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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 84
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 58

Application of Genetic Algorithms to the Implementation of Fractional Electromagnetic Potentials

I.S. Jesus1, J.A. Tenreiro Machado1 and J. Boaventura Cunha2

1Department of Electrical Engineering, Institute of Engineering of Porto, Portugal
2Engineering Department, University of Trás-os-Montes and Alto Douro, Vila-Real, Portugal

Full Bibliographic Reference for this paper
I.S. Jesus, J.A. Tenreiro Machado, J. Boaventura Cunha, "Application of Genetic Algorithms to the Implementation of Fractional Electromagnetic Potentials", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 58, 2006. doi:10.4203/ccp.84.58
Keywords: electrical potential, fractional order systems, genetic algorithm.

A genetic algorithm (GA) is a search process for finding approximate solutions in optimization problems. The GAs are a particular class of evolutionary algorithms that use techniques inspired by evolutionary biology such as inheritance, mutation, natural selection, and crossover, established by the Darwin's theory of evolution [1,2,6].

The concept of differentiation and integration to no integer order, dates from 1695, when Leibniz mentions it in a letter to L'Hopital. Since then, many scientists developed the area and notable contributions have been made, both in theory and in applications [3,4,5,7,8,9,10].

The integer-order differential nature of the potential expressions motivated several authors to propose its generalization in a fractional calculus (FC) perspective. Therefore, a fractional multipole produces at point a potential .

Inspired by the integer-order recursive approximation of fractional-order transfer functions presented previously, we developed a GA for implementing a fractional order potential. In this line of thought, we developed a one-dimensional GA that places recursively n charges    odd   even at the symmetrical positions (with exception of that corresponds to the centre of the n-array of charges where there is a single charge ).

Our goal is to compare the approximate potential , resulting from a number of charges and the corresponding locations, with the desired reference potential . Therefore, in this study the experiments consist of executing the GA several times, in order to generate a combination of positions and charges that lead to an electrical potential with a fractional slope similar to the desired reference potential.

The results show a good fit between the two functions and, moreover, that it is possible to find more than one 'good' solution. Nevertheless, for a given application, a superior precision may be required and, in that case, a larger number of charges must be used. We verify that the position of the charges varies significantly with the number of charges used in the algorithm. Therefore, the charge versus location pattern is not clear and its comparison with a fractal recursive layout is still under investigation.

We conclude that the GA reveals a good balance between the accuracy of the results and the computation time. The GA approach constitutes a step towards the development of a systematic design technique and, consequently, several of its aspects must be further evaluated. Research on the approximation feasibility and its convergence, error variation with the range and the number of charges, improvement when adopting an extended library of primitives rather than, merely, point charges and its extension to the three-dimensional space is presently under development.

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