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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 1, JANUARY 2007
Inclusion of Frequency-Dependent Soil Parameters in
Transmission-Line Modeling
Antonio Carlos Siqueira de Lima, Member, IEEE, and Carlos Portela, Life Senior Member, IEEE
Abstract—This paper presents an analysis of transmission-line
modeling when the soil conductivity and permittivity are treated as
frequency dependent. It also presents a methodology to include this
particular behavior on frequency-dependence realization of transmission lines and cables. The frequency-dependent soil parameters
can be synthesized with a simple first-order model. It demands only
three parameters statistically independent, one being associated
with the low-frequency conductivity as obtained in conventional
modeling and the other two are a function of the frequency dependence in soil conductivity and permittivity. The new formulas used
for the ground return impedances for overhead lines and underground cables are presented. Instead of using an analytical approximation of the infinite integrals involved, a numerical approximation using a Gauss quadrature technique was implemented. These
expressions can be understood as an extension of the well-known
Carson and Pollaczek formulas. Three distinct cases are presented
in order to asses the impact of frequency-dependent soil parameters with respect to the asymmetry of the circuit. The effect of frequency dependence in soil parameters is evaluated using frequency
and time domain. In the frequency domain, modal analysis is applied to assess the overall impact in the ground mode propagation.
To obtain the time-domain response, the line was modeled in the
frequency domain using a phase-domain approach and a numerical Laplace transform was applied.
Index Terms—Electromagnetic transient analysis, power cables,
soil measurements, transmission lines.
I. INTRODUCTION
HERE has been a considerable evolution in the modeling
of frequency dependence of overhead transmission lines
and underground cables. Although the frequency dependence of
the ground and conductors impedances can be more easily dealt
with in frequency-domain analysis, these days there are some
models that can accurately take into account the frequency dependence in time-domain analysis. Independent of whether one
uses the time-domain or frequency-domain approach, one point
of paramount importance remains—the accuracy of the line parameters, namely, the impedance per unit of length and the admittance per unit of length. The former involves the skin effect
on conductors and ground. Usually the ground parameters are
poorly known as they have a wide range variation with some
random phenomena, such as seasonal changes and the amount of
water retained. Semlyen [1] presented a methodology to quan-
T
Manuscript received August 23, 2004; revised March 28, 2006. This work
was supported in part by “Fundação de Amparo à Pesquisa Carlos Chagas,”
FAPERJ, Rio de Janeiro, RJ, Brazil and in part by “Conselho Nacional de Desenvolvimento Científico e Tecnológico,” CNPq, Brazil. Paper no. TPWRD00338-2004.
The authors are with the Federal University of Rio de Janeiro, Rio de Janeiro
21945-970, Brazil (e-mail: acsl@dee.ufrj.br; portelac@ism.com.br).
Digital Object Identifier 10.1109/TPWRD.2006.881582
tify the effect of the variation of soil resistivity , using the anand the line inductance, when the
alytical relation between
soil is treated as a good conductor and the complex plane approximation is assumed [2]. A recent study [3] shows, however,
that except in the case of soil ionization, the soil behaves linand permittivity
are sigearly although its conductivity
nificantly frequency dependent [4]. Therefore, the good conductor assumption does not hold for a wide frequency range.
For instance, in the lightning frequency range, above hundreds
can be of the same order of magnitude of .
of kilohertz,
Furthermore, the usual assumption as a pure conductive soil is
not suitable for induced effects and electromagnetic compatibility (EMC), including equipment and people safety [5].
An analysis of the influence of frequency-dependent soil
parameters was presented in [4] using a quasimodes approach
[6]. The frequency-dependent soil parameters were considered
using the complex plane method where the ideal soil is at
below actual soil in a
a depth
procedure similar to the one presented in [2] but now replacing
by
. This is fully justified by the fact that the
appears directly in the frequency-domain
parameter
Maxwell equations. So any correct result in the assumption of
may be used for
by replacing
by
.
For frequencies up to some megahertz, this usually gives results
close to those obtained with the Carson integral formula (also
by
) and mathematical manipulation is
replacing
faster and easier. Such formulation corresponds to a first-order
approach of asymptotic development of the Carson integral
formula and, in most cases, gives an acceptable error. However,
there may be cases where the error is larger, for instance,
for high-resistivity ground, the maximum error is more than
10%. Therefore, in the present work, a stable and accurate
numerical approximation of the Carson and Pollaczek integral
is used. Another approximation used in [4] is the usage of a
time-domain model based on modal domain with a constant
transformation matrix, namely the quasimodes approach.
