Conference PaperPDF Available

Analysis of grounding problems using Interpolation Element-Free Galerkin method with reduction of computational domain

Authors:

Figures

Content may be subject to copyright.
IEEE TRANSACTIONS ON MAGNETICS, VOL. 54, NO. 3, MARCH 2018 7200304
Proposals for Inclusion of the Electrode Radius in Grounding
Systems Analysis Using Interpolating Element-Free
Galerkin Method
Ursula C. Resende , Rafael Alípio, and Maisa L. F. Oliveira
Federal Center for Technological Education of Minas Gerais, Electrical Engineering Department, MG 35510-000, Brazil
In this paper, the interpolating element-free Galerkin meshless method is presented as a new, simple, and accurate proposal for
the analysis of electrostatic grounding problems. A very important aspect to ensure the precision of this numerical solution is the
correct modeling of the electrode radius. So, two techniques are proposed to include the electrode radius effect in the analysis. The
problem of a vertical electrode introduced in a homogeneous soil is considered as study case. The results of the electrical potential
along the ground surface and grounding resistance calculated using interpolating element-free Galerkin method, for different values
of radius, are compared with those generated by the method of moments.
Index Terms—Element-free Galerkin method (EFGM), grounding problems, meshless method (MM), vertical grounding electrodes.
I. INTRODUCTION
THE grounding system is one of the main elements in
the protection of various power systems. Its function is
fundamentally to provide a low-resistance path for the flow
of fault currents toward the soil and to ensure a smooth
distribution of the electrical potentials developed on the ground
surface (EPGS). The accurate knowledge of the grounding
resistance (RT)and EPGS is very important for grounding
design and allows one to determine the grounding potential
rise [1].
Simplified approaches can be used to determine the RTand
EPGS. However, for proper calculation of practical problems
numerical techniques should be employed. At low frequencies,
there are two basic approaches for analysis of grounding
systems parameters: the integral approach and the differential
approach. When integral equation methods are used, such
as the method of moments (MoM), the problem can be
successfully evaluated [2]. The calculation is relatively fast and
practically independent of the configuration of the grounding
system. However, since the method of images is often used
to take into account the effect of air-earth interface, for non-
homogeneous soils an infinite number of images should be
considered, leading to a very intensive computational demand.
On the other hand, complex structures and inhomogeneities
are very easily evaluated by using differential equation meth-
ods, such as the finite-element method (FEM) [3]. However,
those methods are not so suitable for solving unbounded
problems. Then, it is necessary to extend the discretization
domain by creating a fictitious boundary at some distance
where a boundary condition should be imposed. Another
weakness of the FEM is the process for mesh generation,
which requires considerable computational effort when the
problem involves complex geometries.
Manuscript received June 13, 2017; accepted October 30, 2017. Date of
publication January 4, 2018; date of current version February 21, 2018.
Corresponding author: U. C. Resende (e-mail: resendeursula@cefetmg.br).
Digital Object Identifier 10.1109/TMAG.2017.2771394
In the recent years, a new class of differential equation
methods, meshless methods (MM), which does not require a
mesh structure, has been developed. MM does not require a
mesh structure and the solution is obtained by using only
a cloud of nodes spread throughout the region of interest.
This feature makes MM appropriate to deal with complex
geometries and inhomogeneities. Among the MM available in
the literature, element-free Galerkin method (EFGM) [4], [5],
interpolating EFGM (IEFGM) [6]–[8], and meshless local
Petrov–Galerkin Method [9], for example, have been exten-
sively investigated and proven to be reliable for solving
electromagnetic problems. Especially, the IEFGM produces
interpolating shapes functions, besides being simple, extremely
robust, and have good convergence rates.
Vertical electrodes are one of the simplest and most com-
monly used means of earth termination of electrical systems.
An important aspect, that must be carefully evaluated when
numerical methods are used for solving this kind of problem,
is how the electrode radius is introduced in the mathematical
model of the problem [10]. The correct modeling of this para-
meter is indispensable to ensure the accuracy of the results.
When integral and differential methods are used, this task can
be performed in diverse ways [2], [11], [12]. In this paper, two
techniques are proposed in order to accurately incorporate the
electrode radius in the analysis of an electrostatic grounding
problem using IEFGM.
