Harmonic Balance Finite Element Method

Junwei Lu, Professor, Griffith School of Engineering, Griffith University, Australia.  Professor Lu has developed Harmonic Balance FEM techniques for nonlinear magnetics and Time Domain FEM techniques for wave propagation problems, and has been working in this area since 1985. He has taught numerical techniques in EM, power electronics and electric machines, power transmission and distribution, advanced communications systems since 1993. He holds over 10 international patents related to smart antennas arrays, high frequency transformers and inductor, and other high frequency magnetic devices.  His research interests include Computational Electromagnetics, EMC computer modelling and simulation, high frequency magnetics, smart mobile terminal antennas, MEMS devices, and smart transformer used in renewable energy system and smart grid, and EV technology.Xiaojun Zhao is researcher at the State Key Laboratory of Alternate Electrical Power System with Renewable Energy Source, North China Electric Power University, China. His main research interests are engineering electromagnetic field analysis, DC bias phenomena in power transformers, and modeling properties of magnetic material.Sotoshi Yamada received the B.E. and M.E. degrees from the Department of Electrical Engineering, Kanazawa University, Kanazawa, Japan, in 1972 and 1974, respectively. He received the Dr. Eng. degree from Kyushu University, Fukuoka, Japan, in 1985. From 1974 to 1992, he was with the Department of Electrical and Computer Engineering, Faculty of Engineering, Kanazawa University. He has been Professor at Laboratory of Magnetic Field Control and Applications since 1992 and is engaged in research on power magnetic devices, the numerical electromagnetic field calculation, biomagnetics, etc.

• Preface xiiAbout the Companion Website xv1 Introduction to Harmonic Balance Finite Element Method (HBFEM) 11.1 Harmonic Problems in Power Systems 11.1.1 Harmonic Phenomena in Power Systems 21.1.2 Sources and Problems of Harmonics in Power Systems 31.1.3 Total Harmonic Distortion (THD) 41.2 Definitions of Computational Electromagnetics and IEEE Standards 1597.1 and 1597.2 71.2.1 “The Building Block” of the Computational Electromagnetics Model 71.2.2 The Geometry of the Model and the Problem Space 81.2.3 Numerical Computation Methods 81.2.4 High-Performance Computation and Visualization (HPCV) in CEM 91.2.5 IEEE Standards 1597.1 and 1597.2 for Validation of CEM Computer Modeling and Simulations 91.3 HBFEM Used in Nonlinear EM Field Problems and Power Systems 121.3.1 HBFEM for a Nonlinear Magnetic Field With Current Driven 131.3.2 HBFEM for Magnetic Field and Electric Circuit Coupled Problems 141.3.3 HBFEM for a Nonlinear Magnetic Field with Voltage Driven 141.3.4 HBFEM for a Three-Phase Magnetic Tripler Transformer 141.3.5 HBFEM for a Three-Phase High-Speed Motor 151.3.6 HBFEM for a DC-Biased 3D Asymmetrical Magnetic Structure Simulation 151.3.7 HBFEM for a DC-Biased Problem in HV Power Transformers 16References 172 Nonlinear Electromagnetic Field and Its Harmonic Problems 192.1 Harmonic Problems in Power Systems and Power Supply Transformers 192.1.1 Nonlinear Electromagnetic Field 192.1.2 Harmonics Problems Generated from Nonlinear Load and Power Electronics Devices 212.1.3 Harmonics in the Time Domain and Frequency Domain 252.1.4 Examples of Harmonic