Calculation of Lightning-Induced Overvoltages On

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Calculation of Lightning-Induced Overvoltages onOverhead Lines Based on DEPACT Macromodel

Using Circuit Simulation SoftwareXin Liu, Xiang Cui, Senior Member, IEEE, and Lei Qi

Abstract—One of the simple and accurate methods for calcu-lating the lightning-induced overvoltages is based on the Agrawalet al. field-to-wire model, in which the coupling mechanisms arepresented by distributed voltage sources along the lines. There isa difficulty in calculating the lightning-induced overvoltages withthe built-in transmission line models or other circuit elements ofvarious circuit simulation software such as the PSCAD/EMTDC,EMTP/ATP, PSpice, etc., as the distributed voltage sources dueto the horizontal component of the electric field caused by thelightning channel are in series with the line and not in between twotransmission-line segments. In this paper, a simple circuit approachfor efficient calculation of the lightning-induced overvoltages usingthe circuit simulation software is proposed. The delay extractionpassive compact circuit (DEPACT) macromodel is applied and thetransmission line is divided into a cascade of DEPACT subnetworkswhich can be represented by a cascade of two lossless transmissionlines and a lossy network. By assuming the incident field only cou-ples with the lossless transmission line sections, the distributedequivalent voltage sources due to the horizontal component of theincident electric field can be lumped at the termination of the lines,which makes it convenient for the engineering technologist to cal-culate the lightning-induced overvoltages in various circuit simula-tion software. Furthermore, in order to treat the nonlinear elementsuch as metal oxide arrester (MOA) directly, the DEPACT macro-model is improved and the macromodel of multiconductor trans-mission lines excited by the external electromagnetic field can besolved in the actual phase domain instead of in the modal domain.Compared with the existing method for calculating the lightning-induced overvoltages, the benefit of the proposed algorithm is thatit can be used to calculate the transients of the transmission linemore efficiently. Considering the PSCAD/EMTDC is a powerfulsimulation software which providing lots of equipment models inpower system, such as transformer, generator, and protection mod-ules et al., an implementation procedure in the PSCAD/EMTDC isprovided in this paper, which makes it more convenient for the elec-trical engineering technologists to analyze the lightning-inducedovervoltages problems in the power system. Several numerical cal-culations of the line transient responses are provided and the CPUtime is compared, which indicate the validity and efficiency of the

Manuscript received December 16, 2010; revised May 3, 2011 and June 8,2011; accepted June 23, 2011. This work was supported in part by the NationalNatural Science Foundation of China under Grant No. 51177048 and 50707008,the Fundamental Research Funds for the Central Universities (11MG36 and09TG01), and the Fundamental Research Funds for the Hebei Province Univer-sities under Grant No. Z2011220, respectively.

X. Liu is with the Department of Electrical Engineering, North China ElectricPower University, Baoding, China (e-mail: [email protected]).

X. Cui and L. Qi are with the School of Electrical and Electronic Engineering,North China Electric Power University, Beijing, Beijing 102206, China (e-mail:[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TEMC.2011.2175230

algorithm proposed in the paper. At last, this algorithm is appliedto analyze the influence of the ground wire and the MOA on thelightning-induced overvoltages of the overhead lines.

Index Terms—DEPACT macromodel, electromagnetic coupling,frequency-dependent parameter, lightning-induced overvoltages,multiconductor transmission line, nonlinear circuits, transientresponse.

I. INTRODUCTION

L IGHTNING induced overvoltages may cause damage tothe insulation of the distribution lines or other power equip-

ments. Eriksson et al. have evidenced that the indirect lightningreturn strokes, hitting the ground in the vicinity of overheadlines, constitute a more dangerous cause of damage than directstrikes, because of their more frequent occurrence [1].

Several theoretical models have been introduced in orderto estimate the response induced by external electromagneticfields [2]–[6]. One of the most popular, simple, and accu-rate field-to-wire coupling models for studying the lightning-induced overvoltages (LIOV) is presented by Agrawal et al. [4].In this model, the classical telegrapher’s equations are modified,as a consequence, the equivalent distributed voltage sources dueto the external electromagnetic fields caused by the lightningreturn stroke channel are added to the voltage telegrapher’sequation [4].

Circuit simulation software such as PSCAD/EMTDC,EMTP/ATP, and PSpice et al. has been widely used by theelectrical engineering technologists to study the transients ofthe power system. However, as for the LIOV problem, there isa difficulty in applying the Agrawal et al. model directly withthe built-in transmission line or other circuit models of variouscircuit simulation software, as the voltage source due to thehorizontal component of the electric field caused by the light-ning return stroke channel in the Agrawal et al. model can notbe inserted in series with the line impedance as the impedanceand admittance parameter of the line are embedded within aninaccessible multiport network [7].

Several studies have been carried out to calculate thelightning-induced overvoltages of the overhead line based onthe circuit simulation software. A model of overhead lines ex-cited by electromagnetic fields from a lightning channel has beenestablished in the EMTP/ATP [8], [9], using the MODELS lan-guage in ATP. Another separate lightning-induced overvoltagecode (LIOV-EMTP code) written in a high-level programminglanguage was used for modeling the coupling phenomena, and

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2 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY

the calculated output voltages/currents were then linked to theATP-EMTP externally [10].

However, compared with the upper methods which require theengineering technologists to master the complicated MODELSlanguage or the interface of the high-level programming lan-guages to the ATP-EMTP, the technologists are more conversantwith using and implementing the built-in circuit models/setupsfor transient analysis in the circuit simulation software.

In order to deal with the lightning-induced overvoltages prob-lem using the built-in circuit models/setups in the circuit sim-ulation software, a methodology is proposed by Raul Montanoet al., which allows the technologists to implement the Agrawalet al. model with the available built-in models/setups [7]. How-ever, in this approach, the implementation procedure is some-what complicated, as the distributed voltage source due to thelightning return stroke channel should be inserted into everyconnection node of the transmission lines. When the overheadlines are as long as several kilometers, the transmission linesshould be divided into many segments to obtain the accurateresults because the rise time of the exciting source is only sev-eral microsecond. Especially when a local distributed networkcontaining many lines is studied, inserting a voltage source anda current source at each connection node of the segments willresult in a tedious work. Shinh et al., presented a SPICE macro-model for transient analysis of lossy multicondutor transmissionlines in the presence of incident electromagnetic fields [11].However, the modal presented in [11] is solved in the modaldomain, as for the nonlinear problem, a representation in theactual phase domain is preferred.

