single phase im design 1 (1998)

IEEE Transactions on Energy Conversion, Vol. 3. No. 2, June 1988

OPTIMIZATIOS OF SINGLE-PHASE INDUCTION MOTOR DESIGN PART I: FORMULATION OF THE OPTIMIZATION TECHNIQUE

H. Huang E. F. Fuchs, Senior Member, IEEE Z. Zak*

Department of Electrical and Computer Engineering, University of Colorado, Boulder, Colorado 80309

* Was on sabbatical from VUES, Brno, Czechoslovakia.

Abstract - The optimal design of the motor dimensions, the capacitance of the run capacitor, the winding distribution and the choice of the electrical steel are the most important sources for an improvement of the efficiency of modern single-phase induction motors for given performance and material cost constraints. The formulation of the techniques which realize this optimization is dealt with in Part I of this two part paper, based on nonlinear programming approaches. A comparative study for both the Method of Boundary Search Along Active Constraints and the Han-Powell Method is performed from a theoretical convergence and practical application point of view in order to develope the best method for the optimization of single-phase induction motor designs.

INTRODUCTION

This study on the optimization of single-phase induc-

Formulation of the Optimization Techniques; The Maximum Efficiency and Minimum Cost of an

tion motor design consists of two parts: Part I - Part 11- Optimal Design. The purpose of this project, which was supported by the Electric Power Research Institute, Palo Alto, under contract No.1944-1, is the optimization of single-phase induction motors with respect to efficiency and cost for given performance constraints.

Part I deals with the application of two advanced optimization techniques for single-phase induction motors: the Method of Boundary Search Along Active Constraints [1,2,3] and the Han-Powell Method (4,5,6], and an algorithm for the optimal winding distribution. Part I1 is concerned with the maximum motor efficiency obtained with the above mentioned optimization methods for given performance and cost constraints. The relationships between efficiency, cost, power factor, active material volumes and the capacitance of the run capacitor are studied. In addition, this study makes comments on the limited validity of the model law 171.

Hundreds of thousands of single-phase induction motors are being manufactured every year in this country. To reduce costs of single-phase induction motor designs and fab- rication techniques, motor manufacturers have lost ground in motor efficiency. The recent emphasis on energy conservation, however, demands an improvement of the efficiencies of electrical motors and devices. Therefore, it is necessary to develop analytical techniques for the optimization of the efficiency and cost of single-phase induction motor designs.

In order to assess the merit of such a project, one may assume that the single-phase motor power consumption per household in the United States consists of 3 kW installed capacity (e.g., refrigerators, freezers, dishwashers, fans, etc.). A modest (7%) improvement of the efficiency of such motors maintaining required performance constraints and costs would result in savings of 210 W per household. For 50 million houses

87 SM 609-1 A paper recommended and approved tby the IEEE Rotating Machinery Committee of the IEEE Power Engineering Society fo r presentat ion a t the IEEE/PES 1987 Summer Meeting, San Francisco, California, July 12 - 17, 1987. Manuscript submitted January 28, 1987; made ava i lab le €or pr in t lng May 1, 1987.

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in this country the total power savings will amount to 10,500 million watts which would entail savings of $3.41 lo9, provided one assumes a load factor of 25% and a construction cost for power plants of $1,300 for one thousand watts installed power capacity. On the other hand, the single-phase motor popula- tion within the U.S. is about 720 million machines consuming annually an energy of 10l2 kWh [8]. Even the above mentioned modest improvement in motor efficiency will have a significant impact on the demand of electricity in the U.S. Based on the prevailing cost of electricity in 1981 ($0.047 per kWh) one ar- rives at a reduction of the total operating expenses of $3.290.109 per year.

The optimization of single-phase induction motor designs is a complicated task. In general, it should be based on the following four optimization components:

1. optimization of motor dimensions; 2. optimization of winding distribution; 3. optimization of capacitance of run capacitor; 4. optimization of electrical steel.

The first optimization component is usually formulated as a general nonlinear programming problem:

min f ( ~ ) , ( l a )

such that exists within the n-dimensional feasible region D:

zt D, ( W where D = { E I i! 2 O,gi ( i ! ) 5 0,i = 1 , 2 , . . . , m } .

In the a.bove equations, f(z), g i ( z ) are real-valued scalar functions and vector z comprises the n principal variables for which the optimizs.tion is to be performed. The function f(i) is called the “objective function”, for which the optimal values of 1, result in the minimum (maximum) of f(%), and these optimal values satisfy the given constraints. The objective function may be identified with -7 which is the efficiency with a minus sign in front the minus sign means that the maximum efficiency is required\ or Cnr which is the material cost (including cost of capacitors) of the motor. Inequality constraints s i ( % ) 5 0 include the performance properties of the motor, dimensional restrictions and additional requirements by customers. Note, every constraint can be written as g i ( Z ) 0, since any constraint of the form gi(i?) 2 0 can be changed to the above standard form by introducing a minus sign before the constraint function. In the feasible region D every vector 2: satisfies the given constraints.

The mathematical model for the optimization of the winding distribution is as follows:

ma2 m, P a )

such that

and

min Salr

m13 5 0.03,

where 7 is the efficiency of the motor; Ssl is the copper cross- section of the fullest stator slot; m13 is the ratio between the third and the first harmonics of the magnetomotive force (mmf) of the entire stator winding. The realization of this optimization is based on rules which are established in this study. This is a special nonlinear progamming problem. The third and fourth

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optimization components are mainly addressed in Part 11. To date no publications arc available concerning the

optimization of Single-phase induction motor designs for given performance and ma.terial cost constraints. A brief historical background with respect to the optimization of electric ma- rhines is given in [I]. In this paper the hlrthod of Boundary Search Along Active Constraints for thr solution of the general constrained optimization problem of thrc+phase induction machines was developed. A successful a.ppliration of that method is demonstrated in 121, which resulted in an efficiency increase of comniercially availa.ble thrrr phase induction motors. The above mrthod has the following advant.ages: t,he ga.in of physical insight into the optimized design through plots of the motor characterist,ics within different. planes; the possibility of bring ablr to include functions of discrrtc paranicters (e.g. nurnlwr of turns of windings); t,hrre is no reqiiirenient for a gradient vrctor, Hrssian matrix 19 . etc..

