GA Based ELD

8/3/2019 GA Based ELD

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Elk :SOz emission (tons) of affected unitsBlk : burn (MBtu) of affected unitsEllk :SO, emission (tons) of unaffected unitsBilk : burn (MBtu) of unaffected unitsE d :annual emission allowance (tons) for affected unitsB,,] :baseline burn (MBtu) for affected unitsE,, : surrendered emission allowance (tons)WER : weighted emission rate for unaffected units (tons/TI, : ength of dispatch interval (hrs)pk :power generation (MW)P :vector with components P,, ie(hr, k=l, 2, ...,MH,(Pd : heat rate (MBhdIu)= c,+ b,P* + a,P*'F, : fuel price ($/MBtu)VMCQ :variable operating maintenance and coal handlingEC,4P IPm : ystem load (MW)PLk : ransmission losses (MW)p,P,""(P,"") : lower (upper) generation limits (MW)N :population sizeNvARpc.pm :probabilities for crossover and mutation,P :penalty factor

MBtu)

price ($/MWh):SO, emission rate (tons/MBtu):=1, if the unit is affected; =O, otherwise: =1, if the unit is unaffected;4,otherwise

: output of fixed generalion of hydro-units ( MW )

: length of substrings for each unit output variablerespectively

EDCGP PROBLEM FORMULATIONThe compensating generation option provided in the Actallows underutilizationof the affected units at the expense ofsurrendering SO2 allowances based on the amount ofunderutilization of these units. Since utilities are interested inminimizing the net cost, the EDCGP problem can beexpressed as follows :

U

klMh Ne t c a r t = c P m D k + d * NCt Emission ( l)

subject to:1. Generating capacity limit constraint :e Pu s Pi- , E+k , k=I, 2, ..., M (2)

2. Demand constraint:cPLtPn = Pm + Pu , k=l , 2, ..., MtEoL

(3)

nwhere -Net Emisswn = El& + E, - Ed (4)k1

CBnkL= l

Adjoining the demand constraints(3)onto the original netcost function using the LaGrangian multipliers,&, k=l, 2, ...,M, yields the augmented cost functionM~ ( P , A ) ~ c tosr(p) + ak LIP, (12)k=l

whereAP, = P , + Pu - Pic - Pn

*ka = (a,, A,, ..., AJand P is the vector with components PiL, Ok ,k=l, 2, ...,M.The stationary points of the function SP(P,A)can be found bysolving the following set of nonlinear equations,

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P& = Pm + Pu - Pn , t=l, , ...,M (14)where

a Net = Tk{(F, d y , AECJ H & + VMCH,]a ptkand

Equation set (13,14) is a higher-order nonlinear equation setsince both y, and AEC, in the left-hand side of (13) arefractional functions of generation on all intervals. Thisequation may have more than one solution (stationary points)which satisfy the generating capacity constraint (2). Theproblem is, therefore, multimodal in its feasible solution spacein a nontrivial case.

When the emission price d=O, the EDCGP problembecomes one that minimizes the annual production cost whichcan be easily solved by the SED approach. The solutionsproduced using the SED method may be a goodapproximation of the solutions of the EDCGP problem whend is relatively small. However, the discrepancies betweensolutions using this strategy and the true global optimalsolutions, which can be obtained via the proposed GA , aresignificant when the market price of SO2emission allowancesis relatively high.

GENETIC ALGORITHMS AND IMPLEMENTATIONMovement in a typical GA is accomplished using threeprimary operators: reproduction, crossover. and mutation. Ina sample execution cycle, the reproduction operator selects astring from the previous generation based on the string'sfitness and its probability of propagation to the nextgeneration. Reproduction continuesuntil the population of thenext generation is filed In this paper, a tournament selection

technique with the size of 2 is used in the reproductionprocess. The crossover operator works in conjunction withreproduction. Simple crossover selects a locus position withintwo parent strings and swaps the gene information from thatposition to the end of the string. The mutation operatorchanges random allele values during the reproduction-

crossover phase to increase the diversity of the sample space.The probabilities of crossover and mutation occurrence areuser selectable [4].