To make a more convenient and efficient method to calcu-late the lightning-induced overvoltages using the circuit sim-ulation software, a methodology is presented in this paper.In this method, the delay extraction passive compact circuit(DEPACT) macromodel [12] [13] is adopted because it results ina significantly lower-order macromodel for long lossy-coupledlines, leading to a fast transient simulation. Then, the DEPACTmacromodel is extended to calculate the lightning-induced over-voltages, and is improved to be solved in the phase domain,which makes the treatment of nonlinear problem directly.

As for a lightning-induced overvoltages problem, the trans-mission line is divided into a cascade of DEPACT subnetworkswhich is represented by a cascade of two lossless transmissionlines and a lossy network, what is important is that only severalsegments will provide an accurate result even if a long transmis-sion line is studied. Different from the method of Raul Montanoet al., the distributed equivalent voltage sources due to the hori-zontal component of the incident electric field can be treated andlumped only at the termination of the segmentations, which pro-vides a more convenient way to calculate the lightning-inducedovervoltages in the circuit simulation software. Compared withthe method proposed by Shinh et al., the model presented inthis paper can be solved in the phase domain other than themodal domain, which simplified the treatment of the nonlinearproblems.

The rest of the paper is organized as follows: Section II givesa brief review about the DEPACT macromodel and Agrawalet al. field-to-wire coupling model. Section III describes the

Fig. 1. DPEACT macromodel.

Fig. 2. DPEACT cell.

algorithm for calculating the lightning-induced overvoltagesbased on the DEPACT macromodel in detail. Section IV pro-vides three examples to verify the proposed algorithm and twosimulations are implemented and discussed in Section V.

II. REVIEW OF THE DEPACT MACROMODEL AND THE

AGRAWAL et al. MODEL

A. DEPACT Macromodel

DEPACT macromodel [12], [13] is an algorithm proposedrecently for transient analysis of transmission line interconnectnetworks. Based on the modified Lie product formula, the expo-nential stamp which denotes the transfer function of the trans-mission line can be split into multiple sections, which is shownas (1)

e(A+sB)·l ≈M∏

k=1

e(sB/2M )·le(A/M )·le(sB/2M )·l (1)

where A =[

0 −R−G 0

], B =

[0 −L

−C 0

], R, G, L, and C are

the resistance, conductance, inductance, and capacitance matri-ces parameters per-unit-length (p.u.l) of the transmission lines.

The product terms of (1) can be viewed as a cascade of Msubnetworks as shown in Fig. 1. These subnetworks are referredto DEPACT cells, which can be represented by a cascade of twolossless transmission lines and a lossy network. The structure ofa certain DEPACT cell is shown in Fig. 2, in which the transferfunctions of the lossless transmission line and the lossy networkare e

s B2 M ·l and e

AM ·l , respectively.

The precision of this algorithm has been studied in [12],[13], and a conclusion is obtained that this macromodel of sig-nificant fewer segmentations compared to the circuit using con-ventional lumped segmentations can provide accurate results. Inthis paper, this macromodel is adopted and improved to calculatethe lightning-induced overvoltages of the overhead transmissionlines.

B. Agrawal et al. Model

One of the simple and accurate field-to-wire models forcalculating the lightning-induced overvoltages is the Agrawal


LIU et al.: CALCULATION OF LIGHTNING-INDUCED OVERVOLTAGES ON OVERHEAD LINES BASED ON DEPACT MACROMODEL 3

Fig. 3. Geometry of an N + 1 conductor transmission lines system.

et al. model [4]. Consider an N + 1 conductor transmission linesystem in Fig. 3, the equations of the Agrawal et al. model [4]considering the frequency-dependent parameters in the incidentfields are

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

d

dxUs(x) + RI(x, s) + sL · I(x) + Zg (s) · I(x, s)

= Eincx (x, h)

d

dxI(x) + GUs(x)+sC · Us(x) = 0

(2)

where Us(x) and I(x) are N × 1 vectors of the line scattervoltage and total current at location x in the frequency-domain,Einc

x (x, h) is a N × 1 vector of the horizontal component of theincident electric fields along the x axis at the conductor’s heighth, R, G, L, and Care the resistance, conductance, inductance,and capacitance matrices parameters p.u.l of the lines, and Zg (s)denotes the frequency-dependent lossy ground impedance ma-trix [14].

In order to obtain the total voltage U(x), it is necessary to addthe voltage due to the vertical component of the electric fieldbetween the reference and the ith conductor to the scatteredvoltage, and the equation is given by

U(x) = U s(x) −∫ h

0Einc

z (x, z) · dz (3)

where Eincz (x, z) is the vertical component of the incident elec-

tric field and h is the height of the line.

III. DERIVATION OF AN IMPROVED DEPACT MACROMODEL

FOR LIGHTING-INDUCED OVERVOLTAGES CALCULATION IN

CIRCUIT SIMULATION SOFTWARE

A. DEPACT Macromodel Representation of the Agrawal et al.Model

Based on the DEPACT macromodel, a transmission line canbe divided into M DEPACT cells which can be representedby a cascade of two lossless transmission lines and a lossynetwork [12], [13].

Assuming the incident field only couples with the losslesstransmission line sections, the equivalent circuit according tothe Agrawal et al. model is shown in Fig. 4. The coupling

Fig. 4. DEPACT macromodel representation of the Agrawal et al. model.

between the horizontal component of the electric fields andthe lines is expressed by a series of distributed voltage sourcesalong the lossless transmission line. As seen from Fig. 4, thelossless transmission line sections should be divided into severalsegments in order to insert the voltage source at the connectionnode of each segment, the procedure is somewhat complicated.However, this problem is solved in the Section III-C.

It should be mentioned, if the influence of the finitely conduct-ing ground is considered, the Cooray–Rubinstein formula [15],[16] can be used to calculate the horizontal component of theelectric fields. For the vertical component of the electric field,several authors have shown that it can be calculated with rea-sonable approximation assuming the ground as a perfect con-ductor [14].

B. Treatment of the Lossless Section of a DEPACT Cell

The telegrapher’s equations of the lossless transmission lineilluminated by the external field can be written as

⎧⎪⎪⎨

⎪⎪⎩

dUs(x)dx

+ jωL · I(x) = Eincx (x, h)

dI(x)dx

+ jωCUs(x) = 0.

