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Average-value modeling of high power ac-ac and ac-dc converters for power systems transient studies Ebrahimi, Seyyedmilad 2019

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Average-Value Modeling of High Power AC-AC and AC-DC Converters for Power Systems Transient Studies   by  Seyyedmilad Ebrahimi  B.Sc., Sharif University of Technology, 2010 M.Sc., Sharif University of Technology, 2012   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)  April 2019 ©Seyyedmilad Ebrahimi, 2019 ii The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:  Average-Value Modeling of High Power AC–AC and AC–DC Converters for Power Systems Transient Studies  submitted by Seyyedmilad Ebrahimi  in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical and Computer Engineering  Examining Committee: Dr. Juri Jatskevich Supervisor Dr. Liwei Wang Co-supervisor  Dr. Yu Christine Chen Supervisory Committee Member Dr. Hermann Dommel University Examiner Dr. Yusuf Altintas University Examiner  Additional Supervisory Committee Members: Dr. Jose R. Marti    iii Abstract The ac–ac and ac–dc line-commutated converters are widely used in various high-power applications due to their high reliability and efficiency and low cost. Efficient and accurate computer simulations are necessary to analyze various aspects of power systems in both normal and unbalanced/faulty conditions where using detailed switching models of converters is computationally expensive due to switching. As an alternative, for system-level studies, the so-called parametric average-value modeling (PAVM) technique has been developed to achieve computationally efficient models of power-electronic converters that neglect the switching and capture the averaged dynamics of converters only. In this thesis, the PAVM methodology is extended to three-phase ac–ac class of converter systems. Furthermore, a generalized PAVM (GPAVM) is proposed for ac–dc converters that includes the ac harmonics in thyristor-controlled rectifier models considering their dependency on the line frequency. It is shown that any existing PAVM can be realized as a subset of the proposed GPAVM. Then, the PAVM methodology is extended to rectifiers with internal faults. The new formulation considers the asymmetrical operation of rectifiers by including the ac-side harmonics in both positive and negative sequences as well as dc components that may be present on ac variables. Finally, a new parametric methodology is presented that can provide continuous–detailed models of rectifiers which can also reconstruct the switching details similar to discrete–iv switching–detailed models. However, the proposed parametric–detailed model is continuous and can be simulated with much larger time-steps. Moreover, the new model can be easily converted to a PAVM by disabling the reconstruction of dc details/switching. All the models in this dissertation are verified by extensive experimental measurements and computer studies using detailed models of the subject converters. It is demonstrated that all the proposed new models have excellent accuracy over a wide range of operating conditions while being computationally much faster than the corresponding detailed switching models. It is envisioned that the models and methodologies proposed in this dissertation will receive wide acceptance in the research community and simulation software industry, and will enable the next generation of power systems simulation tools. v Lay Summary Design, study, analysis, and operation of power systems containing ac–ac and ac–dc power converters require many computer studies and simulations. Therein, accurate and efficient models of various electrical components such as electrical machines, transmission lines, and power-electronic converters are essential. However, models of power-electronic converters have always been bottleneck in power systems simulations. In this dissertation, new and computationally efficient models of power-electronic converters are proposed which allow orders-of-magnitude faster simulations and studying larger power systems with many converters and components. The proposed models are shown to provide excellent accuracy over a wide range of operating conditions while providing advantages over existing conventional detailed models. It is envisioned that the models and methodologies proposed in this dissertation will receive wide acceptance in the research community and simulation software industry, and will enable the next generation of power systems simulation tools, used by thousands of engineers around the world. vi Preface Most of the research results presented in this thesis have been already published in or will be submitted to scientific journals and conference proceedings. In all of them, I was responsible for deriving the mathematical formulations, developing all of the proposed models, implementing them in simulation programs, and executing all simulations and experimental studies, as well as writing the draft of each paper. All of these works have been done under the supervision of Dr. Juri Jatskevich who provided guidance, valuable feedback and comments during all the steps and helped in writing and editing the papers. The works have also been co-supervised by Dr. Liwei Wang who provided supervisory feedback and helped though steps in preparing the papers. All other co-authors of papers have also contributed to this research by providing feedback and discussions, editorial comments, and helping to revise the manuscripts as were needed in each case, in compliance with the IEEE Authorship policy. The articles resulted from this doctoral dissertation are listed below: On the basis of Chapter 1:  S. M. Ebrahimi, H. Atighechi, N. Amiri, J. Jatskevich, and J.M. Cano, “Multi-resolution modeling of induction furnace systems with line-commutated rectifier and resonant converter,” in Proc. IEEE Power and Energy Society General Meeting Conf., Denver, USA, 2015, pp. 1–5. vii  S. M. Ebrahimi, N. Amiri, H. Atighechi, J. Jatskevich, and L. Wang, “Performance verification of parametric average-value model of line-commutated rectifiers under unbalanced conditions,” in Proc. IEEE 16th Workshop on Control and Modeling for Power Electron. (COMPEL), Vancouver, Canada, 2015, pp. 1–6. On the basis of Chapter 2:  S. M. Ebrahimi, N. Amiri, H. Atighechi, J. Jatskevich, and L. Wang, “Parametric average-value modeling of high power ac/ac cyclo converters,” in Proc. IEEE 16th Workshop on Control and Modeling for Power Electron. (COMPEL), Vancouver, Canada, 2015, pp. 1–6.  S. M. Ebrahimi, N. Amiri, H. Atighechi, J. Jatskevich, and L. Wang, “Parametric average-value modeling of ac/ac matrix converters,” in Proc. IEEE 17th Workshop on Control and Modeling for Power Electron. (COMPEL), Trondheim, Norway, 2016, pp. 1–7. On the basis of Chapter 3:  S. M. Ebrahimi, N. Amiri, L. Wang, and J. Jatskevich, “Efficient modeling of six-phase PM synchronous machine-rectifier systems in state-variable-based simulation programs,” IEEE Trans. Energy Convers., vol. 33, no. 3, pp. 1557–1570, Sep. 2018.  S. M. Ebrahimi, N. Amiri, Y. Huang, J. Jatskevich, and L. Wang, “Efficient simulation of wind farms using switching reduced models of converters and VBR formulation of six˗phase PM synchronous generators,” in Proc. IEEE Canadian Conf. Elect. Comp. Eng. (CCECE), 2016, Vancouver, Canada, pp. 1–4.   S. M. Ebrahimi, N. Amiri, Y. Huang, M. Chapariha, J. Jatskevich, and L. Wang, “Multi-resolution modeling of variable speed six-phase synchronous generator with regulated 400Hz AC system,” in Proc. IEEE Power and Energy Society General Meeting Conf., Boston, MA, USA, 2016, pp. 1–5. viii On the basis of Chapter 4:  S. M. Ebrahimi, N. Amiri, H. Atighechi, L. Wang, and J. Jatskevich, “Verification of parametric average-value model of thyristor-controlled rectifier systems for variable-frequency wind generation systems,” IEEE Trans. Energy Convers., vol. 31, no. 1, pp. 401–403, Mar. 2016.  S. M. Ebrahimi, N. Amiri, H. Atighechi, Y. Huang, L. Wang, and J. Jatskevich, “Generalized parametric average-value model of line-commutated rectifiers considering ac harmonics with variable frequency operation,” IEEE Trans. Energy Convers., vol. 33, no. 1, pp. 341–353, Mar. 2018.  S. M. Ebrahimi, N. Amiri, J. Jatskevich, and L. Wang, “Parametric average-value modeling of single-phase line-commutated electronic rectifier circuits,” in Proc. IEEE 9th Annual Information Technol. Electron. Mobile. Comm. Conf., Vancouver, Canada, 2018, pp. 1–6. On the basis of Chapter 5:  S. M. Ebrahimi, N. Amiri, Y. Huang, L. Wang, and J. Jatskevich, “Average-value modeling of diode rectifier systems under asymmetrical operation and internal faults,” IEEE Trans. Energy Convers., vol. 33, no. 4, pp. 1895–1906, Dec. 2018.  S. M. Ebrahimi, N. Amiri, L. Wang, and J. Jatskevich, “Parametric average-value modeling of thyristor-controlled rectifiers with internal faults and asymmetrical operation,” IEEE Trans. Power Del., in press, doi: 10.1109/TPWRD.2018.2880616. On the basis of Chapter 6:  S. M. Ebrahimi, N. Amiri, J. Jatskevich, and L. Wang, “Simulation of line-commutated rectifier systems using fixed time-step without zero-crossing events,” in Proc. IEEE 16th Int. Conf. Smart Energy Grid Eng. (SEGE), Oshawa, ON, Canada, 2018, pp. 195–199. ix  S. M. Ebrahimi, N. Amiri, J. Jatskevich, and L. Wang, “Parametric hybrid continuous detailed/AVM model of line-commutated rectifiers,” under preparation. x Table of Contents Abstract ....................................................................................................... iii Lay Summary ............................................................................................... v Preface .......................................................................................................... vi Table of Contents ........................................................................................ x List of Tables ............................................................................................. xvi List of Figures ........................................................................................ xviii List of Abbreviations .......................................................................... xxviii Acknowledgements ................................................................................. xxx Dedication .............................................................................................. xxxii CHAPTER 1: INTRODUCTION .............................................................. 1 1.1 Motivation ............................................................................................... 1 1.2 Literature Review ................................................................................... 5 1.3 Research Objectives ................................................................................ 9 1.3.1 Objective One: Parametric average-value modeling of ac–ac converters   .................................................................................................................. 11 1.3.2 Objective Two: Generalizing PAVM methodology for thyristor-controlled LCCs considering ac harmonics with variable frequency operation . 11 1.3.3 Objective Three: Developing PAVM methodology for LCCs under asymmetrical operation with internal faults ........................................................ 12 1.3.4 Objective Four: Developing versatile continuous detailed models of LCRs using the parametric approach ................................................................... 13 xi 1.4 Platforms for Verifications ................................................................... 14 CHAPTER 2: PARAMETRIC AVERAGE-VALUE MODELING OF AC–AC CONVERTERS ............................................................................. 15 2.1 Parametric Average-Value Modeling of Three-Phase AC–AC Converters ....................................................................................................... 16 2.2 AC–AC Cyclo-Converters ..................................................................... 22 2.2.1 Blocking mode of operation ..................................................................... 24 2.2.2 Circulating current mode of operation ................................................... 25 2.2.3 Verification of the proposed ac–ac PAVM for the three-phase half-bridge cyclo-converters ........................................................................................... 26 2.2.3.1 Steady-state operation ......................................................................... 27 2.2.3.2 Transient operation ............................................................................. 28 2.2.3.3 Computational performance ................................................................ 30 2.2.4 Verification of the proposed ac–ac PAVM for the three-phase full-bridge cyclo-converters ........................................................................................... 31 2.2.4.1 Steady-state operation ......................................................................... 32 2.2.4.2 Transient operation ............................................................................. 34 2.2.4.3 Computational performance ................................................................ 36 2.3 AC–AC Matrix Converters ................................................................... 36 2.3.1 Verification of the proposed ac–ac PAVM for the three-phase matrix converters ................................................................................................................ 40 2.3.1.1 Steady-state and transient operation ................................................. 41 2.3.1.2 Prediction of efficiency and losses ....................................................... 44 2.3.1.3 Computational performance ................................................................ 49 CHAPTER 3: EXTENDING MRF-PAVM METHOD TO TWELVE-PULSE RECTIFIERS CONNECTED TO SIX-PHASE AC SYSTEMS .    ............................................................................................ 51 3.1 Generic Six-Phase Machine-Converter System .................................. 52 xii 3.2 Six-Phase PMSM Modeling .................................................................. 56 3.2.1 qd12 model ............................................................................................... 56 3.2.2 VBR model ............................................................................................... 60 3.2.3 Proposed CPVBR model .......................................................................... 62 3.3 Twelve-Pulse Rectifier Average-Value Modeling ................................ 66 3.4 Computer Studies ................................................................................. 72 3.4.1 Steady-state studies ................................................................................ 74 3.4.2 Transient studies ..................................................................................... 77 3.4.3 Computational performance comparison ............................................... 78 3.4.3.1 Numerical accuracy of detailed models .............................................. 79 3.4.3.2 Performance with non-stiff solver ....................................................... 80 3.4.3.3 Performance with stiff solver .............................................................. 81 3.4.3.4 Performance of AVMs vs detailed models .......................................... 83 CHAPTER 4: GENERALIZED PAVM OF LINE-COMMUTATED RECTIFIERS  ............................................................................................ 85 4.1 Verification of PAVM of Thyristor-Controlled Rectifier Systems for Variable-Frequency Wind Generation Systems ............................................ 86 4.1.1 Computer studies ..................................................................................... 87 4.2 Generalizing the PAVM Methodology for AC-DC Line-Commutated Rectifiers ......................................................................................................... 90 4.2.1 The proposed GPAVM methodology ....................................................... 92 4.2.1.1 Derivation of GPAVM formulation ..................................................... 94 4.2.1.2 Establishing parametric functions ...................................................... 99 4.2.1.3 Implementation of GPAVM ............................................................... 105 4.2.2 Verification of the proposed GPAVM ................................................... 108 4.2.2.1 Diode rectifier variable frequency operation .................................... 110 4.2.2.2 Thyristor rectifier operation .............................................................. 114 xiii 4.2.2.2.1 Transient study at fixed frequency .................................................. 114 4.2.2.2.2 Transient study with variable frequency operation ....................... 117 4.2.2.2.3 Prediction of rectifier efficiency (losses) .......................................... 119 4.2.2.3 Computational performance .............................................................. 121 4.2.2.3.1 Performance compared to MRF-PAVM ........................................... 121 4.2.2.3.2 Performance compared to PAVM ..................................................... 122 CHAPTER 5: PARAMETRIC AVERAGE-VALUE MODELING OF LINE-COMMUTATED RECTIFIERS WITH INTERNAL FAULTS AND ASYMMETRICAL OPERATION ................................................. 124 5.1 Decomposition of Instantaneous Variables into Positive and Negative Sequences ...................................................................................................... 126 5.2 Extended PAVM Methodology for Faulty Diode Rectifier Systems . 134 5.2.1 Formulation of the proposed PAVM ..................................................... 134 5.2.2 Implementation of the proposed PAVM of faulty rectifiers ................ 138 5.2.3 Performance verification of the proposed PAVM of faulty diode rectifiers ................................................................................................................ 142 5.2.3.1 Operation with faulty open-circuited diodes .................................... 143 5.2.3.2 Operation with faulty short-circuited diodes ................................... 150 5.2.3.3 Computational Performance ............................................................. 152 5.3 Extending the Proposed PAVM Methodology of Faulty LCRs to Thyristor-Controlled Rectifiers .................................................................... 154 5.3.1 Formulation of the proposed extended PAVM of faulty thyristor-controlled rectifiers .............................................................................................. 155 5.3.2 Verification of the proposed extended PAVM of thyristor-controlled rectifiers ................................................................................................................ 157   xiv CHAPTER 6: PARAMETRIC HYBRID CONTINUOUS-DETAILED ––AVM OF LINE-COMMUTATED RECTIFIERS .............................. 163 6.1 Verification of LCR Models in Simulations with Fixed Time-Step without Zero-Crossing Events ...................................................................... 165 6.1.1 Simulations without artificial snubbers in detailed models of diodes 166 6.1.2 Using artificial snubbers in detailed models of diodes ........................ 169 6.2 A New Parametric Modeling Technique for Achieving Hybrid Continuous Detailed/AVM Models of Line-Commutated Rectifiers .......... 173 6.2.1 Proposed parametric hybrid continuous detailed/AVM model of line-commutated rectifiers .......................................................................................... 176 6.2.1.1 Derivation of the hybrid detailed/AVM model formulation ............. 178 6.2.1.2 Establishing parametric functions .................................................... 184 6.2.1.3 Implementation of the proposed parametric hybrid continuous detailed/AVM model ............................................................................................... 187 6.2.2 Verification of the proposed parametric hybrid detailed/AVM model of LCRs  ................................................................................................................ 192 6.2.2.1 Line-commutated rectifier with diode switches ............................... 193 6.2.2.1.1 Steady-state operation ...................................................................... 193 6.2.2.1.2 Transient operation .......................................................................... 200 6.2.2.2 Line-commutated rectifier with thyristor switches ......................... 202 6.2.2.3 Line-commutated rectifier with faulty switches .............................. 206 6.2.2.4 Computational performance of the LCR models .............................. 210 CHAPTER 7: CONCLUSIONS AND FUTURE WORKS ................. 221 7.1 Contributions and Their Anticipated Impact .................................... 221 7.1.1 Contributions for Objective 1 ................................................................ 221 7.1.2 Contributions for Objective 2 ................................................................ 222 7.1.3 Contributions for Objective 3 ................................................................ 224 7.1.4 Contributions for Objective 4 ................................................................ 225 xv 7.2 Future Works ...................................................................................... 227 7.2.1 Thrust 1 .................................................................................................. 227 7.2.2 Thrust 2 .................................................................................................. 228 7.2.3 Thrust 3 .................................................................................................. 229 References................................................................................................. 230 Appendices ................................................................................................ 244 Appendix A. Parameters of Case Study System in Section 2.2.3 ............ 244 Appendix B. Parameters of Case Study System in Section 2.2.4 ............ 245 Appendix C. Parameters of Case Study System in Section 2.3.1 ............ 246 Appendix D. Matrices and Scalars Introduced in Chapter 3 ................... 247 Appendix E. Parameters of Case Study System in Section 3.4 ............... 249 Appendix F. Parameters of Case Study System in Section 4.1.1 ............ 250 Appendix G. Parameters of Case Study System in Section 4.2 ............... 251 Appendix H. Parameters of Case Study System in Section 5.2.3 ............ 252 Appendix I. Parameters of Case Study System in Section 5.3.2 ............ 253 Appendix J. Parameters of Case Study System in Section 6.1 ............... 254 Appendix K. Parameters of Case Study System in Section 6.2.2 ............ 255 xvi List of Tables Table 2.1. Computational performance of proposed PAVM and detailed model of half-bridge cyclo-converter for a 3.5-second transient study ........................... 31 Table 2.2. Computational performance of proposed PAVM and detailed model of full-bridge cyclo-converter for a 3.5-second transient study ............................ 36 Table 2.3. The efficiency of MC as predicted by the proposed PAVM and detailed models considering conduction losses only ............................................. 45 Table 2.4. The efficiency of MC as predicted by the subject models considering both conduction and switching losses ................................................ 45 Table 2.5. System power flow as predicted by the subject models considering conduction losses only ............................................................................................ 48 Table 2.6. System power flow as predicted by the subject models considering conduction and switching losses ............................................................................ 49 Table 2.7. Computational performance of the proposed PAVM and detailed model of the matrix converter................................................................................ 50 Table 3.1. 2-norm error in several variables of the subject detailed models for the considered 5-second transient study ............................................................... 80 Table 3.2. Computational performance of the subject detailed and averaged models for the considered 5-second transient study with the non-stiff solver ode45  .......................................................................................................... 81 Table 3.3. Computational performance of the subject detailed and averaged models for the considered 5-second transient study with the stiff solver ode23tb ..   .......................................................................................................... 82 Table 4.1. Capabilities of existing state-of-the-art AVMs of line-commutated rectifiers with respect to the proposed generalized PAVM (GPAVM) ................. 92 Table 4.2. The efficiency of thyristor-controlled rectifier at 60 Hz in different operating conditions as predicted by the subject models ................................... 120 Table 4.3. Computational performance of the subject models of diode rectifier system for the considered 80-second variable frequency study ......................... 122 xvii Table 4.4. Computational performance of the subject models of the thyristor-controlled rectifier for the considered 10-second transient study ...................... 123 Table 5.1. The harmonic content of three-phase voltages obtained from experiments and as predicted by the subject models when diodes D2 and D3 are in fault state (open-circuited) in CCM-1. ............................................................ 149 Table 5.2. The harmonic content of three-phase currents obtained from experiments and as predicted by the subject models when diodes D2 and D3 are in fault state (open-circuited) in CCM-1. ............................................................ 149 Table 5.3. Computational performance of the detailed model and the proposed PAVM of faulty diode rectifiers for the 3-second transient study ...... 153 Table 5.4. Computational performance of the subject models of faulty thyristor-controlled rectifiers for the 2-second transient study ......................... 162 Table 6.1. The harmonic content of phase a voltage (vas) of diode rectifier as predicted by the subject models in DCM mode (Rl = 900Ω) ................................ 198 Table 6.2. The harmonic content of phase a current (ias) of diode rectifier as predicted by the subject models in DCM mode (Rl = 900Ω) ................................ 199 Table 6.3. The harmonic content of phase a voltage (vas) of diode rectifier as predicted by the subject models in CCM-1 mode (Rl = 40Ω) ............................... 199 Table 6.4. The harmonic content of phase a current (ias) of diode rectifier as predicted by the subject models in CCM-1 mode (Rl = 40Ω) ............................... 200 Table 6.5. Computational performance of the subject models of diode rectifier system for the considered 10-second transient case-study ................................. 217 Table 6.6. Computational performance of the subject models of thyristor-controller rectifier system for the considered 10-second transient case-study .. 218 Table 6.7. Computational performance of the subject models of diode rectifier system with faulty switches for the considered 10-second transient case-study ....   ........................................................................................................ 218 xviii List of Figures Figure 1.1. Classification of high-power converters considered for the proposed and future research. ............................................................................................... 10 Figure 2.1. Schematic of a three-phase ac–ac conversion system consisting of a three-phase ac–ac converter connected to the source/load-side ac subsystems. . 15 Figure 2.2. Phasor diagram of ac–ac converter voltages and currents expressed in qd rotating synchronous reference frames: (a) fundamental components of source-side ac variables, and (b) fundamental components of load-side ac variables. .................................................................................................... 17 Figure 2.3. Implementation of the three-phase ac–ac converter PAVM and its input-output interfacing with the source/load-side ac subsystems. ..................... 21 Figure 2.4. Schematic of a single-phase cyclo-converter: (a) half-bridge, and (b) full-bridge.  .......................................................................................................... 23 Figure 2.5. Schematic of a three-phase cyclo-converter: (a) half-bridge, and (b) full-bridge.  .......................................................................................................... 24 Figure 2.6. A generic industrial three-phase ac–ac conversion system consisting of a half-bridge cyclo-converter used as adjustable speed drive. ........ 26 Figure 2.7. Steady-state source-side phase a voltage and current when 10 Hzof   and 0.5M   as predicted by the proposed PAVM and detailed model of half-bridge cyclo-converter. .................................................................................... 27 Figure 2.8. Steady-state load-side phase a voltage and current when 10 Hzof   and 0.5M   as predicted by the proposed PAVM and detailed model of half-bridge cyclo-converter. ........................................................................................... 28 Figure 2.9. Transient response of load-side phase a voltage and current when the converter modulation index is increased from 0.5M  to 0.9M   at 2t  s with 20 Hzof  as predicted by the subject models of half-bridge cyclo-converter. ....... 29 Figure 2.10. Transient response of load-side phase a voltage and current when the output frequency is decreased from 20 Hzof   to 5 Hzof   at 2.5t  s with 0.9M   as predicted by the subject models of half-bridge cyclo-converter. .......... 30 Figure 2.11. A generic industrial three-phase ac–ac conversion system consisting of a full-bridge cyclo-converter used as an adjustable speed drive. .... 32 xix Figure 2.12. Steady-state source-side line ab voltage and phase a current when 5 Hzof   and 0.9M   as predicted by the proposed PAVM and detailed model of full-bridge cyclo-converter. .................................................................................... 33 Figure 2.13. Steady-state load-side phase a voltage and current when 5 Hzof   and 0.9M   as predicted by the proposed PAVM and detailed model of full-bridge cyclo-converter. ....................................................................................................... 33 Figure 2.14. Transient response of load-side phase a voltage and current when the converter modulation index is increased from 0.5M   to 0.9M   at 1t  s with 5 Hzof  as predicted by the subject models of full-bridge cyclo-converter. .......... 34 Figure 2.15. Transient response of load-side phase a voltage and current when the output frequency is increased from 5 Hzof   to 15 Hzof   at 2t  s with 0.9M   as predicted by the subject models of full-bridge cyclo-converter. ....................... 35 Figure 2.16. Simplified diagram of 3 3    matrix converter for variable frequency drive applications. ................................................................................. 37 Figure 2.17. Steady-state system variables when 50 Hzof   and 0.9M   as predicted by the proposed PAVM and detailed model of matrix converter for: (a) converter phase a input voltage, (b) source phase a current, (c) load phase a voltage, and (d) load phase a current. ................................................................... 42 Figure 2.18. Transient response of system variables to the increase of modulation index from 0.5M   to 0.9M   at 0.5t   s when the output frequency is set to 50 Hzof  , as predicted by the proposed PAVM and detailed model of matrix converter for: (a) converter phase a input voltage, (b) source phase a current, (c) load phase a voltage, and (d) load phase a current. ............................................. 43 Figure 2.19. Transient response of system variables to the increase of output frequency from 50 Hzof   to 400 Hzof   at 1t   s with 0.9M  , as predicted by the proposed PAVM and detailed model of matrix converter for: (a) load phase a current, (b) d-axis component of load current, (c) q-axis component of load current, and (d) source phase a current. ............................................................... 44 Figure 2.20. Power flow in the ac–ac energy conversion system of study. ........ 46 Figure 3.1. A generic machine-converter ac–dc power system composed of a six-phase PM synchronous generator connected to a 12-pulse diode rectifier. ... 52 Figure 3.2. The equivalent qd12 circuit of a six-phase PMSM. ....................... 57 xx Figure 3.3. Implementation of the subject models and their interfacing with an arbitrary ac network in abc12 coordinates: (a) qd12 model with indirect interfacing using current sources and snubber circuits; (b) VBR model with direct interfacing using voltage sources; and (c) the proposed CPVBR model with direct interfacing using voltage sources. ............................................................... 65 Figure 3.4. Parametric functions corresponding to dc and fundamental components of ac variables of the 12-pulse diode rectifier over a wide range of operating conditions: (a) ( )iw  , (b) 1( )vw   and (c) ( )  . .............................................. 70 Figure 3.5. Parametric functions depicting the magnitude and phase angle of various harmonics; respectively, in ac voltages of the 12-pulse rectifier over a wide range of operating conditions: (a) ( )nvw   and (b) ( )nv  ..................................... 70 Figure 3.6. Implementation of the proposed extended MRF-PAVM and its interfacing with the dc subsystem and ac subsystem in qd12 coordinates. ........ 72 Figure 3.7. Rectifier dc and ac variables in DCM as predicted by the subject models: (a) dc voltage, (b) dc current, (c) phase a1 voltage, and (d) phase a1 current.  .......................................................................................................... 75 Figure 3.8. The harmonic content of rectifier ac phase a1 voltage and phase a1 current as predicted by the subject models in DCM condition. ............................ 75 Figure 3.9. Rectifier dc and ac variables in CCM as predicted by the subject models: (a) dc voltage, (b) dc current, (c) phase a1 voltage, and (d) phase a1 current.  .......................................................................................................... 76 Figure 3.10. The harmonic content of rectifier ac phase a1 voltage and phase a1 current as predicted by the subject models in CCM condition. ............................ 76 Figure 3.11. Transient response due to load change from DCM to CCM as predicted by the subject models: (a) rectifier dc voltage, (b) rectifier dc current, (c) generator torque, and (d) generator phase a1 current. ................................... 78 Figure 4.1. PM synchronous machine-based wind generation system consisting of a thyristor-controlled-rectifier. ........................................................ 86 Figure 4.2. Implementation of PAVM of the thyristor-controlled rectifier. .... 87 Figure 4.3. The output voltage of thyristor-controlled rectifier versus frequency for different values of thyristor firing angles. ...................................... 88  xxi Figure 4.4. System transient response under variable frequency operation without and with thyristor control as predicted by the detailed model and PAVM of thyristor-controlled rectifier: (a) assumed frequency profile; (b) rectifier dc output voltage; and (c) thyristor firing angle. ....................................................... 89 Figure 4.5. Magnified view of the transient response of system variables as predicted by the detailed model and PAVM of the thyristor-controlled rectifier in open-loop operation for: (a) rectifier dc voltage, (b) rectifier dc current, (c) generator torque, and (d) line current. .................................................................. 89 Figure 4.6. Magnified view of the transient response of system variables as predicted by the detailed model and PAVM of the thyristor-controlled rectifier in closed-loop operation for: (a) rectifier dc voltage, (b) rectifier dc current, (c) generator torque, and (d) line current. .................................................................. 90 Figure 4.7. A generic variable speed ac–dc power system consisting of a line-commutated rectifier. ............................................................................................. 93 Figure 4.8. Phasor diagram of rectifier ac voltages and currents expressed in multiple qd rotating synchronous converter reference frames (denoted by subscript ‘e’) and source reference frame (denoted by subscript ‘r’): (a) fundamental components of ac variables; and (b) n-th harmonics. ..................... 95 Figure 4.9. Pseudo-code for establishing parametric functions of the proposed GPAVM.  ........................................................................................................ 101 Figure 4.10. Parametric functions of the MRF-PAVM [64] for diode rectifiers, PAVM [57] for thyristor-controlled rectifiers, and the proposed GPAVM: (a) dc current; and (b)-(c) magnitude and phase of fundamental components of ac variables, respectively. ......................................................................................... 103 Figure 4.11. Parametric functions of the MRF-PAVM [64] and the proposed GPAVM at 60 Hz versus dynamic impedance and firing angle for the magnitude and phase of 5th and 7th harmonics of ac voltages. ............................................. 104 Figure 4.12. Parametric functions of MRF-PAVM [64] and the proposed GPAVM (with α=0) versus dynamic impedance and line frequency for magnitude and phase of 5th and 7th harmonics of ac voltages. ............................................. 105 Figure 4.13. Implementation of the proposed GPAVM. ................................... 106   xxii Figure 4.14. Experimental setup of power system under study: (1) dc supply for the prime mover; (2) prime mover which emulates wind speed changes; (3) PM synchronous generator; (4) data acquisition system; (5) dc filter inductor; (6) dc filter capacitor and LCC; and (7) resistive load which represents a dc sub-system.  ........................................................................................................ 109 Figure 4.15. Diode rectifier dc- and ac-side variables as predicted by the subject models and experimental results over a range of frequencies from 10 to 90Hz: (a) output dc voltage; (b) magnitude of fundamental component of rectifier input ac voltage; (c) rotor angle; and (d)-(g) magnitude and phase of 5th and 7th harmonics of rectifier input ac voltage, respectively. ........................................................... 112 Figure 4.16. Diode rectifier variables as predicted by the subject models at 10 Hz and CCM-1 condition: (a) rectifier dc voltage (b) rectifier dc current, (c) rectifier phase a voltage, and (d) rectifier phase a current. ............................... 