Journal of Magnetics 19(3), 266-272 (2014) http://dx.doi.org/10.4283/JMAG.2014.19.3.266 © 2014 Journal of Magnetics Optimal Design of Inverse Electromagnetic Problems with Uncertain Design Parameters Assisted by Reliability and Design Sensitivity Analysis Ziyan Ren 1,2 , Doojong Um 2 , and Chang-Seop Koh 2 * 1 School of Electrical Engineering, Shenyang University of Technology, Shenyang 110870, China 2 College of Electrical and Computer Engineering, Chungbuk National University, Chungbuk 361-763, Korea (Received 6 November 2013, Received in final form 10 June 2014, Accepted 11 June 2014) In this paper, we suggest reliability as a metric to evaluate the robustness of a design for the optimal design of electromagnetic devices, with respect to constraints under the uncertainties in design variables. For fast numerical efficiency, we applied the sensitivity-assisted Monte Carlo simulation (S-MCS) method to perform reliability calculation. Furthermore, we incorporated the S-MCS with single-objective and multi-objective particle swarm optimization algorithms to achieve reliability-based optimal designs, undertaking probabilistic constraint and multi-objective optimization approaches, respectively. We validated the performance of the developed optimization algorithms through application to the optimal design of a superconducting magnetic energy storage system. Keywords : inverse electromagnetic problem, Monte Carlo simulation, reliability evaluation, sensitivity analysis 1. Introduction Recent optimal design algorithms of inverse electro- magnetic problems have paid attention to uncertainties in design variables caused by, for example, manufacturing tolerance, uneven material properties, and the imperfect control of operating conditions. These uncertainties often force the deterministic optimal design to violate some constraints, by moving it to the infeasible region [1, 2]. To deal with uncertainties, robust optimal design methods have been developed to improve product quality, by minimizing variations of the system performance [2-4]. However, a reliable algorithm that guarantees the constraint condition in probabilistic terms against uncertain design variables has not yet been popularly presented in the area of electrical engineering. In the fields of mechanical and structural design, the concept of making a trade-off between perfor- mance and reliability has recently been addressed, to increase the robustness of constraint functions [5-7]. In this paper, the reliability of a design is defined as the probability of remaining in the feasible region with respect to a constraint function, when the design is perturbed by uncertainties in the design variables. In mechanical engineer- ing, there have been numerical attempts to evaluate reliability, such as Monte Carlo simulation (MCS), and first-order reliability methods (FORM) [7]. The MCS method is a sampling-based method, and it is known to be accurate only if the number of samples is large enough. Due to the huge computational effort, this method is not practical for engineering problems that normally involve performance analysis by numerical methods, such as the finite element method (FEM). The FORM is an optimization-based method, which calculates the reliability based on the shortest distance from a design to a constraint surface, in a normalized design space. This method, however, is also expensive to apply to an engineering problem, since the reliability calcu- lation itself needs to solve another independent optimi- zation problem. It has been difficult to find a practical and guaranteed reliability calculation algorithm that can be applied to a reliability-based optimal design of electromagnetic devices subject to performance related constraints that involve numerical analysis methods, such as the FEM. In this paper, in order to achieve a constraint-reliable optimal design against uncertain design variables, the sensi- tivity-assisted Monte Carlo simulation method is applied, for numerically efficient reliability calculation. Furthermore, based on the fast reliability calculation, constraint reliability- based and multi-objective reliability-based optimal design algorithms are developed. The validity of the developed algorithms is investigated through applications to an analytic example, and to a superconducting magnetic energy storage ©The Korean Magnetics Society. All rights reserved. *Corresponding author: Tel: +82-43-274-2426 Fax: +82-43-274-2426, e-mail: kohcs@chungbuk.ac.kr ISSN (Print) 1226-1750 ISSN (Online) 2233-6656