Applied Science, Faculty of
Electrical and Computer Engineering, Department of
DSpace
UBCV
Amiri, Navid
Doctor of Philosophy - PhD
University of British Columbia
Design and analysis of power systems has always been reliant on mathematical models of power system components and their implementation in sophisticated electromagnetic transient (EMT) simulation programs. Electric machines are the dominantly used components in models of power grids, and in most simulation programs rotating machines are represented using lumped-parameter models based on magnetically-coupled phase windings. Advancements in semi-conductor devices have allowed conventional three-phase machines as well as multi-phase machines (with more than three electrical phases) to be used with power electronic interfaces in wide range of electromechanical energy conversion applications, including advanced motor drive systems as well as integration of renewable energy. The multi-phase machines, at the expense of their increased complexity, may also provide additional advantages over the conventional three-phase machines, e.g. reduced power per phase, reduced semi-conductor current/voltage rating per phase, increased reliability, fault tolerant operation, etc. However, such multi-phase machines have unique electromagnetic characteristics and complex mathematical models which also become additional challenge in terms of both implementation and numerical performance in simulation programs. The main goal of this dissertation is to propose accurate and numerically efficient models of multi-phase machines which also consider key characteristics such as magnetic saturation and flux harmonics. In this dissertation, the voltage-behind-reactance (VBR) methodology has been used to derive new models for wound-rotor six-phase synchronous machines (including leakage flux cross-coupling and magnetic saturation), five-phase machines (including flux harmonics and inter-harmonic couplings) and also three-phase saturable induction machines considering flux harmonic distortions. The new models possess the advantage of constant-parameter interfacing circuits when implemented in state-variable-based (SVB) simulation programs, which has not been possible prior to this work. Numerous case studies demonstrate the advantageous numerical accuracy and efficiency of the proposed models, and their beneficial impact on simulation performance compared to the alternative existing state-of-the-art machine models of the same class. It is envisioned that due to their numerous advantages the new models will become included in many commonly-used commercial EMT programs, and will benefit thousands of researchers and engineers worldwide.
i Efficient Modelling of Special Purpose Multi‐Phase Electrical Machines for Transient Simulation Programs by Navid Amiri B.Sc., Isfahan University of Technology, 2008 M.Sc., Isfahan University of Technology, 2011 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate and Postdoctoral Studies (Electrical and Computer Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) November 2019 © Navid Amiri, 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: Efficient Modelling of Special Purpose Multi-Phase Electrical Machines for Transient Simulation Programs submitted by Navid Amiri 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. Hermann Dommel Supervisory Committee Member Dr. John Madden University Examiner Dr. Yusuf Altintas University Examiner Additional Supervisory Committee Members: Dr. Yu Christine Chen iii Abstract Design and analysis of power systems has always been reliant on mathematical models of power system components and their implementation in sophisticated electromagnetic transient (EMT) simulation programs. Electric machines are the dominantly used components in models of power grids, and in most simulation programs rotating machines are represented using lumped-parameter models based on magnetically-coupled phase windings. Advancements in semi-conductor devices have allowed conventional three-phase machines as well as multi-phase machines (with more than three electrical phases) to be used with power electronic interfaces in wide range of electromechanical energy conversion applications, including advanced motor drive systems as well as integration of renewable energy. The multi-phase machines, at the expense of their increased complexity, may also provide additional advantages over the conventional three-phase machines, e.g. reduced power per phase, reduced semi-conductor current/voltage rating per phase, increased reliability, fault tolerant operation, etc. However, such multi-phase machines have unique electromagnetic characteristics and complex mathematical models which also become additional challenge in terms of both implementation and numerical performance in simulation programs. The main goal of this dissertation is to propose accurate and numerically efficient models of multi-phase machines which also consider key characteristics such as magnetic saturation and flux harmonics. In this dissertation, the voltage-behind-reactance (VBR) methodology has been used to derive new models for wound-rotor six-phase synchronous machines (including leakage flux cross-coupling and magnetic saturation), five-phase machines (including flux harmonics and inter-harmonic couplings) and also three-phase saturable induction machines iv considering flux harmonic distortions. The new models possess the advantage of constant-parameter interfacing circuits when implemented in state-variable-based (SVB) simulation programs, which has not been possible prior to this work. Numerous case studies demonstrate the advantageous numerical accuracy and efficiency of the proposed models, and their beneficial impact on simulation performance compared to the alternative existing state-of-the-art machine models of the same class. It is envisioned that due to their numerous advantages the new models will become included in many commonly-used commercial EMT programs, and will benefit thousands of researchers and engineers worldwide. v Lay Summary Advanced multi-phase (e.g., five-, six-phase, etc.) electrical machines are being increasingly utilized in energy conversion applications in modern power systems. However, due to their complex and time-varying physical structure, the detailed models of multi-phase electrical machines and their interfacing to external networks has always been a computational bottleneck in transient simulation programs. In this dissertation, new computationally efficient models for several multi-phase machines are proposed. The new models allow direct and constant-parameter interfacing to external networks. The proposed models are validated using extensive simulations and experimental tests and are shown to significantly outperform existing models in terms of simulation speed while providing good numerical accuracy. It is envisioned that the proposed models will become included in many commonly-used electromagnetic transient simulation programs, and thus greatly benefit many researchers and engineers around the world who are working on applications of multi-phase machines and their associated controls. vi Preface Many of the contributions presented in this dissertation have been published in scientific journals, appear in conference proceedings, and have been or will be submitted for peer review. Unless stated otherwise, I derived all the proposed models, implemented all of them in computer programs, executed all simulation studies, and was responsible for writing the first draft of each paper. All this work was done under the supervision of Dr. Juri Jatskevich, who provided research direction, guidance and comments during all of the aforementioned steps. Dr. Jatskevich also helped writing and editing each paper. Other co-authors have also contributed to this research by providing feedback and editorial comments, and by revising the manuscripts. The articles resulting from this doctoral work and additional contributions by the co-authors are listed below: Chapter 2 is based on the following journal and conference papers N. Amiri, S.M. Ebrahimi, J. Jatskevich, and H.W. Dommel, “Voltage-behind-reactance model of six-phase synchronous machine considering stator mutual leakage inductance,” In proc. International Conference on Power System Transients (IPST 2015), 15-18 June 2015, Cavtat, Croatia. N. Amiri, S. M. Ebrahimi, M. Chapariha, J. Jatskevich and H. W. Dommel, “Constant parameter voltage-behind-reactance model of six-phase synchronous machines,” IEEE Trans. on Energy Conversion, vol.32, no.2, pp. 548-559, Jun 2017. Chapter 3 is based on the following journal article N. Amiri, S. M. Ebrahimi, M. Chapariha, J. Jatskevich, and H. W. Dommel, “Voltage-behind-reactance model of six-phase synchronous machines considering stator mutual leakage inductance and main flux saturation,” Elsevier J. Electric Power Syst. Res., vol. 138, pp. 155–164, Sep. 2016. vii N. Amiri, S. M. Ebrahimi, Y. Huang, and J. Jatskevich, “Improved algebraic-loop relaxation in CPVBR models of synchronous machines under power electronic switching,” IEEE Trans. on Energy Conversion. vol.33, no.2, pp. 900-903, Jun. 2018. N. Amiri, S. M. Ebrahimi, M. Chapariha, J. Jatskevich and H. W. Dommel, “Saturable and decoupled constant-parameter VBR model for six-phase synchronous machines in state-variable simulation programs,” IEEE Trans. on Energy Conversion. Early access, doi: 10.1109/TEC.2019.2907268, Mar. 2019. Chapter 4 is based on the following journal article N. Amiri, S. M. Ebrahimi, Y. Huang, J. Jatskevich and S. D. Pekarek, “Constant-parameter voltage-behind-reactance modeling of five-phase synchronous machines with air-gap flux harmonics,” IEEE Trans. on Energy Conversion. Early access, doi: 10.1109/TEC.2019.2942828, Sep. 2019. Chapter 5 is based on the following journal articles N. Amiri, S. M. Ebrahimi, and J. Jatskevich, “Two-axis model of saturable squirrel-cage induction machines including magnetizing flux harmonic distortions,” Journal, under preparation. N. Amiri, S. M. Ebrahimi, and J. Jatskevich, “Constant-parameter voltage-behind-reactance model of saturable induction machines including air-gap flux harmonics,” Journal, under preparation. viii Table of Contents Abstract ......................................................................................................................................... iii Lay Summary ................................................................................................................................. v Preface ............................................................................................................................................ vi Table of Contents ..................................................................................................................... viii List of Tables ................................................................................................................................ xi List of Figures ............................................................................................................................. xii Nomenclature ......................................................................................................................... xviii Acknowledgements ................................................................................................................... xx CHAPTER 1: INTRODUCTION ................................................................................................... 1 1.1 Motivation ..................................................................................................................................... 1 1.2 Literature Review ...................................................................................................................... 6 1.2.1 Coupled Circuit Phase Domain Models ............................................................................ 8 1.2.2 qd models ....................................................................................................................................... 9 1.2.3 Voltage‐Behind‐Reactance Models .................................................................................. 11 1.3 Research Objectives and Contributions ......................................................................... 13 CHAPTER 2: MODELING OF SIX‐PHASE SYNCHRONOUS MACHINES FOR SVB SIMULATION SOFTWARE PACKAGES ................................................................................. 22 2.1 Coupled Circuit Phase Domain Model ............................................................................ 25 2.2 qd0 Model ................................................................................................................................... 30 2.3 VBR Model .................................................................................................................................. 34 2.4 Constant-Parameter VBR model ....................................................................................... 37 2.5 Computer Studies .................................................................................................................... 41 2.5.1 Case Study System .................................................................................................................... 41 ix 2.5.2 Transient Simulation .............................................................................................................. 43 2.5.3 Computational Performance and Accuracy Comparisons .................................... 48 CHAPTER 3: MAGNETIC SATURATION AND DECOUPLED INTERFACING FOR CPVBR MODELS FOR SIX‐PHASE SYNCHRONOUS MACHINES .................................... 52 3.1 Saturable qd0 Model .............................................................................................................. 53 3.2 Saturable Variable-Parameter VBR Model ................................................................... 56 3.3 Saturable Constant-Parameter VBR Model .................................................................. 64 3.4 Saturable Decoupled Constant-Parameter VBR Model ........................................... 70 3.5 Computer Studies .................................................................................................................... 74 3.5.1 Case Study System .................................................................................................................... 74 3.5.2 Transient Simulation .............................................................................................................. 76 3.5.3 Numerical Accuracy Comparison ..................................................................................... 80 3.5.4 Computational Performance Comparison .................................................................... 82 CHAPTER 4: CONSTANT‐PARAMETER VOLTAGE‐BEHIND‐REACTANCE MODEL OF FIVE‐PHASE SYNCHRONOUS MACHINES WITH AIRGAP FLUX HARMONICS ......... 84 4.1 Coupled Circuit Phase Domain Model ............................................................................ 86 4.2 qd0 Model ................................................................................................................................... 91 4.3 Variable-Parameter VBR Model ........................................................................................ 94 4.4 Constant-Parameter VBR Model ....................................................................................... 99 4.5 Model Validation and Computer Studies .................................................................... 103 4.5.1 Steady‐State Study and Model Validation ................................................................. 105 4.5.2 Transient Study ...................................................................................................................... 108 4.5.3 Numerical Accuracy Comparison .................................................................................. 111 4.5.4 Computational Performance Comparison ................................................................. 112 CHAPTER 5: CONSTANT‐PARAMETER VOLTAGE‐BEHIND‐REACTANCE MODEL OF SATURABLE INDUCTION MACHINES INCLUDING AIR‐GAP FLUX HARMONICS . 115 x 5.1 Spatial Flux Harmonics Due to Magnetic Saturation............................................. 116 5.2 qd0 Model ................................................................................................................................ 120 5.3 Variable-Parameter VBR Model ..................................................................................... 124 5.4 Constant-Parameter VBR Model .................................................................................... 131 5.5 Computer Studies ................................................................................................................. 134 5.5.1 Steady‐State Case‐Study 1: Wye‐Connected Stator Windings with Floating Neutral Connection .............................................................................................................. 136 5.5.2 Steady‐State Case‐Study 2: Wye‐Connected Stator Windings with Neutral Connection ................................................................................................................................ 139 5.5.3 Steady‐State Case‐Study 3: Delta‐Connected Stator Windings ....................... 142 5.5.4 Transient Study: Start‐Up and Load‐Change ........................................................... 145 5.5.5 Numerical Accuracy Comparison .................................................................................. 148 5.5.6 Computational Performance Comparison ................................................................. 149 CHAPTER 6: CONCLUSIONS AND FUTURE WORK ........................................................ 151 6.1 Contributions and Anticipated Impact ........................................................................ 152 6.1.1 Contributions .......................................................................................................................... 152 6.1.2 Anticipated Impact ............................................................................................................... 156 6.2 Future Work ............................................................................................................................ 157 References ................................................................................................................................. 160 Appendix A: Parameters for Case Study of Section 2.5.1 .......................................... 168 Appendix B: Parameters for Case Study of Section 3.5.1 .......................................... 169 Appendix C: Parameters for Case Study of Section 4.5.1 .......................................... 170 Appendix D: Parameters for Case Study of Sections 5.5.1–3 ................................... 171 xi List of Tables Table 1–1. Comparison between different types of multi-phase machines versus conventional three-phase machines ..................................................................................................... 4 Table 1–2. Work in this dissertation in relation to most relevant available models in literature ......................................................................................................................................................... 18 Table 2–1. CPU Time and Number of Steps for the 2 Second Study for the Subject Models ............................................................................................................................................................................. 50 Table 2–2. Comparison of 2-Norm Error in the Simulation Results ..................................... 51 Table 2–3. Comparison of the Largest Eigenvalue and Number of State Variables ....... 51 Table 3–1. Comparison of 2-Norm Error Obtained by the Subject Models for the 4 Second Study ................................................................................................................................................. 81 Table 3–2. Computational Performance Comparisons of the Subject Models for the 4 Second Case-Study ..................................................................................................................................... 82 Table 4–1. Comparison of 2-Norm Error of the Simulation Results Over the 2-Second Transient case-Study Obtained by the Subject Models ........................................................... 112 Table 4–2. Computational Performance Comparisons of the Subject Models for the 2 Second Transient ..................................................................................................................................... 114 Table 5–1. 2-Norm Error of the Simulation Results Obtained by the Subject Models Over the 0.5-Second Transient case-Study .............................................................................................. 149 Table 5–2. Computational Performance of the Subject Models for the 0.5 Second Transient Case-Study ............................................................................................................................. 150 xii List of Figures Figure 1–1. The interfacing terminals and windings of the CCPD model for a generic three-phase double-fed induction machine. ..................................................................................... 9 Figure 1–2. Interface and implementation of a qd machine model in a typical SVB simulation program. .................................................................................................................................. 10 Figure 1–3. A voltage-behind-reactance machine model interfacing circuit and terminals for connection with external networks. ....................................................................... 12 Figure 1–4. Procedure used in this dissertation for deriving the CPVBR models for various ac machines; starting with conventional qd models, then to variable-parameter VBR models, and then finally achieving the desired CPVBR models. ................................... 15 Figure 2–1. A simplified cross-section of a salient pole six-phase synchronous machine, assuming a shift angle ζ between the two three-phase stator sets ....................................... 24 Figure 2–2. Transient response of stator phase currents under single phase a1 fault with and without considering mutual leakage inductance effect. ......................................... 29 Figure 2–3. Transformed qd0 equivalent circuit of six-phase synchronous machine in rotor reference frame ............................................................................................................................... 32 Figure 2–4. Implementation of the proposed CPVBR model with constant-parameter RL branch interfacing circuit ................................................................................................................. 41 Figure 2–5. Case study system with a six-phase synchronous generator feeding a rectifier load and a three-phase ac grid ............................................................................................ 42 Figure 2–6. System transient response to a single phase-to-ground fault on phase ‘a1’ as predicted by the subject models: (a) and (b) phase currents for set 1 and set 2, (c) electromagnetic torque, (d) rotor speed, and (e) rectifier dc voltage ................................. 45 xiii Figure 2–7. Magnified view of selected variables showing the details of the fault transient obtained by the subject models for: (a) phase currents of abc1 set; (b) phase currents of abc2 set; (c) stator phase voltages of abc1 set; (d) stator voltages of abc2 set; and (e) electromagnetic torque ................................................................................................... 46 Figure 2–8. Magnified view of the transient response of the subject models compared to the reference solution at the fault moment for (a) stator phase ‘a1’ and (b) ‘a2’ currents and (c) electromagnetic torque ......................................................................................... 47 Figure 3–1. Equivalent circuit of six-phase synchronous machine with main flux saturation in qd0 rotor reference frame ........................................................................................... 55 Figure 3–2. Implementation of the saturable VBR model with magnetically-coupled variable-parameter interfacing RL branches .................................................................................. 63 Figure 3–3. Approximation filter: (a) LPF, and (b) proposed hybrid LPF-BPF ............... 67 Figure 3–4. Implementation of the proposed saturable CPVBR model with magnetically-coupled constant-parameter interfacing RL branches ................................... 69 Figure 3–5. Implementation of the proposed saturable DCPVBR model with magnetically decoupled constant-parameter interfacing RL branches .............................. 73 Figure 3–6. Benchmark system consisting of a six-phase synchronous generator supplying ac grid and diode rectifier load ....................................................................................... 75 Figure 3–7. Assumed magnetization curve depicting unsaturated and saturated regions for two operating points .......................................................................................................................... 77 Figure 3–8. Transient response of several variables due to grid voltage increase from 0.9 to 1.1pu, as predicted by the subject models: (a) a1 phase current, (b) a2 phase current, (c) magnetizing flux, (d) electromagnetic torque, and (e) rotor mechanical speed ................................................................................................................................................................ 78 xiv Figure 3–9. Magnified view of selected variables as predicted by subject models: (a) a1 phase current, (b) a2 phase current, and (c) electromagnetic torque ................................ 79 Figure 3–10. Further magnified view of selected variables as obtained by the subject saturable models for: (a) stator a1 phase current, (b) stator a2 phase current and (c) electromagnetic torque ............................................................................................................................ 80 Figure 4–1. A simplified cross-section of a salient pole five-phase synchronous machine ............................................................................................................................................................................. 86 Figure 4–2. The qd130 equivalent circuit of the five-phase synchronous machine in rotor reference frame, including coupling and cross-coupling between fundamental and third harmonic components ......................................................................................................... 94 Figure 4–3. Implementation of the proposed variable-parameter VBR model of the five-phase synchronous machine with coupled and rotor-position dependent inductances, directly interfaced with the external network ............................................................................... 98 Figure 4–4. Implementation of the proposed CPVBR model of the five-phase synchronous machine with coupled and constant RL branches, directly interfaced with the external network .............................................................................................................................. 103 Figure 4–5. The considered case study system: (a) laboratory setup, (b) its corresponding circuit schematic ....................................................................................................... 105 Figure 4–6. Steady-state results obtained from experimental measurement and the reference model simulation under no-load operating condition for the machine line voltages: (a) vsab and (b) vsac ................................................................................................................ 106 Figure 4–7. Steady-state results obtained from experimental measurement and the reference model simulation, for the machine line voltages and current: (a) vsab, (b) vsac xv and (c) isa, when the machine is loaded with a ten-pulse diode rectifier supplying a 1Ω dc load ........................................................................................................................................................... 108 Figure 4–8. Transient response of several machine variables during load change transient, as predicted by the subject models for: (a) line voltage vsab, (b) line voltage vsac, (c) line current isa, and (d) electromagnetic torque ......................................................... 