II. GROUNDING PROBLEM MODELING
A cylindrical copper electrode with length Le and radius Re,
introduced in a homogeneous soil with conductivity σis the
problem investigated in this paper. The soil geometry is a
semi-hemisphere with radio Rh =10 Le. The electrode is
introduced into the top of the hemisphere in ρ=z=0,
with a constant electric potential Ve. The Laplace equation,
2V=0, can be used for solving the problem and since
it is axisymmetric (V/∂ φ =0) only the ρzplane needs to
be considered. Therefore, the domain of the study is 2-D,
0018-9464 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
Authorized licensed use limited to: CENTRO FED DE EDUCACAO TECNOLOGICA DE MINAS GERAIS. Downloaded on July 29,2020 at 13:19:05 UTC from IEEE Xplore. Restrictions apply.
7200304 IEEE TRANSACTIONS ON MAGNETICS, VOL. 54, NO. 3, MARCH 2018
Fig. 1. Grounding problem.
as illustrated in Fig. 1, and the related Laplace equation is [13]
1
ρ
∂ρ ρV
∂ρ +2V
z2=0(1)
where V, z)is the electric potential, nis the outward normal
unit vector, V/∂n=0onnand V=Vd on dare the
Neumann and Dirichlet boundary conditions, respectively, and
Vd is a constant value calculated by using an analytical process
which approaches the electrode by a point source [14].
The weak form of the problem is obtained by using the
weighted residuals method, multiplying the residue of (1) by
a test function T [3]
TV
∂ρ +V
z·nρd
T
∂ρ
V
∂ρ +T
z
V
zρd=0(2)
where d=dρdz,=ndLe. Over n,the line
integral in (2) is equal to 0 because of the Neumann condition
and over dand Le, this integral is also equal to 0 because
Ve and Vd are constant values imposed directly in the linear
system.
III. IEFGM FORMULATION
In the MM approach, a set of Nnodes is placed in . Each
node Iis a point xI, z)for which a shape function
I(x) is associated. Each shape function is 0 over the whole
domain, except near the corresponding node. Therefore, the
unknown function can be approximated by the following trial
function [15]:
V(x)Vh(x)=
N
I=1
I(xI(3)
where x=, z)and υIis the unknown coefficient of
node I. The discretized electrical potential Vhbelongs to
the finite-dimensional subspace VNspanned by the shape
functions associated with all nodes in ndLe. The
test function T is chosen according to the Galerkin method,
i.e., T also belongs to VNand is expanded as
T(x)Th(x)=
N
J=1
J(x)cJ(4)
where the cJare the coefficients relative to the expansion
of Th, in this paper, cJ=1. IEFGM uses the moving least-
squares method (MLS) in order to construct shape functions.
This process leads to the following local approximation [4]:
Vh(x,xI)=
m
i=1
pi(xI)ai(x)pT(xI)a(x)(5)
where mrepresents the number of monomial terms in the
polynomial basis pT(x)=[1, ρ,z]anda(x) are the unknown
polynomial coefficients. In MLS approximation, the coef-
ficients a(x) are determined by minimizing the following
weighted discrete L2norm:
=
NP
I=1
W(rI)[Vh(x,xI)V(xI)]2(6)
in which W(rI)is the weight function, defined to have
compact support and centered at the node xI
rI=|xxI|
dI(7)
dI=αdc(8)
is the support of the weight function, αis a scaling factor for
the influence domain, dcis the nodal distance associated with
the node I, and NP is the number of nodes (NN) involved
in the local approximation. By minimizing (6), Ican be
determined [4].
Although the MLS generates a smooth approximation for
the unknown functions, its main weakness is the lack of
the Kronecker delta property. An interpolating MLS approx-
imation (IMLS) can be obtained by using singular weight
function (SWF) in the definition of in (6). Thus, it is possible
to obtain shape functions which satisfy the Kronecker delta
property. EFG method using IMLS is called IEFGM [6]–[8].
In this paper, the following SWF is used:
W(rI)=1/rn
I+βn(9)
where β=0.1 is a constant small enough to ensure no division
by 0 and n=10 is a constant whose value is adjusted in order
to improve the approximation accuracy.
If the trial function satisfies the essential boundary condi-
tions and the test function Th(x) vanishes at the boundary, the
Galerkin method leads to the linear system [K][υ]=0, where
KIJ =∂I
∂ρ
∂J
∂ρ +∂J
z
∂J
zρd. (10)
The KIJ elements are calculated by solving the integrals
using a two-point Gaussian quadrature applied to an auxiliary
rectangular cell structure. Therefore, the values of the shape
functions and their derivatives are calculated over the set of
integration points to assemble the linear system [3], [15].
