An Optimization Technique for Real And

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PR OC EEDINGS OF THE IEEE, VOL. 55, NO. 11, NOVEM BER 1967877L. E. Elsgolc, Calculus of Variations. Reading, Mass.: Addison-Wesley, 1962.

[91 H. Goldstein, Classical Mechanics. Reading, Mass.:Addison-Wesley, 1950.[ l o ] R . E.Ka lm an, The theory of optimal control and the calculUS Ofvariations, R esearch Inst. for Advanced Studies, Baltimore, Md, RIASRept. G1-3, 1961.[ l R. E. Bellman, I. Glicksberg, and 0. A. Gross, On he bang-bangcontrolproblem, Quart .Appl .Math., vol. 14, pp. 11-18, 1956. Also,Numerical techniques in optimization, IEEE Internatl Conu. Digest,pp. 26-31, 1967.[ I 2 1 M. Athens and P. L.Falb, Optimal Control. New York: McGraw-Hill, 1966.[ l a ] L. S. Pontryagin, V. G. Boltyanski, R. V. Gamkrelidge, an d E. F.Mischenko, The Mathematical Theory of Optimal Processes. New York:Interscience, 1962.[141 R. A. Rohrer, Synthesis of arbitrarily tapered lossy transmissionlines, presented at Symp. on Generalized Networks, Polytechnic Insti-tute of Brooklyn, April 1966.[151G. Leitmann, Optimization Techniques. New York: AcademicPress, 1962.El6] A. E. Bryson and W. Denham, A steepest-ascent method forsolving optimum programming problems, J . Appl. Mech., Trans. ASME ,vol. 84 , ser. E, pp. 247-257, June 1962.

[ 1 7 ] N . L. Wong, Switching circuit optimization, Masters thesis,Dept. of Elec.Engrg., University of California, Berkeley, May 1967.[ l E 1G . D. Hach tel and R. A. Willoughby , On the computational useof the acobian in circuit analysis, presented at Asilomar Conf. onCircuit and System Theory, Asilomar, Calif, November 1967.[ l g l N. Sato and W. F. Tinney, Techniques for exploiting the sparsityof the network admittance matrix, IEEE Trans. Power and Apparatus,vol. 82 , pp. 944 950 , December 1963.W. F. Tinney and J. W. Walker, Direct solutions of sparse networkequation s by optimally ordered riangular factorization, this issue,p. 1801.[201P. D. Crout, A short method for evaluating determinants andsolving systems of linear equations with real or complex coefficients,Trans. AIEE , vol. 6 0 , pp. 1231-1241,1941.F. Gustavson, W. Liniger, and R. W illoughby, Symbolic genera-tion at an optimal Crout algorithm for sparse systems of linear algebraicequations, to be published.[211 E. . Kuh and R. A. Rohrer, The state-variable approach to net-work analysis, Proc. IEEE, vol. 53, pp. 672-686, July 1965.L 2 * ] C. G . Broyden, A class of methods for solving nonlinear simul-taneous equations, Math. Comput., vol. 19, pp. 577-593, Oc tobe r 1965.[231A. D. Falkoff and K. E. Iverson, The APL terminal system:Instructions for operations, IBM Thomas J. Watson Research Center,Yorktown Heights, N. Y.

An Optimization Technique for Real andReactive Power Allocation

J. F. DOPAZO, MEMBER, I=, 0 . A. KLITIN, G. W. STAGG, SENIOR MEMBER, IEEE,AN D M . WATSONA b s t r a c r - - A n ~ s c h e d a l e f o r r e P l p o w e r g e w f i t i o l l b ~ b y

nL.grrPgirnmethodaudtheaUocationofreactivepowergewfitimisde-terminedbyagrrdientaret8od.Akmmterealdreactivepowerreqire-m e n t s f o r ~ s y s t e m o p e r r t i o e w c o e r p e t e d d t h e t o w p r o d r t i o aeostislniRiAizedwithintheliritrtiorPimpsedbysysteatun&rah&Re-of t loasesiatienofapreakdatedf bss olrwln.Tbecompeterprogrrmprovidesameamtodetermhethe~mmeofavailable real and r e d i v e power gmerath and to plan e um d c d y forfirtarereqnhnents.

p e a t e d s d i r t i o e s O f t h e ~ o r i t e q l n ~ u e m e d t o i a c o r p o r r t e t h e e t r ~. . . .