110 Figure 4–9. Magnified view of selected variables as obtained by the subject five-phase machine models for: (a) line current isa, and (b) electromagnetic torque ...................... 111 Figure 5–1. Cross-section of a squirrel cage induction machine including magnetizing current vector and different saturation zones ............................................................................ 117 Figure 5–2. Airgap flux density including fundamental and third components ........... 118 Figure 5–3. The qd0 equivalent circuit of induction machine including third harmonic component of air-gap flux in synchronous reference frame ................................................. 124 Figure 5–4. Implementation of the proposed variable-parameter VBR model of saturable induction machine including air-gap flux harmonics, and its interfacing with external network using variable-parameter RL branches ..................................................... 130 Figure 5–5. Implementation of the proposed constant-parameter VBR model of saturable induction machine including air-gap flux harmonics, and its interfacing with external network using constant-parameter RL branches .................................................... 134 Figure 5–6. Wye-connected induction machine with floating neutral point supplied by the ac grid .................................................................................................................................................... 136 Figure 5–7. Magnetization curves of the induction machine demonstrating the three considered operating points for: (a) main flux, (b) third harmonic flux ......................... 137 xvi Figure 5–8. Phase variables of wye-connected saturable induction machine with floating neutral point in steady-state at no-load condition under various stator voltage levels: (a) stator phase a voltage vas and (b) stator phase a current ias ........................... 138 Figure 5–9. Phase variables of wye-connected saturable induction machine with floating neutral point in steady-state at full-load condition under various stator voltage levels: (a) stator phase a voltage vas and (b) stator phase a current ias ........................... 139 Figure 5–10. Wye-connected induction machine with neutral wire supplied by the ac grid ................................................................................................................................................................. 140 Figure 5–11. Phase variables of wye-connected saturable induction machine with neutral connection in steady-state at no-load condition under various stator voltage levels: (a) stator phase a voltage vas and (b) stator phase a current ias ........................... 141 Figure 5–12. Phase variables of wye-connected saturable induction machine with neutral connection in steady-state at full-load condition under various stator voltage levels: (a) stator phase a voltage vas and (b) stator phase a current ias ........................... 141 Figure 5–13. Delta-connected induction machine supplied by the ac grid .................... 142 Figure 5– 14. Phase variables of delta-connected saturable induction machine with neutral connection in steady-state at no-load condition under various stator voltage levels: (a) stator phase a voltage vas and (b) stator phase a current ias ........................... 144 Figure 5–15. Phase variables of delta-connected saturable induction machine with neutral connection in steady-state at full-load condition under various stator voltage levels: (a) stator phase a voltage vas and (b) stator phase a current ias ........................... 145 Figure 5–16. Transient response of several variables of the induction machine with wye-connected stator with neutral wire during startup and load change for: (a) stator phase a current ias, (b) electromagnetic torque, and (c) rotor speed ............................... 147 xvii Figure 5–17. Magnified view of machine variables at the moments after load change transient for: (a) stator phase a current ias, and (b) electromagnetic torque ............... 148 xviii Nomenclature In this dissertation, scalars are written using italic fonts, vectors are denoted by bold lowercase characters, and matrices are denoted by bold uppercase characters, respectively. Only general and basic variables are aggregated in this Section (corresponding to the machine models considered in this dissertation). All other variables are defined throughout the text with appropriate explanations. 0 Null matrix 1 Matrix full of 1s i0s Zero-sequence stator current ixs Stator current on phase x ids d-axis stator current im magnetizing current imd d-axis magnetizing current imq q-axis magnetizing current iqs q-axis stator current I Identity matrix J Combined machine-load moment of inertia (in kg∙m2) Ks Park’s three-phase transformation matrix for stator variables Ks-1 Inverse Park’s three-phase transformation matrix for stator variables Lls Stator leakage inductance p Heaviside’s operator ( d / dt ) xix P Number of poles Rs Stator resistance t Time Te Electromagnetic torque Tm Mechanical torque v0s Zero-sequence stator voltage vxs Stator voltage on phase x vds d-axis stator voltage vqs q-axis stator voltage θr Electrical rotor position θs The angle of the synchronously rotating reference frame λ0s Zero-sequence stator flux linkage λxs Stator flux linkage on phase x λds d-axis stator flux linkage λm Main flux λmd d-axis magnetizing flux λmq q-axis magnetizing flux λqs q-axis stator flux linkage ωs Stator operating frequency (in rad/s) ωr Electrical rotor speed (in rad/s) ωm Mechanical rotor speed (in rad/s) xx Acknowledgements I express my most sincere gratitude towards my supervisor, Dr. Juri Jatskevich whom I’ll be forever in debt for his invaluable guidance, vision and knowledge. I would also like to thank Dr. Juri Jatskevich for not only being an outstanding supervisor, but also for being a role model in kindness, patience, dedication to research, and being a supportive and motivating friend throughout all my years here at UBC. I also would like to special gratitude to Dr. Hermann Dommel for sharing with me his endless wisdom and kindness, many discussions, and valuable feedback on my work. I also want to express my appreciation to Drs. Liwei Wang, Mehrdad Chapariha, and Francis Therrien for their help, comments and discussions on the research topic through which we have been related. This dissertation is a continuation of their excellent prior work and very high achievements, on which I was able to stand and build more. I would also like to particularly thank my dear friend, Dr. Milad Ebrahimi, for not only being a great friend and a roommate, but also for his selfless help in providing technical feedback on various efficient models of power electronic converters that I’ve used, and for his editorial help in many papers that we have co-authored together. I would like to thank Dr. Steven Pekarek at Purdue University for providing our research group with several prototypes of advanced five-phase synchronous machines, which greatly helped the modelling and experimental efforts on model validation done in this dissertation. Dr. Pekarek had pioneered the voltage-behind-reactance modelling technique for synchronous machines, and his early publications on this topic have inspired many researchers including myself. xxi I would like to thank Dr. Jatskevich for providing funding for my research through his NSERC Discovery Grant entitled “Next Generation Electromagnetic Transient Simulation Tools”, as well as from other grants that he held. Part of the funding also came from Dr. Hermann Dommel’s NSERC Discovery Grant, for which I am also very thankful to him. I would like to thank our industrial partners, the Power Systems Technology Centre at Manitoba Hydro International Ltd. (formerly known as Manitoba HVDC Research Centre) for their support under NSERC Engage/Plus grant “Modeling of Six-Phase Generators for Renewable Energy Applications,” and later on Typhoon HIL Inc. for providing me with financial support during the final years of my research at UBC under a Grant-In-Aid on “Multi-phase Machine Modeling for Real-time Execution”. In addition, I would also want to thank the UBC’s Graduate Student Initiative (GSI) for the stipends and scholarships that I was receiving during my years at UBC, which have added to my financial comfort. Most of all, I am forever in debt to my family. I am particularly thankful to my parents, Shahbaz and Tayyebeh, for their unwavering support, patience, and trust even through the toughest times of my life. I am thankful to them for raising me in a loving and providing family, and for pushing and encouraging me to try harder. Lastly, I would also like to thank my brothers and sister, Vahid, Farid and Negin for their moral support and for being there for me and instead of me, sharing the family burden during the times when I could not. 1 CHAPTER 1: INTRODUCTION 1.1. Motivation Modern electrical grids are experiencing significant changes due to increasing penetration of distributed energy resources (DERs) and power electronic devices [1]. Consequently, power systems are becoming more complex, while being operated closer to their physical limits. At the same time, rotating electrical machines account for the majority of energy produced and consumed in a typical electrical grid. Although the conventional three-phase machines are the dominant case, with recent advancements in high-power semi-conductor switching devices, there has been an increasing interest in using machines with higher number of phases [2]–[13] (e.g. five, six, seven, nine, twelve, etc.) in power systems using power electronic interfaces in special purpose applications such as aircraft [2], naval and vehicular power systems [3]–[4] as well as renewable energy applications [12]–[13] (e.g., as wind generators). Compared to the conventional and widely-used three-phase machines, machines with higher number of phases offer advantages such as reduced power per 2 phase, reduce rating of power electronic converters per phase, increased reliability, fault tolerant operation, etc., [3]–[4], [8], [10]. Among multi-phase machines, six-phase machines of various types fulfill a special role as they are able to be connected and/or supply two isolated three-phase systems with no additional components [14]–[15] (such as three-winding wye/delta/wye transformers). Apart from the added reliability in power delivery, this allows for feeding different categories of loads with reduced mutual effects using the same electric machine and mechanical interface apparatus. As an example, the isolated dual three-phase setup of the stator allows one to be used to power a traditional three-phase grid consisting of mostly RL loads, while the other three-phase sets can be used to supply power-electronic loads, or the six-phase output to be rectified and used as a source for dc grids. Such setups are common in ships and aircraft grids, which utilize both ac and dc grids. Five-phase electric machines have also become increasingly considered in machine-converter systems for applications such as high-power drives [5], [9], mostly utilized in various types of vehicles. Compared with conventional three-phase machines, five-phase machines offer advantages such as improved mechanical torque pulsations, ability to utilize low-order harmonics to generate useful torque and power, and ability to operate close to full capacity under faulty conditions [8]–[9]. In particular, the ability to utilize low-order current harmonics (e.g., 3rd harmonic) to increase the electromagnetic torque has been a critical research focus, which results in higher torque per-current-peak compared to conventional three-3 phase machines. This capability has been extensively exploited in advanced electrical drives for five-phase machines [9]. Some of the electromagnetic phenomenon which are associated and utilized with multi-phase machine-converter systems, such as magnetic flux harmonic components, can also be considered and exploited in applications involving three-phase machine-converter systems. The prime example of this is the dominantly used squirrel-cage induction machine where magnetic saturation (which is a common phenomenon for such machines) can create harmonic distortions [16]–[20] which can be utilized for sensorless drive and control applications [21]–[26]. Such distortions are due to magnetic saturation of stator teeth [16], which can introduce harmonic distortions in the machine currents and voltages [19]–[20]. Although extensive research has been conducted in modeling saturable three-phase squirrel-cage induction machines, a large number of such research work consider only the fundamental component of the air-gap flux [27]–[33], which is often represented by variable and flux-dependent inductances. While such considerations may be adequate for some system-level simplifications, more detailed models which consider flux harmonic components (due to saturation) are also needed for use in drive applications. Table 1–1 summarizes an overview and comparison between the conventional three-phase machines and advanced multi-phase machines in terms of their complexity and modelling in EMT programs. 4 TABLE 1–1. COMPARISON BETWEEN DIFFERENT TYPES OF MULTI‐PHASE MACHINES VERSUS CONVENTIONAL THREE‐PHASE MACHINES Electrical machine Advantages Disadvantages Conventional Three‐phase machines  Model simplicity for a wide range of applications  Model readily available in most simulation packages  Direct interface to conventional three-phase grids  Widely available in the market at low cost  Susceptible to faults (interrupted operation)  High torque oscillation due to harmonics  Drives using converters requires expensive high power semi-conductor devices  Complex non‐linear models not generally available for advanced drive applications Machines with n number of phases (n>4, n≠3k) Including Five‐Phase machines  High reliability (uninterrupted operation under fault)  Low order harmonics (< n) used for oscillation-free constant torque generation  Low torque oscillation under high frequency harmonics  High torque and power density versus phase current peak  Lower rate, less costly semi-conductor devices used for drive application  Highly complex mathematical model Not readily available in simulation packages  No direct interface to conventional grids Machines with 3k number of phases (k > 1) Including Six‐Phase machines  Direct interface to k isolated conventional three-phase grids  High reliability (uninterrupted operation under fault)  Low dynamics penetration between the k three-phase sets  High torque and power density  Lower rate, less costly semi-conductor devices used for drive application  Complex mathematical model Not readily available in simulation packages Electromagnetic transient (EMT) simulation programs [34]–[44] have been one of the main pillars of power system design and studies in recent decades, where simulations of power system dynamics are done using mathematical models of all system components. The immense computational power of such software packages has allowed large-scale studies to be conducted, contributing to the overall efficiency 5 and reliability of today’s power systems. With the increased size and complexity of modern power systems however, many state-of-the-art simulation programs are approaching their computational limits in which they can be used for studies of practical power and energy systems. As a result, computational efficiency becomes a key factor to modelling power system components in EMT simulation packages. Given the computational complexity of simulating the growing number of machine-converter systems, having access to accurate and numerically efficient machine models become essential for design and analysis of any practical simulations. The transient simulation programs which are commonly used for power system studies can be classified in two groups: 1) nodal-analysis-based electromagnetic transient programs (EMTP-type, e.g. PSCAD [34], EMTP-RV [35], MicroTran [36], ATP [37], RTDS [38], etc.); and 2) state-variable-based (SVB) programs (e.g. MATLAB/Simulink [39], Simscape [40], PLECS [41], ASMG [42], RT-LAB [43], Typhoon HIL [44], etc.). The EMTP approach discretizes the circuit equations at the branch level [45], while the SVB approach creates a set of ordinary differential equations (ODEs) (sometimes in state-space form) [45]–[47] which are then discretized and solved by the ODE solver on the system level. State-variable-based programs naturally handle variable time steps and different integration rules. The EMTP-type programs are very efficient for large networks and typically use a fixed time-step integration rule (e.g. trapezoidal) [45]. On the one hand, a major advantage of the SVB formulation is that its system matrices are independent of the simulation step size. As a result, efficient variable step-size solvers can be used without having to continuously execute the numerically costly state matrix generation algorithm 6 [48]. On the other hand, the conductance matrix of EMTP-type programs is a function of the step size and therefore must be regenerated and re-factored whenever the step size changes. This makes the SVB simulation packages suitable for simulation of system consisting of power electronic switching devices, where it may be desirable to use variable time-step ODE solvers. This dissertation mainly focuses on deriving numerically efficient models of multi-phase rotating machines for their implementation in SVB simulation packages. 1.2. Literature Review Depending on the degree of detail desired and model fidelity, different types of machine models can be used in transient simulations. One way to achieve very precise results is to represent the machines using finite element analysis (FEA) [49]. FEA is computationally prohibitive and requires a large amount of machine parameters that are rarely available to system analysts, making it impractical for system-level studies. Another possibility is to use magnetic equivalent circuits (MECs) [50], [51]. Machine models based on MECs also offer high accuracy while being significantly faster and requiring fewer parameters than the FEA-based models. Nevertheless, the machine parameters for the FEA and MEC models are rarely known by system analysts and the computational cost of these models is prohibitive and impractical for larger scale multi-machine system studies. Therefore, these types of modelling is typically used for offline simulations of a single machine. 7 For system-level studies, so-called general-purpose lumped-parameter machine models have always been most widely used in transient simulation programs. Such models are typically represented by several magnetically coupled stator and rotor windings comprised of equivalent resistances and inductances [27]. The resulting number of differential and algebraic equations is consequently much smaller than for the FEA- and MEC-based models (determined by the total number of coupled windings), and the resulting in numerical cost is also much lower, making such models suitable to power systems studies. Moreover, the machine parameters can be obtained straightforwardly using well-established experimental approaches and standard tests, or from widely available manufacturers data. The relatively low-order general-purpose lumped-parameter models are based on several common simplifying assumptions [27], which result in a loss of modeling accuracy compared to higher fidelity FEA- and MEC-based models. However, the general-purpose lumped-parameter models have been extensively compared with experimental data over the course of several decades and have been shown to yield sufficient accuracy for most system-level studies, e.g., [52], [53]. Several variations of the general-purpose lumped-parameter machine models (including several types of multi-phase machine models [14]–[15], [55]–[60]) have been presented in the literature and implemented in commercial transient simulation packages. Despite their algebraic equivalence under the same modelling assumptions, these models can possess vastly different numerical properties [61]. In particular, several approaches have been developed to interface these machine models with external power networks [61]; wherein a specific lumped-parameter 8 machine model used in a given simulation can significantly affect the numerical accuracy, efficiency, and stability of the entire solution. 1.2.1 Coupled Circuit Phase‐Domain Models General-purpose lumped-parameter machine models were originally derived using physical variables (i.e., in phase coordinates) [27], [62]. Throughout this dissertations, such models are referred to as the coupled circuit phase-domain (CCPD) models. The CCPD models are physically intuitive and use the same set of coordinates as the external power network to which they are connected. For examples, a CCPD model and its phases for the three-phase doubly-fed induction machine is shown in Figure 1–1. However, the mutual inductances between the rotor and the stator (and even the self and mutual stator inductances of salient-pole synchronous machines) vary sinusoidally as a function of the rotor position [27]. This complicates the analysis of machine dynamics and can significantly increase the computational time due to the presence of time-varying branches. Some examples of circuit-based CCPD models for SVB packages are presented in [55]–[56], [58], [63]–[65]. While numerically accurate, these models introduce time-varying inductances in the system matrices. In addition to slowing down the simulation, variable inductances are not available in many commercial EMT programs [40], limiting the range of application of CCPD models particularly in SVB programs. The variable inductance structure of CCPD models, when applied to multi-phase machines [56], [59] also introduces additional computational burden due to increased size of 9 impedance matrix and additional (variable) inductance terms associated with the added stator phases. For these reasons, the CCPD models were rapidly replaced by so-called qd models [see Section 1.2.2], in which the self and mutual inductances are rotor-position-independent. srsrsrasi rrrrrr( )sr rLbsicsiaribricriasvbsvcsvarvbrvcrv Figure 1–1. The interfacing terminals and windings of the CCPD model for a generic three‐phase double‐fed induction machine. 1.2.2 qd Models The so called qd models [14]–[16], [27], [45], [55] are by far the most common machine models that are presently used in power systems transient simulation programs. These models are obtained by applying Park’s transformation [27] to the original CCPD models, yielding a new set of variables in a single or multiple reference frames. Assuming a magnetically-linear machine, the resulting inductances will be fully decoupled and constant, resulting in very efficient numerical implementation [27]. The most efficient and convenient qd models use all stator and rotor flux linkages as the state variables [27]. In the state-variable-based programs, 10 the models have voltage-input and current-output formulation, and can only be directly interfaced to the external network when the input voltage is known [61]. The qd models of multi-phase machines often include multiple reference frames corresponding to additional stator windings (e.g., in six-phase machines [14]) and/or higher order harmonic components (e.g., five-phase machines [54], squirrel-cage induction machines with spatial flux harmonics [16], etc.). Furthermore, utilizing multiple reference frames in qd models of multi-phase machines can lead to various types of cross-coupling (coupling between q- and d-axes) between variables of each frame [14], [58]–[60], due to physical displacement, spatial flux harmonic components, etc., in contrast to the classical decoupled qd models for most conventional three-phase machines [27]. Figure 1–2. Interface and implementation of a qd machine model in a typical SVB simulation program. The major issue regarding qd models in SVB programs is their interface with external power systems represented in phase coordinates. Since qd models have a voltage-input, current-output formulation, their stator currents (converted to phase coordinates) are injected back to the network using voltage-controlled current sources. This input-output interfacing becomes incompatible when the machine is            11 connected to an inductive or ideal current source element [61]. Two indirect interfacing approaches are typically used to generate a proper state-space model [61]. The first and most common one consists of using resistive or capacitive snubbers [40] (as illustrated in Figure 1–2) that are used internally to calculate the interfacing voltage. On the one hand, large resistors and small capacitors add fast modes to the system and thus increase its numerical stiffness [66]. Consequently, smaller step sizes might be necessary to ensure numerical stability and sufficient numerical accuracy [61]. On the other hand, smaller resistors and larger capacitors introduce larger error independently of the step size [61]. The second approach is to use a time-step relaxation [61]. Here also, smaller step sizes may be required to avoid instability and limit numerical error [61]. 1.2.3 Voltage‐Behind‐Reactance (VBR) Models The full-order VBR model [58]–[60], [57]–[76] represents the stator interfacing circuit in phase coordinates, thus allowing it to be directly interfaced to any network (similar to CCPD models). Its equivalent rotor subsystem is represented by state-space equations in the qd reference frame. Therein, transformed (to rotor reference frame in case of synchronous machine) stator currents are used as inputs, while rotor and/or magnetizing flux linkages are chosen as the state variables, resulting in better scaled eigenvalues than CCPD models [67]. Figure 1-3 shows typical implementation and interfacing of VBR models in SVB simulation packages. Such modeling technique can lead to a constant-parameter stator interfacing circuit (unlike CCPD) when the machine is magnetically linear and has a non-salient rotor. However, when dynamic saliency and/or magnetic saturation are/is considered, the 12 stator interfacing circuit will include variable-parameter elements as functions of rotor position and/or machine magnetization [67]. While numerically accurate, this variable-parameter VBR model will still require significant computational resources in order to obtain simulation results and can become a bottleneck in large-scale simulations. Figure 1–3. A voltage‐behind‐reactance machine model interfacing circuit and terminals for connection with external networks. In order to improve upon numerical efficiency, an equivalent constant-parameter VBR (CPVBR) model can be derived using a multitude of methods. Such methods include utilization of fictitious rotor windings [77], using singular perturbations (which create an algebraic loop) [78] in order to eliminate rotor saliency, or introducing numerical approximations [79]. Throughout this dissertation, the latter approach [79] for obtaining CPVBR models is utilized, where variable terms of the VBR model interfacing circuit are separated and calculated as voltage terms, using approximated stator voltages (through specially tuned filters).              13 The improved numerical efficiency of CPVBR models can be especially useful when applied to multi-phase machines, given the fact that alternative models for such machines can lead to increased complexity and interfacing challenges. 1.3. Research Objectives and Contributions The ultimate goal of this research is to improve numerical efficiency of EMT simulation packages, without degrading their numerical accuracy and stability, by proposing new computationally efficient models of electrical machines. This is particularly desirable in view of the changes currently occurring in electrical power systems, which consist of an increasing utilization of power electronic converters, and more advantageous multi-phase machines. All of which require a fast-growing number of simulations for practical design and analysis. As a result, it is expected that significant improvements in numerical efficiency will make it possible to execute a wider range of studies (e.g., electromechanical transients) using SVB and EMTP-type programs in a reasonable amount of time. These case studies would therefore be executed using fewer modelling assumptions, which would also increase the overall simulation precision. In dissertation, the modelling procedure is mainly focused on developing the VBR type machine models that offer efficient constant-parameter interfacing with external power networks. The approach used throughout this dissertation is illustrated and summarized in Figure 1–4. As described in Section 1.2.3, the VBR model is based on the machine qd equivalent circuit, where the magnetizing branch 14 and rotor circuits are replaced with sub-transient inductances and back-emf voltage sources (see top part of Figure 1–4). Moreover, it was mentioned that properties such as saliency, or variable saturation-dependent inductance terms, lead to a computationally expensive interfacing circuit with variable inductances when transformed to phase coordinates (see middle part of Figure 1–4). As illustrated in Figure 1–4 (see bottom part), the CPVBR models are derived based on the VBR models, where the sub-transient inductance matrix (denoted as mL ) in qd coordinates is divided into two parts: 1) an interfacing part that includes a constant and non-salient inductance matrix (denoted by mL ); and 2) a residual part (denoted by _m resL ) that includes all the remaining terms that contain saliency, coupling and/or saturation dependent effects. The new constant and non-salient interfacing part (denoted by mL ), when transformed into stator phase coordinates, results in a constant-parameter interfacing circuit (see bottom part right side of Figure 1–4), while the residual part (denoted by _m resL ) is taken into account separately by computing its effect on the stator voltages and including them in the sub-transient back-emf voltage sources. Considering the effect of residual inductance terms on the stator voltages requires accurate approximation of stator current derivatives, as mentioned in Section 1.2.3. 