dynamic economic emission dispatch using nondominated sorting genetic algorithm-ii

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Electrical Power and Energy Systems 30 (2008) 140–149

Dynamic economic emission dispatch using nondominatedsorting genetic algorithm-II

M. Basu

Department of Power Engineering, Jadavpur University, Kolkata 700098, India

Received 14 August 2006; received in revised form 14 June 2007; accepted 17 June 2007

Abstract

This paper presents nondominated sorting genetic algorithm-II for dynamic economic emission dispatch problem. This problem isformulated as a nonlinear constrained multi-objective optimization problem. Nondominated sorting genetic algorithm-II is proposedto handle dynamic economic emission dispatch problem as a true multi-objective optimization problem with competing and noncom-mensurable objectives. The proposed approach has a good performance in finding a diverse set of solutions and in converging nearthe true pareto-optimal set. Numerical results for a sample test system have been presented to demonstrate the capabilities of the pro-posed approach to generate well-distributed pareto-optimal solutions of dynamic economic emission dispatch problem in one single run.The comparison with the classical technique demonstrates the superiority of the proposed algorithm.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Dynamic economic dispatch; Dynamic economic emission dispatch; Multi-objective optimization; Nondominated sorting genetic algorithm-II;Real-coded genetic algorithm

1. Introduction

Fossil-fueled based electric power generating stationreleases sulphur oxides (SOx), nitrogen oxides (NOx), andcarbon dioxide (CO2) into atmosphere. Atmospheric pollu-tion affects not only humans but also other life-forms suchas animals, birds, fish and plants. It also causes damage tomaterials, reducing visibility as well as causing globalwarming. Due to increasing concern over the environmen-tal considerations, society demands adequate and secureelectricity not only at the cheapest possible price, but alsoat minimum level of pollution. Dynamic economic emis-sion dispatch (DEED) serves to schedule the online gener-ator outputs with the predicted load demands over acertain period of time so as to minimize cost and emissionsimultaneously.

Dynamic economic dispatch (DED) is a method to sche-dule the online generator outputs with the predicted load

0142-0615/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijepes.2007.06.009

E-mail address: [email protected]

demands over a certain period of time so as to operatean electric power system most economically [1–4]. It is adynamic optimization problem taking into account theconstraints imposed on the system operation by generatorramping rate limits. The DED is not only the most accurateformulation of the economic dispatch problem but also themost difficult to solve because of its large dimensionality.Normally, it is solved by dividing the entire dispatch periodinto a number of small time intervals, then a static eco-nomic dispatch has been employed to solve the problemin each interval. Since DED was introduced, several meth-ods have been used to solve this problem.

Several strategies to reduce the atmospheric pollutionhave been proposed and discussed [5]. These include instal-lation of pollutant cleaning, switching to low emissionfuels, replacement of the aged fuel burners with cleanerones, and emission dispatching. The first three optionsrequire installation of new equipment and/or modificationof the existing ones that involve considerable capital outlayand hence they can be considered as long-term options.The emission dispatching option is an attractive alternative

M. Basu / Electrical Power and Energy Systems 30 (2008) 140–149 141

in which both cost and emission is to be minimized. Inrecent years, this option has received much attention [6]since it requires only small modification of the basic eco-nomic dispatch to include emission.

Different techniques have been reported in the literaturepertaining to economic emission dispatch (EED) problem.In [6] linear programming based optimization procedureshas been presented in which the objectives are consideredone at a time. Unfortunately, the EED problem is a highlynonlinear optimization problem. Therefore, classical opti-mization methods that make use of derivatives and gradi-ents, in general, are not able to locate the globaloptimum. Furthermore, this approach does not give anyinformation regarding the trade-off involved. The EEDproblem has been converted to a single objective problemby linear combination of different objectives as a weightedsum [7]. The important aspect of this weighted sum methodis that a set of pareto-optimal solutions can be obtained byvarying the weights. But this requires multiple runs. Fur-thermore, this method cannot be used to find pareto-opti-mal solutions in the problem having a nonconvex pareto-optimal front. In [8] a fuzzy multi-objective optimizationtechnique for the EED problem has been proposed. How-ever, the solutions are sub-optimal and the algorithm doesnot provide systematic framework for directing the searchtoward pareto-optimal front. An evolutionary algorithmbased approach for evaluating the economic impacts ofenvironmental dispatching and fuel switching has been pre-sented in [9]. However, some of nondominated solutionsmay be lost during the search process. A multi-objectivestochastic search technique for the EED problem has beenpresented in [10]. However, this technique is computation-ally involved and time consuming. In addition, the searchbias to some regions may result in premature convergence.

Over the past few years, the studies on evolutionaryalgorithm have shown that these methods can be efficientlyused to eliminate most of the difficulties of classical meth-ods [11–14]. Since they are population-based techniques,multiple pareto-optimal solutions can be found in one sin-gle run.

