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Engineering, Technology & Applied Science Research Vol. 12, No. 4, 2022, 8910-8915 8910
www.etasr.com Allal et al.: Improved Optimization of the Charge Simulation Method for the Calculation of the Electric …
Improved Optimization of the Charge Simulation
Method for the Calculation of the Electric Field
Around Overhead Transmission Lines Using
Statistical Methods
Abdalali Allal
Laboratoire de Recherche en Electrotechnique
Ecole Nationale Polytechnique
Algiers, Algeria
abdelali.allal@univ-djelfa.dz
Ahmed Boubakeur
Laboratoire de Recherche en Electrotechnique
Ecole Nationale Polytechnique
Algiers, Algeria
ahmed.boubakeur@g.enp.edu.dz
Adnan Mujezinović
Faculty of Electrical Engineering
University of Sarajevo
Sarajevo, Bosnia and Herzegovina
am14618@etf.unsa.ba
Received: 21 April 2022 | Revised: 3 June 2022 | Accepted: 4 June 2022
Abstract-In order to decide the appropriate arrangements of
fictitious charges in the charge simulation method, the use of the
Monte Carlo method is proposed for the estimation of the
probability density function of two variables, the radius ratio,
and the angle ratio. Τhe scale and shape parameters of the
Weibull's distribution are determined by the maximum
likelihood estimator. The obtained results are used to calculate
the electric field at arbitrary points in the neighborhood of high
voltage transmission lines. The comparisons between the results
computed by this method, the results calculated by the genetic
algorithm, and those measured, confirm the effectiveness and
accuracy of the proposed method.
Keywords-charge simulation method; Monte Carlo method;
optimization; genetic algorithm; high voltage transmission lines;
electric field calculation
I. INTRODUCTION
Designing any high voltage device and analyzing discharge
phenomena requires complete knowledge of electric and
magnetic field distribution [1]. The potential surface gradient is
a critical design parameter for planning and designing overhead
lines (insulation or discharge) [2, 3]. The electric fields can be
calculated using several analytical and numerical methods. The
most used is the Charge Simulation Method (CSM) [4-9]. The
CSM was introduced in 1969 [4]. Its basic concept is to replace
the distributed charge of conductors and the polarization
charges on the dielectric interfaces with a large number of
fictitious discrete charges. The magnitudes of these charges
have to be calculated so that their integrated effect satisfies the
boundary conditions precisely at a selected number of points on
the boundary. The principle of this method is to simulate an
actual field with a field formed by a finite number of
simulation charges (point and line charges of infinite and semi-
infinite length [11]) placed outside the region where the field is
to be calculated. The values of the discrete charges are
determined by satisfying the boundary conditions at a selected
number of contour points:
���� = ������
� (1)
where [Vb] is the vector of contour point voltages, [Qs] is the
vector of unknown simulation charges, and [P] is the matrix of
potential coefficients calculated by contour points and
simulation charges.
For overhead lines consisting of n parallel conductors
placed above the ground, the elements of the matrix of
potential coefficients are given by the following relation:
0
1
ln
2π
ij
ij
ij
D
P
dε
=
(2)
where ε0 is the electric permittivity of vacuum ≈8.854 10
-12
F/m,
Dij, dij are respectively the distance between the jth point charge
and the image of the ith point charge and the distance between
the jth point charge and the ith point charge.
Based on the Laplace equation (3), the superposition
theorem and image charge theory, the components at an
arbitrary point in the plane y-z plane produced by n point
charges M(y,z) can be calculated by (4) and (5):
Corresponding author: Abdelali Allal
Engineering, Technology & Applied Science Research Vol. 12, No. 4, 2022, 8910-8915 8911
www.etasr.com Allal et al.: Improved Optimization of the Charge Simulation Method for the Calculation of the Electric …
∆ = − ��� (3)
�� ��, �� = ∑ ������ �
����
���
+ ! ������� "
#
$%� (4)
�& ��, �� = ∑ ������ �
&�&�
���
+ ! &'&���� "
#
$%� (5)
where Γ is the reflection coefficient of soil's surface, ($ is the
distance between the arbitrary point M(y,z) and the ith point
charge, yi and zi are the coordinates of the ith contour point, and
Qi is the ith fictitious charge. The reflection coefficient of the
soil can be approximated as ! = −1.