Producing Loads 282.1.5 Harmonics in DC/DC Converter of Isolation Transformer 282.1.6 Magnetic Tripler 332.1.7 Harmonics in Multi-Pulse Rectifier Transformer 352.2 DC-Biased Transformer in High-Voltage DC Power Transmission System 382.2.1 Investigation and Suppression of DC Bias Phenomenon 382.2.2 Characteristics of DC Bias Phenomenon and Problems to be Solved 402.3 Geomagnetic Disturbance and Geomagnetic Induced Currents (GIC) 412.3.1 Geomagnetically Induced Currents in Power Systems 422.3.2 GIC-Induced Harmonic Currents in the Transformer 462.4 Harmonic Problems in Renewable Energy and Microgrid Systems 472.4.1 Power Electronic Devices – Harmonic Current and Voltage Sources 482.4.2 Harmonic Distortion in Renewable Energy Systems 502.4.3 Harmonics in the Microgrid and EV Charging System 522.4.4 IEEE Standard 519-2014 56References 583 Harmonic Balance Methods Used in Computational Electromagnetics 603.1 Harmonic Balance Methods Used in Nonlinear Circuit Problems 603.1.1 The Basic Concept of Harmonic Balance in a Nonlinear Circuit 603.1.2 The Theory of Harmonic Balance Used in a Nonlinear Circuit 633.2 CEM for Harmonic Problem Solving in Frequency, Time and Harmonic Domains 653.2.1 Computational Electromagnetics (CEM) Techniques and Validation 653.2.2 Time Periodic Electromagnetic Problems Using the Finite Element Method (FEM) 663.2.3 Comparison of Time-Periodic Steady-State Nonlinear EM Field Analysis Method 713.3 The Basic Concept of Harmonic Balance in EM Fields 733.3.1 Definition of Harmonic Balance 733.3.2 Harmonic Balance in EM Fields 733.3.3 Nonlinear Medium Description 753.3.4 Boundary Conditions 763.3.5 The Theory of HB-FEM in Nonlinear Magnetic Fields 763.3.6 The Generalized HBFEM 833.4 HBFEM for Electromagnetic Field and Electric Circuit Coupled Problems 853.4.1 HBFEM in Voltage Source-Driven Magnetic Field 853.4.2 Generalized Voltage Source-Driven Magnetic Field 863.5 HBFEM for a DC-Biased Problem in High-Voltage Power Transformers 913.5.1 DC-Biased Problem in HVDC Transformers 913.5.2 HBFEM Model of HVDC Transformer 91References 954 HBFEM for Nonlinear Magnetic Field Problems 964.1 HBFEM for a Nonlinear Magnetic Field with Current-Driven Source 964.1.1 Numerical Model of Current Source to Magnetic Field 974.1.2 Example of Current-Source Excitation to Nonlinear Magnetic Field 994.2 Harmonic Analysis of Switching Mode Transformer Using Voltage-Driven Source 994.2.1 Numerical Model of Voltage Source to Magnetic System 994.2.2 Example of Voltage-Source Excitation to Nonlinear Magnetic Field 1064.3 Three-Phase Magnetic Frequency Tripler Analysis 1074.3.1 Magnetic Frequency Tripler 1074.3.2 Nonlinear Magnetic Material and its Saturation Characteristics 1074.3.3 Voltage Source-Driven Connected to the Magnetic Field 1094.4 Design of High-Speed and Hybrid Induction Machine using HBFEM 1154.4.1 Construction of High-Speed and Hybrid Induction Machine 1154.4.2 Numerical Model of High-Speed and Hybrid Induction Machine using HBFEM, Taking Account of Motion Effect 1174.4.3 Numerical Analysis of High Speed and Hybrid Induction Machine using HBFEM 1264.5 Three-Dimensional Axi-Symmetrical Transformer with DC-Biased Excitation 1314.5.1 Numerical Simulation of 3-D Axi-Symmetrical Structure 1334.5.2 Numerical Analysis of the Three-Dimensional Axi-Symmetrical Model 1364.5.3 Eddy Current Calculation of DC-Biased Switch Mode Transformer 138References 