SIMULTANEOUS MITIGATION OF

SUBSYNCHRONOUS RESONANCE
AND SUBSYNCHRONOUS INTERACTION
USING FULL-SCALE FREQUENCY CONVERTER-
AND DOUBLY-FED INDUCTION GENERATOR-
BASED WIND FARMS

A Thesis

Submitted to the College of Graduate Studies and Research

In Partial Fulfillment of the Requirements

For the Degree of Master of Science

In the Department of Electrical and Computer Engineering

University of Saskatchewan

Saskatoon, Saskatchewan

By

Xuan Gao

© Copyright Xuan Gao, May 2014. All rights reserved

 PERMISSION TO USE

I agree that the Library, University of Saskatchewan, may make this thesis freely available for
inspection. I further agree that permission for copying of this thesis for scholarly purpose may be
granted by the professor or professors who supervised the thesis work recorded herein or, in their
absence, by the Head of the Department or the Dean of the College in which the thesis work was
done. It is understood that due recognition will be given to me and to the University of
Saskatchewan in any use of the material in this thesis. Copying or publication or any other use of
this thesis for financial gain without approval by the University of Saskatchewan and my written
permission is prohibited.

Request for permission to copy or to make any other use of the material in this thesis in whole or
part should be addressed to:

Head of the Department of Electrical and Computer Engineering

University of Saskatchewan
57 Campus Drive
Saskatoon, Saskatchewan S7N 5A9
Canada

i
 ABSTRACT

Subsynchronous Resonance (SSR) is one of the major obstacles for the wide spread of high
degrees (60% and higher) of series capacitor compensation. Recently, a new obstacle, namely
Subsynchronous Interaction (SSI) has been added to the list after the Zorillo Gulf wind farm
incident in Texas in October 2009. SSI is due to the interaction between large Doubly-Fed
Induction Generator (DFIG)-based wind farms and series capacitor compensated transmission
systems.

In integrated power systems incorporating series capacitor compensated transmission lines

and high penetration of wind energy conversion systems, especially DFIG-based wind farms, SSR
and SSI could occur concurrently as a result of some system contingences. Therefore, mitigating
SSR and SSI is an important area of research and development targeting at developing practical
and effective countermeasures.

This thesis reports the results of digital time-domain simulation studies that are carried out
to investigate the potential use of Full-Scale Frequency Converter (FFC) and DFIG-based wind
farms for simultaneous mitigation of SSR and SSI. This is achieved through introducing
supplemental control signals in the reactive power control loops of the grid side converters of the
DFIG and/or the FFC wind turbines. In this context, two supplemental controls designated as
Supplemental Controls 1 and 2 are examined. Supplemental Control 1 introduces a signal in the
grid side converter of the FFC wind turbines to damp both SSR and SSI oscillations. On the other
hand, Supplemental Control 2 introduces a signal in the grid side converter of the FFC wind
turbines for damping SSR oscillations and another signal in the grid side converters of the DFIG
wind turbines for damping SSI oscillations.

Time-domain simulations are conducted on a benchmark model using the ElectroMagnetic

Transients program (EMTP-RV). The results of the investigations have demonstrated that the
presented two supplemental controls are very effective in mitigating the SSR and SSI phenomena
at different system contingencies and operating conditions.

ii
 ACKNOWLEDGEMENTS

Foremost, I would like to express my sincere gratitude to my supervisor Dr. S.O. Faried,
for his enthusiasm, motivation, patience, encouragement and immense knowledge. His guidance
helped me through all the time over my classes, research and the writing of my thesis.

I offer my acknowledgement to my graduate study teachers, Dr. Rajesh Karki, Dr. N.A.
Chowdhury and Dr. Daniel Chen for strengthening my knowledge on reliability, mathematics and
control engineering. I also would like to appreciate Dr. Ulas Karaagac for providing some of the
models used in this thesis.

Besides that, I would like to thank my lab partners: Emmanuel Ogemuno, Qi Zhao, Linh
Pham and Jiping Zhang, for the discussions, inspirations and time we were working together. I
also would like to thank my friend Federica Giannelli for helping me edit the references.

Last but not the lease, I would like to thank my family: my parents Jianzhang Gao and
Xianglan Yin, for supporting me spiritually and financially all through my time living and studying
in University of Saskatchewan.

iii
 TABLE OF CONTENTS

PERMISSION TO USE ............................................................................................................................... i

ABSTRACT ............................................................................................................................................... ii
ACKNOWLEDGEMENTS ....................................................................................................................... iii
TABLE OF CONTENTS ........................................................................................................................... iv
LIST OF FIGURES .................................................................................................................................. vii
LIST OF TABLES ................................................................................................................................. xviii
LIST OF SYMBOLS ............................................................................................................................... xix
1. INTRODUCTION .................................................................................................................................. 1
1.1 Wind Energy ................................................................................................................................ 1
1.2 Series Capacitive Compensation .................................................................................................. 1
1.3 Transmission Line Series Compensation ..................................................................................... 2
1.3.1 Increase the power transfer capability by raising the first swing stability limit ...................... 3
1.3.2 Increase in power transfer ...................................................................................................... 3
1.3.3 Active load sharing between parallel circuits ......................................................................... 4
1.4 Subsynchronous Resonance (SSR)............................................................................................... 5
1.4.1 SSR: basic phenomenon ........................................................................................................ 5
1.5 Doubly-fed Induction Generator Wind Turbine ........................................................................... 8
1.6 Subsynchronous Interaction Phenomenon in DFIG-Based Wind Farms ...................................... 9
1.7 Full-Scale Frequency Converter Wind Turbine.......................................................................... 10
1.8 Research Objective and Scope of the Thesis .............................................................................. 11
2. MATHEMATICAL MODELING OF POWER SYSTEMS INCORPORATING FFC- AND DFIG-
BASED WIND FARMS FOR LARGE DISTURBANCE STUDIES................................................... 13
2.1 Introduction ................................................................................................................................ 13
2.2 System under Study ................................................................................................................... 13
2.3 Power System Modeling ............................................................................................................ 15
2.3.1 Modeling of the synchronous generator ............................................................................... 15
2.3.2 Modeling of the turbine-generator mechanical system ......................................................... 19
2.3.3 Modeling of the transmission line ........................................................................................ 22

iv
 2.3.4 Modeling of transformer ...................................................................................................... 24
2.3.5 Modeling of system loads .................................................................................................... 25
2.3.6 Wind turbine aerodynamic model ........................................................................................ 25
2.3.7 Modeling of the DFIG .......................................................................................................... 26
2.3.8 Modeling of the BtB dc capacitor link ................................................................................. 27
2.3.9 Modeling of PMSG .............................................................................................................. 30
2.3.10 Modelling of the BtB dc capacitor link .............................................................................. 31
2.3.11 FFC Wind turbine control .................................................................................................. 31
2.4 A Sample Case Study: 30% Compensation Degree in Lines 1 and 2 ......................................... 32
2.5 Summary .................................................................................................................................... 41
3. SUPPLEMENTAL CONTROLS OF FFC- AND DFIG-BASED WIND FARMS FOR
SIMULATANEOUS MITIGATION OF SUBSYNCHRONOUS RESONANCE AND
SUBSYNCHRONOUS INTERACTION ............................................................................................. 42
3.1 Introduction ................................................................................................................................ 42
3.2 Supplemental Controls of FFC- and DFIG-Based Wind Farms ................................................. 42
3.3 Performance of Supplemental Control 1 in damping SSR and SSI Oscillations ........................ 48
3.3.1 Effect of the distance between wind farm B and the turbine-generators ............................... 58
3.3.2 Effect of wind farm B rating ................................................................................................ 78
3.3.3 Effect of the fault type .......................................................................................................... 91
3.4 Performance of Supplemental Control 2 in damping SSR and SSI Oscillations ......................... 98
3.4.1 Effect of the fault location .................................................................................................. 101
3.4.2 Effect of the compensation degree ..................................................................................... 108
3.5 Summary .................................................................................................................................. 116
4. SUMMARY AND CONCLUSIONS ................................................................................................. 117
4.1 Summary .................................................................................................................................. 117
4.2 Conclusions .............................................................................................................................. 118
REFERENCES ....................................................................................................................................... 120
APPENDIX A......................................................................................................................................... 123
APPENDIX B ......................................................................................................................................... 128
B.1 Supplemental Control 1 Output Signals ................................................................................... 128
B.2 Supplemental Control 1 Output Signals (Line 3 length = 50 km) ............................................. 129
B.3 Supplemental Control 1 Output Signals (Line 3 length = 100 km) ........................................... 130
B.4 Supplemental Control 1 Output Signals (Line 3 length = 200 km) ........................................... 131
B.5 Supplemental Control 1 Output Signals (Wind farm B rating = 300 MW) ............................... 132

v
B.6 Supplemental Control 1 Output Signals (Wind farm B rating = 400 MW) ............................... 133
B.7 Supplemental Control 1 Output Signals (Double Line-to-Ground Fault) ................................. 134
B.8 Supplemental Control 2 Output Signals ................................................................................... 135
B.9 Supplemental Control 2 Output Signals (Three-phase Fault on Line 4) ................................... 136
B.10 Supplemental Control 2 Output Signals (50% Compensation) ................................................ 137

vi
 LIST OF FIGURES

Figure 1.1: Transient time response of a turbine-generator shaft torsional torque during and
after clearing a system fault on a series capacitive compensated transmission line.
................................................................................................................................. 2
Figure 1.2: Transient time response of a large DFIG-based wind farm terminal voltage (root
mean square) during and after clearing a system fault on a series capacitive
compensated transmission line. .............................................................................. 2
Figure 1.3: A transmission line with a series capacitor. ............................................................ 3
Figure 1.4: Maximum power transmitted over a transmission line as a function of the degree of
series compensation (|Vt| = |Vb| = 1 p.u., XL= 1 p.u.). .............................................. 4
Figure 1.5: Adjusting the power sharing between two parallel lines using a series capacitor. . 5
Figure 1.6: A turbine-generator connected to an infinite-bus system through a series capacitor
compensated transmission line. .............................................................................. 6
Figure 1.7: Schematic diagram of a DFIG wind turbine. .......................................................... 8
Figure 1.8: Schematic diagram of a FFC wind turbine. .......................................................... 11
Figure 2.1: System under study. .............................................................................................. 14
Figure 2.2: Modeling of the synchronous machine in the d-q reference frame. ..................... 16
Figure 2.3: Representation of a typical turbine-generator shaft system. ................................. 19
Figure 2.4: The ith mass of an N-mass spring system. ............................................................. 19
Figure 2.5: A series capacitor-compensated transmission line. ............................................. 22
Figure 2.6: Voltage phasor diagram. ....................................................................................... 23
Figure 2.7: Mechanical power, rotor speed and wind speed relationships.............................. 26
Figure 2.8: Equivalent circuit of the DFIG. ............................................................................ 27
Figure 2.9: Equivalent circuit for the BtB dc capacitor link. .................................................. 28
Figure 2.10: Schematic diagram of a general control scheme of DFIG BtB converters. .......... 29
Figure 2.11: Schematic diagram of a general control scheme of FFC BtB converters. ............ 32
Figure 2.12: Power flow results of bus voltages and transmission line real power flows of the
system under study. ............................................................................................... 34

vii
Figure 2.13: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (30% compensation
degree)................................................................................................................... 35
Figure 2.14: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (30% compensation
degree)................................................................................................................... 36
Figure 2.15: Turbine-generator electrical powers and shaft torsional torques during and after
clearing a 3-cycle, three-phase fault on Line 5 (30% compensation degree). ...... 37
Figure 2.16: Frequency spectrums of the turbine-generator shaft torsional torques during and
after clearing a 3-cycle, three-phase fault on Line 5 (30% compensation degree).
............................................................................................................................... 40
Figure 2.17: Frequency spectrums of the stator current of the DFIG wind turbine. ................. 41
Figure 3.1: Incorporating a supplemental control signal, US, in the reactive power control loop
of the GSC of FFC wind turbine. .......................................................................... 43
Figure 3.2: Incorporating a supplemental control signal, US, in the reactive power control loop
of the GSC of DFIG wind turbine......................................................................... 44
Figure 3.3: Supplemental control 1. ........................................................................................ 45
Figure 3.4: Supplemental control 2. ........................................................................................ 46
Figure 3.5: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is not
activated, wind farm B rating = 200 MW). ........................................................... 49
Figure 3.6: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, wind farm B rating = 200 MW).50
Figure 3.7: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, wind farm B rating = 200 MW).51
Figure 3.8: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is not activated,
wind farm B rating = 200 MW). ........................................................................... 52

viii
Figure 3.9: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is
activated, wind farm B rating = 200 MW). ........................................................... 55
Figure 3.10: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, wind farm B rating = 200 MW). ..... 56
Figure 3.11: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, wind farm B rating = 200 MW). ..... 56
Figure 3.12: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
wind farm B rating = 200 MW). ........................................................................... 57
Figure 3.13: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is not
activated, Line 3 length = 50 km). ........................................................................ 60
Figure 3.14: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, Line 3 length = 50 km). ............ 61
Figure 3.15: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, Line 3 length = 50 km). ............ 61
Figure 3.16: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is not activated,
Line 3 length = 50 km).......................................................................................... 62
Figure 3.17: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is
activated, Line 3 length = 50 km). ........................................................................ 63

ix
Figure 3.18: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, Line 3 length = 50 km). .................. 64
Figure 3.19: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, Line 3 length = 50 km). .................. 64
Figure 3.20: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
Line 3 length = 50 km).......................................................................................... 65
Figure 3.21: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is not
activated, Line 3 length = 100 km). ...................................................................... 66
Figure 3.22: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, Line 3 length = 100 km). .......... 67
Figure 3.23: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, Line 3 length = 100 km). .......... 67
Figure 3.24: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is not activated,
Line 3 length = 100 km)........................................................................................ 68
Figure 3.25: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is
activated, Line 3 length = 100 km). ...................................................................... 69
Figure 3.26: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, Line 3 length = 100 km). ................ 70

x
Figure 3.27: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, Line 3 length = 100 km). ................ 70
Figure 3.28: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
Line 3 length = 100 km)........................................................................................ 71
Figure 3.29: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is not
activated, Line 3 length = 200 km). ...................................................................... 72
Figure 3.30: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, Line 3 length = 200 km). .......... 73
Figure 3.31: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, Line 3 length = 200 km). .......... 73
Figure 3.32: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is not activated,
Line 3 length = 200 km)........................................................................................ 74
Figure 3.33: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is
activated, Line 3 length = 200 km). ...................................................................... 75
Figure 3.34: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, Line 3 length = 200 km). ................ 76
Figure 3.35: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, Line 3 length = 200 km). ................ 76
Figure 3.36: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase

xi
 fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
Line 3 length = 200 km)........................................................................................ 77
Figure 3.37: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is not
activated, wind farm B rating = 300 MW). ........................................................... 79
Figure 3.38: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, wind farm B rating = 300 MW).80
Figure 3.39: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, wind farm B rating = 300 MW).80
Figure 3.40: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is not activated,
wind farm B rating = 300 MW). ........................................................................... 81
Figure 3.41: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is
activated, wind farm B rating = 300 MW). ........................................................... 82
Figure 3.42: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, wind farm B rating = 300 MW). ..... 83
Figure 3.43: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, wind farm B rating = 300 MW). ..... 83
Figure 3.44: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
wind farm B rating = 300 MW). ........................................................................... 84
Figure 3.45: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is not
activated, wind farm B rating = 400 MW). ........................................................... 85

xii
Figure 3.46: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, wind farm B rating = 400 MW).86
Figure 3.47: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, wind farm B rating = 400 MW).86
Figure 3.48: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is not activated,
wind farm B rating = 400 MW). ........................................................................... 87
Figure 3.49: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is
activated, wind farm B rating = 400 MW). ........................................................... 88
Figure 3.50: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, wind farm B rating = 400 MW). ..... 89
Figure 3.51: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, wind farm B rating = 400 MW). ..... 89
Figure 3.52: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
wind farm B rating = 400 MW). ........................................................................... 90
Figure 3.53: Turbine-generator shaft torsional torques during and after clearing a 3-cycle,
double line-to-ground fault on Line 5 (60% compensation degree, supplemental
control 1 is not activated). ..................................................................................... 92
Figure 3.54: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, double line-to-ground fault on Line 5 (60%
compensation degree, supplemental control 1 is not activated). .......................... 93

xiii
Figure 3.55: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, double line-to-ground fault on Line 5 (60%
compensation degree, supplemental control 1 is not activated). .......................... 93
Figure 3.56: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, double line-to-
ground fault on Line 5 (60% compensation degree, supplemental control 1 is not
activated). .............................................................................................................. 94
Figure 3.57: Turbine-generator shaft torsional torques during and after clearing a 3-cycle,
double line-to-ground fault on Line 5 (60% compensation degree, supplemental
control 1 is activated). ........................................................................................... 95
Figure 3.58: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, double line-to-ground fault on Line 5 (60%
compensation degree, supplemental control 1 is activated). ................................. 96
Figure 3.59: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, double line-to-ground fault on Line 5 (60%
compensation degree, supplemental control 1 is activated).................................. 96
Figure 3.60: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, double line-to-
ground fault on Line 5 (60% compensation degree, supplemental control 1 is
activated). .............................................................................................................. 97
Figure 3.61: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 2 is
activated). .............................................................................................................. 98
Figure 3.62: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 2 is activated). ........................................................ 99
Figure 3.63: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 2 is activated). ........................................................ 99
Figure 3.64: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase

xiv
 fault on Line 5 (60% compensation degree, supplemental control 2 is activated).
............................................................................................................................. 100
Figure 3.65: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 4 (60% compensation degree, supplemental control 2 is not
activated). ............................................................................................................ 102
Figure 3.66: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 4 (60% compensation
degree, supplemental control 2 is not activated)................................................. 103
Figure 3.67: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 4 (60% compensation
degree, supplemental control 2 is not activated). ................................................ 103
Figure 3.68: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 4 (60% compensation degree, supplemental control 2 is not activated).
............................................................................................................................. 104
Figure 3.69: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 4 (60% compensation degree, supplemental control 2 is
activated). ............................................................................................................ 105
Figure 3.70: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 4 (60% compensation
degree, supplemental control 2 is activated). ...................................................... 106
Figure 3.71: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 4 (60% compensation
degree, supplemental control 2 is activated). ...................................................... 106
Figure 3.72: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 4 (60% compensation degree, supplemental control 2 is activated).
............................................................................................................................. 107
Figure 3.73: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (50% compensation degree, supplemental control 2 is not
activated). ............................................................................................................ 110

xv
Figure 3.74: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (50% compensation
degree, supplemental control 2 is not activated). ................................................ 111
Figure 3.75: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (50% compensation
degree, supplemental control 2 is not activated). ................................................ 111
Figure 3.76: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (50% compensation degree, supplemental control 2 is not activated).
............................................................................................................................. 112
Figure 3.77: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (50% compensation degree, supplemental control 2 is
activated). ............................................................................................................ 113
Figure 3.78: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (50% compensation
degree, supplemental control 2 is activated). ...................................................... 114
Figure 3.79: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (50% compensation
degree, supplemental control 2 is activated). ...................................................... 114
Figure 3.80: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (50% compensation degree, supplemental control 2 is activated).
............................................................................................................................. 115
Figure B.1: Supplemental control 1 output signals during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, wind farm B rating = 200 MW).
............................................................................................................................. 128
Figure B.2: Supplemental control 1 output signals during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, Line 3 length = 50 km). ...... 129
Figure B.3: Supplemental control 1 output signals during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, Line 3 length = 100 km). .... 130

xvi
Figure B.4: Supplemental control 1 output signals during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, Line 3 length = 200 km). .... 131
Figure B.5: Supplemental control 1 output signals during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, wind farm B rating = 300 MW).
............................................................................................................................. 132
Figure B.6: Supplemental control 1 output signals during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, wind farm B rating = 400 MW).
............................................................................................................................. 133
Figure B.7: Supplemental control 1 output signals during and after clearing a 3-cycle, L-L-G
fault on Line 5 (60% compensation degree). ...................................................... 134
Figure B.8: Supplemental control 2 output signals during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree). ............................................ 135
Figure B.9: Supplemental control 2 output signals during and after clearing a 3-cycle, three-
phase fault on Line 4 (60% compensation degree). ............................................ 136
Figure B.10: Supplemental control 2 output signals during and after clearing a 3-cycle, three-
phase fault on Line 5 (50% compensation degree). ............................................ 137

xvii
 LIST OF TABLES

Table 2.1: Wind speeds and wind farm outputs. .................................................................... 15

