Earth Return Path Impedances of Underground Cables for ... - LabPlan

Earth Return Path Impedances of Underground
Cables for the multi-layer case - A Finite
Element approach
G. K. Papagiannis, Member, IEEE, D. A. Tsiamitros, Non Member, IEEE, G. T. Andreou, As. Member, IEEE,
D. P. Labridis, Senior Member, IEEE, and P. S. Dokopoulos, Member, IEEE
Abstract--The lossy earth return path influences significantly the
electrical parameters of underground power cables, especially in
cases where transient simulation models are of interest. The use
of approximations for the calculation of earth correction terms
proves to be inaccurate at high frequencies or low earth
resistivities. The infinite integral terms representing the earth
influence are high oscillatory in cases of underground cables and
therefore difficult to integrate numerically. Scope of this paper is
to present and compare results, obtained by a novel numerically
stable and efficient integration scheme to those obtained by a
Finite Element Method formulation for several single core cable
configurations and for homogeneous and multi-layered earth.
Significant differences between impedances are recorded,
especially for high frequencies and low earth resistivities.
Index Terms--Power cables, nonhomogeneous earth,
electromagnetic transients, Finite Element Method.
I. INTRODUCTION
N transient simulations, detailed transmission line modeling
I is required. In the case of underground power cables, the
model parameters are strongly influenced by the resistive
earth return path. The influence of the lossy earth on
conductor impedances has been analyzed since 1926. For the
case of overhead lines, correction terms can be calculated
using the most widely accepted Carson’s formulas [1]. Similar
formulas have been developed by Pollaczek [2], applicable
not only to overhead conductor systems but also to cases of
underground isolated conductor systems and to combinations
of both. In these approaches earth is assumed to be semiinfinite
and homogeneous.
Both approaches lead to complicated expressions with
complex infinite integrals. These integrals have been
evaluated by algebraic infinite series. This evaluation has been
adopted by Wedepohl and Wilcox [3] and by Ametani [4],
who proposed general models for wave propagation in power
cables. The formulation of [4] was also adopted in the
CABLE CONSTANTS/PARAMETERS supporting routines
of the well-known Electromagnetic Transients Program
(EMTP) [5].
G. K. Papagiannis, D. A. Tsiamitros, G. T. Andreou, D. P. Labridis and P.
S. Dokopoulos are with the Power Systems Laboratory, Dept. of Electrical &
Computer Engineering, Aristotle University of Thessaloniki, P.O. Box 486,
GR-54124 Thessaloniki, GREECE
Tel: +302310996374, Fax: +302310996302, e-mail: grigoris@eng.auth.gr
In image theory the electromagnetic field for either case of
overhead lines or underground cables results from the
conductors and their images. In the case of underground
cables though, the earth contribution is much more significant
as the actual conductors are in the ground and their images in
the air. This leads to infinite integrals for the self and mutual
earth return impedances, which are different from the
corresponding forms for overhead lines. They include highly
oscillatory functions and therefore are not easy to be
calculated numerically. To overcome these difficulties,
besides infinite series approximations, a simplified closed
form approximation has been proposed [6]. In a more recent
approach [7], a direct numerical integration is proposed but
certain numerical problems, which limit the efficiency of the
method, are reported.
A further complication is due to the fact that the real
ground is composed of several layers of different
electromagnetic properties. Sunde [8] extended the
homogeneous earth solution of [1], [2] and developed new
formulas for the case of a two-layer earth under certain
assumptions. The resulting integral forms for the case of
underground conductors are very complex and were
practically abandoned.
In 1973, Nakagawa [9], [10] proposed a more rigorous and
general solution but only for the case of overhead conductors
above a multi-layered earth model. In this approach each of
the N earth layers could have different electromagnetic
properties. Results obtained by Nakagawa’s model showed
significant differences from the corresponding for the
homogeneous earth case. Nakagawa's earth model is also
included in the CABLE CONSTANTS/PARAMETERS
supporting routine of the EMTP, but only for cases of cable
systems in the air above ground.