113 Figure 4.17. Transient response of thyristor-controlled rectifier from DCM to CCM-1 mode due to change in load resistance from 80 lR    to 4 lR    and thyristor firing angle from 67   to 15   at 60 Hz, as predicted by the subject models: (a) rectifier dc voltage, (b) rectifier dc current, (c) rectifier line voltage, and (d) rectifier phase a current. ......................................................................... 115 Figure 4.18. Thyristor-controlled rectifier ac variables in DCM mode with 80 lR    and 67   at 60 Hz, as predicted by the subject models: (a) rectifier line voltage, and (b) rectifier phase a current. ........................................................... 116 Figure 4.19. Thyristor-controlled rectifier ac variables in CCM-1 mode with 4 lR    and 15   at 60 Hz line frequency, as predicted by the subject models: (a) rectifier line voltage, (b) rectifier phase a current. ............................................. 116 Figure 4.20. Transient response of the system due to variable frequency operation and voltage control activation as predicted by the subject models: (a) line frequency, (b) output dc voltage, (c) rectifier dc current, and (d) rectifier firing angle. The voltage controller is activated at 7.5t  s to regulate the output voltage at 48V dc. ................................................................................................. 118 Figure 4.21. Rectifier ac variables at the moment of activating the voltage controller as predicted by the subject models: (a) rectifier line voltage, and (b) rectifier phase a current. ..................................................................................... 119 Figure 5.1. A generic three-phase ac–dc system consisting of a diode rectifier with faulty valves. ................................................................................................ 125  xxiii Figure 5.2. Phasor diagram of ac voltages and currents in multiple rotating qd reference frames in positive and negative sequences: (a) fundamental components of ac variables in positive sequence; and (b) n-th harmonics in positive sequence; (c) n-th harmonics in negative sequence. ............................. 130 Figure 5.3. Interfacing of the proposed extended PAVM of faulty diode rectifiers with ac- and dc-side subsystems. ......................................................... 138 Figure 5.4. Implementation of the proposed PAVM allowing faulty rectifier states.  ........................................................................................................ 139 Figure 5.5. Transient response of dc-side variables as predicted by the subject models with the LCR initially operating normally in CCM-1 mode, when at t=1s, diodes D2 and D3 fail and become open-circuited: (a) vdc and (b) idc. .................. 144 Figure 5.6. Transient response of ac-side terminal voltages as predicted by the subject models with the LCR initially operating normally in CCM-1 mode, when at t=1s, diodes D2 and D3 fail and become open-circuited: (a) vas , (b) vbs, and (c) vcs.  ........................................................................................................ 144 Figure 5.7. Transient response of ac-side phase currents as predicted by the subject models with the LCR initially operating normally in CCM-1 mode, when at t=1s, diodes D2 and D3 fail and become open-circuited: (a) ias , (b) ibs and (c) ics.   ........................................................................................................ 145 Figure 5.8. Measured and simulated dc-side variables as obtained from experimental setup and predicted by the subject models when the LCR operates with D2&D3 open-circuited: (a) vdc and (b) idc. ..................................................... 146 Figure 5.9. Measured and simulated ac-side terminal voltages as obtained from experimental setup and predicted by the subject models when the LCR operates with D2&D3 open-circuited: (a) vas , (b) vbs , and (c) vcs. ....................... 146 Figure 5.10. Measured and simulated ac-side phase currents as obtained from experimental setup and predicted by the subject models when the LCR operates with D2&D3 open-circuited: (a) ias , (b) ibs and (c) ics. .......................................... 147 Figure 5.11. The harmonic content of ac voltages and currents of LCR with D2&D3 open-circuited, as obtained from experimental setup and the subject models.  ........................................................................................................ 148 Figure 5.12. Transient response of LCR dc-side variables as predicted by the subject models when diode D2 fails to operate (becomes short-circuited) at t=1s with the LCR initially operating in normal condition CCM-1 mode with rl =32.92 Ω: (a) vdc and (b) idc. .............................................................................................. 151 xxiv Figure 5.13. Transient response of LCR ac-side voltages as predicted by the subject models when diode D2 fails to operate (becomes short-circuited) at t=1s with the LCR initially operating in normal condition CCM-1 mode with rl =32.92 Ω: (a) vas, (b) vbs, and (c) vcs. .................................................................................. 151 Figure 5.14. Transient response of LCR ac-side currents as predicted by the subject models when diode D2 fails to operate (becomes short-circuited) at t=1s with the LCR initially operating in normal condition CCM-1 mode with rl =32.92 Ω: (a) ias , (b) ibs and (c) ics. ................................................................................... 152 Figure 5.15. Typical three-phase ac–dc system consisting of a six-pulse thyristor-controlled rectifier with faulty switches. ............................................. 155 Figure 5.16. Transient of dc-side variables as predicted by the detailed model and the proposed extended PAVM of faulty thyristor-rectifiers: (a) idc , (b) vdc , (c) vout , and (d) firing angle α. .................................................................................. 159 Figure 5.17. Transient of ac currents as predicted by the detailed model and the proposed extended PAVM of faulty thyristor-rectifiers: (a) ias, (b) ibs, and (c) ics. ...   ........................................................................................................ 160 Figure 5.18. Magnified view of ac voltages as predicted by the detailed model and the proposed extended PAVM when the gate signal S5 of thyristor T5 is lost at t=0.5s: (a) vas, (b) vbs, and (c) vcs. ...................................................................... 161 Figure 5.19. Magnified view of ac currents as predicted by the detailed model and the proposed extended PAVM when the gate signal S5 of thyristor T5 is lost at t=0.5s: (a) ias, (b) ibs, and (c) ics. ........................................................................ 161 Figure 6.1. Schematic of a generic three-phase ac–dc power system consisting of a six-pulse line-commutated rectifier. ............................................................. 165 Figure 6.2. Transient response of the system variables as obtained by the subject models without artificial snubber circuits with simulation time-step of 0.1 µsec: (a) rectifier phase a current, (b) rectifier phase a voltage, (c) rectifier dc current, (d) rectifier dc voltage, and (e) load dc voltage. .................................... 167 Figure 6.3. Transient response of the system variables as obtained by the subject models without artificial snubber circuits with simulation time-step of 1 µsec: (a) rectifier phase a current, (b) rectifier phase a voltage, (c) rectifier dc current, (d) rectifier dc voltage, and (e) load dc voltage. .................................... 168 Figure 6.4. The considered six-pulse diode rectifier: (a) without snubbers, (b) with artificial snubber circuits. ........................................................................... 169 xxv Figure 6.5. Transient response of the system variables as obtained by the subject models when the artificial snubber circuits are used in the detailed model of diodes with simulation time-step of 50 µsec: (a) rectifier phase a current, (b) rectifier phase a voltage, (c) rectifier dc current, (d) rectifier dc voltage, and (e) load dc voltage. ........................................................................... 171 Figure 6.6. Transient response of the system variables as obtained by the subject models when the artificial snubber circuits are used in the detailed model of diodes with simulation time-step of 300 µsec: (a) rectifier phase a current, (b) rectifier phase a voltage, (c) rectifier dc current, and (d) rectifier dc voltage, and (e) load dc voltage. ........................................................................... 172 Figure 6.7. Typical waveforms of a six-pulse diode rectifier ac- and dc-side variables in CCM-1: (a) rectifier three-phase abc voltages, (b) rectifier three-phase abc currents, (c) rectifier dc voltage, (d) rectifier dc current, (e) rectifier transformed qd-axes voltages, (f) rectifier transformed qd-axes currents. ....... 174 Figure 6.8. A generic three-phase ac–dc power system consisting of a line-commutated rectifier. ........................................................................................... 176 Figure 6.9. Phasor diagram of fundamental components of rectifier ac voltages and currents expressed in synchronously rotating converter and source qd reference frames. .................................................................................................. 179 Figure 6.10. Typical waveform of a six-pulse diode rectifier d-axis current decomposed into average-value and oscillating components: (a) rectifier d-axis current, (b) average-value of rectifier d-axis current, (c) oscillating component of rectifier d-axis current, (d) reconstruction angle rec  used for mapping the angles of samples.   .................................................................................................. 182 Figure 6.11. Pseudo-code for establishing parametric functions of the proposed parametric hybrid continuous detailed/AVM model. .......................................... 187 Figure 6.12. Interfacing of the proposed hybrid parametric detailed/AVM model with the ac- and dc-side subsystems. .................................................................. 189 Figure 6.13. Implementation of the proposed hybrid parametric detailed/AVM model.   .................................................................................................. 190 Figure 6.14. The steady-state waveforms of ac- and dc-side variables of diode rectifier as obtained by the four considered models in DCM condition: (a) rectifier phase a voltage, (b) rectifier phase a current, (c) rectifier dc voltage, and (d) rectifier dc current. ......................................................................................... 194 xxvi Figure 6.15. The steady-state waveforms of ac- and dc-side variables of diode rectifier as obtained by the four considered models in CCM-1 condition: (a) rectifier phase a voltage, (b) rectifier phase a current, (c) rectifier dc voltage, and (d) rectifier dc current. ......................................................................................... 196 Figure 6.16. The harmonic spectrum of diode rectifier ac variables in DCM condition as obtained by the four subject models for: (a) ac phase voltage, (b) ac current.   .................................................................................................. 197 Figure 6.17. The harmonic spectrum of diode rectifier ac variables in CCM-1 condition as obtained by the four subject models for: (a) ac phase voltage, (b) ac current.   .................................................................................................. 197 Figure 6.18. Transient response of the ac- and dc-side variables of diode rectifier as obtained by the four considered models when the rectifier enters CCM-2 from CCM-1 at t=6 s for: (a) rectifier phase a voltage, (b) rectifier phase a current, (c) rectifier dc voltage, and (d) rectifier dc current. ................................................. 201 Figure 6.19. Transient response of the system variables due to dc load variation over a wide range of operating conditions as well as voltage control activation as predicted by the four considered models: (a) resistive load of dc network, (b) rectifier dc voltage, (c) rectifier dc current, and (d) rectifier firing angle. The voltage controller is activated at t = 7s to maintain the output voltage at 60 V dc.   ........................................................................................................ 203 Figure 6.20. The steady-state waveforms of ac-side variables of thyristor-controlled rectifier as obtained by the four considered models in CCM-2 condition: (a) rectifier phase a voltage, and (b) rectifier phase a current. ......... 205 Figure 6.21. The steady-state waveforms of ac-side variables of thyristor-controlled rectifier as obtained by the four considered models in CCM-1 condition: (a) rectifier phase a voltage, and (b) rectifier phase a current. ......... 205 Figure 6.22. The steady-state waveforms of ac-side variables of thyristor-controlled rectifier as obtained by the four considered models in DCM condition: (a) rectifier phase a voltage, and (b) rectifier phase a current. .......................... 206 Figure 6.23. Transient response of rectifier dc-side variables as obtained by the four subject models when the load of LCR is increased by stepping down its resistance from Rl=40Ω to Rl=1Ω at t=3s; diodes D2&D3 fail and become open-circuited: (a) rectifier dc voltage, and (b) rectifier dc current. ........................... 208   xxvii Figure 6.24. Transient response of rectifier ac-side voltages and currents as obtained by the four subject models when the load of LCR is increased by stepping down its resistance from Rl=40Ω to Rl=1Ω at t=3s; diodes D2&D3 fail and become open-circuited: (a)–(c) rectifier abc phase voltage, (d)–(f) rectifier abc phase currents.  .................................................................................................. 209 Figure 6.25. The steady-state waveforms of ac- and dc-side variables of diode rectifier in CCM-1 condition as obtained by the GPAVM and the two proposed hybrid models with 600μs time-step compared to the reference solution: (a) rectifier phase a voltage, (b) rectifier phase a current, (c) rectifier dc voltage, and (d) rectifier dc current. ......................................................................................... 212 Figure 6.26. The steady-state waveforms of ac- and dc-side variables of diode rectifier in CCM-1 condition as obtained by the conventional switching detailed and the proposed hybrid/detailed models run on Opal-RT OP5700 real-time simulator with 20μs time-step: (a) rectifier phase a voltage, (b) rectifier phase a current, (c) rectifier dc voltage, and (d) rectifier dc current. .............................. 214 Figure 6.27. The steady-state waveforms of ac- and dc-side variables of diode rectifier in CCM-1 condition as obtained by the conventional switching detailed and the proposed hybrid/detailed models run on Opal-RT OP5700 real-time simulator with 200μs time-step: (a) rectifier phase a voltage, (b) rectifier phase a current, (c) rectifier dc voltage, and (d) rectifier dc current. .............................. 215   xxviii List of Abbreviations Abbreviation Meaning   AAVM Analytical Average Value Model AC Alternating Current AVM Average Value Model CC Cyclo-Converter CCM Continuous Conduction Mode CCPD Coupled Circuit Phase Domain CPU Central Processing Unit CPVBR Constant Parameter Voltage Behind Reactance D-Axis Direct Axis DC Direct Current DCM Discontinuous Conduction Mode DER Distributed Energy Resources EMF Electro Motive Force EMTP Electro Magnetic Transient Program GPAVM Generalized Parametric Average Value Model HVDC High Voltage Direct Current IGBT Insulated Gate Bipolar Transistor LCC Line Commutated Converter LCR Line Commutated Rectifier MC Matrix Converter MMC Modular Multi-Level Converter MRF Multiple Reference Frame xxix MRF-PAVM Multiple Reference Frame Parametric Average Value Model ODE Ordinary Differential Equation PAVM Parametric Average Value Model PLL Phase Locked Loop PMSM Permanent Magnet Synchronous Machine PSS Power System Stabilizer PWM Pulse Width Modulated Q-Axis Quadrature Axis RL Resistive-Inductive RLC Resistive-Inductive-Capacitive RMS Root Mean Square SPS Sim Power System SVB State Variable Based VBR Voltage Behind Reactance VFD Variable Frequency Drive VSC  Voltage Source Converter        xxx Acknowledgements I would like to express my sincere gratitude and thank to my research supervisor Dr. Juri Jatskevich. I really appreciate his tremendous support, excellent vision, knowledge and inspiration, countless discussions, and valuable guidance in all stages of my Ph.D. I would also like to thank my co-supervisor Dr. Liwei Wang who helped me a lot with his valuable comments and supervisory feedback.  I am also grateful to my research committee members, Dr. Y. Christine Chen, Dr. Jose R. Marti, and Dr. William Dunford for their dedication and time, and all their valuable feedback and comments I received from each of them on my work. I would also thank my university examiners Dr. Hermann Dommel and Dr. Yusuf Altintas as well as my external examiner Dr. Udaya Annakkage for their constructive and interesting comments, feedback and discussions.   I should also thank Dr. Luis Linares from whom I learned many things when I was his Teaching Assistant.  I was also fortunate to receive funding from UBC for the Doctoral Four Year Fellowship (4YF) and the Faculty of Applied Science Graduate Award (GSI). I would also thank Dr. Juri Jatskevich for providing additional funding through xxxi the Natural Science and Engineering Research Council (NSERC) under the Strategic Project Grant entitled “Advanced Integrated AC-DC Systems for Energy Efficient Buildings and Communities in Canada,” and Discovery Grant entitled “Modeling and Analysis of Power Electronic and Energy Conversion Systems,” and other grants.  I would also like to thank my friends and colleagues in Electric Power and Energy Systems research group at UBC who helped me a lot and made my Ph.D. program memorable and enjoyable: Navid Amiri, Hamid Atighechi, Sina Chiniforoosh, Yingwei Huang, Mehrdad Chapariha, Saeed Rezaei, Ali Saket, Francis Therrien, and so many others who are not in this short list.    As the last but not least, I would like to thank all my family members for their support and encouragement. My greatest gratitude goes to my father, my mother and my wife for their continuous care and support, encouragement and unconditional and endless love over thousands of miles away. I also thank my three brothers (Masoud, Meysam, Yaser) for their belief in my success and their moral support in all these years.   xxxii Dedication My Ph.D. dissertation is dedicated to my parents, Seyyed Jalil and Fatemeh, who are the symbols of true love, passion and support in my life, and to my beloved wife, Sepideh, whose love has given a new meaning to my life…   1 CHAPTER 1: INTRODUCTION 1.1 Motivation Ac-dc and ac-ac power electronic converters are extensively used in various medium to high power industrial applications. Depending on the type of semiconductor switches, these converters may be generally classified into Pulse-Width-Modulated (PWM) converters which use power transistors such as Insulated-Gate Bipolar Transistors (IGBTs), or Line-Commutated Converters (LCCs) which utilize diodes or thyristors. The LCCs are widely used in various applications due to their high reliability and efficiency, low cost, and relative simplicity. Particularly, ac–dc LCCs (i.e., diode and thyristor-controlled rectifiers) are extensively utilized in a broad range of applications such as synchronous generator exciters [1], classic high-voltage direct-current (HVDC) transmission systems [2]–[3], vehicular power systems [4], distributed energy resources (DERs) [5],  welding and induction furnaces [6]–[7], front-end rectifier loads, etc. In addition, the LCC rectifiers are very common in machine-converter systems such as marine [8] and aircraft power systems [9]–[11], brushless excitation systems of generators [12]–[13], variable-speed wind turbines [3] 2 where the prime mover speed changes during operation and due to transients, etc. Traditionally, the ac–ac conversion requires two-stage cascaded rectifier-inverter systems with intermediate dc-links to output sinusoidal voltages with adjustable frequency/amplitude from the fixed frequency/amplitude grid voltage. The ac–ac LCCs such as Cyclo-Converters (CCs) [14]–[16] have become popular due to their single-stage energy conversion and are commonly used in very-high power industrial Variable Frequency Drive (VFD) applications. Therein, the motor speed is controlled by adjusting the frequency/amplitude of the output voltage; for example, ship propulsion drives, reversible rolling mill drives, cement tube mill drives, etc. [14]–[16]. The single-stage conversion of ac–ac converters eliminates the need for traditional bulky dc-link capacitors for energy storage (except small ac filters), which reduces cost and size of the system and improves the system transient response. Although the LCCs are very reliable due to their relative simplicity with low cost and high efficiency, they are also significant sources of harmonics into the power systems [15]–[16], which can significantly influence the operation of power system equipment [17] (e.g., transformers, generators, protection systems [18] and relays [19], harmonic compensation measures [20]–[21], etc.). The PWM Matrix Converters (MCs) have also become increasingly popular for ac–ac conversion where a broad range of output frequencies is desirable with low input harmonics [22]–[23]. 3 Design, study and analysis of modern electrical energy systems typically involve many simulations and computer studies. Therefore, with the increasing integration of power-electronic-based systems to electrical energy systems, access to fast and numerically efficient models of power-electronic converters is essential for conducting simulations of large power systems in a reasonable amount of time and computational resources, which is useful for many researchers and practicing engineers around the world who work in the area of electric power and energy systems. The traditional detailed switch-level models of power systems with LCCs can be readily established using the standard library components of any one of commercially available electromagnetic transient (EMT) time-domain simulation tools including either state-variable-based (SVB) programs [24]–[30] such as MATLAB/Simulink [27], Simscape Electrical (SimPowerSystem) [28] and PLECS [29], or the nodal-analysis-based electromagnetic transient programs (EMTP) [31]–[35] such as MicroTran [31], PSCAD/EMTDC [33], EMTP-RV [34]. In detailed models, the switching of all elements is represented in detail where the commutation of each semiconductor device is considered, which provides highly accurate simulations. However, such models typically require significant computational resources for predicting switching times and commutations, and result in long simulation times due to the inherently repeated discrete switching states. Such detailed models become more prohibitive and/or limiting especially in system-level studies when there are a 4 high number of power-electronic-based switching subsystems with many components and interconnection and/or when the simulations are required to run multiple times for designing and optimizing system/controller parameters [36]–[37]. Therefore, application of computationally expensive detailed models of switching converters limits the system size and the number of power electronic devices that can be practically included in power systems for conducting the simulations in a reasonable amount of time. This becomes more crucial for the future modern electrical energy grids with high penetration of power-electronic-based renewable generation and electronic loads compared with traditional power systems. To conquer these challenges, as an alternative for system-level studies where the details of each individual switching can be neglected, the so-called dynamic average-value models (AVMs) have been developed for ac–dc [10]–[12], [38]–[47], [49]–[64] and dc–dc [65]–[69] converters to alleviate the computational burden on simulators. In AVMs, the switching is eliminated and the detailed system variables (currents and/or voltages) are approximated by their corresponding dynamic average values defined over a switching period [38, Chap. 11] where only the lower frequency dynamics of the system are captured, which alleviates the burden associated with handling the discrete switching events. Therefore, the AVMs are computationally very efficient and typically execute orders of magnitude faster than their detailed model counterparts [63]. In addition, despite the detailed models which are discrete and discontinuous, the AVMs are 5 continuous and can be linearized about an operating point of interest for small-signal frequency-domain analysis, which is an important tool for assessing the stability of the systems [57]. A review of average-value-modeling methods and their applications in power system transients can be found in [49], [63].   1.2 Literature Review Depending on the methodology used to derive the average-value relationships among the variables, the AVMs can be classified as analytical AVMs (AAVMs) [11]–[12], [38]–[47], [65]–[68] and parametric AVMs (PAVMs) [10], [50]–[64], [69]. In AAVMs, the key relationships between the averaged ac and dc variables are derived analytically considering specific operating mode – the switching sequence pattern [38, Ch. 11]. The analytical derivation of AAVM for machine/converter systems has been developed in [39]–[41]. However, such models are inaccurate for small-signal analysis in high frequencies. AAVMs which are accurate in both time and frequency domains have been derived in [38, Ch. 11], [42] for thyristor-controlled rectifiers and later applied in [43] for an inductorless machine/converter system. At the next level of advancement, the AAVM methodology of the synchronous machine/rectifier system introduced in [38, Ch. 11], [41]–[42] has been extended to reconstruct the ac-side harmonics [44]–[45]. However, the models in [38, Ch. 11], [41]–[45] consider only the first 6 continuous conduction mode (CCM-1) [48]–[49], while the LCCs can operate in different modes [48]–[49] [i.e. one discontinuous conduction mode (DCM) and three continuous conduction modes (CCMs)]. Significant work is required to derive the AAVMs that are valid in all operating modes (i.e., the analytical expressions should be separately derived for each mode of operation), and it becomes very cumbersome to find all the boundary conditions between operating modes analytically [46], [48]. As a result, such models are developed in [12], [47] for diode rectifier systems only, without considering ac harmonics. Also, the AAVMs typically consider only the LCCs with ideal ON/OFF switches, since it becomes impractical to establish closed-form analytical expressions considering the conduction and switching losses of all semi-conductor devices. Therefore, the typical AAVMs do not capture the converter losses; thus, are not suitable for system-level studies where it is necessary to predict the converter efficiency. Also, the analytical derivation of AAVMs becomes cumbersome for the systems with multiple operating modes where the analytical methods become impractical as the size and complexity of the system increases. Due to these limitations, the computerized parametric AVM (PAVM) methodology has been developed to facilitate the construction of AVMs of diode-rectifier systems [10], [50]–[64], wherein the ac- and dc-side variables are related by explicit algebraic parametric functions which are obtained numerically from the detailed simulation. The PAVM avoids complicated analytical derivations which makes this approach particularly suitable for 7 industrial applications with complex systems containing non-linear components with multiple converter circuits that may include non-idealities/parasitics/losses, for which it would be very challenging (if not impossible) to derive accurate AVMs analytically. The PAVMs have been first introduced in [50]–[51] for diode rectifier systems; however, the parameters of PAVMs in [50]–[51] are constant and independent of loading condition which causes error in both steady-state and dynamic responses of the models.  A full-order PAVM of a synchronous machine diode rectifier system has been introduced in [52], where the numerically calculated parametric functions are dependent on the loading conditions. The model in [52] has proven to be accurate in both time and frequency domains and over a wide range of operating conditions (i.e., in all CCM and DCM operating modes), as well as for variable frequency operation as verified in [53]. A fast procedure has been introduced in [54] for constructing the PAVM parameters based on [52]. The parametric technique has been extended to six-phase line-commutated rectifiers in [55]. The PAVM methodology presented in [50] has also been extended to model the thyristor-controlled rectifier systems in [56], which is valid only for a narrow range of operating conditions.  Subsequently, a full order PAVM of a synchronous-machine-fed thyristor-controlled-rectifier system has been developed in [57] based on [52] and extended to classic LCC HVDC systems in [58]. The PAVMs [52]–[55], [57]–[63] have been shown to be fast and accurate for a wide range of operating condition; however, these models only preserve the 8 fundamental components (i.e., 50/60Hz) of ac variables (i.e., voltages and currents).  Recently, the PAVM methodology has been extended in [64] to include several significant ac-side harmonics for six-pulse diode rectifiers using the multiple-reference-frame (MRF) theory [70], namely the MRF-PAVM [64]. The MRF-PAVM [64] has been shown to accurately reconstruct the ac harmonics of diode rectifiers in steady-state and transients but only at the fixed nominal frequency. The analytical and parametric AVMs of ac–dc rectifiers [10]–[12], [38]–[47], [49]–[64] are typically formulated in qd coordinates and are developed in state-space form, which are compatible and straightforward to implement in SVB simulation programs. To implement AVMs in EMTP-type programs, an indirect interfacing approach has been presented in [49], [59], which uses one time-step delay (approximation/relaxation) in simulation for interfacing [71]–[73]. Such indirect interfacing approach generally requires small integration time-steps to produce stable and sufficiently accurate solutions due to the error introduced by the time-step delay, which is prohibitive in system-level studies where simulations with large time-steps are desirable. To avoid the time-step delay, a direct interfacing technique has been proposed for the AAVMs in [74] (based on AAVM [38, Ch. 11]) which has proven to achieve accurate and stable solutions even with fairly large time-steps (up to 1ms).  9 1.3 Research Objectives The PWM converters use high-frequency switching and generally operate in continuous current mode (CCM), which significantly simplifies their average-value modeling [38, Ch. 11], [63]. For the case of LCCs, it is much more challenging to derive the AVMs due to the nonlinear nature of the commutation process. At the same time, the AVMs of LLCs are very desirable for simulations of many practical systems since these type of converters are very common in high-power applications. Moreover, the AVMs of LCCs are also desirable for modeling the PWM converters (e.g., VSCs, MMCs, etc.) in blocking mode of operation, where the VSC becomes equivalent to a diode bridge. Therefore, the research in this thesis is mainly focused on the line-commutated converters.  To better illustrate the proposed research objectives and their relationship to the existing state-of-the-art average-value models, the classification of high-power converters is depicted in Figure 1.1. Specifically, the existing models of ac–dc LCCs do not cover special operating conditions (e.g., variable frequency, unbalance operation, failure of one or more switches, etc.) which are very desirable for the system-level studies. Moreover, there are no accurate AVMs for the high-power ac–ac converters that are commonly used in industrial applications. As it is commonly assumed in the literature [38, Ch. 11], for the purpose of research in this dissertation, the ac-side subsystem that feeds the LCC is assumed to be balanced and symmetric. This assumption is considered  10                     Figure 1.1. Classification of high-power converters considered for the proposed and future research. acceptable for most practical systems that are designed to be symmetric and operate in conditions that are close to balanced. Considering the unbalanced ac-side subsystems will make the problem of deriving the AVMs extremely challenging due to its complicated impact on the commutation pattern of the DC/AC  PWM Converters Line-Commutated Converters AC/AC  AC/DC  AC/AC  EMTP Models SVB Models Diode Bridges Variable Frequency Including Harmonics Internal Faults Thyristor Bridges Matrix Converters Modular Multilevel Converters Thyristor-Controlled Cyclo Converters Continuous Hybrid Modeling Discretizing PAVM with Harmonics Research in This Thesis Faults Prior Art Suggested Future Work Color Legends: Average-Value Modeling of High Power Electronic Converters 11 switches, and it is left for the future research. Therefore, the research proposed in this dissertation aims at filling the gaps in existing models for the ac–dc and ac–ac converters by pursuing the following objectives: 1.3.1 Objective One: Parametric average-value modeling of ac–ac converters So far, the parametric average-value modeling technique has been developed and studied for dc–dc and ac–dc systems. As part of this research, the PAVM will be extended to ac–ac converters, including line-commutated thyristor-based cyclo-converters (CCs) used in frequency converters; and matrix converters (MCs) that enable direct ac–ac conversion using controllable bi-directional blocking switches. These converters have not been considered in the prior literature since analytical derivation of their AVMs would be extremely complicated. 1.3.2 Objective Two: Generalizing PAVM methodology for thyristor-controlled LCCs considering ac harmonics with variable frequency operation In this objective, the parametric average-value modeling of line-commutated rectifiers will be generalized by considering the following conditions that have not been included or considered in the prior literature: 12 The previously formulated MRF-PAVM of diode rectifiers assumes constant ac-side frequency which is not the case for many applications where the prime mover speed may change in a wide range. Therefore, a generalized formulation will be presented to consider the frequency dependency of ac-side harmonics by appropriately extending the numerically constructed parametric functions.  Next, the variable frequency MRF-PAVM will be extended to thyristor-controlled rectifiers. This will require further extending the parametric functions to include the magnitude and phase of ac harmonics as a function of rectifier dynamic impedance, thyristor firing angle, and frequency. 1.3.3 Objective Three: Developing PAVM methodology for LCCs under asymmetrical operation with internal faults So far, parametric average-value modeling has been developed assuming symmetric operation of rectifiers. Asymmetrical operation (e.g., when a fault of one or more switches occurs) results in non-characteristic harmonics (i.e. 2nd, 4th, …) [75]–[76] in both positive and negative sequences, as well as presence of dc components in ac-side variables. As part of this research, the PAVM methodology will be extended to asymmetric line-commutated rectifiers (both diode bridges and thyristor-controlled rectifiers) with internal faults which can be beneficial in many system-level studies that include analysis of faults in rectifiers for designing fault detection algorithms, compensation methods, etc. 13 This is particularly applicable for studies with classic HVDC systems that are still the dominant technology. 1.3.4 Objective Four: Developing versatile continuous detailed models of LCRs using the parametric approach The switching detailed models of converters provide accurate information of all discrete switching events and harmonics. However, to determine the switching events accurately, the simulation program needs zero-crossing detection algorithms and/or interpolation for exact locating of the switching events, which increases the computational cost of the detailed models and/or forces to use small time-steps. Many commercial simulation programs also require to use artificial snubber circuits with the switches, which also increases the size and complexity of the model. However, the PAVMs are continuous (do not require zero-crossing) and can run orders of magnitude faster than their detailed model counterparts by neglecting the switching ripples of converters. In this objective, based on the parametric modeling methodology, a technique will be presented to achieve continuous-detailed models of LCRs which can reconstruct the switching ripples without the need for zero-crossing and discrete states. Also, the formulation of the model will be versatile so that it can be easily converted to an AVM by setting some parameters equal to zero. This technique will allow the continuous-detailed model to run with much larger time-steps 14 which can be beneficial in system-level studies of large power systems consisting of many power-electronic converters.  1.4 Platforms for Verifications For the purpose of conducting steady-state and transient studies and benchmarking the proposed models compared to the existing models, in this dissertation, various simulation software programs are used as needed in each case. The considered programs include PSCAD/EMTDC [33], MATLAB/Simulink [27], Simscape Electrical (SimPowerSystems) toolbox [28], PLECS toolbox [29], and RT-Lab [106] of Opal-RT real-time simulator.  These programs are widely used by engineers and researchers in industry and academia for simulations of power systems and are proven to provide sufficient accuracy for detailed modeling of switching converters and power-electronic-based systems. Therefore, the detailed models implemented in these programs are suitable as references for benchmarking the new models proposed in this dissertation.  Also, for the purpose of consistency, the simulation studies are conducted in the same simulation and computational environment when the subject models are compared for their numerical and computational efficiency.    