by
Hamid Atighechi
B.Sc. the Isfahan University of Technology, 2006
M.Sc. the Iran University of Science and Technology, 2009
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF
DOCTOR OF PHILOSOPHY in
The Faculty of Graduate Studies
(Electrical and Computer Engineering)
THE UNIVERSITY OF BRITISH COLUMBIA
(Vancouver)
June 2013
© Hamid Atighechi, 2013

High power switching converters such as line-commutated converters (LCC) and high voltage direct current (HVDC) systems are widely used in modern energy grids for interconnection of industrial loads, large motor drives, as well as electronically-interfaced renewable/alternative/distributed energy resources (DER) and storage systems. For design and analysis of systems with power-electronic-based DERs and loads, accurate and efficient computer models are essential. This thesis is focused on dynamic average-value models (AVM) that neglect switching of converter circuits and are established by averaging the variables (currents and voltages) over a prototypical switching interval. The AVMs are continuous (free of switching), allow using larger integration time steps, and typically run much faster than the conventional detailed switching models, which makes them particularly useful for the system-level studies. This thesis considers the parametric AVM framework, and extends this approach to the thyristor-controlled LCCs operating in inverter mode with current source or voltage source control. The proposed modeling methodology is demonstrated on various topologies including the HVDC CIGRE benchmark system. The research is further extended to incorporate the ac side harmonics into the AVM using the multiple reference frame theory. Traditionally, the AVMs are developed using state-variable-based approach. This thesis also presents a new parametric AVM for direct interfacing in nodal-analysis-based electromagnetic transient programs (EMTP), e.g.,
PSCAD/EMTDC, EMTP-RV, and MicroTran. It is expected that the proposed models and ii

interfacing approaches will find their application in widely used transient simulation tools and will be appreciated by many researchers and practicing engineers worldwide. iii

Based on the research contributed in this thesis, several papers have been published, accepted for publication or ready for submission as journal articles and conference publications. My research work and all my publications have been done by me under the supervision of Dr. Juri Jatskevich. The co-authors of publications have provided me with their constructive feedback and comments, and helped with the subsequent iterations and revisions of manuscripts as was necessary in each case.
On the basis of chapter 2:
 H. Atighechi, S. Chiniforoosh, K. Tabarraee, J. Jatskevich, “Dynamic Average-Value
Model of Thyristor-Controlled-Rectifier Synchronous Machine Systems,”
Submitted.
 H. Atighechi, S. Chiniforoosh, J. Jatskevich, “Approximate Dynamic Average-Value
Model for Controlled Line-Commutated Converters,” in Proc. IEEE Canadian
Conference on Electrical and Computer Engineering (CCECE) , Niagara Falls, 2011, pp. 966–970.
 H. Atighechi, S. Amini Akbarabadi, F. Therrien, and J. Jatskevich, “Dynamic
Average-Value Modeling of Synchronous Machine Line-Commutated Converter
Using a Constant Parameter Voltage-Behind Reactance Interfacing Circuit,” iv

accepted for presentation in International Conference on Power Systems
Transients (IPST), Vancouver, 2013.
On the basis of chapter 3:
 H. Atighechi, J. M. Cano, and J. Jatskevich, “Average-Value Modeling of Thyristor
Controlled Line-Commutated Converter Using Voltage and Current Source
Formulation,” accepted for Presentation in IEEE Power and Energy Society
General Meeting (PES), Vancouver, 2013.
 H. Atighechi, F. Therrien, and J. Jatskevich, “Using Current Source Formulation for Dynamic Average-Value Modeling of Inverter Side HVDC System,” in Proc.
IEEE Canadian Conference on Electrical and Computer Engineering (CCECE),
Regina, 2013.
On the basis of chapter 4:
 H. Atighechi, S. Chiniforoosh, J. Jatskevich, A. Davoodi, J. A. Martinez, M. O.
Faruque, V. Sood, M. Saeedifard, J. M. Cano, J. Mahseredjian, D. C. Aliprantis, and
K. Strunz, “Dynamic Average-Value Modeling of CIGRE HVDC Benchmark
System,” Accepted in IEEE Trans. Power Del. 2013.
v

 H. Atighechi, S. Chiniforoosh, J. Jatskevich, “Large-Signal Average Modeling of
HVDC System Transients using Parametric and Analytical Approaches,” submitted.
On the basis of chapter 5:
 H. Atighechi, S. Chiniforoosh, and J. Jatskevich, “Using Multiple Reference Frame
Theory for Considering Harmonics in Dynamic Average-Value Modeling of Line-
Commutated Converter,” to be submitted.
On the basis of chapter 6:
 H. Atighechi, S. Chiniforoosh, J. Jatskevich, “Directly-Interfaced Parametric
Average-Value Models for Line-commutated Rectifier Circuits in EMTP-Type
Solution,” to be submitted.
 H. Atighechi, S. Chiniforoosh, J. Jatskevich, “Discretized Parametric Average-
Value Model of Line-commutated Rectifier based on the trapezoidal integration with localized damping,” under the preparation.
vi

Abstract ................................................................................................................................................................. ii
Preface ................................................................................................................................................................. iv
Table of Contents................................................................................................................................................. vii
List of Tables ......................................................................................................................................................... xi
List of Figures ....................................................................................................................................................... xii
List of Abreviations ............................................................................................................................................ xvii
Acknowledgments ............................................................................................................................................. xviii
Dedication ........................................................................................................................................................... xx
Chapter 1: Introduction ......................................................................................................................................... 1
1.1 Motivation .......................................................................................................................................................... 1
1.2 Literature Review ................................................................................................................................................ 3
1.2.1 Line-Commutated Converter Models ......................................................................................................... 7
1.2.2 HVDC System Models .................................................................................................................................. 8
1.2.3 Multiple Reference Frames Average-Value Models of Line-Commutated Converters ............................... 9
1.2.4 Discretized Average-Value Model of Line-Commutated Converter in EMTP-Type Solution .................... 11
1.3 Research Objectives and Anticipated Impact .................................................................................................... 12
Chapter 2: Dynamic Average-Value Model of Thyristor Controlled Rectifier ........................................................ 16 vii

2.1 Introduction ...................................................................................................................................................... 16
2.2 Detailed Model System ..................................................................................................................................... 18
2.3 Extended Parametric Approach ........................................................................................................................ 20
2.4 Implementation of Parametric Average-Value Model ...................................................................................... 26
2.5 Extended Analytical Average-Value Model ....................................................................................................... 28
2.6 Model Verification............................................................................................................................................. 30
2.6.1 Time- Domain Studies ............................................................................................................................... 31
2.6.2 Time-Domain Transient Study with a Controller ....................................................................................... 36
2.6.3 Frequency-Domain Studies ....................................................................................................................... 38
2.1 Summary of Contributions ................................................................................................................................ 41
Chapter 3: Modeling Bidirectional Line-Commutated Converter Systems ............................................................ 43
3.1 Average-Value Model Formulations ................................................................................................................. 43
3.2 Well-Posedness of Voltage Source AVM (VS-AVM) ........................................................................................... 45
3.2.1 Current Source AVM (CS-AVM) ................................................................................................................. 55
3.2.2 Case Studies .............................................................................................................................................. 57
3.3 Summary of Contributions ................................................................................................................................ 62
Chapter 4: Average Modelling of HVDC System Using Voltage and Current Source Formulations ........................ 63
4.1 Introduction ...................................................................................................................................................... 63
4.2 HVDC System Detailed Model ........................................................................................................................... 64
4.3 HVDC System Dynamic Average-Value Model .................................................................................................. 68 viii

4.3.1 Rectifier Dynamic Average-Value Model .................................................................................................. 70
4.3.2 Inverter Dynamic Average-Value Model ................................................................................................... 74
4.3.3 AC Subsystems .......................................................................................................................................... 77
4.3.4 DC Subsystem ............................................................................................................................................ 80
4.3.5 Control Subsystem .................................................................................................................................... 80
4.4 Computer Studies .............................................................................................................................................. 81
4.4.1 Change in DC Reference Current............................................................................................................... 81
4.4.2 Short Circuit in DC Side ............................................................................................................................. 82
4.5 Summary of Contributions ................................................................................................................................ 87
Chapter 5: Considering AC Side Harmonics in Dynamic Averaging of Line-Commutated Converters .................... 88
5.1 Multiple-Reference Parametric Average-Value Model ..................................................................................... 90
5.2 MRF-PAVM Implementation ............................................................................................................................. 97
5.3 Computer Studies .............................................................................................................................................. 98
5.3.1 Model Verification in Steady-State Study Over Wide Range .................................................................... 99
5.3.2 Transient Study ....................................................................................................................................... 109
5.4 Summary of Contributions .............................................................................................................................. 112
Chapter 6: Direct Interfacing of Parametric Average-Value Models in EMTP Type Solution ............................... 113
6.1 Discretized Parametric Average-Value Model ................................................................................................ 114
6.2 Directly-Interfaced PAVM with Collapsed DC Side .......................................................................................... 116
6.3 Directly-Interfaced PAVM with Snubber in DC Side ........................................................................................ 122
6.4 Computer Studies ............................................................................................................................................ 125 ix

6.5 Damping Technique for Directly-Interfaced Parametric Average-Value Models ............................................ 133
6.5.1 Trapezoidal Integration with Damping ................................................................................................... 134
6.5.2 Localized Damping for Interfacing DI-PAVM ........................................................................................... 136
6.5.3 Case Study ............................................................................................................................................... 138
6.6 Summary of Contributions .............................................................................................................................. 141
Chapter 7: Conclusions and Future Work ........................................................................................................... 143
1.7
Conclusions and Contributions ....................................................................................................................... 143
7.2 Future Work .................................................................................................................................................... 146
References ......................................................................................................................................................... 148
Appendices ........................................................................................................................................................ 162
Appendix A: Synchronous Machine/Rectifier System ............................................................................................ 162
Appendix B: Well-Posedness Study ....................................................................................................................... 164
Appendix C: CS-Formulation Verification .............................................................................................................. 165
Appendix D: CIGRE HVDC System .......................................................................................................................... 166
Appendix E: Harmonic Analysis ............................................................................................................................. 167
Appendix F: Discretized PAVM .............................................................................................................................. 168 x

Table
‎
3-1: Simulation speed for AVM and detailed simulation to t

2 .
25 s ........................................................ 62
Table
‎
4-1: Comparison of simulation speed for different models for time interval from t

0 .
9 s to t

2 s ....... 85
Table
‎
5-1: Comparison of simulation speed for different models for time interval from t

0 to t

2 s .......... 112 xi

Figure
‎
1-1: Example three-phase ac-dc converter system: (a) line-commutated converter (LCC); and (b) pulsewidth-modulated (PWM). ............................................................................................................................. 4
Figure
‎
1-2: Example HVDC system depicting three-phase line-commutated inverter and rectifier. ........................ 5
Figure
‎
1-3: Various average-value modeling methodologies of high-power converters. ...................................... 15
Figure
‎
2-1: Synchronous machine/thyristor-controlled-rectifier system. ............................................................ 17
Figure
‎
2-2: Typical phase terminal voltage and current of controlled-rectifier with firing angle of 30 electrical degrees. ...................................................................................................................................................... 19
Figure
‎
2-3: Generator and converter ac side voltages and currents expressed in rotor and converter reference frames. ....................................................................................................................................................... 21
Figure
‎
2-4: Numerically calculated parametric functions w v
, w i
 
,
  
,
  
and
  
.
................................................................................................................................................................... 25
Figure
‎
2-5: Implementation of parametric average-value models: (a) using rotor position as a reference for generating firing pulses; and (b) using converter terminal voltages for generating firing pulses. ................ 27
Figure
‎
2-6: Analytical average-value model for machine/thyristor rectifier system: (a) equivalent circuit diagram of interfacing voltage-behind-reactance machine model and thyristor rectifier system; and (b) block diagram depicting implementation with algebraic loop. The equation numbering in part (b) refers to the original sources [72] and [15]. .................................................................................................................... 29
Figure
‎
2-7: Steady state output dc voltage: (a) using converter terminal voltages for generating firing pulses; and
(b) using rotor position as a reference for generating firing pulses. ............................................................ 32
Figure
‎
2-8: System response to a change in load in mode 1 as predicted by the detailed model, analytical and parametric AVMs. ....................................................................................................................................... 34 xii

Figure
‎
2-9: System response to a change in firing angle in mode 2 as predicted by the detailed model, analytical and parametric AVMs. ................................................................................................................................ 35
Figure
‎
2-10: Controller used to control load voltage. ........................................................................................... 37
Figure
‎
2-11: Closed-loop system response to a step change in load; detailed and parametric AVM models. ....... 37
Figure
‎
2-12: Output impedance as predicted by various models. ......................................................................... 39
Figure
‎
2-13: Transfer function for the input firing angle to the output dc voltage as predicted by various models.
................................................................................................................................................................... 39
Figure
‎
2-14: Output impedance for controlled system as predicted by various models. ...................................... 41
Figure
‎
3-1: Block diagram depicting AVM implementation of the thyristor-controlled inverter: (a) voltage-source formulation; and (b) current-source formulation. ....................................................................................... 44
Figure
‎
3-2: Stability of the VS-AVM for thyristor-controlled rectifier using

method: (a) stability region in state space; and (b) phase portrait. ..................................................................................................................... 50
Figure
‎
3-3: Stability of the VS-AVM for thyristor-controlled inverter using

method: (a) stability region in state space; and (b) phase portrait. ..................................................................................................................... 52
Figure
‎
3-4: Stability of the VS-AVM for thyristor-controlled inverter using

method L
C

0 .
1 mH : (a) stability region in state space; and (b) phase portrait. ................................................................................ 53
Figure
‎
3-5: Stability of the VS-AVM for thyristor-controlled inverter using

method, L
C

0 .
3 mH : (a) stability region in state space; and (b) phase portrait. ................................................................................ 54
Figure
‎
3-6: Stability region of CS-AVM for inverter. ............................................................................................. 57
Figure
‎
3-7: Inverter side of the HVDC system with current control on rectifier side. ............................................ 58
Figure
‎
3-8: System response to a fast change in input current. ............................................................................. 60
Figure
‎
3-9: System response to a step change in firing angle. .............................................................................. 61 xiii

Figure
‎
4-1: CIGRE HVDC benchmark system circuit diagram. ............................................................................... 65
Figure
‎
4-2: CIGRE HVDC benchmark control subsystem: (a) rectifier control; and (b) inverter control. ............... 66
Figure
‎
4-3: Typical voltages and currents of a 12-pulse rectifier transformer primary winding. .......................... 69
Figure
‎
4-4: Block diagram depicting implementation of the AVM rectifier side: (a) original AVM with two rectifiers separately; and (b) collapsed AVM. ............................................................................................. 74
Figure
‎
4-5: Block diagram depicting implementation of the AVM inverter side. ................................................. 76
Figure
‎
4-6: Block diagram depicting implementation and interfacing of the inverter side ac subsystem: (a) using abc original circuit; (b) using transformed qd 0 -circuit. ........................................................................... 78
Figure
‎
4-7: Inverter side ac subsystem implementation using transformed qd 0 -circuits. .................................. 79
Figure
‎
4-8: Currents and voltages resulting from a change in dc reference current as predicted by various models. ....................................................................................................................................................... 84
Figure
‎
4-9: Currents and voltages during a short circuit in dc side as predicted by various models. .................... 86
Figure
‎
5-1. Typical phase terminal voltage and current of diode rectifier system and their harmonic content. .. 90
Figure
‎
5-2: Rectifier ac side voltages and currents expressed in synchronous, converter [15], and n th
harmonic reference frames. ....................................................................................................................................... 92
Figure
‎
5-3: Block diagram depicting implementation of the MRF-PAVM for the diode rectifier system. ............. 98
Figure
‎
5-4: Steady state regulation characteristic of the considered line-commutated converter as predicted by various models. ........................................................................................................................................ 100
Figure
‎
5-5: System variables as predicted by various models in DCM; detailed, conventional PAVM, and MRF-
PAVM models, respectively. ..................................................................................................................... 102
Figure
‎
5-6: Harmonic content of the phase current and voltage as predicted by various models in DCM. ........ 103
Figure
‎
5-7: System variables as predicted by various models for operation in CCM-1. ...................................... 104 xiv

Figure
‎
5-8: Harmonic content of the phase current and voltage as predicted by various models for the rectifier operation CCM-1. ..................................................................................................................................... 105
Figure
‎
5-9: System variables as predicted by various models in CCM-2; detailed, conventional PAVM, and proposed MRF-PAVM models, respectively. ............................................................................................. 106
Figure
‎
5-10: Harmonic content of the phase current and voltage in rectifier in CCM-2 as predicted by the detailed model and the proposed MRF-PAVM. ......................................................................................... 107
Figure
‎
5-11: Total harmonic distortion of the rectifier phase current and voltage as predicted by the detailed model and the proposed MRF-PAVMs over a wide range of operating conditions. .................................. 109
Figure
‎
5-12: System response to a step change in load as predicted by various models; detailed, conventional
PAVM and proposed MRF-PAVM. ............................................................................................................. 111
Figure
‎
6-1: Thevenin equivalent-circuit diagram of directly-interfaced PAVM in abc variables with collapsed dc side. .......................................................................................................................................................... 115
Figure
‎
6-2: Block diagram of descretized averaged-circuit model with collapsed dc side. .................................. 118
Figure
‎
6-3: Norton equivalent-circuit diagram of directly-interfaced PAVM in abc phase variables with collapsed dc side. ...................................................................................................................................... 121
Figure
‎
6-4: Block diagram of the descretized averaged-circuit model interfaced with resistive snubber in dc side.
................................................................................................................................................................. 124
Figure
‎
6-5: Transient response of the system in dc and ac variables as predicted by the various models using
50
μs
time-step. ..................................................................................................................................... 129
Figure
‎
6-6: Transient response of the system in dc and ac variables as predicted by various models using 500
μs time-step: (a) dc and ac waveforms of the overall transient; and (b) magnified view of the start-up waveforms. ............................................................................................................................................... 131 xv