(4)

Using the similarity transformation, the scatter voltage andtotal current vector Us and I(x) in the phase domain can beexpressed as

Us(x) = Tv Usm (x), I(x) = Ti Im (x) (5)

where Usm and Im (x) are propagating modal scatter voltage and

current vector, respectively, while Tv and Ti are voltage andcurrent transform matrix, respectively. Correspondingly, (4) canbe decoupled and rewritten as

⎧⎪⎪⎨

⎪⎪⎩

−dUsm (x)dx

= jωLm · Im (x) + VF m (x)

−dIm (x)dx

= jωCm · Um (x)

(6)

where

{Lm = T−1

v LTi

Cm = T−1i CTv

, VF m (x) = T−1v Einc

x (x). Lm and

Cm are both diagonal matrices which indicate that there is nocoupling between every modal variables.

Taking a certain modal for example, the corresponding equa-tions are⎡

⎢⎢⎣

dUsm (x)dx

dIm (x)dx

⎤

⎥⎥⎦=[

0 −jωLm

−jωCm 0

][U s

m (x)

Im (x)

]+

[VF m (x)

0

]

(7)



and the solution to (7) can be written as

[U s

m (x)

Im (x)

]=

[φ11(x − x1) φ12(x − x1)

φ21(x − x1) φ22(x − x1)

] [U s

m (x1)

Im (x1)

]

+[

VF T m (x)

IF T m (x)

](8)

where U sm (x1) and Im (x1) denote the modal scatter voltage and

total current at the sending port of the line

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

φ11(x − x1) = cosh γm (x − x1)

φ12(x − x1) = − sinh γm (x − x1) · Zcm

φ21(x − x1) = − sinh γm (x − x1) · Z−1cm

φ22(x − x1) = cosh γm (x − x1)

[VF T m (x)

IF T m (x)

]=

⎡

⎢⎢⎢⎣

∫ x

x1

φ11(x − x′)VF m (x′)dx′

∫ x

x1

φ21(x − x′)VF m (x′)dx′

⎤

⎥⎥⎥⎦

and γm =√

Lm Cm is the propagation constant of the mthmodal.

By setting x = x2 , and δ = x2 − x1 , (8) can be modified as

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Im (x1) − U sm (x1)/Zm = e−γ δ

[Im (x2) − Um (x2)

Zm

]

−e−γ δ

[IF T m (x2) − VF T m (x2)

Zm

]

Im (x2) + U sm (x2)/Zm = e−γ δ

[Im (x1) + Um (x1)

Zm

]

+

[IF T m (x2) + VF T m (x2)

Zm

]

(9)

where U sm (x2) and Im (x2) denote the modal scatter voltage and

total current at the receiving port of the line.Converting (9) into the time-domain, we obtain the following:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

im (x1 , t) −us

m (x1 , t)Zm

= im (x2 , t − τm ) − usm (x2 , t − τm )

Zm

+∫ x2

x1

VF m

(x, t − x − x1

vm

)dx/Zm

im (x2 , t) +us

m (x2 , t)Zm

= im (x1 , t − τm ) +us

m (x1 , t − τm )Zm

+∫ x2

x1

VF m (x, t − (x2 − x/vm ))dx/Zm

(10)

where vm = 1/√

Lm Cm , τm = x2 −x1vm

, and Zm =√

Lm /Cm .

Considering all of the modals, the corresponding equationscan be written as[

imi(x1 , t)

−imi(x2 , t)

]=

[Z−1

m 0

0 Z−1m

][us

mi(x1 , t)

usmi(x2 , t)

]+

[0 −1

−1 0

]

×[

imi(x1 , t − τm )

−imi(x2 , t − τm )

]+

[0 −Z−1

m

−Z−1m 0

]

×[us

mi(x1 , t − τm )

usmi(x2 , t − τm )

]+

[Z−1

m 0

0 −Z−1m

]

×

⎡

⎢⎢⎢⎣

∫ x2

x1

VF m

(x, t − x − x1

vm

)dx

∫ x2

x1

VF m

(x, t − x2 − x

vm

)dx

⎤

⎥⎥⎥⎦. (11)

It is should be mentioned that these lossless sections can besolved in the modal domain using (11) in [12], [13], but asfor the nonlinear problems, a representation in the actual phasedomain is preferred, so that an improved work has been done inthe following.

Using the same transformation matrix Ti and Tv , (11) canbe transformed back to the phase domain, correspondingly, theequations can be written as

[i(x1 , t)

−i(x2 , t)

]=

[TiZ−1

m T−1v 0

0 TiZ−1m T−1

v

][u(x1 , t)

u(x2 , t)

]

+

[0 −1

−1 0

][i(x1 , t − τm )

−i(x2 , t − τm )

]

−[

0 TiZ−1m T−1

v

TiZ−1m T−1

v 0

]

×[u(x1 , t − τm )

u(x2 , t − τm )

]+

[TiZ−1

m 0

0 −TiZ−1m

]

×

⎡

⎢⎢⎢⎣

∫ x2

x1

VF m

(x, t − x − x1

vm

)dx

∫ x2

x1

VF m

(x, t − x2 − x

vm

)dx

⎤

⎥⎥⎥⎦ . (12)

By defining

Y =

[TiZ−1

m T−1v 0

0 TiZ−1m T−1

v

](13a)

[is(x1 , t)

is(x2 , t)

]=

[0 −1

−1 0

] [i(x1 , t − τm )

−i(x2 , t − τm )

]

−[

0 TiZ−1m T−1

v

TiZ−1m T−1

v 0

]

×[us(x1 , t − τm )

us(x2 , t − τm )

](13b)



Fig. 5. Equivalent circuit of the lossless segment in circuit simulation software(three conductor transmission line).

[iincs (x1 , t)

iincs (x2 , t)

]=

[TiZ−1

m 0

0 −TiZ−1m

]

×

⎡

⎢⎢⎢⎣

∫ x2

x1

VF m

(x, t − x − x1

vm

)dx

∫ x2

x1

VF m

(x, t − x2 − x

vm

)dx

⎤

⎥⎥⎥⎦ (13c)

(12) can be realized by the Norton representation of an ac-tive multiport circuit, in which Y is the shunt admittance, and[is(x1 , t)is(x2 , t)

]and

[iincs (x1 , t)iincs (x2 , t)

]are the equivalent current sources.