In the present work, we try to overcome both limitations as the
line is modeled in phase coordinates in the frequency domain,
and the time-domain responses are obtained via a numerical
Laplace transform. In an attempt to produce a thorough analysis
of the impact of soil parameters as a function of the asymmetry
of the transformation matrix, three configurations are considered, namely: a single horizontal circuit, a double vertical circuit, and an underground cable circuit.
II. GROUND MODELING
When the soil is treated as a conductor, as originally developed by Carson for overhead lines and Pollaczek for under-
0885-8977/$20.00 © 2007 IEEE
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DE LIMA AND PORTELA: INCLUSION OF FREQUENCY-DEPENDENT SOIL PARAMETERS
ground cables, we must have a medium which is linear, homogeneous, and isotropic. These assumptions together with the transverse electromagnetic (TEM) or quasi-TEM propagation imply,
in general, that the obtained solution has an acceptable error in
a frequency range up to 1 MHz depending on the soil parameters, line geometry, and circuit length. For a very short line or
very high frequency, above some megahertz, it is safer and more
accurate to consider either a spherical wave approach, where
the line conductors are represented via short-length elements
with adequate boundary conditions, or use a procedure based
on Hertz potentials and some mathematical approximations.
In this work, the impedance formulas are derived directly
from the Maxwell equations, but assuming TEM or quasi-TEM
propagation. The field equations were solved via a double
spatial Fourier integral transform applied to the spatial dimensions. The Appendix provides more details on the formulation
of ground return impedances for an underground cable. For an
overhead line, the procedure is essentially the same, although
not identical. In the following subsection, we present the ground
impedance formulas used in this work.
A. Ground Return Impedances
For overhead lines, the ground return self impedance is given
by
(1)
while the mutual impedance is given by
493
Equations (1) and (2) can be considered an extension of the
Carson impedance formula for complex ground parameters. The
same reasoning is true for underground cables assuming
and
, thus
.
All of the definite integrals in (1)–(4) were evaluated numerically using a Gauss quadrature procedure (a Gauss–Kronrod algorithm was used [7]). The basic idea behind Gaussian quadrature is to evaluate an integral as a linear combination of values
of the integrand calculated at specific points. Thus
(5)
to obtain the best numerical estimate of an integral, we must
.
pick optimal abscissas at which to evaluate the function
The fundamental theorem of Gaussian quadrature states that the
optimal abscissas of the -point Gaussian quadrature formulas
are precisely the roots of the orthogonal polynomial for the same
interval and weighting function. The weights corresponding to
the Gaussian abscissas are calculated via a Lagrange interpowith
lating polynomial for
(6)
This algorithm is an adaptive method in which the error is
estimated based on evaluating the function at some particular
points. Kronrod defined in 1964 [8] how to identify these points
based on a Legendre–Gauss quadrature [9].
B. Frequency-Dependent Soil Parameters
(2)
,
is the horizontal distance
where
is the conductor
between conductor and conductor ,
height, and
and are the soil conductivity and permittivity,
respectively.
For underground cables, the ground return self impedance is
given by
(3)
while the mutual impedance is given by
(4)
where
and
, is the
conductor external radius, is the depth of burial for conductor
,
is the horizontal distance between conductor and ,
, is the same as above, and
,
are Bessel functions.
A simple minimum phase function is used for the realization of the frequency dependence of soil parameters. It demands
three statistically independent elements, one being associated
with the low-frequency conductivity , as obtained by common
measurement procedures and the other two with the frequency
and
[5]. Thus
dependence of both
(7)
where
is the low-frequency soil conductivity in
as obtained by conventional techniques. Despite its simplicity, this
model has proven useful in several soil samples obtained at eight
different sites in Brazil, covering distinct soil and geological
structures. A detailed description of the measurement procedure
is presented in [3], including experimental setup, methods for
soil sampling, and checking physical consistency (i.e., propagation speed at any given frequency below the speed of light).
For the sake of clarity, the basic aspects of this procedure are
summarized here: to minimize the effects of local soil heterogeneity and the samples must keep their structure and humidity.
The measurements depend on whether the soil is clay, sand, or
rock.
After the measurements, the soil samples were fitted to a
curve assuming
(8)
where
is in
and together with is responsible for the
frequency dependence of soil conductivity and permittivity. To
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 1, JANUARY 2007
evaluate the probability density functions associated with paand , Weibull approximations were considered.
rameters
The boundary conditions are related to and not only or
independent as the soil has immittance characteristics, thus the
is treated altogether.
frequency dependence in both and
These conditions were formulated in the frequency domain following the procedures presented in [10].
