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 Zhang 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.

• About the Authors xiPreface xiiiAcknowledgments xv1 Power Converters and Selective Harmonic Elimination 11.1 Introduction to Power Converters 11.1.1 Topologies of Voltage–Source Converters 11.1.2 Topologies of Current–Source Converters 31.2 Modulation Strategies for Power Converters 41.2.1 PWM of Voltage–Source Converters 41.2.2 PWM of Current–Source Converters 71.3 Overview of Selective Harmonic Elimination PWM 71.3.1 Basic Knowledge of SHE and Its Formulation 81.3.2 Overview of SHE Solving Algorithms 101.3.3 Overview of Control Methods with SHE-PWM 131.3.4 Overall SHE-PWM Implementation Procedure 141.3.5 Real-World Implementation Cases of SHE-PWM 161.3.5.1 SHE-PWM in Commercialized Products 161.3.5.2 SHE-PWM in Traction Motor Drives 161.3.5.3 SHE-PWM in High-Power Rectifiers 171.4 Technical Challenges with SHE-PWM 181.4.1 Technical Challenges in SHE Formulation 181.4.2 Technical Challenges in SHE Solving Algorithms 181.4.3 Technical Challenges in SHE Applications 191.5 Outline of the Book 191.5.1 Part 1: Formulation 191.5.2 Part 2: Solving Algorithm 201.5.3 Part 3: Application 21References 21Part I Formulation 292 Principle of SHE 312.1 Fourier Series of Periodic Functions 312.1.1 Half-Wave Symmetric 322.1.2 Quarter-Wave Symmetric 332.2 Principle of SHE Formulations 332.2.1 Fourier Series of SquareWaveform 352.2.2 Three-Level SHE Formulation 352.2.3 Multilevel SHE Formulation 372.2.4 Generalized Odd-Level SHE Formulation 382.2.5 Even-Level SHE Formulation 402.2.5.1 Two-Level SHE Formulation with Three Switching Angles 412.2.5.2 Generalized Even-Level SHE Formulation 422.2.6 SHE Formulation for Asymmetric Multilevel Converters 442.2.7 SHE Formulation with Half-Wave Symmetric 482.3 Summary 51References 513 Unified SHE Formula 533.1 Unified SHE Formulas for Multilevel Converters 533.1.1 Problem with Conventional Multilevel SHE Formula 533.1.2 Unified SHE Formula 543.1.3 Example with Seven Switching Angles 573.1.4 Experimental Results 593.2 Unified SHE Formulas for Asymmetric Multilevel Converters 623.2.1 Asymmetric Multilevel Converters 623.2.2 Unified SHE Formula for Asymmetric Multilevel Converters 623.2.2.1 Unified Asymmetric SHE Models with Fundamental Frequency Modulation 623.2.2.2 Asymmetric SHE Models with High-Frequency Modulation 653.2.3 Example with Distribution Ratio 4:2 683.2.4 Experimental Results 713.3 Optimal Implementation of Unified SHE Formula 723.4 Summary 77References 774 SHE Formulations in Specific Applications 794.1 SHE Formulation with CMV Reduction Ability 794.1.1 Common Mode Voltage 794.1.2 CMV Modeling Based on SHE 794.1.3 SHE Formula with CMV Reduction Ability 814.1.4 Experimental Results 834.2 SHE Formulation for Parallel Converters 864.2.1 Operation Principles of Parallel Converters 864.2.2 SHE Formulation for Parallel Converters 874.2.2.1 SHE Formulation of Individual Converter 874.2.2.2 Combined Formulation of Parallel Converter 874.2.3 Improved Formulation with ZSCC Reduction Ability 894.2.4 Simulation Results 904.3 SHE Formulation for Current Source Converters 924.3.1 Operation Principles of Current Source Converters 924.3.2 SHE Formulations for Current Source Converters 934.3.3 Experimental Results 954.4 Selective Harmonic Mitigation for Grid-Connected Scenarios 974.4.1 Principles of SHM-PWM 974.4.2 Simulation Results 994.5 Summary 100References 101Part II Algorithm 1035 Resultant Elimination Method 1055.1 Introduction 1055.2 Resultant