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Numerically efficient modeling of saturable ac machines for power systems electromagnetic transients… Therrien, Francis 2015

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   NUMERICALLY EFFICIENT MODELING OF SATURABLE AC MACHINES FOR POWER SYSTEMS ELECTROMAGNETIC TRANSIENTS SIMULATION PROGRAMS  by  Francis Therrien  B.Eng., Université de Sherbrooke, 2010    A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  in  The Faculty of Graduate and Postdoctoral Studies  (Electrical and Computer Engineering)    THE UNIVERSITY OF BRITISH COLUMBIA  (Vancouver)  August 2015   © Francis Therrien, 2015  ii  Abstract  Extensive computer simulations are necessary to operate power systems in a stable, secure, and optimal manner. This thesis considers electromagnetic transients (EMT) simulators, which are widely used to study modern power systems of which rotating machines are essential components. In EMT simulations, induction and synchronous machines are usually represented by general-purpose lumped-parameter models, which can be formulated using different sets of coordinates and state variables. While algebraically equivalent, these models’ numerical properties can differ greatly, which in turn can significantly affect the numerical accuracy and efficiency of entire EMT simulations. The ultimate goal of this thesis is to increase the numerical efficiency of EMT simulators without degrading their numerical accuracy and stability. This is achieved by proposing several new machine models with improved numerical properties. Models are presented for both families of EMT simulators, namely nodal-analysis-based (EMTP-type) and state-variable-based (SVB) programs. Moreover, we incorporate the effect of saturation to improve modeling fidelity.  This thesis makes several important contributions to the state of the art. As a first step, the implicit flux correction (FC) method frequently used in SVB programs is reformulated to achieve explicit qd models with main flux saturation. Next, we propose new highly efficient saturable SVB voltage-behind-reactance (VBR) machine models with constant-parameter interfacing circuits. A new and accurate EMTP-type VBR induction machine model with a saturation-independent interfacing circuit is then proposed, thereby avoiding numerically costly re-factorizations of the network’s conductance matrix. Finally, the numerical efficiency of this VBR model is further improved by using multirate techniques. Numerous case studies demonstrate the superior combination of numerical accuracy and efficiency of the proposed models, and their beneficial impact on the speed of EMT simulations. It is envisioned that the proposed models will eventually be included in commercial EMT programs, extending their reach to thousands of engineers worldwide. iii  Preface  Many of the contributions presented in this thesis have been published in scientific journals, appear in conference proceedings, and have been or will be submitted for peer review. Unless stated otherwise, I derived all the proposed models, implemented all of them in computer programs, executed all the studies, and was responsible for writing the first draft of each paper. All of this work was done under the supervision of Dr. Juri Jatskevich, who provided guidance and comments during all of the aforementioned steps. Dr. Jatskevich also helped writing and editing each paper. Other co-authors have also contributed to this research by providing feedback and editorial comments, and by revising the manuscripts. The articles resulting from this doctoral work and additional contributions by the co-authors are listed below.  Chapter 3 is based on the following journal article:  F. Therrien, L. Wang, J. Jatskevich, and O. Wasynczuk, “Efficient explicit representation of AC machines main flux saturation in state-variable-based transient simulation packages,” IEEE Trans. Energy Convers., vol. 28, no. 2, pp. 380–393, Jun. 2013.  The induction machine saturation function presented in Section 3.1.2 was derived by Dr. Jatskevich several years ago but had not been published prior to this paper. The model used to generate the reference trajectories in Section 3.3.1 was implemented by Dr. Liwei Wang.   Chapter 4 is based on the following journal article and conference paper: F. Therrien, M. Chapariha, and J. Jatskevich, “Constant-parameter voltage-behind-reactance induction machine model including main flux saturation,” IEEE Trans. Energy Convers, vol. 30, no. 1, pp. 90–102, Mar. 2015. F. Therrien, M. Chapariha, and J. Jatskevich, “Generalized state-space saturable induction machine model using a voltage-behind-reactance formulation,” in Proc. IEEE Power Energy Soc. General Meeting, Jul. 21–25, 2013, pp. 1–5. iv   Chapter 5 is based on the following journal articles:  F. Therrien, M. Chapariha, and J. Jatskevich, “Pole selection procedure for explicit constant-parameter synchronous machine models,” IEEE Trans. Energy Convers., vol. 29, no. 3, pp. 790–792, Sep. 2014. F. Therrien, M. Chapariha, and J. Jatskevich, “Constant-parameter synchronous machine model including main flux saturation,” under review.  Chapter 6 is based on the following journal article: F. Therrien, L. Wang, M. Chapariha, and J. Jatskevich, “Constant-parameter interfacing of induction machine models considering main flux saturation in EMTP-type programs,” to appear in IEEE Trans. Energy Convers.  Chapter 7 is based on the following journal article: F. Therrien and J. Jatskevich, “A multirate EMTP-type induction machine model,” to be submitted.  Finally, Chapters 1 and 2 of this thesis contain excerpts of the introductory sections of the aforementioned papers.     v  Table of Contents Abstract .................................................................................................................................................... ii Preface ..................................................................................................................................................... iii Table of Contents .................................................................................................................................... v List of Tables .......................................................................................................................................... xi List of Figures ...................................................................................................................................... xiv List of Abbreviations and Acronyms ........................................................................................ xxiii Nomenclature ..................................................................................................................................... xxv Acknowledgments......................................................................................................................... xxxiii CHAPTER 1: INTRODUCTION ............................................................................................................ 1 1.1 Motivation .................................................................................................................................................... 1 1.2 Literature Review ..................................................................................................................................... 6 1.2.1 PD Models ................................................................................................................................................ 7 1.2.2 qd Models................................................................................................................................................. 9 1.2.3 VBR Models .......................................................................................................................................... 12 1.2.4 Magnetic Saturation in EMT Machine Models ..................................................................... 15 1.2.5 Multirate Simulations for EMT Machine Models ................................................................. 18 1.3 Research Objectives and Contributions ........................................................................................ 18 CHAPTER 2: FUNDAMENTALS OF EMT SIMULATIONS AND MACHINE MODELING ...... 24 2.1 SVB Simulations...................................................................................................................................... 24 2.2 EMTP-Type Simulations ...................................................................................................................... 27 2.3 Induction Machines ............................................................................................................................... 30 2.3.1 Machine Equations ........................................................................................................................... 32 vi  2.3.2 Magnetic Saturation ........................................................................................................................ 39 2.3.3 Implementation in SVB Programs ............................................................................................. 45 2.3.4 Implementation in EMTP-Type Programs ............................................................................. 52 2.4 Synchronous Machines ........................................................................................................................ 62 2.4.1 Machine Equations ........................................................................................................................... 65 2.4.2 Magnetic Saturation ........................................................................................................................ 70 2.4.3 Implementation in SVB Programs ............................................................................................. 75 2.5 Numerical Error Assessment ............................................................................................................ 80 CHAPTER 3: EFFICIENT AND EXPLICIT SVB QD MACHINE MODELS WITH MAIN FLUX SATURATION ............................................................................................................... 84 3.1 Induction Machines ............................................................................................................................... 85 3.1.1 Implicit FC Approach ....................................................................................................................... 85 3.1.2 Explicit FC Approach ....................................................................................................................... 87 3.2 Synchronous Machines ........................................................................................................................ 91 3.2.1 Implicit FC Approach ....................................................................................................................... 91 3.2.2 Explicit FC Approach ....................................................................................................................... 92 3.3 Computer Studies .................................................................................................................................. 99 3.3.1 Induction Machines ....................................................................................................................... 100 3.3.2 Synchronous Machines ................................................................................................................ 103 3.3.3 Computational Performance .................................................................................................... 109 CHAPTER 4: CONSTANT-PARAMETER SVB VBR INDUCTION MACHINE MODEL WITH MAIN FLUX SATURATION .................................................................................... 111 vii  4.1 Variable-Parameter SVB VBR Model ............................................................................................ 112 4.2 Constant-Parameter SVB VBR Model ........................................................................................... 116 4.2.1 Derivation of Constant-Parameter Interfacing Circuit ................................................. 117 4.2.2 Numerical Approximation ......................................................................................................... 119 4.3 Error Analysis........................................................................................................................................ 121 4.3.1 Numerical Differentiation Error ............................................................................................. 122 4.3.2 Approximation Error and Machine Parameters .............................................................. 123 4.4 Computer Studies ................................................................................................................................ 125 4.4.1 Induction Motor #1 (IM1) .......................................................................................................... 127 4.4.2 Induction Motor #2 (IM2) .......................................................................................................... 130 4.4.3 Model Accuracy vs. Integration Step Size ............................................................................ 133 4.4.4 Computational Performance .................................................................................................... 134 CHAPTER 5: CONSTANT-PARAMETER SVB VBR SYNCHRONOUS MACHINE MODEL WITH MAIN FLUX SATURATION ........................................................................ 136 5.1 Variable-Parameter SVB VBR Model ............................................................................................ 137 5.1.1 Rotor State Equations .................................................................................................................. 137 5.1.2 Variable-Parameter Interfacing Circuit Equations ........................................................ 140 5.2 Constant-Parameter SVB VBR Model ........................................................................................... 143 5.2.1 Constant-Parameter Interfacing Circuit Equations ....................................................... 143 5.2.2 Numerical Approximation ......................................................................................................... 146 5.3 Pole Selection Procedure .................................................................................................................. 149 5.3.1 q-axis Filter ....................................................................................................................................... 150 viii  5.3.2 d-axis Filter ....................................................................................................................................... 152 5.4 Computer Studies ................................................................................................................................ 154 5.4.1 TF-Based Exciter Model .............................................................................................................. 155 5.4.2 Detailed Exciter Model ................................................................................................................ 160 CHAPTER 6: CONSTANT-PARAMETER EMTP-TYPE VBR INDUCTION MACHINE MODEL WITH MAIN FLUX SATURATION ........................................................................ 165 6.1 SVB VBR Model ..................................................................................................................................... 166 6.2 Variable-Parameter EMTP-Type VBR Model ............................................................................ 168 6.2.1 Model Derivation............................................................................................................................ 168 6.2.2 Simulation Steps ............................................................................................................................. 170 6.3 Constant-Parameter EMTP-Type VBR Model ........................................................................... 172 6.3.1 Constant-Parameter Interfacing Circuit Derivation ...................................................... 172 6.3.2 Simulation Steps ............................................................................................................................. 175 6.4 Computer Studies ................................................................................................................................ 177 6.4.1 Single-Machine System ................................................................................................................ 177 6.4.2 Computational Efficiency ........................................................................................................... 188 6.4.3 Multimachine System ................................................................................................................... 189 CHAPTER 7: MULTIRATE EMTP-TYPE INDUCTION MACHINE MODELS ........................ 194 7.1 Modal Analysis ...................................................................................................................................... 195 7.2 Multirate EMTP-Type VBR Machine Models ............................................................................. 201 7.2.1 Preliminaries ................................................................................................................................... 201 7.2.2 Trapezoidal-Only Model (MR-T/T) ........................................................................................ 204 ix  7.2.3 Trapezoidal/Backward Euler Model (MR-T/BE) ............................................................ 207 7.2.4 Simulation Steps ............................................................................................................................. 210 7.3 Computer Studies ................................................................................................................................ 212 7.3.1 Numerical Accuracy – Single-Machine System ................................................................. 213 7.3.2 Numerical Accuracy – Multimachine System .................................................................... 216 7.3.3 Numerical Efficiency..................................................................................................................... 221 CHAPTER 8: CONCLUSIONS AND FUTURE WORK ................................................................. 223 8.1 Contributions and Anticipated Impact ........................................................................................ 225 8.1.1 Contributions ................................................................................................................................... 225 8.1.2 Anticipated Impact........................................................................................................................ 229 8.2 Future Work ........................................................................................................................................... 231 References .......................................................................................................................................... 234 Appendix A: Parameters for the Case Study of Section 3.3.1 ............................................ 250 Appendix B: Parameters for the Case Studies of Section 3.3.2 ......................................... 251 Appendix C: Parameters for the Case Study of Section 4.4.1 ............................................ 252 Appendix D: Parameters for the Case Study of Section 4.4.2 ............................................ 253 Appendix E: Scalars and Matrices Introduced in Chapter 5 .............................................. 254 Appendix F: Parameters for the Case Study of Section 5.4.1............................................. 255 Appendix G: Parameters for the Case Study of Section 5.4.2 ............................................ 256 Appendix H: Scalars and Matrices Introduced in Chapter 6 ............................................. 257 Appendix I: Parameters for the Case Study of Section 6.4.1.1 .......................................... 259 Appendix J: Parameters for the Case Studies of Sections 6.4.1.2 and 7.3.1 ................. 260 x  Appendix K: Parameters for the Case Studies of Sections 6.4.3 and 7.3.2 ............... 261 Appendix L: Parameters for the Modal Analysis of Section 7.1 ................................... 263 Appendix M: Scalars and Matrices Introduced in Chapter 7 ........................................ 264   xi  List of Tables Table 1–1. Properties of SVB general-purpose lumped-parameter induction machine models. ................................................................................................................................................................... 19 Table 1–2. Properties of SVB general-purpose lumped-parameter synchronous machine models. ................................................................................................................................................................... 19 Table 1–3. Properties of EMTP-type general-purpose lumped-parameter induction machine models (RPD: rotor-position-dependent; SSD: saturation-segment-dependent). ................... 20 Table 3–1. 2-norm relative errors of the synchronous machine’s stator current dsi , magnetizing flux mqλ , and rotor angle δ  for the change of load torque study. ........................ 106 Table 3–2. 2-norm relative errors of the synchronous machine’s stator current dsi  and rotor angle δ  for the change of stator voltage study. .................................................................................... 108 Table 3–3. Numerical performance of the subject synchronous machine models for the change of stator voltage study with different solvers. ...................................................................... 110 Table 4–1. Coefficients lrmu LL Σ  and 1−χ  for various induction machines. ................................ 124 Table 4–2. 2-norm relative errors of stator current asi , electrical rotor speed rω , electromagnetic torque eT , and main flux mλ  for the single-phase voltage sag study (IM1). ................................................................................................................................................................................. 130 Table 4–3. 2-norm relative errors of stator current asi , electrical rotor speed rω , electromagnetic torque eT , and main flux mλ  for the single-phase voltage sag study (IM2). ................................................................................................................................................................................. 133 xii  Table 4–4. Numerical performance of the subject models for the single-phase voltage sag study. ..................................................................................................................................................................... 135 Table 5–1. 2-norm relative errors of stator currents abcsi , field current fdi , and electromagnetic torque eT  for the single-phase-to-ground fault study with a TF-based exciter model. .................................................................................................................................................... 159 Table 5–2. 2-norm relative errors of stator currents abcsi , field current fdi , and electromagnetic torque eT  for the single-phase-to-ground fault study with a TF-based exciter model (without saturation). ......................................................................................................... 160 Table 5–3. Numerical performance of the subject models for the single-phase-to-ground study with a TF-based exciter model. ...................................................................................................... 160 Table 5–4. 2-norm relative errors of stator currents abcsi , field current fdi , and electromagnetic torque eT  for the single-phase-to-ground fault study with a detailed exciter model. ................................................................................................................................................................... 164 Table 5–5. 2-norm relative errors of stator currents abcsi , field current fdi , and electromagnetic torque eT  for the single-phase-to-ground fault study with a detailed exciter model (with fdp0  = 490). ............................................................................................................................... 164 Table 5–6. Numerical performance of the subject models for the single-phase-to-ground study with a detailed exciter model. ......................................................................................................... 164 Table 6–1. CPU time per step of the subject models. ......................................................................... 189 Table 6–2. CPU time of the three-phase fault study with the subject models (1201 time steps). ................................................................................................................................................................... 192 xiii  Table 7–1. Eigenvalues of the 50-hp induction machine (left column) and of the system of Figure 7–1 with the same machine (right column). ........................................................................... 196 Table 7–2. Magnitude of the participation factors of the 50-hp induction machine. ............ 198 Table 7–3. Magnitude of the participation factors of the system of Figure 7–1 with the 50-hp induction machine. .................................................................................................................................... 199 Table 7–4. Magnitude of the algebraic participation factors of the system of Figure 7–1 with the 50-hp induction machine. ..................................................................................................................... 200 Table 8–1. Properties of SVB general-purpose lumped-parameter induction machine models. ................................................................................................................................................................. 228 Table 8–2. Properties of SVB general-purpose lumped-parameter synchronous machine models. ................................................................................................................................................................. 229 Table 8–3. Properties of EMTP-type general-purpose lumped-parameter induction machine models (RPD: rotor-position-dependent; SSD: saturation-segment-dependent). ................. 230  xiv  List of Figures Figure 2–1. State-variable-based representation of a power system in which the linear circuit equations and nonlinear equations are aggregated in different subsystems. ............. 26 Figure 2–2. Series RL circuit represented in (a) continuous time and (b) discrete time. ...... 28 Figure 2–3. Equivalent linear system of nodal equations for EMTP-type simulations. .......... 28 Figure 2–4. Equivalent circuit of an induction machine represented in physical variables. 31 Figure 2–5. Equivalent circuit of a squirrel-cage induction machine represented in qd coordinates. .......................................................................................................................................................... 38 Figure 2–6. Typical saturation curve of an induction machine obtained from the no-load test and the corresponding air-gap line. ................................................................................................... 42 Figure 2–7. Projections of the main flux mλ  and magnetizing current mi  onto the q and d axes of an induction machine. ....................................................................................................................... 43 Figure 2–8. Piecewise-linear representation of a typical saturation curve of an induction machine. ................................................................................................................................................................. 44 Figure 2–9. Block diagram of a Flux-based qd induction machine model with an arbitrary saturation function for implementation in SVB programs. ............................................................... 47 Figure 2–10. Interfacing circuit of Flux-based and Mixed/Current-based qd models for SVB programs. .............................................................................................................................................................. 52 Figure 2–11. Equivalent circuit of a wound-rotor synchronous machine represented in physical variables. .............................................................................................................................................. 63 Figure 2–12. Equivalent circuit of a wound-rotor synchronous machine represented in qd coordinates. .......................................................................................................................................................... 69 xv  Figure 2–13. Typical open-circuit characteristic (OCC, or saturation curve) of a synchronous machine and the corresponding air-gap line. .............................................................. 71 Figure 2–14. Projections of the main flux mλ  and magnetizing current mi  onto the q and d axes of an equivalent isotropic synchronous machine. ....................................................................... 73 Figure 2–15. General behavior of the numerical error of different machine models as a function of the step size (the x axis is presented on a logarithmic scale). ................................... 81 Figure 2–16. Example of the trajectory of an arbitrary variable i  as predicted by a subject model along with a reference curve. The vertical lines indicate the numerical error at each solution point....................................................................................................................................................... 82 Figure 3–1. Main flux saturation characteristic )(1 mF λ−  of an induction machine and its corresponding correction function )( mf λ ................................................................................................. 86 Figure 3–2. Block diagram of the Explicit FC function for induction machines with one rotor circuit ( N  = 1). .................................................................................................................................................... 89 Figure 3–3. (a) Saturation characteristic )(1 miF −  and equivalent explicit function )( ziG  for a 50-hp induction machine; and (b) magnified view of )( ziG  in the saturation region. ............. 90 Figure 3–4. Block diagram of the Explicit FC function with a 2-D look-up table for synchronous machines with 2 q-axis damper windings and 1 d-axis damper winding......... 95 Figure 3–5. Block diagram of the Approximate FC function for synchronous machines with 2 q-axis damper windings and 1 d-axis damper winding. ................................................................. 97 Figure 3–6. Block diagram of the Explicit FC function with 1-D look-up tables for synchronous machines with 2 q-axis damper windings and 1 d-axis damper winding......... 98 xvi  Figure 3–7. Transient in stator current qsi  due to a change in stator voltage as predicted by the subject induction machine models. ................................................................................................... 101 Figure 3–8. Magnified view of the first peak of stator current qsi  due to a change in stator voltage as predicted by the subject induction machine models. ................................................... 101 Figure 3–9. Transient in main flux mλ  due to a change in stator voltage as predicted by the subject induction machine models. ........................................................................................................... 102 Figure 3–10. Transient in electrical rotor speed rω  due to a change in stator voltage as predicted by the subject induction machine models. ........................................................................ 102 Figure 3–11. Transient in stator current dsi  due to a change of load torque as predicted by the subject synchronous machine models. ............................................................................................ 104 Figure 3–12. Transient in magnetizing flux mqλ  due to a change of load torque as predicted by the subject synchronous machine models. ...................................................................................... 104 Figure 3–13. Transient in rotor angle δ  due to a change of load torque as predicted by the subject synchronous machine models. .................................................................................................... 105 Figure 3–14. Magnified view of the first peak of rotor angle δ  due to a change of load torque as predicted by the subject synchronous machine models............................................... 105 Figure 3–15. Transient in main flux mλ  due to a change of load torque as predicted by the subject synchronous machine models. .................................................................................................... 106 Figure 3–16. Transient in stator current dsi  due to a change of stator voltage as predicted by the subject synchronous machine models. ...................................................................................... 107 xvii  Figure 3–17. Magnified view of the first peak of stator current dsi  due to a change of stator voltage as predicted by the subject synchronous machine models. ............................................ 108 Figure 3–18. Transient in rotor angle δ  due to a change of stator voltage as predicted by the subject synchronous machine models. ............................................................................................ 108 Figure 4–1. Variable-parameter VBR (VP-VBR) formulation of the saturable lumped-parameter induction machine model. ...................................................................................................... 116 Figure 4–2. Constant-parameter VBR (CP-VBR) formulation of the saturable lumped-parameter induction machine model. ...................................................................................................... 121 Figure 4–3. One-line diagram of a test system comprised of an induction motor (IM), a cable, a transformer, and a network Thévenin equivalent. ............................................................. 125 Figure 4–4. Transients in stator current asi , electrical rotor speed rω , electromagnetic torque eT , and main flux mλ  during a single-phase voltage sag as predicted by the subject models (IM1). .................................................................................................................................................... 128 Figure 4–5. Magnified view of the first peak of stator current asi  during a single-phase voltage sag as predicted by the subject models (IM1). ..................................................................... 129 Figure 4–6. Saturation curve of machine IM1 and its operating range (denoted by the thick red line) during a single-phase voltage sag. .......................................................................................... 129 Figure 4–7. Transients in stator current asi , electrical rotor speed rω , electromagnetic torque eT , and main flux mλ  during a single-phase voltage sag as predicted by the subject models (IM2). .................................................................................................................................................... 131 Figure 4–8. Magnified view of the first peak of stator current asi  during a single-phase voltage sag as predicted by the subject models (IM2). ..................................................................... 132 xviii  Figure 4–9. Saturation curve of machine IM2 and its operating range (denoted by the thick red line) during the single-phase voltage sag. ...................................................................................... 132 Figure 4–10. 2-norm relative error of stator current asi  as function of constant step sizes for the single-phase voltage sag study as predicted by the (a) CP-VBR-BDF1 and (b) CP-VBR-BDF2 models. ..................................................................................................................................................... 134 Figure 5–1. Variable-parameter VBR (VP-VBR) formulation of the saturable lumped-parameter synchronous machine model. ............................................................................................... 144 Figure 5–2. Constant-parameter VBR (CP-VBR) formulation of the saturable lumped-parameter synchronous machine model. ............................................................................................... 148 Figure 5–3. Pole selection procedure for: (a) the q-axis filter’s qsp0 ; and (b) the d-axis filters’ dsp0  and fdp0 . ..................................................................................................................................................... 151 Figure 5–4. q-axis operational inductance magnitude of the original ( qoL ) and constant-parameter ( qcL ) models. .............................................................................................................................. 152 Figure 5–5. One-line diagram of the test system with the TF-based exciter model............... 155 Figure 5–6. Block diagram of the IEEE AC1A excitation system [95]. ......................................... 156 Figure 5–7. Reference trajectories of stator currents asi , qsi , and dsi , field current fdi , field voltage xfde , and electromagnetic torque eT  during a single-phase-to-ground fault with a TF-based exciter model. ................................................................................................................................ 157 Figure 5–8. Magnified view of the first peak of stator current asi  during a single-phase-to-ground fault as predicted by the subject models with a TF-based exciter model. ................. 158 xix  Figure 5–9. Magnified view of the field current fdi  during a single-phase-to-ground fault as predicted by the subject models with a TF-based exciter model. ................................................. 158 Figure 5–10. One-line diagram of the test system with a circuit-based exciter model. ....... 161 Figure 5–11. Reference trajectories of field current fdi , re-scaled field voltage xfde , and ac exciter currents abcexi  during a single-phase-to-ground fault with a detailed exciter model. ................................................................................................................................................................................. 162 Figure 5–12. Magnified view of the field current fdi  during a single-phase-to-ground fault as predicted by the subject models with a detailed exciter model. ................................................... 163 Figure 5–13. Magnified view of the exciter current bexi  during a single-phase-to-ground fault as predicted by the subject models with a detailed exciter model. ................................... 163 Figure 6–1. One-line diagram of a small test system comprised of an induction motor (IM), a shunt capacitor, a cable, a transformer, and a network Thévenin equivalent. ..................... 177 Figure 6–2. Reference trajectories of stator currents asi , bsi , and csi , electromagnetic torque eT , and main flux mλ  during a single-phase-to-ground fault (IM1). ............................................. 179 Figure 6–3. Magnified view of stator current csi  at the beginning of a single-phase-to-ground fault as predicted by the subject models: (a) with step size t∆  of 100 µs; and (b) with step size t∆  of 500 µs (IM1). .............................................................................................................. 181 Figure 6–4. Magnified view of stator current csi  before the removal of a single-phase-to-ground fault as predicted by the subject models: (a) with step size t∆  of 100 µs; and (b) with step size t∆  of 500 µs (IM1). .............................................................................................................. 182 xx  Figure 6–5. (a) Average 2-norm relative error of stator phase currents si ; and (b) 2-norm relative error of electromagnetic torque eT  as a function of step size t∆  for the single-phase-to-ground fault study as predicted by the subject models (IM1). ................................... 183 Figure 6–6. Reference trajectories of stator currents asi , bsi , and csi , electromagnetic torque eT , and main flux mλ  during a single-phase-to-ground fault (IM2). ............................................. 185 Figure 6–7. Magnified view of stator current csi  at the beginning of a single-phase-to-ground fault as predicted by the subject models: (a) with step size t∆  of 100 µs; and (b) with step size t∆  of 500 µs (IM2). .............................................................................................................. 186 Figure 6–8. Magnified view of stator current csi  before the removal of a single-phase-to-ground fault as predicted by the subject models: (a) with step size t∆  of 100 µs; and (b) with step size t∆  of 500 µs (IM2). .............................................................................................................. 187 Figure 6–9. (a) Average 2-norm relative error of stator phase currents si ; and (b) 2-norm relative error of electromagnetic torque eT  as a function of step size t∆  for the single-phase-to-ground fault study as predicted by the subject models (IM2). ................................... 188 Figure 6–10. One-line diagram of an industrial multimachine power system (Section 6.4.3). ................................................................................................................................................................................. 190 Figure 6–11. Transient in the fault current contributed by the industrial network ainet  during a three-phase fault as predicted by the subject models. .................................................... 191 Figure 6–12. Magnified view of the fault current contributed by the industrial network ainet  during a three-phase fault as predicted by the subject models. .................................................... 191 xxi  Figure 6–13. Average 2-norm relative error of the fault phase current contributed by the industrial network neti  as a function of step size t∆  for the three-phase fault study as predicted by the subject models. ............................................................................................................... 192 Figure 7–1. One-line diagram of a small system comprised of an induction motor (IM) connected to a shunt capacitor and a network Thévenin equivalent. ......................................... 196 Figure 7–2. Grouping of a VBR induction machine model and an external power system into fast and slow subsystems. ............................................................................................................................ 201 Figure 7–3. Illustration of major and minor time steps (with rM  = 4). ...................................... 202 Figure 7–4. (a) Standard trapezoidal rule applied at major time steps only; and (b) expanded trapezoidal rule applied at minor and major time steps (with rM  = 4). ............... 203 Figure 7–5. Reference trajectories of stator currents asi , bsi , and csi , and electromagnetic torque eT  during a single-phase-to-ground fault (single-machine system). ............................ 214 Figure 7–6. Magnified view of the first peak of stator current csi  during a single-phase-to-ground fault as predicted by the MR-T/T and MR-T/BE models (single-machine system). ................................................................................................................................................................................. 215 Figure 7–7. Average 2-norm relative error of the stator phase currents as a function of rM  for the single-phase-to-ground fault study as predicted by the MR-T/T and MR-T/BE models (single-machine system). .............................................................................................................. 216 Figure 7–8. Electrical rotor speed rω  of the 500-hp motor for the start-up study as predicted by the MR-T/T and MR-T/BE models: (a) full trajectory; and (b) magnified view of the first overshoot (single-machine system). .................................................................................. 217 xxii  Figure 7–9. One-line diagram of an industrial multimachine power system (Section 7.3.2). ................................................................................................................................................................................. 218 Figure 7–10. Reference trajectories of the fault currents contributed by the industrial network ainet , binet , and cinet  during a three-phase fault. ................................................................ 219 Figure 7–11. Reference trajectories of (a) the electromagnetic torque eT , and (b) the electrical rotor speed rω  of one of the induction motors connected to bus 17 of the industrial power system during a three-phase fault. ......................................................................... 220 Figure 7–12. Average 2-norm relative error of the fault phase currents contributed by the industrial network ainet , binet , and cinet  for the three-phase fault study as predicted by the MR-T/T and MR-T/BE models. ........................................................................................................................... 220 Figure 7–13. Normalized CPU time of the MR-T/T and MR-T/BE models as a function of rM  for machines with (a) one rotor circuit and (b) two rotor circuits. ............................................. 