Optimal capacitor bank placement and sizing in the IEEE 34-node test model
for loss reduction using Genetic Algorithm
Ubicación óptima y dimensionamiento de banco de capacitores en el modelo de prueba
IEEE 34 nodos para la reducción de pérdidas mediante Algoritmo Genético
Johan Astudillo
Master, Universidad Técnica Estatal de
Quevedo, Quevedo, Ecuador,
Johan.astudillo2016@uteq.edu.ec
https://orcid.org/0000-0002-4599-0488
Maria Loor
Master, Universidad Técnica Estatal de
Quevedo, Quevedo, Ecuador,
Maria.loor2016@uteq.edu.ec
https://orcid.org/0000-0002-1989-0803
Steven Villamar
Master, Universidad Técnica Estatal de
Quevedo, Quevedo, Ecuador,
Steven.villamar2016@uteq.edu.ec
https://orcid.org/0000-0002-8060-4239
Diego Peña
Master, Universidad Técnica Estatal de
Quevedo, Quevedo, Ecuador,
dpena@uteq.edu.ec
https://orcid.org/0000-0003-2108-4306
Johan.astudillo2016@uteq.edu.ec
http://centrosuragraria.com/index.php/revista, Published by: Edwards Deming Institute,
Quito - Ecuador, October - December vol. 1. Num. 11 2021, This work is licensed under
a Creative Commons License, Attribution-NonCommercial-ShareAlike 4.0 International.
https://creativecommons.org/licenses/by-nc-sa/4.0/deed.es
Received June 09, 2021
Approved: July 10, 2021
Abstract
In order to amend disturbances such as: voltage drops,
power losses, among others; because the loads in a
real distribution system are unbalanced and these
affect the quality and reliability of electric service. A
mathematical optimization technique is presented
through a genetic algorithm, where an objective
function is proposed in order to minimize losses in the
system. The objective of the genetic algorithm is to
provide the optimal location and sizing of a capacitor
bank in a distribution system. This method, being
iterative, seeks the most feasible solution by means of
its population, generation, reproduction probabilities
and mutation variables in order to fulfill its objective
function.
Keywords: genetic algorithm (GA), IEEE 34 nodes,
mathematical optimization, iterations, minimization.
36
Resumen
Con la finalidad de enmendar perturbaciones como: caídas de voltaje, pérdidas de potencia, entre otros;
debido a que las cargas en un sistema de distribución real son desbalanceadas y estas afectan la calidad
y confiabilidad del servicio eléctrico. Se presenta una técnica de optimización matemática a través de
un algoritmo genético, donde se plantea una función objetivo con el fin de minimizar perdidas en el
sistema. El objetivo del algoritmo genético es proporcionar la ubicación y dimensionamiento óptimo
de un banco de capacitores en un sistema de distribución. Este método al ser iterativo busca la solución
más factible mediante sus variables de población, generación, probabilidades de reproducción y
mutación para cumplir con su función objetivo.
Palabras clave: algoritmo genético (GA), IEEE 34 nodos, optimización matemática, iteraciones,
minimización.
Introduction
Electrical distribution systems often have disturbances that cause anomalies in the system such
as voltage drops or energy losses, putting at risk the reliability and supply of electrical energy.
Sanchez & Pascual (2021) indicate that for this reason, the implementation of components such
as distributed generation or capacitor banks is of utmost importance to improve the efficiency
and reliability of energy in the electrical system.
By optimally sizing and locating a capacitor bank in the electrical distribution system, it is
possible to contribute to the reduction of active power losses, thus minimizing losses in the
system.
(Simões & Ebecken, 2016) Therefore, in this paper, a heuristic optimization technique will be
used, highlighted as a genetic algorithm that will allow finding a solution or set of solutions for
the possible sizing and location of the capacitor bank. p. 10.