For practically all of the large numbers of soil samples,
covering different soil types and geological structures in a
frequency range up to 2 MHz, this type of formula reproduces
measured frequency dependence within measurement errors
and local soil heterogeneity effects [3], [5]. With this type of
formula, there is statistical independence between the three
parameters , , and . The statistical distribution of paramhas some interesting properties. While the
eters , , and
measurements in
may differ in several orders of magnitude,
has a much narrower spread. As discussed
the term
in [5], for most purposes, it may be acceptable to consider meand , while
is based upon specific
dian values for both
measurements in a first analysis. For the test cases presented
here, we have chosen a specific set of measurements in one site
,
,
in Brazil for which
. Thus, for in S/m and with in rad/s
and
as a new capacitance will appear between the actual ground
and the point where the voltage can be considered zero. However, there are some questions regarding an accurate formula to
take this effect into account and this topic is open for further
research.
The matrix square root and matrix exponential can be calculated using eigenvalue/eigenvector decomposition or directly
in the phase domain via Schur Decomposition. There are some
other possibilities to calculate the matrix square root; for inis a square matrix, its
stance, an iterative mapping [14]. If
square root can be calculated as
where the initial guess for
is an identity matrix. Although
not used in the present work, this iterative procedure is quite
efficient in obtaining the matrix square root. Schür decomposition was used for the evaluation of the matrix square root involved in the line modeling. Also, the nodal admittance was
calculated directly in the phase domain without resorting to
eigenvalue/eigenvector decomposition. The line nodal admitis then given by
tance matrix
(9)
and in a large number of measureThe median values of
ments, covering very different types of soil, are, respectively,
11.7094 mS/m and 0.705519. We must point out that although,
in principle, a better fitting with measurement can be obtained
with a higher order functions, special care must be taken to ensure physical consistency of the model and to avoid the effects
of local heterogeneities in the soil samples. Further details regarding the fitting of soil parameters and the experimental setup
and results can be found in [5] and [11].
III. LINE MODELING
The transmission lines are represented via the well-known
nodal admittance matrix. For a line length , the characteristic
admittance
and propagation matrix , which are calculated
using the line-impedance per unit of length and the line admittance per unit of length , are defined as
(10)
For transmission lines,
can be written as
(11)
is the internal conductor impedance,
is the ideal
where
is the ground return impedance, all
external impedance, and
matrices per unit of length. When Corona is neglected, the insulator surface is assumed clean, and pollution effects and other
insulator losses are negligible, can be assumed, for most purposes, to be proportional to the angular frequency .
For underground cables, all of the impedances per unit of
length are assembled in a matrix according to [12] and [13]. The
inclusion of frequency-dependent parameters should also affect
(12)
where
is the identity matrix.
IV. TEST CASES
For an evaluation of the impact of frequency-dependent soil
parameters in line modeling, three test cases are considered: a
single horizontal circuit, a vertical double circuit, and an underground cable system. The goal is to assess the impact of the soil
parameters as a function of the system asymmetry. In all subis used for conventional soil usage
sequent figures, the label
while
is applied to frequency-dependent soil as defined by
(9). Both the classical soil usage and the complex one use the
numerical integration for the evaluation of the definite integrals
and both approaches represent the line using (12).
Both overhead lines tests use EHS ground wires and all phase
m and conductor diameter
conductors have
29.765 mm. For the double circuit, each phase has a bundle
of four conductors. The underground system consists of three
single-core cables with sheath but without armor. Fig. 1 shows
the configuration tested. Table I shows the conductors data for
and
are the insulathe underground system. In that table,
tion permittivities (core to sheath and sheath to ground), while
and
are, respectively, the sheath and the core conductor
resistivity.
A. Modal Propagation
The modal propagation parameters were calculated using a
complex and frequency-dependent transformation matrix. To
avoid artificial-mode switchover, a switch back procedure based
on [15] was used. For all cases tested, including some tests that
are not shown in the paper, it was found that the inclusion of
the ground parameters frequency dependency has a negligible
effect in the transformation matrix and can only be noted for
frequencies above 1 MHz.
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DE LIMA AND PORTELA: INCLUSION OF FREQUENCY-DEPENDENT SOIL PARAMETERS
495
Fig. 3. Modal damping for horizontal circuit.
Fig. 1. Test cases: (a) horizontal circuit, (b) double vertical circuit, (c) underground cables, and (d) single-core cable.