The Finite Element Method (FEM) is a numerical method
widely used for the solution of electromagnetic field equations
in a region, regardless of the geometric complexity. The
application of FEM in cable parameter calculation was
proposed in [11] and [12] mainly for pipe type cables, where
the discretization area is limited inside the pipe. In a recently
proposed method by some of the authors [13], a suitable FEM
formulation was used in the computation of overhead
transmission line parameters with unbounded discretization
areas. The method is capable of handling cases of terrain

surface irregularities and non-homogeneous, stratified soil,
where most classical methods usually fail.
Scope of this paper is to present a systematic comparison of
results obtained by different approaches for the earth return
impedances of underground single core cable arrangements.
First a novel numerical integration technique is proposed and
used for the direct calculation of the infinite integrals for the
case of homogeneous earth. This technique is based on a
combination of Gauss-Legendre and Gauss-Laguerre methods
with the Lobatto rule [14], in order to overcome efficiently the
problems arising from the oscillative form of the infinite
integrals. Results are compared to those obtained by EMTP
for several cable arrangements, earth resistivities and
frequencies.
The results by the novel numerical integration technique
are also compared against those obtained by the application of
the FEM in the case of homogeneous earth for further
justification.
Finally the FEM formulation is used for the analysis of two
different cases of single core cable arrangements in a twolayered
earth. Six cases of two layer earth models, based on
actual electrical ground measurements are investigated.
Results are compared to those of the homogeneous earth case.
II. EARTH RETURN IMPEDANCES OF UNDERGROUND CABLE
SYSTEMS
A transmission line is generally described by the two
matrix equations (1) and (2), linking the voltages and currents
of the line,
∂
v = −Z′
(ω)
i
(1)
∂z
∂
i = −Y′
(ω)
v
(2)
∂z
where v is the voltage vector with respect to a reference
conductor, i is the current vector and z is the longitudinal
direction along the transmission line. Matrices Ζ΄(ω) and
Υ΄(ω) are the frequency dependent series impedance and
shunt
admittance per unit length matrices, respectively.
For the case of an underground cable system Z΄(ω) may be
considered to consist of two components [4],
( ω) = Z′
i
( ω) + Z ( ω)
Z ′ ′
(3)
where Z i΄(ω) represents the internal impedances of the
conductors in the cable system and Z e΄(ω) accounts for the
influence of the earth return path. The exact form and the
dimensions of the above matrices depend on the number of
conducting elements of each cable, e.g. core, sheath, armor
and on the type and the actual configuration of the cable
system. In the case of single core (SC) cables, the diagonal
elements of Z e΄(ω) represent the impedances of the loops
formed by the outermost tubular conductor of the cables,
either sheath or armor and the earth, while the non-diagonal
elements account for the mutual impedances between the
outermost tubular conductors of each pair of cables and the
e
earth.
Next the cable arrangement of Fig. 1 is considered. It
consists of two SC cables i and j, buried in the ground, which
is considered to be semi-infinite, homogeneous having an
earth resistivity ρ and relative permeability and permittivity of
µ r and ε r respectively.
h
Cable i
d
x
D
Image of cable j
x
y
y
Cable j
air
earth
ρ, µ r , ε r
Fig. 1: Geometric configuration of two SC underground cables.
According to [2], [3], the mutual earth return impedance
between the two underground cables i and j is
where
{ K ( mD) − K ( md ) }
ρm
Z +
2π
2
′
mutual
=
0
0
J m
J
m
∞
2 2
−( h+
y)
a + m
e
jax
= ∫
e da
2 2
−∞ a + a + m
2
= + ( + ) 2
, d x
2 + ( h − y) 2
D x h y
and
K 0
(4)
(5)
= , m=
jωµ
rµ 0
ρ
is the modified complex Bessel function of the second
kind with zero order. The self-impedance formula is derived
from (4) and (5) by replacing x = r and y = h , where r is the
radius of the outermost surface of the cable. By applying some
simple transformations in (5) the following relation results [8]:
2 2
∞ −( h+
y)
a + m
e
J
m
= 2∫
cos( xa)
da (6)
2 2
0 a + a + m
III. NUMERICAL EVALUATION OF THE INFINITE INTEGRAL
The corresponding expression for the mutual earth return
impedances of conductors above ground is [1]:
2 ∞ − ( h+
y)
a
ρm
e
Zm′ =
cos( xa)
da
π
∫ (7)
2 2
a+ a + m
0

Equations (4) and (7) differ in two points. First (4) includes
the Bessel functions, which seem to be no serious problem as
they can be evaluated easily using standard library functions.
