This thesis is presented for the degree of Doctor of Philosophy of The University of Western Australia
Small-Signal Stability Analysis and Wide-Area
Damping Control for Complex Power Systems
Integrated with Renewable Energy Sources
Tat Kei Chau
B.Eng.(Hons), M.Eng.
2019
Power And Clean Energy (PACE) Research Group
Department of Electrical, Electronic and Computer Engineering
School of Engineering
Supervisors:
Prof. Herbert Ho-Ching Iu
Prof. Tyrone Fernando
Prof. Michael Small
Thesis Declaration
I, Tat Kei Chau, certify that:
This thesis has been substantially accomplished during enrollment in this degree.
This thesis does not contain material which has been submitted for the award of any other degree or
diploma in my name, in any university or other tertiary institution.
In the future, no part of this thesis will be used in a submission in my name, for any other degree or
diploma in any university or other tertiary institution without the prior approval of The University of
Western Australia and where applicable, any partner institution responsible for the joint-award of this
degree.
This thesis does not contain any material previously published or written by another person, except
where due reference has been made in the text and, where relevant, in the Authorship Declaration that
follows.
This thesis does not violate or infringe any copyright, trademark, patent, or other rights whatsoever of
any person.
This thesis contains published work and/or work prepared for publication, some of which has been
co-authored.
I acknowledge the support I have received for my research through the provision of the scholarship
funded by the University Western Australia.
Signature:
Date: 24 June 2019
i
Abstract
This thesis presents extensive studies on the dynamic behavior and small-signal stability analysis of
renewable-energy-integrated power systems and the design of wide-area damping controller (WADC)
to mitigate low-frequency oscillations (LFOs) in interconnected power systems. Based on the results
generated from small-signal stability analysis, optimal control parameters are obtained using heuristic
optimization techniques. State-of-the-art forecasting techniques are utilized to estimate the load
demand of the system based on weather information and historical data. Wide-area measurements
acquired by phasor measurement units (PMUs) are employed as input signals of the designed WADC.
The PMU measurements, or synchro-phasors transmitted to the WADC are inherently subject
to noise, measurement error, and other network-related issues such as communication delay, packet
dropout and disorder, which can lead to degradation of the integrity of the measured data. Therefore, a
measurement rectification methodology is proposed in this thesis to tackle the aforementioned problems
in order to improve the quality of the received wide-area signals so as to produce more preferable
LFO mitigation outcomes. In the proposed method, stochastic filtering algorithms are used to remove
unwanted bad data from the PMU measurements and adaptively compensate the communication
delay induced by the networks. Unscented-Kalman-filtering(UKF)-based dynamic state estimators
are used to detect and remove bad data in a decentralized manner, which require only local generator
parameters and local PMU measurements.
In order to demonstrate the performance and the applicability of the proposed methodologies,
realistic IEEE benchmark system models are employed in this thesis to conduct simulation studies.
The adoption of the realistic system model enable this research study to have both academic and
industrial value. The stability analysis results effectively reflect the LFO modes existing in the system,
facilitating the design of WADC. Using the proposed measurement rectification method, the wide-area
input signals with improved quality are obtained and utilized to construct wide-area control signal
using conventional PI-controllers.
ii
Acknowledgments
I would like to express my sincere gratitude to my supervisors Prof. Herbert Iu, Prof. Tyrone Fernando
and Prof. Michael Small for their continuous support of my research project, for their patience,
motivation, expertise and understanding. Their guidance assisted me throughout my Ph.D study.
I would like to thank my fellow researchers in the Power And Clean Energy (PACE) research group,
especially Dr. Samson Yu, Mr. Hunter Guo, Mr. Nestor Vazquez and Mr. Jason Eshraghian, for their
stimulating and focusing discussions, and for their help and support.
I would also like to give special thanks to the staff of the Department of Electrical, Electronics and
Computer Engineering (EECE), the Faculty of Engineering and Mathematic Sciences (EMS), for their
assistance in providing necessary equipment and devices that facilitate my research project.
To my beloved wife Hiu Ling, my beloved twin sons, Brendan and Owen, and my unborn child, I am
grateful for having you all in my life.
This work was supported by the following research funds:
UWA Scholarship for International Research Fees,
CSIRO Chair Studentship, and
Australian Research Council Discovery Project Grant (DP170104426).
iii
AUTHORSHIP DECLARATION: CO-AUTHORED PUBLICATIONS
This thesis contains work that has been published and prepared for publication during my PhD. The
content of the publications are edited and rearranged to achieve high consistency and coherence.
Details of war k 1: Tat Kei Chau, Samson Shenglong Yu, Tyrone Fernando, Herbert Ho-Ching Iu and Michael Small,
"A Load-Forecasting-Based Adaptive Parameter Optimization Strategy of STATCOM Using ANNs for
Enhancement of LFOD in Power Systems", IEEE Transactions on Industrial Informatics, vol.14, no.6,
pp.246:3-2472, 2018. ERA ranking: A
Location in thesis: Chapter 2
Student contribution to work:
Co-author signatures and dates
Details of work 2: Tat Kei Chau, Samson Shenglong Yu, Tyrone Fernando, Michael Small and Herbert Ho-Ching Iu.
"A Novel Control Strategy of DFIG \,Vind Turbines in Complex Power Systems for Enhancement of
Primary Frequency Response ancl LFOD", IEEE Transactions on Power Systems, vol.33, no.2,
pp.1811-1823, 2018. ERA ranking: A*
Location in thesis: Clmpkr ;)
Stnclcnt contribution to work: 8
Co-author sig1rnt11rcs aud dates:
Details of work 3: Samson Shenglong Yu, Tat Kei Chau, Tyrone Fernando, and Herbert Ho-Ching Iu. "An Enhanced
Adaptive Phasor Power Oscillation Damping Approach with Latency Compensation for :t\Iodern Power
Systems". IEEE Transactions on Power Systems, vol.33, no.4, pp.4285-4296, 2018. ERA ranking:
A*
Location in thesis: Chapter 4
Student contribution to work: 40
Co-author signatures and dates:
. Details of work 4: Tat Kei Chau, Samson Shenglong Yu, Tyrone Fernando, Herbert Ho-Ching Iu, Michael Small and
Mark Reynolds, "An Adaptive-Phasor Approach to PMU Measurement Rectification for LFOD
Enhancement", IEEE Transactions on Power Systems, DOI: 10.1109/TPWRS.2019.2907646, 2019.
ERA ranking: A*
Location in thesis: Chapter 5
Student contribution to work:
Co-author signatures and date
·Details of work 5: Tat Kei Chau, Samson Shenglong Yu, Tyrone Fernando, Herbert Ho-Ching Iu and ?viichael Small,
"An Investigation of the Impact of PV Penetration and BESS Capacity on Islanded Microgrids-A
Small-Signal Based Analytical Approach", presented at the 20th IEEE International Conference on
Industrial Technology (ICIT2019), Melbourne, Australia, 2019.
Location in thesis: Chapter 6
St11clc11t contribution to work: 8
Co-,mthor signatures and elates:
V
24-06-2019
24-06-2019
Tat Kei Chau
Date:
Coordinating supervisor signature:
I, Herbert Ho-Ching Iu, certify that the student statements regarding their contribution to each of the
works listed above are correct.
Herbert Ho-Ching Iu
Date:
Vl
OTHER RESEARCH ARTICLES DURING SINCE 2016
6. Accepted Nestor Vazquez, Samson Shenglong Yu, Tatkei Chau, Tyrone Fernando, Herbert
Ho-Ching Iu “A P&O Approach to Decentralized Power Loss Minimization in AC Microgrids”, 2019
IEEE International Symposium on Industrial Electronics, Vancouver, 2019.
7. Published Nestor Vazquez, Samson Shenglong Yu, Tatkei Chau, Tyrone Fernando, Herbert Ho-
Ching Iu “A Fully Decentralized Adaptive Droop Optimization Strategy for Power Loss Minimization
in Microgrids with PV-BESS”, IEEE Transactions on Energy Conversion, vol.34, no.1, 2019.
8. Published Tat Kei Chau, Samson Shenglong Yu, Tyrone Fernando, and Herbert Ho-Ching Iu.
“Demand-side regulation provision from industrial loads integrated with solar PV panels and energy
storage system for ancillary services”, IEEE Transactions on Industrial Informatics , vol.14, no.11 ,
pp.5038-5049, 2018.
9. Published Tat Kei Chau, Samson Shenglong Yu, Tyrone Fernando and Herbert Ho-Ching Iu,
“An Adaptive Optimization Method for LFOD Enhancement in DFIG Integrated Smart Grids”, 2018
IEEE International Symposium on Circuits and Systems (ISCAS), DOI: 10.1109/ISCAS.2018.8351473,
2018.
10. Published Gonzalez, Ander, Ramon Lopez-Erauskin, Johan Gyselinck, Tat Kei Chau, Herbert
Ho-Ching Iu and Tyrone Fernando, “Nonlinear MIMO control of interleaved three-port boost converter
by means of state-feedback linearization”, 2018 IEEE 18th International Power Electronics and Motion
Control Conference (PEMC), 2018.
11. Published Samson Shenglong Yu, Tyrone Fernando, Tat Kei Chau and Herbert Ho-Ching Iu.
“Voltage Control Strategies for Solid Oxide Fuel Cell Energy System Connected to Complex Power Grids
Using Dynamic State Estimation and STATCOM”, IEEE Transactions on Power Systems, vol.32, no.4
, pp.3136-3145, 2017. ERA ranking: A*
12. Published Samson Shenglong Yu, Tat Kei Chau, Tyrone Fernando, Andrey V. Savkin
and Herbert Ho-Ching Iu. “Novel Quasi-Decentralized SMC-Based Frequency and Voltage Stability
Enhancement Strategies using Valve Position Control and FACTS Device”, IEEE Access, vol.5,
pp.946-955, 2016.
vii
List of Figures
Figure 2.1 STATCOM schematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Figure 2.2 STATCOM AC controller and DC controller . . . . . . . . . . . . . . . . . . . 8
Figure 2.3 STATCOM supplementary damping controller . . . . . . . . . . . . . . . . . . 10
Figure 2.4 Parameters tuning procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Figure 2.5 Multi-layer neural network architecture . . . . . . . . . . . . . . . . . . . . . . 12
Figure 2.6 Modified IEEE standard 16-generator, 68-bus power system with STATCOM 15
Figure 2.7 Predicted data and linear regression (R2 = 0.9850) . . . . . . . . . . . . . . . 15
Figure 2.8 Error histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Figure 2.9 7-day load forecasting result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Figure 2.10 Evolution of Gbest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Figure 2.11 Eigenvalues of the power system without and with proposed control strategy . 20
Figure 2.12 Rotor speed deviations of G10, G11, G12, and G13 . . . . . . . . . . . . . . . 21
Figure 3.1 DFIG connected to a multi-area interconnected power system . . . . . . . . . 26
Figure 3.2 Conventional grid side controller . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Figure 3.3 Conventional rotor side controller . . . . . . . . . . . . . . . . . . . . . . . . . 28
Figure 3.4 Proposed RSC in control strategy 1 . . . . . . . . . . . . . . . . . . . . . . . . 30
Figure 3.5 Proposed RSC in control strategy 2 . . . . . . . . . . . . . . . . . . . . . . . . 30
Figure 3.6 Modified IEEE 68-bus, 16-generator power system integrated with DFIG WTGs 33
Figure 3.7 System eigenvalues with untuned PSS and equal weightings in Case 1 and Case 2 35
Figure 3.8 System eigenvalues with optimized PSS and weightings in Case 1 and Case 2 35
Figure 3.9 Part of root loci with varying Kpss . . . . . . . . . . . . . . . . . . . . . . . . 36
Figure 3.10 Evolution of PSO algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Figure 3.11 Simulation results of case study 1 (1) . . . . . . . . . . . . . . . . . . . . . . . 38
Figure 3.12 Simulation results of case study 1 (2) . . . . . . . . . . . . . . . . . . . . . . . 39
Figure 3.13 Simulation results of case study 1 (3) . . . . . . . . . . . . . . . . . . . . . . . 39
Figure 3.14 Simulation results of case study 2(1) . . . . . . . . . . . . . . . . . . . . . . . 40
Figure 3.15 Simulation results of case study 2(2) . . . . . . . . . . . . . . . . . . . . . . . 40
Figure 3.16 Simulation results of case study 2(3) . . . . . . . . . . . . . . . . . . . . . . . 41
Figure 3.17 System eigenvalues without controller in Case 3 . . . . . . . . . . . . . . . . . 42
Figure 3.18 System eigenvalues with optimized controller in Case 3 Control Scenario 1 . . 43
Figure 3.19 System eigenvalues with optimized controller in Case 3 Control Scenario 2 . . 43
Figure 3.20 Simulation results of case study 3(1) . . . . . . . . . . . . . . . . . . . . . . . 44
Figure 3.21 Simulation results of case study 3(2) . . . . . . . . . . . . . . . . . . . . . . . 45
viii
LIST OF FIGURES
Figure 3.22 Migration of modes listed in TABLE 3.7 with varying Kpss . . . . . . . . . . . 46
Figure 4.1 DFIG with proposed EAPPOD controller connected to a multi-area intercon-
nected power system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Figure 4.2 Proposed EAPPOD method coupled with DFIG RSC structure . . . . . . . . 53
Figure 4.3 Flowchart of the proposed signal decomposition method . . . . . . . . . . . . 57
Figure 4.4 Modified IEEE 68-bus, 16-generator power system integrated with DFIG WTGs 59
Figure 4.5 Measured and received signal at control center in Case 1 . . . . . . . . . . . . 61
Figure 4.6 Reconstructed signal with two methods in Case 1 . . . . . . . . . . . . . . . . 62
Figure 4.7 Forgetting factor variations in Case 1 . . . . . . . . . . . . . . . . . . . . . . . 62
Figure 4.8 Compensated signal in Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Figure 4.9 Magnitude of the phasor component . . . . . . . . . . . . . . . . . . . . . . . 64
Figure 4.10 Control singal in Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Figure 4.11 Active power generated by DFIG in Case 1 . . . . . . . . . . . . . . . . . . . 65
Figure 4.12 Active power generated by G10 in Case 1 . . . . . . . . . . . . . . . . . . . . 65
Figure 4.13 Control performances for χ1 in Case 1 . . . . . . . . . . . . . . . . . . . . . . 66
Figure 4.14 Control performances for χ2 in Case 1 . . . . . . . . . . . . . . . . . . . . . . 66
Figure 4.15 Modified 2-area, 4-machine power system with DFIG WTGs . . . . . . . . . . 67
Figure 4.16 On-site measured signal and received signal in Case 2 . . . . . . . . . . . . . . 68
Figure 4.17 Compensated signal in Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Figure 4.18 Magnitude of the phasor component in Case 2 . . . . . . . . . . . . . . . . . . 68
Figure 4.19 Control performances for ∆ω1 in Case 2 . . . . . . . . . . . . . . . . . . . . . 69
Figure 4.20 Active power generated by DFIG with EAPPOD in Case 2 . . . . . . . . . . . 69
Figure 5.1 Overview of the proposed control mechanism . . . . . . . . . . . . . . . . . . . 74
Figure 5.2 Proposed adaptive phasor method for PMU data recovery . . . . . . . . . . . 82
Figure 5.3 Flowchart of the proposed signal decomposition and data restoration method 83
Figure 5.4 2-area 4-machine test system with proposed data rectification and LFOD
enhancement strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Figure 5.5 Probability density of communication latency between control center and G1,
G2 and G4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Figure 5.6 Eigenvalues of the linearized system model without control . . . . . . . . . . . 86
Figure 5.7 Evolution of the PSO algorithm for CDI minimization . . . . . . . . . . . . . 87
Figure 5.8 Decentralized UKF-based DSE and absolute values of normalized deviation
ratios G1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Figure 5.9 Noisy frequency deviation measurement of G1 . . . . . . . . . . . . . . . . . . 89
Figure 5.10 Oscillatory component of frequency deviation from G1 . . . . . . . . . . . . . 89
Figure 5.11 Frequency deviations of G1 and G3 with proposed LFOD enhancer . . . . . . 90
Figure 6.1 DG inverter schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Figure 6.2 f − p droop and v − q droop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Figure 6.3 Current and voltage controllers in DG inverter . . . . . . . . . . . . . . . . . . 96
Figure 6.4 PV-BESS based Virtual Synchronous Generator . . . . . . . . . . . . . . . . . 98
ix
LIST OF FIGURES
Figure 6.5 Multi-inverter microgrid with PV-BESS VSG . . . . . . . . . . . . . . . . . . 101
Figure 6.6 SSSA for the base case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Figure 6.7 Root loci the system eigenvalues when increasing virtual inertia constant of VSG 104
Figure 6.8 Root loci of the system eigenvalues when increasing damping coefficient of VSG 104
Figure 6.9 Root loci and zoom-in of the system eigenvalues when increasing Virtual inertia
constant and damping coefficient of VSG . . . . . . . . . . . . . . . . . . . . 105
Figure 6.10 Root loci and its zoom-in of the system eigenvalues when increasing solar PV
output power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Figure 6.11 Inverter frequency with different VICs . . . . . . . . . . . . . . . . . . . . . . 107
Figure 6.12 Inverter power with different VICs . . . . . . . . . . . . . . . . . . . . . . . . 107
Figure 6.13 Inverter frequency with different BESS capacities . . . . . . . . . . . . . . . . 108
Figure 6.14 Inverter power with different BESS capacities . . . . . . . . . . . . . . . . . . 108
x
List of Tables
2.1 Training data format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Original eigenvalues of interest relating to LFEOs . . . . . . . . . . . . . . . . . . . . . 18
2.3 PSO algorithm parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Control parameters used in the case study . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Original eigenvalues of interest relating to LFEOs without controller . . . . . . . . . . 34
3.2 Resultant eigenvalues of interest relating to LFEOs with optimized controller . . . . . 35
3.3 Important parameters in Case 1 and Case 2 . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 PSO setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5 Operating conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6 Critical mode and damping ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.7 Eigenvalues of interest relating to LFEOs without controller in Case 3 . . . . . . . . . 41
3.8 Important parameters in Case 3 Control Scenario 1 . . . . . . . . . . . . . . . . . . . . 42
3.9 Eigenvalues of interest relating to LFEOs with the proposed controller (Case 3, Scenario 1) 42
3.10 Important parameters in Case 3, Control Scenario 2 . . . . . . . . . . . . . . . . . . . 43
3.11 Eigenvalues of interest relating to LFEOs with the proposed controller (Case 3, Scenario 2) 44
3.12 Eigenvalues of interest relating to LFEOs with the proposed controller in Case 3 Interaction 46
4.1 Important Parameters for Proposed Method in Case 1 . . . . . . . . . . . . . . . . . . 60
4.2 Original eigenvalues of interest relating to LFOs without controller in Case 1 . . . . . 61
4.3 Important Parameters for Proposed Method in Case 2 . . . . . . . . . . . . . . . . . . 67
4.4 Original eigenvalues of interest relating to LFOs without controller in Case 2 . . . . . 67
5.1 Original eigenvalues of interest relating to LFEOs . . . . . . . . . . . . . . . . . . . . . 86
5.2 Optimized parameters used in the proposed controller . . . . . . . . . . . . . . . . . . 86
6.1 Line parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2 Load and Generator Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.3 Inverter and LCL filter parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.4 Load flow of the microgrid when PPV6 = 1.5kW (base case) . . . . . . . . . . . . . . . 102
6.5 Load flow of the microgrid when PPV6 = 0.1W . . . . . . . . . . . . . . . . . . . . . . . 102
6.6 Eigenvalues of interest with base-case settings . . . . . . . . . . . . . . . . . . . . . . . 103
6.7 SSSA numerics with VIC J6 = 8s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.8 SSSA numerics with PBESSmax = 5kW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
xi
Contents
Declaration i
Abstract ii
Acknowledgments iii
Author Declaration iv
List of Figures viii
List of Tables xi
1 Introduction 1
1.1 Research Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.1 Low-Frequency Oscillation and Power System Stabilizer . . . . . . . . . . . . . 1
1.2.2 Phasor Measurement Units and Wide-Area Measurement System . . . . . . . . 2
1.2.3 Wind Power and Doubly-fed Induction Generator . . . . . . . . . . . . . . . . . 2
1.2.4 Inverter-Interfaced Microgrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Thesis Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Load-Forecasting-Based Adaptive Parameter Optimization for LFOD Enhance-
ment in Power Systems 5
2.1 Chapter Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Mathematical Models of Power System and STATCOM . . . . . . . . . . . . . . . . . 7
2.3.1 Power System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.2 STATCOM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.3 Supplementary Damping Controller for STATCOM . . . . . . . . . . . . . . . . 9
2.3.4 System Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Load-oriented Control Parameters Optimization . . . . . . . . . . . . . . . . . . . . . . 11
2.4.1 ANNs-based Machine Learning Technique . . . . . . . . . . . . . . . . . . . . . 11
2.4.2 PSO-based Parameter Optimization . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Simulation and Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5.1 Load Forecasting Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5.2 Modal Analysis of Inter-area Oscillations . . . . . . . . . . . . . . . . . . . . . 16
2.5.3 Online Parameter Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5.4 Control Performances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 A Novel Control Strategy of DFIG Wind Turbines in Complex Power Systems 23
xii
CONTENTS
3.1 Chapter Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Mathematical Model and Conventional Control Strategies of DFIG . . . . . . . . . . . 26
3.3.1 Wind Turbine and Drive Train . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.2 Asynchronous Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.3 Conventional Grid and Rotor Side Controllers . . . . . . . . . . . . . . . . . . . 27
3.3.4 Overall Power System Modeling and Linearization . . . . . . . . . . . . . . . . 28
3.4 Proposed DFIG Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.1 Tie-line Power Deviation Feedback . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.2 TLPD and DFIG-PSS with Optimized Parameters . . . . . . . . . . . . . . . . 30
3.4.3 Secondary Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5.1 Case 1: Load Increase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5.2 Case 2: Load Decrease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5.3 Case 3: New Power System Configuration . . . . . . . . . . . . . . . . . . . . . 41
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 An Enhanced APPOD Approach with Latency Compensation for Modern Power
Systems 49
4.1 Chapter Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Mathematical Model and Conventional Control Strategies of DFIG . . . . . . . . . . . 52
4.4 Proposed EAPPOD for DFIG-Integrated Power Systems . . . . . . . . . . . . . . . . 53
4.4.1 Signal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4.2 Adaptive Latency Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.3 Low-Frequency Oscillation Mitigation Mechanism . . . . . . . . . . . . . . . . . 58
4.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5.1 Case 1 : Modified IEEE New England 68-Bus, 10-Generator Test Power System 59
4.5.2 Case 2 : Modified Two-Area, Four-Generator Test Power System . . . . . . . . 66
4.5.3 Limitation of the Proposed Method . . . . . . . . . . . . . . . . . . . . . . . . 68
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 An Adaptive-Phasor Approach to PMU Measurement Rectification for LFOD
Enhancement 71
5.1 Chapter Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.3 The Overall Measurement Rectification and Control Strategy . . . . . . . . . . . . . . 74
5.4 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.4.1 Model for Decentralized DSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.4.2 Model for small-signal System Stability Analysis . . . . . . . . . . . . . . . . . 76
5.5 DSE-based Measurement Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.5.1 Unscented Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.5.2 Bad Data Detection and Elimination . . . . . . . . . . . . . . . . . . . . . . . . 79
xiii
CONTENTS
5.5.3 Measurement Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.6 AP-Based Data Recovery Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.6.1 Improved Signal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.7 LFOD Enhancer and Parameter Optimization . . . . . . . . . . . . . . . . . . . . . . . 82
5.7.1 LFOD Enhancement Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.7.2 Optimization of Weights and Control Parameters . . . . . . . . . . . . . . . . . 84
5.8 Case Study and Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.8.1 Simulation Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.8.2 SSSA and Parameters Optimization . . . . . . . . . . . . . . . . . . . . . . . . 86
5.8.3 DSE and Local Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.8.4 Signal Decomposition and Restoration . . . . . . . . . . . . . . . . . . . . . . . 87
5.8.5 LFOD Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6 A Stability Analysis of Inverter Interfaced Autonomous Microgrids Integrated
with PV-BESS and VSG 91
6.1 Chapter Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.3 Mathematical Models of an Islanded Multi-Inverter Microgrids . . . . . . . . . . . . . 93
6.3.1 DG Inverter and Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.3.2 LCL filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.3.3 Load modeling and Network Equations . . . . . . . . . . . . . . . . . . . . . . 97
6.4 PV-BESS based Virtual Synchronous Generator . . . . . . . . . . . . . . . . . . . . . 98
6.5 Small Signal Stability Analysis Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.5.1 System State-Space model and Linearization . . . . . . . . . . . . . . . . . . . 99
6.5.2 Proposed Power Flow Analysis for Islanded Microgrids with Secondary Controller100
6.6 Simulation and Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.6.1 Power flow analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.6.2 Small Signal Stability Analysis with Varying Parameters . . . . . . . . . . . . . 102
6.6.3 Time-Domain Simulation with Varying VIC and BESS capacity . . . . . . . . . 104
6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7 Conclusions and Future Work 109
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
xiv
Chapter 1
Introduction
1.1 Research Objective
The aim of this research is to address the small-signal stability challenges on modern power systems
posed by the increasing level of renewable energy penetration and the increased use of power electronic
inverter-interfaced microgrid systems with no/low system inertia. To improve the reliability and enhance
the small-signal stability of such power systems, analytical approaches, state estimation techniques
and associated control methods are investigated, including system linearizion, modal analysis, dynamic
state estimation as well as voltage and frequency control methods. An overview on the topics covered
in this thesis is given in the following section.
1.2 Thesis Overview
1.2.1 Low-Frequency Oscillation and Power System Stabilizer
Inter-area low-frequency oscillations (LFOs) are a common phenomenon that exists in multi-area
interconnected power systems, which is mainly caused by insufficient damping torque among generators
[1]. These oscillations can be triggered by a number of reasons such as line faults, switching of line or
abrupt generation/load changes. Despite the fact that such LFOs have a low oscillatory frequency,
typically between 0.2Hz to 2Hz, the oscillations can potentially cause damages to power systems and
even regional or large-scale power failure [2]. Power system stabilizers (PSSs) are widely adopted in
practical power systems for small-signal stability enhancement through introducing additional damping
torque to the system [3]. PSSs are usually used in conjunction with synchronous generators’ automatic
voltage regulator (AVR) [4] or flexible alternating current transmission systems (FACTs) devices [5, 6]
such as static var compensators (SVC) and thyristor controlled series capacitor (TCSC) [7]. Particularly,
PSS generates a control signal using real-time power system measurements, such as the rate of change
of frequency (RoCoF) of generators, to modulate the reference signal of the embedded controller (or
actuators) of AVR, SVC and TCSC to adaptively produce additional damping torque, hence improving
low-frequency oscillation damping (LFOD) of the power system. In the past decades, extensive research
work has be conducted on the small-signal stability of power systems, including new stability analytical
methods [8, 9], novel PSS designs and associated parameters tuning techniques [10–14], optimal PSS
1
CHAPTER 1. INTRODUCTION
placement methods [15,16], and other robust PSS control strategies [17–19], in order to improve the
damping performance and robustness of the power grid. The challenges of PSS designs and tunning
methods will be discussed in detail in Chapter 2 together with a review of their state-of-the-art
development.
1.2.2 Phasor Measurement Units and Wide-Area Measurement System
Phasor measurement units (PMUs) are widely used in power systems for real-time system monitoring,
stability analysis, power system dynamic state estimation, contingency studies, closed-loop control, etc.
PMUs can continuously report the fundamental frequency and magnitudes and phase angles of the
current and voltage in a power system with very high sampling rate up to 48 samples per cycle for AC
systems [20]. The phasor quantities obtained by PMUs are synchronized using a common time source
such as a clock signal broadcast from global positioning system (GPS) satellites [21], synchronizing
real-time wide-area measurements from geographically dispersed power system components. The
PMU measurements, or sychrophasors, are obtained and transmitted to a wide-area measurement
system (WAMS) through phasor data concentrators (PDCs) in compliance with the IEEE standards
C37.118.1-2011 and C37.118.2-2011, which define the specifications of each measured quantities, the
synchronization requirements, as well as the data formats for real-time communication between PMUs
and PDCs [22, 23]. According to these documents, PMU data can be transferred with any suitable
communication protocols, and packet-based network is mainly used in multi-area applications [24]. In
particular, PMUs are deployed in remote terminals in multi-area power systems to provide instant
information across the system, facilitating the design of various wide-area control strategies. These
control strategies are devised with PMU measurements to enhance the stability of large-scale power
networks in a centralized or decentralized manner [25,26]. Therefore, the accuracy and timeliness of
the PMU measurements are of critical importance in ensuring the performances for such controller
designs. With this backdrop, a new and pressing challenge in power system research has been created:
improving the quality of PMU measurements that are subjected to measurement errors and noises,
time-varying transmission delays, and data loss and disorder.
1.2.3 Wind Power and Doubly-fed Induction Generator
Wind power is a fast growing source of large-scale renewable energy which has been deployed for
electricity generation all around the world to address the energy crisis and environmental concerns. By
the end of 2018, the overall capacity of wind turbine deployment has reached 597 Gigawatts, which
is close to 6% of the global electricity demand [27]. Wind power is projected to be one of the future
energy sources due to its cleanliness and abundant nature. The early designs of wind power generation
system was developed primarily based on a fixed speed wind turbine system, which consists of a
standard squirrel-cage induction generator, coupled mechanically with a multi-stage gear box driven by
an aerodynamically controlled wind turbine and directly connected to the power grids through power
electronic converters [28]. However, such a design is no longer suitable for new developments due to the
stringent technical requirements for grid connection and the increased proportion of power generated by
wind in a power system. Therefore, variable speed wind energy conversion systems (WECS) have been
developed. With the advancement in power electronic converter technology, the variable speed WECS
2
1.3. THESIS CONTRIBUTION
has been implemented in new installations of wind power generation system which uses multiple voltage
source converters to fulfill the grid requirement by providing reactive power and frequency support in
addition to active power. The doubly-fed induction generator (DFIG) is one type of variable speed
WECS, which consists of a small-size back-to-back power converter in the rotor circuit. Numerous
previous research papers are dedicated to the development of DFIG models for power system stability
and dynamic behavior studies which make use of the mechanical models of the wind turbines and
the electromagnetic model of induction generators [29–31]. In order to improve the stability of the
power system, numerous control strategies are devised in the literature to provide additional damping
torque and frequency and voltage supports to power grids using DFIGs [32–36]. Another important
aspect of wind power research is to improve the poor low voltage ride-through (LVRT) capability [37]
of DFIG-based wind turbines during low voltage fault, which is not covered in this study.
1.2.4 Inverter-Interfaced Microgrid
The concept of microgrid was first introduced in [38], intending to address the challenges of reliable
integration of distribution energy resources (DERs) and controllable loads brought by the uptake
of renewable energy sources (RESs) as well as the advancement of energy storage system (ESS)
technologies. Individual microgrid is connected to the main grid via a point of common coupling
(PCC). Microgrids can also be operated in islanded mode when they are disconnected from the main
grid. While microgrids are viewed as a single entity from a power system’s perspective, all components
within a microgrid are coordinated for reliable electricity supply. The introduction of microgrids offers
a smooth transition from the traditional centralized power generation and lossy long-distance power
transmission pattern to a more efficient, smarter, and smaller distributed cluster [39], consisting of
localized DERs, controllable loads and energy storage systems. Since a large number of distributed
generators, such as solar photovoltaics (PV) panels, wind turbines, fuel cells, etc., are used in microgrids,
power electronics converters have become an essential component for interfacing each component in
the microgrid. In order to understand the dynamic behavior and system stability of inverter-interfaced
microgrids, extensive research has been conducted on microgrid modeling and stability analysis [40–43];
also, different control strategies [44,45] and energy management systems [46,47] have been proposed in
the literature aiming to maintain voltage and frequency stability of the microgrid and balance power
sharing among DGs. Similar to the control of traditional power grids, hierarchical architecture is
adopted in the control of microgrids. The control hierarchy of microgrids is divided into three levels:
primary, secondary and tertiary, each of which operates in a different timescale [48]. Existing microgrid
modeling techniques and control strategies will be discussed in details in Chapter 6.
