@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Applied Science, Faculty of"@en, "Electrical and Computer Engineering, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Chiniforoosh, Sina"@en ; dcterms:issued "2012-04-03T17:57:47Z"@en, "2012"@en ; vivo:relatedDegree "Doctor of Philosophy - PhD"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """Power electronic converters are used in a wide range of applications as well as being the enabling technology for interfacing the alternative energy resources and many loads in modern power systems. The methodology of developing the so-called dynamic average-value models (AVMs) for such converters is based on averaging the variables (currents and voltages) within a switching interval resulting in numerically efficient models that are much more suitable than the detailed switching models for system-level studies as well as numerical linearization and the respective small-signal analysis. However, the AVMs available in the literature for line-commutated converters have several limitations such as neglecting the effects of losses, being only valid in certain operational modes and under balanced excitation, as well as employing a simplified representation of the multi-phase transformer in high-pulse-count converters. Moreover, a unified AVM methodology for high-pulse-count converters has not yet been established. In this thesis, a generalized AVM methodology is developed for voltage-source- and rotating-machine-fed multi-pulse line-commutated converters for both classes of transient simulation software packages, i.e., state-variable-based and nodal-analysis-based electromagnetic transient program (EMTP) type. The previously-developed AVM approaches, i.e., analytical and parametric, are extended to the EMTP-type programs, and the indirect and direct methods of interfacing the models with external circuit-network are introduced and compared. For the machine-converter systems, the effects of machine and bridge losses are taken into account in the new AVM. Finally, a generalized dynamic AVM methodology is developed for high-pulse-count converters based on the parametric approach. An effective multi-phase transformer model is developed in transformed (qd0) and phase (abc) variables. An efficient transformer model is also developed, which accurately represents the multi-phase transformer using an equivalent three-phase formulation. The proposed generalized AVM remains valid for all operational modes under balanced and unbalanced excitation. This model is employed for AVM implementation in state-variable-based and EMTP-type programs. Extensive simulation and experimental studies are carried out on several example systems in order to compare the developed AVMs against the detailed and previously-developed average models in time- and frequency-domains. The results demonstrate the great accuracy of the proposed AVMs and a significant improvement compared to the previously-developed models."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/41917?expand=metadata"@en ; skos:note "GENERALIZED DYNAMIC AVERAGE MODELING OF LINE-COMMUTATED CONVERTER SYSTEMS IN TRANSIENT SIMULATION PROGRAMS by Sina Chiniforoosh B.Sc., Shiraz University, 2005 M. Sc., Sharif University of Technology, 2007 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Electrical and Computer Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2012 © Sina Chiniforoosh, 2012 ii Abstract Power electronic converters are used in a wide range of applications as well as being the enabling technology for interfacing the alternative energy resources and many loads in modern power systems. The methodology of developing the so-called dynamic average-value models (AVMs) for such converters is based on averaging the variables (currents and voltages) within a switching interval resulting in numerically efficient models that are much more suitable than the detailed switching models for system-level studies as well as numerical linearization and the respective small-signal analysis. However, the AVMs available in the literature for line-commutated converters have several limitations such as neglecting the effects of losses, being only valid in certain operational modes and under balanced excitation, as well as employing a simplified representation of the multi-phase transformer in high-pulse-count converters. Moreover, a unified AVM methodology for high- pulse-count converters has not yet been established. In this thesis, a generalized AVM methodology is developed for voltage-source- and rotating-machine-fed multi-pulse line-commutated converters for both classes of transient simulation software packages, i.e., state-variable-based and nodal-analysis-based electromagnetic transient program (EMTP) type. The previously-developed AVM approaches, i.e., analytical and parametric, are extended to the EMTP-type programs, and the indirect and direct methods of interfacing the models with external circuit-network are introduced and compared. For the machine-converter systems, the effects of machine and bridge losses are taken into account in the new AVM. Finally, a generalized dynamic AVM methodology is developed for high-pulse-count converters based on the parametric approach. An effective multi-phase transformer model is developed in transformed (qd0) and phase (abc) variables. An efficient transformer model is also developed, which accurately represents the multi-phase transformer using an equivalent three-phase formulation. The iii proposed generalized AVM remains valid for all operational modes under balanced and unbalanced excitation. This model is employed for AVM implementation in state-variable- based and EMTP-type programs. Extensive simulation and experimental studies are carried out on several example systems in order to compare the developed AVMs against the detailed and previously-developed average models in time- and frequency-domains. The results demonstrate the great accuracy of the proposed AVMs and a significant improvement compared to the previously-developed models. iv Preface The research results presented in this thesis have been partly published, accepted for publication, or ready for submission as several journal articles and publications in conference proceedings. In all publications, I was responsible for developing the models, conducting the simulation and experimental studies, compiling the results and conclusions, as well as preparing the manuscripts. My research supervisor, Dr. Juri Jatskevich, has provided supervisory comments and corrections during the process of modeling, conducting the studies, and writing the manuscripts, which I have accommodated in the final manuscripts. The other co-authors have also provided comments, suggestions, and constructive feedback. A version of Chapters 1 and 2 has been published: S. Chiniforoosh, J. Jatskevich, et al., “Definitions and applications of dynamic average models for analysis of power systems,” IEEE Transactions on Power Delivery, vol. 25, no. 4, pp. 2655-2669, Oct. 2010. A version of Chapter 3 has been published: S. Chiniforoosh, A. Davoudi, and J. Jatskevich, “Averaged-circuit modeling of line- commutated rectifiers for transient simulation programs,” in Proc. IEEE International Symposium on Circuits and Systems (ISCAS ‘10), Paris, France, May-Jun. 2010. A version of Chapter 3 is ready for submission: S. Chiniforoosh, H. Atighechi, and J. Jatskevich, “Directly-interfaced dynamic average models for representing line-commutated rectifier systems in EMTP-type programs,” to be submitted. A version of Chapters 1, 2, and 4 has been published: S. Chiniforoosh, H. Atighechi, A. Davoudi, J. Jatskevich, et al., “Dynamic average modeling of front-end diode rectifier loads considering discontinuous conduction mode and unbalanced operation,” IEEE Transactions on Power Delivery, vol. 27, no. 1, pp. 421-429, Jan. 2012. v A version of Chapters 1, 2, and 4 is ready for submission: S. Chiniforoosh, H. Atighechi, and J. Jatskevich, “Steady-state and dynamic impedance of front-end diode rectifier loads as predicted by dynamic average-value models,” to be submitted. A version of Chapter 5 has been accepted for publication: S. Chiniforoosh, A. Davoudi, P. Alaeinovin, and J. Jatskevich, “Dynamic modelling and characterisation of vehicular power system considering alternator iron core and rectifier losses,” to appear in IET Electrical Systems in Transportation, 2012, 25 pages, (Manuscript ID:EST-2011-0031.R1). A version of Chapter 6 is ready for submission: S. Chiniforoosh, H. Atighechi, and J. Jatskevich, “Generalized dynamic average modeling of high-pulse-count rectifier systems in transient simulation programs,” to be submitted. vi Table of Contents Abstract .................................................................................................................................... ii Preface ..................................................................................................................................... iv Table of Contents ................................................................................................................... vi List of Tables ........................................................................................................................... x List of Figures ......................................................................................................................... xi Acknowledgements ............................................................................................................ xviii Dedication .............................................................................................................................. xx Chapter 1: Introduction ........................................................................................................ 1 1.1 Motivation ................................................................................................................. 1 1.2 Literature Review...................................................................................................... 3 1.2.1 Three-Phase (Six-Pulse) Converter Models.......................................................... 7 1.2.2 High-Pulse-Count Converter Models ................................................................... 8 1.3 Research Objectives and Anticipated Impact ......................................................... 10 1.4 Thesis Organization ................................................................................................ 13 Chapter 2: Dynamic Average Modeling in State-Variable-Based Simulators ............... 15 2.1 State-Variable-Based Simulators ............................................................................ 15 2.2 Detailed Analysis .................................................................................................... 18 2.3 Dynamic Average Modeling ................................................................................... 22 2.3.1 Analytical Approach ........................................................................................... 22 2.3.1.1 Classical Reduced-Order Model (AVM-1) ................................................. 25 2.3.1.2 Improved Reduced-Order Model (AVM-2)................................................ 26 2.3.2 Parametric Approach .......................................................................................... 27 2.4 Implementation of the AVMs in SV-Based Programs ........................................... 28 2.5 Small-Signal Frequency-Domain Analysis............................................................. 29 vii Chapter 3: Dynamic AVM Formulation for EMTP-Type Solution ................................ 33 3.1 Electro-Magnetic Transient Programs .................................................................... 33 3.2 Detailed Analysis .................................................................................................... 35 3.3 Dynamic Average Modeling ................................................................................... 40 3.3.1 Analytical AVM for EMTP-type Solution ......................................................... 40 3.3.2 Indirectly-Interfaced Analytical Average-Value Model (IIAAVM) .................. 43 3.3.3 Directly-Interfaced Analytical Average-Value Model (DIAAVM) ................... 45 3.3.3.1 Direct Interface in qd Variables ................................................................ 47 3.3.3.2 Direct Interface in abc Variables ............................................................... 50 3.3.4 Parametric AVM for EMTP-Type Solution ....................................................... 53 3.4 Example of AVM Implementation in PSCAD/EMTDC ........................................ 55 3.5 Comparison of Direct and Indirect Interfacing Methods ........................................ 57 Chapter 4: AVM Verification in SV-Based and EMTP-Type programs ....................... 62 4.1 Example Micro-Wind Turbine Generator System .................................................. 62 4.1.1 Large-Signal Time-Domain Analysis ................................................................. 63 4.1.2 Small-Signal Frequency-Domain Analysis......................................................... 64 4.2 Example 3-Phase (6-Pulse) Front-End Rectifier system ........................................ 66 4.2.1 Steady-State Time Domain Analysis .................................................................. 67 4.2.1.1 Operation in DCM ...................................................................................... 67 4.2.1.2 Operation in CCM....................................................................................... 69 4.2.2 Dynamic Performance under Balanced Conditions ............................................ 71 4.2.2.1 Balanced Operation in DCM ...................................................................... 71 4.2.2.2 Balanced Operation in CCM ....................................................................... 72 4.2.3 Dynamic Performance under Unbalanced Conditions ........................................ 76 4.2.3.1 Unbalanced Operation in DCM .................................................................. 76 4.2.3.2 Unbalanced Operation in CCM .................................................................. 78 viii 4.2.4 System Impedance Analysis ............................................................................... 81 4.2.4.1 Steady-State Analysis ................................................................................. 81 4.2.4.2 Small-Signal Frequency-Domain Analysis................................................. 83 4.2.5 Conclusion .......................................................................................................... 87 Chapter 5: Inclusion of Losses in Machine-Fed Converter Systems .............................. 88 5.1 Introduction ............................................................................................................. 88 5.2 Example Vehicular Power System Architecture..................................................... 89 5.3 System Detailed Model ........................................................................................... 91 5.3.1 Representation of Rotational Losses ................................................................... 93 5.3.2 Verification of Detailed Model ........................................................................... 95 5.3.3 Battery Model ..................................................................................................... 97 5.4 System Dynamic Average-Value Modeling ........................................................... 98 5.4.1 Alternator Model in Transformed qd Coordinates and Variables ..................... 98 5.4.2 Average-Value Modeling of Non-Ideal Rectifier ............................................. 101 5.5 Case Studies .......................................................................................................... 104 Chapter 6: Generalized Dynamic AVM for High-Pulse-Count Converters ................ 110 6.1 Introduction ........................................................................................................... 110 6.2 High-Pulse-Count Converter System Structures .................................................. 110 6.3 Multi-Phase Transformer Modeling ..................................................................... 115 6.3.1 Original Interconnected Transformer Model .................................................... 115 6.3.2 Analytically-Derived Equivalent Compacted Transformer Model ................... 121 6.3.3 Round Shifted Equivalent Model...................................................................... 126 6.3.4 Compensation for Asymmetric Leakages ......................................................... 133 6.4 Detailed Analysis .................................................................................................. 134 6.4.1 Detailed Analysis using Uncompensated Transformer Models........................ 135 6.4.2 Detailed Analysis using Compensated Transformer Models ............................ 139 ix 6.4.3 Operational Mode Analysis .............................................................................. 144 6.5 Dynamic Average Modeling ................................................................................. 151 6.5.1 Reduced-Order Analytical AVM ...................................................................... 151 6.5.2 Generalized Dynamic Average Modeling Methodology .................................. 152 6.5.2.1 Generalized Transformer Model in Transformed Variables ..................... 153 6.5.2.2 Generalized Rectifier Model ..................................................................... 159 6.5.2.3 Implementation in SV-Based Simulators.................................................. 160 6.5.3 Generalized Model with Collapsed Transformer .............................................. 161 6.5.4 Implementation in EMTP-Type Programs........................................................ 164 6.6 Model Verification in Time Domain .................................................................... 170 6.6.1 Steady-State Analysis ....................................................................................... 170 6.6.2 Transient Analysis under Balanced Conditions ................................................ 171 6.6.3 Transient Analysis under Unbalanced Conditions ............................................ 182 6.7 Model Verification in Frequency Domain ............................................................ 188 Chapter 7: Conclusions and Future Work ...................................................................... 192 7.1 Conclusions and Contributions ............................................................................. 192 7.2 Future Work .......................................................................................................... 196 References ............................................................................................................................ 198 Appendices ........................................................................................................................... 208 Appendix A Parameters of the Six-Pulse Converter Example Systems ........................... 208 A.1 Parameters of the 6-Pulse Converter System Considered in Section 3.5 ......... 208 A.2 Parameters of the Micro-Wind Turbine System Considered in Chapter 4 ....... 208 A.3 Parameters of the Front-End Rectifier System Considered in Chapter 4 ......... 208 A.4 Parameters of the Vehicular Power System Considered in Chapter 5 .............. 209 Appendix B Parameters of the 18-Pulse Example System Considered in Chapter 6 ....... 210 x List of Tables Table 1.1 Dynamic average models of the three-phase (six-pulse) rectifier systems. ........... 8 Table 1.2 Dynamic models of high-pulse-count converter systems. ..................................... 9 Table 2.1 Operational modes of the conventional 3-phase (6-pulse) rectifier. .................... 20 Table 4.1 Maximum values of the dc bus voltage and current for different system topologies. ............................................................................................................................... 69 Table 4.2 Steady-state values of the dc bus voltage and current predicted by various models in DCM ....................................................................................................................... 69 Table 4.3 Steady-state values of the dc bus voltage and current predicted by various models in CCM-1. ................................................................................................................... 70 Table 4.4 System eigenvalues predicted by different models in DCM and CCM-1. ......... 74 Table 4.5 Comparison of simulation time steps of different models .................................. 81 Table 4.6 System input impedance in DCM and CCM-1 ................................................... 83 Table 6.1 Operational modes of the conventional 9-phase (18-pulse) rectifier. ............... 147 Table 6.2 Eigenvalues of the 18-pulse rectifier system (without dc capacitor) ................ 178 Table 6.3 Eigenvalues of the 18-pulse rectifier system with dc capacitor ........................ 182 Table 6.4 Comparison of simulation time steps for different models............................... 188 xi List of Figures Figure 1.1 Typical configurations of multi-pulse line-commutated rectifier systems. .......... 4 Figure 2.1 Flowchart of a typical variable-step state-variable-based solver. ..................... 17 Figure 2.2 Simplified circuit diagram of a typical three phase front-end rectifier load system. .................................................................................................................................... 18 Figure 2.3 Typical current and voltage waveforms of the six-pulse rectifier: (a) operation in DCM; and (b) operation in CCM. ........................................................................................... 20 Figure 2.4 (a) Relationship among the variables in the converter and arbitrary reference frames; and (b) Typical waveforms and the respective transformed waveforms in converter reference frame. ...................................................................................................................... 24 Figure 2.5 Block diagrams of AVM implementations in SV-based simulators: (a) analytically-derived (AVM-1 and AVM-2); and (b) parametric (PAVM). ............................ 28 Figure 2.6 System-level impedance-based representation of subsystems interconnected through an ac-dc converter...................................................................................................... 29 Figure 3.1 Flowchart of a typical nodal-analysis-based solver........................................... 34 Figure 3.2 Topological variations of the ac side impedance in a typical three phase front- end rectifier load system (a) series impedance (b) general network with series and parallel branches. ................................................................................................................................. 36 Figure 3.3 Three-phase currents at the bridge terminals for DCM and CCM operation: (a) without shunt filters; and (b) with shunt filters. ...................................................................... 37 Figure 3.4 Three-phase ac currents of the system at the input source terminals in presence of the shunt filters for DCM and CCM operations. ................................................................ 37 Figure 3.5 Circuit diagram of the AVM described by (3-6)-(3.8), (3-20), (3-21). ............. 43 Figure 3.6 Circuit diagram of the IIAAVM using PSCAD-like approach. ........................ 44 Figure 3.7 Circuit diagram of the DIAAVM in qd variables. ........................................... 49 xii Figure 3.8 Equivalent circuit diagram of the DIAVM in abc variables. ........................... 52 Figure 3.9 Indirectly-interfaced parametric average-value model. ..................................... 55 Figure 3.10 Example of IIPAVM implemented in PSCAD: (a) PAVM block together with controllable sources and interfacing ports; and (b) AVM module interfaced with external ac and dc subsystems. .................................................................................................................. 56 Figure 3.11 DC bus waveforms predicted by the models with sµ50 time step. ................ 59 Figure 3.12 Input phase current predicted by the models with sµ50 time step. .................. 59 Figure 3.13 DC bus waveforms predicted by the models with sµ500 time step. .............. 60 Figure 3.14 Input phase current predicted by the models with sµ500 time step. ............... 60 Figure 3.15 Input phase current predicted by the models with sµ1000 time step. ............. 61 Figure 4.1 Example PMSG micro-wind turbine generator system. .................................... 62 Figure 4.2 Waveforms of the example PMSG micro-wind turbine generator system. ....... 64 Figure 4.3 Speed-to-output-voltage transfer function for the example micro-wind turbine system. .................................................................................................................................... 65 Figure 4.4 (a) Regulation characteristic of the system without ac input filter with dc capacitor; and (b) Magnified view showing the performance of models in DCM region. ..... 68 Figure 4.5 Regulation characteristic of the system in CCM operation as predicted by different models. ..................................................................................................................... 70 Figure 4.6 Transient response of the six-pulse rectifier system in DCM predicted by different models. ..................................................................................................................... 72 Figure 4.7 Transient response of the rectifier system with ac filter and without the dc capacitor predicted by different models. ................................................................................. 74 Figure 4.8 Transient response of the six-pulse rectifier system with dc capacitor in CCM predicted by different models. ................................................................................................ 75 Figure 4.9 Transient response of the rectifier system to a change in ac voltages, leading to unbalanced operation in DCM, as observed in the ac currents predicted by various models. 77 xiii Figure 4.10 Transient response of the rectifier system to a change in ac voltages, leading to unbalanced operation in DCM, as observed in the dc bus predicted by various models. ....... 78 Figure 4.11 Transient response of the rectifier system to a change in ac voltages, leading to unbalanced operation in CCM, as observed in the ac currents predicted by various models. 79 Figure 4.12 Transient response of the rectifier system to a change in ac voltages, leading to unbalanced operation in CCM, as observed in the dc bus predicted by various models. ....... 80 Figure 4.13 Phasor diagrams representing unbalanced operation: (a) asymmetric input voltages, (b) positive sequence, (c) negative sequence, and (d) zero sequence components. 82 Figure 4.14 Dc-side impedance dcZ (top) and ac- side impedance qqZ (bottom) of the rectifier system in CCM-1 with input ac filter but without dc capacitor predicted by different models. .................................................................................................................................... 