474 Journal of International Council on Electrical Engineering Vol. 1, No. 4, pp. 474~480, 2011
Application of a Frequency-Domain Partial Element Equivalent
Circuit Method to Tower Surge Response Calculations
Peerawut Yutthagowith*, Akihiro Ametani†, Naoto Nagaoka** and Yoshihiro Baba**
Abstract – This paper presents tower surge response calculations on an actual transmission tower
including ground wires and phase wires. A frequency-domain partial element equivalent circuit (PEEC)
method is applied as a simulation tool. Insulator voltages calculated by the voltage difference of
crossarms and phase wires are compared with experimental results collected from the literature. In
addition, a transmission line (TL) model is adopted with the PEEC method to increase efficiency for
transient surge calculation. The results calculated by the PEEC method with and without the TL model
agree well with the experimental results not only for amplitudes but also for waveshapes.
Keywords: Insulator voltage, Partial element equivalent circuit, Tower surge response
1. Introduction
Direct and indirect lightning strikes to transmission and
distribution systems are ones of the main causes of power
transmission and distribution system failures. An accurate
calculation of a surge over-voltage due to the lightning is
necessary for an economical insulation design of the
transmission and distribution systems.
There are representative methods for evaluating lightning
surge behavior as follows; 1) Experimental measurement on
reduced-scale and simplified models [1], and full-scale and
actual models [2]; 2) A circuit and transmission line theory
assuming simplified geometry [3]-[5]; 3) Numerical
electromagnetic approaches [6]-[8]; and 4) Hybrid method
[9]-[11].
In an experimental measurement, a step like current is
injected by using a pulse generator through a current lead
wire to the top of a tower. To measure voltages on the top of
a tower and across an insulator, a voltage reference wire,
and a voltage divider or a voltage probe are required.
Inherently, a current lead wire, the voltage reference wire,
and the impedance of the voltage divider or the voltage
probe affect measured voltage [9], [10]. It means that some
results of an experiment cannot show actual phenomena
during a lightning strike to a tower in the real system.
In the circuit and transmission line approach such as the
†
Corresponding Author: Dept. of Electrical Engineering, Doshisha
University, Japan (aametani@mail.doshisha.ac.jp)
*
Dept. of Electrical Engineering, King Mongkut’s Institute of
Technology ladkrabang, Thaialnd
(kypeeraw@kmitl.ac.th)
** Dept. of Electrical Engineering, Doshisha University, Japan
Received: May 16, 2011; Accepted: August 30, 2011
EMTP, a tower is represented as a vertical multi-conductor
and several models for estimating the surge impedance of
the vertical conductor were proposed for computation of a
transient voltage. The assumption of the TEM mode
propagation might not be correct for the tower, because the
electromagnetic field around a tower and a pole, when
struck by lightning, changes dynamically during several
round-trips of a traveling wave in the tower or the pole.
To obtain an accurate result, an approach of numerical
electromagnetic analysis (NEA) method such as the method
of moment (MoM) [6] and the finite-difference dimedomain (FDTD) method [7], [8], and a hybrid
electromagnetic-circuit method such as the partial element
equivalent circuit (PEEC) method [9]-[11] are more
appropriate than a circuit and transmission line approach.
The MoM and the FDTD method are quite time
consuming. In the FDTD method, a large calculation
domain is necessary including a grounding system, a
current lead wire and a voltage reference wire which is
normally perpendicular to a grounding system. However, a
voltage reference wire and a current lead wire affect
voltages and currents due to electromagnetic coupling
among them and other objects in the calculated domain. In
the MoM and the FDTD method, an accurate result of a
transient voltage requires integration of electric field.
Therefore, the MoM and the FDTD method is not
computational time efficient method in comparison with the
PEEC method.
As is well known, a transmission line approach assuming
TEM mode propagation is still effective for a horizontal
conductor close to a ground plane, because an electric field
along the line is not significant. For the PEEC method
transforms the mixed potential integral equation (MPIE) to
Peerawut Yutthagowith , Akihiro Ametani , Naoto Nagaoka and Yoshihiro Baba
a circuit domain, circuit elements and also transmission
lines can be implemented readily with the method.
Fortunately, in the case of a lightning surge analysis in an
electrical power system, most of elements can be
represented as horizontal wires above the ground plane.
Therefore, adopting the transmission line model with the
PEEC method is an effective way to increase the efficiency.
The aim of this paper is to propose another efficient
choice of NEA methods for analysis of a tower surge
response.
2. The PEEC Method Incorporating with a
Transmission Line Model
The PEEC method [12] is derived from MPIE and
provides a full wave solution. The first advantage of this
method is that it can incorporate electrical components
based on a circuit theory, such as RLC elements,
transmission lines, cables, transformers, switches, and so on.
The second is that the composition matrix in this method
depends on the configuration of a considered system and a
medium, and does not depend on the sources in the
frequency domain. The third is that a potential on a
conductor is calculated directly from a node potential on a
potential element. Therefore, post processing for calculating
potentials and currents is not required.
Procedures for obtaining solution of the PEEC method as
shown in Fig. 1 start from discretizing geometry structure
into small elements which are composed of current
elements and charge or potential elements. The current
elements and potential elements are interleaved each other.
The rectangular pulse is employed both charge and current
basis functions. Then, Galerkin’s method is applied to
enforce the mixed potential integral equation which is
interpreted as Kirchhoff's voltage law applied to a current
element, and the continuity equation or the charge
conservation equation is applied via Kirchhoff's current law
to a potential element. Whole system equations in the
frequency domain can be written in a matrix form
corresponding to a modified nodal analysis formulation as
shown in (1).
 jP 1  Ya