Recently, multi-objective evolutionary algorithms havebeen applied to solve the EED problem efficiently [15–17].

This paper proposes nondominated sorting genetic algo-rithm-II for dynamic economic emission dispatch (DEED)problem. DEED is an extension of the conventional EEDproblem that takes into consideration the ramp rate limitsof the generating units. This problem is formulated as anonlinear constrained multi-objective optimization prob-lem. Nondominated sorting genetic algorithm-II (NSGA-II) maintains a good spread of solutions and convergesnear the true pareto-optimal set. This proposed approachdoes not require any user-defined parameter for maintain-ing diversity among population members. Due to the diffi-culties of binary representation in dealing with continuoussearch space with large dimensions, the proposed approachhas been implemented using real-coded genetic algorithm(RCGA) [18–20]. In order to show the effectiveness of the

proposed approach, ten-unit test system with nonsmoothfuel cost and emission level functions is used in this paper.Test results are compared to the classical technique.

2. Notation

Pim power output of ith unit at time m

P mini ; P max

i lower and upper generation limits for ith unitPDm load demand at time t

PLm transmission line losses at time m

ai,bi,ci,di,ei cost coefficients of ith unitai,bi,ci,gi,di emission coefficients of ith unitURi,DRi ramp-up and ramp-down rate limits of ith unitN number of generating unitsM number of hours in the time horizon

3. Problem formulation

The present formulation treats dynamic economic emis-sion dispatch problem as a multi-objective mathematicalprogramming problem which attempts to minimize bothcost and emission simultaneously, while satisfying equalityand inequality constraints. The following objectives andconstraints are taken into account in the formulation ofDEED problem.

Objectives:

(i) Cost: The fuel cost function of each thermal genera-tor, considering the valve-point effect [21], isexpressed as the sum of a quadratic and a sinusoidalfunction. The total fuel cost in terms of real poweroutput can be expressed as

f1¼XM

m¼1

XN

i¼1

½aiþbiP imþ ciP 2imþjdi sinfeiðP min

i �P imÞgj�

ð1Þ(ii) Emission: The atmospheric pollutants such as sul-

phur oxides (SOx) and nitrogen oxides (NOx) causedby fossil-fueled generating units can be modeled sep-arately. However, for comparison purposes, the totalemission of these pollutants which is the sum of aquadratic and an exponential function [22] can beexpressed as

f2 ¼XM

m¼1

XN

i¼1

½ai þ biP im þ ciP2im þ gi expðdiP imÞ� ð2Þ

Constraints:

(i) Real power balance constraints: The total real powergeneration must balance the predicted power demandplus the real power losses in the transmission lines, ateach time interval over the scheduling horizonXN

i¼1

P im � P Dm � P Lm ¼ 0 m 2 M ð3Þ

142 M. Basu / Electrical Power and Energy Systems 30 (2008) 140–149

(ii) Real power operating limits:

P mini 6 P im 6 P max

i i 2 N ; m 2 M ð4Þ(iii) Generating unit ramp rate limits:

P im � P iðm�1Þ 6 URi i 2 N ; m 2 M

P iðm�1Þ � P im 6 DRi i 2 N ; m 2 Mð5Þ

4. Determination of generation levels

In this approach, the power loading of first (N � 1) gen-erators are specified. From the equality constraints in Eq.(3) the power level of the Nth generator (i.e. the remaininggenerator) is given by

P Nm ¼ P Dm þ P Lm �XN�1

i¼1

P im m 2 M ð6Þ

The transmission loss PLm is a function of all the genera-tors including that of the dependent generator and it is gi-ven by

P Lm ¼XN�1

i¼1

XN�1

j¼1

P imBijP jm þ 2P Nm

XN�1

i¼1

BNiP im

!

þ BNN P 2Nm m 2 M ð7Þ

Expanding and rearranging, Eq. (6) becomes

BNN P 2Nm þ 2

XN�1

i¼1

BNiP im � 1

!P Nm

þ P Dm þPN�1

i¼1

PN�1

j¼1

P imBijP jm �PN�1

i¼1

P im

!¼ 0 m 2 M

ð8ÞThe loading of the dependent generator (i.e. Nth) can thenbe found by solving Eq. (8) using standard algebraicmethod.

5. Principle of multi-objective optimization

Most real-world problems involve simultaneous optimi-zation of several objective functions. Generally, these func-tions are noncommensurable and often competing andconflicting objectives. Multi-objective optimization havingsuch conflicting objective functions gives rise to a set ofoptimal solutions, instead of one optimal solution becauseno solution can be considered to be better than any otherwith respect to all objectives. These optimal solutions areknown as pareto-optimal solutions.

