null  null
A Small-Scale Standalone Wind Energy
Conversion System Featuring SCIG, CSI and
a Novel Storage Integration Scheme
by
Zuher Alnasir
A thesis
presented to the University of Waterloo
in fulfillment of the
thesis requirement for the degree of
Doctor of Philosophy
in
Electrical and Computer Engineering
Waterloo, Ontario, Canada, 2016
© Zuher Alnasir 2016
Author’s Declaration
I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis,
including any required final revisions, as accepted by my examiners.
I understand that my thesis may be made electronically available to the public.
ii
Abstract
Small-scale standalone wind turbines provide a very attractive renewable energy source for
off-grid remote communities. Taking advantage of variable-speed turbine technology, which
requires a partial- or full-scale power converter, and through integrating an energy storage
system, smooth and fast power flow control, maximum power point tracking, and a highquality power is ensured.
Due to high reliability and efficiency, permanent magnet synchronous generator seems to be
the dominating generator type in gearless wind turbines, employed for off-grid applications.
However, wind turbines using geared squirrel-cage induction generator (SCIG) are still widely
accepted due to their robustness, simplicity, light weight and low cost. Permanent magnet
induction generator, a relatively new induction-based machine, has recently been recognized
in the wind energy market as an alternative for permanent magnet synchronous generator. A
thorough comparative study, among these three generator types, is conducted in this research
in order to enable selection of the most appropriate generator for off-grid wind energy
conversion system (WECS), subject to a set of given conditions. The system based on geared
SCIG has been shown to be the most appropriate scheme for a small-scale standalone WECS,
supplying a remote area.
Different topologies of power electronic converters, employed in WECSs, are overviewed.
Among the converters considered, current source converter is identified to have a great
potential for off-grid wind turbines.
Three current-source inverter-based topologies, validated in the literature for on-grid WECS,
are compared for off-grid WECS application. Feasibility study and performance evaluation are
conducted through analysis and simulation. Among all, the topology composed of three-phase
diode bridge rectifier, DC/DC buck converter, and pulse-width-modulated current-source
inverter (PWM-CSI) is identified as a simple and low-cost configuration, offering satisfactory
performance for a low-power off-grid WECS.
A small-scale standalone wind energy conversion system featuring SCIG, CSI and a novel
energy storage integration scheme is proposed and a systematic approach for the dc-link
inductor design is presented.
iii
In developing the overall dynamic model of the proposed wind turbine system, detailed
models of the system components are derived. A reduced-order generic load model, that is
suitable for both balanced and unbalanced load conditions, is developed and combined with
the system components in order to enable steady-state and transient simulations of the overall
system. A linear small-signal model of the system is developed around three operating points
to investigate stability, controllability, and observability of the system. The eigenvalue analysis
of the small-signal model shows that the open-loop system is locally stable around operating
points 1 and 3, but not 2. Gramian matrices of the linearized system show that the system is
completely controllable at the three operating points and completely observable at operating
points 1 and 3, but not 2.
The closed-loop control system for the proposed wind turbine system is developed. An
effective power management algorithm is employed to maintain the supply-demand power
balance through direct control of dc-link current. The generator’s shaft speed is controlled by
the buck converter to extract maximum available wind power in normal mode of operation.
The excess wind power is dumped when it is not possible to absorb maximum available power
by the storage system and the load. The current source inverter is used to control positive- and
negative-sequence voltage components separately. The feasibility of the proposed WECS and
performance of the control system under variable wind and balanced/unbalanced load
conditions are analyzed and demonstrated through simulation.
Finally, the proposed WECS is modified by removing the dump load and avoiding the
surplus power generation by curtailment of wind power. The operation of the modified system
is investigated and verified under variable wind and load conditions.
iv
Acknowledgements
First and above all, I praise the almighty ALLAH for granting me the capability to proceed
successfully. This thesis would not have been possible without the support of several
individuals who in one way or another contributed their valuable assistance in the completion
of this work.
I would like to express my utmost gratitude and thanks to my academic supervisor, Prof.
Mehrdad Kazerani, who has advised, guided and supported me throughout this research work.
Sincere thanks are also given to my PhD committee members: Prof. Claudio Canizares and
Prof. Magdy Salama from the department of Electrical and Computer Engineering at
University of Waterloo, and Prof. Amir Khajepour from the department of Mechanical and
Mechatronics Engineering at University of Waterloo, for their valuable feedback and
suggestions during my PhD comprehensive examination and seminar. It is my pleasure to
thank Prof. Alireza Bakhshai from the department of Electrical and Computer Engineering at
Queen’s University, for being my external examiner.
Ethically, I should not forget to thank my parents, my wife, and my sisters and brothers for
their moral support during the period of my PhD program.
Finally, I would like to acknowledge the financial support received from the Royal
Commission for Jubail & Yanbu (RCJY) in the Kingdom of Saudi Arabia through the Saudi
culture bureau in CANADA.
v
Dedication
To my Lovely Daughter,
“ASEEL”,
The sunshine of my life
vi
Table of Contents
Author’s Declaration .............................................................................................................. ii
Abstract……………………………………………………………………………………...iii
Acknowledgements ................................................................................................................. v
Dedication ............................................................................................................................... vi
Table of Contents .................................................................................................................. vii
List of Figures .......................................................................................................................... x
List of Tables ........................................................................................................................ xiii
List of Acronyms .................................................................................................................. xiv
List of Symbols .................................................................................................................... xvii
Chapter 1 Introduction .......................................................................................................... 1
1.1
Research Motivations ............................................................................................................. 1
1.2
Literature Review ................................................................................................................... 2
1.2.1
History of Wind Energy ................................................................................................. 3
1.2.2
Conventional and Potential Generator Types used in WECS......................................... 3
1.2.3
Power Electronic Converter Topologies for Standalone WECS .................................... 9
1.2.4
PWM-CSI versus PWM-VSI for Standalone WECS ................................................... 13
1.2.5
Energy Storage in Standalone WECS .......................................................................... 17
1.3
Research Objectives ............................................................................................................. 22
1.4
Thesis Layout ....................................................................................................................... 22
Chapter 2 Selection of Generator Type for Small-Scale WECS ...................................... 24
2.1
Evaluation of Conventional and Evolving Generator Types for Standalone WECS ........... 24
2.1.1
SCIG-WECS versus PMSG-WECS ............................................................................. 26
2.1.2
PMIG versus SCIG and PMSG for Standalone WECS ................................................ 33
2.1.3
Indices for Selecting the Preferred Generator .............................................................. 35
2.2
Summary............................................................................................................................... 38
Chapter 3 Proposed Wind Energy Conversion System .................................................... 39
3.1
Configurations of CSC for Standalone WECS ..................................................................... 39
3.1.1
Topology 1: Diode Rectifier - PWM CSI .................................................................... 40
3.1.2
Topology 2: Diode Rectifier - Buck Converter - PWM CSI ........................................ 42
3.1.3
Topology 3: Back-to-Back CSC ................................................................................... 47
vii
3.2
Comparison of Different CSC-WECS Topologies ............................................................... 50
3.3
Integration of Energy Storage with the CSI ......................................................................... 51
3.4
The Structure of the Proposed WECS .................................................................................. 55
3.4.1
3.5
Systematic Design of the DC-Link Inductor ................................................................ 57
Summary............................................................................................................................... 59
Chapter 4 Dynamic Modeling and Small Signal Analysis of the CSI-WECS ................. 61
4.1
Wind Turbine System ........................................................................................................... 63
4.2
Drive Train System............................................................................................................... 66
4.3
Self-Excited Induction Machine ........................................................................................... 67
4.4
Three-Phase Diode Bridge with DC-Side Capacitive Filter ................................................. 71
4.5
DC/DC Buck Converter ....................................................................................................... 75
4.6
Lead Acid Battery................................................................................................................. 76
4.7
DC/DC Reduced H- Bridge Converter ................................................................................. 78
4.8
Current Source Inverter ........................................................................................................ 80
4.8.1
ABC-Frame Equations ................................................................................................. 80
4.8.2
DQ-Frame Equations .................................................................................................... 83
4.9
Generic Load Model ............................................................................................................. 83
4.9.1
Generic Load Model Proposed in [138] ....................................................................... 84
4.9.2
Reduced-Order Load Model ......................................................................................... 87
4.9.3
Generic Load Model including Unbalanced Load Condition....................................... 88
4.10
DC-Link Model .................................................................................................................... 92
4.11
Dynamic Model of the Proposed Wind Energy Conversion System ................................... 93
4.11.1
State Space Equations................................................................................................... 93
4.11.2
Steady-State Equations ................................................................................................. 95
4.12
Verification of the Overall Model ........................................................................................ 97
4.12.1
Wind Turbine Generation (WTG) Subsystem .............................................................. 98
4.12.2
Energy Storage (ES) Subsystem ................................................................................... 98
4.12.3
Current-Sourced Inverter–Load (CSI-Load) Subsystem .............................................. 99
4.12.4
Starting Operation of the Wind Turbine System ........................................................ 100
4.12.5
Average model versus Switching model for Power Electronics Converters .............. 101
4.12.6
Open-Loop Step Responses of the System ................................................................. 104
4.13
Small-Signal Model and Stability Analysis........................................................................ 109
4.13.1
4.14
Transfer Function and Eigenvalue Analysis ............................................................... 110
System Controllability and Observability .......................................................................... 111
viii
4.14.1
Gramian Matrices ....................................................................................................... 112
4.15
Linearized Model Evaluation ............................................................................................. 113
4.16
Summary............................................................................................................................. 118
Chapter 5 Control System Design for the Proposed PWM-CSI-Based SCIG-WECS . 119
5.1
Overview of the control system of the proposed WECS .................................................... 119
5.2
Closed-Loop Control System ............................................................................................. 121
5.3
DC-link Current Control..................................................................................................... 122
5.3.1
DC-link Current Control Scheme ............................................................................... 122
5.3.2
Parameters Design of DC-link Current PI Controller................................................. 126
5.4
Generator Speed MPPT Control ......................................................................................... 130
5.4.1
MPPT Control Scheme ............................................................................................... 130
5.4.2
Parameter Design of Generator Speed Control Loop ................................................. 132
5.5
Load-Side Control .............................................................................................................. 133
5.5.1
Load-Side Control Scheme ......................................................................................... 133
5.5.2
Parameter Design of Load-Side Control Loop ........................................................... 134
5.6
Simulation Verification ...................................................................................................... 135
5.6.1
System Performance under Various Wind Speed and Load Conditions .................... 136
5.6.2
Effect of System Inertia and Frequency of Wind Speed Variation on MPPT ............ 143
5.6.3
Performance of the Synchronous dq Controller under Various Dynamics of Load ... 144
5.7
Fault Analysis of the CSI-based WECS ............................................................................. 148
5.8
Dump Load-Less Version of the Proposed CSI-based WECS ........................................... 152
5.9
Summary............................................................................................................................. 156
Chapter 6 Conclusions, Contributions, and Future Work ............................................. 158
6.1
Summary and Conclusions ................................................................................................. 158
6.2
Contributions ...................................................................................................................... 160
6.3
Future Work........................................................................................................................ 161
Appendix A ........................................................................................................................... 162
Appendix B ........................................................................................................................... 164
Appendix C ........................................................................................................................... 165
Appendix D ........................................................................................................................... 167
Appendix E ........................................................................................................................... 172
References ............................................................................................................................ 182
ix
List of Figures
Fig. 1.1: Global annual installed Small Wind turbine (SWT)[2]. .......................................................... 3
Fig. 1.2: WRIG-based standalone WECS. ............................................................................................. 4
Fig. 1.3: DFIG-based standalone WECS. ............................................................................................... 5
Fig. 1.4: BDFIG-based standalone WECS. ............................................................................................ 6
Fig. 1.5: WRSG-based direct-drive standalone WECS. ......................................................................... 7
Fig. 1.6: Standalone WECS using direct-drive SRG. ............................................................................. 8
Fig. 1.7: Diode rectifier + dc/dc boost converter + VSI. ....................................................................... 9
Fig. 1.8: Two-level, back-to-back voltage-source converter topology. ................................................ 10
Fig. 1.9: Back-to-back current-source converter topology. .................................................................. 11
Fig. 1.10: Diode Rectifier Bridge + Z-Source inverter......................................................................... 11
Fig. 1.11: Matrix converter. .................................................................................................................. 13
Fig. 2.1: SCIG-based standalone WECS: (a) with generator-side diode bridge rectifier, and (b) with
generator-side voltage-source rectifier. ................................................................................................ 27
Fig. 2.2: PMSG-based direct-drive standalone WECS: (a) with generator-side diode bridge rectifier,
and (b) with generator-side voltage-source rectifier............................................................................. 28
Fig. 2.3: PMIG-based direct-drive standalone WECS: (a) with generator-side diode bridge rectifier,
and (b) with generator-side voltage-source rectifier............................................................................. 34
Fig. 3.1: Block diagram of the proposed standalone WECS. ............................................................... 39
Fig. 3.2:A SCIG-WECS composed of a diode rectifier and a PWM-CSI. ........................................... 41
Fig. 3.3: Simulation responses for topology 1. ..................................................................................... 42
Fig. 3.4: A SCIG-WECS composed of a diode rectifier, a buck converter and a PWM-CSI. ............. 43
Fig. 3.5: Simulation results for topology 2. .......................................................................................... 46
Fig. 3.6: Effect of diode bridge rectifier on generator characteristics at rated speed. .......................... 46
Fig. 3.7: A SCIG-WECS using back-to-back CSC. ............................................................................. 48
Fig. 3.8: MPPT for back-to-back CSC-based WECS. .......................................................................... 49
Fig. 3.9: Stator current and generator electromagnetic torque at rated speed. ..................................... 49
Fig. 3.10: Integration of the energy storage system proposed in [70] with the CSI- based WECS of
topology 2. ............................................................................................................................................ 52
Fig. 3.11: Simulation results for the CSI-based WECS with energy storage system proposed in [70].
.............................................................................................................................................................. 54
Fig. 3.12: Structure of the proposed WECS. ........................................................................................ 55
Fig. 3.13: DC-link inductor voltage and current. ................................................................................. 58
Fig. 4.1: Block diagram of the proposed CSI-based WECS................................................................. 61
x
Fig. 4.2: Performance coefficient versus tip speed ratio for various blade pitch angles. ..................... 64
Fig. 4.3: Turbine power characteristics at zero pitch angle. ................................................................. 64
Fig. 4.4: Wind speed model based on Von-Karman’s method. ............................................................ 66
Fig. 4.5: Two mass model of turbine’s drive train on Turbine side. .................................................... 67
Fig. 4.6: Dq model of Self-excited squirrel cage induction machine. .................................................. 69
Fig. 4.7: Three-phase diode bridge rectifier with a variable dc current. .............................................. 71
Fig. 4.8: Three-phase diode bridge rectifier with a constant dc current. .............................................. 73
Fig. 4.9: Averaged model of CCM diode bridge rectifier with instantaneous commutation................ 74
Fig. 4.10: Dq average model of diode bridge rectifier supplied by SEIG. ........................................... 74
Fig. 4.11: Buck converter circuit. ......................................................................................................... 75
Fig. 4.12: Large-signal nonlinear averaged model of CCM buck converter. ....................................... 76
Fig. 4.13: Electrical model of a battery cell. ........................................................................................ 76
Fig. 4.14: LC filter on battery side. ...................................................................................................... 77
Fig. 4.15: Reduced H-bridge for storage integration. ........................................................................... 78
Fig. 4.16: Large-signal nonlinear averaged model of CCM Full-Bridge Converter. ........................... 79
Fig. 4.17: Three-phase current-sourced inverter feeding a three-phase load........................................ 80
Fig. 4.18: Large-signal nonlinear averaged model of current source inverter. ..................................... 82
Fig. 4.19: Block diagram of the generic load model proposed in [138]. .............................................. 85
Fig. 4.20: Reduced-order model versus the original 4th -order model. ................................................ 87
Fig. 4.21: Generic load model including unbalanced case. .................................................................. 89
Fig. 4.22: Load currents and powers using the extended generic load model. ..................................... 92
Fig. 4.23: Configuration of the dc link. ................................................................................................ 92
Fig. 4.24: Inputs, state variables, and outputs of the proposed WECS................................................. 95
Fig. 4.25: WTG Subsystem. ................................................................................................................. 98
Fig. 4.26: ES Subsystem. ...................................................................................................................... 99
Fig. 4.27: CSI-Load Subsystem............................................................................................................ 99
Fig. 4.28: Starting of the wind turbine under unloaded/loaded conditions. ....................................... 101
Fig. 4.29: Average model versus switching model: diode bridge rectifier waveforms. ..................... 102
Fig. 4.30: Switching and average model waveforms of buck converter............................................. 103
Fig. 4.31: Switching and average model waveforms of recued-H-bridge converter. ......................... 103
Fig. 4.32: Switching and average model waveforms of CSI. ............................................................. 104
Fig. 4.33: Responses of generator rotor speed and active power to 50% step decrease in 𝑑𝑏 . ......... 105
Fig. 4.34: Generator responses to 20% step decrease in 𝑣𝑤 . .............................................................. 106
Fig. 4.35: Responses of dc-link current to 80% step change in 𝑑𝐴 . ................................................... 107
xi
+
Fig. 4.36: Load-side responses to 50% step change in 𝑚𝑖𝑞
................................................................ 108
+
Fig. 4.37: Load-side responses to 50% step change in 𝑚𝑖𝑑
. .............................................................. 108
+
Fig. 4.38: Average and switching model load-side voltage responses to 50% step increase in 𝑚𝑖𝑑
.. 109
Fig. 5.1: Converters and control blocks in the CSI-based WECS. ..................................................... 120
Fig. 5.2: Typical control loop diagram. .............................................................................................. 121
Fig. 5.3: DC-link current control scheme. ......................................................................................... 123
Fig. 5.4: PI tuning of the dc-link current controller at o.p.1. .............................................................. 127
Fig. 5.5: PI tuning of the dc-link current controller at o.p.2. .............................................................. 128
Fig. 5.6: PI tuning of the dc-link current controller at o.p.3. .............................................................. 128
Fig. 5.7: Step/Ramp response of dc-link current at the three operating points................................... 129
Fig. 5.8: Typical turbine power versus wind speed curve. ................................................................. 131
Fig. 5.9: MPPT Controller. ................................................................................................................. 132
Fig. 5.10: Load-side synchronous dq frame control scheme. ............................................................. 134
Fig. 5.11: System behavior under variable balanced load. ................................................................. 139
Fig. 5.12: Effect of generator inductance on stator current and electromagnetic torque. ................... 140
Fig. 5.13: Power management after battery SoC reaches upper limit. ............................................... 141
Fig. 5.14: System behavior under variable unbalanced load. ............................................................. 142
Fig. 5.15: Effects of system inertia and frequency of wind speed variation on MPPT. ..................... 144
Fig. 5.16: Load characteristics under synchronous dq-frame controllers. ......................................... 146
Fig. 5.17: Load-side three-phase voltages and currents under synchronous dq-frame controllers..... 146
Fig. 5.18: Load-side stationary abc frame control scheme using PI controllers. ................................ 147
Fig. 5.19: Performance comparsion of abc-frame and dq-frame PI controllers. ................................ 148
Fig. 5.20: Fault points on the proposed WECS. ................................................................................. 149
Fig. 5.21:System performance during (a) SLG (b) LL (c) TPG faults at secondary side of the ∆/𝑌𝑛
transformer. ........................................................................................................................................ 150
Fig. 5.22: Harmonic Spectrum of CSI’s ac terminal and filtered currents before and during SLG fault.
............................................................................................................................................................ 151
Fig. 5.23: Inverter dc-side current of VSI-and CSI-based WECS during an SLG fault. ................... 152
Fig. 5.24: MPPT/Curtailment Control Scheme. ................................................................................. 153
Fig. 5.25: System behavior under MPPT/Curtailment control. .......................................................... 155
xii
List of Tables
Table 1.1: Cost Comparison of 30kW PMSG-based Wind Turbines with VSI and CSI [93].............. 15
Table 1.2: Advantages and drawbacks of different energy storage technologies. ............................... 18
Table 1.3: Energy storage requirements for a small-scale standalone WECS...................................... 20
Table 1.4: Feasibility of Energy Storage Technologies for small-scale standalone WECS. ................ 21
Table 2.1: Cost Comparison of 30kW PMSG- and SCIG-Based WECSs. .......................................... 32
Table 2.2: SCIG-WECS versus PMSG-WECS. ................................................................................... 33
Table 2.3: Cost of 30kW PMIG-Based WECS. ................................................................................... 35
Table 2.4: Comparison of the geared-drive SCIG, gearless-drive PMSG and gearless-drive
PMIG-WECS configurations................................................................................................................ 37
Table 3.1: THD of SCIG stator current in Topology 2......................................................................... 46
Table 3.2: Comparison of the three standalone CSC-based WECS topologies.................................... 51
Table 3.3: Possible values of 𝑣𝑑 , 𝑣𝑥𝑦 , and 𝑣𝑖𝑛𝑣 . ................................................................................ 57
Table 4.1: Reduced H-bridge converter operating modes. ................................................................... 78
Table 4.2: Dependence of load on voltage magnitude. ........................................................................ 86
Table 4.3: Characteristics of the generic load models. ......................................................................... 88
Table 4.4: Characteristics of extended generic load model. ................................................................. 91
Table 4.5: Input parameters for WTG subsystem used for starting simulation. ................................. 100
Table 4.6: Input, output and parameter values at the operating point for steady-state analysis. ........ 102
Table 4.7: Input, state and output variables at the operating points for linearization. ........................ 114
Table 4.8: Eigenvalues and damping ratios of the linearized system at o.p.1 and o.p.2. ................... 115
Table 4.9: Singular values of Gramian matrices at o.p.1 and o.p.2. ................................................... 115
Table 4.10: Eigenvalues and damping ratios of the linearized system at o.p.3. ................................. 116
Table 4.11: Singular values of Gramian matrices at o.p.3.................................................................. 116
Table 5.1: Small-signal transfer functions of the ES-subsystem. ....................................................... 127
Table 5.2: PI gains of the closed loop system controllers at the three operating points. .................... 135
Table 5.3: Wind speed and load conditions for simulation results of Fig. 5.11 and Fig. 5.14. .......... 136
Table 5.4: Load parameters for simulation results of Fig. 5.16.......................................................... 145
Table 5.5: Wind speed and load conditions for simulation results of Fig. 5.25. ................................ 155
Table A.1: Parameters and operating condition of the 30kW-CSC-SCIG WECS………………......162
Table A.2: Parameters and rated operating conditions of the 20kW-CSI-SCIG-WECS…................163
Table C.1: Switching States for a Three-Phase CSI…………………………………………………166
xiii
List of Acronyms
+ve seq
Positive Sequence
2L-BTB-VSC
Two-Level Back-to-Back Voltage-Source Converter
BDFIG
Brushless Doubly-Fed Induction Generator
BDFRG
Brushless Doubly-Fed Reluctance Generator
BTB-CSC
Back-to-Back Current-Source Converter
CAES
Compressed Air Energy Storage
CCM
Continuous Conduction Mode
CL
Closed Loop
CLTF
Closed-Loop Transfer Function
CSC
Current-Source Converter
CSI-Load
Current Source Inverter-Load
DCM
Discontinues Conduction Mode
DFIG
Doubly-Fed Induction Generator
DOD
Depth of Discharge
DRFOC
Direct Rotor Flux Oriented Control
ES
Energy Storage
ESR
Equivalent Series Resistance
FBES
Flow Battery Energy Storage
FES
Flywheel Energy Storage
HES
Hydrogen Energy Storage
IG
Induction Generator
IGBT
Insulated-Gate Bipolar Transistor
IMC
Indirect Matrix Converter
LAB
Lead Acid Battery
LIES
Lithium Ion battery Energy Storage
LUF
Load Unbalance Factor
MC
Matrix Converter
ML
Multi-Level
MPPT
Maximum Power Point Tracking
NaSES
Sodium Sulphur battery Energy Storage
xiv
NCES
Nickel Cadmium battery Energy Storage
O&M
Operation and Maintenance
o.p.
Operating point
OL
Open Loop
OLTF
Open-Loop Transfer Function
OSF
Open-Switch Fault
PCC
Point of Common Coupling
PHES
Pumped Hydro Energy Storage
PM
Permanent Magnet
PMIG
Permanent-Magnet Induction Generator
PMSG
Permanent-Magnet Synchronous Generator
PWM
Pulse Width Modulation
PWM-CSI
Pulse-Width Modulated Current-Source Inverter
PWM-CSR
Pulse-Width Modulated Current-Source Rectifier
PWM-VSI
Pulse-Width Modulated Voltage-Source Inverter
RMS
Root Mean Square
SCIG
Squirrel-Cage Induction Generator
SEIG
Self-Excited Induction Generator
Seq
Sequence
SES
Super-capacitor Energy storage
SMES
Superconducting Magnetic Energy Storage
SoC
State of Charge
SRG
Switched-Reluctance Generator
SSF
Short-Switch Fault
SVD
Singular Value Decomposition
SWT
Small Wind turbine
THD
Total Harmonic Distortion
-ve seq
Negative Sequence
VSC
Voltage-Source Converter
VSI
Voltage-Source Inverter
VSR
Voltage-Source Rectifier
xv
WECS
Wind Energy Conversion system
WRIG
Wound-Rotor Induction Generator
WRSG
Wound-Rotor Synchronous Generator
WT
Wind turbine
WTG
Wind-Turbine Generation
ZEBRA
Sodium Nickel chloride battery energy storage
ZSI
Z-Source Inverter
SLG
Single Line to Ground
LL
Line to Line
TPG
Three-Phase to Ground
xvi
List of Symbols
The variables that are commonly used in this thesis are provided below. Other variables,
only used in specific sections, are defined in the context where they are used.
In general, the variables used in this thesis are represented as follows:

The sign ∆ followed by a variable represents the small-signal variation of the variable.

In small-signal analysis, the upper-case letter of a variable refer to the quiescent (dc)
component of the variable at the operating point of linearization.

A variable with a suffix ‘𝑎’, ‘𝑏’ and ‘𝑐’ indicates for the corresponding phase a, phase
b, and phase c, respectively.

A variable with a suffix ‘𝑑’ and ‘𝑞’ represents its corresponding direct- and quadratureaxis components in the defined synchronous reference frame, respectively.

The sequence (positive or negative) of a quantity is identified by the sign in the
superscript of the quantity.

A variable with superscript ‘^’ denotes peak of the variable.

A variable with subscript ‘𝑟𝑒𝑓’ denotes reference value of the variable.

A variable with subscript ‘𝑜𝑝𝑡’ denotes the optimum/optimal value of the variable.

A variable with subscript ‘ ss’ denotes the steady-state value of the variable.
𝜌
Air density
𝜆
Turbine’s tip speed ratio
𝛿𝜃
Angle difference
𝛽
Turbine’s blade pitch angle
𝑟
Radius of turbine’s blades
𝑑
Damping of natural modes of the generic load model
𝑃
Pole pair
𝐺
ac gain of the inverter PWM scheme
xvii
𝐴
Turbine’s swept area
∆𝑖𝑑𝑐
dc-link current ripple
𝜔𝑠
Stator angular synchronous frequency
𝜔𝑟
Electrical angular speed of the generator
𝜔𝑜
Oscillation frequency of the generic load model
𝜔𝑚
Mechanical angular speed of the turbine
𝜔𝑔
Mechanical angular speed of the generator
𝜔𝑒
Stator angular electrical frequency
𝜔𝐿𝑜
Nominal load-side frequency
𝜔𝐿
Fundamental angular frequency of the load-side voltage
𝜑𝑑𝑠 , 𝜑𝑞𝑠
𝑑-axis, 𝑞-axis stator fluxes
′
′
𝜑𝑑𝑟
, 𝜑𝑞𝑟
𝑑-axis, 𝑞-axis rotor fluxes referred to the stator
𝜃𝑚
Turbine shaft angle
𝜃𝑔
Generator shaft angle
𝛽𝑝 , 𝛽𝑞
Dependence parameters of load powers on frequency
𝛼𝑝 , 𝛼𝑞
Dependence parameters of load powers on voltage magnitude
𝛼𝑖
Delay angle of the current source inverter
𝑣𝑥𝑦
Output dc voltage of the reduced H-bridge converter
𝑣𝑖𝑛𝑣
Input dc voltage of the current source inverter
𝑣𝑑𝑐
Output dc voltage of diode bridge rectifier
𝑣𝑑
Output dc voltage of the buck converter
𝑣𝑐𝑖𝑑 , 𝑣𝑐𝑖𝑞
𝑣𝑏𝑎𝑡
𝑑-axis, 𝑞-axis voltage of load-side capacitor filter
Battery voltage
xviii
𝑣𝑊
𝑣𝐿𝑑 , 𝑣𝐿𝑞
Wind speed
𝑑-axis, 𝑞-axis load voltage
𝑣𝐿𝐿
Load-side line voltage
𝑣𝐿
Voltage across the dc-link inductance
𝑣𝑔𝑎𝑏𝑐
𝑣𝑐𝑔𝑑 , 𝑣𝑐𝑔𝑞
𝑣𝑐𝑔𝑎𝑏𝑐
𝑣𝑐𝑏
𝑣𝑐𝑖
𝑎𝑏𝑐
3-phase generator voltage
𝑑-axis, 𝑞-axis voltages across excitation capacitor
3-phase voltage across excitation capacitor
dc voltage across capacitor of the battery-side LC filter
3-phase voltages across load-side capacitor filter
𝑣𝐿𝑎𝑏𝑐
3-phase load voltage
𝑛𝑔𝑒𝑎𝑟
Gear box ratio
𝑚𝑖𝑑 , 𝑚𝑖𝑞
𝑚𝑖
𝑑-axis, 𝑞-axis modulation indices of the current source inverter
Modulation index
𝑚𝑖𝑎𝑏𝑐
3-phase modulation indices of the current source inverter
𝑘𝑝 , 𝑘𝑖
Proportional, integral gain of PI regulator
𝑘𝑜𝑝𝑡
Optimum torque constant
𝑖𝑝𝑑 , 𝑖𝑝𝑞
𝑑-axis, 𝑞-axis currents of the primary side of the Δ/𝑌𝑛 transformer
𝑖𝑜𝑑 , 𝑖𝑜𝑞
𝑑-axis, 𝑞-axis components of the fundamental current of the current source
inverter
𝑖𝑖𝑛
Input dc current of the reduced H-bridge converter
𝑖𝑑𝑠 , 𝑖𝑞𝑠
𝑑-axis, 𝑞-axis stator currents
′
′
𝑖𝑑𝑟
, 𝑖𝑞𝑟
𝑑-axis, 𝑞-axis rotor currents referred to the stator
𝑖𝑑𝑐𝑟
Input dc current of the buck converter
xix
𝑖𝑑𝑐𝑔
Generator-side dc current
𝑖𝑑𝑐ℎ
Output dc current of the buck converter
∗∗
𝑖𝑑𝑐
dc-link current reference set according to generator optimum power
∗
𝑖𝑑𝑐
dc-link current reference set according to load demand
𝑖𝑑𝑐
dc-link current
𝑖𝑑𝑐,𝑚𝑖𝑛
Lower limit of the dc-link current
𝑖𝑑𝑐,𝑚𝑎𝑥
Upper limit of the dc-link current
𝑖𝑏𝑎𝑡
𝑖𝐿𝑑 , 𝑖𝐿𝑞
Battery current
𝑑-axis, 𝑞-axis load currents
𝑖𝑝𝑎𝑏𝑐
3-phase primary-side currents of the Δ/𝑌𝑛 transformer
𝑖𝑜𝑎𝑏𝑐
3-phase fundamental currents of the current source inverter
𝑖𝑔𝑎𝑏𝑐
3-phase generator current
𝑖𝑐𝑔𝑑 , 𝑖𝑐𝑔𝑞
𝑖𝐿𝑎𝑏𝑐
𝑑-axis, 𝑞-axis excitation capacitor currents
3-phase load current
𝑓𝑠
Switching frequency of the converters
𝑓𝑟
Resonance frequency
𝑑𝑏
Duty ratio of the buck converter
𝑑𝐴
Duty ratio of the reduced H-bridge converter
𝑌𝑃 , 𝑌𝑄
𝑌𝑄−𝑝𝑑 , 𝑌𝑄−𝑝𝑞
Real, reactive power indices of the generic load model
Power indices of the extended generic load model associated with the negative
sequence current
𝑊𝑜
Observability Gramian matrix
𝑊𝑐
Controllability Gramian matrix
xx
𝑉𝑜
Nominal load voltage
𝑉𝐿,𝑚𝑖𝑛
Minimum instantaneous voltage applied across the dc-link inductor
𝑉𝐿,𝑚𝑎𝑥
Maximum instantaneous voltage applied across the dc-link inductor
𝑇𝑚
Turbine mechanical torque
𝑇𝑒
Generator electromagnetic torque
𝑅𝑠
Stator resistance
𝑅𝑟′
Rotor resistance refereed to the stator
𝑅𝑑𝑐
Resistance of the dc-link reactor
−
−
𝑄𝑝𝑑𝑜
, 𝑄𝑝𝑞𝑜
Nominal powers of the extended generic load model associated with the
negative sequence current
−
−
𝑄𝑝𝑑
, 𝑄𝑝𝑞
Powers of the extended generic load model associated with the negative
sequence current
𝑃𝑜 ,𝑄𝑜
𝑃𝑚
𝑃𝑔 , 𝑄𝑔
Nominal active, reactive powers of the load
Mechanical power captured by the wind turbine
Generator active, reactive power
∗
𝑃𝑏𝑎𝑡
∗
Maximum average power allowed to be absorbed by battery under 𝑖𝑑𝑐
𝑃𝑏𝑎𝑡
Battery power
𝑃𝐿 , 𝑄𝐿
Load active, reactive power
𝐿𝑚
Magnetizing inductance
𝐿𝑙𝑠
Stator leakage inductance
𝐿′𝑙𝑟
Rotor leakage inductance refereed to the stator
𝐿𝑑𝑐
dc-link inductance
𝐿𝑏
Inductance of battery-side LC filter
𝐾𝑠𝑒
Shaft equivalent stiffness factor
xxi
𝐽𝑚
Turbine rotor inertia
𝐽𝑔
Generator rotor inertia
𝐽𝑒𝑞
Equivalent rotor inertia
𝐷𝑠𝑒
Shaft equivalent damping factor
𝐶𝑟
Capacitance of ac capacitor filter of current source rectifier
𝐶𝑝
Turbine’s performance coefficient
𝐶𝑖
Capacitance of ac capacitor filter of current source inverter
𝐶𝑔
Capacitance of ac excitation capacitor of the generator
𝐶𝑑𝑐
Capacitance of dc capacitor filter of diode bridge rectifier
𝐶𝑏
Capacitance of battery-side LC filter
𝑣̅𝑤
Average wind speed
𝑣𝐿𝑑 , 𝑣𝐿𝑞
𝑑-axis, 𝑞-axis synchronous frame load voltage (under balanced load)
xxii
Chapter 1
Introduction
Recently, utilization of wind energy has achieved a rapid growth in Europe, North America
and Asia. Global Wind Energy Council reported that the total capacity of grid-tied wind
energy, installed in 2015 alone, exceeded 63GW worldwide [1] . Small-scale wind turbines
(WTs), mainly employed in off-grid applications, have also received a lot of demand
worldwide [2] . In 2014, the U.S. department of energy (DOE) reported installed capacity of
2.8 MW for small WTs in the United States and exported capacity of 11.2 MW to the global
market [3]. In 2015, Canada installed around 1.5 GW of new wind capacity in 36 projects, 23
of which were deployed to serve off-grid communities, as well as municipal or local
ownership [4] . Asia is still the largest regional market for wind energy. In China alone, the
target is to reach 200GW of wind power capacity by 2020 [5]. A considerable percentage of
this target capacity is expected to be off-grid, due to grid connection issues.
1.1
Research Motivations
According to Global Off-Grid Lighting Association, over 25% of the world’s population,
mostly in developing countries, has no access to electricity [6]. Even in modern countries, there
are remote communities where connection to the main grid is either too expensive or
impractical. For example, there are around 175 off-grid communities throughout Canada [7].
Employing renewable energy sources is the most suitable solution for off-grid applications, if
intermittency is compensated for. At present, standalone small WTs, ranging in power rating
from a few hundred watts to a hundred kilowatts, provide a very attractive renewable energy
source for remote communities. These WTs help in reducing the stress on the grid by supplying
part of the demand without involving the grid, diminish the air pollution [8] and save on fuel
cost by reducing or even eliminating the need for diesel generators, which consume a lot of
air-polluting fossil fuels, have high operating and maintenance costs, and may require
significant additional costs if installed in a remote area, where fuel transportation and refueling
is a complicated mission [9]. Moreover, standalone WTs can be installed wherever wind
resource is adequate and there is no access to the grid, or connection to the grid is very
costly [10], not permitted or difficult due to official approval requirements. In addition to
1
remote communities, standalone small WTs can be used to supply power to boats, recreational
vehicles, cottages, local schools, farms, and small manufacturing facilities.
Although the main principles of operation are the same in on-grid and off-grid wind energy
conversion systems, the absence of grid in the latter case adds to the hardware and control
requirements. In spite of the fact that wind energy is intermittent and cannot be dispatched to
meet the assigned commitment, connection to the grid allows for extracting the maximum
available power from wind resources at any moment of time. In contrast, for an off-grid Wind
Energy Conversion System (WECS) to satisfy the time-varying power demand and maintain
balance of power, an energy storage unit is required to compensate for the power deficit and
absorb the excess power generated from wind. Another issue with off-grid WECS is the
reactive power required by some generator types that has to be supplied by a reactive power
source such as a capacitor bank, synchronous condenser, SVC or STATCOM [11].
To date, voltage-source inverter (VSI) is the dominant topology in both large and small-scale
WECS. Current-source inverter (CSI), on the other hand, has been adopted mainly in mediumvoltage, high power applications. The advantages reported in the literature for CSI, when
substantiated, can make CSI a promising option and possibly a preferred choice for small-scale
standalone WECS. Motivated by the huge demand for off-grid small-scale wind turbines and
potential of CSI to be employed in such turbines, the research presented in this thesis intends
to investigate the feasibility of CSI-based WECS for off-grid applications.
1.2
Literature Review
Compared to fixed-speed WTs, variable-speed WTs produce more energy from the same
wind resource, with less power fluctuations and lower mechanical stress.
Fixed-speed WTs, in general, use squirrel-cage induction generator, with no power
electronic interface [12]-[14]. On the contrary, variable-speed wind turbines enjoy a rather
wide range of options for appropriate generator and power converter types. This section
provides a review on variable-speed WECSs from generator and converter viewpoints. The
section starts with a brief historical data.
2
1.2.1 History of Wind Energy
Wind is a renewable energy source that has been used for a long time in water pumps, wheat
mills and sailing ships. Since1980s, wind has been recognized as an efficient and reliable
source for generating electricity [15]. In the recent years, the utilization of small wind turbines
(SWTs), mainly adopted in off-grid projects, has grown all over the world and is expected to
become more desirable in the future, especially with the development of energy storage
technologies with the required capabilities. Fig. 1.1 illustrates the global annual installed SWT
since 2009 and forecasts the annual growth rate until 2020 [2]. According to the figure, the
market of SWT could subsequently feature a steady growth rate of 20% from 2015 to 2020.
By 2020, a cumulative installed capacity of approximately 2 GW is expected to be achieved.
Fig. 1.1: Global annual installed Small Wind turbine (SWT)[2].
1.2.2 Conventional and Potential Generator Types used in WECS
The following generator types have been employed in the existing wind energy conversion
systems or have been reported in the literature:
1. Wound-Rotor Induction Generator (WRIG)
2. Doubly-Fed Induction Generator (DFIG)
3. Brushless Doubly-Fed Induction Generator (BDFIG)
3
4. Brushless Doubly-Fed Reluctance Generator (BDFRG)
5. Squirrel-Cage Induction Generator (SCIG)
6. Wound-Rotor Synchronous Generator (WRSG)
7. Permanent-Magnet Synchronous Generator (PMSG)
8. Permanent-Magnet Induction Generator (PMIG)
9. Switched-Reluctance Generator (SRG)
Fig. 1.2 shows a simplified configuration for a standalone, variable-speed WRIG-based
WECS. The stator is connected to the PCC (point of common coupling), while the rotor is
connected to a combination of a fixed resistance (𝑅𝑒𝑥𝑡 ) and a power electronic converter that
emulates an adjustable resistance. By varying the value of the resistance seen by the rotor
windings, the generator can run at different operating points. A soft starter is needed in order
to reduce the inrush current at start-up [12]. Standalone WRIG is simply controlled to produce
stable voltages with constant amplitude and frequency even though rotor speed is varied by
several percent [16]-[17] . Due to limited range of speed variation, WRIG has been used for a
long time in fixed- speed WTs, rather than variable-speed WTs [18] .
External variable rotor resistance
Rext
PCC
Transformer
Gear
box
Soft
Starter
LOAD
Battery
Voltage Source
Converter
VAR
Compensator
Fig. 1.2: WRIG-based standalone WECS.
4
Dump Load
WRIG
A DFIG-based standalone WECS is constructed by connecting the stator directly, and the
rotor via a power electronic converter, to the PCC, as shown in Fig. 1.3. The flow of power
through the stator is unidirectional, while the direction of power flow through rotor depends
on the operational mode of generator. If the generator is operating below synchronous speed,
the power is received by the rotor. If the generator is operating above synchronous speed, the
rotor delivers power. The main advantage offered by DFIG is that its rotor power converter is
rated only at 30% of the stator power [15]. This feature makes DFIG a preferred choice in highpower grid-connected WECSs, due to the huge economic gains resulting from reduced sizes
of power converters and filters [19]. Moreover, different control strategies, developed and
investigated in [20]-[22], have demonstrated ease of control of DFIG in standalone wind
energy applications, especially from the voltage regulation point of view. However, the rotor
voltage and current need to be carefully controlled during the initial transients, as they can be
too high to be handled by the reduced-size converters [23].
PCC
DFIG
Transformer
Gear
box
LOAD
DC Link
Load-Side
Converter
Vdc
Battery
Filter
Dump Load
Rotor-Side
Converter
DC/DC Buck-Boost
Converter
Fig. 1.3: DFIG-based standalone WECS.
BDFIG consists of two cascaded wound-rotor induction machines, one for power generation
and the other one for control [24]. BDFIG has two groups of stator windings referred to as
power winding (PW) and control winding (CW). As shown in Fig. 1.4, the PW is directly
5
connected to the PCC, while the CW is connected to the PCC through two back-to-back
reduced-size power converters, i.e., machine-side converter (MSC) and load-side converter
(LSC). BDFIG’s benefits are similar to those of DFIG [24] . Nevertheless, its size is larger,
and the complexity of its assembly and control is higher [19], for the same power rating.
Despite these disadvantages, BDFIG is still attractive for large grid-connected WTs, especially
for off-shore applications, where WTs have to be very reliable and nearly maintenancefree [25],[26].
BDFIG
PCC
Transformer
PW
Gear
box
LOAD
CW
Vdc
Filter
MSC
Battery
LSC
Dump Load
DC Link
DC/DC Buck-Boost
Converter
Fig. 1.4: BDFIG-based standalone WECS.
BDFRG is a different design featuring a reluctance rotor instead of a wound rotor in BDFIG.
Although BDFRG is more efficient and reliable than BDFIG, it still has a complex rotor design
and a large size due to a smaller torque-volume ratio [27]. However, recent improvements in
reluctance rotor design may result in higher future interest in the BDFRG [28].
Amongst the traditional induction generators, SCIG is the smallest in size, lowest in cost and
most robust in structure [29],[30]. As a mature machine in wind energy applications, SCIGbased WT systems have been of interest in many research projects, including simulator design,
emulator set-up, novel power converters as well as control schemes, self-excitation and voltage
build up techniques in standalone applications [31]-[37]. Since SCIG is one of the highly
6
recommended generators for off-grid WTs, its possible configurations will be discussed in
detail in chapter 2 (subsection 2.1.1).
WRSG requires a dc excitation, which can be provided by either an external dc source
through slip rings and brushes or a brushless exciter, involving power electronics and an
auxiliary ac generator. Fig. 1.5 presents a typical standalone WECS based on WRSG. The
generator-side converter (GSC) is responsible for Maximum Power Point Tracking (MPPT),
while the load-side converter (LSC) controls the voltage and frequency at PCC, provided that
the DC link voltage is regulated by implementing a power management strategy that controls
the power transactions of battery and dump load under different load and wind speed
conditions. WRSG-based standalone WECS has been mentioned in [38] as a promising
alternative for serving remote load demands. Different control schemes for stator voltage
regulation in standalone WRSG are described in [39].
DC Excitation
and Control
GSC
DC Link
PCC
LSC
Vdc
Transformer
LOAD
Filter
Dump Load
WRSG
Battery
DC/DC Buck-Boost
Converter
Fig. 1.5: WRSG-based direct-drive standalone WECS.
Unlike WRSG, PMSG is a brushless self-excited synchronous machine. Since 1996, PMSG
has become more attractive than WRSG due to a decrease in the costs of permanent magnets
and power converters [40]. At present, PMSG is known to be the prominent solution in directdrive, small-scale standalone WTs [22],[41]-[49]. Therefore, its possible configurations will
be discussed in detail in chapter 2 (subsection 2.1.1).
7
By adding improved power factor and better performance to the advantages of the SCIG,
PMIG, a relatively new induction generator-based machine, has a very good potential to serve
as a direct-drive generator in grid-connected [50],[51] and isolated WECS [52]. Although
PMIG has been considered for direct-drive WTs since 1999 [53], it has just recently been
recognized in wind energy market. Due to its improved power factor and efficiency, some
manufacturers [54],[55] have started considering PMIG as a good alternative for the highefficiency PMSG, especially for small-scale WECSs. Therefore, its possible configurations
will be discussed in detail in chapter 2 (subsection 2.1.2 ).
SRG is a structurally simple and robust machine. Its stator and rotor are usually made of
steel laminations. The stator consists of a number of salient poles with windings concentrated
around them. The rotor consists of a number of salient poles and has neither windings nor
permanent magnets. Fig. 1.6 shows a typical standalone WECS using direct-drive SRG. The
machine is normally driven by an Asymmetric Half Bridge Converter (AHBC) [56][57]. As a
simple, robust, reliable and inexpensive machine with flexible control, SRG has shown a good
potential to serve as a direct-drive generator in standalone WECS [56],[57] as well as gridconnected WECSs [58]-[60].
AHBC
DC Link
Vdc
Battery
Load-Side
Converter
PCC
Transformer
LOAD
Filter
DC/DC Buck-Boost
Converter
Fig. 1.6: Standalone WECS using direct-drive SRG.
8
Dump Load
SRG
1.2.3 Power Electronic Converter Topologies for Standalone WECS
Currently, voltage-source converter (VSC) is the prominent converter type in WECSs.
However, there are other power electronic configurations deployed in WECSs. This subsection
gives an overview of the power electronic converter topologies that are commonly used, or
have the potential to be used, in standalone WECSs.
A) Diode Rectifier Bridge + DC/DC Boost Converter + VSI
A three-phase diode bridge rectifier and a voltage-source inverter (VSI) with dc link
capacitor in between, is shown in Fig. 1.7. Since the diode rectifier is an uncontrolled converter,
a dc booster is used to achieve maximum power point tracking (MPPT). A buck-boost dc/dc
converter may also be used to handle the extra power available under high wind speed
conditions [61]. Diode rectifier is a simple and cost-effective solution, especially for a
permanent magnet generator-based WECS [62],[63]. The same topology can also be used in
an induction generator-based WECS, but it is less attractive due to the need for an external
source of VAR. Employing semi-controlled rectifiers is also a possibility [64]. One problem
with using a diode rectifier as the generator-side converter is the resulting distortion in the
stator current waveforms, leading to higher losses and torque ripples in the generator.
Diode Bridge
Rectifier
DC Booster
DC-Link
Ld
VSI
D
Sboost
Generation
Side
Cd
Load Side
Fig. 1.7: Diode rectifier + dc/dc boost converter + VSI.
B) Two-Level Back-to-Back Voltage-Source Converter (2L-BTB-VSC)
The drawbacks of using a diode rectifier as the generator-side converter are avoided in the
2L-BTB-VSC system illustrated in Fig. 1.8, where rectification is done by a voltage-source
rectifier (VSR). Compared to the system shown in Fig. 1.7, 2L-BTB-VSC provides more
9
efficient MPPT control, but at a higher cost and control complexity [61]. Indeed, the 2L-BTB
VSC is a well-established technology for WECSs [27],[63], especially in low-voltage, smallpower WT systems, and thus a mature solution for standalone WECSs.
VSR
DC-Link
VSI
Load
Side
Generation
Side
Fig. 1.8: Two-level, back-to-back voltage-source converter topology.
C) Multi-Level Converter
As the power and voltage levels increase, multi-level (ML) converters are preferred.
Compared to the 2L-BTB, ML converters, especially three-level converters, produce much
lower switching losses, as well as lower switch stress, harmonic distortion and 𝑑𝑣/𝑑𝑡 stress
on the generator and transformer [65],[66]. However, due to higher number of switches, higher
cost and control complexity are associated with ML converters. In a standalone WECS, the use
of ML converters may not be justified as the levels of voltage and power are generally low.
The details of classical and advanced converter topologies have been covered in the
literature [67]-[69].
D) Current-Source Converter
Although VSCs dominate the present WECS market, due to their well-developed technology
and fast dynamic response, current-source converter (CSC) can also be a good alternative. CSC
can be constructed as BTB and ML converters. A BTB-CSC is shown in Fig. 1.9. CSC offers
some advantages over VSC [70],[71]. However, CSC has a number of drawbacks that need to
be taken care of in order to make CSC an effective solution [71],[72].
CSC is well-established for high power applications such as medium-voltage industrial
drives. Therefore, references [73],[74] have proposed the employment of CSC in Mega-Watts
on-grid WECSs. However, the use of CSC for a small-scale off-grid WECS has not been
proposed yet, although CSC can also offer some advantages in such an application.
10
DC-Link
CSR
CSI
Generation
Side
Load Side
Fig. 1.9: Back-to-back current-source converter topology.
E) Diode Rectifier Bridge + Impedance-Source Inverter
Impedance-source inverter or Z-source inverter (ZSI) was first proposed in 2003 [75]. A
converter system composed of a three-phase diode bridge rectifier and a ZSI is shown in
Fig. 1.10. Unlike traditional VSI or CSI, ZSI provides the buck-boost feature without an extra
dc/dc buck-boost converter. Moreover, dead times and overlap times, required to prevent short
and open circuit conditions in VSI and CSI, respectively, are not concerns in ZSI.
Compared to the three-stage conversion system shown in Fig. 1.7, number of switches is
reduced by one in the two-stage conversion system shown in Fig. 1.10.
References [76],[77] have shown the potential of ZSI to replace the conventional VSI with
dc booster in standalone WECSs. In [78], a buck-boost dc/dc converter was used to integrate
a storage battery unit with a ZSI-based WECS through one of the ZSI’s capacitors.
Diode Bridge
Rectifier
ZSI
D
Generation
Side
Load Side
Fig. 1.10: Diode Rectifier Bridge + Z-Source inverter.
11
F) Matrix Converter
The presence of bulky dc-link capacitor and inductor in VSC and CSC topologies reduces
their efficiency and shortens their overall lifetime. On the contrary, matrix converter (MC),
which is a direct ac-ac converter, has a higher efficiency, longer life and reduced size due to
the absence of dc-link energy storage devices [79]. The configuration of a matrix converter,
using nine bi-directional switches, is shown in Fig. 1.11. The bi-directional switches allow
voltage blocking and current conduction, irrespective of voltage polarity and current direction.
The switches are controlled in order to produce the desired output voltage magnitude and
frequency on the load side. The output frequency of MC is unrestricted (limited only by the
switching frequency), but its output voltage magnitude is limited to 0.866 of that of the input
voltage.
One of the challenges in MCs is safe commutation in the absence of freewheeling paths.
Connecting two input lines to the same output line causes a short circuit on the input side (i.e.,
generation side), whilst disconnecting one of the output lines causes an open circuit on the
output side (i.e., load side), causing over-voltage spikes. Therefore, safe commutation of MC
has triggered a good deal of research activity [80],[81].
In 2001, a novel MC topology, free of the commutation problems, was proposed by Wei and
Lipo [82]. Such a converter is known as indirect matrix converter (IMC). The idea of this twostage converter is based on a fictitious dc link, although no energy storage element exists
between the supply- and load-side converters.
The employment of MC in an on-grid SCIG-based WECS has been proposed by [83]. IMC
has been used in an on-grid DFIG-based WECS in [84]. Although the use of MC and IMC in
a small-scale off-grid WECS can be a possibility, the commutation problems in MC need to
be carefully addressed, and the absence of dc link component, in both configurations,
complicates the integration of an electrical energy storage system.
12
Bidirectional
Switch
Generation
Side
Load
Side
Fig. 1.11: Matrix converter.
Amongst the power converters which have been employed or have the potential to be
employed in WECSs (i.e., VSC, CSC, ZSI, MC and IMC), VSC is the dominate topology in
both large and small-scale WECSs [63]. The potential of an impedance-source inverter, as a
replacement for the conventional VSI-dc booster combination in standalone WECSs, has been
demonstrated in [76]-[78]. Although the use of MC and IMC, in a small-scale off-grid WECS,
is a possibility, the commutation problems in MC need to be carefully addressed, and the
absence of dc link component, in both configurations, complicates the integration of an
electrical energy storage system. CSC is a reliable technology, but has been mainly proposed
and validated for Mega-Watts-level on-grid WECSs [73],[74]. However, the advantageous
features reported in the literature for CSC, when substantiated, can make CSC a promising
option for small-scale standalone WECSs.
In the following subsection, the advantages of pulse-width modulated current-source inverter
(PWM-CSI) over pulse-width modulated voltage-source inverter (PWM-VSI), for a smallscale WT, are highlighted.
1.2.4 PWM-CSI versus PWM-VSI for Standalone WECS
This subsection conducts a comparison based on reliability, cost, efficiency, and protection
requirements, between PWM-CSI and PWM-VSI for small-scale standalone WECS, assuming
that the inverters use six IGBT switches with six antiparallel diodes (VSI) or series diodes
(CSI), feed the same three-phase load at the same voltage and frequency, have comparable
13
ratings and operate at the same switching frequency and under the same environmental
conditions.
A) Reliability
Reliability of a power electronic converter is usually measured in terms of the rate of failures
leading to converter malfunction [85]. These failures are mainly related to the switching
semiconductor devices, capacitors, inductors and transformers. According to a study reported
in [86], 34% of power electronic system failures are related to semiconductor devices. These
results are consistent with those of another study reported in [87], carried out by 295 different
industrial sectors, showing that semiconductor power devices are the most fragile components
of power electronic converters.
Failures of an IGBT are classified into two classes: open-switch fault (OSF) and short-switch
fault (SSF). OSF can be caused by bond wire lift or rupture [88] and gate drive failure [89].
SSF, on the other hand, can be caused by bond wire rupture, impact ionization, collector
overcurrent, and gate circuit degradation [90]. Moreover, dynamic avalanche of antiparallel
diode can cause SSF in IGBTs [91]. In VSI, SSF is fatal and can lead to potential destruction
of the failed IGBT, the remaining IGBTs, and other components. On the contrary, SSF is not
a serious issue in CSI, but it can degrade the inverter’s performance. On the other hand, OSF
is not fatal for VSI, while it is a critical issue in CSI, because it can lead to dc-link current
interruption, producing high overvoltage transients, and destruction of the failed IGBT as well
as the remaining IGBTs.
Similar to IGBT, a power diode can fail as open-switch or short-switch. The open-switch
failure mode is mainly caused by bond wire lift or rupture, while the short-switch failure mode
can take place as a result of static high voltage breakdown, rise of leakage current, snappy
recovery, reverse recovery dynamic avalanche, and high-power dissipation [92].
Considering the number of causes of faults mentioned above, one can conclude that in IGBTs
and diodes, the probability of short-switch failure mode is higher than that of open-switch
failure mode. SSF in IGBT or its antiparallel diode is fatal in VSI, thus diminishing its
reliability significantly. On the contrary, SSF does not adversely affect the reliability of CSI
unless it lasts for a long period of time.
14
The second most fragile component in a power electronic converter is capacitor [85]. The
results of the study reported in [87] suggest that capacitors are responsible for 18% of power
electronic converter failures, while only 5% of failures are related to inductors. Therefore, the
impact of high failure rate of dc-link capacitor on the reliability of VSI is much higher than
that of low failure rate of dc-link inductor on the reliability of CSI.
Based on the above analysis, one can conclude that IGBT-based CSI is potentially more
reliable than its counterpart, IGBT-based VSI.
B) Cost
Due to the use of series diodes in IGBT-based CSI, it is expected that the cost of the system
be higher than that of IGBT-based VSI. For cost comparison, Table 1.1 shows the prices for
two 30kW-PMSG-WECSs, designed for grid connected applications, using VSI and CSI,
respectively [93]. Even though the capital cost of CSI-based system is higher than that of the
VSI-based system, the difference in the cost is not significant due to lower filtering
requirements of CSI. Unlike PWM-VSI, which requires an L-C or L-C-L filter, only a capacitor
filter is required in CSI. Moreover, the lifetime of the dc-link inductor in CSI is much longer
than that of the dc-link capacitor in VSI [71]. This can result in lower operation and
maintenance (O&M) costs for CSI-based WECS. Overall, capital cost of CSI is slightly higher
than that of VSI, but its O&M costs are lower.
Table 1.1: Cost Comparison of 30kW PMSG-based Wind Turbines with VSI and CSI [93].
Component
IGBT-based PWM-VSI
US$
2,145
IGBT-based PWM-CSI
US$
2,145
Generator
18,015
18,015
Controller + rectifier
3,204
3,204
Inverter (including filter)
10,987
11,583
34,351
34,947
Blades (3, Horizontal axis)
Total
C) Efficiency
In a VSI, the dc-link capacitor has a small Equivalent Series Resistance (ESR) and carries
only the ripple components of current at steady state, whereas in a CSI the dc-link inductor has
15
a larger ESR and carries both dc and ripple components of the dc-link current. Moreover, since
there are always four devices (i.e., two IGBTs and two diodes) conducting in a CSI, its
conduction energy losses are expected to be higher than that of VSI over a specified period of
time. Quantitative analysis performed by [94] has shown that conduction losses of CSI can be
more than 2 times higher than those of VSI. It is noteworthy that conduction losses in CSI can
be reduced if the dc-link current is reduced according to reduction in demand or if reverseblocking (non-punch-through) IGBTs, that do not need series diodes, are employed. The
quantitative analysis in [94] shows that switching losses of a CSI are only 30% of those of a
VSI of comparable ratings. This can be justified by the high commutation voltages experienced
by the VSI switches during turn-on and turn-off processes. Overall, compared to CSI, VSI
features a higher efficiency.
D) Protection Requirements
In sinusoidal PWM VSI, the peak value of the fundamental component of ac-side line
voltage can be expressed as
√3
𝑉̂𝐿𝐿,1 = 2 (𝑚𝑖 )(𝑣𝑑𝑐 )
(1.1)
where 𝑣𝑑𝑐 is the dc-link voltage and 𝑚𝑖 the modulation index (0 < 𝑚𝑖 ≤ 1). To guarantee a
constant 𝑉̂𝐿𝐿,1, the constraint on the dc-link voltage is given as
𝑣𝑑𝑐 ≥
2
√3
𝑉̂𝐿𝐿,1 = 2 𝑉̂∅,1
(1.2)
Similarly, in sinusoidal PWM CSI, the peak value of the fundamental component of ac-side
line current can be expressed as
√3
𝐼̂𝐿,1 = 2 (𝑚𝑖 )(𝑖𝑑𝑐 )
(1.3)
where 𝑖𝑑𝑐 is the dc-link current. Thus, the constraint on the dc-link current is given as
𝑖𝑑𝑐 ≥
2
𝐼̂
√3 𝐿,1
(1.4)
From (1.2), the dc-link voltage in VSI must be at least twice the peak value of the
fundamental component of the ac-side phase voltage, irrespective of the power demand level.
In case of a fault at the ac terminals, a huge transient current will flow through the switches
that are on at the moment of fault occurrence. Therefore, a current limiter is absolutely
16
necessary to protect the switches against overcurrent [95]. In CSI, no current limiter is required
because the current is limited by the dc-link current controller, giving the topology an inherent
current-limiting feature. On the other hand, the dc side of CSI acts as a current source and
hence a fast detection and protection during dc-link current interruption is a must [96].
However, from (1.2) and (1.4), one can notice that unlike the dc-link voltage in VSI, the dclink current in CSI can be reduced if the power demand is reduced. This reduces the
consequences of current interruption at low demands.
Based on the comparisons from the viewpoints of capital cost, overall efficiency, and opencircuit fault protection requirements, IGBT-based PWM-VSI is preferred for small-scale offgrid WECS. On the contrary, IGBT-based PWM-CSI is the winner in terms of reliability,
O&M cost, and short-circuit fault protection requirements. The following advantages offered
by CSI over VSI need to be considered as well when choosing the converter topology for a
small-scale off-grid WECS.
1) CSI has an inherent voltage-boost capability, which is an advantage in WECS application,
where the rectified dc voltage is low at low wind speeds. This feature will help in capturing
wind power at low wind speeds, thus providing a wider range of operation.
2) In a wind turbine, the gearbox, generator and the associated power converter are usually
installed in the nacelle. If the CSI is installed at the bottom of the tower, the length of the
connecting cable can help reduce the size and thus cost of the required dc-link inductor.
Even though based on the above discussions, CSI offers high potentials for small-scale (<
100 kW) off-grid WECS, its performance in such an application has never been investigated.
Therefore, this dissertation focuses on employing CSC in low-power off-grid WECS.
1.2.5 Energy Storage in Standalone WECS
For an off-grid WECS to satisfy time-varying power demand and maintain balance of
power, an energy storage unit is required to compensate for the power deficit and absorb the
excess power generated from wind. Moreover, energy storage improves the quality of power
delivered to the varying load. The energy storage technologies that are feasible for wind energy
integration are: Flywheel Energy Storage (FES), Pumped Hydro Energy Storage (PHES),
Compressed Air Energy Storage (CAES), Super-capacitor Energy storage (SES),
Superconducting Magnetic Energy Storage (SMES), Lead Acid Battery (LAB), Nickel
17
Cadmium battery Energy Storage (NCES) , Lithium Ion battery Energy Storage (LIES),
Sodium Sulphur battery Energy Storage (NaSES), Sodium Nickel chloride battery energy
storage (ZEBRA), Flow Battery Energy Storage (FBES), and Hydrogen Energy Storage
(HES). The details of these storage systems have been covered in the literature [10],[97]-[99].
Table 1.2 summarizes the main advantages and drawbacks of each energy storage technology.
Table 1.2: Advantages and drawbacks of different energy storage technologies.
Class
Topology
Advantageous
Drawbacks
Mechanical
storage
systems
FES
- High power density
- High efficiency
- Long Lifetime
- Environmentally inert
- Low energy density
- Full self-discharge/day
PHES
- Mature technology
- High capacity
- Low capital cost/kWh
- Small self-discharge/day
- Low energy density
- Suitable site requirements
CAES
- Low capital cost/kWh
- Small Self-discharge/day
- Long lifetime
- Suitable site requirements
- Fuel requirement
SES
- High power density
- Long cycle life
- Small environmental impact
- Low energy density
- High self-discharge/day
SMES
- High power density
- Fast response
- Long lifetime
- High efficiency
- Low energy density
- Temperature sensitivity
- High capital cost/kWh
- High self-discharge/day
LAB
- Mature technology
- Low capital cost/kWh
- Short lifetime
- Temperature sensitivity
NCES
- High power density
- Longer lifetime and less
temperature sensitivity compared to
LAB
- Memory effect
- High self-discharge/day
LIES
- High efficiency
- High power density
- Lighter and smaller than NCES
- High capital cost/kWh
- Special charging requirements
to keep voltage and current
within safe limits
NaSES
- High power & energy densities
- High capital cost/kWh
Electrical
storage
systems
Chemical
Storage
systems
18
- Design safety concerns
- High self-discharge/day
- High temperature requirement
(320-340⁰C)
ZEBRA
- Overcharge and discharge
Capability.
- Safer than NaSES
- Lower energy and power
densities compared to NaSES
- More expensive than NaSES
- High self-discharge/day
- High temperature requirement
(250-350⁰C)
FBES
- Decoupled power and energy
capacities
- High power density
- Full discharge capability
-Small self-discharge/day
- Low energy density
- Difficult maintenance
- Complicated design requiring
moving parts such as pumps
HES
- Higher capacity than batteries
- Negligible self-discharge/day
- Immature technology
- Low efficiency
For a standalone WT, installed in a windy location, the storage device is mainly used for
short term power balancing. Such an application requires certain characteristics of the storage
device used. Table 1.3 lists the requirements of energy storage system (ESS) to be employed
in small-scale standalone WECS, supplying a remote area.
In general, PHES and CAES have reduced reliability and slow transient response due to
moving parts. Moreover, they require large areas for installation. Therefore, they are not good
options for small-scale standalone WECS. On the other hand, they are very viable for energy
management in large power (i.e., hundreds of Mega-Watts) applications [98].
On the other hand, batteries and super-capacitors are reliable and fast in response due to the
absence of kinetic components. Among the batteries, Lead Acid Batteries (LABs) feature a
well-established energy storage technology and thus still represent a low-cost option for
standalone WECS [99]. However, they suffer short cycle life. One option for overcoming this
drawback and improving overall performance is integrating LABs with SESs [100].
19
Table 1.3: Energy storage requirements for a small-scale standalone WECS.
Requirement
Comments and justification
High reliability
Wind power is intermittent and energy storage device
must be available to compensate for power mismatch.
Fast response
Storage device should rapidly respond to charge/discharge
commands because wind and load are stochastically
fluctuating.
Short ramping time
Storage device should be fast-response and able to ramp
up and down from no power to rated power and vice versa
within a reasonable time.
High power/energy capability
Short-term power balance requires a storage device with
high power density, whilst energy management
applications require devices with high energy density.
High efficiency
Efficiency of storage device is important, especially when
delivering the whole power demand during turbine’s
shutdown.
Low maintenance requirements Maintenance tends to be very costly in remote areas.
Based on the above discussions, the advantages and drawbacks of storage devices provided
in Table 1.2, and the requirements given in Table 1.3, Table 1.4 assesses the feasibility of each
technology for a small-scale standalone WECS supplying a remote community. According to
the table, LAB and SES seem to have the highest feasibility for low power off-grid
applications. An excellent performance can be obtained by combining the two technologies in
a hybrid storage system. However, in order to reduce the initial cost, LAB is selected for
storage purposes in this research, as a well-established storage technology, offering satisfactory
performance. If cost is not an issue, Lead acid batteries can be replaced by Lithium-Ion
batteries featuring longer cycle life and lighter weight.
20
Table 1.4: Feasibility of Energy Storage Technologies for small-scale standalone WECS.
Technology
Feasibility level
Comments and justification
FES
Medium
-Reduced reliability due to moving parts
PHES
Very Low
- Low reliability due to moving parts
- Long response time
- Site requirements may not be met
CAES
Very Low
- Gas requirement makes it costly
- Low reliability due to moving parts
- Site requirements may not be met
- Long response time
SES
High
- High efficiency
SMES
Medium
- High production cost
- High temperature sensitivity
LAB
High
- Well-established and low-cost technology
NCES
Medium
- Undesired memory effect
- High self-discharge ratio/day
LIES
Medium
- High production cost
- Special charging requirements
NaSES
Medium
- High production cost
- High temperature requirement
ZEBRA
Medium
- More expensive than NaSES
- High temperature requirement
FBES
Low
- Difficult maintenance
- Complicated design, requiring moving parts
HES
Medium
- Immature technology
21
1.3
Research Objectives
Based on the assessment of the state-of-the-art in wind-turbine generators, power electronic
converters, and energy storage technologies, the focus in this work will be on developing a
low-power standalone WECS based on current-source inverter.
The main objectives of this research are:

