fuel constrained economic emission dispatch using nondominated sorting genetic algorithm-ii

lable at ScienceDirect

Energy xxx (2014) 1e16

Contents lists avai

Energy

journal homepage: www.elsevier .com/locate/energy

Fuel constrained economic emission dispatch using nondominatedsorting genetic algorithm-II

M. Basu*

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

a r t i c l e i n f o

Article history:Received 27 September 2013Received in revised form16 October 2014Accepted 20 October 2014Available online xxx

Keywords:Fuel constrained economic emissiondispatchNondominated sorting genetic algorithm-IIEconomic emission dispatch

* Fax: þ91 33 23357254.E-mail address: [email protected].

http://dx.doi.org/10.1016/j.energy.2014.10.0520360-5442/© 2014 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Basu M,Energy (2014), http://dx.doi.org/10.1016/j.en

a b s t r a c t

This paper presents nondominated sorting genetic algorithm-II for solving fuel constrained economicemission dispatch problem of thermal generating units. This is a multi-objective optimization problemwhich includes the standard load constraints as well as the fuel constraints. The generation schedule iscompared to that which would result if fuel constraints are ignored. The comparison shows that fuelconsumed can be adequately controlled by adjusting the power output of various generating units so thatthe power system operates within its fuel limitations and within contractual constraints. It has beenfound that one of the two objectives (i.e. fuel cost and emission level) may be increased while other maybe decreased to serve the same power demand but this may well compensate for the penalty that mightbe otherwise imposed for not maintaining the fuel contract. Numerical results for two test systems havebeen presented and the test results are compared with those obtained from strength pareto evolutionaryalgorithm 2.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Economic emission dispatch involves the allocation of genera-tion among the committed generating units so as to optimize boththe fuel cost and emission level simultaneously while satisfying theseveral operating constraints.

Some power utilities have encountered a new dispatch problem,perhaps more significant than economic emission dispatch prob-lem, because of the sudden concern over fuel shortages. Fuel sup-pliers have imposed increased constraints in their fuel supplycontracts to the point that utilities have been forced to reschedulegeneration on the basis of fuel availability. With the ever increasingproportion of the fuel budget in the total operating cost andincreasing concern over the environmental consideration, eco-nomic emission dispatch [6e9] problem has been popped up.Several papers have been published in the area of fuel scheduling ofthermal units [1e5]. The fuel constrained economic emissiondispatch problem is a multi-objective mathematical programmingproblem which is concerned with the attempt to improve eachobjective simultaneously while satisfying the standard load con-straints and fuel constraints. The fuel constrained economic

Fuel constrained economic eergy.2014.10.052

emission dispatch problemmay be solved by dividing the total timeperiod involved into discrete time increments. Each of the objec-tives is a function of one or more variables from only one time step.Some constraints are made up of variables drawn from one timestep whereas others span two or more time steps.

Over the past few years, several researches have been made onthe development of multi-objective evolutionary search strategies.NSGA II (Nondominating sorting genetic algorithm II) [6], MOEA(multi-objective evolutionary algorithm) [7], multi-objective par-ticle swarm optimization [8,9], FCPSO (fuzzy clustering-basedparticle swarm optimization) [10], hybrid multi-objective optimi-zation algorithm [11], multi-objective fuzzy dominance basedbacterial foraging algorithm [12], multi-objective differential evo-lution algorithm [13,14], multi-objective harmony search algorithm[15], multi-objective q-particle swarm optimization [16] etc.,constitute the pioneering multi-objective approaches that havebeen applied to solve the EED (economic emission dispatch)problem. Multi-objective optimization based on an enhancedfirefly algorithm [17] has been applied to solve reserve constrainedcombined heat and power dynamic economic emission dispatchproblem. An efficient scenario-based and fuzzy self-adaptivelearning particle swarm optimization approach [18] is applied forsolving dynamic economic emission dispatch problem consideringload and wind power uncertainties. Hybrid firefly algorithm [19]has been proposed for economic emission dispatch solutionconsidering wind power penetration.

mission dispatch using nondominated sorting genetic algorithm-II,

Delta:1_given name

mailto:[email protected]

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http://dx.doi.org/10.1016/j.energy.2014.10.052



Nomenclature

Fim fuel delivered to thermal unit i in interval mFmini ; Fmax

i lower and upper fuel delivery limits of ith thermalgenerating unit

FDm fuel delivered in interval mFc total fuel costFe total fuel emissionРim output power of thermal unit i in interval mРmini ; Рmax

i lower and upper generation limits ith thermalgenerating unit

РDm load demand in interval mtm duration of subinterval mVim fuel storage of thermal unit i in interval mVmini ;Vmax

i lower and upper fuel storage limits of ith thermalgenerating unit

V0i initial fuel storage of thermal generating unit i

Fig. 1. Example of valve-point cost function with 5 valves. A e Primary Valve,B e Secondary Valve, C e Tertiary Valve, D e Quaternary Valve, E � Quandary Valve.

M. Basu / Energy xxx (2014) 1e162

This paper proposes NSGA II (nondominating sorting geneticalgorithm II) [23] for solving fuel constrained economic emissiondispatch problem of thermal generating units. This problem isformulated as a nonlinear constrainedmulti-objective optimizationproblem. Due to difficulties of binary representation when dealingwith continuous search space with large dimensions, the proposedapproach has been implemented by using RCGA (real-coded ge-netic algorithm) [20,21]. The proposed method is validated byapplying it to two test systems. Results obtained from the proposedmethod are compared with those obtained from SPEA 2 (strengthpareto evolutionary algorithm 2).

2. Problem formulation

For convenience the entire scheduling period is divided into anumber of sub intervals each having a constant load demand. Thesystem has N thermal generating units over M time intervals. Thefollowing objectives and constraints are taken into account in theformulation of FCEED (fuel constrained economic emissiondispatch) problem.

2.1. Objectives

2.1.1. EconomyThe cost function of a fossil fuel fired generating unit is obtained

from data points taken during “heat run” tests, when input andoutput data are measured as the unit is slowly varied through itsoperating region. Wire drawing effects, occurring as each steamadmission valve in a turbine starts to open, produce a rippling effecton the unit curve. In reality, a sharp increase in fuel loss is added tothe fuel cost curve due to wire drawing effects when steamadmission valve starts to open. This procedure is named as valvepoint effect. To model the effect of valve-points, a recurring recti-fied sinusoid contribution is added to the quadratic function [24],such as the one shown in Fig. 1.

The fuel cost function considering valve-point loading of thegenerating unit is given as

Fc¼XМm¼1

XNi¼1

tmhaiþbiРimþciР

2imþ

��di sinnei�Рmini �Рim

�o��i (1)

where ai, bi and ci are the fuel cost coefficients of the ith generatingunit, and di and ei are the fuel cost coefficients of the ith generatingunit due to valve-point effect.