Lahoz, R. (2004); Ortiz-Quisbert et al., (2016) and Aguado & Cipriano (2009) mention that it
is a systematic heuristic method for solving search and optimization problems that apply the
same principles of biological evolution, selection based on a population, reproduction and
mutation. Then, it is possible to solve the problems of finding the values of the parameters
describing the shape model of the given function. In the genetic algorithm, each vector of
parameters is called a "chromosome", and the set of all parameters that is analyzed to find the
solution is called a "population". For Corso et al. (2016) states that intuitively see as a
population of individuals (chromosomes) in which all compete to find the most act or the most
suitable.
Ortega, Bravo & Ruiz (1997) and Blanco-Kelly et al. (2021) state that the genetic algorithm
works as follows: in an optimization process by natural selection, it starts once an initial
population of chromosomes has been randomly defined, and then evaluates the suitability
(fitness) of each chromosome. If any of the chromosomes represents the optimal level, then the
algorithm stops the search or exploration, and then terminates. Otherwise, for Salazar-Hornig
Astudillo et al.
October - December vol. 1. Num. 10 2021
& Medina (2013), in the reproduction phase, the chromosomes that will be part of the next
generation are selected in such a way that a higher probability in the reproduction rate is added
to the best evaluated chromosomes. The whole process is repeated iteratively until the
algorithm finds the ideal chromosome.
As mentioned by Arahal, Berenguel & Rodríguez (2006); Martí et al., (2014) and Figueroa et
al. (2018) the genetic algorithm is inspired by Darwin's principle of evolution, this algorithm
simulates this principle by evaluating chromosomes of a population. The objective function of
the genetic algorithm is a mathematical function f(x) that represents the chromosome
evaluation function, providing a numerical value that is interpreted as the measure of fitness,
i.e. the suitability of the chromosome against a problem space. Based on this, the objective
function f(x) will determine the evolutionary or adaptive surface over which the genetic
algorithm will scale.
To do so, you must follow the following process:
• Time 0 ß
• Generation of initial population, P0
• Initial stock assessment P0
• As long as (no (completion condition))
• Home
time time +1
ß
;
Selection of the potential solution Mtiempo from the previous population Ptiempo-1;
Modify the potential solution Mtiempo using genetic operators;
Create a new population Ptime from the potential solution Mtime;
Assessment of the current population, Ptime
• End meanwhile;
For the optimization of a distribution network Gutierrez, G (2018) mentions that it is important
to know the behavior of the power flow and losses in the line in order to be based on the
optimization of losses in the distribution network, therefore, the equation in which the losses
of a distribution network transmission line are defined is expressed as follows:
(1)
Where:
Rij: represents the resistance of line i-j.
Pi: represents the active power flow of line i-j.
38
Qi: represents the reactive power flow of line i-j.
Vi: represents the voltage of bus i connecting line i-j.
Then the objective function is expressed as:
!"#$$
%
&'(
)
*2 3 !"#$$%&'()
"#$%&'(
)*+
(2)
In the distribution system for the optimization of losses in the network, first we define the
losses in the distribution line as shown in equation (1), with which we obtain the objective
function in equation (2), which was raised for the resolution of the problem.
Once the optimization objectives are known, the following constraints are considered.
Subject to:
3.456
723.8#59
(3)
0&:;<70&
70&2=5>
(4)
!&(2:;<7!&(
7!&(22=5>
(5)
Soria, Pandolfi & Villagra (2013); Salazar-Hornig & Medina, (2013) define the recombination
operator or also called Crossover as the most important search operator of the genetic
algorithm, this has the function of exchanging the genetic material of a pair of parents
producing successors that normally differ from their parents. On the other hand, Saltos, C
(2000) mentions that the crossover probability (indicated by Pc) is the ratio between the number
of children produced in each generation and the population size (indicated by pop_size), a high
crossover probability allows a greater search of the solution space.
Soria, Pandolfi & Villagra (2013) and Toll et al. (2021) allude that there are three ways to do
vector crossover such are:
One-point crossover - This technique was proposed by Holland and is the simplest crossover
technique, but it is not widely used today because of its disadvantages. Once two individuals
have been selected, their chromosomes are cut at a randomly selected point to generate two
segments in each of them: the head and the tail. The tails are exchanged between the two
individuals to generate new offspring.
