Abstract--Power system stability enhancement with

STATCOM-based damping stabilizers is investigated in this study. Previously, the power system linearized model with STATCOM was introduced by other authors. The linearizing method is considered in this paper. With attention to presence of STATCOM changes operating point, so the initial coefficients should be calculated with new operating point. In this study, The Coordination among the STATCOM internal AC and DC voltage controllers and AC-damping stabilizers has been taken into consideration. This aims to enhance both rotor angle stability and voltage regulation of the power system. The design of STATCOM parameters is considered as an optimization problem and GA is used for searching optimized parameters. These studies are performed for five operating points. The influence of STATCOM on stability is compared with eigenvalues analysis and simulation, in two cases of without STATCOM and with STATCOM. The nonlinear simulation results show the capability of this method for power system dynamic stability improvement.

Index Terms-- Dynamic Stability, Optimization, Genetic Algorithm, STATCOM

I. INTRODUCTION owadays, low frequency oscillations have become the main problem for power system small signal stability.

They restrict the steady-state power transfer limits, which therefore affects operational system economics and security. Using PSS create change in oscillation stability. To increase power system oscillation stability, the installation of supplementary excitation control, power system stabilizer (PSS), is a simple, effective and economical method [1]. The recent advances in power electronics have led to the development of the flexible alternating current transmission systems (FACTS).These devices in addition to main control duties similar to voltage regulation and reactive power injection should be able to damp power oscillations. From the power system dynamic stability viewpoint, the STATCOM provides better damping characteristics than the SVC as it is able to transiently exchange active power with the system, so it can improve oscillation stability better than SVC [2]. The method of phase compensation [3] and damping torque analysis method [4] are conventional methods for design and control of power system stabilizers. Also multivariable design

The authors are with Electrical Engineering K.N.Toosi University of

Technology,Tehran,Iran ( e-mail: msd_zarringhalami@yahoo.com ; Golkar@eetd.kntu.ac.ir ).

method has been performed on coordination between internal AC and DC voltage controllers [5]. In this study, to improve power system dynamic stability and voltage regulation, coordination among AC-damping stabilizer and internal AC and DC voltage controller of STATCOM are performed. The studies are performed on a single machine infinite-bus power system. From power system linearized model is used for the studies [6]. The parameters design is considered as an optimization problem and the genetic algorithm is used for searching optimized parameters. The parameters should be designed so that the eigenvalues of system move to the left part of the j axis as much as possible. The participation factors method is used for identifying of modes type. The eigenvalues analysis and the nonlinear simulation results show the importance of the parameters design for dynamic stability improvement of the system.

II. POWER SYSTEM MODEL The power system is represented by a single machine

infinite-bus system. The system has an installed STATCOM in transmission line as shown in Fig.1.The STATCOM is used at the middle point in transmission line for voltage regulation and power oscillations damping. The system is modeled for low frequency oscillations studies and the linearized power System model is used for this purpose.

Fig. 1. Single machine infinite bus power system with STATCOM

A. Generator The generator is represented by the third-order model

comprising of the electromechanical swing equation and the

Analysis of Power System Linearized Model with STATCOM Based Damping Stabilizer

Masoud Zarringhalami , and M.A.Golkar

N

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generator internal voltage equation. Swing equations can be expressed as:

( ) ( )( ) dofdqq

em

b

TEEE

MDPP+=

==

1

B. Exciter The IEEE Type-ST1 excitation system is considered in this

work. It can be described as:

( ) ( )2 1 ttoA

Afd

Afd VVT

KET

E +=

In equations (1) and (2): eP , qE and tV are related by the following equations:

( )( ) ( )

( ) ( )223

tLqqtLddqt

tLdddqq

tLqtLddqtLqqe

IxIxEV

IxxEEIIxxIEP

+=

+=

+=

Where, AK and AT are the gain and time constant of the exciter respectively; toV is the reference voltage. The terminal voltage tV can be expressed as:

( )tLddqtq

tLqqtd

tqtdt

IxEVIxV

VVV

=

=

+=

4

222

C. STATCOM As shown in Fig.1, The STATCOM consists of a three

phase gate turn-off (GTO) based voltage source converter (VSC) and a DC capacitor. The STATCOM model used in this study is founded well enough for the low frequency oscillation stability problem. The STATCOM is installed through a step-down transformer with a leakage reactance of SDTX . The voltage difference across the reactance produces active and reactive power exchange between the STATCOM and the transmission network. The STATCOM is one of the important FACTS devices and can be used for dynamic compensation of power systems to provide voltage support and stability improvement. The VSC generates a controllable AC voltage given by:

( )5 )sin(cos jcVcVV DCDCO +== For PWM inverter mkc = , where m is the modulation ratio defined by pulse width modulation (PWM), k is the ratio between the AC and DC voltage depending on the converter structure. DCV is the DC voltage, and is the phase defined by PWM. The magnitude and the phase of oV can be controlled through m and respectively. By adjusting the STATCOM

AC voltage oV , the active and reactive power exchange between the STATCOM and the power system can be controlled. Capacitor voltage dynamic has big influences on power system, so capacitor voltage dynamic should be considered. If converter is assumed to be lossless, the exchanged active power between converter and system is equal to the active power that exchange among capacitor and converter ( ACDC PP = ). So with these assumptions the relationship between voltage and current of capacitor, it can be expressed as:

{ } ( )6 ))(sin(cos)( LoqLodDCLooDCDC jIIjcVrealIVrealIV +== Solving the above equation for DCI gives:

( )7 )sincos( LoqLodDC IIcI += With attention to (7) and the equation between voltage and current of capacitor, we have:

( )8 )sincos( LoqLodDC

DC IICcV +=

Where, DCC is the capacitor voltage, DCI is the capacitor current and LodI , LoqI are d and q axis of STATCOM current. From (8) is used in part IV for modeling of capacitor voltage dynamic on the Heffron-Phillips modified model. C.1. STATCOM control system

As shown in Fig.1, the converter and step down transformer in STATCOM can be modeled with a voltage source and a reactance for an operating point. Changing modulation ratio can change the amplitude of the output voltage of converter and so the active power absorbed by the system. By changing the converter voltage angle, reactive power exchanging with system can be controlled. In this study, the P-I controllers are used for voltage regulation.

i. Terminal voltage controller AC voltage controller regulates the voltage of terminal

according to requested reference that it accomplishes through changing of converter output voltage magnitude. cu is input signal for auxiliary damping stabilizer. The terminal voltage controller is introduced in Fig.2.The proposed damping stabilizer of STATCOM is shown in Fig. 3. This stabilizer has a structure similar to PSS. In this stabilizer and cU are the stabilizer input and output signals respectively. This stabilizer is used to create an additional damping signal for STATCOM.

sKIKP ACAC +

refLV

LV

cu

maxVACu

minVACu

c

0c

cF

F

sTK+1

controllerPI dynamicConverter+

++

Fig. 2. AC voltage controller

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c

c

sTsT

2

1

11

++

cACKc

c

sTsT

4

3

11

++

maxcu

mincu

cuW

W

sTsT+1

GainrCompensatolaglead /

Fig. 3. AC-damping stabilizer

ii. Capacitor voltage controller

The inner DC voltage controller regulates the voltage of capacitor. The converter phase angle is calculated according to capacitor voltage of reference with capacitor voltage. The capacitor voltage controller is introduced in Fig.4.