15 CHAPTER 2: PARAMETRIC AVERAGE-VALUE MODELING OF AC–AC CONVERTERS So far, the PAVM technique has been developed for dc–dc and ac–dc converter systems only. In this Chapter, the PAVM method is extended to three-phase ac–ac converter systems as shown in Figure 2.1. These ac–ac conversion systems supply the load at lower/higher ac frequencies with adjustable voltage amplitude that are generally used for motor speed control by providing variable frequency output, for example, ship propulsion drives, reversible rolling mill drives, cement tube mill drives, etc. [14]. The performance of the presented PAVM model of three-phase ac–ac converters is then evaluated on thyristor-controlled ac–ac cyclo-converter systems as well as IGBT-based ac–ac matrix converters and compared to their detailed model counterparts in terms of accuracy, simulation speed and numerical efficiency. AC Source-Side Power SystemAC-ACConverterabcsiabcsvAC Load-Side Power Systemabcliabclv Figure 2.1. Schematic of a three-phase ac–ac conversion system consisting of a three-phase ac–ac converter connected to the source/load-side ac subsystems. 16 2.1 Parametric Average-Value Modeling of Three-Phase AC–AC Converters In the presented PAVM for ac–ac converters, the fundamental frequency components of ac variables (i.e., voltages and currents) at the input terminals (source-side) of the converter are related to the fundamental frequency components of the ac variables at the output terminals (load-side) of the converter. To capture these relationships, the converter source-side ac variables in abc coordinates (i.e., abcsv  and abcsi  in Figure. 2.1) are transformed to the synchronously rotating (with the source-side ac system frequency) converter reference frame, denoted by eqd , as depicted in Figure. 2.2(a). It is also assumed that zero-sequence does not flow through the converter. Here, the subscript “e” is used to denote the converter reference frame, wherein the angular displacement of transformation is chosen to be e , which aligns the eq -axis with 1asv  [52]. Also, 1asv  is the fundamental frequency component of converter source-side phase a voltage and e  is the angular displacement of converter source-side terminal voltages defined as   ,  2e e e idt f     , (2.1) whereif  and e  are the line frequency of the source-side ac electrical system in Hz and rad/s. The relationship between e  and the angle of source-side Thévenin equivalent voltages s  can also be expressed as  s e s    , (2.2) 17 1 , eas qsv vaxiseq axissq es axissd axised s1asissdsisqsi is(a)edsieqsisqsvsdsvoqlvaxisoq oaxisod l1aliil(b)odlioqli1alvodlvval Figure 2.2. Phasor diagram of ac–ac converter voltages and currents expressed in qd rotating synchronous reference frames: (a) fundamental components of source-side ac variables, and (b) fundamental components of load-side ac variables. where s  is the angle by which the Thévenin equivalent source voltages lead the fundamental frequency component of the converter input terminal voltages. Similarly, the converter load-side ac variables in abc coordinates (i.e., abclv  and abcli  in Figure 2.1) are transformed to a reference frame that is synchronously rotating (with the load-side ac system frequency), denoted by oqd , as depicted in Figure. 2.2(b). Here, the subscript “o” is used to denote the load-side (output) reference frame, wherein the angular displacement of transformation is chosen to be o  defined as 18   ,  2o o o odt f     , (2.3) where of  and o  are the line frequency of the load-side ac electrical system in Hz and rad/s. The transformations of source-side ac voltages and currents are carried out using the Park’s transformation matrix sK [70] as  ( )  , ( )e eqds s e abcs qds s e abcs  v K v i K i , (2.4)  ( )  , ( )s sqds s s abcs qds s s abcs  v K v i K i . (2.5) Similarly, the transformations of load-side ac voltages and currents are carried out as  ( )  , ( )o oqdl s o abcl qdl s o abcl  v K v i K i . (2.6) After transformations (2.4) and (2.5), the terms associated with fundamental components of source-side voltages and currents (i.e., 1asv  and 1asi ) will become dc components in eqd  and sqd  coordinates, while all other harmonics become oscillatory/ripples with zero average values. Similarly, by transformations in (2.6), the terms associated with fundamental components of load-side voltages and currents (i.e., 1alv  and 1ali ) will become dc values in oqd  coordinates, while all other harmonics become oscillatory/ripples with zero average values.  For the purpose of deriving the average-value model, the oscillatory components on transformed variables (that are resulted from harmonic components associated with the switching phenomenon) are eliminated from the transformed source-19 side and load-side ac variables by the means of fast dynamic averaging over a switching period as      1 ,     1txt Tx t x d T fT   , (2.7) where the variable x may denote currents and voltages. Here, xf  is equal to if  for source-side variables and is equal to of  for load-side variables. The bar sign (–) above denotes the average value. To relate the magnitude of fundamental components of source-side and load-side ac voltages, parametric function ( )vw   is defined as  ( )oqdlvsqdsw  vv. (2.8) Also, to relate the magnitude of fundamental components of source-side and load-side ac currents, parametric function ( )iw   is defined as  ( )sqdsioqdlw  ii. (2.9) Moreover, the relationships between the angles of fundamental components of source-side ac variables can be defined as parametric function,   1 1( ) tan tans sds dss s sqs qsv iv i                , (2.10) where   1tansdsis sqsii     , (2.11) 20 and is  is the angle of source-side 1asi  current in the source reference frame. Also, the phase angle of output load-side voltages can be captured using a parametric function defined as  1( ) tanodlvl oqlvv      , (2.12) where vl  is the angle of load-side 1alv  voltage in the load-side output reference frame. The parametric functions (2.8)–(2.12) are obtained numerically by briefly simulating the detailed model of the power system of Figure 2.1 over the desired range of system conditions. Thereafter, (2.8)–(2.12) are computed and stored in look-up tables in terms of output frequency of  and the converter modulation index M (as two varying inputs to the system) as well as dynamic admittances of the overall converter switching cell denoted by 1dy  and 2dy , which are constructed using the ac–ac converter input voltages and output currents defined as  1odldsqdsiy v,    2oqldsqdsiy v. (2.13) For the implementation of the ac–ac converter PAVM, the switching converter is replaced with an algebraic block which interfaces with source- and load-side ac subsystems using controlled current and voltage sources, as demonstrated in Figure 2.3. These dependent sources instantaneously relate the ac source-side variables to the ac load-side variables. Here, the ac–ac converter source-side  21 AC Source-Side Power SystemAC-AC Converter PAVMabc-qd0Transformationabc-qd0TransformationsdqsisdqsvodqliodqlvabcliabclvabcsiabcsvofofifMAC Load-Side Power System Figure 2.3. Implementation of the three-phase ac–ac converter PAVM and its input-output interfacing with the source/load-side ac subsystems. terminal voltages abcsv (or sqdsv ) and its load-side currents abcli  (or oqdli ), as well as the output frequency and modulation index of the converter are considered as the inputs to the PAVM. Also, the converter source-side currents abcsi  (or sqdsi ) and its load-side terminal voltages abclv (or oqdlv ) are selected as outputs. The inputs of the PAVM model (i.e., sqdsv  and oqdli ) are used in (2.13) to compute the dynamic admittances. Then, 1dy  and 2dy  are used in conjunction with the two other inputs of the PAVM (i.e., output frequency of  and converter modulation index M) to compute the parametric function (2.8)–(2.12). Thereafter, using the computed parametric functions and input variables of the PAVM (i.e., sqdsv  and oqdli ), the output variables of the PAVM (i.e., oqdlv  and sqdsi ) are computed based on (2.8), (2.12) as    ( ) cos ( )( ) sin ( )o sql v qds vlo sdl v qds vlv wv w     vv, (2.14)  22 and based on (2.9)–(2.11) as     ( ) cos( ) sins oqs i qdl iss ods i qdl isi wi w   ii, (2.15) where is  is computed as  1tan ( )sdsis ssqsvv       . (2.16) After the output variables of the PAVM (i.e., oqdlv  and sqdsi ) are computed in (2.14) and (2.15), they can be transformed into their corresponding abc coordinates using the inverse of transformation in (2.4)–(2.6) to obtain abclv  and abcsi  which are interfaced with the load-side and source-side ac subsystems using dependent three-phase voltage and current sources, respectively as illustrated in Figure 2.3. 2.2 AC–AC Cyclo-Converters The thyristor-controlled cyclo-converters are broadly utilized in many medium and high power industrial applications for ac–ac conversion due to their high efficiency and reliability, and at the same time relative simplicity and low cost compared to the transistor-based converters. A cyclo-converter basically consists of one positive and one negative thyristor-controlled bridge. Schematic of half-bridge and full-bridge single-phase cyclo-converters are shown in Figures 2.4(a) and 2.4(b), respectively. Both positive and negative converters can generate 23 voltages at either positive or negative polarities; although, the positive converter can only supply positive current and the negative converter can only supply negative current. Hence, cyclo-converters can operate in all four quadrants; i.e. first and third quadrants of rectification mode  0vi  and, second and fourth quadrants of inversion mode  0vi  . This topology can also be extended to three-phase converters by using one single-phase cyclo-converter converter for each phase of the three-phase load. The schematic of half-bridge and full-bridge three-phase cyclo-converters are depicted in Figures. 2.5(a) and 2.5(b), respectively.  LoadNegative ConverterPositive ConverterABCNLoadNegative ConverterPositive ConverterABC(a)(b)  Figure 2.4. Schematic of a single-phase cyclo-converter: (a) half-bridge, and (b) full-bridge. 24 Three-Phase LoadAl Bl ClThree-Phase LoadAsBsCsAl Bl ClAsBsCs(a)(b) Figure 2.5. Schematic of a three-phase cyclo-converter: (a) half-bridge, and (b) full-bridge. Generally, the firing angles of both converters are modulated sinusoidally to produce harmonically optimum output voltages. Cyclo-converters are basically controlled in two different modes of operation as explained below. 2.2.1 Blocking mode of operation In this mode of operation, when the load current is positive, positive converter supplies the load voltage and negative converter is disabled. However, when the load current is negative, the positive converter is disabled and load voltage is 25 produced by the negative converter. In practice, a dead-zone is considered at zero-crossings of load current in order to prevent both positive and negative converters conducting and short-circuiting the source. Therefore, the control system will become complex and current dead-zones result in harmonically rich output voltage. However, no bulky inductances are required to limit the short circuit currents. Moreover, higher efficiency is obtained since only one converter operates at a time. 2.2.2 Circulating current mode of operation In this mode of operation, both converters are controlled to produce equal average voltages at all time; therefore:  P N    , (2.17) where P  is the positive converter firing angle and N  is the negative converter firing angle. Since both converters conduct all the time, the output voltage has less harmonics compared to the blocking mode operation due to the continuity of output current. However, due to differences between instantaneous generated voltages of the two converters, high short-circuit circulating currents flow through the converter. In order to limit the circulating currents, bulky inductances are utilized at the outputs of the converter which contribute to size and cost of the cyclo-converter. In addition, since both positive and negative converters operate all the time, efficiency is less than the blocking mode operation. This makes circulating mode operation control less attractive 26 compared to blocking mode operation control. Therefore, only the blocking mode of operation is considered for cyclo-converters in this section. 2.2.3 Verification of the proposed ac–ac PAVM for the three-phase half-bridge cyclo-converters To verify the performance of the proposed ac–ac PAVM, a three-phase ac–ac conversion system is considered, as depicted in Figure 2.6 with parameters summarized in Appendix A. Therein, a 60 Hz three-phase source is feeding a three-phase RL-load (which may represent an inductive motor load) through a half-bridge cyclo-converter with adjustable voltage amplitude and variable frequency in the range of 3–25 Hz. The detailed model of power system has been implemented in MATLAB/Simulink [27] using standard library components of Simscape Electrical (SimPowerSystems) toolbox [28] for the thyristor-controlled converter.  , as asi vthR thL , a ai vcsebseaseLoad lineVThree-Phase Load Figure 2.6. A generic industrial three-phase ac–ac conversion system consisting of a half-bridge cyclo-converter used as adjustable speed drive.   27 2.2.3.1 Steady-state operation Here, the output load-side frequency is chosen to be 10 Hz and the modulation index of the converter is set to 0.5, and the system is in steady-state condition. The ac source-side phase a voltage and current are shown in Figure 2.7 as predicted by the detailed model and proposed PAVM. Also, the phase a of ac load-side variables are depicted in Figure 2.8.   0.5 0.55 0.6 0.65-6000-3000030006000  Time (s)v as (V)Detailed ModelPAVM 0.5 0.55 0.6 0.65-1000-50005001000Time (s)i as (A)  Detailed ModelDetailed Model-Fundamental ComponentPAVM Figure 2.7. Steady-state source-side phase a voltage and current when 10 Hzof   and 0.5M   as predicted by the proposed PAVM and detailed model of half-bridge cyclo-converter. 28 0.5 0.6 0.7 0.8-6000-3000030006000Time (s)Load- v a (V)  Detailed ModelPAVM0.5 0.6 0.7 0.8-1500-1000-50005001000150Time (s)Load- i a (A)  Detailed ModelPAVM Figure 2.8. Steady-state load-side phase a voltage and current when 10 Hzof   and 0.5M   as predicted by the proposed PAVM and detailed model of half-bridge cyclo-converter.  As it can be observed in Figures 2.7 and 2.8, the presented PAVM of ac–ac converters is able to predict the fundamental components of ac source- and load-side variables accurately in steady-state compared to the detailed model of half-bridge cyclo-converter. 2.2.3.2 Transient operation Here, the system is initially operating in steady-state condition with output frequency set to 20 Hz and the converter modulation index to 0.5. At t=2 s the converter modulation index is stepped up to 0.9. The resulting transients are 29 shown in Figure 2.9 for load-side variables. Then at t=2.5 s the output frequency set-point is decreased from 20 Hz to 5 Hz. The resulting transient is shown in Figure 2.10 for load-side variables. As it can be observed in Figures 2.9 and 2.10, the presented PAVM of ac–ac converters is able to predict the transient response of fundamental components of system variables accurately compared to the detailed model of half-bridge cyclo-converter.  1.9 2 2.1-6000-3000030006000Time (s)Load- v a (V)  Detailed ModelPAVM1.9 2 2.1-1500-1000-500050010001500Time (s)Load- i a (A)  Detailed ModelPAVM Figure 2.9. Transient response of load-side phase a voltage and current when the converter modulation index is increased from 0.5M  to 0.9M   at 2t  s with 20 Hzof  as predicted by the subject models of half-bridge cyclo-converter. 30 2.4 2.5 2.6 2.8 3-6000-3000030006000Time (s)Load- v a (V)  Detailed ModelPAVM2.4 2.5 2.6 2.8 3-1500-1000-500050010001500Time (s)Load- i a (A)  Detailed ModelPAVM Figure 2.10. Transient response of load-side phase a voltage and current when the output frequency is decreased from 20 Hzof   to 5 Hzof   at 2.5t  s with 0.9M   as predicted by the subject models of half-bridge cyclo-converter. 2.2.3.3 Computational performance For the purpose of benchmarking the presented PAVM of ac–ac converters for half-bridge cyclo-converters, the 3.5-second transient study presented in section 2.2.3.2 is conducted using the MATLAB solver ode23tb* with maximum time-step of 0.5×10-3s and relative and absolute tolerances set to 0.5×10-3. The simulations are executed on a PC with Intel® Core™ i7-4510U @ 2.00GHz processor.  The computational performance of the proposed PAVM and detailed   *ode23tb is a type of solver used in MATLAB to numerically solve stiff differential equations. This implicit solver is an implementation of trapezoidal rule + backward differentiation formula TR-BDF2 [27], [108], [109].  31 model of half-bridge cyclo-converter is summarized in Table 2.1. As it can be seen from Table 2.1, a significant improvement in the number of time-steps (7,482 vs. 364,832) and speed (0.62s  vs. 85.43s)  is achieved when the detailed model of half-bridge cyclo-converter is replaced with its proposed PAVM.  Table 2.1. Computational performance of proposed PAVM and detailed model of half-bridge cyclo-converter for a 3.5-second transient study System Model Type Number of Steps CPU Time (s) Detailed 364,832 85.43 PAVM 7,482 0.62  2.2.4 Verification of the proposed ac–ac PAVM for the three-phase full-bridge cyclo-converters To verify the performance of the proposed ac–ac PAVM for three-phase full-bridge cyclo-converters, the power system of Figure 2.11 is considered with parameters summarized in Appendix B. Therein, a 60 Hz three-phase source is feeding a three-phase RL-load (which may represent an inductive motor load) through a full-bridge cyclo-converter with adjustable voltage amplitude and variable frequency in the range of 3–25 Hz. The detailed model of power system has been implemented with standard library components for thyristor-controlled converter in MATLAB/Simulink [27] using Simscape Electrical (SimPowerSystems) toolbox [28]. 32 2.2.4.1 Steady-state operation Here, the output load-side frequency is selected to be 5 Hz and the modulation index of the converter is set to 0.9, and the system is in steady-state condition. The ac source-side line voltage and current are shown in Figure 2.12 as predicted by the detailed model and proposed PAVM of full-bridge cyclo-converter. Also, the ac load-side phase a variables are depicted in Figure 2.13. As it can be observed in Figures 2.12 and 2.13, the presented PAVM of ac–ac converters is able to predict the fundamental components of ac source- and load-side variables accurately in steady-state compared to the detailed model of full-bridge cyclo-converter.  Three-Phase Loadasi , a ai vcsebseaseLoad sR sLlineVabsv Figure 2.11. A generic industrial three-phase ac–ac conversion system consisting of a full-bridge cyclo-converter used as an adjustable speed drive. 33 -50000500010000v abs (V)  Detailed ModelPAVM 1.4 1.45 1.5-2000200400Time (s)i as (A)  Detailed ModelDetailed Model-Fundamental ComponentPAVM Figure 2.12. Steady-state source-side line ab voltage and phase a current when 5 Hzof   and 0.9M   as predicted by the proposed PAVM and detailed model of full-bridge cyclo-converter.   -10000-50000500010000Load- v a (V)  Detailed ModelPAVM 1.4 1.5 1.6 1.7-100001000Time (s)Load- i a (A)  Detailed ModelPAVM Figure 2.13. Steady-state load-side phase a voltage and current when 5 Hzof   and 0.9M   as predicted by the proposed PAVM and detailed model of full-bridge cyclo-converter.  34 2.2.4.2 Transient operation Here, the system is initially operating in steady-state condition with output frequency set to 5 Hz and the converter modulation index to 0.5. At t=1 s the converter modulation index is stepped up to 0.9. The resulting transients are shown in Figure 2.14 for load-side variables. Then at t=2 s the output frequency set-point is increased from 5 Hz to 15 Hz. The resulting transient is shown in Figure 2.15 for load-side variables.   -10000-50000500010000Load- v a (V)  Detailed ModelPAVM 0.8 1 1.2 1.4-100001000Time (s)Load- i a (A)  Detailed ModelPAVM Figure 2.14. Transient response of load-side phase a voltage and current when the converter modulation index is increased from 0.5M   to 0.9M   at 1t  s with 5 Hzof  as predicted by the subject models of full-bridge cyclo-converter. 35 -10000-50000500010000Load- v a (V)  Detailed ModelPAVM1.8 2 2.2-100001000Time (s)Load- i a (A)  Detailed ModelPAVM Figure 2.15. Transient response of load-side phase a voltage and current when the output frequency is increased from 5 Hzof   to 15 Hzof   at 2t  s with 0.9M   as predicted by the subject models of full-bridge cyclo-converter. As it can be observed in Figures 2.14 and 2.15, the presented PAVM of ac–ac converters is able to predict the transient response of fundamental components of system variables accurately compared to the detailed model of full-bridge cyclo-converter. It should also be noted that the dead-zones in zero-crossing of load currents in Figures 2.8 – 2.10 and Figures 2.13 – 2.15 are due to blocking control mode of operation in order to prevent source short-circuits when the two positive and negative bridges of cyclo-converters switch their operation.   36 2.2.4.3 Computational performance For the purpose of benchmarking the presented PAVM of ac–ac converters for full-bridge cyclo-converters, the 3.5-second transient study presented in section 2.2.4.2 is conducted using the same computer and program setting as in Section 2.2.3.3. The computational performance of the proposed PAVM and detailed models of full-bridge cyclo-converter are summarized in Table 2.2. As it can be observed from Table 2.2, a significant improvement in the number of time-steps (7,713 vs. 379,425) and speed (0.77s vs. 114.11s) is achieved when the detailed model of full-bridge cyclo-converter is replaced with its proposed PAVM.  Table 2.2. Computational performance of proposed PAVM and detailed model of full-bridge cyclo-converter for a 3.5-second transient study System Model Type Number of Steps CPU Time (s) Detailed 379,425 114.11 PAVM 7,713 0.77   2.3 AC–AC Matrix Converters Matrix converter (MC) is a single-stage ac–ac energy conversion system that consists of an array of m n  bidirectional power switches. Figure 2.16 depicts a generic power system with a 3 3   matrix converter (i.e., m = n = 3). MCs are  37 ,a sifr fllineVaebece11S 12S 13S21S 22S 23S31S 32S 33S,a inifcThree-PhaseLoadAC Filter,a outiabe ,ab inv,ab outv3 3 Matrix Converter,a inv Figure 2.16. Simplified diagram of 3 3    matrix converter for variable frequency drive applications. able to convert the fixed amplitude/frequency ac grid voltage to an adjustable amplitude and controllable frequency (either higher or lower) ac voltage by reconstructing the desired output voltages using different segments of input voltages [22], [23]. To prevent short-circuits on the input terminals and open-circuits on the output terminals (with possible three-phase inductive loads), the following constraints should be satisfied for the switches [23]:   1 2 3 1   1,2,3j j jS S S j    , (2.18) where   1   closed    , 1,2,30   openijS i j . (2.19) 38 The constraints (2.18) allow for only 33=27 switching combinations. Therefore, the input-output relations for phase voltages and currents can be expressed as,  , , , ,, , , ,, , , ,. ,    .a out a in a in a outTb out Ph Ph b in b in Ph Ph b outc out c in c in c outv v i iv v i iv v i i                                     T T , (2.20) where  11 12 1321 22 2331 32 33Ph PhS S SS S SS S S     T , (2.21) and T Ph PhT  is the transpose of the transfer matrix Ph PhT  [23]. The control method adopted in [23] employs simultaneous output voltage and input current space-vector modulation. Using high frequency (HF)-synthesis methodology [23], and assuming sinusoidal average input and output voltages and currents as  ,max,,cos( )2 = V cos( )32cos( )3ia inb in in ic initvv tvt                 ,   ,max,,cos( )2V cos( )32cos( )3o oa outb out out o oc outo otvv tvt                        ,  (2.22)  ,max,,cos( )2I cos( )32cos( )3i ia inb in in i ic ini itii tit                       ,  ,max,,cos( )2I cos( )32cos( )3o o La outb out out o o Lc outo o Ltii tit                             , (2.23) with 2i if  , 2o of  , where if  and of  are the frequencies of input and output terminal variables of MC; respectively, the input-output relation of voltages and currents can be defined as 39  max max3= cos( )2out in iV M V     , (2.24)  max max3= cos( )2in out LI M I     . (2.25) The input-phase to output-line transfer matrix Ph LT  is expressed as [23]  11 21 12 22 13 2321 31 22 32 23 3331 11 32 12 33 13Ph Ld d d d d dd d d d d dd d d d d d            T ,  (2.26)  cos( 30 ) cos( ). cos( 120 30 ) . cos( 120 )cos( 120 30 ) cos( 120 )To o i iPh L o o i io o i it tM t tt t                                 T . (2.27) Here, cos( )L  is the load power factor, cos( )i  is the adjustable input power factor, maxinI  and maxoutI  are the MC input and output peak line currents; respectively, maxinV  and maxoutV  are the converter input and output peak phase voltages; respectively, and M is the modulation index which is a control input ( 0 1M  ). Also, jid  is the duty-cycle of the switch jiS  [23]. Equations (2.24), (2.25) will be satisfied only when the input and output active power of converter are equal. Since the matrix converter switches operate in hard switching condition with high switching frequency, the converter losses (including conduction and switching losses) are not negligible. Considering the voltage drops on switches along with the switching losses makes the derivation of analytical AVMs cumbersome (if not impossible). Therefore, in the next section, the proposed ac–ac PAVM as a numerical methodology is adopted for 40 modeling of the MC to be able to account for converter losses without complicated analytical derivations. 2.3.1 Verification of the proposed ac–ac PAVM for the three-phase matrix converters To demonstrate the applicability of the proposed ac–ac PAVM for modeling non-ideal matrix converters, the generic 3 3   MC system of Figure 2.16 is considered. Here, a three-phase RL load, which can represent an inductive motor, is powered through MC from a three-phase 60 Hz grid. The amplitude of load voltage can be adjusted using the modulation index M, and the output frequency is adjusted by selecting of  in the range of 1–400 Hz. The switching frequency of the matrix converter is chosen to be 20 kHz. Also, an ac filter is used to attenuate the switching ripples at the input terminals. The system parameters and MC switch characteristics are summarized in Appendix C. The studied power system including detailed model and PAVM of MC have been implemented in MATLAB/Simulink using Simscape Electrical [SimPowerSystem (SPS)] toolbox [28]. The detailed model is constructed using standard library components of the SPS toolbox. The PAVM has been realized using algebraic relations (2.1)–(2.16) that are used to replace the switching converter cell with non-switching dependent voltage and current sources, as illustrated in Figure 2.3. 41 2.3.1.1 Steady-state and transient operation To verify the performance of the proposed PAVM of ac–ac converters in steady-state, the output load-side frequency is selected to be 50of   Hz and the modulation index of the MC is set to M=0.9, and the system is in steady-state condition. The ac source- and load-side phase a voltages and currents are shown in Figure 2.17 as predicted by the detailed model and proposed PAVM of MC. As it can be observed in Figure 2.17, the presented PAVM of ac–ac converters is able to predict the fundamental components of ac source- and load-side variables accurately in steady-state compared to the detailed model of the matrix converter. In addition, to investigate the dynamic performance of the proposed ac–ac PAVM, the following transient study has been conducted. The system initially operates with M=0.5 providing 95 V rms at 50 Hz (fundamental frequency component of output voltage) for the three-phase load. At t = 0.5s, the modulation index is increased from 0.5 to 0.9. Then, at t = 1s, the output frequency is increased from 50 Hz to 400 Hz. The corresponding responses captured by the detailed model and the proposed PAVM are illustrated in Figures 2.18 and 2.19 for several variables of interest. Figure 2.18 demonstrates the transient responses for the input and output variables of matrix converter when a step change is applied to the modulation index to increase the load voltage from 95 V rms to 170 V rms. Figure 2.19 shows the transient responses 42 of ac source current and the transformed qd load currents along with its phase a current when the output frequency is changed from 50 to 400 Hz.  Figure 2.17. Steady-state system variables when 50 Hzof   and 0.9M   as predicted by the proposed PAVM and detailed model of matrix converter for: (a) converter phase a input voltage, (b) source phase a current, (c) load phase a voltage, and (d) load phase a current. It should also be noted that the frequency of input terminal variables remains to be constant at 60 Hz. The observations in Figures 2.18 and 2.19 confirm that the proposed ac–ac PAVM predicts the dynamic response of the fundamental 43 components of system variables with great accuracy compared to the detailed model of MC.   Figure 2.18. Transient response of system variables to the increase of modulation index from 0.5M   to 0.9M   at 0.5t   s when the output frequency is set to 50 Hzof  , as predicted by the proposed PAVM and detailed model of matrix converter for: (a) converter phase a input voltage, (b) source phase a current, (c) load phase a voltage, and (d) load phase a current.  44  Figure 2.19. Transient response of system variables to the increase of output frequency from 50 Hzof   to 400 Hzof   at 1t   s with 0.9M  , as predicted by the proposed PAVM and detailed model of matrix converter for: (a) load phase a current, (b) d-axis component of load current, (c) q-axis component of load current, and (d) source phase a current.  2.3.1.2 Prediction of efficiency and losses Here, the ability of the models to capture the efficiency of the matrix converter is considered for two different cases. First, only the conduction loss of the matrix converter is considered for the detailed model and the PAVM. This is achieved by setting the current tail and fall times (see Appendix C) equal to 45 zero, which retains all losses due to on-resistances and voltage drops (see Appendix C) but eliminates the switching losses. The efficiency and power loss of MC as predicted by its detailed model and the proposed PAVM are summarized in Table 2.3. As seen in Table 2.3, the PAVM predicts the converter efficiency with acceptable accuracy (~0.3–0.5% error) compared to the detailed model. Next, both conduction and switching losses are considered. To account for switching losses, typical current tail and fall times are used for switches as presented in Appendix C. The efficiency under different operating conditions are calculated and tabulated in Table 2.4. As can be observed in Table 2.4, the proposed PAVM has an acceptable accuracy (~0.3–0.5% error). This conveys that the proposed PAVM provides an excellent approximation of converter efficiency considering both conduction and switching losses. Table 2.3. The efficiency of MC as predicted by the proposed PAVM and detailed models considering conduction losses only Model fo = 50 Hz fo = 400 Hz M = 0.5 M = 0.9 M = 0.9 Detailed 94.1 % 96.8 % 95.9 % PAVM 93.6 % 96.5 % 95.5 %  Table 2.4. The efficiency of MC as predicted by the subject models considering both conduction and switching losses Model fo = 50 Hz fo = 400 Hz M = 0.5 M = 0.9 M = 0.9 Detailed 93.3 % 96.4 % 95.4 % PAVM 92.8 % 96.1 % 95.0 % 46 It is also worth mentioning that the lower efficiency of MC predicted by the PAVM is due to the fact that this model retains only the fundamental frequency components (e.g., 60 Hz at the input and variable output frequency such as 400 Hz, used here). Hence, other frequency components which are also contributing to delivering power and losses are neglected. For the purpose of discussion, the power flow is illustrated in Figure 2.20. Here, inP  and outP  are total input and output power of matrix converter, respectively, while MClossP  is the amount of power dissipated in the converter due to conduction and/or switching losses. Also, ,1filterlossP  is the power dissipated in the filter resistance due to the fundamental component of line current, and ,hfilterlossP  is the total power loss in the filter due to the remaining harmonic currents. Similarly, ,1loadP  and ,hloadP  are the power portions delivered to the load by the fundamental frequency component and harmonics of the load current, respectively.  MC LoadAC GridsourceP,1filterlossP ,hfilterlossPinP outP,1loadP,hloadPFilter ResistanceMClossP  Figure 2.20. Power flow in the ac–ac energy conversion system of study.  47 The input and output power of MC can be expressed as:  ,1 ,hfilter filterin source loss lossP P P P   , (2.28)  ,1 ,hout load loadP P P  . (2.29) The converter losses can be obtained as follows:   MCloss in outP P P  . (2.30) The total MC losses can be calculated by substituting (2.28), (2.29) in (2.30), which yields     ,1 ,1 ,h ,hMC filter filterloss source load loss load lossP P P P P P     . (2.31) Also, the efficiency of the MC can be written as  ,1 ,h,1 ,hload loadMC outfilter filterin source loss lossP PPP P P P  . (2.32) As opposed to the detailed model where all losses are included, when the PAVM is used, the term  ,h ,hfilterload lossP P  in (2.31) would be equal to zero as only the fundamental component of currents and voltages are considered in PAVM. This makes the power losses to be slightly higher in the PAVM compared to the detailed model, resulting in lower efficiency predicted by the PAVM. Also, the terms ,hloadP  in the numerator and ,hfilterlossP  in the denominator of (2.32) are equal to zero for PAVM which results in lower efficiency predicted as compared to the detailed model.  48 It should also be noted that since the three-phase ac grid voltages (i.e., ,  ,  a b ce e e ) are assumed to be ideal and sinusoidal, only the fundamental components of line currents contribute to the power delivered by the grid. Therefore, sourceP  would be identical when using detailed model or PAVM. For the studied scenarios, the system power flow of Figure 2.20 is tabulated in Table 2.5 when considering conduction losses only, and in Table 2.6 when switching losses are also included.  Table 2.5. System power flow as predicted by the subject models considering conduction losses only Power (W) Model fo = 50 Hz fo = 400 Hz M = 0.5 M = 0.9 M = 0.9 sourceP  Detailed 2,880 8,558 5,685 PAVM 2,880 8,558 5,685 ,1filterlossP  Detailed 32.9 255.8 113 PAVM 33 256 113 inP  Detailed 2,845 8,296 5,568 PAVM 2,847 8,302 5,572 outP  Detailed 2,677 8,030 5,340 PAVM 2,666 8,009 5,319 MClossP  Detailed 168 266 228 PAVM 181 293 253 ,1loadP  Detailed 2,665 8,011 5,318 PAVM 2,666 8,009 5,319    49 Table 2.6. System power flow as predicted by the subject models considering conduction and switching losses Power (W) Model fo = 50 Hz fo = 400 Hz M = 0.5 M = 0.9 M = 0.9 sourceP  Detailed 2,917 8,606 5,717 PAVM 2,916 8,606 5,716 ,1filterlossP  Detailed 33.6 258.6 114.2 PAVM 33 259 114 inP  Detailed 2,880 8,341 5,598 PAVM 2,883 8,347 5,602 outP  Detailed 2,687 8,039 5,339 PAVM 2,676 8,019 5,322 MClossP  Detailed 193 302 259 PAVM 207 328 280 ,1loadP  Detailed 2,676 8,019 5,320 PAVM 2,675 8,020 5,322  2.3.1.3 Computational performance Finally, to benchmark the computational performance of the presented ac–ac PAVM for MC systems, a 1.5-second study comprised of the fragments shown in Figures 2.17 – 2.18 is considered. The detailed model with discrete switching states is executed using variable-step discrete Tustin/Backward Euler solver with the maximum step size set to 10-3. The PAVM is executed using the continuous ode23tb solver with the maximum step size set to 10-3, and the relative and absolute tolerance also set to 10-3. For consistency, all simulations are executed on a PC with Intel® Core™ i7-4510U @ 2.00GHz processor. The computational performance of both PAVM and detailed models are summarized in Table 2.7. As it can be seen in Table 2.7, the detailed switching model 50 requires a large number of time-steps (750,001), which also takes fairly long time (132.21s). At the same time, the presented PAVM of ac–ac converters requires much fewer time-steps (12,219) and offers significant improvement in the simulation speed (2.05s compared to 2.2 min).  Table 2.7. Computational performance of the proposed PAVM and detailed model of the matrix converter System Model Type Number of Steps CPU Time (s) Detailed 750,001 132.21 PAVM 12,219 2.05 51 CHAPTER 3: EXTENDING MRF-PAVM METHOD TO TWELVE-PULSE RECTIFIERS CONNECTED TO SIX-PHASE AC SYSTEMS Many advanced energy conversion systems utilize multi-phase (e.g., six or more phases) electrical machines that feed high-pulse-count (e.g., 12 or more pulses) rectifiers for supplying dc power. Efficient simulation of such power systems in commonly-used state-variable-based or nodal-based EMT programs requires fast and accurate models of electrical machines and power electronic converters with compatible interfaces. As an alternative to the existing qd and voltage-behind-reactance (VBR) machine models, in this chapter, a constant-parameter VBR (CPVBR) model is developed for six-phase permanent magnet synchronous machines to offer computationally efficient interface. For system-level studies where the switching details of rectifiers can be neglected, a multiple-reference-frame parametric-average-value model (MRF-PAVM) is developed for the 12-pulse rectifiers that provides fast simulation and preserves the dominant ac harmonics of interest. The accuracy and numerical efficiency of the proposed machine-converter models are verified using the detailed and alternative existing models.  52 3.1 Generic Six-Phase Machine-Converter System Multi-phase electrical machines (e.g., with six or more phases) and line-commutated rectifiers are often considered in advanced ac–dc energy conversion systems such as vehicles [4], ships [8], aircraft [9]–[11], wind generation [3], classic HVDC systems [2], distributed energy resources [5], exciters [13], etc., due to their increased reliability and high power transfer capability [77]. An example generic configuration considered in this chapter is depicted in Figure 3.1, where a PM synchronous machine (PMSM) is feeding a 12-pulse rectifier connected to a dc subsystem (load). In more conventional configurations, a three-phase generator followed by a transformer with delta/wye secondary windings feeding 12-pulse rectifiers is used to supply dc power with low ripple while reducing the ac-side harmonics [78]. Design and analysis of such power systems require conducting numerous simulations that rely on accurate and numerically efficient models of various components, including electrical machines and power electronic converters. Six-PhasePMSMdcvdci fr fLfc1abcv2abcv1abci2abciPrimeMovereTmT r12-Pulse RectifierDC FilterDC Subsystem Figure 3.1. A generic machine-converter ac–dc power system composed of a six-phase PM synchronous generator connected to a 12-pulse diode rectifier. 