Surrogate worth trade-off method for economic-emission dispatch

Lakhwinder Singh and Jaspreet Singh Dhillon

Abstract-- The economic-emission dispatch (EED) problem is a constraints in classical economic dispatch, which contains anmultiple non-commensurable objective problem that minimizes efficient weights estimation technique. Dhillon and Kothariboth cost and emission together subject to power balance and [6] solved the multiobjective thermal power dispatch problemgenerators capacity constraints. In this paper, multiobjective using £-constraint method. Surrogate worth trade-off approacheconomic-emission dispatch problem is converted into a scalar is applied to find the preferred solution. Hota et al. [7] haveoptimization problem. Scalar optimization problem is solved solved the economic-emission load dispatch through anmany times for a different set of weight pattern to generate non-inferior solutions along with trade-off functions between the interactive fuzzy satisfying method. Dhillon et al have solvedconflicting objectives. Fuzzy decision making theory is exploited stochastic economic emission load dispatch problem [4]to decide the best compromise solution. The optimal solution is considering three objectives viz. expected cost, expectedobtained by considering real and reactive power losses, which emission and expected variation in active power generationare calculated by performing fast decupled load flow analysis. mismatch. Weighted mini-max technique was used to simulateThe validity of the proposed method is demonstrated on 25-bus, the trade-off between the conflicting objectives. Fuzzy set35-lines IEEE system, comprising of five generators. theory was used to choose the optimal operating point. Abido

Index Terms--Economic-emission dispatch; weighting method; [8] has presented a novel approach based on the strengthdecision making; regression analysis pareto evolutionary algorithm to solve environmental

economic power dispatch optimization problem. Fuzzy basedmechanism is employed to extract the best compromise

I. INTRODUCTION solution over the trade-off curve. Brar et al. [9] have usedM\/[ULTIOBJECTIVE optimization methodology is the fuzzy logic based weightage pattern searching to obtain the

technique which simultaneously satisfies multiple solution of multiobjective load dispatch problem. Thecontradictory criteria/goal. The levels of attainment of these evolutionary optimization technique has been employed ingoals are to be expressed in the form of quantitative which the 'preferred' weightage pattern is searched to get theperformance criteria. Some of which can be selected as the 'best' optimal solution in non-inferior domain.optimal objectives. The situation is formulated as amultiobjective optimization problem in which the goal is to The intent of the paper is to solve EED problem which is amaximize or minimize several objective functions multiple non-commensurable objective problem, minimizessimultaneously. For effective operations, the optimal power both cost and emission together subject to power balance andscheduling problem has mainly confined to minimize the generators capacity constraints. In this paper, authors are onlygeneration cost regardless of emissions constraints. With the concerned with measurable volume of environmentallyincrease in the environmental awareness and the passage of harmful emission of NOx and COx from thermal generators.environmental regulations, the clean air act amendments of EED problem is converted into a single objective function1990 [1] has forced utilities to modify their operating using weighting method. Fuzzy decision making theory isstrategies to meet environmental standards set by legislation. exploited to decide the generation schedule whereas weightingIn recent years, emission control has become one of the method is used to generate non-inferior solutions. To accessimportant operational objectives because of having a the indifference band, interaction with the decision maker issignificant impact on the operation of power systems. The obtained via surrogate worth trade-off (SWT) functions of thepollution of the earth's atmosphere caused by three principal objectives. The SWT functions are constructed in thegaseous pollutants, oxides of nitrogen (NOx), oxides of functional space and then transformed into the decision space,sulphur (SOx) and oxides of carbon (COx) from thermal plants, so the SWT functions of objectives relate the decision maker'sis of greater concern to power utility and communities. preferences to non-inferior solutions. Quadratic regressionIdeally, the utilities would like to supply power to its analysis is performed between SWT function of the objectivescustomers with minimum environmental emission as well as and simulated weights to decide the 'optimal' operating point.minimum total fuel cost [2]. The validity of the proposed method is demonstrated on 25-

bus, 35-lines IEEE system, comprising of five generators.Owing to the conflicting and non-commeasurable nature of Results of the proposed method are compared with Ref. [9].

cost and emission control, a trade-off between economy andenvironment needs to be considered in optimization process. II. ECONOMIC-EMISSION PROBLEM FORMULATIONFor the solution of such multiobjective problems differenttechniques have been reported in literature pertaining to The economic-emission problem formulation is aenvironmental-economic dispatch problem [3-4]. Ramanathan mathematical programming problem in which the attempt is to[5] has presented a methodology to include emission minimize the total operating cost and gaseous pollutants of

