I702 zyxwvutsrqponmlk IEEE TRANSACTIONS ON MAGNETICS, VOL. 35, NO. 3, MAY 1999 E mb zyxwvut e d de d S t o chas t i c- D eter m i n i s t ic 0 p t i m i z at i on Met ho d with Accuracy Control Daniel Ioan, Gabriela Ciuprina and Andras Szigeti “Politehnica” University of Bucharest, Numerical Methods Laboratory zyxwv - Electrical Engineering Dept. Spl. Independentei 313, Bucharest, Romania Abstract-A distributed evolutionary strategy is used to solve the optimization problem in a subspace of the searching space. At each cost function evalu- ation, a deterministic method is used to find the op- timal solution in the complementary subspace, with an imposed accuracy. The proposed method is applied to solve TEAM Workshop problem 22 with the control zyxwvu of the numerical error. In this case the stochastic method acts in the geometrical parameter subspace and an em- bedded deterministic technique is used in the 2D cur- rent densities subspace. Indez terms-Optimization, electromagnetic ana- lysis, genetic algorithms, distributed computing. I. INTRODUCTION The optimization of electromagnetic devices is not a simple task and the strategy chosen for optimization has zyxwvuts to be suited for the problem at hand. One class of problems refers to optimization of air-cored coils. At a first glance this type of problems seems to be an easy one, but the attempt of solving them reveals a lot of difficulties. There are mainly two category of optimiz- ation problems with air-cored coils: magnetic resonance and superconducting energy storage. The applications coming from nuclear magnetic reson- ance applications or from magnetic resonance imaging sys- tems require the optimization of coils (usually solenoids) so that a highly homogeneous magnetic field in a certain region of space is obtained [1]-[3]. These types of prob- lems are non-differentiable, non-convex and ill-conditioned [ 13. When just the field analysis is required, closed-form analytical formulae are recommended [a]. Analytical in- tegration of Biot-Savart low over coils volume could also he applied. In the case of superconducting magnetic energy storage (SMES) devices, an imposed magnetic energy has to be stored in superconducting coils so that the level of the stray field in the surrounding region is as small as possible [41-[61. Manuscript received June 3, 1998. D. Ioan, 401-410-6984, fax 401-411-1 190, danielOlmn.pub.ro, This work was supported in part by the Romanian Ministry of http://www.lmn.pub.ro; G. Ciuprina, gabrielaOlmn.pub.ro. Research and Technology under Grant No. 883/A6. The optimization of electromagnetic devices is made by different methods, either stochastic or deterministic. The stochastic methods, such as Simulated Annealing (SA) and Evolutionary Algorithms (EA) , have the capability of find- ing the global optimum, but they usually need large com- putation time and resources. That is why, strategies that mix stochastic and deterministic algorithms in a “locate and identify” manner (i.e. stochastic algorithm followed by a deterministic one) are proposed [5]-[7]. Determin- istic strategies are used sometimes at the beginning of the stochastic procedure [l] for creating good initialization, or during the stochastic algorithm for improving the current solution [8]. The aim of the paper is to describe an efficient and re- liable method for solving problems belonging to the class represented by SMES devices, in which the parameters are of geometrical and physical nature (dimensions and cur- rent densities respectively). For a given geometry, current densities are found by a deterministic optimization so that the energy has the imposed value, while the geometrical parameters are found by a stochastic method (distributed EA). 11. FIELD ANALYSIS A. Evaluataon The optimization of electromagnetic devices spends the most part of the CPU time for the field computation. In order to obtain efficient optimization programs, the method chosen for field computation has to be suitable to the problem. In the case of TEAM Workshop problem 22 [4], all media being linear and nonmagnetic, the most efficient method seems to be the one based on the Biot-Savart- Laplace formula: where R is the windings’ domain and zyx V = VI U 102 where 2)k, k E { 1,2} represents the kth coil cross-section: v k = [Rk - &/2, Rk -k &/2] X [-hk/2, hk/2]. The integration over 0 5 B < 27r yields: 0018-9464/99$10.00 zyxwvut 0 1999 IEEE