modeling alternating current rotating electrical machines
TRANSCRIPT
MODELING ALTERNATING CURRENT ROTATING ELECTRICAL
MACHINES USING CONSTANT-PARAMETER RL-BRANCH
INTERFACING CIRCUITS
by
Mehrdad Chapariha
B.Sc. The Isfahan University of Technology, 2006
M.Sc. The Isfahan University of Technology, 2009
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
in
The Faculty of Graduate and Postdoctoral Studies
(Electrical and Computer Engineering)
THE UNIVERSITY OF BRITISH COLUMBIA
(Vancouver)
November 2013
© Mehrdad Chapariha, 2013
ii
Abstract
Transient simulation programs are used extensively for modeling and simulation of various
electrical power and energy systems that include rotating alternating current machines as
generators and motors. In simulation programs, traditionally, the machine models are
expressed in qd-coordinates (rotational reference frame) and transformed variables, and
the power networks are modeled in abc-phase coordinates (physical variables), which
represents an interfacing problem. It has been shown in the literature that the method of
interfacing machine models and the electric network models plays an important role in
numerical accuracy and computational performance of the overall simulation.
This research considers the state-variable-based simulation programs and proposes a
unified constant-parameter decoupled RL-branch circuit in abc-phase coordinates (with
optional zero-sequence). The proposed circuits are based on voltage-behind-reactance
(VBR) formulation and can be used for interfacing both induction and synchronous
machine models. The new models achieve a direct and explicit interface with arbitrary
external electrical networks, which results in many computational advantages. Extensive
computer studies are presented to verify the proposed models and to demonstrate their
implementation in several commonly-used simulation programs. The new models are
shown to offer significant improvements in accuracy and numerical efficiency over the
existing state-of-the-art models due to their direct interface. It is further envisioned that
the proposed models will receive a wide acceptance in research community and simulation
software industry, and may enable the next generation of power systems simulation tools.
iii
Preface
Some of the research results presented in this thesis have been already published as
journal articles, conference proceedings and/or submitted for peer review. In all
publications, I was responsible for developing the mathematical formulations,
implementing the models, conducting the simulations, compiling the results and
conclusions, as well as preparing majority of the manuscripts. My research supervisor, Dr.
Juri Jatskevich, has provided supervisory comments and corrections during the process of
conducting the studies and writing the manuscripts. My research co-supervisor, Dr.
Hermann W. Dommel, also has provided supervisory feedback, comments, and corrections,
for the research and written manuscripts. The contributions of other co-authors of
published and submitted papers are explained below.
A version of Chapter 2 and 3 has been published. M. Chapariha, L. Wang, J. Jatskevich, H. W.
Dommel, and S. D. Pekarek, “Constant-Parameter RL-Branch Equivalent Circuit for
Interfacing AC Machine Models in State-Variable-Based Simulation Packages," IEEE
Transactions on Energy Conversion, vol. 27, no. 3, pp. 634–645, September 2012. Dr. L.
Wang and Dr. S. D. Pekarek have provided comments and constructive feedback about the
proposed interfacing circuit which is a generalization and improvement of their previous
research work.
A part of Chapter 2 was presented at a conference. M. Chapariha, F. Therrien, J. Jatskevich,
and H. W. Dommel, “Implementation of Induction Machine VBR Model with Optional Zero-
Sequence in SimPowerSystems, ASMG, and PLECS Toolboxes”, In proceedings of
iv
International Conference on Power Systems Transients (IPST 2013), Vancouver, BC,
Canada, 18–20 July 2013. F. Therrien has provided comments, suggestions, and
constructive feedback.
A part of Chapter 3 was presented at a conference. M. Chapariha, F. Therrien, J. Jatskevich,
and H. W. Dommel, “Implementation of Constant-Parameter Directly-Interfaced VBR
Synchronous Machine Models in SimPowerSystems, ASMG, and PLECS Toolboxes”, In
proceedings of Power and Energy Society General Meeting, 2013 IEEE, Vancouver, BC,
Canada, 21–25 July 2013. F. Therrien has provided comments, suggestions, and
constructive feedback and revised the manuscript.
A version of Chapter 4 has been published. M. Chapariha, F. Therrien, J. Jatskevich, and H.
W. Dommel, "Explicit Formulations for Constant-Parameter Voltage-Behind-Reactance
Interfacing of Synchronous Machine Models," IEEE Transactions on Energy Conversion, vol.
28, no. 4, pp. 1053-1063, December 2013. F. Therrien has provided comments, suggestions,
and constructive feedback and revised the manuscript. He has also helped me in the
selection and design of the case-study system and the numerical analysis of the results.
A version of Chapter 5 is has been submitted for peer review. M. Chapariha, F. Therrien, J.
Jatskevich, and H. W. Dommel, "Constant-Parameter Circuit-Based Models of Synchronous
Machine". F. Therrien has provided comments, suggestions, and constructive feedback and
revised the manuscript. He also helped me in selection and design of the case studies.
v
Table of Contents
Abstract .................................................................................................................................................... ii
Preface ..................................................................................................................................................... iii
Table of Contents .................................................................................................................................... v
List of Tables ............................................................................................................................................ x
List of Figures ........................................................................................................................................ xi
List of Abbreviations ........................................................................................................................ xvii
Nomenclature ................................................................................................................................... xviii
Acknowledgments............................................................................................................................. xxii
Dedication ........................................................................................................................................... xxiv
CHAPTER 1: INTRODUCTION ............................................................................................................ 1
1.1 Motivation .................................................................................................................................................... 1
1.2 Background ................................................................................................................................................. 3
1.2.1 Power systems transient simulation tools ................................................................................. 3
1.2.2 Mathematical models of ac rotating machines....................................................................... 4
1.2.3 Voltage-behind-reactance models ................................................................................................ 7
1.2.3.1 Induction machine models ....................................................................................... 7
1.2.3.2 Synchronous machine models ................................................................................ 8
1.3 State-of-the-Art Research ...................................................................................................................... 9
1.4 Research Objectives and Anticipated Impacts ........................................................................... 10
CHAPTER 2: CIRCUIT INTERFACING OF INDUCTION MACHINE MODELS ........................ 15
2.1 Induction Machine Modeling ............................................................................................................. 15
vi
2.1.1 Coupled-circuit phase-domain model ...................................................................................... 15
2.1.2 The qd0 model in arbitrary reference-frame ........................................................................ 19
2.2 Voltage-Behind-Reactance Formulations ..................................................................................... 21
2.2.1 Explicit formulation with zero-sequence branch in the interfacing circuit ............ 27
2.3 Interfacing of Induction Machine Models in State-Variable-Based Simulation
Programs ................................................................................................................................................. 29
2.3.1 Examples of implementation in SimPowerSystems, ASMG, and PLECS toolboxes 30
2.4 Computer Studies .................................................................................................................................. 32
CHAPTER 3: CIRCUIT INTERFACING OF SYNCHRONOUS MACHINE MODELS ................. 40
3.1 Synchronous Machine Modeling ...................................................................................................... 41
3.1.1 Coupled-circuit phase-domain model ...................................................................................... 41
3.1.2 The qd0 model in rotor reference-frame ................................................................................ 44
3.2 Voltage-Behind-Reactance Formulations ..................................................................................... 46
3.2.1 Variable-parameter formulation ............................................................................................... 47
3.2.2 Standard state-space form for the rotor subsystem .......................................................... 50
3.3 Approximation of Dynamic Saliency in VBR Models ................................................................ 53
3.3.1 Additional winding method .......................................................................................................... 54
3.3.2 Singular perturbation method .................................................................................................... 56
3.4 Generalized Constant-Parameter VBR Formulation ................................................................ 60
3.4.1 Possible explicit and implicit formulations ............................................................................ 62
3.5 Interfacing of Synchronous Machine Models in State-Variable-Based Simulation
Programs ................................................................................................................................................. 63
vii
3.6 Implementation of Synchronous Machine VBR Formulations in Simulation Programs
..................................................................................................................................................................... 66
3.6.1 Examples of implementation in SimPowerSystems, ASMG, and PLECS toolboxes 67
3.7 Computer Studies .................................................................................................................................. 69
3.7.1 Verification of the implicit VBR formulation ........................................................................ 69
3.7.1.1 Constant time-step simulation study ................................................................ 70
3.7.1.2 Variable time-step simulation study ................................................................. 71
3.7.2 Accuracy of the approximated constant-parameter VBR formulation ..................... 72
3.7.3 Small machine case-study ............................................................................................................. 75
3.7.4 Large machine case-study ............................................................................................................. 80
CHAPTER 4: NUMERICAL METHODS TO ACHIEVE CONSTANT-PARAMETER VBR
FORMULATIONS ......................................................................................................... 86
4.1 Method of Using Current Derivatives............................................................................................. 86
4.2 Method of Using Algebraic Feed-through..................................................................................... 88
4.3 Algebraic Equivalence of Implicit VBR Formulations .............................................................. 90
4.4 Numerically Efficient Explicit Implementation .......................................................................... 93
4.4.1 Approximation of current derivative ....................................................................................... 93
4.4.2 Relaxation of algebraic loop ........................................................................................................ 95
4.5 Summary of Approximation Techniques ...................................................................................... 98
4.6 Computer Studies .................................................................................................................................. 99
4.6.1 Continuous-time approximation techniques ...................................................................... 102
4.6.2 Discrete-time approximation techniques ............................................................................ 106
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4.6.3 Comparison of models and approximation techniques ................................................. 108
CHAPTER 5: DIRECT INTERFACING OF SYNCHRONOUS MACHINE MODELS FROM
STATOR AND ROTOR TERMINALS .................................................................... 112
5.1 Rotor and Stator Interfacing of Synchronous Machine Models ......................................... 113
5.2 All-Circuit Formulation ...................................................................................................................... 116
5.2.1 Stator voltage equations ............................................................................................................ 116
5.2.2 Rotor voltage equations .............................................................................................................. 119
5.3 Stator-and-Field-Circuit Formulation .......................................................................................... 122
5.3.1 Stator voltage equations ............................................................................................................ 122
5.3.2 Rotor voltage equations .............................................................................................................. 126
5.4 Numerically Efficient Explicit Implementation ........................................................................ 129
5.5 Computer Studies ................................................................................................................................ 130
5.5.1 Single-phase-to-ground fault in the network .................................................................... 134
5.5.2 Diode failure in the exciter system ......................................................................................... 139
CHAPTER 6: SUMMARY OF CONTRIBUTIONS AND FUTURE WORK ............................... 144
6.1 Conclusions and Contributions ...................................................................................................... 144
6.1.1 Objective 1 ......................................................................................................................................... 144
6.1.2 Objective 2 ......................................................................................................................................... 145
6.1.3 Objective 3 ......................................................................................................................................... 145
6.1.4 Objective 4 ......................................................................................................................................... 146
6.2 Potential Impacts of Contributions ............................................................................................... 147
6.3 Future Work ........................................................................................................................................... 148
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6.3.1 Doubly-fed induction machine model with direct interface ........................................ 148
6.3.2 Inclusion of magnetic saturation ............................................................................................ 148
6.3.3 Application to dynamic phasor solution and shifted frequency analysis .............. 149
References .......................................................................................................................................... 150
Appendices ......................................................................................................................................... 157
Appendix A: Parameters of the Induction Machine Model in Section 2.4 ............................. 157
Appendix B: MATLAB Ordinary Differential Equation (ODE) Solvers ................................... 157
Appendix C: Parameters of the Synchronous Machine in Subsections 3.7.1 to 3.7.3 ....... 159
Appendix D: Parameters of the Power System in Subsection 3.7.4 ......................................... 159
Appendix E: Parameters of the Power System in Section 4.6 .................................................... 159
Appendix F: Parameters of the Power System in Section 5.5 .................................................... 160
x
List of Tables
Table 2–1 Comparison of Numerical Efficiency of VBR Formulations for Single-Phase Fault
Study ....................................................................................................................................................................... 39
Table 3–1 Simulation Efficiency for Single-Phase Fault Study for a Synchronous Machine
Models .................................................................................................................................................................... 80
Table 3–2 Numerical Efficiency of CP-VBR Models Versus qd0 with Snubber for the Large
Machine Study ..................................................................................................................................................... 85
Table 4–1 Summary of Approximation Techniques to Achieve the Interfacing Circuit of
Figure 2-2 .............................................................................................................................................................. 99
Table 4–2 Comparison of Continuous-Time Approximation Methods (With Added
Eigenvalue of 1,000) ....................................................................................................................................... 103
Table 4–3 Comparison of Discrete-Time Approximation Methods ............................................. 107
Table 4–4 Eigenvalues of the System when using Different Approximation Techniques .. 110
Table 5–1 Classification of Synchronous Machine Formulations based on Interfacing with
External Inductive Networks....................................................................................................................... 114
Table 5–2 Numerical Performance of the Models for the Single-Phase-to-Ground Fault
Study ..................................................................................................................................................................... 138
Table 5–3 Numerical Performance of the Models for the Exciter Diode Failure Study ....... 143
xi
List of Figures
Figure 2-1 Magnetically coupled circuit model of induction machine. ........................................ 16
Figure 2-2 General interfacing circuit for induction machine VBR formulation. ..................... 28
Figure 2-3 Interfacing of induction machine qd0 models with external electrical network.
................................................................................................................................................................................... 30
Figure 2-4 Interfacing of induction machine VBR models with external electrical network.
................................................................................................................................................................................... 30
Figure 2-5 Example of implementation of the proposed induction machine VBR model in
Simulink using the SimPowerSystems toolbox. ..................................................................................... 31
Figure 2-6 Example of implementation of the proposed induction machine VBR model
equivalent interfacing circuit with external electrical network inside the PLECS Circuit
block. ....................................................................................................................................................................... 32
Figure 2-7 Induction generator connected to the Thévenin equivalent circuit of a network
for the single-phase-to-ground fault transient study. ......................................................................... 33
Figure 2-8 Simulation results for the single-phase-to-ground fault study. From top: source
voltage, source current, machine neutral current, and electromechanical torque. ................. 36
Figure 2-9 Detailed view of source current ics in steady-state excerpt from Figure 2-8. ...... 37
Figure 2-10 Detailed view of source current ics during transient excerpt from Figure 2-8. 37
Figure 2-11 Detailed view of electromagnetic torque Te during transient excerpt from
Figure 2-8. ............................................................................................................................................................. 38
Figure 3-1 Magnetically coupled circuit model of synchronous machine. ................................. 42
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Figure 3-2 Interfacing synchronous machine classical VBR models with external electrical
network-circuit with algebraic feed-through of current. ................................................................... 65
Figure 3-3 Interfacing synchronous machine constant-parameter VBR models with
external electrical network-circuit with algebraic feed-through of current and possible
algebraic loop for voltage. .............................................................................................................................. 65
Figure 3-4 Example of implementation of the synchronous machine CP-VBR model in
SimPowerSystems toolbox. ............................................................................................................................ 68
Figure 3-5 Example of implementation of the synchronous machine CP-VBR model in
PLECS toolbox. .................................................................................................................................................... 68
Figure 3-6 Transient response to the three phase fault predicted by the variable
impedance and the constant-parameters implicit VBR models. ..................................................... 71
Figure 3-7 Detailed view of current iqs from the three phase fault study shown in Figure
3-6; (a) constant time-step solution; and (b) variable time-step solution. ................................. 72
Figure 3-8 Current iqs from the three phase fault study predicted by the approximate
constant-parameters VBR formulation: (a) overall transient; and (b) magnified view of the
window in part (a). ............................................................................................................................................ 73
Figure 3-9 The effect of additional winding resistance on: (a) approximate model error in
iqs; and (b) the system largest eigenvalue magnitude. ......................................................................... 75
Figure 3-10 Synchronous generator connected to an inductive network. The snubbers are
required only for the classical qd0 model. ............................................................................................... 76
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Figure 3-11 Transient response to a single phase to ground fault predicted by qd0 model
with snubber versus approximated constant-parameters VBR and exact implicit VBR
models. ................................................................................................................................................................... 78
Figure 3-12 Detailed view of current ibs in steady-state from Figure 3-11. ............................... 79
Figure 3-13 Detailed view of current ics during the single phase fault from Figure 3-11. .... 79
Figure 3-14 A large synchronous generator connected to an inductive network. The
snubbers are required only for the classical qd0 model. .................................................................... 81
Figure 3-15 Simulation results for large machine single-phase-to-ground fault study as
predicted by CP-VBR models and the conventional qd0 model. ...................................................... 83
Figure 3-16 Detailed view of phase current in steady-state shown in Figure 3-15. ............... 84
Figure 3-17 Detailed view of phase current during transient shown in Figure 3-15. ........... 84
Figure 3-18 Detailed view of electromagnetic torque shown in Figure 3-15. .......................... 84
Figure 4-1 Implementation of explicit constant-parameter VBR model using filter Hi in
continuous-time (high-pass) or discrete-time (backward difference) to approximate the
current derivative. ............................................................................................................................................. 95
Figure 4-2 Implementation of explicit constant-parameter VBR model using filter Hv in
continuous time (low-pass) or discrete time (zero- or first-order-hold) to break the
algebraic loop. ..................................................................................................................................................... 96
Figure 4-3 Approximation of voltage vqs using zero-order-hold (delay) or first-order-hold
(linear prediction). ............................................................................................................................................ 97
xiv
Figure 4-4 Test system consisting of a grounded steam turbine generator connected to the
network via a unit transformer. A single phase-to-ground fault is applied at the machine
terminals. ............................................................................................................................................................ 101
Figure 4-5 Transient response to a single phase-to-ground fault predicted by continuous-
time approximated models. From top to bottom: terminal voltages vabcs, phase a stator
current ias, neutral grounding current ing, field current ifd, and electromagnetic torque Te.103
Figure 4-6 Detailed view of current ias shown in Figure 4-5 for continuous-time
approximated models..................................................................................................................................... 104
Figure 4-7 Detailed view of current ifd shown in Figure 4-5 for continuous-time
approximated models..................................................................................................................................... 104
Figure 4-8 Detailed view of electromagnetic torque Te shown in Figure 4-5 for continuous-
time approximated models. ......................................................................................................................... 105
Figure 4-9 Cumulative error in predicting currents iabcs versus the magnitude of the added
eigenvalue for continuous-time approximated models. ................................................................... 105
Figure 4-10 Detailed view of current ias shown in Figure 4-5 for discrete-time
approximated models..................................................................................................................................... 106
Figure 4-11 Detailed view of current ifd shown in Figure 4-5 for discrete-time
approximated models..................................................................................................................................... 107
Figure 4-12 Detailed view of electromagnetic torque Te shown in Figure 4-5 for discrete-
time approximated models. ......................................................................................................................... 107
Figure 4-13 Cumulative error in predicting currents iabcs versus maximum time-step size
for discrete-time approximated models. ................................................................................................ 108
xv
Figure 5-1 All-circuit constant-parameter VBR synchronous machine model (AC-CP-VBR)
with direct interfacing to arbitrary external ac and dc networks. ................................................ 121
Figure 5-2 Rotor subsystem for the stator-and-field-circuit constant-parameter voltage-
behind-reactance (SFC-CP-VBR) model wherein the damper windings are represented as a
state model and the field winding is made available as an interfacing circuit (the stator
interfacing circuit is the same as in Figure 5-1). .................................................................................. 128
Figure 5-3 Algebraic loops in AC-CP-VBR resulting in an implicit formulation. The H blocks
indicate where the low-pass filters may be inserted to break the algebraic loops. ............... 130
Figure 5-4 A wye-grounded steam turbine generator with a static 12-pulse rectifier-based
exciter system. The stator snubbers are required only for the classical qd0 model and the
field snubber is required only for the CP-VBR model from Chapter 4. ....................................... 132
Figure 5-5 Single-phase-to-ground fault case-study transient responses as predicted by the
considered models. From top to bottom: bus voltages vabc and currents iabc, machine neutral
current ing, three-phase ac exciter current iabcex, rectifier current idc, and electromagnetic
torque Te. ............................................................................................................................................................. 135
Figure 5-6 Magnified fragment from Figure 5-5: phase current ic (in steady-state). ........... 136
Figure 5-7 Magnified fragment from Figure 5-5: phase current ic (during transient). ........ 136
Figure 5-8 Magnified fragment from Figure 5-5: ac exciter phase current icex. ...................... 137
Figure 5-9 Magnified fragment from Figure 5-5: rectified current idc. ....................................... 137
Figure 5-10 Left: the diode bridge connected to the delta transformer in Figure 5-4. Right:
simulation results of the diode currents in pu: 1) after the initial failure (short-circuit) of
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D3; and 2) after D1 and D3 (or the corresponding protective diodes) become open due to
the ensuing excessive currents. .................................................................................................................. 141
Figure 5-11 Transient responses as predicted by the considered models for the static
exciter diode failure case-study. From top to bottom: exciter delta transformer line voltages
vTΔ, delta transformer currents iTΔ, wye transformer line voltages vTY, wye transformer
currents iTY, rectifier output voltage vfd, and rectifier current idc. ................................................. 142
xvii
List of Abbreviations
ac alternating current
AC-CP-VBR all-circuit CP-VBR
ASMG Automated State Model Generator (toolbox)
CC-PD coupled-circuit phase-domain
CPU central processing unit
CP-VBR constant-parameter VBR
DAE differential-algebraic equation
d-axis direct axis
dc direct current
IVBR implicit VBR
IM induction machine
ODE ordinary differential equation
pu per-unit
q-axis quadrature axis
s second(s)
SI the International System (of Units)
SFC-CP-VBR stator-and-field-circuit CP-VBR
SM synchronous machine
SPS SimPowerSystems (toolbox)
UBC the University of British Columbia
VBR voltage-behind-reactance
xviii
Nomenclature
Throughout this thesis, matrix and vector quantities are boldfaced (e.g. abcsv ), and scalar
quantities are italic non-boldfaced (e.g. si0 ). All machine variables are referred to the stator
side using the appropriate turns-ratio.