IV. ELECTRODE RADIUS MODELING
The proper modeling of the electrode radius is a funda-
mental aspect to ensure the accuracy of the results in the
numerical analysis of grounding problems. In this paper, two
techniques are proposed to accurately include this parameter
in IEFGM approach. These techniques employ a uniform
Authorized licensed use limited to: CENTRO FED DE EDUCACAO TECNOLOGICA DE MINAS GERAIS. Downloaded on July 29,2020 at 13:19:05 UTC from IEEE Xplore. Restrictions apply.
RESENDE et al.: PROPOSALS FOR INCLUSION OF ELECTRODE RADIUS 7200304
Fig. 2. Nodes distribution RRA.
Fig. 3. Nodes distribution ERA.
rectangular distribution of nodes over the whole domain of the
problem. The electrode radius effect is introduced by adjusting
the support of the weight function (8) for nodes representing
the electrode. This adjustment allows us to define a region
around the electrode where its effect is significant and should
be considered.
The distribution of nodes in the first technique, real radius
approach (RRA), is performed considering the nodal dis-
tance dcequal to the radius of the electrode Re, as illustrated
in Fig. 2. The electrode structure is represented by a specific
set of nodes. The size of the influence domain of all nodes is
adjusted considering the factor α=1.5 [15].
Although RRA can be used in the analysis of problems
with electrodes of any radius, very small radii lead to a very
high computational effort, since a greater density of nodes
is required. Then, an alternative approach is presented, the
equivalent radius approach (ERA). In this case, the distribution
of nodes is performed by considering dc=10 Re, as illustrated
in Fig. 3. The electrode structure is represented only by
a distribution of nodes over z-axis. The size of influence
domain is adjusted by using α=1.21 for nodes representing
the electrode and α=1.5 for remaining nodes. The value
of 1.21 was determined from a parametric analysis considering
different values of Re.
Fig. 4. ITas a function of Vγ.
V. ELECTRIC CURRENT CALCULATION
The grounding resistance can be obtained by using Ohm’s
law, RT=Ve/IT,whereITis the current that disperses to
the ground calculated by the following process.
1) Calculation of Veverywhere in the problem domain.
2) Definition of an equipotential line γby using the calcu-
lated values of Vand employing search and interpolation
procedures.
3) Over γ, calculation of the electric field, E=−V.
4) Over γ, calculation of the electric current density,
J=σE,whereσis the electric conductivity.
5) Evaluation of IT=S(J·ns)dS, where dS =dφdγis
the surface element and nsis the outward unit normal
vector to the dS. Since the problem is axisymmetric, the
surface integral can be simplified to a line integral
IT=2δγ
(J·ns)dγ. (11)
Since Eis calculated by using derivatives of Vover γ,
one must choose γwith very smooth curvature, far from the
electrode, to guarantee the accuracy in the calculation of the
derivatives. This restriction can be observed in the results
presented in Fig. 4, where the values of IT, obtained by
using IEFGM for different γ, are compared with that obtained
by MoM. The value of the Vover the different γis called Vγ.
The results were produced by using a solid copper electrode
with Le =1m,Re=0.0127 m, Ve =1 V, and a fictitious
homogeneous soil with σ=1 S/m. As can be observed, the
solutions approach each other as γis taken away from the
electrode, (i.e., in regions where Vγis smaller).
VI. NUMERICAL RESULTS
The soil studied in Section V is now used to show the
IEFGM accuracy. The values of Le and Ve remain the same,
but the value of Re varies from 0.00635 to 0.0127 m for
conventional solid copper electrodes and from 0.04 to 0.1 m
for encapsulated electrodes (solid copper electrodes covered by
a concrete layer). The results obtained for EPGS are evaluated
by using the following error equation:
Emed(%)=
n
i=1
VMoM
iVIEFGM
i
VMoM
i
100
n(12)
where VMoM and VIEFGM correspond to the solution obtained
from MoM and IEFGM, respectively, and nis the number of
points where the solution is evaluated.
Authorized licensed use limited to: CENTRO FED DE EDUCACAO TECNOLOGICA DE MINAS GERAIS. Downloaded on July 29,2020 at 13:19:05 UTC from IEEE Xplore. Restrictions apply.