1INTRODUCTION

N THE EARLY approaches to the economic dispatchproblem, transmission losseswereneglected and hegenerating units were loaded within operating limits

at equal incremental costs.[ The development of largeintegrated power systems led to the transmission of largeblocks of power over longer distances, increasing the impor-tance of considering the effectsof transmission losses.Several methods have been developed to take into account

Manuscript received February 17, 1967. This paper was presented a tthe 1967 Power Industry Com puter Ap plications Conference sponsoredby the IEEE Power Group.The auth ors are with the American Electric Power Service Corp ora-tion, New York, N. Y.

the effects of transmission losses in the solution of theeconomic dispatch problem. The most generallyusedmethod employs a precalculated loss formula which ex-presses transmission losses in terms of the power output ofthe generators.[21A loss formula is calculated for a specificconfiguration of the transmission system and a preselectedoperating condition.I3I The derivation of a loss formula isbased on the following assumptions.

1) Bus voltages remain constant in magnitude and angle.2) Individual loads remain a constant complex fraction3) The ratioof reactive to real power generation remainsof the total load.

constant.Loss formulas havebeen applied successfully to those

systems for which the performance approximates theseassumptions. A new loss formula must be calculated, how-ever, when major changes take place in a transmission sys-tem or in its operating conditions.

The method presented in this paper does not require aloss formula. Instead, the us impedance matrix of the net-work and the results from load flow solutions are used toaccount for the ffectsof transmission losses in the economic


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1878 PROCEEDINGS OF THE IEEE, NOVEMBER 1967scheduling ofsystem generati~n.[~l*[~]he method is ap- whereplicable also to determine an economic schedule for reactivepower supply. No assumptions are necessary regarding thecharacteristics or performance of the power system. Anychange in the system configuration or in the operating con-dition can be readily taken intoaccount.

The method of solution consists of an iterative process inwhich alternate real and reactive power loadings are de-termined to minimize the total cost of fuel input to the ys-

aj ,ak are the real components of the elements of theb j , bk are the imaginary components of the elements of

rjk are the real components of the elements of the bus

current vectorthe current vectorimpedance matrix

n is the number of buses.tem. The method of Lagrangian multipliers is employed-to The complex power at bus j isobtain an economic real power schedule which satisfiesheestimated total generation requirements.[61 A gradientmethod isused for calculating reactivepower loadings where

Pj + j Q j = I ~ ~ E j ~ ( c o s, + j sin e j )which reduce transmission losses and, in turn, the genera-tion requirement^.['^*^*] The process is terminated whenthe total cost of fuel input to theystem remains unchangedwithin a specified tolerance. Real and reactive power limitsas well as constraints on system voltages are taken intoaccount during the calculations.

EQUATIONSND SOLUTIONECHNIQUETransmission LossesTo develop the equations from which the real and reac-

tive power schedules are determined, it isnecessary toderive first an expression for transmission losses in terms ofthe real and reactive power impressed at the network buses.The total transmission losses of a power system are

PL + j Q L = (I')*ZBusI (1)where

PL is the total real power lossQ L is the total reactive power lossZBUs s the bus impedance matrix

(Ir)*s the conjugate transpose of I .I is the vector of impressed bus currentsEquation (1 ) can be written

PL + Q L = ( A - B)'(R + j X ) ( A+ jB) (2)where

( A + j B ) = I( R + jx)= ZB,~ .

Then, the total real power loss from 2) isPL = A'RA + B ' X A - A'XB + B R B . (3)

Since B X A is a scalar and the reactance matrix X is sym-metric,

B X A = A'XB.Therefore, (3 ) becomes

PL = A'RA + B'RBor, using index notation,

PL =1 1 ajrjkak + b j r j k b k )j k J, k = 1,2, e . , n (4)

Pj is the real power at bus jQ j is the reactive power at bus jI j is the impressed current at bus j

lEjl is the magnitude of the voltage at bus jB j is the voltage phase angle at bus j with respect to

the reference bus voltage.The real and imaginary components of the impressed buscurrents from ( 5 ) are

1aj =- Pjcos O j + Q j sin e j )lEjl (6 )1b, = - (-PjsinBj + QjcosOj). (7 )P j l

Substituting for aj and b j from ( 6 )and (7) into (4) and letting(6 j - e,)= e j k , then

