A new algorithm for a nonlinearmulticriteria decision-making problem

Prof. P.V. Rao, T.C. Kong, and Prof. V.V. Sastry

Indexing terms: Algorithms, Transformers, Optimisation

Abstract: A multiple-objective function model, formed according to the goal programming techniques ratherthan as a single-objective function model, and subject to a number of constraints, is more realistic for allengineering problems. A new algorithm using the random search technique is presented for obtaining a near-global optimum solution to multiple-objective nonlinear models. The application of these tools is demonstratedfor the optimal design of a 63 kVA distribution transformer.

List of symbols

a = achievement function (vector)bt = bounds on F((x)Fj{x) = constraint equation expressed as a function of

the decision variables associated with G,G, = ith goalJ = total number of decision variablesK = total number of goals, GL, = lower bound on X{

M = number of constraintsMAX AC = maximum number of iterations for one attempt

of searchMTT = maximum number of attempts of searchh = group of negative deviation variablesn, = negative deviation from b{

p = group of positive deviation variablesPi = positive deviation from b{

Ri = pseudorandom number for generation ofnumerical values for x,

5/ = total number of points selected in the complexsearch algorithm

L/r = upper bound on xt-Xt = ith decision variable for a problemx_ = vector of xXB = best point at every stage of the complex searchXCG = centroid calculated from every stage of the

complex searchXw = worst point at every stage of the complex

searchXWN = reflected point from Xw

a = over-reflection(/expansion) coefficient

1 Introduction

The transformer is the most widely used electrical appar-atus of the static type. All the power that is generated in anelectrical form has to be transformed from one voltage/current level to another at various stages. These can beclassified under the following three categories:

(i) power transformers(ii) distribution transformers(iii) instrument transformers.

The application of digital computers to the routine designof transformers and other electrical apparatus has been

Paper 3796C (P7, P9), first received 10th August 1984 and in revised form 3rdJanuary 1985

Prof. Rao and Mr. Kong are with the Department of Electrical Engineering,National University of Singapore, Kent Ridge, Singapore. Dr. Sastry is Professor ofElectrical Engineering, Indian Institute of Technology, Madras, India

described in detail in References 1 and 2. In the earlystages of computerisation of the designs of electricalequipment, the designer's experience had to be judiciouslyused with the incredibly high speed and accuracy of thecomputer, in arriving at the best design. In the present-dayapproach, a reliable mathematical model for the design ofthe electrical apparatus can be developed, and the designproblem can be treated as an exercise in extremism of non-linear inequality and equality constraints. Recently,attempts have been made to utilise the sequential uncon-strained minimisation technique (SUMT) developed in1969 [3, 4] for the design optimisation of inductionmachines and transformers [5, 6, 7, 8].

The principal object of this paper is to outline the pro-cedure for formulating the design optimisation of a powertransformer according to the goal programming approach(GPA) [9]. This approach remains substantially the samefor the other types of transformers mentioned earlier. Fur-thermore, a new algorithm is proposed using the random-search technique with the structure of formulationconforming to goal programming.

2 Nonlinear goal programming: problem definition

Linear programming (LP) has found wide usage as a toolfor the optimisation and analysis of linear systems.However, in many decision problems involving conflictingand incommensurable goals, linear programming wouldnot result in a feasible solution. Such limitations encoun-tered in the usage of linear programming have been resolv-ed by an important technique called goal programming(GP). Formulating a problem according to GP is similarto that of LP. The first step is to define the decision vari-ables; then all the managerial goals must be specified andranked in the order of priority. Even though managementmay not be able to relate the various goals on a cardinalscale, it can usually provide an ordinal ranking to each ofits goals or objectives. Thus a fundamental distinction ofgoal programming is that it provides for the solution ofdecision problems having multiple conflicting and incom-mensurable goals arranged according to management'spriority structure. That is, goals that are ranked in order ofpriority by the decision maker are satisfied sequentially bythe solution algorithm. Lower priority goals are con-sidered only after higher priority goals have been satisfied.Obviously, it is not always possible to achieve every goalto the extent desired by the management.

