IEEE TRANSACTIONS ON MAGNETICS, VOL 49, NO 5, MAY 2013 2105 Multiguiders and Nondominate Ranking Differential Evolution Algorithm for Multiobjective Global Optimization of Electromagnetic Problems Nyambayar Baatar , Minh-Trien Pham , and Chang-Seop Koh , Senior Member, IEEE College of Electrical and Computer Engineering, Chungbuk National University, Cheongju, Chungbuk 361-763, Korea University of Engineering and Technology, Vietnam National University, Hanoi 100000, Vietnam The differential evolution (DE) algorithm was initially developed for single-objective problems and was shown to be a fast, simple algorithm In order to utilize these advantages in real-world problems it was adapted for multiobjective global optimization (MOGO) recently In general multiobjective differential evolutionary algorithm, only use conventional DE strategies, and, in order to optimize performance constrains problems, the feasibility of the solutions was considered only at selection step This paper presents a new multiobjective evolutionary algorithm based on differential evolution In the mutation step, the proposed method which applied multiguiders instead of conventional base vector selection method is used Therefore, feasibility of multiguiders, involving constraint optimization problems, is also considered Furthermore, the approach also incorporates nondominated sorting method and secondary population for the nondominated solutions The propose algorithm is compared with resent approaches of multiobjective optimizers in solving multiobjective version of Testing Electromagnetic Analysis Methods (TEAM) problem 22 Index Terms—Differential evolution, multiguiders, multiobjective optimization, nondominated ranking, Testing Electromagnetic Analysis Methods (TEAM) problem 22 I INTRODUCTION I N ENGINEERING application, optimization problems involving multiple objectives together with constraints are popular Therefore, many multiobjective global optimization (MOGO) algorithms have been proposed In order to apply the DE algorithm for solving MOGO problems, the original scheme has to be modified since the multiobjective problems not consist of single solution Instead, in multiobjective optimization, a set of different solutions should be founded and called Pareto-optimal front There are two issues when designing a multiobjective evolutionary algorithm: population diversity and survivor selection The first issue is directly related to the question of how to guide the search towards the Pareto-optimal front [1] The second one addresses the question of which individual will be kept during the evolution process In the past, a wide variety of evolutionary algorithms (EAs) have been used to solve multiobjective optimization problems [2] However, from the several types of EAs available, few researchers have attempted to extend DE [3] to solve multiobjective optimization problems DE has been very successful in the solution of a variety of continuous (single-objective) optimization problems in which it has shown a great robustness and a very fast convergence These are precisely the characteristics of DE that make it attractive to extend to solve multiobjective optimization problems DE has been adapted to solve MOGO in several ways In the early approaches (PDE [4] and GDE [5]), only the concept of Pareto dominance was used to compare the individuals The candidate replaced its parent only if it (weakly) dominated it Otherwise, it was discarded This is a rather strict demand, especially when the number of objectives is high Many subsequent approaches (PDEA [6], MODE [7], and NSDE-DCS [8]) used nondominated sorting and/or the crowding distance metric to calculate the fitness of individuals Therefore we proposed multiguiders nondominated ranking differential evolution algorithm (MG-NRDE) for solving MOGO problems In mutation step, the proposed method introduces new base vector selection method for constrained multiobjective optimization by adopting multiguiders Additionally, the approach also incorporates nondominated sorting method [9] and secondary population for the nondominated solutions to archive Pareto front solutions The proposed algorithm is compared with recent approaches of multiobjective optimizers in solving multiobjective version of TEAM problem 22 The remainder of this paper is organized as follows Section II provides fundamentals of the MOGO problems and DE algorithm In Section III we described the proposed multiguiders nondominated ranking DE in detail TEAM problem 22 and comparison results are provided in Section IV Finally Section V contains our conclusions II FUNDAMENTALS OF MOGO PROBLEMS AND DE Some fundamentals and basic definitions related to this work are introduced in following subsections A MOGO Problems A general MOGO problem contains a number of conflicting objectives, for example, to be minimized and optional constraints to be satisfied Mathematically, a MOGO problem is formulated as follows: Minimize subject to (1) Manuscript received November 21, 2012; revised December 26, 2012; accepted January 08, 2013 Date of current version May 07, 2013 Corresponding author: C.-S Koh (e-mail: kohcs@cbnu.