104 Journal of Power Electronics, Vol. 12, No. 1, January 2012JPE 12-1-14http://dx.doi.org/10.6113/JPE.2012.12.1.104Optimal Controller Design for Single-Phase PFCRectiers Using SPEA Multi-ObjectiveOptimizationAhmadreza Amirahmadi, Ali Dastfan, and Mohammadreza RaeiDept. of Electrical and Robotic Eng., Shahrood University of Technology, Shahrood, IranDept. of Electrical Eng., Islamic Azad University, Garmsar branch, Garmsar, IranAbstractInthispaperanewmethodforthedesignofasimplePIcontrollerispresentedandithasbeenappliedinthecontrolofa Boost based PFC rectier. The Strength Pareto evolutionary algorithm, which is based on the Pareto Optimality concept, usedinGametheoryliteratureisimplementedasamulti-objectiveoptimizationapproachtogainagoodtransient responseandahigh quality input current. In the proposed method, the input current harmonics and the dynamic response have been assumed asobjective functions, while thePI controllers gains of the PFCrectier (Kpi, Tpi) are design variables.The proposed algorithmgeneratesaset ofoptimal gainscalledaParetoSet correspondingtoaParetoFront, whichisaset ofoptimal resultsfortheobjective functions. All of the Pareto Front points are optimum, but according to the design priority objective function, each onecan be selected. Simulation and experimental results are presented to prove the superiority of the proposed design methodologyover other methods.Key Words: Boost converter, Dynamicresponse, Gametheory, Power-factor-correctionrectier, StrengthParetoEvolutionaryAlgorithm, Total Harmonic DistortionI. INTRODUCTIONPowerFactorCorrection(PFC)RectiersbasedonBoostconverters are one of the most popular topologies provid-inglowinput lineharmonics inaccordancewithharmonicdistortion standards. Control of this type of converter hasreceived considerable attention in the past two decades. Fromthe control point of view, the operation of a PFC rectier canberegardedasatrackingproblem, sincetheoutput voltageshouldfollowthereferencecommandwithagoodtransientbehavior and a low steady state error. Furthermore, the inputcurrent harmonics must remainlow. Normally, alowinputcurrent distortion and a high input power factor are achievedbyemployingahigh-bandwidthcurrent control loopandalow bandwidth voltage control loop [1],[2]. The voltage loopis designed for lowbandwidth to avoid the input currentdistortioncausedbythe output voltage ripple [3]. Sucharectier system exhibits poor dynamic response with respect toinput voltage and load disturbances [4]. In recent years, severaltechniques have been proposed to overcome this problemlike the ripple compensation approach [5]-[12], which can beconsideredastwomethods,compensationbyuseofarippleManuscript received Sep. 30, 2010; revised Nov. 4, 2011Recommended for publication by Associate Editor Byung-Cho Choi.Corresponding Author: amirahmadi@knights.ucf.eduTel: +1-407-683-6503, Shahrood University of TechnologyDept. of Electrical and Robotic Eng., Shahrood University of Technology,Iranestimator [5]-[7] or byuseof alter. Theimplementationofnotchlters[8]-[10], deadzoneADCcontrollers[11]orcomblters[12]havebeenallowedwiththehelpofdigitalcontrollers. The objective of these techniques is to increase thevoltageloopcrossover frequencybyeliminatingthesecondand higher harmonics ripples fromthe control signal. Inanother technique separate bandwidths for the steadystateandtransients has beenconsidered[13]. Thebandwidthofthevoltageloopiskept lowat thesteadystatetoobtainasinusoidal input current. Duringtransients, it isincreasedtohaveagooddynamicresponse. Theoutput voltageerror isusedtodeterminewhethertherectierisinthesteadystateor the transient condition. Although these methods improve thedynamic response of converters without an increase in the linecurrent distortion, they signicantly increase the complexity ofthe control circuit. Also, in digital implementation, the systemrequireslargememorystorage, andsincethedesignof thenew block is based on the converter model, these methods aresensitive to parameter variations.Some control methods like feedforward control of the inputvoltage, load current, duty-ratio and reference current canimprove the output dynamic response, but these methodsrequire more sensors or quantity estimators [3],[4],[14]-[17].Indirect current control of aPFCrectier does not needto measure the input voltage and