Three-Dimensional FDTD Calculation of Lightning-Induced Voltages on a Multiphase Distribution Line...

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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY 1

Three-Dimensional FDTD Calculation ofLightning-Induced Voltages on a Multiphase

Distribution Line With the Lightning Arrestersand an Overhead Shielding Wire

Akiyoshi Tatematsu, Member, IEEE, and Taku Noda, Senior Member, IEEE

Abstract—To suppress the lightning-induced voltages on a dis-tribution line, lightning arresters and/or overhead shielding wirecan be installed, and the effectiveness of these countermeasures areusually studied by simulations. Traditionally, field-to-transmissionline coupling techniques based on the distributed-parameter circuittheory are used for the calculation of the lightning-induced volt-ages. Recently, the finite-difference time-domain (FDTD) methodthat directly and numerically solves Maxwell’s equations was ap-plied to the calculation of the lightning-induced voltages. Com-pared with the conventional methods, the FDTD-based calcula-tion is advantageous in terms of the modeling of inhomogeneousground parameters, 3-D structures, and grounding systems. But,in the previous works, the distribution line was simulated simplyby a single-phase line. Moreover, the representation of lightningarresters in the FDTD method was not yet established. This paperproposes a technique to incorporate the lightning arresters in theFDTD-based lightning overvoltage calculations. In this technique,the voltage–current relationships of the lightning arresters are rep-resented by piecewise linear curves, which can be obtained directlyfrom the data sheets or measured results. For validation purpose,the lightning-induced voltages on a three-phase distribution lineequipped with the lightning arresters and a multipoint-groundedoverhead shielding wire are calculated by the proposed method,and the results are compared with those obtained by the conven-tional method and a very good agreement is found.

Index Terms—Finite-difference time-domain (FDTD) method,lightning protection, lightning-induced voltages, power distribu-tion lines.

I. INTRODUCTION

WHEN a lightning strike occurs nearby a distribution line,the electromagnetic field generated by the return-stroke

current may induce overvoltages on the distribution line [1],[2]. To suppress such overvoltages, lightning arresters and/orshielding wire are installed on a distribution line [3]–[6], andthe effectiveness of these countermeasures is usually studied bysimulations. For accurate simulation results, taking into accountthe lossy ground is one of the most important factors.

Manuscript received March 12, 2013; revised May 17, 2013; accepted June18, 2013.

The authors are with the Central Research Institute of Electric Power In-dustry, Yokosuka-shi 240–0196, Japan (e-mail: [email protected];[email protected]).

Digital Object Identifier 10.1109/TEMC.2013.2272652

Traditionally, the lightning-induced voltages are calculatedbased on the distributed-parameter circuit theory using the fol-lowing two steps [7]–[20]. In the first step, the incident electro-magnetic fields radiated by the return-stroke current are calcu-lated in the absence of the distribution line. Then, in the secondstep, telegrapher’s equations of the distribution line with embed-ded electromagnetic excitation sources obtained in the first stepare solved to find the lightning-induced voltages. In this sec-ond step, the so-called field-to-transmission coupling modelsare used [21]–[24].

Recently, the numerical electromagnetic-field calculationmethods, such as the method of moments [25]–[27], the hybridelectromagnetic model [28], the finite element method [29],the transmission-line modeling method [30], and the finite-difference time-domain (FDTD) method [31], are applied tothe calculation of the lightning-induced voltages [32]–[43]. Theapplication of these methods, in which Maxwell’s equations arenumerically solved, is quite straightforward compared with thefield-to-transmission coupling technique aforementioned, sincethe return-stroke path and the distribution line can be mod-eled simultaneously in the space of analysis. Moreover, thesenumerical electromagnetic field calculation methods have theadvantages of directly modeling nonstraight lightning channels,3-D structures, grounding systems, and nonhorizontal wires.Among these numerical electromagnetic-field calculation meth-ods, the FDTD method is known to require a longer computa-tion time and a larger memory capacity, but these requirementsare not a big issue any more due to the recent progress ofthe high-performance computers, in particular, the developmentof the general-purpose computing on graphics processing units(GPGPU). In addition, the FDTD method is advantageous, sinceit can take into account inhomogeneous electrical parameters ofthe ground, the finite conductivity and the relative permittivity,and the shapes of the ground surface such as mountains andrivers. At this moment, however, the FDTD method was appliedonly to simple simulation cases of single-phase distribution linewithout overhead shielding wires or lightning arresters, or to asingle-phase distribution line with only a shielding wire. Thus,applying the FDTD method to realistic simulation cases of thelightning-induced voltages is important research work.

