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DESIGN OF TOPOLOGY OF MV URBAN DISTRIBUTION
NETWORK WITH USING OF GENETIC ALGORITHMS
Ing. Petr SKALA, Doctoral Degree Programme (2)
Dept. of Electrical Power Engineering, FEEC, BUT
E-mail: [email protected]
Supervised by: Dr. Vladimír Blažek
ABSTRACT
Genetic algorithms are useful tools for solving of many optimization problems. This
paper deals with using of genetic algorithms on MV urban distribution network design. An
algorithm is described which can be used for design of MV distribution network feeders
topology with minimum of costs expended on cables, network losses and non-supply
electrical energy. Algorithm allows using penalty functions for inclusion non-performing
various constrictions. Algorithm functionality is illustrated with a simple example.
1
INTRODUCTION
Genetic algorithms (GA) are, in principle, stochastic optimization processes inspired
with principles of the nature evolution, which search through a given space of feasible
solutions. Their search ability is based on simulation of inheritance and Darwinian struggle
for survival. GA have been proved a very good tools for solving various optimization
problems and have been successfully applied for solving many electrotechnic problems [4,5].
This paper shows that it is possible to use GA also for solving optimization tasks connected
with design of MV urban distribution networks topology.
Due to faster searching for feasible solution space, these algorithms work not only with
one solution but with bigger set of feasible solutions called population. Each real solution has
to be encoded by suitable way. The basic principle, common for all types of genetic
algorithms, is shown in Fig. 1.
Firstly, an initial population is created randomly or by some meaningful method.
Particular individuals (particular solutions) are evaluated by suitable way pursuant to specific
fitness function. Subsequently, the population is submitted to an evolution. The goal of the
evolution is to create new, preferably better, individuals for new population. The first step is
selection that simulates natural selection and chooses individuals from previous population to
new one. The choice is performed according to their fitness function value. The changes in
the evolution process are obtained by the help of a crossover when newly created individuals
get parts of genetic information from each of parents. In order to prevent from sticking in a
local optimum, small changes – mutations - are performed on some new individuals. After
that, individuals of new population are evaluated and whole cycle repeats until specific stop
criterion is not achieved.
begin
t := 0;
Create initial population G(t);
Evaluate G(t);
while (not stop criterion) do
begin
t:=t+1;
Select parents R(t) from G(t-1);
Crossover individuals from R(t), create offsprings set D(t);
Mutation D(t);
Create new population G(t+1):=D(t);
Evaluate G(t+1);
end
end
Fig. 1:
2
GA basic principle
APLICATION OF GENETIC ALGORITHMS IN URBAN DISTRIBUTION
NETWORKS DESIGN
The MV urban distribution networks are designed so that they will comply with
requests for quality and reliability of electric energy supply to LV networks. LV networks are
usually operated as radial or ringed networks.
Let’s suppose that we know a supply concept of supplied area (location of HV/MV
transformer stations or switching stations) and location of MV/LV distribution transformer
stations and MV demands. Furthermore, let’s suppose that values of demands are known. Due
to securing of appropriate reliability of supply to connected consumers, we usually require to
all distribution transformer stations and MV demands can chance to be supply form two sides.
This requirements lead to building-up ringed MV feeders connected to one HV/MV
transformer station or feeders which are led from a HV/MV transformer station to another
one.
The solution of this problem lies in founding the shortest Hamilton’s circle in a
complete undirected graph. Graph nodes correspond to MV networks nodes. Undirected
edges represent practically feasible segments of a MV feeder and are evaluated by
corresponding length of the cable feeder. If we solve this problem by the help of genetic
algorithm, we must feasible solutions suitably to encode. The good way is to encode solutions
by the help of strings whose length is equal to number of network nodes M. The string is
constructed numbers from 1 do M. Each number can be in the string exactly one‘s. For
example, a path 1 – 3 – 5 – 4 – 2 and back to 1 is represented as (1 3 5 4 2) string.
The fitness of particular individuals is determined as a sum of costs expended on cables,
costs on losses, costs on non-supply electrical energy and, in case of need, another costs
connected with network operating. Because GA do not have special requests on a shape of the
fitness function, it is possible to include into evaluation of quality of individuals also nonperforming various constrictions by the help of penalty functions– e.g. overrange permited
voltage drop, overrange permited current of cable etc. For fitness function f we can write
f = N v + N ∆ + N ns + g (∆U max ) + h( I v ,max )
( 2.1 )
where Nv are costs expended on cables, N∆ are costs on network losses, Nns are costs on
non-supply electrical energy, g is penalization for overrange permited voltage drop and h is
penalization for overrange permited current of cables.
We need to use special crossover operator for this type of representation of solutions.
The simplest crossover operator for this, so called natural path representation, is an order
crossover [3]. Another crossover operators are a partially mapped crossover and an edge
recombination crossover.