Figure
‎
6-7: Transient response of the system in dc and ac variables as predicted by various models using 2 ms time-step: (a) dc and ac waveforms of the overall transient; and (b) magnified view of the start-up waveforms. ............................................................................................................................................... 132
Figure
‎
6-8: Discretized equivalent circuit of inductance: (a) Thevenin equivalent; and (b) Norton equivalent. .. 134
Figure
‎
6-9: Discretized equivalent circuit of lumped inductance with damping history term: (a) Thevenin equivalent; and (b) Northon equivalent.................................................................................................... 137
Figure
‎
6-10: Thevenin equivalent-circuit diagram of the locally damped DI-PAVM in abc variables. ............... 138
Figure
‎
6-11: Dc and ac waveforms predicted by the models, with and without localized damping;
 t

2 ms .
................................................................................................................................................................. 140
Figure
‎
6-12: Transient response of DI-PAVM with collapsed dc side as predicted by various models using time step
 t

2 ms , and conventional and localized damping. .................................................................... 141 xvi

Abbreviation
AC
DC
LCC
HVDC
IPP
DER
EMTP
AVM
PAVM
PWM
IHA
VSC
VS-AVM
CS-AVM
MRF
MRF-PAVM
DI-PAVM
Meaning
Alternating Current
Direct Current
Line-Commutated Converter
High Voltage Direct Current
Independent Power Producer
Distributed Energy Resource
Electro Magnetic Transient Program
Average-Value Model
Parametric Average-Value Model
Pulse-Width-Modulation
Iterative Harmonic Analysis
Voltage-Source Converter
Voltage-Source-Average-Value Model
Current-Source-Average-Value Model
Multiple Reference Frames
Multiple Reference Frame Parametric Average-Value Model
Directly Interfaced Parametric Average-Value Model xvii

First and foremost I would like to thank my research supervisor, Dr. Juri Jatskevich, for his inspiration, vision, help, patience and support which made it possible for me to accomplish this research and produce the original contributions that comprise this dissertation. The financial support for this research was made possible through the Natural Science and
Engineering Research Council (NSERC) 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 also I would like to express my gratitude to my research supervisory committee, Dr.
Hermann Dommel, Dr. William Dunford, Dr. Jose Marti, and Dr. KD Srivastava, for the help and advice that I received from them during all these years of my research program at UBC.
I would like to thank my colleagues and fellow researchers in the Electric Power and
Energy Systems research group at UBC. Particularly, I would like to thank Dr. Sina
Chiniforoosh for the countless hours and discussions of many details in my research topic.
My research has extended his Ph.D. work, and I felt very privileged to have Dr. Sina
Chiniforoosh so readily available and willing to see the continuation of research ideas of our group. Also, I owe many thanks to Francis Therrien, Amir Rasuli, Mehrdad Chapariha,
Kamran Tabarraee, and Mehmet Sucu for their time and help in my research and life as a xviii

member of the UBC’s Electric Power and Energy Systems research group. I thank all my dear friends who made my student life at UBC colorful and meaningful.
I also would like to thank all the staff in Electrical and Computer Engineering Department and the Faculty of Graduate Studies for their kind smiles and help with any administrative and/or procedural questions whenever I had them, and I had a lot.
I would like to express my sincere gratitude toward my mother who has been giving me courage during all the years of studying at UBC, her endless love and believe in me and my future success. I also thank all my family members who were always in my heart and provided a strong support for me. I also thank my dearest Hong Zhu who was beside me during all the hard times of writing this dissertation. xix

This thesis is dedicated to my mother, Rababeh Mofazzali, who showed me the true love and passion in life; and to the memory of my father, Morteza Atighechi, who always showed me the way when I was lost; and to my brother, Saeed Atighechi, and my sister, Farahnaz
Atighechi, who never stopped supporting me during all these years despite the distance. xx

Power electronic converters are commonly used in modern power systems and wide variety of applications such as high voltage dc (HVDC) transmission systems, high-power dc supplies, excitation systems of large electric generators, etc. Moreover, the use and application of power electronic converters in power systems has been increasing faster than before due to development of Smart Energy Grid and interconnection of numerous distributed energy recourses (DERs) and independent power producers (IPPs). Modeling and simulation of these power-electronic-based systems are essential for design and analysis of modern electrical energy systems with switching components.
Design and analyzes of such complex systems normally require many computer studies in the time and frequency domains. Thousands of engineers and researchers around the world are involved in conducting various studies and computer simulations of systems containing power electronic converters.
There are many time-domain simulation packages, nodal-analysis-based e lectro m agnetic t ransient p rograms (EMTP-based) [1]-[5], and state-variable-based, [6]-[12], that are extensively used to model the power-electronic-based systems including line-commutated
1

converters considering all the switching details. Although the modern transient simulation tools are constantly improving, their simulation speed and applicability to larger systems with switching converters remain to be among the limiting factors wherein the detailed models of switching converters are often the bottle-neck. In particular, due to inherent switching, the detailed models developed using the time-domain simulation packages are discontinuous, and impossible to linearize for small-signal frequency-domain characterization, which is an important tool for assessing the system’s stability and designing of the required controllers. Moreover, the detailed switching models are computationally expensive for system-level transient studies with many components and subsystems, which often leads to significant increase of the required computing time and in turn limits the size of the system that can be practically studied [13], [14].
In order to overcome these difficulties, the average-value models (AVMs), wherein the effect of fast switching is averaged or neglected with respect to a prototypical switching interval, have been developed. The AVMs are computationally efficient and can execute orders of magnitude faster than the corresponding detailed switching models. Moreover,
AVMs are continuous and can be linearized about any operating point for the small-signal frequency-domain characterization. Since the need for system-level modeling and analysis tools is rapidly increasing, the average-value modeling approach has been recognized by many researchers as a powerful tool to study and analyze complicated power-electronicbased systems when the dynamics of interest is slower than the switching frequency. A comprehensive review of different averaging approaches and their application to the
2

modeling and analysis of power systems transients can be found in reports by the IEEE
Task Force on Dynamic Average Modeling [13], [14].
Depending on the type of semiconductor switches, the high-power converters as it is
shown in Figure ‎ 1-1, may be generally classified into pulse-width-modulated (PWM) (ones
that use power transistors) or the line-commutated (ones that use diodes or thyristors) converters. While the dynamic average modeling of PWM-based converters can be readily established due to continuous nature of the phase currents using the conventional reference frame theory [15], this is generally not the case for the line-commutated converters due to discontinuous phase currents and complicated waveforms of the voltages and currents.
The line-commutated converters such as the one depicted in Figure ‎ 1-1(a), are particularly
common in high-power applications due to low cost and high efficiency of thyristor-based
semiconductor switching devices. Figure ‎ 1-1 shows a generic three-phase topology which results in six pulses per electrical cycle. The line-commutated converter shown in Figure
‎ 1-1, based on the application, may be supplied by a distribution feeder and transformer or
a rotating machine such as synchronous generator.
3

Figure ‎ 1-1: Example three-phase ac-dc converter system: (a) line-commutated converter
(LCC); and (b) pulse-width-modulated (PWM).
Dynamic average-value models for line-commutated converters can be classified in to two main categories: analytical [15]-[23] and parametric [24]-[34]. In the first approach, the average-value model is derived analytically, wherein the variables (currents and voltages) are analytically averaged over a prototypical switching interval [15].
One of the main applications of line-commutated converters is the HVDC systems. HVDC
Classics systems are frequently employed for long distance transmission applications due to their low electrical losses. For shorter distances, the higher costs of the power electronic equipments may still be justified due to other benefits achieved by HVDC such as improved system stability and inter-connection between unsynchronized ac systems. A typical HVDC system is composed of two back-to-back thyristor-controlled line-commutated converters
as shown in Figure ‎ 1-2. Presently, the HVDC Classics technology represents a major source
4

of projects and contracts for the leading manufactures such as ABB, Siemens, and Alstom
Grid.
Figure ‎ 1-2: Example HVDC system depicting three-phase line-commutated inverter and rectifier.
In order to model HVDC systems, it is necessary to develop the desirable model for both the inverter and rectifier sides. The developed HVDC models so far can be classified into three main categories. First, the steady state models [35]-[41] which are very simple to implement but inadequate for predicting the system’s large-signal transients. Second, the dynamic models [42], [43] which can be used to predict the system dynamics with a limited accuracy. Finally, the small-signal models in the frequency-domain [44]-[47] which can be used to predict the system's stability. Such linear or linearized models are advantageous for design of controllers, analysis of converter-machine system interaction, studies involving the sub-synchronous torsional interaction phenomenon, etc. However, the small-signal models are valid only within a small range about an operating point for which they were
5

established (and need to be re-derived when a different operating point is considered) and are not valid for large-signal time-domain studies.
Power-electronic devices such as line-commutated converters are sources of different harmonics injected to the energy grid. The injected harmonics (current and voltage) have a undesirable effect on the system generators, transformers, variable speed drives, etc [48].
The conventional AVMs preserve only the fundamental (60 or 50 Hz) component on the ac side, whereas including at least several of significant (5 th , 7 th , 11 th , etc.) harmonics would be very much desirable for the systems studies involving ac filters and evaluation of losses.
Therefore, it has always been desirable to extend the AVMs to the power quality analysis and harmonic power flow calculations. An analytical AVM of the line-commutated converter has been introduced in [19] and [49], where in the ac side waveforms are reconstructed.
Another challenge comes from the fact that most of previously-proposed average-value models have been formulated in conventional state-variable form. Therefore, these models can be readily implemented in the state-variable-based transient simulators only (e.g.
Matlab/Simulink, ACSL, Easy5 and Eurostag), whereas for modeling of large-scale electric power systems the nodal-analysis-based simulation packages (e.g. EMTP-RV, PSCAD, etc.) in which the discretization is performed at the branch/component level are often preferred due to their numerical efficiency. Reformulation of the established average-value models for the nodal-analysis-based solution is a significant area of research that requires further
6

consideration. Reformulated analytical and parametric AVMs of the line-commutated rectifier systems, on the basis of their interfacing method with the nodal-based simulators
(PSCAD and EMTP-RV) may be classified as indirectly or directly interfaced approaches
[50]. In the following Subsections, a brief literature review of the different approaches of the average-value modeling for the line-commutated converters and their application is provided.
The initial attempts to develop analytical AVM for a synchronous machine/converter
system, such as the one shown in Figure ‎ 1-1, date back to 1960s [17]. For the purpose of
steady state analysis, reduced order models [18] and [22] may be used. However, these reduced order models are inaccurate for small-signal analysis in high frequencies. An analytical AVM which is accurate in both the time- and frequency-domains has been studied in [19], and later extended in [20] for an inductorless machine/converter system.
In general, the analytical derivation of AVM for a switching system is challenging because the analytical expressions should be extracted for each mode of operation separately. A dynamic analytical AVM has been developed in [23] quite recently in which the equations are derived for two modes of operation. The challenges in analytical derivations led to the development of parametric average-value models for machine/diode rectifiers in [24], [25], wherein the rectifier AVM parameters are obtained using a detailed simulation. However,
7

the AVM parameters of the proposed model [24] and [25] are independent of the loading condition which causes error in both the steady state and the dynamic responses of the model. Recently, the parametric approach with constant parameters [24] is extended to model thyristor-controlled rectifier system in [26]. A parametric AVM of a synchronous machine/diode rectifier system in which the AVM parameters are variable based on the loading condition has been introduced in [27]. This model is very accurate in both the time and frequency domains but valid only for diode rectifier. The approach proposed in [27] requires running a detailed simulation to extract the AVM parameters and save them as look-up tables. On the basis of [27], a fast and efficient procedure for extracting the AVM parameters has been introduced in [28]. The parametric approach introduced in [27], has been evolved and used by many researchers [29]-[34] due to its accuracy and being easy to implement.
Developing steady-state and dynamic equivalent models for HVDC transmission systems has been of great interest in the power system research community. Steady-state models for HVDC systems were first proposed in [35], and further improved in [36] and [37]. The effects of harmonics and inter-harmonics in steady-state HVDC models have been discussed in the literature [38]. Impedance mapping and equivalent circuit of HVDC system have been considered in [39]. More recent improvements have been made in [40] by
8

considering the commutation subinterval. The conventional steady-state models discussed in [35]-[40], are straightforward to implement. A simplified model [41] considers the dc and ac side dynamics with some approximations. A dynamic model for the rectifier side of hybrid HVDC system can be found in [42], where the inverter side is modeled as a voltage or current source. In order to predict the HVDC system's stability, the small-signal model has been developed in [43]-[47], which are advantageous for controller design, analysis of interaction among the subsystems, sub-synchronous torsional interaction phenomenon, etc
[46].
If the harmonics can be incorporated into the AVMs, then such models could also be used for faster and more efficient harmonic analysis instead of the traditional detailed models.
For the purpose of modeling, instead of a detailed/switching model, it has been proposed to employ the equivalent harmonics current sources of the most significant harmonics [51].
In this approach, the magnitudes and phases of the current sources in the model are predicted based on the harmonic spectrum of the original switching components. These models are much simpler to implement but they may not provide the desired accuracy. For better accuracy in predicting the injected harmonics, voltage-dependent current-source models have been developed in [52]-[56]. Such models are solved iteratively, and the
9

method is hence known as the Iterative Harmonic Analysis (IHA). The complexity of the
IHA method has led to the development of some analytically-derived models, wherein the relationship between the harmonic voltages and currents are established through a transformation matrix [57] or an admittance matrix [58], [59]. All the above-mentioned models can only be used for steady-state system-level studies.
For the purpose of power system stability analysis, small-signal models have been developed in [60]-[62], wherein the small-signal input impedance of the LCC is analytically derived. However, the common drawback of these models is that they are valid only in a small range (e.g.  5%) about the considered operating point.
In order to overcome the challenges imposed by the switching in power-electronic-based systems, the AVM have been developed in [15]-[32] that are capable of maintaining the system dynamics. The conventional AVMs for LCC preserve only the fundamental (50/60
Hz) component of the ac currents and voltages. Such simplification occurs directly as a result of averaging. In order to employ the AVMs in power quality analysis and harmonic power flow calculations, it is desirable to include at least several significant harmonics (5 th ,
7 th , 11 th , etc.). The analytical AVM of the machine/rectifier system introduced in [18] has been extended in [19] to reconstruct the original switching waveforms from the averagevalue model. More recently, based on the analytical approach introduced in [15], an AVM is formulated in [47] considering ac system harmonics and reconstructing the system variable waveforms. However, extracting the analytical expressions for such complex
10

switching system is challenging in general, and the equations should be derived for each
LCC mode of operation [27], which makes this approach less practical.
The first step to implement the AVMs in EMTP-type programs has been taken in [14] by using indirect interfacing approach [63]-[65], [13]. However, the indirect interfacing technique introduces one step delay in the simulation which causes loss of accuracy and numerical stability for large time-steps. Therefore, in [48], a direct interfacing methodology
[65] has been implemented for interfacing of the analytical AVM introduced in [15] with the external EMTP networks. Direct interfacing technique removes the one-time-step delay and significantly improves the simulation accuracy at large time-steps in comparison to the indirect approach.
Among the modeling approaches, the parametric average-value modeling methodology that was set forth by the UBC researchers appears very promising and extendable to more complex converter systems and topologies. However, at the present time, this approach has only been developed for machine/diode rectifier systems [27], [28]. Therefore, the parametric average-value modeling has been chosen as the framework for the research carried out in this thesis.
11

For the purpose of this research, the methodologies for average-value modeling of high
power ac-dc converters can be broadly depicted in a block diagram shown in Figure ‎ 1-3.
The PWM converters use high-frequency switching and generally operate in continuous current mode, which significantly simplifies their average-value modeling [13], [15]. The research is therefore focused on the line-commutated thyristor-controlled converters (see
green path in Figure ‎ 1-3). In particular, since most of the previous work has been done for
the uncontrolled (diode) rectifiers, herein, this approach is extended to the thyristorcontrolled rectifiers as well as inverters. This extension allows combining the parametric
AVMs of the controllable rectifier and inverter subsystems into an HVDC system.
Another direction of research is the inclusion of major harmonics (e.g. 5th, 7th, and 11th) into the average-value modeling methodology. Such harmonics can be significant in systems with LCCs. The AVMs with such capabilities would then be very useful for studies involving harmonic filters, harmonic power flow, evaluation of losses and efficiency, etc., while the underlying models will be continuous (without switching) and linearizable. This feature is also useful for modeling the flexible ac transmission devices (FACTS) and other power-electronic-based devices in power systems. Moreover, the system analyst would be able to select the desired harmonic(s) that may be needed depending on the study objectives and the desired accuracy without having to default to the conventional switching
(discontinuous) models.
12

Finally, it is also observed that average-value modeling of both the PWM and linecommutated converters can be formulated in a very similar and general (parametric) form that may be used to unify the interfacing of various AVMs with the external power network.
However, since most transient simulation packages are based on the nodal (EMTP-type) solution, it is necessary to formulate the appropriate discretized models and their interface for the EMTP-type programs such as PSCAD/EMTDC, EMTP-RV, MicroTran, and etc. This research extends the previous work of the UBC’s power group on interfacing the AVMs with EMTP-based packages. To summarize, the research objectives of this thesis include the following major steps and directions:
Objective #1: Extend the parametric average-value modeling to thyristor-controlled rectifier systems.
Objective #2: Investigate the challenges in formulating the AVMs for the thyristorcontrolled inverters. Propose appropriate models for representing the voltage source and current source controlled inverter systems.
Objective #3: Combine the results from the previous two objectives to enable the averagevalue modeling of the HVDC systems based on the parametric approach.
Objective #4: Incorporate the ac side harmonics into the parametric average-value modeling approach.
13