Letting

and

Real Power DispatchTo simplify notation, it will be assumed in this derivation

that only one generating unit is onnected at each generatingbus. The economic generation schedule of real power for


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DOPAZO ET A L . : OPTIMIZATIONECHNIQUE FOR POWER ALLOCATION 1879an n-bus system with G generating units can be obtainedfrom the following set of equations

subject to the constraintsn1 , . - P , = O

j = 1

Pi m i nI I, Pimar g = 1,2 , . . * , (13)where

PI, is the real power generation of unit g3 s the incremental cost of generating unit gdPI,- s the incremental loss associated with gener-pL

ating unit g2. is a Lagrangian multiplierPbm in, I,mluare the minimum and maximum generating

limits, respectively, for unit g.The incremental loss associated with generating unit g

can be obtained by taking the partial derivative of (9)withrespect to the real power at bus g.

Substituting for aPJaPI, from (14), the gth equation in (11)becomes

3 + 2~ (ag jPj+ p g j ~ j ) A.dPI, j = 1The incremental cost of unit g can be expressed as afunction of the real power generation PI,. Therefore, (15)

can be solved forPI, (g= 1 ,2 , . . .,G), within the constraintsgiven by (13),for successive values f i ntil (12) is satisfied.This solution provides the optimum real power generationschedule for a specified system configuration and reactivepower loading or a given set of bus voltages.

The steps in the procedure for obtaining an optimum ealpower generation schedule are as follows.

3) Calculate from the current load flow data the co-efficients ajkand B j k .4) Estimate system lambda and calculate the optimumreal power generation schedule.

5 ) Test the summation of generationzG=, against thedesired generation Po and repeat step 4) If the dif-ference between the actual and desired generation isnot within a specified tolerance.6) Solve the load flow to obtain new system conditions.The difference betweenhe scheduled swing enerationPs, determined from the coordination equations, andthe swing generation PsL,obtained from the load flowcalculation, is equal to the differencebetween theestimated and actual transmission losses.

7) Return to Step 2) to recompute the desired generationwith the new estimate of transmission losses.

A solution is obtained when the generation of the swingmachine from the economic dispatch and hat obtainedfrom the load flow calculation are the same within a specifiedtolerance. A flow chart for this solution technique is givenin Fig. 1.Reactive Power Dispatchactive power can be obtained from the equation

Using a gradient method the economic allocation of re-

whereQ(i),Q(i+ )are n-dimensional vectors whose elements

are the net reactive bus powers in iterationi and i + 1, respectivelyk() is a positive factorF)s an n-dimensional gradient vector whoseelements are the partial derivatives of thereal power losses with respecto the eactivepower at each busQmin,6, are n-dimensional vectors whose elementsare the minimum and maximum eactivepower limits, respectively,at each busQ is an n-dimensional vector whose elementsare the reactive power generation at eachbuslEminl,Ernlu[are n-dimensional vectors whose elementsare the minimum and maximum limits, re-spectively,of the bus voltage magnitudes(El is an n-dimensional vector whose elementsare the bus voltage magnitudes.

1) Calculate from load flow data the total received load The factor k can be determined as follows. The changes2) Estimate the transmission losses PL in order to obtain

PR for the powerystem. in reactive powers from (16) arethe total required generation PD . -AQ = Q ( i ) - Q( i + 1 ) = k(i)rp;). (19)


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1880 PROCEEDINGS OF THE IEEE,OVEMBER 1967

FFIOM LOAD PLOW DATA IETERnINE TOTAL RECEIVED LOAD

-SET ITERATION C O m

1I CALCULATE DESIRED GENERATIOW IAALCULATE F R O M LOAD FLOWATA

ISOLVE LOADLOW WITHNE U GENERATION SCHEDULE

ADVANCEITERATION COUNTi+l+i

IDETERnINE LOSSESp~i+l)FROH OAD FLOW

SOLUTION

CALCULATE COSTS AHDPRINT RESULTSFig. 1. Flow chart for real power dispatch.

Assuming that the gradients evaluated in iteration i remainconstant, the change in real power osses A P L forgivenchanges in the reactive bus powers is

APL =@VF" (20)where-Q' is the transpose of the vector whose elements are

equal to the changes in the reactive bus powers.