cos Cosine function
)diag(x A diagonal matrix containing vector x on the main diagonal
abce Sub-transient voltage vector
abce Modified (approximated) sub-transient voltage vector
xfde Scaled excitation voltage equal to fdfdmd vrX / (synchronous machines)
H Inertia constant
si0 Stator zero-sequence current
abcsi Stator actual current vector
ngi Current of the VBR interfacing circuit zero-sequence branch
J Moment of inertia
0L Inductance of the zero-sequence branch of the VBR interfacing circuit
DL Inductance of the three-phase branch of the VBR interfacing circuit
lfdL Field winding leakage inductance (synchronous machines)
lkdjL , Nj 1 The d-axis rotor damper leakage inductances (synchronous machines)
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lkqjL , Mj 1 The q-axis rotor damper leakage inductances (synchronous machines)
lrL Rotor winding leakage inductances (induction machines)
lsL Stator winding leakage inductances (induction and synchronous
machines)
mL Magnetizing inductance (induction machines)
mdL Magnetizing inductance of the d-axis (synchronous machines)
mqL Magnetizing inductance of the q-axis (synchronous machines)
p Heaviside’s operator (differentiation with respect to time)
P Number of poles
BP Base power
0r Resistance of the zero-sequence branch of the VBR interfacing circuit
Dr Resistance of the three-phase branch of the VBR interfacing circuit
fdr Field winding resistance (synchronous machines)
gr External grounding resistance
kdjr , Nj 1 The d-axis rotor damper resistances (synchronous machines)
kqjr , Mj 1 The q-axis rotor damper resistances (synchronous machines)
rr Rotor winding resistance (induction machines)
sr Stator winding resistance (induction and synchronous machines)
s Laplace variable
sin Sine function
xx
BT Base torque
eT Electromagnetic torque
lT Load torque (motor)
MT Mechanical input torque (generator)
abcnv Voltage vector from stator terminals to the point n in VBR interfacing
circuit
abcsv Stator actual voltage vector
drv Rotor voltage transformed to d-axis (induction machines)
dsv Stator voltage transformed to d-axis (induction and synchronous
machines)
fdv Excitation voltage (synchronous machines)
ngv Voltage of the VBR interfacing circuit zero-sequence branch
qrv Rotor voltage transformed to q-axis (induction machines)
qsv Stator voltage transformed to q-axis (induction and synchronous
machines)
gX External grounding reactance
z A discrete random variable
)(x 2-norm cumulative relative error (calculated for all the points of x )
)(x Average 2-norm cumulative relative error of all x elements (three
phases)
xxi
Flux linkage
d Sub-transient flux linkages of the d-axis
q Sub-transient flux linkages of the q-axis
Angular velocity of the rotational reference frame
b Base angular velocity
r Angular velocity of the rotor (electrical)
xxii
Acknowledgments
I would like to express my sincere gratitude to my research supervisor, Dr. Juri Jatskevich,
for his research direction and excellent vision, great inspiration, patient guidance, caring,
and generous support throughout my PhD studies. I also would like to express my very
great appreciation to my research co-supervisor Dr. Hermann W. Dommel for his
exceptional expertise, great help and guidance, admirable manners, and continuous
support. The financial support for this research was made possible through the Natural
Science and Engineering Research Council (NSERC) of Canada under the Discovery Grant
entitled “Modelling and Analysis of Power Electronic and Energy Conversion Systems” and
the Discovery Accelerator Supplement Grant entitled “Enabling Next Generation of
Transient Simulation Programs” lead by Dr. Juri Jatskevich as a sole principal investigator.
I would like to offer my special thanks to the members of my examining committees of the
departmental and university exams, Drs. Ludovic Van Waerbeke, Jose Marti, Farrokh
Sassani, Edmond Cretu, Narayan Kar, William G. Dunford, and Martin Ordonez for
dedication of their valuable time and expertise and assessment of my dissertation.
I would like to thank all the faculty members, instructors, and professors at the Electrical
and Computer Engineering Department whom I attended their classes and lectures.
Specifically I would like to thank Dr. Luis Linares, Dr. William Dunford, Mr. Nathan Ozog,
and my supervisor Dr. Juri Jatskevich for the great teaching experience I gained assisting
them.
xxiii
I wish to acknowledge the help and support of Dr. Liwei Wang and Dr. Steven D. Pekarek.
This thesis would not have been possible without their prior achievements in the area. I am
also particularly grateful for the help of Francis Therrien, for perfecting this research by
dedicating his time, providing valuable feedback, and careful investigations into expanding
the proposed techniques and inclusion of magnetic saturation.
My special thanks are extended to my colleagues and fellow graduate students in the Power
and Energy Systems research group at the University of British Columbia. Particularly, I
would like to thank Jaishankar Iyer, Amir Rasuli, and Sina Chiniforoosh, for all their help
and support in my academic life and teaching and research activities. I also would like to
thank Milad Gougani, Mohammed Talat Khouj, Mehmet Sucu, Hamid Atighechi, Kamran
Tabarraee, Shahrzad Rostamirad, and all the other former and current members of our
research group for their help and good memories we have together during the years at the
University of British Columbia.
I also thank my friends, many of whom are all around the world, for their moral support
and some of the greatest moment of my life that I spend with them.
The last but not the least, I would like to thank my family for their continuous support and
encouragement. My greatest gratitude goes to my parents Mohammad and Ata, for their
unconditional love, to my brother Alireza, for his encouragements, and to my sister Mojgan
for her caring while I was thousands of miles away.
xxiv
Dedication
To My Parents Mohammad and Ata
1
CHAPTER 1: INTRODUCTION
1.1 Motivation
Synchronous and induction machines are responsible for almost all of electromechanical
energy conversion in today’s world. Substantial portion of electrical energy is produced by
synchronous generators run by hydro turbines, steam turbines, or diesel engines. Induction
machines are the workhorse of electric power industries [1]. They are commonly used as
motors with squirrel cage rotors in industrial and commercial applications. More recently,
however, induction machines have been also used as smaller-scale generators in
distributed energy systems and wind turbines in particular [2], [3]. In addition to the
energy conversion, the synchronous machines have a major influence on the power
systems stability, and are also used for the voltage and frequency control at the system
level [4] – [6].
Detailed three-phase transient simulation of small-to-large scale power systems is needed
for various design, testing, and research purposes. Almost all types of large-and-small scale
power systems today include ac rotating machines: utility grids, small community micro-
grids, wind farms, vessel power systems, off-grid telecommunication sites power systems,
electrical drives, etc.
2
Mathematical modeling of rotating electrical machines has been an active area of research
over the century. The complex physical structures of machines and windings in relative
rotation make this modeling quite involved. The mathematical formulation describing the
machine electromechanical phenomena are therefore non-linear and time variant, making
the machine model a bottleneck for many transient simulation programs. A change of
variables to rotational qd0-reference frame is typically used to simplify the modeling and
simulation of electric machines [7]. However, this technique has difficulties when it comes
to interfacing of the machine models with external power networks which are typically
represented in physical variable and abc-phase coordinates [8]. Improving the electric
machine models and their interface to external power networks was shown to have
considerable impact on the overall simulation time and accuracy [8].
General purpose machine models are found as built-in components in almost all simulation
software packages that are used today in industry and academia. These models are usually
represented in qd0-rotational reference frames, and need to be interfaced with the rest of
the network represented in abc-phase coordinates, which represents an interfacing
problem [8].
The traditional approaches used to solve the interfacing problem include using snubbers or
time-delays, which lead to inaccuracies and may cause numerical stiffness and problems
with convergence of solutions. The state-of-the-art solutions to solve the interfacing
problem (without using three-phase snubbers or time-delays) result in models with
variable parameters and/or models that are implicit and require iterative solutions. These
3
solutions require computationally expensive methods and algorithms; therefore they are
not implemented in most software packages today. The preferred machine model of choice
should be computationally efficient, explicit, accurate, and use basic circuit components for
interfacing with remaining power networks without complicating the overall simulation
program solver.
Proposing and developing such models will have significant benefit for the software
developers, who will be able to easily implement the models into their packages, and
thousands of users, who would be able to use, implement, or modify the models. The result
will be equipping the new generation of transient simulation tools with highly accurate and
numerically stable models. These tools can be applied to much larger systems and achieve
considerably faster simulation results without overly-constrained integration time-steps.
1.2 Background
1.2.1 Power systems transient simulation tools
The methods to simulate electrical power systems can be broadly categorized into two
general types:
1) Nodal-analysis-based electromagnetic transient programs (EMTP-type) [9], and
2) State-variable based simulation programs [8].
4
In EMTP-type solution, the circuit and dynamic components are first discretized using a
particular integration rule (e.g. trapezoidal), and then the resultant system of algebraic
equations is solved for every time step [9]. EMTP type simulation programs are widely
used for simulation of transients in power systems. Examples of such programs include:
ATP [10], MicroTran [11], PSCAD [12], EMTP-RV [13], and PSIM [14].
In state-variable-based simulation programs, the entire system is represented as a state-
space model. The resulting system of ordinary differential equations (OEDs) or differential
algebraic equations (DAEs) is discretized at the system level and solved using either fixed
or variable integration techniques. State-variable-based simulation programs are readily
used in simulation of smaller power systems and electrical drive systems. These programs
have the flexibility of using different ODE solvers and settings that may be used to optimize
the simulation performance and time, while allowing the use of the state-space model for
the system-level analysis and control designs either in time or in frequency domains.
Examples of such programs include MATLAB/Simulink [15], [16] and toolboxes such as
PLECS [17], ASMG [18], SimPowerSystems [19], as well as other programs including
Modelica [20], acslX [21], EASY5 [22], and EUROSTAG [23]. In this thesis, only the state-
variable-based simulation programs are considered.
1.2.2 Mathematical models of ac rotating machines
Modeling of ac machines has been an active area of research for many decades. Depending
on the objectives of studies and required accuracy, various models have been proposed in
5
the literature. The general purpose models, which are considered in this thesis, are based
on magnetically coupled circuit representation of the machine’s windings, which leads to a
relatively small number of equations [1]. Such models can be generally represented in
either direct abc-phase coordinates using the physical variables [1], [24] – [27], the so-
called coupled-circuit phase-domain (CC-PD) models; or in rotating qd-coordinates using
the transformed variables (the classical qd0-models) [1], [7]; or the hybrid of those known
as the voltage-behind-reactance (VBR) models [28] – [37].
The commonly used qd0 machine models offer a computationally efficient alternative to
the original lumped parameter coupled-circuit models in phase domain. However, as was
mentioned in section 1.1, the qd0 models suffer from interfacing issues in connection to
inductive networks. In typical state-variable-based programs, these models are
represented as voltage-input current-output subsystems and are interfaced as dependent
current-sources to abc phase-domain systems [8]. Such interfacing method requires the
addition of snubber circuits when the machine model is connected directly to inductive
elements. As a result, the numerical advantages of the qd0 model may deteriorate very
quickly by adding the snubbers. The snubber circuits add error and can make the system
numerically stiff, decreasing the numerical efficiency and stability of the entire simulation.
Since many elements in power systems are inductive (transformers, transmission lines,
etc.), the interfacing problem in simulation of power systems is quite common.
To solve the interfacing problem, the direct implementation of machine model in physical
abc-phase coordinates, the CC-PD models, may be reconsidered [24] – [27]. Although the
6
CC-PD models offer direct interfacing with any external circuit, they are numerically
inefficient due to their rotor-position-dependent (variable) and coupled inductances and
poorly-scaled eigenvalues [28]. In general, when implemented in conventional state-
variable-based simulation packages (e.g. MATLAB/Simulink [15], [16]), the variable
inductances require costly reformulation (or update) of state-space equations at every
time-step. At the same time, the poorly-scaled eigenvalues may also force the integration
solver to use smaller time-steps [28]. These factors generally lead to longer simulation
times.
The inclusion of magnetic saturation generally increases the accuracy of the machine
modes and their range of application to various transient studies and scenarios, but also
comes at the cost of increased complexity of the models. In many power systems studies,
the saturation curves of many machines in the system are not available; therefore such
studies are commonly carried out using the magnetically-linear machine models with
sufficient level of accuracy. For these reasons, many simulation programs have the option
of enabling or disabling the saturation in machine models, wherein the user can decide to
use the magnetically-linear machine models if the saturation curves are not available or to
achieve a faster simulation. This thesis considers only the magnetically-linear machine
models and their interfacing as the primary focus of research. Development of constant-
parameter saturable models with direct interface is left for the future research, and is
currently undertaken by other members of our research group at the University of British
Columbia (UBC).
7
1.2.3 Voltage-behind-reactance models
The VBR models have been derived to have a direct interface of the stator circuits similar to
the CC-PD models. In VBR models [28] – [34], the transformation of variables to the
rotational reference frame [1], [7] is used to increase numerical efficiency, while still
leaving part of the model as RL circuits (instead of pure dependent current-sources). The
changed variables in rotational reference frame are coupled with the circuit part of the
model through dependent voltage-sources behind an impedance (equivalent resistance and
inductance). Many proposed VBR models possess rotor-position-dependent variable-
parameters. However, compared to the original CC-PD models, the VBR models generally
have better scaled eigenvalues and are more numerically efficient [28] – [34].
1.2.3.1 Induction machine models
For induction machine modeling, a simple VBR formulation was presented in [32], which
has a constant and decoupled interfacing circuit. This circuit is algebraically equivalent to
the original CC-PD model and the qd0 model, and it has the combined advantage of
constant-parameter model with direct interface. These properties make the VBR model
[32] superior to the qd0 model when connected to arbitrary networks in abc-physical
coordinate.
In a general case, the induction machines (especially generators) may be grounded for
protection and monitoring purposes [35]. In the VBR model [32] an algebraic loop is
created when the machine is grounded and the zero-sequence is included. The algebraic
8
loop makes this model formulation implicit and complicates its implementation and
numerical solution.
1.2.3.2 Synchronous machine models
Due to asymmetries in the rotor, the development of a constant-parameter VBR model is
especially challenging for the synchronous machines with either round or salient rotor. For
this reason, classical VBR models and many of the most recent ones for synchronous
machines have time-variant interfacing circuits [28], [29], [33], [34].
The pioneering approach to achieve a constant parameter VBR model consists of adding a
fictitious high-frequency damper winding to the model [30]. To minimize error, the
additional winding parameters are chosen specifically as not to affect the typically low
frequencies of interest in the power system under study. However, the added pole in this
method may make the system numerically stiff. The high-frequency winding and the
consequential error can be removed by pushing the effective frequency of the extra
winding to infinity while keeping the constant-parameter property of the formulation [31].
It will be shown in this thesis that the final formulation is algebraically equivalent to the
qd0 model; however, this process creates an algebraically implicit formulation that
requires an iterative solution. It will also be shown that an iterative solution is very
computationally demanding. Additionally, convergence is not guaranteed for implicit
formulations [16].
9
1.3 State-of-the-Art Research
Recently, there has been a considerable amount of research from various groups in the
world focused on the modeling of ac machines for transient simulation programs. A
number of ac machine models have been proposed by the research group at Purdue
University. S. D. Pekarek originally proposed the exact VBR model for synchronous
machines (1998) [28]. This research group has also proposed the additional winding
method for approximation of dynamic saliency (1999) [30] and removal of the error by
singular perturbation (2002) [31].
D. C. Aliprantis at Iowa Sate University, USA, has made considerable contributions with
introducing the arbitrary network rotor representation and inclusion of saturation in the
VBR formulations (2008) [33]. A. C. Cramer, who is with the University of Kentucky, USA,
has continued Aliprantis’ work with introduction of rotor and stator VBR formulation for
the first time (2012) [34].
École Polytechnique de Montréal in Canada has also been active in the transient modeling
of electric machines leaded by U. Karaagac. In recent publications the CC-PD, qd0, and VBR
models are used to increase the efficiency of EMTP-type simulations (2011) – (2013) [38] –
[40].
The University of Glasgow and the University of Manchester are two of the center for
research on modeling of electric machines in the United Kingdom. Their research on VBR
models are focused on the EMTP-type solutions. They have worked on modeling the
10
internal fault in the stator of synchronous machines using the VBR representation (2009)
[41]. In a more recent publication (2012) [42], they claim a new efficient implementation of
machine models for EMTP-type programs.
The Electric Power and Energy Systems research group at the University of British
Columbia is taking an active and leading role in this direction (since 2004). A significant
part of this research effort has been on advance models of electrical machines for the
EMTP-type programs (2006) – (2011) [36], [37], [43] – [45], also considering of magnetic
saturation (2010) – (2013) [44] – [46], and constant conductance matrix implementations
for induction and synchronous machines [43], [47], respectively.
The VBR modeling approach is getting recognition and wide acceptance. Fore example,
recently, a VBR model for five-phase induction machines was presented in [48]. This model,
which is derived for state-variable-based simulation programs, includes the effect of
magnetic saturation with some approximations; however, the presented model is
numerically stiff.
1.4 Research Objectives and Anticipated Impacts
Although the interfacing problem is well-known and various VBR models are already
developed, the qd0 models are still very commonly used for induction and synchronous
machines modeling. The VBR models provide direct interface, but they may have variable
parameters, contain algebraic loops, or be numerically stiff as will be shown in this thesis.
11
Although some simulation tools are designed for implementation of variable elements and
solving algebraic loops [18], many program do not have variable impedance elements [19],
or are not able to solve algebraic loops [17]. Some VBR models are recently included in
programs such as PLECS [17]. However, most programs cannot take advantage of the VBR
formulation with variable parameters and cannot solve implicitly formulated VBR models.
In summary, to achieve new generation of transient simulation tools with highly accurate
and numerically stable models, the objectives of this research are to develop new models
and improve the interfacing of ac machine models specifically in the state-variable-based
simulation programs. More specifically, the following objectives are considered:
Objective 1: Elimination of algebraic loop in induction machine VBR formulation
The state-of-the-art VBR induction machine model has a simple direct interface to arbitrary
networks [32]. Although not very common in practice, in a general case, induction
machines may be grounded for monitoring or protection purposes [35]. As was mentioned
earlier in section 1.2.3.1, the VBR model [32] in such cases imposes an algebraic constraint
to the formulation, and therefore may not be suitable as a general purpose universal
component model for many simulation programs. Removal of this algebraic constraint is
the first objective of this thesis.
12
Objective 2: Unified interfacing circuit for models of ac machines
The pioneering methods to achieve constant-parameter VBR model for the synchronous
machines [30] and [31] do not achieve a simple interfacing equivalent circuit, and therefore
are also difficult to integrate into many simulation programs. The second objective of this
research is to achieve a simple and general constant-parameter interfacing circuit for both
induction and synchronous machine models. In fact, the constant-parameter equivalent
circuit developed in Objective 1 can be used as a reference to develop new formulations for
the synchronous machine model that indeed can have the same equivalent circuit for
interfacing of the machine's stator terminals to the rest of the power network.
Objective 3: Numerically efficient approximations for constant-parameter models
The state-of-the-art existing methods to achieve constant-parameter interface for
synchronous machines [30], [31] are based on modification of the original coupled-circuit
model or result in implicit formulation. Proper selection of the parameters of the modified
circuits and/or dealing with implicit algebraic constraints represents additional
undesirable challenges. Exploring and proposing alternative methods for achieving the
constant-parameter interface with better numerical properties and/or other modeling
advantages is the third objective of this thesis.
13
Objective 4: Direct interfacing of rotor and stator machine terminals
Direct interfacing of machine models with the external ac power networks has been the
concern of most VBR models [28]–[31], [33] . However, if the rotor terminals are also
required for interfacing, for example to study and design the thyristor rectifiers of the
exciter systems [49], [50], the conventional VBR models do not provide a direct interface of
the rotor circuit and therefore are not suitable. A recently published model [34]
incorporates a direct rotor interface into the VBR formulation. However, this model has
rotor-position-dependent (variable) inductances with a 4-by-4 full inductance matrix (3
stator windings and 1 field winding). Therefore, the final objective of this thesis is to
develop a decoupled constant-parameter models that have capability of direct interfacing
of stator and rotor circuits to arbitrary ac side grids and dc exciter systems, respectively.
The outcome of this research will be constant-parameter general purpose ac machine
models especially for state-variable-based simulation programs. The equivalent interfacing
circuit for either induction or synchronous machine will be composed of conventional
circuit elements such as constant and decoupled RL branches, and controllable voltage-
sources. The remaining differential and algebraic equations can also be readily
implemented using the conventional blocks such as integrators, gains, summers, etc. which
are available in most simulation programs.
14
The numerical advantages of the new VBR models with direct interface will offer more
accurate and less computationally demanding solutions than the existing classical and
built-in models. Availability of fast and accurate models will save many hours of time of
uncountable number of power system engineers and researchers. More efficient models
will also enables the users to model larger systems with more details and/or to run
multiple simulations in a fraction of time.
The favourable properties of the new models will facilitate their adoption in various
simulation programs. Software developers and even novice users should be able to
implement such models in their programs and software applications. Moreover, having the
same general structure and the interfacing circuit for both synchronous and induction
machines will make it easier to develop electrical machines components with similar user
interfaces and parameter entry methods.
15
CHAPTER 2: CIRCUIT INTERFACING OF INDUCTION MACHINE MODELS
Induction machines are usually used as motors with squirrel cage rotors. The wound-rotor
machines are also popular especially in renewable energy systems. For the purpose of
modeling, both types of induction machines are represented similarly. Having a
symmetrical stator and rotor structures facilitates mathematical modeling of induction
machines. To provide the reader with the background material helpful in understanding
the proposed advanced models, the classical modeling of induction machines for power
systems transient studies is briefly reviewed. The state-of-the-art VBR model is presented,
and the new explicit general interfacing circuit including zero-sequence is proposed in this
chapter. The numerical efficiency and accuracy of the proposed model is verified by
extensive simulation studies included later in the chapter.