Mathematically, the problem is to find

3c = (xl5 x2, . . . , Xj) so as to minimise

« = {9i{n, P) •• g2(n, P) • • • • > 9K(n, P)}

118 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 3, MAY 1985

such that

G, = Ft(x) + nt - Pi = bi i=l,2,...,M

In other words, according to goal programming, a set of Kgoals (or desired values) is specified for the K objectivefunctions and the problem is one of minimising the devi-ations of the implicit objective functions from the specifiedgoals subject to satisfying the M constraints. Often the Kobjectives are assigned pre-emptive priority levels and theminimisation of the differences between the goals andobjective functions is done in sequential steps. In step 1 thedifference between goals and objective function values cor-responding to priority level 1 are minimised. In step 2, thedifference between goals and objective-function values cor-responding to priority level 2 are minimised, subject to therestriction that the results for the priority level 1 objectivefunction achieved in step 1 are not reduced. The resultsachieved for the level 2 objective function are then addedto the constraints set, and the method continues in thisway until all the priority levels have been dealt with.

One of the methods of solving a nonlinear goal pro-gramming problem is by repeatedly approximating a non-linear problem by a linear problem, which in turn is solvedusing the modified simplex method. This is the approachdescribed by Ignizio [9]. Ignizio, however, points out thedrawback of this approach and proposes another methodwhich does not depend on linear approximation; instead, amodified Hooke and Jeeves pattern search is used to solvethe nonlinear goal programming problem directly. Theproblem is not solved in steps, separately for each prioritylevel, instead the priority levels are used in decidingwhether or not a new solution is better than the presentone in single search sequences. This technique has beenapplied to a large number of real-world problems.

The modified pattern search appears to be satisfactory;however, it has some drawbacks peculiar to the methoditself. In other words, this search depends much on theeducated guesses of the investigator for the step size ofeach variable in order to approach a global solution. Asthe number of variables increases, the guesses for their stepsizes tend to be random. With wrong values for the stepsizes, the method could diverge from the global optimumsolution.

In this paper a method is presented for determining anear global optimum solution to multiple objective, non-linear models, in a systematic way. This method involves amodification to the complex method due to Box [11] usedin nonlinear 'single' objective models so that it can be usedto solve nonlinear goal programming models. It is asequential search technique, and the procedure helps inevolving a global optimum due to the fact that the initialset of points generated for the search are randomly scat-tered throughout the feasible region. To provide the back-ground necessary to appreciate the strategy of the modifiedcomplex method, a brief description of the modifiedcomplex method is given in the following Section.

3 Modified complex algorithm

The incorporation of the goal programming concept intothe basic approach of the complex method has led to thedevelopment of a new algorithm for solving the nonlineargoal programming models. The steps involved in this algo-rithm are described in the following:Step 1: A given problem is formulated as a goal program-

ming model.Step 2: An original complex of SI (^NX + 1) points is

generated, consisting of a starting point and(SI — 1) additional points generated by randomnumbers and bounds for each of the independentvariables such that

where R{ is a set of random numbers uniformlydistributed over the interval 0 to 1, and L, and Ut

are the lower and upper bounds for the Xt.This relation will ensure that the (SI — 1) points

so generated will satisfy the lower and upperbounds of the ith decision variable. However, if agenerated point is found to violate any of theimplicit constraints, that point is discarded and anew point is generated using eqn. 1. This process isrepeated as often as necessary until all the implicitconstraints are satisfied by (SI — 1) points.

Step 3: The achievement function a is evaluated at eachpoint. In goal programming, a is an ordered vectorof a dimension equal to the number of pre-emptivepriorities K within the problem, and it expressesthe level of achievement of each objective setwithin a given priority. The notion of pre-emptivepriorities means that the first group of objectives(or the first element) in a (e.g. gy(n, p)) is preferredto the second group term g2(n, p) and so on.

As we would like to minimise a, the point Xw atwhich the function A(XW) assumes the worst valueis reflected by computing a new point such that

XwN = a(XCG — Xw) + XCG (2)

where XCG is the centroid of the remainingproblem points given by

1SI

(3)

Box recommends an over-relaxation coefficienta ( > l ) = 1 . 3 to compensate for the shrinkingcomplex, which is caused by the move towards thecentroid.

Step 4: If A(XWN) is found to be definitely better thanA(XW) and point XWN is also in the feasible zone, itwill replace Xw and return to step 3. Whereas ifA(XWN) is not superior to A(XW\ the over-relaxation coefficient is reduced to half the originalvalue and a new XWN is computed using eqn. 2.This is repeated, when each XWN does not contrib-ute to a better achievement function until a is lessthan 0.00001, and then step 6 will be executed. It isin this sense that the approach is sequential andnot based on any guess work.

Step 5: So long as a is not less than 0.00001, steps 3 and 4are repeated until the maximum number of iter-ations (MAXAC) has been reached. Thereafter oneproceeds to step 6.

