Numerical Computation of the Electric Field inside a High Voltage

6TH INTERNATIONAL CONFERENCE ON MODERN POWER SYSTEMS MPS2015, 18-21 MAY 2015, CLUJ-NAPOCA, ROMANIA
Numerical Computation of the Electric Field inside a
High Voltage Substation
Florin Miron
Marius Purcar, Calin Munteanu
Department of Electrotechnics and Measurements
Technical University of Cluj-Napoca
Cluj-Napoca, Romania
[email protected]
Department of Electrotechnics and Measurements
Technical University of Cluj-Napoca
Cluj-Napoca, Romania
the superposition principle in any point in the region of
interest.
Abstract – This paper presents a numerical implementation
based on the charge simulation method (CSM) for the calculation
of electric fields in high voltage substations. A comparison
between the numerical implementation and results from
literature are presented on the 110 kV high voltage substation
Cluj-South. Reduced computing time, resources and the basic
programming platform make this application a useful tool for
electric field studies.
The semi-analytical algorithm presented in this paper is
similar to the CSM and it works by dividing the substation
conductors into small line segments on which discrete charges
are placed. These charges are determined so that the potential
on the line segments remains constant. The electric field is
then calculated by considering all of these small line segments
with the formula from [9]:
Keywords- High voltage, electromagnetic field, numerical
application, Pascal, CSM.
I.
Algorithm II.
j
Introduction 2 m
e 3
E
4
Knowledge of the electric field distribution inside high
voltage substations is essential in the design of equipment,
facilities or in studies regarding human safety or
environmental concerns.
Solutions to the problem of calculating the electric field
distribution were developed over the years in the form of
analytical and numerical methods, each with its pros and cons.
For simple physical systems, analytical methods are used due
to their precision and general approach, but in the case of
more complex geometries, such analytical methods become
very hard to apply and a numerical method is applied.
Numerical methods are usually case specific and have less
precision then analytical methods.
The most commonly used numerical methods are the Finite
Difference Method (FDM), the Finite Element Method (FEM),
the Boundary Element Method (BEM) and CSM or
combinations of these methods (FDM-CSM, FEM-CSM,
BEM-CSM) as is the case in [3], [4] and [5]. Due to its
effective computation on large model and open areas the CSM
is preferred when calculating the electric field distribution in
high voltage substations as was done in [7] and [8].
To determine the electric field using the CSM, fictitious
charges are placed outside the region of interest (or inside
equipotential surfaces). The values of these charges are
determined by satisfying the boundary conditions on the
contour points, then the electric field can be calculated using
Lk
n
 r1k
    r
k
k 1
Lk 1
3
1k

r2 k
r23k

dl


(1)
where  k is the electric charge density, Lk and Lk 1 are the
segment limits and rk are the position vectors of the source
segment and its image with regard to the computation point.
Most of the models presented in literature use a software
platform that is not available to the large audience.
Implementing algorithms in such specialized commercial
software does have its advantages as most of them have user
friendly interfaces and intuitive functions making the
“programming” easy. In this paper an open source
programming platform based on Pascal has been used. Hence,
the user can easily access the algorithm and the programming
language.
Another reason for choosing this platform is the flexibility
it provides in solving equations. Most of the specialized
commercial software have predefined functions and solving
algorithms, so tailoring a case specific solution might prove
difficult and resource consuming. In [6], the CSM was
implemented in C++ and optimized so that the CPU
computing time would be reduced. The goal is similar in this
paper, as the time and resources needed by commercial
software to implement this algorithm on large models can be
reduced.
In fig. 1 a schematic view of the algorithm is provided to
better understand the implementing process presented further.
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6TH INTERNATIONAL CONFERENCE ON MODERN POWER SYSTEMS MPS2015, 18-21 MAY 2015, CLUJ-NAPOCA, ROMANIA
where P  is the coefficient matrix, Q  is the matrix of
unknown values of charges and V  is the matrix of given
potential.
To solve this system of equations the Gauss elimination
method is employed. The first step in implementing this
method is rearranging the matrix (swapping rows) so that the
element on the main diagonal (the pivot) is different than zero
for each iteration of the method. This swapping is done only
between the row which contains the pivot and rows beneath it.
If a pivot remains zero after this rearranging then that matrix is
singular and the equation system does not have a solution. The
method comprises of a succession of operations (3) that
transform the coefficient matrix into an upper triangular
matrix. These operations are applied to all the elements on the
rows beneath the pivots as well as in the matrix of potentials
and that of the unknown values.
Pi, j  : Pi, j   Pk , j * Pi, k  / Pk , k 
(3)
where Pi, j  is a matrix element, i and j represent its position
in the matrix on rows and column respectively and k
represents the position of the pivot.
After the matrix transformation, solving the system of
equations is rather simple by using either a recursive function
or a backwards solving function. In this algorithm the lather
solution was used as to reduce time and resources.
The field computation is done in two steps; first the field
generated by each segment of line ELi and its image is
computed taking into account the line charges and phase. The
total field is calculated using the super-position principle.
As the algorithm is written in Pascal, direct operations with
complex numbers, see equation (1), are not possible, so an
indirect approach had to be made so that the functions would
be able to compute such operations. To solve this problem a
library that contains functions which treat complex numbers as
two separate values was referred to and a few alterations were
made to the algorithm to accommodate its use. One of these
alterations involved the operations with matrix elements; the
library could not operate with complex matrix elements so to
fix the problem the matrix was split in two: one matrix which
contained the real part of the numbers and another with the
imaginary part. Functions and procedures were modified to
take this into account and the number of operations doubled.
As the Pascal programming language, does not have
specialized libraries to process large numbers of decimals it
was necessary to monitor the error propagation through the
algorithm to see what the effect was on the results. After this
analysis it can be concluded that the number of segments in
which lines were divided had the most influence on the end
result, so a few tests were made on a line to see how much this
influenced the results. The error seemed to grow exponentially
with the number of line divisions as can be seen from Table 1.
The algorithm presented in this paper consists of 3 parts: data
input, which can be manually entered or by importing it from
an external source (e.g. a text file).
Figure 1. Schematic view of the algorithm In [1] the input data is manually entered proving to be
quite a time consuming task because the data has to be
inserted into the algorithm directly for each of the calculation
steps. In the presented algorithm, data is collected in a text or
dat file and appended automatically when it’s required, using
less system resources. For cases where CAD models are
available this process can be speeded by using a macro to
automatically extract the required data presented in [2].
The second step, the computing functions, comprise the
main body of the algorithm. This part of the algorithm can be
subdivided as in Fig. 1 into functions that calculate the contour
points’ positions, functions that form the system matrixes,
procedures that solve the equation system and functions that
compute the field. The last part outputs the results to file. The
position of the contour points is calculated based on the XYZ
coordinates provided in the input data and the radius of the
line conductor as these contour points will be positioned on
the conductor surface. Each line is divided into a number of
smaller segments, then a fictitious charge is placed in each
segment and boundary conditions are specified so that the
given potential on the each of the segments remains constant.
Computing the potential in each of the contour points
requires the formation of a system of equations in which, the
given potential equals the charge value multiplied with a
coefficient as can be seen from equation (2):
 P11 P12  P1n   Q1  V1 
P
   