2.1 Induction Machine Modeling
2.1.1 Coupled-circuit phase-domain model
To make the reader familiar with the fundamentals of induction machine modeling, the
classical magnetically-coupled circuit-model is shown in Figure 2-1. This model has a
16
relatively small number of equation but is considered to offer sufficient accuracy for
general purpose system-level transient simulations studies [1]. The VBR model is derived
from the coupled-circuit model shown in Figure 2-1.
Figure 2-1 considers a basic three-phase wye-connected induction machine which is
modeled as magnetically coupled circuits. The rotor may be squirrel cage or wound type;
for the squirrel cage, the rotor terminals are short-circuited. Throughout this thesis, all
electrical variables are assumed to be referred to the stator side using appropriate turns-
ratio. The windings are also assumed to be sinusoidally distributed and the effects of
slotting and magnetic saturation are neglected. The corresponding voltage equations in
matrix form are [1]
abcr
abcs
rrT
sr
srss
abcr
abcs
pp
pp
i
i
LrL
LLr
v
v (2–1)
Figure 2-1 Magnetically coupled circuit model of induction machine.
17
where p is the Heaviside’s operator (differentiation with respect to time) and the resistance
and inductance matrices are
ssss rrrdiagr (2–2)
rrrr rrrdiagr (2–3)
mslsmsms
msmslsms
msmsmsls
s
LLLL
LLLL
LLLL
2
1
2
12
1
2
12
1
2
1
L (2–4)
mrlrmrmr
mrmrlrmr
mrmrmrlr
r
LLLL
LLLL
LLLL
2
1
2
12
1
2
12
1
2
1
L (2–5)
rrr
rrr
rrr
srsr L
cos)3
2cos()
3
2cos(
)3
2cos(cos)
3
2cos(
)3
2cos()
3
2cos(cos
L. (2–6)
In the above equations, msL and mrL are the magnetizing inductance of the stator and
rotor winding respectively; lsL and lrL are the leakage inductance of the stator and rotor
winding respectively; and diag[X] is an nn matrix having vector X in its diagonal.
Finally, the electromagnetic torque equation in machine variable is given by
18
abcrsrr
Tabcse
PT iLi
2. (2–7)
The extended form of (2–7) is given in [1, see p. 147].
Throughout this thesis, a simple single rigid body mechanical system is assumed that is a
simplified model of all the masses connected to the shaft including rotor, mechanical loads,
and/or prime movers. If the external load torque is LT , the mechanical angular velocity is
calculated by
Ler TTJ
Pp
2 (2–8)
where J is the inertia of the rotor and connected load and r is the electrical angular
velocity of the rotor. If the system is modeled in per-unit, (2–8) becomes [1]
Leb
r TTH
p
2
1
(2–9)
where b is the base angular velocity and H is the inertia constant (in seconds). H is
defined as the ratio of kinetic energy stored in the rotor at rated speed (in Joules) over the
nominal power (in Watts) as
B
b
B
b
P
J
PT
J
PH
222
2
12
2
1
. (2–10)
19
In (2–10), TB and PB are the base torque and power respectively.
The CC-PD model defined by (2–1) – (2–7) can be implemented in simulation programs
such as PLECS [17] and ASMG [18], that have ability to implement variable inductances.
This model provides direct interface to external circuits on the stator and rotor sides.
However, direct implementation of this model is computationally very expensive and the
model has poorly scaled eigenvalues [32].
2.1.2 The qd0 model in arbitrary reference-frame
A change of variable is commonly used in analysis of ac machines to replace the physical
variables (voltages, currents, and flux linkages) with transformed variables expressed in
rotating reference frame. The matrix that transforms a set of three-phase variables to the
arbitrary reference frame is expressed as [1]
2
1
2
1
2
1
)3
2sin()
3
2sin(sin
)3
2cos()
3
2cos(cos
3
2
sK. (2–11)
Transforming the stator and rotor variables of induction machine described in subsection
2.1.1 to the arbitrary reference frame simplifies the voltage equations to the following [1]
qsdsqssqs pirv (2–12)
dsqsdssds pirv (2–13)
20
ssss pirv 000 (2–14)
qrdrrqrrqr pirv )( (2–15)
drqrrdrrdr pirv )( (2–16)
qrrrr pirv 000 (2–17)
where
mqqslsqs iL (2–18)
mddslsds iL (2–19)
slss iL 00 (2–20)
mqqrlrqr iL (2–21)
mddrlrdr iL (2–22)
rlrr iL 00 (2–23)
The q- and d-axis magnetizing flux linkages are defined as
)( qrqsmmq iiL (2–24)
)( drdsmmd iiL (2–25)
where
msm LL2
3 (2–26)
21
The circuit representation for the above equations is given in [1, see p. 151]. The
electromagnetic torque can be calculated by [1]
)(22
3dsqsqsdse ii
PT (2–27)
which in per-unit is expressed as
)( dsqsqsdsbe iiT . (2–28)
This qd0-model defined by (2–12) – (2–26) in state-variable form is a voltage-input and
current-output system. Therefore, this model is typically interfaced with the external
network represented in abc-phase coordinates using dependent current-sources.
2.2 Voltage-Behind-Reactance Formulations
The voltage-behind-reactance formulation takes advantage of the reference frame
transformation but leaves enough elements in abc phase coordinates to obtain direct
interfacing to arbitrary network. Due to the symmetry of induction machines structure, the
VBR formulation excluding magnetic saturation has constant interfacing circuit [32].
Inclusion of zero-sequence in [32] makes the formulation implicit. In this thesis, the
interfacing circuit of [32] is changed by adding a zero-sequence branch to avoid forming an
algebraic loop when zero-sequence current may exist.
22
Starting from the full-order induction machine model given by (2–1) – (2–7), after
algebraic manipulation the following VBR form is derived [32]
abcabcsabcsabcsabcsabcs p eiLirv (2–29)
where the inductance matrix is
SMM
MSM
MMS
abcs
LLL
LLL
LLL''
L (2–30)
and the entries are defined as
mlsS LLL 3
2 (2–31)
mM LL 3
1. (2–32)
The sub-transient magnetizing inductance is defined as
111
lrmm
LLL . (2–33)
The resistance matrix is given by
SMM
MSM
MMS
abcs
rrr
rrr
rrr''
r (2–34)
23
and the entries are defined as
r
lr
msS r
L
Lrr
2
2''
3
2 (2–35)
r
lr
mM r
L
Lr
2
2''
3
1 . (2–36)
The sub-transient back EMF voltages in (2–29) in the stationary reference-frame are
defined as
Tdqsabc ee ]0[
1''
Ke (2–37)
where
qrlr
mqrq
lr
rmdrq v
L
L
L
rLe
''''
2
'''''' (2–38)
drlr
mdrd
lr
rmqrd v
L
L
L
rLe
''''
2
'''''' . (2–39)
The sub-transient flux linkages of q- and d-axis are defined by
lr
qrmq
LL
(2–40)
lr
drmd
LL
. (2–41)
24
The rotor subsystem state model is defined in terms of the rotor flux linkages as the state
variables, as following
qrdrrmqqrlr
rqr v
L
rp )()( (2–42)
drqrrmddrlr
rdr v
L
rp )()( (2–43)
where the magnetizing fluxes are
qqsmmq iL (2–44)
ddsmmd iL . (2–45)
There are a few possible equations to calculate the developed electromagnetic torque [1].
For this formulation, the torque can be calculated by using the magnetizing fluxes as
)(22
3dsmqqsmde ii
PT . (2–46)
which in per-unit is expressed as
)( dsmqqsmdbe iiT . (2–47)
Note that in (2–29), the inductance matrix (2–30) and resistance matrix (2–34) are
independent of reference frame and constant, which is a consequence of machine
symmetry. These are very desirable numerical properties that make the VBR model more
efficient than the CC-PD model (see VBR-I in [32]). However, using (2–29) directly to
25
interface the machine to the network requires implementation of mutual inductances and
resistances which are not available in the majority of simulation programs.
A further simplification is achieved by diagonalizing the inductance and resistance matrices
(2–30) and (2–34) using the stator current and zero-sequence current as [29]
scsbsas iiii 03 . (2–48)
Taking derivative of (2–34) gives
scsbsas pipipipi 03 . (2–49)
The off-diagonal terms in (2–30) and (2–34) may be eliminated by expressing the stator
currents associated with the off-diagonal entries in terms of the zero-sequence current and
the remaining phase (diagonal) currents. After algebraic manipulations, (2–29) can be
written as
ssabcabcsDabcsDabcs pLrpLr 0000'' 33 iieiiv . (2–50)
where
Tssss iii ][ 0000 i (2–51)
r
lr
msD r
L
Lrr
2
2 (2–52)
mlsD LLL (2–53)
26
r
lr
mM r
L
Lrr
2
2
03
1 (2–54)
mM LLL 3
10 . (2–55)
If the stator winding neutral is floating, then the zero-sequence will not be present and (2–
50) will be simplified. However, for the general purpose interfacing circuit, the zero-
sequence has to be included. The corresponding voltage equation is
ssss pirv 000 (2–56)
where
slss iL 00 (2–57)
that gives the additional state equation
)(1
000 sssls
s irvL
pi . (2–58)
Substitution of (2–58) into (2–50) makes the following VBR formulation (See VBR-III in
[32])
sls
sls
sabcabcsDabcsDabcs
L
L
L
rLrpLr 0
00
00
'' 333 vieiiv
(2–59)
where
27
Tssss vvv ][ 0000 v (2–60)
Implementation of (2–59) requires only constant parameter decoupled RL-branches and
basic built-in circuit elements, which is a major advantage. However, if the zero-sequence is
included, the formulation (2–59) becomes implicit with respect to the stator branch
voltages abcsv through the corresponding zero-sequence voltage sv0 , which are
algebraically related. Therefore, a direct implementation of (2–59) with the zero-sequence
(as may be required in a general case) will result in an algebraic loop if the terminal voltage
abcsv is unknown, e.g. when the machine is in series with inductive elements. Presence of
algebraic loops requires iterative solutions that extensively increase the computational
burden. However, the algebraic loop can be avoided by adding a zero-sequence branch to
the interfacing circuit.
2.2.1 Explicit formulation with zero-sequence branch in the
interfacing circuit
To avoid an implicit formulation, the stator voltage equation (2–50) can be separated into
two parts
ngabcnabcs vvv . (2–61)
In the above equation, abcnv is the phase-to-neutral voltages for the RL interfacing
branches, which is
28
''abcabcsDabcsDabcn pLr eiiv . (2–62)
It is important to distinguish abcnv from the stator phase voltages of the original induction
machine, abcsv . Moreover, the voltage T
ngngngng vvv ][v defines a new branch for
representing the optional zero-sequence of the interfacing circuit, that is
ngngssng piLiripLirv 000000 )3()3( . (2–63)
Here, the variable ngi is the zero-sequence branch current. Equations (2–61), (2–62), and
(2–63) , altogether, define the four constant RL branches depicted in Figure 2-2 that are
needed for direct and explicit interface of machine models with external electrical circuits.
The dependent voltage-sources abce are functions of the branch current and determined by
the rotor subsystem (2–37), (2–38), and (2–39). If the zero-sequence is not needed, the
branch ng is simply removed leaving the neutral point n floating.
Figure 2-2 General interfacing circuit for induction machine VBR formulation.
29
2.3 Interfacing of Induction Machine Models in State-Variable-
Based Simulation Programs
In the state-variable-based environment (e.g. MATLAB/Simulink [15], [16]), the qd0 model
of induction machines (and synchronous machines as will be explained in section 3.5) are
interfaced to the external network (typically represented in abc phase coordinates) using
three-phase voltage-controlled current-sources [8]. These models have voltage-input,
current-output formulation as depicted in Figure 2-3. Whenever it is possible to feed the
machine model from either capacitors or the voltage-sources, for example, this interface
becomes input-output compatible and explicit. However, the direct interfacing of machines
models as current-source in series to inductive branches (which is in fact very common)
represents a challenge. The state model generation algorithm is unable to formulate a
proper state-space equations when the external circuit is inductive [8]. In this case, the
interfacing is typically resolved using a resistive (e.g. see in Figure 2-3) or capacitive
snubber circuit. The snubber leads to numerical disadvantages, namely lose of accuracy
and increased numerical stiffness.
The main goal of the VBR formulation is to achieve a direct interface of the machine model
with the external (inductive) networks as depicted in Figure 2-4. Moreover, to make the
interface very effective and easy to use in most simulation packages, the interfacing circuit
must be composed of very simple and constant-parameter branch elements, which is
achieved for induction machine as presented in section 2.2.
30
Figure 2-3 Interfacing of induction machine qd0 models with external electrical network.
Figure 2-4 Interfacing of induction machine VBR models with external electrical network.
2.3.1 Examples of implementation in SimPowerSystems, ASMG, and
PLECS toolboxes
The formulation proposed in subsection 2.2.1 is easy to implement in almost any available
simulation program. The implementation in the SimPowerSystems (SPS) [19] toolbox is
31
shown in Figure 2-5, wherein the interfacing circuit is shown inside the box. For
compactness, the machine is connected to the Thévenin equivalent circuit of a network;
however, the network could also be represented in detail. In this figure, the machine is
grounded through the resistance gr . If the machine is not grounded, the neutral point is left
unconnected without any modification to the model. To focus on the electrical model, the
mechanical system and the additional inputs and outputs are omitted in the figure. For the
other two toolboxes, PLECS [17] and ASMG [18], the electrical network is simply replaced
with one instance of ASMG-System or PLECS Circuit, respectively. The interfacing circuit
Figure 2-5 Example of implementation of the proposed induction machine VBR model in Simulink
using the SimPowerSystems toolbox.
32
Figure 2-6 Example of implementation of the proposed induction machine VBR model equivalent
interfacing circuit with external electrical network inside the PLECS Circuit block.
implementation in PLECS [17] is shown in Figure 2-6. It is also possible to implement the
control blocks inside the PLECS Circuit instead of using Simulink blocks.
The ASMG version 2 used in this thesis does not have a graphical user interface. It has been
verified that all three implementations give identical results.
2.4 Computer Studies
To assess the numerical efficiency of the proposed explicit VBR model, a simple distributed
generation system is considered here. It is assumed that a 4-pole 50-hp 60Hz induction
machine is connected to a prime mover (e.g. a wind turbine) that in the course of the study
maintains a constant speed of 1.027 pu. The system is illustrated in Figure 2-7, wherein the
33
generator is connected to a Thévenin equivalent circuit that represents the rest of the ac
grid. The machine parameters are taken from [51] and are also summarized in Appendix A.
To emulate a severe unbalanced transient condition, a single-phase-to-ground fault is
assumed in the system close to the generator feeder and the grounding resistance is set to
zero (rg = 0). Initially, the machine is in steady-state, and the fault occurs after one
electrical cycle. The fault is emulated by decreasing voltage of phase a of the equivalent
source to zero (va = 0).
The VBR model is directly connected to the RL network similar to Figure 2-5. However, the
classical qd0 model requires a three-phase snubber for interfacing as shown in Figure 2-7.
To minimize the error, a 10 pu resistive snubber is used here. The snubber adds error and
makes the system numerically stiff.
Figure 2-7 Induction generator connected to the Thévenin equivalent circuit of a network for the
single-phase-to-ground fault transient study.
34
As a reference, the system (including the machine) is represented in the synchronous
reference frame and the whole model is solved using the ode45 solver with the maximum
time-step limited to 1 μs. A list of MATLAB ordinary differential equation (ODE) solvers is
given in Appendix B. The reference model is implemented using basic Simulink blocks.
Having a very simple and linear system, the conversion to the rotational reference frame
would be straightforward in this example.
The implicit VBR-III [32] [(2–59)] model is implemented in the SimPowerSystems (SPS)
toolbox. The SimPowerSystems toolbox allows for solving algebraic loops in Simulink. The
explicit VBR (subsection 2.2.1) with the interface shown in Figure 2-2 is implemented in
the SimPowerSystems, ASMG, and PLECS toolboxes. The three implementations are shown
to give identical results.
Since the built-in qd models in the library of SimPowerSystems and PLECS do not include
the zero-sequence, the qd0 machine model is implemented using basic Simulink blocks.
This model is then interfaced to an instance of PLECS Circuit by means of controlled
current-sources.
To show the consistency of the explicit VBR and the implicit VBR-III models, both models
are run with the MATLAB solver ode45 using the same settings. The maximum and
minimum time-steps are set to 1 ms and 0.1 s, respectively. The models are implemented
in pu and the relative and absolute tolerances of the solver are set to 10−4. For the qd0
35
model with snubbers, the stiffly-stable MATLAB solver ode15s is used with settings that
are identical to those used for the other VBR models.
The simulation results shown in Figure 2-8 demonstrate the consistency between the VBR
models and the reference solution. To show more details, the three fragment windows
highlighted in Figure 2-8 are enlarged and shown in Figure 2-9 to Figure 2-11, respectively.
Figure 2-9 shows phase c stator current in steady-state, where it can be seen that both VBR
models yield exactly the same results. Moreover, the models have chosen identical time-
steps. For the qd0 model, the snubbers sink part of the machine output current and
therefore produce some error. Comparatively, the qd0 model with snubbers has chosen
several times more time-steps and has a visible steady-state error. Figure 2-10 and Figure
2-11, which show phase c stator current and electromagnetic torque during the fault,
confirm that the VBR models have no visible error in transient as well. For the qd0 model
with snubbers, the transient response has some error, although it is less visible due to the
comparatively large fault current. The stiffly-stable solver ode15s has chosen even smaller
time-steps during the transient period (after the fault) than in steady-state.
A quantitative evaluation of the considered transient study is summarized in Table 2–1.
The time-steps and calculation data are obtained from Simulink Profiler [16]. The relative
error is calculated by comparing the predicted trajectory with the reference solution using
the 2-norm error and normalizing the difference [52]. For example, the error for asi
trajectory (including all the solution points) is given by
36
Figure 2-8 Simulation results for the single-phase-to-ground fault study. From top: source voltage,
source current, machine neutral current, and electromechanical torque.
37
Figure 2-9 Detailed view of source current ics in steady-state excerpt from Figure 2-8.
Figure 2-10 Detailed view of source current ics during transient excerpt from Figure 2-8.
%100~
~
)(
2
2
as
asas
asi
iii (2–64)
where asi~
is the reference solution trajectory. The average three-phase current error
)( abcsi , which is shown in Table 2–1, is evaluated using the following
38
Figure 2-11 Detailed view of electromagnetic torque Te during transient excerpt from Figure 2-8.
)()()(3
1)( csbsasabcs iii i . (2–65)
Table 2–1 verifies that both VBR formulations are algebraically identical to the reference. It
also reveals the difference between the computational costs of the two VBR formulations
by comparing the number of sub-transient voltage calculations. Practically, the implicit
VBR-III is significantly slower since it requires iterations in each time-step for the algebraic
loop solution (3865 calculations compared to 764 for the explicit VBR).
The qd0 model is explicit but numerically stiff, and thus it has used several times more
time-steps (989 compared to 110 times for the VBR models). The qd0 model has also more
number of internal current calculations (7,070 times) than the implicit VBR sub-transient
voltage calculations (3,865 times). As shown in Table 2–1, the largest eigenvalue of the qd0
model with snubber is several orders of magnitude bigger than the largest eigenvalue of
the VBR models. The eigenvalues of system are found by linearizing the model around
operating point by using MATLAB/Simulink functions.
39
Based on the study presented in this section, as well as the ones in [32], the explicit VBR
model is suggested as the general purpose model for squirrel-cage induction machines in
state-variable-based simulation programs. This model yields identical results to the
reference and does not require a snubber when connected to an inductive network/system,
while offering high accuracy, numerical stability, and simulation efficiency.
Table 2–1 Comparison of Numerical Efficiency of VBR Formulations for Single-Phase Fault Study
Simulation Parameter
Considered Formulations
Implicit VBR-III Explicit VBR qd0
Number of Major Outputs/Steps calculations
110 110 989
Number of Internal Minor Calculations*
3,865 764 7,070
Current abcsi Prediction Error 0.000 % 0.000 % 2.861 %
Largest eigenvalue –199 ± j118 –199 ± j118 –1.18×105
* This row shows the number of the sub-transient voltage calculations for the VBR models and the
number of injected current calculations for the qd0 model.
40
CHAPTER 3: CIRCUIT INTERFACING OF SYNCHRONOUS MACHINE MODELS
Synchronous machines typically have asymmetrical rotor structure and one field winding.
The filed winding is essential for the control of the machine voltage and stability of power
systems. The field winding and rotor saliency makes the synchronous machine models
more complex than induction machine models. Specifically, the rotor asymmetry makes it
more challenging to achieve the constant-parameter VBR model for the synchronous
machines compared to symmetrical induction machines. In this Chapter, classical modeling
of synchronous machines for power systems transient studies and the formulation of the
models for efficient numerical implementation is provided. The classical voltage-behind-
reactance (VBR) formulation is reviewed for completeness as well. A state-of-the-art VBR
model is used as the basis for deriving the new synchronous machine VBR models with the
same constant-parameter interface as the induction machine VBR models proposed in
Chapter 2. Extensive computer studies presented in this chapter will demonstrate the
accuracy and benefits of the proposed models.