Astudillo et al.
October - December vol. 1. Num. 10 2021
Two-point crossover - Similar to the one-point crossover, the difference is that it generates two
cut points instead of one, where it must be taken into account that none of these cut points
coincide with the end of the chromosomes to ensure that three segments are originated.
Probabilistic crossover - The technique involves the generation of a crossover mask with
binary values, if in one of the positions of the mask there is a 1, the gene located in that position
in one of the offspring is a copy of the first parent, while if there is a 0 the gene is copied from
the second parent. Saltos, C (2000) indicates that the mutation in a genetic algorithm is an
operator that serves to reintroduce lost alleles (value of a gene), i.e. bit positions that converge
to a certain value in a population for this reason in genetic algorithms the crossover is the most
important search operator.
For his part Cuervo, R (2019) points out that the fitness function is a very important part in
genetic algorithms, since it is the one that assesses how good an individual is, if in the
evaluation the individual is good it will pass from generation to generation to achieve the
objective of the algorithm.
There are two ways to optimize a process, which is maximization and minimization, where
according to Amat, J (2019) the fitness function acts differently in both cases, as shown below.
In maximization, the individual will have a higher fitness when the objective function
f(individual) is higher.
In minimization: in this case the individual will have a higher fitness the lower the value of the
objective function, so for minimization problems it can be calculated by -f(individual) or
(1+f(individual))
-1.
Materials and methods
For the development and obtaining of the results, the following research is considered non-
experimental, because it consists only of a technical analysis of the optimization of the radial
distribution system of the IEEE 34 nodes with the objective of minimizing power losses using
the genetic algorithm. For the power loss reduction approach, the objective function to be
minimized and the constraints are formulated; where the same variables will be evaluated for
each iteration so that a solution outside the limits does not converge.
Then the matrices and vectors that will be used for storing the data of the heuristic algorithm
will be defined and initialized, for which it will be necessary to obtain the objects of the nodes
('*.ElmTerm'), load ('*.ElmLod'), lines ('*.ElmLne'), capacitor banks ('*.ElmShnt') and the
system network ('*.ElmNet'), using the AllRelevant command to obtain all the data of the
specified object; for the location and optimal sizing of capacitor banks in the IEEE 34 node
system is complementary the manipulation of the service states, such as the amount of reactive
steps of the capacitor banks for which by means of the AllRelevant command ('*.ElmShnt') can
be enabled and disabled according to the iteration and responses of the genetic algorithm;
40
taking into account that node 800 is not considered for the location of capacitor banks as it is a
reference bus.
Once the active power losses are evaluated through the power flow by means of the Load Flow
Calculation (ComLdf) command, in each iteration the different combinations of capacitor
banks in the nodes evaluated by the genetic algorithm are performed, this should take the best
value, therefore the system will present the best results of power flow, being these the best
power conditions in the lines, these will be the values selected and stored in the fitness vector
that represent the best solution that the GA will provide.
Result
Using the methodology in reference to the previous section. The number of population was
defined to be 100, and a number of 100 iterations was defined. Once the 100 iterations were
completed, the algorithm determined that it is necessary to use 7 capacitor banks. Considering
that the losses before compensation were 206.025 KW shown in figure 1.
Figure 1. Grid-wide data after power flow without reactive compensation.
The solution obtained by the GA to minimize losses was 168.948 kW shown in Figure 2, thus
reducing power losses by 18%, which in power is equivalent to 37.077 kW.
Astudillo et al.
October - December vol. 1. Num. 10 2021
Figure 2. Grid-wide data after power flow with reactive compensation.
Figure 3 shows the graph of the Fitness GA function going from the highest power obtained in
the iterations to the one that was the solution proposed by the heuristic method.
Figure 3. Graph of GA fitness function.