85 Figure 4.15 AC-side impedance qqZ (top) and qdZ (bottom) of the rectifier system in CCM-1 with input ac filter and dc capacitor predicted by different models. ......................... 86 Figure 5.1 A typical vehicular electric power system and its power conversion chain. ..... 90 Figure 5.2 A circuit diagram for automotive alternator-rectifier-battery systems (top) with a typical voltage regulator-exciter (bottom). .......................................................................... 92 Figure 5.3 Measured and simulated detailed responses to speed increase observed in the dc and ac sides. ............................................................................................................................ 96 Figure 5.4 Equivalent circuits of the considered alternator model in q and d axes. ...... 100 Figure 5.5 Parametric functions considering ideal and non-ideal rectifier diodes. .......... 103 Figure 5.6 Block diagram of the overall combined model depicting subsystems and their inputs and outputs. ................................................................................................................ 105 Figure 5.7 System response to the increase in alternator speed as predicted by the detailed and average-value models under fixed excitation. ................................................................ 105 xiv Figure 5.8 Input speed, excitation (field) current and the dc bus voltage predicted by the detailed and the average-value models when the system uses the voltage regulator-exciter. ............................................................................................................................................... 106 Figure 5.9 System response during the engine acceleration from 950 rpm to 4000 rpm predicted by the detailed and proposed average-value models. ............................................ 108 Figure 5.10 System transfer function from the field (excitation) voltage to the dc bus voltage predicted by the detailed and average-value models. ............................................... 109 Figure 6.1 Classification of high-pulse-count rectifier systems. ....................................... 111 Figure 6.2 Typical 9-phase (18-pulse) rectifier example system topology considered in this Chapter. ................................................................................................................................. 113 Figure 6.3 Typical 3- to 9-phase transformer structure for the 18-pulse rectifier example system. .................................................................................................................................. 114 Figure 6.4 Phasor diagrams of the primary and secondary voltages for the 3-to-9-phase transformer of Figure 6.3. ..................................................................................................... 115 Figure 6.5 Equivalent round transformer model depicting three-phase primary winding set and the equivalent secondary segments. ............................................................................... 128 Figure 6.6 The −k th set of extended secondary windings is replaced by its shifted equivalent in the fictitious round equivalent transformer model. ......................................... 130 Figure 6.7 Fictitious round shifted equivalent transformer model. .................................. 132 Figure 6.8 Initialization and steady state of the 18-pulse example system as seen at the dc bus and ac primary side currents predicted by various transformer models. ........................ 137 Figure 6.9 Initialization and steady state of the 18-pulse example system as seen at the secondary side currents predicted by various transformer models. ...................................... 138 Figure 6.10 Initialization and steady state of the 18-pulse example system as seen in the dc bus and ac primary side currents as predicted by compensated transformer models. .......... 140 xv Figure 6.11 Initialization and steady state of the 18-pulse example system as seen in the ac secondary side currents as predicted by compensated transformer models. ......................... 141 Figure 6.12 Comparison of the 18-pulse example system response as seen in the dc bus and ac primary side currents as predicted by symmetric and asymmetric transformer models. ............................................................................................................................................... 142 Figure 6.13 Comparison of the 18-pulse example system response as seen in the ac secondary side currents as predicted by symmetric and asymmetric transformer models. .. 143 Figure 6.14 Nine-phase ac currents of the 18-pulse converter in DCM-type modes (Modes 1-4). ....................................................................................................................................... 145 Figure 6.15 Nine-phase ac currents of the 18-pulse converter in CCM-type modes (Modes 5-11). ..................................................................................................................................... 146 Figure 6.16 Equivalent circuit of the generalized transformer model in transformed qd0 variables. ............................................................................................................................... 158 Figure 6.17 Block diagram of the generalized AVM implementation in SV-based simulators. ............................................................................................................................. 161 Figure 6.18 Equivalent circuit of the collapsed transformer model in transformed qd0 variables. ............................................................................................................................... 162 Figure 6.19 Block diagram of the generalized collapsed AVM implementation in SV-based simulators. ............................................................................................................................. 164 Figure 6.20 Equivalent circuit of the collapsed transformer model in phase ( abc ) variables. ............................................................................................................................... 166 Figure 6.21 Example of the generalized dynamic AVM implementation in PSCAD: (a) details of the PAVM module; and (b) detailes of the equivalent-colapsed transformer model. ............................................................................................................................................... 168 xvi Figure 6.22 Example of the generalized dynamic AVM implementation in PSCAD: (a) interfacing the PAVM and transformer blocks; and (b) AVM module with external ac and dc subsystems. ........................................................................................................................... 169 Figure 6.23 Regulation characteristic of the 18-pulse rectifier system as predicted by various models. ..................................................................................................................... 171 Figure 6.24 Transient response of the 18-pulse rectifier system (without dc filter capacitor) within Mode 5 (7-6 valve) as predicted by various models. ................................................. 174 Figure 6.25 Transient response of the 18-pulse rectifier system (without dc filter capacitor) during a transient from Mode 5 (7-6 valve) to Mode 10 (9-valve) predicted by various models. .................................................................................................................................. 175 Figure 6.26 Response of the 18-pulse rectifier system (without dc filter capacitor) primary and secondary currents during a transient from Mode 5 (7-6 valve) to Mode 10 (9-valve) as predicted by the detailed and proposed GAVM models. ...................................................... 176 Figure 6.27 Response of the 18-pulse rectifier system (without dc filter capacitor) secondary voltages during a transient from Mode 5 (7-6 valve) to Mode 10 (9-valve) as predicted by the detailed and proposed GAVM models. ...................................................... 177 Figure 6.28 Transient response of the 18-pulse rectifier system during a transient from Mode 5 (7-6 valve) to Mode 1 (4-2 valve) in th presence of dc filter capacitor as predicted by various models. ..................................................................................................................... 179 Figure 6.29 Transient response of the 18-pulse rectifier system primary and secondary currents during a transition from Mode 5 (7-6 valve) to Mode 1 (4-2 valve) in the presence of dc capacitor as predicted by detailed and proposed GAVM models. ................................... 180 Figure 6.30 Zoomed-in view of the 18-pulse rectifier system primary and secondary currents during Mode 1 (4-2 valve) operation in the presence of dc capacitor as predicted by detailed and proposed GAVM models. ................................................................................. 181 xvii Figure 6.31 Transient response of the 18-pulse rectifier system to a change in the input voltage leading to unbalanced operation in ac side as predicted by various models. ........... 184 Figure 6.32 Response of the 18-pulse rectifier system primary and secondary currents during a transient leading to unbalanced operation as predicted by detailed and proposed GAVM models. ..................................................................................................................... 185 Figure 6.33 Response of the 18-pulse rectifier system primary and secondary currents in transformed qd0 variables during a transient leading to unbalanced operation as predicted by detailed and proposed GAVM models. ................................................................................. 186 Figure 6.34 Impedance dcZ of the 18-pulse rectifier at dc side in Mode 5 (7-6 valve) without dc capacitor as predicted by various models. .......................................................... 190 Figure 6.35 Impedance qqZ of the 18-pulse rectifier at the ac side in Mode 10 (9-valve) without dc capacitor as predicted by the detailed and proposed GAVM models. ................ 190 Figure 6.36 Impedance qqZ of the 18-pulse rectifier at the ac side in Mode 5 (7-6 valve) in presence of the dc capacitor as predicted by various models. .............................................. 191 xviii Acknowledgements I would like to express my deepest gratitude to my research supervisor, Dr. Juri Jatskevich, for his great vision, immense inspiration, tremendous help, generous support, and endless patience throughout my PhD program. This research was made possible by his continuous encouragement, constant supervision, and invaluable feedback. The financial support for this research was made possible through the Natural Science and Engineering Research Council (NSERC) Discovery Grant entitled “Modelling and Analysis of Power Electronic and Energy Conversion Systems” and the Discovery Accelerator Supplement Grant entitled “Enabling Next Generation of Transient Simulation Programs” lead by Dr. Juri Jatskevich as a sole principle investigator. Special thanks are also owed to the members of my research supervisory committee, Drs. Hermann Dommel and Jose Marti for their help and valuable guidance, and to Drs. William Dunford, Shahriar Mirabbasi, Yusuf Altintas, and Wilsun Xu for being on my examining committee and dedicating their valuable expertise and time. I have received enormous help from all my colleagues and fellow graduate students in the Electric Power and Energy Systems research group at UBC through the years, for which I would like to express my sincere gratitude and appreciation. Particularly, I thank Ali Davoudi and Liwei Wang who have been very inspiring, and have set a great example of dedication and research productivity for me. I would also like to thank Pooya Alaeinovin, Hamid Atighechi, and Mehrdad Chapariha for our countless discussions on various research topics and student life in general. Many thanks also go to Amir Rasuli, Mehmet Sucu, Tom De Rybel, Michael Wrinch, Marcelo Tomim, Arvind Singh, Nathan Ozog, Leon Max Vargas, and all former and current members of the Electric Power and Energy Systems research group at UBC. I would also like to thank all the staff at the Department of Electrical and Computer Engineering and the Faculty of Graduate Studies. xix Finally, I would like to thank my family and friends for their continuous moral support, and their contribution to making my life a pleasant experience. My Mother’s love and support have always been with me even though she was, at times, thousands of miles away. I thank Sally for her love, patience, and for always being there for me. I thank Ehsan Azadi Yazdi, Maziar Eghbalnia, Mahdi Salehi, Bruce Krayenhoff, and all my friends for the wonderful times I spent with them and the tremendous delight they have brought to my life. xx Dedication This thesis, which is by no means a miracle, is dedicated to Sally for showing me that miracles are possible in life. And to my mother, Laleh Chiniforoosh whose love and support have pervaded the years of my life. And to the memory of my beloved father, Bahman Chiniforoosh, whose words of wisdom have been a great motivation for me. 1 Chapter 1: Introduction 1.1 Motivation The recent developments in power systems have been inextricably linked with integration of renewable and clean energy resources and the associated enabling power-electronic-based technology. Power electronic converters are increasingly employed in the power systems with the development of Smart Energy Grid and interfacing of a large number of Independent Power Producers (IPPs) and Distributed Energy Recourses (DERs). Power electronic converters are also utilized in a variety of applications such as High Voltage DC (HVDC) transmission systems, high-power dc supplies, excitation systems of large electric generators, as well as vehicular, naval, and aircraft power systems, etc. Design and analysis of all these modern electrical energy systems extensively relies on modeling and simulation of the power-electronic-based systems with switching components. The procedures for design and analysis of such complex systems typically involve a large number of computer studies in time and frequency domains. Various simulation software packages are commercially available and may be used for modeling the power-electronic-based systems considering all the switching details. These software packages may be generally classified into State-Variable-based (SV-based) [1]- [7] and Nodal-Analysis-based (NA-based) languages [8]- [12]. The Electro-Magnetic Transient Program (EMTP) [8] and many commonly used EMTP-type packages are also based on the nodal or modified nodal equations. With the development of modern simulation tools, the detailed models of power-electronic-based systems, where the switching of all devices is implemented, may be readily implemented in both SV- and NA-based simulation packages. 2 It is therefore possible to develop models of larger systems from a number of smaller subsystems/modules that can be used for the simulation of system transients. However, the detailed switching models are computationally expensive, and result in a significant increase of the required computing time, which in turn limits the size of the system that can be practically implemented in a given package. In addition, the detailed models are discontinuous due to inherent switching, and hence are not directly suitable for linearization and small-signal frequency-domain characterization, e.g., obtaining impedance characteristics and input-output transfer functions. These characteristics are important in assessing the stability of power-electronic-based systems and design of corresponding controllers [13], [14]. Extracting these characteristics from the detailed switching model of the system is possible only through the traditional frequency-sweep method or similar techniques [15], [16], which become extremely time-consuming and challenging especially at low frequencies. These challenges have led to development of the so-called dynamic Average-Value Models (AVMs) which approximate the original system by “neglecting” or “averaging” the effects of fast switching within a prototypical switching interval. The switching effects, i.e., ripples, are often undesirable and small, and do not have a significant effect on longer-term dynamics of the system. The resulting average-value models, therefore, make a useful tool for predicting the slower dynamics of the system below the switching frequency. These models are computationally efficient and could run orders of magnitudes faster than the original switching models, leading to the possibility of efficient system-level simulations and studies. Additionally, since AVMs are time-invariant, they can be numerically linearized about any desired operating point for small-signal analysis, i.e., obtaining local transfer functions and impedances. Obtaining such characteristics from the AVMs is then almost instantaneous in many SV-based simulators, e.g., [4]. 3 Dynamic average models have been successfully used for modeling of vehicular [17], [18] and naval [14], [19] electric power systems as well as the distributed dc power systems of aircraft [20], [21] and spacecraft [22]– [24]. Average-value modeling has also been often applied to variable speed wind energy systems [25]– [32] where the machines are typically interfaced with the grid using the power electronic converters. The need for efficient dynamic AVMs is drastically increasing with the development of the modern power systems and its many components that are interface through power electronics. In order to promote research in this area and disseminate results in the power engineering community, the IEEE Task Force on Dynamic Average Modeling has been assembled under the Working Group on Modeling and Analysis of System Transients Using Digital Programs. The research objectives of this thesis are in line with the overall goals of this Task Force. 1.2 Literature Review For the purpose of this thesis, the high power converters that are commonly encountered in utility applications may be divided into Pulse-Width Modulated (PWM) and the line- commutated converters. This particular research is focused on multi-pulse line-commutated rectifiers which have typical configuration as depicted in Figure 1.1. Depending on the application, the source may be a distribution feeder (or transformer) as in Case I, or a rotating machine (generator) as in Cases II and III. In the simplest case of the conventional three- phase (six-pulse) line-commutated rectifiers, the three-phase ac subsystem may also include an ac filter network which feeds a single three-phase bridge rectifier composed of six diodes. The single rectifier then feeds a dc system (load) through an optional dc filter network. For the converters that are composed of multiple sets of three-phase bridges, the 3-phase/n-phase transformer and possible interconnection/Inter-Phase Transformer (IPT) may be used as depicted in Figure 1.1. 4 Configurations similar to Figure 1.1 are commonly used as the input stage in low- to medium-power variable frequency drives (VFDs) and motor loads that are widely used in industrial and commercial applications [33]- [39]. These loads are often referred to as the front-end rectifier loads [40], [41], and may appear in large numbers in industrial facilities [42], distributed generation [43], as well as vehicular power systems [44]. The ac filter network may take various configurations depending on application and cost constraints, wherein passive or active filters may be considered to reduce the harmonics. Commonly used passive filters include series choke inductors and shunt filters that are tuned to lower harmonics (typically 5th and 7th) [45], [46]. Case I: Case II: Rectifier Bridges AC System (Different Configurations) DC Filter (Optional) DC System 3-Phase Distribution Feeder (or  Transformer) 3-Phase Rotating Machine Case III: Multi-Phase  Rotating Machine Bridge  Interconnection (Series/Parallel )  and Inter-Phase Transformer  (Optional) - + 3-phase n-phase Load Transformer G G C rdc Ldc AC Filter (Optional) Series Shunt - + - + Figure 1.1 Typical configurations of multi-pulse line-commutated rectifier systems. Converters with higher pulse count (e.g. 12-, 18-, and 24-pulse configurations) are also widely used in industrial applications, electric systems of aircrafts, ships, distributed generation systems, etc. [47]- [49]. High-pulse-count converters are generally considered to improve the quality of the output dc voltage as well as the current at the input ac terminals. These converters may also be used to improve the reliability of systems whenever such 5 property is important. As illustrated in Figure 1.1, higher-pulse-count converters may be fed from a distribution feeder (or transformer) or an electric machine. The use of a 3-to-n-phase transformer is inevitable in the first case where the converter is fed from a typically three- phase power network. In the case of a machine-fed high-pulse-count converter, instead of using a multi-phase transformer (as in Case II), the multi-phase voltages may be directly produced by the means of multi-phase machine windings [50], i.e., Case III in Figure 1.1. In all the above cases, additional ac filtering may also be employed, if desired, but it is often not required due to the presence of transformer and/or electric machine. Each of the multiple three-phase output sets of the ac subsystem are fed into a diode bridge. At the dc side, the bridges maybe generally interconnected using various configurations, e.g., series, parallel, or a combination, with or without an inter-phase transformer. Similar to the case of six-pulse converter, the rectifier ultimately feeds a dc subsystem (load) through an optional dc filter network. The objective of average-value modeling is to replace the discontinuous switching cells, the rectifier sub-circuits, with continuous circuit elements and dependent sources that reproduce the averaged behavior of the switching cell within a prototypical switching interval. Dynamic AVMs for line-commutated converters have been generally developed using two main approaches, i.e., analytical [50]- [68] and parametric [69]- [74]. In analytical approach, the system equations are mathematically derived, and the network variables are then averaged over a prototypical switching interval [51]. The first challenge in deriving the AVMs analytically is the existence of the so-called operational modes, e.g., Continuous Conduction Modes (CCM) and Discontinuous Conduction Mode (DCM). In general, an operational mode is defined by a sequence of repeated topologies determined by periodic changes of switches in steady-state operation. The changes in load conditions would lead to a change in the topologies and hence the mode of operation. Therefore, operational modes are functions of the loading/operating conditions. Also, the shapes of the current and 6 voltage waveforms may vary significantly in different operational modes. Thus, deriving the AVMs requires knowledge of the operational modes and the boundary conditions for which the respective averages will be valid. In many cases, the converter might be designed to operate in a certain operational mode in steady-state. Most of the analytically-derived AVMs therefore assume a certain operational mode (typically only one mode) [50], [51], [53]- [59], [63]- [68]. However, in order to accurately predict the transients, the average models should be developed for all possible operational modes. Therefore, there will be an AVM for each operational mode that is in the range of interest. Such models can then be “switched” as the system changes the modes, which makes this approach additionally challenging. In more complicated configurations such as 12-pulse or 18-pulse converters (Figure 1.1), the number and complexity of operational modes significantly increases [62], [75] making the analytical approach of deriving the correct average-value equations even more challenging and less tractable. Using the analytical approach, it is also generally difficult to establish closed-form explicit equations the describe the system of complicated configurations, e.g., machine-fed converter systems (Figure 1.1 Cases II and III) [53]– [55]. In many cases, the final model is implicit and would require iterative solution. In addition to the above challenges, the analytical derivation of AVMs becomes cumbersome or impractical when the model includes parasitics of semiconductor switches, losses or magnetic saturation of electric machinery, etc. To overcome the challenges of the analytical average-value modeling, the parametric approaches have evolved [69]- [74]. In these approaches, a proper state model AVM structure is first defined, and the parameters of the AVM are then extracted numerically from a detailed switch-level model of the system, or possibly a hardware prototype, using methods similar to the “black-box” approach. For this purpose, the averaging concept is numerically applied in order to extract appropriate model parameters. Although establishing the dynamic average-value models using parametric approach requires a detailed model of the system, 7 once established properly, the AVM is valid for a wide range of operating conditions and possibly all desired operational modes. Moreover, this approach may be systematically extended to other converters/configurations, and accounting for the losses and parasitics becomes more practical compared to the analytical approach. 