Execution parameter selection is important in allowing theGA the ability to explore the entire solution space, The initialpopulation is usually created using a random numbergenerator and the subsequent generations are formed from theinitial population information.EncodingThe encoding scheme utilizes the binary alphabet {0,1}.Each chromosomesmng containsan encoding of the solutionfor the unit generation. Considering the difficulty of inclusionof the demand constraint in the cost function as a penaltyterm, a nonlinear mapping technique is used in the decodingprocess.

The first step of decoding is to simply decode achromosome string to a set of discrete integers I P k ' s (iE@JFor example, if a chromosome string representingan encodingof six parameters, IPiL i=l2.3: k=1,2), is expressed as

I " 1 4% I blb@3 I '123 I ' 1 4 I 'I%?? I ffh Iwhere 4, b,, c,, 4, e,, f, = 0, or 1, i=1,2,3, then

IPl1 = a, - 2 0 + + -2, % * 2 =IP2 , = b, * 2O + b, - 2, + b3 - 22IPS, = c1 * 2O + cz - 2' + c3 * z2

IPn = A -20 f 2 - 2 , + A 22IP, = d , - 2O + 4 * 2, + 4 * 22I P ~ cl - 2 0 + e+ - 2 ' +e 3 -2 '

Following the simple decoding process in the first step, anonlinear mapping Q is defined asQc {Pai e k } - { P i cP,=P,+P,-P,} , =l, ..., M

wk

A simple mapping equation could be as follows:P~ = r&


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Table 1: Parameters of the Test System (* The unit is affected)bnit #

123'45678'9'

a F($/MBtu) VMCH($FIwh) I EC(lb/MBtu) Pmm(MW) y(m)0.0 40 40

C-7.9858.6607.6807.6157.549

465.525~~-

0.00129 1.6435 0.75 3.4719 520 67 10.00235 1.6435 0.77 1.4338 95 1760.00677 1.6435 0.77 2.2779 95 1660.01877 1.6435 1.15 1.0168 56 1040.02059 1.6435 1.15 3.3667 56 104

70.656

8.0998.099

161.099

0.00256 2.2185 0.67 2.8369 101 2480.00256 2.2185 0.67 2.8369 101 248

150.464155.902

- 1 - 1 - 1 I 0.0 I 50 I 50

219.514219.514

Fitness FunctionBecause the EDCGP problem is a minimization problemand the constraints need to be included in the objectivefunction, two steps are needed to accomplish fitnessevaluation.

First, a pseudo cost function is defined by including allinequality constraints as penalty terms in the original costfunction. Simple penalty functions are chosen to yieldYPC@seudo cost) = Net Cost + p.AP,.Tk(16)b l rek

where p is a penalty factor, and APk is the deviation of theunit's output from its limits.In the second step, the pseudo cost values are transformedinto fitness values. Since chromosome string's fitnesswill becompared to all other string fitness within the Samepopulation, an absolute measure of optimality is not required.For one generation of population, ifPC,s a pseudo cost valuefor the sbiing i, and PC- A Max{PC,,=12,..."POP},henfitness for this population are defined as

FIT, = PC- - X i ' , =l , 2,...,N (17)Since the fitness function isa raw measure of the solutionvalue, and we desire to obtain the optimal feasible solutionwhich minimizes the original cost, a suitable selection ofpenalty function is very important. If encoding and fitness

function are well defined, accuracy will not suffer.

losses are evaluated using B-Constant method. The systemparameters are listed in Table 1. Execution parameters andpenalty factors for each case are included in Table 2. Figure1 shows the typical iterative process of the proposedalgorithm for d=2000 $/ton.

