R.: ‘Optimisation-based control co-ordination of PSSs

and FACTS devices for optimal oscillations damping in multimachine power

system’, IET Gener. Transm. Distrib., 2007, 1, (4), pp.564-573.

[4] Nguyen, T.T., and Gianto, R.: ‘Neural networks for adaptive control coordination of

PSSs and FACTS devices in multimachine power system’, IET Gener. Transm.

Distrib., 2008, 2, (3), pp.355-372.

AAPPPPEENNDDIIXX JJ PPUUBBLLIICCAATTIIOONNSS

1

Abstract— This paper presents an approach for designing co-

ordinated controllers of power system stabilizers (PSSs) and FACTS devices stabilizers for enhancing small-disturbance stability. The control co-ordination problem is formulated as a constrained optimization with eigenvalue-based objective function without any need for the linear approximation by which the sensitivities of eigenvalues of state matrix to controller parameters are formed. The eigenvalue-eigenvector equations are used as the equality constraints in the optimization. The controller parameters bounds are formulated as the inequality constraints. Simulation results show that the controller design approach is able to provide better damping and small-disturbance stability performance.

Index Terms-- Optimization, co-ordination, design, FACTS, PSS, small-disturbance stability .

I. INTRODUCTION

AMPING of electromechanical oscillations among interconnected synchronous generators is necessary for

secure system operation. Power system stabilizer (PSS) has been used for many years to damp out the oscillations. With increasing transmission line loading over long distances, the use of PSS may in some cases not provide sufficient damping for inter-area oscillations. In such cases, other effective alternatives are needed in addition to PSS. At present, the availability of FACTS devices which have been developed primarily for active- and/or reactive-power flow and voltage control function in the transmission system has led to their use for a secondary function of enhancing the damping of power system oscillations [1], [2].

In particular, FACTS device stabilizers have been proposed to augment the main control systems for the purpose of damping the rotor modes or inter-area modes of oscillation. However, to achieve an optimal performance in terms of small-disturbance stability improvement, the co-ordination between PSSs and FACTS devices controllers is necessary.

A procedure was previously reported in [3] for simultaneous co-ordination of PSSs and FACTS device

T. T. Nguyen and R. Gianto are with the School of Electrical, Electronic and

Computer Engineering at The University of Western Australia, Crawley, Western Australia 6009.

stabilizers to enhance the damping of the rotor modes of oscillation. The procedure [3] determines only the stabilizer gains, based on the approximation that the shift in the rotor mode eigenvalue is linearly related to the increments in stabilizer gains. A separate method using frequency response has to be adopted for the design of stabilizer transfer functions [3].

In this paper, a systematic and optimal control co-ordination design procedure between PSSs and FACTS devices such as static VAr compensator (SVC) and static synchronous compensator (STATCOM) is developed. The controllers design problem is transformed into a constrained optimization problem to search for the optimal settings of controller parameters. The design is based on the minimization of the real parts of any number of eigenvalues, including those of the rotor modes, of the state matrix of the power system for enhancing small-disturbance stability. The alternative design based on the minimization of stabilizer gains with constraints imposed on selected eigenvalues can also be accommodated without any difficulty in the new procedure. The eigenvalue-eigenvector equations form the set of equality constraints in the optimization. Inequality constraints include bounds on controller parameter values, real parts of selected eigenvalues and mode frequencies.

Nonlinear relationships between eigenvalues and controller parameters are fully represented in the optimal co-ordination procedure in which no approximation is needed. Parameters in the stabilizer transfer functions including their gains, and if required, parameters of the main control systems of automatic voltage regulators (AVRs) and FACTS devices are directly included as variables in the optimization. Separate design procedures for stabilizer transfer functions are not required.

The general control co-ordination design procedure is applied to a power system having PSS and FACTS device. The effectiveness of the design procedure in achieving the improvement in the damping of small-disturbance oscillations is presented in the paper.

Application of Optimization Method for Control Co-ordination of PSSs and FACTS Devices to

Enhance Small-Disturbance Stability T. T. Nguyen and R. Gianto

D

2

II. CONTROL CO-ORDINATION USING OPTIMIZATION METHOD

A. Objective Function and Variables

The state-space equation of a power system installed with PSSs and FACTS devices, linearized about a selected operating point, can be compactly written as follows:

uBxAxp ∆∆∆ += (1)

where x is state vector; u is the vector of input reference signals; A is the state matrix which is the function of controller parameters.

The dynamic characteristics of the system are influenced by the locations of eigenvalues of A matrix. Therefore, for the system to have good dynamic characteristic (i.e. good damping), it is necessary to place eigenvalues associated with poorly-damped modes in certain positions in the complex plane so that they have good damping. This can be achieved by solving the tuning problem.

The objective of the tuning problem is to find a set of appropriate controller parameters such that the system damping is improved, i.e., when the selected eigenvalues (poorly-damped modes) have been moved as left most as possible in the complex plane. Therefore, the objective function to be minimized with respect to controller parameters in the control co-ordination design is:

[ ]∑−==

m

iimm zzzKf

1

22121 )Re(),...,,,,...,,,( λλλλ (2)

where: K = vector of controller parameters to be optimized λι = the ith eigenvalue to be placed zi = the eigenvector associated with the ith eigenvalue m = number of selected eigenvalues

The eigenvalues and eigenvectors associated with them are nonlinear functions of parameter vector K. However, closed-form expressions for the functions are, in general, not available. A key feature of the present work is to express the inter-relationship amongst the parameter vector, selected eigenvalues and eigenvectors in the form of eigenvalue-eigenvector equations which are to be satisfied during the optimization process. The equations form a set of equality constraints in the optimization, and the eigenvalues and eigenvectors are treated as variables in addition to those representing the controller parameters.

The variables in the objective function in (2) to be minimized, therefore, comprise selected eigenvalues, eigenvectors and controller parameters. The minimization of the objective function is subject to equality constraints formed from the eigenvalue-eigenvector equations and inequality constraints which represent the bounds required on the selected eigenvalues and controller parameters.

B. Equality Constraints

If λ is an eigenvalue of matrix A and z is an eigenvector associated with λ then [4]:

0=− zAz λ (3)

where z is not equal to 0. Although the state matrix A is real, some or all of its

eigenvalues and eigenvectors can be complex. It is now required to rearrange (3) into a real form for the purpose of including it as a set of constraints in the optimization in which real variables and functions are used.

Defining:

IR

IR

j

jzzz

λλλ +=+=

(4)

Using (4) in (3):

0))(()( =++−+ IRIRIR jzzjjzzA λλ (5)

Separating (5) into the real and imaginary parts gives:

0)(

0)(

=+−=−−

RIIRI

IIRRR

zzAz

zzAz

λλλλ

(6)

Grouping (6) into a vector/matrix form leads to:

0=− CCCC zzA λ (7)

where:

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

UU

UU

z

zz

A

AA

RI

IRC

I

RCC λλ

λλλ; ;

0

0 (8)

If the dimension of the state matrix A is N × N, then the dimension of matrices AC and λC is 2N × 2N. Vector zC has 2N elements, and U is the N × N unit matrix. The real-valued equation in (7) is equivalent to that in complex form in (3).

For m selected eigenvalues, the set of equality constraints to be satisfied are:

m1,2,..., ifor ; 0 ==− CiCiCiC zzA λ (9)

where:

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

UU

UU

z

zz

RiIi

IiRiCi

Ii

RiCi λλ

λλλ; (10)

Equation (9) is the first set of equality constraints to be

satisfied in the optimization process. The second set comes from the eigenvector constraints because the eigenvector associated with an eigenvalue is not unique. Equation (9) has an infinite number of solutions for vector zCi. In particular,

0=Ciz is also a solution which is not a valid eigenvector.

The problem is avoided by imposing a constraint on vector zCi. In the present work, the constraint is imposed on the norm of vector zCi, i.e.:

m1,2,..., ifor ; 1)(2/12

1

2 ==⎟⎠

⎞⎜⎝

⎛∑=

N

kCi kz (11)

3

With the equality constraint (11) imposed on each eigenvector, the trivial and non-valid solution zCi = 0 will be avoided.

C. Inequality Constraints

In the present work, three sets of inequality constraints are used in the optimization process to impose bounds on the eigenvalues and parameter values:

n1,2,...,j ; max,min, =≤≤ jjj KKK (12)

m1,2,...,i ; )Re( , =≤ preii σλ (13)

m1,2,...,i ; max,min, =≤≤ iii ωωω (14)

In (12), Kj is the jth element of controller parameter vector K, the elements of which are controller/stabilizer gains and time constants. In (13), σi,pre is a pre-specified upper limit of the real part of the ith eigenvalue. This value can be determined based on the desired damping ratio. In (14), ωi is the angular frequency given by the imaginary part of the ith eigenvalue. Bounds for these angular frequencies can be determined from the QR method which is used only once for initially estimating the mode frequencies and dampings.

D. Alternative Objective Function

The objective function in (2) is formed in terms of the real parts of selected eigenvalues. Alternatively, the control co-ordination design can be based on the minimization of the weighted sum of stabilizer gains [3]. In this case, the objective function is:

∑==

L

lllmm awzzzKf

12121 ),...,,,,...,,,( λλλ (15)

In (15), al is the positive gain of the lth stabilizer, which is an element of the parameter vector K; L is number of stabilizer gains, and wl is a weighting coefficient assigned to al.

Equality and inequality constraints as developed in sections II.B and II.C are still applicable when the objective function in (15) is minimized.

III. MODEL OF POWER SYSTEM ELEMENTS

A. Power System Configuration

To illustrate the effectiveness and capability of the design procedure developed in section II, simultaneous co-ordination of PSS and FACTS device stabilizer for improving the small-disturbance stability in the test system of Fig.1 is carried out.

Fig. 1. Test system.

The machine at location A is equipped with a PSS, and the

system has an SVC or STATCOM at location B. The part of the system connected to location C is represented by an infinite busbar.

For the purpose of forming the system state-space equation, the following sections derive small-disturbance models for individual items of plant in the system.

B. Generator Model

The synchronous generator at location A in Fig.1 is represented, for small-disturbances, by the fifth-order model [5]:

rmSmmrmmrm VIFAp ∆∆∆Ψ∆Ψ ++= (16)

MKIKKTp rmSmrmmrm /)( 987 ω∆∆ψ∆∆ω∆ −−−= (17)

rmrmp ω∆δ∆ = (18)

where: Ψrm , ωrm , and δrm are rotor flux linkage vector, rotor angular frequency and rotor angle respectively; Vrm is the field voltage; Tm is the rotor input torque; M is the inertia constant; Am, Fm, K7 - K9 are the constant matrices depending on machine parameters (see the Appendix for the expressions of these matrices); and ISm is the stator current vector.

Based on the IEEE Type-ST1 excitation system [6], the automatic voltage regulator (AVR) and PSS shown in Fig.2 are used; and from the general model in [7], the governor together with turbine model shown in Fig.3 is adopted in the present work.

Fig. 2. Excitation system with PSS.

Fig. 3. Governor and turbine system.

The linearized equation system for the excitation, governor

and turbine system, and PSS can be arranged in the following form:

-

-+

+ - Σ

Tg

Tm0 ωref

1G

G

sT1

K

+ 3GsT

1

CHsT1

1

+

Tm -

Σ ωrm

+

Vref

VPSS,min

VPSS,max

Efd min.

VPSS

Efd max

ωrm

Efd

-

-

|VSm|

A

A

sT1

K

+

)sT1)(sT1)(sT1(

)sT1)(sT1(sTK

4P2PP

3P1PPP +++

++

Σ

IS

Pe C B A

Ib Eb VT

COMPENSATOR

VS

4

pmemSmemememem xBVCxAxp ∆∆∆∆ ++= || (19)

rmgmgmgmgm CxAxp ω∆∆∆ += (20)

rmpmpmpmpm pCxAxp ω∆∆∆ += (21)

where: xem = vector of state variables of excitation system xgm = vector of state variables of governor and turbine system xpm = vector of state variables of PSS Aem,Bem,Cem,Agm,Cgm,Apm,Cpm = matrices the elements of which depend on the gains and time constants of the controllers (see the Appendix for the expressions of these matrices).

C. SVC Model

SVC can provide an auxiliary control of active-power flow through a transmission line. The possibility of controlling the transmittable power implies the potential application of this device for damping of power system electromechanical oscillations. In Fig.4 is shown in a block diagram form the control system of an SVC [8] in which Bc represents SVC susceptance. The SVC is equipped with a supplementary damping controller (SDC). The input to the SDC is the transmission line active-power flow Pe. The equations system for the SVC main control system can be arranged as follows:

ssmsmTsmsmsmsm xBVCxAxp ∆∆∆∆ ++= || (22)

where: xsm = vector of state variables of SVC xssm = vector of state variables of SDC Asm,Bsm,Csm = matrices which depend on the gain and time constants of the controller (see the Appendix for the expressions of these matrices).

Fig. 4 . Block diagram of SVC with supplementary damping controller.

D. STATCOM Model

STATCOM stabilizer can also be used to improve the damping of power system [9]. The basic principle of STATCOM is to use a voltage source inverter which generates a controllable ac voltage source behind the transformer leakage reactance (see Fig.5a). The voltage difference across the transformer reactance leads to active- and/or reactive-power flows to the network. The exchange of reactive-power with the network is obtained by controlling the voltage magnitude at the STATCOM terminal, and the exchange of active-power results from the control of the

phase shift between STATCOM terminal voltage and the network voltage VT. The exchange of active-power is used to control the dc voltage.

It can be shown that the state equations for STATCOM in Fig.5 [10] can be written as follows:

gmcmemcmrmcm

rmcmrmcmssmcmcmcmcm

xGxFE

DCxBxAxp

∆∆δ∆ω∆∆Ψ∆∆∆

++++++=

(23)

where:

[ ]TSdccm xVVx φ||= (24)

From (24), the small-disturbance model for the

STATCOM has four equations. The first equation is derived from the capacitor on the dc side (see Fig.5a), and the remaining three equations are derived from the main block diagram of the STATCOM (see Fig.5b). The droop is also included in the model, and input for the droop is the reactive component of the STATCOM current.

Fig. 5.a. STATCOM connection to network.

Fig. 5.b. Block diagram of STATCOM with supplementary damping controller.

E. Supplementary Damping Controller Model

Damping of power oscillations with FACTS devices of either SVC or STATCOM type is effected through power modulation by a supplementary damping controller (SDC). Fig.4 and Fig.5b show the block diagram of this controller

VSC

VS,min

VS,max

Pe

Iq

Vref

k

Vdc

φ

- +

- -

-

+ VDC ref

|V|

Limit min.

|VT|

●

● 2C

2C2C

sT

)sT1(K +

S

K 1C

Droop

Limit max. csT1

1

+

)sT1)(sT1)(sT1(

)sT1)(sT1(sTK

4SS2SSSS

3SS1SSSSSS +++

++

Σ Σ

qpC jIII +=

dtI)C/1(V dcdcdc ∫=

Reactance X

VT = |VT|.ejα

V = k.Vdc.ej(φ+α)

φ

VSC

VS,min

VS,max

- -

+

Pe

Bc

Bc min.

Bc max.

|VT|

Vref

)sT1)(sT1(

)sT1(K

S2S

1SS

+++

)sT1)(sT1)(sT1(

)sT1)(sT1(sTK

4SS2SSSS

3SS1SSSSSS +++

++

Σ

5

installed in SVC and STATCOM respectively. The equation system for the supplementary damping controller can be written as follows:

essmssmssmssm PpCxAxp ∆∆∆ += (25)

where: Assm,Cssm = matrices which depend on the gain and time constants of the controller (see the Appendix for the expressions of these matrices). Pe = input active-power to the controller

IV. STATE-SPACE EQUATIONS OF POWER SYSTEM INSTALLED

WITH FACTS DEVICES

A. Power System Installed with SVC

This section will derive the state equations for the single machine power system equipped with an SVC.

By substituting (A.8) in the Appendix into (16) and (17) to eliminate non-state variable ∆ISm, the generator state equations expressed in terms of state variables only are given in:

smem

rmrmrmrm

xLxL

LLLp

∆∆δ∆ω∆∆Ψ∆Ψ

3130

292827

++++=

(26)

smgm

rmrmrmrm

xLxL

LLLp

∆∆δ∆ω∆∆Ψω∆

3635

343332

++++=

(27)


Non-state variable |∆VSm| in the excitation controller equation (19) is eliminated by substituting (A.9) in the Appendix into (19) to give:

smemem

rmrmrmem

xLxA

LLLxp

∆∆δ∆ω∆∆Ψ∆

40

393837

++++=

(29)

Also, by substituting (27) into (21), the state equation for PSS can be written as:

smpmpm

gmrmrmrmpm

xLxA

xLLLLxp

∆∆∆δ∆ω∆∆Ψ∆

45

44434241

++

+++= (30)

Substituting (A.10) in the Appendix into (22) will result in the state equation for SVC as follows:

ssmsmsm

rmrmrmsm

xBxL

LLLxp

∆∆δ∆ω∆∆Ψ∆

++++=

49

484746

(31)

Using (A.11), (26) - (28), and (31) in (25) leads to the state equation for SVC SDC:

ssmsmgm

emrmrmrmssm

xLxLxL

xLLLLxp

∆∆∆∆δ∆ω∆∆Ψ∆

565554

53525150

++++++=

(32)

Equations (20) and (26) - (32) are the state-space equation

system for the single machine power system in Fig.1 installed

with an SVC. The state matrix A is assembled from the individual coefficient matrices of the state-space equations.

B. Power System Installed with STATCOM

Similar to the previous section, to eliminate non-state variable ∆ISm in (16) and (17), (A.12) in the Appendix is substituted into these equations to give the generator state equations in terms of state variables:

cmem

rmrmrmrm

xMxM

MMMp

∆∆δ∆ω∆∆Ψ∆Ψ

5251

504948

++++=

(33)

cmgm

rmrmrmrm

xMxM

MMMp

∆∆δ∆ω∆∆Ψω∆

5756

555453

++++=

(34)


Substituting (A.13) into (19) will result in state equation for excitation controller described as follows:

cmpmemem

rmrmrmem

xMxMxA

MMMxp

∆∆∆δ∆ω∆∆Ψ∆

6261

605958

+++++=

(36)

Also, by substituting (34) into (21), the state equation for PSS is given in:

cmpmpmgm

rmrmrmpm

xMxAxM

MMMxp

∆∆∆

δ∆ω∆∆Ψ∆

6766

656463

+++

++= (37)

Using (A.14), (33)-(35), and (23) in (25) gives the state equation for STATCOM SDC:

ssmssm

smgmem

rmrmrmssm

xA

xMxMxM

MMMxp

∆∆∆∆

δ∆ω∆∆Ψ∆

+

+++++=

106105104

103102101

(38)

Equations (20), (23) and (33) – (38) are the state-space

equations for the single machine power system in Fig.1 installed with a STATCOM. The state matrix A is also assembled from the individual coefficient matrices of the state space equations.