TABLE I
UNDERGROUND CABLE SYSTEM DATA
Fig. 4. Modal velocities for double vertical circuit.
Fig. 2. Modal velocities for horizontal circuit.
Fig. 2 shows the modal propagation velocities for the horizontal circuit. In that figure, we see that, as expected, mainly the
ground mode is affected. The modal damping is shown in Fig. 3
and again the mode with the higher damping (i.e. ground mode),
is the most affected. A similar behavior was also found for the
double vertical circuit (Figs. 4 and 5). Although not shown in
this paper, tests on other overhead transmission configurations
have also presented similar behavior.
For underground cable, the mismatch is considerably smaller
as can be seen in Fig. 6 for the modal velocities and in Fig. 7
for the modal damping. This is not an unexpected result as the
sheath provides a shielding effect, thus decreasing the impact of
the soil parameters.
Fig. 5. Modal damping for double vertical circuit.
B. Time-Domain Results
To evaluate the impact of the proposed ground parameters
representation, a step voltage was applied at one terminal while
all of the others remained open. Fig. 8 shows the circuit used to
test all line configurations as well as the terminal labeling for
each case.
Fig. 9(a) and (b) shows the voltage at terminal 4 and 6, respectively. At terminal 4, the inclusion of in the line parameters
led only to higher damping. At terminal 6, on the other hand, the
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 1, JANUARY 2007
Fig. 6. Modal velocities for underground cables.
Fig. 7. Modal damping for underground cables.
Fig. 9. Voltage output for the horizontal circuit. (a) Voltage at terminal 4.
(b) Voltage at terminal 6.
voltage, there is nevertheless a sensible difference as can be
seen in Fig. 10(b). The waveforms in the mentioned figure differ
in terms of damping and highest values. This result indicated
that whenever induced voltages are of concern, there should be
greater care regarding the ground parameters.
For the underground system the differences are not so pronounced since the modal propagation and attenuation are quite
similar regardless of the ground representation. Fig. 11 shows
the voltage at terminal 12 of the underground cable system.
V. CONCLUSION
Fig. 8. Circuit used to test the line response. (a) Circuit for horizontal line.
(b) Circuit for the double circuit vertical line and underground cable.
induced voltage presents a distinct behavior and there is higher
damping also but the waveforms present some significant differences. In terms of the highest overvoltage, there is no distinction
among the waveform in the figures since the fastest mode is not
affected by the ground parameters.
The results for the double vertical circuit are shown in
Fig. 10(a) and (b). For the voltage at terminal 7, the result is
essentially the same as in the previous test. For the induced
This work has focused on testing and analyzing the overall
impact of ground parameters (namely, the ground conductivity
and permittivity) on transmission-line modeling and transient
behavior. It has presented extensions of ground return impedances formula for both overhead transmission circuits and underground cable systems.
The transient behavior of any transmission line depends on
ground parameters, which are strongly frequency dependent
and, in principle, must be obtained by measurement. The most
common procedures assume that ground conductivity is constant (frequency independent) and neglect ground permittivity.
These assumptions are quite far from reality and may lead to
important errors in the computed transient of transmission lines.
The time-domain step responses showed that some noticeable
difference can be found in all phases.
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DE LIMA AND PORTELA: INCLUSION OF FREQUENCY-DEPENDENT SOIL PARAMETERS
497
frequency dependence of the ground parameters, a general statistical distribution can be considered.
The ground return impedances were calculated using Gauss
quadrature techniques which cause the overall process to be considerably slow. Nevertheless, the inclusion of the frequency dein the modified Carson and Pollaczek
pendence of and
equations have not caused any extra numerical burden. In fact,
the inclusion of this phenomenon increased the rate of convergence of the numerical integral. The authors believe this result
stresses the fact that higher accuracy can improve the numerical
performance of a model.
The inclusion of the ground permittivity will also affect the
line/cable admittance per unit of length. However, this tends to
be more important in underground cables as the conductors’
geometrical distances are smaller and is a subject for future
research as the present work deals mainly with the impact of
ground conductivity and permittivity frequency dependence on
the line-series impedance per unit of length and their impact on
line behavior.
As for the high-frequency range, above some hundreds of
becomes considerably larger than , it is inkilohertz,
teresting to further test the modeling presented here in surge
cases or in nonuniform transmission lines (i.e., involving tower
impedance and grounding).
APPENDIX
FORMULATION OF GROUND RETURN IMPEDANCES
Fig. 10. Voltage output for double vertical circuit. (a) Voltage at terminal 7.
