00728298 an Evolutionary Algorithm for the Optimal Design of Induction Motors

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3882 IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998

An Evolutionary Algorithm for theOptimal Design of Induction Motors

Jan Pawel Wieczorek, Ozdemir Gol, and Zbigniew Michalewicz

AbstractThis paper describes the application of an evolution-ary algorithm to the design of induction motors. It is shown thatthe use of an evolutionary algorithm offers advantages over otherapproaches. These include a high rate of global convergence andthe ability to handle discrete variables.

Index Terms Evolutionary algorithms, induction motor de-sign, optimization.

I. INTRODUCTION

INDUCTION machines are used in large quantities in ap-

plications varying from household appliances to spacetechnology; hence their design assumes great importance.

Many conflicting criteria have to be reconciled during

design for an acceptable outcome according to a given spec-

ification. Device designers generally attempt to achieve this

by set design procedures, usually with the aid of lumped pa-

rameter circuit models. Commonly, a trial and error approach

is adopted in arriving at a solution which is deemed to best

satisfy such conflicting requirements. However, solutions thus

found are unlikely to be the best possible; especially in view of

commercial pressures which severely curtail the time available

for search.

Lumped parameter circuit models are popular, and intrin-

sically suitable, in the a priori evaluation during the designstage of the external behavior of induction machines. If used

in conjunction with computer-based optimization techniques,

they can vastly improve the design outcomes in that they

facilitate the finding of optimal solutions with a modest

investment of computational effort.

One traditional approach to design optimization of elec-

tromagnetic devices has been to use nonlinear programming

(NLP) techniques, which allow for fast convergence and are

well established in engineering applications. However, discrete

design parameters, such as the number of conductors and the

wire gauge, cannot be drawn into optimization; the designer

needs to perform a number of sub searches, each with a fixed

set of discrete design parameters. The search outcomes are

then compared and judgement is made as to the best possiblesolution. Evidently, it is desirable for this entire process to

Manuscript received September 30, 1997; revised August 10, 1998.J. P. Wieczorek and O. Gol are with the Department of Electrical Engi-

neering, University of South Australia, Mawson Lakes SA 5095, Australia(e-mail: [email protected]).

Z. Michalewicz is with the Department of Computer Science, University ofNorth Carolina, Charlotte, NC 28223 USA (e-mail: [email protected]).

Publisher Item Identifier S 0018-9464(98)08252-1.

take place automatically without intermittent input from the

designer. One way of doing this is to treat discrete variables

as floating values within the optimization subroutine. Once

the search terminates, these sub-optimal floats are rounded

off to the nearest integer (e.g., number of turns) or to the

nearest value determined by availability (e.g., conductor size)

before they are used within the design algorithm. The globality

of the search outcome needs to be checked by initiating the

search from various starting points. This is tantamount to

approximating an actual mixed integer programming (MIP)

problem with an NLP one, which occasionally may resultin pseudo-optimal solutions. Furthermore, the method still

necessitates that the designer exercise judgement in deciding

whether the search outcome represents a valid optimum.

A better way in dealing with the MIP nature of the electro-

magnetic device design would be to use a globally convergent

optimization solver, without having to resort to an NLP

approximation [1]. Discreteness of variables precludes the use

of optimization methods which rely on gradient evaluation;

optimization methods capable of effectively dealing with this

class of problems are still being developed. However, recent

developments in evolutionary optimization promise to lead to

long-sought-for global optimization tools which can handle

MIP problems [2].In recent years, evolutionary algorithms (EAs) have been

recognized as potent tools in optimization. They exhibit a

high rate of global convergence across the broad spectrum of

optimization problems [3]. Since the search progress is based

on function evaluations, no gradient evaluation is required.

These features make EAs eminently suitable for use in the

design optimization of electromagnetic devices; a dedicated

algorithm can be devised to provide a powerful solver. In

achieving this, the EA needs to be enmeshed with a model

which adequately represents the device for design purposes.

The model complexity must be counter-balanced with execu-

tion time, since the number of function evaluations may be

high depending on the number of populations and the size ofeach population.