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thermal power plant while meeting total active and reactive Nb Nbload plus transmission losses with in generator limits. = [A+(Pj ± QiQj) + Bi (QiPj - PQj)] (2)Mathematically the problem is defined as: ix =1

Minimize operating cost Nb NbZlss Z[CJQ'PP ± Q~Q1)D..(Q.P -fPQ )] (3)

F, 2 +biP +C$/h (la) i=l j=1i=1 where Pi =PGi - PDi, Qi = QGi- QDi

Minimize NOx emission R. R.F2(PGi) = ,5E(d2APi +e2iPGi + f2i) Klh lb)Ajj

cos(55); B= sin(i-31)Z=1

Minimize COx emission Cii

cos(i-8j) D = g sin(8i-81)Ng V Vi Jj'~V

F 2c)=(3ic +e3c +3) Kg/h (Ilc)=3(Pi) = 3ipj+ 3iPGi + K(where 3i and 3j are load angles, respectively. V, and Vj are

voltage magnitudes, respectively. Ru is the real component of

Subject to constraints impedance bus matrix. Xu is the reactive component ofimpedance bus matrix [12].

Equality constraints. The total real and reactive power III. WEIGHTING METHODgenerated must meet the total demand plus losses in thesystem. To generate the non-inferior solutions, the EED problem is

converted in scalar optimization problem and is given as:Ng NbL

E PGi = PDi + Ploss (Id) Minimize wkFk (4)i=l i=l ~~~~~~~~~~~~~~~~~~~~k=1

Ng Nb L

L QGi =Y QDi + Qloss (le) Subject to: i) ZWk = 1.0, Wk . 0.0 (5)i=l i=l k=1

Inequality constraints. Each generating unit is restricted ii) Eq. (Id) to (Ig)by its lower and upper limits of real and reactive power where wk are the levels of the weighting coefficients. L is theoutputs to ensure stable operation. total number of objectives. This approach yields meaningful

results to the decision maker when solved many times forpmin <Jp <prmax i 1 different values of wk; k =1, 2,...L. The values of the

Gi Gi i- Gi , 2,..., Ng ( f) weighting coefficients vary from 0 to 1. To find the solution,constrained problem is converted into an unconstrained

QGi < QG < QGi i =1, 2,..., Ng (1 g) problem considering equality and inequality constraints. The

generalized augmented function is formed as:where ai, bi and ci are the cost coefficients. d2 , e2i and f2i are Lthe NOx emission coefficients. d3i, e3i and f3i are the COx L (PGi, QGi., APIAq) = w jFj -

emission coefficients. Ng is number of generators. PGi is the j=1

real power output which is taken as a decision variable. Nb is Ng Nb Ng Nbthe number of buses in the system. QGi is the reactive power AP (PGi - PDi-Poss) -Aq( EQGi - IQDi -QOSS)output. PDi and QDi are real and reactive demands,respectively. PZos and Q10vs are real and reactive power losses N{ {(PG - prini)2+ (PCax - PG, )2 ± (QGx - Q 111 )2 ± (Qmax - Q )2 (6)

in the transmission lines, respectively. PCmin and Q min are thelower limits of real and reactive power generation, where AP, qare Lagrange multipliers, r is penalty factor.

respectively. pmax and Q,mx are the upper limits of real and The Newton-Raphson method is applied to obtain the non-

inferior solutions for simulated weight combinations, toreactive power generation, respectively, civ h eesr cnios.

Power transmission losses: The real and reactive power \FU YDEIONMK Gtransmission losses, P10g. and Qls are given below and theIVFUZ DESONMK Gloss coefficients Aij Bij Cj,. Dij are evaluated from line data Considering the imprecise nature of the decision maker'sperforming load flow analysis. judgment, it is natural to assume that the decision maker (DM)

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may have fuzzy or imprecise goals for each objective function. Regression analysis is performed between Sl. and w to findThe fuzzy sets are defined by equations called membership * *functions. The membership function represents the degree of the optimal value of wj. To find wj, equation satisfyingachievement of the original objective function as a value9 j Slj (Wj ) PD iS solved. AD iS the maximum satisfactionbetween 0 and 1 with ,u(F, )=i as completely satisfactory and S1 (w)=IDi ovd ' stemxmmstsato

/ X .I

. level of the SWT function, which is to be achieved. ThepF,) = 0 as unsatisfactory. Such a linear membership decision maker may choose required band of Sl . The SWTfunction represents the decision maker's fuzzy goal of 1achievement, and at the same time scales the original objective function may vary from zero to infinity. But the value of ADfunctions with different physical units into measure of 0-1. By is maximum in a particular band which satisfies the Eq. (5) totaking account of the minimum and maximum values of each decide the best weight pattern. So, the decision maker have toobjective function together with the rate of increase ofmembership satisfaction, the decision maker must determine crthe membership function p(F, ) in a subjective manner [11].