Selecting the most promising generator type based on a thorough comparative
evaluation.

Selecting an appropriate CSC-based configuration for standalone WECS.

Introducing a low-cost small-scale WECS featuring a CSI and a novel integration
system for Lead Acid battery-based energy storage system.

Developing overall dynamic mathematical model for the proposed system.

Investigating the stability, controllability and observability of the proposed system.

Designing closed-loop controllers to take care of the following tasks:
1. Maximum power point tracking (MPPT) on the generator side;
2. DC-link current control;
3.
Load-side voltage magnitude and frequency control under both balanced and
unbalanced three-phase loads; and
4. Power management among generator, load, dump load and storage battery.

1.4
Proposing a dump load-less version of the proposed system.
Thesis Layout
The rest of this thesis is organized as follows:

The most promising generator type for small-scale standalone WECS supplying a
remote area is identified in Chapter 2.

Chapter 3 investigates various current-source converter topologies for wind power
generation system. The comparison between these topologies leads to the selection of
a simple and low-cost converter configuration, offering satisfactory performance for
low-power low voltage WECS. This chapter introduces the structure of the proposed
system, with a novel scheme for the integration of a battery-based energy storage.
22

Chapter 4 carries out the derivation of dynamic and steady-state models of the proposed
system in the dq reference frame. A generic model suitable for balanced/unbalanced
load conditions is proposed in this chapter. Based on the overall model, a linearized
model is developed to investigate local stability and system performance.

The design of a closed-loop control system is discussed in Chapter 5. The control
system includes the dc-link current control loop, the generator speed control loop, and
the load voltage control loop. The performance of the control system is demonstrated
by simulation.

The main contributions and outcomes of the thesis are summarized in Chapter 6,
followed by suggestions for future research work.
23
Chapter 2
Selection of Generator Type for Small-Scale WECS
In this chapter, the most promising generator type for small-scale standalone WECS,
amongst those mentioned in chapter 1 (subsection 1.2.2), will be identified.
2.1
Evaluation of Conventional and Evolving Generator Types for Standalone WECS
Selection of the right generator type is of key importance to successful capturing of wind
energy under different wind speed conditions, especially at low wind speeds, where the low
power available has to be processed by a high-efficiency conversion system.
Selection of an electrical generator for standalone turbine has been briefly discussed in [101],
where induction and synchronous generators are compared, concluding that the generator for
standalone turbine must be a permanent magnet (PM) machine in order to avoid excitation
requirement. The paper misses to address other issues that need to be considered in addition to
excitation requirements. Reference [62] has reviewed the key technologies of small-scale offgrid wind turbines. However, among all possible machines, the review has focused on PM
generators only. PM generators, especially direct-drive PM synchronous generators, are the
most commonly used electric machine for small-scale WTs [62] and have been of interest to
many researchers as a viable solution for standalone WECS [42],[43],[48],[49]. However, the
attraction to direct-drive PMSG has been based on the criteria of high power density and
reliability only. On the other hand, indirect-drive SCIG has been recommended by [31]-[35]
as a simple, robust, brushless and cost-effective generator for standalone WECS. However, the
attractiveness of such a generator may diminish if efficiency is a main concern.
The above discussion points to the fact that a more comprehensive study should be carried
out leading to selection of the most appropriate generator type for a standalone WECS under
specific conditions. Some principles for generator selection in small off-grid WTs were listed
in [62]. However, some important factors such as control requirements and construction
complexity were not considered. Furthermore, excitation requirement was not an issue in [62],
since the paper has focused on PM generators only.
24
Motivated by lack of a comprehensive and convincing approach to selection of the right
generator for a standalone wind turbine, a thorough study, considering all possible options, has
been conducted in [102]. The study evaluated the nine generator types considered for WECS
in the literature review covered in chapter 1 (subsection 1.2.2); i.e., WRIG, DFIG, BDFIG,
BDFRG, SCIG, WRSG, PMSG, PMIG, and SRG. The evaluation has been conducted on the
basis of efficiency, reliability, cost, operation and maintenance requirements, construction
complexity, control complexity, excitation requirements and noise level associated with each
generator type.
The main observations made based on the study are summarized below.

Wound-Rotor Induction Generator (WRIG): Soft starter requirement, limited speed
range and reduced efficiency due to the power loss in the external resistance are the
major drawbacks of WRIG-based WECS. Moreover, the presence of slip rings and
brushes, requiring regular maintenance and replacement, makes WRIG not an attractive
option for remote area applications, where maintenance is difficult and costly.

Doubly-Fed Induction Generator (DFIG): The feature of reduced power converter
rating offered by DFIG-based WECS may not be a big attraction in the case of
standalone wind turbine systems, where power level is relatively low (ranging from a
few kilowatts to a few hundred kilowatts). Moreover, DFIG has the drawback of
unavoidable use of brushes and slip rings, reducing its reliability and increasing its
maintenance requirements.

Brushless Doubly-Fed Induction Generator (BDFIG): Although brush and slip ring
problems do not exist in BDFIG, giving it an advantage from efficiency point of view,
large size and construction complexity are issues that can defeat its attractiveness for
small WTs. The same drawbacks are present in Brushless Doubly-Fed Reluctance
Generator (BDFRG), despite offering a higher efficiency.

Squirrel-Cage Induction Generator (SCIG): SCIG is a simple, robust, brushless and
cost-effective generator for standalone WECS.

Wound-Rotor Synchronous Generator (WRSG): The need for an external dc source to
excite the rotor winding via brushes and slip rings, or a brushless excitation system
featuring higher complexity and cost, is the main obstacle for adopting WRSG option
in off-grid applications.
25

Permanent-Magnet Synchronous Generator (PMSG): Thanks to its high-efficiency,
PMSG has recently gained widespread acceptance as a viable option in direct-drive,
small-scale standalone WECSs.

Permanent-Magnet Induction Generator (PMIG): Adding an improved power factor
and a better performance to the advantages of the SCIG, PMIG has a very good
potential to serve as a direct-drive generator in isolated WECS.