Then the change in losses s

Taking the partial derivatives of (10) with respect to thereactive bus powers

V P L = 2([a]Q - PIP ) . (23)Substituting from (23) into (22)

Assuming for the purpose of estimating k that the op-timum reactive schedule is obtained in iteration i + 1, thelast term of (24)can be neglected since ,manyf the elementsof the gradient vector VP(i+') are close to zero. Then (24)reduces to

or

The estimated value of k may result in violation of thereactive or voltage constraints at one or more buses duringthe iterative process. If the calculated value of QJ s outsidethe reactive limits for bus j , the minimum or maximum re-active limit is substituted accordingly. If voltage constraintsare violated, the calculation of A B is repeated with a re-duced value of k. By inspection of the busvoltages a reduc-tion factor for k can be determined. Assuming that initeration i + 1 the voltage at bus j exceeds the maximumlimit and its deviation is greater than that of any other bus,the increase in voltage at bus j from iteration i to iterationi + l is

Ap L = k ( i ) ( r p L) ( i ) r p L) . (21) andhe desired changen voltage is


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1882 PROCEEDINGSOFTHE IEEE, NOVEM BER 1967

Fig. 4. Five-bus sample power system.BUS IMPEDANCEMATRIXFORM

ic

SET ITERATION COUNTi = oDETERWINE ECONOMIC

CALCULATE YSTEM

ITERATION COUNT

DETERMINE ECONOMIC tPRINTGENERATION CHEDULE,PRODUCTION COSTS,L O A D FLOW RESULTS

REACTIVE POWER SCHEDULE

iSTOP

Fig. 3. Simplified flow chart for econo mic alloca tionof real and reactive power.

The output of the program gives the results of the finalload flow calculation, the cost for each generating unit, andthe totalsystem production cost.A general flow chart for the economic real and reactivedispatch program is given in Fig. 3.

&SULTS FROMAMPLE SYSTEMSA five-bus typical est system was usedo test the method

for scheduling real and reactive powerand to study the con-vergence characteristics of the optimization process pre-sented in this paper. The test system is hown in Fig. 4. Theimpedance data is given in Table I and system loads aregiven in Table 11. The operating limits and cost data for thethree generating units of the system are given in Table III.

The generation schedule for a base case load flow wasde-termined on the basis of equal incremental costs at terminalsof the generators. Typical voltages were specified for thevoltage regulated buses. The scheduled voltage magnitudes

TABLE 1I M PEDANCES FOR S A M P L E POWER SYSTEM

Bus Codes ~ ImpedanceI1-5 I 0.030+ j0.1032-5 0.080+ j0.2623-5 0.105 + 0.347

3-4 0.106 + 0.4032-3 1 0.033 + O.118Impedance in p e r unit on 100 MVA base.

TABLE I1LOADS FOR SAMPLEOWERYSTEM

Bus IOde ~ Megawatts ~ Megavars

Load

1 I 86 202 , 30 I 12

and the eal and reactive generation obtained from the loadflow calculation are shown in Table IV. The total enerationcost for this case was 1160.8 dollars per hour.

The economic real and reactive generation scheduleswerecalculated for the sample power system using he new com-puter program developed for the optimization method de-scribed in this paper. Minimum cost was obtained on thesixth iteration. Themagnitudes of bus voltages and the ealand reactive powers obtained for each iteration are shownin Table V. The swing generator voltage at bus 5 was heldat 1.04 per unit. The maximum allowable voltage magni-tude of each bus was set at 1.05 per unit. The minimumallowable voltage level was set at 0.95 per unit. No limita-tions were placed on the reactive generating capabilities ofthe generators or condensers. With these constraints themagnitudes of voltages at buses 1 and 4 reached the maxi-mum of 1.05 per unit during the terative solution, as shownin TableV . A comparison of the results obtained from thesixth iteration with those obtained from the base case, givenin Table IV, shows the changes in magnitudes of voltages,real generation, and reactive generation resulting from opti-mization.The gradients calculated in each iteration, for all busesexcept the swing machine bus, are shown in Fig. 5 . Thegradients for buses 2 and 3 reduce to zero since the magni-tudes of voltages at these buses remained within allowablelimits. The gradients for buses 1 and 4 did not reduce to


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DOPAZO ET A L . : OPTIMIZATION TECHNIQUE FOR POWER ALLOCATION 1883TABLE 111

OPERATINGmrrrs AN D COSTS OR GENERATORSF SAMPLE Po- SYSTEMI I Incremental Cost Characteristic

1 Minimumperating 1 Minimumperating Maximum OperatingBus Code Cost (Dolla rs/Ho ur) ~ Limit (Megawatts) ~ Limit (Megawatts) InterceptI (Dollars/Megawatthour)

1 1 240, 50 '4 80 1080 10

2001001002.453.513.89

i 1 O1 O1 o

TABLE IVSCHEDULEDus VOLTAGE AGN ITUDESND GENERATIONFROM BASE CASE OADFLOW

1.049~ 174.8 i 7.3

4 l 0.985 1 68.85 1 1.040 30.8 60.9

2 0.983 0 5.13 1 0.977 0 1 8.0- .6

Swing machine bus 5 .