(b) Voltage at terminal 12.
The ground return impedances are calculated under the
following assumptions: time functions are of harmonic type
, the propagation along the line and surrounding media
type, where is the direction
is exponential of the
of propagation and is the unknown propagation function.
All media are supposed linear, isotropic, and homogeneous.
Thus, the impedances can be found directly from the solution
of Maxwell’s equations, which lead to the following:
(13)
Furthermore
(14)
Fig. 11. Sheath overvoltage due to step voltage in the core of the first cable.
Measurements performed in a large number of soils, covering
different types of soil, have enabled identifying some important
properties of frequency behavior of soil parameters. Namely, it
has been found that it is possible to consider for soil parameters, in what concerns frequency dependence, two independent
terms, one related to soil conductivity at low frequency, that
correspond to the most common measurement procedures, and
others related to frequency dependence of both conductivity and
permittivity, which is not usually considered. This behavior allows deriving fairly accurate ground parameters based only on
measurements of the low-frequency conductivity while for the
If we consider , the electric field in the air and
the electric
field in the ground, (14) can be divided into two equations
(15)
(16)
where
,
is the coordinate of the
is the conductor depth
(the negconductor, thus
ative -axis is the -axis) and is the current in the conductor.
The boundaries conditions are
(17)
(18)
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 1, JANUARY 2007
The third boundary condition is that all fields vanish at infinity
for
). To solve (16), a bidimensional
(e.g.,
and . For
Fourier transform is applied to the coordinates
the -axis
(19)
where
After some manipulation, we obtain for
For the -axis, a Fourier sine transform is applied
(28)
(20)
For the air,
and
are used as coordinates, thus
and
is achieved by applying the
The final solution of
as defined in (28), which
inverse Fourier transform using
gives for
(29)
(21)
while for
(30)
Applying the Fourier transform to
the electric field in the air:
leads to the following for
where
(22)
(31)
(23)
and
,
, is the
depth of burial of the cable. The ground return impedance can
be defined by
Using the sine transform in (22) leads to
For the ground, we have
and as coordinates and
(32)
(24)
By a similar process, we reach the following:
REFERENCES
(25)
Using
and
, where is the outer radius of the
For
conductor, this formula leads to (3) shown in the paper.
The procedure for overhead lines is similar to the one presented here. Although, for overhead lines, the sources are in the
air.
, we can write
(26)
Applying the inverse sine transform, we obtain
(27)
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lines due to inaccurate ground return modeling,” IEEE Trans. Power
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[7] W. Press, S. Teukolsky, W. Vettering, and B. Flannery, Numerical
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DE LIMA AND PORTELA: INCLUSION OF FREQUENCY-DEPENDENT SOIL PARAMETERS
[8] A. S. Kronrod, “Nodes and weights for quadrature formulae. Sixteenplace tables.” Consultants Bureau, 1965.
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499
Antonio Carlos Siqueira de Lima (S’95–M’00) was born in Rio de Janeiro,
Brazil, in 1971. He received the B.Sc., M.Sc., and Ph.D. degrees in electrical
engineering from the Federal University of Rio de Janeiro (UFRJ) in 1995, 1997,
and 1999, respectively.
Currently, he is an Associate Professor with the Electrical Engineering
Department, Federal University of Rio de Janeiro. In 1998, he was a Visiting
Scholar in the Department of Electrical and Computer Engineering, the University of British Columbia, Vancouver, BC, Canada. From 2000 to 2002, he was
with the Brazilian Independent System Operator, ONS, Rio de Janeiro, Brazil,
dealing with electromagnetic transient studies for the Brazilian National Grid.
His main interests are electromagnetic transient analysis in power systems and
analysis of electromagnetic fields.
Carlos Portela (SM’68–LSM’02) received the M.Sc. (Hons.) and Ph.D. degrees in electrical engineering from Instituto Superior Técnico (IST), Lisbon
Technical University (IST-UTL), Lisbon, Portugal, in 1958 and 1963, respectively.
In 1972, he received the “Livre-Docente” from IST-UTL. From 1958 to 1976,
he was an Associate Professor and a Cathedratric Professor with IST-UTL.
During this period, he was responsible for the Portuguese electrical network
studies and planning and electrical network operation. In 1976, he joined the
Federal University of Rio de Janeiro where he was a Professor in the Graduate School of Engineering (COPPE/UFRJ). He has been responsible for major
studies and projects in the electric power and industry sectors in Portugal, Brazil,
and other countries. His interests include research projects in transmission systems and equipment.
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