V. SIMULATION RESULTS

To verify the performance of the proposed method, the algorithm is tested on the power system shown in Fig. 1. Data for this test system together with its initial operating condition is presented in the Appendix. In this test system, the machine is equipped with exciter and PSS. FACTS device (SVC or STATCOM) is located in the middle of the transmission line. The constrained optimization method based on the quasi-Newton algorithm is used in the simulation study. However, other constrained optimization techniques such as the Newton method applied to large and sparse problem can also be used.

In Tables I - III, the initial and optimized values of the controller parameters are given. Controller gains are in pu, and time constants in seconds. The optimized parameters

6

were obtained in the final step where the PSS and FACTS controller were simultaneously co-ordinated. The number of the optimized eigenvalues is one complex-conjugate pair (electromechanical mode).

TABLE I OPTIMAL PARAMETER SETTINGS OF PSS

Initial Optimized

PSS KP = 10; TP = 1 TP1 = 0.2 ; TP2 = 0.1 TP3 = 0.05; TP4 = 0.2

KP = 14.1422; TP = 1.0001 TP1 = 0.2035; TP2 = 0.0928 TP3 = 0.0616; TP4 = 0.1965

TABLE II

OPTIMAL PARAMETER SETTINGS OF PSS AND SVC

Initial Optimized Without SDC PSS :

KP = 10; TP = 1 TP1 = 0.2 ; TP2 = 0.1 TP3 = 0.05; TP4 = 0.2 SVC : KS = 50; TS = 0.05 TS1 = 0.1; TS2 = 1

PSS : KP = 13.5056; TP = 1 TP1 = 0.2013; TP2 = 0.0980 TP3 = 0.0517; TP4 = 0.1987 SVC : KS = 50.2452; TS = 0.0495 TS1 = 0.1, TS2 = 1

With SDC PSS : KP = 10, TP = 1 TP1 = 0.2 , TP2 = 0.1 TP3 = 0.05, TP4 = 0.2 SVC : KS = 50; TS = 0.05 TS1 = 0.1; TS2 = 1 SDC : KSS = 1; TSS = 0.01 TSS1 = 0.2; TSS2 = 0.05 TSS3 = 0.1; TSS4 = 0.2

PSS : KP = 27.1746; TP = 1 TP1 = 0.2044; TP2 = 0.0965 TP3 = 0.0480; TP4 = 0.1957 SVC : KS = 50.1131; TS = 0.0496 TS1 = 0.0981; TS2 = 1.0001 SDC : KSS = 3.6609; TSS = 1 TSS1 = 0.1971; TSS2 = 0.0430 TSS3 = 0.0950, TSS4 = 0.0158

TABLE III

OPTIMAL PARAMETER SETTINGS OF PSS AND STATCOM

Initial Optimized Without SDC PSS :

KP = 10; TP = 1 TP1 = 0.2; TP2 = 0.1 TP3 = 0.05; TP4 = 0.2 STATCOM : KC1 = 1; KC2 = 1 TC2 = 0.1; TC = 0.02

PSS : KP = 11.2742; TP = 1 TP1 = 0.2030; TP2 = 0.0919 TP3 = 0.0626; TP4 = 0.1970 STATCOM : KC1 = 0.2448; KC2 = 1.1648 TC2 = 0.0992; TC = 0.0205

With SDC PSS : KP = 10; TP = 1 TP1 = 0.2; TP2 = 0.1 TP3 = 0.05; TP4 = 0.2 STATCOM : KC1 = 1; KC2 = 1 TC2 = 0.1; TC = 0.02 SDC : KSS = 0.1; TSS = 1 TSS1 = 0.2; TSS2 = 0.05 TSS3 = 0.1; TSS4 = 0.02

PSS : KP = 9.9818; TP = 1 TP1 = 0.2004; TP2 = 0.0987 TP3 = 0.0527; TP4 = 0.1997 STATCOM : KC1 = 0.9050; KC2 = 1.9824 TC2 = 0.1056; TC = 0.0210 SDC : KSS = 2.0604; TSS = 1 TSS1 = 0.2015; TSS2 = 0.0552 TSS3 = 0.0978; TSS4 = 0.0276

In Table IV are given the electromechanical mode

frequencies and dampings for the system before control co-ordination is carried out. The results confirm a significant improvement in damping with PSS. The damping ratio increases from 0.1093 to 0.2027 following the installation of PSS. However, with FACTS device (SVC or STATCOM), the further improvement in damping, if any, is minimal if there is no simultaneous co-ordination of the controllers. Actually, in the case of PSS, SVC and SDC, the result in Table IV

indicates that there is a slight decrease in the damping ratio, in comparison with the case of PSS only.

The results after simultaneous co-ordination and optimization are shown in Table V. Even without FACTS devices, the damping ratio increases to 0.2699 when the PSS parameters are optimized using the procedure described in section II. The installation of an SVC or STATCOM without SDC can lead to reduction in damping as shown in Table V, in comparison with the damping achieved by an optimized PSS.

However, with SDC, either SVC or STATCOM increases the damping ratio substantially when all of the controllers (i.e. PSS, SVC/STATCOM and SDC) are simultaneously co-ordinated. The best result is achieved by the combination of PSS and STATCOM with SDC which gives the highest damping ratio of 0.6361.

The results summarized in Tables IV and V illustrate the importance of the simultaneous co-ordination and optimization of PSS and FACTS controllers in enhancing the damping ratio of the electromechanical mode.

The results also show that the proposed technique can be used for control co-ordination of PSS and FACTS devices, and the performance of the method in solving the problem has also been verified through eigenvalue analysis.

TABLE IV RESULTS OF EIGENVALUE COMPUTATION

(BEFORE OPTIMIZATION)

Real Part

Imaginary Part

f (Hz)

ζ

Without PSS

-1.2216 11.1072 1.77 0.1093

With PSS -2.3939 11.5650 1.84 0.2027 PSS+SVC -2.3991 11.5446 1.84 0.2035 PSS+SVC

+SDC -2.3781 11.5595 1.84 0.2015

PSS+STATCOM -2.6436 11.8121 1.88 0.2184 PSS+STATCOM+SDC -3.0030 11.8751 1.89 0.2452

TABLE V

RESULTS OF EIGENVALUE COMPUTATION (AFTER OPTIMIZATION)

Real

Part Imaginary

Part f

(Hz) ζ

PSS -3.2175 11.4797 1.83 0.2699 PSS+SVC -2.8861 11.6576 1.86 0.2403 PSS+SVC

+SDC -4.3185 11.5575 1.84 0.3500

PSS+STATCOM -3.0606 11.7699 1.87 0.2517 PSS+STATCOM+SDC -9.6907 11.7564 1.87 0.6361

VI. CONCLUSION

In this paper, the control co-ordination of PSSs and FACTS devices such as SVC and STATCOM has been investigated. The control co-ordination problem is solved by constrained optimization. The optimization technique has been successfully applied on a test system. The performance of the proposed technique in solving the problem has also been verified through eigenvalue analysis.

It is found that system damping can be improved by the PSS, and the FACTS controllers can further improve the

7

damping when the controller parameters are properly tuned or co-ordinated. These results show the importance of the control co-ordination of PSS and FACTS controllers and the effectiveness of the proposed technique.

Although a small power system has been used for the purpose of illustration, the co-ordination procedure developed is a general one which can be applied to large multi-machine systems.

VII. APPENDIX

A. System Data (in p.u unless otherwise indicated):

1) Initial Conditions

Generator power : 6242.06023.00 jSGEN +=

Generator voltage : (rad) 0907.01045.10 ∠=SV

FACTS device terminal voltage : (rad) 0479.00512.10 ∠=TV

2) Parameters of Test System Generator:

sHxx

xxxxx

rrrR

mqmd

kqkdfdqd

kqkdfda

3.5;560.1;859.1

;6.1;899.1;999.1;7.1;999.1

;00318.0;00318.0;00107.0;002.0

===

=====

====

Exciter: sTK AA 02.0;50 ==

Governor: sTsTsTK CHGGG 25.0;1.0;25.0;100 31 ====

Transmission Line: 1796.00131.0 jZ L +=

STATCOM: 01.0;1.1;1875.0;15.0 ==== droopkCX dc

B. Expressions for Machine Constant and Controller Matrices

Machine constant matrices:

( ) ( ) rsrrrmrrrm LLRFLRA 11; −− =−= (A.1)

00009

100

00008

1007

)()(

)(

)()()(

)(

rsrTSmSmss

TSm

rsrrsrTSmr

Tsr

TSmr

Tssss

TSmr

rrsrTSmr

IGIIGIK

LLGI

GIGGIK

LGIK

+=

++=

=

−

−

ω

ωω

ω

(A.2)

Excitation main controller:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−=−=

A

Aem

A

Aem

Aem T

KB

T

KC

TA 00;;

1 (A.3)

Governor system controller:

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛−

=

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−

−−

−

=0

0;

110

011

001

1

33

1G

G

gm

CHCH

GG

G

gm

T

K

C

TT

TT

T

A (A.4)

PSS controller:

T

PP

PP

P

PPPpm

PPP

PP

PP

PP

P

P

PPP

PP

P

pm

TT

TKT

T

TKKC

TTT

TT

TT

TT

T

T

TTT

TTT

A

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−=

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

−−

−

=

42

31

2

1

442

32

2

1

4

3

22

1

1

01

.

001

(A.5)

SVC main controller:

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−−

−=

2

1

2

1

22

1

00

00

; 1

01

SS

SS

S

S

sm

SS

SS

S

S

sm

SSS

SS

Ssm

TT

TKT

K

B

TT

TKT

K

C

TTT

TTT

A

(A.6)

Supplementary Damping Controller:

T

SSSS

SSSSSS

SS

SSSSSSssm

SSSSSS

SSSS

SSSS

SSSS

SS

SS

SSSSSS

SSSS

SS

ssm

TT

TTK

T

TKKC

TTT

TT

TT

TT

T

T

TTT

TTT

A

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−=

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

−−

−

=

42

31

2

1

442

32

2

1

4

3

22

1

1

1

01

.

001

(A.7)

C. Network Equations in terms of State Variables

Equations below are the equations for generator current, generator voltage, compensator terminal voltage, and derivative of transmission line active-power in terms of state variables for the power system in Fig.1. Due to space limitation, the derivation of these equations is not presented here.

8

1) Power System Installed with SVC Generator current:

smsrmrmrmSm xLLLLI ∆δ∆ω∆∆Ψ∆ 10987 +++= (A.8)

Magnitude of the generator voltage:

smsrmrmrmSm xLLLLV ∆δ∆ω∆∆Ψ∆ 18171615|| +++= (A.9)

Magnitude of SVC terminal voltage:

smsrmrmrmT xLLLLV ∆δ∆ω∆∆Ψ∆ 22212019|| +++= (A.10)

Derivative of transmission line active-power:

sms

rmrmrme

xpL

pLpLpLPp

∆δ∆ω∆∆Ψ∆

26

252423

+++=

(A.11)

2) Power System Installed with STATCOM Generator current:

φ∆∆δ∆ω∆∆Ψ∆

2221

201918

MVM

MMMI

dc

rmrmrmSm

++++=

(A.12)

Magnitude of generator voltage:


sdcs

rmsrmsrmsSm

MVM

MMMV

2726

252423

||

++++=

(A.13)

Derivative of transmission line active-power:


pMVpM

MpMpMPp

dc

rmrmrme

4746

454443

++++=

(A.14)

VIII. ACKNOWLEDGMENT

The authors gratefully acknowledge the support of the Energy Systems Centre at The University of Western Australia for the research work reported in the paper. They express their appreciation to The University of Western Australia for permission to publish the paper.

IX. REFERENCES [1] M. Noroozian, M. Ghandhari, G. Andersson, J. Gronquist, and I.

Hiskens,”A robust control strategy for shunt and series reactive compensators to damp electromechanical oscillations,” IEEE Trans. Power Delivery, vol. 16, no. 4, pp. 812-817, October 2001.

[2] N. Mithulananthan, C.A. Canizares, J. Reeve, and G.J. Rogers,”Comparison of PSS, SVC, and STATCOM controllers for damping power system oscillations,” IEEE Trans. Power Systems, vol. 18, no. 2, pp. 786-792, May 2003.

[3] P. Pourbeik and M.J. Gibbard,”Simultaneous coordination of power system stabilizers and FACTS device stabilizers in a multimachine power system for enhancing dynamic performance,” IEEE Trans. Power Systems, vol. 13, no. 2, pp. 786-792, May 1998.

[4] T.T. Nguyen, and H.Y. Chan,”Evaluation of modal transformation matrices for overhead transmission lines and underground cables by optimization method,” IEEE Trans. Power Delivery, vol. 17, no. 1, pp. 200-209, January 2002.

[5] T.T. Nguyen,”Eigenvalue methods in multi-machine power systems steady-state stability analysis,” Internal Report, Energy Systems Centre, The University of Western Australia.

[6] IEEE Committee Report,”Excitation system models for power system stability studies,” IEEE Trans. Power Apparatus and Systems, vol. PAS-100, no. 2, February 1981.

[7] IEEE Committee Report,”Dynamic models for steam and hydro turbines in power system studies,” IEEE Trans. Power Apparatus and Systems, vol. PAS-92, no. 6, 1973b.

[8] CIGRE Working Group, Transmission Systems Committee,”Modeling of static shunt var systems (SVS) for system analysis,” Electra, no. 51, pp. 45-74, October 1976.

[9] H.F. Wang,”Phillips-Heffron model of power systems installed with STATCOM and applications,” IEE Proc.-Gener. Transm. Distrib., vol. 146, no. 5, pp. 521-527, September 1999.

[10] CIGRE TF 38.01.08: ”Modeling of power electronics equipment (FACTS) in load flow and stability programs: a representation guide for power system planning and analysis”, 1998.

[11] E.Z. Zhou,”Application of static var compensators to increase power system damping,” IEEE Trans. Power Systems, vol. 8, no. 2, pp. 786-792, May 1993.

X. BIOGRAPHIES

T.T. Nguyen was born in Saigon, Vietnam, in 1956. Currently, he is an Associate Professor at The University of Western Australia. He was an invited lecturer in power system short courses in Thailand and Indonesia (1985), the Philippines (1986), Malaysia (1986, 1990), Singapore (1992), Vietnam (1999, 2000, 2003) and in the inaugural course (1991) in the national series of short course in power system sponsored by the Electricity Supply Association of Australia (ESAA). He was the Director of the ESAA

2000 Residential School in Electric Power Engineering. His interests include power system modeling, analysis and design; power systems control and protection, quality of supply, transmission asset management and applications of neural networks and wavelet networks in power systems. He has published more than 100 papers in international literature, three undergraduate textbooks, and four short course textbooks. Professor Nguyen was awarded the Sir John Madsen Medal of the Institution of Engineers, Australia in 1981 and 1990. He serves on CIGRE International Task Forces. He was the Chairman of the Australasian Universities Power Engineering Conference in 1995.

Rudy Gianto was born in Bandung, Indonesia, in March 1967. He received the BE and ME degree from Tanjungpura University in 1991 and Bandung Institute of Technology in 1995 respectively. Currently, he is working toward the Ph.D. degree at The University of Western Australia under the supervision of Associate Professor T.T. Nguyen.

1

Abstract—This paper develops an optimal procedure for

optimal control co-ordination of controllers of power system stabilizers (PSSs) and FACTS devices for improving small-disturbance stability, particularly the stability of inter-area modes, in multi-machine power systems. The control co-ordination problem is formulated as a constrained optimization by which the objective function formed from selected eigenvalues of the power systems state matrix is minimized. By representing the eigenvalue-eigenvector equations as equality constraints in the optimization, the procedure does not require any special eigenvalue calculation software or eigenvalue calculations at each iteration. Inequality constraints include those for imposing bounds on the controller parameters. The constraints which guarantee that the modes are distinct ones are derived and incorporated in the control co-ordination formulation. Simulation results of a multi-machine power system confirm that the procedure is effective in designing controllers that guarantee and enhance the stability of electromechanical modes.

Index Terms—Control co-ordination, design, FACTS, optimization, PSS, small-disturbance stability, TCSC

I. INTRODUCTION

OLLOWING the restructuring and deregulation of the power supply industry, there has been an increased trend to interconnect separate power systems in forming

electricity markets. This has led to a growing concern about the damping or stability of the inter-area modes of electromechanical oscillations, particularly when there are long-distance tie lines involved in the interconnections. PSSs have previously been proposed for improving the stability of electromechanical modes. However, the use of PSSs only may not be in some cases effective in providing sufficient damping for inter-area oscillations, particularly with increasing transmission line loading over long distances [1].

At present, the application of FACTS devices for the primary purpose of active- and/or reactive-power flow and voltage controls in the transmission system is on the increase. It has been acknowledged that the FACTS device stabilizer, often referred to as supplementary damping controller (SDC),

T. T. Nguyen and R. Gianto are with the School of Electrical, Electronic and

Computer Engineering at The University of Western Australia, Crawley, Western Australia 6009.

which augments the main control system is very effective in damping the rotor modes or inter-area modes, particularly if the FACTS device is of the series form [2-7]. However, proper co-ordination amongst PSSs and FACTS device controllers are required to achieve their optimal performance in rotor mode damping enhancement.

There has been extensive research in design methods for control co-ordination in the context of small-disturbance stability damping enhancement. Most of the methods reported use special eigenvalue/eigenvector calculation software, for example, the software package which implements the QR method [8, 9]. The disadvantages in these methods include the limitations on the size of the power system and/or the need to calculate eigenvalues of the state matrix at each iteration during the control co-ordination.

Recently, a method [10] has been reported for the control co-ordination design where the above disadvantages are removed. In the method, the eigenvalue-eigenvector equations are used as a set of equality constraints in the optimization by which the controllers parameters are determined, and optimal dampings of the specified electromechanical modes are achieved. The method has been developed in the context of, and applied to a single-machine infinite bus system.

Against the above background, the objective of the present paper is to extend and apply the method to multi-machine systems, with a particular reference to the enhancement of inter-area mode damping. The FACTS device considered in the application is the Thyristor-Controlled Series Capacitor (TCSC), which is used in a tie line in a two-area power system. However, the method developed is general and applicable to any multi-area power system. Control co-ordination design of PSSs and the TCSC SDC is carried out in the paper. The correctness and effectiveness of the design procedure are validated by separate eigenvalue calculations and time-domain simulation of the power system.

II. SMALL-DISTURBANCE MODEL OF MULTI-MACHINE

POWER SYSTEM

The starting point is that of the dynamical models appropriate for the frequency range encountered in electromechanical oscillations of synchronous machines together with their controllers, PSSs and FACTS devices. The models lead to a set of differential equations. Given the low frequency in electromechanical modes, the power network is represented in a static form by a set of algebraic equations.

Stability Improvement of Electromechanical Oscillations by Control Co-ordination of PSSs and FACTS Devices in Multi-Machine Systems

T. T. Nguyen and R. Gianto

F

1-4244-1298-6/07/$25.00 ©2007 IEEE.

