IEEE TRANSACTIONS ON MAGNETICS, VOL 49, NO 5, MAY 2013 2061 A Global Optimization Algorithm for Electromagnetic Devices by Combining Adaptive Taylor Kriging and Particle Swarm Optimization Bin Xia , Minh-Trien Pham , Yanli Zhang , and Chang-Seop Koh College of ECE, Chungbuk National University, Chungbuk 361-763, Korea University of Engineering and Technology, Vietnam National University, Hanoi, Vietnam School of Electrical Engineering, Shenyang University of Technology, Liaoning 110870, China This paper presents an efficient optimization strategy which employs adaptive Taylor Kriging and Particle Swarm Optimization (PSO) In this method, the objective function of electromagnetic problem is interpolated by using adaptive Taylor Kriging, in which the covariance parameter is obtained by Maximum Likelihood Estimation (MLE) And then, PSO is used to search for optimal solutions of electromagnetic problem The proposed algorithm is verified its validity by analytic functions and TEAM (Testing of Electromagnetic Analysis Method) problem 22 Index Terms—Adaptive Taylor Kriging, maximum likelihood estimation, particle swarm optimization, TEAM problem 22 I INTRODUCTION P ERFORMANCE ANALYSIS of electromagnetic device usually involves computationally expensive finite element analysis through finding solutions of electromagnetic fields So far, the optimization problems in electromagnetic devices are typified by features that present difficulties to most deterministic search algorithm, such as the existence of multiple local minima PSO with their ability to search more globally is better suited for exploring complicated objective function landscapes The high computational cost of evaluating the objective function in such problems, however, means that directly use of a PSO is often not feasible or is impractical, owing to their general requirement for a large number of objective function evaluations [1], [2] Thus, it is important to consider the possible means of reducing the cost of the analysis One technique that has recently attracted significant attention, called surrogate modeling [3], is the focus of this paper In this technique, the objective function is evaluated indirectly by interpolated functions The approximate function must have low computational cost, high accuracy, and good interpolation performance So far, Kriging, a spatial statistical technique, is now popular for the electromagnetic design optimization The Kriging surrogate model is different from random optimization method, in which the numerical analysis of electromagnetic fields is carried out in each iteration, and it is also different from response surface model (RSM) with fixed parameterized polynomial It is used by the semi-parameterization to construct the response model [4], [5] According to the different drift functions, Kriging models are generally divided into Simple Kriging, Ordinary Kriging and Universal Kriging Universal Kriging is a non-stationary geostatistical method and its drift function is a general linear function Due to complexity of equation calculation, it is seldom inves- Manuscript received November 21, 2012; accepted December 29, 2012 Date of current version May 07, 2013 Corresponding author: C.-S Koh (e-mail: kohcs@chungbuk.ac.kr) Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org Digital Object Identifier 10.1109/TMAG.2013.2238907 tigated Recently, Taylor expansion is used to approximate the drift function of Kriging, which is called Taylor Kriging, and it has the strongest approximation potentials and the best performance [6] In this paper, adaptive Taylor Kriging is developed to simulate the objective function, in which the Gaussian covariance parameters are estimated by Maximum Likelihood Estimation (MLE) [7] And then PSO is employed to get the optimal values of objective function [8] II MLE ASSISTED ADAPTIVE TAYLOR KRIGING A Taylor Kriging Principles The response function of a deterministic computer experiment is given by (1) is the position vector with It is a Kriging model, where dimension, is the regression function, is the unknown vector of regression coefficients, is called a drift function showing the average behavior of response , and is a random error term with Thus, a Kriging model is a combination of a linear regression and a stochastic process with mean and variance Suppose the drift function has continues derivatives up to the th order at point , which is the average of sampling points So the Taylor expansion of the drift function at is given as follows: (2) , and is Lagrange where is a vector between and remainder So , and is the basis function of Taylor expansion The expression is given as follow: (3) Suppose values 0018-9464/$31.00 © 2013 IEEE sample points with corresponding observed , and , and then the function value 2062 IEEE TRANSACTIONS ON MAGNETICS, VOL 49, NO 5, MAY 2013 at the unknown point can be estimated through a linear combination of the observed values by Kriging as follows: (4) are Kriging weights where the coefficients In Kriging, the Best Linear Unbiased Predictor is used to select coefficients by satisfying conditions as follows: 1) The estimator (4) must be unbiased 2) The variance of estimator errors between the estimator and the true response should be minimized The above requirements can be satisfied by minimizing a problem as follows: (5) The variance between (4) can be calculated as: and In order to find the best covariance function, another approach is called Gaussian covariance function, it is trying to find the optimal correlation parameter of the covariance function And the optimal correlation parameter of the covariance function is estimated by using MLE To apply this method, at first, we assume generally the distribution