In this paper, the FDTD method is applied to the calculation ofthe lightning-induced voltages under realistic conditions. First,a technique for representing the lightning arrester in the FDTDmethod is proposed which is based on the approach presented

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in [44] and [45]. In the proposed technique, a lightning arrester ismodeled by a lumped nonlinear resistor, and its voltage–currentrelationship is given by a piecewise linear curve, while the char-acteristics of the nonlinear elements were given by exponentialfunctions or polynomials in the previous works [44]–[48]. Theapplication of a piecewise linear representation enables a morestraightforward modeling of the voltage–current relationshipand makes it a simple procedure to update the electric field atthe lumped nonlinear resistor. With this technique, lightning-induced voltages on a three-phase distribution line with light-ning arresters and a multipoint-grounded overhead shieldingwire are calculated using the FDTD method. For validation pur-pose, the calculated results are compared with those obtained bya traditional method [4], [8] based on the field-to-transmissioncoupling technique.

II. TECHNIQUE FOR REPRESENTING LIGHTNING ARRESTERS IN

THE FDTD METHOD

A. Review of the FDTD Method

In the FDTD method, a rectangular-parallelepiped analy-sis space is divided into numerous smaller rectangular par-allelepipeds, which are often called “cells”. To discretize theelectric and magnetic field in the space, the electric and mag-netic fields are located on the edges and the faces of the cells,respectively. Then, Maxwell’s equations are solved by applyingthe difference method to both space and time derivatives. Thesizes of the cells are set to be smaller than the wavelength toobtain accurate results, and the time step is determined basedon Courant’s condition.

In general, the FDTD method is classified into a uniform-gridtype and a nonuniform-grid type. In the former, the cell sizesare uniform in the analysis space, while the cell sizes can bevaried arbitrarily in the latter. In this Section, the uniform-grid-type FDTD method is adopted to simplify the description of theproposed technique. Note that the proposed technique can alsobe applied to the nonuniform-grid type.

Assuming that the medium of the analysis space is isotropicand nondispersive, the equations used to update the electric andmagnetic fields alternately are derived on the basis of a leapfrogalgorithm as follows [31]:

En+1 = En +Δt

ε∇× Hn+1/2 − Δt

εΔs2 In+1/2 (1)

Hn+3/2 = Hn+1/2 − Δt

μ∇× En+1 (2)

where E,H, I , ε, μ, Δs, and Δt indicate the electric field,the magnetic field, the current, the electrical permittivity, themagnetic permeability, the space step (the size of the cell), andthe time step, respectively. The superscripts n, n + 1/2, and n+ 3/2 represent the times nΔt, (n + 1/2)Δt, and (n + 3/2)Δt,respectively.

B. Representation of the Lightning Arresters

In this Section, on the assumption that the size of the light-ning arrester is much smaller than the wavelength, a technique

Fig. 1. Representation of V–I characteristics of the lightning arrester.