Table 3.1: Transfer functions of Supplemental control 1 (Wind farm B rating = 200 MW,
three-phase fault on Line 5). ................................................................................. 54
Table 3.2: Transfer functions of Supplemental control 1 for Group A (Line 3 length = 50 km).
............................................................................................................................... 58
Table 3.3: Transfer functions of Supplemental control 1 for Group C (Line 3 length = 200 km).
............................................................................................................................... 59
Table 3.4: Transfer functions of Supplemental control 1 for Group B (Wind farm B rating =
400 MW). .............................................................................................................. 78
Table 3.5: Transfer functions of Supplemental control 2 (Wind farm B rating = 200 MW,
three-phase fault on Line 4). ............................................................................... 101
Table 3.6: Transfer functions of Supplemental control 2 (Three-phase fault on Line 5, 50%
compensation). .................................................................................................... 109
Table A.1: Synchronous generator data ................................................................................ 123
Table A.2: Turbine-Generator 1............................................................................................ 124
Table A.3: Transformer data ................................................................................................. 125
Table A.4: Wind farm A and B Parameters .......................................................................... 125

xviii
 LIST OF SYMBOLS

A blade sweep area (m2)

C capacitor
Cf wind turbine blade design constant
Cp power coefficient of the blade
D damping coefficient in p.u.
DE, Dg, DB, DA, DI, DH damping coefficient of the corresponding inertia
d direct axis
ed, eq d- and q- axis components of the stator voltage
efd filed voltage
fn subsynchronous natural frequency
GSC grid side converter
IL1, IL2 line currents
id, iq d- and q- axis components of the stator current
ifd field winding current
i1d d-axis damper winding current
i1q, i2q q-axis damper winding currents
isd, isq d- and q- axis components of the stator current
ids, iqs DFIG q- and d- axis components of the stator current
is, ir steady-state DFIG stator and rotor current
k degree of compensation
K stiffness in p.u.
KEg, KgB, KBA, KAI, KIH stiffness of the connecting shafts
K1, K2, Ki constants

xix
Kshaft speed deviation damping
L inductance
Ls, Lr stator and rotor inductances
Ld, Lq d- and q- axis components of the stator inductance
Lad d-axis magnetizing inductance
Laq q-axis magnetizing inductance
Lffd self-inductance of the field winding
Lls, Llr DFIG stator and rotor linkage inductance
L11d self-inductance of the d-axis damper winding
L11q, L22q self-inductances of the q-axis damper winding
ME, Mg, MB, MA, MI, MH inertia constants of exciter, generator, two low-
pressure turbines, intermediate-pressure turbine and
high-pressure turbine respectively
Mi inertia constant of the ith rotating mass
M mutual inductance between rotor and stator
MPT maximum power tracking point
MSC PSMG side converter
P real power
PA, PB, PI, PH power of the stages of the turbine
PMSG permanent-magnet synchronous generator
PI proportional integral
Pr, Pg active power of the RSC and GSC
Pm Pulse-width-modulation index
Pgrid-Ref real power reference for GSC
PLoad load real power
ρ air density (kgm-3)
Q reactive power
QLoad load reactive power

xx
Qgrid-Ref reactive power reference for RSC
QGSC-Ref reactive power reference for GSC
q quadrature axis
Rω rotor radius of the wind turbine
Ra armature resistance
Rs synchronous generator stator resistance
Rr resistance in the rotor circuits
RSC rotor side converter
RL resistance of the series capacitor compensated
transmission line
Rfd field winding resistance
R1d d-axis damper winding resistance
R1q, R2q q-axis damper winding resistances
rs resistance of the RC snubber circuit
rr resistance of the rotor side
s Laplace transformation operator
T superscript to denote matrix transpose
t time
Te electromagnetic torque
TMECH mechanical torque
TC fault clearing time
TF total simulation time
Tai, Tbi time constants
usd, usq PMSG d- and q- components of the stator voltage
UAC, UDC AC- and DC- voltages
Vdc dc link capacitor voltage
Vdc-Ref reference signal

xxi
VC voltage across the series capacitor of the compensated
transmission line
VCd, VCq voltages across the series capacitor in the d-q reference
frame
VR voltage across the resistance of the series capacitor
compensated transmission line
VL voltage across the inductance of the series capacitor
compensated transmission line
VLd, VLq voltages across the inductance in the d-q reference
frame
vqg, vdg quadrature and direct axis GSC voltages
Vb infinite-bus voltage
Vbd, Vbq d- and q- axis components of the infinite-bus voltages
vs, vr stator and rotor side voltages
Vt generator terminal voltage
Vtd, Vtq d- and q- axis components of the generator terminal
voltages
VRd, VRq voltages across the resistance in the d-q reference frame
vqs, vds quadrature and direct axis GSC voltages
vqr, vdr quadrature and direct axis RSC voltages
Vω wind speed (m/s)
VLoad load voltage
XC series capacitor reactance
XL inductive reactance of the series capacitor compensated
transmission line
Xl synchronous generator leakage reactance
X0 synchronous generator zero-sequence reactance
Z, ZL1, ZL2 impedance of the transmission line
ZLoad load impedence
Ψd, Ψq d- and q- axis components of the stator flux linkages

xxii
Ψfd field winding flux linkage
Ψ1d d-axis damper winding flux linkage
Ψ1q, Ψ2q q-axis damper winding flux linkages
Ψsd, Ψsq PMSG d- and q-axis components of the stator flux
linkage
ΨPM permanent magnet flux
δ generator power (load) angle
δE, δB, δA, δI, δH rotor angles of exciter, two low-pressure turbines,
intermediate-pressure turbine and high-pressure turbine
respectively
λ wind turbine blade tip speed ratio
λs, λr stator and rotor side flux linkages
λdm, λqm d- and q- axis components of the magnetizing flux
linkages
λds, λqs DFIG stator d- and q-axis components of the flux
linkage
λdr, λqr DFIG rotor d- and q-axis components of the flux
linkage
Ωm mechanical angular velocity (rad/s)
ω angular velocity
ωgen turbine-generator rotating mass speeds
ωgen-Ref turbine-generator rotating mass speeds reference
ωE, ωB, ωA, ωI, ωH angular velocity of exciter, two low-pressure turbines,
intermediate-pressure turbine and high-pressure turbine
respectively
ωn natural frequency

ω0 (f0) synchronous frequency (337 rad/sec)

Δωm modal speed deviation matrix

ωm speed deviation matrix of the turbine-generator rotating

masses

ωs, ωr stator and rotor angular frequency

xxiii
θ blade pitch angle

Φi phase compensations

ΔY’ the output of the band pass filters

xxiv
 1. INTRODUCTION

1.1 Wind Energy

Wind energy has become one of the most popular renewable technologies around the world.
Compared to fossil fuels, it is clean, widely distributed, plentiful and environmentally friendly.
World wind generation capacity more than quadrupled between 2000 and 2006, doubling about
every three years. There are now over 200,000 wind turbines operating around the world, with a
total capacity of 282,482 MW as of the end of 2012 [1]. Owing to the rapidly increasing use of
wind power, the aspect of integrating high levels of wind power into the grid is becoming more
and more reality. Examples of large wind farms are the 5000 MW Gansu wind farm in China, the
781.5 MW Roscoe wind farm in Texas, the 845 MW Shepherds Flat wind farm in Oregon, and the
1550 MW Alta wind farm being developed in California [2].

1.2 Series Capacitive Compensation

Series capacitive compensation of power transmission lines using capacitor banks is the
most economical way for increasing power transfer capability and improving system stability,
especially when large amounts of power must be transmitted through long transmission lines.
However, one of the hindering factors for increased utilization of series capacitive compensation
is the potential risk of Subsynchronous Resonance (SSR) and Subsynchronous Interaction (SSI)
[3], [4]. Figure 1.1 shows a typical time response of a turbine-generator shaft torsional torque
(High-Pressure turbine to Low-Pressure turbine shaft section, (HP-LP)) during and after clearing
a fault in a series capacitive compensated transmission system in the presence of the SSR
phenomenon. It is worth noting here that this shaft is designed to withstand a maximum torsional
torque of 0.75 per unit (on the generator base MVA). On the other hand, Figure 1.2 shows a typical
time response of the terminal voltage (root mean square) of a large Doubly-Fed Induction
Generator (DFIG)-based wind farm during and after clearing a fault in a series capacitive
compensated transmission system in the presence of the SSI phenomenon. As it can be seen from
this figure, the wind farm terminal voltage exhibits sustained oscillations.

1
 70
T(HP-LP) (p.u.)

-70
0 1.5 3 4.5
Time (s)

Figure 1.1: Transient time response of a turbine-generator shaft torsional torque during and after
clearing a system fault on a series capacitive compensated transmission line.

800
VWFA (kV)

420

40
0 1 2 3 4
Time (s)

Figure 1.2: Transient time response of a large DFIG-based wind farm terminal voltage (root mean
square) during and after clearing a system fault on a series capacitive compensated
transmission line.

1.3 Transmission Line Series Compensation

The main purpose of series compensation of a transmission line is the virtual reduction of
the line inductive reactance in order to enhance power system stability and increase the loadability
of transmission corridors [5]. The principle is based on the compensation of the distributed line
reactance by the insertion of a series capacitor. The reactive power generated by the capacitor is
continuously proportional to the square of the line current. This means that the series capacitor
has a self-regulating effect. When the system loading increases, the reactive power generated by

2
the series capacitor increases as well. The response of the series capacitor is automatic,
instantaneous and continuous as long as the capacitor current remains within the specified
operating limits. The following are some of the major benefits of incorporating series capacitors
in transmission systems:

1.3.1 Increase the power transfer capability by raising the first swing stability limit

A substantial increase in the stability margin is achieved by installing a series capacitor.

The series compensation will improve the situation in two ways: it will decrease the initial
generator load angle corresponding to a specific power transfer and it will also shift the power-
load angle (P-) characteristic upwards. This will result in increasing the transient stability margin.

1.3.2 Increase in power transfer

The increase in the power transfer capability as a function of the degree of compensation
for a transmission line can be illustrated using the circuit and the vector diagram shown in Figure
1.3. The power transfer on the transmission line is given by:

|𝑉𝑡 ||𝑉𝑏 | |𝑉 ||𝑉 |

𝑃= sin𝛿 = 𝑋 𝑡(1−𝑘)
𝑏
sin𝛿 (1.1)
𝑋𝐿 −𝑋𝑐 𝐿

where k is the degree of compensation defined as

𝑋𝐶
𝑘=
𝑋𝐿

The effect on the power transfer when a constant load angle difference is assumed is shown
in Figure 1.4. Practical compensation degree ranges from 20 to 60 percent and, therefore, doubling
the transmission capability is achievable in practice.

jI(XL-XC)
Vt -jXc Vb
jXL I
Vt Vb
δ

I P

Figure 1.3: A transmission line with a series capacitor.

3
 5

4
Pmax, p.u.
3

1
0 0.2 0.4 0.6 0.8
Degree of series compensation

Figure 1.4: Maximum power transmitted over a transmission line as a function of the degree of
series compensation (|Vt| = |Vb| = 1 p.u., XL= 1 p.u.).
1.3.3 Active load sharing between parallel circuits

When two transmission lines are connected in parallel, the natural power sharing between
them is dictated by their respective impedances. If the two lines are of different configurations
(and consequently of different thermal ratings), their impedances could still be very close.
Therefore, the power transmitted in each line will be similar. The voltage drop in both circuits is
identical, and therefore, the relationship between the line currents 𝐼𝐿1 and 𝐼𝐿2 can be expressed as:

𝐼𝐿1 𝑍𝐿1 = 𝐼𝐿2 𝑍𝐿2 (1.2)

If overloading the lower thermal rating line, (𝐿2 , Figure 1.5) is to be avoided (i.e. 𝐼𝐿2 ≤
𝐼𝐿2𝑚𝑎𝑥 ), then the full power capacity of the other line, 𝐿1 , will never be reached (i.e. 𝐼𝐿1 ≤ 𝐼𝐿1𝑚𝑎𝑥 ).
For example, consider the case when 𝐿1 is a four conductor bundle (quad) circuit configuration,
whereas 𝐿2 has a two conductor bundle (twin) circuit configuration. If the conductors of the two
bundles are identical, then 𝐿1 has twice rating of 𝐿2 . The inductive reactance of the two lines,
however, are very close. If a series capacitor is installed in the higher thermal rating line, both
transmission lines can operate at their maximum capacity when the appropriate degree of
compensation is provided (50% in this case) [6].

4
 -jXc Line L1
jXl1 R1

jXl2
R2 Line L2

Figure 1.5: Adjusting the power sharing between two parallel lines using a series capacitor.

1.4 Subsynchronous Resonance (SSR)

SSR is a dynamic phenomenon in the power system which has certain special
characteristics. The definitions of subsynchronous oscillation and SSR are given by the IEEE as
[7], [8]:

“Subsynchronous oscillation is an electric power system condition where the electric network
exchanges significant energy with a turbine-generator at one or more of the natural
frequencies of the combined system below the synchronous frequency of the system
following a disturbance from equilibrium. The above excludes the rigid body modes of the
turbine-generator rotors.”

“Subsynchronous Resonance (SSR) encompasses the oscillatory attributes of electrical and

mechanical variables associated with turbine-generators when coupled to a series capacitor
compensated transmission system where the oscillatory energy interchange is lightly damped,
undamped, or even negatively damped and growing.”

1.4.1 SSR: basic phenomenon

Consider the simple power system shown in Figure 1.6. It consists of a large turbine-
generator which is connected to an infinite-bus system through a series capacitor compensated
transmission line. The generator is driven by a multi-stage turbine, where the various stages of the
turbine (HP, IP and LP) and the generator rotor (GEN) are coupled by elastic shafts.

The natural resonance frequency for the electrical system is given by

1 𝜔0 𝑋
𝜔𝑛 = = = 𝜔0 √ 𝑋𝐶 rad/s (1.3)
√𝐿𝐶 √(𝜔0 𝐿)(𝜔0 𝐶) 𝐿

5
 or

𝑋
𝑓𝑛 = 𝑓0 √ 𝑋𝐶 Hz (1.4)
𝐿

Infinite Bus

XC

XL

RL

HP IP LP GEN

Figure 1.6: A turbine-generator connected to an infinite-bus system through a series capacitor

compensated transmission line.

where 𝜔𝑛 is the natural frequency, 𝑓0 is the synchronous frequency in Hz, and 𝜔0 = 2𝜋𝑓0 𝑟𝑎𝑑/𝑠,
𝑓0 = 60 𝐻𝑧, XC is the capacitive reactance and XL is the transmission line inductive reactance.

In practice, fn is always below the synchronous frequency f0 since the compensation degrees
of transmission lines are usually less than 100%. For this reason, fn is called the subsynchronous
natural frequency of the electrical system.

The shaft system of the turbine-generator has (N-1) natural torsional frequencies where N
is the number of the rotating masses. These torsional frequencies are functions of the inertia of
the different masses and the stiffness of the connected shafts. Due to the physical properties of the

6
shaft materials and the mechanical design of the turbine-generator shaft system, the torsional
natural frequencies are also subsynchronous. Thus, the basic interaction between the electrical
and mechanical systems is due to the closeness of fn to the natural torsional frequencies of the
turbine-generator shaft system. SSR can occur in the following three forms [3], [8]:

(1) Torsional Interaction: this is due to an interaction and exchange of energy between the series
compensated electrical system and the turbine-generator mechanical system. This can lead to
growing shaft torque oscillations at one of the natural torsional frequencies of the turbine-
generator shaft system. Torsional interaction can occur when the generator is connected to a
series compensated electrical system that has one or more natural frequencies, which are the
synchronous frequency complements of one or more of the spring-mass natural frequencies.
Generally, shaft torques due to torsional interaction can be expected to build up at a relatively
slow rate such that damaging torque levels would not be reached in less than a minute or so.
(2) Induction Generator Effect: this is a pure electrical phenomenon that is due to the fact that,
when subsynchronous currents flow in the armature circuit of a synchronous generator, the
generator appears as a negative-resistance circuit at the prevailing subsynchronous frequencies.
If the apparent resistance is greater than the inherent positive resistance of the circuit at one of
the natural frequencies of the electrical circuit, growing subsynchronous voltages and currents
will be expected in the system and at the generator. This could result in voltages and currents
large enough to be damaging to the generator and power system equipment. In addition, if the
subsynchronous currents in the generator armature are at the frequency corresponding to one
of the turbine-generator spring-mass modes, large oscillatory shaft torques may result. As in
the case of torsional interaction, a relatively slow oscillation growth rate would be expected.
(3) Torque Amplification: this phenomenon occurs when a fault on a series compensated power
system, and its subsequent clearing, results in a high-energy storage in the series capacitor
banks, which then discharge their energy through a generator in the form of a current having a
frequency that corresponds to one of the natural torsional frequencies of the turbine-generator
mechanical system. Unlike torsional interaction and induction generator effect, the growth
rate for torque amplification is high and oscillating shaft torques might be expected to reach a
damaging level within 0.1 second.

7
 The ultimate hazard of SSR is a shaft fraction at full load and rated speed. The damage of
such an occurrence cannot be accurately predicted, but extensive equipment damage could occur
with a safety hazard to personnel. A more likely most-severe hazard would be a crack initiation
at the surface of one of the turbine-generator shafts, indicating fatigue and requiring shaft
replacement, resulting in a unit outage of 90 days or more.

1.5 Doubly-fed Induction Generator Wind Turbine

The basic structure of a doubly-fed induction generator (DFIG) wind turbine is shown in
Figure 1.7. The stator of the induction machine is directly connected to the grid and the wound
rotor windings are connected to the grid through slip rings and an indirect ac-ac converter system
which controls both the rotor and the grid currents. The ac-ac converter system consists of two
three-phase pulse-width modulated (PWM) Voltage-Sourced Converters (VSC) (Rotor-Side
Converter (RSC) and Grid-Side Converter (GSC)) connected by a dc bus. A line inductor and an
ac filter are used at the GSC to improve the power quality. A crowbar is also used to protect the
RSC against over-currents and the dc capacitor against over-voltages [9].