A



R  jL
AT
   I S 
 I   U 
   S
475
elements, L is a matrix of partial inductances of current
elements including the retardation effect, P is a matrix of
partial potential coefficients of potential elements including
the retardation effect,  is a vector of potentials on
potential elements, I is a vector of currents along current
elements, US is a vector of voltage sources, IS is a vector of
external current sources, and Ya is an additional admittance
matrix of linear and non-linear elements.
The equivalent circuit is extracted from threedimensional geometries of a considered structure. An
appropriate solver is employed to obtain solution either in
the time domain or in the frequency domain. The detail of
derivation and formulation of a PEEC method for a thin
wire structure is found in Appendix.
Fig. 1. Procedures in the simulation of PEEC models.
Formulation of a multi-transmission line in the
frequency domain as multi-port network as illustrated in
Fig. 2 is given as the following;
(1)
where A is an incident matrix which expresses the element
connectivity, R is a matrix of series resistances of current
Fig. 2. Multi-transmission line as a multi-port network.
Application of a Frequency-Domain Partial Element Equivalent Circuit Method to Tower Surge Response Calculations
V s  coth() Z c
V    cos ech()
 r 
cos ech()   I s 
,

coth() Z c   I r 
(2)
 2  ZY ,
(3)
Z c   1 Z ,
(4)
where Vs, Vr, Is, and Ir are vectors of voltages and currents
at sending ends and receiving ends, respectively. , Z, Y,
and Zc are a propagation matrix, a series impedance matrix
per unit length, a shunt admittance matrix per unit length,
and a characteristic impedance, respectively.
The impedance matrix on the right hand side of (2) can
be seen as an inversion matrix of the additional admittance
(Ya). Therefore, the transmission line model can be
combined with the PEEC method by adding Ya on the
matrix on the left hand side of (1).
actual system of a 500 kV double circuit transmission line
is illustrated in Fig. 3(a). The experiment system is
composed of three towers with separation distances 450 m
and 560 m between towers, two overhead ground wires, and
six phase wires. The injected current is applied at the
middle tower. To measure the insulator voltages, two
current waveforms with about 0.2 s rise time (fast rise
time current) and 3 s time to crest, 3.4-A crest value (slow
rise time current) are applied as shown in Figs. 3(b) and
3(c). To measure voltages across insulator strings a 10-k
resistive voltage dividers are employed.
Voltage at cross arm position
Insulator voltage
Voltage [V]
476
Voltage on phase wire position
3. Calculation of Tower Surge Response
a)
Voltage at cross arm position
Insulator voltage
Voltage
[V]
In this section, an actual transmission tower was selected
as a test case. The PEEC method is applied to determine the
voltages across insulators of a 500-kV double-circuit
transmission line. The calculated results with and without
adopting a transmission line (TL) model are compared with
experimental results reported by Ishii et al. [2].
Upper phase
Voltage on phase wire position
5
Current [A]
4
3
b) Middle phase
2
1
0
-1
0
0.5
1.0
1.5
2.0
2.5
3
3.5
4.0
Voltage at cross arm position
Insulator voltage
Time [s]
Voltage
[V]
(b) Fast rise time current
5
Voltage [V]
4
3
2
Voltage on phase wire position
1
(a) System configuration
0
-1
Time [s]
0
0.5
1.0
1.5
2.0
2.5
3
3.5
4.0
(c) Slow rise time current
Fig. 3. System configuration and applied currents.
In the actual tower case, the tower is composed of slant
elements and cross arms including overhead ground wires
and phase wires. The configuration of the experiment on an
