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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,
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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 ρ∂V
∂ρ +∂2V
∂z2=0(1)
where V(ρ, 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]
T∂V
∂ρ +∂V
∂z·nρd
−∂T
∂ρ
∂V
∂ρ +∂T
∂z
∂V
∂zρd=0(2)
where d=dρdz,=n∪d∪Le. 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(ρ, 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(x)υI(3)
where x=(ρ, 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 ∪n∪d∪Le. 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=|x−xI|
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
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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
i−VIEFGM
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.
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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.
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