Author name used in this publication: S. L. HoAuthor name used in this publication: G. Z. Ni2006-2007 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Ho, SL

Ni, G

Wong, KF

Yang, S

IEEE Transactions on Magnetics

Author name used in this publication: S. L. HoAuthor name used in this publication: S. Y. YangAuthor name used in this publication: G. Z. NiRefereed conference paper2005-2006 > Academic research: refereed > Refereed conference paperVersion of RecordPublishedVoR allowe

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An efficient multiobjective optimizer based on genetic algorithm and approximation techniques for electromagnetic design

Author name used in this publication: S. L. HoAuthor name used in this publication: G. Z. Ni2006-2007 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishedVoR allowe

IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007 1605An Efficient Multiobjective Optimizer Basedon Genetic Algorithm and ApproximationTechniques for Electromagnetic DesignS. L. Ho1, S. Y. Yang2, G. Z. Ni2, and K. F. Wong1Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong KongCollege of Electrical Engineering, Zhejiang University, Hangzhou 310027, ChinaTo provide an efficient multiobjective optimizer, an approximation technique based on the moving least squares approximation isintegrated into an improved genetic algorithm. In order to use fully, both the a posteriori information gathered from the latest searchednondominated solutions and the a priori knowledge about the search space and individuals, in guiding the search towards more andbetter Pareto solutions, a gradient direction based perturbation search strategy and a preference function based fitness penalizationscheme are proposed. Numerical results are reported to validate the proposed work.Index Terms—Approximation technique, evolutionary computation, genetic algorithm (GA), multiobjective optimization.I. INTRODUCTIONTO MEET THE ever increasing demands in the designautomations of electromagnetic devices, a considerableamount of efforts are dedicated to the development of multi-objective or vector optimizers. A wealth of vector algorithms,such as genetic algorithm (GA) or evolutionary algorithm (EA)[1], [2], simulated annealing algorithm (SA) [3], tabu searchmethod [4] and particle swarm optimization (PSO) method [5],to name but a few, are proposed and used successfully to solvetypical electromagnetic design problems. However, a multiob-jective problem is characterized by the need to optimize severalincommensurable and conflicting objectives simultaneously.In general, the solution of a multiobjective optimal problem isnot a single point but is a set of optimal solutions called Paretooptimal or nondominated solutions.For a multiobjective solver, the following two issues mustbe addressed carefully: 1) means to accomplish the fitness as-signment and selection in order to guide the search towards thePareto optimal front; 2) means to maintain the diversity in thesearched Pareto optimal front. To meet these two goals, mostof the reported efforts are focusing on techniques to extend theavailable scalar optimal methods to obtain some nondominatedpoints with the prescribed diversities in both parameter and ob-jective function spaces.In spite of significant progresses being realized in the devel-opment of multiobjective optimal algorithms, the robustness andefficiency of available vector optimal methods are still unsatis-factory, hence there are still many problems yet to be addressed[6]. Indeed, very few studies have been devoted, hitherto, to thedevelopment of approximating techniques of nondominated setsDigital Object Identifier 10.1109/TMAG.2006.892113in continuous multiobjective optimization problems [7]. How-ever, a proper approximation of the non-dominated set couldprovide a wealth of useful information to guide the search to-wards the finding of more and better Pareto solutions with en-hanced convergence performances. In this regard, an approxi-mation technique is proposed and then integrated into an im-proved vector genetic algorithm (IVGA) in the design of an ef-ficient vector optimizer.II. AN EFFICIENT MULTIOBJECTIVE OPTIMIZERA. An Improved Vector Genetic AlgorithmBoth well established and newly proposed techniques are in-tegrated in the proposed IVGA to help finding the Pareto so-lutions efficiently and to distribute them uniformly along thePareto optimal front.1) Reporting of Nondominated Solutions: As similar to theglobal repository of [8], a Pareto set, , is introduced toreport the so far searched non-dominated solutions in the pro-posed algorithm. Also, the solutions of will be used inthe assignment of the fitness value to an individual in the itera-tion process.2) Assigning Fitness Value for New Individuals: To assessthe quality of an individual in a Pareto optimal sense, somescalar techniques must firstly be designed for multiobjectivefunction optimizations. To favor the selection of individualsnear the Pareto optimal front, and also to distribute themuniformly along the tradeoff surface, other things being equal,the fitness assignment mechanism as proposed by Zitzler andThiele [9] is extended and used in the proposed algorithm.The fitness value of an individual in the current population isassigned in accordance to a two-stage process as follows.Step 1) Determine the strength of the solutions of0018-9464/$25.00 © 2007 IEEEAuthorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on March 14, 2009 at 03:58 from IEEE Xplore.  