Please cite this article in press as: Basu M, Fuel constrained economic eEnergy (2014), http://dx.doi.org/10.1016/j.energy.2014.10.052

2.1.2. EmissionThe most important emissions considered from the fossil-based

generating stations, because of their effects on the environment,are sulfur dioxide (SO2), carbon dioxide (CO2) and nitrogen oxides(NOx). These emissions can be modeled through functions thatassociate emissions with real power production for each unit. Theform of the emission functionmodel depends on the emission type.The emissions of both SO2 and CO2 can be modeled as quadraticpolynomial functions. NOx emissions are more difficult to modelsince they come from different sources and their production isassociated with several factors such as boiler temperature and aircontent. One approach to represent NOx emissions is using acombination of polynomial and exponential terms [25]. Since NOx

emission is more harmful than SO2 and CO2, in this study, nitrogenoxides (NOx) emission is taken as the selected index from theviewpoint of environment conservation.

The total emission level from all the units in the system can beexpressed as

Fe ¼XМm¼1

XNi¼1

tmhai þ biРim þ giР

2im þ si expðqiРimÞ

i(2)

where ai, bi, gi, si and qi are the emission coefficients of ith gener-ating unit. These coefficients are estimated on the basis of unitemissions test results based on using least squares estimation.

2.2. Constraints

2.2.1. Power balance constraintsAt each interval, the total active power generation must balance

the predicted power demand.

XNi¼1

Рim � РDm ¼ 0; m2М (3)

2.2.2. Fuel delivery constraintsAt each interval, summation of fuel delivered to all units must be

equal to fuel supplied by the supplier.


M. Basu / Energy xxx (2014) 1e16 3

XNi¼1

Fim � FDm ¼ 0; m2М (4)

2.2.3. Fuel storage constraintsThe volume of fuel at each unit at the beginning of each interval

plus delivery of fuel to that unit minus the fuel burned at that unitgives the fuel remaining at the beginning of the next interval.

Fig. 2. Computational


Vim ¼ Viðm�1Þ þ Fim � tmhhi þ diРim þ miР

2im

þ��li sinnri�Рmin

i � Рim

�o��i; i2N; m2М(5)

where hi, di and mi are the fuel consumption coefficients of ithgenerating unit and li and ri are the fuel consumption coefficientsof the ith generating unit due to valve-point effect.

flow of NSGA-II.



2.2.4. Generation limitsThe power generated by each unit at each interval should be

within its lower limit Рmin and upper limit Рmax. So that

Рmini � Рim � Рmax

i ; i2N; m2М (6)

2.2.5. Fuel delivery limitsThe fuel delivered to each unit at each interval should be within

its lower limit Fmin and upper limit Fmax. So that

Fmini � Fim � Fmax

i ; i2N; m2М (7)

2.2.6. Fuel storage limitsThe fuel storage limit of each unit at each interval should be

within its lower limit Vmin and upper limit Vmax. So that

Vmini � Vim � Vmax

i ; i2N; m2М (8)

3. Principle of multi-objective optimization

Most of the real-world problems involve simultaneous optimi-zation of several objective functions. These functions are non-commensurable and often competing and conflicting objectives.Multi-objective optimization having such conflicting objectivefunctions gives rise to a set of optimal solutions, instead of oneoptimal solution because no solution can be considered to be betterthan any other with respect to all objective functions. Theseoptimal solutions are known as pareto-optimal solutions.

Generally, multi-objective optimization problem consisting of anumber of objectives and several equality and inequality con-straints can be formulated as follows:

Minimize fiðxÞ i ¼ 1;…::;Nobj (9)

Subject to�gkðxÞ ¼ 0 k ¼ 1; ::::;ΚhlðxÞ � 0 l ¼ 1; ::::; L (10)

where fi is the ith objective function, x is a decision vector.


4. Nondominated sorting genetic algorithm-II

4.1. General approach

Srinivas and Deb [22] introduced NSGA (nondominated sortinggenetic algorithm) to deal with multi-objective optimizationproblems. In this algorithm, nondomination is used as rankingcriterion of solutions, and fitness sharing is used for diversificationcontrol in the search space. Due to high level sensitivity of NSGAperformance to fitness sharing parameters, Deb et al. [23] intro-duced NSGA-II (nondominated sorting genetic algorithm-II), whichis faster and more reliable than its predecessor. Before describingthe Nondominated Sorting Genetic Algorithm-II [23], fast non-dominated sorting procedure, fast crowded distance estimationprocedure and simple crowded comparison operator will bediscussed.

i) Fast nondominated sorting procedure

In order to identify solutions of the first nondominated front in apopulation of size NР, each solution can be compared with everyother solution in the population to find if it is dominated. At thisstage, all individuals in the first nondominated front are found. Inorder to find the individuals in the next nondominated front, thesolutions of the first front are discounted temporarily and eachsolution of the remaining population can be compared with everyother solution of the remaining population to find if it is dominated.Thus all individuals in the second nondominated front are found.This is true for finding third and higher levels of nondomination.

For each solution two entities are calculated: a) dominationcount np, the number of solutions which dominate the solution p,and b) Sp, a set of solutions that the solution p dominates. The al-gorithm for the formation of fast nondominated sort is describedbelow.

Each population is assigned a rank equal to its nondominationlevel or front number (1 is the best level, 2 is the next-best level andso on).

ii) Fast crowded distance estimation procedure

To get an estimate of the density of solutions surrounding aparticular solution in the population, the average distance of two



points on either side of this point along each of the objectives iscalculated. This quantity serves as an estimate of the perimeter ofthe cuboid formed by using the nearest neighbors as the vertices.This is called crowding distance. The crowding-distance computa-tion requires sorting the population according to each objectivefunction value in ascending order of magnitude. Thereafter, for eachobjective function, the boundary populations (populations withsmallest and largest function values) are assigned very high dis-tance value so that boundary points are always selected. All otherintermediate populations are assigned a distance value equal to theabsolute normalized difference in the function values of twoadjacent populations. This calculation is continued with otherobjective functions. The overall crowding-distance value is calcu-lated as the sum of individual distance values corresponding toeach objective. Each objective function is normalized beforecalculating the crowding distance.

The algorithm shown below outlines the crowding distancecomputation procedure of all solutions in a nondominatedset F.

Here, F[i].m refers to the mth objective function value of the ithindividual in the set F. fmax

m and fminm are the maximum and mini-

mum values of the mth objective function.

iii) Crowded-comparison operator

The crowded-comparison operator guides the selection processat the various stages of the algorithm toward a uniformly spread-out pareto-optimal front. Every individual i in the population hastwo attributes:

a) nondomination rank (irank)b) crowding distance (idistance)

i 3 j if irank < jrank or ((irank ¼ jrank) and (idistance > jdistance))Between two populations with differing nondomination ranks,

the population with the lower (better) rank is preferred. If bothpopulations belong to the same front, then the population withlarger crowding distance is preferred.

4.2. Computational flow

Here, real-coded NSGA-II has been used. Generally, the NSGA-IIalgorithm can be described in the following steps:

Step 1) Initialize: Initially, a random parent population Рt of sizeNР is created.