The key difference though is the fact that the exponential
function in the infinite integral in (6) is complex, while in (7),
it is real. As a result the real and imaginary parts of the infinite
integral of (7) are monotonic and non-oscillatory functions [1]
and can be calculated by infinite series, which converge
rapidly [5].
In the approach of Wedepohl [3] the infinite integral of (5)
is divided in parts, each of which is approximated using
infinite series. Simplified expressions of the general formulas
are also proposed for low frequencies and usual cable
configurations.
In the supporting routines CABLE
CONSTANTS/PARAMETERS of EMTP, the earth return
impedances of cables are calculated using Carson’s formula
(7) for overhead conductors, which is evaluated by infinite
series. This approximation is based on the assumption that
a >> m . This is a highly uncertain approximation, which is
obviously false for high frequencies and low earth
resistivities. This approximation is used in [5] together with
the approach of Wedepohl [3] for a single case of two
underground conductors. The results are compared with those
obtained by the numerical integration of (5), using Romberg
extrapolation [14]. The approximation of EMTP shows
significant differences for frequencies greater than 10 kHz,
while Wedepohl’s approach is accurate enough at least up to
100 kHz. In the same reference, numerical efficiency
problems concerning the application of the integration method
are reported.
Finally direct numerical integration is also used in [7].
Several integration methods, such as Simpson rule and
Romberg rule [14] are tried. Problems concerning the
numerical efficiency of all methods are reported, leading to
the adaptation of artificial intelligence techniques for the
evaluation of the infinite integrals.
All above approaches either use approximations, which are
valid only within certain limits, or suffer from numerical
efficiency, which prohibits their generalized application.
Therefore the problem of finding an efficient and accurate
method for the numerical evaluation of the infinite integral in
(6) is still of research interest.
Direct numerical integration of the infinite integral is the
method used in this contribution for the evaluation of (6).
Instead of using the previously mentioned integration
methods,
a novel integration technique is used, based on combinations
of integration methods. More specifically, the Gauss-
Legendre method, a highly accurate numerical integration
method applicable in finite intervals of functions is combined
with two other methods: the Gauss-Laguerre method, which is
best suited for infinite integrals and the Lobatto rule, a very
efficient method for oscillative functions [14]. The selective
implementation of the different integration methods in the
intervals between the roots of
cos( ax)
, for both the real and
the imaginary part of J , leads to a quick and very efficient
integration scheme for the evaluation of (6).
m
IV. THE FINITE ELEMENT APPROACH
The problem of the calculation of the cable series
impedance matrix in (1) could be greatly simplified, assuming
that the per unit length voltage drop V
i
on every conductor is
known for a specific current excitation. The mutual complex
series impedance per unit length Z ′ between conductor i and
another conductor j carrying current I
j
, where all other
conductors are forced to carry zero currents, is then given by:
ij
Vi
Zij
′ = ( i, j = 1, 2, …, n)
(8)
I
j
The self impedance of a conductor may also be calculated
from (8), by setting i = j. In such a case, the following
procedure may be used for the calculation of the cable series
impedance matrix [13]:
• A sinusoidal current excitation of arbitrary magnitude is
applied sequentially to each cable conductor, while the
remaining conductors are forced to carry zero currents.
The corresponding voltages are recorded.
• Using (8), the self and mutual impedances of the j cable
conductor may be calculated. This procedure is repeated n
times, in order to calculate the impedances of n
conductors.
Therefore, the problem is reduced to that of calculating the
actual per unit length voltage drops, when a current excitation
is applied to the conductors. This may be achieved by a
suitable Finite Element Method (FEM) formulation of the
electromagnetic diffusion equation.