As seen from Y, there is no coupling between the sending portand the receiving port, and the circuit of each port can be re-

alized using the network synthesis technology.

[is(x1 , t)is(x2 , t)

]is

the equivalent current source vector related to the voltages andcurrents at the opposite end of the line and at a certain time

delay earlier,

[iincs (x1 , t)iincs (x2 , t)

]is the exciting current source vector

related to the horizontal component of the electric field causedby the lightning return stroke channel. What should be men-tioned is that the influence of the external electromagnetic fieldis lumped at the end of the lines by the integral in (13c), so thesegmentation of the lossless transmission line is unnecessary,which is different from the method in [7]. Furthermore, becausethe segmentations of the entire line using DEPACT macromodelare fewer than the circuit using conventional lumped segmen-tations, the implementation procedure will become much moreconvenient and the transient simulation will be efficient.Taking athree-conductor transmission lines system for example, a mul-tiport equivalent circuit according to (12) is shown in Fig. 5where the resistance Rc11 , Rc22 , Rc33 , Rc12 , Rc23 , and Rc13can be calculated by TiZ−1

m T−1v , is,i(x1 , t) and is,i(x2 , t) are

controlled current sources whose outputs are related to the volt-age and current at the opposite end of the line and at a certaintime delay earlier at the sending and receiving port, respectively.iincs,i (x1 , t) and iinc

s,i (x2 , t) are current sources related to horizon-tal component of the electric fields caused by the lightningreturn stroke, which can be calculated by (13c) using numericalcomputation.

C. Treatment of the Lossy Section of a DEPACT Cell Consid-ering the Frequency-Dependent Ground Return Impedance

Because the transfer function of the lossy section in eachDEPACT cell is e(A/M )·l , the corresponding equations can bewritten as

⎧⎪⎪⎨

⎪⎪⎩

dUs(x)dx

+ R · I(x) + Zg · I(x) = 0

dI(x)dx

+ G · Us(x) = 0

(14)

where R is the resistance matrix parameter per-unit-length ofthe lines, and Zg denotes the frequency-dependent lossy groundimpedance matrix which is composed of the self impedanceZg,ii and mutual impedance Zg,ij . One of the most simple ap-proximate expressions for the self ground impedance Zg,ii wasproposed by Sunde and is given by the following logarithmicfunction [14]

Zg,ii =jωμ0

2πln

(1 + γghi

γghi

)(15)

where γg =√

jωμ0(σg + jωεrg ε0) in which σg and εrg arerespectively the ground conductivity and relative permittivity.The expression for the mutual ground impedance Zgij betweentwo overhead transmission lines can be derived and is givenby [14]

Zg,ij =jωμ0

4πln

((1 + γg (hi + hj/2))2 + (γg (rij /2))2

(γg (hi + hj/2))2 + (γg (rij /2))2

).

(16)The p.u.l conductance matrix parameter G can be neglected be-cause the leakage current is very small for the overhead lines,that is to say, the currents injecting into the sending port areequal to the currents flowing out of the receiving port. Corre-spondingly, the voltage equations of (14) can be representedas (17) by replacing the differential terms with the differenceterms.

Us(x1) − Us(x2) = (RΔx + ZgΔx) · I(x) (17)

where Δx is the length of the DEPACT cell.In order to realize the equivalent circuit of (17), vector fitting

(VF) method [17], [18] and circuit synthesis technology are bothused. First, the ground impedance is approximated as a rationalfunction by the VF method and all the elements of Zg can bewritten as

Zg,ij = Rdc,ij +N∑

f =1

sCij,f

s − Pij,f(18)

where Rdc,ij denotes the direct current resistance and Pij,f

and Cij,f are the pole and numerator polynomial coefficientof (Zg,ij − Rdc,ij ). The elements on the diagonal line of Zg

are related to the self-impedances, which can be modeled by aseries of subnetworks represented by a direct current resistancein series with several networks with same structure. Each ofthese networks is composed of a resistance paralleling with aninductance, in which the value of the resistance is Rii,f = Cii,f

and the value of the inductance is Lii,f = −(Cii,f /Pii,f ). The



Fig. 6. Equivalent circuit of the lossy section in circuit simulation software(three conductor transmission lines). (a) Equivalent circuit representation of thelossy section. (b) Equivalent circuit to obtain the output of the current controlledvoltage source.

elements on the off-diagonal line of Zg denote the couplingeffects between the conductors, which can be modeled by acurrent controlled voltage source while the controlling branchescan also be treated as the way of the diagonal elements. Theequivalent circuit of a three-conductor transmission lines systemaccording to (17) is shown in Fig. 6.

D. Time-Domain Equivalent Circuit for Calculatingthe Lightning-Induced Voltage

Connecting the lossless sections and the lossy sections inaccordance with Fig. 4, a cell circuit describing the Agrawalet al. model can be obtained, which is shown in Fig. 7. Then,the time-domain equivalent circuit for calculating the lightning-induced voltage can be realized by connecting this kind of cellsone by one.

E. Determination of the Segmentation Number

It is obvious that the precision will be improved with theincrease of the segmentation number M , however, the compu-tational amount will be also increased. In order to provide a basisfor the determination of the segmentation number, the relativeerror of the DEPACT macromodel can be estimated by [13]

‖ε‖ =

∥∥∥e(A+j2πfm a x B) −∏M

k=1 Ψk

∥∥∥∥∥e(A+j2πfm a x B)

∥∥ × 100% (19)

where Ψk = es B2 M e

AM e

s B2 M and fmax is the maximum frequency

considered in the transient simulation, which is dependent onthe shortest rise time tr (or decay time tf ) of the excitations. Thecalculating formula of fmax is given by [13]

fmax =0.35

tr (tf ). (20)

Using (19), a diagram describing the relation between the rela-tive error and the segmentation number M can be plotted, andthen the segmentation number can be determined according tothe required accuracy.