1.2.1 Three-Phase (Six-Pulse) Converter Models Various models for the six-pulse converter system have been presented in the literature. Several models that are of particular interest to this research are listed in Table 1.1 together with the features of each model. A great deal of contribution has been made in this field by the research group at Purdue University wherein analytically-derived AVMs have been formulated for voltage-source- [51] and machine-fed [52]– [55] converters. The AVM for this system is typically considered for CCM-1 mode of operation. The model [51], for example, is valid in one mode (i.e. CCM-1) and neglects the resistance on the ac side. Subsequent effort has been made by the research group at Virginia Polytechnic Institute and State University (Virginia Tech) [65], [66] to include the ac side resistance and improve the model dynamics. A constant parameter AVM has been also developed for machine-fed converter in a collaborative work involving the same research group [69]. A particular case of three-phase rectifier feeding constant-voltage loads has been addressed in collaborative work at Massachusetts Institute of Technology (MIT) and the University of Wisconsin-Madison [57], [58], and subsequent collaboration of the research groups at the University of Belgrade and the Swiss Federal Institute of Technology Zurich (ETH) [59]. The development of parametric approach is owed largely to the contributions of the research group at the University of British Columbia (UBC) [70], [71], [72]. The model proposed in [70] relates the averaged dc and ac variables through the parametric algebraic functions that are established numerically for the desired range of operation and modes. A 8 transient study may be carried out using a detailed model of the system from which the parametric functions are readily calculated numerically [71]. Table 1.1 Dynamic average models of the three-phase (six-pulse) rectifier systems. Models Steady- State/ Dynamic Dynamic Order (Full/ Reduced) DCM, CCMs (1, 2, 3) AC Filter ( acr , acL ) DC Filter ( dcr , dcL , C ) Variable AC Inductance (Yes / No) [51] Dynamic Reduced CCM 1 acL dcr , dcL , C No [57], [58] Dynamic Reduced CCM 2 acr , acL C No [59] Dynamic Reduced CCM 2 acL C No [65], [66] Dynamic Reduced CCM 1 acr , acL dcr , dcL , C No [53] Steady-State N/A CCM 1 - dcr , dcL , C Yes [54] Dynamic Reduced CCM 1 - dcr , dcL , C Yes [55], [56] Dynamic Reduced CCM 1 - C Yes [69] Dynamic Full CCM 1 - C Yes [52] Dynamic Reduced CCMs 1, 2, 3 - dcr , dcL , C Yes [70], [71], [72] Dynamic Full CCMs 1, 2, 3 - dcr , dcL , C Yes 1.2.2 High-Pulse-Count Converter Models Several most relevant models of high-pulse-count converters available in the literature, together with their main properties, are summarized in Table 1.2. Most often, the number of pulses is considered to be a multiple of three, e.g., 12-, 18-, and 24-pulse configurations, wherein the system is constructed by several three-phase subsets as depicted in Figure 1.1. However, unusual cases are also encountered, such as the 5-phase (10-pulse) configuration considered in [60], [61]. In the above research, conducted at the University of Missouri- Rolla, extensive analytical effort has been required for derivation of analytical AVMs for all operational modes in both cases of voltage-source- and machine-fed converter systems. However, considering the amount of effort required in [60], such analysis does not seem practical for higher-pulse-count systems with significantly more operational modes. 9 Generally, as the number of pulses increases, the number and complexity of the operational modes will significantly increase. This makes analytical average-value modeling of the high- pulse-count converters very challenging in general. Typically, only the most common mode of operation is considered. In many cases, additional assumptions and/or approximations are employed in order to simplify the model derivation. A common assumption is the idealized operation of the multi-phase transformer in voltage-source-fed converters. Table 1.2 Dynamic models of high-pulse-count converter systems. Models Pulse Count Detailed/ Analytical AVM/ Parametric AVM AC Subsystem (Voltage Source/ Rotating Machine) Dynamic Order (Full /Reduced) Single Mode / Multi Mode Transformer Dynamics (Yes, No) [60], [61] 10-pulse Detailed and Analytical AVM Voltage Source and Rotating Machine Reduced Multi Mode No [47], [76], [77], [78] 12-pulse Detailed Voltage Source Full Multi Mode Yes [62] 12-pulse Analytical AVM Voltage Source Reduced Multi Mode No [63], [64] 12-pulse Analytical AVM Voltage Source Reduced Single Mode No [73]- [75] 12-pulse Detailed Rotating Machine Full Multi Mode N/A [50] 12-pulse Analytical AVM Rotating Machine Reduced Single Mode N/A [73] 12-pulse Parametric AVM Rotating Machine Full Multi Mode N/A [74] 12-pulse Parametric AVM Rotating Machine Full Multi Mode N/A [48], [77], [78] 18-pulse Detailed Voltage Source Full Multi Mode Yes [65]- [68] 18-pulse Analytical AVM Voltage Source Reduced Single Mode No [49], [79] 24-pulse Detailed Voltage Source Full Multi Mode Yes Analytically-derived AVMs for the 6-phase (12-Pulse) voltage-source- and machine-fed converter systems have been developed for common operating conditions in [62]- [64] and [50], respectively. Similar to the case of a 6-pulse converter, in many cases the analysis is limited to only one mode of operation. In [63], [64], to facilitate the derivation, it is additionally assumed that the duration of commutation interval (angle) is small, resulting in an approximate model for a single operating mode. The phase-shifting transformer is also assumed ideal, i.e., the dynamics, magnetizing current, losses, etc. are neglected. Reference [62], however, derives averaged dc side equations for the 12-pulse voltage-source-fed system, in all modes of operations, together with the respective boundary conditions. 10 Obtaining these results required extensive analytical effort. Parametric approach has been considered in [73], [74] for 12-pulse machine-converter systems in an attempt to include a range of operating conditions, and reduce the extent of analytical effort. For the 18-pulse configuration, a reduced-order analytical AVM has been formulated and evaluated in [65]- [68]. 1.3 Research Objectives and Anticipated Impact All above-mentioned average models developed for the line-commutated converters have been reported to provide acceptable results in particular cases investigated therein. In general, however, each model has several limitations. A unified and generalized dynamic AVM methodology for high-pulse-count converters has not been established prior to this thesis. Moreover, the dynamic AVMs available in the literature [50]- [74] have been formulated for the SV-based approach, wherein the discretization of the state equations is done automatically at the system level. These models, therefore, may not be readily implementable in the EMTP-type solution [8], where the discretization is performed at the branch/component level. As these software packages are increasingly used for simulation of power-electronic-based systems, integration of AVMs into these packages is very desirable to achieve efficient simulation of system transients. Developing the AVMs for implementation in these programs is more challenging and has not yet received the needed attention. The ultimate goal of this research is to develop a generalized methodology that can be readily applied to an arbitrary high-pulse-count system (Figure 1.1), in order to develop an accurate and full-order dynamic average representation of the system in both SV- and NA- based simulators. The first steps toward this goal include addressing the fundamental shortcomings of the six-pulse converter AVMs. Therefore, this thesis has the following 11 research objectives for the six-pulse voltage-source-fed, six-pulse machine-fed, and high- pulse-count converter systems: Objective 1: Development of the AVMs for EMTP-type solution Extension of the analytical and parametric approaches to the EMTP-type solution is the first objective of this thesis. For this purpose, indirect and direct methods of interfacing the developed AVMs with the external EMTP circuit network are developed and compared. The performance of the developed AVMs is then investigated and compared in a wide range of operating conditions including DCM and CCM operation, as well as under unbalanced excitation. The dynamic model of front-end rectifiers with unbalanced input voltages is particularly desirable when studying single-phase faults and unbalance voltage sags [80]. Objective 2: Including the effects of machine and rectifier losses In development of AVMs, the switches are typically assumed ideal [50]- [64], [70]- [74], and the effects of machine losses are neglected [50], [52]- [56], [60], [61], [69]- [74]. The effects of diode and machine losses are particularly important in low voltage applications such as vehicular power systems that operate at 12-14 V dc. Taking these effects into account is the second objective of this research. Objective 3: Developing the generalized high-pulse-count AVM methodology For high-pulse-count converters, the existing models have been derived for one particular configuration depicted in Figure 1.1. The third objective of this research is to develop a unified and generalized AVM methodology for all configurations of the high-pulse-count converter systems. Achieving this objective requires the following steps: 12 (a) Accurate full-order representation of a multi-phase transformer The steady-state and dynamic effects of the multi-phase transformer have been fully or partially neglected in the AVMs available in the literature [62]- [68]. The transformer is typically assumed to instantaneously convert the three-phase input voltages to a set of multi- phase voltages that are equal in magnitude and evenly shifted in phase. Due to neglecting parts of the system dynamics, the final model then has a reduced-order formulation. In some cases, [65], in order to improve the accuracy of the model, artificial equivalent leakage inductances and resistances have been computed and placed at the secondary side in series with the ideal multi-phase voltage source. Such oversimplified representation of the transformer ignores important effects such as magnetizing current and dynamics of the transformer. An accurate full-order representation of a multi-phase transformer is required for the generalized AVM methodology. (b) AVM validity in all operational modes and under unbalanced conditions In the available AVMs for high-pulse-count systems, the operation is usually limited to one mode only [50], [62]- [68], [73], [74]. Moreover, all these models have been derived under the assumption of a balanced excitation [50], [60]- [68], and therefore are not suitable to investigate the system performance under any asymmetric faults. It is therefore very desirable to have a generalized AVM methodology that would be valid in all operational modes as well as under unbalanced conditions. Accomplishment of the Objectives 1 through 3 would enable accurate and accelerated simulation of power systems that include a large number of power-electronic-based converters and modules. With the development of Smart Energy Grids, the need for rapid and efficient simulation of such systems is continuously increasing. It is envisioned that such dynamic models may be readily implemented in the future generations of both SV-based and 13 EMTP-type simulation programs as standard built-in library components and models. While conducting transient simulation of large power systems, for each power-electronic-based subsystem, the user would then have an option to choose the dynamic average equivalent instead of the detailed switch-level model. Employing these models for representing the power electronic components will result in significant gains in simulation speeds, set the stage for the new generation of transient simulation tools with enhanced features and capabilities, and enable new computer-aided design tools, etc. Such tools would be used by thousands of researchers and engineers throughout the world saving them countless hours of precious time. 1.4 Thesis Organization This thesis is organized as follows: In Chapter 2, implementation of analytical and parametric AVMs in SV-based programs is considered. This Chapter sets the stage for the discussions in Chapter 3, wherein the AVMs are formulated specifically for implementation in the EMTP-type software packages. Both approaches of constructing the average models, i.e., analytical and parametric, are considered. Two methods for interfacing the developed AVMs with the external circuit-network, i.e., indirect and direct methods, are introduced and compared. In Chapter 4, the performance of developed models in SV-based and NA-based simulators is investigated and compared under various operating conditions including light and heavy loading (DCM and CCM) as well as balanced and unbalanced ac side. For this purpose, a front-end rectifier example system is considered. In Chapter 5, the effects of parasitics and machine losses are taken into account while developing the AVM for machine-fed six-pulse converter systems. An example vehicular power system is considered in this Chapter. A thorough analysis is performed on this system to demonstrate the effects of losses and effectiveness of the new AVM in predicting the 14 entire power conversion chain. It is demonstrated that the results predicted by the proposed AVM provide an excellent match with the experimental set-up and those of the detailed switching model. A significant improvement in accuracy compared to the previously- developed state-of-the-art AVM is also demonstrated. In Chapter 6, a generalized methodology is developed for dynamic average modeling of high-pulse-count converter systems in SV-based and NA-based simulators. An example 400Hz aircraft power system with an 18-pulse topology is considered to demonstrate the new methodology. The proposed methodology is general and could be readily applied to an arbitrary high-pulse-count system where the number of phases is a multiple of three. The steady-state and dynamic effects of transformer are completely taken into account as well as the negative and zero sequence components for unbalanced operation. For this purpose, a generalized transformer model is developed in transformed ( qd ) and phase ( abc ) variables that can be used in AVMs. The effectiveness of the developed methodology is demonstrated through extensive simulation studies in time and frequency domains including asymmetric faults. Finally, in Chapter 7, the conclusions are drawn. The research contributions are also summarized and related to the original objectives of this thesis. The path for future research is then outlined and related to the present efforts undertaken by the UBC’s Electric Power and Energy Systems research group. 15 Chapter 2: Dynamic Average Modeling in State-Variable-Based Simulators 2.1 State-Variable-Based Simulators State Variable (SV) approach is frequently used for analysis of dynamic systems including electric power systems. The examples of software packages include ACSL [1], Easy5 [2], Eurostag [3], and the well-known MATLAB\\Simulink [4]. There are also more specialized tools such as SimPowerSystems (SPS) [5], PLECS [6], and ASMG [7], etc., that come with circuit interfaces and built-in libraries for simulation of transients in power and power- electronic modules [81]. Internally, the program engine assembles system of differential and/or differential algebraic equations (DAEs) that constitute the state-variable-based model of the overall system. Depending upon the features of a given program, the DAEs may be converted into a first-order system of differential equations (ODE) as ),,( ),,( t t uxgy uxfx = =& , (2.1) where x , u , and y denote the vectors of state variables, inputs, and outputs, respectively. Whenever appropriate, the linear time invariant part of the system/circuit may then be represented using a more compact state-space equation: DuCxy BuAxx += +=& , (2.2) 16 where A , B , C , and D are the so-called state-space matrices that are computed for the given topology and parameters of the linear circuit. In some tools, for example SimPowerSystems [5], the circuit part is directly implemented in the form of (2.2), whereas the remaining part of the system e.g. control blocks, mechanical subsystem, etc. are in a more general form (2.1). The time-domain transient responses are then calculated numerically by integrating the state-space equations (2.1), (2.2) using either fixed- or variable-step ODE solvers embedded in the SV program. The formulations (2.1), (2.2) also contain very useful information about the system’s dynamical modes which is often utilized together with numerical linearization for the frequency-domain characterization and design of controllers. The SV-based programs, unlike EMTP-type simulators, allow the use of variable-step integration (solvers) wherein the step-size is dynamically adjusted to satisfy the local accuracy constraints. Figure 2.1 shows a typical time-stepping flowchart of the SV-based programs. As usual, after initialization the program estimates the size for the first time step and the simulation enters the major time-stepping loop. This step-size is limited between the user-defined minimum and maximum values of mint∆ and maxt∆ , respectively. In the next step, the state vector 1+nx is computed together with an error function estimate r and the vector of output variables 1+ny . The local accuracy of the solution in each time-step is assessed based on the value of this error function. Typically, the user can also specify the absolute and relative error tolerances. If the error estimate r exceeds either of the tolerances, the time-step is reduced (for example to a half of the previous t∆ ) and the solver attempts the solution again. To accurately handle the switching discontinuities, the algorithm also computes a special vector of event variables 1+ne . These variables are monitored at each time step for zero-crossing. A topological change in the system is detected whenever one of the variables in 1+ne changes the sign within the time step. If this condition is detected, the 17 time step is reduced and the solution iterates inside the minor time-stepping loop to precisely locate the zero-crossing. After that, the equations are updated accordingly and the simulation proceeds. Hence, very frequent switching will result in more iterations and effective reduction of the time step. This may lead to significant increase in the computation time especially when a large number of switches are present in the network or a high frequency switching is taking place. Estimate time-step ∆tn Compute: ∆tmin ∆tmax Test event vector: zero-crossing events? No en+1 yn+1 Limit step-size Yes No Reduce step-size xn+1State vector || r ||Error function Output vector Event vector Test Accuracy: absolute & relative error tolerances satisfied? No Yes n = n + 1 Yes Ti m e- ste p l oo p Start Simulation Initialization t = tinitial, n = 1 en+1 End Simulation t = tfinal ? Next time-step Reduce step-size Figure 2.1 Flowchart of a typical variable-step state-variable-based solver. 18 2.2 Detailed Analysis Let us start the discussion from a simple case of three-phase (six-pulse) converter system depicted in Figure 2.2, wherein a series filter inductor is considered at the ac side. Topological variations in this system, such as including/excluding the filters at the ac/dc sides, significantly affect the rectifier operation leading to discontinuous (DCM) and continuous (CCM) conduction modes commonly encountered in practical applications. In the simplified circuit diagram of Figure 2.2, the ac network is represented by its Thevenin equivalent voltages, abcse and the series resistance and inductance, thr , thL . The impedance of the optional series ac filter is represented by acr , acL , and the combined equivalent impedance of the ac subsystem is denoted by sr , sL . It should be noted that if the system is fed from a synchronous generator (Figure 1.1 Case II), the machine may be represented using the voltage-behind-reactance formulation [82]– [84] which results in a circuit similar to Figure 2.2 (but possibly with coupled and/or variable equivalent inductances). ias vdc + - S1 vcs +- S2 S3 S4 S5 S6 - + idc vab_s + - rdc - + - + Ldc ecs ebs eas Lth Lth Lth C RL rth rth rth Lac Lac Lac rac rac rac Lsrs + - vC AC Filter (Optional) Combined equivalent AC side impedance Thevenin equivalent of the AC network DC Filter (Optional) Equivalent load Figure 2.2 Simplified circuit diagram of a typical three phase front-end rectifier load system. In Figure 2.2, the dc filter, represented by dcr , dcL , C , is also optional and may be partially present in the system. In many cases, such as battery charging [85], [86] and variable-speed-drive applications [40], [41], the dc filter inductor may be omitted whereas 19 the capacitor is included to smooth the dc bus voltage. In such cases, the ac filter inductor must often be included in order to reduce ac side current harmonics. To represent an equivalent energy dissipating load on the dc side in Figure 2.2, a simple resistor, LR , is connected to the dc bus [85], [87], [88]. In order to study the performance of the front-end rectifier load system of Figure 2.2, the system detailed model, wherein switching of all diodes is represented, maybe readily built in many SV-based simulators. In this system, a discontinuous conduction mode (DCM) operation is typically observed at light load. This mode is frequently encountered in the front-end rectifiers of low- to medium power variable frequency drives [87], [89], where the ac filter is often not used (or is very small) but there is a large capacitor on the dc bus. The corresponding waveforms are illustrated in Figure 2.3 (a). If the angular frequency of the ac source is denoted by eω , there exist six equal switching intervals, sT , within a single electrical cycle defined by e cycleT ω pi2 = . In DCM, each switching interval is then divided into two subintervals. During the conduction subinterval, condt , two diodes are conducting and two of the phases carry the dc bus current in opposite directions. At some point, the line- to-line voltage in these two phases becomes smaller than the dc bus voltage and the current reaches zero and remains zero for the rest of the interval – hence discontinuous mode. This subinterval is denoted by dcmt in Figure 2.3 (a). During dcmt , all diodes are off. The dc load is, in the meanwhile, being fed from the dc capacitor as observed in the dc bus voltage waveform dcv . The switching pattern in DCM is therefore 2-0. If the combined value of the source and ac filter inductances becomes sufficiently large [87], [90], or the load on the dc bus is sufficiently increased, the dc bus current becomes continuous – hence continuous mode. In this case, as the load varies from a light load to a short circuit, three different switching patterns can be observed resulting in three distinct continuous conduction modes (CCM) of operation [91]. The corresponding waveforms of the 20 phase currents and the dc bus voltage are shown in Figure 2.3 (b). The operational modes are summarized in Table 2.1 together with the conduction pattern and the commutation angle. (a) Waveforms for DCM Operation 0 tcond tdcm Ts 0 i dc v d c idc idc v C vdc i as , i bs , i cs i as , i bs , i cs CCM-1 CCM-3 CCM-2 0 0 0 tcond tcom tcom 0 v d c tcom Ts vdc idc idc idc vdc vdc i as , i bs , i cs v d c i as , i bs , i cs v d c (b) Waveforms for CCM Operation Tcycle = 6 Ts Tcycle = 6 Ts Figure 2.3 Typical current and voltage waveforms of the six-pulse rectifier: (a) operation in DCM; and (b) operation in CCM. Table 2.1 Operational modes of the conventional 3-phase (6-pulse) rectifier. Operational Modes Conduction Pattern Commutation Angle DCM 2-0 o0=µ CCM-1 2-3 oo 600 << µ CCM-2 3 o60=µ CCM-3 3-4 oo 12060 << µ 21 Within CCM-1 [see Figure 2.3 (b), top plot], each switching interval is divided into two subintervals referred to as commutation and conduction [51]. During the conduction subinterval, condt , only two diodes conduct; whereas during the commutation subinterval, comt , corresponding to the commutation angle come t⋅= ωµ , the current is being switched from one phase to another; and three diodes conduct. Therefore, a conduction pattern of 2-3 diodes is observed within each 60 electrical degrees, and oo 600 << µ . The mode CCM-2 [see Figure 2.3 (b), middle plot], may be achieved by further increasing the load current. In this mode, the commutation angle µ increases and reaches 60 degrees resulting in disappearance of the conduction subinterval. Consequently, there will be three diodes carrying current throughout the switching intervals. Hence, the conduction pattern becomes just 3 (or 3-3). If the load current is further increased, the commutation angle µ starts to increase resulting in the third mode, CCM-3 [see Figure 2.3 (b), bottom plot]. This changes the switching pattern to 3-4 conducting diodes. Note that this mode contains a topology with 4 simultaneously conducting diodes, which temporarily short-circuits the output as observed in Figure 2.3 (b), (bottom plot). This last mode rarely occurs in practical systems, but may be encountered in rotating machines and brushless exciters with very large inductances [52]. Certain variations in the topology of Figure 2.2 may prevent the occurrence of some operational modes. For instance, without the dc filter inductor, the CCM-3 mode cannot occur and the system could hence only operate in DCM, CCM-1 or CCM-2. Observing the phase currents in DCM depicted in Figure 2.3 (a), it can be concluded that such currents will inject significant harmonics into the network. To avoid operation in DCM, medium and large drives with front-end rectifiers often install additional ac filters (series inductors and shunt filters) to shift the operation to CCM-1 or CCM-2. 22 2.3 Dynamic Average Modeling 2.3.1 Analytical Approach In analytical approach, system equations are mathematically derived, and the network variables are then averaged over a prototypical switching interval [51]. The fast averaging over a prototypical switching interval sT is typically defined as ( )∫ − = t Tts s df T f ττ1 , (2.3) where f is a network variable, e.g., voltage or current, t is a scalar to denote time, and f is the so-called fast average of f . As observed in the waveforms of Figure 2.3, in steady-state, the variables on the dc side, e.g., dci , dcv consist of a constant average value superimposed by some ripple due to switching, which often necessitates the use of large capacitors on the dc link. The averaging concept (2.3) may then be directly applied to the dc variables in the system. However, for ac variables, using the above equation evidently does not yield the desired result. Instead, the ac variables first have to be transformed using an appropriate synchronously rotating 0qd reference frame [51]. The transformation of a three-phase network variable f (i.e., voltage or current) from phase domain ( abc ) to an arbitrary 0qd reference frame is accomplished using the transformation matrix K as [51] abcqd fKf =0 , (2.4) [ ] Tdqqd fff 00 =f , (2.5) [ ] Tcbaabc fff=f , (2.6) 23                         +      −       +      − = 2 1 2 1 2 1 3 2 sin 3 2 sinsin 3 2 cos 3 2 coscos 3 2 piθpiθθ piθpiθθ K . (2.7) The angle θ denotes the instantaneous angle of the reference frame which is expressed as 0at θωθ += , (2.8) in terms of the reference frame angular frequency ω and initial angle 0aθ . To convert 0qd variables back to abc , the inverse transformation may be employed [51]:                         +      +       −      −= − 1 3 2 sin 3 2 cos 1 3 2 sin 3 2 cos 1sincos 1 piθpiθ piθpiθ θθ K . (2.9) In a balanced system, the zero sequence is not present and the third element in (2.5) is hence equal to zero. Also, sometimes the zero sequence is prevented by the system topology as in Figure 2.2, wherein the bridge rectifier only sees the line voltages, and the neutral point of the source appears floating. In such cases, the order of the transformation is reduced to two, and the reference frame is simply referred to as qd reference frame. In a synchronously rotating reference frame, by definition, the angular frequency of the reference frame ω is equal to that of the ac source eω . The so-called converter reference frame [51], a particular case, is typically chosen to facilitate the analysis. In this synchronous reference frame, the reference frame angle θ is aligned with the angle of phase a voltage at the bridge terminal. In this case, due to mathematical properties of the transformation, the d - axis component of the transformed voltages will be zero as depicted in Figure 2.4 (a). The relationship between the voltages in the converter and arbitrary reference frames may be written as [51] 24               − =         a ds a qs cc cc c qs v vv )cos()sin( )sin()cos( 0 φφ φφ . (2.10) In the above equation, the superscripts c and a have been used to denote the variables expressed in the converter and arbitrary reference frames, respectively. Throughout this thesis, the converter reference frame will be the default reference frame of choice. Hence, for compactness of the equations, the superscript c may at times be dropped. The angle between the converter and arbitrary reference frames may also be deduced from Figure 2.4 (a) as [51]         = − a qs a ds c v v1tanφ . (2.11) qa-axis qc-axisvqds c φc θc d -axisad -axisc vqsa iqs a vds a ids a iqdsc φ θa i as , i bs , i cs tcond tcom Ts v d c Time Time Time i ds i qs, vdc iqs ids (a) (b) Figure 2.4 (a) Relationship among the variables in the converter and arbitrary reference frames; and (b) Typical waveforms and the respective transformed waveforms in converter reference frame. The waveforms of the 6-pulse rectifier system for CCM-1 operation together with the transformed ac currents in the converter reference frame have been illustrated in Figure 2.4 25 (b). As seen in this figure, the resulting qd currents in steady-state are composed of a constant (dc) component and a superimposed ripple, quite similar to the dc bus voltage waveform, which allows averaging of these variables according to (2.3). The averaging of these variables using (2.3) removes the switching ripple but preserves the slower (dc) variations of the transformed variables. 