c) Lower phase
Fig. 4. Simulated and measured waveforms of insulator voltages,
voltages on phase wires, and voltages at cross arm
position, the fast rise time current injected.
Measured insulator voltage from [2]
PEEC method without a TL model
PEEC method with a TL model
Peerawut Yutthagowith , Akihiro Ametani , Naoto Nagaoka and Yoshihiro Baba
The same configuration of the experiment is employed in
the PEEC simulations. The simulations involve 256
frequencies upto 5MHz and 5-m element length. The tower
is composed of four main poles of which elements have
0.2-m radius and slant and horizontal elements having 0.1m radius. The cross arms are composed of elements with
0.2 m radius. In simulation, the tower-footing resistance is
represented by a resistance of 17  by connecting four 68 resistors at the bottom of four main poles of the tower.
There
Voltage [V]
Voltage at cross arm position
Insulator voltage
Voltage on phase wire position
a) Upper phase
MPIE to the circuit domain, the current and voltage sources
can be connected directly to voltage nodes. For incurporation of the transmission line model with the PEEC
method in the frequency domain, eight lossless lines are
considered. A perfectly conducting earth is assumed for the
transmission line. Towers and some parts of the
transmission line close to the tower are modeled by the
PEEC method for taking into account the retardation of
electromagnetic fields around the tower. The total number
of elements without adopting the TL model is 2488 and the
total number of elements with adopting the transmission
line model is reduced to 604 including some parts of two
transmission lines with eight conductors, i.e. six phase wire
and two ground wires around the towers.
Figs. 4 and 5 show comparison between experimental
results [2] and PEEC-simulated waveforms. There are two
options of the PEEC simulations. One is calculated by the
PEEC method without the TL model and the other is
calculated by the PEEC method with the TL model. Good
agreements are observed among the simulated and the
experimental results. In all cases, it is observed that the
computation time carried out by the PEEC method with the
TL model is less than 10% of the computation time carried
out by the PEEC method without the TL model.
Voltage
[V]
Voltage at cross arm position
Insulator voltage
477
Voltage on phase wire position
Voltage [V]
b) Middle phase
Voltage at cross arm position
Insulator voltage
4. Conclusion
This paper presents the PEEC simulation for calculations
of a tower surge response. The transmission line model is
adopted with the PEEC method to increase efficiency for
transient surge calculation. The calculated results closely
reproduces a corresponding field-test results, and this
confirms that the accuracy of the PEEC method, when
applied to lightning over-voltage studies, is fairly high.
The PEEC method is another efficient choice to the NEA
for analysis of lightning surges in power systems.
Voltage on phase wire position
c) Lower phase
Fig. 5. Simulated and measured waveforms of insulator voltages,
voltages on phase wires, and voltages at cross arm
position, the slow rise time current injected.
Measured insulator voltage from [2]
PEEC method without a TL model
PEEC method with a TL model
There is no current lead wire and voltage reference wire in
the simulation. Note that the PEEC method interprets the
Appendix
The theoretical derivation of the PEEC method for thin
wire structure starts from considering on a total electric
field on the thin wire. Currents and charge densities are
assumed to be distributed along the contour of wire axis