Objective #5: Developing the discretized formulation of the parametric AVM for direct interfacing with the network solution in EMTP-type transient simulation programs.
14

Figure ‎ 1-3: Various average-value modeling methodologies of high-power converters.
15

The parametric average-value models available in the literature have all been developed for machine/diode rectifier systems. In a wide range of applications , however, controlled line-commutated converters are utilized, which employ controllable switches, such as thyristors. It is therefore advantageous to extend the parametric average-value modeling methodology to thyristor-controlled line-commutated converter systems. In this chapter,
the machine/thyristor controlled LCC shown in Figure ‎ 2-1 is considered. Such systems are
also common in various industrial applications including synchronous machine exciters
[65]-[69]. The shown in Figure ‎ 2-1 is modeled based on the parametric approach wherein
the dc and ac side variables are related with some nonlinear algebraic functions dependent on the applied firing angle and loading condition. On the basis of [28], the AVM parameters are extracted using a fast and efficient procedure wherein the numerical functions are established by applying a slow transient in the load for different firing angles.
16

The advantages of the proposed parametric AVM over the analytically derived AVMs [17]-
[21] of the machine/thyristor controlled LCC system can be summarized as follows:
 Relative ease of implementation.
 Validity and accuracy for all modes of operation.
 Ease of extension to more complex systems, e.g., HVDC
Explicit formulation of the developed AVM, i.e., absence of algebraic loops in the model. As a result of this formulation, the solution of the model at each time step is obtained faster than the analytical AVMs [17]-[21] which require iterative solution at each time-step, due to the algebraic loops.
Figure ‎ 2-1: Synchronous machine/thyristor-controlled-rectifier system.
17

Without loss of generality, a synchronous machine/line-commutated converter benchmark
system [69] is considered here with some modifications as depicted in Figure ‎ 2-1. A
conventional three phase, six-pulse, thyristor-rectifier is used to supply the dc load and regulate the output voltage and current by changing the thyristor firing angle delay. The system includes a synchronous machine, a thyristor-rectifier, a firing pulse generator, a dc filter, and the ac system.
The thyristor firing pulse generator provides the firing pulses based on the commanded firing angle. The methods to produce the firing pulses may be based on the filtered terminal voltages [70] or the sensed rotor position [18], [20]. In the first method, the thyristor firing pulses are generated based on the first harmonic (fundamental) of the line-to-line converter terminal voltages, wherein some filtering is typically required. The typical
waveforms of the line and phase voltages and current are depicted in Figure ‎ 2-2. Here, it is
assumed that switch
S
2
(see Figure ‎
2-1) is fired at point A. As can be seen (see Figure ‎ 2-2,
bottom subplot), the actual phase voltage and phase current may be significantly distorted due to switching. The filtered line-to-line open circuit voltage (which is also the back EMF e bas
), and the terminal voltage (fundamental), v bas
, are superimposed in Figure ‎ 2-2 (top
plot). Using the first firing control method, the thyristor firing instant is delayed by angle  from the reference instance defined by the zero-crossing of v bas
(point B in Figure ‎ 2-2).
18

In the second method, the zero-crossing of the line-to-line open circuit back EMF voltage e bas
is determined based on rotor position  r
. The back EMF e bas
leads the terminal voltage v bas
by the rotor angle  . The reference for the firing angle is then point C, which is shifted by angle 
as shown in Figure ‎ 2-2. Therefore, for this method, the converter
firing angle,  , is defined as
(2-1)      .
Figure ‎ 2-2: Typical phase terminal voltage and current of controlled-rectifier with firing angle of 30 electrical degrees.
19

For values of  smaller than a certain value, the thyristor-rectifier is not controllable by thyristor firing angle delay. This firing angle is referred to as  min
, which denotes the minimum value of  , for which the rectifier output voltage is still controllable. Provided that the applied  is smaller than  min
, the anode to cathode voltages are negative when the thyristors are fired, and consequently the LCC works as a diode rectifier.
The dc filter, composed of an inductor and a capacitor as shown in Figure ‎ 2-1, is used to
reduce the magnitude of ripple in the rectifier output voltage and current. For the purpose of the studies in the following sections, the dc system (load) is represented by an equivalent resistive load that consumes the needed amount of real power (although it is understood that the dc subsystem may have a very complicated configuration in general).
The parametric average-value model presented in this chapter is an extension to the methodology set forth in [27]. Dynamic average-value modelling assumes that all ac and dc variables can be replaced by their fast averages over a prototypical switching interval, T sw
.
This fast average value is defined as f

1
T sw t t


T sw f
  dt
, (2-2)
20

where f ( t )
may represent voltage or current. This definition directly applies to the dc side variables. However, in order to apply (2-2) to the ac variables, they must be first expressed in a qd
synchronously rotating reference frame. Figure ‎ 2-3 depicts the ac side voltages and
currents expressed in the rotor and converter reference frames, respectively. Herein, the angular position of the converter reference frame q
-axis is chosen to be aligned with the phase a voltage [15].
Figure ‎ 2-3: Generator and converter ac side voltages and currents expressed in rotor and converter reference frames.
21

Assuming there is no energy storing element in the converter, for any operating point, the average of dc and ac side variables can be related with algebraic functions as v conv qds
 w v v dc
, and i dc
 w i i conv qds
.
(2-3)
(2-4)
Conditions (2-3) and (2-4) hold for any operating point, because there always exist some weighting coefficients w v
and w i
that will make those equalities true. The power factor angle  between vectors v conv qds
and i conv qds
is defined in Figure ‎ 2-3,
  tan

1



 i r ds i r qs



   tan

1



 i r ds i r qs



 tan

1



 v r ds v r qs



. (2-5)
The generator voltages in rotor reference frame can be calculated based on Figure ‎ 2-3 as


 v v r qs r ds



 w v v dc

 cos sin
 
 


. (2-6)
It is important to note that the existence of w v
and w i
for any operating point further translates into existence of w v
and w i
as functions that cover entire range of operating conditions of the considered converter circuit. However, obtaining closed-form analytical expressions of these functions may not be possible (for the machine-converter systems
[18]-[21]) or practical (for all operating modes). Therefore, the considered approach relies
22

on calculating the functions w v
, w i
and   
numerically using the detailed simulation
[27]. Moreover, since the angles  and  are also related, one may calculate either   
or
  
, which will be sufficient to completely define the ac-dc relationships (2-3)–(2-6). These formulations are referred to as  and  methods, respectively.
The rectifier operating condition may be specified in terms of terminal currents and voltages. For compactness, the terminal voltages and currents are combined into a dynamic impedance z
 v dc i rec qds
. (2-7)
In addition, the thyristor firing angle should be specified, which may be either the angle  or the delay angle 
(see Figure ‎ 2-2), depending on the thyristor firing control method.
Thereafter, it is possible to express the parametric functions as either w v
  
, or alternatively as w v
, w i
and   
.
, w i
and
For implementing the proposed average-value model, w v
, w i
and   
or   
need to be extracted by running the detailed simulation for a range of operating points, changing the firing angle and the loading condition. The corresponding model parameters for some operating points are calculated using (2-3)-(2-6) and stored in look-up tables. The number of operating points for which the look-up tables are extracted is determined based on the
23

desired accuracy (e.g. 15 to 30 points may be sufficient for most studies).
This method may become computationally intense, due to multiple runs of the detailed simulation. However, regarding this fact that the converter circuit is modeled as an algebraic block, the model parameters can be extracted using a single transient study, instead of multiple runs [28].
Herein, the look-up tables w v
, w i

 
and  
 
or   
are extracted by sweeping the load resistance from short circuit to open circuit corresponding to the firing angles of interest.
By sweeping the loading condition from short circuit to open circuit, all the rectifier modes of operation are covered. Besides, the rectifier firing angle delay is changed from minimum to maximum value to cover the whole range of interest. For the considered benchmark
system of Figure ‎ 2-1, these functions have been numerically calculated, and are shown in
Figure ‎ 2-4. As seen in Figure ‎ 2-4, these functions are nonlinear and curve significantly in
the regions of heavier loading conditions, and large values of the firing angle delay  . To specify a diode-rectifier mode, when the switches start conducting as soon as the voltage becomes positive, the firing delay angle is set to  
0

, which is shown in Figure ‎ 2-4
subplots with a highlighted (red) line. These parametric functions may also be equivalently established in terms of the control angle  . Without loss of generality, the numericallycalculated function   
is also depicted in Figure ‎ 2-4 (e), which may be used to
calculate the w v
, w i
and   
, instead of w v
, w i
and   
, respectively.
24

Figure ‎ 2-4: Numerically calculated parametric functions w v and   
.
, w i
,   
,   
25

Assuming that required parametric functions have been calculated and available as look-up
tables, the AVM can be implemented according to Figure ‎ 2-5(a) and (b), wherein the
rectifier is represented as an algebraic block with respective inputs and outputs. This formulation assumes a voltage-source ac side and a current-source dc side, respectively.
The implementation depicted in Figure ‎ 2-5(a) assumes that the firing pulses are generated
based on the rotor position  r
, and therefore, the input firing angle is  . Alternatively, when the firing pulses are generated based on the filtered ac voltages, one may consider
the implementation depicted in Figure ‎ 2-5(b) which uses the firing delay angle
 . When this control method is used, the dynamic effect of filtering the rectifier terminal voltages has to be included. For example, during a voltage transient, the filter will have a delay effect in producing the zero-crossing reference signal for establishing the correct firing angle  .
In the AVM implementation of Figure ‎ 2-5(b), the effect of voltage filter is applied to the
transformed voltages v r qds
in order to reproduce the equivalent effect on the firing angle  during transients. Therefore, based on the transformed voltages,  is calculated as
  tan

1



 v r ds v r qs



. (2-8)
To add the effect of a filter, the applied firing angle to the converter is modified as follows:
26

 '     f
  , (2-9) where  f
is extracted by passing  through the same filter used to extract the fundamental component of the converter terminal voltages. In steady state conditions,  ' becomes equal to  .
Figure ‎ 2-5: Implementation of parametric average-value models: (a) using rotor position as a reference for generating firing pulses; and (b) using converter terminal voltages for generating firing pulses.
27

The analytical approach assumes a certain operating mode, i.e., switching pattern of the phase currents. Subsequently, the average-value relationship between the dc link variables and the ac side variables transferred to qd
reference frame are established analytically
[15]. The analytical AVM introduced in [15] is developed using voltage sources and constant inductances in the ac side. In order to make it compatible to the machine/rectifier system studied in this chapter, a voltage-behind-reactance (VBR) machine model presented in [71], [72] with constant inductances and variable voltage sources based on the rotor dynamics is used in the analytical model.
The analytical expressions for VBR model are achieved by mathematically manipulating the voltage equation of the classical qd 0
model, and making the interfacing branches constant.
The circuit diagram of the VBR model connected to the rectifier system is shown in Figure
‎ 2-6(a). The implemented AVM block diagram of the same circuit is illustrated in Figure
‎ 2-6(b). Connection of the machine VBR model to the rectifier AVM, and also considering
the steady-state voltage drops of the ac circuit in the model, introduces algebraic loops and
makes overall formulation implicit. The algebraic loops shown in Figure ‎ 2-6(b) with
dashed lines are produced due to two reasons. First, the machine VBR model variables are represented in rotor reference frame, as opposed to the rectifier AVM variables, which are
in the converter reference frame. Therefore, as depicted in Figure ‎ 2-6(b), all the input
variables to the rectifier model are transferred to the converter reference frame and all the
28

Figure ‎ 2-6: Analytical average-value model for machine/thyristor rectifier system: (a) equivalent circuit diagram of interfacing voltage-behind-reactance machine model and thyristor rectifier system; and (b) block diagram depicting implementation with algebraic loop. The equation numbering in part (b) refers to the original sources [72] and [15].
29

outputs are transferred to the rotor reference frame on the basis of the rectifier power factor angle  . The rectifier power factor angle 
, as shown in Figure ‎ 2-6(b), is
algebraically calculated based on the input voltages. This relates the inputs and outputs of the rectifier AVM algebraically. The second reason for the existence of algebraic loop is that the commutation angle  has to be calculated algebraically based on the input rectifier voltages, and it is used algebraically to calculate the rectifier output currents. As a result, such implicit model will require iteration at each time-step, and will be less efficient numerically.
The detailed model of the system depicted in Figure ‎ 2-1 has been implemented in
Matlab/Simulink as described in [27]. The proposed parametric AVM has been
implemented in Simulink according to Figure ‎ 2-5 using
 method. The system parameters summarized in the Appendix A are chosen from the test configuration documented in [20].
In order to verify the accuracy of the model, the detailed model, the analytical AVM [15], and the parametric AVM are compared in time- and frequency-domains. The accuracy of the parametric AVM for a machine/diode rectifier has been shown in [27] for modes 1 and
2 operation. Also in [73], it has been shown that the proposed parametric AVM for a machine/diode rectifier is independent of the input frequency. Therefore, in the study presented here,  r
is considered constant and equal to
120
 rad/s. Also, for all studies in
30

this chapter, the system mode 1 operation is assumed, wherein the commutation angle is less than 60 degrees.
The new average-value model is first validated in steady state and then during transient, with and without controller on the applied thyristor firing angle delay.
The accuracy of the proposed average-value model in steady-state is investigated by comparing the performance of the detailed simulation, analytical AVM, and the proposed
parametric AVM of the machine/rectifier system presented in Figure ‎ 2-1 for different firing
angles. In this study, the excitation voltage is equal to 19.5 V and the dc current is also kept constant at 2.46 A by using a PI controller. As it is described in the previous sections, the firing pulses could be generated either based on the rotor position  , or the machine terminal voltages  . The steady-state output voltage of the detailed model and the AVMs versus the firing angles,  and 
, are plotted in Figure ‎ 2-7(a) and Figure ‎ 2-7(b)
respectively. Both the parametric and analytical AVMs show an excellent agreement with the detailed simulation. Here, the AVM results are within less than 2% error with respect to the reference detailed simulation, which is considered sufficiently accurate for all practical
purposes of such models. As it can be seen in Figure ‎ 2-7(a), the dc terminal voltage is
decreasing monotonically by increasing  for the entire range of operation. However, the output voltage is not controllable by  for the values less than  min
, which is close to 30
31

degrees here. It should be noted that the input firing angle  to the analytical AVM and accordingly, 
, is calculated using (2-1). Therefore, as it can be seen from Figure ‎ 2-7(b),
the waveform related to the analytical AVM is plotted for the firing angles between  min and  max
.
Figure ‎ 2-7: Steady state output dc voltage: (a) using converter terminal voltages for generating firing pulses; and (b) using rotor position as a reference for generating firing pulses.
32

In the following time-domain transient study, the system initially operates in steady state with a constant excitation voltage equal to 19.5V and a resistive load of 21  . Since  is dependent on the loading condition (see (2-1)), herein,  is used as the thyristor firing delay angel to study the system response to a step change in the load and firing angle. The initial value for the firing angle is set as  
30

. At t

0 .
02 s
, the load resistance is changed to 3.64  by adding a 4.4  resistor in parallel to the load. The corresponding transient responses in the load voltage v c
, dc current i dc
, and the generator field current i fd
, are shown in Figure ‎ 2-8 for the parametric and analytical AVMs, respectively. As it can be seen in Figure ‎ 2-8, the new AVM with functions
w v
, w i
and   
, and the analytical AVM predict the entire transient very close to the reference detailed model.
Next, at t

2 .
5 s
, the firing angle is changed to 10 degrees. The simulation results for this
study are shown in Figure ‎ 2-9, wherein the AVMs follow the transient response of the
detailed model very accurately. However, by reducing the applied firing angle to 10 degrees, the rectifier operating point is getting close to the mode 2 operation. Therefore, the analytical AVM fails to follow the transient response of the detailed simulation with acceptable accuracy since it is a reduced order model and valid only for the mode 1 operation.
33

Figure ‎ 2-8: System response to a change in load in mode 1 as predicted by the detailed model, analytical and parametric AVMs.
34

Figure ‎ 2-9: System response to a change in firing angle in mode 2 as predicted by the detailed model, analytical and parametric AVMs.
35

Controlled rectifier may be used as a voltage or current source depending on the control objectives. Based on the type of load connected to the dc side, the output voltage or current may need to be controlled by changing the applied firing delay angle to the rectifier switches. Regardless of the load type (with nonlinearity or dynamics) or controller added to the system, the developed average-value model should be accurate in predicting the transient response of the original switching system. Here, the machine/rectifier system
depicted in Figure ‎
2-1 is implemented using the PI controller shown in Figure ‎ 2-10. This
controller simply regulates the load voltage using the firing angle. The controller parameters are summarized in Appendix A. Herein, the thyristor firing pulses are generated based on the rotor position, in order to demonstrate the accuracy of this approach as well. The system is assumed to operate initially in steady state defined by 21  resistive load and firing angle  
64
 . The capacitor voltage is assumed to be kept constant and equal to 40 V. At t

3 s
, the resistive load is changed from 21 to 12  . Since the command voltage is set to 40 V, the firing angle changes in order to keep the capacitor voltage constant. The fast transient response for v c
, i dc
, and i fd
are plotted in Figure ‎ 2-11,
wherein the AVM predicts the behavior of the detailed simulation with great accuracy.
36

Figure ‎ 2-10: Controller used to control load voltage.
Figure ‎ 2-11: Closed-loop system response to a step change in load; detailed and parametric
AVM models.
37