222  xxiii  List of Abbreviations and Acronyms ac Alternating current BDF Backward differentiation formula BDF1 First-order BDF BDF2 Second-order BDF BE Backward Euler CPU Central processing unit d axis Direct axis DAE Differential-algebraic equation dc Direct current DER Distributed energy resource EMT Electromagnetic transients EMTP Electromagnetic transients program EMTP-type Nodal-analysis-based programs based on the original EMTP FACTS Flexible alternating current transmission systems FC Flux correction (function) FEA Finite element analysis HVDC High-voltage direct current IM Induction motor LTE Local truncation error LTI Linear time-invariant MANA Modified augmented nodal analysis MEC Magnetic equivalent circuit MNA Modified nodal analysis n-D n-dimensional (e.g., 2-dimensional) OCC Open-circuit characteristic ODE Ordinary differential equation xxiv  PD Phase-domain (model) q axis Quadrature axis RL Resistor-inductor (circuit) rms Root mean square SG Synchronous generator SPD Symmetric positive definite SPS SimPowerSystems SSFR Standstill frequency response SSR Subsynchronous resonance SVB State-variable-based (programs) TF Transfer function TS Transient stability (programs) UM Universal machine (model) VBR Voltage-behind-reactance (model) xxv  Nomenclature  In this thesis, scalars are written using italic fonts (e.g., asi ), vectors are denoted by bold lowercase characters (e.g., abcsi ), and matrices by bold uppercase characters (e.g., G ). Stator variables are often written using the following vector notation: [ ]Tcsbsasabcs fff=f  [ ]Tsdsqssqd fff 00 =f  [ ]Tdsqsqds ff=f  [ ]Tqsdsdqs 0λλ −=λ  where { }λ,,vif =  (these variables are defined below). Magnetizing and residual fluxes are also frequently represented using the following vectors: [ ]Tmdmqmqd λλ=λ  [ ]Tdqqd resresres λλ=λ . Induction machine rotor variables are expressed as [ ]TcrNbrNarNcrbrarcrbrarabcr fffffffff 222111=f  [ ]TdrNdrdrqrNqrqrqdr ffffff  2121=f  where { }λ,if = . The rotor variables of synchronous machines are defined as [ ]TfdkdNkdkdkqMkqkqqdr fffffff  2121=f  xxvi  where { }λ,,vif = . All rotor variables are referred to the stator side using the proper turns-ratio. All other vectors and matrices are defined explicitly within the dissertation.   The size of the vectors and matrices is usually unambiguous. When it is not the case, the number of rows and columns is appended to the matrix as follows: [ ] baו=A  where a  and b  are the number of rows and columns of the matrix, respectively.   Diagonal or block diagonal matrices of the form =NX000X000XA21 are often condensed as [ ]NXXXA 21diag=  where 1X , 2X , and NX  may be scalars, vectors, or matrices.  Space vectors are used in parts of Chapters 2 and 3. They are denoted by underlined variables, e.g., qd fff ι+=  where df  and qf  are the space vector’s projections on the direct and quadrature axes, respectively.  Only the basic variables are aggregated in this section. All other variables are defined throughout the thesis. The variables compiled in this nomenclature section are separated in three categories: i) those that pertain (directly or indirectly) to both induction xxvii  and synchronous machines; ii) those that refer only to induction machines; and iii) those that refer only to synchronous machines.  i) General variables  0  Null matrix 1  Matrix full of 1s bf  Base frequency (in Hz) G  Network’s conductance matrix (EMTP-type) H  Combined machine-load moment of inertia (in s) si0  Zero-sequence stator current  asi  Stator current on phase a  bsi  Stator current on phase b csi  Stator current on phase c dsi  d-axis stator current hi  Current injection vector (EMTP-type) mi  magnetizing current mdi  d-axis magnetizing current mqi  q-axis magnetizing current ngi  Neutral-to-ground current qsi  q-axis stator current I  Identity matrix J  Combined machine-load moment of inertia (in kg∙m2) sK  Park’s transformation matrix for stator variables 1−sK  Inverse Park’s transformation matrix for stator variables lsL  Stator leakage inductance rM  Ratio of st∆  over ft∆  (multirate simulations) )(⋅O  Big O notation xxviii  p  Heaviside’s operator ( dtd / ) P  Number of poles BP  Base active power sr  Stator resistance BS  Base apparent power t  Time BT  Base torque eT  Electromagnetic torque mT  Mechanical torque sv0  Zero-sequence stator voltage asv  Stator voltage on phase a bsv  Stator voltage on phase b csv  Stator voltage on phase c dsv  d-axis stator voltage nv  Unknown nodal voltages (EMTP-type) ngv  Neutral-to-ground voltage qsv  q-axis stator voltage BV  Base voltage (rms line-to-line) BHV  Transformer high-side base voltage (rms line-to-line) BLV  Transformer low-side base voltage (rms line-to-line) HV  Transformer high-side nominal voltage (rms line-to-line) LV  Transformer low-side nominal voltage (rms line-to-line) 0Z  Zero-sequence impedance 1Z  Positive-sequence impedance t∆  Simulation step size (unique step size) ft∆  Simulation step size of the fast subsystem (multirate simulations) st∆  Simulation step size of the slow subsystem (multirate simulations) xxix  )(xε  2-norm relative error of x  (which might be a single- or three-phase  variable) eθ  Angle of the terminal voltages rθ  Electrical rotor position  ι  Imaginary number s0λ  Zero-sequence stator flux linkage asλ  Stator flux linkage on phase a bsλ  Stator flux linkage on phase b csλ  Stator flux linkage on phase c dsλ  d-axis stator flux linkage mλ  Main flux  mdλ  d-axis magnetizing flux mqλ  q-axis magnetizing flux qsλ  q-axis stator flux linkage φ  Flux vector angle bω  Base frequency (in rad/s) eω  Operating frequency (in rad/s) rω  Electrical rotor speed (in rad/s)  ii) Induction machines-only variables  .,...,1, Nziarz =   Current on phase a of the z th rotor circuit .,...,1, Nzibrz =  Current on phase b of the z th rotor circuit .,...,1, Nzicrz =  Current on phase c of the z th rotor circuit .,...,1, Nzidrz =  d-axis current of the z th rotor circuit .,...,1, Nziqrz =  q-axis current of the z th rotor circuit xxx  j  Operating segment of the piecewise-linear main flux saturation  function rK  Park’s transformation matrix for rotor variables (neglecting zero- sequence) invrK  Inverse Park’s transformation matrix for rotor variables (neglecting  zero-sequence) DjL  Dynamic inductance of the j th piecewise-linear main flux saturation  segment  .,...,1, NzLlrz =  Leakage inductance of the z th rotor circuit mL  Magnetizing inductance (general) msL  Stator magnetizing inductance (general) muL  Magnetizing inductance (unsaturated) ( )mmL λ  Magnetizing inductance (saturated) N  Number of three-phase rotor circuits .,...,1, Nzrrz =   Resistance of the z th rotor circuit α   Damping coefficient of the damped trapezoidal rule θ  Angular displacement of the arbitrary reference frame .,...,1, Nzarz =λ   Flux linkage on phase a of the z th rotor circuit .,...,1, Nzbrz =λ  Flux linkage on phase b of the z th rotor circuit .,...,1, Nzcrz =λ  Flux linkage on phase c of the z th rotor circuit .,...,1, Nzdrz =λ  d-axis flux linkage of the z th rotor circuit .,...,1, Nzqrz =λ  q-axis flux linkage of the z th rotor circuit dresλ  d-axis residual flux jresλ  Residual flux of the j th piecewise-linear main flux saturation  segment qresλ  q-axis residual flux ω  Angular velocity of the arbitrary reference frame (in rad/s) xxxi   iii) Synchronous machines-only variables  xfde   Re-scaled field winding voltage abcexi   Exciter three-phase ac currents aexi   Exciter ac current on phase a bexi   Exciter ac current on phase b cexi   Exciter ac current on phase c fdi   Field winding current .,...,1, Nzikdz =   Current of the z th d-axis damper winding .,...,1, Mzikqz =   Current of the z th q-axis damper winding lfdL  Field winding leakage inductance .,...,1, NzLlkdz =  Leakage inductance of the z th d-axis damper winding .,...,1, MzLlkqz =  Leakage inductance of the z th q-axis damper winding mdL  d-axis magnetizing inductance (general) mduL  d-axis magnetizing inductance (unsaturated) ( )mmdL λ  d-axis magnetizing inductance (saturated) mqL  q-axis magnetizing inductance (general) mquL  q-axis magnetizing inductance (unsaturated) ( )mmqL λ  q-axis magnetizing inductance (saturated) M  Number of q-axis damper windings N  Number of d-axis damper windings fdr  Field winding resistance .,...,1, Nzrkdz =   Resistance of the z th d-axis damper winding .,...,1, Mzrkqz =   Resistance of the z th q-axis damper winding FS   Saliency factor fdv   Field winding voltage .,...,1, Nzvkdz =   Voltage of the z th d-axis damper winding (= 0) xxxii  .,...,1, Mzvkqz =   Voltage of the z th q-axis damper winding (= 0) δ   Rotor angle fdλ   Field winding flux linkage .,...,1, Nzkdz =λ   Flux linkage of the z th d-axis damper winding .,...,1, Mzkqz =λ   Flux linkage of the z th q-axis damper winding xxxiii  Acknowledgments  I owe a debt of gratitude to my research supervisor, Dr. Juri Jatskevich. His vision, knowledge, and constructive comments were essential to the realization of this dissertation. I would also like to thank Dr. Jatskevich for encouraging me to pursue a doctoral degree. Above all, it is his passion for research and dedication to his students that stood out the most in the last four years.  I also want to express my appreciation to Drs. Liwei Wang and Mehrdad Chapariha. This dissertation is a continuation of their excellent research. They also provided me with valuable guidance and feedback during my time at UBC. In particular, I would like to thank Mehrdad for our numerous discussions, for providing me with a very handy dissertation template, and for listening to my countless rambles and complaints. I would also like to thank Drs. Hamid Atighechi and Oleg Wasynczuk for their comments on some of the papers written during my graduate studies.  I am grateful to Drs. Hermann Dommel, Christine Chen, José Martí, William Dunford, and Wayne Nagata for generating interesting discussions during my final examinations, and for providing me with feedback about this thesis. I am particularly indebted to my external examiner, Dr. Brian K. Johnson, for very diligently reading this lengthy thesis and pointing out several mistakes and sentences lacking clarity.   I would also like to thank my friends from the Power Lab for making my time here more enjoyable: Alexey Beskaravayny, Arash Alimardani, Arash Tavighi, Hamed Ahmadi, Hamid Atighechi, Mehrdad Chapariha, Soroush Amini, and Yingwei Huang. This list is obviously far from exhaustive.   I was very fortunate to receive excellent funding from different organizations during my graduate studies. I would like to thank the Fonds de Recherche du Québec – Nature et xxxiv  Technologies (FRQNT) for awarding me masters and doctoral scholarships; the Institute for Computing, Information and Cognitive Systems (ICICS) for their Graduate Scholarship; the Natural Sciences and Engineering Research Council (NSERC) for awarding me Canada Graduate Scholarships (CGS) at the masters and doctoral level; and UBC for the Four Year Doctoral Fellowship (4YF) and Graduate Student Initiative (GSI) stip-ends/scholarships. I would also like to acknowledge ICICS and UBC for the financial support provided to present papers at conferences. Finally, I would like to thank Dr. Jatskevich for providing additional funding through his NSERC Discovery Grant entitled “Next generation electromagnetic transient simulation tools”, as well as from other NSERC grants.  Last but not least, I am indebted to my family for supporting me from the other side of the continent. I am particularly thankful to my sister Amélie for pushing me to be better, and to my parents Sylvane and Alain for raising me in a stable and unconditionally loving environment, and for being role models of dedication, hard work, honesty, and curiosity.       1  CHAPTER 1:  INTRODUCTION 1.1     Motivation  Electrical power systems are evolving quickly. Due to a combination of environmental concerns and economical factors, renewable intermittent energy sources are replacing older fossil-fuel-based and nuclear power plants at a rapid pace. For instance, the electric energy produced by solar panels in the United States increased by an astonishing 110% from September 2013 to September 2014 [1]; 63% of the new electric capacity installed in the European Union in 2013 came from wind turbines and solar panels [2]; and as of 2013, wind and solar sources accounted for 22% of the total electric capacity of the European Union [2]. Unlike more traditional power plants, these distributed energy resources (DERs) are frequently interfaced with electric grids using power electronic converters. The number and type of loads supplied by the onboard power systems of aircraft, ships, and automobiles is also expanding [3]. These are but a few examples of the changes occurring in today’s bulk and isolated power systems, which open the door for new and potentially harmful interactions between components. Thousands of engineers and researchers around the world develop computer models and run various simulations to ensure that power systems are being properly designed and operated in a stable, secure, and optimal way. Numerically accurate and efficient computer simulations are therefore more necessary than ever.  Several types of simulation tools can be used depending on the study’s objectives [4]. This dissertation focuses on the so-called electromagnetic transients (EMT) programs [5]–[15]. EMT programs offer high flexibility and superior modeling accuracy [16], and as 2  such are typically the preferred time-domain tools to investigate complex, higher frequency phenomena. Some of the studies commonly executed with EMT tools include insulation coordination, control tuning, overvoltage calculations, and detailed short-circuit analyses [16]. These studies are often repeated numerous times with different sets of parameters [17], [18]. Owing to their high modeling fidelity, EMT programs are also able to very accurately simulate slower electromechanical transients (e.g., rotor angle transient stability [19]). However, due to the high computational cost typically associated with EMT programs, these slower phenomena are almost exclusively simulated using numerically faster – but less accurate – transient stability (TS) programs [20], [21]. It is known that TS programs can have difficulty accurately representing high-voltage direct current (HVDC) applications and flexible alternating current transmission systems (FACTS) [22].  Industry-grade EMT programs are tailored for either offline or real-time analyses. They are typically divided in two categories: nodal-analysis-based tools derived from the original electromagnetic transients program EMTP [5] (hereinafter referred to as EMTP-type programs) [6]–[11], and state-variable-based (SVB) [12]–[15] programs. EMTP-RV [6], PSCAD [7], ATP [8], and MicroTran [9] are some of the most widely used offline EMTP-type programs; RTDS [10] and HyperSim [11] are commercial examples of real-time EMTP-type simulators. Offline SVB simulators include the MATLAB/Simulink [23] toolboxes PLECS [12], SimPowerSystems [13], and ASMG [14]; real-time SVB simulators are also commercialized by OPAL-RT [15].  EMTP-type programs discretize the circuit equations at the branch or component level (typically using the trapezoidal integration rule), and combine the difference equations to create a system of nodal equations [5]; SVB tools typically create sets of ordinary differential equations (ODEs) in state-space form (differential-algebraic equations (DAEs) are also sometimes used) [24], [25]. A major advantage of the SVB formulation is that its system matrices are independent of the simulation step size. As a result, efficient 3  variable step-size solvers can be used without having to continuously execute the numerically costly state matrix generation algorithm [26]. On the other hand, the conductance matrix of EMTP-type programs is a function of the step size: it must therefore be regenerated and re-factored whenever the step size changes. While matrix re-factorizations can be greatly accelerated using proper sparsity-oriented methods [27] or in some cases partial re-factorization schemes [28], they nevertheless account for a large part of the computational cost of EMTP-type programs. Consequently, to achieve numerically efficient solutions (and for other practical reasons, see [16]), the leading EMTP-type programs [6]–[9] assume a fixed step size during simulations. As for the advantages of EMTP-type programs, perhaps the most important one is its high numerical efficiency for large power systems (when compared to SVB programs) [26]. Moreover, the procedure to construct EMTP-type conductance matrices is simpler and faster than the SVB state matrix generation algorithms [24], [25].  Using a fixed step size is necessary but not sufficient to achieve a constant conductance matrix in EMTP-type programs; another necessary condition is that all equivalent branches forming the conductance matrix be time-independent (such branches are herein referred to as constant-parameter elements). This is the case for resistances, inductances, and capacitances that are already constant in the continuous domain. However, saturable branches and position-dependent coupled inductances (e.g., as in many rotating machine models [29]) typically introduce time-varying elements in the conductance matrix. Re-factoring the conductance matrix at every time step has been reported to increase simulation time by a factor of 2 to 4 in practical power systems [30], [31]. The presence of time-varying branches also requires updating the system matrices in SVB programs [32]. The overhead associated with the presence of variable-parameter elements in SVB programs fluctuates greatly with the case study and the updating scheme used. For instance, with the state-of-the-art toolbox PLECS, the replacement of time-varying elements with constant-parameter branches is shown in [33] to reduce simulation 4  time by an order of magnitude for a small machine-converter system. While the gains in simulation speed depend on a plethora of factors, considerably faster simulations can be achieved when the conductance matrix (EMTP-type) and system matrices (SVB) remain constant at every time step.  The step size also has a considerable impact on power system simulations. Increasing the step size decreases the CPU time; however, due to the nature of numerical integration/discretization, it also introduces more numerical error, perhaps even resulting in numerical instability [34]. The maximum permissible step size depends on several factors such as the integration rule (order, implicit versus explicit, etc.), the modes of the system, the disturbance, and the degree of numerical error deemed acceptable [34]. The interface of external components (e.g., rotating machines and power electronics converters) with the power network is of particular importance since it can sometimes further constrain the maximum step size that can be used [35].  In most transient simulation programs, the same step size is used for the whole power system, i.e., a single-rate simulation scheme is employed. However, it is very common for power system models to exhibit different time scales [36]. Consequently, parts of the network are simulated using prohibitively small step sizes, needlessly increasing the CPU time. Another possible way to improve numerical efficiency is to use a multirate framework, in which different sections of the system are simulated with step sizes better related to their local modes [36]–[39]. Despite the promising results presented in [38] and [39], state-of-the-art offline EMTP-type programs such as EMTP-RV and PSCAD have yet to embrace multirate techniques.  Synchronous machines are the biggest source of electrical energy in bulk power systems [19]. They are also used to provide reactive power support, either as dedicated synchronous condensers or in addition to their generating duties, and to drive some industrial loads [19]. Moreover, synchronous generators are often the main source of 5  energy in islanded power systems such as those of remote communities, aircraft, ships, and automobiles. Induction machines are most widely used to convert electrical energy into mechanical energy for a variety of industrial, commercial, and residential applications. In fact, it is mentioned in [19] that on average 60 to 70% of the electrical energy in a typical bulk power system is consumed by motors, in particular induction motors. Induction generators are also prevalent in wind farms [40]. Consequently, induction and synchronous machines are present in practically every power system. In this dissertation, due to their dominant presence in today’s power systems, only the so-called three-phase wound-rotor synchronous machine (i.e., with a field winding) and the three-phase squirrel-cage induction machine are considered. As explained in more detail in Section 1.2, state-of-the-art rotating machine models can have a significant impact on the numerical performance of SVB and EMTP-type simulations.  In rotating electrical machines such as induction and synchronous machines, electrical energy is converted to mechanical energy (and vice versa) through a magnetically coupled field [29]. At low flux levels, it can be rightfully assumed that the relationship between magnetizing flux (flux density) and current (magnetic field) is almost linear. However, practical rotating machines typically exhibit minor-to-moderate saturation near their nominal operating point [19], [29], [41], rendering the flux-current relationships nonlinear. The level of saturation becomes more pronounced when the machine currents and/or voltages are increased, which frequently occurs as a result of system-level disturbances. Moreover, magnetic saturation is essential to the operation of self-excited induction generators [42]. Consequently, representation of machine magnetic saturation can significantly increase the modeling accuracy of EMT simulations. 6  1.2     Literature Review Depending on the degree of detail desired, different types of machine models can be used in transient simulations. One way to achieve very precise results is to represent machines using finite element analysis (FEA) [43]. FEA is computationally prohibitive and requires a large amount of machine parameters that are rarely available to system analysts, making it impractical for system-level studies. Another possibility is to use magnetic equivalent circuits (MECs) [44], [45]. Machine models based on MECs also offer high accuracy while being significantly faster and requiring fewer parameters than FEA-based models. Nevertheless, the machine parameters are rarely known by system analysts and the computational cost of the machine models may become prohibitive for larger scale multimachine system studies. The most widely used machine models for EMT simulations are the so-called general-purpose lumped-parameter models [29]. Most – if not all – built-in machine models in industry-grade EMT programs [6]–[15] fall into this category. General-purpose lumped-parameter machine models are represented by a few stator and rotor windings comprised of equivalent resistances and inductances [29]. The resulting number of differential equations is consequently much smaller than for FEA- and MEC-based models, resulting in better numerical efficiency. Moreover, the machine parameters can be obtained straightforwardly using tried and tested experimental approaches. Induction machine parameters are usually determined either using a combination of the simple dc, no-load, and locked-rotor tests [46], or from widely available manufacturers data (rated slip, rated current, breakdown torque, etc.) [47], [48]. Synchronous machine parameters can be obtained using several approaches [49], including the standard short-circuit test [29], standstill frequency response (SSFR) testing [29], and load rejection tests [50]. Since similar models are used in transient stability programs [19], these parameters are well known by power system analysts. 7  The relatively low-order general-purpose lumped-parameter models are based on several common simplifying assumptions [29], which results in a loss of modeling accuracy compared to higher fidelity FEA- and MEC-based models. However, general-purpose lumped-parameter models have been extensively compared with experimental data over the course of several decades and have been shown to yield sufficient accuracy for most system-level studies, e.g., [24], [41], [46], [48], [51]. Several variations of the general-purpose lumped-parameter machine models have been presented in the literature and implemented in commercial EMT packages. Despite their algebraic equivalence under the same modeling assumptions [52], these models can possess vastly different numerical properties [35]. In particular, several approaches have been developed to interface these different machine models with power networks [35]. The specific lumped-parameter machine model used in a given simulation can significantly affect the numerical accuracy, efficiency, and stability of the entire solution.  Three main types of algebraically equivalent general-purpose lumped-parameter models are usually considered: coupled-circuit phase-domain (PD), qd, and voltage-behind-reactance (VBR) models [35]. Their properties will be reviewed in Sections 1.2.1 to 1.2.3, respectively; incorporation of saturation in these models will be discussed in Section 1.2.4; and a few words will be said in Section 1.2.5 on the use of multirate techniques for the simulation of general-purpose lumped-parameter machine models. 1.2.1 PD Models  General-purpose lumped-parameter machine models were originally derived using physical variables (i.e., in abc coordinates) [29], [53]; such models are hereinafter referred to as PD models. PD models are physically intuitive and use the same set of coordinates as external power systems. However, the mutual inductances between the rotor and the stator (and even the self and mutual stator inductances of salient-pole synchronous 8  machines) vary sinusoidally as a function of the rotor position [29]. This complicates the analysis of machine dynamics and can significantly increase CPU time due to the presence of time-varying branches. For these reasons, PD models were rapidly replaced by so-called qd models [see Section 1.2.2], in which the self and mutual inductances are rotor-position-independent.  However, as explained in more detail in Section 1.2.2, indirect interfacing approaches are required to solve qd models along with external power networks represented in abc coordinates [35]. Such indirect approaches can be cumbersome and considerably deteriorate the numerical accuracy, efficiency, and stability of the system’s solution [35]. Unlike qd models, PD models can be interfaced directly with the rest of the power system, achieving a simultaneous machine-network solution [35]. Consequently, the numerical accuracy and stability of PD models are higher than those of qd models [35].   The 1990s saw a greater concern about numerical instability in EMTP-type simulations [31], [54]–[57]. At the same time, researchers started developing SVB programs with automated state matrix generation algorithms [14], [24], [25]. This prompted some EMT specialists to advocate replacing qd models by PD models.   Some examples of circuit-based PD models for SVB packages are presented in [24], [58], [59]. While numerically accurate, these models introduce time-varying inductances in the system matrices. In addition to slowing down simulations, variable inductances are not available in every commercial program [13], limiting the range of application of SVB PD models.   EMTP-type PD models can be separated in two groups: in the first category, the discretized time-varying stator and rotor windings are inserted directly into the conductance matrix [56], [60]; in the second, a three-phase Thévenin (or Norton) equivalent of the machine seen from its stator terminals is inserted into the network’s 9  conductance matrix [31], [57], [59], [61]–[64]. The main advantage of the first approach is that the terminal voltages of all the windings are solved simultaneously with the rest of the network, providing direct rotor- and stator-network interfaces. This is beneficial when the rotor voltages are unknown, e.g., with wound-rotor induction machines or synchronous machines whose field winding is connected to an equivalent circuit. An advantage of the second category is that the size of the conductance matrix is reduced, decreasing the computational cost of the solution of the nodal equation. The most important advantage of the second approach from a numerical point of view is that for symmetrical induction machines, the 3-by-3 interfacing matrix is rotor-position-independent [61], allowing for a numerically efficient solution of the full system. For synchronous machines, the stator interfacing matrix will typically be rotor-position-dependent due to the so-called discrete dynamic saliency [62]; however, numerically accurate constant-parameter models can be achieved using stator current prediction [60], [64], or by introducing a higher frequency damper winding [62]. The Type-58 model in ATP is based on the model proposed in [31]. A hybrid of the first and second category was also presented in [65], in which a four-phase Thévenin equivalent of the machine seen from the stator and field windings is derived. This model has a rotor-position-dependent interfacing circuit [65]. 1.2.2 qd Models  As proposed by Park in his seminal 1929 paper, an algebraically equivalent synchronous machine model with rotor-position-independent inductances can be achieved by transforming the original stator windings of the PD models to decoupled windings rotating along with the rotor [29], [66]. Since these fictitious orthogonal windings are located on the so-called quadrature and direct axes [29], these models are commonly referred to as qd models (the terms dq or dq0 models are also sometimes used in the literature). Researchers later proposed rotor-position-independent qd induction machine models using a reference frame either fixed to the stator [67], rotating with the magnetic 10  field [68], or rotating with the rotor [69]. Finally, a rotor-position-independent qd induction machine model rotating with an arbitrary reference frame was proposed in [70]. The qd models are highly efficient since their q- and d-axis windings are decoupled; their voltages, currents, and fluxes can be constant in steady state; and they are comprised of constant-parameter branches (assuming magnetic linearity) [29]. Consequently, qd models make up the majority of the built-in machine models in today’s industry-grade EMT simulators.   When implementing qd models in SVB programs, it is common to select either all the winding flux linkages [13], [29], [71], [72] or all the winding currents [29], [42], [73]–[75] as state variables. The winding flux linkages model is the most numerically efficient since its state matrices contain the fewest coefficients; however, unlike the winding currents model, the variables of interests (such as the stator and field currents) must often be calculated algebraically [76]. It is also possible to derive qd models with other combinations of state variables [76]–[80]. These models will typically simulate faster than the winding currents ones but slower than the winding flux linkages models. Hereinafter SVB qd models whose state vector comprises all the winding flux linkages will be referred to as Flux-based models; all other qd models will be lumped in the Mixed/Current-based category. SVB qd models are typically implemented using state-space equations, as is the case of the built-in models of SimPowerSystems and PLECS. While less common, it is also possible to represent some Mixed/Current-based qd models using circuit elements [81].   The major issue regarding qd models in SVB programs is their interface with external power systems represented in abc coordinates. Since qd models have a voltage-input, current-output formulation, their stator currents (converted to abc coordinates) are injected back to the network using voltage-controlled current sources, potentially resulting in incompatible input-output interfacing when the machine is connected to an inductive or ideal element [35]. Two indirect interfacing approaches are typically used to generate a 11  proper state-space model [35]. The first and most common one consists of using resistive or capacitive snubbers [13]. On the one hand, large resistors and small capacitors add fast modes to the system and thus increase its numerical stiffness [34]. Consequently, smaller step sizes might be necessary to ensure numerical stability and sufficient numerical accuracy [35]. On the other hand, smaller resistors and larger capacitors introduce more numerical error independently of the step size [35]. The second approach is to use a time-step relaxation [35]. Here also, smaller step sizes may be required to avoid instability and limit numerical error [35].   Several approaches have been proposed to interface qd machine models with power networks in EMTP-type programs [35]. A very common approach is to reduce the discrete model to a three-phase constant-parameter Thévenin equivalent in abc coordinates (these models are hereinafter referred to as a Thévenin prediction-based qd models) [5], [35], [82]. Such constant-parameter models are achieved at the cost of predicting speed voltage terms at every time step, decreasing the numerical accuracy and stability of the models at larger step sizes [31], [35]. For synchronous machines, the field voltage (if unknown) and the stator currents (due to the machine’s anisotropicity [5]) must also be predicted, further decreasing numerical accuracy [35]. Examples of such models include MicroTran’s Type-50 and ATP’s Type-59 models [35]. A variable-parameter Thévenin qd model that does not require predicting speed voltages or stator currents was briefly considered in [30]. This model was shown to be only slightly more accurate than the constant-parameter Thévenin prediction-based qd model [82] and significantly less accurate than PD models [30].  Another indirect interfacing method is used in PSCAD [7], [83], which is sometimes referred to as the Norton current source approach [35]. Therein, the qd model is solved separately from the rest of the power system, and the resulting stator currents (converted to abc coordinates) are injected back to the main network using current sources [83]. This approach involves a time-step relaxation, which decreases numerical accuracy and may 12  trigger numerical instability [35]. Constant-parameter shunt resistances and additional compensating current sources are used to improve numerical stability [83]. Nevertheless, this interfacing approach may still result in poor numerical accuracy with larger step sizes [35].  The qd models can also be interfaced with the rest of the network using compensation-based methods [35], [84], [85]. In this approach, the machine equations are solved along with a reduced-order Thévenin representation of the rest of the linear network [35]. The benefits of this approach are that it yields a simultaneous machine-network solution (due to the superposition principle), and that the power system’s conductance matrix is independent of the machine equations. However, artificial distributed-parameter lines are often needed to interface multiple machines [85], introducing noticeable numerical error and/or requiring the use of prohibitively small step sizes [35]. The well-known universal machine (UM) model [85] is interfaced this way in ATP. Alternatively, a higher dimensional Thévenin equivalent of the network could be used to interface multiple machines together, which is both computationally costly and complicated, making this approach less practical. Overall, compensation-based interfacing methods are ill suited when the number of external devices (in this case, the machine models) is large and when the same modified equations must be solved repetitively [28]. 1.2.3 VBR Models  As explained in [61, Section VII], machines have long been represented using an equivalent voltage source behind a reactance, in particular for transient stability studies [19], [29]. In many cases, these models are of lower order than the general-purpose lumped-parameter models commonly used for EMT simulations. Such lower order models are not considered in this thesis. Here, following the naming convention established by previous authors [35], VBR models refer to a subset of the general-purpose lumped-parameter models whose electrical state variables comprise the stator currents in abc 13  coordinates, and rotor and/or magnetizing fluxes in qd coordinates [35]. The equivalent stator of a VBR model can therefore be represented using circuit elements, thereby providing a direct interface between the machine and the power network (similarly to PD models) [35].  The first SVB VBR model was proposed in [86]. This synchronous machine model was shown to be less numerically stiff [34] than the equivalent PD model, thus requiring fewer simulation time steps [86]. However, due to the anisotropicity of synchronous machines and in particular dynamic saliency, the interfacing circuit of the original VBR model comprised rotor-position-dependent inductances and resistances [86]. As shown in [87], the equivalent stator resistance can be made constant by simply transferring the time-varying terms to the subtransient voltage source. The rotor-position dependency of the stator inductance can also be eliminated by negating dynamic saliency with the addition of a fictitious high-frequency damper winding [88]. This latter approach [88] requires compromising between numerical accuracy and efficiency. On the one hand, increasing the frequency of the damper winding decreases the numerical error; on the other hand, it increases the numerical stiffness and thus forces the use of smaller step sizes. It is also possible to push the frequency of the fictitious damper winding to infinity using singular perturbation theory [89], resulting in an algebraically exact model with a rotor-position-independent interfacing circuit. However, this approach can create algebraic loops, which are undesirable for the following three reasons: some commercial SVB toolboxes cannot solve them [12]; they decrease the robustness of the solution; and they add significant computational overhead. A different approach to achieve a rotor-position-independent interfacing circuit for SVB VBR synchronous machine models was recently proposed in [90]. It consists of moving the time-varying part of the stator inductance into the voltage source. Various continuous- and discrete-time filters can be used to form a proper state-space model and eliminate potential algebraic loops [90]. The models presented in [88] and [90] have a better combination of numerical accuracy and efficiency than snubber-14  interfaced qd models [33], [91]. Since induction machines are usually considered symmetric, the SVB VBR induction machine models have rotor-position-independent interfacing circuits without the need for numerical artifices [59]. Finally, a unified SVB VBR interfacing circuit comprised only of decoupled elements is presented in [91].   Discrete VBR models were also derived for EMTP-type programs [61], [92]–[94]. These models were shown to be significantly more accurate than Thévenin qd models. When derived without any approximation, the interfacing circuits of EMTP-type VBR induction and synchronous machine models are rotor-position-dependent. It is shown in [61] that the interfacing circuit of VBR induction machine models varies only slightly as a function of the rotor speed: it can therefore be assumed constant without a significant decrease in numerical accuracy [61]. A rotor-position-independent interfacing circuit can also be achieved for synchronous machines by predicting stator currents [94].  The aforementioned VBR synchronous machine models [86]–[92], [94] are only interfaced with external power systems through their stator terminals: they do not include a circuit-based interface for their field winding. This is not a limitation when, as is common, the voltage applied at the field winding is either constant or the output of a transfer function (TF)-based equivalent excitation system [95]. However, it is sometimes necessary and/or convenient to model the exciter using circuit components, for example to analyze diode failures in brushless exciters [81], [96], or to study the impact of field discharge resistors and exciter breakers [97], [98]. These exciter models may result in an incompatible interface with the VBR model’s field terminal. A snubber located at the field terminal or a time-step relaxation might be needed, forcing the use of smaller step sizes and/or decreasing numerical accuracy. To obtain a model that can be interfaced seamlessly with all external circuits, a SVB VBR model representing both stator and field windings in circuit form (with rotor-position-dependent inductances) was first proposed in [96]. A 15  variation of this model possessing a rotor-position-independent interfacing circuit [81] was later achieved by extending the approach presented in [90]. 1.2.4 Magnetic Saturation in EMT Machine Models  Magnetic saturation in rotating machines is a very complex phenomenon, which would ideally be represented using high-fidelity modeling approaches such as FEA and MECs. However, the computational cost of these methods is usually considered too heavy for system-level EMT simulations. Consequently, researchers have spent considerable time developing approaches to adequately incorporate the effect of magnetic saturation into general-purpose lumped-parameter models. These approaches have been verified experimentally on numerous occasions and shown to improve modeling accuracy when compared to unsaturated (i.e., magnetically linear) models, e.g., [74], [99]–[101].  In EMT simulations, the leakage and main fluxes are generally considered separately. In this thesis, only saturation of the main flux is considered; stator and rotor leakage flux saturation (e.g., [48], [71], [101]) is neglected. Moreover, only the fundamental component of the main flux is considered; its saturation harmonics (e.g., [41], [102]) are also neglected. The impact of these assumptions on modeling accuracy will be discussed in Sections 2.3.2 and 2.4.2 along with details on how to incorporate main flux saturation in EMT models. This section reviews the impact of main flux saturation modeling on the numerical properties of general-purpose lumped-parameter machine models. 1.2.4.1 SVB machine models  Methods for implementing main flux saturation in SVB qd induction machine models vary as a function of the state variables. Main flux saturation can be accounted for in Mixed/Current-based qd models by using techniques based on the generalized flux space vector approach proposed in [77], [78]. These techniques result in the presence of 16  saturation-dependent steady-state magnetizing and dynamic inductances [42], [73], [77], [78], [103], which render the state matrices of the qd model time-dependent. Cross-saturation terms further introduce coupling between the q and d axes. Similar observations are made for synchronous machine models [72], [74]–[76], [104], with the only noteworthy distinction being the absence of cross saturation when main flux saturation is only considered along the d axis [104].  Incorporation of main flux saturation is different in Flux-based qd models since the steady-state magnetizing inductance is never explicitly differentiated. A possibility is to update the steady-state magnetizing inductance as a function of the main flux computed in the previous time step [46], thereby introducing a time-step relaxation. A more elegant method is the so-called flux correction (FC) approach, which initially calculates the unsaturated magnetizing fluxes and then corrects them according to the saturation characteristic [29], [71], [72], [105]. The FC approach is very straightforward, easy to append to the unsaturated model (as an option that can be turned on/off), and is therefore very convenient to use in SVB programs. Its main downside is that its formulation is implicit, i.e., it contains an algebraic loop. An explicit FC function was presented in [106] for synchronous machines. While significantly more efficient, this function is based on crude approximations, which introduce non-negligible numerical error.   The methods used to incorporate main flux saturation in Mixed/Current-based qd models should be readily extendable to PD models. The magnetizing current magnitude, which can be used to adjust the magnetizing inductances, can be calculated from phase variables as explained in [107]. It is recalled that the equivalent windings of unsaturated SVB PD models are already coupled and time-varying. Representation of main flux saturation therefore does not change the core numerical properties of these models.  The effect of main flux saturation is typically represented in SVB VBR models similarly to as in Mixed/Current-based qd models [96], [108], [109]. All existing saturable 17  three-phase VBR models have saturation-dependent interfacing circuits [96], [108], [109]. Their interfacing circuits are also function of the angular displacement of their transformed reference frame. Straightforward representation of main flux saturation in the constant-parameter unsaturated VBR models [59], [88], [89], [90] would also render the interfacing circuit inductance saturation-dependent (and thus rotor-position-dependent). Variations of the models proposed in [96] and [109] have been recently added to PLECS’ built-in component library. 1.2.4.2 EMTP-type machine models  The approach to represent main flux saturation is similar in most EMTP-type PD [64], qd [64], [110], and VBR models [94], [111], [112]; consequently, its effect on their interfacing circuits is also similar. Main flux saturation is typically represented using a piecewise-linear function made of a limited number of segments [5]. With this type of representation, the main flux can be considered as the sum of a residual flux plus the magnetizing current multiplied by the piecewise dynamic inductance (i.e., the slope of the segment) [5]. Consequently, the interfacing circuits of the saturable PD, Thévenin qd, and VBR models depend on the operating segment of the saturation curve, and re-factorization of the network’s conductance matrix is necessary every time that this operating segment changes. These models are referred to as having saturation-segment-dependent interfacing circuits. While saturation-segment-dependent models might not always increase the overall computational cost significantly (depending on the case study), they can be particularly inconvenient for real-time simulators wherein the number of available flops per time step is limited. The network’s conductance matrix is independent of machine saturation when the qd models are interfaced using the Norton current source approach or compensation-based methods [35]. There also exist other approaches to incorporate main flux saturation in PD, Thévenin qd, and VBR models, but they result in saturation-dependent conductance matrices (as opposed to saturation-segment-dependent) [57], [60]. 18  1.2.5 Multirate Simulations for EMT Machine Models  Multirate frameworks have already been proposed for the simulation of mid- to long-range dynamics [36] and electromagnetic transients [37]–[39], [113]. In particular, successful implementations of multirate schemes have been reported in SVB [37] and EMTP-type [38], [39], [113] environments. The second test system in [37] includes synchronous and induction machines that are modeled using a VBR formulation [86]. Their rotor and mechanical subsystems are part of the slower subsystem, while the stator interfacing circuits are included in the faster subsystem. This partitioning is based on heuristics [37]. To the best of the author’s knowledge, the use of a multirate framework for EMTP-type machine models has yet to have been proposed in the literature. 