1.3 Thesis Contribution
The main contributions of this thesis include:
Development of a simulation framework for small-signal stability analysis and dynamic behavior
study of complex power systems integrated with DFIG wind turbines and FACTs devices.
Development of load-forecasting-based parameter tuning method for power system stabilizers.
3
CHAPTER 1. INTRODUCTION
Control strategy design for enhancement of low-frequency oscillation damping of DFIG-integrated
complex power system using the rotor-side-controller embedded in DFIG control structure.
Design of PMU measurements-based power oscillation damping control strategy using an adaptive
phasor approach. A new algorithm is developed to handle network induced communication delay,
packet dropout, and packet disorder to improve the quality of the recovered PMU measurements
for damping performance improvement.
Development of the measurement rectification method to validate PMU measurements and
eliminate bad data using decentralized dynamic state estimation algorithms.
Mathematical reformulation of inverter-interfaced microgrids and modification of power flow
problem for such microgrids to facilitate small-signal stability studies, modal analysis, and
time-domain simulations.
1.4 Thesis Organization
This thesis contains chapters edited from a number of peer-reviewed journal publications completed
during my PhD period. The full publication list can be found in the Authorship Declaration section.
The organization of this thesis is given as follows.
Chapter 1 presents an introduction of the thesis, where its objective and contribution are stated,
with thesis overview presented.
In chapter 2, a novel load-oriented parameter optimization method is presented, where a PSO
algorithm is employed to optimize PSS parameters using forecasted load demand generated by ANNs.
The mathematical model of a doubly-fed induction generator is presented in Chapter 3. A novel
control method is proposed and verified using the model in order to improve the frequency response
and small-signal stability of the power system integrated with a DFIG-based wind farm. The proposed
method is validated by software simulations using the presented DFIG model together with the IEEE
68-bus test system model.
In Chapter 4, an enhanced adaptive phasor power oscillation damping controller is proposed to
mitigate the LFO in a multi-area power system using wide-area measurements obtained by PMUs. The
communication latency is compensated with the proposed method to achieve better LFO mitigation.
A DSE-based measurement rectification method is presented in Chapter 5, where decentralized
UKF filtering algorithms are used to eliminate bad data from PMU measurements. The rectified
measurements are then used for LFO enhancement with PI-controller-based PSSs.
Chapter 6 presents a small-signal stability analysis framework for islanded inverter-interfaced
microgrids.
Finally, a conclusion is drawn in Chapter 7, where the main findings of this thesis are stated and
possible future work is discussed.
4
Chapter 2
Load-Forecasting-Based Adaptive
Parameter Optimization for LFOD
Enhancement in Power Systems
ABSTRACT
This chapter presents a load-oriented control parameter optimization strategy for STATCOM to
enhance low-frequency oscillation damping (LFOD) and improve stability of overall complex power
systems. Frequency deviations of generators of interest are employed as the input signals of the designed
supplementary damping controller of STATCOM. In order to obtain the optimal load-oriented control
parameters, a day-ahead load-forecasting scheme is devised, using artificial neural network (ANN)
learning techniques. The ANN is trained by a set of data over a 4-year period, and then the control
parameters are optimized using particle swarm optimization (PSO) technique by minimizing the critical
damping index (CDI). The proposed control strategy is implemented in an IEEE standard complex
power system, and the numerical results demonstrate that the low-frequency oscillations (LFOs) of the
power system can be effectively mitigated using the proposed controller. Compared to conventional
robust controller with universal parameters, this novel load-oriented optimal control strategy shows
its superiority in alleviating LFOs and enhancing the overall stability of the power system. Since the
proposed control scheme aims to adaptively adjust the controller parameters in correspondence to load
variations, this study is envisaged to have practical utilizations in industrial applications.
5
CHAPTER 2. LOAD-FORECASTING-BASED ADAPTIVE PARAMETEROPTIMIZATION FOR LFOD ENHANCEMENT IN POWER SYSTEMS
2.1 Chapter Foreword
The content of this chapter is based on following academic paper:
Tat Kei Chau, Samson Shenglong Yu, Tyrone Fernando, Herbert Ho-Ching Iu and Michael Small,
“A Load-Forecasting-Based Adaptive Parameter Optimization Strategy of STATCOM Using ANNs for
Enhancement of LFOD in Power Systems”, IEEE Transactions on Industrial Informatics, vol.14, no.6,
pp.2463-2472, 2018.
In this chapter, a load-forecast-based PSS parameters optimization method is proposed in order
to enhance the low-frequency oscillation damping performance of a multi-machine power systems.
A modified IEEE 68-bus benchmark system is employed to verify the applicability of the proposed
parameter tuning method.
2.2 Introduction
Inter-area low-frequency oscillations (0.2 to 2 Hz), mainly caused by damping torque deficiency, are one
of the major concerns of inter-area power systems [49]. Numerous researchers have been dedicated to
designing novel control strategies to mitigate LFOs in interconnected multi-area power systems. In [15],
an optimal power system stabilizers (PSSs) tuning and placement strategy is proposed to improve the
damping ratio of the power system. In [12], a low-frequency oscillation damping enhancement scheme
is proposed using energy storage and PSO techniques. Low-frequency electromechanical oscillations
(LFEOs) are analyzed using a non-stationary synthetic signal in [8]. However, generic automatic voltage
regulator-PSSs or (AVR)-PSSs cannot cope with LFOs that are caused by non-electromechanical
factors of generators, and hence a popular member of flexible alternating current transmission system
(FACTS) device family, static synchronous compensator (STATCOM) is employed to improve the
power system LFO damping. In order to achieve LFOD enhancement, STATCOM is normally equipped
with a supplementary controller, which is constructed by a group of cascading lead-lag compensator
and a wash-out filter [9]. Conventionally, local signals are employed as the input of the supplementary
damping controller, which limits its efficiency on improving inter-area oscillation damping. To tackle
this issue, in [13], an adaptive LFOD controller is designed using STATCOM and energy storage.
Wide-area control signals are utilized to enhance damping control performances of inter-area LFOD,
which resolves the drawback of traditional PSSs where only local-area control signals are employed.
Furthermore, authors in [50] proposed a novel LFOD enhancement method for power systems with
doubly-fed induction generator (DFIG)-based wind power plants where the proposed control strategy
is integrated with DFIG control structure.
Ideally, the optimal control parameters should adapt in accordance with daily load demands in order
to generate the best control performances. Since future loads are always unknown, such load-oriented
control parameter optimization for STATCOM controller strategy has not yet been thoroughly studied
in literature. With the advent of artificial neural network learning technology, we now can train the
ANN regression model using historical available load data and utilize the trained model to predict future
load information [51]. ANN has a wide range of utilizations in both theoretical studies and industrial
applications. For instance, in [52], ANN has recently been used on FPGAs for back-propagation
learning. In [53], authors make use of ANNs to control shunt active power filter, and in [54], ANNs
6
2.3. MATHEMATICAL MODELS OF POWER SYSTEM AND STATCOM
are leveraged to enhance the stability of an adaptive speed-sensorless induction motor. Other recent
academic articles on industrial applications of ANNs are reported in [55, 56] and references therein.
Recent work on adaptive control methodologies can be found in [57] where an observer-based adaptive
fuzzy control was proposed for nonstrict-feedback stochastic nonlinear systems and [58] where a fuzzy
control scheme is devised for nonstrict feedback systems with unmodeled dynamics.
In most established applications, STATCOMs with PSS-like supplementary controllers are imple-
mented with parameters obtained with nominal loading conditions, where the robustness is tested,
see [10,11,14]. However the lack of considerations in load variations may impede the control mechanisms
to exert their best performances. To overcome this issue, in this chapter, we propose a novel control
parameter tuning and optimization strategy for STATCOM to enhance the stability of overall power
system, taking into account time-varying loading conditions. To predict the future unknown load data,
we employ the ANN machine-learning methodology to procure the load demands for the following day,
based on which control parameters are optimized and updated adaptively. The ANN model is retrained
when the actual load data arrive in order to prepare for the next forecasting process. The proposed
parameter tuning method is then implemented in an IEEE standard complex power system to validate
its functionality, and the simulation results indicate the superiority of the proposed optimization
method over the traditional one in producing more effective damping performances. The adaptivity of
the proposed parameter optimization strategy also overcomes the inflexibility and inaccuracy of the
traditional look-up table method. The novel parameter optimization method is also easy to implement,
and thus demonstrates its industrial potentials.
The rest of the chapter is organized as follows. In Section 2.3, the mathematical model of the
power system and STATCOM will be briefly discussed, followed by a concise discussion of system
linearization. Load-forecasting scheme and parameter tuning method will be presented in Section 2.4.
A case study is conducted in Section 2.5 where the performances of proposed controller with different
control parameters will be compared and analyzed.
2.3 Mathematical Models of Power System and STATCOM
2.3.1 Power System Model
In an N -bus, n-generator power system, suppose all synchronous generators have the same basic
dynamical characteristics. The differential equations for ith generator are shown in [59], wherein E′d,
E′q, Ψ1d, Ψ2q, δ, Ω, Efd, Rf and VR are identified as dynamic states.
2.3.2 STATCOM Model
STATCOM is a member of the FACTS device family, used to regulate the AC voltage by supplying or
absorbing reactive power. The mathematical model of STATCOM in [60] is adopted in this study and
briefly discussed as follows. The configuration of the STATCOM model, as shown in Fig. 2.1 consists
of a two-level voltage source converter (VSC), which is physically built with a two-level converter using
an array of PWM-driven self-commutating power electronic switches. It can be modeled as a Load
Tap Changing (LTC) transformer with its primary side connected to a small-rating capacitor bank and
7
CHAPTER 2. LOAD-FORECASTING-BASED ADAPTIVE PARAMETEROPTIMIZATION FOR LFOD ENHANCEMENT IN POWER SYSTEMS
the secondary winding connected to the AC terminal through a series-connected step-up transformer
impedance Zstats and we denote, 1/Zstats = Gstats + jBstats .
As shown in Fig. 2.1 (b), the magnitude of output voltage of VSC, V statac has the following
relationship with the DC side voltage V statdc ,
V statac = k mstat
a V statdc , (2.1)
where term mstata is the pulse width modulation index, αstat is the phase angle of the AC-side voltage
of the converter, and constant k =√
3/2 is directly proportional to the modulation index. The model
also includes a conductance Gstatsw to account for the switching losses on the DC side.
ControllerController
(a) (b)
Figure 2.1: STATCOM schematics
Fig. 2.2 shows the controller models of the STATCOM for the regulation of AC bus-bar voltage
and also the DC side voltage of the capacitor. AC bus voltage magnitude is controlled through the
modulation index mstata , and phase angle αstat determines the active power P that flows into the
converter, which consequently controls the DC voltage magnitude by charging and discharging the
capacitor. Both controllers are subject to converter current limits.
Supplementary damping controller
(a)
(b)
Figure 2.2: (a) STATCOM AC controller (b) STATCOM DC controller
The p.u. differential and algebraic equations (DAEs) describing the dynamics of the control
8
2.3. MATHEMATICAL MODELS OF POWER SYSTEM AND STATCOM
strategies and power balance of the STATCOM model are shown as follows [60],
mstat = Kpac(V′ref − Vbus) + xstatac , (2.2)
xstatac = Kiac(V′ref − Vbus), (2.3)
αstat = Kpdc(Vdcref − V statdc ) + xstatdc , (2.4)
xstatdc = Kidc(Vdcref − V statdc ), (2.5)
V statdc =
1
Cstatdc V statdc
(VbusIbus cos (θbus − γbus)
−Gstatsw (V statdc )2 − I2
busRs). (2.6)
0 =
P − VbusIbus cos (θbus − γbus)
Q− VbusIbus sin (θbus − γbus)
P − V 2busG
stats + kV stat
dc VbusGstats cos(θbus − αstat)
+kV statdc VbusB
stats sin(θbus − αstat)
Q+ V 2busB
stats − kV stat
dc VbusBstats cos(θbus − αstat)
+kV statdc VbusG
stats sin(θbus − αstat)
,
(2.7)
where Kpac , Kiac , Kpdc , Kidc are respectively the proportional and integral gains for the AC side and
DC side controllers of STATCOM, Vbus∠θbus, Ibus∠γbus are the voltage and current of the bus-bar to
which STATCOM is connected, and xstatac and xstatdc are intermediate variables. It should be noted that
in Fig. 2.2 (a), a supplementary controller is designed and incorporated to the generic STATCOM AC
side controller model, Vref is modulated by the input signal acquired from the supplementary control
scheme Vsup , and becomes V ′ref , which follows the relationship as shown below,
V ′ref = Vref + Vsup. (2.8)
The design of this controller and its parameter tuning will be discussed in the next subsection.
2.3.3 Supplementary Damping Controller for STATCOM
A supplementary controller is incorporated to the STATCOM model to enhance the damping per-
formance for the power system [51]. The supplementary controller utilized in this study is a 2nd
order lead-lag type controller as shown in Fig. 2.3, where Ksup is the controller gain, TW is the
wash-out time constant and T11, T12, T21 and T22 are the lead-lag time constants. We introduce
a set Υ = [υ1, υ2, · · · , υν ] to represent the frequency deviation signals of the generators of interest.
The selected generator bus-bars have the highest participation factor to critical modes, which can be
obtained based on eigenvalue sensitivity analysis, and thus υj = ∆Ωj , j ∈ 1, 2, · · · ν, where ν is the
number of generators of interest. We introduce another set W = [w1, w2, · · · , wν ] consisting of the
weight of each input signal. The weighted sum of the frequency deviation signals of the generators of
interest Λ, is employed as the input signal to the supplementary damping controller of STATCOM.
9
CHAPTER 2. LOAD-FORECASTING-BASED ADAPTIVE PARAMETEROPTIMIZATION FOR LFOD ENHANCEMENT IN POWER SYSTEMS
Wash outLead-lag
Figure 2.3: STATCOM supplementary damping controller
The following equations are used to describe the dynamics inside the STATCOM supplementary
damping controller,
Λ =ν∑j=1
wjυj , (2.9)
x1sup =
1
TW(KsupΛ− xsup1 ), (2.10)
x2sup =
1
T12(KsupΛ− xsup1 − xsup2 ), (2.11)
x3sup =
1
T12T22
(KsupT11Λ− T11x
sup1 (2.12)
− (T11 − T12)xsup2 − T12xsup3
), (2.13)
Vsup =1
T12T22
(KsupT11T21Λ− T11T21x
sup1
+ (T12 − T11)T21xsup2 + (T22 − T21)T12x
sup3
), (2.14)
where xsup1 , xsup2 , and xsup3 are intermediate states, Vsup is the output supplementary voltage signal to
be sent to the PSS, and the rest are time constants and the PSS gain. In the STATCOM with the
proposed supplementary controller, parameters Kpac , Kiac , wj , j ∈ 1, 2, · · · , ν, Ksup, T11, T12, T21
and T22 are to be tuned by the PSO algorithm, using the ANNs-based load-forecasting strategy in
Section 2.4.
2.3.4 System Linearization
In order to perform small signal analysis and wide-area power system stability study, system linearization
is required in this study. Due to the complexity of the power system in question, only a brief explanation
is provided here. Following the established models in 2.3.1 and 2.3.2, the overall STATCOM-integrated
power system can be written in the following compact form [59],
X = f(X,U),
0 = g(X,U), (2.15)
X =
[xgen
xstat
], U =
[ugen
ustat
], (2.16)
10
2.4. LOAD-ORIENTED CONTROL PARAMETERS OPTIMIZATION
where f(·) and g(·) are respectively the state differential and algebraic functions of the system model,
X is the system dynamic state vector consisting of generator state vector xgen and STATCOM state
vector xstat and U is the system input vector, comprised of generator input vector ugen and STATCOM
input vector ustat. The following equations demonstrate the elements in state and input vectors for an
n-generator, m-STATCOM power system,
xgen = [xgen1, xgen2
, · · · , xgenn]T ,
xgeni= [E′di,Ψ1di, E
′qi,Ψ2qi, δi,Ωi, Efdi, Rfi, VRi]
T , (2.17)
for i ∈ 1, 2, · · ·n,
xstat = [xstat1 , xstat2 , · · · , xstatm ]T ,
xstatl = [V statdcl
,mstatal
, xstatacl, αstatl , xstatdcl
, xsup1l, xsup2l
, xsup3l]T , (2.18)
for l ∈ 1, 2, · · ·m, where term l indicates the lth STATCOM, and xsup1l, xsup2l
and xsup3lare intermediate
variables in (2.10)-(2.12). The following equations illustrate the structure of the input vector,
ugen = [ugen1, ugen2
, · · · , ugenn]T ,
ugeni= [TMi, Vrefi]
T , (2.19)
for i ∈ 1, 2, · · ·n,
ustat = [ustat1 , ustat2 , · · · , ustatm ]T ,
ustatl = [V statrefl
, V statdcrefl
]T , (2.20)
for l ∈ 1, 2, · · ·m. For small signal stability analysis of the power system, the following linearized
system equation can be obtained [61],
∆X = A∆X +B∆U, (2.21)
where A and B are coefficients of the linearized system. For detailed steps and derivations of linearizing
nonlinear power systems, refer to [59,61].
2.4 Load-oriented Control Parameters Optimization
In order to obtain the optimized control parameters for STATCOM, a load-oriented parameter tuning
strategy is proposed and implemented in this study. Fig. 2.4 shows the overall parameter-tuning
procedure, which will be explained in this section.
2.4.1 ANNs-based Machine Learning Technique
ANN, a type of machine learning technique, used for non-parametric regression problems, is employed in
this study to forecast day-ahead load demands. The regression model can be used to make predictions
11
CHAPTER 2. LOAD-FORECASTING-BASED ADAPTIVE PARAMETEROPTIMIZATION FOR LFOD ENHANCEMENT IN POWER SYSTEMS
InputW
b+
W
b+ output
Hidden Layer Output Layer
PSO+ANNs Algorithmcomputation unit
ANNs
Test data
STATCOM‐based damping control
unit
Control parameters
Figure 2.4: Parameters tuning procedure
from the predictors with priori knowledge of the predictor-response (input-output) relationship from
the training data. During the training process, the learning algorithm attempts to find the best model
by minimizing the difference between the observed response and the predicted values. The collected
historical data are usually separated into two parts: most of the data are used for the training purpose
and a small portion of the data are used for model validation. In this study, the MATLAB® Neural
Network ToolboxTM is utilized to obtain the regression model for the load forecasting scheme. As
shown in Fig. 2.5, the ANN learning mechanism makes use of a multilayer feed-forward neural network
for load prediction, which consists of a hidden layers of sigmoid neurons followed by an output layer of
linear neurons [62], where details of ANN are presented. The training data for load forecasting consists
Input
W
b+
W
b+ output
Hidden Layer Output Layer
PSO+ANNs Algorithmcomputation unit
ANNs
Test data
STATCOM‐based damping control
unit
Control parameters
Sigmoid function
Linear function
Weighting + biasing
Weighting + biasing
Figure 2.5: Multi-layer neural network architecture
of historical load data, weather conditions and weekday/non-weekday/holiday [62] information. The
input signal of the neural network is comprised of eight factors, considered as predictors, as shown in
TABLE 2.1. In particular, “Drybulb” and “DewPoint” are the hourly temperature conditions, are
provided by a third-party, a meteorological observatory. The input “Hour” is the current time and
“Weekday” is the day of a week. “IsWorkingDay” is a boolean entry indicating whether a day is a
working day or a holiday. “PrevWeekSameHourLoad” and “PrevDaySameHourLoad” are the historical
load data of a chosen bus. “ActualLoad” is the output of the training mechanism, used as the response
in this fitting problem. Using historical predictor and response data, the training process attempts to
seek a set of optimal values for W and b in ANN by minimizing the mean square error (MSE). For
detailed explanations of the ANN-based nonlinear regression method, refer to [62] and [63].
Table 2.1: Training data format
Predictors ResponseDryBulb, DewPoint, Hour, Weekday, IsWorkingDay, PrevWeek-SameHourLoad, PrevDaySameHourLoad
ActualLoad
12
2.4. LOAD-ORIENTED CONTROL PARAMETERS OPTIMIZATION
2.4.2 PSO-based Parameter Optimization
Particle swarm optimization is a population-based stochastic optimization technique [64]. Initially, a
group of particles are randomly generated, which move around a given search space based on their
current positions and velocities. Then the fitness of each particle is evaluated by a particular objective
function. The velocity and position are updated with the following equations. The updated velocity
of pth particle in gth generation is denoted as velp(g), and position of pth particle in gth generation is
denoted as posp(g). Hence,
velp(g + 1) = τ · velp(g) + c1 · rand1 ·(Pbest(g)− posp(g)
)+ c2 · rand2 ·
(Gbest(g)− posp(g)
), (2.22)
posp(g + 1) = posp(g) + velp(g), (2.23)
where τ is the inertia weight. Inertia weight is a parameter that balances the global search (exploration)
and local search (exploitation) processes, and is normally set between 0.8 ∼ 1.2 to ensure the
lowest failure rates [65]. Terms c1 and c2 are acceleration factors and usually set to 1 ∼ 2, and
0 < rand1, rand2 < 1 are two positive random values. Term Pbest and Gbest indicate the best position
for a particular particle in the past generations and the global best position for the entire particle
swarm in the past generations, respectively.
In this particular study, the objective of the optimization is to minimize the critical damping index
(CDI), which is acquired using the following equations,
ξq =−Realλq|λq|
, for q ∈ 1, 2, · · · , 9n+ 8m (2.24)
ξcrit = minξ1, ξ2, · · · , ξ9n+8m, (2.25)
CDI = 1− ξcrit, (2.26)
where ξcrit is the critical damping ratio for a power system. For an n-generator, m-STATCOM power
system, with given differential equations in Section 2.3, a 9n+ 8m-dimension state-space can be formed
and thus the same number of eigenvalues can be obtained from system linear analysis. Damping ratio
ξ is a dimensionless value between 0 ∼ 1, which is used to characterize the decay of oscillations under
external disturbances. Critical damping ratio can be obtained through linear analysis of the overall
power system, which is explained in detail in [61].
The following algorithm (Algorithm 1) details the proposed parameter tuning procedures based
on the one-day-ahead predicted load. In order to clarify the time line, we denote the current day “Day
D”. The parameter optimization problem is formulated as follows.
minimize CDI
13
CHAPTER 2. LOAD-FORECASTING-BASED ADAPTIVE PARAMETEROPTIMIZATION FOR LFOD ENHANCEMENT IN POWER SYSTEMS
subject to:
Kminpac ≤ Kpac ≤ Kmax
pac , Kminiac ≤ Kiac ≤ Kmax
iac ,
Kminsup ≤ Ksup ≤ Kmax
sup , Tmin11 ≤ T11 ≤ Tmax11 ,
Tmin12 ≤ T12 ≤ Tmax12 , Tmin21 ≤ T21 ≤ Tmax21 ,
Tmin22 ≤ T22 ≤ Tmax22 , 0 ≤ wj ≤ 1, (2.27)
andν∑j=1
wj = 1 (2.28)
Algorithm 1 Load-Oriented Online Parameter Tuning
Import historical load statistics and acquire the hourly weather data for Day D+1 from local meteorologicalobservatory;
Schedule the power generation of each generator for Day D+1 using predicted load information obtainedby proposed method;
Perform power flow analysis using the data of Day D+1 from the last step and obtain the operating pointof the system;
Acquire the linearized model of the overall power system;
Implement PSO algorithm proposed in Section 2.4.2 to obtain the optimized controller parameters;
Configure the control system with the optimized control parameters from the last step on Day D+1;
When the actual load data become available at the end of Day D+1, together with historical data, retrainthe neural network and go to the first step to procure the control parameters for Day D+2 and daysthereafter.
2.5 Simulation and Numerical Results
In this study, an IEEE standard New York/New England 16-generator 68-bus test system is employed
as the base system, as shown in Fig. 2.6. A STATCOM is employed in this study, which is connected to
bus 39. Sub-transient model described in [59] is used for the the simulation of synchronous generators.
The IEEE DC1A AVR systems are incorporated in G1∼ G9, G11 and G12, and manual excitation
mechanism is utilized for the rest of the generators. The proposed supplementary controller of
STATCOM is utilized to mitigate the inter-area oscillations and enhance the stability of the power
system. All parameters for machines and AVR are taken from [66]. The simulation is performed in
MATLAB® 2015b coding environment on a desktop computer with Intel® Core i7-4790, 3.6GHz CPU
and 64-bit Windows®7 operating system. The simulation results are observed, acquired and analyzed
using MATLAB built-in algebraic-differential-equation solvers in continuous time.
2.5.1 Load Forecasting Performance
For simplicity and without loss of generality, the load forecasting strategy is applied on bus 17 which
bears the largest load in the given power system. Note that to perform a comprehensive parameter
training study for the entire power system, load data of every load bus need to be predicted, i.e., each
14
2.5. SIMULATION AND NUMERICAL RESULTS
G77
23
6
G6
22
21
68
24
20
194
G4
5
G5
G33
62
65
63
66
67
37
64
58
G22
59
60
57
56
52
55
G9
9
29
28
26
27
25
1
54
G1
8
G8
ModifiedIEEE68‐bustestsystem
Area1
61
13G13
17
12G12
36
30 34
43
44
39 45
35
51
50
33
32
11
G11
4938
46
10
G10
31
53
47 48
40
18
16G16
Area2
Area5
Swing
42
15
G15
Area3
G14
14
41
Area4
STATCOM
Figure 2.6: Modified IEEE standard 16-generator, 68-bus power system with STATCOM
load bus has its own ANN model for load forecasting. The ANNs are trained by the back-propagation
algorithm using a 4-year-long realistic dataset with the method described in Section 2.4.1. The dataset
used in this study is a properly scaled historical load demand characteristics of a real power system.
In the case study, 80% of the data are used for the model training, 15% are reserved for the model
validation, and the remaining 5% are utilized for load forecasting.
4 4.5 5 5.5 6 6.5 7 7.5
×103
4
4.5
5
5.5
6
6.5
7
7.5×103
Target (actual load, MW)
Loaddem
ands(M
W)
Predicted load data Linear regression of predicted load data Perfect match
Figure 2.7: Predicted data and linear regression (R2 = 0.9850)
Fig. 2.7 shows predicted load data with proposed forecasting method and the linear regression of the
trained model. In the figure, the dashed line represents a perfect match where the actual and predicted
load demand are identical, i.e., actual load=predicted load. The dots are the predicted load data, and
15
CHAPTER 2. LOAD-FORECASTING-BASED ADAPTIVE PARAMETEROPTIMIZATION FOR LFOD ENHANCEMENT IN POWER SYSTEMS
their close proximity to the dashed line indicates an accurate load-forecasting result. The solid line
indicates the linear regression of the predicted load data. Simply by observation, one can tell that
the linear regression is very close to the perfect-match line. Note that the coefficient of determination
R2 = 0.9850, which indicates a good fitness of the linear regression line for the predicted data points.
Fig. 2.8 is the error histogram for the model validation, and approximately 7500 test instances have
the smallest error of −43.74 and 5 instances have the largest error of −3053.44. The characteristic of
test-instance against prediction error is roughly subject to normal distribution.
−3,053
−2,752
−2,451
−2,150
−1,849
−1,548
−1,247
−946
−645
−344
−43
257558
8591,160
1,461
1,762
2,063
2,364
2,664
0
2
4
6
8×103
← Zero error
Error (MW)
Testinstances
Test error histogram
Figure 2.8: Error histogram
To demonstrate the performance of the ANNs-based load-forecasting method, a 7-day test data
are taken from the reserved data. Fig. 2.9 (a) shows the load-forecasting result and Fig. 2.9 (b)
demonstrates the prediction error during the period.
2.5.2 Modal Analysis of Inter-area Oscillations
As mentioned in Section 2.3.4, system linearization is required to perform the modal analysis. With
given configuration of the power system and using the operating point of Day 4, after arduous
mathematical computation, system linearization yields matrix A with a dimension of 137× 137 and
B with a dimension of 137 × 34. The reason for choosing Day 4 is that Day 4 has the largest load-
forecasting error, as shown in Fig. 2.9, and thus if the proposed controller generates satisfactory
control performances for Day 4, it will work for any other given periods. The eigenvalues λ of the
original linearized system and electromechanical modes with relevant contributory states that have
high participation factors, i.e., dominant states, are illustrated in TABLE 2.2. Terms f and ξ represent
the oscillation frequency and the damping ratio of a particular mode, respectively. Four inter-area and
ten local-area modes are identified, and the 5th local-area mode is the critical mode in the studied
power system. Detailed descriptions and mathematical derivations on small signal stability analysis
and system modes acquisition are documented in [61].
Based on the participation factor analysis, frequency deviations of generators 10, 11, 12 and 13 are
the dominant states relating to the critical mode, which are thus used to construct the input signals
16
2.5. SIMULATION AND NUMERICAL RESULTS
1 2 3 4 5 6 74
5.5
7
7.5×103
Days
Load
dem
ands(M
W)
Actual Forecast
(a) 7-day predicted load with trained ANN
1 2 3 4 5 6 7−0.5
0
0.5
1×103
Days
Error(M
W)
(b) 7-day prediction error
Figure 2.9: 7-day load forecasting result
for the supplementary control of STATCOM. Specifically using the following equations,
υ1 = ∆Ω10, υ2 = ∆Ω11, υ3 = ∆Ω12, υ4 = ∆Ω13, (2.29)
Λ =4∑j=1
wjυj . (2.30)
2.5.3 Online Parameter Tuning
Three integral steps are implemented in the proposed parameter tuning method: (1) ANN model
training, (2) ANN model validation, and (3) load forecasting using the ANN model. Four-year historical
data are utilized to train and validate the ANN model, which is then used to predict the load data
on a day-ahead basis. When the new data arrive, we incorporate the data to modify (retrain) the
ANN model to prepare for the prediction for the following day, see Algorithm 1. With the estimated
future load demand of the next day, the control parameters used by PSS-STATCOM are optimally
tuned. The predicted load data are used for parameter tuning of the proposed supplementary damping
controller for STATCOM, using PSO algorithm. In this study, we do not consider any load control
mechanisms. Therefore, the existing control system in the power system will not affect the load forecast
results. The tuning procedure has already been discussed in Section 2.4.2, and the parameters for
17
CHAPTER 2. LOAD-FORECASTING-BASED ADAPTIVE PARAMETEROPTIMIZATION FOR LFOD ENHANCEMENT IN POWER SYSTEMS
Table 2.2: Original eigenvalues of interest relating to LFEOs
Eigenvalue λ f(Hz) ξ(%) Dominant states Oscillation modes
−0.03± j0.80 0.31 3.76 ∆Ω and ∆δ of G13, G14, G15, G16 Inter-area Mode 1−0.14± j2.61 0.42 5.25 ∆Ω and ∆δ of G12, G13, G14, G15 Inter-area Mode 2−0.20± j3.49 0.55 5.74 ∆Ω and ∆δ of G3, G6, G9, G14, G16 Inter-area Mode 3−0.45± j4.42 0.70 10.16 ∆Ω and ∆δ of G13, G14 Inter-area Mode 4−0.39± j6.09 0.97 6.44 ∆Ω and ∆δ of G2, G3, G5, G9 Local Mode 1−0.42± j6.43 1.02 6.50 ∆Ω and ∆δ of G2, G3, G4, G5, G6 Local Mode 2−0.57± j7.37 1.17 7.70 ∆Ω and ∆δ of G4, G5, G6, G7 Local Mode 3−0.56± j7.65 1.22 7.27 ∆Ω and ∆δ of G2, G3 and ∆ψ1d of G2 Local Mode 4
−0.08± j7.87† 1.25 0.99 ∆Ω and ∆δ of G10, G11, G12, G13 Local Mode 5−0.44± j7.99 1.27 5.54 ∆Ω and ∆δ of G1, G8 Local Mode 6−1.51± j8.37 1.33 17.79 ∆Ω and ∆δ of G10, G11, G12 Local Mode 7−0.91± j9.53 1.52 9.48 ∆Ω and ∆δ of G4, G5, G6, G7 Local Mode 8−0.90± j9.73 1.55 9.17 ∆Ω and ∆δ of G1, G8 and ∆E′d and ∆ψ1d of G8 Local Mode 9−0.71± j9.61 1.53 7.32 ∆Ω and ∆δ of G4, G6, G7 and ∆E′d of G6, G7 Local Mode 10† Critical mode
PSO algorithm configuration are listed in TABLE 2.3. The initial inertia weight τ ini is set to 0.9 and
converges to its final value τfin = 0.4. Terms c1 and c2 are the acceleration coefficients.
