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TitleCigreTB_543
TagsAlternating Current Electrical Substation Inductance Transmission Line Lightning
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Total Pages106
Document Text Contents
Page 1

543








Guideline for Numerical Electromagnetic
Analysis Method and its Application to Surge

Phenomena





Working Group
C4.501









June 2013

Page 2

GUIDE FOR NUMERICAL
ELECTROMAGNETIC ANALYSIS
METHODS: APPLICATION TO SURGE
PHENOMENA AND COMPARISON WITH
CIRCUIT THEORY-BASED APPROACH

Cigré WG C4.501

Members

A. Ametani, Convenor (JP), M. Paolone, Secretary (IT), K. Yamamoto, Secretary (JP), Prof. Maria

Teresa Correia de Barros (PT), A. M. Haddad (UK), J.L. He (CN), M. Ishii (JP), T. Judendorfer (AT), W.

Jung-Wook (KR), A. J. F. Keri (US), C. A. Nucci (IT), Dr. Marjan Popov (NL), F. Rachidi (CH), M.

Rubinstein (CH), E. Shim (KR), J. Smajic (CH), K. Tanabe (JP), L. Tang (US), P. Yutthagowith (TH)



Corresponding member

Y. Baba (JP)





Copyright © 201 3

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Disclaimer notice

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any responsibility, as to the accuracy or exhaustiveness of the information. All implied warranties
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ISBN : 978-2-85873-237-1

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Guide for numerical electromagnetic analysis methods: application to surge phenomena and comparison with circuit

theory-based approach





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By comparing GSI with (4.2.7) and (4.2.8), it can be seen that if the kernel of the GSI can be approximated by a
sum of complex exponentials, GSI can be approximated by a sum of expressions just like the left side in (4.2.7) or
(4.2.8). The Prony method is often used to approximate the kernel [120]. Now, GPOF method is a more popular
method that has advantages over the Prony method in both computation and noise sensitivity [121-123].

Usually generalized pencil-of-function (GPOF) method [122] is used to extract features of functions in target
identification. Now, it is introduced to calculate the GSI. It can approximate declining functions by a sum of complex
exponentials series as following:


=

=
M

i

ua
i

iebuf
1

)( (4.2.9)

where ia and ib are complex values.

First, GPOF method solves a generalized eigenvalue problem to get ia by singular value decomposition (SVD).

Then, ib can be obtained by solving a linear least squares problem. With the sum of complex exponentials, GSI
can be approximated into a sum of simple expressions. For example, by using GPOF method with (4.2.7), (4.2.1)
can be approximated into the following expression:

( )

k

kA





=



∞ −′−

=

=

M

i i

rjk

i

0
0

zuzu

r

e
b

π

Idlµ

dρJGe
π

Idlµ

i

1

1

1
1

1

12

4

2
λλλ

(4.2.10)

where 22 )( ii azr −+= ρ .

By using the same method, (4.2.2) to (4.2.6) can also be approximated.

After obtaining the vector potential A, the electromagnetic field can be calculated from the following formulas:

)](
1

[
2

AAAE ⋅∇∇+−=∇−−=
k

jj ωφω (4.2.11)

AB ×∇= (4.2.12)

where φ is the scalar potential which can be obtained from A according to the Lorentz condition. It can be seen
that the electromagnetic field can also be calculated easily from the approximated vector potential A.

4.3 Partial Element Equivalent Circuit (PEEC)

Procedures for obtaining solution of the PEEC method start from discretizing geometry structure into small cells or
elements which are composed of current cells and charge or potential cells. The current cells and potential cells
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 cell, and the continuity equation or the charge conservation equation is applied via
Kirchhoff's current law to a potential cell. Whole system equations in the frequency domain can be written in a
matrix form corresponding to a modified nodal analysis (MNA) formulation as shown in eq. (4.3.1).









=







Φ









+−
+−

S

S
T

a

j

j

U

I

ILRA

AYP

ω
ω 1

(4.3.1)

Page 54

Guide for numerical electromagnetic analysis methods: application to surge phenomena and comparison with circuit

theory-based approach





Page 53

where A is an incident matrix which expresses the cell connectivity, R is a matrix of series resistances of current
cells, L is a matrix of partial inductances of current cells including the retardation effect, P is a matrix of partial
potential coefficients of potential cells including the retardation effect, Φ is a vector of potentials on potential cells,
I is a vector of currents along current cells, 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 three-dimensional geometries of a considered structure. An appropriate
solver is employed to obtain solution either in the time domain or in the frequency domain. Figure 4.3 shows the
procedures of the PEEC simulation. The detail of derivation and formulation of a PEEC for a thin wire structure is
found in [124-127].



Figure 4.3: Procedures in the simulation of PEEC models.


In [124-127], Yutthagowith et al. have developed a full-wave PEEC method based on the thin wire structure in the
frequency domain which has been successfully applied to lightning studies. The calculated results by the full-wave
PEEC method have shown good agreement in comparison with the results calculated by the MoM and the FDTD
methods, as well as with measured results. Moreover, in some specific cases, the computational efficiency of the
method was found to be higher than those of the MoM and the FDTD method, because post processing for
calculating voltages and currents in a given system is not required. Moreover, the PEEC method can readily
incorporate electrical components based on a circuit theory, such as resistive, inductive, and capacitive elements,
transmission lines, cables, transformers, switches, and so on.

In this section, an effective way to increase the efficiency of the PEEC method in terms of computation time, which
consists of the appropriate combination of the PEEC method and the transmission line theory, is presented.

4.3.1 Formulation of the quasi-static PEEC method in the time domain

The formulation of the quasi-static PEEC method in the time domain can be derived by using (4.3.1) and is given in
eq. (4.3.2).









=







Φ









+−
+−

S

S

dt
d

T
adt

d

U

I

ILRA

AYP 1

(4.3.2)

Page 105

Guide for numerical electromagnetic analysis methods: application to surge phenomena and comparison with circuit

theory-based approach





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