2.3.1.1 Classical Reduced-Order Model (AVM-1) In the classical model [51], the state equation describing the dynamics of the dc bus current is formulated as dcs Cdcdcse dc LL virLE dt id + −      +− = 2 3233 ω pipi , (2.12) where E is the rms value of the phase voltage, and Cv is the average dc filter capacitor voltage. To establish the average −q and −d axes components of the phase currents, the dc current is assumed constant and equal to its average value during the switching interval. Due to this assumption, the current dynamics disappear resulting in a reduced-order model. The phase currents are then expressed during conduction and commutation subintervals, and are averaged, respectively. The result of this procedure yields the following equations: ( ) ( )( ),12cos2 4 31cos23 6 5 sin 6 5 sin32 , −+−−             +      − − = µ ωpi µ ωpi pipiµ pi sese dc c comqs L E L E ii (2.13) ( )( ),22sin2 4 3 sin23 6 5 cos 6 5 cos 32 , µµ ωpi µ ωpi piµpi pi ++−             −−     − = sese dc c comds L E L E ii (2.14) 26             +−     − = 6 5 sin 6 7 sin32 , piµpi pi dc c condqs ii , (2.15)             −      + − = 6 7 cos 6 5 cos 32 , pipiµ pi dc c condds ii , (2.16) The final averaged ac side currents are obtained as c condqs c comqs c qs iii ,, += , (2.17) c condds c comds c ds iii ,, += , (2.18) Finally, the commutation angle is expressed as         −= − dc se i E L 3 21cos 1 ωµ . (2.19) The model defined by (2.12)–(2.19) is referred to as AVM-1. 2.3.1.2 Improved Reduced-Order Model (AVM-2) A similar model has been derived in [65], where instead of assuming a constant value for the dc current during the switching interval, this current is assumed to change as ( )       −⋅+= 20 µθθ kii dcdc , (2.20) where 0dci is the average value of dci during the commutation period, and k is the derivative edc ddi θ during this period of time. The effect of ac side resistance has been also partly taken into account. The resulting model has the following form: , 332 2 sin1233 2 3 2 344 3 3 0 2 Cdcdcses se s dc dcdc e ss e virLr L rE dt diLrLr −      ++ − −      − −=             + − + − + +− ω pipi µpiµµ ωpi ω µpi pi µpi ω pi µµpi (2.21) 27 ( )       −−+      +−+−= 4 3 4 2cos cos 23 3 sin3cos32 0 µµ ωpi piµ pi µ pi se dc c qs L Ekii , (2.22) ( )       −−+        − ++−= 24 2sin sin23 3 33 cos 3 sin32 0 µµµ ωpi piµ pi µ pi se dc c ds L Ekii . (2.23) The model defined by (2.20)–(2.23) is referred to as AVM-2. This model uses the same commutation angle formulation expressed by (2.19). 2.3.2 Parametric Approach In the parametric model [70], the rectifier switching cell is considered as an algebraic block that relates the dc voltage and current dcv , dci to the ac voltages and currents c qdsv and cqdsi through the respective parametric functions as ( ) dccqds v.α=v , (2.24) ( ) cqdsdci i.β= , (2.25) where ( ).α and ( ).β are algebraic functions of the loading conditions. To complete this model, the angle between the vectors cqdsv and c qdsi is calculated based on Figure 2.4 (a) as ( )         −         = −− a qs a ds a qs a ds v v i i 11 tantan.φ . (2.26) Deriving closed-form analytical expressions for ( ).α , ( ).β , and ( ).φ is impractical. Instead, these functions may be extracted using the simulation and expressed in terms of dynamic impedance of the switching cell defined as c qds dcvz i = . (2.27) The model based on (2.24)-(2.27) is referred to as PAVM. 28 2.4 Implementation of the AVMs in SV-Based Programs Both analytical (AVM-1 [51] and AVM-2 [65]) and parametric (PAVM [70]) AVMs for the three-phase line-commutated rectifier have been formulated as state models. Block diagrams for the final implementations of these approaches are depicted in Figure 2.5 (a) and (b), respectively. These AVMs are straightforward to implement in SV-based simulators using the respective dependent sources and a set of differential and/or algebraic equations that relate the dc and transformed ac variables. - + vdcids iqs vqs idsvds idc - + - + x = f (x , i qd s ) . - + - + b) a) - + vdc idc vqs vds x = f (x , v q ds ) . x = idc - - iqs -- - - x = f (x , vqds , vdc ) -- iqds , idc = g (x , vqds , vdc)- -- - - - - - - - - vqds , idc = g ( iqds , vdc)- -- - - - Figure 2.5 Block diagrams of AVM implementations in SV-based simulators: (a) analytically-derived (AVM-1 and AVM-2); and (b) parametric (PAVM). 29 2.5 Small-Signal Frequency-Domain Analysis In addition to time-domain studies, the detailed and dynamic average models of line- commutated converters are frequently used for small-signal frequency-domain analysis. To illustrate the small-signal impedance-based analysis of the ac-dc converters [14], [92]– [97], let us first consider the general case of a controlled ac-dc converter connected between System 1 and System 2 as shown in Figure 2.6. The control input (denoted by d ) may be the firing angle, for instance, in thyristor controlled converters, or the duty cycle in PWM converters. This converter is first assumed to be operating in a steady-state determined by the fixed control input Dd = and a set of balanced ac input voltages. Averaged over a switching interval, the dc variables dci , dcv will then have constant quiescent values dcI , dcV , respectively, that define the operating point. If the input voltages are kept intact, and a small- signal perturbation dˆ , much smaller in amplitude compared to the quiescent value d , is superimposed on the control input, the control-to-output transfer function of the converter ( )sH may then be defined as )(ˆ )(ˆ)( sd sv sH dc= , (2.28) where dcv̂ is the small-signal perturbation around the average output dc voltage dcV due to the changes in the control input dˆ . d AC/DC ConverterSystem 1 Zqd Zdc idc vdc + - System 2 vabcs iabcs Figure 2.6 System-level impedance-based representation of subsystems interconnected through an ac- dc converter. 30 Next, the control input is assumed to be kept constant ( Dd = ), and a small-signal perturbation is then considered around the quiescent operating point caused by, for instance, the small-signal voltage dcv̂ injected at the dc bus. In uncontrolled line-commutated converters, the small-signal perturbation at the dc bus may be implemented in practice by means of a resistor that is frequently switched on and off at the dc bus [54]. This small-signal injection is going to introduce a small perturbation in other network variables about their quiescent operating points. The small-signal output impedance looking from the dc side may then be expressed as )(ˆ )(ˆ)( si sv sZ dc dc dc = , (2.29) where dciˆ is the small-signal perturbation around dcI caused by the injected signal. The frequency of the injection signal may be slowly swept in the desired frequency range and in each point the magnitude and phase of the impedance are obtained based on (2.29). The impedance looking from the dc side is essentially the impedance of System 1 as mapped through the switching cell of the ac-dc converter to the dc side; and is determined by the converter and the three phase ac System 1. Therefore, the impedance dcZ will not only depend on the dynamics of System 1 but also the properties of the switching converter itself, e.g., switching strategy, modulation, controls, mode of operation, etc. Evaluating impedance of the system looking from the ac side requires special consideration because the ac variables are inherently time-variant even in steady state. To extend the impedance-based approach to the ac systems, the ac variables have to be viewed in a system of coordinates where they appear constant in steady state. This extension has been made in [14], where this approach was developed for the stability analysis of ac-dc electric power systems. Therefore, the physical variables on the ac side, abcsv and abcsi , are transferred into a synchronous reference frame. The resulting variables qdsv and qdsi have 31 the required property of being constant in steady state. For this reason, the appropriate impedances (as well as the averaging) are expressed in terms of these transformed variables. Specifically, assuming that the system operates in a steady state operating point determined by the external inputs and control signal d , the small-signal perturbations can be applied. Neglecting the zero sequence, there are two axes to be considered for calculating the appropriate impedance on the ac side denoted here by qdZ . For example, a small-signal perturbation qsv̂ is first considered around the steady-state value qsV . The response of the system in terms of currents generally will be seen in both q and d axes, respectively. This will give the first row of the transfer matrix (2.30). Next, a small-signal perturbation dsv̂ is considered around dsV . Similarly, the system response in terms of currents will be seen in both axes. Calculating these transfer functions will give the second row of the impedance transfer matrix qdZ . The final form of impedance transfer matrix is therefore:                 =      = == == 0ˆ0ˆ 0ˆ0ˆ )(ˆ )(ˆ )(ˆ )(ˆ )(ˆ )(ˆ )(ˆ )(ˆ qq dd vd d vq d vd q vq q dddq qdqq qd si sv si sv si sv si sv ZZ ZZ Z . (2.30) In a SV-based simulator, the dynamic AVM will typically have the form of (2.1). Assuming a small-signal perturbation about the steady-state operating point as uUu xXx ˆ ˆ += += , (2.31) where x̂ and û are small-signal disturbances about the operating point values X and U , the linearized equations are then obtained as ,ˆ ˆ ˆ ˆ ˆ ),,( ˆ ),,( ˆ ˆ ˆ ˆ ˆ ˆ ),,( ˆ ),,( ˆ uDxCu u UXg x x UXgy uBxAu u UXf x x UXf x += ∂ ∂ + ∂ ∂ = += ∂ ∂ + ∂ ∂ = tt tt& , (2.32) 32 which has the form of (2.2). Once the state-space matrices Aˆ , Bˆ , Cˆ , and Dˆ of the linearized system are calculated, obtaining any transfer function of the system becomes a straightforward and almost instantaneous procedure. In a very general case, the input-output transfer function matrix )(sH (considering all inputs and outputs) may be written as DBAICH ˆˆ)ˆ(ˆ)( 1 +−= −ss . (2.33) In most SV-based simulation packages, e.g., [4], this procedure can be done automatically making a powerful tool for system-level analysis and design of controllers. A subset of (2.33) will then contain any particular transfer function (impedance) (2.28)–(2.30). 33 Chapter 3: Dynamic AVM Formulation for EMTP-Type Solution 3.1 Electro-Magnetic Transient Programs The programs based on nodal analysis [8], [98] (or modified nodal analysis) include EMTP-type and Spice-type programs [99]. The underlying solution approach is based on discretizing the differential equations for each circuit component using a particular integration rule. The EMTP [8] uses an implicit trapezoidal rule for discretization and formulating the network nodal equation that has the following general form: hn IGV = . (3.1) Here, G is the network nodal conductance matrix, and the vector hI includes the so-called history current sources and independent current sources injected into the nodes. The nodal voltages nV are unknown and calculated by solving (3.1) at every time step. As the G matrix is usually sparse, specially designed techniques such as optimal reordering schemes and partial LU factorization [100], [101] are often used to solve this linear system instead of inverting the matrix G directly. In a typical EMTP formulation which uses a fixed time-step t∆ , topological changes due to switching events require special consideration since such changes may happen inside a time-step. A trivial solution is to reduce the step size until a sufficient accuracy of solution is achieved. This would result in using very small time steps and increases the computational burden of the overall simulation. Significant effort has been made to design efficient 34 algorithms for handling switching events [102]- [106]. Such methods typically include interpolation and/or extrapolations within a time step. Start Simulation Topological Change? Initialization t = tinitial and history terms Solve/Obtaine: Nodal voltages & network variables Invoke appropriate algorithm t = t + ∆t End Simulation Yes Yes No t = tfinal ? Ti m e- ste p l oo p Update network configuration No Update: G matrix Calculate instant of change Update network solution (interpolation/extrapolation) (interp. between t and t - ∆t) Next time-step Figure 3.1 Flowchart of a typical nodal-analysis-based solver. Figure 3.1 shows the time-stepping procedure in a typical EMTP. After the initialization, the simulation enters the major time-stepping loop which continues to execute until the end of simulation when finalt is reached. In each time-step, the network G matrix and history terms are updated as needed, the nodal equations are solved, and the nodal voltages are calculated. At each time step, the network variables are computed to perform the test for the changes in topology. If a change in topology within a given time step is detected (turning on/off of a diode, thyristor, transistor, etc.) the present time step is not accepted and a special 35 algorithm has to be evoked. A typical algorithm interpolates between the solution points to precisely locate the time instance of the switching event within the time step. Then, the network equation has to be solved considering the shorter subinterval before the switching instance. Thereafter, the system is updated for the new topology and solved again. For re- synchronizing back with the existing time step, several solutions have been proposed in the literature using interpolation/extrapolation [102]– [107]. In many EMTP languages, the so- called Critical Damping Algorithm (CDA) [108], [109] is invoked to suppress the artificial numerical oscillations due to the insufficient damping of the trapezoidal integration. This procedure is then applied whenever there is a switching in the system until the end of simulation. 3.2 Detailed Analysis In most EMTP-type programs, detailed models of line-commutated rectifier systems may be readily implemented using standard library components. Here, particularly, to demonstrate the effects of the ac side impedance topology on the system performance, a front-end rectifier system is considered with two possible configurations. The corresponding snapshots of the system detailed models implemented in PSCAD/EMTDC [11] are depicted in Figure 3.2 (a) and (b). In Figure 3.2 (a), the impedance on the ac side consists only of a series RL connection. This represents the Thevenin equivalent impedance of the ac power system combined with the optional series choke inductor. If present in the system, the latter typically has the dominant value between the two. In Figure 3.2 (b), a general filter network is considered composed of two series RL connections on both sides of the parallel RLC branches that each represent a shunt filter tuned to a specific harmonic (in this case 5th and 7th). As seen in this figure, when both series and shunt filters are included, the series choke inductor is typically broken into two unequal parts (such as 67% and 33%) and inserted on both sides of the parallel branches [46]. 36 The mode of operation at a fixed given load (for example defined by Ω= 70LR in Figure 3.2) is essentially determined by the value of the equivalent series inductance present on the ac side. In both Figure 3.2 (a) and (b), excluding the additional series choke inductors would then result in an inductance value small enough to force the system into DCM operation under normal loading conditions. The three-phase currents in DCM are shown for both configurations in the top plots of Figure 3.3 (a) and (b). (a) (b) Figure 3.2 Topological variations of the ac side impedance in a typical three phase front-end rectifier load system (a) series impedance (b) general network with series and parallel branches. 37 -40 0 40 i ab cs = i a bc _i n ( A) Without Series Choke Inductor (DCM): -20 0 20 i ab cs = i a bc _i n ( A) With Series Choke Inductor (CCM): W ith ou t S hu nt F ilt er s Time (s) 0.970 0.985 1 i ab cs (A ) -20 0 20 W ith S hu nt F ilt er s Time (s) 0.970 0.985 1 i ab cs (A ) -40 0 40 Without Series Choke Inductor (DCM): With Series Choke Inductor (CCM): (a) (b) Figure 3.3 Three-phase currents at the bridge terminals for DCM and CCM operation: (a) without shunt filters; and (b) with shunt filters. i ab c_ in (A ) -20 0 20 W ith S hu nt F ilt er s Time (s) 0.970 0.985 1 i ab c_ in (A ) -40 0 40 Without Series Choke Inductor (DCM): With Series Choke Inductor (CCM): Figure 3.4 Three-phase ac currents of the system at the input source terminals in presence of the shunt filters for DCM and CCM operations. As dictated by the system topology, the currents at the source and bridge terminals, denoted in Figure 3.2 by abcsi and inabc _i , are identical in absence of shunt filters, i.e., Figure 3.2 (a). In this case, the high-harmonic-content currents of Figure 3.3 (a) (top plot) are directly fed to the network. In presence of the shunt filters, i.e., Figure 3.2 (b), the source 38 currents are depicted in Figure 3.4 (top plot). These waveforms show that a somewhat lower harmonic content is achieved in presence of the shunt filters of Figure 3.2 (b). Since these filters are essentially tuned for the 5th and 7th harmonics, their effects on the system operation are normally intended to be minimal except ensuring that the corresponding harmonic content flows through the shunt path instead of the network. The mode of operation and other characteristics of the system are then expected to remain intact as long as the dc load and equivalent series inductive component on the ac side do not vary significantly. The bridge currents are then expected to be almost identical in presence and absence of a well- designed set of shunt filters. This is indeed endorsed by the close agreement observed between the top plots in Figure 3.3 (a) and (b) that have been obtained in the absence and presence of the shunt filters, respectively. The DCM operation discussed above is most frequently encountered in the front-end rectifiers of the low- to medium power variable frequency drives [87]- [89] where the series ac filter is often not used (or is very small) but there is a large capacitor on the dc bus. This mode is essentially associated with the needle-shape waveforms with high harmonic content that are typically challenging and expensive to filter. As observed in Figure 3.4 (top plot), even in presence of the shunt filters of Figure 3.2 (b), the source current has a seemingly high harmonic content. This problem, of course, can be alleviated using additional shunt filters tuned to the next present harmonics until the desired THD criteria are satisfied. It should be noted, however, that this would further increase the size, cost, and complexity of the already complicated filter network of Figure 3.2 (b). With the additional series choke inductors, in both configurations of Figure 3.2 (a) and (b), the dc bus current becomes continuous resulting in CCM operation. As the load increases, CCM-1 and CCM-2 operation may both be observed. However, the less common CCM-3 operation is prevented in the configurations of Figure 3.2 (a) and (b) due to the absence of dc link filter inductor as is commonly the case in front-end rectifier variable frequency drives. 39 The three-phase ac currents for the most common type of CCM operation (CCM-1) corresponding to the same operating point as the previous study ( Ω= 70LR ) are illustrated in Figure 3.3 (a) and (b) (bottom plots) in the absence and presence of the shunt filters, respectively. Comparing these plots to the respective top plots, it is observed that, in both cases, the harmonic contents of the bridge currents automatically reduce to some extent when the system enters CCM. Also, comparing the two bottom plots of Figure 3.3 (a) and (b), it is seen that the bridge currents are essentially similar in absence and presence of the shunt filters, except minor differences in the ripple shape. In the absence of shunt filters, the bridge currents of Figure 3.3 (a) (bottom plot) are directly injected at the source. However, as depicted in Figure 3.4 (bottom plot), the source currents are significantly improved in presence of the shunt filters especially compared to those of the DCM operation (top plot). In summary, the detailed analysis presented above demonstrates that, on the one hand, the value of the equivalent series inductance on the ac side essentially determines the mode of operation at a given dc load. On the other hand, the effects of the parallel branches (shunt filters) are mainly confined to the currents drawn from the source network which are important in system-level studies concerning the overall performance of the power system. As discussed in the previous Chapter, one crucial step in dynamic average modeling, in general, is relating the ac currents at the bridge terminals to the current injected to the dc bus. This step clearly relies on the mode of operation which determines the shape of the waveforms and hence the values of the respective averages. Therefore, for the purpose of dynamic average modeling, the currents injected to the bridge terminals are of the main interest. From this point of view, the series component of the ac inductance has the dominant impact. The system performance may then be typically investigated for two cases, without and with the series ac inductor filter, regardless of the details of the shunt filter branches. 40 3.3 Dynamic Average Modeling As described in the previous Chapter, the AVMs for line-commutated converters based on both analytical and parametric approaches have been formulated as state models. Significant additional effort is required to reformulate the dynamic average models and interface them with the overall circuit network for the EMTP-type solution. In general, both analytical and parametric approaches may be extended to the EMTP-type solution. These extensions are set forth in the next two subsections. 3.3.1 Analytical AVM for EMTP-type Solution In the circuit of Figure 2.2, if the ac and dc filter inductors together with the diode bridge are considered as the rectifier block, and the input voltage to this block (from the external ac subsystem) is denoted by qsv in the converter reference frame, the dynamics of dc bus can be represented by the following state equation [51]: rec Cdcrecqsdc L viRv dt id −− = pi 33 . (3.2) Here, the parameters recR and recL are defined in terms of the original system parameters as: eacdcrec LrR ωpi 3 += , acdcrec LLL 2+= , (3.3) Integrating both sides in (3.2) yields: dtviRvidL t tt Cdcrecqs t tt dcrec ∫∫ ∆−∆−         −−= pi 33 , (3.4) where t∆ is the chosen integration time step. Next, discretization of (3.4) is carried out using the trapezoidal rule (as commonly done for the EMTP): 41 ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )ttvtvt ttititRttvtvt ttitiL CC dcdcrecqsqs dcdcrec ∆−+∆− ∆−+∆−∆−+∆ =∆−− 2 22 33 pi (3.5) Upon rearranging terms, the following equation is obtained: ( ) ( ) ( ) ( )titvtv R ti dchCqs eq dc , 331 +        −= pi , (3.6) where eqR is defined as t LRR recreceq ∆ += 2 . (3.7) In (3.6), the so-called history term, ( )ti dch, , is a function of the network variables at the previous time step. In particular, ( ) ( ) ( ) ( )       ∆−−∆−+∆−         −= ttvttv R tti R R ti Cqs eq dc eq rec dch pi 33121 , . (3.8) In the approach set forth in [51], the algebraic equations describing the ac side are obtained by averaging the q - and d -component currents in the commutation and conduction sub-intervals, and combining/adding the result. Omitting the intermediate steps, the result is: condqscomqsqs iii ,, += , (3.9) conddscomdsds iii ,, += , (3.10) ( ) ( )( ),2cos1 4 3 1cos 3 6 5 sin 6 5 sin32 , µ ωpi µ ωpi pipiµ pi −−−−             +      − − = eac qs eac qs dccomqs L v L v ii (3.11) 42 ( )( ),22sin 4 3 sin 3 6 5 cos 6 5 cos 32 , µµ ωpi µ ωpi piµpi pi ++−             −−     − = eac qs eac qs dccomds L v L v ii (3.12)             +−     − = 6 5 sin 6 7 sin32 , piµpi pi dccondqs ii (3.13)             −      + − = 6 7 cos 6 5 cos 32 , pipiµ pi dccondds ii , (3.14) where µ is the so-called commutation angle written as:         −= − qs dceac v iL 3 21cos 1 ωµ . (3.15) Here, all qd variables are expressed in the converter reference frame. The above equations are first mathematically simplified using trigonometric identities in order to facilitate the next steps in the derivation of the model. After some effort, the simplified expressions for (3.11)- (3.14), are obtained as follows: µ pi sin3 , dccomqs ii = , (3.16) µ ωpi µ ωpi ω pi qseac qs eacqs dc eaccomds vL v Lv iLi 2 3 sin 2 332 2 , +−         = , (3.17) dc qs dc eacdccondqs i v iLii pi ω pi µ pi 322 sin3 2 , +         − − = , (3.18) µ pi ω pi sin332 2 , dc qs dc eaccondds i v iLi +         − = . (3.19) Substituting the above equations in (3.9) and (3.10) yields the final equations for qsi and dsi : 43 dc qs dc eacqs i v iLi pi ω pi 322 2 +         − = , (3.20) ( )µµ ωpi µ pi −−= sin 2 3 sin3 qs eac dcds vL ii . (3.21) 3.3.2 Indirectly-Interfaced Analytical Average-Value Model (IIAAVM) The AVM described by (3.6)-(3.8) and (3-20), (3-21) is of the form depicted in Figure 3.5, where the dc and ac sides have been represented by Norton-equivalent inter-dependent current sources. The presence of these dependent sources, as well as the ac side being represented in qd variables, makes it challenging to implement this model directly in EMTP wherein the ac network is normally represented in phase variables, i.e., abc . vqs vds ih,dc - + - + - + vdc - - -Reqf (vqds) - g (idc , vqds) -- h (idc , vqds) -- idc - Figure 3.5 Circuit diagram of the AVM described by (3-6)-(3.8), (3-20), (3-21). Similar to what has been done in interfacing the qd machine models with the abc networks in EMTP; one solution is to introduce a time-step delay between the dc and ac subsystems. This approach is in fact the so-called indirect interfacing [110], [111] of the model with the network. This method of interfacing has been used to interface all the machine models in the PSCAD/EMTDC software [11], [110], [111]. Adopting this method, the ac and dc subsystems become essentially decoupled and the dependent current sources 44 become independent sources. The values of these sources are calculated, similar to the history sources, according to the solution of the network at the previous time step. Also, since the solution of the network at the previous time steps are readily available, the transformation of the network variables between abc and qd becomes a straightforward operation and the final model is conveniently interfaced with the ac network in abc phase variables (coordinates). In particular, the values of the input three-phase voltages are transformed into the converter reference frame using the appropriate transformation matrix (2.7). The value of the dependent current source at the dc side is then readily obtained as a function of qsv . Similarly, the values of the dependent current sources at the ac side are readily obtained using the solution of the network at the previous time step. These current sources are then transformed back to abc applying (2.9), and injected into the ac network as ainji _ , binji _ , and cinji _ . However, the time-step delay may cause unfavorable numerical oscillations and convergence problems. To alleviate the problem, the use of an interfacing circuitry has been proposed for the machine models in [111]. A modified circuit includes a resistor zr and the so-called compensating current source compi . This circuit is inserted between the ac network and the input port of the AVM in each phase. The final interfacing equivalent circuit of the AVM using this indirect approach is illustrated in Figure 3.6. rz ias (t) iinj_a icomp_a External Network rz ibs (t) iinj_b icomp_brz ics (t) iinj_c icomp_c Figure 3.6 Circuit diagram of the IIAAVM using PSCAD-like approach. 