The Han-Powrll Mrthorl differs from the direct search met hc )(is ant1 unrons t rained tranft mnatic ni search iriethc d s , in that a n('w opthizittioil a1gorit.liiii was (lewloptd t1ia.t solves thc ronstrainrtl optimization problrm neither directly nor by unronstrainrtl transforilia tion [ -l ,5,6].

Both al)ovc. Incntionrd methods a.re applied to the optimization of the singlr-phase induction motor designs of this paper. Thr rrasons for employing two methods are:

1. to compare opt,imal results of both methods; 2. to find the most suitable optimization method for the de-

sign of single-phase induction motors. The following design quantities are chosen as the prin-

cipal variables [lj of the optimization: 21 -stator bore diameter; 22 -stator core length; 2 3 -number of turns in series of main-phase winding; x4 - number of turns in series of auxiliary-phase winding; z5 -wire diameter of main-phase winding; 2 6 -wire diameter of auxiliary-phase winding; x7-average rotor slot width; 2 8 -stator slot height; x9 -rotor slot height; 2 1 0 -stator yoke height.

(3)

The performance constraints g; for the single-phase induction motor as studied in this two-part paper (Eq.1 of Part XI) are:

1. 2. 3. 4. 5.

6. 7.

power factor at rated load pf 2 0.724; locked-rotor current to rated current ratio Z L R 5 6.000; starting torque to rated torque ratio tat 2 2.390; break-down torque to rated torque ratio 1, 2 1.800; temperature rise according to the insulation used for the windings AT 5 65°C; rated slip s 5 3%; material cost (except cost of capacitors) C, 5 $120.33.

(4)

THECONVERGENCEOFTHEMETHODOFBOUNDARY SEARCH ALONG ACTIVE CONSTRAINTS

The Method of Boundary Search Along Active Con- straints has extensively been used in the past but no proof of convergence has been published to date. In proving the optimum convergence of the above method one has to ascertain that the iterative decomposion of the optimization problem (Eq.1) into suboptimizations [l] indeed leads to the optimal solution of the given constrained optimization problem In particular, one has to answer the questions:

whether the search along the active constraints within all planes of the n-dimensional spare R" generates a convergent sequrnrr, and

2. whrthrr the limit point of the convergent sequence is identical with the optimal solution of the constrained optimization problem.

1

Vl'hrn thr optimization method at hand procrctls from the ith to the ( i+ 1) th siibo~,tiInizaticrm, thr It11 sul)opt1- mal solution vector is replaced by the (i t 1)th suboptirnal ver- tor only if thc latter results in a smaller value of the obJective function. Assume the feasible region of the problem (Eq.1) is bounded by finite values of the principal variables, then the algorithm will generate a convergent sequence in niost cases and the limit point of this convergent sequrnce should be thc optimum point for all planes passing through this point. The question whether the limit point of this convergent sequence is identical with the optimal solution is dealt with in two ways:

A. Convergence Theorem

Suppose that the objective function is pseudoconvex and all constraints g l (E) for z 1,2,. . . , m arc quasiconvcx, then there is only one optimal solution of the optimization p r o b lem (Eq.1) in R" 191. Furthermore, suppose that the vector T *

is the optimal solution in R", then it must be thr optimal point of any possible subspaces of R". This will be explained for the special case of a three-dimensional space RS in the following example. Example: Suppose I* is, for given constraints, the optimal point in Rs as shown in Fig.1. Then, no better solution thaIi Z * can exist in any two-dimensional spaces, i.e. the planes xy,xz,and yz. To understand this, assume the solution i ig on plane xv is better. This in turn means that there is a better point than T *

within Rs contradicting the original assumption that T * I S the optimum within Rs.

Fig.1 Illustration of example.

After knowing that the optimum solution z* within the n-dimensional space R" also is the optimal solution of all possible subspaces of R", that is

whrrr op nieans optimum, one has to answrr thc qiicstion whr- ther

op E * zri all R2 > . . . 2 o p zn ul l R" > o p in R", (56)

is triir. If it is true then thr limit point of thr convergent sequcnre of thr method undrr considrration IS identical with the optimal s~ lu t ion of the ronstrajned optimization problrrn (Eq.l) . The answer to this qurstion is based on the following theorem, the proof of which is giwn in the Appendix and in Ref. [ lo]. Theorcm.

If

1. 5' is thr optimal point within all IC dinicnsional subspaces of R" for the optimization problem (Eq.1);

t T is the transpose S p d " .


35 1

the objective function f ( z ) is a pseudoconvex function, i.e.

(cl ~ 5 2 ) ~ ~ v f ( Z 2 ) 2 0 implies f ( z ! > f ( Z Z ) for all cl, zz existing in the open convex set S within R"; the constraint functions g , ( z ) f o r z = 1,2 , . . . ,m are quasiconvex functions, i.e. g t ( z l ) i g , ( f z ) , then one obtains (51 - ~ 2 ) the number of variables n , the dimension of the subspaces k and the number of active constraints ml at z' satisfy the

T v g , ( s z ) 5 0 for all c ~ , z Z existing in x, and

relation: k < n

the gradients of the active constraints at E' are linearly independent, then %* is the optimum of the optimization problem (Eq.1) in R". If all the above conditions except Eq.6b are satisfied, F* can be the optimum of the optimization problem (Eq.1) in R" by chance only.

One should mention for the case k = 1 that the one- dimensional search by fixed step size may fail on a shape ridge of the objective function.