IV. METHOD VALIDATION

This section presents numerical examples to demonstratethe computational aspects of the proposed algorithm. ThePSCAD/EMTDC circuit simulation software is adopted forcalculating the lightning-induced overvoltages of the overheadtransmission line based on the method proposed in this paper.In order to demonstrate the accuracy and efficiency of the pro-posed method, the methods used in [19]–[22] is adopted, whichare the most widespread methods for solving the lightning-induced overvoltages problems, meanwhile, in these methods,the FDTD scheme is adopted to solve the equations of Agrawalet al.’s model. In the following of this section, three examples areconsidered and they are performed on a PC with an INTEL P42.4-GHz CPU. The first example deals a lightning-induced volt-age calculation with a single transmission line above a perfectconducting ground. In the second example, a three-conductoroverhead lines system above a perfect conducting ground is ana-lyzed. In the third example, the influence of the lossy frequency-dependent ground is considered, and the lightning-induced volt-ages are calculated. The methods for calculating the electricfields are the same as the existing methods in [15], [16], [20]for each example (analytical formula for the perfect conduct-ing ground case [20] and Cooray-Rubinstein formula for theloosy ground case [15], [16]), then the lumped current sourcescan be obtained by the numerical computation of the integral in(13c), in which only some summation, time delay, and low ordermatrix multiplication operation are used, so their CPU time isnot considered. In a word, the CPU time for the electric fieldsand lumped sources calculation is not considered and only thetransient computation time is compared.

A. Example 1

In this example, the line is 1 km long, 10 m high and is ter-minated by its characteristic impedance, which is shown inFig. 8. A perfect conducting ground is assumed. The return-stroke model adopted for the presented study is the modifiedtransmission line model, proposed and tested by the authorsin [14], [20]–[22], in which the lightning current is allowed todecrease with the height while propagating the channel upward,the current distribution along the channel can be expressed by

⎧⎪⎪⎨

⎪⎪⎩

i(z′, t) = i

(0,

t − z′

v

)exp

(−z′

λ

)t ≥ z′

v

i(z′, t) = 0t ≤ z′

v

(21)

where v is the return stroke velocity and is assumed to be 1.3 ×108 m/s, and λ is the decay constant which takes into account theeffect of the vertical distribution of charge stored in the coronasheath of the leader and subsequently discharge during the returnstroke phase, its value has been determined to be in the rangeof 1-2 km [20], assumed to be 1.7 km in this paper. The striking



Fig. 7. Equivalent circuit representation for the calculation of lightning-induced voltage.

Fig. 8. Vertically configured three-phase line for analysis.

point is considered equidistant from the line terminations and adistance of 50 m from the line center. The channel-base returnstroke current has a peak value of 12 kA and a maximum time-derivative of 40 kA/μs [20], typical of subsequent return strokes.

The horizontal component of the electric fields at the locationof 50-300 m from the line center caused by the lightning returnstroke channel are calculated, which is shown in Fig. 9. Theshortest rise time of the result is about 1 μs, so a 0-1 MHzfrequency band should be considered. Using (19), the relativeerror diagram can be plotted, which is shown in Fig. 10. Wewill find out that the relative error is about 0.66% when the lineis divided into four segments, that is to say, the length of eachsegment shorter than 250 m will give a good accuracy. In thisexample, the 1-km-long transmission line is divided into foursegments. But as for the existing methods, the length of eachsegment should be shorter than 10 m according to the rule ofwave propagation, so that the 1-km-long line will be split intoas least 100 sections, which is much larger than the proposedmethod. Therefore, the method proposed in this paper is moreefficient and convenient.

In this example, the electric fields along the transmission lineare calculated with an interval Δx of 10 m, and the time stepΔt is adopted as 0.01 μs according to their rise time, as a result,the segmentation for the electric field calculation along a 1-kmtransmission line is 100 while the segmentation for the induced

Fig. 9. Horizontal component of the electric fields at different locations causedby the lightning return stroke channel.

Fig. 10. Relative error of DEPACT model.

voltage calculation is 4. The current sources according to the ex-ternal fields can be calculated and lumped at the terminations ofthe lossless lines in each section, with a numerical computationof the integral in (13c).The transient voltages at the terminal endof the line using the proposed method are shown in Fig. 11 andthe result obtained from [20] is also provided with the dottedline of blue color. As we can see, the results are in excellentagreement. The transient computation time is compared, andthe FDTD method adopted in [20] to solve the Agrawal et al.model takes 0.347 s while the proposed method takes 0.115 s,thus providing a speed-up of 3.



Fig. 11. Comparison of lightning-induced voltage of line at terminal end.

Fig. 12. Geometry used for calculation of voltages induced by a lightningreturn stroke on multiconductor overhead lines.

B. Example 2

In this example, three transmission lines are placed at the dif-ferent heights above a perfect conducting ground which has beenused in other studies on the subject [21], as shown in Fig. 12.The radius of each conductor is 9.14 mm and the length of thelines is assumed to be 1 km. Each conductor is terminated witha resistance equal to its characteristic impedance determined inthe absence of the other conductors. The return-stroke modeland the peak of the channel-base current adopted are the sameas example 1.

The lightning induced electric fields along the transmissionline are calculated by the method in [20] with Δx = 10 m andΔt = 0.01 μs. Using the rectangular numerical integral, time de-lay and low order matrix operations, the lumped current sourcesdescribing by (13c) can be obtained. Based on the proposedmethod, by dividing all of the three 1-km-long lines dividedinto four segments and connecting the lumped current sourcesat the corresponding port terminations as Fig. 7, the lightninginduced overvoltages can be calculated. Fig. 13 shows the in-duced voltages calculated using the proposed method at theline extremities for a stroke location 50 m from the line center

Fig. 13. Lightning-induced voltage at the terminal end of three-line conductorsof a vertically configured three-phase line (solid lines: proposed method; dottedlines: from [21]).

Fig. 14. Three-phase overhead line configuration for the calculation oflightning-induced voltages.

and equidistant to the line terminations, while the correspond-ing results obtained from [21] are the dotted lines shown inthe same figure. The max deviation of the peak value of allphase’s voltage is 0.72 kV, about 0.75% of the highest line’svoltage’s peak value, which demonstrates the accuracy of theproposed method for dealing with multiconductor transmissionlines (MTLs). In addition, simulating the field-to-circuit modelusing FDTD method adopted in [21] takes 6.28 s (only the tran-sient computation time), while the propose method takes only0.38 s, which indicates that the proposed method becomes moreefficient when MTLs are considered.