7200304 IEEE TRANSACTIONS ON MAGNETICS, VOL. 54, NO. 3, MARCH 2018
TAB L E I
ITAND RT—ENCAPSULATED ELECTRODES
TAB L E I I
ITAND RT—CONVENTIONAL ELECTRODES
Fig. 5. EPGS for Re =0.00635 m using ERA proposal.
Using the techniques proposed in this paper, the NN scat-
tered in the domain is defined as a function of Re, as shown
in Figs. 2 and 3. As the analysis was performed for different
values of Re, different values of NN were used, as presented
in Tables I and II, together with the results obtained. In these
analyses, the number of integration points (NPI) used was such
that the NPI/NN ratio was approximately equal to 4. As can be
observed the values of RTobtained from IEFGM and MoM
are very close to each other, which validate and demonstrate
the accuracy of proposed techniques. The maximum values
of Emed obtained were equal to 5.32% for RRA and 1.80%
for ERA. For Re =0.00635 m, the EPGS result is presented
in Fig. 5.
VII. CONCLUSION
The problem of a vertical electrode introduced in a homo-
geneous soil was evaluated using a numerical method different
from those traditionally used. As the correct modeling of
the electrode radius is a very important aspect to ensure the
convergence of the numerical solution, the main aim of this
paper was to propose two techniques to accomplish this task
for IEFGM approach. The radius effect was introduced by
adjusting the influence domain size of the nodes representing
the electrode. The results obtained from IEFGM for different
values of radius were compared with those generated by
MOM and the proximity between these solutions confirmed
the validity and accuracy of the proposed techniques.
ACKNOWLEDGMENT
This work was supported in part by FAPEMIG, in part by
CAPES, in part by CNPq, and in part by CEFET-MG.
REFERENCES
[1] IEEE Guide for Safety in AC Substation Grounding,
IEEE Standard 80-2013, 2013.
[2] F. Dawalibi and D. Mukhedkar, “Parametric analysis of ground-
ing grids,” IEEE Trans. Power App. Syst., vol. PAS-98, no. 5,
pp. 1659–1668, Sep./Oct. 1979.
[3] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for
Electromagnetics. Piscataway, NJ, USA: IEEE Press, 1997.
[4] T. Belytschko, Y. Y. Lu, and L. Gu, “Element-free Galerkin methods,”
Int. J. Numer. Methods Eng., vol. 37, no. 2, pp. 229–256, 1994.
[5] A. Manzin, D. Ansalone, and O. Bottauscio, “Numerical modeling
of biomolecular electrostatic properties by the element-free Galerkin
method,” IEEE Trans. Magn., vol. 47, no. 5, pp. 1382–1385, May 2011.
[6] G. N. Marques, J. M. Machado, S. L. Verardi, S. Stephan, and
A. J. Preto, “Interpolating EFGM for computing continuous and discon-
tinuous electromagnetic fields,” Int. J. Comput. Math. Elect. Electron.
Eng., vol. 26, no. 5, pp. 1411–1438, 2007.
[7] U. D. C. Resende, E. H. da Rocha Coppoli, and M. M. Afonso,
“A meshless approach using EFG interpolating moving least-squares
method in 2-D electromagnetic scattering analysis,” IEEE Trans. Magn.,
vol. 51, no. 3, Mar. 2015, Art. no. 7200704.
[8] E. H. R. Coppoli, R. C. Mesquita, and R. C. Silva, “Induction machines
modeling with meshless methods,” IEEE Trans. Magn., vol. 48, no. 2,
pp. 847–850, Feb. 2012.
[9] W. L. Nicomedes, R. C. Mesquita, and F. J. S. Moreira, “A meshless
local Petrov–Galerkin method for three-dimensional scalar problems,”
IEEE Trans. Magn., vol. 47, no. 5, pp. 1214–1217, May 2011.
[10] T. Noda and S. Yokoyama, “Thin wire representation in finite difference
time domain surge simulation,” IEEE Trans. Power Del., vol. 17, no. 3,
pp. 840–847, Jul. 2002.
[11] B. Nekhoul, C. Guerin, P. Labie, G. Meunier, R. Feuillet, and
X. Brunotte, “A finite element method for calculating the electromag-
netic fields generated by substation grounding systems,” IEEE Trans.