This paper is organized as follows. The next section intro-

duces induction machine design as an optimization problem.

Section III presents the proposed algorithm together with

experimental results. Section IV concludes the paper.

II. INDUCTIONMOTOR DESIGN: AN OPTIMIZATION PROBLEM

The suitability of EAs in induction motor design optimiza-

tion will now be demonstrated. In doing so, the so-called

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WIECZOREK et al.: AN EVOLUTIONARY ALGORITHM 3883

Fig. 1. Exact equivalent circuit model of induction machine.

exact equivalent circuit model of the machine, depicted

in Fig. 1, will be used. This model is basically a per phase

representation of a balanced polyphase machine in the fre-

quency domain, comprising six elements, or model parameters.

It elegantly represents the electromagnetic interactions taking

place in an induction motor, albeit with substantial simplifica-

tions and idealizations. Overall, the model is popular and well

understood among engineers and, despite its shortcomings,offers a reasonably good prediction accuracy with modest

computational effort.

The six model parameters shown in Fig. 1 are predomi-

nantly dependent upon machine topology, although the ma-

terials used also have a significant effect, especially if the

machine is driven into saturation. For instance, the stator

leakage reactance (one of the six model parameters) is

obtained as a complex function of a number of quantities

(design parameters) in the form

(1)

where represent, respectively, the

number of turns, core length, stator bore diameter, stator slotheight, stator slot width, etc. [4][6]. For simplicity, not all

quantities which affect are shown in (1); an example is

that of skewing of slots. In order to optimize performance,

these design parameters need to be selected in such a way

as to produce a value for for the best possible overall

performance. Similar considerations apply to the remaining

model parameters which too are complex functions of many

factors related to machine topology and material properties.

A more detailed description of how the model parameters

were obtained in this case is deemed to be outside the scope

of this discussion due to both the complexity of the design

algorithm (which resulted in some 60 pages of MATLAB

code) and intellectual property considerations. The high levelprogramming language MATLAB has been adopted as the

programming platform on account of its numerous advantages

including rapid code development, ease of debugging, and

excellent graphical interface.

In the case presented here, 11 design parameters as specified

in Table I have been drawn into the optimization process

as variables with a view to obtaining values for the six

model parameters (as per Fig. 1) for optimal performance. The

selection of parameters was based on authors experience in

induction machine design. These 11 parameters are known

to have the most significant effect on the performance of an

TABLE IDESIGN PARAMETERS WITH THEIR DOMAINS

induction motor. The inclusion in the optimization process of

further design parameters results in a slight improvement in

the returned optimal value of the objective function. However,

the cost in terms of computational effort is disproportionately

high. Table I also shows the practicable domains for the design

parameters.

The first two of the above, i.e., the number of conductors

per stator slot and the stator wire gauge, are discrete variables;

the former is an integer, whereas the latter assumes a discretevalue from a list of available wire gauges. All of the above

parameters are subject to a set of design constraints. Such con-

straints include threshold values for power factor, efficiency,

starting current, torque, slip, cost of materials, and machine

size and weight.

III. AN EVOLUTIONARY ALGORITHM

An evolutionary algorithm has been developed to optimize

the design of an induction motor. The algorithm uses bi-

nary representation; the individuals are formed by appending

strings, delineating the eleven design parameters drawn into

the process. The algorithm is based on a roulette wheelselection, single-point crossover, bit mutation, and elitism. The

population size, bit string length and the number of generations

can be selected depending on the requirements and available

computer resources. Each parameter can be varied within its

domain, the boundaries of which are user-specified.To enable the EA to handle the constraints, a nonstationary

penalty function was used [7]. In this case, an evaluation

function devised in the form

if

if (2)


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3884 IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998

is used, where is an objective function (to be maximized)

and is the penalty function which is defined by

gen (3)

with

if constraints not violated

otherwise (4)

In the above equations, x represents the vector of design

parameters as per Table I. is a constant, chosen in a

way that amplifies the constraint violations so as to make

them significant enough for the EA to confine the search

to the feasible solution space; value of is related to the

value of objective function. gen is the current generation

number, represents a constraint violation within that

generation with being the total number of constraints;

is expressed as a per unit value to render constraints

comparable with one another. is the value resultingfrom the current evaluation whereas is the user-defined

constant. In general, constraints are defined as

for for Type 1 constraints

for for Type 2 constraints

(5)