Table 2.3: PSO algorithm parameters
Population Dimension c1, c2 Iterations τ ini τfin
120 11 1.5 200 0.4 0.9
Fig. 2.10 shows the gradual convergence of CDI (see (2.24)) using PSO algorithm and a continual
improvement of the fitness of Gbest over 200 iterations can be noticed. The critical damping ratio
reduces over time and settles after the 140th iteration, which indicates the production of an optimal
set of parameters. TABLE 2.4 demonstrates three sets of control parameters, (1) Universal control
0 40 80 120 160 200
0.9628
0.9629
0.9629
Iterations
CDI
Critical damping index
Figure 2.10: Evolution of Gbest
parameters where the weights are equally assigned to all the frequency deviations; (2) Optimized
STATCOM control parameters using proposed load-oriented optimization strategy; and (3) Tuned
control parameters using actual load data. It is easy to identify that the last set is the best control
parameters, as they are obtained with actual load data. However, in real-world applications, this set of
parameters cannot be obtained due to the absent knowledge of future loading conditions. This set of
parameters are only for the purpose of comparison study, which will also be discussed in Section 2.5.4.
They are incorporated in the controller to make a comparison for control performances of proposed
18
2.5. SIMULATION AND NUMERICAL RESULTS
control strategy. The convergence problem of PSO algorithms is not discussed in the chapter as it is
not the focus of this study. Note that PSO algorithms are not the only option for PSS tuning problems.
Other heuristic algorithms such as genetic algorithms can also be used to solve such problems.
Table 2.4: Control parameters used in the case study
Ksup T11 T12 T21 T22 Kpac Kiacw1 w2 w3 w4
Universal control parameters 10 0.9 0.01 0.9 0.01 1 10 0.25 0.25 0.25 0.25
Tuned with forecast load data 28.95 1.50 0.1094 1.50 0.1115 5.3762 66.8748 0.2848 0.1311 0.0001 0.6540Tuned with actual load data 35.89 1.3817 0.0963 1.2992 0.1500 7.46 102.21 0.2463 0.1640 0.0001 0.5896
2.5.4 Control Performances
With the optimized control parameters using predicted load data, eigenvalues of the power system are
shown in Fig. 2.11 (b), whereas Fig. 2.11 (a) illustrates the eigenvalues of the original power system
without the proposed supplementary controller for STATCOM. It is clear that without the proposed
control strategy, the critical mode, shown in TABLE 2.2, has a damping of 0.99%. An eigenvalue
with positive real part can also be noticed in Fig. 2.11 (a), which leads to instability of the power
system under external disturbances, and this can also be seen in time-domain simulation results. In
comparison, the designed supplementary controller for STATCOM with control parameters optimized
by predicted load data issues a much better damping performance for the critical mode with a 3.7%
damping ratio, and it also eliminates the eigenvalue on the right-hand side of the complex plane, which
enhances the stability of the power system under abnormal operating conditions. This result also
agrees well with the critical damping index shown in Fig. 2.10 where the final critical damping ratio is
approximately 0.037(= 1− CDI). Also note that in Fig. 2.11, only eigenvalues with positive imaginary
part are shown due to the symmetric nature of conjugate pairs of eigenvalues.
The time-domain simulation lasts for 60 seconds and the system operates at steady state during the
first one second. When t = 1s, an outage occurs and the transmission line between bus 34 and bus 35
is disconnected. The disconnection remains unresolved for the rest of the simulation. Figs. 2.12 (a), (c),
(e), (g) show the change of frequency deviations of generators G10 ∼ G13 over the entire simulation
time. As discussed above, without the proposed supplementary controller, the frequency deviations of
the generators will not settle, causing endless fluctuations, whereas the supplementary controller with
different sets of control parameters are all capable of regulating the frequency deviations to zero, capable
of stabilizing the system. In order to further compare control performances, Figs. 2.12 (b), (d), (f), (h)
provide better observations, where only system responses with controllers are shown. Apparently, the
STATCOM supplementary controller incorporated with parameters optimized by actual loads issues
the best control result, which nevertheless cannot be realized in practice as future load demands cannot
be accurately known. As evidenced in the Figs. 2.12 (b), (d), (f), (h), the designed control system with
parameters optimized by the predicted load data produces noticeable improvements on the control
performances than universal-parameter-based controller. Particularly a shorter settling time and fewer
and lower-amplitude oscillations can be observed in the system responses with the proposed optimized
controller. From the simulation results shown in Figs. 2.12, the best control outcome is achieved using
the actual load data, where forecast error is absent. The damping performance is degraded with the
parameters tuned based on the predicted load demand due to the forecast error.
19
CHAPTER 2. LOAD-FORECASTING-BASED ADAPTIVE PARAMETEROPTIMIZATION FOR LFOD ENHANCEMENT IN POWER SYSTEMS
−2 −1.5 −1 −0.5 0 0.50
3
6
9
12
← Critical mode
0.99% damping
← 3% damping line
Real part
Imaginarypart
Eigenvalues Electromechanical modes
(a) Eigenvalues of the original power system
−2 −1.5 −1 −0.5 0 0.50
3
6
9
12
← Critical mode
3.7% damping
← 3% damping line
Real part
Imaginary
part
(b) Eigenvalues of the power system with optimized control parameters
Figure 2.11: Eigenvalues of the power system without and with proposed control strategy
2.6 Conclusion
In this chapter, a novel parameter optimization strategy is proposed for STATCOM to mitigate
low-frequency oscillations for complex power systems under external disturbances. The proposed
load-oriented parameter optimization method is based on PSO algorithm and utilizes one-day-ahead
predicted load data procured by an ANN-based learning mechanism. The proposed control strategy
with optimized load-oriented control parameters has provided noticeable enhancements for LFOD for
inter-connected, multi-machine complex power systems, and has demonstrated its superiority over
the conventional robust controller. The designed control strategy infers the possibility of industrial
implementation, with which power system operators are able to adaptively adjust control parameters
for better control performances with varying loads to maintain the stability of power systems. Future
work may involve developing coordinated control strategies for renewable-energy-integrated complex
power systems, using the devised control and parameter optimization strategy.
20
2.6. CONCLUSION
0 20 40 60−0.04
−0.020
0.02
0.04
time (s)
∆Ω
10
(p.u
.)
(a)
No ctrl. Ctrl. with universal para.
Ctrl. with predicted-load-based para. Ctrl. with real-load-based para.
20 30 40 50−0.01
−0.005
0
0.005
0.01
time (s)
∆Ω
10
(p.u
.)
(b)
0 20 40 60−0.04
−0.020
0.02
0.04
time (s)
∆Ω
11
(p.u
.)
(c)
20 30 40 50
−0.005
0
0.005
time (s)
∆Ω
11
(p.u
.)
(d)
0 20 40 60−0.05−0.03
0
0.030.05
time (s)
∆Ω
12
(p.u
.)
(e)
20 30 40 50−0.008
−0.004
0
0.004
0.008
time (s)
∆Ω
12
(p.u
.)
(f)
0 20 40 60−0.04
−0.020
0.02
0.04
time (s)
∆Ω
13
(p.u
.)
(g)
20 30 40 50−0.008
−0.004
0
0.004
0.008
time (s)
∆Ω
13
(p.u
.)
(h)
Figure 2.12: Rotor speed deviations of G10, G11, G12, and G13
21
CHAPTER 2. LOAD-FORECASTING-BASED ADAPTIVE PARAMETEROPTIMIZATION FOR LFOD ENHANCEMENT IN POWER SYSTEMS
22
Chapter 3
A Novel Control Strategy of DFIG
Wind Turbines in Complex Power
Systems for Enhancement of Primary
Frequency Response and LFOD
ABSTRACT
In this chapter, we propose a novel control strategy for doubly fed wind turbine generators (DFWTG)
in complex power systems to improve the primary frequency response and enhance low-frequency
oscillation damping of power systems. The main innovation in the new control scheme dwells in the
novel control schemes for rotor side controller (RSC) of DFWTG. Weighted frequency deviations of
local synchronous generator (SG) bus-bars are utilized as input signals to a dedicated power system
stabilizer (PSS), specifically designed for the RSC of DFWTG, with parameters optimized by particle
swarm optimization (PSO). The newly devised RSC with conventional DFWTG control structure
is capable of ameliorating primary frequency response of the power system. To eliminate the area
control error (ACE), a secondary control scheme is incorporated, which makes use of the spinning
reserve of selected synchronous generators through automatic generation control (AGC). Tie-line power
deviations are employed as control signals in both primary and secondary control schemes, on the
purpose of further enhancing the primary and secondary frequency regulations and also maintaining
the obligation of power transmissions among adjoining areas. Simulation results demonstrate the
superiority of the proposed DFWTG control methods in enhancing primary frequency response and
also suppressing low-frequency oscillations (LFO) of the power system over the conventional strategy.
23
CHAPTER 3. A NOVEL CONTROL STRATEGY OF DFIG WIND TURBINES INCOMPLEX POWER SYSTEMS
3.1 Chapter Foreword
This chapter is based on the following academic paper:
Tat Kei Chau, Samson Shenglong Yu, Tyrone Fernando, Herbert Ho-Ching Iu and Michael Small,
“A novel control strategy of DFIG wind turbines in complex power systems for enhancement of primary
frequency response and LFOD”, IEEE Transactions on Power Systems, vol.33, no.2, pp.1811-1823,
2018.
In the previous chapter, a load-forecast-based parameter tuning method is proposed, which uses
the predicted load demand obtained from a trained ANN to tune the parameters of a PSS attached to
a STATCOM. In this chapter, we propose a RPSS, a PSS-like damping controller, based on the rotor
side converter of a DFIG, and the parameters of the RPSS is tuned using the PSO algorithm discussed
in Chapter 2.
3.2 Introduction
Low-frequency oscillations are one of the concerning issues in complex power systems that can
cause potential damages to power system components and even regional or large-scale power failure.
Low-frequency oscillation study is typically conducted through modal analysis, a technique used
to investigate the oscillatory behavior of power systems. Modal analysis further sub-categorizes
system modes into electromechanical and non-electromechanical oscillatory modes. Conventional
PSSs (CPSSs), composed of a series of multi-stages lead-lag compensators, are widely used to provide
extra damping for electromechanical modes of power systems, by modulating the reference voltage
of automatic voltage regulators (AVRs) of synchronous generators [1]. Recent work relating to PSS
designs can is reported in [67] and [68], where main innovations lie in the acquisition and construction
of PSS control signals using assorted estimation algorithms or/and cutting-edge measuring technologies.
However, in multi-area interconnected power systems, only utilizing conventional AVR-PSSs may
not be able to provide sufficient damping for low-frequency oscillations. In these cases, coordinated
flexible alternating current transmission systems (FACTS) power oscillation damping controller (POD)
and AVR-PSSs may be a feasible solution [7]. For instance, in [69] the authors use FACTS devices
to enhance wide-area damping considering communication time delays, in [70] a coordinated design
strategy is proposed for multiple robust FACTS damping controllers, and in [71], a coordinated PSS
and SVC damping controller is designed to improve probabilistic small signal stability of power system
integrated with wind farm. Despite the simplicity of theoretically devising such control schemes, the
extremely high costs of FACTS devices, to some extent, may impede the implementation of proposed
damping controllers in real-world applications.
Maintaining and enhancing the stability of power systems integrated with renewable energy power
generation units has become a major concern to increase renewable power penetration. Doubly fed
induction generators (DFIGs) are most widely-used in wind power plants, and numerous recent research
papers have dedicated to the improvement of their robustness and stability. For DFIG wind turbines
connected to an infinite bus, extensive work on controller design and stability enhancement can be
found in [72–74], where the authors mainly work toward improving the low voltage ride through (LVRT)
capability, active and reactive power regulation, and inertia support under abnormal situations. Despite
24
3.2. INTRODUCTION
the tangible innovations seen from these references, the feasibility and applicability of the proposed
methods in large-scale power systems are not discussed, and only infinite-bus structure is incorporated.
Other researchers on the other hand, have been concentrating on the study of integrating DFWTGs
with complex power systems. In order to achieve LFO damping enhancement, authors in [75] utilize
frequency deviations of SGs from exogenous areas to form the control signal of the proposed PSS that
resides at the reactive power regulation loop of DFIG. To achieve the same purpose, electrical power
produced by local DFWTGs [76], aggregated electrical power produced by inter-area DFWTGs [77] and
frequency deviations of local SGs [78] are utilized to construct the input signals to drive the proposed
PSSs, which are located at voltage regulator [76] and active power control loop [77, 78]. Another
research work relating to this topic focuses on decentralized dynamic state estimation (DSE) [79]
and DSE-based control [80] for DFWTGs connected to IEEE standard complex power systems with
realistic specifications. Other research work on wind-integrated power systems can be found in [81, 82]
and references therein. Despite the great work that has been done in this area, there has been no
reported work on control strategy design to achieve simultaneously both primary frequency response
enhancement and LFO mitigation. This became the motivation of this particular study.
In this chapter, we propose a novel control strategy to enhance the primary frequency response and
mitigate low-frequency oscillations of DFWTGs-integrated complex power systems, using the rotor
side controller and power converters embedded in DFIGs. Frequency deviations of local synchronous
generators are utilized to construct the input signal of the dedicated rotor-side PSS, which is integrated
with the voltage control loop. Due to the fact that generators contribute differently to low-frequency
electromechanical oscillation modes, the feedback signal of local frequency deviations shall have different
weightings. To achieve this, particle swarm optimization (PSO) technique, a widely used optimization
algorithm in power systems [10], is employed to find the optimal weightings of frequency deviations of
local SGs. With the same method parameters used in the control strategy are optimized. Additionally,
tie-line power deviations (TLPD) are also employed to modify the power regulation scheme in the
RSC of conventional DFIG. Different from secondary control, which also requires TLPD feedback,
this control strategy is aimed to alleviate the frequency fluctuations after a disturbance, reducing the
amplitudes of frequency swings and improving the values of nadir/zenith points. Therefore, the primary
frequency response is enhanced by the newly designed power regulator in RSC with the utilization
of TLPD signal, and the improvement of LFOD is achieved by the proposed PSS using frequency
deviation signals. This design makes possible the simultaneous enhancement of both primary frequency
response and LFOD of local generators. A secondary control scheme is also implemented in the power
system, in order to eliminate frequency biases of SGs and maintain power export/import obligations of
each area. This purpose is achieved by assigning the control duty to a generator in each area, and
making use of the spinning reserve of the assigned generators during abnormal operating conditions.
System linear analysis will demonstrate the significant amelioration in critical damping of the power
system with the proposed control strategy, and time-domain simulation results will illustrate the
noticeable enhancement of primary frequency response and improvement of low-frequency oscillations.
Different to conventional SG AVR-PSSs and FACTS-PSSs, the designed control strategy does not
require additional FACTS devices connected to the transmission line, or to be implemented on SGs
with AVR; but rather, the dedicated PSS is connected to a modified rotor side controller of conventional
25
CHAPTER 3. A NOVEL CONTROL STRATEGY OF DFIG WIND TURBINES INCOMPLEX POWER SYSTEMS
DFIG structure. At the end of the simulation study, the interactions among the proposed DFIG-PSS
control structure and the nearby SG-PSSs are also investigated in depth. This design aligns well with
the current trend of incorporating renewable energy sources to power systems, and also poses economic
advantages due to the absence of FACTS devices.
The rest of the paper is organized as follows. Mathematical model, conventional control schemes of
DFIG and power system modeling are briefly presented in Section 3.3. The proposed control strategy
is detailed in Section 3.4. Then, in Section 3.5, two distinct case studies are performed with simulation
results analyzed and compared. Finally, a conclusion is drawn in Section 3.6.
3.3 Mathematical Model and Conventional Control Strategies of
DFIG
In this section, a brief description of DFIG mathematical model is presented, followed by a concise
introduction of conventional DFIG controllers. Fig. 3.1 illustrates the structure of doubly fed induction
generator wind turbine system, integrated with a complex power system. Topology of the interconnected
power network will be presented later in Section 3.5.
GearBox
AG
Rotorsidecontrollerconverter DCLink
Gridsidecontrollerconverter
PitchDrive
Pitchanglecontroller
Multi‐AreaInterconnectedPowerSystem
RPSS
Figure 3.1: DFIG connected to a multi-area interconnected power system
3.3.1 Wind Turbine and Drive Train
Mechanical torque Tm and mechanical power Pm harnessed by the wind turbine are calculated as
follows [83],
Tm = −Pmωr
, (3.1)
Pm = 0.5ρAV 3wCp(λ, β), (3.2)
λ =ωrR
Vw(3.3)
where Vw is wind speed, ρ is the air density, A is the area swept by the turbine blade as it rotates, and
R is the blade radius. Term Cp(λ, β), the mechanical power coefficient, is a function of the blade tip
ratio λ and wind turbine pitch angle β. For detailed information and mathematical model of the pitch
26
3.3. MATHEMATICAL MODEL AND CONVENTIONAL CONTROL STRATEGIESOF DFIG
angle controller, see [84]. A simplified 2-mass model of the turbine drive train is adopted in this study
and the overall mathematical expression is shown as follows [85],
dωrdt
=1
2Hg(Ts − Te − Fωr), (3.4)
dθtdt
= ωb(ωt − ωr), (3.5)
dωtdt
=1
2Ht(Tm − Ts), (3.6)
where Ts is the shaft torque, Te is the electrical torque, Hg and Ht are the equivalent gear-box and
turbine inertia, ωb is the base angular speed, F is the friction factor, and θt and ωt are respectively the
turbine angle and speed.
3.3.2 Asynchronous Generator
The mathematical model of an asynchronous generator (AG) in synchronous reference frame can be
expressed in the following equations [79],
dΦdr
dt= ωb(Vdr + (ωs − ωr)Φqr −RrIdr), (3.7)
dΦqr
dt= ωb(Vqr − (ωs − ωr)Φdr −RrIqr), (3.8)
dΦds
dt= ωb(Vds + ωsΦqs −RsIds), (3.9)
dΦqs
dt= ωb(Vqs − ωsΦds −RsIqs), (3.10)
where ωs = 1 is the synchronous angular speed, subscripts d and q represent the direct and quadrature
components of rotor flux Φr, stator flux Φs, rotor voltage Vr, stator voltage Vs, rotor current Ir and
stator current Is.
3.3.3 Conventional Grid and Rotor Side Controllers
Conventional grid side controller is illustrated in Fig.3.2. The upper left superscript κ indicates the
control operation is in stator voltage reference frame, i.e., stator voltage is κVs = |Vs|∠0o. Term Zcc is
the common coupling impedance, connecting the grid and the GSC, which can also be seen in Fig. 3.1.
The conventional GSC is used to regulate the voltage of DC-link capacitor and also reactive power
Figure 3.2: Conventional grid side controller
27
CHAPTER 3. A NOVEL CONTROL STRATEGY OF DFIG WIND TURBINES INCOMPLEX POWER SYSTEMS
injected into the GSC by controlling quadrature component of Ig. For detailed mathematical model
and explanation of GSC, refer to [79].
Conventional rotor side controller is utilized to regulate output electrical power and terminal
bus-bar voltage, which follows the working principle described in Fig. 3.3. RSC is performed in mutual
Figure 3.3: Conventional rotor side controller
flux reference frame, denoted as upper left superscript m. Term Xs is a series-connected reactance
that links the common coupling point to the DFIG terminal bus-bar, VWF and Vref are the actual and
nominal voltage of the DFIG terminal bus-bar. Active power reference Pref is obtained through the
Maximum Power Point Tracking (MPPT) algorithm, power loss Ploss accounts for both electrical and
mechanical power losses during the mechanical-electrical power conversion process, and electrical power
Pe is measured at the terminal bus-bar with the knowledge of the bus voltage and the current injected
to the bus-bar. The acquisition of the equivalent rotor impedance Z′r is reported in [85] and [80]. It is
noteworthy that according to MPPT algorithm, under a given wind speed, any rotor angular speed
that is not the optimal speed renders a deficient kinetic energy extraction from the ambient wind.
However, a lower rotational speed can lead to a higher electrical power injection to the grid, and vice
versa. This phenomenon will be observed in the case study, and has been explained with the concept
“virtual inertia” in [81]. For rationale behind the control strategy and detailed mathematical models,
see [73,79]. The DC-link voltage can then be calculated as [79],
dVDCdt
=1
CDC
Pr − PgVDC
, (3.11)
where Pr and Pg represent the power entering the RSC and exiting the GSC and CDC is the equivalent
capacitance of the DC-link structure.
3.3.4 Overall Power System Modeling and Linearization
In order to perform small signal analysis and wide-area power system stability study, system linearization
is required in this study. In previous reported study, we have been able to model a power system
integrated with DFIG wind farm using the following compact form [79],
d
dtX = f(X,U),
0 = g(X,U), (3.12)
28
3.4. PROPOSED DFIG CONTROL STRATEGIES
X =
[xSG
xDFIG
], U =
[uSG
uDFIG
], (3.13)
where f(·) and g(·) are respectively the state differential and algebraic functions of the system model,
X is the system dynamic state vector consisting of synchronous generator state vector xSG and DFIG
state vector xDFIG and U is the system input vector, comprised of synchronous generator input vector,
uSG and DFIG input vector uDFIG. The components of each vector can be found in [79, 80]. For
small signal stability analysis of the power system, the following linearized system equation can be
obtained [59,61],
∆X = A∆X +B∆U, (3.14)
where A and B are coefficients of the linearized system. For detailed steps and derivations of linearizing
nonlinear power systems for small-signal stability analysis (SSSA), refer to [59,61]. The entire power
system can now be linearized at certain operating point to perform SSSA, and the results will later be
used in parameter and weighting optimizations for the PSS design.
3.4 Proposed DFIG Control Strategies
Building on the conventional DFIG control mechanism, the proposed control method is now presented
in this section. The first two subsections provide a detailed discussion on the primary frequency
response enhancement scheme, and the third subsection presents the automatic generation control
with brevity. Note that all the controller designs are based on the nonlinear model of DFIG wind
turbine power generation units connected to complex power system, and the linearized model is used
for SSSA, which is utilized to acquire the optimized control parameters that maximize the critical
damping ratio. Control schemes to be discussed in this section will be implemented in doubly fed
wind turbine generators connected to a complex power grid, with different scenarios implemented and
compared in Section 3.5.
3.4.1 Tie-line Power Deviation Feedback
In order to enhance primary frequency response, and improve stability of large power systems, the
most direct way is to feed tie-line power deviations to a power control unit [59], which in case of DFIG,
is the RSC. Fig. 3.4 illustrates the working mechanism of the TLPD-feedback control scheme.
Suppose DFIG wind farm is installed in area b and there are k adjacent areas, named a1, a2, · · · ,ak, between which area b transmits power. Then ∆Ptl in Fig. 3.4 is calculated as follows,
∆Ptl =i=k∑i=1
(P schtli − Ptli), (3.15)
where Ptli is the power flowing between area b and area ai and P schtli is the scheduled power transmission
between the two areas. This design is aimed to store extra mechanical power in DFIG rotor when less
29
CHAPTER 3. A NOVEL CONTROL STRATEGY OF DFIG WIND TURBINES INCOMPLEX POWER SYSTEMS
Figure 3.4: Proposed RSC in control strategy 1
power is needed, or release more electrical power to the grid in case of an increased power demand [81],
so as to improve the primary frequency response.
3.4.2 TLPD and DFIG-PSS with Optimized Parameters
The RSC in the second proposed DFIG control method shown in Fig. 3.5 makes use of TLPD employed
by the first proposed control scheme, and also the frequency deviations of local synchronous generator
bus-bars, which are fed to a dedicated rotor side PSS (RPSS), whose parameters are optimized by
using particle swarm optimization algorithm. Suppose there are n synchronous generators in the area
where DFIG wind farm is installed, and in this strategy, weighting coefficients of frequency deviations
are employed, then the input signal of the RPSS ΥRPSS is constituted as follows,
ΥRPSS =
n∑i=1
υRi ∆fi, (3.16)
i=n∑i=1
υRi = 1, (3.17)
where ∆fi represents the ith synchronous generator’s frequency deviation, and υRi is the normalized
weighting coefficient of this frequency deviation.
Figure 3.5: Proposed RSC in control strategy 2
30
3.4. PROPOSED DFIG CONTROL STRATEGIES
The following equations can be used to describe the dynamical behavior of the proposed power
system stabilizer,
dxRPSS1
dt=
1
TRW
(KRPSSΥRPSS − xRPSS1 ), (3.18)
dxRPSS2
dt=
1
TR12
(KRPSSΥRPSS − xRPSS1 − xRPSS
2 ), (3.19)
dxRPSS3
dt=
1
TR12T
R22
(KRPSST
R11ΥRPSS
− TR11x
RPSS1 − (TR
11 − TR12)xRPSS
2 − TR12x
RPSS3
), (3.20)
V Rsup =
1
TR12T
R22
(KRPSST
R11T
R21ΥRPSS
− TR11T
R21x
R1 + (TR
12 − TR11)TR
21xR2 + (TR
22 − TR21)TR
12xR3
), (3.21)
where xRPSS1 , xRPSS
2 , and xRPSS3 are intermediate states, is the output supplementary voltage signal
produced by the PSS and sent to the voltage regulator in the proposed RSC, and the rest are time
constants and the PSS gain. Apparently, there are 5 + n parameters that need to be optimized: KRPSS,
TR11, TR
12, TR21, TR
22, and υR1 to υR
n . Parameter constraints are shown below (with simple notations for
generality),
KminPSS ≤ KPSS ≤ Kmax
PSS , Tmin11 ≤ T11 ≤ Tmax11 ,
Tmin12 ≤ T12 ≤ Tmax12 , Tmin21 ≤ T21 ≤ Tmax21 , (3.22)
Tmin22 ≤ T22 ≤ Tmax22 .
The objective of PSO parameter optimization algorithm used in this study is to maximize the critical
damping ratio (CDR) or to minimize the critical damping index (CDI = 1− CDR) of the linearized
state space model formed by connecting DFIG wind farm and complex power systems comprised of a
number of synchronous generators. Now suppose the linearized system is expressed by (3.14), where
A(N ×N) is the system matrix after linearization, and λi is the ith eigenvalue of A, then the damping
ratio of this eigenvalue is defined as [1],
ξi =−realλi|λi|
, i ∈ 1, 2, · · ·N, (3.23)
and critical damping ratio ξcrit (≥ 0) and critical mode λcrit are
ξcrit = minξi, (3.24)
λcrit = λargminξi i ∈ 1, 2, · · ·N. (3.25)
In Section 3.5, two scenarios of this control strategy will be considered: equal frequency weightings
and optimized frequency weightings. The latter is predicated on the fact that system states contribute
differently to low-frequency oscillations of the power system, so that the method with optimized
weightings shall generate a more satisfactory control performance. The weighting optimization is
31
CHAPTER 3. A NOVEL CONTROL STRATEGY OF DFIG WIND TURBINES INCOMPLEX POWER SYSTEMS
achieved through linear analysis and PSO algorithm, which will be presented in Section 3.5.
3.4.3 Secondary Control
A brief discussion is presented for the secondary control strategy. A widely-used valve position control
is adopted in this study, which is capable of instantaneous response to active power imbalance and
correcting frequency to nominal and area control error to zero. The following equations describe the
dynamics of this control method for generator g,
dTMg
dt=
1
TCHg(−TMg + PSVg) , (3.26)
dPSVg
dt=
1
TSVg
(−PSVg + PCVg −
1
RDg
(ωgωs− 1
)), (3.27)
where TCHg and TSVg are time constants, PSVg represents the power for a specific valve position, RDg
is droop constant and PCVg is the control signal that drives the change of the output mechanical power.
More details on this secondary control method can be found in [61]. In this study, the aforementioned
secondary control method is employed and implemented to a selected generator of each area, and we
assume that all the selected generators have sufficient capacities to provide active power needed under
certain abnormal conditions. It is noteworthy that due to the essential difference amongst LFOD,
primary and secondary frequency enhancements and the fact that the selected generators only perform
secondary control in that area, there is no conflicts in frequency regulations caused by isochronous
frequency control.
3.5 Case Studies
In this section, all the proposed control strategies will be implemented in a DFIG-integrated modified
IEEE 68-bus power system in order to enhance primary frequency response and also realize post-fault
frequency restoration. Fig. 3.6 demonstrates the network topology of the power system in question,
where 16 synchronous generators are located in 5 interconnected areas. To demonstrate the functionality
of the proposed controller and without the loss of generality, DFIG wind farm is connected to bus 69,
then to bus 31 in area 1, whose bus-bar is marked in color red. Bus-bar 16 is the swing bus of the
power system of interest. Generators 9, 13, 14, 15 and 16 marked in color green, in their corresponding
areas are selected to perform AGC as secondary control method. The configuration and data of this
test system are detailed in [66].
Additionally, generator 1∼ 9, 11 and 12 are equipped with IEEE DC1A exciter, generator 10 has
static exciter and generators 13, 14, 15 and 16 have manual exciters. All generator parameters are
adopted from [66] (SG) and [80] (DFIG), and detailed mathematical model of the entire system can be
found in [59,61]. The simulation runs for 100s. In the first 2 seconds, the power system operates at
steady state, and when t = 2s, sudden load decrease and increase faults are introduced in this study.
Both load changes occur on bus 33, marked in color blue, a load bus with 1.12p.u. load demand in
normal situations. Given the power system network configuration, fixed variables and parameters
in the proposed control strategies can now be assigned. For tie-line power deviation, the following
32
3.5. CASE STUDIES
G77
23
6
G6
22
21
68
24
20
194
G4
5
G5
G33
62
65
63
66
67
37
64
58
G22
59
60
57
56
52
55
G9
9
29
28
26
27
25
1
54
G1
8
G8
ModifiedIEEE68‐bustestsystem
Area2
61
13G13
17
12G12
36
30
34
43
44
39
45
35
51
50
33
32
11
G11
4938
46
10
G10
31
53
47 48
40
18
16G16
Area1
Area4
Swing
42
15
G15
Area3
G14
14
41
Area5
WTGs
69 Case 3
Case 1, 2
Figure 3.6: Modified IEEE 68-bus, 16-generator power system integrated with DFIG WTGs
equation holds,
Ptl = Ptl,area1 = Parea1→2 + Parea1→3 + Parea1→4 + Parea1→5
= P61→60 + P53→54 + P53→27 + P50→18 + P49→18 + P40→41, (3.28)
and then ∆Ptl can be calculated accordingly.
Based on the network topology of the power system in question, G10, G11, G12 and G13 are local
to the DFIG wind power plant, and hence frequency deviations of these generators are employed in the
controller design and parameter tuning algorithm to fulfill the control purposes of this study. Local
generator frequency deviations can be substituted as follows,
∆f1 = ωs − ω10, ∆f2 = ωs − ω11,
∆f3 = ωs − ω12, ∆f4 = ωs − ω13. (3.29)
For the state space model of the power system in question, vector xSG in (3.13) has 154 states and
xDFIG has 14 states, which include 3 PSS states described earlier in Section 3.4.2. Input vector uSG,
consisting of mechanical torque reference value and voltage reference value has 32 elements, and uDFIG,
containing voltage reference value, wind speed, and turbine pitch angle has 3 elements. Note that in
the proposed control strategy, we only employ measurements of local generator bus-bars and tie-line
power measured in Area 1, both of which can be realized by using phasor measurement units (PMUs)
located at specific bus-bars [59]. Since only local PMU measurement data are required in the controller
design, the time delay issue can be disregarded [86]. However, if wide-area control signals are to be
employed, the time delay must be considered, which however is outside the scope of this research work.
