Understanding Electromagnetic Transients in Power Systems

Luiz Cera Zanetta, Jr., PhD, is a Senior Member of IEEE and a full professor at University of São Paulo. His numerous publications and R&D projects for utilities have played a key role in advancing and solidifying the area of electromagnetic transient analysis and equipment specifications for major power plant projects and long-distance interconnections in Brazil. His interests range from electromagnetic transients to Flexible AC Transmission Systems, including dynamic stability analysis, in research domains that are challenging for achieving consistency. Currently his primary interest is to contribute to the enhancement of engineering education.

• About the Author xviiPreface xix1 Transients in Elementary Circuits and the Laplace Transform 11.1 Introduction 11.2 Laplace Transform 21.2.1 Definition 21.2.2 Some Transforms and Their Elementary Properties 21.2.3 Inversion of the Laplace Transform 51.3 The Convolution Integral 71.4 RL Circuit 81.4.1 RL Circuit with Sinusoidal Voltage Source 91.4.2 RL Circuit with DC Voltage Source 131.5 Series RLC Circuit 151.5.1 RLC Circuit with Sinusoidal Voltage Source 151.5.2 LC Circuit 201.6 Resonance at the Nominal Frequency 271.7 Analysis of Simple Networks with More Than One Loop 281.7.1 Inductive and Capacitive Elements with Initial Conditions 291.7.2 Network Analysis 30References 342 Traveling Waves in Single-Phase Lines 352.1 Introduction 352.2 Basic Equations 382.2.1 Transmission Line with Losses 382.2.2 Lossless Transmission Line 402.3 Voltage and Current Relations and Surge Impedance of a Lossless Transmission Line 442.4 Traveling Waves in Discontinuities – Reflected and Refracted Waves 452.4.1 A Generic Impedance at the Line Terminal 462.4.2 Analysis of Discontinuities Using the Thévenin Equivalent 552.5 Nonlinear Elements 582.6 Lattice Diagram 632.7 Sine Voltage Waves 66References 673 Traveling Waves in Multiphase Lines 693.1 Introduction 693.2 Elements of Matrix Algebra 703.2.1 Calculation of the Exponential Matrix e Ax 703.2.2 Modal Decomposition 713.2.3 Properties of Symmetric and Balanced Matrices 733.2.4 Diagonalization of the Product of Symmetrical Matrices 733.3 Phase Domain 753.3.1 Multiphase Line 753.3.2 Relationship Between Voltages and Currents – Matrix of Characteristic Impedances 783.3.3 Lossless Transmission Line 793.3.4 Traveling Waves in Multiphase Lines with Discontinuities 813.3.5 Thévenin Equivalent in Multiphase Circuits 833.4 Modal Domain 843.4.1 Modal Analysis 843.4.2 Analysis of the Propagation Modes 863.4.3 Basic Models in the Modal Domain 913.4.4 Traveling Waves in Discontinuities 93References 1064 Numerical Solution of Electromagnetic Transients 1094.1 Introduction 1094.2 Single-Phase Models 1104.2.1 Inductance Model 1104.2.2 Capacitance Model 1114.2.3 Resistance Model 1124.2.4 RL Circuit 1124.2.5 Single-Phase Transmission Line Models 1134.3 Transient Solution Using Nodal Analysis 1204.4 Nonlinear Elements 1284.4.1 Resistive Elements 1284.4.2 Inductive Elements 1314.4.3 Conversion of the Saturation Curve 1344.5 Representation of Switches 1384.6 Multiphase Models 1394.6.1 Three-Phase Inductive Circuit with Mutual Inductances 1394.6.2 Three-Phase Circuit with Resistances and Inductances 1414.6.3 Three-Phase Capacitive Circuit 1424.6.4 Three-Phase Transmission Lines 1434.7 Comments on Numerical Errors 147References 1525 Electrical Parameters Dependence on Frequency 1535.1 Introduction 1535.2 Elements for Mathematical Modeling 1545.2.1 Fitting of Rational Functions 1555.2.2 Convolution Integral by the Recursive Method 1575.3 Modal Domain Approach 1605.3.1 Convolution with the Propagation Function 1625.3.2 Convolution with the Characteristic Admittance 1665.4 Frequency-Dependent Transformation Matrix 1685.5 Model of the Transmission Line with the Nodal Admittance Matrix 1715.5.1 Inverse Fourier Transform 1715.5.2 State-Space Model of the Transmission Line 1735.5.3 Norton’s Equivalent 1745.5.4 Calculation of the Nodal Admittance Matrix in Frequency