41
3.1 Synchronous Machine Modeling
3.1.1 Coupled-circuit phase-domain model
To familiarize the reader with basics of synchronous machine modeling for transient
studies of power system, the classical model is reviewed. Similarly to the induction
machines, synchronous machines are represented by lumped-parameter magnetically-
coupled circuit-based models. Such general-purpose models have the advantage of
simplicity while yielding a relatively small number of equations [1]. A synchronous
machine may have a round- or salient-pole rotor and usually has one field winding. A basic
three-phase wye-connected synchronous machine is considered here. It is assumed that
the machine has M damper windings in the q-axis and N damper windings plus a field
winding in the d-axis. The magnetically coupled-circuit representation of the synchronous
machine model is shown in Figure 3-1. For simplicity of equations, in this subsection only,
it is assumed that the machine has two damper windings in q-axis, and one damper
winding in d-axis. The standard assumption of sinusoidal distribution of winding is also
used here. All variables are referred to the stator side. The voltage equations in terms of
machine physical variable are [1]
qdr
abcs
rrT
sr
srss
qdr
abcs
pp
pp
i
i
LrL
LLr
v
v
3
2 (3–1)
where the voltage and current vectors are defined as
42
Figure 3-1 Magnetically coupled circuit model of synchronous machine.
Tcsbsasabcs ffff (3–2)
Tkdfdkqkqqdr ffff 121f (3–3)
where f can represent v or i. The resistance and inductance matrices are
ssss rrrdiagr (3–4)
121diag kdfdkqkqr rrrrr (3–5)
43
)3
2(2cos)(2cos2
1)
3(2cos
2
1
)(2cos2
1)
32(2cos)
3(2cos
2
1
)3
(2cos2
1)
3(2cos
2
12cos
rBAlsrBArBA
rBArBAlsrBA
rBArBArBAls
s
LLLLLLL
LLLLLLL
LLLLLLL
L
(3–6)
mdlkdmd
mdmdlfd
mqlkqmq
mqmqlkq
r
LLL
LLL
LLL
LLL
1
2
1
00
00
00
00
L (3–7)
)3
2sin()
3
2sin()
3
2cos()
3
2cos(
)3
2sin()
3
2sin()
3
2cos()
3
2cos(
sinsincoscos
rmdrmdrmqrmq
rmdrmdrmqrmq
rmdrmdrmqrmq
sr
LLLL
LLLL
LLLL
L. (3–8)
In the above equations, mqL and mdL are the magnetizing inductance of the q- and d-axis
respectively; lsL is the leakage inductance of the stator, and ljL is the leakage inductance of
the rotor windings. Here, the index j denotes the windings, 1kq , 2kq , fd , and 1kd . The
electromagnetic torque equation in machine variables is given by
qdrsr
r
Tabcsabcslss
r
Tabcse L
PT iLiiILi ][)(][)(
2
1
2 . (3–9)
The extended form of the above equation is given in [1. see p. 198]. The mechanical system
can be modeled similar to induction machine discussed in Chapter 2 by using (2–8) or (2–
9).
44
Equations (3–1) – (3–9) define the classical CC-PD model. For direct interfacing to arbitrary
networks, this model may be implemented in simulation programs that have ability to
include variable inductances such as PLECS [17] and ASMG [18]. The CC-PD model is
computationally expensive and is less numerically efficient than the VBR models [28] (also
see section 3.2).
3.1.2 The qd0 model in rotor reference-frame
The time-varying inductances from the voltage equations are eliminated if the variables are
transformed to the rotor reference-frame by the well-known Park’s transformation [7]. The
transformation matrix is given by (2–11), where dtd rr / . Using Park’s transformation,
the equations become more compact and the rotor-position-dependency is removed. Here,
considering simplicity of equations unlike subsection 3.1.1, the general case with M damper
windings in q-axis and N damper windings in d-axis is considered. The change of variables
applied to (3–1) results in Park’s voltage equations
qsdsrqssqs pirv (3–10)
dsqsrdssds pirv (3–11)
ssss pirv 000 (3–12)
Mjpirv kqjkqjkqjkqj ,1, (3–13)
fdfdfdfd pirv (3–14)
Njpirv kdjkdjkdjkdj ,1, (3–15)
45
where
mqqslsqs iL (3–16)
mddslsds iL (3–17)
slss iL 00 (3–18)
MjiL mqkqjlkqjkqj ,1, (3–19)
fdfdlfdfd iL (3–20)
NjiL mdkdjlkdjkdj ,1, (3–21)
The magnetizing fluxes are
)(1
M
j
kqjqsmqmq iiL (3–22)
)(1
N
j
kdjfddsmdmd iiiL (3–23)
The circuit representation for the above equations is given in [1, see p. 202] and is not
included here. The electromagnetic torque can be calculated by
)(22
3qsdsdsqse ii
PT (3–24)
which in per-unit becomes
)( qsdsdsqsbe iiT . (3–25)
46
The mechanical system is similar to the one Chapter 2 for induction machines, however
since synchronous machines are usually used as generators, equation (2–8) becomes [1]
eMr TTJ
Pp
2 (3–26)
where TM is the mechanical input torque from the prime mover. Equation (3–26) in per-
unit becomes
eMb
r TTH
p
2
1
(3–27)
Similar to the induction machine model, the classical qd0 synchronous machine model in
state-variable form has voltage-input and current-output. This model is therefore
interfaced to abc networks by the means of voltage-controlled current-sources.
3.2 Voltage-Behind-Reactance Formulations
The synchronous machine voltage-behind-reactance model [28] partially solves the
interfacing issue by changing the model structure to branches consisting of voltage-sources
behind series RL elements. However, the asymmetry of the rotor of synchronous machine
makes their VBR formulation more complex compared to the induction machines
presented in Chapter 2. This asymmetry is not only due to the possible uneven paths for
the magnetic flux in the q- and d-axis (salient-pole rotor vs. round-pole rotor), but also due
47
to the presence of the rotor field and damper windings, which together result in unequal
parameters of the sub-transient equivalent circuits known as dynamic saliency.
3.2.1 Variable-parameter formulation
The original algebraically-exact voltage-behind-reactance synchronous machine model was
proposed in [28]. Unless there is no dynamic saliency, in general, this formulation has
variable impedance interfacing circuit. The resistance matrix of the interfacing circuit can
be made constant by transferring its time-variant components into the sub-transient
voltage-source [29]. The final stator voltage equation is
'''' ])([ abcsabcsrabcsabcssabcs pr eiLiv . (3–28)
where p denotes Heaviside’s operator (differentiation with respect to time). The sub-
transient inductance matrix is
)3
2(2cos)(2cos2
1)
3(2cos
2
1
)(2cos2
1)
32(2cos)
3(2cos
2
1
)3
(2cos2
1)
3(2cos
2
12cos
rBAlsrBArBA
rBArBAlsrBA
rBArBArBAls
abcs
LLLLLLL
LLLLLLL
LLLLLLL
L
(3–29)
where
3
mqmdA
LLL
(3–30)
48
3
mqmdB
LLL
. (3–31)
The sub-transient magnetizing inductances are determined by
1
1
11
M
j lkqjmqmq
LLL (3–32)
1
1
111
N
j lkdjlfdmdmd
LLLL . (3–33)
The sub-transient inductance of the q- and d-axis are
mqlsq LLL (3–34)
mdlsd LLL . (3–35)
The inductance matrix (3–29) is rotor-position dependent and time variant if 0BL or
mdmq LL (or dq LL ). The inequality of sub-transient inductances is referred to as
dynamic saliency. In general, the sub-transient magnetizing inductances mqL and mdL are
not equal even if the rotor is round due to influence of field and damper windings [see (3–
32) and (3–33)].
The sub-transient sources are given by
Tdqsabc ee ]0[
1 Ke (3–36)
49
where sK is Park’s transformation matrix from stationary abc to rotor qd0 reference
frame given by (2–11) [1]. The sub-transient voltages for this model are
qsmq
M
j lkqj
kqjM
j
kqjq
lkqj
kqjmqdrq iL
L
r
L
rLe
2
12
12
)(
(3–37)
dsmd
N
j lkdj
kdj
lfd
fdfdd
lfd
fdmdfd
lfd
mdN
j
kdjd
lkdj
kdjmdqrd iL
L
r
L
r
L
rLv
L
L
L
rLe
2
1222
12
)()(
.
(3–38)
The sub-transient flux linkages are defined by
M
j lkqj
kqjmqq
LL
1
(3–39)
N
j lkdj
kdj
lfd
fdmdd
LLL
1
. (3–40)
The rotor state equations are
MjL
rp mqkqj
lkqj
kqjkqj ,1,)( (3–41)
fdmdfdlfd
fdfd v
L
rp )( (3–42)
NjL
rp mdkdj
lkdj
kdjkdj ,1,)( . (3–43)
50
The magnetizing fluxes are calculated by
qqsmqmq iL (3–44)
ddsmdmd iL . (3–45)
Electromagnetic torque in SI units is calculated using the same equation as induction
machine given by (2–46).
This VBR formulation has two parts. The first part defines the interfacing circuit given by
(3–28) and the variable coupled inductance matrix (3–29). The second part includes the
sub-transient voltages (3–37) and (3–38) that are the link between the rotor subsystem
[with state equation given by (3–41) – (3–43)] and the interfacing circuit.
While the numerical properties of the classical VBR model are improved compared to the
CC-PD model (subsection 3.1.1), the variable mutual inductances of (3–29) are still a
concern that will be considered in this chapter and the following one.
3.2.2 Standard state-space form for the rotor subsystem
For an easier implementation, the rotor subsystem state equations can be written in a
standard state-space form:
DUCe
BUAλλ
)(''rqd
p
(3–46)
51
where the state vector is
kdNkdfdkqMkqT 11λ (3–47)
and the inputs are
fddsqsT viiU . (3–48)
The matrices A and B are defined as
22)1(
)1(11
A0
0AA
MN
NM (3–49)
221)1(
211
B0
0BB
N
M (3–50)
where the sub-matrices are defined by
1
1
1
21
2
2
22
2
12
2
1
1
21
1
11
1
11
lkqM
mq
lkqM
kqM
lkqlkqM
mqkqM
lkqlkqM
mqkqM
lkqMlkq
mqkq
lkq
mq
lkq
kq
lkqlkq
mqkq
lkqMlkq
mqkq
lkqlkq
mqkq
lkq
mq
lkq
kq
L
L
L
r
LL
Lr
LL
Lr
LL
Lr
L
L
L
r
LL
Lr
LL
Lr
LL
Lr
L
L
L
r
A (3–51)
52
1
1
1
1
21
2
2
22
2
12
2
2
2
1
1
21
1
11
1
1
1
21
22
lkdN
md
lkdN
kdN
lkdlkdN
mdkdN
lkdlkdN
mdkdN
lfdlkdN
mdkdN
kdNlkd
mdkd
lkd
md
lkd
kd
kdlkd
mdkd
lfdlkd
mdkd
kdNlkd
mdkd
kdlkd
mdkd
lkd
md
lkd
kd
lfdlkd
mdkd
lkdNlfd
mdfd
lkdlfd
mdfd
lkdlfd
mdfd
lfd
md
lfd
fd
L
L
L
r
LL
Lr
LL
Lr
LL
Lr
LL
Lr
L
L
L
r
LL
Lr
LL
Lr
LL
Lr
LL
Lr
L
L
L
r
LL
Lr
LL
Lr
LL
Lr
LL
Lr
L
L
L
r
A (3–52)
lkqM
mqkqM
lkq
mqkq
lkq
mqkqT
L
Lr
L
Lr
L
Lr
2
2
1
1
11B (3–53)
00012
2
1
1
22
lkdN
mdkdN
lkd
mdkd
lkd
mdkd
lfd
mdfdT
L
Lr
L
Lr
L
Lr
L
Lr
B . (3–54)
If G and H are defined as
M
j lkqj
kqjmq
L
rLG
12
2
(3–55)
N
j lkdj
kdjmd
lfd
fdmd
L
rL
L
rLH
12
2
2
2
. (3–56)
then )( rT C and D will become
53
2
22
2
22
21
1
11
2
2
22
2
2
2
12
1
1
1
)(
lkdN
mdkdN
lkdNlkdN
mdr
lkd
mdkd
lkdlkd
mdr
lkd
mdkd
lkdlkd
mdr
lfd
mdfd
lfdlfd
mdr
lkqM
mqr
lkqM
mqkqM
lkqM
lkq
mqr
lkq
mqkq
lkq
lkq
mqr
lkq
mqkq
lkq
rT
L
Lr
L
H
L
L
L
Lr
L
H
L
L
L
Lr
L
H
L
L
L
Lr
L
H
L
L
L
L
L
Lr
L
G
L
L
L
Lr
L
G
L
L
L
Lr
L
G
C
(3–57)
lfd
md
L
LH
G
0
00
D (3–58)
3.3 Approximation of Dynamic Saliency in VBR Models
The inductance matrix (3–29) will be constant if BL [(3–31)] is equal to zero. This is
achieved if the sub-transient inductances are equal, that is dq LL (or mdmq LL ) which is
not true in a general case. The sub-transient inductances are related to the operational
impedances [30]. In particular, in very high frequencies range, the leakage inductances of
the rotor windings (circuits) will dominate the resistances. Consequently, the value of the
operational impedances in the q- and d-axis equivalent circuits (normalized with respect to
54
frequency) will become equal to the sub-transient inductances qL and dL , respectively.
The traditional approximation based on averaging qL and dL was shown to reduce the
model accuracy however other accurate approximation methods have been suggested in
the literature [28], [30].
To better explain the proposed models, and for consistency of derivations, the methods for
removing the dynamic saliency are briefly explained here.
3.3.1 Additional winding method
A pioneering technique based on addition of an artificial damper winding was proposed in
[30]. This method is based on adding one extra damper winding to the rotor circuit with
the purpose of enforcing numerical equality of qL and dL . In doing so, the added winding
should have sufficiently high resistance as not to impact the low-frequency operational
impedance of the rotor circuit. But otherwise, the added winding does not have any
physical meaning with respect to the original parameters of a given machine. Since the sub-
transient inductance in d-axis is typically smaller, the additional winding is normally added
to the q-axis equivalent circuit. Therefore, denoting this added winding as (M +1), its
leakage inductance is calculated based on (3–32) and (3–33) as
1
)1(
11
mqmdMlkq
LLL . (3–59)
55
This also adds one more state equation to (3–41), where 1,1 Mj .
To distinguish the sub-transient quantities of the approximate model that uses the (M +1)th
winding, triple-prime sign (''' ) instead of double-prime sign ('' ) is used here. Thus,
including this additional winding gives the desired result
mdmdmq LLL (3–60)
which makes 0BL in (3–29) and results in a constant inductance matrix
MSS
SMS
SSM
abcs
LLL
LLL
LLL
L (3–61)
where
mdlsS LLL 3
2 (3–62)
mdM LL 3
1. (3–63)
Thus, the voltage equation (3–28) becomes
''abcsabcs
SMM
MSM
MMS
abcssabcs p
LLL
LLL
LLL
r eiiv
. (3–64)
Here, the subscript M in ML should not be confused with the winding indexing.
56
For either round rotor or salient pole synchronous machines, the parameters are such that
qd LL . However, for the sub-transient inductances (reactances) this relationship is
typically reversed, i.e. dq LL (or dq XX ). A table with typical parameters for the
synchronous generators (turbo and hydro), condensers, and motors is found in [5, see p.
40, Table 2], wherein the condition dq XX holds for the low, average, and high ranges of
typical parameters. The same conclusion can be made for the synchronous machine
parameters found in [6], as well as in [1] (where for the hydro generator, dq XX ). The
condition dq XX follows from the fact that the field winding leakage inductance is
typically small, which makes the dX smaller even when qd XX . Based on this
observation, the additional artificial winding would be normally added to the q-axis
equivalent circuit and its leakage inductance calculated using (3–59) will have positive
value (see Appendix C and Appendix D). However, if for some reason one has dq XX ,
then the winding can be added to the d-axis instead in order to remove the numerical
saliency and achieve a constant-parameter voltage equation as given by (3–64).
3.3.2 Singular perturbation method
The artificial winding adds an additional state variable and a fast dynamic mode. The
disadvantageous effect of this winding can be “undone” at the expense of making this
formulation implicit, which is achieved using the singular perturbation [31].
57
It is observed that when the resistance of additional winding is approaching infinity, the
largest eigenvalue of the system is also going to infinity. In this method, the aim is to
remove the fast state from the model. To adequately separate the fast dynamic due to
additional winding from the other dynamics of the model, a new state variable is defined as
the voltage across the additional winding resistance as [31]
)1()1()1( MkqMkqMkq irv (3–65)
which yields to the following state equation after required algebraic manipulation
M
j
mqkqj
lkqj
kqjmdqssqdsdrqs
d
mdMkq
lkqM
mdMkq
L
rLireiLv
L
Lv
L
Lpv
12)1()1( )()(1
(3–66)
where )1()1( / MkqMkq rL . The new sub-transient voltage of the q- and d-axis become
qs
M
j lkqj
kqjmdMkq
Mlkq
mdM
j
kqjq
lkqj
kqjmddrq i
L
rLv
L
L
L
rLe
1
2
2)1(
)1(12
)( (3–67)
ds
N
j lkdj
kdj
lfd
fdmdfdd
lfd
fdmdfd
lfd
mdN
j
kdjd
lkdj
kdjmdqrd i
L
r
L
rL
L
rLv
L
L
L
rLe
122
2
21
2)()(
(3–68)
where the q-axis sub-transient flux is defined by
M
j Mlkq
Mkq
lkqj
kqjmdq
LLL
1 )1(
)1( . (3–69)
58
If the winding resistance approaches infinity, goes to zero which makes (3–66) an
algebraic equation
0)1()1( MkqMkq pvthenrif . (3–70)
Finding )1( Mkqv from (3–66) and substituting qe from (3–67) after simplification gives
)(1
1
)1(
2
)1()1( drdsdrqs
d
md
Mlkqd
md
Mlkq
mdMkq iLv
L
L
LL
L
L
Lv
M
j
qssd
md
d
md
lkqj
mdkqjM
j
mqkqjd
md
lkqj
mdkqjir
L
L
L
L
L
Lr
L
L
L
Lr
12
2
12
1)(1 (3–71)
where the approximate magnetizing flux is defined as
qqsmdmq iL . (3–72)
To remove the additional winding parameters from the voltage equations, it is considered
that
mqmqMkq thenrif )1( . (3–73)
Therefore, it can be assumed that mqmq , which considering (3–44) and (3–72) gives
qqsmdmqq iLL )( . (3–74)
Combining (3–39) with the above equation yields
59
M
j lkqj
kqjmqqsmdmqq
LLiLL
1
)(
(3–75)
which is the flux linkage equation independent of additional winding parameters.
In summary, this method uses the same number of state-variables as the variable
impedance VBR model, but nonetheless it has a constant-parameter interface as (3–64).
The sub-transient flux linkage of q-axis are given by (3–75) and the sub-transient voltage
equations are defined by (3–67) and (3–71), and (3–68).
If the machine is connected to an infinite bus, then the bus voltage qsv is readily available,
thus (3–71) and consequently (3–67) do not require the solution for the branch voltages
and the overall formulation becomes explicit. However, in a general case, voltage qsv is
calculated from the stator branch voltages abcsv that is solved together with the external
network which has the sub-transient voltages abce as input. This implicit relationship
between abcsv and abce forms an algebraic loop that has to be solved iteratively by the
program solver.
It should be pointed out that because the effect of additional (artificial) winding has been
removed (up to infinite frequency), this formulation becomes algebraically equivalent to
the original VBR model with rotor-position-dependant inductances. Further algebraic
derivations and computer studies confirm this conclusion.
60
3.4 Generalized Constant-Parameter VBR Formulation
The interfacing circuit introduced in Chapter 2 (see Figure 2-2) consists of constant
parameters decoupled RL branches and includes the zero-sequence. These are much
desired properties for simple and direct interface of ac electric machine models to arbitrary
networks. In this section, the constant-parameter VBR formulations (achieved by adding an
artificial winding or using singular perturbation) are modified to present the same
interfacing circuit as it was achieved for the induction machine. Here, again the off-diagonal
elements of inductance matrix are removed by incorporation of the zero-sequence current
given by (2–48). More specifically, considering (3–64) and defining
MSD LLL (3–76)
MLL 0 (3–77)
yields the following voltage equation
''00 )3( abcsabcsDabcssabcs pLpLr eiiiv (3–78)
Consideration of zero-sequence for modeling of synchronous machines is more important
than it was in case of induction machines. In many instances of synchronous generators,
the neutral point of the stator winding is grounded through external impedance for
protection and monitoring purposes. Therefore, it is important to develop a general
purpose interfacing circuit with zero-sequence that will be fully capable of predicting
61
unbalanced faults and grounding currents for different grounding classes (low/high
resistance/impedance, etc.) and protection requirements [53].
Similar to the induction machine formulation, a fourth branch for representing zero-
sequence is expressed using (2–61), where
abcabcsDabcssabcn pLr eiiv (3–79)
The zero-sequence branch is defined by the voltage equation
ngsng piLipLv 000 )3( . (3–80)
Unlike the induction machine circuit, the zero-sequence branch here does not have a
resistive part. Instead, the resistance is left in the sub-transient voltage-sources defined by
(3–37) and (3–38). It can be pointed out that it is possible to move equal parts of resistance
from the sub-transient voltage-sources to the voltage equation, which would add a resistive
component to the zero-sequence branch and change the value of the interfacing circuit
resistances. However, this step does not eliminate the direct feed-through of the stator
currents (through sub-transient voltage equation) due to unequal resistances of the rotor
q- and d-axis circuits. Such feed-through will require a current-dependent voltage-source
which is allowed in most simulation programs.
Eventually, (2–61), (3–79), and (3–80) define the four constant and decoupled RL-branches
depicted in Figure 2-2. This circuit is used to interface the synchronous machine model
with the external network. The circuit parameters are
62
sD rr (3–81)
mdlsD LLL (3–82)
00 r (3–83)
mdM LLL 3
10 . (3–84)
In summary, the sub-transient voltage-sources defined by (3–37) and (3–38) with the
interfacing circuit given in Figure 2-2 and state equations (3–41), (3–42), and (3–43) define
the constant-parameter synchronous machine VBR formulation.