2

For a small-disturbance stability consideration, the combined set of differential-algebraic equations is linearized about a specified operating point to give:

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

∆w

∆x

JJ

JJxp∆

43

21

0 (1)

where x is the vector of state variables; w is the vector of non-state variables; J1, J2, J3 and J4 are Jacobian submatrices obtained by linearizing system equations, and p is time derivative operator.

Eliminating the non-state variables leads to:

xA∆xp∆ = (2)

In (2), 31

421 JJJJA −−= which is the system state matrix

based on which small-disturbance stability investigation and control co-ordination design of power system controllers can be carried out.

III. MODEL OF TCSC

Thyristor Controlled Series Capacitor (TCSC) is a FACTS device that can provide fast and continuous changes of transmission line impedance, and can regulate power flow on the line. The possibility of controlling the transmittable power implies the potential application of this device for the improvement of power oscillations damping [1, 4, 11].

In Fig.1 is shown in a block diagram form the control system of a TCSC [2, 6, 11]. In the figure, XC is the reactance of TCSC. The TCSC control block diagram contains Proportional-Integral (PI) controller block, SDC block and the block that represents the TCSC thyristor firing delays.

Fig. 1. Block diagram of TCSC with supplementary damping controller.

In Fig.1, the PI block is the TCSC main controller. The power flow control is usually implemented with a slow controller which is typical for a PI controller with a large time constant. The SDC block provides a modulation for power oscillation damping or small-disturbance stability improvement control. The SDC block contains a washout, lead-lag blocks and a limiter. The washout block is used to make the controller inactive to the input signal dc offset. The lead-lag blocks are needed to obtain the necessary phase-lead characteristics.

The transmission line active-power flow is the most commonly used input signal for SDC [2]. Therefore, in this paper, it is proposed to use the active-power flow as an input to the SDC.

It can be shown that the equations system for the TCSC main control system in Fig.1 can be arranged as follows:

eTCSCeTCSC

SDCTCSCTCSCTCSCTCSC

PpDPC

xBxAxp

∆+∆+∆+∆=∆

(3)

where: xTCSC = vector of state variables of TCSC xSDC = vector of state variables of SDC ATCSC, BTCSC, CTCSC, DTCSC = matrices which depend on the gains and time constants of the controllers (see the Appendix for the expressions of these matrices) Pe = input active-power to the controller

Equation (3) is derived by examining the transfer functions of the PI controller block and the block that represents the TCSC thyristor firing delays. It can be shown also that the equations system for the supplementary damping controller can be written as follows:

eSDCSDCSDCSDC PpCxAxp ∆+∆=∆ (4)

where: ASDC,CSDC = matrices which depend on the gain and time constants of the controllers (see the Appendix for the expressions of these matrices).

IV. CONTROL CO-ORDINATION DESIGN

In the present work, the controllers design problem is transformed into a constrained optimization problem to search for the optimal settings of controller parameters. The design is based on the minimization of the real parts of any number of eigenvalues. Therefore, the objective function to be minimized with respect to controller parameters in the control co-ordination design is [10]:

[ ]∑−==

m

iimm zzzKf

1

22121 )Re(),...,,,,...,,,( λλλλ (5)

SDC

Pref

XF

XS,min

XC

Pe

+

Pe -

XS,max

XC,min. XS

XC,max

+

+ C

C

sT1

K

+

)sT(1sT)(1sT(1

)sT)(1sT(1sTK

S4S2S

S3S1SS

+++

++

Σ

⎟⎟⎠

⎞⎜⎜⎝

⎛ +

FsT

FsT1FK Σ

3

where: K = vector of controller parameters to be optimized λι = the ith eigenvalue to be placed zi = the eigenvector associated with the ith eigenvalue m = number of selected eigenvalues

The following are the two sets of equality constraints to be satisfied in the optimization process. The first set of equality constraints are the eigenvalue-eigenvector equations, and the second set are the constraints imposed on zCi to avoid the trivial and non-valid solution zCi = 0.

m1,2,..., ifor ; 0 ==− CiCiCiC zzA λ (6)

m1,2,..., ifor ; 1)(2/12

1

2 ==⎟⎠

⎞⎜⎝

⎛∑=

N

kCi kz (7)

where:

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

UU

UU

z

zz

A

AA

RI

IRC

I

RCC λλ

λλλ; ;

0

0 (8)

In (8), zR and zI are the real and imaginary parts of z respectively, λR and λI are the real and imaginary parts of λ respectively. If the dimension of the state matrix A is N × N, then the dimension of matrices AC and λC is 2N × 2N. Vector zC has 2N elements, and U is the N × N unit matrix

Three sets of inequality constraints will be used in the optimization process to impose bounds on the controller parameter values, damping ratios and mode frequencies:

n1,2,...,j ; =≤≤ max,jjmin,j KKK (9)

n2,..., 1,i ; )Re(

)Re(22

=ς≥+

− desi,

ii

i

ωλ

λ (10)

m1,2,...,i ; =≤≤ maxi,imini,

ωωω (11)

In (9), Kj is the jth element of controller parameter vector K. In (10), ζi,des is a desired damping ratio of the ith mode. In (10) and (11), ωi is the angular frequency given by the imaginary part of the ith eigenvalue. The angular frequency range [ωi,min, ωi,max] is selected to cover the rotor mode frequency or the inter-area mode frequency.

V. PREVENTION AGAINST CONVERGENCE TO THE SAME MODES

In a single-machine infinite bus system discussed in [10], there is only one electromechanical mode of oscillation. However, for the case of a multi-machine power system considered in the present paper, there are multi-modes of electromechanical oscillations, and depending on the number of areas in the power system, there can be more than one inter-area mode.

The rotor mode frequencies or eigenvalues can be very close to one another. This leads to the possibility of the optimization converging to the same mode twice or more times. Therefore, it is essential to augment with additional constraints the co-ordination design procedure described in Section IV to ensure that distinct modes are used in the optimization. In the following is the derivation of the additional constraints for achieving the purpose.

If the angular frequencies of the ith and kth modes (for i ≠ k) are different, then the two modes are necessarily distinct ones. Based on this property, and to make the provision for the situation the two mode frequencies are very close to each other, the following inequality constraint is proposed:

εωω >− ki (12)

In (12), ε is a small positive value (for example, 10-3 rad/s) specified in the optimization procedure. For the optimization solution algorithm where derivatives are required, it is preferable to use the following constraint which is equivalent to (12):

( ) 22εωω >− ki (13)

Based on (13), the set of inequality constraints in Section IV is now extended to include the following constraints for distinct modes:

( ) 22εωω >− ki (14)

for i = 1, 2, ….., m k = 1, 2, ….., m and i ≠ k The additional set of constraints in (14) will prevent the optimization from converging to the same mode twice or more times.

Although angular frequencies (i.e. the imaginary parts of eigenvalues) have been used in the constraints in (14), it is possible to adopt instead the real parts of eigenvalues or combinations of both the real and imaginary parts to form the constraints for distinct modes.

VI. SIMULATION RESULTS

A. Initial Investigation

To verify the performance of the proposed method, the algorithm is tested on a modified two-area power system shown in Fig.2 [12]. It is a 4 generator, 12 bus system with a total connected load of 2734 MW. The two areas are connected by three AC tie lines.

The synchronous generators in the system are represented by the fifth-order model [13]. Based on the IEEE Type-ST1 excitation system [14], the automatic voltage regulator (AVR) model is adopted in the present work; and from the general

4

model in [15], the governor together with turbine model are used in the test system.

Fig. 2. Test system.

In the initial investigation, PSSs and FACTS devices are not included. The eigenvalues results and participation factors are given in Table I for the three electromechanical modes. The damping ratio of the inter-area mode 3 is very poor. It is only 0.0271. Stabilization measure is, therefore, required for improving the damping of the inter-area oscillation.

TABLE I PARTICIPATION FACTORS

Gen. Mode 1

(local mode) λ = -0.7337 ± j6.5606 f = 1.04 Hz; ζ = 0.11

Mode 2 (local mode)

λ = -0.7248 ± j6.8685 f = 1.09 Hz; ζ = 0.10


λ = -0.1264 ± j4.6665 f = 0.74 Hz; ζ = 0.03

1 0.5068 -0.0051 0.2991 2 -0.0010 0.6676 0.1588 3 0.0117 0.4242 0.3759 4 0.6023 0.0224 0.1944

B. Applications of PSSs and TCSC

The participation factors in Table I indicate that it is most effective to install PSSs in generators 1 and 3 in relation to inter-area mode damping enhancement. The PSS model in [16] is adopted in this paper. In addition, for the primary purpose of power flow controls in the system, a FACTS device, i.e. a TCSC, is installed in the long transmission line between nodes N9 and N10. An opportunity is then taken to equip the TCSC installed with an SDC to provide a secondary function for damping improvement of the low-frequency electromechanical mode.

In Table II are given the electromechanical mode eigenvalues, frequencies, and damping ratios for the system before control co-ordination is carried out. The controller parameters are not optimized. Typical values for them are used in the evaluation which gives the results in Table II. There are some damping improvements for both the local and inter-area modes when PSSs are installed. However, without

optimizing the controller parameters, the improvements in dampings offered by PSSs and TCSC SDC are minimal.

The results in Table II confirm the need to co-ordinate properly the controller parameters if maximum dampings are to be achieved. The co-ordination procedure described in Sections IV and V when applied to the test system of Fig.2 leads to the results of Table III which show the eigenvalues after the optimization of controller parameters.The limiting values of controller parameters used in the design are given in the Appendix. The desired minimum damping ratios for local and inter-area modes in the design are 0.3 and 0.1 respectively.

TABLE II RESULTS OF EIGENVALUE COMPUTATION

(NON-OPTIMIZED CONTROLLER PARAMETERS)

Eigenvalues f (Hz) ζ Without Stabilizer -0.7337 ± j6.5606

-0.7248 ± j6.8685 -0.1264 ± j4.6665

1.04 1.09 0.74

0.1111 0.1049 0.0271

PSSs -0.8006 ± j6.8822 -0.8267 ± j6.5739 -0.1602 ± j4.6646

1.10 1.05 0.74

0.1155 0.1248 0.0343

PSS + TCSC -0.8039 ± j6.8965 -0.8270 ± j6.5762 -0.1693 ± j4.9244

1.10 1.05 0.78

0.1158 0.1248 0.0344

TABLE III RESULTS OF EIGENVALUE COMPUTATION (OPTIMIZED CONTROLLER PARAMETERS)

Eigenvalues f (Hz) ζ

PSSs -2.4700 ± j6.5509 -2.4636 ± j6.5392 -0.6097 ± j4.8551

1.04 1.04 0.77

0.3528 0.3526 0.1246

PSSs + TCSC -3.0158 ± j6.7083 -3.0424 ± j6.6870 -1.1479 ± j4.8887

1.07 1.06 0.78

0.4100 0.4141 0.2286

It is interesting to note, even without the TCSC, the damping ratios increase when the PSS parameters are optimized using the proposed method. The installation of the TCSC with optimal parameters further improves the dampings as confirmed by the results in Table III. The optimal controller parameters are given in the Appendix.

C. Time-Domain Simulations

Although the results of the design given in Section VI.B have been confirmed by eigenvalue calculations, it is desirable to investigate the performance of the designed controllers in the time-domain under a large disturbance. The disturbance is a three-phase fault near node N8 on the line between nodes N8 and N11. The fault is initiated at time t = 0.1 second, and the fault clearing time is 0.1 second.

The improvement in performance is quantified by comparing the time-domain responses in Figs.3-5. As the critical mode is the inter-area mode, the responses used in the comparisons are those of the relative voltage phase angle transients between nodes N9 and N10 of the tie line having TCSC. From the responses, it can be seen that, without

N1 N5 N3

N4

N7

N11

N10 N9

N2

N8

TCSC

N12

N6

5

damping controllers (PSSs and/or FACTS device), the system oscillation is poorly damped and takes a considerable time to reach a stable condition (see Fig. 3). With the PSSs and TCSC installed, the oscillation is damped more quickly and settled down after about 5 – 6 seconds (see Fig.5).

Fig. 3. System transient (without stabilizer).

Fig. 4. System transient (with PSSs only).

Fig. 5. System transient (with PSSs and TCSC).

VII. CONCLUSION

This paper has developed a procedure for optimal control co-ordination design of PSSs and FACTS devices in a multi-machine power system. The control co-ordination problem is solved through the application of constrained optimization method. The key advantages of the procedure include the removal of the limitations on the size of the power system in the design imposed by some eigenvalue evaluation software, and the need to calculate eigenvalues at each iteration, which is time consuming, in the design.

The optimization technique has been implemented in MATLAB software and successfully applied to a test system. The performance of the proposed technique in solving the problem has also been verified through both eigenvalue calculations and time-domain simulations.

VIII. APPENDIX

A. Expression for Controller matrices:

TCSC main controller:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−=

0 ; 00

000

0 ; 1

00

FTCSC

C

CTCSC

F

F

TCSC

CC

CTCSC

KD

T

KB

T

KC

TT

KA

(A.1)

Supplementary Damping Controller:

T

SS

SSS

S

SSSSDC

SSS

SS

SS

SS

S

S

SSS

SS

S

SDC

TT

TTK

T

TKKC

TTT

TT

TT

TT

T

T

TTT

TTT

A

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−=

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−⎟⎟⎠

⎞⎜⎜⎝

⎛ −

−−

−

=

42

31

2

1

442

32

2

1

4

3

22

1

1

1

01

.

001

(A.2)

B. System Data:

Unless otherwise indicated, impedances, admittances, powers, and voltages are in pu on 100 MVA, and time constants in seconds.

0 1 2 3 4 5 6 7 8 9 10 -0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

time, s

rela

tive

volta

ge p

hase

ang

le, r

ad

Non-Optimized Optimized

0 1 2 3 4 5 6 7 8 9 10 -0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

time, s

rela

tive

volta

ge p

hase

ang

le, r

ad

Non-Optimized Optimized

0 1 2 3 4 5 6 7 8 9 10 -0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

time, s

rela

tive

volta

ge p

hase

ang

le, r

ad

6

TABLE B.1

TRANSMISSION LINE DATA

Line Node Impedance Shunt Admitance 1 N3 – N5 0.0010 + j0.0120 0 2 N2 – N6 0.0010 + j0.0120 0 3 N1 – N8 0.0010 + j0.0120 0 4 N4 – N7 0.0010 + j0.0120 0 5 N5 – N12 0.0025 + j0.0250 j0.150 6 N6 – N9 0.0010 + j0.0100 j0.030 7 N6 – N12 0.0013 + j0.0125 j0.075 8 N7 – N10 0.0010 + j0.0100 j0.030 9 N7 – N11 0.0013 + j0.0125 j0.075

10 N8 – N11 0.0013 + j0.0125 j0.075 11 N9 – N10 0.0074 + j0.0734 j0.990

TABLE B.2 SYSTEM INITIAL CONDITIONS

(VOLTAGE, POWER GENERATION AND LOAD DEMAND) Generation Load Node Voltage

PGEN QGEN PLOAD QLOAD

1 1.05 ∠ 0o 7.1962 0.5429 0 0

2 1.05 ∠ -1.95o 7 1.1727 0 0

3 1.05 ∠ 12.02o 7 0.6231 0 0

4 1.05 ∠ -9.66o 7 1.0830 0 0

5 1.0392 ∠ 7.64o 0 0 0 0

6 1.0329 ∠ -6.33o 0 0 0 0

7 1.0340 ∠ -14.04o 0 0 0 0

8 1.0402 ∠ -4.51o 0 0 0 0

9 1.0327 ∠ -13.75o 0 0 11.59 2.12

10 1.0324 ∠ -21.59o 0 0 15.75 2.88

11 1.0344 ∠ -9.27o 0 0 0 0

12 1.0301 ∠ -1.67o 0 0 0 0

TABLE B.3 GENERATOR CONSTANTS


Ra 0.00028 0.00028 0.00028 0.00028 xd 0.2 0.2 0.2 0.2 xq 0.19 0.19 0.19 0.19

xmd 0.178 0.178 0.178 0.178 xmq 0.168 0.168 0.168 0.168 xkd 0.50 0.50 0.50 0.50 xkq 0.2218 0.2218 0.2218 0.2218 xfd 0.1897 0.1897 0.1897 0.1897 rkd 1 1 1 1 rkq 0.001471 0.001471 0.001471 0.001471 rfd 0.000063 0.000063 0.000063 0.000063

H (s) 63 54 54 63

TABLE B.4 EXCITATION SYSTEM CONSTANTS


KA 200 200 200 200 TA 0.02 0.02 0.02 0.02

TABLE B.5

TURBINE AND GOVERNOR CONSTANTS

Gen. 1 Gen. 2 Gen. 3 Gen. 4 KG 10 10 10 10 TG1 0.25 0.25 0.25 0.25 TG3 0.1 0.1 0.1 0.1 TCH 0.25 0.25 0.25 0.25

C. Optimal Controller Parameter Values

TABLE C.1

OPTIMAL CONTROLLER PARAMETERS OF PSSS

Initial Optimized PSS

(in generator 1) KP = 1; TP = 1 TP1 = 0.2 ; TP2 = 0.1 TP3 = 0.05; TP4 = 0.2

KP = 12.6503; TP = 1.1442 TP1 = 0.2236; TP2 = 0.1651 TP3 = 0.0441; TP4 = 0.1160

PSS (in generator 3)

KP = 1; TP = 1 TP1 = 0.2 ; TP2 = 0.1 TP3 = 0.05; TP4 = 0.2

KP = 19.8027; TP = 1.1932 TP1 = 0.0547; TP2 = 0.0584 TP3 = 0.1387; TP4 = 0.1648

TABLE C.2 OPTIMAL CONTROLLER PARAMETERS OF PSSS AND TCSC

Initial Optimized

PSSs + TCSC PSS (in generator 1) : KP = 1, TP = 1 TP1 = 0.2 , TP2 = 0.1 TP3 = 0.05, TP4 = 0.2 PSS (in generator 3) : KP = 1, TP = 1 TP1 = 0.2 , TP2 = 0.1 TP3 = 0.05, TP4 = 0.2 TCSC : KF = 0.01; TF = 1 KC = 0.1; TC = 0.01 SDC : KS = 0.1; TS = 1 TS1 = 0.2; TS2 = 0.1 TS3 = 0.05; TS4 = 0.2

PSS (in generator 1) : KP = 9.2505; TP = 1.0216 TP1 = 0.4859; TP2 = 0.2163 TP3 = 0.0475; TP4 = 0.1096 PSS (in generator 3) : KP = 7.2189, TP = 1.6989 TP1 = 0.0493 , TP2 = 1.0237 TP3 = 0.3144, TP4 = 0.0897 TCSC : KF = 0.0126; TF = 1.0119 KC = 0.1004; TC = 0.0217 SDC : KS = 0.0258; TS = 1.0334 TS1 = 0.2542; TS2 = 0.0564 TS3 = 0.7106; TS4 = 0.0933

TABLE C.3 LIMITING VALUES OF CONTROLLER PARAMETERS

Parameter Limit

Gain (KP) 1 - 20 PSSs Time Constants (TP, TP1 – TP4) 0.01 – 10 s

Gain (KF , KC) 0.01 - 1 TCSC main controller Time Constants (TF , TC) 0.01 – 0.03 (TF)

0.01 – 10 (TC) Gain (KS) 0.01 - 1 SDC

Time Constants (TS, TS1 – TS4) 0.01 – 10 s

IX. ACKNOWLEDGMENT

The authors gratefully acknowledge the support of the Energy Systems Centre at The University of Western Australia for the research work reported in the paper. They

7

express their appreciation to The University of Western Australia for permission to publish the paper.