Generally, multi-objective optimization problem con-sisting of a number of objectives and several equality andinequality constraints can be formulated as follows:

Minimize f iðxÞ i ¼ 1; . . . ;N obj ð9Þ

Subject togkðxÞ ¼ 0 k ¼ 1; . . . ;K

hlðxÞ 6 0 l ¼ 1; . . . ; L

�ð10Þ

where fi is the ith objective function, x is a decision vectorthat represents a solution.

6. Nondominated sorting genetic algorithm-II

6.1. General approach

Before describing the nondominated sorting geneticalgorithm-II (NSGA-II) algorithm, fast nondominatedsorting procedure, fast crowded distance estimation proce-dure and simple crowded-comparison operator have beendiscussed.

(i) Fast nondominated sorting procedure: In order toidentify solutions of the first nondominated front ina population of size NP, each solution can be com-pared with every other solution in the population tofind if it is dominated. At this stage, all individualsin the first nondominated front are found. In orderto find the individuals in the next nondominatedfront, the solutions of the first front are discountedtemporarily and the above procedure is repeated.This argument is true for finding third and higher lev-els of nondomination.

(ii) Fast crowded distance estimation procedure: To getan estimate of the density of solutions surrounding aparticular solution in the population, the averagedistance of two points on either side of this pointalong each of the objectives is calculated. This quan-tity serves as an estimate of the perimeter of thecuboid formed by using the nearest neighbors asthe vertices. This is called crowding distance. Thecrowding distance computation requires sorting thepopulation according to each objective functionvalue in ascending order of magnitude. Thereafter,for each objective function, the boundary popula-tions (populations with smallest and largest functionvalues) are assigned an infinite distance value. Allother intermediate populations are assigned a dis-tance value equal to the absolute normalized differ-ence in the function values of two adjacentpopulations. This calculation is continued with otherobjective functions. The overall crowding distancevalue is calculated as the sum of individual distancevalues corresponding to each objective. Each objec-tive function is normalized before calculating thecrowding distance.

(iii) Crowded-comparison operator: The crowded-com-parison operator guides the selection process at thevarious stages of the algorithm toward a uniformlyspread-out pareto-optimal front. Every populationhas two attributes: nondomination rank and crowd-ing distance. Between two populations with differingnondomination ranks, the population with the lower(better) rank is preferred. If both populations belongto the same front, then the population with largercrowding distance is preferred.


6.2. Computational flow

Generally, the algorithm can be described in the follow-ing steps:

(Step 1) Initialize: Initially, a random parent populationof size NP is created.

(Step 2) Fast nondominated sorting of parent population:The population is sorted based on the nondomi-nation. Each population is assigned a rank equalto its nondomination level or front number (1 is

No

Yes

No

Yes

No

Yes

No

Yes

Yes

No

Start

Front =1

Is population classified ?

Tournament selection

Crossover and Mutation

gen = gen +1

Population filled?

Combine parent population and child population

Front =1

Is combined population classified ?

C Select Np population members from

nondominated fronts in order of their ranking

C

Is child feasible?

Is gen < max_gen?

Select the first member of the first front

Stop

Set no of population =Np and gen = 0

Fig. 1. Computational

the best level, 2 is the next-best level and so on).Calculate the crowding distance of populationsin each nondomination level and sort populationsin descending order of crowding distance.

(Step 3) Tournament selection: Select two individuals atrandom. Compare their front number and crowd-ing distance. Select the better one and copy it tothe mating pool.

(Step 4) Crossover and mutation: The simulated binarycrossover (SBX) and polynomial mutation [18]have been used in the present work as explained

Identify nondominated individuals

Sort individuals according to descending order of crowding distance

Front = Front + 1

Identify nondominated individuals

alculate crowding distance of each individual

Sort individuals according to descending order of crowding distance

alculate crowding distance of each individual

Front = Front + 1

flow of NSGA-II.


in Appendix A. The crossover probability ofpc = 0.9 and a mutation probability of pm = 1/n(where n is the number of decision variables) areused. Here, distribution indexes [18] for crossoverand mutation operators as gc = 10 and gm = 10are used respectively. A child population of sizeNP is created.

(Step 5) Combine the parent population and child popula-tion. The size of combined population is 2NP.