To implement a constant parameter VBR using the additional winding technique, the
standard state-space A, B, C, and D matrices given in subsection 3.2.2 can be used.
However, using the singular perturbation method requires modification of the output
equations to incorporate the additional winding voltage )1( Mkqv into the sub-transient
voltages.
3.4.1 Possible explicit and implicit formulations
The approximation of dynamic saliency using additional winding described in subsection
3.3.1 (and used before in this section to achieve a generalized interfacing circuit) achieves
the explicit implementation. The additional winding inductance is given by (3–59), but the
user must choose the winding resistance. In general, choosing this resistance very large
will improve the accuracy of this approximation at the expense of making the system
63
numerically stiffer. Otherwise, this method gives zero error in steady-state since the effect
of damper windings in this condition diminishes.
As mentioned in subsection 3.3.2, by removing the fast state using singular perturbation
method the algebraically exact model is achieved but an algebraic loop is added that makes
the formulation implicit. Additional iterations will be required to solve the algebraic loop.
Therefore, this approach can make the solution computationally very expensive.
3.5 Interfacing of Synchronous Machine Models in State-Variable-
Based Simulation Programs
Interfacing of synchronous machine qd0 models in state-variable-based simulation
programs is similar to the induction machines case shown in Figure 2-3 in Chapter 2.
However, due to asymmetrical rotor structure and dynamic saliency, interfacing of
synchronous machine VBR models is more complex as shown in Figure 3-2 for the classical
VBR formulation with variable parameters and Figure 3-3 for the constant-parameter VBR
formulation presented earlier in this chapter.
Specifically, without approximation of dynamic saliency, as shown in Figure 3-2, the
synchronous machine VBR models will contain coupled and variable inductances (and even
resistances [28]), which precludes forming the linear time invariant (LTI) state-space
model with constant matrices A, B, C and D. A constant resistance matrix is obtained by
algebraic feed-through of the stator currents to the sub-transient voltage sources [29] [see
64
(3–37) and (3–38)], i.e. rqdsqdrqd g ,i,λe . This algebraic feed-through, however, does
not generally represent any issue for simulation programs since the current-controlled
voltage-source can be easily connected with inductive branches.
A basic inductive branch equation that is solved within the external circuit-system has the
form edtLidriv / . Therefore, an algebraic feed-through is being formed within the
circuit solver from the input controlled voltage-sources e to the output branch voltages v.
In some constant-parameter VBR formulations as illustrated in Figure 3-3, to calculate the
remaining machine state-variables, in addition to the branch (stator) currents i, the (stator)
branch voltages v may also be required for the following reasons:
To approximate the dynamic saliency [31] (also see Chapter 4, section 4.2);
To include the zero-sequence [32];
To include saturation [33]; etc.
If the sub-transient voltages in machine subsystem are calculated in terms of the rotor
state-variables and the stator currents, i.e. rqdsqdrqd g ,i,λe , then the algebraic loop is
not formed as it is the case in [28], [33]. However, in these models the interfacing branches
have variable impedances in a general case. Conversely, whenever (in addition to the rotor
states) the branch voltages v are required for calculating the controlled voltage-sources e,
i.e. rqdsqdsqdrqd g ,v,i,λe , an algebraic loop is being formed as depicted in Figure 3-3.
65
Figure 3-2 Interfacing synchronous machine classical VBR models with external electrical network-
circuit with algebraic feed-through of current.
Figure 3-3 Interfacing synchronous machine constant-parameter VBR models with external electrical
network-circuit with algebraic feed-through of current and possible algebraic loop for voltage.
The existence of such algebraic loop makes the overall system of equations of the external
circuit and the machine subsystem implicit, which generally results in invoking an iterative
66
solution process and leads to increase of computing time [16], [54]. It is therefore very
desirable firstly to have a constant-parameter decoupled interfacing circuit such that the
LTI state-space model with constant matrices A, B, C and D can be formed; and secondly,
to have an explicit formulation that does not require the branch voltages for the machine
subsystem.
3.6 Implementation of Synchronous Machine VBR Formulations in
Simulation Programs
Constraints specific to individual toolboxes may affect the implementation of the entire
system model and simulation accuracy, especially when other components such as
switches and transformers are used. For example, in SimPowerSystems (SPS) [19], the
ideal transformers are only allowed in simple cases when no series inductive elements
exists. In PLECS, the ideal transformers are also available, but this toolbox cannot solve
algebraic loops and prevents the use of constant time step ODE solvers if switches (such as
diodes) are used in the system. There is no ideal switch model in SimPowerSystems, and
typically some snubber circuits are required. The PLECS toolbox is the only toolbox that
presently has built-in VBR machine models, but these models have variable inductances.
The ASMG toolbox can implement the variable impedances for a machine of arbitrary
configuration. The SimPowerSystems toolbox does not have variable impedance elements.
67
The rotor model for the explicit formulation could be implemented using the standard
state-space A, B, C, D matrices given in subsection 3.2.2, where an additional winding is
added to the system. For the implicit formulation, A and B matrices stay the same as in the
original VBR formulation (without the extra winding). However, the output equations are
given by (3–67), (3–71), (3–68), and (3–75). Therefore, C and D matrices should be
modified accordingly.
3.6.1 Examples of implementation in SimPowerSystems, ASMG, and
PLECS toolboxes
An example of implementing the explicit VBR model in SimPowerSystems V5.1 toolbox [19]
is shown in Figure 3-4, wherein the rotor subsystem is implemented using conventional
Simulink blocks. To clearly show the electrical circuit interfacing, some of the outputs and
the mechanical system are not shown. For the ASMG toolbox [18], the electrical circuit of
Figure 3-4 is replaced with one instance of the ASMG-System, wherein the circuit may be
defined using text (or graphical representation in the newer version of this toolbox). The
algebraic feed-through in the rotor subsystem causes false detection of algebraic loops in
SimPowerSystems if any voltmeter block is used. For the ASMG toolbox, the model settings
are changed to allow algebraic feed-through. A false detection of algebraic loops needlessly
calls a routine that may significantly decrease simulation speed.
68
Figure 3-4 Example of implementation of the synchronous machine CP-VBR model in
SimPowerSystems toolbox.
Figure 3-5 Example of implementation of the synchronous machine CP-VBR model in PLECS toolbox.
69
An example of implementing the CP-VBR model in the PLECS toolbox Version 3.3.1 [17] is
shown in Figure 3-5. To prevent false detection of algebraic loops, the rotor subsystem is
included inside the PLECS Circuit and it is implemented using its basic components.
3.7 Computer Studies
A 125-kW synchronous machine is considered in the first three following subsections. The
parameters of this machine are summarized in Appendix C. In the last subsection (3.7.4), a
large 555-MVA power systems generator is considered to show the accuracy of models for
different machine sizes. The parameters of the larger machine are given in Appendix D. All
considered models have been implemented using MATLAB/Simulink [16] and the PLECS
[17], ASMG [18], or SimPowerSystems [19] toolboxes. The simulation studies are
conducted on a personal computer with 2.53GHz Intel CPU.
3.7.1 Verification of the implicit VBR formulation
The 125-kW machine is originally modeled with one damper winding in the q-axis and one
damper winding in the d-axis. Wherever appropriate, one artificial damper winding 2kq is
added to the q-axis to approximate the dynamic saliency.
In the following study, a single machine infinite bus system is assumed in order to verify
the consistency of the implicit voltage-behind-reactance (IVBR) model discussed in
subsection 3.3.2. The machine is assumed to initially operate as a generator in a steady-
70
state defined by the field excitation voltage 15.2/ fdfdmdxfd vrXe pu and mechanical
torque 5.0mT pu. A three phase fault is applied at time 05.0t s to the machine terminals.
The fault is subsequently removed after 3 electrical cycles (50 ms). The reference solution
is produced by the conventional qd0 model implemented using standard Simulink blocks
and run with very small time-step of s1 to ensure high accuracy. The classical VBR model
is implemented in the ASMG toolbox [18].
3.7.1.1 Constant time-step simulation study
To verify the consistency among the explicit/exact VBR model with variable inductance
(VBR) (subsection 3.2.1) and the implicit model using singular perturbation (IVBR)
(subsection 3.3.2), these models were run with large time-step of 1 ms using the MATLAB
solver ode4. The corresponding transient responses predicted by various models are
shown in Figure 3-6. As can be seen in Figure 3-6, all models produce responses that are
visibly consistent and very close to each other. A magnified view of the current qsi shown
in Figure 3-7 (a) reveals that at such a large time-step there is a slight difference between
the subject VBR models and the reference solution. However, as can also be seen in Figure
3-7 (a), the exact variable-parameter explicit VBR and IVBR produce exactly the same
solution point by point. This is expected result since these models are algebraically
equivalent despite their different implementation as explained in subsection 3.3.2.
71
Figure 3-6 Transient response to the three phase fault predicted by the variable impedance and the
constant-parameters implicit VBR models.
3.7.1.2 Variable time-step simulation study
To compare the model efficiency, the same transient study was also run using variable
integration time-step. To achieve reasonably accurate results, here the ode15s (see
Appendix B) was used with relative and absolute error tolerances set to 10−4, and the
maximum and minimum time-step set to 10−3 and 10−7 seconds, respectively. The result of
this study looks very similar to Figure 3-6. For comparison, the same magnified view of the
current qsi is also shown in Figure 3-7 (b), which shows that both VBR models take about
72
Figure 3-7 Detailed view of current iqs from the three phase fault study shown in Figure 3-6; (a)
constant time-step solution; and (b) variable time-step solution.
the same number of time-steps to satisfy the required tolerance and the solution is visibly
very close to the reference.
3.7.2 Accuracy of the approximated constant-parameter VBR
formulation
To evaluate the accuracy of the approximate explicit model (AVBR) that has constant
parameter interfacing circuit, and specifically the choice of resistance in the additional
artificial winding 2kq , the same three phase fault study is considered here. For
73
conciseness, only the current qsi and its magnified view are shown in Figure 3-8. Moreover,
it was found that among other variables, the current qsi has the largest error due to the
extra winding. The AVBR model is run with additional winding resistance 2kqr set to 1 pu
and 10 pu As is seen in Figure 3-8 (a), when 2kqr is equal to 1 pu, there is a noticeable
difference between the predicted transient in current qsi . This difference becomes much
smaller when the resistance is increased to 10 pu.
Figure 3-8 Current iqs from the three phase fault study predicted by the approximate constant-
parameters VBR formulation: (a) overall transient; and (b) magnified view of the window in part (a).
74
To systematically evaluate the impact of winding resistance 2kqr on the solution accuracy,
the resistance value is changed from 10−3 to 103 pu and the solution error is calculated.
Here also the normalized 2-norm cumulative error defined by (2–64) in section 2.4 is
considered. To ensure consistency with the reference solution and since the encountered
error is primarily due to the saliency approximation, a time-step of s1 was used here as
well. The result of this evaluation is shown in Figure 3-9 (a). For example, for 2kqr equal to
10 pu, the cumulative error in qsi for the chosen time span is 1.7%. As can be seen in Figure
3-9 (a), increasing the additional winding resistance 2kqr improves the solution accuracy.
This accuracy improvement comes at the price of increasing the magnitude of the largest
eigenvalue of the system which results in numerical stiffness. In Figure 3-9 (b), the largest
eigenvalue calculated as a function of resistance 2kqr is also plotted. For comparison, the
largest eigenvalue of the conventional (time invariant) qd0 model and the exact VBR model
are shown in that figure as well. Similar to Chapter 2, section 2.4, the eigenvalues of system
are found by using MATLAB/Simulink functions by linearizing the models around the
steady-state operating point. The eigenvalues of qd0 model are the smallest and the most
favourable from numerical point of view if snubbers are not required. The exact VBR model
is time varying and uses the stator branch currents as the independent variables, which
results in higher magnitude of the largest eigenvalue. However, using 2kqr as large as 10 pu
in the approximate VBR model raises the largest eigenvalue to the 104 range
75
Figure 3-9 The effect of additional winding resistance on: (a) approximate model error in iqs; and (b)
the system largest eigenvalue magnitude.
3.7.3 Small machine case-study
To demonstrate the improvement achieved by the proposed constant-parameter
interfacing circuit, the synchronous machine is connected to a source with 5% inductive
impedance as depicted in Figure 3-10. The considered source with inductive impedance
76
Figure 3-10 Synchronous generator connected to an inductive network. The snubbers are required
only for the classical qd0 model.
represents the Thévenin equivalent of the external power system which is implemented as
part of the power network using the transient simulation program and its circuit-interface
(and therefore should not be combined with the machine’s stator leakages). To verify the
models in a commonly-used commercially available tool, the electrical circuit has been
implemented in the SimPowerSystems toolbox [19].
As explained in section 3.5 and also in [8], in order to interface the conventional qd0 model
with the external network (that is represented as a physical circuits in abc
variables/coordinates), a snubber circuit has to be used. Due to lack of zero-sequence in
the built-in qd0 model in SimPowerSystems, the classical qd0 model has been implemented
using standard Simulink blocks and interfaced with controlled current-sources as shown in
Figure 2-3. Aiming for reasonably high accuracy (on the order of 1%), a 100 pu resistive
snubber is used as shown in Figure 3-10. In order to emulate a severe unbalanced
condition, the machine’s neutral was assumed grounded and a single phase fault is applied
77
to phase a as depicted in Figure 3-10. This study also verifies the applicability of the
proposed interfacing circuit for modeling the unbalanced conditions with the zero-
sequence, which may be needed particularly for modeling synchronous generators [53].
The reference solution has been obtained using the exact VBR model (with variable
inductances, implemented in the ASMG toolbox [18]) that was run with time-step of s1 to
ensure very high accuracy. Calculating the cumulative error using (2–64), it was found that
the qd0 model with snubber results in about 1.2% error in the phase current csi . It was
further determined that the same level of accuracy (cumulative error) can be achieved by
the approximate VBR model if 2kqr is chosen equal to 15 pu. The transient response of the
system resulting from applying and removing this fault is shown in Figure 3-11.
The magnified view of the phase current trajectory before and during the fault is shown in
Figure 3-12 and Figure 3-13, respectively. The qd0 model has some steady-state error due
to the snubbers, which can be readily seen in Figure 3-12. However, the approximate VBR
model has zero steady-state error since the effect of the damper windings disappears in
steady-state. Moreover, since the transient phenomenon is predominantly low frequency,
the accuracy of predicting the fault currents also remains very good as can be seen in
Figure 3-13. Also, as shown in Figure 3-12 and Figure 3-13, the IVBR model has even
smaller error (about 0.1% cumulative) in transient and steady-state.
To compare the numerical efficiency of considered models, the MATLAB solver ode15s
with relative and absolute error tolerances set to 10−4, and the maximum and minimum
78
Figure 3-11 Transient response to a single phase to ground fault predicted by qd0 model with
snubber versus approximated constant-parameters VBR and exact implicit VBR models.
time-step set to 10−3 and 10−7 seconds respectively is used here (as in sub-subsection
3.7.1.2). The time-steps and CPU time taken by each model together with the magnitude of
their respective largest eigenvalues are summarized in Table 3–1 As can be seen in this
table, the 100 pu snubber interface of the qd0 model results in significant numerical
stiffness and the largest eigenvalue of 1.2×106. At the same time, the proposed interfacing
circuit and the approximate VBR model with 2kqr equal to 15 pu results in the largest
eigenvalue of 2.4×104, which is significantly smaller. This in turn explains very large
number of steps and the CPU time required by the conventional qd0 model as compared to
79
Figure 3-12 Detailed view of current ibs in steady-state from Figure 3-11.
Figure 3-13 Detailed view of current ics during the single phase fault from Figure 3-11.
the proposed approximate VBR model. This is also consistent with Figure 3-12 and Figure
3-13, wherein it can be seen that the stiffer model requires smaller time-steps to satisfy the
same error tolerances under the variable step integration.
Table 3–1 also verifies that the implicit VBR (IVBR) model is equivalent to the exact VBR
model in terms of accuracy. The IVBR model gives accurate results and even requires fewer
time-steps as shown in Table 3–1. However, in this model, an algebraic loop is formed
80
Table 3–1 Simulation Efficiency for Single-Phase Fault Study for a Synchronous Machine Models
Performance measure qd0 with 100 pu
snubber
(section 3.1.2)
Approximate VBR
with 15 pu rkq2
(section 3.3.1)
Implicit VBR
(section 3.3.2)
Number of Time-Steps 1871 557 400
Simulation Time 0.35 s 0.20 s 0.47 s
Cumulative Error of ics 1.2% 1.2% 0.1%
Largest Eigenvalue –1.2×106 –2.4×104 –1.7×102
because the sub-transient voltages abce (in addition to the rotor states) also depend on
the branch voltages abcsv (e.g. [31]). Therefore, the high accuracy of the IVBR model comes
at the price of iterative solution and generally results in longer CPU time per step or longer
overall simulation (computing) time as seen in the third row of Table 3–1. At the same
time, the AVBR model is explicit, non-iterative, and for the same local error tolerances
requires less than half of the CPU time as compared to the IVBR model. Therefore,
considering the number of integration time-steps by itself is not an accurate measure for
comparing the efficiency of models.
3.7.4 Large machine case-study
The studies presented so far were using a relatively small machine (125-kW, Appendix A).
To verify the model for larger scale power systems machines, and also to compare
81
implementation results in different simulation programs side-by-side, a larger synchronous
generator (555-MVA, Appendix C) connected to a network is implemented using three
different toolboxes PLECS [17], ASMG [18], and SimPowerSystems (SPS) [19]. Figure 3-14
shows the test system. Here, the network including a short transmission line and a unit
transformer modeled as a Thévenin equivalent circuit with impedance ZS = (2+j16)%. The
mechanical input torque is 0.9 pu, and the excitation voltage xfde is 2.35 pu. The generator
is grounded through a 0.3 pu resistance. A single-phase-to-ground fault occurs in the
source behind impedance ZS.
To obtain a constant-parameter VBR (CP-VBR) model, a winding is added to the q-axis. Its
inductance is calculated using (3–59) and its resistance is chosen to be 2.0 pu to get
approximately 1% cumulative error in the stator current. Since the SimPowerSystems and
PLECS built-in qd0 models do not include the stator zero-sequence circuit, the qd0 model
was implemented using standard Simulink blocks, and then that model was interfaced
Figure 3-14 A large synchronous generator connected to an inductive network. The snubbers are
required only for the classical qd0 model.
82
using controlled current-sources in the SimPowerSystems toolbox. A 15 pu three-phase
resistive snubber was used therein. This snubber gives a similar stator current cumulative
error (around 1%) as the CP-VBR model mentioned above. As a reference, the entire
system implemented in conventional Simulink blocks (without any snubbers) is solved
using the ode45 solver (see Appendix B) with a very small time-step of s0.1 t .
The qd0 model with snubber and CP-VBR models (implemented in different toolboxes) are
numerically stiff. Thus, the ode15s solver is selected for the studies. The minimum and
maximum time-step settings are 1 ms and 0.1 µs, respectively. The initial time-step is set to
the minimum time-step. All models are represented in pu and the relative and absolute
tolerances of the solver are set to 10−4. The simulation results of the considered fault study
are shown in Figure 3-15, where it can be seen that all models produce consistent transient
trajectories. To show more details, three fragments of Figure 3-15 are enlarged and shown
in Figure 3-16 to Figure 3-18, where it can be seen that the CP-VBR models use much fewer
time-steps than the qd0 model with snubber. Figure 3-16 shows that unlike the qd0 model,
the CP-VBR models have no error in steady-state.
A more quantitative evaluation of this simulation study is presented in Table 3–2. As
defined by (2–64) in section 2.4, the 2-norm relative error is chosen as the mean to
evaluate the accuracy of the variables. For the stator three-phase current iabcs, the average
error of the three currents is given in Table 3–2. This table shows that both formulations
have similar stator current cumulative errors. However, the CP-VBR models are faster and
require less than one third the numbers of time-steps of the qd0 model. The largest
83
eigenvalue of the qd0 model is about 46 times bigger than it is in the CP-VBR models, which
shows that this model is significantly stiffer. There are also considerable differences
between the run times of the different toolboxes, highlighting the numerical efficiency of
their respective algorithms. The error in electromagnetic torque eT is larger for the qd0
model due to the snubber that adds error to both q- and d-axis.
Figure 3-15 Simulation results for large machine single-phase-to-ground fault study as predicted by
CP-VBR models and the conventional qd0 model.
84
Figure 3-16 Detailed view of phase current in steady-state shown in Figure 3-15.
Figure 3-17 Detailed view of phase current during transient shown in Figure 3-15.
Figure 3-18 Detailed view of electromagnetic torque shown in Figure 3-15.
85
Table 3–2 Numerical Efficiency of CP-VBR Models Versus qd0 with Snubber for the Large Machine
Study
Implementation
Performance measures
Time-steps Error iabcs Error Te Run Time Max. |λ| *
qd0 with snubber SPS 797 1.074 % 1.332 % 250 ms 74,111
CP-VBR SPS 251 1.069 % 0.559 % 188 ms 1,594
CP-VBR ASMG 255 1.054 % 0.559 % 73 ms 1,594
CP-VBR PLECS 262 1.052 % 0.558 % 180 ms 1,594
* λ = Eigenvalue
86
CHAPTER 4: NUMERICAL METHODS TO ACHIEVE CONSTANT-PARAMETER VBR FORMULATIONS
In this chapter, alternative formulations to achieve the constant-parameter VBR interfacing
circuit of Figure 2-2 are proposed and verified by simulations in commercial simulation
programs. Although these formulations achieve the same goal as the added damper
winding described in Chapter 3, section 3.3, their derivation and application appear very
intuitive and easy-to-use, which can offer additional advantages. The method presented in
this chapter is based on mathematical equalization of q- and d-axis fictitious circuits
putting the unequal parts to the sub-transient voltage-sources. This method is also used in
the next chapter to extend direct interfacing to the rotor circuit. The proposed method is
validated by extensive computer studies presented at the end of this chapter. The same
generic mechanical system as the previous chapters is considered here.
