Harmonic Modeling of Voltage Source Converters using Basic Numerical Methods

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 modeling 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 his Ph.D. degree in Electrical Engineering from the University of Toronto, Canada, and he is a Senior Member of the Institute of Electrical and Electronics Engineers (IEEE).Ramadhani Kurniawan Subroto, Postdoctoral Researcher, Department of Electrical Engineering, Technical University of Denmark, Denmark. Dr. Subroto received his Ph.D. degree in Electrical Engineering from National Taiwan University of Science and Technology, Taiwan in 2021. His research interests include power converter control, power system control, energy storage control, model predictive control, sliding mode control, and harmonics modeling of power converter. Victor Andrean, received his M.Sc. degree from the Department of Electrical Engineering at National Taiwan University of Science and Technology, Taipei City, Taiwan, in 2019. Victor is currently working as a data scientist for HedgeDesk, CA, USA. 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. degrees in Electrical Engineering from the National Taiwan University of Science and Technology in 2018 and 2020, respectively.

• Preface xiiiAcknowledgments xviiList of Symbols xix1 Fundamental Theory 11.1 Background 11.2 Definition of Harmonics 21.3 Fourier Series 21.3.1 Trigonometric Form 31.3.2 Phasor Form 41.3.3 Exponential Form 41.4 Waveform Symmetry 51.4.1 Even Symmetry 51.4.2 Odd Symmetry 61.4.3 Half-Wave Symmetry 61.5 Phase Sequence of Harmonics 81.6 Frequency Domain and Harmonic Domain 81.7 Power Definitions 91.7.1 Average Power 91.7.2 Apparent and Reactive Power 91.8 Harmonic Indices 111.8.1 Total Harmonic Distortion (THD) 111.8.2 Total Demand Distortion (TDD) 121.8.3 True Power Factor 121.9 Detrimental Effects of Harmonics 131.9.1 Resonance 131.9.2 Misoperations of Meters and Relays 171.9.3 Harmonics Impact on Motors 181.9.4 Harmonics Impact on Transformers 181.10 Characteristic Harmonic and Non-Characteristic Harmonic 191.11 Harmonic Current Injection Method 211.12 Steady-State vs. Transient Response 211.13 Steady-State Modeling 221.14 Large-Signal Modeling vs. Small-Signal Modeling 241.15 Discussion of IEEE Standard (STD) 519 251.16 Supraharmonics 302 Power Electronics Basics 372.1 Some Basics 372.2 Semiconductors vs. Wide Bandgap Semiconductors 382.3 Types of Static Switches 402.3.1 Uncontrolled Static Switch 402.3.2 Semi-Controllable Switches 412.3.3 Controlled Switch 422.4 Combination of Switches 442.5 Classification Based on Commutation Process 452.6 Voltage Source Converter vs. Current Source Converter 463 Basic Numerical Iterative Methods 493.1 Definition of Error 493.2 The Gauss–Seidel Method 503.3 Predictor-Corrector 523.4 Newton’s Method 553.4.1 Root Finding 553.4.2 Numerical Integration 563.4.3 Power Flow 573.4.4 Harmonic Power Flow 613.4.5 Shooting Method 633.4.6 Advantages of Newton’s Method 673.4.7 Quasi-Newton Method 693.4.8 Limitation of Newton’s Method 713.5 PSO 714 Matrix Exponential 734.1 Definition of Matrix Exponential 744.2 Evaluation of Matrix Exponential 754.2.1 Inverse Laplace Transform 754.2.2 Cayley–Hamilton Method 764.2.3 Padé Approximation 784.2.4 Scaling and Squaring 804.3 Krylov Subspace Method 804.4 Krylov Space Method with Restarting 834.5 Application of Augmented Matrix on DC-DC Converters 864.6 Runge–Kutta Methods 905 Modeling of Voltage Source Converters 955.1 Single-Phase Two-Level VSCs 955.1.1 Switching Functions 955.1.2 Switched Circuits 975.2 Three-Phase Two-Level VSCs 995.3 Three-Phase Multilevel Voltage Source Converter 1125.3.1 Multilevel PWM 1125.3.2 Diode Clamped Multilevel VSCs 1145.3.3 Flying Capacitor Multilevel VSCs 1205.3.4 Cascaded Multi-Level VSCs 1285.3.5 Modular Multi-Level VSC 1406 Frequency Coupling Matrices 1496.1 Construction of FCM in the Harmonic Domain 1496.2 Construction of FCM in the Time Domain 1557 General