Several different emission prices were used in thesimulations. Test results using the GA approach and the SEDare included in Table 3.Table 2: Execution Parameters

P P

I'Two important conclusions can be drawn from Table 3.First, in the case that d=O, the net cost obtained using theproposed GA and using the SED are very close. This shows

that the standard economic w a t c h is sufficient to find theoptimal solution of a unimodal problem. On the other hand,as the emission allowance price increases, the differencebetween the annual net cos ts of the GA and the SEDbecomesmore noticeable. This establishes the ability of geneticalgorithms to solve the EDCGP problem and complicatedmultimodal power system optimization problems in general.

CONCLUSIONS AN D FURTHER RESEARCHSIMULATION RESULTSThe algorithm was tested on a 9-unit system. Of these,two are hydro-units, and their outputs are fixed.Transmission

The results presented in this paper show that the GA ispowerful optimization method which can be used to solve theproblem of economic dspatch underthecompensating genera-

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Table 3: Performance of the GA and the SED ADDroaches

E,, (tom)Prod.Cost (10' $)Net Cost (lo3$)

6 t (103 $)

E.,=60,000 ons; B,,=32,000.000 MBtu

6,343 6 3 4 3 6,473 6,343 6,477 6343 6,474 6,34385,458 85,457 85,501 85,457 85,513 85,457 85,544 85,45785,458 85,457 77,537 77,624 69.574 69,790 53,624 54,123

- 0 -87 -216 -499t 6 = (Net Cost),, - (Net Cost)sED

tion plan provided in the Clean Air Act of 1990. Theadvantage of the proposed GA lies in its ability to handle ill-structured nonlinear optimization problems. Selection of [13penalty functions is crucial to solution accuracy. The simpleencoding skill, combined with the proposed nonlinear [2]mapping technique. proves to be very reliable. A key factorin the GA's tendency to find near global optimal solutions lies [3]in its ability to distinguish the fitness of optimal solutions,thus the application of other penalty functions could yieldimprovement in solution accuracy and should be investigatedfurther. [4168,058~66.00064,000

62.00060.000

REFERENCESA. J. Wood and B. F. Wollenberg, Power Generation,Operation, and Control, John Wiley & Sons, N. Y.Congressional Amendment to the Constitution, HR.3030/S. 1490, 1990.A. A. El-Keib, H. Ding, G. Frazier, and H. Ma,"Environmentally Constrained Economic DispatchConsidering the UnderutiIization Provision of the CIeanAir Act of 1990," The University ofAlabama, Ber Report,D. E. Goldberg, Genetic Algorithm in Search,Optimization & Machine Learning, Addison-Wesley,Reading. MA, 1989.J. J. Grefensteae, "Genetic Algorithms for the TravelingSalesman Problem," Proc. International Con$ o n GeneticAlgorithms and Their Applications, 1985.X . Yin and N. Gennay, "Investigation on Solving theLoad Flow Problem by Genetic Algorithms," ElectricPower Systems Research, 22 , 1991.V. Ajarapu and 2. Albanna, "Application of GeneticBased Algorithms toOptmalCapacitor Placement," Proc .of the First International Sym p. on Application ofNeuralNetworks to Power Systems.

NO. 571-220, August 1992.

56.000- [8] J. Lansbeny and L. Wozniak. "Optimal HydrogeneratorGovemor Turning with a Genetic Algorithm," IEEEIPES1992 Winter Meeting, New York, NY, Jan. 1992.4,000- -52.000 I I I I I I I ~ [91 T. Haida and Y. Akimoto, "Genetic Algorithm Approach

0 10 20 30 40 50 60 70 BO to Voltage Optimization," Proc . of he First InternationalSymp. on Application of Neural Networks to PowerSystems.[10]H. Ding, A. A. El-Keib, and R E. Smith, "OptimalClustering of Power Networks Using Gnetic Algorithms,"Electric Po wer System Research (In Press).

GENERATIONS

Figure 1: Performanceof the GA Approach (d=-/ton)

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GA Based ELD