6.3.3 Computational Results and Analysis 243 -- 6.4 HBFEM Modeling of the DC-Biased Transformer in GIC Event 245 -- 6.4.1 GIC Effects on the Transformer 245 -- 6.4.2 GIC Modeling and Harmonic Analysis 248 -- 6.4.3 GIC Modeling Using HBFEM Model 249 -- 6.5 HBFEM Used in Renewable Energy Systems and Microgrids 253 -- 6.5.1 Harmonics in Renewable Energy Systems and Microgrids 253 -- 6.5.2 Harmonic Analysis of the Transformer in Renewable Energy Systems and Microgrids 254 -- 6.5.3 Harmonic Analysis of the Transformer Using a Voltage Driven Source 256 -- 6.5.4 Harmonic Analysis of the Transformer Using a Current-Driven Source 258 -- References 261 -- Appendix 263 -- Appendix I & II 263 -- Matlab Program and the Laminated Core Model for Computation 263 -- Appendix III 265 -- FORTRAN-Based 3D Axi-Symmetrical Transformer with DC-Biased Excitation 265 -- Index 267.

5.3.2 DC-Biasing Magnetization Curve 180 -- 5.3.3 Fixed-Point Harmonic-Balanced Theory 182 -- 5.3.4 Electromagnetic Coupling 184 -- 5.3.5 Validation and Discussion 184 -- 5.4 Hysteresis Model Based on Neural Network and Consuming Function 188 -- 5.4.1 Introduction 188 -- 5.4.2 Hysteresis Model Based on Consuming Function 189 -- 5.4.3 Hysteresis Loops and Simulation 191 -- 5.4.4 Hysteresis Model Based on a Neural Network 194 -- 5.4.5 Simulation and Validation 196 -- 5.5 Analysis of Hysteretic Characteristics Under Sinusoidal and DC-Biased Excitation 199 -- 5.5.1 Globally Convergent Fixed-Point Harmonic-Balanced Method 199 -- 5.5.2 Hysteretic Characteristic Analysis of the Laminated Core 202 -- 5.5.3 Computation of the Nonlinear Magnetic Field Based on the Combination of the Two Hysteresis Models 206 -- 5.6 Parallel Computing of HBFEM in Multi-Frequency Domain 210 -- 5.6.1 HBFEM in Multi-Frequency Domain 210 -- 5.6.2 Parallel Computing of HBFEM 212 -- 5.6.3 Domain Decomposition 212 -- 5.6.4 Reordering and Multi-Coloring 213 -- 5.6.5 Loads Division in Frequency Domain 214 -- 5.6.6 Two Layers Hybrid Computing 217 -- References 217 -- 6 HBFEM and Its Future Applications 222 -- 6.1 HBFEM Model of Three-Phase Power Transformer 222 -- 6.1.1 Three-Phase Transformer 222 -- 6.1.2 Nonlinear Magnetic Material and its Saturation Characteristics 223 -- 6.1.3 Voltage Source-Driven Model Connected to the Magnetic Field 224 -- 6.1.4 HBFEM Matrix Equations, Taking Account of Extended Circuits 225 -- 6.2 Magnetic Model of a Single-Phase Transformer and a Magnetically Controlled Shunt Reactor 231 -- 6.2.1 Electromagnetic Coupling Model of a Single-Phase Transformer 231 -- 6.2.2 Solutions of the Nonlinear Magnetic Circuit Model by the Harmonic Balance Method 233 -- 6.2.3 Magnetically Controlled Shunt Reactor 235 -- 6.2.4 Experiment and Computation 237 -- 6.3 Computation Taking Account of Hysteresis Effects Based on Fixed-Point Reluctance 240 -- 6.3.1 Fixed-Point Reluctance 240 -- 6.3.2 Computational Procedure in the Frequency Domain 242.