VI. THE SWT ALGORITHM

F <F mi SWT analysis includes trade-off analysis which isi11 performed between the cost and each of the other objectives.IF max - F A stepwise procedure to compute best compromise solution is

t(Fj) = max _Fmin ;Fi <.F < Frm (7) given below:Fi

* F > F max Step 1. Input the system data consisting of line data, fuel costcurve and emission curve coefficients, limits onactive and reactive power generations and demand

where Fimi and Fimax are minimum and maximum values of etc.

ith objective function in which the solution is expected. Fig. 1 Step 2. Compute loss coefficients Aij, Bij, Cij, Dij, real andillustrates a typical shape of membership function. thus reactive power losses Ploss and Qloss by

performing load flow using decoupled Newton

pti(Fi) Raphson method.Step 3. Find the minimum and maximum values for each of

1 < the objectives, Frnif and Fm ; j = 1,2,..., L. Thisis carried out by giving full weightage to oneobjective and neglecting other objectives, becausesome objectives are of conflicting nature.

Step 4. Generation of trade-off functions: Generate non-\_____ >inferior solutions by solving Eq.(6), using Newton-

0 F mi Frnax Raphson method.

Fig. 1: Linear membership function 4.1. Set the index to weightage, j = 24.2. Non-inferior solutions are obtained by varying

V. SURROGATE WORTH TRADE-OFF FUNCTION the weightage to prime objective, w1 and

It is assumed that there exists a surrogate worth trade-off weightage to other objectives, w1 (j . 1) and

function for a decision maker that can predict his/her by fixing the weightage to rest of thebehaviour and interest. Let the DM's surrogate worth trade-off objective, wk (k X1 and k Xj); k = 2, 3...., L.function be defined for each objective depending on the The weights are varied and fixed in such aimportance of the objective function, F1 to other objective manner that their sum remains equal to 1.0function Fj ; j = 2 ..., L. Slj indicates the ratio of the degree 4.3. Compute membership functions for objectives,of satisfaction of objective function F1 with respect to Fj and it F, and Fj, using Eq.(7).is defined as 4.4. Check j > L, if yes GOTO 5, else j = j +1 and

GOTO step 4.2 and repeat.

ut (F / ) (8) Step 5. Generation of SWT function: The SWT functions lis obtained by exploiting Eq.(8) for all the non-inferior solutions generated in step 4.

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Step 6. Find functions Slj (wj); j = 2,..., L by performing Table 2: Non-inferior solutions when w3 is fixedregression analysis or by interpolation method. Theweights, w = *are evaluated by finding the roots Weights Objectives Membership SWTweights, Wj = Wj are evaluated by findlng the roots Functionsof equation, S11(w1)=ID' where AD is w1 W2w 3 F, s/h F2 Kg/h (FI) p(F2) S12satisfaction level of the SWT function. The roots ofequation should not be negative and imaginary 0.7 0.1 0.2 4951.223 849.8408 0.988682 0.592334 1.669128

because weights are positive or may be zero 0.6 0.2 0.2 4958.955 815.4244 0.953792 0.689829 1.382650following Eq. (5) (w;> 0; i =1, 2, ..., L). So, the 0.5 0.3 0.2 4972.339 785.5348 0.893399 0.774499 1.153518

value of AD is selected in such a manner that Eq.(5) 0.4 0.4 0.2 4992.036 759.9167 0.804517 0.847071 0.949764must also be satisfied. 0.3 0.5 0.2 5019.096 738.6018 0.682413 0.907452 0.752009

0.2 0.6 0.2 5055.126 721.9232 0.519832 0.954698 0.544498Step 7. The best set of decision vector is found by solving the

folwn rbe 0.1 0.7 0.2 5102.589 710.6037 0.305662 0.986764 0.309761following problem0.0 0.8 0.2 5165.386 705.9683 0.022295 0.999895 0.022297