15 s dqs λsr lsLqdsvqdsi qdrisl dqr λ rrlrLqdrvmqdLs dqs λsr lsLqdsvqdsimqdL( , )qd qds qdre i λqdr qdr qds qdr  λ Aλ Bi vmqdL abce sr ( , ,...)abc r m L( , ,...)abc r m LmqdL_mqd resLabce sr abcLqdsv qdsλ qdeqdsiqdspiqdeabcsvabcsiabcsvabcsi Figure 1–4. Procedure used in this dissertation for deriving the CPVBR models for various ac machines; starting with conventional qd models, then to variable‐parameter VBR models, and then finally achieving the desired CPVBR models. The main contributions of this dissertation are the new general purpose lumped-parameter models (CPVBR) for multi-phase machines. In order to capture more dynamic details, the modeling approach focuses on taking into account some of the most dominant effects (e.g., magnetic couplings and cross couplings between variables in different planes, magnetic saturation, spatial harmonics in magnetic flux distribution, etc.), specific to the considered machine types, while possessing 16 enhanced numerical properties. This will help achieving the ultimate objective of improving the numerical efficiency of EMT simulators without adversely affecting their numerical accuracy and stability. The research is primarily focused on the machine models for the SVB offline and real-time EMT programs, whereas its extension to the nodal-based EMTP-type programs is considered for the future research. In order to better position the contributions with respect to the state-of-the-art, the work and models that have been accomplished in this dissertation are compared against the most relevant existing models in the prior literature in terms of various important features, as summarized in Table 1–2. The first and second objectives of this dissertation focus on six-phase synchronous machine models. The qd and PD models of six-phase synchronous machines has been derived and proposed in [14] and [15], which also considered mutual leakage inductances. In this dissertation, the CPVBR modeling approach with direct interface has been proposed for the six-phase synchronous machine model to improve upon the main limitations of the existing models (i.e., indirect interface in qd models and variable parameters in PD models). As a continuation of this research path, the proposed CPVBR model for six-phase machines was extended to include the magnetic saturation in a directly interface CPVBR model. Furthermore, a decoupled CPVBR model for the six-phase machine was proposed in order to completely eliminate the magnetic coupling between the three-phase sets, and as a result simplifying the RL-branch interfacing circuit even more, and enabling very easy and straightforward implementation of this model in a wide range of commonly-used EMT simulation programs. 17 The third objective of this dissertation focuses on efficient and accurate modelling of five-phase synchronous machines. Models of five-phase synchronous machines have been available as either qd [9], [55]–[56] (with indirect interface), or PD models [9], [55] (with variable parameters), as summarized in Table 1–2. The available state-of-the-art models of five-phase SMs [55] have considered harmonic components of air-gap flux and were mostly derived for the PM type machines. However, with increased use of efficient five-phase machines (including wound-rotor types) connected to power electronics converters, more detailed models are required. In this dissertation, first, the five-phase machine model in [55] is extended to include both rotor circuits and cross-coupling terms between the fundamental and harmonic components of the air-gap flux contributing to more accurate representation of machine’s dynamics. Then, the CPVBR modeling approach is applied to the wound-rotor five-phase synchronous machine to derive a numerically efficient and directly interfaced model which is able to predict complex dynamics for various applications. These new features of the proposed models are also highlighted in Table 1–2. The fourth and final objective of this dissertation focuses on increasing the model fidelity of modelling the conventional saturable three-phase induction machine. The main flux magnetic saturation has been implemented in various types of induction machine models including qd models [28], [32], PD [61], [63], and VBR models [65], [68], [70]. In all of the listed models, the effect of magnetic saturation is considered to only affect the peak of a sinusoidally distributed flux density waveform. However, it has been shown in [16]–[20] that magnetic saturation can 18 create harmonic distortions in the flux density waveform with significant effects on machine voltages and currents. Thus far, the saturable induction machine models including the flux harmonics have been presented in [18]–[20], which are all the PD-type models. In this dissertation, a new qd model for saturable induction machine with flux harmonics is proposed using the modelling approach [33]. Then, a directly interfaced and efficient CPVBR model is proposed for saturable induction machine with flux harmonics, which enables easy and efficient implementation of such models in many simulation programs. These new features of the proposed three-phase induction machine models are also highlighted in Table 1–2. TABLE 1–2. WORK IN THIS DISSERTATION IN RELATION TO MOST RELEVANT AVAILABLE MODELS IN LITERATURE Machine Type Model Ref. Stator Interface Circuit Parameters Magnetic Saturation Flux Harmonics Explicit Formulation Six‐phase SM qd [14]-[15] Indirect N/A No No Yes My work Indirect N/A Yes No Yes PD [14]-[15] Direct Variable No No Yes My work Direct Variable Yes No Yes VBR My work Direct Constant Yes No Yes My work Decoupled Direct Constant Yes No Yes Five‐Phase SM qd [9],[55]-[56] Indirect N/A No Yes Yes My work Indirect N/A No Yes, Incudes Cross couplings Yes PD [9]-[55] Direct Variable No Yes Yes My work Direct Variable No Yes, Incudes Cross couplings Yes VBR My work Direct Constant No Yes, Incudes Cross couplings Yes 19 Three‐Phase IM qd [28] Indirect N/A Yes No No [32] Indirect N/A Yes No Yes My work Indirect N/A Yes Yes Yes PD [63] Direct Variable No No Yes [61] Direct Variable Yes No Yes [18]-[20] Direct Variable Yes Yes Yes VBR [65],[68],[70] Direct Variable Yes No Yes My work Direct Constant Yes Yes Yes Thus, based on the gap in the currently available models and the need for further improvements in this field, as summarized in Table 1–2, the research of this dissertation has the following objectives: Objective 1: Developing new model for six‐phase synchronous machines with constant‐parameter direct interfacing considering mutual leakage fluxes Dual plane qd0 models have been used to simulate six-phase machines due to its constant parameter structure compare to CCPD counterparts. The qd0 models however suffer from indirect interfacing and their need for auxiliary snubber circuits which can cause numerical stiffness and/or stability issues. The first objective in this research is to propose a constant-parameter explicit and direct interface VBR model for six-phase synchronous machines which also takes into account mutual flux leakages between the two stator three-phase sets. The proposed model is shown to provide an excellent combination of numerical efficiency and accuracy, compared to all other prior existing models of the same class. 20 Objective 2: Developing fully decoupled interfacing circuits for models of six‐phase synchronous machines including main flux saturation. Due to shared magnetizing branch in a six-phase machine model, the stator three-phase sets have a natural magnetic coupling which results in mutual inductance terms in interfacing impedance matrices. This limits implementation of such models to software packages that permit mutual impedance elements. In the second objective, based on the developed CPVBR model, further simplifications are carried out to provide a fully decoupled branches for simple implementation. Moreover, main flux saturation is also considered as one of the most common phenomenon with a prominent effect on machine dynamics. Objective 3: Developing numerically efficient models for five‐phase synchronous machines including air‐gap flux spatial harmonics, with constant‐parameter interfacing circuit. Air-gap flux harmonics in five-phase synchronous machines are normally utilized for improving machine dynamics in terms of electromagnetic torque development. This results in increased complexity of the variable machine’s self and mutual inductances which is the main bottleneck in CCPD models of five-phase machines. Multi-reference frame qd models have been used to simplify five-phase machine models using constant-parameter inductances, which comes at a cost of interfacing challenges (as explained in 1.2.2). As the third objective of this thesis, the 21 VBR modeling approach is applied to qd models of five-phase machines in order to provide a numerically efficient and constant-parameter interfacing circuit, while maintaining numerical accuracy. Objective 4: Developing numerically efficient models for saturable three‐phase squirrel‐cage induction machines considering air‐gap flux harmonics, with constant‐parameter interfacing circuit. The magnetic saturation in electrical machines, including squirrel-cage induction machines, can introduce spatial air-gap flux harmonics. Such flux harmonics have pronounced effects on in common squirrel-cage induction machines which are often operated in a partially saturated region. For the sake of simplicity, most modeling approaches applied to induction machines (i.e. CCPD, qd0 and VBR models) only consider the fundamental component of the machine flux. However, there has been a recent demand for models that consider spatial flux harmonics due to development of advanced machine drive algorithms. So far, the CCPD and qd modeling approaches have been used to model spatial flux harmonics due to saturation, which come with their own limitations (as explained in Sections 1.2.1 and 1.2.2, respectively). Therefore, the fourth objective of this dissertation is to utilize the VBR modeling approach with constant-parameter interfacing, in order to provide a new accurate and numerically efficient model for induction machines including the air-gap flux harmonics. 22 CHAPTER 2: MODELING OF SIX‐PHASE SYNCHRONOUS MACHINES This chapter considers the six-phase synchronous machines, for which the coupled-circuit phase-domain (CCPD) and conventional qd0 models have been used for steady- state and transient analysis. The CCPD model is generally derived based on the magnetically-coupled circuits of the stator windings and rotor equivalent model. This base model can be directly interfaced with any external network, provided that the simulation program where the model is implemented allows implementation of the rotor-position-depended (variable) inductances. The rotor is typically represented using several equivalent damper windings and a field winding, all of which result in increasing size of the machine inductance matrix. Moreover, since the machine inductance matrix is variable, the resulting CCPD model becomes computationally expensive as its parameters have to be recalculated at each time step in the simulation program. The traditional qd0 model for six-phase machines with two sets of abc stator windings can be obtained using the classical rotor reference frame transformation 23 [24], which results in constant parameter equivalent circuits in two qd planes. The mutual leakage inductance between the two sets of abc stator windings also creates a cross-coupling between the two qd planes, respectively [14]. Similar to the conventional three-phase qd0 models [27], the six-phase qd0 model is difficult to interface with the external inductive networks, and such interface in state-variable-based simulation programs is often resolved using artificial snubber circuits [61]. The modeling approach presented in this chapter considers the state-variable transient simulation programs such as PLECS [41], Simscape [40] block set in MATLAB/Simulink [39], and Opal RT [43]. The voltage-behind-reactance (VBR) formulation derived for three-phase machines [67] is extended here to the six-phase synchronous machines. The resulting VBR model has six-phase interfacing stator circuit in abc1abc2 coordinates, and the rotor subsystem is in implemented in dual qd012 planes. However, similar to the CCPD, such variable parameter VBR model will also have the rotor-position-depended inductances, and will be computationally expensive. To achieve a very efficient interfacing, a new constant-parameter voltage-behind-reactance (CPVBR) model for six-phase synchronous machines is developed using the approach set forth in [79]. The proposed six-phase CPVBR model is shown to produce very accurate simulation results and offer advantages over the existing CCPD, qd0, and the recently presented variable parameter VBR models in terms of simulation efficiency and numerical accuracy. Moreover, due to simple constant RL-branch interfacing circuit, the new model can be easily implemented in most commonly available state-variable-based simulation programs and toolboxes. 24 Figure 2–1. A simplified cross‐section of a salient pole six‐phase synchronous machine, assuming a shift angle ζ between the two three‐phase stator sets. To set the stage for the new model, a six-phase synchronous machine with two sets of sinusoidally distributed symmetric three-phase windings on the stator is assumed. In this chapter linear magnetization is considered. A simplified cross sectional view is shown in Figure 2–1. The stator windings are denoted by the two sets abc1 and abc2, respectively, and the sets are assumed to be displaced by angle ζ. A salient rotor is assumed to have one damper winding along the q-axis, and one damper and a field winding along the d-axis [14]. All parameters are assumed to be referred to the stator side, and the motor sign convention is used. 1a axis2a axisq axisd axis 2a1c ‘2c ‘1b2b1a ‘2a ‘1c2c1‘b2b ‘ 1ar2afdkdkqkq ‘kd ‘fd ‘25 2.1 Coupled Circuit Phase Domain Model Based on Figure 2–1, the CCPD model can be can be expressed using the following voltage and flux equations: 1 1 1 12 2 2 20 00 00 0abc s s s abc sabc s s s abc sqdr rr qdrp                     v r λv r λv r λ, (2–1) 1 1 1 1 2 1 12 2 1 2 2 2 21 2 9 9( ) ( ) ( )( ) ( ) ( )( ) ( )abc s s s r s s r s r r abc sabc s s s r s s r s r r abc sqdr rs r rs r rr qdr                          λ L L L iλ L L L iλ L L L i. (2–2) Here, the Heaviside’s operator is used to denote the time derivative, p = d/dt. In (2–2), the stator self and mutual inductances are defined as 11 1 112 2cos(2 ) cos(2 ) cos(2 )2 3 2 32 2( ) cos(2 ) cos(2 ) cos(2 )2 3 3 22 2cos(2 ) cos(2 ) cos(2 )2 3 2 3A Als A B r B r B rA As s r B r ls A B r B rA AB r B r ls A B rL LL L L L LL LL L L L LL LL L L L L                                         L (2–3) 1 2 1 21 2 1 2 1 21 2 1 22 2cos( ) cos(2 ) cos( ) cos(2 )3 32 2 2( ) cos( ) cos(2 ) cos( ) cos(2 )3 3 32 2 2cos( ) cos(2 ) cos( )3 3 3la a A B r la b A B rs s r la c A B r la a A B rla b A B r la c A BL L L L L LL L L L L LL L L L L L                                          Lcos(2 )r   1 21 21 22 2cos( ) cos(2 )3 32cos( ) cos(2 )32cos( ) cos(2 )3la c A B rla b A B rla a A B rL L LL L LL L L                     (2–4) 26 1cos( ) sin( ) sin( )2 2 2( ) cos( ) sin( ) sin( )3 3 32 2 2cos( ) sin( ) sin( )3 3 3mq r md r md rs r r mq r md r md rmq r md r md rL L LL L LL L L                        L (2–5) The rotor self-inductance matrix is: 0 000lkq mqrr lkd md mdmd lfd mdL LL L LL L L      L . (2–6) The remaining sub-matrices are 2 2 1 1( ) ( )s s r s s r   L L , (2–7) 2 1( ) ( )s r r s r r   L L , (2–8) 2 1 1 2( ) ( )Ts s r s s r L L , (2–9) 1 12( ) ( )3Trs r s r r L L , (2–10) 2 22( ) ( )3Trs r s r r L L , (2–11) where LA and LB are defined as 3md mqAL LL , 3md mqBL LL , (2–12) The mutual leakage inductances in (2–4) are denoted by Lla1a2, Lla1b2, …, Llc1c2. These inductances are the result of magnetic flux that does not cross the air gap but couples the three-phase stator sets. It should be noted that considering mutual 27 leakage inductance does not mean that any additional terms are being added to the total leakage inductance of each phase of the machine. The mutual leakage inductance is only part of the total leakage inductance per phase, which is shared by the two three-phase sets. Assuming that every stator phase has the same winding function, the following relation between the mutual leakage inductances can be established [14] 1 2 1 2 1 2la a lb b lc cL L L  , 1 2 1 2 1 2la b lb c lc aL L L  , 1 2 1 2 1 2la c lb a lc bL L L  (2–13) Finally, the electromagnetic torque is expressed as 1 1 1 1 2 16 62 2 1 2 2 2( ) ( )1( ) ( ) ( )2 2Tabc s s s r s s r abc se r lsabc s s s r s s r abc sPT L                                 i L L iIi L L i 1 12 2( )( )Tabc s s r rqdrabc s s r r               i L ii L . (2–14) The CCPD model derived in (2–1)–(2–14) has 9×9 inductance matrix. This model can be interfaced with arbitrary networks connected to both stator and rotor circuits. However, according to (2–3)–(2–4), the inductance matrices are variable depending on the rotor position. In order to demonstrate the importance of modelling the mutual leakage inductance in the six-phase machines, a transient study has been carried out using the machine presented in [58]. In this study, the three-phase stator sets are isolated; one is supplying a three-phase grid; and the other one is feeding a diode rectifier, respectively. The model considers both cases: with 33% of leakage mutual inductance; and without mutual leakage inductance. The simulation results for a 28 single phase fault (at t = 0.5s) on phase a1 (of the set connected to the three-phase grid) is demonstrated in Figure 2–2. As it is observed in Figure 2–2, the mutual inductance can have a significant effect on the shape and peak of the machine’s current wave forms. While the current in the phase under fault is not affected very much (see Figure 2-2, first subplot), the current in the healthy phases are affected considerably according to their values of the mutual leakage inductance with the faulted phase (see Figure 2-2, second to six subplots). 29 Figure 2–2. Transient response of stator phase currents under single phase a1 fault with and without considering mutual leakage inductance effect. -1000010002000i a1s (A) 33% Mutual leakageNo Mutual leakage-300-1500150300i b1s (A) 33% Mutual leakageNo Mutual leakage-300-1500150300i c1s (A) 33% Mutual leakageNo Mutual leakage-300-1500150300i a2s (A) 33% Mutual leakageNo Mutual leakage-300-1500150300i b2s (A) 33% Mutual leakageNo Mutual leakage0.45 0.5 0.55-300-1500150300i c2s (A) 33% Mutual leakageNo Mutual leakageTime (s) 30 2.2 qd0 Model The qd0 model is derived by transforming the corresponding CCPD model to the rotor reference frame [27], wherein one transformation per three-phase set is required. The transformation can be expressed in matrix form as 012 1 2( )qd r abc abcf K f , (2–15) where ( ) diag [ ( ), ( )]r r r    K K K , (2–16) 1 2 1 1 1 2 2 2[ , , , , , ]Tabc abc a b c a b cf f f f f ff , (2–17) 012 1 1 01 2 2 02[ , , , , , ]Tqd d q d qf f f f f ff . (2–18) Here, f may represent voltage, current, and flux variables, and K is the Park’s 3×3 transformation matrix [27]: 2 2cos( ) cos( ) cos( )3 32 2 2( ) sin( ) sin( ) sin( )3 3 31 1 12 2 2r r rr r r r                   K . (2–19) The machine voltage and flux equations (2–1)-(2–2) can be transformed to the rotor reference frame using (2–15), which results (2–20)–(2–37). Based on the transformed equations (2–20)–(2–37), an equivalent circuit for the machine qd0 model can be derived as depicted in Figure 2–3. The corresponding stator voltage equations in rotor reference frame are: 31 1 1 1 1q s s q s r d s q sv r i p     , (2–20) 1 1 1 1d s s d s r q s d sv r i p     , (2–21) 01 01 01s s s sv r i p  , (2–22) 2 2 2 2q s s q s r d s q sv r i p     , (2–23) 2 2 2 2d s s d s r q s d sv r i p     , (2–24) 02 02 02s s s sv r i p  . (2–25) The rotor voltage equations are: 0 kq kq kqr i p  , (2–26) 0 kd kd kdr i p  , (2–27) fd fd fd fdv r i p  . (2–28) 32 1d sv1q sv1q si2q si2d si2q sv2d svsrsr1r d s 2r d s 1r q s 2r q s lsLlsLmqLlmLkqilkqL kqrlqdLlqdL1d si sr lsLlmL fdifdrlfdLmdLkdrlkdLfdvsr lsLkdi01sisr 02L1 2la cL02si sr01sv 02sv1 2la aL1 2la bL01L Figure 2–3. Transformed qd0 equivalent circuit of six‐phase synchronous machine in rotor reference frame. The stator flux linkages are: 1 1 2 1 2( )q s mq lm q s q s l q s ldq d sL i i L i L i      , (2–29) 1 1 2 1 2( )d s md lm d s d s l d s ldq q sL i i L i L i      , (2–30) 01 01 01 1 2 1 2 1 2 01 02( )( )s s la a la b la c s sL i L L L i i      , (2–31) 2 1 2 2 1( )q s mq lm q s q s l q s ldq d sL i i L i L i      , (2–32) 2 1 2 2 1( )d s md lm d s d s l d s ldq q sL i i L i L i      , (2–33) 02 02 02 1 2 1 2 1 2 01 02( )( )s s la a la b la c s sL i L L L i i      . (2–34) 33 The rotor flux linkages can be expressed as kq mq lkq kqL i   , (2–35) kd md lkd kdL i   , (2–36) fd md lfd fdL i   . (2–37) In (2–29)–(2–37), the magnetizing fluxes are defined as 1 2( )mq mq q s q s kqL i i i    , (2–38) 1 2( )md md d s d s kd fdL i i i i     . (2–39) Note that due to the angle displacement ζ between the stator three-phase sets and the mutual leakage flux, the stator magnetic axes d1 and q2 becomes coupled. Similarly, the axes q1 and d2 are also coupled. This coupling is also depicted in Figure 2–3 and it can be calculated as [14] 1 2 1 2 1 22 2cos cos cos3 3lm a a a b a cL L L L                 , (2–40) 1 2 1 2 1 22 2sin sin sin3 3ldq a a a b a cL L L L                 . (2–41) The electromagnetic torque of the machine can also be calculated using the following equation 1 2 1 21 2 1 23 ( ) ( )2 2 ( ) ( )e q s q s md d s d s kd fdd s d s mq q s q s kqPT i i L i i i ii i L i i i        , (2–42) Equations (2–20)–(2–42) and equivalent circuit in Figure 2–3 define the qd0 model for six-phase synchronous machine [14]. Note that this qd0 state model has 34 the rotor and stator fluxes as the state variables and currents as its outputs. Because of that, it cannot be directly interfaced to an inductive or power electronic network without snubbers [61]. 2.3 VBR Model The derivation of VBR formulation starts from the qd0 model. For completeness, this model is also briefly presented here. The rotor fluxes and stator currents are selected as the state variables. Substituting the rotor currents in (2–38)–(2–39) with (2–35)–(2–37), the magnetizing fluxes can be obtained as 1 2kqmq mq q s q slkqL i iL       , (2–43) 1 2fdkdmd md d s d slkd lfdL i iL L        . (2–44) Here, the sub-transient inductances are: 11 1mqmq lkqLL L      , (2–45) 11 1 1mdmd lkd lfdLL L L       . (2–46) Using (2–26)–(2–28), the rotor currents (2–35)–(2–37), and magnetizing fluxes (2–43)–(2–44), the following state equations for machine rotor fluxes can be obtained: 35 1 21 ( )kq mqkq kq mq q s q slkq lkqr Lp L i iL L             , (2–47) 1 21 ( )fdkd mdkd kd md d s d slkd lkd lfdr Lp L i iL L L                   , (2–48) 1 21 ( )fd md kdfd fd md d s d s fdlfd lfd lkdr Lp L i i vL L L                    . (2–49) Substituting the stator fluxes in (2–20)–(2–25) with (2–29)–(2–34), and using (2–43)–(2–44) for the magnetizing fluxes and (2–47)–(2–49) for the rotor state equation, the stator voltage equation can be re-written in the following matrix form 1 11 1 1 11 11 1 1 1122 22 2 2 22 22 2 2 2d s d sq s q s q s q sq s q sd s d s d s d ss qd rd s d sq s q s q s q sq s q sd s d s d s d sv i i ev i i er pv i i ev i i e                                                              L (2–50) where the sub-transient inductance matrix is defined as 120000 4 4l mq lm mq lm ldql md lm ldq md lmqdmq lm ldq l mq lmldq md lm l md lmL L L L L LL L L L L LL L L L L LL L L L L L                       L (2–51) and the sub-transient back-emf voltages are 21 1 1 22 21 ( )kq mq mq kq mqq s r d kq q s q slkq lkq lkqr L L r Le i iL L L             , (2–52) 36 2 22 21 1 3 2 2 3 2 2fd md fd md fd mdkd md kd md kd mdd s r q kd fdlkd lkd lkd lfd lfd lfd lfd lkdr L r L r Lr L r L r LeL L L L L L L L                            221 22 2 ( )fd mdkd md d s d s fdlkd lfdr Lr L i i vL L       , (2–53) 22 2 1 22 21 ( )kq mq mq kq mqq s r d kq q s q slkq lkq lkqr L L r Le i iL L L             , (2–54) 2 22 22 2 3 2 2 3 2 2fd md fd md fd mdkd md kd md kd mdd s r q kd fdlkd lkd lkd lfd lfd lfd lfd lkdr L r L r Lr L r L r LeL L L L L L L L                            221 22 2 ( )fd mdkd md mdd s d s fdlkd lfd lfdr Lr L Li i vL L L        , (2–55) In (2–52), (2–55), the sub-transient fluxes are defined as: 1 2 mqq q kqlkqLL     , (2–56) 1 2 md mdd d kd fdlkd lfdL LL L        . (2–57) Applying the inverse transformation to (2–50) yields the voltage equation in six-phase abc1abc2 coordinates, which can be written as   1 2 1 2 1 2 1 21 2 1 2 1 2( ) ( )abc abc s s abc abc s abc abc r abc abc sabc abc r abc abc s abc abc sr pp   v i L iL i e . (2–58) Here, the variable inductance matrix is: 11 2 12( ) ( ) ( )abc abc r r dq r   L K L K . (2–59) 37 Equation (2–59) represents the desired interfacing circuit model that can be represented using six RL branches behind the current-controlled voltage sources as depicted in Figure 2–4. Unlike the qd0 machine model, the VBR model can be easily interfaced with any external network without the need for snubber circuits. Compared to the CCPD model, the size of the variable inductance matrix decreases from 9×9 to 6×6, which reduces the computational burden. The electromagnetic torque (2–42) is also valid for the VBR model. The VBR model defined by (2–47)–(2–55) is an exact equivalent to the CCPD defined in Section 2.1 and the qd0 model presented in Section 2.2, as it was derived using only algebraic manipulations and no approximations. Moreover, the rotor-position-dependent inductance matrix in (2–58) implies that the interfacing branches will have variable and coupled inductances, which complicates the implementation of this model as well as making it computationally expensive [67]. 2.4 Constant‐Parameter VBR Model The variable inductances in VBR machine model are the result of different sub-transient inductances along rotor q‐ and d‐axes in (2–50), making the inductance matrix (2–51) variable when it is transformed back to abc1abc2 coordinates (2–59). In order to make the interfacing equation (2–58) constant parameter (and also the corresponding interfacing circuit), one can modify (2–50) so that stator voltages would have the same sub-transient inductances in both q‐ and d‐axes. The extra terms resulted from making the q‐ and d‐axes sub-transient inductances equal are 38 then added to the sub-transients back emf voltages, as shown in (2–60) 1 11 1 11 11 1 1122 22 2 22 22 2 2d s d sq s q s q sq s q sd s d s d ss qd rd s d sq s q s q sq s q sd s d s d sv i iv i ir pv i iv i i                                                   L 1 2 11 2 11 2 21 2 2( )( )( ) ( )( )q s q s q sr q s q s d smd mqq s q s q sr q s q s d sp i i ei i eL Lp i i ei i e                         , (2–60) where the modified matrix 12qdL in (2–60) is 120000 4 4l md lm md lm ldql md lm ldq md lmqdmd lm ldq l md lmldq md lm l md lmL L L L L LL L L L L LL L L L L LL L L L L L                       L (2–61) and has equal values in q‐ and d-axes. Also, based on (2–60), the new sub-transient back-emf voltage vector can be defined containing the leftover saliency terms from the q-axis sub-transient inductance: 1 21 11 21 11 22 21 22 2( )( )( ) ( )( )q s q sq s q sr q s q sd s d smd mqq s q sq s q sr q s q sd s d sp i ie ei ie eL Lp i ie ei ie e                                          . (2–62) Based on (2–61)–(2–62), the interfacing circuit equation becomes:  1 2 1 2 1 2 1 2 1 2abc abc s s abc abc s abc abc abc abc s abc abc sr p    v i L i e , (2–63) 39 where the matrix 1 2abc abcL is constant and obtained using (2–61) as 11 2 12( ) ( )abc abc r dq r  L K L K . (2–64) It should be noted that the leftover term at the end of (2–62) compensates for the sub-transient saliency and contains the derivatives of the stator currents in q-axis ( 1q si and 2q si ) and their resulting rotational induced voltages on the machine d-axis. The rotational induced voltage terms in (2–62) are easily calculated using the rotor speed and the stator currents. However, obtaining the stator current derivatives along q-axis is more challenging. Based on (2–29) and (2–32), it is possible to calculate the sum of q-axis stator currents derivatives as:      1 2 1 2 2 11 22q s q s q s q s mq lqd d s d slm lp i i p L i iL L            . (2–65) However, (2–65) also contains the q-axis stator flux linkages and the d-axis stator currents. Substituting (2–30) and (2–33), and using (2–20)–(2–25), after some algebraic manipulations, (2–65) can be re-written in terms of stator currents, rotor fluxes, and voltages, expressed as 1 2 2 2( ) .2( )lq s q sl lm mq ldqLp i iL L L L         1 2 2 1( )ldqs q s q s d s d slLr i i i iL           1 2 2 1( )ldqr d s d s q s q slLL          + 1 2 1 2 2 12 1 ( ) ( )mq kq mq ldqkq mq q s q s q s q s d s d slkq lkq lkq lL r L LL i i v v v vL L L L                     .