4.1 Method of Using Current Derivatives
To obtain an equivalent formulation, considering (3–16) to (3–18), (3–44), (3–45), (3–34),
and (3–35), the stator voltage equations given by (3–10) to (3–12) are rewritten as [28]
87
)()( qqsqddsdrqssqs iLpiLirv (4–1)
)()( ddsdqqsqrdssds iLpiLirv (4–2)
)( 000 slssss iLpirv . (4–3)
To match the interfacing circuit of Figure 2-2, (4–1) and (4–2) are rewritten as
qqsddsdrqssqs eiLpiLirv (4–4)
ddsdqsdrdssds eiLpiLirv (4–5)
where
qqsdqdrq piLLpe )( (4–6)
dqsdqrqrd piLLe )( . (4–7)
Following the classical VBR model derivation procedure [28], and considering (3–39) and
(3–40), the terms qp and dp in (4–6) and (4–7) are replaced by the rotor state
equations (3–41) to (3–43) to get
M
j
kqjmq
lkqj
kqjmqqsdqdrq
L
rLiLLpe
12
)()( (4–8)
fd
lfd
mdN
j
kdjmd
lkdj
kdjmdqsdqrqrd v
L
L
L
rLiLLe
12
)()(
88
)(2 fdmd
lfd
fdmd
L
rL
. (4–9)
Finally, transforming (4–4), (4–5), and (4–3) into stationary abc-phase coordinates gives
the branch voltage equation (3–78). The interfacing circuit of the proposed model is thus
identical to Figure 2-2, and its parameters are defined by (3–81) to (3–84). However, a
direct and straightforward implementation of this model in a state-variable-based
simulation package requires the current derivative term qsdq iLLp )( in order to calculate
qe using (4–6). This, in turn, leads to a non-proper state model. Approximation techniques
will be presented in subsection 4.4.1 to resolve this issue.
The electromagnetic torque equation is not changed and the torque is calculated using the
same equations given by (3–24) or (3–25) in Chapter 3. Mechanical system also stays
unchanged.
4.2 Method of Using Algebraic Feed-through
If the machine’s terminal voltages are known (i.e. as external inputs or state variables), the
stator current derivative qspi in (4–6) is calculated using state variables and inputs from
the circuit subsystem. This results in an algebraically exact VBR model (without any
approximations). In order to express qspi in terms of inputs and states, (4–4) and (4–8) are
first combined to obtain
89
M
j
kqjmq
lkqj
kqjmqqdrdsdrqssqs
qqs
L
rLLiLirv
Lpi
12
)(/)(1
. (4–10)
Substituting (4–10) into (4–8) again yields
)()()(1
2 qssqsq
dqM
j
kqjmq
lkqj
kqjmq
dsdqdrq
dq irv
L
LL
L
rLiLL
L
Le
. (4–11)
Finally, after replacing (4–8) by (4–11) and transforming the sub-transient voltages to abc-
phase coordinates (similar to section 4.1), the second VBR formulation is completed.
In a general case, the terminal voltage abcsv is unknown (e.g. machine connected to an
inductive network). In such cases, this VBR formulation (including the circuit model)
creates an implicit set of equations that results in an algebraic loop which contains input
voltage qsv and sub-transient voltages abce : in this loop, branch voltage abcsv , which is an
output of the circuit subsystem, is a function of the rotor subsystem output abce , while
abce itself is a function of abcsv (both having algebraic feed-through). It is further shown in
the next section (section 4.3) that this formulation is algebraically equivalent to the model
derived using singular perturbation approach [31]. The techniques to break this algebraic
loop will be described in subsection 4.4.2.
90
4.3 Algebraic Equivalence of Implicit VBR Formulations
An interesting (but not obvious) fact that has contributed to the idea of section 4.1 of this
chapter is that the constant-parameter singular perturbation VBR model [31] and the
implicit VBR model from subsection 3.3.2 (herein referred to as IVBR1) are algebraically
equivalent to the implicit constant-parameter model derived in section 4.2 (herein referred
to as IVBR2). The interfacing circuits and the remaining state equations of IVBR1 and
IVBR2 are identical; however, their sub-transient voltages are formulated differently.
Therefore, it remains to show that the q- and d-axis sub-transient voltages of IVBR1 and
IVBR2 are in fact equivalent.
For consistency with Chapter 3, the variables altered/approximated by the fictitious
)1( M th q-axis damper winding are denoted by the triple prime symbol (''' ). The q-axis
sub-transient flux of IVBR1 is given by (3–69). The q-axis magnetizing flux is given by (3–
72), and the sub-transient back EMF voltages are given by (3–67) and (3–68).
Since )1()1( / MkqMkq rL ,
0)1( thenrif Mkq (4–12)
Therefore, the voltage induced in the additional winding given by (3–66) can be expressed
as
91
M
j
mqkqj
lkqj
kqjmd
qsmdMlkq
mdMkq
L
rLpiL
L
Lv
12
1
)1()1( )(1 . (4–13)
Based on (3–59), the following equalities are derived
mq
md
Mlkq
md
L
L
L
L
)1(
1 (4–14)
)1(
Mlkq
mdmq
mdmqL
LLLL . (4–15)
By substituting (3–44) and (3–74) into (3–67), after some algebraic manipulations, qe is
rewritten as
M
j
kqjmq
lkqj
kqjmd
Mlkq
mdMkqdrq
L
rL
L
Lve
12
)1()1( )( . (4–16)
Substituting (4–14) into (4–13), and considering (3–73), expression for )1( Mkqv simplifies
to
qsmd
M
j
mqkqj
lkqj
kqjmd
md
mq
Mkq piLL
rL
L
Lv
12)1( )( . (4–17)
Substituting (4–17) into (4–16) and rearranging the resulting terms yields
M
j
kqjmq
lkqj
kqjmq
qsMlkq
mdmq
drqL
rLpi
L
LLe
12
)1(
)(
92
M
j
mdmqMlkq
mdmq
kqjmq
lkqj
kqjLL
L
LL
L
r
1 )1(2
)()( . (4–18)
Substituting (4–15) into (4–18) further yields
M
j
kqjmq
lkqj
kqjmq
qsmdmqdrqL
rLpiLLe
12
)()( . (4–19)
Based on (3–32) and (3–33), we have
dqmdmq LLLL (4–20)
This proves that (4–19) is identical to (4–8), i.e., the q-axis sub-transient voltages of IVBR1
and IVBR2 are equal.
Next, it is required to show that the d-axis sub-transient voltages of IVBR1 and IVBR2 are
equal. Therefore, (3–74) is substituted into (3–68) to get
fdlfd
mdN
j
kdjd
lkdj
kdjmd
qsmdmqrqrd vL
L
L
rLiLLe
12
)()(
dsmd
N
j lkdj
kdj
lfd
fd
fdd
lfd
fdmdiL
L
r
L
r
L
rL 2
1222
)(
. (4–21)
After substituting (3–45) into (4–22) and some algebraic manipulations, de simplifies to
93
N
j
kdjmd
lkdj
kdjmd
qsmdmqrqrdL
rLiLLe
12
)()(
)(2 fdmd
lfd
fdmd
fdlfd
md
L
rLv
L
L
. (4–22)
Considering (4–20), this result proves that (4–22) is equal to (4–9). Comparing (4–19) and
(4–22) for IVBR1 with (4–8) and (4–9) for IVBR2, respectively, it is concluded that these
two models are algebraically equivalent.
4.4 Numerically Efficient Explicit Implementation
4.4.1 Approximation of current derivative
Exact numerical calculation of current derivative qspi in (4–8) requires infinitely small
time-steps. However, qspi may be approximated with reasonable accuracy using filters or
other numerical techniques. Specifically, in continuous-time domain, the derivative of a
signal may be approximated using a first-order high-pass filter [16] such as
0
0)(ps
spsH i
. (4–23)
94
which has a pole at 0p . This filter adds a pole to the overall system. If 0p is sufficiently
large, the filter approximation will have high accuracy. However, the added pole, with
magnitude around 0p , could make the overall system numerically stiff.
Alternatively, considering a fixed time-step t , the derivative may be estimated by using
the previous and current values of the stator current (i.e. backward difference), yielding the
following discrete-time filter
t
zzH i
11)( . (4–24)
where 1z denotes a unit delay. For variable time-steps, qspi may be approximated using
the first-order backward differentiation formula
1
1)()()(
nn
nqsnqsnqs
tt
tititpi . (4–25)
Here, the subscripts n and 1n denote the current and previous time-steps, respectively.
The approximations (4–24) and (4–25) may add very little computational cost to the VBR
model. Moreover, unlike (4–23), they do not add an extra continuous state variable to the
system. The implementation of the proposed constant-parameter VBR model interfaced
with an external circuit-system in a typical state-variable-based transient simulation
program (e.g. Simulink) is depicted in Figure 4-1. Here, the interfacing circuit of Figure 2-2
may be implemented in a circuit program or toolbox (PLECS [17], ASMG [18],
95
Figure 4-1 Implementation of explicit constant-parameter VBR model using filter Hi in continuous-
time (high-pass) or discrete-time (backward difference) to approximate the current derivative.
SimPowerSystems [19], etc.) by using basic circuit elements (i.e. resistors, inductors, and
controlled voltage-sources). The machine’s rotor state model, the reference frame
transformations, and the high-pass filter iH are implemented using conventional library
components (gains, summations, integrators, functions, etc.).
4.4.2 Relaxation of algebraic loop
An alternative implementation to current derivative estimation was proposed in section
4.2, wherein the sub-transient voltage abce evaluated in the rotor subsystem is function of
the machine’s terminal voltage qsv that forms an algebraic loop. To break this algebraic
96
Figure 4-2 Implementation of explicit constant-parameter VBR model using filter Hv in continuous
time (low-pass) or discrete time (zero- or first-order-hold) to break the algebraic loop.
loop, the voltage qsv can be approximated using a filter vH . A block diagram depicting this
implementation is shown in Figure 4-2.
In continuous-time domain, vH can be implemented using a first-order low-pass filter such
as
0
0)(ps
psHv
, (4–26)
where 0p is the filter pole. Similarly to the method of subsection 4.4.1, a large value of 0p
will make the approximation more accurate at the expense of making the overall system
numerically stiffer.
97
A zero-order-hold can also be used to break this algebraic loop. In discrete-time domain,
this results in the following filter
1)( zzHv . (4–27)
It is also possible to approximate qsv with higher order filters, for example using a first-
order-hold filter
212)( zzzHv . (4–28)
where a constant time-step is assumed. The zero-order-hold simply introduces a one-time-
step delay, whereas the first-order-hold implies linear prediction based on the two
previous values. An example of using the zero- and first-order-hold filters to approximate a
sample rising signal ( qsv ) is depicted in Figure 4-3.
Figure 4-3 Approximation of voltage vqs using zero-order-hold (delay) or first-order-hold (linear
prediction).
98
If the time-step is not constant, the discrete-time approximations of qsv by the zero- and
first-order-hold filters respectively become
)()( 1 nqsnqs tvtv . (4–29)
)()(1)( 221
11
21
1
nqs
nn
nnnqs
nn
nnnqs tv
tt
tttv
tt
tttv . (4–30)
Using (4–27) to (4–30), as opposed to (4–26), the system numerical stiffness is not changed
and no extra continuous state variables are added, which minimizes the additional
computational cost.
4.5 Summary of Approximation Techniques
Based on the material presented in the last two subsections (4.4.1 and 4.4.2), the proposed
approximation techniques can be classified into two groups
i. Continuous-time approximations
ii. Discrete-time approximations
A summary of this classification is presented in Table 4–1. Type (i) approximations include
the use of continuous-time filters to obtain either the current derivative, (4–23), or the
voltage algebraic loop relaxation, (4–26). It can also include the addition of the fictitious
damper winding [30] (see Chapter 3, subsection 3.3.1). All these techniques increase the
99
number of continuous state variables by one, and could also make the overall system
numerically stiff if the filter pole (or the added winding resistance) is made very large in an
attempt to achieve high accuracy. Type (ii) approximation techniques, i.e. (4–24), (4–25),
and (4–27) – (4–30), are based on discrete-time filters. Such methods therefore do not add
continuous state variables and barely increase the computational cost. No discrete-time
form is available for the additional damper winding method.
Table 4–1 Summary of Approximation Techniques to Achieve the Interfacing Circuit of Figure 2-2
Approximation
Technique
Current Derivative
Approximation
Voltage Algebraic
Loop Relaxation
Circuit-Level
Approximation
Continuous-Time High-Pass Filter Low-Pass Filter Fictitious Damper
Winding
Discrete-Time Backward Difference Zero-/First-
Order-Hold N/A
4.6 Computer Studies
In order to validate and assess the accuracy of the proposed explicit formulations (Table 4–
1), the system shown in Figure 4-4 is considered in this section. This network contains a
555-MVA steam turbine synchronous generator whose neutral point is grounded through a
reactor gX for protection purposes [53]. The generator is connected to a large network
100
(represented by an infinite bus) through a Wye/Delta unit transformer. The machine and
network parameters are summarized in Appendix E. The generator circuit model [4]
includes two damper windings in the q-axis, and one damper winding and a field winding
in the d-axis. The magnetizing reactances mqX and mdX are very close (1.61 pu and 1.66
pu). However, due to the dominant effect of the damper and field windings in the sub-
transient period, the sub-transient reactances mqX and mdX are not equal (0.10 pu vs. 0.08
pu). Approximation techniques are therefore required to obtain the constant-parameter
interfacing circuit of Figure 2-2.
In the considered transient study, the generator is assumed to operate initially in steady-
state condition with 9.0eT pu and 35.2xfde pu, while the voltage of the infinite bus is set
to its nominal value of 1 pu. At 0175.0t s, a single phase-to-ground fault is applied at the
machine’s terminals. This study was chosen as to emulate a highly unbalanced condition
when qsi and qsv vary greatly to test the proposed approximation techniques in an
unfavourable condition.
All considered models and approaches have been implemented using MATLAB/Simulink
[15], [16] and the PLECS toolbox [17]. The general-purpose MATLAB solver ode45 (see
Appendix B) is used to run all simulation studies with relative and absolute error
tolerances both set to 10−3, and the maximum and minimum time-steps set to 10–3 and 10–7
seconds, respectively. Since all models are consistent, the results of all discrete-time
approximation methods converge if the time-step is small enough. For the purpose of
101
comparison, the reference solution is obtained by simulating the study with a very small
step size of 710t seconds. All simulation studies are conducted on a personal computer
(PC) with a 2.53GHz Intel CPU.
To evaluate the accuracy of machine models, the predicted stator currents are compared to
a reference. Here, the 2-norm (cumulative) relative error [defined by (2–64) in section 2.4]
of the stator currents asi , bsi , and csi for each approximation technique is computed. As
the error may be different in each phase, an average error is considered in this chapter as a
more consistent measure of accuracy.
Figure 4-4 Test system consisting of a grounded steam turbine generator connected to the network
via a unit transformer. A single phase-to-ground fault is applied at the machine terminals.
102
4.6.1 Continuous-time approximation techniques
The selection of the pole (or cut-off frequency) of the continuous-time filters (4–23) and
(4–26) (and similarly, the resistance of the additional winding approach) involves a trade-
off between accuracy and numerical stiffness (see subsections 4.4.1 and 4.4.2). Therefore,
to objectively compare the considered approaches, the filter pole and the additional
winding resistance values are chosen as to yield the same largest eigenvalue magnitude
( 1000~ ) during the fault condition. The corresponding pole and winding resistance values
are summarized in the first column of Table 4–2. The system eigenvalues will be shown in
subsection 4.6.3.
The single phase-to-ground fault study is first executed using the current filter (4–23), the
voltage filter (4–26), and the additional winding approach. The predicted trajectories of
abcsv , asi , ngi , fdi , and eT are shown in Figure 4-5. Therein, it is observed that all three
approximation techniques give visibly accurate solutions. To see the difference among the
three approaches, portions of Figure 4-5 are enlarged and reproduced in Figure 4-6 to
Figure 4-8. The magnified views in Figure 4-6 to Figure 4-8 demonstrate that in this case all
techniques yield acceptable solutions. However, the voltage filter and the additional
winding approaches are noticeably more accurate than the current filter technique.
The corresponding cumulative errors )( abcsavg i are summarized in Table 4–2, where
similar observations are made. In particular, the error of the voltage filter (0.68%) is
103
Table 4–2 Comparison of Continuous-Time Approximation Methods (with Added Eigenvalue of 1,000)
Number of Time-Steps
Error εavg(iabcs)
Simulation Time
Additional Winding (rkq3 = 1.247 pu) 126 0.79 % 175 ms
Current Filter (p0 = 947 ) 126 2.96 % 186 ms
Voltage Filter (p0 = 1031) 126 0.68 % 179 ms
Figure 4-5 Transient response to a single phase-to-ground fault predicted by continuous-time
approximated models. From top to bottom: terminal voltages vabcs, phase a stator current ias, neutral
grounding current ing, field current ifd, and electromagnetic torque Te.
104
Figure 4-6 Detailed view of current ias shown in Figure 4-5 for continuous-time approximated models.
Figure 4-7 Detailed view of current ifd shown in Figure 4-5 for continuous-time approximated models.
slightly smaller than that of the additional winding approach (0.79%). Table 4–2 also
contains the number of time-steps and the overall CPU processing time taken by each
model which are very similar for all considered models.
To further investigate the effect of the added pole on the model accuracy, the cumulative
error )( abcsavg i has been calculated for different largest eigenvalue magnitudes (from
1,000 to 10,000). The corresponding results are shown in Figure 4-9 which demonstrate
105
Figure 4-8 Detailed view of electromagnetic torque Te shown in Figure 4-5 for continuous-time
approximated models.
Figure 4-9 Cumulative error in predicting currents iabcs versus the magnitude of the added eigenvalue
for continuous-time approximated models.
that all continuous-time approximation techniques are consistent, as their cumulative error
decreases when increasing their largest eigenvalue magnitude. Overall, the voltage filter
has the best accuracy, closely followed by the additional winding method.
106
4.6.2 Discrete-time approximation techniques
Here, the same single phase-to-ground fault study as subsection 4.6.1 is repeated using the
discrete-time approximations: first-order numerical derivative (4–24); the zero-order-hold
(4–27); and the first-order-hold (4–28), which are implemented using memory blocks. All
models produced results that are visibly very similar to Figure 4-6. The magnified views of
the predicted trajectories of asi , fdi , and eT are shown in Figure 4-10 to Figure 4-12,
respectively. Similarly to the continuous-time approaches, all three discrete-time
approximations yield acceptable and visibly close results, with some preference given to
the first-order-hold of qsv method.
A quantitative assessment of this study is summarized in Table 4–3 that shows the smallest
error, 78.0)( abcsavg i %, is in the same range as those obtained using the most accurate
continuous-time techniques (Table 4–2). The numerical efficiency and the overall
computational times of all approaches are almost identical.
Figure 4-10 Detailed view of current ias shown in Figure 4-5 for discrete-time approximated models.
107
Figure 4-11 Detailed view of current ifd shown in Figure 4-5 for discrete-time approximated models.
Figure 4-12 Detailed view of electromagnetic torque Te shown in Figure 4-5 for discrete-time
approximated models.
Table 4–3 Comparison of Discrete-Time Approximation Methods
No. of Time-Steps Error εavg(iabcs) Simulation Time
Backward Difference of iqs 126 2.64 % 177 ms
Zero-Order-Hold of vqs 126 1.71 % 170 ms
First-Order-Hold of vqs 126 0.78 % 172 ms
108
Figure 4-13 Cumulative error in predicting currents iabcs versus maximum time-step size for discrete-
time approximated models.
Next, the study is repeated using different values of maximum time-step ranging from 10 µs
to 2 ms. The resulting cumulative error )( abcsavg i as a function of time-step is plotted in
Figure 4-13. This figure shows that all three discrete-time approaches are consistent, and
that the first-order-hold approximation demonstrates the best accuracy.
4.6.3 Comparison of models and approximation techniques
All proposed approximation techniques work with the same VBR model, i.e. they share the
same interfacing circuit of Figure 2-2 and the same rotor state equations. This makes it
simple to switch between the approaches. To get more insight into the achieved results, the
eigenvalues of the power system of Figure 4-4 have been calculated for different VBR
formulations by linearizing the system around the operating point; the results are
summarized in Table 4–4. As it can be seen in Table 4–4, unlike discrete-time
109
approximations, the continuous-time approximations add a large eigenvalue (~1,000) to
the system, making it stiffer.
Despite the difference in numerical stiffness of models with discrete- and continuous-time
approximation techniques, Table 4–2 and Table 4–3 show that all models used
approximately the same number of time-steps. This is because in this specific case, the step
size is predominantly determined by the maximum allowed time-step rather than by
numerical stability constraints [55].
The filters are placed on the q-axis voltage (or current), which in the rotor reference frame
becomes constant in steady-state. Therefore a reasonably good accuracy in general and
zero steady-state error in particular is expected since the filters add no error in steady-
state.
When the original system is not stiff, it appears beneficial to use discrete-time
approximations, since such methods have negligible additional computational cost and do
not change the number of continuous state variables nor the numerical stiffness. The
discrete-time approximation techniques also have the advantage of not requiring any
parameter settings.
The discrete-time filters might not produce good results in some situations, for example:
when using stiff solvers [16] and/or if the time-step is too large for a given transient study
when the approximated variables have high-frequency components; and if the dynamic
saliency is considerable (e.g. 5.1/ mdmq XX ).
