00717696 multiobjective optimal design of three-phase induction motor using improved evolution...

8/13/2019 00717696 Multiobjective Optimal Design of Three-phase Induction Motor Using Improved Evolution Strategy

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2980 IEEE TRANSACTIONS O N MAGNETICS VOL. 34, NO. 5, SEPTEMBER 1998

ul t io~ject iveOptimal Design of Three-phase Induction MotorUsing Im proved Evolution StrategyMin-Kyu Kim, Cheol-Gyun Lee, Hyun-Kyo Jung

F.A. Research Institute, Samsung Electronics Co., Ltd., Kyunggi-Do 442-742, KOREAResearch Institute Hyundai Heavy Industries Co .,Ltd.,Kyunggi-Do 449-910, KOREASchool of Electrical Engineering, Seoul National University, Seoul 151-742, KOREAAbstract This paper presents the multiobjective optimaldesign of induction motor for Electric Vehicle(EV) using amodified evolution strategy(ES). The O ptimization process is

performed by using (1+1) evolution strategy(ES). Theconventional ES algorithm falls into a trap of local minima withhigh probability in solving the optimization problem that hasmany design variables. To overcom e this problem, ES algorithms modified by introducing shaking technique. To verify itsvalidity, the proposed m ethod is applied to an induction motordesign.

Index terms - Optimization method s, evolution strategy,induction motorsI. INTRODUCTION

Multiobjective optimization deals with two or moreobjective functions. In the multiobjective optimizationproblem, the solution which maximizes one objective maynot usually maximize any other objectives. New conceptcalled noninferiority can serve a optimality for themultiobjective problems. A feasible solution to multiobjectiveproblem is noninferior if there exists no other feasiblesolution that yields an improvement in one objective withoutcausing a degradation in at least one of other objectives. Thenoninferior set includes many alternatives, which cannot beselected obviously in general. The noninferior solutionselected as a preferred alternative is called the bestcompromise solution. Several techniques are suggested tosolve the multiobjective problems: the weighting method, theconstraint method and the multiobjective simplexmethod[ 1],[2]. A mong the multiobjective optimizationmethods, the weighting method is used in this paper. Themethod is to combine all the objective functions into oneweighted objective function by weighting.In the area of design of motor for electric vehicles(EV),important because of the limitation of electric power frombatteries. So the aim of the design of induction motors for EVis to achieve lighter motor and higher efficiency, whichbecomes a multiobjective design problem.Optimization process for searching optimal point isperformed by using the (1+1) evolution strategy(ES). ES is anon-deterministic method that can find a global minimum.

the weight a d he ef f ic iency of the tr ction otor is very

Manuscript received November 3, 1997.Min-Kyu Kim, 82-33 1-200-2427, fax 82-331-200-2434,mkkimasr t f sec.samsung co.kr

But in case of the optimization problems that have manydesign variables, such as the design of induction motor, theconventional ES algorithm falls into a trap of local minimawith high probability. To overcome this problem, the shakingtechnique is introduced.In the induction motor design procedure, the maindimensions are selected as design variables and the otherdimensions that have little effect on the objective function asconstants. T-equivalent circuit is used for the performanceanalysis of the designed motor. The equivalent circuitanalysis using lumped parameter is sometimes imprecise,especially in the rotor part. To reduce the error of calculationof rotor leakage reactance, the parameters of rotor arecorrected utilizing the results of finite elem ent analysis.

11. MULTIOBJECTIVEPTIMIZATIONTo solve the multiobjective problems the weightingmethod is used. In case of a two-objective problem, both

objectives, Z, an dZ ,, can be weighted using weightingvalues, w1 and w 2 , espectively so that

maximize z = [ Z , ,Z , ] (1)(2)

becomesmaximize Z ( w,w 2 = w, Z, w 2 Z ,

And an objective function can be divided by a positivenumber without altering the solution. After divide (2) by w,w 2 w, an be redefined as w. hen, (2) can be written asfollows :

maximize Z ( w )= Z , w . Z ,where w [ 0, 00 ] (3)

Because it is difficult to realize according to the w n totalrange, objective function is reformed for covering the totalrange. The final objective function is represented by (4).

Z ( w ) = w.Z, X>+ l-w).Z, X) (4)where w = [ 0, 1 ]X : a set of design variables and a convex setEquation (4) is rewritten as (5) in case of a fixed w.

0018-9464/98 10.00 998 IEEE


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.*0 . 8

Lq.* 0.60 . 40 . 2

e,

Z(W) U [ Z , z 5 ) = X p if F, 5 Fpwhere Nup : number o f updating of objective function

a = .85 if Nup Cul obj x 0.2The algorithm fo r finding th e best Compromise solution are step 4 ( Annealing ) changin g the mutation step length.as follows :

- - - *-.'e

_ _ _ _ _~ Improved ES

Convent ional ES

step : Set i=O, j=O and W O . Solve (4) to find X o .step 2 : Increase w by specified value : w' = w J 6 .