In this paper the single-phase induction machine is assumed to have m = 7 performance constraints and n = 10 variables and for the Method of Boundary Search Along Active Constraints k = 2, m l 5 k - 1 = 1 must hold to guarantee that the limit point of the convergent solution is the optimum in R", according to the above theorem. In other words only if the optimal solution lies on one active constraint (ml = 1) or the optimal solusion is insight of the feasible region (mi = 0), then it is guaranteed that the solution of the optimization method at hand converges to the optimum solution within RIO. Con- sequently, this does not hold for cases wherr ml > 1. However, if ml > 1, it is still possible for the limit point of the convergent sequence to be optimum in R" by chance but therr is no guarantee. Unfortunately, in many practical cases the optimal so!ution is located at a vertex of the frasible region which is formed by two or niore active ronstraints (ml 2 2). This is a disadvantage for the optimization method at hand, althollgh it yields in most cases an improvement of the objective function (e.g. efficiency).

B. Geometric Interpretation

The same conclusion of the preceding subsection can be reached employing a graphical representation of the gradients of the objective and constraint functions, v f a n d v g r for z = 1,2 , . . . , ml, respectively. Fig.2 illustrates the feasible descent direction set in the three-dimensional space R* where the optimal solution is located on one active constraint (ml = 1) or on a vertex formed by two active constraints (ml = 2). The feasible descent direction set is defined as a set of directions for which the product of the gradient of the objective function o f ' with the direction d is negative, that is V f T d < 0, and simultaneously the product of the gradients of the constraints V g a T , f o r z = 1 ,2 , . . . ,m , with the descent direction d must satisfy the conditions v g t T d < 0 f o r z = 1 , 2 , . . . , m 191. For ml = 2 the feasible descent direction set will be a cone having an acute angle, because V f T d < 0, v g I T d < 0 and V g z T d < 0 imply that any angle between a feasible descent direction vector and the gradient vectors f , V g 1 , V g z must be obtuse. This configuration is shown in Fig.2a provided that the feasible descent direction set is not empty, i.e. there exists at least one feasible descent direction d satisfying the above inequalities. If the feasible direction set is empty then point 0 represents the optimum solution provided the objective function f is pseudoconvex, and constraint functions g, ,a = 1,2 , . . . , m are quasiconvex and all constraints are satisified. For a cone with an acute angle it is possible that none of the existing three planes xy, xz, yz intersects the cone representing the feasible descent direction set. In this case the limit point of the convergent sequence will be at point 0 which is a pseudo-optimum and will not be identical with the optimum solution as shown in Fig.2af. If at least one of the planes intersects with the cone then a better point than point 0 can be found. Fig.2aII indicates such a better solution (point 0' ) than that of point 0 within or

~~

on the boundary of the frasiblr rrgion of the xy plane. Such a configuration is purely accidental for ml - 2 and cannot be guaranteed. However, for ml ~ 1 at least one of the planes will intersert the frasiblr drsrcnt direction srt as shown in Fig.% and the limit point of thr ronvrrgrnt seqiienrr will be identical with the optiiniim of R'.

BOUNDARY OF THE FEASIBLE REGION IN PLANE x z

DIRECTION

BOUNDARYOF

FEASIBLE REGION I N PLANE x y

a1

BOUNDARY OF THE FEASIBLE REGION IN PLANE xz t' I

BOUNDARY OF

a11

BOUNDARY OF THE FEASIBLE REGION

BOUNDARY OF THE 4' I IN PLANE XL FEASIBLE REGION I N PLANE Y Z \ I A

BOUNDARY OF THE FEASIBLE REGION IN

0

b Fig.2 Feasible descent, direction set in R8.

APPLICATION OF THE HAN POWELL METHOD TO INDUCTION MOTOR DESIGN OPTIMIZATION

Being a general purpose algorithm, the Han-Powell Method has several desirable features 1121. 1. it is globally convergent because the search direction is a

descent direction for the absolute-value penalty function; 2. the iterative process converges superlinearly in the neighbor-

hood of the optimal solution, if the step length corresponds to unity;

3. like all quasi-Newton methods, only first-order information is employed to achieve superlinear convergence;

4. being based on recursive quadratic programmming, the me-


352

thod is suitable for inequality as well as for eiiiiality cori- stmints.

The design of single-phase induction motors is 30

complicated that i t is not possible to formulate the objective function f and Constraint funrtions g z in closed forms. In order t o obtain the first-order information reqiiired for the programming of the Han-Powell hlethod. the forward finite-difference approximatiorisfor \ 7 f , vgt.7 1 , 2 , . . . ,ni. areemployed, that is

(7)

where p i s rit1ic.r f or y,,i ~ 1 . 2 ; . . , m : h i s the step size, r, is thej th unit rector. Note that the step size h for the finite difference approximations should be nioderate: too large steps result in a loss of accuracy of the difference expressions, while too small steps may generate ineaninglcss gradients, v f (k), o g t ( i ) because of dorninatirig subtractire cancellat,ion errors. In other words for a very sniall stcp size,, f ( T 4 he , ) f (5) and yz(? 7

h c , ) g 2 ( T ) . For a proper selection of the step size h coni- putational experiments are performrd. Such tests yield, for an accuracy of 0.5.10 for f and g , of the single-phase niot,or design of this paper, an optimal step size of h :. 0.5.10 - 4 . m a x ( 1 . , , 1.0) p.11..

In addition to the motor performance constraints Eq.4, some dimensional restrictions have to be imposed to ascertain that the optimal design is physically rrasonable: for example, a negativr slot widt.h is not possible. Also certain values of the variables niay result in indefinite expressions within thr design program and therefore must be avoided. With these additional constraints which fortunately are all linear for induction motors, the Han-Powell Method finds the proper direction d l by the quadratic programming to avoid indefinite expression prob- lems. This forces the motor design to stay in a region, including the feasible region formed by all performance and dimensional restrictive constraints, which does not generate indefinite mathematical expressions. This is due to the following property [6] where the direction dl yields