for representing the lightning arrester by a lumped nonlinearresistor in the FDTD method is derived based on the techniquefor incorporating nonlinear devices such as diodes in the FDTDmethod [44], [45]. In general, this assumption is satisfied whenthe FDTD method is applied to the lightning-induced voltagecalculation. While, the voltage–current relationship (V–I) of thediode is modeled by an exponential function [44]–[46] and theoutput of an active current source simulating the diode is givenby a polynomial [47], [48] in the previous works, the V–I charac-teristics of the lumped nonlinear resistor are more directly repre-sented by several specified points here, as shown in Fig. 1. Thesepoints can be obtained from data sheets or measured results ofthe V–I characteristics. In Fig. 1, Ik and Vk represent the cur-rent and voltage at the kth specified point, respectively, and thetotal number of points is denoted by K. Note that the specifiedpoints must satisfy the conditions Vk < Vk+1 and Ik < Ik+1 .Using the piecewise linear function defined by these discretepoints, the V–I characteristics of the nonlinear resistor are givenby the following equations:

1) In case of V n+1/2 < V1

V n+1/2 = R0

(In+1/2 − I0

)+ V0 (3)

2) In case of Vk ≤ V n+1/2 < Vk+1(1 ≤ k ≤ K−3)

V n+1/2 = Rk (In+1/2 − Ik ) + Vk (4)

3) In case of VK−2 ≤ V n+1/2

V n+1/2 = RK−2(In+1/2 − IK−2) + VK−2 (5)

where Rk is given by the following equation:

Rk =Vk+1 − Vk

Ik+1 − Ik. (6)

In the aforementioned equations, the V–I characteristics forvoltages smaller than V0 and larger than VK−1 are represented bya linear extrapolation of (I0 , V0) and (I1 , V1), and (IK−2 , VK−2)and (IK−1 , VK−1), respectively.

From the aforementioned equation, the following equation isobtained:

In+1/2 =V n+1/2

Rk−

(Vk

Rk− Ik

). (7)

Substituting the relationship V n+1/2 = (En+1 + En )Δs/2into the aforementioned equation, the following equation is


TATEMATSU AND NODA: THREE-DIMENSIONAL FDTD CALCULATION OF LIGHTNING-INDUCED VOLTAGES 3

obtained:

In+1/2 =(En+1 + En )Δs

2Rk−

(Vk

Rk− Ik

). (8)

In the same way with the technique in [45], substituting (8)into (1) gives the equation used to update the electric field lo-cated at the lumped nonlinear resistor

En+1 =1 − C

1 + CEn +

Δt

(1 + C)ε(∇× H)

+Δt

(1 + C)εΔs2

(Vk

Rk− Ik

)(9)

where C = Δt/(2Rk εΔs).Since (9) is dependent only on the electric field across the

lightning arrester and the four magnetic fields surrounding thelightning arrester, more lightning arresters can be handled in thesame procedure.

C. Update of the Electric Field Located Acrossthe Lightning Arrester

Since the V–I relationship of the lumped nonlinear resistorrepresenting the lightning arrester is given by a piecewise linearfunction, the electric field across the lightning arrester is updatedusing (9) in the following simple procedure without numericalcalculation techniques such as the Newton–Raphson method.Step 1: Update each electric field using (9) with k = 0 with

the assumption that V n+1/2 satisfies the conditionV n+1/2 < V1 , then go to Step 2.

Step 2: If the calculated V n+1/2 satisfies the assumption in Step1, the calculated electric field is the correct solution.Otherwise, go to Step 3 with k = 1.

Step 3: If k = K−2, go to Step 5. Otherwise, update each elec-tric field using (9) with the value of k with the assump-tion that V n+1/2 satisfies the condition Vk ≤ V n+1/2 <Vk+1 , then go to Step 4.

Step 4: If V n+1/2 satisfies the assumption in Step 3, the cal-culated electric field is the correct solution. Otherwise,add one to k and go to Step 3.

Step 5: Update each electric field using (9) with k = K−2with the assumption that V n+1/2 satisfies the conditionVK−2 ≤ V n+1/2 , then go to Step 6.

Step 6: If V n+1/2 satisfies the assumption in Step 5, the calcu-lated electric field is the correct solution.

In this procedure, the electric field across the lightning arrestercan be obtained by calculating (9) K−1 times at the most. Fig. 2shows the flowchart for updating the electric field across thelightning arrester in the aforementioned procedure.