An underground power cable arrangement, consisting of
n parallel conductors, is assumed to be long enough to ignore
end effects. Furthermore, if the current density vector is
supposed to be in the z direction, the problem becomes twodimensional
and it is confined in the x-y plane, in which the
conductors’ cross sections lie. The above two-dimensional
diffusion problem is described by the system of equations
[15],
where
2 2
1 ⎡∂
Az
∂ A ⎤
z
⎢ + j A
2 2 ⎥− ωσ
z
+ J
µµ
0 r ⎣∂x
∂y
⎦
z sz z
sz
= 0 (9)
− jωσ
A + J = J
(10)
∫∫ JdS
z
= Ii
, i=1, 2, ... , n (11)
Si
A
z
is the z direction component of the magnetic vector
potential (MVP).
In (10) the total current density
two components,
J z
is decomposed in

J
z
= Jez + Jsz
(12)
where J
ez
is the eddy current density and J sz
the source
current density, given by (13) and (14) respectively.
J =− jωσ
A
(13)
ez
J
sz
z
=−σ∇Φ (14)
FEM is applied for the solution of (9) and (10) with the
boundary conditions of (11). Values for J on each
conductor i of conductivity σ
i
are then obtained and equation
(8) takes the form
V J
sz
σ
i
i i
Zij
= = ( i, j = 1, 2, …, n)
(15)
I I
j
j
linking properly electromagnetic field variables and
equivalent circuit parameters.
A distinct advantage of the FEM is its capability of
handling cases of geometrical or electromagnetic ground
irregularities of almost any kind, where conventional methods
fail.
V. NUMERICAL RESULTS
The proposed numerical integration scheme was applied in
the following cases of SC cable arrangements in homogeneous
ground.
First the case of a shallowly located SC cable system of
Fig. 2 was examined [12] in a horizontal (a) and a vertical (b)
cable arrangement. Cable is in a depth h=1,2 m with a spacing
−3
s=0,25m. The core radius is = 23,4 ∗10
m, while the outer
radius is
r S
= 48,4 ∗10
−3
sz i
m. The conductance of the cable core
is σ = 5, 88235∗10 7 S/m and µ = 1 for the ground.
r C
r
air
earth
h r c
r c r c
h
r c
air
earth
compared to the corresponding obtained by EMTP and FEM
for earth resistivities from 2 Ωm up to 1000 Ωm and for a
frequency range of 1 Hz to 10 MHz. For each case the relative
differences of the form of (16) were calculated
ZNum
− ZEMTP
Relative difference (%) = ∗ 100 (16)
Z
EMTP
In Fig. 3 the relative differences for the magnitude of the
self impedance of the horizontal cable arrangement of Fig. 2
are shown. Differences up to 12% occur, especially for low
earth resistivities and high frequencies.
difference(%)
40%
30%
20%
10%
ρ=2 Ωm ρ=10 Ωm
ρ=50 Ωm ρ=100 Ωm
ρ=200 Ωm ρ=300 Ωm
ρ=500 Ωm ρ=1000 Ωm
0%
1,E+00 1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06 1,E+07
frequency(Hz)
Fig. 3: Differences in Z 11 between numerical integration and EMTP.
In Fig. 4 the corresponding differences for the magnitude
of the mutual impedance of the cable arrangement are shown.
The differences in this case are due only to the different
approaches in earth return impedance calculation and are
higher reaching up to 25%.
difference(%)
40%
30%
20%
10%
ρ=2 Ωm ρ=10 Ωm
ρ=50 Ωm ρ=100 Ωm
ρ=200 Ωm ρ=300 Ωm
ρ=500 Ωm ρ=1000 Ωm
r s
r s
r s
s
a
s
Fig. 2: Horizontal (a) and vertical (b) SC cable arrangements in homogeneous
ground.
Series impedances were calculated for this configuration
using the proposed integration scheme and they were
s
s
b
r s
r c
r c
r s
r s
0%
1,E+00 1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06 1,E+07
frequency(Hz)
Fig. 4: Differences in Z 12 between numerical integration and EMTP.
Next the case of a SC core cable with core and sheath, as in
Fig. 5 is considered. Original cable data from [3] are
reproduced in Table I. The cable arrangements are similar to
those in Fig. 1 with h=0,75m, s=0,15m and µ
r
= 1 for both
media.
