Unified Selective Harmonic Elimination for Power Converters - Formulation, Algorithm, and Application

Comprehensive reference detailing key aspects of SHE, enabling readers to formulate different kinds of SHE equations, effectively solve the nonlinear SHE equations, and grasp key aspects of SHE applications. Unified Selective Harmonic Elimination for Power Converters focuses on the three main challenges of selective harmonic elimination (SHE)-the mathematical modeling of fundamental and harmonic components using the pre-defined waveform, accurately solving SHE equations and obtaining the complete switching angle solution trajectory, and implementing SHE on multilevel converters and industrial drives-with information on how to fully leverage the strength of SHE techniques in power converters. The book covers the basics of the SHE method and reviews state-of-the-art research towards SHE, such as unified SHE formulations for multilevel converters, algebraic switching angle solving algorithms for SHE equations, and optimal implementations of SHE in multilevel converters and electric drives. The book delves into model predictive SHE control for PMSM with simulation and experimental results and explains how to achieve common mode voltage reduction and capacitor voltage balance in multilevel converters. Concepts are supported by original MATLAB/Mathematica/Maple codes. This book includes information on:

• Detailed derivation steps on Fourier series of square waveform and traditional SHE equations
• Unified SHE formulations for symmetric and asymmetric multilevel converters, and different SHE equations for various scenarios
• Advanced SHE solving algorithms including the resultant elimination method, the Groebner Bases-based method, symmetric polynomials, and Newton identities
• Online implementations of SHE based on both algebraic algorithms and intelligent algorithms
• Advanced capacitor voltage balancing methods under SHE for multilevel converters
• Basic and advanced closed-loop controller and model-predictive control under SHE for industrial drives

This book is a good reference for engineers and researchers in the area of power electronics, with particular interest to those involved in renewable power generation, high-power energy storage, and high-power drives.