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.2013.2240285 and are the where is the vector of design variables, numbers of the objectives and constraints, respectively In practical applications, there is no solution that can miniobjectives simultaneously As a result, mulmize all of the 0018-9464/$31.00 © 2013 IEEE 2106 IEEE TRANSACTIONS ON MAGNETICS, VOL 49, NO 5, MAY 2013 tiobjective optimization problems tend to be characterized by a family of alternatives that must be considered equivalent in the absence of information concerning the relevance of each objective relative to the others [1] These alternatives are referred to as Pareto optimal solutions A multiobjective optimization problem, on the other hand, has a set of optimal solutions which every member is not dominated by others All the members are optimal from the viewpoint of one (or more) objective(s), but none of them is optimal for all the objectives The choice among the Pareto-optimal solutions belongs to a designer’s decision [1], [9] The target of a MOGO algorithm, therefore, is to converge to the true Pareto-front and to provide a good distribution of the solutions on the entire Pareto-front B Deferential Evolution Algorithm The differential evolution algorithm is a novel parallel direct search method, which utilizes parameter vectors as a population , for each generation The crucial idea behind DE is a scheme for generating trial parameter vectors DE generates new parameter vectors by adding a weighted difference vector between two population members to a third member Currently, there are several variants of DE The particular variant described throughout this section is the “DE/rand/1/bin” scheme Each individual ) is a -dimensional vector with parameter values determined randomly and uniformly in the search space (2) For each target vector a mutant vector ) is generated by mutation operator where , and are randomly chosen indexes Note that indexes have to be different from each other, is , and are difference vector called base vector index, and indexes After mutation, the binominal crossover is applied to generate the trial vector The specified process is shown in (3) if otherwise In order to provide good Pareto front, the suggested algorithm incorporates nondominated ranking and multiguiders methods The proposed multiguiders nondominated ranking DE (MGNRDE) algorithm keeps two populations: the main population which is the target population (used to search Pareto optimum solutions) and external population (to archive nondominated solutions and provide guiders) Additionally, in mutation step, we considered feasibility of solution when we select the guiders; this action will be taking into account in case of performance constraint problems Furthermore, when number of external solutions exceeds its maximum value, an improved pruning method [10] is used to remove the solutions with small crowding distance, one by one until number of solutions equals to its maximum value Step 1: Initialize the target population • Generate target population with randomly uniform, and set the iteration counter • Evaluate all objective function and constraint values, and apply nondominated sorting to rank the all individuals in the current population, and calculate crowding distance in each rank • Store nondominated solutions into the external archive , Calculate the crowding distance in objective space (objective crowding distance) for all members in Step 2: Generate mutant populations • Randomly select the first guider (in conventional DE would be called base vector) from the top 10% of solution with the big crowding distances • Second guider required only when the fist guider is not extreme solution [1] • Between the two nondominated solutions beside in Pareto-front, as shown in Fig 1, the one with bigger crowding distance is selected as the second guider And the feasibility of the guiders must be considered And then generated by using (5) or (5) (3) where a trial vector is generated from the mutant vector and its target vector based on probabilistic parameter selection Cr is a user-specified crossover factor in the range [0,1) and is a randomly chosen integer in the range [1, ] to ensure that the trial vector will differ from its correby at least one parameter sponding target vector if else III PROPOSED MULTIGUIDERS NONDOMINATED RANKING DE ALGORITHM (4) The fitness value of each trial vector is compared to that of its corresponding target vector in the current population, and one with better fitness will be selected for next iteration The selection operation is expressed in (4) is -th mutant vector and are guiders and are randomly selected difference is number of population are vector random numbers in [0, 1]; however, is the uniform distribution and is Gaussian distribution , is an extreme solution when the Step 3: Crossover operation • Generate trial vector using binominal crossover expressed in (3) Step 4: Selection operator • Combine target population and trial population into population • Update the number of feasible solutions — If : And apply nondominated ranking number method for all feasible solutions to select of individuals for the next iteration where BAATAR et al.: MULTIGUIDERS AND NONDOMINATE RANKING DEA FOR MOGO OF ELECTROMAGNETIC PROBLEMS 2107 Fig Selection of two guiders and population ranking Fig SMES Configuration and Design variables — If : the combination of feasible solutions and solutions with lowest sum of constraint violation solutions will be selected the next iteration — In case of unconstrained problem: From the combined population, individuals with lower rank will be will be survived for the next iteration — Note that Rank1 solutions in the Fig are the Pareto front solutions in the current iteration — If the number of solutions in the Rank1 is greater than , we apply improved pruning method to remove the solutions with small crowding distance [10] Step 5: Update nondominated solutions • External archive absorbs superior current nondominated solutions and eliminates inferior ones using constrained domination • If the number of solutions in A is bigger than predetermined value, reduce it by repeating following process: — calculate the crowding distance for all members in ; — remove the nondominated solutions which have small crowding distance by using pruning method [3] Step 6: Termination check • If termination condition (maximum number of iteration) is not satisfied, go to Step 2, otherwise terminate IV OPTIMIZATION RESULT In order to validate the proposed algorithm, we adopted three parameter version of multiobjective TEAM problem 22 reported in [11] The TEAM problem 22 is an optimal design of a superconducting magnetic energy storage device (SMES) (SMES) configuration shall