45 3.3.3 Directly-Interfaced Analytical Average-Value Model (DIAAVM) The indirect method of interfacing may lead to reduced accuracy and potential problems with numerical stability of the overall solution especially at larger time steps. This has been demonstrated in [112] for the machine models interfaced using the indirect approaches. It is therefore desirable to achieve a direct interface between the EMTP network and the AVM so that the time-step delay is eliminated and a simultaneous solution of the two subsystems is achieved. This has been set forth in this subsection. Equations (3.20) and (3.21) describe the qd components of the input ac currents as nonlinear functions of the dc bus current and the qd components of the input ac voltage. Based on (3.20), (3.21) let us define the following nonlinear functions: ( ) qs dc qdc v i vif 2 1 , = , (3.22) ( ) µsin,2 dcqdc ivif = , (3.23) ( ) µsin,3 qsqdc vvif = , (3.24) ( ) µqsqdc vvif =,4 , (3.25) where µ itself is, of course, a nonlinear function of dci and qsv defined by (3.15). These functions should be first linearized within a time step t∆ . For example, the function 1f at time t may be written as ( ) )()( 111 tv v f ti i f tf qs ttqs dc ttdc ∆−∆− ∂ ∂ + ∂ ∂ = . (3.26) Using a similar approach, the functions 1f through 4f are linearized to have the following form: ( ) dcqs iBvAtf 111 += , (3.27) ( ) dcqs iDvCtf 222 += , (3.28) 46 ( ) dcqs iDvCtf 333 += , (3.29) ( ) dcqs iDvCtf 444 += , (3.30) with the coefficients, obtained after extensive effort, as follows: 2 1 SA −= , (3.31) SB 21 = , (3.32) S L S SSL C eac eac ω ω 3 3 2 2 23 2 +− −      = , (3.33) S L S SLS D eac eac ω ω 3 3 43 2 2 2 +−       − = , (3.34) S L S SC eacω 32 3 +− = , (3.35) S L S SL D eac eac ω ω 3 1 3 2 2 3 +− +      − = , (3.36) S L S S ttC eacω µ 3 )( 2 4 +− −∆−= , (3.37) S L S D eacω 3 1 2 4 +− = . (3.38) In the above equations the variable S itself is defined as 47 ttqs dc v iS ∆−         = . (3.39) Substituting (3.27)-(3.39) into (3.20), (3.21), simplifying, and rearranging terms, yields the final linearized equations for qd components of the ac side currents: dcqsqs iBvAi += , (3.40) dcqsds iDvCi += , (3.41) with the final coefficients A , B , C , and D defined as 22 SLA eacωpi = , (3.42) pi ω pi 324 +−= SLB eac , (3.43) ( ) S L S S L SSL L ttC eac eac eac eac ω ωpipipi ω ωpi µ 3 332 2 3 2 23 +− −− + ∆− = , (3.44) S L S SSL D eac eac ω pi ω pi 3 344 2 2 +− + − = . (3.45) Equations (3.40)-(3.45) together with the discretized dc bus equation (3.6)-(3.8) form the basis for the AVM that may be directly interfaced with the ac network. 3.3.3.1 Direct Interface in qd Variables Although it is not very common, in some cases, the whole ac network itself may be expressed in qd variables. The direct interfacing of the AVM in this case is simplified. For the sake of completeness, this special case is considered here before the general case of direct 48 interface in abc variables is discussed. For this purpose, first the discretized dc bus equation (3.6) is rewritten as: dchCqsdc ivFvEi ,++= , (3.46) where the new parameters E , and F have been defined as eqR E pi 33 = , (3.47) eqR F 1−= . (3.48) Substituting (3.46) into (3.40) and (3.41) yields: qhCqsqs ivBvAi ,+′+′= , (3.49) dhCqsds ivDvCi ,+′+′= , (3.50) where the new coefficients and history terms are expressed as BEAA +=′ , (3.51) BFB =′ , (3.52) DECC +=′ , (3.53) DFD =′ , (3.54) dchqh iBi ,, = , (3.55) dchdh iDi ,, = , (3.56) In all above equations, the ac variables have been expressed in the converter reference frame. As shown in Chapter 1, in this frame, 0=dsv , and qsv can be written, according to (2.10), as c a dsc a qsqs vvv φφ sincos += , (3.57) where the superscript a denotes variables in the arbitrary reference frame. Also, cφ is the so- called converter angle that defines the position of the converter reference frame q -axis. 49 Derivation of the directly-interfaced AVM in the arbitrary reference frame requires substitution of (3.57) into (3.46), (3.49), and (3.50), and the subsequent transformation of the ac currents into the arbitrary reference frame. The final model may then be written in the following compact matrix form: qdhbrnqdavm −− −= IIVG , (3.58)                 ′ +′′+ ′ ′+′ ′− ′′ −′′−′= − cccccc cccccc cc qdavm CACADB CACADB EEF φφφφφφ φφφφφφ φφ 2sin 2 sincos2sin 2 cossin sin2sin 2 2sin 2 cossincos sincos 22 22G , (3.59)           + −= − cdhcqh cdhcqh dch qdh ii ii i φφ φφ cossin sincos ,, ,, , I , (3.60)           = a ds a qs C n v v v V ,           = a ds a qs dc br i i i I . (3.61) iqsids idc External AC Network (in qd) External DC Network ih,dc Gavm-qd Figure 3.7 Circuit diagram of the DIAAVM in qd variables. If the external ac network is expressed in qd variables, it can then be directly interfaced with the above model in qd domain as illustrated by the equivalent circuit of Figure 3.7. 50 This is particularly suitable for the simulation of smaller systems which may include several machine-rectifier modules together with the dc subsystems. Direct interface is conveniently achieved between the AVMs and the machine models expressed in qd variables. 3.3.3.2 Direct Interface in abc Variables In general, the external ac network is represented in abc phase variables. In order to achieve a direct interface in abc , additional effort is required to transform (3.58) back to the abc phase variables. For this purpose, first, qsv is written, employing (2.9), in terms of the phase voltages as ( )321 coscoscos3 2 θθθ csbsasqs vvvv ++= , (3.62) where ce φθθ −=1 , (3.63) 3 2 2 piφθθ −−= ce , (3.64) 3 2 3 piφθθ +−= ce , (3.65) and eθ is the electrical angle of the source. Transforming the AVM back to abc variables then includes substitution of (3.62) into (3.46), (3.49), and (3.50), rearranging terms, and finally employing the inverse qd transformation using the transformation matrix (2.9). After some effort, the final model is written in the following compact matrix form: abchbrnabcavm −− −= IIVG , (3.66)             = − 44434241 34333231 24232221 14131211 GGGG GGGG GGGG GGGG abcavmG ,               = − ch bh ah dch abch i i i i , , , , I , (3.67) 51             = cs bs as C n v v v v V ,             = c b a dc br i i i i I . (3.68) The elements of abcavm−G , and abch−I are expressed as FG =11 , 11 cos3 2 − = mm EG θ , 111 sincos −− ′+′= kkk DBG θθ , 4,3,2, =km , (3.69) ( )111 sincoscos3 2 −−− ′+′= iijij CAG θθθ , 4,3,2, =ji , (3.70) 1,1,, sincos θθ dhqhah iii += , (3.71) 2,2,, sincos θθ dhqhbh iii += , (3.72) 3,3,, sincos θθ dhqhch iii += , (3.73) The final AVM (3.66) is expressed as four Norton-equivalent pairs of current source- conductance branches to represent the dc and ac terminals of the converter. The Thevenin equivalent of the developed AVM, if desired, may also be readily obtained by multiplying both sides of (3.66) by 1− −abcavmG . This leads to the following equations: abchbrabcavmn − − − −= eIGV 1 , (3.74) abchabcavm ch bh ah dch abch e e e e − − −− =               = IGe 1 , , , , , (3.75) The final equivalent circuit of interfacing the Thevenin equivalent AVM defined by (3.74) is depicted in Figure 3.8. 52 - + - + - + eh,c eh,b eh,a - + eh,dc Gavm-abc -1 ia ib ic idc External AC Network External DC Network Figure 3.8 Equivalent circuit diagram of the DIAVM in abc variables. Implementing the AVMs (3.66) and (3.75) are straightforward for EMTP-type solution. However, since the original model was developed in the converter reference frame, the final implementation of the models still requires knowledge of the angle cφ which defines the position of the converter reference frame q -axis. An approximate model may be readily obtained employing the synchronously-rotating reference frame aligned with the source angle instead of the converter reference frame. In many cases, the deviation angle between these two reference frames (due to the impedance connected between the source and the converter) is small and the approximation may be valid for normal operation. If additional accuracy is desired, a simple prediction-correction algorithm may be employed to obtain cφ more precisely. It should also be noted that, in the final directly-interfaced average model of Figure 3.8, the conductance matrix abcavm−G may generally change at different time steps due to the nonlinearity of the system. This would typically require triangularization of the system G matrix which is of course associated with some additional computational time. However, in the EMTP-type software packages, typically the refactorization is performed only on the time-variant portion of the system G matrix to minimize the computational burden [100], [101]. In general, the additional burden imposed by this step depends on the overall system 53 and how the EMTP-type software package handles the refactorization of the G matrix. Further investigation of this effect is therefore outside the scope of this thesis. 3.3.4 Parametric AVM for EMTP-Type Solution The parametric approach may also be considered for developing AVMs suitable for EMTP-type solution, especially with the intention to alleviate the burden imposed by the extensive mathematical derivations. Similar to the case of analytically-derived AVM, in general, both indirect and direct methods may be used for interfacing the developed model with the EMTP network. However, to avoid laborious mathematical derivations associated with formulating the directly-interfaced parametric average-value model, here, only the indirect method of interfacing is considered to interface the parametric AVM with the network. According to the approach set forth in [70], parametric functions are considered to relate the dc voltage and current dcv , dci to the ac voltages and currents cqdsv and cqdsi , and also establish the angle between the vectors cqdsv and cqdsi . Based on Figure 2.5 (b), the rectifier block takes the dc bus voltage dcv , and the ac currents cqdsi as the input, and provides the dc current dci and ac voltages cqdsv as the output. To formulate and interface the model at dc and ac ports with external circuit networks, the following has to be accomplished. First, rearranging terms in (2.26), the angle δ may be defined as: ( ).tantan 11 φδ −         =         = −− a qs a ds a qs a ds i i v v . (3.76) Next, using (2.24) and (3.76), the ac voltages in arbitrary reference frame may be written as 54 ( ) ( ) ( ) ( )           = 0 sin. cos. 0 δα δα dc dc a sqd v v v . (3.77) The third element in the above vector is set to zero because the rectifier sees the line voltages according to the topology of Figure 2.2, hence the zero sequence is not transferred. More general configurations, wherein the zero sequence is also allowed to exist at the ac side, will be considered in Chapter 6. The inverse 0qd transformation (2.9) is now applied, and the ac voltages are computed as follows: ( ) ( ) ( ) ( )           = − 0 sin. cos. 1 δα δα dc dc abc v v Kv . (3.78) To calculate the dc bus current, the ac phase currents at the input port of the rectifier block first have to be transformed into arbitrary reference frame according to (2.7). Next, using (2.25), the dc bus current is computed as ( ) abcdci iK.β= . (3.79) Using the indirect method of interfacing, the rectifier block may now be interfaced with the circuit networks at the ac and dc ports. For this purpose, the dependent voltage sources are placed at the ac side whose values are calculated according to (3.78) at each time step. A dependent current source is also placed at the dc port whose value is determined according to (3.79). At each time step, the rectifier block uses the dc bus voltage and ac phase currents as the inputs. These values are then employed in the above calculations. Also, the dynamic impedance z , is computed based on the inputs according to abc dcvz iK = . (3.80) The values of parametric functions, corresponding to the operating point defined by z , are then obtained from the pre-stored look-up tables and will be used along with the inputs for 55 calculating the values of the dependent sources. The final Indirectly-Interfaced Parametric Average-Value Model (IIPAVM) together with the interfacing circuitry is illustrated in Figure 3.9. -+ -+ -+ vc vb va idc External AC Network External DC Network(3.76)-(3.80) - + vdc- - ia ic ib Figure 3.9 Indirectly-interfaced parametric average-value model. 3.4 Example of AVM Implementation in PSCAD/EMTDC To illustrate the use of AVM in EMTP-type packages, an example implementation of the IIPAVM in PSCAD/EMTDC [11] is shown in Figure 3.10. The block shown in Figure 3.10 (a) contains (3-76)–(3-80) and the transformations between qd and abc coordinates. The subsystem of Figure 3.10 (a) is encapsulated into a single module-block that is then interconnected using its nodes with the external network as shown in Figure 3.10 (b). Such AVM block can then replace the detailed switching rectifier module within a larger ac and dc network (which may include ac filters, e.g. shunt harmonic filters, etc). Figure 3.10 (b) also depicts the timed breaker blocks that are used to implement the system changes for particular transient studies that will be presented in the next Chapter. 56 (a) (b) Figure 3.10 Example of IIPAVM implemented in PSCAD: (a) PAVM block together with controllable sources and interfacing ports; and (b) AVM module interfaced with external ac and dc subsystems. 57 In the next Chapter, extensive simulation studies will be carried out on the analytical and parametric AVMs developed in Chapters 2 and 3 in both SV-based and EMTP-type simulators. The purpose of these studies is to compare the different AVM implementations in different software packages in terms of predicting the system performance in different operating modes as well as under balanced and unbalanced ac side. Before presenting such thorough analyses, it is useful to further compare the indirect and direct methods of interfacing introduced earlier. This is the purpose of the simulation studies presented in the next section which will conclude the present Chapter. 3.5 Comparison of Direct and Indirect Interfacing Methods To better demonstrate different properties of the indirectly-interfaced and directly- interfaced AVMs formulated in the previous sections, simulation studies are carried out as follows. For each model, a small integration time-step ( µs50 ) is used first to validate the model against the detailed switch-level reference model of the system. Then, larger integration time steps ( ms5.0 and ms1 ) are used to compare the performance of the models. The detailed (switch-level) model of the system (Figure 2.2) has been implemented in PSCAD/EMTDC [11]. The example system parameters are provided in Appendix A.1. The average-value models are implemented using an EMTP-type algorithm written in Matlab. Initially, at 0=t , the system is at zero initial conditions when the three-phase input source is switched on. Then, at st 02.0= , the load resistance is switched from its initial value, Ω= 2loadR , to Ω= 1loadR . Finally, the load resistance is switched back to Ω= 2loadR at st 04.0= . Figure 3.11 illustrates the waveforms of the dc bus, dci , dcv , as predicted by the detailed and average-value models. The waveforms of the phase a current asi predicted by the detailed and average-value models are superimposed in Figure 3.12 (a). A magnified view of the first cycle is illustrated in Figure 3.12 (b) for clarity. Studies of Figure 3.11 and 58 Figure 3.12 are obtained using the typical EMTP time step of sµ50 . As seen in Figure 3.11, provided that the chosen time step is sufficiently small, both methods of interfacing the AVMs lead to convergent results. This is expected as both models have been analytically derived from the same averaged equations. However, as seen in Figure 3.12 (b), the IIAAVM has noticeable numerical oscillations at start-up. As explained previously, this is due to the time delay between the ac and dc systems. Except for the start-up oscillations, the above- mentioned figures show that both AVMs can produce acceptable match with the detailed switch-level model of the system, provided that the integration time steps is sufficiently small. Next, the effect of increasing the time-step size is evaluated. For this purpose, the results of the previous study (Figure 3.11 and Figure 3.12) as predicted by the DIAAVM are chosen as the reference solution (labeled as “Ref” in the following figures). The same study is carried out with the time-step of sµ500 , and the results are superimposed in Figure 3.13 and Figure 3.14 with magnified views for clarity. These results demonstrate that the accuracy of the IIAAVM somewhat degrades compared to the DIAAVM. The error is especially well- pronounced in Fig. Figure 3.13 (b), where IIAAVM predicts an out-of-phase response. At the same time, the DIAAVM follows the reference with exceptional accuracy even at such a large time step. This numerical behavior of the indirect and direct interfacing methods becomes even more pronounced using a time-step size of sµ1000 , which is shown in Figure 3.15. 59 0 0.02 0.04 0.06 0 60 120 180 Time (s) Detailed IIAAVM,  DIAAVM i d c (A ) 0 0.02 0.04 0.06 0 100 200 300 Detailedv dc ( V) IIAAVM,  DIAAVM Figure 3.11 DC bus waveforms predicted by the models with sµ50 time step. -200 -100 0 100 200 0 0.02 0.04 0.06 Time (s) i a s (A ) Detailed IIAAVM, DIAAVMSee Fig. 3.12 (b) (a) 0 0.008 0.016 -200 -100 0 100 200 Time (s) Detailed IIAAVM DIAAVM i a s (A ) (b) Figure 3.12 Input phase current predicted by the models with sµ50 time step. 60 0 0.02 0.04 0.06 0 60 120 180 0 0.02 0.04 0.06 Time (s) 0 100 200 300 i d c (A ) v d c (V ) IIAAVM, DIAAVM Ref Ref See Fig. 3.13 (b) (a) 0.019 0.0205 0.022 0.0235 100 160 220 Ref DIAAVM IIAAVM Time (s) v d c (V ) (b) IIAAVM, DIAAVM Figure 3.13 DC bus waveforms predicted by the models with sµ500 time step. -200 -100 0 100 200 0 0.02 0.04 0.06 Time (s) Ref, DIAAVM IIAAVM i a s (A ) See Fig. 3.14 (b) (a) 0.0315 0.033 0.0345 0.036 -190 -160 -130 -100 Ref DIAAVM IIAAVM Time (s) i a s (A ) (b) Figure 3.14 Input phase current predicted by the models with sµ500 time step. 61 -200 -100 100 200 0 0.02 0.04 0.06 Time (s) Ref, DIAAVM IIAAVM i a s (A ) See Fig. 3.15 (b) 0 (a) (b) 0.0315 0.033 0.0345 0.036 -190 -160 -130 -100 Ref DIAAVM IIAAVM Time (s) i a s (A ) Figure 3.15 Input phase current predicted by the models with sµ1000 time step. 62 Chapter 4: AVM Verification in SV-Based and EMTP-Type programs 4.1 Example Micro-Wind Turbine Generator System To demonstrate the properties and benefits of the average-value models, a micro-wind turbine generator system is considered first as depicted in Figure 4.1. The system parameters are summarized in Appendix A.2. Such systems may be used to generate power for telecommunication equipment in the remote areas where an electric grid is not easily accessible. ias vdc + - S1vas +- S2 S3 S4 S5 S6 PMSG as bs cs idc vab_s+ - Rectifier Filter dc-dc Converter Load Controller PWM PID Compensator Pulse-Width Modulator + - Wind Turbine vref vCf + - Lf Cf vout + - Cb Lb iout Figure 4.1 Example PMSG micro-wind turbine generator system. A typical low-cost system consists of a wind turbine coupled to a Permanent Magnet Synchronous Generator (PMSG) through a gearbox. The generator feeds a line-commutated rectifier circuit, followed by an L-C filter, and a dc-dc converter which is connected to the dc bus. The controller adjusts the duty-cycle and regulates the output voltage to the appropriate 63 level, typically 28V dc. This system contains mechanical components, rotating machine, line- commutated ac-dc converter, dc-dc converter, and a controller; and is therefore considered a suitable example system. 4.1.1 Large-Signal Time-Domain Analysis The example system is first modeled in detail considering the switching of all diodes and transistor. The average-value model is then developed using the PAVM for the line- commutated converter, and the circuit-averaging [133] for the dc-dc converter. In the computer study considered in this section, the micro-turbine system is subjected to the speed change shown in Figure 4.2 (top plot). The transient responses as observed in the output voltage, output current, dc-link capacitor voltage, and the duty cycle are also shown in Figure 4.2. It is observed that the AVM of the system predicts the behavior of the detailed model with excellent accuracy. To achieve accurate results for this study using the variable-step solver ODE23 in Matlab/Simulink [4], the detailed model required a total of 386,680 time-steps whereas the AVM results were obtained by only 625 time-steps in total. Relatively small time steps were required in the detailed model in order to capture all the switching events due to high frequency switching of the boost converter as well as the switching of the rectifier diodes. However, the AVM does not have switching and can be executed with much larger time steps. This demonstrates an increase of simulation efficiency of 1,951 times to obtain the results of Figure 4.2. 64 0.5 1.5 2.5 0.5 1.5 2.5 0.5 1.5 2.5 0.5 1.5 2.5 0.5 1.5 2.5 Time (s) 500 1500 2500 Sp ee d (r pm ) 20 28 36 v o ut ( V) 1.8 2.4 3 i ou t ( A) 20 35 v C f ( V) 0.4 5 d 0 0.8 Detailed AVM Detailed AVM Detailed AVM Detailed AVM Time (s) Time (s) Figure 4.2 Waveforms of the example PMSG micro-wind turbine generator system. 4.1.2 Small-Signal Frequency-Domain Analysis It is also desired to compare the detailed and average-value models in portraying the small- signal frequency-domain characteristics of the system. For this purpose, a small-signal analysis is performed around the steady-state operating point corresponding to the shaft speed of 1500 rpm. First, the control loop is removed and the duty ratio of the transistor gate signal is fixed to 0.35 to obtain the output voltage of 28 V on the dc bus. The open-loop small-signal input-output transfer function )(sH may then be expressed as )(ˆ )(ˆ)( sn sv sH out= , (4.1) 65 where outv̂ is the change in the output dc bus voltage due to the small-signal perturbation n̂ in the shaft speed. This transfer function includes the effects of both ac-dc and dc-dc stages of the system; it is useful to designing the controller, but very difficult to derive analytically. It is therefore a suitable measure of comparing the detailed and average-value models. This transfer function has been extracted using both models and the results are superimposed in Figure 4.3. Since the input is the mechanical speed, particularly lower frequency dynamics are of significant importance. Nevertheless, the results in Figure 4.3 are illustrated for up to one-half of the lowest switching frequency present in the system, which corresponds to the diode rectifier stage, (i.e., 300 Hz). As the switching frequency is approached further from this point, it would be normal to observe deviations between the results obtained from the detailed and average-value models. In the whole frequency range depicted in Figure 4.3, however, the results demonstrate an excellent match between the detailed and average-value models. AVM H( s) - M ag ni tu de ( db ) H( s) - P ha se ( de g.) Detailed Frequency (Hz) 100 10 1 10 2 -100 -60 -20 -200 -100 0 10 0 10 1 10 2 DetailedAVM Figure 4.3 Speed-to-output-voltage transfer function for the example micro-wind turbine system. 66 To obtain the results of the frequency-domain analysis of Figure 4.3 from the detailed model (crosses), the frequency sweep method has been employed which becomes a very time-consuming procedure especially at low frequency. However, the results of Figure 4.3 using the AVM (solid lines) may be produced very rapidly using the automated linearization function offered by Matlab/Simulink [4]. At the same time, the results demonstrate a close agreement between the detailed and average-value models in the whole depicted frequency range. 4.2 Example 3-Phase (6-Pulse) Front-End Rectifier system Next, a typical medium power frond-end rectifier example system operating at 480V level is considered. The layout of the system topology is similar to Figure 2.2 with the exception that the ac and dc filtering network may take various configurations. The system parameters are summarized in Appendix A.3. Extensive time- and frequency-domain simulation studies are carried out to compare the performance of various models under different operating conditions. Three simulation software packages are chosen to conduct the studies, namely Matlab\\Simulink [4], PSCAD/EMTDC [11], and EMTP-RV [12]. These packages have been used to implement the detailed and dynamic average-value models of the system. In general, it is observed that the results predicted by all simulation packages are essentially identical, provided the time step and solver properties have been selected appropriately. For the analyses presented in this section, in addition to the detailed switch-level models of the system, three dynamic average-value models have been considered based on AVM-1, AVM-2, and PAVM. Regardless of the benefits and challenges of the approaches, all methodologies, if applied correctly, should lead to similar results in predicting the averaged dynamic behavior of the detailed rectifier system below the switching frequency. The studies that follow consider two topological cases, with and without series filter, and do not consider any shunt filter branches. This choice is based on the conclusions drawn after 67 the detailed analysis performed in the beginning of Chapter 3. As demonstrated there, if shunt filters are included, the currents drawn from the network are smoother than the bridge currents, since some harmonic content flows through the parallel branches. Other than this, the performance of the system is quite similar in these cases from a dynamic average modeling point of view. 4.2.1 Steady-State Time Domain Analysis The steady state regulation characteristic provides a suitable measure for comparing the performance of various models under different loading conditions. The regulation characteristics may be obtained by varying the load from open circuit to short circuit and recording the values of the average dc bus voltage and current. These values are denoted by dcv and dci , respectively. For the purpose of plotting the regulation characteristics, the voltages and currents are typically normalized by their corresponding maximum values, i.e., open-circuit and short-circuit, as predicted by the detailed model. In order to investigate the operation and performance of various models in DCM and CCM, the considered rectifier system is assumed without and with the series ac inductor filter, respectively. 