C. Example 3

In this example, the lightning-induced voltage of a 1 km-longhorizontally-configured three-phase line is studied, whose con-figuration is the same as [22], which is shown in Fig. 14. Theconductors are located at the same height above ground h =10 m and each conductor is terminated on a resistance equal toits characteristic impedance determined in absence of the otherconductors. The influence of the lossy ground is considered, theground conductivity is 0.001 S/m and ground relative permittiv-ity is assumed to be equal to 10. The return-stroke model and



TABLE ICONSTANTS, NUMERATOR POLYNOMIAL COEFFICIENTS OF THE ELEMENTS OF

THE GROUND IMPEDANCE

the peak of the channel-base current adopted are the same asexample 1 and 2. The lighting stroke is located at 50 m fromphase B and equidistant to the line terminations.

Taking advantage of the VF method, the self and mutualground impedances can be approximated by the following ex-pression with the same poles

Zg,ij = Rdc,ij +N∑

f =1

sCij,f

s − Pij,f(22)

where s is the complex frequency variable of Laplace trans-formation, Rdc,ij is the constant term, Cij,f and Pij,f are thef th numerator polynomial coefficient and pole, respectively.In this example, all of the elements of the ground impedancematrix are approximated by (22) with N = 8, and the corre-sponding coefficients of (22) are listed in Table I. Taking theself ground impedance of phase A (Zg,AA ) for example, themagnitude-frequency characteristics and the phase-frequencycharacteristics are shown in Fig. 15, using both (15) and theapproximation expression (22). As we can see, they representan excellent agreement.

It should be mentioned, for a certain MTL system, all of theelements of the ground impedance matrix should have the samepoles, so the poles are listed in a specific table individually.Furthermore, because the height of the three phase lines are thesame with each other, we will get Zg,AA = Zg,BB = Zg,C C

and their constant term Rdc,ij , numerator polynomial coeffi-cients Cij,f are same with each other, that is why we have listedthe Rdc,ii and Cii,f in the same column of Zg,AA , Zg,BB andZg,C C . As for Zg,AB , Zg,BA , Zg,BC , Zg,C B and Zg,AC , Zg,C A ,because the symmetry of the ground impedance matrix, themutual impedance of every two lines are equal to each other(Zg,AB = Zg,BA , Zg,BC = Zg,C B , Zg,AC = Zg,C A ), so theircorresponding constant term Rdc,ij , numerator polynomial co-efficients Cij,f are listed in the same column of Zg,ij .

Using the method described in Section III-C, the equivalentcircuit can be obtained whose network structure is the same as

Fig. 15. Magnitude-frequency characteristics and phase-frequency charac-teristics of the self-ground impedance (Zg ,A A ) of phase A. (a) Magnitude-frequency characteristics. (b) Phase-frequency characteristics.

Fig. 6 and the corresponding p.u.l circuit parameters are shownin Table II.

The Cooray–Rubinstein formula [15], [16] is used to calculatethe electric fields along the lines in order to consider the effectof loosy ground, and the space interval and the time step are10 m and 0.01 μs, respectively. The current source exciting thetransmission lines in each section is calculated using (13c), in thesame way as example 2. By setting the length of each segmentequal to 250 m, the 1 km long transmission lines are all dividedinto 4 segments, while the segmentations of the existing methodare 100. The induced voltage at the terminal end of phase Ais calculated using the proposed method and shown in Fig. 16together with the result from [22]. The deviation between theirpeak values is about 1 kV, about 2.8% of the voltage’s peak value,which indicates the applicability of the proposed method in thepaper. The simulation CPU time (only the transient computationtime) of the proposed method and the FDTD method adoptedin [22] are 0.52 s and 12.88 s, the proposed method provides aspeed-up 24.77.



TABLE IIPER-UNIT-LENGTH CIRCUIT PARAMETERS OF THE EQUIVALENT CIRCUIT FOR

THE GROUND IMPEDANCE ACCORDING TO FIG. 6

Fig. 16. Lightning-induced voltage at the termination of phase A consideringthe lossy ground (σ = 0.001 S/m, εr = 10).

V. SIMULATION AND DISCUSSION

In this section, a 35 kV overhead line over a lossy groundis studied. The influence of the ground wire and the metal-oxide arrester (MOA) to the lightning induced voltages are bothdiscussed.

A. Influence of Ground Wire

Two typical 35 kV overhead line configurations are shown inFig. 17(a) and (b). The only difference between these two con-figurations is that two ground wires are placed on the top of thepole, shown as Fig. 16(b). The model type of the phase conduc-tors and ground wires are LGJ-70/35 and GJ-25, respectively.There are nine poles along the 1-km lines and the distance be-tween each other is 100 m. The conductors are terminated withtheir chararcteristic impedance in order to eliminate reflections.The lossy ground is considerate, and the ground conductivityand relative permittivity are 0.01 S/m and 10, respectively. Thelightning return stroke field is calculated using the modified

Fig. 17. Two 35 kV power line configurations. (a) Configuration I: Withoutground wire. (b) Configuration II: With ground wire.

transmission line return-stroke model and the horizental com-ponent of the electric field above the imperfectly conductingground is calculated by the Cooray–Rubinstein formula [15],[16]. The striking point is considered equidistant from the lineterminations and at a distance of 50 m from the line centre. Thereturn-stroke model and the peak of the channel-base currentadopted are the same as example 1. The impedance of each poleis ignored which means that the potential of the ground wires atevery pole location is equal to that of ground potential.

Fig. 18(a) and (b) shows the lightning induced voltages at theline center of the three conductors for the two configurations andthe corresponding peaks are listed in Table III. It can be seenthat because of the presence of the ground wires, the peaks ofthe induced voltages are reduced significantly. That is becausethe potential of the ground wire is close to the ground potentialand there is a coupling effect between the ground wire and eachphase conductor.

B. Influence of Arrester

The metal oxide arrester (MOA) protects the insulation ofequipment or overhead lines in electrical systems against in-ternal and external overvoltages. In China, MOAs have beenadopted to 35 kV overhead transmission lines in some areasto reduce the damage caused by lightning. They exhibit an ex-tremely high resistance during normal operation and a very lowresistance during transient overvoltages [23]–[26]. That is, theV–I characteristic of the device is nonlinear. In this paper, theMOA is considered as a nonlinear resistor whose V–I character-istic can be represented by an arbitrary number of exponentialsegments, where each segment has a constraint equation definedby (23)

u = Aiα . (23)

In our analysis, the simulated system consists of three-phaseoverhead lines with a length of 1 km, and matched at bothends. Nine poles were assumed, resulting in a span length of100 m. The MOAs are equiped on every pole. We chose three



Fig. 18. Comparison of voltage induced on the three line conductors of the twoconfigurations. (a) Voltage induced on the three line conductors of configurationI (without ground wire). (b) Voltage induced on the three line conductors ofconfiguration II (with ground wire).