Magn., vol. 31, no. 3, pp. 2150–2153, May 1995.
[12] E. B. Joy and R. E. Wilson, “Accuracy study of the ground grid analysis
algorithm,” IEEE Trans. Power Del., vol. PWRD-1, no. 3, pp. 97–103,
Jul. 1986.
[13] U. C. Resende, R. S. Alípio, M. L. Oliveira, and S. T. M. Gonçalves,
“Analysis of grounding problems using EFG-IMLS meshless method,”
in Proc. Int. Symp. Electromagn. Fields Mechatronics, Elect. Electron.
Eng. (ISEF), Valence, Spain, 2015.
[14] M. L. Oliveira, R. B. Macedo, U. C. Resende, and R. S. Alípio, “Analysis
of grounding problems using interpolation element-free Galerkin method
with reduction of computational domain,” in Proc. Int. Microw. Opto-
electron. Conf. (IMOC), Águas de Lindóia, São Paulo, Brazil, 2017.
[15] G. R. Liu, MeshFree Methods: Moving Beyond the Finite Element
Method. Boca Raton, FL, USA: CRC Press, 2009.
Authorized licensed use limited to: CENTRO FED DE EDUCACAO TECNOLOGICA DE MINAS GERAIS. Downloaded on July 29,2020 at 13:19:05 UTC from IEEE Xplore. Restrictions apply.
Article
In this paper, the interpolating element-free Galerkin meshless method is presented as a new, simple, and accurate proposal for the analysis of electrostatic grounding problems. A very important aspect to ensure the precision of this numerical solution is the correct modeling of the electrode radius. So, two techniques are proposed to include the electrode radius effect in the analysis. The problem of a vertical electrode introduced in a homogeneous soil is considered as study case. The results of the electrical potential along the ground surface and grounding resistance calculated using interpolating element-free Galerkin method, for different values of radius, are compared with those generated by the method of moments.
Article
Full-text available
This paper describes a new numerical procedure for analysing earthing grids buried in horizontally stratified multilayer earth. The procedure is very efficient and general. The total number of layers and the total number of metallically disconnected earthing grids are completely arbitrary. A single earthing grid can be positioned in several layers. The procedure is based on an integral equation formulation. Earthing grid conductors are subdivided into segments and the average potential method is used. Efficiency and generality of the computation procedure are based on the successful application of numerical approximations of two kernel functions of the integral expression for the potential distribution within a single layer which is caused by a point current source. Each kernel function of the observed layer is approximated using a linear combination of 15 exponential functions. Extension from the point source to a segment of the earthing grid conductor is done by integrating the potential contribution due to a line of point current sources along the segment axis. This computational procedure gives highly accurate results in a short execution time. (C) 1998 John Wiley & Sons, Ltd.
Article
Full-text available
In this paper, the element-free Galerkin method with special interpolating shape functions is used to solve an electromagnetic scattering problem. These shape functions make use of the interpolating moving least-squares method that guarantees the Kronecker delta property. This allows enforcing the absorbing boundary condition directly in the discrete system, consisting of a more practical way when compared with other methods, such as the Lagrange multipliers. The technique is applied to the analysis of electromagnetic scattering generated by an infinite dielectric cylinder illuminated by a TMZ plane wave and a sensibility analysis is done in which the numerical results are compared against the analytical solutions.
Article
Full-text available
Meshless Methods, also called Meshfree Methods, are a class of numerical methods to solve partial differential equations. The main characteristic of these methods is that they do not need a mesh like the one used in the Finite Element Method. In this sense mesh-less methods are very useful for modeling moving structures, such as electric machines, without a remeshing process. In this work the Element-Free Galerkin Method is used to simulate a three phase induction motor model including, for the first time, the field circuit coupling transient equations and the rotor movement.
Article
Full-text available
In this paper, we apply a meshless method based on local boundary integral equations (LBIEs) to solve electromagnetic problems. The discretization process is carried out through the use of special basis functions that, unlike the Finite Element Method, are not confined to an element and do not require the support of an underlying mesh. The approach herein developed can be applied to general three-dimensional scalar boundary value problems arising in electromagnetism.
