Electromagnetic Computation Methods for Lightning Surge Protection Studies
Description
Presents current research into electromagnetic computation theories with particular emphasis on Finite-Difference Time-Domain Method
This book is the first to consolidate current research and to examine the theories of electromagnetic computation methods in relation to lightning surge protection. The authors introduce and compare existing electromagnetic computation methods such as the method of moments (MOM), the partial element equivalent circuit (PEEC), the finite element method (FEM), the transmission-line modeling (TLM) method, and the finite-difference time-domain (FDTD) method. The application of FDTD method to lightning protection studies is a topic that has matured through many practical applications in the past decade, and the authors explain the derivation of Maxwell's equations required by the FDTD, and modeling of various electrical components needed in computing lightning electromagnetic fields and surges with the FDTD method. The book describes the application of FDTD method to current and emerging problems of lightning surge protection of continuously more complex installations, particularly in critical infrastructures of energy and information, such as overhead power lines, air-insulated sub-stations, wind turbine generator towers and telecommunication towers.
- Both authors are internationally recognized experts in the area of lightning study and this is the first book to present current research in lightning surge protection
- Examines in detail why lightning surges occur and what can be done to protect against them
- Includes theories of electromagnetic computation methods and many examples of their application
- Accompanied by a sample printed program based on the finite-difference time-domain (FDTD) method written in C++ program
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Persons
Professor Yoshihiro Baba, Associate Professor, Department of Electrical Engineering, Doshisha University, Kyoto, Japan
Professor Baba received his PhD from the University of Tokyo. He was Visiting Scholar at the Lightning Laboratory at the University of Florida, USA, and is currently Editor of the IEEE Transactions on Power Delivery. His areas of interest include computational electromagnetics, electromagnetic compatibility and lightning protection.
Professor Vladimir A. Rakov, Department of Electrical and Computer Engineering, University of Florida, USA
Professor Rakov studied for his PhD at Tomsk University in Russia and has written extensively on the subject of lightning in numerous international journals and conference proceedings. He is A Fellow of the IEEE and a Fellow of the American Meteorological Society, amongst others, and his areas of interest cover lightning, lightning protection and atmospheric electricity.
Content
Preface xi
1 Introduction 1
1.1 Historical Overview of Lightning Electromagnetic-Field and Surge Computations 1
1.2 Overview of Existing Electromagnetic Computation Methods 2
1.2.1 Method of Moments 2
1.2.2 Partial-Element Equivalent-Circuit Method 4
1.2.3 Finite-Element Method 4
1.2.4 Transmission Line Modeling Method 4
1.2.5 Constrained Interpolation Profile Method 5
1.2.6 Finite-Difference Time Domain Method 6
1.3 Summary 7
References 7
2 Lightning 11
2.1 Introduction 11
2.2 Thundercloud 12
2.2.1 Formation of Thunderclouds 12
2.2.2 Mechanism of Cloud Electrification 14
2.3 Lightning Discharges 15
2.3.1 Categories of Lightning Discharges 15
2.3.2 Classification of Cloud-to-Ground Lightning Discharges 15
2.3.3 Downward Negative Lightning Discharges to Ground 16
2.3.4 Positive Lightning Discharges 23
2.3.5 Upward Lightning Discharges 23
2.3.6 Rocket-Triggered Lightning Discharges 25
2.4 Lightning Electromagnetic Fields 26
2.4.1 Measured Lightning Return-Stroke Electromagnetic Fields 26
2.4.2 Mathematical Expressions for Calculating Electric and Magnetic Fields 29
2.5 Lightning Surges 31
2.5.1 Surges Due to Direct Lightning Strike 31
2.5.2 Surges Induced by a Nearby Lightning Strike 32
2.5.3 Surges Coming from Grounding Due to Its Potential Rise 33
2.6 Lightning Surge Protection 34
2.6.1 Insulation Coordination 34
2.6.2 Protection against Direct Lightning Strikes 35
2.6.3 Back-Flashover Phenomena 37
2.6.4 Lightning Surge Protection Measures 38
2.7 Summary 40
References 41
3 The Finite-Difference Time Domain Method for Solving Maxwell's Equations 43
3.1 Introduction 43
3.2 Finite-Difference Expressions of Maxwell's Equations 44
3.2.1 3D Cartesian Coordinate System 44
3.2.2 2D Cylindrical Coordinate System 49
3.3 Subgridding Technique 51
