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Power conversion and control of wind energy systems
First Edition Bin Wu Digital Instant Download
Author(s): Bin Wu, Yongqiang Lang, Navid Zargari, Samir Kouro
ISBN(s): 9781118028988, 1118028988
Edition: First
File Details: PDF, 20.17 MB
Year: 2011
Language: english

POWER CONVERSION
AND CONTROL OF WIND
ENERGY SYSTEMS

R. Abhari
J. Anderson
G. W. Arnold
F. Cañavero
IEEE Press
445 Hoes Lane
Piscataway, NJ 08855
IEEE Press Editorial Board
Lajos Hanzo, Editor in Chief
M. El-Hawary
B-M. Hammerli
M. Lanzerotti
D.Jacobson
O. P. Malik
S. Nahavandi
T. Samad
G. Zobrist
Kenneth Moore, Director of IEEE Book and Information Services (BIS)

POWER CONVERSION
AND CONTROL OF WIND
ENERGY SYSTEMS
BinWu
Yongqiang Lang
Navid Zargari
Samir Kouro
IEEE
PRESS
SERIES |
GROWER
ENGINEERING
♦IEEE
IEEE Press
WILEY
A JOHN WILEY & SONS, INC., PUBLICATION

Copyright © 2011 by the Institute of Electrical and Electronics Engineers, Inc.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey. All rights reserved.
Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by
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Library of Congress Cataloging-in-Publication Data:
Power conversion and control of wind energy systems / Bin Wu ... [et al.],
p. cm. — (IEEE Press series on power engineering ; 74)
Summary: "The book covers a wide range of topics on wind energy conversion and control from the
electrical engineering aspect. It includes wind generators and modeling, power converters and modulation
schemes, operating principle of fixed and variable speed wind turbines, advanced generator control
schemes, active and reactive power controls of individual wind and is a valuable reference book for
academic researchers, practicing engineers, and other professionals. The book can also be used as a
textbook for graduate level and final year undergraduate-level courses"— Provided by publisher.
Includes bibliographical references and index.
ISBN 978-0-470-59365-3 (hardback)
1. Wind energy conversion systems. I. Wu, B. (Bin), 1957-
TK1541.P685 2011
621.31'2136—dc22 2010045226
Printed in the United States of America.
oBook ISBN: 978-1-118-02900-8
ePDFISBN: 978-1-118-02899-5
ePub ISBN: 978-1-118-02898-8
10 9 8 7 6 5 4 3 2 1

CONTENTS
Preface xi
List of Symbols xiii
Acronyms and Abbreviations xxiii
1. Introduction 1
1.1 Introduction 1
1.2 Overview of Wind Energy Conversion Systems 2
1.2.1 Installed Capacity and Growth Rate 2
1.2.2 Small and Large Wind Turbines 3
1.2.3 Stand-Alone and Grid-Connected Applications 3
1.2.4 On-Land and Offshore Applications 4
1.2.5 Costs of Wind Energy Conversion Systems 8
1.3 Wind Turbine Technology 9
1.3.1 Horizontal- and Vertical-Axis Wind Turbines 10
1.3.2 Fixed-and Variable-Speed Turbines 12
1.3.3 Stall and Pitch Aerodynamic Power Controls 12
1.4 Wind Energy Conversion System Configurations 13
1.4.1 Fixed-Speed WECS without Power Converter Interface 14
1.4.2 Variable-Speed Systems with Reduced-Capacity Converters 14
1.4.3 Variable-Speed Systems with Full-Capacity Power Converters 17
1.5 Grid Code 20
1.5.1 Fault Ride-Through Requirements 20
1.5.2 Reactive Power Control 21
1.6 Summary 22
References 22
2. Fundamentals of Wind Energy Conversion System Control 25
2.1 Introduction 25
2.2 Wind Turbine Components 26
v