(2–66) which is then used to calculate the new sub-transient back-emfs in (2–62). 40 However, since (2–66) requires stator voltages 1q sv , 2q sv , 1d sv , and 2d sv , (2–62) and (2–66) contain algebraic feed-through from stator voltages to the sub-transient back-emfs. At the same time, stator branch voltages (2–63) are also directly/algebraically dependent on the sub-transient back-emfs, which creates an algebraic loop and yields the formulation implicit. This algebraic loop may be relaxed [79] by introducing an intermediate variable sv that contains all stator voltages in (2–66) as 1 2 2 1( )ldqs q s q s d s d slLv v v v vL    , (2–67) and passing it through a low pass filter 11s sfv vs   . (2–68) Thereafter, sv is used in (2–66) instead of 1q sv , 2q sv , 1d sv , and 2d sv to calculate the stator currents derivatives. Similar to the three-phase VBR model, the filter pole (-1/τf) should be chosen to give an accurate estimate of vs up to certain frequency without introducing large eigenvalue into the system. Interested reader will find the detailed procedure with example in [81]. Finally, based on (2–63), the interfacing circuit of the six-phase CPVBR is shown in Figure 2–4. 41 n1Networkrsrsrsrsrsrs K(θr)K(θr)-1n2Rotor state-space modelEq: (47)-(49), (52)-(55),1 2abcs abcsLSix-Phase synchronous machine CPVBR model1sabcv(62)012sqdv012sqdi(Eq: B5)1sabce2sabce1sae1sabci2sabcv2sabci1sbe1sce2sae2sbe2sce Figure 2–4. Implementation of the proposed CPVBR model with constant‐ parameter RL branch interfacing circuit. 2.5 Computer Studies 2.5.1. Case Study System To verify the proposed CPVBR model, the benchmark system [14] is used where a six-phase synchronous machine with 30 degrees shift angle ζ between the two three-phase stator sets is considered. The machine is assumed to operate as a generator driven by a turbine prime mover that supplies mechanical torque. For the purpose of investigating the models’ interfacing properties, the generator is connected to a network consisting of a six-pulse rectifier through set abc2 and a three-phase ac system through set abc1, as shown in Figure 2–5. The neutral points of the two wye-connected three-phase sets are isolated and floating. The three-phase uations: (2-47)-(2-49), (2-52)-(2-55), (2-62) 42 ac grid is represented by its Thévenin equivalent impedance gridZ and source gabcv . The remaining system parameters are summarized in Appendix A. 3-Phase AC grid60Hz 340VRloadLdcCdcvdcSix-Phase Syncronous MachineTmTurbineThree-Phase rectifierSingle-phasefault at t = 0.5 son phase a1gabcvcableZcableZgridZ2sabci1sabci Figure 2–5. Case study system with a six‐phase synchronous generator feeding a rectifier load and a three‐phase ac grid. The power system of Figure 2–5 has been implemented in MATLAB/Simulink [39] software using PLECS [41] toolbox. For the purpose of comparison, four machine models have been implemented: qd0 model, CCPD model, basic VBR model (with variable inductances), and the proposed CPVBR model. For consistency, all subject models have been executed on a PC with Intel® Core™ i7-4700 @ 2.4GHz processor using variable time step solver ode45 with relative and absolute error tolerances set to 10-4. To assess the numerical accuracy of the subject models, the reference solution has been obtained using the CCPD model with relative and absolute error tolerances set to a very small value of 10-7 and a maximum step size limited to 20µs. Since the qd0 model does not allow direct connection to external inductive networks, according to methodology in [59], six un-grounded snubber resistors 43 connected in parallel with the machine stator terminals have been used to interface this model with the network assumed in Figure 2–5. It should be noted that choosing larger snubber resistors improves accuracy of such interfacing at the cost of having larger system eigenvalues and lower numerical efficiency. Here, the snubber resistors are selected to be 120 Ω (which corresponds 100 pu, with the machine’s nominal power and voltage as taken as the base values), to limit the snubber branch current to approximately 1% of the nominal stator current. For the proposed CPVBR model, the filter pole (-1/τf) in (2–68) should be sufficiently large in order to have good accuracy of this approximation in steady state and fast transients up to certain frequency. Since the machine is connected to a 60Hz grid and a diode-rectifier, the stator voltage and current experience constantly excited transients with significant 5th and 7th harmonics due to rectifier switching. Therefore, even in steady state, the stator voltages in qd0 frame [i.e. 1q sv , 2q sv , 1d sv , and2d sv , and consequently sv in (2–68)] contain the frequency of 360 Hz associated with the 7th harmonic. Hence, according to the methodology set forth in [81], to achieve a solution accuracy within 1% at 360 Hz, the pole (-1/τf) is selected to be -2000. Choosing a larger pole for the low-pass filter improves accuracy at the cost of making the model numerically stiffer. 2.5.2. Transient Simulation The considered transient study assumes that the six-phase synchronous generator is operating in steady-state delivering 100 kW of total power to its loads. The prime mover turbine is supplying input mechanical torque of 500 Nm, which is 44 assumed constant during the study. The machine’s three-phase set abc1 provides approximately 40 kW to the ac grid. The set abc2 is feeding 60 kW into a rectifier supplying dc system represented by a 2.5 Ω resistive load. At t = 0.5 s, a single phase-to-ground fault occurs on phase a1 of the generator as depicted in Figure 2–5. The corresponding transients as predicted by the subject models are shown in Figure 2–5 for a 2 s simulated study. Figure 2–6 shows that the fault mostly affects the set abc1, due to containing the faulty phase. Meanwhile, due to the mutual leakage inductances between the two sets, the fault disturbance also propagates into the set abc2 causing the reduction and ripple of the rectifier dc voltage as shown in Figure 2–6 (e). Figure 2–7 shows magnified view of the generator phase voltages, currents, and output torque during the fault transient. As can be observed in Figure 2–7, all the four subject models predict the transients that are visibly very close to the reference. To examine the performance of models even further, the magnified views from Figure 2–7 at the fault moment are shown in Figure 2–8, which reveals a noticeable difference among the models in terms of number of steps taken and accuracy. As it can be seen in Figure 2–8, the qd0 model visibly deviates from the reference solution and takes significantly larger number of time steps as compared to other models. This deficiency of qd0 model is attributed to the artificial interfacing snubbers that draw the additional small currents introducing solution error, as well as making the model numerically stiff and cause the variable step solver to choose much smaller time steps. 45 Figure 2–6. System transient response to a single phase‐to‐ground fault on phase a1 as predicted by the subject models: (a) and (b) phase currents for set 1 and set 2, (c) electromagnetic torque, (d) rotor speed, and (e) rectifier dc voltage. -1000-20002001000i sabc1 (A) -50005001500T e (N.m) -1000-20002001000i sabc2 (A) 17851,8001815 r (rpm) 0 .5 1 1.5 2320350380Time (s)v dc (V) isa1isb1isc1isa2isb2isc2See Fig. 6be)a)b)c)d)See Fig. 6aSee Fig. 6eSee Fig. 2–7(a) See Fig. 2–7(b) See Fig. 2–7(e) i abc1s (A) i abc2s (A) 46 Figure 2–7. Magnified view of selected variables showing the details of the fault transient obtained by the subject models for: (a) phase currents of abc1 set; (b) phase currents of abc2 set; (c) stator phase voltages of abc1 set; (d) stator voltages of abc2 set; and (e) electromagnetic torque. -1000-50005001000i sabc1 (A) -200-1000100200i sabc2 (A) 0.48 0.485 0.49 0.495 0.5 0.505 0.51 0.515 0.520500100015002000Time (s)T e (N.m) -2000200v sabc1 (V) -2000200v sabc2 (V) Referenceqd modelCCPDBasic VBRCPVBRa)b)c)d)In a) - d):e)Phase a1, a2 Phase c1, c2See Fig. 7aSee Fig. 7bSee Fig. 7cPhase b1, b2See Fig. 2–8(a) See Fig. 2–8(b)See Fig. 2–8(c)i abc2s (A) i abc1s (A) v abc2s (V) v abc1s (V) 47 Figure 2–8. Magnified view of the transient response of the subject models compared to the reference solution at the fault moment for (a) stator phase ‘a1’ and (b) ‘a2’ currents and (c) electromagnetic torque. At the same time, the CCPD, basic VBR and CPVBR models are directly interfaced and achieve highly accurate results with significantly fewer time steps as can be seen in Figures 2–8(a–c). 0.5062 0.5064 0.5066 0.5068 0.507 0.507211101115112011251130Time (s)i sa1 (A) 0.5078 0.508 0.5082 0.5084 0.5086 0.5088210215220225Time (s)i sa2 (A) 0.5048 0.505 0.5052 0.5054 0.5056 0.50581900191019201930Time (s)T e (N.m) Referenceqd modelCCPDBasic VBRCPVBRReferenceqd modelCCPDBasic VBRCPVBRReferenceqd modelCCPDBasic VBRCPVBRqd Model (high number of data points)b)a)c)i a1s (A) i a2s (A) 48 2.5.3. Computational Performance and Accuracy Comparison To further assess the computational efficiency of the subject models, the CPU time taken and the number of time steps for the 2 s transient study are summarized in Table 2–1. As can be seen in Table 2–1, the qd0 model takes the highest number of steps (484,570) and longest CPU time (51s). This performance is due to high stiffness created by the artificial snubbers. The directly interfaced CCPD, basic VBR and CPVBR models are solved with considerably fewer number of time steps (10,536), (8,427) and (9,464), respectively. The CCPD and basic VBR models have rotor-position-dependent (variable) inductances of the interfacing circuit, which puts a computational burden on the solver due to re-calculating the system impedance matrix at each time step. However, the inductance matrix of the basic VBR model is 6×6, whereas this matrix is 9×9 for the CCPD model. This results in a CPU time of 14.51s for the basic VBR and 16.62s for the CCPD model. Finally, the CPVBR model achieves the fastest solution due to its constant parameter interfacing circuit, taking only 3.3s. However, the superior performance of CPVBR model is achieved at a cost of approximating the voltage in (2–68) to relax the algebraic loop, which introduces a small numerical error in the solution. To compare the numerical accuracy of the subject models, the errors in the simulation results for stator phase a1 current isa1 and electromagnetic torque Te are summarized in Table 2–2. Without loss of generality, the 2-norm cumulative error evaluated over the 2 s solution interval has been considered for comparing the accuracy of the solutions produced by the subject models with respect to the reference solution trajectories. The 2-norm error of the solution trajectory x with 49 respect to reference solution xref is defined as [82] 222( ) ( )( )refrefx t x tex t . (2–69) According to Table 2–2, the qd0 model has the largest numerical error (1.4% for the stator phase a1 current and 1.8% for the output torque) due to the artificial snubbers. The CCPD and basic VBR produce most accurate results (0.01% error for stator phase a1 current for both models, and 0.006% and 0.009% for the torque calculated by the CCPD and basic VBR models, respectively). These small errors are mainly due to the solver discretization. Table 2–2 also verifies that the CPVBR model achieves a very good accuracy (0.7% error in stator phase a1 current and 0.73% in torque). It is worth mentioning that the performance of CPVBR model is a compromise between accuracy and stiffness (i.e. the model can be made more accurate by increasing the low-pass filter pole, which comes at a cost of numerical stiffness and slows down the simulation). To see the impact on model stiffness, the largest eigenvalue and number of state variables of the overall power system with each of the subject machine models are summarized in Table 2–3. Table 2–3 shows that the qd0 model has the largest number of state variables (44) as well as largest eigenvalue (-9.07 × 105) due to the snubbers. The effect of this eigenvalue can be clearly seen on the number of time steps and CPU time taken by the qd0 model. The CCPD model has the minimum number of state variables (35) with the largest system eigenvalue (-1,191). The basic VBR model uses the same number of state variables (35) but improves the eigen-structure compared to the CCPD model, which results in reduction of the largest 50 eigenvalue (-447) and making this model more efficient. The first order low-pass filter in the CPVBR model adds one state variable compared to the CCPD and basic VBR models (36 states). The largest eigenvalue in the CPVBR model (-1,783) is associated with the low-pass filter pole selection in (2–68). The value of the selected pole is affected by the maximum transient frequency in which the accuracy of the model should remain within acceptable range (under 1% error in this case). Since the machine is connected to a diode rectifier, the large pole in the CPVBR model corresponds to the maximum desired frequency of 360Hz according to [81]. However, in spite of having a larger eigenvalue, the constant parameter interface of the CPVBR model significantly reduces its computational cost by eliminating the network impedance matrix re-calculations at every time step. The result is superior simulation performance of the CPVBR model compared to the CCPD and basic VBR models in terms of CPU time. TABLE 2–1. CPU TIME AND NUMBER OF STEPS FOR THE 2 SECOND STUDY FOR THE SUBJECT MODELS Model Simulation Time Number of steps qd0 model with snubbers 51 s 484,570 CCPD model (with variable inductance) 16.62 s 10,536 VBR model (with variable inductance) 14.51 s 8,427 CPVBR model 3.3 s 9,464 51 TABLE 2–2. COMPARISON OF 2‐NORM ERROR IN THE SIMULATION RESULTS Model Stator Current a1 error Torque error qd0 model with snubbers 1.4 % 1.8 % CCPD model (with variable inductance) 0.01 % 0.006 % VBR model (with variable inductance) 0.01 % 0.009 % CPVBR model 0.7 % 0.73 TABLE 2–3. COMPARISON OF THE LARGEST EIGENVALUE AND NUMBER OF STATE VARIABLES Model Largest eigenvalue Number of state variables qd0 model with snubbers -9.07 × 105 44 CCPD model (with variable inductance) -1,191 35 VBR model (with variable inductance) -447 35 CPVBR model - 1,783 36 52 CHAPTER 3: MAGNETIC SATURATION AND DECOUPLED INTERFACING FOR CPVBR MODELS FOR SIX‐PHASE SYNCHRONOUS MACHINES This chapter extends the work in chapter 2, and derives a CPVBR model for salient-pole six-phase synchronous machines while considering the main flux saturation. The resulting model has the desired constant-parameter stator interfacing circuit, which already possesses numerical advantages. But since this circuit is composed of magnetically-coupled RL branches, this model still may not be practical to implement in some common simulation programs. To further simplify the interfacing, a magnetically-decoupled CPVBR (DCPVBR) formulation is proposed that achieves both decoupled and constant-parameter stator interfacing circuit, which is obtained at a cost of slight increase in numerical error. To the best of our knowledge, a six-phase machine model with such interfacing circuit has not been reported in the prior literature. We present extensive studies that verify the proposed model and demonstrate its computational advantages over the alternative existing qd0 and VBR models in terms of simulation performance and numerical 53 efficiency. Due to their constant parameter interface, it is envisioned that the proposed CPVBR and DCPVBR (in particular) models will find wide application in many commonly-used simulation packages where the benefits of their simple interface can be exploited. Here, a wound-rotor six-phase synchronous machine with saturable main flux is considered. A salient rotor with one field winding and one damper winding on the d-axis and one damper winding on the q-axis is assumed. The stator is assumed to have two identical and sinusoidally distributed three-phase winding sets which are wye-connected and electrically displaced by angle ζ. It is assumed that all variables and parameters are referred to the stator side and motor sign convention is used for the machine variables. 3.1 Saturable qd0 Model The main flux saturation has been considered for qd0 and VBR models in [83]. The resulting qd0 equivalent circuit is depicted in Figure 3–1, which includes the effect of saturation. Based on Figure 3–1, the voltage equations for the two three-phase stator sets are 1 1 1 1q s s q s r d s q sv r i p     , 1 1 1 1d s s d s r q s d sv r i p     , (3–1) 01 01 01s s s sv r i p  , (3–2) 2 2 2 2q s s q s r d s q sv r i p     , 2 2 2 2d s s d s r q s d sv r i p     , (3–3) 02 02 02s s s sv r i p  , (3–4) where p = d/dt is the Heaviside’s derivative operator. Also, the rotor voltage 54 equations can be written as 0 kq kq kqr i p  , 0 kd kd kdr i p  , fd fd fd fdv r i p  . (3–5) In (3–1)–(3–4), the stator fluxes are 1 1 2 1 2( )q s mq lm q s q s l q s ldq d sL i i L i L i      , (3–6) 1 1 2 1 2( )d s md lm d s d s l d s ldq q sL i i L i L i      , (3–7) 01 01 01 1 2 1 2 1 2 01 02( )( )s s la a la b la c s sL i L L L i i      , (3–8) 2 1 2 2 1( )q s mq lm q s q s l q s ldq d sL i i L i L i      , (3–9) 2 1 2 2 1( )d s md lm d s d s l d s ldq q sL i i L i L i      , (3–10) 02 02 02 1 2 1 2 1 2 01 02( )( )s s la a la b la c s sL i L L L i i      . (3–11) The rotor fluxes in (3–5) can also be written as kq mq lkq kqL i   , kd md lkd kdL i   , fd md lfd fdL i   . (3–12) In (3–6)–(3–11), lmL and lqdL are the transformed direct and cross-coupling mutual leakage inductances, which are defined in terms of actual leakage inductances, shared between the individual phases of each three-phase set in phase domain (i.e., 1 2la aL , 1 2la bL , 1 2la cL ) [14] and expressed as 1 2 1 2 1 22 2cos cos cos3 3lm la a la b la cL L L L                 , (3–13) 1 2 1 2 1 22 2sin sin sin3 3lqd la a la b la cL L L L                 . (3–14) The zero sequence leakage inductance is also obtained as 01 02 1 2 1 2 1 2( )ls lm la a la b la cL L L L L L L      . (3–15) 55 1d sv1q sv1q si2q si2d si2q sv2d svsrsr1r d s 2r d s 1r q s 2r q s lsLlsL( )mq mL lmLkqilkqL kqrlqdLlqdL1d si sr lsLlmL fdifdrlfdL( )md mL kdrlkdLfdvsr lsLkdi01sisr 02L1 2la cL02si sr01sv 02sv1 2la aL1 2la bL01L Figure 3–1. Equivalent circuit of six‐phase synchronous machine with main flux saturation in qd0 rotor reference frame. Also, based on the equivalent circuit in Figure 3–1, in (3–6)–(3–12) the q- and d-axes components of the magnetizing flux are expressed as 1 2( )( ) ( )mq mq m q s q s kq mq m mqL i i i L i      , (3–16) 1 2( )( ) ( )md md m d s d s kd fd md m mdL i i i i L i       . (3–17) It is noted that the magnetizing inductances ( )mq mL  and ( )md mL  are expressed as functions of the main magnetizing flux. To model the magnetic saturation based on an equivalent isotropic machine [84]–[85], the magnetizing flux and current vectors are expressed as    2 22 2, m mq F md m F mq mdS i S i i      . (3–18) 56 In (18), FS is the saliency factor defined as F mqu mduS L L , (3–19) where mquL and mduL are the unsaturated magnetizing inductances in q- and d-axes. The magnetizing inductances (which include the effect of saturation) are then defined as 2( ) , ( ) ( )( )m mmd m mq m F md mm d mL L S Li F      , (3–20) where (.)dF is the non-linear magnetization curve on d-axis and is obtained through standard open-circuit tests [84], [86]–[87]. To complete the model, the machine electromagnetic torque can also be expressed as [59] 1 2 1 23 ( ) ( )2 2e sq sq md sd sd mqPT i i i i       . (3–21) Equations (3–1)–(3–17), (3–21) form the qd0 model of the saturable six-phase synchronous machine, where the stator and rotor fluxes are the state-variables, and the stator and rotor voltages are the inputs. The outputs of the model are the stator and rotor currents, which are represented as controlled current sources in simulation programs for interfacing. 3.2 Saturable Variable‐Parameter VBR Model The VBR model is derived based on the qd0 model by choosing rotor fluxes as state variables and stator currents as inputs to the rotor subsystem, similar to the approach presented in [38]. Implementing saturation in the VBR model also requires 57 variable magnetizing inductances ( )mq mL  and ( )md mL  . The method used here is based on the flux correction approach [33], which uses intermediate variables zqi and zdi that are defined in terms of the state variables (rotor fluxes) and inputs (stator currents) as 1 2 kq mqzq q s q s mqlkq lkqi i i iL L      , (3–22)  1 2 md lkd lfdfdkdzd d s d s mdlkd lfd lkd lfdL Li i i iL L L L       . (3–23) According to [33], a direct relation between the magnetizing flux and current vector magnitudes and the magnitude zi can be defined based on their components in q- and d-axis (3–18) and (3–22)–(3–23), respectively and expressed as  2 2 2 lkd lfdz zq zd m m mlkd lfdL Li i i i GL L          , (3–24) where the saliency correction variable α [which is different from saliency factor (3–19)] is obtained using the machine’s parameters and magnetization data as  lkd lfdm mlkd lfdmm F F m lkqL Li L LHi S S L       . (3–25) By writing mi as a function of m using the d-axis saturation function (.)dF , both zi and α can be defined as functions of magnetizing flux vector magnitude m based on (3–24) and (3–25), respectively, as 58 ( )( )z mmi GH  . (3–26) Here, the functions G(.) and H(.) denote the non-linear relation between zi and m , and between α and m , which can be computed and implemented as simple 1-D look-up tables [33]. As seen in (3–24), computing zi requires the variable α which can be expressed as a function of zi based on (3–26) as  1( )zH G i  . (3–27) This leads to an implicit formulation for zi that contains an algebraic loop as  2 2 2 2 1 2 2( )z zq zd z zq zdi i i H G i i i     . (3–28) In order to form an explicit formulation for calculating zi , an estimated value of zi (denoted by *zi ) is defined as * 2 2 2z u zq zdi i i  , (3–29) which uses a constant parameter u that is defined using the unsaturated magnetization data [33] ( mui , mu provided in (.)dF ) as lkd lfdmu mulkd lfdumu F F mu lkqL LiL Li S S L        . (3–30) The estimate variable *zi can then be used instead of zi in (3–27) to calculate α as  1 *( )zH G i  , (3–31) 59 which is then used in (3–28) to form an explicit formulation for calculating zi . The value of zi is then used to calculate m based on (3–26) (i.e. 1( )m zG i  ) which is then used to compute the magnetizing inductances in (3–20). This results in magnetizing inductance being defined as a function of zi (which is itself a function of state-variables and inputs) as  121( )( ) , ( ) ( )( )zmd z mq z F md zd zG iL i L i S L iF G i  . (3–32) Replacing rotor currents in (3–12) with the ones from (3–16)–(3–17), and using the magnetizing inductances from (3–32) the magnetizing flux components can be defined as 1 2( ) ( ) kqmq mq z q s q slkqL i i iL       , (3–33) 1 2( ) ( ) fdkdmd md z d s d slkd lfdL i i iL L        , (3–34) where ( )mq zL i and ( )md zL i are the saturable sub-transient inductances defined as 1 11 1 1( ) , ( )( ) ( )lkd lfdmq z md zmq z lkq md z lkd lfdL LL i L iL i L L i L L                 , (3–35) The rotor state equations can then be formulated based on the rotor voltage equations (3–5), rotor flux equations (3–12), and magnetizing fluxes (3–33)–(3–34), as 1 2( ) 1 ( )( )kq mq zkq kq mq z q s q slkq lkqr L ip L i i iL L             , (3–36) 60 1 2( ) 1 ( ) ( )kd md zkd kdlkd lkdfdmd z d s d slfdr L ipL LL i i iL            , (3–37) 1 2( ) 1 ( ) ( )fd md zfd fd fdlfd lfdkdmd z d s d slkdr L ip vL LL i i iL             . (3–38) The VBR interfacing equation is obtained by writing (3–1), (3–3) as 11 1 111 1 122 2 222 2 2d sq s q s q s mqq sd s d s d s mds l rd sq s q s q s mqq sd s d s d s mdv i iv i ir p pv i iv i i                                                         L , (3–39) where Ll is defined as 0000lm ls lm lqdlm ls lqd lmllm lqd lm lslqd lm lm lsL L L LL L L LL L L LL L L L          L . (3–40) The derivatives of magnetizing fluxes can be expressed in terms of rotor fluxes and stator currents using (3–32)–(3–33) as 1 21 2( )( , ) ( )kqlkqmq q s q smi z mfdmd d s d s kdlkd lfdpLi ip i p pi i pL L                        L λ , (3–41) where 61 11 0( , ) ( , ) 1 10( , ) ( , ) ( , ) ( , )lkqmi z m mi z mlkd lfdmqq z m mqd z mmqd z m mdd z mLi iL LL i L iL i L i                  L λ Γ λλ λλ λ, (3–42) with ( , )mi z miΓ λ being defined as [44]. 24 2 2 222 2 2( ) ( )1 1 1( ) ( ) ( )( , )( ) ( )1 1 1( ) ( ) ( )mq mq mdd m d mm md z F m mq z m md z F mmi z mmq mdd m d m mdm md z F m m md z m md zdF dFd L i S L i d L i SidF dFd L i S d L i L i                                          Γ λ . (3–43) In (3–43), the vector term mλ denotes the two magnetizing flux components in q- and d-axis, which are computed as a function of zi , inputs and state-variables as shown in (3–33)–(3–34). Using (3–41) in (3–39), and substituting rotor flux derivatives (3–36)–(3–38), the interfacing equation can be rewritten as 11 1 1 111 1 1 11222 2 2 222 2 2 2( , )d sq s q s q s q sq sd s d s d s d ss qd z m rd sq s q s q s q sq sd s d s d s d sv i i ev i i er i pv i i ev i i e                                                         L λ , (3–44) where 12 ( , )qd z miL λ is defined as 62 12 ( , )( , ) ( , ) ( , ) ( , )( , ) ( , )( , ) ( , )( , ) ( , )qd z mls mqq z m lm mqd z m mqq z m lm mqd z m lqdmqd z m ls mdd z m lm mqmqq z m lm mqd z m lqdmqd z m lqd mdd z m lmiL L i L L i L i L L i LL i L L i L LL i L L i LL i L L i L                  L λλ λ λ λλ λλ λλ λ 4 4( , ) ( , )( , ) ( , )( , ) ( , )d z m lqd mdd z m lmls mqq z m lm mqd z mmqd z m ls mdd z m lmi L L i LL L i L L iL i L L i L          λ λλ λλ λ. (3–45) and the sub-transient fluxes are  1 11 1122 22 2( , )q s q sd s d sl qd z mq s q sd s d siiiii                   L L λ , (3–46) while the sub-transient back-emf voltages are defined as 1 111 11 2 22 22 2 22 22( , )kqd s d sq slkqq s q sd sr mi z md s d sq s fdkdq s q s lkd lfdd speLeie ppL Le                                                  I L λI . (3–47) Applying the inverse Park’s transformation [59] to (3–44), results in the interfacing stator voltage equation in phase coordinates as   1 2 1 2 1 2 1 21 2 1 2 1 2( , , ) ( , , )abc abc s s abc abc s abc abc r z m abc abc sabc abc r z m abc abc s abc abc sr p ii p   v i L λ iL λ i e , (3–48) where the saturable and rotor-position-dependent interfacing inductance matrix is 11 2 12( , , ) ( ) ( , ) ( )abc abc r z m r dq z m ri i   L λ K L λ K . (3–49) 63 1a se1b se1c se2a se2b se2c sesrsrsrsrsrsr1 2 ( , , )abc abc r z miL λ( )rK1( )rK 12qd si12qd se1 2abc abc se1 2abc abc si Figure 3–2. Implementation of the saturable VBR model with magnetically‐coupled variable‐parameter interfacing RL branches. In (3–48), ( )rK is the six-phase Park’s transformation matrix [59] from abc1abc2 phase coordinates to rotor reference frame qd01qd02. The state equations (3–36)–(3–38) (with stator currents as input and sub-transient back-emfs (3–47) as output) form the VBR model of the saturable six-phase synchronous machine. Also, the electromagnetic torque is calculated similar to the qd0 model using (3–21). It should be noted that for transforming stator voltages to phase coordinates, the zero sequence (3–2) and (3–4) are also considered since the neutral points may be used. This VBR model can be implemented using controlled voltage sources connected in series with RL branches in phase coordinates, as depicted in Figure 3–2. This model can be directly interfaced to an arbitrary external network without artificial snubber circuits. However, as seen in Figure 3–2, the magnetically coupled 64 interfacing inductances are variable due to saturation and dependency on the rotor position. 