If .-I, then go to step 4.step 3 :Compare U [ Z l i , Z,']with U [ Z I J + , 2Jc1]

j = i+ l, increase i by 1 and go to step 2.increase by 1 and go to step 4.

step : I f j = 0, stop. :X o s the best compromise solution.If j 2 1 the best com prom ise solution is in betweenstep 5 : If 6IS,,, , lop. : Else reduce 6 and repeat from

If U[Z, ' ,Z , ' ] U[z1J+1,z2J+1],If U[Z,',Z,'] >U[z1 +1,z2 +1],

wJ-l nd wJ+'

step 1 between wJ-l nd w'+*The solution that maximizes U over the interval is the bestcompromise solution.

111. MODIFIEDVOLUTION STRATEGY ALGORITHMWhen we search the noninferior set with variable w forbest-compromise solution, the optimization algorithm to findthe global maximum value is needed. The (1+1) evolutionstrategy that could find the global maximum is used as theoptimization algorithm. Evolution strategy combines twooptimization algorithms, i.e., simulated annealing and genetic

algorithm, which shows the characteristics of the fastconver gence as a non-deterministic method[3][4].In the case of the optiinization problem such as the optimaldesign of induction motor which has many design variables,the conventional ES algorithm falls into the trap of localminima with high probability. To resolve this problem, ESalgorithm is improved by introducing shaking technique asfollows :step ( Initialization ) calculating the initial individual asstep 2 ( Reproduction ) producin g offsprin gs by reproductionparent of first generation.

X, = X,, t a, * R I 6 )where a, : mutation step length

RI : nominal probability function over [-1,1]X : parent generation vector1 ffgpring generation vectorF , F, : object function of X and X,

principle of fitness.X,= X, f F, > F p increase Nup

P

P Pstep 3 ( Selection ) selecting superior individual by the

a = a .0.85 if Nup Cal obj x 0.2where Cul obj : otal calculation number of objective func.step 5( Shaking ) calculating the objective function of randomindividuals in whole evolution window. According tothe convergence, frequency of shaking is varied.When random individual is superior to parent, replacethe old parent by individual as parent of nextgeneration. The number of Shaking is defined byfollowing equation.

7 )NConvergenceShaking No.=i

i = l NNConvergence= C

where N : numb er of design variablesstep 6 ( Termination ) If the termination criterion issatisfied, stop this proced ures. Otherwise, go to step 2 .

The convergence characteristics of the conventional ESand the improved ES in case of five design variables arecompared in Fig. 1. Sinc function is chosen as test function.In the improved ES, the convergence speed is improved by25 . According to the variation o f the number of designvariables, the probability of convergence to global maximumis compared in Table I. When the number of design variable

0 50 100 150 200 250 300 350 400No. of Generation

Fig. 1. Comparison of convergence

TABLE ICOMPARISON OF CONVERGENCE PROBABILITYN Shaking ES Conventional ES3 100 984 1003 IO06 1007 918 73

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becomes more than five, the proposed ES algorithm shows ahigher convergence probability than conventional one. Fromthe above results, we know that the improved ES algorithm isfast and robust. Therefore for the optimization problem suchas the design of induction motor which has many designvariables, the proposed method can be very useful.IV. INUDCTIONOTORDESIGN ROCEDURE

The design procedure is based on the set of the designvariables and the motor specifications. At first, a set forfinding a feasible machine must be selected. The objectivefunction becomes very sensitive to the selection of designvariables.A . Synthesis

Synthesis includes the selection of optimization variablesand feasible design. If the search is made without anyreference to optimization criteria, it is called a synthesis[3].The synthesis is a procedure for producing the feasiblemachine, on the basis of a set of values for the designvariables, other design data and the moto r specification.The design variables for induction motor are chosen asconsisting of four flux densities at teeth and yokes for thestator and rotor, one current density in stator winding andthree geometric variables. Three geometric variables are thedepth of stator slot, the ratio of rotor slot bottom width torotor teeth width and the ratio of rotor slot top radius to rotorslot bottom radius. Therefore the number of design variablesis eight. The number of pole P, supply voltage U l , frequencyf a r e given. In Fig. 2 , a cross section of induction motor isrepresented.From the rotor design variables, the stack length L, and thenominal slip s [ 5 ] are given by

The st tor is dosigned from dcsign vmiablcs and rotor data.

Fig. 2. Cross section of induction motor

AdjustN\ ~Tp2r--lectric Equivalent Circuit

NO

Fig. 3 . Flow chart of the synthesisAir gap length and stator slot opening are dictated bymechanical considerations. The important quantities to beconsidered are stator flux densities. The height of the statoryoke and the width of a stator teeth are determined from thestator yoke flux density and stator teeth flux density.The flow chart of the synthesis routine is presented in Fig.3. The feedback loop in Fig. 3is to meet output power Pofrom the change of the number of series turns per phase N,.After that, the ratio of starting torque to nominal torque ischecked. T-equivalent circuit is used for performancecalculation of the designed motor[6].B Parameter Correction