Here, for consistency with the reference studies performed in [27], the system load is assumed resistive and equal to 10.74
 . The firing angle is also considered as 30 degrees.
The AVM has been implemented for both cases of with and without the effect of the filter used to extract the fundamental components of the converter terminal voltages for generating the thyristor firing pulses. The magnitude and phase of the rectifier output
impedance as predicted by various models are shown in Figure ‎ 2-12 for the frequency
range of 1 to 200 Hz. For the frequencies closer to the switching frequency, 360Hz, the results become distorted due to the interaction between the injected small-signal currents
and the rectifier switching, and it is outside the validity range of the AVMs. Figure ‎ 2-12
shows that the output impedance changes very noticeably by changing the thyristor delay firing angle (here, from  
0

to  
30

). This figure also reflects the importance of considering the filter used to extract the fundamental components of the converter voltages in the AVM. As seen in this figure, neglecting the filter introduces a noticeable error in the predicted magnitude and phase of the output impedance, especially in the lowfrequency range, 7 to 30 Hz.
38

Figure ‎ 2-12: Output impedance as predicted by various models.
Figure ‎ 2-13: Transfer function for the input firing angle to the output dc voltage as predicted by various models.
39

Next, a transfer function from the input firing angle to the averaged output dc voltage may
be particularly useful. Figure ‎ 2-13 depicts this transfer function, extracted at the same operating point considered in the previous study. As seen in Figure ‎ 2-13, the AVM captures
the frequency-domain characteristics of the original switching system with superior accuracy.
The magnitudes and phases of the rectifier output impedance of the closed loop system
studied in section 2.6.2 are plotted in Figure ‎ 2-14, for both detailed and average models.
The detailed and AVM results match almost perfectly as observed in Figure ‎ 2-14.
Comparison between the output impedances of the open loop study, plotted in Figure ‎ 2-12,
and closed loop study shows the superior performance of the PI controller for low
frequencies. As it can be seen in Figure ‎ 2-14, the magnitude of the output impedance has
reduced, which shows that low frequency components of the rectifier output voltage have been regulated. However, the PI controller does not have any effect in the high frequency range close to the switching frequency. This is due to the low-pass filter in the controller loop. In order to reduce the effect of this filter, a more advanced controller should be applied which is fast enough for high frequencies and stable for low load conditions.
However, designing such a controller is outside of the scope of this work.
40

Figure ‎ 2-14: Output impedance for controlled system as predicted by various models.
In this chapter, the parametric AVM approach has been extended to model thyristor controlled LCC in the rectifier mode of operation. This has been achieved by constructing parametric functions that in addition to the converter switching cell dynamic impedance also include the thyristor firing angle. For implementation of the proposed extended parametric AVM, the required parametric functions are readily calculated using the detailed switching model and stored as 3-dimensional look-up tables. The number of points for which the parametric functions are extracted is determined based on the desired
41

accuracy, wherein even 15 to 30 points may be sufficient for most studies. The performance of the new AVM has been verified against the detailed simulation and the classical analytical AVM of a synchronous machine/thyristor-controlled-rectifier system. It has been shown that the new AVM is very accurate in predicting the transient responses in a wide range of operating conditions and thyristor firing angle [from very light load (close to open-circuit) to very heavy load (close to short-circuit)], which has not been achieved by the previously established models.
42

In Chapter 2, the new parametric AVM of thyristor controlled LCC working in the rectifier mode was developed. In many industrial applications such as HVDC system, the line commutated converter is operating in inverter mode. Therefore, in the next step, it is necessary to develop the AVM that is capable to operate as an inverter. In this Chapter, the parametric AVM of the LCC working as an inverter is developed based on two formulations:
(i) voltage source, and (ii) current source. Here, the application of these formulations is further classified on the basis of their well-posedness in different modes of operation.
The converter of Figure ‎ 1-1(a) can also operate as an inverter when the firing angle
 
90
 . The corresponding AVM may be formulated based on (2-3)–(2-7) using either
voltage-source or current-source approach as depicted in Figure ‎ 3-1. Moreover, the
functions w v
, w i
,   
and/or   
have to be re-calculated for inverter
operation. In Figure ‎ 3-1 (a), the algebraic block representing converter AVM outputs the ac
voltages and is therefore referred to as voltage-source AVM formulation (VS-AVM). In
43

Figure ‎ 3-1 (b), the algebraic block representing converter AVM outputs the ac currents and
is therefore referred to as current-source AVM formulation (CS-AVM).
Figure ‎ 3-1: Block diagram depicting AVM implementation of the thyristor-controlled inverter: (a) voltage-source formulation; and (b) current-source formulation.
Depending on the control mode, one of these AVM formulations may be chosen. Using the voltage source formulation in one of the state-variable-based simulation programs (such as
Matlab/Simulink), the ac system can be formulated as a proper state space model.
44

However, since the rectifier subsystem outputs the dc current, in order to add the dynamic effect of the dc inductance
L
to the model, the numerical differentiation is required. At the f same time, using the current source formulation, the dc-link filter is modeled using a proper state model without any difficulty, unless
L
is equal to zero. However, numerical f differentiation is still inevitable in the ac subsystem model since the converter cell outputs the converter currents.
The parametric AVM of the LCC (rectifier and inverter) system shown in Figure ‎ 3-1 (a) is
extracted based on (2-3)–(2-7), which should result in a proper state model that is built by the simulation program and can be symbolically expressed as a system of nonlinear differential equations d x
 f dt
, (3-1) where x
is the vector of state variables, and u
is the vector of inputs. The relationships (2-
3)–(2-7) that are defined by parametric functions may be directly integrated into (3-1), or equivalently, expressed as companion algebraic equations. The Jacobian of (3-1) is
A

 f
 x
(3-2)
45

In order for the AVM to be well-posed, the following conditions must be satisfied for any steady state operating point defined by x ss
, u ss
and achievable by the respective original switching model: d x
 f
 x ss
, u ss


0
, and dt
Re
 

0
, for all  i
 eig

A x ss
(3-3)
(3-4)
Conditions (3-3) and (3-4) ensure that the parametric AVM will achieve the same steady state as the original switching model. Moreover, the AVM should also track the transient response in order to properly capture the system’s dynamics close to the converter switching frequency. However, as it will be shown, despite (2-3)–(2-7) being always satisfied by the numerically-constructed parametric functions, this only ensures condition
(3-3), but (3-4) may not always hold.
To demonstrate the conditions (3-3) and (3-4), we first consider the converter circuit of
Figure ‎ 1-1, where the dc system is replaced with a voltage source
E dc
. The state equations describing the ac side in the synchronous reference frame aligned with the phase a
source voltage (hence “rotor” superscript is used herein) can be expresses in the following form di r qs dt

 r
C
L
C i r qs
 i r ds
 e

1
L
C e r qs

1
L
C v r qs
, (3-5)
46

di r ds dt

 r
C
L
C i r ds
 i r qs
 e

1
L
C e r ds

1
L
C v r ds
(3-6) where  e
is the speed of synchronous reference frame, and e r qs
and e r ds
are the equivalent ac sources also expressed in the same reference frame. Note that e r ds

0
. Using (2-7) we have di r qs dt

 r
C
L
C i r qs
 i r ds
 e

1
L
C e r qs

1
L
C w v cos
  v dc di r ds dt

 r
C
L
C i r ds
 i r qs
 e

1
L
C e r ds

1
L
C w v sin
  v dc
.
where rotor angle can be calculated based on Figure ‎ 2-3 as
  tan

1


 i r ds i r qs


  .
The converter dc voltage and currents can be written as v dc

E dc

R f i dc i dc
 w i i
    qs ds
2
.
Substituting (3-9) - (3-11) into (3-7) and (3-8), we obtain a system that uses angle  ,
(3-7)
(3-8)
(3-9)
(3-10)
(3-11)
47

di r qs dt

 r
C
L
C i r qs
 i r ds
 e

1
L
C e r qs

E dc
L
C w v z ,

R f
L
C w v
,
 w i z , cos
 tan

1



 i r ds i r qs



 
,
 cos
 tan

1



 i r ds i r qs



  i
   
2
,

E dc
, di r ds dt

 r
C
L
C i r ds
 i r qs
 e

1
L
C e r ds

E dc
L
C w v z ,

R f
L
C w v
,
 w i z , sin
 tan

1


 i r ds i r qs


  sin
 tan

1


 i r ds i r qs


  i
   
2
E dc
.
(3-12)
(3-13)
An equivalent system of equations can also be obtained if one uses  as a lookup table instead of  . Assuming that   
is available, we get di r qs dt

R f
L
C

 r
C
L
C i r qs
 i r ds
 e

1
L
C e r qs w v
    cos
    

E dc i r qs w v
  cos
    
E
L
C
    ds
2 dc di r ds dt

R f
L
C

 r
C
L
C i r ds
 i r qs
 e

1
L
C e r ds w v
    sin
    

E dc i r qs w v
  sin
    
E dc
L
C
   
2
.
(3-14)
(3-15)
48

The AVMs defined by (3-12)–(3-13) (  method), or (3-14)–(3-15) (  method), have the general form of (3-1) where the state and input vectors are x

 i r qs i r ds

T
and u

 e r qs e r qs
E dc


T
.
To investigate the conditions (3-3)–(3-4), we first assume a thyristor-controlled rectifier that operates with firing angle  
30
 and dc voltage,
E dc

2 .
46 kV. The system parameters are summarized in Appendix B. The VS-AVM is implemented using (3-12)–(3-
13),  method. The Jacobian matrix and its eigenvalues have been numerically calculated for a wide range of operating conditions of this rectifier circuit, and the results are shown
in Figure ‎ 3-2. In particular, Figure ‎ 3-2(a) shows the state-space region in which all
eigenvalues have negative real part – stable region (depicted in white inside the investigated region). The steady state operating point x ss
is also shown as the stable
equilibrium point. Moreover, the corresponding phase portrait depicted in Figure ‎ 3-2(b)
also shows that for the given inputs, all state-space trajectories converge to this steady state equilibrium point x ss
, which demonstrates that the considered VS-AVM of rectifier circuit is well-posed.
49

Figure ‎ 3-2: Stability of the VS-AVM for thyristor-controlled rectifier using  method: (a) stability region in state space; and (b) phase portrait.
50

The same investigation is performed for the thyristor-controlled converter operating as an inverter. The VS-AVM is implemented according to (3-12)–(3-13) using  method. The new operating point is defined by the firing angle  
140
 and
E dc
 
825
V. The Jacobian matrix and its eigenvalues have been numerically calculated for a wide range of operating
conditions, and the results are shown in Figure ‎ 3-3. As it can be seen in Figure ‎ 3-3(a), the
considered range of operating conditions is not entirely stable, and the assumed steady state equilibrium point x ss
is now outside of the stable region. The phase portrait depicted
in Figure ‎ 3-3(b) further reviles that the state trajectories around this point would move the
state away. Next, the thyristor-controlled inverter system is implemented according to (3-
14)–(3-15) using 
method. Figure ‎ 3-4 shows the corresponding region of stability and the phase portrait. As it can be seen in Figure ‎ 3-4(a) the considered steady state operating
point is now inside stable region, and the region itself has different shape. Moreover, the phase portrait also shows that this is a stable equilibrium, and the entire state trajectory with different initial conditions will easily converge.
51

Figure ‎ 3-3: Stability of the VS-AVM for thyristor-controlled inverter using  method: (a) stability region in state space; and (b) phase portrait.
52

Figure ‎ 3-4: Stability of the VS-AVM for thyristor-controlled inverter using  method
L
C

0 .
1 mH
: (a) stability region in state space; and (b) phase portrait.
53

Figure ‎ 3-5: Stability of the VS-AVM for thyristor-controlled inverter using  method,
L
C

0 .
3 mH
: (a) stability region in state space; and (b) phase portrait.
54

As it can be seen by looking at Figure ‎
3-3 and Figure ‎ 3-4, using
 method, the stable region is larger than the stable region resulted from  method. Further investigation has shown that the region of stability is dependent on the systems parameters. For example, as it can
be seen by looking at Figure ‎ 3-5, by increasing
L
C
in the ac side and using  method the equilibrium point is not stable anymore and the state trajectories do not converge to that.
Based on these studies for the inverter, it can be concluded that the parametric VS-AVM formulation is not well-posed for some operating conditions and system parameters that are otherwise well predicted by the original switching model. The numerically-constructed parametric functions always satisfy (2-3)–(2-7) and condition (3-3), but (3-4) may not always hold.
3.2.1 Current Source AVM (CS-AVM)
To extend the parametric AVMs, the current-source of Figure ‎ 3-1 (b) is considered here. In
this formulation, instead of (2-7), the converter switching cell operating condition is determined in terms of equivalent dynamic admittance y

1 z
 i dc v qds
, (3-16)
55

Moreover, according to Figure ‎ 3-1 (b), the ac side can now be interfaced with the current-
source inverter AVM block based on indirect approach, using snubber circuits (typically large resistors or small capacitors).
Using the snubber approach, the state equations (3-5)–(3-6) for the ac side may be kept, wherein the input voltages will be calculated as the voltages across the snubbers. An equivalent interfacing may also be achieved if the ac side inductor branches are represented using transfer function with added very fast pole. As the dc inductance cannot
be neglected in this case, the CS-AVM of the switching system shown in Figure ‎ 1-1(a) with
the dc filter inductance
L f

0 .
5968
H is considered, and the CS-AVM will have three state variables. The final state model will have the same general form (3-1), but with the input vector u

 e r qs e r ds i dc _ sys


T
. The system parameters are summarized in Appendix B, and the firing angle  
140
 is assumed.
In order to investigate the validity of (3-3) and (3-4), the same system as before is considered with the dc filter inductor (
L f

0 .
5968
H) added to the dc side, and the firing angle  
140
 is assumed. For the considered model, the stability region has been first investigated for the considered operating point, as well as for a large range of state space
similar to the studies of Figure ‎ 3-5, all predicting stable operation. Next, the stability region
is investigated with respect to the system parameters, specifically the ac line inductance
L
C
, and the operating condition defined by dynamic admittance y
. Figure ‎ 3-6 shows the
56

region reachable by the corresponding detailed model, which is also entirely stable for the proposed CS-AVM. Based on these investigations, the CS-AVM formulation for the inverter model is compliant with condition (3-4), and therefore should result in convergent parametric AVM.
Figure ‎ 3-6: Stability region of CS-AVM for inverter.
3.2.2 Case Studies
To validate the proposed CS-AVM formulation, the inverter subsystem of the HVDC system
shown in Figure ‎ 3-7 is considered. The system parameters are summarized in Appendix C.
57

Since in HVDC systems it is assumed that the dc current is controlled by the rectifier side, the dc subsystem herein is represented by a dc current source,
I dc _ rec
. The detailed simulation is executed using the Automated State Model Generator (ASMG) toolbox [12], and the parametric AVM is simulated in Matlab/Simulink using standard library components. The accuracy of the developed AVM is verified by comparing it with the detailed model in time- and frequency- domain studies.
Figure ‎ 3-7: Inverter side of the HVDC system with current control on rectifier side.
Here, the system is assumed to initially operate in steady state. The peak value of the ac source voltage is 175.6 kV, the dc current
I dc _ rec
is equal to 3 kA, and the thyristor firing
58

angle  is set to 135°. At t

1 s, the dc current is abruptly changed to 11 kA. The simulated responses of current i dc
and voltage v dc
for this study are shown in Figure ‎ 3-8. As it can be seen in Figure ‎ 3-8, the AVM follows the transient response of the detailed simulation with
an excellent accuracy.
Next, at t

2 s, the firing angle is changed to  
125
 . The transient response predicted
by the new AVM and detailed simulation are superimposed in Figure ‎ 3-9. As it can be seen in Figure ‎ 3-9, the results of the AVM are again in excellent agreement with the reference.
A comparison of the CPU time and number of simulation steps for the AVM and detailed
simulation is denoted in Table 1. By looking at Table ‎ 3-1, it can be concluded that the AVM
is approximately 30 times faster than the detailed simulation.
59

Figure ‎ 3-8: System response to a fast change in input current.
60

Figure ‎ 3-9: System response to a step change in firing angle.
61

Table ‎ 3-1: Simulation speed for AVM and detailed simulation to t

2 .
25 s
Model
Detailed (Variable step solver)
AVM (Variable step solver)
Step size,
2 ms
 t
(max)
2 ms (max)
CPU time, s
5.061
0.154
Time steps
18935
1168
In this chapter, we have shown that although the parametric AVM methodology as it has been formulated prior to this thesis and applied to the rectifier circuits, the same formulation may not be result in a well-posed model when considering the inverter operation. This discovery has led to the new formulations proposed in this chapter for modeling the thyristor-controlled inverters using either voltage-source or current-source control modes. To demonstrate the new AVM formulations, the proposed methodology has been applied to the HVDC inverter side operating in current-source mode. For the considered example system, it has been shown that the proposed parametric AVM is very accurate in predicting the transient responses, and at the same time is computationally very efficient (32 times faster than the detailed simulation). These contributions set the stage for modeling the complete HVDC systems at the level of accuracy and computational efficiency that were not achieved by the previously established models.
62

HVDC systems are frequently employed for long distance transmission due to their lower electrical losses. Since this research work is focused on the high power LCCs, in this thesis we consider only the HVDC Classic type thyristor-based converter technology. An HVDC system is composed of two back to back connected converters and a control system.
Depending on the controlling mode, one of LCCs’ is operating as a rectifier and the other one as an inverter. The LCCs used in HVDC systems often utilize the 12-pulse converter topology (two 6-pulse LCCs connected in series). Therefore, for modeling the HVDC systems, the parametric approach should be extended to model the 12-pulse thyristor controlled LCCs.
Herein, a benchmark system is considered to demonstrate the dynamic average modeling methodology developed in this thesis and its advantages over a conventional detailed switching model. The average-value models for the rectifier and inverter sides are set forth
63

and developed based on the parametric approach [27], [28] and the AVMs presented in
Chapters 2 and 3.
The CIGRE HVDC benchmark system, first proposed in [74], is a mono-polar 500 kV, 1000
MW dc link which employs 12-pulse converters on rectifier and inverter sides. The detailed modeling of this system in PSCAD/EMTDC and Matlab/Simulink has been presented in
[75]. The circuit diagram of this system is illustrated in Figure ‎ 4-1, and the corresponding
parameters are summarized in Appendix D. The system is composed of two ac sides, each represented by the equivalent supplying/receiving network. The rectifier and inverter ac networks are represented by R-L circuits to represent weak grids on each side. In order to absorb the harmonics generated by the converter as well as providing the converter with reactive power, multi-harmonic ac filters are placed on both ac sides. The dc subsystem is represented by the T-equivalent circuit of the transmission line combined with the smoothing reactors that are connected at both sides in order to reduce the ripple in dc current. The 12-pulse converters are constructed by a series connection of two 6-pulse converters. At each ac side, (rectifier and inverter), the 3-phase transformers with Y- and
D-connected secondary windings are employed to produce the desired 30-degrees shifted voltages (equivalent to 6-phase) that feed the two 6-pulse converters, respectively, thus
resulting in a 12-pulse operation. As depicted in Figure ‎ 4-1, each converter is controlled
through the respective control module.
64