Elimination Theory 1055.3 Solving Procedure for SHE Equations 1095.4 Parallel Resultant Elimination Method 1115.4.1 Principle of Polynomial Interpolation 1115.4.2 Algorithm Description and Its Parallelization 1125.5 Results and Discussions 1145.5.1 Computational Results 1145.5.2 Optimal Solutions Versus Modulation Index in Full Range 1175.5.3 Experimental Results 1195.6 Summary 123References 1246 Groebner Basis-Based Method 1256.1 Groebner Basis Theory 1256.1.1 Concept of Ideal and Basis 1256.1.2 Concept of Groebner Basis 1276.2 Solving Procedure of SHE Equations 1286.3 Computation Results and Verifications 1316.3.1 Computation Results for Single-Phase Converters 1326.3.2 Computation Results for Three-Phase Converters 1356.3.3 Experimental Verifications 1386.4 Comparative Analysis 1396.4.1 Comparison with Resultant Elimination Method 1406.4.2 Comparison with Numerical and Intelligent Methods 1426.5 Summary 143References 1447 Degree Reduction of SHE Equations 1477.1 Symmetric Polynomials 1477.1.1 Concept of Elementary Symmetric Polynomials 1477.1.2 Solving Procedure of SHE Equations Based on Symmetric Polynomials 1497.1.3 Computation Results 1537.1.4 Experimental Results 1587.2 Newton’s Identities 1597.2.1 Definitions of Newton’s Identities 1597.2.2 Solving Procedure of SHE Equations Based on Newton Identities 1607.2.3 Computation Results 1647.2.4 Experimental Results 1647.2.5 Performance Evaluations and Comparative Analysis 1667.3 Summary 171References 1728 Online Implementation of SHE 1738.1 Introduction 1738.2 Algebraic-Numerical Hybrid Algorithm 1738.2.1 Offline Transformation of SHE Equations 1748.2.2 Online Implementation Stage 1768.2.2.1 Online Solving Procedure 1778.2.2.2 Case Study 1788.2.3 Algorithm Evaluations and Verifications 1798.2.3.1 Computing Results 1808.2.3.2 Online Calculation Performance on MCUs 1818.2.3.3 Algorithm Comparison with Other Solving Methods 1828.2.3.4 Experimental Verifications 1848.3 Summary 187References 187Part III Application 1899 Selective Harmonic Elimination PWM in Multilevel Converters 1919.1 General Requirements of Multilevel Converters 1919.2 Overview of Existing Capacitor Voltage Balancing Methods Under SHE-PWM 1939.3 Capacitor Voltage Balancing Control Methods for Multilevel Converters Under SHE-PWM 1949.3.1 Self-Balancing Control 1949.3.1.1 Operation Principles 1949.3.1.2 Results and Discussions 1969.3.2 Charge Amount Regulation 1979.3.2.1 Operation Principles 1979.3.2.2 Results and Discussions 2019.3.3 Switching Angle Modifications 2029.3.3.1 Operation Principles 2029.3.3.2 Results and Discussions 2049.3.4 Redundant State Adjustment 2049.3.4.1 Operation Principles 2049.3.4.2 Results and Discussions 2089.3.5 Space Voltage Vector Adjustment 2109.3.5.1 Operation Principles 2109.3.5.2 Results and Discussions 2119.3.6 Composite SHE-MPC Method 2149.3.6.1 Operation Principles 2149.3.6.2 Results and Discussions 2179.4 Summary 218References 21910 SHE-Based Closed-Loop Controller Design 22310.1 Controller Design for PMSM Drives Under SHE-PWM 22310.1.1 Mathematical Model of PMSMs 22310.1.2 Classical Control Strategies of PMSM 22610.1.2.1 Field-Oriented Control 22610.1.2.2 Direct Torque Control 22710.1.3 Control Strategies Based on SHE-PWM 22810.1.3.1 Inner Current Loop 22910.1.3.2 Outer Speed Loop 23010.1.3.3 SHE-PWM Modulation 23010.2 Advanced Closed-Loop Controller Design Under SHE-PWM 23110.2.1 Analysis of SHE-PWM Field-Oriented Control Issues 23110.2.2 Improved SHE-PWM Field-Oriented Control Method Based on Dead Zone 23310.2.2.1 Principle of Dead