VI CONTENTS
2.2.1 Turbine Blade 27
2.2.2 Pitch Mechanism 29
2.2.3 Gearbox 30
2.2.4 Rotor Mechanical Brake 32
2.2.5 Generator 33
2.2.6 Yaw Drive 33
2.2.7 Tower and Foundation 35
2.2.8 Wind Sensors (Anemometers) 36
2.3 Wind Turbine Aerodynamics 37
2.3.1 Power Characteristic of Wind Turbines 38
2.3.2 Aerodynamic Power Control: Passive Stall, Active Stall, and 39
Pitch Control
2.3.3 Tip Speed Ratio 42
2.4 Maximum Power Point Tracking (MPPT) Control 43
2.4.1 MPPT with Turbine Power Profile 45
2.4.2 MPPT with Optimal Tip Speed Ratio 46
2.4.3 MPPT with Optimal Torque Control 46
2.5 Summary 47
References 47
3. Wind Generators and Modeling 49
3.1 Introduction 49
3.2 Reference Frame Transformation 50
3.2.1 abcldq Reference Frame Transformation 51
3.2.2 abclaß Reference Frame Transformation 53
3.3 Induction Generator Models 55
3.3.1 Construction 55
3.3.2 Space-Vector Model 56
3.3.3 dq Reference Frame Model 58
3.3.4 Simulation Model 59
3.3.5 Induction Generator Transient Characteristics 61
3.3.6 Steady-State Equivalent Circuit 65
3.4 Synchronous Generators 71
3.4.1 Construction 72
3.4.2 Dynamic Model of SG 75
3.4.3 Steady-State Equivalent Circuits 80
3.5 Summary 85
References 85
4. Power Converters in Wind Energy Conversion Systems 87
4.1 Introduction 87
4.2 AC Voltage Controllers (Soft Starters) 88
4.2.1 Single-Phase AC Voltage Controller 89
4.2.2 Three-Phase AC Voltage Controller 92

Vil
4.3 Interleaved Boost Converters 97
4.3.1 Single-Channel Boost Converter 99
4.3.2 Two-Channel Interleaved Boost Converter 103
4.3.3 Multichannel Interleaved Boost Converters 107
4.4 Two-Level Voltage-Source Converters 112
4.4.1 Sinusoidal PWM 112
4.4.2 Space Vector Modulation 116
4.4.3 Harmonic Analysis 124
4.5 Three-Level Neutral Point Clamped Converters 125
4.5.1 Converter Configuration 126
4.6 PWM Current Source Converters 131
4.6.1 Current Source Inverter Topology 132
4.6.2 Selective Harmonic Elimination 132
4.6.4 PWM Current Source Rectifier 140
4.7 Control of Grid-Connected Inverter 142
4.7.1 Voltage Oriented Control (VOC) 144
4.7.2 VOC with Decoupled Controller 146
4.7.3 Operation of Grid-Connected Inverter with VOC and Reactive 148
Power Control
4.8 Summary 152
References 152
5. Wind Energy System Configurations 153
5.1 Introduction 153
5.2 Fixed-Speed WECS 154
5.2.1 Single-Speed WECS 154
5.2.2 Two-Speed WECS 155
5.3 Variable-Speed Induction Generator WECS 157
5.3.1 Wound Rotor Induction Generator with External Rotor 157
Resistance
5.3.2 Doubly Fed Induction Generator WECS with Reduced- 158
Capacity Power Converter
5.3.3 SCIG Wind Energy System with Full-Capacity Power 160
Converters
5.4 Variable-Speed Synchronous Generator WECS 162
5.4.1 Configuration with Full-Capacity Back-to-Back Power 163
Converters
5.4.2 Configuration with Diode Rectifier and dc/dc Converters 164
5.4.3 Configuration with Distributed Converters for Multiwinding 167
Generators
5.5 Summary 169
References 170
CONTENTS