of sampled response values is Gaussian distribution For two samples and , the Gaussian covariance function is defined as follows (9) is a unknown covariance paramwhere eter vector, which influences the effect of covariance function along direction If the variance of is , the covariance between two samples is: as defined in (1) and (10) In order to estimate the best correlation parameter , MLE is used, and the likelihood function for N sampled response values, which follows Gaussian distribution is defined as: (6) Since rived: , then constraint function is de(7) Finally, because the basic functions are obtained by the Taylor expansion, this model is called Taylor Kriging, and the corresponding equations are formed by Lagrange multipliers as follows: (11) where , the design matrix , and the correlation function matrix So, for simplicity, maximizing the logarithm of likelihood functions is defined as follows (8) Therefore, the Kriging weights can be obtained from (7) and (8) Taylor Kriging is one of Universal Kriging However, if the order of Taylor expansion is given, the basis function in its drift function can be obtained, so Taylor Kriging overcome the difficult that the base function in Universal Kriging needed to be specified B Covariance Function In the Taylor Kriging equations, the covariance function is not defined yet In Kriging models, selecting a proper covariance function is a crucial problem The research on covariance function mainly focuses on two approaches; one is the known correlation function such as the spherical model, the thin elastic plates model, the cubic model and so on It aims to find the best correlation function which best approximate the distribution of sampled response values However, in real engineering problem, with the given correlation parameter of the covariance function, the existence covariance function with fixed coefficient maybe not the best approximation for the distribution of sampled response value (12) Then the maximum likelihood estimators of and are (13-a) (13-b) By substituting the mean and variance into (12), and eliminating the constants, the maximize likelihood function becomes function only of correlation parameter as follows (14) In order to maximize this problem, because of difficulty of likelihood function, global version of PSO is used, which is known as the accuracy and fast convergence After the optimal parameter is found, the covariance function is defined Therefore, Taylor Kriging model is completely defined Any unknown response of point will be calculated by (4) XIA et al.: A GLOBAL OPTIMIZATION ALGORITHM FOR ELECTROMAGNETIC DEVICES 2063 Fig Flow chart of global optimization algorithm III GLOBAL OPTIMIZATION ALGORITHM EMPLOYING MULTIPLE ITERATION AND GRADUAL REFINEMENT When evolutionary algorithms are used to solve optimization problems, explicit fitness functions may not exist With the help of interpolation capability, the Taylor Kriging model is applied to build surrogate fitness function to guide further searching of optimal solution In the optimization strategy, the Latin Hypercube sampling (LHS) technique is used to obtain sample points in the design space Based on the optimal result obtained from previous iteration, the design space is reduced and new sample points of current iteration are gradually inserted to the approximate objective function, so that the efficiency and simulation accuracy will be improved The proposed optimization algorithm is summarized as follows: Step 1: Define the initial design space and generate initial sampling points by LHS in the whole design space Step 2: Calculate corresponding objective values Step 3: Construct the response surface by Taylor Kriging model Step 4: Find the current optimal point by PSO, and check the convergence, stop and output the result when the error of the current optimal point and the previous one is very small (less than ) Step 5: Reduce design space by adaptive factor of 0.618 around the current optimal point [4] Step 6: Generate new sampling points by the LHS in the reduced design space, and go to Step In the algorithm, the iteration repeats until the optimal point converges The flow chart is shown in Fig IV ANALYTIC EXAMPLES A Mathematical Examples The true accuracy of a surrogate model can only be determined if the true function, which it is attempting to approximate, is also known and is available for comparison Almost all electromagnetic optimization design problems are non-analytic, meaning it is impossible to measure the accuracy of any surrogate model constructed Therefore, in this paper, one analytic test problem is selected as: Fig Optimization process of analytic function (a) true response surface; (b) initial 25 sampling points; (c) 2nd iteration; (d) 3rd iteration; (e) Kriging response surface; (f) normalized root mean squared error for different sampling points (15) where the true global maximum exists at with the function value of 8.1061 Firstly, Generating 25 initial sampling points in the whole design space by LHS, and the best parameter vector is found by MLE, because of two variables, Therefore, the Gaussian covariance function is confirmed Fig shows the constructed response surface by Taylor Kriging model and the distribution of sample points at each iteration At the initial iteration, 25 sample points are obtained by the LHS in the whole design space, and corresponding response values are calculated by the sample points And then the Kriging response surface is constructed to give an optimum (0, 1.4658) with , as shown in Fig 2(b) Then, the design space is adaptively reduced, and and 13 additional sampling points are inserted the design space, respectively, as shown in Fig 2(c) and (d) After three iterations, a