1.3 Research Objectives and Contributions  The ultimate goal of this research is to improve the numerical efficiency (CPU time) of EMT simulators without degrading their numerical accuracy and stability, which we do by proposing improved machine models. This is particularly desirable in view of the changes currently occurring in electrical power systems, which require a fast-growing number of simulations. Moreover, it is expected that significant improvements in numerical efficiency will make it possible to execute a wider range of studies (e.g., electromechanical transients) using SVB and EMTP-type programs in a reasonable amount of time. These case studies would therefore be executed using fewer modeling assumptions, which would increase the overall simulation precision.  The properties of the existing general-purpose lumped-parameter SVB induction and synchronous machine models discussed in Section 1.2 are summarized in Tables 1–1 and 1–2, respectively; those of the EMTP-type induction machine models are compiled in Table 1–3. The desirable properties are denoted by bold fonts in these tables. It is recalled19  Table 1–1. Properties of SVB general-purpose lumped-parameter induction machine models.   Main Flux Saturation Stator Interface Circuit Parameters Explicit Formulation Flux-based qd [29], [70] No Indirect N/A Yes Flux-based qd [71] Yes Indirect N/A No Mixed/Current-based qd [29]  No Indirect Constant * Yes Mixed/Current-based qd [42], [73] Yes Indirect Variable * Yes PD [59] No Direct Variable Yes VBR [59] No Direct Constant Yes VBR [109] Yes Direct Variable Yes * When the qd model is implemented using circuit elements. Table 1–2. Properties of SVB general-purpose lumped-parameter synchronous machine models.  Main Flux Saturation Stator Interface Field Interface Circuit Parameters Explicit Formulation Flux-based qd [29], [66] No Indirect Indirect N/A Yes Flux-based qd [29], [105] Yes Indirect Indirect N/A No Flux-based qd [106] Yes Indirect Indirect N/A Yes ** Mixed/Current-based qd [29] Yes Indirect Direct * Constant * Yes Mixed/Current-based qd [74], [76] Yes Indirect Direct * Variable * Yes PD [24], [58] No Direct Direct Variable Yes VBR [86], [87] No Direct Indirect Variable Yes VBR [88], [90] No Direct Indirect Constant Yes VBR [89] No Direct Indirect Constant No VBR [108], [109] Yes Direct Indirect Variable Yes VBR [81] No Direct Direct Constant Yes VBR [96] Yes Direct Direct Variable Yes * When the qd model is implemented using circuit elements. ** As shown in Section 3.3.2, this model creates significant numerical error. that in an SVB environment, indirect machine-network interfacing (e.g., using snubbers) can force the use of smaller step sizes, decrease numerical accuracy, and even result in20   Table 1–3. Properties of EMTP-type general-purpose lumped-parameter induction machine models (RPD: rotor-position-dependent; SSD: saturation-segment-dependent).  Main Flux Saturation Stator Interface * Interfacing Circuit Parameters ** Thévenin qd [30], [82]  No Indirect Constant Thévenin qd [30] No Indirect Variable (RPD) Thévenin qd [64], [110]  Yes Indirect Variable (SSD) Norton current source qd [35], [83] No Indirect Constant Norton current source qd [35], [83] Yes Indirect Constant Compensation-based qd [35], [85] No Indirect Constant Compensation-based qd [35], [85] Yes Indirect Constant PD [61] No Direct Constant PD [64] Yes Direct Variable (SSD) VBR [93] No Direct Variable (RPD) VBR [61] No Direct Constant VBR [111] Yes Direct Variable (RPD/SSD) VBR [94] Yes Direct Variable (SSD) * The distinction between direct and indirect interfacing is blurrier for EMTP-type models. The definitions used here are based on [35]. While it could be argued that the Thévenin qd models are directly interfaced with the power network, it is recalled that their numerical accuracy is significantly worse than that of PD and VBR models. ** The interfacing circuit parameters refer to the elements added to the network’s conductance matrix.  numerical instability. System matrices must also be continuously updated when time-varying parameters are present, which adds significant computational overhead. Moreover, implicit models are less robust, less general, and more numerically costly than explicit models. Finally, Flux-based qd models are more efficient than Mixed/Current-based models. For EMTP-type simulations, indirect interfaces typically introduce additional numerical error and/or are less general than direct interfacing approaches. Rotor-position-dependent and/or saturation-segment-dependent interfacing circuit parameters increase computational cost and can be ill suited for real-time simulators.   As seen in Tables 1–1 and 1–2, there is no state-of-the-art SVB induction or synchronous machine model with main flux saturation that possesses all of the following desirable properties: a direct interface, constant circuit parameters, and an explicit 21  formulation. In the same vein, Table 1–3 demonstrates that there is no EMTP-type induction machine model with a true constant-parameter direct interfacing circuit when main flux saturation is taken into account. Consequently, the existing general-purpose lumped-parameter machine models can significantly deteriorate the numerical performance of EMT simulators, and as such are sometimes referred to as simulation bottlenecks. The main contribution of this thesis is to propose and present new general-purpose lumped-parameter machine models that account for main flux saturation while possessing enhanced numerical properties. This will help achieving the ultimate objective of improving the numerical efficiency of EMT simulators without adversely affecting their numerical accuracy and stability. Objective 1: Creating explicit SVB Flux-based qd induction and synchronous machine models with main flux saturation.  Despite a growing recognition of the beneficial properties of PD and VBR models, qd models frequently remain the default “go-to” models in SVB simulations. In particular, Flux-based qd models offer the best numerical efficiency when a compatible machine-network interface is achieved, e.g., when the machine is connected to a shunt capacitance [59]. As seen in Tables 1–1 and 1–2, there is a Flux-based qd model that takes into account main flux saturation while having an explicit formulation [106]; however, this model introduces significant numerical error. The first objective of this thesis is therefore to develop an explicit Flux-based qd model that takes into account main flux saturation (and cross saturation). It is achieved by reformulating the original implicit FC approach as to remove the algebraic loop. The proposed Flux-based qd induction and synchronous machine models retain the advantages of the original FC approach, including its high numerical accuracy.  22  Objective 2: Developing an explicit SVB induction machine model with constant-parameter direct interfacing including main flux saturation.  As seen in Table 1–1, there is no explicit SVB induction machine model with a constant-parameter direct interface that also takes into account main flux saturation. The second objective of this dissertation is to present a VBR model that possesses all of these desirable properties. The proposed saturable induction machine model therefore provides a combination of numerical accuracy and efficiency that had yet to be attained.  Objective 3: Developing an explicit SVB synchronous machine model with constant-parameter stator and field interfacing circuits including main flux saturation.   Table 1–3 shows that there is no explicit SVB synchronous machine model with a constant-parameter direct interface that also takes into account main flux saturation. This holds whether the field winding is represented in circuit form or not. The third objective is to propose an explicit VBR synchronous machine model with constant-parameter stator and field interfacing circuits that includes main flux saturation. Similarly to the induction machine model from Objective 2, the proposed model provides an unmatched combination of numerical accuracy and efficiency. Objective 4: Formulating a constant-parameter EMTP-type induction machine model with a direct interface including main flux saturation.  It is shown in Table 1–3 that all saturable directly interfaced EMTP-type induction machine models have a saturation-segment-dependent (and sometimes also rotor-position-dependent) interfacing circuit. This thesis’ fourth objective is to derive an EMTP-type VBR induction machine model with a true constant-parameter interfacing circuit that also incorporates the effect of main flux saturation. The proposed model achieves 23  considerable computational gains, and is particularly well suited for real-time simulators since it does not require matrix re-factorizations. Objective 5: Deriving multirate EMTP-type machine models.  Multirate simulation techniques can help improving the numerical efficiency of simulations. While such frameworks have been proposed for EMTP-type applications in a few publications (e.g., [38], [39], and [113]), they have yet to be implemented in industry-grade software. The fifth and final objective of this dissertation is to propose multirate VBR machine models that can be integrated seamlessly to state-of-the-art single-rate EMTP-type packages as well as future multirate EMTP-type programs. The proposed models retain the interfacing properties of the original single-rate machine model from which they are derived, herein the saturable EMTP-type induction machine model from the fourth objective. The multirate models decrease the computational overhead while introducing little additional numerical error.   All the proposed models have been implemented in industry-grade offline EMT simulation packages and verified against other state-of-the-art machine models. As the case studies included in this dissertation demonstrate, the proposed models help improving the numerical efficiency of EMT simulations without affecting numerical accuracy.  24  CHAPTER 2:  FUNDAMENTALS OF EMT SIMULATIONS AND MACHINE MODELING  To set the stage for the derivation of the proposed models, a review of the fundamentals of EMT simulations and machine modeling is provided in this section. We start by briefly reviewing the basics of the SVB and EMTP-type simulation approaches. We then present the basic induction and synchronous machine dynamic equations, while stating the assumptions used to obtain them. Various methods for representing magnetic saturation in lumped-parameter models are discussed. The approaches used to incorporate the effect of saturation in the subject induction and synchronous machine models are clearly presented and justified. For convenience and consistency, we also present the equations of the saturable PD and qd machine models that will be used in Chapters 3 to 7 to benchmark the proposed models. We close this chapter by specifying how we will assess the numerical accuracy of the subject models. 2.1 SVB Simulations  A short discussion on the fundamentals of SVB simulators is provided in this section. Further details on specific aspects can be found in numerous references such as [24], [25], [34], [114], and [115].  Dynamic systems can often be represented by the following system of first-order ODEs  25   ( )tp ,,uxfx =  (2–1)  ( )t,,uxgy =  (2–2) where x  is the state vector, u  represents the inputs, t  denotes time, and y  is the output vector. Equations (2–1) and (2–2) are often referred to as the general form of a state-space system. In some cases, it might also be beneficial [25] or necessary [34] to add algebraic constraints, transforming the ODEs into a system of DAEs.  For linear and piecewise-linear systems, (2–1) and (2–2) can be represented in standard linear state-space form as  BuAxx +=p  (2–3)  DuCxy +=  (2–4) where  A , B , C , and D  are the well-known state-space matrices. The eigenvalues of A  represent the system’s dynamic modes.  Equations (2–1)–(2–2) or (2–3)–(2–4) can be solved using various numerical integration methods [34]. MATLAB/Simulink’s ODE suite is a particularly powerful example [23]. It has fixed and variable step-size solvers. It also contains explicit and implicit solvers, including some which are A- and L-stable [23], [34]. As noted in the introduction, the state-space matrices are independent of the integration step size.  State-of-the-art Simulink-based SVB circuit simulators such as PLECS [114] and SimPowerSystems [115] contain extensive built-in component libraries and have easy-to-use graphical user interfaces. The libraries comprise both circuit (inductors, transformers, etc.) and control (summation blocks, rate limiters, etc.) components. System analysts can simply interconnect the various circuit and control components to form the desired case system [116]. The built-in state model generation algorithms then create the differential26  Linear Circuit Equationspx1 = Ax1 + Bu1y1 = Cx1 + Du1Nonlinear Equationspx2 = f(x2,u2,t)y2 = g(x2,u2,t)y1u2u1y2 Figure 2–1. State-variable-based representation of a power system in which the linear circuit equations and nonlinear equations are aggregated in different subsystems. equations automatically [24], [25]. Consequently, the tedious task of manually generating the state-space equations is avoided. Nevertheless, the automated process of formulating (2–1)–(2–2) and/or (2–3)–(2–4) remains numerically costly [32].  In SVB circuit simulators, it is common to formulate the circuit equations as a separate subsystem using (2–3) and (2–4), and the typically nonlinear controls as another subsystem using the more general (2–1) and (2–2). This grouping can be visualized in Figure 2–1. An instructive example is the VBR machine model, wherein the interfacing circuit is combined with the rest of the power network, while the rotor and mechanical dynamics are considered as a control subsystem [116]. In the constant-parameter VBR case, the subsystems are interconnected using current and/or voltage meters, and controlled voltage sources [116]. Other types of partitioning (some using DAEs) are also possible, as explained in [25]. It is emphasized that significant computational savings can be achieved when the state-space matrices A , B , C , and D  remain constant for most – if not all – of the simulation duration [33], [88].   For the purpose of this thesis, SVB programs implicitly refer to circuit-based SVB simulators such as ASMG, PLECS, and SimPowerSystems. Consequently, it is assumed that27   the system analyst does not need to explicitly formulate and assemble the differential equations of the circuit elements. 2.2 EMTP-Type Simulations  A very brief overview of EMTP-type simulations for the purpose of this thesis is presented in this section. A much more detailed review of EMTP-type simulations is given in [5]. The origins of EMTP-type programs can be traced back to a 1969 paper written by H. W. Dommel [117]. The solution method consists of first discretizing the network components individually, typically using the trapezoidal rule [5]. For instance, discretization of the simple resistor-inductor (RL) system of Figure 2–2 (a) with the trapezoidal rule yields  )()()( histeq titvGti km +=  (2–5) where t∆  is the fixed integration step size, )()()( τττ mkkm vvv −=  (where τ  is any integer multiple of t∆ ), the conductance eqG  is   1eq2 −∆+=tLRG , (2–6) and the history current source )(hist ti  is    ∆−∆−−∆−= )(2)()( eqhist ttitLRttvGti km . (2–7)  The equivalent circuit of the discrete single-phase RL system is shown in Figure 2–2 (b). After all the network components have been discretized [5], they can be grouped together to form the following linear system of nodal equations [117] 28  vk vmL Ri(a) Continuous timevk(t)2Lihist(t)i(t) ∆tR+(b) Discrete timevm(t) Figure 2–2. Series RL circuit represented in (a) continuous time and (b) discrete time. =G vn ihabca b cvavcvbGeqihistaihistcihistb Figure 2–3. Equivalent linear system of nodal equations for EMTP-type simulations.  )()( tt hn iGv =  (2–8) where G  is the network’s conductance matrix, nv  represents the unknown nodal voltages (it is assumed in this section that all nodal voltages are unknown, although known nodal voltages can be considered fairly easily [117]), and hi  is the current injection vector. Equation (2–8) can also be visualized in Figure 2–3, wherein it is shown how to insert the conductance matrix ( eqG ) and history current sources ( aihist , bihist , and cihist ) of an arbitrary grounded three-phase component [61]. In particular, it is seen from Figure 2–3 that the matrix G  is made of the conductance submatrices of all the individual components 29  [5], [61]. It is common for the sparse matrix G  to be symmetric positive definite (SPD) [118].  The nodal voltages nv  are found by solving (2–8) at every time step. This is typically done by factoring G  into triangular matrices with sparsity-oriented methods (e.g., the Cholesky decomposition with the minimum degree algorithm) and finding nv  using backward-forward substitution [27]. If all the network components are time-independent and G  remains constant throughout the simulation, it is sufficient to factor G  only once at the beginning of the simulation. The same sparse triangular matrices (factors) can then be used to find nv  at every time step using backward-forward substitution. As discussed in the introduction, a constant matrix G  has been shown to improve simulation speed by a factor of 2 to 4 on practical systems [30], [31]. Otherwise, G  must be re-factored or partially re-factored [28] every time that one of its coefficient changes. It is also possible to extend (2–8) to take into account various ideal elements [6], yielding the more general equation  )()( tt bAx =  (2–9) where G , nv , and hi  are elements of A , x , and b , respectively. This is referred to by some as modified nodal analysis (MNA) [119], and by others as modified augmented nodal analysis (MANA) [16]. The same solution method as for (2–8) applies. However, A  typically will not be SPD, precluding the use of more efficient factorization schemes such as the Cholesky decomposition [118].   Equations (2–8) and (2–9) are systems of linear equations. Some nonlinear circuit components can be included into (2–8) or (2–9) using piecewise-linear representations, or as current injections with a time-step delay (e.g., transformer magnetization in PSCAD). Compensation-based methods [84] are also widely used to interface nonlinear components 30  in EMTP-type programs such as ATP. A system-wide Newton-Raphson approach is used in EMTP-RV to solve nonlinear components. The simpler piecewise-linear representation is usually sufficient to account for machine nonlinearities caused by magnetic saturation. 2.3 Induction Machines  This section presents the basics of the general-purpose lumped-parameter induction machine models. These models are based on several assumptions [29], which include symmetrical stator and rotor windings with sinusoidal distributions, and a smooth air gap. The equivalent circuit of a two-pole wye-connected induction machine represented in physical variables (abc coordinates) and with N  rotor circuits is depicted in Figure 2–4. The equivalent rotor windings of squirrel-cage induction machines are short-circuited.  The rotors of squirrel-cage induction machines have a distributed nature. Depending on the specifics of the physical machine and the case study, a lumped-parameter model with one equivalent rotor circuit (i.e., a three-phase rotor winding) might provide sufficient modeling accuracy [29]. For instance, PLECS and SimPowerSystems’ built-in induction machine models each have one equivalent rotor circuit.   However, as noted in [120], these models might not be adequate when the slip varies over a wide range. In particular, they are usually insufficient to represent the dynamics of motors with deep rotor bars and double-cage rotors (NEMA design classes B and C, respectively [121]). Lumped-parameter models with two equivalent rotor circuits are usually considered accurate enough to represent induction motors of all NEMA design classes for the purpose of system-level EMT simulations [120]. EMTP-RV and PSCAD are examples of industry-grade EMT programs whose built-in models can comprise two rotor circuits.  31  rspλasiasvas+ -pλbs+-pλcs+-rsrsibsicsvcsvbsrr1pλar1iar1+-pλbr1+-pλcr1+-rr1rr1ibr1icr1rr2pλar2iar2+-pλbr2+-pλcr2+-rr2rr2ibr2icr2rrNpλarNiarN+-pλbrN+-pλcrN+-rrNrrNibrNicrN. . .{ {StatorRotor  Figure 2–4. Equivalent circuit of an induction machine represented in physical variables.  Some researchers have also suggested using higher order lumped-parameter equivalent rotor circuits [122], [123]. The objective behind these models is typically to better evaluate the higher frequency content of machine-converter systems, for example by appropriately representing the skin effect [123]. Up to four rotor circuits can be modeled using PSCAD’s built-in induction machine model. A different approach to achieve a similar goal is presented in [124]. Therein, the rotor is represented by a single (saturable) inductance in series with an arbitrary linear time-invariant (LTI) TF. The relatively involved experimental procedure to determine these machine parameters is presented in [125]. It is also recalled that an LTI system can be represented using circuit elements.  32   In this dissertation, a lumped-parameter induction machine model with an arbitrary number of rotor circuits is considered. All parameters are implicitly referred to the stator; this holds for all induction and synchronous machine models throughout this thesis. Motor sign convention is followed. To simplify notation, it is assumed that there is no rotor end-ring common resistance [42], and that the magnetic coupling between all magnetically aligned windings is identical. It is emphasized that these two assumptions are not necessary to derive the proposed models but are used to facilitate the reading of this document. Moreover, as shown in [120], it is sometimes possible to convert models with unequal magnetic coupling into algebraically equivalent models with equal magnetic coupling using linear circuit theory. 2.3.1 Machine Equations 2.3.1.1 Voltage equations in physical variables  The voltage equations in physical variables (abc coordinates) of a general-purpose lumped-parameter squirrel-cage induction machine model with N  equivalent rotor circuits are given by [29]  +=abcrabcsabcrabcsabcrsabcs pλλiiR00R0v (2–10) where abcsv , abcsi , and abcsλ  are the stator voltages, currents, and flux linkages in abc coordinates, respectively; and abcri  and abcrλ  are the rotor currents and flux linkages in abc coordinates, respectively. The diagonal resistance matrices are defined as  [ ]ssss rrrdiag=R  (2–11)  [ ]abcrNabcrabcrabcr RRRR 21diag=        and     [ ]abcrzabcrzabcrzabcrz rrrdiag=R  (2–12) where sr  is the stator resistance and rzr  is the resistance of the z th rotor circuit. 33   The stator and rotor flux linkages abcsλ  and abcrλ  are related to the stator and rotor currents abcsi  and abcri  by [29]  ( ) =abcrabcsabcrTabcsrabcsrabcsabcrabcsiiLLLLλλ. (2–13) The inductance matrices appearing in (2–13) are expressed as  +−−−+−−−+=mslsmsmsmsmslsmsmsmsmslsabcsLLLLLLLLLLLL5.05.05.05.05.05.0L  (2–14)  =abcrdNabcroabcroabcroabcrdabcroabcroabcroabcrdabcrLLLLLLLLLL21 (2–15)  .,...,1,5.05.05.05.05.05.0NzLLLLLLLLLLLLmslrzmsmsmsmslrzmsmsmsmslrzabcrdz =+−−−+−−−+=L  (2–16)  −−−−−−=msmsmsmsmsmsmsmsmsabcroLLLLLLLLL5.05.05.05.05.05.0L  (2–17)  °−°+°+°−°−°+=−rrrrrrrrrmsabcwsr Lθθθθθθθθθcos)120cos()120cos()120cos(cos)120cos()120cos()120cos(cosL  (2–18)  [ ] Nabc wsrabc wsrabc wsrabcsr 33×−−−= LLLL   (2–19) where lsL  is the stator leakage inductance, lrzL  is the leakage inductance of the z th rotor circuit, msL  is the stator magnetizing inductance, and rθ  is the electrical rotor position. 34  2.3.1.2 Voltage equations in qd coordinates  It is customary to transform (2–10) to an arbitrary reference frame [29]. The relationships between abc variables and the resulting qd variables are expressed as [29]  abcsssqd fKf =0  (2–20)  abcrrqdr fKf =  (2–21) where f  may represent voltages, currents, or fluxes, and the transformation matrices sK  and rK  are defined as  °+°−°+°−=5.05.05.0)120sin()120sin(sin)120cos()120cos(cos32 θθθθθθsK  (2–22)  [ ])120cos()120cos()cos(32°+−°−−−=− rrrwrq θθθθθθK  (2–23)  [ ])120sin()120sin()sin(32°+−°−−−=− rrrwrd θθθθθθK  (2–24)  ,3NNwrwrwrr×−−−=γγγγK000K000KK          { }.,dq=γ    (2–25)  [ ]TTrdTrqr KKK =  (2–26) where the angle θ  represents the angular displacement of the arbitrary reference frame. Here, the transformation matrices sK  and rK  have been scaled as to preserve the amplitude of the waveforms [29]. It is also convenient to define the inverse relationships between abc and qd variables, i.e., 35   sqdsabcs 01fKf −=  (2–27)  qdrrabcr fKf inv= . (2–28) Matrices 1−sK  and invrK  are expressed analytically as [29]  °+°+°−°−=−1)120sin()120cos(1)120sin()120cos(1sincos1θθθθθθsK  (2–29)  [ ]Trrrwrq )120cos()120cos()cos(inv °+−°−−−=− θθθθθθK  (2–30)  [ ]Trrrwrd )120sin()120sin()sin(inv °+−°−−−=− θθθθθθK  (2–31)  NNwrwrwrr×−−−=3invinvinvinvγγγγK000K000KK,          { }.,dq=γ    (2–32)  [ ]invinvinv rdrqr KKK = . (2–33) The matrix that transforms rotor variables from qd to abc coordinates is denoted by invrK  instead of 1−rK , since rK  is herein a non-square matrix due to the absence of zero-sequence rotor components in squirrel-cage machines.  Transforming (2–10) to qd coordinates using (2–20)–(2–33) yields [29]  qsdsqssqs pirv λωλ ++=  (2–34)  dsqsdssds pirv λωλ +−=  (2–35)  ssss pirv 000 λ+=  (2–36)  ( ) .,...,1,0 Nzpir qrzdrzrqrzrz =+−+= λλωω  (2–37) 36   ( ) .,...,1,0 Nzpir drzqrzrdrzrz =+−−= λλωω  (2–38) where qsv , dsv , and sv0  are the q-axis, d-axis, and zero-sequence stator voltages, respectively; qsi , dsi , and si0  are the q-axis, d-axis, and zero-sequence stator currents, respectively; qsλ , dsλ , and s0λ  are the q-axis, d-axis, and zero-sequence stator flux linkages, respectively; qrzi  and drzi  are the q-axis and d-axis currents of the z th rotor circuit, respectively; qrzλ  and drzλ  are the q-axis and d-axis flux linkages of the z th rotor circuit, respectively; ω  is the angular velocity of the arbitrary reference frame; and rω  is the electrical rotor speed. The stator zero-sequence equation (2–36) can be omitted when the machine’s stator has a floating-wye connection.  The flux linkages in qd coordinates are defined as [29]  mqqslsqs iL λλ +=  (2–39)  mddslsds iL λλ +=  (2–40)  slss iL 00 =λ  (2–41)  .,...,1, NziL mqqrzlrzqrz =+= λλ  (2–42)  .,...,1, NziL mddrzlrzdrz =+= λλ  (2–43)  Finally, the q- and d-axis magnetizing fluxes mqλ  and mdλ  are expressed as [29]  +≡= ∑=Nzqrzqsmmqmmq iiLiL1λ  (2–44)  +≡= ∑=Nzdrzdsmmdmmd iiLiL1λ  (2–45) where mqi  and mdi  are the q- and d-axis magnetizing currents, and the magnetizing inductance mL  is related to msL  by 37   mms LL 5.1= . (2–46)  The use of qd coordinates also greatly simplifies the calculation of the electromagnetic torque eT  [29]. When considering main flux saturation, a particularly convenient way to evaluate eT  is to use  ( )dsmqqsmde iiPT λλ −=43  (2–47) since mqλ  and mdλ  are usually evaluated at every time step.   The equivalent circuit of the general-purpose lumped-parameter squirrel-cage induction machine model represented in qd coordinates is sketched in Figure 2–5. 2.3.1.3 Mechanical equations  Induction machines and their mechanical loads are usually represented as a single rigid body in EMT simulations. Consequently, the mechanical equations are given by [29]  rrp ωθ =  (2–48)  rme pPJTT ω=−2  (2–49) where eT  and mT  are the developed electromagnetic and mechanical torque, respectively, J  is the moment of inertia of the machine and its mechanical load (in kg∙m2), and P  is the number of poles. While not considered in this dissertation, it is possible to extend (2–48) and (2–49) as to emulate a multimass system. This can be useful for wind generator systems and/or systems with long shafts, for which a single rigid body representation might not always be sufficient [126]. In power system applications, it is frequent to give the moment of inertia in seconds ( H ). The relationship between J  and H  is given by 38  iqs+ -rsLlsimqvqs+-ωλds+-rr1 Llr1(ω−ωr)λdr1 iqr1+-rr2 Llr2(ω−ωr)λdr2 iqr2+-rrN LlrN(ω−ωr)λdrN iqrNLmids+-rsLlsimdvds+-ωλqs+ -rr1 Llr1(ω−ωr)λqr1 idr1+ -rr2 Llr2(ω−ωr)λqr2 idr2+ -rrN LlrN(ω−ωr)λqrN idrNLmi0s rsLlsv0s+- Figure 2–5. Equivalent circuit of a squirrel-cage induction machine represented in qd coordinates.  BbTJPHω=221  (2–50) where bω  and BT  are the base frequency and torque, respectively. The base torque BT  can be related to the base active power BP  by 39   bBBPPTω=2. (2–51) 2.3.2 Magnetic Saturation  To improve the modeling accuracy of the general-purpose lumped-parameter induction machine model, it is customary to consider some or all of its equivalent inductances as saturable. In practice, which inductances could remain magnetically linear and which ones should be saturable depends on various factors such as the considered system, the study objectives, the variables of interest, the availability of saturation functions, and the desired modeling and simulation precision.  It has long been recognized that the self-excitation process of islanded induction generators can be modeled very accurately while only considering saturation of the main flux. Experiments validating this assumption have been reported in many papers [42], [73], [127], [128]. EMTP-type studies have also been conducted to analyze the self-excitation phenomenon of distributed induction generators following their disconnection from the main grid [128], [129]. Additionally, [73] and [127] demonstrated experimentally that representation of main flux saturation is sufficient to predict machine behavior following sudden load changes. Field-oriented control schemes can also be significantly affected by the level of main flux saturation [29], [130], [131]. This is particularly true in the field weakening region, wherein the main flux can be considerably smaller than its rated value [131], [132]. In the aforementioned examples, it is implied that the winding currents stay relatively small (e.g., less than 150% of their nominal values). Since leakage inductances do not heavily saturate at such currents, neglecting leakage flux saturation has a limited impact on accuracy in these scenarios.  On the one hand, neglecting leakage flux saturation may introduce considerable errors in situations where the winding currents are several times larger than their nominal 40  values, for example when simulating faults near the machine terminals or uncontrolled start-up transients. Reference [101] shows that when simulating the start up of a 0.25-hp motor with a magnetically linear model, the peak current is underestimated by almost half while the start-up duration is approximately doubled. Incorporating main flux saturation has only a limited effect on the start-up transient performance of the model, whereas the fully nonlinear model (with saturation of all inductances) predicts the complete transient with great accuracy [101]. Similar conclusions are reached in [133] using transient simulations with a different motor, and in [134] through sensitivity analyses.  On the other hand, it is also possible to use constant re-fitted leakage inductances and still obtain very reasonable results for transients generating large winding currents. For example, in [48] the start-up times of a 6600-hp induction motor modeled with and without leakage flux saturation differ by only approximately 5%. However, it is mentioned that the machine has minimal leakage saturation, and more importantly, that the parameters of the magnetically linear model have been selected (or re-fitted) as to yield the same start-up torque as the saturable model [48]. In other words, while remaining constant (i.e., saturation-independent), the machine parameters were modified to more accurately reproduce the start-up transient of the physical machine. This idea was further developed in [46], wherein an online identification procedure is presented. Using a least-squares-based approach, the parameters of an induction machine model with main flux saturation but without leakage flux saturation are determined from short-circuit and start-up transient tests (including steady-state intervals). The resulting model is shown to yield good agreement with the physical motor during transient and steady-state conditions [46]. Consequently, [46] demonstrates that the large discrepancies during start ups observed in [101] and [133] can be alleviated by selecting constant rotor and stator parameters that differ from the standard parameters obtained by the locked-rotor and no-load tests [46, Table I]. Although it is understood that such re-fitted models cannot be as accurate as fully 41  nonlinear models for all possible operating regions, they nevertheless expand the range of application of models with constant leakage inductances.  For these reasons, and bearing in mind the limitations stated above, the proposed induction machine models only consider saturation of the main flux. This is a fairly common assumption for system-level studies. For instance, PLECS and SimPowerSystems’ built-in induction machine models only allow main flux saturation. Moreover, only the fundamental component of the main flux is considered. In practice, there also exist main flux saturation harmonics, which rotate through the air gap along with the fundamental component of the main flux [41], [102]. These harmonics contribute to the developed electromagnetic torque and to rotor losses [102]. However, they have considerably smaller magnitudes than the fundamental component [102]. The procedures to determine the saturation functions of these harmonics are also more complicated than for the fundamental component. Consequently, representation of main flux saturation harmonics remains confined to a few advanced lumped-parameter models, e.g., [41], [102], and is not considered in the built-in models of state-art-of-art EMT packages such as EMTP-RV, PLECS, PSCAD, and SimPowerSystems. Hereinafter, the term “main flux” refers only to its fundamental component.  The main flux saturation characteristic of an induction machine can be obtained experimentally from the standard no-load test. Specifically, neglecting stator and core losses, the measured rms stator line current NLasI ,  and rms line-to-neutral voltage NLasV ,  are related to the main flux mλ  and the magnetizing current mi  by  ( )bNLaslseNLasmILVωωλ ,,2 −≈  (2–52)  NLasm Ii ,2≈  (2–53)  42  imλ m  Air-gap lineSaturation curve Figure 2–6. Typical saturation curve of an induction machine obtained from the no-load test and the corresponding air-gap line.  where eω  is the operating electrical frequency. The relationship between mλ  and mi  is defined as  )( mm Fi λ=  (2–54) or, conversely,  )(1 mm iF −=λ . (2–55) The saturation curve )(1 miF −  of a typical induction machine is shown in Figure 2–6 along with the air-gap line.  Using (2–54) or (2–55), one can also define the so-called steady-state saturable magnetizing inductance as mmmm iL λλ =)( . Consequently, (2–44) and (2–45) can be rewritten for saturable machines as  +≡= ∑=Nzqrzqsmmmqmmmq iiLiL1)()( λλλ  (2–56) 43  λmdφimdimq λmqλmq axisd axisim Figure 2–7. Projections of the main flux mλ  and magnetizing current mi  onto the q and d axes of an induction machine.  +≡= ∑=Nzdrzdsmmmdmmmd iiLiL1)()( λλλ . (2–57)   For smooth-air-gap symmetrical induction machines, the main flux mλ  and magnetizing current mi  are aligned and are related to their orthogonal projections by  22 mdmqm λλλ +=  (2–58)  22 mdmqm iii += . (2–59) Consequently, the effect of cross saturation is taken into account [77]. Equations (2–58) and (2–59) can be visualized in the vector diagram of Figure 2–7, wherein φ  represents the flux vector angle.  For EMT simulations, )( miF  (or )(1 miF − ) can be represented using several types of functions. For instance, main flux saturation is represented within PLECS and SimPowerSystems’ built-in models using arctangent [104] and polynomial functions, respectively. Differentiable functions are particularly useful for SVB models which include the dynamic inductance mm didλ , e.g., Mixed/Current-based qd [42], [74], [76] and VBR [96], [108], [109] models. Piecewise-linear functions can also be (and are) used in such44  λmλres1λres2 imLD1 LD2 λresjλres(j+1)LDj LD(j+1)  Figure 2–8. Piecewise-linear representation of a typical saturation curve of an induction machine.  situations, for example by computing the dynamic inductance using Lagrange polynomial interpolation for non-uniformly spaced points [135]. This approach is used throughout this thesis.  The main flux saturation characteristic is usually represented in EMTP-type programs by a piecewise-linear function. An example of a piecewise-linear saturation characteristic with an arbitrary number of segments is reproduced in Figure 2–8. The main flux mλ  and magnetizing current mi  are related by  jmDjm iL resλλ +=  (2–60) where DjL  and jresλ  are the dynamic inductance (i.e., the slope) and residual flux, respectively, of the j th piecewise-linear segment depicted in Figure 2–8. Using (2–58)–(2–60) and Figure 2–8, the orthogonal projections of mλ  can also be written as  qmqDjmq iL resλλ +=  (2–61)  dmdDjmd iL resλλ +=  (2–62) wherein the residual fluxes qresλ  and dresλ  are expressed as 45   mmqjjq λλλφλλ resresres cos ==  (2–63)  mmdjjd λλλφλλ resresres sin == . (2–64)  An important advantage of this type of representation is that its equivalent circuit parameter ( DjL ) is a function of the operating segment as opposed to the steady-state magnetizing inductance ( )( mmL λ ). Consequently, for machine models that are interfaced through the conductance matrix, e.g., PD, Thévenin qd, and VBR models, re-factorizations are only needed when the saturation curve operating segment j  changes. 2.3.3 Implementation in SVB Programs  Implementation of the standard saturable qd lumped-parameter models in SVB programs is presented in this section. SVB PD models were not implemented for this thesis and as such are not discussed here. The equations of the existing and proposed SVB VBR models will be presented in Chapters 4 and 6. 2.3.3.1 qd models  As explained in the introduction, this thesis considers two groups of qd models. We will first consider Flux-based qd models. In particular, the Flux-based model derived in this section is formulated as to allow the use of any saturation function – or even no saturation function, i.e., a magnetically linear model. As for Mixed/Current-based qd models, several combinations of state variables are possible. The generalized flux space vector approach [77] will be used to derive a Mixed/Current-based single-rotor-circuit qd induction machine model whose state vector comprises qsi , dsi , qsλ , and dsλ .  46   Flux-based qd model  Unsaturated Flux-based qd models can be reformulated as to consist solely of state variables and inputs, i.e., by algebraically eliminating the winding currents and magnetizing fluxes appearing in (2–34)–(2–38). However, this is not practical for saturated models. Here, only the winding currents are algebraically eliminated; the magnetizing fluxes mqλ  and mdλ  are considered as the output of an arbitrary saturation function whose input is the machine’s state variables. It will be seen in Chapter 3 that the various FC functions (including the proposed one) can be seamlessly connected to this model.  The winding currents can be formulated as a function of state variables and magnetizing fluxes by reformulating (2–39)–(2–43) as follows [29]:  lsmqqsqs Liλλ −=  (2–65)  lsmddsds Li λλ −=  (2–66)  lsss Li 00λ=  (2–67)  .,...,1, NzLilrzmqqrzqrz =−=λλ (2–68)  .,...,1, NzLilrzmddrzdrz =−=λλ  (2–69)  Equations (2–65)–(2–69) can then be substituted into (2–34)–(2–38). Solving for the time derivative of the flux linkages in the resulting expressions yields  ( ) dsmqqslssqsqs Lrvp ωλλλλ −−−=  (2–70) 47  Electrical State Equations(2–70)–(2–74)SaturationFunctionMechanical State Equations(2–48), (2–49)ElectromagneticTorque(2–47), (2–65),(2–66)λqds, λqdrλmqdλqds, λqdrTeTm ωrvqd0sω Figure 2–9. Block diagram of a Flux-based qd induction machine model with an arbitrary saturation function for implementation in SVB programs.  ( ) qsmddslssdsds Lrvp ωλλλλ +−−=  (2–71)  slssss Lrvp 000 λλ −=  (2–72)  ( ) ( ) .,...,1, NzLrp drzrmqqrzlrzrzqrz =−−−−= λωωλλλ  (2–73)  ( ) ( ) .,...,1, NzLrp qrzrmddrzlrzrzdrz =−+−−= λωωλλλ  (2–74)  The Flux-based SVB qd induction machine model is comprised of the following subsystems: the electrical state equations (2–70)–(2–74); the mechanical state equations (2–48) and (2–49); the electromagnetic torque calculated using (2–47), (2–65), and (2–66); and any saturation function computing the magnetizing fluxes as a function of the state variables, e.g., the FC functions in Chapter 3. The expression “saturation function” is herein used for generality: it is also possible to simply relate the magnetizing fluxes and state variables linearly [29]. A block diagram depicting this model is presented in Figure 2–9. 48   Mixed/Current-based qd model  The generalized flux space vector approach is a technique to derive saturable lumped-parameter induction machine models with almost any combination of independent state variables [77], [78]. It is shown in this section how to derive a saturable machine model with one rotor circuit using this approach. To follow the same convention as in the original references, the basic qd equations (2–34)–(2–43), (2–56), and (2–57) are rewritten in compact space vector form [107] as  sssss pirv λλιω ++=  (2–75)  ( ) 11110 rrrrr pir λλωωι +−+=  (2–76)  mslss iL λλ +=  (2–77)  mrlrr iL λλ += 111  (2–78)  ( )1)()( rsmmmmmm iiLiL +≡= λλλ  (2–79) where underlined variables denote space vectors. Space vectors are related to scalars in qd coordinates by qd fff ι+=  [107] where ι  is the imaginary number (the symbol ι  is used in this thesis since the more common variable j  is already used to identify piecewise-linear saturation segments). To closely follow the derivation presented in [77], the stator zero-sequence component is omitted. It could however be added to the model seamlessly.   The first step consists in selecting the desired state variables (in space vector form) [77]. Without loss of generality, si  and sλ  are herein chosen as state variables. All auxiliary algebraic variables appearing in (2–75) and (2–76) can then be expressed as a function of the chosen state variables using (2–34)–(2–43), (2–56) and (2–57) [77]:  sssmmsmmlsr cicLiLLi λλλλ 12111 )(1)(1 +≡++−=  (2–80) 49   sssmmlrsmmlslrlsr cicLLiLLLL λλλλλ 2221111 )(1)(1 +≡++++−= . (2–81)  The next step is to define a generalized flux space vector λ  (which may or may not have a physical meaning) as the linear combination of state vectors [77],  ss bia λλ +=  (2–82)  such that the coefficients a  and b  be saturation-independent and λ  and mi  be aligned, i.e.,  miΛ=λ . (2–83)  For the chosen set of state variables, λ  can be conveniently defined as [77]  mslss iL λλλ ≡−= . (2–84) Consequently, for this specific model, the inductance Λ  appearing in (2–83) is equal to )( mmL λ . This is not the case for every combination of state variables [77].  The machine differential equations must then be expressed as a function of state variables and inputs. No additional step is needed for the stator equation since (2–75) is already expressed in the desired form. As for the rotor differential equation, (2–80) and (2–81) must be substituted into (2–76). Using the coefficients 11c , 12c , 21c , and 22c  for compactness, and applying the product rule, (2–76) is rewritten as  ( ) ( )( ) ( ) ( ) ssssssrssr pcipcpcipcciccicr λλλωωιλ 222122212221121110 +++++−++= . (2–85) Equation (2–85) is straightforward except for the presence of the time derivatives of 21c  and 22c . Using the chain rule, 21pc  and 22pc  can be reformulated as [77]  )1()1(2121 ΛΛ= pddcpc      and     )1()1(2222 ΛΛ= pddcpc . (2–86) 50  Using standard calculus operations, the time derivative of Λ1  can be developed as [77]  ( )qqdd ppp λλλλλ +Λ−Λ=Λ 211'1)1(  (2–87) where mmmddiddi λλ ≡=Λ '1  . Moreover, from the generalized flux space vector theory, we have that [77]  qqsqs ddciddc γλλ =Λ+Λ )1()1(2221          and       ddsds ddciddc γλλ =Λ+Λ )1()1(2221 . (2–88) For the chosen set of state variables, it is found that 1lrL=γ . Consequently, substituting (2–86)–(2–88) into (2–85), combining with (2–75), and converting to qd components yields the following Mixed/Current-based qd model:  xBxAu svsv p +=  (2–89) where  [ ]Tdsqs vv 00=u      and      [ ]Tqsqsdsqs ii λλ=x  (2–90)  +++−−+−++−=ddlrqdlrddlrlslrlsqdlrlsqdlrqqlrqdlrlsqqlrlslrlssvLLLLLLLLLLLLLLLLLLLLLLLL11111111111110000100A  (2–91)  ( ) ( )( ) ( ) ΛΛ+−−Λ+−−Λ+−Λ−−Λ+−−=111aux11aux111110000rlrrlsrrlrrrrlsrsssvrLLrLLrLLrrrωωωωωωωωωωB  (2–92) 51  and  Λ++= 11auxlrlslrlsLLLLL  (2–93)  µµ 22 sin'1cos11 Λ+Λ=qqL     and      µµ 22 sin1cos'11 Λ+Λ=ddL (2–94)  µµ sincos1'11 Λ−Λ=qdL,     λλµ d=cos ,      and       λλµ q=sin . (2–95)  Equation (2–89) can be easily transformed to standard state-space form as follows:  ( )xBuAx svsvp −= −1 . (2–96) This requires factoring the state matrices continuously since svA  contains the saturation-dependent terms qqL , ddL , and qdL . The latter is sometimes referred to as a cross-saturation term.  Interfacing/solution  Flux-based and Mixed/Current-based qd models can be implemented in SVB programs using basic simulation blocks or state-space matrices as shown in Figure 2–9 and (2–96), respectively. The Mixed/Current-based qd model whose state vector comprises the stator and rotor winding currents can also be implemented using circuit elements in qd coordinates; however, since its equivalent inductances are saturation-dependent and cross saturation introduces coupling between the q and d axes, there is no significant advantage in doing so.  The qd models have a voltage-input, current-output formulation. Consequently, they are usually interfaced with external networks using voltage-controlled current sources [35]. An example of this interfacing approach is presented in Figure 2–10. Machines in52  External Networkqd modelvabcsiabcs Figure 2–10. Interfacing circuit of Flux-based and Mixed/Current-based qd models for SVB programs. series with inductive or switching elements may require the use of snubbers to form a proper state-space model. Snubbers are a source of numerical error and can increase the numerical stiffness of the system [35]. It is also possible to interface qd models with external networks using a time-step relaxation [35]. This approach, which can cause numerical instability when using larger step sizes, is not considered in this thesis. 2.3.4 Implementation in EMTP-Type Programs  Implementation of the standard saturable PD and qd lumped-parameter models in EMTP-type programs is presented in this section. The existing and proposed EMTP-type VBR models will be considered in Chapters 6 and 7. 