VIII CONTENTS
6. Fixed-Speed Induction Generator WECS 173
6.2 Configuration of Fixed-Speed Wind Energy Systems 174
6.2.1 Wind Turbine 174
6.2.2 Gearbox 175
6.2.3 Generator 175
6.2.4 Soft Starter 176
6.2.5 Reactive Power Compensation 176
6.2.6 Main Features and Drawbacks 177
6.3 Operation Principle 177
6.3.1 Fixed-Speed Operation of SCIG 177
6.3.2 Two-Speed Operation of Fixed-Speed WECS 179
6.4 Grid Connection with Soft Starter 180
6.5 Reactive Power Compensation 184
6.6 Summary 188
References 189
7. Variable-Speed Wind Energy Systems with Squirrel Cage 191
Induction Generators
7.2 Direct Field Oriented Control 192
7.2.1 Field Orientation 192
7.2.2 Direct FOC for SCIG Wind Energy Systems 193
7.2.3 Rotor Flux Calculator 195
7.2.4 Dynamic and Steady-State Analysis of Direct FOC WECS 197
7.3 Indirect Field Oriented Control 204
7.3.1 Principles of Operation of Indirect FOC 204
7.3.2 Steady-State Analysis of Indirect FOC SCIG Wind Energy 206
System
7.4 Direct Torque Control 210
7.4.1 Principle of Direct Torque Control 210
7.4.2 Switching Logic 211
7.4.3 Stator Flux and Torque Calculator 214
7.4.4 Transient Analysis of SCIG WECS with DTC 215
7.4.5 Steady-State Analysis of SCIG WECS with DTC 217
7.5 Control of Current Source Converter Interfaced WECS 223
7.5.1 Introduction 223
7.5.2 Control ofCSC WECS with Variable a and Fixed m 225
7.5.3 Steady-State Analysis of CSC WECS 228
7.6 Summary 235
References 235
8. Doubly Fed Induction Generator Based WECS 237
8.2 Super-and Subsynchronous Operation of DFIG 238

CONTENTS IX
8.3 Unity Power Factor Operation of DFIG 240
8.3.1 Steady-State Equivalent Circuit of DFIG with Rotor-Side 240
Converter
8.3.2 Torque-Slip Characteristics of DFIG WECS 246
8.3.3 Steady-State Analysis of DFIG WECS with PFS = 1 249
8.3.4 Simplified Calculations 251
8.4 Leading and Lagging Power Factor Operation 252
8.5 Stator Voltage Oriented Control of DFIG WECS 254
8.5.1 Principle of Stator Voltage Oriented Control (SVOC) 254
8.5.2 System Block Diagram 259
8.5.3 Dynamic Performance of DFIG WECS 261
8.5.4 Steady-State Performance of DFIG WECS 265
8.6 DFIG WECS Start-Up and Experiments 269
8.7 Summary 272
References 273
9. Variable-Speed Wind Energy Systems with Synchronous 275
Generators
9.2 System Configuration 276
9.3 Control of Synchronous Generators 277
9.3.1 Zero ¿-Axis Current (ZDC) Control 277
9.3.2 Maximum Torque Per Ampere (MTPA) Control 279
9.3.3 Unity Power Factor (UPF) Control 281
9.3.4 Comparison ofZDC, MTPA and UPF Controls 282
9.4 SG Wind Energy System with Back-to-Back VSC 289
9.4.1 Nonsalient SG WECS with ZDC and Optimal Torque Control 289
9.4.2 Transient and Steady-State Analysis of Nonsalient SG WECS 291
9.4.3 Salient-Pole SG WECS with MTPA and Rotor Speed Feedback 294
Controls
9.4.4 Transient and Steady-State Analysis of Salient-Pole SG WECS 296
9.4.5 Grid-Side MPPT Control Scheme 301
9.5 DC/DC Boost Converter Interfaced SG Wind Energy System 302
9.6 Reactive Power Control of SG WECS 304
9.7 Current Source Converter Based SG Wind Energy Systems 308
9.7.1 CSC Wind Energy Systems with Firing Angle Control 308
9.7.2 CSC Wind Energy System with Reactive Power Control 312
9.8 Summary 315
References 315
Appendix A. Per-Unit System 317
Appendix B. Generator Parameters
B. 1 Squirrel Cage Induction Generators
319
319

X CONTENTS
B.2 Doubly Fed Induction Generators 322
B.3 Synchronous Generators 324
Appendix C. Problems and Solutions 327
Index 449
IEEE Press Series on Power Engineering