The current paper aims to optimize CSM parameters using
stochastic optimization employing the Monte Carlo Method
(MCM). This optimization is based on estimating the
Probability Density Functions (PDFs) of the polar coordinates
of the simulation charges where the relative mean square error
of the voltage on the conductor surface is less than a threshold.
To ensure the accuracy of this method, a comparison is made
with the measured values of the electric field at arbitrary points
near high voltage transmission lines with standard dimensions
of the tower on the 40kV line SS Sarajevo 10 –SS Sarajevo 20
[10].
II. RELATED WORK
Since 1969, when the CSM was used for the first time [4],
it has been applied and developed in many cases. Some of the
main contributions, in chronological order, are:
Authors in [12] used CSM combined with the
Rosenbloom’s method to solve the potential distribution of the
rod-plane. Authors in [13] calculated the field distribution for
multi-phase AC sources or in configurations including volume
resistance. Authors in [5] simulated the sheathed three cores
belted power cable using the complex fictitious charges.
Authors in [14] combined CSM with the Genetic Algorithm
(GA) to optimize the CSM for a 2D electrode system with an
asymmetrical structure. Authors in [15] used CSM-GA to
calculate the electric field of a 35kV Vacuum Interrupter (VI).
GA has been utilized to compute the electric field [16], to
model the horizontal sphere gap [17], and to model the
horizontal sphere gap above the ground plane [18]. Authors in
[19] calculated the electric field around the head of a
transmission tower and its composite insulators by coupling
CSM with BEM. Authors in [20] used an optimization strategy
to arrange the simulated charges in the thin electrode. Authors
in [21] combined CSM with GA to solve the inverse problem
in electric-fields of high voltage insulators. Authors in [22]
combined CSM with Hashing integrated Adaptive GA
(HAGA) to the contour design of support insulators. Authors in
[23] used CSM combined with the gold section method to
calculate the conductors' surface electrical field of ±800kV
UHVDC transmission lines. Authors in [24] used CSM-GA to
enhance the computation precision of electric fields associated
with plate‐type electrostatic separators. An adapting Particle
Swarm Optimization (PSO) combined with CSM was used for
calculating the field distribution with non-axial symmetry
resulting from a floating spherical conductor between the
spheres in [25]. Authors in [26] improved the calculation
accuracy of the electric fields associated with electrostatic plate
separators by using CSM-GA. For the optimization of high
voltage electrode surfaces, authors in [27] used CSM combined
with a Biogeography-based algorithm. Authors in [28] used
CSM-PSO for sphere-plane gaps. 3D calculation of electric
field intensity under transmission lines with CSM-PSO and
CSM-GA was conducted in [29]. Authors in [30] made a
comparison between the performance of PSO, GA, and Grey
Wolf Optimizer (GWO) in 3D quasi-static modeling of the
electric field produced by High Voltage (HV) overhead power
lines. To optimize the ion flow field calculation, authors in [31]
used CSM combined with the Flux Tracing Method (FTM).
III. THE PROPOSED ALGORITHM
A. Intoduction
The proposed algorithm is based on Stochastic
Optimization (SO) methods. The SO methods generate and use
random variables [32]. They are used in many areas, including
aerospace, medicine, transportation, finance, electrical
engineering, and many more science and engineering fields.
SO can rely on sampling methods such as MCM [33], Latin
hypercube sampling [34], or the Quasi-Monte Carlo Method
(QMCM) [35]. The algorithm aims to optimize the location of
fictitious charges by generating a bivariate distribution of N×N
random variables 〈Cr, Ca〉 which are respectively the ratio
between ,- and ,
, .- and .
according to (6)-(8). As shown in
Figure 1, the contour points are arranged at equal distances on
the perimeter of the conductor and are determined by their
polar coordinates rc and θ
k
b according to (7)-(8). The simulation
charges are also arranged at an equal distance on the perimeter
of a virtual circle inside and are determined by their polar
coordinates ,- and .-/.