In (5), and allude to the existence of two different

types of inequality constraint in an engineering problem of

this kind, namely:

Type 1: There are constraints which constitute thepermissible lower limits. For instance, high values for

power factor are desired for good performance in an

induction motor (the higher the better); hence the need for

defining an acceptable lower limit. If the user specified

constraint for the power factor were 0.85, then anything

less than that would be a violation. Thus, for a returned

value of 0.83, the numerator in (4) would be 0.02; a

positive value, indicating a violation.

Type 2: There are constraints which constitute the

permissible upper limits. For instance, stator current

density may not exceed certain values on account of

permissible machine operating temperatures; hence the

need for defining an acceptable upper limit. If the userspecified constraint for the stator current density were

5 A/mm , then anything more than that would be a

violation. Thus, if the algorithm returns a current density

value of 6 A/mm , the numerator in (4) would be 1;

this time a negative value indicating a violation.

The latter type of constraint explains the use of the absolute

notation in (4).

On the other hand, constraints cannot be equal to zero for

practical reasons. For instance, zero current as a constraint

would mean that there is no current in the windings when the

machine operates at full load; a very desirable but impossible

situation. Similarly, a zero constraint value for the stator slot-

filling factor would imply that no windings have been fitted

into stator slots. To avoid such absurdities, a provision is

made for the algorithm to block any constraint from being

active if the corresponding . At the beginning of

program execution, the designer specifies the constraints to be

present during optimization. If a nonzero value is entered for a

constraint, this constraint is present during optimization; if the

value is zero, the constraint is disabled, i.e., it is not present

during optimization. This offers the user a choice by blocking

selected constraints from being active. Consequently, division

by zero will never occur in (4) during algorithm execution.

If a constraint is not violated, then the corresponding

becomes zero, implying that there is no penalty associated

with this constraint. If a constraint is violated, then the

corresponding value of violation in per unit terms, as defined

in (4), is introduced into the calculation of in (3). For

example, if the current density in stator conductors resulting

from a design evaluation is 6 A/mm and the user-specified

constraint for it is 5 A/mm , then the resulting violation will

be equal toabs . Thus, can only assumeeither positive values in the case of nonfeasible solutions or

is equal to zero in cases when none of the constraints is

violated.

The constraints are somewhat flexible in that the devised

penalty function allows for slight constraint violations. This

means that the final solution resulting from the optimization

may lay slightly outside the feasible space. The use of such

flexibleconstraints facilitates the finding of optima, which may

be located just outside of a specified feasible region. Often,

designers will be prepared to relax some of the constraints

in order to gain significant improvements in the value of

the objective function (e.g., machine cost or efficiency). If

such an infeasible design cannot be accepted, the designer canforce a reduction in the constraint violation by multiplying the

relevant by a suitably selected constant to increase such

a constraints weight.

In the case presented, efficiency was selected as the ob-

jective function to be maximized subject to the following ten

nonlinear inequality constraints:

maximum stator slot filling factor 0.69;

maximum magnetic flux density in stator teeth 1.7 T;

maximum magnetic flux density in stator yoke 1.55 T;

maximum magnetic flux density in rotor yoke 1.5 T;

maximum current density in stator conductors 6 A;

maximum per unit starting current 7; minimum per unit pull-out torque 2.2;

minimum per unit starting torque 1.3;

maximum full-load slip 0.03;

minimum full-load power factor 0.87.

Fig. 2 illustrates the program structure. Execution of the

program starts with the specification of the nonvariant design

parameters such as the number of poles, materials used,

and magnetization characteristics of core materials. This is

followed by the selection and specification of the constraints,

as well as the number of generations, bit-string length, and

population size.


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Fig. 2. Program structure.