Please cite this article in press as: Basu M, Fuel constrained economic emiEnergy (2014), http://dx.doi.org/10.1016/j.energy.2014.10.052

Step 2) Fast Nondominated Sorting of parent population: Thepopulation is sorted based on the nondomination. Each popu-lation is assigned a rank equal to its nondomination level orfront number (1 is the best level, 2 is the next-best level and soon).Step 3) Tournament Selection: Select two individuals at random.Compare their front number and crowding distance. Select thebetter one and copy it to the mating pool.Step 4) Crossover and Mutation: The SBX (Simulated BinaryCrossover) and polynomial mutation [20] have been used in thepresent work as explained in appendix-1. The crossover prob-ability of pc ¼ 0.9 and a mutation probability of pm ¼ 1/n (wheren is the number of decision variables) are used. Here, distribu-tion indexes [20] for crossover and mutation operators ashc ¼ 10 and hm ¼ 10 are used respectively. A child population Qt

of size NР is created.Step 5) Combine the parent population and child population.Combined population Rt ¼ Рt∪Qt. The size of combined popu-lation is 2NР.

Step 6) Fast Nondominated Sorting of combined population: Thecombined population is sorted according to nondomination.Since all parent and child population members are included,elitism is ensured. Now, populations belonging to the bestnondominated set F1 are of best populations in the combinedpopulation and must be emphasized more than any otherpopulation in the combined population. Let the size of F1 besmaller thanNР, all members of the set F1 are chosen for the newpopulation. The remaining members of the new population arechosen from subsequent nondominated fronts in the order oftheir ranking. Thus, population members from the set F2 arechosen next, followed by solutions from the set F3 and so on.This procedure is continued until no more sets can be accom-modated. If the set Fl is the last nondominated set beyond whichno other set can be accommodated. In general, the count ofsolutions in all sets from F1 to Flwould be larger than populationsize NР. To choose exactly NР population members, populationmembers of the last front Fl are sorted using the crowded-comparison operator in descending order and best populationmembers are chosen to fill all population slots. The new popu-lation Рtþ1 of size NР is formed. The new population is now usedfor tournament selection, crossover and mutation to create anew population Qtþ1 of size NР.Step 7) Stopping rule: The process is stopped after a fixednumber of generations. Check for stopping criteria. If it issatisfied then go to Step 8 else copy new population to parentpopulation and go to Step 3.Step 8) Select the first population member of the first front.

ssion dispatch using nondominated sorting genetic algorithm-II,


Step 9) Stop

The algorithm is given below.

The computational flow chart of NSGA-II is shown in the Fig. 2.

4.3. Best compromise solution

Once the Pareto optimal set is obtained, at the end of the NSGA-II, it is necessary to choose one solution from all non-dominatedsolutions that represents the best compromise according to therequirements of the decision maker. Due to the imprecise nature ofthe DM (decision maker's) judgment, it is natural to assume that

Fig. 3. Linear membership function.

Table 1Specification for test system 1.

Cases Condition

Case 1 Only load constraints are considered and fuel constraints are ignored.Case 2 Load constraints and fuel constraints are considered and all units

have sufficient coal.

Case 3 Load constraints and fuel constraints are considered and there is fuelshortage at unit 4.

Case 4 Load constraints and fuel constraints are considered and there is fuelshortage at unit 5.


the DM may have fuzzy or imprecise nature goals of each objectivefunction. Hence, the membership functions are introduced to rep-resents the goals of each objective function; each membershipfunction is defined by the experiences and intuitive knowledge ofthe decision maker.

In this study, a simple linear membership function depicted inFig. 3 and given by Equation (11) is considered for each of theobjective functions.

mi ¼

8>>>>><>>>>>:

0; if fi � fmaxi

fmaxi � fi

fmaxi � fmin

i

if fmini < fi < fmax

i

1; if fi � fmini

(11)

where, fmini and fmax

i are the minimum and the maximum value ofthe ith objective function among all non-dominated solutions,respectively. Themembership function mi is varied between 0 and 1,where mi ¼ 0 indicates the incompatibility of the solution with theset, while mi ¼ 1 means full compatibility.

For each non-dominated solution k, the normalized member-ship function mk is calculated as follows:

mk ¼PNobj

i¼1 mkiPМndk¼1

PNobj

i¼1 mki

(12)

Input parameters

РD1 ¼ 700 MW,РD2 ¼ 800 MW, РD3 ¼ 650 MWРD1 ¼ 700 MW,РD2 ¼ 800 MW, РD3 ¼ 650 MWFD1 ¼ FD2 ¼ FD3 ¼ 7000 tonV01 ¼ 2000 ton, V0

2 ¼ 5000 ton, V03 ¼ 5000 ton, V0

4 ¼ 8000 ton, V05 ¼ 8000 ton

РD1 ¼ 700 MW, РD2 ¼ 800, РD3 ¼ 650FD1 ¼ FD2 ¼ FD3 ¼ 7000 tonV01 ¼ 2000 ton, V0

2 ¼ 5000 ton, V03 ¼ 5000 ton, V0

4 ¼ 500 ton, V05 ¼ 8000 ton

РD1 ¼ 700 MW,РD2 ¼ 800 MW, РD3 ¼ 650 MWFD1 ¼ FD2 ¼ FD3 ¼ 7000 tonV01 ¼ 2000 ton, V0

2 ¼ 5000 ton, V03 ¼ 5000 ton, V0

4 ¼ 8000 ton, V05 ¼ 500 ton


Table 2Dispatch solution for case 1 of test system 1 without considering fuel constraints.

Interval Economic dispatch (RCGA) Emission dispatch (RCGA) Economic emission dispatch (NSGA-II) Economic emission dispatch (SPEA 2)

Generation(MW)

Objective Generation(MW)



Objective

1 Р1 ¼ 48.7070 Cost ¼ 1,057,760 $ Р1 ¼ 74.9805 Cost ¼ 1,127,095 $ Р1 ¼ 73.3965 Cost ¼ 1,087,943 $ Р1 ¼ 71.9981 Cost ¼ 1,086,006 $Р2 ¼ 124.9947 Р2 ¼ 108.2674 Р2 ¼ 122.7345 Р2 ¼ 111.9303Р3 ¼ 174.9226 Emission ¼ 847,189 lb Р3 ¼ 147.1719 Emission ¼ 561,342 lb Р3 ¼ 157.9995 Emission ¼ 624,113 lb Р3 ¼ 165.6123 Emission ¼ 624,565 lbР4 ¼ 52.6383 Р4 ¼ 207.8208 Р4 ¼ 126.7651 Р4 ¼ 149.7298Р5 ¼ 298.7373 Р5 ¼ 161.7594 Р5 ¼ 219.1045 Р5 ¼ 200.7295