( C( r ) ). The boundary condition on the surface of the thin
wire as illustrated in Fig. A1 is that the total tangential
electric field shall be as;
Application of a Frequency-Domain Partial Element Equivalent Circuit Method to Tower Surge Response Calculations
478
 
s 
 t 
 i 
 J( r )
,
s  E ( r )  s  ( E ( r )  E ( r ))  s 


E s   jA   ,
(A1)

 

A( r ) 
4



where s is a unit tangential vector along C( r ) , s' is a

unit tangential vector on the wire surface, the E i is the

incident electric field and E s is the scattered electric field
which represents the reaction of the wire to an incident field
corresponding to (A1).
 

s ( r ' )  s'

r'

C( r )

r
Origin po int
1
4ˆ

 e  R
ds' ,
R
(A2-2)

 e  R
ds' ,
R
(A2-3)
 s' I ( r ' )
c
l ( r' )
c

where ds' is a small distance along C( r ) , the
 
ˆ     /( j ) , R  r  r ' and  in (A3) is the
 
s( r )

C( r )

( r ) 
(A2-1)

r'
propagation constant of the considered medium.

r

Origin po int
Fig. A-1. Problem geometry.
j (   j ) .
(A3)
The time dependence of the variables in the frequency
domain is exp(jt). , ,  and  are conductivity,
permittivity, and permeability of the medium and the
angular frequency, respectively.
Substituting (A2) into (A1), (A4) is obtained:

    I( r )
    e  R
d
s  Ei( r )  s 
 j
 s  s' I ( r' ) R ds'  ds  0 , (A4)
l
4 c
Fig. A2. Configuration of elements in the medium domain.
(a) Elements of the first group and (b) elements of
the second group.
The scattered field is calculated by volume current
densities and volume charge densities which can be
 
expressed in terms of a magnetic potential vector A( r ) and


an electric scalar potential  ( r ) at a point r . Assuming
 

thin wires, A( r ) and  ( r ) can be written in forms of a

current along a conductor I ( r ' ) and line charge

density  l ( r ' ) as given in (A2) and illustrated in Fig. A1:
where l is length conductivity of a conductor. Eqaution (4)
is called the mixed potential integral equation.
As illustrated in Fig. A2, the conductors in the media are
segmented into two groups. The first group in Fig. A2 a)
has N elements and the second group in Fig. A2 b) has M
elements. The elements in two groups are interleaved each
other. The configurations of elements in the first group and
the second group are used to calculate series impedances
and shunt admittances, respectively.
From (A4), an element m with length Lm for nodes k and
l in the first group is considered.
Rearranging and integrating (A4) along the segment m
from a point k to a point l, yields
 s'  l    e R
d



ds
j

  s  s' I ( r' ) R ds' ds
4 s'  k c
k ds

s'  l
l
  
I( r )
 
ds   s  E i ( r )ds ,
l
s'  k
l
k
(A5-1)
Peerawut Yutthagowith , Akihiro Ametani , Naoto Nagaoka and Yoshihiro Baba
l  k  lk   j

 s'  l    e R
  s  s' I ( r' ) R ds' ds
4 s'  k c
s'  l

I( r )
s'  k
l

l
  
ds   s  E i ( r )ds .
(A5-2)
k
The first term on the right hand side in (A5) can be
expressed as follows
j
N
 s'  l    e R
s  s' I ( r ' )
ds' ds'   Z Lmn I Ln  , (A6-1)


4 s'  k c
R
n 1
Z Lmn  
j
e R cos 
dln dlm .


4 L L
R
479
and l) of the element m, and k is an average voltage of the
node k. lm is a local distance along element m employed for
integration of (A6) and (A7). Subscript m and n indicate
elements m and n in the first group, respectively, and
subscript k and i corresponds elements k and i in the second
group, respectively. R is the distance between the element
m and element n and cos is a cosine function of an angle
between the element m and n corresponding to Fig. A2.
From those formulations, an element can be represented
in the form of an equivalent circuit as illustrated in Fig. A3.
The series impedance (Rij and Lij) and the voltage source
(Vjj) are derived from (A5) and (A6). The shunt admittance
(Gi, Gj, Ci, and Cj) and the current sources (Ii and Ij) are
derived from (A8).
(A6-2)
m n
The current along each small element is assumed to be
constant. Also, the propagation times between any point on
the element n and any other point on the element m are
assumed to be identical. (A6-2) can be derived from (A6-1).
The second and the third terms represent resistance and
an additional voltage source of the element.
The different voltages on both ends of each element are
calculated by (A5) and (A6).
Consider the element of which center is at node k in the
second group as shown in Fig. A2 b). Assuming a constant
charge density along the element and conservation of
charge, the charge density can be calculated by (A7):
l  ( I Lm  I Lo  I Lp ) /( jLk )  ITi /( jLk ) ,
(A7)
where Li is total length of element k.
The average potential along the element at node k can be
calculated by (A8):
k 