X k + l = l k t a d ] , (8)

and d l is obtained from the following quadratic programming:

and is always a descent direction of the merit function

m

I = 1

that is, the following relation is always true except if x k is the optimum,

with a properly chosen a , where a denotes thr line search step length. s is the vector of the principal variables consisting of all components as listed in Eq.3 except 2 3 and 2 4 which are both integer values and cannot casily be included in the optimization algorithm of the Han-Powell Method. The subscript k repre- sentes the kth iteration. g ( 5 ) is the vector of the performance and additional restriction constraints. B is an approximation of the Hessian matrix, which is positive definite. Sinre these a.d- ditional constraints a.re linear, they will not be violated in the procedure of the line search. Owing to the above-mentioned property, whenever a motor design sta.ys in the region which docs not yield indefinite ma.thematica1 expressions, it will a p p o a c h the feasible region instead of leaving the region not gen- c>ra.ting indefinite mathematical expressions. This is becaiisr a continuous decrease of thr merit function forces the motor design into the direction in whirh all the constraints are satisfied. For this piirpose twelve addit.iona1 constra.ints me introduced in the optimization program at hand. After normalization, they

Cp(zc + a d ] ) < i P ( z k ) , (11)

arc:

where

<

c1 ~- T b u s e 9 /3 .banC.1 ,

C2 ’- dshafl/Xhnzel, C3 = 0 . 9 T x b , , , l / ( ~ h , , , 7 Z ~ ) ,

c4 = K x h a , e g / ( I b a x r 7 Z Z ) ,

c.5 =- K X h a s e 9 / [ Z Z I h a m r 7 ( 1 r / 2 2 ) 1 1

c6 T h 2 n / [ 2 2 I b a s c 7 ( 1 t T / z 2 ) ! >

z1 = number of .stator s lots , 22 = number of rotor. s lots ,

d s h a f l = d iame t r r of s h a f t , h2, , = height of rotor slot opcnzng. ill = uveragc stator slot w id th .

Notr, Xhasl, f o r - 1,2 ,5 , . . . , 10 arc thr haws for thc principal variables and are to bc the values of the. principal variables per- taining to the starting point. of thr optiinizat,ion. Constraints Eqs. 12 and 13 are thc restrictions for thr principal variab1c.s. The values 0.8 p.u.and 0.6 p.u.are chosen froin experirnee. The smaller value 0.6 p.ii.for the restriction of the stator core length x 2 indicates that the optimal drsigri of the single-phase motor will have a short iron core length. Constraint Eq.14 rrstrict,s the st,ator t,ooth width, coiistraint Eq.15 limits the rot,or yoke height, constraint Eq.16 restricts the rotor tooth width arid coli- s traint Eq. 17 restricts t he rotor tc )oth 1~ )t torn wit1 th.

The Han-Powell Method requirrs that the iiorrii of thr Lagrange multiplier vector 1 1 X ‘ I < M. Ha11 has shown that if thew arc only convrx inclii:~lit~v constraints aiid t,herr is a point 5 surh that all gZ(s) < 0, t.hcm 1 1 X i l < 00 141. N o fiirt,hrr theorem is available. For the singlr- phase motor design of t,his paper the performance constraiiits are assumed t o IK qiiasiicon- vex fiinctions. Therrforc, thrre is no thvorrtical guarantre for 1 1 X 1 1 < 00. Experinierit,ally, hoivrrcr, ~1 X l i < 00 is always found t,o be satisfied.

C!)hlBINATION OF BOTH OPTIhlIZATION METHODS

From the preceding disrussioiis it is apparent t,hat both optimization methods h a w advantagrs and disadvantagrs. The Method of Boundary Search Along Active Constraints can easily handle the optimization of integer principal variables (e.g. 1 3 , 2 4 of Eq.3) and gives an cxcellrnt physical insight into the optimization process. On the other hand the Hari-Powell hlrthod is globally convergent. Thrrefore, in this study thc two methods are combined resulting in a hybrid mcthod incorporating the advantages of both methods. The procedure is: first optimize the continuous variables by Han-Powell Method; thrn keep the continuous variables a.s const,ant,s and optimize the discrete variables by the Method of Boundary Search Along Active Constraints; finally, keep the discrete v a r i a l h as constants and optimize the continuous variables by Han-Powell Llethod oncr more. The Method of Boundary Search Along Active Coli- straints is a.lso used for plotting. To reduce computing time, the successive quadratic prograinmirig siil)roiitinc “VF02AD” / 5 ] by Powell is modified by eliminating unnecessary gradient calculations in the line search.

OPTIMIZATIONS OF THE RUN CAPACITQR AND THE EZEXTRICAL STEEL-

The capacitance of the run capacitor aiid the s p r - eific loss and B-H characteristics could have been treated likr


353

principal variables. However, the discrete nature of the rapaci- tance of oil filled capacitors and the characteristics of electrical steels lead the authors to believe that such a treatment may not br advisable. Instead, the optimization of the dimensions of the motor and the winding distribution is performed for given run capacitor values C,,, and for given electrical steel characteristics. This ensures that availablr run capacitor values and elertrical sterls are treated as parameters influencing the opti mal design. Here, one should Inrntlon that the poor B-H char- arteristics of low-loss electrical steels can be compensated by choosing a higher valiir of thr ra1)arItancc. of the run capacitor. This is to sag that the increased Inagnctizing current rrquirr nicmts of lowloss electriral stet4 ran be rancelled by the run raparitor rurrent and tlie power fartor will l w niainly drtrr- niinrd 1)y thc caparitanrc, of thr run raparitor. Drtails of thc.sc t w o optimimtion components will I)c disciissrd in Part I1 1111.

DESLGN PRQGRAM

The purposr of the design program is to simiilatr thr prrforrrianre of single-phasr indurtion motors. Since i t IS not possible to find closed fornis for the o1,jective and constraint functioIi\ f and y,, respertively, thev appear as iiiiplirit functions of thr motor design. Thcrcforr, thr only way to rvaluatr these functions is to use an iterative method. A design program as shown in Fig.3 In block form must br available. For given rx- citations which are the principal variables of the optimization problem, the design program must have unique and bounded responses.