Because the equivalent circuit method uses lumpedparameters, it takes short computational time but it cannot getlocally distributed characteristics. Whereas the finite elementmethod(FEM) can deal with the distributed characteristics ofinduction motors but it requires much computational time andlarge memory capacity. So in this paper the merits of bothmethods are combined and utilized[7l.[SlThe entire characteristics of induction motor are calculatedby equivalent circuit method. Then the equivalent circuitparameters of rotor part are recalculated by finite elementanalysis using the rotor current from equivalent circuitmethod. From the result of FEM calculation, the equivalentcircuit components for rotor part are changed. Therefore itenables improved estimates of the equivalent circuitcomponents to be obtained, taking magnetic saturation anddeep bar effect into account[8]. FEM calculation is performedonly in the rotor part for a fast execution time, because therotor part has stronger effect than stator on the characteristics


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Eauivalent Circuit Parameters

rEM Calculation of Rotor PartIew Parameters of Rotor PartFig. 4. Flow chart of parameter correction.of the induction motor. The rotor model for FEM consists ofa sector corresponding to a single rotor slot pitch applyingperiodic boundary condition. The flowchart of parametercorrection is represented in Fig. 4 .

TABLE ICOMPARISON OF INITIAL VALUES TO OPTIMAL VALUESDesign Variable Initial Value Optimal ValueWsbr Wtr 1 o 0.602Stator currentdensity[A/mm2] 6.0 2.948Stator slot depth [mm] 50 26.998BYS [TI 1.7 1.342Bts Btr 0.8 0.988Btr [TI 1.7 15 27Byr [TI 1.7 1.056Rotor slot depth ratio 1.7 2.196Weight (Power Density) [Kg] 69.8(0.215) 47.4(0.316)Efficiency [ I 84.0 90.5

From the result of the algorithm explained at section 11, thebest compromise solution is determined when the w is 0.124.In that case, the efficiency is 90.5[ ] an d power density is0.316[kW/kg] which means the weight of motor is 47.4[kg].Table I1shows the comp arisons between the initial model andoptimized model. It is noted that the variables related to theweight are decreasing for low weight and those related to theefficiency such as flux density are decreasing for highefficiency.

V . SAMPLEESIGN VI. CONCLUSIONThe 1SKw, 4-pole, three-phase squirrel-cage inductionmotor for EV is designed as a sample design. The ratedfrequency is 100[H z] and voltage is 17O[V]. Also, the ratio o fmaximum torque to nominal torque is set 2.5 as a constraint.Low er limit of efficiency is 90[0/,] and that of power d ensityis 0.3[kW/kg]. The efficiency has to be maximized and theweight be minimized in the case of the design of traction

motor for EV. To solve. the problem in terms of maximum,the weight of motor is transformed into the power density.Therefore Z I is efficiency and Z2 is power density. Becausethe number of design vxiables is eight, ES proposed in thispaper should be used for optimization algorithm.Fig. 5 shows the non-inferior solutions as the weightingwvaries. Generally the non-inferior solutions have the relationof tradeoff each other. Ihe efficienc y changes from 54.02[ ]to 93.43[ ] and the power density does from 0.068[kW/kg]to 1.24[kW/kg].0 9 5 ~~_1

0 6 50 6 0 L

0 0 0 0 2 0 0 4 0 0 6 0 0 8 0 1 0 0 1 2 0Power densi ty [kWkg]

Fig 5 Noninferior solutions

For the design of high efficient and light-weight inductionmotor, the multiobjective optimal design method of inductionmotor employing the weighting method and the bestcompromise solution search technique is suggested. To getaround the trap of local minima, ES algorithm is modified byintroducing shaking technique. And the parameter correctionusing finite element analysis is carried out to obtain the moreaccurate motor parameters[S]. Through the results fromsample motor design, it is found that the proposed method isuseful for multiobjective optimal design of an inductionmotor.

REFERENCES[ Jared L. Cohon, Multiobjective programming andplanning,ACADEMIC Press, New York, U.S.A, 1978.[ 2 ] Andrezj Osyczka, Multicriterion optimization in engineering withFORTRANprograms, ELLIS HORWO OD LIMITED, England, 1984.[3] Hans-Paul Schwefel, Optimization, John Wiley Sons, Great Britain,1981.[4] D. E. Goldberg, Genetic algorithms and rule learning in dynamicsystem control, Proc. Intl. Con on Genetic Algorithm and theirApplications, pp. 8-15, July 1985.[5] M . Nurdin, M. Poloujadoff and Faure, Synthesis of squirrel cagemotors : A key to optimization, IEEE Trans.on Energy Conversion,Vol. 6, NO. 2, 1991, pp. 327-335.[ 6 ] C. G. Veinott, Theory and design of small induction motor,Mcoraw-Hill BOOK company, New Tow U.S.A, 19s9[7] S. Williamson, and M. C. Begg, Analysis of cage induction motors - Acombined fields and circuit approach, IEEE Trans. onMagnetics,V01.21, NO. 6, 1985, pp. 2396-2399.[8] S. Williamson, and M. J. Robinson, Calculation of cage inductionmotor equivalent circuit parameters using finite elements, IEEProceedings-B, Vol. 138, No. 5, pp. 264-276, September 1991.

00717696 multiobjective optimal design of three-phase induction motor using improved evolution...

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