The capacitor banks to be used according to the GA will be located at nodes 814, 836, 844,
844, 848, 860, 864 and 890, as shown in Table 1, which also details the reactive power of each
capacitor injected to the system, and the reactive steps of the capacitor banks being the
maximum 30.
Figure 4 shows illustratively the location of the capacitor banks in service, marked with a blue
box in the IEEE 34-node system, modeled in DIgSILENT PowerFactory software.
179,974
168,948
168
169,25
170,5
171,75
173
174,25
175,5
176,75
178
179,25
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
POWER LOSSES [KW].
NUMBER OF ITERATIONS
FITNESS GA
42
Table 1. Optimal Capacitor Bank Location proposed by the GA
Node
Capacitor
Power
[MVAr]
[MVAr
Act. Step
864
C23
0.030
2
860
C27
0.090
6
836
C28
0.345
23
890
C30
0.060
4
844
C32
0.045
3
848
C33
0.135
9
814
C6
0.120
8
Total
7
0,825
Figure 4. Location of capacitor banks that injected reactive power to the system.
Considering that the main objective function is to reduce losses in the system and it is not a
multiobjective function, but reducing losses also improves the voltage profile of each phase as
shown in Figures 5, 6 and 7, which represent the voltage per phase in per unit of each of the
nodes of the distribution network.
The uncompensated system is shown in blue on the curve and the compensated system with
the 7 capacitor banks is shown in red.
Astudillo et al.
October - December vol. 1. Num. 10 2021
Figure 5. Voltage profile per node in phase A.
Figure 6. Voltage profile per node in phase B.
Figure 7. Voltage profile per node in phase C.
The approximate value when improving the voltage is 0.03 Pu as a maximum value in most of
its nodes, such voltage profile values are within the restrictions.
To obtain a better validation of the method used, it would be to choose an alternative software
for the development and comparison of the results or to perform the same procedure using
another heuristic method.
Another factor to consider would be the execution of multi-objective functions to obtain the
minimization of losses in addition to improving the voltage profile in the system as well as the
power factor, thus obtaining more feasible and efficient results in the location and sizing of
capacitor banks in a distribution system.
0,87
0,9
0,85
0,87
0,89
0,91
0,93
0,95
0,97
0,99
1,01
1,03
1,05
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
VOLTAGE (P.U)
NODES
VOLTAGE PROFILE PHASE A
Uncompensated Compensated
0,88
0,91
0,85
0,87
0,89
0,91
0,93
0,95
0,97
0,99
1,01
1,03
1,05
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
VOLTAGE (P.U)
NODES
VOLTAGE PROFILE PHASE B
Uncompensated Compensated
0,88
0,91
0,85
0,87
0,89
0,91
0,93
0,95
0,97
0,99
1,01
1,03
1,05
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
VOLTAGE (P.U)
NODES
PHASE C VOLTAGE PROFILE
Uncompensated Compensated
44
Conclusions
The implementation of the genetic algorithm yields the best solution of the system. For this
purpose, the set of feasible solutions that limits the random search of the capacitor banks was
proposed, this is detailed by means of a matrix where the node number and the reactive steps
in each capacitor bank that predetermines the range of solutions are denoted. A preliminary
analysis of the system should be performed so that the maximum capacity of the capacitor
banks does not exceed the load capacity per node in the system.
The implementation of the genetic algorithm determines that the best solution for the 34-node
IEEE system is the contribution of 7 capacitor banks with an injection of 825 kVar of reactive
power to the system.
The algorithm solution was based on an objective function and constraints, opting for the best
solution based on mutation and replications; the algorithm consists of 100 iterations, this being
the population of the GA. Before implementing the algorithm the losses of the system were
206,025kw, optimizing the distribution system the losses in the network are 168,948kw this
indicates that the system reduced the losses in total of 37,077 kw equivalent to a percentage of
18% of losses.
It is determined that the optimal sizing and placement of the capacitor bank in the IEEE 34
node system raises the voltage level in each of the phases, with a maximum value of 0.3 pu,
which shows an improvement in the voltage profile.
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