2.3.4.1 PD model  To derive the equations of the EMTP-type PD model [64], it is convenient to rewrite the flux linkage-current relationship assuming a piecewise-linear saturation function as defined by (2–60)–(2–64). Modifying (2–13) accordingly yields  ( ) +=abcrabcsabcrabcsabcrjTabcsrjabcsrjabcsjabcrabcsresresλλiiLLLLλλ (2–97) where 53   0res1res qd ssabcs λKλ −=      and      qd rrabcr resinvres λKλ =  (2–98)  [ ]TT qdqd s 0res0res λλ =       and       qdNNNNqdr res1111res λ1001λ =××××  (2–99) and where 1  is a full matrix (or vector) filled with 1s. The inductance matrices appearing in (2–97) are defined as  +−−−+−−−+=DjlsDjDjDjDjlsDjDjDjDjlsabcsjLLLLLLLLLLLL5.05.05.05.05.05.0L  (2–100)  =abcrdNjabcrojabcrojabcrojabcjrdabcrojabcrojabcrojabcjrdabcrjLLLLLLLLLL21 (2–101)  .,...,1,5.05.05.05.05.05.0NzLLLLLLLLLLLLDjlrzDjDjDjDjlrzDjDjDjDjlrzabcrdzj =+−−−+−−−+=L  (2–102)  −−−−−−=DjDjDjDjDjDjDjDjDjabcrojLLLLLLLLL5.05.05.05.05.05.0L  (2–103)  °−°+°+°−°−°+=−rrrrrrrrrDjabcwsrj Lθθθθθθθθθcos)120cos()120cos()120cos(cos)120cos()120cos()120cos(cosL  (2–104)  [ ] Nabc wsrjabc wsrjabc wsrjabcsrj 33×−−−= LLLL  . (2–105)  Discretizing (2–10) and substituting (2–97) into the resulting equation gives 54   ( ) +++=)()()()()()()(ttttktktkktabcrhabcshabcrabcsabcrjabcrTabcsrjabcsrjabcsjsabcseeiiLRLLLR0v (2–106) where tk ∆= 2  and  ( )∆−∆−−+∆−∆−−∆−−∆−−−+ ∆−−=)()()()()()()()()()()(resresresresttttkttkttttkttkttkkttttabcrabcsabcrabcsabcrabcsabcrjabcrTabcsrjabcsrjabcsjsabcsabcrhabcshλλλλiiLRLLLR0vee. (2–107) Throughout this dissertation, the subscripts of all saturation-segment-dependent discrete scalars and matrices end with j . Moreover, to emphasize coefficient time dependency, the suffixes “ )(t ” and “ )( tt ∆− ” are added to all discrete scalars and matrices that depend on the angle or speed (of the rotor and/or the reference frame) of the current and previous time step, respectively.  Equation (2–106) could be inserted directly into the network’s conductance matrix G  [57]; however, it would introduce time-dependent elements. We can instead reduce (2–106), which yields  )()()( eq ttt abchabcsabcjabcs eiRv +=  (2–108) where  ( ) ( )Tabcsrjabcrjabcrabcsrjabcsjsabcj tkktkk )()( 1eq LLRLLRR −+−+=  (2–109)  ( ) )()()()( 1 tktktt abcrhabcrjabcrabcsrjabcshabch eLRLee −+−= . (2–110)  The interfacing matrix abcjeqR  might appear at first to be time-dependent since it is constructed using rotor-position-dependent matrices. However, as pointed out in [61], the 55  trigonometric functions cancel out each other due to the machine’s symmetry. Consequently, abcjeqR  is only saturation-segment-dependent.   The rotor currents abcri  are necessary to evaluate the next time step’s history terms. Solving for the rotor currents in (2–106) yields  ( ) ( )  ++−= − )()()()( 1 tttkkt abcrhabcsTqdsrjabcrjabcrabcr eiLLRi . (2–111)  Finally, discretizing the mechanical equations (2–48) and (2–49) gives  ( ) ( ))()(2)()(2)()( 11 ttTtTJPkttTtTJPkttt mmeerr ∆−+−∆−++∆−= −−ωω  (2–112)  ( ))()()()( 1 tttkttt rrrr ∆−++∆−= − ωωθθ . (2–113)  Interfacing/solution  The necessary steps to simulate the PD model at time step t  are described below. It is assumed that the machine’s interfacing resistance/conductance matrix (corresponding to the operating piecewise-linear saturation segment) has already been inserted inside the network’s conductance matrix G . 1) Predict the electrical rotor speed )(trω  and the magnetizing fluxes )(tmqdλ  using linear extrapolation:  )2()(2)(~ ttfttftf ∆−−∆−=  (2–114) where { }mqdrf λ,ω=  and the tilde symbol “~” indicates predicted values.  2) Calculate the electrical rotor position )(trθ  using the trapezoidal integration rule: 56   ( ))()(~)()( 1 tttkttt rrrr ∆−++∆−= − ωωθθ . (2–115) 3) Calculate the main flux )(tmλ  using (2–58). 4) Form )(res tabcsλ  and )(res tabcrλ  using (2–63), (2–64), (2–98), and (2–99). 5) Compute )(tabcshe  and )(tabcrhe  using (2–107), and construct )(tabche  using (2–110). 6) Convert (2–108) to a Norton equivalent and insert its current source into the equivalent current injection vector )(thi  of the EMTP-type system. 7) Find the network’s unknown nodal voltages using (2–8) or (2–9). 8) Using the newly found stator voltages )(tabcsv , solve for the stator currents )(tabcsi  in (2–108), and convert them to qd coordinates using (2–20). 9) Compute )(tabcri  using (2–111) and convert the rotor currents to qd coordinates using (2–21). 10) Update the magnetizing fluxes )(tmqdλ  using (2–61) and (2–62), and re-compute )(tmλ  using (2–58). 11) If the operating piecewise-linear segment j  of the saturation characteristic changes, go to step 12). Otherwise, skip to step 14). 12) Update j , and re-evaluate )(res tabcsλ , )(res tabcrλ , )(tabcri , )(tqdri , )(tmqdλ , and )(tmλ . 13) Insert the new interfacing circuit resistance matrix abcjeqR  into the network’s conductance matrix G  and re-factor it. 57  14) Evaluate the electromagnetic torque )(tTe , electrical rotor speed )(trω , and electrical rotor position )(trθ  using (2–47), (2–112), and (2–113), respectively.  As shown in (2–107), )(tabcshe  and )(tabcrhe  are function of )( ttabcsrj ∆−L . It is very important to construct these history terms using the matrix )( ttabcsrj ∆−L  computed with the electrical rotor position rθ  predicted in step 3) of the previous time step, instead of the corrected position from step 14). As shown in [63], using the corrected position may lead to a significant decrease in numerical accuracy with larger step sizes. This is because the stator and rotor currents are solved at each time step using equations that are linearized around the electrical rotor position predicted in step 3). 2.3.4.2 qd model  For the purpose of deriving EMTP-type qd induction machine models, it is convenient to first write (2–34)–(2–38) and (2–39)–(2–43) in matrix-vector notation. Assuming a piecewise-linear saturation function as in Section 2.3.4.1, and choosing the rotor reference frame, i.e., rωω = , the machine differential equations can be expressed as  ++=qdrsqddqsrqdrsqdqdrssqd p λλ0λiiR00R0v 000 ω  (2–116)  ( ) +=qdrqdsqdrsqdqdrjTqdsrjqdsrjqdsjqdrsqdres0res000λλiiLLLLλλ (2–117) where  [ ]rNrrrNrrqdr rrrrrr  2121diag=R  (2–118)  ( ) ( )[ ]lsDjlsDjlsqdsj LLLLL ++= diag0L  (2–119) 58   =××××××NNNNNNDjqdsrj L1111110001001L  (2–120)  [ ] [ ]lrNlrlrlrNlrlrNNNNDjqdrj LLLLLLL  2121diagdiag += ×× 11L . (2–121)  Since Thévenin prediction-based qd models are prone to numerical oscillations and error accumulation, the qd model considered in this thesis is discretized using the damped trapezoidal rule [5], [30], [136]. Integrating the arbitrary first-order ODE ),( tyfpy =  with the damped trapezoidal rule yields  ),(1),(1)()( ttyfttyftttyty ∆−+∆++∆+∆−=ααα (2–122) where α  is the damping factor and 0 ≤ α  ≤ 1. The damped trapezoidal rule is actually the standard trapezoidal rule when α  = 1, and is equivalent to Backward Euler (BE) when α  = 0. Smaller values of α  provide better damping, but introduce more discretization error.  Discretizing (2–116) with the aforementioned damped trapezoidal rule and substituting (2–117) into the resulting equation gives [5], [64]  ( ) +++=)()()()('''')( 0000tttttqdrhqdshqdrsqdqdrjqdrTqdsrjqdsrjqdsjssqdeeiiLRLLLR0vαααα (2–123) where  ( )∆−∆−−+ ∆−++∆−∆−−−−−+ ∆−−=)()(')()(')()()()('''')()()(res0resres0res0000tttttttttttttttttqdrqdsqdrqdsdqsrdqsrqdrsqdqdrjqdrTqdsrjqdsrjqdsjssqdqdrhqdshλλλλ0λ0λiiLRLLLR0veeαααωωααααααα (2–124) 59  where ( ) t∆+= αα 1' . Since the rotor is assumed short-circuited, (2–123) can be reduced to the three-phase Thévenin equivalent  )()()( 000eq0 ttt qdhsqdqdjsqd eiRv +=  (2–125) where  ( ) ( )Tqdsrjqdrjqdrqdsrjqdsjsqdj LLRLLRR '''' 100eq αααα −+−+=  (2–126)  ( ) )('')()( 100 ttt qdrhqdrjqdrqdsrjqdshqdh eLRLee −+−= αα . (2–127)  Since the history terms 0qdshe  and 0qdrhe  are function of the rotor currents, it is necessary to compute these currents at every time step. Solving for )(tqdri  in (2–123) yields  ( ) ( ) ( )  ++−= − )('')( 01 ttt qdrhsqdTqdsrjqdrjqdrqdr eiLLRi αα . (2–128)  Interfacing/solution  As explained in the introduction and in [35], there are several ways to interface qd models with the rest of the network in EMTP-type programs. Only the so-called Thévenin prediction-based qd model is considered in this section [35]. This approach consists of developing a three-phase Thévenin equivalent of the machine model in abc coordinates and incorporating it into the network nodal equations (2–8) or (2–9) as shown in Figure 2–3.  A Thévenin equivalent of the machine model in qd coordinates is given in (2–125). Converting it to abc coordinates using (2–20) and (2–27) yields  )()()( eq ttt abchabcsabcjabcs eiRv +=  (2–129) 60  where  )()()( 01 ttt qdhsabch eKe −= . (2–130) Since 0eqqdjR  is a diagonal matrix that can be expressed in the form of  [ ]211diag0eq rrr jjqdj =R  (2–131) where jr1  is a saturation-segment-dependent scalar and 2r  is a constant, the resistance submatrix abcjeqR  will have the following structure  =SjMjMjMjSjMjMjMjSjabcjrrrrrrrrreqR  (2–132) where 2rrr ajSj += , ajMj rr 5.0−= , and ( )( )213/2 rrr jaj −= . The interfacing matrix abcjeqR  is therefore saturation-segment-dependent but rotor-position-independent.  The necessary steps to simulate the qd model are similar to those of the PD model presented in Section 2.3.4.1. Here also, it is assumed that abcjeqR  has already been inserted into the network’s conductance matrix G . The simulation steps at an arbitrary time step t  are as follows: 1) Predict the electrical rotor speed )(trω  and the magnetizing fluxes )(tmqdλ  using linear extrapolation as shown in (2–114). 2) Predict the speed voltage term )()( tt dqsrs λu ω≡  using a three-point linear predictor with smoothing [5]:  )3(75.0)2(5.0)(25.1)(~ ttttttt ssss ∆−−∆−+∆−= uuuu . (2–133) 61  3) Calculate the electrical rotor position )(trθ  using the trapezoidal integration rule as shown in (2–115). 4) Calculate the main flux )(tmλ  using (2–58). 5) Form )(0res tqd sλ  and )(res tqd rλ  using (2–63), (2–64), and (2–99). 6) Compute )(0 tqdshe  and )(tqdrhe  using (2–124), construct )(0 tqdhe  using (2–127), and evaluate )(tabche  using (2–130). 7) Convert (2–129) to a Norton equivalent and insert its current source into the equivalent current injection vector )(thi  of the EMTP-type system. 8) Find the network’s unknown nodal voltages using (2–8) or (2–9). 9) Using the newly found stator voltages )(tabcsv , solve for the stator currents )(tabcsi  in (2–129), and convert )(tabcsv  and )(tabcsi  to qd coordinates using (2–20). 10) Compute )(tqdri  using (2–128). 11) Update the magnetizing fluxes )(tmqdλ  using (2–61) and (2–62), and re-compute )(tmλ  using (2–58). 12) If the operating piecewise-linear segment j  of the saturation characteristic changes, go to step 13). Otherwise, skip to step 15). 13) Update j , and re-evaluate )(0res tqd sλ , )(res tqd rλ , )(tqdri , )(tmqdλ , and )(tmλ . 14) Insert the new interfacing circuit resistance matrix abcjeqR  into the network’s conductance matrix G  and re-factor it. 62  15) Evaluate the electromagnetic torque )(tTe , electrical rotor speed )(trω , and electrical rotor position )(trθ  using (2–47), (2–112), and (2–113), respectively. 16) Update the speed voltage term )(tsu  using  −−+=0)()()()()()( ttiLttiLtt mqqslsmddslsrs λλωu . (2–134) 2.4  Synchronous Machines  The fundamentals of synchronous machine modeling for EMT simulations are presented in this section. Lumped-parameter wound-rotor synchronous machine models are usually represented by a three-phase symmetrical stator winding, a few damper windings, and one field winding [29]. All these windings are sinusoidally distributed. Some of the damper windings are aligned with the field winding on the so-called direct axis, while the remaining damper windings are displaced by 90° and aligned on the quadrature axis [29]. The equivalent circuit of a two-pole synchronous machine represented in physical variables with M  damper windings on the q axis and N  damper windings on the d axis is reproduced in Figure 2–11 [29]. This model is suitable for both salient-pole and round-rotor synchronous machines.  In salient-pole synchronous machines, the damping currents typically stay near the rotor periphery. A single equivalent damper winding on each axis is usually sufficient to represent this effect [29]. In round-rotor machines, the damping currents can flow both inside (iron) and outside (physical dampers) the rotor. Models with two q-axis damper windings and one d-axis damper winding are frequently used to represent round-rotor synchronous machines [29]. PLECS’ and SimPowerSystems’ built-in salient-pole63  rspλasiasvas+ -pλbs+-pλcs+-rsrsibsicsvcsvbs{ {StatorRotorrkq1pλkq1-+ikq1rkq2pλkq2-+ikq2. . .rkqMpλkqM-+ikqMrkd1pλkd1+ -ikd1rkd2pλkd2+ -ikd2 . . .rkdNpλkdN+ -ikdNrfdpλfd+ -ifd+ -vfd Figure 2–11. Equivalent circuit of a wound-rotor synchronous machine represented in physical variables. synchronous machine models have one damper winding on each axis, whereas their built-in round-rotor machine models have an additional q-axis damper winding. The synchronous machine model available in PSCAD’s library has 1 d-axis rotor winding and up to 2 damper windings on the q axis.  There also exists some literature on higher order lumped-parameter rotor models. For instance, it is shown in [137] that third-order rotor models (with 3 and 2 damper windings on the q and d axes, respectively) predict different torsional dynamics than lower order models. Reference [138] demonstrates that second-order rotor models (with 2 and 1 damper windings on the q and d axes, respectively) poorly predict the asynchronous braking torque and asynchronous starting. It is also concluded in [139] that even higher order d-axis rotor models (with 3 to 5 damper windings) might be necessary to properly represent the rotor impedance from 0.01 to 200 Hz. Moreover, models with more than 1 d-axis damper winding and/or 2 q-axis damper windings have also been used to study 64  machine-converter systems, e.g., [24], [104], [108]. EMTP-RV’s built-in model allows up to 2 d-axis and 3 q-axis damper windings.   Lumped-parameter synchronous machine models are sometimes derived by assuming that all windings on the same magnetic axis have identical magnetic coupling. This assumption is very common in the SVB community (e.g., [24], [29], [80], and [86]) and as such is reflected in PLECS and SimPowerSystems’ built-in models. The resulting compact notation facilitates the formulation and implementation of synchronous machine models. However, it was shown in [140] that differential leakage inductances, which represent the coupling difference between individual rotor and stator windings, are necessary to properly evaluate rotor quantities. Alternatively, differential leakage inductances can be sometimes eliminated using linear circuit theory [140], [141], which requires modifying the value of the magnetizing inductances and complicates the representation of main flux saturation. EMTP-RV’s and PSCAD’s synchronous machine models have provisions to incorporate differential leakage inductances. As with induction machines, it is also possible to directly represent rotors using arbitrary linear TFs [96], [109], [142], [143]. This approach preserves the physical meaning of the magnetizing inductances, but its notation is rather involved.  A lumped-parameter wound-rotor synchronous machine model with M  damper windings on the q axis and N  damper windings on the d axis is considered in this thesis. The equations are derived using motor sign convention. To facilitate reading, it is assumed that all windings on a given magnetic axis have identical magnetic coupling. Here again, we stress that this assumption is not necessary for the derivation of the proposed models. 65  2.4.1 Machine Equations 2.4.1.1 Voltage equations in physical variables  The voltage equations in physical variables of the general-purpose lumped-parameter synchronous machine model are [29]  +=qdrabcsqdrabcsrsqdrabcs p λλiiR00Rvv (2–135) where abcsv , abcsi , and abcsλ  are the stator voltages, currents, and flux linkages in abc coordinates, respectively; qdrv , qdri  and qdrλ  are the rotor voltages, currents, and flux linkages in qd coordinates, respectively; and  [ ]ssss rrrdiag=R  (2–136)  [ ]fdkdNkdkdkqMkqkqr rrrrrrr  2121diag=R     (2–137) where sr  is the stator resistance, kqzr  and kdzr  are the resistances of the z th q- and d-axis damper windings, respectively, and fdr  is the field winding resistance.  The stator and rotor flux linkages abcsλ  and qdrλ  are related to the stator and rotor currents abcsi  and qdri  by [29]  ( ) =qdrabcsqdrTabcsrabcsrabcsqdrabcsiiLLLLλλ32 . (2–138) The inductance matrices appearing in (2–138) are expressed as 66  °−−+−−°+−−−−°+−+°−−−°+−−°−−−−+=)1202cos()2cos(5.0)1202cos(5.0)2cos(5.0)1202cos()1202cos(5.0)1202cos(5.0)1202cos(5.0)2cos(rBAlsrBArBArBArBAlsrBArBArBArBAlsabcsLLLLLLLLLLLLLLLLLLLLLθθθθθθθθθL (2–139)  [ ]qdrdqdrqqdr LLL diag=  (2–140)  +++=mqlkqMmqmqmqmqlkqmqmqmqmqlkqqdrqLLLLLLLLLLLL21L  (2–141)  ++++=mdlfdmdmdmdmdmdmdlkdNmdmdmdmdmdmdmdmdlkdmdmdmdmdmdlkdqdrdLLLLLLLLLLLLLLLLLLLLLLLL21L  (2–142)  Mrrrrrrrrrmqabcsrq L×°+°+°+°−°−°−=3)120cos()120cos()120cos()120cos()120cos()120cos(coscoscosθθθθθθθθθL  (2–143)  ( )13)120sin()120sin()120sin()120sin()120sin()120sin(sinsinsin+×°+°+°+°−°−°−=Nrrrrrrrrrmdabcsrd LθθθθθθθθθL  (2–144)  [ ]abcsrdabcsrqabcsr LLL =  (2–145) where lsL  is the stator leakage inductance, lkqzL  and lkdzL  are the leakage inductances of the z th q- and d-axis damper windings, respectively, lfdL  is the field winding leakage inductance, mqL  and mdL  are the q- and d-axis magnetizing inductances, respectively, and rθ  is the electrical rotor position. The equivalent inductances AL  and BL  are defined as 67   ( )mdmqA LLL += 31       and      ( )mqmdB LLL −= 31 . (2–146) 2.4.1.2 Voltage equations in qd coordinates  The rotor variables in (2–135) are expressed in qd coordinates that rotate along with the rotor. It is therefore convenient to express the stator variables in the same reference frame. This can be done by using the transformation presented in (2–20) with rθθ ≡ . Applying this transformation to (2–135) yields [29]  qsdsrqssqs pirv λλω ++=  (2–147)  dsqsrdssds pirv λλω +−=  (2–148)  ssss pirv 000 λ+=  (2–149)  .,...,1, Mzpirv kqzkqzkqzkqz =+= λ  (2–150)  .,...,1, Nzpirv kdzkdzkdzkdz =+= λ  (2–151)  fdfdfdfd pirv λ+=  (2–152) where qsv , dsv , and sv0  are the q-axis, d-axis, and zero-sequence stator voltages, respectively; qsi , dsi , and si0  are the q-axis, d-axis, and zero-sequence stator currents, respectively; qsλ , dsλ , and s0λ  are the q-axis, d-axis, and zero-sequence stator flux linkages, respectively; kqzv , kdzv , and fdv  are the voltages of the z th q-axis damper winding, z th d-axis damper winding, and field winding, respectively; kqzi , kdzi , and fdi  are the currents of the z th q-axis damper winding, z th d-axis damper winding, and field winding, respectively; kqzλ , kdzλ , and fdλ  are the flux linkages of the z th q-axis damper winding, z th d-axis damper winding, and field winding, respectively; and rω  is the electrical rotor speed. Since damper windings are short-circuited, kqzv  and kdzv  are equal to 68  0. These voltages will henceforth be omitted to simplify notation. The flux linkages in qd coordinates are expressed as [29]  mqqslsqs iL λλ +=  (2–153)  mddslsds iL λλ +=  (2–154)  slss iL 00 =λ  (2–155)  .,...,1, MziL mqkqzlkqzkqz =+= λλ  (2–156)  .,...,1, NziL mdkdzlkdzkdz =+= λλ  (2–157)  mdfdlfdfd iL λλ +=  (2–158) and the magnetizing fluxes mqλ  and mdλ  are given by [29]  +≡= ∑=Mzkqzqsmqmqmqmq iiLiL1λ  (2–159)  ++≡= ∑=Nzfdkdzdsmdmdmdmd iiiLiL1λ  (2–160) where mqi  and mdi  are the q- and d-axis magnetizing currents, respectively, and mqL  and mdL  are the q- and d-axis magnetizing inductances, respectively. The electromagnetic torque eT  can also be computed using (2–47). The equivalent circuit of the general-purpose lumped-parameter synchronous machine model represented in qd coordinates is reproduced in Figure 2–12. 69  iqs+ -rsLlsimqvqs+-ωrλdsrkq1 Llkq1 ikq1rkq2 Llkq2 ikq2rkqM LlkqM ikqMLmqids+-rsLlsimdvds+-ωrλqsrkd1 Llkd1 ikd1rkd2 Llkd2 ikd2rkdN LlkdN ikdNLmdi0s rsLlsv0s+-+ -rfd Llfdvfd ifd Figure 2–12. Equivalent circuit of a wound-rotor synchronous machine represented in qd coordinates. 2.4.1.3 Mechanical equations  The mechanical equations of a synchronous machine represented by a single rigid body are given by (2–48) and (2–49). The rotor angle δ  is sometimes of interest. It is defined as [29]  er θθδ −=  (2–161) 70  where eθ  is the angle of the terminal voltage.  A multimass representation can sometimes be necessary, in particular to study subsynchronous resonance (SSR) [137], [144]. While all the studies in this thesis have been executed using a single rigid body representation, the proposed models can be easily extended to incorporate multiple masses by modifying (2–48) and (2–49) [5]. 2.4.2 Magnetic Saturation  Synchronous machine saturation is known to affect steady-state and transient power system stability [49], [110] as well as various EMTs [110]. Representation of main flux saturation in general-purpose lumped-parameter synchronous machine models has received considerable attention for several decades, owing to the challenge of developing an approach that offers sufficient modeling accuracy yet remains simple and general enough for most synchronous machines.   The simplest saturation function to generate is the open-circuit characteristic (OCC) [19], which is obtained by measuring the field current and the stator voltage when the stator is open circuited. Since the main flux and magnetizing current are aligned with the d axis under these conditions, this function is also often referred to as the d-axis saturation characteristic (e.g, [76], [145]). It is herein defined as  )( mddmd Fi λ=  (2–162) or  )(1 mddmd iF −=λ  (2–163) where mdλ  is related to the measured rms stator line-to-neutral voltage NLasV ,  and mdi  to the measured field current fdI  (properly referred to the stator side) by 71  imdλ md Air-gap lineOCC (Saturation curve) Figure 2–13. Typical open-circuit characteristic (OCC, or saturation curve) of a synchronous machine and the corresponding air-gap line.   bNLasmdVωλ ,2=  (2–164)  fdmd Ii ≈ . (2–165) A typical OCC is shown in Figure 2–13.  In lumped-parameter salient-pole synchronous machine models, it is fairly common to only represent main flux saturation along the d axis, based on the premise that the q-axis path consists mostly of air [19]. This approach is used in PSCAD and SimPowerSystems. However, it was shown experimentally in [145] and [146] that q-axis saturation and cross saturation can have a noticeable impact even for salient-pole synchronous machines.  As for round-rotor machines, magnetic saturation is usually taken into account by applying the OCC to the main flux mλ  [19], which is defined as  22 mdmqm λλλ += . (2–166) 72  Consequently, saturation is accounted for along both axes. The effect of cross saturation is also taken into account [80]. The built-in round-rotor synchronous machine models available in EMTP-RV, PLECS, and SimPowerSystems use this type of representation. However, the q-axis path might saturate significantly more than the d-axis’ [19], [147], making the use of the OCC for all load angles questionable.  Over the years, numerous researchers have proposed advanced methods to improve the representation of magnetic saturation in lumped-parameter synchronous machine models, e.g., [19], [75], [142], [145], [146], [148]. The common challenge with these approaches remains the determination of the proper saturation parameters. For example, while modeling accuracy can certainly benefit from the knowledge of the q-axis saturation curve, it is in most cases significantly more complicated to obtain than the OCC [145], [146]. Consequently, it is debatable whether system analysts have the necessary information to properly use such advanced saturable models.   Another difficulty in representing magnetic saturation in synchronous machine models is that the main flux mλ  and magnetizing current mi  are not necessarily aligned [75]. To simplify saturation modeling, an equivalent isotropic machine in which mλ  and mi  are always aligned is considered in this dissertation [75], [76], [80], [112], [149]. The main flux and magnetizing current of this equivalent machine are related to their q- and d-axis projections by  22mdFmqm Sλλλ +=  (2–167)  ( ) 22 mdmqFm iiSi +=  (2–168) where FS  is the saliency factor. These relationships are illustrated in Figure 2–14. 73  λmdφimdSFimqλmqλmq axisd axisimSF Figure 2–14. Projections of the main flux mλ  and magnetizing current mi  onto the q and d axes of an equivalent isotropic synchronous machine.  The saliency factor FS  can be defined in several ways. For instance, it is function of the magnetizing current in [75]. Whereas this two-factor approach can increase modeling accuracy, it requires extensive measurements and/or the use of FEA [75]. In this thesis, we opt for the simpler single saturation factor approach [76], [80], [112], [149], wherein FS  is constant and defined as  mdumquF LLS =  (2–169) where mquL  and mduL  are the unsaturated q- and d-axis magnetizing inductances, respectively. Having defined an equivalent isotropic machine, the OCC (2–162) can be applied to the main flux as   )( mdm Fi λ=  (2–170) thereby taking into account q- and d-axis saturation as well as cross saturation. The reader is referred to the end of Section 2.3.2 for details on how to implement (2–170) in simulation environments. Finally, the steady-state saturable d-axis magnetizing inductance is defined as  74   mmmmd iL λλ =)(  (2–171) while its q-axis counterpart is given by   )()( 2 mmdFmmq LSL λλ = . (2–172) Consequently, we can reformulate (2–159) and (2–160) for saturable machines as  +≡= ∑=Mzkqzqsmmqmqmmqmq iiLiL1)()( λλλ  (2–173)  ++≡= ∑=Nzfdkdzdsmmdmdmmdmd iiiLiL1)()( λλλ . (2–174)  The single saturation factor approach is suitable for both round-rotor and salient-pole synchronous machines, and it becomes identical to the traditional approach used for round-rotor machines [19] when mdumqu LL = . PLECS’ built-in synchronous machine models use this technique to represent saturation. The single saturation factor approach offers a good compromise between modeling accuracy and simplicity. If higher accuracy is desired, more advanced means of accounting for saturation such as those of [75], [142], [145], [146], and [148] might be used.  The literature on leakage flux saturation is much less abundant for synchronous machines than for induction machines, in part because leakage inductances saturate negligibly under nominal conditions and that synchronous machines are rarely subjected to high winding currents for more than a few cycles. Nevertheless, it is shown in [150] that during a steady-state short circuit, the value of the stator leakage inductance can decrease by up to 25%. However, as pointed out in the discussion of [150], it is not obvious whether stator leakage inductances saturate in a similar way during transient conditions. As a result, synchronous machine leakage flux saturation is neglected in this dissertation. This assumption is consistent with the state of the art, since the built-in synchronous machine 75  models of EMTP-RV, PLECS, PSCAD, and SimPowerSystems all assume unsaturated stator and rotor leakage inductances. 2.4.3 Implementation in SVB Programs  Implementation of the standard saturable qd lumped-parameter synchronous machine models in SVB programs is presented in this section. As with induction machines, SVB PD synchronous machine models are not considered in this thesis and thus are not presented here, while implementation of VBR models is explained in Chapter 5. Finally, EMTP-type synchronous machine models are not considered in this thesis. 2.4.3.1 qd models  Similarly to Section 2.3.3.1, we first derive the equations of a Flux-based qd synchronous machine model for an arbitrary saturation function. Then, using the generalized flux space vector approach, we present a Mixed/Current-based qd model using the single saturation factor approach presented in Section 2.4.2.  Flux-based qd model  A Flux-based qd synchronous machine model can be obtained by algebraically eliminating the winding currents from (2–147)–(2–152). This is achieved by first reformulating (2–153)–(2–158) as [29]  lsmqqsqs Liλλ −=  (2–175)  lsmddsds Li λλ −=  (2–176)  lsss Li 00λ=  (2–177) 76   .,...,1, MzLilkqzmqkqzkqz =−=λλ (2–178)  .,...,1, NzLilkdzmdkdzkdz =−=λλ  (2–179)  lfdmdfdfd Liλλ −= . (2–180)  The resulting winding currents are then substituted into (2–147)–(2–152), which gives [29]  ( ) dsrmqqslssqsqs Lrvp λωλλλ −−−=  (2–181)  ( ) qsrmddslssdsds Lrvp λωλλλ +−−=  (2–182)  slssss Lrvp 000 λλ −=  (2–183)  ( ) .,...,1, MzLrp mqkqzlkqzkqzkqz =−−= λλλ  (2–184)  ( ) .,...,1, NzLrp mdkdzlkdzkdzkdz =−−= λλλ  (2–185)  ( )mdfdlfdfdfdfd Lrvp λλλ −−= . (2–186)  The Flux-based SVB qd synchronous machine model is comprised of the following subsystems: the electrical state equations (2–181)–(2–186); the mechanical state equations (2–48) and (2–49); the electromagnetic torque calculated using (2–47), (2–175), and (2–176); and any saturation function computing the magnetizing fluxes as a function of 77  the state variables. This model can be implemented similarly to the induction machine model as shown in Figure 2–9.  Mixed/Current-based qd model  A synchronous machine model with main flux saturation is derived in this section using the generalized flux space vector approach [80]. Without loss of generality, the model has one damper winding on each axis and its state variables are si , fdi , and mλ . As in Section 2.3.3.1, underlined variables indicate space vectors, i.e., qd fff ι+= . The stator zero-sequence component is omitted for compactness. It can be appended to the model with little effort. Finally, main flux saturation is accounted for using the single saturation factor approach explained at the end of Section 2.4.2.  As a first step, the auxiliary algebraic variables appearing in (2–147)–(2–152) must be expressed as a function of electrical state variables [80]. First, the stator and field winding flux linkages are formulated as a function of si , fdi , and mλ  as in (2–153), (2–154), and (2–158). Then, using (2–173) and (2–174), the damper winding currents are expressed as   qsmmqmqkq iLi −= )(1 λλ (2–187)  fddsmmdmdkd iiLi −−= )(1 λλ . (2–188) Finally, the damper winding flux linkages are formulated as a function of state variables by substituting (2–187) and (2–188) into (2–156) and (2–157), respectively, yielding  mqqsmmqmqlkqkq iLL λλλλ +−= )(11  (2–189) 78   mdfddsmmdmdlkdkd iiLL λλλλ +−−= )(11 . (2–190)  Next, we must define a generalized flux space vector λ  as the linear combination of state variables [80],    ( ) ( )( ) qdmqqqsqFmddfddqsd biaScibia ιλλλιλλ +≡++++= 1  (2–191) such that λ  and mi  (see (2–168)) are aligned, i.e.,  miΛ=λ . (2–192) Since mqλ  and mdλ  are state variables, we choose 0=== qdd aba  and 1== qd bc , which yields )( mmdL λ≡Λ . As stated in [80], the equivalent inductance Λ  is equal to )( mmdL λ  for any possible set of state variables when modeling synchronous machines with non-smooth air gaps.  We then substitute the auxiliary algebraic variables defined in (2–153), (2–154), (2–158), and (2–187)–(2–190) into (2–147), (2–148), and (2–150)–(2–152), which gives  ( ) qsmddslsrqssqs piLirv λλω +++=  (2–193)  ( ) dsmqqslsrdssds piLirv λλω ++−=  (2–194)  +++−−= )(1)()(01111mmqlkqmqmqmmqlkqqslkqqsmmqmqkq LLppLLpiLiLr λλλλλλ (2–195) ( ) ++++−−−= )(1)()(01111mmdlkdmdmdmmdlkdfddslkdfddsmmdmdkd LLppLLpipiLiiLr λλλλλλ (2–196)  mdfdlfdfdfdfd ppiLirv λ++= . (2–197) 79   The time derivative of the magnetizing inductances )( mmqL λ  and )( mmdL λ  must be calculated in order to formulate (2–195) and (2–196) in state-space form [80]. It is recalled that the time derivative of the generalized inductance Λ  has been expressed in terms of the generalized flux λ  and its q and d projections in (2–87). Consequently, we can first substitute (2–87) into (2–195) and (2–196) using the aforementioned relationships between )( mmqL λ , )( mmdL λ , and Λ . The generalized fluxes qλ  and dλ  are then replaced by state variables using (2–191). Equations (2–193)–(2–197) can thus be expressed in the form of (2–89) or (2–96) where [80]  [ ]Tfddsqs vvv 00=u     and      [ ]Tmdmqfddsqs iii λλ=x  (2–198)  +−−+−=ddlkdqdlkdlkdlkdqdlkqqqlkqlkqlfdlslssvLLLLLLLLLLLLLL111111110100100010000100A  (2–199)  Λ−−Λ−−−=1112110000000000000kdkdkdFkqkqfdrslsrrlsrssvrrrSrrrrLLrωωωωB  (2–200) and   Λ+Λ= µµ 222 sin'1cos111Fqq SL      and      µµ 22 sin1cos'11 Λ+Λ=ddL (2–201) 80   µµ sincos1'111 Λ−Λ=Fqd SL,   mmddλλλλµ ≡=cos ,   and   mmqFqS λλλλµ 1sin ≡=  (2–202)  mmmddiddi λλ ≡=Λ '1 . (2–203)  Similarly to induction machines, the state-space matrices are saturation-dependent due to the terms qqL , ddL , and qdL . The latter is also sometimes referred to as a cross-saturation term.  Interfacing/solution  Flux-based and Mixed/Current-based qd synchronous machine models can be interfaced to external power networks as described in Section 2.3.3.1. 2.5  Numerical Error Assessment  Several new machine models are proposed in this thesis in order to improve the speed of EMT simulations. Since the majority of these models are based on numerical approximations that introduce some degree of numerical error, it is crucial to properly assess their overall numerical accuracy. The most foolproof method would be to obtain the analytical solutions of the original and proposed models, from which we could derive error bounds. However, due to the complexity of these nonlinear time-varying dynamic models, such analytical solutions are unavailable. Consequently, we assess the numerical accuracy of the subject models by comparing their simulated trajectories with reference solutions.  It is common in the engineering community to compare computer models with experimental measurements. A drawback of this approach in the context of this work is that the modeling and numerical errors become superimposed, making it difficult to distinguish between the two. A different method is used in this thesis [30], [88], [92]. In81  Step sizeNumerical error  Model CModel B Model A Figure 2–15. General behavior of the numerical error of different machine models as a function of the step size (the x axis is presented on a logarithmic scale). order to solely focus on the numerical accuracy of the subject models, a reference solution that is virtually devoid of numerical error is considered. This numerical-error-free reference solution is achieved by simulating an equivalent (i.e., based on the exact same physical modeling assumptions) algebraically exact model with very small step sizes, thereby taking advantage of the consistency of such models. The concept of consistency can be visualized in Figure 2–15. Therein, Models A and B are termed consistent since their numerical error vanishes as the step size goes towards zero; however, Model C is not consistent since it introduces numerical error even when the step size tends towards zero.  An infinite amount of case studies can be executed to compare the proposed models; similarly, several machine and network variables can be analyzed in each case study. For the sake of conciseness, only a few studies and variables are considered in each chapter. In particular, much attention is devoted to developing case studies in which the proposed models operate under stressed conditions; in other words, case studies and machine parameters have been chosen as to intentionally introduce a large amount of numerical error. Moreover, variables that are most often of interest for system analysts and that tend to have significant numerical error are analyzed. 82  Time ReferenceSubject modeli1 i2im–1imim+1iNsi1refi2refim–1imim+1 iNsrefrefref ref Figure 2–16. Example of the trajectory of an arbitrary variable i  as predicted by a subject model along with a reference curve. The vertical lines indicate the numerical error at each solution point.  Based on the above discussion, two complementary approaches are used in this document to assess the numerical accuracy of the subject models. The first one consists of plotting the transient trajectories of a few machine variables as predicted by the subject models, along with their equivalent reference solutions. This method offers a qualitative overview of the numerical precision of the considered models. An example is shown in Figure 2–16, wherein the vertical lines indicate the numerical error of the subject model at each solution point.  The second and more quantitative approach consists of calculating the 2-norm (cumulative) relative error [118] of the predicted transient trajectories. The 2-norm relative error of the example from Figure 2–16 is computed using  ( )( )∑∑==−=ssNmmNmmmiiii12ref12ref)(ε ×100% (2–204) 83  where mi  is the predicted solution at the m th time step, refmi  is the reference solution at that same time, and sN  is the total number of time steps. Equation (2–204) is usually written in a more formal fashion. For example, the 2-norm relative error of the stator current on phase x is defined as  2ref2ref)(xsxsxsxsiiii −=ε ×100% (2–205) where refxsi  and xsi  are the reference and calculated (predicted) solution trajectories, respectively, over the duration of interest (the full study unless stated otherwise). For three-phase quantities, a more general assessment can be obtained by taking the average of the single-phase errors. For stator currents, this gives  3)()()()( csbsassiiii εεεε ++= . (2–206)  84  CHAPTER 3:  EFFICIENT AND EXPLICIT SVB QD MACHINE MODELS WITH MAIN FLUX SATURATION  The properties of the general-purpose lumped-parameter qd induction and synchronous machine models are well known [29]: their q- and d-axis windings are decoupled; their voltages, currents, and fluxes can be constant in steady state; they only comprise constant-parameter branches (assuming magnetic linearity); etc. As a result of these properties, qd models are widely available as built-in components in industry-grade SVB power system transient simulators such as PLECS and SimPowerSystems.   Lumped-parameter qd models can be formulated using various combinations of state variables [77]. The most compact and efficient formulations are achieved when the electrical state vector comprises all the winding flux linkages [29]. These formulations are herein referred to as Flux-based qd models. A common, elegant, and straightforward means of incorporating main flux saturation in Flux-based qd models is the original FC approach [29], [71], [105]; however, this method is implicit, i.e., it creates algebraic loops. Algebraic loops cannot be solved in all industry-grade SVB programs; they decrease the robustness of the solution; and they significantly increase the model’s computational cost.  The objective of this chapter is to propose explicit FC functions for Flux-based qd models that retain the advantages of the original implicit FC method. An explicit FC function is first derived for the simpler case of a symmetrical induction machine. Two new explicit FC functions are then proposed for anisotropic synchronous machines. Case studies 85  demonstrate that the proposed FC functions introduce virtually no numerical error and are significantly more efficient than the original implicit FC functions. 3.1 Induction Machines 3.1.1 Implicit FC Approach  The basic idea of the FC approach is to define a correction function )( mf λ  as the difference between the saturated (OCC) and unsaturated (air-gap line) curves as shown in Figure 3–1 [29]. Extending this approach to q and d axes, the corrected (saturated) flux linkages in each axis become [29]  )(1mqNzqrzqsmumq fiiL λλ −+= ∑= (3–1)  )(1mdNzdrzdsmumd fiiL λλ −+= ∑= (3–2) where muL  is the unsaturated magnetizing inductance and the corresponding projections of the correction function )( mf λ  are  )()( mmmqmq ff λλλλ =  (3–3)  )()( mmmdmd ff λλλλ = . (3–4)  Substituting (2–39), (2–42), and (3–3) in (3–1), inserting (2–40), (2–43), and (3–4) in (3–2), and solving for mqλ  and mdλ , we obtain [29]  86  imλm f(λm)f(λm)λm= F-1(im)Unsaturated fluxSaturated flux Figure 3–1. Main flux saturation characteristic )(1 mF λ−  of an induction machine and its corresponding correction function )( mf λ .  )(1mmmqmuaquNz lrzqrzlsqsaqumq fLLLLL λλλλλλ −+= ∑= (3–5)  )(1mmmdmuaduNz lrzdrzlsdsadumd fLLLLL λλλλλλ −+= ∑= (3–6) where  11111−=++== ∑Nz lrzlsmuaduaqu LLLLL . (3–7)  A Flux-based qd induction machine model that takes into account main flux saturation can be obtained by replacing the saturation function/block of Figure 2–9 by (3–5) and (3–6) [29]. However, mqλ  and mdλ , which are auxiliary algebraic variables, are dependent on )( mf λ . Therefore, (3–5) and (3–6) are implicit nonlinear equations, which cause algebraic loops. It is thus very desirable to reformulate the saturation model such that it becomes free of algebraic loops, which will improve the computational efficiency. 87  3.1.2 Explicit FC Approach  It is possible to reformulate the FC method by introducing intermediate variables. In particular, considering the q axis first, substituting (2–39) and (2–42) in (2–44), mqi  can be rewritten as a function of state variables ( qsλ  and qrzλ , Nz ,...,1= ) and the magnetizing flux linkage mqλ  as   +−+= ∑∑==Nz lrzmqlsmqNz lrzqrzlsqsmq LLLLi11λλλλ. (3–8) Similarly, for the d axis, substituting (2–40) and (2–43) in (2–45), mdi  is rewritten as   +−+= ∑∑==Nz lrzmdlsmdNz lrzdrzlsdsmd LLLLi11λλλλ . (3–9) Introducing the intermediate variables zqi  and zdi , (3–8) and (3–9) can be reformulated as  lmqmqNz lrzqrzlsqszq LiLLiΣ=+=+≡ ∑ λλλ1 (3–10)  lmdmdNz lrzdrzlsdszd LiLLiΣ=+=+≡ ∑ λλλ1 (3–11) where  1111−=Σ += ∑Nz lrzlsl LLL . (3–12)  Based on (3–10), (3–11), and the vector diagram of Figure 2–7, we can establish the relationships 88   lmmzqlmmlmqmqLiiLiLiΣΣΣ+=++= λλλφcos  (3–13)  lmmzdlmmlmdmdLiiLiLiΣΣΣ+=++= λλλφsin  (3–14) where φ  is the flux angle defined in Figure 2–7. Applying the Pythagorean trigonometric identity to (3–13) and (3–14), after some manipulations, we obtain  122=+++ ΣΣ lmmzdlmmzqLiiLiiλλ  (3–15) and  lmmzdzqz Liiii Σ+=+≡ λ22 . (3–16)  It is now clear that there is a unique mapping between the state-variable-dependent zi  [see (3–10) and (3–11)] and the pair ( mi , mλ ) [see (3–16)]. Moreover, based on (3–16) and (2–54), it is possible to define the explicit relationship between zi  and the saturated main flux mλ . This new function is denoted as  )( zm iG=λ , (3–17) which can be readily calculated from the measured data points (2–54) using (3–16), and subsequently implemented using a one-dimensional (1-D) look-up table. While it is not necessary for the simulation, the magnetizing current mi  can also be evaluated using (3–16) if desired. Finally, using (3–16) and Figure 2–7, the resulting saturated q- and d-axis magnetizing fluxes can be computed using  zzqmmq iiλλ =      and      zzdmmd iiλλ = . (3–18) 89  izqizdizq + izd2 2 G(iz)λmizλqs    λqr1Lls     Llr1+λds    λdr1Lls     Llr1+λm izqizλmdλmqλqr1λqs λds λdr1 λm izdiz  Figure 3–2. Block diagram of the Explicit FC function for induction machines with one rotor circuit ( N  = 1).  A block diagram depicting the algebraic equations of the Explicit FC method is shown in Figure 3–2 for a machine with one rotor circuit ( N  = 1). As it can be seen, as opposed to the method described in Section 3.1.1, mqλ  and mdλ  are defined in terms of intermediate variables zqi  and zdi , which are computed from the state variables according to (3–10) and (3–11), yielding an explicit formulation. Consequently, an efficient explicit Flux-based qd induction machine model can be achieved by inserting the FC function sketched in Figure 3–2 into Figure 2–9.  To better understand the proposed approach, the function )( ziG  and the saturation characteristic )(1 miF −  are plotted in Figure 3–3 (a) for a 50-hp induction machine [see Appendix A for the machine parameters], wherein we observe that )( ziG  appears very linear, with only a small change in the slope in the saturated region. For clarity, this segment is magnified and plotted in Figure 3–3 (b). To explain this phenomenon, we can solve for the winding currents in (2–39), (2–40), (2–42), and (2–43), and substitute the results into (2–56) and (2–57), which yields 90  0 500 1000 1500 2000 250000.20.40.60.81.0λ m (Wb)  G(iz)0 10 20 30 40 50 60 70F -1(im)iz (A)im (A)2000 2200 2400 2600 28000.80.91.01.1λ m (Wb)iz (A)  a) G(iz)b) Laquizba(a)(b)See (b) Figure 3–3. (a) Saturation characteristic )(1 miF −  and equivalent explicit function )( ziG  for a 50-hp induction machine; and (b) magnified view of )( ziG  in the saturation region.  zqmaqNz lrzqrzlsqsmaqmq iLLLL )()(1λλλλλ =+= ∑= (3–19)  zdmadNz lrzdrzlsdsmadmd iLLLL )()(1λλλλλ =+= ∑= (3–20) 91  where   1111)(1)()(−=++== ∑Nz lrzlsmmmadmaq LLLLL λλλ . (3–21) Therefore, )( ziG  is equivalent to the product of )( maqL λ  (or )( madL λ ) and zi . According to (3–21), )( maqL λ  and )( madL λ  appear as the parallel combination of the leakage inductances and the saturation-dependent magnetizing inductance. Since the magnetizing inductance of a rotating machine is typically much larger than its leakage inductances, )( maqL λ  and )( madL λ  will change very little from no saturation to full saturation. As a result, it may visually appear that )( ziG  remains almost linear through the operating region. However, the magnified plot in Figure 3–3 (b) shows a small level of nonlinearity.  3.2 Synchronous Machines 3.2.1 Implicit FC Approach  A flux correction function )( mf λ  similar to the one depicted in Figure 3–1 is first derived for synchronous machines using (2–170). It is then possible to write the corrected (saturated) magnetizing flux for each axis as [29]  )(1mqFMzkqzqsmqumq fSiiL λλ −+= ∑= (3–22)  )(1mdfdNzkdzdsmdumd fiiiL λλ −++= ∑= (3–23) where 92   )()( mmFmqmq fSf λλλλ =  (3–24)  )()( mmmdmd ff λλλλ = . (3–25)  Substituting (2–153), (2–156), and (3–24) in (3–22), inserting (2–154), (2–157), (2–158), and (3–25) in (3–23), and solving for mqλ  and mdλ , we obtain  )(1mmmqmquaquMz lkqzkqzlsqsaqumq fLLLLL λλλλλλ −+= ∑= (3–26)  )(1mmmdmduadulfdfdNz lkdzkdzlsdsadumd fLLLLLL λλλλλλλ −++= ∑= (3–27) where  11111−=++= ∑Mz lkqzlsmquaqu LLLL     and     111111−=+++= ∑lfdNz lkdzlsmduadu LLLLL . (3–28)  A Flux-based qd synchronous machine model that takes into account main flux saturation can be formed using (2–181)–(2–186) and (3–26)–(3–27) [29]. As in Section 3.1.1, the saturable model has an implicit formulation. 3.2.2 Explicit FC Approach  The methodology to achieve an explicit FC function for synchronous machines involves similar steps to those described in Section 3.1.2. In particular, considering the q axis first, mqi  is expressed as a function of state variables ( qsλ  and kqzλ , Mz ,...,1= ) and mqλ  by substituting (2–153) and (2–156) into (2–159), yielding 93   +−+= ∑∑==Mz lkqzmqlsmqMz lkqzkqzlsqsmq LLLLi11λλλλ. (3–29) For the d axis, mdi  is rewritten as a function of state variables ( dsλ , kdzλ , Nz ,...,1= , and fdλ ) and mdλ  by substituting (2–154), (2–157), and (2–158) into (2–160), which results in  ++−++= ∑∑== lfdmdNz lkdzmdlsmdlfdfdNz lkdzkdzlsdsmd LLLLLLi λλλλλλ11. (3–30)  Then, introducing the intermediate variables zqi  and zdi , (3–29) and (3–30) are reformulated as  lqmqmqMz lkqzkqzlsqszq LiLLiΣ=+=+≡ ∑ λλλ1 (3–31)  ldmdmdlfdfdNz lkdzkdzlsdszd LiLLLiΣ=+=++≡ ∑ λλλλ1 (3–32) where  1111−=Σ += ∑Mz lkqzlslq LLL     and       11111−=Σ ++= ∑lfdNz lkdzlsld LLLL . (3–33) It can be observed that generally ldlq LL ΣΣ ≠ . From the vector diagram of Figure 2–14 and (3–31)–(3–32), we obtain  lqmFFmzqlqmFFmlqmqmqLSSiiLSSiLiΣΣΣ+=++= λλλφcos  (3–34)  ldmmzdldmmldmdmdLiiLiLiΣΣΣ+=++= λλλφsin  (3–35) 94  where φ  is the angle depicted in Figure 2–14. Using the Pythagorean trigonometric identity, we can express (3–34) and (3–35) as  122=+++ ΣΣ ldmmzdlqmFFmzqLiiLSSiiλλ . (3–36)  Unlike in (3–15), the denominators of the two left-hand side terms in (3–36) are different due to the saliency factor FS  and the inequality between lqLΣ  and ldLΣ . Therefore, unlike for induction machines in Section 3.1.2, (3–36) cannot be solved easily for mi  and mλ . Consequently, several methods to obtain the desirable explicit formulation are described here. 3.2.2.1 2-D Explicit FC  Based on (3–31), (3–32), and (3–36), it can be observed that for any pair of intermediate variables ( zqi , zdi ), there exists a unique pair of ( mi , mλ ), as they are related by the saturation function (2–170). This unique mapping can be implemented using a two-dimensional (2-D) look-up table with zqi  and zdi  as inputs and mλ  as the output. This can be expressed symbolically as  ),(2 zdzqDm iiG=λ . (3–37)  Assuming that this function (look-up table) is known, the corresponding Explicit FC formulation with a 2-D look-up table can be realized according to Figure 3–4. Therein, the intermediate variables zqi  and zdi  are calculated directly from the state variables (stator and rotor flux linkages). The main flux mλ  and magnetizing current mi  are then evaluated using the functions (3–37) and (2–170), respectively. The q- and d-axis magnetizing fluxes are finally computed using a combination of (3–34), (3–36), and the projections depicted in Figure 2–14 as follows: 95  λqs    λkq1    λkq2Lls    Llkq1    Llkq2+ +λds    λkd1    λfdLls    Llkd1    Llfd+ +λkq1λqs λkq2λkd1λds λfdFd (λm)λm imizqizdSF λm izqim /SF + SFλm /LΣlqλm izdim  + λm /LΣldλmdλmqG2D (izq,izd) Figure 3–4. Block diagram of the Explicit FC function with a 2-D look-up table for synchronous machines with 2 q-axis damper windings and 1 d-axis damper winding.  