3.3 Saturable Constant‐Parameter VBR Model It is desirable to obtain a CPVBR model by reformulating the sub-transient inductances in (3–45) to achieve constant and equal parameters in both q- and d-axes. This is achieved by reforming the sub-transient inductances for both q- and d-axes using the constant/unsaturated magnetizing inductance of d-axis, as 12 12( , ) ( , )dq z m qd resqd z mi i   L λ L L λ . (3–50) In (50) 12qdL is a constant sub-transient inductance matrix defined as 124 40000ls mdu lm mdu lm lqdls mdu lm lqd mdu lmqdmdu lm lqd ls mdu lmlqd mdu lm ls mdu lmL L L L L LL L L L L LL L L L L LL L L L L L                        L (3–51) which has equal inductance terms in both q- and d-axes using only the machine unsaturated sub-transient magnetizing inductance in d-axis [59] expressed as 11 1 1mdumdu lkd lfdLL L L       . (3–52) Moreover in (3–50), ( , )resqd z miL λ contains all the remaining terms corresponding to the saliency and variable inductances (due to saturation) and is defined as 65 4 42 2 2 22 2 2 2 4 4( , ) ( , )( , ) ( , ) ( , ) res z m res z mresqd z mres z m res z mmdui iii iL              L λ L λL λ L λ L λI II I, (3–53) where 2 2( , ) ( , )( , ) ( , ) ( , )mqq z m mqd z mres z mmqd z m mdd z mL i L iiL i L i        λ λL λ λ λ . (3–54) The constant and non-salient 12qdL is then used as the interfacing inductance matrix and the remaining inductive terms in ( , )resqd z miL λ corresponding to magnetic saturation and rotor saliency are calculated and included as voltages in sub-transient back-emfs. This is achieved by writing (3–44) as 1 11 1 11 11 1 1122 22 2 22 22 2 2 + ( , )d s d sq s q s q sq s q sd s d s d ss qd rd s d sq s q s q sq s q sd s d s d sresqd z mv i iv i ir pv i iv i ii                                                        LL λ11 111 122 222 2d sq s q sq sd s d srd sq s q sq sd s d si ei epi ei e                               , (3–55) where 1 11 12 22 2( , )d s q sq s d sresqd z md s q sq s d siiiii                   L λ . (3–56) By defining the new sub-transient back-emfs as 66 11 1 111 1 122 2 222 2 2( , )d sq s q s q sq sd s d s d sresqd z m rd sq s q s q sq sd s d s d se e ie e ii pe e ie e i                                                      L , (3–57) the stator voltage equation in (3–55) can be expressed as 1 11 1 1 11 11 1 1 1122 22 2 2 22 22 2 2 2d s d sq s q s q s q sq s q sd s d s d s d ss qd rd s d sq s q s q s q sq s q sd s d s d s dv i i ev i i er pv i i ev i i e                                                            Ls      . (3–58) It is noted that calculating the new sub-transient back emfs in (3–57) requires derivatives of stator currents. These derivatives can be calculated using the stator voltage equation (3–44) as  11 1 11 11 1 11222 2 222 2 21 1 2 2( , ) d sq s q s q sq sd s d s d sqd z m rd sq s q s q sq sd s d s d sTs q s d s q s d si v ei v ep ii v ei v er i i i i                                                      L. (3–59) Using (3–59) in (3–57) results in an algebraic loop since the sub-transient back-emfs in (3–57) will have a direct relation with the stator voltages in (3–59), and at the same time, the stator voltages themselves are algebraically related to sub-transient back-emfs in (3–58). An explicit formulation is obtained by applying the algebraic loop relaxation method [88], where a tuned filter (consisting of low-pass and band-pass filters) is used to approximate the stator voltage as 67 1 11 12 22 2( )q s q sd s d sq s q sd s d sv vv vF sv vv v                  , (3–60) which can then be used in (3–59). The pole selection for the filter transfer function F(s) is done according to the procedure set forth in [81], [88], so that it provides an accurate approximation of stator voltages up to the required frequency. Using a low-pass filters (LPFs) for F(s) has been shown to provide good approximation accuracy in power system simulations where the ac variables are mainly sinusoidal (i.e. 50/60Hz with small/no harmonics). However, in simulations of machine-converter systems where high-frequency ac harmonics exist due to periodic switching of converters, maintaining the same approximation accuracy results in reduced numerical efficiency, since stiff poles are needed for the LPF. In order to provide acceptable accuracy in such case study scenarios, a new method for algebraic-loop relaxation in CPVBR models for simulation of machine-converter systems is proposed. The proposed method uses tunable band-pass filters (BPF) to provide highly accurate approximation of the ac harmonics, whose frequencies have direct relationship with the machine speed, to provide numerical advantages without making the system stiff. Figure 3–3. Approximation filter: a) LPF , and b) proposed hybrid LPF‐BPF. ∑ ∑ ( )H s( )G sqsv qsv+++ _( )H sqsv qsva) b)LPFLPFBPF68 The resulting filter is composed of the conventional LPF (denoted here with H(s)), which is tuned to achieve sufficient accuracy for low frequency transients [81], and an additional BPF (denoted here with G(s)) (that is tuned to pass the desired high frequency), as shown in Figure 3–3(b). The LPF and BPF have the following form 01( ) 1H s s  , 02 20 02( ) 2sG ss s    . (3–61) The hybrid filter yields transfer function  ( ) ( ) ( ) ( ) ( )F s H s G s H s G s   . (3–62) The pole of LPF H(s) (i.e., 1/ 0 ) can be chosen independently using the procedure in [81] to satisfy the accuracy in low frequency transients, which does not make the model stiff. Thereafter, in BPF G(s), 0 is set as the frequency of dominant oscillations in stator voltages. The damping ratio ζ is chosen to achieve a critically damped characteristic. Finally, to obtain the interfacing equations for the CPVBR model, (3–58) is transformed to phase coordinates using the inverse Park’s transformation as  1 2 1 2 1 2 1 2 1 2abc abc s s abc abc s abc abc abc abc s abc abc sr p   v i L i e . (3–63) In (3–63), 1 2abc abcL is a constant-parameter matrix calculated as 11 2 12( ) ( )abc abc r dq r  L K L K . (3–64) Equations (3–21), (3–24), (3–31)–(3–38), (3–47), (3–53), (3–57), (3–59) and (3–64) form the proposed CPVBR model of six-phase synchronous machine. This model is implemented using controlled voltage sources in series with magnetically-coupled but constant RL branches, as shown in Figure 3–4. As seen in Figure 3–4, the 69 1a se1b se1c se2a se2b se2c sesrsrsrsrsrsr1 2abc abcLCPVBR coupled interfacing circuitExternal networkEquations (3-24), (3-31)-(3-38), (3-47)(3-53), (3-57), (3-59) F(s)State-space model( )rK1( )rK12qd si12qd sv12qd sv12qd se1 2abc abc se1 2abc abc si 1 2abc abc sv Figure 3–4. Implementation of the proposed saturable CPVBR model with magnetically‐coupled constant‐parameter interfacing RL branches. proposed CPVBR model achieves a direct interfacing with the external network (without the need for artificial snubbers). The main advantage of this model (over the one depicted in Figure 3–2) is that its interfacing circuit has constant parameters. However, the interfacing inductance matrix 1 2abc abcL includes coupling terms among all branches. Therefore, implementation of this model is subject to availability of multi-phase coupled inductors in a given simulation program, which may be available in some programs (e.g. [41]) but not others (e.g. [40]). 70 3.4 Saturable Decoupled Constant‐Parameter VBR Model Here, a decoupled CPVBR (DCPVBR) model is derived for the six-phase synchronous machine to achieve magnetically-decoupled interfacing inductances in addition to having constant-parameters. However, due to the shared magnetizing flux between the two three-phase sets, this requires eliminating the mutual inductance terms between the two sets in the interfacing inductance matrix. This is done by re-formulating the sub-transient inductance matrix in (3–45), so that it has constant and equal terms in both q- and d-axes (based on unsaturated sub-transient inductances) as well as zero mutual terms between the three-phase sets. For this purpose, 12 ( , )dq z miL λ is decomposed as following 12 12 12( , ) ( , )dq z m sqd mqd z mi i   L λ L L λ , (3–65) where 12 ( , )mqd z miL λ is defined as 12 ( , )( , ) ( , ) ( , ) ( , )( , ) ( , ) ( , ) ( , )( , ) ( , ) ( ,mqd z mmqq z m mdu mqd z m mqq z m lm mqd z m ldqmqd z m mdd z m mdu mqd z m ldq mdd z m lmmqq z m lm mqd z m ldq mqq z miL i L L i L i L L i LL i L i L L i L L i LL i L L i L L i                L λλ λ λ λλ λ λ λλ λ λ4 4) ( , )( , ) ( , ) ( , ) ( , )mdu mqd z mmqd z m ldq mdd z m lm mqd z m mdd z m mduL L iL i L L i L L i L i L               λλ λ λ λ (3–66) and the constant decoupled sub-transient matrix 12sqdL is 120 0 00 0 0 , ( )0 0 00 0 0DDsqd D ls mdu lmDDLLL L L LLL          L . (3–67) 71 Using (3–65), the stator voltage equation (3–44) can be expressed in terms of 12 ( , )mqd z miL λ and 12sqdL as 1 11 1 11 11 1 1122 22 2 22 22 2 212 (d s d sq s q s q sq s q sd s d s d ss sqd rd s d sq s q s q sq s q sd s d s d smqd zv i iv i ir pv i iv i ii                                                      LL11 111 122 222 2, )d sq s q sq sd s d sm rd sq s q sq sd s d si ei epi ei e                               λ, (3–68) where 1 11 1122 22 2( , )d s q sq s d smqd z md s q sq s d siiiii                   L λ . (3–69) Next, the new sub-transient back-emf voltages are defined to include the terms corresponding to 12 ( , )mqd z miL λ , which contains saliency, variable saturation, and coupling terms, as 11 1 111 1 11222 2 222 2 2( , )d sq s q s q sq sd s d s d smqd z m rd sq s q s q sq sd s d s d se e ie e ii pe e ie e i                                                     L λ . (3–70) Based on (3–68), equation (3–66) can be rewritten as 72 1 11 1 11 11 1 1122 22 2 22 22 2 21 d s d sq s q s q sq s q sd s d s d ss sqd rd s d sq s q s q sq s q sd s d s d sq sv i iv i ir pv i iv i ie                                                       L1 2 2Td s q s d se e e   . (3–71) Calculating the new sub-transient back-emfs in (3–70) requires the derivatives of stator currents, which are obtained from (3–59). Similar to CPVBR model, the stator voltages are required for calculating the stator current derivatives in (3–59), which are needed to calculate the sub-transient back emfs in (3–70). Since the stator voltages in (3–71) have direct algebraic relation with the sub-transient back emfs in (3–70), an algebraic loop is created, making the stator voltage equation implicit. The explicit formulation is achieved according to the procedure explained in Section 3.3, where the algebraic loop is relaxed using filter approximation of stator voltages as in (3–60) with similar pole selection procedure. Finally, to obtain the DCPVBR model interfacing equations, (3–71) is transformed to phase coordinates using the six-phase Park’s inverse transformation matrix 1( )rK as   1 2 1 2 1 2 1 23 31 23 3 6 6 abc abc s s abc abc s D abc abc s abc abc sMabc abc sMr L pp       v i i eL 0 i0 L, (3–72) where 3 33mduMLL 1 . (3–73) 73 1a se1b se1c se2a se2b se2c sesrsrsrsrsrsrDLDLDLDLDLDL3mduL( )rK1( )rK12qd si12qd sv12qd sv12qd se1 2abc abc se1 2abc abc si 1 2abc abc sv3mduL Figure 3–5. Implementation of the proposed saturable DCPVBR model with magnetically decoupled constant‐parameter interfacing RL branches. Equations (3–72), (3–31)–(3–38), (3–47), (3–59), (3–60), (3–66), (3–70) and (3–21) form the proposed DCPVBR model of the six-phase synchronous machine, which also includes the effect of saturation with constant-parameter and decouple interfacing circuit, as shown in Figure 3–5. As seen in Figure 3–5, the resulting interfacing circuit is composed of constant and decoupled RL branches and controlled voltage sources. It is also worth mentioning that the interfacing circuit includes constant-parameter inductances in the neutral branch of each three-phase set (represented in (3–72) by ML ) which is a direct result of decoupling stator phases in the same three-phase set as explained in [70]. Neutral branch inductance is used only when external circuits are connected to the neutral points and zero sequence stator currents are present. As a result of the simple and decoupled interfacing circuit, this model can be conveniently implemented in most simulation 74 packages using simple and standard RL branches available in their built-in component libraries. 3.5 Computer Studies 3.5.1 Case Study System To investigate the numerical properties, the subject models, are evaluated on a case system shown in Figure 3–6 [59]. Therein, a 4-pole 100kVA six-phase synchronous generator is considered, wherein a 30 degrees phase shift between the two three-phase sets (ζ = 30°) is assumed. The generator is assumed to be driven by a turbine. The three-phase sets are assumed to be wye-connected with isolated and floating neutrals (resulting in no zero-sequence currents). The machine parameters are summarized in in Appendix B along with the saturation data. As depicted in Figure 3–6, the generator is interfaced to a three-phase grid (represented by voltage source vabcg ) through one of its sets abc1, while the second stator set abc2 is connected to a six-pulse diode rectifier that supplies a dc load (here represented by rload). Also, the three-phase lines represented by Rline and Lline are used to interface the abc1 and abc2 sets to external networks. The system of Figure 3–6 has been implemented in MATLAB/Simulink [39] and PLECS toolbox [41] using four different models (i.e., qd0, variable-parameter VBR, CPVBR, and DCPVBR). To interface the qd0 model with the external networks, the parallel snubber resistors have been used. The snubber resistance has been chosen to be 385Ω in order to limit the snubber branch currents to 1% of the nominal stator current. 75 dcCdcLloadrabcgvlineL2abc1abcmTlineR Figure 3–6. Benchmark system consisting of a six‐phase synchronous generator supplying ac grid and diode rectifier load. In the proposed CPVBR models, the poles of the filter F(s) (which consists of a first-order low-pass and a second-order band-pass filter [88]) are tuned to provide an accurate approximation of the stator voltages [i.e., 1q sv , 2q sv , 1d sv , and 2d sv in (3–60)] during steady-state and transients. Since one side of machine is connected to a six-pulse diode rectifier, the stator voltages and currents are also under constant switching transients containing dominant 5th and 7th harmonics. These 5th and 7th harmonics correspond to 360Hz oscillations in voltages and currents when transformed to the qd0 reference frame. Therefore, according to [81], [88], to limit the solution error to 1%, the pole of the low-pass filter is chosen to be –2300, and the band-pass filter is tuned for 2262 rad/s (corresponding to 360 Hz) with critically damping (factor = 0.7). Simulation studies are executed on a PC with Intel® Core™ i7-4700 @ 2.4GHz processor. For the purpose of consistency, all models are run using MATLAB variable time-step solver ode45 with relative and absolute error tolerances set to 10–4. For accuracy comparisons, a reference solution is obtained by simulating the variable-76 parameter VBR model (which does not have artificial snubbers or approximation filters) with small error tolerances of 10-7 and maximum time-step size of 10-5 s. 3.5.2 Transient Simulation The effect of including saturation in the models is demonstrated first. The generator is assumed to be supplied with constant mechanical torque of Tm = 500 Nm. Initially, the three-phase grid voltage is set to 333V (corresponding to 0.9 pu), and the field winding is excited with a voltage corresponding to 1xfd r mdu fd fde L v r  = 339V [27]. Under this operating condition, the generator is supplying 52 kW at 0.96 PF to the three-phase grid, and 42 kW to the diode rectifier load (represented by rload = 5.5 Ω). Then, at t = 1 s, the grid voltage is increased to 407V (1.1 pu), and the simulations are continued until t = 4 s. This 4-second transient study is used to evaluate the performance of the considered models. To further visualize the change of operating point in the this study, the machine’s magnetization characteristics used in the saturable model is also depicted in Figure 3–7, where the two operating points before and after the grid voltage increase are highlighted (by “×” and “О”). As seen in Figure 3–7, an increase in applied voltage moves the operating point further into the saturated region. Transient response of several variables as predicted by the subject models are shown in Figure 3–8. For comparison, transient responses produced the magnetically-linear machine model [59] are also presented in Figure 3–8. As can be seen in Figure 3–8(a)-(c), the increase of grid voltage vabcg results in increase of the stator currents iabc1s and iabc2s, as well as increase of the main magnetizing flux. It is 77 also observed that in the saturable model the amount of flux increase is smaller due to saturation and reduced effective magnetizing inductance compared to the linear model. Figure 3–7. Assumed magnetization curve depicting unsaturated and saturated regions for two operating points. Finally, Figure 3–8(d)–(e) show the transient response in electromagnetic torque and mechanical speed. Since the input power remains the same in the new steady state, the output real power also redistributes (rectifier power to 57 kW, and grid power to 37 kW, respectively). To have a closer look at the effect of saturation on machine variables, a magnified view of the machine stator phase currents a1 and a2 as well as the electromagnetic torque is presented in Figure 3–9. As seen in Figure 3–9, the current flowing through stator phase a1 (which is connected to the grid) [see Figure 3–9(a)] increases in magnitude approximately by 30% compared to the model without saturation; and the current in phase a2 that feeds the rectifier [see Figure 3–9(b)] is not significantly impacted by the including/excluding the effect of saturation. Also, 0 90000.8650.980im (A) m (Wb)saturatedmagnetization|vgabc|=1.1pu|vgabc|=0.9pu337 412unsaturatedmagnetization78 small differences are visible in electromagnetic torque as observed in Figure 3–9(c). For more detailed analysis, a further magnified view of the stator phase currents a1 and a2 as well as the electromagnetic torque are illustrated in Figure 3–10. Figure 3–8. Transient response of several variables due to grid voltage increase from 0.9 to 1.1pu, as predicted by the subject models: (a) a1 phase current, (b) a2 phase current, (c) magnetizing flux, (d) electromagnetic torque, and (e) rotor mechanical speed. 0200400i sa1 (A)0150i sa2 (A)0.870.941.01λ m (Wb)00250.50.751.0T e (kN.m)1 1.14 1.28 1.42 1.56 1.7186188190Time (s)ω m (rad/s)(II) With saturation(I) Without saturationd)c)b)e)a)(I) (II)See Fig. 8(c)See Fig. 8(b)See Fig. 8(a)---See Fig. 3–9(a)See Fig. 3–9(b)Se Fig. 3–9 )i a1s (A) i a2s (A) 79 Therein, the results obtained with the subject models are compared to the reference solution trajectory to demonstrate the accuracy of various models. As it can be observed in Figure 3–10, the qd0 model has a considerable deviation (i.e. due to interfacing snubbers). Figure 3–9. Magnified view of selected variables as predicted by subject models: (a) a1 phase current, (b) a2 phase current, and (c) electromagnetic torque. In contrast, the variable parameter VBR model, the proposed CPVBR and DCPVBR models all demonstrate very good accuracy, with the last two models having small deviations from the reference solution (which is due to approximations used in these models). 0150i sa1 (A)0120i sa2 (A)1.69 1.7 1.71470500530Time (s)T e (N.m)(II) With saturation:DCPVBRVBRqd0(I) Without saturation(I)(II)(II)See Fig. 9(c)c)(II)CPVBRb) (I)See Fig. 9(b)(I)See Fig. 9(a)a)--ee Fig. 10 ig. 10(ee Fig. 10See Fig. 3– )e Fig. 3–10(b)See Fig. 3– )i a1s (A) i a2s (A) 80 3.5.3 Numerical Accuracy Comparison To more precisely evaluate accuracy of each model, the numerical error with respect to the reference solution in the stator phase a1 current 1a si and electromagnetic torque eT as obtained by the subject models are compared. The 2- Figure 3–10. Further magnified view of selected variables as obtained by the subject saturable models for: (a) stator a1 phase current, (b) stator a2 phase current and (c) electromagnetic torque. 1.6993 1.6997 1.7001150160170Time (s)i sa1 (A)1.6877 1.6881 1.6885110115120Time (s)i sa2 (A)1.6987 1.6991 1.6995505515525Time (s)T e (N.m)c)Reference VBRb)a)CPVBR DCPVBRqd0i a1s (A) i a2s (A) 81 norm cumulative errors [82] of the solution trajectories over the 4-second study are summarized in Table 3–1. As seen in Table 3–1, the qd0 model has the largest numerical error (i.e., 0.92% in 1a si and 0.85% in eT ), which is due to the current in the snubber circuit. This error is below 1% due to the fairly large value of the snubber resistors, which was the choice. The variable parameter VBR model is shown to be the most accurate (i.e., 0.01% error in 1a si and 0.01% error in eT ), since it does not use approximations in deriving the interfacing circuit. The small in this model are mostly due to the numerical discretization with larger time-steps compared to the reference solution. It is also seen in Table 3–1 that the proposed CPVBR model is able to achieve very good accuracy (i.e., 0.19% error in 1a si and 0.21% error in eT ). Also, the proposed DCPVBR model is able to predict the solution with very good accuracy (i.e., 0.38% error in 1a si and 0.29% error in eT ); however, it introduces slightly more error compared to the CPVBR model due to additional approximation for decoupling the RL interfacing branches. TABLE 3–1. COMPARISON OF 2‐NORM ERROR OBTAINED BY THE SUBJECT MODELS FOR THE 4 SECOND STUDY Model Stator phase a1 current (ia1s) Electromagnetic torque (Te) qd0 model (with snubbers) 0.92 % 0.85 % VBR model (with variable inductances) 0.01 % 0.01 % Proposed CPVBR model 0.19 % 0.21 % Proposed DCPVBR model 0.38 % 0.29 % 82 3.5.4 Computational Performance Comparison Here, the computational performance of the subject models are compared for the 4-second transient presented in Section 3.5.1. For this purpose, the total CPU time taken for simulation, the number of time steps, and the average CPU time per-step are summarized in Table 3–2 for all subject models. Since snubbers in qd0 model and approximations in CPVBR and DCPVBR models make these models numerically stiffer, their corresponding eigenvalues are also compared in Table 3–2. TABLE 3–2. COMPUTATIONAL PERFORMANCE COMAPRISONS OF THE SUBJECT MODELS FOR THE 4 SECOND CASE‐STUDY Model Largest eigenvalue Number of steps CPU time per step CPU Time qd0 model (with snubbers) –1.02×106 1,704,804 118 µs 202 s VBR model (with variable inductances) –698 16,780 1955 µs 32.8 s Proposed CPVBR model –1,922 19,627 352 µs 6.9 s Proposed DCPVBR model –3,912 36,050 197 µs 7.1 s As can be observed in Table 3–2, the qd0 model takes the longest CPU time (202 s) and requires the largest number of time steps (1,704,804) due to high stiffness of the model (largest eigenvalue of –1.02×106) caused by large snubber resistors. However, the qd0 model requires the smallest CPU time per-step (118 µs) due its simplicity. Meanwhile, the variable parameter VBR model is not stiff (largest eigenvalue of –698) and requires significantly fewer time-steps (16,780) and less CPU time (32.8 s) compared to the qd0 model. However, the variable interfacing inductance matrix 83 necessitates additional calculations at every time-step, leading to the larger average CPU time per-step (1955 μs). It can also be observed in Table 3–2 that the proposed CPVBR model significantly outperforms the qd0 and VBR models in terms of CPU time (6.9 s) while taken slightly larger number of time-steps (19,627) compared to the VBR model. The faster simulation time of CPVBR model is due its constant interfacing circuit which also results in smaller CPU time per-step (352 µs). The largest eigenvalue in the CPVBR model (–1,922) is associated with the pole of the filter used in approximation (3–59). Moreover, the proposed DCPVBR model offers similar speed compared to the CPVBR model (i.e., 7.1 s vs. 6.9 s). However, the DCPVBR model requires larger number of time-steps (36,050 vs. 19,627) due to its increased stiffness (largest eigenvalue of –3,912 vs. –1,922). This increased stiffness of the DCPVBR model is attributed to a larger portion of the interfacing matrix being approximated (64). It is also noted that due to its simpler interface, the DCPVBR model requires smaller average CPU time per-step compared to the CPVBR model (i.e., 197 µs vs. 352 µs). 84 CHAPTER 4: CONSTANT‐PARAMETER VOLTAGE‐BEHIND‐REACTANCE MODEL OF FIVE‐PHASE SYNCHRONOUS MACHINES WITH AIR‐GAP FLUX HARMONICS In this chapter, the VBR modeling approach is adopted for a salient rotor five-phase synchronous machine and a variable-parameter VBR model of salient rotor five-phase synchronous machines is developed for the first time, which considers air-gap flux harmonics up to third component. Moreover, the modeling approach considers rotor circuits, including damper and field windings. Based on the developed variable-parameter VBR model, a numerically efficient CPVBR model of five-phase synchronous machines is proposed using algebraic-loop relaxation method in [71]. This model offers a simple and constant-parameter RL branch interfacing circuit in abcde phase coordinates. The modeling approach considers cross-couplings between the fundamental and third harmonic components of the machine inductances in its formulation, which improves the ability of the model to 85 predict machine dynamics accurately with respect to experimental measurements obtained from hardware setups. The proposed CPVBR model of five-phase synchronous machine is validated by extensive experimental measurements and simulation studies, and is shown to offer accurate results with superior simulation performance compared to the CCPD, qd0 and the proposed VBR models. It is envisioned that the proposed CPVBR model will become a valuable asset in many simulation packages for design and analysis of machine-converter systems consisting of five-phase synchronous machines (e.g., various drive applications where third harmonic current injection is done for improved machine dynamics [9]). Here, a salient pole five-phase synchronous machine is considered where the stator phases are displaced by 72 degrees. Without loss of generality, the machine rotor is assumed to have one damper winding in each of the q- and d-axes as well as a field winding in the d-axis. A cross-section of the subject five-phase machine is depicted in Figure 4–1. In addition, up to third harmonic component for the stator winding functions and air-gap functions are considered, which is proven to provide acceptable accuracy in representing machine dynamics [55]. It is also assumed that the third harmonic component of air-gap and winding functions may not necessarily be aligned with the fundamental component. All machine parameters are referred to the stator side and motor sign convention is used. For consistency with [27] and prior literature on the subject (e.g., [71]), matrices, vectors and instantaneous quantities/variables are denoted using bold upper-case, bold lower-case and lower case italic letters, respectively. 86 abed  cabc dekqkd fd kdfdrkq Figure 4–1. A simplified cross‐section of a salient pole five‐phase synchronous machine. 