The mutation in problems with natural path representation usually works so that
individuals from population are chosen with determined small probability and chosen
individuals undergo the mutation. One of possibilities performing the mutation is the change
of two randomly selected elements of the string.
3
EXAMPLE OF USING AND COMPUTATION RESULTS
Applicability of genetic algorithms for design of MV urban distribution network
topology can be illustrated by a simple example of MV urban network contained one
110/22 kV transformer station and nine 22/0,4 kV distribution transformer stations which
should be supplied from one ringed cable feeder (see Fig. 2).
Fig. 2:
Testing network arrangement and resulting optimal solution
To simplification of situation, we will use for evaluating of solutions only the first two
terms of fitness function (2.1). Thus
f = N v + N ∆ = cv .l v + c ∆ .∆P
( 3.1 )
where cv is weighted coefficient of total feeder length, lv is total feeder length for
corresponding solution, c∆ is weighted coefficient of losses in feeder and ∆P are losses in
feeder at supplying from two sides for corresponding solution. Both weighted coefficients are
elected pursuant to preliminary calculations for solved network so that losses in feeders were
only auxiliary criterion during decision-making between two solutions with the same length
of feeders. There was used cv = 1 m-1 and c∆ = 10-4 W-1 in mentioned illustrative case.
In mentioned example, the initial population was created randomly and a tournament
selection was used in the selection. For conservation of the best individual in the population
elitism principle was applied that means the best individual was copied directly to population
of next generation without crossover and mutation. The order crossover was elected as the
crossover operator [3]. The mutation was performed mutual exchange of randomly selected
nodes. Computation was stopped as soon as optimal solution that had been known for this
problem in advance was achieved.
in 1. generation
in 8. generation
55
minimum value
mean value
in 16. generation
in 24. generation
V alue of fitness function
50
45
40
35
30
0
Value of fitness function of the best solution in p.u. [-]
20
25
1.9
1.5
Mean value of fitness function in p.u. [-]
1.45
1.8
1.4
1.7
1.35
1.6
1.3
1.5
1.25
1.2
1.4
1.15
1.3
1.1
1.2
1.05
Fig. 4:
10
15
Generation
One run of algorithm and its history
Fig. 3:
1
0
5
10
20
30
Generation
40
50
1.1
0
10
20
30
Generation
40
50
a)
b)
Samples of courses of fitness of the best solution in the population and average
fitness of solution in the population during iterative processes
Fig. 4 illustrates the population behavior during iterative process. Fig. 4 a) shows
successive drop of fitness function value of the best solution in given population during
iterative process. Using of the elitism principle has ensured that displayed courses have
permanently non-increasing character. Subsequently, we can see corresponding courses of
average values of fitness functions of solution in the population during iterative process in
Fig. 4b). Although displayed courses have large fluctuation, it is evident it has happened
successive improving of quality of whole population during “evolution”. This process is
caused by recombination operators (crossover and mutation) that can create poor quality
offsprings from qualitatively good parents.
4
CONCLUSION
Genetic algorithms are useful tools for solving many optimization problems. They can
be used for optimization of MV urban distribution network topology. This problem can be
defined as a searching of the shortest Hamilon’s circle in complete undirected graph. Edges of
the graph are evaluated appropriated lengths of the lines. Because genetic algorithms don’t
have special requests on a shape of fitness function, it is possible to include to optimization
not only the request on finding of solution with minimal costs expended on feeders but also
with costs on losses, costs on non-supply electrical energy, an overrange permited voltage
drop penalty, an overrange permited cable current penalty etc. It is suitable, by reason of
effective searching through feasible solutions space and computation acceleration, to use for
encoding of solution so called natural path representation, in this case. If we work with this
representation, we must use special recombination operators.
Applicability of genetic algorithm for mentioned problem was demonstrated on the
simple example of MV cable network. The algorithm was elaborated in Matlab whereas it
was used the order crossover and mutation implemented mutual change of randomly selected
nodes. The elitism principle was applied due to preservation of the best solution in the
population. The convergence speed of used method depends on number of network nodes.
Made computations show that for larger networks (networks with 40 and more nodes) it is
better to use some cleverer and more complicated crossover method, for example edge
recombination crossover.
REFERENCES
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Planning of Urban Distribution Networks of Medium Voltage. In: IEEE Transactions on
Power Systems. 2002, vol. 7, no. 3, pp. 879-884.
[3] Mařík V., Štěpánková O., Lažanský J. a kol.: Umělá inteligence 3. Academia, Praha,
2001. ISBN 80-200-0472-6.
[4] Miranda V., Ranito J.V., Proenca L.M: Genetic Algorithms in Optimal Multistage
Distribution Network Planning. In: IEEE Transactions on Power Systems. 1994, vol. 9,
no. 4, pp. 1927-1933.
[5] Momoh J.A.: Electric Power System Applications of Optimization. Marcel Dekker, New
York, 2001. ISBN 0-8247-91058-3.