Control Approaches of a VSC 1797.1 Reference Frame 1797.1.1 Stationary-abc Frame 1797.1.2 Stationary-𝛼𝛽 Frame 1807.1.3 Synchronous-dq Frame 1817.1.4 Phase-Locked Loop 1827.2 Control Strategies 1837.2.1 Vector-Current Controller 1837.2.2 Direct Power Controller 1867.2.3 DC-bus Voltage Controller 1887.2.4 Circulating Current Controller 1898 Generalized Steady-State Solution Procedure for Closed-Loop Converter Systems 1938.1 Introduction 1938.2 Generalized Procedure 1938.2.1 Step 1: Determine How and Where to Break the Loop 1958.2.2 Step 2: Check if the Calculation Flows of the Broken System are Feasible 1958.2.3 Step 3: Determine What Domain of Each Component in the System Should be Modeled 1968.2.4 Step 4: Formulate the Mismatch Equations 1978.2.5 Step 5: Iterate to Find the Solution 1978.3 Previously Proposed Methods Derived from the Proposed Solution Procedures 1978.3.1 Steady-State Methods Derived from Loop-Breaking 1 Method 1978.3.2 Steady-State Methods Derived from Loop-Breaking 2 Method 1988.4 The Loop-Breaking 3 Method 2009 Loop-Breaking 1 Method 2059.1 A Typical Two-Level VSC with AC Current Control and DC Voltage Control 2059.2 Loop-Breaking 1 Method for a Two-Level VSC 2069.2.1 Block 1 2089.2.2 Current Controller Block 2089.2.3 Voltage Controller Block 2109.3 Solution Flow Diagram 2109.3.1 Initialization 2129.3.2 Jacobian Matrix 2129.3.3 Number of Modulating Voltage Harmonics to be Included 22810 Loop-Breaking 2 Method for Solving a VSC 24510.1 Modeling for a Closed-Loop DC-DC Converter 24510.1.1 Model of the Buck Converter 24510.1.2 Constraints of Steady-State 24710.1.3 Switching Time Constraints 24810.1.4 Solution Flow Diagram 24810.2 Two-Level VSC Modeling: Open-Loop Equations 25210.2.1 Steady-State Constraints 25610.2.2 Switching Time Constraints 25710.2.3 Solution Flow Diagram 26010.2.4 Initialization 26010.2.5 Jacobian Matrix 26010.2.6 Discussions of Results 26910.3 Comparison Between the LB 1 and LB 2 Methods 27010.3.1 Case #1: Balanced System 27010.3.2 Case #2: Unbalanced System with AC Waveform Exhibiting Half-Wave Symmetry 27010.3.3 Case #3: Unbalanced System, No Waveform Symmetry 27210.4 Large-Signal Modeling for Line-Commutated Power Converter 27210.4.1 Discontinuous Conduction Mode 27310.4.2 Continuous Conduction Mode 28210.4.3 Steady-State Constraint Equations 28410.4.4 General Comments 29111 Loop-Breaking 3 Method 29311.1 OpenDSS 29311.2 Interfacing OpenDSS with MATLAB 29411.3 Interfacing OpenDSS with Harmonic Models of VSCs 29912 Small-Signal Harmonic Model of a VSC 31512.1 Problem Statement 31512.2 Gauss–Seidel LB 3 and Newton LB 3 31612.2.1 Current Injection Method 31612.2.2 Norton Circuit Method 31712.3 Small-Signal Analysis of DC-DC Converter 32012.4 Small-Signal Analysis of a Two-Level VSC 32512.4.1 Approach from Section 12.3 32512.4.2 Simpler Approach 32613 Parameter Estimation for a Single VSC 33513.1 Background on Parameter Estimation 33513.2 Parameter Estimator Based on White-Box-and-Black-Box Models 33713.3 Estimation Validations 33913.3.1 Experimental Validation 34013.3.2 PSCAD/EMTDC Validation 34314 Parameter Estimation for Multiple VSCs with Domain Adaptation 34914.1 Introduction of Deep Learning 34914.2 Domain Adaptation 35114.3 Parameter Estimation for Multiple VSCs 35214.4 Notations for DA 35314.5 Supervised Domain Adaptation for Regression 35514.6 Supervised Domain Adaptation for Classification 35614.7 Test Setup 35814.7.1 Data Generator 35914.7.2 Data Preprocessing 35914.8 Performance Metrics 36114.8.1 R square (Regression) 36114.8.2 Mean Absolute Percentage Error, MAPE (Regression) 36114.8.3 Accuracy (Classification) 36214.8.4 F1 score (Classification) 36214.9 Test Results 36314.9.1 Classification Task on Multiple VSC 36314.9.2 Regression Task on Multiple VSC 36314.10 Software for Running the Codes 37014.11 Implementation of Domain Adaptation 37014.11.1 Data Generation 37014.11.2 Regression 37214.11.3 Classification network 375References 379Index 389