TABLEVRESULTSROM TEST CALCULATION FOR SAMPLE POWER SYSTEM

Bus Voltage Magnitude (Per Unit) for Each Iteration2 1 5 6I ,

1 Real Power Generation Megawatts) for Each teration

3 1 0 054.2 54.7 55.0 55.1 55.248.0 47.1 46.6 , 46.4 i 46.3~

Reactive Power Generation (Megavars) for Each IterationBusCode , l 1 2 1 3 1 ~ 5 1 61 i 8.53.13.6 42 ~ 6.88 ~ 8 .00 ~ 8.66s I 46.43 39.96 36.44 1 33.93 i 28.44 31.793 7.22 ~ 8.50 i 9.214 - .36 I 1.01 I 2.25 1.16 I -0.61 0.19

zero because of the voltage constraints imposed at thesebuses.

The operation cost for the optimum schedules is 1 152.2dollars per hour compared to 1160.8 dollars per hour forthe base case. The changes in cost during the iterativecalculation are shown in Fig. 6.

0 2 4 6 8 10I T E R A T I O N S

0 2 4 6 8 10I T E R A T I O N S

Fig. 5. Variation of gradients for the sample power system.

Similar tests were performed with and without constraintson he sample system for various voltages at the swingmachine bus. The results of these tests are summarized inFig. 7. Curve A shows changes in operating costs for variousswing bus voltages without other voltage or reactive powerconstraints. Curve B shows the results with constraints onall bus voltages. The results in curve C include, in additionto voltage constraints, the effect of a reactive power con-straint at bus 3.In addition, real power dispatch calculations were per-

formed for the American Electric Power (AEP) System.The network comprised 176 buses, 268 lines and 25 regu-lated buses. The system generation included 36 steamgenerating units installed in 15 major power plants. Theunit incremental heat rate characteristics were representedby straight line segments with an average of five segmentsper unit.The convergence characteristic of the process for a


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1884 PROCEEDINGS OF THE IEEE.NOVEMBER 1967

SWING MACHINE BUS VOLTAGE-PER UNITFig. 7. Effects of constraints on system production cost for the samplepower system. Curve A-no constraints. Curve &voltage constraints0 . 9 5 5 lEjlI 1.05. Curve C-voltage constraints 0.95I Ejl I 1.05 an d

10I T E R A T I O N S

Fig. 6 . Variation of system produc tion cost for the sample power system. reactive power constraint at bus 3, Qmin Q = 0.

2NUMBER OF ESTIMATES OF SYSTEM LAMBDA

Fig. 8. Conv ergence cha racteristic of the method for real power dispatch of the American Electric Power System.

typical calculation for theAEP System is shown in Fig. 8.This graph shows the number of estimates calculated forsystem lambda to obtain an economic generation schedulethat satisfied the real power requirements. Three load flowcalculations were required to account for the effectoftransmission losses. The initial system lambda was esti-mated from a lambda-power curve whichwas approxi-mated by a straight line through theorigin. The initial busvoltages were set to l .O+ jO . The average number of itera-

tions in the solution of the coordination equations for agiven estimate of lambda was three. The number of itera-tions for the hree load flow solutions were 204,65, and 27,respectively. The voltage tolerances for the load flow solu-tion were O.OOO1 per unit. The tolerance for the change inthe swing machine power was 2 megawatts.

The solution required 1.5 minutes on an IBM System/360Model 50. This excludes the input and outputimeaswell asthe time to form the bus impedance matrix.


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PROCEEDI NGS OF TH EEEE, VOL. 55, NO. 11, NOVEMBER 1967885CONCLUSIONS

An optimization technique has been presented for theeconomic allocation of realand reactive power applying themethod of Lagrangian multipliers and a gradient method.The optimization procedure uses solutions of networkequations to account for the effects of transmission losses.A precalculated transmission loss formula is not required.