Restrictions apply.1606 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007For each solution in , its strength is pro-portional to the number of individuals in the currentpopulation which are dominated by it, i.e.,(1)where is the number of individuals in the currentpopulation which are dominated by the solution ;is the size of the population. The fitness value ofthe solution is the inverse of its strength.Step 2) Fitness assignment of individuals in populationThe fitness value of an individual in the popula-tion is computed by summing the strengths of allPareto set members which dominate individual .Mathematically(2)where , , means that solutiondominates solution .3) A Priori Penalization of the Fitness Value: In most of theavailable multiobjective optimizers, only the information gath-ered on the so far searched solutions, the a posteriori informa-tion, is used to guide the search towards more promising solu-tions. However, if some a priori knowledge about the searchspace and individuals is available and could be used, the searchefficiency and quality of the solution of the optimizer couldbe improved. Hence a preference function, that includes the apriori information about an individual as proposed in [10], isused to penalize the fitness value to guide the search to findbetter Pareto solutions. For the paper to be self-contained, theconcept of the preference function will be described firstly.For illustrative simplicity, one considers the minimization ofan optimal problem involving two decision parametersand two objective functions. Mathemat-ically, if the contour lines of and in the parameter space ex-hibiting different convexities are intersecting at a feasible pointas shown in Fig. 1(a), there will be at least one point in thefeasible space which is better than point for both objectivefunctions. By repeatedly and symmetrically reducing the valuesof both functions and , it is possible to locate a Pareto op-timal point at the tangential point between the contour lines ofthe two objectives, as shown in Fig. 1(b). In order to charac-terize the feature of an intersection point of the contour lines ofthe two objectives in the parameter space, one defines the pref-erence function as(3)where is the Euclidean norm.The preference function reaches its minimum functionvalue of zero at all Pareto optimal points. In other words, thepreference function provides a “gauge” to measure the “close-ness” of a feasible point to the nondominated solutions of theoptimal problems. Therefore, to intensify searches around theFig. 1. Illustration of intersection points for a design problem comprising oftwo decision parameters and two objective functions.points which are “close” to a Pareto solution, the fitness valueof individuals in the current population is penalized by using(4)where is the th individual; is a constant pre-defined by the user.When the number of objective functions is more than 2, thepreference functions are defined in a “pairwise” sense for everytwo objectives, and the power of the penalized function of (4),i.e., , is the weighted sum of the values of all possiblepreference functions pairwisely defined. From (4), it is clearthat the smaller the preference function, the smaller is the fit-ness being devalued. Consequently, the proposed penalizationscheme of the fitness value of an individual will force the searchtowards more and better Pareto solutions.4) Fitness Sharing: To produce a uniform distribution of thesearched Pareto solutions, not only in the objective but also inthe parameter spaces, the fitness sharing concept is introduced.In order to reduce the implementation complexity, a simple fit-ness sharing function is proposed. Mathematically, the fitnesssharing function is defined as(5)where is the point density around the spec-ified point in the -space; is the number of the totalsolutions which are available in the current generation; andare, respectively, the objective and parameter spaces.To compute the point density of a specified point, a hyper-sphere with the specific point as the centre is constructed andthe number of the solutions which fall inside this sphere is usedas a measure of its fitness sharing function. The fitness value ofan individual, , is the weighted sum of its fitness and fitnesssharing function values, i.e.,(6)where and are two weighting constants.5) Selection of New Generation: At the end of every gen-eration, a new population is selected from all solutions whichare available in the current generation using a Roulette wheelAuthorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on March 14, 2009 at 03:58 from IEEE Xplore.  