Windmill
Stator power IL, PL
Grid
Gearbox
Slip rings Step down

Transformer
ir
ig

Vdc
Rotor power
Crow-bar
Rotor-Side Converter Grid-Side Converter

Figure 1.7: Schematic diagram of a DFIG wind turbine.

8
 The control of the DFIG is realized by controlling the RSC and GSC using vector control
techniques. The function of the RSC is to control the active and reactive powers delivered to the
grid, and to follow a tracking characteristic to adjust the generator speed for optimal power
generation depending on the wind speed. On the other hand, the function of the GSC is to keep
the dc bus voltage constant and to support the grid with reactive power during system faults [4].
Details on DFIG wind turbine controls are given in [10].

The main advantage of the DFIG is the low cost of its converters as their rating is typically
25% to 30% of the DFIG rated power. As a result, the cost of the converters and electromagnetic
interference (EMI) filters is also reduced.

1.6 Subsynchronous Interaction Phenomenon in DFIG-Based Wind Farms

Recent studies have identified the vulnerability of DFIG wind turbines to subsynchronous
resonance interaction (SSI). This has been confirmed in October 2009 by the Zorillo Gulf wind
farm incident in Texas which can be regarded as the first event of SSI between a DFIG-based wind
farm (485 MW) and a series capacitor compensated transmission line [4], [10]. As a result of
clearing a fault on a transmission line connected to the Zorillo Gulf wind farm, the wind farm
became radially connected to the grid through a series capacitor compensated transmission line
(the Rio Hondo line: 345 kV, 80 miles and a 50% compensation degree). Severe subsynchronous
growing currents and voltages were recorded and damage occurred in the DFIG-wind turbine
electrical and control systems.

Comprehensive studies of the SSI phenomenon using linearized techniques and eigenvalue
analysis have attributed the “main cause” of the problem to the fast rotor current controller of the
Rotor Side Converter (RSC) of the DFIG wind turbine [10]. Such a controller tends to move the
subsynchronous electrical modes to the right-hand side of the s-plane causing system instability
when its gain is increased. In other words, the rotor current controller changes the rotor resistance
in such a way that it is seen from the stator side as a negative resistance. This occurs over a wide
range of slip frequencies and can be classified as an Induction Generator Effect (IGE).

It is worth noting here that the Torsional Interaction (TI) phenomenon in DFIG wind
turbine shaft systems is of a little concern. This is due to the fact that the natural torsional
frequencies of oscillations of the shaft system (turbine, turbine-blades, gear box, generator rotor)

9
are in the range of 1 Hz to 3 Hz. Very high and impractical degrees of series capacitive
compensation (90% and higher) are required to excite these torsional modes.

1.7 Full-Scale Frequency Converter Wind Turbine

Full-scale frequency converter (FFC) wind turbines are becoming popular in wind farms,
since they can meet the stringent grid code requirements more easily than DFIG turbines. This is
because FFC wind turbine has full control capability for real and reactive power output, with the
generator decoupled from the grid.

The FFC wind turbine employs a permanent–magnet synchronous generator (PMSG) that
has a large number of poles; hence a gear box is not required. This is also known as a direct-drive
wind turbine generator where the synchronous machine rotates at the slow turbine speed and
generates electrical power with frequency well below that of the gird (the synchronous frequency).
The increased generator weight is offset by the absence of the gearbox. Further, the reliability and
maintenance considerations for a gearbox are eliminated. Hence, this concept is particularly
attractive for offshore locations [11].

Figure 1.8 shows a FFC PMSG wind turbine connected to the grid through an ac-ac
converter system. Depending on the size of the wind turbine, the PMSG side converter (MSC)
can be either a diode rectifier or a VSC. On the other hand, the GSC is typically a VSC. In the
studies conducted in this thesis, the back-to-back (BtB) VSC topology is adopted. In addition, the
gearbox can be eliminated, such that [12].

10
 Machine Side Converter Grid-Side Converter
(MSC) (GSC)

PMSG Vdc Grid

Figure 1.8: Schematic diagram of a FFC wind turbine.

Similar to the DFIG, the control of the FFC is realized by controlling the MSC and GSC
using also vector control techniques [4]. The MSC controls the active power delivered by the
PMSG, and follows a tracking characteristic to adjust the PMSG speed for optimal power
generation depending on wind speed. The function of GSC is maintaining the dc bus voltage at
its desired level, i.e. transmitting the active power delivered to the dc link by the MSC. GSC also
controls the reactive power delivered to the grid.

1.8 Research Objective and Scope of the Thesis

In integrated power systems incorporating series capacitor compensated transmission lines

and high penetration of wind energy conversion systems, especially DFIG-based wind farms, SSR
and SSI could occur concurrently as a result of some system contingences. Therefore, mitigating
SSR and SSI is an important area of research and development targeting at developing practical
and effective countermeasures.

As most large wind farms in North America employ DFIG and FFC wind turbines, their
voltage-sourced converter-based back-to-backs (BtBs) offer independent control of the real and
reactive power. The use of these control capabilities have been recently proposed for damping
power swings as well as inter-area oscillations [13]-[16]. Very little research has been reported on
utilizing DFIG- and FFC-based wind farms for damping SSR [17], [18] and virtually, no research
has been reported on simultaneous mitigation of SSR and SSI using large wind farms.

11
 The main objective of this research work is to investigate the potential use of FFC- and
DFIG-based wind farms for simultaneous mitigation of SSR and SSI. This is achieved through
introducing supplemental control signals in the reactive power control loops of the grid side
converters of the DFIG wind turbines or the grid side converter of the FFC wind turbines. In this
context, two supplemental controls designated as Supplemental Controls 1 and 2 are examined.
Supplemental Control 1 introduces a signal in the grid side converter of the FFC wind turbines to
damp both SSR and SSI oscillations. On the other hand, Supplemental Control 2 introduces a
signal in the grid side converter of the FFC wind turbines for damping SSR oscillations and another
signal in the grid side converters of the DFIG wind turbines for damping SSI oscillations.

The thesis is organized in four chapters, a list of references section and two appendices.
Chapter 1 introduces some fundamental benefits of series compensation of transmission lines.
Brief introductions to SSR and SSI are also presented. The objective of the research is also
presented in this chapter.

In Chapter 2, the system used for the investigations conducted in this thesis is described
and the detailed dynamic models of its individual components are also presented in this chapter.
The results of the digital time-domain simulations of a case study for the system during a three-
phase fault are presented at the end of this chapter.

Chapter 3 demonstrates the effectiveness of the presented two supplemental controls in

damping SSR and SSI oscillations through time-domain simulation studies. The supplemental
controls performance at different system contingencies and operating conditions is also
investigated.

Chapter 4 summarizes the research described in this thesis and presents some conclusions.

The data of the systems under investigations are given in Appendix A. Supplemental
controls 1 and 2 output SSR and SSI signals for the case studies reported in Chapters 2 and 3 are
given in Appendix B.

12
 2. MATHEMATICAL MODELING OF POWER
SYSTEMS INCORPORATING FFC- AND DFIG-
BASED WIND FARMS FOR LARGE
DISTURBANCE STUDIES

2.1 Introduction

In this chapter, the system used for the studies reported in this thesis is described and the
mathematical models of its various components are presented. A digital time-domain and a
frequency spectrums simulation of the system during a three-phase fault are presented at the end
of this chapter.

2.2 System under Study

The system used in the investigations of this thesis is shown in Figure 2.1. It consists of a
generating station and two wind farms designated as wind farms A and B. The generating station
and wind farm A are connected to each other through two transmission lines (Lines 3 and 4) as
well as to an infinite-bus system through two series capacitor compensated transmission lines
(lines 1 and 2, each with a 60% compensation degree except for the sample case study in Section
2.4 where the compensation degree in both lines is 30%). Wind farm B is connected to the
generating station and wind farm A through lines 3, 4 respectively and to the infinite-bus system
through two transmission lines (lines 5 and 6). The generating station comprises two turbine-
generators, G1 (600 MVA, 22 kV) and G2 (700 MVA, 22 kV). The shaft system of G1 comprises
high and low-pressure turbines (HP1, LP1), the generator rotor and an exciter (EXC). The shaft
system of G2 comprises high and low-pressure turbines (HP2, LP2) and the generator rotor. The
frequencies of the natural torsional modes of oscillations of G1 and G2 shaft systems are 24.65 Hz,
32.39 Hz, 51.1 Hz and 24.65 Hz, 44.99 Hz respectively. The data of G1 and G2 are taken from
the IEEE second benchmark model for computer simulation of SSR. Wind farm A comprises 300
× 1.5 MW DFIG wind turbines. Wind farm B is a FFC wind farm of 100 × 2 MW wind turbines.

13
 L C
R T1
Bus A
Line 1, 500 km LP1 HP1

EXC G1

Line 3,
T2 LP2 HP2
150 km
G2

Line 6, Line 5,
150 km 150 km Bus B

∞ T3

500 kV transmission GSC MSC G

network

Wind Farm B

Line 4,
50 km

Bus C
T4 Wind Farm A
IG
R1 L1
C1
Il, Pl
Line 2, 500 km GSC RSC

Figure 2.1: System under study.

14
 The operating wind speeds and power outputs of the two wind farms are given in Table
2.1. The medium voltage collector grid is represented with its equivalent PI circuit model [19].
Faults are assumed to occur on Lines 4 and 5 and to be cleared by circuit breaker operations at
both ends of the line.

Table 2.1: Wind speeds and wind farm outputs.

Wind farm Operating speed and output power

A (DFIG-based wind farm) Wind speed = 11 m/s, ≈ 425 MW

B (FFC-based wind farm) Wind speed = 15 m/s, ≈ 200 MW

2.3 Power System Modeling

The nonlinear differential equations of the system under study are derived by developing
individually the mathematical models which represent the various components of the system,
including the synchronous generator, the generator turbine, the transmission line, the transformer,
the system load, the wind turbine aerodynamic model, the DFIG-based wind turbine and its BtB
converter controllers, as well as the FFC-based wind turbine and its BtB converter controllers.
Knowing the mutual interaction among these models, the system of differential equations can be
formed.

2.3.1 Modeling of the synchronous generator

In a conventional synchronous machine, the stator circuit consisting of a three-phase

winding produces a sinusoidally space distributed magnetomotive force. The rotor of the machine
carries the field (excitation) winding which is excited by a dc voltage. The electrical damping due
to the eddy currents in the solid rotor and, if present, the damper winding is represented by three
equivalent damper circuits; one on the direct axis (d-axis) and the other two on the quadrature axis
(q-axis). The performance of the synchronous machine can be described by the equations given
below in the d-q reference frame (Figure 2.2) [20]. In these equations, the convention adopted for
the signs of the voltages and currents is that v is the impressed voltage at the terminals and that the
direction of positive current i corresponds to generation. The sign of the currents in the equivalent
damper windings is taken positive when they flow in a direction similar to that of the positive field
current.

15
 q-axis

i2q

𝜔𝑟 , elec. Rad/sec

i1q

iq

eq

d-axis
id ifd i1d
ed efd

Figure 2.2: Modeling of the synchronous machine in the d-q reference frame.

With time t expressed in seconds, the angular velocity 𝜔 expressed in rad/s ( 𝜔0 =

377𝑟𝑎𝑑/sec) and the other quantities expressed in per unit, the stator equations become:

1 𝑑Ψ𝑑 𝜔
𝑒𝑑 = 𝜔 − 𝜔 Ψ𝑞 − 𝑅𝑎 𝑖𝑑 (2.1)
0 𝑑𝑡 0

1 𝑑Ψ𝑞 𝜔
𝑒𝑞 = 𝜔 + 𝜔 Ψ𝑑 − 𝑅𝑎 𝑖𝑞 (2.2)
0 𝑑𝑡 0

The rotor equations:

1 𝑑Ψ𝑓𝑑
𝑒𝑓𝑑 = 𝜔 + 𝑅𝑓𝑑 𝑖𝑓𝑑 (2.3)
0 𝑑𝑡

1 𝑑Ψ1𝑑
0=𝜔 + 𝑅1𝑑 𝑖1𝑑 (2.4)
0 𝑑𝑡

1 𝑑Ψ1𝑞
0=𝜔 + 𝑅1𝑞 𝑖1𝑞 (2.5)
0 𝑑𝑡

16
 1 𝑑Ψ2𝑞
0=𝜔 + 𝑅2𝑞 𝑖2𝑞 (2.6)
0 𝑑𝑡

The stator flux linkage equations:

Ψ𝑑 = −𝐿𝑑 𝑖𝑑 + 𝐿𝑎𝑑 𝑖𝑓𝑑 + 𝐿𝑎𝑑 𝑖1𝑑 (2.7)

Ψ𝑞 = −𝐿𝑞 𝑖𝑞 + 𝐿𝑎𝑞 𝑖1𝑞 + 𝐿𝑎𝑞 𝑖2𝑞 (2.8)

The rotor flux linkage equations:

Ψ𝑓𝑑 = 𝐿𝑓𝑓𝑑 𝑖𝑓𝑑 + 𝐿𝑎𝑑 𝑖1𝑑 − 𝐿𝑎𝑑 𝑖𝑑 (2.9)

Ψ1𝑑 = 𝐿𝑎𝑑 𝑖𝑓𝑑 + 𝐿11𝑑 𝑖1𝑑 − 𝐿𝑎𝑑 𝑖𝑑 (2.10)

Ψ1𝑞 = 𝐿11𝑞 𝑖1𝑞 + 𝐿𝑎𝑞 𝑖2𝑞 − 𝐿𝑎𝑞 𝑖𝑞 (2.11)

Ψ2𝑞 = 𝐿𝑎𝑞 𝑖1𝑞 + 𝐿22𝑞 𝑖2𝑞 − 𝐿𝑎𝑞 𝑖𝑞 (2.12)

The electromagnetic torque equation:

𝑇𝑒 = Ψ𝑑 𝑖𝑞 − Ψ𝑞 𝑖𝑑 (2.13)

The overall differential equations which describe the transient performance of the
synchronous machine are given by the following matrix equation:

𝑉𝑡𝑑
𝑑𝑋𝑠𝑦𝑛
[ ] = [𝐴𝑡𝑠𝑦𝑛 ][𝑋𝑠𝑦𝑛 ] + [𝐵𝑡𝑠𝑦𝑛 ] [ 𝑉𝑡𝑞 ] (2.14)
𝑑𝑡
𝑒𝑓𝑑

where

[𝑋𝑠𝑦𝑛 ] = [𝑖𝑑 𝑖𝑞 𝑖𝑓𝑑 𝑖1𝑞 𝑖1𝑑 𝑖2𝑞 ]𝑇

[𝐴𝑡𝑠𝑦𝑛 ] = [𝐿]−1 [𝑄𝑡]

[𝐵𝑡𝑠𝑦𝑛 ] = [𝐿]−1 [𝑅𝑡]

17
  Ld 0 Lad 0 Lad 0
0  Lq 0 Laq 0 Laq
 Lad 0 L ffd 0 Lad 0
[𝐿 ] = (2.15)
0  Laq 0 L11q 0 Laq
 Laq 0 Lad 0 L11d 0
[ 0  Laq 0 Laq 0 L22 q ]

0 Ra  Lq 0  Laq 0  Laq
 Ld 0 Ra  Lad 0  Lad 0
0 0 0 R fd 0 0 0
[𝑄𝑡] =
0 0 0 0 R1q 0 0
0 0 0 0 0 R1d 0
[ 0 0 0 0 0 0 R2 q ]

0 0 0
0 0 0
[𝑅𝑡] = 0 0 0
0 0 0
0 0 0
[ 0 0 0 ]

Here, the superscript T means matrix transpose.

The synchronous machine swing equation can be written as:

𝑀 𝑑𝜔
= 𝑇𝑀𝐸𝐶𝐻 − 𝑇𝑒 (2.16)
𝜔0 𝑑𝑡

𝑑𝛿
= 𝜔 − 𝜔0 (2.17)
𝑑𝑡

In the above two equations (2.16 and 2.17), 𝜔 is in radians per second, the inertia constant
M is in seconds, and the load angle 𝛿 is in radians, 𝜔0 is the synchronous frequency (377 rad/sec)
and the mechanical and electrical torques 𝑇𝑀𝐸𝐶𝐻 and 𝑇𝑒 are in per unit.

In developing the equations of multi-machine systems, the equations of each synchronous

machine expressed in its own d-q reference frame which rotates with its rotor must be expressed
in a common reference frame. Usually, a reference frame rotating at synchronous speed is used as

18
the common reference. Axis transformation equations are used to transform between the
individual machine (d-q) reference frames and the common (R-I) reference frame [20].

2.3.2 Modeling of the turbine-generator mechanical system

Figure 2.3 illustrates a typical representation of the mechanical system of a large turbine-
generator which consists of a high-pressure turbine (HP), an intermediate-pressure turbine (IP),
two low-pressure turbines (LPA & LPB), the generator rotor (GEN) and the exciter (EXC). Such
a system can be modeled by a six-mass-spring system [21].

TH TI TA TB Te

KIH KAI KBA KgB KEg

EXC

HP IP LPA LPB GEN EXC

Figure 2.3: Representation of a typical turbine-generator shaft system.

Assuming that M is the inertia constant in seconds, D is the damping coefficient in p.u.
torque/p.u. speed for each rotating mass and K is a stiffness in p.u. torque/rad for each shaft section,
the equations of the ith mass of an N-mass spring system shown in Figure 2.4 are given by

Ti

Ki-1,i Ki,i+1
Mass i-1 Mass i Mass i+1

Mi, Di, δi, ωi

Figure 2.4: The ith mass of an N-mass spring system.