Switched-Reluctance Generator (SRG): SRG has the potential to become a good
solution for direct-drive WECS in off-grid applications. Although application of SRG
in wind energy systems was proposed in the early 1990s, its performance evaluation
has been limited to simulation and some laboratory tests, with no field implementation.
Thus, when compared with PMSG- and SCIG-WECS, SRG-WECS is still considered
to be in early stages of development.
Based on the above remarks, SCIG, PMSG and PMIG seem to be the most suitable generator
types for standalone WECS. However, PMIG is relatively an immature machine, when
compared to PMSG and SCIG. Therefore, the advantages and drawbacks of PMSG and SCIG
will be compared in more details, leaving PMIG to be visited next.
2.1.1 SCIG-WECS versus PMSG-WECS
The comparison in this subsection will be on the basis of topology, efficiency, reliability,
control complexity, cogging torque, noise, and cost.
A) Topology
PMSG -based WECS offers an advantage over SCIG-based WECS in terms of possibility of
eliminating the need for gearbox. Thus, they are called gearless-PMSG and geared-SCIG,
respectively. Fig. 2.1 and Fig. 2.2 show typical topologies for SCIG- and PMSG-based
standalone WECS, respectively. Since PMSG is self-excited, a three-phase diode rectifier can
be used as the generator-side converter, as shown in Fig. 2.2(a). In contrast, a VAR
compensator, such as a capacitor bank, is required to excite the SCIG if a three-phase diode
rectifier is to be used, as in Fig. 2.1(a). In both cases, a chopper (e.g., a DC/DC boost converter)
is required to control the speed of the generator shaft in order to achieve MPPT. Alternatively,
full generator control can be obtained by using pulse-width modulated voltage source rectifier
26
(PWM-VSR), as shown in Fig. 2.1(b) and Fig. 2.2(b). This eliminates the need for selfexcitation capacitors for SCIG, as the required reactive power is supplied by the power
electronic converter itself. In all topologies shown in Fig. 2.1 and Fig. 2.2, two-level pulse
width modulated voltage-source inverters (PWM-VSI) are used as the load-side converters.
Excitation
Capacitor Bank
Generator-Side
Control
Load-Side
Control
PCC
DC Link
SCIG
Gear
box
Transformer
Vdc
Chopper
PWM-VSI
LC Filter
Dump Load
Diode Bridge
Rectifier
LOAD
Battery
DC/DC Buck-Boost
Converter
(a)
Generator-Side
Control
Load-Side
Control
PCC
DC Link
SCIG
Gear
box
Transformer
Vdc
LOAD
PWM-VSI
LC Filter
Battery
DC/DC Buck-Boost
Converter
Dump Load
PWM-VSR
(b)
Fig. 2.1: SCIG-based standalone WECS: (a) with generator-side diode bridge rectifier, and
(b) with generator-side voltage-source rectifier.
27
Generator-Side
Control
PMSG
Load-Side
Control
PCC
DC Link
Transformer
Vdc
Chopper
PWM-VSI
LC Filter
Dump Load
Diode Bridge
Rectifier
LOAD
Battery
DC/DC Buck-Boost
Converter
(a)
Generator-Side
Control
PMSG
Load-Side
Control
PCC
DC Link
Transformer
Vdc
PWM-VSI
LC Filter
Dump Load
PWM-VSR
LOAD
Battery
DC/DC Buck-Boost
Converter
(b)
Fig. 2.2: PMSG-based direct-drive standalone WECS: (a) with generator-side diode bridge
rectifier, and (b) with generator-side voltage-source rectifier.
The fact that in the system of Fig. 2.2(a) a diode rectifier can be used without the need for
self-excitation capacitors, is considered a big advantage for PMSG-WECS over SCIG-WECS.
Indeed, it is a trend to use a diode rectifier and a boost dc/dc converter with PMSG-WECS, as
a simple and cost-effective option [62],[63].
28
B) Efficiency
Due to presence of permanent magnets in PMSG, it is not necessary to supply magnetizing
current to the stator for a constant air-gap flux. Therefore, the stator current is only responsible
for producing the torque component and hence PMSG, when compared to SCIG, will operate
at a higher PF, leading to higher efficiency. SCIG, in contrast, needs to be connected to an
external VAR source, in order to establish the magnetic field across the air gap. This results in
a low power factor and efficiency. In general, induction generators are less efficient than
synchronous generators with comparable ratings [103].
C) Reliability
Reliability of a wind turbine can be measured by frequency and duration of failures in the
system [104]. The gearbox requires regular maintenance and is not immune to failure. If it
fails, the repair required is a major task. Studies have shown that the gearbox has a very long
downtime per failure when compared with other components of WECS [105]. Thus, the
elimination of gearbox in direct-drive PMSG-based WECS can significantly improve the
reliability of the system. However, direct-drive systems feature higher number of failures in
generator and power electronic converters [106] due to direct transfer of wind turbine rotor
torque fluctuations to the generation side; however, the downtime of direct-drive systems due
to power electronics or generator failures is definitely much shorter than those of gearbox in
indirect drive systems.
Although gearless design is an advantage for PMSG-based WECS over SCIG-based WECS,
the fact that the reliability of PMSG can be affected by permanent magnet’s demagnetization
and change of characteristics under harsh environmental conditions (such as high
temperatures), is considered a serious disadvantage.
As far as the generator type is concerned, real data has shown that synchronous generatorbased turbines suffer higher failure rates than those using induction generators [107].
D) Control Complexity
In variable-speed WECS, the generator shaft speed is controlled to achieve MPPT, which is
of key importance in wind energy systems.
29
SCIG is one of the simplest machines in terms of control requirements. Control techniques
suitable for SCIG, such as direct field oriented, indirect field oriented and direct torque control,
are very well-known and well-established. In contrast, one of the drawbacks of PMSG is its
control complexity, which is caused by the fact that the magnet excitation cannot be varied and
hence the output voltage of PMSG will vary with load. This problem can be solved by
capacitive VAR compensation or an electronic voltage controller, adding to the control
complexity. Zero d-axis current, maximum torque per ampere and unity power factor, are three
common methods of PMSG control [15].
E) Cogging Torque and Noise
In PMSG, the interaction between the magnets of the rotor and the slots of the stator
generates an undesirable torque, called cogging torque, which causes fluctuations in torque and
speed of the shaft. Cogging torque results in vibration and noise in the machine, especially at
low speed and hence it can negatively affect the cut-in speed of the PMSG turbine [108].
Unlike PM synchronous machines, the phenomenon of cogging torque is not significant in
induction machines [109]. However, a geared-SCIG-based wind turbine has another source of
noise as a result of presence of gearbox in the drive train [110]. In summary, both gearlessPMSG and geared-SCIG WECS have a source of noise, which is not so important if the turbine
is installed far away from the community. However, the cogging torque of PMSG does always
matter, as it affects the cut-in speed and hence the total kWh production of the wind turbine,
leading to a lower capacity factor. Nevertheless, cut-in speed for SCIG-based wind turbine is
also restricted by the generator threshold speed, below which the machine excitation is not
possible. Thus, capacity factor is negatively affected by limitation of cut-in speed in both
PMSG and SCIG wind turbines.
F) Cost
Compared to the geared-SCIG system, the gearless-PMSG system saves on the cost of
gearbox. However, the multi-pole structure adds to the cost of gearless-drive PMSG system.
Moreover, PM generators are generally more expensive than induction generators due to the
high price of magnets.
30
The combination of a diode rectifier and a dc/dc converter, shown in Fig. 1.7, is less
expensive than a switch-mode voltage-sourced rectifier, shown in Fig. 1.8. The former
configuration is commonly used in small-scale, standalone PMSG systems [62]. If the same
configuration is to be used with SCIG, there will be an extra cost due to the need for external
exciter. However, capital cost comparison should be conducted, considering all system
components.
For cost comparison purposes, a 30kW wind turbine is selected as an example for small wind
turbines in off-grid applications. Such a turbine can supply power to a small village, a large
farm or a small enterprise, when equipped with an energy storage system. Table 2.1 shows the
prices for a gearless-drive PMSG-WECS and a geared-drive SCIG-WECS with similar power
ratings (i.e., 30kW) [111]-[113]. The comparison reveals the cost advantage of geared-SCIG
turbine with respect to gearless-PMSG turbine. The combined cost of SCIG and gearbox is
around 50% of PMSG cost. Although the price difference depends on power rating and varies
from one manufacture to another, and from one country to another, the price ratio between
geared SCIG and gearless PMSG systems are currently significant due to the involvement of
PM materials in the latter system.
Operation and maintenance (O&M) cost is another contributor to a WECS overall cost.
O&M cost includes costs of regular inspection, repair, spare parts and insurance [114]. When
comparing geared-SCIG and gearless-PMSG systems, the O&M is mainly associated with
gearbox and generator. The O&M cost for geared-SCIG is expected to be relatively high due
to the presence of gearbox, which requires regular maintenance and expensive spare parts if a
repair is needed [105]. On the other hand, the gearless-PMSG’s O&M cost is due to high rate
of failures in generator and power electronic converters [106], but it is still much lower than
the gearbox maintenance cost. Insurance of a wind turbine is also counted as a part of O&M
expenses. The insurance of a geared-SCIG turbine is considerably affected by the gearbox. The
cost of replacing a gearbox can reach 10% of the original construction cost of the wind
turbine [115], which defeats the advantage of low capital cost in a geared-SCIG wind turbine.
On the other hand, the insurance cost is generally proportional to capital cost and hence a
gearless-PMSG turbine’s insurance is negatively affected by its high capital cost, which is
expected to increase further in future due to unreliable supply of permanent magnet material
in the global market. In summary, although the presence of gearbox in a geared-SCIG turbine
31
adds to the O&M expenses, its overall cost, including capital cost, is still lower than that of a
gearless-PMSG wind turbine.
Table 2.1: Cost Comparison of 30kW PMSG- and SCIG-Based WECSs.
PMSG-WECS [111]
US$
3,890
SCIG-WECS [112]
US$
2,120
Gearbox
None
4,838
Generator
13,400
1,400
Controller
(including rectifier,
dump load and inverter)
8,500
8,630
Lead Acid Batteries (144 kWh) [113]
Total
8,400
34,190
8,400
25,388
Component
Blades (3- Horizontal axis)
Based on the comparison from the viewpoints of efficiency, reliability (particularly the
length of gearbox downtime), and external excitation requirements, the direct-drive PMSG
system represents the preferred topology for small-scale, standalone WECS. On the other hand,
based on the comparison from the viewpoints of reliability (particularly the failure rate of
generator and power converters), machine size and weight, control simplicity, and overall cost,
the indirect-drive SCIG system wins against the direct-drive PMSG system. Moreover, PMSG
might face a real problem in future due to shortage and monopoly of permanent magnet supply.
The resources of permanent magnets, especially the Neodymium type, are almost entirely
limited to China. This fact is raising concerns about shortage of PM supply in the near future
as a result of considerable increase in demand that is expected due to proliferation of Hybrid
Electric Vehicles and Electric Vehicles that commonly use PM synchronous machines for their
traction motors [116].
The main advantages and drawbacks of the geared SCIG- and gearless PMSG-based wind
energy conversion systems are summarized in Table 2.2.
32
Table 2.2: SCIG-WECS versus PMSG-WECS.
Topology
Common properties
Indirect-drive SCIG
Direct-drive PMSG
- Brushless machine
- No windings in rotor
- Full active and reactive power control
- Good control bandwidth
Advantages
- Robust operation
- Low cost
- Low generator maintenance
- Ease of control
- Gearless
- Self excited
- High PF operation
- High efficiency
- No rotor copper loss
Disadvantages
- Gear box losses and
maintenance
- Need for external excitation
- Low efficiency
- Magnet cost
- PM Demagnetization
- Large size
- Complex control
- Cogging torque
2.1.2
PMIG versus SCIG and PMSG for Standalone WECS
As mentioned above, in spite of its advantageous features, SCIG suffers from low power
factor and low efficiency, as the machine requires magnetizing current from a source of
reactive power. If part of the magnetic flux is supplied within the machine, the magnetizing
current will be reduced and hence the power factor will be improved. This can be achieved by
incorporating permanent magnets within a cage-rotor IG. Such a configuration is called
permanent-magnet induction generator (PMIG). The stator of the PMIG is similar to that of
the conventional IG, but its rotor design is different. PMIG has two rotor parts: a squirrel-cage
rotor and a PM rotor. As the squirrel-cage rotor is partially excited from the PM rotor, the
reactive power required from an external source is reduced. Moreover, PMIG can be directly
driven without a gearbox. In other words, PMIG, to some extent, combines the advantages of
SCIG and PMSG. Fig. 2.3 shows the possible configurations of standalone WECS using
PMIG. Compared to SCIG-WECS, shown in Fig. 2.1, the gearbox is no longer an essential
33
component. Moreover, the size of the capacitor bank in PMIG-WECS shown in Fig. 2.3(a), is
considerably smaller than that in SCIG-WECS shown in Fig. 2.1(a).
Reduced-size
Capacitor Bank
Generator-Side
Control
PMIG
Load-Side
Control
PCC
DC Link
Transformer
Vdc
Chopper
PWM-VSI
LC Filter
Dump Load
Diode Bridge
Rectifier
LOAD
Battery
DC/DC Buck-Boost
Converter
(a)
Generator-Side
Control
PMIG
Load-Side
Control
PCC
DC Link
Transformer
Vdc
PWM-VSI
LC Filter
Dump Load
PWM-VSR
LOAD
Battery
DC/DC Buck-Boost
Converter
(b)
Fig. 2.3: PMIG-based direct-drive standalone WECS: (a) with generator-side diode bridge
rectifier, and (b) with generator-side voltage-source rectifier.
In recent years, some manufacturers [54],[55] have started considering PMIG as a good
alternative for PMSG, especially for small-scale wind turbines. However, the construction of
PMIG is complex due to its double-rotor design which also increases the effect of cogging
torque in the machine. Another drawback of PMIG is the increase in cost due to magnet
installation. Table 2.3 shows the cost information for a 30kW PMIG-WECS. Based on the
34
information given in Table 2.1, PMIG is slightly less expensive than PMSG, but much more
costly than SCIG of the same power rating.
Table 2.3: Cost of 30kW PMIG-Based WECS.
PMIG-WECS[54]
US$
3,709
Component
Blades ( 3 - Horizontal)
Gearbox
None
Generator
12,516
Controller
(including rectifier, dump load and inverter)
Lead Acid Batteries (144 kWh) [113]
8,225
Total
32,850
2.1.3
8,400
Indices for Selecting the Preferred Generator
Based on the discussions in the previous subsections, three wind generator configurations,
namely geared-SCIG, gearless-PMSG and gearless-PMIG, were selected among all
configurations for comparison purposes. Compared to SCIG and PMSG, PMIG is relatively
new to the wind energy market. The focus of the comparison is on the generator and the
associated drive train. Therefore, the three configurations are assumed to:
1. have identical rotor blades;
2. have similar generator-side rectifiers (i.e., a three-phase diode bridge rectifier in
addition to a dc/dc converter);
3. have similar three-phase inverters;
4. have similar types and ratings of energy storage units;
5. be subjected to the same environmental conditions;
6. have comparable kW ratings;
7. be designed for off-grid application over their entire life time; and
8. be land-based wind turbines.
35
According to the discussions made in the previous subsections, Table 2.4 compares the three
recommended systems in terms of different indices. The indices are set up in order, starting
with the most important index for a small-scale, off-grid WECS supplying a remote
community, where failure in the supply system is a critical issue. Therefore, the top priority is
given to the reliability of the system, followed by its continuous O&M cost, while the lowest
priority is given to construction complexity, that is reflected in topology’s size and weight
(increasing transportation and installation costs), and noise level, assuming that the turbine is
not very close to the community that is supplied. Due to difficulties in giving an accurate
quantitative analysis (i.e., in terms of numbers or percentages), a qualitative comparison is
performed based on the discussions conducted in the previous subsections. For each index,
each system is assigned a number (1, 2 or 3) to show its rank for that index with respect to the
other two systems. If two systems are assigned the same number for a specific index, they are
at the same level for that index. As shown in the table, geared-SCIG system is prominent in
58.3% of the indices whilst gearless-PMSG system dominates in 41.7% of the indices. Also,
gearless-PMIG is similar to the gearless-PMSG in 60% of its advantages. Therefore, gearedSCIG system prevails in terms of number of indices. However, gearless-PMSG dominates in
three of the top-priority indices, namely duration of failure, gearbox O&M cost and generation
efficiency. Nevertheless, geared-SCIG is also dominant in three of the top priority indices,
namely frequency of failure, generator O&M cost, and capital cost. In order to achieve more
accurate results, the weight of an index, according to its order, should be included in the
comparison. Considering the order of each index (i) and rank of each generator (R) provided
in Table 2.4, the credit of each generator ( C ) is obtained from (2.1).
𝐶 = ∑12
1
𝑖=1[(𝑖)(𝑅𝑖 )]
(2.1)
It has been found that SCIG scores the highest credit, while the lowest credit is gained by
PMIG. Taking SCIG as base, the relative credit of PMSG and PMIG are 89% and 74%,
respectively. Therefore, the geared-SCIG proves to be the most suitable for small-scale offgrid WECS, provided that its reliability and efficiency can be improved, while maintaining the
advantage of lowest overall cost.
36
Table 2.4: Comparison of the geared-drive SCIG, gearless-drive PMSG and gearless-drive
PMIG-WECS configurations.
Order
of
index
Index
name
Details of
index
Geared
-SCIG
Gearless
-PMSG
Gearless
-PMIG
Best
option
Comments and justifications
Duration of
failure
2
1
1
1
2
2
2
Frequency of
failure
PMSG
&
PMIG
SCIG
Geared-SCIG suffers a very
long downtime per gearbox
failure
Direct-drive WT suffers
higher failure rate
3
Gearbox
2
1
1
Generator
1
2
2
PMSG
&
PMIG
SCIG
1
3
2
SCIG
3
1
2
PMSG
3
1
2
PMSG
1
2
2
SCIG
1
2
3
SCIG
1
2
3
SCIG
Drive train
2
1
1
Generator
1
2
2
1
Reliability
O&M Cost
4
Cost of
generator and
gearbox
Accounts for
gearbox and
Generator loss
5
Capital Cost
6
Efficiency
7
Excitation
requirements
Reactive power
source
8
Magnet problems
Demagnetization & security
9
Control simplicity
10
Construction
simplicity
11
Number of
poles, diameter
size, and rotor
design
Noise level
12
PMSG
&
PMIG
SCIG
No gearbox in direct-drive
WT
Generator failures are costly
in direct-drive WTs.
PM machines are expensive
due to magnets.
Gearless PMSG has neither
gearbox nor rotor copper
losses. It also operates at high
PF.
SCIG is fully externally
excited.
PMIG is partially externally
excited.
PMSG is fully internally
excited.
SCIG has no magnets.
SCIG is simple in control
while fixed magnet excitation
in PM machines complicates
their controls.
Direct-drive PMSG is large
and heavy due to multiplepole construction.
PMIG is complicated due to
double rotor design.
PMSG & PMIG have no
gearbox noise.
SCIG has no significant
cogging torque.
ORDER OF INDEX (1 TO 12) DENOTES DEGREE OF SIGNIFICANCE/PRIORITY (1: HIGHEST PRIORITY).
RANK OF SYSTEM FOR AN INDEX (1, 2 OR 3) DENOTES SUPERIORITY (1: THE BEST OPTION).
37
2.2
Summary
This chapter gave an analytical evaluation of nine generator types available in wind market
and reported in the literature, for small-scale standalone wind turbine applications.
Gearless-drive PMSG-based and geared-drive SCIG-based systems were concluded to be
the most desirable solutions among different configurations considered. These two preferred
generator types were compared with each other. Construction, efficiency, reliability, control
complexity, cogging torque and cost including capital cost as well as operation and
maintenance cost of the topology were the criteria for comparison. In terms of efficiency and
reliability, the direct-drive PMSG system was found to be the best option. In particular, the
direct-drive PMSG with diode rectifier was found to be currently the most preferred topology
for small-scale, standalone WECS, as it is less expensive compared to direct-drive PMSG with
back to back converter. However, in terms of construction, cogging torque, control simplicity
and overall cost, geared-SCIG WECS prevails.
The candidacy of PMIG to replace PMSG in a direct-drive wind turbine was discussed. As
an induction machine with improved performance, PMIG has a very good potential to be
another alternative for PMSG in small-scale WECS. However, similar to PMSG, PMIG is
suffering from issues regarding magnet cost, PM demagnetization, and insecurity of future PM
supply.
Finally, the three generation systems, namely geared SCIG, gearless PMSG and gearless
PMIG systems were compared with one another, as they are suggested by the discussion to be
the top candidates in today’s market. A group of indices were used as basis for a qualitative
comparison. The system based on geared-SCIG was shown to be the most appropriate scheme
for a small-scale standalone WECS, supplying a remote area.
The material of this chapter has appeared in the published journal paper [102].
38
Chapter 3
Proposed Wind Energy Conversion System
Based on the objectives stated in Chapter 1 and evaluation of wind generators conducted in
Chapter 2, a wind energy conversion system composed of a geared-drive squirrel-cage
induction generator and a current-source converter, integrated with a Lead Acid battery storage
unit, is proposed in this chapter. The block diagram of the proposed standalone WECS is shown
in Fig. 3.1 .In order to assess the feasibility of the proposed system, a number of possible CSCbased configurations that have been proposed for grid-connected WECS will be investigated
in this chapter.
MPPT Algorithm &
Controllers
Wind
Turbine
Rotor
Shaft &
Gearbox
Squirrel
Cage
Induction
Generator
CSC-based Power
Converters
ThreePhase
Load
Lead Acid Battery-based
Energy Storage System
Fig. 3.1: Block diagram of the proposed standalone WECS.
3.1
Configurations of CSC for Standalone WECS
In the following subsections, three CSC-based configurations, reported in the
literature [73],[74] for grid-connected WECS, will be investigated for standalone WECS. For
simulation purposes, the systems are built in Matlab/Simulink environment. In all Simulink
models, the current-source inverter is controlled by Sinusoidal PWM technique and the loadside voltage is regulated in the load-side-oriented dq synchronous frame; i.e., the reference of
d-axis and q-axis load voltages are 𝑣𝐿𝑑,𝑟𝑒𝑓 = 1 pu and 𝑣𝐿𝑞,𝑟𝑒𝑓 = 0, respectively.
39
3.1.1 Topology 1: Diode Rectifier - PWM CSI
Fig. 3.2 shows a geared-self-excited-SCIG-based WECS using a diode rectifier and a PWM
CSI. The generator is excited by a three-phase capacitor bank. The variable generated ac
voltage is rectified by the three-phase diode bridge rectifier. The dc capacitor filter (𝐶𝑑𝑐 ) assists
in smoothing the rectifier output voltage. The dc-link reactor (𝐿𝑑𝑐 ) acts as a current source for
the PWM-CSI. The size of 𝐿𝑑𝑐 is selected to reduce ripple in the dc-link current (𝑖𝑑𝑐 ) to an
acceptable level. Typically, 𝐿𝑑𝑐 is designed to have a size between 0.6 and 1.2 pu [15]. In
topology 1, the rectification is performed through a line-commutated bridge rectifier.
Therefore, the switching frequency is the line frequency and hence a bulky 𝐿𝑑𝑐 is required. The
goal of the PWM-CSI is to produce three-phase line currents at a fixed frequency. The output
C-filter (𝐶𝑖 ) absorbs the switching harmonics produced by the inverter and defines the output
voltage required at the load bus. The 𝐶𝑖 combined with cable/load inductance forms a secondorder LC filter, improving the quality of voltage delivered to the load. The C-filter design
depends on the inverter switching frequency, the LC filter resonance frequency, the allowable
line current Total Harmonic Distortion (THD), and the load type [15]. Typically, 𝐶𝑖 is in range
of 0.3 to 0.6 pu for a switching frequency of a few hundred hertz [15], assuming that the
frequency of the lowest harmonic injected by the PWM-CSI is higher than the resonant
frequency (𝑓𝑟 = 1/2 √𝐿𝐶) of the Load-side LC filter. For switching frequencies on the order
of kilo Hertz, 𝐶𝑖 is considerably reduced. The PWM-CSI, controlled by synchronous dq-axis
reference frame PI regulators, regulates the output voltage by varying the modulation index
(𝑚𝑖 ), while the frequency is set at the desired value (i.e., 50 or 60 Hz) in open-loop control. It
has to be noted that the minimum value of dc-link current 𝑖𝑑𝑐 is given by (1.4). Since the CSI
controls the load-side voltage and frequency, 𝑖𝑑𝑐 is left without control. Therefore, the
minimum 𝑖𝑑𝑐 cannot be guaranteed and MPPT cannot be achieved in this configuration.
40
Excitation
Capacitor Bank
Rotor
Blades
DC-Link
→
PWM-CSI
S1
Gear
Box
S3
S5
Load Bus
→
SCIG
Cdc
LOAD a
LOAD b
LOAD c
S2
S4
Ci
S6
C-Filter
PI
Controller
PI
Controller
-+
dq/abc
-+
PWM
Generator
abc/dq
Fig. 3.2:A SCIG-WECS composed of a diode rectifier and a PWM-CSI.
Simulation of this system is carried out on a 30kW/320V standalone WECS feeding a threephase balanced RL load at 380V/220V. The load is assumed to have a constant impedance Z,
determined from the nominal phase-to-phase voltage (i.e., 1pu), as well as the specified active
(P) and reactive (Q) power values. In the process of finding load flow solution, the load
impedance is kept constant. The effective P and Q are, therefore, varying proportionally to the
square of the bus voltage. The system’s parameters are given in Appendix A(Table A.1). The
rated wind speed is 12m/s. The diode-bridge provides no control over the generator torque or
speed. The CSI controls the load-side voltage and frequency. Due to the lack of control over
the dc-link current or generator, the dc-link current rises uncontrollably, resulting in high
reactive power absorbed from the generation side, which prevents generator’s flux from
building up and causes the generator voltage to collapse in 40 milliseconds, as shown in
Fig. 3.3. At this moment, the generator produces zero torque and the shaft over speeds. The
minimum 𝑖𝑑𝑐 is reached at t = 0.01s and 𝑖𝑑𝑐 keeps increasing until t = 0.04 s. The stored energy
in dc reactor feeds the CSI. However, the load-side voltage collapses once the reactor’s stored
energy is fully depleted at t = 0.07 s.
41
Fig. 3.3: Simulation responses for topology 1.
3.1.2 Topology 2: Diode Rectifier - Buck Converter - PWM CSI
In order to control the generator, the system of Fig. 3.2 has to be modified by inserting a dcdc buck converter between the diode rectifier and the CSI, as shown in Fig. 3.4. Compared to
topology 1, a smaller dc-link choke 𝐿𝑑𝑐 can be used due to the high switching frequency of the
buck converter. The buck converter serves as a current booster that provides decoupling
between the generation side and the CSI. Hence, the generator-side dc current (𝑖𝑑𝑐𝑔 ) is
decoupled from the CSI-side dc current (𝑖𝑑𝑐 ). Based on optimal value of turbine’s tip-speed
ratio (𝜆𝑜𝑝𝑡 ), the buck converter is used to achieve MPPT by regulating the generator shaft
speed (𝜔𝑔 ) at the corresponding optimum value (𝜔𝑔,𝑜𝑝𝑡 ) at each wind speed (𝑣𝑊 ) [73]. In the
speed-control loop, 𝑛𝑔𝑒𝑎𝑟 and 𝑟 denote the gear box ratio and radius of the turbine,
respectively. Details of this control loop will be discussed in chapter 5 (subsection 5.4.1).
42
Excitation
Capacitor Bank
Rotor
Blades
→→
DC-Link
→
PWM-CSI
S1
Gear
Box
S3
S5
Load Bus
Sbuck
→
SCIG
Cdc
D
LOAD a
LOAD b
LOAD c
S2
S4
Ci
S6
C-Filter
PWM
Generator
dq/abc
PI
Controller
PI
Controller
-+
+
PWM
Generator
-+
PI
Controller
abc/dq
Fig. 3.4: A SCIG-WECS composed of a diode rectifier, a buck converter and a PWM-CSI.
Although the buck converter adds a degree of freedom to the system control, it increases the
system’s power loss. All the power transferred from generation to load side has to pass through
the buck converter, resulting in high conduction losses. In fact, the buck switch needs to
withstand a higher stress, compared to those in the CSI. Thus, special care should be practiced
in the selection process, from the viewpoints of proper sizing and reliability.
The PWM-CSI regulates the output voltage by varying the modulation index provided that
the minimum 𝑖𝑑𝑐 is guaranteed. However, the inverter-side dc-link current 𝑖𝑑𝑐 is not controlled
in this configuration. Therefore, it may fall below its minimum value under low wind speed
and/or heavy load conditions.
To evaluate the performance of this configuration, simulation of a 30kW/320V standalone
WECS, feeding a three-phase RL balanced load at 380V/220V, is carried out. The system’s
parameters are given in Appendix A (Table A.1). The system is run under variable wind speed
and rated load. The buck converter is controlled by a PI regulator in order to adjust the rotor
speed to the reference speed (𝜔𝑔,𝑜𝑝𝑡 ) where maximum power, available at each wind speed, is
captured. Simulation results are shown in Fig. 3.5. Except for responses after t = 1 s, MPPT
controller works successfully by tracking the reference speed. At rated wind speed (12m/s)
and with balanced excitation, the SCIG produces the rated power (30 kW) at rated line voltage
(320V) and frequency (60Hz). Because no effort is made to control the inverter input dc current
43
(𝑖𝑑𝑐 ), it varies as the generator output does. Until t = 0.3 s, the minimum 𝑖𝑑𝑐 required for the
rated active load of 30kW, is guaranteed at the rated wind speed (12m/s). A unity modulation
index produces the rated voltage (380V/220Vrms) at the load bus. At t = 0.3 s, the wind speed
goes below the rated value; hence, 𝑖𝑑𝑐 falls below its minimum value and 𝑚𝑖 saturates at its
upper limit of 1. As a result, the output voltage cannot be maintained at the desired level. At t
= 1 s, the wind speed goes above the rated value. The MPPT controller is trying to extract the
optimum wind power which exceeds the demand of the system. Because this topology has no
mechanism to store or dump the excess power, the entire generated power will be transferred
to the load irrespective of the demand. However, the CSI is trying to keep rated voltage across
the constant-impendence load, which requires only rated current to flow in the load. In other
words, the CSI works against the MPPT controller. As a result of this contradiction, the excess
power is temporarily stored in the dc link reactor causing the dc-link current to rise
uncontrollably, resulting in generator loss of excitation, as well as overvoltage and overcurrent
at the load. In fact, with no mechanism of energy storage and/or dc-link current control, the
system’s behaviour under high wind speeds and/or light load is highly unpredictable and
unsafe. On the contrary, if generated power is less than the demand, the load voltage will be
lower than the desired value, which can also be harmful to the load.
One problem with using a diode rectifier as the generator-side converter is the resulting
distortion in the stator current waveforms, leading to higher harmonic losses and torque ripples
in the generator. Fig. 3.6 shows the generator stator current and electromagnetic torque at rated
operating speed. It is noteworthy that harmonic distortion varies with generated frequency. The
THD of the generator current at different wind speed is given in Table 3.1. The table shows
high THDs in the generator current, especially at low wind speed corresponding to lowfrequency operation. In order to reduce the harmonic distortion, an L filter is typically installed
on the generator side.
44
(a) Generator-Side Characteristics
(b) DC-Link-Side Characteristics
45
(c) Load-Side Characteristics
Fig. 3.5: Simulation results for topology 2.
Fig. 3.6: Effect of diode bridge rectifier on generator characteristics at rated speed.
Table 3.1: THD of SCIG stator current in Topology 2.
Wind Speed (m/s)
%THD
11
56.3
12
45.2
13
32.7
46
3.1.3 Topology 3: Back-to-Back CSC
The drawbacks of using a diode bridge rectifier has been significantly reduced in the system
illustrated in Fig. 3.7, where rectification is performed by a PWM-Current-Source Rectifier
(PWM-CSR). Such a topology eliminates the need for the excitation capacitor bank because
the reactive power, required by the SCIG, can be supplied from the PWM-CSR. However,
capacitor bank (𝐶𝑟 ) is still required at the ac side of the PWM-CSR in order to filter out the
switching harmonics in the line current and assist in the commutation of the rectifier switching
devices. Nevertheless, size of 𝐶𝑟 is much smaller than that required for generator excitation in
diode bridge rectifier configuration. 𝐶𝑟 , combined with cable/generator inductance, forms a
second-order LC filter, reducing the harmonic injected to the generator. 𝐶𝑟 ’s design depends
on PWM-CSR switching frequency, LC resonance, permitted line current THD, and generator
type [15]. Similar to 𝐶𝑖 , 𝐶𝑟 is designed under the assumption that the frequency of the lowest
harmonic injected by the PWM-CSR is higher than the resonant frequency of the generatorside LC filter. The PWM-CSR is used to harvest the maximum power available from the wind,
through regulating the generator torque or speed. The higher the CSR switching frequency is,
the smaller the dc choke that is required. Compared with the case where a diode bridge rectifier
is used, the dynamic performance of the generator is greatly improved by employing a PWM
rectifier. The induction generator can be controlled by direct field oriented, indirect field
oriented or direct torque control schemes. Direct rotor flux oriented control (DRFOC),
implemented in Fig. 3.7, is one of the most common schemes used in WECS. The idea is to
control the rotor flux (𝜑𝑟 ) and electromagnetic torque (𝑇𝑒 ) independently. 𝜑𝑟 is regulated to
align with the d-axis rotor flux (i.e., 𝜑𝑟 = 𝜑𝑑𝑟 , 𝜑𝑞𝑟 = 0) , while 𝑇𝑒 is regulated to trace the
generator’s optimum torque (𝑇𝑒,𝑜𝑝𝑡 ) at each wind speed, leading to active power flow control.
The reference q-axis stator current (𝑖𝑞𝑠,𝑟𝑒𝑓 ), required to produce optimum torque, is obtained
based on equations (4.7) and (4.29 (c)) presented in Chapter 4. Rotor flux estimator block uses
stator voltage (𝑣𝑔𝑎𝑏𝑐 ) and current (𝑖𝑔𝑎𝑏𝑐 ) to solve for stator flux, which can then be used along
with stator current to calculate the rotor flux magnitude and angle. Further details on DRFOC
can be found in [117].
47
→
PWM-CSR
Rotor
Blades
DC-Link
→
PWM-CSI
S1i
Gear
Box
S1r
S3r
S3i
S5i
Load Bus
S5r
→
SCIG
LOAD a
LOAD b
LOAD c
Cr
S4r
S6r
C-Filter
PWM
Generator
PWM
Generator
Rotor Flux
Estimator
[117]
Ci
S6i
dq/abc
dq/abc
PI
Controller
PI
Controller
-+
S2r
S4i
-+
C-Filter
S2i
abc/dq
÷
PI
-+
Fig. 3.7: A SCIG-WECS using back-to-back CSC.
To demonstrate the high performance of this topology, simulation of a 30kW/320V
standalone WECS, feeding a three-phase RL balanced load at 380V/220V, is carried out. The
system’s parameters are given in Appendix A (Table A.1). Fig. 3.8 shows that MPPT is
achieved successfully by tracking the optimum electromagnetic torque at different wind
speeds. As noticed in Fig. 3.9, the quality of generator’s stator current is significantly improved
(i.e., the current is nearly sinusoidal) when compared to topologies using a diode bridge
rectifier as the generator-side converter. This will considerably reduce the generator harmonic
losses and improve quality of generator torque. However, the advantage of lower generator
harmonic contents is diminished by the losses of CSR switches. Since the DC-link provides an
energy buffer between the generator-side and load- side converters, allowing for separate
control of the converters on the two sides, the load-side characteristics, for this topology, are
similar to those of topology 2. The 𝑖𝑑𝑐𝑟 is regulated by the PWM-CSR; however, the 𝑖𝑑𝑐 is still
uncontrolled and hence it varies as the generator output does. As a result, minimum 𝑖𝑑𝑐 is not
guaranteed for the rated load at wind speeds below the rated value, leading to possibility of dclink current collapse and failure to supply the load.
48
Fig. 3.8: MPPT for back-to-back CSC-based WECS.
Fig. 3.9: Stator current and generator electromagnetic torque at rated speed.
49
3.2
Comparison of Different CSC-WECS Topologies
Since it implements a diode bridge rectifier as the generator-side converter, the first topology
(Fig. 3.2) provides no control over the generator, making MPPT impossible. Moreover, the
two control degrees of freedom offered by CSI are used to control the voltage magnitude and
frequency at the load bus. Thus, the dc-link current is left without control and hence the system
behaviour is not predictable. This makes the first topology not applicable for standalone
WECS. However, this configuration can be used for grid-connected WECS because the
frequency is set by the grid and hence CSI can be used to control power factor at grid interface
(through reactive power compensation, providing voltage support to the grid that might be
weak at the point of connection) and dc-link current, according to a reference dictated by MPPT
controller [74].
Adding a dc/dc buck converter to the output of generator-side diode bridge rectifier in the
second topology (Fig. 3.4), introduces an additional control degree of freedom. It makes
generator-side dc current controllable and MPPT achievable as long as the generated power
doesn’t exceed the demand of the system. However, the minimum load-side dc current is still
not guaranteed at all wind speeds. Another problem associated with this topology is the
nonlinear characteristics of the diode bridge rectifier, introducing high harmonic distortion to
the generator winding currents, leading to high harmonic losses and torque ripples in the
generator. Nevertheless, this harmonic distortion can be attenuated by installing an L filter in
series with generator.
The back-to-back CSC configuration (Fig. 3.7) improves the generator performance
significantly, but with a complex control, as well as higher cost and converter losses due to
higher number of switching devices. Moreover, although the MPPT is achievable, the
minimum value of the inverter input dc current, required to maintain the output voltage, is not
guaranteed.
A brief comparison of the three topologies described above, is given in Table 3.2. As a
simple and low-cost configuration, offering satisfactory performance, topology 2 (i.e., diode
bridge rectifier - buck converter – PWM CSI) is selected as the base for detailed studies in this
thesis.
50
Table 3.2: Comparison of the three standalone CSC-based WECS topologies.
Comparison item
Diode Rectifier +
PWM CSI
Converter cost
Lowest
Control Degrees of freedom
Diode Rectifier +
Buck Converter
+ PWM CSI
Low
Back-to-back
CSC
High
2
3
Generator-side harmonics
High
High
Low
Excitation Capacitor
Required
Required
Not required
MPPT
Not applicable
Achievable
Achievable
Minimum 𝑖𝑑𝑐 requirement
Not applicable
Not guaranteed
Not guaranteed
Dynamic performance
Not applicable
Medium
High
3.3
4
Integration of Energy Storage with the CSI
Storage integration within VSI-based WECS has received a great deal of attention from both
researchers [32],[47],[65] and manufacturers [113]. However, since the CSI-based WECSs are
usually employed for on-grid applications, storage integration has not been a real concern,
neither in the literature nor in the wind market. Reference [70] designed a simple energystorage circuit to be added to a standalone small-scale generation system. The purpose of the
circuit was to handle the turn-on transient events in the load fed through a CSI. This is done
by temporarily increasing the system input power as loading increases. A capacitor was used
as the temporary storage device. To evaluate the feasibility of such a circuit for a permanent
storage purposes, it was implemented in topology 2, as shown in Fig. 3.10, but the capacitor
bank was replaced by a storage battery. The idea is to charge the battery during zero states of
the CSI. This is done by modifying the gating pattern of PWM-CSI in order to open all the
switches during the zero states only, such that the dc-link current is forced to flow through the
diode 𝐷𝑐 and charge the battery. To ensure that 𝐷𝑐 will not conduct during CSI’s active states,
the battery nominal voltage (𝑣𝑏𝑎𝑡 ) should be higher than the peak inverter input voltage (𝑣𝑖𝑛𝑣 ),
51
which is equal to the peak line voltage at unity modulation index. This condition can be stated
as 𝑣𝑏𝑎𝑡 > 𝑉̂𝐿𝐿 .
A Lead Acid Battery (LAB) is used in this simulation. Considering a daily demand of 𝐷𝐿
kWh for 𝑢 back-up days with no wind power (determined based on the wind profile at the
installation site and the load profile), battery rated voltage of 𝑉𝑏𝑎𝑡,𝑟𝑎𝑡𝑒𝑑 , depth of discharge of
𝐷𝑂𝐷 and temperature factor of 𝑘𝑇 , the required Ampere-hour (Ah) capacity of battery bank
can be found as
(𝐷 )(𝑢)(𝑘𝑇 )
𝐿
𝐴ℎ = (𝐷𝑂𝐷)(𝑉
(3.1)
𝑏𝑎𝑡,𝑟𝑎𝑡𝑒𝑑 )
For 𝐷𝐿 = 30𝑘𝑊ℎ, 𝑢 = 1, 𝑉𝑏𝑎𝑡,𝑟𝑎𝑡𝑒𝑑 = 650𝑉, 𝐷𝑂𝐷 = 50%, and 𝑘𝑇 = 1 (at 25C), the Ah
capacity of the battery bank is around 92 Ah. Thus, 54 standard commercial 12V, 92Ah
batteries are connected in series to produce a total storage capacity of 59.6 kWh.
Hysteresis
-+
Excitation
Capacitor Bank
Rotor
Blades
→→
Gear
Box
DC-Link
Sbuck
→
+
PWM-CSI
S1
S3
S5
Load Bus
Dc
Sc
→
SCIG
Cdc
D
LOAD a
LOAD b
LOAD c
S2
S4
Ci
S6
C-Filter
Modified
PWM
Generator
dq/abc
PI
Controller
-+
PWM
Generator
PI
Controller
-+
PI
Controller
abc/dq
+
Fig. 3.10: Integration of the energy storage system proposed in [70] with the CSI- based
WECS of topology 2.
Simulation responses are shown in Fig. 3.11. The system starts running at rated wind speed
(12m/s), feeding rated RL load. The demand is reduced to 70%, 50% and 20% of the rated load
52
at t = 1 s, 1.5 s, and 3 s, respectively. At t = 3.3 s, the rated load is applied again. The wind
speed is changed to 13 and 14 at t = 2 s and 2.5 s, respectively. MPPT control is achieved
successfully, as illustrated in Fig. 3.11. The figure also demonstrates a good power distribution
among wind generator, battery and load during transients and steady-state conditions. If wind
power is lower than the power demand, as in the start-up period (t = 0 s to t =0.16 s), the battery
compensates the shortage in power through switch 𝑆𝑐 and hence the dc-link current (𝑖𝑑𝑐 ) is
guaranteed not to fall below its reference. Although the diode 𝐷𝑐 offers no control over the dclink current, the fact that the battery is periodically charged during zero states of the CSI can
help in reducing dc-link current if it exceeds its reference, when the available wind power is
higher than the load demand. This can be seen in the period from t= 1 s to 2 s. Since wind
power is higher than demand during most of the simulation time, the battery’s SoC is set to
low value (i.e., 25%) initially so that the battery has enough room for charging. As the excess
of wind power increases during the period t = 2 s to 3.5 s, the 𝑖𝑑𝑐 can get much higher than the
reference and the storage circuit will not be able to maintain it. Even if a controlled switch is
used instead of the diode 𝐷𝑐 , the maximum value of 𝑖𝑑𝑐 is still unregulated because the charging
process can only take place during the zero states of the inverter. Even though 𝑖𝑑𝑐 exceeds its
reference, this case is still accepted for the load-side because the modulation index will adjust
the CSI output current. However, as 𝑖𝑑𝑐 increases further, lower values of 𝑚𝑖 (< 0.5) results in
distortion of the output voltage waveform, as can be noticed from t = 3 s to t = 3.3 s. Moreover,
2
high 𝑖𝑑𝑐 produces higher 𝑖𝑑𝑐
𝑅𝑑𝑐 and inverter losses. Therefore, this storage circuit may be
used in a standalone WECS only if the load demand and wind speed ranges are restricted to
produce small value of extra power that can be handled by the frequent battery charging, taking
place during zero states of the inverter. However, restricting wind power conflicts with the
MPPT objective. To overcome the shortcomings of the energy storage system proposed in [70],
a novel scheme for integration of a battery energy storage system with the CSI-based WECS
of topology 2 is proposed in the following section.
53
Fig. 3.11: Simulation results for the CSI-based WECS with energy storage system proposed
in [70].
54
3.4
The Structure of the Proposed WECS
In this work, the duality of CSI and VSI topologies has been taken advantage of to come up
with a novel scheme for the integration of a battery-based energy storage system with the
proposed CSI-based WECS. In standalone VSI-based WECS, the voltage across the dc-link
capacitor is regulated by connecting a battery bank across the dc-link filter capacitor via a
bidirectional buck-boost dc/dc converter [49], with bidirectional-current and unipolar-voltage
capabilities, and controlling the converter to manage the power exchange between the battery
and the dc-bus. In CSI-based WECS to exchange power between the battery and the dc bus,
an interfacing converter with bipolar-voltage and unidirectional-current capabilities is
required. As shown in Fig. 3.12, this requirement has been fulfilled by employing a full-bridge
dc/dc converter (H-bridge) with reduced number of switches, which is simply referred to as
→
reduced H-bridge.
→
DC-link Current
Control
SA
DB
SB
→
DA
→→
Rotor
Blades
Gear
Box
+
Sbuck
-
+
DC-Link
+
→
+
Sd
PWM-CSI
S1
S3
S5
+ -
∆ /Yn
→
→
SCIG
Load Bus
LOAD a
LOAD b
LOAD c
-
Excitation
Capacitor Bank
-
→
Cg
S2
S4
S6
Ci
C-Filter
-
Dump load
circuit
MPPT CONTROL
Load-Side Control
Dump Load
Control
Fig. 3.12: Structure of the proposed WECS.
55
n
As shown in Fig. 3.12, the proposed SCIG-based WECS employing a PWM CSI consists of
the following components:
-
Rotor blades,
-
Geared-drive, self-excited squirrel-cage induction generator,
-
Three-phase diode bridge rectifier,
-
DC/DC buck converter,
-
DC-Link inductor,
-
IGBT PWM three-phase current-source inverter,
-
C-Filter (𝐶𝑖 ),
-
Delta/star (∆/𝑌𝑛) transformer, providing a path for zero sequence current, isolating
the load from the system, and protecting the motoring load from common mode
voltage,
-
Y-connected three-phase load,
-
Storage battery integrated with the dc-link via a reduced H-bridge dc/dc converter,
and
-
Dump load.
The main objectives of the controller design are:
1. To achieve maximum power point tracking (MPPT),
2. To achieve effective control coordination among the wind generator, battery, and
dump load to maintain the dc-link current at the desired value, and
3. To maintain balanced voltages, with constant magnitude and frequency, at the load
bus.
A standalone WECS, employing a diode rectifier, a dc/dc boost converter, a VSI and a
bidirectional buck-boost dc/dc converter as the interface of the storage battery with the system,
requires two inductors (one for the boost converter and one for the buck-boost converter), and
a dc-link capacitor. In the proposed CSI-WECS, in contrast, the dc-link inductor (𝐿𝑑𝑐 ) is shared
by the buck converter, reduced H-bridge and CSI, resulting in reduction of the system size,
weight and cost. A systemic procedure for design of 𝐿𝑑𝑐 is introduced in the following
subsection.
56
3.4.1 Systematic Design of the DC-Link Inductor
DC-link inductor (𝐿𝑑𝑐 ) is sized according to the specified upper limit for the ripple in the dclink current at steady state. The study of the voltage that is placed across the dc-link inductor
(𝑣𝐿 ), under different operating conditions, is the key to the design of 𝐿𝑑𝑐 . Applying KVL to
dc-link loop in Fig. 3.12, 𝑣𝐿 can be expressed as
𝑣𝐿 = 𝑣𝑑 + 𝑣𝑥𝑦 − 𝑣𝑖𝑛𝑣
(3.2)
where 𝑣𝑑 is the buck converter unfiltered output voltage, 𝑣𝑥𝑦 the voltage inserted by the
reduced H-bridge in the dc-link and 𝑣𝑖𝑛𝑣 the CSI’s dc-side voltage. The instantaneous values
assumed by 𝑣𝑑 , 𝑣𝑥𝑦 and 𝑣𝑖𝑛𝑣 depend on the operating conditions of buck converter, reduced Hbridge and CSI, respectively, and are given in Table 3.3, where 𝑣𝑑𝑐 , 𝑣𝑏𝑎𝑡 and 𝑣𝐿𝐿 represent the
rectified dc voltage, the battery voltage, and the load-side line voltage, respectively. For
simplicity, the instantaneous value of the rectified voltage has been approximated by its
average value.
Table 3.3: Possible values of 𝑣𝑑 , 𝑣𝑥𝑦 , and 𝑣𝑖𝑛𝑣 .
Voltage
Value
𝑣𝑑
𝑣𝑑𝑐
0
𝑣𝑥𝑦
𝑣𝑏𝑎𝑡
−𝑣𝑏𝑎𝑡
0
𝑣𝑖𝑛𝑣
𝑣𝐿𝐿
0
Operation condition
𝑆𝑏𝑢𝑐𝑘 is ON
𝑆𝑏𝑢𝑐𝑘 is OFF
Battery is discharging
Battery is charging
Battery is neither charging nor
discharging (freewheeling state)
CSI is in one of the active states
CSI is in one of zero (shoot-through)
states
The maximum instantaneous voltage that is applied across 𝐿𝑑𝑐 (𝑉𝐿,𝑚𝑎𝑥 ) is
𝑉𝐿,𝑚𝑎𝑥 = 𝑣𝑑𝑐,𝑚𝑎𝑥 + 𝑣𝑏𝑎𝑡 + 𝑉̂𝐿𝐿
where
57
(3.3)
3√2
𝑣𝑑𝑐,𝑚𝑎𝑥 =
𝜋
𝑉𝑔,𝐿𝐿,𝑚𝑎𝑥
(3.4)
and 𝑉𝑔,𝐿𝐿,𝑚𝑎𝑥 is the maximum rms value of SCIG line voltage.
On the other hand, the minimum instantaneous voltage that is applied across the dc-link
inductor (𝑉𝐿,𝑚𝑖𝑛 ) is
𝑉𝐿𝑚𝑖𝑛 = 0 − 𝑣𝑏𝑎𝑡 − 𝑉̂𝐿𝐿
(3.5)
As shown in Fig. 3.13, the dc-link inductor current rises and falls in response to application of
positive and negative voltages across the dc-link inductor, respectively. The extreme case for
the range of variation of 𝑣𝐿 corresponds to the rare condition where 𝑉𝐿,𝑚𝑎𝑥 and 𝑉𝐿,𝑚𝑖𝑛 are
applied across dc-link inductor successively. Under this condition, the maximum rise, during
∆𝑡1, and maximum fall, during ∆𝑡2 , of the dc-link current can be obtained from (3.6) and (3.7),
respectively, as
∆𝑖𝑑𝑐,𝑟𝑖𝑠𝑒,𝑚𝑎𝑥 =
∆𝑖𝑑𝑐,𝑓𝑎𝑙𝑙,𝑚𝑎𝑥 =
𝑉𝐿,𝑚𝑎𝑥
𝐿𝑑𝑐
𝑉𝐿,𝑚𝑖𝑛 
𝐿𝑑𝑐
∆𝑡1
(3.6)
∆𝑡2
(3.7)
∆t1→
→
→
0
∆t2 →
t
Fig. 3.13: DC-link inductor voltage and current.
At steady state, the inductor current is repetitive and the values of inductor current rise and fall
are equal. Therefore,
∆𝑖𝑑𝑐,𝑟𝑖𝑠𝑒,𝑚𝑎𝑥 = ∆𝑖𝑑𝑐,𝑓𝑎𝑙𝑙,𝑚𝑎𝑥 = ∆𝑖𝑑𝑐
(3.8)
Assuming the same switching frequency (𝑓𝑠 ) for the buck converter, reduced H-bridge and
CSI, one can write
58
1
∆𝑡1 + ∆𝑡2 = 𝑓
𝑠
(3.9)
Based on (3.6) - (3.9), the worst-case steady-state ripple in dc-link inductor current can be
expressed as
∆𝑖𝑑𝑐 =
𝑉𝐿,𝑚𝑖𝑛 
𝑉

𝐿𝑑𝑐 𝑓𝑠 (1+ 𝐿,𝑚𝑖𝑛 )
𝑉𝐿,𝑚𝑎𝑥
(3.10)
The inductor current ripple is normally expressed relative to the average value of the inductor
current. Thus,
∆𝑖𝑑𝑐
𝑖𝑑𝑐
=
𝑉𝐿,𝑚𝑖𝑛 
𝑉

𝐿𝑑𝑐 𝑓𝑠 (1+ 𝐿,𝑚𝑖𝑛 ) 𝑖𝑑𝑐
𝑉
𝐿,𝑚𝑎𝑥
(3.11)
Typically, dc-link current ripple in CSI is limited to 15% [15]. From (3.11), the minimum
inductance required to satisfy this limit is
𝐿𝑑𝑐𝑚𝑖𝑛 =
𝑉𝐿,𝑚𝑖𝑛 
𝑉𝐿,𝑚𝑖𝑛 
) (0.15 𝑖𝑑𝑐 )
𝑓𝑠 (1+
𝑉𝐿,𝑚𝑎𝑥
(3.12)
It can be noticed from (3.12) that the size of dc-link inductor can be reduced by increasing the
switching frequency. Moreover, the higher the dc-link current is, the smaller the inductance
required to guarantee not exceeding the specified current ripple limit will be. However, it is
useful to minimize the dc-link current so as to reduce the ohmic losses of the dc-link inductor
as well as the losses of the current source inverter. The minimum dc-link current based on
which the dc-link inductor is designed will be specified in Chapter 5 when dc-link current
control system is designed.
3.5
Summary
In this chapter, the feasibility of employing CSI in off-grid WECS was investigated. The
three CSC topologies, validated in the literature for on-grid WECS, were evaluated for off-grid
WECS application. Since the first topology (i.e., diode bridge rectifier - PWM CSI) provides
no control over generator and dc-link current, it is not applicable for an off-grid WECS. Adding
an intermediate dc/dc buck converter to the first topology, the second topology (i.e., diode
bridge rectifier - buck converter – PWM CSI) makes MPPT achievable. One problem
59
associated with diode bridge rectifier-based topologies is the high harmonic distortion in the
generator winding current. This problem was overcome by employing PWM-CSR in the third
topology (i.e., Back-to-back PWM CSC), leading to significant improvement in the generator
performance, but with complex control and at higher cost. As a simple and low-cost
configuration, offering satisfactory performance, the second topology was selected as the base
for further study in this research.
Although the feasibility of the selected topology was demonstrated, the minimum value of
the CSI input dc current, required to maintain desired output voltage, was not guaranteed at
wind speeds below rated value. Moreover, a storage unit was required to be integrated within
the structure of the system in order to achieve effective power balance. A simple storage circuit,
proposed in the literature for temporary storage purposes, was implemented in the second
topology, but the capacitor bank was replaced by a storage battery. The validity of such a
simple storage circuit, for standalone turbine, was demonstrated only when the load and wind
speed ranges are restricted to produce small value of extra power; this implies restricting wind
power and conflicts with the MPPT objective. Therefore, a novel scheme for integration of a
battery energy storage system with the CSI-based WECS of the second topology, was
proposed. The dc-link inductor, shared by three converters, was systematically designed.
60
Chapter 4
Dynamic Modeling and Small Signal Analysis of the CSI-WECS
Fig. 4.1 shows the block diagram of the proposed system. In this chapter, detailed models of
mechanical and electrical components are developed and combined to form an overall model.
Rotor
Blades
Shaft &
Gearbox
SelfExcited
SCIG
Lead Acid
Battery
Reduced
H-Bridge
Converter
3-phase
Diode
Bridge
Buck
Converter
Balanced/
Unbalanced
Load
DC-Link
3-phase
Current
Source
Inverter
Delta/Star
Isolation
Transformer
Generic
Load Model
Fig. 4.1: Block diagram of the proposed CSI-based WECS.
Forced-commutated power converter devices operate at high frequencies and hence a very
small simulation time step is required to produce a sufficiently accurate simulation of switching
transients. Therefore, simulating the entire system using a real switching model will demand a
long simulation time and a large memory size. Moreover, the discontinuities and nonlinearities
associated with the switching action of the power converters make it complicated to apply
classical control methods for system analysis.
When switching ripples are of interest or detailed transient information is needed, detailed
switching models are unavoidable; otherwise, average models can provide adequate
information in a low frequency range [118]. High-frequency switching harmonics are not
presented in the average model, but the dynamics of the system is preserved. Average model,
which predicts low-frequency behaviour of the actual switching model, can be simulated much
faster than switching model. Assuming that the smallest time constant of a converter system is
larger than the switching period (𝑇𝑠 ) by at least an order of magnitude, it is a very good
approximation to average the converter variable quantities over 𝑇𝑠 . As a result, the switching
action is eliminated from the model and hence system’s discontinuity is no longer a concern.
61
In other words, the average model is continuous and hence can be linearized around an
operating point. The linearized model is used to investigate local stability, observability and
controllability of the system. The linearized model also provides the basis for closed-loop
control system design in the following chapter.
Typically, the process of modeling starts with deriving three-phase equations for electrical
circuits in abc frame. These relations are then transformed into their corresponding dq-frame
equations. The transformation matrix is given in (4.1), where 𝑥 can be any system variable
such as three-phase voltage, current, and flux linkage, and 𝜔𝑡 is the angular position of the
rotating 𝑑𝑞 frame in radians.
2𝜋
2𝜋
cos(𝜔𝑡)
cos(𝜔𝑡 − 3 )
cos(𝜔𝑡 + 3 ) 𝑥𝑎
𝑥𝑑
2
𝑥
[𝑥 ] = 3 [
2𝜋
2𝜋 ] [ 𝑏 ]
𝑞
−sin (𝜔𝑡) −sin(𝜔𝑡 − 3 ) −sin(𝜔𝑡 + 3 ) 𝑥𝑐
(4.1)
Since the proposed WECS has some passive components, which can be described using
differential equations, an additional term is brought to the transformation, as seen in (4.2),
where 𝜔 is the angular speed of the 𝑑𝑞 frame.
𝑥𝑎
𝑑 𝑥𝑑
0
[𝑥𝑏 ] ⇨ [𝑥 ] + [
𝑑𝑡
𝑑𝑡
𝑞
𝜔
𝑥𝑐
𝑑
−𝜔 𝑥𝑑
][ ]
0 𝑥𝑞
(4.2)
In this chapter:
1) The models of system components are derived and combined to develop a nonlinear
dynamic model of the proposed standalone CSI-based WECS in 𝑑𝑞 reference frame. Detailed
models of the aerodynamic conversion, drive train, self-excited induction generator, Lead Acid
battery, and power-electronic converters are presented and combined with a reduced-order
generic load model to enable transient and steady-state analyses of the overall system;
2) The behavior of the system is investigated by simulating its operation at start-up and in
response to a step change in the input;
3) A small-signal linear model is developed by linearizing the nonlinear dynamic equations
around steady-state selected operating points; and
62
4) The linearized model is employed to investigate the local stability of the system at the
steady-state operating point. Also, some of the system properties, such as controllability and
observability are investigated.
4.1
Wind Turbine System
The mechanical power captured by a wind turbine (𝑃𝑚 ) can be obtained from
𝜌𝐴
𝑃𝑚 =
2
𝑣𝑤3 𝐶𝑝 (𝜆, 𝛽)
(4.3)
where 𝜌 is the air density (1.225 𝑘𝑔/𝑚3 at 15C and at sea level), 𝐴 the turbine’s swept area
in 𝑚2 (𝐴 = 𝜋𝑟 2 , with 𝑟 the radius of blades in 𝑚), 𝑣𝑤 the wind speed in 𝑚/𝑠, 𝐶𝑝 the wind
turbine performance coefficient, 𝜆 the tip speed ratio, and 𝛽 the blade pitch angle in degrees.
𝜆, defined as the ratio of rotor blade tip speed to wind speed, is
𝜆=
𝜔𝑚 𝑟
(4.4)
𝑣𝑤
where 𝜔𝑚 is the angular speed of the turbine in 𝑟𝑎𝑑/𝑠.
𝐶𝑝 is a function of 𝜆 and 𝛽, as given in (4.5) [119].
116
𝐶𝑝 (𝜆, 𝛽) = 0.5176 ( 𝜆 − 0.4𝛽 − 5) 𝑒
𝑖
−
21
𝜆𝑖
+ 0.0068𝜆
(4.5)
In (4.5),
1
𝜆𝑖
1
0.035
= 𝜆+0.08𝛽 − 𝛽3 +1
(4.6)
According to Betz law, the theoretical limit of 𝐶𝑝 is 0.59. Practically, 𝐶𝑝 varies between 0.2
and 0.5 in the modern turbines [15]. The 𝐶𝑝 − 𝜆 characteristics for different 𝛽 values are
illustrated in Fig. 4.2 for a three-blade horizontal-axis WT. The maximum 𝐶𝑝 (𝐶𝑝𝑚𝑎𝑥 ), for 𝛽 =
0, is obtained at 𝜆 = 8.1. This particular 𝜆 is known as optimal 𝜆 (𝜆𝑜𝑝𝑡 ). For a variable-speed
WT, with the pitch angle fixed at zero, the target is to maintain 𝜆 at its optimal value
corresponding to maximum power capture.
63
Cp- Curves at differenet Pitch Angles 
Maximum power coefficient (Cpmax ) at zero 
Power coefficient C p
0.5
 =0
0.4
 =5
0.3
 =10
0.2
 =15
0.1
 =25
 =20
0
-0.1
0