X. REFERENCES [1] M. Noroozian, M. Ghandhari, G. Andersson, J. Gronquist, and I.

Hiskens,”A robust control strategy for shunt and series reactive compensators to damp electromechanical oscillations,” IEEE Trans. Power Delivery, vol. 16, no. 4, pp. 812-817, October 2001.

[2] A.D. Del Rosso, C.A. Canizares, and V.M. Dona,”A study of TCSC controller design for power system stability improvement,” IEEE Trans. Power Systems, vol. 18, no. 4, pp.1487-1496, November 2003.

[3] B. Chaudhuri, and B.C. Pal,”Robust damping of multiple swing modes employing global stabilizing signals with a TCSC,” IEEE Trans. Power Systems, vol. 19, no. 1, pp. 499-506, February 2004.

[4] D. Jovcic, and G.N. Pillai,”Analytical modeling of TCSC dynamics,” IEEE Trans. Power Delivery, vol. 20, no. 2, pp. 1097-1104, April 2005.

[5] Q. Liu, V. Vittal, and N. Elia,”LVP supplementary damping controller design for a Thyristor Controlled Series Capacitor (TCSC) device,” IEEE Trans. Power Systems, vol. 21, no. 3, pp. 1242-1249, August 2006.

[6] N. Martins, H.J.C.P. Pinto, and J.J. Paserba,”Using a TCSC for line scheduling and system oscillation damping – small signal and transient stability studies,” Proc. IEEE/PES Winter Meeting, Singapore, January 2000.

[7] Y.L. Abdel-Magid, and M.A. Abido,”Robust coordinated of excitation and TCSC-based stabilizers using genetic algorithms,” Electric Power Systems Research, vol. 69, Issues 2-3, pp. 129-141, May 2004.

[8] P. Pourbeik and M.J. Gibbard,”Simultaneous coordination of power system stabilizers and FACTS device stabilizers in a multimachine power system for enhancing dynamic performance,” IEEE Trans. Power Systems, vol. 13, no. 2, pp.473-1479, May 1998.

[9] L.J. Cai and I. Erlich,”Simultaneous coordinated tuning of PSS and FACTS controller for damping power system oscillations in multi-machine systems,” IEEE Bologna PowerTech Conference, Italy, June 2003.

[10] T.T. Nguyen, and R. Gianto,”Application of optimization method for control co-ordination of PSSs and FACTS devices to enhance small-disturbance stability,” Proc. IEEE PES 2005/2006 T&D Conference & Exposition, pp. 1478-1485, May 2006.

[11] CIGRE TF 38.01.08,”Modeling of power electronics equipment (FACTS) in load flow and stability programs: a representation guide for power system planning and analysis”, 1999.

[12] K.R. Padiyar,” Power system dynamics stability and control,” John Wiley & Sons (Asia) Pte Ltd, Singapore, 1996.

[13] W.D. Humpage, J.P. Bayne, and K.E. Durrani,”Multinode-power-system dynamic analysis,” Proc. IEE, Vol 119, no. 8, pp. 1167-1175, August 1972.

[14] IEEE Std 421.5-2005,” IEEE recommended practice for excitation system models for power system stability studies,” 2005.

[15] IEEE Working Group,” Dynamic models for fossil fueled steam units in power system studies,” IEEE Trans. Power Systems, vol. 6, no. 2, pp. 753-761, May 1991.

[16] N. Mithulananthan, C.A. Canizares, J. Reeve, and G.J. Rogers,”Comparison of PSS, SVC, and STATCOM controllers for damping power system oscillations,” IEEE Trans. Power Systems, vol. 18, no. 2, pp. 786-792, May 2003.

XI. BIOGRAPHIES

T.T. Nguyen was born in Saigon, Vietnam, in 1956. Currently, he is an Associate Professor at The University of Western Australia. He was an invited lecturer in power system short courses in Thailand and Indonesia (1985), the Philippines (1986), Malaysia (1986, 1990), Singapore (1992), Vietnam (1999, 2000, 2003) and in the inaugural course (1991) in the national series of short course in power system sponsored by the Electricity Supply Association of Australia (ESAA). He was the Director of the ESAA

2000 Residential School in Electric Power Engineering. His interests include power system modeling, analysis and design; power systems control and protection, quality of supply, transmission asset management and applications of neural networks and wavelet networks in power systems. He has published more than 100 papers in international literature, three undergraduate textbooks, and four short course textbooks. Professor Nguyen was awarded the Sir John Madsen Medal of the Institution of Engineers, Australia in 1981 and 1990. He serves on CIGRE International Task Forces. He was the Chairman of the Australasian Universities Power Engineering Conference in 1995.

Rudy Gianto was born in Bandung, Indonesia, in March 1967. He received the BE and ME degree from Tanjungpura University in 1991 and Bandung Institute of Technology in 1995 respectively. Currently, he is working toward the Ph.D. degree at The University of Western Australia under the supervision of Associate Professor T.T. Nguyen.

Optimisation-based control coordination of PSSsand FACTS devices for optimal oscillations dampingin multi-machine power system

T.T. Nguyen and R. Gianto

Abstract: An optimal procedure for designing co-ordinated controllers of power systemstabiliser and flexible ac transmission system devices is developed for achieving and enhancingsmall-disturbance stability in multi-machine power systems. A constrained optimisationapproach is applied for minimising an objective function formed from selected eigenvalues ofthe power systems state matrix. The eigenvalue–eigenvector equations associated with theselected modes form a set of equality constraints in the optimisation. There is no need forany standard eigenvalue calculation routines, and the use of sparse Jacobian matrix in thecase of large system for forming the eigenvalue–eigenvector equations leads to the sparsityformulation. Inequality constraints include those for imposing bounds on the controllerparameters. Constraints which guarantee that the modes are distinct ones are derived andincorporated in the control coordination formulation using the property that eigenvectors associ-ated with distinct modes are linearly independent. The robustness of the controllers is achievedvery directly through extending the sets of equality constraints and inequality constraintsin relation to selected eigenvalues and eigenvectors associated with the state matrices ofpower systems with loading conditions and/or network configurations different from that ofthe base case. Simulation results of a multi-machine power system confirm that the procedureis effective in designing controllers that guarantee and enhance the power systemsmall-disturbance stability.

List of symbols

A system state matrix

B system input matrix

x vector of state variables

u vector of input reference signals

l eigenvalue

lR real part of l

lI imaginary part of l

z eigenvector associated with l

z damping ratio

f frequency

v angular frequency

AC matrix derived from A matrix

lC real matrix formed from lR and lI

zC real vector formed from the eigenvectorassociated with l

zR real part of z

zI imaginary part of z

K vector of controller parameters

# The Institution of Engineering and Technology 2007

doi:10.1049/iet-gtd:20060065

Paper first received 19th February and in revised form 5th July 2006

The authors are with Energy Systems Centre, School of Electrical, Electronicand Computer Engineering, The University of Western Australia, 35 StirlingHighway, Crawley, Western Australia 6009 Australia

E-mail: [email protected]

564

m number of selected eigenvalues

U unit matrix

ci, i ¼1, 2, . . . , m scalar coefficients in the linear combination

of eigenvectors

C vector of the scalar coefficients, cis

CR, CI real, and imaginary parts of C

Z eigenvectors matrix

CC vector of CR and CI

ZR, ZI real, and imaginary parts of Z

ZC real matrix formed from ZR and ZI

D small change notation used in the linearisa-tion process

p time-derivative operator

T vector or matrix transpose

min, max,des, REF, 0 minimum, maximum, desirable, reference,

and initial values, respectively

s Laplace transform operator

k ratio between ac and dc voltages

m1, m2 modulation indices used in the pulse-width-modulation scheme for the shunt and seriesconverters

C1, C2 phases used in the pulse-width-modulationscheme for the shunt and series converters

n number of controller parameters to beoptimised

N number of state variables

IET Gener. Transm. Distrib., 2007, 1, (4), pp. 564–573

1 Introduction

Small-disturbance stability, particularly in the contextof positive damping of electromechanical modes oroscillations among the interconnected synchronous genera-tors in a power system, constitutes one of the essential cri-teria for secure system operation. Power system stabilisers(PSSs) together with their coordination have been devel-oped for enhancing system stability. However, the use ofPSSs only may not be, in some cases, effective in providingsufficient damping for inter-area oscillations, particularlywith increasing transmission line loading over longdistances [1].Drawing on the availability of flexible ac transmission

system (FACTS) devices at present, which have been devel-oped primarily for active- and/or reactive-power flow andvoltage control functions in the transmission system, moreeffective measures have been proposed for improvingsystem damping [1, 2]. Specifically, the FACTS deviceprimary or main control systems are augmented with sup-plementary controllers or stabilisers for the purpose ofdamping the rotor modes or inter-area modes of oscillations.There have been numerous publications reporting or

proposing methods for designing PSSs and/or FACTSdevice stabilisers to achieve damping improvements. Thepreviously published methods can be classified into thefollowing categories.In the first category, the control strategy is based on

Lyapunov function or energy function [1, 3]. Input signalsand control laws are derived for FACTS devices. Theyoffer robust and decentralised control structure. However,some issues have been identified in Ghandhari et al. [3]for further research. One of them is the inclusion of detaileddynamic models for synchronous generators and loads, andtransmission system with losses. The other is related to theeffects of modelling on the control laws.In parallel with the work using Lyapunov functions,

active research on control coordination has been carriedout [4–7]. In control coordination methods, which belongto the second category, detailed dynamic models for genera-tors and loads can be represented directly. Parameters of allof the supplementary controllers are identified in acoordinated manner to achieve optimal damping of electro-mechanical modes. In general, the coordination iseigenvalue-based in the context of multiple FACTS control-lers and/or PSSs and detailed representation for the powersystem.The key issues that remain to be addressed, in relation to

those methods in the second category, are:

1. Robustness of the controllers designed. Effective andefficient techniques are needed for obtaining robust control-lers, particularly with respect to changes in power systemconfigurations.2. Sparsity formulation. The control coordination pro-cedure needs to take into account the sparsity in thepower system Jacobian matrix, and at the same time avoidthe separate eigenvalue calculations at each iteration inthe control coordination. This is an important requirement,particularly in the context of large power system.

In addition to the research published regarding the abovecategories, there have been publications reporting the appli-cations of the residue method [8], eigenvalue-distance mini-misation technique [9], LMI (linear matrix inequality)approach [10] and multiple-model adaptive control strategy[11] for designing damping controller(s) of FACTSdevice(s). The methods in Majumder et al. [10] and

IET Gener. Transm. Distrib., Vol. 1, No. 4, July 2007

Chaudhuri et al. [11] require approximation or simplifica-tion where the order of the power system is significantlyreduced. The technique proposed in Sadikovic et al. [8]was applied to place one eigenvalue using one FACTSdamping controller. The approach proposed in Chaudhuriet al. [9] is also applicable to only one FACTS device. Itcan also be said that the design methods in Sadikovicet al. [8], Chaudhuri et al. [9], Majumder et al. [10] andChaudhuri et al. [11] are eigenvalue-based.

The present paper focuses on the eigenvalue-based controlstrategy. The objective is to develop a new optimisation-based control coordination of multiple PSSs and FACTSdevices, which addresses the above-mentioned two issues(1) and (2). Furthermore, the control coordination caninclude any number of modes.

The coordination procedure proposed draws on con-strained optimisation in which the eigenvalue-based objec-tive function is minimised to identify the optimal controllerparameters. A key advance is that there is no need for anyspecial software to calculate eigenvalues. In the methodproposed, the nonlinear relationships among eigenvaluesand controller parameters are expressed as eigenvalue–eigenvector equations associated with the electromechani-cal modes selected in the coordination. These equationsare included directly in the optimisation in the form ofequality constraints. Therefore for a large power system,the method lends itself to sparsity formulation in whichthe sparse Jacobian matrix is used directly in forming theeigenvalue–eigenvector equations. Sparse optimisationtechnique based on the Newton algorithm [12] then pro-vides a fast and efficient solution method for the coordi-nation problem in large power systems. The algorithmdoes not require separate eigenvalue calculations at eachiteration during the control coordination.

Special constraints in addition to those representing theeigenvalue–eigenvector equations are derived in the paperto guarantee that the modes are distinct ones in the optimis-ation process. By comparison, it is quite difficult, if notinfeasible, to apply sequentially the deflating procedure[13] when several modes are to be considered simul-taneously in the design.

The present work also addresses the issue of robustness inthe control coordination design through extending the set ofconstraints. The additional constraints are those related toeigenvalue–eigenvector equations and eigenvalues of thepower systems with changes in configurations and/or loaddemands.

Although the principal application is in the optimalcontrol coordination, the procedure developed can also beadapted for calculating selected eigenvalues and eigenvec-tors associated with the electromechanical modes, forknown controller parameters. The use of the QR method(orthogonal-triangular decomposition method) that is notsuitable for large power systems is avoided altogether.

Results of the control coordination design applied to arepresentative power system with PSSs and FACTSdevices, together with the verification by time-domainsimulation of the nonlinear power system, confirm the effec-tiveness of the design procedure proposed to achieveenhancement in system damping.

2 Optimisation-based control coordination

2.1 Objective function and variables

The state-space equation of a power system installed withPSSs and FACTS devices, linearised about a selected

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operating point, can be compactly written as follows

pD x ¼ AD xþ BDu ð1Þ

In (1), p is the time-derivative operator, that is, p ¼ d/dtwhere t is the independent time variable. Other variablesand quantities in (1) are defined in List of symbols.

The objective of the optimisation is to find a set of appro-priate controller parameters such that the system damping ismaximised or improved, that is, when the selected eigen-values (poorly damped modes) have been moved as leftmost as possible in the complex plane subject to controllerparameter constraints as given in Section 2.3. Therefore, theobjective function to be minimised with respect to control-ler parameters in the control coordination design is [14]

f ðK;l1; l2; . . . ;lm; z1; z2; . . . ; zmÞ ¼ �Xmi¼1

½ReðliÞ�2

ð2Þ

where K is vector of controller parameters to be optimised,li the ith eigenvalue to be placed, zi the eigenvector associ-ated with the ith eigenvalue, and m number of selectedeigenvalues.

The relationships among eigenvalues, eigenvectors andcontrollers parameters are given by the eigenvalue–eigenvector equations. The equations form a set of equalityconstraints that are to be satisfied while minimising theobjective function in (2).

2.2 Equality constraints

If l is an eigenvalue of matrix A and z is an eigenvectorassociated with l, then [15]

Az � lz ¼ 0 ð3Þ

where z is not equal to 0.Although the state matrix A is real, some or all of its

eigenvalues and eigenvectors can be complex. It is nowrequired to rearrange (3) into a real form to include it as aset of constraints in the optimisation in which real variablesand functions are used.

Separating (3) into real and imaginary parts, and group-ing them into a vector/matrix form leads to [16]

ACzC � lCzC ¼ 0 ð4Þ

where

AC ¼A 0

0 A

� �ð5Þ

zC ¼zRzI

� �ð6Þ

lC ¼lRU �lIU

lIU lRU

� �ð7Þ

The variables and quantities used in (4)–(7) are defined inList of symbols. The matrix AC as defined in (5) is thesame as that given in Nguyen and Chan [16] except thatthe imaginary part of matrix A in the present paper iszero. The real-valued equation in (4) is equivalent to thatin complex form in (3).

For m selected eigenvalues, the set of equality constraintsto be satisfied is

ACzCi� lCi

zCi¼ 0 for i ¼ 1; 2; . . . ;m ð8Þ

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Equation (8) is the first set of equality constraints to be sat-isfied in the optimisation process. The second set comesfrom the eigenvector constraints. As eigenvector associatedwith an eigenvalue is not unique, (8) has an infinite numberof solutions for vector zCi

. In particular, zCi¼ 0 is also a sol-

ution, which is not a valid eigenvector. The problem isavoided by imposing a constraint on vector zCi

. In thepresent work, the constraint is imposed on the norm ofvector zCi

, that is

X2Nk¼1

z2CiðkÞ

!1=2

¼ 1 for i ¼ 1; 2; . . . ;m ð9Þ

With the equality constraint (9) imposed on each eigenvector,the trivial and non-valid solution zCi

¼ 0 will be avoided.In Section 4, the modification of the eigenvalue–

eigenvector equation in (3) is described to provide a sparsityformulation for large power systems.

2.3 Inequality constraints

Three sets of inequality constraints will be used in theoptimisation process to impose bounds on the eigenvaluesand parameter values

Kj;min � Kj � Kj;max for j ¼ 1; 2; . . . ; n ð10Þ

�ReðliÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ReðliÞ2þ v2

i

q � zi;des for i ¼ 1; 2; . . . ;m ð11Þ

vi;min � vi � vi;max for i ¼ 1; 2; . . . ;m ð12Þ

In (10), Kj is the jth element of controller parameter vectorK. In (11), zi,des is a desired damping ratio of the ith mode.In (11) and (12), vi is the angular frequency given by theimaginary part of the ith eigenvalue.

2.4 Selection of modes for design

The normal practice is to investigate the dampings of indi-vidual rotor modes prior to control coordination. For initia-lisation, the initial FACTS controllers and PSSs parametersare set to representative values within specified lower andupper limits. The eigenvalues and eigenvectors obtainedfrom the initial calculations will be used as starting valuesrequired in the optimisation procedure. The formulationdescribed in this paper provides this analysis facility forthe investigation.Using the results of the investigation, rotor modes that are

unstable or lightly damped are selected for subsequentdesign of FACTS controllers and PSSs. Once the designbased on the selected modes has been carried out, rotormode dampings will be evaluated again to confirm whetherall of the rotor modes have adequate damping ratios. If oneor more rotor modes do not have adequate dampings, thenthe control coordination will have to be repeated, with theadditional modes (unstable or lightly-damped) included.The design procedure can be an iterative one (first option).An alternative is to include all of the rotor modes in thecontrol coordination design at the outset to avoid the possi-bility of iterations referred to in the first option.

2.5 Robust controller design

The method can also handle any system outage (e.g. trans-mission circuit outage or generator outage) or operatingconditions leading to changes in the number of statevariables.


To achieve a robust controller design, the sets of equalityconstraints to be satisfied in addition to (8) and (9) are, foreach contingency case

ACzCi� lCi

zCi¼ 0 for i ¼ 1; 2; . . . ;m ð13Þ

and

X2Nk¼1

z2CiðkÞ

!1=2

¼ 1 for i ¼ 1; 2; . . . ;m ð14Þ

Also, the sets of inequality constraints for each contingencycase to be satisfied in addition to (11) and (12) are

�ReðliÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ReðliÞ2þ v2

i

q � zi;des for i ¼ 1; 2; . . . ;m ð15Þ

and

vi;min � vi � vi;max for i ¼ 1; 2; . . . ;m ð16Þ

The symbol ¯ above the variables and quantities in (13) –(16) represents those for contingency cases.It is not necessary to assume that the pre-contingency and

the post-contingency systems would have identical modes.The selection of modes for including in contingency casesand the initialisation process can also be based on theprocedure described in Section 2.4.