Zone-Based SHE-PWM Field-Oriented Control 23310.2.2.2 Dead Zone Threshold Configuration Based on Current Ripple Model 23410.3 Simulation Results 23710.4 Summary 240References 24111 Model Predictive Control with SHE-PWM 24311.1 Hybrid Predictive Control with SHE-PWM 24311.1.1 Principle of FCS-MPC 24411.1.2 Implementation of the Switching Strategy 24511.1.2.1 Initialization of the PI controller for Voltage Estimation Based on SOGI 24511.1.2.2 The Impact of SOGI Filtering on the Stability of the Modulationb Index 24711.1.2.3 Analysis of the Control Loop Model Based on SOGI 24811.1.2.4 System Switching Criteria 25011.2 Model Predictive Pulse Pattern Control 25111.2.1 Relation Between Modulation Index and Flux Amplitude 25211.2.2 Reference Trajectory of Stator Flux 25311.2.3 Speed and Torque Outer-Loop Control 25411.2.4 Flux Inner-Loop Control 25411.3 Model Predictive SHE Control 25811.3.1 Standard of Choosing SHE Optimal Solutions 25811.3.2 Revision Limitations of SHE-PWM Switching Angles 26211.4 Simulation and Experimental Results 26511.4.1 Simulation Results 26511.4.2 Speed Variation Simulation 27311.4.3 Experimental Results 27411.5 Summary 282References 282Index 285

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 Zhang 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.

• About the Authors xiPreface xiiiAcknowledgments xv1 Power Converters and Selective Harmonic Elimination 11.1 Introduction to Power Converters 11.1.1 Topologies of Voltage–Source Converters 11.1.2 Topologies of Current–Source Converters 31.2 Modulation Strategies for Power Converters 41.2.1 PWM of Voltage–Source Converters 41.2.2 PWM of Current–Source Converters 71.3 Overview of Selective Harmonic Elimination PWM 71.3.1 Basic Knowledge of SHE and Its Formulation 81.3.2 Overview of SHE Solving Algorithms 101.3.3 Overview of Control Methods with SHE-PWM 131.3.4 Overall SHE-PWM Implementation Procedure 141.3.5 Real-World Implementation Cases of SHE-PWM 161.3.5.1 SHE-PWM in Commercialized Products 161.3.5.2 SHE-PWM in Traction Motor Drives 161.3.5.3 SHE-PWM in High-Power Rectifiers 171.4 Technical Challenges with SHE-PWM 181.4.1 Technical Challenges in SHE Formulation 181.4.2 Technical Challenges in SHE Solving Algorithms 181.4.3 Technical Challenges in SHE Applications 191.5 Outline of the Book 191.5.1 Part 1: Formulation 191.5.2 Part 2: Solving Algorithm 201.5.3 Part 3: Application 21References 21Part I Formulation 292 Principle of SHE 312.1 Fourier Series of Periodic Functions 312.1.1 Half-Wave Symmetric 322.1.2 Quarter-Wave Symmetric 332.2 Principle of SHE Formulations 332.2.1 Fourier Series of SquareWaveform 352.2.2 Three-Level SHE Formulation 352.2.3 Multilevel SHE Formulation 372.2.4 Generalized Odd-Level SHE Formulation 382.2.5 Even-Level SHE Formulation 402.2.5.1 Two-Level SHE Formulation with Three Switching Angles 412.2.5.2 Generalized Even-Level SHE Formulation 422.2.6 SHE Formulation for Asymmetric Multilevel Converters 442.2.7 SHE Formulation with Half-Wave Symmetric 482.3 Summary 51References 513 Unified SHE Formula 533.1 Unified SHE Formulas for Multilevel Converters 533.1.1 Problem with Conventional Multilevel SHE Formula 533.1.2 Unified SHE Formula 543.1.3 Example with Seven Switching Angles 573.1.4 Experimental Results 593.2 Unified SHE Formulas for Asymmetric Multilevel Converters 623.2.1 Asymmetric Multilevel Converters 623.2.2 Unified SHE Formula for Asymmetric Multilevel Converters 623.2.2.1 Unified Asymmetric SHE Models with Fundamental Frequency Modulation 623.2.2.2 Asymmetric SHE Models