Harmonic Modeling of Voltage Source Converters using Basic Numerical Methods

One of the first books to bridge the gap between frequency domain and time-domain methods of steady-state modeling of power electronic converters

Harmonic Modeling of Voltage Source Converters using Basic Numerical Methods presents detailed coverage of steady-state modeling of power electronic devices (PEDs). This authoritative resource describes both large-signal and small-signal modeling of power converters and how some of the simple and commonly used numerical methods can be applied for harmonic analysis and modeling of power converter systems. The book covers a variety of power converters including DC-DC converters, diode bridge rectifiers (AC-DC), and voltage source converters (DC-AC).

The authors provide in-depth guidance on modeling and simulating power converter systems. Detailed chapters contain relevant theory, practical examples, clear illustrations, sample Python and MATLAB codes, and validation enabling readers to build their own harmonic models for various PEDs and integrate them with existing power flow programs such as OpenDss.

This book:

• Presents comprehensive large-signal and small-signal harmonic modeling of voltage source converters with various topologies
• Describes how to use accurate steady-state models of PEDs to predict how device harmonics will interact with the rest of the power system
• Explains the definitions of harmonics, power quality indices, and steady-state analysis of power systems
• Covers generalized steady-state modeling techniques, and accelerated methods for closed-loop converters
• Shows how the presented models can be combined with neural networks for power system parameter estimations

Harmonic Modeling of Voltage Source Converters using Basic Numerical Methods is an indispensable reference and guide for researchers and graduate students involved in power quality and harmonic analysis, power engineers working in the field of harmonic power flow, developers of power simulation software, and academics and power industry professionals wanting to learn about harmonic modeling on power converters.

Ryan Kuo Lung Lian, Professor, Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan. He has been working in power system modelling for more than 10 years. His research interests are in power quality analysis, energy management systems, renewable energy systems, real time simulation, and power electronic control systems. Dr. Lian received the B.A.Sc. (Hons.), M.A.Sc., and Ph.D. degrees in electrical engineering from the University of Toronto, Toronto, ON, Canada, and he is a Senior member of the Institute of Electrical and Electronics Engineers (IEEE).

Ramadhani Kurniawan Subroto, Lecturer, Department of Electrical Engineering, Brawijaya University. His research interest includes control design of DC/DC converter, adaptive control, power electronics modeling, harmonics modeling, and sliding mode control.

Bing Hao Lin, Associate Researcher, Green Energy and Environment Research Laboratories, Industrial Technology Research Institute, Hsinchu, Taiwan. He received his B.Sc. and M.Sc. degree in Electrical Engineering from the National Taiwan University of Science and Technology in 2018 and 2020, respectively.

Table of Contents

1 Fundamental Theory 5

1.1 Background 5

1.2 Definition of Harmonics 7

1.3 Fourier Series 7

1.3.1 Trigonometric Form 7

1.3.2 Phasor Form 9

1.3.3 Exponential Form 10

1.4 Waveform Symmetry 11

1.4.1 Even Symmetry 11

1.4.2 Odd Symmetry 11

1.4.3 Half-wave symmetry 12

1.5 Phase Sequence of Harmonics 14

1.6 Frequency Domain and Harmonic Domain 15

1.7 Power Definitions 15

1.7.1 Average Power 15

1.7.2 Apparent and Reactive Power 16

1.8 Harmonic Indices 19

1.8.1 Total Harmonic Distortion (THD) 19

1.8.2 Total Demand Distortion (TDD) 20

1.8.3 True Power Factor 20

1.9 Detrimental Effects of Harmonics 21

1.9.1 Resonance 21

1.9.2 Misoperations of Meters and Relays 27

1.9.3 Harmonics Impact on Motors 28

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1.9.4 Harmonics Impact on Transformers 28

1.10 Characteristic Harmonic and Non-Characteristic Harmonic 29

1.11 Current Injection Method 32

1.12 Steady-State v.s. Transient Response 33

1.13 Steady-State Modeling 34

1.14 Large-Signal Modeling v.s. Small-Signal Modeling 37

1.15 Discussion on IEEE Standard (STD) 519 38

1.16 Supraharmonics 45

2 Power Electronics Basics 52

2.1 Some Basics 53

2.2 Semiconductors v.s Wide Bandgap Semiconductors 55

2.3 Types of Static Switches 56

2.3.1 Uncontrolled static switch 56

2.3.2 Semi-controllable switches 58

2.3.3 Controlled Switch 58

2.4 Combination of Switches 63

2.5 Classification Based on Commutation Process 63

2.6 Voltage Source Converter vs. Current Source Converter 65

3 Basic Numerical Iterative Methods 69

3.1 Definition of Error 70

3.2 Gauss-Seidel 71

3.3 Predictor-Corrector 73

3.4 Newton’s Method 77

3.4.1 Root Finding 78

3.4.2 Numerical Integration 79

3.4.3 Power Flow 80

3.4.4 Harmonic Power Flow 85

3.4.5 Shooting Method 87

3.4.6 Advantages of Newton’s Method 93

3.4.7 Quasi-Newton Method 95

3.4.8 Limitation of Newton’s Method 97

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3.5 Particle Swarm Optimization 97