load current, so this isusually the basis for new control methods for PFC rectiers.Optimal Controller Design for Single-Phase PFC Rectiers Using SPEA Multi-Objective Optimization 105In this method and in the resistance emulation method aproportional-integral (PI)-typevoltagecontrollerisusedandall the modied methods take into consideration this part of thecontroller [4],[18]-[20]. In the indirect current control scheme,liketheothercontrol methodsforaPFCrectier, theinputcurrent quality and the dynamic response of the PFC rectierareconictingobjectives. Asaresult, whenoneobjectiveisimporoved, the other is degraded. In this case of simultaneousoptimization, there is no single optimal solution, but a set ofoptimal solutions. Thesesolutionsareoptimal inthewidersense that no other solutions in the search space are superiorto them when all objectives are considered.Inthis paper, thedesigningof asimplePI controller isproposed which is able to gain a good transient responseaswell asahighqualityinput current viaamulti-objectiveoptimization approach.II. MULTI-OBJECTIVE OPTIMIZATIONThe Game theory concept is applicable to a multi-objectiveoptimizationprobleminitsownoriginal statuswithout theneed for modifying or combining the objectives, unlike othermethodologieswhichcombinethedesiredgoalsoftheopti-mizationproblem,constructascalarfunctionandthenuseacommon scalar optimization approach to resolve the problem[21]. The major problem with these methodologies, which arecalledplainaggregatingapproaches, istheunavailabilityofanystraightforwardmethodfor combiningtheobjectivesorgoalsoftheproblemwhentheyarenot constant quantities.For experimental problems, the Game theory concept requiresan evolutionary algorithm to solve MOPs because they processaset ofsolutionsinparallel. Oneofthebest EAstoreachglobally optimum results is SPEA. This evolutionary algorithmcoversall ofthesolutionswhichmethodslikeHajelasandLins Genetic Algorithm (HLGA) [22], Niched Pareto GeneticAlgorithm (NPGA) [23], Vector Evaluated Genetic Algorithm(VEGA) [24], and Non-dominated Sorting Genetic Algorithm(NSGA) [25] offer and combines several of their features in aunique manner.A simple mathematical denition of a multi-objective opti-mization problem can be considered as (1).Minimize y =f (x) = ( f1(x), f2(x), ..., fk(x))Subject to x = (x1, x2, ......xn) X& y = (y1, y2, ....yk) Y(1)where x, X, y, and Y are the decision vector, parameter space,objective vector, and objective space, respectively.Objective vectors that cannot be improved in one dimensionwithout degradationinanother are found, andtheir corre-spondingdecisionvectors are calledsolutions for a multi-objective optimization problem. To describe them mathemati-cally, can be said that vector a dominate vector bif:i = {1, 2, ..., k} : fi(a) fi(b)j {1, 2, ..., k} : fj(a) > fj(b)(2)All decision vectors which are not dominated by any of theother decision vectors of a given set are called non-dominated.Everynon-dominatedsolutionisregardedasoptimal inthesense of the Pareto Optimality concept or is called Pareto Op-timal. Obviously, any Pareto Optimal solution is comparativelythe most optimal one in terms of at least one of the objectivefunctions.The set of all non-dominated solutions is called the ParetoOptimal Set andtheset of thecorrespondingvaluesof theobjective functions is called the Pareto Optimal Front [26].A. Strength Pareto Evolutionary Algorithm (SPEA)The SPEA because of its high diversity and fast convergenceis one of the most popular evolutionaryalgorithms amongsimilar approaches [26].The ow of the algorithm can be summarized as followingeight stages.1Randomize a prime population P within the allowedboundaries andconsider anexternal non-dominatedsetPND.2Copy the non-dominated members of P into PND.3Delete the members of PNDwhich are dominated.4If the number of PND members exceeds a given maximumN, prune PNDby means of clustering.5Compute the tness of all the individuals in P as well asin PND.6Choose the individuals from P+PND, until the mating poolis lled. Binary tournament selection with replacement canbe used.7Apply crossover and mutation operators as usual.8Ifthemaximumnumberofgenerationsisdone, endthealgorithm, otherwise return to step 2The Fitness assignment technique is done in two steps [26].1Every individual (i) of PND is considered a strength 0Si