Step 6: At this step, all the 'complex' points are discardedexcept XB at which the achievement function Aassumes the best value. Calculation of XB is con-sidered to be the completion of one attempt ofsearch. To further improve the value of A, overA(XB), the next attempt will be taken by returningto step 2, at which instance XB will be used as afeasible starting point for 'the new complex gener-ated by means of eqn. 1.

Step 7: The process is terminated when a specified numberof sequential attemptsjsay, MTT) gives the sameachievement function A(XB).

IEE PROCEEDINGS, Vol. 132, Pt. C, No. 3, MAY 1985 119

All these steps are illustrated in the form of a flow chart asshown in Fig. 1. The effectiveness of this new algorithm

pick starting point(feasible)

generate point incomplex of SI points

discardthe point

violation^-^cneck F

evaluate achievement _function at each point (A)

compare A and determinewith worst A )

Xg is startingpoint

compute XWNumber of

iterations=MAXAC ?

yes

Fig. 1 Algorithm flow chart

was tested in relation to a number of engineering problemsof which the power-transformer design optimisation is atypical case.

4 Design optimisation of a distributiontransformer

For the optimal design of a distribution transformer, thefollowing decision variables: xl5 x2, x3, x4 and x5 are tobe determined:

(i) core diameter = xx(ii) secondary turns = x2(iii) primary current density = x3(iv) secondary current density = x4(v) number of primary layers = x5.

The decision variables xx and x2 control the magneticloading of the transformer, while x3 and x4 control theelectric loading. Consequently, the cost of iron is con-trolled by the former variables and the latter variablescontrol the cost of aluminium (or copper) required for thetransformer. The decision variable x5 affects the reactance,and consequently the regulation, of the transformer. Theother design parameters can be either treated as constantsor expressed in terms of the above quantities for a givenconfiguration of the transformer.

The implicit constraint functions for any problem areusually national standards of a country and are

(a) no-load losses (iron losses)(b) full-load losses (copper losses)(c) percentage impedance.

120

For a 63 kVA distribution transformer, the no-load andfull-load losses are limited to 200 W and 1300 W, respec-tively, and the percentage impedance is constrained to 4.5.

The multiple objective functions according to their pre-emptive priority levels are

% reactance 1Flux density 2Copper losses 3Iron losses 4Boundary limits on the core diameter 5Boundary limits on the secondary

winding turns 6Boundary limits on the primary

current density 7Boundary limits on the secondary

current density 8Boundary limits on the number of

primary layers 9Cost of the transformer 10

Such a formulation of a 63 kVA distribution transformerin respect of the constraints and the achievement functionis given in Appendix 8.

4.1 Optimisation resultsThe results derived, based on the pattern search accordingto goal programming approach for discrete step sizes aswell as for different initial vectors, are presented in Table 1.From this one observes the sensitivity of the search tech-nique for the step size. In addition, it is normally difficultto arrive at a near-global optimum. Out of the six startingpoints only two yielded acceptable solutions.

The results obtained from the modified complex algo-rithm, as presented in this paper, are shown in Table 2 forthe same initial vectors. A glance at the optimal resultsconclusively proves the superiority of this approach overthe earlier method. In every case, one obtains a near-globaloptimum solution with violation of the constraints.

Yet another interesting feature of the goal programmingapproach is Pareto optimality and the information thedecision maker (DM) gets from it. Such data relating tothe first starting vector (see Table 2) are presented in Table3 from which one can observe the following aspects:

(i) If the limit on the iron loss of the transformer isincreased with a corresponding decrease in the copper loss,a substantial saving in the capital cost is achieved.

(ii) If the limit on the full load losses is decreased withthe percentage impedance remaining the same, the capitalcost will be slightly higher than the optimal value. Thistype of design can be identified under the energy efficientcategory.

5 Conclusions

The feasibility of the goal programming approach to anonlinear model such as the design of a power transformerusing a modified complex algorithm as well as a patternsearch is discussed. The superiority of the random searchin deriving a global optimum of a power-transformerdesign problem is fully established. A large number ofPareto-optimal points are automatically generated beforearriving at a global optimum. Finally, the effectiveness ofthis algorithm in solving a number of other problems inthe domain of nonlinear digital control systems has alsobeen checked and the results are reported elsewhere.