Chapter
A campaign of measurements on some of the RAλs rebroadcasting stations of typical design has made it possible to determine the principal characteristics of the over voltages of atmospheric origin that affect stations, as well as the places where they assume dangerous values. It thus became practicable to determine which, in principle, are the optimum precautions to be taken at stations of that type, to reduce the over voltages to tolerable values. Individual tests are, however, necessary for those stations that are particularly endangered by lightning flashes, or that are exceptional on account of the importance of the service or of the complexity of the circuitry.
Article
Purpose This paper proposes an interpolating approach of the element‐free Galerkin method (EFGM) coupled with a modified truncation scheme for solving Poisson's boundary value problems in domains involving material non‐homogeneities. The suitability and efficiency of the proposed implementation are evaluated for a given set of test cases of electrostatic field in domains involving different material interfaces. Design/methodology/approach The authors combined an interpolating approximation with a modified domain truncation scheme, which avoids additional techniques for enforcing the Dirichlet boundary conditions and for dealing with material interfaces usually employed in meshfree formulations. Findings The local electric potential and field distributions were correctly described as well as the global quantities like the total potency and resistance. Since, the treatment of the material interfaces becomes practically the same for both the finite element method (FEM) and the proposed EFGM, FEM‐oriented programs can, thus, be easily extended to provide EFGM approximations. Research limitations/implications The robustness of the proposed formulation became evident from the error analyses of the local and global variables, including in the case of high‐material discontinuity. Practical implications The proposed approach has shown to be as robust as linear FEM. Thus, it becomes an attractive alternative, also because it avoids the use of additional techniques to deal with boundary/interface conditions commonly employed in meshfree formulations. Originality/value This paper reintroduces the domain truncation in the EFGM context, but by using a set of interpolating shape functions the authors avoided the use of Lagrange multipliers as well as of a penalty strategy. The resulting formulation provided accurate results including in the case of high‐material discontinuity.
Chapter
Since the publication of the first edition [see the review in Zbl 0823.65124] of this book about eight years ago, much progress has been made in the development of the finite element method for the analysis of electromagnetics problems, especially in five areas. The first is the development of higher-order vector finite elements, which make it possible to obtain highly accurate and efficient solutions of vector wave equations. The second is the development of perfectly matched layers as an absorbing boundary condition. Although the perfectly matched layers were intended primarily for the time-domain finite-difference method, they have also found applications in the finite element simulations. The third is perhaps the development of hybrid techniques that combine the finite element and asymptotic methods for the analysis of large, complex problems that were unsolvable in the past. The fourth is further development of the finite element-boundary integral methods that incorporate fast integral solvers, such as the fast multipole method, to reduce the computational complexity associated with the boundary integral part. The last, but not least, is the development of the finite element method in the time domain for transient analysis. As a result of all these efforts, the finite element method has gained more popularity in the computational electromagnetics community and has become one of the preeminent simulation techniques for electromagnetics problems. In this second edition, we have updated the subject matter and introduced new advances in finite element technology. Three new chapters have been added: Absorbing boundary conditions; Finite element analysis in the time domain: The method of moments and fast solvers.
Article
An element-free Galerkin method which is applicable to arbitrary shapes but requires only nodal data is applied to elasticity and heat conduction problems. In this method, moving least-squares interpolants are used to construct the trial and test functions for the variational principle (weak form); the dependent variable and its gradient are continuous in the entire domain. In contrast to an earlier formulation by Nayroles and coworkers, certain key differences are introduced in the implementation to increase its accuracy. The numerical examples in this paper show that with these modifications, the method does not exhibit any volumetric locking, the rate of convergence can exceed that of finite elements significantly and a high resolution of localized steep gradients can be achieved. The moving least-squares interpolants and the choices of the weight function are also discussed in this paper.
Article
This paper develops a basic idea which allows low frequency electromagnetic fields generated by cylindrical conductors, with very small radii compared with their length, to be calculated. The proposed method can be applied to both power overhead conductors and those buried in a conducting environment. The paper presents a direct application for this concept which consists in calculating the electromagnetic fields created by a substation grounding system following an accidental short circuit or a lightning stroke. The finite element method (FEM) with several formulations is used. Currents in the cylindrical conductors which make up the earth network grid are taken into account without volume meshing; simple line discretization of these is carried out. Electromagnetic fields generated by leakage currents and currents induced underground are taken into account. Furthermore, open boundaries of both air and earth environments are processed by introducing a spatial transformation. This application for an earth network enables to show the advantages of FEM processing compared with the method traditionally used in antenna theory for this kind of problem