3.4 Absorbing Boundary Conditions 55
3.5 Representation of Lumped Sources and Lumped Circuit Elements 57
3.5.1 Lumped Voltage Source 57
3.5.2 Lumped Current Source 57
3.5.3 Lumped Resistance 59
3.5.4 Lumped Inductance 59
3.5.5 Lumped Capacitance 60
3.6 Representation of Thin Wire 61
3.7 Representation of Lightning Return-Stroke Channel 63
3.7.1 Lightning Return-Stroke Channel 63
3.7.2 Excitations 66
3.8 Representation of Surge Arresters 67
3.9 Summary 69
References 70
4 Applications to Lightning Surge Protection Studies 73
4.1 Introduction 73
4.1.1 Overview 73
4.1.2 Lightning Electromagnetic Fields at Close and Far Distances 73
4.1.3 Lightning Surges on Overhead Power TL Conductors and Towers 75
4.1.4 Lightning Surges on Overhead Distribution and Telecommunication Lines 76
4.1.5 Lightning Electromagnetic Environment in Power Substations 77
4.1.6 Lightning Surges in Wind-Turbine-Generator Towers 77
4.1.7 Lightning Surges in Photovoltaic Arrays 78
4.1.8 Lightning Electromagnetic Environment in Electric Vehicles 78
4.1.9 Lightning Electromagnetic Environment in Airborne Vehicles 78
4.1.10 Lightning Surges and the Electromagnetic Environment in Buildings 79
4.1.11 Surges on Grounding Electrodes 79
4.2 Electromagnetic Fields at the Top of a Tall Building Associated with Nearby Lightning Return Strokes 80
4.2.1 Introduction 80
4.2.2 Methodology 81
4.2.3 Analysis and Results 85
4.2.4 Summary 96
4.2.5 Appendix: Comparison of Fields in the Absence of a Building Computed Using the FDTD Method and Thottappillil et al.'s (2001) Analytical Expressions 96
4.2.6 Appendix: Enhancement Factors Due to the Presence of Hemisphere or Rectangular Building in a Uniform Static Electric Field 97
4.3 Influence of Strike Object Grounding on Close Lightning Electric Fields 100
4.3.1 Introduction 100
4.3.2 Methodology 103
4.3.3 Analysis and Results 105
4.3.4 Discussion 122
4.3.5 Summary 128
4.3.6 Appendix: Comparison of Fields Due to a Lightning Strike to Flat Ground Calculated Using the FDTD Method in the 2D Cylindrical Coordinate System and Thottappillil et al.'s (2001) Analytical Expressions 128
4.4 Simulation of Corona at Lightning-Triggering Wire: Current, Charge Transfer, and Field Reduction Effect 129
4.4.1 Introduction 129
4.4.2 General Approach 135
4.4.3 Model 136
4.4.4 Analysis and Results 141
4.4.5 Discussion 145
4.4.6 Summary 149
4.4.7 Appendix: Geometry of a Wire Corona Sheath 149
4.5 On the Interpretation of Ground Reflections Observed in Small-Scale Experiments Simulating Lightning Strikes to Towers 151
4.5.1 Introduction 151
4.5.2 Current Pulses Propagating along a Conical Conductor Excited at Its Apex or Base 153
4.5.3 FDTD Simulation of Small-Scale Experiments 157
4.5.4 Interpretation of Ground Reflections Arriving at the Tower Top 162
4.5.5 TL Representation of a Tall Object on the Ground Plane 164
4.5.6 Summary 169
4.5.7 Appendix: FDTD Representation of Tower Models 170
4.6 On the Mechanism of Attenuation of Current Waves Propagating along a Vertical Perfectly Conducting Wire above Ground: Application to Lightning 171
4.6.1 Introduction 171
4.6.2 Incident Current (Iinc), Incident E-field (Einc): Analytical Solution 174
4.6.3 Total Current (Itot), Total E-field (Etot): Numerical Solution 176
4.6.4 Scattered Current (Iscat), Scattered E-field (Escat): Iscat = Itot - Iinc, Escat = -Einc 179
4.6.5 Dependences of Current Attenuation on the Source Length, Conductor Thickness, and Frequency 181
4.6.6 Nonuniform TL Approximation 184
4.6.7 Summary 186
4.6.8 Appendix: Incident E-field for Two Parallel Vertical Phased Current Source Arrays-Analytical Solution 187
4.6.9 Appendix: Total Current for Horizontal Configurations-Numerical Solution 188
4.6.10 Appendix: Comparison of FDTD Simulation with an Analytical Solution 190
4.6.11 Appendix: E-field Structure around a Vertical Nonzero-Thickness Perfect Conductor 191
4.6.12 Appendix: Vertical E-field Produced by an Electrically-Short Vertical Dipole 192
4.7 FDTD Simulation of Lightning Surges on Overhead Wires in the Presence of Corona Discharge 193
4.7.1 Introduction 193
4.7.2 Modeling 195
4.7.3 Results and Discussion 199
4.7.4 Summary 209
4.8 FDTD Simulation of Insulator Voltages at a Lightning-Struck Tower Considering the Ground-Wire Corona 212
4.8.1 Introduction 212
4.8.2 Methodology 212
4.8.3 Analysis and Results 215
4.8.4 Summary 224
4.9 Voltages Induced on an Overhead Wire by Lightning Strikes to a Nearby Tall Grounded Object 224
4.9.1 Introduction 224
4.9.2 Methodology 228
4.9.3 Analysis and Results 231
4.9.4 Discussion 238
4.9.5 Summary 240
4.9.6 Appendix: Testing the Validity of the FDTD Calculations against Experimental Data (Strikes to Flat Ground) 242