PREFACE
Wind energy is clean and sustainable. It is one of the fastest growing renewable energy
resources. The conversion of wind kinetic energy into electric energy is of a multidis-
ciplinary nature, involving aerodynamics, mechanical systems, electric machines,
power electronics, control theory, and power systems. In the past, a number of books
have addressed some of these subjects. This book explores the power conversion and
control of wind energy conversion systems (WECS) from the electrical engineering
perspective. It provides a comprehensive and in-depth analysis of wind generators,
system configurations, power converters, control schemes, and dynamic/steady-state
performance of various practical wind energy systems.
The book contains nine chapters. Chapter 1 provides a market survey and an
overview of wind turbine technology, wind energy system classifications, costs, and
grid codes for wind power integration. Chapter 2 introduces the fundamentals and con-
trol principles of wind energy systems, including wind turbine components, aerody-
namics, stall and pitch controls, and maximum power point tracking schemes. Chapter
3 presents commonly used wind generators, including squirrel cage induction genera-
tors, doubly fed induction generators, and synchronous generators. The dynamic and
steady-state models of these generators are also derived to facilitate the analysis of
wind energy systems in the subsequent chapters.
Chapter 4 discusses various power converters and PWM schemes used in wind en-
ergy systems. Both voltage and current source converters are presented with an em-
phasis on high-power wind energy system. Chapter 5 presents a general overview of
configurations and characteristics of major practical WECS. Chapter 6 focuses on
fixed-speed, inductor generator based wind energy systems; important issues such as
grid connection, two-speed operation, and reactive power compensation are discussed.
Chapter 7 deals with wind energy systems with variable-speed, squirrel cage induc-
tion generators (SCIG), in which typical system configurations and advanced control
schemes such as field oriented controls (FOC) and direct torque control (DTC) are
elaborated. Chapter 8 discusses doubly fed induction generator (DFIG) systems, where
the subsynchronous and supersynchronous modes of operation are investigated. Chap-
ter 9 is dedicated to variable-speed, synchronous generator wind systems, in which
various control schemes, including zero d-axis current (ZDC) control, maximum
xi

XII PREFACE
torque per ampere (MTPA) control, and unity power factor (UPF) control, are ana-
lyzed in detail.
To help the reader understand the principle and operation of various wind energy
conversion systems, we have developed more than 30 case studies in the body of the
book and more than 100 solved problems in Appendix C—Problems and Solutions.
Therefore, this book is not only for academic researchers and practicing engineers as a
reference book, but also suitable for graduate students and final-year undergraduate
students as a textbook.
We would like to express our sincere appreciation to Mr. Venkata Yaramasu, our
Ph.D. student in the Laboratory for Electric Drive Applications and Research
(LEDAR) at Ryerson University, for his great assistance in preparing the appendices
and the first three sections of Chapter 4. We would like to thank our postdoctoral fel-
lows and graduate students in LEDAR, in particular, Dr. Victor F. Liu, Dr. Moya J.
Dai, Mr. Ning Zhu, and Mr. Ehsan Al-Nabi, for their kind assistance in preparing the
manuscript. Our special thanks go to Wiley/IEEE Press editors for their great help and
support. We also wish to acknowledge the support and inspiration of our families dur-
ing the preparation of this book.
BINWU
YONGQIANG LANG
NAVID ZARGARI
SAMIR KOURO
Toronto, Ontario, Canada
January 2011

LIST OF SYMBOLS
A sweep area of turbine rotor blades
C, filter capacitor of current source inverter
Cp power coefficient of blade
Cp.max maximum power coefficient of blade
Cr filter capacitor of current source rectifier
D duty cycle of boost converter
Z>! coefficient (Z), = Ls Lr - L„) in the model of induction generators
E DC capacitor voltage of three-level NPC converter
battery voltage in DC link circuit
/ fundamental frequency of inverter output voltage
fcr frequency of carrier wave in sinusoidal pulse width modulation
fg frequency of the grid
fm frequency of modulating wave in sinusoidal pulse width modulation
f„ cut-off frequency of a first-order low-pass filter
fs stator frequency of induction or synchronous generators
fsp sampling frequency in space vector modulation
fsw switching frequency of solid-state switching device
fsw.inv equivalent switching frequency of inverter
phase-a current of AC voltage controller
phase-a current of the grid
phase-a rotor current of induction generator
phase a stator current of induction or synchronous generator
phase-a PWM current of current source inverter
phase-a PWM current of current source rectifier
phase-6 current of AC voltage controller
phase-6 current of the grid
ibr phase-è rotor current of induction generator
ibs phase-c stator current of induction or synchronous generator
ibwi phase-Z) current of current source inverter
phase-c current of AC voltage controller
icid d-axis filter capacitor current in current source inverter
<7-axis filter capacitor current of current source inverter
'■awr
l
ciq
XIII