Restrictions apply.HO et al.: EFFICIENT MULTIOBJECTIVE OPTIMIZER BASED ON GENETIC ALGORITHM AND APPROXIMATION TECHNIQUES 1607selection mechanism according to the fitness values of the so-lutions. The new populations are generally composed of threedifferent individuals, i.e., those generated by means of selec-tion, crossover, and mutation operations; those that come fromthe current population; and those that are stored in the externalPareto optimal set.B. Approximation and Utilization of Nondominated SetTo fully use the information gathered from the so far searcheddiscrete nondominated solutions, a continuous approximation ofthese discrete points at the end of each generation is constructedusing a moving least squares approximation based response sur-face model [11]. As better nondominated points may be found,in the gradient direction of this approximated presentation froma geometrical perspective, an intensifying searching phase is ac-tivated by perturbing a specific point selected from this approx-imation on its gradient direction, in order to obtain such solu-tions. Essentially, the proposed intensifying searching phase is:1) Select a point from the approximated presentation, using aRoulette wheel selection rule according to the point den-sities of the searched discrete nondominated solutions, inthe objective function space.2) Perturb the specific point on its gradient direction, and de-termine the objective function values of the newly per-turbed point using(7)where is the value of the th objective function at thespecific point; is the approximated continuous contour ofthe so far searched nondominated solutions in the objectivespace; is a small perturbation value of the th prede-fined by the user.3) Calculate the decision parameter values of the new pertur-bation point according to their objective function values.As i) the number of the objective functions is generallysmaller than that of the decision parameters, and ii) oneimage in the objective space may relate to multiple pointsin the decision parameter space, the values of the deci-sion parameters of the new perturbation point is determinedfrom(8)where and are the values of theth objective function obtained, respectively, in step (2)and from a response surface of the corresponding objec-tive function as proposed in this paper.4) Check if the newly calculated decision parameters are inthe feasible space. If the answer is “Yes,” go to next step;Otherwise, reduce the perturbation value and return tostep 2.5) Determine if the stop criterion is satisfied. If the answer is“Yes,” stop this search phase; Otherwise, go to step 1.Fig. 2. Schematic diagram of the multisectional pole shoes.The initial value of the perturbation value, , is automat-ically adjusted during the iterative process in such a way thatit jumps to the maximum value at the beginning of the searchand reaches the minimum value at the end of the search. Typ-ically, the maximum and minimum values of the perturbationparameter of are set, respectively, to 10% and 5% of the“averaged” value of the th objective function over some sam-pling points. Moreover, from the aforementioned description itis clear that: 1) since the selection probability for a point inthe approximated presentation of the so far searched nondom-inated solutions is inversely proportional to the point density,the sparser the points in the objective space are, the higher arethe probabilities for the points are to be selected. Therefore, thediversity of the nondominated solutions is guaranteed; 2) com-pared with a pure stochastic perturbation method, the proposedgradient based one can reach a promising point in the maximumdescending direction, resulting in an enhancement in search ef-ficiency. Consequently, a high performance multiobjective op-timizer could be expected.III. NUMERICAL EXAMPLETo critically compare the performances of the proposed al-gorithm with other vector optimal techniques, the geometricaldesign of the multisectional pole arcs of a large hydrogenerator[3] is used as a case study. Mathematically, this multiobjectivedesign problem is formulated as(9)where is the amplitude of the fundamental component ofthe flux density in the air gap, is the distortion factor of asinusoidal voltage of the machine at the no-load, telephone har-monic factor (THF) is the direct axis transient reactance ofthe short circuit ratio (SCR) generator.The decision parameters of this problem are the centerpositions and radii of the multisectional arcs of the pole shoes(Fig. 2). In the numerical implementation, is directlycomputed from the finite element solution of the no-load elec-tromagnetic field of the machine, and the other performanceparameters of (9) are derived based on these finite elementsimulations.Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on March 14, 2009 at 03:58 from IEEE Xplore.  Restrictions apply.1608 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007TABLE IALGORITHM PARAMETERS USED BY THE PROPOSEDAND A STANDARD GENETIC ALGORITHMFig. 3. Searched Pareto solutions by a SA based method.Fig. 4. Searched Pareto solutions by the proposed algorithm.Fig. 5. Searched Pareto solutions by a standard generic algorithm.This problem is solved by using, respectively, the proposedIVGA, a standard GA and a SA algorithm. The parameters usedby the proposed and the standard GA are given in Table I. Theparameters of the SA algorithm used in this paper are the sameas those of [3].In the numerical experiments, for the geometrical optimaldesign of a 300 MW hydro-generator, every algorithm is runindependently three times, and the “averaged” iterative numberof the three runs for the proposed, the SA, and the standardGA algorithms are, respectively, 1576, 1784, and 2045. Thesearched Pareto fronts using three different algorithms areshown in Figs. 3–5. By comparing these searched Pareto fronts,it is obvious that the standard GA algorithm cannot find someparts of the Pareto solutions, i.e., those in the back-right cornerand in the front-right sub-regions, which can however befound by either the proposed or the SA algorithms because theexploitation search ability of a standard GA is not as strong asits exploration searching ability. In summary, the performancesof the proposed algorithm are superior to its ancestor, the stan-dard GA algorithm, in terms of search efficiency and solutionquality; and it is comparable to the SA based algorithm, asdemonstrated by this primary numerical experiment as reportedin this paper.IV. CONCLUSIONAn efficient multiobjective optimizer is proposed and itsperformances are evaluated by numerical experiments on apractical inverse vector optimal problem. The primary numer-ical results have demonstrated the robustness and feasibility ofthe proposed algorithm in the study of engineering applications.Therefore, the proposed work offers an alternative algorithmfor multiobjective optimizations of electromagnetic devices.ACKNOWLEDGMENTThis work was supported by the Research Grant Council ofthe Hong Kong Special Administrative Region, China, underProject PolyU 5242/04E.REFERENCES[1] S. L. Avila, A. C. Lisboa, and L. Krahenbuhl et al., “Sensitivity anal-ysis applied to decision making in multiobjective evolutionary opti-mization,” IEEE Trans. Magn., vol. 42, pp. 1103–1106, 2006.[2] D. A. G. Vieira, R. L. S. Adriano, J. A. Vasconcelos, and L. Krahen-buhl, “Treating constraints as objectives in multiobjective optimizationproblems using niched Pareto genetic algorithm,” IEEE Trans. Magn.,vol. 40, pp. 1188–1191, 2004.[3] S. L. Ho, S. Y. Yang, H. C. Wong, and G. Z. Ni, “A simulated an-nealing algorithm for multiobjective optimizations of electromagneticdevices,” IEEE Trans. Magn., vol. 39, pp. 1285–1288, 2003.[4] S. L. Ho, S. Y. Yang, G. Z. Ni, and H. C. Wong, “A tabu method tofind the Pareto solutions for multiobjective optimal problems in elec-tromagnetics,” IEEE Trans. Magn., vol. 38, pp. 1013–1016, 2002.[5] U. Baumgartner, C. Magele, and W. Renhart, “Pareto optimalityand particle swarm optimization,” IEEE Trans. Magn., vol. 40, pp.1172–1175, 2004.[6] C. A. Coello Coello, “An updated survey of GA-based multiobjectiveoptimization techniques,” ACM Comput. Surveys, vol. 32, pp. 109–143,2000.[7] J. Martin, C. Bielza, and D. R. Insua, “Approximating nondominatedsets in continuous multiobjective optimization problems,” Naval Re-search Logistics, vol. 52, pp. 469–480, 2005.[8] C. A. Coello Coello and M. S. Lechuga, “MOPSO: a proposal for mul-tiple objective particle swarm optimization,” in Proc. Congr. Evolu-tionary Computation, 2002, pp. 1051–1056.[9] E. Zitzler and L. Thiele, “Multiobjective evolutionary algorithms: acomparative case study and the strength Pareto approach,” IEEE Trans.Evol. Comput., vol. 3, pp. 257–271, 1999.[10] P. Di Barba, “Multiobjective design optimization: a microeconomics-inspired strategy applied to electromagnetics,” Int. J. Appl. Electro-magn. Mechan., vol. 21, pp. 101–117, 2005.[11] S. L. Ho, S. Y. Yang, P. H. Ni, and H. C. Wong, “Development of anefficient global optimal design technique-a combined approach of MLSand SA algorithm,” COMPEL, vol. 21, pp. 604–614, 2002.Manuscript received April 24, 2006 (e-mail: eeslho@polyu.edu.hk).Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on March 14, 2009 at 03:58 from IEEE Xplore.  Restrictions apply.

An efficient multiobjective optimizer based on genetic algorithm and approximation techniques for electromagnetic design

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