Figure ‎ 4-1: CIGRE HVDC benchmark system circuit diagram.
65

Figure ‎ 4-2: CIGRE HVDC benchmark control subsystem: (a) rectifier control; and (b) inverter control.
66

In the considered CIGRE HVDC benchmark system, the ac network is weak (short circuit ratio of 2.5 at rated frequency of 50 Hz) [75], which also presents some challenges for the
controls. Here, in Figure ‎ 4-1 the power is assumed to flow from left to right, which
determines the rectifier and inverter sides, respectively. But otherwise, the controllers can reverse the power flow by appropriately changing the firing angles on each side and changing the ac voltage magnitudes of the two sources. The details of the rectifier and
inverter controllers are depicted in Figure ‎ 4-2, and the corresponding parameters are
summarized in Appendix D. For the rectifier side, a proportional-plus-integral (PI) current control (PI-1) is realized with the reference denoted by i dc _ ref _ lim
. This reference current is provided by the inverter controller. At the inverter side, the controller is composed of a voltage dependent current controller (PI-2) and the extinction angle  -control, respectively. The reference value for the rectifier controller, i dc _ ref _ lim
, is determined by the inverter controller in order to ensure limited current during under-voltage conditions.
Under normal operation, the externally provided reference value i dc _ ref
is sent to the output. This value determines the power flow through the HVDC system. Under fault conditions, the reference value i dc _ ref _ lim
is limited based on the dc link voltage v dci
and current i dci
at the inverter side, as shown in Figure ‎ 4-2.
During normal operation, the inverter is controlled using the extinction angle  which determines the difference between the turning off angle of the valves and the angle of 180 degrees. The lower limit for this angle is determined based on the protection constraints to
67

allow enough time for the valves to turn off before the end of a half-cycle. The  -control
(PI-3) adjusts the value of extinction angle. However, the inverter control may be switched from the  -control strategy to the current-control mode that limits the current to some specified reference value. The firing angle for the inverter is then established based on the output of these control modules and the respective gate signals are constructed by the
Pulse Generators depicted in Figure ‎ 4-1.
Herein, the AVM is developed in both Matlab/Simulink and PSCAD/EMTDC. In order to model a complex system such as CIGRE HVDC benchmark, it is necessary to divide it up into smaller parts. Hence, the main circuit of HVDC system is divided into seven subsystems: two line-commutated converters (rectifier and inverter sides) which are modeled using the parametric average-value modeling approach; two ac subsystems; two controller subsystems; and the dc transmission line subsystem. For convenience, each subsystem is implemented as a separate module. These modules are then interconnected to form the overall HVDC system.
The parametric AVM approach is generally independent of the simulator platform. For the purpose of dynamic average-value modeling, all the ac and dc variables are replaced by their fast averages over a prototypical switching interval
T sw
as (2-2). Due to the 12-pulse
68

operation,
T sw

1
12
T s
, where
T s
represents the period of the ac waveform as depicted in
Figure ‎ 4-3. The switching interval
T sw
and primary voltages and currents of the
transformer in the rectifier side are illustrated in Figure ‎ 4-3. As seen in Figure ‎ 4-3, the
transformed ac voltages v qdrp
and currents i qdrp appear dc in the qd
synchronous reference frame and have ripple with period of
T sw
.
Figure ‎ 4-3: Typical voltages and currents of a 12-pulse rectifier transformer primary winding.
69

In this Section, the parametric average-value modeling methodology for 6-pulse diode rectifier is extended to the 12-pulse rectifier side of CIGRE HVDC benchmark system.
Herein, the 12-pulse rectifier is composed of two 6-pulse line-commutated converters. In the AVM developed here, each converter may be modeled separately to represent the corresponding dc and ac side variables. The ac side converter voltages and currents for both 6-pulse rectifiers can be expressed in converter reference frame [15].
The parametric AVM assumes that the rectifier may be viewed as a multi-port switching cell, wherein the transformed ac side and dc side variables can be related through some algebraic functions. Herein, since there are two 6-pulse bridges, the two sets of parametric functions relating the ac and dc voltages are defined as:




 v conv qdrs ,
1
1 v conv qdrs ,
2
2








 w v , 1 w v , 2 v v dcr , 1 dcr , 2



. (4-1)
Considering the series connection of the rectifier bridges, the ac and dc side currents are also related as w i , 1 i conv 1 qdrs , 1
 w i , 2 i conv qdrs
2
, 2
 i dcr
. (4-2)
70

where w v
, w i
and  are algebraic functions of loading conditions and thyristor firing angle, and “conv1” and “conv2” refers to the converter reference frame for each converter, respectively. The power factor angles between vectors v conv qdrs
and i conv qdrs
for each converter are defined as




1
2


 tan

1


 i i se drs , 1 se drs , 2 i i se qrs , 1 se qrs , 2



 tan

1


 v v se drs , 1 se drs , 2 v se qrs , 1 v se qrs , 2



. (4-3)
Here, the superscript “ se ” refers to the sending-end equivalent voltage.
Conditions (4-1) and (4-2) hold for any operating point, since functions w v
, w i

 
, and
  
exist for the entire range of operation. The considered approach relies on calculating functions w v
, w i
, and   
numerically by running the detailed simulation similar to conventional 6-pulse converters [27]. The loading condition of each rectifier may be specified in terms of terminal currents and voltages. For compactness, the terminal voltages and currents are combined into dynamic impedances defined as


 z z
2
1








 v v dcr dcr
, 1
, 2 i se qdrs , 1 i se qdrs , 2




. (4-4)
Assuming that AVM parameters for each rectifier are extracted and saved as lookup tables, the AVM of the rectifier side of CIGRE HVDC benchmark system can be implemented
71

according to the diagram depicted in Figure ‎ 4-4. For example, if the two AVMs are constructed (one for each bridge), then the system can be implemented according to Figure
‎ 4-4(a), where each bridge is modeled as an algebraic module that relates the
corresponding dc and ac side variables transferred to a qd
rotating reference frame.
Alternatively, taking into account the symmetry between the bridges and their series
connection, the two sub-models can be collapsed into one as depicted in Figure ‎ 4-4(b). In
particular, assuming that the dc voltages for both rectifiers have the same average-value, we have v dcr , 1
 v dcr , 2
 v dcr
2
Therefore, (4-1) and (4-2) can be simplified as
(4-5) v conv 1 qdrs , eq
 w veq v dcr i dcr
 w ieq i conv 1 qdrs , eq
(4-6)
(4-7) where w veq

1
2 w v , 1

1
2 w v , 2
and w ieq
 w i , 1
 w i , 2
. Following the same approach, the equivalent power factor angle  eq
may be defined as
 eq
 tan

1 i se drs , eq i se qrs , eq
 tan

1 v se drs , eq v se qrs , eq
(4-8)
72

Using (4-6)-(4-8), the collapsed AVM of the rectifier side can be implemented as shown in
Figure ‎ 4-4(b). The equivalent dynamic impedance is defined similarly, using appropriate
equivalent dc voltage and ac currents in (4-4). The firing angle 
R
is also provided by the
respective controller (see Figure ‎ 4-2). It is worth mentioning that the collapsed AVM of
Figure ‎ 4-4(b) provides more simplicity in comparison to the general formulation of (4-1)-
(4-3) when the rectifiers are symmetrical with respect to each other.
The Pulse Generator module in Figure ‎ 4-1 produces the necessary thyristor firing pulses
which are generally determined using the filtered converter ac voltages or their fundamental components. For example, the generation of firing pulses may be implemented based on the zero-crossing of the respective ac voltages (fundamental components). Any such filters will generally have some response delay which should be taken into account when constructing the AVM where all voltages and currents are continuous and appear as dc in appropriate synchronous reference frame.
73

Figure ‎ 4-4: Block diagram depicting implementation of the AVM rectifier side: (a) original
AVM with two rectifiers separately; and (b) collapsed AVM.
The inverter side of CIGRE HVDC benchmark can be modeled using an approach similar to
the rectifier side. As depicted in Figure ‎ 4-1, the 12-pulse inverter is composed of two 6-
pulse bridges connected in series. It is also important to keep in mind that the rectifier ac side (sending-end) and the inverter ac side (receiving-end) may have difference phase and even frequency. Therefore, to distinguish between the reference frame coordinates associated with the sending and receiving ac sides (represented by Thevenin equivalent
74

sources as depicted in Figure ‎ 4-1, the variables in the respective reference frames will be
denoted by the superscript “ se ” and “ re ”, respectively.
If the inverters’ AVMs are represented separately, one may consider two converter reference frames with the angular difference of 30 degrees. Assuming that there are no energy-storing-elements in the inverter circuit, the algebraic relationships relating the dc and ac variables similar to (4-1)-(4-3) can be readily established. The corresponding
implementation will be similar to Figure ‎ 4-4(a), except reverse direction of the power and
variables will be required. For the inverter side, the  -control is also required to calculate the extinction angle. In the detailed simulation, the extinction angle  is calculated based on voltages and currents on the valves of the bridges. However, in the AVM, the switching variables are replaced by their respective fast averages. Therefore,  is extracted running a detailed simulation similar to w v
, w i
, and  , and saved for the future use.
Similar to the rectifier side, due to the symmetry in the system, the two AVMs of the inverters can be collapsed and represented using simplified equivalent relationships similar to (4-5)-(4-8). Implementation of the equivalent (collapsed) inverter AVM is shown
in Figure ‎ 4-5. The firing angle

I
is also provided by the respective controller (see Figure
‎
4-2). For the inverter AVM shown in Figure ‎ 4-5, as opposed to the rectifier AVM shown in
Figure ‎ 4-4, the inverter ac currents
i re qdis , eq
and dc voltage v dci
are assumed to be the outputs of the inverter average switching cell, whereas the ac voltages v re qdis , eq
and dc
75

current i dci
are the inputs, respectively. Here, the inverter operating condition is determined in terms of an equivalent dynamic admittance defined as: y eq

1 z eq
 i dci v re qdis , eq
(4-9)
Interfacing the AVM depicted in Figure ‎ 4-5, as it is explain in Chapter 3, may require special
consideration to achieve the input-output compatibility with other modules of the overall
system. The desired interfacing for the AVM of Figure ‎ 4-5 can be achieved using snubber
circuits (typically large resistors or small capacitors), or equivalently may also be achieved if the ac side inductor branches are represented using proper transfer function with a very fast pole [27].
Figure ‎ 4-5: Block diagram depicting implementation of the AVM inverter side.
76

The ac subsystems at the rectifier and inverter sides are composed of an equivalent ac networks, the ac filters, and the phase-shifting transformers. For the purpose of dynamic average modeling, the ac subsystem can be modeled using approaches as depicted in
Figure ‎ 4-6. In the first approach shown in Figure ‎ 4-6(a), the ac side is modeled as a
network circuit in physical abc
variables. In this case, interfacing the ac subsystem model with the AVM of the converter may require interfacing circuitry [14]. This approach is suitable for either PSCAD/EMTDC or SimPowerSystems in Matlab/Simulink. However, the ac subsystem can also be implemented in transformed synchronous qd coordinates/variables as a qd 0
-circuit shown in Figure ‎ 4-7. If this is possible, then the
interface of the converter AVM with the ac network becomes simpler as depicted in Figure
‎ 4-6(b), since no
abc
to qd
transformations of the interfacing variables would be required.
Moreover, due to the symmetry of the transformer and the collapsed AVM representation of the converter, the transformer is represented here only in terms of its equivalent series
impedance as shown in Figure ‎ 4-7. The ac subsystem of the rectifier side is very similar and
can be modeled using any of the approaches depicted in Figure ‎
4-6 and Figure ‎ 4-7, which is
not shown due to similarity.
77

Figure ‎ 4-6: Block diagram depicting implementation and interfacing of the inverter side ac subsystem: (a) using abc original circuit; (b) using transformed qd 0
-circuit.
78

Figure ‎ 4-7: Inverter side ac subsystem implementation using transformed qd 0
-circuits.
79

For the purpose of modeling in this study, the dc line is represented as an equivalent T circuit with two smoothing inductors (one on each side), and one equivalent capacitor, as
depicted in Figure ‎ 4-1. If the dc subsystem is implemented as a circuit, it can be readily
realized in most transient simulators such as PSCAD/EMTDC and SimPowerSystems in
Matlab/Simulink, wherein appropriate interfacing with the rectifier and inverter AVMs will be required [65]. Additionally, the dc network could be modeled in state space form using proper transfer functions to represent the inductors and capacitor as explained in [27]. For example, for interfacing with the rectifier AVM which requires the input dc voltage v dcr
as an input, the smoothing inductor in the rectifier dc side can be represented using a proper transfer function as in [27]. Alternatively, a snubber circuit (typically large resistor or small capacitor) can be added to the dc terminal of the rectifier [65].
Since the dynamic average modeling is only changing the implementation of the switching part of the system, the developed AVM of the entire CIGRE HVDC benchmark should be valid for a wide range of operating conditions as well as different controllers. Thus, the
control system depicted in Figure ‎ 4-2 is considered here for the AVM as well for all
subsequent studies for consistency with the original detailed model of the system.
80

Here, the detailed model of the CIGRE HVDC benchmark system is implemented using
SimPowerSystem toolbox (using discrete blocks for faster simulation) in Matlab/Simulink.
The AVM of the same system is also implemented in PSCAD/EMTDC and Matlab/Simulink using standard library blocks. In order to verify the accuracy of the proposed AVMs, the results of the detailed model are compared with the results of the developed AVMs in two transient studies.
In the first study, a fast change in the dc current reference is applied. First, it is assumed that the HVDC system is operating in a steady state and under nominal conditions with the dc current of 2000A. At t

1 s
, the command value of the dc current i dc _ ref
is reduced to
1600A. After t

1 .
4 s
, the current reference is set back to 2000A. The transient responses produced by the detailed switching model and the AVMs implemented in Matlab/Simulink
and PSCAD/EMTDC are shown in Figure ‎ 4-8. As it can be seen in Figure ‎ 4-8, the reduction
in the reference current on the rectifier side causes the current drop on both the rectifier and inverter sides, which also results in respective increase in the dc voltage. The ac
voltages on both sides are also depicted in Figure ‎ 4-8 (last two subplots), wherein the
corresponding changes are also visible but less pronounced due to the scale. As expected, the results of AVMs in Matlab/Simulink and PSCAD/EMTDC follow the transient response of the detailed simulation with good accuracy but without the switching. As the switching is
81

averaged over a prototypical switching interval, the AVMs can be run with larger time-step without losing the accuracy of results and therefore execute faster than the corresponding detailed model. In order to demonstrate this point for the same study, both AVMs and detailed simulation are run with the maximum possible time-step to achieve acceptable accuracy. The detailed simulation is run using a fixed-step discrete solver as the model is developed using discrete blocks from the SimPowerSystem library. For this model, the time step of at least
10
 s
was required to maintain the accuracy. The AVM implemented in
PSCAD/EMTDC required a
110
 s
time step. The AVM in Matlab/Simulink is run using a variable-step solver ODE23tb which dynamically adjusts the time-step size. For this model, the maximum time step was set to
2 ms
, and the average step size used by the solver was
1 .
6 ms
. The corresponding CPU time for different models is summarized in Table I. To remove the overhead due to initialization in each program, the CPU time is measured from t

0 .
9 s
to t

2 s
. As it can be seen in Table I, the AVMs run much faster than the detailed simulation.
In the following study, a short circuit happens in the dc line. The HVDC system is again assumed to operate in a steady state and under the nominal conditions with the dc current of 2000A. Then, at t

2 s
, a resistive short circuit (0.4 pu) is applied in the middle of the dc line. The short circuit is then cleared at t

2 .
5 s
and the system recovers following a transient. The dc currents and voltages, and the phase voltage of the transformers primary
82

sides as predicted by various models are shown in Figure ‎ 4-9. As can be seen from Figure
‎ 4-9, during the short circuit, the dc currents of the rectifier and inverter drop significantly
due to the controller action that limits the value of the fault currents by reducing the dc reference current and also increasing and decreasing the applied firing angles of the rectifier and inverter sides, respectively. During the short circuit, as the dc voltage drops, the output current of the rectifier is limited to the minimum value due to the voltagedependent current control on the rectifier side. However, on the inverter side, the firing angle is reduced to the minimum value and therefore inverter current is close to zero. The dc voltages at the inverter and rectifier ends of the transmission line are also undergoing significant drop. After the fault is cleared, the system recovers and following a transient
response all voltages and currents return to their pre-fault values as can be seen in Figure
‎ 4-9. Throughout the whole transient study, which is determined by the HVDC system
dynamics and the action of its both controllers, the results predicted by the AVM remain in very good agreement with respect to the reference solution provided by the detailed switching model.
83

Figure ‎ 4-8: Currents and voltages resulting from a change in dc reference current as predicted by various models.
84

Table ‎ 4-1: Comparison of simulation speed for different models for time interval from t