4.1.1 Numerical Model of Current Source to Magnetic Field 97 -- 4.1.2 Example of Current-Source Excitation to Nonlinear Magnetic Field 99 -- 4.2 Harmonic Analysis of Switching Mode Transformer Using Voltage-Driven Source 99 -- 4.2.1 Numerical Model of Voltage Source to Magnetic System 99 -- 4.2.2 Example of Voltage-Source Excitation to Nonlinear Magnetic Field 106 -- 4.3 Three-Phase Magnetic Frequency Tripler Analysis 107 -- 4.3.1 Magnetic Frequency Tripler 107 -- 4.3.2 Nonlinear Magnetic Material and its Saturation Characteristics 107 -- 4.3.3 Voltage Source-Driven Connected to the Magnetic Field 109 -- 4.4 Design of High-Speed and Hybrid Induction Machine using HBFEM 115 -- 4.4.1 Construction of High-Speed and Hybrid Induction Machine 115 -- 4.4.2 Numerical Model of High-Speed and Hybrid Induction Machine using HBFEM, Taking Account of Motion Effect 117 -- 4.4.3 Numerical Analysis of High Speed and Hybrid Induction Machine using HBFEM 126 -- 4.5 Three-Dimensional Axi-Symmetrical Transformer with DC-Biased Excitation 131 -- 4.5.1 Numerical Simulation of 3-D Axi-Symmetrical Structure 133 -- 4.5.2 Numerical Analysis of the Three-Dimensional Axi-Symmetrical Model 136 -- 4.5.3 Eddy Current Calculation of DC-Biased Switch Mode Transformer 138 -- References 139 -- 5 Advanced Numerical Approach using HBFEM 141 -- 5.1 HBFEM for DC-Biased Problems in HVDC Power Transformers 141 -- 5.1.1 DC Bias Phenomena in HVDC 141 -- 5.1.2 HBFEM for DC-Biased Magnetic Field 142 -- 5.1.3 High-Voltage DC (HVDC) Transformer 160 -- 5.2 Decomposed Algorithm of HBFEM 165 -- 5.2.1 Introduction 165 -- 5.2.2 Decomposed Harmonic Balanced System Equation 166 -- 5.2.3 Magnetic Field Coupled with Electric Circuits 169 -- 5.2.4 Computational Procedure Based on the Block Gauss-Seidel Algorithm 170 -- 5.2.5 DC-Biasing Test on the LCM and Computational Results 172 -- 5.2.6 Analysis of the Flux Density and Flux Distribution Under DC Bias Conditions 176 -- 5.3 HBFEM with Fixed-Point Technique 178 -- 5.3.1 Introduction 178.

2.2.2 Characteristics of DC Bias Phenomenon and Problems to be Solved 40 -- 2.3 Geomagnetic Disturbance and Geomagnetic Induced Currents (GIC) 41 -- 2.3.1 Geomagnetically Induced Currents in Power Systems 42 -- 2.3.2 GIC-Induced Harmonic Currents in the Transformer 46 -- 2.4 Harmonic Problems in Renewable Energy and Microgrid Systems 47 -- 2.4.1 Power Electronic Devices ¿́¿ Harmonic Current and Voltage Sources 48 -- 2.4.2 Harmonic Distortion in Renewable Energy Systems 50 -- 2.4.3 Harmonics in the Microgrid and EV Charging System 52 -- 2.4.4 IEEE Standard 519-2014 56 -- References 58 -- 3 Harmonic Balance Methods Used in Computational Electromagnetics 60 -- 3.1 Harmonic Balance Methods Used in Nonlinear Circuit Problems 60 -- 3.1.1 The Basic Concept of Harmonic Balance in a Nonlinear Circuit 60 -- 3.1.2 The Theory of Harmonic Balance Used in a Nonlinear Circuit 63 -- 3.2 CEM for Harmonic Problem Solving in Frequency, Time and Harmonic Domains 65 -- 3.2.1 Computational Electromagnetics (CEM) Techniques