+

controllerPI dynamicConverter

sKIKP DCDC +

F

F

sTK+1

refDCV

DCV maxVDCu

minVDCu

0

++

Fig. 4. Capacitor voltage controller

III. LINEARIZED MODEL

A. Currents Calculation For creating power system linearized model should be calculated the output currents of Generator and STATCOM. In here, the output currents of Generator and STATCOM are calculated respectively:

i. Generator Current : With attention to Fig.1 , It can be written :

( )9 SDT

OtLtLttL

SDT

OLtLLOtLLB jx

VIjxVIjx

VVIIII ===

( )10 BLBLBtLtLt VIjxIjxV ++= The generator current ( tLI ), in d-q is expressed as:

( )11 1

sincos1

dSDT

LB

SDT

LBtLLBtL

DCSDT

LBBq

SDT

LB

tLd

xxx

xxxxx

cVxxVE

xx

I

++++

+

=

( )12 1

cossin

qSDT

LB

SDT

LBtLLBtL

DCSDT

LBB

tLq

xxx

xxxxx

cVxx

VI

++++

+=

ii. STATCOM Current : With attention in Fig.1 , the LoI can be calculated to this form :

( )13 SDT

OtLtLtLO jx

VIjxVI =

The STATCOM current ( LoI ), in d-q reference is expressed as:

( )14 sinSDT

DCtLd

SDT

tLd

SDT

qLOd x

cVIx

xxxE

I +

=

( )15 cos tLqSDT

tLq

SDT

DCLOq Ix

xxx

cVI+

=

B. Heffron-Phillips Modified Model The power system linearized model with STATCOM is

introduced in Fig.5. The operating point of system changes with STATCOM connection to power system. Accordingly in this state, the model should be linearized in new operating

point. In the design of stabilizer parameters, the linearized model around operating point is usually employed. With linearizing (3) and using (11), (12),(14) and (15), the differential equations (16) can be calculated:

( )

++++=

++++=

++++=

++++=

ddcDCqDC

vvcDCvDCqt

qqcDCqDCqq

ppcDCpDCqe

KcKVKEKKV

KcKVKEKKVKcKVKEKKEKcKVKEKKP

987

65

34

21 16

With using equations (16) and to considerate, inputs

( MT , refV ) and converter controller signals ( c , ), the state equation of power system can be extended to form of below:

( )

00

0

00

10

0000

17

00

10

10

0

0000

987

65

34

21

+

=

cT

V

KKTKK

TKK

TK

TK

TK

MK

MK

M

VEE

KKKTKK

TTKK

TKK

MK

TTK

TK

MK

MK

MD

MK

VEE

m

ref

ddc

A

vA

A

vcA

A

A

dpo

q

dpo

qc

ppc

DC

fd

q

A

vDCA

AA

A

A

A

qDC

dpodpodpo

pDC

b

DC

fd

q

Using vector representation, the above equation can be written as: uBxAx += The system has 5 state variables. 4 new state variables are added to system with consideration of STATCOM controllers. Three new state variables are added to the system with consideration of damping stabilizer of STATCOM, so the system has 12 state variables totally. The AC damping stabilizer and STATCOM internal controller parameters should be designed so that the eigenvalues of system move to the left part of the j axis as much as possible and the small signal stability of system has been improved. In Table I the related state variables of power system is introduced.

Fig. 5.The Power System Linearized Model with STATCOM

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TABLE I CLASSIFFICATION OF SYSTEM STATE VARIABLES

uBxAx xx += The State variables : [ ]Tfdq EEx = 1 Generator and Exciter [ ]TDCVx = 2 Capacitor Dynamic [ ]Tc xcxx = 3 STATCOM Internal Voltage Controllers [ ]Tccc uxxx = 214 AC Damping Stabilizer [ ] 1124321 = xxxxx Total of State Variables

C. Initial Conditions Calculation Considering the operating point of system, the initial

conditions can be calculated. Initial conditions are used for calculating the coefficients of the Heffron-Phillips modified model. In this study is used power flow for calculation of initial conditions [7]. MATLAB program is used for the initial conditions calculation [8]. In here the generator bus is considered as a PV bus. These studies are performed for 5 operating points. The studies are performed without STATCOM and with STATCOM in 5 operating points. The amount of Q in each operating point is calculated with Power Flow program. The considered operating point and the results of power flow have been shown in Table II. Also the calculated initial conditions with STATCOM are shown in Fig.6.