53 For modeling six-phase electrical machines, conventionally, the qd12 models that are expressed in two sets of qd rotor reference frames/coordinates have been used [79]. However, since the qd12 models are interfaced with external network using controlled current sources, interfacing them with the detailed switching models of power electronic converters in state-variable-based simulation programs [27]–[29] requires artificial snubbers which make the system numerically stiff* [73]. This may considerably reduce the simulation speed and necessitates the use of stiff/implicit solvers in the simulation programs. In addition to the inaccuracy introduced by the fictitious snubber circuits, the numerical instability of the simulation becomes a challenge. It has been also shown in [80] that inaccurate synchronous machine models can negatively influence power system stabilizer (PSS) and damping controller performance. As an alternative to the qd12 model, the so-called voltage-behind-reactance (VBR) model has been recently presented for six-phase wound-rotor synchronous machines in [81]. The VBR formulation [82]–[84] eliminates the need of snubbers for interfacing with external networks and increases the simulation speed even with non-stiff/explicit solvers. However, similar to coupled-circuit phase domain (CCPD) synchronous machine models [38, Ch. 5], the interfacing inductance matrix of the conventional VBR model is rotor-position-dependent   *A system of ordinary differential equations may be referred to as being “numerically stiff” if it simultaneously has very small and very large eigenvalues [27]. Stiffness is also associated with the simultaneous presence of very slow and very fast dynamics. 54 due to rotor dynamic saliency [81]–[83], and it has to be re-computed at every time-step. This requires special considerations and additional computational effort in most simulation programs, for example, PLECS [29]. Recently, a constant-parameter VBR (CPVBR) formulation has been presented in [85] for six-phase wound-rotor synchronous machines, which increases the simulation speed by keeping a constant interfacing inductance matrix. It has also been envisioned in [86]–[88] that computationally efficient models of machines and converters are essential for application in real-time simulators to allow using larger time-steps and avoid numerical instability. The interested reader is referred to publications by the IEEE Task Force on Interfacing Techniques for Simulation Tools, such as [73] and [89], for more information on interfacing. It is noted that the VBR and CPVBR models are developed based on the well-established classical qd models. Such classical qd models have been used and verified experimentally in many publications, for example in [90], [91]. The work presented in this chapter assumes that the reference qd12 model (without including main and leakage flux saturations) has sufficient accuracy for the system-level studies, while there are numerous methodologies for including magnetic saturation, losses, etc., for example as in [81], [92]–[94] that can be considered as necessary. This chapter extends the prior works [64] and [79] on machine-converter modeling with the focus on achieving numerically efficient interfacing of the 55 machine and converter subsystems. The presented machine-converter models include several ac harmonics of interest that are pertained to the six-phase 12-pulse system depicted in Figure 3.1. This Chapter extends the prior works and makes the following additional contributions:  Using the approach set forth in [85] for six-phase wound-rotor synchronous machines, a CPVBR model is proposed for six-phase PMSMs. The explicit formulation of the proposed six-phase PMSM CPVBR model achieves a constant 6×6 interfacing inductance matrix, which eliminates the need for re-calculating inductances at each time-step.  An extended MRF-PAVM is proposed for 12-pulse rectifiers considering six-phase ac-side power system where both lower-order harmonics (5th and 7th) and also higher-order harmonics (11th and 13th) are present and can be dominant depending on the operating condition and system parameters. The formulation of the proposed extended MRF-PAVM allows direct interfacing with six-phase machine models.  Using comprehensive steady-state and transient studies, the advantages of the proposed machine-converter models in terms of accuracy and numerical efficiency in different combinations of models/interfaces are demonstrated.  It is shown that the proposed PMSM CPVBR model outperforms the existing/alternative qd12 and VBR counterparts and has the best simulation performance when either stiff or non-stiff ODE solvers are used.  It is also demonstrated that in system-level studies, where only the fundamental components and several dominant harmonics of ac waveforms are required, the proposed MRF-PAVM can be very effective by increasing the simulation speed, either with stiff or non-stiff ODE solvers.  56 Depending on the study objectives, interfacing requirements, the required details of waveforms, and the type of ODE solver used, the user may choose the best combination of the proposed machine and rectifier models. 3.2 Six-Phase PMSM Modeling The six-phase PMSM considered in Figure 3.1 is assumed to have two sets of three-phase windings on stator with a spatial shift equal to   and a damper cage on the rotor. For the modeling purpose, first, the machine variables in abc12 coordinates (i.e., stator voltages 12abcv  and stator currents 12abci ) are transformed to qd12 coordinates as  012 12 012 12( ) ,     ( ) ,sqd r abc sqd r abc  v K v i K i  (3.1) where r  is the rotor angular position and ( )rK  is a modified Park’s transformation matrix defined as (D.5) in Appendix D. Due to floating neutrals of the stator windings, the zero sequence variables in (3.1) for the system of Figure 3.1 are not considered. Also, generator sign convention is considered for machine variables. 3.2.1 qd12 model The equivalent qd12 circuit of six-phase PMSM is demonstrated in Figure 3.2 [79]. Therein, the equivalent rotor damping is modeled by two damper windings, 57 one on q12-axis and one on d12-axis. For notational compactness, the stator and rotor voltages, currents, and fluxes are defined in matrix form as  12 1 2 1 2Tsqd sq sq sd sdv v v v   v , (3.2)  12 1 2 1 2Tsqd sq sq sd sd      λ , (3.3)  12 1 2 1 2Tsqd sq sq sd sdi i i i   i , (3.4)  Tkqd kq kd    λ ,    Tkqd kq kdi i   i . (3.5) Here, the subscripts ‘s’ and ‘k’ denote stator and damper winding variables, respectively [79]. Also, the resistances and leakage inductances of damper windings and stator leakage inductances on q- and d-axes are defined as   0 0,     0 0kq lkqkqd lkqdkd lkdr Lr L          r L , (3.6)  lL1sdi(q12-axis)lLsrsrkdrlkdLlmLmdLpmmdL1r sq 2r sq kdi2sdi1sdv2sdvlL1sqi srsrkqrlkqLlmLmqL1r sd 2r sd kqi2sqi1sqv2sqv(d12-axis)lLmdimqilqdLlqdL Figure 3.2. The equivalent qd12 circuit of a six-phase PMSM. 58  0000l lm lm lqdlm l lm lqdlmlqd l lm lmlqd lm l lmL L L LL L L LL L L LL L L L         L , (3.7) where   1 2 1 2 1 22 2cos cos cos3 3lm a a a b a cL L L L                , (3.8)   1 2 1 2 1 22 2sin sin sin3 3lqd a a a b a cL L L L                , (3.9) with 1 2a aL , 1 2a bL , 1 2a cL  as the stator mutual leakage inductances [79]. The magnetizing inductances and currents in q- and d-axes are defined as  1 21 20,  0sq sq kqmq mqmqd mqd pmmd md sd sd kdmdi i iL iL i i i iL                      L i , (3.10) where pm  is the flux of the PMSM’s permanent magnet. Based on circuits of Figure 3.2, stator and rotor fluxes can be expressed as  2 1 2 112 122 1 2 1 4 2sqd mqd mqd lm sqd       1 0λ L i L i0 1, (3.11)  kqd mqd mqd lkqd kqd λ L i L i . (3.12) The stator and rotor voltages can also be written as  2 2 2 212 12 12 122 2 2 2 4 4sqd s sqd r sqd sqdr p         0 Iv i λ λI 0, (3.13)  2 1 kqd kqd kqdp   0 r i λ , (3.14) 59 where sr  is the resistance of stator windings, and p d dt  is the Heaviside’s operator. In the qd12 model, the stator and rotor fluxes (i.e., 12sqdλ  and kqdλ ) are chosen as the state-variables, stator voltages 12sqdv  are the inputs, and stator and rotor currents (i.e., 12sqdi  and kqdi ) are the outputs of the model, respectively [79]. Therefore, the state-space equations of the qd12 model can be compactly written as  12 1212 12 121 2 12sqd sqdqd qd qdpm sqdkqd kqdp            λ λA B B vλ λ, (3.15)   12 1212 12 121 2 12sqd sqdqd qd qdpm sqdkqd kqd           i λC D D vi λ, (3.16) where the state-space matrices are derived in (D.1)–(D.4) in Appendix D. The electromagnetic torque of the machine can also be calculated using (D.6).  Equations (3.15)–(3.16) form the state-space model of the six-phase PMSM in the qd12 reference frame. Implementation of this model is illustrated in Figure 3.3(a) with its interface to an arbitrary network in abc12 coordinates. Since the model computes currents as its outputs, controlled current sources are used in the interfacing circuit. Moreover, in order to interconnect with an external network that may be inductive or include switching power electronics, the shunt snubbers are required for establishing the terminal/interfacing voltages. This interfacing is sometimes referred to as indirect [73, Sec. IV-A]. 60 3.2.2 VBR model In the VBR model, the rotor fluxes kqdλ  are chosen as the state variables. The stator currents 12abci  (or equivalently 12sqdi ) are the inputs to the VBR model, which are calculated along with the stator voltages 12abcv  (or equivalently 12sqdv ) by the simulation program. Therefore, replacing kqdi  in (3.14) using (3.10) and (3.12), the state equation for rotor fluxes are obtained as  1 2 12kqd kqd pm sqdp   λ Aλ B B i , (3.17)    00kqlkq mqkdlkd mdrL LrL L       A , (3.18)   10kdlkd mdrL L     B  , 1 2 1 221 2 1 22 4mqkqlkqmdkdlkdLrLLrL          1 0B0 1, (3.19) where mqL  and mdL  are the sub-transient magnetizing inductances defined as  11 1mqlkq mqLL L       ,   11 1mdlkd mdLL L     . (3.20) Replacing 12sqdλ  in (3.13) using (3.10)–(3.11), the stator voltages can be expressed as  2 2 2 212 12 12 12 122 2 2 2 4 4sqd s sqd sqd r sqd sqdr p             0 Iv i L i L i eI 0. (3.21) 61 Here, 12sqde  consists of the sub-transient back-emfs defined as  12 1 2 1 2 1 2 12Tsqd q q d d kqd pm sqde e e e          e Cλ D D i , (3.22)    2 1 2 12 1 2 14 2mq kq mdrlkq lkdlkq mqmq md kdrlkq lkd lkd mdL r LL LL LL L rL L L L             1 1C1 1, (3.23)   2 112 14 1mdrmdmd kdlkd lkd mdLLL rL L L       1D1, (3.24)  22 2 2 2222 2 2 24 4mqkqlkqmdkdlkdLrLLrL                   1 0D0 1.  (3.25) Also, L  in (3.21) is the sub-transient inductance matrix defined as  0000q q l lqdq l q lqdlqd d d llqd d l dL L L LL L L LL L L LL L L L               L , (3.26) where  q l lm mqL L L L     ,   d l lm mdL L L L    . (3.27) In order to interface the VBR model with an arbitrary network, (3.21) should be transformed back to abc12 coordinates using the inverse of modified Park’s matrix 1( )rK , which yields 62   12 12 12 12( )abc s abc abc r abc abcr p     v i L i e , (3.28) where the interfacing inductance matrix ( )abc rL is  1( ) ( ) ( )abc r r r   L K L K . (3.29) When the stator voltages (3.21) are transformed to abc12 coordinates, having unequal inductances in q12- and d12-axes in L  (3.26) (which is generally the case due to machine salient structure) results in a 6×6 variable (rotor-position-dependent) interfacing matrix ( )abc rL  in (3.29) [81]. Equations (3.17)–(3.20) with (3.22)–(3.29) form the VBR model, which can be directly interfaced (without snubbers) with an arbitrary network in abc12 coordinates as illustrated in Figure 3.3(b). Therein, controlled voltage sources are used to inject the sub-transient back emfs. The interfacing equivalent circuit has to be implemented using six branches composed of resistors and variable coupled inductors that have to be available in the simulation programs as basic circuit elements (e.g., [29] permits such elements, but [28] does not). The branch currents are then calculated by the simulation program and passed to the VBR model as the inputs. 3.2.3 Proposed CPVBR model To achieve a constant-parameter VBR formulation (with constant inductances in abc12 coordinates), L  in (3.26) is split into two parts as 63    2 2 2 22 2 2 2 4 4mq mdL L            1 0L L0 0 , (3.30) where  0000d d l lqdd l d lqdlqd d d llqd d l dL L L LL L L LL L L LL L L L               L . (3.31) The new matrix L  (3.31) has equal inductances in q12- and d12-axes. Then, (3.21) is rewritten by replacing L  with (3.30), resulting in  2 2 2 212 12 12 12 122 2 2 2 4 4sqd s sqd sqd r sqd sqdr p             0 Iv i L i L i eI 0, (3.32) where       2 2 2 2 2 112 12 12 1 22 2 2 2 2 14 4 4 1sqd sqd r mq md sqd mq md sq sqL L L L p i i                        0 0 1e e i1 0 0.(3.33) Here, the derivative of the sum of 1sqi  and 2sqi  in (3.33) is provided in (3.34)–(3.36).           1 2 2 11 2 2 11 2 2 21 2 2 11 22221lqdsq sq sd sdllqds sq sq sd sdllsq sqlqdl l lm mq lqdr sd sd sq sqlmq kq mqkq mq sq sqlkq lkqLv v v vLLr i i i iLLp i iLL L L L LLL r LL i iL L                                                        ,(3.34) 64        1 2 1 2 1 22 22 2 md mdsd sd md lm l sd sd kd pm lqd sq sqlkd mdL LL L L i i L i iL L                      ,(3.35)       2 1 1 2 1 2sq sq l sq sq lqd sd sdL i i L i i      . (3.36) Therein, the derivate expression contains the term    1 2 2 1lqdsq sq sd sdlLv v v vL      which makes 12sqdv  in (3.32) implicitly dependent on itself. This implicit dependency of (3.32)–(3.34) on 12sqdv  creates an algebraic loop in the model [85]. To relax (break) this algebraic loop, the term corresponding to qd12 voltages in (3.34) is approximated (predicted) using a low-pass filter as      12 1 2 2 111lqdsqd sq sq sd sdlLv v v v vs L          . (3.37) After that, the output voltage of filter ( 12sqdv ) will be used in (3.34) instead. A detailed procedure for selecting the low-pass filter time-constant   in (3.37) can be found in [95]. Transforming the stator voltages in (3.32) to abc12 coordinates using 1( )rK , the interfacing equation of the proposed CPVBR model is obtained as   12 12 12 12abc s abc abc abc abcr p     v i L i e , (3.38) where the interfacing inductance matrix abcL  is  1( ) ( )abc r r  L K L K . (3.39) Here, abcL  is a 6×6 constant-parameter matrix. 65 Implementation of the proposed CPVBR model is illustrated in Figure 3.3(c). According to Figure 3.3(c), the machine stator terminals are interfaced with the external arbitrary ac network in abc12 coordinates using six branches with constant RL elements and dependent voltage sources. The branch currents and voltages are calculated by the simulation program and passed to the CPVBR model as inputs. 12abci12sqdi12abcv12sqdvkqdλSnubber CircuitRequiredEq. (16)12sqdλ1( )rKInterfacing Circuit( )rKqd12 State-Space Model12abce sr ( )abc rL12abci12sqdiEq. (17)kqdλ12sqdeEq. (22)1( )rK( )rKVBR State-Space ModelInterfacing Circuit12abce sr abcL12abci12sqdi12sqde12abcv12sqdvEq. (37)Eq. (17)kqdλ12sqdeEq. (22)Eq. (33)Eqs.(34)-(36)1 1( )sq sqp i i1( )rK( )rK( )rKProposed CPVBR State-Space ModelInterfacing Circuit(b)(a)(c)Eq. (15)12sqdv Figure 3.3. Implementation of the subject models and their interfacing with an arbitrary ac network in abc12 coordinates: (a) qd12 model with indirect interfacing using current sources and snubber circuits; (b) VBR model with direct interfacing using voltage sources; and (c) the proposed CPVBR model with direct interfacing using voltage sources. 66 3.3 Twelve-Pulse Rectifier Average-Value Modeling In the proposed extended PAVM methodology, the relationships between the dc and ac (expressed in qd coordinates) variables of the rectifier are defined with respect to their averaged values within the prototypical switching interval as  1( ) ( )swtt Tx t x dT   . (3.40) Here, x  denotes currents and voltages, swT  is the switching period equal to  1 6 ef  and ef  is the line frequency. Then, the relationships between the averaged ac (in qd12 coordinates) and dc variables can be captured using the following parametric functions as  ,1 ,21 11 2( ) ,     ( )dc dci isqd sqdi iw w   i i,  (3.41)  1 11 21 1,1 ,2( ) ,     ( )sqd sqdv vdc dcw wv v   v v. (3.42) Also, the angles between ac voltages and currents are captured by defining the parametric functions as  1 11 11 11 1 11 1( ) tan tansd sdsq sqv iv i                , (3.43)  1 11 12 22 1 12 2( ) tan tansd sdsq sqv iv i                . (3.44) 67 The parametric functions (3.41)–(3.44) form the key relationships of the PAVM of 12-pulse rectifier where only the dc and fundamental components of ac variables are considered In order to reproduce the ac harmonics, an MRF-PAVM is proposed, where additional parametric functions are used to capture and represent several dominant harmonics of interest. The relationship between the magnitude of harmonic voltages and the 12-pulse rectifier dc voltage are captured using the parametric functions defined as  1 2,1 ,2( ) ,     ( )n nsqd sqdn nv vdc dcw wv v   v v, (3.45) where n  is the harmonic order. Also, the phase angles of harmonic voltages are captured with respect to the fundamental components of voltages of each set. Since the angular frequency of harmonics are n  times faster than the fundamental components, in order to lock the harmonic voltages to their corresponding fundamental components, the following parametric functions are defined as  11 11 1,1 11 1( ) tan tannn sd sdv nsq sqv vnv v                , (3.46)  11 12 2,2 12 2( ) tan tannn sd sdv nsq sqv vnv v                . (3.47) The operating condition of the rectifier may be specified by the dynamic impedances, herein defined as   68  ,1 ,21 11 2,     dc dcd dsqd sqdv vz z i i. (3.48) The parametric functions (3.41)–(3.47) are computed numerically by running the detailed simulation over the desired range of operating conditions. These parametric functions are saved in look-up tables in terms of ,1dz  and ,2dz  (3.48). In the case of 12-pulse rectifier symmetry, as shown in Figure 3.1, the parametric functions associated with fundamental components of ac variables and dc variables (3.41)–(3.44) would be equal for both sets as  ,1 ,2( ) ( ) ( )i i iw w w     ,  (3.49)  1 1 1,1 ,2( ) ( ) ( )v v vw w w     ,  (3.50)  1 2( ) ( ) ( )       . (3.51) Similarly, the parametric functions corresponding to harmonic voltages (3.45)–(3.47) would also become equal as  ,1 ,2( ) ( ) ( )n n nv v vw w w     , (3.52)  ,1 ,2( ) ( ) ( )n n nv v v       . (3.53) Moreover, since the dynamic impedances (3.48) would also be equal, the operating condition of the rectifier can be captured by only one dynamic impedance as   ,1 ,2d d dz z z  . (3.54) 69 It is worth mentioning that in this chapter the ac currents and dc voltage of rectifier are the inputs to the AVMs, and therefore these variables are used for defining the dynamic impedances. However, depending on the interfacing requirements and the corresponding formulations, different pairs of dc and ac variables can be appropriately selected for defining the dynamic impedances or admittance (e.g., [96] where dc current and ac voltages are used due to interfacing requirements). For the symmetric power system of Figure 3.1 (with parameters summarized in Appendix E), the parametric functions corresponding to the dc and fundamental components of ac variables are shown in Figures 3.4(a)–(c) for ( )iw  ,1( )vw   and ( )  , respectively. As it can be observed in Figure 3.4, the non-linear relationships between the dc and fundamental components of the ac variables are tabulated in one-dimensional lookup tables with respect to dynamic impedance dz . Also, the parametric functions corresponding to the dominant ac harmonics (up to 13th) are illustrated in Figures 3.5(a)–(b) over a wide range of operating conditions for magnitude ( nvw ) and phase (nv ) of harmonic voltages, respectively. As it can be observed in Figure 3.5(a), for the considered 12-pulse rectifier system connected to the six-phase machine, both lower-order (i.e., 5th and 7th) and higher-order (i.e., 11th and 13th) harmonics are present on the ac voltages 70 and their corresponding parametric functions are included in the proposed extended MRF-PAVM model; hence  5,7,11,13n .  Figure 3.4. Parametric functions corresponding to dc and fundamental components of ac variables of the 12-pulse diode rectifier over a wide range of operating conditions: (a) ( )iw  , (b) 1( )vw   and (c) ( )  .  Figure 3.5. Parametric functions depicting the magnitude and phase angle of various harmonics; respectively, in ac voltages of the 12-pulse rectifier over a wide range of operating conditions: (a) ( )nvw   and (b) ( )nv  .  71 To implement the developed extended MRF-PAVM, dependent voltage and current sources are used to interface with the ac and dc subsystems as depicted in Figure 3.6. Therein, the ac currents 12sqdi  and the dc voltage dcv  are the inputs to the MRF-PAVM, where ,1dz  and ,2dz  are computed based on (3.48) and used as inputs to the lookup tables. The fundamental components of the ac voltages are calculated as     1 1 1 11 ,1 1 1 ,1 1( ) cos ,     ( ) sinsq v dc sd v dcv w v v w v     , (3.55)     1 1 1 12 ,2 2 2 ,2 2( ) cos ,     ( ) sinsq v dc sd v dcv w v v w v     , (3.56) where   1 1 11 1 1 1tan ( )sd sqi i    , (3.57)   1 1 12 2 2 2tan ( )sd sqi i    . (3.58) The harmonic voltages for the first three-phase set are calculated based on (3.45) and (3.46) as   1 ,1 1 ,1( ) cos ( )n n nsq v dc vv w v n     , (3.59)   1 ,1 1 ,1( ) sin ( )n n nsd v dc vv w v n     . (3.60) Similar equations can be written for the second three-phase set based on (3.45) and (3.47) as   2 ,2 2 ,2( ) cos ( )n n nsq v dc vv w v n     , (3.61)   2 ,2 2 ,2( ) sin ( )n n nsd v dc vv w v n     . (3.62) 72 Finally, the interfacing ac voltages and dc current that become the outputs of the MRF-PAVM are calculated as     1112 12 12( 1)nsqd sqd r sqdnn   v v K v , (3.63) and  1 1,1 1 ,2 2( ) ( )dc i sqd i sqdi w w   i i . (3.64) For interfacing the current source dci  with the dc subsystem, the filter inductor fL  is modeled using a transfer function as in [52, see Eq. (9)] with a small time-constant f . dcv12sqdiBased on Eqs. (3.55)-(3.64)12sqdvdci12-Pulse Rectifier MRF-PAVMDC Filter and DC SubsystemSix-Phase PMSM qd12 Model Figure 3.6. Implementation of the proposed extended MRF-PAVM and its interfacing with the dc subsystem and ac subsystem in qd12 coordinates.   3.4 Computer Studies The machine-converter system of Figure 3.1 has been implemented in MATLAB/Simulink [27] using PLECS toolbox [29] with system parameters summarized in Appendix E. For constructing the detailed model of 12-pulse rectifier, the standard library switching diode bridges are used. Three versions 73 of detailed switching machine-rectifier models are considered: qd12-Detailed, VBR-Detailed, and CPVBR-Detailed, where the six-phase PMSM is modeled using the qd12, VBR, and CPVBR formulations as presented in sections 3.2.1, 3.2.2, and 3.2.3, respectively. For interfacing the qd12 model with the detailed model of 12-pulse rectifier, parallel snubber resistors of 350Ω are selected to ensure maximum error in stator current to be on the order of 1% at nominal loading condition. The VBR and CPVBR models have a direct interface of the stator circuit terminals to the rectifier bridges and do not require snubbers. In order to limit the solution error of the CPVBR model to less than 1%, the time constant of the low-pass filter in (3.37) is selected to be τ = 0.17 [95].  Also, two versions of the average-value models are considered: qd12-PAVM and qd12-MRF-PAVM, where the qd12 model of six-phase PMSM is interfaced with the PAVM and the proposed extended MRF-PAVM models of the 12-pulse rectifier using (3.55)–(3.58), (3.64) and (3.55)–(3.64), respectively. Without loss of generality, in all the models the dc subsystem has been represented as an equivalent variable resistive load rl. Also, the parametric functions of the two AVMs (i.e., PAVM and MRF-PAVM) have been obtained using the fast procedure presented in [54] by simulating the detailed model of the power system in Figure 3.1 with a fast sweep of the load resistance over a wide range of operating conditions (from rl =1Ω to rl =1000Ω). It should also be noted that this is a one-time process, and once the parametric functions are calculated and saved in lookup tables, they can be used for various studies including transients. 74 In the subsequent subsections, the performance of all the considered models is verified in steady-state and in transients. 3.4.1 Steady-state studies Here, the generator speed is kept constant corresponding to the line frequency ef =60 Hz. The steady-state waveforms of rectifier variables predicted by the subject models are shown in Figures 3.7 and 3.9 for DCM condition (with rl =1000Ω) and CCM nominal condition (with rl =10Ω) [48], respectively; with their associated harmonic contents depicted in Figures 3.8 and 3.10. As it can be observed in Figures 3.7 – 3.10, the proposed CPVBR-Detailed model accurately follows the steady-state results of the qd12-Detailed and VBR-Detailed models. Also, using artificial snubbers [73] in the qd12-Detailed model results in an error as can be seen in Figures 3.7(a) and 3.9(a). Meanwhile, the proposed qd12-MRF-PAVM provides excellent accuracy in predicting the average values of the dc variables as well as the fundamental components and the harmonics of the ac variables as compared to the rectifier detailed switching models. However, the qd12-PAVM can only predict the average values of dc and the fundamental components of ac variables. It should also be noted that in Figure 3.10, the 11th and 13th harmonics are dominant in the voltage (compared to the 5th and 7th harmonics), while the 5th and 7th harmonics are larger in the current (compared to the 11th and 13th  75  Figure 3.7. Rectifier dc and ac variables in DCM as predicted by the subject models: (a) dc voltage, (b) dc current, (c) phase a1 voltage, and (d) phase a1 current.  Figure 3.8. The harmonic content of rectifier ac phase a1 voltage and phase a1 current as predicted by the subject models in DCM condition. 76  Figure 3.9. Rectifier dc and ac variables in CCM as predicted by the subject models: (a) dc voltage, (b) dc current, (c) phase a1 voltage, and (d) phase a1 current.  Figure 3.10. The harmonic content of rectifier ac phase a1 voltage and phase a1 current as predicted by the subject models in CCM condition. 77 harmonics). This is due to the fact that the impedance of the electrical machine increases very rapidly with the harmonic frequency.  3.4.2 Transient studies Here, the dynamic performance of the proposed machine-converter models is investigated for the system transient by changing rl from 1000Ω to 100Ω (DCM to CCM) at t=0.5s. The transient responses of several system variables are shown in Figure 3.11 for all the subject models. As can be observed in Figure 3.11, the proposed CPVBR-Detailed model accurately follows the transient results of system variables compared to the qd12-Detailed and VBR-Detailed models. Moreover, the error in generator torque predicted by the qd12-Detailed model, as can be observed in Figure 3.11(c), is a result of the error introduced to the stator currents due to the artificial snubbers. Meanwhile, the proposed qd12-MRF-PAVM very accurately follows the transient behavior of detailed switching models, as can be observed in Figures 3.11(a)–(b) for dc variables and also in Figure 3.11(d) for the ac variables including the harmonics. Moreover, the ac-side voltage and current harmonics predicted by the proposed qd12-MRF-PAVM result in an accurate prediction of the generator torque ripple (compared to VBR/CPVBR-Detailed switching models), as demonstrated in Figure 3.11(c), where the qd12-PAVM only predicts the average value of the torque. 78  Figure 3.11. Transient response due to load change from DCM to CCM as predicted by the subject models: (a) rectifier dc voltage, (b) rectifier dc current, (c) generator torque, and (d) generator phase a1 current.  3.4.3 Computational performance comparison To benchmark the subject models, the following 5-second transient study has been carried out. The system initially operates with half of the nominal load (with rl =20Ω). At t=2.5 s the load is switched to its nominal value corresponding to rl =10Ω. For consistency, all the models are run on a PC with Intel® Core™ 79 i7-4510U @ 2.00GHz processor. To evaluate the effect of different interfacing methods and numerical stiffness, the explicit (non-stiff) ode45* and implicit (stiff) ode23tb solvers have been used for all the subject models with relative and absolute tolerances set to 10-3. The maximum time-step is also set to 0.5 ms for the qd12/VBR/CPVBR-Detailed models, and 1.0 ms and 0.2 ms for the qd12-PAVM and qd12-MRF-PAVM models, respectively. 3.4.3.1 Numerical accuracy of detailed models To verify the accuracy of the proposed CPVBR model of the six-phase PMSM compared to the existing qd12 and VBR models, the solutions obtained by the qd12/VBR/CPVBR-Detailed models are first compared with a reference solution. The reference solution is obtained using the VBR model ran with a very small maximum time-step as well as absolute and relative errors, all set to 10-5. The VBR model does not introduce a solution approximation (error) such as the qd12 model due to artificial snubbers or the CPVBR model due to algebraic-loop relaxation filter. Therefore, its solution trajectory obtained with a very small time-step is considered to be the reference. The cumulative 2-norm errors [97] for the subject models computed for several variables are summarized in Table 3.1 for the 5-second transient study.    *ode45 is a type of solver used in MATLAB to numerically solve non-stiff differential equations. This in an explicit solver based on Runge-Kutta formula of orders 4 and 5 [27], the Dormand-Prince (4,5) pair [110]. 80  As it can be observed in Table 3.1, the VBR model ran with larger time-step and relaxed tolerances introduces very small error compared to the reference (~0.08%). The errors of the qd12 model in torque and stator current are on the order of 1%, which are consistent with the selected interfacing snubbers.  Meanwhile, the proposed CPVBR model offers higher accuracy compared to the qd12 model in torque (0.21% vs. 0.85%) and stator current (0.23% vs. 1.15%).  Table 3.1. 2-norm error in several variables of the subject detailed models for the considered 5-second transient study  Model 2-Norm Error Machine - Rectifier torque eT  stator current 1asi  dc current dci  VBR-Detailed 0.08 % 0.07 % 0.07 % qd12-Detailed 0.85 % 1.15 % 0.18 % CPVBR-Detailed 0.21 % 0.23 % 0.18 %  3.4.3.2 Performance with non-stiff solver Here, the computational performance of the subject models with non-stiff (ode45) solver is investigated, and the results are summarized in Table 3.2. As it can be observed in Table 3.2, the qd12-Detailed model is the slowest (~11 min) due to the stiffness introduced by the snubber circuit. The VBR-Detailed model improves the simulation speed (23.66 s). Meanwhile, the proposed CPVBR-Detailed model outperforms both the qd12-Detailed and VBR-Detailed models in terms of the CPU time (7.06 s CPVBR vs. 23.66 s VBR vs. 663.19 s qd12). 81 Moreover, the CPVBR-Detailed and VBR-Detailed models take almost the same number of steps (38,741 and 24,850 respectively) which are much less than the number of steps required by the qd12 model (8,666,686).  Table 3.2. Computational performance of the subject detailed and averaged models for the considered 5-second transient study with the non-stiff solver ode45 Model Number of steps CPU Time (s) Machine - Rectifier qd12-Detailed 8,666,686 663.19 (~11 min) VBR-Detailed 24,850 23.66 CPVBR-Detailed 38,741 7.06 qd12-MRF-PAVM 14,649 2.92 qd12-PAVM 12,851 0.81  3.4.3.3 Performance with stiff solver Here, the computational performance of the subject models with stiff (ode23tb) solver is investigated and the results are summarized in Table 3.3.  As it can be observed in Table 3.3, using stiff solver reduces the computational burden caused by the large snubber resistors in the qd12-Detailed model, which leads to faster simulation speed compared to VBR-Detailed model (13.42 s qd12 vs. 33.92 s VBR). This is expected since unlike the VBR-Detailed model with variable inductors, the qd12-Detailed model has constant parameters. Meanwhile, the proposed CPVBR-Detailed model outperforms both qd12- and VBR-Detailed models in terms of the CPU time (8.49 s CPVBR vs. 13.42 s qd12 vs. 33.92 s 82 VBR). The faster performance of the CPVBR-Detailed model is due to its constant parameter interfacing circuit (compared to VBR-Detailed) and avoiding snubber circuits that introduce stiffness (compared to qd12-Detailed). Moreover, the CPVBR-Detailed and VBR-Detailed models take almost the same number of steps (56,891 and 54,966, respectively) which are less than the number of steps required by the qd12 model (137,345).  As it can be concluded based on Tables 3.2 and 3.3, in applications where the switching details of the rectifier have to be considered, the proposed CPVBR-Detailed machine-rectifier model achieves the best performance with either stiff or non-stiff solvers, assuming that the accuracy of CPVBR model of PMSM is acceptable.   Table 3.3. Computational performance of the subject detailed and averaged models for the considered 5-second transient study with the stiff solver ode23tb Model Number of steps CPU Time (s) Machine - Rectifier qd12-Detailed 137,345 13.42 VBR-Detailed 54,966 33.92 CPVBR-Detailed 56,891 8.49 qd12-MRF-PAVM 16,760 2.40 qd12-PAVM 5,029 0.21  83 3.4.3.4 Performance of AVMs vs detailed models As it can be observed in Table 3.2, the CPU times associated with qd12-MRF-PAVM and qd12-PAVM models with ode45 non-stiff solver are 2.92 s (in 14,649 steps) and 0.81 s (in 12,851 steps), respectively. Also, Table 3.3 shows that it takes 2.40 s (in 16,760 steps) and 0.21 s (in 5,029 steps) to run the qd12-MRF-PAVM and qd12-PAVM models with ode23tb stiff solver, respectively. Overall, the qd12-PAVM and qd12-MRF-PAVM models offer much faster simulations compared to the three detailed models (i.e., qd12/VBR/CPVBR-Detailed) by eliminating the discrete switching states of the rectifier. However, the qd12-PAVM model is only capable of predicting the dc and fundamental components of ac variables. Meanwhile, the proposed qd12-MRF-PAVM also captures the dominant ac harmonics that constitute much of the details of the ac waveforms.  It is also worth mentioning that the lookup tables of the presented MRF-PAVM have been constructed using a 14-s simulation of the power system of Figure 3.1 over the desired range of loading conditions (see Section 3.3) using the CPVBR-Detailed model. With the software settings presented in Section 3.4.3, it took approximately 49 s of CPU time to obtain the parametric functions. This time may be taken into consideration when choosing between the presented MRF-PAVM and the detailed models. Since computing parametric functions are a one-time process, the difference between the required CPU times for the detailed model and MRF-PAVM in simulations can determine the minimum 84 number of studies that make the MRF-PAVM preferable in terms of simulation time (which depends on the specific studies and length of each simulation). Therefore, for system-level studies with many long transient studies, wherein capturing the system fast dynamics and accurate waveforms (without switching) is sufficient, the proposed qd12-MRF-PAVM of the machine-converter system may be the best choice with either stiff or non-stiff solvers. 85 CHAPTER 4: GENERALIZED PAVM OF LINE-COMMUTATED RECTIFIERS The thyristor-controlled line-commutated rectifiers (LCRs) are commonly used in high power industrial applications as well as variable frequency/speed rectifiers in aircraft, ships, variable-speed wind turbines, etc., where the prime mover and generator speed is expected to change and the thyristor firing control is used to regulate the dc-link voltage. This Chapter first extends the previous work on PAVM of thyristor-controlled rectifiers [57] by providing a numerical verification that although the thyristor-controlled rectifier parametric functions are calculated at a fixed/nominal frequency, the resulting PAVM is valid for variable speed and variable frequency operation in steady-state and transients including the thyristor firing angle control dynamics. This represents an important feature of PAVMs and its direct application to many practical systems with variable speed, which is not obvious from [57]. Next, the PAVM methodology is generalized by formulating the ac harmonics for thyristor-controlled rectifiers and considering their dependency on line frequency. 86 4.1 Verification of PAVM of Thyristor-Controlled Rectifier Systems for Variable-Frequency Wind Generation Systems Here, without loss of generality, a variable speed permanent magnet (PM) synchronous-machine-based wind generation system is considered as depicted in Figure 4.1. In the PAVM method [57], the thyristor-controlled rectifier switching cell is substituted by an algebraic block as depicted in Figure 4.2. The algebraic block utilizes numerically constructed functions which are defined in terms of averaged transformed ac variables (in qd coordinates) and dc variables [57]. The parametric functions are acquired by simulating the detailed model of the system for the desired range of operating conditions and storing them in two-dimensional lookup tables in terms of the rectifier firing angle   and the rectifier dynamic impedance [57].   abcvabciDC NetworkoutvdcvdciPLLFiring Pulse GeneratorDC Voltage ControllerReference DC Voltagefr flfcGear Box Figure 4.1. PM synchronous machine-based wind generation system consisting of a thyristor-controlled-rectifier.  87 AC Side Power SystemThyristor Rectifier PAVMdcvdciabcsiabcsvDC Side Power System(Firing Angle)qdsvqdsiabc-qd0Transformation Figure 4.2. Implementation of PAVM of the thyristor-controlled rectifier.   It is noted that the functions in PAVM technique [57] are obtained at fixed/nominal line frequency (i.e., 50/60 Hz). It is shown that although these functions [57, Eqs. 3–5] are obtained at a fixed nominal frequency, the PAVM is valid for transient studies over a wide range of operating conditions including variable speed/frequency operation, with and without closed-loop control of thyristor firing angle. 4.1.1 Computer studies Here the system of Figure 4.1 is considered for study with PMSM parameters summarized in [98] and dc side parameters presented in Appendix F. Here, the dc network is represented as a resistive load of 30 lr   . The rectified output voltage versus frequency is illustrated in Figure 4.3 for different rectifier firing angles, verifying the accuracy of PAVM in predicting the detailed model steady-state behavior for a wide range of frequencies and operating conditions. In order to verify the transient performance of PAVM in variable frequency wind generation scenario, the system is assumed to initially operate in steady-state at 88 60Hz with thyristor firing angle set to 15 degrees. The operating frequency is assumed to change according to the profile depicted in Figure 4.4 (a). To emulate a very severe change in wind speed and verify the PAVM accuracy under abrupt conditions, at 6 st  , the frequency is stepped from 50 to 40 Hz. For this instance, the magnified view of transient in several variables of interest is shown in Figure 4.5. At 11 st  , the dc voltage controller is activated to maintain the output voltage at 300 volts by controlling the thyristor firing angle, where the magnified view of transient in several variables of interest at this moment is shown in Figure 4.6. The results shown in Figures 4.4 – 4.6 verify that the PAVM is capable of accurately capturing the steady-state and transient response of system variables under variable frequency operations with/without control dynamics even under abnormally fast transients (as validated in Figures 4.4 and 4.5 for 6 st  ).  0 20 40 60 800200400Frequency (Hz)v out (V)       Detailed Model    PAVM =30o=60o=0o=75o Figure 4.3. The output voltage of thyristor-controlled rectifier versus frequency for different values of thyristor firing angles. 89 40506070f (Hz) 200300400v out (V)   Detailed ModelPAVM0 5 6 10 11 15153045Time (s) (deg.)   