OBJECTIVE FUNCTION (COST OR EFFICIENCY)

DESIGN PROGRAM CONSTRAINT FUNCTIONS

V A R l (7) V A R l (8) V A R I (9)

EXCITATIONS R E S P O N S E S

Fig.3 The simulation of the single-phase induction motor

of this study is realized by using the rotating field theory corn- bined with the method of symmetrical components. The good agreement between the theoretical and the experimental results has been documented in Ref.3.

As mentioned before, [ z l x z . ..zI”]* represents the vector consisting of principal variables. There are several possibilities to choose the principal variables (Eq.3) of the optimization problem. However, the selection mainlv depends on the following conditions:

1. the principal variables must br independent of each other; 2. they havr to rrpresrnt well thr motor dimensions, 3. the selcrtion of such variables should minimize the corn-

puting time for the prrformanrr ralculation of thr motor. Based on tlir above ronsiderations, onc rhooses $3, zq

to be thc Iiiirnhers of tiiriis of the main and aiixiliarv-phase windings. It is not advisable to rhoosc thr maximum air gap flux density B, as a prinripal variable brcausr thr design program uses B, as a criterion to discontinue the iterations, provided a rcrtain accuracy is reached. Also, thrrr arr two possibilitirs for rhoosmg t5, T ~ , 2,. They rould be cithrr the rurrrnt densities or thr diameters of the conductors of the stator and rotor windings. But the former choice would require many iterations within the drsign program for the exact determination of the diameters of thr conductors and rrsult in vrry high romputing costs.

In the design program, all secondary variables ll,2] ran be uniquely determined by thr chosen principal variables exrept the constants, which have to be initialized at the very beginning of the computation. This prevents the intermediate solutions from leaving the feasible region when the Method of Boundary Search Along Active Constraints is employed. A procedure for reentering the feasible region is therefore not needed in this single-phase motor optimization.

Block diagram of design program.

QJPTIMAL WINDING DIST-RIBUTIQN

The optimal winding distribution proLlem is a special nonlinear programming problem. The name “special” refers to a realization of thr optimization by a particular mrthod based on rules gained during thc course of this study rather than from any other general nonlinear programming approach.

The mathematical formulation is prrsented in the IN- TRODUCTION. The purpose of the optimization of thc winding distribution is to find a distribution which results in the highest efficiency for the same amount of artivr material cost as that of the original motor design, provided that the motor performance constraints are satisfied. A sinusoidaltt winding distribution always has the lowest harmonic content, but it cer- tainly has two disadvantages:

1. a motor design with a sinusoidal distribution cannot reach

2. it uses more iron material since the copper cross-section

A deiailed analysis based on the inspection of vari- ous winding distributions of single-phase motors, shows that the optimal winding distribution with respect to efficiency must lie in between the sinusoidal distribution and that with a constant copper cross-section within all stator slots [3]. A comparison of the properties of windings shows that the optimal winding distribution must be closer to the sinusoidal distribution than to that with a constant copper cross-section, that is,

the highest efficiency which it otherwise could havr;

within stator slots is nonuniform.

1. the sinusoidal winding distribution should be chosen as a starting point for the winding optimization, and

2. the search direction should bc from the starting point to the winding distribution with a constant ropper cross-srction.

Thr nonuniform distribution of turns within tlic stator slots must hr made more uniform in order to arrive at a minimuin stator dot rross-scvtion for a givm content of the third stator mmf harmonic. Such a more iiniforni distri1)ution of turns within the stator slots ran 1~ achieved by moving individual turns from coils with small coil pitches to roils with larger spans. This redistribution increases thr fundamcntal as well as the higher harmonir contents. There is a region, starting from the sinusoidal distribution, in which the efficieny increases, caused by tlie increases of the funtlaIrirnta1 winding factors of the main and auxiliary- phasc windings. For rontiniied further redistributions, the efficiency may start to decrease due to the increased lengths of the two endwindings. The determination of the optimal winding with respect to efficiency requires the ralculation of the efficiency for all intermediate feasible designs. The algorithm for the winding optimization is as follows: Specify the number of total attempts, n, for moving one turn to the ith roil from the (i-1)th coil and start from the largest pitch. Then

1. obtain the sinusoidal distribution; 2. set the number of attempts 1 = 0; 3. move 1 turn from the (i-1)th coil to the ith coil;

3 = 1 t 1 and go to ( 5 ) , otherwise, go to (6); 4. if (Ssi)J+l < (S.i) , ,(m13),+~ < 0.03,(71),, > (vI1, then

, 5. if j 5 n, go to (3), otherwise go to (6); 6. if i<2, stop, otherwise go to (2).

After finishing the procedure for the main- phase winding, repeat it for the auxiliary- phase winding.

Thc rrsiilts of thr winding redistri1)lition for tlir 2 lip singlc- phase Inarhine of Eq.l in Part I1 is illristratrd ill Tables 1 and 2. Tablr 1 shows the parameters of thc sinusoidal distribution, whilr Table 2 gives thr parameters of the oI’tirna1 winding distrilnition. An rxprrimental comparison of thr original and ___.

t t A sinusoidal winding distribution is defined as a winding, where the number of turns within a finite number of slots is determined by a sine function.


354

Tablr 1 ~ -.- - _ _ _ _ _ _ _ ~

No. of coils pcAr pole q c A F , qcBl Copper cross-srrtion in slots

Slot NO. 1 2 3 4 5 6 7 8 9

No. of turns per pole N,A, NP* No. of turns per phase in series

NA, N n Effective no. of turns per phase

in series N A , j j , N B , f j

Winding fartors for harmonics:

3rd (a3 ( ~ 3

5th (a5 ( ~ 5

Content of higher harmonics mi3 [%I m 1 5 I % ]

1st ( A 1 (SI

7th F A 7 (I37

wL17 I % ] _____ _ _ _ _ ~- ~ - . . * Number of turns

4 4 Main -f Auxi . : Tot a1 20.1.3 -26.0 18.1.3 + 7 . 1 ~ 3 0 . 4 1 3 4 . 3 + 11 . 1-27.9

7.1.3 + 1 2 . 1 ~ 2 1 . 1 2 . 7 . 1-14.0

7.1.3 t 1 2 . 1 ~ 2 1 . 1 13.1.3 t 11 . 1k27.9 18.1.3 + 7 . 1130.4 20q.3** -26.0

58 37

116 148

92 121

0.79371 0.82629 0.00000 -0.01351

-0.00492 -0.05190 0.00767 0.08696

0.000 0.397 0.458

.