110
Table 4–4 Eigenvalues of the System when using Different Approximation Techniques
(a) Continuous-Time Methods (b) Discrete-Time Methods
Additional
Winding [30]
(Subsection 3.3.1)
+23.578 ± j39.426 (6.27 Hz)
–0.536
–3.391
–6.850 ± j15.790 (2.51 Hz)
–11.401
–47.124
–68.871 ± j36.410 (5.79 Hz)
–999.684
N/A
Current
Derivative
Approximation
(Subsection 4.4.1)
+23.450 ± j39.762 (6.33 Hz)
–0.544
–3.480
–7.096 ± j15.881 (2.53 Hz)
–11.401
–47.124
–68.713 ± j36.251 (5.77 Hz)
–999.782
+23.636 ± j40.363 (6.42 Hz)
–0.544
–3.481
–7.071 ± j15.863 (2.52 Hz)
–11.401
–47.124
–69.822 ± j36.658 (5.83 Hz)
Voltage Algebraic
Loop Relaxation
(Subsection 4.4.2)
+26.231 ± j30.574 (4.87 Hz)
–0.563
–3.737
–8.843 ± j15.777 (2.51 Hz)
–11.321
–47.124
–69.196 ± j42.156 (6.71Hz)
–999.849
+26.215 ± j30.614 (4.87 Hz)
–0.560
–3.735
–8.821 ± j15.757 (2.51 Hz)
–11.321
–47.124
–69.214 ± j42.114 (5.70 Hz)
111
The continuous-time approximation approaches are suggested for cases when the original
system is stiff from the beginning due to the natural components and time constants
present in the system. In those cases, the added eigenvalue is controlled by selecting the
filter pole, which then could be increased without making the overall system any stiffer
while achieving the needed approximation accuracy or more.
112
CHAPTER 5: DIRECT INTERFACING OF SYNCHRONOUS MACHINE MODELS FROM STATOR AND ROTOR TERMINALS
The VBR models were originally introduced to facilitate the stator interface of synchronous
machines to arbitrary networks [28]. The rotor interface is neglected in most VBR
publications. In this chapter, two new models for synchronous machine with direct
interface of rotor and stator terminals are introduced. In the first model, all of the rotor and
stator windings are represented by an equivalent circuit. The whole model is represented
only with constant-parameter RL impedance and voltage-source branches (VBR form). This
model offers direct interfacing of any rotor and/or stator windings. In the second model,
the damper windings are represented using standard state-space equations, while the field
and stator windings remain in circuit form allowing a direct interface with external ac
power network and dc exciter network, respectively.
The approximation techniques used in this chapter are similar to those presented in
Chapter 4. Similar to the previous chapter, the electromagnetic torque equations will not
change and the same generic mechanical system will be considered (see subsection 3.1.2).
With respect to the state-of-the-art models that will be summarized in the following
section, this chapter proposes new VBR models that combine the following characteristics:
113
1) Direct stator interfacing;
2) Direct rotor interfacing; and
3) Constant-parameter equivalent interfacing circuit.
5.1 Rotor and Stator Interfacing of Synchronous Machine Models
The interfacing properties of various models for synchronous machines transient studies
are summarized in Table 5–1. As depicted in this table, the straightforward implementation
of the CC-PD model (CC-PD) [1], [24], [26], [27] achieves a direct interface to arbitrary
networks from the stator and rotor sides, but due to its rotor-position-dependent
inductances, this model has variable parameters and is generally more computationally
expensive.
The classical qd0 model possesses constant-parameter elements. Its stator circuit requires
an indirect interface with external inductive networks (i.e. it requires snubbers), but its
rotor field winding, if represented as a circuit, can be directly interfaced with the external
dc exciter circuit. This model is therefore shown in Table 5–1 as having a direct interface on
the rotor side.
The VBR models [28] – [31], [33], [34] have been derived to achieve a direct interface of the
stator terminals with an arbitrary network on the ac side. The original/classical VBR
formulation [28] possesses rotor-position-dependent stator interfacing resistance and
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Table 5–1 Classification of Synchronous Machine Formulations based on Interfacing with External
Inductive Networks
Formulation Stator Interface Rotor Interface Circuit Parameters
Classical CC-PD [1] Direct Direct Variable
Classical qd0 [1], [7] Indirect Direct Constant
Classical VBR [28], [29] Direct Indirect Variable
Stator VBR [30], [31], Chapter 3, & 4 Direct Indirect Constant
Stator VBR [33] Direct Indirect Variable
Stator/Rotor VBR [34] Direct Direct Variable
Chapter 5’s VBR Direct Direct Constant
inductance matrices. It was later shown in [29] how to obtain a constant resistance matrix.
Overall, the VBR models presented in [28] and [29], and also the more recent models
proposed in [33] and [34], have stator interfacing branches with variable parameters
(similar to the CC-PD model). Simulation of variable elements is possible in some programs
(such as PLECS [17] and ASMG [18]). However, it is very computationally costly and should
be avoided if possible.
The variable inductances of the stator interfacing circuit were first made constant by
equating the sub-transient inductances (i.e. neglecting dynamic saliency) using: a) an
115
additional fictitious damper winding [30]; and b) singular perturbation [31] as discussed in
subsections 3.3.1 and 3.3.2, respectively. A generalized VBR model and a unified interfacing
circuit for both induction and synchronous machines was proposed in Chapter 2 and 3.
Mathematical methods and several numerically advantageous continuous- and discrete-
time approximation techniques are also proposed in Chapter 4 to obtain the same unified
stator interfacing equivalent circuit of Chapter 2 and 3. All these models achieve constant-
parameter interfacing of the machine's stator terminals as shown in Table 5–1.
The synchronous machine VBR models [28] – [31], [33] and the ones presented in Chapter
3 and 4 were initially introduced to allow direct network-machine interfacing at the stator
terminals, while assuming that a voltage-source representing the exciter is connected to
the rotor field winding. In these models, to interface the rotor to an arbitrary network, e.g.
an ac excitation systems [4], a controlled current-source is used. This also creates an
incompatible interface if the field winding terminals are connected to inductive elements or
a switching converter, therefore requiring a snubber. To obtain direct rotor interfacing, the
model presented in [33] was recently extended to also represent the field winding using a
VBR formulation [34]. Therein, the resulting equivalent interfacing circuit is defined by
four coupled and variable RL branches, which achieves a direct interface of both stator and
rotor terminals with external circuits as summarized in Table 5–1.
116
5.2 All-Circuit Formulation
The aim of this section is to derive a synchronous machine model consisting solely of basic
circuit elements with constant parameters. Such model is easy to implement in traditional
simulation packages while offering a direct interface to external circuits. The model can
also be used to study special cases, e.g. broken damper windings and machines with
excitable auxiliary windings [56].
5.2.1 Stator voltage equations
Inserting (3–44) and (3–45) into (3–16) and (3–17), respectively, and taking the derivative
of the result gives
qqsqqs piLpp (5–1)
ddsdds piLpp (5–2)
where the q- and d-axis sub-transient inductances are defined by (3–34) and (3–35). Then,
using (3–13) to (3–15), the time derivatives of the rotor sub-transient flux linkages are
written as [28]
M
j lkqj
kqjkqjkqj
mqqL
irvLp
1
(5–3)
N
j lkdj
kdjkdjkdj
lfd
fdfdfd
mddL
irv
L
irvLp
1
(5–4)
117
Inserting (3–16), (3–17), (3–22), and (3–23) into (3–10) and (3–11) and substituting (5–1),
(5–2), (5–3), and (5–4) into the resulting equations, after algebraic manipulation, the q-
and d-axis stator voltages are rewritten as
qsqsqdsdrqssqs eiLpiLirv (5–5)
dsdsdqsqrdssds eiLpiLirv (5–6)
where the sub-transient voltages qse and dse are defined as
M
j lkqj
kqjkqjkqj
mq
N
j
kdjfdmdrqsL
irvLiiLe
11
(5–7)
N
j lkdj
kdjkdjkdj
lfd
fdfdfd
md
M
j
kqjmqrdsL
irv
L
irvLiLe
11
(5–8)
The q- and d-axis steady-state inductances, similar to (3–34) and (3–35), are defined as
mqlsq LLL (5–9)
mdlsd LLL . (5–10)
As shown in Chapter 4, to obtain constant-parameter stator interfacing branches in
stationary abc-phase coordinates, it is convenient to rearrange (5–5) and (5–6) as follows
qsqsddsdrqssqs eiLpiLirv (5–11)
dsdsdqsdrdssds eiLpiLirv (5–12)
118
where the new sub-transient voltages are
qsdq
M
j lkqj
kqjkqjkqj
mq
N
j
kdjfdmddsddrqs iLLpL
irvLiiLiLLe )()(
11
(5–13)
N
j lkdj
kdjkdjkdj
lfd
fdfdfd
md
M
j
kqjmqqsdqrdsL
irv
L
irvLiLiLLe
11
)( (5–14)
Using (5–5), after algebraic manipulation, the derivative term qsdq iLLp )( in (5–13) can
be expressed as
)()( qsdsdrqssqsq
dq
qsdq eiLirvL
LLiLLp
(5–15)
Equation (5–13) can then be rewritten by substituting qse in (5-15) by (5–7) and inserting
the resulting equation into (5–13), which simplifies to
M
j lkqj
kqjkqjkqj
q
dmq
N
j
kdjfdq
dmdds
q
ddrqs
L
irv
L
LLii
L
LLi
L
LLe
11
)1(
)( qssqsq
dqirv
L
LL
(5–16)
Finally, transforming (5–11), (5–12), and (3–12) to abc-phase coordinates yields the
constant-parameter stator interfacing circuit presented in Chapter 3 and 4 and shown in
Figure 2-2 and the upper part of Figure 5-1. The parameters of the stator circuit are given
by (3–81), (3–82), and (3–84).
119
5.2.2 Rotor voltage equations
Solving for stator currents qsi and dsi in (3–16) and (3–17), substituting the result in (3–
22) and (3–23), after some algebraic manipulations, the magnetizing fluxes md are
rearranged as
qsmqls
mqM
j
kqjmqlsmqLL
LiLL
1
|| (5–17)
dsmdls
mdN
j
kdjfdmdlsmdLL
LiiLL
1
|| (5–18)
where 111|| baba LLLL . Inserting (5-17) and (5-18) into (3–19) to (3–21) and taking
the derivative of the resulting equations gives
MjpLL
LiLLiLpp qs
mqls
mqM
a
kqamqlskqjlkqjkqj ,,2,1,||1
(5–19)
dsmdls
mdN
a
kdafdmdlsfdlfdfd pLL
LiiLLiLpp
1
|| (5–20)
NjpLL
LiiLLiLpp ds
mdls
mdN
a
kdafdmdlskdjlkdjkdj ,,2,1,||1
(5–21)
Incorporating (3–10), (3–11), and (5-19) – (5-21) into (3–13) – (3–15), after algebraic
manipulation, yields
120
MjeiLLpipLirv qr
M
a
kqamqlskqjlkqjkqjkqjkqj ,,2,1,)||(1
(5–22)
dr
N
a
kdafdmdlsfdlfdfdfdfd eiiLLpipLirv
1
)||( (5–23)
NjeiiLLpipLirv dr
N
a
kdafdmdlskdjlkdjkdjkdjkdj ,,2,1,)||(1
(5–24)
where
)( dsrqssqsq
mqqr irv
L
Le (5–25)
)( qsrdssdsd
mddr irv
L
Le . (5–26)
In summary, the all-circuit constant-parameter voltage-behind-reactance machine model,
herein referred to as AC-CP-VBR, is shown in Figure 5-1. The constant-parameter circuit
representation based on (5-22) to (5-26) is shown in Figure 5-1 (lower part). The sub-
transient algebraic voltage equations are calculated by (5–16) [or (5–13)], (5–14), (5–25),
and (5–26) in addition to qs and ds given by (3–16), (3–17), (3–22), and (3–23). As
shown in Figure 5-1, the voltage terminals of the damper windings are available for
connection with arbitrary external circuits.
The calculation of qse using (5–13) will require the approximation of a time derivative (see
Chapter 4, section 4.4.1). Alternatively, if qse is calculated using (5–16), the resulting model
121
Figure 5-1 All-circuit constant-parameter VBR synchronous machine model (AC-CP-VBR) with direct
interfacing to arbitrary external ac and dc networks.
122
is algebraically identical to the classical qd0 model as no approximations are made.
However, this may introduce an algebraic constraint (loop) in the q-axis, which needs to be
relaxed to achieve a numerically efficient explicit model (see Chapter 4, section 4.4.2).
Several possible discrete- and continuous-time methods to approximate the time derivative
and break such algebraic loops have been presented in Chapter 4, section 4.4.
5.3 Stator-and-Field-Circuit Formulation
In most studies, the damper windings of synchronous machines are all short-circuited. As a
result, they are not interfaced with external networks, and as such there is no numerical
advantage gained by representing them using circuit elements. In fact, representing the
dynamics of all damper windings in a standard state-space form with flux linkages as state
variables (as opposed to the AC-CP-VBR model presented earlier in section 5.2) further
increases the model efficiency, as will be shown in section 5.5.
5.3.1 Stator voltage equations
After manipulating (3–19) and (3–21), the damper winding currents can be written as
MjL
ilkqj
mqkqj
kqj ,,2,1,
(5–27)
NjL
ilkdj
mdkdj
kdj ,,2,1,
(5–28)
123
Substituting (5–27) and (5–28) into (3–13) and (3–15), respectively, the damper windings
state equations become
MjvL
rp kqjkqjmq
lkqj
kqjkqj ,,2,1,)( (5–29)
NjvL
rp kdjkdjmd
lkdj
kdjkdj ,,2,1,)( (5–30)
Substituting (5–28) into (3–23), and after algebraic simplification, the d-axis magnetizing
flux is written as
N
j lkdj
kdj
fddsmdmdL
iiL1
(5–31)
where
1
1
11
N
j lkdjmdmd
LLL . (5–32)
Inserting (3–39) and (5–29) into (5–1) also gives
M
j lkqj
kqjmqkqjmqqsqqs
L
rLiLpp
12
)( . (5–33)
Substituting (3–17) into (3–10), replacing md with (5–31), and inserting (5–33) into the
resulting equation give the q-axis stator voltage equation
124
qsqsqdsdrqssqs eiLpiLirv (5–34)
where
mdlsd LLL (5–35)
and
M
j lkqj
kqjmqkqjmq
N
j lkdj
kdj
fdmdrqsL
rL
LiLe
12
1
)( . (5–36)
To find the d-axis stator voltage equation, (3–40) is first substituted into (5–2). The flux
linkage time derivatives are then replaced by using (3–14) and (5–30), which yields
N
j lkdj
kdjmdkdj
lfd
fdfdfdmddsdds
L
r
L
irvLiLpp
12
)( . (5–37)
Inserting (3–16) and (3–44) into (3–11), and substituting the resulting equation into (5–
37), after some algebraic manipulation, the d-axis stator voltage equation becomes
dsdsdqsqrdssds eiLpiLirv (5–38)
where
N
j lkdj
kdjmdkdjmd
lfd
fdfdfdmdqrds
L
rL
L
irvLe
12
)()( . (5–39)
125
Following the approach set forth in Chapter 4, section 4.2 and in this chapter, subsection
5.2.1, the stator voltage equations (5–34) and (5–38) are rearranged as
qsqsddsdrqssqs eiLpiLirv (5–40)
dsdsdqsdrdssds eiLpiLirv , (5–41)
where the new sub-transient voltages are
qsdq
M
j lkqj
kqjmqkqjmq
N
j lkdj
kdj
fdmddsddrqs iLLpL
rL
LiLiLLe )(
)()(
12
1
(5–42)
lfd
fdfdfdmd
N
j lkdj
kdjmdkdjmdqqsdqrds
L
irvL
L
rLiLLe
12
)()(
(5–43)
Similar to AC-CP-VBR, in order to replace the term qsdq iLLp )( in (5–42) with state
variables and voltages, qsqiLp is first expressed using (5–34) and (5–36) and then inserted
into (5–42). After some algebraic manipulations, (5–42) becomes
M
j lkqj
kqjmqkqj
q
dmq
N
j lkdj
kdj
fdq
dmdds
q
qddrqs
L
r
L
LL
Li
L
LLi
L
LLLe
12
1
)()(
))(( qssqsq
dqirv
L
LL
. (5–44)
126
When converted to abc-coordinates, (5–40) and (5–41) along with (3–12) form the same
constant-parameter stator interfacing circuit as the one in the upper part of Figure 5-1. The
parameters of this interfacing circuit are also defined by (3–81), (3–82), and (3–84).
5.3.2 Rotor voltage equations
The state equations for the q- and d-axis rotor damper windings are given by (5–29) and
(5–30), respectively. Therein, mq and md are computed using (3–44) and (3–39), and (5–
31), respectively. The implementation of the damper windings state model is shown in
Figure 5-2 (left side). To allow for a direct interface with external networks, i.e. exciter
circuit, the field winding has to be represented as a circuit with input terminals.
To obtain the field circuit, (3–17) is solved for ids, and then the result is substituted into (5–
31) which following algebraic manipulation gives
N
j lkdj
kdj
ls
dsfdmdmd
LLiL
1
(5–45)
where
1
1
111
N
j lkdjlsmdmd
LLLL . (5–46)
Substituting (5–45) into (3–20), taking the derivative of the results, and inserting (5–30)
give
127
N
j lkdj
kdjmdkdjmdds
ls
mdfdlfdfd
L
rL
L
LpiLpp
12
)( (5–47)
where
mdlfdlfd LLL . (5–48)
In the next step, (3–11) is solved for dsp and substituted in (5–47), and the resulting
equation is then inserted into (3–14). This procedure, after some algebraic simplifications,
gives
fdfdlfdfdfdfd eiLpirv (5–49)
where
N
j lkdj
kdjmdkdj
ls
qsrdssdsmdfd
L
r
L
irvLe
12
)( . (5–50)
The resulting constant-parameter field winding interfacing circuit is depicted in Figure 5-2
(right side).
The complete model is obtained by replacing the rotor circuit in Figure 5-1 (bottom) with
Figure 5-2 and updating the sub-transient voltage equations. For this model, the sub-
transient voltages are given by (5–44) [or (5–42)], (5–43), and (5–50). The algebraic
equations (3–44), (3–39), and (5–31) are used to compute the magnetizing fluxes. This
128
stator-and-field-circuit constant-parameter VBR model is herein referred to as SFC-CP-
VBR.
The SFC-CP-VBR model is algebraically equivalent to the classical qd0 model when qse is
calculated using (5–44). However, an algebraic loop may be introduced when the model is
interfaced to an external network. This algebraic loop may be relaxed to make an efficient
explicit model. Similarly to AC-CP-VBR model, if qse is calculated using (5–42), a current
derivative must be approximated, as will further be explained in section 5.4.
Figure 5-2 Rotor subsystem for the stator-and-field-circuit constant-parameter voltage-behind-
reactance (SFC-CP-VBR) model wherein the damper windings are represented as a state model and
the field winding is made available as an interfacing circuit (the stator interfacing circuit is the same
as in Figure 5-1).
129
5.4 Numerically Efficient Explicit Implementation
The AC-CP-VBR and SFC-CP-VBR models have constant parameters. However, when
connected to inductive external networks, these models will contain algebraic loops in both
the d- and q-axis. The algebraic loops are illustrated in Figure 5-3 for the AC-CP-VBR
formulation. Similar algebraic loops exist for the SFC-CP-VBR model. An algebraic loop of
similar nature was also encountered in Chapter 4, section 4.2 in the q-axis.
Models containing algebraic loops typically require additional iterations within each time-
step that considerably add to the computational cost [16]. To achieve a numerically
efficient solution (with acceptable accuracy but with less computational cost), it is possible
to break the algebraic loops using discrete- or continuous-time low-pass filters (see
Chapter 4, subsection 4.4.2). For example, a first-order low-pass filter (LPF) may be
considered as given by (4–26). By increasing the pole’s magnitude, the filter becomes
faster, improving the accuracy of the approximation. However, a very fast filter can make
the system numerically stiff.
Inserting a low-pass filter anywhere in the loop will relax the algebraic constraint.
However, a more accurate solution can be obtained when the approximated variable is
varying slowly. Herein, qsv and fdv are assumed to be the slowest variables in the q- and d-
axis loops, respectively (see Figure 5-3). Therefore, in the proposed models, the low-pass
filters are inserted as to relax these variables accordingly.
130
Figure 5-3 Algebraic loops in AC-CP-VBR resulting in an implicit formulation. The H blocks indicate
where the low-pass filters may be inserted to break the algebraic loops.
The algebraic loop in the q-axis of AC-CP-VBR (see Figure 5-3, left) does not exist if qse is
evaluated using (5–13) [or (5–42) for the SFC-CP-VBR model], which requires the current
derivative qspi . In this case, the derivative can be approximated using a high-pass filter
(see Chapter 4, subsection 4.4.1).
5.5 Computer Studies
To verify the proposed models, a test system consisting of a large synchronous generator
with a static exciter, as shown in Figure 5-4, is considered. Without loss of generality, in
this case-study, a simplified model of a potential-source controlled-rectifier excitation
system [5, see sec. 8.3] is considered, wherein the ac side of the exciter is represented by an
inductive Thévenin equivalent. Further details of the excitation system are omitted, as the
131
focus of the studies is on the accuracy of various machine models in predicting the
electrical transients on the stator and rotor terminals of the machine.
The system parameters are summarized in Appendix F. A single per-unit system based on
the machine’s nominal power and stator voltage is used for the generator and all external
circuits. The steam turbine generator is assumed to be driven by a constant 0.8 pu
mechanical torque in steady-state. The Thévenin voltage-source on the stator side has a
frequency of 60 Hz with nominal voltage. On the exciter side, a 12-pulse rectifier fed from a
three-phase 60 Hz voltage-source generates a field current of about 1.325 pu. The
generator is assumed to be grounded through a grounding resistor (rg = 0.2 pu) as depicted
in Figure 5-4.