𝑀𝑖 𝑑𝜔𝑖 𝐷
= 𝑇𝑖 + 𝐾𝑖−1,𝑖 (𝛿𝑖−1 − 𝛿𝑖 ) − 𝐾𝑖,𝑖+1 (𝛿𝑖 − 𝛿𝑖+1 ) − 𝜔 𝑖 (𝜔𝑖 − 𝜔0 ) (2.18)
𝜔0 𝑑𝑡 0

𝑑𝛿𝑖
= 𝜔𝑖 − 𝜔0 (2.19)
𝑑𝑡

19
where
𝐾𝑖−1,𝑖 |𝑖=1 = 0, 𝐾𝑖,𝑖+1 |𝑖=𝑁 = 0, (2.20)

As an example, when Equations (2.18) to (2.20) are applied to the linear six-mass-spring
system of Figure 2.3, the shaft system equations are written as:

𝑀𝐸 𝑑𝜔𝐸 𝐷
= 𝐾𝐸𝑔 (𝛿 − 𝛿𝐸 ) − 𝜔𝐸 (𝜔𝐸 − 𝜔0 )
𝜔0 𝑑𝑡 0

𝑑𝛿𝐸
= 𝜔𝐸 − 𝜔0
𝑑𝑡

𝑀𝑔 𝑑𝜔 𝑔 𝐷
= −𝑇𝑒 + 𝐾𝑔𝐵 (𝛿𝐵 − 𝛿) − 𝐾𝐸𝑔 (𝛿 − 𝛿𝐸 ) − 𝜔 (𝜔 − 𝜔0 )
𝜔0 𝑑𝑡 0

𝑑𝛿
= 𝜔 − 𝜔0
𝑑𝑡

𝑀𝐵 𝑑𝜔𝐵 𝜔0 𝐷𝐵
= 𝑃 + 𝐾𝐵𝐴 (𝛿𝐴 − 𝛿𝐵 ) − 𝐾𝑔𝐵 (𝛿𝐵 − 𝛿) − (𝜔 𝐵 − 𝜔 0 )
𝜔0 𝑑𝑡 𝜔𝐵 𝐵 𝜔0

𝑑𝛿𝐵
= 𝜔𝐵 − 𝜔0 (2.21)
𝑑𝑡

𝑀𝐴 𝑑𝜔𝐴 𝜔 𝐷
= 𝜔 0 𝑃𝐴 + 𝐾𝐴𝐼 (𝛿𝐼 − 𝛿𝐴 ) − 𝐾𝐵𝐴 (𝛿𝐴 − 𝛿𝐵 ) − 𝜔𝐴 (𝜔𝐴 − 𝜔0 )
𝜔0 𝑑𝑡 𝐴 0

𝑑𝛿𝐴
= 𝜔𝐴 − 𝜔0
𝑑𝑡

𝑀𝐼 𝑑𝜔𝐼 𝜔0 𝐷
= 𝑃𝐼 + 𝐾𝐼𝐻 (𝛿𝐻 − 𝛿𝐼 ) − 𝐾𝐴𝐼 (𝛿𝐼 − 𝛿𝐴 ) − 𝜔𝐼 (𝜔𝐼 − 𝜔0 )
𝜔0 𝑑𝑡 𝜔𝐼 0

𝑑𝛿𝐼
= 𝜔𝐼 − 𝜔0
𝑑𝑡

𝑀𝐻 𝑑𝜔𝐻 𝜔 𝐷
= 𝜔 0 𝑃𝐻 − 𝐾𝐼𝐻 (𝛿𝐻 − 𝛿𝐼 ) − 𝜔𝐻 (𝜔𝐻 − 𝜔0 )
𝜔0 𝑑𝑡 𝐻 0

𝑑𝛿𝐻
= 𝜔𝐻 − 𝜔0
𝑑𝑡

The overall shaft equations are given by the following matrix equation

𝑑𝑋𝑚𝑠
[ ] = [𝐴𝑡𝑚𝑠 ][𝑋𝑚𝑠 ] + [𝐵𝑡𝑚𝑠 ][𝑈𝑡𝑚𝑠 ] (2.22)
𝑑𝑡

20
where

[𝑋𝑚𝑠 ] = [𝛿𝐸 𝛿 𝛿𝐵 𝛿𝐴 𝛿𝐼 𝛿𝐻 𝜔𝐸 𝜔 𝜔𝐵 𝜔𝐴 𝜔𝐼 𝜔𝐻 ]𝑇

[𝑈𝑡𝑚𝑠 ] = [𝜔0 𝑃𝐻 𝑃𝐼 𝑃𝐴 𝑃𝐵 𝑇𝑒 ]𝑇

[𝐴𝑡𝑚𝑠 ] = [06×6 𝐼6×6

]
𝐴𝑠1 𝐴𝑠2

 K Eg K Eg 
 0 0 0 0 
 ME ME 
 K Eg K gB  K Eg K gB 
  0 0 0 
 Mg Mg Mg 
 K gB K BA  K gB K gB 
 0  0 0 
 As1  0  MB MB MB 

 0 K BA K AI  K BA K AI
0  0 
 MA MA MA 
 
 0 K AI K HI  K AI K HI 
0 0 
 MI MI MI 
 
 0 0 0 0
K HI

K HI 
 MH M H 

 DE 
 M 0 0 0 0 0 
 E

 Dg 
 0  0 0 0 0 
 Mg 
 DB 
 0 0  0 0 0 
 As 2   
MB
DA 
 0 0 0  0 0 
 MA 
 DI 
 0 0 0 0  0 
 MI 
 DH 
 0 0 0 0 0  
 MH 
(2.23)

21
  161 061 061 061 061 061 
 D 
 E 0 0 0 0 0 
 ME 
 
 Dg 0 
0 0 0 0 
 Mg Mg 
 
 DB 02 
 Btms    M B
0 0 0 0 
B M B 
 DA 02 
 0 0 0 0 
 MA B M B 
 D 02 
 I 0 0 0 0 
 MI I M I 
 
 DH 02
0 0 0 0 
 M H H M H 

Here, the [𝐼𝑛×𝑛 ] is an n by n identity matrix, 0𝑚×𝑛 is an m by n matrix with all elements
zero, and −16×1 is a 6 by 1 matrix with all elements -1.

2.3.3 Modeling of the transmission line

A series capacitor-compensated transmission line may be represented by the RLC circuit

shown in Figure 2.5 [21]. In the voltage phasor diagram shown in Figure 2.6, the rotor angle 𝛿 is
the angle (in elec. rad) by which the q-axis leads the infinite-bus (reference) voltage 𝑉𝑏 . The
differential equations for the circuit elements, after applying Park’s transformation [21], can be
expressed in the d-q reference frame by the following matrix expressions.

i RL XL XC

Infinite-Bus
System

GEN
Vt VR VL VC Vb

Figure 2.5: A series capacitor-compensated transmission line.

22
 q-axis

Vtq

Vbq Vt


Vb

Vtd
Vbd d-axis

Figure 2.6: Voltage phasor diagram.

The voltage across the resistance:

𝑉 𝑅 0 𝑖𝑑
[𝑉𝑅𝑑 ] = [ 𝐿 ][ ] (2.24)
𝑅𝑞 0 𝑅𝐿 𝑖𝑞

The voltage across the inductance:

𝜔 𝑋𝐿 𝑑𝑖𝑑
𝑉 0 − 𝜔 𝑋𝐿 𝑖 0
𝜔
[ 𝐿𝑑 ] = [ 𝜔 0
] [𝑖𝑑 ] + [ 0 𝑑𝑡
𝑋𝐿 ] [𝑑𝑖𝑞 ] (2.25)
𝑉𝐿𝑞 𝑋 0 𝑞 0
𝜔0 𝐿
𝜔0 𝑑𝑡

The voltage across the capacitor:

𝑑𝑉𝐶𝑑
𝜔0 𝑋𝐶 0 𝑖 0 𝜔 𝑉𝐶𝑑
[𝑑𝑉𝑑𝑡𝐶𝑞 ] = [ ] [ 𝑑] + [ ][ ] (2.26)
0 𝜔0 𝑋𝐶 𝑖𝑞 −𝜔 0 𝑉𝐶𝑞
𝑑𝑡

The overall equations of the transmission line can be written as

23
 𝑑𝑉𝐶𝑑
𝑑𝑡 𝑑𝑖𝑑
𝑑𝑉𝐶𝑞 𝑉𝐶𝑑 𝑖
𝑑𝑡
𝑑𝑡
= [𝐴𝑡𝑡] [𝑉 ] + [𝑅𝑡1] [𝑑𝑖𝑞
] + [𝑅𝑡2] [𝑖𝑑 ] + [𝐵𝑡𝑡][𝑉𝑏 ] (2.27)
𝐶𝑞 𝑞
𝑉𝑡𝑑 𝑑𝑡
[ 𝑉𝑡𝑞 ]

where

 0
 0
[𝐴𝑡𝑡] =
1 0
[ 0 1]

0 0
0 0
[𝑅𝑡1] = X L 0
0
XL
0
[ 0 ]

0 X C 0
0 0 X C
[𝑅𝑡2] = 
RL  X (2.28)
0 L

XL RL
[ 0 ]

0
[𝐵𝑡𝑡] = [ 0 ]
sin𝛿
cos𝛿

2.3.4 Modeling of transformer

The three-phase transformer is constructed by using three single-phase transformers

connected in Delta (LV side)/Y ground (HV side). The connection type, both sides of the voltage,
as well as the winding resistance and winding reactance are represented in the model.

24
2.3.5 Modeling of system loads

The system loads are modeled in these studies by constant impedances. The formula,
which is used in calculating the load impedances, is given by [22]:
|𝑉𝐿𝑜𝑎𝑑 |2
𝑍𝐿𝑜𝑎𝑑 = 𝑃 (2.29)
𝐿𝑜𝑎𝑑 −𝑗𝑄𝐿𝑜𝑎𝑑

where

𝑍𝐿𝑜𝑎𝑑 = load impedence

𝑉𝐿𝑜𝑎𝑑 = load voltage

𝑃𝐿𝑜𝑎𝑑 = load real power

𝑄𝐿𝑜𝑎𝑑 = load reactive power

2.3.6 Wind turbine aerodynamic model

The dynamic output mechanical torque of the wind turbine is expressed as [10]

1
𝑇𝑀𝐸𝐶𝐻 = 2 𝜌A𝑅𝜔 𝐶𝑝 𝑉𝜔2 /𝜆 (2.30)

where 𝜌 is the air density (kgm−3 ), A is the blade sweep area (m2 ), Rω is the rotor radius of wind
turbine (m), and 𝑉𝜔 is the wind speed (m/s). 𝐶𝑝 is the power coefficient of the blade which is a
function of the blade pitch angle 𝜃 and the tip speed ratio 𝜆 according to the following equation:

𝑅𝜔 𝐶𝑓
1 𝑅𝜔 𝐶𝑓
𝐶𝑝 = 2 ( 𝜆
− 0.022𝜃 − 2) 𝑒 −0.255 𝜆 (2.31)

where 𝐶𝑓 is the wind turbine blade design constant and the tip speed ratio λ is given by

Ω𝑚 𝑅𝜔
𝜆= (2.32)
𝑉𝜔

where Ω𝑚 is the mechanical angular velocity (rad/s).

The power rotating speed, wind speed and the pitch angle relationships are illustrated in
Figure 2.7.

25
 1.6
1.4 14 m/s

1.2 13 m/s
1
12 m/s
Pm (p.u)

0.8
11 m/s
0.6 10 m/s
9 m/s
0.4 8 m/s
0.2 7m/s

0 6 m/s

-0.2
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
ωm (p.u)
Figure 2.7: Mechanical power, rotor speed and wind speed relationships.

2.3.7 Modeling of the DFIG

Figure 2.8 shows the equivalent circuit of a DFIG in the synchronous qd reference frame,
where the q-axis leads the d-axis by 90˚. The stator and rotor voltage equations in the qd reference
frame can be written as follows:

𝑑𝜆𝑠
𝑣𝑠 = 𝑟𝑠 𝑖𝑠 + 𝑗𝜔𝑠 𝜆𝑠 +
𝑑𝑡
{ 𝑑𝜆𝑟
(2.33)
𝑣𝑟 = 𝑟𝑟 𝑖𝑟 + 𝑗(𝜔𝑠 − 𝜔𝑟 )𝜆𝑟 + 𝑑𝑡

where 𝑣𝑠 = 𝑣𝑞𝑠 − 𝑗𝑣𝑑𝑠 and 𝑣𝑟 = 𝑣𝑞𝑟 − 𝑗𝑣𝑑𝑟 . The flux linkage expressions are given as follows:

𝜆 = 𝐿𝑠 𝑖𝑠 + 𝑀𝑖𝑟
{ 𝑠 (2.34)
𝜆𝑟 = 𝐿𝑟 𝑖𝑟 + 𝑀𝑖𝑠

where 𝐿𝑠 = 𝐿𝑙𝑠 + 𝑀 , 𝐿𝑟 = 𝐿𝑙𝑟 + 𝑀 , 𝜆𝑠 = 𝜆𝑞𝑠 − 𝑗𝜆𝑑𝑠 , 𝜆𝑟 = 𝜆𝑞𝑟 − 𝑗𝜆𝑑𝑟 , 𝑖𝑠 = 𝑖𝑞𝑠 − 𝑗𝑖𝑑𝑠 and 𝑖𝑟 =
𝑖𝑞𝑟 − 𝑗𝑖𝑑𝑟 .

26
 rs Lls Llr rr
+ +
+ +

vs vr

Figure 2.8: Equivalent circuit of the DFIG.

From Equations 2.33 and 2.34, a set of differential equations with the stator and rotor
currents are state variables and the stator and rotor voltage as inputs can be established. While the
rotor voltages are determined by the RSC control scheme, the stator voltages are determined by
the network interface.

The electromagnetic torque 𝑇𝑒 can be expressed as follows:

𝑇𝑒 = 𝜆𝑞𝑚 𝑖𝑑𝑟 − 𝜆𝑑𝑚 𝑖𝑞𝑟 (2.35)

where 𝜆𝑞𝑚 and 𝜆𝑑𝑚 are, respectively, the q-and d-axes magnetizing flux linkages defined as

𝜆𝑞𝑚 = 𝜆𝑞𝑠 − 𝑖𝑞𝑠 𝐿𝑙𝑠 (2.36)

𝜆𝑑𝑚 = 𝜆𝑑𝑠 − 𝑖𝑑𝑠 𝐿𝑙𝑠 (2.37)

2.3.8 Modeling of the BtB dc capacitor link

The dynamics of the BtB dc capacitor link can be described with the help of the equivalent
circuit shown in Figure 2.9 as

𝑑𝑣𝑑𝑐
𝐶 = 𝑃𝑟 − 𝑃𝑔 (2.38)
𝑑𝑡

where

27
 𝑃𝑟 = 𝐾1 (𝑣𝑞𝑟 𝑖𝑞𝑟 + 𝑣𝑑𝑟 𝑖𝑑𝑟 )
{ (2.39)
𝑃𝑔 = 𝐾2 (𝑣𝑞𝑔 𝑖𝑞𝑔 + 𝑣𝑑𝑔 𝑖𝑑𝑔 )

In Equation 2.39, 𝐾1 and 𝐾2 are constants, 𝑃𝑟 , 𝑃𝑔 are the active powers of the RSC and the
GSC respectively, 𝑣𝑞𝑟 , 𝑣𝑑𝑟 are the quadrature- and direct- axes RSC voltage respectively and 𝑣𝑞𝑔 ,
𝑣𝑑𝑔 are the quadrature- and direct- axes GSC voltage respectively.

Pr Pdc Pg
+ +
+
Vr vdc C Vg
_
_ _

Figure 2.9: Equivalent circuit for the BtB dc capacitor link.

The frequency converters are modeled by a fundamental frequency approach, which is

appropriate and sufficient for power system dynamic studies. Assuming an ideal DC-link voltage
and an ideal pulse-width modulation [23], the relation between AC- and DC- voltages can be
expressed as follows [12]

|𝑈AC | = 𝐾0 𝑃𝑚 𝑈DC (2.40)

𝑃𝑚 , which donates the pulse-width-modulation index, is limited (0 < 𝑃𝑚 ≤ 1) to avoid

saturation effects. The factor 𝐾0 depends on the modulation method, e.g. rectangular or sinusoidal
modulation. In case of sinusoidal modulation, which is standard in power application, the factor
𝐾0 is defined as

√3
𝐾0 = 2 (2.41)
√2

The DC-link capacitor provides an intermediate energy storage, which decouples the
generator-side converter and the grid-side converter. The size of the DC-link Capacitor is selected
based on a trade-off between voltage ripples, lifetime and fast control of the DC-link [12], [24].

The control of the DFIG wind turbine is achieved by controlling the RSC and GSC utilizing
vector control techniques. Vector control allows decoupled control of both the real and reactive

28
power. The idea is to use a rotating reference frame based on an AC flux or voltage and then to
project the currents on this rotating frame. Such projections are usually referred to as the d- and
q- components of their respective currents. For flux-based rotating frames, changes in the q-
component leads to real power changes, while changes in the d- component leads to reactive power
changes. In voltage-based rotating frames (90˚ ahead of flux-based frames), the effect is the
opposite.

Control of the DFIG BtB Converters

RSC GSC
AC DC
DC AC

mq md md mq
iqr iqg
idr PI PI PI PI idg

iqr-Ref idr-Ref idg-Ref iqg-Ref

Stage-1

Pgrid Vdc
PI PI Qgrid PI PI QGSC

Pgrid-Ref
Qgrid-Ref Vdc-Ref QGSC-Ref
Stage-2
Maximum
power tracking ωgen
point

Figure 2.10: Schematic diagram of a general control scheme of DFIG BtB converters.

Figure 2.10 shows a general control scheme for the DFIG BtB converters [25]-[27]. In
such a scheme, the RSC operates in the stator flux reference while the GSC operates in the stator

29
voltage reference frame. The q-axis current of the RSC is used to control the real power while the
d-axis current is used for reactive power control. On the other hand, the d-axis current for the GSC
is used to control the dc link voltage to a constant level while the q-axis current is used for reactive
power control.

As illustrated in Figure 2.10, both RSC and GSC are controlled by a two-stage controller.
The first stage consists of very fast current controllers regulating the rotor currents to references
values that are specified by slower power controllers (Stage-2). In normal operation, the aim of
the RSC is to control independently the real and reactive power on the grid while the GSC has to
maintain the dc link capacitor at a set value regardless of the magnitude and direction of the rotor
power and to guarantee converter operation with unity power factor (zero reactive power). The
reference Pgrid−Ref for the real power is given by the maximum power tracking point (MPT)
lookup table as a function of the optimal generator speed. The reference Q grid−Ref for the reactive
power of the RSC can be set to a certain value or to zero according to whether or not the DFIG is
required to contribute with reactive power. The reactive power reference for the GSC, Q GSC−Ref
is “usually” set to zero. This means that the GSC exchanges only real power with the grid and,
therefore, the transmission of reactive power controllability of the GSC can be useful during the
process of voltage reestablishment after clearing a system fault. The reference signal Vdc−Ref is
set to a constant value that depends on the size of the converter, the stator/rotor voltage ratio and
the modulation factor of the power converter [27].

2.3.9 Modeling of PMSG

The equations of a PMSG can be expressed directly from the equations of a DC excited
synchronous generator, with the simplification that PMSG does not have damper windings. The
voltage equations of the generator [20], expressed in the dq-reference frame, can be expressed as
follows

𝑒𝑑 = 𝑅𝑠 𝑖𝑑 − 𝜔𝜓𝑞 + 𝜓̇𝑑 (2.42)

𝑒𝑞 = 𝑅𝑠 𝑖𝑞 + 𝜔𝜓𝑑 + 𝜓̇𝑞 (2.43)

With the stator flux components

30
 𝜓𝑑 = 𝐿𝑑 𝑖𝑑 + 𝜓𝑃𝑀 (2.44)

𝜓𝑞 = 𝐿𝑞 𝑖𝑞 (2.45)

where ed and eq, id and iq, Ld and Lq are the d- and q-components of the stator voltage, of the stator
current and of the stator inductance, respectively. 𝑅𝑠 and 𝛹PM stand for stator resistance and
permanent magnet flux, respectively. In stability studies, the stator transients can be neglected.

The electromagnetic torque equation:

𝑇𝑒 = 𝜓𝑑 𝑖𝑞 − 𝜓𝑞 𝑖𝑑 (2.46)

2.3.10 Modelling of the BtB dc capacitor link

As shown in Figure 2.11, the dynamic of the BtB capacitor link is very similar to that
discussed in subsection 2.3.8 if the symbol “r” representing the “rotor” in RSC is replaced by the
symbol “m” represent “MSC”.

2.3.11 FFC Wind turbine control

Similar to DFIG, the control of the FFC is achieved by controlling the MSC and GSC
utilizing vector control techniques [4]. The MSC controls the active power delivered by the PMSG,
and follows a tracking characteristic to adjust the PMSG speed for optimal power generation
depending on wind speed. The function of GSC is maintaining the dc bus voltage to the dc link
by the MSC. GSC also controls the reactive power delivered to the grid [12], [28].