III. CALCULATION OF THE LIGHTNING-INDUCED VOLTAGES ON

A MULTIPHASE DISTRIBUTION LINE WITH THE LIGHTNING

ARRESTERS AND AN OVERHEAD SHIELDING WIRE

A. Calculation Arrangement

In this Section, applying the proposed technique describedearlier, lightning-induced voltages on a three-phase distribu-tion line with the lightning arresters and a multipoint-grounded

Fig. 2. Flowchart to update the electric field across the lumped nonlinearresistor representing the lightning arrester.

overhead shielding wire, similar to realistic conditions, werecalculated by the FDTD method. For validation purpose, thecalculated results were compared with those obtained by theconventional circuit-theory-based calculation method [4], [8].In the conventional method, the lightning-induced voltages werecalculated based on the Rusck model [49]. Rusck’s equations,where scalar and vector potential due to the return stroke wereobtained by the analytical formula, were solved by the differencemethod.

Fig. 3 shows the calculation arrangement used here. The vol-ume of the analysis space was 1400 m × 650 m × 600 m.The bottom surface of the analysis space was assumed to be aperfectly conducting ground plane and the other surfaces weretreated as absorbing boundaries using Liao’s formulation of thesecond-order to assume an open space [50]. A return-strokemodel and the three-phase distribution line were set up in theanalysis space. In this Section, the following four cases of thedistribution-line arrangement were considered:

1) case 1: the distribution line without the shielding or thelightning arresters;

2) case 2: the distribution line with the multipoint-groundedoverhead shielding wire and without the lightningarresters;

3) case 3: the distribution line without the shielding wire andwith the lightning arresters;



Fig. 3. Calculation arrangement of the lightning-induced voltages on a three-phase distribution line above a perfectly-conducting ground plane.

4) case 4: the distribution line with both the multipoint-grounded overhead shielding wire and the lightningarresters.

Figs. 4 and 5 represent side views of the calculation arrange-ment used in the cases 1 and 4, respectively.

The return stroke was located at a position 50 m away fromthe center of the distribution line. The return-stroke model wasvertical against the ground surface and represented by the trans-mission line (TL) model composed of a phased array of currentsources [51], and its propagation speed was set to 120 m/μs.The return-stroke current had a triangular waveform of whichwavefront time, wavetail time, and peak value are 1 μs, 50 μs,and 100 kA, respectively. The output I(z, t) of a current sourceat a position z and time t was represented by I(0, t − z/v),where v was a propagation speed and I(0, t) was a current at thelower end of the return-stroke model. Although the TL modelis applied in this paper, other engineering return-stroke modelssuch as MTLL [52] and MTLE [53], or antenna theory (electro-magnetic) model to represent a lightning channel as a wire canalso be applied to the FDTD-based surge calculation. In [35],we used the antenna theory model that included a series of in-ductances along the channel to control the propagation speed ofthe current, and showed the relationship between the inductanceand the resulting propagation speed.

The three-phase distribution line was simulated by three thinwires parallel to the ground plane. The distance between theneighboring phase lines was 0.6 m, and the radius and heightof the distribution line were 4 mm and 10 m, respectively. Thelength of the distribution line was 1400 m, and both the ends ofthe distribution line were connected to the absorbing boundariesto assume no reflection at both the ends. The three-phase lineswere called “Phases A, B, and C”, in the ascending order of thedistance from the return-stroke model. In the FDTD method, thethin wire was placed along the sides of the cells and representedby setting the values of the electric fields corresponding to thewire to zero. The radius of the distribution line was simulatedby the thin-wire representation technique proposed in [54]. Asshown in Figs. 4 and 5, a local-coordinate l was determined

on the distribution line, where l = 0 and 1400 m correspond toboth the ends of the distribution line, respectively. As mentionedpreviously, both the ends of the distribution line were connectedto the absorbing boundaries for simplicity here. However, sincelumped-parameter elements of resistors, inductors, and capaci-tors can be taken into account in the FDTD method [45], lumpedelements, for example, lumped capacitors to model transform-ers, can be easily connected to the ends and other positions ofthe distribution line.