2 Р1 ¼ 74.8560 Р1 ¼ 74.9881 Р1 ¼ 71.5928 Р1 ¼ 73.5603Р2 ¼ 124.9843 Р2 ¼ 119.3675 Р2 ¼ 122.6324 Р2 ¼ 119.8816Р3 ¼ 174.9952 Р3 ¼ 165.1740 Р3 ¼ 171.5753 Р3 ¼ 173.6553Р4 ¼ 125.2769 Р4 ¼ 232.9771 Р4 ¼ 194.1488 Р4 ¼ 171.1374Р5 ¼ 299.8876 Р5 ¼ 207.4933 Р5 ¼ 240.0507 Р5 ¼ 261.7655

3 Р1 ¼ 34.3568 Р1 ¼ 74.9018 Р1 ¼ 65.0902 Р1 ¼ 73.7655Р2 ¼ 123.2650 Р2 ¼ 100.3692 Р2 ¼ 93.1135 Р2 ¼ 119.7038Р3 ¼ 173.1910 Р3 ¼ 139.9243 Р3 ¼ 148.3967 Р3 ¼ 167.5600Р4 ¼ 40. 2155 Р4 ¼ 193.7269 Р4 ¼ 121.5782 Р4 ¼ 107.1962Р5 ¼ 278.9726 Р5 ¼ 141.0778 Р5 ¼ 221.8214 Р5 ¼ 181.7745


where Мnd is the number of non-dominated solutions and Nobj isthe number of objective functions. The function mk can beconsidered as a membership function of non-dominated solutionsin a fuzzy set, where the solution having the maximum mem-bership in the fuzzy set is considered as the best compromisesolution.

5. Simulation results

The proposed method has been applied to two test systems. Inorder to show the effectiveness of the proposed NSGA-II approach,SPEA 2 has been applied to solve the problem. All the algorithmsi.e. NSGA-II, SPEA 2, and RCGA, used in this paper for solving fuelconstrained economic emission dispatch problem are imple-mented by using MATLAB 7.0 on a PC (Pentium-IV, 80 GB,3.0 GHz).

5.1. Test system 1

This test system considers five coal-burning generating unitswhich remain on line for a 3-week period. All the generator data

Fig. 4. Cost and emission convergence for case 1 of test system 1.


containing coefficients of cost and coal consumption, fuel deliverylimits, fuel storage limits, load demand and fuel delivered duringthe scheduling period are given in the Table A.1 and Table A.2 in theappendix-2.

Here, four cases are considered. In case 1 economic dispatch,emission dispatch and economic emission dispatch are donewithout considering fuel constraints. In case 2 economic fueldispatch, emission fuel dispatch and economic emission fueldispatch are done considering fuel constraints when all the unitshave sufficient coal. Case 3 and case 4 are purposely structured toshow the interaction of the fuel deliveries and different types ofdispatch of the generating units when there is fuel shortage. Incase 3 there is fuel shortage at unit 4 and in case 4 there is fuelshortage at unit 5. Table 1 shows specification for this testsystem.

In order to explore the extreme points of the trade-off surface,fuel cost and emission objectives are minimized individually byusing RCGA (real coded genetic algorithm) for all the four cases. Incase of RCGA, the population size, maximum number of iterations,crossover and mutation probabilities have been selected as 100,

Fig. 5. Pareto-optimal front of the last iteration for case 1 of test system 1.


Table 3Fuel constrained dispatch solution for case 2 of test system 1 with initial fuel storage (tons) V0

1 ¼ 2000, V02 ¼ 5000, V0

3 ¼ 5000, V04 ¼ 8000, V0

5 ¼ 8000.

Interval Economic dispatch (RCGA) Emission dispatch (RCGA) Economic emission dispatch (NSGA-II) Economic emission dispatch (SPEA2)

Generation(MW)

Fuel delivered(tons)







Objective

1 Р1 ¼ 50.3862 F1 ¼ 778.1 Cost ¼ 1,057,804 $ Р1 ¼ 74.8411 F1 ¼ 700.3 Cost ¼ 1,127,496 $ Р1 ¼ 66.9647 F1 ¼ 914.1 Cost ¼ 1,084,709 $ Р1 ¼ 64.3156 F1 ¼ 644.0 Cost ¼ 1,087,806 $Р2 ¼ 124.4792 F2 ¼ 932.4 Р2 ¼ 106.5009 F2 ¼ 899.9 Р2 ¼ 113.0.3389 F2 ¼ 757.4 Р2 ¼ 122.6326 F2 ¼ 994.0Р3 ¼ 174.4406 F3 ¼ 1534.5 Emission ¼

850,138 lbР3 ¼ 145.1616 F3 ¼ 1447.0 Emission ¼

561,511 lbР3 ¼ 154.9983 F3 ¼ 1039.0 Emission ¼

631,164 lbР3 ¼ 170.3805 F3 ¼ 1923.3 Emission ¼

620,638 lbР4 ¼ 50.9235 F4 ¼ 1329.0 Р4 ¼ 210.4224 F4 ¼ 2950.7 Р4 ¼ 150.9667 F4 ¼ 2894.3 Р4 ¼ 141.3882 F4 ¼ 2949.9Р5 ¼ 299.7705 F5 ¼ 2426.0 Р5 ¼ 163.0741 F5 ¼ 1002.1 Р5 ¼ 213.7314 F5 ¼ 1395.3 Р5 ¼ 201.2832 F5 ¼ 488.8

2 Р1 ¼ 74.8391 F1 ¼ 569.4 Р1 ¼ 74.9405 F1 ¼ 998.2 Р1 ¼ 74.7246 F1 ¼ 622.4 Р1 ¼ 74.2791 F1 ¼ 945.7Р2 ¼ 124.9490 F2 ¼ 918.2 Р2 ¼ 121.4427 F2 ¼ 415.1 Р2 ¼ 124.0558 F2 ¼ 569.9 Р2 ¼ 117.8608 F2 ¼ 939.0Р3 ¼ 174.8974 F3 ¼ 1755.2 Р3 ¼ 163.5206 F3 ¼ 1266.0 Р3 ¼ 174.2851 F3 ¼ 1531.4 Р3 ¼ 174.1277 F3 ¼ 1726.0Р4 ¼ 125.5740 F4 ¼ 2558.6 Р4 ¼ 235.2615 F4 ¼ 2488.1 Р4 ¼ 162.7982 F4 ¼ 2482.9 Р4 ¼ 191.1961 F4 ¼ 1981.4Р5 ¼ 299.7405 F5 ¼ 1198.7 Р5 ¼ 204.8347 F5 ¼ 1832.6 Р5 ¼ 264.1364 F5 ¼ 1793.3 Р5 ¼ 242.5362 F5 ¼ 1407.8