4.1 Couple Circuit Phase Domain Model The CCPD model of the five-phase synchronous machine is derived by expressing the machine stator and rotor voltages in terms of the machine currents using the self and mutual inductances between the stator and rotor windings and their corresponding resistances as [55] 8 88 1 8 1 8 100abcdes abcdes abcdessqdr qdr qdrrp                     v i λrv i λr , (4–1) 8 88 1 8 1( ) ( )( )abcdes abcdess r sr rqdr qdrrs r r               λ iL Lλ iL L . (4–2) Here, rs and rr are the stator and rotor resistance matrices as 87 5 5s sr r I , 0 00 00 0kqr kdfdrrr      r , (4–3) where I is identity/unit matrix. Also, in (4–1) lower-case p is the Heaviside’s operator used to denote the time derivative d/dt. Moreover, in (4–2) the stator inductance matrices are obtained using the approach in [55], considering all coupling and cross-coupling terms between the machine air-gap and winding functions for both fundamental and third harmonic components. The stator inductance matrix in (4–2) is expressed as ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )a r ab r ac r ad r ae rab r b r bc r bd r be rs r ac r bc r c r cd r ce rad r bd r cd r d r de rae r be r ce r de r e rL L L L LL L L L LL L L L LL L L L LL L L L L                              L . (4–4) where ( )a rL  , ( )ab rL  and ( )ac rL  are defined as 1 3 1 33 13 13( ) cos(2 ) cos(6 ) sin(6 ) 2 cos(2 ) 2 sin(2 ),a r ls A A B r B rC r D r E rL L L L L LL L L           1 3 13 313132 6 2( ) cos( ) cos( ) cos(2 )5 5 56 6 cos(6 ) sin(6 )5 56 2 cos(2 ) cos(2 )5 56 2 sin(2 ) sin(2 ) ,5 5ab r A A B rB r C rD r rE r rL L L LL LLL                            88 1 3 13 313134 12 4( ) cos( ) cos( ) cos(2 )5 5 512 12 cos(6 ) sin(6 )5 512 4 cos(2 ) cos(2 )5 512 4 sin(2 ) sin(2 ) .5 5ac r A A B rB r C rD r rE r rL L L LL LLL                            (4–5) and the inductances of other stator phases are defined as ( ) ( 2 5), ( ) ( 4 5),( ) ( 6 5), ( ) ( 8 5),( ) ( 5), ( ) ( 3 5),( ) ( 2 5), ( ) ( 2 5),( ) ( 3 5), ( )b r a r c r a rd r a r e r a rad r ac r ae r ab rbc r ab r bd r ac rbe r ac r cd rL L L LL L L LL L L LL L L LL L L                                      ( 4 5),( ) ( 4 5), ( ) ( 6 5).ab rce r ac r de r ab rLL L L L          (4–6) In (4–4), the rotor angle r is considered to be aligned with the fundamental component of the machine self-inductance in stator phase a. Using a similar approach, the mutual inductances between rotor and stator windings in (4–2) are also defined as ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )Takq r bkq r ckq r dkq r ekq rsr r akd r bkd r ckd r dkd r ekd rafd r bfd r cfd r dfd r efd rL L L L LL L L L LL L L L L                   L . (4–7)  2( ) ( )5Trs r sr r L L . (4–8) where ( )akq rL  and ( )akd rL  are defined as 89         13 1 13 313 3 1313 1 13 313 3 13( ) cos( ) cos(3 ) sin( ) sin(3 ),( ) sin( ) sin(3 ) cos( ) cos(3 ).akq r m mq r m mq rmqd r mqd mqd rakd r m md r m md rmqd r mqd mqd rL L L L LL L LL L L L LL L L                 (4–9) and are used to derive mutual inductances between the rotor windings and stator phases b, c, d and e as ( ) ( 2 5), ( ) ( 4 5),bx r ax r cx r ax rL L L L         ( ) ( 6 5), ( ) ( 8 5), ( , ),dx r ax r ex r ax rL L L Lx kq kd         ( ) ( ), ( , , , , ).yfd r ykd rL L y a b c d e   (4–10) In (4–5), (4–9) and (4–12) Lls, Llkq, Llkd and Llfd are the stator and rotor leakage inductances. Moreover, the following relations between inductances in (4–5), (4–9) are defined as    1 1 1 1 1 10.2 , 0.2A md mq B md mqL L L L L L    ,    3 3 3 3 3 30.2 , 0.2A md mq B md mqL L L L L L    , 3 3 13 13 13 130.4 , 0.4 , 0.4C mqd D m E mqdL L L L L L   . (4–11) In (4–11), 1AL and 1BL are the average and saliency level of the machine inductance corresponding to the fundamental component, and 3AL and 3BL denote similar quantities (i.e., average and saliency level) corresponding to the third harmonic component, respectively. Also, 3CL , 3DL , and 13EL are inductive terms corresponding to cross-coupling for third harmonic component, direct coupling 90 between fundamental and third harmonic components, and cross-coupling between fundamental and third harmonic components, respectively. Finally, the rotor inductances are defined as 0 00 ,0kqr kd mdmd fdLL LL L      L (4–12) where 13 1 313 1 32 , 2 ,, .kq lkq m mq mqmd m md mdkd lkd md fd lfd mdL L L L LL L L LL L L L L L         (4–13) The electromagnetic torque of the five-phase machine is defined in terms of the machine currents and spatial derivatives of machine inductances as    5 5( ) ( )1( ) 2 2T Ts r sr re r abcdes ls abcdes abcdes qdrr rPT L                   L Li I i i i .(4–14) where the upper-case P is the number of machine poles. The equations (4–1)–(4–12) and (4–14) constitute the CCPD model of the five-phase synchronous machine which can be implemented and interfaced directly to any arbitrary network. However, it is noted that the interfacing inductance matrix in (4–2) is variable and rotor-position-dependent (indicated as functions of r ). This requires re-calculation of the inductance matrix at each time-step, which creates a significant computational burden due to the size and complexity of the matrix. 91 4.2 qd0 Model The qd0 model of the subject five-phase synchronous machine is derived by transforming variables in the machine CCPD model to the rotor reference frame as 130 5 ( )qd ph r abcdef K f , (4–15) where 5 ( )ph rK is Park’s five-phase transformation matrix [54] defined as 52( ) 52 4 4 2cos( ) cos( ) cos( ) cos( ) cos( )5 5 5 52 4 4 2sin( ) sin( ) sin( ) sin( ) sin( )5 5 5 56 2 2 6cos(3 ) cos(3 ) cos(3 ) cos(3 ) cos(3 )5 5 5 56 2sin(3 ) sin(3 ) sin(35 5ph rr r r r rr r r r rr r r r rr r r                                  K2 6) sin(3 ) sin(3 )5 51 2 1 2 1 2 1 2 1 2r r                .(4–16) and [ , , , , ]Tabcde a b c d ef f f f ff , (4–17) 130 1 1 3 3 0[ , , , , ]Tqd q d q df f f f ff . (4–18) Here, abcdef and 130qdf may represent voltage, current, or flux vectors in stator phase and rotor reference frame, respectively. It is noted that q1-d1 frame corresponds to fundamental component of variables, while q3-d3 frame corresponds to the 3rd harmonics. For clarity and distinction from conventional three-phase machine model, the qd0 model of the five-phase synchronous machine is denoted as qd130 in order to indicate coordinates in both fundamental and third harmonic 92 components. Transforming the machine voltage and flux equations in (4–1) and (4–2) using (4–15), the fundamental and third harmonic components of stator voltage can be expressed in rotor reference frame as 1 1 1 1q s s q s q s r d sv r i p     , (4–19) 1 1 1 1d s s d s d s r q sv r i p     , (4–20) and 3 3 3 33q s s q s q s r d sv r i p     , (4–21) 3 3 3 33d s s d s d s r q sv r i p     , (4–22) 0 0 0s s s sv r i p  , (4–23) where r is the synchronous speed in rad/s. Also, the rotor voltages can be expressed as 0 kq kq kqr i p  , 0 kd kd kdr i p  , fd fd fd fdv r i p  . (4–24) In (4–19)–(4–24), the stator fluxes are defined as 1 1 1q s mq ls q sL i   , 1 1 1d s md ls d sL i   (4–25) 3 3 3q s mq ls q sL i   , 3 3 3d s md ls d sL i   (4–26) 0 0s ls sL i  , (4–27) and the rotor fluxes are 1 3kq mq mq lkq kqL i     , (4–28) 1 3kd md md lkd kdL i     , 1 3fd md md lfd fdL i     . (4–29) In (4–25)–(4–29), the magnetizing flux components are 93 1 11 11 13 1 11 133 13 33 3 13 334 4 4 23 3mq q skqmd m m d s m mkdT Tmq m m q s m mfdmd d siiiiiii                                         L L L L αL L L L . (4–30) where 111100mqmmdLL    L , 13 131313 13m mqdmmqd mL LL L    L , (4–31) 3 3333 3mq mqdmmqd mdL LL L    L , 1 0 00 1 1    α. (4–32) The electromagnetic torque of the machine can be derived using the machine stator currents and magnetizing fluxes as  1 1 1 1 3 3 3 35 3 32 2e q s md d s mq q s md d s mqPT i i i i       . (4–33) The machine voltage and flux equations in (4–19)–(4–32) form the qd130 model of the five-phase synchronous machine, where its equivalent circuit in rotor reference frame is depicted in Figure 4–2. As seen in Figure 4–2, the qd130 model [(4–19)–(4–33)] is implemented using stator and rotor fluxes as state variables, stator and rotor voltages as inputs, and stator and rotor currents as outputs. This model is interfaced with external circuits using controlled current sources, which require snubber circuits in case of inductive and power electronic networks. 94 fdifdrlfdL1mdLkdrlkdLfdvkdi0sisr0svlsL3mdL1r q s sr lsL33 r q s sr lsL1d sv3d sv1d si3d si 13mL1r d s sr lsL33 r d s sr lsL1q sv3q sv1q si3q si1mqL3mqL13mL13mqdL3mqdL13mqdLkqilkqL kqr Figure 4–2. The qd130 equivalent circuit of the five‐phase synchronous machine in rotor reference frame, including coupling and cross‐coupling between fundamental and third harmonic components. 4.3 Variable‐Parameter VBR Model Based on the qd130 equivalent circuit, the variable-parameter VBR model of the synchronous five-phase machine is derived by replacing the rotor circuit with sub-transient inductances in conjunction with back-emf voltage sources, while representing the stator circuit in stator phases. Therein, the rotor fluxes are state 95 variables, stator currents and rotor voltages are inputs, and the sub-transient back-emf voltages are the output of the model. To obtain the rotor state equation for the VBR model, rotor flux derivatives are expressed in terms of rotor currents, voltages and magnetizing fluxes based on (4–24) as 113300mqkq kqmdkd r lr kdmqfd fd fdmdpv                                               r Γ β , (4–34) where 1 0 00 1 00 0 1lkqlr lkdlfdLLL      Γ , 1 0 1 00 1 0 10 1 0 1      β . (4–35) Substituting rotor currents in (4–30) from (4–28)–(4–29), the magnetizing flux vector in (4–34) can be written in terms of rotor fluxes and stator currents as 1 11 13 33 3mq q skqmd d sm A kdmq q sfdmd d siiii                               L L , (4–36) where the sub-transient inductance matrix mL is defined as 1 1 13 1311 1 13 1311 13 11 134 431 31 3 313 33 13 334 2 4 431 31 3 3mq mqd mq mqdmdq md mdq mdm m m mm lrT Tmq mqd mq mqdm m m mmdq md mdq mdL L L LL L L LL L L LL L L L                                   L L L LL I αΓ βL L L L. (4–37) 96 and AL is a 4×3 matrix expressed as 111 13 11 134 413 33 13 33m m m mA lr lrT Tm m m m               L L L LL I αΓ β αΓL L L L . (4–38) Furthermore, writing the magnetizing flux vector in (4–34) in terms of state variables and inputs using (4–36), the state equation is derived as  113 33300q skq kqd skd r lr A kd mq sfd fd fdd siipivi                                             r Γ I βL βL . (4–39) Substituting the stator flux in stator voltage equations (4–19)–(4–23) with the ones defined in (4–25)–(4–27), the stator voltage equations in q- and d-axes for fundamental and third harmonic components can be written as 1 1 1 1 111 1 1 13 3 5 5 3 3 333 3 3 30 0 0 0 0q s q s q s d s mqq sd s d s d s mdq s s q s ls q s r d s mqq sd s d s d s mds s sv i iv i iv r i L p i pv i iv i i                                                                      I , (4–40) where stator fluxes are calculated using (4–25)–(4–26). In (4–40), the magnetizing flux derivatives in q- and d-axes can be calculated using (4–36) as 1 11 13 33 3mq q skqmd d sm A kdmq q sfdmd d siip p pii                               L L , (4–41) Replacing magnetizing flux derivatives in (4–40) with (4–41), and after algebraic 97 manipulations, the stator voltage equations can be written in the VBR form as 1 1 1 11 1 1 14 4 4 13 3 3 31 4 5 53 3 3 30 0 01 10 q s q s q s qd s d s d s dls mq s s q s q s qlsd s d s d s ds s sr d qv i i ev i i eLv r i p i eLv i i ev i i                                                               I L 003 3 0 Td q  , (4–42) where the sub-transient fluxes are defined as  1 11 14 43 33 3q q sd d sls mq q sd d siiL ii                    I L , (4–43) and the sub-transient back-emfs are 1 111 113 333 33d s dqkqq s qdr A kdd s dqfdq s qdeepee                                        L . (4–44) In (4–44), the rotor flux derivatives are calculated using the state equation in (4–39) in terms of state variables (rotor fluxes) and inputs (stator currents and rotor voltages). Applying Park’s inverse five-phase transformation defined as 15 13015 5( )5( ) ( )2abcde ph r qdTph r ph r f K fK K , (4–45) the stator interfacing circuit of the VBR model can be finally derived in abcde phase coordinates as 98 aebecedeeesr( )abcde rL5 ( )ph rK15 ( )ph rK 13qd si13qdeabcdeeabcdesisrsrsrsr Figure 4–3. Implementation of the proposed variable‐parameter VBR model of the five‐phase synchronous machine with coupled and rotor‐position dependent inductances, directly interfaced with the external network. ( )abcdes s abcdes abcde r abcdes abcder p   v i L i e , (4–46) where the variable-parameter sub-transient inductance matrix is defined as 4 4 4 115 51 4 5 5( ) ( ) ( )ls mabcde r ph r ph rlsLL         I L 0L K K0 . (4–47) The electromagnetic torque in the VBR model is identical to the qd130 model and can be directly calculated from (4–33). The state model equations in (4–36)–(4–44), and stator interfacing equation (4–46) constitute the proposed variable-parameter VBR model of five-phase synchronous machine. This model is implemented as shown in Figure 4–3, which can be directly interfaced with external network without requiring any auxiliary snubber circuits. However, it is noted that the variable-parameter inductance matrix in (4–47) is still rotor-position-dependent, which can have adverse impact on the simulation performance since the machine inductances need to be re-calculated at every time-step. 99 4.4 Constant‐Parameter VBR Model To achieve a more computationally efficient VBR model, a constant-parameter interfacing circuit is required. Variable-parameter inductances in phase coordinates are due to salient and coupling terms in the machine inductance matrix in rotor qd coordinates (i.e., equation (4–37)). Thus, to achieve constant parameter interfacing, salient and coupling inductance terms from the machine inductance matrix in rotor qd coordinates are separated and their effects are considered as voltage terms, included in the sub-transient voltage sources. The sub-transient inductance matrix in (4–37) is re-formulated into two parts expressed as _m m m res   L L L , (4–48) where the non-salient and magnetically decoupled inductance matrix mL is defined based on d-axis inductances as 11330 0 00 0 00 0 00 0 0mdmdmmdmdLLLL        L , (4–49) and the residual inductance matrix _m resL (including saliency and coupling terms) is defined as 1 1 1 13 131 13 13_31 31 3 3 331 31 300mq md mqd mq mqdmdq mdq mdm resmq mqd mq md mqdmdq md mdqL L L L LL L LL L L L LL L L                     L . (4–50) The non-salient and decoupled inductance matrix mL is used to derive a new 100 constant-parameter interfacing inductance matrix in phase coordinates, and the residual inductance matrix _m resL is represented as modified voltage sources and included into the sub-transient back-emf voltages. For this purpose, by substituting mL in (4–42) with the one in (4–48) using (4–49) and (4–50), the stator voltage equation in (4–42) is rewritten as 1 1 11 1 14 4 4 13 3 31 43 3 30 0 01 1 3 31 0 q s q s q sd s d s d sls mq s s q s q slsd s d s d ss s sTr d q d qqv i iv i iLv r i p iLv i iv i ie e                                                    I L 001 3 31 1 11 114 4 _ 4 13 3 31 4 3 330 ,0 0Td q dq s d dq qd sls m resrq s d dlsq qd se eiiLp iLi                                            I L 00 (4–51) where the new sub-transient fluxes are  1 11 14 43 33 3q q sd d sls mq q sd d siiLii                   I L . (4–52) Defining the new sub-transient back-emf voltages as 1 11 1 11 11 1 1_3 33 3 33 33 3 3d dq q q sq qd d d sm res rd dq q q sq qd d d se e ie e ipe e ie e i                                                                L , (4–53) the stator voltage equation can be written as 101 1 1 1 11 1 1 14 4 4 13 3 3 31 43 3 3 30 0 0 0 q s q s q s qd s d s d s dls mq s s q s q s qlsd s d s d s ds s srv i i ev i i eLv r i p i eLv i i ev i i                                                              I L 001 1 3 3 0 Td q d q        . (4–54) As can be seen in (4–53), calculating the new sub-transient back-emfs requires stator current derivatives in q- and d-axes for both fundamental and third harmonic components which can be calculated using the stator voltage equation in (4–42) as  11 1 11 11 1 14 433 3 333 3 31 1 dq s q s q sqd s d s d sls m s rdq s q s q sqd s d s d sq d qi v ii v ip L ri v ii v ie e e                                                   I L3 3 Tde  . (4–55) It is observed that the stator current derivatives in (4–55) have a direct algebraic relation with stator voltages. Also, it is noted that the stator voltages in (4–54) have a direct algebraic relation with the new sub-transient back-emf voltages defined in (4–53), which are themselves calculated using stator current derivatives in (4–55). As a result, an algebraic-loop is created and the stator voltage formulation becomes implicit. In order to achieve explicit formulations, the algebraic relaxation method [71] is applied where a tuned filter is used to provide accurate approximations of stator voltages as 102 1 11 13 33 3( )q s q sd s d ssq s q sd s d sv vv vF sv vv v                    v , (4–56) which can then be replaced with stator voltages in (4–55) for calculating stator current derivative. The pole selection and tuning procedure for the filter ( )F s is done according to the approach presented in [81], [88]. Finally, the stator interfacing equation in abcde phase coordinates is derived applying Park’s inverse five-phase transformation (4–45) to stator voltage equation in (4–54) which results in abcdes s abcdes abcde abcdes abcder p   v i L i e . (4–57) Here, the sub-transient interfacing inductance matrix in phase coordinates abcdeL is constant-parameter (as a result of mL being non-salient and decoupled) and is defined as 4 4 4 115 51 4 5 5( ) ( )ls mabcde ph r ph rlsLL        I L 0L K K0 . (4–58) The proposed CPVBR model of the five-phase synchronous machine is implemented using the rotor state equation defined in (4–39), new sub-transient back-emfs in (4–53) and stator interfacing equation in (4–57), shown in Figure 4–4. 103 aebecedeeesrabcdeL5 ( )ph rK15 ( )ph rK 13qd si13qdeabcdeeabcdesisrsrsrsrabcdesv13qd svsv Figure 4–4. Implementation of the proposed CPVBR model of the five‐phase synchronous machine with coupled and constant RL branches, directly interfaced with the external network. 4.5 Model Validation and Computer Studies In order to validate the proposed models, an experimental prototype 12-pole five-phase synchronous machine with wye-connected stator windings and one rotor field winding is considered. The laboratory setup and its circuit schematic are demonstrated in Figure 4–5. Therein, the five-phase machine is used as a generator, which supplies a ten-pulse diode rectifier load through five identical lines represented with rline resistances. Mechanical power is provided using a dc motor which drives the five-phase machine at constant mechanical speed. The system parameters including the five-phase machine, line and diode-rectifier are presented in Appendix. The five-phase machine CCPD, conventional qd130, variable-parameter VBR and CPVBR models presented in Sections 4.1–4.4, respectively are implemented in 104 MATLAB/Simulink [39] using PLECS toolbox [41]. For consistency, all simulations are conducted on a PC with Intel® Core™i7-8700K @ 3.7GHz processor using the MATLAB ode45 variable time-step solver with both absolute and relative error tolerances set to 10-4. For comparison purposes, a reference solution is obtained using the CCPD model, which is simulated at a very small time-step of 1μs. In order to interface the qd130 model with the ten-pulse diode rectifier, snubber resistors are connected in parallel to each of the stator phases. The snubber resistance value is chosen according to the machine operating condition (detailed in the following case studies) to limit the snubber branch current to 1% of the machine phase current. In the proposed CPVBR model, the filter F(s) consists of a low-pass filter and a band-pass filter as presented in [81], [88], and is used to provide an accurate approximation [i.e., equation (4–56)] of stator voltages in the rotor reference frame. Since the machine is connected to a ten-pulse five-phase diode rectifier, dominant 7th, 9th, 11th and 13th harmonic components are always present on the machine ac variables, which result in oscillations with a frequency corresponding to 10 r on transformed variables in rotor reference frame. Therefore, for accurate approximation of stator voltages in (4–56), the band-pass filter in F(s) is designed to have a resonance frequency equal to 10 r and to be critically damped (i.e., damping factor equal to 0.7). Moreover, to preserve approximation accuracy at low frequency machine dynamics up to 100Hz [81], the low pass filter pole in F(s) is chosen to be –1300. The pole selection procedures are explained in detail in [81], [88]. 105 4.5.1 Steady State Study and Model Validation In order to validate the ability of the of the proposed mathematical models in predicting correct machine dynamics, the reference solution is compared to experimental measurements from a laboratory five-phase synchronous machine. Comparisons are carried out under both no-load and loaded operating conditions for different machine variables. Figure 4–5. The considered case study system: (a) experimental laboratory setup, (b) its corresponding circuit schematic. a)b)a) 106 Under no-load conditions, the five-phase machine is rotating at a constant speed of 2000 rpm (200 Hz stator frequency) and the field winding is supplied by a 4.24 V voltage source (0.53V when referred to stator side). Waveforms for line voltages vabs and vacs under no-load condition obtained from experimental measurements and simulations of the reference model are presented in Figure 4–6. As it can be seen in Figure 4–6, the reference model is able to produce accurate results well matched to experimental measurements. Specifically, it is observed that the five-phase machine provides line voltages with two different amplitudes as expected in a balanced five-phase system. Figure 4–6. Steady‐state results obtained from experimental measurement and the reference model simulation under no‐load operating condition for the machine line voltages: (a) vabs and (b) vacs. -24-1201224Time (s)v sab (V)0.322 0.324 0.326 0.328-50-2502550Time (s)v sac (V)(II) Reference model simulation(I) Experimental measurmentb)(I)(II)(II)(I)a)v abs (V) v acs (V) 107 It is also worth mentioning that the waveforms obtained by experiments contain higher order harmonic components which are not considered in the subject models. Such high order harmonics have much smaller amplitudes compared to the dominant fundamental and third harmonic component (thus having minimal impact on waveforms seen in Figure 4–6), and are generally neglected for most applications [9]. To compare the simulation results with experimental measurements under loading conditions, the machine is set under the same prime mover and excitation conditions as Figure 4–6, and the stator phases deliver a total power of 510 W to a ten-pulse diode rectifier supplying a 1 Ω dc load. Waveforms for line voltages vabs and vacs and line current ias obtained from hardware measurements as well as the reference model simulation are shown in Figure 4–7. As observed in Figure 4–7, the reference model is able to provide highly accurate predictions with respect to experimental measurements even under non-linear loading conditions (e.g., diode rectifiers which include switching and commutations), by considering the fundamental and third harmonic components (and their cross-couplings) in the five-phase machine model. 108 Figure 4–7. Steady‐state results obtained from experimental measurement and the reference model simulation, for the machine line voltages and current: (a) vabs, (b) vacs and (c) ias, when the machine is loaded with a ten‐pulse diode rectifier supplying a 1Ω dc load. 4.5.2 Transient Study For the purpose of numerical accuracy and computational performance comparison between the subject machine models (presented in Section 4.1–4.4), a 2 second transient simulation study is conducted. Therein, the machine is under the same prime mover and excitation conditions as Figure 4–6, and the stator phases is connected to a ten-pulse diode rectifier which is initially supplying a relatively small 0.322 0.324 0.326 0.328-12-60612Time (s)i sa (A)-30-1501530v sac (V)-20-1001020v sab (V)(I)(II)(II)(I)(I) (II)a)c)b)(II) Reference model simulation(I) Experimental measurmentv abs (V) v acs (V) i as (A) 109 dc load (160 W corresponding to rload = 15 Ω). At t = 1 s, the rectifier load is increased to 510 watts (corresponding to rload = 1 Ω) using a step function. In this transient study, the snubber resistors in the qd130 model is chosen as 160 Ω (to limit the current in the snubber branch to 1% of the phase current under the subject loading conditions). Also, the filter F(s) in the proposed CPVBR model is set to have a resonant frequency of 10ωr = 12,566 rad/s (corresponding to 2000 Hz oscillations on machine variables in rotor reference frame) to attain accurate approximations in (4–56). Transient response of the machine stator line voltages vabs and vacs, line current ias and electromagnetic torque Te, obtained by the subject models are presented in Figure 4–8. As seen in Figure 4–8, increasing the rectifier load leads to more distorted stator line voltages (Figures 4–8(a)–(b)) as a result of increased rectifier current with high harmonic content (Figure 4–8(c)) and commutation. It is also observed in Figure 4–8(d) that an increase in rectifier load results in higher electromagnetic torque. Figure 4–8 also shows that all the subject models are able to produce similar and consistent simulation results with respect to the reference model solution. In order to provide a more detailed comparison, magnified views of stator line current ias and electromagnetic torque Te after load change transient are presented in Figure 4–9. It can be seen in Figure 4–9 that the qd130 model has a visible deviation from the reference solution, which is caused by the excess current flowing into the snubber branches. The CCPD and the proposed variable-parameter VBR models both are able to produce nearly identical results to the reference 110 Figure 4–8. Transient response of several machine variables during load change transient, as predicted by the subject models for: (a) line voltage vabs, (b) line voltage vacs, (c) line current ias, and (d) electromagnetic torque. solution with almost no numerical error (due to not having any approximations or snubber circuits). In addition, Figure 4–9 shows that the proposed CPVBR model can obtain highly accurate results compared to the reference solution, with negligible 1 1.005 1.01 1.0150123Time (s)T e (N.m)-20-1001020i sa (A)-502502550v sac (V)-26-1301326v sab (V)CPVBRqd130 CCPDReferenceb)c)d)a)See Fig. 9(b)See Fig. 9(a)VBRSe Fig. 4–9(a)Se Fig. 4–9(v abs (V) v acs (V) i as (A) 111 numerical error due to the filter approximations. Figure 4–9. Magnified view of selected variables as obtained by the subject five‐phase machine models for: (a) line current ias, and (b) electromagnetic torque. 4.5.3 Numerical Accuracy Comparison To quantitatively assess the numerical accuracy, numerical errors in stator current and electromagnetic torque as predicted by the subject models are calculated with respect to the reference solution and are summarized in Table 4–1. For all models, the numerical error is obtained using the cumulative 2-norm error calculation presented in (2–69). 1.0171 1.01725 1.01743.253.353.45Time (s)T e (N.m)1.01905 1.0192 1.0193517.71818.3Time (s)i sa (A)CPVBRVBRCCPDReference qd130b)a)i as (A) 112 As can be seen in Table 4–1, the qd130 model has the largest numerical error (0.97% in stator current and 0.85% in torque), which is expected due to presence of snubber resistors. The CCPD and the proposed variable-parameter VBR models both are shown to have negligible numerical errors (0.008% and 0.007% in stator current, respectively, and both 0.006% in torque), which are mainly associated with solver discretization. Table 4–1 also shows that the proposed CPVBR model is able achieve decent accuracy with small errors (0.26% in stator current and 0.20% in torque) which are resulted from filter approximations. TABLE 4–1. COMPARISON OF 2‐NORM ERROR OF THE SIMULATION RESULTS OVER THE 2‐SECOND TRANSIENT CASE‐STUDY OBTAINED BY THE SUBJECT MODELS Model Stator phase a current (ias) Electromagnetic torque (Te) qd130 model (with snubbers) 0.97 % 0.85 % CCPD model (with variable inductances) 0.008 % 0.006 % Proposed VBR model (with variable inductances) 0.007 % 0.006 % Proposed CPVBR model 0.26 % 0.20 % 4.5.4 Computational Performance Comparison The computational performance of the implemented subject models are compared in terms of the largest eigenvalue, number of time-steps, average CPU time per-step, and the total required CPU time as summarized in Table 4–2. As can be seen in Table 4–2, the qd130 model has the largest eigenvalue (-1.11×106, i.e., significant stiffness) associated with snubber resistances, which results in the largest amount of time-steps (685,185). This leads to a prohibitive total CPU time (49.6 s), despite of 113 the relatively small average CPU time per-step (72 μs). Table 4–2 also shows that both the CCPD model and the proposed variable-parameter VBR model have significantly smaller eigenvalues (-262 and -260, respectively), which enables the solver to obtain simulation results with much fewer time-steps (26,616 and 26,546 respectively). It is also observed that both the CCPD and the proposed variable-parameter VBR models require significantly larger average CPU time per-step (725 μs and 682 μs, respectively), since both models have variable-parameter interfacing inductances which need to be re-computed at each time-step. However, the total amount of required CPU times (19.3 s and 18.1 s, respectively) are still smaller than the qd310 model due to their need for much smaller number of time-steps. Furthermore, it is seen in Table 4–2 that the proposed CPVBR model has a larger eigenvalue (-8,745) compared to the CCPD and variable-parameter VBR models (and requires 45,156 time-steps), which is due to the tuned filter that is used for approximations. However, despite requiring a larger number of time-steps, it is noted that the constant-parameter interfacing circuit in the proposed CPVBR model significantly reduces the model complexity and the required CPU time per step (140 μs), which yields the smallest total required CPU time (down to only 6.3 s). Considering the small numerical error (less than 0.3%) as shown in Table 4–1, it can be concluded that the proposed CPVBR models possess a superior combination of simulation performance and numerical accuracy over other subject models. 