33
CHAPTER 3. A NOVEL CONTROL STRATEGY OF DFIG WIND TURBINES INCOMPLEX POWER SYSTEMS
3.5.1 Case 1: Load Increase
In this case, DFIG wind farm is connected to bus 69, and a sudden load increase occurs on bus 33
at t = 2s, ∆P33 = 0.5p.u, which makes P33 = 1.62p.u. The TLPD control scheme can easily be
implemented, and thus the procedure is omitted, with simulation result shown. We now use the
steady-state system as an operating point and perform linear analysis. After arduous computation, a
dimension of 168× 168 matrix A is obtained, and then small signal stability analysis can be explored.
Since the SSSA reflects intrinsic characteristics of the power system and is not affected by any faults
implemented, the analysis is suitable for the investigation of both cases. TABLE 3.1 lists the original
eigenvalues of interest in relation to low frequency electromechanical oscillations (LFEO). Dominant
states are determined by participation factor (PF ) of each state in each mode, an index that measures
the eigenvalue sensitivity of a specific state, which is calculated as follows [1],
PFki =∂λi∂akk
, (3.30)
where λi is the ith mode, i.e., ith eigenvalue of the linearized system, akk is the kth diagonal item of the
linearized system matrix A, and then PFki is the participation factor of the kth state in the ith mode.
Table 3.1: Original eigenvalues of interest relating to LFEOs without controller
Eigenvalue λ f(Hz) ξ(%) Dominant states
−0.087± j8.469 1.348 1.025 ∆ω and ∆δ of G10−0.155± j3.839 0.611 4.033 ∆ω and ∆δ of G13−0.102± j2.506 0.399 4.052 ∆ω and ∆δ of G13 and G9−0.140± j3.435 0.547 4.071 ∆ω and ∆δ of G14 and G16−0.268± j6.109 0.972 4.388 ∆ω and ∆δ of G9−0.372± j7.834 1.247 4.746 ∆ω and ∆δ of G1 and G8−0.251± j4.996 0.795 5.012 ∆ω and ∆δ of G14 and G15−0.371± j7.059 1.123 5.245 ∆ω and ∆δ of G12−0.341± j6.386 1.123 5.336 ∆ω and ∆δ of G2 and G3−0.65± j11.263 1.793 5.763 ∆ω and ∆δ of G11−0.508± j7.645 1.217 6.635 ∆ω and ∆δ of G2 and G3−0.682± j9.608 1.529 7.078 ∆ω and ∆δ of G1 and G8−0.530± j7.372 1.173 7.171 ∆ω and ∆δ of G5 and G6−0.871± j9.714 1.546 8.932 ∆ω and ∆δ of G6 and G7−0.867± j9.513 1.514 9.079 ∆ω and ∆δ of G4 and G5
From TABLE 3.1, it is easy to tell that the critical damping ratio is 1.025% and dominant states
are the electromechanical states (rotor angular speed ω and rotor angle δ) of generator 10. Our design
purpose is now to increase the critical damping ratio using PSO. Based on the above analysis, one
may consider using a higher weighting for ∆f10 and lower weightings for frequency deviation of other
local generators. However, let us implement the proposed strategies with a set of conventionally used
parameters. Fig. 3.7 shows the eigenvalues of the linearized system with untuned parameters and
equal weightings for frequency deviations of all generator bus-bars. It is clear that with this set of
parameters, the critical damping ratio of the studied complex power system increases to 1.3%.
This critical damping ratio can be further enhanced by using optimized parameters and weighting
coefficients. Fig. 3.8 demonstrates the system eigenvalues when using this control strategy, and it
is evident that the damping ratio of critical mode has increased to 3.8%. Note that only a part of
34
3.5. CASE STUDIES
−1 −0.8 −0.6 −0.4 −0.2 0 0.20
2
4
6
8
10
12
← Critical mode
1.3% damping
← 4% damping line
Real part
Imaginarypart
Eigenvalues
Electromechanical modes
Figure 3.7: System eigenvalues with untuned PSS and equal weightings in Case 1 and Case 2
the complex plane is shown for easy observation, and there are no eigenvalues on the right half-plane
found in this simulation. Also only eigenvalues with positive imaginary part are shown here due to the
symmetrical nature of complex conjugate pairs.
−1 −0.8 −0.6 −0.4 −0.2 0 0.20
2
4
6
8
10
12
← Critical mode 3.8% damping
← 4% damping line
Real part
Imaginarypart
Eigenvalues
Electromechanical modes
Figure 3.8: System eigenvalues with optimized PSS and weightings in Case 1 and Case 2
Table 3.2: Resultant eigenvalues of interest relating to LFEOs with optimized controller
Eigenvalue λ f(Hz) ξ(%) Dominant states
−0.096± j2.503 0.398 3.838 ∆ω and ∆δ of G13 and G14−0.318± j8.282 1.318 3.838 ∆ω and ∆δ of G10−0.153± j3.835 0.61 3.992 ∆ω and ∆δ of G13−0.141± j3.434 0.547 4.103 ∆ω and ∆δ of G14 and G16−0.268± j6.109 0.972 4.388 ∆ω and ∆δ of G9−0.355± j7.752 1.234 4.57 ∆ω and ∆δ of G1 and G10−0.251± j4.996 0.795 5.019 ∆ω and ∆δ of G14 and G15−0.379± j7.055 1.123 5.365 ∆ω and ∆δ of G12 and G13−0.345± j6.385 1.016 5.391 ∆ω and ∆δ of G2 and G3−0.651± j11.229 1.787 5.792 ∆ω and ∆δ of G11−0.509± j7.645 1.217 6.637 ∆ω and ∆δ of G2 and G3−0.683± j9.613 1.530 7.092 ∆ω and ∆δ of G1 and G8−0.530± j7.372 1.173 7.171 ∆ω and ∆δ of G5 and G6−0.871± j9.714 1.546 8.932 ∆ω and ∆δ of G6 and G7−0.867± j9.513 1.514 9.079 ∆ω and ∆δ of G4 and G5
TABLE 3.2 shows the eigenvalues of the system with the optimized control strategy. From this
table, it is clear that the critical damping ratio has risen from 1.025% to 3.838%. The comparison
between TABLE 2.2 and TABLE 3.2 further demonstrates the functionality and superiority of the
35
CHAPTER 3. A NOVEL CONTROL STRATEGY OF DFIG WIND TURBINES INCOMPLEX POWER SYSTEMS
proposed controller in damping critical modes of the studied power system.
Important parameters used in this case study are shown in TABLE 3.3. In order to further verify
the selected KRpss shown in TABLE 3.3 (2), a root loci study is performed and a proportion of the root
loci are shown in Fig. 3.9, where KRpss min = 1× 10−5 and KR
pss crit = 790. It is noteworthy that the
critical value of KRpss is KR
pss crit, and if KRpss is greater than this value, an eigenvalue (whose trajectory
is marked in color green) will locate in the right-half of complex plane, causing instability of the system.
This study has proven the robustness of the optimized KRpss in the given condition shown in TABLE 3.5.
It is worth mentioning that in Fig. 3.9 some modes migrate toward the right hand side of the complex
plane as KRpss increases, leading to reduced instability of the system. This phenomenon is caused by
the fact that the objective of the optimization process is to minimize critical damping ratio, not all
modes’ damping ratios. All modes in a complex power system cannot be damped with a single PSS,
and the modes traveling toward the instability region with this control mechanism can be mitigated by
introducing additional PSSs at corresponding bus-bars. This point will be shown and explained again
in Section 3.5.3.
−0.8 −0.6 −0.4 −0.2 00
2
4
6
8
Real part
Imag
inarypart
Figure 3.9: Part of root loci with varying Kpss (‘x’: KRpss = KR
pss min, ‘o’: KRpss = KR
pss crit)
It is also noteworthy that the optimized weighting coefficients display consistency with the partic-
ipation factors obtained through linear analysis, and frequency deviation weighting of generator 10
dominates, which indicates that low-frequency oscillations of this power system are mainly caused by
the insufficient damping torque of generator 10.
Table 3.3: Important parameters in Case 1 and Case 2
(1) Untuned parameters
KRpss 20 TR
11 0.5 TR12 0.2 TR
21 0.3 TR22 0.4
υR1 0.25 υR
2 0.25 υR3 0.25 υR
4 0.25
(2) Optimized parameters
KRpss 80 TR
11 1 TR12 0.2054 TR
21 0.2912 TR22 0.2053
υR1 0.99997 υR
2 0.00001 υR3 0.00001 υR
4 0.00001
PSO parameters used in this study, adopted and modified from [87], are listed in TABLE 3.4.
Fig. 3.10 shows the evolution of PSO algorithm over 100 iterations. The critical damping index (recall
CDI=1-CDR) reduces gradually, and finally settles at 96.2%. Therefore the resultant CDR is 3.8%,
36
3.5. CASE STUDIES
Table 3.4: PSO setting
Max. inertia weight 0.9 Acceleration constant I 2 Population 30
Min. inertia weight 0.2 Acceleration constant II 2 Max. iteration 1000
which can also be seen in Fig. 3.8. To study the robustness of the obtained PSS parameters, in this
case study we have implemented three different operating conditions and performed SSSA for them,
see TABLE 3.5. TABLE 3.6 shows the functionality of the proposed RPSS, detailing corresponding
operating conditions, critical modes, and critical damping ratios. Note that the optimization process
was carried out at the operating point specified as “Base” case. The numerical result has proven
the PSS parameters we obtained are robust, and in all cases with the proposed RPSS, the critical
damping ratios of the power system are improved. Alternatively, a new set of objective functions can
be proposed to ensure the robustness of the PSS settings in early stages of the optimization processes,
which however is not considered in this study. For details, see [76], [10] and [14].
Table 3.5: Operating conditions
Cases Base Heavy Light
Total Gen (MW) 17787 21491 14230
Total Load (MW) 17621 20803 13874
Table 3.6: Critical mode and damping ratio
Base Heavy Light
RPSS λcrit, ξcrit λcrit, ξcrit λcrit, ξcrit
No −0.087±j8.469, 1.0% 0.018±j8.492, −0.2% −0.157±j4.157, 3.8%
Yes −0.318±j8.282, 3.2% −0.266±j8.299, 3.8% −0.157±j4.157, 3.8%
0 20 40 60 80 1000.96
0.965
0.97
0.975
0.98
0.985
Iterations
CDI
Critical damping index
Figure 3.10: Evolution of PSO algorithm
Time-domain simulation results are shown in Fig. 3.11-Fig. 3.13. Note that the secondary control
scheme is implemented simultaneously with the above-mentioned control strategies. Specifically
speaking, Fig. 3.11 (a) and (b) demonstrate the frequency variations of generator bus-bars 10 and 11 in
case study 1 with different control strategies. It is evident that without any primary control strategy,
the frequencies oscillate throughout the entire simulation, and are severely infected with low-frequency
oscillations. It has been reported that persistent low-frequency oscillations in power systems could
lead to irreversible damage to power system components and even large-scale power failures [1]. With
untuned parameters and tie-line power control, the frequencies still experience some oscillations but
eventually settle down within the simulation time frame. This result also displays congruity with
37
CHAPTER 3. A NOVEL CONTROL STRATEGY OF DFIG WIND TURBINES INCOMPLEX POWER SYSTEMS
the system linear analysis performed earlier, where the system with controllers that have optimized
parameters bears a higher damping ratio. On the other hand, when using optimized controller without
tie-line power feedback, the oscillations are suppressed within 20 seconds, but the first and subsequent
swings have higher amplitudes. The best primary frequency response is generated when utilizing
the optimized control parameters and tie-line power deviation signals. Compared to other control
methods, the frequency response with this optimal controller demonstrates a higher nadir point with a
smoother transience in the first swing, and discernibly smaller amplitudes in subsequent swings. With
the secondary control strategy operating alongside the primary control scheme, frequencies of local
generators are all able to be restored to their nominal values. Simulation results for frequencies of
generator 12 and 13 are omitted here due to lack of space. As a matter of fact, their frequencies do not
oscillate so much as those of other local generators which have been shown, and thus this omission
does not affect our analysis.
In terms of the tie-line deviation of area 1, control results can be observed in Fig. 3.12 (a). The
optimized control strategy displays superior capability of restoring the tie-line power transmission
obligation in area 1, and shows the least oscillations and fastest settling time. Fig. 3.12 (b) shows
the significant reduction in fluctuation of tie-line power deviation of area 2. In fact, tie-line power
deviations of all areas are zeroed with substantial improvement on low-frequency oscillation mitigations.
0 20 40 60 80 1000.9994
0.9998
1.0002
1.0006
time (s)
ω10(p.u.)
No control Ctrl. with untuned para.+TLPD
Ctrl. with opt. para. Ctrl. with opt. para.+TLPD
(a)
0 20 40 60 80 1000.9994
0.9996
0.9998
1
1.0002
1.0004
time (s)
ω11(p.u.)
(b)
Figure 3.11: Simulation results of case study 1: Frequency of (a) generator 10 and (b) generator 11
Fig. 3.13 (a) illustrates active power variations at DFIG bus-bar during the simulation, and it is
clear that with the improved control strategy, DFIG wind farm is able to inject more power to the grid
in this increase-load case. As mentioned earlier, the DFIG mechanical components cannot harness
more kinetic energy than what the MPPT algorithm indicates, but the wind turbine generators are
capable of injecting additional electrical power and torque to the grid [81]. By observing Fig. 3.13, it
can be seen that when the rotor speed of DFIG wind turbine decreases, the DFWTG system is able to
produce more electrical power, i.e., release more power [81], than at steady state. Furthermore, with
only conventional DFIG controllers, the rotor speed and electrical power injected to the grid remain
38
3.5. CASE STUDIES
0 20 40 60 80 100−0.5−0.4
−0.2
0
0.2
time (s)
Ptl,area1(p.u.)
(a)
0 20 40 60 80 100−0.15
−0.050
0.1
0.2
time (s)
Ptl,area2(p.u.)
(b)
Figure 3.12: Simulation results of case study 1: TLPD of (a) area 1 and (b) TLPD of area 2
0 20 40 60 80 1000.55
0.65
0.75
time (s)
PDFIG
(p.u.)
(a)
0 20 40 60 80 1001.1
1.13
1.16
1.19
1.22
time (s)
ωr(p.u.)
(b)
Figure 3.13: Simulation results of case study 1: (a) Active power at DFIG WTG terminal bus-bar and (b) DFIGrotor speed
unchanged, and this explains the insufficiency in frequency support as noticed in earlier analysis. When
TLPD feedback is absent, the power curve displays converse characteristics and this also corroborates
the results seen in frequency and tie-line power restorations in Fig. 3.11 and Fig. 3.12. Another
simulation is carried out and shows DFIG rotor speed settles at around 400s, due to the slow dynamical
response of DFIG mechanical components. This result is not shown in the simulation time frame
of this study. Note that for a more comprehensive study of modal analysis, the observability and
controllability of modes should be investigated on a mathematical level. However, as far as this study
concerns, the observability is implied by the fact that low-frequency oscillations can be detected by
observing the frequency responses, and controllability is also proven by the fact that the damping of
the modes can be improved using the proposed controller [1]. A complete mathematical derivation of
mode observability matrix and mode controllability matrix can be found in [1].
39
CHAPTER 3. A NOVEL CONTROL STRATEGY OF DFIG WIND TURBINES INCOMPLEX POWER SYSTEMS
3.5.2 Case 2: Load Decrease
In the second case study, we implement a scenario where sudden decrease in load demand on bus
33 occurs at t = 2s, ∆P33 = −0.5p.u. (thus P33 = 0.62p.u). Fig. 3.14 and Fig. 3.15 demonstrate
the simulation results in this scenario. Apart from the differences in oscillation magnitudes, this
set of simulation results display a converse dynamic characteristics of frequency and tie-line power
shown in the first scenario. The proposed optimized PSS with TLPD control strategy is able to
significantly alleviate the low-frequency oscillations of the power system during abrupt load changes,
and simultaneously enhance the primary frequency response, indicated by a lower zenith point in the
first swing and smaller amplitudes in subsequent swings, which is shown in Figs. 3.14 (a) and (b).
With the implementation of secondary control, frequencies and tie-line power can be restored to their
nominal values, as shown in Figs. 3.14 and Figs. 3.15. In this scenario, using the optimized control
scheme, DFIG WTGs produces less power with a higher rotor speed during the fault, which can be
seen in Figs. 3.16.
0 20 40 60 80 1000.9996
0.9999
1.0002
1.0005
1.0008
time (s)
ω10(p.u.)
(a)
0 20 40 60 80 1000.9996
0.9999
1.0002
1.0005
time (s)
ω11(p.u.)
(b)
Figure 3.14: Simulation results of case study 2: Frequency of (a) generator 10 and (b) generator 11
0 20 40 60 80 100−0.1
0.1
0.3
0.5
time (s)
Ptl,area1(p.u.)
(a)
0 20 40 60 80 100−0.2
−0.1
0
0.1
time (s)
Ptl,area2(p.u.)
(b)
Figure 3.15: Simulation results of case study 2: (a) TLPD of (a) area 1 and (2) area 2
40
3.5. CASE STUDIES
0 20 40 60 80 1000.45
0.5
0.55
0.6
0.65
0.7
time (s)
PDFIG
(p.u.)
(a)
0 20 40 60 80 1001.17
1.2
1.23
1.26
1.28
time (s)
ωr(p.u.)
(b)
Figure 3.16: Simulation results of case study 2: (a) Active power at DFIG WTG terminal bus-bar and (b) DFIGrotor speed
3.5.3 Case 3: New Power System Configuration
In order to further test the functionality and capability of the proposed DFIG-PSS control strategy,
we introduce a new configuration of the DFWTG-connected power system and connect the DFIG
wind farm to bus 32, as shown in Fig. 3.6. The operating condition is the same as that of Case 2
where a sudden decrease of load demand occurs on bus 33, at t = 2s. The new connection leads to a
new set of analytical result of small signal stability analyses. TABLE 3.7 and Fig. 3.17 demonstrate
the eigenvalues of interest of the power system without any controller implemented. Evidently, the
critical damping is 0.8% and the critical mode is an electromechanical mode whose dominant states
are the rotation-related states of G10 and G8. We now incorporate two control scenarios regarding
the proposed RPSS method, utilizing two and four frequency deviations in (3.29) respectively, and
also an interaction study where the interactions among proposed DFIG-PSS and local SG-PSSs are
investigated.
Table 3.7: Eigenvalues of interest relating to LFEOs without controller in Case 3
Eigenvalue λ f(Hz) ξ(%) Dominant states
−0.067± j8.309 1.3224 0.8 ∆ω and ∆δ of G10, G8−0.187± j6.122 0.9744 3.05 ∆ω and ∆δ of G9−0.141± j3.452 0.5494 4.08 ∆ω and ∆δ of G16, G14−0.175± j3.904 0.621 4.47 ∆ω and ∆δ of G13, G9−0.553± j12.331 1.963 4.48 ∆ω and ∆δ of G1, G8−0.114± j2.519 0.401 4.52 ∆ω and ∆δ of G13, G14, G15−0.533± j11.21 1.784 4.75 ∆ω and ∆δ of G11−0.348± j7.061 1.124 4.92 ∆ω and ∆δ of G12−0.252± j4.996 0.795 5.04 ∆ω and ∆δ of G15, G14−0.337± j6.429 1.023 5.24 ∆ω and ∆δ of G3, G2−0.479± j7.678 1.222 6.22 ∆ω and ∆δ of G2, G3−0.533± j8.483 1.350 6.27 ∆ω and ∆δ of G8, G10−0.499± j7.375 1.174 6.74 ∆ω and ∆δ of G5, G6−0.804± j9.709 1.545 8.25 ∆ω and ∆δ of G7, G6
41
CHAPTER 3. A NOVEL CONTROL STRATEGY OF DFIG WIND TURBINES INCOMPLEX POWER SYSTEMS
−1 −0.8 −0.6 −0.4 −0.2 0 0.20
2
4
6
8
10
12
← Critical mode
0.8% damping
← 4% damping line
Real part
Imaginarypart
Eigenvalues
Electromechanical modes
Figure 3.17: System eigenvalues without controller in Case 3
Case 3, Control Scenario 1
In this control scenario, we only utilize ∆f1 and ∆f2 in (3.29) to realize the LFOD enhancement of
the critical mode. TABLE 3.8 details the optimized parameters used in the designed RPSS control
method, where due to the stipulated control requirement υR3 and υR
4 are 0. From this table, it is easy
to deduce that frequency deviation of G10 plays a major role in causing the critical mode, however
there is no predominant contribution made by either of them, which differs from the situation in Case
1 and Case 2.
Table 3.8: Important parameters in Case 3 Control Scenario 1
KRpss 60.7 TR
11 0.9406 TR12 0.2763 TR
21 0.9188 TR22 0.1128
υR1 0.7280 υR
2 0.2727 υR3 0 υR
4 0
TABLE 3.9 and Fig. 3.18 demonstrate the resulting eigenvalues with the control scheme in this
scenario. It is evident that the critical damping ratio has increased significantly from 0.8%, as shown
in TABLE 3.7 and Fig. 3.17, to 2.7%.
Table 3.9: Eigenvalues of interest relating to LFEOs with the proposed controller (Case 3, Scenario 1)
Eigenvalue λ f(Hz) ξ(%) Dominant states
−0.224± j8.144 1.296 2.748 ∆ω and ∆δ of G10, G8−0.187± j6.122 0.974 3.05 ∆ω and ∆δ of G9, G5−0.142± j3.452 0.549 4.107 ∆ω and ∆δ of G16, G14−0.104± j2.52 0.401 4.109 ∆ω and ∆δ of G13, G14−0.172± j3.901 0.621 4.4 ∆ω and ∆δ of G13−0.572± j12.35 1.966 4.624 ∆ω and ∆δ of G1, G8−0.253± j4.996 0.795 5.048 ∆ω and ∆δ of G15, G14−0.373± j7.057 1.123 5.281 ∆ω and ∆δ of G12−0.344± j6.427 1.023 5.342 ∆ω and ∆δ of G3, G2−0.647± j11.292 1.797 5.721 ∆ω and ∆δ of G11, G10−0.478± j7.678 1.222 6.219 ∆ω and ∆δ of G2, G3−0.538± j8.512 1.355 6.308 ∆ω and ∆δ of G8, G10−0.499± j7.375 1.174 6.744 ∆ω and ∆δ of G5, G6−0.804± j9.709 1.545 8.25 ∆ω and ∆δ of G7, G6
42
3.5. CASE STUDIES
−1 −0.8 −0.6 −0.4 −0.2 0 0.20
5
10
← Critical mode
2.7% damping
← 4% damping line
Real part
Imaginarypart
Eigenvalues
Electromechanical modes
Figure 3.18: System eigenvalues with optimized controller in Case 3 Control Scenario 1
Case 3, Control Scenario 2
In this control scenario, we make use of all four frequency deviations in local areas of DFIG in (3.29)
to achieve the control purpose. The table below (TABLE 3.10) shows the optimized parameters used
in the proposed PSS controller in Case 3. Different from TABLE 2.4, it is clear that in this control
scenario, there is no predominant contributions made by a single generator; rather, frequency deviations
of both G10 and G13 play important roles in the low-frequency oscillation phenomena. Using the same
procedures in Case 1 and Case 2, the critical gain, robustness and validity of the optimized parameters
utilized in the proposed DFIG-PSS control structure can be proven.
Table 3.10: Important parameters in Case 3, Control Scenario 2
KRpss 80 TR
11 0.6746 TR12 0.1437 TR
21 1 TR22 0.1463
υR1 0.6729 υR
2 0.0549 υR3 0.0001 υR
4 0.2721
With the proposed controller, the eigenvalues of interest are shown in TABLE 3.11 and Fig. 3.19.
Obviously, the critical damping ratio of the power system has increased from 0.8% to 3.05%, displaying a
better damping performance than in Case 3 Control Scenario 1 where only two frequency deviations are
considered. This simulation result further solidifies the capacity of the proposed DFIG-PSS controller
in enhancing low-frequency oscillation damping of the power system.
−1 −0.8 −0.6 −0.4 −0.2 0 0.20
2
4
6
8
10
12
← Critical mode
3.1% damping
← 4% damping line
Real part
Imaginarypart
Eigenvalues
Electromechanical modes
Figure 3.19: System eigenvalues with optimized controller in Case 3 Control Scenario 2
43
CHAPTER 3. A NOVEL CONTROL STRATEGY OF DFIG WIND TURBINES INCOMPLEX POWER SYSTEMS
Table 3.11: Eigenvalues of interest relating to LFEOs with the proposed controller (Case 3, Scenario2)
Eigenvalue λ f(Hz) ξ(%) Dominant states
−0.243± j7.936 1.263 3.05 ∆ω and ∆δ of G10, G8−0.339± j11.073 1.76 3.05 ∆ω and ∆δ of G11−0.187± j6.122 0.974 3.05 ∆ω and ∆δ of G9−0.0957± j2.526 0.402 3.79 ∆ω and ∆δ of G13, G14−0.149± j3.904 0.621 3.82 ∆ω and ∆δ of G13−0.142± j3.452 0.549 4.11 ∆ω and ∆δ of G16, G14−0.563± j12.35 1.966 4.56 ∆ω and ∆δ of G1−0.252± j4.996 0.795 5.04 ∆ω and ∆δ of G15, G14−0.372± j7.036 1.12 5.28 ∆ω and ∆δ of G12−0.344± j6.421 1.022 5.35 ∆ω and ∆δ of G3, G2−0.478± j7.678 1.222 6.22 ∆ω and ∆δ of G2, G3−0.564± j8.511 1.355 6.61 ∆ω and ∆δ of G8, G10−0.499± j7.375 1.174 6.74 ∆ω and ∆δ of G5, G6−0.804± j9.709 1.545 8.25 ∆ω and ∆δ of G7, G6
Time-domain simulation results are shown in Fig. 3.20 and Fig. 3.21, and similar comments on
control performances can be drawn as in Case 1 and Case 2. The proposed DFIG-PSS control structure
is able to mitigate the low-frequency oscillations by reducing the fluctuations in the frequency curves
and also enhance the primary frequency response by moderating the amplitudes of the first and
subsequent swings. This case study has verified the functionality of the proposed controller and the
parameter optimization method, demonstrating its flexibility on various system configurations.
0 20 40 60 80 100
0.9998
1
1.0002
1.0004
time (s)
ω10(p.u.)
(a)
0 20 40 60 80 1000.9998
1
1.0002
1.0004
time (s)
ω11(p.u.)
(b)
Figure 3.20: Simulation results of case study 3: Frequency of (a) generator 10 and generator 11
44
3.5. CASE STUDIES
0 20 40 60 80 100
0
0.2
0.4
time (s)
Ptl,area1(p.u.)
(a)
0 20 40 60 80 100
0.5
0.6
0.7
time (s)
PDFIG
(p.u.)
(b)
Figure 3.21: Simulation results of case study 3: (a) TLPD of area 1 and (b) Active power at DFIGWTG terminal bus-bar
Case 3, Interactions among DFIG-PSS and Local SG-PSSs
In this subsection, the interactions among the newly designed DFIG-PSS control mechanism and
PSSs mounted with local synchronous generators are studied. To perform this study, three PSSs are
introduced and equipped on all generators in Area 1, but G13 due to its manual excitation system. All
parameters of PSSs are adopted from [88] where investigations into PSSs in the IEEE 68-bus test system
have been reported in detail. The philosophy and methodology of performing this interaction study
are based on articles [89, 90]. Note that this interaction study only incorporates damping enhancement
analyses, whereas primary response improvement study is omitted.
With this new configuration, a new SSSA needs to be performed. TABLE 3.12 shows the eigenvalues
of the power system without any controllers, incorporated with additional PSSs. Apparently, with the
newly added PSSs on G10, G11 and G12, the local modes are improved significantly, in comparison
with modes shown in TABLE 3.7.
We now implement the proposed control method, but with a different objective function in the
optimization process. The new objective function is to ameliorate critical mode contributed solely
or partially by local generators, which as shown in bold in TABLE 3.12 is the electromechanical
inter-area mode contributed mainly by G13, G9 and G6. In order to test the impact of DFIG-PSS on
electromechanical modes relating to local generators, gain KRpss varies between KR
pss min = 1 × 10−5
and KRpss max = 1000. Note that KR
pss max is not the critical gain as no eigenvalue is located in the
instability region.
Fig. 3.22 depicts the poles’ movements as KRpss changes. It is easy to detect that all electromechanical
modes contributed solely or partially by local generators’ rotational dynamics are improved as KRpss
increases, whereas modes dominantly contributed by foreign generators are not noticeably influenced
with changing DFIG-PSS gains. This interaction study further verifies the ameliorative effects on
modes generated by the proposed DFIG-PSS strategy, and also proves the installation of the control
45
CHAPTER 3. A NOVEL CONTROL STRATEGY OF DFIG WIND TURBINES INCOMPLEX POWER SYSTEMS
Table 3.12: Eigenvalues of interest relating to LFEOs with the proposed controller in Case 3Interaction
Eigenvalue λ f(Hz) ξ(%) Dominant states
−0.187± j6.122 0.974 3.05 ∆ω and ∆δ of G9−0.153± j3.909 0.622 3.899 ∆ω and ∆δ of G13, G9, G6−0.141± j3.454 0.55 4.079 ∆ω and ∆δ of G16, G14−0.116± j2.517 0.401 4.619 ∆ω and ∆δ of G13, G14−0.58± j12.262 1.952 4.721 ∆ω and ∆δ of G1−0.253± j4.996 0.795 5.052 ∆ω and ∆δ of G15, G14−0.347± j6.439 1.025 5.374 ∆ω and ∆δ of G3, G2−0.479± j7.678 1.222 6.222 ∆ω and ∆δ of G2, G3−0.499± j7.375 1.174 6.744 ∆ω and ∆δ of G5, G6−0.632± j8.446 1.344 7.462 ∆ω and ∆δ of G8−0.804± j9.709 1.545 8.252 ∆ω and ∆δ of G7, G6−0.828± j9.534 1.517 8.656 ∆ω and ∆δ of G4, G5−1.783± j5.812 0.925 29.333 ∆ω and ∆δ of G10, G11−2.415± j5.801 0.923 38.442 ∆ω and ∆δ of G10, G11
−3 −2.5 −2 −1.5 −1 −0.5 00
5
10
Real part
Imaginarypart
Figure 3.22: Migration of modes listed in TABLE 3.7 with varying Kpss (‘x’: KRpss = KR
pss min, ‘o’:
KRpss = KR
pss max)
structure does not adversely affect the modes relating to local generators. Note that only eigenvalues
listed in the above table (TABLE 3.7) have been plotted, with non-electromechanical and far-left
modes omitted.
3.6 Conclusion
In this chapter, a novel DFIG-PSS control structure is proposed to simultaneously improve the primary
frequency response and low-frequency oscillation damping of a DFIG-integrated complex power system.
The conventional power regulation method in DFIG rotor side controller is modified to receive tie-line
power deviation signals to enhance the primary frequency response, and the proposed PSS with weighted
frequency deviations of local generators is integrated with the voltage regulator in the conventional
DFIG rotor side controller. Control parameters and weighting coefficients are optimized using particle
swarm optimization algorithm to achieve the highest critical damping ratio of the system. Through
multiple case studies, the proposed control method has shown significant improvement on low-frequency
oscillation damping and primary frequency response of the multi-area interconnected complex power
systems integrated with wind farms over conventional DFIG control strategies, demonstrating high
46
3.6. CONCLUSION
tolerance of external disturbances and system configurations. The interaction study verifies that the
proposed DFIG-PSS control method is beneficial to the system modes contributed solely or partially
by the electromechanical dynamics of local generators which are equipped with AVR-PSSs. The newly
designed control scheme is integrated with the embedded rotor side controller and power converters of
DFIG, and does not require additional AVR or FACTS devices, which poses its economic advantages.