Domain 1765.5.5 Frequency-Dependent Network Equivalents-FDNEs 1765.6 Transmission Line Parameters 1775.6.1 Internal Impedance of the Conductor 1775.6.2 Matrix of Series Impedance with Carson’s Corrections 1785.6.3 Matrix of Series Impedance with a Complex Ground Return Plane 1795.6.4 Matrix of Capacitances 180References 1806 Elements of Power Electronics 1856.1 Introduction 1856.2 LCC – Line Commutated Converters 1866.2.1 Rectifier Bridge without Commutation Angle 1876.2.2 Rectifier Bridge with Commutation Angle 1896.2.3 Inverter Bridge 1926.2.4 Fourier Analysis of Current in Six-Pulse Bridges 1946.3 Thyristor Controlled Reactors and Switched Capacitors 1986.4 Power Electronics – with VSC 2026.4.1 Voltage Source Converters – VSC in Transmission Systems 2026.4.2 Application of VSC in Renewable Generation 2076.5 VSC Elements 2086.5.1 Converter Bridges 2086.5.2 Gate Drivers 2106.6 MMC – Modular Multilevel Converter 2126.7 Converter Control 2176.7.1 Transformation abc/αβ and αβ/dq 2176.7.2 PLL – Phase-Locked Loop 2226.7.3 Elementary Control 2266.8 VSC Models 2276.8.1 Switching Models 2286.8.2 Averaged Switch Models 2286.8.3 Simple Source Models 232References 2337 Phasor Domain Analysis and Temporary Overvoltages 2357.1 Introduction 2357.2 Line Energization and Load Rejection 2357.2.1 Line Energization 2367.2.2 Load Rejection 2457.3 Faults 2517.4 Open Phases in Transmission Lines 2577.4.1 Introduction 2577.4.2 Network Modeling 2597.4.3 Model for Single-Phase Autoreclosure 2717.4.4 Model for Stuck Breaker Analysis 2777.4.5 Single-Phase Autoreclosure 2777.5 Voltages Induced in Parallel Circuits 2787.5.1 General Considerations 2787.5.2 Model for the Capacitive Coupling Between Circuits 2787.5.3 Circuits with Reactive Compensation 2817.5.4 Comments on Resonance Analysis in Parallel Circuits 2867.6 Frequency Response Analysis 2907.6.1 Introduction 2907.6.2 Modeling the Network Elements 2907.6.3 Harmonic Flow 2927.6.4 Harmonics of Transformers 2937.6.5 Harmonics of Converters and Filtering 2947.7 Temporary Overvoltages with Transformers 3017.7.1 Transformer Energization and Load Rejection 3017.7.2 Ferroresonance 302References 3148 Switching Surges 3178.1 Introduction 3178.2 General Considerations 3188.3 Line Energization and Line Autoreclosure 3208.3.1 Energization 3208.3.2 Autoreclosure 3258.3.3 Residual Voltage for Tripolar Opening 3288.3.4 Preinsertion Resistor 3348.4 Faults 3428.4.1 AC Systems 3428.4.2 dc Transmission Line 3448.5 Fault Clearing 3468.6 Load Rejection 3478.7 Transformer Energization 3488.8 Controlled Switching 3538.8.1 Opening and Closing Switching 3548.8.2 Switching of Reactive Compensation and Transmission Lines 3578.9 VFTO – Very Fast Transient Overvoltages 3608.9.1 Disconnector Operation in Gas-Insulated Substations 3608.9.2 GIS Components Modeling 362References 3649 Lightning Surges 3679.1 Introduction 3679.2 Data to Calculate Lightning Surges 3699.2.1 Lightning Current 3699.2.2 Wavefront and Tail Time 3719.2.3 Ground Flash Density 3739.2.4 Topography and Soil Resistivity 3739.3 Models for Overvoltage Calculations 3749.3.1 Lines and Cables 3749.3.2 Towers 3749.3.3 Tower Grounding 3779.3.4 Substation Equipment 3809.3.5 Lightning Stroke Attachment 3809.3.6 Dielectric Strength of the Insulation 3829.4 Transmission Line Analysis 3829.4.1 Lightning Strokes 3839.4.2 Direct Stroke 3839.4.3 Back-Flashover 3839.4.4 Line-Arrester Application 3939.4.5 Induced Overvoltages in Transmission Lines 4029.4.6 Underground Cables 4109.4.7 Corona 4119.5 Substations Studies 4139.5.1 Air Insulated Substations 4159.5.2 Gas Insulated Substations-GIS 419References 42210 Transients in Systems with Shunt Capacitors 42710.1 Introduction 42710.2 High-Frequency Current and Voltage Transients 42710.2.1 Energization of Shunt-Capacitor Banks 42810.2.2 Restrike and Trapped Charge 43110.2.3 Overvoltages and Arresters 43310.2.4 Voltage Amplification 43710.2.5 Lightning Surges 43710.3 Back-to-Back Shunt Capacitor 43910.3.1 Transient Inrush Currents 43910.3.2 Back-to-Back Energization 44010.3.3 