The initial population of (in our experiments, 200) in-

dividuals is generated next. Each chromosome consists of

11 genes representing randomly selected values for the 11

variables from within specified domains. Every gene is madeup of 20 bits, including those representing the two discrete

design parameters (the number of conductors per stator slot

and the stator wire gauge). These discrete design parameters

require modification for each population in order to only

assume the permissible discrete values. In the case of the

number of conductors per stator slot the above is achieved

by rounding off the random number to the nearest integer.

The other real number, delineating the stator wire gauge, is

used after rounding off to the nearest integer as a pointer to

the table of standard wire gauges setting the stator conductor

size for the subsequent design evaluation. All other variables

are continuous and do not require any special treatment.

Both the probability of crossover and mutation rate are userdefinable; in the case presented in these experiments they

were 0.9 and 0.005, respectively. The design is evaluated

for every individual of a population; the algorithm terminates

after performing the specified number (in our experiments,

20) of generations. At this point, the designer is offered the

option to view the performance characteristics for the proposed

design.

Table II contains the results of eight consecutive program

executions. Due to the random nature of genetic algorithms,

the computation time of a single execution varies slightly,

with the approximate average value being just over three hours

on a desktop computer with a Pentium 120 MHz processor.

Values for the eleven variables are presented for each run,

together with the values of wire gauge and airgap flux

density with which the optimal value for the objective

function (Efficiency) has been achieved. The algorithm has

returned an acceptable solution every time, which is indicated

by a goodvalue for the objective function with no constraint

violations. The optimized efficiency is within a narrow band

with a mean of 92.316%, which is entirely satisfactory for this

frame size. The ensuing air-gap magnetic flux density values

are included in the table to indicate that the algorithm

naturally selects values that have been historically established

as being suitable for this frame size.

Fig. 3 depicts examples of performance characteristics dis-

played at the program termination, in this case for run number

8 from Table II. The solid curves represent the optimized

performance, whereas the curves with broken lines represent

the performance of the existing motor calculated using the

same design algorithm. Test results for the existing motor

are also superimposed (shown as ). The inclusion in the

graphical representation of constraints together with the fullload operating point (shown as ) is most useful as it allows

for a better interpretation of the results; constraints (shown as

dotted lines) are as follows.

Line current versus speed characteristic: horizontal line

represents the maximum starting current constraint; the

vertical line represents the maximum slip constraint at

full load.

Shaft torque versus speed characteristic: the upper hori-

zontal line represents the minimum pull-out torque; the

lower horizontal line represents the minimum starting

torque constraints. The vertical line depicts the maximum

permissible slip at full load as in the first characteristic.

Efficiency versus shaft torque: no constraints are indi-cated.

Power factor versus shaft torque characteristic: minimum

acceptable power factor at full load constraint is shown.

This facility may prompt the designer to review constraint

specification and may wish to relax some of the constraints

toward improving the value of the objective function.

It is observed that the performance of the existing machine

does not satisfy the criteria set in this case. For example, the

constraints ofmaximum starting currentand minimum power

factor at full load are both violated (sub-plots 1 and 4 of

Fig. 3). The performance of the optimized machine is seen

to be improved respectably when compared with the existing

machine. The motor used as a benchmark for algorithm

verification is the product of a long evolutionary design

process. The use of the EA-based optimization approach

enables similar or better results to be obtained without the

benefit of accumulated design experiences over a hundred

years! A further significant aspect is that of minimal need

for interactive design input.

IV. CONCLUSION

It has been shown that EA based algorithms constitute a

viable and powerful tool for the optimal design of induc-


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The EAs intrinsic tendency to search for global optima

requires some caution in the implementation of design al-

gorithms. For example, in attempting to explore the feasible

space, EA may select a set of parameters for which magnetic

properties ( - curves, iron losses) are unknown; a situation

which may occur when the value of magnetic flux density

exceeds, due to selection by the algorithm of very narrow

teeth, the available values. Such problems are overcome by

a range of measures including the allocation of a low fitness

value each time such an anomaly is encountered.