5
opt
10
15
Tip-speed ratio 
Fig. 4.2: Performance coefficient versus tip speed ratio for various blade pitch angles.
Fig. 4.3 shows the steady-state mechanical power-speed curves for a variable-speed, fixedpitch WT in per unit. The curves show the nonlinear relationship between the mechanical
power and wind speed. The red dashed line links the optimum power points at different wind
speeds. The rotor speed corresponding to the maximum power at each wind speed can be found
based on the optimal tip speed ratio from (4.4). Fig. 4.3 is obtained for base wind speed of 12
𝑚/𝑠 and rated power of 1 pu at 1 pu rotational speed.
Fig. 4.3: Turbine power characteristics at zero pitch angle.
64
Alternative to speed regulation, torque regulation can be applied in order to achieve MPPT
for a WT. Considering (4.3) and (4.4), the optimum mechanical torque can be expressed by
𝑃
1
(4.7), provided that 𝑇𝑚 = 𝜔𝑚 = 2 𝜌𝐴 𝑟 𝐶𝑝 𝑣𝑤2 /𝜆 .
𝑚
2
𝑇𝑚𝑜𝑝𝑡 = 𝐾𝑜𝑝𝑡 𝜔𝑚
In (4.7), 𝐾𝑜𝑝𝑡 =
𝜌𝐴
2
𝐶𝑝𝑚𝑎𝑥 (𝜆
𝑟
𝑜𝑝𝑡
(4.7)
)3 .
As can be seen from (4.3), calculation of wind power requires knowledge of instantaneous
wind speed. Wind speed is a stochastic variable signal and its behaviour is difficult to predict,
especially over short-time period (i.e., seconds, minutes or few hours). Different Statistic wind
speed models were summarized in [120]. One of commonly used wind speed models is the
spectral density-based model, which describes the variation of wind speed as an overlapping
of different frequency components. The basic model of the turbulent component, used in the
Von-Karman power spectrum model [121] is given in (4.8).
𝑆(𝜔) =
0.475 𝜎2 (𝐿/𝑣̅𝑤 )
𝜔𝐿
𝑣𝑤
[1+ ( ̅ )2 ]5/6
(4.8)
In (4.8), 𝜔 is the circular frequency obtained by multiplying 𝑣̅𝜔 with the spatial frequency Ω
(rad/m), 𝑣̅𝑤 the mean wind speed (typically, determined over 10-minute period), 𝜎 the
turbulence intensity, and 𝐿 the turbulence length scale in feet. For low-altitude model (altitude
< 1000 feet) [122],
ℎ
𝐿 = (0.177+0.000823ℎ)1.2
(4.9)
where ℎ is the height in feet at which the wind speed signal is of interest.
The turbulence intensity 𝜎 is found from (4.10), where 𝑣𝑤20 is the wind speed at 20 feet (6
meters). Typically, 𝑣𝑤20 is 7.72 m/s, 15.43 m/s, and 23.15 m/s for light, moderate, and severe
turbulence, respectively [122].
0.1 𝑣𝑤
20
𝜎 = (0.177+0.000823ℎ)
0.4
65
(4.10)
The amplitude (𝐴𝑖 ) of wind speed fluctuation (Amplitude of the 𝑖 𝑡ℎ harmonic), at discrete
frequency of 𝑓𝑖 (𝑖 = 1,2,3, . . 𝑀), where 𝑀 is the number of samples, is expressed by
(4.11) [121].
2
𝐴𝑖 (𝜔𝑖 ) = 𝜋 √0.5 [𝑆(𝜔𝑖 ) + 𝑆(𝜔𝑖+1 )](𝜔𝑖+1 − 𝜔𝑖 )
(4.11)
The instantaneous value of wind speed 𝑣𝑤 (𝑡) can be described as a sum of average wind speed
and fluctuation of the wind, as given by (4.12), where 𝜓𝑖 is a uniformly-distributed random
phase angle in the domain [𝜋, −𝜋].
𝑣𝑊 (𝑡) = 𝑣̅𝑤 + ∑𝑀
𝑖=1 𝐴𝑖 cos (𝜔𝑖 𝑡 + 𝜓𝑖 )
(4.12)
Based on (4.8), a spectral density function is shown in Fig. 4.4(a). The figure is obtained for
M = 55 , 𝑣̅𝜔 = 10𝑚/𝑠, 𝑣𝑤20 = 7.72 𝑚/𝑠, and ℎ = 18 𝑚.The low-pass filtered fluctuation of
wind speed is shown in Fig. 4.4(b).
Fig. 4.4: Wind speed model based on Von-Karman’s method.
4.2
Drive Train System
The drive train is essentially represented by a three-mass model which is composed of three
masses accounting for the turbine’s rotor, the gearbox and the generator [123]. The masses are
linked by two shafts. The moments of inertia of the gearbox and the shafts are assumed to have
small influence on the system behaviour compared to moment of inertia of the turbine rotor
and generator. As a result, the three-mass model can be reduced to a two-mass model, shown
in Fig. 4.5. It consists of two masses linked by an equivalent shaft. In this model, only the gear
ratio (𝑛𝑔𝑒𝑎𝑟 ) of the gearbox is considered. In Fig. 4.5, 𝑇𝑚 , 𝐽𝑚 and 𝜔𝑚 are the turbine torque in
66
𝑁𝑚, moment of inertia in 𝑘𝑔𝑚2 and angular speed in 𝑟𝑎𝑑/𝑠, respectively, 𝑇𝑒 , 𝐽𝑔 and 𝜔𝑔 the
generator torque in 𝑁𝑚, moment of inertia in 𝑘𝑔𝑚2 and angular speed in 𝑟𝑎𝑑/𝑠, respectively,
𝐾𝑠𝑒 and 𝐷𝑠𝑒 the shaft equivalent stiffness in 𝑁𝑚/𝑟𝑎𝑑 and damping factor in 𝑁𝑚/𝑟𝑎𝑑/𝑠,
respectively, and 𝜃𝑚 and 𝜃𝑔 the turbine and generator shaft angles in 𝑟𝑎𝑑, respectively.
Fig. 4.5: Two mass model of turbine’s drive train on Turbine side.
The dynamics of the drive train can be described by the following differential equations [123],
where 𝛿𝜃 = 𝜃𝑚 − 𝜃𝑔 /𝑛𝑔𝑒𝑎𝑟 and 𝛿𝜔 = 𝜔𝑚 − 𝜔𝑔 /𝑛𝑔𝑒𝑎𝑟 .
𝑑
𝑑𝑡
𝑑
𝑑𝑡
1
𝜔𝑚 = 𝐽 [𝑇𝑚 − 𝐷𝑠𝑒 𝛿𝜔 − 𝐾𝑠𝑒 𝛿𝜃]
𝑚
1
𝜔𝑔 = 𝐽 [ 𝑛
𝑔
1
𝑔𝑒𝑎𝑟
𝐷𝑠𝑒 𝛿𝜔 + 𝑛
𝑑(𝛿𝜃)
𝑑𝑡
4.3
1
𝑔𝑒𝑎𝑟
= 𝛿𝜔
𝐾𝑠𝑒 𝛿𝜃 − 𝑇𝑒 ]
(4.13)
(4.14)
(4.15)
Self-Excited Induction Machine
Induction machine operates based on induction or transformer action between the stator and
the rotor conductors. An electromagnetic torque is produced due to the interaction between
stator and rotor magnetic fields. The difference between the rotor speed and the speed of
67
rotating magnetic field of stator (synchronous speed), normalized to synchronous speed,
defines slip (s) as:
s=
𝜔𝑠 −𝜔𝑔
𝜔𝑠
=
𝜔𝑒 −𝜔𝑟
𝜔𝑒
(4.16)
where 𝜔𝑠 is the synchronous angular speed in 𝑟𝑎𝑑/𝑠, 𝜔𝑒 the stator electrical angular frequency
in 𝑟𝑎𝑑/𝑠 (𝜔𝑒 = 𝑃𝜔𝑠 ), 𝑃 the number of pole pairs, 𝜔𝑔 the rotor shaft angular speed in 𝑟𝑎𝑑/𝑠,
and 𝜔𝑟 the rotor electrical angular speed in 𝑟𝑎𝑑/𝑠 (𝜔𝑟 = 𝑃𝜔𝑔 ).
An induction generator needs to be continuously excited by a source of reactive power in
order to generate voltage and supply active power. In grid-connected wind turbines, this
reactive power is typically supplied by the grid. In standalone wind turbines, it can be supplied
by either a power electronic converter or an external source of reactive power, such as switched
capacitors. In this case, the generator is called self-excited induction generator (SEIG). In
SEIG, the machine should operate in saturation region at the intersection point between the
magnetization curve of the machine and the impedance of the excitation capacitor [124]. Either
initial capacitor voltage or machine’s residual magnetism is required for the flux to build up
and the machine to operate as a generator. A combination of machine’s parameters, shaft speed,
excitation level, and loading condition determines the generated voltage magnitude and
frequency. The minimum capacitor needed to generate the rated voltage at rated speed and no
load condition is obtained by (4.17) [29] where 𝐿𝑚𝑢𝑛𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑒𝑑 is the magnetizing inductance
before saturation.
𝐶𝑚𝑖𝑛 = 𝜔2 𝐿
𝑟
1
𝑚𝑢𝑛𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑒𝑑
(4.17)
Amongst various models developed to analyze the transient and steady-state performances
of self-excited induction machines [125], the well-known dq model, shown at no load in
Fig. 4.6, is used in this work. In the figure, 𝑅𝑠 is the stator resistance, 𝐿𝑙𝑠 the stator leakage
inductance, 𝐿𝑚 the magnetizing inductance, 𝑅𝑟′ the rotor resistance refereed to the stator, 𝐿′𝑙𝑟
the rotor leakage inductance refereed to the stator, 𝐶𝑔 the excitation capacitance, 𝑖𝑑𝑠 and 𝑖𝑞𝑠
′
′
the 𝑑-axis and 𝑞-axis stator currents, respectively, 𝑖𝑚 the magnetizing current, 𝑖𝑑𝑟
and 𝑖𝑞𝑟
the
𝑑-axis and 𝑞-axis rotor currents referred to the stator, respectively, 𝑣𝑐𝑔𝑑 and 𝑣𝑐𝑔𝑞 the 𝑑-axis
and 𝑞-axis voltage of the excitation capacitor, respectively, 𝜑𝑑𝑠 and 𝜑𝑞𝑠 the 𝑑-axis and 𝑞-axis
68
′
′
stator flux, respectively, 𝜑𝑑𝑟
and 𝜑𝑞𝑟
the 𝑑-axis and 𝑞-axis rotor flux referred to the stator,
respectively, and 𝑝 the derivative operator (𝑑/𝑑𝑡).
(a) d-axis circuit
-+
+
-
+
+
-
-
+ -
(b) q-axis circuit
+ -
+
-
+
+
-
-
-+
Fig. 4.6: Dq model of Self-excited squirrel cage induction machine.
The mathematical model of SEIG in an arbitrary dq reference frame is described by the set of
differential equations (4.18) - (4.21). All electrical variables and parameters indicated by the
prime signs are referred to the stator.
𝑑𝜑𝑑𝑠
𝑣𝑐𝑔𝑑 = 𝑅𝑠 𝑖𝑑𝑠 +
𝑑𝑡
𝑑𝜑𝑞𝑠
𝑣𝑐𝑔𝑞 = 𝑅𝑠 𝑖𝑞𝑠 +
′
0 = 𝑅𝑟′ 𝑖𝑑𝑟
+
′
0 = 𝑅𝑟′ 𝑖𝑞𝑟
+
′
𝑑𝜑𝑑𝑟
𝑑𝑡
𝑑𝑡
− 𝜔𝜑𝑞𝑠
(4.18)
+ 𝜔𝜑𝑑𝑠
(4.19)
′
− (𝜔 − 𝜔𝑟 )𝜑𝑞𝑟
(4.20)
′
+ (𝜔 − 𝜔𝑟 )𝜑𝑑𝑟
(4.21)
′
𝑑𝜑𝑞𝑟
𝑑𝑡
Equations (4.22) – (4.25) give the air gap flux linkages, with 𝐿𝑠 = 𝐿𝑚 + 𝐿𝑙𝑠
and
𝐿′𝑟 = 𝐿𝑚 + 𝐿′𝑙𝑟 .
′
𝜑𝑞𝑠 = 𝐿𝑠 𝑖𝑞𝑠 + 𝐿𝑚 𝑖𝑞𝑟
(4.22)
′
𝜑𝑑𝑠 = 𝐿𝑠 𝑖𝑑𝑠 + 𝐿𝑚 𝑖𝑑𝑟
(4.23)
′
′
𝜑𝑞𝑟
= 𝐿′𝑟 𝑖𝑞𝑟
+ 𝐿𝑚 𝑖𝑞𝑠
(4.24)
69
′
′
𝜑𝑑𝑟
= 𝐿′𝑟 𝑖𝑑𝑟
+ 𝐿𝑚 𝑖𝑑𝑠
(4.25)
Equations (4.18)-(4.21) can be adapted for stationary, rotor and stator angular frequency
reference frames, by substituting 0, 𝜔𝑟 , and 𝜔𝑒 , respectively, for 𝜔. The dq equations in stator
reference frame (i.e., 𝜔 = 𝜔𝑒 ) are given by (4.26), where 𝑘𝑠 = 𝐿2𝑚 − 𝐿𝑠 𝐿′𝑟 .
𝑖𝑞𝑠
𝑑 𝑖𝑑𝑠
1
=𝑘
′
𝑑𝑡 𝑖𝑞𝑟
𝑠
′
[𝑖𝑑𝑟 ]
𝜔𝑒 (𝐿2𝑚
[
𝑅𝑠 𝐿′𝑟
− 𝐿𝑠 𝐿′𝑟 ) − 𝜔𝑟 𝐿2𝑚
−𝑅𝑠 𝐿𝑚
𝜔𝑟 𝐿𝑚 𝐿𝑠
−𝐿′𝑟
1
0
+ [
𝑘𝑠 𝐿𝑚
0
0
−𝐿′𝑟
0
𝐿𝑚
𝐿𝑚
0
−𝐿𝑠
0
𝜔𝑒 (𝐿𝑠 𝐿′𝑟 − 𝐿2𝑚 ) + 𝜔𝑟 𝐿2𝑚
𝑅𝑠 𝐿′𝑟
−𝜔𝑟 𝐿𝑚 𝐿𝑠
−𝑅𝑠 𝐿𝑚
−𝑅𝑟′ 𝐿𝑚
−𝜔𝑟 𝐿𝑚 𝐿′𝑟
𝑅𝑟′ 𝐿𝑠
2
𝜔𝑒 (𝐿𝑚 − 𝐿𝑠 𝐿′𝑟 ) + 𝐿𝑠 𝐿′𝑟 𝜔𝑟
𝑖𝑞𝑠
𝜔𝑟 𝐿𝑚 𝐿′𝑟
𝑖𝑑𝑠
−𝑅𝑟′ 𝐿𝑚
′
𝜔𝑒 (𝐿𝑠 𝐿′𝑟 − 𝐿2𝑚 ) − 𝐿𝑠 𝐿′𝑟 𝜔𝑟 𝑖𝑞𝑟
′
𝑅𝑟′ 𝐿𝑠
] [𝑖𝑑𝑟 ]
𝑣𝑐𝑔𝑞
0
𝐿𝑚 𝑣𝑐𝑔𝑑
][
]
0
0
−𝐿𝑠
0
(4.26)
The reactive power, required to excite the machine, is provided by a star-connected capacitor
bank. The capacitor dq currents are given by (4.27) and (4.28).
𝑑
𝑖𝑐𝑔𝑑 = 𝐶𝑔 𝑑𝑡 𝑣𝑐𝑔𝑑 − 𝐶𝑔 𝜔𝑒 𝑣𝑐𝑔𝑞
(4.27)
𝑑
𝑖𝑐𝑔𝑞 = 𝐶𝑔 𝑑𝑡 𝑣𝑐𝑔𝑞 + 𝐶𝑔 𝜔𝑒 𝑣𝑐𝑔𝑑
(4.28)
The electromagnetic torque (𝑇𝑒 ) generated by SCIG can be obtained by the relations given
in (4.29).
3𝑃
𝑇𝑒 =
{
(𝑖𝑞𝑠 𝜑𝑑𝑠 − 𝑖𝑑𝑠 𝜑𝑞𝑠 )
2
3𝑃𝐿𝑚
′
′
(𝑖𝑞𝑠 𝑖𝑑𝑟
− 𝑖𝑑𝑠 𝑖𝑞𝑟
)
2
3𝑃𝐿𝑚
′
′
(𝑖𝑞𝑠 𝜑𝑑𝑟
− 𝑖𝑑𝑠 𝜑𝑞𝑟
)
2𝐿
𝑟
70
(𝑎)
(𝑏)
(𝑐)
(4.29)
4.4
Three-Phase Diode Bridge with DC-Side Capacitive Filter
In the proposed WECS, shown in Fig. 3.12, the SEIG generates a variable ac voltage which
is converted into dc voltage through a three-phase diode bridge rectifier. The rectifier’s output
voltage is smoothed by a capacitor of finite capacitance. Therefore, the dc-side voltage is not
ripple free. The output dc current of the rectifier (𝑖𝑑𝑐𝑟 ) is boosted by a buck converter.
Therefore, 𝑖𝑑𝑐𝑟 alternates between zero and 𝑖𝑑𝑐ℎ depending on the state of the buck switch
(𝑆𝑏𝑢𝑐𝑘 ). In other words, the dc side of the bridge rectifier is equivalent to a current source when
the buck switch, 𝑆𝑏𝑢𝑐𝑘 , is on, and an open circuit when 𝑆𝑏𝑢𝑐𝑘 is off. Equivalent circuit diagram
of the topology is shown in Fig. 4.7.
D1
D3
D5
→
→
→
+
→
↓
→
D2
D4
D6
-
Three-Phase SEIG
Fig. 4.7: Three-phase diode bridge rectifier with a variable dc current.
In Fig. 4.7, the SEIG is represented by a three-phase ac voltage supply, with the following time
functions:
𝑉̂𝑐𝑔 sin(𝜔𝑒 𝑡)
𝑣𝑐𝑔𝑎
[𝑣𝑐𝑔𝑏 ] = [𝑉̂𝑐𝑔 sin(𝜔𝑒 𝑡 − 2𝜋/3)]
𝑣𝑐𝑔𝑐
𝑉̂𝑐𝑔 sin(𝜔𝑒 𝑡 + 2𝜋/3)
(4.30)
Depending on the values of the ac-side inductor 𝐿𝑔 and dc-side capacitor 𝐶𝑑𝑐 , and/or loading
conditions, the topology shown in Fig. 4.7 can operate in one of the two different modes:
Discontinues Conduction Mode (DCM) and Continuous Conduction Mode (CCM) [126].
In order to derive the average model of diode bridge rectifier in Fig. 4.7, the following
assumptions are made:
71
1. The self-excited IG generates three-phase balanced sinusoidal voltages.
2. The series inductance 𝐿𝑔 is negligible. This assumption is justifiable for low-power
induction generator-based turbines. 𝐿𝑔 represents the cable and generator-side L filter. Real
data show that the stator inductance in low-voltage, low-power induction generator (<
100 𝑘𝑊), is relatively high (i.e., 0.1 - 0.25 pu) [127],[128] compared with those in Mega-Watt
IGs. The combination of the stator inductance and the excitation capacitance can provide
acceptable harmonic attenuation to the generator current. Therefore, no external L filter is
required to be added in the generator side and hence some cost is saved. Thus, 𝐿𝑔 is mainly
attributed to the cable inductance. Reactance of low-voltage cables (up to 1kV) is in the range
of 0.1 – 0.75 Ω/mile at 60Hz (i.e., 0.27 – 2 mH/mile) [129]. In a wind turbine, the generator
and its converter are placed in the nacelle and hence the length of cable connection between
the generator and the diode-bridge is very short (around 1-2 meters) and cable inductance is
too small (< 2.5 H) and can be neglected.
3. The dc-side filter capacitor is large (0.3-1.5 pu), but not significant enough to force the
bridge into DCM operation. When the 𝑆𝑏𝑢𝑐𝑘 is turned off, the diode bridge output current (𝑖𝑑𝑐𝑔 )
flows through 𝐶𝑑𝑐 . Hence, 𝑖𝑑𝑐𝑔 never falls to zero. In other words, the bridge is always
operating at CCM with instantaneous commutation (i.e., the commutation angle is very small
and hence approximated to zero).
4. The voltage across the dc-side capacitor (𝑣𝑑𝑐 ) has very slow dynamics and hence 𝑣𝑑𝑐 may
be considered constant with respect to state variables that vary under the influence of high
switching frequency of the converters.
5. All harmonics are neglected.
6. As noticed from Fig. 3.12, the reduced H-bridge terminals are connected in series with the
dc link inductor during non-dumping periods (i.e., 𝑖𝑑𝑐ℎ = 𝑖𝑑𝑐 ). Thus, the average value of the
rectifier output current 𝑖𝑑𝑐𝑔 is equal to the average value of 𝑖𝑑𝑐𝑟 which depends on the duty
ratio of buck converter 𝑑𝑏 , i.e., 𝑖𝑑𝑐𝑔 = 𝑖𝑑𝑐𝑟 = 𝑑𝑏 𝑖𝑑𝑐ℎ = 𝑑𝑏 𝑖𝑑𝑐 . For certain values of 𝑑𝑏 and
𝑖𝑑𝑐 , the average output current of the rectifier is constant.
72
Based on the assumptions made above, the circuit of Fig. 4.7 can be approximated by that
shown in Fig. 4.8.
ThreePhase
SEIG
D3
D5
+
→
→
→
→
D1
D2
D4
D6
Fig. 4.8: Three-phase diode bridge rectifier with a constant dc current.
In Fig. 4.8, the average output voltage is found from (4.31), where 𝑉̂𝑐𝑔 is the peak of generator
phase voltage.
2
𝜋/6
𝑣𝑑𝑐 = 2𝜋/6 ∫0
3√3
√3 𝑉̂𝑐𝑔 cos 𝜔𝑡 𝑑(𝜔𝑒 𝑡) = 𝜋 𝑉̂𝑐𝑔
(4.31)
The rms value of the fundamental component of the ac line current can be found by
(4.32) [130].
𝐼𝑠1𝑟𝑚𝑠 =
√6
𝑑 𝑖
𝜋 𝑏 𝑑𝑐
(4.32)
From (4.31) and (4.32), diode bridge rectifier is represented by its average output, as shown in
Fig. 4.9. In the figure, the supply voltages and ac-side currents are approximated by their
fundamental components, assuming negligible harmonic contents. 𝜃𝑖 is phase angle between
fundamental component of current and supply voltage. In the switching model, the
fundamental current is slightly leading the supply voltage due to dc-side capacitor effect.
Hence, a small part of the reactive power absorbed by the induction generator is supported by
the 𝐶𝑑𝑐 . For the average model, however, the effect of 𝐶𝑑𝑐 is neglected and the reactive power
required for generator excitation is completely supplied by the capacitor bank. Under this
assumption, the fundamental component of line current is in phase with the supply voltage
(i.e., 𝜃𝑖 = 0).
73
In Fig. 4.9, the effect of load side on the dc average voltage (𝑣𝑑𝑐 ) is neglected. This
assumption is valid as 𝑣𝑑𝑐 is mainly dependent on the ac-side applied voltage.
→
→
+
+
-
→
-
→
Fig. 4.9: Averaged model of CCM diode bridge rectifier with instantaneous commutation.
By transforming the three-phase supply voltages and currents into dq frame rotating at
angular electrical speed of the generator, 𝜔𝑒 , the abc-based average model, shown in Fig. 4.9,
is redrawn in dq frame in Fig. 4.10. Assuming that the d-axis is initially aligned with the stator
terminal phase voltage and diode bridge rectifier input current is in phase with that voltage,
3√3
2√3
𝑣𝑐𝑔𝑑 = 𝑉̂𝑐𝑔 , 𝑣𝑐𝑔𝑞 = 0, 𝑖𝑞 = 0. Thus, from (4.31) and (4.32), 𝑣𝑑𝑐 = 𝜋 𝑣𝑐𝑔𝑑 and 𝑖𝑑 = 𝜋 𝑑𝑏 𝑖𝑑𝑐 .
→
→
→
-
-
+
+
+
-
→
+
→
→
→
Fig. 4.10: Dq average model of diode bridge rectifier supplied by SEIG.
74
Applying KCL to Fig. 4.10, the dynamics of capacitor voltage is described by (4.33) and (4.34).
𝑑
𝑑𝑡
1
𝑔
𝑑
𝑣
𝑑𝑡 𝑐𝑔𝑞
4.5
1 2√3
𝑣𝑐𝑔𝑑 = − 𝐶 𝑖𝑑𝑠 − 𝐶
𝑔
𝜋
𝑑𝑏 𝑖𝑑𝑐 + 𝜔𝑒 𝑣𝑐𝑔𝑞
(4.33)
1
= − 𝐶 𝑖𝑞𝑠 − 𝜔𝑒 𝑣𝑐𝑔𝑑
(4.34)
𝑔
DC/DC Buck Converter
In the proposed system (Fig. 3.12), the dc/dc buck converter is placed between the diode
bridge rectifier and the PWM-CSI. As mentioned in the previous section, except during
dumping periods, the converter output current 𝑖𝑑𝑐ℎ is equal to the dc-link current 𝑖𝑑𝑐 which is
kept continuous and regulated via the large dc-link reactor and the control implemented by the
H-bridge interfacing the storage battery bank. Hence, the buck converter is operating in
Continuous Conduction Mode (CCM). The buck converter employed in the proposed system
is shown in Fig. 4.11.
+
→+
→
Sbuck
-
Fig. 4.11: Buck converter circuit.
In Fig. 4.11, the average values of 𝑖𝑑𝑐𝑟 and 𝑖𝑑𝑐 are related through the duty cycle of the buck
switch, 𝑑𝑏 .
𝑖𝑑𝑐𝑟 = 𝑑𝑏 𝑖𝑑𝑐
(4.35)
The converter is boosting the current and hence bucking the voltage. Thus,
𝑣𝑑 = 𝑑𝑏 𝑣𝑑𝑐
(4.36)
Based on (4.35) and (4.36), the equivalent circuit of the converter is shown in Fig. 4.12.
75
→
+
→
→
+
+
-
-
-
Fig. 4.12: Large-signal nonlinear averaged model of CCM buck converter.
4.6
Lead Acid Battery
Fig. 4.13 shows one of the common models of lead-acid batteries reported in the
literature [131],[132]. As shown in the figure, the voltage-current characteristics of a battery is
modelled by a controlled voltage source 𝐸𝑜𝑐 and series resistance 𝑅𝑠𝑒𝑟𝑖𝑒𝑠 . As given in (4.37)
and (4.38) [132], 𝐸𝑜𝑐 and 𝑅𝑠𝑒𝑟𝑖𝑒𝑠 are functions of battery’s state of charge SoC.
→
+
+
Fig. 4.13: Electrical model of a battery cell.
𝐸𝑜𝑐 = 𝐸𝑜 − 𝐾𝐸 (1 − 𝑆𝑜𝐶)
𝑅𝑠𝑒𝑟𝑖𝑒𝑠 = 𝑅𝑜 (1 + 𝐴0 (1 − 𝑆𝑜𝐶))
where
76
(4.37)
(4.38)
𝑄
𝑆𝑜𝐶 = 1 − 𝑄 𝑒
(4.39)
𝑜
𝑡
𝑄𝑒 = ∫0 𝑖𝑏𝑎𝑡 𝑑𝑡
(4.40)
In (4.37)-(4.40), 𝐸𝑜 is the open-circuit voltage at 100% SoC, 𝐾𝐸 and 𝐴𝑜 constants, 𝑅𝑜 the series
resistance at 100% SoC, 𝑄𝑜 and 𝑄𝑒 the battery rated Ampere-hour (Ah) capacity and extracted
Ah, respectively, and 𝑖𝑏𝑎𝑡 the battery current. It is assumed that the battery packs are kept in a
temperature-controlled environment and hence the effect of temperature variation is not
modelled. From Fig. 4.13, the battery terminal voltage is given by (4.41), where a positive 𝑖𝑏𝑎𝑡
implies battery is being discharged.
𝑣𝑏𝑎𝑡 = 𝐸𝑜𝑐 − 𝑅𝑠𝑒𝑟𝑖𝑒𝑠 𝑖𝑏𝑎𝑡
(4.41)
In the proposed WECS (Fig. 3.12), the reduced H-bridge terminals are connected in series
with the dc link inductor. Thus, the battery-side current 𝑖𝑖𝑛 is composed of pulses of magnitude
𝑖𝑑𝑐 (discharging) or −𝑖𝑑𝑐 (charging), separated by zero-current periods (freewheeling). A lowpass L-C filter is used to smooth out the battery current (See Fig. 4.14). Design of LC filter is
based on a specified cut-off frequency and damping ratio, as explained in Appendix B.
→
→
→
Reduced
H- Bridge
Fig. 4.14: LC filter on battery side.
Based on Fig. 4.14, the dynamic of the battery-side filter is described by (4.42) and (4.43).
𝑑
𝑖
𝑑𝑡 𝑏𝑎𝑡
𝑑
𝑣
𝑑𝑡 𝑐𝑏
1
= 𝐿 (𝑣𝑏𝑎𝑡 − 𝑣𝑐𝑏 )
𝑏
1
= 𝑐 (𝑖𝑏𝑎𝑡 − 𝑖𝑖𝑛 )
𝑏
77
(4.42)
(4.43)
4.7
DC/DC Reduced H- Bridge Converter
Fig. 4.15 shows the dc/dc H-bridge converter, with reduced number of switches and diodes,
employed as the interface between the storage battery bank and the dc link. The terminal
current of the H-bridge is equal to 𝑖𝑑𝑐 ; hence, the load of the H-bridge can be represented by a
dc current source.
→
SA
DB
+
Battery
with LC
filter
-
SB
-
+
DA
↓
Fig. 4.15: Reduced H-bridge for storage integration.
The reduced H-bridge can be controlled by either PWM bipolar voltage or PWM unipolar
voltage switching scheme. Unipolar voltage switching scheme provides frequency doubling
effect, resulting in higher-quality waveforms. In this switching scheme, the two switches S𝐴
and S𝐵 of the converter are controlled independently. The switch and diode in each leg can
never conduct at the same time; hence, short circuiting of the dc source (i.e., the battery) is
always avoided. Table 4.1 shows the possible output voltage values at different states of the
two switches.
Table 4.1: Reduced H-bridge converter operating modes.
𝑺𝑨
𝑺𝑩
𝒗𝒙𝒚
𝒊𝒊𝒏
0
0
−𝑣𝑐𝑏
−𝑖𝑑𝑐
0
1
0
0
Freewheeling state through 𝑆𝐵 and 𝐷𝐴
1
0
0
0
Freewheeling state through 𝐷𝐵 and 𝑆𝐴
1
1
𝑣𝑐𝑏
𝑖𝑑𝑐
Mode
Charging through 𝐷𝐵 and 𝐷𝐴
Discharging through 𝑆𝐴 and 𝑆𝐵
78
Let 𝑑𝐴 be the duty ratio of switch 𝑆𝐴 . The average values of output voltage 𝑣𝑥𝑦 and input
current 𝑖𝑖𝑛 can be expressed by (4.44) and (4.45), respectively.
𝑣𝑥𝑦 = (2𝑑𝐴 − 1)𝑣𝑐𝑏
(4.44)
𝑖𝑖𝑛 = (2𝑑𝐴 − 1)𝑖𝑑𝑐
(4.45)
Based on (4.44) and (4.45), the equivalent circuit diagram of the converter is shown in
Fig. 4.16.
→
+
→
→
+
+
-
-
-
Fig. 4.16: Large-signal nonlinear averaged model of CCM Full-Bridge Converter.
Substituting (4.41) and (4.45) in (4.42) and (4.43), respectively, the dynamic of storage-side
LC filter is described by the following differential equations.
𝑑
1
𝑖𝑏𝑎𝑡 = (𝐸𝑜 − 𝐾𝐸 (1 − 𝑆𝑜𝐶) − 𝑅𝑜 (1 + 𝐴0 (1 − 𝑆𝑜𝐶))𝑖𝑏𝑎𝑡 − 𝑣𝑐𝑏 )
𝑑𝑡
𝐿𝑏
𝑑
𝑣
𝑑𝑡 𝑐𝑏
1
= 𝑐 (𝑖𝑏𝑎𝑡 − (2𝑑𝐴 − 1)𝑖𝑑𝑐 )
𝑏
(4.46)
(4.47)
From (4.39) and (4.40), the dynamic of SoC is described by
𝑑
1
𝑆𝑜𝐶 = − 𝑄 𝑖𝑏𝑎𝑡
𝑑𝑡
(4.48)
𝑜
The dynamic of SoC is very slow with respect to other states. Therefore, for stability analysis,
carried out around an operating steady-state point, SoC is assumed constant.
79
4.8
Current Source Inverter
In this section, the current source inverter is modelled under balanced and unbalanced load
conditions.
In the proposed WECS, shown in Fig. 3.12, the inverter dc side is supplied by the power
received from a wind turbine, augmented by a storage system. This combination can represent
a dispatchable distributed energy resource unit, producing a controllable current (i.e., 𝑖𝑑𝑐 ), and
hence can be represented by a dc current source, as shown in Fig. 4.17. In the figure, the threephase current-sourced inverter is feeding a three-phase load. The relationship between the acside and dc-side voltages and currents of CSI are determined by the switching functions of the
three legs of the inverter. The switching constraints and states in the operation of CSI are
explained in Appendix C. The switching actions of the inverter generate high-frequency
harmonics, which are significantly reduced by the C-filter (𝐶𝑖 ), and hence can be neglected, as
far as the fundamental components of output-side currents are concerned.
Extended Generic Load Model (Fig. 4.21)
S1
+
S3
S5
∆ /Yn
↑
b
c
S2
S4
→
→
→
→
→
→
a
S6
Balanced/
Unbalanced
load
Ci
C-Filter
n
Fig. 4.17: Three-phase current-sourced inverter feeding a three-phase load.
4.8.1
ABC-Frame Equations
Under three-phase balanced load condition, the inverter supplies the load with three-phase
balanced voltages and currents. Under a PWM scheme, the fundamental components of the acside currents are related to the dc-side current as in (4.49).
[𝑖𝑜 ]𝑎𝑏𝑐 = 𝐺 [𝑚𝑖 ]𝑎𝑏𝑐 𝑖𝑑𝑐
In (4.49),
80
(4.49)
𝑚𝑖𝑎
𝑚𝑖 sin(𝜔𝐿 𝑡 − 𝛼𝑖 )
= [𝑚𝑖𝑏 ] = [ 𝑚𝑖 sin(𝜔𝐿 𝑡 − 𝛼𝑖 − 2𝜋/3) ]
𝑚𝑖𝑐
𝑚𝑖 sin(𝜔𝐿 𝑡 − 𝛼𝑖 + 2𝜋/3)
[𝑚𝑖 ]𝑎𝑏𝑐
(4.50)
where 𝑚𝑖 and 𝛼𝑖 are the modulation index (0 < 𝑚𝑖 < 1) and delay angle of the inverter,
respectively, 𝐺 the ac gain of the corresponding PWM scheme (𝐺 =
√3
2
for sinusoidal PWM)
and 𝜔𝐿 the fundamental frequency of the load-side voltage (i.e., 100𝜋 or 120𝜋 𝑟𝑎𝑑/𝑠 for 50Hz
or 60Hz, respectively). Ignoring the inverter losses, the power balance of the ac-side and dcside of the inverter gives:
𝑣𝑖𝑛𝑣 𝑖𝑑𝑐 = [𝑖𝑜 ]𝑇𝑎𝑏𝑐 [𝑣𝑐𝑖 ]
where 𝑣𝑖𝑛𝑣 is the average dc-input voltage and [𝑣𝑐𝑖 ]
𝑎𝑏𝑐
𝑎𝑏𝑐
(4.51)
the three-phase capacitor voltages,
defined as
𝑉̂𝑐𝑖 sin(𝜔𝐿 𝑡 + 𝜃𝑣𝑐 )
[𝑣𝑐𝑖 ]
𝑎𝑏𝑐
= [ 𝑉̂𝑐𝑖 sin(𝜔𝐿 𝑡 + 𝜃𝑣𝑐 − 2𝜋/3) ]
𝑉̂𝑐𝑖 sin(𝜔𝐿 𝑡 + 𝜃𝑣𝑐 + 2𝜋/3)
(4.52)
𝑐
Substituting (4.49) into (4.51), yields
𝑣𝑖𝑛𝑣 = 𝐺 [𝑚𝑖 ] 𝑇𝑎𝑏𝑐 [𝑣𝑐𝑖 ]
𝑎𝑏𝑐
(4.53)
Applying KCL on the ac-side of the inverter, the dynamics of C-filter is described by:
𝑑
𝑑𝑡
[𝑣𝑐𝑖 ]
𝑎𝑏𝑐
1
1
𝑖
𝑖
= 𝐶 [𝑖𝑜 ]𝑎𝑏𝑐 − 𝐶 [𝑖𝑝 ]𝑎𝑏𝑐
(4.54)
where [𝑖𝑝 ]𝑎𝑏𝑐 is the transformer three-phase primary-side current. Based on (4.49) and (4.53),
the equivalent circuit diagram of the inverter is shown in Fig. 4.18.
81
+
→
+
_
_
+
→
→
→
_
Fig. 4.18: Large-signal nonlinear averaged model of current source inverter.
Under three-phase unbalanced load condition, the three phase currents on the transformer
secondary side are unbalanced, with positive (+ve), negative (-ve), and zero sequence (seq)
components. Since the zero seq component is trapped in the Δ winding of the transformer, the
primary-side three-phase currents (i.e., 𝑖𝑎𝑝 , 𝑖𝑏𝑝 and 𝑖𝑐𝑝 ) contain only +ve and –ve seq
components. Now, every three-phase quantity in the ac-side of the inverter is divided into
three-phase +ve and –ve seq symmetrical components, rotating at the fundamental frequency
(𝜔𝐿 ), but with opposite phase sequence. In other words, [𝑚𝑖 ]𝑎𝑏𝑐 , [𝑖𝑜 ]𝑎𝑏𝑐 , [𝑣𝑐𝑖 ]
𝑎𝑏𝑐
, and [𝑖𝑝 ]𝑎𝑏𝑐
are divided into [𝑚𝑖+ ]𝑎𝑏𝑐 and [𝑚𝑖− ]𝑎𝑏𝑐 , [𝑖𝑜+ ]𝑎𝑏𝑐 and [𝑖𝑜− ]𝑎𝑏𝑐 , [𝑣𝑐+𝑖 ]𝑎𝑏𝑐 and [𝑣𝑐−𝑖 ]𝑎𝑏𝑐 , and [𝑖𝑝+ ]𝑎𝑏𝑐
and [𝑖𝑝− ]𝑎𝑏𝑐 , respectively, where the +ve and –ve seq components can be defined by (4.55) and
(4.56), respectively, by replacing the variable 𝑥 with the quantity under consideration.
[𝑥 + ]𝑎𝑏𝑐
𝑋̂ + sin(𝜔𝐿 𝑡 − 𝜃𝑥+ )
= [ 𝑋̂ + sin(𝜔𝐿 𝑡 − 𝜃𝑥+ − 2𝜋/3) ]
𝑋̂ + sin(𝜔𝐿 𝑡 − 𝜃𝑥+ + 2𝜋/3)
(4.55)
[𝑥 − ]𝑎𝑏𝑐
𝑋̂ − sin(𝜔𝐿 𝑡 − 𝜃𝑥− )
= [ 𝑋̂ − sin(𝜔𝐿 𝑡 − 𝜃𝑥− + 2𝜋/3) ]
𝑋̂ − sin(𝜔𝐿 𝑡 − 𝜃𝑥− − 2𝜋/3)
(4.56)
82
4.8.2 DQ-Frame Equations
For balanced load, where only +ve seq, three-phase voltages and currents exist, the control
design of the CSI is based on a synchronously-rotating dq frame. By transforming the abc
voltages and currents into dq frame rotating at 𝜔𝐿 , the dq-axis equations of the inverter are
obtained as
+
+
𝑚𝑖𝑑
𝑖𝑜𝑑
[ + ] = 𝐺 [ + ] 𝑖𝑑𝑐
𝑖𝑜𝑞
𝑚𝑖𝑞
(4.57)
+ +
+ +
𝑣𝑖𝑛𝑣 = 1.5 𝐺 (𝑚𝑖𝑑
𝑣𝑐𝑖𝑑 + 𝑚𝑖𝑞
𝑣𝑐𝑖𝑞 )
+
+
𝑣𝑐𝑖𝑑
0
1 𝑖
[ + ] = [ 𝑜𝑑
]−[
+
𝑑𝑡 𝑣
𝐶𝑖 𝑖𝑜𝑞
𝜔𝐿
𝑐𝑖𝑞
𝑑
+
+
−𝜔𝐿 𝑣𝑐𝑖𝑑
1 𝑖𝑝𝑑
][ + ] − [ + ]
𝐶𝑖 𝑖
0
𝑣𝑐𝑖𝑞
𝑝𝑞
(4.58)
(4.59)
+
+
where 𝑚𝑖𝑑
and 𝑚𝑖𝑞
are the equivalent 𝑑- and 𝑞-axis modulation indices of the inverter,
+
+
+
+
respectively, 𝑣𝑐𝑖𝑑
and 𝑣𝑐𝑖𝑞
the d- and q-axis capacitor voltages, respectively, and 𝑖𝑝𝑑
and 𝑖𝑝𝑞
the 𝑑- and 𝑞-axis primary-side transformer currents, respectively.
For unbalanced load, the +ve and –ve seq abc components are transformed into dq axis
frames rotating at 𝜔𝐿 and −𝜔𝐿 , respectively. The resulting equations are given in (4.60)(4.62). Note that the sequence of a quantity is identified by the sign in its superscript.
+
+
𝑚𝑖𝑑
𝑖𝑜𝑑
+
+
𝑖𝑜𝑞
𝑚𝑖𝑞
=
𝐺
−
− 𝑖𝑑𝑐
𝑖𝑜𝑑
𝑚𝑖𝑑
−
−
[𝑖𝑜𝑞
]
[𝑚𝑖𝑞 ]
(4.60)
+ +
+ +
− −
− −
𝑣𝑖𝑛𝑣 = 1.5 𝐺 (𝑚𝑖𝑑
𝑣𝑐𝑖𝑑 + 𝑚𝑖𝑞
𝑣𝑐𝑖𝑞 + 𝑚𝑖𝑑
𝑣𝑐𝑖𝑑 + 𝑚𝑖𝑞
𝑣𝑐𝑖𝑞 )
(4.61)
+
+
𝑣𝑐𝑖𝑑
𝑖𝑜𝑑
0
+
+
𝜔𝐿
𝑑 𝑣𝑐𝑖𝑞
1 𝑖
= 𝐶 𝑜𝑞
− −[ 0
𝑑𝑡 𝑣 −
𝑖 𝑖𝑜𝑑
𝑐𝑖𝑑
−
−
0
𝑣
[𝑖𝑜𝑞
]
[ 𝑐𝑖𝑞 ]
4.9
−𝜔𝐿
0
0
0
0
0
0
−𝜔𝐿
+
+
𝑖𝑝𝑑
0 𝑣𝑐𝑖𝑑
+
+
0 𝑣𝑐𝑖𝑞
1 𝑖
] − − 𝐶 𝑝𝑞
−
𝜔𝐿 𝑣𝑐𝑖𝑑
𝑖 𝑖𝑝𝑑
−
−
0 [𝑣𝑐𝑖𝑞 ]
[𝑖𝑝𝑞
]
(4.62)
Generic Load Model
In a standalone energy system, non-linearity and variability (including frequent switching)
of the load have significant adverse impact on the system’s performance and may jeopardize
the stability of voltage/frequency control. Therefore, load modeling is an essential part of
83
stability analysis and controller design for a standalone system. The stability analysis carried
out in [133] has been based on an RL load. Similarly, control schemes developed in [134]
and [135] have assumed RL and RLC loads, respectively. A nonlinear load has been presented
by a current source in [136]. The effect of induction motor loads on system’s stability has been
discussed in [137]. However, due to the diversity of residential loads, a generic load model,
proposed by [138], is used in this thesis. The model is, in essence, capable of emulating actual
loads of different transient and steady-state characteristics and is appropriate for simulation
studies of stability and dynamic performance of a standalone system. The model has been
successfully used to emulate RL and induction motor loads [138],[139]. However, the model
has been mainly developed and validated for balanced load conditions. In the next subsections,
the generic load model proposed in [138] will be described first. Then, the order of the model
will be reduced by minimal realization. Finally, the reduced-order model will be extended to
include unbalanced load condition.
4.9.1 Generic Load Model Proposed in [138]
The structure of the model is illustrated in Fig. 4.19. The load is modeled by three dependent
current sources. The control signals for this model are obtained from dq/abc transformation of
the load d- and q-axis currents, 𝑖𝐿𝑑 and 𝑖𝐿𝑞 , that are dynamically determined based on load
voltage dq components, 𝑣𝐿𝑑 and 𝑣𝐿𝑞 . As shown in Fig. 4.19, the angle of transformation (𝜃𝐿 )
and angular velocity (𝜔𝐿 ) of the load voltage vector are obtained by implementing a phaselocked loop (PLL). In this model, the dynamics of the load is described by a set of state-space
equations as:
𝑋̇𝐿 = 𝐴𝐿 𝑋𝐿 (𝑡) + 𝐵𝐿 [
𝑌𝐿 (𝑡) = 𝐶𝐿 𝑋𝐿 (𝑡) = [
𝑣𝐿𝑑 (𝑡)
]
𝑣𝐿𝑞 (𝑡)
𝑖𝐿𝑑 (𝑡)
]
𝑖𝐿𝑞 (𝑡)
(4.63)
(4.64)
where 𝑋𝐿 is the vector of state variables, 𝑌𝐿 the vector of outputs, 𝑣𝐿𝑑 and 𝑣𝐿𝑞 the d- and q-axis
components of load voltage (as inputs), 𝑖𝐿𝑑 and 𝑖𝐿𝑞 the d- and q-axis components of load
current (as outputs), and 𝐴𝐿 , 𝐵𝐿 , 𝐶𝐿 time-invariant matrices determining the dynamic and
steady-state characteristics of the load.
84
Fig. 4.19: Block diagram of the generic load model proposed in [138].
By solving (4.63) and (4.64), the load currents are obtained, as given by (4.65), where 𝑋𝐿 (0)
denotes the initial state variables vector.
[
𝑖𝐿𝑑 (𝑡)
𝑣𝐿𝑑 (𝑡)
𝑡
] = 𝐶𝐿 [ 𝑒 𝐴𝐿𝑡 𝑋𝐿 (0) + ∫0 𝑒 𝐴𝐿(𝑡−𝜏) 𝐵𝐿 [
] 𝑑𝜏]
𝑖𝐿𝑞 (𝑡)
𝑣𝐿𝑞 (𝑡)
(4.65)
From (4.63) and (4.64), the steady-state currents are:
[
𝑣𝐿𝑑
𝑖𝐿𝑑𝑠𝑠
] = (−𝐶𝐿 𝐴−1
𝐵𝐿 ) [𝑣 𝑠𝑠 ]
𝐿
𝑖𝐿𝑞𝑠𝑠
𝐿𝑞𝑠𝑠
(4.66)
With a load-side voltage-oriented synchronous reference frame, 𝑣𝐿𝑞𝑠𝑠 = 0. The steady-state
real and reactive powers of the load are given by:
𝑃
[ ]=
𝑄
3
𝑣
2 𝐿𝑑𝑠𝑠
[
𝑖𝐿𝑑𝑠𝑠
]
−𝑖𝐿𝑞𝑠𝑠
(4.67)
Substituting (4.66) in (4.67), yields
[
𝑣𝐿𝑑
𝑃
3
] = 2 𝑣𝐿𝑑𝑠𝑠 (−𝐶𝐿 𝐴−1
𝐵𝐿 ) [ 𝑠𝑠 ]
𝐿
−𝑄
0
85
(4.68)
Assuming the product (𝐶𝐿 𝐴−1
𝐿 𝐵𝐿 ) to be a function of load-side frequency, 𝜔𝐿 , 𝑃 and 𝑄 can
be expressed as:
𝑃 = 𝑃𝑜 (
𝑣𝐿𝑑𝑠𝑠 𝛼
𝜔
) 𝑝 (𝜔 𝐿 )𝛽𝑝
𝑉𝑜
𝐿𝑜
(4.69)
𝑣𝐿𝑑𝑠𝑠 𝛼
𝜔
) 𝑞 (𝜔 𝐿 )𝛽𝑞
𝑉𝑜
𝐿𝑜
𝑄 = 𝑄𝑜 (
(4.70)
where 𝑃0 and 𝑄0 stand for real and reactive power at nominal load voltage (𝑉0) and nominal
load frequency (𝜔𝐿𝑜 ). The parameters 𝛽𝑝 and 𝛽𝑞 characterize the dependence of load powers
on frequency, while the parameters 𝛼𝑝 and 𝛼𝑞 characterize the dependence of load powers on
voltage magnitude depending on load type, as given in Table 4.2. In this model, constantimpedance load type is assumed, i.e., 𝛼𝑝 = 𝛼𝑞 = 2. The matrices 𝐴𝐿 , 𝐵𝐿 , 𝐶𝐿 are given in
Table 4.3. In the table, 𝑑 denotes damping of natural modes of the load model in 𝑠 −1 (𝑑 is
inversely proportional to the settling time, 𝑡𝑠 ), 𝜔𝑜 the oscillation frequency in 𝑟𝑎𝑑/𝑠, and 𝑌𝑃
and 𝑌𝑄 , given in (4.71) and (4.72), the real and reactive power indices of the load, respectively.
The dynamic characteristics of the load is determined by 𝑑 and 𝜔𝑜 which, in turn, are equal to
the real and imaginary parts of the eigenvalues of matrix 𝐴𝐿 . On the other hand,
𝑃𝑜 , 𝑄𝑜 , 𝛼𝑝 , 𝛼𝑞 , 𝛽𝑝 , 𝛽𝑞 , 𝑉𝑜 and 𝜔𝐿𝑜 determine the steady-state characteristics of the load.
2 1
𝑌𝑃 = 3
2 1
𝑌𝑄 = 3
𝜔
𝑉𝑜2
𝑉𝑜2
(𝜔 𝐿 )𝛽𝑝 𝑃𝑜
(4.71)
𝐿𝑜
𝜔
(𝜔 𝐿 )𝛽𝑞 𝑄𝑜
(4.72)
𝐿𝑜
Table 4.2: Dependence of load on voltage magnitude.
𝜶𝒑
𝜶𝒒
Constant Impedance
2
2
Constant Current
1
1
Constant Power
0
0
Load type
Examples
Incandescent Lighting, electric stoves, and
water heaters
Controlled-current motor drives
Induction motor drives and controlled power
supplies
86
4.9.2 Reduced-Order Load Model
As shown in Table 4.3, the model proposed by [138] is a 4th -order state-space system with
two uncontrollable modes (i.e., the controllability matrix is not full rank). By applying minimal
realization technique, the matrices 𝐴𝐿 , 𝐵𝐿 and 𝐶𝐿 are modified into 𝐴𝐿𝑚 , 𝐵𝐿𝑚 and 𝐶𝐿𝑚 and the
model order is reduced to 2. The properties of the reduced-order model versus the 4th-order
model are given in Table 4.3. The states 𝑥1𝑚 and 𝑥2𝑚 in the reduced-order model are fictitious
states resulting from the minimal realization process. The reduced-order model has the same
inputs and outputs as the 4th -order model. In the 4th -order model, 𝑌𝑄 assumes a negative
value [138]. In the reduced-order model, in order to comply with the common sign convention
for load reactive power, 𝑌𝑄 has been assumed to be positive and hence multiplied by -1. Using
both models, Fig. 4.20 gives the responses of load currents to connection of a balanced load at
t = 0.1 s for the load parameters given in Table 4.3. The figure shows that the characteristics
of the reduced-order model are identical to those of the original 4th -order model.
Fig. 4.20: Reduced-order model versus the original 4th -order model.
87
Table 4.3: Characteristics of the generic load models.
4th - order Model [138]
𝑋𝐿 = [𝑖𝐿𝑑
𝑥2
𝑖𝐿𝑞
Reduced-order Model
𝑥4 ]𝑇
𝑋𝐿𝑚 = [𝑥1𝑚 𝑥2𝑚 ]𝑇
𝑥2 = 𝑑𝑖𝐿𝑑 /𝑑𝑡, 𝑥4 = 𝑑𝑖𝐿𝑞 /𝑑𝑡
Inputs :
𝑈𝐿 = [𝑣𝐿𝑑
𝑣𝐿𝑞 ]𝑇
Outputs :
𝑌𝐿 = [𝑖𝐿𝑑
𝑖𝐿𝑞 ]𝑇
0
−(𝑑2 + 𝜔𝑜2 )
𝐴𝐿 = [
0
0
1
2𝑑
0
0
0
0
0
0
]
0
1
−(𝑑2 + 𝜔𝑜2 ) 2𝑑
Eigenvalues: 𝜆1 = 𝜆2 = 𝑑 + 𝑗𝜔𝑜 ,
𝜆3 = 𝜆4 = 𝑑 − 𝑗𝜔𝑜
0
𝑌𝑃 (𝑑 + 𝜔𝑜2 )
𝐵𝐿 = [
0
𝑌𝑄 (𝑑 2 + 𝜔𝑜2 )
2
𝐶𝐿 = [
1 0
0 0
Steady-state outputs: [
2
2
𝐴𝐿𝑚 = [0 −(𝑑 + 𝜔𝑜 )]
1
2𝑑
Eigenvalues: 𝜆1 = 𝑑 + 𝑗𝜔𝑜 ,
𝜆2 = 𝑑 − 𝑗𝜔𝑜
0
0
0]
0
𝐵𝐿𝑚 = [
0 0
]
1 0
𝐶𝐿𝑚 = [
𝑖𝐿𝑑𝑠𝑠
𝑌𝑃 𝑣𝐿𝑑𝑠𝑠
]=[
]
𝑖𝐿𝑞𝑠𝑠
𝑌𝑄 𝑣𝐿𝑑𝑠𝑠
[
1 0
]
0 0
0 𝑌𝑃 (𝑑 2 + 𝜔𝑜2 )
]
0 −𝑌𝑄 (𝑑 2 + 𝜔𝑜2 )
𝑖𝐿𝑑𝑠𝑠
𝑌𝑃 𝑣𝐿𝑑𝑠𝑠
]=[
]
𝑖𝐿𝑞𝑠𝑠
−𝑌𝑄 𝑣𝐿𝑑𝑠𝑠
Parameters for Fig. 4.20: 𝑑 = 5 𝑠 −1 , 𝜔𝑜 = 75 𝑟𝑎𝑑/𝑠, 𝑉𝑜 = 310.3 𝑉, 𝜔𝐿𝑜 =
377 𝑟𝑎𝑑/𝑠, 𝑃𝑜 = 20𝑘𝑊, 𝑄𝑜 = 10 𝑘𝑉𝑎𝑟, 𝑌𝑝 = 0.1385 𝑊/𝑉 2 , 𝑌𝑄 = 0.0692 𝑉𝑎𝑟/𝑉 2 ,
𝛼𝑝 = 𝛼𝑞 = 2, 𝛽𝑝 = 𝛽𝑞 = 0.
4.9.3 Generic Load Model including Unbalanced Load Condition
In Fig. 3.12, the proposed WECS supplies the load via a ∆/𝑌𝑛 transformer. As mentioned
in subsection 4.8.1, under three-phase unbalanced load condition, the primary-side three-phase
currents (i.e., 𝑖𝑎𝑝 , 𝑖𝑏𝑝 and 𝑖𝑐𝑝 ) contain only +ve and –ve seq components; hence, each
component can be described by the reduced-order model of Table 4.3. In other words, the
combination of unbalanced load and the ∆/𝑌𝑛 transformer, placed inside the dashed rectangle
in Fig. 4.17, can be replaced by six dependent current sources (3 for each sequence), as shown
in Fig. 4.21.
88
→
→
→
→
→
→
∆ /Yn
→
→
→
→
→
→
Balanced/
Unbalanced
load
n
.
Fig. 4.21: Generic load model including unbalanced case.
In the presence of load imbalance, the function of the three-phase inverter is to maintain the
symmetry of the three-phase output voltage applied to the load; hence, no –ve seq voltage is to
−
−
be present across the capacitor in steady state, i.e., 𝑣𝑐𝑖𝑑
= 𝑣𝑐𝑖𝑞
= 0. In other words, the CSI
𝑠𝑠
𝑠𝑠
with capacitor filter represent a three-phase voltage source that supplies balanced voltages to
an unbalanced three-phase load.
Based on (4.66)-(4.72), the steady-state active and reactive powers and power indices
associated with +ve seq voltage and +ve seq current are given by (4.73) - (4.75).
[
+
𝑖𝑝𝑑
𝑠𝑠
+
𝑣
[
]
2 𝑐𝑖𝑑𝑠𝑠 −𝑖 +
𝑝𝑞𝑠𝑠
𝑃+
]=
𝑄+
3
2 1
𝑌𝑃+ = 3
𝜔
𝑉𝑜2
2 1
𝑌𝑄+ = 3
𝑉𝑜2
(𝜔 𝐿 )𝛽𝑝 𝑃𝑜+
𝐿𝑜
𝜔
(𝜔 𝐿 )𝛽𝑞 𝑄𝑜+
𝐿𝑜
(4.73)
(4.74)
(4.75)
−
Since 𝑣𝑐𝑖𝑑
= 0, steady-state active and reactive powers produced by the multiplication of
𝑠𝑠
–ve seq voltage and –ve seq currents will be equal to zero.
89
The interference between +ve seq voltage and –ve seq current produces a power component
oscillating at twice the load-side frequency. This oscillating power is referred to as negative
seq reactive power (𝑄 ± ) and is given by (4.76) [140], where 𝑉𝑐+𝑖 , 𝜃𝑣+𝑐 , 𝐼𝑝− and 𝜃𝑖−𝑝 present
𝑖
+ve seq capacitor voltage and –ve seq load current,
magnitudes and phase angles of
respectively.
𝑄 ± = −3 𝑉𝑐+𝑖 𝐼𝑝− cos (2𝜔𝐿 𝑡 + 𝜃𝑣+𝑐 − 𝜃𝑖−𝑝 )
𝑖
(4.76)
In this thesis, the peak value of 𝑄 ± (i.e., 3 𝑉𝑐+𝑖 𝐼𝑝− ), denoted by 𝑄 − , will be adopted to obtain
expressions for power indices, allowing to extend the reduced-order load model to unbalanced
load case. In synchronous dq reference frame, 𝑄 − is expressed as:
3
+
−
−
−
−
𝑄 − = 2 𝑣𝑐𝑖𝑑
(𝑖𝑝𝑑
+ 𝑗𝑖𝑝𝑞
) = 𝑄𝑝𝑑
+ 𝑗 𝑄𝑝𝑞
(4.77)
−
−
In (4.77), 𝑄𝑝𝑑
and 𝑄𝑝𝑞
represent the reactive powers associated with the –ve seq currents and
are counterparts of 𝑃+ and 𝑄 + , respectively, that are associated with the +ve seq currents.
In parallel with (4.73)-(4.75), the steady-state d-axis and q-axis powers and power indices
associated with +ve seq voltage and -ve seq currents are given by (4.78)-(4.80).
[
−
𝑄𝑝𝑑
− ]=
𝑄𝑝𝑞
−
(𝜔 𝐿 )𝛽𝑝 𝑄𝑝𝑑𝑜
(4.79)
𝜔
𝑉𝑜2
2 1
𝑌𝑄−𝑝𝑞 = 3
(4.78)
𝑣+ [
2 𝑐𝑖𝑑𝑠𝑠
2 1
𝑌𝑄−𝑝𝑑 = 3
−
𝑖𝑝𝑑
𝑠𝑠
− ]
−𝑖𝑝𝑞
𝑠𝑠
3
𝑉𝑜2
𝐿𝑜
𝜔
(𝜔 𝐿 )
𝐿𝑜
𝛽𝑞
−
𝑄𝑝𝑞𝑜
(4.80)
Based on the above equations, the reduced-order model introduced in Table 4.3 is extended,
as shown in Table 4.4, to describe the dynamics of the load currents under unbalanced load
conditions. In Table 4.4, 𝑑 and 𝜔𝑜 determine the dynamics of both +ve and –ve seq currents,
−
−
while 𝑃𝑜+ , 𝑄𝑜+ , 𝑄𝑝𝑑𝑜
, 𝑄𝑝𝑞𝑜
, 𝛽𝑝 , 𝛽𝑞 , 𝑉𝑜 and 𝜔𝐿𝑜 correspond to the steady-state currents. Using the
extended model, Fig. 4.22 gives the responses of load currents to connection of an unbalanced
load to the load bus at t = 0.1 s for the load parameters given in Table 4.4.
90
It can be seen that the extended model is valid for both balanced and unbalanced load
conditions. In case of balanced load, the differential equations associated with –ve seq currents
are eliminated; hence, the system becomes identical to the reduced-order model shown in
+
−
−
+
Table 4.3 (i.e., 𝑖𝑝𝑑
= 𝑖𝑝𝑞
= 0, 𝑖𝑝𝑑
= 𝑖𝐿𝑑 , 𝑖𝑝𝑞
= 𝑖𝐿𝑞 ).
The load unbalance factor (LUF) is defined as the ratio of magnitude of –ve seq current to
that of +ve seq current, i.e.,
− )2 +(𝑖 − )2
(𝑖𝑝𝑑
𝑝𝑞
𝐿𝑈𝐹 = √(𝑖 +
+ 2
2
𝑝𝑑 ) +(𝑖𝑝𝑞 )
2
− ) +(𝑄− )
(𝑄𝑝𝑑𝑜
𝑝𝑞𝑜
=√
2
(𝑃𝑜+ ) +(𝑄𝑜+ )
2
(4.81)
2
Table 4.4: Characteristics of extended generic load model.
+
+
−
− ]𝑇
𝑋𝐿𝑀 = [𝑥1𝑚
𝑥2𝑚
𝑥1𝑚
𝑥2𝑚
State variables
+
𝑈𝐿 = [𝑣𝑐𝑖𝑑
Inputs
+
𝑌𝐿𝑀 = [𝑖𝑝𝑑
Outputs
𝐴𝐿𝑀
Model Matrices
𝐶𝐿𝑀
𝑇
+
−
−
𝑖𝑝𝑞
𝑖𝑝𝑑
𝑖𝑝𝑞
]
𝑇
0 −(𝑑 2 + 𝜔𝑜2 ) 0
0
2𝑑
0
0
= [1
]
2
0
0
0 −(𝑑 + 𝜔𝑜2 )
0
0
1
2𝑑
𝑇
𝐵𝐿𝑀 = [1 0, 0 0, 1 0, 0 0]
0
0
= 0
𝑌𝑃+ (𝑑 2 + 𝜔𝑜2 )
−𝑌𝑄+ (𝑑 2 + 𝜔𝑜2 )
0
[0
Steady- state currents
+
𝑣𝑐𝑖𝑞
]
0
0
0
0
0
0
− (𝑑 2
𝑌𝑄𝑝𝑑
+ 𝜔𝑜2 )
0 −𝑌𝑄−𝑝𝑞 (𝑑 2 + 𝜔𝑜2 )]
+
+
+
+
𝑖𝑝𝑑
= 𝑌𝑃+ 𝑣𝑐𝑖𝑑
, 𝑖𝑝𝑞
= −𝑌𝑄+ 𝑣𝑐𝑖𝑑
𝑠𝑠
𝑠𝑠
+
+
−
−
𝑖𝑝𝑑
= 𝑌𝑄−𝑝𝑑 𝑣𝑐𝑖𝑑
, 𝑖𝑝𝑞
= −𝑌𝑄−𝑝𝑞 𝑣𝑐𝑖𝑑
𝑠𝑠
𝑠𝑠
Parameter for Fig. 4.22
𝑑 = 10 𝑠 −1 , 𝜔𝑜 = 377 𝑟𝑎𝑑/𝑠, 𝑉𝑜 = 310.3𝑉, 𝑃𝑜+ = 20𝑘𝑊,
𝑌𝑃+ = 0.1385 𝑊/𝑉 2 , 𝑄𝑜+ = 10 𝑘𝑉𝑎𝑟, 𝑌𝑄+ = 0.0692 𝑉𝑎𝑟/𝑉 2 ,
−
𝑄𝑝𝑑𝑜
= −5.08 𝑘𝑉𝑎𝑟, 𝑌𝑄−𝑝𝑑 = −0.0352 𝑉𝑎𝑟/𝑉 2 ,
−
𝑄𝑝𝑞𝑜
= −6.04 𝑘𝑉𝑎𝑟, 𝑌𝑄−𝑝𝑞 = −0.0418 𝑉𝑎𝑟/𝑉 2
𝛼𝑝 = 𝛼𝑞 = 2, 𝛽𝑝 = 𝛽𝑞 = 0.
91
Fig. 4.22: Load currents and powers using the extended generic load model.
4.10 DC-Link Model
In the proposed system, the generator side combined with storage side is linked with the load
side through the dc-link inductor, as shown in Fig. 4.23.
Fig. 4.23: Configuration of the dc link.
92
Based on Fig. 4.23, the dynamics of dc-link current is given by:
𝑑
1
𝑖 =𝐿
𝑑𝑡 𝑑𝑐
𝑑𝑐
(𝑣𝑑 + 𝑣𝑥𝑦 − 𝑣𝑖𝑛𝑣 )
(4.82)
where 𝑣𝑑 , 𝑣𝑥𝑦 , and 𝑣𝑖𝑛𝑣 were defined in (4.36), (4.44), and (4.61), respectively.
4.11 Dynamic Model of the Proposed Wind Energy Conversion System
In the previous sections, the dynamic equations of different components of the proposed
system were derived. Using these equations, the dynamic and steady-state models of the
proposed WECS are developed in this section.
4.11.1 State Space Equations
In general, a linear-time invariant system is described by the following set of state-space
equations:
𝑋̇ = 𝐴 𝑋 + 𝐵 𝑈
𝑌 =𝐶𝑋+𝐷𝑈
(4.83)
where 𝑋 , 𝑈, and 𝑌 are the state variables vector, inputs vector, and output vector of the system,
respectively. 𝐴, 𝐵, 𝐶, and 𝐷 are the matrices for multi-input, multi-output (MIMO) system. By
combining all the equations derived in pervious sections, the entire system of the proposed
CSI-based WECS is described by a 20th–order state-space system given in (4.84). The
derivation process of the overall system model is presented in Appendix D. The dynamics of
the system is described by 20 first-order differential equations: 3 for the wind turbine shaft, 6
for the self-excited induction generator, 2 for the battery-side low-pass filter, 4 for the CSIside capacitor filter, 4 for the extended generic load model, and 1 for the dc link. In case of
balanced load condition, the four differential equations associated with the –ve seq components
of CSI-side filter capacitor voltages and transformer primary-side currents are eliminated;
hence, the order of the overall system is reduced to 16.
93
𝑑
1
30 𝑣𝑤3 2.35 𝑣𝑤2 (−21 𝑣𝑤+0.74)
𝜔𝑟
𝜔𝑚 = [𝜌 𝜋𝑟 2 ( ( 2 −
)𝑒 𝜔𝑚 𝑟
+ 0.0034 𝑟 𝑣𝑤 )𝑣𝑤 − 𝐾𝑠𝑒 𝛿𝜃 − 𝐷𝑠𝑒 (𝜔𝑚 −
)]
𝑑𝑡
𝐽𝑚
𝜔𝑚 𝑟
𝜔𝑚
𝑃 𝑛𝑔𝑒𝑎𝑟
𝑑
𝑃
1
1
𝜔𝑟
3𝑃𝐿𝑚
′
′
𝜔𝑟 = [
𝐾𝑠𝑒 𝛿𝜃 +
𝐷𝑠𝑒 (𝜔𝑚 −
)−
(𝑖𝑞𝑠 𝑖𝑑𝑟
− 𝑖𝑑𝑠 𝑖𝑞𝑟
)]
𝑑𝑡
𝐽𝑔 𝑛𝑔𝑒𝑎𝑟
𝑛𝑔𝑒𝑎𝑟
𝑃 𝑛𝑔𝑒𝑎𝑟
2
𝑑
𝜔𝑟
(𝛿𝜃) = 𝜔𝑚 −
𝑑𝑡
𝑃 𝑛𝑔𝑒𝑎𝑟
…
…
𝑑
1
′
′
𝑖 = [𝑅 𝐿′ 𝑖 + (𝜔𝑒 (𝐿𝑠 𝐿′𝑟 − 𝐿2𝑚 ) + 𝜔𝑟 𝐿2𝑚 )𝑖𝑑𝑠 −𝑅𝑟′ 𝐿𝑚 𝑖𝑞𝑟
+ 𝜔𝑟 𝐿𝑚 𝐿′𝑟 𝑖𝑑𝑟
−𝐿′𝑟 𝑣𝑐𝑔𝑞 ]
𝑑𝑡 𝑞𝑠 𝑘𝑠 𝑠 𝑟 𝑞𝑠
𝑑
1
′
′
𝑖 = [(𝜔𝑒 (𝐿2𝑚 − 𝐿𝑠 𝐿′𝑟 ) − 𝜔𝑟 𝐿2𝑚 ) 𝑖𝑞𝑠 + 𝑅𝑠 𝐿′𝑟 𝑖𝑑𝑠 −𝜔𝑟 𝐿𝑚 𝐿′𝑟 𝑖𝑞𝑟
−𝑅𝑟′ 𝐿𝑚 𝑖𝑑𝑟
−𝐿′𝑟 𝑣𝑐𝑔𝑑 ]
𝑑𝑡 𝑑𝑠 𝑘𝑠
𝑑 ′
1
′
′
𝑖𝑞𝑟 = [−𝑅𝑠 𝐿𝑚 𝑖𝑞𝑠 −𝜔𝑟 𝐿𝑚 𝐿𝑠 𝑖𝑑𝑠 + 𝑅𝑟′ 𝐿𝑠 𝑖𝑞𝑟
+ (𝜔𝑒 (𝐿𝑠 𝐿′𝑟 − 𝐿2𝑚 ) − 𝐿𝑠 𝐿′𝑟 𝜔𝑟 )𝑖𝑑𝑟
+ 𝐿𝑚 𝑣𝑐𝑔𝑞 ]
𝑑𝑡
𝑘𝑠
𝑑 ′
1
′
′
𝑖 = [𝜔 𝐿 𝐿 𝑖 −𝑅 𝐿 𝑖 + (𝜔𝑒 (𝐿2𝑚 − 𝐿𝑠 𝐿′𝑟 ) + 𝐿𝑠 𝐿′𝑟 𝜔𝑟 )𝑖𝑞𝑟
+ 𝑅𝑟′ 𝐿𝑠 𝑖𝑑𝑟
+ 𝐿𝑚 𝑣𝑐𝑔𝑑 ]
𝑑𝑡 𝑑𝑟 𝑘𝑠 𝑟 𝑚 𝑠 𝑞𝑠 𝑠 𝑚 𝑑𝑠
𝑑
1
𝑣 = − 𝑖𝑞𝑠 − 𝜔𝑒 𝑣𝑐𝑔𝑑
𝑑𝑡 𝑐𝑔𝑞
𝐶𝑔
𝑑
1
2√3
𝑣𝑐𝑔𝑑 = − (𝑖𝑑𝑠 +
𝑑 𝑖 ) + 𝜔𝑒 𝑣𝑐𝑔𝑞
𝑑𝑡
𝐶𝑔
𝜋 𝑏 𝑑𝑐
…
…
𝑑
1
𝑖𝑏𝑎𝑡 = (𝐸𝑜 − 𝐾𝐸 (1 − 𝑆𝑜𝐶) − 𝑅𝑜 (1 + 𝐴0 (1 − 𝑆𝑜𝐶))𝑖𝑏𝑎𝑡 − 𝑣𝑐𝑏 )
𝑑𝑡
𝐿𝑏
𝑑
1
𝑣 = (𝑖 − (2𝑑𝐴 − 1)𝑖𝑑𝑐 )
𝑑𝑡 𝑐𝑏 𝑐𝑏 𝑏𝑎𝑡
…
…
𝑑 +
1
+
+
+ )
𝑣 = (𝐺𝑚𝑖𝑑 𝑖𝑑𝑐 − 𝑌𝑃+ (𝑑 2 + 𝜔𝑜2 )𝑥2𝑚
+ 𝜔𝐿 𝑣𝑐𝑖𝑞
𝑑𝑡 𝑐𝑖𝑑 𝐶𝑖
𝑑 +
1
+
+
+
𝑣𝑐𝑖𝑞 = (𝐺𝑚𝑖𝑞
𝑖𝑑𝑐 + 𝑌𝑄+ (𝑑 2 + 𝜔𝑜2 )𝑥2𝑚
) − 𝜔𝐿 𝑣𝑐𝑖𝑑
𝑑𝑡
𝐶𝑖
𝑑 −
1
−
−
−
𝑣 = (𝐺𝑚𝑖𝑑
𝑖𝑑𝑐 − 𝑌𝑄−𝑝𝑑 (𝑑 2 + 𝜔𝑜2 )𝑥2𝑚
) − 𝜔𝐿 𝑣𝑐𝑖𝑞
𝑑𝑡 𝑐𝑖𝑑 𝐶𝑖
𝑑 −
1
−
−
−
𝑣𝑐𝑖𝑞 = (𝐺𝑚𝑖𝑞
𝑖𝑑𝑐 + 𝑌𝑄−𝑝𝑞 (𝑑 2 + 𝜔𝑜2 )𝑥2𝑚
) + 𝜔𝐿 𝑣𝑐𝑖𝑑
𝑑𝑡
𝐶𝑖
…
…
𝑑 +
+
+
𝑥 = −(𝑑 2 + 𝜔𝑜2 )𝑥2𝑚
+ 𝑣𝑐𝑖𝑑
𝑑𝑡 1𝑚
𝑑 +
+
+
𝑥 = 𝑥1𝑚
+ 2𝑑𝑥2𝑚
𝑑𝑡 2𝑚
𝑑 −
+
−
𝑥 = −(𝑑 2 + 𝜔𝑜2 )𝑥2𝑚
+ 𝑣𝑐𝑖𝑑
𝑑𝑡 1𝑚
𝑑 −
−
−
𝑥 = 𝑥1𝑚
+ 2𝑑𝑥2𝑚
𝑑𝑡 2𝑚
…
{
𝑑
1 3√3
1
1.5
+
+
+ +
−
−
− −
(2𝑑𝐴 − 1)𝑣𝑐𝑏 −
𝑖𝑑𝑐 =
𝑑𝑏 𝑣𝑐𝑔𝑑 +
𝐺 (𝑚𝑖𝑑
𝑣𝑐𝑖𝑑
+ 𝑚𝑖𝑞
𝑣𝑐𝑖𝑞 + 𝑚𝑖𝑑
𝑣𝑐𝑖𝑑
+ 𝑚𝑖𝑞
𝑣𝑐𝑖𝑞 )
𝑑𝑡
𝐿𝑑𝑐 𝜋
𝐿𝑑𝑐
𝐿𝑑𝑐
(4.84)
94
The schematic diagram of the proposed system’s overall dynamic model is illustrated in
Fig. 4.24. The model is characterized by 7 inputs 𝑈, 20 state variables 𝑋, and 6 outputs 𝑌. The
vectors 𝑋 and 𝑈 are:
′
′
𝑋 = [ 𝜔𝑚 𝜔𝑟 𝛿𝜃 𝑖𝑞𝑠 𝑖𝑑𝑠 𝑖𝑞𝑟
𝑖𝑑𝑟
+
𝑣𝑐𝑖𝑑
+
𝑣𝑐𝑖𝑞
−
𝑣𝑐𝑖𝑑
−
𝑣𝑐𝑖𝑞
+
𝑥1𝑚
+
𝑥2𝑚
𝑣𝑐𝑔𝑞 𝑣𝑐𝑔𝑑
−
𝑥1𝑚
−
𝑥2𝑚
𝑖𝑏𝑎𝑡 𝑣𝑐𝑏
𝑖𝑑𝑐 ]𝑇
+
+
−
− 𝑇
𝑈 = [ 𝑣𝑤 𝑑𝑏 𝑑𝐴 𝑚𝑖𝑑
𝑚𝑖𝑞
𝑚𝑖𝑑
𝑚𝑖𝑞
]
(4.85)
(4.86)
The state variables to be controlled are selected as the output of the system, i.e.,
+
+
−
−
𝑌 = [ 𝜔𝑟 𝑣𝑐𝑖𝑑
𝑣𝑐𝑖𝑞
𝑣𝑐𝑖𝑑
𝑣𝑐𝑖𝑞
𝑖𝑑𝑐 ]𝑇
(4.87)
The system matrices 𝐴 and 𝐵 are provided in Appendix E.
Fig. 4.24: Inputs, state variables, and outputs of the proposed WECS.
4.11.2 Steady-State Equations
The steady-state equations of the system are obtained by setting the right-hand sides of the
equations given in (4.84) to zero, resulting in (4.88).
95
𝜔𝑚 =
𝜌𝐴 𝐶𝑝 𝑣𝑤3
′
′
3𝑃𝐿𝑚 (𝑖𝑞𝑠 𝑖𝑑𝑟
− 𝑖𝑑𝑠 𝑖𝑞𝑟
)𝑛𝑔𝑒𝑎𝑟
𝑛𝑔𝑒𝑎𝑟 3𝑃𝐿𝑚
′
′
𝛿𝜃 =
(
(𝑖𝑞𝑠 𝑖𝑑𝑟
− 𝑖𝑑𝑠 𝑖𝑞𝑟
))
𝐾𝑠𝑒
2
𝜔𝑟 = 𝑃 𝑛𝑔𝑒𝑎𝑟 𝜔𝑚
…
…
1
′
′
[−(𝜔𝑒 (𝐿𝑠 𝐿′𝑟 − 𝐿2𝑚 ) + 𝜔𝑟 𝐿2𝑚 )𝑖𝑑𝑠 +𝑅𝑟′ 𝐿𝑚 𝑖𝑞𝑟
− 𝜔𝑟 𝐿𝑚 𝐿′𝑟 𝑖𝑑𝑟
+𝐿′𝑟 𝑣𝑐𝑔𝑞 ]
𝑅𝑠 𝐿′𝑟
1
′
′
𝑖𝑑𝑠 =
[−(𝜔𝑒 (𝐿2𝑚 − 𝐿𝑠 𝐿′𝑟 ) − 𝜔𝑟 𝐿2𝑚 ) 𝑖𝑞𝑠 + 𝜔𝑟 𝐿𝑚 𝐿′𝑟 𝑖𝑞𝑟
+𝑅𝑟′ 𝐿𝑚 𝑖𝑑𝑟
+𝐿′𝑟 𝑣𝑐𝑔𝑑 ]
𝑅𝑠 𝐿′𝑟
1
′
′
𝑖𝑞𝑟
= ′ [𝑅𝑠 𝐿𝑚 𝑖𝑞𝑠 + 𝜔𝑟 𝐿𝑚 𝐿𝑠 𝑖𝑑𝑠 − (𝜔𝑒 (𝐿𝑠 𝐿′𝑟 − 𝐿2𝑚 ) − 𝐿𝑠 𝐿′𝑟 𝜔𝑟 )𝑖𝑑𝑟
− 𝐿𝑚 𝑣𝑐𝑔𝑞 ]
𝑅𝑟 𝐿𝑠
1
′
′
𝑖𝑑𝑟
= ′ [−𝜔𝑟 𝐿𝑚 𝐿𝑠 𝑖𝑞𝑠 +𝑅𝑠 𝐿𝑚 𝑖𝑑𝑠 − (𝜔𝑒 (𝐿2𝑚 − 𝐿𝑠 𝐿′𝑟 ) + 𝐿𝑠 𝐿′𝑟 𝜔𝑟 )𝑖𝑞𝑟
− 𝐿𝑚 𝑣𝑐𝑔𝑑 ]
𝑅𝑟 𝐿𝑠
1
𝑣𝑐𝑔𝑑 = −
𝑖
𝜔𝑒 𝐶𝑔 𝑞𝑠
𝑖𝑞𝑠 =
1
2√3
𝑑 𝑖 )
(𝑖𝑑𝑠 +
𝜔𝑒 𝐶𝑔
𝜋 𝑏 𝑑𝑐
…
…
= 𝐸𝑜 − 𝐾𝐸 (1 − 𝑆𝑜𝐶) − 𝑅𝑜 (1 + 𝐴0 (1 − 𝑆𝑜𝐶))𝑖𝑏𝑎𝑡
𝑖𝑏𝑎𝑡 = (2𝑑𝐴 − 1)𝑖𝑑𝑐
…
…
1
+
+
+
𝑣𝑐𝑖𝑞 = −
𝑖𝑑𝑐 − 𝑖𝑝𝑑
(𝐺𝑚𝑖𝑑
)
𝜔𝐿 𝐶𝑖
1
+
+
+
𝑣𝑐𝑖𝑑
=
𝑖𝑑𝑐 − 𝑖𝑝𝑞
(𝐺𝑚𝑖𝑞
)
𝜔𝐿 𝐶𝑖
1
−
−
−
𝑣𝑐𝑖𝑞
=
𝑖𝑑𝑐 − 𝑖𝑝𝑑
(𝐺𝑚𝑖𝑑
)
𝜔𝐿 𝐶𝑖
1
−
−
−
𝑣𝑐𝑖𝑑
=−
𝑖𝑑𝑐 − 𝑖𝑝𝑞
(𝐺𝑚𝑖𝑞
)
𝜔𝐿 𝐶𝑖
…
…
+
+
𝑖𝑝𝑑
= 𝑌𝑃+ 𝑣𝑐𝑖𝑑
𝑣𝑐𝑔𝑞 =
𝑣𝑐𝑏
+
+
𝑖𝑝𝑞
= −𝑌𝑄+ 𝑣𝑐𝑖𝑑
+
−
𝑖𝑝𝑑
= 𝑌𝑄−𝑝𝑑 𝑣𝑐𝑖𝑑
+
−
𝑖𝑝𝑞
= −𝑌𝑄−𝑝𝑞 𝑣𝑐𝑖𝑑
…
3√3
+ +
+ +
− −
− −
{ 𝜋 𝑑𝑏 𝑣𝑐𝑔𝑑 + (2𝑑𝐴 − 1)𝑣𝑐𝑏 = 1.5𝐺 (𝑚𝑖𝑑 𝑣𝑐𝑖𝑑 + 𝑚𝑖𝑞 𝑣𝑐𝑖𝑞 + 𝑚𝑖𝑑 𝑣𝑐𝑖𝑑 + 𝑚𝑖𝑞 𝑣𝑐𝑖𝑞 )
(4.88)
96
The twenty equations in (4.88) can be used to calculate the steady-state values of the system
variables according to an arbitrary set of system inputs. Among the system inputs stated in
(4.86), wind speed (𝑣𝑤 ) is the only uncontrolled input. The remaining inputs depend on the
+
specific control algorithms employed. Among the state variables stated in (4.85), 𝑣𝑐𝑖𝑞
is equal
to zero in load-side voltage-oriented synchronous frame. Similarly, 𝑣𝑐𝑔𝑞 is equal to zero in
−
generator-voltage-oriented synchronous frame. As mentioned in subsection 4.9.3, 𝑣𝑐𝑖𝑑
and
−
𝑣𝑐𝑖𝑞
are both equal to zero at steady-state operation, since the objective of load-side controllers
is to compensate for voltage imbalance at load-side bus and hence no –ve seq voltage
components exist across the output C-filter.
The steady-state values are obtained under the following assumptions:

The wind turbine is operated under maximum power point tracking scheme.

The mechanical losses are assumed to be 1% of the turbine’s rated power [141]; hence,
they are neglected.

Due to the very high efficiency of power electronic devices, their conduction and
switching losses are neglected.

The generator-side cable losses are neglected.

The transformer losses are considered as part of the load demand.
4.12 Verification of the Overall Model
The purpose of this section is to study the behavior of the system’s output variables in
response to variations of the input variables. Due to system complexity, it is decomposed into
three subsystems: Wind-Turbine Generation (WTG) subsystem, Energy Storage (ES)
Subsystem, and Current-Sourced Inverter-Load (CSI-Load) subsystem. The verification is
conducted for each subsystem separately. State-space matrices of each subsystem as well as
the entire system are given in Appendix E. The structure of the three subsystems are described
in the following subsections.
97
4.12.1 Wind Turbine Generation (WTG) Subsystem
The structure of the WTG subsystem is shown in Fig. 4.25. The input variables are the
uncontrolled wind speed (𝑣𝑤 ), the adjustable control variable of the buck converter (𝑑𝑏 ), and
the dc-link current (𝑖𝑑𝑐 ), which is regulated by the ES subsystem. The dc current required from
the generator (i.e., 𝑖𝑑𝑐𝑟 = 𝑑𝑏 𝑖𝑑𝑐 ) represent the load of the WTG subsystem. The dc voltage
produced by the buck converter (𝑣𝑑 = 𝑑𝑏 𝑣𝑑𝑐 ) can be taken as the output of the subsystem.
Thus, with respect to the combination of ES and CSI-Load subsystems, the WTG subsystem
can be looked at as a variable dc power supply. The active power produced by the generator
(i.e., 𝑃𝑔 ) can also be considered as output of the subsystem.
→
Gear
Box
Averaged-Value
Diode Bridge
Rectifier.
SCIG
Excitation
Capacitor Bank
+
-
Averaged-Value
buck converter.
+
(Fig. 4.12)
-
→
(Fig. 4.9)
→
→
Rotor
Blades
Cg
Fig. 4.25: WTG Subsystem.
4.12.2 Energy Storage (ES) Subsystem
As illustrated in Fig. 4.26, the input variables to ES subsystem are the variable dc voltage
applied by WTG subsystem (𝑣𝑑 ), the adjustable control variable of the reduced H-bridge
converter (𝑑𝐴 ), and the average dc-side voltage of the current-sourced inverter (𝑣𝑖𝑛𝑣 ), which is
reflected from the load side. Since the function of ES subsystem is to regulate the dc-link
current, this current represents the output of the subsystem. Assuming a lossless system, 𝑣𝑑
and 𝑣𝑖𝑛𝑣 can be approximated by 𝑃𝑚 /𝑖𝑑𝑐 and 𝑃𝐿 /𝑖𝑑𝑐 , respectively, where 𝑃𝑚 and 𝑃𝐿 are the
wind turbine mechanical power and the load power, respectively.
98
→
→
AveragedValue
Reduced
H-Bridge
→
(Fig. 4.16)
+
→
-
+
+
Fig. 4.26: ES Subsystem.
4.12.3 Current-Sourced Inverter–Load (CSI-Load) Subsystem
Fig. 4.27 illustrates the structure of the CSI-Load subsystem. The combination of the WTG
and ES subsystems are assumed to be a dispatchable unit, and hence can be represented by a
conditioned energy source, supplying a regulated dc current to the CSI-Load subsystem. The
+
+
−
−
modulation indices of the current source inverter (i.e., 𝑚𝑖𝑑
, 𝑚𝑖𝑞
, 𝑚𝑖𝑑
, 𝑚𝑖𝑞
) are the adjustable
input variables to the CSI-Load subsystem. The function of the current source inverter is to
control the voltage magnitude and frequency at the load bus under balanced and unbalanced
+
+
−
−
load. Therefore, the dq capacitor voltages (i.e., 𝑣𝑐𝑖𝑑
, 𝑣𝑐𝑖𝑞
, 𝑣𝑐𝑖𝑑
, 𝑣𝑐𝑖𝑞
) represent the output of
the CSI-Load subsystem.
→
+
AveragedValue CSI.
Extended Generic
Load Model.
(Fig. 4.18)
(Fig. 4.21)
Ci
C-Filter
Fig. 4.27: CSI-Load Subsystem.
99
In the following subsections, the starting operation of the system will be simulated first.
Then, inputs and outputs of average and switching models of power electronic converters will
be compared. Finally, the dynamic characteristics of the system will be investigated by
examining its step responses. Simulation results are obtained based on switching and average
models of a 20kW wind turbine system. Both models are built in Matlab/Simulink environment
based on the system’s parameters provided in Appendix A (Table A.2). In the following
simulations, the wind speed is considered as a constant signal. The realistic wind speed model
described in section 4.1 will be implemented in chapter 5.
4.12.4 Starting Operation of the Wind Turbine System
The process of self-excitation in induction machines has been studied for over eighty years
(since 1935) [142] and well-illustrated in the literatures through simulation and
experiments [29],[124],[143]. When an induction machine, excited by an appropriate capacitor
across its stator terminals, is driven by an external prime mover, a voltage will appear at its
terminals. The starting operation of the WTG subsystem is investigated with the input
parameters given in Table 4.5.
Table 4.5: Input parameters for WTG subsystem used for starting simulation.
Wind speed (m/s)
12
Buck duty cycle
𝑑𝑏 = 0.63
DC-link current (A)
𝑖𝑑𝑐 = 51
In Fig. 4.28, two starting scenarios of the generator are simulated. In the first scenario,
simulation starts with zero shaft speed, with no load connected to the generator (i.e., 𝑑𝑏 = 0).
Once the shaft speed reaches the rated speed of 1 pu, rated load is connected to the generator
by switching the buck duty cycle to 0.63. The self-excitation process starts at t = 1.95 s, after
which the generated voltage builds up until it reaches steady-state value, where full excitation
is achieved. This scenario is referred to as unloaded starting. In the second scenario, simulation
starts with zero shaft speed, with rated load connected to the generator (i.e., 𝑑𝑏 = 0.63).
Compared to the unloaded starting, the loaded starting requires a longer time for self-excitation
process to start and thereby for full excitation to take place. This scenario is referred to as
loaded starting. From the simulation responses, one can notice that the loaded starting scenario
100
experiences very high power transients (with a power overshoot of around 90%) compared
with the unloaded starting (with a power overshoot of around 30%).
Fig. 4.28: Starting of the wind turbine under unloaded/loaded conditions.
4.12.5 Average model versus Switching model for Power Electronics Converters
In this subsection, the inputs and outputs of average models of power electronic converters,
namely, diode-bridge rectifier, buck converter, reduced H-bridge converter and CSI, are
compared with those of switching models. The system is running at the steady-state operating
point described in Table 4.6.
Fig. 4.29 illustrates the ac-side current (phase a) and dc-side voltage of diode bridge rectifier.
One can notice a very small phase shift between the average model current and fundamental
component of switching model current. This is caused by the effect of the dc-side capacitor
(𝐶𝑑𝑐 ) which was neglected in developing the average model. In other words, the average model
101
line current is in phase with the supply phase voltage, while the fundamental component of
switching model current is leading the supply phase voltage.
The waveforms, illustrated in Fig. 4.30, Fig. 4.31, and Fig. 4.32, show that the average
models of power electronic converters represent to a very good approximation the
corresponding switching models, with the distinct difference that high-frequency switching
harmonics are not present in the average models.
Table 4.6: Input, output and parameter values at the operating point for steady-state analysis.
𝑣𝑤 = 12 𝑚/𝑠
𝑑𝑏 = 0.63
𝑑𝐴 = 0.108
+
𝑚𝑖𝑑
= 0.1945
+
𝑚𝑖𝑞
= 0.233
−
𝑚𝑖𝑑
= −0.049
−
𝑚𝑖𝑞
= 0.059
𝜔𝑟 = 379.5 𝑟𝑎𝑑/𝑠
𝑃𝑔 = 20 𝑘𝑊
𝑄𝑔 = −11.5 𝑘𝑉𝑎𝑟
𝑖𝑑𝑐 = 51 𝐴
𝑃𝑏𝑎𝑡 = −16 𝑘𝑊
𝑃𝐿+ = 4 𝑘𝑊
𝑄𝐿+ = 2 𝑘𝑉𝑎𝑟
−
𝑄𝑝𝑑
= 1.02 𝑘𝑊
−
𝑄𝑝𝑞
= 1.21 𝑘𝑉𝑎𝑟
𝑑 = 100 𝑠 −1
𝜔𝑜 = 75 𝑟𝑎𝑑/𝑠
+
𝑣𝑐𝑖𝑑
= 310.3 𝑉
+
𝑣𝑐𝑖𝑞
=0
−
−
𝑣𝑐𝑖𝑑
= 𝑣𝑐𝑖𝑞
=0
𝑣𝑏𝑎𝑡 = 393.3𝑉
𝑆𝑜𝐶 = 50%
𝑓𝑠 = 5.1 𝑘𝐻𝑧
Fig. 4.29: Average model versus switching model: diode bridge rectifier waveforms.
102
Fig. 4.30: Switching and average model waveforms of buck converter.
Fig. 4.31: Switching and average model waveforms of recued-H-bridge converter.
103
Fig. 4.32: Switching and average model waveforms of CSI.
4.12.6 Open-Loop Step Responses of the System
This subsection aims to investigate the dynamic behaviour of the system by simulating its
step response to different input variables. This purpose is served by illustrating the following
responses.
-
The generator rotor speed (𝜔𝑟 ) response to step change in buck converter duty ratio
(𝑑𝑏 );
-
The generated active power (𝑃𝑔 ) response to step change in 𝑑𝑏 ;
-
The dc-link current (𝑖𝑑𝑐 ) response to step change in reduced H-bridge duty ratio (𝑑𝐴 );
and
-
+
+
−
−
The responses of load-side dq voltages (𝑣𝑐𝑖𝑑
, 𝑣𝑐𝑖𝑞
, 𝑣𝑐𝑖𝑑
, and 𝑣𝑐𝑖𝑞
) to step changes in +ve
+
+
seq modulation indices (𝑚𝑖𝑞
and 𝑚𝑖𝑑
).
104
Although wind speed (𝑣𝑤 ) is an uncontrollable disturbance input, the system dynamic response
to 𝑣𝜔 is important. It should be noted that the pitch angle is taken to be a stall control against
wind gusts. In other words, in the system under study, pitch angle is fixed at its optimal value
of zero at and below rated wind speed. Above rated wind speed, the turbine operates under
passive-stall control.
Before applying any step change, the system operates at its steady-state operating point given
in Table 4.6. In the following figures, a step change to a specific input is applied while the
remaining inputs are maintained fixed.
The responses shown in Fig. 4.33 are related to step change in the buck converter duty ratio,
𝑑𝑏 . The generator was running at optimum rotor speed, and hence capturing the maximum
wind power at rated wind speed (i.e., 𝑣𝜔 , 𝜔𝑟 , and 𝑃𝑔 are all equal to 1 pu). At t = 0.4 s, a 50%
step decrease in 𝑑𝑏 is applied. As shown in the figure, the rotor speed increases until it settles
down to a new steady-state value (i.e., 1.2 pu). The generated active power oscillates for 0.1
second before it settles at 0.85 pu. Since 𝜔𝑟 deviates from its optimum value, the generated
active power decreases.
Fig. 4.33: Responses of generator rotor speed and active power to 50% step decrease in 𝑑𝑏 .
105
Wind speed (𝑣𝑤 ) is an uncontrollable input and the system response to a step change in wind
speed shows the system performance under disturbance variation. The step responses of the
generator, illustrated in Fig. 4.34, are obtained by decreasing 𝑣𝑤 by 20% at t = 0.4 s. As a result
of reduced wind speed, less mechanical power is produced by the wind turbine. Since the active
current absorbed from the generator (i.e., 𝑑𝑏 𝑖𝑑𝑐 ) is kept fixed, the rotor slows down and the
generated voltage is reduced in order to balance the generated electrical power with the input
mechanical power. The figure shows that a rather small decrement in 𝑣𝑤 causes a dramatic
decrease in 𝑃𝑔 . This is because of the cubic relationship between these variables (i.e., 𝑃𝑔 ∝ 𝑣𝑤3 ).
As a result of significant reduction in the generated active power, the reactive power absorbed
by the generator is significantly reduced.
Fig. 4.34: Generator responses to 20% step decrease in 𝑣𝑊 .
106
In Fig. 4.35, an 80% step increase/decrease is applied to the duty ratio of reduced H-bridge
(𝑑𝐴 ), at t = 0.4 s. The dc-link current steps up/down by around 28% / 18.5% to the new steadystate values, revealing stable operation. A smooth transition between the old and the two new
steady-state conditions is noticed.
Fig. 4.35: Responses of dc-link current to 80% step change in 𝑑𝐴 .
+
+
The responses of load-side dq voltages to 50% step change in 𝑚𝑖𝑞
and 𝑚𝑖𝑑
are illustrated in
Fig. 4.36 and Fig. 4.37, respectively. Before t = 0.4 s, the system operates at its steady-state
operating point given in Table 4.6. At this operating point, the load-side voltages are balanced;
+
+
−
−
hence, 𝑣𝑐𝑖𝑑
= 1 pu, while 𝑣𝑐𝑖𝑞
= 𝑣𝑐𝑖𝑑
= 𝑣𝑐𝑖𝑞
= 0 in synchronous dq reference frame. When a
step change is applied to a particular modulation index, while the remaining indices are kept
+
−
+
−
fixed, voltage imbalance is introduced. As a result, load-side currents (𝑖𝑝𝑑
, 𝑖𝑝𝑞
, 𝑖𝑝𝑑
and 𝑖𝑝𝑞
)
+
+
−
−
will change, leading to deviations in 𝑣𝑐𝑖𝑑
, 𝑣𝑐𝑖𝑞
, 𝑣𝑐𝑖𝑑
and 𝑣𝑐𝑖𝑞
from the desired values. In all
cases, the frequency experiences transients within ± 0.5 Hz before it returns to 60 Hz. The
simulation responses shown in Fig. 4.36 and Fig. 4.37 are obtained based on the average model.
+
The switching model responses of dq voltages to 50% step increase in 𝑚𝑖𝑑
are given in
Fig. 4.38.
The figures shown in this subsection demonstrate that with respect to the switching model,
the average model doesn’t represent switching harmonics, but the dynamics resulting from
control system and power system interaction are preserved to a high accuracy.
107
+
Fig. 4.36: Load-side responses to 50% step change in 𝑚𝑖𝑞
.
+
Fig. 4.37: Load-side responses to 50% step change in 𝑚𝑖𝑑
.
108
Fig. 4.38: Average and switching model load-side voltage responses to 50% step increase in
+
𝑚𝑖𝑑
.
4.13 Small-Signal Model and Stability Analysis
The dynamic model developed in subsection 4.11.1 and given in (4.84) consists of twenty
nonlinear first-order differential equations. In order to study the stability of the system, a smallsignal model needs to be developed by linearizing the nonlinear equations around a quiescent
operating point so that the system can be treated as a linear system with regard to very small
disturbances. The stability analysis of the linearized system investigates its capability to return
to a stable operating point after a disturbance, causing a small change in one or more of the
system’s state variables. It should be emphasized that a linear model derived at an operating
point is valid only for small perturbations of the system around that operating point.
The small-signal model of the proposed WECS can be described by:
𝑑
𝑑𝑡
∆𝑋 = 𝐴′ ∆𝑋 + 𝐵 ′ ∆𝑈
where
109
(4.89)
′
′
∆𝑋 = [∆𝜔𝑚 ∆𝜔𝑟 ∆𝛿𝜃 ∆𝑖𝑞𝑠 ∆𝑖𝑑𝑠 ∆𝑖𝑞𝑟
∆𝑖𝑑𝑟
∆𝑣𝑐𝑔𝑞 ∆𝑣𝑐𝑔𝑑
+
∆𝑖𝑏𝑎𝑡 ∆𝑣𝑐𝑏 ∆𝑣𝑐𝑖𝑑
+
∆𝑣𝑐𝑖𝑞
−
∆𝑣𝑐𝑖𝑑
−
∆𝑣𝑐𝑖𝑞
+
∆𝑥1𝑚
+
∆𝑥2𝑚
−
∆𝑥1𝑚
−
∆𝑥2𝑚
+
+
−
− 𝑇
∆𝑈 = [∆𝑣𝜔 ∆𝑑𝑏 ∆𝑑𝐴 ∆𝑚𝑖𝑑
∆𝑚𝑖𝑞
∆𝑚𝑖𝑑
∆𝑚𝑖𝑞
]
∆𝑖𝑑𝑐 ]
𝑇
(4.90)
(4.91)
In (4.89), 𝐴′ and 𝐵 ′ are the Jacobian matrices, evaluated at the steady-state operating point.
These matrices are provided in Appendix E for WTG subsystem, ES subsystem, CSI-Load
subsystem, and the overall system. The symbol ∆ followed by a variable represents smallsignal perturbation of the variable. The quiescent (dc) component of a variable at the operating
point, at which the small-signal model is derived, is represented with capitalized letters. The
products of small perturbations of quantities are very small, and hence can be neglected. During
the process of small-signal derivation, the cross saturation effect of the SEIG is ignored and
hence the mutual inductance of the machine is assumed fixed.
4.13.1 Transfer Function and Eigenvalue Analysis
The time-invariant linearized system described in (4.89) relates the input ∆𝑈 and the state
variables ∆𝑋 in the time domain. For a particular output vector ∆𝑌, the small-signal equation
is:
∆𝑌 = 𝐶′∆𝑋 + 𝐷′∆𝑈
(4.92)
The direct transfer function matrix 𝐺(𝑠) from the input to the output in s domain is given as
∆𝑌(𝑠)
𝑁(𝑠)
𝐺(𝑠) = ∆𝑈(𝑠) = 𝐷(𝑠) = 𝐶′(𝑠𝐼 − 𝐴′ )−1 𝐵′ + 𝐷′
(4.93)
where 𝑁(𝑠) and 𝐷(𝑠) are the equivalent numerator and denominator polynomials of the
transfer function, respectively. For a MIMO system, 𝐺(𝑠) is a matrix. Based on the output and
input selection, different transfer functions 𝐺𝑖𝑗 (𝑠) can be obtained corresponding to the 𝑖 𝑡ℎ row
and 𝑗 𝑡ℎ column of 𝐺(𝑠).
The zeros and poles of 𝐺(𝑠) are the roots of 𝑁(𝑠) and 𝐷(𝑠), respectively. The zeros of 𝐺(𝑠)
change based on the output variable selection; however, the poles of 𝐺(𝑠) are associated with
matrix 𝐴′ only, regardless of output variable selection. The poles of the linearized system must
satisfy the following equation:
110
|𝑠𝐼 − 𝐴′| = 0
(4.94)
Equation (4.94) is referred to as the characteristic equation of the system matrix 𝐴′. The roots
of this equation, namely the eigenvalues of matrix 𝐴′ , indicate the behaviour of the system at
a steady-state operating point. An eigenvalue can be real or complex number. A complex or
real eigenvalue corresponds to an oscillatory or a non-oscillatory mode of the system. Every
pole of 𝐺(𝑠) is an eigenvalue of 𝐴′. Pole locations or eigenvalues are commonly used to test
the stability of the system. If every eigenvalue of 𝐴′ has a negative real part, the oscillation
magnitude of zero-input responses (i.e., responses driven by initial conditions) of the linearized
system will decrease exponentially with time and hence the system is asymptotically stable
within a small region surrounding the equilibrium point (i.e., locally stable). If one or more
distinct (non-repeated) eigenvalues of 𝐴′ has zero real part and the remaining eigenvalues have
negative real part, the system is not asymptotically stable, but is marginally stable. On the
contrary, one or more eigenvalues with positive real part indicate that the system impulse
response, driven by any finite initial conditions, will blow up exponentially (i.e., increases in
magnitude as time increases). Even if there is no positive eigenvalue, but there are repeated
eigenvalues with zero real part, the system impulse response will still blow up, although more
slowly. In both cases, the system is considered unstable. It should be noted that eigenvalues
instability results hold true if no pole/zero cancellation exists. In other words, even if 𝐴′ has
some eigenvalues with zero or positive real part, the system may still be stable if the poles
equivalent to those eigenvalues are cancelled with zeros.
Damping ratio (𝜁) of an eigenvalue 𝜆 = 𝜎 ± 𝑗𝜔, computed as 𝜁 = −𝜎/√(𝜎 2 + 𝜔 2 ) , is
another indicator of stability. A negative 𝜁 indicates an unstable eigenmode, while a stable
eigenmode has a positive 𝜁. If 𝜁 of an eigenmode approaches unity, the eigenmode becomes
less oscillatory and tends to be more stable. On the contrary, if 𝜁 of an eigenmode approaches
zero, the eigenmode becomes more oscillatory and tends to be less or critically stable.
4.14 System Controllability and Observability
Before a closed-loop control scheme is designed for a system, it must be made sure that the
system is controllable. Controllability is defined as the ability to drive a system from any state
to another desired state in a finite period of time. If a system state cannot be influenced by any
111
of the system inputs, then the mode associated with that state is uncontrollable. If all the modes
of a system are controllable, the system is called completely controllable.
Before a state estimation/measurement scheme is designed for a system, it must be confirmed
that the system is observable. Observability is defined as the ability to deduce the initial state,
∆𝑋(0), of an unforced system (i.e., ∆𝑈 = 0), by observing the output over a finite period of
time. If a system state does not contribute in the system output, then the mode associated with
that state is unobservable. If all the modes of a system are observable, the system is called
completely observable.
In this section, controllability and observability of the linearized system are investigated
based on Gramian matrices.
4.14.1 Gramian Matrices
Controllability and observability of a continuous, linear, time- invariant, and asymptotically
stable system are determined by solving the following Lyapunov equations [144], where 𝑊𝑐
and 𝑊𝑜 are the controllability and observability Gramian matrices, respectively.
𝐴𝑊𝑐 + 𝑊𝑐 𝐴𝑇 + 𝐵𝐵 𝑇 = 0
(4.95)
𝐴𝑇 𝑊𝑜 + 𝑊𝑜 𝐴 + 𝐶 𝑇 𝐶 = 0
(4.96)
The system is controllable and observable if 𝑊𝑐 and 𝑊𝑜 are full rank matrices, which can be
checked by applying singular value decomposition (SVD) [144]. SVD of 𝑊𝑐 and 𝑊𝑜 at an
operating point can be used to indicate system controllability and observability, respectively.
If all SVDs of 𝑊𝑐 have non-zero values, the system is completely controllable. Similarly, if
all SVDs of 𝑊𝑜 have non-zero values, the system is completely observable. The size of the
numerical error which is likely introduced by matrix computation can be found from the
condition number calculated by dividing the maximum SVD over minimum SVD of the
matrix. While a small condition number indicates a well-conditioned system, a large condition
number indicates ill-conditioned system.
Even if matrix 𝐴 has a positive 𝜆, the Lyapunov equations (4.95) and (4.96) may still have
solutions. In fact, they have unique solutions if and only if 𝜆 (𝐴) + 𝜆( 𝐴̅ ) ≠ 0, where 𝐴̅ is a
matrix with entries equal to complex conjugates of 𝐴 [145].
112
4.15 Linearized Model Evaluation
In order to investigate stability, controllability and observability of the system represented
by the developed mathematical model, a 20kW wind turbine system is considered. The rating
and parameters of the studied turbine are given in Appendix A (Table A.2). For linearization,
three operating points are selected based on the operating mode of the energy storage
subsystem. The operating points are defined as follows:
a) Operating point 1(o.p.1): Charging Mode
At this operating point, the system is operated under rated wind speed (i.e., 𝑃𝑔,𝑜𝑝𝑡 =
20 𝑘𝑊) and 20% rated load. Therefore, 20% of the generated power under MPPT
(𝑃𝑔,𝑜𝑝𝑡 ) is supplied to the load (i.e., 𝑃𝐿 = 4 𝑘𝑊) and the remaining 80% of 𝑃𝑔,𝑜𝑝𝑡 is
stored in the storage battery (i.e., 𝑃𝑏𝑎𝑡 = −16 𝑘𝑊).
b) Operating point 2 (o.p.2): Freewheeling mode
At this operating point, the system is operated under rated wind speed (i.e., 𝑃𝑔,𝑜𝑝𝑡 =
20 𝑘𝑊) and rated load condition (i.e., 𝑃𝐿 = 20 𝑘𝑊) . Therefore, 𝑃𝑔,𝑜𝑝𝑡 is totally
transferred to the load (assuming a lossless system) and the average power absorbed or
delivered by the storage battery is equal to zero.
c) Operating point 3(o.p.3): Discharging mode.
At this operating point, the system is operated under 66.7% rated wind speed (i.e.,
29.7% of rated generated power or 𝑃𝑔,𝑜𝑝𝑡 = 5.94𝑘𝑊) and 50% load (𝑖. 𝑒. , 𝑃𝐿 =
10 𝑘𝑊). The demand is higher than 𝑃𝑔,𝑜𝑝𝑡 and hence the storage battery compensates
for the shortage (i.e., 𝑃𝑏𝑎𝑡 = 4.06 𝑘𝑊).
The three operating points selected for linearization are given in Table 4.7. As mentioned in
section 4.13, all variables are represented by capitalized letters to indicate quiescent (dc)
component of the variable at the selected operating point for linearization.
Based on the analysis described in previous sections, the eigenvalues, damping ratios, and
SVDs of Gramian controllability/observability matrices of the small-signal model linearized
113
around each of the three defined operating points are given in Table 4.8, Table 4.9, Table 4.10,
and Table 4.11.
Table 4.7: Input, state and output variables at the operating points for linearization.
Operating point 1: Rated wind speed and 20% load (Charging Mode)
𝑉𝜔 = 12 𝑚/𝑠
+
𝑀𝑖𝑞
= 0.233
𝐷𝑏 = 0.63
−
𝑀𝑖𝑑
= −0.049
𝐷𝐴 = 0.108
−
𝑀𝑖𝑞
= 0.059
+
𝑀𝑖𝑑
= 0.1945
𝐼𝑑𝑐 = 51 𝐴
𝑊𝑚 = 1.518 𝑟𝑎𝑑/𝑠
𝑊𝑟 = 379.5 𝑟𝑎𝑑/𝑠
𝛿𝜃 = 3 𝑟𝑎𝑑
𝐼𝑞𝑠 = −19.8 𝐴
𝐼𝑑𝑠 = −35.5 𝐴
′
𝐼𝑞𝑟
= 4.4 𝐴
′
𝐼𝑑𝑟
= 37.5 𝐴
𝑉𝑐𝑔𝑞 = 0
𝑉𝑐𝑔𝑑 = 375.6 𝑉
𝐼𝑏𝑎𝑡 = −40 𝐴
𝑉𝑐𝑏 = 393.3𝑉
𝑆𝑜𝐶 = 50%
+
𝑉𝑐𝑖𝑑
= 310.3 𝑉
+
𝑉𝑐𝑖𝑞
=0
−
𝑉𝑐𝑖𝑑
=0
−
𝑉𝑐𝑖𝑞
=0
−
𝐼𝑝𝑑
= −10.92/5 𝐴
−
𝐼𝑝𝑞
= 12.97/5 𝐴
+
𝐼𝑝𝑑
= 42.97/5 𝐴
𝑌𝑃+ = 0.1385/5
+
𝐼𝑝𝑞
= −21.47/5 𝐴,
𝑌𝑄+ = 0.0692/5
𝑌𝑄−𝑝𝑑 = −0.0352/5
𝑌𝑄−𝑝𝑞 = −0.0418/5
Operating point 2: Rated wind speed and rated load (Freewheeling mode)
𝑉𝜔 = 12 𝑚/𝑠
𝐷𝑏 = 0.58
𝐷𝐴 = 0.5
+
𝑀𝑖𝑑
= 0.894
+
𝑀𝑖𝑞
= −0.143
−
𝑀𝑖𝑑
= −0.227
−
𝑀𝑖𝑞
= 0.270
𝐼𝑑𝑐 = 55.5 𝐴
𝑊𝑚 = 1.518 𝑟𝑎𝑑/𝑠
𝑊𝑟 = 379.5 𝑟𝑎𝑑/𝑠
𝛿𝜃 = 3 𝑟𝑎𝑑
𝐼𝑞𝑠 = −19.8 𝐴
𝐼𝑑𝑠 = −35.5 𝐴
′
𝐼𝑞𝑟
= 4.4 𝐴
′
𝐼𝑑𝑟
= 37.5 𝐴
𝑉𝑐𝑔𝑞 = 0
𝑉𝑐𝑔𝑑 = 375.6 𝑉
𝐼𝑏𝑎𝑡 = 0
𝑉𝑐𝑏 = 395.5 𝑉
𝑆𝑜𝐶 = 60%
+
𝑉𝑐𝑖𝑑
= 310.3 𝑉
+
𝑉𝑐𝑖𝑞
=0
−
𝑉𝑐𝑖𝑑
=0
−
𝑉𝑐𝑖𝑞
=0
+
𝐼𝑝𝑑
= 42.97 𝐴
+
𝐼𝑝𝑞
= −21.47 𝐴
−
𝐼𝑝𝑑
= −10.92 𝐴
−
𝐼𝑝𝑞
= 12.97 𝐴
𝑌𝑃+ = 0.1385
𝑌𝑄+ = 0.0692
𝑌𝑄−𝑝𝑑 = −0.0352
𝑌𝑄−𝑝𝑞 = −0.0418
Operating point 3: 66.7% of rated wind speed and 50% load (Discharging mode)
𝑉𝜔 = 8 𝑚/𝑠
𝐷𝑏 = 0.82
𝐷𝐴 = 0.69
+
𝑀𝑖𝑑
= 0.894
+
𝑀𝑖𝑞
= 0.161
−
𝑀𝑖𝑑
= −0.227
−
𝑀𝑖𝑞
= 0.270
𝐼𝑑𝑐 = 27.75 𝐴
𝑊𝑚 = 1.012 𝑟𝑎𝑑/𝑠
𝑊𝑟 = 253 𝑟𝑎𝑑/𝑠
𝛿𝜃 = 1.33 𝑟𝑎𝑑
𝐼𝑞𝑠 = −12𝐴
𝐼𝑑𝑠 = −25 𝐴
′
𝐼𝑞𝑟
= 3.3 𝐴
′
𝐼𝑑𝑟
= 25.6 𝐴
𝑉𝑐𝑔𝑞 = 0
𝑉𝑐𝑔𝑑 = 158 𝑉
𝐼𝑏𝑎𝑡 = 10.5 𝐴
𝑉𝑐𝑏 = 388 𝑉
𝑆𝑜𝐶 = 40 %
+
𝑉𝑐𝑖𝑑
= 310.3 𝑉
+
𝑉𝑐𝑖𝑞
=0
−
𝑉𝑐𝑖𝑑
=0
−
𝑉𝑐𝑖𝑞
=0
−
𝐼𝑝𝑑
= −10.92/2 𝐴
−
𝐼𝑝𝑞
= 12.97/2 𝐴
+
𝐼𝑝𝑑
= 42.97/2 𝐴
+
𝐼𝑝𝑞
= −21.47/2 𝐴,
𝑌𝑃+ = 0.1385/2
𝑌𝑄+ = 0.0692/2
𝑌𝑄−𝑝𝑑 = −0.0352/2
114
𝑌𝑄−𝑝𝑞 = −0.0418/2
Table 4.8: Eigenvalues and damping ratios of the linearized system at o.p.1 and o.p.2.
Operating point 1
Eigenvalue
Damping ratio
𝜆1 = −9731
1.0
𝜆2 = −4208
1.0
𝜆3,4 = −47.27 ± 𝑗1854
0.0255
𝜆5,6 = −44.27 ± 𝑗1478
0.0299
𝜆7,8 = −103 ± 𝑗191
0.4746
𝜆9,10 = −100 ± 𝑗75
0.8
𝜆11 = −113
1.0
𝜆12,13 = −111 ± 𝑗111
0.7071
𝜆14 = −23.22
1.0
𝜆15 = −20.51
1.0
𝜆16 = −0.016
1.0
𝜆17 = −0.016
1.0
𝜆18 = −0.0005
1.0
1.0
𝜆19 = −4.69 𝑥 10−9
1.0
𝜆20 = −7.32 𝑥 10−10
Operating point 2
Eigenvalue
Damping ratio
𝜆1 = −9721
1.0
𝜆2 = −4243
1.0
𝜆3,4 = −47.3 ± 𝑗1854
0.0255
𝜆5,6 = −44.33 ± 𝑗1478
0.0299
𝜆7,8 = −100 ± 𝑗188
0.4696
𝜆9,10 = −100 ± 𝑗75
0.8
𝜆11 = −114
1.0
𝜆12,13 = −111 ± 𝑗111
0.7071
𝜆14 = −23.14
1.0
𝜆15 = −20.53
1.0
𝜆16 = −0.016
1.0
𝜆17 = −0.016
1.0
𝜆18 = −0.0011
1.0
-1.0
𝜆19 = 1.3 𝑥 10−6
1.0
𝜆20 = −4.15𝑥 10−9
Table 4.9: Singular values of Gramian matrices at o.p.1 and o.p.2.
Singular values of
the matrix
Rank of the matrix
Condition no.
Operating point 1
𝑊𝑐
𝑊𝑜
1.361 𝑥 1023
1.054 𝑥 1017
1.168 𝑥 1017
2.119 𝑥 1014
16
3.785 𝑥 10
7.758 𝑥 108
13
1.175 𝑥 10
1.029 𝑥 108
3.400𝑥 106
1.139 𝑥 1013
9
3.309 𝑥 10
3.276 𝑥 102
9
1.610 𝑥 102
1.027 𝑥 10
6
4.082 𝑥 10
1.031 𝑥 102
2.997 𝑥 101
5.388 𝑥 105
5
2.613 𝑥 101
1.644 𝑥 10
4
2.070 𝑥 101
3.118 𝑥 10
4.643
3.842 𝑥 103
3.148
1.876 𝑥 103
2.268
1.122 𝑥 103
2
1.136
9.989 𝑥 10
2
1.804
𝑥 10−1
1.695 𝑥 10
7.482 𝑥 10−2
1.221 𝑥 102
1
1.080 𝑥 10−3
6.672 𝑥 10
5.617 𝑥 10−4
3.595 𝑥 101
2.233 𝑥 10−5
4.711 𝑥 10−1
Operating point 2
𝑊𝑐
𝑊𝑜
1.391 𝑥 1023
1.049 𝑥 1017
7.385 𝑥 1022
2.352 𝑥 1016
19
1.438 𝑥 1012
4.006 𝑥 10
18
8.185 𝑥 10
9.469 𝑥 109
2.117 𝑥 107
1.717 𝑥 1016
13
3.400 𝑥 106
7.447 𝑥 10
13
5.555 𝑥 10
2.492 𝑥 105
13
1.383 𝑥 10
1.312 𝑥 102
1.294 𝑥 1013
5.719 𝑥 101
12
1.572 𝑥 10
2.083 𝑥 101
9
8.145
1.563 𝑥 10
7
1.121
4.198 𝑥 10
5.715 𝑥 10−1
3.723 𝑥 107
1.213 𝑥 10−1
3.389 𝑥 107
7
9.223 𝑥 10−2
1.521 𝑥 10
3.061 𝑥 10−2
5.142 𝑥 106
5
1.061 𝑥 10−2
2.896 𝑥 10
4
8.412 𝑥 10−4
2.310 𝑥 10
0
2.239 𝑥 103
2
0
9.611 𝑥 10
20
Completely
Controllable
2.889 𝑥 1023
20
Completely
Controllable
1.447 𝑥 1020
20
Completely
observable
4.720 𝑥 1021
115
18
Not Completely
observable
-
Table 4.10: Eigenvalues and damping ratios of the linearized system at o.p.3.
Eigenvalue
Damping ratio
𝜆1 = −9731
𝜆2 = −4208
𝜆3,4 = −50.15 ± 𝑗1468
𝜆5,6 = −42.96 ± 𝑗1091
𝜆7,8 = −79.73 ± 𝑗170
𝜆9,10 = −100 ± 𝑗75
𝜆11 = −118
𝜆12,13 = −111 ± 𝑗111
𝜆14 = −22.98
𝜆15 = −3.66
𝜆16 = −0.016
𝜆17 = −0.016
𝜆18 = −2.51𝑥10−4
𝜆19 = −2.368 𝑥 10−8
𝜆20 = −7.61𝑥10−9
1.0
1.0
0.0341
0.0393
0.4246
0.8
1.0
0.7071
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Table 4.11: Singular values of Gramian matrices at o.p.3.
𝑊𝑐
Singular values of the
matrix
Rank of the matrix
Condition no.
22
𝑊𝑜
7.777 𝑥 10
5.537 𝑥 1015
1.946 𝑥 1015
1.276 𝑥 1013
7.331 𝑥 1012
3.603 𝑥 1012
3.546 𝑥 1012
6.150 𝑥 108
3.124 𝑥 107
1.119 𝑥 107
8.758 𝑥 106
2.785 𝑥 106
1.585 𝑥 106
9.929 𝑥 105
1.564 𝑥 105
3.040 𝑥 104
1.881 𝑥 104
1.938 𝑥 103
2.922 𝑥 102
2.561 𝑥 10−1
2.102 𝑥 1017
8.833 𝑥 1012
1.107 𝑥 1010
3.730 𝑥 109
3.400 𝑥 106
1.608 𝑥 103
2.500 𝑥 101
2.259 𝑥 101
9.568
7.199
8.697 𝑥 10−1
1.112 𝑥 10−1
5.103 𝑥 10−2
4.909 𝑥 10−2
3.804 𝑥 10−3
2.173 𝑥 10−3
1.098 𝑥 10−3
1.797 𝑥 10−4
1.750 𝑥 10−4
1.177 𝑥 10−5
20
Completely Controllable
20
Completely observable
3.036 𝑥 1023
116
1.785 𝑥 1022
Based on the results obtained in Table 4.8, Table 4.9, Table 4.10, and Table 4.11, the
stability, controllability, and observability of the system at the three operating points, defined
in Table 4.7, are evaluated as follows:
-
Stability