l
crq
XIV LIST OF SYMBOLS
phase-c current of the grid
phase-c rotor current of induction generator
filter capacitor current vector of current source rectifier
crd ¿-axis filter capacitor current in current source rectifier
g-axis filter capacitor current in current source rectifier
phase-c stator current of induction or synchronous generator
phase-c PWM current of current source inverter
'dc DC link current
% DC link reference current
*ci DC link reference current of current source inverter
%r DC link reference current of current source rectifier
dg ¿-axis current of the grid
dg ¿-axis reference current of the grid
dr ¿-axis rotor current of induction generator
% ¿-axis rotor reference current of DFIG
ds ¿-axis stator current of induction or synchronous generator
% ¿-axis stator reference current of induction or synchronous generator
^wi ¿-axis PWM reference current of current source inverter
dwr ¿-axis PWM current of current source rectifier
%r ¿-axis PWM reference current of current source rectifier
g current of the grid
:
gl gate signal for thyristor Tx in AC voltage controller
gl gate signal for thyristor T2 in AC voltage controller
lgi gate signal for thyristor T3 in AC voltage controller
:
g4 gate signal for thyristor T4 in AC voltage controller
!g5 gate signal for thyristor T5 in AC voltage controller
gate signal for thyristor T6 in AC voltage controller
input current of boost converter
¿1 current through inductor Lx in boost converter
¿2 current through inductor L2 in boost converter
output current of boost converter
<?-axis current of the grid
<¡f-axis reference current of the grid
ç-axis rotor current of induction generator
<3>-axis rotor reference current of DFIG
¿/-axis stator current of induction or synchronous generator
<7-axis stator reference current of induction or synchronous generator
<7-axis PWM reference current of CSI
g-axis PWM current of current source rectifier
ç-axis PWM reference current of current source rectifier
rotor current of induction generator or doubly fed induction generator
rotor current vector of induction generator
Kf magnitude (length) of reference current vector Tref
r
ref reference current vector of current source converter
ref,max maximum magnitude of reference current vector íref
'ig
i*
lg
Iqr
;'*
l
qr
l
qs
i*
l
qs
i*
1
qwi
'•qwr
i*
1
qwr

LIST OF SYMBOLS XV
-^awl.max
Lr
he
Ids
h
h
In
hi
ILB
fLß.max
L
L
lr
ïr
Is
b
-* wr
J
k
A T
* I
stator current of induction or synchronous generator
stator current vector of induction or synchronous generator
PWM reference current of current source inverter
PWM current of current source rectifier
PWM current vector in current source rectifier
PWM reference current in current source rectifier
a-axis current
/3-axis current
rms value of grid phase-a current
rms value of the nth-order harmonic current in current source inverter
maximum rms fundamental-frequency current in current source inverter
filter capacitor current in current source rectifier (complex)
average value of DC link current
rms value of d-axis stator current of induction or synchronous generator
field current of synchronous generator
rms value of grid phase current
average value of boost converter input current
current space vector in current source converter (k = 0, 1,..., 6)
average value of inductor Lx current in boost converter
average value of inductor L2 current in boost converter
inductor boundary current between the CCM and DCM
maximum inductor boundary current
magnetizing current of induction generators (complex)
average value of output current of boost converter
boundary output current of boost converter
maximum boundary output current
rms value of ^r-axis stator current of induction or synchronous generator
rms value of rotor current of induction generator
rotor current of induction generator (complex)
rms value of stator current of induction or synchronous generator
stator current of induction generator (complex)
PWM current of current source rectifier (complex)
moment of inertia
number of harmonics to be eliminated by SHE scheme
/ 2PL„
armature constant IKT =
I 2LX
coefficient of boost converter K, =
Km
L
Ld
Ldc
'-'eq
coefficient Km
1
m,R
"m,R
inductance of each interleaved boost converter
c/-axis self inductance of synchronous generator
DC link inductance
equivalent inductance of rotor-side converter in DFIG