L

Minimize wy 1] i=l Step-II: When weights W1 and w3 are varied regularly with

Subject to: Eq. (Id) to (ig) step of 0.1 and weight W2 is fixed at 0.2, so that their sum

VII. TEST SYSTEM AND RESULTS remains equal to one. Corresponding values of non-inferiorsolutions of objectives, F1 and F, their membership

The validity of the proposed method is illustrated on a 25- 1.. ~~~functions and the SWT functions iS shown in Table 3. Thebus, 35-lines power system [10] comprising five generator SWT functionidi n ic regressio

system [6]. Minimum and maximum values of the objectivesare obtained by using step 3 of algorithm and depicted in analysis and is given below:Table 1. S13= 1.85507 -2.21425 W3 - 0.0419799 2

Table 1: Minimum and maximum values of objectives Similarly by solving above equation, the optimal value of

F,min=4948.7150 $/h F7lmi=705.9313 Kg/h ]nin 8.60063 Kg/h weight W3 is obtained as 0.383. The SWT function is set to

Fn =5170.3270 $/h Fax=1058.940 Kg/h Fnax 22.3411 Kg/h

Table 3: Non-inferior solutions when w2 is fixedTo obtain the solution, two steps are considered and are

given below: |Weights Objectives |Membership |SWT|Functions

Step-I: When w3 is fixed at 0.2, WI and w2 are varied w F, s/h F3Kg/h p(Fl) p(F swith step of 0.1 WI 2 W3 3 13

Step-Il: When W is fixed at 0.2, W1 and w are varied 0.7 0.2 0.1 4956.906 13.5155 0.963039 0.642363 1.499215

with2step of 0.1 0.6 0.2 0.2 4958.956 13.6379 0.953787 0.633454 1.5056950.5 0.2 0.3 4961.918 13.7988 0.940423 0.621744 1.512556

Step-I: When weights W1 and W2 are varied regularly with 0.4 0.2 0.4 4966.451 14.0202 0.919967 0.605637 1.519006

step of 0.1 and weight W3 is fixed at 0.2, so that their sum 0.3 0.2 0.5 4973.929 14.3448 0.886223 0.582012 1.522689

remains equal to one. Non-inferior solutions of objectives, F] 0.2 0.2 0.6 4987.707 14.8699 0.824053 0.543801 1.5153580.1 0.2 0.7 5017.972 15.8723 0.687484 0.470862 1.460056

and F and correspondmg membership funchons along with2 . . ~~~~~~~~~~~~~0.00.2 0.8 5111.193 18.5706 0.266836 0.274514 0.972034

the SWT functions are shown in Table 2. By performingquadratic regression analysis, the SWT function is givenbelow: The value of weight W; is calculated from Eq.(5) i.e. w

S = 1.28798+ 1 69267 - 2.39746 W2 * *12 * * 4 * = 1.0 - (1122+ w3 ) and is 0.475 at same satisfaction level.

By slvin abve euaton, he otiml vaue f weght The optimal decision vector is obtained by solving step 7 of* the algorithm. The obtained optimal values of weights, fuel

112 is obtained as 0.142, when satisfaction level of SWT cost, NOx emission, CQ. emission along with their

function, A'D iS set to 1.0. membership functions of proposed method and Ref. [9] areshown in Table 4. It has been observed from the comparison

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of results that overall membership function obtained in VIII. BAND OF SWT FUNCTIONproposed method is more than the overall membershipfunction obtained in Ref. [9]. The 'best' power generation Analysis to select the band of SWT is carried out byschedule of the proposed method is shown in Table 5. giving variation to ID from 0.4 to 1.3. The value of ID isTable 4: Comparison of Results to be selected in such a manner that roots of the equation for

all w. j = 2, 3...., L are real and positive as well as rootsCost NOx Cox J' '($\h) emission emission should satisfy the condition of normalization of weights i.e.

(Kg/h) (Kg/h) L

Weights 0.475 0.142 0.383 1Wj = 1.0, w1 0-.0. Both the conditions areProposed j=lmethod Objectives 4957.007 822.4789 13.3512

satisfied when the value of AD is more than or equal to 0.4Membership 0.962584 0.669845 0.654317

and is less than 1.3. When the value of ,D is less than 0.4Weights 0.660 0.170 0.170DWeights_ 0.660 0.170 0.170

or greater than 1.3, the above mentioned conditions are notRef.[9] Objectives 4955.571 827.0858 13.3627 satisfied, so the value of AD lying in this range is

Membership 0.969063 0.656794 0.653476 neglected. Thus, the desired range of I'D is more than 0.4

and less than 1.3. The variation of AD with respect toobtained degree of membership function of final generation

Table 5: The 'best' power generation schedule schedule is shown in Fig. 2. It has been observed from Fig.