Future work may involve implementing the proposed control mechanism to complex power systems
integrated with renewable energy sources, taking into consideration wide-area control signals, and also
communication delay compensation.
47
CHAPTER 3. A NOVEL CONTROL STRATEGY OF DFIG WIND TURBINES INCOMPLEX POWER SYSTEMS
48
Chapter 4
An Enhanced Adaptive Phasor Power
Oscillation Damping Approach with
Latency Compensation for Modern
Power Systems
ABSTRACT
In this chapter, a novel enhanced adaptive phasor power oscillation damping (EAPPOD) strategy is
proposed, which is capable of compensating time-varying data transmission latencies between phasor
measurement unit (PMU) sites and the control center, and also mitigating the low-frequency oscillations
(LFO) of inter-area signals. The proposed method can handle general communication delay-related
problems and fulfill LFO mitigation tasks in modern power systems, and in this study this method is
integrated with the doubly-fed induction generator (DFIG) rotor-side control (RSC) scheme to achieve
the control purpose. The control signal is produced for the purpose of minimizing the amplitude of
the phasor component disaggregated from the measured signal, using a novel signal decomposition
algorithm. It is then transmitted to the active power regulation scheme in the DFIG RSC structure to
modulate the power reference value, so as to realize LFO mitigation. Improving upon the recently
established APPOD method, the EAPPOD strategy incorporates a series of integral newly designed
methods, including average assignment, phase tracking and magnitude attenuation, to overcome the
limitations of the APPOD method operating in varying-latency situations, and consequently to achieve
a better LFOD performance. The newly proposed EAPPOD method will thus benefit both online
power system monitoring and LFOD enhancement.
49
CHAPTER 4. AN ENHANCED APPOD APPROACH WITH LATENCYCOMPENSATION FOR MODERN POWER SYSTEMS
4.1 Chapter Foreword
The content of this chapter is mainly based on and modified from the following academic papers:
Shenglong Yu, Tat Kei Chau, Tyrone Fernando, and Herbert Ho-Ching Iu. “An Enhanced
Adaptive Phasor Power Oscillation Damping Approach With Latency Compensation for Modern Power
Systems” IEEE Transactions on Power Systems, vol.33, no.4, pp.4285-4296, 2017.
In the previous chapters, power-system-stabilizer-based damping controllers and the associated
parameters tuning method are discussed based on the assumption that real-time power system
information such as rotor frequencies of generators are directly accessible from the control center, and
the communication delays are ignored. However, in practical power systems, such information/data
is usually transmitted to the control center through packet-based communication networks, and the
associated network-related communication issues can lead to data degradation. Therefore, data pre-
processing is often required before any further manipulations. To address this problem, in this chapter,
an enhanced adaptive phasor power oscillation damping approach is proposed, which uses an adaptive
algorithm to compensate communication delays in order to achieve better LFOD control outcomes.
4.2 Introduction
Power grids around the globe are currently experiencing an unprecedented transformation as the
amount of electrical energy generated by renewable energy sources increases at a dramatical speed.
With this backdrop, it is of critical importance to ensure the stability of the new power systems
integrated with renewables. Among all the instability problems, inter- and local- area low-frequency
oscillations are a long-lasting issue that undermines the stability of power systems and may cause
structural damages and power outages in electric grids. A large number of academic articles have
proposed diverse methodologies in tackling this issue. In [91] an analysis is performed on inter-area
oscillation, which provides a comprehensive perspective in this research field. Power system stabilizers
(PSSs) coupled with automatic voltage regulators (AVRs) have been widely applied to synchronous
generators for the purpose of LFOD enhancement [1,67,68]. However, researchers have discovered that
not all system modes can be damped by conventional PSSs, which led to the invention of a control
structure comprised of Flexible Alternating Current Transmission System (FACTS) devices and a
series of auxiliary damping controllers that resemble the structure of PSSs. Recent work relating to
FACTS devices is dedicated to the modification and optimization of internal parameters and controllers
of FACTS to improve local-/inter- area LFOD, see [92]. Nevertheless, there has not been adequate
attention paid to the communication delay compensation which is of especial necessity when dealing
with wide-area oscillation mitigation, although research work reported in [69] and [70] has considered,
but has not yet compensated, the communication delay. Data transmission delay is normally controlled
within tens of milliseconds with contemporary technology, however longer communication latency
cannot be ignored when designing oscillation suppressors [93]. The communication delays can be
constant, time-varying or even random, resulting from the combination of network-induced delay,
packet dropout or disordering [94,95].
Phasor power oscillation damping is a methodology first introduced in [96], where the authors
posited that POD can be performed by utilizing an estimated phasor component disaggregated from
50
4.2. INTRODUCTION
a measured signal that reflects its oscillatory characteristics. An important step in fulfilling the
phasor-POD strategy is decomposing the oscillatory signal into an average part and a phasor part,
which was achieved by using a low-pass filter in [96]. After acquiring the phasor component of the
signal, a control system was constructed with the existing thyristor controlled series compensation
(TCSC) controller, a member of FACTS device family, for the purpose of minimizing the magnitude of
the phasor component. In [96], however, no detailed investigation was documented on communication
delay compensation. Inspired by this study, researchers in [93] proposed an adaptive phasor POD
method to adaptively compensate for communication latencies, and the control signal is then generated
and utilized as the input signal to TCSC to achieve the same control purpose. Thereafter, authors
in [97] put forward a modified recursive least square (RLS) algorithm to decompose the oscillatory
signal for online estimation of LFO, where however the latency compensation again was not performed
due to a different purpose of using POD. Since [93], the study of adaptive latency compensation
from a phasor sense seemed stagnant, which is evidenced by the lack of publications germane to this
methodology.
There are a number of research articles that have made attempts to mitigate LFOs in DFIG-wind-
farm-connected electric grids, see [75–77] for instance. There has been no reported research work on
using phasor-POD strategy on a wind-farm-integrated power system for LFOD enhancement. In this
study, for the first time the philosophy of phasor-POD is applied to a complex power system integrated
with a DFIG wind farm, which is capable of compensating time-varying data transmission latency and
suppressing LFOs that exist in inter-area signals of interest. The compensated measurements are then
employed in the design of the oscillation suppressor along with the control mechanisms embedded in
DFIG structure. The main contribution of this study is the proposition of an Enhanced Adaptive
Phasor Power Oscillation Damping strategy, in order to particularly tackle the issue of distortions in the
compensated signal caused by the variation of communication delays. When communication delay is
deemed constant, an improved recursive least square method with an adaptive forgetting factor (AFF)
is devised to perform the signal decomposition procedure, whereas when communication delay changes,
a series of new techniques, including phase tracking and magnitude attenuation, are proposed and
implemented to cope with the disruption of data flow, and consequently to achieve a better performance
of signal restoration and delay compensation. Particularly, the design of the phase tracking method is
to overcome the disadvantage of conventional phasor estimation method that impedes the estimation
process when there is no new data being received [93] and produces inaccurate estimates when an
incoherent flow of data arrive. The magnitude attenuation, on the other hand, is to reflect the decay of
magnitude in the phasor component when communication latencies vary, an important issue that has
been disregarded in the APPOD proposed in [93]. Since the phasor component reflects the oscillatory
characteristics of the signal in question, the control signal is produced for minimizing the amplitude of
the phasor component disaggregated from the measured signal. Furthermore, the proposed EAPPOD
algorithm is integrated to the RSC of the DFIG structure to modify the active power regulator, and
does not require any additional power electronic equipment including AVR-PSS and FACTS devices, as
documented in [9]. Different from the established method in [93] which modifies power factor through
changing reactive power by using TCSC, the proposed method in this study modulates the active
power reference inside the power regulation loop in the RSC of DFIG structure. when LFO does not
51
CHAPTER 4. AN ENHANCED APPOD APPROACH WITH LATENCYCOMPENSATION FOR MODERN POWER SYSTEMS
exist in the power system, the reference value of DFIG’s active power regulator follows the output of
the MPPT algorithm which reverts to the original control strategy of DFIG, hence creating least effect
to the operation of DFIG and also the power system.
The remainder of the chapter is organized as follows. In Section 4.3, the basic description of DFIG
and its conventional control schemes are presented briefly. The proposed EAPPOD strategy with
DFIG RSC is detailed in Section 4.4. Section 4.5 presents two distinct case studies with simulation
results analyzed and compared. Finally, a conclusion is drawn in Section 4.6.
4.3 Mathematical Model and Conventional Control Strategies of
DFIG
GearBox
AG
Rotorsidecontrollerconverter DCLink
GridsidecontrollerconverterPitch
Drive
Pitchanglecontroller
Multi‐AreaPowerSystem
Computation
EAPPODcontrolcenter bus
bus
areaarea
IOS PMU
PMU
Figure 4.1: DFIG with proposed EAPPOD controller connected to a multi-area interconnected powersystem
Fig. 4.1 depicts the structure of doubly fed induction generator wind turbine. As shown in the
figure, the DFIG is connected to a multi-area power system, and multi-area measurements are employed
as inputs of the newly proposed EAPPOD controller. The control signal generated by the EAPPOD
controller is then fed to the rotor side controller of the DFIG. In this section, brief descriptions of
DFIG mathematical model and conventional DFIG control strategies are presented.
In this study, a single-mass model of the turbine drive train is adopted, and the overall mathematical
expression is shown as follows [80,98],
dωrdt
=1
2Heq(Γm − Γe − Fωr), (4.1)
where Γe is the electrical torque, Heq is the equivalent generator inertia, and F is the friction factor.
Γm is the mechanical torque harnessed by the wind turbine which can be calculated using (3.1)-(3.3).
The mathematical model of an asynchronous generator (AG) is detailed in Chapter 3.
As discussed in Chapter 3, the conventional DFIG control structure comprises rotor side control
and grid side control schemes. The block digram of the conventional RSC is shown in the lower half
of Fig. 4.2. Together with the newly proposed EAPPOD control method. the RSC scheme will be
52
4.4. PROPOSED EAPPOD FOR DFIG-INTEGRATED POWER SYSTEMS
discussed in Section 4.4.
4.4 Proposed EAPPOD for DFIG-Integrated Power Systems
Fig. 4.1 in Section 4.3 shows the general operating principle of the proposed enhanced adaptive phasor
power oscillation damping method. Fig. 4.2 demonstrates the detailed control mechanism coupled
with conventional DFIG control strategy. The proposed LFOD control strategy incorporates signal
decomposition, adaptive latency compensation, and LFO mitigation functionalities into the existing
DFIG RSC structure. For detailed explanations and mathematical models of DFIG control methods,
see [99] and [100].
In order to improve the recently proposed APPOD strategy in [93], a number of novel features are
designed and implemented in the proposed EAPPOD method. Particularly speaking, two new methods
are designed and implemented in the “signal decomposition” block to overcome the data absence when
latency increases and arrival of incoherent data when delay decreases. Two additional variables are
introduced in the “adaptive latency compensation” block to generate a better performance of signal
restoration. Furthermore, on the LFOD front, the compensated signal is sent to the control mechanism,
instead of the delayed signal as reported in [93]. All the new features will be detailed in this section.
The superiority of the proposed EAPPOD over the established APPOD in signal reconstruction and
LFOD enhancement will be illustrated in Section 4.5.
Control signal
Clock
Frequency adaptation
EAPPOD-based LFOD controller
Conventional DFIG RSC
Signal decomposition
Latency compensation
Phase shift
Power converter
PWM
Figure 4.2: Proposed EAPPOD method coupled with DFIG RSC structure
53
CHAPTER 4. AN ENHANCED APPOD APPROACH WITH LATENCYCOMPENSATION FOR MODERN POWER SYSTEMS
4.4.1 Signal Decomposition
The signal decomposition method employed in this study is based on an adaptive phasor POD concept
and it is predicated on the assumption that a measured signal χ(t), a function of time, can be
represented as a space-phasor and has the following relations [97],
χ(t) = χav(t) + χosc(t), (4.2)
χosc(t) = ReχphejΩt
= χph,d cos(Ωt+ α)− χph,q sin(Ωt+ α), (4.3)
where χav and χosc are the estimated average and oscillatory component of the measured signal
respectively. Terms χph,d and χph,q denote the direct-quadrature components of the estimated phasor
of the oscillatory signal, Ω is the oscillatory frequency, i.e., frequency of LFO in this study, and angle
α will be calculated later in this section.
The adaptivity of the EAPPOD is expressed through two key features, one of which is the adaptation
of the estimated frequency Ω. Although the oscillatory frequency can be approximated by using the
acquired eigenvalues of identified modes through linear analysis of the system, the actual frequency at
which LFO oscillates is difficult to directly obtain, especially when the oscillations are contributed by
multiple modes. Therefore, in this study, the estimated frequency Ω comprised of an initial assignment
of frequency Ω0 and a frequency correction element ∆Ω which is estimated by a “frequency adaptation”
mechanism, as shown in Fig. 4.2 [93]. To obtain a reasonable initial frequency, small signal analysis
(SSA) can be performed, which indicates modal information and oscillations frequency of each mode.
The frequency obtained from SSA can be employed as an initial assignment for Ω0.
With constant communication latency, at kth iteration, a modified Recursive Least Square (RLS)
estimation algorithm with an adaptive forgetting factor or AFF-RLS method, inspired by [97], can
be utilized to perform the signal decomposition task. Based on (4.2) the estimated state vector Υ is
derived using measured signal χ[k] and observation matrix ψ[k] as follows [101],
Υ[k] = Υ[k − 1] +K[k]ε[k], (4.4)
ε[k] = χ[k]− ψ[k]Υ[k − 1], (4.5)
and the estimated state vector Υ[k] and the observation matrix ψ[k] are formed by
Υ[k] = [χav[k], χph,d[k], χph,q[k]]T , (4.6)
ψ[k] =
1
cos(Ω[k]t+ α[k])
− sin(Ω[k]t+ α[k])
T
. (4.7)
54
4.4. PROPOSED EAPPOD FOR DFIG-INTEGRATED POWER SYSTEMS
In (4.4), the gain matrix K[k] is obtained with the following equation,
K[k] =Q[k − 1]ψT [k]
υ[k] + ψ[k]Q[k − 1]ψT [k], (4.8)
Q[k] =I −K[k]ψ[k]
υ[k]Q[k − 1], (4.9)
where Q is the covariance matrix, I is the identity matrix with an appropriate dimension, and υ[k] is
the forgetting factor (0 < υ < 1), which is obtained by the following relation,
υ[k] = υss − (υss − υtr)e−KffCff[k]Ts , (4.10)
where υss and υtr are the steady state and transient forgetting factors respectively, and Cff is a counter
variable, obtained by
Cff[k] =
Cff[k − 1] + 1 if |ε[k]| < εth,
0 otherwise,(4.11)
where εth is an error threshold value. Term 1/Kff in (4.10) is an appropriately chosen mean lifetime,
which is 5 in this study. This lifetime is for the control scheme to decide when to reset the forgetting
factor.
With the obtained χph,d[k] and χph,q[k], the magnitude and phase angle of the estimated phasor
component of the oscillatory signal (χph[k]α[k]) can then be calculated accordingly.
The above RLS-based signal decomposition and reconstruction method is applicable in the situations
where constant smooth data flow is available, i.e., latency is considered constant. However, a large
variation in communication delay can lead to sudden changes in the resultant gain and covariance
calculated in the above algorithm, due to either the lack of data when latency increases or incoherent data
flow when it decreases. This will cause an inaccurate reconstruction of the measured signal, which may
lead to ineffective latency compensation. Therefore, two newly proposed methods – average assignment
and phase tracking–are implemented in the “signal decomposition” block, and magnitude attenuation
method in the “latency compensation” block, so as to produce a more favorable reconstructed and
compensated signal. The algorithm below depicts the improved signal decomposition strategy at kth
iteration, which is branded as the enhanced adaptive phasor power oscillation damping method. Two
new variables are introduced in the algorithm and will be used in the “latency compensation” block: a
counter variable Cda to obtain the number of iterations of data absence and an attenuation factor Fatt
to account for magnitude decay of the phasor component.
Proposed Signal Decomposition Algorithm
Step 1: Data Detection
At the control center, data receiver detects if a new set of time-stamped PMU data have arrived at current iteration.
If new data are detected, store the new PMU data and their time stamps. Go to Step 2.
If no new data are detected, indicating an increased latency in data transmission, then assign the PMU measurement
and communication delay of the last iteration to the current one. Increment counter variable Cda located at a
dedicated accumulator, i.e., Cda[k] = Cda[k− 1] + 1. The counter variable will be used in the “latency compensation”
block. Go to Step 4.
55
CHAPTER 4. AN ENHANCED APPOD APPROACH WITH LATENCYCOMPENSATION FOR MODERN POWER SYSTEMS
Step 2: Communication Latency Calculation
Reset the counter variable, i.e., Cda[k] = 0.
Calculate communication latency Td[k] and the change in time delay, i.e., ∆Td[k] = Td[k]− Td[k − 1], using the
stored time stamps in Step 1 and the satellite-synchronized local clock.
If ∆Td[k] < −Tth, indicating a significant reduced time delay in data transmission, where term Tth(> 0) is a
delay-time variation threshold value determining the necessity of estimation correction (which is 50ms in this
study), then proceed to Step 4. Estimation correction encompasses both phase tracking and average assignment
procedures.
Otherwise, the communication delay is considered constant. Go to Step 3.
Step 3: Constant Latency Method
Perform RLS-based signal decomposition algorithm as has been shown in (4.4)∼(4.10) and obtain χav[k], χph,d[k]
and χph,q[k].
Go to Step 6.
Step 4: Average Assignment
If ∆Td[k] = 0, i.e., Td[k] = Td[k − 1] (from Step 1 last bullet point), it indicates there are no new data detected
and thus communication latency has increased. Assign the estimated average component from last iteration to the
current one, i.e., χav[k] = χav[k − 1].
If ∆Td[k] < 0, inferring a decreased communication delay, then the average component is calculated as χav[k] =
χ[k]− χosc[k − 1], with (4.2).
Proceed to Step 5.
Step 5: Phase Tracking
In case of varied communication latency, a phase tracking method is implemented to cope with the data absence or
incoherence issues.
Assign the magnitude of estimated phasor component from last iteration to the current estimate, i.e., χph[k] =
χph[k − 1]. The inaccuracy of this approximation will be compensated in the “latency compensation” block with
the attenuation factor.
Manipulate angle α[k] using the following equation,
α[k] = α[k − 1]− (Ω[k]Ts + Ω[k]∆Td[k]). (4.12)
It is obvious that when communication latency increases, ∆Td[k] = 0 due to the value retainment in Step 2, while
when latency reduces, a negative ∆Td[k] will participate in (4.12).
χph,d[k] and χph,q[k] can then be obtained with χph[k] and α[k] through a simple trigonometric relation.
Go to Step 6.
Step 6: Results Collection and Transmission
Store counter variable Cda[k] and resultant estimates χav[k], χph,d[k] and χph,q[k].
Calculate magnitude attenuation factor Fatt[k] (< 0) with the following equation,
Fatt[k] =1
Natts
k∑i=k−Natt
s +1
log
∣∣∣∣ χph[i]
χph[i− 1]
∣∣∣∣ , (4.13)
where Natts [k] is the number of samples used in the FIR filter for the calculation of the attenuation factor. The
existence of the attenuation factor Fatt is postulated and extrapolated on the premises that the phasor component
of the measured signal is subject to an approximate exponential decay [102].
Transmit Cda[k], χav[k], χph,d[k], χph,q[k] and Fatt[k] to the “latency compensation” block.
Increment k.
56
4.4. PROPOSED EAPPOD FOR DFIG-INTEGRATED POWER SYSTEMS
Go to Step 1 to perform the next iteration with the stored results.
Note that the threshold value Tth is application-based, which depends on the subsequent manipula-
tion of the the recovered signals. When the latency variation is larger than tens of milliseconds, if not
appropriately compensated, it will jeopardize the LFOD control scheme [93]. Fig. 4.3 is a simplified
representation of the proposed signal decomposition method. Note that for the purpose of this study,
only the distortions in measured signals caused by communication delay variations are considered.
Other data quality issues resulting from the combinations and interactions of the many components of
the end-to-end synchrophasor measurement and data delivery process are disregarded [103].
Data detection
Data arrived?
YReset
Calculate
Latency deceased?
Y
Average assignment
NLatency increased
Phase tracking
RLS‐based algorithm
Increment
Δ
, ,
Latency constant
N
Figure 4.3: Flowchart of the proposed signal decomposition method
4.4.2 Adaptive Latency Compensation
The second feature manifesting the adaptivity of the control scheme lies in its capability of compen-
sating time-varying communication latencies With the newly introduced counter variable Cda and
attenuation factor Fatt in Section 4.4.1, the following relations are used to compensate for time-varying
communication latencies,
α = α+ Ω (Td + CdaTs) , (4.14)
χph ≈ χpheFattTd , (4.15)
χph,d + jχph,q = χph∠α, (4.16)
57
CHAPTER 4. AN ENHANCED APPOD APPROACH WITH LATENCYCOMPENSATION FOR MODERN POWER SYSTEMS
and χph,d and χph,q can then be calculated accordingly. It is obvious that when latency is constant,
i.e., Cda = 0, then (4.14) reverts to APPOD method by the rotation of reference frames. The
proposed EAPPOD method can not only rotate the reference frame adaptively in response to various
communication delays, but also be able to overcome the inaccuracies and distortions in the APPOD
strategy when latency changes. Magnitude χph will be utilized in the oscillation damping mechanism.
4.4.3 Low-Frequency Oscillation Mitigation Mechanism
The goal of LFO mitigation mechanism in this study is to minimize the magnitude of the estimated
phasor component of the measured signal, i.e., χph, which is calculated in Section 4.4.2. Therefore the
resultant signal % in Fig. 4.2 is acquired with the following equation,
% = Kphp (χref
ph − χph) +Kphi
∫(χref
ph − χph), (4.17)
where the reference value χrefph = 0. The error signal leads to a further phase shift by angle % [96], and
based on the derivation before, it is easy to obtain the following relation,[χ′ph,d
χ′ph,q
]=
[cos % − sin %
sin % cos %
][χph,d
χph,q
]. (4.18)
Then the control signal to be transmitted to the DFIG RSC structure Ppod is obtained by transforming
χ′ph∠α′ = χ′ph,d + jχ′ph,q back to the stationary d− q reference frame, which can be achieved with the
following relation,
Ppod =[
cos(Ωt+ α′) − sin(Ωt+ α′)] [ χ′ph,d
χ′ph,q
], (4.19)
and this signal resembles a power deviation signal that modulates the active power regulation scheme.
As shown in Figs. 4.2, the RSC is performed in mutual flux reference frame to regulate active power
and the terminal voltage of DFIG wind power generator, which has been detailed in [99, 100] and [73].
The output signals are the d− q components of the rotor side voltage.
4.5 Case Study
The proposed EAPPOD method is theoretically able to implement in a wide range of actuators
including TCSCs [104], flywheels [91], FACTS devices [69], and DFIG which is studied in this chapter.
As shown in Fig. 4.2, in the control mechanism, the inter-area oscillation signal (IOS) is transmitted
to EAPPOD control center where the measurement is recovered from communication latencies and a
control signal is produced and transmitted to DFIG. The control signal resembles power surplus or
deficiency upon the occurrence of electrical disturbances, which is used to modulate the output power
of DFIG RSC structure to alleviate the oscillatory characteristics of the IOS.
In this section, the proposed EAPPOD strategy is implemented in two case studies: (1) a modified
IEEE 68-bus power system integrated with a DFIG wind farm, and (2) a modified Kundur’s two-area,
four-generator test system with DFIG WTGs for LFOD enhancement. Two cases are implemented to
58
4.5. CASE STUDY
test the applicability of the proposed DFIG-integrated EAPPOD structure for power systems with
low and high DFIG-based wind power penetrations. The wind energy penetration however does
not affect the performance of the proposed EAPPOD method in latency compensation. Note that
DFIG-integrated power system is only an example of incorporating the proposed EAPPOD method,
which can also be coordinated with other actuators such as FACTS devices, e.g., TCSC [104]. Modal
analysis is conducted based on eigenvalue sensitivity among the states relating to generators’ rotational
mechanisms. The study is carried out in MATLAB® 2015b coding environment, on a desktop computer
with Intel® Core i7-4790 CPU, 8G RAM, and Microsoft® Windows 7 64-bit operating system.
4.5.1 Case 1 : Modified IEEE New England 68-Bus, 10-Generator Test Power
System
G77
23
6
G6
22
21
68
24
20
194
G4
5
G5
G33
62
65
63
66
67
37
64
58
G22
59
60
57
56
52
55
G9
9
29
28
26
27
25
1
54
G1
8
G8
ModifiedIEEE68‐bustestsystem
Area2
61
13G13
17
12G12
36
30
34
43
44
39
45
35
51
50
33
32
11
G11
4938
46
10
G10
31
53
47 48
40
18
16G16
Area1
Area4
Swing
42
15
G15
Area3
G14
14
41
Area5
WTGs
69
Figure 4.4: Modified IEEE 68-bus, 16-generator power system integrated with DFIG WTGs
Fig. 4.4 demonstrates the network topology of the power system in question, where 16 synchronous
generators are located in 5 interconnected areas and the DFIG wind farm is connected to bus 69, which
is then connected to bus 31. Bus-bar 16 is the swing bus of the power system of interest. The nominal
output power of the DFIG wind farm is 63 MW, and the wind energy penetration in the mix of power
generation in local area (Area 1) is approximately 1%. The DFIG wind generator has sufficient capacity
to provide certain amount of power to cope with the post-fault oscillations in certain PMU signals,
caused by the small disturbance introduced in this study. A larger disturbance, if introduced, would
lead to the engagement of other generators present in the power system to provide additional power
for suppressing the oscillations. For detailed description of the test system, including parameters and
mathematical models, see [88].
Since bus 46 and bus 67 are load buses not generator buses, modal analysis cannot directly indicate
which generators contribute how much to their oscillations. It is however admitted that these oscillations
are a result of the propagation of generators’ electromechanical interactions. In other words, the
59
CHAPTER 4. AN ENHANCED APPOD APPROACH WITH LATENCYCOMPENSATION FOR MODERN POWER SYSTEMS
oscillations of angular signals of load buses are caused by the superposition of multiple generator
modes. The model adopted in this study can sufficiently represent the power system to be studied.
All generator parameters are adopted from [88] (SG) and [80] (DFIG), and detailed mathematical
model of synchronous generators can be found in [59]. The simulation runs for 30s. In the first 1
second, the power system operates at steady state, and when t = 1s, a sudden decrease happens
in the load connected to bus 33. The nominal value of the load is 1.12 p.u. and it decreases by
0.5p.u. PMUs transmit time-stamped measurement data to the control center at a sampling rate
of Ts = 10ms. The synchronous generators in the test system are already equipped with secondary
control mechanisms, which are able to restore the frequencies of bus-bars to their nominal values during
electrical disturbances. In this section, however, only the LFOD enhancement functionality of the
EAPPOD method is illustrated, and the secondary control strategies are omitted, which can be found
in [59].
To perform the simulation study, without loss of generality, two inter-area oscillation signals (IOS
in Fig. 3.1) are defined as χ1 = θ46 − θ16 and χ2 = θ67 − θ16, where θ is the phase angle of voltage
of corresponding bus-bars, and θ46 is from a load bus local to the DFIG wind farm, θ67 is from an
external area, and θ16 is the phase angle of the swing bus. The delay compensation and control of χ1
will be discussed in detail, whereas χ2 will only be briefly mentioned in Section 4.5.1 due to lack of
space.
Table 4.1: Important Parameters for Proposed Method in Case 1
Kphp 0.1 Kph
i 3 χav[0] 0
χph,d[0] 0 χph,q[0] 0 Ω0 2π · 0.4υss 0.9875 υtr 0.8975 Kff 20
TABLE 4.1 presents important parameters used in the proposed EAPPOD method. The following
table (TABLE 4.2) demonstrates the modes for the given test power system, and it can be seen that
this power system is stable. Since the purpose of this study is to damp low-frequency oscillations in
inter-area oscillatory signals, only a disturbance, not electric fault, is introduced in this study to test
the functionality of the proposed EAPPOD strategy. Therefore, instability issue is not discussed in
this study. Detailed procedures of performing small signal analysis are fully documented in [50].
Signal Reconstruction in Case 1
As explained above, χ is a series of time-stamped PMU measurements with data transmission delay
from their origin to the EAPPOD control center. To simplify the study, it is assumed that the time
represented by the “clock” block in Fig. 4.2 is synchronized throughout the entire power system, and
all PMUs are satellite-synchronized.
Fig. 4.5 demonstrates the latency of data transmission in this study, with detailed delay time periods
presented. A wide range of time-varying communication latencies are implemented to test and verify
the functionality of the proposed EAPPOD, and all delay time durations are reasonable and applicable
in real-world applications [93]. In order to demonstrate the functionality of the proposed latency
compensation strategy, all variations in communication delay is well above the threshold value Tsh in
the aforementioned algorithm. It is easy to see that when latency increases data absence occurs during
60
4.5. CASE STUDY
Table 4.2: Original eigenvalues of interest relating to LFOs without controller in Case 1
Eigenvalue λ f(Hz) ξ(%) Dominant states
−0.087± j8.469 1.348 1.025 ∆ω and ∆δ of G10−0.155± j3.839 0.611 4.033 ∆ω and ∆δ of G13−0.102± j2.506 0.399 4.052 ∆ω and ∆δ of G13 and G9−0.140± j3.435 0.547 4.071 ∆ω and ∆δ of G14 and G16−0.268± j6.109 0.972 4.388 ∆ω and ∆δ of G9−0.372± j7.834 1.247 4.746 ∆ω and ∆δ of G1 and G8−0.251± j4.996 0.795 5.012 ∆ω and ∆δ of G14 and G15−0.371± j7.059 1.123 5.245 ∆ω and ∆δ of G12−0.341± j6.386 1.123 5.336 ∆ω and ∆δ of G2 and G3−0.65± j11.263 1.793 5.763 ∆ω and ∆δ of G11−0.508± j7.645 1.217 6.635 ∆ω and ∆δ of G2 and G3−0.682± j9.608 1.529 7.078 ∆ω and ∆δ of G1 and G8−0.530± j7.372 1.173 7.171 ∆ω and ∆δ of G5 and G6−0.871± j9.714 1.546 8.932 ∆ω and ∆δ of G6 and G7−0.867± j9.513 1.514 9.079 ∆ω and ∆δ of G4 and G5
the latency built-up period at the control center, which is marked with dotted line, whereas when time
delay decreases, a sudden incoherent data flow will arrive at the control center. As mentioned in the
Introduction, data transmission delays can be time-varying due to network-induced delay and packet
dropout or disordering [94,95]. In this study we implement a wide range of data transmission latencies
and extreme cases of delay time and variation in delay time are incorporated to test the functionality
of the proposed EAPPOD method. The communication delays incorporated in this study can reflect
realistic situations and the range of latency variations is sufficiently large to represent extreme cases
in practical situations [93]. The proposed signal decomposition-reconstruction method is designed to
resolve the inaccuracy of signal estimates obtained from previously established adaptive phasor POD
method.
0 5 10 15 20 24 30
0
0.2
0.4
0.6
0.8
Td=20ms Td=400ms Td=100ms Td=800ms Td=40ms Td=300ms
latency
increases
latency
increaseslatency
increases
latency
decreases
latency
decreases
time (s)
χ1(o)
Signal measured at PMU site Delayed signal at control center
Figure 4.5: Measured and received signal at control center in Case 1
Fig. 4.6 shows the reconstructed signal using APPOD method proposed in [93] and the EAPPOD
in this study. From the simulation result, it is evident that the EAPPOD method is able to better
restore the delayed signal than the APPOD with both constant and changing communication latencies.