with High-Frequency Modulation 653.2.3 Example with Distribution Ratio 4:2 683.2.4 Experimental Results 713.3 Optimal Implementation of Unified SHE Formula 723.4 Summary 77References 774 SHE Formulations in Specific Applications 794.1 SHE Formulation with CMV Reduction Ability 794.1.1 Common Mode Voltage 794.1.2 CMV Modeling Based on SHE 794.1.3 SHE Formula with CMV Reduction Ability 814.1.4 Experimental Results 834.2 SHE Formulation for Parallel Converters 864.2.1 Operation Principles of Parallel Converters 864.2.2 SHE Formulation for Parallel Converters 874.2.2.1 SHE Formulation of Individual Converter 874.2.2.2 Combined Formulation of Parallel Converter 874.2.3 Improved Formulation with ZSCC Reduction Ability 894.2.4 Simulation Results 904.3 SHE Formulation for Current Source Converters 924.3.1 Operation Principles of Current Source Converters 924.3.2 SHE Formulations for Current Source Converters 934.3.3 Experimental Results 954.4 Selective Harmonic Mitigation for Grid-Connected Scenarios 974.4.1 Principles of SHM-PWM 974.4.2 Simulation Results 994.5 Summary 100References 101Part II Algorithm 1035 Resultant Elimination Method 1055.1 Introduction 1055.2 Resultant Elimination Theory 1055.3 Solving Procedure for SHE Equations 1095.4 Parallel Resultant Elimination Method 1115.4.1 Principle of Polynomial Interpolation 1115.4.2 Algorithm Description and Its Parallelization 1125.5 Results and Discussions 1145.5.1 Computational Results 1145.5.2 Optimal Solutions Versus Modulation Index in Full Range 1175.5.3 Experimental Results 1195.6 Summary 123References 1246 Groebner Basis-Based Method 1256.1 Groebner Basis Theory 1256.1.1 Concept of Ideal and Basis 1256.1.2 Concept of Groebner Basis 1276.2 Solving Procedure of SHE Equations 1286.3 Computation Results and Verifications 1316.3.1 Computation Results for Single-Phase Converters 1326.3.2 Computation Results for Three-Phase Converters 1356.3.3 Experimental Verifications 1386.4 Comparative Analysis 1396.4.1 Comparison with Resultant Elimination Method 1406.4.2 Comparison with Numerical and Intelligent Methods 1426.5 Summary 143References 1447 Degree Reduction of SHE Equations 1477.1 Symmetric Polynomials 1477.1.1 Concept of Elementary Symmetric Polynomials 1477.1.2 Solving Procedure of SHE Equations Based on Symmetric Polynomials 1497.1.3 Computation Results 1537.1.4 Experimental Results 1587.2 Newton’s Identities 1597.2.1 Definitions of Newton’s Identities 1597.2.2 Solving Procedure of SHE Equations Based on Newton Identities 1607.2.3 Computation Results 1647.2.4 Experimental Results 1647.2.5 Performance Evaluations and Comparative Analysis 1667.3 Summary 171References 1728 Online Implementation of SHE 1738.1 Introduction 1738.2 Algebraic-Numerical Hybrid Algorithm 1738.2.1 Offline Transformation of SHE Equations 1748.2.2 Online Implementation Stage 1768.2.2.1 Online Solving Procedure 1778.2.2.2 Case Study 1788.2.3 Algorithm Evaluations and Verifications 1798.2.3.1 Computing Results 1808.2.3.2 Online Calculation Performance on MCUs 1818.2.3.3 Algorithm Comparison with Other Solving Methods 1828.2.3.4 Experimental Verifications 1848.3 Summary 187References 187Part III Application 1899 Selective Harmonic Elimination PWM in Multilevel Converters 1919.1 General Requirements of Multilevel Converters 1919.2 Overview of Existing Capacitor Voltage Balancing Methods Under SHE-PWM 1939.3 Capacitor Voltage Balancing Control Methods for Multilevel Converters Under SHE-PWM 1949.3.1 Self-Balancing Control 1949.3.1.1 Operation Principles 1949.3.1.2 Results