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ms is the mechanical pole spread or the maximum time between thefirst and last pole contacts to close mechanically. Since the breakercontacts cani close electrically prior to mechanical close due to theprestrike phenomena, this must be factored into the computerstudies. At first glance it would appear that the prestrike time shouldbe added directly to the maximum mechanical pole span to give themaximum electrical pole span. Thus a breaker with a 0.15-cyclemechanical pole span and 0.25-cycle maximuim prestrike time wouldappear to have a maximum electrical pole span of 0.4 cycles. Thefallacy of this reasoning is that it fails to recognize that if the firstpole to close can prestrike, so cain the remaining two poles. Thevoltage across all three breaker poles must therefore be consideredin relation to the maximum physical contact separation at anlyinstant to determine at which point the three phase3 electricallyclose. While a description of the detailed method used in the ANA-COM studies is bevond the scope of this discussion, suffice it to saythat the maximu-m electrical pole spans obtained using this methodwith the breaker just described usually falls in the range of 0.2 to0.3 cycles rather than the 0.4 cycle that might be expected.Mr. Webb mentioned the application of resistor breakers on FHV

cable circuits. While cables were not covered in the paper the authorsrecognize the importance of the problem an-d agree with Mr. Webbthat resistors should be evaluated for this application. Limiteddata are available on switching surge strength of cables but the in-vestigation of the overall problem should be considered.Mr. Webb was correct irn the assumption that the tertiaries in our

analysis were delta coinected. It is gratifying to know his investi-gations confirm our own.While Mr. Simmons agreed with the authors' statement that

most breakers in service at 345-kV and below have no preinsertionresistors, he states that many breakers on order do include resistors.

The authors do not know what many means but it is considerablyless than 50 percent of the applications. In any event, the authors'original statement was Imade to enmphasize the fact that in service ex-perience has not demonstrated the need for resistors.Mr. Simmons refers J. Sabath et al. (reference [2] of the paper)

to prove that voltages above 3.0 p.u. are possible with high-sideswitching. This work discusses the compuiter studies for the SouthernCalifornia Edison 500-kV system and as mentioned previouslycomputer studies have showni voltages tip to 5.0 p.u. The operatingand field test data however have shown maximum voltages of 3.0p.u. As to whether a 2.8 p.u. design level without resistors will beacceptable for 345-kV lines of. the future, Mr. Beehler in his dis-cussion evidently feels that they will.In answer to Mr. Simmons' question on whether lightning,

switching surge or 60-eyele coinsiderations dictated the minimumnumber of insulators deemed practical, all three factors were con-sidered both from a theoretical standpoint as well as from an actualutility use standpoint. The values listed in the table represeint thelowest number of insulators in use or contemplated in new con-tructions. As mentionied in the paper, up-rating of existing lineswas niot considered. To cover this condition as well as to allow evalu-ation of new lines with insulation levels lower than given in the table,the data given in the tables and figures can be iut-erpolated toinvestigate the necessity of surge suppression resistors.In closing this discussion, the authors would like to take exception

to Mr. Simmons' closing statement that "utility planning stuidiesleave little question concerning the need for switching surge controlrequirements when switching EYV lines with low-side breakers."As indicated by the data in the paper and the comments of thediscussors it is very rare that utility studies will show that pre-insertion resistors are needed for this application.

Optimum Design of Electrical Machines0. W. ANDERSEN, MEMBER, IEEE

Abstract-A unified method of designing electrical machines ispresented, utilizing the capabilities of modern, high-speed digitalcomputers. The objective is to meet all performance requirementsat minimum cost. The general approach is described by means oftwo examples, a program for design of power transformers and aprogram for design of hydroelectric generators.

INTRODUCTION

A UNIFIED method of designing electrical machinesis presented, utilizing the capabilities of modern,

high speed computers. The objective is to find the optimumdesign assuming certain design practices, meeting allperformance requirements and other limitations at mini-mum cost. For a large transformer or rotating machinethe cost includes the cost of losses.

Paper 31 TP 66-869, recommended and approved by the RotatingMachinery Committee of the IEEE Power Group for presentation atthe IEEE SuLmmer Power Meeting, New Orleans, La., July 10-15,1966. Manuscript submitted February 9, 1966; made available forprinting May 9, 1966.The author is with the Norwegian Institute of Technology,

Trondheim, Norway.