(Step 6) Fast Nondominated Sorting of combined popula-tion: The combined population is sorted accord-ing to nondomination and crowding distance.Since all parent and child population membersare included, elitism is ensured. Now, populationsbelonging to the best nondominated set F1 are ofbest populations in the combined population andmust be emphasized more than any other popula-tion in the combined population. If the size of F1

is smaller than NP, all members of the set F1 arechosen for the new population. The remainingmembers of the new population are chosen fromsubsequent nondominated fronts in the order oftheir ranking. Thus, population members fromthe set F2 are chosen next, followed by solutionsfrom the set F3 and so on. This procedure is con-tinued until no more sets can be accommodated.If the set Fl is the last nondominated set beyondwhich no other set can be accommodated. In gen-eral, the count of solutions in all sets from F1 to Fl

would be larger than population size NP. Tochoose exactly NP population members, popula-tion members of the last front Fl are sorted using

Table 1Hourly generation (MW) schedule, cost (·106 $) and emission (·105 lb) obtain

Hour P1 P2 P3 P4 P5 P6

1 150.0003 135.2160 90.8166 83.0824 175.8707 124.22 150.0000 135.0005 137.5988 62.7808 178.3749 158.73 150.0021 135.0001 184.7400 112.7577 223.0310 157.64 209.3793 135.0004 241.0086 150.2570 238.3450 159.95 193.8733 135.0003 313.1606 162.2772 223.9264 138.26 219.1637 194.7078 337.2860 201.2667 204.8291 147.17 204.0349 255.9436 326.5886 244.3597 224.2474 148.08 215.0954 299.4115 303.6719 266.9448 234.2255 143.69 263.3450 303.2617 339.9995 299.9937 242.9995 159.910 336.7451 336.8279 339.9998 300.0000 242.9999 159.911 370.5632 395.3005 340.0000 299.9998 242.9999 159.912 369.7308 444.7153 339.9994 300.0000 243.0000 160.013 340.0234 388.3938 340.0000 300.0000 243.0000 159.914 279.6662 345.3625 308.9811 273.5422 242.9104 159.915 215.0916 289.6284 301.8692 297.4123 241.0544 158.416 150.0006 214.2817 240.9584 248.4114 222.9947 148.317 196.8637 136.3887 190.8215 242.6391 230.2764 159.918 234.0913 216.0244 211.8307 254.2934 227.6584 159.919 276.3832 219.4764 269.3099 294.6170 242.3806 159.920 331.0557 293.7675 334.3439 299.9888 242.9614 159.921 261.5277 305.1080 339.9738 299.9999 242.9997 159.922 208.8596 225.1090 261.1452 288.3805 224.8644 123.523 150.0006 153.4765 188.7817 242.6190 181.4841 104.724 150.0005 135.0000 129.6401 239.1916 157.2371 122.9

the crowded-comparison operator in descendingorder and best population members are chosento fill all population slots. The new populationof size NP is formed.

(Step 7) Stopping rule: The process can be stopped after afixed number of generations, or when no signifi-cant improvement in the solution occurs. In thispaper, NSGA-II is run for a fixed number of gen-erations. Check for stopping criteria. If it is satis-fied then go to Step 8 else copy new population toparent population and go to Step 3.

(Step 8) Select the first population member of the firstfront.

(Step 9) Stop.

The computational flow chart of NSGA-II is shown inthe Fig. 1.

7. Simulation results

The ten-unit test system with nonsmooth fuel cost andemission level functions is used in this paper to demon-strate the performance of the proposed method. Thedemand of the system was divided into 24 intervals. Unitdata was modified from [4] and can be found in TablesA.1 and A.2 in Appendix B.

The proposed algorithm has been implemented in MAT-LAB 7 on a PC (Pentium-IV, 80 GB, 3.0 GHz). The popu-lation size and the maximum number of generations havebeen selected as 20 and 100, respectively for the systemunder consideration. The crossover and mutation probabil-ities have been selected as 0.9 and 0.2, respectively. At first,

ed from dynamic economic dispatch

P7 P8 P9 P10 Cost Emission

571 129.7777 71.1322 52.1124 43.4211 2.5168 3.1740484 129.6114 87.5146 49.4353 43.4207299 104.2200 117.4920 58.2987 43.4214297 108.4817 104.4145 52.0572 43.4216124 129.9782 100.5003 79.9997 43.4212314 129.5921 119.9996 79.9997 43.4217799 129.7872 119.9995 59.7353 43.4220991 129.8331 120.0000 79.3114 43.4218958 129.9646 120.0000 79.9993 54.9894996 129.9992 120.0000 79.9999 55.0000999 129.9995 120.0000 79.9999 54.9999000 129.9997 119.9998 79.9999 54.9994998 130.0000 120.0000 79.9997 54.9997998 129.9735 119.9998 79.9862 54.9177122 129.7510 106.7044 52.0616 43.4218505 129.7757 119.9997 79.9337 43.4208995 129.6885 120.0000 69.7003 43.4214999 129.9549 120.0000 79.9999 43.4231994 129.7165 119.9994 79.9991 43.4244993 129.9301 119.9993 79.9998 54.9947958 129.9459 119.9998 79.9994 54.9994583 129.9754 119.9985 52.0701 43.4214961 129.6118 119.9994 50.0175 43.4211155 99.6194 91.6786 40.7427 43.4205


fuel cost and emission objectives are minimized individu-ally using real-coded genetic algorithm in order to explorethe extreme points of the trade-off surface and evaluate thediversity characteristics of the pareto-optimal solutionsobtained by the proposed algorithm.