The results obtained for a realistic device such as the

induction motor presented here substantiate the usefulness of

evolutionary optimization methods in electromagnetic device

design. The algorithm is capable of finding near-global optima

with a considerable level of confidence. The accuracy of con-

vergence can be improved further by additional enhancements

to the EA solver. Improvements may include the use of floating

point representation of individuals (including integer repre-

sentation for two discrete variables), ranking or tournament

selection, etc. Also, it is possible to accelerate the convergence

by means of hybridization [8]; a near-optimal solution foundby the EA can be pursued by a suitable gradient method

such as the sequential quadratic programming for quick and

accurate convergence.

REFERENCES

[1] O. Gol and J. P. Wieczorek, Use of MATLAB in induction motor de-sign optimization, in Proc. 1st Australian MATLAB Conf., Melbourne,Australia, 1996, pp. EE1824.

[2] , Application of nonlinear programming techniques in elec-tromagnetic device design, in The Role of Mathematics in Modern

Engineering, A. K. Easton and J. M. Steiner, Eds. Lund, Sweden:Chartwell-Bratt, 1996, pp. 417424.

[3] Z. Michalewicz, Genetic Algorithms Data Structures = EvolutionPrograms, 3rd ed. Berlin, Germany: Springer-Verlag, 1996.

[4] W. Nurnberg, Die Asynchronmaschine, 2nd ed. Berlin, Germany:Springer-Verlag, 1963.

[5] W. Schuisky, Induktionsmaschinen. Vienna, Austria: Springer-Verlag,1957.

[6] O. Gol, Induction machine models for design, in Proc. Int. Conf.Evolution Modern Aspects of Induction Mach., Turin, Italy, 1986, pp.510.

[7] J. A. Joines and C. R. Houck, On the use of nonstationary penaltyfunctions to solve nonlinear constrained optimization problems withGAs, in Proc. 1st IEEE Int. Conf. Evolutionary Computation, Z.Michalewicz, D. Schaffer, H.-P. Schwefel, D. Fogel, and H. Kitano,Eds. Piscataway, New Jersey; Orlando, FL, 1994, vol. 2, pp. 579584.

[8] H. Myung, J.-H. Kim, and D. B. Fogel, Preliminary investigationsinto a two-stage method of evolutionary optimization on constrainedproblems, in Proc. 4th Annu. Conf. Evolutionary Programming, J. R.McDonnell, R. G. Reynolds, and D. B. Fogel, Eds. Cambridge, MA:MIT Press, 1995, pp. 449463.

Jan Pawel Wieczorek received the Master of Engineering Science degree inelectrical engineering from the Technical University of Lodz, Poland. He iscurrently working towards the Ph.D. degree.

His current research is in the field of electrical machines. His interestsare focused on a range of areas related to electrical machines such aselectromagnetic device design, magnetic materials, modeling and simulation,energy efficiency in electromagnetic energy conversion processes, and the useof mathematical optimization in engineering.

Ozdemir Gol received the MESc degree from the Technical University ofIstanbul, Turkey, the M.E. degree from the University of Melbourne, Australia,and the Ph.D. degree from the University of Adelaide, Australia, all inelectrical engineering.

He has had extensive experience in the field of electrical machines, bothin industry and as an academic. He is currently Head of the ElectricalEngineering Department and Head of Electrical Machines and Drives ResearchGroup at the University of South Australia. His interests include dynamicmodeling of electrical machines using numerical methods, design and analysisof novel electromechanical energy conversion devices, and application ofmathematical techniques to design optimization of electromagnetic devices.

Zbigniew Michalewicz received the M.Sc. degree from the Technical Uni-versity of Warsaw, Poland, in 1974 and the Ph.D. degree from the Instituteof Computer Science, Polish Academy of Sciences, Poland, in 1981.

He is Professor of Computer Science at the University of North Carolina

at Charlotte. His current research interests are in the field of evolutionarycomputation.

Dr. Michalewicz was the General Chairman of the First IEEE InternationalConference on Evolutionary Computation, Orlando, FL, June 1994. He isa member of the editorial board of several publications including IEEETRANSACTIONS ON EVOLUTIONARYCOMPUTATIONand the Journal of AdvancedComputational Intelligence.

00728298 an Evolutionary Algorithm for the Optimal Design of Induction Motors

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