TABLE IIIPEAK VALUES AND PROTECTIVE RATIO OF THE INDUCED VOLTAGES ON THREE

LINE CONDUCTORS

configurations to study the influence of the MOA to the light-ning induced overvoltages. In the first configuration, there is noMOA placed on any phase, which is same as Fig. 17(a); in thesecond configuration, the MOAs are placed on Phase A and B oneach pole, while in the third configuration, the MOAs are placedon the all the phases on each pole; the latter two configurationsare shown in Fig. 19. The parameters of the overhead lines arethe same as the configuration of Fig. 17(a). The striking pointis considered equidistant from the line terminations and at adistance of 50 m from the pole, assuming a typical channel-basereturn stroke current has a peak value of 35 kA, which is shownin Fig. 20, and a maximum time-derivative of 120 kA/μs. TheMOA is considered as a nonlinear resistance which is describedby three exponential segments and the corresponding equation

Fig. 19. Two 35 kV power line configurations. (a) Configuration II (MOAplaced on Phase B). (b) Configuration III (MOA placed on Phase A).

Fig. 20. Channel-base return stroke current for a typical negative subsequentreturn stroke.

TABLE IVPEAK VALUES OF INDUCED VOLTAGE WITH AND WITHOUT MOA FOR

CONFIGURATION I, II, AND III

is shown in (24).

u =

⎧⎪⎨

⎪⎩

146.46 × 103 · i0.2 0 mA ≤ u < 1 mA

51.97 × 103 · i0.05 1 mA ≤ u < 1 A

51.97 × 103 · i0.06 1 A ≤ u < 6 kA.

(24)

The lightning-induced voltages at the middle point of the lineare presented in Fig. 20 and their peaks are listed in Tables IV.As seen form Fig. 21(b), it indicates that the placement of MOAcan not only limit the voltage of the phases with MOA but alsoreduce the voltage of other phases. The reason is the same asthe shielding effect of the ground wire. Due to the coupling ef-fect between each conductor, any phase’s voltage reduction willresult in the voltage reduction of other phases. However, only



Fig. 21. Comparison of voltage induced on a three-phase line with and withoutMOA. (a) Without MOA. (b) MOA on phase A and C. (c) MOA on three phases.

equiping MOAs on two phases can not give reliable protection,as a insulation flashover will occur on the phase without MOAunder a large lightning current. So the equipment of MOAs onall the three phases is necessary, as seen from Fig. 21(c).

VI. CONCLUSION

The DEPACT macromodel is applied to calculate thelightning-induced overvoltages considering the frequency-dependent ground impedance. The frequency-dependent groundimpedance is approximated by the vector fitting method and aequivalent circuit is realized by the circuit synthesis technology.By assuming the incident field only couples with the losslesstransmission line sections, the distributed equivalent voltage

sources along the line due to the horizontal component of theincident electric field are lumped at the termination of the lines,which makes it convenient for the engineering technologist tocalculate the lightning-induced voltage in the circuit simulationsoftware. Furthermore, in order to treat the nonlinear elementdirectly, the DEPACT macromodel is improved and solved inthe actual phase domain instead of in the modal domain.

Compared with existing methods, the proposed algorithmprovides a conveninet and efficient way for the calculation thelightning-induced ovrevoltages of the overhead transmissionline in the circuit simulation software.

Using the proposed algorithm, a 1-km 35 kV overhead lineabove an imperfectly conducting ground is studied, the influ-ence of the ground wire and the MOA to the lightning inducedovervoltages are both discussed. We find that because of thecoupling effect between the ground wires to the other conduc-tors, the voltage of the phase conductors can be reduced; theplacement of the MOA is an effective way to reduce the over-voltages of the overhead lines, and the equipment of MOAs onthe three phases is necessary.

REFERENCES

[1] A. J. Eriksson, D. V. Meal, and M. F. Stringfellow, “Lightning-inducedovervoltages on overhead distribution lines,” IEEE Trans. Power App.Syst., vol. PAS-101, no. 4, pp. 960–969, Apr. 1982.

[2] S. Rusck, “Induced lightning overvoltages on power-transmission lines:With special reference to the overvoltage protection of low voltage net-works,” in Thesis, Royal Inst. Technol., Stockholm, Sweden, 1957.

[3] P. Chowdhuri and E. T. B. Gross, “Voltage surges induced on overheadlines by lightning strokes,” in Proc. IEE, Dec. 1967, vol. 114, no. 12,pp. 1899–1907.

[4] A. K. Agrawal, H. J. Price, and S. H. Gurbaxani, “Transient responseof multiconductor transmission lines excited by a nonuniform electro-magnetic field,” IEEE Trans. Electromagn. Compat., vol. 22, no. 2,pp. 119–129, May 1980.

[5] C. D. Taylor, R. C. Satterwhite, and C. W. Harrison, Jr., “The responseof a terminated two-wire transmission line excited by a nonuniform elec-tromagnetic field,” IEEE Trans. Antennas Propag., vol. AP-13, no. 6,pp. 987–989, Nov. 1965.

[6] F. Rachidi, “Formulation of field to transmission line coupling equations interms of magnetic excitation field,” IEEE Trans. Electromagn. Compat.,vol. 35, no. 3, pp. 404–407, Aug. 1993.

[7] R. Montano, N. Theethayi, and V. Cooray, “An efficient implementation ofthe Agrawal et al. model for lightning-induced voltage calculations usingcircuit simulation software,” IEEE Trans. Circuits Syst. I, Reg. Papers,vol. 55, no. 9, pp. 2959–2965, Oct. 2008.

[8] H. K. Høidalen, “Calculation of lightning-induced overvoltages usingMODELS,” presented at the Int. Conf. Power Syst. Transients, Budapest,Hungary, 1999.

[9] H. K. Høidalen, “Calculation of lightning-induced voltages in MODELSincluding lossy ground effects,” presented at the Int. Conf. Power SystemTransient, New Orleans, LA, 2003.

[10] A. Borghetti, A. Gutierrez, C. A. Nucci, M. Paolone, E. Petrache, andF. Rachidi, “Lightning-induced voltages on complex distribution systems:Models, advanced software tools and experimental validation,” J. Elec-trostatics, vol. 60, pp. 163–174, May 2004.

