Multiobjective Optimization of an Electrostatic Micromotor Using the Multiple Cuts Ellipsoid Method

A. C. Lisboa1, D. A. G. Vieira1, J. A. Vasconcelos1, R. R. Saldanha1 and R. H. C. Takahashi2

1Department of Electrical Engineering, Federal University of Minas Gerais (UFMG) 2Department of Mathematic, Federal University of Minas Gerais (UFMG)

{adriano,douglas}@cpdee.ufmg.br

Abstract— This paper addresses the multiobjective optimization of an electrostatic micromotor using the Multiple Cuts Ellipsoid Method (MCEM). The MCEM is an extension of the classical ellipsoid optimization method to employ multiple cuts, instead of the classical method single cut. The results show the efficiency of the proposed approach.

I. INTRODUCTION

In the last years the multiobjective design of electromagnetic devices was largely studied using both stochastic [1] and deterministic optimization methods [2]. The choice of the right algorithm depends on the problem nature. The present paper deals with the multiobjective optimization of an electrostatic micromotor [3] aiming at maximizing the average torque while minimizing the torque ripple. Since the objective of minimizing the torque ripple does not have continuous derivatives, search direction methods are very prone to fail. In such a case methods based on the idea of exclusion of half-spaces should be favored, since they are capable to deal with sub-gradients. The micromotor multiobjective design is solved here using the MCEM [4].

II. THE MULTIPLE CUTS ELLIPSOID METHOD (MCEM)

The MCEM is an extension of the classical ellipsoid optimization method [4], which employs multiple cuts, instead of the classical method single cut. The use of multiple cuts leads to a natural extension of the classical algorithm, which deals only with single objective problems, to one capable to deal directly with multiobjective optimization problems without any scalarization procedure. This novel algorithm can also be used in single objective, constrained or not, problems, leading to a higher convergence rate than the classical ellipsoid method without losing any of its theoretical characteristics which guarantee the method robustness. Given a set of gradient evaluations, the objective gradients, the method applies a central cut for one of them. To the following ones it is calculated their equivalent deep or shallow cuts, in such a way to guarantee the algorithm convergence properties. These cuts are iteratively applied generating a smaller ellipsoid that contains the intersection region between the previous ellipsoid and the points that dominate the previous design point.

III. ELECTROSTATIC MICROMOTOR DESIGN

The electrostatic micromotor with 8 poles at rotor and 12 at stator, as shown in Fig. 1, is considered [3]. Its optimization parameters are the rotor and stator tooth width and the slot

radius, τr, τs and Rslot, respectively. The optimization problem consists in maximizing the average torque while minimizing the torque ripple level, in order to get a square torque distribution along rotor rotation.

Fig. 1 - Micromotor configuration.

IV. RESULTS

In Fig. 2 is an optimal torque distribution as a function of the rotor position, obtained with the MCEM. The motor analysis was performed by Finite Element Method (FEM), where the torque was post-processed using the Maxwell stress tensor. In the full paper more details will be depicted.

0 10 20 30 40 50 60 70 80 90-8

-6

-4

-2

0

2

4

6

8x 10-12

θ (º)

τ (N

m)

Fig. 2 – Optimal torque distribution.

V. REFERENCES

[1] Douglas A. G. Vieira, Ricardo L. S. Adriano, João A. Vasconcelos, and Laurent Krähenbühl, “Treating Constraints as Objectives in Multiobjective Optimization Problems Using Niched Pareto Genetic Algorithm”, IEEE Trans. on Magnetics, vol. 40, no. 2, March 2004.

[2] A. C. Lisboa, D. A. G. Vieira, J. A. Vasconcelos, R. R. Saldanha and R. H. C. Takahashi, “Multi-objective Shape Optimization of Broad-band Reflector Antennas Using the Cone of EfficientDirections Algorithm”, IEEE Trans. on Magnetics, accepted paper.

[3] P. Di Barba, A. Savini and S. Wiak, “2D numerical simulation of electrostatic micromotor torque”, in Proc. IEEE 2nd Int. Conf. Computation in Electromagnetics, pp. 227-330, April 1996.

[4] N.Z. Shor, “Cut-off method with space extension in convex programming problems”, Cybernetics, vol. 12, pp. 94–96, 1977.

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