3 Р1 ¼ 31.4473 F1 ¼ 411.8 Р1 ¼ 74.7648 F1 ¼ 945.6 Р1 ¼ 73.9756 F1 ¼ 411.4 Р1 ¼ 68.6797 F1 ¼ 795.0Р2 ¼ 122.4461 F2 ¼ 872.2 Р2 ¼ 102.7998 F2 ¼ 361.9 Р2 ¼ 116.6737 F2 ¼ 620.4 Р2 ¼ 105.4940 F2 ¼ 698.0Р3 ¼ 174.7436 F3 ¼ 1781.6 Р3 ¼ 139.8154 F3 ¼ 1714.0 Р3 ¼ 174.3426 F3 ¼ 1946.2 Р3 ¼ 166.9206 F3 ¼ 1532.6Р4 ¼ 41.0864 F4 ¼ 1318.9 Р4 ¼ 190.4028 F4 ¼ 1878.9 Р4 ¼ 110.0962 F4 ¼ 1774.1 Р4 ¼ 111.2280 F4 ¼ 2787.8Р5 ¼ 280.2767 F5 ¼ 2615.5 Р5 ¼ 142.2173 F5 ¼ 2099.6 Р5 ¼ 174.9119 F5 ¼ 2247.8 Р5 ¼ 197.6777 F5 ¼ 1186.6

Fig.7.Pareto-optim

alfront

ofthe

lastiteration

forcase

2of

testsystem

1.

M.Basu

/Energy

xxx(2014)

1e16

8Pleasecite

thisarticle

inpress

as:Basu

M,Fuel

constrainedeconom

icem

issiondispatch

usingnondom

inatedsorting

geneticalgorithm

-II,Energy

(2014),http://dx.doi.org/10.1016/j.energy.2014.10.052

100,

0.9an

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Table2sh

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than

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Fig.6.Cost

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tests
s
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Fig.4dep

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3.Th

e


1 ¼ 2000, V02 ¼ 5000, V0

3 ¼ 5000, V04 ¼ 500, V0

5 ¼ 8000.


Generation(MW)








Objective

1 Р1 ¼ 46.2967 F1 ¼ 490.8 Cost ¼ 1,058,819 $ Р1 ¼ 74.8447 F1 ¼ 546.6 Cost ¼ 1,092,863 $ Р1 ¼ 74.8300 F1 ¼ 860.9 Cost ¼ 1,073,682 $ Р1 ¼ 51.7241 F1 ¼ 937.0 Cost ¼ 1,073,649 $Р2 ¼ 124.1988 F2 ¼ 944.8 Р2 ¼ 117.2953 F2 ¼ 962.3 Р2 ¼ 118.2417 F2 ¼ 839.7 Р2 ¼ 122.5560 F2 ¼ 728.0Р3 ¼ 174.8694 F3 ¼ 1586.1 Emission ¼

832,642 lbР3 ¼ 170.0582 F3 ¼ 1924.2 Emission ¼

602,331 lbР3 ¼ 172.9781 F3 ¼ 1647.1 Emission ¼

677,937 lbР3 ¼ 172.3057 F3 ¼ 1997.9 Emission ¼

678,686 lbР4 ¼ 57.8008 F4 ¼ 1088.1 Р4 ¼ 140.4073 F4 ¼ 2998.2 Р4 ¼ 116.7264 F4 ¼ 2990.8 Р4 ¼ 116.5391 F4 ¼ 2694.8Р5 ¼ 296.8343 F5 ¼ 2890.1 Р5 ¼ 197.3945 F5 ¼ 568.7 Р5 ¼ 217.2239 F5 ¼ 661.4 Р5 ¼ 236.8751 F5 ¼ 642.2

2 Р1 ¼ 74.0376 F1 ¼ 750.3 Р1 ¼ 74.9631 F1 ¼ 884.7 Р1 ¼ 74.8626 F1 ¼ 662.4 Р1 ¼ 74.9554 F1 ¼ 546.0Р2 ¼ 124.4470 F2 ¼ 553.6 Р2 ¼ 124.9673 F2 ¼ 996.7 Р2 ¼ 124.8258 F2 ¼ 926.7 Р2 ¼ 123.5521 F2 ¼ 991.7Р3 ¼ 174.6488 F3 ¼ 1865.4 Р3 ¼ 170. 9546 F3 ¼ 1721.6 Р3 ¼ 173.2441 F3 ¼ 933.7 Р3 ¼ 174.7013 F3 ¼ 1803.8Р4 ¼ 128.0763 F4 ¼ 2596.9 Р4 ¼ 180.4024 F4 ¼ 2998.6 Р4 ¼ 153.8389 F4 ¼ 2257.7 Р4 ¼ 149.5281 F4 ¼ 2306.1Р5 ¼ 298.7903 F5 ¼ 1233.8 Р5 ¼ 248.7126 F5 ¼ 398.4 Р5 ¼ 273.2286 F5 ¼ 2219.5 Р5 ¼ 277.2631 F5 ¼ 1352.5

3 Р1 ¼ 38.6909 F1 ¼ 669.2 Р1 ¼ 74.6438 F1 ¼ 660.2 Р1 ¼ 50.6470 F1 ¼ 977.1 Р1 ¼ 64.0207 F1 ¼ 880.3Р2 ¼ 106.0992 F2 ¼ 732.6 Р2 ¼ 107.8920 F2 ¼ 992.6 Р2 ¼ 119.1599 F2 ¼ 900.4 Р2 ¼ 119.8319 F2 ¼ 594.4Р3 ¼ 174. 5749 F3 ¼ 1168.9 Р3 ¼ 151.6888 F3 ¼ 1990.5 Р3 ¼ 170.0571 F3 ¼ 1977.8 Р3 ¼ 172.9764 F3 ¼ 1729.0Р4 ¼ 51.0046 F4 ¼ 2612.0 Р4 ¼ 147.4801 F4 ¼ 2994.7 Р4 ¼ 83.8227 F4 ¼ 1910.6 Р4 ¼ 99.5791 F4 ¼ 2980.6Р5 ¼ 279.6304 F5 ¼ 1817.2 Р5 ¼ 168.2953 F5 ¼ 362.1 Р5 ¼ 226.3133 F5 ¼ 1234.1 Р5 ¼ 193.5918 F5 ¼ 815.7

Fig.9.Pareto-optim

alfront

ofthe

lastiteration

forcase

3of

testsystem

1.

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Pleasecite

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M,Fuel

constrainedeconom

icem

issiondispatch

usingnondom

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geneticalgorithm

-II,Energy


Thedisp

atchsolu

tionsfor

case2are

summ

Cost

and

emission

convergen

ceobtain

edsh

own

inFig.

6.Th

edistribu

tionof

20non

do

obtain

edin

thelast

iterationof

prop

osedNSG

Acase

2is

dep

ictedin

Fig.7.Itis

seenfrom

Tablecon

omic

disp

atch,cost

is1,057,80

4$an

dem

iButcost

increases

to1,127,496

$an

dem

is561,511

lbfor

thecase

ofem

issiondisp

atch.In

emission

disp

atchby

usin

gNSG

A-II,

costis

1is

more

than

1,057,804$an

dless

than

1,127,4is

631,164

lbwhich

isless

than

850,138lb

561,511lb.