[11] G. S. Shinh, N. M. Nakhla, R. Achar, M. S. Nakhla, A. Dounavis, andI. Erdin, “Fast transient analysis of incident field coupling to multicon-ductor transmission lines,” IEEE Trans. Electromagn. Compat., vol. 48,no. 1, pp. 57–73, Feb. 2006.

[12] N. M. Nakhla, A. Dounavis, M. S. Nakhla, and R. Achar, “Delay-extraction-based sensitivity analysis of multiconductor transmission lineswith nonlinear terminations,” IEEE Trans. Microw. Theory Tech., vol. 53,no. 11, pp. 3520–3530, Nov. 2005.

[13] N. M. Nakhla, “Analytical algorithm for macromodeling and sensitivityanalysis of high-speed interconnects,” M. S. dissertation, Univ. Carleton,Ontario, Canada, 2005.



[14] F. Rachidi, C. A. Nucci, M. Ianoz, and C. Mazzetti, “Influence of a lossyground on lightning-induced voltages on overhead lines,” IEEE Trans.Electromagn. Compat., vol. 38, no. 3, pp. 250–264, Aug. 1996.

[15] V. Cooray, “Horizontal fields generated by return strokes,” Radio Sci.,vol. 27, pp. 529–537, Jul./Aug. 1992.

[16] M. Rubinstein, “An approximate formula for the calculation of the hori-zontal electric field from lightning at close, intermediate and long ranges,”IEEE Trans. Electromagn. Compat., vol. 38, no. 3, pp. 531–535, Aug.1996.

[17] B. Gustavsen and A. Semlyen, “Rational approximation of frequencydomain responses by vector fitting,” IEEE Trans. Power Del., vol. 14,no. 3, pp. 1052–1061, Jul. 1999.

[18] W. Hendrickx and T. Dhaene, “A discussion of “rational approximation offrequency domain responses by vector fitting,” IEEE Trans. Power Syst.,vol. 21, no. 1, pp. 441–443, Feb. 2006.

[19] L. Qi, “Research of the multi-conductor transmission lines with the finitedifference-time domain method,” M. S. thesis, Dept. Electric. Eng., NorthChina Electrical Univ., Hebei, China, 2002.

[20] C. A. Nucci, F. Rachidi, M. Ianoz, and C. Mazzetti, “Lightning inducedvoltages on overhead lines,” IEEE Trans. Electromagn. Compat., vol. 35,no. 1, pp. 75–86, Feb. 1993.

[21] F. Rachidi, C. A. Nucci, M. Ianoz, and C. Mazzetti, “Response of multi-conductor power lines to nearby lightning return stroke electromagneticfields,” IEEE Trans. Power Del., vol. 12, no. 3, pp. 1404–1411, Jul. 1997.

[22] F. Rachidi, C. A. Nucci, and M. Ianoz, “Transient analysis of multicon-ductor lines above a lossy ground,” IEEE Trans. Power Del., vol. 14,no. 1, pp. 294–302, Jan. 1999.

[23] C. A. Christodoulou, F. A. Assimakopoulou, I. F. Gonos, and I. A. Stathop-ulos, “Simulation of metal oxide surge arresters behavior,” presented atthe Conf. Power Electron. Specialists 2008, Rhodes, Greece, Jun. 15–Jun.19, 2008, pp. 1862–1866.

[24] F. Heidler, “Traveling current source model for LEMP calculation,” inProc. 6th Int. Symp. Tech. Exhibition on Electromagn. Compat., Zurich,Switzerland, Mar. 1985, pp. 157–162.

[25] K. P. Mardira and T. K. Saha, “A Simplified Lightning Model for MOSurge Arresters,” in Proc. Australasian Universities Power EngineeringConf., Melbourne, Australia, Sep. 29 – Oct. 3, 2002.

[26] A. Bayadi, N. Harid, K. Zehar, and S. Belkhiat, “Simulation of metal oxidesurge arrester dynamic behavior under fast transients,” in Proc. Int. Conf.Power Syst. Transients 2003, New Orlean, LA, Sep. 28–Oct. 3, 2003.

Xin Liu was born in Tianjin, China, in 1980. Hereceived the B.S. degree from North China ElectricPower University (NCEPU), Baoding, Hebei, China,in 2003, and M.S degree from Huazhong Universityof Science and Technology (HUST), Wuhan, Hubei,China, in 2006.

He is currently a Lecturer of electrical andelectronic engineering at NCEPU, Baoding, Heibei,China, and working toward the Ph.D. degree atNCEPU, Beijing, China. His research interests in-clude electromagnetic compatibility (EMC) on power

systems, electromagnetic pulse (EMP) interaction with transmission lines, andhigh-voltage equipment modeling.

Xiang Cui (M’97–SM’98) was born in Baoding,China, in 1960. He received the B.Sc. and M.Sc.degrees in electrical engineering from North ChinaElectric Power University (NCEPU), China, in 1982and 1984, respectively, and the Ph.D. degree in accel-erator physics from China Institute of Atomic Energy,China, in 1988.

He is currently a Professor and the Head of theElectromagnetic Fields and Electromagnetic Com-patibility Laboratory at NCEPU. He is the author orcoauthor of more than 200 journal articles. His cur-

rent research interests include computational electromagnetics, electromagneticenvironment and electromagnetic compatibility in power systems, and insula-tion and magnetic problems in high voltage apparatus.

Prof. Cui is a standing council member of the China Electrotechnical Society,a fellow of the IET, and a member of CIGRE C4.02.01 Working Group (Elec-tromagnetic Compatibility in power systems). He is also an Associate Editor ofIEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY and a member ofthe Editorial Advisory Board of the International Journal for Computation andMathematics in Electrical and Electronic Engineering (COMPEL).

Lei Qi was born in Henan, China, in 1978. He re-ceived the B.S., M.S., and Ph.D. degrees in electricalengineering from North China Electric Power Uni-versity, Baoding, Hebei, China, in 2000, 2003, and2006, respectively.

He is currently an Associate Professor of elec-trical and electronic engineering at North ChinaElectric Power University, Beijing, China. His re-search interests include electromagnetic (EM) fieldnumerical computation, EM compatibility on powersystems, and ultrahigh-voltage power transmission

technology.

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Calculation of Lightning-Induced Overvoltages On