This can be better observed with the magnified curves shown in Fig. 4.6 (a)∼(d) which illustrate
the estimation and reconstruction results when delay time changes. The differences between the two
reconstructed signals will lead to more noticeable discrepancies in the latency compensation results,
which will be discussed in Section 4.5.1. As mentioned in Section 4.4.1, in the “signal decomposition”
61
CHAPTER 4. AN ENHANCED APPOD APPROACH WITH LATENCYCOMPENSATION FOR MODERN POWER SYSTEMS
0 5 10 15 20 24 30
0
0.2
0.4
0.6
0.8
(a)
(b)
(c)
(d)
time (s)
χ1(o)
Signal at control center Signal Recons. with APPOD
Signal Recons. with EAPPOD
(a) (b) (c) (d)
Figure 4.6: Reconstructed signal with two methods in Case 1
block the EAPPOD implements an AFF-RLS mechanism in constant-latency situations, and phase
tracking and average assignment methods when communication delay varies. After being decomposed,
the acquired signal at the control center is then reconstructed with (4.2). To mimic the method
employed in [93], a constant forgetting factor is used (υ = υss = 0.9755). Fig. 4.7 illustrates the
variations of the forgetting factor υ in this study. This result displays good consistency with the study
in [97] where the concept of adaptive forgetting factor was introduced, but realized with a different
strategy.
0 5 10 15 20 25 30
0.85
0.9
0.95
(s)
υ
Figure 4.7: Forgetting factor variations in Case 1
Communication Latency Compensation in Case 1
As detailed in Section 4.4.2, the proposed EAPPOD is designed to compensate time-varying communi-
cation latencies and restore the measured signal with a more favorable performance than the established
method. The adaptive feature of the proposed method enables the latency-compensation mechanism to
rotate the reference frame with an adaptive angle corresponding to the estimated frequency of the LFO
and delay periods. With additional angle and magnitude manipulations in the “latency compensation”
block in Section 4.4.2, using data absence counter Cda and attenuation factor Fatt, the EAPPOD
method is seen to be able to cope better with the situations where communication latency varies.
Fig. 4.8 illustrates the differences between the original signal measured at the PMU site and the two
62
4.5. CASE STUDY
compensated signals. Figs. 4.8 (a)∼(d) are zoomed-in curves of the comparison. It is evident that when
0 5 10 15 20 24 30
0
0.2
0.4
0.6
0.8
(a)
(b)
(c)(d)
time (s)
χ1(o)
Signal measured at PMU site Compensated signal with APPOD
Compensated signal with EAPPOD
(a) (b) (c) (d)
Figure 4.8: Compensated signal in Case 1
latency is constant, both APPOD and EAPPOD produce a comparable signal restoration performance,
whereas when communication delay changes the superiority of EAPPOD becomes prominent. Particu-
larly, when delay rises, as shown in Figs. 4.8 (a) and (c), with the phase and magnitude manipulations
on the extracted phasor component (see (4.14) and (4.15)), the compensated signal is able to roughly
track the original measurement though measurements are unavailable. Similarly, Figs. 4.8 (b) and (d)
demonstrate the cases where latency reduces, and with the same stratagem, the EAPPOD is able to
eliminate the spikes and deviations that exist in APPOD. The result for APPOD displays consistency
with the one reported in [93]. It is worth mentioning that there are still noticeable offsets between the
original and the compensated signals, which exist due to the lack of sophistication in the arrangement
of the average component χav when delay changes. As a matter of fact, the magnitude of the average
component also varies over time, which however is not considered in this study, due to the fact that
the average component is not utilized in the POD-based LFOD enhancement method, and thus further
mathematical and computational costs can be saved.
Fig. 4.9 shows the differences in estimating the phasor part of the signal. This simulation result
further solidifies that the proposed EAPPOD is able to generate a better compensated signal than
APPOD, especially when delay periods change. Given the control method designed in this study, a
more accurate compensated signal produced by EAPPOD is expected to have a positive influence on
the LFOD control mechanism. It is noteworthy that the phasor component of the original signal shown
in Fig. 4.9 is disaggregated from the original signal using the RLS algorithm with a constant (not
adaptive) forgetting factor, as detailed in [93]. This further proves that the deviations and spikes of
the APPOD-generated signal come from the latency compensation method employed in the previously
published work. It is thus evident that the EAPPOD equipped with a number of new features is
capable of compensating continuously varying data communication latencies with a higher accuracy.
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CHAPTER 4. AN ENHANCED APPOD APPROACH WITH LATENCYCOMPENSATION FOR MODERN POWER SYSTEMS
0 5 10 15 20 24 30
−0.4
−0.2
0
0.2
0.4
(a)
(b)
(c) (d)
time (s)
χph1(o)
χph of original signal χph of APPOD signal
χph of EAPPOD signal
(a) (b) (c) (d)
Figure 4.9: Magnitude of the phasor component
LFOD Enhancement in Case 1
As discussed in Section 4.4.3, the LFOD enhancement is achieved by modulating the reference value of
active power inside the active power regulator located at DFIG RSC during external disturbances,
and the control signal is generated for the purpose of minimizing the magnitude of the extracted
phasor component of the measured inter-area signal, which resembles a power modulation signal that
participates in the control mechanism of DFIG RSC. Fig. 4.10 shows the control signal generated
by APPOD and EAPPOD. As mentioned in Section 4.5.1 the EAPPOD method has shown a more
favorable delay compensation performance, which directly affects the construction of the control signal.
Fig. 4.11 shows the variation in the active power generated by DFIG during the simulation with
the modulated reference power value in DFIG RSC. Without any control strategy, DFIG wind farm
experiences slight oscillations when the electrical disturbance occurs. With the power modulation of
both APPOD and EAPPOD, the output power of the doubly fed wind turbine generators fluctuates
and finally subsides within the simulation time window. Evidently, the EAPPOD method leads to
lower active power fluctuations and smoother transitions when communication delay varies, which
further demonstrates the superiority of the proposed EAPPOD method over the previously established
method. Fig. 4.11 also shows that DFIG over a period of time generates more electrical power than
its nominal value 0.63 p.u. The DFIG mechanical components cannot harness more kinetic energy
than what the MPPT algorithm indicates, but the wind turbine generators are capable of injecting
additional electrical power and torque to the grid. This phenomenon is explained with the introduction
of “virtual inertia” [81]. However, certain limitations need to be imposed on the amount of kinetic
energy that can be drawn from the rotor, which is bound by two main factors – (1) the current limiter
in the RSC of DFIG as shown in Fig. 4.2, and (2) the adverse effects of increasing the amount of
energy exported from the rotor. For the second point, there has been in-depth research work on the
impacts of virtual inertia of DFIG on the stability of power systems, see [98, 105, 106]. Particularly
in [98], researchers have shown that an increasing virtual inertia can cause a reducing damping ratio
in a power system. In other words, if the amount of kinetic energy drawn from the DFIG rotor is
64
4.5. CASE STUDY
excessively high, it may cause further oscillations in the power system, after compensating the effects
of the initial disturbance, especially for weakly interconnected power systems.
0 5 10 15 20 25 30−0.2
−0.1
0
0.1
0.2
(s)
Ppod1(p.u.)
Control signal by APPOD Control signal by EAPPOD
Figure 4.10: Control singal in Case 1
0 5 10 15 20 25 30
−0.7
−0.65
−0.6
−0.55
(s)
Pdfig1
e(p.u.)
System response APPOD method
EAPPOD method
Figure 4.11: Active power generated by DFIG in Case 1
0 5 10 15 20 25 30
4.95
5
5.05
(s)
PG10
System response APPOD method
EAPPOD method
Figure 4.12: Active power generated by G10 in Case 1
Fig. 4.12 presents the active power generated by G10, a generator located in proximity to DFIG. It
is clear that with two control schemes, the power produced by G10 does not show significant differences,
which indicate that the proposed EAPPOD strategy does not excessively interact with local modes.
This result is also drawn in [104], a practical implementation of the work performed in [93] issued by
ABB.
Fig. 4.13 demonstrates the differences in the two methods for LFO mitigation. Without any control
strategy, the measured signal fluctuates and does not settle down within the simulation time window,
whereas the system with either control method has shown significant improvement on LFO alleviation,
which subsides at a faster rate. A closer observation of APPOD and EAPPOD reveals that the newly
65
CHAPTER 4. AN ENHANCED APPOD APPROACH WITH LATENCYCOMPENSATION FOR MODERN POWER SYSTEMS
proposed EAPPOD is able to “correct” the anomalies existing in the APPOD curve due to its superior
latency compensation performance. It is also apparent that EAPPOD is capable of weakening the
LFO in a shorter time frame and has lower amplitudes since the third zenith point, which is about
when the communication latency starts varying.
0 5 10 15 20 25 30−0.1
0
0.5
0.9
(s)
χ1(o)
System response APPOD method
EAPPOD method
Figure 4.13: Control performances for χ1 in Case 1
Another case with χ2 is also implemented to demonstrate the functionality of the proposed EAPPOD
and its comparison with APPOD. Fig. 4.14 shows the control performances of both strategies, and it is
evident that the EAPPOD also produces a better LFOD enhancement. This case is simple to further
verify the effectiveness of the proposed EAPPOD method, which however is not a usual arrangement
as both control targets – bus 67 and bus 16 – are located in external areas of DFIG wind farm, and a
more common way can be installing another EAPPOD-DFIG system in the area local to bus 67, as
reported in [78].
0 5 10 15 20 25 30−9.8
−9.5
−9
−8.7
(s)
χ2(o)
System response APPOD method
EAPPOD method
Figure 4.14: Control performances for χ2 in Case 1
4.5.2 Case 2 : Modified Two-Area, Four-Generator Test Power System
Fig. 4.15 shows the topology of the modified 2-area, 4-machine test power system, which was first
introduced by Kundur in [1] for the purpose of small signal stability analysis in power systems, where
all relevant data are available. DFIG WTGs are connected to bus 11, which is connected to bus 10
through a transformer. In this setup, the DFIG-based wind power penetration is 22.6%, with the
total power generation of the power system being 2819.73 MW, for the purpose of demonstrating the
applicability of EAPPOD in a power system with higher wind power penetration. The simulation
runs for 30s. In the first 1 second, the test power system operates at steady state, and when t = 1s, a
sudden decrease happens in the load connected to bus 7. The nominal value of the load is 9.67 p.u.
66
4.5. CASE STUDY
1
G1
5 6 7 8 9 10
G3
2
G2
4
3
Area1 Area2
Swing
WTGs
11
G4
Figure 4.15: Modified 2-area, 4-machine power system with DFIG WTGs
and it decreases by 0.5p.u. In case 2, the inter-area signal of interest is the rotational speed of G1,
denoted as ω1.
Table 4.3: Important Parameters for Proposed Method in Case 2
Kphp 0.1 Kph
i 0.5 χav[0] 0
χph,d[0] 0 χph,q[0] 0 Ω0 2π · 0.4υss 0.9575 υtr 0.8375 Kff 20
Table 4.4: Original eigenvalues of interest relating to LFOs without controller in Case 2
Eigenvalue λ f(Hz) ξ(%) Dominant states
−0.904± j6.775 1.078 0.132 ∆ω and ∆δ of G1, G2−1.490± j6.694 1.065 0.217 ∆ω and ∆δ of G4, G3−1.801± j2.957 0.471 0.520 ∆ω and ∆δ of G1, G30.078± j2.384 0.379 −0.033 ∆ω and ∆δ of G1, G2, G4−0.076± j0.043 0.007 0.871 ∆ω of DFIG
TABLE 4.4 demonstrates the modes associated with generators’ electromechanical dynamics.
Fig. 4.16 shows the variations of the measured signal and the signal received at control center with
time-varying communication delays. Latencies implemented in Case 1 are employed in this case.
Communication Latency Compensation in Case 2
Similar discussions presented in Case 1 apply for Case 2, and the compensated measurement signal is
shown in Fig. 4.17. Evidently, the proposed EAPPOD also works in this situation for time-varying
communication delay compensation.
Signal decomposition is also performed in Case 2, and Fig. 4.18 shows the oscillatory part of the
signal which has been disaggregated from the compensated signal of interest, i.e., ∆ωph1 .
LFOD Enhancement in Case 2
Control signal needed in this Case is obtained according to the control mechanism discussed in
Section 4.4.3, which is fed to the DFIG RSC structure as illustrated in Fig. 4.2. The control result is
demonstrated in Fig. 4.19 (a) with its magnified version in Fig. 4.19 (b), where the oscillation part of
the figure is enlarged from t = 3s. Evidently, the proposed EAPPOD is able to suppress the oscillations
67
CHAPTER 4. AN ENHANCED APPOD APPROACH WITH LATENCYCOMPENSATION FOR MODERN POWER SYSTEMS
0 5 10 15 20 24 300
0.2
0.4
0.6
Td=20ms Td=400ms Td=100ms Td=800ms Td=40ms Td=300ms
latency
increases
latency
increases
latency
increases
latency
decreases
latency
decreases
latency
increases
time (s)
∆ω1
(rad
/s)
Signal measured at PMU site Delayed signal at control center
Figure 4.16: On-site measured signal and received signal in Case 2
0 5 10 15 20 25 300
0.2
0.4
0.6
(s)
∆ω1
(rad
/s)
Signal measured at PMU site Compensated signal with EAPPOD
Figure 4.17: Compensated signal in Case 2
in the measured signal with a shorter settling time and thus demonstrates its applicability for cases
where DFIG-based wind power penetration is at a high percentage in a power system.
Fig. 4.20 shows the variation in the active power generated by DFIG with proposed EAPPOD
method in Case 2. Same sign convention is adopted and a negative power indicate power generation,
and power is converted into per unit system, with the base apparent power being 100 MVA.
4.5.3 Limitation of the Proposed Method
The proposed enhanced adaptive phasor power oscillation damping method is able to restore most
inter-area measurements under varying communication delay periods, using a series of newly introduced
methods, including average assignment and phase tracking. This is based on the fact that most inter-
0 5 10 15 20 25 30
0
0.1
0.2
(s)
∆ωph
1(r
ad/s
)
Figure 4.18: Magnitude of the phasor component in Case 2
68
4.6. CONCLUSION
0 5 10 15 20 25 300
0.2
0.4
0.6
(s)
∆ω1
(rad
/s)
System response EAPPOD method
(a)
4 6 8 10 12 14 16 18 20 22 24 26 28 300.55
0.6
0.65
0.7
(s)
∆ω1
(rad
/s)
(b)
Figure 4.19: Control performances for ∆ω1 in Case 2
0 5 10 15 20 25 30−6.44
−6.42
−6.4
−6.38
(s)
Pdfig2
e(p.u.)
Figure 4.20: Active power generated by DFIG with EAPPOD in Case 2
area oscillatory signals are assumed or observed as exponential decay signal [93,102]. The limitation of
the proposed method is that if the oscillatory signal cannot be approximated as exponential decay, then
the phase tracking and average assignment (including magnitude attenuation) will not be applicable.
This rarely happens in power system studies and only when the oscillations are contributed by a
number of modes with no dominant participation by generators’ electromechanical interactions, this
situation may happen. This limitation however does not impede the investigation for the purpose of
this study.
4.6 Conclusion
In this chapter, an EAPPOD strategy is proposed to compensate time-varying communication latencies
and mitigate the LFOs that exist in inter-area signals under external disturbances of a complex power
system integrated with a DFIG wind farm. Through extensive simulations, the proposed EAPPOD has
been proven to be capable of restoring the delayed signal with a higher accuracy than the established
69
CHAPTER 4. AN ENHANCED APPOD APPROACH WITH LATENCYCOMPENSATION FOR MODERN POWER SYSTEMS
method, which facilitates power system online monitoring. On the LFOD control front, the newly
designed control strategy leverages the internal power regulation structure inside DFIG RSC and
achieves the control purpose by minimizing the magnitude of the phasor component decomposed from
the compensated IOS. The time-domain simulation results have demonstrated the superiority of the
EAPPOD strategy in LFO suppression caused by small signal stability issues in the power system. Two
distinct case studies have shown the applicability of the proposed EAPPOD method to power systems
with both relatively low wind power penetration and higher wind power penetration, manifesting
its capability of enhancing the stability of DFIG-WTG-integrated power systems. Furthermore, the
EAPPOD-DFIG structure poses economic advantages due to the absence of costly FACTS devices
that have been widely used in conventional local-/inter- area LFOD enhancement methods.
70
Chapter 5
An Adaptive-Phasor Approach to
PMU Measurement Rectification for
LFOD Enhancement
ABSTRACT
In this chapter, we propose an integral data rectification strategy for phasor measurement units (PMUs)
in multi-area power systems, comprising local data processing and central data recovery modules. The
local data processing is designed for the purpose of detecting and eliminating false PMU measurements,
which is performed in a decentralized manner, based on our previously developed dynamic state
estimation (DSE) technique; whereas the data recovery is performed in a centralized manner at the
control center, based on a newly proposed adaptive-phasor (AP) approach. The recovered PMU
measurements are then utilized in a low-frequency oscillation damping (LFOD) enhancement scheme,
which is achieved by a modified proportional-integral power system stabilizer (PI-PSS) mechanism
embedded in the automatic voltage regulator (AVR) structure of a synchronous generator. Control
parameters of the PI-PSS are optimized by maximizing the critical damping ratio of the power system.
This study is intended to make a contribution to the need of high-quality data transmission in modern
power grids with contemporary measuring technologies.
71
CHAPTER 5. AN ADAPTIVE-PHASOR APPROACH TO PMU MEASUREMENTRECTIFICATION FOR LFOD ENHANCEMENT
5.1 Chapter Foreword
The content of this chapter is mainly based on and modified from the following academic papers:
Tatkei Chau, Shenglong Yu, Tyrone Fernando, Herbert Iu, Michael Small and Mark Reynolds,
“An Adaptive-Phasor Approach to PMU Measurement Rectification for LFOD Enhancement”, IEEE
Transactions on Power Systems, DOI: 10.1109/TPWRS.2019.2907646, 2019.
In the previous chapter, an EAPPOD with time latency compensation is proposed, where an
adaptive algorithm is used to manipulate the PMU data based on the time stamp embedded in the
PMU data. Although the algorithm can be used to compensate time delay, it has a limited ability to
recover missing data using the two newly proposed average assignment and phase tracking techniques.
In this chapter, in addition to the time delay compensation techniques proposed in the previous
chapter, a DSE-based data rectification methodology is proposed, which uses a decentralized DSE
algorithm to detect and recover corrupted frequency measurement data based on local voltage and
current measurements to further improve the quality of data.
5.2 Introduction
With the introduction of IEEE C37.118.1-2011, which defines phasor measurement units and associated
terminologies and specifications, PMUs have been widely deployed in multi-area power systems to
provide instant information, facilitating the design of various wide-area control strategies. These
control strategies are devised with PMU measurements to enhance the stability of large-scale power
networks. The accuracy and timeliness of the PMU measurements, or synchrophasors, are thus of
critical importance in ensuring the performances of designed controllers. With this backdrop, a new
and pressing challenge in power system research has been created: improving the quality of PMU
measurements that are subjected to measurement errors and noises, time-varying transmission delays,
and data loss and disorder.
Bad data is one of the sources responsible for the quality degradation of obtained power system
information, which can either be caused by data corruption during transmission or incorrect PMU
measurements. The network-induced data errors can be easily detected with error detection methods,
such as well-established cyclic redundancy check (CRC) code with information embedded in the
data packets [107]; and subsequent error correction or data abandonment take place to rectify the
received data. The bad data due to erroneous measurements caused by software/hardware flaws,
on the other hand, are not easy to identify [108]. In recent literature, only a few researchers have
focused on bad PMU measurements, see for example [109], where however a system-level analysis,
incorporating dynamical behavior of the power system, is absent. In [99], the authors proposed a bad
data detection method for false voltage and current measurements during state estimation processes,
but data transmission delay was not considered therein.
In addition to bad data at the PMU sites, the synchrophasors are also subjected network-induced
time-varying transmission delays, and data loss and disorder. In this study, we particularly focus on
designing a data recovery strategy for low-frequency oscillations (LFOs)-borne measurement signals,
which widely exist in power systems. In power system analysis, the frequency of LFOs can be estimated
through small-signal stability analysis (SSSA) of the linearized system at a given operating point. The
72
5.2. INTRODUCTION
trend of LFOs in power systems generally follow an exponential decay characteristic [102]. These
features have made possible the adaptive-phasor-based data recovery methodology. The AP approach
was first proposed in [96], where the authors posited that an oscillatory signal can be disaggregated
into an average component and an oscillatory component. Then in [93] an AP-based power oscillation
damping (APPOD) method is proposed to adaptively compensate communication latencies that exist
in PMU measurement signals, and the control signal is then generated and utilized to enhance the
low-frequency oscillation damping (LFOD) of the power system. However, the APPOD method
proposed in [93] fails to cope with frequently-varying communication delays. An enhanced APPOD
approach proposed in [110] managed to compensate the frequently-changing and large time-varying
time delay, but bad data issue has not been considered therein.
Power system stabilizers (PSSs) have been widely applied in interconnected power systems for
the purpose of low-frequency oscillation damping (LFOD) enhancement [1, 67, 68], with local and
wide-area measurement signals. The communication latency issue has been considered in these studies
through dedicated lead-lag compensators, which are specially designed to cope with short delays in
the measurements. However, when the communication delay is more than tens of milliseconds, the
control performances will be significantly undermined [93]. Therefore, it is imperative to devise a
delay compensation strategy to cope with time-varying data transmission latencies, and the delay-
compensation function, if embedded in the PSS structure, can substantially reduce the requirement of
the design and tuning of the lead-lag compensators, and a simple PI-PSS can be utilized to fulfill the
LFOD enhancement duty.
Based on the above discussion and literature review, it is of great necessity for power system
operators to know the validity of local PMU measurements, eliminate bad data and recover delayed,
missing and disordered measurement data at the control center. This study intends to offer a feasible
solution to this need by developing a comprehensive measurement rectification (MR) mechanism. In
this study, an integral local data processing and central data recovery scheme is proposed. The data
processing procedure is to detect and eliminate bad data and verify PMU measurements, whereas the
data recovery scheme is to compensate data transmission delays, and restore missing and disordered
data. The main contributions of this chapter are listed as follows:
(i) The decentralized data processing strategy is proposed based on our previous work in the area
of decentralized dynamic state estimation (DSE) as reported in [99] and [111]; the proposed method
can achieve fast and precise bad data detection and measurement verification;
(ii) The proposed AP-based data restorer treats data loss and data disorder as increased and
decreased communication delays; and measurement data is then restored by compensating the commu-
nication delays. The proposed data recovery technique is capable of handling noisy measurements,
time-varying communication delays, and bad-data-induced and network-induced data loss and/or
disorder.
(iii) The processed and restored remote frequency signals are fed into a simple PI-PSS structure,
which is embedded in the AVR of a pre-specified generator in the power system, to achieve LFOD
enhancement. The control parameters of the PI-PSS are optimized at an operating point with the
objective of improving the system critical damping ratio.
The remainder of the chapter is organized as follows. In Section 5.3, a general description of the
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CHAPTER 5. AN ADAPTIVE-PHASOR APPROACH TO PMU MEASUREMENTRECTIFICATION FOR LFOD ENHANCEMENT
overall data processing and restoration strategy is presented. The models of the power system in
question are expressed mathematically in Section 5.4. The descriptions of the proposed DSE-based
decentralized data processing method is briefed in Section 5.5. The improved AP-based data recovery
strategy is presented in Section 5.6. The design and parameter tuning procedures of the proposed
LFOD enhancer are discussed in Section 5.7. A case study is presented in Section 5.8. Finally, a
conclusion is drawn in Section 5.9.
5.3 The Overall Measurement Rectification and Control Strategy
As mentioned in the Introduction, the proposed overall measurement processing and restoration and
LFOD enhancement strategy, as shown in Fig. 5.1, comprises (i) a DSE-based bad data detection
and elimination and measurement verification mechanism, collocated with PMUs at each generator
bus-bar, (ii) a data recovery block, which is able to decompose oscillatory PMU measurements and
restore the measurement data after time-varying communication delays, and data loss and disorder,
with dedicated channels for each synchrophasor, (iii) an LFOD enhancer (PI-PSS) connected to the
AVR structure of a selected synchronous generator (GM in Fig. 5.1), and (iv) a parameter optimization
block to obtain the optimal set of parameters for the PI-PSS.
1
G1
DSE-based bad data
detection and elimination
with system Model 1
Measurement verification
∠ ∠
Data transmission
2
n
Power
system
G2
GnPMU
Data recovery
Channel 1 Channel 2 Channel n
GM
Parameter optimization
with system model 2 PI-PSS
Contr
ol
sign
al
, , …,
AVR
Bad-data-induced data loss
Network-induced communication delays
and data loss and disorder
Figure 5.1: Overview of the proposed control mechanism
The following assumptions are made in this study:
(i) All clocks at the PMUs and the centralized controller are satellite-synchronized; we do not
consider any time discrepancies in this study;
74
5.4. SYSTEM MODELING
(ii) The data transmission is realized with generic communication channels, which have varied
network-induced delay and packet dropouts or disorder [94]; we do not consider dedicated communication
channels;
(iii) The parameter optimization of the control mechanism is performed only once at one operating
point, with a known operating condition; and
(iv) Measurement noises of all PMUs are subject to Gaussian distribution, which is adopted
from [99].
5.4 System Modeling
The dynamic behavior of the power system is mathematically described with the differential-algebraic
equation (DAE) formulation throughout this study, which can be written in a compact form as
follows [59],
d
dtX = f(X,V,U),
0 = g(X,V,U), (5.1)
where f(·) and g(·) are respectively the differential and algebraic functions of the system nonlinear
state-space model, and X, V and U are respectively the system dynamic state, algebraic state and
input vectors. For the purpose of this study, the standard form in (5.1) is re-written into two different
formulations (detailed in Subsections 5.4.1 and 5.4.2) in order to provide suitable models to achieve the
following two purposes: (i) decentralized dynamic state estimation for bad data detection, elimination
and measurement verification; and (ii) small-signal stability analysis for parameter optimization for
the PI-PSS control structure.
5.4.1 Model for Decentralized DSE
In order to fulfill purpose (i), the DAE set in (5.1) is reformulated as follows. Since the DSE
algorithms are executed at the generator buses in a decentralized manner, where the knowledge of
the external system is absent, the system is modeled into several decoupled sub-systems, with only
local measurements being the model’s pseudo-inputs and pseudo-outputs. In this study, the voltage
and current phasors (magnitudes and phase angles) measured by the PMUs are employed to be
the pseudo-inputs and pseudo-outputs of each decoupled sub-system respectively. For a n-generator
power system, we lump all voltage phasors in a pseudo-input vector U and all current phasors in a
pseudo-output vector Y. Then a multi-input multi-output (MIMO) system can be formulated in the
following form,
d
dtX = f(X,V,U),
Y = g(X,V,U), (5.2)
where U = [u1, u2, · · · , un]T , Y = [y1, y2, · · · , yn]T with ui = [Vi, θi]T and yi = [Ii, γi]
T , for i ∈1, · · · , n.
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CHAPTER 5. AN ADAPTIVE-PHASOR APPROACH TO PMU MEASUREMENTRECTIFICATION FOR LFOD ENHANCEMENT
In order to facilitate the decentralized dynamic state estimation process, the decoupled sub-systems
with noisy PMU measurements are discretized as follows,
xi[k] = fi(xi[k − 1], Vi[k − 1], θi[k − 1], µi[k − 1]),
yi[k] = gi(xi[k], Vi[k], θi[k], µi[k]) + νi[k], (5.3)
The diacritic (·) indicates raw noisy PMU measurements. Voltage and current measurement noise,
µ2×1 and ν2×1, are distributed normally with zero mean and covariance Q and R as follows,
µ[k] ∼ N([
0]2×1
, Q), (5.4)
ν[k] ∼ N([
0]2×1
, R), (5.5)
and
Q =
[σ2µ1
0
0 σ2µ2
],R =
[σ2ν1
0
0 σ2ν2
]. (5.6)
5.4.2 Model for small-signal System Stability Analysis
Similar to Section 5.4.1, the power system model in (5.1) can be re-formulated for small-signal analysis
using system linearization techniques. As reported in our previous study [50], after the elimination of
algebraic variables, the state-space representation of the linearized power system can be obtained and
expressed in the following form,
∆X = A∆X +B∆U2, (5.7)
where ∆U2 = [∆u21,∆u22, · · · ,∆u2n]T is the input vector, which is formed by the mechanical power
set-points and the AVR reference values (if equipped) of each generator unit, i.e., ∆u2i = [∆Tmi ,∆Vrefi ]T
for i ∈ 1, · · · , n, and A and B are respectively the system state and input matrices of the linearized
system. For more details, please visit reference [1].
The system formulations in (5.2) and (5.7), named Model 1 and Model 2, will be used throughout
this chapter for decentralized data manipulation and centralized stability analysis respectively.
5.5 DSE-based Measurement Processing
The decentralized measurement processing algorithm, consisting of bad data detection and elimination
and measurement verification, is developed based on the DSE-based bad data detection method reported
in [99]. In this study, the bad data will first be detected and eliminated in the DSE mechanism to
avoid the generation of erroneous estimates, and then the estimates obtained with valid measurements
are employed to verify the frequency measurement signals. The verified frequency measurements are
transmitted to the central controller, so that the subsequent control procedures can fulfill their duties.
76
5.5. DSE-BASED MEASUREMENT PROCESSING
5.5.1 Unscented Kalman Filter
The unscented transformation (UT) is a method to estimate statistics of a random variable subjected
to a given nonlinear transformation [112]. Let us assume that υ is a τ dimensional random variable
distributed normally with mean υ and covariance Pυυ. If υ undergoes a nonlinear transformation
ψ = Υ(υ), then UT can provide the estimation of the mean ψ and covariance Pψψ of ψ. A set of 2τ + 1
points called sigma points (ς) with mean υ and covariance Pυυ are chosen to estimate the mean (ψ)
and covariance (Pψψ) of the transformed points using the following equations [113],
ς0 = υ, (5.8)
ςr = υ +(√
(τ + λ)Pυυ
)r; r ∈ 1, 2, · · · , τ, (5.9)
ςr+τ = υ −(√
(τ + λ)Pυυ
)r; r ∈ 1, 2, · · · , τ, (5.10)
where(√
(τ + λ)Pυυ
)ris the rth row or column of the matrix square root of (τ + λ)Pυυ. Also
λ = a2(τ + κ)− τ is a scaling parameter where a is a factor which specifies the spread of the sigma
points, and κ = 0 is the second scaling parameter. Furthermore, mean and covariance of ψ are
approximated based on the following corresponding weights,
W 0m =
λ
(λ+ τ)(5.11)
W 0c =
λ
(λ+ τ)+(1− a2 + b
), (5.12)
W rm = W r
c =1
2(λ+ τ); r ∈ 1, 2, · · · , 2τ, (5.13)
where b is a factor to incorporate prior knowledge of the distribution of υ, e.g., b = 2 for normal
distributions. The mean and covariance of the random variable ψ can be calculated using the following
equations,
ψr = Υ(ςr); r ∈ 0, 1, · · · , 2τ, (5.14)
ψ =2τ∑r=0
W rmψ
r, (5.15)
Pψψ =2τ∑r=0
W rc (ψr − ψ)(ψr − ψ)T . (5.16)
Now consider the decoupled dynamic of the kth iteration (5.3). As discussed before, PMU measurement
noise is assumed to have a normal distribution with zero mean. If we assume the covariances of
measurement noise in pseudo inputs are constant then the state vector x[k] and measurement noise
µ[k] can be considered as a new augmented state vector, X[k] = [x[k] µ[k]]T , where X[k] ∈ R(n+2)×1 is
the augmented state vector. Predicted augmented state vector X[k] and its covariance can be obtained
77
CHAPTER 5. AN ADAPTIVE-PHASOR APPROACH TO PMU MEASUREMENTRECTIFICATION FOR LFOD ENHANCEMENT
as follows,
X[k] =[x[k]
[0]2×1
]T, (5.17)
PXX [k] =
[Pxx[k] Pxµ[k]
Pxµ[k] Q
]. (5.18)
Using the unscented transformation, state estimation of generator can be implemented using the
following filtering algorithm,
Step 0:Initialization
Set κ = 0 and let a = 10−3, b = 2.
Select initial value of the state vector x[0] and select it as x[0].