.
/ = �0/#1 �2 − 1� , 2 = 1 34 5- (6)
,- = 67 ,
(7)
.-/ = .
/ + 68 . �0:1 (8)
where rb is the radius of the conductor, θ
k
b the angle of the k
th
contour point, NC the number of contour points, rc the radius of
the virtual circle that contains simulation charges, θ
k
c the angle
of the k
th
fictitious charge, and Cr and Ca the radius and angle
ratios ranging between 0 and 1.
Fig. 1. Arrangement of contour points, fictitious charges, and test point.
Engineering, Technology & Applied Science Research Vol. 12, No. 4, 2022, 8910-8915 8912
www.etasr.com Allal et al.: Improved Optimization of the Charge Simulation Method for the Calculation of the Electric …
B. The Algorithm
For each iteration, we have a set of coordinates of fictitious
charges, denoted by �67$ , 68$ � such that:
The set of all coordinates is: ℇ � 〈67 , 68 〉 and the set of
acceptable coordinates is ℇ< � 〈6=7 , 6=8 〉. The work is carried out
in two steps.
1) Step 1: Extraction of the Set of Solutions ℇ<
Data: The electrical and geometric parameters of the line
such as the potential of point contour Vb, the number of
conductor nc, the number of fictitious charges Nc, the objective
function threshold Fobjthreshold,, the number of iterations N
which generate N random (Cr,Ca) pairs.
From i=1 to N
Do
Calculate Nc*nc coordinates of fictitious charges with (6)
and (7).
Calculate the potential created by these fictitious charges
with (2).
Calculate the fictitious charges with (1).
Calculate the potential created by these fictitious charges:
� > � � ��> �����
Calculate the objective function Fobj:
?@AB � 1C D� >
�
�
Compare Fobj with Fobjthreshold
If Fobj > Fobjthreshold then (Cr,Ca) is rejected.
Else add (Cr,Ca) to the ℇ<
2) Step 2: Statistical Study
The followed steps are:
1. Establish the histograms of Cr and Ca
2. Estimate the Weibull law parameters A and B with the
Maximum Likelihood Estimator (MLE).
3. Calculate the mean and standard deviation of Cr and Ca
The above algorithm is executed for a simple geometry
problem (Figure 2) where nc=2, Nc=3, N=100,
Fobjthreshold=4×10
-12
, h=11m, d=2m, rb=7cm, and V=400kV.
After the iterations are completed, there are 26 accepted
bivariates (Cr,Ca) and their histograms are shown in Figures 3
and 4. It is quite obvious that the greatest PDF is concentrated
around 0.95 for Cr and 0.49 for Ca. From the obtained results, it
should be noted that the shapes of the two histograms are
asymmetrical. The obtained data of the first histogram are
grouped near the upper limit and incline to the left towards the
lower values. On the other hand, in the second histogram, the
data are grouped towards the center, which leads to estimating
the two parameters of the Weibull distribution as follows.
Fig. 2. Histogram geometry problem.
Fig. 3. Histogram of Cr.
Fig. 4. Histogram of Ca.
The Weibull distribution is used in reliability studies, for
example, to study the voltage breakage of electric circuits [36].
The Weibull distribution has two parameters, denoted in the
following equation:
E�F|H, I� � I. H�J F J��K �L
M
NO
P
(10)
where A> 0 is the scale parameter and B > 0 is the shape
parameter of the distribution.
The Maximum Likelihood Estimator (MLE) [37] estimates
the Weibull parameters A and B. The results are given in Table
I and the estimate distributions are shown in Figures 5 and 6.
TABLE I. ESTIMATED WEIBULL PARAMETERS
Data A B
67 Relative radius 0.968882 31.02033
68 Relative angle 0.507651 27.0408
Engineering, Technology & Applied Science Research Vol. 12, No. 4, 2022, 8910-8915 8913
www.etasr.com Allal et al.: Improved Optimization of the Charge Simulation Method for the Calculation of the Electric …
Fig. 5. Estimate distribution of Cr.