List of contents

About the Authors xi
Preface xiii
Acknowledgments xv
1 Power Converters and Selective Harmonic Elimination 1
1.1 Introduction to Power Converters 1
1.1.1 Topologies of Voltage-Source Converters 1
1.1.2 Topologies of Current-Source Converters 3
1.2 Modulation Strategies for Power Converters 4
1.2.1 PWM of Voltage-Source Converters 4
1.2.2 PWM of Current-Source Converters 7
1.3 Overview of Selective Harmonic Elimination PWM 7
1.3.1 Basic Knowledge of SHE and Its Formulation 8
1.3.2 Overview of SHE Solving Algorithms 10
1.3.3 Overview of Control Methods with SHE-PWM 13
1.3.4 Overall SHE-PWM Implementation Procedure 14
1.3.5 Real-World Implementation Cases of SHE-PWM 16
1.3.5.1 SHE-PWM in Commercialized Products 16
1.3.5.2 SHE-PWM in Traction Motor Drives 16
1.3.5.3 SHE-PWM in High-Power Rectifiers 17
1.4 Technical Challenges with SHE-PWM 18
1.4.1 Technical Challenges in SHE Formulation 18
1.4.2 Technical Challenges in SHE Solving Algorithms 18
1.4.3 Technical Challenges in SHE Applications 19
1.5 Outline of the Book 19
1.5.1 Part 1: Formulation 19
1.5.2 Part 2: Solving Algorithm 20
1.5.3 Part 3: Application 21
References 21
Part I Formulation 29
2 Principle of SHE 31
2.1 Fourier Series of Periodic Functions 31
2.1.1 Half-Wave Symmetric 32
2.1.2 Quarter-Wave Symmetric 33
2.2 Principle of SHE Formulations 33
2.2.1 Fourier Series of SquareWaveform 35
2.2.2 Three-Level SHE Formulation 35
2.2.3 Multilevel SHE Formulation 37
2.2.4 Generalized Odd-Level SHE Formulation 38
2.2.5 Even-Level SHE Formulation 40
2.2.5.1 Two-Level SHE Formulation with Three Switching Angles 41
2.2.5.2 Generalized Even-Level SHE Formulation 42
2.2.6 SHE Formulation for Asymmetric Multilevel Converters 44
2.2.7 SHE Formulation with Half-Wave Symmetric 48
2.3 Summary 51
References 51
3 Unified SHE Formula 53
3.1 Unified SHE Formulas for Multilevel Converters 53
3.1.1 Problem with Conventional Multilevel SHE Formula 53
3.1.2 Unified SHE Formula 54
3.1.3 Example with Seven Switching Angles 57
3.1.4 Experimental Results 59
3.2 Unified SHE Formulas for Asymmetric Multilevel Converters 62
3.2.1 Asymmetric Multilevel Converters 62
3.2.2 Unified SHE Formula for Asymmetric Multilevel Converters 62
3.2.2.1 Unified Asymmetric SHE Models with Fundamental Frequency Modulation 62
3.2.2.2 Asymmetric SHE Models with High-Frequency Modulation 65
3.2.3 Example with Distribution Ratio 4:2 68
3.2.4 Experimental Results 71
3.3 Optimal Implementation of Unified SHE Formula 72
3.4 Summary 77
References 77
4 SHE Formulations in Specific Applications 79
4.1 SHE Formulation with CMV Reduction Ability 79
4.1.1 Common Mode Voltage 79
4.1.2 CMV Modeling Based on SHE 79
4.1.3 SHE Formula with CMV Reduction Ability 81
4.1.4 Experimental Results 83
4.2 SHE Formulation for Parallel Converters 86
4.2.1 Operation Principles of Parallel Converters 86
4.2.2 SHE Formulation for Parallel Converters 87
4.2.2.1 SHE Formulation of Individual Converter 87
4.2.2.2 Combined Formulation of Parallel Converter 87
4.2.3 Improved Formulation with ZSCC Reduction Ability 89
4.2.4 Simulation Results 90
4.3 SHE Formulation for Current Source Converters 92
4.3.1 Operation Principles of Current Source Converters 92
4.3.2 SHE Formulations for Current Source Converters 93
4.3.3 Experimental Results 95
4.4 Selective Harmonic Mitigation for Grid-Connected Scenarios 97
4.4.1 Principles of SHM-PWM 97
4.4.2 Simulation Results 99
4.5 Summary 100
References 101
Part II Algorithm 103
5 Resultant Elimination Method 105
5.1 Introduction 105
5.2 Resultant Elimination Theory 105
5.3 Solving Procedure for SHE Equations 109
5.4 Parallel Resultant Elimination Method 111
5.4.1 Principle of Polynomial Interpolation 111
5.4.2 Algorithm Description and Its Parallelization 112
5.5 Results and Discussions 114
5.5.1 Computational Results 114
5.5.2 Optimal Solutions Versus Modulation Index in Full Range 117
5.5.3 Experimental Results 119
5.6 Summary 123
References 124
6 Groebner Basis-Based Method 125
6.1 Groebner Basis Theory 125
6.1.1 Concept of Ideal and Basis 125
6.1.2 Concept of Groebner Basis 127
6.2 Solving Procedure of SHE Equations 128
6.3 Computation Results and Verifications 131
6.3.1 Computation Results for Single-Phase Converters 132
6.3.2 Computation Results for Three-Phase Converters 135
6.3.3 Experimental Verifications 138
6.4 Comparative Analysis 139
6.4.1 Comparison with Resultant Elimination Method 140
6.4.2 Comparison with Numerical and Intelligent Methods 142
6.5 Summary 143
References 144
7 Degree Reduction of SHE Equations 147
7.1 Symmetric Polynomials 147
7.1.1 Concept of Elementary Symmetric Polynomials 147
7.1.2 Solving Procedure of SHE Equations Based on Symmetric Polynomials 149
7.1.3 Computation Results 153
7.1.4 Experimental Results 158
7.2 Newton's Identities 159
7.2.1 Definitions of Newton's Identities 159
7.2.2 Solving Procedure of SHE Equations Based on Newton Identities 160
7.2.3 Computation Results 164
7.2.4 Experimental Results 164
7.2.5 Performance Evaluations and Comparative Analysis 166
7.3 Summary 171
References 172
8 Online Implementation of SHE 173
8.1 Introduction 173
8.2 Algebraic-Numerical Hybrid Algorithm 173
8.2.1 Offline Transformation of SHE Equations 174
8.2.2 Online Implementation Stage 176
8.2.2.1 Online Solving Procedure 177
8.2.2.2 Case Study 178
8.2.3 Algorithm Evaluations and Verifications 179
8.2.3.1 Computing Results 180
8.2.3.2 Online Calculation Performance on MCUs 181
8.2.3.3 Algorithm Comparison with Other Solving Methods 182
8.2.3.4 Experimental Verifications 184
8.3 Summary 187
References 187
Part III Application 189
9 Selective Harmonic Elimination PWM in Multilevel Converters 191
9.1 General Requirements of Multilevel Converters 191
9.2 Overview of Existing Capacitor Voltage Balancing Methods Under SHE-PWM 193
9.3 Capacitor Voltage Balancing Control Methods for Multilevel Converters Under SHE-PWM 194
9.3.1 Self-Balancing Control 194
9.3.1.1 Operation Principles 194
9.3.1.2 Results and Discussions 196
9.3.2 Charge Amount Regulation 197
9.3.2.1 Operation Principles 197
9.3.2.2 Results and Discussions 201
9.3.3 Switching Angle Modifications 202
9.3.3.1 Operation Principles 202
9.3.3.2 Results and Discussions 204
9.3.4 Redundant State Adjustment 204
9.3.4.1 Operation Principles 204
9.3.4.2 Results and Discussions 208
9.3.5 Space Voltage Vector Adjustment 210
9.3.5.1 Operation Principles 210
9.3.5.2 Results and Discussions 211
9.3.6 Composite SHE-MPC Method 214
9.3.6.1 Operation Principles 214
9.3.6.2 Results and Discussions 217
9.4 Summary 218
References 219
10 SHE-Based Closed-Loop Controller Design 223
10.1 Controller Design for PMSM Drives Under SHE-PWM 223
10.1.1 Mathematical Model of PMSMs 223
10.1.2 Classical Control Strategies of PMSM 226
10.1.2.1 Field-Oriented Control 226
10.1.2.2 Direct Torque Control 227
10.1.3 Control Strategies Based on SHE-PWM 228
10.1.3.1 Inner Current Loop 229
10.1.3.2 Outer Speed Loop 230
10.1.3.3 SHE-PWM Modulation 230
10.2 Advanced Closed-Loop Controller Design Under SHE-PWM 231
10.2.1 Analysis of SHE-PWM Field-Oriented Control Issues 231
10.2.2 Improved SHE-PWM Field-Oriented Control Method Based on Dead Zone 233
10.2.2.1 Principle of Dead Zone-Based SHE-PWM Field-Oriented Control 233
10.2.2.2 Dead Zone Threshold Configuration Based on Current Ripple Model 234
10.3 Simulation Results 237
10.4 Summary 240
References 241
11 Model Predictive Control with SHE-PWM 243
11.1 Hybrid Predictive Control with SHE-PWM 243
11.1.1 Principle of FCS-MPC 244
11.1.2 Implementation of the Switching Strategy 245
11.1.2.1 Initialization of the PI controller for Voltage Estimation Based on SOGI 245
11.1.2.2 The Impact of SOGI Filtering on the Stability of the Modulationb Index 247
11.1.2.3 Analysis of the Control Loop Model Based on SOGI 248
11.1.2.4 System Switching Criteria 250
11.2 Model Predictive Pulse Pattern Control 251
11.2.1 Relation Between Modulation Index and Flux Amplitude 252
11.2.2 Reference Trajectory of Stator Flux 253
11.2.3 Speed and Torque Outer-Loop Control 254
11.2.4 Flux Inner-Loop Control 254
11.3 Model Predictive SHE Control 258
11.3.1 Standard of Choosing SHE Optimal Solutions 258
11.3.2 Revision Limitations of SHE-PWM Switching Angles 262
11.4 Simulation and Experimental Results 265
11.4.1 Simulation Results 265
11.4.2 Speed Variation Simulation 273
11.4.3 Experimental Results 274
11.5 Summary 282
References 282
Index 285