and Discussions 1969.3.2 Charge Amount Regulation 1979.3.2.1 Operation Principles 1979.3.2.2 Results and Discussions 2019.3.3 Switching Angle Modifications 2029.3.3.1 Operation Principles 2029.3.3.2 Results and Discussions 2049.3.4 Redundant State Adjustment 2049.3.4.1 Operation Principles 2049.3.4.2 Results and Discussions 2089.3.5 Space Voltage Vector Adjustment 2109.3.5.1 Operation Principles 2109.3.5.2 Results and Discussions 2119.3.6 Composite SHE-MPC Method 2149.3.6.1 Operation Principles 2149.3.6.2 Results and Discussions 2179.4 Summary 218References 21910 SHE-Based Closed-Loop Controller Design 22310.1 Controller Design for PMSM Drives Under SHE-PWM 22310.1.1 Mathematical Model of PMSMs 22310.1.2 Classical Control Strategies of PMSM 22610.1.2.1 Field-Oriented Control 22610.1.2.2 Direct Torque Control 22710.1.3 Control Strategies Based on SHE-PWM 22810.1.3.1 Inner Current Loop 22910.1.3.2 Outer Speed Loop 23010.1.3.3 SHE-PWM Modulation 23010.2 Advanced Closed-Loop Controller Design Under SHE-PWM 23110.2.1 Analysis of SHE-PWM Field-Oriented Control Issues 23110.2.2 Improved SHE-PWM Field-Oriented Control Method Based on Dead Zone 23310.2.2.1 Principle of Dead Zone-Based SHE-PWM Field-Oriented Control 23310.2.2.2 Dead Zone Threshold Configuration Based on Current Ripple Model 23410.3 Simulation Results 23710.4 Summary 240References 24111 Model Predictive Control with SHE-PWM 24311.1 Hybrid Predictive Control with SHE-PWM 24311.1.1 Principle of FCS-MPC 24411.1.2 Implementation of the Switching Strategy 24511.1.2.1 Initialization of the PI controller for Voltage Estimation Based on SOGI 24511.1.2.2 The Impact of SOGI Filtering on the Stability of the Modulationb Index 24711.1.2.3 Analysis of the Control Loop Model Based on SOGI 24811.1.2.4 System Switching Criteria 25011.2 Model Predictive Pulse Pattern Control 25111.2.1 Relation Between Modulation Index and Flux Amplitude 25211.2.2 Reference Trajectory of Stator Flux 25311.2.3 Speed and Torque Outer-Loop Control 25411.2.4 Flux Inner-Loop Control 25411.3 Model Predictive SHE Control 25811.3.1 Standard of Choosing SHE Optimal Solutions 25811.3.2 Revision Limitations of SHE-PWM Switching Angles 26211.4 Simulation and Experimental Results 26511.4.1 Simulation Results 26511.4.2 Speed Variation Simulation 27311.4.3 Experimental Results 27411.5 Summary 282References 282Index 285

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 Zhang 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.

• About the Authors xiPreface xiiiAcknowledgments xv1 Power Converters and Selective Harmonic Elimination 11.1 Introduction to Power Converters 11.1.1 Topologies of Voltage–Source Converters 11.1.2 Topologies of Current–Source Converters 31.2 Modulation Strategies for Power Converters 41.2.1 PWM of Voltage–Source Converters 41.2.2 PWM of Current–Source Converters 71.3 Overview of Selective Harmonic Elimination PWM 71.3.1 Basic Knowledge of SHE and Its Formulation 81.3.2 Overview of SHE Solving Algorithms 101.3.3 Overview of Control Methods with SHE-PWM 131.3.4 Overall SHE-PWM Implementation Procedure 141.3.5 Real-World Implementation Cases of SHE-PWM 161.3.5.1 SHE-PWM in Commercialized Products 161.3.5.2 SHE-PWM in Traction Motor Drives 161.3.5.3 SHE-PWM in High-Power Rectifiers 171.4 Technical Challenges with SHE-PWM 181.4.1 Technical Challenges in SHE Formulation 181.4.2 Technical Challenges in SHE Solving Algorithms 181.4.3 Technical Challenges in SHE Applications 191.5 Outline of the Book 191.5.1 Part 1: Formulation 191.5.2 Part 2: Solving Algorithm 