Optimization methods

 

 

Date

30/05/2007

Author

F. Moussouni, T. V. Tran, S. Brisset, P. Brochet

Affiliation

L2EP – EC Lille – France

Email

fouzia.moussouni@ec-lille.fr, tran.tuan-vu@ec-lille.fr, stephane.brisset@ec-lille.fr

Method

Genetic Algorithm

References

[1]  J. C. Spall, Introduction to stochastic search and optimization, A John Wiley & Sons, Inc., Hoboken, New Jersy, 2003.

[2]  Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, ISBN 3-540-58090-5 Springer-Verlag Berlin Heidelberg New York, 1994.

Description of the method

GA is a stochastic method based on the Darwin theory of evolution. They are considered as a global optimization method. In particular, genetic algorithms manage very well combinatorial and mixed problems. Unlike gradient search methods, GA is less susceptible to be trapped in local optima [1], [2].

In the optimization of the safety isolating transformer problem, each individual in the population is considered as a combination of three chromosomes (Fig. 1). The genome encoding of the three chromosomes is a discrete value (Fig. 1).

 

                              Fig. 1. Genome encoding of chromosomes

 

As the reproduction operators depend on the encoding and on the problem [5], specific crossover and mutation operators are implemented, in this study.

At the new generation, the mutation operator makes random changes on some individuals in order to jump outside local optima. In this study, a bit flip mutation is used.

Bit lip mutation is a two-step process. First, the algorithm creates a random binary vector, then, selects genes of an individual for mutation, where the vector is a 1. In the second step, the algorithm replaces each selected gene by a new gene selected uniformly from its data base (fig. 2).

Fig. 1. Bit flip mutation: Muted genomes are selected uniformly from the data base

Crossover combines two parents, to form children, for the next generation. Then a scattered crossover is used.

This type of crossover creates a random binary vector. So, the genes are selected from the first parent where the vector is a 1, and from the second one where the vector is a 0, and combines the genes to form the first child, and verse versa to form the second one (see fig. 2).

Fig. 2. Scattered crossover: Parents are selected uniformly from the data base

For GA parameters, the following values are used: population size N = 60, maximum number of generations T = 100, crossover probability 0.9, binary tournament selection operator is used. If the objective value is not improved over 50 generations the algorithm stops.

As optimization of the safety isolating transformer is a non-linear constrained problem the external penalty approach is used. The objective function and non-linear constraints are combined to formulate the following sub-problem:

(1)

 

Where f(X) = Mtot (X), is the original objective function to be optimized, gi(X) are the m=7 inequality constraints (see equation (1)). The parameter µ is a penalty factor updated at each generation k as follow:

                                            (2)

 

a is a positive parameter set to 10. µ0 is the upper limit of µ(k), given by the users and must be positive and very large.

Publication of the method

 

 

 

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