A generic 2 MW, 60 Hz FFC model of EMTP-RV is used in this thesis. The model includes
a pitch control, dc chopper and over/under voltage protections. The wind turbine drive system is
represented with its two-mass model. The FFC converters are also modeled with their AVMs.

31
 Control of the FFC BtB Converters

MSC GSC
AC DC
DC AC

mq md md mq
iqr iqg
idr PI PI PI PI idg

iqr-Ref idr-Ref idg-Ref iqg-Ref

Stage-1

Pgrid Vdc
PI PI Qgrid PI PI QGSC

Pgrid-Ref
Qgrid-Ref Vdc-Ref QGSC-Ref
Stage-2
Maximum
power tracking ωgen
point

Figure 2.11: Schematic diagram of a general control scheme of FFC BtB converters.

2.4 A Sample Case Study: 30% Compensation Degree in Lines 1 and 2

In the studies conducted in this thesis, the ElectroMagnetic Transient Program (EMTP-RV)
is used for modeling the various system components and producing the time-domain simulation
results [29]. Moreover, faults are assumed to occur at t = 0.1 second.

Figure 2.12 shows the power flow results for the system bus voltages and transmission line
real power flows. The real and reactive powers, terminal voltages and dc capacitor voltages of
wind farms A and B during and after clearing a 3-cycle, three-phase fault on Line 5 are shown in

32
Figures 2.13 and 2.14 respectively. The turbine-generator electrical powers and shaft torsional
torques during and after clearing the same fault are shown in Figure 2.15. Moreover, the frequency
spectrums (obtained by using Fast Fourier Transform (FFT) analysis) of the turbine-generator shaft
torsional torques and the stator current of the DFIG wind turbine are illustrated in Figures 2.16 and
2.17 respectively.

The following observations can be made from examining these figures:

1. The system is stable after fault clearing as the low-frequency oscillations in the turbine-
generator electrical powers are damped.
2. It can be seen from Figure 2.15 that the turbine-generator shaft torsional torques are
not sinusoidal with a single frequency but contain contributions from all the torsional
modes. Moreover, it can be noticed from the same figure that the shaft section between
the generators and low-pressure turbines are subjected to the highest stresses.
3. Except for the (GEN1-EXC) shaft torsional torque which exhibits slowly growing
oscillations, the other turbine-generator shaft torsional torques are either slightly
damped or exhibit sustained oscillations.
4. The frequency spectrums of the turbine-generator shaft torsional torques identify the
four natural torsional modes of oscillations of G1 and G2 shaft systems with
frequencies 24.65 Hz, 32.39 Hz, 44.99 Hz and 51.1 Hz.
5. At this compensation degree, wind farm A does not exhibit subsynchronous interaction
as the oscillations in its real output power and terminal voltages are damped after fault
clearing.

33
 VA = 1.03∠17.8˚
716 MW
R L C
T1
Bus A
Line 1, 500 km LP1 HP1

EXC G1
Line 3,
150 km T2 LP2 HP2

G2
266 MW

Line 6, Line 5,
150 km 150 km Bus B
VB = 1.03∠15.0˚

∞ 0.1 MW 301MW
T3 FFC-Based Wind Farm
L
GSC MSC G
500 kV
transmission Wind Farm B
network
Line 4, 162 MW
50 km

Bus C VC = 1.02∠14.5˚
580 MW DFIG-Based Wind Farm
T4
IG
R1 L1
C1
Il, Pl
Line 2, 500 km GSC RSC

Wind Farm A

Figure 2.12: Power flow results of bus voltages and transmission line real power flows of the
system under study.

34
 2
PWFA (p.u.)

0.75

-0.5
0 1.5 Time (s) 3 4.5
1
QWFA (p.u.)

-0.25

-1.5
0 1.5 Time (s) 3 4.5
1.2
VWFA (p.u.)

0.6

0
0 1.5 Time (s) 3 4.5
1.2
Vdc_WFA (p.u.)

0.8
0 1.5 Time (s) 3 4.5

Figure 2.13: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (30% compensation
degree).

35
 1.5
PWFB (p.u.)

0.5

-0.5
0 1.5 3 4.5
Time (s)
1
QWFB (p.u.)

-0.5

-2
0 1.5 3 4.5
Time (s)
1.4
VWFB (p.u.)

0.7

0
0 1.5 3 4.5
Time (s)
1.2
VdcWFB (p.u.)

0.8
0 1.5 3 4.5
Time (s)

Figure 2.14: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (30% compensation
degree).

36
 1.6
PG1 (p.u.)

0.8

0
0 1.5 3 4.5
Time (s)
0.3
T(GEN1-EXC) (p.u.)

-0.3
0 1.5 3 4.5
Time (s)
0.3
T(GEN1-EXC) (p.u.)

-0.3
2 3
Time (s)
3
T(LP1-GEN1) (p.u.)

-1
0 1.5 3 4.5
Time (s)

Figure 2.15: Turbine-generator electrical powers and shaft torsional torques during and after
clearing a 3-cycle, three-phase fault on Line 5 (30% compensation degree).

37
 3

T(LP1-GEN1) (p.u.)
1

-1
2 3
Time (s)
2
T(HP1-LP1) (p.u.)

0.5

-1
0 1.5 3 4.5
Time (s)
2
T(HP1-LP1) (p.u.)

0.5

-1
2 3
Time (s)
1.6
PG2 (p.u.)

0.8

0
0 1.5 3 4.5
Time (s)

Figure 2.15: Continued.

38
 3
T(LP2-GEN2) (p.u.)

-1
0 1.5 3 4.5
Time (s)
3
T(LP2-GEN2) (p.u.)

-1
2 3
Time (s)
1.5
T(HP2-LP2) (p.u.)

0.5

-0.5
0 1.5 3 4.5
Time (s)
1.5
T(HP2-LP2) (p.u.)

0.5

-0.5
2 3
Time (s)

Figure 2.15: Continued.

39
 0.14
Amplitude (p.u.)
0.07 T(GEN1-EXC)

0
0 20 40 60 80
Frequency (Hz)

0.4
Amplitude (p.u.)

0.2 T(LP1-GEN1)

0
0 20 40 60 80
Frequency (Hz)

0.5
Amplitude (p.u.)

0.25 T(HP1-LP1)

0
0 20 40 60 80
Frequency (Hz)

0.4
Amplitude (p.u.)

0.2 T(LP2-GEN2)

0
0 20 40 60 80
Frequency (Hz)

0.3
Amplitude (p.u.)

0.15 T(HP2-LP2)

0
0 20 40 60 80
Frequency (Hz)

Figure 2.16: Frequency spectrums of the turbine-generator shaft torsional torques during and after
clearing a 3-cycle, three-phase fault on Line 5 (30% compensation degree).

40
 600
Amplitude (p.u.)

300 I Stator

0
0 20 40 60 80 100 120
Frequency (Hz)

Figure 2.17: Frequency spectrums of the stator current of the DFIG wind turbine.

2.5 Summary

This chapter introduces the system used for the studies reported in this thesis and presents
the mathematical models of its various components. A digital time-domain simulation of a case
study of the system during a three-phase fault is also presented. The results of this case study show
the existence of subsynchronous resonance and the absence of subsynchronous interaction. In the
next chapter, supplemental controls are presented for simultaneous mitigation of the two
phenomena.

41
3. SUPPLEMENTAL CONTROLS OF FFC- AND DFIG-
BASED WIND FARMS FOR SIMULATANEOUS
MITIGATION OF SUBSYNCHRONOUS
RESONANCE AND SUBSYNCHRONOUS
INTERACTION

3.1 Introduction

In this chapter, two supplemental controls of FFC- and DFIG-based wind farms for
simultaneous mitigation of SSR and SSI are introduced. The effectiveness of the two presented
controllers in mitigating both phenomena is investigated through several time-domain simulation
case studies for different wind farm ratings and system contingences.

3.2 Supplemental Controls of FFC- and DFIG-Based Wind Farms

SSR and SSI damping is achieved through the modulation of the reactive powers of the
FFC and DFIG wind turbines. This is attained by introducing supplemental control signals, US,
in the reactive power control loop of their grid side converters of as shown in Figures 3.1 and 3.2
respectively. In this context, two supplemental controls designated as Supplemental controls 1
and 2 are proposed.

In Supplemental control 1, shown in Figure 3.3, SSR and SSI damping is attained by adding
the supplementary control signal US = USSR + USSI in the reactive power control loop of the GSC
of the FFC wind turbines before the PI regulator of the inner control loop as shown in Figure 3.1.
On the other hand, in Supplemental control 2, shown in Figure 3.4, SSR and SSI damping is
attained by adding supplementary control signals US = USSR and US = USSI in the reactive power
control loops of the GSC of the FFC and DFIG wind turbines respectively before the PI regulator
of the inner control loop as shown in Figures 3.1 and 3.2.

42
 Control of the FFC BtB Converters

MSC GSC
AC DC
DC AC

mq md md mq
iqr iqg
idr PI PI PI PI idg

Stage-1

iqr-Ref +
idr-Ref idg-Ref ∑ Us
+
iqg-Ref
Qgrid Vdc
Pgrid PI PI PI PI
QGSC

Pgrid-Ref
Qgrid-Ref Vdc-Ref QGSC-Ref
Stage-2
Maximum
power tracking ωgen
point

Figure 3.1: Incorporating a supplemental control signal, US, in the reactive power control loop
of the GSC of FFC wind turbine.

43
 Control of the DFIG BtB Converters

RSC GSC
AC DC
DC AC

mq md md mq
iqr iqg
idr PI PI PI PI idg

Stage-1
+
iqr-Ref idr-Ref idg-Ref ∑ Us
+
iqg-Ref
Qgrid Vdc
Pgrid PI PI PI PI
QGSC

Pgrid-Ref
Qgrid-Ref Vdc-Ref QGSC-Ref
Stage-2
Maximum
power tracking ωgen
point

Figure 3.2: Incorporating a supplemental control signal, US, in the reactive power control loop
of the GSC of DFIG wind turbine.

44
 SSR Supplemental Control

∆ωm,0
Lead-Lag Networks

∆ωm,1 +
Lead-Lag Networks +

∆ωm,i Lead-Lag Networks + +

USSR

∆ωm,N-1 Lead-Lag Networks

SSI Supplemental Control

_ PLine 2 USSR
+
+ 𝑠𝐾𝑤 +
 1 + 𝑠𝐾𝑤 BPF Lead-Lag Networks 
PLine 2 ref
USSI

iqg-Ref +
∑ PI UTotal_max
UTotal +

UTotal_min

Figure 3.3: Supplemental control 1.

45
 SSR Supplemental Control

∆ωm,0 Lead-Lag Networks

UTotal_max
∆ωm,1 Lead-Lag Networks +
+
+
∑ ∑
PI
USSR +
∆ωm,i Lead-Lag Networks + +
UTotal_min

∆ωm,N-1 Lead-Lag Networks iqg-Ref

SSI Supplemental Control

PLine 2 UTotal_max
_
𝑠𝐾𝑤 +
+ Lead-Lag
 1 + 𝑠𝐾𝑤 BPF Networks  PI
PLine 2 ref
USSI
+
UTotal_min

iqg-Ref

Figure 3.4: Supplemental control 2.

46
 The SSR supplemental control, shown in Figures 3.3 and 3.4 has N-channels (N is the
number of turbine-generator shaft system rotating masses) that employ the modal speeds as control
signals. These modal speeds are derived from the turbine-generator rotating mass speeds as [30],
[31]:

[∆𝜔𝑚 ] = [𝑄 ]−1 [∆𝜔] (3.1)

where, m is the modal speed deviation matrix, Q is the eigenvector matrix and  is the speed
deviation matrix of the turbine-generator rotating masses. The rotating mass speeds can be
obtained using torsional monitor. Each modal speed as presented is separately phase and gain
adjusted to provide damping for its corresponding torsional mode. The phase compensations are
provided as:

1+𝑠𝑇𝑎𝑖
∅𝑖 = , i = 0,1, … , N − 1 (3.2)
1+𝑠𝑇𝑏𝑖

The values of the matrices Q for the two turbine-generator shaft systems are given in
Appendix A. On the other hand, the SSI supplemental control, shown in Figures 3.3 and 3.4, is an
m-stage lead-lag compensation controller incorporating wash-out (high-pass) and band-pass (BPF)
filters and utilizes Line 2 real power flow as a control signal.

In Supplemental control 1, a central controller located in wind farm B accepts the remote
input signals and sends an output signal US to each FFC wind turbine. On the other hand, in
Supplemental control 2, there are two central controllers located in wind farms A and B. The
central controller in wind farm A accepts a local input signal (Line 2 real power flow) and sends
output signal US = USSI to each DFIG wind turbine. The other central controller accepts the remote
signals (turbine-generator shaft rotating mass speeds) and sends an output signal US = USSI to each
FFC wind turbine. In the investigations reported in this thesis, it is presumed the accessibility of
a wide-area network of synchronized phasor measurement units where the supplemental control
input signals can be downloaded at the controller in real time without delay. Nevertheless,
incorporating the effect of the time delay in computing the phasor quantities and the variable
communication network latency for a controller that uses remote synchrophasor data is achievable
[32].

A trial-and-error approach is adopted in the investigations conducted in this thesis for

47
finding appropriate controller gains and time constants that result in an acceptable oscillations
damping. The fine-tuning of the controller parameters is achieved by performing repetitive time-
domain simulations to minimize the objective functions

𝑇
𝐶𝑆𝑆𝐼 = ∫𝑇 𝐹 (∆Y′)2 dt (3.3)
𝐶

and

𝑇 2
𝐶𝑆𝑆𝑅 = ∫𝑇 𝐹 ∑𝑁−1
𝑖=0 (∆𝜔𝑚,𝑖 ) 𝑑𝑡 (3.4)
𝐶

where TC is the fault clearing time, TF is the total simulation time and ΔY’ is the output of the band
pass filters in the controller. The design of optimal supplemental controls using nonlinear control
techniques, such as indirect adaptive control, is out of scope of this thesis.

3.3 Performance of Supplemental Control 1 in damping SSR and SSI Oscillations

Figure 3.5 shows the turbine-generator shaft torsional torque time responses during and
after clearing a 3-cycle, three-phase fault on line 5 for the case when the supplemental control is
not activated. Figures 3.6 and 3.7 illustrate respectively the time responses of wind farms A and
B active and reactive powers, terminal voltage and the BtB dc voltage for the same case. It can be
seen from Figure 3.5 that, at the 60% compensation degree of lines 1 and 2, the turbine-generator
shaft torsional torques endure torsional instability. It can also be seen from Figures 3.6 and 3.7
that SSI, which is due to the induction generator effect, is clearly present in wind farm A and that
the adverse effects of the simultaneous existence of SSR and SSI extend their impact to the FFC
wind farm (wind farm B) performance.

For a clear insight of the excited SSR and SSI mode components and the effectiveness of
the supplemental controllers in decreasing these components, Figure 3.8 shows the frequency
spectrums of the stator current of the DFIG wind turbines and the turbine-generator shaft torsional
torques. It can be seen from Figure 3.8 that the stator current contains frequency components
(electrical modes) of 10.37 Hz, 14.8 Hz, 35.09 Hz, 60 Hz, 84.83 Hz, 109.86 Hz and 115.36 Hz.
The complements (60 – 10.37 = 49.625 Hz, 60 – 14.8 = 45.2 Hz, 60 – 35.09 =24.91 Hz) of the
three electrical modes excite SSI mode (24.5 Hz) as well as the three shaft torsional modes (24.65
Hz, 44.99 Hz, 51.1 Hz).

48
 1
T(GEN1-EXC)
(p.u.)
0

-1
0 1.5 Time (s) 3 4.5
80
T(LP1-GEN1)
(p.u.)

-80
0 1.5 Time (s) 3 4.5
30
T(HP1-LP1)
(p.u.)

-30
0 1.5 Time (s) 3 4.5
60
T(LP2-GEN2)
(p.u.)

-60
0 1.5 Time (s) 3 4.5
20
T(HP2-LP2)
(p.u.)

-20
0 1.5 Time (s) 3 4.5

Figure 3.5: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is not
activated, wind farm B rating = 200 MW).

49
 2.2
PWFA (p.u.)

-0.2
0 1.5 Time (s) 3 4.5
1.3
QWFA (p.u.)

-0.1

-1.5
0 1.5 Time (s) 3 4.5
1.5
VWFA (p.u.)

0.75

0
0 1.5 Time (s) 3 4.5
1.1
Vdc_WFA (p.u.)

0.9
0 1.5 Time (s) 3 4.5

Figure 3.6: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, wind farm B rating = 200 MW).

50
 1.2
PWFB (p.u.)

0.5

-0.2
0 1.5 Time (s) 3 4.5
1
QWFB (p.u.)

-0.5

-2
0 1.5 Time (s) 3 4.5
1.5
VWFB (p.u.)

0.75

0
0 1.5 Time (s) 3 4.5
1.2
VdcWFB (p.u.)

0.9

0.6
0 1.5 Time (s) 3 4.5

Figure 3.7: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, wind farm B rating = 200 MW).

51
 600
Amplitude (p.u.)

300 I Stator

0
0 20 40 60 80 100 120
Frequency (Hz)
40
Amplitude (p.u.)

I Stator
20 Zoom In

0
0 20 40 60 80 100 120
Frequency (Hz)
0.3
Amplitude (p.u.)

0.15 T(GEN1-EXC)

0
0 20 40 60 80
Frequency (Hz)
30
Amplitude (p.u.)

15 T(LP1-GEN1)

0
0 20 40 60 80
Frequency (Hz)

Figure 3.8: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase fault
on Line 5 (60% compensation degree, supplemental control 1 is not activated, wind
farm B rating = 200 MW).

52
 12
Amplitude (p.u.)

6 T(HP1-LP1)

0
0 20 40 60 80
Frequency (Hz)
20
Amplitude (p.u.)

10 T(LP2-GEN2)

0
0 20 40 60 80
Frequency (Hz)
8
Amplitude (p.u.)

4 T(HP2-LP2)

0
0 20 40 60 80
Frequency (Hz)

Figure 3.8: Continued.

53
 Figure 3.9 shows turbine-generator shaft torsional torque time responses during and after
clearing the same fault for the case when Supplemental control 1 is activated. Figures 3.10 and
3.11 illustrate respectively the time responses of wind farms A and B active and reactive powers,
terminal voltage and the BtB dc voltage for the same case. Moreover, Figure 3.12 shows the
frequency spectrums of the stator current of the DFIG wind turbines and the turbine-generator
shaft torsional torques for the same study case. Furthermore, the transfer functions of
Supplemental control 1 are given in Table 3.1. The comparison between the two groups of figures
(Figures 3.5, 3.6, 3.7, 3.8) and (Figures 3.9, 3.10, 3.11, 3.12) establishes the effectiveness of
Supplemental control 1 in damping the torsional torques in all turbine-generator shaft sections as
well as in mitigating SSI in wind farm A. It can also be seen from Figures 3.10 that the dc chopper
protection of FFC limits the dc bus voltage at 1.1 p.u.

Table 3.1: Transfer functions of Supplemental control 1 (Wind farm B rating = 200 MW, three-
phase fault on Line 5).