Calculated Initial Conditions

0.11872

0.23023

0.329180.373 0.37352

0.99293 0.97314 0.944270.98152

1.0877

0.029708

0.11553

0.24892

0.50894

0.86449

0.19787

0.38371

0.548640.62166 0.62253

1.0018 1.0078 1.0189

1.1342

1.347

0.23908

0.47289

0.69741

0.836140.89942

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 2 3 4 5Operating Point

Valu

e (P

U)

Vtd Vtq ItLd ItLq E'q delta

Fig.6. Calculated Initial Conditions in Operating Points (with STATCOM)

TABLE II POWER FLOW RESULTS WITHOUT STATCOM AND WITH STATCOM

Operating Point Parameters 1 2 3 4 5

tV 1 1 1 1.05 1.15 tP 0.2 0.4 0.6 0.8 1

( )STATCOMWithouttQ 0.012 0.048 0.112 0.281 0.569

( )STATCOMWithtQ 0.006 0.024 0.054 0.267 0.708 ( )STATCOMWithoV 1.002 1.007 1.016 1.003 0.967

( ) radSTATCOMWith 0.061 0.120 0.181 0.242 0.304

IV. PROBLEM FORMULATION

A. Objective Function The objective function consists of two individual objective

functions with weighed coefficients. The advantage of this function is consideration of all modes in stability evaluation. In other word, it can move the eigenvalues to the left part of j axis with parameters optimization. Also, the damping of oscillation mode can be increased. Flowchart of the proposed objective function is introduced in Fig.7. The objective function can be expressed in 3 parts:

i.In the first part, two coefficients 1 and 2 are defined

( 21 > ). If thi eigenvalue of system became as an electromechanical mode, the coefficient 1=K and else 2=K . The K is a weight coefficient that is used in part of ii and iii for Objective Function.

ii.In the second part, the real parts of eigenvalues are evaluated. The area in the left j axis is divided to 1n vertical parallel bands. The amount of 1n is calculated with (18). In here 01 =a , 51 =b and 1.01 =S are supposed. So if the real part of eigenvalues become more negative with parameters changing, the amount of 1J can be increased more.

iii. In the third part, the damping ratios of eigenvalues are

evaluated. Like to second part, the left area j axis is divided to 2n radial part. The amount of 2n is calculated with (18).In here 02 =a , 22 =b and 01.02 =S are supposed. To limit maximum overshoot, the parameters of the stabilizer may be selected with evaluation of 2J .

( )18 1,2i ==i

iii s

abn

iv. The single objective problem described maybe converted to

a multiple objective problem by assigning distinct weights to each objective [9]. With attention to flowchart in Fig.7 the objective function consists of two individual objective functions with weighted coefficient w, so a unified objective Function J is created. . The parameters of stabilizer may be selected to minimize the following objective function :

( ) ( )19 21 wJJJ +=

In Fig.8 the considered area for eigenvalues evaluation by objective function has been shown. As shown in Fig.8 the left area of j axis is divided to small areas. With this procedure we can allocate a mark for each eigenvalue with attention to its place in j plane. So the objective function can evaluate stability easily.

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Fig.7 .Flow chart of proposed Objective Function

Fig.8.The S-plane divided regions via objective Function

B. Optimization Problem In this study, the proposed objective function J is minimized. The problem constraints are the stabilizer optimized parameter bounds. Therefore, the design problem can be formulated as the following optimization problem:

{ } ( )

maxmin

maxmin

20 :

xxx

xxx

TTTKKK

JMinimize

The proposed approach employs genetic algorithm to solve this optimization problem and search for optimal set of STATCOM parameters.

C. Electromechanical Mode Identification The state equations of the linearized model can be used for

eigenvalues calculation of matrix A. Out of these eigenvalues, there is a mode of oscillation related to machine inertia. To ensure an effective operation of the stabilizers, it is extremely important to identify the eigenvalues associated with the electromechanical mode. In this study, the participation factors method [10] is used.