Detailed ModelPAVM(a)(c)See Fig. 5See Fig. 6DC VoltageController Activation(b) Figure 4.4. System transient response under variable frequency operation without and with thyristor control as predicted by the detailed model and PAVM of thyristor-controlled rectifier: (a) assumed frequency profile; (b) rectifier dc output voltage; and (c) thyristor firing angle.   4812  i dc (A) 250300350v dc (V)   Detailed ModelPAVM102030  T (N.m) 5.95 6 6.05 6.1-10010  Time (s)i a (A)  Figure 4.5. Magnified view of the transient response of system variables as predicted by the detailed model and PAVM of the thyristor-controlled rectifier in open-loop operation for: (a) rectifier dc voltage, (b) rectifier dc current, (c) generator torque, and (d) line current.  90 102030  T (N.m) 200300400v dc (V)   Detailed ModelPAVM5913  i dc (A) 10.95 11 11.05 11.1 11.15-10010  Time (s)i a (A)  Figure 4.6. Magnified view of the transient response of system variables as predicted by the detailed model and PAVM of the thyristor-controlled rectifier in closed-loop operation for: (a) rectifier dc voltage, (b) rectifier dc current, (c) generator torque, and (d) line current.    4.2 Generalizing the PAVM Methodology for AC-DC Line-Commutated Rectifiers The PAVMs of line-commutated rectifiers in [52]–[63] are accurate in all CCM and DCM operating modes; however, these models only consider the fundamental components of ac variables. When considering only the dc and fundamental components of ac variables, these PAVMs are also shown to be accurate in variable frequency operation for diode bridges in [53] and for thyristor-controlled rectifiers in previous section 4.1. Recently, the PAVM methodology has been extended to include the ac harmonics for diode rectifiers using the multiple reference frame theory, namely the MRF-PAVM [64]. The 91 MRF-PAVM [64] is shown to reconstruct the ac harmonics in steady-state and transients but only at the fixed nominal frequency.  In this section, the PAVM methodology is generalized by including the ac harmonics considering their non-linear dependency on the line frequency for thyristor-controlled rectifiers. Using extensive computer simulations and measurements from the experimental machine-converter system, it is demonstrated that the proposed generalized PAVM (GPAVM) reconstructs the detailed model waveforms (including harmonics) with excellent accuracy under various loading conditions and operating modes as well as variable frequency operation. Also, the proposed GPAVM is shown to be highly accurate in predicting the efficiency of non-ideal converters with conduction losses*.  The term “generalized” in GPAVM refers to the comprehensive features of the proposed model as summarized in Table 4.1 compared to prior related average-value models. Specifically, it is shown that any previous PAVM is just a special case of the proposed GPAVM, and they can be obtained by using only a subset of parametric functions used in GPAVM. This makes the GPAVM a useful simulation asset applicable to many systems with variable speed/frequency operations.   *Losses of converter are calculated here by subtracting the output power from the input power of the non-ideal converter.  92 Table 4.1. Capabilities of existing state-of-the-art AVMs of line-commutated rectifiers with respect to the proposed generalized PAVM (GPAVM) Model Thyristor Operation DCM Mode CCM Modes Harmonics Variable Frequency Prediction of Losses 1 2 3 Fundamental ac & dc  Harmonics AAVM [38], [41], [42] √ N/A √ N/A N/A N/A √ N/A N/A AAVM [44], [45] √ N/A √ N/A N/A √ √ √ N/A AAVM [12], [47] N/A N/A √ √ √ N/A √ N/A N/A PAVM [52]–[55], [59]–[63] N/A √ √ √ √ N/A √ N/A √ PAVM [57], [58] √ √ √ √ √ N/A √ N/A √ MRF-PAVM [64] N/A √ √ √ √ √ √ N/A √√ Proposed GPAVM √ √ √ √ √ √ √ √ √√ Note:  The N/A implies that the model does not have the capability; √ indicates the capability; and √√ indicates capability with higher accuracy.  4.2.1 The proposed GPAVM methodology In this section, a generic variable speed ac–dc power system is considered as depicted in Figure 4.7. The system consists of an ac subsystem that may be a rotating machine (e.g., in renewable energy generation, vehicular or aircraft generator, etc.). The ac-side subsystem may also be a conventional ac network represented by its Thévenin equivalent sources and impedances as shown in Figure 4.7. A voltage-behind-reactance machine model may also be interfaced to the LCC using equivalent interfacing circuit with sources and impedances [99].   93 abcsvabcsidcvdciLCCPLLFiring Pulse GeneratorDC Voltage Controller Reference DC Voltager eDC NetworkAC-Side Sub-SystemGear BoxVariable Speed Wind3-Phase GeneratorThevenin Equivalent SystemoutvfR fLfCDC FilterDC-Side Sub-System Figure 4.7. A generic variable speed ac–dc power system consisting of a line-commutated rectifier.  The dc-side subsystem consists of a filter to smoothen the rectifier output voltage, which supplies the dc network (load). The LCC rectifier depicted in Figure 4.7 may be controlled by thyristor firing angle delay, which may be determined with respect to machine rotor angle (or the angle of ac source equivalent voltages) r , or with respect to the angle of rectifier ac-side terminal voltages e  [41], [42], [57]. The angle e  is defined as    ,  2e e e edt f     , (4.1) where ef  and e  are the line frequency of the electrical system in Hz and rad/s. The relationship between r   and e  can also be expressed as  r e    , (4.2) where   is the angle by which the generator back-emf voltages (or the Thévenin equivalent source voltages) lead the fundamental component of the rectifier terminal voltages. Generally, for p-pulse rectifiers, the ac voltages and currents will contain n-th order harmonics, where 94     1, 1, 1,...  ,  1,2,3,...n ip ip i    . (4.3) For the purpose of this methodology, the converter ac voltages abcsv  and currents abcsi  in Figure 4.7 can be expressed using their corresponding Fourier series as   1 ,  cosn n n nas as as as e vnv v v V n    , (4.4)   1 ,  cosn n n nas as as as e ini i i I n    , (4.5) where nasV  and nasI  are the amplitudes of the n-th harmonics of voltages and currents, nv  and ni  are their corresponding phase angles, respectively. For notational convenience, the phase a voltage fundamental component is considered as the reference, which yields 1 0v  . Assuming symmetry, similar formulations can be obtained for other phases, nbsv , nbsi  and ncsv ,  ncsi   by replacing e  in (4.4)–(4.5) with  2 3e  , respectively. 4.2.1.1 Derivation of GPAVM formulation To capture the relationship between the dc and the fundamental components of ac variables of the rectifier, the ac variables in abc coordinates (i.e., abcsv  and abcsi ) are first transformed to the synchronously rotating converter reference frame, denoted by 1eqd , as depicted in Figure 4.8(a). It is also assumed that zero-sequence does not exist in the power system of Figure 4.7. In this section, the subscript “e” is used to denote the converter reference frame, wherein the  95 nasvaxisneq enaxisned nvnnasi,n edsi,n eqsi,n edsv,n eqsvni1 1,, eas qsv v1 axiseq 1 axisrq er1 axisrd 1 axised 11asi1,rdsi1,rqsi1i(a)(b)1,edsi1,eqsi1,rqsv1,rdsv Figure 4.8. Phasor diagram of rectifier ac voltages and currents expressed in multiple qd rotating synchronous converter reference frames (denoted by subscript ‘e’) and source reference frame (denoted by subscript ‘r’): (a) fundamental components of ac variables; and (b) n-th harmonics.  angular displacement of transformation is chosen to be e , which aligns the 1eq -axis with 1asv  [52], as depicted in Figure 4.8(a). The transformations of voltages and currents are carried out using the Park’s transformation matrix nsK [64] as  1, 1 1, 1( )  , ( )e eqds s e abcs qds s e abcs  v K v i K i . (4.6) After transformations, the terms in (4.4)–(4.5) associated with 1asv  and 1asi  will become dc components in 1eqd  coordinates, while all other harmonic terms become oscillatory with zero average values. For the purpose of deriving the average-value model, the oscillatory components resulted from the switching phenomenon are eliminated from the transformed ac variables and also from the 96 rectifier dc-side variables by the means of fast dynamic averaging over a switching period 1 eT pf  as      1 tt Tx t x dT   , (4.7) where the variable x may denote currents and voltages. The bar sign (–) above denotes the average value. In converter reference frame, the angle of ac current 1asi  with respect to 1eq -axis is equal to 1i , as depicted in Figure 4.8(a). Also, the angle by which 1asi  is displaced from 1asv  is the power factor angle which can be expressed as,  1 1i   . (4.8) For the fundamental components of ac variables in the converter reference frame, the following relationships exist [38]  1, 1, 1, , 0e e eqs qds dsv v v , (4.9)     1, 1, 1 1, 1, 1cos  , sine e e eqs qds i ds qds ii i  i i . (4.10) At any moment, there exist parametric functions 1( )iw   and 1( )vw   that relate the magnitudes of ac fundamental components in (4.4)–(4.5) and the dc variables as  1 1 111( )  , ( )as v dc as dciV w v I iw  . (4.11) Since 1asV  is equal to 1,eqdsv  and 1asI  is equal to 1,eqdsi , these parametric functions can be defined as 97  1,1 11,( )  ,   ( )eqdsdcv iedc qdsiw wv   vi. (4.12) Moreover, the displacement angle between the fundamental components of ac variables can also be defined as a parametric function,   1,1 11,( ) tanedseqsii       . (4.13) To generalize the methodology and capture the relationship between the dc variables and any desired n-th harmonic of ac variables,abcsv and abcsi  are transformed to the n-th harmonic converter reference frame neqd  that rotates n-times faster than 1eqd , as illustrated in Figure 4.8(b). The transformations are similar to (4.6) but with the angle  en as  , ,( )  , ( )n e n n e nqds s e abcs qds s e abcsn n  v K v i K i . (4.14) Similarly, the terms in (4.4)–(4.5) associated with harmonics nasv  and nasi  will become dc components in neqd  coordinates, while all other terms become oscillatory with zero average value [where the oscillatory terms are averaged-out by the means of fast averaging (4.7)]. In the neqd  reference frame, the angles nv  and ni  in (4.4)–(4.5) are the angles of harmonics nasv  and nasi  with respect to neq -axis as depicted in Figure 4.8(b). These angles can be expressed as  , ,1 1, ,tan  , tann e n en nds dsv in e n eqs qsv iv i               . (4.15) 98 Also, the harmonic current nasi  is displaced from the harmonic voltage nasv  by the angle defined as  n n nv i    . (4.16) Moreover, the following relationships can be written for the magnitude of harmonics of the transformed ac variables:     ,e , , ,cos  , sinn n e n n e n e nqs qds v ds qds vv v  v v , (4.17)     , , , ,cos  , sinn e n e n n e n e nqs qds i ds qds ii i  i i . (4.18) Similarly, at any moment there exist parametric functions ( )niw   and ( )nvw   that relate the magnitudes of ac harmonics in (4.4)–(4.5) and the dc variables as  1( )  ,   ( )n n nas v dc as dcniV w v I iw  . (4.19) Since nasV  is equal to,n eqdsv  and nasI  is equal to ,n eqdsi , these parametric functions for harmonics are defined as  ,,( )  , ( )n eqdsn n dcv in edc qdsiw wv   vi. (4.20) Moreover, the angle of harmonic voltages ( )nv   and the angle of harmonic currents with respect to voltages ( )n   are also defined as parametric functions:  , , ,1 1 1, , ,( ) tan  ,  ( ) tan tann e n e n en nds ds dsv n e n e n eqs qs qsv v iv v i                           . (4.21) 99 Comparing (4.20)–(4.21) with the Fourier series coefficients in (4.4)–(4.5), the rectifier ac variables can be reconstructed from the dc variables using the parametric functions as    1( ) cos ( )mn nas v dc e vnv w v n     , (4.22)     1cos ( ) ( )( )mn ndcas e vnn iii nw            , (4.23) where m is the highest desirable harmonic to be included in the model. 4.2.1.2 Establishing parametric functions Generally, it is impractical to derive the parametric functions ( )niw  , ( )nvw  , ( )nv   and ( )n   analytically for LCC rectifiers under various operating conditions, especially when the losses are considered. Therefore, the GPAVM uses a computer-aided technique [54] to establish the parametric functions from a brief set of studies using the detailed model of the considered system.  It is convenient to specify the converter operating point in terms of its terminal currents and voltages.  Depending on the AVM formulation, one may consider using different combinations of the ac and dc voltages and currents. For example, if the ac voltages and the dc current will be the inputs into the AVM, then these variables can be used to define the so-called dynamic impedance dz  with respect to fundamental components. However, since in many applications the ac subsystem may be a rotating machine model that requires voltages as the 100 inputs, in this section the converter dynamic impedance is defined using the ac currents fundamental component and dc voltage as  1,dcdeqdsvz i. (4.24) Assuming that the GPAVM will have ac voltages and dc current as the outputs, as in [57], the parametric functions relating the fundamental components of ac voltages and currents are defined in terms of impedance dz  and thyristor firing angle   as  1,1 11,( , )  , ( , )eqdsdci d v dedcqdsiw z w zv  vi, (4.25)  1,1 11,( , ) tanedsd eqsizi       . (4.26) The parametric functions for harmonic voltages and their phase angles are also specified as   ,( , , )  , 1n eqdsnv d edcw z f nv  v, (4.27)   ,1,( , , ) tan  , 1n en dsv d e n eqsvz f nv       . (4.28) Adding ef  and   as the arguments to the parametric functions (4.27)–(4.28) enables the proposed model to consider the non-linear dependency of the ac harmonics on the line frequency as well as the thyristors firing angle. 101 To establish parametric functions (4.24)–(4.28), the detailed model of the power system of Figure 4.7 is simulated over the desired range of frequencies (ef ), thyristor firing angles ( ) and loading conditions ( dz ). The pseudo-code presented in Figure 4.9 (Algorithm 1) can be used to numerically obtain the parametric functions. Therein, ,minef , ,maxef , min  and max  are the minimum and maximum desired frequency and firing angle, with ,stepef  and step  as their resolution in parametric functions, respectively.    Figure 4.9. Pseudo-code for establishing parametric functions of the proposed GPAVM.  Algorithm 1. Establishing parametric functions using detailed simulation. 1.  for ,mine ef f  to ,maxef  step ,stepef  do 2.     for min   to max  step step  do 3.          Initialize the detailed model 4.          Start simulation, change dc load over the desired range             5.          Compute dynamic impedance (4.24) 6.          if ef nominal frequency then 7.             Compute and process parametric functions (4.25)–(4.26) 8.             Save (4.25)-(4.26) in 2-D lookup tables 9.          end if 10.        Compute and process parametric functions (4.27)–(4.28) 11.        Save (4.27)-(4.28) in 3-D lookup tables 12.        End simulation            13.     end for 14.  end for 102 The computed functions (4.25)–(4.26) are stored in two-dimensional lookup tables (identical to PAVMs [57]), and the functions (4.27)–(4.28) are stored in three-dimensional lookup tables, respectively. To demonstrate the result of Algorithm 1 for the system of Figure 4.7 with a 6-pulse LCC (with parameters given in Appendix G), the parametric functions of the MRF-PAVM for diode rectifiers [64], the PAVM for thyristor-controlled rectifiers [57], and the proposed GPAVM are presented in Figure 4.10 for the dc variables and the fundamental components of ac variables. As can be seen in Figure 4.10, the proposed GPAVM uses parametric functions 1( )iw  ,1( )vw   and 1( )   that are identical to the PAVM [57] for thyristor-controlled rectifiers; whereas, the parametric functions of MRF-PAVM [64] for diode rectifiers can be realized by setting the firing angle   equal to zero. This demonstrates the consistency of the new GPAVM with the prior related models in terms of the parametric functions for the fundamental components for diode and thyristor rectifiers.  103  Figure 4.10. Parametric functions of the MRF-PAVM [64] for diode rectifiers, PAVM [57] for thyristor-controlled rectifiers, and the proposed GPAVM: (a) dc current; and (b)-(c) magnitude and phase of fundamental components of ac variables, respectively.  It is also possible to verify the consistency of the new GPAVM parametric functions for the 5th and 7th harmonics at the nominal fundamental frequency (i.e., 60 Hz source) with the prior model MRF-PAVM [64]. Figure 4.11 shows the parametric functions for harmonic voltages 5vw ,7vw , and their angles 5v , 7v . As it can be seen in Figure 4.11, the MRF-PAVM parametric functions can be realized as sub-sets of the proposed GPAVM parametric functions by setting 0  .   104  Figure 4.11. Parametric functions of the MRF-PAVM [64] and the proposed GPAVM at 60 Hz versus dynamic impedance and firing angle for the magnitude and phase of 5th and 7th harmonics of ac voltages.  To demonstrate the dependency of harmonics on the line frequency, the parametric functions of MRF-PAVM [64] and the new GPAVM (with 0  ) for the 5th and 7th harmonics are illustrated in Figure 4.12. As it can be seen in Figure 4.12, the proposed GPAVM considers the frequency dependency of harmonics, while the MRF-PAVM functions are constructed only at nominal frequency (60 Hz). 105  Figure 4.12. Parametric functions of MRF-PAVM [64] and the proposed GPAVM (with α=0) versus dynamic impedance and line frequency for magnitude and phase of 5th and 7th harmonics of ac voltages.  4.2.1.3 Implementation of GPAVM Once the parametric functions (4.24)–(4.28) are established, the switching rectifier is replaced by a non-switching circuit composed of dependent voltage and current sources that are algebraically related and interfaced with the ac and dc subsystems [52]. It is worth mentioning that depending on the required 106 input-output interfaces, the proposed GPAVM can be formulated differently for relating the ac and dc variables. Moreover, the rectifier ac-side subsystem can be represented either in abc phase coordinates or in transformed qd coordinates.  For the later case, the qd reference frame may be fixed at the ac source (if the ac subsystem is represented by its Thévenin equivalent voltage sources) or on the rotor (if the ac subsystem is a rotating machine), as can be seen from Figure 2.8(a). Therein, the subscript “r” is used to denote the rotor (or source) reference frame. In this section, without loss of generality and for the convenience of interfacing the LCC rectifier with models of electrical machines, the rectifier terminal voltages abcsv (or rqdsv ) and the rectifier dc current dci  are considered as the outputs. Also, the source currents abcsi (or rqdsi ) and the rectifier dc voltage dcv , as well as the thyristor firing angle  , are considered to be the inputs. The corresponding implementation of the GPAVM is illustrated in Figure 4.13.  1,rqdsiabcsvabcsidcvef1iwnvwnvdzReference DC VoltageDC Voltage ControllerFiring Angle Set PointAC-SideSub-SystemrEq. (24)Eq. (32)Eq. (6)Eqs. (30)-(31)Eq. (29)Functions (27)-(28) n m 1n ip Functions (27)-(28) 1n ip Functions (27)-(28)1vw1dcirqdsvEq. (35)ePLLEqs. (1),(2)Functions (25)-(26)1( )e i DC-Side Sub-SystemEq. (36)Eq. (34)Eq. (33)Eqs. (30)-(31)Eqs. (30)-(31) 1n  Figure 4.13. Implementation of the proposed GPAVM.  107 A phase-locked-loop (PLL) may be used to obtain the line frequency ef  as well as e  from the rectifier terminal voltages if these variables are used for thyristor control. Alternatively, if the control is implemented with respect to the rotor position r ,  the converter reference frame angle e  and frequency can also be computed using (4.1)–(4.2). The magnitude of the fundamental component of the current 1,rqdsi  is derived using (4.6) and used to compute the dynamic impedance dz  based on (4.24), which also requires dcv . Here, it is assumed that dcv  is calculated by the dc-side subsystem using the current dci , as explained in [52, see Fig. 4(b) and (8)-(9)]. The firing angle   is an input that may be set manually (open-loop) or computed by the dc voltage controller (closed-loop). The sub-model for each harmonic requires the arguments dz ,   and ef  as the inputs to the parametric functions 1( )iw  , ( )nvw  ,1( )   and ( )nv  . Based on input dcv  and calculations with parametric functions, the fundamental voltages are computed (in converter reference frame) as  1, 1( , )eqs v d dcv w z v ,1, 0edsv  . (4.29) Similarly, the harmonic voltages are calculated as     ,e ( , , ) cos ( , , )  , 1n n nqs v d e dc v d ev w z f v z f n    , (4.30)     ,e ( , , ) sin ( , , )  , 1n n nds v d e dc v d ev w z f v z f n    . (4.31) The total rectifier voltages are reconstructed from the fundamental and harmonic voltages as 108    1,1( 1)me n n eqds s e qdsnn  v K v . (4.32) For interfacing with the rectifier ac-side subsystem, assuming that the source voltages and currents are in source (rotor) reference frame (i.e. rqdsv  and rqdsi ), the GPAVM output voltages can be transformed as        cos sinsin cosr eqds qds      v v , (4.33) where   is calculated based on Figure 4.8(a) as  1,1 11,( , ) tanrdsd rqsizi         . (4.34) For interfacing with the rectifier ac-side subsystem in abc coordinates, the GPAVM output voltages are calculated as  11 ( ) rabcs s r qds   v K v . (4.35) Finally, the dc interfacing output current is calculated as  1 1,( , ) edc i d qdsi w z  i .  (4.36) 4.2.2 Verification of the proposed GPAVM For the purpose of model verification, a representative variable frequency wind generation system as depicted in Figure 4.7 is considered. The power system consists of a synchronous generator-turbine whose speed/frequency changes according to the assumed wind speed profile. To verify the proposed GPAVM methodology, a reduced-scale experimental setup (with parameters 109 summarized in Appendix G) has been implemented as shown in Figure 4.14. Therein, the LCC is connected to a PM synchronous generator whose speed and the resulting output voltages are controlled by a prime mover which emulates the wind speed variations. The detailed model has been implemented in PLECS [29] blockset in MATLAB/Simulink [27], where the coupled-circuit phase-domain (CCPD) model [38, Chap. 5] is used for the PM synchronous machine. The LCC rectifier and the dc subsystem are also implemented using the PLECS built-in library components. In the detailed model, the LCC conduction losses are represented    Figure 4.14. Experimental setup of power system under study: (1) dc supply for the prime mover; (2) prime mover which emulates wind speed changes; (3) PM synchronous generator; (4) data acquisition system; (5) dc filter inductor; (6) dc filter capacitor and LCC; and (7) resistive load which represents a dc sub-system. 110 by the ON resistance Ron and voltage drop Von of the switches. A piece-wise linear approximation is used to derive Ron and Von from the measured voltage-current characteristic of the considered diodes. For the purpose of comparison, the MRF-PAVM [64] for diode rectifiers and PAVM [57] for thyristor-controlled rectifiers have been implemented in MATLAB/Simulink [27] as well. The AVMs of the LCC rectifier are interfaced with conventional qd machine model [38, Chap. 5]. 4.2.2.1 Diode rectifier variable frequency operation Here, the proposed GPAVM is verified against the experimental setup, the detailed model, the PAVM and the MRF-PAVM for the diode rectifier operation. The dc network is also represented by an equivalent variable resistive load, lR . Variable frequency operation is considered by varying the prime mover speed corresponding to electrical frequency range from 10 to 90Hz. The dc subsystem equivalent resistance is set to 32.92 lR    resulting in rectifier operating in CCM-1 mode [49]. The rectifier dc output voltage, the ac fundamental component as well as the 5th and 7th harmonics (magnitude and phase) are illustrated in Figure 4.15 as obtained from the measurements and predicted by the subject models. As it can be observed in Figures 4.15(a)–(c), the generated dc and fundamental component of ac voltages linearly increase with the speed/frequency of the generator, which also results in an increase of real power and rotor angle  . All 111 subject models predict this trend very well and have very good agreements with the experimental results (green cross “x”). The detailed switching model also predicts the nonlinear behavior of harmonics and their angles (black circle “o”). However, the magnitude and phase of ac harmonics predicted by the MRF-PAVM [64] are linear with frequency and only valid at the nominal 60 Hz, while the proposed GPAVM is accurate for the entire considered range and remains consistent with the detailed model and experimental results. As it can be observed in Figure 4.15, the harmonic magnitude and phase errors associated with MRF-PAVM are very large (even more than 100% in magnitude and 180° shift in phase), while the proposed GPAVM does not introduce error in the magnitude and phase of ac harmonics. To see the impact of the harmonic errors on the final waveforms, the rectifier dc and ac variables in CCM-1 mode at 10 Hz line frequency for the subject models are shown in Figure 4.16. As it can be seen in Figure 4.16, the errors in harmonics of the MRF-PAVM [64] lead to noticeably incorrect waveforms of ac variables, while the proposed GPAVM provides an excellent approximation of the detailed model and experimental results. In the meantime, the PAVM accurately predicts the fundamental components of ac variables; however, it cannot predict their harmonics. The excellent accuracy of the proposed GPAVM has been verified under various operating conditions (all CCM and DCM modes) and frequencies (below and above nominal frequencies), but are not shown here due to space limitation. 112  Figure 4.15. Diode rectifier dc- and ac-side variables as predicted by the subject models and experimental results over a range of frequencies from 10 to 90Hz: (a) output dc voltage; (b) magnitude of fundamental component of rectifier input ac voltage; (c) rotor angle; and (d)-(g) magnitude and phase of 5th and 7th harmonics of rectifier input ac voltage, respectively.  113  Figure 4.16. Diode rectifier variables as predicted by the subject models at 10 Hz and CCM-1 condition: (a) rectifier dc voltage (b) rectifier dc current, (c) rectifier phase a voltage, and (d) rectifier phase a current.        114 4.2.2.2 Thyristor rectifier operation A 10-second computer study has been carried out to verify the proposed GPAVM in various operating conditions compared to the existing PAVM of thyristor-controlled rectifiers and their detailed models in steady-state and transient. 4.2.2.2.1 Transient study at fixed frequency The rectifier is initially operating in DCM mode at 60 Hz with thyristors firing angle set to 67   providing 48V dc for the equivalent dc load with 80 lR   . At 1t   s, the load is decreased to 4 lR    and the rectifier firing angle is stepped down to 15   and the rectifier enters CCM-1 mode. The transients for several system variables of interest are shown in Figure 4.17. The corresponding rectifier ac variables are also illustrated in Figures 4.18 – 4.19 in steady-state. As it can be observed in Figures 4.17 – 4.19, compared to the PAVM which is only capable of capturing the fundamental components, the presented GPAVM reconstructs the ac variables (including their harmonics) in steady-state and transients with excellent accuracy in all operating modes. Also, as shown in Figure 4.17, the proposed GPAVM is able to predict the average value of dc variables very precisely similar to the PAVM of the thyristor-controlled rectifier.     115  Figure 4.17. Transient response of thyristor-controlled rectifier from DCM to CCM-1 mode due to change in load resistance from 80 lR    to 4 lR    and thyristor firing angle from 67   to 15   at 60 Hz, as predicted by the subject models: (a) rectifier dc voltage, (b) rectifier dc current, (c) rectifier line voltage, and (d) rectifier phase a current.  116  Figure 4.18. Thyristor-controlled rectifier ac variables in DCM mode with 80 lR    and 67   at 60 Hz, as predicted by the subject models: (a) rectifier line voltage, and (b) rectifier phase a current.    Figure 4.19. Thyristor-controlled rectifier ac variables in CCM-1 mode with 4 lR    and 15   at 60 Hz line frequency, as predicted by the subject models: (a) rectifier line voltage, (b) rectifier phase a current.   117 4.2.2.2.2 Transient study with variable frequency operation The system is initially operating in steady-state at 60 Hz line frequency with 5   and 10 lR   . At 3t  s, the line frequency starts to fluctuate due to wind speed variations as depicted in Figure 4.20(a). The dc output voltage of rectifier starts fluctuating with the same profile as shown in Figure 4.20(b) since the rectifier firing angle is kept constant (the dc voltage controller is not activated). At 7.5t  s, the dc voltage controller is activated to maintain the output voltage at 48V dc, which controls the rectifier firing angle as shown in Figure 4.20(d). The resulting transients of several variables of interest are also depicted in Figure 4.20. A magnified view of several ac variables is presented in Figure 4.21. As it can be observed in Figures 4.20 – 4.21, the proposed GPAVM is able to very precisely reconstruct the fundamental components and harmonics of ac variables as well as the average values of dc variables in variable frequency operation.  It is worth mentioning that in case there are controllers in the ac-side subsystem that are sensitive to harmonics, using the GPAVM instead of the original detailed model should not negatively affect the controllers as the GPAVM provides very similar waveforms (up to the number of considered harmonics). For the controllers on the dc-side subsystem, the rectifier switching ripples are typically not considered when designing the control loops, and the controllers are typically designed using linearized models (e.g., average-value linearized/analytical models). This has been demonstrated in in Figure 4.20(d) where the same PI controllers (parameters summarized in Appendix G) have 118 been used for the detailed model, the PAVM, and the proposed GPAVM to control the dc-side voltage with thyristor firing angle control. The performance of GPAVM with controllers has also been verified in other operating modes, including CCM-2, but these results have not been included due to space limitation.   Figure 4.20. Transient response of the system due to variable frequency operation and voltage control activation as predicted by the subject models: (a) line frequency, (b) output dc voltage, (c) rectifier dc current, and (d) rectifier firing angle. The voltage controller is activated at 7.5t  s to regulate the output voltage at 48V dc. 119  Figure 4.21. Rectifier ac variables at the moment of activating the voltage controller as predicted by the subject models: (a) rectifier line voltage, and (b) rectifier phase a current.  4.2.2.2.3 Prediction of rectifier efficiency (losses) The proposed GPAVM is also verified in predicting the efficiency (and losses) of the non-ideal thyristor-controlled rectifiers in several loading conditions and firing angles at 60 Hz. The computed converter efficiencies are summarized in Table 4.2 for the detailed model, the PAVM, and the GPAVM.  As shown in Table 4.2, by including the dominant ac harmonics (here 5th and 7th), the GPAVM predicts the efficiency of thyristor-controlled rectifier more accurately than the existing PAVM. This is due to the fact that the PAVM considers the energy dissipated in the line resistance only due to the fundamental components of ac currents, whereas the GPAVM considers also the power loss caused by the ac harmonics. Meanwhile, the PAVM and the GPAVM predict identical output dc power for rectifier since both reconstruct ripple-free 120 dc voltages and currents. Generally, the PAVM predicts slightly lower efficiency compared to GPAVM since it predicts higher power at the rectifier input terminal (due to not considering the power loss of harmonics), which is consistent with the data in Table 4.2. Therefore, the superior accuracy of GPAVM in predicting the converter efficiency would be more noticeable when the converter operates in nominal/high-current conditions with pronounced distorted voltages due to thyristor operation. For example, for 67   and 4 lR    the absolute errors associated with GPAVM and PAVM are 0.07% and 1.86%, respectively. However, the relationships between each harmonic and its contribution to the losses under different operating conditions/modes are very complicated/nonlinear, and beyond the scope of this section. It should also be mentioned that the GPAVM predicts slightly lower efficiency compared to the detailed model, as it can be observed in Table 4.2. This is due to neglecting the output power transmitted by the dc-side ripples as a result of averaging the dc variables.  Table 4.2. The efficiency of thyristor-controlled rectifier at 60 Hz in different operating conditions as predicted by the subject models Model 67  deg.  15  deg.  5  deg. 80 lR    4 lR     4 lR     10 lR    0.1 lR    Detailed 97.15% 91.30%  94.28%  96.87% 75.02% GPAVM 96.96% 91.23%  94.27%  96.86% 74.98% PAVM 96.38% 89.44%  93.66%  96.45% 74.81%  121 4.2.2.3 Computational performance Since the proposed GPAVM requires additional (or higher dimensional) lookup tables to properly implement the harmonics, the computational performance of GPAVM is investigated to evaluate how the increased complexity affects the model performance compared to MRF-PAVM and PAVM. Therefore, the detailed model, the MRF-PAVM [64], the PAVM [57], and the proposed GPAVM have been executed on a PC with Intel® Core™ i7-4510U @2.00GHz processor using the MATLAB/Simulink [27] ode23tb solver with the maximum step-size set to 10-3 s, and the relative and absolute tolerances also set to 10-3. Moreover, since MRF-PAVM [64] (for diode rectifiers with harmonics) and PAVM [57] (for thyristor rectifiers without harmonics) have different features, separate comparisons to these models are made.  4.2.2.3.1 Performance compared to MRF-PAVM Here, the same study as in Section 4.2.2.1 is used. In this study, the line frequency slowly increases from 10 to 90 Hz using a ramp function over a period of 80 seconds. The computational performance of the subject models is summarized in Table 4.3. As it can be seen in Table 4.3, the detailed model is the slowest, as expected. At the same time, both GPAVM and MRF-PAVM have very similar performance in terms of CPU time (11.19 s vs. 10.53 s), the number of time-steps (110,325 vs. 106,576), and the CPU time per-step (101 vs. 99 µs). These results can be explained by the fact that both models include the same 122 number of harmonics and therefore have very similar computational cost; however, the GPAVM considers the frequency dependency of ac harmonic in its parametric functions. Overall, the proposed GPAVM executes about 9 times faster than the detailed model. The computational performance of PAVM has also been summarized in Table 4.3, with results of 5.69 s CPU time, 80,006 time-steps, and 71 µs CPU time per-step. These results show that including harmonics in MRF-PAVM and GPAVM comes at a cost of increased computational complexity and CPU time compared to PAVM, where only the fundamental components of ac variables are computed. Table 4.3. Computational performance of the subject models of diode rectifier system for the considered 80-second variable frequency study Model CPU Time (s) Number  of steps CPU Time per Step (µs) Detailed 103.90 348,426 298 MRF-PAVM 10.53 106,576 99 Proposed GPAVM 11.19 110,325 101 PAVM 5.69 80,006 71  4.2.2.3.2 Performance compared to PAVM Here, the same study as in Sections 4.2.2.2.1 and 4.2.2.2.2 is used. The computational performance of the subject models for the 10-second transient study is summarized in Table 4.4. As expected, the detailed model is the slowest model due to the effort required to reproduce the switching. The PAVM considers only the fundamental components and therefore is faster than GPAVM 123 in terms of simulation speed (0.96 s vs. 2.51 s). The increased computational cost of the GPAVM is associated with inclusion of harmonics, which has higher cost per time-step (93 µs vs. 124 µs) as well as leading to a larger number of time steps (10,417 vs. 20,238) since the higher frequency harmonics generally require smaller time-steps. At the same time, the GPAVM executes approximately 13 times faster than the detailed model while preserving much of the details (dominant harmonics) in the ac waveforms. Table 4.4. Computational performance of the subject models of the thyristor-controlled rectifier for the considered 10-second transient study Model CPU Time (s) Number of steps CPU Time per Step (µs) Detailed 32.49 66,736 487 PAVM 0.96 10,417 93 Proposed GPAVM 2.51 20,238 124       124 CHAPTER 5: PARAMETRIC AVERAGE-VALUE MODELING OF LINE-COMMUTATED RECTIFIERS WITH INTERNAL FAULTS AND ASYMMETRICAL OPERATION Analysis and study of power systems consisting of LCRs under unbalanced conditions are of crucial importance [49], [100]. The unbalanced condition may occur due to various reasons, e.g., asymmetric faults on any of the equipment and lines/cables, unbalanced loads, etc. Another type of asymmetrical operation may occur due to internal faults on the switching valves of rectifiers (i.e., diodes or thyristors) [101]–[103], which in addition to the characteristic harmonics (i.e., 5th and 7th) may also introduce non-characteristic ac harmonics (i.e., 2nd, 3rd, 4th, etc.) as well as dc components in ac variables [75], [76] due to the unbalanced operation. Presence of such harmonics can significantly influence the operation of other equipment [17], including protection systems and relays [18], [19], and harmonic compensation measures [20], [21]. The PAVMs of LCRs have been proven accurate in all CCM and DCM operating modes [49]–[64], and give reasonable results even under moderately unbalanced conditions introduced in the three-phase sources of ac subsystem 125 [49]. However, to the best of our knowledge, none of the AVMs presented in the literature are able to capture the asymmetrical operation of rectifiers due to the fault of switches.  In this Chapter, the PAVM methodology is for the first time extended to asymmetric LCRs with internal faults. This is achieved by decomposing the unbalanced ac-side variables into two balanced sets of positive and negative sequences and including both characteristic and non-characteristic harmonics as well as dc components/offsets. The proposed model is validated by extensive simulations and experimental measurements on a reduced-scale laboratory machine-diode rectifier system of Figure 5.1. It is shown that the new PAVM is able to reconstruct the experimental and detailed model waveforms under various operating modes, loading conditions, and asymmetries (i.e., internal fault configurations). Then, the proposed methodology is extended to thyristor-controlled rectifiers in subsequent sections.  abcsvabcsidcvdciSix-Pulse LCRDC NetworkAC-Side Subsystem3-Phase Generator3- Phase Grid ThéveninEquivalent CircuitoutvfLfCDC FilterDC-Side SubsystemfrPrime MoverthLthr4D 2D5D3D6D1D Figure 5.1. A generic three-phase ac–dc system consisting of a diode rectifier with faulty valves.   