Modal speeds Transfer function

G1, Δωm0 0
𝑠 + 300
G1, Δωm1 𝐺𝜔1 (𝑠) = 2.5
𝑠+1
𝑠 + 1000
G1, Δωm2 𝐺𝜔2 (𝑠) = −2.5
𝑠+1
𝑠 + 1000
G1, Δωm3 𝐺𝜔3 (𝑠) = 2.5
𝑠+1
G2, Δωm0 0
𝑠 + 250
G2, Δωm1 𝐺𝜔1 (𝑠) = 25
𝑠+1
𝑠 + 1250
G2, Δωm2 𝐺𝜔2 (𝑠) = 25
𝑠+1
𝑠
Washout filter 𝐺 (𝑠 ) =
𝑠 + 10
62.83𝑠
Band-Pass filter 𝐺 (𝑠 ) = 2
𝑠 + 62.83𝑠 + 35500
𝑠 + 250
Lead-Lag compensator 𝐺 (𝑠 ) =
𝑠+1
UTotal_max, UTotal_min 0.33, -0.33

54
 T(GEN1-EXC) (p.u.) 0.2 4

T(LP1-GEN1) (p.u.)
0 1

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

2 3
T(HP1-LP1) (p.u.)

T(LP2-GEN2) (p.u.)
0.5 1.1

-1 -0.8
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

1.5
T(HP2-LP2) (p.u.)

0.5

-0.5
0 1.5 3 4.5
Time (s)

Figure 3.9: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
wind farm B rating = 200 MW).

55
 1.8 1

QWFA (p.u.)
PWFA (p.u.)

0.8 -0.1

-0.2 -1.2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.2 1.1

Vdc_WFA (p.u.)
VWFA (p.u.)

0.6 1

0 0.9
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.10: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, wind farm B rating = 200 MW).
1.2 1
PWFB (p.u.)

QWFB (p.u.)

0.5 -0.5

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.3 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.65 0.9

0 0.6
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.11: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, wind farm B rating = 200 MW).

56
 600 40

Amplitude (p.u.)
I Stator I Stator
Amplitud (p.u.)
Zoom In
300 20

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.03 0.15
T(GEN1-EXC)

Amplitude (p.u.)
T(LP1-GEN1)
Amplitude (p.u.)

0.015 0.075

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)

0.2 0.1
Amplitude (p.u.)

T(HP1-LP1) T(LP2-GEN2)
Amplitude (p.u.)

0.1 0.05

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.06
Amplitude (p.u.)

T(HP2-LP2)

0.03

0
0 20 40 60 80
Frequency (Hz)

Figure 3.12: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
wind farm B rating = 200 MW).

57
3.3.1 Effect of the distance between wind farm B and the turbine-generators

The effect of the distance between wind farm B and the turbine-generators on the
performance of Supplemental control 1, especially in mitigating SSR, is examined by changing
the length of transmission Line 3. The results of this study are presented in the following three
groups of figures:

Group A: Line 3 = 50 km, (Figures 3.13, 3.14, 3.15, 3.16, 3.17, 3.18, 3.19, 3.20)

Group B: Line 3 = 100 km, (Figures 3.21, 3.22, 3.23, 3.24, 3.25, 3.26, 3.27, 3.28)

Group C: Line 3 = 200 km, (Figures 3.29, 3.30, 3.31, 3.32, 3.33, 3.34, 3.35, 3.36)

In each group, the turbine-generator shaft torsional torques, wind farms A and B active and
reactive powers, terminal voltage and the BtB dc voltage as well as the frequency spectrums of the
stator current of the DFIG wind turbines and the turbine-generator shaft torsional torques are
shown for the cases when Supplemental control 1 is disabled and activated respectively. Moreover,
the transfer functions of Supplemental control 1 in Groups A and C are given in Tables 3.2 and
3.3 respectively. The transfer functions of Supplemental control 1 in Group B are the same as those
in Table 3.1.

Table 3.2: Transfer functions of Supplemental control 1 for Group A (Line 3 length = 50 km).

Modal speeds Transfer function

G1, Δωm0 0
𝑠 + 300
G1, Δωm1 𝐺𝜔1 (𝑠) = 15
𝑠+1
𝑠 + 1000
G1, Δωm2 𝐺𝜔2 (𝑠) = −2.5
10𝑠 + 1
𝑠 + 500
G1, Δωm3 𝐺𝜔3 (𝑠) = 2.5
𝑠+1
G2, Δωm0 0
10𝑠 + 250
G2, Δωm1 𝐺𝜔1 (𝑠) = 25
𝑠+1
𝑠 + 1250
G2, Δωm2 𝐺𝜔2 (𝑠) = 25
𝑠 + 0.1
𝑠
Washout filter 𝐺 (𝑠 ) =
𝑠 + 10

58
 62.83𝑠
Band-Pass filter 𝐺 (𝑠 ) =
𝑠 2 + 62.83𝑠 + 35500
𝑠 + 250
Lead-Lag compensator 𝐺 (𝑠 ) =
𝑠+1
UTotal_max, UTotal_min 0.33, -0.33

Table 3.3: Transfer functions of Supplemental control 1 for Group C (Line 3 length = 200 km).

Modal speeds Transfer function

G1, Δωm0 0
10𝑠 + 300
G1, Δωm1 𝐺𝜔1 (𝑠) = 2.5
𝑠+1
𝑠 + 1000
G1, Δωm2 𝐺𝜔2 (𝑠) = −2.5
𝑠+1
10𝑠 + 300
G1, Δωm3 𝐺𝜔3 (𝑠) = 2.5
𝑠+1
G2, Δωm0 0
50𝑠 + 250
G2, Δωm1 𝐺𝜔1 (𝑠) = 25
10𝑠 + 1
0.1𝑠 + 1250
G2, Δωm2 𝐺𝜔2 (𝑠) = 25
𝑠+1
𝑠
Washout filter 𝐺 (𝑠 ) =
𝑠 + 10
62.83𝑠
Band-Pass filter 𝐺 (𝑠 ) = 2
𝑠 + 62.83𝑠 + 35500
𝑠 + 250
Lead-Lag compensator 𝐺 (𝑠 ) =
𝑠+1
UTotal_max, UTotal_min 0.33, -0.33

The comparisons between the two sets of figures in Group A, B and C (Set 1: Supplemental
control 1 is disabled and Set 2: Supplemental control 1 is activated; e.g. Figures 3.13, 3.14, 3.15,
3.16 and Figures 3.17, 3.18 3.19 and 3.20 in Group A) demonstrate the effectiveness of the
supplemental control in mitigating SSR and SSI when it is located at different distances from the
turbine-generators and wind farm A.

59
 T(GEN1-EXC) (p.u.) 3 250

T(LP1-GEN1) (p.u.)
-0.25 0

-3.5 -250
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

100 220

T(LP2-GEN2) (p.u.)
T(HP1-LP1) (p.u.)

0 0

-100 -220
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

70
T(HP2-LP2) (p.u.)

-70
0 1.5 3 4.5
Time (s)

Figure 3.13: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is not
activated, Line 3 length = 50 km).

60
 PWFA (p.u.) 4 3

QWFA (p.u.)
0.5 -0.5

-3 -4
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
2 1.2

Vdc_WFA (p.u.)
VWFA (p.u.)

1 0.85

0 0.5
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.14: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, Line 3 length = 50 km).

2 3
PWFB (p.u.)

QWFB (p.u.)

-0.4 -1.5

-2.8 -6
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
2 1.2
VdcWFB (p.u.)
VWFB (p.u.)

1 0.75

0 0.3
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.15: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, Line 3 length = 50 km).

61
 600 40
Amplitude (p.u.) I Stator
I Stator

Amplitude (p.u.)
Zoom In
300 20

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
1 100
Amplitude (p.u.)

Amplitude (p.u.)
T(GEN1-EXC) T(LP1-GEN1)

0.5 50

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)

40 80
Amplitude (p.u.)
Amplitude (p.u.)

T(HP1-LP1) T(LP2-GEN2)

20 40

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
30
Amplitude (p.u.)

T(HP2-LP2)

15

0
0 20 40 60 80
Frequency (Hz)

Figure 3.16: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is not activated,
Line 3 length = 50 km).

62
 T(GEN1-EXC) (p.u.) 0.3 5.5

T(LP1-GEN1) (p.u.)
0 1.25

-0.3 -3
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

3 6
T(HP1-LP1) (p.u.)

T(LP2-GEN2) (p.u.)
0.5 1.5

-2 -3
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

2.2
T(HP2-LP2) (p.u.)

0.6

-1
0 1.5 3 4.5
Time (s)

Figure 3.17: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
Line 3 length = 50 km).

63
 2 1
PWFA (p.u.)

QWFA (p.u.)
0.9 -0.1

-0.2 -1.2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.2 1.1

Vdc_WFA (p.u.)
VWFA (p.u.)

0.6 1

0 0.9
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.18: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, Line 3 length = 50 km).
1.2 1
PWFB (p.u.)

QWFB (p.u.)

0.5 -0.5

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.3 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.65 0.9

0 0.6
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.19: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, Line 3 length = 50 km).

64
 600 40
Amplitude (p.u.) I Stator I Stator

Amplitude (p.u.)
Zoom In
300 20

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.08 0.5

Amplitude (p.u.)
T(GEN1-EXC) T(LP1-GEN1)
Amplitude (p.u.)

0.04 0.25

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.4 0.2
Amplitude (p.u.)
Amplitude (p.u.)

T(HP1-LP1) T(LP2-GEN2)

0.2 0.1

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.2
Amplitude (p.u.)

T(HP2-LP2)

0.1

0
0 20 40 60 80
Frequency (Hz)

Figure 3.20: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
Line 3 length = 50 km).

65
 T(GEN1-EXC) (p.u.) 1.5 150

T(LP1-GEN1) (p.u.)
-0.25 0

-2 -150
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

50 100
T(HP1-LP1) (p.u.)

T(LP2-GEN2) (p.u.)
0 0

-50 -100
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

35
T(HP2-LP2) (p.u.)

-35
0 1.5 3 4.5
Time (s)

Figure 3.21: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is not
activated, Line 3 length = 100 km).

66
 2.5 2
PWFA (p.u.)

QWFA (p.u.)
0.75 -0.5

-1 -3
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.8 1.15
VWFA (p.u.)

Vdc_WFA (p.u.)
0.9 0.95

0 0.75
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.22: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, Line 3 length = 100 km).
1.2 1
PWFB (p.u.)

QWFB (p.u.)

0.5 -0.75

-0.2 -2.5
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.6 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.8 0.85

0 0.5
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.23: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, Line 3 length = 100 km).

67
 600 40
I Stator

Amplitude (p.u.)
Amplitude (p.u.) I Stator
Zoom In
300 20

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.5 50
T(GEN1-EXC) T(LP1-GEN1)

Amplitude (p.u.)
Amplitude (p.u.)

0.25 25

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
20 40
T(HP1-LP1) T(LP2-GEN2)
Amplitude (p.u.)

Amplitude (p.u.)

10 20

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
12
Amplitude (p.u.)

T(HP2-LP2)

0
0 20 40 60 80
Frequency (Hz)

Figure 3.24: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is not activated,
Line 3 length = 100 km).

68
 T(GEN1-EXC) (p.u.) 0.3 5.5

T(LP1-GEN1) (p.u.)
0 1.25

-0.3 -3
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

3 6
T(HP1-LP1) (p.u.)

T(LP2-GEN2) (p.u.)
0.5 1.5

-2 -3
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

2.2
T(HP2-LP2) (p.u.)

0.6

-1
0 1.5 3 4.5
Time (s)

Figure 3.25: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
Line 3 length = 100 km).

69
 PWFA (p.u.) 2 1

QWFA (p.u.)
0.9 -0.1

-0.2 -1.2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.2 1.1

Vdc_WFA (p.u.)
VWFA (p.u.)

0.6 1

0 0.9
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.26: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, Line 3 length = 100 km).
1.2 1
PWFB (p.u.)

QWFB (p.u.)

0.5 -0.5

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.3 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.65 0.9

0 0.6
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.27: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, Line 3 length = 100 km).

70
 600 40
Amplitude (p.u.)

Amplitude (p.u.)
I Stator I Stator
Zoom In
300 20

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.05 0.25
T(GEN1-EXC) T(LP1-GEN1)

Amplitude (p.u.)
Amplitude (p.u.)

0.025 0.125

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.25 0.1
T(LP2-GEN2)
Amplitude (p.u.)

T(HP1-LP1)
Amplitude (p.u.)

0.125 0.05

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.1
T(HP2-LP2)
Amplitude (p.u.)

0.05

0
0 20 40 60 80
Frequency (Hz)

Figure 3.28: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
Line 3 length = 100 km).

71
 T(GEN1-EXC) (p.u.) 1 70

T(LP1-GEN1) (p.u.)
0 0

-1 -70
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

30 50
T(HP1-LP1) (p.u.)

T(LP2-GEN2) (p.u.)
0 0

-30 -50
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

15
T(HP2-LP2) (p.u.)

-15
0 1.5 3 4.5
Time (s)

Figure 3.29: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is not
activated, Line 3 length = 200 km).

72
 2.2 1.2
PWFA (p.u.)

QWFA (p.u.)
1 -0.2

-0.2 -1.6
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.5 1.1

Vdc_WFA (p.u.)
VWFA (p.u.)

0.75 1

0 0.9
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.30: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, Line 3 length = 200 km).
1.2 1
PWFB (p.u.)

QWFB (p.u.)

0.5 -0.5

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.3 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.65 0.95

0 0.7
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.31: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, Line 3 length = 200 km).

73
 600 40
Amplitude (p.u.) I Stator I Stator

Amplitude (p.u.)
Zoom In
300 20

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.3 30
T(GEN1-EXC)
Amplitude (p.u.)

T(LP1-GEN1)

Amplitude (p.u.)
0.15 15

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
12 20
T(HP1-LP1) T(LP2-GEN2)
Amplitude (p.u.)

Amplitude (p.u.)

6 10

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
6
T(HP2-LP2)
Amplitude (p.u.)

0
0 20 40 60 80
Frequency (Hz)

Figure 3.32: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is not activated,
Line 3 length = 200 km).

74
 T(GEN1-EXC) (p.u.) 0.3 5.5

T(LP1-GEN1) (p.u.)
0 1.25

-0.3 -3
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

3 6
T(HP1-LP1) (p.u.)

T(LP2-GEN2) (p.u.)
0.5 1.5

-2 -3
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

2.2
T(HP2-LP2) (p.u.)

0.6

-1
0 1.5 3 4.5
Time (s)

Figure 3.33: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
Line 3 length = 200 km).

75
 2.2 1
PWFA (p.u.)

QWFA (p.u.)
1 -0.1

-0.2 -1.2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.2 1.1

Vdc_WFA (p.u.)
VWFA (p.u.)

0.6 1

0 0.9
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.34: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, Line 3 length = 200 km).
1.2 1
PWFB (p.u.)

QWFB (p.u.)

0.5 -0.5

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.3 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.65 0.9

0 0.6
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.35: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, Line 3 length = 200 km).

76
 600 40

Amplitude (p.u.)
I Stator I Stator
Amplitude (p.u.)

Zoom In
300 20

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.06 0.5
T(GEN1-EXC) T(LP1-GEN1)

Amplitude (p.u.)
Amplitude (p.u.)

0.03 0.25

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.2 0.3
T(HP1-LP1) T(LP2-GEN2)
Amplitude (p.u.)

Amplitude (p.u.)

0.1 0.15

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.15
T(HP2-LP2)
Amplitude (p.u.)

0.075

0
0 20 40 60 80
Frequency (Hz)

Figure 3.36: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
Line 3 length = 200 km).

77
3.3.2 Effect of wind farm B rating

The effect of wind farm B rating on the performance of Supplemental control 1, is explored
at two different ratings, namely 300 MW and 400 MW. The results of this study are presented in
the following two groups of figures:

Group A: Wind farm B rating = 300 MW, (Figures 3.37, 3.38, 3.39, 3.40, 3.41, 3.42, 3.43, 3.44)

Group B: Wind farm B rating = 400 MW, (Figures 3.45, 3.46, 3.47, 3.48, 3.49, 3.50, 3.51, 3.52)

Table 3.4: Transfer functions of Supplemental control 1 for Group B (Wind farm B rating = 400
MW).

Modal speeds Transfer function

G1, Δωm0, G2, Δωm0 0

0.1𝑠 + 300
G1, Δωm1 𝐺𝜔1 (𝑠) = 2.5
𝑠+1
0.01𝑠 + 1000
G1, Δωm2 𝐺𝜔2 (𝑠) = −25
10𝑠 + 1
𝑠 + 700
G1, Δωm3 𝐺𝜔3 (𝑠) = 2.5
𝑠+1
𝑠 + 250
G2, Δωm1 𝐺𝜔1 (𝑠) = 25
𝑠+1
0.01𝑠 + 500
G2, Δωm2 𝐺𝜔2 (𝑠) = 2.5
𝑠+1
𝑠
Washout filter 𝐺 (𝑠 ) =
𝑠 + 10
62.83𝑠
Band-Pass filter 𝐺 (𝑠 ) =
𝑠 2 + 62.83𝑠 + 35500
𝑠 + 250
Lead-Lag compensator 𝐺 (𝑠 ) =
𝑠+1

UTotal_max, UTotal_min 0.33, -0.33

78
 T(GEN1-EXC) (p.u.) 0.8 70

T(LP1-GEN1) (p.u.)
0 0

-0.8 -70
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

30 50

T(LP2-GEN2) (p.u.)
T(HP1-LP1) (p.u.)

0 0

-30 -50
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

20
T(HP2-LP2) (p.u.)

-20
0 1.5 3 4.5
Time (s)

Figure 3.37: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is not
activated, wind farm B rating = 300 MW).

79
 2 1
PWFA (p.u.)

QWFA (p.u.)
0.9 -0.1

-0.2 -1.2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.4 1.1

Vdc_WFA (p.u.)
VWFA (p.u.)

0.7 1

0 0.9
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.38: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, wind farm B rating = 300 MW).
1.2 1
PWFB (p.u.)

QWFB (p.u.)

0.5 -0.5

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.3 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.65 0.925

0 0.65
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.39: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, wind farm B rating = 300 MW).

80
 600 40
Amplitude (p.u.) I Stator I Stator

Amplitude (p.u.)
Zoom In
300 20

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.3 30
T(GEN1-EXC) T(LP1-GEN1)

Amplitude (p.u.)
Amplitude (p.u.)

0.15 15

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
12 20
T(HP1-LP1) T(LP2-GEN2)
Amplitude (p.u.)

Amplitude (p.u.)

6 10

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
8
T(HP2-LP2)
Amplitude (p.u.)

0
0 20 40 60 80
Frequency (Hz)

Figure 3.40: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is not activated,
wind farm B rating = 300 MW).

81
 T(GEN1-EXC) (p.u.) 0.2 3.5

T(LP1-GEN1) (p.u.)
0 1

-0.2 -1.5
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

2 3
T(HP1-LP1) (p.u.)

T(LP2-GEN2) (p.u.)
0.5 1.1

-1 -0.8
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

1.5
T(HP2-LP2) (p.u.)

0.5

-0.5
0 1.5 3 4.5
Time (s)

Figure 3.41: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
wind farm B rating = 300 MW).

82
 1.8 1
PWFA (p.u.)

QWFA (p.u.)
0.8 0

-0.2 -1
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.2 1.1

Vdc_WFA (p.u.)
VWFA (p.u.)