XVI LIST OF SYMBOLS
Lg grid-side line inductance
Llr rotor leakage inductance of induction or synchronous generator
Lh stator leakage inductance of induction or synchronous generator
LL inductance of three-phase RL load
Lm magnetizing inductance of induction generator
Lq ^r-axis self inductance of synchronous generator
Lr rotor self-inductance of induction generator
Ls stator self-inductance of induction or synchronous generator
ma amplitude modulation index of SPWM or SVM
m
a,ma maximum modulation index of SPWM or SVM
rtif frequency modulation index in sinusoidal pulse-width modulation
m, modulation index of current source inverter
m
i,nax maximum modulation index of current source inverter
mmax maximum modulation index
mr modulation index of current source rectifier
m
r,max maximum modulation index of current source rectifier
nm generator speed in rpm
nM turbine speed in rpm
nr rotor speed of induction or synchronous generator in rpm
N number of channels of interleaved boost converter
NP number of pulses per half-cycle of current source converter
p derivative operator I p = —
at
P number of pole pairs
system active power
Pac ac-side active power of converter
Pag air-gap power of induction generator
Pcus stator copper loss of induction generator
Pcur rotor copper loss of induction generator
PF power factor
PFS stator power factor
PFL load power factor
Pdc DC power of converter
Pg active power delivered to the grid
Pm mechanical power of generator
PM mechanical power captured by turbine
PL load active power
Pr rotor power of DFIG
Prot rotational power losses of generator
Ps stator active power of generator
Pw wind power
Q reactive power
Qg grid-side reactive power
Q* grid-side reactive power reference

LIST OF SYMBOLS XVII
ôg,max maximum grid-side reactive power
QGSC reactive power reference of grid-side converter in DFIG
Qi load reactive power
Qs stator reactive power of DFIG
Q* stator reactive power reference
rgb gearbox conversion ratio (gear ratio)
rT radius of turbine rotor (blade length)
R load resistance of boost converter
resistance of simplified generator-side converter
Req equivalent resistance of rotor-side converter in DFIG
Rr rotor winding resistance of induction generator
Rs stator winding resistance of induction or synchronous generator
RL resistance of three-phase RL load
s slip of induction generator
s
T,mm slip at the maximum torque
S Laplace operator
apparent power of system
Sg grid-side apparent power
ion turn-on time of boost converter
toñ turn-off time of boost converter
T0 dwell time for voltage vector VQ and current vector T0 in space vector modu-
lation
Ta dwell time for voltage vector Vx in space vector modulation
Tb dwell time for voltage vector V2 in space vector modulation
Tj dwell time for current vector 7, in space vector modulation
T2 dwell time for current vector 72 in space vector modulation
Te electromagnetic torque of induction and synchronous generators
T* electromagnetic torque reference
TL load torque
Tm mechanical torque from generator shaft
TM mechanical torque generated by turbine
Tmax maximum torque
Ts switching period of solid switching device
sampling period of space vector modulation
va phase-a voltage
v* phase-a reference voltage
vab inverter line-to-line voltage between phase a and phase b
vAB line-to-line supply voltage of three-phase AC voltage controller
vabx fundamental frequency component of inverter line-to-line voltage vab
vabs line-to-line stator voltage or rectifier PWM voltage
vabs, fundamental component of line-to-line voltage vabs
vag phase-a grid voltage
v* phase-a reference voltage of PWM inverter
van phase-a load voltage of three-phase AC voltage controller
vaN phase-a load voltage of two-level voltage source converter