Itis

seenfrom

Tables2

and

3that

resuecon

omic

disp

atch,em

issiondisp

atchan

de

disp

atchare

almost

sameboth

forcase

1i.e.w

fuel

constrain

tsan

dcase

2i.e.

when

allun

coal.Thedisp

atchsolu

tionsfor

case3are

show

n

Fig.8.Cost

andem

issionconvergence

forcase

3of

arizedin

Table3

fromcase

2are

minated

solution

s-II

andSPEA

2for

e3that

incase

ofssion

is850,138

lb.sion

decreases

tocase

ofecon

omic

,084,709$which

96$an

dem

issionan

dmore

than

ltsobtain

edfrom

conom

icem

issionith

outcon

siderin

gits

have

sufficien

t

inTable

4.Fig.

8

testsystem

1.

9

Table 5Fuel constrained dispatch solution for case 4 of test system 1 with initial fuel stor

Interval Economic dispatch (RCGA) Emission dispatch Economic emission dispatch (SPEA2)

Generation(MW)



Fu(t

Generation(MW)


Objective

1 Р1 ¼ 74.5460 F1 ¼ 670.7 Cost ¼ 1,072,842 $ Р1 ¼ 74.7600 F1 $ Р1 ¼ 74.1570 F1 ¼ 982.8 Cost ¼ 1,096,108 $Р2 ¼ 124.3769 F2 ¼ 967.1 Р2 ¼ 112.7311 F2 Р2 ¼ 97.8061 F2 ¼ 357.4Р3 ¼ 173.9435 F3 ¼ 1546.7 Emission ¼

686,620 lbР3 ¼ 143.8396 F3 Р3 ¼ 174.2494 F3 ¼ 1875.0 Emission ¼

613,024 lbР4 ¼ 119.3743 F4 ¼ 824.1 Р4 ¼ 211.8778 F4 Р4 ¼ 144.0560 F4 ¼ 1265.6Р5 ¼ 207.7592 F5 ¼ 2991.4 Р5 ¼ 156.7915 F5 Р5 ¼ 209.7315 F5 ¼ 2519.2

2 Р1 ¼ 71.3752 F1 ¼ 952.5 Р1 ¼ 74.4660 F1 Р1 ¼ 52.4535 F1 ¼ 133.4Р2 ¼ 124.9989 F2 ¼ 810.3 Р2 ¼ 112.4141 F2 Р2 ¼ 119.7050 F2 ¼ 830.8Р3 ¼ 173.8995 F3 ¼ 1918.0 Р3 ¼ 161.8784 F3 Р3 ¼ 169.7750 F3 ¼ 1023.5Р4 ¼ 146.6616 F4 ¼ 328.2 Р4 ¼ 243.5720 F4 Р4 ¼ 212.4764 F4 ¼ 2087.6Р5 ¼ 283.0648 F5 ¼ 2990.9 Р5 ¼ 207.6695 F5 Р5 ¼ 245.5901 F5 ¼ 2924.7

3 Р1 ¼ 72.1086 F1 ¼ 765.7 Р1 ¼ 74.3946 F1 Р1 ¼ 62.1006 F1 ¼ 928.8Р2 ¼ 115.2357 F2 ¼ 893.1 Р2 ¼ 98.6743 F2 Р2 ¼ 117.2436 F2 ¼ 674.1Р3 ¼ 172.5911 F3 ¼ 1962.1 Р3 ¼ 145.1968 F3Р4 ¼ 67.4497 F4 ¼ 381.2 Р4 ¼ 196.9188 F4Р5 ¼ 222.6150 F5 ¼ 2997.8 Р5 ¼ 134.8154 F5

Fig.11.Pareto-optim

alfront

ofthe

lastiteration

forcase

4of

testsystem

1.

10Pleasecite

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M,Fuel

constrainedeconom

icem

issiondispatch

usingnondom

inatedsorting

geneticalgorithm

-II,Energy


age (tons) V01 ¼ 2000, V0

2 ¼ 5000, V03 ¼ 5000, V0

4 ¼ 8000, V05 ¼ 500.

(RCGA) Economic emission dispatch (NSGA-II)

el deliveredons)



Objective

¼ 781.4 Cost ¼ 1,131,378 $ Р1 ¼ 74.1566 F1 ¼ 583.8 Cost ¼ 1,096,467¼ 684.0 Р2 ¼ 123.8994 F2 ¼ 588.3¼ 1065.3 Emission ¼

563,327 lbР3 ¼ 159.8493 F3 ¼ 1595.1 Emission ¼

599,093 lb¼ 2387.1 Р4 ¼ 137.6202 F4 ¼ 1602.2¼ 2082.2 Р5 ¼ 204.4744 F5 ¼ 2630.6¼ 778.6 Р1 ¼ 67.5505 F1 ¼ 575.4¼ 908.6 Р2 ¼ 120.6256 F2 ¼ 735.3¼ 1501.6 Р3¼174.5772 F3 ¼ 1662.6¼ 900.2 Р4 ¼ 202.4951 F4 ¼ 1205.1¼ 2911.0 Р5 ¼ 234.7515 F5 ¼ 2821.6¼ 457.4 Р1 ¼ 72.7024 F1 ¼ 680.5¼ 866.6 Р2 ¼ 115.5119 F2 ¼ 827.2

distribu

tionof

20non

dom

inated

solution

sobtain

edin

the

lastiteration

ofproposed

NSG

A-II

andSPEA

2for

case3is

dep

ictedin

Fig.9.Itis

seenfrom

Table4that

under

econom

icdisp

atch,cost

is1,058,819

$an

dem

issionis

832,642lb.

Butin

caseof

emission

disp

atch,cost

increases

to1,092,863

$an

dem

issiondecreases

to602,331

lb.Incase

ofeconom

icem

issiondisp

atchby

usin

gNSG

A-II,

costis

1,073,682$which

ismore

than

1,058,819$an

dless

than

1,092,863$an

dem

issionis677,937

lbwhich

isless

than

832,642lb

andmore

than

602,331lb.

Table5

summarizes

thedisp

atchsolu

tionsfor

case4.

Fig.10dep

ictscost

andem

issioncon

vergence

obtained

fromcase

4.The

distribu

tionof

20non

dom

inated

solution

sobtain

edin

the

lastiteration

ofprop

osedNSG

A-II

andSPEA

2for

case4is

show

nin

Fig.11.Itisseen

fromTable

5thatin

caseofecon

omicdisp

atch,cost

is1,072,842

$an

dem

issionis

686,620lb.

Butcost

increases

to¼ 969.6 Р3 ¼ 136.0506 F3 ¼ 1268.7 Р3 ¼ 157.3089 F3 ¼ 1999.9¼ 2877.4 Р4 ¼ 150.9101 F4 ¼ 1527.7 Р4 ¼ 142.3451 F4 ¼ 1022.4¼ 1829.0 Р5 ¼ 174.8250 F5 ¼ 2695.9 Р5 ¼ 171.0018 F5 ¼ 2374.8

Fig.10.Cost

andem

issionconvergence

forcase

4of

testsystem

1.