The differences of (16) were calculated for the above cable
arrangement using the previous earth resistivities and
frequencies. In Fig. 6 and Fig. 7 these differences are shown
for the case of vertical configuration, for the real and the
imaginary parts of the mutual impedance between the cable

sheaths respectively. Differences of almost 25% are recorded
The results of the numerical integration were also checked
against the corresponding by FEM for all above cases.
Differences calculated by a formula similar to (16) were less
than 1,5% in all cases over the whole range of frequencies
sheath
conductor
r 1
r 2
r 3
r 4
impedance matrices of cables buried in a two-layered earth.
The two horizontal arrangements, discussed previously, were
considered.
difference (%)
40%
30%
20%
10%
ρ=2 Ωm ρ=10 Ωm
ρ=50 Ωm ρ=100 Ωm
ρ=200 Ωm ρ=300 Ωm
ρ=500 Ωm ρ=1000 Ωm
.
difference (%)
insulation
Fig. 5: Single Core cable with core and sheath.
Core radius
1
TABLE I.
DATA OF THE SC CABLE OF FIG. 5
Main insulation radius r
2
Sheath radius r
3
Outer insulation radius 4
r 2
Dc resistance of copper core
Dc resistance of lead sheath
40%
30%
20%
10%
ρ=2 Ωm ρ=10 Ωm
ρ=50 Ωm ρ=100 Ωm
ρ=200 Ωm ρ=300 Ωm
ρ=500 Ωm ρ=1000 Ωm
1, 27 ∗10 − m
2, 28∗10 −2 m
2,54∗10 −2 m
r
2
2,79∗10 − m
0,034 Ω/km
0,436 Ω/km
0%
1,E+00 1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06 1,E+07
frequency (Hz)
Fig. 6: Real part of the mutual impedance of cable sheaths. Differences
between numerical integration and EMTP.
. The numerical integration scheme proved to be absolutely
numerically stable. The computation time for the numerical
integration was less than 5s for a set of 280 resistivity and
frequency combinations using an Intel Pentium IV PC at
1,7 GHz.
The case of a multi-layered earth was considered next.
Sunde’s extension [8] refers only to conductors over, or on the
surface of a multi-layered earth. For the case of underground
conductors no solution for the electromagnetic field exists.
Therefore the FEM was used for the calculation of the
0%
1,E+00 1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06 1,E+07
frequency(Hz)
Fig. 7: Imaginary part of the mutual impedance of cable sheaths. Differences
between numerical integration and EMTP.
Six different two-layered earth models were investigated,
based on actual grounding parameter measurements [16]. The
corresponding data for the resistivities of the first ρ 1
and
second ρ
2
layer and for the depth d of the first layer are
shown in Table II. The second layer is of infinite extend.
TABLE II.
TWO-LAYERED EARTH MODELS
ρ 1 (Ωm) ρ 2 (Ωm) d(m)
CASE I 372,729 145,259 2,690
CASE II 246,841 1058,79 2,139
CASE III 57,344 96,714 1,651
CASE IV 494,883 93,663 4,370
CASE V 160,776 34,074 1,848
CASE VI 125,526 1093,08 2,713
Cable impedances obtained from FEM for the frequency
range of 50 Hz to 1 MHz were compared to the corresponding
from the proposed numerical integration scheme, under the
assumption of homogeneous earth. The differences calculated
by a formula similar to (16) are shown in Fig. 8 concerning
the magnitude of the mutual impedance between the cable
sheaths of the SC cable in Fig. 5. The homogeneous earth
model was assumed to have the resistivity ρ
1
of the first earth
layer. Differences of up to 40% are encountered, even at
power frequency, especially in cases of great divergence
between the resistivities of the two layers. The differences
seem to be
minimized near the frequencies, for which the penetration
depth p given from (17) approaches the depth of the first
layer.
1
p = (17)
π f µσ
0

Finally, in Fig. 9 the differences between FEM and
numerical integration in the results for the magnitude of the
self impedance are shown. The horizontal SC cable
arrangement in Fig. 2 was examined for the earth models IV
50%
40%
case Ι
case ΙΙΙ
case ΙΙ
case ΙV
CONSTANTS/PARAMETERS support routine, due to the
assumptions implemented in the latter calculation procedure.