Table 1 shows hourly generation schedule, cost andemission obtained from dynamic economic dispatch usingRCGA. Table 2 summarizes hourly generation schedule,



1 174.8080 188.6501 101.9869 132.3613 95.2927 81.92 191.7091 154.8541 120.9425 141.3736 142.6534 112.93 258.4501 200.4815 149.2758 131.4198 132.1808 135.44 304.0630 209.5215 161.9455 148.1058 175.2617 144.55 305.1895 219.7379 175.2079 166.9474 188.9595 139.16 292.3681 202.3881 226.6530 207.0497 224.2657 159.57 324.0116 216.7012 249.9903 205.0014 241.6878 159.88 331.1859 233.9533 296.6265 233.2594 202.7085 159.99 337.1272 282.6307 312.9842 274.7691 242.9793 159.910 361.0516 334.9448 318.7049 299.2637 243.0000 160.011 384.1888 381.7184 339.9953 299.9993 243.0000 160.012 406.0858 408.3973 340.0000 300.0000 243.0000 159.913 369.5026 366.1170 335.3231 297.6455 242.9997 159.914 336.9573 329.8223 296.8850 252.0269 235.4009 160.015 320.0531 293.1593 238.9675 217.5926 225.0799 159.916 258.5153 277.7038 171.0818 193.2007 231.1082 143.717 198.1038 261.0200 223.7778 166.5723 191.4545 112.718 257.1188 305.6274 233.6932 191.9455 173.1577 158.919 305.4218 310.0993 241.8483 230.3257 206.2804 159.220 346.9707 354.7377 301.7281 256.9630 242.8116 159.921 342.0109 318.4550 300.9892 246.9846 242.8425 159.922 266.1163 255.3410 222.7147 200.0106 228.5893 140.923 195.1313 235.1226 149.0955 158.3756 188.8071 107.224 181.2974 193.3034 119.4186 166.6463 147.2636 100.9



1 150.0014 136.2495 106.4747 116.6711 80.5442 125.82 150.5465 135.0310 173.5647 96.2554 117.3027 135.43 155.3053 174.6394 190.6634 117.5803 166.4208 159.14 199.6724 156.9451 237.0530 157.7959 187.6836 135.05 201.3435 138.3629 291.4233 166.3230 220.3843 159.46 183.5267 217.8569 292.2957 205.3736 232.9672 160.07 261.5049 157.3354 331.6192 218.0883 242.6555 159.98 273.5036 225.9981 295.8563 253.0238 242.0839 159.99 293.5230 286.0555 328.5824 298.5848 242.9963 159.910 317.8649 358.8123 337.3030 299.6257 242.9879 159.911 379.1188 386.7769 340.0000 300.0000 242.9960 159.912 394.3017 420.1542 339.9972 300.0000 243.0000 160.013 354.7062 376.6550 337.1296 299.9995 242.9996 159.914 289.1482 305.0699 320.5566 292.3232 242.8261 159.915 251.8416 235.1787 274.9380 292.5701 235.6232 159.916 181.1091 215.9403 210.0928 244.5697 225.3100 157.917 150.2536 206.1057 247.1464 228.2928 225.8783 119.118 161.6819 249.6347 285.4126 246.1945 228.4605 143.219 213.9291 272.6255 302.4649 279.7436 229.6403 159.920 291.8890 328.9087 338.1126 299.9838 242.9943 160.021 273.1907 308.4240 327.4411 297.6824 242.9925 159.922 211.5307 229.6167 252.5434 256.1307 222.4579 136.323 150.0128 176.4459 180.7401 209.1211 173.3931 123.324 150.3320 135.0028 135.3240 178.9356 151.9618 131.0

cost and emission obtained from dynamic emission dis-patch using RCGA. Results obtained from proposedNSGA-II are given in Table 3. It is seen from Tables 1and 2 that cost is 2.5168 · 106$ under dynamic economicdispatch but it increases to 2.6563 · 106$ under dynamicemission dispatch and emission obtained from dynamiceconomic dispatch is 3.1740 · 105lb but it decreases to3.0412 · 105lb under dynamic emission dispatch. Table 3