The test system of Fig. 4 has been implemented using MATLAB/Simulink [15], [16] and the
PLECS toolbox [17]. For a numerically efficient and algebraic-loop-free implementation of
the AC-CP-VBR and SFC-CP-VBR models, the first-order low-pass filters with poles of
−5,000 and −1,000, respectively, have been used. For conciseness, only these voltage filters
are herein considered. However, similar to Chapter 4, discrete-time filters are applicable
here as well.
The existing constant-parameter models, namely qd0 [1] and the stator constant-
parameter VBR (CP-VBR) with voltage loop relaxation (Chapter 4), were also implemented
for comparison. Since PLECS’ built-in qd0 model does not include zero-sequence and for
consistency with the other models, a custom qd0 model has been created using
132
conventional circuit elements. A circuit-based implementation of the qd0 model (as
opposed to the more traditional state-space model) is required for direct connection of its
field winding with the exciter. Such implementation is shown in [1, see p. 202]. The stator
is also implemented as a circuit which is interfaced to the external power system using
controlled current-sources [8]. Since the external ac network (its Thévenin equivalent) is
inductive, the qd0 model requires a three-phase interfacing snubber, as shown in Figure
5-4. To have reasonable accuracy, a 50 pu stator snubber is chosen for the qd0 model. This
value of snubber results in a system eigenvalue with a magnitude of 2.61×105, as shown in
Table 5–2.
Figure 5-4 A wye-grounded steam turbine generator with a static 12-pulse rectifier-based exciter
system. The stator snubbers are required only for the classical qd0 model and the field snubber is
required only for the CP-VBR model from Chapter 4.
133
The CP-VBR model (see Chapter 4, section 4.2) has a direct interface with the network
through its stator terminals, but its field winding is represented as a standard state-space
model with a voltage-input current-output formulation. It is therefore interfaced to the
external rotor circuit using controlled current-sources. In this case, a resistive (or
capacitive) snubber must be connected to the field winding, as shown in Figure 4-4. To
have reasonable accuracy, a 0.030 pu snubber (50 times larger than the field winding
resistance) is chosen for the CP-VBR model of Chapter 4. This introduces an eigenvalue
with a magnitude of 1.35×106. Without the snubber, the largest eigenvalue magnitude of
the system would be smaller than 1,000.
Finally, the CC-PD model [1] is used to generate a reference solution. This model requires
no snubber at either sets of terminals, but it has variable coupled inductances resulting in
significant computational cost. To compare the accuracy of all the subject models, similar to
the other chapters, the 2-norm cumulative relative error is considered which is given by
(2–64).
Since the CC-PD, AC-CP-VBR, and SFC-CP-VBR models are not very stiff, MATLAB’s default
solver ode45 has been used for simulating these models. The qd0 and CP-VBR models
require snubbers and therefore are stiff. For these two models, the stiffly-stable solver
ode15s is used instead. A list of MATLAB ordinary differential equation (ODE) solvers is
given in Appendix B. For consistency among all simulation studies, the relative and
absolute error tolerances are set to 10−4, and the maximum and minimum time-steps are
set to 10−3 and 10−7 s, respectively. To produce an accurate reference solution, the CC-PD
134
model was used with a very small time-step of 10−6 seconds and both error tolerances were
set to 10−6. For consistency, all simulations were run on a personal computer with a
2.53GHz Intel CPU and Windows XP operating system. In all studies, the system is assumed
to operate in steady-state prior to a transient event.
5.5.1 Single-phase-to-ground fault in the network
To emulate a severe unbalanced condition at the machine stator terminals, a single-phase-
to-ground fault is applied at the source. The fault is implemented by setting the phase a
voltage to zero at the end of the first cycle ( 60/1t s). The corresponding source voltages
abcv and currents abci , the machine grounding current ngi , the three-phase exciter current
abcexi , the rectified exciter current dci , and the electromagnetic torque eT predicted by the
various models are presented in Figure 5-5. As it can be seen in Figure 5-5, all the models
give visibly close results. To provide a better comparison, the magnified views of abci ,
abcexi , and dci are plotted in Figure 5-6 to Figure 5-9.
A summary of the models numerical performance is also given in Table 5–2. Analyzing
Figure 5-6 to Figure 5-9 and the second column of Table 5–2 shows that the stiff models
(i.e. qd0 and CP-VBR) need considerably more time-steps (2,384 and 2,739) than the non-
stiff models (i.e. CC-PD, AC-CP-VBR, and SFC-CP-VBR) (less than 1,000). The CC-PD model
also uses a small number of steps (922) while producing the most accurate results.
However, due to its variable parameters, this model is by far the slowest (6.772 s). It was
135
Figure 5-5 Single-phase-to-ground fault case-study transient responses as predicted by the
considered models. From top to bottom: bus voltages vabc and currents iabc, machine neutral current ing,
three-phase ac exciter current iabcex, rectifier current idc, and electromagnetic torque Te.
also determined that using the explicit solver ode45 for the qd0 and CP-VBR models (which
are stiff) leads to significantly more time-steps and longer simulation times.
136
Figure 5-6 Magnified fragment from Figure 5-5: phase current ic (in steady-state).
Figure 5-7 Magnified fragment from Figure 5-5: phase current ic (during transient).
Figure 5-6 shows that as opposed to the other models, qd0 predicts steady-state stator
currents with a noticeable error. Similarly, as it can be seen in Figure 5-8 and Figure 5-9,
the stator CP-VBR model (Chapter 4, section 4.2) also has a noticeable error in field
winding current. Based on these figures and Table 5–2 (third and fourth columns from left),
137
Figure 5-8 Magnified fragment from Figure 5-5: ac exciter phase current icex.
Figure 5-9 Magnified fragment from Figure 5-5: rectified current idc.
it can be concluded that the presence of snubbers in the stator or rotor side increases the
errors in iabc or iabcex, respectively.
The two fastest models are AC-CP-VBR and SFC-CP-VBR, both of which do not require
snubbers and have direct stator and field windings interfacing. These two models use the
138
Table 5–2 Numerical Performance of the Models for the Single-Phase-to-Ground Fault Study
Model
Number of
Time-Steps
Error εavg Largest
Eigenvalue
Magnitude
max|λ|
Simulation
Time Stator Source
Current iabc
Exciter Source
Current iabcex
Coupled-Circuit
Phase-Domain
(CC-PD)
922 0.03 % 0.01 % 831 6.772 s
Classical QD0 With
Stator Snubber
(qd0)
2,384* 2.63 % 0.08 % 2.61×105 0.642 s
Stator CP-VBR
(Chapter 4) With
Rotor Snubber
2,739* 1.97 % 1.52 % 1.35×106 0.954 s
All-Circuit
(AC-CP-VBR) 969 2.51 % 0.39 % 5,000 0.619 s
Stator-And-Field-
Circuit
(SFC-CP-VBR)
875 1.95 % 0.23 % 1,000 0.547 s
* For the numerically stiff qd0 and CP-VBR models, MATLAB’s ode15s solver is used. The other
models are solved with ode45.
139
low-pass filter (4–26) to relax algebraic loops. As Table 5–2 shows, the SFC-CP-VBR model
offers better accuracy since its damper windings are implemented as a state-space model
and do not contribute directly to the sub-transient voltage dre algebraic loop. Therefore,
low-pass filter poles with small magnitudes can be chosen (~1,000) which merely affect
the system’s stiffness. At the same time, the AC-CP-VBR model requires a filter pole with a
much larger magnitude (~5,000) to achieve a comparable accuracy. Having the smallest
eigenvalues, in the case studies with lower frequencies, the SFC-CP-VBR model would be
able to choose significantly larger time-steps than the other models.
5.5.2 Diode failure in the exciter system
When system disturbances (such as faults) in the stator circuit of a synchronous generator
happen, the field current fluctuates violently and can become negative and threaten the
transient blocking voltage of rectifiers of a static exciter. A computer model can be used for
prediction of over-voltages and over-currents and to design the required remedies [49]. In
case of brushless exciters, testing the diode rectifiers may not be easy. However, detecting
diode failure from analyzing exciter waveforms is possible [34]. Designing such detection
system also depends on computer simulations. To validate the proposed models in these
situations, where the field winding and exciter variables are of particular interest, a diode
failure scenario is simulated.
A three-leg diode bridge is shown in Figure 5-10 (left). In this study, the diode D3 in the
wye transformer rectifier (see Figure 5-4) fails and becomes short-circuited when the
140
reverse voltage is at its peak (at 0153.0t s). This exposes D3 and D1 to excessive currents
(more than 100 pu), which burn both of them or their protective fuses [50]. It is assumed
that the current continues to flow until it goes to zero, after which the diodes become open.
Figure 5-10 (right) shows the simulated diode currents for this scenario. The times of the
equivalent short-circuit and open-circuit events are also shown in Figure 5-10 (right) by
the circled numbers 1 and 2, respectively.
Simulation results of the exciter and field variables predicted by various models are
presented in Figure 5-11. This figure shows the delta transformer (secondary) voltages vTΔ
and currents iTΔ, the wye transformer (tertiary) voltages vTY and currents iTY, and the
rectifier output voltage vfd and current idc. The stator voltages and currents for this study
are not affected much and therefore are not shown.
A summary of the numerical performance of all considered models is given in Table 5–3. As
seen in the second and fifth columns of Table 5–3 (from left), the models proposed in this
chapter need fewer time-steps and are faster than the stiff models with snubbers. The CC-
PD model requires the fewest time-steps (668), but is the slowest due to its variable
inductances. Table 5–3 (third and fourth columns from left) and Figure 5-11 indicate that
all considered models have excellent accuracy except for the CP-VBR model. The resistive
snubber of CP-VBR model (see Figure 5-4) adds a noticeable current to the rectifier current
( dci ), as seen in Figure 5-11 (last subplot).
141
Figure 5-10 Left: the diode bridge connected to the delta transformer in Figure 5-4. Right: simulation
results of the diode currents in pu: 1) after the initial failure (short-circuit) of D3; and 2) after D1 and
D3 (or the corresponding protective diodes) become open due to the ensuing excessive currents.
At the same time, the qd0 model with the stator interfacing snubber accurately predicts the
field and exciter quantities since its field winding is directly interfaced and the three-phase
snubber depicted in Figure 5-4 mostly affects its stator variables. This study demonstrates
that the models with direct interface of the rotor field winding predict the rotor-exciter
variables more accurately in steady-state and during transients.
142
Figure 5-11 Transient responses as predicted by the considered models for the static exciter diode
failure case-study. From top to bottom: exciter delta transformer line voltages vTΔ, delta transformer
currents iTΔ, wye transformer line voltages vTY, wye transformer currents iTY, rectifier output voltage
vfd, and rectifier current idc.
143
Table 5–3 Numerical Performance of the Models for the Exciter Diode Failure Study
Model
Number of
Time-Steps
Error εavg Simulation
Time Stator Source
Current iabc
Exciter Source
Current iabcex
Coupled-Circuit
Phase-Domain
(CC-PD)
668 0.02 % 0.01 % 5.198 s
Classical QD0 With
Stator Snubber
(qd0)
1,966* 0.02 % 0.09 % 1.069 s
Stator CP-VBR
(Chapter 4) With
Rotor Snubber
2,243* 1.59 % 0.63 % 1.115 s
All-Circuit
(AC-CP-VBR)
724 0.04 % 0.01 % 0.802 s
Stator-And-Field-
Circuit
(SFC-CP-VBR)
695 0.02 % 0.01 % 0.757 s
* For the numerically stiff qd0 and CP-VBR models, MATLAB’s ode15s solver is used. The other
models are solved with ode45.
144
CHAPTER 6: SUMMARY OF CONTRIBUTIONS AND FUTURE WORK
6.1 Conclusions and Contributions
In this thesis, initially a general constant-parameter interfacing circuit for synchronous and
induction machines models with voltage-behind-reactance (VBR) form was developed.
Based on the interfacing circuit, several formulations for ac machines were presented to
obtain numerically efficient VBR models. The proposed formulations have direct interface
and are demonstrated to be more accurate and numerically efficient than the existing
classical qd0 models, and are easier to implement into different simulation programs as
well. A new approach for obtaining constant-parameter models of synchronous machines
was proposed. This methodology is used to achieve constant-parameter VBR interface for
both the rotor and stator terminals of the machine. With respect to the initial objectives of
this research, the contributions of this thesis can be summarized as follows:
6.1.1 Objective 1
In Chapter 2, the model presented in [32] was extended by including the zero-sequence in
the interfacing circuit. The new model has constant-parameters, is explicit, and is easy to
implement in commonly used simulation programs. The computer study showed that the
145
proposed model requires the least number of calculations (764) compared to the state-of-
the-art VBR model (3,865) and the classical qd0 model (7,070) for the same transient study
and similar level of accuracy, which represents a substantial improvement in simulation
efficiency. This contribution addresses Objective 1 of this thesis.
6.1.2 Objective 2
In Chapter 3, the state-of-the-art methods [30], [31] for obtaining the constant-parameter
synchronous machine VBR model were used to formulate a new synchronous machine
model with the same unified interfacing circuit as was developed for the induction
machines in Chapter 2. It was demonstrated that the new explicit model has very good
accuracy (1.2%, for the case study) and is faster than the classical qd0 model (number of
time steps 557 versus 1,871, respectively). This result addresses the Objective 2 of this
thesis. It was also shown that the singular perturbation method [31] is actually
algebraically equivalent to the classical qd0 formulation. However, the model obtained
using the singular perturbation is in fact implicit and therefore may become
computationally very expensive to solve and difficult to implement in many commonly
used simulation programs.
6.1.3 Objective 3
Chapter 4 extended the work presented in Chapter 3 and presented new approximation
approaches to obtain the same constant-parameter VBR interfacing circuit for synchronous
146
machine models. The proposed approach artificially makes the machine's sub-transient
reactance equal while maintaining all properties of the original machine model up to the
desired level of accuracy specified by the user. This new methodology is based on either
continuous or discrete filters that can be used to approximate either current derivatives or
the voltage in order to break algebraic loops. The proposed continuous- and discrete-time
approximations result in computationally efficient implementation of the proposed models
(see section 4.4). The new approximation methods (while achieving the same goal as the
added fictitious damper winding) are very intuitive and easy to implement, do not require
complex analysis or fitting procedures, and are shown to give a high degree of accuracy. As
can be seen in section 4.6, the cumulative error remains below 1% at a fairly large time-
step of mst 1 . This accuracy is acceptable for most power systems transient studies.
This chapter satisfies the Objective 3 of this thesis.
6.1.4 Objective 4
Chapter 5 presents two new explicit synchronous machine models (AC-CP-VBR, section
5.2) and (SFC-CP-VBR, section 5.3) with constant-parameters and direct interfacing of the
machine's stator and rotor terminals with external circuits. The proposed models are
considerably less stiff than the conventional indirectly-interfaced qd0 and constant-
parameter stator VBR models (Chapter 4, section 4.2). The AC-CP-VBR model goes beyond
just the state-variable-based simulation programs since it is represented solely by
basic/conventional and constant circuit elements. This model is easy to implement in
different simulation programs and is practical for special cases where the equivalent
147
damper windings may not be short-circuited or additional field windings are required. In
the second model, SFC-CP-VBR, to increase numerical accuracy and efficiency, the short-
circuited damper windings are represented in state-space form (with flux linkages as state
variables). Single-phase-to-ground fault and the exciter rectifier diode failure case studies
have been conducted to validate and compare the proposed models. It is shown that both
models have an excellent combination of accuracy and numerical efficiency. The new
models are demonstrated to be about 10 times faster than the CC-PD model and achieve
similar accuracy with qd0 models, while having a few orders of magnitude lower stiffness
ratio. These results completely address the last objective of this thesis (Objective 4).
6.2 Potential Impacts of Contributions
It is further envisioned that the proposed constant-parameter VBR interfacing circuits and
approximation techniques will find wide application in many state-variable-based transient
simulation programs. Having the same general structure and interfacing circuit for both
synchronous and induction machines will make it easier to develop customized electrical
machines components, their user interfaces and parameter entry. Software developers and
users should be able to easily implement such models in their programs. Therein,
depending on the solver selection and the machine/system parameters, the user can select
the type of approximation that gives the best numerical results. The proposed fast and
accurate machine models will enables the users to model larger systems with more details
148
that is not possible or practical with the present tools and will save countless number of
hours spend on running simulations by many power engineers, researchers, and students.
6.3 Future Work
The research directions listed below are already being considered by the members of
Electric Power and Energy System research group at the University of British Columbia.
6.3.1 Doubly-fed induction machine model with direct interface
A stator-fed induction machine (with squirrel cage rotor or when the rotor circuit is
connected to voltage-sources) has a simple VBR representation shown in Chapter 2.
However, having direct interface for both rotor and stator terminals is required for doubly-
fed induction machines. The methods proposed in Chapter 4 and 5 can be investigated for
development of a new model for induction machine with constant-parameter decoupled
interfacing circuit for both rotor and stator terminals.
6.3.2 Inclusion of magnetic saturation
For more accurate prediction of power system transients, the effect of magnetic saturation
should be considered. Magnetic saturation results in change of inductances (especially
magnetizing inductances of the q- and d-axis) as the magnetizing flux changes. The non-
linear (variable) part of the magnetizing inductances can be also incorporated in the sub-
149
transient voltage-source using similar approximation approaches as was proposed in
Chapter 4.
6.3.3 Application to dynamic phasor solution and shifted frequency
analysis
Simulation of power system using dynamic phasors approach and shifted frequency
solution has been considered in [57] – [60]. More specifically, using the VBR formulation
with variable parameters has been investigated recently in [61], [62] with promising
results. However, it should be relatively straightforward to extend the proposed constant-
parameter interfacing equivalent circuits to the dynamic phasor and shifted frequency
solution approaches. Therein, additional computational saving can be achieved due to
constant and decoupled interfacing circuit parameters of the models presented in this
thesis.
150
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Appendices:
Appendix A: Parameters of the Induction Machine Model in Section 2.4
50-hp, 60 Hz induction machine [51]
4-pole, 3-phase, 460 V, 1705 rpm, rs = 0.087 Ω, Xls = 0.302 Ω, Xm = 13.08 Ω, rr = 0.228 Ω,
Xlr = 0.302 Ω.
Appendix B: MATLAB Ordinary Differential Equation (ODE) Solvers [15], [63]
Solver Type of
Problem
Accuracy Method MATLAB’s Suggestion
ode45 Non-stiff Medium One-step explicit Runge-
Kutta (4,5) (the
Dormand-Prince pair)
The first solver to try
ode23 Non-stiff Low One-step explicit Runge-
Kutta (2,3) (the Bogacki-
Shampine pair)
May be more efficient than
ode45 at crude tolerances and
for moderately stiff systems
ode113 Non-stiff Low to
high
Multi-step variable order
Adams-Bashforth-
Moulton PECE
May be more efficient than
ode45 at stringent tolerances
and when the evaluation of
the ODE file function is
expensive
158
Solver Type of
Problem
Accuracy Method MATLAB’s Suggestion
ode15s Stiff Low to
medium
Multi-step variable order
based on the numerical
differentiation formulas
NDFs; optionally uses the
backward differentiation
formulas (BDFs)
Try whenever ode45 fails or is
very inefficient, or whenever
the system is stiff; can solve
DAEs
ode23s Stiff Low One-step modified
Rosenbrock formula of
order 2
May be more efficient than
ode15s at crude tolerances;
can be more effective than
ode15s for some stiff
problems
ode23t Moderately
stiff
Low One-step Trapezoidal Use for moderately stiff
problem where numerical
damping is not needed; can
solve DAEs
ode23tb Stiff Low Implicit, TR-BDF2 May be more efficient than
ode15s at crude tolerances
ode15i Fully
implicit
BDFs To solve fully implicit
differential equations
159
Appendix C: Parameters of the Synchronous Machine in Subsections 3.7.1 to
3.7.3
125-kW, 60 Hz synchronous machine (in pu) [30]
rs = 0.00515, Xls = 0.0800, Xmq = 1.00, Xmd = 1.77, rkq1 = 0.0610, rfd = 0.00111, rkd1 = 0.024,
Xlkq1 = 0.330, Xlfd = 0.137, Xlkd1 = 0.334, H = 2.5 s, 248.0qX and 0921.0dX .
Appendix D: Parameters of the Power System in Subsection 3.7.4
555-MVA steam turbine generator (in pu) [4, sec 8.3]:
24 kV, 2-pole, 3600 r/min, 0.9 p.f., 60 Hz, rs = 0.003, Xls = 0.15, Xmq = 1.61, Xmd = 1.66,
rkq1 = 0.00619, rkq2 = 0.02368, rfd = 0.0006, rkd1 = 0.0284, Xlkq1 = 0.7252, Xlkq2 = 0.125,
Xlfd = 0.165, Xlkd1 = 0.1713, H = 5.6 s, 25.0qX , and 23.0dX .
Appendix E: Parameters of the Power System in Section 4.6
555-MVA steam turbine generator (in pu) [4, sec. 8.3]:
The machine is the same as subsection 3.7.4 which is given in Appendix D.
Grounding reactance (in pu, on the machine base):
Xg = 0.3.
Unit transformer impedance (in pu, on the machine base):
Xt = 0.16 and rt = 0.02.
160
Appendix F: Parameters of the Power System in Section 5.5
555-MVA steam turbine generator (in pu) [4, sec. 8.3]:
The machine is the same as subsection 3.7.4 (and section 4.6) which is given in Appendix D.
Grounding resistance (in pu, on the machine base):
Rg = 0.2.
Thévenin impedance of the network (in pu, on the machine base):
Xs = 14% and X/R = 14.
Rotor-exciter Ydy transformer impedance (in pu, on the transformer base):
10 MVA, 24:0.144:0.144 kV, XPS = XPT = XST = 8% and XPS /RPS = XPT/RPT = XST /RST =13.33.