The advent of ever faster digital computers has made itpossible to synthesize in great detail hundreds of alternativedesigns to given specifications for any type of electricalmachine in a matter of seconds or a few minutes. This canbe done in such a way that the computer systematicallyworks in the direction of better designs, and finally endsup with one which for all practical purposes is the optimum.Although the optimum usually is quite flat, it is often

possible by such a technique to reduce the cost severalpercent from that of a design made by an experienceddesigner using slide rule or less sophisticated computerprograms. The potential economic benefits from taking fulladvantage of present day computers in this field are there-fore very substantial.

It is also possible to meet unusual performance require-ments, where experienced designers have great difficultiesusing more conventional methods. Whenever a design doesnot meet all the requirements, the computer itself can findout how corrective action is to be taken, without thisspecifically being built into the program.

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The evolution of such a program for the design of 3-phase 2-winding power transformers is described in thefollowing. The program was written in Algol for the Univac1107 computers at the Norwegian Institute of Technology,Trondheim, and at the Norwegian Computing Center,Oslo. It was the first in a series of transformer programsmade in cooperation with A/S National Industri, Dram-men. Subsequent programs also cover distribution trans-formers and different types of 3-winding transformers.The first step is to make a simple performance program,

where no design choices are made. The performance pro-gram is incorporated into a synthesis program, which pro-duces one alternative design based on a set of independentvariables (core diameter, core height, etc.). Finally, thesynthesis program is combined with an optimizing routineto a complete optimizing program producing a largenumber of alternative designs, changing the independentvariables systematically toward lower and lower cost,moving closer to or staying within the required perform-ance characteristics.An outline is also given of a similar program for hydro-

electric generators, showing how the same approach canbe used for rotating machines. In fact the basic ideaspresented here can be applied to a wide range of engineer-ing problems.

TRANSFORMER PERFORMANCE PROGRAM

The transformer for which the performance is to becalculated is shown schematically on Fig. 1. The perform-ance program takes as input all information regardingrating, dimensions, winding data, cooling system, etc.With an execution time of about half a second, thecomputer produces one sheet of output containing:

1) record of input data2) headings for information which may be put in by

hand, such as customer, drawing numbers, etc.3) all the required performance data, such as losses at

different load points, temperature rises of windings andoil, short-circuit reactance, short-circuit forces and stresses,magnetizing current, and impulse voltage distribution.

The sheet is arranged in such a way that test resultscan also be put in by hand. For transformers which areactually built, it serves as a permanent design record.The performance program remains an independent pro-

gram. As such it can be used for recalculating old designs,using up to date and uniform calculating methods, so thatthese become more useful as references. It can also be usedif the designer wants to make small changes in a designwhich comes out of the optimizing program.

TRANSFORMER SYNTHESIS PROGRAM

The performance program removes a lot of the cal-culating chores from the designer, but unless more sophis-ticated programs are made, he is still required to make allthe design decisions. The purpose of the synthesis programis to relieve him of most of this work also. However, a fewbasic decisions remain to be done by the designer.

Fig. 1. Three-phase power transformer.

The design process starts with the choice of some values,which, subject to certain restrictions, are considered to beindependent variables. As far as the core is concerned (Fig.1) these can be the diameter, the height, and the fluxdensity. After the windings are designed, the pitch comesout as a dependent variable. Assuming that the cross sec-tions of leg and yoke are standardized, the core is com-pletely described by these four variables. The synthesis ofthe transformer core really only consists of finding thepitch.

Theoretically, the pitch could have been chosen as anindependent variable and the height as a dependent vari-able. But this would have made the synthesis of the wind-ings very difficult, and must be discarded as impractical.The main windings are concentric disk windings or

helical windings. The choice of independent variables forthese may be less obvious than for the core. However, if theaxial copper dimension is chosen as one independent vari-able, this fixes the winding axially. Further, if the tem-perature rise oil-copper is chosen as another independentvariable, this together with a permissible value of percentstray load loss fixes the winding radially. It is assumed thatinsulation clearances and cooling ducts are standardized,based on criteria built into the program.The value chosen for the axial copper dimension cannot

be used without adjustment. The total number of turns isdetermined by the cross section of the core and the desiredflux density. The number of turns radially can, with somereservations, only be integers. The number of disks, andthereby the axial copper dimension, can therefore assumeonly certain values. The required adjustment can be quitelarge, especially for the low-voltage windinig.