114 TABLE 4–2. COMPUTATIONAL PERFORMANCE COMAPRISONS OF THE SUBJECT MODELS FOR THE 2 SECOND TRANSIENT CASE‐STUDY Model Largest eigenvalue Number of steps CPU time per step CPU Time qd130 model (with snubbers) –1.11×106 685,185 72 µs 49.6 s CCPD model (with variable inductances) –262 26,616 725 µs 19.3 s Proposed VBR model (with variable inductances) –260 26,546 682 µs 18.1 s Proposed CPVBR model –8,745 45,156 140 µs 6.3 s 115 CHAPTER 5: CONSTANT‐PARAMETER VOLTAGE‐BEHIND‐REACTANCE MODEL OF SATURABLE INDUCTION MACHINES INCLUDING AIR‐GAP FLUX HARMONICS In this chapter, a CPVBR model for saturable induction machines is proposed which considers the air-gap flux spatial harmonics (up to the dominant third harmonic components). The proposed model is based on the qd0 model of saturable induction machines with flux harmonics, which is itself the result of transforming the CCPD model presented in [20]. The proposed CPVBR model of saturable induction machines achieves constant-parameter RL interface using the algebraic relaxation method in [71] with low-pass filters. The new CPVBR model is compared against the conventional qd0 and the variable parameter VBR models in terms of accuracy and simulation performance and is shown to provide accurate results while offering significant improvements in numerical efficiency and simulation speed. Additionally, the constant-parameter interfacing branches make the proposed CPVBR model able 116 to be conveniently implemented in most EMT simulation packages with no interfacing issues. Here, the saturable three-phase squirrel cage induction machine with air-gap flux harmonics (up to third component) in [20] is considered. All machine parameters are referred to the stator side and motor sign convention is used. 5.1 Spatial Flux Harmonics Due to Magnetic Saturation In conventional models of saturable electrical machines, it is often assumed that magnetic flux density in the air-gap maintains a sinusoidally distributed along the stator circumference, and that only the amplitude of the magnetizing flux is affected by the saturation. This assumption permits many simplifications in the modeling, and it is based on considering the same relative level of saturation through-out the entire flux path. However, it has been shown that flux distribution can become considerably distorted as a result of different saturation levels in different parts of the machine, which creates spatial harmonics in the magnetizing flux distribution [16]–[20]. One of the most common causes of the air-gap flux spatial harmonics is a somewhat deeper magnetic saturation in the stator slots compared to the stator core [16], and at the regions where the magnetic field is at its peak amplitude, as demonstrated in Figure 5–1. As a result, under saturation, the magnetic flux density will have a flattop waveform [16], containing odd spatial harmonics as illustrated in Figure 5–2 by the 3rd harmonic. It is also worth noting that in a squirrel-cage induction machine, the peak of the magnetic flux density is aligned with the machine magnetizing current vector. 117 Figure 5–1. Cross‐section of a squirrel cage induction machine including magnetizing current vector and different saturation zones. In this chapter, the spatial harmonics are considered up to third component only since it has the most dominant effect (i.e. the 5th and the higher harmonics have much smaller magnitudes). Thus, the two flux components (i.e. the 1st and the 3rd harmonic components) are defined based on the magnetizing current vector angle and magnitude, as depicted in Figure 5–2. In Figure 5–2, the fundamental and third harmonic components of magnetic flux density are defined [20] as 1 13 31 3( , ) ( )cos( )( , ) ( )cos(3 3 )( , ) ( , ) ( , )m m mm m mm m m m m mB i B iB i B iB i B i B i         , (5–1) where  is the physical angle around the rotor circumference, and mi and  are the magnitude and angle of the magnetizing current vector, respectively. It is noted that abcabcmi0as 2 3cs  2 3bs 118 the peaks of fundamental and third harmonic components of flux density are aligned with the magnetizing current vector as a result of non-salient rotor and stator in the induction machine. Figure 5–2. Saturated air‐gap flux density including fundamental and third components. Using the magnetic flux distribution functions defined in Figure 5–2, the stator and rotor flux linkage components can be defined separately. Based on the stator winding functions (up to third harmonic component), the following magnetizing flux linkages can be defined for each of the phases [20] as 31 31cos( ) cos33wmxs m xs mwkk        , (5–2) where 1wk and 3wk are the stator’s winding factors for fundamental and third harmonic components, x corresponds to each of the stator three a, b and c phases, and xs corresponds to the angle of the each phase’s winding on the stator (i.e., 0, 2π/3 and -2π/3 for phases a, b, and c, respectively). Also, the fundamental and third harmonic components of magnetizing flux magnitudes 1m and 3m are defined as a function of the magnetizing flux density, and as a result, magnetizing current as 0Flux density 2|B3||B1|Bm3(im, )Bm1(im , )Bm(im , )119   1 1 1 13 3 3 3( ) ( )( ) ( )m m mm m mU B i F iU B i F i   , (5–3) where functions U1(.) and U3(.) are dependent on the machine’s geometric properties. In standard characterization tests [20], functions F1(.) and F3(.) can be directly extracted and often stored as look up tables, relating each of the magnetizing flux components in the squirrel-cage induction machine to the magnetizing current amplitude. In the actual subject squirrel-cage induction machine, the rotor consists of many conductors. Therefore, as a result of only considering up to third flux harmonic component, the rotor is represented as two separate three-phase winding sets, corresponding to fundamental and third harmonic flux components, rotating at the electrical frequencies r and 3 r , respectively [20]. As a result, rotor flux linkages for both sets of rotor windings are defined as 1 1 13 3 3cos( )1 cos( 3 )3my m ymy m y         , (5–4) where y corresponds to rotor three phase windings ar, br and cr. Angles 1y and 3y correspond to each of the rotor winding component phases with respect to the stationary frame and are defined as [20] 1 1 13 3 3, 2 3, 2 33 , 3 2 3, 3 2 3ar r br r cr rar r br r cr r                       , (5–5) where r is the rotor electrical angle. Based on the defined flux linkage components, a complete coupled circuit 120 phase domain model for the squirrel-cage induction machine is derived and presented in [20] and validated using extensive experimental tests. 5.2 qd0 Model The qd0 model of saturable induction machines with spatial flux harmonics is derived in this dissertation for the first time. This is achieved by transforming the machine CCPD model [20] to (in this case) synchronously rotating reference frame. Variables corresponding to the three-phase stator winding and the rotor winding fundamental component are transformed by the same qd0 reference frame using Park’s transformation [27], as 0 ( )qd s s abcf K f , (5–6) [ , , ]Tabc a b cf f ff , 0 0[ , , ]Tqd q df f ff . (5–7) where ( )s sK is Park’s three-phase transformation matrix [27] and s is the qd0 reference frame angle. Variables corresponding to the rotor winding representing the third harmonic flux component are transformed to a qd0 plane rotating with three-time higher frequency (i.e.,3 s ) as 03 3(3 )qd s s abcrf K f , (5–8) where 3 3 3 3[ , , ]Tabcr ar br crf f ff , 03 3 3 03[ , , ]Tqd q df f ff . (5–9) Applying the qd0 transformations in (5–6) and (5–8), to the machine voltage equations in [20] the following stator and rotor voltage equations can be defined as 121 qs s qs qs s dsv r i p     , (5–10) ds s ds ds s qsv r i p     , (5–11) 0 0 0s s s sv r i p  , (5–12)  0 r qr qr s r drr i p       , (5–13)  0 r dr dr s r rqr i p       , (5–14)  3 3 30 3r qr qr s r drr i p       , (5–15)  3 3 30 3r dr dr s r qrr i p       , (5–16) where s and r are the electrical reference frame angular speed and rotor electrical frequency, respectively. It is worth mentioning that the zero sequence rotor voltage equations are omitted since zero sequence current does not flow in a squirrel-cage type rotor. In (5–10)–( 5–16), stator currents are defined as  1qs ls qs mqi L    ,  1ds ls ds mdi L    , (5–17)  10 0 0s ls s mi L    , (5–18) and rotor currents for fundamental and third harmonic components are expressed as  1qr lr qr mqi L    ,  1dr lr dr mdi L    , (5–19)  13 3 3qr lr qr mqi L    ,  13 3 3dr lr dr mdi L    . (5–20) In (5–17)–( 5–20), variables mq , md , 3mq , 3md are the fundamental and third harmonic components of the machine magnetizing flux in q- and d-axes corresponding to both fundamental and third harmonic component planes, 122 respectively. Noticing that the magnetizing current and flux vectors in induction machines are aligned, the fundamental and third harmonic components of the machine magnetizing flux (i.e., mq , md , 3mq , 3md , resulting from transforming (5–2), (5–4), using (5–6), (5–8)) in q- and d-axes are defined as 1 sin( )mq m m   , 1 cos( )md m m   , (5–21) 3 3 sin(3 )mq m m   , 3 3 cos(3 )md m m   , (5–22)  30 31sin 3 3wm m s mwkk     , (5–23) In (5–21)–( 5–23) m is the magnetizing current vector angle in qd0 reference frame which is defined as 1tan mqm smdii           . (5–24) In (5–24), the machine’s main magnetizing current components in q- and d-axes are defined as mq qs qri i i  , md ds dri i i  . (5–25) 2 2m mq mdi i i  . (5–26) In (5–21)–( 5–23) the fundamental and third harmonic components of the machine’s magnetizing flux amplitudes (i.e., 1m , 3m ) are functions of the machine magnetizing current magnitude (defined in (5–3)). Also, it can be observed in (5–3) that both the fundamental and third harmonic components of the machine magnetizing flux are defined in terms of the magnitude of fundamental component magnetizing current, therefore the third harmonic component of the magnetizing 123 flux can be written in terms of fundamental component as  13 3 1 1( )m mF F  . (5–27) The electromagnetic torque produced by the machine is comprised of two components, corresponding to the fundamental and third harmonic components of the air-gap flux, respectively, which are defined [20] as  1 32 2e md qs mq dsPT i i         , (5–28)  3 3 3 3 392 2e mq dr md qrPT i i         , (5–29) 1 1 3e e eT T T  . (5–30) Electromagnetic torque in (5–28) is resulting from the fundamental component of the magnetizing flux and stator currents in q- and d- axes. Also, the third harmonic component of the magnetizing flux links with the rotor third harmonic component sub-circuit which produces an additional electromagnetic torque in (5–29) in the same direction as of the fundamental component, contributing to the total produced torque (5–30). The machine equivalent circuit in qd0 reference frame can be derived using voltage equations defined in (5–10)–( 5–16) which is depicted in Figure 5–3. 124 s ds sr lsLqsvqsimqp lrLqri3mqp( )s r dr  33( )s r dr  rrrr3qrilrLs qs sr lsLdsvdsimdp lrLdri3mdp( )s r qr  33( )s r qr  rrrr3drilrL0sisr0svlsL0mpmqimdi Figure 5–3. The qd0 equivalent circuit of induction machine including third harmonic component of air‐gap flux in synchronous reference frame. 5.3 Variable‐Parameter VBR Model The proposed VBR model formulation is derived based on the qd0 model of the machine, presented in Section 5.1 by choosing the rotor fluxes for both fundamental and third harmonic components in q- and d-axes as the state variables, while using the stator currents as the inputs to the system. Rotor fluxes and stator currents are then used to replace the magnetizing branches of the machine, as well as rotor circuits with sub-transient inductances and voltage sources which are considered as outputs to the rotor state-space model. According to the machine rotor voltage equations defined in (5–13)–( 5–16), and writing the rotor currents in terms of the rotor and magnetizing fluxes from (5–125 19)–( 5–20), the derivatives of the rotor flux components can be expressed as    1qr r lr qr mq s r drp r L          , (5–31)    1dr r lr dr md s r qrp r L          , (5–32)    13 3 3 33qr r lr qr mq s r drp r L          , (5–33)    13 3 3 33dr r lr dr md s r qrp r L          . (5–34) In order to form a complete set of state-space equations for rotor currents, the magnetizing flux components (i.e., mq , md , 3mq , 3md ) in (5–31)–( 5–34) must be expressed in terms of state variables and inputs (i.e., rotor flux components and stator currents). Substituting rotor currents in (5–25) with the ones defined in (5–19), the machine magnetizing current and fundamental flux components can be written in terms of state variables and inputs as 1 1mq lr mq qs lr qri L i L     , (5–35) 1 1md lr md ds lr dri L i L     . (5–36) Defining the intermediate current components zqi and zdi in terms of rotor fluxes and stator currents [33] as 1zq qs lr qri i L   , 1zd ds lr dri i L   , (5–37) the magnetizing current and flux components in (5–35)–( 5–36) can be expressed as 1mq lr mq zqi L i  , (5–38) 1md lr md zdi L i  . (5–39) Since in an induction machine the magnetizing current and flux vectors are 126 aligned, their magnitudes can be expressed in terms of intermediate current magnitude zi [33] based on (5–38)–(5–39) as 1 1m lr m zi L i  , (5–40) where 2 2z zq zdi i i  . (5–41) Expressing the magnetizing current in (5–40) in terms of magnetizing flux using (5–3), the intermediate current zi can be expressed as a non-linear function (denoted here as G(.)) of the machine magnetizing flux as 1 11 1 1 1( ) ( )z m lr m mi F L G      , (5–42) Using the fact that the intermediate current vector (defined using zqi and zdi ) is also aligned with the magnetizing current and flux vectors [33], the magnetizing flux components in (5–21)–(5–23) can be expressed in terms of zi using (5–27) and (5–42) as 1( )sin( )mq z zG i  , 1( )cos( )md z zG i  , (5–43)   1 13 3 1 ( ) sin(3 )mq z zF F G i   , (5–44)   1 13 3 1 ( ) cos(3 )md z zF F G i   , (5–45)      1 130 3 110( ) sin 3 3 ( )sin 3 3wm z s zwz s zk F F G ikF i      , (5–46) where 127 1tan zqz mzdii       . (5–47) Since the intermediate currents zqi and zdi are functions of the state variables and inputs (as expressed in (5–37)), the magnetizing flux components in (5–43)–(5–45) can be used in (5–31)–(5–34) directly to form the final form of the rotor state equations. To obtain the VBR formulation for the stator voltage equations in (5–10)–(5–12), the stator fluxes in (5–17)–(5–18) are written in terms of stator currents and magnetizing flux components, resulting in a new stator voltage equations as 0 0 0 0qs qs qs mqds s ds ls ds md ss s s mv i iv r i L p i pv i i                                      E , (5–48) where sE is the stator rotational voltage terms, defined as   0s ls sd mds s ls sq mqL iL i         E . (5–49) The magnetizing flux derivatives in (5–48) can be obtained based on (5–43)–( 5–46) and can be expressed as 0 3 20 00( ) 0( , ) 3 ( ) cos(3 3 )mqmqd z zqmdm z s zdm z s s zip piF i                             L iL i , (5–50) where sub-transient matrices mqdL and 0mL are defined as 128 1 11 12 21 11 12 2( )( ) ( )sin( ) ( ) cos( ) sin( ) ( ) cos( ) ( ) ( )cos( ) ( ) sin( ) cos( ) ( ) sin( )mqd zzq zqzd zdz zz z z z z zz z z z z zzq zqzd zdz zz z z z z zz z z z z zi ii idG i dG iG i G idi i i di i ii ii idG i dG iG i G idi i i di i i                  L i.(5–51) 00 2000 2( ) sin(3 3 ) 3 ( ) cos(3 3 )( , ) ( ) sin(3 3 ) 3 ( ) cos(3 3 )Tzqz zds z z s zz z zm z szqz zds z z s zz z zidF i iF idi i iidF i i F idi i i                  L i . (5–52) The vector term zi in (5–51)–(5–52) denotes the two components of the intermediate currents in q- and d-axes which are functions of inputs and state-variables as expressed in (5–37). Using (5–37) and (5–50), and noting that machine has constant leakage inductances, the magnetizing flux component derivatives can be expressed as 10 3 20 00( ) 0 .( , ) 3 ( ) cos(3 3 )mqmqd z qs qrmd lrm z s ds drm z s s zip p L piF i                                          L iL i (5–53) Magnetizing flux derivatives (5–53) can then be used in (5–48) to obtain a new equation for the stator voltage components as 10 3 20 0 00( )( , )0 0 .3 ( ) cos(3 3 )qs qs qsmqd z qs qrds s ds ls ds s lrm z s ds dss s sz s s zv i iiv r i L p i p L piv i iF i                                                           L iE L i (5–54) 129 The stator voltage equations (5–54) can then be written in the VBR from and expressed as 00 0 0 0( , )0qs qs qs s sd qsds s ds qd z s ds s sq dss s s sv i i ev r i p i ev i i e                                                   L i , (5–55) where the sub-transient matrix 0( , )qd z sL i is defined as 2 2 2 100 3 3( )( , ) ( , )ls mqd zqd z sm z s lsLL       I L i 0L i L i , (5–56) the sub-transient fluxes in q- and d-axes are defined as  2 2 ( )qs qsls mqd zds dsiLi           I L i , (5–57) and the sub-transient back-emf voltages as 10 3 20 0( ) .( , )3 ( ) cos(3 3 )qs s dsmqd z qrds s s qs lrm z s drs z s s zee L pe F i                                      L iE L i (5–58) In (5–58), the rotor flux derivatives can also be written in terms of the inputs and state-variables by directly using their definitions (5–31)–( 5–34). Applying Park’s inverse transformation [27] to (5–55) the stator interfacing equations in abc phase coordinates are obtained in the VBR form as  ( , )abcs s abcs abc z s abcs abcsr p    v i L i i e , (5–59) where the variable sub-transient inductance matrix ( , )abc z sL i (which is a function of saturation and synchronous frame angle) is defined as 1 0( , ) ( ) ( , ) ( )abc z s s qd z s s    L i K L i K . (5–60) 130 The VBR model of the saturable induction machine including air-gap flux harmonics can be implemented using state-equations (5–31)–( 5–47) with rotor fluxes as state variables and stator currents as inputs. Model outputs are the sub-transient back-emf voltages in (5–58). The electromagnetic torque in the proposed VBR model is calculated using identical equations to the qd0 model expressed in (5–28)–(5–30). It is worth mentioning that the third harmonic rotor current components required in (5–29) are calculated using the third harmonic rotor and magnetizing fluxes as presented in (5–20). The interfacing circuit of the VBR model is realized according to (5–59), consisting of controlled voltage sources connected in series with RL branches which can be interfaced with any arbitrary network without auxiliary snubber circuits, as illustrated in Figure 5–4. However, in the proposed VBR model, the interfacing inductances are variable (as shown in (5–60)) due to saturation and synchronous frame angle-dependent terms, which can create computational bottleneck. asebsecsesrsrsr( , )abc z sL i( )sK1( )sK0qd si0qd seabcseabcsi Figure 5–4. Implementation of the proposed variable‐parameter VBR model of saturable induction machine including air‐gap flux harmonics, and its interfacing with external network using variable‐parameter RL branches. 131 5.4 Constant‐Parameter VBR Model To achieve a constant-parameter interfacing circuit for the proposed VBR model, the interfacing inductance matrix in qd0 coordinates (defined in (5–56)) is re-formulated and separated in to two parts: a constant and non-salient part (denoted by 0qdL ); and a residual part (denoted by 0 ( , )qd res z sL i ) containing all the variable and saliency terms. The re-formulated sub-transient inductance is thus expressed as 0 0 0( , ) ( , )qd z s qd qd res z s    L i L L i . (5–61) In (5–61), 0qdL is derived based on the unsaturated machine magnetizing inductance (denoted by muL ) with equal and constant terms in both q- and d-axes and is defined as   2 2 2 101 2 3 3ls muqdlsL LL       I 0L 0 , (5–62) where 11 1mulr muLL L     . (5–63) Moreover, 0 ( , )qd res z sL i includes the saturation-dependent saliency and synchronous frame angle-dependent terms, expressed as   2 2 2 100 3 3( )( , ) ( , ) 0ls mu mqd zqd res z sm z sL L         I L i 0L i L i . (5–64) The constant and non-salient sub-transient matrix defined in (5–62) is then used as the new interfacing inductance matrix, while the variable part of the sub-transient inductance matrix in (5–64) is taken into account and considered as voltage 132 terms included in the sub-transient back-emf voltage outputs. As a result, the stator voltage equations (5–55) can be re-written as 0 00 0 0 0 0( , )0qs qs qs s ds qs qsds s ds qd ds s qs ds qd res z s dss s s s sv i i e iv r i p i e p iv i i e i                                                               L L i . (5–65) Defining the new sub-transient fluxes as 0 ( , )qs qsqd res z sds dsii          L i , (5–66) and the new sub-transient back-emf voltages as 00 0 0( , )0qs qs s ds qsds ds s qs qd res z s dss s se e ie e p ie e i                                          L i , (5–67) the stator voltage equation (5–65) can be written with a constant and non-salient inductance matrix (i.e., 0qdL ) as   00 0 0 00s ds dsqs qs qs qsds s ds qd ds s qs qs dss s s sv i i ev r i p i ev i i e                                                       L . (5–68) As can be seen in (5–67), in order to calculate the new sub-transient back-emf voltages, the stator current derivatives are required, which can be calculated using the stator voltage equation (5–55), and expressed as 100 0 0 0( , )0qs qs qs s ds qsds qd z s ds s ds s qs dss s s si v i ep i v r i ei v i e                                                       L i . (5–69) 133 It is noted that since current derivatives in (5–69) have a direct relation with stator voltages, using them (i.e., current derivatives) to obtain sub-transient back-emfs in (5–67) can lead to an algebraic loop due to voltage equations in (5–68) having a direct relation with the sub-transient back-emfs in (5–69). In order to obtain an explicit formulation for stator voltage equations, the algebraic loop relaxation method introduced in (2–68) is used, where a low-pass filter (represented by the transfer function H(s)) is utilized to approximate the stator voltages in (5–69) as 0 0( )qs qsds dss sv vv H s vv v                . (5–70) The low-pass filter pole is tuned and chosen using the method in [39] in order for the filter to provide an accurate approximation of the stator voltages up to a certain transient frequency. Ultimately, the interfacing circuit in abc phase coordinates is obtained applying the inverse Park’s transformation to the stator voltage equations (5–68), which yields abcs s abcs abc abcs abcsr p   v i L i e , (5–71) where the constant-parameter interfacing inductance matrix is obtained as 1 023 3 32( ) ( ) 3 3 323 3 3mu mu mulsmu mu muabc s s qd s s lsmu mu mulsL L LLL L LLL L LL                        L K L K . (5–72) The proposed CPVBR model can be implemented using the rotor flux state-134 equations (5–31)–(5–47), the sub-transient inductances (5–51)–(5–52), (5–62)–(5–64), the output back-emf voltages (5–58) and (5–67), and the stator interfacing equation (5–71). Also, the electromagnetic torque can be calculated using the same equation as defined for the qd0 model (5–28)–(5–30). The model uses controlled voltage sources (corresponding to sub-transient back-emf outputs) connected in series with constant-parameter RL branches and can be interfaced with an arbitrary network without requiring snubber circuits, as shown in Figure 5–5. Moreover, it is worth mentioning that the proposed CPVBR model uses the stator phase voltage approximation in (5–70) in addition to the phase currents as the input signal (which was the only input in the previous VBR model). asebsecsesrsrsrabcL( )sK1( )sK0qd si0qd sv0qd sv0qd seabcseabcsi abcsv Figure 5–5. Implementation of the proposed constant‐parameter VBR model of saturable induction machine including air‐gap flux harmonics, and its interfacing with external network using constant‐parameter RL branches. 5.5 Computer Studies In order to verify the two proposed saturable induction machine VBR and CPVBR models, the three-phase 3hp 208V 4-pole squirrel-cage induction machine used in [12] is considered here. Moreover, the magnetic saturation is also considered 135 which causes the air-gap flux harmonics. The machine parameters are summarized in Appendix D. It is noted that the selected induction machine is extensively validated in [20] using hardware measurements, and the available models are shown to predict the experimental results accurately. The three induction machine models presented in this chapter (i.e., qd0, VBR, and CPVBR presented in Sections 5.2–5.4, respectively) are implemented in MATLAB Simulink [39] using PLECS blocksets [41]. The simulations are run on a PC with Intel® Core™ i7-8700K @ 3.7GHz processor using MATLAB ode45 variable time-step solver with relative and absolute error tolerances set to 10-4. For the purpose of numerical accuracy comparisons, a reference solution is obtained using the variable-parameter VBR model presented in Section 5.3 with the maximum time-step size of 1μs. It is noted that the VBR model does not require snubber circuits (used in qd0 model) or filter approximations (used in CPVBR model) which contribute to errors; and therefore, the results obtained by the VBR model with a very small time-step are used as the reference solution. In order to implement the qd0 model, the snubber resistors are connected in parallel to each of the machine phases. This is due to the fact that qd0 models use controlled current sources as the interfacing circuit, and therefore cannot be connected in series with inductive networks. Moreover, the snubber resistances are used to provide a zero sequence equivalent circuit for modeling of the third harmonic components in the stator variables. The value of the snubber resistors in the implemented qd0 model is chosen to be 650Ω (corresponding to 100pu) in order to keep the snubber branch current under 1% of the machine nominal current. Also, in the CPVBR model, the low-pass filter (i.e., H(s)) pole is chosen according to the 136 procedure in [81] in such a way that the filter is able to provide an accurate approximation of stator voltages. Therefore, for an error threshold of 1% at fitting frequency [81] of 100 Hz, the low-pass filter pole is set to (-3300). To see the effect of saturation including the air-gap flux harmonics, three steady-state case studies are considered where the reference model is simulated with three different stator winding configurations: under no-load and under full-load, each at three different stator voltage levels. In order to conduct computational performance and accuracy comparison between the models presented in Section 5.2–5.4, all subject models are simulated in a start-up and load change transient case study. 