2.6 Prevention against convergence to the sameeigenvalues

The following procedure is developed to guarantee that dis-tinct modes are used in the optimisation process even whentheir frequencies or eigenvalues are close to one another.The set of eigenvectors associated with distinct modes

must be linearly independent. Therefore if

c1z1 þ c2z2 þ � � � þ cmzm ¼ 0 ð17Þ

then

c1 ¼ c2 ¼ � � � ¼ cm ¼ 0 ð18Þ

In (17) and (18), cis (for i ¼ 1, 2, . . ., m) are the scalar coef-ficients in linear combination in (17), and zis are linearlyindependent eigenvectors.Rewriting (17) into vector/matrix form

Z � C ¼ 0 ð19Þ

where

Z ¼ z1 z2 � � � zm� �

ð20Þ

C ¼ c1 c2 � � � cm� �T

ð21Þ

Equation (19) is rearranged into a real form, as required inthe optimisation

ZC � CC ¼ 0 ð22Þ

where

ZC ¼ZR �Z I

Z I ZR

� �ð23Þ

CC ¼CR

CI

� �ð24Þ

In (23), ZR and ZI are the real and imaginary parts of Z,respectively and, in (24), CR and CI are the real and imagin-ary parts of C, respectively.


The number of linear equations in (22) is greater than thenumber of coefficients in vector CC. If the pseudo-inverse ofZC exists, then there is a unique solution for CC which isequal to a zero vector. This is the condition for the set ofeigenvectors fzi for i ¼ 1, 2, . . ., mg being linearly indepen-dent. For developing the constraint corresponding to thiscondition, matrix DC is defined in

DC ¼ ZTCZC ð25Þ

If DC in (25) is non-singular, then the psuedo-inverse of ZC

exists.When DC is non-singular, there exists matrix EC that

satisfies the following constraint

DCEC ¼ U ð26Þ

or

P2mj¼1

DCði; jÞECð j; kÞ ¼ 1 if i ¼ k

0 if i = k

8<: ð27Þ

In (26), U is a 2m � 2m unit matrix and, in (27), DC(i, j) isan element of matrix DC and EC(i, j) an element of matrixEC. The set of individual constraints in (27) is then includedin the optimisation where EC(i, j)’s are the additionalvariables.

2.7 Constrained minimisation methods

The formulation of the optimal control coordinationproblem in Sections 2.1–2.6 is a general one. In principle,a number of standard constrained minimisation algorithmscan be applied to solve the problem formulated. Forexample, the quasi-Newton algorithm or sequential quadra-tic programming [17, 18] is directly applicable.

3 System state matrix

Combining the state equations for synchronous machines,including their controllers and, where applicable, PSSs andFACTS devices, with network equations in algebraic form,leads to the following vector/matrix equation describingthe small-disturbance model of a multi-machine powersystem

pD x

0

� �¼

J1 J 2

J3 J 4

� �Dx

Dw

� �ð28Þ

where w is the vector of non-state variables and J1, J2, J3 andJ4 are Jacobian submatrices obtained by linearising systemequations.

Eliminating the non-state variables leads to

pD x ¼ ADx ð29Þ

In (29), A ¼ J12 J2J421J3, which is the system state matrix

needed for evaluating the dynamic characteristic of thepower system. This matrix is the function of controllers(PSSs and FACTS devices) parameters.

4 Sparsity formulation

The sparsity in the Jacobian submatrices J1–J4 in (28) canbe directly taken into account in the formulation. Thisoffers an important advantage for large power systems.The modification required in sparsity formulation isdescribed as following: instead of eliminating the non-statevariables to form the A matrix in (29), the equality con-straint based on eigenvalue–eigenvector equation in (3) is

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modified to, using (28)

J 1 J2

J 3 J4

� �z

y

� �¼ l

z

0

� �ð30Þ

In (30), the eigenvector z is augmented with vector y toinclude the non-state variables.

The modification in (30) can be directly included in theformulation of Section 2. The advantage is that of preser-ving the sparse structure in the matrix coefficients whichare used in the Newton algorithm [12].

5 Advantages of the method proposed

5.1 Selection of modes in the control coordination

A method for optimisation and coordination of dampingcontrols based on time-domain approach using a postulateddisturbance was reported in Lei et al. [7]. However, theresults depend on the nature of disturbances used to excitethe system, and the controller robustness might be compro-mised [7]. The method does not provide the flexibility ofselecting the electromechanical modes for optimisation.These problems do not arise in the new method proposedin this paper.

In general, the control coordination design is an iterativeprocess, particularly when the location of the measurementunit yielding the feedback signal and even of the FACTSdevice itself is to be determined for achieving optimaldamping enhancement. The advantage of the mode selec-tion provided by the proposed method can be exploited inthe design. Different arrangements for supplementarydamping controller (SDC) and/or PSS input signals andFACTS device locations can lead to different modes beingconsidered. The design procedure developed which allowsmodes to be selected directly is applied repeatedly fordifferent combinations of specified input signals and/orFACTS device locations, with the objective of determiningthe optimal combination.

5.2 Elimination of eigenvalue shift approximation

In Pourbeik and Gibbard [4], a scheme for simultaneouscoordination of PSSs and FACTS device stabilisers basedon linear programming and eigenvalue analysis was devel-oped. Central to the scheme is the approximation by whichthe shifts in eigenvalues are formed as linear functions ofthe changes in stabilisers gains. A drawback is that the accu-racy of the predicted shift in an eigenvalue diminishes as thechanges in stabiliser gains become large. Another disadvan-tage of the scheme in Pourbeik and Gibbard [4] is the require-ment of a separate procedure using frequency response forthe design of stabiliser transfer functions. In the proposedmethod, these drawbacks or disadvantages are eliminated.

5.3 Simultaneous coordination

A scheme was reported in Ramirez et al. [6] for coordinat-ing FACTS-based stabilisers, using the method ofclosed-loop characteristic polynomial and eigenvalueassignment. The scheme solves the problem of coordinatingthe stabilisers sequentially, that is, in a pre-specifiedsequence, rather than simultaneously. For a given powersystem, a pre-specified sequence used in the coordinationmay not lead to the optimal results. According to Ramirezet al. [6], a compromise should be established among thestabilisers to avoid them penalising each other. Methodsreported in Sadikovic et al. [8], Chaudhuri et al. [9] and

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Chaudhuri et al. [11] are applicable only to one FACTScontroller.The approach of the present paper offers simultaneous

coordination of multiple controllers, without any need tospecify a sequence or compromise in the design.

5.4 Preserving the matrix sparse structure

On the basis of information presented, it appears that themethods reported in Pourbeik and Gibbard [4] and Cai andErlich [5] draw on the calculations of the eigenvalues ofthe A matrix by the QR algorithm, which does not exploitthe sparsity structure in power system Jacobian matrices.The method based on the closed-loop characteristic poly-

nomial in Ramirez et al. [6] requires the A matrix to beformed explicitly. This will destroy the sparsity structureof the Jacobian matrix.As described in Section 4, the method proposed takes into

account fully the sparsity in the Jacobian matrix of a largepower system.There are other algorithms such as the modified-Arnoldi

algorithm [19] that provide eigenvalue calculations and takeadvantage of the Jacobian matrix sparsity. In principle,these algorithms can also be applied to the methods reportedin Pourbeik and Gibbard [4] and Cai and Erlich [5].However, the control coordination design using these algor-ithms needs to calculate separately the eigenvalues at everyiteration where controller parameters are updated. This canbe time-consuming. The control coordination design pro-posed in this paper uses the equality constraints providedby the eigenvalue–eigenvector equations in the optimis-ation and avoids separate calculations of eigenvalues ateach iteration. Eigenvalues together with optimal controllerparameters are available at the convergence of theoptimisation process.

6 Unified power flow controller (UPFC) model

The UPFC is a versatile FACTS controller, which has awide range of control functions for the improvement ofpower system performance [20, 21]. Fig. 1 shows thegeneral structure of the UPFC [20, 21]. The UPFC combinestwo voltage source converters linked by a dc bus.In Figs. 2a and b, the dynamic models are shown for the

shunt and series converter controllers, respectively [20, 21].In addition to the main controllers, there is an SDC, theoutput of which is the input to the shunt converter controlleras shown by the dashed box in Fig. 2a. The purpose of thissupplementary controller is to improve the damping of elec-tromechanical modes. It is also possible to use an SDC inconjunction with the series converter controller.The SDC structure shown in Fig. 2a is a representative

one for the study adopted in this paper. The same structure

Fig. 1 UPFC block diagram


Fig. 2 Control block diagram for UPFC

a Shunt partb Series part

is also used for the PSSs where the inputs are generator rotorspeeds. However, the formulation is a general one, and anyparticular SDC and/or PSS structures can be selected andincluded directly.Bounds are imposed on the SDC and PSSs parameters in

the design. However, if required, further constraints on thecontroller parameters can be derived and included in thedesign to avoid any possibility of the pole-zero cancellationin the controller transfer functions.

7 Simulation results

7.1 Initial investigation

The system study and design is based on the two-area powersystem as shown in Fig. 3 [22]. In the initial investigation,PSSs and FACTS devices are not included. The optimisa-tion procedure in Section 2 is adapted to selectivelyevaluate eigenvalues, and left and right eigenvectors associ-ated with the electromechanical modes. The results aregiven in Table 1 for the three electromechanical modes.For presentation purpose, the participation factors in thetable have been scaled by 100.


The damping ratio of the inter-area mode 3 is verypoor. It is only 0.0335. Stabilisation measure is thereforerequired for improving the damping of the inter-areaoscillation.

Fig. 3 Two-area 230 kV system

Total connected load ¼ 2734 MWSynchronous machines model: fifth-order model [23]Excitation systems model: based on IEEE Type-ST1 system [24]Turbine and governor model: adopted from [25]

569

570

Table 1: Participation factors

Generator Mode 1

(local mode)

l ¼ 20.7381+ j 7.0030

f ¼ 1.11 Hz; z ¼ 0.1048

Mode 2

(local mode)

l ¼ 20.7467+ j 6.7649

f ¼ 1.08 Hz; z ¼ 0.1097

Mode 3

(inter-area mode)

l ¼ 20.1569+ j 4.6861

f ¼ 0.75 Hz; z ¼ 0.0335

1 20.8558 0.0020 20.5085

2 21.2171 0.0022 20.2462

3 0.0015 0.1878 20.3836

4 20.0140 0.1772 20.3427

7.2 Applications of PSSs and UPFC

On the basis of participation factors in the inter-area modeas shown in Table 1, two PSSs are installed in generators1 and 3. A UPFC with an SDC is installed in the systemas shown in Fig. 3.

In Table 2, the electromechanical mode eigenvalues,frequencies and damping ratios for the system are givenbefore control coordination is carried out. The controllerparameters are not optimised. Typical values for themwithin the practical limits as shown in Fig. 4 are used inthe evaluation, which gives the results shown in rows 2and 3 of Table 2. There are some damping improvementsfor both local and inter-area modes when PSSs are installed(see the damping ratios in rows 1 and 2 of Table 2).However, without proper coordination, the combined useof the PSSs and UPFC with an SDC hardly provides anyfurther damping improvements, as confirmed by theresults in rows 2 and 3 of Table 2.

The optimal control coordination procedure described inSection 2 is now applied to further enhance the dampings ofthese modes. The limiting values of controller parametersare given in Fig. 4. The desired minimum damping ratiosfor local and inter-area modes in the design are 0.3 and0.1, respectively.

The eigenvalues after simultaneous coordination areshown in Table 3. Even without FACTS device, thedamping ratios increase when the PSS parameters are opti-mised (see the damping ratios in row 2 of Table 2 and row 1of Table 3). The installation of the UPFC with optimal par-ameters further improves substantially the dampings as con-firmed by the results in rows 1 and 2 of Table 3. The optimalcontroller parameters are given in Fig. 4.

7.3 Robust controller design

Changes in system operating condition or system configur-ation can have an adverse impact on the performance in

Table 2: Results of eigenvalues computation(non-optimised controller parameters)

Row number Stabiliser Eigenvalues f, Hz z

1 Without

stabiliser

20.7381+ j 7.0030 1.11 0.1048

20.7467+ j 6.7649 1.08 0.1097

20.1569+ j 4.6861 0.75 0.0335

2 PSSs 21.2497+ j 7.0725 1.13 0.1740

21.4371+ j 6.8537 1.09 0.2052

20.3109+ j 4.7061 0.75 0.0659

3 PSSsþUPFC 21.2621+ j 7.0617 1.12 0.1759

21.4327+ j 6.8495 1.09 0.2047

20.2486+ j 3.5017 0.56 0.0695

terms of dampings provided by the optimal controllersthat have been designed for one particular operating con-dition and system configuration. For example, with theloss of transmission line between buses 8 and 10 of thesystem as shown in Fig. 3, the dampings of the electrome-chanical modes, particularly the inter-area mode, arereduced to those of row 1 of Table 4 when the optimal con-troller parameters determined in Section 7.2 are used for thesystem under the contingency. The damping of the inter-area mode is reduced from 0.2215 to 0.1222 (see thedamping ratios in row 2 of Table 3 and row 1 of Table 4).The robust controller design procedure described in

Section 2.5 is now applied to achieve robustness of the con-trollers with respect to the change in system configuration.In general, a number of contingency cases are to be includedin the design. The constraints described in Section 2.5 arethen applied for each and every contingency considered.The steps are those of the design where only the base-casesystem configuration is used, except that the set of equalityand inequality constraints are now extended in the robustdesign. The procedure for the selection of the modes forthe design together with the initialisation process isdescribed in Sections 2.4 and 2.5.To illustrate the effectiveness of the robust controller

design technique, the power system of the base case inSection 7.2 and that in the contingency where the trans-mission line between nodes 8 and 10 is lost are includedsimultaneously in the design.The robust controllers designed lead to good damping

ratios for the systems both of the base case and the contin-gency case, as confirmed in Tables 3 and 4 (see row 2 ofTable 3, and rows 2 and 3 of Table 4). The damping ratioof the inter-area mode is now 0.1716, when the transmissionline between nodes 8 and 10 is lost in comparison with0.1222 if the controller design is confined to the system ofthe base case only (see rows 1 and 3 of Table 4). Therobust controller parameters are given in Fig. 4.

7.4 Validation based on time-domain simulations

In order to validate the performance of the coordinatedcontrol, time-domain simulations are carried out for thesystem with and without damping control. The disturbanceis a three-phase fault near bus 8 on the line between buses 8and 10. The fault is initiated at time t ¼ 0.1 s, and the faultclearing time is 0.1 s.In Fig. 5a the system transient responses are shown,

assuming that there is no transmission circuit outage follow-ing the fault clearance. With the loss of the line betweenbuses 8 and 10 after the fault clearance, the responses aregiven in Fig. 5b.As the focus is on the inter-area mode oscillation, the

relative voltage phase angle transient between buses 9 and10 is used for forming the responses as shown in Figs. 5a


Fig. 4 PSS and SDC Transfer functions with optimal parameters

a Controller design using base case (non-robust design)b Robust controller designThe upper bound of all of PSS/SDC gains is 20 pu on 100 MVA basePSS/SDC time constants for individual blocks are limited in the range [0.01 s, 10 s]

and b. From the responses, it can be seen that, withoutdamping controllers (PSSs and/or FACTS device), thesystem oscillation is poorly damped and takes a consider-able time to reach a stable condition. With the PSSs andUPFC installed, the oscillation is damped more quicklyand settled down after about 4–5 s (see Fig. 5a).In Table 5 is given the comparison in terms of inter-area

mode frequency and damping between those obtained fromthe time-domain simulation results as shown in Figs. 5a andb and those from the calculations of eigenvalues. The closecomparison as shown in Table 5 confirms the validity of theprocedure for the optimal control coordination.The outputs from the SDC of the UPFC are given in

Fig. 5c for the cases with optimal controller parameters con-sidered in Figs. 5a and b. The SDC output amplitude is

Table 3: Results of eigenvalues computation (optimalcontroller design with base-case system configuration)

Row no. Stabilisers Eigenvalues f (Hz) z

1 PSSs 22.4615+ j6.9306 1.10 0.3347

22.5446+ j6.8954 1.10 0.3462

20.4978+ j4.8252 0.77 0.1026

2 PSSsþUPFC 23.0664+ j6.6835 1.06 0.4170

23.1407+ j6.6565 1.06 0.4267

20.8010+ j3.5267 0.56 0.2215


limited to a band of 10%, whereas those of PSSs arelimited to 5%. In the simulations, the outputs of SDC andPSSs do not exceed their respective limits.

The responses shown in Fig. 5c indicate that the SDCoutput frequencies are the same as the inter-area mode fre-quencies, and SDC output amplitudes in the system with thecontingency are substantially higher than those in the base-case system configuration in the transient period followingthe fault disturbance. In the design procedure described inSection 2, the SDC parameters are constrained to therange of practical values as shown in Fig. 4.

There might be a concern that the SDC and/or PSSoutput signals would be saturated if their limits are verylow. To quantify the effect of controller output signal satur-ation on damping, the SDC output limit is reduced to 1.5%.The reduction leads to saturation. The time-domain simu-lation result obtained with output limit reduction indicatesthat the change in the damping is minimal in comparisonwith those when a higher output limit of 10% is used.

However, it is conceivable that there would be caseswhere SDC and/or PSS output saturation can lead to amore significant effect on the rotor mode dampings.Possible solutions include:

† increase the output limits. The controller parametersobtained in the initial design remain, and the SDC and/orPSS output limits are increased until satisfactory dampings

Table 4: Comparisons between robust and non-robust controller designs

Row number Case Eigenvalues f, Hz z

1 Base-case controller design applied to

contingency case (non-robust design)

23.1375+ j 6.6111 1.05 0.4288

22.1866+ j 6.3632 1.01 0.3250

20.5785+ j 4.6977 0.75 0.1222

2 Robust design applied to base case 24.2649+ j 7.6056 1.21 0.4891

24.2853+ j 7.6968 1.22 0.4865

20.6777+ j 2.9906 0.48 0.2210

3 Robust design applied to contingency

case

24.2381+ j 7.5423 1.20 0.4899

23.2188+ j 6.1848 0.98 0.4617

20.7966+ j 4.5721 0.73 0.1716

571

are achieved. Time-domain simulations are used to evaluatethe impact of output limit values on mode dampings.† revising the control coordination design. The SDC and/or PSS output limits remain as initially specified. However,the control coordination design is repeated with lower con-troller gain limits and/or lower specified mode dampingratios. The revised design is then tested by time-domainsimulation to confirm whether output signal saturation that

Fig. 5 System transients following the disturbance

a Voltage phase angle of bus 9 relative to that of bus 10 (comparisonbetween non-optimised and optimised design for base case)b Voltage phase angle of bus 9 relative to that of bus 10 (comparisonbetween non-robust and robust designs)c Supplementary control signal outputs related to a and b

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causes unacceptable deterioration of mode dampings stillexists. The design process can be an iterative one, depend-ing on the outcome of the test.