0 .
9 s
to t

2 s
Model
Detailed –SimPowerSystems
(Discrete Blocks)
PAVM- Simulink
(Variable step solver)
PAVM- PSCAD/EMTDC
Step size,
10  s
 t
2
1.6 ms (max) ms (avg.)
110  s
CPU time, s
13.911
0.5290
0.62
85

Figure ‎ 4-9: Currents and voltages during a short circuit in dc side as predicted by various models.
86

In this chapter, the parametric average-value modeling approach presented in Chapters 2 and 3 has been extended to model the 12-pulse line-commutated rectifier and inverter.
Based on the new AVMs for 12-pulse rectifier and inverter subsystems, the AVM of the
CIGRE HVDC Benchmark system has been developed for the first time in both statevariable-based simulation languages such as Matlab/Simulink as well as EMTP-type packages such as PSCAD/EMTDC. On the basis of the work accomplished in this chapter, a significant gain in simulation speed (26 times faster) and time-steps (up to 2ms for SV programs) are demonstrated for the CIGRE HVDC Benchmark system.
87

Although line-commutated converters such as the one shown in Figure ‎ 1-1(a) provide
numerous advantages [76], [77], they are also a significant source for current harmonic in the power systems. As an example, the phase current waveform of the line-commutated
rectifier of Figure ‎
1-1(a) together with its harmonic content is illustrated in Figure ‎ 5-1. As it can be seen from Figure ‎ 5-1, the current waveform is significantly distorted due to high
content of harmonics produced by the switching in the system.
The injected harmonic currents can result in harmonics in the common bus voltage. In general, the harmonics in current and voltage increase losses and have an adverse impact on the system generators, transformers, variable speed drives, etc [48].
To provide some guidelines for dealing with harmonics generated by static powerelectronic converters and preventing any damage to the power system equipments, a number of standards have been set, e.g. [80], which set the limitations on maximum current and voltage harmonics in a particular system. Therefore, engineers commonly conduct various feasibility studies to predict the amount and spectrum of injected harmonics from the equipment that is under consideration for installation in the power grid. Presently, such
88

studies may only be carried out using the conventional simulation techniques and detailed switching models (which are computationally expensive).
The conventional dynamic AVMs, based on either the analytical and parametric approaches, are only capable of predicting the fundamental component of the ac voltages and currents. This is due to the fact that the ac variables are first transferred to a qd rotating reference frame with the rotational speed equal to the fundamental frequency. The transformed waveforms are then composed of a dc component (which has a constant average in steady-state) superimposed by some ripple due to harmonics. Subsequently, all the ripple in these transformed qd
variables are filtered (averaged out). Hence, the averaged transformed qd
variables become pure dc, and transferring them back to phase variables generates only the fundamental component of the ac waveforms (which is also
illustrated in Figure ‎ 5-1).
Extending the parametric AVMs for the transient studies with harmonics would potentially enable a new family of computational tools for the power quality and feasibility studies, where traditionally only the detailed switching models could be used. To the best of our knowledge, this has not been achieved or proposed prior to this thesis.
89

Figure ‎ 5-1. Typical phase terminal voltage and current of diode rectifier system and their harmonic content.
The conventional parametric AVM (PAVM), introduced in [27], is a single-reference framebased model, and therefore incapable of including the effects of harmonics in the ac side. In this Chapter, the parametric methodology is extended to consider the ac system harmonics using the multiple reference frames theory. In general, the qd
rotating reference frame is not limited to the system fundamental frequency, and its rotational speed could be chosen according to any system harmonic. The multiple reference frames theory, was introduced
90

in [78] to analyze symmetrical induction machines under presence of unbalanced stator voltages or harmonic content.
The multiple reference frame PAVM (MRF-PAVM) developed in this section utilizes multiple rotating coordinate systems in order to extract the fast average of variables related to each harmonic in the ac side. These rotating reference frames are depicted with
respect to each other in Figure ‎ 5-2. The synchronous reference frame, wherein the
q
-axis is aligned with phase a of the ac system Thevenin equivalent voltage source, is rotating according to the fundamental frequency in the modeling, and referred to by the superscript
“ s ”. However, for simplicity, the converter reference frame, superscript “ c ”, is defined wherein the angular position of the rotating reference frame q
-axis is considered to be aligned with the phase a
voltage [15]. Herein, the n th harmonic of the synchronous reference frame is referred with superscript “ n ”. The harmonics such as 7 th and 13 th that
rotate in the same direction as the fundamental component (see Figure ‎ 5-2) are known as
the positive sequence harmonics. The harmonics such as the 5 th and 11 th rotate in the opposite direction as the fundamental, and are called negative sequence harmonics. The triple harmonics (all multiples of 3) such as 3 rd and 9 th do not rotate because their respective three-phase components are aligned with each other. These harmonics comprise the zero–sequence component, whose current is blocked by the bridge. A particular emphasis should be placed on the sequence for different harmonics. These positive- and negative-sequence harmonics play an essential role in the operation of ac
91

electric machinery. In synchronous machines, for example, the positive-sequence harmonics act to rotate field and rotor in the proper direction; whereas, the negativesequence harmonics act against the direction of the rotation.
Figure ‎ 5-2: Rectifier ac side voltages and currents expressed in synchronous, converter
[15], and n th harmonic reference frames.
92

Assuming that the ac side has only periodic integer harmonics (no sub harmonics, etc.), each ac variable can be represented based on its harmonic components as f abcs

 n

M f n abcs
, (5-1) where n
represents the harmonic number of each component. The positive and negative superscript signs are used to differentiate between the positive- and negative-sequence harmonics. As a result,
M

1 ,


5 , 7 , ...,
  
2 k

1
 k

2 , 3 ...

.
In order to calculate the fast average of the ac variables for each harmonic component over a switching interval, first, (5-1) should be transferred to the n th -harmonic synchronous
reference frame as shown in Figure ‎ 5-2:
f n qds

K n s
 n

M f n abcs
. (5-2)
As it is denoted in Figure ‎ 5-2, the positive- and negative-sequence-harmonic reference
frames rotate in opposite directions.
The transformation matrix for each harmonic reference frame is defined as
93

K n s

2
3







 cos sin
1
 
 
2
 cos sin n
 s n
 s


2
3

2
3

1 2 cos
 n
 s sin n
 s


2
3

2
3

1 2
 









, (5-3) where  s
, as shown in Figure ‎ 5-2, is the angular position of the synchronous reference
frame. The rotational speed of the n th -harmonic reference frame is n
times faster than the conventional synchronously rotating reference frame (which corresponds to the fundamental component). Moreover, the direction of rotation alternates for different harmonics, starting with the positive direction for the fundamental component. For instance,
5 th harmonic is negative sequence, and
7 th harmonic is positive-sequenceharmonics. If desired, (5-1) may be transformed to the n th -harmonic reference frame using the frame-to-frame transformation matrix according to: f n qds
 s
K n f s qds
, wherein s
K n 


 cos sin



 n n


1
1



 s s

 sin
  n

1
 cos
  n

1
 

 s s  



.
(5-4)
(5-5)
Upon transferring (5-1) into the n th -harmonic rotating reference frame, the n th harmonic component will have a dc value in steady state, as opposed to all other harmonic
94

components which will be oscillatory with a zero average value. Adopting the above concepts of multiple reference frames established for various harmonics, the average-value of each variable in the n th -harmonic rotating reference frame may then be defined by the generalized fast-average expression as f n qds

1
T sw t t


T sw f n qds
  dt
, (5-6) where f n qds
( t )
represents the ac voltage or current transferred to the n th -harmonic rotating reference frame.
The dynamic average model can now be formulated employing the parametric approach.
For this purpose, the fast average of dc and ac side variables are related through algebraic functions. These relationships are established at each operating point for each harmonic, with the ac variables transferred to the respective reference frame. Specifically, w v , n
 v dc v n qds
, n

M
, (5-7) and w i , n
 i n qds i dc
, n

M
. (5-8)
95

The above algebraic functions are extracted and tabulated for all the system operating points and the desired harmonics (herein, 1 st , 5 th and 7 th ). This can be done by running a detailed simulation using a large signal transient, similar to the procedure introduced in
[28].
In order to solve the ac network, it is required to calculate the rectifier voltages for each
harmonic. According to the relationship between the vectors shown in Figure ‎ 5-2, the
converter terminal voltages can be calculated as


 v n qs
 v n ds



 w v , n v dc

 cos sin
 
 


, n

M
, (5-9) where,  n
is also extracted and tabulated directly from a detailed simulation. For the fundamental component,  may also be calculated as
  tan

1


 i s ds i s qs


   tan

1


 v s ds v s qs


. (5-10)
In general, the converter operating point is directly determined by the transferred power to the dc load. The consumed power in the dc load is calculated based on the converter dc terminal voltage and current. These dc variables are typically dependent on the ac fundamental components. Therefore, regardless of the harmonic content in the ac side, similar to the conventional PAVM [27], the rectifier operating condition is defined as
96

z
 v dc i c qds
. (5-11)
After extracting the AVM parametric functions for all the desired ac harmonics, the MRF-
PAVM is implemented according to the block diagram shown in Figure ‎ 5-3. As it can be seen in Figure ‎ 5-3, the whole system is divided into three subsystems: the ac system, which
could be in abc
phase variables or in qd
rotating reference frame; the dc system; and the
rectifier subsystem. As it is illustrated in Figure ‎ 5-3, the rectifier subsystem is composed of
different sub-modules for the fundamental and other-order harmonics of the ac variables.
The blocks labeled as
AVM _ 1
to
AVM _ n
algebraically relate the dc and ac variables of
the corresponding harmonic. Figure ‎ 5-3 shows that in the proposed implementation the
harmonics other than the fundamental component, do not have effect on the predicted dc waveforms. The dc current i dc
is the input for the dc subsystem model is calculated according to the fundamental ac current inside the subsystem
AVM _ 1
. According to this implementation, both the proposed MRF-PAVM and conventional PAVM produce the same dc waveforms.
97

Figure ‎ 5-3: Block diagram depicting implementation of the MRF-PAVM for the diode rectifier system.
In this Chapter, the detailed model of the diode rectifier system shown in Figure ‎ 1-1(a) is
simulated using the ASMG toolbox [12] in Matlab/Simulink [9]. The system parameters are summarized in Appendix E. The AVMs, PAVM and MRF-PAVM, of the same system are implemented in Matlab/Simulink using standard library blocks. In order to verify the accuracy of the proposed MRF-PAVM, the results of the detailed model as well as the conventional PAVM are compared with the results of the proposed MRF-PAVM in timedomain studies.
98

First, the MRF-PAVM is validated in steady-state operations for CCM and DCM from very light dc load (close to open circuit) to very heavy dc load (close to short circuit). The CCM results are shown for the first and second rectifier modes of operation. In the second step, it is shown that the new MRF-PAVM can accurately follow the transient response of the original switching system.
The LCC may work under a wide range of loads, from open to short circuit. On the basis of switching pattern and loading condition, different modes of operation may be defined. Just based on the load, the rectifier operation may be roughly classified as light and heavy. The
LCC with typical system parameters normally operates with the light load, wherein the commutation angle is less than 60 degrees and 2 or 3 switches are conducting during each conduction and commutation subintervals, respectively [14]. However, in some cases with a very light load (and depending on the system parameters), the dc current may be discontinuous. Therefore, in general, the LCC modes of operation in the light load may be defined as discontinuous current mode (DCM) and continuous current (CCM). The conduction and commutation pattern defined by 2 or 3 switches is therefore referred to as the CCM-1. Under heavy loads, the commutation angle may become higher than 60 degrees, and three switches will be conducting at each time. This mode mostly is called CCM-2, and it may happen for systems with sufficiently large commutating inductance in dc filter inductance, or the conditions that are close to short circuit or fault. The corresponding
99

steady state regulation characteristic obtained from the detailed simulation and PAVM is
plotted in Figure ‎ 5-4, wherein the different modes of operation are illustrated.
In the following studies, the proposed MRF-PAVM is validated in steady-state for three modes of operation. The chosen operating points are marked on the regulation
characteristic plotted in Figure ‎ 5-4.
Figure ‎ 5-4: Steady state regulation characteristic of the considered line-commutated converter as predicted by various models.
100

First, it is assumed that the system is operating in DCM with a very light load defined by the dc load resistance of 905  . The resulting dc and ac currents and voltages are shown in
Figure ‎ 5-5 for the detailed simulation and the AVMs. As it can be observed in Figure ‎ 5-5, in
DCM, the phase voltage is very close to the sinusoidal waveform but the phase current has very pronounced distortions in the form of 5 th and 7 th harmonics. In the dc side, both the conventional PAVM and MRF-PAVM predict results and that perfectly match the detailed
simulation (see Figure ‎ 5-5). For the ac side, as it is shown in Figure ‎ 5-5, the results of the
proposed MRF-PAVM are visually close to the actual current and voltage waveforms. The
MRF-PAVM results for the ac side could become even closer to the actual waveforms if more accuracy is required by including more of the higher order harmonics. Extending the
MRF-PAVM to the higher order harmonics requires smaller simulation time step. However,
as it is shown in Figure ‎ 5-5, for the ac side, the conventional PAVM can only predict the
response in terms of the fundamental component. The extracted harmonic content of the
rectifier phase current and voltage is shown in Figure ‎ 5-6 for the MRF-PAVM and detailed simulation. As it can be seen in Figure ‎ 5-6, for the considered 1
st , 5 th , and 7 th harmonics in the AVM, the results of the MRF-PAVM match the detailed simulation very well.
Next, the load is changed to 150  , and the converter operation changes to CCM-1, which is the most common operating mode for the LCCs. The current and voltage waveforms of
the detailed and AVMs are shown in Figure ‎ 5-7. As the Figure ‎ 5-7 shows, both AVMs have
the same results for the dc side that match the detailed simulation. For the ac side, the
101

proposed MRFPAVM predicts the actual ac waveforms much better while the conventional
PAVM can predict only the first harmonic of the ac variables.
Figure ‎ 5-5: System variables as predicted by various models in DCM; detailed, conventional PAVM, and MRF-PAVM models, respectively.
102

The harmonic content of the detailed simulation and the MRF-PAVM in CCM-1 is plotted in
Figure ‎ 5-8, wherein the harmonics predicted by MRF-PAVM are in very good agreement
with the results produced by the detailed simulation.
Figure ‎ 5-6: Harmonic content of the phase current and voltage as predicted by various models in DCM.
103

Figure ‎ 5-7: System variables as predicted by various models for operation in CCM-1.
104

Figure ‎ 5-8: Harmonic content of the phase current and voltage as predicted by various models for the rectifier operation CCM-1.
Finally, the rectifier operation is changed to the heavy mode CCM-2, which is achieved by the dc load resistance of 1 
. The corresponding simulation results are shown in Figure
‎ 5-9, which demonstrates that the MRF-PAVM may be considered sufficiently accurate in
reproducing ac waveforms as well as predicting the average of system variables even under heavy loads. The phase current and voltage harmonic content for the CCM-2 extracted from
the MRF-PAVM and detailed simulation are shown in Figure ‎ 5-10. This figure shows that
the MRF-PAVM is also capable of predicting the harmonic content of the detailed simulation very well. In CCM-2, the phase current is very close to the sinusoidal. This fact is
verified in Figure ‎ 5-10, wherein the fundamental component is dominant.
105

Figure ‎ 5-9: System variables as predicted by various models in CCM-2; detailed, conventional PAVM, and proposed MRF-PAVM models, respectively.
106

Figure ‎ 5-10: Harmonic content of the phase current and voltage in rectifier in CCM-2 as predicted by the detailed model and the proposed MRF-PAVM
.
Considering studies presented in Figure ‎
5-5 through Figure ‎ 5-10, it can be concluded that
the proposed MRF-PAVM is sufficiently accurate in predicting the original switching system response in the range of operations from very light to heavy loads. For the purpose of power quality and feasibility analysis of the proposed connections to the energy grid, it is necessary to consider the total harmonic distortion (THD) level [79], which is defined as
THD

 n n

M

1
F n
2
,
F
1
2
(5-12)
107

where
F
is the
RMS
value of current or voltage. To assess how the proposed AVM can be used for the power-quality-focused studies, the THD of ac voltage and current is obtained for a wide range of load from open to short circuit using detailed simulation and proposed
MRF-PAVM considering system harmonics. The results are plotted in Figure ‎ 5-11. As it can be seen in Figure ‎ 5-11, considering only the 5
th and 7 th harmonics, the MRF-PAVM can
predict the system THD fairly accurate. However, as it is shown in Figure ‎ 5-11, although
considering the 5 th harmonic only increases the simulation speed, this also reduces the accuracy in predicting the THD of the terminal voltages and currents. For instance, according to [80], the maximum permissible voltage THD is 5 % for a typical industrial
power system. However, based on Figure ‎ 5-11, it can be noticed that the level of THD in the
LCC terminal voltage is far above the standards and therefore, injected harmonics are not negligible. The proposed MRF-PAVM will then be capable of predicting the effect of various harmonic filters and other measures applied in the system without the needed for the detailed switching model.
108

Figure ‎ 5-11: Total harmonic distortion of the rectifier phase current and voltage as predicted by the detailed model and the proposed MRF-PAVMs over a wide range of operating conditions.
For the purpose of this study, it is assumed that the system is initially operating in steady state with resistive load of 20  . At t