4.2.1.1 Operation in DCM Figure 4.4 (a) depicts the regulation characteristics obtained from the detailed and average- value models in the absence of series ac inductor filter and presence of the dc bus capacitor. The calculated values of the short-circuit currents and open-circuit voltages are summarized in Table 4.1. It should be noticed that without the ac filter inductor and only the network Thevenin impedance, the short circuit current would be extremely high, 1982.3A. In practice, the system with such configuration and parameters typically operates at lighter loads (mostly in the DCM or CCM-1) and the operation under heavy load current is impractical. A magnified view of the practical lighter-load region is provided in Figure 4.4 (b). As seen in 68 this figure, the DCM extends up to the 34.5A of the average dc current. The regulation characteristics predicted by the analytical models, AVM-1 and AVM-2, give a lower voltage and deviate from the one predicted by the detailed switching model. Observing the results in Figure 4.4 (a), it is also evident that the short-circuit current is incorrectly predicted by the analytical models since these models are not valid for heavy-loading conditions in CCM-2 and below. At the same time, the PAVM predicts the same characteristic as the detailed model over the entire region from open to short circuit. 0 1 2 0 0.5 1 Detailed  & PAVM AVM-1 See Fig. 4.4 (b) idc idc,sc vdc vdc, oc AVM-2 ( idc = 1982.3 A ) 0 0.05 0.1 0.9 0.95 1 Detailed  & PAVM AVM-1 DCM  Operation Region idc idc,sc vdc vdc, oc idc = 34.5A AVM-2 ( idc = 99.1 A ) (a) (b) Figure 4.4 (a) Regulation characteristic of the system without ac input filter with dc capacitor; and (b) Magnified view showing the performance of models in DCM region. To better compare the accuracy attained by the average models, the steady state values of the dc bus voltage and currents calculated by these models are summarized in Table 4.2 for two operating points in DCM. The values computed by the detailed model are assumed as reference. As it can be seen in this table, under heavier load, Ω= 35LR , closer to the CCM (but still in DCM), both analytical models AVM-1 and AVM-2 under-estimate the dc voltage and current by about 1.3%. Under a lighter load in DCM, Ω= 70LR , the error increases to above 2%, which may still be reasonable for some studies. Note that the error in dc voltage and current is the same since all models assume the same load resistance. At the same time, 69 the PAVM predicts the steady state dc voltages and current in DCM with extremely good accuracy (up to third digit). Table 4.1 Maximum values of the dc bus voltage and current for different system topologies. Configuration Open-Circuit Voltage, ocdcv , Short-Circuit Current, scdci , Without acr , acL 669 V 1982.3 A With acr , acL 648.2 V 98.8 A Table 4.2 Steady-state values of the dc bus voltage and current predicted by various models in DCM Models Ω= 35LR Ω= 70LR Detailed Model (Reference) Aidc 6625.18= Vvdc 1879.653= Aidc 4354.9= Vvdc 4708.660= AVM-1 Aidc 4260.18= Vvdc 9111.644= %2672.1=error Aidc 2366.9= Vvdc 5652.646= %1070.2=error AVM-2 Aidc 4160.18= Vvdc 5613.644= %3208.1=error Aidc 2341.9= Vvdc 3869.646= %1335.2=error PAVM Aidc 6618.18= Vvdc 1625.653= %0038.0=error Aidc 4353.9= Vvdc 4708.660= %0011.0=error 4.2.1.2 Operation in CCM If the series ac inductor filter is added to the system, the value of the short circuit current would become 98.8A as shown in Table 4.1. Moreover, the DCM operation will be limited to a very small region at the extremely light load close to an open circuit at the dc bus. This will also improve the ac current waveforms making them similar to the bottom plot of Figure 3.3 (a) and reducing the harmonic content. Figure 4.5 illustrates the regulation characteristic for this case, wherein the CCM-1 extends to about 35% of the short-circuit dc current and 72% of the open circuit voltage. In this range of CCM-1, which is the most practical range, the characteristics predicted by the analytical models AVM-1 and AVM-2 match very closely the characteristic predicted by the detailed model. The characteristics predicted by analytical 70 models then start to deviate when the system enters CCM-2, and finally significantly over- estimate the average dc current all the way to the short circuit current which is higher by 180%. However, the PAVM matches the detailed model over the entire region. 0 1 2 0 0.5 1 Detailed & PAVM AVM-1 CCM-1  Operation  Region idc idc,sc vdc vdc, oc AVM-2 ( idc = 98.8 A ) Figure 4.5 Regulation characteristic of the system in CCM operation as predicted by different models. Table 4.3 Steady-state values of the dc bus voltage and current predicted by various models in CCM-1. Models Ω= 35LR Ω= 70LR Detailed model (Reference) Aidc 6247.16= Vvdc 8654.581= Aidc 7308.8= Vvdc 1575.611= AVM-1 Aidc 7864.16= Vvdc 5247.587= %9726.0=error Aidc 8055.8= Vvdc 3853.616= %8556.0=error AVM-2 Aidc 7161.16= Vvdc 0649.585= %5498.0=error Aidc 7848.8= Vvdc 9369.614= %6185.0=error PAVM Aidc 6442.16= Vvdc 5479.582= %1173.0=error Aidc 7864.8= Vvdc 0515.615= %6368.0=error To further compare the accuracy attained by the average models, the steady-state values of the dc bus voltage and currents calculated by these models are summarized in Table 4.3 for two operating points in CCM-1. The values computed by the detailed model are assumed as reference. As seen in this table, under heavier load, Ω= 35LR , closer to the CCM-2 (but still in CCM-1), the AVM-1 and AVM-2 over-estimate the dc voltage and current by 0.97% 71 and 0.55%, respectively. Under a lighter load in CCM-1, Ω= 70LR , these errors are still below 1% which is deemed acceptable. The PAVM also predicts the steady state dc voltages and currents in CCM-1 with great accuracy. 4.2.2 Dynamic Performance under Balanced Conditions 4.2.2.1 Balanced Operation in DCM Next, the rectifier system is assumed to operate without the ac filter but with the dc capacitor as to enable the DCM. In the following study, the system initially operates in steady-state in DCM with load Ω= 35LR . At st 3.0= , the load resistance is stepped to Ω= 70LR , which forces the system deeper into the DCM operation at a lighter load. The transient responses obtained by the considered detailed and average models are shown in Figure 4.6. The analytical models AVM-1 and AVM-2 predict the dc current and voltage with reasonable accuracy, including the change in operating conditions. The ac current fundamental component is also predicted by these models with sufficient accuracy. The PAVM shows a somewhat higher damping than the other models but a superior prediction of the dc voltage. 72 Time (s) -40 20 600 650 0.285 0.315 0.345 i d c ( A) 80 700 v d c ( V) -50 0 50 i a s ( A) Detailed PAVM AVM-2 AVM-1 Figure 4.6 Transient response of the six-pulse rectifier system in DCM predicted by different models. 4.2.2.2 Balanced Operation in CCM First, in order to compare the transient responses of the models, particularly the effects of the model dynamic order (full vs. reduced), the series inductor filter is added to the system 73 and the dc capacitor is removed from the dc bus. This topology does not allow the dc bus current to become discontinuous and hence prevents the DCM operation. The system will therefore operate in CCM for all loading conditions. In the next study, the system is assumed to operate initially under a very light load, close to open circuit condition. Then, at st 3.0= , the load resistance is stepped to Ω= 1.13LR which forces the dc bus current to increase. The corresponding transient responses predicted by the detailed and average models are shown in Figure 4.7. As seen in this figure, the new operating point is achieved in CCM-1 and all models predict the new steady state with reasonable accuracy, i.e., the average responses go through the ripple of the waveforms predicted by the detailed model as shown in Figure 4.7. This result is expected and agrees with the steady state analysis presented earlier. Taking a close look at Figure 4.7, however, it is observed that the responses predicted by the detailed model and PAVM include an overshoot. This overshoot in response is possible because these two models have second order (without the dc capacitor). The AVM-1 and AVM-2, however, are both first-order (reduced-order) models and hence incapable of predicting this effect. These properties are evident from the eigenvalues of the system that have been extracted using numerical linearization and are summarized in Table 4.4 for DCM and CCM operation. In the right column, corresponding to the same operating point as the study of Figure 4.7, it is observed that the AVM-1 and AVM-2 each have the first order corresponding to a real eigenvalue. The PAVM, however, has a pair of complex conjugate eigenvalues corresponding to an underdamped oscillatory transient response. The oscillations are not well observed in Figure 4.7 due to the high damping dictated by the large real part of these eigenvalues. The left column in Table 4.4 corresponds to the DCM operation with the dc filter capacitor that increases the system order by one making the AVM-1 and AVM-2 of the second and the PAVM of the third order. 74 Time (s) 15 0 30 Detailed PAVM 0.28 0.30 0.32 i d c ( A) 45 AVM-2 AVM-1 Figure 4.7 Transient response of the rectifier system with ac filter and without the dc capacitor predicted by different models. Table 4.4 System eigenvalues predicted by different models in DCM and CCM-1. Model DCM, Ω= 35LR (w/o ac filter, with dc capacitor) CCM-1, Ω= 1.13LR (with ac filter, w/o dc capacitor) PAVM 35580− , j9.140854.463 ±− j3.5298.737 ±− AVM-1 j9.14126.118 ±− 8.1081− AVM-2 j6.14199.128 ±− 1.832− Next, to investigate the performance within different modes in CCM operation, the dc capacitor is added to the system. As a result, the order of the system is increased by one. In the following study, the system is assumed to start from zero initial conditions in CCM-1 with the load Ω= 9.11LR . At st 05.0= , the load is stepped to Ω= 2LR forcing the system into the CCM-2. The resulting transient response predicted by various models is shown in Figure 4.8. This figure clearly shows that although AVM-1 and AVM-2 have been derived for CCM-1, their transient response does not exactly follow the dynamic response of the detailed model. This is due to the reduced-order formulation of the models. 75 0 70 0 350 0 0.05 0.1 i d c ( A) 140 700 v d c ( V) -120 0 120 i a s ( A) Time (s) Detailed PAVM AVM-2 AVM-1 Figure 4.8 Transient response of the six-pulse rectifier system with dc capacitor in CCM predicted by different models. The improved model AVM-2 indeed predicts the transient slightly better than the classical model AVM-1, but the effects of reduced-order formulation are still visible. The PAVM, due 76 to full-order formulation, perfectly follows the initialization transient including the overshoot oscillations. At st 05.0= , the rectifier system undergoes another transient and transitions into CCM-2. Both AVM-1 and AVM-2 clearly do not follow this transition very well and predict higher dc voltage and current. This is expected as the models were developed assuming CCM-1 operation. However, the response of the full-order PAVM model is very much consistent with the transient and new steady state predicted by the detailed model since the model has been developed for a range of operating conditions. 4.2.3 Dynamic Performance under Unbalanced Conditions 4.2.3.1 Unbalanced Operation in DCM For implementing the DCM operation, the ac input filter is removed and the dc capacitor is added to the rectifier system. In the following study, the rectifier system initially operates under a balanced condition with resistive load Ω= 20LR . This operating point is close to the boundary between DCM and CCM-1. At st 08.0= , a phase shift of 45 degrees in the c- phase voltage, cse , is introduced making the three phases asymmetric. This change in input voltage throws the rectifier system into unbalanced operation in DCM. Next, at st 11.0= , the load is stepped to Ω= 50LR making the DCM operation even lighter. The corresponding transient responses, as seen in the ac and dc sides, are shown in Figure 4.9 and Figure 4.10, respectively. As seen in these figures, the heavy asymmetry of the input ac voltages leads to a pronounced change in the conduction pattern of the rectifier diodes, making the ac currents particularly spiky and uneven among the phases. It can be observed in Figure 4.10 that the models AVM-1 and AVM-2 do not predict this condition well by producing larger ripple in dc voltage and current. At the same time, PAVM appears to predict the dc variables with much greater accuracy closely resembling the peaks and fluctuation produced by the detailed simulation of the rectifier system. 77 Time (s) -200 0 0.05 0.10 0.15 200 i a s (A ) -200 0 200 i cs ( A) -200 0 200 i b s (A ) Detailed PAVM Figure 4.9 Transient response of the rectifier system to a change in ac voltages, leading to unbalanced operation in DCM, as observed in the ac currents predicted by various models. 78 Time (s) -200 50 0 600 0.05 0.1 0.15 i d c ( A) 300 1200 v d c ( V) Detailed PAVM AVM-2 AVM-1 Figure 4.10 Transient response of the rectifier system to a change in ac voltages, leading to unbalanced operation in DCM, as observed in the dc bus predicted by various models. 4.2.3.2 Unbalanced Operation in CCM To implement the CCM, the input ac filter is added to the system, and the dc capacitor is removed. In the following study, the rectifier is assumed to start in CCM-1 with balanced three-phase source and a resistive load Ω=15LR . At st 03.0= , the magnitude of cse is reduced by half leading to unbalanced operation among the rectifier phases. Then, at st 06.0= , the load is stepped to Ω= 5LR , which changes the mode to CCM-2. The resulting ac phase currents predicted by the models are illustrated in Figure 4.11. The corresponding dc bus current and voltage are shown in Figure 4.12. 79 Time (s) -100 0 0.01 0.04 0.1 100 i a s (A ) -100 0 100 i cs ( A) -100 0 100 i b s (A ) 0.07 Detailed PAVM- Simulink PAVM- PSCAD PAVM- EMTP-RV Figure 4.11 Transient response of the rectifier system to a change in ac voltages, leading to unbalanced operation in CCM, as observed in the ac currents predicted by various models. For clarity of the figures, only the responses of the detailed and PAVM models have been superimposed. The results of PAVM obtained using PSCAD, EMTP-RV, and Simulink are 80 all superimposed for comparison, and are essentially identical. It is observed that the PAVM predicts the unbalanced operation of the dc and ac variables quite well. 0 45 Detailed PAVM- Simulink 0 300 0.01 0.04 0.1 i d c ( A) 90 600 v d c ( V) Time (s) PAVM- PSCAD 0.07 PAVM- EMTP-RV Figure 4.12 Transient response of the rectifier system to a change in ac voltages, leading to unbalanced operation in CCM, as observed in the dc bus predicted by various models. The time steps needed for simulation by various models in different software packages are also tabulated in order to compare these models in terms of numerical efficiency. The number of time steps taken by the considered models for the study of Figure 4.12 is summarized in Table 4.5. In PSCAD, the detailed model is run using the typical EMTP time step of sµ50 . In Simulink, a variable-step solver ODE15s has been used which dynamically adjusts the time-step size and may therefore require fewer time steps as seen in Table 4.5. 81 Here, the maximum allowable time step of ms1 and both absolute and relative tolerances of 1e-4 have been used. As seen in Table 4.5, the PAVMs implemented in PSCAD, EMTP-RV, and Simulink require much fewer time steps and are therefore significantly faster than the respective detailed models. For consistency, the PAVM-Simulink model was run with the same solver, ODE15s, step size limits and tolerances as the corresponding detailed model. To obtain a similar result, the PAVM in PSCAD and EMTP-RV had to use a time step of sµ200 and sµ300 , respectively. Table 4.5 Comparison of simulation time steps of different models Model Step Size ( t∆ ) Number of Time Steps Detailed - PSCAD sµ50 2,001 Detailed - Simulink ms1 (max) 1,042 PAVM - PSCAD sµ200 501 PAVM – EMTP-RV sµ300 334 PAVM - Simulink ms1 (max) 247 4.2.4 System Impedance Analysis 4.2.4.1 Steady-State Analysis Let us first consider the system of Figure 2.2 fed from a balanced three-phase source and operating in a steady-state condition defined by a fixed load resistance LR . As discussed earlier, the harmonic content of the three-phase currents depends on the mode of operation which is in turn determined by this operating point and the overall equivalent series inductance present at the ac side. Performing a Fourier analysis on these currents and obtaining the fundamental component, the ac input impedance of the system acZ as seen from the input terminals, may be readily established by dividing the respective phasors of voltage and current. 82 To investigate the case of unbalanced operation, it is assumed that the magnitude of c- phase source voltage cse is reduced by half leading to asymmetry among the rectifier phases. The phasor diagrams for the input voltages in this case are depicted in Figure 4.13 (a). Since the input source voltages are known, the respective positive, negative, and zero sequence components may be readily obtained applying the appropriate transformation matrix [113]. The phasor diagrams corresponding to these symmetric components are also depicted in Figure 4.13 (b)-(d). It was clarified earlier that the zero sequence currents will not exist in the system due to topology even though the zero sequence is present in the input phase voltages of Figure 4.13. Under these conditions, the input currents are asymmetric consisting of the positive and negative sequences. The symmetric components of the fundamental currents may be obtained numerically by performing a Fourier analysis. The input impedances of the system corresponding to the positive and negative sequences may then be established from the respective phasors as ±±± ± ± ± ∠−∠=∠= asasac as as ac ieZ i e Z , , (4.2) eas ebs ecs eas + ecs+ ebs+ eas - ecs - ebs - 60o eas0ebs0ecs0 (a) (b) (c) (d) Figure 4.13 Phasor diagrams representing unbalanced operation: (a) asymmetric input voltages, (b) positive sequence, (c) negative sequence, and (d) zero sequence components. The values of the above impedances have been tabulated in Table 4.6 for balanced and unbalanced operation in DCM and CCM. For DCM operation, the series ac filter inductor is 83 removed from the system but the dc capacitor is included. For CCM operation with the series ac filter inductor, two separate cases (with and without the dc capacitor) are considered. Table 4.6 System input impedance in DCM and CCM-1 Ω= 70LR DCM CCM-1 (with Cap) CCM-1 (w/o Cap) acZ Ω∠ o2.104.36 Ω∠ o9.188.40 Ω∠ o9.165.40 + acZ Ω∠ o3.54.30 Ω∠ o2.2840 Ω∠ o3.165.40 − acZ Ω∠ o3.51.6 Ω∠ o7.349.8 Ω∠ o5.31.34 Ω= 35LR DCM CCM-1 (with Cap) CCM-1 (w/o Cap) acZ Ω∠ o1.124.18 Ω∠ o3.246.21 Ω∠ o1.245.21 + acZ Ω∠ o1.14.15 Ω∠ o1.309.21 Ω∠ o2.235.21 − acZ Ω∠ o1.11.3 Ω∠ o9.408.5 Ω∠ o1.78.16 4.2.4.2 Small-Signal Frequency-Domain Analysis Next, the small-signal output and input impedances as predicted by the average models are compared to the reference (the detailed model). For this purpose, the rectifier system is first assumed to work in the steady state operating point defined by LR . Based on the discussions of Section 2.5, the small-signal impedance of the system looking into the dc bus is then considered as dc dc dc i vZ ˆ ˆ − = . (4.3) The resulting impedances calculated using different models are shown in Figure 4.14 (top plots) for CCM operation defined by Ω= 1.13LR with the ac inductor filter but without the dc capacitor. For the purpose of extracting this impedance from the detailed model the data points for the magnitude and phase (crosses) have been obtained using the frequency sweep 84 technique. However, for the average models, the results may be obtained almost instantaneously using the numerical linearization offered by Matlab/Simulink [4]. As expected, all models predict inductive-type impedance, with the analytically-derived reduced-order models AVM-1 and AVM-2 being less accurate than the full-order PAVM. Extracting the positive and negative sequence impedances looking into the ac terminals and mapping the impedances for various harmonics can be extremely useful for analyzing the impact of the rectifier loads on the ac network, the system stability and power quality [14], [92]- [97]. The necessary considerations for evaluating such small-signal characteristics were discussed in Section 2.5. Adopting that approach, the final 4-by-4 impedance transfer matrix is then defined by (2.30). To give an example, the (1,1) element of the transfer matrix (2.30), the impedance qqZ , has been calculated for all considered models, and the result is shown in Figure 4.14 (bottom plots). As seen in this figure, overall, the system has an inductive-type response with the detailed switching model and the PAVM clearly showing a higher order response and a close agreement as compared to the AVM-1 and AVM-2, which are reduced- order models. This figure also illustrates that the AVM-2 does show an improvement over the classical model AVM-1 especially in predicting the magnitude in the high frequency range. As discussed earlier, the impedance seen from the ac side, in addition to including the ac input filter, also maps the dc side load impedance as appeared through the rectifier switching cell. This impedance therefore depends not only on dynamics of the load and filters, but also on the operational mode of the rectifier. To demonstrate the effects of the dc side on the ac side impedance, the dc capacitor is added to the rectifier system and the ac side impedances are calculated using the detailed and average models. The impedances qqZ and qdZ , i.e., the first row of the transfer matrix (2.30), are shown in Figure 4.15. The second row impedances are obtained similarly and would essentially reflect the same properties of the modes. The effects of the dc capacitor can be clearly seen in all plots of Figure 4.15. This effect is 85 particularly pronounced in the range of high frequencies in qqZ . This figure also shows that the full-order PAVM predicts the phase and magnitude of both impedances with great accuracy. f (Hz) 5 20 Detailed PAVM 35 AVM-2 AVM-1 0 30 60 10-1 101 102100 Z d c - M ag ni tu de (d B) Z d c - P ha se (d eg .) 90 f (Hz) 20 24 Detailed PAVM 28 AVM-2 AVM-1 0 40 101 102100 Z q q - M ag ni tu de (d B) Z q q - P ha se (d eg .) 80 Figure 4.14 Dc-side impedance dcZ (top) and ac- side impedance qqZ (bottom) of the rectifier system in CCM-1 with input ac filter but without dc capacitor predicted by different models. 86 f (Hz) 5 20 Detailed PAVM 35 AVM-2 AVM-1 -60 30 101 102100 Z q q - M ag ni tu de (d B) Z q q - P ha se (d eg .) 120 5 25 45 -450 -300 Z q d - M ag ni tu de (d B) Z q d - P ha se (d eg .) -150 f (Hz)101 102100 Detailed PAVM AVM-2 AVM-1 Figure 4.15 AC-side impedance qqZ (top) and qdZ (bottom) of the rectifier system in CCM-1 with input ac filter and dc capacitor predicted by different models. 87 4.2.5 Conclusion In this Chapter, the performance of analytical and parametric dynamic average models, implemented in SV-based and NA-based programs, was compared through extensive simulation studies in time and frequency domains under various loading conditions as well as for the cases of balanced and unbalanced excitation. Generally, it is concluded that the results obtained by different software simulation packages are essentially identical, given the time step and other solver properties are appropriately adjusted. Also, it is concluded that, although analytically-derived models AVM-1 and AVM-2 possess reduced dynamic order, they can be effectively used for the transient studies in CCM-1 (and in the vicinity of this mode). The PAVM model is shown to have great accuracy over a much wider range of operating conditions covering both DCM and CCM. It is also demonstrated that such models can accurately predict the small-signal impedance of the system as seen from either ac or dc sides. Based on this conclusion, the parametric approach is considered in next Chapter as the basis for developing a dynamic AVM for machine-fed six-pulse converters taking into account the effects of losses in the diode bridge as well as the machine losses. 88 Chapter 5: Inclusion of Losses in Machine- Fed Converter Systems 5.1 Introduction The previous Chapters were focused on the voltage-source-fed line-commutated converter systems (Case I in Figure 1.1) such as the variable frequency drives that are fed from the power network. As discussed in Chapter 1, another category of the line-commutated converter systems is the rotating-machine-fed systems (Cases II and III in Figure 1.1) that are widely encountered in applications such as the distributed generation systems and the electric systems of vehicles. In the design stages of vehicular power systems, particularly, numerous computer simulations are typically run many times for concept evaluation, prototyping of new power train, design optimization, evaluation of energy and fuel efficiency under various driving conditions, reliability assessment, etc [18]. Accurate dynamic models are then required for large-signal time-domain system-level transient studies as well as small-signal frequency- domain stability analysis and controller design. This becomes a problem, especially when the alternator-rectifier system is just a part of the larger, more complex and diverse vehicular system containing electrical and mechanical components with a wide dynamic range and spread of time scales. Moreover, for multi-objective parameter optimization studies [114], for instance, simulations of such systems may run multiple times (in a loop) and the overall computing time could become quite significant. Therefore, employing dynamic average 89 models instead of the detailed switching models is extremely beneficial for such analyses conducted on vehicular power systems. As discussed in Chapter 1, however, the existing average models for machine-converter systems typically neglect the losses in the system. The main sources of losses in such systems are diode conduction losses in the bridge and the machine rotational losses. The effects of these losses are particularly important in lower voltage applications such as vehicular power systems that typically operate at 12-14 V dc. In such cases, the typical diode drops are significant with respect to the output voltage value and their effects are hence pronounced. Moreover, typical car alternators have a low efficiency due their particular structure [115], [116] which demands for appropriate consideration of the machine losses to achieve accurate system modeling. The focus of this Chapter is the dynamic average modeling for machine-fed converters taking various sources of losses into account. For this purpose, the parametric methodology [70], is extended to include the machine rotational losses as well as the diode bridge conduction losses. First, a brief description of the considered vehicular power system structure is presented. An example vehicular power system is then considered with the parameters provided in Appendix A.4. The detailed and dynamic average modeling of the system including various losses is presented next. The developed models are verified using experimental data and simulation studies. 