Separate tap windings usually consist of only one wireradially, with a radial dimension often determined bymechanical considerations. It is therefore assumed thatthey can be designed, if they are present, without any needfor choice of independent variables.To sum up, the independent variables are:

1) core diameter2) core height3) flux density4) axial copper dimension for lov-voltage winding5) temperature rise oil-copper for low-voltage winding6) axial copper dimension for high-voltage winding7) temperature rise oil-copper for high-voltage winding.

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The synthesis program has the performance programincluded in it, and produces an output sheet exactly likethe one described for this program. The advantage in usingthe synthesis program is that it is no longer necessary forthe designer to specify all the dimensions and windingdata. Instead he is only required to choose values for theseven independent variables, and the program makes therest of the design for him. However, the synthesis programis only considered a stepping stone on the way to the com-plete optimizing program, and does not remain an in-dependent program after this is made.

TRANSFORMER OPTIMIZING PROGRAMThe design that is produced by the synthesis program is

one that not necessarily meets all the performance require-ments. The windings were designed to the required tem-perature rise, but there may be other requirements as well,such as minimum and maximum limits for the short-circuitreactance and a maximum limit for the impulse voltagestress. These values depend on the original choice of in-dependent variables. Therefore the designer might have torun the synthesis program several times, each time chang-ing his input data, before the design is satisfactory as faras performance is concerned.However, there will always be different combinations of

the independent variables which meet the same require-ments. The designer must try to come as close as possible tothe one combination that does it at the lowest cost. Evenwith a synthesis program such as the one described here,this could be a very laborious task.The obvious next step is to have such a search done

automatically. The time it takes to synthesize a completedesign, including calculation of cost and of the most im-portant performance characteristics, is only about 0.15seconds. Scanning several hundred alternatives is thereforequite feasible.The first task is to define which design represents the

optimum. For this a criterion function is chosen:

Y = cost of transformer + capitalized cost of losses +penalty for not meeting performance requirements.

Figure 2 shows how the penalty is arrived at for designshaving a short-circuit reactance outside the permissiblerange. It consists of a small fixed percentage of the totalcost, plus a percentage proportional to the difference be-tween the actual and the permissible reactances. Similarpenalties are applied to designs which do not meet otherperformance requirements.The optimum design is the one with the minimum value

of the criterion function. For this design the penalty willbe zero, if at all possible. This means that all the perform-ance requirements are met. Furthermore it will have theminimum cost.The only remaining problem is to make a routine which

within limits for each of the variables finds a combinationthat for practical purposes gives the lowest possible valueof Y. Such a routine will be described in the sectionwhich follows.

°/o penalty

Fig. 2. Cost penalty.

OPTIMIZING ROUTINE MONICAAs the name suggests, Monica is a Monte Carlo routine,

but it also has features similar to those of the gradientmethod.The gradient method uses the partial derivatives of the

function with respect to the different variables to establishthe most favorable direction in which to move (directionof steepest descent). The method was rejected mainlybecause it requires a continuous function. Therefore, dur-ing the design synthesis it would not have been permissibleto round off dimensions, numbers of turns and so on, andhave each alternative represent a practical design. Theadjustments would have to be done only for the final designrepresenting the optimum, and might have to be drasticenough to make the whole optimization an illusion.A simple Monte Carlo approach would be to try out a

large number of alternatives at random, with equal prob-ability for each of the variables to assume values at anypoint within its permissible range. With a large enoughnumber of alternatives the one with the lowest value of Ycould be said to represent the optimum. The main objec-tion to this method is that the search is too haphazard andtherefore too wasteful of computer time. However, it couldbe used to find a starting point for a more concentratedsearch.Monica starts the search at a point which could be

found by the method just described. It could also be anyother point within the permissible range, for example, theaverage of minimum and maximum values for all thevariables. The value of Y for this first alternative can becalled Y1.To find a new combination of variables for a second

alternative, the routine proceeds as follows. Each variableis assigned a value which can be called a unit vector. Forthe core diameter for example it could be 20 mm. By meansof random numbers with a flat distribution between -1and + 1, the new value of core diameter is then arbitrarilychosen within the range + 20 mm from the starting point.Similarly the unit vector for the core height could be chosenas 100 mm. The new core height will then assume a valuewith equal probability anywhere within the range i 100mm from the starting point. All the seven variables areassigned new values in this way before a second alternativeis tried. Of course the routine will have to make sure thatthe specified minimum and maximum limits for the vari-ables are not exceeded, and make adjustments if necessary.