and Validation 65 -- 3.2.2 Time Periodic Electromagnetic Problems Using the Finite Element Method (FEM) 66 -- 3.2.3 Comparison of Time-Periodic Steady-State Nonlinear EM Field Analysis Method 71 -- 3.3 The Basic Concept of Harmonic Balance in EM Fields 73 -- 3.3.1 Definition of Harmonic Balance 73 -- 3.3.2 Harmonic Balance in EM Fields 73 -- 3.3.3 Nonlinear Medium Description 75 -- 3.3.4 Boundary Conditions 76 -- 3.3.5 The Theory of HB-FEM in Nonlinear Magnetic Fields 76 -- 3.3.6 The Generalized HBFEM 83 -- 3.4 HBFEM for Electromagnetic Field and Electric Circuit Coupled Problems 85 -- 3.4.1 HBFEM in Voltage Source-Driven Magnetic Field 85 -- 3.4.2 Generalized Voltage Source-Driven Magnetic Field 86 -- 3.5 HBFEM for a DC-Biased Problem in High-Voltage Power Transformers 91 -- 3.5.1 DC-Biased Problem in HVDC Transformers 91 -- 3.5.2 HBFEM Model of HVDC Transformer 91 -- References 95 -- 4 HBFEM for Nonlinear Magnetic Field Problems 96 -- 4.1 HBFEM for a Nonlinear Magnetic Field with Current-Driven Source 96.

-- Preface xii -- About the Companion Website xv -- 1 Introduction to Harmonic Balance Finite Element Method (HBFEM) 1 -- 1.1 Harmonic Problems in Power Systems 1 -- 1.1.1 Harmonic Phenomena in Power Systems 2 -- 1.1.2 Sources and Problems of Harmonics in Power Systems 3 -- 1.1.3 Total Harmonic Distortion (THD) 4 -- 1.2 Definitions of Computational Electromagnetics and IEEE Standards 1597.1 and 1597.2 7 -- 1.2.1 ¿́¿The Building Block¿́¿ of the Computational Electromagnetics Model 7 -- 1.2.2 The Geometry of the Model and the Problem Space 8 -- 1.2.3 Numerical Computation Methods 8 -- 1.2.4 High-Performance Computation and Visualization (HPCV) in CEM 9 -- 1.2.5 IEEE Standards 1597.1 and 1597.2 for Validation of CEM Computer Modeling and Simulations 9 -- 1.3 HBFEM Used in Nonlinear EM Field Problems and Power Systems 12 -- 1.3.1 HBFEM for a Nonlinear Magnetic Field With Current Driven 13 -- 1.3.2 HBFEM for Magnetic Field and Electric Circuit Coupled Problems 14 -- 1.3.3 HBFEM for a Nonlinear Magnetic Field with Voltage Driven 14 -- 1.3.4 HBFEM for a Three-Phase Magnetic Tripler Transformer 14 -- 1.3.5 HBFEM for a Three-Phase High-Speed Motor 15 -- 1.3.6 HBFEM for a DC-Biased 3D Asymmetrical Magnetic Structure Simulation 15 -- 1.3.7 HBFEM for a DC-Biased Problem in HV Power Transformers 16 -- References 17 -- 2 Nonlinear Electromagnetic Field and Its Harmonic Problems 19 -- 2.1 Harmonic Problems in Power Systems and Power Supply Transformers 19 -- 2.1.1 Nonlinear Electromagnetic Field 19 -- 2.1.2 Harmonics Problems Generated from Nonlinear Load and Power Electronics Devices 21 -- 2.1.3 Harmonics in the Time Domain and Frequency Domain 25 -- 2.1.4 Examples of Harmonic Producing Loads 28 -- 2.1.5 Harmonics in DC/DC Converter of Isolation Transformer 28 -- 2.1.6 Magnetic Tripler 33 -- 2.1.7 Harmonics in Multi-Pulse Rectifier Transformer 35 -- 2.2 DC-Biased Transformer in High-Voltage DC Power Transmission System 38 -- 2.2.1 Investigation and Suppression of DC Bias Phenomenon 38.