5.5.1. Steady‐State Case‐Study 1: Wye‐Connected Stator Windings with Floating Neutral Connection In this steady-state case-study, the stator windings are wye-connected with floating neutral point and are connected to a three-phase grid (represented by three phase ideal voltage sources vg) through identical lines (represented with rline and Lline) as shown in Figure 5–6. asvliner lineL abcgvIMasiabc Figure 5–6. Wye connected induction machine with floating neutral point supplied by the ac grid. 137 Figure 5–7. Magnetization curves of the induction machine demonstrating the three considered operating points for: (a) main flux, (b) third harmonic flux. The simulation studies are done under three different voltage levels, corresponding to nominal operating point [phase rms voltage |vg| = 120V (1pu)], slight magnetic saturation [phase rms voltage |vg| = 144V (1.2pu)], and deep saturation [phase rms voltage |vg| = 186V (1.55pu)], respectively. The considered magnetic operating points are shown in Figure 5–7, depicting the machine main and third harmonic magnetization curves, all corresponding to the experimental results reported in [20]. The simulation results for the stator phase a voltage and current in steady-state operation in both no-load and full-load (12Nm mechanical load) conditions are depicted in Figures 5–8 to 5–9, respectively. As it can be seen in Figures 5–8 to 5–9, increasing the stator voltage level leads to excessive stator current as a result of magnetic saturation. Moreover, it is observed 00.150.30.450.60.75 m (Wb)0 5 10 15 20 25 3000.0150.030.0450.060.075im (A) m3 (Wb)UnsaturatedMagnetizationa)b)|vg| = 1.55 pu|vg| = 1.2 pu|vg| = 1.0 pu|vg| = 1.0 pu|vg| = 1.2 pu|vg| = 1.55 pu138 that deeper levels of saturation at higher stator voltage levels creates visible third harmonic distortions in stator phase voltages which is a result of magnetizing flux spatial harmonics of the same order (i.e., third). However, due to the floating neutral point in the stator winding connection, the stator current does not contain any third harmonic which corresponds to the zero sequence. Furthermore, it can be seen in Figure 5–9 that when the machine is under full-load, deeper levels of saturation lead to larger lagging phase shifts in stator currents which is due to higher levels of reactive power being supplied to the saturated machine. It is worth mentioning that the simulation results presented in Figures 5–8 to 5–9 are also consistent with experimental measurements reported in [20]. Figure 5–8. Phase variables of wye‐connected saturable induction machine with floating neutral point in steady‐state at no‐load condition under various stator voltage levels: (a) stator phase a voltage vas and (b) stator phase a current ias. -250-1250125250v as (V)0.405 0.41 0.415 0.42-20-1001020Time (s)i as (A)(III)(III)(I)(I)(II)(II)a)b)Reference model:(I): | vg| = 1.0 pu(III): | vg| = 1.55 pu(II): | vg| = 1.2 pu139 Figure 5–9. Phase variables of wye‐connected saturable induction machine with floating neutral point in steady‐state at full‐load condition under various stator voltage levels: (a) stator phase a voltage vas and (b) stator phase a current ias. 5.5.2. Steady‐State Case‐Study 2: Wye‐Connected Stator Windings with Neutral Connection In this second case-study, the same steady-state simulation of the induction machine as in the previous case (Section 5.5.1) is repeated with the addition of a neutral wire (also represented by rline and Lline) connected between the stator wye-connected windings and the wye-connected three-phase voltage sources, as shown in Figure 5–10. The steady-state simulation results for the machine stator phase a voltage and current under both no-load and full-load operations are presented in Figures 5–11 to 5–12, respectively. As it can be expected and seen in Figures 5–11 to 5–12, increased stator voltage levels lead to considerable increase in stator current -250-1250125250v as (V)0.405 0.41 0.415 0.42-20-1001020Time (s)i as (A)(III)(I)(II)a)b)Reference model:(III)(I)(II)(I): | vg| = 1.0 pu(III): | vg| = 1.55 pu(II): | vg| = 1.2 pu140 due to main flux saturation. However, as seen in Figures 5–11(a) and 5–12(a), the addition of the neutral wire between the machine and the three-phase sources has largely eliminated third harmonic distortions from the stator phase voltage due to sources being purely sinusoidal and lines having small impedances. asvliner lineL abcgvasi Figure 5–10. Wye connected induction machine with neutral wire supplied by the ac grid. The added neutral wire also allows the third harmonic stator current to flow as a result of third harmonic back-emfs in the stator windings. As can be seen in Figures 5–11(a)–(b), the third harmonic component of the stator current has a considerable amplitude since it only has a small impedance in its path, corresponding to the machine’s stator and lines impedances. Analysis of the full-load operating condition in Figure 5–12 also shows that increased level of saturation results in larger lagging phase shift of the fundamental component of the stator currents, which also leads to a larger phase shift between the fundamental and the third harmonic components. It is also noted that simulation results in Figures 5–11 to 5–12 are consistent with the hardware measurements presented in [20]. 141 Figure 5–11. Phase variables of wye‐connected saturable induction machine with neutral connection in steady‐state at no‐load condition under various stator voltage levels: (a) stator phase a voltage vas and (b) stator phase a current ias. Figure 5–12. Phase variables of wye‐connected saturable induction machine with neutral connection in steady‐state at full‐load condition under various stator voltage levels: (a) stator phase a voltage vas and (b) stator phase a current ias. -250-1250125250v as (V)0.405 0.41 0.415 0.42-20-1001020Time (s)i as (A)(I)(II)a)b)Reference model:(III)(III)(II) (I)(I): | vg| = 1.0 pu(III): | vg| = 1.55 pu(II): | vg| = 1.2 pu-250-1250125250v as (V)0.405 0.41 0.415 0.42-20-1001020Time (s)i as (A)(I)(II)a)b)Reference model:(III)(I)(II)(III)(I): | vg| = 1.0 pu(III): | vg| = 1.55 pu(II): | vg| = 1.2 pu142 5.5.3. Steady‐State Case‐Study 3: Delta‐Connected Stator Windings For the third steady-state case-study, the stator windings are configured as delta (Δ) and are connected to the three phase sources through the same three-phase line described before, as illustrated in Figure 5–13. In this case, in order to magnetize the machine in the same operating points shown in Figure 5–7, the hree new grid phase voltage levels are considered, consisting of nominal operating point [phase rms voltage |vg| = 69V (1pu)], slight magnetic saturation [phase rms voltage |vg| = 82.8V (1.2pu)], and deep saturation [phase rms voltage |vg| = 107V (1.55pu)], respectively. Also, the same loading condition (12Nm mechanical load) is considered for full-load operation. asvliner lineL abcgvasi Figure 5–13. Delta‐connected induction machine supplied by the ac grid. The steady-state simulation results for stator phase a voltage and current under various stator voltages are presented in Figures 5–14 to 5–15 for no-load and full-load operating conditions, respectively. As can be seen in both of Figure 5–14(a) and 5–15(a), the stator voltages remain purely sinusoidal since the sources are harmonic-free and no zero sequence third harmonic current flows through the three-phase lines. However, as observed in Figures 5–14(b) and 5–15(b), as the stator 143 voltage level increases, the stator phase currents become larger in amplitude and start becoming distorted by the third harmonic component that is circulating through the Δ-connected stator windings. Moreover, similar to the cases presented in Section 5.51–5.5.2, the increased level of saturation results in larger lagging phase shift of the fundamental component of stator currents and also leads to a larger phase shift between the fundamental and the third harmonic components, respectively. It is also worth mentioning that in case of a Δ-connected stator winding, the third harmonic distortions have a higher amplitude compared to the case with wye-connected stator including neutral connection. This is due to the fact that the total impedance in the third harmonic component’s path is smaller in the Δ-connected stator (only consisting of machine’s impedances), whereas in the wye-connected case, the line impedances are also included. Ultimately, it should be stated that simulation results shown in Figures 5–14 to 5–15 are also consistent with the experimental measurements presented in [20]. 144 Figure 5– 14. Phase variables of delta‐connected saturable induction machine with neutral connection in steady‐state at no‐load condition under various stator voltage levels: (a) stator phase a voltage vas and (b) stator phase a current ias. -250-1250125250v as (V)0.905 0.91 0.915 0.92-20-1001020Time (s)i as (A)a)b)Reference model:(II)(III)(I)(II)(III)(I)(I): | vg| = 1.0 pu(III): | vg| = 1.55 pu(II): | vg| = 1.2 pu145 Figure 5–15. Phase variables of Δ‐connected saturable induction machine with neutral connection in steady‐state at full‐load condition under various stator voltage levels: (a) stator phase a voltage vas and (b) stator phase a current ias. 5.5.4. Transient Study: Start‐up and Load Change In order to validate the presented induction machine models in Section 5.2–5.4 (i.e., qd0, proposed VBR and proposed CPVBR), a start-up and load change transient simulation is conducted. In this study, the machine stator is connected in a wye configuration which also includes a neutral wire as depicted in Figure 5–10. The machine is initially at stand still and no mechanical load is applied to the rotor. At t = 0s, the stator is energized by a three-phase voltage source with a phase voltage equal to 186V (1.55pu) and the rotor begins to accelerate. After the machine reaches the steady-state speed, at t = 0.3s a mechanical torque of 12Nm (i.e., nominal load) is -250-1250125250v as (V)0.905 0.91 0.915 0.92-20-1001020Time (s)i as (A)a)b)(I)(II)(III)Reference model:(I)(II)(III)(I): | vg| = 1.0 pu(III): | vg| = 1.55 pu(II): | vg| = 1.2 pu146 applied to the shaft and the simulation continues to run until t = 0.5s. The transient simulation results for the phase a current, electromagnetic torque, and mechanical speed as obtained by all subject models are presented in Figure 5–16. As it can be seen in Figure 5–16, all the subject models are able to produce similar and consistent results with respect to the reference solution. In Figure 5–16(a), it is observed that the machine draws high currents during the start-up with minimal harmonic distortions. This is due to fact that magnetizing flux inside the machine does not change instantly and reaches saturated levels (corresponding to 1.55pu stator phase voltage) with a time constant. After the machine reaches steady-state, the third harmonic distortion begin to appear on the stator phase currents due to the existence of neutral wire. It is also observed that the load change transient causes a phase shift between the fundamental and the third harmonic component of the stator phase current. Furthermore, Figures 5–16(b)–(c) show the dynamic behavior of the machine electromagnetic torque and speed during this transient study. To better analyze the performance of each subject model in the conducted transient study, a magnified view of the stator phase a current and electromagnetic torque right after the load change transient is presented in Figure 5–17. As it is seen in Figure 5–17, the qd0 model has a slight deviation from the reference model due to excess current in the snubber branches. The proposed VBR model (with variable parameters) is also shown to be able to provide highly accurate results compared to the reference model due to not having filter approximations or large auxiliary snubber resistors. Also, the proposed CPVBR model is demonstrated to be able to 147 obtain results with comparable high accuracy to the VBR model due to accurate tuning of the approximation filter. Figure 5–16. Transient response of several variables of the induction machine with wye‐connected stator with neutral wire during startup and load change for: (a) stator phase a current ias, (b) electromagnetic torque, and (c) rotor speed. -120-60060120i as (A)-4504590135Time (s)T e (N.m) 0 0.1 0.2 0.3 0.4050100150200Time (s) m (rad/s)061218185189Reference qd0 VBR CPVBRa)b)c)See Fig. 17(b)See Fig. 17(a)e Fig. 5– (a) See Fig. 5–17(b)148 Figure 5–17. Magnified view of machine variables at the moments after load change transient for: (a) stator phase a current ias, and (b) electromagnetic torque. 5.5.5. Numerical Accuracy Comparison To assess the numerical accuracy of the subject models, the numerical errors in the stator phase a current and the electromagnetic torque, obtained by all implemented models are calculated with respect to the reference solution and are summarized in Table 5–1. The numerical errors in all cases are based on the 2-norm error calculation presented in (2–69). As can be seen in Table 5–1, the qd0 model has the largest numerical error (0.22% in stator current and 0.19% in torque) due to the snubber resistors. The proposed VBR model provides results with almost no numerical error (0.004% in 0.3345 0.3355 0.336516182022Time (s)i as (A)0.325 0.326 0.32716.116.1216.1416.16Time (s)T e (N.m) Reference qd0 VBR CPVBRb)a)149 stator current and 0.006% in torque), which is mostly due to the solver discretization. Also, the proposed CPVBR model is able to provide accurate results with only a slight numerical error (0.13% in stator current and 0.14% in torque) which is due to filter approximation. TABLE 5–1. 2‐NORM ERROR OF THE SIMULATION RESULTS OBTAINED BY THE SUBJECT MODELS OVER THE 0.5‐SECOND TRANSIENT CASE‐STUDY Model Stator Phase a Current (ias) Electromagnetic Torque (Te) qd0 model (with snubbers) 0.22 % 0.19 % Proposed VBR model (with variable inductances) 0.004 % 0.006 % Proposed CPVBR model 0.13 % 0.14 % 5.5.6. Computational Performance Comparison The computational performance and numerical efficiency of the subject models in the presented 0.5-second transient study presented in Section 5.5.4 are compared in terms of their largest eigenvalue, total number of time-steps, average CPU time per-step, and total required CPU time, as tabulated in Table 5–2. As it can be seen in Table 5–2, the qd0 model has the largest eigenvalue (1.85×105) due to its large snubber resistors. This results in the qd0 model requiring the most number of time-steps (110,969) to obtain simulation results, which despite low amount of required average CPU time per step (63.5μs), leads to the longest required CPU time (7.05s). Table 5–2 also shows that the proposed VBR model has the smallest 150 eigenvalue (1146) of all subject models which results in the least number of time-steps (424) to obtain the results. However, due to its variable interfacing inductances, the VBR model requires the largest amount of average CPU time per-step (342μs), despite being able to obtain solutions faster in less CPU time (0.145s) compared to the qd0 model. Finally, it is observed in Table 5–2 that the proposed CPVBR model has a larger eigenvalue (5244) compared to the VBR model, which leads to slightly more time steps (771). This is due to the filter pole that is used for stator voltage approximation in (5–70). Meanwhile, due its constant-parameter interfacing structure, the new CPVBR model requires significantly smaller amount of average CPU time per-step (90.8μs) and the least total CPU time (0.07s), compared to the qd0 and VBR models. This observations highlights the benefits achieved the constant parameter interface of the new CPVBR model compared to alternative state-of-the-art models. TABLE 5–2. COMPUTATIONAL PERFORMANCE OF THE SUBJECT MODELS FOR THE 0.5 SECOND TRANSIENT CASE‐STUDY Model Largest Eigenvalue Number of Steps Average CPU Time Per‐Step CPU Time qd0 model (with snubbers) 1.85×105 110,969 63.5 µs 7.05 s Proposed VBR model (with variable inductances) 1146 424 342 µs 0.145 s Proposed CPVBR model 5244 771 90.8 µs 0.07 s 151 CHAPTER 6: CONCLUSIONS AND FUTURE WORK In this dissertation, the focus was on formulating computationally efficient models of electrical machines for SVB simulation packages. Electrical machines by nature have variable electromagnetic structure due to rotation of windings and also effects of magnetic saturation. To achieve fast simulation, the machine models are preferred to have constant-parameter structure, should not be numerically stiff, and should be able to directly interface with arbitrary external networks. Traditionally, the machine models have been implemented either using the well-known qd0 modeling approach (which has constant-parameter structure, yet require artificial interfacing snubber circuits which can cause numerical stiffness) or the CCPD modeling approach (which has direct interface, but variable-parameter structure). Recently, the VBR modeling approach has been gaining attention and is being currently utilized to represent electrical machines in some popular EMT simulation packages. With the advancing of power-electronic converters, more advanced multi-phase machines are being increasingly considered for energy conversion in industrial applications and renewable energy generation. Despite their advantages, multi-phase machines often have unique characteristics which result in complex and 152 computationally expensive mathematical models when implemented using traditional approaches (i.e., CCPD and qd0) in EMT simulation programs. Prior to this dissertation, the VBR and CPVBR models have only been developed for traditional three-phase machines (i.e., induction and synchronous). 6.1. Contributions and Anticipated Impact 6.1.1. Contributions In this dissertation, the VBR modeling approach has been expanded to multi-phase (i.e. six- and five-phases) electrical machines along with consideration of their unique electromagnetic characteristics such as magnetic saturation and flux harmonics. The new models advance the state-of-the-art and offer significant computational and practical advantages compared to existing alternative models when implemented in typical SVB EMT simulation programs. Overall, the contributions made by this research work are summarized as follows: Contribution #1 The first contribution is presented in Chapter 2, where a constant-parameter VBR (CPVBR) model for six-phase synchronous machines has been proposed for the first time. The desirable constant-parameter RL branch interfacing circuit is achieved by removing the dynamic saliency in the sub-transient inductances and appropriately compensating the sub-transient back emf sources. This new formulation also involves an approximation and a relaxation of algebraic loop using 153 a low-pass filter in order to yield the formulation explicit. However, the resulting CPVBR possesses significant advantages over the alternative/existing models (i.e. CCPD, basic VBR, and qd0) in terms of its interfacing and numerical efficiency, which comes only at a slight reduction of accuracy as compared to the directly interfaced models (CCPD and basic VBR). According to the transient simulation results (demonstrated in Section 2.5.2), the proposed CPVBR model for the six-phase synchronous machine was able to achieve high accuracy (numerical errors under 1%), while at the same time reducing the total CPU time significantly (by a factor of 4 to 5 compared to VBR and CCPD models), resulting in major improvement in simulation speed. Contribution #2 The second contribution is presented in Chapter 3, where the magnetic saturation is considered in CPVBR model for six-phase synchronous machines. Moreover, in order to further simplify the model interface, a decoupled CPVBR (DCPVBR) formulation has been proposed, which achieves simultaneously constant-parameter and magnetically-decoupled RL branches for its interface. This type of interface may be preferred in many commonly used transient simulation programs. Using extensive simulation studies, the proposed CPVBR and DCPVBR models have been verified and demonstrated to offer advantages over the existing qd0 and VBR models in terms of simulation speed, numerical efficiency, and interfacing 154 compatibility. In the transient studies demonstrated in Section 3.5.2, both the directly interfaced CPVBR and DCPVBR models are shown to take significantly less CPU time (both providing a reduction of CPU time by a factor of 5) compared to the VBR model while providing accurate results with numerical errors well under 1%. Moreover, the proposed DCPVBR model was shown to be able to reduce the cost of computations per time-step by a factor of 2 compared to the CPVBR model in addition to providing a fully decoupled interfacing circuit which is compatible with a wide range of simulation software packages. Contribution #3 As the third contribution presented in Chapter 4, a new constant-parameter VBR model has been achieved for five-phase synchronous machines. The rotor circuits such as damper and field windings are considered. Moreover, the model considers air-gap flux harmonics up to third component as well as their cross-couplings with the fundamental component. A simple and easy-to-implement interfacing circuit is achieved with constant-parameter RL branches using approximation of the machine saliency and cross-couplings. The proposed model has been validated by experimental measurements and is shown to predict the machine’s waveforms accurately under various conditions (from no-load to heavy-load) as well as when connected to a diode-rectifier with switching and commutations. Computer simulations have been conducted to validate the accuracy and computational performance of the new CPVBR model against the existing conventional models of 155 five-phase synchronous machines. It is demonstrated that the proposed CPVBR model is able to provide accurate results, while significantly outperforming the conventional qd0, CCPD, and VBR models in terms of numerical efficiency and simulation speed. According to the transient simulation study (demonstrated in Section 4.5.2), the proposed CPVBR model for the five-phase synchronous machine was able to achieve accurate simulation results with numerical errors well under 1%, while at the same time reducing the total CPU time significantly (by a factor of 3 compared to VBR and CCPD models) which resulted in a major improvement in simulation speed. Contribution #4 The fourth and last contribution is a new model for saturable three-phase induction machines considering the air-gap flux harmonics, using the qd0 and the VBR approaches. In Chapter 5, a new constant-parameter VBR (CPVBR) model has been proposed for saturable induction machines with air-gap flux harmonics, wherein the magnetizing flux spatial harmonics caused by saturation and their effects are considered. The explicit formulation is achieved through an algebraic-loop relaxation method using tuned low-pass filters. The proposed CPVBR model is validated through extensive steady-state and transient studies, under various operating conditions and system configurations, and is shown to provide highly accurate results while significantly outperforming the existing state-of-the-art models in terms of simulation speed and numerical efficiency. According to the 156 transient simulation study (demonstrated in Section 5.5.4), the proposed CPVBR model for the three-phase induction machine was able to reduce the simulation computational cost per each time-step by a factor of 4 (compared to the variable-parameter VBR model) which has resulted in a reduction of total required CPU time by a factor of 2. Simulation accuracy has been shown to be maintained with numerical errors remaining well under an acceptable threshold of 1%. In addition, the interfacing circuit of the proposed CPVBR model benefits form constant-parameter RL branches connected in series with controlled voltage sources which makes it very convenient for implementation in most EMT simulation packages with no issues in interfacing with arbitrary external networks. 6.1.2. Anticipated Impact Prior to this research, the available options for modeling multi-phase machines are either based on CCPD or qd0 approaches. When such machines are simulated in conjunction with power-electronic converters, the high number of switching events and discrete states can significantly reduce simulation performance (e.g., variable parameter nature of CCPD model that needs to be recalculated at each time step) and even cause instability (e.g., due numerically stiff qd0 models as a result of large snubber impedances). The proposed new models have constant parameter interfaces and numerically efficient structure, which makes them ideal for including in many commonly-used EMT simulation packages (or ideal replacements of existing models if there are such) while offering appreciable computational gains. It is 157 envisioned that the new efficient models presented in Chapters 2–5 will greatly benefit many researchers and engineers around the world who are working on applications of multi-phase machines and their controls, and require fast and accurate EMT simulations for system-level studies. 6.2. Future Work To conclude this dissertation, three directions of potential extensions of the future work are discussed: Direction #1 In this dissertation, efficient models have been proposed only for symmetrical machines which could also include air-gap flux harmonics. The modeling methodology can be further extended to include models of machines with structural asymmetries, such as rotor eccentricities, and internal winding faults. These problems are common challenges in modelling of electrical machines of all sizes, and are active research topics with applications in power system protection and electric drives. Developing machine models with such capabilities for real-time ETM simulators can be of great use for testing of various protection schemes and control algorithms since creating faults in actual healthy machines and devices is prohibitively costly especially for high power utility-scale equipment. 158 Direction #2 One of the most dominant class of electrical machines driven by power-electronic converters is the permanent magnet synchronous machines (PMSMs), which are now becoming increasingly considered as high power motors and generators in renewable energy applications. However, computationally efficient models of PMSMs considering flux harmonics, saturation, and de-magnetization effects are still hardly available. Currently, a brute force modeling approach for such machines is being utilized based on either the CCPD or qd0 models and large look-up tables generated from either experimental measurements or finite-element design tools. Such models are not only computationally costly but impose numerical stiffness and stability issues when implemented in conventional EMT simulators. The CPVBR modeling approach presented in this dissertation can be applied to this class of machines in order to obtain fast and efficient models that are more suitable for most EMT simulation programs. Direction #3 With availability of more powerful computers and processors, the real-time transient simulation of power and power-electronic systems of small and large scales with interfacing to the actual hardware (hardware-in-the-loop) becoming more and more attractive option, and in many cases (e.g. HVDC systems, testing of controllers and protection relays, etc.) may be the only solution for design and operation of such systems. Such simulation platforms are becoming a key part in designing and tuning 159 controls and prototyping devices. 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Energy Convers., vol. 33, no. 2, pp. 900–903, Jun. 2018. 168 Appendix A: Parameters for Case Study of Section 2.5 WOUND‐ROTOR SIX PHASE SYNCHRONOUS MACHINE PARAMETERS [14] Parameters Value Rated Phase Voltage / Power 240 V / 100 kVA Rated Rotor Speed / Torque 1800 rpm / 530 N.m Lmd, Lmq , Lls 3 mH , 1.4 mH , 150 µH Llfd , Llkq , Llkd 120 µH , 180 µH , 140 µH rs , rrkq , rrkd , rfd 16 mΩ , 2.5 mΩ , 2.3 mΩ , 1.6 mΩ La1a2 , La1b2 , La1c2 43 µH , -43 µH , 0 H Rcable , Lcable, Rgrid , Lgrid 0.05 Ω , 50 µH, 0.05 Ω , 50 µH Rload , Ldc, Cdc 2.5 Ω , 10 mH, 6 mF 169 Appendix B: Parameters for Case Study of Section 3.5 SIX‐PHASE SYNCHRONOUS MACHINE PARAMETERS [14] Parameters Value Rated Phase Voltage / Power 240 V / 100 kVA Rated Rotor Speed / Torque 1800 rpm / 530 Nm Lmdu, Lmqu , Lls 3 mH , 1.4 mH , 150 µH Llfd , Llkq , Llkd 120 µH , 180 µH , 140 µH rs , rrkq , rrkd , rfd 16 mΩ , 25 mΩ , 23 mΩ , 16 mΩ Lla1a2 , Lla1b2 , Lla1c2 43 µH , -43 µH , 0 H MAGNETIC SATURATION DATA λm (Wb) 0.04 0.27 0.56 0.74 0.84 0.9 0.93 0.96 1.00 im (A) 14.4 97.9 207 280 325 355 372 392 430 λm (Wb) 1.05 1.1 1.17 1.23 1.25 1.26 1.27 1.28 im (A) 536 691 925 1131 1200 1234 1269 1304 Saturation data obtained using the approach in [85] as           22 2 22 arctan arctan ln 1 ln 1dm m T T m T T T TdT T T m T a mTMiM M                        , where  1 10.5d ls mduM L L   ,  1 10.5a ls mduM L L   , 40T  , 1.02T  Line impedance: 0.05 lineR   , 50 μHlineL  . Rectifier dc filter: 10 mHdcL  , 6 mFdcC  . 170 Appendix C: Parameters for Case Study of Section 4.5 LABORATORY FIVE‐PHASE SYNCHRONOUS MACHINE PARAMETERS Machine parameters measured at ifd = 16 A (referred to stator) Parameters Value Lmq1, Lmd1 , Lmq3, Lmd3 0.4 mH , 1.35 mH , 90 µH , 54 µH Lm13 , Lmqd3 , Lmqd13 -84 µH , -6.1 µH , 13.2 µH Lls , Llfd 150 µH , 120 µH rs , rfd 0.19 Ω , 0.033 Ω Line resistance: 0.04 liner   . Ten‐pulse diode rectifier: Harris semiconductor: RHRG7560, 0.091 onR   , 0.637 VonV  . Rectifier dc filter: 12 mHfL  , 0.6 fr   , 470 μFfC  . 171 Appendix D: Parameters for Case Study of Section 5.5 INDUCTION MACHINE PARAMETERS [20] Parameters Value Rated Voltage / Power / Poles 208 V / 3 hp / 4 Lmu , Lls , Llr 62.7 mH , 2.8 mH , 2.7 mH rs , rr 0.515 Ω , 0.527 Ω kw1 , kw3 0.9022 , 0.3349 MAGNETIC SATURATION DATA [20] im (A) 0 3 4 5 6.5 8 10 λm (Wb) 0 0.188 0.249 0.307 0.385 0.447 0.502 λm3 (mWb) 0 0.1 0.2 0.4 1.6 6 17.6 im (A) 11.5 13 15 19 22.5 26 30 λm (Wb) 0.534 0.563 0.597 0.639 0.669 0.693 0.715 λm3 (mWb) 26.6 33 41 54.8 63.2 69.8 75.3 Line impedance: 0.04 liner   , 300 μHlineL  .
Thesis/Dissertation
eng
Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Efficient modelling of special purpose multi-phase electrical machines for transient simulation programs
Text