Restrike 44210.3.4 Faults 44210.4 Three-Phase Circuits 45610.4.1 Stored Charges in Ungrounded Shunt Capacitors 45610.4.2 Trapped Charges in Grounded Shunt Capacitors 46010.4.3 Reclosing and Restrike in Three-phase Circuits 46010.5 High-Frequency Requirements for Substation Equipment 46510.5.1 Circuit Breakers 46610.5.2 Current Transformers 46810.5.3 Shunt Capacitors 47010.5.4 Surge Arrester 470References 47011 Transients in Systems with Series Capacitors 47311.1 Introduction 47311.2 Protection Schemes for Series Capacitor Banks 47411.2.1 Protection by Spark Gaps 47511.2.2 Protection by Metal Oxide Varistor 47611.3 Protection Schemes Performance 47711.3.1 Triggering Levels for Spark Gaps 47711.3.2 Reinsertion Overvoltages 47811.3.3 Protection Schemes with MOV 48311.4 Complementary Studies 490References 49312 Transient Recovery Voltage 49512.1 Introduction 49512.1.1 Fault Currents 49512.1.2 Extinction of the Fault Current 49612.2 Transient Recovery Voltage 49712.2.1 Steady-State Component and Transient Component 49712.2.2 Opening Sequence for the Circuit Breaker Poles 49812.3 Calculation of the Transient Recovery Voltage 49912.3.1 Current Injection Method and Principle of Superposition 49912.3.2 Calculation with Electromagnetic Transient Programs 50112.4 TRV in Single Phase Inductive Circuits 50212.4.1 Current Interruption in Inductances 50212.4.2 Inductance and Capacitance 50412.4.3 Transient Recovery Voltage with Transmission Lines 50912.5 Calculation of the TRV in Three-Phase Circuits 51212.5.1 Three-phase Ungrounded Fault in the Transmission Line 51312.5.2 Three-Phase Ungrounded Fault in the Substation Bus 51612.5.3 Rate of Rise of the Recovery Voltage – RRRV 51712.5.4 Analysis with Symmetrical Components 52012.5.5 Traveling Waves 52512.5.6 TRV Analysis in the Frequency Domain 53012.6 Short Line Fault 53412.6.1 Time Domain Analysis 53412.6.2 Analysis with Two-Port Network 54012.7 TRV in Systems with Series Capacitors 54112.8 Electric Arc 54312.8.1 Cassie’s Model 54512.8.2 Mayr’s Model 54612.8.3 Stability of the Electric Arc for Small Currents 54712.9 Comments on Asymmetrical Faults and ITRV 54712.9.1 Asymmetrical Current 54712.9.2 Initial Transient Recovery Voltage 54812.10 Standards for Transient Recovery Voltage 549References 55113 Surge Arrester 55313.1 Introduction 55313.2 Overvoltage Control – Basic Concepts 55413.2.1 Analysis Using the Thévenin Equivalent Circuit 55413.2.2 Three-Phase Transmission Line 55713.3 Types and Characteristics of Surge Arresters 55813.3.1 Silicon–Carbide Surge Arrester 55813.3.2 Metal Oxide Surge Arrester (MOSA) 55913.4 Surge Arrester Application 56313.4.1 Rating Selection 56413.4.2 Protection Levels and Insulation Coordination 56513.5 Performance of Surge Arresters 56713.5.1 Simplified Model of the Surge Arrester 56713.5.2 Arrester Energy Dissipation 56813.5.3 Arrester and Switching Surges 57813.5.4 Surge Arrester and Fast-Front Overvoltages 580References 59214 Insulation Coordination of Transmission Lines and Substations 59314.1 Introduction 59314.2 Basic Probabilistic Concepts 59414.2.1 Elementary Concepts 59414.2.2 Probability Density Function and Distribution Function 59514.2.3 Function of Random Variable 60014.2.4 Joint Probability Density Function and Distribution with Two Random Variables 60114.3 Insulation Strength 60214.3.1 Impulse Tests for Lightning and Switching Surges 60314.3.2 Self-Restoring and Non-Self-Restoring Insulation 60314.3.3 Withstand Levels for Self-Restoring Insulation 60614.4 Insulation Coordination Methods 61014.4.1 Deterministic Method 61214.4.2 Statistical Method 61214.4.3 Simplified Statistical Method 61614.4.4 Further Comments on Slow-Front and Fast-Front Overvoltages 61614.5 Insulation Coordination of Substations 61714.5.1 Power-Frequency Voltage 61814.5.2 Fast-Front Overvoltages 61814.5.3 Slow-Front Overvoltages 62014.6 Insulation Coordination of Transmission Lines 62514.6.1 Insulation Coordination for Lightning Surges 62714.6.2 Insulation Coordination for Switching Surges 645References 650Index 653