Augment initial value of the state vector with PMU measurement noise mean, i.e., X[0] =[x[0]
[0]2×1
]T.
Initiate covariance of the augmented state vector X[0] as PXX [0] =
[Pxx[0] 0
0 Q
].
Set k = 1.
Step 1: Time Update,
Consider υ = X[k − 1] and Pυυ = PXX [k − 1] in (5.8).
Generate 2(n+2)+1 sigma points according to (5.8), i.e., ς[k−1] =[ς0[k − 1] · · · ς2(n+2)[k − 1]
].
Associate weights according to (5.11) and (5.13), i.e.,
Wm =[W 0m W 1
m · · · W2(n+2)m
],
Wc =[W 0c W 1
c · · · W2(n+2)c
].
Calculate transferred points according to (5.14), i.e., Xr[k] = f(ςr[k − 1], u[k − 1]
).
Calculate mean X[k] and covariance PXX of the transferred points X[k], according to (5.15) and
(5.16) with ψ replaced by X and ψ replaced by X.
Step 2: Measurement Update,
Calculate measurement update based on the transferred sigma points X[k] obtained from Step 1,
i.e., Y [k] = g(X[k], u[k]
).
Calculate mean Y [k] according to (5.15) with ψ = Y
Calculate covariance of Y [k], PY Y , according to (5.16), i.e.,
PY Y = R+2(n+2)∑r=0
W rc
(Y r[k]− Y [k]
)(Y r[k]− Y [k]
)T.
78
5.5. DSE-BASED MEASUREMENT PROCESSING
Calculate cross-covariance PXY as,
PXY =2(n+2)∑r=0
W rc
(Xr[k]− X[k]
)(Y r[k]− Y [k]
)T.
Step 3: Filtering,
Calculate filter gain K[k] as, K[k] = PXY P−1Y Y
.
Update predicted states based on PMU measurement as,
X[k] = X[k] +K[k](y[k]− Y [k]
).
Calculate covariance PXX [k] as,
PXX [k] = PXX −K[k]PY YKT [k].
Step 4:
Reset PXX [k] to
[Pxx[k] Pxµ[k]
Pxµ[k] Q
].
Reset X[k] to[x[k]
[0]2×1
]T.
Increment k and goto Step 1.
At the end of each filtering algorithm iteration X[k] provides on-line estimate of generator augmented
state vector X[k]. Estimation of the state vector x[k] can be extracted from the augmented state
vector X[k] according to (5.17).
5.5.2 Bad Data Detection and Elimination
In practice, the quantities measured by PMU may, in additional to noises, have gross errors that
deviate significantly from their actual values. These measurements, if used, will result in unacceptable
discrepancies in the measurement verification process. Therefore, bad data must be detected and
eliminated from PMU measurement pool. Before performing this technique, the concept of normalized
innovation ratio [114] is introduced as follows,
%y,1 =y1[k]− Y1[k]√PY Y (1,1)[k]
and %y,2 =y2[k]− Y2[k]√PY Y (2,2)[k]
. (5.19)
Terms %y,1 and %y,2 indicate the extent to which predicted measurements diverge from the actual
PMU output measurements, Y [k] is expectation of the output at the kth instance, and PY Y [k] is the
auto-covariance of the output at the kth instance. When the absolute value of ratios exceed certain
thresholds %th,1 and %th,2 respectively, the data will be classified as bad data. The bad data can exist
in pseudo-input vector [V , θ]T and/or output measurement vector [I , γ]T .
The modified UKF algorithm with bad detection methodology is shown as follows,
Step 1: Perform first two steps of UKF algorithm.
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CHAPTER 5. AN ADAPTIVE-PHASOR APPROACH TO PMU MEASUREMENTRECTIFICATION FOR LFOD ENHANCEMENT
Step 2: Acquire normalized innovation ratios according to (5.19).
Step 3:
(1) If %y,1 < %th,1 and %y,2 < %th,2, go to Step 4.
(2) If %y,1 > %th,1 and %y,2 < %th,2, y1[k] = Y1[k] and go to Step 4.
(3) If %y,1 < %th,1 and %y,2 > %th,2, y2[k] = Y2[k] and go to Step 4.
(4) If %y,1 > %th,1 and %y,2 > %th,2, store current vector u[k] and use the latest uncorrupted input
vector u[k − 1] to acquire a new set of %y,1 and %y,2.
(5) If %y,1 < %th,1 and %y,2 < %th,2, set u[k] = u[k − 1]. Go to Step 4.
(6) If %y,1 > %th,1 and %y,2 < %th,2, y1[k] = Y1[k], set u[k] = u[k − 1]. Go to Step 4.
(7) If %y,1 < %th,1 and %y,2 > %th,2, y2[k] = Y2[k], set u[k] = u[k − 1]. Go to Step 4.
(8) If %y,1 > %th,1 and %y,2 > %th,2, then y1[k] = Y1[k] and y2[k] = Y2[k], continue to the next
step.
(9) Use stored u[k] to acquire a new set of %y,1 and %y,2 with the output obtained in Step 3.8.
(10) If %y,1 < %th,1 and %y,2 < %th,2, keep u[k] and go to Step 4.
(11) If %y,1 > %th,1 and %y,2 > %th,2, set u[k] = u[k − 1]. Go to Step 4.
Step 4: Perform last two steps of UKF algorithm.
Bad data can exist in pseudo-input vector or/and output vector. Faulty data in either or both
of pseudo-input elements will affect %y,1 and %y,2 significantly, whereas bad data present in output
measurement only affects its corresponding normalized innovation ratio, i.e., corrupted y1[k] only
affects %y,1. If there are no bad data, the detection scheme will stop at Step 3.1 and continue to Step
4. If bad data are in either of the output measurement elements (I , γ), it is to be detected in Step
3.2 or Step 3.3, and then faulty output measurement is discarded and replaced with the predicted
output. If bad data are in either or both of pseudo-input items (V , θ) and output measurement is
robust, then Step 3.4 will be triggered and detection stops and returns at Step 3.5. This means u[k] is
corrupted and so u[k− 1] replaces it, and continue onto the next iteration. If bad data exist in (i) both
pseudo-input and output measurement vectors with one or two elements in both of them infected or
(ii) both of output measurement elements are corrupted with reliable pseudo-input vector, Step 3.4 will
be triggered but the provenance of faulty data are unclear. At Step 3.4, we tentatively postulate there
are bad data in pseudo-input vector, so current pseudo-input vector u[k] is stored and u[k − 1] is used
to generate a new, reliable predicted output vector. Step 3.6 or Step 3.7, if activated, indicates that
bad data exist in pseudo-input and one of the measured output elements. Therefore, this pseudo-input
vector sample is abandoned and replaced by u[k− 1] and faulty measured output element is eliminated
and renewed with predicted output. If Step 3.8 is true, it can be inferred that bad data exist in both
items of output measurement but the accuracy of pseudo-input is unknown. Proceed to next step to
80
5.6. AP-BASED DATA RECOVERY METHOD
make further judgment. If Step 3.10 is satisfied there is no bad data in pseudo-input vector and u[k] is
preserved for the next iteration. Lastly, if Step 3.11 is triggered, it indicates the presence of bad data
dwells in current pseudo-input sample and hence, u[k] is ditched and u[k − 1] takes its place. It should
be noted that there is no necessity to tell which one of pseudo-input elements is corrupted as when
either or both of them are faulty, the predicted output is highly erroneous and thus the whole vector is
discarded.
5.5.3 Measurement Verification
As shown in Fig. 5.1, the frequency estimates acquired with valid voltage and current phasors
measurements are used to verify the frequency measurements; false frequency measurements are
eliminated, with the correct ones sent to the remote control center. In particular, when the difference
between the measured and the estimated frequency, ω and ω, of a given generator bus is less than a
certain error threshold εth, the verified measurement will be used to compute the remote frequency
deviation signal χ. Measurements that fail to pass the verification are discarded, and no data is sent
to the data center for at this instance, causing bad-data-induced data loss. The remote measurement
from the ith generator bus can be expressed as follows,
χi[k] =
ωi[k]− ωsync, if |ωi[k]− ωi[k]| < εth
∅, otherwise(5.20)
for i ∈ 1, · · · , n, and the remote measurement vector χ is formed by a collection of remote frequency
deviation signals sent from the generator buses, i.e., χ = [χ1, χ2, · · · , χn]. Note that the removed data
will be treated as missing data in the data center.
5.6 AP-Based Data Recovery Method
In this study, we propose an adaptive-phasor-based delay compensation method with measurement
rectification to achieve the measurement data recovery purpose. In order to address missing data
issue, which is caused either by bad measurement of the PMUs or network-induced reasons, as well
as data disorder issue, we consider missing data as increased communication delay and data disorder
as decreased communication delay, so that the proposed method can cope with all these situations.
The proposed AP-based data recovery method is only aware of the consistency of the received data,
and when the data flow is smooth, we consider the data transmission process has a constant delay;
otherwise if data is empty or data flow is in consistent, communication latency is deemed changing.
5.6.1 Improved Signal Decomposition
As discussed in Chapter 4, the fundamental assumption of this method is that when a power system is
subjected to small disturbances the low-frequency oscillatory signals χ(t) can be decomposed into an
average component and an oscillatory component [97], which is shown as follows,
χ(t) = χav(t) + χosc(t), χosc(t) = ReχphejΩt, (5.21)
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CHAPTER 5. AN ADAPTIVE-PHASOR APPROACH TO PMU MEASUREMENTRECTIFICATION FOR LFOD ENHANCEMENT
where χav and χosc are the average and oscillatory components of decomposed signal. Terms χph and
Ω are respectively the phasor and angular frequency of the oscillatory signal, i.e., the frequency of LFO
signal in this study. Then through adaptively rotating the reference frame of the phasor component
based on the time latency Td, the time-varying latencies can be compensated. Fig. 5.2 shows the overall
AP-based signal restoration method, which consists of three major components: signal decomposition,
frequency adaptation and signal recovery.
Clock
Frequency adaptation
Improved RLS
Average assignment and
Phase tracking
Signal decomposition
Error compensation
Signal recovery
Ω ⋅
∆
ΥSystem model 2
Parameter optimization PI-PSS
Adaptive phasor-based signal restoration
Phasor-only LFOD enhancer
AVR′
Synchronous Generator
Communication delay
Missing data
Data disorder
Ω
Ω∆Ω
Figure 5.2: Proposed adaptive phasor method for PMU data recovery
The flowchart of the signal decomposition and latency compensation scheme is depicted in Fig. 5.3.
As shown in the flowchart, the improved signal decomposition method is comprised of two major
parts: the recursive least square (RLS) estimation with adaptive forgetting factor (AFF), or AFF-RLS
method, when communication latency is considered constant, and the “average assignment” and “phase
tracking” methods when latency varies.
5.7 LFOD Enhancer and Parameter Optimization
5.7.1 LFOD Enhancement Mechanism
In this study, a PI-based PSS is adopted from [115] to mitigate LFOs as shown in Fig. 5.2, where
the weighted sum of the rectified PMU measurements Υpss is employed as the input signal, which is
acquired with the following equation,
Υpss =n∑i=1
wiχi, (5.22)
82
5.7. LFOD ENHANCER AND PARAMETER OPTIMIZATION
Data detection
Data arrived?
Compute 1). communication delay 𝑇 𝑘 , and 2). delay variation of the two instances Δ𝑇 𝑘
𝑇 𝑘 𝑇 𝑘 1 0
Instance 𝑘
Reduced or constant communication delay
Y
N
Missing data orincreased communication delay
Latency decreased?
Y
N
Improved RLS
Constant communication latency
Assign 𝑇 𝑘 𝑇 𝑘 1 , so that Δ𝑇 𝑘 0
𝜒 𝑘 𝜒 𝑘 𝑅𝑒 𝝌𝒑𝒉 𝑘 1 ⋅ 𝑒
Update the angle of the phasor component as𝛼 𝑘 𝛼 𝑘 1 Ω 𝑘 𝑇 𝛥𝑇
Phase tracking
Reset counter 𝐶 𝑘 0 Increment counter 𝐶 𝑘 𝐶 𝑘 1 1
Average assignment
Assign phasor magnitude as 𝝌𝒑𝒉 𝑘 𝝌𝒑𝒉 𝑘 1
Phasor part of the decomposed signal 𝝌𝒑𝒉 𝑘 𝝌𝒑𝒉 𝑘 ∠𝛼 𝑘
Signal decomposition
Calculate attenuation factor
𝐹 𝑘1
𝑁 log𝝌𝒑𝒉 𝑖
𝝌𝒑𝒉 𝑖 1
Estimate the phase angle of 𝝌𝒑𝒉 𝑘 as𝛼 𝑘 𝛼 𝑘 Ω 𝑘 𝑇 𝑘 𝐶 𝑇Update oscillation frequency Ω 𝑘
Estimate the magnitude of 𝝌𝒑𝒉 𝑘 as𝝌𝒑𝒉 𝑘 𝜒 𝑘 ⋅ 𝑒
Delay compensation
Restored measurement 𝑘
Increment 𝑘
LFOD controller
𝜒 𝑘𝜒 𝑘 1
Figure 5.3: Flowchart of the proposed signal decomposition and data restoration method
where χi and wi are respectively the ith restored PMU measurement calculated in Section 5.6.1 and the
ith optimal normalized weighting coefficient. The AVR control input Vsup is generated by the PI-PSS
controller as follows,
Vsup = Kpssp Υpss +Kpss
i
∫Υpss, (5.23)
and this supplementary voltage signal modulates the voltage regulator of the selected synchronous
generator.
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CHAPTER 5. AN ADAPTIVE-PHASOR APPROACH TO PMU MEASUREMENTRECTIFICATION FOR LFOD ENHANCEMENT
5.7.2 Optimization of Weights and Control Parameters
Given the system Model 2 in (5.7), a linearized system model for small-signal stability analysis can be
obtained, and the critical damping ratio ξcrit is a function of the parameters to be optimized, including
the weighting coefficients wi and the parameters of the PI-PSS, i.e., Kpssp and Kpss
i . A particle swarm
optimization (PSO) algorithm is used to optimize these parameters by minimizing the critical damping
index (CDI) as follows [50,116],
minimize CDI = 1− ξcrit, (5.24)
subject to
Kpssi,min ≤ K
pssi ≤ Kpss
i,max, Kpssp,min ≤ Kpss
p ≤ Kpssp,max,
0 ≤ wj ≤ 1, for j ∈ 1, 2, · · · , n. (5.25)
5.8 Case Study and Simulation Results
In this section, a 2-area, 4-machine power system modified from [1] is employed to test the functionality
of the measurement rectification and LFOD enhancement strategies proposed in the study. The test
system runs in steady state until t = 1.0s when a sudden load decrease, i.e., P7 = P ini7 − 0.1p.u.,
happens at the load connected to bus 7. Modal analysis is also conducted for parameter optimization
based on eigenvalue sensitivity of the generators’ electromechanical modes. The case study includes
a thorough comparison of control performances of using unprocessed PMU data with bad data and
the restored data, and also demonstrates the restored data and data measured by the PMU before
transmission. The dynamic states estimation and bad data detection results are also included in order
to validate the performance of the decentralize DSE algorithm implemented at remote PMUs.
1
G1
5 6 7 8 9 10
2
G2
4
3
Area 1 Area 2
SwingG4
Data 1
Data 2
Data 3
Data 4
Data
Tra
nsm
issio
n
Data
Resto
ratio
n
Centralized
PI-PSS
AVR
Parameter optimization
PMUs
System model 2
Bad data detection and
removal
Measurement validation
Measurement 1
Decentralized
Figure 5.4: 2-area 4-machine test system with proposed data rectification and LFOD enhancementstrategies
84
5.8. CASE STUDY AND SIMULATION RESULTS
5.8.1 Simulation Conditions
As shown in Fig. 5.4, processed PMU measurements are transmitted to the data center after bad data
removal and measurement verification processes in a decentralized manner, and the control center
preforms the data recovery and generates the control signal to perform the LFOD enhancement duty.
In this particular study, four PMUs are individually installed in all generator buses, and the control
center is placed at bus 3. All PMUs, except the one that is locally installed at bus 3, are connected to
the control center via communication links subjected to random time-varying latencies, data loss and
data disorder. Fig. 5.5 depicts the distributions of the time-varying communication latencies used in
the simulation, which are generated with the generalized Pareto distribution model [117] with empirical
PMU latencies seen in practical situations [93]. Note that the mean values of the communication
latencies are proportional to the geographical distance between the control center and the PMUs, and
the communication latency between the control center and the PMU located in bus 3 ignored.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.240
0.05
0.1
0.15
0.2
Time(s)
(a)
0 0.1 0.2 0.3 0.40
0.1
0.2
0.3
Time(s)
(b)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.1
0.2
0.3
Time(s)
(c)
Figure 5.5: Probability density of communication latency between control center and (a) G1, (b) G2and (c) G4
85
CHAPTER 5. AN ADAPTIVE-PHASOR APPROACH TO PMU MEASUREMENTRECTIFICATION FOR LFOD ENHANCEMENT
5.8.2 SSSA and Parameters Optimization
With the linearized system Model 2 in (5.7) obtained by SSSA, three electromechanical modes are
identified through modal analysis, and the eigenvalues of interests are shown in Fig. 5.6, where Mode
1 located in the first quadrant of the complex plane is the critical mode of the system with the lowest
damping ratio ξcrit = −0.97%, which can cause system instability. The detailed results of the modal
analysis are listed in the TABLE 5.1. The oscillation frequency of the critical mode is found to be
0.43Hz, which is used as the initial guess of the frequency of the oscillatory component used in the
RLS algorithm, i.e., Ω0 = 0.43Hz.
−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.20
5
10
← 10% damping ratio line
mode 1→
critical modedamping ratio:−0.97%
mode 2
mode 3
Real part
Imaginary
part
Eigenvalues
Electromechanical Mode
Figure 5.6: Eigenvalues of the linearized system model without control
Table 5.1: Original eigenvalues of interest relating to LFEOs
Oscillationmodes
Eigenvalue λ f(Hz) ξ(%) Dominant states
Mode 1† 0.03± j2.70 0.43 −0.97 ∆Ω and ∆δ of G1, G3
Mode 2 −0.75± j6.72 1.07 11.16 ∆Ω and ∆δ of G1, G2
Mode 3 −0.80± j6.92 1.10 11.54 ∆Ω and ∆δ of G3, G4†Critical mode
As mentioned in Section 5.7.2, the linearized system model is also utilized for optimizing the weights
and control parameters using the PSO algorithm, and the optimal results generated by the optimization
algorithm is listed in the TABLE 5.2. The evolution of the PSO algorithm is shown in Fig. 5.7. As
shown in the figure, the best CDI value rapidly decreases from 0.93 during the first 25 iterations and
stays at 0.88 till iteration = 80. The settings for the PSO algorithm are adopted from [116].
Table 5.2: Optimized parameters used in the proposed controller
Kpssp Kpss
i w1 w2 w3 w4
53.3951 0.3512 0.3405 0.3898 0.2624 0.0073
5.8.3 DSE and Local Data Processing
Figs. 5.8 (a)-(b) demonstrate the selected state estimation results generated by the decentralized
UKF-based DSE algorithm from G1. It is evident that the UKF algorithm is able to track the dynamic
states using local voltage and current measurements infected with bad data obtained from PMUs. The
86
5.8. CASE STUDY AND SIMULATION RESULTS
0 10 20 30 40 50 60 70 80
0.9
0.95
1
Iterations
CDI
Iteration 0 indicates the CDI of the original system, as in Fig. 5.6 (1.0097)
Figure 5.7: Evolution of the PSO algorithm for CDI minimization
bad data detection functionality has also been examined by injecting random perturbations into the
measurement signals at selected time instances, i.e., t = 2, 4, 6, 8s. As shown in Figs. 5.8(c)-(d),
outstanding readings are found from the deviation ratios at G1 at t = 2, 4, 6, 8s, indicating bad data
are detected, which are then removed. See [80] for the discussion on classifying bad data with the
deviation ratios.
As mentioned in Section 5.5.2, after all bad data is removed with the DSE algorithm and the
frequency measurements are verified with the estimated signals, the frequency deviation signals are
computed and sent to the control center from the remote units.
5.8.4 Signal Decomposition and Restoration
The frequency deviation signal of G1 is shown in Fig. 5.9, where the solid line shows the signal captured
from the PMU, the dotted line is the uncompensated signal retrieved from the control center and the
remaining curve represents the compensated signal to be fed into the PI-PSS controller. Comparing the
original signal to the uncompensated signal from the enlarged graphics in Figs. 5.9 (a)-(d), distortions
can be observed from the uncompensated signal, which are caused by data loss, data disorder and
time-varying delays. Evidently, the AP-based signal recovery algorithm is able to restore the distorted
frequency deviation signal to be input into the PI-PSS structure for the best control outcome.
Fig. 5.10 demonstrates the decomposed oscillatory component of the frequency deviation signal
of G1 with and without the proposed measurement rectification method. It is noteworthy that the
extracted oscillatory signal oscillates at the frequency obtained through the modal analysis. The result
also indicates the functionality of signal decomposition strategy proposed in Section 5.6.1.
5.8.5 LFOD Enhancement
Fig. 5.10 also shows the difference between the oscillatory component of the restored signal with and
without LFOD enhancer. It is clear that with the proposed AP-LFOD enhancement method, the
oscillatory component of the measured signal subsides substantially faster than without the proposed
control scheme. In addition, Figs. 5.11 (a) and (b) demonstrate the differences in the LFO mitigation
with and without the proposed measurement rectification method (marked with MR in the figures).
Without any control strategies enforced, the frequency deviation signals fluctuate within increasing
amplitudes due to system instability, as shown in Fig.5.6, whereas the system with the AP-LFOD
87
CHAPTER 5. AN ADAPTIVE-PHASOR APPROACH TO PMU MEASUREMENTRECTIFICATION FOR LFOD ENHANCEMENT
0 1 2 3 4 5 6 7 8 9 10 11 120.948
0.949
0.95
Time(s)
Eq′ 1(p.u.)
Theoretical Estimated
(a)
0 1 2 3 4 5 6 7 8 9 10 11 12
1
1.0002
1.0004
Time(s)
ω1(p.u.)
(b)
0 1 2 3 4 5 6 7 8 9 10 11 120
100
200
300
Time(s)
|%y,1|
(c)
0 1 2 3 4 5 6 7 8 9 10 11 120
200
400
600
800
Time(s)
|%y,2|
(d)
Figure 5.8: (a)-(b) Decentralized UKF-based DSE and (c)-(d) absolute values of normalized deviationratios G1
enhancer and the proposed MR methodology has shown significant improvement on LFO alleviation.
It can also be seen that LFOD controller with MR performs more favorably than without MR, which
88
5.9. CONCLUSION
0 1 2 3 4 5 6 7 8 9 10 11 12
0
2
4
×10−4
(a)
(b)
(c) (d)
Time(s)
∆ω1
(p.u
.)
Raw received Raw sent
Restored received
(a) (b) (c) (d)
Figure 5.9: Noisy frequency deviation measurement of G1
0 1 2 3 4 5 6 7 8 9 10 11 12
−0.5
0
0.5
1
×10−4
Time(s)
χosc1(p.u.)
AP-LFOD Enchancer with MR No control
Figure 5.10: Oscillatory component of frequency deviation from G1
suppresses unnecessary oscillations before the frequency signal settles, hence a better primary frequency
response. This reveals that the newly proposed measurement rectification method can effectively
eliminate the unwanted PMU measurements and ensures the AP-LFOD enhancer have valid input
data, which in turn enhances the stability of the power system.
5.9 Conclusion
In this chapter, a PMU measurement rectification strategy, comprising local measurement processing
and central data recovery functions, is proposed based on the adaptive phasor concept. The recovered
PMU measurements are imported into a dedicated central PI-PSS mechanism LFOD enhancement
realization. The proposed method is intended to improve the quality of synchrophasors obtained from
PMU so that the designed controller can generate satisfactory control performances, which would
otherwise be compromised when using unprocessed data as demonstrated in the case study. Time
domain studies have shown the functionality of the proposed measurement rectification strategy, and
89
CHAPTER 5. AN ADAPTIVE-PHASOR APPROACH TO PMU MEASUREMENTRECTIFICATION FOR LFOD ENHANCEMENT
0 4 8 12 16 200
1
2
3
×10−4
Time(s)
∆ω1
(p.u
.)
No control LFOD with MR
LFOD w/o MR
(a)
0 4 8 12 16 200
1
2
3
×10−4
Time(s)
∆ω3
(p.u
.)
(b)
Figure 5.11: Frequency deviations of (a) G1 and (b) G3 with proposed LFOD enhancer
also the effectiveness of the LFOD enhancer that improves the small-signal stability of the overall
system.
90
Chapter 6
A Stability Analysis of
Inverter-Interfaced Autonomous
Microgrids Integrated with PV-BESS
and VSG
ABSTRACT
In this chapter, a comprehensive small signal stability analysis (SSSA) framework is developed for solar
photovoltaics (PV) and battery energy storage system (BESS) integrated autonomous microgrids. This
model incorporates dispatchable Distributed Generators (DG), solar PV energy source and BESS, and
also a virtual synchronous generator (VSG)-based frequency regulator. The proposed SSSA analytical
model investigates the stability of a microgrid when varying PV output power, BESS output power,
virtual inertia coefficient (VIC) and the virtual damping coefficient (VDC) in the VSG are applied to
the microgrid system. The development of this analytical model utilizes the latest microgrid modeling
methodologies where dynamics of power converters, dynamics of loads, primary (droop) control method,
secondary control method, and VSG are systematically integrated. In this work, both modal analysis
and time-domain simulation present a full picture of the stability of the microgrid with a diverse range
of parameters. The proposed SSSA framework model developed in this study, with great scalability
and applicability, can serve as a useful tool for microgrid establishment and analyses in determining
the optimal uptake of solar PV energy, sizing of BESS, and also the optimal control parameters for the
purpose of maximizing the stability of a microgrid.
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CHAPTER 6. A STABILITY ANALYSIS OF INVERTER INTERFACEDAUTONOMOUS MICROGRIDS INTEGRATED WITH PV-BESS AND VSG
6.1 Chapter Foreword
The content of this chapter is mainly based on and modified from the following academic papers:
Tatkei Chau, Shenglong Yu, Tyrone Fernando, Herbert Iu and Michael Small, “An Investigation
of the Impact of PV Penetration and BESS Capacity on Islanded Microgrids–A Small-Signal Based
Analytical Approach”, presented at 2019 IEEE International Conference on Industrial Technology
(ICIT2019), Melbourne, Australia, 2019.
Previously, damping controllers and PMU measurement rectification methods are proposed to
enhance the low-frequency oscillation damping of multi-machine power systems. In such power systems,
the instability is mainly caused by damping torque deficiency among rotational power generators.
Unlike traditional power systems, microgrids are powered by a variety of distributed energy resources
such as DGs, renewables and BESSs which are interfaced by power electronic converters. Therefore,
in this chapter, a stability analysis framework is proposed to analyze the small-signal stability of
inverter-interfaced microgrids systems.
6.2 Introduction
With the electric power systems transforming into a more sustainable state, an increasing penetration of
renewable energy is being integrated into the electric grid. Generally speaking, renewable energy power
sources require power converters to convert electricity from DC to AC to form distributed microgrids.
The stability of the newly formed microgrids has become the main concern of power converters designer.
Among different stability studies in power systems, small-signal stability analysis, is a long-lasting and
important topic because of its close relations to power system damages [1]. Researchers have reported
the impact of the penetration level of renewable energy on the small-signal stability of power systems.
Extensive research efforts have been spent on the small-signal stability on power system integrated with
wind turbine generators [118–120]: authors in [118] analyzed the small signal stability of doubly-fed
induction generator (DFIG) wind turbines under a range of operation modes; authors in [119] studied
the stability of permanent magnet synchronous generator (PMSG)-based offshore wind turbines; and
authors in [120] investigated the low-frequency stability of grid-tied DFIG wind turbines with a newly
designed identification method. Impact of solar PV energy has been explored in [121,122], where the
authors assessed the small-signal stability of various penetration of solar PV energy on a large-scale
power system in the Western Interconnected Power Network in U.S. These are the only analytical and
practical research work on SSSA of diverse solar PV levels in a traditional power system. However, for
inverter-based microgrids with solar PV energy, such study has not been carried out in the literature.
The introduction of VSG, or VIC-based control methodology, is to enhance the primary frequency
response of inverter-interfaced microgrids, where a control structure is formulated to emulate the
dynamic characteristic of a synchronous generator [123]. Unlike traditional power systems, where the
electricity is mainly generated by synchronous generators, distributed generation with renewable energy
such as wind turbine and PV can provide no or very little inertia support to the main grid when the
system is subjected to external disturbance [124]. To enhance the primary frequency response, authors
in [124–127] have proposed novel control methods by means of VSG and VIC. Particularly, [125]
proposed a novel VIC emulation strategy through formulating DC-link capacitors to provide inertia
92
6.3. MATHEMATICAL MODELS OF AN ISLANDED MULTI-INVERTERMICROGRIDS
response; in [126], a VIC formation was devised for DFIG-based wind turbine with input-to-state
stability (ISS) analyzed; in [124], a similar approach was employed and DFIG wind turbines were
controlled in coordination with dispatchable generators for primary frequency response enhancement;
and an advanced control strategy for power systems with PMSG wind turbines was proposed in [127] to
provide inertia support. Despite the fact that the concept of VIC and VSG has been recently used in
multiple applications with renewable energy sources, there have not been a notable amount of research
articles carrying out investigations on PV-BESS integrated inverter-interfaced microgrids. The SSSA
studies for PV-BESS integrated microgrids with VSG control methodology are thus still lacking in the
research field.
The difficulty in performing SSSA for PV-BESS integrated, inverter-interfaced microgrids mainly
lies in a completely different and more complicated mathematical model, compared to the model
used in traditional power systems, that needs to be employed to accurately represent the dynamics of
microgrids. Authors in [42] are the first team that proposed the modeling approach for inverter-based
microgrids; another major contribution following this original paper is [128] where load models were
for the first time incorporated into the inverter-based microgrid model; and most recently research
team of [129] detailed the secondary control strategy in inverter-based islanded microgrids.
Aiming to fill the void of this aspect in current microgrid research, this study has the following
contributions: (i) A comprehensive microgrid model is built by means of utilizing the latest useful
microgrid modeling techniques in [42, 128, 129] with a newly proposed power flow method, where
primary control, secondary control, dynamics of inverters and loads, and VSG are integrated into the
microgrid with PV-BESS. The establishment of such microgrid model will benefit future advanced
studies on control methodology development. (ii) A small signal stability analytical framework is
established based on the model built in (i) for PV-BESS integrated modern microgrids, which has
great applicability due to the fact that the developed mathematical model can realistically represent
the real-world microgrids. (iii) Both modal analysis and time-domain simulations are carried out and
presented in detail in this study, which will qualitatively and quantitatively illustrate the effects of PV
output power, BESS capacity, VIC, and VDC of VSG. This will help determine the optimal uptake
of a microgrids with solar PV devices, size of BESS, and also control parameters of the controllers
employed in the microgrid.
The remainder of the chapter is organized as follows. In Section 6.3, a mathematical model for
inverter-based microgrids is detailed. The implementation of VSG into the PV-BESS structure is
described in Section 6.4. Section 6.5 presents the SSSA formulation and power flow analysis of the
microgrid. Simulation experiments with 2 cases are shown in Section 6.6 where the effects of a range of
parameters inside of the microgrids on the small signal stability are demonstrated through both modal
analysis and time-domain simulations. A conclusion is drawn in Section 6.7.
6.3 Mathematical Models of an Islanded Multi-Inverter Microgrids
In this section, detailed descriptions of the mathematical model of each electrical component in a
microgrid, from the inverter controller to the network and loads, are presented. Each subsection
will present a general model of an electrical component, and at the end the overall microgrid will be
described by a set of differential-algebraic equations in order to conduct subsequent modal analysis
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CHAPTER 6. A STABILITY ANALYSIS OF INVERTER INTERFACEDAUTONOMOUS MICROGRIDS INTEGRATED WITH PV-BESS AND VSG
and time-domain simulations.