1395 Advanced Numerical Approach using HBFEM 1415.1 HBFEM for DC-Biased Problems in HVDC Power Transformers 1415.1.1 DC Bias Phenomena in HVDC 1415.1.2 HBFEM for DC-Biased Magnetic Field 1425.1.3 High-Voltage DC (HVDC) Transformer 1605.2 Decomposed Algorithm of HBFEM 1655.2.1 Introduction 1655.2.2 Decomposed Harmonic Balanced System Equation 1665.2.3 Magnetic Field Coupled with Electric Circuits 1695.2.4 Computational Procedure Based on the Block Gauss-Seidel Algorithm 1705.2.5 DC-Biasing Test on the LCM and Computational Results 1725.2.6 Analysis of the Flux Density and Flux Distribution Under DC Bias Conditions 1765.3 HBFEM with Fixed-Point Technique 1785.3.1 Introduction 1785.3.2 DC-Biasing Magnetization Curve 1805.3.3 Fixed-Point Harmonic-Balanced Theory 1825.3.4 Electromagnetic Coupling 1845.3.5 Validation and Discussion 1845.4 Hysteresis Model Based on Neural Network and Consuming Function 1885.4.1 Introduction 1885.4.2 Hysteresis Model Based on Consuming Function 1895.4.3 Hysteresis Loops and Simulation 1915.4.4 Hysteresis Model Based on a Neural Network 1945.4.5 Simulation and Validation 1965.5 Analysis of Hysteretic Characteristics Under Sinusoidal and DC-Biased Excitation 1995.5.1 Globally Convergent Fixed-Point Harmonic-Balanced Method 1995.5.2 Hysteretic Characteristic Analysis of the Laminated Core 2025.5.3 Computation of the Nonlinear Magnetic Field Based on the Combination of the Two Hysteresis Models 2065.6 Parallel Computing of HBFEM in Multi-Frequency Domain 2105.6.1 HBFEM in Multi-Frequency Domain 2105.6.2 Parallel Computing of HBFEM 2125.6.3 Domain Decomposition 2125.6.4 Reordering and Multi-Coloring 2135.6.5 Loads Division in Frequency Domain 2145.6.6 Two Layers Hybrid Computing 217References 2176 HBFEM and Its Future Applications 2226.1 HBFEM Model of Three-Phase Power Transformer 2226.1.1 Three-Phase Transformer 2226.1.2 Nonlinear Magnetic Material and its Saturation Characteristics 2236.1.3 Voltage Source-Driven Model Connected to the Magnetic Field 2246.1.4 HBFEM Matrix Equations, Taking Account of Extended Circuits 2256.2 Magnetic Model of a Single-Phase Transformer and a Magnetically Controlled Shunt Reactor 2316.2.1 Electromagnetic Coupling Model of a Single-Phase Transformer 2316.2.2 Solutions of the Nonlinear Magnetic Circuit Model by the Harmonic Balance Method 2336.2.3 Magnetically Controlled Shunt Reactor 2356.2.4 Experiment and Computation 2376.3 Computation Taking Account of Hysteresis Effects Based on Fixed-Point Reluctance 2406.3.1 Fixed-Point Reluctance 2406.3.2 Computational Procedure in the Frequency Domain 2426.3.3 Computational Results and Analysis 2436.4 HBFEM Modeling of the DC-Biased Transformer in GIC Event 2456.4.1 GIC Effects on the Transformer 2456.4.2 GIC Modeling and Harmonic Analysis 2486.4.3 GIC Modeling Using HBFEM Model 2496.5 HBFEM Used in Renewable Energy Systems and Microgrids 2536.5.1 Harmonics in Renewable Energy Systems and Microgrids 2536.5.2 Harmonic Analysis of the Transformer in Renewable Energy Systems and Microgrids 2546.5.3 Harmonic Analysis of the Transformer Using a Voltage Driven Source 2566.5.4 Harmonic Analysis of the Transformer Using a Current-Driven Source 258References 261Appendix 263Appendix I & II 263Matlab Program and the Laminated Core Model for Computation 263Appendix III 265FORTRAN-Based 3D Axi-Symmetrical Transformer with DC-Biased Excitation 265Index 267