* * Copper cross-srrtion of wire in mm2 I windings, respectivcly.

Subscripts A and B represent the main and aiixiliary-phasc

Table 2 Optimal Winding . .

No. of coils per pole qcAl , qcB1 ~-~ Di ~~

Copper cross-section in slots Slot No. 1

2 3 4 5 6 7 8 9

No. of turns per pole N p a , N p ~ No. of turns per phase in srrics

NA, N n Effective no. of tiirns per phase

in series N a e j j , NB,j j

Winding factors for harmonics: 1st ( A i ( S I

3rd ( A 3 ( B 3

5th F A 5 (B5

Content of higher harmonirs 7th FA7 ( 8 7

m 1 3 [”/U] m i 5 I%] mi7 [%)I

riljution. ~~ ~ ~-~~ .

4 4 Maini Auxi.-Total 22.1.3 -28.6 16.1.3 + 5 . 1=25.8 14.1.3 i 1 0 . 1=28.2 6.1.3 i 1 3 . 1=20.8

2 . 9 . 1 ~ 1 8 . 0 6.1.3 t 1 3 . 1k20.8

14.1.3 + 1 0 . 1 ~ 2 8 . 2 16.1.3 + 5.1-25 .8 22?L.3** =28.6

58 37

116 148

93 126

0.80300 0.84800 -0.02986 -0.14865 0.02123 -0.00417

-0.06204 -0.09328

2.600 0.243 1.317

~ ~ ~~~~~~~~

Nnnilwr of tiirns

Siil)srript,s A and B rrprrsent the main and auxiliary-phasr * * Copprr cross-srction of wire in nim2 t xTintliiigs, rcq)rrtively.

thr optinid winding distrilmtions for thc rsamplr motor with r c y r r t t o a11 increase of efiricncy will I,r prrsrntrd in Part I1

This spccial nonlinrar progmmnring problem ran bc cornlinrd with the grnrral programming probleiii of th r o p tiniization of the niotor dimensions either in “parallrl”, or in “srrics”. In “parallel” means that the two optiniizations arc performed simultaneously. A computational comparison of the both comlinations indirates that the “parallel” arrangement is computationally more efficient than the ‘‘series’’ approach.

The ratio m13 is a mrasure for the third stator harmonic mmf component in air gap. The constraint m 1 3 5 0.03 is used as one of the criteria for t,he optimal winding distribtution and this value is based on experience. The ratio m 1 3 is derived [3] as follows: If t,he rurrent in the main-phase winding is

1111.

i A = IAmazsinwt, (18) then for given reference roodinates, the v location x and at a time t , can be written as

th harmonic at a

fA , , (x , t ) = a ~ , , N ~ ( ~ , s i n v x s i n w t , (19)

where aAv - :<;;;NA[A,. N A , a p ~ and EA,, arc the niimlwr of turns per phase in series, the number of parallel branrhes and the winding factor of the main-phase winding, respectively. I’ is the number of pole pairs.

Similarly, for the auxiliary-phase winding one can write

i~ = IBmazSZn(Wt+ 4)l (20)

f n , , ( x , t ) = a e u N ~ ( ~ , s i n v ( x - xo)sin(wt i $), (21)

where xo .and 4,. arc the spatial shift between the two windings and the time shift between the two currents, respectively.

After some manipulation one obtains for the resultant v-th mmf harmonic:

Fu(z,t) ~L aj ,s in(vx ~ w t i $ j ) t ab,sin(i>x t w t t $ b ) , (22)

where $r, & are p h s e shifts 131. The first tcrm of thr above eqiia.tion represrnts the forward rotating mmf wave, while the second term corresponds t,o the backward rota.ting mmf wave. 111 Eq.22 t,lie aInplitiidrs a r r defined as

1 u j , , J ~ A , , t aL~3t, + % ~ , , u B ~ c o . ~ ( ~ ~ T o i 4), (23)

(Lht, t ~ u z ~ a i a2B1, i 2aAl,aH,,co.5(17x(l 4) (24) - - _ _ _ ~ -~ - ~

Note that with inrreasing time the aniplltudes of tlir resul tant harmonic time vectors at a location along thr circumfrr ence of the air gap oscillate between their minimum (F,,,,, aiu - abu) and maximum (F,,,, = aj,, t ab,,) values. The medium value of this nonuniform resultant harmonic time ver- tor can be found from the area spanned by the elliptical lo- cus of the v-th harmonic time vector surh that the equality k a Z m e d u = kFumznFumaz holds. The medium valne of the w t h mmf harmonic time vector is then

with zu = ~ 5.


355

_ _ ~ _ _ _ OPTIMIZATION OF _ _ _ _ ~ - . ~ A PRODUCTION LIT, -~ E OF MOTORS

The optimization techniques formulated in this study not only can deal with a particular design of a single-phase induction motor but also can handle designs of an entire production line of single-phase Indnction m.)tors. In the latter situ- ation, the samr iron core lamination may be used for several horsepower ratings with thr samr perforniancc constraints except the cost constraint. There are thr following three possibilities fibr this kind of drsign. Possibility 1 The shaft liright of thr niotor is not presrribcd. 111 this case, thc objective function of Eqs. 1 can l ~ e identified with

where qt is the efficiency of the ith motor within the produc tion line; m, is the total number of motors within the given production line with the sanie iron corr laminations; and C 1s

a weighting factor, 0 < mation in Eq.27 is the average effiricncy which is expected to be as high as possible, while the second part is the restriction that each motor within the production line has an efficiency as close as possible to the highest average efficiency. All constraint functions are the same for all motor designs involved, except the cost constraint must be the sum o: the cost constraints of all the individual motors of the production line. The principal variables x ~ , 23, 24, x5r 26 of Eq.3 should be replaced by principal variables which specify the stator core lengths, the numbers of turns in series and the wire diameters of main and auxiliary- phase windings of all the motors within the production line, respectively. The rrmaining principal variables of Eq.3 are the same. Possibility 2 : The shaft height of the motor is given. In this case, the only difference from possibility 1. is that since the outer diameter of the stator core Do is fixed, the stator bore diameter x1 has to be eliminated from the principal variable vectc-r of the optimization and determined by