1 s
, the load is changed to 5  . The corresponding
simulation results are shown in Figure ‎ 5-12. The waveforms plotted in Figure ‎ 5-12 clearly
demonstrate that the proposed MRF-PAVM can follow the transient response of the detailed simulation as accurate as the conventional PAVM for a large disturbance in the system beside the fact that it is advantageous in predicting the ac variables waveforms by
109

including the harmonics into the model. Before the step change in the load, the rectifier is operating in CCM-1. After the step change in the load, the operating mode changes to CCM-
2, close to the short circuit. The conducted transient study shows that the propose MRF-
PAVM can accurately follow the transient response of the switching mode in a wide range of operating conditions.
Furthermore, since the AVMs are continuous, the simulation time-step could be chosen larger and therefore such models can run faster than the corresponding detailed model. To support this claim, the same transient study is run using the detailed simulation and
PAVMs with the maximum possible time-step to achieve acceptable accuracy. For consistency, all simulations are using fixed-step solver ODE 3. The CPU time and time-steps
for each simulation are summarized in Table ‎ 5-1. As it can be seen in Table ‎ 5-1, the PAVMs
are running with larger time step and execute faster than the detailed simulation. However, considering system harmonics, the MRF-PAVM requires smaller time steps, and consequently slower than the conventional PAVM which has only the fundamental component of the ac variables.
110

Figure ‎ 5-12: System response to a step change in load as predicted by various models; detailed, conventional PAVM and proposed MRF-PAVM.
111

Table ‎ 5-1: Comparison of simulation speed for different models for time interval from t

0 to t

2 s
Model
Detailed
PAVM (1st, 5th &7th Harmonics)
PAVM (1st Harmonic)
Step size
100
500

 s s
1.6
m s
Simulation time (s)
1.0113
0.377
0.191
In this chapter, the ac side harmonics have been incorporated into the parametric AVM approach using multiple reference frame theory. This extension makes it possible to use the parametric AVMs for the power quality and feasibility analysis with transients; whereas in the past such studies would require the use of computationally expensive detailed switching models. To demonstrate the proposed concept, herein only the 5 th and
7 th harmonics have been included in the simulation studies as these are typically are the dominant harmonics contributing to the power quality issues with the LCCs. The accuracy of the developed PAVM in predicting the desired ac harmonics has been established by conducting steady-state and large-signal transient studies. It has also been demonstrated that that new MRF-PAVM significantly increases the simulation speed and efficiency in comparison to the detailed simulation, although it runs slower than the conventional PAM which includes only the fundamental 60/50 Hz component. This difference is due to the fact that predicting higher order harmonics also requires smaller integration time step.
112

AVMs are typically formulated in state-space, which is suitable and straightforward to implement in state-variable-based programs. However, in EMTP-type programs [3], [4],
[81], implementation of AVMs developed in state-space is challenging due to the discretization and interfacing. In addition, the AVMs are typically developed in qd synchronous reference frame, but the terminal variables of the switching system are in abc physical variable and coordinates. Hence, the interfacing of AVM with the external EMTP network requires additional attention. In general, the interfacing approaches may be categorized as indirect and direct [14]. The direct interfacing significantly improves the simulation accuracy at large time-steps in comparison to the indirect approach due to removing of the time-step delay. Therefore, in this Chapter, the parametric AVM introduced in [27], which appears very promising and extendable to more complex converter systems and topologies, is discretized using the trapezoidal rule and directly interfaced to the external EMTP networks using two different approaches. The developed methodologies are validated for all the ac-dc converters including the LCC and PWM converters. Developing
113

AVM of the LCC is much more complicated and challenging than for the PWM converters.
Therefore, herein, the direct interfacing approach has been applied to the PAVM of the LCC.
The established average-value models in state-space and qd
coordinates are required to be reformulated for the nodal-analysis-based simulators (PSCAD and EMTP-RV), wherein each circuit component is presented based on the discretized formulation using a particular integration rule (such as trapezoidal or backward Euler). The trapezoidal and backward
Euler integrations are formulated respectively as y n

1
 y n

 t  f n

1

2 f n
 y n

1
 y n
  t
 f n

1
(6-1)
(6-2)
In EMTP solution, the network is solved based on the nodal-analysis formulation as
GV n

I h
, (6-3) where,
G
represents the system nodal conductance matrix, and
I h
is composed of history terms and independent current-sources in the network.
Recently, analytical and parametric AVMs of the line-commutated rectifier systems have been reformulated and interfaced with the EMTP-type simulators using an indirect
114

interfacing approach (which involves a time-step relaxation) [14]. However, the indirect interfacing generally requires very small integration time-step in order to produce sufficiently accurate solution, and leads to poor accuracy and/or numerical instability when large time-steps are used. Here, a direct interfacing approach that achieves a
simultaneous and numerically stable solution is developed. As it is shown in Figure ‎ 6-1, the
directly interfaced parametric AVM (DI-PAVM) is interfaced into the discretized power network by adding 4 elements to the
G
matrix and history terms to
I h
using the Thevenin
equivalent circuit depicted in Figure ‎ 6-1. Therefore, the DI-PAVM is solved together with
the external EMTP network at each time-step using (6-3) without any delay.
Figure ‎ 6-1: Thevenin equivalent-circuit diagram of directly-interfaced PAVM in abc variables with collapsed dc side.
115

In order to interface the PAVM directly with the ac network model in abc
phase variables, it is necessary to express the ac voltages based on the rectifier ac currents (or vice versa) using (2-3)-(2-7). Substituting (2-5) into (2-6), the qd
components of the ac terminal voltages in the synchronous reference frame are

 v qs v ds
 
 


 w v v dc





 cos sin tan

1
( i ds i qs tan

1
( i ds i qs
 
 
 
)
 
 
 
)





 


. (6-4)
In the PAVM as shown in Figure ‎ 6-2, the dc voltage which is the input of the converter AVM
block is determined by the dc sub-system circuit. Here, the dc voltage is calculated based on
the dc side Thevenin or Norton equivalent circuit. Figure ‎ 6-2 shows the PAVM described by
(2-3)-(2-7), (6-4) wherein the dc sub-system Norton equivalent circuit is considered as
part of the rectifier AVM subsystem. The dc voltage in the block diagram in Figure ‎ 6-2 is
calculated as v dc

R eq
 i dc
 i h , dc

. (6-5)
Substituting (6-6) and (6-5) into (6-4), and using  method, results in
116

v qs
 
 w v
   
R eq i qs
2
 
 i ds
2
  cos
 tan

1
( i ds i qs
 
    ) w v
R eq i h , dc cos
 tan

1
( i ds i qs
 
    )

, v ds
 
 w v
   
R eq w v
R eq i h , dc sin
 tan i qs
2

1
( i ds i qs
 
 i ds
2
  sin
 tan

1
( i ds i qs
 
    )

.
 
    )


After some extensive effort, (6-6) can be written as v qs
   w v
   
R eq i qs
2
   i ds
2
  


 w v i qs i qs
2
   i ds
2
  cos

R eq i h , dc cos
 t , t
 i qs
2 i ds
   i ds
2
  sin
 t

 
 v ds
   w v
   
R eq


 i qs
 i qs
2
 
 i ds
2
  w v
R eq i h , dc sin
 
.
sin
 t i qs
2
   i ds
2
  
 i qs
2 i ds
 
 i ds
2
  cos
 

 
 wherein
  tan

1 i ds i qs
 
 
 
.
(6-6)
(6-7)
(6-8)
117

In (6-8), the ac side voltage-sources shown in Figure ‎ 6-2, are represented as nonlinear
functions of ac currents. Hence, these functions are required so that the model can be linearized in a time-step  t
as
Figure ‎ 6-2: Block diagram of descretized averaged-circuit model with collapsed dc side. v qs w v
  w i

R eq w v sin
 t
 
  t
  t
  t
 t
  t
   

R eq
 cos
   t
  t
   
 w v v ds w v

R eq t
 i h

 t
, dc
 w v cos

  t
 tan
 t


1
(
R eq i ds i qs

 t t


 t
  t
  t
  t
 cos
 
 t
 t
R eq

  sin
 t

  t

) ,
 t
  t
   
 t
  t
   

 w v
 t
  t

R eq i h , dc sin
 tan

1
( i ds i qs

 t t


 t
 t

  t
  t

) .
(6-9)
Equation (6-9) describes the discretized AVM in qd
which can be directly interfaced with the ac network. The final implemented model can be represented in the general matrix form as
118

V n

Z avm
 qd
I br
 e h
 qd
, (6-10) where
Z avm
 qd





R eq
R eq cos sin




 t
  t
 
 t
  t
 
R eq
R eq sin cos

 
 
 t
  t
  t
  t
 



 w v
 t
  t
  t
  t

, e h
 qd






 w v w v
 t
  t

R eq i h , dc
 t
  t

R eq i h , dc cos
 sin tan

1
( i ds i qs

 t t


 t
 t

  tan

1
( i ds i qs

 t t


 t
 t

  t t


 t
 t


)

)








,
I br

 i qs i ds

T , and
V n

 v qs v ds

T .
(6-11)
(6-12)
(6-13)
It is worth mentioning that using  instead of  method results in different equations. In this case, the discretized ac equations can be expressed as follows, v qs i qs
 
 w v i qs

 t t


 t
 t

  t
  t

R eq i qs
2
 t
  t

 i ds
2
 t
  t
 cos
   t
  t

)


 
 w v
 t
  t
  t
  t

R eq cos
   t
  t
  v d i qs i qs i qs
2
2
2


 t t i ds
 w v i qs

 t w i
 t

 t
  t


 t

R eq i ds sin
 



 t i ds

2
2

 t t

 t
 t
  t

)
 t

 i ds i qs
 t i ds
  t
 t
  t


 i ds
2
 t
  t
 i ds

 

 

 w v w v

 t t


 t
 t


R eq
R eq i h , dc i h , dc cos
 
 w v
 t
  t
  t
  t

R eq sin

 

 t
 sin
 
 t
 t


 t
  t
 
.

,
 t
 


(6-14)
119

Defining the following nonlinear functions as
A
 i qs
 t
  t
 i qs
2
 t
  t

 i ds
2
 t
  t

,
B
 i qs
2
 t i ds
  t
 t



 t i ds

2
 t
  t

,
(6-15)
(6-16) the AVM Impedance and history terms matrixes are written as
Z avm
 qd




R eq
R eq cos sin

 
 
 t t


 t
 t


)
)


A
A
R eq
R eq cos sin

 
 
 t
  t
  t
  t
 
B
B



 w v
 t
  t
  t
  t

, e h
 qd



 w v w v

 t t


 t
 t


R eq
R eq i i h , dc h , dc cos sin

 
  t
 t
  t
 
  t
 



.
(6-17)
(6-18)
In a case that the ac system is developed in qd
, the developed PAVM can be directly interfaced to the ac circuit. However, ac system model developed in qd
is not a common practice in most of system studies wherein the network model is prepared in abc
phase variables. Therefore, it is necessary to transform the discretized PAVM to the abc
phase coordinates. Applying the inverse transformation [15] and after some effort, the Theveninequivalent of the AVM can be represented as
V n

Z avm
 abc
I br
 e h
 abc
,
Z avm
 abc

 
3

3
, e h
 abc

 e h , a e h , b e h , c

T ,
(6-19)
(6-20)
120

V n

 v as v bs v cs

T , and
I br

 i as i bs i cs

T . (6-21)
Then, in order to make the PAVM easy to interface in EMTP-type simulators such as
PSCAD/EMTDC, the developed PAVM can be expressed based on the Norton-equivalent of the current source-conductance branches by multiplying both sides of (6-19) by
Z

1 as bellow.
Z

1 avm
 abc
V n

I br

I h
 abc
,
I h
 abc





I
I
I h , a h , b h , c





Z

1 avm
 abc e h
 abc
.
(6-22)
(6-23)
The interfacing of the Norton-equivalent PAVM with the ac system is shown in Figure ‎ 6-3.
Figure ‎ 6-3: Norton equivalent-circuit diagram of directly-interfaced PAVM in abc
phase variables with collapsed dc side.
121

In the directly interfaced PAVM developed in the Subsection 6.2, the dc voltage in (6-4) is calculated based on the equivalent-circuit of the dc system. Therefore, the dc side is collapsed and its equivalent-circuit is mapped into the ac side equations. Although, this
PAVM is very suitable for the applications when the dc system is isolated from the entire ac network such as machine drives or induction furnaces, etc., in some applications such as
HVDC systems the dc side of the rectifier is interconnected to the other parts of the ac network and other converters. Therefore, the discretized PAVM should be capable of being interfaced with both the ac and dc networks.
As it is shown in Figure ‎ 6-2, the rectifier is modeled in the dc side as a dependent current
source and therefore the dc voltage cannot be calculated independent of the dc system
equivalent-circuit. In order to overcome this challenge shown in Figure ‎ 6-4, a resistive
snubber is added to the rectifier output. In order to reduce the interfacing error in the dc side, the snubber resistance should be chosen sufficiently large. Using this snubber, the
rectifier dc voltage can be calculated independent of the dc network variables (see Figure
‎ 6-4) as
v dc

R sb w i i qs
2
 
 i ds
2
 
 i
Load
. (6-24)
Substituting (6-24) in (6-4) and linearizing the resulting equation, the discretized PAVM based on the  method can be written as
122

v qs i qs
 
 w v i qs

 t t


 t
 t

  t
  t

R sb i qs
2
 t
  t

 i ds
2
 t
  t
 cos
   t
  t
 

 
 w v
 t
  t
  t
  t

R sb cos
   t
  t
  v d i qs
  i qs
2
2

 t i ds w v i qs


 t
 t
 t
 t




 t
 t
 t i ds

 t
  t

 i ds
2
 t
  t

 

2
 t t

  t
 t

 i ds
R sb i qs
 
 w v
 t
  t

R sb sin
   t
  t
 
 cos
   t

 
 w v
 t
  t
  t
  t

R sb
 t sin


 i
Load
  t

 
 t

,


 i qs
2 i ds
 t
  t

 t
  t

 i ds
2
 t
  t
 i ds
 
 w v
 t
  t

R sb sin
   t
  t
  i
Load
 
, v dc w i
 t
 


 t w i

 t
R sb
  t i qs

R sb
2
 t
 i qs
 t
  t
 i ds i qs
2
 t

 t
  t

 t

 i ds
2
 t i ds
  t


 t

 i ds
2
 t
  t
 i qs

R sb i
Load
 
.
(6-25)
As it can be seen in (6-25), the PAVM does not have any history terms and the impedance matrix is generated as
..
Z avm
 qd




 w v w v w i
 t

 t t



 t
 t
 t

 
 
R sb
 t
 t


 t
 
  t t


 t


R
R sb sb t t
A


 t
 t


R
R sb sb cos sin
 
  w v
...
w v w i



 t
 w v w v
R
 t
 t sb

 t t

 w i
R
 sb
 t
 t


B
R
R
 t sb cos sin

 
 
 t t

 sb cos sin
 
   t
  t
 t
  t
 
 




 t
 t




B
B
 t
A
...
  t
 
A
...
(6-26)
123

where the coefficients
A
and
B
are defined in (6-15) and (6-16). By transforming the discretized PAVM into abc
phase variables, the Thevenin-equivalent of the model can be written as
V n

Z avm
 abc
I br
,
Z avm
 abc

 
4

4
,
V n

 v as v bs v cs v dc

T , and
I br

 i as i bs i cs i
Load

T
(6-27)
(6-28)
(6-29)
The Norton-equivalent can then be represented as
Z

1 avm
 abc
V n

I br
. (6-30)
The Thevenin-equivalent of this DI-PAVM is the same as the diagram shown in Figure ‎ 6-1,
wherein the converter is connected to the EMTP external network as a discretized component with multiple terminals.
Figure ‎ 6-4: Block diagram of the descretized averaged-circuit model interfaced with resistive snubber in dc side.
124

Although using resistive snubber in the dc side solves the interfacing problem of the PAVM with the dc system as described, it also introduces interfacing losses to the model and causes some numerical oscillations for the large time-steps. To mitigate this problem, the interfacing is modified by adding a compensating current source in parallel to the snubber
as shown in Figure ‎ 6-4. This current source is calculated in each step as
i comp
( t )

V dc
( t

R sb
 t )
. (6-31)
Using this modified circuit brings the ability to reduce the resistance of the snubber as the resistive loss introduced by the snubber is compensated, and smaller value for the resistive snubber prevents numerical oscillations in large time-steps.
It should be mentioned that all discretized PAVMs are developed and formulated in synchronous reference frame, but the concepts of the parametric approach and discretization technique are general, and the PAVMs can be reformulated for any arbitrary rotating reference frame such as the converter reference frame.
Here, the LCC system shown in Figure ‎ 1-1 with parameters summarized in Appendix F is
simulated using the directly interfaced PAVMs proposed and formulated in the previous
125

sections. In order to verify the accuracy and demonstrate the performance of the DI-
PAVMs, these models are compared with the detailed simulation and the analytical directly interfaced AVM introduced in [51] for different time-steps  t
. Here, the detailed simulation is implemented in PSCAD/EMTDC, and all the AVMs are implemented based on an EMTPtype algorithm written in MATLAB.
In the first study, the integration time-step is considered to be small,
50 μs . The system is starting up from zero initial condition. At t