4.9.7 Appendix: Comparison with Rusck's Formula (Strikes to Flat Ground) 243
4.9.8 Appendix: Testing the Validity of the FDTD Calculations against Experimental Data (Strikes to a Tall Object) 245
4.10 3D-FDTD Computation of Lightning-Induced Voltages on an Overhead Two-Wire Distribution Line 247
4.10.1 Introduction 247
4.10.2 Methodology 249
4.10.3 Analysis and Results 252
4.10.4 Summary 260
4.11 FDTD Simulations of the Corona Effect on Lightning-Induced Voltages 260
4.11.1 Introduction 260
4.11.2 Methodology 261
4.11.3 Analysis and Results 263
4.11.4 Discussion 269
4.11.5 Summary 277
4.12 FDTD Simulation of Surges on Grounding Electrodes Considering Soil Ionization 277
4.12.1 Introduction 277
4.12.2 Representation of Soil Ionization and De-ionization 278
4.12.3 Analysis and Results 279
4.12.4 Conclusions 288
4.13 Summary 288
References 288
Appendix: 3D-FDTD Program in C++ 299
Index 311
1
Introduction
1.1 Historical Overview of Lightning Electromagnetic-Field and Surge Computations
Lightning return-stroke electromagnetic fields have been calculated using analytical expressions, derived for a vertical lightning channel (e.g., Uman et al. 1975). Effects of finite ground conductivity on lightning electromagnetic fields have also been studied using analytical expressions (e.g., Rachidi et al. 1996). These analytical expressions are still being used. Lightning-induced voltages on an overhead power distribution line or telecommunication line have been calculated using an engineering model of the lightning return stroke (e.g., Uman et al. 1975) and a field-to-wire coupling model (e.g., Rachidi 1993). Horizontal electric fields above a finitely conducting ground, which are needed for calculating lightning-induced voltages, have been evaluated using approximate expressions such as the Cooray-Rubinstein formula (Rubinstein 1996). Note that the Cooray-Rubinstein formula is given in the frequency domain, although its time-domain counterparts also exist. Lightning surges due to a direct lightning strike to an overhead power transmission or distribution line have been analyzed using distributed-circuit simulation methods such as the electromagnetic transients program (EMTP) (Dommel 1969). EMTP and other similar programs are still widely used in lightning surge simulations.
Around 1990, electromagnetic computation methods were first applied to lightning electromagnetic and surge simulations. One of the advantages of electromagnetic computation methods, in comparison with circuit simulation methods, is that they allow a self-consistent full-wave solution for both the transient current distribution in a 3D conductor system and resultant electromagnetic fields, although they are computationally expensive. Podgorski and Landt (1987) applied the method of moments (MoM) in the time domain (Van Baricum and Miller 1972; Miller et al. 1973) to analyze the lightning current along a tall object struck by lightning. Grcev and Dawalibi (1990) applied the MoM in the frequency domain (Harrington 1968) to analyze the surge characteristics of a grounding electrode. Since then, the MoM in the frequency domain has been frequently used in lightning surge simulations (e.g., Baba and Rakov 2008 and references therein).
Tanabe (2001) applied the finite-difference time domain (FDTD) method (Yee 1966), which is one of the electromagnetic computation methods, to studying the surge characteristics of a grounding electrode. Baba and Rakov (2003) used the FDTD method to compute lightning electromagnetic fields. More than 60 journal papers and a large number of conference papers, which use the FDTD method in lightning electromagnetic-field and surge simulations, have been published during the last 15 years (e.g., Baba and Rakov 2014 and references therein). Interest in using the FDTD method continues to grow. The FDTD method is presently the most widely used electromagnetic computation method in lightning electromagnetic-field and surge simulations.
Other electromagnetic computation methods such as the finite-element method (FEM) (e.g., Sadiku 1989), the partial-element equivalent-circuit (PEEC) method (Ruehli 1974), the hybrid electromagnetic model (HEM) (Visacro and Soares 2005), the transmission line matrix or modeling (TLM) method (Johns and Beurle 1971), and the constrained interpolation profile (CIP) method (Takewaki et al. 1985) have been recently applied to analyzing lightning electromagnetic fields and surge simulations (e.g., Yutthagowith et al. 2009 (PEEC); Silveira et al. 2010 (HEM); Smajic et al. 2011 (FEM); Tanaka et al. 2014a (TLM); Tanaka et al. 2014b (CIP)).