1. Note, the first part of thr sum

_ _ _ ~ -

21 = DO - 2.3 -- x10, (28)

where z g , 2 1 0 are defined in the INTRODUCTION. Possibility 3 : Add an additional power rating to an existing production line. In this case, the objective function should be the same as that of a particular design . Then only 22, 23, x4, 25, 26 will be principal variables and the remainders z,, 2 = 1,7,8,9,10 will be considered as constants

CONCLUSIONS

The optimization of single-phase induction motor dr- signs contains four important components. Earh of them has its own contribution to the improwment of the cfficirncy of thr motor

The entire optimization is lmwd on the two pr( :ram- mings: the gencral constrained xionlinear programming ii,ld thr special constrained nonlinear programming, which are iormu- lated in this paper. I t is lwtter for both programmings to be connected in "parallel" in order t o minimize coniputirig timr.

The general programmmg can br rtdizecl bv two 01)- tiinization methods: the Metliod of Boiindary Sc.arch Along Ac- tive Constraints and the Han-Powell Method. It is shown by thc comparative study of this paper th.it both mrthods havr advan tages and tlisadvantagrs. A novel hybrid optimization nieth1)tl incorporating the advantages of both methods has been devzl oped in this studu and is an excellent choice for the general prt )gramming.

It has bern proven that the Method of Boundary Search Along Active Constraints can guarantee that the limit point of the convergent sequence is the optimal solution, if the optimum !ies on one actite constraint only.

The optimization techniqaes formulated in this study not only can deal with a particular drsign of a single-phase induction motor Luj also can handle designs of an entire production line of single-phase induction motors.

ACKNOWLEDGEMENTS ._____.__-.-__

The authors would like to thank Mr. J . C. White of the Electric Power Research Institute, Palo Alto, for his fre- quent discussions and Professor R. Byrd of the Computer Sci- ence Department of the University of Colorado for his advice on the optimization.

APPENDIX

Proof ,,f Theorem in Section "The Convergence of the Method of Boundary Search Along Active Constraints" :

is an optimum in all (nl - 1) dimensional subspaces of R"' for the optimization problem (Eq.l), therefore the Kuhn-Tucker 191 conditions are satisfied:

. _____ ________

First assume k = n - 1. Becwise

w - , X * ) l r , - - c o n s t -

m

where J 1,2;. . ,n1. of(i?*)l,. ronr, drnotcs the following cqxration: with zJ , - C O T L S ~ U ~ ~ , find thr gradient of f(z ) arid lct 3' s* The same applies t o v L ( . c * , A * ) l r j u n d vy,(s*) IZ, The active constraints In all (n ' 1) dinirnslorial \llbsI>it'.eS of R n l arc thr same at point L* Without loss of generality, one can assume the first ml ronstraintr to br activr Thc~rrfore A,,, + I * , A,,, + 2 * ,

Am* are zero for all ( t i l 1) dimensional su1,spacc~s. Because 1 - 1 2,. . . , n i , there arr n' sets for E ~ s . A-1. In each set, Eq. A-la contains (nl 1) quntions in which there always exist (n' - 2) equations identical to those of any neigh1)oring set. If ml 5 n l - 2 , then A I * , A Z * ..., A,,' willbe uniquefor all these sets, provided that the gradients of the active constraints at E' are linearly independent. That means A 1 * Az *, . . . , Am, satisfy the Kuhn-Tucker conditions for the n' dimensional space R"' :

At*gt(x*) - 0, ~ O T all z 1 , 2 ; . . , m , ( A -2b)

A,' 2 0, for all 2 : 1 , 2 , . . ,m. ( A - 2c) -*

Therefore ( z* , A )-is a Kuhn-Tucker pair in R"'. Choose any point z within the feasible region D, then

g l ( z ) 5 g l ( z * ) , f O T 2 E I(r*) 1 (2 : gz(z*) = 0)

Si.ire the constraints gz(z*) . z ~ 1,2,. . . ,m, are quasiconvex,

Note that A,* = 0 for z $! 1(s*). It follows that

m

(z - Z * ) T { X A,' Tj? gz(s*)} 5 0 I = 1


356

Gsirig Eq. .4-2a and the above eqna.tion, one obtains

(s ~ 2)7‘ v f(F*) > 0

Brcausc. thv ol)jc.ctivc function f ( ~ ) is a pseudoconvex function, f ( . r ) f i r * ) .

Koiv assiinic. k ~ TL 2 a.nd go through thz same 1)rowdiirc~. on(^ can arrive the same conclIisioI1 that if all the aI)ov(~ 5 conditions arc satisfied, E* is the optirrium for (n-1) (limciisional spaces and thus for the whole prc)Lleni. The same concliision can be ol)t,aincd for k ~ 2, 6 - n ~ 3, . . ., etc., a s long as k

On thc. other hand, if Eq.61) is not satisfied, i.e., mi > ,k 1. thcw thrrc, is no giiara.ntre t,hat one can obtain unique v.aliies for A I * , Ax*; ’ . , X,,,I * satisfying all thr sets of Eq.(A-la). Therefore, i n this c a w , r * can be the optinlum by chanre only. E Z ( L I I L I J ~ ( 1 :

S1ipp)sc f, gt < 0, i ~ - - 1, 2 , . . . , 8, satisfy the conditions of thc theorem, , I .- 3, and k - 2. nl I : IC ~ 1 ~ 1. There are 3 two-dimensional siil)spaces 1x1 z21T, 1x1 1 3 I T , 1x2 x3lT in RS, i.e. 1 3 . 1 1 2 rjIT. Because I* = [ x 1 * r z 2 * r z 3 * 1 T is an optimal point for all two-dimensional subspaces, ?* must satisfy the Kuhii-Tucker conditions in all these subspaces. Then one obtains the following three sets of equations:

1 > m i .