About the author

Kehu Yang is a Professor with the School of Artificial Intelligence, China University of Mining and Technology-Beijing, Beijing, China, and a member of the IEEE. Mingzhe Wu is a Lecturer with the School of Mechanical and Electrical Engineering, China University of Mining and Technology-Beijing, Beijing, China. Qi ZhangM is a Lecturer with the School of Artificial Intelligence, China University of Mining and Technology-Beijing, Beijing, China. Chenxu Wang is a Ph.D. with the School of Artificial Intelligence, China University of Mining and Technology-Beijing, Beijing, China.

Product details

• Authors: Chengxu Wang, Mingzhe Wu, Kehu Yang, Kehu (China University of Mining and Technol Yang, Qi Zhang
• Publisher: Wiley, John and Sons Ltd
• Languages: English
• Product format: Hardback
• Release: 01.02.2026
• EAN: 9781394314324
• ISBN: 978-1-394-31432-4
• No. of pages: 304
• Subjects: Natural sciences, medicine, IT, technology > Technology > Electronics, electrical engineering, communications engineering; TECHNOLOGY & ENGINEERING / Power Resources / General, TECHNOLOGY & ENGINEERING / Power Resources / Electrical, Electrical Engineering, Alternative and renewable energy sources and technology, Energy technology and engineering, Energy, power generation, distribution and storage, Electronics & Communications Engineering