The computer program provides the option to obtainonly a real power schedule or both real and reactive powerallocation. The real powerdispatch obtained by the methodpresented provides the following benefits.

A solution can be obtained which reflectscurrent sys-tem operating conditions as well as any hanges in thenetwork due to ine additions or outages.The results include, in addition o the economic load-ing of each generating unit, complete voltage andpower flow information associated with the genera-tion schedule.

The method presented can be used in planning studies todetermine operating costs for both present and proposedgeneration and transmission facilities. The method can beadapted also to on-line control of generation.[] This wouldprovide the opportunity to check voltage and line loadingsbefore initiating generation changes. Security analyses canbe performed also by simulating generator and line outages.

In addition, the elements of the coefficient matrices aand /3 can be used to determine a transmission loss formula

based on the usual assumptions. This would be an auto-matic means with an on-line control computer to revise theloss formula for current ystem changes.

The real and reactive power allocation option of the com-puter program provides a method of planning economicreactive capability. It also provides the potential of on-linecontrol of reactive power scheduling.

REFERENCES[ I M. J. Steinberg and T. H. mith, Economy Loading of Power Plantsand Electric Systems.New York :Wiley, 1943.

Yo rk: W iley, 1958.[2 1 L. K. Kirchmayer, Economic Operation of Power Systems. NewH. H. Happ, J. . Hohenstein, L. K. Kirchmayer, and G . W . Stagg,Direct alculation of transmission loss formula-11, ZEEE TransPower Apparatus and Systems, vol. 8 3, pp . 702-707, July 1964.[41 H. E. Brown, C. E. Person, L. K. Kirchmayer, and G. W . Stagg,Digitalcalculation of three-phase shortcircuits by matrixmethod,Trans. AZEE (Power Apparatus andSystems),ol. 74, pp. 1394-1 397,195515 ] A. F. Glimn an d G. W. Stagg, Automatic calculation of loadflows, Trans. AZEE (Power Apparatusand Systems), vol. 76, pp . 817-823,1957.16 R. Courant, Differential and Integral Calculus, vol. 2. New YorkInterscience, 1949.[I H. A. Spang, 111, A review of minimization techniques for non-linear functions,SZAM Rev., vol. 4, pp. 34S3 63,October 1962.[* I L.P. Smith, Mathematical Methodr fo r Scientists and EngineersNew Yor k: Dover, 1953.[I G. W. tagg and E. L. Wizemann, Computer program fo r loadflow study handles ten-system interconnection,Electrical World, August1, 1960.[I G. W. Stagg, J. F. Dopazo, M. Watson, J. M. Crawley, G. R.Bailey and E. F. Alderink, A time-sharing on-line control system foreconomic operationof a power system, roc. Znstrum. SOC . Am., ctober1966.

Optimization in Engineering DesignALLAN D. WAREN, MEMBER, IEEE, LEON S. LASDON, MEMBER, IEEE, ANDDANIEL F. SUCHMAN, MEMBER, IEEE

Manu script received July 5, 1967 ; revised August 30, 1967.A. D. Waren is with the Departmentof Electrical Engineering,Cleve-land State University, Cleveland, Ohio. At the time this paper was sub-mitted he was with Bell Teleph one Laborato ries, Inc., M urray H ill, N. .,for the summer.L. S. Lasdon is with th e Op erations Re search D epartment andystemsResearch Center, Case Institutef Technology (now alled Case-Westem-Reserve University), Cleveland, Ohio.D. F. Suchman is with the Ordnance Division, Clevite Corporation,

spch problem ar e brie6y described. Two reasonably complexexamples,w a t e r ~ r s y s t ~ P n d t h e s e e o e d d e s c ~ t b e ~ o f . r r i d e b r n d c rm e r .lrPingthesemethods,Prepresented:tbetirstd&ibthed&ofm&-

E I. INTRODUCTIONNGINEERING design sviewed here as a three-stage iterative process: 1) the selection or modifica-tion of a structure for the system and the identification of the design variables in this structure, 2) the assignment of numerical values to thesedesign variables, 3)evaluation of the resulting design and the decision as towhich of steps 1) or 2), if either, must be repeated.

Computer-aided design generally refers to the useofdigital computer programs in the analysis stage and hasgreatly reduced the time required to evaluate proposed

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An Optimization Technique for Real And