4 Matrix Exponential 100

4.1 Definition of Matrix Exponential 102

4.2 Evaluation of Matrix Exponential 103

4.2.1 Inverse Laplace Transform 103

4.2.2 Cayley-Hamilton Method 104

4.2.3 Pad´e approximation 107

4.2.4 Scaling and Squaring 109

4.3 Krylov Subspace Method 110

4.4 Krylov Space Method with Restarting 114

4.5 Application of Augmented Matrix on DC-DC Converters 115

4.6 Runge-Kutta Methods 125

5 Modeling of Voltage Source Converters 130

5.1 Single-Phase Two-Level VSCs 130

5.1.1 Switching Functions 130

5.1.2 Switched Circuits 134

5.2 Three-Phase Two-Level VSCs 134

5.3 Three-Phase Multilevel Voltage Source Converter 153

5.4 Multilevel PWM 153

5.4.1 Diode Clamped Multilevel VSCs 156

5.4.2 Flying Capacitor Multilevel VSCs 165

5.4.3 Cascaded Multi-Level VSCs 174

5.4.4 Modular Multi-Level VSC 195

6 Frequency Coupling Matrices 203

6.1 Construction of FCM in Harmonic Domain 206

6.2 Construction of FCM in Time Domain 212

7 General Control Approach of a VSC 246

7.1 Reference Frame 246

7.1.1 Stationary-abc Frame 246

7.1.2 Stationary-αβ Frame 247

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7.1.3 Synchronous-dq Frame 248

7.1.4 Phase-Locked-Loop 249

7.2 Control Strategies 251

7.2.1 Vector-Current Controller 251

7.2.2 Direct Power Controller 255

7.2.3 DC-bus Voltage Controller 257

7.2.4 Circulating Current Controller 259

8 Iterative Techniques for Closed-Loop System 263

8.1 Introduction 263

8.2 Generalized Procedure 264

8.2.1 Step 1: Determine how and where to break the loop 264

8.2.2 Step 2: Check if the calculation flows of the broken system are feasible 267

8.2.3 Step 3: Determine what domain each component in the system should be

modeled 267

8.2.4 Step 4: Formulate the mismatch equations 268

8.2.5 Step 5: Iterate to find the solution 268

8.3 Previously Proposed Methods Derived From the Proposed Solution Procedures 269

8.3.1 Steady-State Methods Derived From Loop-Breaking 1 Method 269

8.3.2 Steady-State Methods Derived From Loop-Breaking 2 Method 270

8.4 The Loop-Breaking 3 Method 272

9 Loop Breaking 1 Method 280

9.1 A Typical Two-level VSC with AC Current Control and DC Voltage Control 281

9.2 Loop-Breaking 1 Method for a Two-Level VSC 281

9.2.1 Block 1 284

9.2.2 Current Controller Block 284

9.2.3 Voltage Controller Block 286

9.3 Solution Flow Diagram 287

9.3.1 Initialization 288

9.3.2 Jacobian Matrix 288

9.3.3 Number of Modulating Voltage Harmonics to be Included 307

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10 Loop-Breaking 2 Method for Solving a VSC 327

10.1 Modeling for a Closed-loop DC-DC Converter 327

10.1.1 Model of the Buck Converter 328

10.1.2 Constraints of Steady-State 329

10.1.3 Switching Time Constraints 331

10.1.4 Solution Flow Diagram 331

10.2 Two-Level VSC Modeling 338

10.3 Open-Loop Equations 338

10.3.1 Steady-State Constraints 343

10.3.2 Switching Time Constraints 344

10.3.3 Solution Flow Diagram 348

10.3.4 Initialization 348

10.3.5 Jacobian Matrix 348

10.3.6 Discussions of Results 357

10.4 Comparison Between the LB 1 and LB 2 Methods 360

10.4.1 Case # 1: Balanced System 360

10.4.2 Case # 2: Unbalanced System With AC Waveform Exhibiting Half-Wave

Symmetry 360

10.4.3 Case # 3: Unbalanced System, No Waveform Symmetry 361

10.5 Large-Signal Modeling for Line-Commutated Power Converter 361

10.5.1 Discontinuous Conduction Mode 365

10.5.2 Continuous Conduction Mode 377

10.5.3 Steady State Constraint Equations 379

10.5.4 General Comments 389

11 Loop Breaking 3 Method 390

11.1 OpenDSS 390

11.2 Interfacing OpenDss with MATLAB 391

11.3 Interfacing OpenDss with Harmonic Models of VSCs 398

12 Small-Signal Model of a VSC 420

12.1 Problem Statement 420

12.2 Gauss-Seidel LB 3 and Newton LB 3 421

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12.2.1 Current Injection Method 422

12.2.2 Norton Circuit Method 424

12.3 Small-Signal Analysis of DC-DC Converter 426

12.4 Small-Signal Analysis of a Two-Level VSC 434

12.4.1 Approach From Section 12.3 434

12.4.2 Simpler Approach 435

13 Parameter Estimation for a Single VSC 444

13.1 Background on Parameter Estimation 444

13.2 Parameter Estimator based on White-Box-and-Black-Box Models 447

13.3 Estimation Validations 451

13.3.1 Experimental Validation 451

13.3.2 PSCAD/EMTDC Validation 456

14 Parameter Estimation for Multiple VSCs 461

14.1 Estimation for a Wide Range of Parameter Values 462

14.2 Introduction of Deep Learning 463

14.3 Introduction of domain adaptation 465

14.4 The Black-Box Model based on DNN 466

14.4.1 Modeling Data Generator 468

14.4.2 Data Preprocessing 469

14.4.3 Deep Neural Network Model 471

14.4.4 DNN Training 478

14.4.5 Testing Result for the DNN Model without DA 481

14.5 Implementation of Domain Adaptation 486

14.6 Deep SDA 490

14.6.1 Visualization 495

14.7 Testing Result for the DNN Model with DA 497

14.7.1 PE of Single VSC 497

14.7.2 PE of Multiple VSCs 497

References 497

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