converged optimal point with is obtained and corresponding Kriging response surface is shown in Fig 2(e) In other to assess its accuracy, the normalized root mean squared error (NRMSE) will be used for comparing the surrogate model with the true function, and it is defined as follows (16) 2064 IEEE TRANSACTIONS ON MAGNETICS, VOL 49, NO 5, MAY 2013 TABLE I COMPARISON OF OPTIMIZATION RESULTS FOR TEAM PROBLEM 22 The results and comparison with other methods are given in Table I Fig Configuration of the SMES device where is the predicted value at by Taylor Kriging, is the true value at , and the mesh of test points about Fig 2(f) shows NRMSE for different number of sample points From the results, the estimation error of the NRMSE decreases as the number of sample points increases Cross-validation becomes more reliable, resulting in a higher accuracy of Kriging interpolation This verifies Kriging claim to be a very flexible for highly nonlinear functions B TEAM Workshop Problem 22 TEAM workshop problem 22 is classified as three-parameter and eight-parameter problem Fig shows the design parameters of TEAM workshop problem 22 In this paper, we consider the three design parameters problem The objective function of the problem takes into account both the energy requirement (E should be as closed as possible to 180 MJ) and the minimum stray field requirement ( evaluated along 22 equidistant points along line a and line b in Fig as small as possible) Therefore, the single objective function is defined as follows [9] (17) where as: MJ, and T and is defined (18) Limits of three design parameters are specified in [4] This optimization problem has several feasible regions, and nonlinear multi-objective function that is consist of the error of energy to be stored and stray field It is not easy to find optimum solution of this problem using typical optimization techniques Therefore, proposed global optimization by Taylor Kriging and PSO is employed to resolve the difficulties Electromagnetic analysis of TEAM problem is performed by MAXWELL 12.0 27 sampling points generating by LHS is used as initial sample set We can find a global optimum which is superior to other methods such as GA and simulated annealing (SA) as given in Table I The global optimum is and corresponding objective function value is 0.0870 The objective function value of optimum design is decreased by 65.66% compared with initial design V CONCLUSION In this paper, a global optimization strategy employing multiple iterations and gradual refinement is proposed The Taylor Kriging model with Gaussian covariance function is used as interpolation function approximate to the objective function Then the best parameter of Gaussian covariance function is successfully found by MLE, and the optimal point of problem or objective function is obtained Through the applications to numerical example and TEAM problem 22, the proposed optimization strategy is computationally efficient with less sampling points and higher computation efficiency ACKNOWLEDGMENT This work was supported by the Basic Science Research Program through NRF of Korea funded by the Ministry of Education, Science, and Technology (2011-0013845) REFERENCES [1] L Wang and D Lowther, “Selection of approximation models for electromagnetic device optimization,” IEEE Trans Magn., vol 42, no 4, pp 1227–1230, Apr 2006 [2] G Hawe and J Sykulski, “Considerations of accuracy and uncertainty with Kriging surrogate models in single-objective electromagnetic design optimization,” IET Sci Meas Technol., vol 1, no 1, pp 37–47, 2007 [3] N V Queipo, R T Haftka, W Shyy, T Goel, R Vaidyanathan, and P K Tucker, “Surrogate-based analysis and optimization,” Progr Aerosp Sci., vol 41, pp 1–28, 2005 [4] Y Zhang, H S Yoon, P S Shin, and C S Koh, “A robust and computationally efficient optimal design algorithm of electromagnetic devices using adaptive response surface method,” J Elect Engrg Technol., vol 3, no 2, pp 207–212, 2008 [5] S Koziel, I Couckuyt, and T Dhaene, “Reliable low-cost co-Kriging modeling of microwave devices,” presented at the Int Microwave Symp (IMS2012), 2012 [6] H P Liu and S Maghsoodloo, “Simulation optimization based on Taylor Kriging and evolutionary algorithm,” Appl Soft Comput., vol 11, no 4, pp 3451–3462, Jun 2011 [7] L Lebensztajn, C A R Marretto, and M C Costa, “Kriging: A useful tool for electromagnetic device optimization,” IEEE Trans Magn., vol 40, no 2, pp 1196–1199, Mar 2001 [8] M T Pham, M H Song, and C S Koh, “Coupling particles swam optimization for multimodal electromagnetic problems,” J Elect Engrg Technol., vol 5, no 3, pp 423–430, 2010 [9] P G Alotto et al., in SMES Optimization Benchmark: TEAM Workshop Problem 22, 2005 [Online] Available: http://www.igte.tugraz.ac.at/archive/team/index.htm [10] R H C Takahashi, J A Vasconcelos, J A Ramirez, and L Krahenbuhl, “A multiobjective methodology for evaluating genetic operators,” IEEE Trans Magn., vol 39, no 3, pp 1321–1324, 2003 [11] F Campelo, F G Guimaraes, H lgarashi, and F A Ramirez, “A clonal selection algorithm for optimization in electromagnetic,” IEEE Trans Magn., vol 41, no 5, pp 1736–1739, 2005 ... Flow chart of global optimization algorithm III GLOBAL OPTIMIZATION ALGORITHM EMPLOYING MULTIPLE ITERATION AND GRADUAL REFINEMENT When evolutionary algorithms are used to solve optimization problems,... R Vaidyanathan, and P K Tucker, “Surrogate-based analysis and optimization, ” Progr Aerosp Sci., vol 41, pp 1–28, 2005 [4] Y Zhang, H S Yoon, P S Shin, and C S Koh, A robust and computationally... is attempting to approximate, is also known and is available for comparison Almost all electromagnetic optimization design problems are non-analytic, meaning it is impossible to measure the accuracy