According to the Kuhn-Tucker conditions, A I * 2 0. Since one always can find one idential equation within two neighboring sets, X I ’ sa.tisfirs Eqs. A-3, A-4 and A-5. Therefore, (z*, 1’) is a Kuhn-Tucker pair in RS. Furthermore, z* is an nptimum in RS because f, g; satisfy the conditions of the theorem. But if mi = 2, one cannot get unique positive XI’, A 2 * from Eqs. A-3, A-4 and A-5, that is, one cannot guarantee that z* is a Kuhn-Tucker point in RS. Example 2 :

~ 1 , 2 , ” . ,8 , satisfy the conditions of thr theorem, nl ~~ 4 , and IC = 3. mi 7 3 ~ 1 - 2. There are 4 three- dimensional subspaces:

[ZI 12 x 3 I T , 1x1 13 1 4 ] ~ , 1x2 13 141 , 111 12 141 i.e. 1x1 x 2 13 ~ ~ 1 ~ . Because ?* ‘ z I * x2* 1 3 * zq*l”’ is the optimal point within a11 tlirc.r-diiriensiona1 subspaces, s* must satisfy the Kuhii-Tucker conditions in all these siilxpaces. Then one obtains the following four sets of rqiiations:

.- -~

Suppose f, gi 0, i

.T 7’ . zn R4,

Of 1 A]’;;: 1 X 2 * i ) 9 2 a r , ~ (’,

ijr * aq, Xz*;;: ~ (), a*, + A I a i 2 ( A 9) I g4 4 XI*;:; + X 2 * 2 2 - 0 , 4

Acrording to the Kuliii-Tucker conditions, X I * > 0, X z * > 0. Since one can always find two identical rquations within any two neighboring sets, A:* , X2 ’ satisfy Eqs. A-6, A-7, A-8, A-9 and are unique. Therefore, ( k * , A’) is a Kuhn-Tucker pair in R4. Furthermcm f’ is the optimum in R4 berause f,g; satisfy the ronditions of the theoreni. But if m1 ~ 3, one cannot obtain iiniqiie positive X I * , X 2 * , A 3 * from Eqs. A-6, A-7, A- 8 and A-9, that is, one cannot guarantee that x * is a Kiihn- Tucker point in R4. This example implies tha.t if a direct three- dimensional search were used in the Method of Boundary Search Along Active Constraints thrn the convergent solution would br identical with the optimal solution in R” for ml ~ 2.

R EFERENCES ll] J . Appelbaum, E. F. Fuchs and J . C. M’hitr, “Optimization

of Three-phase Induction Motor Design. Part I: Formula- tion of the Optimization Technique,” IEEE Summer Power Meeting, Mrxiro City, hlexico, July 20-25, 1986. Paper No. 86 SM 486-5.

121 J . Apprlbauni, I. A. Khan, E. F. Fuchs and J . C. White, “Optimization of Three-Pha.se Induction Motor Design. Part 11: Thr Efficiency and Cost of an Optimal Design,” IEEE Summer Powrr hlerting, Mexico City, Mexico, July 20-25, 1986. Paprr N0.86 SM 487-3.

131 E. F. Fiichs, H. Hiiang, A. Vandrnpiit., J . Hiill, Z. Zak, J . Apprlbauni and hl . Erlicki, “Optiniixation of Induct,ion Mo- t,or Efficirncy, I’oliinic. 2: Single-Phasr Indurtion Motors.” Final Report, Electric Powrer Rrscarrh Institiitr, EPRI EL-

141 S. P. Han, “A G1ol)ally Convergent, hlrtliotl for Nonlinear 4 i r % c c n r , 1987.

Pra ypinniing,” J ( nirnal of Optiinim tic ni Tlicv )ry ancl Applicat~ions, Vo1.22, N0.3, July, 1977.

151 h l . J . D. Powell, “A Fast Algorithm for Nonlinearly Ccn- strained ( 11) t,imixa t i c )n C alcula tic ms ,” 1 9 7 7 D~~r!dec C( nfc renrc on Niiinerical Analysis, Springer-Vrrlag, Serirs on Lecture-Notcs~in hIathematics, pp. 144-157.

161 hl . J. D. Powrll, “Algorithms for Nonlinear Constraints that Use La.graI1gia.u Functions” ~ Mathrniatiral . ~ ~ ~~ ~ ~ . Progra.mming, __ Vol.14, (1978).

Mijglichkeiten und Grenzen fiir einr Verbesserung (Efficiency of Electrical Machines ~~ ~ Possibilities and Lirnita.tions for an Improvement),” Siemens Energietechnik, 2. Jahrgang, Heft 7, 1980, pp.271-276.

181 U.S. Department of Energy, “Classification and Evaluation of Electric Motors and Pumps,” Feb. 1980.

191 M. Avriel, Nonlinear Programming, Prentice Hal1,1976.

~~~~~ ~

171 H. Auinger, et al. “Wirkungsgratl Elrktrischrr Maschinen-

[ 101 H. Huang, “Optimization of Single-Pha.se Induction Ma- chines with Resnect to Efficicncv for Given Performance and Cost Constrainis,” Doctoral Tji&isSis, University of Colorado, Boulder, 1987.

Ill] H. Huang, E. F. Fuchs and J . C. V\’hitc, “Optimization of Single-Phase Induction hlotor Design. Part 11: The Ma.xi- mum Efficiency and Minimum Cost of an Optirna.1 Design,” IEEE Summer Power Meeting, 1987, San Francisco, July 12-17, 1987. A companion paper.

Addison-Wesley Publishing Company, 1984. j12] D. G. Lurnhergcr, Linear and Nonlinrar Programming,

.. . ~ ~~ ~ ~~ - ..... - ~~~ ~


single phase im design 1 (1998)

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