ed from dynamic emission dispatch


054 101.8174 58.9972 65.5667 54.9981 2.6563 3.0412941 91.3413 65.0233 57.0712 54.9944245 91.6170 80.8921 55.0538 53.6335997 83.8826 84.1953 77.8534 54.9990208 105.4556 88.9561 77.7773 54.9997350 128.1540 105.3209 77.4094 54.9999666 129.8212 95.3891 79.9999 54.9994957 126.4413 117.4108 79.9956 54.9998992 129.9993 119.9998 79.9364 54.9997000 129.9999 119.9990 79.9998 54.9996000 129.9999 120.0000 80.0000 54.9998999 129.9999 119.9999 80.0000 55.0000999 129.9993 119.9996 80.0000 55.0000000 129.9997 120.0000 79.9850 54.9999935 127.1143 119.9984 80.0000 54.9984125 105.8116 97.4078 66.7549 54.9998035 118.3832 119.7930 74.5319 54.9994515 119.9959 109.1758 74.2030 54.9998088 129.9532 118.8013 79.9813 54.9999994 129.9994 119.9960 80.0000 54.9998996 129.7368 119.9992 79.9999 54.9999649 111.5130 118.3139 79.8428 54.9999914 100.6326 99.4842 76.5431 54.9994625 77.7187 103.9420 64.6983 54.9961

ed from proposed NSGA-II algorithm


117 126.5755 109.7029 58.5082 45.0902 2.5226 3.0994467 127.0207 85.6996 63.6581 47.9270082 100.5033 112.9911 54.7811 54.8996569 125.5319 107.6597 80.0000 54.9946123 129.5840 94.2504 64.1587 54.9835000 129.9088 119.9837 80.0000 54.9875766 129.9405 119.9854 79.9917 54.9958625 129.9930 119.9759 79.9947 54.9951985 129.9994 119.9943 79.9997 54.9984956 129.9946 119.9996 79.9976 55.0000968 129.9972 119.9992 80.0000 54.9992000 129.9987 120.0000 79.9989 54.9985963 130.0000 119.9995 79.9923 54.9972934 129.9831 119.9986 79.9983 54.9963984 129.9955 119.9980 79.9881 54.9980172 129.5989 119.9945 58.8393 54.9810597 129.3882 90.9849 68.0453 54.9893779 129.9286 119.3702 58.1235 54.9930981 129.7931 120.0000 71.9376 54.9942000 129.9966 119.9991 79.9970 54.9977971 129.9969 119.9952 80.0000 54.9998280 129.9463 119.9529 63.8169 54.9992792 129.6407 119.6210 49.5205 52.3605109 129.8147 118.2211 24.0029 54.6686

Table 4Hourly generation (MW) schedule, cost (·106 $) and emission (·105 lb) obtained from scalar optimization problem when w = 0.5

Hour P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 Cost Emission

1 150.0135 135.0651 171.7208 177.0318 109.0171 104.8753 62.4477 56.0262 43.3355 46.1546 2.5251 3.12462 152.5518 141.8248 197.6778 172.4214 120.9106 75.5416 90.6767 80.6330 47.0437 53.38903 152.1059 135.0004 223.4787 219.7224 123.5018 115.5938 108.3501 110.0332 44.0937 54.86134 176.6697 136.0753 209.3639 230.2175 164.8874 157.7644 129.4583 119.6680 62.6775 54.96515 151.1029 199.4557 250.4702 257.4377 152.4703 136.4911 129.6317 119.6600 68.4070 54.97256 160.0629 237.7694 303.1455 248.9000 182.9644 159.9868 129.7634 119.4516 79.9129 54.98477 187.7623 218.2595 301.3198 280.1853 223.1004 159.9986 129.8860 119.9790 79.9819 54.99568 179.2158 282.1391 305.4484 280.4146 242.7467 159.9820 129.9963 119.9981 79.9886 54.98389 242.8601 333.4279 333.2437 297.1885 242.9959 159.9992 129.9994 119.9939 80.0000 54.999310 289.0439 386.0399 338.5638 299.9781 243.0000 159.9974 129.9900 119.9998 79.9984 54.994511 364.7733 401.0962 339.9983 299.9994 242.9935 160.0000 130.0000 119.9999 80.0000 54.993912 400.5420 413.9404 339.9985 300.0000 242.9965 159.9972 129.9933 120.0000 79.9985 54.998513 344.0327 386.0448 338.3820 299.9996 242.9956 159.9951 130.0000 119.9940 80.0000 54.999414 267.0046 350.3788 297.9603 291.9196 242.9843 159.9936 129.9995 119.9798 79.9976 54.997015 231.5481 300.9842 266.9934 248.7273 242.4454 159.9934 129.9932 119.9994 79.9689 54.998616 182.2297 226.8151 224.8229 244.6361 233.7497 140.1220 114.6138 113.2758 63.4115 54.985017 152.6113 220.8881 197.5207 212.0895 220.7505 144.3481 129.9994 119.2979 68.0436 54.997518 193.5284 250.8139 193.3813 258.6399 238.5981 158.0453 129.9900 119.9780 79.6916 54.987619 212.4401 287.4515 268.7397 282.7456 238.8859 159.9980 130.0000 119.9983 80.0000 54.999620 292.3302 334.6683 335.3944 296.6009 242.9853 159.9980 129.9933 119.9988 79.9976 54.992721 290.7706 305.0372 314.2357 296.9546 242.9827 159.9794 130.0000 119.9986 80.0000 54.999322 215.1713 231.8134 260.5426 248.2308 207.0094 130.9662 129.9995 119.0021 79.9885 54.997723 150.0303 159.3560 191.5382 216.4360 188.6074 115.4146 129.9932 116.9824 54.5237 54.154524 150.0465 137.0307 148.4868 178.7665 168.1235 129.8655 114.6138 93.3625 53.7269 54.9779