be optimized with respect to the following objectives • The stored energy in the device should be 180 MJ • The magnetic field must not violate a certain physical condition which guarantees superconductivity • The stray field (measured at a distance of 10 meters from the device) should be as small as possible Multiobjective TEAM problem 22 is expressed in (6) Minimize subject to (6) Fig Critical curve of the superconductor The true critical curve is in continuous black The dotted line represents the linear approximation to the previous curve used as quench condition limit in this paper TABLE I VARIABLE RANGES AND VALUES USED where the reference stored energy is MJ and are current density and magnetic flux density in the -th coil TEAM problem 22 is composed of two coils with opposite current densities The first coil is charged to store the energy and the second should be designed to diminish the high magnetic stray caused by the first coil The classical configuration of the SMES device can be obtained from Fig The superconducting material should not violate the quench condition that links together the value of the current density and the maximum value of magnetic flux density, as shown in Fig The critical curve has been approximated by constraints in (6) The range of the three design variables which defines the size and position of the outer coil of the device and other fixed parameters are shown in Table I Optimal result of proposed algorithm is compared with those from MultiGuiders Cross-search MOPSO [1] and Gaussian MOPSO [12] The optimization parameters are set exactly same as in [1] for all algorithms, i.e., 30 individuals, maximum number of iteration 200, and external archive of 100 2108 IEEE TRANSACTIONS ON MAGNETICS, VOL 49, NO 5, MAY 2013 V CONCLUSION Fig Pareto-front obtained by MGC-MOPSO, G-MOPSO, and MG-NRDE TABLE II COMPARISONS OF EXTREME SOLUTIONS In this paper, the multiguiders nondominated ranking differential evolution algorithm (MG-NRDE) is developed for multiobjective optimization problems The proposed algorithm is compared with recent approaches of multiobjective optimizers in solving multiobjective version of TEAM problem 22 Our proposed approach was able to produce results that competitive with respect to other approaches such us MGC-MOPSO and G-MOPSO The comparison results show the advantageous behavior of the proposed algorithm in locating Pareto optimal solutions regarding the multiple objectives without missing the extreme solutions Furthermore, a better uniformly distribution of solutions was obtained by the proposed algorithm ACKNOWLEDGMENT This work was supported by the Basic Science Research Program through the NRF of Korea funded by the Ministry of Education, Science, and Technology (2011-0013845) TABLE III COMPARISONS OF SPACING METRIC USING CROWDING DISTANCE Fig shows the obtained Pareto front solutions of G-MOPSO, MGC-MOPSO and MG-NRDE respectively In order to show the difference more clearly, every four others of Pareto front solution is shown in the figures The solutions obtained by MG-NRDE are more uniform distribution than those of MGC-MOPSO and G-MOPSO Furthermore, it is also proven in comparison of spacing metric by using crowding distances of Pareto front solutions Graphically we can see the proposed MG-NRDE obtained better uniformly distributed Pareto front solutions while G-MOPSO and MGC-MOPSO obtained Pareto fronts with some crowding and discontinuity But we can see the lower part of Pareto front obtained by proposed method is not showing the good results than other methods except the extreme solution However, the proposed MG-NRDE algorithm shows better result in the middle and upper part of the Pareto front Numerically, we compared the extreme solutions in Table II Comparison of space metric using crowding distance is shown in the Table III It is revealed that Pareto front obtained by MG-NRDE has better distribution than these who obtained by G-MOPSO and MGC-MOPSO Additionally, the proposed MG-NRDE has smaller standard deviation in the Pareto front It can be said that the proposed algorithm can find better solutions regarding the multiple objectives REFERENCES [1] M T Pham, D Zhang, and C S Koh, “Multi-guider and cross-searching approach in multiobjective particle swarm optimization for electromagnetic problems,” IEEE Trans Magn., vol 48, no 2, pp 539–542, Feb 2012 [2] C A C Coello, D A Veldhuizen, and G B Lamont, Evolutionary Algorithms for Solving Multi-Objective Problems Norwell, MA, USA: Kluwer, 2002 [3] R Storn and K Price, “Differential evolution—A simple and efficient adaptative scheme for global optimization over continuous spaces,” Int Comput Sci., Berkeley, CA, USA, Tech Rep TR-95-012, 1995 [4] H A Abbass, R Sarker, and C Newton, “PDE: A Pareto-frontier differential evolution approach for multiobjective optimization problems,” in Proc Congr Evol Comput., 2001, vol 2, 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Congr Evol., Comp., 2006, pp 1179–1186 [11] P Alotto et al., “SMES optimization benchmark extended—Introducing uncertainties and pareto optimal solutions into TEAM22,” IEEE Trans Magn., vol 44, no 6, pp 106–109, Jun 2008 [12] L S Coelho, H V Ayala, and P Alotto, “A multiobjective Gaussian particle swarm approach applied to electromagnetic optimization,” IEEE Trans Magn., vol 46, no 8, pp 3289–3292, Aug 2010 ... select of individuals for the next iteration where BAATAR et al.: MULTIGUIDERS AND NONDOMINATE RANKING DEA FOR MOGO OF ELECTROMAGNETIC PROBLEMS 2107 Fig Selection of two guiders and population ranking. .. the multiguiders nondominated ranking differential evolution algorithm (MG-NRDE) is developed for multiobjective optimization problems The proposed algorithm is compared with recent approaches of. .. Selection operator • Combine target population and trial population into population • Update the number of feasible solutions — If : And apply nondominated ranking number method for all feasible