lqmFFmzqmFmq LSSiiSΣ+= λλλ      and     ldmmzdmmd LiiΣ+= λλλ . (3–38)  The disadvantages of this method are twofold. First, not all simulation packages provide a convenient way of implementing multidimensional look-up tables. The second disadvantage relates to the difficulty of generating ),(2 zdzqD iiG , which is done during the initialization stage. Assuming that the vectors zqi  and zdi  have a  and b  components, respectively, ba ⋅  values of mλ  must be calculated by solving nonlinear equations using an iterative approach, which may be time-consuming. 3.2.2.2 Approximate FC  An explicit but approximate FC method using only 1-D look-up tables was presented in [106]. For completeness, its main steps are briefly summarized here. In particular, (3–36) can be algebraically manipulated as to yield  96   ldmmzdzqfcz Liiiki Σ+=+≡ λ222  (3–39) where  ++=ΣΣlqmFFmldmmfc LSSiLik λλ . (3–40) It is observed that due to the saliency factor FS  and the asymmetry of the equivalent leakage inductances lqLΣ  and ldLΣ , the coefficient fck  is function of mλ , making the underlying equations implicit. However, assuming mqlq LL 11 >>Σ  and mdld LL 11 >>Σ  [106], (3–40) can be simplified to obtain the following approximate constant coefficient:  =ΣΣldFlqfc LSLk~ . (3–41) Using (3–41), the intermediate variable zi  is then reformulated as  ldmmzdzqfcz Liiiki Σ+≈+≡ λ222~  (3–42) from which a unique mapping between zi  and mλ  becomes apparent. This new mapping can be defined symbolically by the function  )( zm iG=λ . (3–43)  A block diagram showing the implementation of the Approximate FC function is sketched in Figure 3–5. 3.2.2.3 1-D Explicit FC  To avoid the errors introduced by (3–41) and (3–42) [106], a corrected methodology is proposed here. First, it is noted that (3–39) and (2–170) capture the unique97  λqs    λkq1    λkq2Lls    Llkq1    Llkq2+ +λds    λkd1    λfdLls    Llkd1    Llfd+ +λkq1λqs λkq2λkd1λds λfdFd (λm)λm imizqizdSF λm izqim /SF + SFλm /LΣlqλm izdim  + λm /LΣldλmdλmq~kfc izq + izd2 22 G(iz)iz Figure 3–5. Block diagram of the Approximate FC function for synchronous machines with 2 q-axis damper windings and 1 d-axis damper winding. relation between mλ  and zi . Using (3–40), this can be extended to obtain the mapping between fck  and zi , herein defined by the function   )( zfc iHk = . (3–44)  Equation (3–44) can be easily implemented using a 1-D look-up table. However, (3–44) cannot be used directly since zi  is also a function of fck  [see (3–39)], which would again result in an implicit formulation. Instead, the approach presented here first uses the unsaturated constant coefficient defined as  ++=ΣΣlqmuFFmuldmumufcu LSSiLik λλ  (3–45) where ( muλ , mui ) is usually the first pair of ( mλ , mi ) given by (2–170). Then, another intermediate variable *zi  is calculated using 98  λqs    λkq1    λkq2Lls    Llkq1   Llkq2+ +λds    λkd1     λfdLls    Llkd1    Llfd+ +λkq1λqs λkq2λkd1λds λfd(kfcuizq) + izd  2 2 H(iz ) (kfcizq) + izd2 2 G(iz) Fd (λm)iz iz λm imkfcizqizdSF λm izqim /SF + SFλm /LΣlqλm izdim  + λm /LΣldλmdλmq** Figure 3–6. Block diagram of the Explicit FC function with 1-D look-up tables for synchronous machines with 2 q-axis damper windings and 1 d-axis damper winding.  222* zdzqfcuz iiki += . (3–46) If the flux mλ  is in the unsaturated region, the variable *zi  will be close to the exact zi  as defined by (3–39). However, in the saturated region and more generally, it will not be exact. Thus, another step is needed, which is achieved by evaluating fck  using  )( *zfc iHk = . (3–47)  It is then possible to compute zi  using (3–39) with this new and almost exact fck . After that, the saturated main flux mλ  is computed with (3–43), and the projections mqλ  and mdλ  are calculated using (2–170) and (3–38). A block diagram that shows the implementation of the proposed Explicit FC approach with 1-D look-up tables for modeling main flux saturation in synchronous machines is depicted in Figure 3–6. As it can be seen, this approach requires three one-dimensional look-up tables (or two look-up tables if (2–170) is a fitted function), but otherwise, the formulation is very straightforward and similar to those depicted in Figures 3–4 and 3–5. 99   Finally, as it was done for the induction machine, the magnetizing fluxes can be expressed in terms of state variables. Substituting (2–153), (2–154), (2–156), (2–157), and (2–158) into (2–173) and (2–174), we obtain  zqmaqMz lkqzkqzlsqsmaqmq iLLLL )()(1λλλλλ ≡+= ∑= (3–48)  zdmadlfdfdNz lkdzkdzlsdsmadmd iLLLLL )()(1λλλλλλ ≡++= ∑= (3–49) where  1111)(1)(−=++= ∑Mz lkqzlsmmqmaq LLLL λλ  (3–50)  11111)(1)(−=+++= ∑lfdNz lkdzlsmmdmad LLLLL λλ . (3–51) Therefore, the relation between mqλ  and zqi , and also the one between mdλ  and zdi , will again be almost linear – similar to that depicted in Figure 3–3. Consequently, the function )( ziG  defined for synchronous machines in (3–43) will also be almost linear due to the fact that )( maqL λ  and )( madL λ , as defined by (3–50) and (3–51), will change very little from no to full saturation.  3.3 Computer Studies  Three case studies are presented in this section to validate and compare the proposed Explicit FC functions. For consistency, all studies were executed on a personal computer with a 2.83-GHz Intel CPU running Windows XP. 100  3.3.1 Induction Machines  For the purpose of validation, the proposed Explicit FC induction machine model [see Section 3.1.2] has been implemented in MATLAB/Simulink along with the Implicit FC model [see Section 3.1.1] and the built-in qd induction machine models from PLECS and SimPowerSystems (SPS). All these models except PLECS’ are Flux-based qd models, i.e., they use winding flux linkages as state variables. It is also important to note that based on the results of [151], PLECS’ model does not include dynamic cross saturation. A reference solution is obtained by implementing in the same environment a Mixed/Current-based qd model based on the generalized flux space vector approach (with qsi , dsi , qsλ , and dsλ  as state variables, see Section 2.3.3.1 and [77]). The dynamic inductance is included in the reference model [77]. To obtain a very accurate reference solution, the reference model was solved using the ode5 solver with a very small fixed step size t∆  of 1 µs. For comparison, all other models were solved using the same solver but with a larger step size t∆  of 1 ms. For consistency, all models were implemented in the synchronous reference frame. 3.3.1.1 Change of stator voltage  In this study, a 50-hp squirrel-cage induction motor modeled with one rotor circuit [29] [see Appendix A] is connected to an infinite bus. Initially, the machine operates in steady state with a 150 N∙m mechanical load. At t  = 0.02 s, the bus voltage is increased from 0.8 to 1 pu. This fast electrical transient should allow the observation of the static and dynamic cross-saturation effects.  The predicted transient responses of qsi , mλ , and rω  are plotted in Figures 3–7 to 3–10. As it can be seen in Figure 3–7, the models that include dynamic cross saturation (i.e., Implicit FC, Explicit FC, and SPS) predict very similar q-axis stator currents. A magnified view of the first peak of qsi  is shown in Figure 3–8. Here, the small error between the101  0 0.05 0.10 0.15–150–100–500Time (s)i qs (A)  a) Referenceb) Implicit FCc) Explicit FCd) SPSe) PLECSSee Figure 3–8 Figure 3–7. Transient in stator current qsi  due to a change in stator voltage as predicted by the subject induction machine models. 0.0260 0.0265 0.0270 0.0275 0.0280–175–170–165–160Time (s)i qs (A)  a) Referenceb) Implicit FCc) Explicit FCd) SPSe) PLECS Figure 3–8. Magnified view of the first peak of stator current qsi  due to a change in stator voltage as predicted by the subject induction machine models. reference model and the Implicit FC, Explicit FC, and SPS models is mostly due to the larger step size (i.e., discretization error). However, PLECS’ model, in which dynamic cross saturation is neglected, shows a small but noticeable error. This error vanishes in steady state. Similar observations can be made for the transient response of the main flux mλ  102  0 0.05 0.10 0.150.80.91.0Time (s)λ m (Wb)  a) Referenceb) Implicit FCc) Explicit FCd) SPSe) PLECS Figure 3–9. Transient in main flux mλ  due to a change in stator voltage as predicted by the subject induction machine models. 0 0.05 0.10 0.150.9450.9500.9550.960Time (s)ωr (pu)  a) Referenceb) Implicit FCc) Explicit FCd) SPSe) PLECS Figure 3–10. Transient in electrical rotor speed rω  due to a change in stator voltage as predicted by the subject induction machine models. shown in Figure 3–9. Moreover, based on Figures 3–3 and 3–9 (and the saturation data in Appendix A), it can be observed that the machine operates significantly in the saturated region. Finally, Figure 3–10 shows the transient electrical rotor speed rω , which is accurately predicted by all models.  103  3.3.2 Synchronous Machines  The proposed Explicit FC synchronous machine model using 1-D look-up tables [see Section 3.2.2.3] has been implemented in MATLAB/Simulink along with the Implicit FC [see Section 3.2.1] and Approximate FC [see Section 3.2.2.2] models. For benchmarking with other existing models, the saturable synchronous machine model from the commercial toolbox SPS has been implemented as well. As in the induction machine study, all these models are Flux-based qd models. The reference model is again generated using the generalized flux space vector method (with qsi , dsi , fdi , mqλ , and mdλ  as state variables, see Section 2.4.3.1 and [80]), which includes dynamic cross saturation. The reference model is solved using the ode5 solver with a small step size t∆  of 10 µs. All other models were simulated with the same solver but using a step size t∆  of 1 ms.  To obtain a more general case, a salient-pole synchronous generator [29] is considered. The machine parameters and saturation curve are summarized in Appendix B. To keep the focus on the numerical accuracy of the machine models, the subject generator is assumed to be connected to an infinite bus. 3.3.2.1 Change of load torque  In this study, the machine is assumed to initially operate in steady state under nominal excitation and terminal voltages in idle mode (without mechanical torque). At t  = 0.2 s, a mechanical torque of 1.2 pu is applied. Following a relatively slow transient, the machine becomes heavily loaded, and the projection of the main flux onto the q axis and the effect of cross saturation become significant. The corresponding transient responses in d-axis stator current dsi , q-axis magnetizing flux mqλ , rotor angle δ , and main flux mλ  predicted by the subject models are plotted in Figures 3–11 to 3–15. As it can be seen in these figures, the SPS model predicts responses that are visibly different from the reference trajectories and those of all the other models because the SPS salient-pole synchronous104  0 0.5 1.0 1.5 2.0 2.5 3.0 3.5050001000015000Time (s)i ds (A)  a) Referenceb) Implicit FCc) Explicit FCd) Approx. FCe) SPSa, b, ced Figure 3–11. Transient in stator current dsi  due to a change of load torque as predicted by the subject synchronous machine models. 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5–20–100Time (s)λ mq (Wb)  a) Referenceb) Implicit FCc) Explicit FCd) Approx. FCe) SPSa, b, cde Figure 3–12. Transient in magnetizing flux mqλ  due to a change of load torque as predicted by the subject synchronous machine models. machine model considers saturation along the d axis only. Since this model neglects q-axis saturation and cross saturation, it predicts higher rotor angle δ  even in steady state, which results in higher flux projection on the q axis and higher d-axis stator currents. SPS’ model is the only model considered in this thesis that has different modeling assumptions from105  0 0.5 1.0 1.5 2.0 2.5 3.0 3.50153045Time (s)δ (Deg.)  a) Referenceb) Implicit FCc) Explicit FCd) Approx. FCe) SPSa to deSee Figure 3–14 Figure 3–13. Transient in rotor angle δ  due to a change of load torque as predicted by the subject synchronous machine models. 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.6848495051Time (s)δ (Deg.)  a) Referenceb) Implicit FCc) Explicit FCd) Approx. FCe) SPSea, b, cd Figure 3–14. Magnified view of the first peak of rotor angle δ  due to a change of load torque as predicted by the subject synchronous machine models. the one used to generate the reference trajectories. All other models that utilize the FC approach are much closer to the reference solution, with the Approximate FC method giving slightly different results. It can also be observed from Figure 3–15 and the saturation data given in Appendix B that the machine is saturated in both steady state and transient.  106  0 0.5 1.0 1.5 2.0 2.5 3.0 3.544464850Time (s)λ m (Wb)  a) Referenceb) Implicit FCc) Explicit FCd) Approx. FCe) SPSd a, b, ce Figure 3–15. Transient in main flux mλ  due to a change of load torque as predicted by the subject synchronous machine models. Table 3–1. 2-norm relative errors of the synchronous machine’s stator current dsi , magnetizing flux mqλ , and rotor angle δ  for the change of load torque study.  Implicit FC Explicit FC Approx. FC SPS dsi  (%) 0.079 0.072 5.752 27.43 mqλ  (%) 0.017 0.017 0.954 6.514 δ  (%) 0.020 0.020 0.551 3.964   The 2-norm relative errors of dsi , mqλ , and δ  with respect to the reference solution are summarized in Table 3–1. As it can be seen in Table 3–1, the SPS model produces by far the largest errors due to neglecting q-axis saturation and cross saturation. The Approximate FC method also gives noticeable errors (5.752%, 0.954%, and 0.551%, respectively) due to approximations in the calculation of zi . The Implicit FC method gives very small errors for all three variables (0.079%, 0.017%, and 0.020%, respectively). These errors are almost identical to the errors obtained with the proposed Explicit FC approach. They are essentially caused by the large integration step size. 107  0 0.5 1.0 1.5 2.0 2.5 3.0 3.5–5000–250002500Time (s)i ds (A)  a) Referenceb) Implicit FCc) Explicit FCd) Approx. FCe) SPSea, b, cdSee Figure 3–17 Figure 3–16. Transient in stator current dsi  due to a change of stator voltage as predicted by the subject synchronous machine models. 3.3.2.2 Change of stator voltage  In the following study, the machine is assumed to initially operate in steady state under nominal excitation and terminal voltages, and rated mechanical torque. At t  = 0.2 s, the stator terminal voltage is increased from 1 to 1.05 pu, thus pushing the machine deeper into the saturated region. Similar fast electrical transients have been used in [76], [80], and [152] for investigating synchronous machine saturation and are considered appropriate and informative. The corresponding transient responses of the stator current dsi  and the rotor angle δ  are shown in Figures 3–16 to 3–18. Specifically, a magnified view of the first peak of dsi  from Figure 3–16 is shown in Figure 3–17, whereas the rotor angle δ  is plotted in Figure 3–18. As it can be seen in Figures 3–16 and 3–18, the SPS model predicts different steady states before and after the transient as a consequence of neglecting q-axis saturation and cross saturation. The Approximate FC model also shows noticeable differences in dsi  and δ  with respect to the reference trajectories. At the same time, the Explicit FC and Implicit FC models predict results that are almost indistinguishable from the reference. The 2-norm relative errors with respect to the reference solution for this transient study are summarized in Table 3–2, which shows that the Explicit FC and Implicit FC models have108  0.2055 0.2060 0.2065 0.2070 0.2075 0.2080–5800–5600–5400–5200Time (s)i ds (A)  a) Referenceb) Implicit FCc) Explicit FCd) Approx. FC Figure 3–17. Magnified view of the first peak of stator current dsi  due to a change of stator voltage as predicted by the subject synchronous machine models. 0 0.5 1.0 1.5 2.0 2.5 3.0 3.528293031Time (s)δ (Deg.)  a) Referenceb) Implicit FCc) Explicit FCd) Approx. FCe) SPSeda, b, c Figure 3–18. Transient in rotor angle δ  due to a change of stator voltage as predicted by the subject synchronous machine models. Table 3–2. 2-norm relative errors of the synchronous machine’s stator current dsi  and rotor angle δ  for the change of stator voltage study.  Implicit FC Explicit FC Approx. FC SPS dsi  (%) 1.588 1.584 19.47 96.86 δ  (%) 0.004 0.004 0.387 5.924 109  small and almost identical errors. Again, such small errors are mostly introduced by the larger step size. 3.3.3 Computational Performance  The presented studies demonstrated that the Explicit FC and Implicit FC models predict saturation phenomena with the best numerical accuracy during slow and fast transients. To investigate the computational efficiency of the subject models, the transient study of Section 3.3.2.2 (change of synchronous machine stator voltage) is repeated using fixed-step (ode5) and variable-step (ode15s and ode45) solvers. For all variable-step solvers, the maximum and minimum step sizes are set to 10-2 s and 10-8 s, respectively, and the absolute and relative error tolerances are set to 10-4. The CPU time, the number of time steps, and the number of times the saturation function was called for this study are summarized in Table 3–3. When the fixed-step solver (ode5) is used, the number of calls to the saturation function for all explicit models (i.e., Explicit FC, Approximate FC, and SPS) is the same (24 001). However, the Implicit FC model contains an algebraic loop, which requires iterations and thus many more calls to the saturation function (303 964), resulting in the longest computational time (1.1 s). The proposed Explicit FC model runs as fast as the Approximate FC model, and is about 5 times faster than the Implicit FC model while achieving the same numerical accuracy [see Table 3–2]. The same trend is observed when the variable-step solvers (ode15s and ode45) are used. Since the same error tolerances are used for all models, the number of time steps taken by each model when using ode15s (about 2600) and ode45 (about 900) is consistent and similar for each model. However, the Implicit FC model requires about 12 to 15 times more calls to the saturation function, which results in slower simulation times. The SPS model is also explicit and takes approximately the same number of time steps and calls to the saturation function as the other explicit models (i.e., Approximate FC and Explicit FC). However, this model is slower (perhaps due to its interface/overhead), while also neglecting q-axis saturation and cross110  Table 3–3. Numerical performance of the subject synchronous machine models for the change of stator voltage study with different solvers.  Solver Implicit FC Explicit FC Approx. FC SPS CPU time (s) ode5 1.100 0.217 0.213 0.766 ode15s 0.475 0.113 0.113 0.483 ode45 0.387 0.084 0.083 0.403 No. of time steps ode5 4001 4001 4001 4001 ode15s 2624 2621 2621 2601 ode45 915 906 905 881 No. of calls to saturation function ode5 303 964 24 001 24 001 24 001 ode15s 112 313 7969 7971 7899 ode45 94 555 6338 6341 6167  saturation. At the same time, the proposed Explicit FC model preserves the numerical efficiency of the Approximate FC model and the numerical accuracy of the Implicit FC model while correctly including q-axis saturation and cross saturation.   111  CHAPTER 4:  CONSTANT-PARAMETER SVB VBR INDUCTION MACHINE MODEL WITH MAIN FLUX SATURATION  A major downside of SVB qd models is that they must be interfaced with external power networks represented in abc coordinates using controlled current sources. Consequently, it can be impossible to generate a proper state-space model when the machine is in series with inductive branches. For these cases, fictitious snubbers or time-step relaxations must therefore be used, which have an adverse impact on the numerical accuracy and/or efficiency of the solution [35].  Unlike qd models, VBR models can be interfaced directly to power systems represented in abc coordinates. However, representation of main flux saturation renders their interfacing circuits function of the level of saturation and of the angular displacement of the transformed reference frame. While providing high numerical accuracy due to their direct interface, these VBR models significantly increase the computational cost of the solution and cannot be implemented in some widely used SVB simulators that do not allow variable inductances.  The objective of this chapter is to propose a constant-parameter SVB VBR induction machine model that accounts for main flux saturation. It is achieved by transferring the time-varying part of the variable-parameter model’s interfacing circuit inductance to its voltage source, and approximating its time derivative using numerical techniques. Case 112  studies in conjunction with an error analysis highlight the superior combination of numerical accuracy and efficiency of the proposed model. 4.1 Variable-Parameter SVB VBR Model  The equations of the variable-parameter VBR induction machine model with main flux saturation (VP-VBR) are presented in this section.   For magnetically linear VBR models, the state variables of the equivalent rotor subsystem are typically chosen to be the rotor flux linkages [59], [86]. When modeling main flux saturation, it is convenient to also consider the magnetizing fluxes mqλ  and mdλ  as state variables [108]. Consequently, one pair of q- and d-axis rotor flux linkages is removed from the set of state variables. Without loss of generality, in this chapter, 1qrλ  and 1drλ  are excluded from the state vector and instead are considered as auxiliary algebraic variables.  Defining )(1)( mmmm L λλ ≡Γ , differentiating (2–56) and (2–57) with respect to time, and using the chain and quotient rules yields  =mdmqmqdmimdmq piip λλ)(λΓ  (4–1) where  Γ+Γ−Γ−Γ−Γ+Γ−=)()()()()()()(222222mmmmdmmmmmmdmqmmmmmmdmqmmmmmmmmqmmmmmqdmiddiddiddiddiλλλλλλλλλλλλλλλλλλλλλΓ . (4–2) Based on (2–56) and (2–57), the left-hand side of (4–1) is explicitly defined as 113   ∑=+= Nz drzqrzdsqsmdmqiipiipiip1. (4–3) The rotor currents in (4–3) must be written as a function of state and input variables. As a first step, the rotor currents can be expressed in terms of rotor flux linkages and magnetizing fluxes by algebraically manipulating (2–42) and (2–43), giving  .,...,1,1 NzLiimdmqdrzqrzlrzdrzqrz=−=λλλλ (4–4) Substituting (4–3) and (4–4) into (4–1) consecutively and solving for mqdpλ  yields the state equation  += ∑=Nz drzqrzlrzdsqsmqdmimdmqLpiipp1'' )( λλλλλL  (4–5) where  1221'''''''''' 1)()(−×=+=≡ ∑ IλΓλL Nz lrzmqdmimddmqdmqdmqqmqdmi LLLLL. (4–6)  To formulate a standard state-space model, (4–5) must also be rewritten as a function of state and input variables. First, inserting (2–39), (2–40), and (4–5) into (2–34) and (2–35) and solving for qdspi  gives  −−−−−×++= −dqmqmddsqsslslssdsqsmddlsmqdmqdmqqlsdsqseeiirLLrvvLLLLLLiipλλωωω1'''''''' (4–7) where 114   = ∑= drzqrzNz lrzmddmqdmqdmqqdqLpLLLLeeλλ1''''''''. (4–8) Next, solving (2–37) and (2–38) for qrzpλ  and drzpλ  yields  ( ) .,...,1, Nziirpqrzdrzrdrzqrzrzdrzqrz=−−−−=λλωωλλ (4–9) where the rotor currents are defined using (4–4). Due to the choice of state variables, (4–9) is only integrated for Nz ,,2…= . Finally, in order to obtain an explicit model, 1qrλ  and 1drλ  must be written in terms of state and input variables. Substituting (4–4) into (2–56) and (2–57), and solving for 1qrλ  and 1drλ   yields   +−−+Γ= ∑∑== mdmqNz drzqrzlrzdsqsmdmqNz lrzmmlrdrqrLiiLL λλλλλλλλλ22111 11)( . (4–10)  The next step is to define the equation for the stator interfacing circuit. Inserting (2–39), (2–40), and (4–5) into (2–34) and (2–35) yields ++++−−+−+=''''''''''''''''''''dsqsdsqsmddlsmqdmqdmqqlsdsqsmqdmddlsmqqlsmqddsqssdsqseeiipLLLLLLiiLLLLLLiirvvω (4–11) where  +−+−−=dqmqmddsqsmqdmddmqqmqddsqseeiiLLLLeeλλωω''''''''''''. (4–12) Adding zero-sequence [(2–36) and (2–41)] to (4–11) and applying the inverse Park’s transformation (2–29) yields the machine-network interfacing equation 115   ( ) ''''33 ),( abcsabcsmqdslsabcssabcs pLr eiλLIiv +++= × θ  (4–13) where  [ ]Tdsqssabcs ee 0''''1'' −= Ke . (4–14) The inductance matrix ),('' mqds λL θ  is defined as   +−−+−++−−−−−−=)120()()120()()120()120()120()120()(15.05.05.015.05.05.01),(''θθθθθθθθθθLLLLLLLLLLamqds λL  (4–15) where the function )(ϕL  is   )2sin()2cos()( ϕϕϕ cb LLL +=  (4–16) and { } 120,120, +−= θθθϕ . Finally, the equivalent inductances aL , bL , and cL  are defined as  3''''mddmqqaLLL+= ,    3''''mddmqqbLLL−= ,    and     ''32mqdc LL = . (4–17) A schematic of the VP-VBR model is presented in Figure 4–1.  The resulting interfacing circuit (4–13) has a constant diagonal resistance matrix; however, its inductance matrix is full and dependent on the angle θ  and the level of saturation. From (4–15)–(4–17), it can be observed that ),('' mqds λL θ  is constant when the following three conditions are satisfied:  tLL mddmqq ∀==+ .const'''' κ  (4–18)  tLL mddmqq ∀=− 0''''  (4–19) 116  (4–5), (4–9),(4–12), (4–15)EquivalentRotorSubsystemLlsI3×3 +Ls(θ,λmqd)rs+-rs+-rs+-iasibsicsneas''ebs''ecs''vasvbsvcsabciabcsvabcsvqdsiqdseqds''θωrKsKsθVariable-Parameter Interfacing CircuitKs-1''ωLs(θ,λmqd)''θ Figure 4–1. Variable-parameter VBR (VP-VBR) formulation of the saturable lumped-parameter induction machine model.  tLmqd ∀= 0''  (4–20) where κ  is an arbitrary time-independent scalar. Conditions (4–18)–(4–20) are respected in the magnetically linear symmetrical VBR induction machine model [59]. However, when saturation is considered, it can be seen from (4–2) and (4–6) that the three conditions are not always met. 4.2 Constant-Parameter SVB VBR Model   To obtain the desired constant-parameter (CP-VBR) model, it is necessary to satisfy conditions (4–18)–(4–20) at all time. This is achieved if )('' mqdmi λL  becomes a constant scalar matrix, i.e., a diagonal matrix with identical non-zeros. 117  4.2.1 Derivation of Constant-Parameter Interfacing Circuit  The first step consists of substituting (4–3) into (4–1) and separating )( mqdmi λΓ  into constant and time-dependent terms as  ∆+Γ=+ ∑= mdmqmqdmismdmqmuNz drzqrzdsqs ppiipiip λλλλ )(1λΓ  (4–21) where mumu L1≡Γ  is the unsaturated magnetizing inverse inductance, and ∆Γ+Γ−Γ−Γ−∆Γ+Γ−=∆)()()()()()()(222222mmmmdmmmmmmdmqmmmmmmdmqmmmmmmmmqmmmmmqdmisddiddiddiddiλλλλλλλλλλλλλλλλλλλλλΓ      (4–22) where mummmm Γ−Γ=∆Γ )()( λλ . Substituting (4–4) into (4–21) and solving for the mqdpλ  vectors multiplied by time-independent elements yields  ∆−+= ∑= mdmqmqdmisNz drzqrzlrzdsqsmumdmq pLpiipLp λλλλλλ )(1'' λΓ  (4–23) where  1''1−Σ+Γ=lrmumu LL       and      ∑=Σ=Nz lrzlr LL 111 . (4–24) Substituting (4–5) into the right-hand side of (4–23) yields +∆−+= ∑∑==Nz drzqrzlrzdsqsmqdmimqdmisNz drzqrzlrzdsqsmumdmqLpiipLpiipLp1''1'' )()( λλλλλλλLλΓ . (4–25) 118  Inserting (2–39), (2–40), and (4–25) into (2–34) and (2–35) then gives  ( ) ( )+++−++=''''''''dsqsdsqsmulsqsdsmulsdsqssdsqseeiipLLiiLLiirvvω  (4–26) where  +∆−+−+−=∑∑==Nz drzqrzlrzdsqsmqdmimqdmisNz drzqrzlrzqsdsmumqmddsqsLpiipLpiiLee1''1'''''')()( λλλλωλλωλLλΓ. (4–27)  Combining (4–26) with (2–36) and (2–41) and applying the inverse Park’s transformation results in the following constant-parameter machine-network interfacing equation  ''''33 )( abcsabcscslsabcssabcs pLr eiLIiv +++= ×  (4–28) where the new constant inductance matrix ''csL  is defined as  −−−−−−=15.05.05.015.05.05.0132 ''''mucs LL . (4–29) The constant-parameter interfacing circuit (4–28) has been achieved without modifying the rotor subsystem state equations (4–5) and (4–9).  The final step consists of modifying (4–28) to achieve an interfacing circuit comprised only of decoupled RL branch elements [91]. Assuming a wye-connected (grounded or floating neutral) machine, the terminal stator voltage abcsv  is decomposed into two vectors following the procedure set forth in [91]: 119   ngabcnabcs vvv +=  (4–30) where  ''abcsabcsDabcssabcn pLr eiiv ++=  (4–31) and [ ]Tngngngng vvv=v . The voltage ngv  is defined as  ngng piLv 0=  (4–32) where csbsasng iiii ++= . Finally, the newly introduced inductances DL  and 0L  are related to lsL  and ''muL  by  ''mulsD LLL +=      and     3''0muLL −= . (4–33)  Equations (4–30)–(4–33) define the decoupled RL branch interfacing circuit [91, Fig. 1]. Up to this point, the CP-VBR model is algebraically equivalent to the qd and VP-VBR models. 4.2.2 Numerical Approximation  The procedure presented in Section 4.2.1 is analogous to moving the time-varying elements of the algebraically exact inductance matrix ),('' mqds λL θ  defined in (4–15) into the subtransient voltage source ''abcse . As a consequence, the term [ ]Tdsqsqds ee '''''' ≡e  is a function of the current time derivative qdspi .   One method to calculate qdspi  in (4–27) is to rewrite it in terms of state and input variables using (4–7), as is done in (4–5). However, since ''qdse  is an output of the equivalent rotor subsystem, this approach results in an algebraic feedthrough between ''qdse  and qdsv . In the case where qdsv  is an output of the external circuit subsystem (e.g., if the machine is 120  connected to an inductive branch), qdsv  will be algebraically related to ''qdse . Consequently, the state model will have an algebraic loop containing ''qdse  and qdsv  [90]. It is recalled that models with algebraic loops generally require special iterative solvers that are not available in some simulation programs such as PLECS, reduce simulation robustness, and increase the overall computational time.  An alternative approach is to approximate qdspi  in (4–27) using numerical techniques. Specifically, qdspi  is herein approximated using backward differentiation formulas (BDFs) [34], [90]. A similar approach was originally presented in [90] to achieve an explicit constant-parameter magnetically linear VBR synchronous machine model. Therein, due to dynamic saliency, only qspi  needed to be approximated; in addition, qspi  was multiplied by a constant value [90]. This is in contrast to the proposed model, wherein due to saturation, qdspi  is multiplied by a time-dependent coupled matrix [see (4–27)].  Assuming a variable-step solver for generality, the first-order backward differentiation formula (BDF1) is   )()1()()(~mmqdsmqdsmqdstp∆−=−iii  (4–34) where the superscript (m ) indicates the value at the m th time step, and )1()()( −−=∆ mmm ttt . Here, the tilde symbol “~” denotes an approximation as opposed to the exact value. If a more accurate approximation is desired, the second-order BDF (BDF2) can also be used:  ( ) ∆+∆−∆∆−∆+∆+∆−=−−−−−−)1()()2()1()1()()1()()()1()()( 11~mmmqdsmqdsmmmmmmqdsmqdsmqdstttttttpiiiii . (4–35)  Replacing qdspi  in (4–27) by (4–34) or (4–35), an explicit constant-parameter VBR induction machine model with main flux saturation is obtained. A diagram depicting the121  (4–5),(4–9),(4–27)EquivalentRotorSubsystemLDrs+-LDrs+-LDrs+-L0iasibsicsing neas''ebs''ecs''gvasvbsvcsabciabcsvabcsvqdsiqdseqds''θω-ωrKsKsBDF iqdspiqds~θConstant-Parameter Decoupled Interfacing CircuitKs-1 Figure 4–2. Constant-parameter VBR (CP-VBR) formulation of the saturable lumped-parameter induction machine model. proposed CP-VBR model is shown in Figure 4–2, wherein the interfacing circuit defined by (4–30)–(4–33) is contained within the shaded area. In general, other approximations (e.g., [90]) could also be used instead of (4–34) or (4–35).  4.3 Error Analysis  If the solver uses a high-order integration rule and the integration step size is sufficiently small, it can be assumed that the numerical error in the proposed model will mostly come from the approximation of qdsmqdmimqdmismu pL iλLλΓ )()( '''' ∆  in (4–27) using BDFs [(4–34) or (4–35)]. This overall error is thus determined by two major factors: i) the BDFs, and ii) the magnitude of the saturation-dependent matrix )()( '''' mqdmimqdmismuL λLλΓ∆ .  122  4.3.1 Numerical Differentiation Error  The error associated with the BDFs can be related to the integration step size. Assuming a constant step size t∆ , the local truncation error (LTE) )( 1mBDFd  [34] of BDF1 is defined as  )()1()()(1mqdsmqdsmqdsmBDF ptd iii−∆−=− (4–36) where as in Section 4.2.2 the superscript ( m ) indicates the value at the m th time step. Similarly, the LTE of BDF2 is given by  ( ) ( ) )()2()()1()()( 2 212 mqdsmqdsmqdsmqdsmqdsmBDF pttd iiiii −−∆−−∆= −− . (4–37) Inserting Taylor series in (4–36) and (4–37), after some algebraic manipulations, we obtain  )(22)(2)(1 tOptd mqdsmBDF ∆+∆−= i  (4–38)  )(33)(32)(2 tOptd mqdsmBDF ∆+∆−= i . (4–39) The error introduced by the BDF approximations is thus proportional to t∆  for BDF1 and 2t∆  for BDF2, respectively, which also proves the consistency [34] of these approaches. Moreover, assuming a purely sinusoidal signal and using the synchronous reference frame, it can be seen that (4–34) and (4–35) will not introduce error in steady state. Finally, the error will be maximized in situations where the second and/or third time derivatives of qdsi  are large, for example during fast and/or large disturbances. 123  4.3.2 Approximation Error and Machine Parameters  Substituting (4–27) into (4–26), it is observed that the term )()( '''' mqdmimqdmismuL λLλΓ∆  is a saturation-dependent inductance matrix in series with the constant inductance ''muls LL + . The differential equation related to ''muls LL +  is converted to abc coordinates and is solved together with the rest of the ac network by the program solver; however, the equation related to )()( '''' mqdmimqdmismuL λLλΓ∆  is solved inside the rotor subsystem using BDFs, which introduce additional numerical error [see (4–38) and (4–39)]. Therefore, smaller values of )()( '''' mqdmimqdmismuL λLλΓ∆  should result in smaller overall error, as the effect of the approximated saturation-dependent term will be minimized with respect to the remaining constant inductance ''muls LL + . The ratio of these two inductances   ''2'''' )()()(mulsmqdmimqdmismumqd LLL+∆=λLλΓλζ  (4–40) is therefore considered as an indication of how accurate the proposed approximation is expected to be.   To get more insight into the expected model numerical accuracy, it is particularly useful to understand how )( mqdλζ  depends on machine parameters. This analysis can be simplified by neglecting cross saturation in (4–40). The reasoning behind this assumption is presented in [151], wherein it is shown that cross saturation has a marginal effect on the accuracy of induction machine models with stator currents and rotor and/or magnetizing fluxes as state variables. Therefore, equating mmdi λ  to )( mm λΓ  in (4–40) [151], after some algebraic manipulations, )( mqdλζ  can be rewritten as   )()( sat mm K λχλζ ⋅=  (4–41) 124  Table 4–1. Coefficients lrmu LL Σ  and 1−χ  for various induction machines.  Machine  [29] [71] [29] [29] [29] [78] [120] [120] [120] Rating (hp) 3 5 50 500 2250 10 14.7 121 845 No. of rotor circuits 1 1 1 1 1 2 2 2 2 lrmu LL Σ  34.7 14.3 43.3 44.8 57.7 214 52.5 72.4 94.5 1−χ  72.3 31.6 89.6 92.6 118 999 135 200 285   mulrlrmulsmuLLLLLL+⋅+=ΣΣ''''χ  (4–42)  )()(1)(satmmlrmulrmummm LLLLLLK λλλ++−=ΣΣ . (4–43)  Based on (4–42) and (4–43), it becomes apparent that )( mλζ  depends on both the machine parameters and the saturation level. In particular, since for induction machines lrmu LL Σ>> , we can assume that )(sat mK λ  mostly depends on the saturation level, and thus that the effect of machine parameters is essentially captured by χ . It can also be seen from (4–43) that for unsaturated conditions )(sat mK λ  = 0, and that )(sat mK λ  monotonically increases as a function of the level of saturation. Finally, since lsL  and ''muL  typically have the same order of magnitude, the value of χ  is predominantly defined by the simple ratio lrmu LL Σ .   The values of lrmu LL Σ  and 1−χ  for several machines with a diverse range of parameters and power ratings are summarized in Table 4–1. As this table shows, lrmu LL Σ  and 1−χ  are fairly correlated and generally increase with machine size. These coefficients are also typically larger for machines modeled with two rotor circuits instead of one. Finally, based on (4–41)–(4–43) and the magnitude of these coefficients in Table 4–1, it can be concluded that )( mλζ  should remain relatively small even for machines with less125  IMCableThév. Imp.a-gRsn Figure 4–3. One-line diagram of a test system comprised of an induction motor (IM), a cable, a transformer, and a network Thévenin equivalent. favorable parameters under heavy saturation. This is a direct result and benefit of the VBR formulation which uses the subtransient inductance, wherein the magnetizing inductance )( mmL λ  appears in parallel with the typically much smaller equivalent rotor leakage inductance lrLΣ . 4.4 Computer Studies  The proposed CP-VBR model [see Section 4.2] and the algebraically exact VP-VBR model [see Section 4.1] have been implemented in MATLAB/Simulink using the PLECS toolbox. Two versions of the proposed model are considered. In the first one, the current time derivative qdspi  in (4–27) is computed using (4–34) (i.e., BDF1); in the second one, qdspi  is computed using (4–35) (i.e., BDF2). The resulting formulations are referred to as CP-VBR-BDF1 and CP-VBR-BDF2, respectively. To provide a meaningful comparison, a Flux-based qd model is also implemented, wherein saturation is represented using the Explicit FC function presented in Section 3.1.2. All machine models are simulated in the synchronous reference frame. For consistency, all simulations are executed on a PC with a 2.83-GHz Intel CPU running Windows XP.  In the considered case study, a low-voltage wye-connected induction motor (IM) is connected to a medium-voltage distribution network (represented by a three-phase Thévenin equivalent) through a cable and a transformer. The single-line diagram of the network is shown in Figure 4–3. The motor is initially assumed to operate in steady state 126  with applied base mechanical torque. The source voltage sV  is set to 1.1 pu as to emulate a strongly saturated operation and thus increase the numerical error as explained in Section 4.3.2. While this voltage is high for grid-connected induction motors, it is not uncommon for machine-converter systems to operate at even higher voltages [41]. At t  = 0.04 s, a single-line-to-ground fault is assumed to occur upstream in the network, resulting in an unbalanced voltage sag in the test system. The voltage sag is modeled by decreasing the voltage on phase a of the equivalent Thévenin source to 0.5 pu. The fault is cleared 100 ms later, and sV  is restored back to 1.1 pu. This study has been chosen because of the highly unbalanced currents and severe transients occurring as a result of the voltage sag. Consequently, according to Section 4.3.1, the approximation error will be maximized under such conditions, thus validating the model in its less accurate operating region.  The network of Figure 4–3 has been implemented using PLECS’ built-in lumped-parameter circuit components. Since the motor is in series with an inductive cable, only the VBR models can be interfaced directly to the external network among the test case models [35]. The qd model is represented using controlled current sources and is interfaced using fictitious resistive snubbers snR  [35]. To demonstrate the effect of snubbers, two values (large and small) of snR  have been considered. The corresponding models are referred to as qd1 (which is numerically stiff) and qd2 (which is less stiff), respectively.   A reference solution is obtained by solving the saturable VP-VBR model using Simulink’s general-purpose solver ode45 with the following stringent settings: maximum and minimum step sizes of 10-5 and 10-8 s, respectively, and absolute and relative error tolerances of 10-5. To achieve a fair comparison, due to the numerical stiffness of the qd1 model, all subject models are simulated using the stiffly stable solver ode15s. Typical settings are used: maximum and minimum step sizes of 10-3 and 10-7 s, respectively, and absolute and relative error tolerances of 10-4.  127   To verify the proposed approach, a few motors with different parameters are considered. 4.4.1 Induction Motor #1 (IM1)  The subject transient study is first executed using a 10-hp machine (IM1) with two rotor circuits [78]. The machine and system parameters are summarized in Appendix C. For this machine, the ratio of unsaturated magnetizing inductance over equivalent rotor leakage inductance lrmu LL Σ  (214) and the coefficient 1−χ  (999) are very large [see Table 4–1]. Therefore, as shown in Section 4.3.2, the proposed CP-VBR model should yield excellent numerical accuracy. Snubber resistances snR  of 250 Ω and 5 Ω are used with the qd1 and qd2 models, respectively.  The simulated stator current asi ,  electrical rotor speed rω , electromagnetic torque eT , and main flux mλ  are shown in Figure 4–4, wherein the unbalanced conditions created by the single-phase voltage sag are apparent. Only phase a of the stator current is presented since it has the largest peaks during the disturbance. To better visualize the performance of the models, a magnified view of the first peak of asi  is also reproduced in Figure 4–5. As observed in Figures 4–4 and 4–5, all models except qd2 accurately predict the machine behavior in both steady-state and transient conditions. The inaccuracy of qd2 is caused by its fairly small snubber.   In Section 4.3.2, the level of saturation was found to affect the solution accuracy. Specifically, the more saturated the machine is, the larger the numerical error should be. To verify that IM1 is well saturated, the dynamic values of mλ  (taken from the reference solution of Figure 4–4) are depicted on the steady-state saturation curve in Figure 4–6. The air-gap line is also shown in Figure 4–6 to better visualize the level of saturation at different operating points. Analyzing Figures 4–4 and 4–6, it is seen that IM1 is well128  –4004080i as (A)0.950.970.99ωr (pu) a) Referenceb) VP-VBRc) CP-VBR-BDF1d) CP-VBR-BDF2e) qd1 (Rsn = 250 Ω)f) qd2 (Rsn = 5 Ω)–50050T e (N·m)0 0.05 0.10 0.15 0.20 0.25 0.301.251.501.75Time (s)λ m (Wb)a to effa to eSee Figure 4–5 Figure 4–4. Transients in stator current asi , electrical rotor speed rω , electromagnetic torque eT , and main flux mλ  during a single-phase voltage sag as predicted by the subject models (IM1). saturated in steady state ( mλ  = 1.77 Wb), and that it saturates even further ( mλ  = 1.83 Wb) shortly after the fault. The machine remains saturated during the rest of the sag.   To quantify the accuracy achieved by the various models, the 2-norm (cumulative) relative errors of asi , rω , eT , and mλ  with respect to their reference trajectories are summarized in Table 4–2. As this table shows, all three VBR models yield errors below129  0.048 0.049 0.050 0.051 0.052–40–39–38–37–36Time (s)i as (A)  a) Referenceb) VP-VBRc) CP-VBR-BDF1d) CP-VBR-BDF2e) qd1 (Rsn = 250 Ω)f) qd2 (Rsn = 5 Ω)fa to e Figure 4–5. Magnified view of the first peak of stator current asi  during a single-phase voltage sag as predicted by the subject models (IM1).   0 1 2 3 4 5 6 700.51.01.52.02.5im (A)λ m (Wb)  Air-gap lineSaturation curveLowest saturationlevelHighest saturationlevelSteady-state saturation level Figure 4–6. Saturation curve of machine IM1 and its operating range (denoted by the thick red line) during a single-phase voltage sag. 0.1% for all variables, and no significant difference is observed between CP-VBR-BDF1 and CP-VBR-BDF2. The qd1 model is also highly accurate, as its 2-norm errors are all smaller than 0.21%. However, as observed in Figures 4–4 and 4–5 and quantified in Table 4–2, the qd2 model is much less accurate with errors of 5.578% for asi  and 7.895% for eT .  130  Table 4–2. 2-norm relative errors of stator current asi , electrical rotor speed rω , electromagnetic torque eT , and main flux mλ  for the single-phase voltage sag study (IM1).  VP-VBR CP-VBR-BDF1 CP-VBR-BDF2 qd1 ( snR  = 250 Ω) qd2 ( snR  = 5 Ω) asi  (%) 0.035 0.056 0.052 0.153 5.578 rω  (%) 0.001 0.002 0.002 0.004 0.208 eT  (%) 0.033 0.060 0.055 0.205 7.895 mλ  (%) 0.004 0.010 0.009 0.090 4.445 4.4.2 Induction Motor #2 (IM2)  The same voltage sag transient study is repeated here using a 5-hp machine (IM2) [71]. The machine and system parameters are summarized in Appendix D. For this machine, the very low values of lrmu LL Σ  (14.3) and 1−χ  (31.6) indicate that the proposed approximation method should be less accurate. In this study, snR  is set to 300 Ω and 3 Ω for the qd1 and qd2 models, respectively.   The predicted stator current asi , electrical rotor speed rω , electromagnetic torque eT , and main flux mλ  are presented in Figure 4–7. A magnified view of the first peak of asi  after the clearing of the upstream fault is also shown in Figure 4–8. Despite the unfavorable machine parameters, all models except qd2 again predict electrical and mechanical variables with sufficient numerical accuracy.   The dynamic values of mλ  during the study [see Figure 4–7] are depicted on the steady-state saturation curve in Figure 4–9 along with the air-gap line. The machine IM2 is also well saturated in steady state ( mλ = 0.473 Wb). Higher values of flux are reached at the beginning and the end of the voltage sag ( mλ = 0.488 and 0.484 Wb, respectively). The motor also remains saturated throughout the rest of the sag. 131  –3003060i as (A) a) Referenceb) VP-VBRc) CP-VBR-BDF1d) CP-VBR-BDF2e) qd1 (Rsn = 250 Ω)f) qd2 (Rsn = 5 Ω)0.940.950.96ωr (pu)–2502550T e (N·m)0 0.05 0.10 0.15 0.20 0.25 0.300.350.400.45Time (s)λ m (Wb)a to ea to effSee Figure 4–8 Figure 4–7. Transients in stator current asi , electrical rotor speed rω , electromagnetic torque eT , and main flux mλ  during a single-phase voltage sag as predicted by the subject models (IM2).  