Every eigenvalue of the small-signal model linearized at o.p.1 and o.p.3 has
negative real part and hence the linearized system is asymptotically stable at these
two operating points. However, eigenmodes corresponding to λ3,4 and λ5,6 are very
oscillatory modes with very small damping ratios (close to zero) and the system
tends to be critically stable. Moreover, eigenvalues λ16 to λ20 result in a large
settling time and a slow open-loop system. The speed of responses as well as
stability margins are to be taken care of in the closed loop control design.

The real eigenvalue 𝜆19 is positive at o.p.2; thus, its corresponding eigenmode will
blow up exponentially, although very slowly, as 𝜆19 is almost at the origin. This
indicates an unstable open loop system around o.p.2. It is the function of closed
loop controller to stabilize the system at this operating point.
-
Controllability
The singular values of controllability matrix 𝑊𝑐 under the three operating points
have non-zero values; therefore, the system is completely controllable. However,
the singular values of 𝑊𝑐 are distributed over a wide range, the indicator of an illconditioned system close to rank deficiency. This is shown by the large condition
number of 𝑊𝑐 for each of the three operating points.
-
Observability
The singular values of observability matrix 𝑊𝑜 under o.p.1 and o.p.3 have non-zero
values; therefore, the system is completely observable. However, some of the SVDs
are too small (almost zero), the indicator of hardly-observable modes that almost
don’t contribute to any of the desired outputs. At o.p.2, 𝑊𝑜 has two zero SVDs;
hence, the system is not completely observable. During freewheeling mode of
operation (o.p.2), the average power exchange by the storage subsystem is zero. In
117
other words, the energy storage subsystem doesn’t participate in the system
performance at this operating point. Therefore, the eigenmodes associated with
battery-side LC filter, namely, 𝑖𝑏𝑎𝑡 and 𝑣𝑐𝑏 , don’t contribute to any of the system
outputs selected in (4.87); hence, they are unobservable modes. It should be
mentioned that, 𝑊𝑐 and its SVDs depend on the output matrix of the state-space
system (Matrix 𝐶); hence, they change based on the output variable selection.
4.16 Summary
This chapter focused on developing the dynamic mathematical model of the proposed CSIbased wind turbine system. Detailed models of the aerodynamic conversion, drive train, selfexcited induction generator, Lead Acid battery, and power electronic converters were
presented. A reduced-order generic load model including balanced/unbalanced load was
developed. By combining the state equations of the system components, the overall model of
the system was described by seven inputs, six outputs, and twenty state variables.
To verify the dynamic model, the starting operation of the system was simulated under noload and rated-load scenarios. Then, the behaviour of the system was investigated by
simulating its step response. For comparison purposes, simulation results were shown for both
switching and average models. The results demonstrated that with respect to the switching
model, the average model doesn’t represent switching harmonics (as expected), but the
dynamics resulting from control system and power system interaction is preserved to a high
degree of accuracy.
A linearized model of the system was developed around three operating points. The
eigenvalue analysis of the linearized model showed that the open-loop system is locally stable
around operating points 1 and 3, but not 2. Gramian method was employed to evaluate the
controllability and observability of the system at the three operating points. Based on SVDs,
the system is completely controllable at the three operating points, completely observable at
operating points 1 and 3, and not completely observable at operating point 2, at which the
eigenmodes of the battery-side LC filter don’t contribute to any of the system outputs selected.
118
Chapter 5
Control System Design for the Proposed PWM-CSI-Based SCIG-WECS
In Chapter 4, a small-signal linear model for the proposed standalone WECS was developed.
The stability analysis showed that the open-loop system was stable around operating points 1 and
3, but not 2; therefore, the objective of the closed-loop controller design is to stabilize the system
and track a class of desired references.
In this chapter:
1) The control objectives of the proposed standalone CSI-based WECS are defined;
2) The structures of the required control loops are designed;
3) The proportional-integral gains of the PI controllers are designed by using a tuning tool available
in Matlab/Simulink control toolbox. This tool systematically tunes PI controller for a given plant
transfer function or a Simulink model of the plant;
4) The expected performance of the proposed WECS will be verified by simulation;
5) The performance of the proposed WECS under faults will be evaluated; and
5) A dump load-less version of the proposed WECS will be presented.
5.1
Overview of the control system of the proposed WECS
The arrangement of the components of the CSI-based WECS, together with the corresponding
control blocks, is shown in Fig. 5.1. The system has six control variables, namely, the buck
converter duty ratio 𝑑𝑏 , the reduced H-bridge duty ratio 𝑑𝐴 , and the equivalent dq-axis modulation
+
+
−
−
indices in the load-side frequency frame 𝑚𝑖𝑑
, 𝑚𝑖𝑞
, 𝑚𝑖𝑑
, and 𝑚𝑖𝑞
. The state variables required to
be regulated are the rotor shaft speed 𝜔𝑟 or 𝜔𝑔 (𝜔𝑟 = 𝑃𝜔𝑔 ), the dc-link current 𝑖𝑑𝑐 , and the dq+
+
−
−
axis voltages in the load-side synchronous frame 𝑣𝑐𝑖𝑑
, 𝑣𝑐𝑖𝑞
, 𝑣𝑐𝑖𝑑
and 𝑣𝑐𝑖𝑞
.
119
→
→
→
ES Subsystem
Dc-Link Current Control
(Fig. 5.3)
Fig. 3.12
+
→
Load
Bus
Threephase
PWMCSI
(Fig. 3.12)
+
-
→
→
Extended Generic Load Model
(Fig. 4.21)
∆ /Yn
↑↑↑
Reduced
H- Bridge
CSI-Load Subsystem
→
Balanced/
Unbalanced
load
→
Ci
→
Geared-drive self-excited
SCIG Wind Turbine.
(Fig. 3.12)
Buck Converter
(Fig. 3.12)
Cdc
Load-Side Control
(Fig. 5.10)
Dump
Load
Circuit
(Fig.3.12)
n
C-Filter
+
-
Generator Speed
Control
(Fig. 5.9)
WTG Subsystem
Fig. 5.1: Converters and control blocks in the CSI-based WECS.
The main tasks assigned to the closed loop control system are:
1) DC-link current regulation, resulting in power management among the wind turbine generator,
the storage battery bank, the load, and the dump load;
2) Generator speed control to achieve MPPT; and
3) Load-side voltage magnitude and frequency control under both balanced and unbalanced threephase loads.
Since the dc-link inductor provides an energy buffer between the load-side converter (i.e., CSI)
and the combination of generator and dc-link current converters (i.e., buck and H-bridge), the loadside converter can be controlled irrespective of dynamic of generator speed and dc-link current.
Moreover, due to the mechanical inertia of the turbine, time constant of the generator-speed
controller is much longer than the electrical time constant of the dc-link current controller. By
proper design of the dc-link current controller, the dc-link current can be regulated at a rate much
faster than that of the generator speed. Therefore, dc-link current can be assumed constant when
designing the control loop for the generator speed. In other words, a fast (high bandwidth) and
stable control of the dc-link current can decouple the controllers of the generator speed, dc-link
current, and load-side voltage/frequency; hence, as shown in Fig. 5.1, the system is decomposed
120
into three subsystems (i.e., WTG, ES, and CSI-Load), allowing the classical controller design
techniques for SISO systems to be applied.
In the following section, a brief description of closed loop control system is given. Then, the
structure of the control loops implemented for the proposed CSI- WECS is explained.
5.2
Closed-Loop Control System
Fig. 5.2 shows a general closed-loop control system where 𝑥 is the controlled variable, and 𝜀
the error between the reference value 𝑥 ∗ and the feedback 𝑥 ′ . The CL controller is composed of
three main blocks: controller transfer function 𝐶(𝑠), plant transfer function 𝐺(𝑠), and feedback
transfer function 𝐹(𝑠).
-+
Plant
Controller
Feedback
Fig. 5.2: Typical control loop diagram.
In Fig. 5.2, 𝐶(𝑠) is the PI controller, defined in s domain by (5.1), where 𝑘𝑝 and 𝑘𝑖 are the
proportional and integral gains of the controller, respectively.
𝐶(𝑠) = 𝑘𝑝 +
𝑘𝑖
𝑠
(5.1)
𝐺(𝑠) is the plant transfer function of the linearized system.
𝐹(𝑠) is the feedback transfer function. In this work, 𝐹(𝑠) is assumed to be unity. However, in an
actual setup, it has a gain that is determined by the sensor. In addition, 𝐹(𝑠) can represent a low
pass filter that might be required to suppress the noise of the measured value.
The open-loop transfer function (OLTF ) is defined as
121
𝑂𝐿𝑇𝐹 =
𝑥 ′ (𝑠)
𝜀(𝑠)
= 𝐶(𝑠) 𝐺(𝑠) 𝐹(𝑠)
(5.2)
The closed loop transfer function (CLTF ) is
𝑥(𝑠)
𝐶(𝑠) 𝐺(𝑠)
𝐶𝐿𝑇𝐹 = 𝑥 ∗(𝑠) = 1+𝐶(𝑠) 𝐺(𝑠) 𝐹(𝑠)
(5.3)
Setting the denominator of the 𝐶𝐿𝑇𝐹 equal to zero yields
1 + 𝐶(𝑠) 𝐺(𝑠) 𝐹(𝑠) = 1 + 𝑂𝐿𝑇𝐹 = 0
(5.4)
Equation (5.4) is equivalent to the characteristics equation given in (4.94). In this sense, the open
loop transfer function can be used as indicator of stability of the system.
The design of the PI controller parameters can be carried out in time or frequency domain.
Matlab/Simulink control toolbox provides PI Tuning tools for such a purpose. In general, the PI
parameters are tuned to balance performance (response time) and robustness (stability margins) of
the controlled system. It should be mentioned that, in practice, the PI gains are commonly tuned
on a trial-and-error basis [146].
5.3
DC-link Current Control
This section develops a control algorithm for the dc-link current regulation, followed by the
design of the PI-controller parameters.
5.3.1 DC-link Current Control Scheme
The operation of CSI requires a regulated current in the dc link which is shared by the generator,
storage and load-side converters. DC-link current is controlled through power management
between the battery and dump load. The battery is required to absorb the excess power generated
from the wind, simply referred to as excess power, during high-wind speed and/or low-load
periods, and compensate for the shortage of power during low-wind speed and/or high-load periods
or when the wind turbine is not operating. If the battery state-of-charge (SoC) reaches its upper
limit, the excess power generated from the wind should be dissipated in the dump load. To achieve
this objective, the control scheme shown in Fig. 5.3 is implemented. The reduced H-bridge is
controlled by PWM unipolar voltage switching scheme. The control is done by comparing the
measured 𝑖𝑑𝑐 with the dc current reference (𝑖𝑑𝑐,𝑟𝑒𝑓 ), processing the error by a PI controller, and
producing two control signals, 𝑣𝑐𝑜𝑛𝑡𝑟𝑜𝑙 and −𝑣𝑐𝑜𝑛𝑡𝑟𝑜𝑙 , that are compared with a triangular carrier
122
signal to produce the gating signals for the converter switches. To guarantee a longer operating
life for the battery, the battery’s SoC is maintained between 25% and 75%. This is done by
multiplying the 𝑣𝑐𝑜𝑛𝑡𝑟𝑜𝑙 signal with a control signal 𝑥 which assumes a value of 0, when SoC is
beyond limits, and a value of 1, when SoC is within limits.
SoC%
75%
AND
OR
Maximum
-+
-1
Limiter
1
Rate
Limiter
PI
Controller
PWM
Generator
NOT
PWM
Generator
NOT
-+
0
2
÷
Limiter
SoC%
75%
SoC%
25%
NAND
NAND
AND
Fig. 5.3: DC-link current control scheme.
The reference of the dc-link current must satisfy the minimum current requirements for both
generator- and load-side converters. Considering the load-side converter, it is desired to reduce the
2
dc-link current as the power demand is reduced, in order to reduce 𝑖𝑑𝑐
𝑅𝑑𝑐 and CSI losses, and
avoid CSI’s low modulation indices which increase harmonic distortion in the output voltage. As
∗
a result, the dc-link current reference, 𝑖𝑑𝑐
, is determined as follows:
1) On a per-phase basis, the CSI’s ac-side line current (𝑖𝐿,𝑜 ) is equal to the sum of the filter
capacitor current and the primary-side line current (𝑖𝐿,𝑝 ) of the ∆/𝑌𝑛 transformer, which is related
to the secondary-side line current (𝑖𝐿,𝑠 ) through the ∆/𝑌 transformation ratio 𝑁.
2) The fundamental component of the current in the C-filter is considerably smaller than that of
the transformer primary-side; hence, it can be neglected. Moreover, the high-order harmonic
components of the transformer primary-side current can be neglected as they are significantly
reduced by the C-filter. Hence, the rms value of transformer primary-side current, 𝐼𝐿,𝑝,𝑟𝑚𝑠 , can be
123
assumed to be equal to the rms value of the fundamental component of the ac-side line current,
𝐼𝐿,𝑜,1,𝑟𝑚𝑠 , i.e.,
𝐼𝐿,𝑝,𝑟𝑚𝑠 = 𝐼𝐿,𝑜,1,𝑟𝑚𝑠
(5.5)
From (1.3), when 𝑚𝑖 = 1,
√3
𝐼𝐿,𝑜,1,𝑟𝑚𝑠 = 2√2 (𝑖𝑑𝑐 )
(5.6)
∗
Based on (5.5) and (5.6), and considering the turns-ratio 𝑁 of the ∆/𝑌𝑛 transformer, 𝑖𝑑𝑐
can be
obtained for the case of balanced load as
∗
𝑖𝑑𝑐
=
2√2
√3
𝐼𝐿,𝑝,𝑟𝑚𝑠 =
2√2
𝑁
𝐼𝐿,𝑠,𝑟𝑚𝑠
(5.7)
∗
Equation (5.7) states that 𝑖𝑑𝑐
can be set to different values based on either measured primary-side
∗
current (𝐼𝐿,𝑝,𝑟𝑚𝑠 ) or secondary-side current (𝐼𝐿,𝑠,𝑟𝑚𝑠 ). To account for unbalanced load, 𝑖𝑑𝑐
is set
based on the highest 𝐼𝐿,𝑝,𝑟𝑚𝑠 of the three phases so that a high-enough 𝑖𝑑𝑐 that suits all phases is
guaranteed. The modulation index corresponding to the phase with the largest rms current is 1,
whilst it is less than 1 for the other two phases. Even though it is known that as the load current
decreases 𝑖𝑑𝑐 can be decreased, there should be a minimum 𝑖𝑑𝑐 based on which the dc-link inductor
is designed (see equation (3.12)). In this work, the minimum demand is assumed to be 20% of
the rated load; hence, only 20% of the rated dc-link current will be enough to satisfy the load.
However, in order to reduce the size of the bulky dc-link inductor, it is desired to increase the
minimum limit for the dc-link current, which, nevertheless, implies higher dc-link and inverter
losses. As a trade-off between dc-link inductor size reduction and dc-link efficiency improvement,
the minimum reference for 𝑖𝑑𝑐 is set at 40% of the rated 𝑖𝑑𝑐 in order to avoid modulation indices
below 0.5. This current is referred to as 𝑖𝑑𝑐,𝑚𝑖𝑛 . On the other hand, there should be an upper limit
for 𝑖𝑑𝑐 to avoid exceeding the current limit of power IGBTs. This current is referred to as 𝑖𝑑𝑐,𝑚𝑎𝑥 .
Load kVA should be managed so that the required dc-link current will never exceed this upper
limit. 𝑖𝑑𝑐,𝑚𝑎𝑥 has to be lower than the battery current limit.
As noticed from Fig. 5.1, the reduced H-bridge terminals are connected in series with the dc link
inductor during non-dumping periods (i.e., 𝑖𝑑𝑐ℎ = 𝑖𝑑𝑐 ). Thus, the battery-side current (𝑖𝑖𝑛 ) during
charging or discharging is composed of pulses of magnitude 𝑖𝑑𝑐 or −𝑖𝑑𝑐 , separated by zero-current
periods. Therefore, the battery average power (𝑃𝑏𝑎𝑡,𝑎𝑣𝑔. ) can be calculated using (5.8), where 𝑣𝑏𝑎𝑡
is the battery terminal voltage, and 𝑖𝑏𝑎𝑡,𝑎𝑣𝑔. and 𝑣𝑥𝑦,𝑎𝑣𝑔. the average values of the filtered battery
124
current and reduced-H-bridge output voltage, respectively, with a conversion efficiency of 100%
assumed.
𝑃𝑏𝑎𝑡,𝑎𝑣𝑔. = (𝑣𝑏𝑎𝑡 )(𝑖𝑏𝑎𝑡,𝑎𝑣𝑔. ) = (𝑣𝑥𝑦,𝑎𝑣𝑔. )(𝑖𝑑𝑐 )
(5.8)
At high wind speeds and low- or no-load conditions, the excess power to be absorbed by the
∗
battery is high, requiring a high 𝑖𝑑𝑐 . However, 𝑖𝑑𝑐
is set by (5.7) according to load demand. Due
to the fact that the maximum of 𝑣𝑥𝑦,𝑎𝑣𝑔. does not exceed 𝑣𝑏𝑎𝑡 , the maximum average power that
∗
∗
∗
the battery is allowed to absorb under 𝑖𝑑𝑐
is 𝑣𝑏𝑎𝑡 𝑖𝑑𝑐
. This particular power is referred to as 𝑃𝑏𝑎𝑡
.
∗
∗
If the excess power exceeds 𝑃𝑏𝑎𝑡
, the dc-link current will be forced to go from 𝑖𝑑𝑐
, which is set
low to reduce inverter and dc-link losses, to a higher value required by the battery to absorb
whatever excess power is available. This implies a conflict between improving inverter and dclink efficiencies and MPPT. Because wind power is intermittent, and hence may not be available
when it is needed most, this work gives the priority to achieving maximum power and storing all
excess power as long as the battery bank has free capacity and its current rating allows. However,
the advantage of reducing 𝑖𝑑𝑐 at low load should be preserved as long as it doesn’t contradict
∗
MPPT. In other words, the reference of 𝑖𝑑𝑐 is initially set at 𝑖𝑑𝑐
required by the load. This will
∗
allow the battery to absorb excess power below or equal to 𝑃𝑏𝑎𝑡
. If excess power goes beyond
∗
∗
𝑃𝑏𝑎𝑡
, an 𝑖𝑑𝑐 higher than 𝑖𝑑𝑐
is required. In this case, the reference of 𝑖𝑑𝑐 is set as required by the
battery to absorb the entire excess wind power, as long as the battery current limit and SoC upper
∗∗
limit are not exceeded. This current is denoted as 𝑖𝑑𝑐
. Assuming a lossless system, equation (5.9)
∗∗
can be used to find 𝑖𝑑𝑐
, where 𝑃𝑚,𝑜𝑝𝑡 is the steady-state optimum wind power, obtained from wind
power-shaft speed look-up table based on Fig. 4.3, and 𝑃𝐿 is the load power. It should be noted
∗
that excess power will not go beyond 𝑃𝑏𝑎𝑡
unless 𝑃𝑚,𝑜𝑝𝑡 ≫ 𝑃𝐿 ; then, 𝑃𝑚,𝑜𝑝𝑡 − 𝑃𝐿 ≈ 𝑃𝑚,𝑜𝑝𝑡 .
∗∗
𝑖𝑑𝑐
=
𝑃𝑚,𝑜𝑝𝑡 − 𝑃𝐿
𝑣𝑏𝑎𝑡
≈
𝑃𝑚,𝑜𝑝𝑡
𝑣𝑏𝑎𝑡
(5.9)
The mechanism of determining the dc-link current reference (𝑖𝑑𝑐,𝑟𝑒𝑓 ) is incorporated in the dclink current control scheme of Fig. 5.3 by employing an automatic software switch, which by
∗
∗
default assumes position 1, corresponding to 𝑖𝑑𝑐
. If 𝑖𝑑𝑐 exceeds 𝑖𝑑𝑐
plus permitted ripple, the switch
∗∗
assumes position 2, corresponding to 𝑖𝑑𝑐
. A limiter is used to ensure the dc-link current reference
∗∗
always remains between 𝑖𝑑𝑐,𝑚𝑖𝑛 and 𝑖𝑑𝑐,𝑚𝑎𝑥 . As noticed from Fig. 5.3, 𝑖𝑑𝑐
is prevented from
exceeding 𝑖𝑑𝑐,𝑚𝑎𝑥 by engaging the dump load, thus limiting the power absorbed by the battery to
125
𝑖𝑑𝑐,𝑚𝑎𝑥 𝑣𝑏𝑎𝑡 . Moreover, a falling/rising-rate limiter block is used in order to change 𝑖𝑑𝑐,𝑟𝑒𝑓
gradually to the desired value; this is because the dc choke will not allow fast changes in 𝑖𝑑𝑐 .
5.3.2 Parameters Design of DC-link Current PI Controller
As described in subsection 5.3.1, the dc-link current is set based on the load demand (i.e.,
∗
∗∗
𝑖𝑑𝑐,𝑟𝑒𝑓 = 𝑖𝑑𝑐
) or the optimal wind power (i.e., 𝑖𝑑𝑐,𝑟𝑒𝑓 = 𝑖𝑑𝑐
). In other words, the response time of
the dc-link current controller should be fast enough to respond to load variation as well as wind
speed variation. For aggregated residential load, combining a group of houses, a considerable
change in demand can occur in the range of minutes. On the other hand, wind speed may vary very
dramatically in a matter of few seconds [147]. Therefore, the bandwidth of the dc-link current
controller should be higher than the frequency of wind speed variation. In general, the higher the
bandwidth, the better the performance of the controller. In practice, however, rejection error of
high-frequency disturbances and the ripple threshold for control signal set limits on the controller
bandwidth [146].
In order to show the system performance under various wind and load conditions during short
intervals of time, the dc-link control bandwidth is selected in range of tens of hertz, enabling very
short rise time as well as response time. The parameters of PI controller of dc-link current
controller are designed to satisfy the following time-domain specifications:
• The step response settling time of less than 0.05 second;
• The step response steady-state error of zero; and
• The overshoot/undershoot of less than 10%.
The frequency-domain specifications are as follows:

Controller Bandwidth ≥ 40 Hz (250 rad/s); and

Stability Phase Margin ≥ 60𝑜 .
Based on transfer functions (provided in Table 5.1) derived from the small-signal model of ES
subsystem (given in Appendix E), Matlab/Simulink PI tuning tools are used to tune the PI
parameters of the dc-link controller at the three operating points of linearization described in
chapter 4 (Table 4.7). The Tuning function can also be achieved based on the Simulink model of
the ES subsystem.
126
Table 5.1: Small-signal transfer functions of the ES-subsystem.
Operating point
𝒊
Transfer Function (𝑮𝒊𝒅𝒄 = 𝟐𝒅𝒅𝒄−𝟏)
𝑨
o.p.1
o.p.2
o.p.3
2.55 𝑥 104 𝑠2 +3.562 𝑥 108
𝑠+1.051 𝑥 1012
𝑠3 +1.432 𝑥 104 𝑠2 +4.626 𝑥 107 𝑠+1.567 𝑥 1010
9.752 𝑥 104 𝑠2 +1.36 𝑥 109 𝑠+3.994 𝑥 1012
𝑠3 +1.394 𝑥 104 𝑠2 +4.095 𝑥 107 𝑠
2.523 𝑥 104 𝑠2 +3.516 𝑥 108 𝑠+1.031 𝑥 1012
𝑠3 +1.362 𝑥 104 𝑠2 +3.64 𝑥 107 𝑠−1.335 𝑥 1010
Fig. 5.4 shows the tuning results of the dc-link current (𝑖𝑑𝑐 ) controller at o.p.1. The tuning
time/frequency domain results and step responses of the closed loop 𝑖𝑑𝑐 controller show that the
system meets the design conditions specified above. Same procedures are carried out at o.p.2 and
o.p.3, with the tuning results given in Fig. 5.5 and Fig. 5.6, respectively. The resulting 𝑘𝑝 and 𝑘𝑖
are displayed in Table 5.2.
Fig. 5.4: PI tuning of the dc-link current controller at o.p.1.
127
Fig. 5.5: PI tuning of the dc-link current controller at o.p.2.
Fig. 5.6: PI tuning of the dc-link current controller at o.p.3.
128
Based on the small-signal transfer functions and PI parameters given in Table 5.1 and Table 5.2,
respectively, Fig. 5.7 shows the small-signal step responses of dc-link current from one operating
point to a new steady-state value. In Fig. 5.7(a), the system was operating at o.p.1 ( 𝑖𝑑𝑐,𝑜.𝑝.1 =
51 𝐴) before a 25% step increase is applied at t = 0.5 s. The new steady-state value of 𝑖𝑑𝑐 is reached
in about 0.02 s, with an overshoot of about 9.3%. These values match the time-domain
characteristics obtained from the PI tuning tools (see Fig. 5.4). Because the dc choke will not allow
step change in 𝑖𝑑𝑐 , the reference 𝑖𝑑𝑐 is applied as ramp signal, as in Fig. 5.7(b), resulting in slower
response time and less overshoot. Similar observations are noticed from Fig. 5.7(c) - (f) where step
change and ramp signal are applied to 𝑖𝑑𝑐 at o.p.2 and o.p.3.
Fig. 5.7: Step/Ramp response of dc-link current at the three operating points.
129
5.4
Generator Speed MPPT Control
This section describes control algorithm for the generator speed followed by the design of the
PI-controller parameters.
5.4.1 MPPT Control Scheme
In a variable-speed wind turbine, the generator is controlled to extract the maximum power
available from the wind. Different techniques for maximum power point tracking, reported in the
literature, have been reviewed in [148]. They can be categorized under perturbation and
observation (P&O) based techniques and Look-up table based techniques. P&O techniques are
simple and require no prior knowledge of the system parameter. However, due to its slow
operation, P&O technique may not be efficient for wind turbines, where the dynamics of wind is
very fast (i.e., wind speed changes quite fast in a matter of seconds) [147]. On the other hand, the
look-up table-based techniques, such as power or torque-signal feedback, and tip-speed ratio
(TSR) techniques, are commonly used in wind turbines although they require speed sensors and a
pre-programmed look-up table of the turbine data. Among the look-up table techniques, TSR
technique can provide the fastest control action because it depends on direct measurement of the
wind speed and sets the control reference instantaneously, resulting in more energy
production [147],[149]. In this chapter, therefore, the TSR technique is implemented to achieve
MPPT.
Fig. 5.8 shows a typical power-wind speed curve for a variable-speed wind turbine. The curve is
divided into three regions. From cut-in to rated wind speed (i.e., region 1), the turbine is controlled
at the maximum performance coefficient (𝐶𝑝𝑚𝑎𝑥 ) corresponding to the optimum tip speed ratio
(𝜆𝑜𝑝𝑡 ); thus, maximum power available from wind is extracted. At wind speeds higher than the
rated value (i.e., region 2), the turbine is aerodynamically controlled in order to limit the extracted
power to the rated value. Such a control is carried out either by stall or pitch regulation mechanism.
For a fixed-pitch wind turbine, the blades are aerodynamically designed to achieve passive stall.
Moreover, the rotor speed of a fixed-pitch turbine can be adjusted by furling control or electronic
brakes. In order to protect the turbine in region 3, it must be stopped at cut-out speed. Furling
control can significantly reduce the rotor speed, but it may not be able to stop it. Therefore, a
mechanical, electronic or hydraulic brake is used to bring the rotor to rest after its speed is reduced
130
by furling. The brake can also be used to ramp off the generated wind power in emergency cases
such as sudden disconnection of load while the storage unit is not capable of absorbing the
generated power. Even in the presence of brake, a dump load is necessary to complete the
protection scheme, especially during transients.
Fig. 5.8: Typical turbine power versus wind speed curve.
In the proposed WECS, shown in Fig. 5.1, the main function of the generator-side converter
(i.e., buck converter) lies in region 1. The converter, as shown in Fig. 5.9, is controlled to extract
maximum power from wind by regulating the generator shaft speed at the optimum value
corresponding to the present value of wind speed. The same concept has been applied in [73],[74]
to control a direct-drive PMSG in a grid-connected WECS. The control is based on keeping the
tip speed ratio at the optimal value. As revealed from equation (4.4), at a fixed tip speed ratio, the
turbine’s rotational speed is linearly related to the wind speed (i.e., 𝜔𝑚  𝑣𝑊 ). Therefore, as shown
in Fig. 5.9, the reference signal for generator angular speed (𝜔𝑔,𝑜𝑝𝑡 ) is produced based on the
optimal tip speed ratio (𝜆𝑜𝑝𝑡 ), the measured wind speed (𝑣𝑤 ), gearbox ratio (𝑛𝑔𝑒𝑎𝑟 ) to account for
the speed level conversion, and the turbines radius (𝑟). The error between 𝜔𝑔,𝑜𝑝𝑡 and the measured
speed 𝜔𝑔 is amplified by a PI controller, leading to the generation of gating pulses for the buck
switch, 𝑆𝑏𝑢𝑐𝑘 , through PWM process. As a result, the rectifier output current (𝑖𝑑𝑐𝑟 ), the SCIG stator
131
current (𝑖𝑑𝑠 ), the SCIG counter torque (𝑇𝑒 ) and finally the shaft speed (𝜔𝑔 ) are adjusted to achieve
maximum power.
PI
Controller
PWM
Generator
+
Fig. 5.9: MPPT Controller.
5.4.2 Parameter Design of Generator Speed Control Loop
As mentioned in section 5.1, the generator speed (𝜔𝑔 ) control loop is designed to be much
slower than that of the dc-link current (𝑖𝑑𝑐 ) control loop. In view of this, the design of 𝜔𝑔 controller
does not take into account the dc-link dynamics based on the assumption that 𝑖𝑑𝑐 is quickly and
robustly regulated by the ES-subsystem. Accordingly, the PI parameters of generator speed
controller are designed to satisfy the following time-domain specifications:
• The step response settling time of less than 0.5 second;
• The step-response steady-state error of zero; and
• The overshoot/undershoot of less than 10%.
The frequency-domain specifications are as follows:

Controller Bandwidth ≥ 4 Hz (25 rad/s) (One-tenth of that in dc-link current control
loop); and

Stability Phase Margin ≥ 60𝑜 .
By applying a procedure similar to that for dc-link current controller design, PI parameters of
the generator-speed controller are tuned by Matlab/Simulink tuning tools based on the Simulink
model of the linearized WTG subsystem. The resulting 𝑘𝑝 and 𝑘𝑖 are displayed in Table 5.2.
132
5.5
Load-Side Control
This section describes the control algorithm for the load-side voltage/frequency, followed by the
design of the PI-controller parameters.
5.5.1 Load-Side Control Scheme
The unbalanced load currents lead to unbalanced line voltages at the load bus. Voltage imbalance
can cause serious problems to three-phase loads, especially motor loads. Therefore, it is
compulsory to compensate for the voltage imbalance at the load bus so that the voltage unbalance
factor (VUF), defined as the ratio of fundamental component of negative-seq voltage to that of
positive-seq voltage, does not exceed the permissible limit of 1%.
In the proposed WECS, shown in Fig. 5.1, the ∆/𝑌𝑛 transformer isolates the zero-seq component
of load current. Therefore, as shown in Fig. 5.10, the unbalanced voltage across the C-filter (𝑣𝑐𝑖 𝑎𝑏𝑐 )
is decomposed into symmetrical positive-seq (𝑣𝑐+𝑖 𝑎𝑏𝑐 ) and negative-seq (𝑣𝑐−𝑖 𝑎𝑏𝑐 ) components based
on transformation matrices given in (5.10) and (5.11), respectively.
𝑣𝑐+𝑖 𝑎
1 1
[𝑣𝑐+𝑖 𝑏 ] = [𝑎2
3
𝑎
𝑣𝑐+𝑖 𝑐
𝑎
1
𝑎2
𝑎2 𝑣𝑐𝑖 𝑎
𝑎 ] [𝑣𝑐𝑖 𝑏 ]
1 𝑣𝑐𝑖 𝑐
(5.10)
𝑣𝑐−𝑖 𝑎
1 1
−
[𝑣𝑐𝑖 𝑏 ] = [ 𝑎
3 2
𝑣𝑐−𝑖 𝑐
𝑎
𝑎2
1
𝑎
𝑎 𝑣𝑐𝑖 𝑎
𝑎2 ] [𝑣𝑐𝑖 𝑏 ]
1 𝑣𝑐𝑖 𝑐
(5.11)
where 𝑎 = 𝑒 𝑗2𝜋/3 .
The concept of sequence decomposition has been used in [150],[151] to control a VSI supplying
an unbalanced load. The big advantage of sequence decomposition is that it allows decoupled
control of the positive and negative sequence voltage components. However, it delays the
measured voltage by one-fourth of the period at the fundamental frequency [151]. In Fig. 5.10,
the ability of synchronous dq control to achieve zero steady-state error is utilized for each
+
symmetrical component. The positive-seq d-axis voltage (𝑣𝑐𝑖𝑑
) is compared with the desired value
+
−
−
of 1 pu and the error is processed in a PI controller. All other voltages (i.e., 𝑣𝑐𝑖𝑞
, 𝑣𝑐𝑖𝑑
, 𝑣𝑐𝑖𝑞
) are
kept at zero value. This guarantees achieving a positive seq balanced voltage. The dq-frame output
signals of the PI controllers are transformed into abc frame and the resulting signals of the same
133
phases are added together to produce the modulating signals (𝑚𝑖𝑎𝑏𝑐 ) to be applied to the PWM
generator to generate gating pulses for the CSI switches. The frequency is set at the desired value
(i.e., 50 or 60 Hz) in open-loop control.
dq/abc
PI
+
++-
abc/dq
PI
PWM
Generator
To CSI
switches
Negativesequence
Calculator
abc/dq
++-
Positivesequence
Calculator
PI
dq/abc
PI
Fig. 5.10: Load-side synchronous dq frame control scheme.
5.5.2 Parameter Design of Load-Side Control Loop
As mentioned in section 5.1, the dc-link inductor provides an energy buffer between CSI-Load
subsystem and the combination of WTG and ES subsystems. In addition, with very fast dc-link
current (𝑖𝑑𝑐 ) control loop, 𝑖𝑑𝑐 is considered as constant input to the CSI-Load subsystem. As a
result, the load-side voltage and frequency can be controlled irrespective of wind speed variation.
The parameters of PI controller of load-side controller are designed to satisfy the following timedomain specifications:
• The step response settling time of less than 0.1 second;
• The step-response steady-state error of zero; and
• The overshoot/undershoot of less than 10%.
The frequency-domain specifications are as follows:

Controller Bandwidth ≥ 20 Hz (125 rad/s) (Half of that in dc-link current loop); and

Stability Phase Margin ≥ 60𝑜 .
134
Employing the same Matlab/Simulink tool that was used for tuning dc-link current and
generator-speed controllers, 𝑘𝑝 and 𝑘𝑖 of the load-side controllers are obtained as given in
Table 5.2. In the table, 𝑘𝑝+ and 𝑘𝑖+ are the gains of PI controllers of positive sequence dq voltages
and 𝑘𝑝− and 𝑘𝑖− the gains of PI controllers of negative sequence dq voltages.
Table 5.2: PI gains of the closed loop system controllers at the three operating points.
Generator speed
controller
𝑘𝑝
5.6
𝑘𝑖
DC-link current
controller
𝑘𝑝
𝑘𝑖
o.p.1
42.7024
284.0791
0.00098308
4.4427
o.p.2
39.4928
268.823
0.0029066
0.13213
o.p.3
9.3191
119.4989
0.15879
11.0886
Load-Side Controllers
𝑘𝑝+
𝑘𝑖+
𝑘𝑝−
𝑘𝑖−
0.24761
0.01211
0.22753
0.04352
0.45507
0.03541
5.6081
5.5194
5.1534
5.0621
10.3068
9.8675
Simulation Verification
In this section, the closed-loop control systems for the proposed SCIG-CSI-WECS are simulated
in Matlab/Simulink environment for a 20kW standalone WECS using a 460Vrms SCIG and the
extended generic load model requiring a regulated voltage of 380V/220Vrms at 60 Hz. The
system’s parameters are given in Appendix A (Table A.2). First, the system performance is
examined under variable turbulence-free wind speed and well-damped three-phase load
conditions. Next, the effect of system inertia and frequency of wind speed variation on MPPT is
studied. Finally, the performance of the synchronous dq controller under load conditions with
various dynamics and steady-state characteristics will be evaluated.
In the following simulations, the closed-loop PI controllers are tuned based on gain-scheduling
given in Table 5.2 for the three operating points selected for linearization. However, in order to
illustrate the performance of the control schemes under various conditions, the system is also
simulated at other operating points. At each operating point, Matlab/Simulink tuning tools is used
to linearize the system Simulink model, tune the controller gains, and reconcile the gain values to
provide smooth transition between operating conditions.
135
Unless otherwise specified, the simulation results of this section are obtained based on switching
model of the entire CSI-based WECS.
In general, loads in remote areas are composed of 80% constant-impedance and 20% constantpower components [152]. Since the load is dominantly of constant-impedance type, the simulation
results presented in this section are based on the assumption of 100% constant-impedance loads.
5.6.1 System Performance under Various Wind Speed and Load Conditions
The performance of the system is examined under variable wind speed and balanced/unbalanced
three-phase load as follows:
A) Variable Balanced Load
The system is operated under variable wind speed and balanced three-phase load, according to
the wind speed and load profiles given in Table 5.3. The wind speed is considered a constant signal
at its average value in each sub-period. The rated load is 6.67 kW and 3.33 kVar per phase (i.e.,
−
−
three-phase powers are 𝑃+ = 20𝑘𝑊, 𝑄 + = 10 𝑘𝑉𝑎𝑟, and 𝑄𝑝𝑑
= 𝑄𝑝𝑞
= 0). Throughout the
simulation period, the load profile has the same dynamic characteristics (𝑑 = 100 𝑠 −1 and 𝜔𝑜 =
75 𝑟𝑎𝑑/𝑠). The initial SoC of the battery is 50%. Simulation results are shown in Fig. 5.11.
Table 5.3: Wind speed and load conditions for simulation results of Fig. 5.11 and Fig. 5.14.
Time (s)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
Wind speed (m/s)
12
12
12
11
11
11
10
10
9
9
8
8
7
% of rated load
0
20
100
100
110
90
90
70
70
50
50
20
20
136
DC-link current reference (A)
∗∗
𝑖𝑑𝑐
= 51
∗∗
𝑖𝑑𝑐 = 51
∗
𝑖𝑑𝑐 = 55.5
∗
𝑖𝑑𝑐
= 55.5
∗
𝑖𝑑𝑐 = 61
∗
𝑖𝑑𝑐
= 50
∗
𝑖𝑑𝑐 = 50
∗
𝑖𝑑𝑐
= 39
∗
𝑖𝑑𝑐 = 39
∗
𝑖𝑑𝑐 = 27.75
∗
𝑖𝑑𝑐
= 27.75
∗
𝑖𝑑𝑐 = 𝑖𝑑𝑐,𝑚𝑖𝑛 = 22.2
∗
𝑖𝑑𝑐
= 𝑖𝑑𝑐,𝑚𝑖𝑛 = 22.2
As shown in Fig. 5.11(a), MPPT controller behaves as expected by tracking the shaft speed
reference. Generator voltage and frequency vary as wind speed changes. At t = 0 s, the system is
exposed to rated wind speed of 12 m/s under no-load condition. The SCIG produces the rated
power of 20 kW at rated voltage of 460 Vrms and frequency 60 Hz. As shown in Fig. 5.11(b),
∗
since no load is connected, the dc-link current reference is initially set to 𝑖𝑑𝑐
= 𝑖𝑑𝑐,𝑚𝑖𝑛 , which is
enough for the CSI to produce the small ac-side current required by the filter capacitors to define
∗
∗
the load voltage. However, because the wind power exceeds 𝑃𝑏𝑎𝑡
, the 𝑖𝑑𝑐 surpasses the 𝑖𝑑𝑐
at t =
∗∗
0.03s; hence, the dc-link current reference is switched to the 𝑖𝑑𝑐
required for the battery to absorb
the entire generated power. At t = 0.5 s, 20% of the rated load is to be supplied, but the excess
∗
∗∗
power still exceeds 𝑃𝑏𝑎𝑡
; hence, 𝑖𝑑𝑐 is still regulated at 𝑖𝑑𝑐
. Very low modulation index (𝑚𝑖 ) is
∗∗
noticed before t =1 s because 𝑖𝑑𝑐
is much higher than what is required by the load during this
∗
period. From t = 1 s to t = 7 s, the difference between the wind power and demand is below 𝑃𝑏𝑎𝑡
;
∗
thus, the dc-link current is regulated at the 𝑖𝑑𝑐
, set according to the load; hence, modulation index
∗
is close to unity except after t = 5.5 s, when 𝑖𝑑𝑐
is maintained at 40% of rated 𝑖𝑑𝑐 even though the
load is only 20% of the rated value. Since the dc-link inductor provides an energy buffer between
the generator-side and load-side converters, load-side voltage and frequency are controlled
irrespectively of wind speed variations. This task is achieved by the CSI controller, which
maintains the load-bus voltage magnitude at 380 V/220 Vrms and frequency at 60 Hz, with a
maximum deviation of within ±0.2 Hz during load change. During simulation period, the highest
THD of the line voltage at the load bus is about 4% which is below the common permissible limit
of 5%. Similarly, the THD of line current is about 1% (implying a nearly sinusoidal current).
Thus, a very high waveforms-quality is obtained at the load side with the help of the output Cfilter. As shown in Fig. 5.11(c), the battery bank either absorbs or delivers power according to the
difference between wind power and load demand. Since the battery SoC is below the upper limit
during charging, no power needs to be consumed by the dump load. From Fig. 5.11(b), one can
notice that the dc-link current reference is adjusted gradually to the desired value because the dc
choke will not allow a step change in the dc-link current.
137
(a) Generator-Side Characteristics
138
(b) DC-link and Load-Side Characteristics
(c) Power Management
Fig. 5.11: System behavior under variable balanced load.
139
One problem with using a diode rectifier as the generator-side converter is the resulting
distortion in the stator current waveforms, leading to higher harmonic losses and torque ripples in
the generator. This can be seen in Fig. 5.12(a) where a low generator inductance of 0.05 pu is
assumed. However, as mentioned in chapter 4 (section 4.4), low-power induction generator
features relatively high inductance (i.e., 0.1 - 0.25 pu) which helps reducing the harmonics of the
generator current as can be seen in Fig. 5.12(b) where a higher generator inductance of 0.15 pu is
assumed. In this case, the generator experiences torque ripples within 10% versus 35% for the low
generator inductance. It is worth mentioning that the quality of the generator current can be
significantly improved by adding an external L filter on generator side, but at additional cost.
Fig. 5.12: Effect of generator inductance on stator current and electromagnetic torque.
Fig. 5.13 shows the simulation results for the case where the battery SoC is initially set very
close to its upper limit (i.e., 75%), wind is blowing at the rated speed (12 m/s) and the load is at
90% of its rated value. At start, the battery supplies the load and continues to deliver power until
the wind turbine produces the whole demand at t = 0.16 s. After this point, the battery is charged
due to the availability of excess power. When battery SoC reaches its limit at t = 0.61 s, the battery
140
cannot be charged further; hence, the excess power is consumed by the dump load. The dc-link
current is maintained at the desired value under all conditions.
Fig. 5.13: Power management after battery SoC reaches upper limit.
B) Variable Unbalanced Load
The system is operated under the same conditions as in Table 5.3, but with unbalanced load. The
rated values of three-phase load +ve seq active power, +ve seq reactive power, -ve seq d-axis
−
power, and –ve seq q-axis power are: 𝑃 + = 20𝑘𝑊, 𝑄 + = 10 𝑘𝑉𝑎𝑟, 𝑄𝑝𝑑
= −5.08𝑘𝑉𝑎𝑟, and
−
𝑄𝑝𝑞
= −6.04𝑘𝑉𝑎𝑟, respectively; hence, the load unbalance factor is 35%. In remote communities,
the load unbalance factor can be much higher than 35%, but this doesn’t affect the generality of
the load control scheme shown in Fig. 5.10. Simulation results are shown in Fig. 5.14. Since the
dc-link current is controlled by the ES subsystem, the effect of unbalanced load on the WTG
subsystem is negligible. Thus, the generator-side characteristics, under this case, are similar to
those in the case of balanced load (see Fig. 5.11(a)). Load phase currents are not equal; hence,
different modulation indices are produced for the three phases. Similar to the balanced-load case,
the load-bus voltage magnitude is maintained at 380 V/220 Vrms and the frequency at 60 Hz, with
a maximum deviation of within ±0.2 Hz during load change. As a result of employing dq
synchronous frame PI controllers, rms values of line voltages exhibits almost zero imbalance. VUF
is varying with load. The highest value noticed for VUF is 0.2% which is far below the permissible
141
limit of 1%. During simulation period, 4.35%, 4.1%, and 2.3% are the highest THDs detected in
the line voltages 𝑣𝑎𝑏 , 𝑣𝑏𝑐 , and 𝑣𝑐𝑎 , respectively, which are still below the common permissible
limit of 5%. Similarly, the highest THDs of load currents are 1%, 0.73%, and 0.81% for 𝑖𝑎𝑝 , 𝑖𝑏𝑝 ,
and 𝑖𝑐𝑝 , respectively, implying good sinusoidal currents.
Fig. 5.14: System behavior under variable unbalanced load.
142
5.6.2 Effect of System Inertia and Frequency of Wind Speed Variation on MPPT
The simulation results shown in Fig. 5.11(a) were obtained based on a turbulence-free wind
speed profile. As described in chapter 4 (section 4.1), wind speed is a stochastic variable that can
be described as the sum of average wind speed and the fluctuations about the average value. Wind
speed fluctuations have a significant effect on achieving MPP. In MPPT speed controller, shown
in Fig. 5.9 and described in subsection 5.4.1, the generator speed is controlled to track its optimal
value at a given wind speed. Due to system inertia, there is a transient stage between any two
steady-state generator speeds corresponding to two wind speed values. If the wind speed varies
rapidly, the MPPT controller may not be able to track the optimum generator speed and hence the
instantaneous maximum power will not be delivered. The effects of wind turbine inertia and rate
of wind speed change on the MPPT operation and average generated power were investigated
in [153]. It was shown that the average of the maximum power lost in a wind turbine increases
with inertia and frequency of wind speed fluctuations. Fig. 5.15 shows the generator responses
when the proposed system is run under two different low-pass filtered wind speed profiles
generated by Von-Karman model at a height of 18 meters. The wind speed rate of change (𝑑𝑣𝑤 /𝑑𝑡)
in Fig. 5.15(b) is higher than 2 times that in Fig. 5.15(a). As illustrated in the figure, the MPPT
controller under wind speed profile with lower 𝑑𝑣𝑤 /𝑑𝑡 performs much better than in the case of
higher 𝑑𝑣𝑤 /𝑑𝑡. The effect of system inertia is also shown in the figure for the two wind speed
profiles. In Fig. 5.15(a), the optimum speed is tracked less precisely for 𝐽 = 0.5 𝑘𝑔. 𝑚2 than for
𝐽 = 0.1 𝑘𝑔. 𝑚2 . As a result, over the displayed period, the average of the power lost for 𝐽 =
0.5 𝑘𝑔. 𝑚2 is twice that for 𝐽 = 0.1 𝑘𝑔. 𝑚2 . Under a higher rate of wind speed change
(Fig. 5.15(b)), the average power lost becomes worse for both inertia values; in particular, 𝐽 =
0.5 𝑘𝑔. 𝑚2 results in an average power reduction of 18% versus 4% for a lower rate of wind speed
change (Fig. 5.15(a)). It should be mentioned that in order to respond to the fast change of wind
speed in Fig. 5.15, the PI parameters of the MPPT controller are tuned under very high bandwidth.
143
Fig. 5.15: Effects of system inertia and frequency of wind speed variation on MPPT.
5.6.3 Performance of the Synchronous dq Controller under Various Dynamics of Load
In subsection 5.6.1, the system was examined under load profiles with same dynamic
characteristics. In this subsection, Fig. 5.16 illustrates the performance of the synchronous dq
controller under load conditions with various dynamics (natural damping (𝑑) and oscillation
frequency (𝜔𝑜 )) and steady-state characteristics, as given in Table 5.4. The nominal values of +ve
144
seq active power, +ve seq reactive power, -ve seq d-axis power, and –ve seq q-axis power are:
−
𝑃𝑜+ = 20𝑘𝑊, 𝑄𝑜+ = 10 𝑘𝑉𝑎𝑟, 𝑄𝑝𝑑𝑜
= −5.08𝑘𝑉𝑎𝑟,
and
−
𝑄𝑝𝑞𝑜
= −6.04𝑘𝑉𝑎𝑟,
respectively.
Because harmonics are not of interest in this subsection, the simulation responses shown here are
obtained based on the average model of the system.
Table 5.4: Load parameters for simulation results of Fig. 5.16.
Time
(s)
𝒅
(𝒔−𝟏 )
𝝎𝒐
(𝒓𝒂𝒅/𝒔)
% of 𝑷+
𝒐
% of 𝑸𝒐+
−
% of 𝑸𝒑𝒅𝒐
−
% of 𝑸𝒑𝒒𝒐
𝒊𝒅𝒄,𝒓𝒆𝒇
(A)
0
0
0
0
0
0
0
∗∗
𝑖𝑑𝑐
= 51
0.5
10
37
60
60
0
0
∗
𝑖𝑑𝑐
= 33.3
1.5
5
75
30
30
50
50
∗
𝑖𝑑𝑐
= 50
3.0
15
120
10
10
50
50
∗
𝑖𝑑𝑐
= 55.5
Before connecting any load, the system operates at rated wind speed and the dc-link current is
∗∗
regulated at 𝑖𝑑𝑐
, required for the battery to absorb the entire generated power. At t = 0.5 s, a
balanced load is connected to the load bus. Therefore, no –ve seq voltage or current exists. The
+ve seq dq load currents oscillate at a frequency of about 37 rad/s before settling down at 0.9 s
(i.e., 𝑡𝑠 = 0.4 𝑠). The dq load voltages experience very small undershoots and overshoots before
+
+
settling at the desired values of 𝑣𝑐𝑖𝑑
= 310.3 𝑉 (1pu) and 𝑣𝑐𝑖𝑞
= 0 𝑉. At t = 1.5 s, an unbalanced
load is connected to the load bus; adding more +ve seq demand and introducing a –ve seq reactive
power. At t = 3 s, another unbalanced load is switched on. As can be seen from the figure, the –ve
seq voltages are compensated by the synchronous dq controller, and therefore the voltage
imbalance caused by the unbalanced load is corrected. Only during transients, a disturbance can
+
be noticed. Since 𝑣𝑐𝑖𝑑
is almost fixed throughout the simulation, the +ve and –ve seq powers follow
the patterns of their associated currents. Throughout the simulation interval, except before t = 0.5
∗
s, the dc-link current is regulated at 𝑖𝑑𝑐
, set according to the load. Fig. 5.17 shows the three-phase
load currents and voltage (only peak portion) before and after connecting the unbalanced load at t
= 1.5 s. The ability of synchronous dq control to achieve zero steady-state error can be clearly seen
under balanced (before t = 1.5 s) and unbalanced (right before t = 2.4 s) load conditions. During
load change, the voltage experiences transients within ± 5%.
145
Fig. 5.16: Load characteristics under synchronous dq-frame controllers.
Fig. 5.17: Load-side three-phase voltages and currents under synchronous dq-frame controllers.
146
Although synchronous dq-frame-based controllers, shown in Fig. 5.10, provide very satisfactory
performance due to achieving zero steady-state amplitude and phase errors, the intensive
computations, required to extract the symmetrical components from the measured asymmetrical
voltages, complicates the control approach and introduce time delays in the feedback signal. A
simpler control scheme with no time delay can be obtained by implementing stationary abc-frame
PI controllers as shown in Fig. 5.18. In this scheme, the modulating signals of the CSI’s three legs
are separately produced to balance the voltage at load bus under unbalanced conditions.
Nevertheless, three-phase stationary frame PI controllers, in general, result in steady-state
amplitude and phase errors when regulating ac quantities [154]. This can be seen in Fig. 5.19 where
a steady-state three-phase load voltage under a load condition given in Table 5.4 is illustrated.
Although the stationary abc-frame PI controller provides very good voltage regulation under
unbalanced load (see Fig. 5.19(a)), the phase voltages still exhibits some imbalance (see
Fig. 5.19(b)). The VUF is 0.88%. Moreover, Fig. 5.19(c) shows a phase shift between the regulated
voltage and its reference. On the contrary, voltage responses regulated by the synchronous dq
controller are almost identical to the reference in magnitude and phase. The VUF is 0.19% which
is much less than that of stationary frame PI controller and far below the permissible limit of 1%.
Fig. 5.18: Load-side stationary abc frame control scheme using PI controllers.
147
Fig. 5.19: Performance comparsion of abc-frame and dq-frame PI controllers.
5.7
Fault Analysis of the CSI-based WECS
As highlighted in chapter 1 (section 1.2.4), one of the advantages of CSI-based systems over
VSI-based systems is that the dc-link current is directly regulated; hence, the ac-side current may
not face a sharp rise during faults on the ac side of the inverter. This section studies the behaviour
of the proposed system under different types of faults in the ac side of the inverter. In order to
conduct this study, protection breakers, which typically open during faults to protect the inverter
and the loads, are assumed to be deactivated. The neutral of the ∆/𝑌𝑛 transformer is usually
grounded for safety reasons. Three types of fault, i.e., Single Line to Ground (SLG), Line to Line
(LL), and Three-Phase to Ground (TPG) faults are considered.
In Fig. 5.20, the Wind Turbine Generation (WTG) subsystem, the Energy Storage (ES)
subsystem, and the CSI-Load subsystem are controlled by the MPPT controller, shown in Fig. 5.9,
the dc-link current controller, shown in Fig. 5.3, and the load-side controller shown in Fig. 5.10,
respectively.
148
DC-link Control
(Fig. 5.3)
DC-Link
ES Subsystem
WTG Subsystem
→
Current-Source
Inverter
Load-Side
control
(Fig. 5.10)
→
∆ /Yn
x
f2
x
f1
→
3 phase
Unbalanced
RL load
n
Ci-Filter
MPPT Control
(Fig. 5.9)
Fig. 5.20: Fault points on the proposed WECS.
Before applying a fault, the system is operating at a steady state operating point corresponding
to rated wind speed and is supplying a 3-phase unbalanced RL load (20 kW at 380V/220V,
pf=0.89). The values of resistances and inductances for the unbalanced load are 4.8 Ω/3.7 mH,
3.66 Ω/14.56 mH, and 6.93 Ω/3.57 mH for phases a,b and c, respectively. Fig. 5.21(a) displays
the system responses to an SLG fault applied to phase a on the secondary side of the transformer
(point f1 in Fig. 5.20) from t = 0.1 s to 0.2 s. Due to the fault, voltage 𝑣𝐿𝑎 drops to zero. The dclink current (𝑖𝑑𝑐 ) experiences a small increase in the ripples which does not exceed 8% of the
current. The dc-link current controller tracks the reference current in less than 15 ms after the fault
is cleared at 0.2 s. Since phase a is short circuited to ground, no current is supplied to phase a of
the load during the fault. The CSI’s unfiltered terminal current of phase a (𝑖𝑜𝑎 ) undergoes over
modulation, resulting in low order harmonics. Hence, the currents delivered to the other healthy
load phases are no longer sinusoidal. This test clearly demonstrates the inherent over-current
protection built in CSI that limits the currents on both dc and ac sides of the inverter. Moreover,
the stable control of the dc-link current keeps the WTG subsystem at its normal operation,
generating the rated wind power (i.e., 1 pu). During the fault, the load-side demand is reduced by
70% and hence the excess wind power is absorbed by the battery. Similar conclusions are made
for LL and TPG faults, shown in Fig. 5.21(b) and (c), respectively. In all types of fault, the dc-link
current is limited, resulting in a limited magnitude of the ac current delivered to the load.
149
Fig. 5.21:System performance during (a) SLG (b) LL (c) TPG faults at secondary side of the ∆/𝑌𝑛
transformer (𝑣𝐿 𝑎𝑏𝑐 : Load three-phase voltage, 𝑖𝑑𝑐 : DC-link current, 𝑖𝑜𝑎 : CSI phase-a unfiltered
output current, 𝑖𝐿 𝑎𝑏𝑐 : Load three-phase current, and 𝑃: System powers).
Harmonic spectrums of the CSI terminal unfiltered and filtered currents (𝑖𝑜𝑎 and 𝑖𝑎 ), before and
during the SLG fault, are shown in Fig. 5.22. From Fig. 5.22(a) and (b), it is clear that no major
low-order harmonics exists before fault and the CSI output capacitor filters the switching
harmonics of 𝑖𝑎 . On the other hand, as shown in Fig. 5.22(c) and (d), low-order harmonics are
presented in 𝑖𝑜𝑎 during fault and cannot be filtered by the CSI output capacitor as it is mainly
designed for high-order harmonics attenuation.
150
Fig. 5.22: Harmonic Spectrum of CSI’s ac terminal and filtered currents before and during SLG
fault.
Due to the linear relationships between the voltages on the primary and secondary sides of the
Delta/Star transformer, system responses to LL fault applied on the primary side (point f2 in
Fig. 5.20) are similar to those of SLG fault applied on the secondary side of the transformer.
Likewise, system responses to three-phase (TP) fault applied on the primary side of the transformer
are similar to those of TPG on the secondary side of the transformer; hence, these responses are
not shown here.
In Fig. 5.23, the performance of a VSI-based standalone WECS and a CSI-based standalone
WECS with comparable ratings (i.e., 20 kW) are compared when an SLG fault is applied at t =
0.1s on the secondary side of the transformer. In VSI-based WECS, the function of the dc-bus
151
controller is to regulate the dc-side voltage, whereas in CSI-based WECS, the dc-bus controller
regulates the dc-side current. From the figure, one can observe that the dc-side current of VSI
(average value) is allowed to vary in a wide range and reach almost 7 pu (In practice, a current
limiter is employed to limit this current). On the contrary, the dc-side current in CSI has very
limited variations (within 1.08 pu).
Fig. 5.23: Inverter dc-side current of VSI-and CSI-based WECS during an SLG fault.
5.8
Dump Load-Less Version of the Proposed CSI-based WECS
In the proposed CSI-WECS, shown in Fig. 5.1, the dc-link current is controlled using power
management between the battery and dump load (see Fig. 5.3). The battery is charged or
discharged based on the power mismatch between the wind generation and the demand. If the
battery state-of-charge (SoC) reaches its upper limit, the excess power generated from the wind is
dissipated in the dump load (see Fig. 5.13).
In this section, the system of Fig. 5.1 is modified by removing the dump load and implementing
wind power curtailment through the generator-side converter when the combination of load and
energy storage is not capable of absorbing the extra wind power. To achieve this objective, the
MPPT/curtailment control scheme shown in Fig. 5.24 is implemented.
152
MPPT Loop
-+
PI
Controller
1
PWM
Generator
AND
SoC%
75%
PI
Controller
2
+
Curtailment Loop
Fig. 5.24: MPPT/Curtailment Control Scheme.
The MPPT/Curtailment control scheme contains two control loops: generator-speed loop and
dc-link current control loop. The generator-speed loop controls the generator in order to extract
the maximum power in normal mode of operation (i.e., MPPT mode). This loop is identical with
the MPPT control loop shown in Fig. 5.9 and described in subsection 5.4.1. If the wind power,
generated under MPPT mode, exceeds the demand and the battery SoC reaches 75%, the dc-link
current controller, shown in Fig. 5.3 and described in subsection 5.3.1, will block the charging
mode of the battery and hence 𝑖𝑑𝑐 will exceed 𝑖𝑑𝑐,𝑟𝑒𝑓 , assuming a new value depending on the
level of supply-demand power mismatch. In order to prevent such a scenario, the function of buck
converter, as shown in Fig. 5.24, is switched from shaft speed controller to current regulator so
that the power absorbed form the generator is adjusted to maintain 𝑖𝑑𝑐 at 𝑖𝑑𝑐,𝑟𝑒𝑓 . The mechanism
of switching between shaft speed control (MPPT control mode) and dc-link current regulation
(curtailment control mode) is carried out by employing an automatic software switch, which by
default assumes position 1, corresponding to MPPT mode. If battery SoC exceeds its upper limit,
and 𝑖𝑑𝑐 exceeds 𝑖𝑑𝑐,𝑟𝑒𝑓 plus the permitted ripple, the switch assumes position 2, implying
curtailment mode. It should be noted that the SoC condition for transition from speed control to
153
current regulation is set to the upper limit (𝑆𝑜𝐶 ≥ 75%), assuming a very small mechanical
system inertia and hence possibility of almost instantaneous transition. In practice, however, the
system inertia should be considered by setting the SoC condition slightly less than its upper limit.
To study the performance of the MPPT/Curtailment control scheme for the dump load-less
SCIG-CSI-WECS, the 20kW standalone WECS, described in Appendix A (Table A.2), is simulated
under variable wind speed and unbalanced three-phase load, according to the wind speed and load
profiles given in Table 5.5. The rated values of three-phase load powers are 𝑃+ = 20𝑘𝑊, 𝑄 + =
−
−
10 𝑘𝑉𝑎𝑟, 𝑄𝑝𝑑
= −5.08𝑘𝑉𝑎𝑟, and 𝑄𝑝𝑞
= −6.04𝑘𝑉𝑎𝑟; hence, the load unbalance factor is 35%.
Simulation results are shown in Fig. 5.25. In order to verify the successful operation of the
MPPT/Curtailment control scheme, the initial SoC is set very close to the upper limit of 75%. At
t = 0 s, the system is exposed to rated wind speed of 12 m/s. Since no load is connected, the dc∗∗
link current reference is set to the 𝑖𝑑𝑐
required for the battery to absorb the entire generated power.
∗
At t = 0.5 s, 20% of the rated load is to be supplied, but the excess power still exceeds 𝑃𝑏𝑎𝑡
; hence,
∗∗
𝑖𝑑𝑐 is still regulated at 𝑖𝑑𝑐
. From t = 1 s to t = 8 s, the difference between the wind power and
∗
∗
demand is below 𝑃𝑏𝑎𝑡
; thus, the dc-link current is regulated at the 𝑖𝑑𝑐
. Throughout the entire
simulation interval, except from t = 4.7 s to t = 6.07 s, battery SoC is below 75%. Therefore, the
generator controller tracks the 𝜔𝑔,𝑜𝑝𝑡 in order to achieve MPPT. The battery bank either absorbs
or delivers power according to the difference between wind power and demand. When battery SoC
reaches its upper limit at t = 4.7 s, the battery cannot be charged further; hence, the generator-side
controller is switched to curtailment mode, where the shaft speed deviates from 𝜔𝑔,𝑜𝑝𝑡 in order to
limit the generated power to the demand level. At t = 5.5 s, the load is decreased further and the
curtailment controller readjusts generated power accordingly. The curtailment mode continues
until t = 6.07s when the demand exceeds the generated wind power, forcing the battery to supply
the deficit, and hence battery SoC drops below its upper limit. As a result, the generator controller
reactivates the MPPT control mode. The dc-link current is maintained at the desired value under
all conditions with current ripples within the typical limit of 15%. It can be seen that the dc-link
current ripple during the curtailment interval is slightly higher than those during MPPT intervals.
154
Table 5.5: Wind speed and load conditions for simulation results of Fig. 5.25.
Time (s)
0
0.5
1.0
1.5
2.0
2.5
3.0
5.5
6.0
Wind speed (m/s)
12
12
12
11
11
11
10
10
10
% of rated load
0
20
100
100
110
60
40
20
70
DC-link current reference (A)
∗∗
𝑖𝑑𝑐
= 51
∗∗
𝑖𝑑𝑐 = 51
∗
𝑖𝑑𝑐 = 55.5
∗
𝑖𝑑𝑐
= 55.5
∗
𝑖𝑑𝑐 = 61
∗
𝑖𝑑𝑐 = 33.3
∗
𝑖𝑑𝑐
= 22.2
∗
𝑖𝑑𝑐 = 𝑖𝑑𝑐,𝑚𝑖𝑛 = 22.2
∗
𝑖𝑑𝑐
= 39
Fig. 5.25: System behavior under MPPT/Curtailment control.
155
Because the wind power-shaft speed curve is nearly parabolic (see Fig. 4.3), there exists two
operating shaft speeds (one lower and one higher than the optimum speed) at which wind power
curtailment can be achieved. During curtailment interval, the buck converter regulates 𝑖𝑑𝑐 by
limiting the converter input current (𝑑𝑏 𝑖𝑑𝑐 ) by reducing buck duty ratio (𝑑𝑏 ); hence, increasing
shaft speed. As a result, as noticed in Fig. 5.25, the generator operates at the speed higher than the
optimum speed. However, in order to reduce the mechanical stress on the shaft, it is desirable for
the turbine shaft to operate at the lower speed that results in the targeted power. One way to achieve
this is to use the generator–side converter (buck converter) to drive the machine into the low speed
(obtained from wind power-shaft speed look-up table based on Fig. 4.3) that results in generating
power as required by the load. However, this requires a prior knowledge of the short-term load
demand which is time-varying and difficult to identify.
5.9
Summary
In this chapter, the closed-loop control system of the proposed CSI-based WECS was developed.
The open-loop system is stable around operating points 1 and 3 but not 2; therefore, the objective
of the closed-loop controller is to stabilize the system and track a class of desired references for
the dc-link current and turbine speed, and control the voltage magnitude and frequency at the load
bus.
Considering a fast and stable control for the dc-link current, the load-side and the generatorside converters are decoupled; therefore, the wind turbine system was decomposed into three
subsystem, allowing the classical control theories for a SISO system to be applied.
An efficient control algorithm for the reduced H-bridge converter was developed to regulate the
dc-link current through power management between battery bank and dump load. The fact that
CSI’s dc-link current can be reduced under light-load conditions was utilized to reduce dc-link and
CSI power losses. However, this advantage could not be benefited from under certain conditions
where a high dc-link current was required by the battery bank to absorb high excess power. The
generator’s shaft speed was controlled by the buck converter to extract maximum available wind
power. The load-side voltage magnitude was regulated at the desired value by controlling the
current-sourced inverter based on concept of positive and negative sequence decomposition. The
156
parameters of the system PI controllers were tuned based on certain time and frequency domain
specifications.
The performance of the closed-loop control system was verified under various wind speed as
well as balanced/unbalanced load conditions, with different dynamic and steady-state
characteristics. In all simulation results, a fast dc-link current control was demonstrated.
Irrespective of load dynamic and steady-state characteristics, the synchronous dq frame controllers
maintained balanced voltage at the load bus, with limited disturbances during transients.