201.5.3 Part 3: Application 21References 21Part I Formulation 292 Principle of SHE 312.1 Fourier Series of Periodic Functions 312.1.1 Half-Wave Symmetric 322.1.2 Quarter-Wave Symmetric 332.2 Principle of SHE Formulations 332.2.1 Fourier Series of SquareWaveform 352.2.2 Three-Level SHE Formulation 352.2.3 Multilevel SHE Formulation 372.2.4 Generalized Odd-Level SHE Formulation 382.2.5 Even-Level SHE Formulation 402.2.5.1 Two-Level SHE Formulation with Three Switching Angles 412.2.5.2 Generalized Even-Level SHE Formulation 422.2.6 SHE Formulation for Asymmetric Multilevel Converters 442.2.7 SHE Formulation with Half-Wave Symmetric 482.3 Summary 51References 513 Unified SHE Formula 533.1 Unified SHE Formulas for Multilevel Converters 533.1.1 Problem with Conventional Multilevel SHE Formula 533.1.2 Unified SHE Formula 543.1.3 Example with Seven Switching Angles 573.1.4 Experimental Results 593.2 Unified SHE Formulas for Asymmetric Multilevel Converters 623.2.1 Asymmetric Multilevel Converters 623.2.2 Unified SHE Formula for Asymmetric Multilevel Converters 623.2.2.1 Unified Asymmetric SHE Models with Fundamental Frequency Modulation 623.2.2.2 Asymmetric SHE Models with High-Frequency Modulation 653.2.3 Example with Distribution Ratio 4:2 683.2.4 Experimental Results 713.3 Optimal Implementation of Unified SHE Formula 723.4 Summary 77References 774 SHE Formulations in Specific Applications 794.1 SHE Formulation with CMV Reduction Ability 794.1.1 Common Mode Voltage 794.1.2 CMV Modeling Based on SHE 794.1.3 SHE Formula with CMV Reduction Ability 814.1.4 Experimental Results 834.2 SHE Formulation for Parallel Converters 864.2.1 Operation Principles of Parallel Converters 864.2.2 SHE Formulation for Parallel Converters 874.2.2.1 SHE Formulation of Individual Converter 874.2.2.2 Combined Formulation of Parallel Converter 874.2.3 Improved Formulation with ZSCC Reduction Ability 894.2.4 Simulation Results 904.3 SHE Formulation for Current Source Converters 924.3.1 Operation Principles of Current Source Converters 924.3.2 SHE Formulations for Current Source Converters 934.3.3 Experimental Results 954.4 Selective Harmonic Mitigation for Grid-Connected Scenarios 974.4.1 Principles of SHM-PWM 974.4.2 Simulation Results 994.5 Summary 100References 101Part II Algorithm 1035 Resultant Elimination Method 1055.1 Introduction 1055.2 Resultant Elimination Theory 1055.3 Solving Procedure for SHE Equations 1095.4 Parallel Resultant Elimination Method 1115.4.1 Principle of Polynomial Interpolation 1115.4.2 Algorithm Description and Its Parallelization 1125.5 Results and Discussions 1145.5.1 Computational Results 1145.5.2 Optimal Solutions Versus Modulation Index in Full Range 1175.5.3 Experimental Results 1195.6 Summary 123References 1246 Groebner Basis-Based Method 1256.1 Groebner Basis Theory 1256.1.1 Concept of Ideal and Basis 1256.1.2 Concept of Groebner Basis 1276.2 Solving Procedure of SHE Equations 1286.3 Computation Results and Verifications 1316.3.1 Computation Results for Single-Phase Converters 1326.3.2 Computation Results for Three-Phase Converters 1356.3.3 Experimental Verifications 1386.4 Comparative Analysis 1396.4.1 Comparison with Resultant Elimination Method 1406.4.2 Comparison with Numerical and Intelligent Methods 1426.5 Summary 143References 1447 Degree Reduction of SHE Equations 1477.1 Symmetric Polynomials 1477.1.1 Concept of Elementary Symmetric Polynomials 1477.1.2 Solving Procedure of SHE Equations Based on Symmetric Polynomials 1497.1.3 Computation Results 1537.1.4 Experimental