The 2-norm relative errors of asi , rω , eT , and mλ  with respect to the reference trajectories are summarized in Table 4–3. All models except qd2 provide accurate results with errors smaller than 1%. However, there is now a noticeable difference between the two CP-VBR formulations. Specifically, CP-VBR-BDF2 yields more accurate results than CP-VBR-BDF1 (e.g., 0.094% error in asi  compared to 0.754%) due to its second-order approximation of qdspi . Nevertheless, the numerical accuracy attained by the CP-VBR-132  0.145 0.146 0.147 0.148–46–44–42–40Time (s)i as (A)  a) Referenceb) VP-VBRc) CP-VBR-BDF1d) CP-VBR-BDF2e) qd1 (Rsn = 300 Ω)f) qd2 (Rsn = 3 Ω)a to e f Figure 4–8. Magnified view of the first peak of stator current asi  during a single-phase voltage sag as predicted by the subject models (IM2).   0 5 10 15 20 2500.20.40.6im (A)λ m (Wb)  Air-gap lineSaturation curveLowest saturationlevelHighest saturationlevelSteady-state saturation level Figure 4–9. Saturation curve of machine IM2 and its operating range (denoted by the thick red line) during the single-phase voltage sag. BDF1 model remains satisfactory for most practical cases. This study demonstrates that highly accurate results can be achieved with the proposed constant-parameter model even for well-saturated machines with unfavorable parameters.  133  Table 4–3. 2-norm relative errors of stator current asi , electrical rotor speed rω , electromagnetic torque eT , and main flux mλ  for the single-phase voltage sag study (IM2).  VP-VBR CP-VBR-BDF1 CP-VBR-BDF2 qd1 ( snR  = 300 Ω) qd2 ( snR  = 3 Ω) asi  (%) 0.069 0.754 0.094 0.473 3.640 rω  (%) 0.002 0.009 0.002 0.017 0.371 eT  (%) 0.074 0.273 0.086 0.256 4.406 mλ  (%) 0.013 0.068 0.014 0.086 3.694 4.4.3 Model Accuracy vs. Integration Step Size  It is also useful to validate the accuracy of the proposed models as a function of a constant step size t∆ . For this purpose, the same voltage sag transient study is repeated here using the proposed CP-VBR-BDF1 and CP-VBR-BDF2 models. To focus only on the machine models, the whole ac network (Thévenin equivalent, transformer, and cable) is replaced by an infinite bus. A third motor (IM3), whose value of lrmu LL Σ  is in the middle range (43.3), is also considered. The parameters of the 50-hp IM3 are summarized in Appendix A. To enforce a fixed step size using PLECS, all simulations are executed using Simulink’s ode45 solver with relaxed error tolerances.  The resulting 2-norm relative error of asi  using CP-VBR-BDF1 and CP-VBR-BDF2 is plotted in Figure 4–10 for the three machines. As it can be observed, )( asiε  increases with the step size, which was predicted in Section 4.3.1. The errors also converge towards zero for small step sizes, which demonstrates the consistency [34] of the proposed method. At the same time, )( asiε  grows faster for machines with lower ratios lrmu LL Σ . This emphasizes the correlation between model accuracy and machine parameters noted in Section 4.3.2 and observed in Sections 4.4.1 and 4.4.2. It can also be seen in Figure 4–10 that as the error increases, CP-VBR-BDF2 is clearly more accurate than CP-VBR-BDF1. For example, for the machine IM2 and t∆  = 500 µs, the CP-VBR-BDF2 model yields an error twice as small as the CP-VBR-BDF1 model (0.46% and 0.92%, respectively). Another134  00.51.01.52.0  CP-VBR-BDF1 (IM1)CP-VBR-BDF1 (IM2)CP-VBR-BDF1 (IM3)10-5 10-4 10-300.51.01.52.0Step size (s)  CP-VBR-BDF2 (IM1)CP-VBR-BDF2 (IM2)CP-VBR-BDF2 (IM3)(a)(b)ε(ias) (%)ε(ias) (%) Figure 4–10. 2-norm relative error of stator current asi  as function of constant step sizes for the single-phase voltage sag study as predicted by the (a) CP-VBR-BDF1 and (b) CP-VBR-BDF2 models. important point is that )( asiε  remains relatively small (below 2%) even at a fairly large integration step size of one millisecond with machine IM2 having unfavorable parameters (i.e., low lrmu LL Σ ). In particular, in this study, )( asiε  never exceeds 1% for the CP-VBR-BDF2 model. 4.4.4 Computational Performance  To evaluate the models' computational performance, the resulting number of time steps, CPU time, and average CPU time per time step from the studies of Sections 4.4.1 and 4.4.2 are summarized in Table 4–4. For a given machine, all three VBR models take similar numbers of time steps. However, the CPU time of VP-VBR is much higher than that of the CP-VBR model. For example, for IM1, the VP-VBR model takes more than 4 times longer than CP-VBR-BDF1 or CP-VBR-BDF2 (836 ms versus 179 ms or 184 ms, respectively). This135  Table 4–4. Numerical performance of the subject models for the single-phase voltage sag study.  VP-VBR CP-VBR-BDF1 CP-VBR-BDF2 qd1 qd2 IM1 No. time steps 603 567 578 2528 741 CPU time (ms) 836 179 184 422 131 CPU time per time step (µs) 1386 316 318 167 177 IM2 No. time steps 777 816 793 3602 1308 CPU time (ms) 1079 184 197 433 158 CPU time per time step (µs) 1389 225 248 120 121  is explained by the presence of variable inductances in VP-VBR. Table 4–4 also illustrates that using BDF2 instead of BDF1 to approximate qdspi  adds little overhead. Additionally, the computational cost of the qd model per time step is much less than that of the CP-VBR model (approximately 172 µs versus 317 µs for the small system with IM1), which is due in part to its efficient implementation of saturation using the Explicit FC function [see Section 3.1.2]. However, to obtain an accurate solution, the large snubber required to interface the qd model considerably increases the system’s numerical stiffness, which in turn results in the use of smaller step sizes (even with a stiffly stable solver) and increased computational burden. The proposed CP-VBR model (in particular the CP-VBR-BDF2 implementation) therefore offers the best combination of numerical accuracy and efficiency.  136  CHAPTER 5:  CONSTANT-PARAMETER SVB VBR SYNCHRONOUS MACHINE MODEL WITH MAIN FLUX SATURATION  As mentioned at the beginning of Chapter 4, VBR models are a valuable alternative to conventional qd models in SVB simulations since they can be interfaced directly with external power networks. However, representation of main flux saturation causes their interfacing circuits to be time-dependent, which adds significant computational overhead and reduces their range of application.  In state-of-the-art SVB VBR induction machine models, only the stator windings are represented in circuit form. This is the case, for instance, of the approximate but numerically accurate constant-parameter SVB VBR induction machine model with main flux saturation proposed in Chapter 4. This approach is sufficient for squirrel-cage induction machines, since their equivalent rotor windings are not interfaced with external circuitry. However, only representing the stator windings in circuit form can be insufficient for SVB wound-rotor synchronous machine models, as it can result in an incompatible interface between the field winding and the exciter model. An algebraically exact SVB VBR synchronous machine model with main flux saturation and whose stator and field windings are represented in circuit form was recently proposed in [96]. Its interfacing circuit is time-dependent not only due to the effect of main flux saturation, but also because of dynamic saliency and stator-field magnetic coupling.  137   A constant-parameter SVB VBR synchronous machine model that accounts for main flux saturation and employs a circuit-based representation of the stator and field windings is proposed in this chapter. This new formulation is achieved by extending the so-called stator-and-field-circuit VBR model presented in [81] to account for main flux saturation while maintaining a constant-parameter interfacing circuit. The proposed model contains low-pass filters that are inserted to obtain an explicit formulation. An approach to select the parameters of the filters with the objective of controlling numerical accuracy is also proposed. Computer studies are used to demonstrate the validity of the parameters selection approach, and to highlight the excellent combination of numerical accuracy and efficiency of the proposed model.  5.1 Variable-Parameter SVB VBR Model  This section presents the equations of the saturable stator-and-field-circuit variable-parameter VBR model (VP-VBR) with main flux saturation [12], [96]. 5.1.1 Rotor State Equations  The state variables of the VP-VBR model’s state-space rotor subsystem are the magnetizing fluxes mqλ  and mdλ , and the rotor fluxes kqzλ  ( .,...,2 Mz = ) and kdzλ  ( .,...,2 Nz = ). This means that 1kqλ  and 1kdλ  are auxiliary algebraic variables.  Differentiating (2–173) and (2–174) with respect to time gives  =mdmqmqdmimdmq piip λλ)(λΓ  (5–1) where 138   Γ+Γ−Γ−Γ−Γ+Γ−=)()()()()()()(222222242mmdmmdmmdmmmFmdmqmmdmmmFmdmqmmdmmmmqmFmqmmdmmmqdmiddiSddiSddiSddiλλλλλλλλλλλλλλλλλλλλλΓ  (5–2) and where the reluctances )( mmq λΓ  and )( mmd λΓ  are equal to )(1 mmqL λ  and )(1 mmdL λ , respectively.  Substituting (2–153), (2–154), and (2–156)–(2–158) into (2–173) and (2–174), the magnetizing currents can be rewritten in terms of fluxes as  ∑=−+−=Mz lkqzmqkqzlsmqqsmq LLi1λλλλ (5–3)  ∑=−+−+−=Nz lkdzmdkdzlfdmdfdlsmddsmd LLLi1λλλλλλ . (5–4)  Inserting (5–3) and (5–4) into the left-hand side of (5–1) and solving for mqdpλ  yields  ++=∑∑==Σ Nz lkdzkdzMz lkqzkqzlfdfdlsdslsqsmqdmdmqLpLpLpLpLpp11''0)(λλλλλλλλL  (5–5) where  1''1001)()(−ΣΣΣ +=ldlqmqdmimqd LLλΓλL  (5–6) and 139   1111−=Σ += ∑Mz lkqzlslq LLL     and     11111−=Σ ++= ∑Nz lkdzlfdlsld LLLL . (5–7)  The right-hand side of (5–5) can be computed as follows. First, using (2–147), (2–148), and (2–152)–(2–154), qspλ , dspλ , and fdpλ  can be formulated as a function of states and inputs (stator and field currents and voltages, and electrical rotor speed) as  ( )mddslsrqssqsqs iLirvp λωλ +−−=  (5–8)  ( )mqqslsrdssdsds iLirvp λωλ ++−=  (5–9)  fdfdfdfd irvp −=λ . (5–10)  The rotor flux linkage time derivatives can also be expressed as a function of rotor and magnetizing fluxes by inserting (2–156) and (2–157) into (2–150) and (2–151), yielding  ( ) .,...,1, MzLrp mqkqzlkqzkqzkqz =−−= λλλ  (5–11)  ( ) .,...,1, NzLrp mdkdzlkdzkdzkdz =−−= λλλ  (5–12)  Since (5–11) and (5–12) are not used as state equations for z  = 1, it is necessary to write 1kqλ  and 1kdλ  as a function of states and inputs. Substituting (2–156) and (2–157) into (2–173) and (2–174) and solving for 1kqλ  and 1kdλ  yields  −−= ∑=Mz lkqzkqzqsmmqmqlkqkq LiLL2''11 )(λλλλ  (5–13)  −−−= ∑=Nz lkdzkdzfddsmmdmdlkdkd LiiLL2'''11 )(λλλλ  (5–14) 140  where )('' mmqL λ  and )(''' mmdL λ  are defined as  11'' 1)(1)(−=+= ∑Mz lkqzmmqmmq LLL λλ  (5–15)  11''' 1)(1)(−=+= ∑Nz lkdzmmdmmd LLL λλ . (5–16)  In summary, the state equations of the VP-VBR model’s rotor subsystem are given by (5–5), (5–11), and (5–12).  5.1.2 Variable-Parameter Interfacing Circuit Equations  The derivation of the interfacing circuit equations starts by inserting (2–153), (2–154), and (2–158) into (2–147), (2–148), and (2–152), yielding  ( ) mqqslsmddslsrqssqs ppiLiLirv λλω ++++=  (5–17)  ( ) mddslsmqqslsrdssds ppiLiLirv λλω +++−=  (5–18)  mdfdlfdfdfdfd ppiLirv λ++= . (5–19) To form a proper circuit model, the time derivatives of the magnetizing fluxes must be eliminated from (5–17)–(5–19). A possibility is to substitute (5–5); however, the right-hand sides of (5–17)–(5–19) would be function of stator and field voltages due to the presence of qspλ , dspλ , and fdpλ , which would in turn require solving a 3-by-3 system of equations. Instead, we choose to reformulate mqdpλ  so that it becomes function of qspi , dspi , and fdpi . This is achieved by redefining (5–3) and (5–4) as  ∑=−+=Mz lkqzmqkqzqsmq Lii1λλ (5–20) 141   ∑=−++=Nz lkdzmdkdzfddsmd Liii1λλ  (5–21) and substituting (5–20) and (5–21) into (5–1), yielding  ++=∑∑==Nz lkdzkdzMz lkqzkqzfddsqsmqdmimdmqLpLpipiipp11''' 0)(λλλλλL  (5–22) where the modified inductance matrix is  111'''''''''''''''1001)()(−==+=≡∑∑Nz lkdzMz lkqzmqdmimddmqdmqdmqqmqdmiLLLLLLλΓλL . (5–23)  The machine-network interfacing equation (in qd coordinates) is then obtained by inserting (5–22) into (5–17)–(5–19), and combining the results with (2–149) and (2–155) as  ( )+++=''''''00'''000 )()(fdsqdfdsqdmqdsfqdfdsqdmqdrcfdsqdeipiv reiλLiλLRvωω  (5–24) where  [ ]fdsssc rrrrdiag=R  (5–25)  −−−=0000000000)(''''''''''''''''''mqdmddmqdddqqmqdmqdLLLLLLrλLω  (5–26) 142   =''''''''''''''''''''''''''''''0000000)(fddmddmqdlsmddddmqdmqdmqdqqmqdsfqdLLLLLLLLLLλL  (5–27)  [ ]Tdsqssqd ee 0''''''''' 0 =e . (5–28) The modified subtransient inductances appearing in (5–26) and (5–27) are defined as  '''''' mqqlsqq LLL += ,    '''''' mddlsdd LLL += ,    and    '''''' mddlfdfdd LLL +=  (5–29) while the modified subtransient voltages are given by  +−−−+−=∑∑==Nz lkdzkdzMz lkqzkqzmddmqdmddmqdmqdmqqdsqsmqdmddmqdmddmqqmqdmqmdrfddsqsLpLpLLLLLLiiLLLLLLeee11'''''''''''''''''''''''''''''''''''''''''''''0 λλλλω . (5–30)  The time derivatives of the rotor fluxes can be computed using (5–11)–(5–14). Finally, applying Park’s transformation to the stator variables in (5–24) [96, see (66) and (67)] yields the final machine-network interfacing equation of the VP-VBR model:  ( )+++=''''''''' ),(fdabcsfdabcsmqdrvsflcsffdabcscfdabcseipiveiλLLiRvθ  (5–31) where  [ ]lfdlslslslcsf LLLLdiag=L  (5–32) 143   °+°−°+°+°−°−°−°+°−°++−−−−−−=0)120()120()()120(5.1)120()()120()120(5.1)()120()120()(5.1)120()120()(00005.05.005.05.005.05.0),(222211121112111''''''rrrrrrrrrrrrrrrmddaaaaaaaaamqdrvsfLLLLLLLLLLLLLLLLLLLLLLLLLθθθθθθθθθθθθθθθθ λL (5–33)  [ ]Tdsqssabcs ee 0''''''1''' −= Ke . (5–34) The inductances necessary to compute (5–33) are defined as  )2sin()2cos()(1 ϕϕϕ cb LLL +=      and     )sin()cos()(2 ϕϕϕ dc LLL +=  (5–35)  3''''''mddmqqaLLL+=     and     3''''''mddmqqbLLL−=  (5–36)  '''32mqdc LL =      and    '''32mddd LL =  (5–37) where { } 120,120, +−= rrr θθθϕ .  A block diagram of the VP-VBR model is shown in Figure 5–1. 5.2 Constant-Parameter SVB VBR Model 5.2.1 Constant-Parameter Interfacing Circuit Equations  Analyzing (5–31), (5–33), and (5–35)–(5–37), it is observed that the interfacing circuit of the VP-VBR model is independent of the rotor position and the level of saturation144  (5–5), (5–11),(5–12), (5–30)RotorState-SpaceSubsystemrs+-rs+-rs+-iasibsicseas'''ebs'''ecs'''nvasvbsvcsabcvqdseqds'''ωrKsθrStator and Field Interfacing CircuitKsifdvfdefd'''+-efd''' rfdfd–Llcsf +Lvsf(θr,λmqd)'''ifd vfd fd+iqdsKsθrθrLvsf(θr,λmqd)'''θr-1 Figure 5–1. Variable-parameter VBR (VP-VBR) formulation of the saturable lumped-parameter synchronous machine model. if aL  is constant and bL  = cL  = dL  = 0. Tracing these terms back to (5–24), it can be seen that a constant-parameter model can be achieved if ''' 0sfqdL  is a diagonal matrix of the form [ ]χβαα LLLLdiag=L  (5–38) where αL , βL , and χL  are constants. This structure can be achieved by transferring select terms to the subtransient voltage sources and modifying the speed voltage terms accordingly [81], [90]. Applying this approach to (5–24) yields the following machine-network interfacing equation 145   ( )+++=''''''''00'''000fdsqdfdsqdsfuqdfdsqdurcfdsqdeipiv reiLiLRvωω  (5–39) where  −−+=00000000000000''''''mddulsmddulsuLLLLrωL  (5–40)  [ ]'''''''''''' 0 diag mddulfdlsmddulsmddulssfuqd LLLLLLL +++=L  (5–41)  [ ]Tdsqssqd ee 0'''''''''''' 0 =e . (5–42) The newly introduced inductance '''mdduL  refers to the unsaturated value of '''mddL , while the modified voltage sources ''''qse , ''''dse , and ''''fde  are expressed as  ∆∆−++−+−=∑∑==fddsqsmddmddmqdmddmddmqdmqdmqdmddumqqNz lkdzkdzMz lkqzkqzmddmqdmddmqdmqdmqqdsqsmddumddumqmdrfddsqsiiipLLLLLLLLLLLpLpLLLLLLiiLLeee''''''''''''''''''''''''''''''11''''''''''''''''''''''''''''''''''''00000 λλλλω (5–43) where  ''''''''' mddumddmdd LLL −=∆ . (5–44)  Finally, converting the stator variables of (5–39) to abc coordinates yields the machine-network interfacing equation of the constant-parameter VBR model (CP-VBR): 146   ++='''''''''''fdabcsfdabcscsffdabcscfdabcseipiveiLiRv (5–45) where the constant inductance matrix '''csfL  is expressed as  −−−−−−+=5.1000015.05.005.015.005.05.0132 ''''''mddulcsfcsf LLL  (5–46) and ''''abcse  is computed similarly to (5–34). If desired, (5–45) can be decoupled by introducing a zero-sequence branch to the stator [91], resulting in  +++=''''''''0 0 fdabcsngfdabcsDfdabcscfdabcsepipiveiLiLiRv (5–47) where  [ ]Tngngngng iii=i     and    csbsasng iiii ++=  (5–48)  [ ]DfDsDsDsD LLLLdiag=L     and    [ ]0diag 0000 LLL=L   (5–49)  '''mddulsDs LLL += ,    '''mddulfdDf LLL += ,   and    3'''0mdduLL −= . (5–50) The state equations of the rotor subsystem are still given by (5–5), (5–11), and (5–12). 5.2.2 Numerical Approximation  The modified subtransient voltages (5–43) contain current time derivatives, resulting in an improper state model. To form a proper state model, it is possible to approximate them using high-pass filters as in [90] and Section 4.2.2. A more accurate 147  option is to eliminate the time derivatives algebraically [81], [90]. This is done by solving for qspi , dspi , and fdpi  in (5–24), yielding −−−−−=∑∑==Nz lkdzkdzMz lkqzkqzmddmqdmddmqdmqdmqqmqmdrfddsqsfdslsrlsrsfddsqsqdsffddsqsLpLpLLLLLLiiirrLLrvvviiip11'''''''''''''''''''''00000λλλλωωωΓ (5–51) where the symmetric matrix '''qdsfΓ  is given by  1'''''''''''''''''''''''''''333231232221131211'''−=ΓΓΓΓΓΓΓΓΓ≡fddmddmqdmddddmqdmqdmqdqqqdsfLLLLLLLLLΓ . (5–52)  Substituting (5–51) into (5–43), it is observed that the modified subtransient voltages ''''qse , ''''dse , and ''''fde  (which are outputs of the rotor state-space subsystem) are now function of the terminal voltages. This can result in algebraic loops when the terminal voltages are neither states nor inputs, e.g., when the machine windings are connected to inductive elements [90]. It is recalled that algebraic loops are undesirable as they cause a sharp decrease in numerical robustness and a noticeable increase in computational cost [90]. Certain commercial programs are also unable to solve algebraic loops, e.g., PLECS.   Using an approach similar to [81], [90], algebraic loops can be prevented by approximating the terminal voltages found in (5–43) and (5–51) using first-order low-pass filters of the form  zzz pspsH00)(+=  (5–53) 148  (5–5), (5–11),(5–12), (5–43)RotorState-SpaceSubsystemLDs rs+-rs+-rs+-iasibsicsing neas''''ebs''''ecs''''gvasvbsvcsabciabcsvabcsvqdsiqdseqds''''ωrKsKsθrConstant-Parameter Decoupled Stator Interfacing CircuitLDsLDsL0srfd+ -ifd efd''''vfdfd+ LDffd–Hqdsvqds~Hfdifdvfdvfd~efd''''Constant-Parameter Field Interfacing Circuitθrθr(Outputs ofthe mechanicalsubsystem)Ks-1 Figure 5–2. Constant-parameter VBR (CP-VBR) formulation of the saturable lumped-parameter synchronous machine model. where zp0−  is the filter pole and { }fddsqsz ,,= , which correspond to the filters of qsv , dsv , and fdv , respectively. A block diagram of the CP-VBR model is shown in Figure 5–2. Therein, the tilde (~) superscript indicates filtered values. 149  5.3 Pole Selection Procedure  The low-pass filters (5–53) introduce numerical error, which can be alleviated by selecting large values of zp0 . However, this will increase stiffness, and possibly force the use of smaller integration step sizes to ensure numerical stability and/or sufficient accuracy. A procedure to select the smallest values of zp0  that achieve a given accuracy up to a defined frequency is presented here.   An approach to obtain a magnetically linear constant-parameter VBR synchronous machine model by the introduction of a fictional q-axis damper winding was proposed in [88]. The value of this winding’s leakage inductance is directly function of the machine’s original q- and d-axis subtransient inductances, whereas its resistance can be chosen freely as a means to control numerical accuracy and stiffness. As explained in [88], a good way to visualize and control the error introduced by this fictitious damper winding is to compare the operational impedance (inductance) of the original and approximate models.  The technique presented in [88] is herein extended to the proposed CP-VBR model. Two major differences arise. First, the fictitious damper winding of the model from [88] only affects the frequency response of the q axis, whereas the filters of the proposed CP-VBR model distort both the q and d axes. Second, the method presented in [88] assumes a magnetically linear model, thereby resulting in LTI TFs. Since the proposed CP-VBR model includes main flux saturation, its TFs will be nonlinear. Moreover, there is also coupling between the q and d axes due to the cross-saturation term '''mqdL .  However, as explained in [152], dynamic cross saturation can be neglected while only incurring a small loss in numerical accuracy (for this specific model). Consequently, for the purpose of selecting the filter poles, the inductance '''mqdL  is set to 0, thereby effectively decoupling the q and d axes. Moreover, at higher frequencies, the operational inductances converge towards the subtransient inductances, which depend only lightly on 150  saturation [91]. Because the proposed model contains low-pass filters, only the higher frequencies will be distorted. Therefore, saturation has a limited effect on the CP-VBR model’s numerical accuracy, and the pole selection procedure can be executed only once (at any level of saturation).  It will first be shown how to compute qsp0 , and later how to evaluate dsp0  and fdp0 . 5.3.1 q-axis Filter  The q-axis operational inductance, defined as the ratio of qsλ  over qsi , is related to stator currents and voltages by [29], [88]  qsqsqsq ivssLrsZ =+≡ )()(  (5–54) wherein Heaviside’s operator p  is replaced by Laplace’s operator s  to follow standard convention [29]. The original q-axis operational inductance )(sLqo  can be obtained as follows. First, consider an unsaturated machine and set rω  = 0. Then, insert (5–30) into (5–24) (considering only the q axis), and replace all other variables by q-axis stator voltages and currents using (2–147), (2–153), and (5–11). Finally, rearrange the resulting equation in the form of (5–54), yielding  )()()()()(2121'''sssLsLsL lsqqqo αααα−−= . (5–55)  Under the same assumptions, the operational inductance of the CP-VBR model )(sLqc  is found by substituting (5–51) into (5–43) (using the filtered value of qsv ), and inserting the resulting equation into (5–39), which after elimination of internal variables gives 151  Choose ffit, εfit, andinitial value of p0qs |Lqo(ffit)|-|Lqc(ffit)||Lqo(ffit)|ec =|ec| < εfit ?Increment p0qs NoYesStartEndChoose ffit, εfit, andinitial value of p0ds |L21o(ffit)|-|L21c(ffit)||L21o(ffit)|ec=|ec| < εfit ?Increment p0ds NoYesStartEnd(a) q-axis (b) d-axisp0fd = p0ds Figure 5–3. Pole selection procedure for: (a) the q-axis filter’s qsp0 ; and (b) the d-axis filters’ dsp0  and fdp0 .  ( )( )( )( ) )()()()()()(12021102110spsspssLsLpssLqsqslsDsqsqc αβαβααβα−−+−+= . (5–56) The functions )(1 sα  and )(2 sα  along with the coefficients 1β  and 2β  are given in (E1)–(E2) in Appendix E.   The smallest value of qsp0  yielding a relative error smaller than fitε  up to a fit frequency fitf  (in terms of the magnitude of the q-axis operational inductance) can be found using the iterative procedure presented in the flow chart of Figure 5–3 (a). The magnitude of the original and approximate q-axis operational inductances )(sLqo  and )(sLqc  of a sample machine are plotted in Figure 5–4. The parameter qsp0  was evaluated using  fitε  = 1% and fitf  = 100 Hz. 152  10–3 10–2 10–1 100 101 102 1030.20.30.40.50.60.7Frequency (Hz)Magnitude of the q-axis Operational Inductance (pu)  |Lqo||Lqc| Figure 5–4. q-axis operational inductance magnitude of the original ( qoL ) and constant-parameter ( qcL ) models. 5.3.2 d-axis Filter  The machine’s d axis can be described by a two-port network whose terminals are connected to the stator and field windings [153]. From circuit theory, it is known that the input-output characteristic of a two-port network can be represented using equivalent network parameters, e.g., impedance, admittance, or hybrid parameters. Admittance parameters, i.e.,   ≡=fddsfddsdfddsvvsYsYsYsYvvsii)()()()()(22211211Y  (5–57) are sometimes referred to as short-circuit parameters since they are determined by shorting one pair of terminals. Specifically, )(11 sY  and )(21 sY  are obtained with fdv  = 0, and )(12 sY  and )(22 sY  are found with dsv  = 0. It is therefore convenient to work with admittance parameters in this analysis since the effect of dsp0  and fdp0  is decoupled. 153   The original d-axis admittance parameters of the machine can be obtained similarly to the q-axis operational inductance. Using the same assumptions (i.e., neglecting saturation and setting rω  to 0), )(sdoY  is found by inserting (5–30) into (5–24) (this time only considering the d axis) and eliminating internal variables using (2–148), (2–152), (2–154), (2–158), and (5–12), yielding  [ ] ( )[ ]224316453 )()()()()()()( ×− −−= IY sssssssdo αααα αα . (5–58)  Similarly, to compute the admittance matrix )(sdcY  of the CP-VBR model, (5–51) is inserted into (5–43) (using the filtered values of dsv  and fdv ), and the resulting equation is substituted into (5–39). After elimination of the internal variables, )(sdcY  can be formulated as  ( )[ ] ( )[ ]4432231644573 )()()()()()()()( ββαββα ssssssssdc αααα −−−−= ×− FIY . (5–59) The functions and coefficients appearing in (5–58) and (5–59) are defined in (E3)–(E7) in Appendix E.  Here, four TFs can be considered to select dsp0  and fdp0  (two each). An option is to apply the method used in Section 5.3.1 to all four TFs, and subsequently select the most constraining results. However, this may be unnecessary.  It can be seen that at higher frequencies, where the d-axis circuit is predominantly inductive, the order of magnitude of the four TFs is similar. Since fdv  is typically a few orders of magnitude smaller than the stator voltage, it can be concluded that its high-frequency content – which is distorted by fdH  – has a marginal effect on dsi  and fdi . Consequently, there is no need to determine fdp0  meticulously. Moreover, )(11 sY  and )(21 sY  are both function of dsp0  (but not of fdp0 ). Our experiments with several machines have shown that for any given frequency, the relative error of )(21 sY  is usually greater than 154  that of )(11 sY . As a result, a larger value of dsp0  (i.e., more constraining) is obtained by applying the approach of Section 5.3.1 to )(21 sY  instead of )(11 sY .   The proposed approach to determine dsp0  and fdp0  is presented in Figure 5–3 (b). The value of dsp0  is obtained by comparing the magnitude of )(21 sL o  and )(21 sL c , which correspond respectively to the original and approximate short-circuit inductances. The short-circuit inductance )(21 sL z  is related to )(21 sY z  by  )(1)(2121ssYsLzz =  (5–60) where { }coz ,= . Finally, fdp0  is assumed equal to dsp0 , which is a conservative assumption but does not worsen stiffness. 5.4 Computer Studies  The proposed saturable CP-VBR [see Section 5.2], VP-VBR [see Section 5.1], and qd [see Section 3.2.2.3] models have been implemented in MATLAB/Simulink using the PLECS toolbox. As shown in Chapters 2 and 3, the efficient qd model is implemented using state-space equations, i.e., without circuit elements. This means that it cannot be interfaced directly with some exciter circuits. A per unit representation is used in all models and studies. All parameters are given on the generator’s base power. As with the rotor circuit, the exciter parameters are referred to the machine’s stator, resulting in very small voltages. For the sole purpose of visualization, the field winding voltage (or exciter output voltage) will be re-scaled using ( ) fdfdmdxfd vrXe =  [29].  Unless otherwise stated, the subject models are simulated using Simulink’s explicit ode45 solver (maximum and minimum step sizes of 1 ms and 0.1 µs, respectively, and relative and absolute error tolerances of 10-4). Reference solutions are obtained using the155  Thév. Imp.Rsn-sSGa-gVsType AC1AExciter [95]Vc Vref+-iabcs ifdvqs + vds2 2vqd0sXfo1TransducerTRs + 11 Figure 5–5. One-line diagram of the test system with the TF-based exciter model. VP-VBR model with the same solver but with much stricter settings: maximum and minimum step sizes of 1 µs and 10 ns, respectively, and relative and absolute tolerances of 10-5. All simulations are executed on a PC with Windows XP and a 2.83-GHz Intel CPU. 5.4.1 TF-Based Exciter Model   In the first study, a 120-MVA wye-grounded synchronous generator (SG) [83] is connected through a step-up transformer (Xfo1) to a Thévenin representation of a transmission system as shown in Figure 5–5. The machine and network parameters are summarized in Appendix F. The source voltage is set to 1 pu.  The SG is fed by an ac excitation system with a diode bridge rectifier [19], [95]. The exciter is represented using the TF-based IEEE AC1A model [95, Sections 4 and 6.1]. Its block diagram is presented in Figure 5–6. The sample parameters [95, Annex H.5] are used, with the exception of the overexcitation and underexcitation limiters which are omitted. These parameters are reproduced in Appendix F. The reference voltage refV  (controlling the voltage at the stator terminals) is set to 1.05 pu.  As explained in Section 2.3.3.1 and [35], resistive snubbers are added to the stator terminals ( sR −sn ) of the qd model to achieve a compatible machine-network interface. To highlight the resulting compromise between numerical accuracy and efficiency, two156  VcVref + – KATAs + 1KFsTFs + 1–1TEXsVESE(VE)VEKEKDKCifdVEFEX(IN)ΠIN+–+ +++ ifdrfdXmdexfd vfd0VRMAXVRMIN Figure 5–6. Block diagram of the IEEE AC1A excitation system [95]. snubbers are considered: sR −sn  = 10 pu and sR −sn  = 1 pu. The resulting formulations are referred to as qd1 and qd2, respectively. Since the TF-based exciter model outputs a voltage, no snubber is required at the field terminals of the machine. The filter poles of the CP-VBR model are found using the procedure described in Section 5.3 with fitf  = 100 Hz and fitε  = 1%. They are qsp0  = 2240 and dsp0  = fdp0  = 4900.  In this case study, the SG initially operates in steady state with nominal torque. At t  = 0.05 s, a single-phase-to-ground fault occurs on phase a of the stator terminals. A small fault resistance of 0.01 pu is considered. It is assumed that the fault is cleared after 100 ms.   The reference trajectories of the stator currents asi , qsi , and dsi , the field current fdi , the field voltage xfde , and the electromagnetic torque eT  are presented in Figure 5–7. The peak value of asi  (which contains a dc offset [29]) is more than 9 times larger than its nominal value. As expected, Figure 5–7 demonstrates that the TF-based exciter model is devoid of high frequencies, i.e., the switching harmonics are neglected.  A magnified view of the first peak of the stator current asi  as predicted by the subject models is reproduced in Figure 5–8. The qd2 model is shown to predict the phase and magnitude of asi  with noticeable errors, which are caused by the presence of a small snubber ( sR −sn  = 1 pu). All other models calculate asi  with much better accuracy. An157  –505i as (pu)–202i qs (pu)024i ds (pu)34i fd (pu)2.02.5e xfd (pu)0 0.05 0.10 0.15 0.20–303Time (s)T e (pu)See Figure 5–8See Figure 5–9 Figure 5–7. Reference trajectories of stator currents asi , qsi , and dsi , field current fdi , field voltage xfde , and electromagnetic torque eT  during a single-phase-to-ground fault with a TF-based exciter model. 158  0.0570 0.0575 0.0580 0.0585 0.0590 0.0595 0.06008.59.09.5Time (s)i as (pu)  a) Referenceb) VP-VBRc) CP-VBRd) qd1e) qd2abedc Figure 5–8. Magnified view of the first peak of stator current asi  during a single-phase-to-ground fault as predicted by the subject models with a TF-based exciter model.  0.106 0.108 0.110 0.112 0.1142.53.0Time (s)i fd (pu)  a) Referenceb) VP-VBRc) CP-VBRd) qd1e) qd2eca, b, d Figure 5–9. Magnified view of the field current fdi  during a single-phase-to-ground fault as predicted by the subject models with a TF-based exciter model.   enlarged view of the field current fdi  during the fault condition is also presented in Figure 5–9. The CP-VBR model introduces some numerical error, which could be alleviated if necessary by increasing the magnitude of the filter poles. The qd2 model also introduces some error in fdi . 159  Table 5–1. 2-norm relative errors of stator currents abcsi , field current fdi , and electromagnetic torque eT  for the single-phase-to-ground fault study with a TF-based exciter model.  VP-VBR CP-VBR qd1 qd2 abcsi  (%) 0.001 1.031 1.923 20.27 fdi  (%) 0.001 1.227 0.371 2.391 eT  (%) 0.001 1.435 1.388 14.52   To evaluate the models’ accuracy, the 2-norm relative errors of abcsi , fdi , and eT  for t  = 0 to 0.25 s are presented in Table 5–1. As explained in Section 2.5, the 2-norm error of abcsi  refers to the mean of the single-phase errors of asi , bsi , and csi . The VP-VBR model is virtually errorless since it has no approximations and a higher order solver with small error tolerances is used. The 2-norm errors of the proposed saturable CP-VBR model vary between 1 and 1.5%, e.g., 1.031% for abcsi , which is adequate for most studies. These results also support the validity of the pole selection procedure. The numerical accuracy of the qd1 model is similar to that of CP-VBR. Finally, as seen in Figure 5–8, the 2-norm errors of qd2 are very large, e.g., 14.52% for eT .  The pole selection procedure presented in Section 5.3 is executed assuming an unsaturated machine, based on the hypothesis that saturation has a limited effect at the higher frequencies where the filters introduce numerical error. To validate this assumption, the above case study is re-executed using magnetically linear models. The resulting 2-norm relative errors of the CP-VBR model are collected in Table 5–2 (the reference solution also assumes an unsaturated model, since only the numerical errors are of interest). Tables 5–1 and 5–2 show that the errors of the magnetically linear and saturated models have the same order of magnitude, e.g., 1.267% and 1.435% for eT . This corroborates the assumption used in Section 5.3. 160  Table 5–2. 2-norm relative errors of stator currents abcsi , field current fdi , and electromagnetic torque eT  for the single-phase-to-ground fault study with a TF-based exciter model (without saturation).  CP-VBR abcsi  (%) 0.863 fdi  (%) 0.942 eT  (%) 1.267 Table 5–3. Numerical performance of the subject models for the single-phase-to-ground study with a TF-based exciter model.  VP-VBR CP-VBR qd1 qd2 qd1 qd2  (with ode45) (with ode15s) No. time steps 3034 4960 45092 7756 9187 4599 CPU time (s) 3.46 1.32 8.35 1.55 3.82 1.55   The number of integration time steps and CPU time taken by each model for this case study (simulated from t  = 0 to 3 s) are summarized in Table 5–3. The VP-VBR model takes the fewest number of steps (3034). However, due to its variable-parameter circuit, it is more than 2.5 times slower than the CP-VBR model (3.46 s vs. 1.32 s, respectively). Table 5–3 demonstrates that the good accuracy of the qd1 model [see Table 5–1] is achieved at the expense of a high computational cost (8.35 s with ode45). Its CPU time can be decreased if a stiffly stable solver (e.g., ode15s) is used; however, it remains slower than both VBR models. Finally, the qd2 model, which has poor numerical accuracy, is nevertheless slower than the proposed CP-VBR model. 5.4.2 Detailed Exciter Model  In this section, the TF-based ac exciter is replaced with a detailed circuit-based potential-source static excitation system [19]. The new system is shown in Figure 5–10. The simplified control system consists of a voltage transducer, a PI controller, a slew rate161  Thév. Imp.Rsn-sSGa-giabcsRsn-fifdXfo1Xfo2iabcexplpfs + plpfkp ski+SmaxSminRate limiter6-PulseGener-atorβss + βs + ωbp2 2vqs + vds2 2vqd0sVrefααmaxαmin+- Figure 5–10. One-line diagram of the test system with a circuit-based exciter model. limiter, and SimPowerSystems’ built-in 6-pulse generator [13]. The parameters are summarized in Appendix G.   The qd models now require snubbers at the stator and field terminals. For qd1, the field snubber fR −sn  is set to fdr100 ; for qd2, we have fR −sn  = fdr10 . The VBR models do not require snubbers at either set of terminals.  The fault study from Section 5.4.1 is repeated with the new system/exciter model. The reference trajectories of the field current fdi , field voltage xfde , and three-phase ac side exciter currents abcexi  are presented in Figure 5–11. The trajectories of the stator currents and electromagnetic torque are similar to those of Figure 5–7 and are not presented here. As seen in Figure 5–11, the field voltage is highly oscillatory; however, the current fdi  still contains only lower frequencies. This demonstrates the low impact of the higher frequency content of fdv .  An enlarged view of fdi  is reproduced in Figure 5–12. This time, the qd1 model exhibits a small but noticeable error (similarly to the CP-VBR model), which is a result of its162  2.53.03.5i fd (pu)036e xfd (pu)0 0.05 0.10 0.15 0.20 0.25–404Time (s)i abcex (pu)See Figure 5–12See Figure 5–13 Figure 5–11. Reference trajectories of field current fdi , re-scaled field voltage xfde , and ac exciter currents abcexi  during a single-phase-to-ground fault with a detailed exciter model. field snubber. The qd2 model is considerably worse. Figure 5–13 contains a magnified view of the phase b exciter current bexi . Both VBR models predict bexi  very accurately. This is not the case of the qd models, which exhibit noticeable phase shifts.  The 2-norm relative errors for this case study (for t  = 0 to 0.25 s) are summarized in Table 5–4. These results confirm the observations made in Figures 5–12 and 5–13. In particular, the numerical errors of the CP-VBR model are very similar for both case studies [see Tables 5–1 and 5–4]. Conversely, the 2-norm error of fdi  as predicted by qd1 is considerably worse in the second case study (1.645%) than in the first (0.371%). 163  0.130 0.132 0.134 0.136 0.1382.53.0Time (s)i fd (pu)  a) Referenceb) VP-VBRc) CP-VBRd) qd1e) qd2ecda, b Figure 5–12. Magnified view of the field current fdi  during a single-phase-to-ground fault as predicted by the subject models with a detailed exciter model.    0.126 0.130 0.134 0.138 0.142–303Time (s)i bex (pu)  a) Referenceb) VP-VBRc) CP-VBRd) qd1e) qd2eda, b, c Figure 5–13. Magnified view of the exciter current bexi  during a single-phase-to-ground fault as predicted by the subject models with a detailed exciter model.    To validate the lesser impact of fdp0  on numerical accuracy, this case study has been repeated with the CP-VBR model using a reduced value of fdp0  (490 instead of 4900). The resulting 2-norm relative errors are presented in Table 5–5. As hypothesized in Section 5.3.2, the 2-norm errors remain almost unchanged, e.g., 1.035% instead of 1.040% for abcsi . 164  Table 5–4. 2-norm relative errors of stator currents abcsi , field current fdi , and electromagnetic torque eT  for the single-phase-to-ground fault study with a detailed exciter model.  VP-VBR CP-VBR qd1 qd2 abcsi  (%) 0.001 1.040 1.913 21.60 fdi  (%) 0.006 1.132 1.645 7.012 eT  (%) 0.007 1.531 1.635 16.74 Table 5–5. 2-norm relative errors of stator currents abcsi , field current fdi , and electromagnetic torque eT  for the single-phase-to-ground fault study with a detailed exciter model (with fdp0  = 490).  CP-VBR abcsi  (%) 1.035 fdi  (%) 1.151 eT  (%) 1.532 Table 5–6. Numerical performance of the subject models for the single-phase-to-ground study with a detailed exciter model.  VP-VBR CP-VBR qd1 qd2 qd1 qd2  (with ode45) (with ode15s) No. time steps 21111 21198 177193 37736 45271 43390 CPU time (s) 69.0 6.97 42.6 9.52 13.8 9.35   Finally, Table 5–6 contains the number of integration steps and CPU time of the subject models for t  = 0 to 3 s. Unlike in the first case study, the two VBR models use a similar number of steps (around 21000), showing that the higher stiffness of the CP-VBR model has no impact in this system. As a result, the CP-VBR model is almost 10 times faster than the VP-VBR model (6.97 and 69 s, respectively), which is a substantial improvement. The CP-VBR model is also almost twice as fast as the qd1 model (with ode15s), while introducing less error [see Table 5–4]. Moreover, the CP-VBR model is faster than the less accurate qd2 model (which requires more than 9 s). Consequently, the proposed CP-VBR model clearly exhibits the best combination of numerical accuracy and efficiency. 165  CHAPTER 6:  CONSTANT-PARAMETER EMTP-TYPE VBR INDUCTION MACHINE MODEL WITH MAIN FLUX SATURATION  Thévenin prediction-based qd models are frequently used to represent induction machines in EMTP-type programs since such models have a rotor-position-independent interfacing circuit and require little CPU time compared to other state-of-the-art models [35]. However, these models tend to be numerically inaccurate when using larger integration step sizes due to the need for predicting speed voltage terms at every time step. Significantly better numerical accuracy can be obtained with PD and VBR models [35]. While slightly less efficient than qd models, PD and VBR induction machine models can also be formulated to have rotor-position-independent interfacing circuits [61].  The effect of main flux saturation can be incorporated into EMTP-type models straightforwardly [64], [111]. While this increases modeling accuracy, it also renders the interfacing circuits of PD, Thévenin qd, and VBR models saturation-segment-dependent. Consequently, the network’s conductance matrix G  must be re-factored every time that the operating segment of any piecewise-linear saturation function changes. This can increase the solution’s computational burden and make such models unsuitable for real-time simulators.  A new VBR induction machine model with a true constant-parameter model (i.e., rotor-position- and saturation-segment-independent) that takes into account main flux saturation is proposed in this chapter. The model is achieved by transferring the rotor-166  position- and saturation-segment-dependent parts of the original interfacing circuit resistance into the equivalent voltage source. Through computer studies, the proposed model is shown to be virtually as accurate as the original PD and VBR models. It is also demonstrated that the proposed model can significantly reduce simulation time for multimachine power systems.  6.1 SVB VBR Model  To set the stage for the derivation of the proposed model, the equations of the state-space VBR induction machine model with main flux saturation are recalled [111]. Unlike in Chapter 4, the SVB model considered here assumes that main flux saturation is represented using a piecewise-linear function [see (2–60)–(2–64)]. Moreover, the state variables of the rotor subsystem are all the rotor winding flux linkages, i.e., the magnetizing fluxes mqλ  and mdλ  are auxiliary algebraic variables.   The machine-network interfacing equation is [111]  '''' abcsabcsabcsjabcssabcs p eiLiRv ++=  (6–1) where sR  is defined in (2–11) and   =SjMjMjMjSjMjMjMjSjabcsjLLLLLLLLL''L  (6–2) with  ajlsSj LLL += ,    ajMj LL 21−= ,    and    ''32Djaj LL =  (6–3) 167  and  11'' 11−=+= ∑Nz lrzDjDj LLL . (6–4) As explained in Chapter 2, the subscript j  refers to the j th piecewise-linear segment of the saturation curve depicted in Figure 2–8. It is added to all saturation-segment-dependent scalars and matrices. The subtransient voltages ''abcse  in (6–1) are defined as  [ ]Tdqsabcs ee 0''''1'' −= Ke  (6–5) while the q- and d-axis subtransient voltage sources [ ]Tdqqd ee '''''' ≡e  are given by  qdjqdrjqdsjqd a res321'' λAλAie ++= . (6–6) The rotor flux linkage state equation is expressed as [111]  qdjqdrjqdsjqdrp res321 λBλBiBλ ++= . (6–7) Finally, the magnetizing fluxes mqdλ , which are used to determine the saturation level, can be expressed as  ( ) qdDjDjqdrqdsDjmqd LLL res''4'' λλBiλ ++= . (6–8) The terms ja1 , j2A , j3A , j1B , j2B , j3B , and 4B  are defined in (H1)−(H10) in Appendix H. It is noted that j3A  contains the time derivative of the flux angle φ , which is defined in Figure 2–7. To follow the convention set forth in [111], its time derivative is denoted by φω . It will be discussed in Section 6.3.2 how to evaluate φω . 168  6.2 Variable-Parameter EMTP-Type VBR Model 6.2.1 Model Derivation   The derivation of the variable-parameter EMTP-Type VBR model is recalled in this section [111]. As stated in Section 2.3.4, all discrete scalars and matrices that depend on the angle or speed (of the rotor and/or the reference frame) contain the suffix )(τ  where τ  indicates the time step. Such scalars and matrices are often referred to as time-dependent for conciseness. Moreover, as for the state-space model of Section 6.1, the subscript j  is added to all discrete saturation-segment-dependent scalars and matrices. Following discretization of (6–1), the stator voltages at the current time step t  are given by [111]  ( ) )()()()( '''' tttkt shabcsabcsabcsjsabcs eeiLRv +++=  (6–9) where it is recalled that tk ∆= 2  and  ( ) )()()()( '''' ttttttkt abcsabcsabcsabcsjssh ∆−−∆−+∆−−= veiLRe . (6–10) Similarly, discretizing the rotor states (6–7) yields ( ) ( ))()()()()()()()()()( resres21 tttttttttttttt qdqdjqdrjjqdsqdsjqdr ∆−++∆−∆−+∆−+= λλDλFFiiEλ . (6–11) The definitions of jE , j1F , j2F , and jD  are given in (H11) and (H12) in Appendix H. Therein, it can be seen that by choosing rωω =  (i.e., using the rotor reference frame), the matrices found in (6–11) become independent of the rotor speed. This observation will become useful in Section 6.3.1. 169   To interface the machine model with the power network, (6–9) must be written as a function of )(tabcsv , )(tabcsi , and history terms, which requires eliminating )('' tabcse . Substituting (6–11) into (6–6), )('' tqde  can be rewritten after algebraic manipulation as  )()()()('' tttt qdqdsjqd hiHe +=  (6–12) where the history term vector )(tqdh  is expressed as )()()()()()()()()()( res2res12 tttttttttttttt qdjqdjqdrjjqdsjqd ∆−++∆−∆−+∆−= λPλPλFNiMh . (6–13) The definitions of jH , jM , jN , j1P , and j2P  are summarized in (H13) and (H14) in Appendix H.   Next, (6–12) must be converted to abc coordinates. This yields [111]  )()()()('' tttt rabcsjabcs eiKe +=  (6–14) where the three-phase history voltage source )(tre  is given by  [ ]TTqdsr ttt 0)()()( 1 hKe −= . (6–15)  Using (H11)−(H13), it is observed that jH  [see (6–12)] can be expressed as [93], [111]  ( ) −≡−+= −××jjjjjjNNjjj hththhtktat1221112222221 )()()()()( BBIAIH . (6–16) Since jH  is skew-symmetric, jK  can be formulated as follows [93], [111]: 170   =jjjjjjjjjjktktktkktktktkkt132213321)()()()()()()(K  (6–17) where  jj hk 11 32= ,     )(333)( 212 thhtk jjj −−= ,     and     )(333)( 213 thhtk jjj +−= . (6–18)  Finally, substituting (6–14) into (6–9) yields the machine-network interfacing equation  )()()()( eq tttt habcsjabcs eiRv +=  (6–19) where the equivalent resistance matrix jeqR  is given by  )()( ''eq tkt jabcsjsj KLRR ++=  (6–20) and the three-phase history voltage source )(the  is expressed as  )()()( ttt rshh eee += . (6–21)  The interfacing equation (6–19) contains the saturation-segment- and time-dependent resistance matrix (6–20), which together with the history voltage source (6–21) defines the variable-parameter model thereafter referred to as VP-VBR.  6.2.2 Simulation Steps  The necessary steps to simulate the VP-VBR model at time step t  are given below.  1) Predict the electrical rotor speed )(trω , the flux angle derivative )(tφω , and the magnetizing fluxes )(tmqdλ  using linear extrapolation, i.e., with (2–114). 171  2) Calculate the electrical rotor position )(trθ  using the trapezoidal integration rule and the predicted rotor speed, i.e., evaluate (2–115). 