shows that cost is 2.5226 · 106$ which is more than2.5168 · 106$ and less than 2.6563 · 106$ and emission is3.0994 · 105lb which is less than 3.1740 · 105lb and morethan 3.0412 · 105lb. The distribution of 20 nondominatedsolutions obtained in the last generation of proposedapproach is shown in Fig. 2. Results of the proposedapproach are compared to the classical technique.

For comparison purposes, the problem has been con-verted to a scalar optimization problem by linear combina-tion of cost and emission objective as follows:

Minimize wf1 þ ð1� wÞkf2 ð11Þwhere k is the scaling factor and w is a weighting factorwhich is varied. The problem is minimized using real-codedgenetic algorithm. To generate 20 nondominated solutions,

3.07 3.08 3.09 3.1 3.11 3.12 3.13 3.14

x 105

2.52

2.521

2.522

2.523

2.524

2.525

2.526

2.527

2.528

2.529x 10

6

Cos

t ($

)

Emission (lb)

Fig. 2. Pareto-optimal front of the proposed approach in the lastgeneration.

the algorithm has been applied 20 times by varying w. Ta-ble 4 provides hourly generation schedule, cost and emis-sion when w is set to 0.5, i.e. equal weightage is given forboth cost and emission. The pareto-optimal front of thescalar optimization problem is shown in Fig. 3. ComparingFig. 2 and Fig. 3, it can be concluded that the 20 solutionsshown in Fig. 2 that represent the results of the proposedapproach have been obtained in one single run while thesolutions shown in Fig. 3 have been obtained in 20 separateruns. It is worth mentioning that the run time per solutionof the scalar optimization problem was 18 min and 25.363 swhile that of the proposed approach to produce 20 solu-tions was 20 min 11.475 s. Comparing Tables 3 and 4 it isseen that the proposed approach provides better cost andemission compared to the classical technique.

3.04 3.06 3.08 3.1 3.12 3.14 3.16 3.182.5

2.52

2.54

2.56

2.58

2.6

2.62

2.64

2.66x 10

6

Co

st ($

)

Emission (lb)x 10

5

Fig. 3. Pareto-optimal front of linear combination in 20 separate runs.


8. Conclusion

In this paper nondominated sorting genetic algorithm-II has been applied to solve dynamic economic emissiondispatch problem. The problem has been formulated asmulti-objective optimization problem with competing fuelcost and emission objectives. Numerical result shows thatthe proposed approach can obtain better solution andrequires less CPU time than the classical technique. Alsoin the proposed approach multiple pareto-optimal solu-tions can be found in one simulation run. Since the pro-posed approach does not impose any limitation on thenumber of objectives, it can be extended to include moreobjectives.

Appendix A

A.1. Simulated Binary Crossover (SBX) operator

The procedure of computing child populations c1 and c2

from two parent populations y1 and y2 under SBX operatoras follows:

1. Create a random number u between 0 and 1.2. Find a parameter c using a polynomial probability dis-

tribution as follows:

c ¼ðuaÞ1=ðgcþ1Þ if u 6 1

a

ð1=ð2� uaÞÞ1=ðgcþÞ; otherwise

(

where a ¼ 2� b�ðgcþ1Þ and b is calculated as follows:

b ¼ 1þ 2

y2 � y1

min½ðy1 � ylÞ; ðyu � y2Þ�

Here, the parameter y is assumed to vary in [yl, yu].Here, the parameter gc is the distribution index forSBX and can take any non-negative value. A small valueof gc allows the creation of child populations far awayfrom parents and a large value restricts only near-parentpopulations to be created as child populations.