Results 1587.2 Newton’s Identities 1597.2.1 Definitions of Newton’s Identities 1597.2.2 Solving Procedure of SHE Equations Based on Newton Identities 1607.2.3 Computation Results 1647.2.4 Experimental Results 1647.2.5 Performance Evaluations and Comparative Analysis 1667.3 Summary 171References 1728 Online Implementation of SHE 1738.1 Introduction 1738.2 Algebraic-Numerical Hybrid Algorithm 1738.2.1 Offline Transformation of SHE Equations 1748.2.2 Online Implementation Stage 1768.2.2.1 Online Solving Procedure 1778.2.2.2 Case Study 1788.2.3 Algorithm Evaluations and Verifications 1798.2.3.1 Computing Results 1808.2.3.2 Online Calculation Performance on MCUs 1818.2.3.3 Algorithm Comparison with Other Solving Methods 1828.2.3.4 Experimental Verifications 1848.3 Summary 187References 187Part III Application 1899 Selective Harmonic Elimination PWM in Multilevel Converters 1919.1 General Requirements of Multilevel Converters 1919.2 Overview of Existing Capacitor Voltage Balancing Methods Under SHE-PWM 1939.3 Capacitor Voltage Balancing Control Methods for Multilevel Converters Under SHE-PWM 1949.3.1 Self-Balancing Control 1949.3.1.1 Operation Principles 1949.3.1.2 Results and Discussions 1969.3.2 Charge Amount Regulation 1979.3.2.1 Operation Principles 1979.3.2.2 Results and Discussions 2019.3.3 Switching Angle Modifications 2029.3.3.1 Operation Principles 2029.3.3.2 Results and Discussions 2049.3.4 Redundant State Adjustment 2049.3.4.1 Operation Principles 2049.3.4.2 Results and Discussions 2089.3.5 Space Voltage Vector Adjustment 2109.3.5.1 Operation Principles 2109.3.5.2 Results and Discussions 2119.3.6 Composite SHE-MPC Method 2149.3.6.1 Operation Principles 2149.3.6.2 Results and Discussions 2179.4 Summary 218References 21910 SHE-Based Closed-Loop Controller Design 22310.1 Controller Design for PMSM Drives Under SHE-PWM 22310.1.1 Mathematical Model of PMSMs 22310.1.2 Classical Control Strategies of PMSM 22610.1.2.1 Field-Oriented Control 22610.1.2.2 Direct Torque Control 22710.1.3 Control Strategies Based on SHE-PWM 22810.1.3.1 Inner Current Loop 22910.1.3.2 Outer Speed Loop 23010.1.3.3 SHE-PWM Modulation 23010.2 Advanced Closed-Loop Controller Design Under SHE-PWM 23110.2.1 Analysis of SHE-PWM Field-Oriented Control Issues 23110.2.2 Improved SHE-PWM Field-Oriented Control Method Based on Dead Zone 23310.2.2.1 Principle of Dead Zone-Based SHE-PWM Field-Oriented Control 23310.2.2.2 Dead Zone Threshold Configuration Based on Current Ripple Model 23410.3 Simulation Results 23710.4 Summary 240References 24111 Model Predictive Control with SHE-PWM 24311.1 Hybrid Predictive Control with SHE-PWM 24311.1.1 Principle of FCS-MPC 24411.1.2 Implementation of the Switching Strategy 24511.1.2.1 Initialization of the PI controller for Voltage Estimation Based on SOGI 24511.1.2.2 The Impact of SOGI Filtering on the Stability of the Modulationb Index 24711.1.2.3 Analysis of the Control Loop Model Based on SOGI 24811.1.2.4 System Switching Criteria 25011.2 Model Predictive Pulse Pattern Control 25111.2.1 Relation Between Modulation Index and Flux Amplitude 25211.2.2 Reference Trajectory of Stator Flux 25311.2.3 Speed and Torque Outer-Loop Control 25411.2.4 Flux Inner-Loop Control 25411.3 Model Predictive SHE Control 25811.3.1 Standard of Choosing SHE Optimal Solutions 25811.3.2 Revision Limitations of SHE-PWM Switching Angles 26211.4 Simulation and Experimental Results 26511.4.1 Simulation Results 26511.4.2 Speed Variation Simulation 27311.4.3 Experimental Results 27411.5 Summary 282References 282Index 285