Application of FEM in cases of underground cables in a
two-layered earth showed differences 15% - 40% from those
obtained with the assumption of homogeneous earth. These
differences cannot be disregarded. They justify the need for
the development of a mathematical model for multi-layered
earth in the case of underground conductors as well as
methods for its numerical evaluation.
case V
case VI
difference(%)
30%
20%
10%
0%
1,E+00 1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06
frequency(Hz)
Fig. 8: Two-layered earth case. Differences in the magnitude of the mutual
impedance.
and VI. In these models the earth resistivities of the first and
second layer have their maximum values. FEM results are
compared to those corresponding for homogeneous earth
having the resistivity of either the first or the second layer.
difference (%)
50%
40%
30%
20%
10%
caseIV, ρ=ρ1
case VI,ρ=ρ1
case IV, ρ=ρ2
case VI,ρ=ρ2
0%
1,E+00 1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06
frequency (Hz)
Fig. 9: Two-layered earth case. Differences in the magnitude of the self
impedances.
VI. CONCLUSIONS
The problem of the calculation of the earth return
impedances of underground SC power cables was addressed
in this contribution. The available approaches for the
numerical evaluation of the infinite integrals and the
corresponding assumptions and limitations are reported.
A novel numerical integration scheme, based on proper
combinations of integration methods, is proposed. The scheme
proved to be numerically stable and efficient in all examined
cases and lead to results whose validity was justified using a
proper FEM formulation. These results showed differences
10%-25% against those computed from EMTP CABLE
VII. REFERENCES
[1] J. R. Carson, ‘Wave propagation in overhead wires with ground return’,
Bell Syst. Tech. J., Nr. 5, pp.539-554, 1926.
[2] F. Pollaczek, ‘Ueber das Feld einer unendlich langen
wechselstromflossenen Einfachleitung’, Elektrische Nachrichtentechnik
vol.3, pp. 339-359, 1926.
[3] L. M. Wedepohl, D. J. Wilcox, ‘ Transient analysis of underground
power-transmission systems. System model and wave propagation
characteristics’, Proc. IEE, vol. 120, No. 2, pp. 253-260, 1973.
[4] A. Ametani, ‘ A general formulation of impedance and admittance of
cables’, IEEE Trans. on P.A.S , vol. PAS-99, No 3, pp. 902-910, 1980.
[5] H. W. Dommel, ‘Electromagnetic Transients Program Reference
Manual’, Boneville Power Administration, Portlant OR, 1986.
[6] O. Saad, G. Gaba, M. Giroux, ‘ A closed-form approximation for ground
return impedance of underground cables’, IEEE Trans. on Power
Delivery, vol. PWRD-11, No 3, pp. 1536-1545, 1996.
[7] T. T. Nguen, ‘ Earth return path impedances of underground cables –
Pt.1 Numerical integration of infinite integrals, Pt. 2 Evaluations using
neural networks’, Proc. IEE – Generation, Transm., Distr., vol. 145, No
6, pp. 621-633, 1998.
[8] E. D. Sunde, ‘ Earth Conduction effects in transmission systems 2nd
Edition’, Dover Publications, 1968, p. 113.
[9] M. Nakagawa, A. Ametani, K. Iwamoto, ‘Further studies on wave
propagation in overhead transmission lines with earth return: Impedance
of stratified earth’, Proc. IEE, vol. 120, No 12, pp.1521-1528, 1973
[10] M. Nakagawa, K. Iwamoto, ‘ Earth return impedance for the multi-layer
case’, IEEE Trans. on P.A.S , vol. PAS-95, No 2, pp. 671 - 676, 1976
[11] S. Cristina, M. Feliziani, ‘ A finite element technique for multiconductor
cable parameters calculation’, IEEE Trans. on Magnetics, vol. 25, No. 4,
pp. 2986-2988, 1989.
[12] Y. Yin, H. W. Dommel, ‘ Calculation of frequency dependent
impedances of underground power cables with finite element method’,
IEEE Trans. on Magnetics, vol. 25, No. 4, pp. 3025-3027, 1989.