126 5.1 Decomposition of Instantaneous Variables into Positive and Negative Sequences Without loss of generality, this section assumes an example three-phase ac–dc system as shown in Figure 5.1, which uses a conventional three-phase six-pulse rectifier. The dc-side subsystem may include a low-pass filter (composed of RLC branches) and a dc network. The ac-side subsystem may be composed of either a three-phase generator or a power grid represented by its Thévenin equivalent network. The angle of the rectifier terminal voltages is assumed as the reference for the fundamental frequency ac component as   e edt   , 2e ef  , (5.1) where ef  and e  are the fundamental electrical frequency of the ac network in Hz and rad/s, respectively. Also, the generator rotor angle (or the angle of three-phase Thévenin equivalent sources) is denoted by r , which is related to e  as [52]  r e    , (5.2) where   is the angle between the machine’s back-emf (or the grid’s Thévenin source) voltages and the fundamental components of the LCR terminal voltages. In healthy rectifiers, the ac voltages and currents are symmetric and contain  n = (6k±1)-th harmonics with  1,2,3,...k . Therefore, the 5th, 7th, etc. ac harmonics will be present in the ac-side subsystem. However, in asymmetric 127 (faulty) six-pulse LCRs, the ac variables will be unbalanced and may contain all harmonic components [75], [76] as   0,1,2,3,4,5,6,7,...n , (5.3) where n is the harmonic order. In (5.3), n=0 represents the dc components that may be present during the asymmetrical operation. Hence, in the most general case of a faulty rectifier in Figure 5.1, the unbalanced ac voltages abcsv  and currents abcsi  can be expressed by their Fourier series as   0 0 0 ,   ,   ,n n nas as bs bs cs csn n nv v v v v v          (5.4)  0 0 0 ,   ,   .n n nas as bs bs cs csn n ni i i i i i          (5.5) In (5.4) and (5.5), each phase may contain different harmonics and their waveforms may no longer be identical with ±120º phase shifts. For the purpose of this section, the instantaneous unbalanced three-phase voltages and currents in (5.4) and (5.5) are decomposed into two balanced three-phase sets of positive and negative sequences and possible dc components based on the approach presented in [104] as 128       0,pos ,neg,0 0, pos ,pos neg ,neg10,pos ,pos0 0,negcos cos cos2cos32cos3n nas asas dcn n n nas dc v dc e v e vnbs dc n ne vbs dc v dcv vvv V V n V nvV nv VV                         1,neg0,pos ,pos0 0,1neg ,neg2cos32cos32cos3 2cos3n n ne vcs dc n ne vcs dc v dcn n ne vnvV nv VV n                                   , (5.6)       0,pos ,neg,0 0, pos ,pos neg ,neg10,pos ,pos0 0,negcos cos cos2cos32cos3cn nas asas dcn n n nas dc i dc e i e inbs dc n ne ibs dc i dcni iii I I n I niI ni II                         1,neg0,pos ,pos0 0,1neg ,neg2os32cos32cos3 2cos3n ne ics dc n ne ics dc i dcn n ne iniI ni II n                                   . (5.7) Here, posnV  and posnI  are the amplitudes of the n-th voltage and current harmonics in positive sequence, respectively, with ,posnv  and ,posni as their corresponding phase angles. Similarly, negnV  and negnI  are the amplitudes of the n-th voltage and current harmonics in the negative sequence, respectively, with ,negnv  and ,negni  as 129 their corresponding phase angles. Also, it is assumed that the dc offsets in voltages and currents comply with 0 0 0, , , 0as dc bs dc cs dcv v v    and 0 0 0, , , 0as dc bs dc cs dci i i   , and therefore are represented in terms of their amplitudes 0dcV  and 0dcI  and angles 0,v dc  and 0,i dc , respectively. For notational consistency with Section 4.2.1 of previous Chapter 4, the fundamental component of phase a voltage in the positive sequence is chosen as the reference, which yields 1,pos 0v  . It should also be noted that the zero sequence ac variables do not exist in the system of Figure 5.1. In order to obtain the parameters of decompositions (5.6)–(5.7), the ac variables in abc phase coordinates (i.e.,abcsv  and abcsi  in Figure 5.1) are transformed to multiple rotating qd reference frames. For computing the n-th harmonic of ac variables in positive sequence, the variables in (5.6) and (5.7) are transformed to ,posneqd  frame [rotating with ( )en  angle]. For n=1, 1,poseq  is the synchronous (converter) reference frame with angular displacement equal to e  rotating in the positive direction. Following the common convention, the quadrature axis of the converter reference frame denoted by 1,poseq  is chosen to be aligned with 1,posasv , as shown in Figure 5.2(a). Here, the subscript “e” indicates the converter reference frame. The phasor diagram of ac voltages and currents in ,posneqd  frames are illustrated in Figure 5.2(b). 130 ,posnasv,posneqen,posned,posnvposn,posnasi,,posn edsi,,posn eqsi,,posn edsv,,posn eqsv,posni1 1,,pos ,pos,eas qsv v1,poseqrqerrd1,posed1pos1,posasi1,,posrdsi1,posi(a)(b)1,,posedsi1,,poseqsi1,,posrqsv1,,posrdsv1,,posrqsi,negnasv,negneqen,negned ,negnvnegn ,negnasi,,negn edsi,,negn eqsi,,negn edsv,,negn eqsv,negni(c) Figure 5.2. Phasor diagram of ac voltages and currents in multiple rotating qd reference frames in positive and negative sequences: (a) fundamental components of ac variables in positive sequence; and (b) n-th harmonics in positive sequence; (c) n-th harmonics in negative sequence. These transformations are done using the Park’s transformation matrix sK  [38] with ( )en  argument for positive sequence n-th harmonic reference frame ,posneqd  as  , ,,pos ,pos( ) ,   ( )n e n eqds s e abcs qds s e abcsn n  v K v i K i . (5.8) With transformations in (5.8), the n-th harmonic components in positive sequence become dc values in ,posneqd  coordinates, while all other components of 131 positive and negative sequences become oscillatory (i.e., ripples) with zero average value. The ripples on the ac variables in ,posneqd  coordinates can then be removed using averaging as     1 tt Tx t x dT   , (5.9) where T  is equal to 1 ef . Also, x denotes the average value of variable x (which can be current or voltage). After transforming (5.6) and (5.7) using (5.8) and applying (5.9), the average values of n-th harmonic components of positive sequence ac voltages and currents in ,posneqd  coordinates can be written as [38]     ,e ,,pos pos ,pos ,pos pos ,poscos  , sinn n n n e n nqs v ds vv V v V    , (5.10)     , ,,pos pos ,pos ,pos pos ,poscos  , sinn e n n n e n nqs i ds ii I i I    . (5.11) Using (5.10), (5.11) and based on Figures 5.2(a) and 5.2(b), posnV , posnI , ,posnv  and ,posni  in (5.6), (5.7) can be obtained from the transformed variables in ,posneqd coordinates as  ,,pos, 1pos ,pos ,pos ,,pos= , tann edsn n e nqds v n eqsvVv       v , (5.12)  ,,pos, 1pos ,pos ,pos ,,pos , tann edsn n e nqds i n eqsiIi        i . (5.13) 132 Here, ,posnv  and ,posni are the phase angles of positive sequence n-th harmonic ac voltage ,posnasv  and current ,posnasi  in ,posneqd  reference frame, respectively, as shown in Figure 5.2(b). It should also be mentioned that 1,posv  is assumed to be zero (due to the chosen reference), as shown in Figure 5.2(a). In the negative sequence, the n-th harmonics of ac variables are computed by using the ,negneqd  frame [rotating with ( )en  angle, corresponding to the speed of n-th harmonic in the negative direction]. The phasor diagram of ac voltages and currents in ,negneqd  frame are illustrated in Figure 5.2(c). These transformations are expressed as  , ,,neg ,neg( ) ,   ( ) ,n e n eqds s e abcs qds s e abcsn n    v K v i K i  (5.14) with ( )en  as argument of sK . With transformations in (5.14), the n-th harmonic components in negative sequence become dc values in ,negneqd  coordinates. Similarly, after applying (5.14) and (5.9) to ac variables in (5.6), (5.7), the average values of n-th harmonic components of negative sequence ac voltages and currents in ,negneqd  reference frame can be expressed as     ,e ,,neg neg ,neg ,neg neg ,negcos  , sinn n n n e n nqs v ds vv V v V   , (5.15)     , ,,neg neg ,neg ,neg neg ,negcos  , sinn e n n n e n nqs i ds ii I i I   . (5.16) 133 Using (5.15), (5.16) and based on Figure 5.2(c), negnV , negnI , ,negnv  and ,negni  in (5.6), (5.7) can be obtained as  ,,neg, 1neg ,neg ,neg ,,neg= , tann edsn n e nqds v n eqsvVv      v , (5.17)  ,,neg, 1neg ,neg ,neg ,,neg , tann edsn n e nqds i n eqsiIi       i , (5.18) where ,negnv  and ,negni  are the phase angles of negative sequence n-th harmonic ac voltage ,negnasv  and current ,negnasi  in ,negneqd  reference frame, respectively, as shown in Figure 5.2(c). For computing the dc offsets (n=0), the ac variables in (5.6), (5.7) are transformed to the stationary reference frame 0,dceqd  corresponding to angle   0en  ,  as  0, 0,, ,(0) ,   (0)e eqds dc s abcs qds dc s abcs v K v i K i . (5.19) With (5.19), the offsets on three-phase ac variables become dc values in the stationary reference frame and all other components (all harmonics in both positive and negative sequences) become oscillatory. By applying (5.9), the average values are obtained as      0,e 0 0 0,e 0 0, , , ,cos ,   sinqs dc dc v dc ds dc dc v dcv V v V    , (5.20)     0, 0 0 0, 0 0, , , ,cos ,   sine eqs dc dc i dc ds dc dc i dci I i I    . (5.21) 134 Therefore, using (5.20), (5.21) and based on Figure 5.2(b) (with setting n=0),0dcV , 0dcI , 0,v dc  and 0,i dc  in (5.6), (5.7) can be obtained from the transformed variables in stationary reference frame as  0,,0 0, 0 1, , 0,,= , taneds dcedc qds dc v dc eqs dcvVv       v , (5.22)  0,,0 0, 0 1, , 0,, , taneds dcedc qds dc i dc eqs dciIi        i . (5.23)  5.2 Extended PAVM Methodology for Faulty Diode Rectifier Systems In the proposed methodology, the average values of dc-side variables are related to the average values of transformed ac-side variables (5.10)–(5.11), (5.15)–(5.16) and (5.20)–(5.21) in their qd reference frames through so-called parametric functions [52]. 5.2.1 Formulation of the proposed PAVM The relationship between the averaged dc-side variables (i.e., dcv  and dci ) and the magnitudes of transformed ac-side positive sequence variables (5.12), (5.13) can be represented using parametric functions ,pos ( )nvw   and ,pos ( )niw   defined as 135  ,,pos,pos ,pos ,,pos( )  ,   ( )n eqdsn n dcv i n edc qdsiw wv   vi. (5.24) The phase angles of voltages and currents in (5.12), (5.13) are captured using parametric functions ,posnv  and ,posni  defined as  , ,,pos ,pos1 1,pos ,pos, ,,pos ,pos( ) tan  , ( ) tann e n eds dsn nv in e n eqs qsv iv i                   . (5.25) Also, the phase displacement of 1,posasi  with respect to 1,posasv  is captured by the parametric function 1pos ( )   defined as  1,,pos1 1 1 1pos ,pos ,pos 1,,pos( ) tanedsv i eqsii           , (5.26) which is the power factor angle of the LCR. Similarly, the relationship between the average values of LCR dc-side variables and the magnitudes of the transformed ac-side negative sequence variables (5.17), (5.18) can be obtained using parametric functions ,neg ( )nvw   and ,neg ( )niw   defined as  ,,neg,neg ,neg ,,neg( )  ,   ( )n eqdsn n dcv i n edc qdsiw wv   vi. (5.27) The phase angles of negative sequence ac voltages and currents are expressed using parametric functions as  , ,,neg ,neg1 1,neg ,neg, ,,neg ,neg( ) tan  , ( ) tann e n eds dsn nv in e n eqs qsv iv i                 . (5.28) 136 The relationship between the averaged dc-side variables and the dc offsets in ac variables (5.22), (5.23) is captured using parametric functions 0, ( )v dcw   and 0, ( )i dcw   defined as  0,,0 0, , 0,,( )   ,   ( )eqds dc dcv dc i dc edc qds dciw wv   vi. (5.29) To capture the distribution of dc offsets into each phase as in (5.6), (5.7), the associated angles 0,v dc  and 0,i dc  are obtained using parametric functions defined as  0, 0,, ,0 1 0 1, ,0, 0,, ,( ) tan  , ( ) tane eds dc ds dcv dc i dce eqs dc qs dcv iv i                   . (5.30) Since it is generally complicated (if not impossible) to derive functions (5.24)–(5.30) analytically, these parametric functions are constructed numerically using detailed model simulations of the system of Figure 5.1 over a wide range of considered operating conditions [54]. The simulations are run for various operating conditions including different configurations of faults of LCR switches. The LCR loading condition is specified by a dynamic impedance defined in terms of its terminal currents and voltages as  1,,posdcd eqdsvz i. (5.31) The parametric functions (5.24)–(5.30) are computed and stored in appropriate lookup tables in terms of the dynamic impedance (5.31). The construction time of 137 lookup tables depends on the desired range of operating conditions as well as the number of fault configurations that need to be considered for studies. It is also noted that this is a one-time process, and once the parametric functions are calculated and saved in lookup tables, they can be used for various studies including transients. Finally, based on the stored parametric functions (5.24)–(5.30), the ac voltages abcsv  and currents abcsi  can be readily reconstructed using dc-side variables as      0 0, , ,pos ,pos ,neg ,neg10 0, , ,pos ,pos ,neg( )cos ( ) ( )cos ( ) ( )cos ( )2 2( )cos ( ) ( )cos ( ) ( )cos3 3mn n n nas dc v dc v dc dc v e v v e vnn n nbs dc v dc v dc dc v e v vv v w v w n w nv v w v w n w n                                             ,neg10 0, , ,pos ,pos ,neg ,neg12( )32 2 2( )cos ( ) ( )cos ( ) ( )cos ( )3 3 3mne vnmn n n ncs dc v dc v dc dc v e v v e vnv v w v w n w n                                                                    ,(5.32)       0, ,pos ,neg01, ,pos ,neg0, ,pos0, ,poscos ( ) cos ( ) cos ( )( ) ( ) ( )2 2cos ( ) cos ( )3 3( ) ( )n nmi dc e i e ias dc dc n nni dc i ini dc e ibs dc dc ni dc in ni i iw w wni i iw w                                                 ,neg1 ,neg0, ,pos ,neg01, ,pos ,neg2cos ( )3( )2 2 2cos ( ) cos ( ) cos ( )3 3 3( ) ( ) ( )ne imnn in ni dc e i e imcs dc dc n nni dc i inwn ni i iw w w                                                              . (5.33) In (5.32) and (5.33), m is the highest order considered harmonic. It is noted that the highest harmonic order and the magnitudes of all harmonics depend on specific rectifier system configurations and system parameters. Therefore, m is 138 selected as a compromise between the simulation speed and accuracy of reconstructing the ac variables, which is determined by specific study requirements. 5.2.2 Implementation of the proposed PAVM of faulty rectifiers In the proposed PAVM, the detailed switching model of LCR is replaced with continuous dependent current and voltage sources as shown in Figure 5.3. For interfacing with external subsystems and consistency with Section 4.2.1.3 of previous Chapter 4, in Figure 5.3, the LCR ac currents abcsi  and dc voltage dcv  are chosen as the inputs to the PAVM block, and the ac voltages abcsv  and dc current dci  are the outputs. The implementation of PAVM allowing faulty LCR switches is demonstrated in Figure 5.4. Here, the input Rectifier State specifies whether the model represents a normal or a faulty condition. In case of a faulty condition, this input   abcsvabcsir dciAC-SideSubsystemDC-Side SubsystemdcvInterfacing CircuitLCR PAVMRectifier State Figure 5.3. Interfacing of the proposed extended PAVM of faulty diode rectifiers with ac- and dc-side subsystems. 139 1,rqdsiabcsvabcsi 1,posiw,posnvw,posnvdzrEq. (31)Eq. (41)Eq. (8)Eq. (34)1,posvw1posdcieqdsvEq. (45)ePLLEqs. (2),(26)Parametric Functions (24)-(30)Eq. (46)Eq. (42)Eq. (44)Eqs. (35),(36)1,pos( )e i dcv,negnvw,negnvEqs. (37),(38)rqdsvRectifier StateNormalD2 OpenD2 &D3 OpenD2 Shortenenenen0,dcvw0,dcvEqs. (39),(40) Figure 5.4. Implementation of the proposed PAVM allowing faulty rectifier states.   also specifies the type of fault configuration (e.g., D2 open, or D2&D3 open, D2 shorted, etc.), which enables the appropriate lookup tables as shown in Figure 5.4. It is noted that the Rectifier State input is used similarly in both the detailed model and the proposed PAVM. The ac currents abcsi  are the inputs and are used in (5.8) to obtain the amplitude of fundamental frequency positive sequence component 1, ,poseqdsi , which is then used with dcv  in (5.31) to compute dz . The dynamic impedance is used to compute the values of parametric functions in (5.24)–(5.30) corresponding to all the considered harmonic components in positive sequence, negative sequence, and dc offsets. Based on (5.24)–(5.25) and using the input dc voltage dcv , the 140 fundamental frequency positive sequence ac voltages of LCR in 1,poseqd coordinates are computed as   1, 1 1,,pos ,pos ,pos( )  ,   0e eqs v d dc dsv w z v v  . (5.34) Also, the n-th harmonic voltages in ,posneqd  coordinates are computed as     ,e,pos ,pos ,pos( ) cos ( )  , 1n n nqs v d dc v dv w z v z n  , (5.35)     ,e,pos ,pos ,pos( ) sin ( )  , 1n n nds v d dc v dv w z v z n   . (5.36) Similarly, the n-th harmonic voltages in the negative sequence ,negneqd  frame are calculated based on (5.27)–(5.28) as   ,e,neg ,neg ,neg( ) cos ( )n n nqs v d dc v dv w z v z , (5.37)   ,e,neg ,neg ,neg( ) sin ( )n n nds v d dc v dv w z v z . (5.38) Also, the dc offset in ac variables in the stationary reference frame are computed based on (5.29)–(5.30) as   0,e 0 0, , ,( ) cos ( )qs dc v dc d dc v dc dv w z v z , (5.39)   0,e 0 0, , ,( ) sin ( )ds dc v dc d dc v dc dv w z v z  . (5.40) The total three-phase ac voltages in the synchronous converter reference frame are calculated by combining the fundamental frequency and harmonic voltages in both positive and negative sequences plus the dc offsets as       1 ,1 ,pos0,,dc 1 ,1,neg0n ems e qdse eqds s e s qdsn ens e qdsnn                 K vv K K vK v. (5.41) 141 In (41), the converter reference frame angle e  may be obtained from the LCR terminal voltages/currents using a phase-locked-loop (PLL). Here, since the ac currents are measured, the output of PLL is the phase of fundamental components of ac currents 1,pos( )e i  as shown in Figure 5.4. Based on (5.26) where 1 1pos ,pos( ) i    , the output of PLL is added to 1pos ( )   to obtain e .  Also, e  can be acquired from the ac subsystem using generator rotor angle r  and (5.2), where   is obtained based on (5.2), (5.26) and Figure 5.2(a) as  1,,pos1 1pos 1,,pos( ) tanrdsd rqsizi        . (5.42) The interfacing output voltages in abc coordinates are then calculated as    1( ) eabcs s e qdsv K v . (5.43) In case the ac subsystem is represented in the source (rotor) qd reference frame with angular displacement r  as shown in Figure 5.2(a) (denoted by subscript “r”), the LCR voltages eqdsv  can be transformed to rotor (Thévenin source) reference frame as        cos sinsin cosr eqds qds      v v . (5.44) Also, the relationship between the voltages in abc coordinates and the qd rotor (Thévenin source) reference frame is   1( ) rabcs s r qdsv K v . (5.45) 142 Finally, the output current for interfacing with the dc subsystem is calculated as  1 1,,pos ,pos( )edc i d qdsi w z i . (5.46) It is also noted that for interfacing the current source dci  with the dc-side subsystem, the dc filter inductor fL  is modeled using a transfer function as in [52, see Eq. (9)] with a small time-constant (  as given in Appendix H). 5.2.3 Performance verification of the proposed PAVM of faulty diode rectifiers In order to verify the proposed PAVM, the same reduced-scale machine-rectifier system of Figure 5.1, with parameters summarized in Appendix H, has been set up in the laboratory. The same experimental setup was used in Section 4.2.2 (see Figure 4.14). The ac subsystem is composed of a three-phase permanent magnet synchronous machine (PMSM) operating as a generator. The dc subsystem consists of a low-pass RLC filter connected to a resistive load rl, which represents the dc network.  The proposed PAVM (i.e., Figures 5.3 and 5.4) has been implemented in MATLAB/Simulink [27] and directly interfaced with the qd model of PMSM [38, Chap. 5]. For the studies presented in this section, the dominant components up to 7th harmonic (m=7) are considered. For comparison, the detailed model of the system has been implemented using semiconductor devices available in standard library of PLECS [29]. In the detailed model, the ac-side of LCR is 143 interfaced with coupled-circuit phase-domain (CCPD) model of PMSM [38, Chap. 5]. Here, the CCPD model is selected to avoid artificial snubber circuits that are required for interfacing the qd models with switching rectifiers. Artificial interfacing snubbers introduce error and stiffness and can result in numerical instability at large time-steps [73]. The dc subsystem is implemented using standard RLC branches in PLECS. The diode conduction losses are also taken into account by considering the forward voltage drop Von and resistance Ron (which are calculated/estimated using the voltage-current characteristic of diodes).  5.2.3.1 Operation with faulty open-circuited diodes Here, it is assumed that the system of Figure 5.1 is initially in steady-state and the LCR is operating under normal condition (i.e., all healthy diodes) in CCM-1 mode [49] where the dc network load is set to rl =32.92 Ω. The PMSM generator is assumed to operate at a constant speed corresponding to 60 Hz electrical frequency. The generator back-emf voltages are very close to ideal and for the modeling purposes are assumed to consist of fundamental frequency positive sequence only. In the following study, without loss of generality, at t=1s an internal fault is applied and the diodes D2 and D3 (as labeled in Figure 5.1) are assumed to fail and become open circuit (i.e., in PAVM the input Rectifier State changes from Normal to D2&D3 open). The transient responses of several dc- and ac-side variables are demonstrated in Figures 5.5 – 5.7, as predicted by the proposed PAVM and the detailed model.  144  Figure 5.5. Transient response of dc-side variables as predicted by the subject models with the LCR initially operating normally in CCM-1 mode, when at t=1s, diodes D2 and D3 fail and become open-circuited: (a) vdc and (b) idc.    Figure 5.6. Transient response of ac-side terminal voltages as predicted by the subject models with the LCR initially operating normally in CCM-1 mode, when at t=1s, diodes D2 and D3 fail and become open-circuited: (a) vas , (b) vbs, and (c) vcs.  145  Figure 5.7. Transient response of ac-side phase currents as predicted by the subject models with the LCR initially operating normally in CCM-1 mode, when at t=1s, diodes D2 and D3 fail and become open-circuited: (a) ias , (b) ibs and (c) ics.  As it can be observed in Figure 5.5, the PAVM is capable of following the dynamic behavior of the dc variables predicted by the detailed model. It is also verified that under transient, the proposed PAVM can effectively predict the asymmetric distorted ac voltages and currents as illustrated in Figures 5.6 and 5.7, respectively. To verify the predicted waveforms, Figures 5.8 – 5.10 also show the measured results obtained from the experimental setup where diodes D2 and D3 have been disconnected, and compare them to the detailed model and PAVM. Figure 5.8 demonstrates the impact of the asymmetrical operation on the faulty LCR output dc  146  Figure 5.8. Measured and simulated dc-side variables as obtained from experimental setup and predicted by the subject models when the LCR operates with D2&D3 open-circuited: (a) vdc and (b) idc.   Figure 5.9. Measured and simulated ac-side terminal voltages as obtained from experimental setup and predicted by the subject models when the LCR operates with D2&D3 open-circuited: (a) vas , (b) vbs , and (c) vcs. 147  Figure 5.10. Measured and simulated ac-side phase currents as obtained from experimental setup and predicted by the subject models when the LCR operates with D2&D3 open-circuited: (a) ias , (b) ibs and (c) ics.  voltage and current waveforms, which become heavily distorted. As it can be observed in Figure 5.8, the detailed model precisely reconstructs the measurements. Meanwhile, the proposed PAVM is able to accurately predict the average values of both dc variables. Figures 5.9 and 5.10 show the measured and simulated ac terminal voltages and phase currents of faulty LCR, respectively. It is seen in Figures 5.9 and 5.10 that the ac variables become unbalanced/asymmetric. Moreover, Figure 5.10 shows that phase b can carry only negative current (since diode D3 is open), and phase c can conduct only positive current (since diode D2 is open). In addition, Figures 5.9 and 5.10 verify the good accuracy of the proposed PAVM in 148 reconstructing the distorted voltages and currents under faulty conditions by considering just up to 7th harmonics in both negative and positive sequences in addition to the dc offsets. The harmonic contents of the ac voltages and currents are illustrated in Figure 5.11, with their associated amplitudes and phase angles summarized in Tables 5.1 and 5.2. As it can be observed in Figure 5.11 and Tables 5.1 – 5.2, the proposed PAVM can accurately predict all the considered positive and negative sequence components (characteristic and non-characteristic harmonics) and the dc offsets compared to the detailed model and experimental results.   Figure 5.11. The harmonic content of ac voltages and currents of LCR with D2&D3 open-circuited, as obtained from experimental setup and the subject models.     149 Table 5.1. The harmonic content of three-phase voltages obtained from experiments and as predicted by the subject models when diodes D2 and D3 are in fault state (open-circuited) in CCM-1. Harmonic Order Positive Sequence Experiments/ Detailed Model Proposed PAVM Negative Sequence 0-dc (non-characteristic) 1.52 97.12  1.53 96.82  1st (fundamental) 62.33 0  62.37 0  0.518 37.88  0.516 38.35  2nd (non-characteristic) 7.624 49.42  7.631 49.45  2.834 61.23  2.846 61.34  3rd (non-characteristic) 0.525 5.64  0.523 5.59  2.224 132.4  2.226 133.2  4th (non-characteristic) 1.08 105.6  1.08 105.5  5.118 93.27  5.127 93.92  5th (characteristic) 0.454 24.54  0.45 24.84  2.637 34.36  2.627 35.24  6th (non-characteristic) 2.468 69.6  2.452 69.91  1.761 65.29  1.742 65.27  7th (characteristic) 2.298 118  2.275 119.4  1.511 118.1  1.475 118.6    Table 5.2. The harmonic content of three-phase currents obtained from experiments and as predicted by the subject models when diodes D2 and D3 are in fault state (open-circuited) in CCM-1. Harmonic Order Positive Sequence Experiments/ Detailed Model Proposed PAVM Negative Sequence 0-dc (non-characteristic) 2.035 82.89  2.046 83.24  1st (fundamental) 2.758 21.3  2.732 21.37  0.190 67.28  0.189 67.04  2nd (non-characteristic) 1.343 41.06  1.467 48.81  0.544 38.18  0.546 38.04  3rd (non-characteristic) 0.048 251  0.048 250.8  0.294 37.08  0.295 38.07  4th (non-characteristic) 0.096 46.57  0.096 46.92  0.519 0.143  0.520 0.513  5th (characteristic) 0.065 137.2  0.064 136.8  0.219 56.84  0.219 56.07  6th (non-characteristic) 0.212 189.2  0.211 189.2  0.117 156.3  0.115 158  7th (characteristic) 0.148 143.5  0.146 142.4  0.089 150.4  0.084 149   150 5.2.3.2 Operation with faulty short-circuited diodes Depending on the manufacturing technology of semiconductor switches (e.g., press-pack diodes, etc.) and other conditions, they may also become short-circuited when faulted. In order to verify the proposed PAVM methodology in predicting the short circuit faults, the following transient study is conducted assuming that the healthy LCR is initially operating in steady-state in CCM-1 mode with rl =32.92 Ω. At t=1s, the diode D2 is assumed to fail and becomes short-circuited (i.e., in PAVM the input Rectifier State changes from Normal to D2 short). To avoid damage of the experimental equipment due to high currents, this study has been conducted using the proposed PAVM and detailed model only (without measurements, with the detailed model being the reference). The transient responses of dc- and ac-side variables of LCR as predicted by the subject models are shown in Figures 5.12 – 5.14. As it can be seen in Figure 5.12, the PAVM can accurately predict the dynamics of average values for the dc variables when a short circuit fault occurs. The transient responses in Figures 5.13 and 5.14 show the asymmetric/distorted ac voltages and currents. These results also validate the effectiveness and accuracy of the proposed PAVM in reconstructing the dc and ac variables for the short-circuit type of faults in the LCR. 151  Figure 5.12. Transient response of LCR dc-side variables as predicted by the subject models when diode D2 fails to operate (becomes short-circuited) at t=1s with the LCR initially operating in normal condition CCM-1 mode with rl =32.92 Ω: (a) vdc and (b) idc.   Figure 5.13. Transient response of LCR ac-side voltages as predicted by the subject models when diode D2 fails to operate (becomes short-circuited) at t=1s with the LCR initially operating in normal condition CCM-1 mode with rl =32.92 Ω: (a) vas, (b) vbs, and (c) vcs. 152  Figure 5.14. Transient response of LCR ac-side currents as predicted by the subject models when diode D2 fails to operate (becomes short-circuited) at t=1s with the LCR initially operating in normal condition CCM-1 mode with rl =32.92 Ω: (a) ias , (b) ibs and (c) ics.  In addition to the studies performed in Sections 5.2.3.1 and 5.2.3.2, the effectiveness and accuracy of the developed PAVM methodology have been verified under various operating conditions (including DCM) with different fault configurations (e.g., faults on other combinations of diodes), but are not included here due to space limitation. 5.2.3.3 Computational Performance The computational performance of the proposed PAVM is compared to the detailed model. For consistency, both PAVM and detailed models are simulated 153 in MATLAB/Simulink [27] using ode23tb solver with maximum time-step of 10-3 s and relative/absolute tolerance of 10-3. The simulations are executed on a PC with Intel® Core™ i7-4510U @2.00GHz processor. A 3-second transient study is performed on the system of Figure 5.1 that is initially operating in CCM-1 with rl =32.92 Ω (similar to section 5.2.3.1). At t=1s, the diodes D2 and D3 become open-circuited, and at t=2s, the dc load resistance is increased to rl =1282 Ω (corresponding to DCM when there is no fault). The computational performance of the subject models is summarized in Table 5.3.  As it can be observed in Table 5.3, the proposed PAVM is computationally more efficient compared to the detailed model in terms of simulation speed (1.93 s vs. 7.19 s for CPU time) and number of time-steps (3,967 vs. 21,971), while achieving very similar waveforms for ac variables and providing accurate prediction for average values of dc variables. Moreover, it is seen that the average time-step of simulation is ~5.5 times larger when the proposed PAVM of LCR is used instead of its detailed model. This is due to the fact that in the simulation with the detailed model the time-step should be fairly small to predict the exact switching events of LCR; however, with the proposed PAVM the step-size is limited to the highest harmonic included in the model. Table 5.3. Computational performance of the detailed model and the proposed PAVM of faulty diode rectifiers for the 3-second transient study Model CPU Time (s) Number of Steps Average Time-Step (µs) Detailed 7.19 21,971 137 Proposed PAVM 1.93 3,967 756 154 5.3 Extending the Proposed PAVM Methodology of Faulty LCRs to Thyristor-Controlled Rectifiers Faults can also occur in thyristor-controlled rectifiers, for example, due to loss of their gate signals, which introduce unbalanced conditions and have a negative impact on power system equipment and their operation [103]. In this section, the faulty diode rectifier PAVM presented in Section 5.2 is extended and generalized for thyristor-controlled rectifiers with internal faults by incorporating the firing angle of thyristors into the PAVM formulations for various fault configurations. The diode rectifier PAVM of Section 5.2 is also a special case of the generalized PAVM proposed here by setting firing angle to zero. Using numerous simulation studies, the extended PAVM is verified on an example ac–dc system shown in Figure 5.15, consisting of a thyristor-controlled rectifier with internal faults in switches. It is demonstrated that the proposed PAVM can predict the unbalanced operation with excellent accuracy under various loading conditions of the thyristor-controlled rectifier. Moreover, the proposed model captures the dynamics of thyristor firing angle with both open-loop and closed-loop control in faulty conditions. Compared to its detailed model, the presented PAVM is an efficient and fast simulation tool that can be beneficial in many system-level studies which include analysis of faults in thyristor-controlled rectifiers for designing fault detection algorithms, compensation methods, etc.  155 abcsvabcsidcvdciSix-Pulse Thyristor-Controlled Rectifier4T 2T5T3T6T1TFiring Signal GeneratorDC Voltage ControllerReference DC VoltageAC-SideSub-System5S3S1S2S6S4S 1 2 6, ,...,S S SPLLreoutvfLfCDC FilterfrDC NetworkDC-Side Sub-SystemFiring Angle (Open Loop) Figure 5.15. Typical three-phase ac–dc system consisting of a six-pulse thyristor-controlled rectifier with faulty switches.  5.3.1 Formulation of the proposed extended PAVM of faulty thyristor-controlled rectifiers It is assumed that the ac-side variables abcsv  and abcsi  in Figure 5.15 are transformed to appropriate n-th harmonic qd reference frames in positive and negative sequences, and their average values are obtained according to Section 5.1. In the proposed extended PAVM, the averaged transformed ac-side variables in positive and negative sequences are related to average values of dc-side variables (i.e., dcv  and dci ) using parametric functions which are formulated in terms of the rectifier dynamic impedance (5.31) as well as the firing angle of thyristors  , as an additional argument of the parametric functions. Therefore, the modified parametric functions that relate the magnitude of ac voltage harmonics in positive and negative sequences to dc voltage are defined as 156  , ,,pos ,neg,pos ,neg( , )  ,   ( , )n e n eqds qdsn nv d v ddc dcw z w zv v  v v, (5.47) where  1,2,3,4,5,6,7,...n  is the harmonic order [75]. Similarly, the magnitude of ac current harmonics in positive and negative sequences are related to dc current using parametric functions defined as  ,pos ,neg, ,,pos ,neg( , )  ,   ( , )n ndc dci d i dn e n eqds qdsi iw z w z  i i. (5.48) Also, the magnitude of dc offsets on ac voltages and currents are related to the dc voltage and dc current, respectively, using modified parametric functions defined as  0,,0 0, , 0,,( , )   ,   ( , )eqds dc dcv dc d i dc d edc qds dciw z w zv  vi. (5.49) The phase angles of positive and negative sequence voltages and currents are also tabulated with parametric functions as  , ,,pos ,neg1 1,pos ,neg, ,,pos ,neg( , ) tan  ,   ( , ) tann e n eds dsn nv d v dn e n eqs qsv vz zv v                  , (5.50)  , ,,pos ,neg1 1,pos ,neg, ,,pos ,neg( , ) tan  ,   ( , ) tann e n eds dsn ni d i dn e n eqs qsi iz zi i                  . (5.51) The dc offsets on ac voltages and currents are also distributed among different phases in correspondence with their phase angles which are captured with parametric functions as   0, 0,, ,0 1 0 1, ,0, 0,, ,( , ) tan  ,   ( , ) tane eds dc ds dcv dc d i dc de eqs dc qs dcv iz zv i                   . (5.52) 157 The relationship between phase angles of ac voltages and currents is considered by defining a parametric function as  1 1 1pos ,pos ,pos( , )d v iz     . (5.53) In order to obtain the parametric functions (5.47)–(5.53), the system of Figure 5.15 is briefly simulated with the detailed model of the thyristor-controlled rectifier for the desired range of operating conditions (i.e., various loads and firing angles of thyristors) under the considered faulty situations. Then, the parametric functions (5.47)–(5.53) are computed numerically and stored in two-dimensional lookup tables in terms of the rectifier dynamic impedance (5.31) and the thyristors firing angle   for various fault configurations. It is noted that the lookup table construction is a one-time process that can be done by the simulation program (and possibly automated), and once established they can be used for many transient studies. Finally, the implementation of extended PAVM is similar to Figures 5.3 and 5.4, which replaces the detailed switching model with dependent current and voltage sources, and the model now includes   as an additional input to the lookup tables. 5.3.2 Verification of the proposed extended PAVM of thyristor-controlled rectifiers In order to investigate the accuracy of the extended PAVM, the same three-phase PM synchronous generator ac–dc system of Figure 5.15 is considered here 158 with the addition of thyristor-control, for consistency with Section 5.2.3. The detailed model of this system has been experimentally verified in Section 5.2.3.1. The dc network is represented by a resistive load rl and an RLC low-pass filter with parameters summarized in Appendix I. To demonstrate the features of the extended PAVM, studies spanning a wide range of operating conditions with open and closed loop control (here, a PI controller with parameters summarized in Appendix I) are presented. Also, herein up to 7th harmonic components (in both positive and negative sequences) as well as the dc offsets are included in the PAVM.  Initially, the system is assumed to operate under normal condition in steady-state where the generator speed is kept constant corresponding to the electrical frequency of 60 Hz, and the firing angle of thyristors is set to α=32º providing 48V DC for the dc network with rl =6 Ω. At t=0.5s, the gate signal S5 is lost due to an internal fault and the thyristor T5 becomes open-circuited (as indicated in Figure 5.15). At t=1s, the dc load is changed to rl =30 Ω to emulate a light loading condition. Next, at t=1.5s, the dc voltage controller is activated to regulate the output voltage to 48V DC by controlling the firing angle of thyristors. The resulting transient responses of several dc- and ac-side variables are depicted in Figures 5.16 and 5.17, respectively.  As it can be observed in Figure 5.16, there is significant ripple in both dc voltage and current due to missing the thyristor T5 operation. It is also verified in the magnified areas of Figure 5.16 that the extended PAVM accurately 159  Figure 5.16. Transient of dc-side variables as predicted by the detailed model and the proposed extended PAVM of faulty thyristor-rectifiers: (a) idc , (b) vdc , (c) vout , and (d) firing angle α.  predicts the dynamics of average values of dc-side variables in faulty situations under different loading conditions and thyristor firing angle control operation which is sufficient for system-level studies. Figure 5.17 illustrates the transient observed in three-phase ac currents, which become unbalanced and asymmetric after the fault. The magnified view of ac voltages and currents at the moment of fault are depicted in Figures 5.18 and 5.19, respectively. As it can be observed in Figures 5.18 and 5.19, the extended PAVM has excellent agreement with the detailed model in predicting the 160 distorted and asymmetric three-phase voltages and currents of the faulty thyristor rectifier and gives a reasonable prediction of ac waveforms for studying the impact of the faulty rectifier on the ac-side network in system-level studies. The extended PAVM has been also verified under many other operating conditions, but the results are not included here due to limited space.    Figure 5.17. Transient of ac currents as predicted by the detailed model and the proposed extended PAVM of faulty thyristor-rectifiers: (a) ias, (b) ibs, and (c) ics.   161  Figure 5.18. Magnified view of ac voltages as predicted by the detailed model and the proposed extended PAVM when the gate signal S5 of thyristor T5 is lost at t=0.5s: (a) vas, (b) vbs, and (c) vcs.  