M.Basu

/Energy

xxx(2014)

1e16

Table 6Specification for test system 2.

Cases Condition Input parameters

Case 1 Only load constraints are considered and fuel constraints are ignored. РD1 ¼ 1400 MW,РD2 ¼ 1600 MW, РD3 ¼ 1300 MWCase 2 Load constraints and fuel constraints are considered and all units have

sufficient coal.РD1 ¼ 1400 MW,РD2 ¼ 1600 MW, РD3 ¼ 1300 MWFD1 ¼ FD2 ¼ FD3 ¼ 14000 tonV01 ¼ 2000, V0

2 ¼ 5000, V03 ¼ 5000, V0

4 ¼ 8000, V05 ¼ 8000,

V06 ¼ 2000, V0

7 ¼ 5000, V08 ¼ 5000, V0

9 ¼ 8000, V010 ¼ 8000

Case 3 Load constraints and fuel constraints are considered and there is fuelshortage at unit 2

РD1 ¼ 1400 MW,РD2 ¼ 1600 MW, РD3 ¼ 1300 MWFD1 ¼ FD2 ¼ FD3 ¼ 14000 tonV01 ¼ 2000, V0

2 ¼ 500, V03 ¼ 5000, V0

4 ¼ 8000, V05 ¼ 8000,

V06 ¼ 2000, V0

7 ¼ 5000, V08 ¼ 5000, V0

9 ¼ 8000, V010 ¼ 8000

Table 7Dispatch solution for case 1 of test system 2 without considering fuel constraints.

Interval Economic dispatch (RCGA) Emission dispatch (RCGA) Economic emission dispatch(NSGA-II)

Economic emission dispatch(SPEA 2)

Generation(MW)




Objective

1 Р1 ¼ 49.6700 Cost ¼ 2,115,312 $ Р1 ¼ 75.0000 Cost ¼ 2,254,749 $ Р1 ¼ 60.0667 Cost ¼ 2,180,098 $ Р1 ¼ 62.8150 Cost ¼ 2,187,249 $Р2 ¼ 125.0000 Р2 ¼ 108.4258 Р2 ¼ 75.9957 Р2 ¼ 99.0488Р3 ¼ 175.0000 Emission ¼

1,702,439 lbР3 ¼ 148.0901 Emission ¼

1,122,981 lbР3 ¼ 175.0000 Emission ¼

1,278,305 lbР3 ¼ 170.1243 Emission ¼

1,270,616 lbР4 ¼ 48.7606 Р4 ¼ 208.7384 Р4 ¼ 150.2407 Р4 ¼ 148.3592Р5 ¼ 300.0000 Р5 ¼ 163.4668 Р5 ¼ 196.4266 Р5 ¼ 248.7428Р6 ¼ 50.4765 Р6 ¼ 74.9942 Р6 ¼ 57.8382 Р6 ¼ 45.8290Р7 ¼ 125.0000 Р7 ¼ 106.8783 Р7 ¼ 117.4547 Р7 ¼ 102.6019Р8 ¼ 175.0000 Р8 ¼ 147.2972 Р8 ¼ 167.7686 Р8 ¼ 144.1110Р9 ¼ 51.1239 Р9 ¼ 208.4608 Р9 ¼ 153.1728 Р9 ¼ 133.4928Р10 ¼ 299.9690 Р10 ¼ 158.6482 Р10 ¼ 246.0359 Р10 ¼ 244.8752

2 Р1 ¼ 75.0000 Р1 ¼ 75.0000 Р1 ¼ 74.3981 Р1 ¼ 69.7539Р2 ¼ 125.0000 Р2 ¼ 119.5158 Р2 ¼ 116.0600 Р2 ¼ 109.9665Р3 ¼ 175.0000 Р3 ¼ 160.3909 Р3 ¼ 168.9857 Р3 ¼ 162.8381Р4 ¼ 123.1548 Р4 ¼ 233.6421 Р4 ¼ 205.2961 Р4 ¼ 178.0559Р5 ¼ 299.9855 Р5 ¼ 201.1709 Р5 ¼ 227.5774 Р5 ¼ 288.1677Р6 ¼ 75.0000 Р6 ¼ 75.0000 Р6 ¼ 57.5327 Р6 ¼ 67.0694Р7 ¼ 124.9717 Р7 ¼ 122.7651 Р7 ¼ 110.8068 Р7 ¼ 109.8927Р8 ¼ 174.9907 Р8 ¼ 163.9101 Р8 ¼ 169.6052 Р8 ¼ 150.5354Р9 ¼ 126.9569 Р9 ¼ 232.5510 Р9 ¼ 192.2565 Р9 ¼ 198.4050Р10 ¼ 299.9404 Р10 ¼ 216.0541 Р10 ¼ 277.4814 Р10 ¼ 265.3153

3 Р1 ¼ 33.8680 Р1 ¼ 74.7722 Р1 ¼ 46.3045 Р1 ¼ 65.6957Р2 ¼ 125.0000 Р2 ¼ 100.9603 Р2 ¼ 117.7248 Р2 ¼ 91.6457Р3 ¼ 175.0000 Р3 ¼ 139.4649 Р3 ¼ 162.7273 Р3 ¼ 135.3582Р4 ¼ 40.0698 Р4 ¼ 192.1170 Р4 ¼ 137.7063 Р4 ¼ 152.6960Р5 ¼ 283.5515 Р5 ¼ 142.4954 Р5 ¼ 212.7022 Р5 ¼ 182.8921Р6 ¼ 32.1340 Р6 ¼ 75.0000 Р6 ¼ 52.3889 Р6 ¼ 72.4818Р7 ¼ 120.9186 Р7 ¼ 98.7891 Р7 ¼ 98.3618 Р7 ¼ 94.1453Р8 ¼ 174.7231 Р8 ¼ 140.4028 Р8 ¼ 161.6639 Р8 ¼ 135.7746Р9 ¼ 40.0000 Р9 ¼ 195.7691 Р9 ¼ 102.2068 Р9 ¼ 165.1651Р10 ¼ 274.7349 Р10 ¼ 140.2293 Р10 ¼ 208.2136 Р10 ¼ 204.1456

Fig. 12. Cost and emission convergence for case 1 of test system 2. Fig. 13. Pareto-optimal front of the last iteration for case 1 of test system 2.


Please cite this article in press as: Basu M, Fuel constrained economic emission dispatch using nondominated sorting genetic algorithm-II,Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.10.052


1 ¼ 2000, V02 ¼ 5000, V0

3 ¼ 5000, V04 ¼ 8000, V0

5 ¼ 8000, V06 ¼ 2000, V0

7 ¼ 5000, V08 ¼ 5000, V0

9 ¼ 8000, V010 ¼ 8000.


