# Harmonic Balance Finite Element Method: Applications in Nonlinear Electromagnetics and Power Systems

# Harmonic Balance Finite Element Method: Applications in Nonlinear Electromagnetics and Power Systems

ISBN: 978-1-118-97577-0

Aug 2016, Wiley-IEEE Press

304 pages

## Description

**The first book applying HBFEM to practical electronic nonlinear field and circuit problems **

• Examines and solves wide aspects of practical electrical and electronic nonlinear field and circuit problems presented by HBFEM

• Combines the latest research work with essential background knowledge, providing an all-encompassing reference for researchers, power engineers and students of applied electromagnetics analysis

• There are very few books dealing with the solution of nonlinear electric- power-related problems

• The contents are based on the authors’ many years’ research and industry experience; they approach the subject in a well-designed and logical way

• It is expected that HBFEM will become a more useful and practical technique over the next 5 years due to the HVDC power system, renewable energy system and Smart Grid, HF magnetic used in DC/DC converter, and Multi-pulse transformer for HVDC power supply

• HBFEM can provide effective and economic solutions to R&D product development

• Includes Matlab exercises

## Related Resources

### Instructor

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

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

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

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

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