B ¼

0:000049 0:000014 0:000015 0:000015 0:000016

0:000014 0:000045 0:000016 0:000016 0:000017

0:000015 0:000016 0:000039 0:000010 0:000012

0:000015 0:000016 0:000010 0:000040 0:000014

0:000016 0:000017 0:000012 0:000014 0:000035

0:000017 0:000015 0:000012 0:000010 0:000011

0:000017 0:000015 0:000014 0:000011 0:000013

0:000018 0:000016 0:000014 0:000012 0:000013

0:000019 0:000018 0:000016 0:000014 0:000015

0:000020 0:000018 0:000016 0:000015 0:000016

26666666666666666664

3. The intermediate populations are calculated as follows:

cp1 ¼ 0:5½ðy1 þ y2Þ � cðjy2 � y1jÞ�cp2 ¼ 0:5½ðy1 þ y2Þ þ cðjy2 � y1jÞ�

Each variable is chosen with a probability pc and theabove SBX operator is applied variable-by-variable.

A.2. Polynomial mutation operator

A polynomial probability distribution is used to create achild population in the vicinity of a parent populationunder the mutation operator. The following procedure isused:

1. Create a random number u between 0 and 1.2. Calculate the parameter d as follows:

d¼ ½2uþð1�2uÞð1�/Þðgmþ1Þ�1

ðgmþ1Þ�1; if u60:5

1�½2ð1�uÞþ2ðu�0:5Þð1�/Þðgmþ1Þ�1

ðgmþ1Þ; otherwise

(

where u ¼ min½ðcp�ylÞ;ðyu�cpÞ�ðyu�ylÞ

The parameter gm is the distri-bution index for mutation and takes any non-negativevalue.

3. Calculate the mutated child as follows:

c1 ¼ cp1 þ dðyu � ylÞc2 ¼ cp2 þ dðyu � ylÞ

The perturbance in the population can be adjusted byvarying gm and pm with generations as given below:

gm ¼ gm min þ gen

pm ¼1

nþ gen

genmax

1� 1

n

� �

where gm min is the user-defined minimum value for gm,pm is the probability of mutation, and n is the numberof decision variables.

Appendix B

See Tables A.1 and A.2.The transmission loss formula coefficients are:

0:000017 0:000017 0:000018 0:000019 0:000020

0:000015 0:000015 0:000016 0:000018 0:000018

0:000012 0:000014 0:000014 0:000016 0:000016

0:000010 0:000011 0:000012 0:000014 0:000015

0:000011 0:000013 0:000013 0:000015 0:000016

0:000036 0:000012 0:000012 0:000014 0:000015

0:000012 0:000038 0:000016 0:000016 0:000018

0:000012 0:000016 0:000040 0:000015 0:000016

0:000014 0:000016 0:000015 0:000042 0:000019

0:000015 0:000018 0:000016 0:000019 0:000044

37777777777777777775

Tab

leA

.1G

ener

ato

rch

arac

teri

stic

s

Un

itP

max

iP

min

ia

ib

ic i

di

e ia i

b ic i

g id i

UR

iD

Ri

MW

MW

$/h

$/M

Wh

$/(M

W)2

h$/

hra

d/M

Wlb

/hlb

/MW

hlb

/(M

W)2

hlb

/h1/

MW

MW

/hM

W/h

115

047

078

6.79

8838

.539

70.

1524

450

0.04

110

3.39

08�

2.44

440.

0312

0.50

350.

0207

8080

213

547

045

1.32

5146

.159

10.

1058

600

0.03

610

3.39

08�

2.44

440.

0312

0.50

350.

0207

8080

373

340

1049

.997

740

.396

50.

0280

320

0.02

830

0.39

10�

4.06

950.

0509

0.49

680.

0202

8080

460

300

1243

.531

138

.305

50.

0354

260

0.05

230

0.39

10�

4.06

950.

0509

0.49

680.

0202

5050

573

243

1658

.569

636

.327

80.

0211

280

0.06

332

0.00

06�

3.81

320.

0344

0.49

720.

0200

5050

657

160

1356

.659

238

.270

40.

0179

310

0.04

832

0.00

06�

3.81

320.

0344

0.49

720.

0200

5050

720

130

1450

.704

536

.510

40.

0121

300

0.08

633

0.00

56�

3.90

230.

0465

0.51

630.

0214

3030

847

120

1450

.704

536

.510

40.

0121

340

0.08

233

0.00

56�

3.90

230.

0465

0.51

630.

0214

3030

920

8014

55.6

056

39.5

804

0.10

9027

00.

098

350.

0056

�3.

9524

0.04

650.

5475

0.02

3430

3010

1055

1469

.402

640

.540

70.

1295

380

0.09

436

0.00

12�

3.98

640.

0470

0.54

750.

0234

3030

Table A.2Load demands

Hour PD (MW) Hour PD (MW) Hour PD (MW)

1 1036 9 1924 17 14802 1110 10 2022 18 16283 1258 11 2106 19 17764 1406 12 2150 20 19725 1480 13 2072 21 19246 1628 14 1924 22 16287 1702 15 1776 23 13328 1776 16 1554 24 1184


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