IEE PROCEEDINGS, Vol. 132, Pt. C, No. 3, MAY 1985

Table 1: Optimisation data derived using pattern search for a 63 kVA distribution transformer

a Core diameter, mm (x,)Secondary turns (x2)Primary current density. A/mm2 (x3)Secondary current density. A/mm2 (x4)Number of primary layers (x5)Flux density, Wb/m2

Copper loss, WIron loss, W% reactanceTotal cost, $

b Core diameter, mm (x,)Secondary turns (x2)Primary current density. A/mm2 (x3)Secondary current density. A/mm2 (x4)Number of primary layers (x5)Flux density, Wb/m2

Copper loss, WIron loss, W% reactanceTotal cost, $

I

10484

1.701.3

20.01.97

1221.6274

4.92779

104100

1.31.5

15.01.66

1181.4224

2.95947

F

117.883.61

1.5381.691

20.01.56

1299.38200.5*

4.50915.32

104.4101

1.581.06

16.481.62

1300.1*200

4.5976

I

100105

1.21.2

161.71

1096231

3.551000

11290

1.31.5

171.59

1146211

3.45977

F

99.82109.63

1.5081.02

16.031.64

1299.8199.8

4.49998

111.790.35

1.6091.35

18.251.59

1300200

4.49923

I

10893

1.31.7

181.65

1219.1217.8

3.68910

10893

1.31.7

181.65

1219217.8

3.68910

F

1009717

1.5941.315

181.8

1300238*

4.49836

109.9192.35

1.4441.518

19.0171.61

1300200

4.51905.5

* Secondary; I = initial values; F = final valuesa Stepsize: Ax, = 5.0, Ax2 = 0.5, Ax3 = 0.1, Ax4 = 0.1, Ax5 = 2.0fcStepsizeiAx, =0.5, Ax2 = 0.5, Ax3 = 0.1, Ax4 = 0.1, Ax5 = 0.5

Table 2: Optimisation results based on modified complex algorithm for a 63 kVA transformer

Core diameter, mm (x,)Secondary turns (x2)Primary current density. A/mm2 (x3)Secondary current density. A/mm2 (x4)Number of primary layers (x5)Flux densityCopper lossIron loss% reactanceTotal cost

Core diameter (x,)Secondary twins (x2)Primary current density (x3)Secondary current density (x4)Number of primary layers (x5)Flux densityCopper lossesIron loss% reactanceTotal cost

I

10484

1.71.3

201.97

1221.61274

4.92779

11290

1.31.5

171.59

1146211.9

3.45977

F

108.893.87

1.4491.564

19.141.61

1299.9199.9

4.49902

109.892.4

1.4181.659

19.611.61

1299200

4.50900

I

100105

1.21.2

161.71

1096231

3.551000

8080

0.50.5

10—————

F

109.492.99

1.401.676

19.671.61

1300200

4.50900

109.992.29

1.4131.671

19.651.61

1300200

4.49901

I

10893

1.31.7

181.65

1219217.8

3.68910

F

108.194.77

1.3831.662

19.551.62

1299.8200

4.49900

I = initial values; F = final values

Table 3: Pareto-optimal values for a 63 kVA transformer

S/number Total cost Iron loss Copper loss % impedance Remarks

859.94881.29894.95902.38915.77920.36922.59927.45

235.48221.97223.79199.97205.46198.93199.79208.33

1239.611266.311239.021299.991278.361292.431289.601259.93

4.50)4.49 \4.52 J4.494.49-)4.51 (4.50 (4.49 J

Savings in capital cost if thebounds on the specification arechangedOptimal design for the given specification

Energy efficient designs canbe derived

Initial vector for this data: x , =104, x2 = 84, x3 = 1.7, x4 = 1.3, x5 = 20Note: In the domain of multicriteria, decision making pareto-optimal solutions are defined as acceptablesolutions in which there is improvement in at least one of the goals possibly at the expense of one or moreof the other goals. With reference to solution 4 given in the above table, each of the pareto-optimal solu-tions shows an improvement in one of the goals.

IEE PROCEEDINGS, Vol. 132, Pt. C, No. 3, MAY 1985 121

6 Acknowledgments

The authors wish to thank Prof. S.C. Choo, Head of theEE Department, National University of Singapore, Singa-pore for the facilities provided. V.V. Sastry wishes toacknowledge, in addition, encouragement received fromProf. P.V. Indiresan, Director, Indian Institute of Tech-nology, Madras, India.