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Also the routine makes sure that at least one of thevariables is changed a full unit vector, so that the secondalternative is not too close to the first.Now two possibilities exist. The value Y2 of the func-

tion for the second alternative is either less than Y1, or itis not less than Y1. With Y2 < Y1, the second alternativecan be called a successful step and becomes the new start-ing point. With Y2 > Y1, chances are that with differentsigns for all the changes in the variables, the change in Ywould also be of a different sign. As an example, assumethat the diameter was changed +10 mm and the height-50 mm between the first and the second alternative, andthat this resulted in Y2 > Y1. For the third alternativeMonica would then try to change the diameter -10 mmand the height +50 mm, using the first alternative as thestarting point, and with a good probability this wouldresult in Y3 < Y1.As long as successful steps are the result, the changes in

the variables remain constant. The values -10 mm for thediameter and + 50 mm for the height would therefore betried again as many times as the value of Y decreases.This, of course, can only be done a few times. Sooner orlater the minimum in this direction will be reached.M\onica will then again resort to random numbers to try adifferent direction. If it is successful, it continues. If not,it changes to the opposite direction and tries there.

In this initial stage of the optimization the unit vectorsare fairly large. After a while it becomes more difficult tofind lower values of Y. Maybe from a certain point 42different directions are all unsuccessful, corresponding to21 sets of random numbers (three times the number ofvariables). The routine still does not give up, however,but starts the final stage of the optimization, reducing thevalues of the unit vectors. For the core diameter the unitvector could be reduced from 20 mm to 10 mm, for thecore height from 100 mm to 30 mm, and so on. This willprobably result in more successful steps. Only when again42 different directions are all unsuccessful, the optimizationis considered to be completed.There are two reasons for dividing up the optimization in

two stages. Having larger initial vectors moves theoptimization more quickly to the vicinity of the optimum,where a finer search can then be made. It also reduces theprobability of getting caught in a local minimum, since thesearch initially is done in a fairly large area around thestarting point. Especially since the function is discon-tinuous, there is always a danger that this might happen,unless care is taken to avoid it. As an added precaution,more than one starting point can be used.

OPTIMUM DESIGN oF HYDROELECTRIC GENERATORSMonica is a general purpose optimizing routine and can

be used virtually unchanged in optimizing programs forany type of electrical machine. The performance calcula-tions are usually straightforward, especially if conventionalmethods are considered to be satisfactory. The only prob-lem is to decide on a set of independent variables and tomake the synthesis program.

The synthesis of a rotating machine is perhaps moredifficult than that of a transformer, but is greatly simpli-fied by the expediency of applying a cost penalty whencertain performance requirements and other limitations arenot met. In the description of the transformer program is anexplanation of how the computer was led to believe that adesign was expensive when in fact it did not meet one ormore of the performance requirements. The optimizingroutine then automatically took corrective action, withoutthis action having to be built specifically into the program.For the routine to know which way to go, it is onlynecessary to have less penalty the closer to the permissiblerange the characteristics are.An outline of a synthesis program for hydroelectric

generators is given briefly in the following, so that bymeans of another example the general approach may bebrought out more clearly. Figure 3 shows the basic partsthat will be referred to.As independent variables the following may be chosen:

Stator variables

1) inside diameter2) slot width3) ratio slot width over minimum tooth width4) ampere-turns per centimeter loading5) watts per cm2 surface loading of the winding6) flux density in the teeth7) flux density in the yoke

Rotor variables

8) flux density in the pole base9) per unit thickness of the field coil10) watts per cm2 surface loading of the field coil.

The synthesis starts with establishing the proper numberof parallel circuits in the stator winding, based on variables2, 3, and 4. The choice depends on whether the manu-facturer prefers a single turn bar winding or a multiturncoil winding. Only certain numbers are permissibie de-pending on the number of poles.An approximate number of stator slots can be found on

the basis of variables 1, 2, and 3. This also has to be ad-justed, due to the requirements of winding balance and thedivision of the core into segments. The ratio slot width overminimum tooth width will therefore come out somewhatdifferent from the desired value. This will also be true forthe ampere-turns per centimeter loading, especially if themachine voltage is specified by the customer. However,the inside diameter and the slot width remain unchanged.The next step is to establish the cross section of the slot

including the depth, arrangement and dimensions of thestrands, and insulation allowances. The insulation is givenby the voltage. The division of the conductor into parallelstrands is determined largely by the permissible eddy cur-rent loss. The slot depth is found by trial and error until itgives the desired value of watts per cm2 surface loading.The surface can be taken as the circumference of the slotbelow the wedge times unit length.

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Fig. 3. Hydroelectric generator.