D. Genetic Algorithm The genetic algorithm is used for optimization of system

parameters. This algorithm has been programmed in MATLAB software based on continuous search algorithm [11]. The algorithm considers many search points simultaneously, so the probability of converging to local minimums is decreased. For all of operating points the Generation and Population are considered, 50 and 1000 respectively. Also Crossover and Mutation are considered, 0.6 and 0.02 respectively.

V. RESULT AND DISCUSSION

A. Setting of Parameters In papers the analysis and assessment of STATCOM for power system stability has been studied [12]. In here the coordinated design of AC-damping stabilizer and internal PI voltage controllers of STATCOM is performed and the results illustrate that the power system stability have been improved. The proposed approach has been implemented on a weakly connected power system as shown in Fig.1. The detailed data of the power system used in this study is given in Appendix. With attention to block diagram in Fig.4, the damping stabilizer has 6 parameters. The parameters of cT2 and cT4 are considered equal to 10 and only cT1 , cT3 and cACK are optimized. The STATCOM internal AC and DC voltage controllers have 6 parameters. The parameters of FK and FT have been supposed before and only 4 parameters are optimized in the STATCOM internal AC and DC voltage controllers. In this study, design of AC damping stabilizer and the STATCOM voltage controller parameters is accomplished in coordinated form. The final setting of the optimized parameters for five operating points has been given in Table III. The eigenvalues of electromechanical oscillation mode for each operating point have been shown in Table IV.

TABLE III OPTIMAL PARAMETER SETTING

Operating Point Parameters 1 2 3 4 5

DCKP 4.26 4.03 3.73 3.51 3.69 DCKI 86.68 69.48 43.17 22.39 42.52 ACKP 7.51 13.59 4.89 3.82 5.20 ACKI 45.77 76.42 34.52 32.51 40.56

cACK 4.73 8.34 11.21 7.39 8.79

cT1 2.91 1.32 1.25 3.73 2.35

cT3 4.93 6.45 7.89 1.33 9.23

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TABLE IV EIGENVALUES OF SYSTEM

Without STATCOM With STATCOM Operating

Point Oscillation

Mode (EM) Oscillation

Mode (EM)

1 -0.166 7.249i 0.023 -0.770 7.535i 0.102 2 -0.161 7.271i 0.022 -0.818 6.531i 0.124 3 -0.142 7.242i 0.019 -1.004 4.732i 0.207 4 -0.115 7.419i 0.015 -1.261 3.012i 0.386 5 -0.098 7.812i 0.012 -1.267 2.072i 0.522

B. Nonlinear Simulation Results Nonlinear simulation has been used for checking the design

accuracy. This work has been accomplished with a disturbance creation in system. In here, a 10% pulse disturbance in input mechanical torque is considered. The simulation time equal to 5 seconds is considered. The results in two cases, Without STATCOM and with STATCOM for five operating points have been shown in Fig.9 to Fig.16. Case i. Without STATCOM: The time responses of variables ( , tV and eP ) for a disturbance in mechanical torque have been introduced in Fig.9 to Fig.11. In here, with attention to Table IV, the amounts of (damping ratio) and electromechanical oscillation frequency can be calculated. For example in last operating point the oscillation frequency is equal to 1.243 Hz and is equal to 0.012. Also we observe with increase in output active power generator (changing operating point), the amount of is decreased. Because amount of power angle is increased with active power increasing .So Power System stability is decreased. Case ii. With STATCOM: The system response ( , ,

tV , DCV and eP ) for a Pulse disturbance in mechanical torque have been introduced in Fig.12 to Fig.16. In here, like to previous case, with attention to Table IV, the amounts of (damping ratio) and electromechanical oscillations frequency can be calculated. For example in last operating point the oscillation frequency is equal to 0.329 Hz and is equal to 0.522. In comparison to case of without STATCOM, we can observe the amount of is increased and oscillation frequency is decreased. So the STATCOM increases power system stability, but the oscillation frequency is changed with STATCOM. Also we can observe with increase in output active power generator, the amount of is increased because the amount of power angle has been increased with increase of active power .But the STATCOM can control the amount of output reactive power of generator. So in this case with attention to increase of power angle the amount of generator output reactive power is controlled with STATCOM. In result, the dynamic stability is improved. These results with comparison two cases can be observed.