In the cases 2 and 4, the multipoint-grounded overhead shield-ing wire with a radius of 2.5 mm was located at a position1 m higher than the distribution line conductors. The overheadshielding wire was grounded via a lumped-element resistor of20 Ω at the interval of 200 m, i.e., at the positions of l = 100,300, 500,. . ., 1300 m. In the cases 3 and 4, the lightning arresterswere installed on all the phase lines of the distribution line at thepositions of l = 100, 300, 500,. . ., 1300 m. The total numberof the installed lightning arresters was 21. The V–I characteris-tics of the lightning arresters were specified in detail using 16discrete points as shown in Fig. 6 and Table I.

As mentioned previously, the FDTD method is classified intoa uniform-grid type and a nonuniform-grid type. In the for-mer, the cell sizes are uniform in the analysis space, while thecell sizes can be varied arbitrarily in the latter. In this Section,the nonuniform-grid type was used to reduce the calculationtime and required memory capacity in the lightning-induced-voltage calculations, and the analysis space was divided into938 × 461 × 505 nonuniform cells. Although the thin-wire-representation technique has been proposed for the uniform-gridtype in [54], the sizes of the three cells adjacent to the thin wirewere kept uniform to apply the thin-wire representation tech-nique to the nonuniform-grid type [55]. In this calculation, thesizes of the three cells adjacent to the thin wires were all set to0.2 m and those of the other cells became gradually larger withdistance from the thin wires. The sizes of the other cells rangedfrom 0.3 to 2 m. The time discretization was 0.2 ns, which wasset to half of Courant’s condition to avoid numerical instabilitydue to the thin wire representation technique [56]. Note that, inthe FDTD calculation, a GPU-based parallel code was appliedto a 16-GPU(Tesla C2070)-based computer. For example, thecalculation time in the case 4 was 0.31 h.

B. Calculated Results

Under the previously described condition, we calculated thelightning-induced voltages on the distribution line for the cases1–4 by the FDTD method and the conventional circuit-theory-based calculation method [4], [8]. In the FDTD method, thelightning-induced voltages were obtained by integrating the ver-tical electric fields between the distribution line and the groundsurface. Figs. 7–10 show the lightning-induced voltages on thephase A of the three-phase distribution line at the positions ofl = 600, 400, and 200 m in the cases 1–4, respectively. Thepositions of l = 600, 400, and 200 m are 100, 300, 500 m awayfrom the midpoint of the distribution line closest to the strik-ing point, respectively. Although the calculated results of thephases B and C are not illustrated here, the waveforms of the



Fig. 4. Side view of the calculation arrangement used in the case 1 above the perfectly conducting ground plane. (a) Side view (ZX). (b) Side view (YZ).

Fig. 5. Side view of the calculation arrangement used in the case 4 above the perfectly conducting ground plane. (a) Side view (ZX). (b) Side view (YZ).

Fig. 6. V–I characteristics of the lightning arrester.

lightning-induced voltages on the phases B and C are similarto the ones on the phase A as shown in Figs. 7–10. The peakvalues of the lightning-induced voltages on the three phases A,B, and C calculated by both the methods in the cases 1–4 aresummarized in Table II.

In the case 1, the differences between the peak values of thevoltages obtained by the two methods are within 0.8%. In thecase 2, as well known, the lightning-induced voltages are sup-pressed by the overhead shielding wire in comparison to thosein the case 1. The differences between the peak values calcu-lated by the two methods at the position of l = 600 m are within1.1%. In the cases 3 and 4, the lightning-induced voltages arefurther reduced by the lightning arresters, and the peak valuescalculated by the FDTD method at the position of l = 600 magree well with those calculated by the conventional methodwithin the difference of 5.8%, but the differences between thetwo methods in the cases 3 and 4 are larger than those in the

TABLE ISPECIFIED POINTS TO REPRESENT THE V–I CHARACTERISTICS OF THE

LIGHTNING ARRESTER

cases 1 and 2. Unlike the FDTD method, in the circuit-theory-based calculation method, the vertical grounding wire betweenthe grounding resistance and the lightning arresters is modeledby a lumped-element inductor. Therefore, the main reasons ofthese larger differences in the cases 3 and 4 are considered tobe as follows. The voltage differences occur due to the dif-ferent model of the vertical wire and such voltage differencespropagate through the distribution line. In addition, the differ-ent model of the vertical wire has an influence on the surgepropagation, i.e., the reflection at the vertical wire.