6.3.1 DG Inverter and Controller
A typical DG-inverter structure comprises a power source, voltage source converters, an LCL filter and
internal controllers for the power electronics converters. To simplify the study without losing generality,
we assume the power source is able to produce the required amount of power for the islanded microgrid
under both steady-state operation and transient operation caused by small disturbances. In this study,
a multiple-stage control loop is adopted from [42], which is illustrated in Fig. 6.1. This control scheme
consists of three controllers–the power, voltage and current controller.
Powersource
Node
Microgrid
DCAC
Inverter
Powercontroller LPF
∗Voltagecontroller
Currentcontroller
∗
∗
Figure 6.1: DG inverter schematic
Power Controller
Voltage and frequency references are generated by the power controller based on the filtered local
active, reactive power, current and voltage measurements from the LCL filter, which are obtained with
(6.1)-(6.2) and the droop characteristics (6.3)-(6.4) for active and reactive power sharing of inverters.
As shown in Fig. 6.1, the calculated instantaneous active and reactive output power, po and qo,
passes through a low pass filter with a cut-off frequency ωc to obtain the power quantities corresponding
to the fundamental components, Po and Qo. The instantaneous power quantities are calculated with
the measured output voltage and current from the LCL filter, vdo , vqo ,ido and iqo, using the following
equations,
po = idovdo + iqov
qo, qo = idov
qo − iqovdo , (6.1)
Po = ωc (po − Po) , Qo = ωc (qo −Qo) , (6.2)
Droop control is adopted to mimic the governor characteristic of a synchronous generator, which
regulates the power sharing between inverter-interfaced power sources. As shown in Fig. 6.2, a change
in active or reactive power will lead to a corresponding change in power generation by the inverter-based
94
6.3. MATHEMATICAL MODELS OF AN ISLANDED MULTI-INVERTERMICROGRIDS
Secondaryresponse
Activepower
Frequency
Secondaryresponse
Reactivepower
Voltage
| |
| |
| |
(a) (b)
Figure 6.2: f − p droop and v − q droop
power sources, which will alter the inverter frequency and voltage, based on the following relations,
ωk = ωNLk + Ωc −mpkPok , (6.3)
v∗dok= V NL
k − nqkQok , v∗qok= 0, (6.4)
θ = ωk (6.5)
where subscript k is used to denote the number of node in the multi-inverter microgrid in this study.
Term ωk is the inverter frequency, ωNLk and V NL
k are the no-load frequency and voltage, mpk and nqkare the f − p and v − q droop coefficients, and θ is the phase angle of the inverter voltage. Term Ωc is
the control command set by the secondary controller for frequency restoration, which can be calculated
with the following equations,
Ωc = Kpfk(ωsp − ωk) +Kifkxfk , (6.6)
xfk = ωsp − ωk. (6.7)
where Kpfk and Kifk are the proportional and integral gains of the PI-controller in the secondary
controller and xfk is a intermediate variable that accumulates the frequency deviation. The input error
of the PI-controller is the difference between the frequency setpoint ωsp and the inverter frequency ωk.
The equation reduces to (6.8) for the inverters without secondary control mechanism,
ωk = ωNLk −mpkPok . (6.8)
Since the microgrid is not connected to the grid, one of the inverters is selected as the reference node
for the entire microgrid, and the angle difference between the rotating reference frames of the kth
inverter and the common reference frame is denoted by δk, which is computed by
δk =
∫(ωk − ωcom) . (6.9)
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CHAPTER 6. A STABILITY ANALYSIS OF INVERTER INTERFACEDAUTONOMOUS MICROGRIDS INTEGRATED WITH PV-BESS AND VSG
Current and Voltage Controllers
As shown in Fig. 6.3, both current and voltage controllers consist of two standard PI-controllers to
individually control the d − q components of voltage and current. As mentioned earlier, reference
Currentcontroller
Inverter
0
Voltagecontroller
‐ ‐
‐‐
∗
∗∗
∗∗
∗
‐‐
Figure 6.3: Current and voltage controllers in DG inverter
values of the d and q-axis output voltage v∗dokand v∗qok are generated by the power controller, and the
outputs of the PI-controllers in the voltage controller are then added together with feed-forward terms
KF idok
and KF iqok , to generate the references of the inverter currents i∗dik and i∗qik , where KF is the
feed-forward gain and idok and iqok are the d− q components of the output current of the LCL filter as
shown in Fig. 6.1. Similarly, the generated current references are fed into current controller to generate
the reference values for the inverter voltage v∗dikand v∗qik
with the similar approach used in the voltage
controller. The mathematical expressions of the voltage and current controllers are summarized as
follows,
i∗dik = KFkidok − ω
NLk Cfkv
qok
+Kpvk
(v∗dok− vdok
)+Kivkx
dvk
(6.10)
i∗qik = KFkiqok + ωNL
k Cfkvdok
+Kpvk
(v∗qok− vqok
)+Kivkx
qvk
(6.11)
v∗dik= −ωNL
k Lfkiqik
+Kpck
(i∗dik − i
dik
)+Kickx
dck, (6.12)
v∗qik= +ωNL
k Lfkidik
+Kpck
(i∗qik − i
qik
)+Kickx
qck, (6.13)
where Lf and Cf are respectively the inductance and capacitance of the LCL filter as shown in
Fig. 6.1 and Fig. 6.5, Kpv, Kiv, Kpc and Kic are respectively the proportional and integral gains of the
PI-controllers in the voltage and current loops, and xdqvi and xdqci are respectively the integrals of the
voltage and current errors between output voltage and the inverter current and their reference values,
xd,qvk = v∗d,qok− vd,qok
, (6.14)
xd,qck = i∗d,qik− id,qik
. (6.15)
To simplify the study, we assume the inverter is able to produce the required voltage without over-
modulation, i.e., vdik = v∗dikand vqik = v∗qik
[42].
96
6.3. MATHEMATICAL MODELS OF AN ISLANDED MULTI-INVERTERMICROGRIDS
6.3.2 LCL filter
The following differential equations summarizes the dynamics of the output LCL filter connected to an
inverter (Fig. 6.5 shows the topology),
Cfdvdokdt
= idik − idok
+ ωkCfvqok, (6.16)
Cfdvqokdt
= iqik − iqok− ωkCfvdok , (6.17)
Lfdidikdt
= vdik − vdok− rf idik + ωkLf i
qik, (6.18)
Lfdiqikdt
= vqik − vqok− rf iqik − ωkLf i
dik, (6.19)
Lcdidokdt
= vdok − vdk − rcidok + ωkLci
qok, (6.20)
Lcdiqokdt
= vqok − vqk − rciqok − ωkLci
dok, (6.21)
where vdk, vqk is the d− q components of voltage at kth node, Lc is the coupling inductance in the LCL
filter, and rf and rc are respectively the parasitic resistance of the filtering inductor and coupling
inductor.
6.3.3 Load modeling and Network Equations
In this study, all electric loads are modelled as R − L loads, instead of assuming constant power
consumptions or impedances as in large-scale power system studies, see [50,99]; and the transmission
lines are modeled with R−L impedances as well. Therefore, the loads and transmission line impedances
will vary with the system frequency. The following equations describe the relations between current
and voltage of transmission lines,
Llinekj
diDlinekj
dt= vDk − vDj −Rlinekj i
Dlinekj
+ ωLlinekj iQlinekj
, (6.22)
Llinekj
diQlinekj
dt= vQk − v
Qj −Rlinekj i
Qlinekj
− ωLlinekj iDlinekj
, (6.23)
where Llinekj and Rlinekj are the inductance and resistance of the transmission line connecting node
k and j, ilinekj is the current flowing from node k to node j, vk is the voltage of node k, and D −Qrepresents the direct and quadrature components of the common reference frame in the islanded
microgrid.
The following equations describe the relations between current and voltage at load nodes,
Lloadk
diDloadk
dt= vDk −Rloadk
iDloadk+ ωLloadk
iQloadk, (6.24)
Lloadk
diQloadk
dt= vQk −Rloadk
iQloadk− ωLloadk
iDloadk, (6.25)
where Lloadkand Rloadk
are the inductance and resistance of the load connected to node k, and iloadkis
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CHAPTER 6. A STABILITY ANALYSIS OF INVERTER INTERFACEDAUTONOMOUS MICROGRIDS INTEGRATED WITH PV-BESS AND VSG
the current flowing into the load connected to node k. In microgrid modeling, the current and voltage
at each node also have the following relation,
vD,Qk R−1N = iD,Qoj − iD,Qloadj
+N∑
k=1,k 6=jiD,Qlinekj
, (6.26)
where RN is the virtual resistance connecting each node to the ground. The introduction of the virtual
resistance is to ensure the numerical stability when conducting the simulation experiments for the
microgrid [42]. The value of RN is chosen to be sufficiently large, so that the calculated voltage at
each node will not differ significantly from its actual value.
6.4 PV-BESS based Virtual Synchronous Generator
A Virtual Synchronous Generator control method is implemented into the inverter connected to the
PV-BESS system to synthesize the frequency response from a synchronous generator in order to provide
inertia support to the islanded microgrid [130]. Fig. 6.4 depicts the working principle of the control
system of the VSG, where local measurements are fed into a VSG block together with the input power
set-point generated by the virtual governor to generate the frequency reference of the inverter based on
the swing equation [131],
ωrkJkdωrkdt
= PPVk + PBESS
k − Pok −D(ωrk − ωg), (6.27)
where ωrk is the frequency of virtual rotor, ωg is the frequency of the islanded microgrid, Jk is the
virtual inertia constant, and D is the virtual damping coefficient. The total power generated by PV
and BESS, PPVk +PBESS
k , is the virtual shaft power determined by the virtual governor based on (6.28),
PBESSk =
1
mpk
(ωNLrg − ωk)− PBESS
k , (6.28)
which is adopted and modified from [132].
Powersource
LCLfilter
Node
PWM
VSGVirtualgovernor
Q‐VDroop
Invertercontrol dq
abc
∗ ∗ ∗
∗
∗
∑
∑
‐∗
Figure 6.4: PV-BESS based Virtual Synchronous Generator
98
6.5. SMALL SIGNAL STABILITY ANALYSIS MODEL
6.5 Small Signal Stability Analysis Model
In this section, a brief discussion on SSSA model formulation will be presented, where a Differential
Algebraic Equation (DAE) formulation is used, and system linearization is performed to realize the
SSSA for microgrids.
6.5.1 System State-Space model and Linearization
The nonlinear mathematical model describing a microgrid can be written in the following compact
form [1,42],
X = f (X,Υ,V,U) ,
0 = g1 (X,Υ,V) , (6.29)
0 = g2 (X,V) ,
where X is the dynamic state vector of the microgrid, including the dynamics of the inverters, DGs,
transmission lines and loads; V is the vector containing the voltage magnitudes and angles of all
nodes; Υ represents inverter algebraic variables; and U is the control input vector comprising droop
coefficients, PV output power and no-load frequency and voltage for droop curves. Function f(·) is
the system state-space function, g1(·) = 0 is the inverter algebraic equation set, and g2(·) = 0 is the
network equation set.
System SSSA requires system linearization, with the compact form in (6.29), the system matrix
Asys can be calculated symbolically as follows,
∆X = Asys∆X +Bsys∆U, (6.30)
where
Asys = A1 −B1 ·D−11 · C1 −B2 ·D−1
4 · C2, (6.31)
and
A1 =∂f
∂X, B1 =
∂f
∂Υ, B2 =
∂f
∂V,
C1 =∂g1
∂X, C2 =
∂g2
∂X, D1 =
∂g1
∂Υ, D4 =
∂g2
∂V, (6.32)
and the calculation of the input matrix of the linearized system Bsys is omitted. With the acquired
system matrix, the effects of varying parameters on the system stability can be observed and analyzed.
See [1] for more details on the procedures of linearizing a power system, where the system is significantly
different from the one used in this study, but the principles are similar.
99
CHAPTER 6. A STABILITY ANALYSIS OF INVERTER INTERFACEDAUTONOMOUS MICROGRIDS INTEGRATED WITH PV-BESS AND VSG
6.5.2 Proposed Power Flow Analysis for Islanded Microgrids with Secondary Con-
troller
The steady-state solution of the islanded droop-based microgrid can be obtained by carrying out
the modified power flow analysis [128] using Newton-Raphson method. However, when a centralized
secondary frequency regulator is deployed in an islanded microgrid, its steady-state frequency settles
to a preset setpoint programmed in the secondary controller instead of free running according solely
to the droop characteristics (6.8). Therefore, it is necessary for us to further modify the algorithm
in [128] in order to obtain the power flow solution when a secondary controller is present in the islanded
microgrid. The governing equations of the modified power flow algorithm with sceondary control is
summarized as follows,
X i+1 = J−1i ∆P i + ∆X i, (6.33)
where X , ∆P and ∆X are the modified unknown vector, power mismatch vector and correction vector
respectively, and J is Jacobian matrix of the power flow problem. The modified unknowns of the power
flow problem are given as follows,
X =[ω′ θ |V |
]T, (6.34)
where θ and |V | are respectively the voltage phase angles and magnitudes for all nodes as stated in
original formulation of a power flow problem [1], and ω′ is a vector that contains the modified no-load
frequencies of the nodes with secondary control as mentioned in (6.6),
ω′ = ωNL + Ωc, (6.35)
and the formulation of the modified power mismatch vector, ∆P , can be found in [128]. Note that the
active power generation of each inverter node has now become
PGk=
1
mpk
(ω′k − ωk
), (6.36)
and RN is modeled as shunt impedance. Corresponding modifications also need to be made when
computing the Jacobian matrix. After solving the power flow problem, X ss is obtained, from which the
initial condition of voltage magnitudes and angles of all nodes Vss can be extracted. The initial values
of state variables Xss and inverter algebraic variables Υss can be solved by setting the differential
equations in (6.29) equal to zero [59].
6.6 Simulation and Numerical Results
In this study, a 6-node microgrid system is employed, which is adopted and modified from [42]. Fig. 6.5
demonstrates the system topology. Nodes 4, 5 and 6 are generator nodes, which are connected to a DG,
BESS and PV-BESS respectively; nodes 1, 2 and 3 are load nodes (modeled as RL loads). TABLEs 6.1,
6.2 and 6.3 show respectively the line parameters, load and generator settings, and the inverter and
100
6.6. SIMULATION AND NUMERICAL RESULTS
LCL parameters. Both modal analysis and time-domain simulations are conducted in this section. The
simulation is performed in MATLAB® 2016b coding environment on a desktop computer with Intel®
Core i7-4790, 3.6GHz CPU and 64-bit Windows®7 operating system. The modal analysis is carried
out through system linearization, eigenvalue analysis and eigenvalue sensitivity study, whereas the
time-domain simulation is performed by using MATLAB built-in algebraic-differential-equation solvers
in continuous time.
InverterInverter
Inverter
1
2
3
4
5
6
BESS
DG
PV
BESS
LCLfilter
LCLfilter
LCLfilter
Figure 6.5: Multi-inverter microgrid with PV-BESS VSG
Table 6.1: Line parameters
From To R(Ohm) L(mH)
1 2 0.43 0.3181 4 0.3 0.352 3 0.15 1.8432 5 0.2 0.253 6 0.05 0.05
Table 6.2: Load and Generator Settings
Node mp nq ωNL |V NL| R L(rad/s/kW) (V/kVar) (rad/s) (V) (Ohm) (mH)
1 - - - - 6.950 12.23 - - - - 5.014 9.44 9.4× 10−5 1.3× 10−3 120π 220 - -5 9.4× 10−5 1.3× 10−3 120π 220 - -6 9.4× 10−5 1.3× 10−3 120π 220 - -
6.6.1 Power flow analysis
Power flow analysis is the first step to understand and observe a power system. In this work, a novel
microgrid power flow approach, as stated in Section 6.5.2, is employed, and the power flow results
are shown in TABLEs 6.4 and 6.5 with PPV6 = 1.5kW and 0.1kW respectively. In the microgrid of
interest, node 4 is used as the reference node, and its voltage angle is always assumed to be 0o. The
secondary control mechanism is installed on the generator connected to node 4, i.e., the DG generator.
The nominal frequency of this microgrid is 1 p.u. for the purpose of this study. In real-world situations
the nominal frequency is generated by the tertiary control level, which is not incorporated in this study.
101
CHAPTER 6. A STABILITY ANALYSIS OF INVERTER INTERFACEDAUTONOMOUS MICROGRIDS INTEGRATED WITH PV-BESS AND VSG
Table 6.3: Inverter and LCL filter parameters
ωbase 120π rad/s Vbase 220V Sbase 1kVACf 50µF rc 0.03Ω Lc 0.35mHrf 0.1Ω Lf 1.35mH KF 0.75Kpv 0.05 Kiv 390 Kpc 10.5Kic 16× 103 Kif 10 Kpf 1ωc 10ωnom J 2s D 17p.u.
The secondary control is implemented to maintain this nominal frequency by varying the no-load
frequency of the DG droop characteristics, i.e., ωNL4 , as illustratively shown in Fig. 6.2(a).
The PV output power in the two cases is 1.5kW and 1× 10−4kW respectively. It is assumed that
during steady-state operations, the BESS does not charge and discharge so as to make more efficient
use of power supplied by the BESS, and also extend the longevity of BESS. This makes active power
generated by node 5 always zero in power flow analysis, and active power supplied from node 6 equal
to PV output power. As shown in the two tables, with different solar PV output power, the power flow
results show two different operating statuses of the microgrid, where active and reactive power load
demands vary, and power losses change from case 1 to case 2. During steady-state analysis, it is already
clear that distinct solar PV power penetration causes different operating statuses for a microgrid. This
phenomenon will be observed multiple times in the following studies.
Table 6.4: Load flow of the microgrid when PPV6 = 1.5kW (base case)
Node Voltage Generation LoadV (p.u.) θ(o) P (kW) Q(kVAr) P (kW) Q(kVAr)
1 0.9868 −1.1922 −− −− 2.3579 1.56042 0.9791 −3.0783 −− −− −− −−3 0.9813 −4.5542 −− −− 3.0995 2.19064 1.0087 ∗0.0000 4.2515 −1.4716 −− −−5 0.9843 −3.7307 0.0000 2.6554 −− −−6 0.9840 −4.6859 1.5000 2.7144 −− −−
Total: 5.7515 3.8982 5.4574 3.7510
Table 6.5: Load flow of the microgrid when PPV6 = 0.1W
Node Voltage Generation LoadV (p.u.) θ(o) P (kW) Q(kVAr) P (kW) Q(kVAr)
1 0.9890 −2.0210 −− −− 2.3688 1.56762 0.9707 −5.0383 −− −− −− −−3 0.9782 −7.9574 −− −− 3.0802 2.17704 1.0178 ∗0.0000 6.1583 −3.0195 −− −−5 0.9780 −5.9677 0.0000 3.7257 −− −−6 0.9840 −4.6859 0.0001 3.4503 −− −−
Total: 6.1584 4.1565 5.4490 3.7446
6.6.2 Small Signal Stability Analysis with Varying Parameters
Base Case
Using the proposed SSSA framework and applying to the microgrid model, the eigenvalues of the
system can be acquired, which are shown in Fig. 6.6. This result is for the microgrid with base-case
102
6.6. SIMULATION AND NUMERICAL RESULTS
settings, which are shown in TABLEs 6.1 ∼ 6.3 with PPV6 = 1.5kW. TABLE 6.6 demonstrates the
−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0
·103
−2
0
2
·103
0.96
0.880.74 0.6 0.48 0.34 0.22 0.12
0.96
0.880.74 0.6 0.48 0.34 0.22 0.12
Real part
Imaginarypart
Figure 6.6: SSSA for the base case
eigenvalues associated with generators, controllers and LCL filters and of low damping ratios in the
microgrid with the base-case setup, where frequencies and dominant states of the modes are presented.
Table 6.6: Eigenvalues of interest with base-case settings
λ ξ f(Hz) Dominant states
−117.0± j1573 7.41(ξcrit) 250 vdqo6 , iDQline23
−144.0± j1934 7.43 307 vdqo6 , iDQline23
−4.972± j44.68 11.05 7.11 ω6 and PBESS6
−57.70± j465.8 12.29 74.1 vdqo of inverter 4, 5, 6−297.9± j2322 12.73 370 vdqo , idqo of inverter 4, 5−349.6± j2685 12.91 427 vdqo , idqo of inverter 4, 5−3.463± j15.69 21.55 2.50 PBESS
6 , xdqv , δ of inverter 5, 6
Combining TABLE 6.6 and Fig. 6.6, it is safe to state that with the base-case setup of the microgrid,
the SSSA indicates a stable system. Now we are in the position to investigate how changing parameters
affect the stability of the microgrid.
Impact of VSG parameters and PV-BESS sizes
With varying VSG parameters, including VIC and VDC, the movements of eigenvalue root loci are
shown in Fig. 6.7 and Fig. 6.8, where Fig. 6.7 corresponds to the root loci with increasing VIC, and
Fig. 6.8 corresponds to increasing VDC of the VSG. It is clear that when VIC and VDC vary, some
system modes approach to a more stable region, some move towards the unstable region, and the rest
may make a turn with a particular VIC or VDC. Based on the analytical process and numerical results,
it is possible for us to determine the best control parameters to maximize the system stability.
System stability also has a close relation with the PV output power and the capacity of the BESS
connected to PV. Root loci reflecting such variations are shown in Fig. 6.9 and Fig. 6.10. Observing the
root loci, similar conclusion can be drawn, and some modes move to the right-hand side, some to the
left-hand side, and some make a turn and move towards a different direction as PV output power and
BESS increase. This analysis can serve as a tool in determining the optimal PV power and BESS sizes
for the purpose of maximizing the stability of microgrids with distributed solar PV devices and BESSs.
103
CHAPTER 6. A STABILITY ANALYSIS OF INVERTER INTERFACEDAUTONOMOUS MICROGRIDS INTEGRATED WITH PV-BESS AND VSG
−500 −450 −400 −350 −300 −250 −200 −150 −100 −50 0
0
200
400
Real part
Imaginary
part
−14 −10 −5 0−40
−20
0
20
40
Real part
Imaginary
part
Figure 6.7: Root loci of the system eigenvalues when increasing VIC (from 0.002 to 64s ).
−400 −350 −300 −250 −200 −150 −100 −50 0
0
200
400
Real part
Imaginarypart
−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0−20
−10
0
10
20
Real part
Imaginary
part
Figure 6.8: Root loci of the system eigenvalues when increasing VDC (from 0.0017 to 272 p.u. ).
6.6.3 Time-Domain Simulation with Varying VIC and BESS capacity
In this subsection, time-domain simulations are carried out on two distinct cases where different sets of
VIC and BESS capacity are employed. The microgrid system operates in steady-state for the first 50
seconds, and at t = 50s, a disturbance is introduced and the resistance of the load at node 3 reduces
by 17%, i.e., Rload3 = Rbase caseload3
× 83%. The time-domain simulation results will further illustrate the
difference in system stability when changing microgrid parameters.
104
6.6. SIMULATION AND NUMERICAL RESULTS
−400 −350 −300 −250 −200 −150 −100 −50 0
0
200
400
Real part
Imaginary
part
−40 −35 −30 −25 −20 −15 −10 −5 0 5 10
−20
0
20
Real part
Imaginary
part
Figure 6.9: (a) Root loci and (b) its zoom-in of the system eigenvalues when increasing BESS capacity(from 0.0098 to 1000kW).
−400 −350 −300 −250 −200 −150 −100 −50 0
0
200
400
Real part
Imaginarypart
−14 −12 −10 −8 −6 −4 −2 0−20
−10
0
10
20
Real part
Imaginary
part
Figure 6.10: (a) Root loci and (b) its zoom-in of the system eigenvalues when increasing solar PVoutput power (from 0.0001 to 12kW).
Varying VIC
As already discussed in the root loci movement in Fig. 6.7, increasing VIC, i.e., J6 in this particular
study will cause the decrease in the Critical Damping Ratio (CDR) ξcrit and some eigenvalues may
even move to the right half of the complex plane, leading to system instability. In this subsection,
two VIC values, J6 = 2s and J6 = 8s are chosen in the simulation study, so that the system in the
two cases are still stable but has highly distinguishable responses. TABLE 6.7 shows the CDR of
the system and the participation factor of each state when J6 = 8s. Table depicting similar data for
105
CHAPTER 6. A STABILITY ANALYSIS OF INVERTER INTERFACEDAUTONOMOUS MICROGRIDS INTEGRATED WITH PV-BESS AND VSG
J6 = 2s is shown in TABLE 6.6. It is obvious that the CDR has reduced from 7.411 in TABLE 2.2
to 6.641 in TABLE 6.7 as J6 step-changes from 2s to 8s. Fig. 6.11 and Fig. 6.12 demonstrate the
time-domain system responses with different VIC values. Congruous with the analysis in TABLE 6.6
and TABLE 6.7, the microgrid has two distinct primary frequency responses. When J6 = 2s, the system
has a higher CDR and shows a better primary response, with lower oscillation magnitudes, shorter
settling time and a mitigated first nadir/zenith point in each sub-figure of Fig. 6.11 and Fig. 6.12.
Varying BESS Sizes
The BESS capacity also has an impact on the stability of the overall microgrid. Through SSSA shown
in Fig. 6.9, we have already seen that as BESS capacity rises, the CDR of the system increases, leading
to a more stable system. In this time-domain simulation, we incorporate two BESS capacities and
observe the system responses when subjected to the specified disturbance. TABLE 6.8 depicts the
SSSA numerics when the BESS capacity is PBESSmax = 5 kW, a lower value than the nominal, which is
PBESSmax = 10kW. Comparing TABLE 6.8 with TABLE 6.6, it is easy to identify that with a smaller
BESS capacity, the system CDR reduces from 7.411 to 3.203, i.e., smaller PESS capacity leads to less
stability. Time-domain simulation in Fig. 6.13 and Fig. 6.14 the identical system stability change as
shown in the above table; with a reduced BESS capacity, the microgrid becomes less stable under
electrical disturbances.
Table 6.7: SSSA numerics with VIC J6 = 8s
State PBESS6
ω6 xdv6 Po6 xdv6 δ6 xdv5 xqv5PF 0.244 0.149 0.103 0.102 0.097 0.095 0.07 0.067†λcrit = −1.148± j17.243 ξcrit = 6.641 fcrit = 2.744Hz
Table 6.8: SSSA numerics with PBESSmax = 5kW
State PBESS6 Po6 δ6 xdv6 xqv6 xdv5 xqv5 ω6
PF 0.179 0.140 0.131 0.127 0.119 0.085 0.082 0.052†λcrit = −0.545± j17.00 ξcrit = 3.203 fcrit = 2.707Hz
6.7 Conclusion
In this chapter, a mathematical framework is developed for the small signal stability analysis of
inverter-interfaced microgrids with solar PV energy sources and battery storage systems. Applying
this analytical framework to a typical islanded microgrid with multiple loads, distributed generators
and renewable energy sources, it has become possible for us to qualitatively and quantitatively observe,
interpret and test the impact of changing parameters on the stability of the microgrid. These factors
include solar PV energy uptake in a microgrid, virtual inertia coefficient, damping coefficient of the
VSG, and the capacity of the energy storage system. Modal analysis and time-domain simulations have
demonstrated such impact, which has proven the applicability of the proposed SSSA framework for
microgrid research. Future work may include expanding the proposed framework to a more complex
inverter-based microgrids with a variety of generation types, and proposing feasible control approaches
106
6.7. CONCLUSION
−0.4
−0.2
0
0.15·10−4
∆ω4
(p.u
.)
J6 = 2s J6 = 8s
−1
−0.5
0·10−4
∆ω5
(p.u
.)
49 51 53 55 57 59
−2
−1
0
1·10−4
∆ω6
(p.u
.)
Figure 6.11: Inverter frequency with different VICs
4.2
4.4
4.6
4.8
5
P4(kW
)
J6 = 2s J6 = 8s
0
0.2
0.4
P5(kW
)
49 51 53 55 57 59
1.5
1.55
1.6
1.65
P6(kW
)
Figure 6.12: Inverter power with different VICs
for microgrids with renewable energy sources based on the analytical and numerical results acquired
from this study.
107
CHAPTER 6. A STABILITY ANALYSIS OF INVERTER INTERFACEDAUTONOMOUS MICROGRIDS INTEGRATED WITH PV-BESS AND VSG
−0.4
−0.2
0
0.15·10−4
∆ω4
(p.u
.)
PBESSmax6
=10kW PBESSmax6
=5kW
−1
−0.5
0
·10−4
∆ω5
(p.u
.)
49 51 53 55 57 59−3
−2
−1
0
1·10−4
∆ω6
(p.u
.)
Figure 6.13: Inverter frequency with different BESS capacities
4.2
4.4
4.6
4.8
5
P4(kW
)
PBESSmax6
=10kW PBESSmax6
=5kW
0
0.2
0.4
P5(kW
)
49 51 53 55 57 59
1.5
1.55
1.6
P6(kW
)
Figure 6.14: Inverter power with different BESS capacities
108
Chapter 7
Conclusions and Future Work
7.1 Conclusions
This thesis investigates the small-signal stability and dynamical behavior of complex power systems
integrated with renewable energy sources and FACTs devices. The mathematical models of wind
turbines and STATCOM are studied together with the IEEE standard test system model, and the
models are reformulated to provide suitability for the implementation of low frequency oscillation
damping controllers and the proposed dynamic state estimation-based PMU measurement rectification
methodology.
Small-signal stability and modal analysis techniques are used to identify low-frequency oscillatory
modes existing in the power system through linearizing the reformulated system models. A PSS-like
supplementary damping controller is designed for STATCOM and tuned with a PSO algorithm using
the load forecast obtained with ANNs in order to improve the critical damping ratio of the test system
used in the study.
A control strategy for DFIGs is proposed using the rotor-side-controller embedded in the control
structure to improve primary frequency response and the small-signal stability of the system. Modal
analysis demonstrates that the proposed RPSS can effectively increase the critical damping of system,
and time-domain simulation has shown satisfactory improvement on both primary frequency response
and small-signal stability enhancement.
Also, an adaptive phasor-based PI-PSS is proposed for damping enhancement using PMU mea-
surements, and a measurement rectification method is proposed in order to eliminate bad PMU
measurements and compensate communication delay for a better damping performance. Decentralized
dynamic state estimators, which only make use of the local voltage and current PMU measurements,
are designed to screen out the bad measurements and remove them before being fed into the proposed
damping enhancer.
On the microgrid front, the mathematical models of the power inverters, PV-BESS and VSG are
studied and reformulated to investigate the small-signal stability of islanded microgrids and qualitatively
and quantitatively observe, interpret and test the impact of changing parameters and the sizes of
renewable energy sources on the stability of the microgrid.
109
CHAPTER 7. CONCLUSIONS AND FUTURE WORK
7.2 Future Work
The following future work is recommended to improve and solidify the damping control design and
tunning methods developed in this thesis. From the point of view of the parameter tuning method of
PSS, load forecast is considered to be the main tool used to reflect changes on the system operating
condition, but the changes on renewable energy generation have not yet been considered and utilized
in the tunning algorithm. When renewable energy sources are integrated into a power systems, the
total power generated is no longer solely dependent on scheduled power output from central generators,
and the output power of the RESs also need to be forecasted for the calculation of system operating
conditions. Therefore, incorporating forecasts of renewable energy generation in the parameter tunning
method can be the next task. In this thesis, we have thus far only considered solar PV panels or
wind turbines in our stability studies, but the less common RESs such as fuel cells and biomass
generators have not yet been considered. Therefore, investigating potential stability challenges posed
by integrating such devices into a complex power system would be a good topic to discuss in the near
future as it is beneficial to practical power system design engineers for preparing future uptake of such
RESs. In terms of microgrid, after the analysis of stability of the inverter-interfaced microgrid, control
strategies need to be devised in order to enhance voltage and frequency stability as well as maintaining
the balance of power sharing within the microgrid, which would be a valuable direction to undertake
future research.
110
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