5.2 Example Vehicular Power System Architecture The most common mass-produced vehicles are based on the power system architecture shown in Figure 5.1. A typical configuration includes an alternator-rectifier system which is driven by the internal combustion engine (ICE). In most cases, the alternator-rectifier system consists of a synchronous machine with wound field winding and a diode bridge which rectifies the three-phase output voltage to supply the internal electrical system with 12-14 V 90 dc (or a dual 12- 42 V dc). The rectifier is usually packaged inside the alternator, whereas an LC filter may or may not be included. Since the engine speed varies over a wide range, the field current is adjusted, by means of a closed-loop voltage regulator, to obtain the desired voltage level. This system supplies power to the automotive dc subsystems and energy storage elements (e.g., batteries [117]), as depicted in Figure 5.1. Alternator Exciter DC System (Low Power Loads) IC AltEngine Energy Storage Rectifier LV Battery DC Bus (12...14 V) Pm Pac PdcProt_loss PCu Pfd Pbr Figure 5.1 A typical vehicular electric power system and its power conversion chain. The claw-pole, also known as “Lundell”, alternator is the most common type of synchronous machine in automotive applications. This alternator is similar to standard wound rotor synchronous machine in functionality but has a different internal structure. Concentrated stator windings (1 slot/pole/ phase) are also used to minimize cost. The claw- type rotor geometry and concentrated stator windings lead to voltage and/or current harmonics. Precise dynamic characterization of Lundell machines conventionally relies on the detailed finite element (FE) analysis [118]- [120] or the magnetic equivalent circuits (MEC) [121] (which is not available in commonly used simulation packages), both of which are computationally very expensive [122]. As a compromise between model complexity and computational efficiency, a fourth-order coupled-circuit model of a claw-pole alternator has been proposed in [123], where harmonic effects introduced by rotor saliency, concentrated 91 stator windings and stator slots, as well as magnetic saturation [124] have been included. The resulting model is shown to be more accurate but still considerably complicated compared to standard models of synchronous machines [51]. For simulation of automotive alternator-rectifier systems, and particularly from a system- level point of view, accurate representation of the alternator structural details may be of less interest. Therefore, the standard synchronous machine models in physical ( abc ) variables, that are numerically efficient and considerably less complicated, may be used to provide a reasonable accuracy. However, due to their special structure, the 14V Lundell- alternator/diode-rectifier systems are well known to have a relatively low efficiency [115], [116] and the amount of iron core losses is particularly significant in such machines. To achieve accurate system modeling, appropriate representation of the losses present in the system is then of critical importance. In the next sections, detailed and dynamic average modeling of the claw-pole Lundell- alternator-based automotive power system systems is performed with the special focus on effectively including the effects of the losses in order to accurately predict the power chain in the system. 5.3 System Detailed Model In the vehicular power system of Figure 5.1, the power conversion chain from input mechanical power, mP at the alternator shaft, to output electric power, dcP at the dc bus, is also illustrated. As shown in this figure, the important losses in the alternator include the stator copper loss, CuP , combined rotational and core loss, lossrotP _ , and the excitation (field) winding loss fdP . Other types of loss in the alternator include mechanical losses (friction, windage effect, etc.) as well as the stray losses which may be challenging to quantify independently. The approach used in this Chapter lumps these losses into the 92 rotational losses while characterizing the losses in the alternator. Therefore, all these rotational, stray, and iron core losses of the alternator are combined and represented by lossrotP _ in Figure 5.1. Finally, the difference between the electrical power at the ac and dc terminals, denoted by acP and dcP respectively, is equal to the diode bridge conduction loss denoted by brP . A circuit diagram of the alternator-rectifier-battery system of Figure 5.1 is depicted in Fig. 2 (top). The assumed voltage regulator-exciter, Fig. 2 (bottom), is used to regulate the dc voltage through the field excitation adjustment. The system parameters are summarized in Appendix A.4. In detailed model, the alternator synchronous machine is represented in terms of physical ( abc ) variables. ias vdc + - vfd S1vas +- S2 S3 S4 S5 S6 - + Alternator as bs cs kfd ifd idc vab_s + - Cin rb voc BatteryRectifier RL Load ia2s va2s +- a2s b2s c2s iload ibat 1 τ1s +1 vref0 Vmax 1 τ2s +1 vdcvfd sKp + Ki Voltage Regulator-Exciter rc rc rc Figure 5.2 A circuit diagram for automotive alternator-rectifier-battery systems (top) with a typical voltage regulator-exciter (bottom). 93 5.3.1 Representation of Rotational Losses To represent the iron core losses, which have resistive nature (due to eddy currents) and are significant in this type of machine, the approach considered here consists of introducing an additional short-circuited three-phase windings ( 2abcs ) to the stator side such that its resistance cr would dissipate the real power. The machine model therefore becomes similar to a six-phase synchronous machine model [125] with one set of windings short circuited. The classical representation of iron losses typically uses a single resistor that can only represent the losses at one particular frequency of operation. Including an additional set of winding, however, allows for inclusion of losses over a range of frequencies. Using the curve-fitting techniques, the winding parameters may be then obtained such that the loss characteristics predicted by the model would match the actual loss characteristics over a winde range of frequencies. The adopted concept is similar to using RL ladder network in order to model the core losses over a range of frequencies in magnetic and electric equivalent circuits [126], [127]. Since the purpose of this additional winding is to represent the losses, its angular displacement is not relevant. Without loss of generality, the winding 2abcs is then assumed to be in the same angular position as the main stator winding abcs . This choice of the winding location significantly simplifies magnetic coupling among the windings making it easier to determine the corresponding parameters. The parameters of additional winding 2abcs are calculated based on the experimental measurements of the alternator losses. To measure the losses, a variable-speed dc machine is employed as a prime mover while the alternator stator terminals are kept open-circuited. For a fixed excitation current of Ai fd 2= , the open-circuit stator phase voltage and the mechanical torque on the shaft are measured while the shaft speed is varied. The corresponding measured losses are summarized in Appendix A.4. As mentioned earlier, this 94 measurement combines the rotational, stray, and iron core loss into a single lumped loss characteristic. Since the stator back emf voltages induced in winding 2abcs vary linearly with speed, the variation of losses with respect to the back emf is quite similar in shape to the variation of losses with respect to the shaft speed. In particular, the measured characteristic is then approximated by a quadratic expression as ( ) c a alossrot r EEP 2 _ = , (5.1) where aE is the back-emf induced in the additional winding 2abcs . Since the additional windings 2abcs are similar to the stator windings abcs , their back emf are equal to the open-circuit stator voltages. The value of cr is then obtained using the least square curve- fitting method to achieve the best fit for the measured loss characteristic (5.1). The result is given in Appendix A.4. The final machine model in phase coordinates is then described by the corresponding voltage, flux linkage and torque equations as           +           − −           =           = fd sabc abcs fd sabc abcs fd c s fd sabc abcs dt d irv λ 222 00 00 00 0 λ λ i i r r v v , (5.2) ( ) ( ) ( ) ( )           − −           =           fd sabc abcs fdrrsrrs rsrsss rsrsss fd sabc abcs iL 221 21 2 i i LL LLL LLL λ λ θθ θ θ λ , (5.3) ( ) ( )[ ] fdrsr r T sabcabcse i PT ′′ ∂ ∂ +      = θ θ Lii 22 , (5.4) where abcsv , abcsi and abcsλ are the stator phase voltage, current and flux linkage vectors, and fdv , fdi and fdλ are rotor-winding voltage, current and flux linkage, respectively. Also, sabc2v , sabc2i and sabc2λ are the voltage, current and flux linkage of the second set 95 of stator winding. The diagonal matrices sr and cr contain the stator winding resistance and the resistance corresponding to the combined losses, respectively. The field winding resistance and inductance are denoted by fdr and fdL . Matrices sL and ( )rsr θL represent the stator winding self- and mutual-inductances and the stator-to-rotor mutual inductances respectively. The angle rθ denotes the rotor position. The detailed expressions for these inductance matrices may be found in [51]. Note that since the additional stator winding is chosen to have the same angular displacement as the regular stator windings, the inductance matrix is considerably simplified compared to a six-phase synchronous machine with shifted sets of windings [125]. Matrix 21ssL contains the mutual inductances between the two sets of stator windings and may be expressed similarly. Such a choice of winding displacement also simplifies the expression for electromagnetic torque (5.4). In (5.4), P denotes the number of poles. In ( )rsr θL′ and fdi′ , all quantities are referred to the stator side. 5.3.2 Verification of Detailed Model Next, the detailed model of the alternator-rectifier system is experimentally verified. For this purpose, the field winding is supplied with a constant dc source of Vv fd 4.6= , and a resistive load, Ω= 5.11LR , is directly connected to the dc bus. A dc machine, with adjustable speed, is coupled to the alternator to emulate the changes in engine speed. A transient study is carried out in which the alternator speed is varied as shown in Figure 5.3 (top plot). The recorded speed waveform is also used as the synchronous machine speed in the detailed model simulation studies. The resulting measured and predicted dc bus voltage, dcv as well as measured and simulated waveforms of the stator phase current, asi , and stator line-to-line voltage, sabv _ , are illustrated in Figure 5.3. For better comparison between the measured and simulated 96 waveforms, the dc bus voltages, dcv , and stator currents, asi , are superimposed and a zoomed-in view is also provided in Figure 5.3 (two bottom plots). An excellent match between corresponding waveforms in Figure 5.3 implies a good accuracy of the detailed model, which is considered acceptable for the purpose of dynamic average modeling methodology presented in the later sections. 0.2 0.4 0.6 0.8 1 500 1500 2500 1000 2000 Sp ee d (r pm ) 10 0 20 30 10 0 20 30 v d c (V ) v d c (V ) See  Zoomed-in Views Below MeasuredDetailed Model -3 -3 i as ( A) 1 0 2 3 1 0 2 3 -1 -2 -1 -2 i as ( A) MeasuredDetailed Model 0.2 0.4 0.6 0.8 10.2 0.4 0.6 0.8 1 Time (s) -40 -40 v a b_ s (V ) 20 0 40 20 0 40 -20-20 v ab _s ( V) MeasuredDetailed Model Time (s) Measured Detailed Model 0.820 0.823 0.826 0.829 Time (s) -4 i as ( A) 4 2 6 0 -2 -6 Measured Detailed Model 0.843 0.844 0.845 0.846 0.847 0.848 14 v d c (V ) 30 26 34 22 18 Time (s) Zoomed-in Views: See  Zoomed-in Views Below Figure 5.3 Measured and simulated detailed responses to speed increase observed in the dc and ac sides. 97 5.3.3 Battery Model To complete the system depicted in Figure 5.1 with the energy storage, the following methodology was considered to model the battery. While the proposed methodology is independent of battery type, a commercial lead-acid battery for automotive application is considered in this paper with the details summarized in Appendix A.4. The battery is represented by its equivalent circuit as depicted in Figure 5.2. It is well understood that the battery open circuit terminal voltage, ocv , and the equivalent internal resistance, br , vary significantly depending upon its state of charge [128]. A state-of-charge model is employed here, where ocv and br are approximated by polynomial functions of the state of discharge. A similar battery representation has been used previously in the modeling of electric vehicles [129] and [130]. In [130], accurate results were obtained using fifth-degree polynomials ( )∑ = −+= 5 1 0 1 j jjoc Qv αα , (5.5) ( )∑ = −+= 5 1 0 1 j jjb Qr ββ , (5.6) where Q is the relative state of charge (SoC), and the coefficients jα and jβ are extracted experimentally. The relative SoC is expressed as ( ) ττ diQQQ t batinit ∫ ⋅ += 0 max3600 1 , (5.7) where ( )τbati is the battery charging current, initQ is the initial relative SoC, and maxQ is the battery capacity (Ah). According to (5.7) and Figure 5.2, the current drawn from the battery is simply bati− . 98 5.4 System Dynamic Average-Value Modeling 5.4.1 Alternator Model in Transformed qd Coordinates and Variables The developed AVM employs the classical qd rotor reference frame to model the alternator in transformed coordinates and variables. The general theory of qd model of a six- phase machine with two sets of arbitrary shifted stator windings has been set forth in [125]. However, since in the detailed model developed in the previous section, the two sets of windings are aligned, the complexity of the model [125] significantly reduces. In particular, to obtain the transformation matrix for representing the variables in the so-called rotor reference frame, the general reference frame transformation matrix (2.7) is adopted with the angle set to the rotor angle rθ :                         +      −       +      − = 2 1 2 1 2 1 3 2 sin 3 2 sinsin 3 2 cos 3 2 coscos 3 2 piθpiθθ piθpiθθ rrr rrr sK . (5.8) The abc variables for each set of the three-phase windings are transferred into the rotor reference frame. Applying transformation (5.8) to (5.2)–(5.4) and algebraically manipulating the results, the final state model can be compactly expressed as follows:       +−= qsds b r qssb qs vir dt d ψ ω ω ω ψ . (5.9)       ++= dsqs b r dssb ds vir dt d ψ ω ω ω ψ . (5.10)       +−= sqsd b r sqcb sq vir dt d 222 2 ψ ω ω ω ψ . (5.11) 99       ++= sdsq b r sdcb sd vir dt d 222 2 ψ ω ω ω ψ . (5.12) ( )fdfdfdbfd virdt d ′+′′−= ′ ω ψ . (5.13) Here, bω is the base angular frequency, and rω denotes the rotor angular speed. Without loss of generality, in (5.9)-(5.13), the flux linkages ( λ ) and the inductances ( L ) have been replaced by flux linkages per second (ψ ) and the reactances ( x ), respectively, according to the following relationships: λωψ b= , Lx bω= . (5.14) The expressions for currents are derived from the flux linkage equations as: ( ) ls mqqs qs x i ψψ − −= . (5.15) ( ) ls mdds ds x i ψψ −−= . (5.16) ( ) sl mqsq sq x i 2 2 2 ψψ − −= . (5.17) ( ) sl mdsd sd x i 2 2 2 ψψ − −= . (5.18) ( ) lfd mdfd fd x i ′ −′ =′ ψψ . (5.19) 1 2 111 −         ++= sllsmq aq xxx x . (5.20) 1 2 1111 −         ′ +++= lfdsllsmd ad xxxx x . (5.21)         += sl sq ls qs aqmq xx x 2 2ψψψ . (5.22) 100         ′ ′ ++= lfd fd sl sd ls ds admd xxx x ψψψψ 2 2 . (5.23) The torque equation in the rotor reference frame becomes [ ]sdsqsqsddsqsqsds b e iiii PT 2222 1 22 3 ψψψψ ω −+−                  = . (5.24) The model (5.9)–(5.24) is relatively simple and very convenient to implement. Finally, Figure 5.4 shows the equivalent circuit of the alternator model in qd coordinates. This final model is used for the developed average-value model. - + rc ωr ψq2s ωb - + rs ωr ψqs ωb r'fd vds + - v'fd + - ids id2s i'fd -+ rc ωr ψd2s ωb -+ rs ωr ψds ωb xls vqs + - iqs iq2s ωb xl2s ωb xmq ωb xmd ωb xls ωb xl2s ωb x'lfd ωb Figure 5.4 Equivalent circuits of the considered alternator model in q and d axes. 101 5.4.2 Average-Value Modeling of Non-Ideal Rectifier The parametric approach discussed previously relates the averaged rectifier dc voltage and current, dcv and dci , to transformed ac variables rec qdsv and rec qdsi through (2.24) and (2.25). For the case of an ideal bridge which was assumed in [70], the first relationship is dc c qds v)(⋅= αv , (5.25) where (.)α is an algebraic function of the loading conditions. However, in low-voltage automotive applications, the losses due to the diode forward-voltage drop can be significant (this is simply because the voltage drop of the two conducting diodes in series, which may be on the order of 1.2 to 1.5V, will be more noticeable in proportion to the output voltage, here about 12V) and it is imperative to take such losses into account. In the case of detailed switching model, the non-ideal diode can be represented by a series connection of an ideal diode, a resistor and/or a constant voltage source in order to account the typically series losses in the bridge. The series losses (the total diode forward voltage drops) would cause an unknown equivalent drop in the output average voltage denoted by drpeqv , . Considering this effect, (5.25) can be extended for the case of non-ideal diodes as ( ) dcdc dc drpeq drpeqdc c qds vv v v vv )(1)()( , , ⋅′=        −⋅=−⋅= αααv , (5.26) Here, (.)α ′ is a new parametric function which also includes the effect of diode voltage drops on the dc bus voltage. The dc and ac side currents are related similarly. For the case of an ideal bridge: c qdsdci i)(⋅= β , (5.27) where (.)β is another parametric function of the loading conditions. If it is desired to extend the above equation to include the effects of shunt losses in the rectifier, the parametric 102 function (.)β should be modified to )(⋅′β according to the unknown equivalent shunt current sheqi , as: c qdssheqdc ii i)(, ⋅=− β , (5.28) c qds c qdsc qds sheq sheq c qdsdc i ii ii i i )(1)()( , , ⋅′=           +⋅=+⋅= βββ , (5.29) Although representing shunt losses may be important for high voltage applications (e.g. HVDC) where the snubbers and voltage balancing circuits cause additional losses, such losses are typically negligible in the low voltage rectifies due to good insulation and small reverse current of the diodes. Therefore, (5.27) may be considered sufficient for the automotive system considered in this paper. To complete the model, the angle between vectors cqdsv and cqdsi is expressed based on Figure 2.4, setting the arbitrary reference frame to rotor reference frame:         −         =−         =⋅ r qs r ds r qs r ds r qs r ds v v i i i i arctanarctanarctan)( δφ . (5.30) Instead of deriving analytical expressions for (.)α ′ , (.)β ′ and (.)φ , these functions can be extracted numerically from the detailed simulation. For compactness, these functions are expressed in terms of dynamic impedance of the rectifier switching cell defined as c qds dcvz i = . (5.31) A transient study is carried out in which the load resistance is slowly changed in a wide range from 0.1 Ω to 100 Ω . The numerical functions (.)α ′ , (.)β ′ , and (.)φ together with the impedance, z , are then calculated for each point using (5.26), (5.29), and (5.30). These functions are stored in look-up tables and used in the average-value model implementation. 103 Using the modified switching model with non-ideal diodes to extract the average-value model parameters, the effects of diode non-idealities and rectifier losses are automatically included into the numerically calculated parametric functions (.)α ′ , (.)β ′ and (.)φ . To demonstrate the improvement achieved by including the rectifier losses, the parametric functions (.)α ′ , (.)β ′ and (.)φ have been extracted for both ideal and non-ideal rectifier. The results for both cases are superimposed in Figure 5.5. As this figure demonstrates, the effect of forward voltage drop is more pronounced at heavy loads as the impedance z (or the output dc voltage) decreases. For completeness, the values of parametric functions are also given in Appendix A.4 for the non-ideal rectifier. 0 30 60 90 0 30 60 90 0 30 60 90 Operating point Impedance (z = vdc / ||iqds||), Ω 0.50 1.10 0.90 1.30 0.70 α 1.50 0.90 0.96 0.94 0.92 β 0.00 0.15 0.10 0.20 0.05 0.25 φ Ideal Diode Non-ideal Diode Ideal Diode Non-ideal Diode Ideal Diode Non-ideal Diode Figure 5.5 Parametric functions considering ideal and non-ideal rectifier diodes. 104 5.5 Case Studies The new AVM developed in the previous section has been implemented in Matlab\\Simulink [4] and verified in a number of studies presented here. The block diagram of the overall combined model of the vehicular system is depicted in Figure 5.6. This figure also shows the input and output variables of the corresponding subsystems. In the following study, it is assumed that the alternator field winding is supplied from a constant dc source of Vv fd 4.6= . Similar to Section 5.2, a resistive load of Ω= 5.11LR is assumed to be directly connected to the dc bus. To verify the considered detailed and the average-value models, the same speed variation as in the studies of Figure 5.3 is assumed here. The corresponding simulation results are superimposed in Figure 5.7. It should be noted that this study has also been considered for verifying the detailed model with the experimental results presented in Figure 5.3. As seen in Figure 5.7, the transient responses predicted by the proposed average- value model precisely follow the averaged behavior of the detailed model (and measured waveforms from Figure 5.3) throughout the whole speed variation region. In order to demonstrate the improvement achieved by the proposed average-value model with respect to the previously established AVM [70] that assumes an ideal/lossless rectifier, the results predicted by AVM [70] are also provided in Figure 5.7 (right plots). As seen in these plots, the AVM [70] overestimates the dc bus voltage and current because the bridge losses are not considered. The output (dc) power predicted by this model is also significantly higher. At the same time, the proposed AVM predicts non-zero rectifier losses (see Figure 5.7 bottom subplot) and remains in close agreement with the detailed model throughout the transient study. 105 Battery Model  (5.5)-(5.7) and DC System vdc Proposed Average Model (5.26), (5.29)-(5.31) Alternator Model (5.9)-(5.24) Voltage Regulator-Exciter vfd vdc idc vqds iqds vrefωr Figure 5.6 Block diagram of the overall combined model depicting subsystems and their inputs and outputs. Time (s) 0.2 0.4 0.6 0.8 1 500 1500 2500 1000 2000 Sp ee d (rp m ) 10 0 20 30 v d c (V ) 0.2 0.4 0.6 0.8 1 Proposed AVM Detailed 1.4 0.8 2.0 2.6 i dc ( A) AVM [70] Detailed 0.2 0.4 0.6 0.8 1 12 8 16 20 v q s , v d s (V ) 4 0 0.2 0.4 0.6 0.8 1 vqs-Proposed AVM vqs-Detailed 0.2 0.4 0.6 0.8 1 1 0 2 3 i qs , i ds ( A) -1 iqs-Proposed AVM iqs-Detailed 2.0 1.9 2.1 2.2 i fd ( A) 1.8 0.2 0.4 0.6 0.8 1 Proposed AVMDetailed ids-Proposed AVMids-Detailed vds-Detailed vds-Proposed AVM Time (s) 80 40 0 6 3 0 P d c (W ) P b r (W ) 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 AVM [70] Proposed AVM AVM [70] Proposed AVM Proposed AVM AVM [70] Figure 5.7 System response to the increase in alternator speed as predicted by the detailed and average-value models under fixed excitation. 106 Next, the voltage regulator-exciter is added to the system to regulate the dc bus voltage. The reference voltage, refv , is set to 14 Volts. The same speed increase of Figure 5.3 is considered here. Figure 5.8 shows the transient responses in excitation/field current and the dc bus voltage obtained by the detailed and the proposed average-value models as a result of the considered speed increase. As can be seen in Figure 5.8, the transient response of the closed loop alternator-rectifier system is predicted very well by the proposed AVM. Time (s) 0.2 0.4 0.6 0.8 1 500 1500 2500 1000 2000 Sp ee d (r pm ) 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.0 2.0 1.5 2.5 i fd ( A) 10 6 14 18 v d c (V ) AVMDetailed AVMDetailed Figure 5.8 Input speed, excitation (field) current and the dc bus voltage predicted by the detailed and the average-value models when the system uses the voltage regulator-exciter. In the next study, the battery is connected to the dc bus in parallel with a variable resistor that represents the equivalent electric loads in the vehicle. The reference voltage refv is set 107 to 13.55 V to maintain the desired charging current of 2.6A into the battery. The value of this current is adjustable and it will determine how fast the battery will charge during the driving of the vehicle. The battery is assumed to have initial SoC of 95%. The study carried out demonstrates the system dynamics during the engine acceleration. For this purpose, the alternator is accelerated from 950 rpm (i.e., the idle engine speed) to 4000 rpm. The corresponding voltage and current waveforms at the dc bus are illustrated in Figure 5.9, together with the battery current and the relative SoC in percent. Overall, an excellent match is observed between the system responses predicted by the average-value and the detailed models, apart from the switching ripple present in the detailed model. To demonstrate the effects of various losses present in the system and the effectiveness of the proposed AVM in predicting the complete power conversion chain, the alternator input mechanical power, mP , and the rectifier output dc power, dcP , have also been plotted in Figure 5.9. For better clarity, in the right corner bottom plot, various losses in the system have been superimposed and compared. In this subplot, only the average model results are shown, whereas the detailed model responses are similar and omitted for clarity reason only. As seen in Figure 5.9, the unique feature of the proposed average model is that it can predict the whole power conversion chain in the system equivalent to using the detailed model. As observed in this study, the amount of losses is significant and consistent with typical Lundell-Alternator-Rectifier systems previously described in the literature [115], [116]. This application also clearly demonstrates the need for inclusion of the losses in the system-level models. Note that even the diode bridge losses are significant since the current drawn from the dc bus can be quite high in such automotive systems. The numerical efficiency of the proposed model is evaluated next. The computer studies were carried out using Matlab\\Simulink software ran on a personal computer (PC) with a 2.4 MHz AMD 3800+ processor. To achieve accurate results with the detailed model, it was found that the variable-step solver ODE15s with a maximum allowable time step of 0.0001s 108 was needed. To obtain the results of Figure 5.9, the detailed simulation took 10.28s of CPU time requiring a total of 79,512 time steps. The AVM does not have switching and can execute with much larger time steps. With the maximum allowable time step of 0.01s, using the same solver settings, the AVM simulation took 0.14s of CPU time requiring only 1,645 time-steps. This demonstrates almost two orders of magnitude improvement in simulation speed for obtaining essentially identical system-level transient results. 