Figure 5.19. Magnified view of ac currents as predicted by the detailed model and the proposed extended PAVM when the gate signal S5 of thyristor T5 is lost at t=0.5s: (a) ias, (b) ibs, and (c) ics. 162 The computational performance of the detailed switching model and the extended PAVM of faulty thyristor-controlled rectifiers for the 2-second study presented in Figure 5.16 – 5.19 is summarized in Table 5.4, with identical solver and error tolerances settings as in Section 5.2.3.3. In the detailed model, handling of switching events by the simulation program has a computational overhead (i.e., reduced time-step, iterations, etc.). However, in PAVM, the discrete switching is removed and the time-step is limited by the highest order included harmonic. Therefore, as seen in Table 5.4, the extended PAVM allows using larger average time-steps (560 µs vs. 207 µs) and requires fewer steps (3,575 vs. 9,649). It is also noted that although the PAVM is more complicated than the detailed model (as it requires additional variables to represent harmonics), it is still significantly faster than its detailed model counterpart (1.21 s vs. 4.73 s of CPU time), while providing sufficient accuracy up to 7th harmonic, as seen in Figures 5.16 – 5.19.  Table 5.4. Computational performance of the subject models of faulty thyristor-controlled rectifiers for the 2-second transient study Model CPU Time (s) Number of Steps Average Time-Step (µs) Detailed 4.73 9,649 207 Proposed PAVM 1.21 3,575 560  163 CHAPTER 6: PARAMETRIC HYBRID CONTINUOUS-DETAILED ––AVM OF LINE-COMMUTATED RECTIFIERS In the detailed models of line-commutated converters, to determine the switching events accurately, the simulation program [24]–[35] needs zero-crossing detection algorithms [105] and/or interpolation for exact locating of the switching events, which increases the computational cost of the detailed models and/or forces to use small time-steps. Many commercial simulation programs (e.g., [28], [29], [33]) also require to use artificial snubber circuits in parallel with the switches, which also increases the size and complexity of the model.   In this Chapter, first it is shown how much the detailed models of LCRs are dependent on zero-crossing detection, and it is demonstrated that the detailed switching model, when implemented without zero-crossing detection and interpolation, loses numerical accuracy and stability for large time-steps. Moreover, the performance of GPAVM of LCRs (presented in Chapter 4, Section 4.2.1) is investigated for simulations with fixed time-steps in system-level studies that require several harmonics, but where it is also desirable to use fairly large time-steps without internal iterations (i.e., locating zero-crossing events, interpolations, etc.) to achieve fast and reliable simulation results. 164 It is shown that the developed GPAVM retains good accuracy at fairly large fixed time-steps (on the order of several hundreds of micro-seconds) without the need for switching events and artificial snubbers.  Thereafter, a new methodology for modeling power-electronic converters is presented based on the parametric approach and the resultant model is able to reproduce detailed model results precisely, including the switching ripples which are neglected in AVMs. The new proposed parametric detailed model is continuous (does not have discrete switching states) and does not require zero-crossing detection for producing the ac harmonics and switching ripples; therefore, it can be simulated with large time-steps as opposed to conventional discrete switching detailed models of LCRs which need fairly small time-steps to produce accurate and reliable results. It is also worth mentioning that the proposed parametric continuous detailed model can be easily simplified into a parametric AVM (with superior performance compared to the existing state-of-the-art PAVMs, as will be shown) by setting OFF some parameters in the model.   It is envisioned that the presented models can be very effective simulation assets for studies where the power system may include a large number of electronic loads and LCRs, and where large fixed time-steps are desirable to reduce the simulation time and computational complexity. For example, these features may become very useful for online and real-time simulation tools for distribution systems consisting of a large number of electronic loads where several ac harmonics of interest are also needed to be considered. 165 6.1 Verification of LCR Models in Simulations with Fixed Time-Step without Zero-Crossing Events In this Section, the six-pulse diode rectifier system of Figure 6.1 is considered, which may represent the front-stage of typical electronic loads in a distribution system. The rectifier ac terminals are interfaced with the Thévenin equivalent of the ac subsystem. The dc terminals of LCR are connected to a low-pass filter and supply the dc subsystem. Here, the dc network is represented as a resistive load rl to consume the required energy.  Also, to represent conduction losses, the ON resistance Ron and forward voltage drop Von of the diodes are considered. The system parameters are summarized in Appendix J. In order to investigate the effect of time-step on simulation results, the system of Figure 6.1 has been implemented using the detailed model of rectifier in PSCAD [33] without zero-crossing detection algorithms (note that zero-crossing detections and snubber circuits can be enabled/disabled independently), and also the corresponding GPAVM (section 4.2.1) has been implemented in MATLAB/Simulink [27].  abcsvabcsiSix-Pulse RectifierDC NetworkAC SubsystemThree-PhaseThévenin Equivalent CircuitfLfCLow-Pass FilterDC SubsystemfrsLsrdcvdcioutvasebsecse Figure 6.1. Schematic of a generic three-phase ac–dc power system consisting of a six-pulse line-commutated rectifier. 166 In the following transient study, the LCR system loaded with rl = 65Ω (corresponding to CCM-1 mode) starts up with zero initial conditions. Then, at t = 70ms, the dc load is stepped up by decreasing the resistive load to rl = 10Ω (corresponding to CCM-2 mode) and the simulation is continued until t = 120ms. 6.1.1 Simulations without artificial snubbers in detailed models of diodes To see the loss of accuracy in the detailed model without the zero-crossing and interpolation, the diodes in PSCAD use only Ron and Von to represent the conduction losses, but without artificial snubber circuits. The system of Figure 6.1 is simulated using both the detailed model in PSCAD and the GPAVM in MATLAB/Simulink (run using fixed-step solver ode2). The reference solution is obtained by the detailed model using a very small time-step 0.01 µs. The transient response of several variables as predicted by the subject models are shown in Figures 6.2 and 6.3. In Figure 6.2, the time-step is set to 0.1µs for PSCAD detailed model and GPAVM. As it can be observed in Figure 6.2, both the GPAVM and the PSCAD detailed models predict the system variables accurately. In Figure 6.3, the time-step is increased to 1µs (which is still very small). As it can be observed in Figure 6.3, due to zero-crossing error in discrete switching states, the detailed model of PSCAD entirely fails to predict the correct solution for most variables; whereas, the GPAVM does not have 167 switching events and maintains very good accuracy in reconstructing the dc and ac waveforms using just a few considered harmonics.   Figure 6.2. Transient response of the system variables as obtained by the subject models without artificial snubber circuits with simulation time-step of 0.1 µsec: (a) rectifier phase a current, (b) rectifier phase a voltage, (c) rectifier dc current, (d) rectifier dc voltage, and (e) load dc voltage.    168  Figure 6.3. Transient response of the system variables as obtained by the subject models without artificial snubber circuits with simulation time-step of 1 µsec: (a) rectifier phase a current, (b) rectifier phase a voltage, (c) rectifier dc current, (d) rectifier dc voltage, and (e) load dc voltage.        169 6.1.2 Using artificial snubbers in detailed models of diodes To improve numerical stability of handling the switching events, the artificial snubber circuits have been used in the detailed models of diodes in PSCAD with typical software default values (provided in Appendix J), as shown in Figure 6.4(b). It is worth mentioning that the use of snubber circuits makes the rectifier model more complicated as seen in Figure 6.4(b) compared to Figure 6.4(a), which will create a computational bottleneck in system-level studies with multiple converters.  (a) (b)snubrsnubCsnubrsnubC Figure 6.4. The considered six-pulse diode rectifier: (a) without snubbers, (b) with artificial snubber circuits.    170 The same transient study as in Section 6.1.1 is performed again with time-step set to 50µs (as the PSCAD typical/default value) and 300µs (as a large time-step). The corresponding results are shown in Figures 6.5 and 6.6, respectively. As it can be observed in Figure 6.5, although the performance of the detailed model in PSCAD is improved, numerical oscillations still exist in system variables, especially in ac and dc voltages (since the zero-crossing is still disabled). Figure 6.6 also demonstrates that the PSCAD detailed model loses its accuracy with large time-steps even when the damping snubbers are used. At the same time, the GPAVM is shown to produce accurate results and is capable of reconstructing the waveforms of ac and dc currents and voltages even using time-steps as large as 300µs.  It is also worth mentioning that selection of snubber circuit parameters for the diodes (switches) of detailed model in PSCAD depends on the system and studies considered, and may require some expertise and/or trial and error approach. Meanwhile, the GPAVM is envisioned as an enabling methodology for simulation of power systems consisting of LCRs and electronic loads while including the dominant harmonics and permitting fairly large time-steps without the need for zero-crossing and artificial snubber circuits.  171  Figure 6.5. Transient response of the system variables as obtained by the subject models when the artificial snubber circuits are used in the detailed model of diodes with simulation time-step of 50 µsec: (a) rectifier phase a current, (b) rectifier phase a voltage, (c) rectifier dc current, (d) rectifier dc voltage, and (e) load dc voltage. 172  Figure 6.6. Transient response of the system variables as obtained by the subject models when the artificial snubber circuits are used in the detailed model of diodes with simulation time-step of 300 µsec: (a) rectifier phase a current, (b) rectifier phase a voltage, (c) rectifier dc current, and (d) rectifier dc voltage, and (e) load dc voltage.        173 6.2 A New Parametric Modeling Technique for Achieving Hybrid Continuous Detailed/AVM Models of Line-Commutated Rectifiers  Typical waveforms of a six-pulse diode rectifier of Figure 6.1 operating in CCM-1 mode are shown in Figure 6.7 for ac- and dc-side variables of the rectifier. As it can be observed from Figures 6.7(a) and 6.7(b), the three-phase abc variables of the rectifier are composed of fundamental components and harmonics as formulated in (4.3) – (4.5). Also, the dc-side variables are composed of dc average values and oscillating components (i.e., ripples) as shown in Figures 6.7(c) and 6.7(d). In PAVM [52] and MRF-PAVM [64] techniques, the ac-side variables are transformed to synchronously rotating qd coordinates, depicted in Figures 6.7(e) and 6.7(f). As it can be observed in Figures 6.7(e) and 6.7(f), the ac variables become dc variables in qd coordinates with some oscillatory components (i.e., ripples), similar to dc-side variables of the rectifier. It is noted that the magnitude and phase of average values of qd variables correspond to amplitude and phase of fundamental components of ac variables in abc coordinates, while the ripples on qd variables correspond to the sum of all harmonic components of variables in abc coordinates. 174  Figure 6.7. Typical waveforms of a six-pulse diode rectifier ac- and dc-side variables in CCM-1: (a) rectifier three-phase abc voltages, (b) rectifier three-phase abc currents, (c) rectifier dc voltage, (d) rectifier dc current, (e) rectifier transformed qd-axes voltages, (f) rectifier transformed qd-axes currents.     175 In [52] and [64], to obtain the fundamental components of ac variables, the ripples of the qd variables illustrated in Figures 6.7(e) – 6.7(f) are removed by averaging them over a prototypical switching period T corresponding to angle β as shown in Figure 6.7. For, p-pulse rectifiers, T and β are   1 2   ,       ,      2 e eeT f T Tpf p       , (6.1) where ef  and e  are the frequency of ac network in Hz and rad/s. By obtaining the average values of qd variables in Figures 6.7(e) and 6.7(f), the fundamental components of ac variables in Figures 6.7(a) and 6.7(b) can be reconstructed. However, to reconstruct the harmonics of ac variables of Figures 6.7(a)–6.7(b), in MRF-PAVM [64] and GPAVM (Section 4.2.1), the ac-side variables of the rectifier are transformed to multiple qdn coordinates that rotate n-times faster than the synchronous reference frame, corresponding to n-th order harmonics. With transformations to qdn coordinates, the n-th harmonic components become dc values and all other components become ripples, which are then removed using the averaging. By obtaining the average values of qdn variables, the n-th harmonic components of ac variables in Figures 6.7(a) – 6.7(b) can be reconstructed. To achieve a more accurate reconstruction of ac variables, more harmonic components need to be considered. Basically, this process is to reconstruct the ripples of qd variables in Figures 6.7(e) – 6.7(f) which correspond to the sum of harmonic components of abc variables in Figures 6.7(a) – 6.7(b). By including more harmonic components and qdn transformations in MRF-PAVM 176 and GPAVM, the ripples on qd variables are reconstructed more accurately at the cost of more complexity of the model and numerical burden on the simulator introduced by each qdn transformation.  6.2.1 Proposed parametric hybrid continuous detailed/AVM model of line-commutated rectifiers  In this Section, a typical three-phase ac–dc power system is considered as shown in Figure 6.8. The system consists of an ac subsystem that may be a rotating electrical machine or a conventional ac network represented by its Thévenin equivalent circuit.  The dc-side subsystem may consist of an optional low-pass filter to smoothen the rectifier output voltage, which supplies a dc network. The LCR depicted in Figure 6.8 may be an uncontrolled diode bridge or a thyristor-controlled rectifier. In case thyristor switches are used, the output abcsvabcsidcvdciSix-Pulse LCR4T 2T5T3T6T1TFiring Signal GeneratorDC Voltage ControllerReference DC VoltageAC-SideSub-SystemPLLreoutvfLfCOptional DC FilterfrDC NetworkDC-Side Sub-SystemFiring Angle (Open Loop) Figure 6.8. A generic three-phase ac–dc power system consisting of a line-commutated rectifier.  177 voltage of rectifier can be controlled by thyristor firing angle delay, which may be determined with respect to machine rotor angle (or the angle of ac source equivalent voltages) r , or with respect to the angle of rectifier ac-side terminal voltages e  . The angle e  is defined as    ,  2e e e edt f     , (6.2) where ef  and e  are the line frequency of the electrical system in Hz and rad/s. The relationship between r   and e  can also be expressed as  r e    , (6.3) where   is the angle by which the generator back-emf voltages (or the Thévenin equivalent source voltages) lead the fundamental component of the rectifier terminal voltages. Generally, for p-pulse rectifiers, the ac voltages and currents will contain n-th order harmonics, where     1, 1,...  ,  1,2,3,...n ip ip i    . (6.4) For the purpose of this section, the converter ac voltages abcsv  and currents abcsi  in Figure 6.8 can be decomposed into fundamental and harmonic components expressed as  1 has as asv v v  ,  1 has as asi i i  , (6.5) where the fundamental components can be written as     1 1 1 1 1cos ,   cosas e as e iv V i I     , (6.6) 178 and the total harmonic components can be written as the sum of the Fourier series terms as   h nas asnv v ,  h nas asni i . (6.7) Here, 1V  and 1I  are the amplitudes of the fundamental components of voltages and currents, respectively. For notational convenience, the phase a voltage fundamental component is considered as the reference and 1i  is the phase angle of the fundamental component of phase a current. Also, nasv  and nasi  are the n-th harmonic components of phase a voltage and current, respectively. Assuming symmetry, similar formulations can be written for other phases b and c considering ±120 shifts. 6.2.1.1 Derivation of the hybrid detailed/AVM model formulation To capture the relationship between the dc and the fundamental components of ac variables of the rectifier, the ac variables in abc coordinates (i.e., abcsv  and abcsi ) are transformed to the synchronously rotating converter reference frame, denoted by eqd , as depicted in Figure 6.9. It is also assumed that zero-sequence variables do not exist in the power system of Figure 6.8. In this section, the subscript “e” is used to denote the converter reference frame, wherein the angular displacement of transformation is chosen to be e , which aligns the eq -axis with 1asv  [52], as depicted in Figure 6.9. 179 1 , eas qsv vaxiseq axisrq eraxised 1asirdsirqsi1iedsieqsirqsvrdsvaxisrd  Figure 6.9. Phasor diagram of fundamental components of rectifier ac voltages and currents expressed in synchronously rotating converter and source qd reference frames.  The transformations of voltages and currents are carried out using the Park’s transformation matrix sK [64] as  ( )  ,   ( )e eqds s e abcs qds s e abcs  v K v i K i . (6.8) As demonstrated in Figures 6.7(e) – 6.7(f), the transformed qd variables are composed of dc average values as well as ripples, which can be formulated as  + ,   e e e e e eqds qds qds qds qds qds  v v v i i i . (6.9) In (6.9), eqdsv  and eqdsi  are the average value components of transformed qd voltages and currents, which are corresponding to fundamental frequency components of ac voltages and currents in abc coordinates (i.e., 1asv  and 1asi ), respectively.  Also, eqdsv  and eqdsi  are the oscillatory components (ripples) of the transformed qd voltages and currents, which correspond to the total sum of harmonic components of ac voltages and currents in abc coordinates (i.e., hasv  and hasi ), respectively. Moreover, the dc-side variables of the rectifier (i.e., dci  and dcv ), 180 demonstrated in Figures 6.7(c) – 6.7(d), can be expressed in terms of their average values and ripples as  + ,   dc dc dc dc dc dcv v v i i i   , (6.10) where dcv  and dci  are the average value components of dc voltages and currents, and dcv  and dci  are their oscillatory components (ripples).   In order to capture the fundamental components of ac variables, similar to [52] and [64], the oscillatory components in (6.9) (i.e., eqdsv  and eqdsi ) are removed by averaging them over a switching period T (6.1) as      1 tt Tx t x dT   , (6.11) where the variable x may denote currents and voltages. The bar sign (–) above denotes the average value. Also, the ripples of rectifier dc-side variables (i.e., dcv  and dci ) are removed by applying (6.11) to (6.10). In converter reference frame, the angle of ac current 1asi  with respect to eq -axis is equal to 1i , as depicted in Figure 6.9. Also, the angle by which 1asi  is displaced from 1asv  is the power factor angle which can be expressed as,  1 1i   . (6.12) For the fundamental components of ac variables in the converter reference frame, according to Figure 6.9, the following relationships exist [38]   ,   0e e eqs qds dsv v v , (6.13) 181     1 1cos  , sine e e eqs qds i ds qds ii i  i i . (6.14) At any moment, the magnitudes of ac fundamental components in (6.6) and the average values of dc variables in (6.10) can be related using parametric functions 1( )iw   and 1( )vw   as  1 1 1 1( )  ,     ( )v dc dc iV w v i w I    . (6.15) Since 1V  is equal to eqdsv  and 1I  is equal to eqdsi  [38], the relationship between dc variables and fundamental components of ac variables can be expressed as  1 1( )  ,      ( )e eqds v dc dc i qdsw v i w   v i . (6.16) Moreover, the displacement angle between the fundamental components of ac variables can also be defined as a parametric function,   1 1( ) tanedseqsii        . (6.17) To capture the relationship between the dc variables and harmonics of ac variables of the rectifier, as opposed to [64] that uses multiple qdn transformations (for several ac harmonics), here the oscillatory components of ac variables eqdsv  and eqdsi  are directly constructed from the transformations in (6.8). Typical waveform for transformed qd variables are shown in Figures 6.7(e) – 6.7(f). As an example, the d-axis current dsi  is depicted in Figure 6.10 decomposed into average value and oscillatory components, according to (6.9). As it can be observed from Figure 6.10 (c), the oscillatory components of transformed qd variables are periodic over the prototypical switching period (i.e., 182 angle β). Since the oscillatory components of qd variables are corresponding to the total sum of ac harmonics in abc coordinates, it is realized that the ac harmonics can be directly reconstructed (without multiple qdn transformations) using samples of oscillatory components as demonstrated in Figure 6.10 (c). In this technique, since the oscillatory components are periodic, by capturing sufficient samples over the angle β only, the ripples of qd variables can be reproduced with high accuracy. As a result, the sum of all the harmonics of ac   Figure 6.10. Typical waveform of a six-pulse diode rectifier d-axis current decomposed into average-value and oscillating components: (a) rectifier d-axis current, (b) average-value of rectifier d-axis current, (c) oscillating component of rectifier d-axis current, (d) reconstruction angle rec  used for mapping the angles of samples. 183 variables in abc coordinates (not only several selected harmonics as in [64]) can be reproduced with high accuracy. To relate the oscillatory components of qd variables (correspondingly sum of all harmonics of abc variables) to average values of dc variables of the rectifier, at any moment, parametric functions can be used to relate each sample of oscillatory components over angle β to dc variables as   [ ] ( )  ,     [ ] ( )e q e dqs v dc ds v dcv k w v v k w v    , (6.18) for qd voltages, and as  ( ) [ ] ,     ( ) [ ]q e d edc i qs dc i dsi w i k i w i k    , (6.19) for qd currents. Here, k is the k-th sample of oscillatory qd components (see Figure 6.10(c)) where     1,2,3,...,k m , (6.20) and m is the number of samples to be used for reconstruction of waveforms. Assuming equidistant samples, the resolution of sampling is  m  radians over the angle β. It is noted that selecting larger values of m (i.e., more samples) results in a more accurate reconstruction of harmonics. It is noted that the oscillatory components of dc-side variables of the rectifier (dcv  and dci ) are also periodic over the angle β as illustrated in Figures 6.7(c) – 6.7(d). Therefore, by defining appropriate parametric functions, the presented 184 technique can also be used to obtain relationships between average values of ac-side qd variables and oscillatory components of dc-side variables as  ( ) [ ] ,     [ ] ( )e dc dc eqds v dc dc i qdsw v k i k w   v i , (6.21) where similar to Figure 6.10(c), [ ]dcv k  and [ ]dci k  are the k-th samples of dc-voltage ripples and dc-current ripples over the angle β, respectively. 6.2.1.2 Establishing parametric functions It would be very challenging, if not impossible, to obtain the relationships (6.16) – (6.21) analytically for LCRs under various operating conditions, especially when losses are considered. Therefore, in the proposed approach, a computer-aided technique is used to establish the parametric functions in (6.16) – (6.21) from a brief set of simulations using the detailed model of the considered system.  Similar to the conventional PAVM methodology [52], the operating point of the rectifier is specified in terms of its terminal currents and voltages by the so-called dynamic impedance defined as  dcd eqdsvz i. (6.22) Based on (6.16), to relate the amplitude of fundamental components of ac voltages and currents to average values of dc-side variables, similar to the GPAVM, two-dimensional parametric functions are defined in terms of the dynamic impedance dz  and thyristor firing angle   as 185  1 1( , )  ,     ( , )eqds dcv d i d edc qdsiw z w zv  vi. (6.23) Also, based on (6.17), similar to the GPAVM, relationships between the angles of fundamental components of ac voltages and currents are captured using a two-dimensional parametric function in terms of dynamic impedance dz  and thyristor firing angle   as  1( , ) tanedsd eqsizi        . (6.24) To obtain the relationships between the harmonic components of ac-side and average values of dc-side variables, parametric functions are defined based on (6.18) – (6.20) as  ( , , )  ,   ( , , )e eqsq d dsv d rec v d recdc dcv vw z w zv v     , (6.25) for ac voltages, and  ( , , )  ,   ( , , )q ddc dci d rec i d rece eqs dsi iw z w zi i     , (6.26) for ac currents, respectively. The parametric functions (6.25) and (6.26) are three-dimensional in terms of the rectifier dynamic impedance dz  and thyristor firing angle   as well as a reconstruction angle rec  defined as  0 ,      0Trec e recdt      . (6.27) The angle rec  is used to map the samples of ripples to their corresponding angle over the interval β, as illustrated in Figure 6.10(d). It is also realized that 186  [ ]rec k km    . (6.28) In the same way, the oscillatory components of dc-side variables can be related to average values of ac-side qd variables based on (6.21) by defining three-dimensional parametric functions as   ( , , )  ,   ( , , )eqdsdc dc dcv d rec i d rec edc qdsiw z w zv    vi. (6.29) It is worth mentioning that the parametric functions in (6.29) that are used to reconstruct the ripples of dc-side variables, will give a model that reproduces similar results for dc-side variables as a detailed model (i.e., includes the ripples of dc-side variables of rectifier); hence achieving a parametric continuous detailed model. However, if needed to neglect the dc-side ripples, the parametric functions (6.29) can be set to zero, which makes the proposed approach equivalent to the parametric average-value model of the rectifier.  To establish the parametric functions (6.23) – (6.26) and (6.29), the power system of Figure 6.8 is simulated for various thyristor firing angles ( ) over the desired range of loading conditions (dz ) by changing the dc network load (here represented as an resistive load lR ), using the detailed model of the rectifier (which may include non-idealities, losses, etc.). The computed functions (6.23), (6.24) are stored in two-dimensional lookup tables in term of dz  and  , while the parametric functions (6.25), (6.26) and (6.29) are stored in three-dimensional lookup tables in term of dz ,  , and reconstruction angle rec (6.28). The 187 numerical procedure of obtaining these lookup tables is presented as a pseudo-code in Figure 6.11 (Algorithm 1). Therein, min , ,minlR  and max , ,maxlR  are the minimum and maximum considered firing angle and dc load resistance, respectively, with step  , ,steplR  as their resolution in parametric functions.   Figure 6.11. Pseudo-code for establishing parametric functions of the proposed parametric hybrid continuous detailed/AVM model.   6.2.1.3 Implementation of the proposed parametric hybrid continuous detailed/AVM model Once the parametric functions (6.23)–(6.26), (6.29)  are constructed as lookup tables, the switching rectifier is replaced by a continuous non-switching circuit Algorithm 1. Establishing parametric functions using detailed simulation. 1.     for min   to max  step step  do 2.         for ,minl lR R  to ,maxlR  step ,steplR  do 3.              Initialize the detailed model 4.              Start simulation 5.             Establish rec  in (6.27)–(6.28) for mapping the angles of sample points 6.              Compute dynamic impedance dz  in (6.22) 7.              Compute and process parametric functions (6.23)–(6.26), (6.29) 8.              Save (6.23), (6.24) in 2-D lookup tables in terms of dz  and   9.              Save (6.25), (6.26), (6.29) in 3-D lookup tables in terms of dz ,   and rec  10.            End simulation            11.        end for 12.     end for 188 composed of dependent voltage and current sources that are algebraically related and interfaced with the ac and dc subsystems [52], as shown in Figure 6.12. Moreover, the rectifier ac-side subsystem can be represented either in abc phase coordinates or in transformed qd coordinates.  The qd reference frame may be fixed at the ac source (if the ac subsystem is represented by its Thévenin equivalent voltage sources) or on the rotor (if the ac subsystem is a rotating machine), as can be seen from Figure 6.9. Therein, the subscript “r” is used to denote the rotor (or source) reference frame. In this Section, the rectifier terminal voltages abcsv (or rqdsv ) and the rectifier dc current dci  are considered as the outputs. Also, the source currents abcsi (or rqdsi ) and the rectifier dc voltage dcv , as well as the thyristor firing angle  , are considered to be the inputs. The corresponding implementation of the proposed parametric hybrid detailed/AVM model is illustrated in Figure 6.13. Therein, the input “Detailed/AVM” determines whether the hybrid model is used as a detailed model (i.e., input=1) or as the AVM (i.e., input=0). A phase-locked-loop (PLL) may be used to obtain e  from the rectifier terminal voltages. Alternatively, when a rotating machine-converter system is used, the rotor position r  is available as the input, and the converter reference frame angle e  can be computed using (6.2)–(6.3). The magnitude of the fundamental component of the current eqdsi  is derived through (6.8), (6.14) and used to compute the dynamic impedance dz  based on (6.22). The firing angle   is an 189 input that may be set manually (open-loop) or computed by the dc voltage controller (closed-loop). The sub-model for computing the fundamental components of ac variables requires the arguments dz  and   as the inputs to the parametric functions (6.23)–(6.24) (i.e., 1( )iw  ,1( )vw  , ( )  ). Based on input dcv  and calculations with parametric functions, the fundamental voltages are computed (in converter reference frame) as  1( , )eqs v d dcv w z v ,       0edsv  . (6.30) To consider the harmonic components of ac variables, the oscillatory components of qd voltages should be added to their average values in (6.30). The sub-model for computing the oscillatory components of qd voltages requires the arguments dz ,   and rec  as the inputs to the parametric functions in (6.25)  (i.e., ( )qvw  , ( )dvw  ), which are used with the input dcv  to compute the oscillatory components of qd voltages  (in converter reference frame) as  ( , , )e qqs v d rec dcv w z v  ,       ( , , )e dds v d rec dcv w z v  . (6.31) abcsvabcsir dciAC-SideSubsystemDC-Side SubsystemdcvInterfacing CircuitLCR ModelDetailed / AVM Figure 6.12. Interfacing of the proposed hybrid parametric detailed/AVM model with the ac- and dc-side subsystems. 190 abcsvdzrdci Parametric Functions (6.23)-(6.24)Eq. (6.22)dcvrecabcsiEqs. (6.2),(6.3), (6.8)-(6.11), (6.34)eqdi Eqs.(6.36)-(6.38)Detailed / AVM Parametric Functions (6.25) Parametric Functions (6.29)dciwdvwqvw1vw1iweEq. (6.27)Eqs.(6.30)-(6.35) Figure 6.13. Implementation of the proposed hybrid parametric detailed/AVM model.  The total rectifier qd voltages (which correspond to the sum of fundamental and total harmonic components of abc voltages) are obtained by adding the average-value and ripples of qd voltages as  e e eqs qs qsv v v  ,       e e eds ds dsv v v  . (6.32) For interfacing with the rectifier ac-side subsystem, assuming that the source voltages and currents are in source (rotor) reference frame (i.e. rqdsv  and rqdsi ), the output voltages in (6.32) can be transformed as        cos sinsin cosr eqds qds      v v , (6.33) where   is calculated based on Figure 6.9 as  1,1 11,( , ) tanrdsd rqsizi         . (6.34) For interfacing with the rectifier ac-side subsystem in abc coordinates, the output voltages of the model can be calculated as 191   1( ) rabcs s r qdsv K v . (6.35) To compute the average-value components of dc-side variables of the rectifier, the parametric function 1(.)iw  in (6.23) with arguments dz and   is used with the computed magnitude of input currents eqdsi  as  1( , ) edc i d qdsi w z  i .  (6.36) Also, to consider the ripples of dc-side variables (similar to a detailed model), the parametric function (.)dciw  in (6.29) with arguments dz ,   and rec  is used with the computed magnitude of input currents eqdsi  to compute the oscillatory components of dc-side current as  ( , , )dc edc i d rec qdsi w z   i .  (6.37) Finally, the total dc interfacing output current is calculated as  dc dc dci i i  .  (6.38) It is noted that in Figure 6.13 the parametric function (.)dciw  is multiplied by the input “Detailed/AVM”. This allows the presented hybrid model to be used as either a detailed model (i.e., input “Detailed/AVM” = 1 to consider the dc-side ripples) or as an AVM (i.e., input “Detailed/AVM” = 0 to neglect the dc-side ripples) of LCR.  192 6.2.2 Verification of the proposed parametric hybrid detailed/AVM model of LCRs   To investigate the performance of the proposed parametric continuous hybrid detailed/AVM model of LCRs, the generic ac–dc power system of Figure 6.8 is considered for the study. Here, the ac-side subsystem consists of a PM synchronous generator-turbine whose speed is regulated to output constant three-phase sinusoidal ac voltages at 60Hz. To show the generality of the proposed hybrid detailed/AVM, the line-commutated rectifier is considered both with diode and thyristor switches, including their conduction losses represented by the ON resistance Ron and voltage drop Von. Moreover, LCRs with faulty switches are considered. The dc-side subsystem consists of an optional low-pass filter (here a dc capacitor fC ) connected to a dc network which is represented by a resistive load lR . The parameters of the system are summarized in Appendix K.   The power system of Figure 6.8 has been implemented using four different models of the LCR. The conventional switching detailed model has been implemented in MATLAB/Simulink environment [27] which uses standard library components of SimScape/SimPowerSystem blockset [28] for switches of LCR. Also, therein the CCPD model [38, Chap. 5] is considered for the PM synchronous machine to interface with the detailed switching model of the 193 rectifier in abc coordinates. Also, as the most state-of-art existing average-value model of LCRs, the GPAVM [Section 4.2.1] is implemented in MATLAB/Simulink [27] considering up to 7th harmonics for ac variables, for the purpose of comparison. In addition, the presented parametric continuous hybrid detailed model (i.e., input “Detailed/AVM” = 1) as well as the proposed hybrid AVM (i.e., input “Detailed/AVM” = 0) are implemented in MATLAB/Simulink [27] using (6.22) – (6.38). For the three latter models, the LCR models are directly interfaced with the qd model [38, Chap. 5] of the PM synchronous machine. 6.2.2.1 Line-commutated rectifier with diode switches  Here, it is assumed that the six-pulse line-commutated rectifier in Figure 6.8 is composed of six non-ideal diodes with conduction loss parameters (i.e., Von and Ron) presented in Appendix K. The accuracy of the subject models is investigated in reconstructing the detailed model waveforms under various loading conditions in steady-state and transients. 6.2.2.1.1 Steady-state operation First, the equivalent resistive load of the dc network is chosen to be 900 lR    to operate the diode rectifier under very light loading condition resulting in DCM mode [49]. The ac- and dc-side variables of the rectifier are depicted in Figure 6.14 as obtained by the four considered models. As it can be observed in Figure 6.14(a), the ac-side voltage is very close to sinusoidal due to light loading 194 condition and the four subject models predict consistent waveforms. However, it is observed in Figure 6.14(b) that the GPAVM is accurate up to the considered harmonic order (i.e., 7th harmonic) while the two proposed hybrid/AVM and hybrid/detailed models provide identical waveforms as the switching detailed   Figure 6.14. The steady-state waveforms of ac- and dc-side variables of diode rectifier as obtained by the four considered models in DCM condition: (a) rectifier phase a voltage, (b) rectifier phase a current, (c) rectifier dc voltage, and (d) rectifier dc current. 195 model. This is achieved by considering sufficient samples of the transformed qd variables over a switching period as described in Section 6.2.1.1. Also, as it can be observed from Figures 6.14(c) and 6.14(d), the GPAVM and proposed hybrid/AVM are both able to accurately predict the average values of dc-side variables of the rectifier. At the same time, the proposed hybrid/detailed model can reconstruct the ripples of the dc variables very accurately, visibly identical to the switching detailed model.  Next, the equivalent resistive load of the dc network is chosen to be 40 lR    to operate the diode rectifier in CCM-1 mode [49]. The ac- and dc-side variables of the rectifier are depicted in Figure 6.15 as obtained by the subject models. As it can be observed in Figures 6.15(a) – 6.15(b), the ac-side voltage and current are accurately predicted by the GPAVM up to the considered harmonics; whereas, the two proposed hybrid/AVM and hybrid/detailed models provide identical waveforms as the switching detailed model, and even the sharp non-linear discontinuities of waveforms are reconstructed although the proposed models are continuous. Also, Figures 6.15(c) and 6.15(d) verify that the proposed hybrid/AVM can accurately predict the average values of dc-side voltage and current in CCM-1 mode, similar to the GPAVM. In addition, the proposed hybrid/detailed model can reproduce the ripples of the dc variables similar to the switching detailed model. 196  Figure 6.15. The steady-state waveforms of ac- and dc-side variables of diode rectifier as obtained by the four considered models in CCM-1 condition: (a) rectifier phase a voltage, (b) rectifier phase a current, (c) rectifier dc voltage, and (d) rectifier dc current.  The harmonic content of the ac-side variables in Figure 6.14 for DCM mode and in Figure 6.15 for CCM-1 mode as obtained by the four subject models are shown in Figures 6.16 and 6.17, respectively. The corresponding data of  197  Figure 6.16. The harmonic spectrum of diode rectifier ac variables in DCM condition as obtained by the four subject models for: (a) ac phase voltage, (b) ac current.   Figure 6.17. The harmonic spectrum of diode rectifier ac variables in CCM-1 condition as obtained by the four subject models for: (a) ac phase voltage, (b) ac current.  