From these results, the validity of the technique for represent-ing the lightning arresters and the three phase distribution line



Fig. 7. Calculated results of the lightning-induced voltages on the phase A inthe case 1 without the shielding wire or the lightning arresters above the perfectlyconducting ground plane. (a) FDTD method. (b) Conventional method.

Fig. 8. Calculated results of the lightning-induced voltages on the phase Ain the case 2 with the only multipoint-grounded overhead shielding wire abovethe perfectly conducting ground plane. (a) FDTD method. (b) Conventionalmethod.

with the multipoint-grounded overhead shielding wire using thenonuniform grid in the FDTD method is confirmed.

C. Inclusion of the Finite Ground Conductivity

In the calculation of the lightning-induced voltages, the finiteconductivity of the ground is one of the most important factors.In [35]–[37], the applicability of the FDTD method to the calcu-lation of the lightning-induced voltages in the case of the groundwith a finite conductivity has been confirmed. It is well knownthat the numerical model with the nonlinear characteristics may

Fig. 9. Calculated results of the lightning-induced voltages on the phase Ain the case 3 with the only lightning arresters above the perfectly conductingground plane. (a) FDTD method. (b) Conventional method.

Fig. 10. Calculated results of the lightning-induced voltages on the phase Ain the case 4 with both the multipoint-grounded overhead shielding wire andthe lightning arresters above the perfectly conducting ground plane. (a) FDTDmethod. (b) Conventional method.

cause the numerical instability. In this Section, in order to con-firm the numerical stability of the FDTD method applied to thepractical arrangement of the distribution line above the lossyground, we calculate the lightning-induced voltages in case ofa ground with a finite conductivity.

The calculation arrangement used in this section is as follows.The height of the analysis space shown in Fig. 3 was changedfrom 600 to 700 m, and the bottom space of the analysis spacewith a thickness of 100 m was treated as the ground with a



TABLE IISUMMARY OF THE PEAK VALUES OF THE LIGHTNING-INDUCED VOLTAGES IN

THE CASE OF THE PERFECTLY CONDUCTING GROUND

relative permittivity of 10 and a finite conductivity of 0.001 S/m.The arrangements of the distribution line, the return-strokemodel, the lightning arrester, and the shielding wire were thesame with those used in the cases 3 and 4 in the previous sec-tion. Note that the lightning arresters and the shielding wirewere not connected with the lumped-element resistor but with

Fig. 11. Side view of the calculation arrangement used in the case 4 above thefinite conductivity ground.

Fig. 12. Calculated results of the lightning-induced voltages and currents atthe lightning arrester of the phase A at the position of l = 700 m in the cases3 and 4 with the lightning arresters and the overhead shielding wire above thefinite-conductivity ground. (a) Case 3. (b) Case 4.

the grounding electrodes as shown in Fig. 11. This figure showsthe calculation arrangement used in the case 4. The volume ofthe grounding electrode was 4.2 m × 4.2 m × 8 m. When astep current with a rise time of 1 μs and a peak value of 1 Awas injected into the grounding electrode, its potential rise atthe time of 4 μs was 31.6 V. In the same way with the perfectly-conducting-ground case, the nonuniform grid was applied andthe analysis space was divided into 938 × 461 × 505 nonuni-form cells. The time discretization was set to 0.2 ns.