0.4 0.8 1.2 1.6 0 2000 4000 1000 3000 Sp ee d (rp m ) 13.45 13.65 13.55 i dc ( A) 16 11 21 v d c (V ) 5000 0.4 0.8 1.2 1.6 0.4 0.8 1.2 1.6 0.4 0.8 1.2 1.6 i ba t ( A) 2 -2 6 AVMDetailed AVM Detailed AVM Detailed Time (s) 0.4 0.8 1.2 1.6 Q (% ) 95.003 95.002 95.004 95.005 AVM Detailed 320 400 360 220 160 280 0.4 0.8 1.2 1.6 0.4 0.8 1.2 1.6 0.4 0.8 1.2 1.6 80 0 160 Time (s) Pm , AVM M ec ha ni ca l P ow er (W ) Ou tp ut D C Po we r ( W ) Va rio us L os se s ( W ) Diode Bridge Loss ( Pbr= Pac- Pdc ) Machine Field ( Pfd ) Machine Combined Losses (Prot_loss ) Pm , Detailed Pdc , Detailed Pdc , AVM Figure 5.9 System response during the engine acceleration from 950 rpm to 4000 rpm predicted by the detailed and proposed average-value models. 109 Another very useful feature of the developed AVM is its application to the small-signal analysis of the considered system, which can be achieved very efficiently using numerical linearization (available in all state-variable-based simulation tools such as Simulink [4]). This type of analysis cannot be conveniently performed using the conventional detailed switching model. To demonstrate this feature, the control-to-output transfer function of the system, from the field (excitation) voltage to the dc bus output voltage, has also been extracted using both detailed and average models. Other system-level transfer functions (e.g., from the shaft speed to the output voltage, etc.) can also be similarly obtained by numerical linearization of the developed AVM. A small-signal analysis has been performed around an operating point corresponding to the engine speed of 1520 rpm and the output voltage of 13.55V. The small signal perturbation is then only injected in the field voltage and the speed is kept constant. Figure 5.10 shows the Bode diagrams of the resulting system transfer function. As this figure demonstrates, the average model predicts the small-signal characteristics of the system with a superior accuracy. 100 -60 -40 -50 -160 -40 -100 102101 AVM Detailed M ag ni tu de ( db ) 20 Ph as e (d eg ) 100 102101 AVM Detailed -30 Frequency (Hz) Figure 5.10 System transfer function from the field (excitation) voltage to the dc bus voltage predicted by the detailed and average-value models. 110 Chapter 6: Generalized Dynamic AVM for High-Pulse-Count Converters 6.1 Introduction As discussed in Chapter 1, several AVMs for converter systems with the number of pulses more than six have been developed in the literature, most existing models are derived for a specific topology and have a limited range of accuracy. The objective of this Chapter is to develop a methodology that could be applied to any network- or rotating-machine-fed high- pulse-count converter system. 6.2 High-Pulse-Count Converter System Structures The general structure of a high-pulse-count converter system, wherein the number of phases is considered to be a multiple of three is depicted in Figure 6.1. The most-commonly- used numbers of pulses are 12, 18, and 24, corresponding to 6-, 9-, and 12- phase ac subsystems, respectively. Depending on the application, the system may be fed from a three- phase power network, represented by its Thevenin equivalent, as in Case I, or a single rotating machine (generator) as in Cases II and III. In the first two cases, the three-phase voltages at the input are transformed into multi-phase voltages at the output, using a 3-to-n- phase transformer. Alternatively, in Case III, these multi-phase voltages are directly produced by means of a multi-phase generator. In any case, the resulting multi-phase voltages are fed into several bridge rectifiers that are interconnected at their dc sides through 111 an optional Inter-Phase Transformer (IPT) followed by an optional dc filter network. The bridges, in general, maybe interconnected in various configurations, i.e., series, parallel, or a series/parallel combination. The overall system ultimately feeds a dc system (load) as depicted in Figure 6.1. Case I: Case II: Rectifier Bridges AC System DC Filter (Optional) DC System 3-Phase Distribution Feeder (or  Transformer) 3-Phase Rotating Machine Case III: Multi-Phase  Rotating Machine Bridge  Interconnection (Series/Parallel )  and Inter-Phase  Transformer  (Optional)iap - + ecs ebs eas Ls Ls Ls rs rs rs 3-phase n-phase ibp icp idc1 vdc1 + - idc2 vdc2 + - idcn vdcn + - Load Transformer G G iasex-1 iasex-nset iasex-2 iasex-1 iasex-nset idc vdc + - C rdc Ldc iabc,sex-1 iabc,sex-nset - + - + Figure 6.1 Classification of high-pulse-count rectifier systems. The particular emphasis of this Chapter is on modeling of the multi-phase transformer associated with the first two cases. In the ac subsystem of Figure 6.1, the secondary voltages are divided into several sets of three-phase voltages that are supposed to be equally shifted in phase. In the case of a 6-phase (12-pulse) converter, the two sets of 3-phase voltages at the output are typically required to be shifted by 30 electrical degrees. In Cases I and II in Figure 6.1, such voltages may then be readily obtained using a wye-delta connection on the secondary side of a conventional 3-phase transformer with two sets of secondary windings. The displacement angle of 60 degrees, instead of 30 degrees, is also sometimes considered in 12-pulse converter systems, especially for machine-fed systems [74], i.e., Case III in Figure 6.1. 112 In converters with higher numbers of pulses than 12, the output 3-phase sets are typically shifted according to the following: °=°=°= 120136013602 setph disp nnp θ , (6.1) where p , phn , and setn are used to denote the number of pulses, phases, and three-phase sets, respectively. In general, the desired shift is obtained using the appropriate interconnection of the windings at the secondary side of a 3-phase transformer with several segments of secondary windings. The terminology used throughout this thesis is that, each of the three-phase groups of windings on the secondary side of such transformer is denoted by “segment” because these windings are interconnected in order to produce the final “sets” of extended secondary windings. However, one of these final sets of extended secondary windings is typically composed of only one base segment. In the simplest and exceptional case of 6-phase transformer discussed above, both final sets are each composed of only one segment of the secondary windings; hence no distinction has to be made between the sets and segments therein. In general, however, the number of segments segn is equal to or greater than the number of final three-phase sets setn . For the purpose of discussions, an example 400Hz aircraft power system with the 9-phase (18-pulse) rectifier topology shown in Figure 6.2 and the parameters summarized in Appendix B is considered in this thesis. The presented analyzes, however, can be readily extended to any multiple of three phases. As seen in Figure 6.2, a three-phase source feeds the 3-to-9-phase transformer which in turn is connected to three bridges at the secondary side. The displacement angle between each set of voltages at the secondary side of the transformer is 40 electrical degrees as dictated by (6.1). Without loss of generality, herein the bridges are connected in parallel without an inter-phase transformer and feed the dc system 113 through an optional capacitor as the dc filter. The dc load is represented by its equivalent resistance denoted by LR . Several configurations of the magnetic core and connections may be considered for the 3- to-9-phase transformer of Figure 6.2 [131], [78]. For example, in the configuration depicted in Figure 6.3 [131], the transformer has a conventional three-leg structure with a primary set of windings and five secondary segments of windings. The primary and secondary voltages corresponding to the configuration of Figure 6.3 are illustrated in Figure 6.4 (top diagrams) with the three sets of evenly shifted secondary voltages shown separately (bottom diagrams). In this figure, it is illustrated how the winding connections on the secondary side result in the shifted sets of voltages. According to these diagrams, in order for the secondary voltages to be equal in magnitude and evenly shifted in phase by 40 degrees, it is necessary that [131]: 1152 742.0120sin 40sin ssss nnnn = ° ° == . (6.2) 1143 395.0120sin 20sin ssss nnnn = ° ° == . (6.3) vdc + - idciap- + - + - + ecs ebs eas Ls Ls Ls rs rs rs 3ph 9ph C RL -iabc,sex-1 ibp icp idc1 idc3 vdc1 + - -iabc,sex-3 vdc3 + - Figure 6.2 Typical 9-phase (18-pulse) rectifier example system topology considered in this Chapter. 114 vap + - iap ias1 1 O 5 5' 2' O' 9 9' 6' O\" vbp + - ibp ibs1 4 O 8 8' 5' O' 3 3' 9' O\" vcp + - icp ics1 7 O 2 2' 8' O' 6 6' 3' O\" ias2 ias3 ias4 ias5 ibs2 ibs3 ibs4 ibs5 ics2 ics3 ics4 ics5 np : ns1 : ns2 : ns3 : ns4 : ns5 : Figure 6.3 Typical 3- to 9-phase transformer structure for the 18-pulse rectifier example system. 115 2 2' 8' O' 5'5 8 O 1 7 4 vap vbp vcp vasex-2 6' 3' O\" 3 6 99' 40o 40ovas1=vasex-1 vcs1= vcsex-1 v bs 1= v bs ex -1 vas3 vbs3 vcs3 -vcs2 -vbs2 -vas2 vasex-2 vbsex-2 vcsex-2 vas4 vbs4 vcs4 -vas5 -vbs5 -vcs5 vasex-3 vbsex-3 vcsex-3 vasex-1 vasex-3 vbsex-1 vbsex-2 vbsex-3 vcsex-1 vcsex-2 vcsex-3 Figure 6.4 Phasor diagrams of the primary and secondary voltages for the 3-to-9-phase transformer of Figure 6.3. 6.3 Multi-Phase Transformer Modeling 6.3.1 Original Interconnected Transformer Model Let us consider a general case of 3-to-n-phase transformer with a three-leg structure, similar to that depicted in Figure 6.3. The transformer is typically composed of a primary set of windings, specified by the subscript p , and segn secondary segments of windings denoted by the subscript ks , segnk ...,,2,1= . Each leg of the transformer corresponds to one 116 of the three phases a , b , and c , and the three windings within each of the three-phase primary set/secondary segment are assumed identical. This transformer may then be mathematically described by the voltage and flux linkage equations corresponding to the primary set and each of the secondary segments of windings. These equations can be written in a compact matrix form as pabcpabcppabc dt d ,,, λiRv += , (6.4) ∑ = += seg kk n k sabcpspabcpppabc 1 ,,, iLiLλ , (6.5) segksabcsabcssabc nkdt d kkk ,...,2,1,,,, =+= λiRv , (6.6) seg n i sabcsspabcpssabc nk seg iikkk ,...,2,1, 1 ,,, =+= ∑ = iLiLλ , (6.7) In the above equations, xabc,v , xabc,i , and xabc,λ are the vectors including the three-phase voltages, currents, and flux-linkages, respectively, corresponding to the winding x which could be the primary set ( p ) or any of the secondary segments ( ks ). The matrix xR is a diagonal matrix including the resistance xr of the same winding set/segment x . Also, xxL is an inductance matrix including the values of self- and mutual- inductances among the windings within the same set/segment x , while the matrix xyL contains the values of mutual inductances between the windings from two different set/segments x and y which can each be the primary set ( p ) or a secondary segment ( ks ). Due to symmetric properties of the transformers [51], [49] ( ) Txyyx LL = , kspyx ,, = . (6.8) 117 Some secondary segments may also be chosen to be identical to each other, in terms of the number of turns, etc., as it is the case in the 3-to-9-phase transformer of Figure 6.3. In such cases, a few of the above inductance matrices would be identical. In order for the transformer detailed model to be implemented according to (6.4)-(6.8), all necessary resistances and inductances have to be established. For each of the windings in any set/segment, the self-inductance includes the magnetizing and leakage parts. The mutual inductance between two arbitrary windings from any set/segment would always include a magnetizing term. However, if these windings are on the same leg of the transformer, an additional leakage term may also be included [49]. Determining the inductance values for the transformer parameters is generally outside the scope of this thesis. These values may be established according to the geometry and magnetic characteristics of the core as well as the number of turns per each winding, and in some cases, the values of all self and mutual inductances are readily available from design [49], [79]. Alternatively, the short-circuit impedances may be extracted from the hardware prototype by performing tests, and the representation of the transformer in terms of resistances and inductances may then be conveniently obtained according to the step-by-step procedure provided in Chapter 6 of [8]. To facilitate the derivation of the transformer models for the purpose of this thesis, several fundamental parameters are defined upon which the inductance and resistance values may be established in a systematic way. Without loss of generality, the self inductance of b -phase primary winding (wound on the middle leg) may be expressed as blpbmbpp LLL ,,, += , (6.9) where blpL , and bmL , denote the leakage and magnetizing terms. Some degree of symmetry is always present in the geometry of the core. In the three-leg structure depicted in Figure 6.3, symmetry is present around the middle leg of the transformer (corresponding to phase b ). Such symmetry is typically present in all cases, even when the core structure has more 118 legs, such as the five leg structure considered in [49]. Incorporating this symmetry, for the primary windings on the side legs, phases a and c , the self inductance is equal, and may be expressed in a similar way to (6.9) as alpamapp LLL ,,, += . (6.10) The mutual inductance between the middle-leg primary winding and any side-leg primary windings is denoted by apbpL , and the mutual inductance between the two side-leg primary windings is denoted by apcpL . These values may be readily established as a fraction of bmL , and amL , . For instance, in the three-leg structure of Figure 6.3, clearly we have: bmapbp LL ,1ξ−= , (6.11) amapcp LL ,2ξ−= , (6.12) where typically 5.01 =ξ but 5.02 <ξ , since the magnetizing flux in the middle-leg winding is equally divided between the two side-leg windings according to the core symmetry, whereas the magnetizing flux in a side winding sees a lower reluctance for closing through the middle leg compared to the farther side leg. Using the above fundamental parameters, other inductances may be established using the appropriate turn ratio. In particular, the self inductance of any other winding i may then be expressed as aliam p i aii LL n n L ,, 2 , +         = , (6.13) blibm p i bii LL n n L ,, 2 , +         = , (6.14) for the cases when the winding is on the side and middle legs, respectively. The leakage terms in these cases are represented by aliL , and bliL , , respectively. Also, in and pn denote the number of turns in the winding i and primary winding, respectively. 119 Next, in order to establish the mutual inductance values, we consider the mutual inductances between the primary windings ( p ) and the first segment of the secondary windings ( 1s ). The values for this type of mutual inductance are denoted by bpbsL and apasL for the cases that both windings are on the middle leg, and on one of the side legs, respectively. The magnetizing parts of these inductances may be readily written in terms of the turn ratio and the primary magnetizing values defined above. However, these inductances also include a leakage term which, in general, is not equal to the leakage of the primary windings since a fraction of the leakage flux solely links an individual winding. Adopting this concept, the mutual inductance between any two windings i and j , in general, has to be written for four cases: i) both windings are on the middle leg, ii) both windings are on one of the side legs, iii) the two windings are each on one side leg, and finally, iv) only one of these windings is on the middle leg. Mathematically, for each case we have: ( )blpbm p ji bbjibbij LL n nn LL ,3,2,, ξ+== , (6.15) ( )alpam p ji aajiaaij LL n nn LL ,4,2,, ξ+== , (6.16) ( )am p ji apcp p ji acjiacij L n nn L n nn LL ,222,, ξ−=== , (6.17) ( )bm p ji apbp p ji abjiabij L n nn L n nn LL ,122,, ξ−=== , (6.18) where the coefficients 3ξ and 4ξ determine the fraction of the leakage flux linking both windings. Also, (6.12) and (6.11) have been employed for obtaining the last terms in (6.17) and (6.18). Finally, to establish the winding resistances, the primary winding resistance is defined by pr . Typically, the resistance of any other winding i is then expressed using the turn ratio as: 120 p p i i r n n r 2         = . (6.19) If the number of turns for all windings is known, and the values of the fundamental parameters amL , , bmL , , alpL , , blpL , as well as the coefficients 1ξ through 4ξ are specified, all the self and mutual inductances as well as resistance values may be readily established using (6.13)-(6.19), and the transformer model (6.4)-(6.7) can be constructed. Once the transformer model is established, the next challenge in implementation is the wiring of the transformer at the secondary side. Without loss of generality, we assume that the primary of the transformer is Y-connected. Also, as shown in the case of 9-phase transformer in Figure 6.3, one of the secondary winding segments is typically Y-connected. As a result, the voltages and currents of this segment and those of the primary winding set will be in phase. This segment, 1s , forms the first set of the final extended secondary windings and its three-phase set of voltages is hence chosen as the reference. For a typical 3- to-n-phase transformer, the ( 1−segn ) remaining segments of the secondary windings, are interconnected in order to form the ( 1−setn ) remaining sets of final extended secondary windings responsible for producing ( 1−setn ) sets of three-phase voltages that are all equal in magnitude but evenly shifted in phase with respect to the reference set produced by 1s . Various configurations may be used to achieve such a goal depending on a particular application [131], [78]. Developing such configurations is outside the scope of this research. Regardless, all configurations, if applied correctly, would lead to similar results in terms of producing the above-mentioned sets of voltages. The above transformer model includes the details of the original interconnected windings of the real transformer and is hence called the original interconnected transformer model. This model is expected to provide sufficient accuracy in most practical cases where the core losses and saturation are not significant. However, the high number of windings as well as 121 the complicated interconnection among the secondary segments makes this model challenging to implement as well as numerically inefficient. Moreover, this model as is, cannot be conveniently transformed into 0qd equivalent circuit for use in the dynamic average model of the system. The ultimate goal of the next sections to develop a simplified model based on the above detailed representation without loss of generality and accuracy. It should also be noted that, since typically one of the final extended secondary sets is composed of only one secondary segment, while the other sets include a series connection of more than one segment, in the real transformer, the values of leakage inductance and resistance are not equal among the final extended secondary sets of windings. In practice, this will introduce more asymmetry in the performance of transformer, in addition to the asymmetry imposed by the core geometry. The transformer model developed above, however, is quite general, and can include all these effects. 6.3.2 Analytically-Derived Equivalent Compacted Transformer Model In order to facilitate the implementation and increase the numerical efficiency of the model, several secondary segment currents may be analytically expressed in terms of other currents, as dictated by the topological structure of the transformer. Thereafter, replacing these equations, the new parameters may be calculated analytically for each extended secondary set of windings. This step results in a simpler model that is easier to implement as well as numerically more efficient, because the interconnected wiring at the secondary side is essentially removed, and the number of windings at the secondary is reduced to the number of sets multiplied by three. Such an approach has been adopted in [49] to simplify a transformer model that produces 12-phase voltages for a 24-pulse converter. The approach significantly depends on a particular configuration of the transformer secondary wiring, and is challenging to generalize. It is, however, beneficial in significantly simplifying the model while completely preserving the generality and accuracy of the model. 122 To demonstrate this analytically-derived equivalent model, let us consider the example system of 18-pulse converter (Figure 6.2) with the transformer structure of Figure 6.3. The primary winding p and the secondary segments 1s through 5s are numbered from top to bottom. According to the topology dictated in Figure 6.3 as indicated by the numberings at the winding ends, we have: 23 csas ii −= , (6.20) 23 asbs ii −= , (6.21) 23 bscs ii −= , (6.22) 45 csas ii −= , (6.23) 45 asbs ii −= , (6.24) 45 bscs ii −= , (6.25) The above equations may be written in the following matrix form: 23 ,, sabcsabc iTi = , (6.26) 45 ,, sabcsabc iTi = , (6.27) wherein the mapping matrix T has been introduced as           − − − = 010 001 100 T . (6.28) Next, we consider the voltages across the final extended secondary sets, noting that the first secondary set is directly produced by the 1s segment. The notation used for the final extended secondary sets is kexs − , setnk ,...,2,1= . It is clear that, for the example system, segn and setn are equal to five and three, respectively. According to the topology illustrated in Figure 6.3, we have: 232 csasas vvv ex −= − , (6.29) 123 232 asbsbs vvv ex −=− , (6.30) 232 bscscs vvv ex −=− , (6.31) 543 csbsas vvv ex −=− , (6.32) 543 ascsbs vvv ex −=− , (6.33) 543 bsascs vvv ex −=− . (6.34) In matrix form, we may write: 11 ,, sabcsabc ex vv =− , (6.35) 232 ,,, sabcsabcsabc ex vTvv +=− , (6.36) 453 , 1 ,, sabcsabcsabc ex vTvTv − −= − . (6.37) In the above equation, the inverse of the mapping matrix T can be written as:           − − − = − 001 100 010 1T . (6.38) It should be noticed that TTT =−1 . Considering the currents in the extended secondary windings: 11 ,, sabcsabc ex ii =− , (6.39) 32 ,, sabcsabc ex ii =− , (6.40) 43 , 1 , sabcsabc ex iTi − −= − . (6.41) Rearranging terms in (6.26), (6.27), (6.39)-(6.41), all secondary segment currents may be written in terms of the extended secondary set currents: 11 ,, − = exsabcsabc ii , (6.42) 22 , 1 , − − = exsabcsabc iTi , (6.43) 124 23 ,, − = exsabcsabc ii , (6.44) 34 ,, − −= exsabcsabc iTi , (6.45) 35 , 1 , − − = exsabcsabc iTi . (6.46) In the last equation, the following equation has been employed to simplify the expression: 12 − =− TT . (6.47) The above identity can be proven using (6.28) and (6.38). Next, (6.42)-(6.46) are substituted into (6.4)-(6.7), and the resulting equations are substituted into (6.35)-(6.37). This results in the following model: pabcpabcppabc dt d ,,, λiRv += , (6.48) ∑ = −− += set kexkex n k sabcpspabcpppabc 1 ,,, iLiLλ , (6.49) setkexsabcsabcssabc nkdt d kexkexkex ,...,2,1,,,, =+= −−−− λiRv , (6.50) set n i sabcsspabcpssabc nk set iexiexkexkexkex ,...,2,1, 1 ,,, =+= ∑ = −−−−− iLiLλ . (6.51) In general, the new inductance and resistance matrices may be expressed in terms of the original model (6.4)-(6.7). For the example system considered here, after extensive analytical work, these matrices are written as 11 ssex RR = − , (6.52) 422 sssex RRR += − , (6.53) 423 sssex RRR += − , (6.54) 11 ssex LL = − , (6.55) 11 424244222 −− +++= − TLLTLTLTL sssssssssex , (6.56) 125 1111 424244223 −−−− −−+= − TLTTLTTLTTLTL sssssssssex , (6.57) 111 pspsps exex LLL == −− , (6.58) ( ) 4222 1 psps T psps exex LTLLL +== − −− , (6.59) ( ) TLTLLL 4233 1 psps T psps exex −== − −− , (6.60) ( ) 41211221 1 ssss T ssss exexexex LTLLL +== − −−−− , (6.61) ( ) TLTLLL 41211331 1 ssss T ssss exexexex −== − −−−− , (6.62) ( ) TLTLTLTTLTLL 434242522332 11 ssssssss T ssss exexexex −+−== −− −−−− . (6.63) The above model is more effective than the original model, but the analytical effort in obtaining it is extensive and highly depends on the wiring of the secondary side. However, the generality of the model, with respect to the original model, is preserved including the asymmetry between the mutual inductances across different transformer legs as well as the asymmetry between the leakage inductance and resistance of the final secondary sets. The above-mentioned asymmetric properties of the transformer will affect the shape of the secondary voltages causing asymmetric ripple around the average value of the dc side variables. However, in the dynamic average models, due to the averaging over the switching interval, the ripples are averaged out, and the overall effect of these asymmetric properties is less important. Therefore, in the next step, we will consider removing these asymmetric properties while keeping the effects of these features in an average sense. In doing so, the modeling is considerably simplified while maintaining the desired accuracy of the final model. First, we consider the asymmetry in core geometry which results in different mutual inductances among the windings on different transformer legs. 126 6.3.3 Round Shifted Equivalent Model Earlier in this Chapter, we assumed a general case wherein the symmetry in the core exists solely around the middle leg. In many cases, additional symmetry either exists in the core, or is approximated in a way that the interactions among the coils on any two of the transformer legs appear identical. In this case, the distinction between the side and middle legs, introduced earlier, no longer has to be made. In other words, self inductance of the primary coils (6.9) and (6.10) are equal to each other, and in (6.11) and (6.12), the coefficients 5.021 == ξξ . For the sake of dynamic average modeling, even if such symmetry does not exist, the values of appL , and bppL , may be averaged, and used for all the coils. As will be shown later, such averaging provides sufficient accuracy for the purpose of dynamic average modeling. The self inductance of the primary windings is then written as: lpmpp LLL += . (6.64) The self inductance of any other winding i may then be expressed as ( )lpm p i pp p i ii LL n nL n nL +         =         = 22 . (6.65) Moreover, in establishing the mutual inductances, no distinction has to be made between the four cases considered earlier in (6.15)-(6.18). The distinction, instead, is made between the two cases where both windings are on the same leg and otherwise. In a compact form, the mutual inductance between any two windings is then conveniently expressed as ( )lpm p ji jiij LcLc n nn LL 212 +== . (6.66) In the above equation, the coefficient 1c is equivalent to 1ξ , 2ξ , and the coefficient 2c is equivalent to 3ξ , 4ξ . Hence, if both windings are on the same leg, 11 =c , 12