Satisfactory performance of the MPPT controller under fixed and turbulent wind speed profiles
was demonstrated. In the process, negative effects of wind speed rate of change and system inertia
on MPPT were observed.
The performance of the proposed system under faults on the ac side was examined. The built-in
inherent over-current protection that limits the currents on both dc and ac sides of the inverter, was
demonstrated. This is an exclusive feature specific to CSI, thanks to the direct control of dc-link
current.
A dump load-less version of the CSI-based system was proposed. The dump load was removed
and the generator-side converter was used as a shaft-speed controller during MPPT mode and as a
dc-link current regulator during curtailment mode. Successful operation of the MPPT/Curtailment
controller under different wind speed and load conditions was demonstrated through simulation.
The core material of this chapter has appeared in the published journal paper [155] and has been
included in the accepted conference paper [156].
157
Chapter 6
Conclusions, Contributions, and Future Work
In this thesis, a small-scale, standalone wind energy conversion system featuring SCIG, CSI
and a novel storage integration scheme was proposed as an attractive renewable energy solution
for off-grid communities.
In the following sections, the work presented in this thesis is summarized, the main contributions
are highlighted and some items for future work are suggested.
6.1
Summary and Conclusions
This dissertation focuses on small-scale (< 100 kW) wind turbines as very attractive renewable
energy source for off-grid applications, especially in remote communities.
In chapter 1, a review of variable-speed wind energy conversion systems (WECSs) from
generator and converter viewpoints was provided. Amongst the power converters which have been
employed or have the potential to be employed in WECSs, voltage-sourced inverter (VSI)-based
power electronic converters are the dominant topology in both large- and small-scale WECSs. On
the other hand, current-sourced-inverter (CSI)-based power electronic converters have mainly
been adopted in Megawatt-level on-grid WECSs. In order to assess the possibility of employing a
CSI-based power electronic converter in off-grid low power wind turbines, a comparison between
Pulse-width modulated CSI and VSI for small-scale standalone WECS, based on reliability, cost,
efficiency, and protection requirements was conducted. Even though based on the comparison,
CSI offers high potentials for small-scale off-grid WECS, its performance in such an application
has never been investigated. Therefore, CSI-based power converter was selected as the base of this
research. A brief review on the energy storage technologies that are feasible for wind energy
integration was also conducted. As a low-cost mature storage technology, offering satisfactory
performance, lead acid battery was selected for storage purposes.
Motivated by lack of a comprehensive and convincing approach to selection of the right
generator for a standalone wind turbine, a thorough study, considering all possible options, was
conducted in chapter 2. Amongst the different generator types considered in the study, three wind
generator configurations, namely geared-drive squirrel-cage induction generator (geared-SCIG),
158
direct-drive permeant-magnet synchronous generator (gearless-PMSG) and direct-drive
permanent-magnet induction generators (gearless-PMIG) were identified as the most promising
technologies for standalone wind turbines. Therefore, the three configurations were qualitatively
compared with one another in terms of reliability, cost, efficiency, excitation requirements, control
simplicity, construction simplicity, and noise level. The system based on geared-SCIG was shown
to be the most appropriate scheme for a small-scale standalone WECS, supplying a remote area.
In chapter 3, the three CSI-based topologies, validated in the literature for on-grid WECS, were
evaluated for off-grid WECS application. The first topology (i.e., diode bridge rectifier - PWM
CSI) provides no control over generator and dc-link current; hence, it is not applicable for an offgrid WECS. The second topology (i.e., diode bridge rectifier - buck converter – PWM CSI) makes
the generator control achievable. The main drawback of this topology is that it causes high
harmonic distortion in the generator stator winding current. This drawback is avoided by
employing current-sourced rectifier in the third topology (i.e., back-to-back PWM CSC), leading
to considerable improvement in the generator performance, but at a higher cost and with higher
complexity in control. As a simple and low-cost configuration, offering satisfactory performance,
the second topology (i.e., diode bridge rectifier - buck converter – PWM CSI) was selected as the
base for further study in this research. The duality of CSI and VSI topologies was taken advantage
of to come up with a novel scheme for the integration of a battery-based energy storage system
with the proposed CSI-based WECS. An H-bridge, with reduced number of switches and diodes,
was employed as the interfacing converter. In the proposed system, the dc-link inductor is shared
among the generator-side, storage-side and load-side converters, resulting in reduction of the
system size, weight and cost.
Chapter 4 focused on developing the dynamic mathematical model of the proposed CSI-based
wind turbine system. Detailed models of the system components were presented. A reduced-order
generic load model suitable for balanced/unbalanced loads was developed. By combining the state
equations of the system components, as well as the generic load model, the overall model of the
system was described by seven inputs, six outputs, and twenty state variables. A small-signal
model of the system was developed around three operating points. The eigenvalue analysis of the
small-signal model showed that the open-loop system is locally stable around operating points 1
and 3, but not 2. Based on Gramian matrices, the system was found completely controllable at the
159
three operating points, completely observable at operating points 1 and 3, but not completely
observable at operating point 2.
Chapter 5 presented the design of the closed-loop control system. An efficient control algorithm
for the reduced H-bridge converter was developed to regulate the dc-link current, considering the
minimum dc-link current requirements for both generator and load sides. The generator shaft speed
was controlled to extract maximum available wind power. The load-side voltage was regulated at
the desired value by synchronous dq frame controllers, resulting in zero-steady-state error. The
parameters of the PI controllers were tuned using Matlab/Simulink control tools based on certain
time- and frequency-domain specifications. The performance of the closed-loop control system
was verified under various wind speed and balanced/unbalanced load conditions of different
dynamic and steady state characteristics. The performance of the proposed system under ac-side
faults was also examined. The inherent over-current protection capability, built in CSI, that limits
fault currents on both dc and ac sides of the inverter, was demonstrated. Finally, a dump load-less
version of the CSI-based system was proposed and the successful operation of the
MPPT/Curtailment controller was demonstrated.
6.2
Contributions
The main contributions of this research are as follows:
1) A critical analytical study of standalone wind energy conversion systems from
generator viewpoint was conducted. This study has resulted in a journal publication [102]
based on the review of 148 references, and serves as an excellent source of information
and a helpful guide for researchers and practitioners involved in standalone wind turbine
systems.
2) A small-scale, standalone wind energy conversion system featuring SCIG, CSI and a
novel energy storage integration scheme was proposed. The dc-link inductor shared by
three power electronic converters was systematically designed. Switching and average
Simulink models of the system were developed.
3) An efficient power management algorithm to maintain supply-demand balance
through direct control of the dc-link current was developed. The fact that the CSI’s dc-
160
link current can be reduced at light load was utilized to improve the dc-link and CSI
efficiencies.
4) A reduced-order generic load model suitable for balanced/unbalanced load
conditions was developed. The model is capable of emulating actual loads of different
transient and steady-state characteristics and is appropriate for simulation studies of
stability and dynamic performance of a standalone system.
5) A dump load-less version of the developed CSI-based WECS was proposed.
The work presented in this thesis has been published in [102],[155] and [156].
6.3
Future Work
The following items are suggested for future research.
1) Experimental Verification
A laboratory experimental platform of the proposed wind energy conversion system can be
set up and used to verify the validity of the simulation results.
2) A transformer-less version of the proposed CSI-based WECS.
Although the Delta/Star isolation transformer provides a zero-sequence current path for
unbalanced load, it is bulky and costly. In order to eliminate the need for the transformer, a path
for the load neutral current must be provided by other means. A simple and low-cost solution was
implemented in [157] for a CSI-based grid connected PV system by connecting the neutral
terminal of the grid to the midpoint of the PV-side smoothing dc capacitor. For the proposed
WECS, however, such a configuration will generate significant voltage ripple in the generatorside dc capacitor, causing more toque ripples and reducing the quality of the generated power.
3) Sensor-less control of SCIG in the proposed WECS
The MPPT control loop requires an anemometer to measure wind speed and a position sensor to
measure the rotor speed. However, mechanical sensors increase the capital cost, as well as
maintenance costs, and reduce the reliability of the turbine system. A sensor-less control algorithm
for the SCIG-WECS will bring great benefits to the system.
161
Appendix A
Parameters and Operation Conditions of the Systems used in Simulation
Table A.1 and Table A.2 give the parameters and operating conditions for the 30 kW-CSC-based
WECS and the 20kW-CSI-based WECS, respectively.
Table A.1: Parameters and operating condition of the 30kW-CSC-SCIG WECS.
Requirement
Wind Turbine
Topology 1
Topology 2
Cut-in wind speed : 5 m/s
Rated wind speed : 12 m/s
Cut-out wind speed : 20 m/s
Rated power = 30 kW
Topology 3
3 stages
Ratio: 5:1 per stage
Gearbox
Squirrel cage Induction
30 kW, 320V, 4 pole , 60 Hz, 1812 rpm,
63A, 158 Nm, pf =0.86
Generator
No-Load Excitation 400 µF (0.45 pu)
Capacitor
400 µF (0.45 pu)
None
180 µF (0.2 pu)
Input filter (𝐶𝑟 )
None
None
Generator-side
converter
Diode bridge rectifier
Diode bridge rectifier
PWM- IGBT-CSR
+ dc/dc buck converter, 𝑓𝑠 = 5.1 kHz
𝑓𝑠 = 2.1 kHz
DC Choke
18 mH (2.3 pu)
6.2 mH (0.8 pu)
Rated dc current
Inverter
84A
84A
84A
PWM-IGBT-CSI, 𝑓𝑠 = 5.1 kHz
125 µF (0.14 pu)
Output Filter (𝐶𝑖 )
Rated Load
3.2 mH (0.41 pu)
3-phase balanced RL load (33.5 kVA at 380V/220V, pf = 0.89)
162
Table A.2: Parameters and rated operating conditions of the 20kW-CSI-SCIG-WECS.
𝑃𝑚 = 23 𝑘𝑊
Turbine and drive train
𝑟 = 6.25 𝑚
𝜌 = 1.225 𝑘𝑔/𝑚3
𝑣𝑤𝑟𝑎𝑡𝑒𝑑 = 12 𝑚/𝑠
𝑣𝑤𝑐𝑢𝑡−𝑜𝑢𝑡 = 20 𝑚/𝑠
𝜔𝑚 = 14.5 𝑟𝑝𝑚
𝜆𝑜𝑝𝑡 = 8.1
𝐶𝑝𝑚𝑎𝑥 = 0.48
𝐽𝑚 = 0.23 𝑘𝑔. 𝑚2
𝐾𝑠𝑒 = 4.4 𝑥 103 𝑁𝑚/𝑟𝑎𝑑
(0.3 𝑝𝑢 of 𝑇𝑚 /𝛿𝜃)
𝑃𝑔 = 20 𝑘𝑊
𝑇𝑒 = 105 𝑁. 𝑚
𝑅𝑠 = 0.1325 Ω
𝐿𝑚 = 63.69 𝑚𝐻
𝛽 = 0°
Gear box : 3 stages
Ratio : 5:1 per stage
(𝑛𝑔𝑒𝑎𝑟 = 125)
𝐷𝑠𝑒 = 1.6 𝑥 103 𝑁𝑚/𝑟𝑎𝑑/𝑠
(1 𝑝𝑢 of 𝑇𝑚 / 𝑝𝑢 𝛿𝜔)
4 pole-Self-excited induction generator
𝑉𝑠𝑡𝑎𝑡𝑜𝑟
𝑓 = 60 𝐻𝑧
= 460 𝑉𝐿𝐿 𝑟𝑚𝑠
𝑝𝑓𝑔 = 0.86
𝐽𝑔 = 0.07 𝑘𝑔. 𝑚2
𝐿𝑙𝑠 = 3.7 𝑚𝐻
𝐹 = 0.0576 𝑁. 𝑚. 𝑠
𝑣𝑤𝑐𝑢𝑡−𝑖𝑛 = 5 𝑚/𝑠
𝑅𝑟′
= 0.1242 Ω
𝐶𝑔 = 140 µ𝐹/𝑝ℎ𝑎𝑠𝑒
(0.48 pu)
𝜔𝑔 = 1812 𝑟𝑝𝑚
𝐼𝑠𝑡𝑎𝑡𝑜𝑟 = 29 𝐴𝑟𝑚𝑠
𝐿′𝑙𝑟 = 3.7 𝑚𝐻
𝐶𝑑𝑐 = 100 µ𝐹
(0.34 pu)
Lead acid battery bank ( 408V 300Ah) at 100% SoC
(Obtained by combining 34 units in series, each 12V 300Ah, 𝐸𝑜 = 12.4 𝑉, 𝐾𝐸 = 1.1 𝑉 , 𝑅𝑜 =
1 𝑚Ω at 25°C).
𝐴𝑜 = 5 𝑚Ω
𝐷𝐿 = 60 𝑘𝑊ℎ
𝑢=1
𝑘𝑇 = 1
𝑅𝐿 = 50 𝑚Ω
𝑅𝑐 = 10 𝑚Ω
𝐿𝑏 = 6.6 µ𝐻
𝐶𝑏 = 3.7 𝑚𝐹
𝐿𝑑𝑐
DC-Link and output C-filter
= 16.2 𝑚𝐻 (0.67 pu)
𝑖𝑑𝑐 = 55.5 𝐴
𝑓𝑠 = 5.1 𝑘𝐻𝑧
𝑉𝑜 = 310.3 𝑉
𝑃𝑜+ = 20𝑘𝑊
𝑌𝑃+ = 0.1385 𝑊/𝑉 2
Extended Generic Load Model (rated load)
𝜔𝐿𝑜 = 377𝑟𝑎𝑑/𝑠
𝛼𝑝 = 𝛼𝑞 = 2
𝑄𝑜+ = 10 𝑘𝑉𝑎𝑟
𝑌𝑄+ = 0.0692 𝑉𝑎𝑟/𝑉 2
−
𝑄𝑝𝑑𝑜
= −5.08𝑘𝑉𝑎𝑟
𝑌𝑄−𝑝𝑑 = −0.0352 𝑉𝑎𝑟/𝑉 2
163
𝐶𝑖 = 125 µ𝐹/𝑝ℎ𝑎𝑠𝑒
(0.42 pu)
𝛽𝑝 = 𝛽𝑞 = 0
−
𝑄𝑝𝑞𝑜
= −6.04𝑘𝑉𝑎𝑟
−
𝑌𝑄𝑝𝑞 = −0.0418 𝑉𝑎𝑟/𝑉 2
Appendix B
Design of Battery-Side L-C filter
Resonant frequency and damping ratio of the L-C filter are given by (B.1) and (B.2),
respectively, where 𝑅𝐿 and 𝑅𝐶 represent the equivalent series resistances (ESRs) of filter inductor
and capacitor, respectively.
1
𝑓𝑟 = 2𝜋√𝐿
𝜉=
𝑅𝐿 +𝑅𝑐
2
𝑏
𝐶𝑏
𝐶
√𝐿𝑏
𝑏
(B.1)
(B.2)
The frequency of the dominant unwanted component to be suppressed is the ripple frequency of
the current on the battery-side of the reduced H-bridge, which is the same as the switching
frequency (𝑓𝑠 ). Typically, 𝑓𝑟 is selected to be one decade below 𝑓𝑠 (𝑓𝑟 = 0.1 𝑓𝑠 ) and the damping
ratio is set equal to 1/√2. Solving (B.1) and (B.2) yields 𝐿𝑏 and 𝐶𝑏 . It should be noticed that the
ripple frequency of the current on the battery-side of the reduced H-bridge converter is effectively
doubled under PWM unipolar voltage switching scheme.
Based on (B.1) and (B.2) and the parameters of the 20kW WECS given in Table A.2 (i.e., 𝑅𝐿 =
50 𝑚Ω, 𝑅𝑐 = 10 𝑚Ω, 𝑓𝑠 = 5.1 𝑘𝐻𝑧) the battery-side LC filter parameters are obtained as
𝐿𝑏 = 6.6 µ𝐻 and 𝐶𝑏 = 3.7 𝑚𝐹.
164
Appendix C
Switching States of Current Source Inverter
The schematic diagram of a three-phase CSI, feeding a three-phase load, is shown in Fig. C.1.
S1
S3
S5
a
↑
Threephase
load
b
c
S2
S4
S6
C-Filter
Fig. C.1: Three phase CSI feeding three-phase load.
In order to satisfy KVL and KCL in three-phase CSI, two constraints for CSI switching must be
always met. The dc link of CSI acts as a current source and cannot be open-circuited. Thus, at least
one top switch and one bottom switch must be on at any instant of time. On the other hand, in
order to produce defined three-phase current waveforms at the ac side, at most one top switch and
one bottom switch must be on at any instant of time. In other words, one and only one top switch
and one and only one bottom switch must be on at any instant of time. This statement applies to
current source rectifier as well. In practice, the switching of devices in a converter is not
instantaneous. Therefore, an overlap between the on-periods of outgoing and incoming switches
is necessary to prevent interruption of the dc-link current. The filter capacitors on the ac side will
facilitate commutation of switches, besides filtering the switching harmonics.
Based on the constraints mentioned above, there are nine valid switching states in the operation
of CSI, as shown in Table C.1. States 1 to 6 are active states, where current flows from dc side to
the load. On the other hand, states 7 to 9 are zero states, where the dc link current (𝑖𝑑𝑐 ) freewheels
165
through one of the inverter legs. Thus, no power transfers from dc side to load during these states.
They are also known as shoot-through states. The inverter must change state according to a certain
pattern in order to generate a set of ac line current. The resulting ac currents consists of discrete
values, that are 𝑖𝑑𝑐 , 0 or −𝑖𝑑𝑐 , as noticed in Table C.1.
Table C.1: Switching States for a Three-Phase CSI.
Active
States
Zero States
𝒊𝒐𝒃
𝒊𝒐𝒄
𝑖𝑑𝑐
−𝑖𝑑𝑐
0
𝑆1&𝑆6
𝑖𝑑𝑐
0
−𝑖𝑑𝑐
3
𝑆3 &𝑆2
−𝑖𝑑𝑐
𝑖𝑑𝑐
0
4
𝑆3 &𝑆6
0
𝑖𝑑𝑐
−𝑖𝑑𝑐
5
𝑆5 &𝑆2
−𝑖𝑑𝑐
0
𝑖𝑑𝑐
6
𝑆5 &𝑆4
0
−𝑖𝑑𝑐
𝑖𝑑𝑐
7
8
9
𝑆1&𝑆2
𝑆3 &𝑆4
𝑆5 &𝑆6
0
0
0
0
0
0
0
0
0
State no.
ON Switches
1
𝑆1&𝑆4
2
𝒊𝒐𝒂
166
Appendix D
Overall Dynamic Model Equations of the Proposed WECS
The overall dynamic model of the proposed wind turbine system contains equations of the
mechanical drive train, self-excited induction generator, battery-side LC filter, current-source
inverter supplying the extended generic load model, and dc-link inductor. The equations for each
component, as well as the overall system, are derived in the following sections.
D.1 Equations of the 2-Mass Drive Train
From equations (4.13)-(4.15), one gets:
𝑑
𝑑𝑡
𝑑
1
𝜔𝑚 = 𝐽 [𝑇𝑚 − 𝐷𝑠𝑒 𝛿𝜔 − 𝐾𝑠𝑒 𝛿𝜃]
𝑚
1
1
𝜔 =𝐽 [𝑛
𝑑𝑡 𝑔
𝑔
𝑔𝑒𝑎𝑟
𝐷𝑠𝑒 𝛿𝜔 + 𝑛
1
𝑔𝑒𝑎𝑟
𝐾𝑠𝑒 𝛿𝜃 − 𝑇𝑒 ]
𝜃𝑔
𝜔𝑔
𝑑(𝛿𝜃) 𝑑
= (𝜃𝑚 −
) = 𝛿𝜔 = 𝜔𝑚 −
𝑑𝑡
𝑑𝑡
𝑛𝑔𝑒𝑎𝑟
𝑛𝑔𝑒𝑎𝑟
(D.1)
(D.2)
(D.3)
where 𝑇𝑚 , 𝑇𝑒 , and 𝐶𝑝 , given in (D.4)-(D.6), are the turbine mechanical torque, the electromagnetic
torque of the induction generator, and the turbine’s performance coefficient, respectively.
𝜌𝜋𝑟 2
𝑇𝑚 = 2 𝜔 𝑣𝑤3 𝐶𝑝 (𝜆, 𝛽)
(D.4)
𝑚
𝑇𝑒 =
3𝑃𝐿𝑚
2
′
′
(𝑖𝑞𝑠 𝑖𝑑𝑟
− 𝑖𝑑𝑠 𝑖𝑞𝑟
)
116
𝐶𝑝 (𝜆, 𝛽) = 0.5176 ( 𝜆 − 0.4𝛽 − 5) 𝑒
𝑖
−
21
𝜆𝑖
(D.5)
+ 0.0068𝜆
(D.6)
Equations (D.7) and (D.8) give relations for 𝜆𝑖 and 𝜆.
1
𝜆𝑖
1
0.035
= 𝜆+0.08𝛽 − 𝛽3 +1
𝜆=
𝜔𝑚 𝑟
𝑣𝑤
(D.7)
(D.8)
For the wind turbine system under study, it is assumed that the rotor pitch angle is fixed at zero
(i.e., 𝛽 = 0). Hence, equation (D.7) becomes
167
1
𝜆𝑖
1
= 𝜆 − 0.035
(D.9)
Substituting (D.9) in (D.6) yields
21
60
𝐶𝑝 (𝜆, 0) = ( 𝜆 − 4.69) 𝑒 (− 𝜆 +0.74) + 0.0068𝜆
(D.10)
Substituting 𝜆 from (D.8) in (D.10) yields
𝐶𝑝 (𝜆, 0) = (
21 𝑣𝑤
60 𝑣𝑤
𝜔𝑚 𝑟
(−
+0.74)
− 4.69) 𝑒 𝜔𝑚 𝑟
+ 0.0068
𝜔𝑚 𝑟
𝑣𝑤
(D.11)
Substituting 𝐶𝑝 from (D.11) in (D.4) gives
𝜌𝜋𝑟 2
60 𝑣𝑤
𝑇𝑚 = 2 𝜔 𝑣𝑤3 [( 𝜔
𝑚
𝑚
𝑟
− 4.69) 𝑒
(−
21 𝑣𝑤
+0.74)
𝜔𝑚 𝑟
+ 0.0068
𝜔𝑚 𝑟
𝑣𝑤
]
(D.12)
Substituting (D.12) and (D.5) in (D.1) and (D.2), respectively, yields
𝑑
1
3
30 𝑣𝑤
𝑚
𝑚
𝜔 = 𝐽 [𝜌 𝜋𝑟 2 ( ( 𝜔2
𝑑𝑡 𝑚
𝑟
−
2
2.35 𝑣𝑤
𝜔𝑚
)𝑒
(−
21 𝑣𝑤
+0.74)
𝜔𝑚 𝑟
𝜔𝑟
𝐷𝑠𝑒 (𝜔𝑚 − 𝑃 𝑛
𝑔𝑒𝑎𝑟
𝑑
𝑑𝑡
𝑃
𝜔𝑟 = 𝐽 [ 𝑛
𝑔
1
𝑔𝑒𝑎𝑟
𝐾𝑠𝑒 𝛿𝜃 + 𝑛
1
𝑔𝑒𝑎𝑟
𝐷𝑠𝑒 (𝜔𝑚 −
+ 0.0034 𝑟 𝑣𝑤 )𝑣𝑤 − 𝐾𝑠𝑒 𝛿𝜃 −
)]
𝜔𝑟
𝑃 𝑛𝑔𝑒𝑎𝑟
)−
3𝑃𝐿𝑚
2
′
′
(𝑖𝑞𝑠 𝑖𝑑𝑟
− 𝑖𝑑𝑠 𝑖𝑞𝑟
)]
(D.13)
(D.14)
The mechanical angular speed of the induction generator shaft (𝜔𝑔 ) is related to the electrical
angular speed (𝜔𝑟 ) through the pole pair (𝑃) of the machine, i.e., 𝜔𝑔 = 𝜔𝑟 /𝑃. Thus, from (D.3),
𝑑
𝑑𝑡
𝜔𝑟
(𝛿𝜃) = 𝜔𝑚 − 𝑃 𝑛
𝑔𝑒𝑎𝑟
(D.15)
D.2 Equations of Self-Excited Induction Generator with Diode Bridge Rectifier
From dq state space matrix equation given in (4.26), the dq stator and rotor currents of the selfexcited induction generator are
𝑑
𝑖
𝑑𝑡 𝑞𝑠
1
′
= 𝑘 [𝑅𝑠 𝐿′𝑟 𝑖𝑞𝑠 + (𝜔𝑒 (𝐿𝑠 𝐿′𝑟 − 𝐿2𝑚 ) + 𝜔𝑟 𝐿2𝑚 )𝑖𝑑𝑠 −𝑅𝑟′ 𝐿𝑚 𝑖𝑞𝑟
+
(D.16)
𝑠
′
𝜔𝑟 𝐿𝑚 𝐿′𝑟 𝑖𝑑𝑟
−𝐿′𝑟 𝑣𝑐𝑔𝑞 ]
𝑑
𝑖
𝑑𝑡 𝑑𝑠
1
′
′
= 𝑘 [(𝜔𝑒 (𝐿2𝑚 − 𝐿𝑠 𝐿′𝑟 ) − 𝜔𝑟 𝐿2𝑚 ) 𝑖𝑞𝑠 + 𝑅𝑠 𝐿′𝑟 𝑖𝑑𝑠 −𝜔𝑟 𝐿𝑚 𝐿′𝑟 𝑖𝑞𝑟
−𝑅𝑟′ 𝐿𝑚 𝑖𝑑𝑟
−𝐿′𝑟 𝑣𝑐𝑔𝑑 ] (D.17)
𝑠
168
𝑑
1
′
′
𝑖 ′ = 𝑘 [−𝑅𝑠 𝐿𝑚 𝑖𝑞𝑠 −𝜔𝑟 𝐿𝑚 𝐿𝑠 𝑖𝑑𝑠 + 𝑅𝑟′ 𝐿𝑠 𝑖𝑞𝑟
+ (𝜔𝑒 (𝐿𝑠 𝐿′𝑟 − 𝐿2𝑚 ) − 𝐿𝑠 𝐿′𝑟 𝜔𝑟 )𝑖𝑑𝑟
+
𝑑𝑡 𝑞𝑟
𝑠
𝑑 ′
𝑖
𝑑𝑡 𝑑𝑟
(D.18)
𝐿𝑚 𝑣𝑐𝑔𝑞 ]
1
′
′
= 𝑘 [𝜔𝑟 𝐿𝑚 𝐿𝑠 𝑖𝑞𝑠 −𝑅𝑠 𝐿𝑚 𝑖𝑑𝑠 + (𝜔𝑒 (𝐿2𝑚 − 𝐿𝑠 𝐿′𝑟 ) + 𝐿𝑠 𝐿′𝑟 𝜔𝑟 )𝑖𝑞𝑟
+ 𝑅𝑟′ 𝐿𝑠 𝑖𝑑𝑟
+
𝑠
(D.19)
𝐿𝑚 𝑣𝑐𝑔𝑑 ]
where 𝑘𝑠 = 𝐿2𝑚 − 𝐿𝑠 𝐿′𝑟 .
From (4.33) and (4.34), the dynamics of the excitation capacitor voltage is described by
𝑑
𝑣
𝑑𝑡 𝑐𝑔𝑑
1
= − 𝐶 (𝑖𝑑𝑠 +
2√3
𝜋
𝑔
𝑑
𝑑𝑏 𝑖𝑑𝑐 ) + 𝜔𝑒 𝑣𝑐𝑔𝑞
1
𝑣 = − 𝐶 𝑖𝑞𝑠 − 𝜔𝑒 𝑣𝑐𝑔𝑑
𝑑𝑡 𝑐𝑔𝑞
𝑔
(D.20)
(D.21)
D.3 Battery-Side LC Filter Equations
From (4.46) and (4.47), the dynamic of battery-side LC filter is described by
𝑑
1
𝑖𝑏𝑎𝑡 = (𝐸𝑜 − 𝐾𝐸 (1 − 𝑆𝑜𝐶) − 𝑅𝑜 (1 + 𝐴0 (1 − 𝑆𝑜𝐶))𝑖𝑏𝑎𝑡 − 𝑣𝑐𝑏 )
𝑑𝑡
𝐿𝑏
𝑑
𝑣
𝑑𝑡 𝑐𝑏
=
1
𝑐𝑏
(𝑖𝑏𝑎𝑡 − (2𝑑𝐴 − 1)𝑖𝑑𝑐 )
(D.22)
(D.23)
D.4 Equations of Current Source Inverter Supplying the Extended Generic Load Model
From dq state space matrix equation given in (4.62), the dynamic of the CSI capacitive filter is
described by
𝑑
𝑑𝑡
𝑑
𝑑𝑡
𝑑
1
1
𝑖
𝑖
1
1
𝑖
𝑖
1
1
𝑖
𝑖
1
1
𝑖
𝑖
+
+
+
+
𝑣𝑐𝑖𝑑
= 𝐶 𝑖𝑜𝑑
+ 𝜔𝐿 𝑣𝑐𝑖𝑞
− 𝐶 𝑖𝑝𝑑
+
+
+
+
𝑣𝑐𝑖𝑞
= 𝐶 𝑖𝑜𝑞
− 𝜔𝐿 𝑣𝑐𝑖𝑑
− 𝐶 𝑖𝑝𝑞
−
−
−
𝑣 − = 𝐶 𝑖𝑜𝑑
− 𝜔𝐿 𝑣𝑐𝑖𝑞
− 𝐶 𝑖𝑝𝑑
𝑑𝑡 𝑐𝑖𝑑
𝑑
−
−
−
𝑣 − = 𝐶 𝑖𝑜𝑞
+ 𝜔𝐿 𝑣𝑐𝑖𝑑
− 𝐶 𝑖𝑝𝑞
𝑑𝑡 𝑐𝑖𝑞
From (4.60), the fundamental dq-axis output currents of CSI are:
169
(D.24)
(D.25)
(D.26)
(D.27)
+
+
𝑖𝑜𝑑
= 𝐺 𝑚𝑖𝑑
𝑖𝑑𝑐
(D.28)
+
+
𝑖𝑜𝑞
= 𝐺 𝑚𝑖𝑞
𝑖𝑑𝑐
(D.29)
−
−
𝑖𝑜𝑑
= 𝐺 𝑚𝑖𝑑
𝑖𝑑𝑐
(D.30)
−
−
𝑖𝑜𝑞
= 𝐺 𝑚𝑖𝑞
𝑖𝑑𝑐
(D.31)
From the dq state space matrix of the extended generic load model given in Table 4.4, the dynamics
of the load model is described by the following equations.
𝑑 +
𝑥
𝑑𝑡 1𝑚
+
+
= −(𝑑 2 + 𝜔𝑜2 ) 𝑥2𝑚
+ 𝑣𝑐𝑖𝑑
𝑑
𝑥+
𝑑𝑡 2𝑚
𝑑 −
𝑥
𝑑𝑡 1𝑚
(D.32)
+
+
= 𝑥1𝑚
+ 2𝑑 𝑥2𝑚
(D.33)
+
−
= −(𝑑 2 + 𝜔𝑜2 ) 𝑥2𝑚
+ 𝑣𝑐𝑖𝑑
(D.34)
𝑑 −
𝑥
𝑑𝑡 2𝑚
−
−
= 𝑥1𝑚
+ 2𝑑 𝑥2𝑚
(D.35)
From the same table (Table 4.4), the dq outputs currents of the extended generic load model are
+
+
𝑖𝑝𝑑
= 𝑌𝑃+ (𝑑 2 + 𝜔𝑜2 ) 𝑥2𝑚
(D.36)
+
+
𝑖𝑝𝑞
= −𝑌𝑄+ (𝑑 2 + 𝜔𝑜2 ) 𝑥2𝑚
(D.37)
−
−
𝑖𝑝𝑑
= 𝑌𝑄−𝑝𝑑 (𝑑 2 + 𝜔𝑜2 ) 𝑥2𝑚
(D.38)
−
−
𝑖𝑝𝑞
= −𝑌𝑄−𝑝𝑞 (𝑑 2 + 𝜔𝑜2 ) 𝑥2𝑚
(D.39)
Substituting (D.28)-(D.31) and (D.36)-(D.39) in (D.24)-(D.27), yields
𝑑 +
1
+
+
+ )
𝑣𝑐𝑖𝑑 = (𝐺𝑚𝑖𝑑
𝑖𝑑𝑐 − 𝑌𝑃+ (𝑑 2 + 𝜔𝑜2 )𝑥2𝑚
+ 𝜔𝐿 𝑣𝑐𝑖𝑞
𝑑𝑡
𝐶𝑖
(D.40)
𝑑 +
1
+
+
+
𝑣𝑐𝑖𝑞 = (𝐺𝑚𝑖𝑞
𝑖𝑑𝑐 + 𝑌𝑄+ (𝑑 2 + 𝜔𝑜2 )𝑥2𝑚
) − 𝜔𝐿 𝑣𝑐𝑖𝑑
𝑑𝑡
𝐶𝑖
(D.41)
𝑑 −
1
−
−
−
𝑣𝑐𝑖𝑑 = (𝐺𝑚𝑖𝑑
𝑖𝑑𝑐 − 𝑌𝑄−𝑝𝑑 (𝑑 2 + 𝜔𝑜2 )𝑥2𝑚
) − 𝜔𝐿 𝑣𝑐𝑖𝑞
𝑑𝑡
𝐶𝑖
(D.42)
𝑑 −
1
−
−
−
𝑣𝑐𝑖𝑞 = (𝐺𝑚𝑖𝑞
𝑖𝑑𝑐 + 𝑌𝑄−𝑝𝑞 (𝑑 2 + 𝜔𝑜2 )𝑥2𝑚
) + 𝜔𝐿 𝑣𝑐𝑖𝑑
𝑑𝑡
𝐶𝑖
(D.43)
170
D.5 DC-Link Equations
From (4.82), the dynamics of the dc-link current is described by
𝑑
𝑖
𝑑𝑡 𝑑𝑐
1
=𝐿
𝑑𝑐
(𝑣𝑑 + 𝑣𝑥𝑦 − 𝑣𝑖𝑛𝑣 )
(D.44)
where 𝑣𝑑 , 𝑣𝑥𝑦 , and 𝑣𝑖𝑛𝑣 were defined in (4.36), (4.44), and (4.61), respectively, as
𝑣𝑑 = 𝑑𝑏 𝑣𝑑𝑐
(D.45)
𝑣𝑥𝑦 = (2𝑑𝐴 − 1)𝑣𝑐𝑏
(D.46)
+ +
+ +
− −
− −
𝑣𝑖𝑛𝑣 = 1.5 𝐺 (𝑚𝑖𝑑
𝑣𝑐𝑖𝑑 + 𝑚𝑖𝑞
𝑣𝑐𝑖𝑞 + 𝑚𝑖𝑑
𝑣𝑐𝑖𝑑 + 𝑚𝑖𝑞
𝑣𝑐𝑖𝑞 )
(D.47)
In (D.45), 𝑣𝑑𝑐 = 3√3 (𝑣𝑐𝑔𝑑 )/𝜋.
Substituting (D.45)-(D.47) in (D.44), yields
𝑑
1
𝑖 =𝐿
𝑑𝑡 𝑑𝑐
𝑑𝑐
3√3
𝜋
1
1.5
+ +
+ +
𝑑𝑏 𝑣𝑐𝑔𝑑 + 𝐿 (2𝑑𝐴 − 1)𝑣𝑐𝑏 − 𝐿 𝐺 (𝑚𝑖𝑑
𝑣𝑐𝑖𝑑 + 𝑚𝑖𝑞
𝑣𝑐𝑖𝑞 +
𝑑𝑐
(D.48)
𝑑𝑐
− −
− −
𝑚𝑖𝑑
𝑣𝑐𝑖𝑑 + 𝑚𝑖𝑞
𝑣𝑐𝑖𝑞 )
D.6 Complete Dynamic Equations
By combining equations (D.13)-(D.23), (D.32)-(D.35), (D.40)-(D.43), and (D.48), the complete
dynamic model of the proposed WECS can be summarized in the form of 20 state equations, as
given in (4.84).
171
Appendix E
Small-Signal Models for CSI-SCIG-WECS
The following sections give the details of the small-signal models of the Wind-Turbine
Generation (WTG) subsystem, Energy Storage (ES) subsystem, Current Source Inverter-Load
(CSI-Load) subsystem, and the entire system.
In small-signal model matrices, capital letter of a variable (input or state variable) indicates
the steady-state value of the variable at the selected operating point for linearization.
E.1 WTG Subsystem
1) State-Space Equations
𝑑
𝑋 = 𝐴𝑔 𝑋𝑔 + 𝐵𝑔 𝑈𝑔
𝑑𝑡 𝑔
(E.1)
where
′
′
𝑋𝑔 = [𝜔𝑚 𝜔𝑟 𝛿𝜃 𝑖𝑞𝑠 𝑖𝑑𝑠 𝑖𝑞𝑟
𝑖𝑑𝑟
𝑣𝑐𝑔𝑞 𝑣𝑐𝑔𝑑 ]𝑇
(E.2)
𝑈𝑔 = [𝑣𝑤 𝑑𝑏 𝑖𝑑𝑐 ]𝑇
−
𝐷𝑠𝑒
𝐷𝑠𝑒
𝐽𝑚
𝐽𝑚 𝑃 𝑛𝑔𝑒𝑎𝑟
𝑃 𝐷𝑠𝑒
𝐽𝑔 𝑛𝑔𝑒𝑎𝑟
1
0
𝐴𝑔 =
[
−
−
−
𝐾𝑠𝑒
𝐷𝑠𝑒
𝑃 𝐾𝑠𝑒
2
𝐽𝑔 𝑛𝑔𝑒𝑎𝑟
1
𝐽𝑔 𝑛𝑔𝑒𝑎𝑟
𝑃 𝑛𝑔𝑒𝑎𝑟
0
0
𝐽𝑚
−
3𝑃2 𝐿
0
0
𝑚
4 𝐽𝑔
−𝑎45
0
0
0
−
0
0
0
0
0
0
0
𝑅𝑠 𝐿𝑚
𝑘𝑠
𝜔𝑟 𝐿 𝑚 𝐿 𝑠
𝑘𝑠
1
−
𝐶𝑔
0
0
′
𝑖𝑞𝑟
3𝑃2 𝐿
𝑎45
−
𝑘𝑠
𝜔𝑟 𝐿 𝑚 𝐿 𝑠
−
𝑘𝑠
𝑅𝑠 𝐿𝑚
𝑘𝑠
0
−
0
𝑖𝑑𝑠
−
3𝑃2 𝐿
1
𝐶𝑔
172
−
𝑅𝑟′ 𝐿𝑚
𝑘𝑠
𝜔𝑟 𝐿𝑚 𝐿′𝑟
𝑘𝑠
𝑅𝑟′ 𝐿𝑠
𝑘𝑠
−𝑎67
𝑚
4 𝐽𝑔
0
𝑅𝑠 𝐿′𝑟
−
𝑚
4 𝐽𝑔
0
𝑘𝑠
0
𝑚
4 𝐽𝑔
𝑅𝑠 𝐿′𝑟
0
0
′
𝑖𝑑𝑟
0
0
0
0
3𝑃2 𝐿
(E.3)
𝑖𝑞𝑠
0
𝜔𝑟 𝐿𝑚 𝐿′𝑟
−
𝑘𝑠
𝑅𝑟′ 𝐿𝑚
𝑘𝑠
𝑎67
𝑅𝑟′ 𝐿𝑠
𝑘𝑠
0
0
0
0
0
0
−
𝐿′𝑟
𝑘𝑠
0
𝐿𝑚
𝑘𝑠
0
−
(E.4)
𝐿′𝑟
𝑘𝑠
0
0
𝐿𝑚
𝑘𝑠
0
0
0
− 𝜔𝑒
0
0
𝜔𝑒
0
]
0
0
0
0
𝐵𝑔 = 0
0
0
0
0
0
0
0
0
0
0
0
0
𝑏11
[
−
0
0
0
0
0
0
0
0
2√ 3
𝜋 𝐶𝑔
(E.5)
𝑑𝑏
]
In (E.4) and (E.5),
1
1
𝑠
𝑠
𝑅 𝑖
−𝑣
𝑠
𝑎45 = 𝑘 (𝜔𝑒 (𝐿𝑠 𝐿′𝑟 − 𝐿2𝑚 ) + 𝜔𝑟 𝐿2𝑚 ),𝑎67 = 𝑘 (𝜔𝑒 (𝐿𝑠 𝐿′𝑟 − 𝐿2𝑚 ) − 𝐿𝑠 𝐿′𝑟 𝜔𝑟 ), 𝜔𝑒 = 𝐿 𝑖 𝑑𝑠+𝐿 𝑐𝑔𝑑
,
𝑖′
𝑏11 =
3
30 𝑣𝜔
𝜌 𝜋𝑟 2
[( 𝜔2
𝐽𝑚
𝑟
𝑚
−
2
2.35 𝑣𝜔
𝜔𝑚
)𝑒
(−
21 𝑣𝑤
+0.74)
𝜔𝑚 𝑟
𝑠 𝑞𝑠
𝑚 𝑞𝑟
+ 0.0034 𝑟 𝑣𝜔 ].
2) Small-Signal Equations
𝑑
∆𝑋 = 𝐴𝑔′ ∆𝑋𝑔 + 𝐵𝑔′ ∆𝑈𝑔
𝑑𝑡 𝑔
(E.6)
where
′
′
∆𝑋𝑔 = [∆𝜔𝑚 ∆𝜔𝑟 ∆𝛿𝜃 ∆𝑖𝑞𝑠 ∆𝑖𝑑𝑠 ∆𝑖𝑞𝑟
∆𝑖𝑑𝑟
∆𝑣𝑐𝑔𝑞 ∆𝑣𝑐𝑔𝑑 ]𝑇
∆𝑈𝑔 = [ ∆𝑣𝑤 ∆𝑑𝑏 ∆𝑖𝑑𝑐 ]𝑇
𝐷𝑠𝑒
′
𝑎11
𝑃 𝐷𝑠𝑒
−
𝐽𝑔 𝑛𝑔𝑒𝑎𝑟
1
0
𝐴𝑔′
=
0
0
0
[
0
0
−
𝐽𝑚 𝑃 𝑛𝑔𝑒𝑎𝑟
−
𝐿2𝑚
𝐼
𝑃 𝐾𝑠𝑒
2
𝐽𝑔 𝑛𝑔𝑒𝑎𝑟
1
𝐽𝑔 𝑛𝑔𝑒𝑎𝑟
𝑃 𝑛𝑔𝑒𝑎𝑟
−
𝐿𝑚 𝐿𝑠
𝑘𝑠
𝐿𝑚 𝐿𝑠
𝑘𝑠
𝐿𝑚 𝐿′𝑟 ′
𝐼
𝑘𝑠 𝑑𝑟
𝐿𝑚 𝐿′𝑟 ′
−
𝐼𝑞𝑟
𝑘
+
𝑠
𝐼𝑑𝑠 −
𝐼𝑞𝑠 +
0
0
𝐿𝑠 𝐿′𝑟 ′
𝐼𝑑𝑟
𝑘
𝑠
𝐿𝑠 𝐿′𝑟
𝑘𝑠
′
𝐼𝑞𝑟
0
𝐽𝑚
𝐷𝑠𝑒
𝑘𝑠 𝑑𝑠
𝐿2
− 𝑚 𝐼𝑞𝑠
𝑘
𝑠
𝐾𝑠𝑒
−
3𝑃2 𝐿𝑚
2 𝐽𝑔
′
𝐼𝑑𝑟
0
0
3𝑃2 𝐿𝑚
3𝑃2 𝐿𝑚
′
𝐼𝑞𝑟
2 𝐽𝑔
2 𝐽𝑔
(E.8)
0
𝐼𝑑𝑠
−
3𝑃2 𝐿𝑚
2 𝐽𝑔
𝐼𝑞𝑠
0
0
0
0
0
0
′
𝑎44
′
𝑎45
′
𝑎46
𝑊𝑟 𝐿𝑚 𝐿′𝑟
0
′
𝑎54
′
𝑎55
′
𝑎56
−
0
′
𝑎64
′
𝑎65
′
𝑎66
′
𝑎67
0
′
𝑎74
′
𝑎75
′
𝑎76
𝑅𝑟′ 𝐿𝑠
0
0
′
𝑎84
′
𝑎94
′
𝑎85
′
𝑎95
′
𝑎86
′
𝑎96
0
0
173
(E.7)
𝑘𝑠
𝑅𝑟′ 𝐿𝑚
𝑘𝑠
𝑘𝑠
0
0
0
0
0
0
−
𝐿′𝑟
𝑘𝑠
′
𝑎49
0
′
𝑎59
𝐿𝑚
′
𝑎69
𝑘𝑠
0
′
𝑎79
0
𝑊𝑒
′
𝑎89
′
𝑎99
]
(E.9)
′
𝑏11
0
0
0
0
′
𝐵𝑔 =
0
0
0
[
0
0
0
0
0
0
0
0
0
−
0
0
0
0
0
0
0
0
2√3
𝐼
𝜋 𝐶𝑔 𝑑𝑐
−
(E.10)
2√3
𝐷
𝜋 𝐶𝑔 𝑏 ]
In (E.9) and (E.10),
1
′
𝑎11
= 𝐽 [−𝐷𝑠𝑒 + 𝜌 𝜋𝑟 2 (−
𝑚
3
2.35 𝑉𝜔
𝑊𝑚
)𝑒
21 𝑉𝑤
(−
+0.74)
𝑊𝑚 𝑟
4
60 𝑉𝜔
3
𝑊𝑚
𝑟
+
3
2.35 𝑉𝜔
2
𝑊𝑚
𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑
)], 𝑊𝑒 = 𝐿
)𝑒
1
𝑎′ 54 = 𝑘 [−
′ )
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
(𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑 )(𝐿2𝑚 −𝐿𝑠 𝐿′𝑟 )𝐿𝑠 𝐼𝑞𝑠
𝑠
′ )
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
1 𝑅𝑠 𝐼𝑞𝑠 (𝐿2𝑚 −𝐿𝑠 𝐿′𝑟 )
′
𝑠
𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟 )
𝑎′ 55 = 𝑘 [ (𝐿
𝑠
′
𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
𝑠
1
+ 𝑅𝑠 𝐿′𝑟 ], 𝑎′ 56 = 𝑘 [ −
1
𝑘𝑠
[−
1
′
𝑎76
= 𝑘 [−
𝑠
𝑠
′
𝑎86
=
2
′ )
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
2
2
−𝑊𝑟 𝐿𝑚 𝐿′𝑟 ],
− 𝑅𝑠 𝐿𝑚 ],
′
(𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑 )(𝐿𝑠 𝐿′𝑟 −𝐿2𝑚 )𝐿𝑚 𝐼𝑑𝑟
′ )
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
′
+𝑊𝑟 𝐿𝑚 𝐿𝑠 ], 𝑎75
=
+
2
1
𝑔
′
, 𝑎89
=
′ (𝐿2 −𝐿 𝐿′ )
1 𝑅𝑠 𝐼𝑞𝑟
𝑚
𝑠 𝑟
𝑘𝑠
[
(𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑 )(𝐿2𝑚 −𝐿𝑠 𝐿′𝑟 )
′
+ 𝐿𝑚 ], 𝑎84
= −𝐶 +
(𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑 )𝑉𝑐𝑔𝑑 𝐿𝑚
2
+ 𝑅𝑟′ 𝐿𝑠 ] ,
′
(𝐿2 −𝐿𝑠 𝐿′𝑟 )𝐼𝑑𝑟
′ ],
𝑠
𝑠 𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟 )
′ )2
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
′
𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟 )
− 𝑊𝑟 𝐿2𝑚 ],
1
′
(𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑 )(𝐿2𝑚 −𝐿𝑠 𝐿′𝑟 )𝐿𝑚 𝐼𝑞𝑟
′
(𝐿𝑠 𝐿′𝑟 −𝐿2𝑚 )𝐼𝑞𝑟
,
′
− 𝑊𝑟 𝐿𝑠 𝐿′𝑟 ], 𝑎69
= 𝑘 [(𝐿 𝑚𝐼
′
(𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑 )(𝐿2𝑚 −𝐿𝑠 𝐿′𝑟 )𝐿𝑠 𝐼𝑞𝑟
1
′
𝑎79
= 𝑘 [(𝐿
′ )
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
𝑠
′ )
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
2
′
𝑠 (𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟 )
′
(𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑 )(𝐿𝑠 𝐿′𝑟 −𝐿2𝑚 )𝐿𝑠 𝐼𝑑𝑟
1
1 (𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑 )(𝐿𝑠 𝐿′𝑟 −𝐿2𝑚 )
′ )
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
𝑠
(𝐿𝑠 𝐿′𝑟 −𝐿2𝑚 )𝐼𝑑𝑠
′ )
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
′
−𝑊𝑟 𝐿𝑚 𝐿𝑠 ] , 𝑎66
= 𝑘 [−
′
𝑎67
=𝑘 [
′ )
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
(𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑 )(𝐿2𝑚 −𝐿𝑠 𝐿′𝑟 )𝐿𝑚 𝐼𝑞𝑠
𝑠
𝑠
′ )
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
], 𝑎′49 = − 𝑘
′ )
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
1
(𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑 )(𝐿𝑠 𝐿′𝑟 −𝐿2𝑚 )𝐿𝑠 𝐼𝑑𝑠
+ 𝑊𝑟 𝐿2𝑚 ],
(𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑 )(𝐿2𝑚 −𝐿𝑠 𝐿′𝑟 )
+
𝑚
𝑠
2
′
− 𝐿′𝑟 ], 𝑎64
= 𝑘 [−
)
′
(𝐿𝑠 𝐿′𝑟 −𝐿2𝑚 )
1 𝑅𝑠 𝐼𝑑𝑟
′
𝑎65
=𝑘 [
′
𝑎74
=
2
(𝐿𝑠 𝐿′𝑟 −𝐿2𝑚 ) 𝐼𝑞𝑠
1
′
𝑎59
= 𝑘 [(𝐿
𝑚
(𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑 )(𝐿𝑠 𝐿′𝑟 −𝐿2𝑚 )𝐿𝑚 𝐼𝑑𝑠
𝑠
30 𝑉 4
1
+ 𝑤2 (21𝜌 𝜋𝑟 𝑉𝜔 ( 𝑊 2 𝜔
−
𝑟
, 𝑎′44 = 𝑘 [𝑅𝑠 𝐿′𝑟 −
1 𝑅𝑠 𝐼𝑑𝑠 (𝐿𝑠 𝐿′𝑟 −𝐿2𝑚 )+(𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑 )(𝐿𝑠 𝐿′𝑟 −𝐿2𝑚 )
′ )
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
𝑠
1
21 𝑉𝑤
+0.74)
𝑊𝑚 𝑟
1
′
𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
𝑎′45 = 𝑘 [
𝑎′46 = 𝑘 [−𝑅𝑟′ 𝐿𝑚 −
(−
2𝑉𝑐𝑔𝑑 −𝑅𝑠 𝐼𝑑𝑠
′
𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
′ )
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
(𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑 )𝑉𝑐𝑔𝑑 𝐿𝑠
′ )2
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
′
, 𝑎94
=−
174
′ )
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
−𝑅𝑠 𝐿𝑚 ],
+ 𝑊𝑟 𝐿𝑠 𝐿′𝑟 ],
′
, 𝑎85
= − (𝐿
(𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑 )𝑉𝑐𝑔𝑞 𝐿𝑠
′ )
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
2
,
𝑉𝑐𝑔𝑑 𝑅𝑠
′
𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟 )
,
],
1
′
𝑎95
= −𝐶 +𝐿
𝑔
′
𝑏11
𝑅𝑠 𝑉𝑐𝑔𝑞
′
𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
1
′
, 𝑎96
=−
3
= 𝐽 [0.0068 𝜌 𝜋𝑟 𝑉𝜔 + 𝜌
3
2.35 𝑉𝜔
𝑊𝑚
𝑚
)𝑒
21 𝑉𝑤
(−
+0.74)
𝑊𝑚 𝑟
(𝑅𝑠 𝐼𝑑𝑠 −𝑉𝑐𝑔𝑑 )𝑉𝑐𝑔𝑞 𝐿𝑚
′ )2
(𝐿𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
120 𝑉 3
𝜋𝑟 2 ( 𝑊 2 𝑟𝜔
𝑚
−
2
7.05 𝑉𝜔
𝑊𝑚
′
, 𝑎99
= −𝐿
)𝑒
(−
𝑉𝑐𝑔𝑞
′
𝑠 𝐼𝑞𝑠 +𝐿𝑚 𝐼𝑞𝑟
21 𝑉𝑤
+0.74)
𝑊𝑚 𝑟
1
,
30 𝑉 4
− 𝑊 (21𝜌 𝜋𝑟 ( 𝑊 2 𝜔𝑟 −
𝑚
𝑚
)] .
E.2 ES Subsystem
1) State-Space Equations
𝑑
𝑋 = 𝐴𝑏 𝑋𝑏 + 𝐵𝑏 𝑈𝑏
𝑑𝑡 𝑏
(E.11)
𝑋𝑏 = [𝑖𝑏𝑎𝑡 𝑣𝑐𝑏 𝑖𝑑𝑐 ]𝑇
(E.12)
𝑈𝑏 = [ 𝑣𝑑 𝑣𝑖𝑛𝑣 𝑑𝐴 ]𝑇
(E.13)
where
𝑎10,10
−
1
𝐴𝑏 =
𝑐𝑏
[
0
1
0
𝐿𝑏
0
1
𝐿𝑑𝑐
−
𝑐𝑏
0
0
𝐵𝑏 = [ 1
𝐿𝑑𝑐
−
1
𝐿𝑑𝑐
(E.14)
(2𝑑𝐴 − 1)
(2𝑑𝐴 − 1)
0
0
𝑅𝑜
1
0
0
0
]
0
]
(E.15)
In (E.14), 𝑎10,10 = − 𝐿 ((1 + 𝐴0 (1 − 𝑆𝑜𝐶)).
𝑏
2) Small-Signal Equations
𝑑
∆𝑋 = 𝐴′𝑏 ∆𝑋𝑏 + 𝐵𝑏′ ∆𝑈𝑏
𝑑𝑡 𝑏
(E.16)
∆𝑋𝑏 = [∆𝑖𝑏𝑎𝑡 ∆𝑣𝑐𝑏 ∆𝑖𝑑𝑐 ]𝑇
(E.17)
∆𝑈𝑏 = [∆𝑣𝑑 ∆𝑣𝑖𝑛𝑣 ∆𝑑𝐴 ]𝑇
(E.18)
where
175
𝑎10,10
−
1
𝐴′𝑏 =
𝟎
𝐿𝑏
𝟎
𝑐𝑏
[
1
1
𝟎
𝐿𝑑𝑐
𝐵𝑏′ =
−
0
0
0
[𝐿𝑑𝑐
𝑐𝑏
(2𝐷𝐴 − 1)
(2𝐷𝐴 − 1)
0
1
1
−
0
]
0
−
2
𝑐𝑏
2
1
𝐿𝑑𝑐
(E.19)
𝐿𝑑𝑐
𝐼𝑑𝑐
𝑉 𝑐𝑏
(E.20)
]
E.3 CSI-Load Subsystem
1) State-Space Equations
𝑑
𝑋 = 𝐴𝑐 𝑋𝑐 + 𝐵𝑐 𝑈𝑐
𝑑𝑡 𝑐
(E.21)
where
+
+
−
−
+
𝑋𝐶 = [ 𝑣𝑐𝑖𝑑
𝑣𝑐𝑖𝑞
𝑣𝑐𝑖𝑑
𝑣𝑐𝑖𝑞
𝑥1𝑚
+
𝑥2𝑚
−
𝑥1𝑚
−
𝑥2𝑚
]𝑇
(E.22)
𝑇
+
+
−
−
𝑈𝐶 = [𝑖𝑑𝑐 𝑚𝑖𝑑
𝑚𝑖𝑞
𝑚𝑖𝑑
𝑚𝑖𝑞
]
𝐴𝐶 =
0
−𝜔𝐿
0
𝜔𝐿
0
0
0
0
0
0
0
−𝜔𝐿
0
0
𝜔𝐿
0
1
0
1
[ 0
0
0
0
0
0
0
0
0
0
0
0
0
(E.23)
0 −𝑌𝑃+ (𝑑 2 + 𝜔𝑜2 )/𝐶𝑖
0 𝑌𝑄+ (𝑑 2 + 𝜔𝑜2 )/𝐶𝑖
0
0
0
0
0
0
0 −𝑌𝑄−𝑝𝑑 (𝑑2 + 𝜔𝑜2 )/𝐶𝑖
0
0
𝑌𝑄−𝑝𝑞 (𝑑 2 + 𝜔𝑜2 )/𝐶𝑖
0
0
0
1
0
0
2
−(𝑑 + 𝜔𝑜2 )
2𝑑
0
1
0
0
0
2
−(𝑑 +
2𝑑
0
0
𝜔𝑜2 )
176
(E.24)
]
1
𝐶𝑖
1
𝐶𝑖
1
𝐶𝑖
1
𝐵𝐶 =
𝐶𝑖
[
𝐺𝑚+
𝑖𝑑
0 0 0
0
𝐺𝑚+
𝑖𝑞
0 0 0
0
𝐺𝑚−
𝑖𝑑
0 0 0
0
(E.25)
𝐺𝑚−
𝑖𝑞
0 0 0
0
0
0
0
0
0
0
0
0
0
0
0
0]
0
0
0
0
0
0
0
0
2) Small-Signal Equations
𝑑
∆𝑋 = 𝐴′𝑐 ∆𝑋𝑐 + 𝐵𝑐′ ∆𝑈𝑐
𝑑𝑡 𝑐
(E.26)
where
+
+
−
−
+
∆𝑋𝐶 = [ ∆𝑣𝑐𝑖𝑑
∆𝑣𝑐𝑖𝑞
∆𝑣𝑐𝑖𝑑
∆𝑣𝑐𝑖𝑞
∆𝑥1𝑚
+
∆𝑥2𝑚
−
∆𝑥1𝑚
−
∆𝑥2𝑚
]𝑇
(E.27)
𝑇
+
+
−
−
∆𝑈𝑐 = [∆𝑖𝑑𝑐 ∆𝑚𝑖𝑑
∆𝑚𝑖𝑞
∆𝑚𝑖𝑑
∆𝑚𝑖𝑞
]
𝐴′𝑐 =
0
−𝜔𝐿
0
𝜔𝐿
0
0
0
0
0
0
0
−𝜔𝐿
0
0
𝜔𝐿
0
1
0
1
[ 0
0
0
0
0
0
0
0
0
0
0
0
0
0 −𝑌𝑃+ (𝑑 2 + 𝜔𝑜2 )/𝐶𝑖
0 𝑌𝑄+ (𝑑 2 + 𝜔𝑜2 )/𝐶𝑖
0
0
0
0
0
0
− (𝑑 2
0 −𝑌𝑄𝑝𝑑
+ 𝜔𝑜2 )/𝐶𝑖
0
0
𝑌𝑄−𝑝𝑞 (𝑑 2 + 𝜔𝑜2 )/𝐶𝑖
0
0
0
1
0
0
−(𝑑 2 + 𝜔𝑜2 )
2𝑑
0
1
0
0
0
2
−(𝑑 +
2𝑑
0
0
𝜔𝑜2 )
177
(E.28)
(E.29)
]
1
𝐶𝑖
1
𝐶𝑖
1
𝐵𝑐′ = 𝐶𝑖
1
𝐶𝑖
[
𝐺𝑀+
𝑖𝑑
1
𝐶𝑖
𝐺 𝐼𝑑𝑐
𝐺𝑀+
𝑖𝑞
0
𝐺𝑀−
𝑖𝑑
0
1
𝐶𝑖
0
0
0
𝐺 𝐼𝑑𝑐
0
0
𝐺 𝐼𝑑𝑐
0
1
0
𝐶𝑖
𝐺𝑀−
𝑖𝑞
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
𝐶𝑖
(E.30)
𝐺 𝐼𝑑𝑐
0
0
0
0
]
E.4 The Entire System
1) State-Space Equations
𝑑
𝑋 = 𝐴𝑋 + 𝐵𝑈
𝑑𝑡
(E.31)
where
′
′
𝑋 = [ 𝜔𝑚 𝜔𝑟 𝛿𝜃 𝑖𝑞𝑠 𝑖𝑑𝑠 𝑖𝑞𝑟
𝑖𝑑𝑟
𝑣𝑐𝑔𝑞 𝑣𝑐𝑔𝑑 𝑖𝑏𝑎𝑡 𝑣𝑐𝑏
+
+
−
−
+
+
−
−
𝑣𝑐𝑖𝑑
𝑣𝑐𝑖𝑞
𝑣𝑐𝑖𝑑
𝑣𝑐𝑖𝑞
𝑥1𝑚
𝑥2𝑚
𝑥1𝑚
𝑥2𝑚
𝑖𝑑𝑐 ]𝑇
(E.32)
+
+
−
− 𝑇
𝑈 = [ 𝑣𝑤 𝑑𝑏 𝑑𝐴 𝑚𝑖𝑑
𝑚𝑖𝑞
𝑚𝑖𝑑
𝑚𝑖𝑞
]
(E.33)
𝑏11
0
0
0
0
0
0
0
0
0
𝐵=
0
0
0
0
0
0
0
0
0
[0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
178
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0]
(E.34)
(E.35)
179
2) Small-Signal Equations
𝑑
𝑑𝑡
∆𝑋 = 𝐴′ ∆𝑋 + 𝐵 ′ ∆𝑈
(E.36)
where
∆𝑖𝑏𝑎𝑡
′
′
∆𝑋 = [∆𝜔𝑚 ∆𝜔𝑟 ∆𝛿𝜃 ∆𝑖𝑞𝑠 ∆𝑖𝑑𝑠 ∆𝑖𝑞𝑟
∆𝑖𝑑𝑟
∆𝑣𝑐𝑔𝑞 ∆𝑣𝑐𝑔𝑑
+
+
−
−
+
+
−
−
∆𝑣𝑐𝑏 ∆𝑣𝑐𝑖𝑑 ∆𝑣𝑐𝑖𝑞 ∆𝑣𝑐𝑖𝑑 ∆𝑣𝑐𝑖𝑞 ∆𝑥1𝑚 ∆𝑥2𝑚 ∆𝑥1𝑚
∆𝑥2𝑚
∆𝑖𝑑𝑐 ]𝑇
+
+
−
− 𝑇
∆𝑈 = [∆𝑣𝜔 ∆𝑑𝑏 ∆𝑑𝐴 ∆𝑚𝑖𝑑
∆𝑚𝑖𝑞
∆𝑚𝑖𝑑
∆𝑚𝑖𝑞
]
′
𝑏11
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2√3
−
𝐼
𝜋 𝐶𝑔 𝑑𝑐
0
0
0
0
2
− 𝐼𝑑𝑐
𝑐𝑏
0
0
0
0
𝐵′ =
[
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
𝐺
𝐼
𝐶𝑖 𝑑𝑐
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
𝐺
𝐼
𝐶𝑖 𝑑𝑐
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3√3
𝑉
𝜋𝐿𝑑𝑐 𝑐𝑔𝑑
0
0
0
0
0
0
0
0
1.5
+
−
𝐺 𝑉𝑐𝑖𝑑
𝐿𝑑𝑐
0
0
0
0
1.5
+
−
𝐺 𝑉𝑐𝑖𝑞
𝐿𝑑𝑐
0
0
0
0
1.5
−
−
𝐺 𝑉𝑐𝑖𝑑
𝐿𝑑𝑐
0
(E.38)
0
0
0
0
0
0
0
0
𝐺
𝐼
𝐶𝑖 𝑑𝑐
2
𝑉
𝐿𝑑𝑐 𝑐𝑏
180
(E.37)
𝐺
𝐼
𝐶𝑖 𝑑𝑐
0
0
0
0
1.5
−
−
𝐺 𝑉𝑐𝑖𝑞
𝐿𝑑𝑐
]
(E.39)
(E.40)
181
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