Fig.9. Time response of

Fig.10. Time response of

tV

Fig.11. Time response of

eP

Fig.12. Time response of

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Fig.13. Time response of

Fig.14. Time response of

tV

Fig.15. Time response of

eP

Fig.16. Time response of

DCV

VI. CONCLUSION In this study, the coordination among STATCOM internal

AC and DC voltage controller and AC-damping stabilizer was presented and discussed for power system dynamic stability improvement. The design problem was formulated as an optimization problem and the GA was used for searching optimized parameters. The results of coordinated design show dynamic stability improvement. Also In second case with nonlinear simulation has been shown that the oscillations are damped properly.

VII. APPENDIX

TABLE V THE POWER SYSTEM DATA RELATING TO FIG.1

POWER SYSTEM GENERATOR EXCITER LINE

M

D do

T dx

dx

qx

AK

AT

tLx

LBx

6 2 5.044 1 0.3 0.6 10 0.1 0.3 0.3

STATCOM CAPACITOR,TRANSFORMER CONVERTER STABILIZER

DCC

DCV

SDTX

c

FK

FT WT

cT2 cT4

1 1 0.15 Power flow 1 0.05 5 10 10

VIII. REFERENCES [1] P. Kundur, M. Klein, G. J. Rogers, and M. S. Zywno, "Application of

Power System Stabilizers for Enhancement of Overall System Stability" , IEEE Trans.PWRS, Vol. 4, No. 2, 1989, pp. 614-626.

[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. PWRS, Vol. 18, No. 2, 1993, pp. 786-792.

[3] H.F.Wang , "The phase compensation method to design FACTS-based stabilizer. Part I:single-machine infinite-bus power systems" , Adv. Model. Anal., 1998.

[4] H.F.Wang , " Applications of damping torque analysis to STATCOM control " , Elsevier, Electrical Power and Energy Systems 22 (2000) 197204.

[5] H.F.Wang , "Interactions and Multivariable Design of STATCOM AC and DC Voltage Control" , Elsevier , Electrical Power and Energy Systems, Vol. 25, 2003, pp. 387-394.

[6] H.F.Wang , "Phillips-Heffron model of power systems installed with STATCOM and applications" , IEE Proc-Gener. Transm. Distrib. , Vol. 146, No. 5, September 1999 , pp.521-527.

[7] E.Acha , C.R. Fuerte-Esquivel , H.Ambriz-Perez , C.Angeles-Camacho , "FACTS: Modelling and Simulation in Power Networks " , Chapter 5 , John Wiley & Sons Ltd , Copyright 2004.

[8] Y.N.Yu , "Electric Power System Dynamics", Chapter 3 , Academic Press, 1983.

[9] Y.L.Abdel-Magid , M.A.Abido , "Optimal Multiobjective Design of Robust Power System Stabilizers Using Genetic Algorithms", IEEE Trans. on Power Systems, VOL. 18, NO. 3, AUGUST 2003 , pp. 1125-1132.

[10] P.Kundur , "Power System Stability and Control" , Part 3: Chapter 12 , New York , USA , Mc-Graw-Hill , 1994.

[11] Randy L. Haupt , Sue Ellen Haupt , "Practical Genetic Algorithms", chapter 2 , second edition , Copyright 2004 by John Wiley & Sons.

[12] M.A.Abido , "Analysis and assessment of STATCOM-based damping stabilizers for power system stability enhancement" , Elsevier , Electric Power Systems Research 73 , 2005, pp.177185.

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