In the following section, we briefly introduce each of these electromagnetic computation methods.
1.2 Overview of Existing Electromagnetic Computation Methods
1.2.1 Method of Moments
The MoM in the time domain (Van Baricum and Miller 1972; Miller et al. 1973) has been used to analyze responses of thin-wire conducting structures to external transient electromagnetic fields. The entire conducting structure is modeled by a combination of cylindrical wire segments whose radii are much smaller than the wavelengths of interest. The so-called electric-field integral equation for a perfectly conducting thin wire in air (shown in Figure 1.1) is given below, assuming that current I and charge q are confined to the wire axis (thin-wire approximation) and that the boundary condition on the tangential electric field on the surface of the wire (this field must be equal to zero) is fulfilled:
Figure 1.1 Thin-wire cylindrical segment for method of moment (MoM)-based computation. Current is confined to the wire axis, and the tangential electric field on the surface of the wire is set to zero.
(1.1)where C is an integration path along the wire axis; Einc denotes the incident electric field that induces current I; r and t denote the observation location (a point on the wire surface) and time, respectively; r´ and t´ denote the source location (a point on the wire axis) and time, respectively; R = r - r´; s and s´ denote the distance along the wire surface at r and that along the wire axis at r´, respectively; s and s´ denote unit vectors tangential to path C in Eq. (1.1) at r and r´, respectively; µ0 is the permeability of vacuum; and c is the speed of light. By numerically solving Eq. (1.1), which is based on Maxwell's equations, the time-dependent current distribution along the wire structure excited by external field is obtained.
The thin-wire time domain (TWTD) code (Van Baricum and Miller 1972), developed at the Lawrence Livermore National Laboratory, is based on the MoM in the time domain. One of the advantages of the time-domain MoM is that it can incorporate nonlinear effects such as the lightning attachment process (Podgorski and Landt 1987) or the back-flashover at an overhead-power transmission line tower struck by lightning (Mozumi et al. 2003), although it does not allow lossy ground and wires buried in lossy ground to be incorporated. Other representative applications of the time domain MoM to lightning electromagnetic-field or surge simulations are found in Moini et al. (1998, 2000), Kato et al. (1999), Kordi et al. (2002, 2003), Pokharel and Ishii (2007), and Bonyadi-Ram et al. (2008).
The MoM in the frequency domain (Harrington 1968) has been widely used in analyzing responses of thin-wire conducting structures to incident electromagnetic fields. In order to obtain the time-varying responses, Fourier and inverse Fourier transforms are employed. The electric-field integral equation derived for a perfectly conducting thin wire in air, as shown in Figure 1.1, in the frequency domain is given by
(1.2)where ? is the angular frequency, µ0 is the permeability of vacuum, and e0 is the permittivity of vacuum. Other quantities in Eq. (1.2) are the same as those in Eq. (1.1). Current distribution along the thin-wire conducting structure can be obtained by numerically solving Eq. (1.2). Note that triangular and/or rectangular patches based on a surface-current formulation could also be used in the MoM in the frequency domain.
This method allows lossy ground and wires in lossy ground to be incorporated into the model. The numerical electromagnetic codes such as NEC-2 (Burke and Poggio 1980) and NEC-4 (Burke 1992), developed at the Lawrence Livermore National Laboratory, are based on the MoM in the frequency domain. Representative applications of the MoM in the frequency domain to lightning electromagnetic-field or surge simulations are found in Grcev and Dawalibi (1990), Baba and Ishii (2000, 2003), Pokharel et al. (2003, 2004), Shoory et al. (2005, 2010), Geranmayeh et al. (2006), Pokharel and Ishii (2007), Miyazaki and Ishii (2008a, 2008b), Sheshyekani et al. (2008), Aniserowicz and Maksimowicz (2011), Khosravi-Farsani et al. (2013), and Miyamoto et al. (2015). The MoM is the second most frequently used electromagnetic computation method in lightning electromagnetic-field and surge simulations.
1.2.2 Partial-Element Equivalent-Circuit Method
The PEEC method (Ruehli 1974) provides a full-wave solution to Maxwell's equations. The method is applicable to both time (Wang et al. 2010) and frequency domains. A significant difference from the MoM is that the conductor system subject to analysis is transformed into its equivalent circuit. Although the PEEC method is based on exact field theory, it was originally developed not for electromagnetic-field computations but for the analysis of interconnect and packaging structures. In the 1990s, field retardation, external field excitation, and the treatment of dielectric materials were incorporated (Ruehli and Heeb 1992). This method has been recently employed in lightning-surge simulations (e.g., Yutthagowith et al....