XVIII LIST OF SYMBOLS
v ^ phased supply voltage of three-phase AC voltage controller
var phase-a rotor voltage of DFIG
v*r phase-a reference rotor voltage
varl fundamental frequency component of phase-a rotor voltage of DFIG
vas phase-a stator voltage of induction and synchronous generator
v *s phase-a reference voltage of PWM rectifier
vasl fundamental frequency component of phase-a stator voltage
vaZ phase-a terminal voltage of three-level NPC inverter
vb phase-6 voltage
v£ phase-6 reference voltage
vbc line-to-line voltage between phase b and phase c
vBC line-to-line supply voltage between phase B and phase C of AC voltage
controller
vbg phase-è grid voltage
v% phase-è reference voltage of PWM inverter
vb„ phase-è load voltage of three-phase AC voltage controller
vbN phase-¿> load voltage of two-level voltage source converter
vBN phase-S supply voltage of three-phase AC voltage controller
vbr phase-è rotor voltage of DFIG
vfr phase-ft rotor reference voltage
vbs phase-¿> stator voltage of induction and synchronous generator
vfs phase-6 reference voltage for PWM rectifier
vbz phase-6 terminal voltage of three-level NPC inverter
vc phase-c voltage
v* phase-c reference voltage
vca line-to-line voltage between phase c and phase a
vCA line-to-line supply voltage between phase C and phase A
vci capacitor voltage in current source inverter
v* phase-c reference voltage of PWM inverter
vcg phase-c voltage of the grid
vcn phase-c load voltage of three-phase AC voltage controller
vc^ phase-c load voltage of two-level voltage source converter
v*r phase-c rotor reference voltage
v„ phase-c stator voltage of induction and synchronous generator
v*s phase-c reference voltage for PWM rectifier
vcr triangular carrier wave in sinusoidal pulse-width modulation
phase-c rotor voltage of DFIG
vcZ phase-c terminal voltage of three-level NPC inverter
vdc DC link voltage
v% reference DC link voltage
vdcl DC voltage of boost converter
vdc2 DC voltage of boost converter
v%¡ inverter-side DC voltage reference
vdcr DC output voltage of current source rectifier
vdg d-axis voltage of the grid

LIST OF SYMBOLS
vdi ¿-axis voltage of grid-tied inverter
v% ¿-axis reference voltage of PWM inverter
vdr ¿-axis rotor voltage
v% ¿-axis rotor reference voltage
vds ¿-axis stator voltage
v% ¿-axis reference voltage of PWM rectifier
vg grid phase voltage
Vg grid voltage vector
vgl gating signal of switch S¡
vg2 gating signal of switch S2
v, input voltage of boost converter
output voltage of grid-tied inverter
v¿1 inductor voltage
vma phase-a modulating wave in SPWM scheme
vmb phase-è modulating wave in SPWM scheme
vmc phase-c modulating wave in SPWM scheme
vqg o-axis voltage of the grid
vqi a-axis voltage of grid-tied inverter
v* g-axis reference voltage of PWM inverter
vqr g-axis rotor voltage
v*r ^-axis rotor reference voltage
vqs qr-axis stator voltage
v*s ^r-axis reference voltage of PWM rectifier
vr rotor voltage of DFIG
Vr rotor current vector
vref magnitude (length) of reference voltage vector v?ref
v'jef reference voltage vector
v
ref,max maximum magnitude of reference voltage vector
vs supply voltage of single-phase AC voltage controller
stator voltage of induction or synchronous generator
% stator voltage vector
vsl voltage across switch 5] in boost covnerter
vw wind velocity/speed
va a-axis voltage
Vß /3-axis voltage
Va peak value of inverter phase-a voltage
Vabn the nth-order rms harmonic voltage of inverter line-to-line voltage
^aM.max maximum fundamental rms voltage of inverter line-to-line voltage
^at.max maximum rms voltage of inverter line-to-line voltage vab
Vag rms value of grid phase-a voltage
Van rms value of load phase-a voltage
VaN rms inverter terminal voltage (phase-a)
VbN rms inverter terminal voltage (phase b)
VcN rms inverter terminal voltage (phase-c)
Vcr peak value of carrier wave in sinusoidal pulse-width modulation

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