Generation Load Injected Voltage 2 that the overall maximum satisfaction is achieved whenPower Profile value of SWT function is set to 1.0.

Bus PG QG PD QD P Q V 3

1 0.2091 .0632 .00 .00 .20912 .06323 1.020 .00 1

2 1.1964 .0113 .00 .00 1.1964 .01130 .9690 .2323 0.9S 0.83 1.1284 .1441 .00 .00 1.1284 .14414 .9950 .1572 X

(0 0.74 0.9063 -.0236 .00 .00 .90627 -.02356 .9600 .0702 a)0.65 0.8303 -.1250 .00 .00 .8303 -.1250 .9600 .1082 E 0 5

6 0.00 0.00 .15 .05 -.15 -.05 .9622 .1512 0.47 0.00 0.00 .15 .05 -.15 -.05 .9566 .1078 = 0.3

8 0.00 0.00 .25 .00 -.25 .00 .9572 .0885 0.2I

9 0.00 0.00 .15 .05 -.15 -.05 .9494 .0585 o 0.1°10 0.00 0.00 .15 .05 -.15 -.05 .9565 .0561 0

11 0.00 0.00 .05 .00 -.05 .00 .9610 .0415 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

12 0.00 0.00 .10 .00 -.10 .00 .9595 .0678 Satisfaction level of SWT13 0.00 0.00 .25 .08 -.25 -.08 .9646 .1418

14 0.00 0.00 .20 .07 -.20 -.07 .9513 -.0182 Fig. 2: Band of SWT Function15 0.00 0.00 .30 .10 -.30 -.10 .9541 -.0419

16 0.00 0.00 .30 .10 -.30 -.10 .9705 -.0431 IX. CONCLUSIONS17 0.00 0.00 .60 .20 -.60 -.20 .9641 .0425

18 0.00 0.00 .15 .05 -.15 -.05 .9532 .0334 In power system operation planning, there exist multiple19 0.00 0.00 .15 .05 .15 .05 .96 19 .0584 objectives to be attained, which conflict with each other and

are subjected to a mutual interface. It means that any one20 0.00 0.00 .25 .08 -.25 -.08 .9475 .0199 objective can be improved only at the expense of one or more21 0.00 0.00 .20 .07 -.20 -.07 .9509 -.0186 of the other objectives. In economic-emission dispatch22 0.00 0.00 .20 .07 -.20 -.07 .9571 -.0568 problem it is realized that cost and emission are conflicting23 0.00 0.00 .15 .05 -.15 -.05 .9894 -.0446 with each other. The solution set of the problem is non-23 0.00 0.00 .15 .05 -.15 -.05 .9894 -.0446 inferior due to conflicting nature of the obJectives and has24 0.00 0.00 .15 .05 -.15 -.05 .9706 -.1003 been obtained through weighting method. Fuzzy decision25 0.00 0.00 l.25 .08 l-.25 -.08 1.98 19 l-.0987 making methodology is exploited to decide the 'best'

generation schedule. Regression analysis iS performedbetween SWT functions and simulated weights to decide the'optimal' operating point. Cost and emission are calculated atthe 'optimal' values of the weights. The proposed method

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provides interface between the decision maker and Electrical Power & Energy Systems, vol. 25, pp. 179-mathematical model through SWT. It is a simple one and 185, 2003.needs few non-inferior solutions and hence gives [12] Lakhwinder Singh, J. S. Dhillon and R. C. Chauhan,computational efficiency. The validity of the proposed method "Evaluation of best weight pattern for multiple criteriais demonstrated on 25-bus, 35-lines IEEE system, comprising load dispatch", Electric Power Components andof five generators. Results of the proposed method are Systems, Vol. 34, No.1, pp. 21-35, Jan. 2006.compared with Ref. [9] which are better in terms of overallmembership satisfaction achieved.

BIOGRAPHIES

X. REFERENCES

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[2] M.T.Tsay, "Applying the multi-objective approach for He is Head of Electrical & Electronicsoperation strategy of cogeneration systems under Engineering Department from 2001. Heenvironmental constraints", Electrical Power & Energy obtained his B.E. (Electrical) and M.E.Systems, vol. 25, pp. 219-226, 2003. (Power Systems) from Guru Nanak Dev

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