The stator synthesis is completed by finding the lengthbased on the desired flux density in the teeth, and the out-side diameter based on the desired flux density in the yoke.This is done on the basis of an assumed waveshape offlux density in the air gap.

Initially the minimum length of the air gap must beguessed at on the basis of the desired short-circuit ratio.The pole chord or pole arc may be taken as a fixed percent-age of the pole pitch. The surface of the pole tip towardsthe air gap can be formed as part of a cylinder, with a

fixed ratio maximum to minimum air gap, or it can be of a

different shape depending on the preference of the manu-

facturer. The minimum depth of the pole tip can at thi@stage also be taken as a constant, and the amortisseurwinding can be designed with a fixed bar diameter and a

spacing depending on the stator slot pitch. This, then,completely describes the pole tip.The width of the pole body is primarily determined by

the desired flux density in the pole base, and the depthprimarily by the thickness and the watts per cm2 surfaceloading of the field coil. However, the two are also inter-related, and trial and error will be required to arrive at thefinal dimensions.

It may be conveniient to have the thickness of the fieldcoil specified in a per unit system, so that the value one

corresponds to the surface of the coil, being flush with thepole tip.At this stage it will be required to check the short-circuit

ratio and if necessary make corrections to the air gap, eachtime repeating the complete synthesis of the pole and thefield coil.The rotor synthesis is completed by finding the depth of

the rim from the required inertia and the permissible stressat overspeed.Now all the active parts of the machine have been

determined. Some of the performance requirements were

taken care of during the design synthesis, such as short-circuit ratio and rotor inertia. Others depend on the originalchoice of variables, and are taken care of by applying a costpenalty. This can be done for the transient reactance andfor the temperature rises in stator and rotor. The latter areto a large extent determined by the original values ofsurface loading for the two windings, but other factors arealso of importance.

Cost penalties can also be used to correct undesirablefeatures of the design, such as the field coils being tooclose together between the poles.For practical reasons the program only checks one com-

bination of punching materials in stator, poles, and rim.In cases of doubt, the program will have to be run severaltimes with different combinations.

CONCLUSION

The transformer program has been in use since Septem-ber 1965, and the soundness of the general approach hasbeen proved by experience.The method is not only effective, but also simple to pro-

gram. Designs which do not meet certain requirementssimply receive a cost penalty, without the corrective actionhaving to be programmed specifically. This is taken careof by the optimizing routine, which systematically changesthe independent variables toward lower and lower cost, atthe same time forcing the penalty to become zero.The optimizing routine is also made with simplicity in

mind. By means of random numbers it checks a new designin an arbitrary direction from the starting point. If thecost is lower, it moves in that direction. If the cost is higher,it tries the opposite direction. Unlike the gradient methodit does not require a continuous function, so all the alter-natives can be rounded off to practical designs.

ACKNOWLEDGMENT

The author gratefully acknowledges permission to pub-lish this paper from A/S National Industri, Drammen,Norway, which cooperated in the transformer programs.

BIBLIOGRAPHY[1] Godwin, G. L., "Optimum machine design by digital computer,"

AIEE Trans. (Power Apparatus and Systems), vol. 78, pp. 478-488, August 1959.

[2] Herzog, G. W., 0. W. Andersen, J. Scrimgeour, and W. S. Chow,"The application of digital computers to rotating-machine de-sign," AIEE Trans. (Power Apparatus and Systems), vol. 78, pp.814-819, October 1959.

[3] Inkinen, L., "Die optimale Transformatorenberechnung aufDigitalrechenmaschinen," Electrotech u Maschinenbau, Heft19/20, pp. 464-469, October 1963.

[4] MacKinnon, L. J., "Power transformer design and estimatecost program with an IBM 650 digital computer," AIEETrans. (Power Apparatus and Systems), vol. 77, pp. 1262-1266,February 1958.

[51 Sharpley, W. A., and J. V. Oldfield, "The digital computerapplied to the design of large power transformers," Proc. IEE(London), vol. 105, pt. A, pp. 112-125, 1958.

[6] Williams, S. B., P. A. Abetti, and E. F. Magnusson, "Applicationof digital computers to transformer design," AIEE Trans. (PowerApparatus and Systems), vol. 75, pp. 728-735, August 1956.

[7] Williams, S. B., P. A. Abetti, and H. J. Mason, "Completedesign of power transformers with a large-size digital com-puter," AIEE Trans. (Power Apparatus and Systems), vol. 77,pp. 1282-1291, February 1958.

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