7 References

1 OLDFIELD, J.V., McDONALD, D., and HUMPHREY DA VIES,M.W.: 'Transformers design with digital computers', Proc. IEE, 1956,103B, (Suppls. 1-3), pp. 54-58

2 SHARPLY, W.A, and OLDFIELD, J.V.: 'The digital computerapplied to the design of large power transformers', Proc. IEE, 1958,105A, pp. 112-125

3 FIACCO, A.V., and McCORMICK, G.P.: 'Nonlinear programming:sequential unconstrained minimisation technique (SUMT)' (JohnWiley, New York, 1969)

4 BRACKEN, J., and McCORMICK, G.P.: 'Selected applications ofnon-linear programming' (John Wiley, New York, 1969)

5 RAMARATNAM, R, and DESAI, B.G.: 'Optimisation of polyphaseinduction motor design: a non-linear programming approach', IEEETrans., 1971, PAS-90, p. 570

6 RAMARATNAM, R., DESAI, B.G., and SUBBA RAO, V.: 'Compa-rative study of minimisation techniques for optimisation of inductionmotor design'. T73, pp. 116-1 presented at IEEE Winter PowerMeeting, New York, 1973

7 APPELBAUM, J.: 'Optimisation of induction motor design', ETZ-A,September 1974

8 SRIDHARA RAO, G., SASTRY, V.V. and RAO, P.V.: 'Comparisonof non-linear programming techniques for the optimal design oftransformers', Proc. IEE, 1979, 104, (12), p. 1225

9 IGNIZIO, P.: 'Goal programming and extensions' (Lexington Books,Lexington, 1976)

10 RAO, P.V, SASTRY, V.V., SRIDHARA RAO, G., and SASIDHARARAO, P.: 'Computer aided optimal designs for power transformers'.Technical report prepared for M/s. Transformers and Switch GearLtd, Madras, February 1981

11 BOX, M.J.: 'A new method of constrained optimisation and a com-parison with other methods', Computer J., 1965,8, pp. 42-52

12 KONG, T.C.: 'Non-linear goal programming by the modifiedcomplex algorithm, and its applications'. B.Eng. Project report, 1983/84, National University of Singapore, Singapore

13 VILAS WUWONGSI, SHIGENOBU KOBAYASHI, SHIN-ICHII WAI and ATSUNOBU ICHIKAWA: 'Optimal design of linearcontrol systems by an interactive optimisation method', Comput. lnd.,1983, 4, pp. 371-394

Appendix: Problem formulation for a distributiontransformer

p

Is

Vs

TB =

= primary-phase current= secondary-phase current= secondary-phase voltage= primary/secondary turns ratio

K x 4.0 x 106

= flux density =3.14159 x Kw x 4.44 x / x x2 x x\

Kw = coreflll factorPru = primary radial length = Ip/x3 + Ax

V, = volts/turnW^! = window leg lengthWsc = secondary conductor width7 .̂ = secondary conductor thicknessSru = radial depth of the secondary winding

Dms = mean diameter of the secondary windingDHl = diameter at the centre of separation between

primary and secondary windingsDmp = mean diameter of the primary windingGps = gap between the primary and secondary windingsMd = centre mean distanceX = % reactance

A 2 x x2 x Is x DGPS Pr

Hi 10 30

W

Ws

LcuWF

s

p = weight of the primary winding= 3.0 x 2.7* x nx2 x DmpAcp T x 10"4

= weight of the secondary winding= 3.0 x 2.7* x n x x2 x Dms x A c s x 10~ 4

= I2R loss = At(Wp x x\ + Ws x xl)= weight of iron

= ^ x A5 x 7.7 x xi(3Wlt + 4DH1 + 2xJ

Lfe = iron loss= (bo + biBf+b2 x B2f+---+bxoB

10) x Wf

F = Cost of the transformer = Wf x A6 + (Wp +

* 1

* 1x2

x2

x3

x9

x9X5x5

BB

+ nl+ n2

+ n3

+ «4+ n5

+ n6

+ n7

+ n8

+ n9

+ nu

+• " n

-r-»i2x + w13x + n14

- P l

-Pl— p 3

-P*- P s- P e-Pi

-Ps- P 9) — Pio- p t l

- Pl2- P l 3- Pl4

= 120= 80= 120= 80

2.01.22.01.22.01.21.851.004.514.485

Lcu+ nl5 -pls =1300he + "i6-Pi6= 200F + "IT - Pn = 400

Achievement function

a=minimise(Pi + ni\

{(p13

(P3 +n14),), (p5 6), 12

(p7

p1 5 p 1 6

(p9 +

Note:(i) Au A2, ..., A-, are constants which vary depending

upon the type of transformer and the per-unit costs ofmaterials.

(ii) b0, blt . . . , bl0 are the coefficients of a polynomialcurve fit for the iron loss variation in respect of fluxdensity for a chosen lamination material.

(iii) Aluminium coils are used in this design.(iv) Only the cost of the transformer is considered in this

case, although for larger sizes one could choose F as thecapitalised cost to include the cost of losses.

122 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 3, MAY 1985