3) Calculate the main flux )(tmλ  using (2–58). 4) Compute )(res tqdλ  using (2–63) and (2–64). 5) Compute )(tqdh  using (6–13). Evaluate )(tshe  and )(tre  using (6–10) and (6–15), respectively, and combine to form )(the  using (6–21). 6) Evaluate the variable-parameter resistance matrix )(eq tjR  using (6–20), and insert it into the network’s conductance matrix G . 7) Convert (6–19) to a Norton equivalent and insert its current source into the equivalent current injection vector )(thi  of the EMTP-type system. 8) Find the network’s unknown nodal voltages using (2–8) or (2–9). 9) Using the stator voltages )(tabcsv , solve for the stator currents )(tabcsi  in (6–19), and convert them to qd coordinates using (2–20). 10) Compute )('' tabcse  using (6–14) and )(tqdrλ  using (6–11). 11) Compute the corrected magnetizing fluxes )(tmqdλ  using (6–8), and evaluate the corrected main flux )(tmλ  using (2–58). 12) Evaluate the flux angle )(tφ  using (2–63) or (2–64) and correct its time derivative )(tφω  using  ttttt∆∆−−=)()()( φφωφ . (6–22) 172  13) If the operating piecewise-linear segment j  of the saturation characteristic changes, go to step 14). Otherwise, skip to step 15). 14) Update j, and re-compute )(res tqdλ , )(tqdh , )(tre , )('' tabcse , )(tqdrλ , )(tmqdλ , )(tmλ , )(tφ , and )(tφω . 15) Compute the electromagnetic torque )(tTe  using (2–47), and the corrected electrical rotor speed )(trω  and position )(trθ  using (2–112), and (2–113), respectively. 6.3 Constant-Parameter EMTP-Type VBR Model 6.3.1 Constant-Parameter Interfacing Circuit Derivation   To avoid having to re-factor the network’s conductance matrix G  during simulations, the machine’s interfacing resistance matrix jeqR  should be constant. However, as shown in (6–20), the resistance matrix of the VP-VBR model is function of the rotor speed (due to jK ) and the saturation segment (due to ''abcsjL  and jK ). Specifically, it can be seen from (6–16)−(6–18) that the rotor-speed-dependent terms of jK  originate from oj2B  and oj2a . Their definitions are recalled here for convenience:  ( ) NNroj ×−= IB ωω2  (6–23)  [ ]11211''2 −−−= lrNlrlrDjroj LLLL ωa . (6–24) As stated in Section 6.2, the speed dependency of oj2B  can be easily eliminated by using the rotor reference frame ( rωω = ). It was also demonstrated in [61] that for magnetically linear VBR induction machine models, oj2a  can be replaced by a null vector when forming the resistance matrix eqjR  without incurring any significant loss in accuracy.  173   Furthermore, the only saturation-segment-dependent term in both ''abcsjL  and jK  is ''DjL , which is defined in (6–4) as the parallel combination of the magnetizing and rotor leakage inductances. Since the rotor leakage inductances are much smaller than the magnetizing inductance, relatively large changes of DjL  due to saturation result in considerably smaller variations of ''DjL . Consequently, the magnitude of ''abcsjL  and jK  is marginally affected by the level of saturation.   As a result, the interfacing resistance matrix jeqR  can be written as the sum of a constant part consteq−R  and a considerably smaller saturation-segment- and rotor-position-dependent part jeqR∆ :  )()( eqconsteqeq tt jj RRR ∆+= −  (6–25) where consteq−R  is constructed as follows:  const'' constconsteq KLRR ++= −− csabs k . (6–26) In (6–26), '' const−csabL  is built using (6–2)−(6–4) with unsaturated coefficients, i.e., with DjL  evaluated at j  = 1, which corresponds to the unsaturated magnetizing inductance [see Figure 2–8]. The matrix constK  results from computing (6–17) using the rotor reference frame, setting 0a =oj2 , and evaluating DjL  at Sj = , where S  refers to the last piecewise-linear segment of Figure 2–8. It is given by  =455545554constkkkkkkkkkK  (6–27) where the scalars 4k  and 5k  are defined in (H15) and (H16) in Appendix H.   A constant-parameter interfacing circuit could potentially be achieved by replacing )(eq tjR  with consteq−R  in (6–19). Since the magnitude of consteq−R  is much larger than that 174  of jeqR∆ , only a small numerical error would be introduced. A more accurate approach is to substitute (6–25) into (6–19) and transfer )()(eq tt abcsj iR∆  into the subtransient voltage source. The interfacing equation of the resulting constant-parameter VBR (CP-VBR) model is thus  )()()( consteq ttt hRiabcsabcs eiRv += −  (6–28) where  )(~)()()( eq tttt abcsjhhRi iRee ∆+= . (6–29) The matrix consteq−R  is symmetric. Therefore, unlike the VP-VBR model, the proposed model does not preclude the use of more efficient symmetry-optimized factorization schemes such as the Cholesky decomposition [118].  Unless an iterative nonlinear solution method is used, the stator currents )(~ tabcsi  required to compute )(thRie  must be predicted at every time step (the “~” symbol indicates predicted/approximated values). Here, a three-point linear predictor with smoothing [5] is applied to the qd stator currents:  )3(43)2(21)(45)(~ 0000 ttttttt sqdsqdsqdsqd ∆−−∆−+∆−= iiii . (6–30) The resulting currents are then converted to the abc reference frame using (2–27). Three-point linear prediction of qd stator currents has been successfully used in PD [64], qd [5], [82], and VBR [94] synchronous machine models to eliminate the dependency on rotor position and/or speed (but not saturation).   Moreover, to reflect the true operating point from the previous time step, the stator history term )(tshe  should be rewritten using (6–14) and (6–28) as 175   ( ) ( ))()()(~)()()( '''' constconstttttttktttktabcsrabcscsjabjabcscsabssh∆−−∆−+∆−∆−∆+∆−−+=−veiLKiLKRe  (6–31) where   )()( const tt jj KKK ∆+=  (6–32)  '''' const'' csjabcsabcsjab LLL ∆+= − . (6–33)  Finally, it is emphasized that no terms have been neglected in the proposed model: all saturation-segment- and rotor-position-dependent terms are transferred from the resistance matrix to the subtransient voltage source.  6.3.2 Simulation Steps  The necessary steps to simulate the CP-VBR model at time step t  are given below.  1) Predict the electrical rotor speed )(trω , the flux angle derivative )(tφω , and the magnetizing fluxes )(tmqdλ  using linear extrapolation, i.e., with (2–114). 2) Calculate the electrical rotor position )(trθ  using the trapezoidal integration rule and the predicted rotor speed, i.e., evaluate (2–115). 3) Predict the stator currents )(0 tsqdi  using (6–30) and convert them to abc coordinates using (2–27). 4) Calculate the main flux )(tmλ  using (2–58). 5) Compute )(res tqdλ  using (2–63) and (2–64). 176  6) Compute )(tqdh  using (6–13). Evaluate )(tshe  and )(tre  using (6–31) and (6–15), respectively, and combine to form )(the  using (6–21). 7) Compute )(tjK  and )(tjK∆  using (6–17) and (6–32), respectively, construct jeqR∆  using (6–25), and form the three-phase history voltage source )(thRie  using (6–29). Convert (6–28) to a Norton equivalent and insert its current source into the equivalent current injection vector )(thi  of the EMTP-type system. 8) Find the network’s unknown nodal voltages using (2–8) or (2–9). 9) Using the stator voltages )(tabcsv , solve for the stator currents )(tabcsi  in (6–28), and convert them to qd coordinates using (2–20). 10) Compute the corrected magnetizing fluxes )(tmqdλ  using (6–8), and evaluate the corrected main flux )(tmλ  using (2–58). 11) Evaluate the flux angle )(tφ  using (2–63) or (2–64) and correct its time derivative )(tφω  using (6–22). 12) If the operating piecewise-linear segment j  of the saturation characteristic changes, go to step 13). Otherwise, skip to step 14). 13) Update j , and re-compute )(res tqdλ , )(tqdh , )(tre , )(tjK , )(tjK∆ , )(tqdrλ , )(tmqdλ , )(tmλ , )(tφ , and )(tφω . 14) Compute the electromagnetic torque )(tTe  using (2–47), and the corrected electrical rotor speed )(trω  and position )(trθ  using (2–112), and (2–113), respectively. 177  IMCableThév. Imp.a-g  Figure 6–1. One-line diagram of a small test system comprised of an induction motor (IM), a shunt capacitor, a cable, a transformer, and a network Thévenin equivalent. 6.4  Computer Studies  The proposed CP-VBR model has been implemented in C and directly interfaced with PSCAD’s computational engine. To establish meaningful comparisons, the PD [see [93] and Section 2.3.4.1], Thévenin prediction-based qd [see [5] and Section 2.3.4.2], and VP-VBR [see [111] and Section 6.2] models were also implemented in the same environment. As explained in Section 2.3.4.2, the qd model is integrated using the damped trapezoidal rule for increased numerical stability [5], [30]. To achieve efficient implementations, the saturation-segment-dependent scalars and matrices of all models are pre-computed during their initialization stages. The qd and VBR models are implemented in the rotor reference frame. Finally, the subject models are also compared with PSCAD’s built-in induction machine model, which is based on the Norton current source interfacing approach [7], [83]. This model is hereinafter referred to as Norton-SS. For consistency, all simulations are executed on a PC with a 2.83-GHz Intel CPU running Windows XP. 6.4.1 Single-Machine System  As a first validation step, the small system shown in Figure 6–1 has been implemented in PSCAD. This network consists of a low-voltage induction motor (IM) with a power factor correction shunt capacitor bank, a short cable (represented by lumped elements), an ideal delta/wye-grounded step-down transformer with leakage inductances, and a Thévenin equivalent representing a medium-voltage system. This system has been 178  chosen for two main reasons: i) due to its small size, it is easier to correlate the numerical errors with the machine models; and ii) the interaction of the capacitor bank with the machine and the network excites higher frequency modes (i.e., supersynchronous electrical resonance). This test system also permits the use of fairly large step sizes, at which the difference in numerical accuracy between the subject models can become more pronounced.  In this study, the motor initially operates in steady state with nominal load torque. At t  = 0.015 s, a single-phase-to-ground fault occurs in the medium-voltage network. The fault is cleared a tenth of a second later. This case study was chosen as it validates the numerical properties of the proposed model in difficult conditions. In particular, this study creates fast stator current transients, which introduces more prediction error [see (6–30)]. The fault removal also results in a steep increase of the main flux and thus fast changes between operating saturation segments. To further increase the level of saturation, and therefore the magnitude of jeqR∆ , the voltage source of the Thévenin equivalent is set to 1.1 pu. Finally, to properly validate the proposed model, two motors are considered.  6.4.1.1 Induction machine #1 (IM1)  The first machine (IM1) is a 230-V, 5-hp motor modeled with one rotor circuit [71]. The machine and network parameters are summarized in Appendix I. This machine is of particular interest due to its large rotor leakage inductance with respect to its magnetizing inductance [see Chapter 4], which increases the magnitude of jeqR∆  (relatively to jeqR ) and thus amplifies the stator current prediction error.  The reference trajectories of the stator currents abcsi , electromagnetic torque eT , and main flux mλ  for this study are shown in Figure 6–2. As can be seen, this study excites the natural resonance of the system (approximately 500 Hz). Based on the trajectory of mλ  179  –50050i as (A)–50050i bs (A)–40040i cs (A)–50050T e (N·m)0 0.05 0.10 0.15 0.200.30.40.5λ m (Wb)Time (s)See Figure 6–3See Figure 6–4 Figure 6–2. Reference trajectories of stator currents asi , bsi , and csi , electromagnetic torque eT , and main flux mλ  during a single-phase-to-ground fault (IM1).  180  in Figure 6–2 and the saturation data given in Appendix I, it is observed that the machine operates in the saturated region.  This case study has been simulated using all subject models with step sizes ranging from 50 µs to 1 ms. For this given scenario, artificial damping (α  = 0.95) is added to the qd model in order to attenuate the numerical oscillations occurring with larger step sizes (≥ 500 µs ) [5], [30]. Despite already containing internal damping resistances, additional shunt resistances snR  located at the stator terminals are mandatory to obtain converging solutions when using the Norton-SS model with t∆  = 250 µs ( snR  = 100 Ω) and 500 µs ( snR  = 50 Ω). No artificial damping is required for the PD and VBR models.  Magnified views of csi  following the fault occurrence and just before its removal are shown in Figures 6–3 and 6–4, respectively. Each figure contains two parts: part (a) is computed with t∆  = 100 µs, whereas part (b) is evaluated with t∆  = 500 µs. The first peak of csi , which includes supersynchronous electrical resonance, is predicted satisfactorily by all models with t∆  = 100 µs. For t∆  = 500 µs, the PD, qd, and VBR models accurately predict the fundamental component of the solution. However, the higher frequency content is poorly predicted, mostly due to using the low-order trapezoidal integration rule with a step size of only roughly twice the inverse of the Nyquist frequency. The solutions computed with the Norton-SS model are overall less accurate for both step sizes.  As shown in Figure 6–4 (a), with t∆  = 100 µs, all models except Norton-SS yield visually similar solutions before the fault removal. With t∆  = 500 µs, [see Figure 6–4 (b)], the PD and VP-VBR models also give very accurate solutions, which shows that these models are less prone to error accumulation. The CP-VBR model is slightly less accurate, which is mostly due to its large rotor leakage inductance. At the same time, the solution given by the Norton-SS model significantly diverges from the reference trajectory, and the qd model exhibits sustained numerical oscillations. The use of a smaller damping181  20253035i cs (A)  a) Referenceb) CP-VBRc) PDfa to e(a)0.019 0.020 0.021 0.022 0.023 0.024 0.025 0.02620253035Time (s)i cs (A)  d) qde) VP-VBRf) Norton-SS(b)fbc, ead Figure 6–3. Magnified view of stator current csi  at the beginning of a single-phase-to-ground fault as predicted by the subject models: (a) with step size t∆  of 100 µs; and (b) with step size t∆  of 500 µs (IM1). coefficient α  could further attenuate these oscillations at the expense of additional loss of numerical accuracy.  To get a more comprehensive insight into the numerical accuracy of the subject models, the average of the single-phase stator current errors )( siε  and the electromagnetic torque error )( eTε  as a function of t∆  are presented in Figure 6–5. In concordance with Figures 6–3 and 6–4, the numerical accuracy of the proposed CP-VBR model is slightly worse than that of the PD and VP-VBR models at larger step sizes. Their errors for the182  61014i cs (A)  a) Referenceb) CP-VBRc) PD0.101 0.102 0.103 0.104 0.105 0.106 0.10761014Time (s)i cs (A)  d) qde) VP-VBRf) Norton-SSfa to efa, c, eeb(a)(b) Figure 6–4. Magnified view of stator current csi  before the removal of a single-phase-to-ground fault as predicted by the subject models: (a) with step size t∆  of 100 µs; and (b) with step size t∆  of 500 µs (IM1). bigger step sizes are fairly large (e.g., over 5% in si  and eT  with t∆  = 1 ms), which is due to the combination of the severity of the disturbance, the resonance, and the low-order integration rule. The qd model offers similar numerical accuracy to the PD and VBR models at step sizes of up to 250 µs, after which the error resulting from its approximations (e.g., prediction of speed voltages) dominates the discretization error. The Norton-SS model is last in terms of numerical accuracy for all step sizes, likely due to the time-step delay in its interfacing circuit [83]. Further tests with smaller step sizes have shown that all models183  50 100 250 500 10000510152025∆t (µs)ε(Te) (%)  010203040ε(is) (%)  a) CP-VBRb) PDc) qdd) VP-VBRe) Norton-SSeeccab, dab, d(a)(b) Figure 6–5. (a) Average 2-norm relative error of stator phase currents si ; and (b) 2-norm relative error of electromagnetic torque eT  as a function of step size t∆  for the single-phase-to-ground fault study as predicted by the subject models (IM1). (including Norton-SS) are consistent, i.e., all models converge to the exact/reference solution as the step size goes towards zero. 6.4.1.2 Induction machine #2 (IM2)  To further investigate the range of application of the proposed model, the study of Section 6.4.1.1 is repeated using a 460-V, 500-hp induction machine (IM2) modeled with two rotor circuits. Its equivalent circuit parameters were extracted from catalogue data using the approach proposed in [47] (with a typical magnetizing saturation curve). The corresponding parameters are summarized in Appendix J along with those of the network. Artificial damping (α  = 0.95) is again necessary to attenuate numerical oscillations in the qd model. Additional damping resistances are required to preserve numerical stability when using the Norton-SS model with t∆  = 100 µs ( snR  = 15 Ω) and t∆  = 250 µs ( snR  = 5 Ω). 184   The reference trajectories of abcsi , eT , and mλ  for this study are presented in Figure 6–6. It is observed that the interaction between the machine and the network creates electrical resonance of approximately 860 Hz. It is also seen from Figure 6–6 and the saturation data in Appendix J that the machine is well saturated. A magnified view of the first peak of csi  following the occurrence of the fault as predicted by the subject models is shown in Figure 6–7. As seen in the upper part of Figure 6–7, all models yield fairly accurate trajectories with t∆  = 100 µs. For the reasons explained in Section 6.4.1.1, the error becomes more noticeable with t∆  = 500 µs [see Figure 6–7 (b)]. In particular, the Norton-SS model is again much less accurate than the other models. However, unlike in Section 6.4.1.1, the CP-VBR model is as numerically accurate as the PD and VP-VBR model, since the rotor leakage inductances of IM2 are smaller. An enlarged view of csi  before the removal of the fault is also shown in Figure 6–8. As the supersynchronous resonance is essentially damped by this point, all models except Norton-SS predict csi  with almost no numerical error when using t∆  = 100 µs [see Figure 6–8 (a)]. However, as seen in Figure 6–8 (b), the qd and Norton-SS models predict significantly erroneous waveforms with t∆  = 500 µs. This is unlike the PD and VBR models, which introduce very little error even with this fairly large step size.   The errors )( siε  and )( eTε  are shown in Figure 6–9 as functions of the step size. As observed in Figures 6–7 and 6–8, the errors of the CP-VBR, PD, and VP-VBR models are very close. This emphasizes that the prediction of stator currents [see (6–30)] in the CP-VBR model yields significantly less error than the numerical discretization (i.e., trapezoidal integration rule). The solution given by the qd model starts diverging from those of the PD and VBR models at t∆  = 250 µs and becomes unusable for t∆  = 500 µs, as the errors are larger than 10%. The Norton-SS model also has limited numerical accuracy, e.g., 2-norm error of 8.3% for eT  with t∆  = 250 µs.  185  –400004000i as (A)–200002000i bs (A)–300003000i cs (A)–500005000T e (N·m)0 0.05 0.10 0.15 0.200.500.751.00λ m (Wb)Time (s)See Figure 6–7See Figure 6–8 Figure 6–6. Reference trajectories of stator currents asi , bsi , and csi , electromagnetic torque eT , and main flux mλ  during a single-phase-to-ground fault (IM2). 186  2000240028003200i cs (A)  a) Referenceb) CP-VBRc) PD0.022 0.023 0.024 0.025 0.026 0.027 0.0282000240028003200Time (s)i cs (A)  d) qde) VP-VBRf) Norton-SS(a)(b)b to efafadb, c, e Figure 6–7. Magnified view of stator current csi  at the beginning of a single-phase-to-ground fault as predicted by the subject models: (a) with step size t∆  of 100 µs; and (b) with step size t∆  of 500 µs (IM2).  The two studies presented in Section 6.4.1 demonstrate that the numerical accuracy of the CP-VBR model is very similar to that of the algebraically exact PD and VP-VBR models even in worst-case scenarios (large rotor leakage inductance, high source voltage, large step size, etc.). The studies also highlighted that the CP-VBR model is significantly more accurate than the qd and Norton-SS models in such situations. These worst-case scenarios may be uncommon in practical EMTP-type simulations, wherein considerably smaller step sizes (e.g., 100 µs and less) are frequently used. As observed in Figures 6–6 and 6–9, the numerical inaccuracies of the qd model vanish when using smaller step sizes.187  –1000–5000i cs (A)  a) Referenceb) CP-VBRc) PD0.095 0.096 0.097 0.098 0.099 0.100 0.101 0.102 0.103–1000–5000Time (s)i cs (A)  d) qde) VP-VBRf) Norton-SSfa to efdab, c, e(a)(b) Figure 6–8. Magnified view of stator current csi  before the removal of a single-phase-to-ground fault as predicted by the subject models: (a) with step size t∆  of 100 µs; and (b) with step size t∆  of 500 µs (IM2). Consequently, for case studies where such small step sizes are necessary, the PD, qd, and VBR models may be used interchangeably without affecting numerical accuracy. This is not the case of the Norton-SS model. Despite the simplicity of the case studies included in this section, the Norton-SS model is significantly less accurate than the other models even for small step sizes [see Figures 6–6 and 6–9]. Consequently, of the subject models, only CP-VBR possesses a true constant-parameter interfacing circuit while introducing minimal numerical error, which are desirable properties independently of the step size used or the188  50 100 250 500 1000010203040∆t (µs)ε(Te) (%)  010203040ε(is) (%)  a) CP-VBRb) PDc) qdd) VP-VBRe) Norton-SSeca, b, deca, b, d(a)(b) Figure 6–9. (a) Average 2-norm relative error of stator phase currents si ; and (b) 2-norm relative error of electromagnetic torque eT  as a function of step size t∆  for the single-phase-to-ground fault study as predicted by the subject models (IM2). complexity of the network. The Norton-SS model will not be considered further in this chapter. 6.4.2 Computational Efficiency  An important property of a machine model is its numerical efficiency, particularly: i) its computational cost per time step; and ii) its impact on the simulation speed of the entire system. To first quantify the cost per time step, the CP-VBR, PD, qd, and VP-VBR models were implemented using standard C++ code, compiled using Microsoft Visual Studio 2008, and executed in standalone mode (i.e., outside of PSCAD).   The CPU times per step of the four models with one and two rotor circuits are summarized in Table 6–1. The qd model requires the smallest CPU time, whether one or189  Table 6–1. CPU time per step of the subject models.  qd PD VP-VBR CP-VBR CPU Time (µs) 1 Rotor Circuit 0.329 0.374 0.478 0.422 2 Rotor Circuits 0.384 0.462 0.552 0.483  two rotor circuits are considered; it is followed by the PD model. The CPU time per step of the proposed CP-VBR model is around 32% greater than that of the qd model. The VP-VBR model is numerically costlier than the CP-VBR model by 13 to 15%, which is mostly due to the need to invert the 3-by-3 rotor-speed- and saturation-segment-dependent resistance matrix jeqR  at every time step. The effect of each model on the simulation speed of power systems will be investigated in Section 6.4.3. 6.4.3 Multimachine System  To analyze the numerical properties of the proposed model in a more practical scenario, a typical multimachine industrial power system [154] has been implemented in PSCAD. The system’s one-line diagram is depicted in Figure 6–10. It is comprised of 19 induction motors (IMs) with nominal voltages between 460 and 4000 V. All machine models include main flux saturation. The lines, transformers, and cables are modeled using lumped-parameter elements. Transformer saturation is also considered in this study. The machine and network parameters are summarized in Appendix K.  With the source voltage set to 75 kV, all motors initially operate in steady state driving the mechanical loads indicated in Figure 6–10. A three-phase fault is assumed to occur between the Thévenin equivalent of the utility and the industrial network [see Figure 6–10] at t  = 0.04 s. The fault vanishes 100 milliseconds later, before any protection device has reacted. The study is executed using the CP-VBR, PD, qd, and VP-VBR models. As in Section 6.4.1, this case study allows the use of larger step sizes. 190  IM3 IM2800 hp 400 hpIM3 IM3750 hp 750 hpIM41750 hpIM3 IM2600 hp 500 hpIM5500 hpIM6 IM61800 hp 1950 hpIM6 IM62250 hp 1500 hpIM3 IM3750 hp 750 hpIM3 IM2800 hp 400 hpIM3750 hpIM62250 hpIM3800 hpabc-ginet1 inet2inet = inet1 + inet240213 4925 12 5 263928 17 296111419152016217273013182431363237 Figure 6–10. One-line diagram of an industrial multimachine power system (Section 6.4.3). 191  0 0.05 0.10 0.15 0.20–100001000Time (s)i neta (A)  a) Referenceb) CP-VBRc) PDd) qde) VP-VBRSee Figure 6–12 Figure 6–11. Transient in the fault current contributed by the industrial network ainet  during a three-phase fault as predicted by the subject models. 0.091 0.092 0.093 0.094 0.095 0.0960100200Time (s)i neta (A)  a) Referenceb) CP-VBRc) PDd) qde) VP-VBRdab, c, e Figure 6–12. Magnified view of the fault current contributed by the industrial network ainet  during a three-phase fault as predicted by the subject models.  To get an overview of the numerical accuracy of each model, the fault current on phase a contributed by the industrial network ( ainet ) is considered [see Figure 6–10]. Its trajectory, computed with a step size of 500 µs, is presented in Figure 6–11; a magnified view of the fault current is shown in Figure 6–12. All models produce a relatively small192  50 100 250 5000510∆t (µs)ε(inet) (%)  a) CP-VBRb) PDc) qdd) VP-VBRca, b, d Figure 6–13. Average 2-norm relative error of the fault phase current contributed by the industrial network neti  as a function of step size t∆  for the three-phase fault study as predicted by the subject models. Table 6–2. CPU time of the three-phase fault study with the subject models (1201 time steps).  qd PD VP-VBR  CP-VBR CPU Time (s) 1.150 1.416 4.625 0.788  numerical error during the first peak of the fault. Figure 6–11 shows that whereas the error of the PD and VBR models decreases quickly as a function of time, the qd model’s error stays relatively constant throughout the fault condition. These results are consistent with the case studies of Section 6.4.1.  The average 2-norm error of the fault current )( netiε  contributed by the industrial network for this case study is plotted in Figure 6–13 for different step sizes. The proposed model yields similar error to the PD and VP-VBR models, whereas the qd model is significantly less accurate as the step size increases.  Finally, the CPU times for this study with t∆  = 250 µs are summarized in Table 6–2. The use of CP-VBR gives by far the fastest solution – almost 1.5 and 1.8 times faster than with qd and PD, respectively, and almost 6 times faster than with VP-VBR. These results highlight the significant computational gains that can be achieved on multimachine systems 193  by using models with true constant-parameter interfacing circuits (CP-VBR) as opposed to models with saturation-segment-dependent (PD and qd) or saturation- segment- and rotor-speed-dependent (VP-VBR) interfacing circuits. In particular, these results demonstrate that re-factoring (or not) the network’s conductance matrix G  can have a much greater impact on the overall simulation efficiency and speed than the cost per time step of the individual machine models.  It is also noted that an additional VBR model with a rotor-speed-independent but saturation-segment-dependent interfacing circuit can be achieved using the approach proposed in [61], i.e., neglecting rω  while forming the interfacing matrix of the VP-VBR model. The impact of this model on the simulation speed of power systems would be similar to that of the qd and PD models, and based on the error analysis presented in [61], its numerical accuracy would also be comparable to that of the PD and VP-VBR models.  One must also be aware that the results of Table 6–2 are strongly system-dependent. For example, the numerical efficiency of the PD and qd models could be improved in systems with fewer machines and smaller step sizes, since there would be proportionally fewer changes between saturation segments and thus fewer re-factorizations of the network’s conductance matrix. In the same vein, the PD and qd models would also benefit from longer steady-state conditions. At the same time, the numerical advantages of CP-VBR compared to other models would be further amplified when using continuous saturation functions [111] or piecewise-linear saturation curves with a larger number of segments. The simulation environment, in this case PSCAD, also impacts the numerical efficiency of the overall simulation.  194  CHAPTER 7:  MULTIRATE EMTP-TYPE INDUCTION MACHINE MODELS   The EMTP-Type CP-VBR model presented in Chapter 6 possesses better numerical accuracy and stability than the traditional Thévenin prediction-based qd model. Moreover, due to its rotor-position- and saturation-segment-independent interfacing circuit, the CP-VBR model can improve the simulation speed of the whole system by avoiding re-factorizations of the network’s conductance matrix G . The main disadvantage of the CP-VBR model is that it internally requires more flops per time step than the qd model. For example, ignoring the solution of the network nodal equation (2–8) or (2–9), it was demonstrated in Table 6–1 that the CP-VBR model requires around 32% more CPU time than the Thévenin prediction-based qd model.   Multirate schemes have already been proposed to improve the speed of power system transient studies, e.g., [36]–[39], [113]. Taking advantage of the diverse time scales exhibited by many power systems, sections of the network are simulated using different step sizes and potentially different integration rules. However, the state-of-the-art EMTP-type programs still simulate the whole network with a unique step size.  In the final research chapter of this thesis, the concept of multirate simulations is applied for the first time to EMTP-type machine models. In particular, it is shown using modal analysis that the stator subsystem of VBR induction machine models can be combined with the power network to form a fast subsystem, while the rotor and mechanical subsystems can be grouped together into a slower subsystem. Using this 195  partitioning, the interfacing properties of single-rate VBR models are preserved (including constant-parameter interfacing circuits), and the resulting multirate models can be used with existing single-rate EMTP-type programs with no modifications to their computational engines. Two new multirate VBR models are proposed in this chapter, which are derived using a different combination of integration rules. Computer studies demonstrate that the proposed multirate models can require less CPU time than the traditional qd models while being almost as accurate as the state-of-the-art single-rate PD and VBR models.  7.1 Modal Analysis  In the continuous-time multirate method proposed in [37], the VBR machine models were separated into fast and slow subsystems based on engineering intuition. The objective of this section is to mathematically justify the partitioning of VBR induction machine models into two such subsystems. A 50-hp induction machine is herein considered. Its parameters are taken from [29] and reproduced in Appendix L. For simplicity, the machine is considered unsaturated (i.e., j  = 1 and qdresλ  = 0) and is modeled with only one rotor circuit. The rotor reference frame is considered throughout this chapter.  As a first step, the eigenvalues of the state-space VBR model presented in Section 6.1 are computed for the subject machine. Specifically, the electrical equations (6–1) and (6–7), and the mechanical equations (2–48) and (2–49) are rearranged in the standard nonlinear state-space form of (2–1). The state vector x  of the VBR model is defined as  [ ]TrrTqdrTabcs θωλix =  (7–1) 196  Table 7–1. Eigenvalues of the 50-hp induction machine (left column) and of the system of Figure 7–1 with the same machine (right column). Machine only Machine and network 2,1λ  = –268.3 ± ι143.5 4,3λ  = 76.9 ± ι143.6 5λ  = –108.6 6λ  = –14.9 7λ  = 0.007 2,1λ  = –218.3 ± ι118.8 4,3λ  = 64.3 ± ι118.7 5λ  = –92.5 6λ  = –14.4 7λ  = 0.015 9,8λ  = –59.1 ± ι3108 11,10λ  = –54.0 ± ι3108 13,12λ  = –45.7 ± ι3565  IMThév. Imp.iL+-vC Figure 7–1. One-line diagram of a small system comprised of an induction motor (IM) connected to a shunt capacitor and a network Thévenin equivalent. and the input vector u  is comprised of the stator voltages abcsv . Linearizing the state-space model around a given steady-state operating point yields  uBxAx ∆+∆=∆ opopp . (7–2) Herein, the operating point is defined by nominal values of load and stator voltages. The computed eigenvalues of opA  are summarized in the first column of Table 7–1.  As a second step, the system presented in Figure 7–1 is considered, wherein the same machine is connected to a power factor correction shunt capacitor and a network Thévenin equivalent. The network parameters are given in Appendix L. For this system, the state vector becomes 197   [ ]TTabcCTabcLrrTqdrTabcs viλix θω=  (7–3) where abcLi  and abcCv  represent the Thévenin equivalent currents and the capacitor voltages, respectively. The resulting eigenvalues (computed in steady state under nominal load and source voltage) are reproduced in the right column of Table 7–1.  It is observed in Table 7–1 that the first seven eigenvalues of the machine-network system are similar to those of the machine-only system (e.g., 2,1λ  = –268.3 ± ι143.5 and –218.3 ± ι118.8, respectively). It is also seen that the imaginary components of the eigenvalues added by the network ( 8λ  to 13λ ) are more than 20 times larger than those of the machine. This means that higher frequency oscillatory modes appear due to the network, and thus that smaller integration step sizes will be required to accurately simulate the system of Figure 7–1 in a single-rate environment.   However, Table 7–1 does not imply that the state variables of the machine model are not affected by the faster modes. Therefore, simulating the network with one step size and the machine with a larger one might not produce accurate results. A systematic approach to quantify the effect of separate modes on each state (and vice versa) is to calculate their participation factors [19], [155], [156]. The participation factor of the n th mode and the m th state is defined as [155]  mnmnmn Ψ= φρ  (7–4) where mnφ  and mnΨ  refer to the m th element of the n th right and left eigenvectors nφ  and nΨ , respectively. These eigenvectors are related to the linearized state matrix opA  and the n th eigenvalue ( nλ ) by the following relationships:  nnnop φφ λ=A  (7–5)  nTnopTn λΨΨ =A . (7–6) 198  Table 7–2. Magnitude of the participation factors of the 50-hp induction machine.  2,1λ  4,3λ  5λ  6λ  7λ  asi  0.23 0.12 0.33 3e-3 4e-4 bsi  0.22 0.12 0.33 2e-3 2e-3 csi  0.23 0.12 0.33 2e-3 1e-4 qrλ  0.19 0.33 0 0.19 0.17 drλ  0.19 0.32 0 0.77 0.78 rω  0.01 0.02 0 1.00 5e-4 rθ  0.01 0.03 0 0.95 0.04  The eigenvectors are typically normalized such that [155]  1=iTn φΨ    if  in = . (7–7) As a result, we have  11=∑=λρNmmi    and    11=∑=λρNnin  (7–8) for any integer 1 ≤ i  ≤ λN , where λN  is the number of states or eigenvalues (all eigenvalues are assumed to be distinct).   A large value of mnρ  indicates that the n th mode and the m th state are strongly correlated. Conversely, small values of mnρ  imply that the given mode and state are weakly coupled. In the extreme, mnρ  = 0 means that the n th mode and the m th state are completely independent.  The magnitude of the participation factors for the machine only and the system of Figure 7–1 are summarized in Tables 7–2 and 7–3, respectively. For the machine-only case, the complex conjugate eigenvalues ( 1λ  to 4λ ) and the electrical states ( abcsi  and qdrλ ) are199  Table 7–3. Magnitude of the participation factors of the system of Figure 7–1 with the 50-hp induction machine.  2,1λ  4,3λ  5λ  6λ  7λ  9,8λ  11,10λ  13,12λ  asi  0.15 0.08 0.17 3e-3 6e-4 0.05 0.05 0.08 bsi  0.15 0.09 0.17 2e-3 2e-3 0.05 0.05 0.08 csi  0.16 0.08 0.17 2e-3 1e-3 0.05 0.05 0.08 qrλ  0.19 0.33 0 0.21 0.20 4e-4 4e-4 0 drλ  0.19 0.31 0 0.73 0.75 4e-4 4e-4 0 rω  0.02 0.02 0 1.00 1e-3 2e-5 1e-5 0 rθ  0.01 0.03 0 0.95 0.05 2e-6 2e-6 0 aLi  0.07 0.04 0.16 1e-3 3e-4 0.11 0.11 0.09 bLi  0.07 0.04 0.16 7e-4 8e-4 0.11 0.11 0.09 cLi  0.07 0.04 0.16 1e-3 7e-4 0.11 0.11 0.09 aCv  4e-4 2e-4 7e-6 8e-7 3e-7 0.17 0.17 0.17 bCv  4e-4 2e-4 7e-6 4e-7 7e-7 0.17 0.17 0.17 cCv  4e-4 2e-4 7e-6 6e-7 6e-7 0.17 0.17 0.17  strongly correlated since their participation factors are greater than 0.1. The mechanical states ( rω  and rθ ) are strongly linked to the real eigenvalue 6λ , but only weakly coupled to the complex eigenvalues (< 0.04).  For the system of Figure 7–1 [see Table 7–3], it is observed that the fast modes ( 8λ  to 13λ ) and the states of the external network ( Li  and Cv ) are highly correlated, whereas the rotor flux linkages and the mechanical states are virtually independent from the fast eigenvalues (< 0.001). However, the stator currents are shown to be significantly affected by both slower ( 1λ  to 4λ ) and faster ( 8λ  to 13λ ) modes. As a result, stator currents are hereinafter referred to as fast variables, whereas the rotor flux linkages and mechanical states are termed slow variables. This observation will serve as the basis of the multirate models proposed in the next section. 200  Table 7–4. Magnitude of the algebraic participation factors of the system of Figure 7–1 with the 50-hp induction machine.  2,1λ  4,3λ  5λ  6λ  7λ  9,8λ  11,10λ  13,12λ  mqλ  0.19 0.32 0 0.21 0.20 4e-4 4e-4 0 mdλ  0.19 0.30 0 0.72 0.73 4e-4 4e-4 0   In the VBR model presented in Section 6.1, the magnetizing fluxes mqdλ  are defined by (6–8) as the combination of fast ( abcsi ) and slow ( qdrλ ) state variables. This relationship can be written in matrix form as  Txλ =mqd  (7–9) where the element found in the y th row and z th column of T  is defined as yzt . Since mnρ  is an indicator of the relative participation of nλ  in the m th state [155], the participation of nλ  in the y th algebraic variable can be approximated as  ∑==λρσNmmnymyn t1. (7–10) These so-called algebraic participation factors are compiled in Table 7–4 for the system of Figure 7–1. It is seen from Tables 7–3 and 7–4 that mqdλ  is affected by the system modes similarly to qdrλ . Consequently, mqdλ  can be considered as a pair of slow variables. When incorporating saturation, the same conclusion can be reached for qdresλ  since, as shown in (2–63) and (2–64), it is only function of mqdλ .  201  External Network&Stator EquationsMechanical& Rotor Equationsλqdr, λresqdiabcsFast Subsystem (∆tf) Slow Subsystem (∆ts) Figure 7–2. Grouping of a VBR induction machine model and an external power system into fast and slow subsystems. 7.2 Multirate EMTP-Type VBR Machine Models 7.2.1 Preliminaries  The VBR model can be viewed as a combination of three coupled sets of nonhomogeneous differential equations: the stator equation (6–1), the rotor dynamics (6–7), and the mechanical equations (2–48) and (2–49). Based on the conclusions of Section 7.1, an efficient and accurate multirate scheme can be devised by grouping the rotor and mechanical equations together to form a slow subsystem, while the stator equation is combined with the external power network to create a fast subsystem. Figure 7–2 illustrates this concept. An important advantage of this grouping is that the stator equation of the multirate VBR models is solved simultaneously with the external network.  In this dissertation, it is assumed that the external power system is solved using a unique step size, as in state-of-the-art commercial EMTP-type programs like EMTP-RV and PSCAD. However, the proposed machine model can also be incorporated into more general multirate frameworks [38], [39]. In such cases, the external network of Figure 7–2 would refer to the subsystem of the grid to which the machine is connected.  202  t0xx x xxxxMinor Time StepMajor Time Stepxt-∆ts t-3∆tf t-2∆tf t-∆tft-5∆tf∆tf t+∆tf t+2∆tf  Figure 7–3. Illustration of major and minor time steps (with rM  = 4).  Here, the fast subsystem is solved using a small step size ft∆ , whereas the slow subsystem is computed with a larger step size st∆ . For simplicity, it is assumed that st∆  is an integer multiple of ft∆ , with their ratio defined as   fsr ttM ∆∆= . (7–11)  The time steps where only the fast subsystem’s solution is evaluated are referred to as minor time steps, whereas those where the solution of both subsystems are computed are termed major time steps. This nomenclature can be visualized in Figure 7–3.  As seen in Figure 7–2, the fast subsystem has slow inputs ( qdrλ  and qdresλ ), which are only computed at major time steps. In order to evaluate the solution of the fast subsystem at minor time steps, it is necessary to predict these slow variables. This is done using linear extrapolation. Conversely, the slow subsystem has fast inputs ( abcsi ). Due to the possibility of fast oscillations, the snapshots of abcsi  at stt ∆−  and t  might not be representative of its trajectory between these two points. To get a more accurate solution, the so-called expanded trapezoidal rule [39] is applied to the stator currents (here in qd coordinates):  ( ) ∑∫−=∆−∆−∆+∆−+∆≈11)()()(2)(rsMifqdsfsqdsqdsftttqds tittttttd iiii ττ . (7–12)  203  (a) StandardTrapezoidal Rule(b) ExpandedTrapezoidal Rulet-∆ts t t-∆ts tt-3∆tf t-2∆tf t-∆tf  Figure 7–4. (a) Standard trapezoidal rule applied at major time steps only; and (b) expanded trapezoidal rule applied at minor and major time steps (with rM  = 4). The difference between the standard and expanded trapezoidal rules is presented in Figure 7–4, wherein the gains in accuracy that can be achieved by using the latter with fast-changing signals are made evident.  The modes affecting the slow and fast subsystems might be of different orders of magnitude, as in the example of Section 7.1. However, increasing the value of rM  beyond a certain point may yield insignificant computational gains due to the finite computational cost of each subsystem. An option to decrease the overall computational cost is to discretize the slow variables using a numerically cheaper first-order integration rule, e.g., Backward Euler (BE). Since slow variables are inputs and states of the fast and slow subsystems, respectively, a decrease in computational cost will be observed. Based on this idea, two multirate models will be presented in this section. In the first, both subsystems are discretized using the trapezoidal rule (MR-T/T). In the second, the fast variables are discretized using trapezoidal and the slow variables using BE (MR-T/BE). It is recalled that in EMTP-type programs [5], the external power systems are typically discretized using the trapezoidal rule. 204  7.2.2 Trapezoidal-Only Model (MR-T/T) 7.2.2.1 Slow subsystem  The rotor dynamics (6–7) are discretized by applying the standard trapezoidal rule to the slow variables ( qdrλ  and qdresλ ) and the expanded trapezoidal rule to qdsi . This yields after algebraic manipulation  )()(1)( ttMt rqdsjrqdr λeiEλ +=  (7–13) where  ( ))()()()()(21)(resres2111sqdqdjsqdrjjMisqdsfqdsjrrttttttttitMtr∆−++∆−+∆−+∆−= ∑−=λλDλFFiiEeλ . (7–14) The definitions of jE , j1F , j2F , and jD  are given in (M1)–(M2) in Appendix M. The coefficient sk  is defined as st∆2 . It can be seen from (7–13) and (7–14) that )(res tqdλ  is needed to evaluate )(tqdrλ . In traditional single-rate VBR models, )(tmqdλ  is predicted at the beginning of every time step using linear extrapolation, allowing to evaluate )(res tqdλ  using (2–63) and (2–64). Corrected values of )(tmqdλ  are then calculated at the end of every time step using (6–8). Here, since magnetizing fluxes are slow variables, the predicted and corrected values of )(tmqdλ  along with )(res tqdλ  are only evaluated at major time steps.  Integrating (2–48) and (2–49) using the trapezoidal rule, the discretized mechanical equations are defined as  ( ) ( ))()(2)()(2)()( 11 smmsseessrr ttTtTJPkttTtTJPkttt ∆−+−∆−++∆−= −−ωω  (7–15) 205   ( ))()()()( 1 srrssrr tttkttt ∆−++∆−= − ωωθθ . (7–16) The electromagnetic torque can be computed using (2–47). 7.2.2.2 Fast subsystem  The stator equation (6–1) is part of the fast subsystem. Discretizing it using the trapezoidal rule yields  ( ) )()()()( '''' tttkt shabcsabcsabcsjfsabcs eeiLRv +++=  (7–17) where ff tk ∆= 2 and  ( ) )()()()( '''' fabcsfabcsabcsjfsfabcssh ttttkttt ∆−+∆−−+∆−−= eiLRve . (7–18) Substituting (6–5) into (7–17), and combining the result with (7–13) gives the machine-network interfacing equation  ( ) )()()()()()( 01'' tttttkt shssabcsjabcsjfsabcs eeKiKLRv ++++= −  (7–19) where  ( )[ ]TTss tt 0)()(0 ee =  (7–20)  )()()()()( res32 ttttt qdjrjs λAeAe += λ . (7–21) The definition of jK  is given in (M3)–(M5) in Appendix M. The vector )(tse  can be evaluated directly when t  coincides with a major time step, since )(res tqdλ , )(res sqd tt ∆−λ , and )( sqdr tt ∆−λ would be known [see (7–14)]. For minor time steps, )(tse  is extrapolated linearly using 206   ( ))()()()(~ modmodmodmod sfsfsrfss ttzttztMztztt ∆−∆−−∆−+∆−= eeee  (7–22) where it is recalled that the tilde symbol (~) indicates predicted values and where  ( ) rf Mttz modmod ∆= , (7–23) i.e., the remainder of ( )ftt ∆  over rM .  Returning to the machine-network interfacing equation, (7–19) can be rewritten as the three-phase Thévenin equivalent  )()()()( eq tttt habcsjabcs eiRv +=  (7–24) where  )()( ''eq tkt jabcsjfsj KLRR ++=  (7–25)  )()()()( 01 tttt shssh eeKe += − . (7–26)  Finally, the approach to achieve a numerically efficient constant-parameter interfacing circuit presented in Chapter 6 can also be applied to (7–24). Transferring the time- and saturation-segment-dependent part of )(eq tjR  into the subtransient voltage source gives  )()()()( consteq tttt hRihabcsabcs eeiRv ++= −  (7–27) where  )(~)()( eq ttt abcsjhRi iRe ∆=  (7–28)  consteqeqeq )()( −−=∆ RRR tt jj . (7–29) 207  The stator currents are predicted in qd coordinates using a three-point linear predictor with smoothing as defined in (6–30) [5]. The stator history term )(tshe  must also be rewritten using (6–5) and (7–13) as  ( ) ( ))()()()(~)()()(01''''constfabcsfsfsfabcscsjabffjfabcsabcsjfsshttttttttkttttkt∆−−∆−∆−+∆−∆−∆−∆+∆−−+=− veKiLKiLKRe (7–30) where  const)()( KKK −=∆ tt jj  (7–31)  '' const'''' −−=∆ csabcsjabcsjab LLL . (7–32) The matrices constK , '' const−csabL , and consteq−R  can be constructed as detailed in Section 6.3.1. 7.2.3 Trapezoidal/Backward Euler Model (MR-T/BE) 7.2.3.1 Slow subsystem  For this model, (6–7) is discretized by applying the expanded trapezoidal rule to qdsi  and BE to qdrλ  and qdresλ . This results in  )()(21)( ttMt TBrqdsTBjrqdr λeiEλ +=  (7–33) where  )()(1)()(221)(res111tttttttitMtqdTBjsqdrTBjsMisqdsfqdsTBjrTBrrλDλFiiEe+∆−∆+∆−+∆−= ∑−=λ. (7–34) 208  The definitions of TBjE , TBj1F , and TBjD  are given in (M6)–(M7) in Appendix M. The prediction and evaluation of )(res tqdλ  and )(tmqdλ  follows the pattern described in Section 7.2.2.1.   