Fig. 6. Estimate distribution of Ca.
The estimated distributions shown in Figures 6 and 7 agree
with the histograms obtained in Figures 3 and 4. The estimated
distributions have mean and variance μr=0.9519, σr=0.0015 for
the relative radii Cr, and μr=0.9519, σr=0.0015, μa=0.4975,
σa=5.2887 10
-4
for the relative angular distribution Ca. The
optimum position of the coordinates of the fictitious charges
obtained are Cr Є [μr–σr, μr+σr] and Ca Є [μa –σa, μa+σa] with
respective mean [0.9505; 0.9534] and [0.4970; 0.4980]. For
example, for Cr=0.95 and Ca=0.49 the optimum arrangement of
the fictitious charges is shown in Figure 7.
Fig. 7. Optimum arrangement Cr=0.95 and Ca=0.49.
To verify this approach, an example of the electric field
calculation around an overhead- transmission line, already
treated in the literature [10] is examined below.
IV. RESULT COMPARISON
The considered example is a 400kV line with a horizontal
conductor configuration, as shown in Figure 2. Measurements
were taken at 1m height according to recommendations [38],
and were performed in the middle of the range between the two
adjacent transmission line towers. The electric field is
calculated without considering the effect of conductor end, arc
sag, and the influence of the tower. In addition, the
electromagnetic fields caused by the overhead transmission
lines can be approximated by quasi-static fields [39], where
quasi-static field displacement current and changes in the
magnetic flux are negligible, so the electric field has exactly the
same characteristics as the static one. It is assumed that the
component of the electric field vector in the x direction is equal
to zero, and the electric field vector in an arbitrary point,
caused by the n point charges, can be calculated using (7) and
(8).
The application of CSM with a relative radius Cr=0.95 and
relative angle Ca=0.497 normally leads to optimum results
close to the measured results, but to ensure its effectiveness it
must be compared with another method already used with
CSM. The choice fell on the GA because it is the most used
with CSM as shown above. Also, it has been widely used in the
field of electrical engineering [40-43]. Figure 8 illustrates the
graphs of the electric fields calculated by this method, by
CSM-GA, and the measured values (the values are listed in
Table II). It can be seen that the field calculated with the
proposed method is closer to the measured field than the field
calculated with CSM-GA.
Fig. 8. Distribution of the electric field.
TABLE II. COMPARISON OF MEASURED AND CALCULATED RESULTS
Distance
(m)
Measured
(kV/m)
CSM-MCM CSM-GA
0 4.13 17.69% 02.50%
5 4.45 02.50% 06.86%
10 5.93 01.44% 03.17%
10.8 6.09 01.11% 03.50%
15 5.81 03.20% 07.66%
20 3.84 01.75% 06.26%
25 2.30 00.99% 03.61%
30 1.39 04.69% 00.04%
35 0.81 16.11% 10.90%
Mean 04.38% 04.95%
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V. CONCLUSION
A new approach for optimizing the charge simulation
method using the MCM has been presented in this paper. The
proposed algorithm aims to determine the PDFs of the classes
of polar coordinates for which the error is minimal. The two
PDFs follow Weibull's law. The PDF of relative radius (Cr) is
asymmetric and concentrated near 1, while the PDF of the
relative angle (Ca) is symmetric and is slightly centered at 0.49.
The proposed algorithm offers excellent flexibility and
accuracy in determining the optimal locations of simulation
charges. Accurate results are achieved for the electric field
calculation around the overhead transmission lines. In addition,
the solution is not a single element (Ca,Cr) like the results of
other methods, but a range distributed according to the Weibull
distribution whose parameters are calculated. This work aims at
an optimal calculation of the electric field by CMS by
arranging the fictitious charges so that they are very close to the
edges of the conductor, and each imaginary charge mediates
two consecutive contour points. The main contribution of this
work is direct optimization without going through optimization
methods.
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