[13] G. K. Papagiannis, D. G. Triantafyllidis, D. P. Labridis, ‘ A One-Step
Finite Element Formulation for the Modeling of Single and Double
Circuit Transmission Lines’, IEEE Trans. On Power Systems, vol 15, Nr.
1, pp.33-38, 2000.
[14] P. Davies, P. Rabinowitch, ‘ Methods of Numerical Integration – 2nd
Edition’, Academic Press, 1984, pp. 104, 223
[15] D. Labridis, P. Dokopoulos: ‘Finite Element Computation of Field,
Losses and Forces in a Three-Phase Gas Cable with Non-Symmetrical
Conductor Arrangement’, IEEE Trans. on Power Delivery, Vol. PWRD-
3, No.4, pp. 1326-1333, 1988.
[16] J. L. del Alamo: ‘A Comparison among different techniques to achieve
an optimum estimation of electrical grounding parameters in two-layered
earth’, IEEE Trans. on Power Delivery, vol. PWRD-8, No. 4, pp. 1890-
1899, 1993.
VIII. BIOGRAPHIES
Grigoris Papagiannis (S’79-M’88) was born in Thessaloniki, Greece, on
September 23, 1956. He received his Dipl.-Eng. Degree and the Ph.D. degree
from the Department of Electrical Engineering at the Aristotle University of
Thessaloniki, in 1979 and 1998 respectively.
Since 1981 he has been working as research assistant and since 1998 as
Lecturer at the Power Systems Laboratory of the Department of Electrical and
Computer Engineering at the Aristotle University of Thessaloniki, Greece. His

special interests are power systems modeling and computation of
electromagnetic transients.
Dimitrios Tsiamitros was born in Kozani, Greece, on March 19, 1979. He
received his Dipl.-Eng. Degree from the Department of Electrical and
Computer Engineering at the Aristotle University of Thessaloniki, in 2001.
Since 2001 he has been a postgraduate student at the Department of Electrical
and Computer Engineering at the Aristotle University of Thessaloniki. His
special interests are power system modeling and computation of
electromagnetic transients.
Georgios Andreou (S' 98-A' 02) was born in Thessaloniki, Greece, on August
16, 1976. He received his Dipl.-Eng. Degree from the Department of
Electrical and Computer Engineering at the Aristotle University of
Thessaloniki, in 2000.
Since 2001 he has been a postgraduate student at the Department of Electrical
and Computer Engineering at the Aristotle University of Thessaloniki. His
special interests are power system analysis and powerline communications.
Dimitris Labridis (S' 88-M' 90-SM' 00) was born in Thessaloniki, Greece, on
July 26, 1958. He received the Dipl.-Eng. Degree and the Ph.D. degree from
the Department of Electrical Engineering at the Aristotle University of
Thessaloniki, in 1981 and 1989 respectively.
During 1982-2001 he has been working at first as a research assistant and later
as Lecturer and as Assistant Professor at the Department of Electrical
Engineering at the Aristotle University of Thessaloniki, Greece. Since 2001 he
has been Associate Professor at the same Department. His special interests are
power system analysis with special emphasis on the simulation of
transmission and distribution systems, electromagnetic and thermal field
analysis, numerical methods in engineering, artificial intelligence applications
in power systems and powerline communications.
Petros S. Dokopoulos (M’77) was born in Athens, Greece, in September
1939.He received the Dipl. Eng. degree from the Technical University of
Athens in 1962 and the Ph.D. degree from the University of Brunswick,
Germany, in 1967. During 1962–1967 he was with the Laboratory for High
Voltage and Transmission at the University of Brunswick, Germany, during
1967–1974 with the Nuclear Research Center at Julich, Germany, and during
1974–1978 with the Joint European Torus. Since 1978 he has been Full
Professor at the Department of Electrical Engineering at the Aristotle
University of Thessaloniki, Greece.
He has worked as consultant to Brown Boveri and Cie, Mannheim, Germany,
to Siemens, Erlagen, Germany, to Public Power Corporation, Greece and to
National Telecommunication Organization and construction companies in
Greece. His scientific fields of interest are dielectrics, power switches, power
generation (conventional and fusion), transmission, distribution and control in
power systems.
Prof. Dokopoulos has 81 publications and 7 patents on these subjects.