For the mechanical system, integrating (2–48) and (2–49) using BE gives  ( ))()(2)()( tTtTJPtttt messrr −∆+∆−= ωω  (7–35)  )()()( ttttt rssrr ωθθ ∆+∆−= . (7–36) 7.2.3.2 Fast subsystem  Before discretizing (6–1), it is convenient to separate ''abcse  between slow and fast variables, yielding  ( ) '' slow''1 −+++= abcsabcsabcsjabcsabcjsabcs p eiLiARv  (7–37) where  ( ) TTqdssabcs = −−− 0'' slow1'' slow eKe  (7–38)  qdjqdrjqds res32'' slow λAλAe +=− . (7–39) Matrix abcj1A  is equal to sjs a KIK 3311 ×− . Integrating the fast variables ( abcsv  and abcsi ) using the trapezoidal rule and '' slow−abcse  using BE, (7–37) becomes  ( ) )()(2)()( '' slow1'' tttkt TBshabcsabcsabcjabcsjfsabcs eeiALRv ++++= −  (7–40) where 209   ( ) )()()( 1'' fabcsabcjabcsjfsfabcsTBsh ttkttt ∆−+−+∆−−= iALRve . (7–41) Substituting (7–33) into (7–38) allows to rewrite (7–40) as  ( ) )()()(2)()()( 01'' tttttkt TBshTBssabcsTBjabcsjfsabcs eeKiKLRv ++++= −  (7–42) where  ( ) TTTBsTBs tt = 0)()(0 ee  (7–43)  )()()()()( res32 ttttt qdjTBrjTBs λAeAe += λ . (7–44) The definition of TBjK  is given in (M8)–(M10) in Appendix M. Similarly to the MR-T/T model, )(tTBse  is only defined at major time steps. For minor time steps, )(tTBse  is also predicted using linear extrapolation [see (7–22)].  Following the approach presented in Section 6.3.1 and used in Section 7.2.2.2, (7–42) can be modified to achieve a constant-parameter machine-network interfacing circuit:  )()()()( consteq tttt TBhRiTBhabcsTBabcs eeiRv ++= −  (7–45) where  )(~)()( eq ttt abcsTBjTBhRi iRe ∆=  (7–46)  )()()(2)( 01 tttt TBshTBssTBh eeKe += −  (7–47) and  TBTBjTBj tt consteqeqeq )()( −−=∆ RRR  (7–48) 210   )()( ''eq tkt TBjabcsjfsTBj KLRR ++= . (7–49) Moreover, as in Sections 6.3.1 and 7.2.2.2, )(tTBshe  must be redefined as  ( ) ( ))()(~)()( ''1'' constconst1fabcsfabcsabcsjfabcjfabcsabcsfabcsTBshttttkttkt∆−−∆−∆−∆+∆−−+=−−viLAiLARe (7–50) where  abcabcjabcj const111 −−=∆ AAA . (7–51) The matrix TB consteq−R  is formed similarly to consteq−R  in Sections 6.3.1 and 7.2.2.2, while abcconst1−A  is constructed by evaluating sjs a KIK 3311 ×−  with Sj = , where S  refers to the last piecewise-linear segment of Figure 2–8. The stator currents )(tabcsi  are predicted as explained in Sections 6.3.1 and 7.2.2.2. 7.2.4 Simulation Steps  The simulation steps of the proposed models at the time step t  are listed below. Curly brackets “{ }” are used to denote variables and equations specific to the MR-T/BE model. 1) Linearly predict the electrical rotor speed )(trω  using   ( ))()()()(~ modmodmodmod sfrfrrfrr ttzttztMztztt ∆−∆−−∆−+∆−= ωωωω ,   if 0mod ≠z  (7–52) or  )2()(2)(~ srsrr ttttt ∆−−∆−= ωωω ,   if 0mod =z  (7–53) where the definition of modz  is given in (7–23). 211  2) Evaluate the electrical rotor position )(trθ  using (7–16) {(7–36)} (replacing st∆  therein by the difference between the actual time step and the previous major time step).  3) At major time steps only: predict the magnetizing fluxes )(tmqdλ  and the flux angle derivative )(tφω  using linear extrapolation, i.e., with (2–114), evaluate the main flux )(tmλ  using (2–58), and compute )(res tqdλ  using (2–63) and (2–64). 4) Compute )(tshe  { )(tTBshe } using (7–30) {(7–50)}. 5) At major time steps only: calculate )(trλe  { )(tTBrλe } using (7–14) {(7–34)}, evaluate )(tse  { )(tTBse } using (7–21) {(7–44)}, and compute )(0 tse  { )(0 tTBse } using (7–20) {(7–43)}. 6) At minor time steps only: linearly predict )(tse  { )(tTBse } using the values of se  { TBse } evaluated at the previous two major time steps, i.e., (7–22) for )(tse , and form )(0 tse  { )(0 tTBse } using (7–20) {(7–43)}. 7) Compute )(the  { )(tTBhe } using (7–26) {(7–47)}. 8) Evaluate )(tjK  { )(tTBjK } using (M5) {(M10)}, compute )(tjK∆  using (7–31) (for the MR-T/T model only), and form the resistance matrices )(eq tjR  and )(eq tjR∆  { )(eq tTBjR  and )(eq tTBjR∆ } using (7–25) and (7–29) {(7–49) and (7–48)}, respectively. 9) Predict the stator currents )(0 tsqdi  using (6–30), convert )(~0 tsqdi  to abc coordinates using (2–27), and evaluate )(thRie  { )(tTBhRie } using (7–28) {(7–46)}. 10) Convert (7–27) {(7–45)} to a Norton equivalent and insert its current source into the equivalent current injection vector )(thi  of the fast subsystem. 212  11) Find the fast subsystem’s unknown nodal voltages using (2–8) or (2–9). 12) Using the stator voltages )(tabcsv , evaluate the stator currents )(tabcsi  using (7–27) {(7–45)} and convert to qd coordinates using (2–20). The remaining steps are executed at major time steps only:  13) Compute )(tqdrλ  using (7–13) {(7–33)}, correct the magnetizing fluxes )(tmqdλ  and the main flux )(tmλ  using (6–8) and (2–58), respectively, evaluate the flux angle )(tφ  using (2–63) or (2–64), and recalculate the flux angle time derivative )(tφω  using (6–22) (wherein t∆  is replaced by st∆ ). 14) Using )(tmλ , check if the operating segment j  of the saturation curve changes. If so, go to 15). Otherwise, skip to 16). 15) Update j , re-compute )(res tqdλ , )(trλe  { )(tTBrλe }, )(tse  { )(tTBse }, )(tjK  and )(tjK∆  (for the MR-T/T model only), )(tqdrλ , )(tmqdλ , )(tmλ , )(tφ , and )(tφω . 16) Evaluate the electromagnetic torque )(tTe  using (2–47), and correct the electrical rotor speed )(trω  and position )(trθ  using (7–15) and (7–16) {(7–35) and (7–36) }, respectively. 7.3 Computer Studies  To assess the numerical accuracy of the proposed multirate models, the MR-T/T and MR-T/BE models have been implemented in C and directly interfaced with PSCAD’s computational engine [see Sections 7.3.1 and 7.3.2]. For the purpose of comparison, reference trajectories have been obtained by solving the original variable-parameter VBR model [111] in the same environment with a very small step size of 1 µs.  213   To properly evaluate their computational efficiency [see Section 7.3.3], the proposed models have also been implemented in C++ and compiled using Microsoft Visual Studio 2008. The standard Thévenin prediction-based qd model [see [5] and Section 2.3.4.2], which is the single-rate model that requires the fewest flops per time step [see Table 6–1], has also been implemented in the same environment to provide an insightful comparison. Main flux saturation is considered in each study. All saturation-segment-dependent coefficients and matrices are computed during the initialization stage. The case studies are executed on Windows XP using a PC with a 2.83-GHz CPU. 7.3.1 Numerical Accuracy – Single-Machine System  To assess numerical accuracy at the machine model level, the single-machine system shown in Figure 6–1 is considered first. This system has been implemented in PSCAD using lumped-parameter elements. The parameters of the two-rotor-circuit 500-hp motor and its external network are summarized in Appendix J. 7.3.1.1 Electromagnetic transient  In this study, the source voltage is initially set to 1.1 pu while the machine operates in steady state under nominal load. A single-phase-to-ground fault is then emulated by setting the source’s phase a voltage to 0 from t  = 0.015 s to t  = 0.115 s. As in the example of Section 7.1, the combination of the inductive elements and shunt capacitor creates higher frequency oscillatory modes. Consequently, a moderately small step size is required to accurately simulate this disturbance.   Figure 7–5 shows the reference curves of the stator currents asi , bsi , and csi , and of the electromagnetic torque eT . Resonance of approximately 500 Hz is observed immediately after the fault occurrence and its removal. 214  –400004000i as (A)–400004000i bs (A)–400004000i cs (A)0 0.05 0.10 0.15 0.20–500005000T e (N·m)Time (s)See Figure 7–6 Figure 7–5. Reference trajectories of stator currents asi , bsi , and csi , and electromagnetic torque eT  during a single-phase-to-ground fault (single-machine system).  To achieve numerically accurate solutions, the step size of the fast subsystem (i.e., the external network and the stator) ft∆  is set to 50 µs. The case study is then executed with the proposed models for values of rM  ranging from 1 to 20, i.e., with st∆  varying from 50 to 1000 µs. 215  0.023 0.024 0.025 0.026 0.0272500275030003250Time (s)i cs (A)  a) Referenceb) MR-T/T (∆tf = 50 µs, Mr = 1)c) MR-T/BE (∆tf = 50 µs, Mr = 1)d) MR-T/T (∆tf = 50 µs, Mr = 10)e) MR-T/BE (∆tf = 50 µs, Mr = 10)f) MR-T/T (∆tf = 500 µs, Mr = 1) Figure 7–6. Magnified view of the first peak of stator current csi  during a single-phase-to-ground fault as predicted by the MR-T/T and MR-T/BE models (single-machine system).  A magnified view of the first peak of csi  during the fault as predicted by the proposed models is reproduced in Figure 7–6. It is observed that the models are able to very accurately predict the supersynchronous resonance with rM  = 1 and 10 [curves b) to e)]. This supports the proposed partitioning. The predicted trajectory of csi  using the MR-T/T model with ft∆  = st∆  = 500 µs is also shown in Figure 7–6 [curve f)]. As expected, the higher frequency components are poorly predicted.  For further analysis, the average 2-norm error of the single-phase stator currents, )( siε , is plotted in Figure 7–7 as a function of rM . It is seen that the MR-T/T model is slightly more accurate than MR-T/BE for smaller values of rM . However, as rM  increases, the error of MR-T/T grows faster than that of MR-T/BE. It is also observed that the 2-norm error remains under 1% for rM  ≤ 8, and is less than 3.3% for rM  = 20. 7.3.1.2 Electromechanical transient  Overall, Figure 7–7 shows that for a given rM , the MR-T/T and MR-T/BE models predict the fast-changing stator currents with similar numerical accuracy. However, the216  4 8 12 16 200123Mrε(i s) (%)  MR-T/TMR-T/BE1 Figure 7–7. Average 2-norm relative error of the stator phase currents as a function of rM  for the single-phase-to-ground fault study as predicted by the MR-T/T and MR-T/BE models (single-machine system). MR-T/BE model should predict rotor and mechanical variables less precisely than the MR-T/T model since they are discretized using a lower order integration rule.  To verify this hypothesis, the uncontrolled start-up transient of the 500-hp machine connected to the system of Figure 6–1 is simulated. The complete trajectory of its electrical rotor speed rω  is presented in Figure 7–8 (a), while a magnified view of its highest peak is reproduced in Figure 7–8 (b). When simulating the fast subsystem with ft∆  = 50 µs, the MR-T/T model predicts rω  with virtually no error whether rM  = 1 or 10. It is not the case of the MR-T/BE model: with ft∆  = 50 µs and rM  = 10, the rotor speed is delayed by approximately 2 electric cycles with respect to the reference solution. A similar delay is observed with MR-T/T and ft∆  = st∆  = 500 µs. In summary, the two models have similar numerical accuracy for faster electromagnetic transients; however, the MR-T/T model simulates electromechanical transients more precisely. 7.3.2 Numerical Accuracy – Multimachine System  To quantify the numerical accuracy of the proposed models in a larger scale system, the 19-motor industrial power system simulated in Section 6.4.3 is considered. For reactive217  0 1 2 3 4 500.51.0Time (s)ωr (pu)  a) Referenceb) MR-T/T (∆tf = 50 µs, M  = 1)c) MR-T/BE (∆tf = 50 µs, M  = 1)4.6 4.7 4.8 4.9 5.00.960.981.001.02Time (s)ωr (pu)  d) MR-T/T (∆tf = 50 µs, M  = 10)e) MR-T/BE (∆tf = 50 µs, M  = 10)f) MR-T/T (∆tf = 500 µs, M  = 1)See (b)(b)(a)rrrrre, fca, b, d Figure 7–8. Electrical rotor speed rω  of the 500-hp motor for the start-up study as predicted by the MR-T/T and MR-T/BE models: (a) full trajectory; and (b) magnified view of the first overshoot (single-machine system).  power compensation, a 2-MVAR shunt capacitor is added to the secondary of each of the two main transformers (which introduces fast oscillatory modes). Two 150-kW resistive loads are also added to the same buses. The updated one-line diagram of the industrial power system is shown in Figure 7–9. The machine and network parameters can still be found in Appendix K.  The source voltage is set to 75 kV and all motors initially operate in steady state driving the mechanical loads indicated in Figure 7–9. At t  = 0.02 s, a three-phase fault with a resistance of 0.1 Ω occurs at the head of the network. The fault disappears after 2218  IM3 IM2800 hp 400 hpIM3 IM3750 hp 750 hpIM41750 hpIM3 IM2600 hp 500 hpIM5500 hpIM6 IM61800 hp 1950 hpIM6 IM62250 hp 1500 hpIM3 IM3750 hp 750 hpIM3 IM2800 hp 400 hpIM3750 hpIM62250 hpIM3800 hpabc-ginet1 inet2inet = inet1 + inet240213 4925 12 5 263928 17 2961114191520162172730131824313632372 MVAR125 kW 2 MVAR 125 kW Figure 7–9. One-line diagram of an industrial multimachine power system (Section 7.3.2). 219  –100001000i neta (A)–100001000i netb (A)0 0.05 0.10 0.15–100001000Time (s)i netc (A) Figure 7–10. Reference trajectories of the fault currents contributed by the industrial network ainet , binet , and cinet  during a three-phase fault. electrical cycles. The reference trajectory of the fault current contributed by the industrial power system ( neti ) is presented in Figure 7–10. Supersynchronous resonance of approximately 600 Hz is observed during and after the fault. To give further insight, the electromagnetic torque eT  and the electrical rotor speed rω  of one of the motors connected to bus 17 [see Figure 7–9] are also plotted in Figure 7–11. The electromagnetic torque also contains high-frequency components, whereas in accordance with Section 7.1, rω  is essentially free of higher frequencies.  Due to the magnitude of the higher frequency resonance and the low damping of the system [see Figure 7–10], the fast subsystem must be simulated with a very small step size.220  –101x 104T e (N·m)0 0.05 0.10 0.150.9850.9900.995Time (s)ωr (pu) Figure 7–11. Reference trajectories of (a) the electromagnetic torque eT , and (b) the electrical rotor speed rω  of one of the induction motors connected to bus 17 of the industrial power system during a three-phase fault. 4 8 12 16 202345Mrε(inet) (%)  MR-T/TMR-T/BE1 Figure 7–12. Average 2-norm relative error of the fault phase currents contributed by the industrial network ainet , binet , and cinet  for the three-phase fault study as predicted by the MR-T/T and MR-T/BE models. To achieve adequate numerical accuracy, ft∆  is set to 25 µs. The average 2-norm error of neti , denoted )( netiε , is presented in Figure 7–12 for several values of rM  ranging from 1 to221   20. Changing rM  from 1 to 8 barely increases the error of the MR-T/T (3.34% vs. 2.91%) and MR-T/BE (2.68% vs. 2.15%) models. This means that for smaller values of rM , the majority of the error comes from the discretization of the fast subsystem. As shown in Figure 7–12, the slow subsystem contributes more to the total error for larger values of rM . However, even for rM  = 20, the error )( netiε  remains relatively small with MR-T/T (4.26%) and MR-T/BE (4.65%), which demonstrates the high accuracy of the proposed multirate models. 7.3.3 Numerical Efficiency  To evaluate the computational savings offered by the proposed multirate models, the fault study of Section 7.3.1.1 is executed in a compiled environment for various values of rM . To focus only on the machine models, the external network of Figure 6–1 is replaced by an infinite bus. Furthermore, machine models with one and two rotor circuits are considered [see Appendices D and J, respectively, for their parameters].   The CPU times for the two proposed models are displayed in Figure 7–13. The CPU times are normalized with respect to the MR-T/T model with rM  = 1, which is identical to the numerically efficient and accurate single-rate CP-VBR model proposed in Chapter 6. For further comparison, the CPU time of the single-rate qd model is also shown in Figure 7–13. The qd model is solved using the same step size ( t∆ ) as that of the fast subsystem of the multirate models ( ft∆ ). It is recalled that the qd model is the most numerically efficient single-rate machine model (when excluding the solution of (2–8) or (2–9)).  As seen in Figure 7–13 (b), computational savings of around 45% can be achieved using the MR-T/BE model. Moreover, both models are more efficient than the single-rate qd model when rM  > 2 for machines represented with two rotor circuits.  Figure 7–13 also demonstrates that the MR-T/BE model is slightly more efficient than the MR-T/T model for all values of rM .  222  0.500.751.00NormalizedCPU Time  MR-T/TMR-T/BE2 4 6 8 10 12 14 16 18 200.500.751.00MNormalizedCPU Timeqd model with ∆t = ∆tfqd model with ∆t = ∆tf1 Rotor Circuit2 Rotor Circuits(a)(b)r  Figure 7–13. Normalized CPU time of the MR-T/T and MR-T/BE models as a function of rM  for machines with (a) one rotor circuit and (b) two rotor circuits.  As hypothesized in Section 7.2.1, the slope of the CPU time curve is much steeper for smaller values of rM . For example, for the MR-T/T model with one rotor circuit, the CPU time decreases by around 20% when rM  is changed from 1 to 2, whereas increasing rM  from 18 to 20 improves the numerical efficiency by less than 0.5%. By combining this observation with Figures 7–7 and 7–12, it can be concluded that in general, using values of rM  greater than 8 or so yields very little benefit.  Figure 7–13 also shows that the proposed models offer additional computational savings for machines modeled with two rotor circuits instead of one, since its slow subsystem has a higher computational cost. By analogy, further improvements may be expected when the mechanical system is represented using multiple lumped masses [126].  223  CHAPTER 8:  CONCLUSIONS AND FUTURE WORK  Vast quantities of computer simulations are necessary to ensure that power systems operate in an optimal, stable, and secure manner. Several types of simulation tools and environments have been developed throughout the years, each targeted for the investigation of specific classes of phenomena. This thesis focuses on the so-called EMT simulators. EMT simulators, which are herein separated between EMTP-type and SVB programs, are very flexible and offer high modeling accuracy, which permits the precise simulation of complex higher frequency phenomena. A major limitation of EMT simulators, and in particular of SVB programs, is their high computational cost. This can impose restrictions on numerous factors, such as the size of the system studied, the number of times the study is repeated (e.g., during parametric sweeps), the simulation length, and the type of phenomena investigated. The computational efficiency of EMT programs is further worsened when time-varying elements are present, and when prohibitively small step sizes are required to ensure adequate numerical accuracy and even numerical stability.  Induction and synchronous machines are found in the majority of power systems, either as motors, generators, or condensers. These machines typically operate under some level of magnetic saturation. Consequently, more precise simulations can be achieved by accounting for saturation in machine models. Rotating machines are most widely represented in EMT programs by general-purpose lumped-parameter models, of which several formulations (e.g., PD, qd, and VBR models) have been proposed through the years. While these PD, qd, and VBR models are algebraically equivalent (assuming the same modeling assumptions), they can affect the numerical accuracy, efficiency, and stability of 224  the network solution in very different ways. In particular, the interface of these machine models with external power networks is known to have a significant impact on the numerical properties of the simulation.  From a numerical perspective, the ideal SVB machine model would have a direct and constant-parameter interface with external power networks, and would be formulated explicitly (i.e., with no algebraic loops). Such models have previously been derived for magnetically linear induction and synchronous machines using VBR formulations [59], [81]; however, prior to this dissertation, the existing saturable VBR models had variable-parameter interfacing circuits [96], [108], [109]. As for the more traditional qd models, they are connected to external networks using controlled current sources. This sometimes results in incompatible input-output interfacing, e.g., when machines are connected to inductive branches, thus requiring snubbers or time-step relaxations. Nevertheless, qd models remain widely used in EMT programs, and can be very efficient when a compatible interface is achieved. A common and convenient means of accounting for saturation in numerically efficient (interfacing issues aside) Flux-based qd models is the FC approach [29], [71], [105]. However, standard FC functions contain algebraic loops, which cannot be solved in all SVB packages, can cause divergence, and significantly reduce computational efficiency.   Similarly, the desired EMTP-type induction machine model would have a direct constant-parameter interface. Several magnetically linear PD and VBR models possess these properties [61], [64]; however, incorporating the effect of saturation renders their interfacing circuits saturation-segment-dependent [64]. 225  8.1 Contributions and Anticipated Impact 8.1.1 Contributions  The global objective of this thesis is to propose new SVB and EMTP-type general-purpose lumped-parameter models in order to improve the numerical efficiency of EMT simulations without adversely affecting numerical accuracy and stability. The five specific contributions of this thesis are summarized below. Contribution #1  The first contribution was presented in Chapter 3. It consists of simple, elegant, and numerically efficient means of representing main flux saturation in Flux-based SVB qd induction and synchronous machine models. The proposed saturation functions were achieved by reformulating the traditional implicit FC functions in order to render them explicit. The computer studies presented in Section 3.3 demonstrated that the proposed explicit FC functions are as numerically accurate as the original implicit FC functions. Furthermore, it was shown in Section 3.3.3 that the proposed explicit saturation functions can reduce the CPU time required to solve Flux-based qd models by a factor of 4 to 5 when compared to implicit FC functions. Contribution #2  The second contribution, a constant-parameter SVB VBR induction machine model with main flux saturation (CP-VBR), was presented in Chapter 4. The proposed model was achieved by transferring the time-varying part of the interfacing circuit inductance of the original variable-parameter VBR model (VP-VBR) into the voltage source. The stator current time derivative appearing inside the voltage source of the CP-VBR model was then approximated using BDFs to achieve a proper state model. The error resulting from this 226  approximation was analyzed in Section 4.3 and shown to be function of the machine parameters, the saturation level, the transient, and the integration step size. Computer studies presented in Sections 4.4.2 and 4.4.3 showed that the model remains very accurate in worst-case scenarios (numerical errors of less than 1 to 2%). It was also shown in Section 4.4.4 that the proposed CP-VBR model requires roughly 4 times less CPU time than the VP-VBR model when simulating a small single-machine system with the SVB toolbox PLECS. Contribution #3  The next contribution, presented in Chapter 5, is a constant-parameter VBR synchronous machine model with main flux saturation (CP-VBR). The interfacing circuit of the proposed model comprises the stator and field windings, thereby concurrently providing a direct interface with external power networks and circuit-based excitation systems. The CP-VBR synchronous machine model was derived similarly to the CP-VBR induction machine model, i.e., by transferring time-varying inductances to voltage sources. However, the proper state model was this time achieved using continuous-time low-pass filters. A procedure to select the poles of the filters in order to control numerical accuracy and efficiency was proposed in Section 5.3. The case studies presented in Sections 5.4.1 and 5.4.2 demonstrated the superior combination of numerical accuracy and efficiency of the proposed CP-VBR model. In particular, the CP-VBR model was shown to require one order of magnitude less simulation time than the VP-VBR model for a fault study on a single-machine system modeled with a circuit-based exciter. Contribution #4  The penultimate contribution was presented in Chapter 6. Therein, the original variable-parameter EMTP-type VBR induction machine model with main flux saturation (VP-VBR) was reformulated as to obtain a model with a truly constant interfacing 227  resistance matrix (CP-VBR). The proposed CP-VBR model, which requires predicting stator currents, was shown in Section 6.4 to be virtually as numerically accurate as the saturation-segment-dependent PD model and the saturation-segment- and rotor-position-dependent VP-VBR model, and considerably more precise than the traditional saturation-segment-dependent Thévenin prediction-based qd model. A three-phase fault study on a multimachine industrial power network was executed in PSCAD to validate the proposed model and compare its computational efficiency. As reported in Section 6.4.3, the CPU time for this study was reduced by factors of approximately 1.5, 1.8, and 6 when using the CP-VBR model instead of the qd, PD, and VP-VBR models, respectively. The model’s constant-parameter interface can be particularly useful for real-time simulators, wherein conductance matrix re-factorizations must sometimes be avoided at all cost. Contribution #5  Finally, two multirate EMTP-type induction machine models (MR-T/T and MR-T/BE) were proposed in Chapter 7. Using modal analysis, it was first demonstrated that VBR induction machine models can be separated into a fast subsystem, comprising the stator equations, and a slower subsystem, made of the rotor and mechanical equations. The fast subsystem can be aggregated with the network equations surrounding the machine model. The MR-T/T model was achieved by discretizing the original state-space VBR model with main flux stauration using the trapezoidal rule with distinct step sizes for each subsystem; the MR-T/BE model was derived by discretizing the saturable VBR model’s fast subsystem with the trapezoidal rule, and its slow subsystem using BE and a different step size. Similarly to the single-rate CP-VBR model proposed in Chapter 6, both models were formulated as to have constant-parameter stator interfacing circuits. The case studies presented in Sections 7.3.1 and 7.3.2 demonstrated that considerably larger integration step sizes can often be used for the rotor subsystem without introducing significant228  Table 8–1. Properties of SVB general-purpose lumped-parameter induction machine models.   Main Flux Saturation Stator Interface Circuit Parameters Explicit Formulation Flux-based qd [29], [70] No Indirect N/A Yes Flux-based qd [71] Yes Indirect N/A No Proposed Flux-based qd [Ch. 3] Yes Indirect N/A Yes Mixed/Current-based qd [29]  No Indirect Constant * Yes Mixed/Current-based qd [42], [73] Yes Indirect Variable * Yes PD [59] No Direct Variable Yes VBR [59] No Direct Constant Yes VBR [109] Yes Direct Variable Yes Proposed VBR [Ch. 4] Yes Direct Constant Yes * When the qd model is implemented using circuit elements. additional numerical error, while it was shown in Section 7.3.3 that the proposed models can require roughly half the computational time of the single-rate CP-VBR model.  The numerical properties of the state-of-the-art SVB and EMTP-type machine models were compiled in Tables 1–1 to 1–3. In order to further summarize and emphasize the contributions of this thesis, these tables are herein reproduced as Tables 8–1 to 8–3, to which are now added the machine models proposed in Chapters 3 to 7. The following contributions are highlighted: • As shown in Tables 8–1 and 8–2, we proposed, implemented, and validated the first numerically accurate and explicit saturable Flux-based qd machine models for SVB programs. • As also shown in Tables 8–1 and 8–2, we proposed, implemented, and validated the first saturable machine models combining direct machine-network interfaces, constant interfacing circuit parameters, and explicit formulations. 229  Table 8–2. Properties of SVB general-purpose lumped-parameter synchronous machine models.  Main Flux Saturation Stator Interface Field Interface Circuit Parameters Explicit Formulation Flux-based qd [29], [66] No Indirect Indirect N/A Yes Flux-based qd [29], [105] Yes Indirect Indirect N/A No Flux-based qd [106] Yes Indirect Indirect N/A Yes ** Proposed Flux-based qd [Ch. 3]  Yes Indirect Indirect N/A Yes Mixed/Current-based qd [29] Yes Indirect Direct * Constant * Yes Mixed/Current-based qd [74], [76] Yes Indirect Direct * Variable * Yes PD [24], [58] No Direct Direct Variable Yes VBR [86], [87] No Direct Indirect Variable Yes VBR [88], [90] No Direct Indirect Constant Yes VBR [89] No Direct Indirect Constant No VBR [108], [109] Yes Direct Indirect Variable Yes VBR [81] No Direct Direct Constant Yes VBR [96] Yes Direct Direct Variable Yes Proposed VBR [Ch. 5] Yes Direct Direct Constant Yes * When the qd model is implemented using circuit elements. ** As shown in Section 3.3.2, this model creates significant numerical error. • As demonstrated in Table 8–3, we also proposed, implemented, and validated the first EMTP-type induction machine models with main flux saturation that have direct stator interfaces and constant-parameter interfacing circuits. 8.1.2 Anticipated Impact  The machine models proposed in Chapters 3 to 7 were implemented directly in industry-grade offline EMT programs using their user-defined modeling interfaces. Moreover, the proposed models essentially require the same input parameters (winding resistances and reactances, main flux saturation curve, etc.) as the existing built-in machine230  Table 8–3. Properties of EMTP-type general-purpose lumped-parameter induction machine models (RPD: rotor-position-dependent; SSD: saturation-segment-dependent).  Main Flux Saturation Stator Interface * Interfacing Circuit Parameters ** Thévenin qd [30], [82]  No Indirect Constant Thévenin qd [30] No Indirect Variable (RPD) Thévenin qd [64], [110]  Yes Indirect Variable (SSD) Norton current source qd [35], [83] No Indirect Constant Norton current source qd [35], [83] Yes Indirect Constant Compensation-based qd [35], [85] No Indirect Constant Compensation-based qd [35], [85] Yes Indirect Constant PD [61] No Direct Constant PD [64] Yes Direct Variable (SSD) VBR [93] No Direct Variable (RPD) VBR [61] No Direct Constant VBR [111] Yes Direct Variable (RPD/SSD) VBR [94] Yes Direct Variable (SSD) Proposed VBR [Ch. 6] Yes Direct Constant Proposed multirate VBR [Ch. 7] Yes Direct Constant  models. The only exceptions are the SVB CP-VBR synchronous machine model proposed in Chapter 5, which requires filter parameters, and the multirate EMTP-type models from Chapter 7, which need the integration step size of the slow subsystem st∆ . The pole selection procedure presented in Section 5.3 can be easily implemented within EMT programs. Users could therefore be prompted to enter the more intuitive fit frequency fitf  and error tolerance fitε , and the program would automatically compute the filter parameters. As for the multirate models, reasonable values of st∆  could be suggested as default values, e.g., 250 or 500 µs. Advanced users could modify these values if need be.  It is consequently envisioned that the proposed machine models will eventually complement or replace the built-in models currently available in industry-grade EMT programs. As demonstrated in the numerous case studies contained in this dissertation, the proposed models can significantly reduce the time required to execute case studies while 231  offering high numerical accuracy and stability. While the CPU times reported in this thesis are typically small (often in the order of seconds), it is emphasized that the proposed models were mostly validated on small-scale systems. However, when using large-scale networks, it is not uncommon for EMT simulations to take several minutes. For example, a transformer energization case study done using EMTP-RV was shown in [157] to take more than 20 minutes. Consequently, considerable computational gains can be obtained using machine models that, for instance, allow larger step sizes or avoid matrix re-factorizations. Moreover, the CPU times reported in this dissertation are for a single run. In practice, it frequently takes a considerable amount of runs to initialize the test system and finally achieve the desired solution. Monte Carlo simulations and parametric sweeps, which can require hundreds or even thousands of runs, are also frequently executed using EMT programs. Additionally, higher efficiency can result in economical gains for real-time simulations, since for instance fewer processors may be needed for a given system.   As demonstrated in this dissertation, the proposed models can considerably improve the numerical efficiency of EMT simulators. As a result, the time spent by analysts running a variety of simulations will decrease without compromising on numerical accuracy. The proposed models will also help expanding the type of studies that can be executed using very precise EMT simulators, such as the investigation of electromechanical transients. 8.2 Future Work  To conclude this thesis, four potential extensions to the proposed work are discussed.  232  Task #1    Induction machines with squirrel-cage rotors were considered in this thesis. Consequently, only the stator windings were represented using circuit elements in the CP-VBR induction machine models proposed in Chapters 4, 6, and 7. Using these models to simulate wound-rotor induction machines, which are very common in wind farms [40], can therefore result in incompatible rotor-network interfacing. A future research task would be to derive induction machine models with constant-parameter stator and rotor interfacing circuits that would be suitable for both squirrel-cage and wound-rotor induction machines. This has yet to be achieved for either magnetically linear or saturable induction machines.  Task #2  The proposed models only consider saturation of the main flux. While leakage flux saturation is usually an afterthought in general-purpose lumped-parameter synchronous machine models [see Section 2.4.2], it is fairly more common in induction machine models. The effect of leakage flux saturation in induction machines was thoroughly discussed in Section 2.3.2, wherein several justifications were presented for neglecting leakage flux saturation in the proposed models. However, it is clear that representing stator and/or rotor leakage flux saturation can only improve the fidelity of general-purpose induction machine models and extend their range of applications. The second future research task would be to derive constant-parameter SVB and EMTP-type VBR induction machine models that account for saturation of the main and leakage fluxes. Task #3  Single-rate and multirate constant-parameter EMTP-type VBR induction machine models were presented in Chapters 6 and 7, respectively. The third future task would be to derive analogous models for synchronous machines. This would be of particular interest 233  when studying transmission networks, which include numerous synchronous generators. Moreover, the computational savings of the multirate models would be particularly pronounced for synchronous machines represented with higher order rotor circuits [139] and/or multiple masses [144]. Task #4  Models of conventional three-phase induction and synchronous machines were presented in this dissertation, as these rotating machines remain ubiquitous in today’s power systems. However, owing to improvements in machine drive technology, multiphase machines (i.e., with more than 3 phases) are becoming more prevalent in a variety of applications [158]. The advantages of multiphase machines include improved fault tolerance, reduced current per phase/leg, and lower space-harmonic content [158]. A five-phase VBR induction machine model was recently proposed in [159]. The fourth and final research task would be to develop constant-parameter SVB and EMTP-type VBR multiphase induction and synchronous machine models. 234  References [1] “Electric Power Monthly with Data for September 2014,” U.S. Energy Information Administration, U.S. Department of Energy, Washington, DC, Nov. 2014. [Online]. Available: http://www.eia.gov/electricity/monthly/epm_table_grapher.cfm?t=epmt_1_20_b. 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Machine Saturation Curve: mλ  (Wb)  0.38 0.48 0.60 0.68 0.75 0.79 0.82 0.86 0.92 mi  (A) 11.20 14.33 18.33 21.31 24.42 26.74 29.07 33.21 40.78 mλ  (Wb)  0.96 1.01 1.06 1.10      mi  (A) 46.21 53.21 60.33 66.10       251  Appendix B:    Parameters for the Case Studies of Section 3.3.2 Synchronous Machine Parameters [29]: BS  = 325 MVA, BV  = 20 kV, P  = 64, bf  = 60 Hz, J  = 35.1∙106 J∙s2, sr  = 0.00234 Ω, lsX  = 0.1478 Ω, fdr  = 0.0005 Ω, lfdX  = 0.2523 Ω, 1kqr  = 0.01675 Ω , 1lkqX  = 0.1267 Ω, 1kdr  = 0.01736 Ω, 1lkdX  = 0.1970 Ω, mqX  = 0.4433 Ω, mdX  = 0.8989 Ω. Machine Saturation Curve: mdλ  (Wb)  10.8 22.5 31.0 35.3 38.2 40.9 42.5 45.3 47.9 50.3 mdi  (kA) 4.514 9.498 13.26 15.26 16.71 18.20 19.21 21.34 23.65 25.93  252  Appendix C:    Parameters for the Case Study of Section 4.4.1 Induction Machine Parameters [78]: BP  = 7.5 kW, BV  = 658 V, P  = 2, bf  = 50 Hz, J  = 0.04 J∙s2, sr  = 1.969 Ω, lsL  = 10.2 mH, 1rr  = 1.093 Ω, 1lrL  = 8.98 mH, 2rr  = 5.727 Ω, 2lrL  = 4.07 mH, mL  = 598 mH.  Machine Saturation Curve: mλ  (Wb)  1.237 1.579 1.674 1.754 1.812 1.851 1.881 1.903 1.924 mi  (A) 2.121 3.182 3.677 4.243 4.808 5.303 5.798 6.222 6.718  Transformer: BS  = 50 kVA, BHV  = 12.47 kV, BLV  = 658 V, 0Z  = 1Z  = 1.5 + ι2%.  Cable:  0Z  = 0.15 + ι0.15 Ω, 1Z  = 0.05 + ι0.05 Ω. Thévenin Equivalent: BV  = 12.47 kV, 0Z  = 0.75 + ι1.95 Ω, 1Z  = 0.3 + ι0.6 Ω.  253  Appendix D:    Parameters for the Case Study of Section 4.4.2 Induction Machine Parameters [71]: BP  = 5 hp, BV  = 230 V, P  = 4, bf  = 60 Hz, J  = 0.11 J∙s2, sr  = 0.4122 Ω, lsX  = 1.1 Ω, 1rr  = 0.4976 Ω, 1lrX  = 1.1 Ω, mX  = 15.7 Ω. Machine Saturation Curve: mλ  (Wb)  0.147 0.295 0.398 0.454 0.486 0.522 0.535 0.543 0.553 mi  (A) 3.536 7.071 10.61 14.41 17.68 24.75 28.28 31.82 35.82  Transformer: BS  = 25 kVA, BHV  = 12.47 kV, BLV  = 230 V, 0Z  = 1Z  = 1.5 + ι2%. Cable:  0Z  = 0.15 + ι0.15 Ω, 1Z  = 0.05 + ι0.05 Ω. Thévenin Equivalent: BV  = 12.47 kV, 0Z  = 0.75 + ι1.95 Ω, 1Z  = 0.3 + ι0.6 Ω.  254  Appendix E:    Scalars and Matrices Introduced in Chapter 5  The functions )(1 sα  and )(2 sα  and the coefficients 1β  and 2β  introduced in Section 5.3.1 are expressed as  ( )∏=+=Mzlkqzkqz sLrs11 )(α        and      ( )∑ ∏= ≠= +=MzMzyylkqykqylkqzkqzmqqu sLrLrLs1 ,1'''2 )(α  (E1)  ''''''1 qquddu LL=β     and     12 1 ββ −= .    (E2)  The functions and coefficients used in Section 5.3.2 are defined as  ( )∏=+=Nzlkdzkdz sLrs13 )(α        and      ( )∑ ∏= ≠= +=NzNzyylkdykdylkdzkdzmddu sLrLrLs1 ,1'''4 )(α  (E3)  ++=sLrsLsLsLrsfddufdmddumdduddus''''''''''''5 )(α  (E4)  [ ]sLrsLrs lfdfdlss ++= diag)(6α        and      [ ]sLrsLrs fddufdddus ''''''7 diag)( ++=α  (E5)  ΓΓΓΓ=33322322''''''3 00mddumdduLLβ        and      3224 ββ −= ×I  (E6)  [ ]fds rrdiag35 ββ =        and      [ ])()(diag)( sHsHs fdds=F . (E7)  The inductances '''qquL , '''mqquL , '''dduL , '''fdduL , and '''mdduL  refer to the unsaturated values of '''qqL , '''mqqL , '''ddL , '''fddL , and '''mddL , respectively. 255  Appendix F:    Parameters for the Case Study of Section 5.4.1 Synchronous Machine Parameters [83]: BS  = 120 MVA, BV  = 13.8 kV, bf  = 60 hz, H  = 3.42 s , sr  = 0.0037 pu, lsX  = 0.17 pu, fdr  = 0.0007 pu, lfdX  = 0.152 pu, 1kqr  = 0.0035 pu , 1lkqX  = 0.0835 pu, 1kdr  = 0.0035 pu, 1lkdX  = 0.122 pu, mqX  = 0.472 pu, mdX  = 0.936 pu. Machine Saturation Curve: mdλ  (pu)  0.70 0.80 0.90 0.95 1.00 1.10 1.20 mdi  (pu) 0.75 0.88 1.04 1.15 1.29 1.60 1.98  Transformer Xfo1 (on the machine base): HV  = LV  = 1 pu, 0Z  = 1Z  = 0.005 + ι0.12 pu. Thévenin Equivalent (on the machine base): 0Z  = 0.00233 + ι0.03 pu, 1Z  = 0.00233 + ι0.02 pu . AC1A Exciter Parameters: RT  = 0.02, AK  = 400, AT  = 0.02, RMAXV  = 6.03, RMINV  = –5.43, EXT  = 0.80, CK  = 0.20, DK  = 0.38, EK  = 1.0, FK  = 0.03, FT  = 1.0.  AC1A Exciter Functions:  ( ) ( )>≤<−≤<−≤−=10175.01732.175.0433.075.0433.0577.012NNNNNNNNEXIIIIIIIIF  EV  0.0 3.14 4.18 )( EE VS  0.0 0.88 1.04   256  Appendix G:    Parameters for the Case Study of Section 5.4.2 Synchronous Machine Parameters: see Appendix F. Machine Saturation Curve: see Appendix F. Transformer Xfo1 (on the machine base): see Appendix F. Thévenin Equivalent (on the machine base): see Appendix F. Excitation Transformer Xfo2: BS  = 1 MVA, HV  = 13.8 kV, LV  = 0.0025 HV , 0Z  = 1Z  = 0.01 + ι0.10 pu . Low-pass Filter: lpfp  = 100. PI Controller: pk  = ik  = 200, minα  = 10, maxα  = 150. Rate Limiter: minS  = –100, maxS  = 100. Band-pass Filter: β  = 30, bpω  = 377 rad/s.  257  Appendix H:    Scalars and Matrices Introduced in Chapter 6  The scalars and matrices in (6–6) are  12''1 cLa Djj =     (H1)  −=djojojdjj22222 aaaaA       where      −=djojojdjj aaaa33333A     (H2)  [ ]11211''2 −−−= lrNlrlrDjroj LLLL ωa  (H3)  −−−= 21''22212''21111''''2lrNrNlrNDjlrrlrDjlrrlrDjDjdj LrcLLLrcLLLrcLLL a  (H4)  12''3 cLLaDjDjdj = ,     ( )φωω −=DjDjoj LLa''3 ,     and      ∑==Nz lrzrzLrc121  (H5) where dtdφωφ = .  The matrices in (6–7) are defined as  =djdjj111 b00bB        and       −=djojojdjj22222 BBBBB   (H6)  jDjj L 131 BB =      and        TlrNrNlrrlrrDjdj LrLrLrL = 2211''1b  (H7)  =oNNjjoNjoNNjojdjoNjojojddjbbbbbbbbb2122221112112B        and       ( ) NNroj ×−= IB ωω2  (H8) 258   −= 1''lrzDjlrzrzdzzj LLLrb       and       lrzlryDjryoyzj LLLrb''= .    (H9)  The matrix used in (6–8) is expressed as  =dd444 b00bB      and       [ ]lrNlrlrd LLL 214 =b . (H10)  The matrices used in (6–11) are  ( ) 12221 −× −= jNNj k BIF      and       ( )jNNj k 2222 BIF += ×  (H11)  jjj 11 BFE =      and      jjj 31 BFD = . (H12)  The matrices used in (6–13) are  jjj a MIH += ×221 ,     jjj EAM 2= ,     and      jjj 12 FAN =  (H13)  jjj 321 APP +=      and       jjj DAP 22 = . (H14)  The scalars 4k  and 5k  from (6–27) are defined as  ( ) dSdSNNdSS kah 11221const1 bBIa −×− −+=  (H15)  ( ) const14 3/2 −= hk     and      ( ) const15 3/1 −−= hk  (H16) where the subscript S  refers to the last piecewise-linear segment of the saturation curve depicted in Figure 2–8.  259  Appendix I:    Parameters for the Case Study of Section 6.4.1.1 Induction Machine Parameters (IM1) [78]: see Appendix D. Machine Saturation Curve: see Appendix D. Transformer: see Appendix D. Cable: 0Z  = 0.3 + ι1.5 Ω, 1Z  = 0.1 + ι0.5 Ω. Capacitor: C  = 100 µF. Thévenin Equivalent: see Appendix D. 260  Appendix J:    Parameters for the Case Studies of Sections 6.4.1.2 and 7.3.1 Induction Machine Parameters (IM2) (Baldor M25502S-4, fitted using [47]): BP  = 500 hp, BV  = 460 V , P  = 2, bf  = 60 Hz, J  = 15 J∙s2, sr  = 1.646 mΩ, lsX  = 0.045 Ω, 1rr  = 3.291 mΩ, 1lrX  = 0.0925 Ω, 2rr  = 20.19 mΩ, 2lrX  = 0.045 Ω, mX  = 1.952 Ω. Machine Saturation Curve: mλ  (Wb)  0.48 0.68 0.79 0.86 0.96 1.06 1.10 mi  (A) 96.037 142.82 179.21 222.57 306.69 404.32 442.99  Transformer: BS  = 1 MVA, BHV  = 12.47 kV, BLV  = 460 V, 0Z  = 1Z  = 1 + ι5%. Cable: 0Z  = 3 + ι3.01 mΩ, 1Z  = 1 + ι1.25 mΩ. Capacitor: C  = 1.5 mF. Thévenin Equivalent: see Appendix D.  261  Appendix K:    Parameters for the Case Studies of Sections 6.4.3 and 7.3.2 Induction Machines Parameters:  IM2: see Appendix J.  IM3: BP  = 800 hp, BV  = 460 V, P  = 4, bf  = 60 Hz, J  = 141.2 J∙s2, sr  = 2.29 mΩ, lsX  = 0.0218 Ω , 1rr  = 4.59 mΩ, 1lrX  = 0.224 Ω, 2rr  = 5.56 mΩ, 2lrX  = 0.0218 Ω, mX  = 1.763 Ω .  IM4: BP  = 1750 hp, BV  = 4160 V, P  = 6, bf  = 60 Hz, J  = 125 J∙s2, sr  = 0.140 Ω, lsX  = 1.541 Ω , 1rr  = 1.665 Ω, 1lrX  = 1.541 Ω, 2rr  = 0.0919 Ω, 2lrX  = 1.500 Ω, mX  = 64.30 Ω.  IM5: BP  = 500 hp, BV  = 2300 V, P  = 4, bf  = 60 Hz, J  = 11.06 J∙s2, sr  = 0.262 Ω, lsX  = 1.206 Ω , 1rr  = 0.187 Ω, 1lrX  = 1.206 Ω, mX  = 54.02 Ω.  IM6: BP  = 2250 hp, BV  = 2300 V, P  = 4, bf  = 60 Hz, J  = 63.87 J∙s2, sr  = 0.029 Ω, lsX  = 0.226 Ω, 1rr  = 0.022 Ω, 1lrX  = 0.226 Ω, mX  = 13.04 Ω. Machines Saturation Curves: IM2 See Appendix J. IM3 mλ  (Wb)  0.48 0.58 0.68 0.79 0.86 0.96 1.06 1.10 mi  (A) 103.18 125.00 153.43 192.53 239.11 332.71 434.38 475.92 IM4 mλ  (Wb)  4.174 5.150 5.913 6.870 7.478 8.348 9.217 9.565 mi  (A) 24.600 30.650 36.580 45.905 57.005 79.325 103.56 113.46 IM5 mλ  (Wb)  2.40 3.00 3.40 3.95 4.30 4.80 5.30 5.50 mi  (A) 16.836 21.100 25.035 31.415 39.016 54.289 70.877 77.656 IM6 mλ  (Wb)  2.40 3.00 3.40 3.95 4.30 4.80 5.30 5.50 mi  (A) 69.745 89.000 103.72 130.14 161.63 224.90 293.62 321.70 262  Transformers Parameters:  Nodes (To/From) BS  (MVA) BHV  (kV) BLV  (kV) Leakage Reactance (pu) Grounding Resistance (pu) Winding Config. 12 / 3 15 69 13.8 0.08 1.57 Dyn11 2 / 4 15 69 13.8 0.08 1.57 Dyn11 25 / 28 1.5 13.8 0.48 0.0575 0 Dyn11 12 / 17 1.5 13.8 0.48 0.0675 0 Dyn11 5 / 39 1.725 13.8 4.16 0.06 0 Dyn11 26 / 29 1.5 13.8 0.48 0.0575 0 Dyn11 6 / 11 1.5 13.8 2.4 0.055 0 Dyn11 14 / 19 3.75 13.8 2.4 0.055 0 Dyn11 15 / 20 3.75 13.8 2.4 0.055 0 Dyn11 13 / 18 1.5 13.8 0.48 0.0575 0 Dyn11 27 / 30 1.5 13.8 0.48 0.0575 0 Dyn11 16 / 21 0.75 13.8 0.48 0.0575 – Dd 31 / 36 2.5 13.8 2.4 0.0575 0 Dyn11 32 / 37 1 13.8 0.48 0.0575 0 Dyn11  Transformers Saturation Curves (based on PSCAD parameters [7]): for all transformers, the air core reactance is set to twice the leakage reactance, the knee voltage to 1.10 pu, and the magnetizing current to 1%. Lines (on a 100 MVA base):  Nodes (To/From) BV  (kV) 10 ZZ =  (pu)  Nodes (To/From) BV  (kV) 10 ZZ =  (pu) 40 / 1 69 0.014 + ι0.034  4/ 15 13.8 0.084 + ι0.026 40 / 2 69 0.014 + ι0.034  4 / 7 13.8 0.022 + ι0.010 3 / 9 13.8 0.044 + ι0.016  7 / 27 13.8 0.042 + ι0.015 9 / 25 13.8 0.395 + ι0.051  27 / 13 13.8 0.145 + ι0.019 9 / 12 13.8 0.036 + ι0.005  7 / 16 13.8 0.256 + ι0.033 3 / 5 13.8 0.070 + ι0.009  4 / 24 13.8 0.100 + ι0.064 3 / 26 13.8 0.146 + ι0.019  24 / 31 13.8 0.074 + ι0.009 3 / 6 13.8 0.026 + ι0.011  24 / 32 13.8 0.104 + ι0.014 6 / 14 13.8 0.189 + ι0.057      Thévenin Equivalent (on a 100 MVA base): BV  = 69 kV, 0Z  = 0.019 + ι0.191 pu, 1Z  = 0.0045 + ι0.1 pu . 263  Appendix L:    Parameters for the Modal Analysis of Section 7.1 Induction Machine Parameters: see Appendix A. Capacitor: C  = 202.8 µF. Thévenin equivalent: BV  = 460 kV, 0Z  = 1Z  = 0.0567 + ι0.2836 Ω.  264  Appendix M:    Scalars and Matrices Introduced in Chapter 7  The scalars and matrices ja1 , j2A , j3A , j1B , j2B , j3B , and 4B  used below are defined in Appendix H.  The matrices used in the discrete MR-T/T model (Section 7.2.2) are  ( ) 12221 −× −= jNNsj k BIF      and       ( )jNNsj k 2222 BIF += ×  (M1)  jjj 11 BFE =      and       jjj 31 BFD =  (M2)  jjj EAM 2=      and       jrjjjjjMahhhhMI 12211221+=−×  (M3)  jj hk 11 32= ,     jjj hhk 212 333−−= ,      and      jjj hhk 213 333+−=  (M4)  =jjjjjjjjjjkkkkkkkkk132213321K . (M5)  Those used in the discrete MR-T/BE model (Section 7.2.3) are  12221 2−× −= jNNsTBjkBIF  (M6)  jTBjTBj 11 BFE =      and      jTBjTBj 31 BFD =  (M7)  TBjjTBj EAM 2=      and       TBjrjTBjTBjTBjTBjMahhhhMI 12211221 +=−×  (M8)  TBjTBj hk 11 32= ,       TBjTBjTBj hhk 212 333−−= ,      and        TBjTBjTBj hhk 213 333+−=  (M9) 265   =TBjTBjTBjTBjTBjTBjTBjTBjTBjTBjkkkkkkkkkt132213321)(K . (M10)  

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