In these both solutions, the control coordination iscombined with time-domain simulation to achieve optimalcontrollers design subject to their output signal limits.

8 Conclusions

Through the application of constrained optimisationmethod, this paper has formulated a procedure for optimalcontrol coordination design of multiple PSSs and FACTSdevices in a multi-machine power system. The formulationhas been implemented in MATLAB software and validatedby using nonlinear time-domain simulations. The keyadvances made by the control coordination designprocedure developed include:

† sparsity formulation. The formulation exploits the spar-sity in the Jacobian matrix. This is of particular benefit incontrol coordination for very large power systems. A par-ticular feature of the formulation is that separate eigenvaluecalculations are not needed at each iteration in the con-strained optimisation.† the constraints developed for always obtaining distinctmodes in the optimisation procedure. Mode frequencieswhich are the same or similar to one another impose nodifficulty in the coordination process.† approximation by which eigenvalues are linearly relatedto controller parameters is not required.† robustness in the optimal controller design. Critical con-tingencies and/or system load changes can be includedstraightforwardly in the design.

As the design is based on small-disturbance stability theorythat does not deal explicitly with controller limits, it has tobe used in conjunction with nonlinear time-domain simu-lation if controller output signal limitations are to be con-sidered. Furthermore, the control coordination designprocedure described in this paper is in the context of fixedcontroller parameters and off-line design. Future work isbeing undertaken to extend the control coordination toonline applications where controller parameters will beadaptive to the prevailing power system operatingcondition.

9 Acknowledgments

The authors gratefully acknowledge the support of theEnergy Systems Centre at The University of WesternAustralia for the research work reported in this paper.They express their gratitude to The University of WesternAustralia for permission to publish the paper.

Table 5: Comparison of inter-area mode frequency and damping

Case From eigenvalues calculation From time-domain simulation

f, Hz Damping f, Hz Damping

Without stabiliser 0.75 0.16 0.75 0.17

PSSsþUPFC (non-optimised) 0.56 0.25 0.54 0.25

PSSsþUPFC (optimised) 0.56 0.80 0.57 0.81

Robust Design (base case) 0.48 0.68 0.50 0.66

Robust Design (contingency) 0.73 0.80 0.75 0.81

Non-robust design 0.75 0.58 0.75 0.57


10 References

1 Noroozian, M., Ghandhari, M., Andersson, G., Gronquist, J., andHiskens, I.A.: ‘A robust control strategy for shunt and seriesreactive compensators to damp electromechanical oscillations’,IEEE Trans. Power Deliv., 2001, 16, (4), pp. 812–817

2 Mithulananthan, N., Canizares, C.A., Reeve, J., and Rogers, G.J.:‘Comparison of PSS, SVC, and STATCOM controllers for dampingpower system oscillations’, IEEE Trans. Power Syst., 2003, 18, (2),pp. 786–792

3 Ghandhari, M., Andersson, G., and Hiskens, I.A.: ‘Control Lyapunovfunctions for controllable series devices’, IEEE Trans. Power Syst.,2001, 16, (4), pp. 689–694

4 Pourbeik, P., and Gibbard, M.J.: ‘Simultaneous coordination of powersystem stabilizers and FACTS device stabilizers in a multimachinepower system for enhancing dynamic performance’, IEEE Trans.Power Syst., 1998, 13, (2), pp. 473–479

5 Cai, L.J., and Erlich, I.: ‘Simultaneous coordinated tuning of PSS andFACTS controller for damping power system oscillations inmulti-machine systems’. IEEE Bologna PowerTech Conf., Bologna,Italy, June 2003

6 Ramirez, J.M., Davalos, R.J., and Coronado, I.: ‘Use of an optimalcriterion for coordinating FACTS-based stabilizers’, IEE Proc.,Gener. Transm. Distrib., 2002, 149, (3), pp. 345–351

7 Lei, X., Lerch, E.N., and Povh, D.: ‘Optimization and coordination ofdamping controls for improving system dynamic performance’, IEEETrans. Power Syst., 2001, 16, (3), pp. 473–480

8 Sadikovic, R., Korba, P., and Andersson, G.: ‘Application of FACTSdevices for damping of power system oscillations’. IEEE PowerTech2005, St. Petersburg, Russia, June 2005

9 Chaudhuri, B., Korba, P., and Pal, B.C.: ‘Damping controller designthrough simultaneous stabilization technique’. Proc. WAC 2004,2004, vol. 15, pp. 13–18

10 Majumder, R., Chaudhuri, B., El-Zobaidi, H., and Jaimoukha, I.M.:‘LMI approach to normalised H-infinity loop-shaping design ofpower system damping controllers’, IEE Proc., Gener. Transm.Distrib., 2005, 152, (6), pp. 952–960

11 Chaudhuri, B., Majumder, R., and Pal, B.C.: ‘Application ofmultiple-model adaptive control strategy for robust damping ofinterarea oscillations in power system’, IEEE Trans. Contr. Syst.Tech., 2004, 12, (5), pp. 727–736


12 Sun, D.I., Ashley, B., Brewer, B., Hughes, A., and Tinney, W.F.:‘Optimal power flow by Newton approach’, IEEE Trans. PowerAppar. Syst., 1984, 103, (10), pp. 2864–2879

13 Semlyen, A., and Wang, L.: ‘Sequential computation of the completeeigensystem for the study zone in small signal stability analysis oflarge power system’, IEEETrans.Power Syst., 1988,3, (2), pp. 715–725

14 Nguyen, T.T., and Gianto, R.: ‘Application of optimization method forcontrol co-ordination of PSSs and FACTS devices to enhancesmall-disturbance stability’. Proc. IEEE PES 2005/2006Transmission and Distribution Conference & Exposition, 2006ISBN 0-7803-9194-2 (CD-ROM Version)

15 Wilkinson, J.H.: ‘The algebraic eigenvalue problem’ (OxfordUniversity Press, Oxford, 1965)

16 Nguyen, T.T., and Chan, H.Y.: ‘Evaluation of modal transformationmatrices for overhead transmission lines and underground cables byoptimization method’, IEEE Trans. Power Deliv., 2002, 17, (1),pp. 200–209

17 Mayne, D.Q., and Polak, E.: ‘Feasible directions algorithms foroptimization problems with quality and inequality constraints’,Math. Program, 1976, 11, pp. 67–80

18 Panier, E.R., and Tits, A.L.: ‘On combining feasibility, descent andsuperlinear convergence in inequality constrained optimization’,Math. Program, 1993, 59, pp. 261–276

19 Wang, L., and Semlyen, A.: ‘Application of sparse eigenvaluetechniques to the small signal stability analysis of large powersystems’, IEEE Trans. Power Syst., 1990, 5, (2), pp. 635–642

20 CIGRE TF 38.01.08: ‘Modeling of power electronics equipment(FACTS) in load flow and stability programs: a representation guidefor power system planning and analysis’ 1999

21 Dong, L.Y., Zhang, L., and Crow, M.L.: ‘A new control strategy forthe unified power flow controller’. IEEE PES Winter Meeting,2002, vol. 1, pp. 562–566

22 Padiyar, K.R.: ‘Power system dynamics stability and control’ (JohnWiley & Sons (Asia) Pte Ltd, Singapore, 1996)

23 Humpage, W.D., Bayne, J.P., and Durrani, K.E.: ‘Multinode-power-system dynamic analyses’, Proc. IEE, 1972, 119, (8), pp. 1167–1175

24 IEEE Std 421.5-2005: ‘IEEE recommended practice for excitationsystem models for power system stability studies, 2005

25 IEEE Working Group: ‘Dynamic models for fossil fueled steam unitsin power system studies’, IEEE Trans. Power Syst., 1991, 6, (2),pp. 753–761

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Published in IET Generation, Transmission & DistributionReceived on 21st September 2006Revised on 4th December 2007doi: 10.1049/iet-gtd:20070125

ISSN 1751-8687

Neural networks for adaptive controlcoordination of PSSs and FACTS devices inmultimachine power systemT.T. Nguyen R. GiantoEnergy Systems Centre, School of Electrical, Electronic and Computer Engineering, The University of Western Australia, 35Stirling Highway, Crawley, Western Australia 6009, AustraliaE-mail: [email protected]

Abstract: The paper develops a new design procedure for online control coordination which leads to adaptivepower system stabilisers (PSSs) and/or supplementary damping controllers of flexible ac transmission system(FACTS) devices for enhancing the stability of the electromechanical modes in a multimachine power system.The controller parameters are adaptive to the changes in system operating condition and/or configuration.Central to the design is the use of a neural network synthesised to give in its output layer the optimalcontroller parameters adaptive to system operating condition and configuration. A novel feature of the neural-adaptive controller is that of representing the system configuration by a reduced nodal impedance matrixwhich is input to the neural network. Only power network nodes with direct connections to generators andFACTS devices are retained in the reduced nodal impedance matrix. The system operating condition isrepresented in terms of the measured generator power loadings, which are also input to the neural network.For a representative power system, the neural network is trained and tested for a wide range of credibleoperating conditions and contingencies. Both eigenvalue calculations and time–domain simulations are used inthe testing and verification of the dynamic performance of the neural-adaptive controller.

Nomenclaturem number of neural network inputsM number of neural network outputsT vector or matrix transposeVT1, VT2 AC terminal voltageIsh, Ise shunt, series currentVsh, Vse shunt, series voltageIshp, Ishq p and q components of shunt currentIsep, Iseq p and q components of series currentVshp, Vshq p and q components of shunt voltageVsep, Vseq p and q components of series voltagek ratio between AC and DC voltagesm1, m2 pulse-width modulation ratios for shunt,

series converters

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C1, C2 pulse-width modulation phases for shunt,series converters

VDC DC capacitor voltages Laplace transform operatorVSDC output signal from supplementary

damping controllerPe line active powerVREF voltage referencedroop slope of the voltage-current characteristicVDC,REF DC voltage referenceVshp0,Vshq0

p and q components of shunt voltageinitial value

PREF,QREF

active-, reactive-power reference

Isep,REF,Iseq,REF

p and q components of series currentreference

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Vsep0,Vseq0

p and q components of series voltageinitial value

VPSS output signal from power system stabiliservr input signal to PSS (rotor speed)

1 IntroductionFollowing the restructuring of the power supplyindustry and increased trend of interconnecting powersystems, the damping of electromechanical modes ofoscillations among the interconnected synchronousgenerators, including the inter-area modes, is agrowing concern, and constitutes one of the essentialcriteria for secure system operation.

It is acknowledged that power system stabilisers(PSSs) and/or FACTS devices with supplementarydamping controllers (SDCs) can enhance or maintainthe stability of the electromechanical modes. In thiscontext, there has been extensive research in theapplication of PSSs and/or SDCs of FACTS devices,particularly their control coordination, for achievingoptimal damping of electromechanical modes,including inter-area modes in a power system [1–6].In [1–6], the control coordination design proceduresin offline environment which lead to fixed-parametercontrollers have been reported. However, it is, ingeneral, accepted that there are disadvantagesassociated with fixed-parameter controllers, even withthose obtained by robust design.

If the design is based on one particular power systemoperating condition and configuration [1], it is possiblethat the performances of the controllers willdeteriorate under other operating conditions orconfigurations. There have been publications [2–8]reporting research on offline robust design of dampingcontrollers with fixed parameters, taking into accountthe variation of power system operating conditionand/or configuration. In [4], a linear matrix inequality(LMI) approach to normalised H1 loop shaping wasproposed for robust control design of power systemdamping controllers with fixed parameters to ensure aminimum damping ratio for inter-area modes.

However, there remains a key disadvantage withfixed-parameter controllers. It is, in general, notpossible to achieve maximum damping performancefor each and every operating condition or contingencywhen the controller parameters are fixed.

Based on the Lyapunov function and modellingapproximation [9, 10], robust control laws togetherwith decentralised control structure have been derivedfor FACTS devices to achieve damping ofelectromechanical oscillations.

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More recently, adaptive control techniques have beenapplied for power system damping controller design. In[11, 12], neural networks and radial basis functionnetworks were proposed for implementing PSS in asingle-machine infinite bus system. Control coordinationamong different PSSs in multimachine power systemand/or SDCs has not been considered. Furthermore,the changes in system configuration because ofcontingencies, which have a significant impact onelectromechanical mode dampings, have not beendiscussed in the design procedure.

The use of neural networks is extended to SDCs ofFACTS devices in [13, 14]. The adaptive thyristor-controlled series capacitor (TCSC) controller wasdesigned for a single-machine infinite bus system in[13]. Transmission line power flows were used asneural network inputs. The design procedure has nottaken into account the control coordination andcontingencies arising in a larger system withmultimachines. The approach in [14] proposed a staticVAr compensator (SVC) damping controller based ona neuro-identifier and neuro-controller to be trainedonline. The disadvantages include the application oftrial-and-error technique for forming the cost functionin the neuro-controller training, the possibility ofconvergence difficulty encountered in training, andhow to choose the order of the neuro-identifier. Thelevels of electromechanical mode dampings requiredcannot be specified in the proposed approach.

Multiple-model adaptive control strategy wasproposed in [15] for robust damping of inter-areaoscillations. The plant models need to be simplifiedand linearised with reduced order for controllerdesign and tuning. There is another issue related tothe choice of the appropriate number of plant models,particularly for large systems with a wide range ofdisturbances and responses. A self-tuning controllerfor one TCSC is proposed in [16]. It is based on alinear model with time-varying coefficients identifiedonline to represent the power system. A procedureremains to be developed for determining anappropriate model order, given that the number ofelectromechanical modes with low or negativedampings depends on system operating condition and/or configuration.

The above review indicates that there remain two keyissues that need to be addressed in relation to the designof adaptive PSSs and SDCs:

1. Optimal control coordination It is required to achieveonline control coordination of multiple PSSs and/orSDCs in a multimachine power system. Therequirement is to maximise the damping ratio forelectromechanical modes for each and every crediblesystem operating condition or configuration.

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2. Representation of power system configuration The optimalcontroller parameters depend importantly on powersystem configuration. There is a need to representdirectly and systematically the change in systemconfiguration in online tuning and coordination ofmultiple controllers.

The present paper develops an adaptive controlcoordination scheme for PSSs and SDCs that addressesthe above two issues. The scheme is based on the useof a neural network which identifies online theoptimal controller parameters. The inputs to theneural network include the active and reactive powersof the synchronous generators which represent thepower loading on the system, and elements ofthe reduced nodal impedance matrix for representingthe power system configuration. It is, therefore notrequired to form and store a range of system modelsfor subsequent online use.

The use of the reduced nodal impedance matrix isa novel feature in the scheme proposed by whichany power system configuration can be representedvery directly and systematically. The matrix is formedfor only power network nodes that have directconnections to synchronous generators and FACTSdevices. The reduced nodal impedance matrix isderived very efficiently from the power system nodaladmittance matrix and sparse matrix operations. Theremaining inputs to the neural network in terms ofgenerator powers are available from measurements.

The neural network is trained and tested offline with awide range of credible power system operating conditionsand configurations. For all of the tests considered, thecontroller parameters obtained from the trained neuralnetwork are verified by both eigenvalue calculations andtime–domain simulations, which confirm that gooddampings of the rotor modes are achieved.

2 Representing systemconfiguration by reduced nodalimpedance matrix2.1 Concept

In addition to active- and reactive-power loadings on thepower system, the optimal parameters of PSSs andSDCs of FACTS devices depend importantly onsystem configuration. In designing the adaptivecontrollers, it is required to represent power systemconfiguration which is variable. One option is to use aset of discrete variables to describe the power systemtopology. However, this option is not a practical oneas it will lead to a very large number of combinations,particularly for a large power system.


The present paper proposes to use the nodalimpedance matrix confined to the controller locationsto represent the effects of system configuration oncontroller parameters. The matrix elements are inputto the neural network-based adaptive controller.

2.2 Forming reduced nodal impedancematrix

The steps of forming the reduced nodal impedancematrix are given in Table 1.

3 Power system dampingcontrollersThe power system damping controllers include PSSs andSDCs of FACTS devices which are installed for theprimary function of power flow and voltage control.The unified power flow controller (UPFC) [19, 20]can be seen as a general form of FACTS devices. InTable 2 and Fig. 1 are shown the structures of theUPFC together with an SDC and PSS.

4 Development of neuralnetwork-based adaptive controller4.1 General concept of neural network

The relationship among the optimal controller parametersand power system operating condition including systemconfiguration is, in general, a nonlinear one. Thepresent paper draws on the key property of themultilayer feedforward neural network, which is that ofthe nonlinear multi-variable function representation [22].The neural network is used for the mapping betweenthe power system configurations and/or operatingconditions and optimal controller parameters.

In Fig. 2a is shown the general structure of themultilayer feedforward neural network adopted in thepresentwork. The structure description is given in Table 3.

4.2 Overall structure

In Fig. 2b is shown the overall structure of which theneural-adaptive controller described in Section 4.1 is apart. For online tuning of the parameters of PSSs andFACTS device main controllers together with SDCs,the inputs required are, as shown in Fig. 2b:

† circuit-breaker and isolator status data;

† power network branch parameters;

† generator active and reactive powers.

The output of the trained neural network in responseto the changes in the input determined by the changes in


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Table 1 Forming the reduced nodal impedance matrix

Step number Description

1 Forming the power system configuration from circuit-breaker and isolator status data [17]

2 Forming system nodal admittance matrix. The system configuration determined in step 1 is used inconjunction with the network branch parameters stored in the power system database to form thesystem nodal admittance matrix

3 Reducing the system nodal admittance matrix formed in step 2 to the nodal impedance matrix for thepower system nodes that have direct connections to generators and SDCs. This is achieved throughsparse matrix operations and lower and upper (LU) matrix factorisation

4 Online modification of the reduced nodal impedance matrix. The LU matrix factorisation in step 3 of thesystem nodal admittance matrix is performed only once in an offline mode for the system configurationof the base case (i.e. full system). The results of the factorisation are then stored for subsequent use inthe online mode. A scheme based on the compensation technique reported in [18] is adopted to formthe reduced nodal impedance matrix for any contingency, using the stored results of the base-casefactorisation, and only a minimal amount of computation which does not involve the refactorisation isrequired. The scheme is suitable for online application of the adaptive controllerWith the present advances in LU factorisation techniques, it is possible that the online full refactorisationcan be carried out to form the reduced nodal impedance matrix, without using the compensationmethod. This also allows load models in the form of admittance to be represented in the system nodaladmittance matrix

/

circuit-breaker status data and/or generator active andreactive powers gives the updated optimal parametersfor the PSSs and FACTS device main controllerstogether with SDCs. The feedback inputs to thesecontrollers are generator speeds and transmission lineactive powers, as in the case of fixed-parametercontrollers.