 21 P22  P2 n  Q2   V2 
(2)
 
        

   
 Pn1 Pn 2  Pnn  Qn  Vn 
TABLE I.
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ERROR PROPAGATION
Line divisions
Error propagation
10
0.0001%
100
0.001%
6TH INTERNATIONAL CONFERENCE ON MODERN POWER SYSTEMS MPS2015, 18-21 MAY 2015, CLUJ-NAPOCA, ROMANIA
Line divisions
Error propagation
1000
1%
The number of segments in which lines are divided
influence the accuracy of the field values compared with the
implementation in [1].
III.
CAD model The model on which the field computation is conducted is
presented in fig. 2 and it was built in SolidWorks by the
authors in [2] at a 1:1 scale using equipment technical
descriptions and onsite measurements of the 110kV high
voltage substation Cluj-South.
Figure 3a. Electric field distribution – computation module in [1]
Figure 2. SolidWorks CAD model of the 110 kV Cluj-South substation
The model contains 12 cells and a total of 372 lines.
Coordinates are extracted for the start and end of all the lines
using a macro. Each line is divided in 2 to 140 segments
which total 4870 segments.
IV.
results An analysis of the results from computing the electric field
strength with the algorithm from [1], case 1, and the algorithm
presented in this paper, case 2, showed good correspondence
as can be seen from fig. 3a and 3b. High values for the field
are located in the vicinity of bus-bars circuit breakers and
disconnectors.
Figure 3b. Electric field distribution – presented algorithm
Computation of the electric field was conducted at a height
of 1.8 m above ground level. Fig. 3a shows the results
obtained with the module from [1] which finished calculations
in 4 hours and 15 minutes. For the algorithm presented in the
paper, the results are represented in fig. 3b, the computing
time for this case taking 1 hour and 50 minutes, which is 43%
of the time from the first case. A more detailed analysis was
conducted by extracting values along parallel lines, at 20, 40
and 60 m, of the OX axis of the model at 5 m intervals. These
results are shown in fig. 4, 5 and 6 where E M and E P represent
the electric field strength for case 1 and case 2 respectively.
As can be seen from fig. 4, 5 and 6, small differences appear
between the two cases. This computing error comes from the
precision loss in case 2 due to the programming language.
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6TH INTERNATIONAL CONFERENCE ON MODERN POWER SYSTEMS MPS2015, 18-21 MAY 2015, CLUJ-NAPOCA, ROMANIA
[1]. Use of Pascal language, reduced time and resources and
fast data input make this algorithm a useful tool when
computing the electric field strength in HV substations.
Figure 4. Electric field strength probed at 20 m
Figure 5. Electric field strength probed at 40 m
Figure 6. Electric field strength probed at 60 m
Solutions for this problem can be applied either by storing
all the variables in vectors and build functions to apply the
needed operations on a decimal level. Another solution is to
include another step to compensate for this precision loss by
adjusting the boundary conditions and computing the field
strength distribution again. This compensation and
recalculation would continue until a condition is satisfied.
Both of these corrections would require additional computing
time and resources.
I.
Conclusions Computation of the electric field strength on the CAD
model in fig. 2 was conducted with both the algorithm
presented in this paper and with the computation module
presented in [1]. An analysis of the results shows good
correspondence between the two. When comparing the
computing time, the algorithm presented in this paper was
57% faster than the module presented in [1]. Tests showed that
the number of segments in which lines are divided influence
the accuracy of the results compared to the implementation in
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6TH INTERNATIONAL CONFERENCE ON MODERN POWER SYSTEMS MPS2015, 18-21 MAY 2015, CLUJ-NAPOCA, ROMANIA
II.
Acknoledgment This paper is supported by the Human Resources
Development Program POSDRU/159/1.5/S/137516 financed
by the European Social Fund and by the Romanian
Government.
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