small-signal stability analysis and wide-area damping

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*


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*


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.


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 3.14 Simulation results of case study 2(1) . . . . . . . . . . . . . . . . . . . . . . . 40



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



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


3.2 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


4.2 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


5.2 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


6.2 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


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


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


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


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


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


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


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


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


−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


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


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


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


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


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


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


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


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


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


−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


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


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


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


−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


−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


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)


−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


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


Figure 3.19: System eigenvalues with optimized controller in Case 3 Control Scenario 2

43


Table 3.11: Eigenvalues of interest relating to LFEOs with the proposed controller (Case 3, Scenario2)


−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


Table 3.12: Eigenvalues of interest relating to LFEOs with the proposed controller in Case 3Interaction


−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


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


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


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


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


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


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


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


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


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


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


−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


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.

63


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


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


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)


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)


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


−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


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


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



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

73


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.

75


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


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.

79


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)

81


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


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


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.

83


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


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


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


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


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.

91

CHAPTER 6. A STABILITY ANALYSIS OF INVERTER INTERFACEDAUTONOMOUS MICROGRIDS INTEGRATED WITH PV-BESS AND VSG



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

93


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


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)

95


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


6.3.2 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

97


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


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


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


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


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


−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


−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


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


−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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