4.3 Training procedure for neural-adaptive controller

In Table 4 are given the stages required in offline trainingprocedure for the neural-adaptive controller.


4.4 Neural network testing and sizing

In addition to forming the training data set, a separatetesting data set is also required. The procedure fortesting data generation is similar to that of trainingwhere the optimisation-based control coordinationmethod in [6] is used.

The trained neural network in Section 4.3 isthen tested with the testing data set. Theinteraction among the training, testing and sizingthe neural network is explained in the flowchart ofFig. 3.

Table 2 UPFC and PSS structures

Controller type Structure description

UPFC Fig. 1a shows the general structure of the UPFC [19, 20]. The UPFC combines two voltage sourceconverters linked by a DC bus

In Figs 1b and 1c are shown the dynamic models for the shunt and series converter controllers,respectively [19, 20]

In addition to the main controllers, there is an SDC the output of which is input to the shunt convertercontroller as shown by the dashed box in Fig. 1b

The input to the SDC is the active-power flow in the transmission line controlled by the UPFC seriesconverter. However, the other forms of input signal such as the phase difference between thetransmission line terminal voltages can also be used for SDC input. The purpose of this supplementarycontroller is to improve the damping of electromechanical modes

It is also possible to use an SDC in conjunction with the series converter controller

PSS In Fig. 1d is shown the structure for the PSS [21] in which the input is the generator rotor speed andthe output is fed to the excitation controller


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Figure 1 Block diagram of UPFC and PSS

a UPFC block diagramb UPFC shunt converter control block diagramc UPFC series converter control block diagramd PSS control block diagram

5 Simulation results5.1 Power system structure

The system in the study is based on the two-area 13-buspower system of Fig. 4a [23]. Each of the four


synchronous generators in the system is representedby a fifth-order dynamical model described in theappendix. Initial investigations have been carried outfor the system. The investigations confirm that theinter-area mode has poor damping. Stabilisation


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measure based on PSSs and FACTS device controllerswith SDCs as discussed in [6] is, therefore proposedfor improving the damping of the inter-area mode inthe power system. However, these controllers willalso enhance local mode dampings.

Stability analysis of the power system without anyPSSs and FACTS devices indicates that, among thefour generators in Fig. 4a, participation factors of theinter-area mode in generators G1 and G3 are greaterthan those in the other two generators. On thisbasis, it is proposed to install PSSs for generators G1and G3 only. The other two generators (generatorsG2 and G4) do not have PSSs. The PSSs forgenerators G1 and G3 have adaptive parameters. ThePSSs have the structure described in Section 3 withrotor speed inputs. A FACTS device, that is a UPFCwith an SDC, is installed at node N13 in line L16.Also, it is proposed to use the line active power as

Figure 2 Neural network and neural-adaptive controller

a Input and output structure of the neural networkb Block diagram of the system with neural-adaptive controllerP1, P2, . . .., Pn: Real and imaginary parts of the elements of thereduced nodal impedance matrixPnþ1, Pnþ2, . . .., Pm: Active- and reactive- power of generatorsa1, a2, . . .., am, Optimal controller parametersf: Activation function

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the input to the SDC which has the structure givenin Fig. 1b. For this system with four generators,there are three swing modes (two local modes andone inter-area mode) of low-frequency oscillations.All of these electromechanical modes are representedin the control coordination and the design of theneural-adaptive controller described in the nextsection.

5.2 Design of the proposed neural-adaptive controller

5.2.1 Neural network training and test data: Thekey requirement is to design a neural controller that hasthe capability of generalising with high accuracy fromthe training cases. This requirement is achievedthrough the neural network training, testing and sizingreferred to in Sections 4.3 and 4.4 based on theselection of the training and testing data sets. Theneural network training set should be representative ofthe cases described by credible system contingenciesand changes in system operating conditions.

The possible contingencies of the system in Fig. 4a forline(s) outages, load and power generation variations areshown in Tables 5 and 6, respectively. Both single-lineand double-line outages are considered in thepostulated contingencies where there is no loss of anygenerator, and the two areas remain connected. Theinput and output pairs for neural network training andtesting cases are generated from the combinations ofthese contingencies and operating conditions.

For the system in Fig. 4a, the number of neuralnetwork inputs, as determined on the basis of Section4.1, is 38. In this paper, the parameters of both themain controller and SDC of the UPFC are to betuned online to achieve the maximum benefit in termsof damping. Therefore 26 linear neurons are neededin the output layer (6 for each PSS controller and 14for the UPFC controller).

The load demands together with their power factors(PFs) at nodes N9 and N10 are varied in therepresentative range between minimum and maximumvalues. Power-flow solutions with the specified loaddemands give the range of active and reactive powersat generator nodes as shown in Table 6. It has beentaken that the load demands at nodes N9 and N10follow similar patterns. However, any differentpatterns of load demand variations, for example, inareas in different time zones, when they arise, can beincluded in the data set without difficulty.

For each contingency, the procedure described inSection 2 and power-flow studies are used for formingthe neural network input data in the training case.The optimal controller parameters are also


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Table 3 Neural network structure

Layeridentification

Layer description

Input layer There are two separate sets of nodes in the input layer in Fig. 2a

The first set has n nodes the inputs to which are obtained from the real and imaginary parts of thereduced nodal impedance matrix as formed in Section 2. These inputs represent power systemconfiguration. If there are Nc generator nodes and FACTS device nodes, the number of input nodes inthe first set is Nc2þ Nc, when the symmetry in the nodal impedance matrix is exploited

The second set of inputs comprises active and reactive powers of each and every generator. Thereforeif there are Ng generators in the power system, there will be 2Ng input nodes in the second set. Theseinputs in the second set represent power system operating condition

The total number of inputs is Nc2þ Ncþ 2Ng

Hidden layer The number of hidden layers, the number of nodes in each hidden layer and the weightingcoefficients of the connections between the nodes in the structure of Fig. 2a are to be determined byneural network training, and verified by testing which will be discussed in Sections 4.3 and 4.4

Output layer The nodes in the output layer of the neural network structure in Fig. 2a give the optimal values of theparameters of PSSs and FACTS device control systems, including the SDCs

It is possible to exclude the FACTS device main controllers from the adaptive control coordination.However, to achieve maximum benefit in terms of damping, both FACTS device main controller andSDC are included in the adaptive control coordination in the present work

The structure in Fig. 2a assumes that there are M controller parameters to be tuned online. On thisbasis and with the controllers in Fig. 1, the output parameters from the neural network in Fig. 2a aredescribed as follows.

(a) PSS

PSS gain (denoted by KPSS)

Time constants of PSS washout block (denoted by TPSS)

Time constants of PSS lead-lag blocks (denoted by TPSS1, TPSS2, TPSS3 and TPSS4)

(b) SDC

SDC gain (denoted by KSDC)

Time constants of SDC washout block (denoted by TSDC)

Time constants of SDC lead-lag blocks (denoted by TSDC1, TSDC2, TSDC3 and TSDC4)

(c) UPFC main controller

Shunt converter controller gains (denoted by Ksh1 and Ksh2)

Shunt converter controller time constants (denoted by Tsh1 and Tsh2)

Series converter controller gains (Kse1 and Kse2)

Series converter controller time constants (Tse1 and Tse2)

determined for each case using the method described in[6]. These optimal controller parameter values are usedas the specified network output data.

In applying the optimal control coordination [6] fortraining and test data generation, the sum of the squaresof the real parts of all of the eigenvalues of theelectromechanical modes is maximised, with the


constraints that the minimum damping ratio of the localmodes is to be 0.3, and that of the inter-area mode 0.1.

The cases generated from Tables 5 and 6 are sub-divided into the training set and test set. For thetraining set, line outage cases 1.1–1.4, 1.6–1.9,1.11–1.20, 1.22–1.27, 1.29–1.34 and 1.36–1.42together with load demand variations in cases 2.1–2.5,


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Table 4 Neural-adaptive controller training procedure

Stage intraining

Task required

1 Training set generationThe training set is generated using the optimisation-based control coordination method in [6]. A briefdescription of the method is given in the following.

The method is based on a constrained optimisation in which the objective function formed from the realpart of eigenvalues of selected modes is minimised. The method does not require any special eigenvalue/eigenvector calculation software. Eigenvalue-eigenvector equations are represented in terms of equalityconstraints in the optimisation. Based on the linear independence of eigenvectors, additional equalityconstraints are derived and included in the optimisation to guarantee distinct modes at the convergence.Inequality constraints related to minimum damping ratios required and controller parameter limits arerepresented in the control coordination.

For a given power system, a wide range of credible operating conditions and configurations which includethose arising from contingencies is considered in the training data generation. For the ith training case,the pair of specified input and output vectors is fpi, tig. Based on the structure in Fig. 2a, the input vectorpi is:

pTi ¼ (p1i, p2i, . . . , pmi); i ¼ 1, 2, . . . ,N

in which N is the total number of training cases.

The target output vector ti for the ith training case is the optimal controller parameters vector for thepower system with the operating condition and configuration specified by the input vector pi.

2 Training error minimisationThe requirement in the training is to minimise the difference between the target output vector ti andresponse of the neural network in Fig. 2a. For N training cases, it is proposed to minimise the followingmean square error

F(x) ¼l

N

XNi¼1

(ti � ai)T(ti � ai)

where ai is the neural network response which has the following form, based on the structure in Fig. 2a

aTi ¼ (a1i, a2i, . . . aMi); i ¼ 1, 2, . . . , N

Vector x is the vector of weighting coefficients of the connections in the neural network to be identified.Minimising the error function F(x) with respect to x gives the weighting coefficient vector. In the presentwork, the Levenberg-Marquardt algorithm which is a second-order method with a powerful convergenceproperty is adopted for minimising F(x).

3 Verifying convergence in trainingOne of the criteria for the convergence in training is that the error function F(x) has to be less than aspecified tolerance.

In addition to the training performance expressed in terms of error function F(x), the controllerparameters obtained from the trained neural network are also used for calculating the damping ratios ofthe rotor modes, which are then compared with the optimal damping ratios obtained at the stage oftraining data generation.

The convergence in training is confirmed when both the error function F(x) and the damping ratiocomparison satisfy the specified tolerances.

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2.7–2.10 and 2.12–2.16 are selected. The remainingcases of line outages and load demand variations inTables 5 and 6 are used for the test set.

5.2.2 Training, testing and sizing the neuralnetwork: In the present work, the neural network isinitially assumed to have one hidden layer and the


number of hidden nodes is taken to be 5. The size of theneural network is then adjusted according to theprocedure described in Section 4.4.

The performance goals specified in terms of the errorfunction F(x) of 0.004 (for training) and 0.006 (fortesting) are used. The maximum differences between the


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Figure 3 Flowchart for training, testing and sizing of the neural network

MSE: Mean Square Error

optimal damping ratio and the damping ratio calculatedusing neural network outputs of 0.03 (for training) and0.05 (for testing) are also used as the performance goals.Maximum number of epoch of 100 is specified for thenetwork training. Several network sizes (i.e. number ofhidden neurons) are investigated to achieve theperformance goals. Based on the investigation, it is foundthat the network with ten hidden neurons in one hiddenlayer satisfies the convergence criteria. On this basis, thetrained and tested neural network is used in theapplication mode, and its dynamic performance isevaluated by simulation in the following section.

5.3 Dynamic performance of the systemin the study

Table 7 shows the comparison of modal responsecharacteristics (electromechanical mode eigenvalues,frequencies and damping ratios) between non-adaptiveand adaptive controllers of the system in Fig. 4a for arange of contingencies and operating conditions. For


non-adaptive controller, the controller parametersderived from the base case design are used for all ofthe contingency cases and load changes.

The base case (referred to as case 1 inTable 7) is thatwiththe full system in Fig. 4a, and load demands at nodesN9 andN10 being 10þ j2 and 13þ j2.5 pu, respectively. Thecomparison in Table 7 for case 1 confirms that thedamping ratios for the electromechanical modes achievedby the neural-adaptive controller are closely similar tothose obtained from the fixed-parameter controllers (i.e.non-adaptive) designed with the system configuration andoperating condition specified in the base case. In theoffline training of the neural-adaptive controller, the basecase has not been included in the training set. Thecomparison for case 1 can, therefore be seen as a neural-adaptive controller testing.

In case 2 of Table 7, the load demands at nodes N9 andN10 increase to 15þ j7 and 16þ j8 pu, respectively,whereas the system configuration remains as that of the


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Figure 4 Test system and transients

a Two-area 230 kV systemb Relative voltage phase angle transients for case A disturbancec Relative voltage phase angle transients for case B disturbanced Relative speed (G2–G1) transients for case A disturbancee Relative speed (G2–G1) transients for case B disturbanceIn the system of Fig. 4a:Total connected load ¼ 2300 MWExcitation systems model: based on IEEE Type-ST1 system [27]Turbine and governor model: adopted from [28]

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base case. With non-adaptive controllers, the dampingratios of the electromechanical modes decrease noticeablyin comparison with those in the base case. However, withthe neural-adaptive controller, the damping ratios aremaintained at the levels similar to those of the base case.


Further comparisons in cases 3–6 of Table 7 focus oncontingencies where one or two transmission circuitsare lost. The load demands are those in the base case.In case 3, where there is an outage of transmissionline L5 in Fig. 4a, there is a substantial reduction in


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the inter-area mode damping in comparison with thebase case. The decreases in the local mode dampingare non-uniform. The local mode associated with

generators G2 and G3 is affected severely in terms ofdamping, given that these generators are electricallyclose to the outage location. The damping ratio of this

Table 5 Line(s) outages cases

No. Single-line outages Double-line outages

No. Lines No. Lines No. Lines

1.1 line L5 1.11 lines L5 and L9 1.22 lines L7 and L12 1.33 lines L11 and L13










1.21 lines L7 and L11 1.32 lines L10 and L13 1.43 lines L13 and L14

Table 6 Variations of load and power generation

No. Load demand, pu Power generation, pu

Node N9 Node N10 Slack bus (node N1) PV bus

Load PF Load PF PGEN QGEN PGEN QGEN

2.1 8þ j2 0.97 11þ j3 0.96 5.4–6.8 21.0–20.9 4.5–5.0 22.0–21.1

2.2 8þ j2 0.97 12þ j3 0.97 5.4–6.8 21.0–20.9 4.5–5.0 22.0–21.1

2.3 9þ j8 0.75 11þ j9 0.77 5.0–6.4 2.1–2.2 5.0–5.5 1.6–3.3

2.4 9þ j8 0.75 12þ j9 0.80 5.0–6.4 2.1–2.2 5.0–5.5 1.6–3.3

2.5 10þ j5 0.89 13þ j6 0.85 6.5–7.9 0.9–1.0 5.5–6.0 0.1–1.3

2.6 10þ j5 0.89 14þ j6 0.91 6.5–7.9 0.9–1.0 5.5–6.0 0.1–1.3

2.7 11þ j6 0.88 13þ j7 0.92 6.0–7.4 1.3–1.4 6.0–6.5 0.7–2.1

2.8 11þ j6 0.88 14þ j7 0.89 6.0–7.4 1.3–1.4 6.0–6.5 0.7–2.1

2.9 12þ j8 0.83 15þ j9 0.86 7.7–9.0 3.0–3.1 6.5–7.0 2.0–4.2

2.10 12þ j8 0.83 16þ j9 0.87 7.7–9.0 3.0–3.1 6.5–7.0 2.0–4.2

2.11 13þ j4 0.96 15þ j5 0.95 7.1–8.5 0.7–0.8 7.0–7.5 20.2–1.1

2.12 13þ j4 0.96 16þ j5 0.95 7.1–8.5 0.7–0.8 7.0–7.5 20.2–1.1

2.13 14þ j7 0.89 17þ j8 0.90 7.5–8.8 2.9–3.1 8.0–8.5 2.0–4.0

2.14 14þ j7 0.89 18þ j8 0.91 7.5–8.8 2.9–3.1 8.0–8.5 2.0–4.0

2.15 15þ j2 0.99 17þ j3 0.98 6.9–8.2 0.1–0.2 8.5–9.0 2 0.7–0.4

2.16 15þ j2 0.99 18þ j3 0.99 6.9–8.2 0.1–0.2 8.5–9.0 2 0.7–0.4

pu on 100 MVA

he Institution of Engineering and Technology 2008 IET Gener. Transm. Distrib., 2008, Vol. 2, No. 3, pp. 355–372/doi: 10.1049/iet-gtd:20070125

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Table 7 Dynamic performances of controllers

No Case Non-adaptive controller Adaptive controller

Eigenvalues Freq.,Hz

Damp.ratio

Eigenvalues Freq.,Hz

Damp.ratio

1 base caseload (pu): node N9:10þ j2.0; node N10:13þ j2.5

22.3699+ j7.0642a 1.12 0.3181 22.3089+ j7.3413a 1.17 0.3000

22.3448+ j6.6088b 1.05 0.3344 22.3925+ j7.1246b 1.13 0.3183

20.5329+ j3.5437c 0.56 0.1487 20.5036+ j3.3037c 0.53 0.1507

2 load-change caseload (pu): node N9:15þ j7node N10: 16þ j8

21.8169+ j7.5759a 1.21 0.2332 22.3145+ j7.2747a 1.16 0.3032

21.8685+ j7.1286b 1.13 0.2535 22.2683+ j7.0094b 1.12 0.3079

20.3723+ j3.1591c 0.50 0.1170 20.5303+ j3.1685c 0.50 0.1651

3 line L5 out 22.3783+ j7.3294a 1.17 0.3086 22.2690+ j6.9820a 1.11 0.3091

21.4468+ j6.6625b 1.06 0.2122 22.0757+ j6.6838b 1.06 0.2966

20.2630+ j3.3211c 0.53 0.0789 20.5093+ j3.4126c 0.54 0.1476

4 lines L7 and L11 out 21.7872+ j6.8207a 1.09 0.2535 22.0100+ j6.6457a 1.06 0.2895

21.5108+ j6.5671b 1.05 0.2242 22.0145+ j6.4854b 1.03 0.2966

20.3246+ j3.2679c 0.52 0.0988 20.4865+ j3.2230c 0.51 0.1493

5 line L13 out 22.5048+ j7.4695a 1.19 0.3179 22.3543+ j7.4433a 1.18 0.3016

22.5559+ j7.3784b 1.17 0.3273 22.4456+ j7.3200b 1.17 0.3169

20.1622+ j3.7914c 0.60 0.0427 20.5619+ j3.3340c 0.53 0.1662

6 lines L5 and L14 out 22.5297+ j7.5291a 1.20 0.3185 22.4497+ j7.3964a 1.18 0.3144

21.5138+ j6.6691b 1.06 0.2214 21.9943+ j6.5613b 1.04 0.2908

20.1575+ j3.5010c 0.66 0.0449 20.4610+ j3.0515c 0.49 0.1494

aLocal mode associated with generators G1 and G4; bLocal mode associated with generators G2 and G3; cInter-area mode

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mode is reduced to 0.2122, compared with 0.3344 inthe base case. The damping of the local modeassociated with generators G1 and G4 is hardlyaffected by this outage. Its damping ratio is now0.3086 in comparison with 0.3181 of the base case.With the adaptive controller, the damping ratios of allof the electromechanical modes are only marginallyaffected by the outage, in comparison with those inthe base case, as indicated in Table 7.