0.02
s
, the resistive load is changed from
2
 to
1
 . Then, at t

0.04
s
, the load is changed back to
2
 . For this study, the waveforms resulted from the detailed simulation, directly interfaced PAVM with the collapsed dc side
(DI-PAVM1), directly interfaced PAVM with the resistive snubber in the dc side (DI-
PAVM2), and the directly interfaced analytical AVM [50] are plotted in Figure ‎ 6-5. As it is shown in Figure ‎ 6-5, all the AVMs follow the transient response of the detailed switching
simulation almost perfectly.
Next, the performance of the proposed PAVMs is studied for larger time-steps. First, the same study as is carried out with the time-step of
500
μs . The simulation results of the
different AVMs are compared with the detailed simulation in Figure ‎ 6-6(a), which
demonstrates the accuracy of the AVMs in predicting the transient response of the detailed simulation. The compensation current connected in parallel with the resistive snubber is considered based on the dc voltage calculated in the previous time step. The effect of this may be presented in the startup for large time-steps and small value of the snubber
126

resistance. For instance, a magnified view of the startup trajectories of i as
and v as
is shown
in Figure ‎ 6-6(b). As it is shown in Figure ‎ 6-6(b), there is a small error and oscillations in the
startup predicted by the PAVM with the resistive snubber (DI-PAVM 2) in dc side for a large time-step.
Finally, for the same case study, the simulation results are plotted in Figure ‎ 6-7(a) for a
very large time-step of
2 ms
. In Figure ‎ 6-7(a), the AVMs are compared with the reference
results which are obtained using the time-step of
50
μs
. As it is illustrated in Figure ‎ 6-7(a),
the AVMs still predict the response of the detailed simulation with acceptable accuracy. For more clarification, the magnified view of v
Load
and i as
during the transient is illustrated in
Figure ‎ 6-7(b). Based on Figure ‎ 6-7(b), it can be verified that the accuracy of the proposed
DI-PAVMs does not degrade by increasing the size of the time-step. However, due to the trapezoidal integration rule, for very large time-steps such as
2 ms
, some numerical oscillations may be introduced to v dc
using the analytical AVM and DIPAVM with the collapsed dc side (DI-PAVM1) during the initialization and fast transient which will be damped after a while. This can be seen in v dc
curve plotted in Figure ‎ 6-7. Using the
resistive snubber in the dc side, these numerical oscillations are damped as can be seen from v dc
curve in Figure ‎ 6-7.
127

In order to suppress these steady numerical oscillations introduced to the simulation by the trapezoidal integration, a damping technique is developed on the basis of the methodology introduced in [82].
128

Figure ‎ 6-5: Transient response of the system in dc and ac variables as predicted by the various models using
50
μs time-step.
129

130

Figure ‎ 6-6: Transient response of the system in dc and ac variables as predicted by various models using
500
μs time-step: (a) dc and ac waveforms of the overall transient; and (b) magnified view of the start-up waveforms.
131

Figure ‎ 6-7: Transient response of the system in dc and ac variables as predicted by various models using
2 ms
time-step: (a) dc and ac waveforms of the overall transient; and (b) magnified view of the start-up waveforms.
132

Damping Technique for Directly-Interfaced Parametric Average-Value
Models
The trapezoidal integration used in the previous Section 6.3 is the dominant method for the
EMTP-type solution as it is proven to be simple, numerical stable for stiff systems and also self-starting [81]-[86]. However, the trapezoidal integration also appears to be oscillatory.
Therefore, during the past decades, many researches have been trying to solve this weak point of trapezoidal integration. One of the most significant techniques is the so called critical damping adjustment (CDA) introduced in [84], [85]. Another method is trapezoidal integration with damping introduced in [82], [83]. In this method, the
G
matrix is required to be changed or updated, which makes it challenging to be implemented in software packages. This issue is solved by the methodology introduced in [86], wherein the timestep size is flexible and changing during the simulation. Nevertheless, it is challenging to apply this methodology in simulators with fixed simulation time-step such as
PSCAD/EMTDC and EMTP-RV. Herein, on the basis of the trapezoidal with damping technique [82], a methodology is developed which does not require any change or update in the
G
matrix.
133

Using the trapezoidal integration, all the system elements can be presented as pure
resistive and history voltage source or nodal current injection as shown in Figure ‎ 6-8. In
order to solve the steady numerical oscillations, (6-1) is replaced with the following equation as y n

1
 y n

 t
2
 
1
   f n

1


1
   f n

(6-32) where  is the damping ratio [82]. The discretized model extracted using (6-32) for system elements such as inductor is equivalent as an ideal inductor in parallel with a shunt resistor
R

2 L
  t
. The trapezoidal rule with damping formulation adds some dissipation to the solution [82]. By varying  from zero to one, the integration is changing from ordinary trapezoidal to Backward Euler.
Figure ‎ 6-8: Discretized equivalent circuit of inductance: (a) Thevenin equivalent; and (b)
Norton equivalent.
134

For instance, an equivalent discretized circuit of an inductor using trapezoidal integration
as shown in Figure ‎ 6-9 is calculated as [81]
R

2 L
 t e h
 
  v
 t
  t


2 L
 t i
 t
  t

(6-33)
(6-34)
However, using the trapezoidal integration with damping for the same inductor results in
[81]
R

 t
2 L

1
  
, e h
 
1
1



 v
 t
  t


 t

2
1
L
    t
  t

.
(6-35)
(6-36)
As it can be seen in (6-35) and (6-36), for the EMTP-type transient simulation programs wherein the discretization is implemented on the component level, each system component can be reformulated on the basis of the trapezoidal integration with damping.
Although the trapezoidal integration with damping technique is very advantageous to solve steady numerical oscillation in the solution, most existing EMTP-type transient simulators are developed based on the ordinary trapezoidal integration. Therefore, it requires the network conductance matrix
G
to be updated in order to use the trapezoidal integration with damping for the solution. Hence, herein, a damping technique is used on the basis of
135

the trapezoidal integration with damping method in which the
G
matrix is not required to be changed or updated, and damping is implemented locally.
In order to add dissipation to the solution, each component should be reformulated based on the trapezoidal integration with damping as it is presented in (6-35) and (6-36) for an inductor. However, using the reformulated components with damping in any of the existing
EMTP-type transient simulation programs such as PSCAD/EMTDC, the network conductance matrix
G
is required to be changed. Hence, the reformulated components based on the trapezoidal integration with damping should be written in a suitable form. For simplicity, the procedure is explained on an inductor using (6-35) and (6-36).
In order to avoid changes in the
G
matrix, (6-35) is approximated as (6-33), and the history term calculated in (6-35) can be written as e h
 
  v
 t
  t


2 L
 t i
 t
  t

 e hd
  where e hd

2

1
  v
 t
  t


 t

L

1
    t
  t

(6-37)
(6-38)
136

Equation (6-38) is combined with the history term calculated in (6-34) using the ordinary trapezoidal integration plus one extra term, which is called as correction term. Therefore,
as it is shown in Figure ‎ 6-9, in order to damp the oscillations, in each time-step of the
solution, all the system elements are presented as a resistor and history voltage source, and the damping history term which is added as a voltage source e hd
or current source i hd
.
Figure ‎ 6-9: Discretized equivalent circuit of lumped inductance with damping history term:
(a) Thevenin equivalent; and (b) Northon equivalent.
Using this damping technique in EMTP type programs, there will be a correction history matrix added to the nodal-analysis formulation as
GV n

I h

I hd
, (6-39)
137

where
I hd
is composed of damping history terms. Applying this damping technique to
Thevenin equivalent-circuit diagram of DI-PAVM in abc
variables shown in Figure ‎ 6-1, all
the system element as shown in Figure ‎ 6-10 have one added voltage source for damping.
These damping terms are calculated at each time-step in the converter AVM sub-system and added to the history terms in the simulation for damping of the oscillation.
Figure ‎ 6-10: Thevenin equivalent-circuit diagram of the locally damped DI-PAVM in abc variables.
Here, the same study as in Section 6.2.1 is conducted on the system shown in Figure ‎ 1-1(a)
using both the proposed parametric DI-PAVMs with the time-step  t

2 ms
, and damping
138

ratio  
0 .
2
. As it is shown in Figure ‎ 6-11, the oscillation introduced in the dc voltage by
the DI-PAVMs (especially with the collapsed dc side) is damped very well using the
trapezoidal with localized damping. Moreover, as it is shown in Figure ‎ 6-6, the DI-PAVM
with the snubber in the dc side may introduce an error or oscillation in the start-up for large time-steps and small value of the snubber resistance. However, by using the damping technique, it is possible to chose larger snubber resistance which reduces the error in the
start-up. This reduction in the start-up error can be clearly seen in Figure ‎ 6-11.
The trapezoidal rule with localized damping is advantageous over the conventional trapezoidal as it is not required to change or update the
G
matrix. In general, the simulation results using both damping techniques are the same in steady state. However, during the transient, due to difference in the
G
matrix, the predicted transient response of the system is slightly different using this two damping methods. This fact is depicted for the
DI-PAVM with the collapsed dc side in Figure ‎ 6-12, wherein the simulation results are
compared using these two damping methods.
The study shown in Figure ‎ 6-11 demonstrates that the directly interfaced PAVMs, unlike
the indirectly interfaced AVM [14] or the other AVMs developed in the state-space, can be used for system level studies with very large time-steps and with damped oscillations.
139

Figure ‎ 6-11: Dc and ac waveforms predicted by the models, with and without localized damping;  t

2 ms
.
140

Figure ‎ 6-12: Transient response of DI-PAVM with collapsed dc side as predicted by various models using time step  t

2 ms
, and conventional and localized damping.
In this chapter, we have investigated the problem of interfacing the AVMs with the external networks in the EMTP solution. As an outcome of this research, two discretized equivalent
141

circuits for direct interfacing of the parametric AVMs have been proposed for the first time.
It has been demonstrated that the directly interfaced PAVMs can perform very well even at fairly large time-steps (~0.5 …1 ms). Moreover, for efficient interfacing in EMTP-type packages and removing the possible artificial numerical oscillations caused by the trapezoidal integration, the new DI-PAVM with local damping has been proposed. This new
DI-PAVM model with damping is shown to be capable of producing fairly good/accurate results even at a much larger time-step (~2 …3 ms). This achievement also sets the stage for the future generation of the EMTP-type transient simulation programs, in which the user can potentially automatically replace the detailed switching models with their AVM equivalents, wherein it will be desirable for the system-level-studies.
142

This thesis is focused on dynamic average-value modeling of high power switching converters. Averaging of the switching details makes the AVMs computationally efficient and also continuous. Due to the importance of line-commutated converters in high power applications, the research conducted here is mainly focused on the development of a general parametric AVM of the line-commutated converters for various applications. The motivation behind this research has been described in Chapter 1.
In Chapter 2, the parametric approach introduced in [27] for the state-variable-based simulation languages is extended to model the thyristor controlled line-commutated converters in the rectifier mode of operation. The contribution of this chapter addresses
Objective 1 stated in Introduction of this thesis. The extended PAVM enables modeling of more complicated systems with closed-loop control such as rectifier side of HVDC system, wind turbines, and synchronous machine exciter. In this chapter, on the basis of the extended parametric approach, the new average-value model of machine/thyristorcontrolled-rectifier system is presented for two methods of generating thyristor firing pulses: (i) using converter terminal voltages, and (ii) using the machine rotor position. The resulting model is continuous and computationally efficient, and can be used for large-
143

signal time-domain studies as well as linearization for small-signal frequency-domain analyses. The proposed AVM is verified against the original switching model as well as analytical AVM in large-signal time-domain studies as well as small-signal frequencydomain transfer functions characteristics, and is shown to produce expected good results.
In Chapter 3, based on the parametric approach and the PAVM introduced in Chapter 2, the thyristor controlled LCC model is reformulated for the inverter operation as was the focus of Objective 2 . The voltage-source and current-source average-value modeling (VS-AVM and CS-AVM) formulations are proposed based on the well-posedness of the underlying state equations. It is shown that the proposed CS-AVM will have advantages for modeling the thyristor controlled LCC inverter, whereas a more conventional VS-AVM may not be convergent. The contributions of this chapter enable the average-value modeling of HVDC systems with bidirectional energy flow.
Chapter 4 addresses Objective 3 . Specifically, the PAVMs developed in Chapters 2 and 3 for rectifier and inverter modes, are extended to the 12-pulse converters. The dynamic AVM of the CIGRE HVDC benchmark system has been developed and demonstrated for the first time in state-variable-based simulation languages such as Matlab/Simulink as well as
EMTP-type packages such as PSCAD/EMTDC. The accuracy of the developed AVMs is verified against the detailed simulation in large-signal time-domain transients including the dc reference current command change and a short circuit implemented on the dc
144

transmission line. It has been shown that by utilizing the developed PAVM, a significant gain can be archived in simulation time.
All conventional AVMs only predict the fundamental component of the ac variables. In
Chapter 5, a new parametric average-value model of diode rectifier system which includes the main harmonics (5 th and 7th) in ac system has been proposed. The LCC is modeled on the basis of the parametric approach in which the nonlinear parametric functions relate the fundamental and harmonic components of the ac voltages and currents to the dc side variables. The introduced MRF-PAVM (aligned with Objective 4 ) enables to use the parametric AVM for the transients and power quality analysis, which has not been achieved. The developed MRF-PAVM can also be used by utility companies for feasibility studies to evaluate the impact of the LCC-equipment connections to the energy grid.
In Chapter 6, two discretized parametric average-value models of the LCC are developed for the EMTP-type solution, as outlined in Objective 5 . In the first one, the dc side is collapsed and the dc equivalent circuit is mapped into the ac equations. This approach is useful mostly in a case when the dc side is isolated from the EMTP external network. In the second approach, the AVM is interfaced with the dc system using a resistive snubber and injecting a compensation current connected in parallel with the snubber. The proposed models are interfaced directly with the external ac networks, which provide the simultaneous solution of the system variables. It is shown that the proposed models are numerically stable and accurate even for very large simulation time-steps such as
2 ms
.
145

Moreover, a new methodology is developed to damp the artificial numerical oscillations caused by the trapezoidal integration without introducing changes or update in the overall network conductance
G
matrix.
The modeling techniques and contributions in this thesis will be useful to many researchers and practicing engineers in the area of electric power and energy systems who are dealing with applications of power electronic components and modules as well as using modern digital simulation tools. The generalized parametric AVMs developed in this thesis can be easily added as modules to the libraries of different simulation packages such as state-variable-based and EMTP-type programs.
For more precise simulations and extending the AVMs to the power quality analysis and harmonic power flow calculations including at least several of significant (5th, 7th, 11th, etc.) harmonics has always been desirable. In this thesis, the contributions from a number of (significant) harmonics of interest are considered in the parametric AVM of the diode rectifier using multiple reference frame theory. However, in most LCC applications such as
HVDC systems, the power is transferred bidirectional using thyristor switches. Therefore, one of the immediate directions for the future research would be to extend the multiple reference frame theory approach of Chapter 5 to the thyristor-controlled LCCs with bidirectional power flow and as HVDC systems.
146

The directly interfaced PAVMs proposed in this thesis for EMTP type solution have been developed and demonstrated for the diode rectifier systems. The extension of this approach to the thyristor-controlled LCCs operating in rectifier and inverter mode is therefore appears to be very feasible. The next logical direction would be to include the harmonics for the directly interfaced PAVMs of the LCCs with bi-directional power flow and as HVDC systems.
Other HVDC technologies based on multi-level modular converters (MMC) are gaining wide acceptance and will be considered for developing the computationally efficient dynamic average models. Those and other research topics will be considered in the very near future by other graduate students in the Electric Power Energy Systems research group at UBC.
147

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[Online]. Available: http://www.microtran.com
[2] (2007). Alternative Transients Programs (ATP-EMTP, ATP User Group). [Online].
Available: http://www.emtp.org
[3] (2005). PSCAD/EMTDC (Ver. 4.0 On-Line Help Manitoba HVDC Research Centre and
RTDS Technologies Inc. Winnipeg, MB, Canada).
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161

Synchronous machine parameters:
US Electrical Motors, 5 HP, 230Volt, 215T Frame, 1800 rpm, rated field current 1.05A, custom-made for university lab.
 b

2
  
60 rad/s r s

0 .
382
 x mq r

1

9.3871


5 .
07
 r

2

1 .
06
 r

3

0 .
447
 r
 kd 1

140
 r kd 2 r
 kd 3 r
 fd


1 .
19


1 .
58

0 .
112
 stator-to-field turns ratio
P

4 (four poles) x ls

0.4222
 x md x
 lkq 2

14.8158


1.3195
 x
 lkq 2

1.3195
 x
 lkq 3

9.8772
 x
 lkd 1 x
 lkd 2 x
 lkd 3 x
 lfd



3.7209
1.8510
0.5768



1.7002


N s
N fd

0 .
0269
L f

1 .
19 mH
; r f

0 .
32
 ;
C f

2.28

F
162

 
2

10

2
; K p

10 ; K i

100 ;
 max

90 ;
 min

0
The averaging window is
1 6
of the line frequency.
163

System parameters:
L f

0
( mH
),
R f

0 .
32
(  ), r
C

0 .
1
 . e as
 v m cos(
 t )
,
2

60 rad / s
, v m

391 .
4
(
V
),
164

System parameters:
2

60 rad / s
, v m

175 .
6 (kV),
L
C

24
( m H), r
C

0 .
1
 ,
L
Line

0 .
5968
(H),
R
Line

0 .
32
(

),
C
Line

26
(μF).
165

CIGRE HVDC Benchmark System Parameters:
Table A. 1: CIGRE HVDC system parameters used for simulation studies
Controller Parameters:
System parameters
Ac voltage base
Transf. tap (HV side)
Voltage source
Nominal dc voltage
Nominal dc current
Transformer impedance
System frequency
Minimum angle
Rectifier
345 kV
1.01 pu
1 .
088

22 .
18

500 kV
2 kA
0.18 pu
50 Hz
 
15

Inverter
230 kV
0.989 pu
0 .
935
 
23 .
14

500 kV
2 kA
0.18 pu

50 Hz

15


1
 
2
 
3

0 .
0003 , k

0 .
01 , a

10

6
, b

0 .
0101 , k
P 1

1 .
0989 , k
I 1

100 .
63 , k
P 2

0 .
63 , k
I 2

65 .
61 , k
P 3

0 .
7506 , k
I 3

20 ,

R max

80

,

Imin

100

,

Imax

150

.
166

Testing system parameters:
V line

480 V , f e

60 Hz , r
C

0 .
2

, L
C

10 mH , r dc

0 .
5

, L dc

1 .
33 mH , C f

500

F .
167

Parameters of the front-end rectifier example system:
V rms

208 V
, f e

60 Hz
, r th
 r ac

0 .
05
 ,
L th

45

H
,
L ac

50

H
, r dc

0 .
5
 ,
L dc

1 .
33 mH
.
168