198 harmonic content for ac voltage and current are also presented in Tables 6.1 and 6.2, respectively for DCM mode, and summarized in Tables 6.3 and 6.4 for CCM-1 mode, respectively. As it can be seen from Figures 6.16 – 6.17 and Tables 6.1 – 6.4, the GPAVM can accurately reconstruct the harmonics of ac variables, but up to the considered 7th harmonic components. However, the proposed modeling methodology in Section 6.2.1 allows the hybrid/AVM and hybrid/detailed models to accurately reconstruct all the harmonics of ac variables (shown in Figures 6.16 – 6.17 up to 19th harmonic components due to limited space), similar to the switching detailed model of LCR.   Table 6.1. The harmonic content of phase a voltage (vas) of diode rectifier as predicted by the subject models in DCM mode (Rl = 900Ω) Harmonic Order Switching  Detailed Model GPAVM Proposed Hybrid Detailed/AVM 1st 66.560 66.560 66.550 5th 1.3723.79 1.3724.09 1.3722.76 7th 1.423.11 1.423.33 1.412.98 11th 0.5793.81 0 0.5692.07 13th 0.4331.97 0 0.4229.71 17th 0.4688.75 0 0.4486.67 19th 0.3644.81 0 0.3543.35      199 Table 6.2. The harmonic content of phase a current (ias) of diode rectifier as predicted by the subject models in DCM mode (Rl = 900Ω) Harmonic Order Switching Detailed Model GPAVM Proposed Hybrid Detailed/AVM 1st 0.139-11.88 0.139-11.92 0.139-12.08 5th 0.099119.0 0.099119.2 0.098117.8 7th 0.06991.4 0.06991.6 0.06890.8 11th 0.020-170.5 0 0.019-171.0 13th 0.012113.9 0 0.012112.4 17th 0.010-176.6 0 0.010-179.6 19th 0.007128.6 0 0.007125.3     Table 6.3. The harmonic content of phase a voltage (vas) of diode rectifier as predicted by the subject models in CCM-1 mode (Rl = 40Ω) Harmonic Order Switching Detailed Model GPAVM Proposed Hybrid Detailed/AVM 1st 63.110 63.130 63.080 5th 7.369.54 7.379.43 7.329.61 7th 3.07-139.2 3.07-139.6 2.99-138.8 11th 2.60-65.33 0 2.57-65.12 13th 2.40127.1 0 2.38125.9 17th 1.37165.4 0 1.31163.2 19th 1.2516.96 0 1.2315.87      200 Table 6.4. The harmonic content of phase a current (ias) of diode rectifier as predicted by the subject models in CCM-1 mode (Rl = 40Ω) Harmonic Order Switching Detailed Model GPAVM Proposed Hybrid Detailed/AVM 1st 2.736-12.92 2.719-12.96 2.721-12.87 5th 0.596103.6 0.597103.5 0.592103.0 7th 0.237-50.89 0.236-51.38 0.234-51.04 11th 0.10226.17 0 0.10025.14 13th 0.084-140.6 0 0.081-139.8 17th 0.033-98.54 0 0.032-99.03 19th 0.029103.1 0 0.029104.9  6.2.2.1.2 Transient operation Here, a 10-second transient study is conducted to investigate the performance of the subject models in transient operation. It is assumed that the system of Figure 6.8 is initially in steady-state and the rectifier is operating in CCM-1 mode supplying 40 lR   . At t=6s, the load of the dc network is increased by stepping down the resistive load to 1 lR    (corresponding to CCM-2 mode), and the simulation is continued up to 10 seconds. The transient response of the ac- and dc-side variables of LCR at the moment of transient are shown in Figure 6.18. As it can be observed from Figures 6.18(a) – 6.18(b), the GPAVM and the two proposed hybrid/AVM and hybrid/detailed models are able to accurately capture the transient of the system consistent with the switching detailed model. However, the GPAVM can reconstruct the waveforms up to the considered  201   Figure 6.18. Transient response of the ac- and dc-side variables of diode rectifier as obtained by the four considered models when the rectifier enters CCM-2 from CCM-1 at t=6 s for: (a) rectifier phase a voltage, (b) rectifier phase a current, (c) rectifier dc voltage, and (d) rectifier dc current.   harmonic components (here 7th harmonic), while the proposed hybrid models achieve identical results to the switching detailed model. This is better 202 illustrated in the magnified area of Figure 6.18(a). Moreover, Figures 6.18(c) and 6.18(d) verify the performance and accuracy of the hybrid/AVM model in capturing the dynamic average values of dc-side variables of rectifier identical to the GPAVM. Meanwhile, the proposed hybrid/detailed model is also able to reconstruct the oscillatory components (ripples) of dc-side variables similar to the switching detailed model, as shown in Figures 6.18(c) – 6.18(d).  6.2.2.2 Line-commutated rectifier with thyristor switches Here, it is assumed that controlled thyristor switches are used for six-pulse LCR in Figure 6.8 with parameters presented in Appendix K. During the course of a 10-second transient study, the accuracy of the subject models is investigated in reconstructing the switching detailed model waveforms under various loading conditions including the dynamics of firing angle control. Initially, the thyristor firing angle is set to 15   and the system is operating at steady-state condition with the LCR operating in CCM-2 mode supplying the dc load with 1 lR   . At t=2.5s, the dc load starts to decrease by increasing the dc resistive load lR  with the profile shown in Figure 6.19(a). As the dc load decreases the rectifier enters the CCM-1 mode, while the firing angle of thyristors is kept constant at 15  . Then at t=7s, the dc voltage controller (a PI controller with parameters presented in Appendix K) is activated to regulate the rectifier output dc voltage at a set-point of 60 V dc by controlling the firing angle 203 of thyristors, as shown in Figure 6.19(d), and the rectifier enters into DCM mode of operation. The transient responses of several system variables are shown in Figure 6.19.   Figure 6.19. Transient response of the system variables due to dc load variation over a wide range of operating conditions as well as voltage control activation as predicted by the four considered models: (a) resistive load of dc network, (b) rectifier dc voltage, (c) rectifier dc current, and (d) rectifier firing angle. The voltage controller is activated at t = 7s to maintain the output voltage at 60 V dc. 204 As it can be observed from Figures 6.19(b) and 6.19(c), the proposed hybrid/AVM is able to accurately predict the average values of dc-side variables of thyristor-controlled rectifier similar to its GPAVM. In addition, the proposed hybrid/detailed model can capture the ripples of dc-side variables similar to the switching detailed model, which is better demonstrated in magnified areas of Figure 6.19(c). It is also verified in Figures 6.19(b) – 6.19(d) that the two proposed hybrid models are valid and provide accurate results during the closed-loop control operation of thyristor rectifier with firing angle regulation, as depicted in Figure 6.19(d) where the results of the proposed models are consistent with GPAVM and the switching detailed model of LCR.  The ac-side variables of thyristor-controlled rectifier under CCM-2, CCM-1 and DCM conditions encountered during the 10-second transient study while changing the dc load over a wide range are illustrated in Figures 6.20 – 6.22, respectively. As it can be observed in Figures 6.20 – 6.22, the GPAVM can reconstruct the distorted ac waveforms up to the considered harmonic order. This is better realized from Figure 6.20(a), where the results of GPAVM are visually different from the detailed model which is due to not including higher order harmonics, although being accurate for the considered harmonic components. However, the two proposed hybrid/AVM and hybrid/detailed models are both able to provide accurate results that perfectly match the results of the switching detailed model of LCR.  205  Figure 6.20. The steady-state waveforms of ac-side variables of thyristor-controlled rectifier as obtained by the four considered models in CCM-2 condition: (a) rectifier phase a voltage, and (b) rectifier phase a current.  Figure 6.21. The steady-state waveforms of ac-side variables of thyristor-controlled rectifier as obtained by the four considered models in CCM-1 condition: (a) rectifier phase a voltage, and (b) rectifier phase a current. 206  Figure 6.22. The steady-state waveforms of ac-side variables of thyristor-controlled rectifier as obtained by the four considered models in DCM condition: (a) rectifier phase a voltage, and (b) rectifier phase a current. 6.2.2.3 Line-commutated rectifier with faulty switches To verify the generality and applicability of the proposed modeling technique to line-commutated rectifiers with internal faults and asymmetrical operation, here an LCR with faulty switches is considered. It is noted that when an internal fault occurs inside the rectifier, all harmonic components (i.e., 2nd, 3rd, 4th, 5th, 6th, 7th,...) will be present [75]. As a result, the oscillatory components of transformed qd ac variables as well as the ripples of dc-side variables of LCR will be periodic with a frequency equal to the line frequency ef . Therefore, the angle β in (6.1) should be selected as 2  for this case. Also, the parametric functions (6.23)–(6.26), (6.29) should be acquired for various fault configurations 207 which are selected in the model with an input “Rectifier-State” that determines whether the rectifier is in normal condition or in fault state, and in case of faulty condition it also specifies the type of fault (i.e., which/how the switches are faulted). For the purpose of comparison, the PAVM of faulty rectifiers presented in Section 5.2 has also been implemented in MATLAB/Simulink [27], which considers components up to 7th harmonics in both positive and negative sequences, as well as the dc-offsets on ac variables. It is assumed that the LCR in Figure 6.8 is composed of six diode switches {D1, D2, …, D6} (see Figure 6.8 for the numbering order of switches) and is operating at steady-state condition in CCM-1, supplying the dc load with 40 lR   . At t=3s, the load of LCR is increased by stepping down the dc load resistance to 1 lR    and a fault occurs inside the LCR and diodes D2&D3 become open-circuited (in the PAVM [Section 5.2] and proposed hybrid models, the “Rectifier-State” input changes from “Normal” to “D2&D3 open” state). The simulation is continued up to 10 seconds.  The transient responses of dc-side and ac-side variables of rectifier at the moment of internal fault are depicted in Figures 6.23 and 6.24, respectively, as predicted by the switching detailed mode, the PAVM [section 5.2], and the two proposed hybrid AVM/detailed models. As it can be observed from Figure 6.23, the dc-side variables of rectifier become heavily distorted due to the internal fault and the oscillatory components of dc variables are periodic with the line frequency (i.e., 60Hz). Also, 208 it is observed that the proposed hybrid/AVM can predict the dynamics of average values of dc-side variable precisely, similar to the PAVM. In addition, the proposed hybrid/detailed model can also reconstruct the ripples of dc-side variables identical to the switching detailed model of LCR even under asymmetrical conditions with internal faults. It is also observed from Figure 6.24 that the internal fault causes asymmetrical operation of LCR and the three-phase voltages and currents are heavily distorted and not symmetric with ±120 shifts anymore. The amplitudes of three-phase voltages and currents are not also equal anymore and the three-phase ac variables possess dc-offsets. It is also noted that phase b carries only negative current (due to D3 being open) and phase c carries only positive current (due to D2 being open).  Figure 6.23. Transient response of rectifier dc-side variables as obtained by the four subject models when the load of LCR is increased by stepping down its resistance from Rl=40Ω to Rl=1Ω at t=3s; diodes D2&D3 fail and become open-circuited: (a) rectifier dc voltage, and (b) rectifier dc current. 209  Figure 6.24. Transient response of rectifier ac-side voltages and currents as obtained by the four subject models when the load of LCR is increased by stepping down its resistance from Rl=40Ω to Rl=1Ω at t=3s; diodes D2&D3 fail and become open-circuited: (a)–(c) rectifier abc phase voltages, (d)–(f) rectifier abc phase currents.  210 It is verified in Figure 6.24 that the PAVM [see Section 5.2] provides a reasonable prediction of asymmetric waveforms of the LCR under faulty conditions, but up to the considered 7th harmonic. However, the two proposed hybrid detailed/AVM models provide similar results as the switching detailed model of faulty LCR by including enough samples for oscillatory components of qd variables as described in Section 6.2.1.1. 6.2.2.4 Computational performance of the LCR models In order to benchmark the two proposed continuous parametric hybrid/AVM and hybrid/detailed models of line-commutated rectifiers (for diode bridges, thyristor-controlled rectifiers as well as LCRs with faulty switches), the computational performance of the considered models is investigated in terms of the time-step, CPU time, etc. For consistency, all the subject models are executed on a PC with Intel® Core™ i7-4510U @2.00GHz processor using the MATLAB/Simulink [27] ode3 fixed-step solver. In addition, the considered models are verified on a real-time simulator (Opal-RT OP5700 [106]) run with ode3 fixed-step solver. It is noted that the Opal-RT OP5700 does not allow CCPD model of synchronous machine with variable inductances. Therefore, the switching detailed models of rectifiers are interfaced with the qd model of machine [38, Chap. 5] in abc coordinates using snubber circuits (resistors here) [73], which 211 are selected here to be 685 Ω in order to limit the error of stator current to maximum 1% at nominal loading condition [85]. The maximum possible time-steps of simulations associated with each of the subject models are selected to achieve the same acceptable accuracy. The discrete switching detailed models of LCRs are run with 20µs due to the need for accurate zero-crossing detection. However, the GPAVM, the PAVM of faulty LCRs, and the two proposed hybrid models can run with larger time-steps. This is due to the fact that these models are continuous and do not require zero-crossing detection. The maximum possible time-step of simulation to achieve acceptable results with the GPAVM, PAVM of faulty LCRs, hybrid/AVM and hybrid/detailed model is 600µs in case of the diode rectifier, and 400µs in case of the thyristor-controlled rectifier (due to the parameters of the PI controller for dc voltage regulation). For instance, the diode rectifier waveforms in CCM-1 obtained by the GPAVM and the two proposed hybrid AVM/detailed models with 600µs time-step are shown in Figure 6.25 and compared with the reference solution. As it can be observed in Figures 6.25(a)–(b), all the three considered models can reconstruct the reference waveforms with reasonable accuracy. Also, Figures 6.25(c)–(d) verify that the GPAVM and proposed hybrid/AVM are able to capture the average values of dc-side variables of rectifier accurately. 212 In addition, it is observed in Figures 6.25(c)–(d) that the proposed hybrid/detailed model can accurately predict the reference solution at the 600μs intervals and the samples produced by the hybrid/detailed model exactly land on the reference solution.   Figure 6.25. The steady-state waveforms of ac- and dc-side variables of diode rectifier in CCM-1 condition as obtained by the GPAVM and the two proposed hybrid models with 600μs time-step compared to the reference solution: (a) rectifier phase a voltage, (b) rectifier phase a current, (c) rectifier dc voltage, and (d) rectifier dc current.  213 It is also seen that the considered models do not require zero-crossing detection (see Figures 6.25(c)–(d)), as opposed to the conventional event-driven switching detailed model of rectifier that requires to accurately locate the discrete zero-crossing points (see the sharp points in Figure 6.25(d) that correspond to zero-crossings of ac currents) to provide accurate results, which in turn requires small time-steps. To see the effect of time-step size on the predicted waveforms of the models, the conventional event-driven switching detailed model of diode rectifier and its corresponding proposed hybrid/detailed model are loaded and executed on the Opal-RT OP5700 real-time simulator with both small and large time-step sizes. Figure 6.26 shows the produced results of the two models with 20μs time-step, where the results of the two detailed models are consistent. It should be mentioned that the slight difference between the predicted results by the switching detailed model and the proposed hybrid/detailed model for the rectifier dc voltage, shown in Figure 6.26(c), is due to the error introduced to the solution of conventional switching detailed model due to the snubber resistors required for interfacing; whereas, the proposed hybrid/detailed model retains its accuracy (due to not requiring artificial snubbers). 214  Figure 6.26. The steady-state waveforms of ac- and dc-side variables of diode rectifier in CCM-1 condition as obtained by the conventional switching detailed and the proposed hybrid/detailed models run on Opal-RT OP5700 real-time simulator with 20μs time-step: (a) rectifier phase a voltage, (b) rectifier phase a current, (c) rectifier dc voltage, and (d) rectifier dc current. Figure 6.27 illustrates the waveforms obtained by the two detailed models run with 200μs. As it can be observed in Figure 6.27, the conventional detailed model entirely fails to provide accurate waveforms due to the error of zero-crossing detection at large time-steps; however, the proposed hybrid/detailed model obtains accurate results (compared with 20μs waveforms) due to being continuous and not requiring zero-crossing detection.  215  Figure 6.27. The steady-state waveforms of ac- and dc-side variables of diode rectifier in CCM-1 condition as obtained by the conventional switching detailed and the proposed hybrid/detailed models run on Opal-RT OP5700 real-time simulator with 200μs time-step: (a) rectifier phase a voltage, (b) rectifier phase a current, (c) rectifier dc voltage, and (d) rectifier dc current.  The same 10-second transient studies presented in Sections 6.2.2.1.2, 6.2.2.2 and 6.2.2.3 have been conducted on the ac–dc system of Figure 6.8 with a diode rectifier, with a thyristor-controlled rectifier and with a diode rectifier including faulty valves, respectively, and the computational performance indices are summarized in Tables 6.5 – 6.7, respectively. Also, the number of steps required to perform the simulations, presented in Tables 6.5 – 6.7, can be calculated as 216 the 10-second simulation time divided by the associated time-step size in each case. As it can be observed from Tables 6.5, the offline PC CPU times associated with the considered models of diode rectifier system are: 8.26s switching detailed model, 0.65s GPAVM, 0.27s hybrid/AVM and 0.31s hybrid/detailed model. Table 6.6 shows that the PC CPU times required to execute the subject models of thyristor-controlled rectifier system are: 22.48s switching detailed model; 0.96s GPAVM; 0.42s hybrid/AVM and 0.47s hybrid/detailed model. Also, it is seen in Table 6.7 that to simulate the diode rectifier system with internal faults on PC, the CPU times associated with the subject models are: 9.38s switching detailed model, 2.81s PAVM, 0.37s hybrid/AVM and 0.42s hybrid/detailed model. Therefore as seen in Tables 6.5–6.7, the GPAVM, the PAVM of faulty LCRs, the proposed hybrid/AVM and hybrid/detailed models of LCRs can run much faster than their switching detailed model counterparts. Also, the two proposed hybrid models of LCRs are 2–2.4 times faster than their existing GPAVM counterparts (for healthy/normal diode and thyristor rectifiers), and are 6.8–7.6 faster than their existing PAVM of LCRs with internal faults, while providing more accurate and closer results to the switching detailed model results as demonstrated in Figures 6.14 – 6.24. Moreover, the proposed continuous hybrid/detailed model can run almost 27 times (in case of diode rectifier), 48 217 times (in case of thyristor rectifier) and 23 times (in case of faulty diode rectifier) faster than their corresponding discrete switching detailed models of LCR.    It is worth mentioning that the slow simulation speed of PAVM of faulty LCRs compared to the GPAVM of healthy/normal LCRs is due to the fact that in faulty conditions, the PAVM should include all harmonic components (i.e., 2nd, 3rd, 4th, 5th, 6th and 7th harmonics) in both positive and negative sequences as well as the dc-offsets of ac variables, which increases its computational complexity and results in slower simulation speed compared to GPAVM (which needs to include only 5th and 7th harmonics in one sequence only).   Table 6.5. Computational performance of the subject models of diode rectifier system for the considered 10-second transient case-study Model Time-Step Size Offline PC CPU Time Number  of steps Opal-RT OP5700 CPU Utilization Detailed 20 µs 8.26 s 500,000 4.87 % GPAVM 600 µs 0.65 s 16,667 0.24 % Proposed Hybrid/AVM 600 µs 0.27 s 16,667 0.19 % Proposed Hybrid/Detailed 600 µs 0.31 s 16,667 0.21 %      218 Table 6.6. Computational performance of the subject models of thyristor-controller rectifier system for the considered 10-second transient case-study Model Time-Step Size Offline PC CPU Time Number  of steps Opal-RT OP5700 CPU Utilization Detailed 20 µs 22.48 s 500,000 12.89 % GPAVM 400 µs 0.96 s 25,000 0.54 % Proposed Hybrid/AVM 400 µs 0.42 s 25,000 0.43 % Proposed Hybrid/Detailed 400 µs 0.47 s 25,000 0.47 %   Table 6.7. Computational performance of the subject models of diode rectifier system with faulty switches for the considered 10-second transient case-study Model Time-Step Size Offline PC CPU Time Number  of steps Opal-RT OP5700 CPU Utilization Detailed 20 µs 9.38 s 500,000 4.75 % PAVM [Section 5.2] 600 µs 2.81 s 16,667 0.83 % Proposed Hybrid/AVM 600 µs 0.37 s 16,667 0.31 % Proposed Hybrid/Detailed 600 µs 0.41 s 16,667 0.36 %   In addition, as seen in Table 6.5, the switching detailed model of the diode rectifier requires 4.87% of CPU resources when running on OP5700 real-time simulator. However, the CPU utilization of OP5700 can be drastically decreased by application of the GPAVM (0.24%), hybrid/AVM (0.19%) and hybrid/detailed models (0.21%). 219  Also, as observed in Table 6.6, the detailed model of thyristor-controlled rectifier requires 12.89% of CPU resources of OP5700 real-time simulator. However, it is decreased to 0.54%, 0.43% and 0.47% when using GPAVM, hybrid/AVM and hybrid/detailed models, respectively. Moreover, Table 6.7 shows that the detailed model of faulty diode rectifier requires 4.75% of OP5700 CPU resources. However, the CPU utilization of OP5700 can be drastically decreased by application of the PAVM of faulty LCRs (0.83%), hybrid/AVM (0.31%) and hybrid/detailed (0.36%) models. It is also observed from Tables 6.5 – 6.7 that the proposed hybrid/AVM and hybrid/detailed models of diode and thyristor rectifiers and also faulty diode bridges require less computational resources compared to their corresponding GPAVMs, and faulty PAVM, respectively. It is also worth mentioning that the accuracy of the GPAVM of diode/thyristor rectifiers (shown in Figures 6.14 – 6.22) and/or PAVM of faulty rectifiers (shown in Figures 6.23 – 6.24) can be increased by including more harmonic components in the models. However, this would come at the cost of higher complexity of the models, thus slower simulations.  Overall, the proposed hybrid/AVM and hybrid/detailed models are demonstrated to be much faster and more efficient than the conventional switching detailed models of line-commutated rectifiers. Also, the proposed hybrid/AVM and hybrid/detailed models can consider faulty states of rectifiers 220 and are also faster and computationally more efficient than their corresponding GPAVM and PAVM. The user can select the model to be an average-value model to neglect the dc-side ripples (e.g., in system-level studies), or a detailed model to capture the dc-side ripples if needed (e.g., in real-time power-electronic applications) using the Detailed/AVM input of the proposed hybrid model; meanwhile, being more efficient and faster than the existing average-value models and conventional discrete switching detailed models of LCRs. Due to these advantageous properties, it is envisioned that the two proposed hybrid/AVM and hybrid/detailed models can be enabling new tools for efficient and practical simulation of large power systems for analysis, design and system-level studies. 221 CHAPTER 7: CONCLUSIONS AND FUTURE WORKS 7.1 Contributions and Their Anticipated Impact The research presented in this dissertation has been focused on dynamic average-value modeling of ac–ac and ac–dc power-electronic converters. In average-value modeling techniques, the effects of fast switching are averaged/neglected, and only the lower frequency dynamics of the system are captured which alleviates the burden associated with handling the discrete switching events. Therefore, the AVMs are computationally more efficient and typically execute orders of magnitude faster than their detailed model counterparts. The contributions of presented dissertation in this research area with respect to the considered objectives are listed below. 7.1.1 Contributions for Objective 1 As part of this research, in Chapter 2, for the first time, the PAVM technique has been extended to ac–ac converters which are widely used in many industrial applications to supply the load at lower/higher ac frequencies with adjustable voltage amplitude. This has been done by appropriately formulating and relating 222 the variables of the two ac-side subsystems in terms of the ac–ac converter modulation index, output frequency, and two new defined dynamic impedance/admittances. The dynamic impedance/admittances of the ac–ac converter are two-dimensional, accounting for magnitude and phase (i.e., real and reactive power, equivalently), as opposed to one-dimensional dynamic impedances for ac–dc rectifiers. The performance of the developed PAVM methodology of ac–ac converters has been demonstrated on line-commutated thyristor-based cyclo-converters as well as force-commutated matrix converters that enable direct ac–ac conversion using controllable bi-directional blocking switches. The presented simulation results verified excellent accuracy of the proposed ac–ac PAVM in predicting the steady-state and transient responses of system variables compared to their detailed model, while achieving much superior computational performance and simulation speed (i.e., 138~148 times faster than detailed models of half-bridge and full-bridge cyclo-converters, and ~65 times faster than detailed model of matrix converters in the considered case-studies). The accuracy of the proposed PAVM has also been verified in predicting the losses and efficiency of ac–ac converters with non-ideal switches.  7.1.2 Contributions for Objective 2 In Chapters 3–4, the previous works on PAVM [52] and MRF-PAVM [64] of LCRs have been extended and the parametric average-value modeling technique 223 is generalized by including the ac harmonics in models of thyristor-controlled rectifier systems as well as considering their nonlinear dependency (both magnitude and phase) on the line frequency. This has been done by appropriately formulating and extending the parametric functions to include the dynamics of firing angles of thyristors and including the line-frequency as a new argument/dimension in the parametric functions governing the formulation of harmonics.  It has been shown that any previous PAVM is a special case of the proposed generalized PAVM (GPAVM), which can be obtained by using only a subset of parametric functions used in the GPAVM. The added computational cost of the proposed GPAVM is insignificant compared to the previous MRF-PAVM that also considers harmonics for diode rectifiers (i.e., the simulation speeds are almost equal). Moreover, the GPAVM is capable of predicting the losses/efficiency of non-ideal LCRs. The proposed GPAVM has been verified against the detailed simulation and the experimental measurements from machine-fed rectifier system and has been shown to be very accurate over a wide range of operating conditions with fixed and variable frequency while offering superior numerical efficiency (i.e., 9.3~12.9 faster) compared to the detailed switching model. Therefore, the proposed GPAVM can be an accurate, fast and efficient simulation asset for studies of power systems that include either uncontrolled or controlled line-commutated rectifiers.  224 7.1.3 Contributions for Objective 3 Simulation and analysis of ac–dc power systems operating under unbalanced/asymmetrical conditions due to faulty converters are very important for system-level studies and have not been achieved by any AVM in the prior literature. In Chapter 5, the PAVM methodology has been extended to the asymmetrical operation of line-commutated rectifiers due to internal faults. This has been done by decomposing the asymmetric waveforms of faulty rectifiers into characteristic (i.e., 5th, 7th, …) and non-characteristic (i.e., 2nd, 3rd, 4th, etc.) harmonics as well as the dc components present on ac variables in both positive and negative sequences, and properly formulating them in terms of rectifier dynamic impedance and firing angle in their associated reference frame. The developed PAVM of faulty rectifiers has been verified against the experimental results and detailed model simulations, and demonstrated to accurately capture the unbalanced and asymmetric ac and dc waveforms under various faulty conditions of LCRs, while being computationally more efficient (i.e., ~3.8 faster) than the switching detailed models. Therefore, it is envisioned that the proposed extended PAVM may become useful for efficient studies of large-scale ac–dc systems that include analysis of faulty line-commutated rectifiers.  225 7.1.4 Contributions for Objective 4 In the detailed models of line-commutated converters, to determine the switching events accurately, the simulation program needs zero-crossing detection algorithms and/or interpolation for exact locating of the switching events, which increases the computational cost of the detailed models and/or forces them to use small time-steps. In Chapter 6, first it has been shown how much the switching detailed models of LCRs are dependent on zero-crossing detection, and when implemented without zero-crossing detection and interpolation, they lose their numerical accuracy and stability for large time-steps.  Then, a new methodology for modeling line-commutated rectifiers has been presented based on the parametric approach where the resultant model is able to reproduce detailed waveforms of currents and voltages precisely, including the switching ripples which are neglected in AVMs. This has been done by properly decomposing the rectifier ac-side harmonics (transformed to qd coordinates) as well as dc-side variables into average values and oscillatory components. The oscillatory components of ac-side qd variables (which correspond to the total harmonics in abc coordinates) and the oscillatory terms of dc-side variables are then formulated in terms of the rectifier dynamic impedance and firing angle by capturing sufficient samples from them over a prototypical switching interval. 226 The new proposed parametric detailed model is continuous (does not have discrete switching states) and does not require zero-crossing detection for producing the ac waveforms and switching ripples. Using extensive computer studies and with tests on real-time simulators, it has been demonstrated that the new proposed parametric detailed model can be simulated using fairly large time-steps (400~600 μs) as opposed to conventional discrete switching detailed models of all types of LCRs (i.e., diode bridges, thyristor-controlled rectifier as well as LCRs with internal faults) which typically need fairly small time-steps (~20 μs) to produce accurate and reliable results. Also, the proposed parametric continuous detailed model can be easily converted to a parametric AVM (by disabling the reconstruction of details of dc waveform in the model) which has been shown to achieve superior performance (i.e., 2.3~7.6 times faster) compared to the existing state-of-the-art PAVMs. These beneficial numerical features may become very useful for online and real-time simulation tools for power systems that include a large number of power-electronic loads and need to consider ac harmonics.  All the models and methodologies proposed in this dissertation are envisioned to enhance existing EMT programs and enable new simulation tools that will become useful for many researchers and practicing engineers around the world who work in the area of electric power and energy systems. The new generation of models and simulation tools will allow conducting much faster simulations of 227 large power systems in a reasonable amount of time using less computational resources (and computer hardware) which is especially desirable for applications with the increasing use of power-electronic converters. Moreover, it is envisioned that such models can be generated automatically by the proposed algorithms and corresponding software during the compilation process without introducing additional complexity for the user.  7.2 Future Works The research presented in this dissertation can be continued in several possible directions. Three potential extensions that are currently being considered by the members of the Electric Power and Energy Systems group at the University of British Columbia (UBC) that have shown promising results are discussed here. 7.2.1 Thrust 1 The GPAVM presented in Chapter 4 is easy and straightforward for implementation in state-variable-based (SVB) simulation packages (off-line and real-time). However, many EMT simulation packages are based on the nodal EMTP-type solution. In order to directly interface the GPAVM with the external network in EMTP-type programs, the dc- and ac-side equations must be discretized and linearized, such that the final model may be formulated in a 228 Norton or Thevenin equivalent structure. Therefore, this task aims to obtain a discretized form of the GPAVM with its Thévenin equivalent dc- and ac-side interfacing impedances by numerically discretizing its equations including ac harmonics and assembling the resultant resistance/conductance sub-matrices that will be inserted into the EMTP nodal equation. The preliminary results are reported in [107] and are shown to be promising. 7.2.2 Thrust 2 The PAVMs of LCRs (diode bridges and thyristor-controlled rectifiers) with internal faults presented in Chapter 5 assume that the fault is introduced to the model (both PAVM and detailed model) with a fault signal which specifies the moment of fault and its type. However, the failure of commutation of the converter/rectifier switches may occur due to external abnormal conditions, for example, due to unbalanced ac sources or faults on ac lines (e.g., line-to-ground faults). This phenomenon needs further investigation and appropriately formulating the model to correctly and automatically include the effect of the unbalanced and asymmetric external systems on the operation of converter system and commutations of its switches, which is already being considered by our research group.   229 7.2.3 Thrust 3 The modeling methodology presented in Chapter 6 provides parametric detailed models of line-commutated rectifiers which are continuous, and it has been shown to allow much larger simulation time-steps than in traditional discrete switching detailed models. This is desirable for system-level studies of large power systems that consist of many switching converters. The presented methodology can be extended to other types of power-electronic converters [e.g., force-commutated IGBT-based converters and PWM voltage source converters (VSCs)]. 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Parameters of Case Study System in Section 2.2.3 Half-bridge cyclo-converter system parameters: Three-phase grid system: 6600 VlineV  , 0.05 thR   , 0.2 mHthL  , 60 Hzef  . Three-Phase RL load: 5 LoadR    , 5 mHLoadL  .              245 Appendix B. Parameters of Case Study System in Section 2.2.4 Full-bridge cyclo-converter system parameters: Three-phase grid system:  63000 VlineV  , 0.1 sR   , 0.26 mHsL  , 60 Hzef  , Trans. Ratio 63000:6600 . Three-Phase RL load: 5 LoadR    , 5 mHLoadL  .                246 Appendix C. Parameters of Case Study System in Section 2.3.1 Three-phase source and ac filter parameters: 380 VlineV  , 60 Hzif  , 0.5 fr   , 1 mHfl  , 20 μFfc  . Three-phase RL load:  10 LoadR    , 3 mHLoadL   , 1~ 400 Hzof  . Switch parameters: 0.05 onr    , 1.5 Vonv  , current tail time = 2 μs , current 10% fall time = 1 μs . Anti-parallel diode parameters: 0.05 onr    , 1.5 Vonv  .            247 Appendix D. Matrices and Scalars Introduced in Chapter 3  2 2 2 2 2 24 4 4 212 12 2 2 2 2 22 4 6 62 2 2 2 2 2 6 6sqdr akqdr           × × ×× ×× × ××× × ×0 I 0I 0A I 0 0 L0 r0 0 0,  (D.1)  2 14 4 4 2 4 42 112 1 121 22 4 2 4 6 46 66 1,  01sqd qdakqdr                 × × ××0I 0 I1B L B0 r 0, (D.2)  2 12 112 1 12 1 121 2 6 46 1 ,     ,     01qd qd qda a          01C L D L D 0 ,  (D.3) where    6 60 00 00 00 00 0 00 0 0l lm mq lm mq lqd mqlm mq l lm mq lqd mqlqd l lm md lm md mdalqd lm md l lm md mdmq mq lkq mqmd md lkd mdL L L L L L LL L L L L L LL L L L L L LL L L L L L LL L L LL L L L                     L .  (D.4) 248     2 2cos cos cos 0 0 03 32 20 0 0 cos cos cos3 32 2sin sin sin 0 0 02 3 3( )3 2 20 0 0 sin sin sin3 31 1 10 0 02 2 21 1 10 0 02 2 2r r rr r rr r rrr r r                                                                     K6 6        .  (D.5)       1 2 1 2 1 2 1 234pme md sq sq sd sd kd mq sd sd sq sq kqmdPT L i i i i i L i i i i iL                . (D.6)            249 Appendix E. Parameters of Case Study System in Section 3.4 Six-phase machine parameters [79], [85]: Rated phase voltage: 240 V, rated power: 100 kVA, poles: 4, rated speed: 1800 rpm, PM flux: 1.05 wbpm  , 6  ,  16 msr   , 2.37 mΩkdr  , 2.5 mΩkqr  ,  1 2 43 μHa aL  , 1 2 43 μHa bL   , 1 2 0 μHa cL  , 75 μHlL  ,  140 μHlkdL  , 180 μHlkqL  , 3 mHmdL  , 1.4 mHmqL  . DC filter parameters: 0.1 fr   , 0.5 mHfL  , 1000 μFfC  , 52 10f  .            250 Appendix F. Parameters of Case Study System in Section 4.1.1 DC sub-system parameters: 0.1 fr    10 mHfl   1000 μFfc   30 lr   .251 Appendix G. Parameters of Case Study System in Section 4.2 PM synchronous machine parameters: FANUC ac servo: A06B-0315-B063#0008, number of poles: 8, voltage constant: 0.177 V.s/rad , at 60 Hz, voltage: 90 V , current: 7.6 A , power: 1.2 kW , 0.748 sr   , 6.06 mHdL  , 7.6 mHqL  .  LCC-diode bridge: Harris semiconductor: RHRG7560, conduction loss parameters: 0.091 onR   , 0.637 VonV  . DC filter parameters: 0.535 fR   , 12.21 mHfL  , 470 μFfC  . PI controller parameters: 0.5Pk  , 100Ik  .         252 Appendix H. Parameters of Case Study System in Section 5.2.3 PM synchronous machine parameters: Presented in Appendix G. LCC-diode bridge: Presented in Appendix G. DC filter parameters: 0.63 fR   , 12.38 mHfL  , 470 μFfC  , 410  .               253 Appendix I. Parameters of Case Study System in Section 5.3.2 PM synchronous machine parameters: Presented in Appendix G. LCC-diode bridge: Presented in Appendix G. DC filter parameters: 0.63 fR   , 12.38 mHfL  , 2700 μFfC  . DC voltage PI controller parameters: 0.5Pk  , 100Ik  .             254 Appendix J. Parameters of Case Study System in Section 6.1 Three-phase Thévenin equivalent circuit: 46.95 V(rms)asE  , 60 Hzef  , 1.49 sr   , 12.12 mHsL  . LCC-diode bridge: Presented in Appendix G. DC filter parameters: 0.57 fr   , 12.21 mHfL  , 470 μFfC  . Snubber circuit parameters: 5000 snubr   , 50 nFsnubC  .            255 Appendix K. Parameters of Case Study System in Section 6.2.2 PM synchronous machine parameters: Presented in Appendix G. LCC-diode bridge: Presented in Appendix G. DC filter parameters: 470 μFfC  . DC voltage PI controller parameters: 0.1Pk  , 100Ik  . 

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