The response characteristics of the threeelectromechanical modes in case 4 where there aredouble outages of transmission lines L7 and L11 aregiven in Table 7. The modal damping ratios with non-adaptive controllers are now substantially lower thanthose of the base case. In comparison, the adaptivecontrollers are able to restore the damping ratios tothe levels which are nearly equal to those of the basecase, even though the contingency of case 4 has notbeen included in the offline training of the adaptivecontroller.


The outage of transmission line L13 in case 5 ofTable 7 affects the damping of the inter-area modevery severely when the non-adaptive controllers areused. The damping ratio of 0.1487 in the base case isnow reduced to 0.0427 in the outage case 5.However, the outage does not affect the local modedampings to any significant extent, relative to those inthe base case. This response characteristic is consistentwith the topology of the power system in Fig. 4awhere transmission line L13 has the primary functionof interconnecting the two areas. The robustness ofthe adaptive controller in this outage case is confirmedby the results of Table 7. The controller parametersdetermined by the trained neural network are able toadapt to the new system configuration for maintainingthe modal damping ratios at the levels similar to thosein the base case.

Double outages of transmission lines L5 and L14 arethen considered in case 6 of Table 7. As expected, theadditional outage of transmission line L14 which


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interconnects the two areas affects mainly the dampingof the inter-area mode when non-adaptive controllersare used. Comparisons among the damping ratios ofthe inter-area mode achieved by the non-adaptivecontrollers in cases 1, 3 and 6 confirm the effect ofthe outage of transmission line L14 on the inter-area mode damping. With adaptive controllerparameters, the adverse effects of the outages incase 6 are largely countered, as indicated in thedamping ratios results of Table 7. The levels ofelectromechanical mode dampings are almost the sameas those in the base case.

5.4 Time–domain simulations

In order to further validate the performance of theproposed neural-adaptive controller, time–domainsimulations are carried out for the selectedcontingency cases (i.e. line L13 outage and lines L5and L14 outage). The time-step length of 50 ms isadopted for the simulations. The descriptions of theline(s) outage cases and the disturbances used toinitiate the transients for each case are given in Table 8.

In Figs. 4b–4e are shown the system transientsfollowing the disturbances. As the focus is on theinter-area mode oscillation, relative voltage phaseangle transient between nodes N9 and N10 is used informing the responses in Figs. 4b and 4c. From theresponses, it can be seen that, with non-adaptivecontroller, the system oscillation is poorly damped andtakes a considerable time to reach a stable condition.With the proposed neural-adaptive controller, thesystem reaches steady-state condition in 6–7 ssubsequent to the disturbance for the contingencycases considered (Figs. 4b and 4c). Further comparisonin terms of the transients in the rotor speed ofgenerator G2 relative to that of generator G1 aregiven in Figs. 4d and 4e. The comparison confirms thenoticeable improvement in electromechanicaloscillation damping when the adaptive controller isused.

In Figs. 5a–5d are also shown the plots of twocontroller parameters (i.e. PSSs and SDC gains)during the transient period following the disturbance.The plots of PSSs and SDC gains for line L13 outageare shown in Figs 5a and 5c respectively, whereas the


plots of PSSs and SDC gains for lines L5 and L14outage are shown in Figs 5b and 5d, respectively.

There are rapid changes in the controller gains in theinitial transient period following the fault and faultclearance, because of the transients in generatorpowers. To facilitate the adaptation of the controllerparameters in the initial transient period typicallywithin the range up to about 6 s, the option ofkeeping the inputs to the neural network representinggenerator powers at the base-case values, andchanging only the inputs derived from the reducednodal impedance matrix can be used. This option isbased on the result of the study given in Table 7 ofSection 5.3 which confirms that the overall damping ismore substantially affected by system configurationthan generator loadings. In Figs. 5e and 5f are shownthe relative voltage phase angle transient and SDCgain transient for disturbance case B in Table 8,respectively, using the option described in the above.The damping of the transient in Fig. 5e is similar tothat in Fig. 4c, whereas the transient in the controllerparameter in Fig. 5f is substantially reduced incomparison with that in Fig. 5d, which will facilitatethe implementation of the adaptive controller.

In practice, there will be some time delay in thecommunication channel before the inputs to theneural-adaptive controller which represent the powersystem configuration can be updated, following adisturbance. Studies have been carried out to quantifythe performance of the neural-adaptive controllerwhen there is the time delay.

In [24], a signal transmission delay of 0.75 s hasbeen proposed in the design of H1 dampingcontrollers using remote signals. A time delay up to2 s is, therefore considered in the present work forevaluating the effect on the neural-adaptive controllerperformance. With signal transmission delaysrepresented in the inputs to the neural-adaptivecontroller, the system transient responses fordisturbance case B described in Table 8 arere-evaluated and shown in Figs. 5e–5h.

Time delays of 1 and 2 s in relation to the updating ofsystem topology after fault clearance have been adoptedin the study. The comparisons made of the inter-area

Table 8 Descriptions of line(s) outage cases and disturbances

Case Outage description Disturbance description

A line L13 has to be disconnected to clear the fault Three-phase fault near node N13 on line L13. The fault isinitiated at time t ¼ 0.1 s, and the fault clearing time is 0.1 s

B line L5 is initially taken out for maintenance thenline L14 has to be disconnected to clear the fault

Three-phase fault near node N13 on line L14. The fault isinitiated at time t ¼ 0.1 s, and the fault clearing time is 0.1 s


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Figure 5 System transients

a PSSs gain transients for case A disturbanceb PSSs gain transients for case B disturbancec SDC gain transient for case A disturbanced SDC gain transient for case B disturbancee Relative voltage phase angle transients for case B disturbance (effects of time delay)f SDC gain transients for case B disturbance (effects of time delay)g Relative speed (G4–G1) transients for case B disturbance (effects of time delay)h Relative speed (G3–G2) transients for case B disturbance (effects of time delay)

mode responses of Fig. 5e, and the local mode responsesof Figs. 5g and 5h indicate that the effect of the timedelay is to reduce only slightly the electromechanical


mode dampings. However, in relation to signaltransmission delay and/or communication channelfailure, the neural-adaptive controller developed in the


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paper offers a key advantage in comparison with othercontroller designs using remote signals [7]. Whenthere is a loss of communication channel or substantialtime delay, the neural-adaptive controller will revertback to the fixed-parameter controller, with sub-optimal damping. In the delay period/loss ofcommunication channel, the PSSs and SDCs still havelocal input signals (rotor speed/power), and theyoperate normally to give continuous non-zero outputswhich contribute to the system damping. Othercontrollers which depend totally on remote inputsignals will not be able to function without thecommunication channel.

5.5 Possible improvements

Table 9 shows the range of optimal controller parametervariation for different operating conditions and systemconfigurations described in Tables 5 and 6. Results inthe table show that the range of variation in thecontroller gains is wider than that in the controllertime constants. This indicates that the controller gainsare more sensitive to system changes than the timeconstants. Therefore to simplify the adaptivecontroller and its training, it is possible to adapt onlythe controller gains to the prevailing system condition,


and keep the controller time constants at the constantvalues determined in the base case.

It is also found out from the investigation that thelocal modes are more affected by PSSs, whereas theinter-area mode is more affected by the SDC. In otherwords, SDCs are more important if only the inter-area modes are to be considered. Therefore if thedamping ratios of the local modes are high in the basecase, it is possible to include only the SDCs in theneural-adaptive controller design, and to have fixed-parameter PSSs designed in the base case.

In order to check whether a smaller number of neuralnetwork inputs can be used in the adaptive controller,representation of the system configuration with areduced nodal impedance matrix of a lower dimensionis investigated. In the investigation, only power systemnodes with direct connections to generators with PSSsand FACTS devices are retained. The neural networkwith a smaller number of inputs is then trained andtested using the test cases described in Section 5.2.1.Based on the outcome of the investigation, it is foundthat the neural network with a reduced number ofinputs can also provide acceptable results. Furtherreduction in the number of inputs is also possible by

Table 9 Range of optimal controller parameter variation for different operating conditions and system configurations

Controller type Parameters Range

Symbol Type

PSS KPSS gain 4–20 pu

TPSS time constant 0.80–1.33 s

TPSS1 time constant 0.16–0.24 s




SDC KSDC gain 0.1–1.0 pu

TSDC time constant 0.16–0.24 s

TSDC1 time constant 0.16–0.24 s




UPFC main controller Ksh1, Ksh2 gain 0.1–1.0 pu

Tsh1, Tsh2 time constant 0.05–0.16 s

Kse1, Kse2 gain 0.01–0.10 pu

Tse1, Tse2 time constant 0.16–0.24 s

pu on 100 MVA


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Table 10 Comparisons between the neural adaptive controller size and ultra-large-scale neural network capability

Neural-adaptive controller size requirements Ultra-large-scale neural network capability[25]

The dimension of the reduced nodal impedance matrix is 110 � 110.Because of the symmetry in the impedance matrix, only 6105 elements arerequired to represent the power system configuration. The impedancematrix elements are, in general, complex numbers. However, in atransmission system (which is the focus of the present paper), theparameters of transmission circuits are dominated by the reactances. Thismeans that it is possible to discount the real parts of the nodal impedancematrix, for the purpose of representing the system configuration

Multi-processor technology (a cluster of 196processors)

1.73 million weighting coefficients

Nine million training patterns

Computational speed of 163.3 GFlops/s

In addition to 6105 elements (in real numbers, following the removal of thereal parts of the nodal impedance matrix) used for representing the powersystem configuration, there are 200 input values for representing generatoractive and reactive powers. Therefore in this example of the system having100 generators each of which has a PSS, and 10 FACTS devices, the totalnumber of input nodes of the neural-adaptive controller is about 6300

Cost: about 150 000 US dollars (in 2000). Itis highly likely that the cost at present ismuch lower, given that the cost of computerhardware is decreasing, whereas thecomputing capability (in terms of memoryand processing speed) is increasing

Based on the controller output parameters in Table 3, the total number ofoutput nodes of the neural-adaptive controller is about 750

discounting the real parts of the reduced nodalimpedance matrix elements, given that the parametersof the transmission circuits are dominated by reactances.

By applying the above measures, the size of the neuralnetwork and its training can be greatly simplified andkept to be minimal.

5.6 Discussion on large power systemapplication

Drawing on the measures for improvements in Section5.5 and the development of ultra-large-scale neuralnetwork reported in [25], it is feasible to meet therequirements of large power system application interms of neural network size and response time. Forthe purpose of illustration, it is taken in the discussionthat a large power system has 100 generators and 10FACTS devices with each generator having a PSS. InTable 10 are shown the comparisons between theneural adaptive controller requirements and theavailable capability of the ultra-large-scale neuralnetwork. The comparisons confirm that the capabilityexceeds the requirements by a large margin.

6 ConclusionsAn adaptive control algorithm and procedure have beenderived and developed for online tuning of the PSSs andSDCs of FACTS devices. The procedure is based on theuse of a neural network which adjusts the parameters ofthe controllers to achieve system stability and maintainoptimal dampings as the system operating conditionand/or configuration change. A particular contributionof the paper is that of representing the power system


configuration in terms of a reduced nodal impedancematrix, which is formed using sparse matrixoperations. This allows any variation of systemconfiguration to be included and input to the neural-adaptive controller.

The neural-adaptive controller trained for arepresentative power system with a UPFC has beencomprehensively tested to verify its dynamicperformance. Both eigenvalue calculations and time–domain simulations are applied in the testing andverification. Many comparative studies have beencarried out to quantify the improved performance ofthe adaptive controller proposed in comparison withthat achieved with fixed-parameter controllers.

The results confirm that the decrease in systemdampings arising from the use of fixed-parametercontrollers when system operating condition changeswill be removed, and the maximum or optimal dampingis regained by the proposed neural-adaptive controller.

7 AcknowledgmentsThe authors gratefully acknowledge the support of theEnergy Systems Centre at The University of WesternAustralia for the research work reported in the paper.They express their appreciation to The University ofWestern Australia for permission to publish the paper.

8 References

[1] POURBEIK P., GIBBARD M.J.: ‘Simultaneous coordination ofpower system stabilizers and FACTS device stabilizers in a


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multimachine power system for enhancing dynamicperformance’, IEEE Trans. Power Syst., 1998, 13, (2),pp. 473–479

[2] SADIKOVIC R., KORBA P., ANDERSSON G.: ‘Application ofFACTS devices for damping of power systemoscillations’. IEEE PowerTech 2005, St. Petersburg,Russia, June 2005

[3] CHAUDHURI B., KORBA P., PAL B.C.: ‘Damping controllerdesign through simultaneous stabilization technique’.Proc. WAC 2004, 2004, vol. 15, pp. 13–18

[4] MAJUMDER R., CHAUDHURI B., EL-ZOBAIDI H., ET AL.: ‘LMIapproach to normalised H-infinity loop-shaping design ofpower system damping controllers’, IEE Proc. Gener.Transm. Distrib., 2005, 152, (6), pp. 952–960

[5] NGUYEN T.T., GIANTO R.: ‘Application of optimizationmethod for control co-ordination of PSSs and FACTSdevices to enhance small-disturbance stability’. Proc.IEEE PES 2005/2006 Transmission and DistributionConf. and Exposition, Dallas, Texas, May 2006,pp. 1478–1485

[6] NGUYEN T.T., GIANTO R.: ‘Optimisation-based control co-ordination of PSSs and FACTS devices for optimaloscillations damping in multimachine power system’, IETGener. Transm. Distrib., 2007, 1, (4), pp. 564–573

[7] CHAUDHURI B., PAL B.C.: ‘Robust damping of multipleswing modes employing global stabilizing signalswith a TCSC’, IEEE Trans. Power Syst., 2004, 19, (1),pp. 499–506

[8] PAL B.C., COONICK A.H., CORY B.J.: ‘Robust damping of inter-area oscillations in power systems with superconductingmagnetic energy storage devices’, IEE Proc. Gener.Transm. Distrib., 1999, 146, (6), pp. 633–639

[9] NOROOZIAN M., GHANDHARI M., ANDERSSON G., ET AL.:‘A robust control strategy for shunt and seriesreactive compensators to damp electromechanicaloscillations’, IEEE Trans. Power Deliv., 2001, 16, (4),pp. 812–817

[10] GHANDHARI M., ANDERSSON G., HISKENS I.A.: ‘Control LyapunovFunctions for Controllable Series Devices’, IEEE Trans.Power Syst., 2001, 16, (4), pp. 689–694

[11] SEGAL R., KOTHARI M.L., MADNANI S.: ‘Radial basis function(RBF) network adaptive power system stabilizer’, IEEETrans. Power Syst., 2000, 15, (2), pp. 722–727

[12] CHATURVEDI D.K., MALIK O.P., KALRA P.K.: ‘Generalised neuron-based adaptive power system stabiliser’, IEE Proc. Gener.Transm. Distrib., 2004, 151, (2), pp. 213–218


[13] HSU Y.Y., LUOR T.S.: ‘Damping of power systemoscillations using adaptive thyristor-controlled seriescompensators tuned by artificial neural networks’IEE Proc. Gener. Transm. Distrib., 1999, 146, (2),pp. 137–142

[14] CHANGAROON B., SRIVASTAVA S.C., THUKARAM D., ET AL.:‘Neural network based power system damping controllerfor SVC’, IEE Proc. Gener. Trans. Distrib., 1999, 146, (4),pp. 370–376

[15] CHAUDHURI B., MAJUMDER R., PAL B.C.: ‘Application ofmultiple-model adaptive control strategy for robustdamping of interarea oscillations in powersystem’ IEEE Trans. Contr. Syst. Tech., 2004, 12, (5),pp. 727–736

[16] SADIKOVIC R., KORBA P., ANDERSSON G.: ‘Self-tuning controllerfor damping of power system oscillations with FACTSdevices’. 2006 PES General Meeting, Montreal, Canada,June 2006

[17] HUMPAGE W.D., WONG K.P., NGUYEN T.T.: ‘PROLOG network-graph generation in system surveillance’, Electr. PowerSyst. Res., 1985, 9, pp. 37–48

[18] ALSAC O., STOTT B., TINNEY W.F.: ‘Sparsity-orientedcompensation methods for modified network solutions’,IEEE Trans. Power Appar. Syst., 1983, 5, (102),pp. 1050–1060

[19] CIGRE TF 38.01.08: ‘Modeling of power electronicsequipment (FACTS) in load flow and stability programs:a representation guide for power system planning andanalysis’, 1999

[20] DONG L.Y., ZHANG L., CROW M.L.: ‘A new control strategy forthe unified power flow controller’. IEEE PES WinterMeeting, 2002, vol. 1, pp. 562–566

[21] MITHULANANTHAN N., CANIZARES C.A., REEVE J., ET AL.:‘Comparison of PSS, SVC, and STATCOM controllers fordamping power system oscillations’, IEEE Trans. PowerSyst., 2003, 16, (2), pp. 786–792

[22] DEMUTH H.B., BEALE M.: ‘Neural network toolbox user’sguide: for use with MATLAB (version 4)’ (The Math WorksInc., 2004)

[23] PADIYAR K.R.: ‘Power system dynamics stabilityand control’ (John Wiley & Sons (Asia) Pte Ltd,Singapore, 1996)

[24] CHAUDHURI B., MAJUMDER R., PAL B.C.: ‘Wide-areameasurement-based stabilizing control of power systemconsidering signal transmission delay’, IEEE Trans. PowerSyst., 2004, 19, (4), pp. 786–792


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[25] ABERDEEN D., BAXTER J., EDWARDS R.: ‘92¢/MFlops/sultra-large-scale neural network training on a PIIIcluster’. Supercomputing, ACM/IEEE 2000 Conf.,November 2000

[26] HUMPAGE W.D., BAYNE J.P., DURRANI K.E.: ‘Multinode-power-system dynamic analyses’, Proc. IEE, 1972, 119, (8),pp. 1167–1175

[27] IEEE STD 421.5-2005: ‘IEEE recommended practice forexcitation system models for power system stabilitystudies’, 2005

[28] IEEE WORKING GROUP‘Dynamic models for fossil fueledsteam units in power system studies’, IEEE Trans. PowerSyst., 1991, 6, (2), pp. 753–761

9 AppendixEach of the synchronous generators of the system inFig. 4a is represented by the fifth-order nonlinear


model in the d–q axes having the rotor frame of [26]

C†

rm ¼ AmCrm þ FmISm þ V rm (1)

v†

rm ¼ (Tm � Te)=M (2)

d†

rm ¼ vrm (3)

where Crm, vrm, and drm are vector of rotor fluxlinkages established by the field winding and damperwindings, rotor angular frequency and rotor angle,respectively; Vrm is the rotor voltage vector, Tm andTe are the mechanical rotor input and electricaltorques, respectively; M is the machine inertiaconstant; Am and Fm are the matrices depending onmachine parameters and ISm is the stator currentvector. For small-disturbance study, a standardlinearisation process is applied to (1)–(3).


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