0.6 1

0 0.9
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.42: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, wind farm B rating = 300 MW).
1.2 1
QWFB (p.u.)
PWFB (p.u.)

0.35 -0.7

-0.5 -2.4
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.3 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.65 0.95

0 0.7
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.43: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, wind farm B rating = 300 MW).

83
 600 40
I Stator

Amplitude (p.u.)
Amplitude (p.u.) I Stator
Zoom In
300 20

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.03 0.1
T(GEN1-EXC)
Amplitude (p.u.)

T(LP1-GEN1)

Amplitude (p.u.)
0.015 0.05

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.15 0.1
Amplitude (p.u.)

T(HP1-LP1) T(LP2-GEN2)
Amplitude (p.u.)

0.075 0.05

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.05
T(HP2-LP2)
Amplitude (p.u.)

0.025

0
0 20 40 60 80
Frequency (Hz)

Figure 3.44: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
wind farm B rating = 300 MW).

84
 T(GEN1-EXC) (p.u.) 0.6 50

T(LP1-GEN1) (p.u.)
0 0

-0.6 -50
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

20 40
T(HP1-LP1) (p.u.)

T(LP2-GEN2) (p.u.)
0 0

-20 -40
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

15
T(HP2-LP2) (p.u.)

-15
0 1.5 3 4.5
Time (s)

Figure 3.45: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is not
activated, wind farm B rating = 400 MW).

85
 1.8 1
PWFA (p.u.)

QWFA (p.u.)
0.8 0

-0.2 -1
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.2 1.1
VWFA (p.u.)

Vdc_WFA (p.u.)
0.6 1

0 0.9
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.46: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, wind farm B rating = 400 MW).
1.2 1
QWFB (p.u.)
PWFB (p.u.)

0.35 -0.5

-0.5 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.2 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.6 1

0 0.8
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.47: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is not activated, wind farm B rating = 400 MW).

86
 600 40
Amplitude (p.u.) I Stator I Stator

Amplitude (p.u.)
Zoom In
300 20

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.3 30
T(GEN1-EXC) T(LP1-GEN1)

Amplitude (p.u.)
Amplitude (p.u.)

0.15 15

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
12 28
T(HP1-LP1) T(LP2-GEN2)
Amplitude (p.u.)

Amplitude (p.u.)

6 14

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
8
T(HP2-LP2)
Amplitude (p.u.)

0
0 20 40 60 80
Frequency (Hz)

Figure 3.48: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is not activated,
wind farm B rating = 400 MW).

87
 T(GEN1-EXC) (p.u.) 0.2 3.5

T(LP1-GEN1) (p.u.)
0 1

-0.2 -1.5
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

2 3
T(HP1-LP1) (p.u.)

T(LP2-GEN2) (p.u.)
0.5 1.1

-1 -0.8
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

1.5
T(HP2-LP2) (p.u.)

0.5

-0.5
0 1.5 3 4.5
Time (s)

Figure 3.49: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
wind farm B rating = 400 MW).

88
 1.8 1
PWFA (p.u.)

QWFA (p.u.)
0.8 0

-0.2 -1
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.2 1.1

Vdc_WFA (p.u.)
VWFA (p.u.)

0.6 1

0 0.9
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.50: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, wind farm B rating = 400 MW).
1.2 1
PWFB (p.u.)

QWFB (p.u.)

0.35 -0.7

-0.5 -2.4
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.3 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.65 0.95

0 0.7
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.51: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 1 is activated, wind farm B rating = 400 MW).

89
 600 40
Amplitude (p.u.) I Stator I Stator

Amplitude (p.u.)
Zoom In
300 20

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.02 0.1
Amplitude (p.u.)

T(GEN1-EXC) T(LP1-GEN1)

Amplitude (p.u.)
0.01 0.05

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.1 0.08
Amplitude (p.u.)

T(HP1-LP1) T(LP2-GEN2)
Amplitude (p.u.)

0.05 0.04

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.16
T(HP2-LP2)
Amplitude (p.u.)

0.08

0
0 20 40 60 80
Frequency (Hz)

Figure 3.52: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 1 is activated,
wind farm B rating = 400 MW).

90
 In each group, the turbine-generator shaft torsional torques, wind farms A and B active and
reactive powers, terminal voltage and the BtB dc voltage as well as the frequency spectrums of the
stator current of the DFIG wind turbines and the turbine-generator shaft torsional torques are
shown for the cases when Supplemental control 1 is disabled and activated respectively. Moreover,
the transfer functions of Supplemental control 1 in Group A are the same as those in Table 3.1.
The transfer functions of Supplemental control 1 in Group B are given in Tables 3.4.

The comparisons between the two sets of figures (Set 1: Supplemental control 1 is disabled
and Set 2: Supplemental control 1 is activated) in Groups A and B show the effectiveness of the
supplemental control in mitigating SSR and SSI at different ratings of wind farm B.

3.3.3 Effect of the fault type

The effect of the fault type on the performance of Supplemental control 1 is examined by
applying a double line-to-ground fault at the same fault location (Line 5). The results of this
study are presented in the following two groups of figures:

Group A: Supplemental control 1 is disabled, (Figures 3.53, 3.54, 3.55, 3.56)

Group B: Supplemental control 1 is activated, (Figures 3.57, 3.58, 3.59, 3.60)

In each group, the turbine-generator shaft torsional torques, wind farms A and B active and
reactive powers, terminal voltage and the BtB dc voltage as well as the frequency spectrums of the
stator current of the DFIG wind turbines and the turbine-generator shaft torsional torques are
shown. Moreover, the transfer functions of Supplemental control 1 in Group B are the same as
those in Table 3.1.

The comparisons between the two sets of figures in Groups A and B (Set 1: Supplemental
control 1 is disabled and Set 2: Supplemental control 1 is activated) demonstrate the effectiveness
of the supplemental control in mitigating SSR and SSI during and after clearing an unsymmetrical
fault. It is worth noting here that the comparison between Figures 3.5, 3.6, 3.7 and Figures 3.53,
3.54, 3.55 shows that the responses of the turbine-generator shaft torsional torques, wind farm A
and B real and reactive powers as well as terminal voltages are “almost” the same during the three-
phase and double line-to-ground faults.

91
 T(GEN1-EXC) (p.u.) 1 80

T(LP1-GEN1) (p.u.)
0 0

-1 -80
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

30 60
T(HP1-LP1) (p.u.)

T(LP2-GEN2) (p.u.)
0 0

-30 -60
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

20
T(HP2-LP2) (p.u.)

-20
0 1.5 3 4.5
Time (s)

Figure 3.53: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, double
line-to-ground fault on Line 5 (60% compensation degree, supplemental control 1
is not activated).

92
 2.2 1.2
PWFA (p.u.)

QWFA (p.u.)
1 -0.15

-0.2 -1.5
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.5 1.1

Vdc_WFA (p.u.)
VWFA (p.u.)

0.75 1

0 0.9
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.54: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, double line-to-ground fault on Line 5 (60%
compensation degree, supplemental control 1 is not activated).
1.2 1
PWFB (p.u.)

QWFB (p.u.)

0.5 -0.5

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.4 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.7 0.95

0 0.7
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.55: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, double line-to-ground fault on Line 5 (60%
compensation degree, supplemental control 1 is not activated).

93
 600 40
I Stator I Stator

Amplitude (p.u.)
Amplitude (p.u.)
Zoom In
300 20

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.3 30
T(GEN1-EXC) T(LP1-GEN1)
Amplitude (p.u.)

Amplitude (p.u.)
0.15 15

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
12 20
T(HP1-LP1) T(LP2-GEN2)
Amplitude (p.u.)

Amplitude (p.u.)

6 10

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
8
T(HP2-LP2)
Amplitude (p.u.)

0
0 20 40 60 80
Frequency (Hz)

Figure 3.56: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, double line-to-
ground fault on Line 5 (60% compensation degree, supplemental control 1 is not
activated).

94
 T(GEN1-EXC) (p.u.) 0.2 4

T(LP1-GEN1) (p.u.)
0 1

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

2 3
T(HP1-LP1) (p.u.)

T(LP2-GEN2) (p.u.)
0.5 1.1

-1 -0.8
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

1.5
T(HP2-LP2) (p.u.)

0.5

-0.5
0 1.5 3 4.5
Time (s)

Figure 3.57: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, double
line-to-ground fault on Line 5 (60% compensation degree, supplemental control 1
is activated).

95
 1.8 1
PWFA (p.u.)

QWFA (p.u.)
0.8 -0.1

-0.2 -1.2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.2 1.1

Vdc_WFA (p.u.)
VWFA (p.u.)

0.6 1

0 0.9
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.58: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, double line-to-ground fault on Line 5 (60%
compensation degree, supplemental control 1 is activated).
1.2 1
PWFB (p.u.)

QWFB (p.u.)

0.5 -0.5

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.3 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.65 0.9

0 0.6
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.59: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, double line-to-ground fault on Line 5 (60%
compensation degree, supplemental control 1 is activated).

96
 600 40
I Stator I Stator

Amplitude (p.u.)
Amplitude (p.u.)
Zoom In
300 20

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.03 0.2
T(GEN1-EXC) T(LP1-GEN1)
Amplitude (p.u.)

Amplitude (p.u.)
0.015 0.1

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.2 0.1
Amplitude (p.u.)

T(HP1-LP1) T(LP2-GEN2)
Amplitude (p.u.)

0.1 0.05

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.06
T(HP2-LP2)
Amplitude (p.u.)

0.03

0
0 20 40 60 80
Frequency (Hz)

Figure 3.60: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, double line-to-
ground fault on Line 5 (60% compensation degree, supplemental control 1 is
activated).

97
3.4 Performance of Supplemental Control 2 in damping SSR and SSI Oscillations

Figure 3.61 shows the turbine-generator shaft torsional torque time responses during and
after clearing a 3-cycle, three-phase fault on line 5 for the case when Supplemental control 2 is
activated. Figures 3.62 and 3.62 illustrate respectively the time responses of wind farms A and B
active and reactive powers, terminal voltage and the BtB dc voltage for the same case. Moreover,
Figure 3.64 shows the frequency spectrums of the stator current of the DFIG wind turbines and
the turbine-generator shaft torsional torques for the same study case. Furthermore, the transfer
functions of Supplemental control 2 are the same as those in Table 3.1. The comparison between
the two groups of figures (Figures 3.5, 3.6, 3.7, 3.8) and (Figures 3.61, 3.62, 3.63, 3.64) establishes
the effectiveness of Supplemental control 2 in damping the torsional torques in all turbine-
generator shaft sections as well as in mitigating SSI in wind farm A.

0.2 T(LP1-GEN1) 4
T(GEN1-EXC)

(p.u.)
(p.u.)

0 1

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
2 3
T(HP1-LP1) (p.u.)

T(LP2-GEN2)
(p.u.)

0.5 1.1

-1 -0.8
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.5
T(HP2-LP2) (p.u.)

0.5

-0.5
1.5 3 0
4.5
Time (s)
Figure 3.61: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (60% compensation degree, supplemental control 2 is
activated).

98
 1.8 1
PWFA (p.u.)

QWFA (p.u.)
0.8 -0.1

-0.2 -1.2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.2 1.1

Vdc_WFA (p.u.)
VWFA (p.u.)

0.6 1

0 0.9
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.62: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 2 is activated).
1.2 1
PWFB (p.u.)

QWFB (p.u.)

0.5 -0.5

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.3 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.65 0.9

0 0.6
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.63: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (60% compensation
degree, supplemental control 2 is activated).

99
 600 40
I Stator I Stator

Amplitude (p.u.)
Amplitude (p.u.)

Zoom In
300 20

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.03 0.3
T(GEN1-EXC)

Amplitude (p.u.)
T(LP1-GEN1)
Amplitude (p.u.)

0.015 0.15

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.2 0.14
Amplitude (p.u.)

Amplitude (p.u.)

T(HP1-LP1) T(LP2-GEN2)

0.1 0.07

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.06
T(HP2-LP2)
Amplitude (p.u.)

0.03

0
0 20 40 60 80
Frequency (Hz)

Figure 3.64: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, supplemental control 2 is activated).

100
3.4.1 Effect of the fault location

The effect of the fault location on the performance of Supplemental control 2 is examined
by applying a three-cycle, three-phase fault on Line 4. The results of this study are presented in
the following two groups of figures:

Group A: Supplemental control 2 is disabled, (Figures 3.65, 3.66, 3.67, 3.68)

Group B: Supplemental control 2 is activated, (Figures 3.69, 3.70, 3.71, 3.72)

In each group, the turbine-generator shaft torsional torques, wind farms A and B active and
reactive powers, terminal voltage and the BtB dc voltage as well as the frequency spectrums of the
stator current of the DFIG wind turbines and the turbine-generator shaft torsional torques are
shown. Moreover, the transfer functions of Supplemental control 2 in Group B are given in Table
3.5.

Table 3.5: Transfer functions of Supplemental control 2 (Wind farm B rating = 200 MW, three-
phase fault on Line 4).

Modal speeds Transfer function

G1, Δωm0 0
0.1𝑠 + 1
G1, Δωm1 𝐺𝜔1 (𝑠) = 20000
𝑠 + 100
0.1𝑠 + 100
G1, Δωm2 𝐺𝜔2 (𝑠) = −2.5
𝑠+1
𝑠 + 2000
G1, Δωm3 𝐺𝜔3 (𝑠) = 25
𝑠 + 0.1
G2, Δωm0 0
0.1𝑠 + 5
G2, Δωm1 𝐺𝜔1 (𝑠) = 20000
𝑠 + 100
𝑠+5
G2, Δωm2 𝐺𝜔2 (𝑠) = −15000
𝑠+1
𝑠
Washout filter 𝐺 (𝑠 ) =
𝑠 + 10
62.83𝑠
Band-Pass filter 𝐺 (𝑠 ) = 2
𝑠 + 62.83𝑠 + 35500
𝑠 + 250
Lead-Lag compensator 𝐺 (𝑠 ) =
𝑠+1
UTotal_max, UTotal_min 0.33, -0.33 (same for both FFC and DFIG wind turbines)

101
 T(GEN1-EXC) (p.u.) 0.2 5

T(LP1-GEN1) (p.u.)
0 1.25

-0.2 -2.5
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

2.3 6
T(HP1-LP1) (p.u.)

T(LP2-GEN2) (p.u.)
0.4 1.5

-1.5 -3
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

2
T(HP2-LP2) (p.u.)

0.5

-1
0 1.5 3 4.5
Time (s)

Figure 3.65: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 4 (60% compensation degree, supplemental control 2 is not
activated).

102
 2.6 3
PWFA (p.u.)

QWFA (p.u.)
0.05 -4.5

-2.5 -12
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
3 1.3

Vdc_WFA (p.u.)
VWFA (p.u.)

1.5 0.8

0 0.3
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.66: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 4 (60% compensation
degree, supplemental control 2 is not activated).
1.2 1
PWFB (p.u.)

QWFB (p.u.)

0.6 -0.5

0 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.2 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.6 1

0 0.8
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.67: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 4 (60% compensation
degree, supplemental control 2 is not activated).

103
 600 100
I Stator I Stator

Amplitude (p.u.)
Amplitude (p.u.)

Zoom In
300 50

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.1 1.4
Amplitude (p.u.)

Amplitude (p.u.)
T(GEN1-EXC) T(LP1-GEN1)

0.05 0.7

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.6 3
Amplitude (p.u.)
Amplitude (p.u.)

T(HP1-LP1) T(LP2-GEN2)

0.3 1.5

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.8
Amplitude (p.u.)

T(HP2-LP2)

0.4

0
0 20 40 60 80
Frequency (Hz)

Figure 3.68: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 4 (60% compensation degree, supplemental control 2 is not activated).

104
 T(GEN1-EXC) (p.u.) 0.2 5

T(LP1-GEN1) (p.u.)
0 1.5

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

2 5
T(HP1-LP1) (p.u.)

T(LP2-GEN2) (p.u.)
0.5 1.5

-1 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

1.5
T(HP2-LP2) (p.u.)

0.5

-0.5
0 1.5 3 4.5
Time (s)

Figure 3.69: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 4 (60% compensation degree, supplemental control 2 is
activated).

105
 2.6 1.2
PWFA (p.u.)

QWFA (p.u.)
1.2 -0.3

-0.2 -1.8
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.5 1.1

Vdc_WFA (p.u.)
VWFA (p.u.)

0.75 1

0 0.9
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.70: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 4 (60% compensation
degree, supplemental control 2 is activated).
1.2 1
PWFB (p.u.)

QWFB (p.u.)

0.6 -0.5

0 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.3 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.65 0.9

0 0.6
0 1.5
3 4.5 0 1.5 3 4.5
Time (s) Time (s)
Figure 3.71: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 4 (60% compensation
degree, supplemental control 2 is activated).

106
 600 100
I Stator I Stator

Amplitude (p.u.)
Amplitude (p.u.)
Zoom In
300 50

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.05 0.3
T(GEN1-EXC)

Amplitude (p.u.)
Amplitude (p.u.)

T(LP1-GEN1)

0.025 0.15

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.3 0.4
Amplitude (p.u.)

Amplitude (p.u.)

T(HP1-LP1) T(LP2-GEN2)

0.15 0.2

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.12
T(HP2-LP2)
Amplitude (p.u.)

0.06

0
0 20 40 60 80
Frequency (Hz)

Figure 3.72: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 4 (60% compensation degree, supplemental control 2 is activated).

107
 As it can be seen from Figure 2.1, clearing a system fault on line L4 “virtually” isolates
wind farm A from wind farm B and the turbine-generators. Wind farm A is now radially connected
to the infinite-bus system through a series capacitor compensated transmission line (Line 2). This
is a favorable condition for the occurrence of SSI. As it can be seen from Figure 3.66, wind farm
A real and reactive powers exhibit severe oscillations after fault clearing that result in the tripping
of the wind farm by the protective system at t = 0.75 seconds.

Regarding the turbine-generator shaft torsional torques, Figure 3.65 shows that these
torques exhibit either decaying, sustained or growing oscillations. These oscillations, however,
have no adverse impact on the performance of wind farm B as it can be seen from Figure 3.67.

The comparison between the figures in Groups A and B demonstrates the effectiveness of
Supplemental control 2 in mitigating SSR and SSI after clearing a three-phase fault at a different
location.

3.4.2 Effect of the compensation degree

The effect of the compensation degree of transmission Lines 1 and 2 on the performance
of Supplemental control 2 is examined by changing it from 60% to 50%. Moreover, the
disturbance is a three-cycle, three-phase fault on Line 5. The results of this study are presented in
the following two groups of figures:

Group A: Supplemental control 2 is disabled, (Figures 3.73, 3.74, 3.75, 3.76)

Group B: Supplemental control 2 is activated, (Figures 3.77, 3.78, 3.79, 3.80)

In each group, the turbine-generator shaft torsional torques, wind farms A and B active and
reactive powers, terminal voltage and the BtB dc voltage as well as the frequency spectrums of the
stator current of the DFIG wind turbines and the turbine-generator shaft torsional torques are
shown. Moreover, the transfer functions of Supplemental control 2 in Groups B are given in Table
3.6.