Under the aforementioned conditions, the lightning-inducedvoltages of the three-phase distribution line were calculated bythe FDTD method. The waveforms of the lightning-inducedvoltages across the lightning arrester and the current flowingthrough the lightning arrester installed in the phase A at theposition of l = 700 m, which is the midpoint of the distributionline, in the cases 3 and 4 are shown in Fig. 12. For comparison,



the results for the perfectly-conducting-ground plane calculatedby the FDTD method are also shown in Fig. 12. In the case ofperfectly-conducting-ground plane, the overhead shielding wirewas grounded via a lumped-element resistor of 31.6 Ω. In thecalculated results, no numerical oscillation is observed, and theapplicability of the FDTD method to the practical arrangementof distribution lines with lightning arresters and overhead shield-ing wire above the lossy ground was confirmed. It is reasonablethat the voltages are suppressed by the lightning arresters inboth the cases and the currents get larger due to the effect ofthe finite-conductivity ground in comparison with those for theperfectly-conducting ground.

IV. CONCLUSION

In this paper, in order to apply the FDTD method to the calcu-lation of lightning-induced voltages on a distribution line underthe realistic conditions, the technique to represent a lightningarrester in the FDTD method has been proposed based on thetechnique for incorporating lumped nonlinear elements such asdiodes. In the proposed technique, a lightning arrester is mod-eled by a nonlinear resistor whose voltage–current relationshipis represented directly by a piecewise linear curve. With thistechnique, lightning-induced voltages on a three-phase distribu-tion line with 21 lightning arresters and a multipoint-groundedoverhead shielding wire have been calculated using the FDTDmethod. The calculated results agree well with those obtainedby a traditional method based on the field-to-transmission linecoupling technique. Although it is known that FDTD simula-tions with nonlinear elements are prone to numerical instability,no numerical instability has been observed in the presented sim-ulation results.

The presented results demonstrate the applicability of theFDTD method to the analysis of mitigation measures againstlightning-induced overvoltages on the distribution lines un-der realistic conditions. Compared to conventional approacheswhich are based on the distributed-parameter circuit theory, theproposed approach presents a number of advantages, namelythe straightforward taking into account of the ground electricalproperties (e.g., inhomogeneities), the presence of 3-D nearbystructures (towers, buildings), the presence of grounding sys-tems, nonvertical lightning channel, etc.

ACKNOWLEDGMENT

The authors would like to thank Prof. F. Rachidi from theSwiss Federal Institute of Technology, Lausanne, Switzerlandfor his invaluable comments on this paper.

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Akiyoshi Tatematsu (M’04) was born in Aichi,Japan, in 1975. He received his B.S., M.S., andPh.D. degrees in electrical engineering from KyotoUniversity, Kyoto, Japan in 1999, 2001, and 2004,respectively.

He has been with the Central Research Instituteof Electric Power Industry, Yokosuka, Japan, since2004. He was a Postdoctoral Fellow at the SwissFederal Institute of Technology, Lausanne, Switzer-land from 2012 to 2013. He has been engaged in thestudy of electromagnetic field calculation and surge

analysis.Dr. Tatematsu is a member of the IEE of Japan.

Taku Noda (M’97–SM’08) was born in Osaka,Japan, in 1969. He received the B. Eng., M. Eng.,and Ph.D. degrees from Doshisha University, Kyoto,Japan in 1992, 1994, and 1997, respectively.

In 1997, he joined the Central Research Instituteof Electric Power Industry (CRIEPI), Tokyo, Japan.He is currently a Senior Research Scientist with theElectric Power Engineering Research Laboratory ofCRIEPI, Yokosuka, Japan. He also serves as the Edi-tor of the IEEE TRANSACTIONS ON POWER DELIVERY

and the Lecturer at the Shibaura Institute of Technol-ogy, Tokyo, Japan. He was a Visiting Scientist at the University of Toronto,Toronto, ON, Canada from 2001 to 2002 and was an Adjunct Professor atDoshisha University from 2005 to 2008. His research interest includes electro-magnetic transient analysis of power systems and numerical electromagneticfield computations for surge analysis.

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Three-Dimensional FDTD Calculation of Lightning-Induced Voltages on a Multiphase Distribution Line...

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