ITi e R
1
1

ds'dlk (A8-1)
 ( r )dlk 



Lk L
4   j Lk L c Li
R
k
k
M
k   ZTki I Ti ,
(A8-2)
i 1
Z Tki 
1
4   j Lk Li

Lk Li
e  R
dli dlk ,
R
(A8-3)
where lk is the potential difference between both ends (k
Fig. A3. Equivalent circuit model of a PEEC element.
Equations (A6) and (A8) apply to all element. The
coupling effects among elements and the propagation
effects are included in this method. The potential difference
and the potential average of the elements are written in
forms of matrices which are composed by node potentials
(potentials at the ends of an element), the conduction
currents along conductors, the transverse currents and
incident electric fields. The voltage of each node can be
calculated from the equations employing nodal analysis
approaches in the frequency domain at each frequency
through the considered frequency range. The frequency step
in the PEEC method depends on the maximum considering
time and the frequency range depends on the circuit
condition. The length of a small element shall be much
smaller than the minimum wavelength which is inversely
proportional to the maximum frequency of interest.
The ground effect can be taken into account by
employing the image methods. From applying the modified
inverse fast Fourier transform [13], all quantities in the time
domain are found. The accuracy of this method is
dependent on element length and the number of frequencies
to be used.
480
Application of a Frequency-Domain Partial Element Equivalent Circuit Method to Tower Surge Response Calculations
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Peerawut Yutthagowith received the
B. Eng and M. Eng degrees from
Chulalongkorn University, Bangkok,
Thailand in 1998 and 2002, and the
Ph.D. degree from Doshisha University,
Kyoto, Japan, in 2010. He joined with
King Mongkut's Institute of Technology, Ladkrabang as a Lecturer in 2007. Dr. Peerawut is
also a member of International Council on Large Electric
Systems (CIGRE) WGC4.501. His research interests are in
area of a high voltage equipment modeling and electromagnetic transients in power systems.
Akihiro Ametani received the Ph.D.
degree from UMIST, Manchester, U.K.,
in 1973. He was with UMIST from
1971 to 1974, and Bonneville Power
Administration to develop EMTP for
summer from 1976 to 1981. He has
been a Professor at Doshisha University
since 1985 and was a professor at the Chatholic University
of Leaven, Belgium in 1988. He was the Director of the
Institute of Science and Engineering from 1996 to 1998 and
Dean of Library and Computer/Information Center from
1998 to 2001. Dr. Ametani is a Chartered Engineer in the
U.K., a Distinguished Member of CIGRE, a Fellow of IET
and IEEE.
Naoto Nagaoka received the B.Sc.,
M.Sc., and Dr. Eng. Degrees from
Doshisha University, Kyotanabe, Japan,
in 1980, 1982 and 1993, respectively.
He joined the Faculty of Engineering at
Doshisha University in 1985, and has
been a Professor since 1999. Since
2008, he has been the Dean of the Student Admission
Center of the same University. Dr. Nagaoka is a member of
IET and IEEE.
Peerawut Yutthagowith , Akihiro Ametani , Naoto Nagaoka and Yoshihiro Baba
Yoshihiro Baba received the B.S.,
M.S., and Ph.D. degrees in electrical
engineering from the University of
Tokyo, Japan, in1994, 1996, and 1999,
respectively. He is an Associate Professor in the Department of Electrical
Engineering, Doshisha University,
Kyoto, Japan. From April 2003 to August 2004, he was a
Visiting Scholar at the University of Florida, on sabbatical
leave from Doshisha University. From 2009, he has served
as an Editor of IEEE Transaction on Power Delivery.
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