The comparison between Figures 3.5, 3.8 and Figures 3.73, 3.76 shows that changing the
series capacitive compensation degrees results in changing the contributions of the torsional modes
to the induced torsional torques in the different shaft sections of the turbine-generators. As it can

108
be seen from Figure 3.73, some of the turbine-generator shaft torsional torques exhibit growing
oscillations that indicate the presence of SSR. On the other hand, Figure 3.74 clearly shows the
absence of SSI in wind farm A. Moreover, Figure 3.75 shows that the adverse impact of SSR is
not extended to affect the performance of wind farm B. The comparison between the figures in
Groups A and B demonstrates the effectiveness of the supplemental control in mitigating SSR at
the 50% compensation degree of Lines 1 and 2.

Table 3.6: Transfer functions of Supplemental control 2 (Three-phase fault on Line 5, 50%
compensation).

Modal speeds Transfer function

G1, Δωm0 0

𝑠 + 2000
G1, Δωm1 𝐺𝜔1 (𝑠) = 2.5
𝑠+1
𝑠 + 1000
G1, Δωm2 𝐺𝜔2 (𝑠) = −2.5
𝑠+1
𝑠 + 1000
G1, Δωm3 𝐺𝜔3 (𝑠) = 2.5
𝑠+1

G2, Δωm0 0

𝑠 + 500
G2, Δωm1 𝐺𝜔1 (𝑠) = 25
𝑠+1
𝑠 + 1250
G2, Δωm2 𝐺𝜔2 (𝑠) = 25
𝑠+1
𝑠
Washout filter 𝐺 (𝑠 ) =
𝑠 + 10
62.83𝑠
Band-Pass filter 𝐺 (𝑠 ) =
𝑠 2 + 62.83𝑠 + 35500
𝑠 + 250
Lead-Lag compensator 𝐺 (𝑠 ) =
𝑠+1
0.33, -0.33 (same for both FFC and DFIG wind
UTotal_max, UTotal_min
turbines)

109
 T(GEN1-EXC) (p.u.) 0.2 4

T(LP1-GEN1) (p.u.)
0 1

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

2.3 5
T(HP1-LP1) (p.u.)

T(LP2-GEN2) (p.u.)
0.4 1.5

-1.5 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

2
T(HP2-LP2) (p.u.)

0.5

-1
0 1.5 3 4.5
Time (s)

Figure 3.73: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (50% compensation degree, supplemental control 2 is not
activated).

110
 2 1
PWFA (p.u.)

QWFA (p.u.)
0.9 -0.1

-0.2 -1.2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.2 1.1
VWFA (p.u.)

Vdc_WFA (p.u.)
0.6 1

0 0.9
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.74: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (50% compensation
degree, supplemental control 2 is not activated).
1.2 1
PWFB (p.u.)

QWFB (p.u.)

0.5 -0.5

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.3 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.65 1

0 0.8
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.75: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (50% compensation
degree, supplemental control 2 is not activated).

111
 600 40
I Stator I Stator

Amplitude (p.u.)
Amplitude (p.u.)
Zoom In
300 20

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.12 1.2
T(GEN1-EXC) T(LP1-GEN1)
Amplitude (p.u.)

Amplitude (p.u.)
0.06 0.6

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.6 2
T(HP1-LP1) T(LP2-GEN2)
Amplitude (p.u.)

Amplitude (p.u.)

0.3 1

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.6
T(HP2-LP2)
Amplitude (p.u.)

0.3

0
0 20 40 60 80
Frequency (Hz)

Figure 3.76: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (50% compensation degree, supplemental control 2 is not activated).

112
 T(GEN1-EXC) (p.u.) 0.2 4

T(LP1-GEN1) (p.u.)
0 1

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

2.8 4
T(HP1-LP1) (p.u.)

T(LP2-GEN2) (p.u.)
0.5 1.5

-1.8 -1
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

2
T(HP2-LP2) (p.u.)

0.5

-1
0 1.5 3 4.5
Time (s)

Figure 3.77: Turbine-generator shaft torsional torques during and after clearing a 3-cycle, three-
phase fault on Line 5 (50% compensation degree, supplemental control 2 is
activated).

113
 2.2 1
PWFA (p.u.)

QWFA (p.u.)
1 -0.5

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.2 1.1

Vdc_WFA (p.u.)
VWFA (p.u.)

0.6 1

0 0.9
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.78: Wind farm A real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (50% compensation
degree, supplemental control 2 is activated).
1.2 1
PWFB (p.u.)

QWFB (p.u.)

0.5 -0.5

-0.2 -2
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)
1.3 1.2
VdcWFB (p.u.)
VWFB (p.u.)

0.65 0.9

0 0.6
0 1.5 3 4.5 0 1.5 3 4.5
Time (s) Time (s)

Figure 3.79: Wind farm B real and reactive powers, terminal voltage and dc capacitor voltage
during and after clearing a 3-cycle, three-phase fault on Line 5 (50% compensation
degree, supplemental control 2 is activated).

114
 600 40
I Stator I Stator

Amplitude (p.u.)
Zoom In
Amplitude (p.u.)

300 20

0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Frequency (Hz) Frequency (Hz)
0.05 0.5
T(GEN1-EXC) T(LP1-GEN1)

Amplitude (p.u.)
Amplitude (p.u.)

0.025 0.25

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.2 0.4
T(HP1-LP1) T(LP2-GEN2)
Amplitude (p.u.)

Amplitude (p.u.)

0.1 0.2

0 0
0 20 40 60 80 0 20 40 60 80
Frequency (Hz) Frequency (Hz)
0.16
T(HP2-LP2)
Amplitude (p.u.)

0.08

0
0 20 40 60 80
Frequency (Hz)

Figure 3.80: Frequency spectrums of the stator current of the DFIG wind turbine and the turbine-
generator shaft torsional torques during and after clearing a 3-cycle, three-phase
fault on Line 5 (50% compensation degree, supplemental control 2 is activated).

115
3.5 Summary

In this chapter, two supplemental controls of FFC- and DFIG-based wind farms for
mitigating SSR and SSI are presented. The effectiveness of these controllers are investigated
through several case studies of time-domain simulations for different fault type, fault location,
transmission line compensation degrees as well as the rating and location of the FFC-based wind
farm. The main conclusions drawn from the results of these studies are presented in the next
chapter.

116
 4. SUMMARY AND CONCLUSIONS

4.1 Summary

Transmission system capability limitations and the ways to overcome them are challenging
problems facing power system engineers. The reasons for transmission system limitations extend
from steady-state and thermal considerations to transient and dynamic stability of the power
system. It can be due to unfavorable power flow pattern in the transmission system where some
of the transmission lines may be very close to their thermal limits while other lines may have
unnecessarily large thermal margins. Other reasons are transient and dynamic stability
considerations, which impose limits on the power that can be transmitted while ensuring that the
power system will be able to regain a new stable state following any expected disturbance. Due
to these limitations, transmission lines are often loaded to levels below of their thermal capability
and, in order to increase their loadability limit, some measures must be adopted.

Series capacitive compensation is the most economical way for increasing the transmission
capacity and improving power system transient stability. However, subsynchronous resonance
(SSR) is one of the major obstacles for the wide spread of high degrees (60% and higher) of series
capacitor compensation. Recently, a new obstacle, namely subsynchronous interaction (SSI) has
been added to the list after the Zorillo Gulf wind farm incident in Texas in October 2009. SSI is
due to the interaction between large doubly fed induction generator (DFIG)-based wind farms and
series capacitor compensated transmission systems.

In integrated power systems incorporating series capacitor compensated transmission lines

and high penetration of wind energy conversion systems, especially DFIG-based wind farms, SSR
and SSI could occur concurrently as a result of some system contingences. One solution to SSR
and SSI is to avoid those series compensation degrees that excite the two phenomena. This is,
however, a difficult to implement due to the large number of variables that impact SSR and SSI
and may not be cost effective in view of reduced system capacity. Therefore, mitigating SSR

117
and SSI is an important area of research and development targeting at developing practical and
effective countermeasures.

This thesis investigates the potential use of variable speed wind energy conversion systems
(full-scale frequency converter (FFC) and doubly-fed induction generator (DFIG)-based wind
farms) for concurrent mitigation of SSR and SSI. SSR and SSI damping is achieved by
introducing supplemental control signals in the reactive power control loops of the grid side
converters of the FFC and DFIG wind turbines of large wind farms.

Chapter 1 introduces some fundamental benefits of series compensation of transmission

lines. Brief introductions to SSR and SSI are also presented. The objective of the research is also
presented in this chapter.

In Chapter 2, the system used for the investigations conducted in this thesis is described
and the detailed dynamic models of its individual components are also presented in this chapter.
The results of the digital time-domain simulations of a case study for the system during a three-
phase fault are presented at the end of this chapter.

Chapter 3 demonstrates the effectiveness of two supplemental controls in damping SSR

and SSI oscillations through time-domain simulation studies. The supplemental controls
performance at different system contingencies and operating conditions is also investigated.

4.2 Conclusions

The studies conducted in this thesis yield the following conclusions for the system under study:

1. Subsynchronous resonance and subsynchronous interaction do not necessarily occur

simultaneously, at all high degrees (50% and higher) of transmission line series capacitive
compensation.
2. The turbine-generator shaft torsional torques are not sinusoidal with a single frequency but
contain contributions from all the torsional modes. Moreover, the shaft section between
the generators and low-pressure turbines are subjected to the highest stresses.
3. Changing the series capacitive compensation degrees results in changing the contributions
of the torsional modes to the induced torsional torques in the different shaft sections of the
turbine-generators.

118
 4. The adverse impacts of SSR are more severe than those of SSI. Although FFC wind
turbines are “supposed” to be immune to SSI because their back-to-back converters
virtually isolate them from the system dynamics, the studies have shown that SSR can
cause FFC wind turbines to exhibit induction generator (IG) effect.
5. The electrical subsynchronous resonance frequency of the system (fn) is a function of the
capacitive reactances of Lines 1 and 2 and the equivalent inductive reactance of the system.
The SSI mode is the complement of this frequency (60 – fn). In all the reported case
studies in this thesis except for the interruption of Line 4, SSI mode is found to be 24.5 Hz.
In the case of the interruption of Line 4, SSI mode is 30.5 Hz. This is due to the change of
fe from 35.5 Hz to 29.5 Hz.
6. The proposed supplemental controls, namely Supplemental controls 1 and 2 have shown
to be effective in mitigating SSR and SSI at different series capacitive compensation
degrees, fault types and locations as well as wind farm B (FFC) ratings. Supplemental
control 2 is, however, more effective in mitigating SSR and SSI in the case of system faults
on Line 4 followed by subsequent line interruption as this action virtually splits the system
into two subsystems.

This thesis is a stepping-stone in the direction of more research on the simultaneous

mitigation of subsynchronous resonance and subsynchronous interaction. It is hoped that the
research work documented in this thesis would provide useful guidance for conducting more
studies and analyzing other technical issues that might be impacted by the simultaneous existence
of the two phenomena.

119
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122
 APPENDIX A

DATA OF THE SYSTEM UNDER STUDY

A.1 Synchronous Generators

Table A.1: Synchronous generator data

G1 G2

Rating, MVA 600 700

Rated voltage, kV 22 22

Armature resistance, ra , p.u. 0.0045 0.0045

Leakage reactance, xl , p.u. 0.14 0.12

Direct-axis synchronous reactance, xd , p.u. 1.65 1.54

Quadrature-axis synchronous reactance, xq , p.u. 1.59 1.50

Direct-axis transient reactance, x’d , p.u. 0.25 0.23

Quadrature-axis transient reactance, x’q , p.u. 0.46 0.42

Direct-axis subtransient reactance, x”d , p.u. 0.20 0.18

Quadrature-axis subtransient reactance, x”q , p.u. 0.20 0.18

Direct-axis transient open-circuit time constant, T’do , s 4.50 3.70

Quadrature-axis transient open-circuit time constant, T’qo ,s 0.67 0.43

Direct-axis subtransient open-circuit time constant, T”do , s 0.04 0.04

Quadrature-axis subtransient open-circuit time constant, T”qo ,s 0.09 0.06

Zero-sequence reactance, xo , p.u. 0.42 0.36

123
A.2 Turbine-Generator Mechanical Shaft Data

Table A.2: Turbine-Generator 1

Damping D Spring Constant

Mass Shaft Section Inertia M (p.u.)
(p.u./p.u. speed) k (p.u./rad)

EXC 5.1953028 0.001382

GEN1-EXC 3.7414773

GEN1 661.91839 0.176111

LP1-GEN1 83.497161

LP1 1167.2677 0.310564

HP1-LP1 42.7159

HP1 187.49671 0.04988

Turbine-Generator 2

Damping D Spring Constant

Mass Shaft Section Inertia M (p.u.)
(p.u./p.u. speed) k (p.u./rad)

GEN2 1078.3791 0.0573

LP2-GEN2 114.03409

LP2 1192.9066 0.0634

HP2-LP2 145.15423

HP2 353.93443 0.0188

124
A.3 Transformers

Table A.3: Transformer data

T3 T4
T1 T2
(Wind farm (Wind farm
(Generator 1) (Generator 2)
A) B)
Rating, MVA 600 700 525 225

Rated voltage, kV 22/500 22/500 34.5/500 34.5/500

Resistance, rT , p.u. 0.0012 0.0028 0.005 0.005

Leakage reactance, xT , p.u. 0.12 0.28 0.15 0.15

A.4 Wind Farm A (DFIG-Based Wind Farm) and Wind Farm B (FFC-Based Wind
Farm)
Table A.4: Wind farm A and B Parameters

Wind Farm A Wind Farm B

Number of wind turbine generators 300 100

System frequency, Hz 60 60

Rated capacity of each wind farm generator, MVA 1.67 2.22

Generator rated voltage, kV 0.575 0.575

DC nominal voltage, V 1150 1150

Number of poles 6 6

Average wind speed, m/s 11 15

Rated capacity of turbine, MW 1.5 2.0

125
A.5 Modal Speed Calculation Data
Turbine-Generator 1

∆𝜔𝑚 = (∆𝜔0 ∆𝜔1 ∆𝜔2 ∆𝜔3 )𝑇

∆𝜔 = (∆𝜔𝐻𝑃 ∆𝜔𝐿𝑃 ∆𝜔 ∆𝜔𝐸𝑋𝐶 )𝑇

0.5  0.6306 0.8959  0.0005
0.5  0.1629  0.2514 0.0012 
Q1  
0.5 0.4611 0.1872  0.0097
 
0.5 0.6026 0.3148 1 

Turbine-Generator 2

∆𝜔𝑚 = (∆𝜔0 ∆𝜔1 ∆𝜔2 )𝑇

∆𝜔 = (∆𝜔𝐻𝑃 ∆𝜔𝐿𝑃 ∆𝜔)𝑇

 0.9333  0.6592 0.5774

Q2   0.3499  0.3872 0.5774
 0.0808 0.6446 0.5774
 

A.6 DFIG Data

1.5 MW, 0.575 kV, 60 Hz
Rs = 0.023 p.u., Lls = 0.18 p.u., Lmd = 2.9 p.u., Lmq = 2.9 p.u.,
𝑝.𝑢.
R’r = 0.016 p.u., L’lr = 0.16 p.u., Llr = 0.16 p.u., Ht = 4 sec, Hg = 0.71 sec, Kshaft = 1.2 𝑟𝑎𝑑
,

BtB dc capacitor = 0.01 F, dc voltage = 1.15 kV.

A.7 FFC Data

2 MW, 0.575 kV, 60 Hz
Rs = 0.002 p.u., Xl = 0.05 p.u., Xo = 0.05 p.u., Xd = 0.6 p.u., Xq = 0.6 p.u.,
𝑝.𝑢.
Ht = 4 sec, Hg = 1 sec, Kshaft = 0.3 ,
𝑟𝑎𝑑

BtB dc capacitor = 0.01 F, dc voltage = 1.15 kV.

A.8 Transmission Lines

126
 All transmission lines have the same series impedance and shunt admittance per unit
length.
ZT.L.series= 0.0118+j0.3244 Ω/km
YT.L.shunt= 5.0512 μs/km
Transmission voltage = 500 kV

A.9 System Loads

Total Loads (L) = 300 MW + 150 MVAR

127
 APPENDIX B

SUPPLEMENTAL CONTROL OUTPUT SIGNALS FOR THE CASE

STUDIES
Figure B.1 to B.10 illustrate supplemental control 1 and 2 output signals USSR, USSI and US
for the case studies reported in Chapters 2 and 3.

B.1 Supplemental Control 1 Output Signals

0.5
USSR (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

0.5
USSI (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

0.5
US (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

Figure B.1: Supplemental control 1 output signals during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, wind farm B rating = 200 MW).

128
B.2 Supplemental Control 1 Output Signals (Line 3 length = 50 km)

1.5
USSR (p.u.)

-1.5
0 1.5 Time (s) 3 4.5

0.5
USSI (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

0.5
US (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

Figure B.2: Supplemental control 1 output signals during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, Line 3 length = 50 km).

129
B.3 Supplemental Control 1 Output Signals (Line 3 length = 100 km)

0.5
USSR (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

0.5
USSI (p.u.)

-0.5
0 1.5 3 4.5
Time (s)

0.5
US (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

Figure B.3: Supplemental control 1 output signals during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, Line 3 length = 100 km).

130
B.4 Supplemental Control 1 Output Signals (Line 3 length = 200 km)

1.5
USSR (p.u.)

-1.5
0 1.5 Time (s) 3 4.5

0.5
USSI (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

0.5
US (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

Figure B.4: Supplemental control 1 output signals during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, Line 3 length = 200 km).

131
B.5 Supplemental Control 1 Output Signals (Wind farm B rating = 300 MW)

0.5
USSR (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

0.5
USSI (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

0.5
US (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

Figure B.5: Supplemental control 1 output signals during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, wind farm B rating = 300 MW).

132
B.6 Supplemental Control 1 Output Signals (Wind farm B rating = 400 MW)

0.4
USSR (p.u.)

-0.4
0 1.5 Time (s) 3 4.5

0.5
USSI (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

0.5
US (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

Figure B.6: Supplemental control 1 output signals during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree, wind farm B rating = 400 MW).

133
B.7 Supplemental Control 1 Output Signals (Double Line-to-Ground Fault)

0.4
USSR (p.u.)

-0.4
0 1.5 Time (s) 3 4.5

0.5
USSI (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

0.5
US (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

Figure B.7: Supplemental control 1 output signals during and after clearing a 3-cycle, L-L-G fault
on Line 5 (60% compensation degree).

134
B.8 Supplemental Control 2 Output Signals

0.4
USSR (p.u.)

-0.4
0 1.5 Time (s) 3 4.5

0.5
USSI (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

Figure B.8: Supplemental control 2 output signals during and after clearing a 3-cycle, three-phase
fault on Line 5 (60% compensation degree).

135
B.9 Supplemental Control 2 Output Signals (Three-phase Fault on Line 4)
0.4
USSR (p.u.)

-0.4
0 1.5 3 4.5
Time (s)
0.5
USSI (p.u.)

-0.5
0 1.5 Time (s) 3 4.5

Figure B.9: Supplemental control 2 output signals during and after clearing a 3-cycle, three-phase
fault on Line 4 (60% compensation degree).

136
B.10 Supplemental Control 2 Output Signals (50% Compensation)
USSR (p.u.) 0.4

-0.4
0 1.5 3 4.5
Time (s)

0.5
USSI (p.u.)

-0.5
0 1.5 3 4.5
Time (s)

Figure B.10: Supplemental control 2 output signals during and after clearing a 3-cycle, three-
phase fault on Line 5 (50% compensation degree).

137