Generation(MW)








Objective

1 Р1 ¼ 55.4326 F1 ¼ 432.7 Cost ¼2,115,339 $

Р1 ¼ 74.6003 F1 ¼ 17.0 Cost ¼ 2,255,937 $ Р1 ¼ 59.5047 F1 ¼ 293.3 Cost ¼ 2,186,266 $ Р1 ¼ 57.8699 F1 ¼ 982.6 Cost ¼ 2,187,380 $Р2 ¼ 124.9783 F2 ¼ 817.8 Р2 ¼ 109.8706 F2 ¼ 1000.0 Р2 ¼ 121.0789 F2 ¼ 930.7 Р2 ¼ 98.0756 F2 ¼ 711.1Р3 ¼ 175.0000 F3 ¼ 1412.8 Emission ¼

1,705,350 lbР3 ¼ 154.1717 F3 ¼ 2000.0 Emission ¼

1,125,132 lbР3 ¼ 160.7524 F3 ¼ 1284.8 Emission ¼

1,275,961 lbР3 ¼ 157.6822 F3 ¼ 1422.3 Emission ¼

1,283,791 lbР4 ¼ 47.7339 F4 ¼ 3000.0 Р4 ¼ 210.2204 F4 ¼ 2632.6 Р4 ¼ 154.4364 F4 ¼ 1862.0 Р4 ¼ 172.6089 F4 ¼ 2542.3Р5 ¼ 299.9991 F5 ¼ 2999.0 Р5 ¼ 161.9010 F5 ¼ 2561.2 Р5 ¼ 187.2767 F5 ¼ 1852.7 Р5 ¼ 229.3905 F5 ¼ 1819.4Р6 ¼ 47.9198 F6 ¼ 493.2 Р6 ¼ 71.3700 F6 ¼ 893.2 Р6 ¼ 50.1314 F6 ¼ 624.9 Р6 ¼ 73.7117 F6 ¼ 543.5Р7 ¼ 124.9998 F7 ¼ 873.5 Р7 ¼ 110.6421 F7 ¼ 3.6 Р7 ¼ 110.5439 F7 ¼ 68.1 Р7 ¼ 96.3066 F7 ¼ 162.7Р8 ¼ 175.0000 F8 ¼ 1962.0 Р8 ¼ 146.2438 F8 ¼ 2000.0 Р8 ¼ 160.4361 F8 ¼ 1790.2 Р8 ¼ 136.0807 F8 ¼ 1609.5Р9 ¼ 48.9865 F9 ¼ 0 Р9 ¼ 208.0683 F9 ¼ 2026.3 Р9 ¼ 197.6984 F9 ¼ 2763.5 Р9 ¼ 188.3622 F9 ¼ 1532.0Р10 ¼ 299.9501 F10 ¼ 2009.0 Р10 ¼ 152.9119 F10 ¼ 866.1 Р10 ¼ 198.1411 F10 ¼ 2529.7 Р10 ¼ 189.9117 F10 ¼ 2674.5

2 Р1 ¼ 74.8972 F1 ¼ 907.9 Р1 ¼ 74.6609 F1 ¼ 864.3 Р1 ¼ 62.7225 F1 ¼ 559.3 Р1 ¼ 41.7474 F1 ¼ 0Р2 ¼ 124.9988 F2 ¼ 5.7 Р2 ¼ 120.8477 F2 ¼ 400.8 Р2 ¼ 125.0000 F2 ¼ 1000.0 Р2 ¼ 125.0000 F2 ¼ 552.4Р3 ¼ 174.9942 F3 ¼ 1874.1 Р3 ¼ 165.6299 F3 ¼ 2000.0 Р3 ¼ 162.0203 F3 ¼ 1781.2 Р3 ¼ 157.6466 F3 ¼ 0Р4 ¼ 125.2755 F4 ¼ 2999.4 Р4 ¼ 238.1010 F4 ¼ 2035.6 Р4 ¼ 147.6511 F4 ¼ 507.8 Р4 ¼ 154.8707 F4 ¼ 2189.0Р5 ¼ 300.0000 F5 ¼ 3000.0 Р5 ¼ 203.3887 F5 ¼ 782.0 Р5 ¼ 240.4762 F5 ¼ 2577.1 Р5 ¼ 287.2517 F5 ¼ 2732.1Р6 ¼ 75.0000 F6 ¼ 0 Р6 ¼ 75.0000 F6 ¼ 901.6 Р6 ¼ 75.0000 F6 ¼ 514.8 Р6 ¼ 68.2847 F6 ¼ 799.9Р7 ¼ 125.0000 F7 ¼ 260.2 Р7 ¼ 118.4838 F7 ¼ 615.2 Р7 ¼ 110.8484 F7 ¼ 1000.0 Р7 ¼ 123.2466 F7 ¼ 1000.0Р8 ¼ 175.0000 F8 ¼ 1981.1 Р8 ¼ 163.9530 F8 ¼ 2000.0 Р8 ¼ 175.0000 F8 ¼ 1329.0 Р8 ¼ 162.3762 F8 ¼ 1786.2Р9 ¼ 124.9284 F9 ¼ 911.5 Р9 ¼ 231.2312 F9 ¼ 1937.0 Р9 ¼ 219.6618 F9 ¼ 2879.6 Р9 ¼ 214.1380 F9 ¼ 2597.5Р10 ¼ 299.9059 F10 ¼ 2059.9 Р10 ¼ 208.7039 F10 ¼ 2463.6 Р10 ¼ 281.6198 F10 ¼ 1851.1 Р10 ¼ 265.4382 F10 ¼ 2342.8

3 Р1 ¼ 32.2989 F1 ¼ 606.1 Р1 ¼ 73.1558 F1 ¼ 1000.0 Р1 ¼ 67.0586 F1 ¼ 257.5 Р1 ¼ 41.3935 F1 ¼ 341.3Р2 ¼ 119.1383 F2 ¼ 1000.0 Р2 ¼ 99.1936 F2 ¼ 593.0 Р2 ¼ 84.5325 F2 ¼ 500.7 Р2 ¼ 118.6675 F2 ¼ 206.4Р3 ¼ 175.0000 F3 ¼ 1548.2 Р3 ¼ 139.9260 F3 ¼ 1322.4 Р3 ¼ 163.4698 F3 ¼ 1033.0 Р3 ¼ 166.4952 F3 ¼ 1679.9Р4 ¼ 40.0000 F4 ¼ 271.8 Р4 ¼ 195.4711 F4 ¼ 2626.0 Р4 ¼ 136.6050 F4 ¼ 2731.2 Р4 ¼ 81.2083 F4 ¼ 1767.1Р5 ¼ 281.8965 F5 ¼ 2294.2 Р5 ¼ 147.6313 F5 ¼ 835.1 Р5 ¼ 237.0478 F5 ¼ 2934.1 Р5 ¼ 216.1248 F5 ¼ 1890.0Р6 ¼ 32.5553 F6 ¼ 949.2 Р6 ¼ 72.6139 F6 ¼ 1000.0 Р6 ¼ 30.5466 F6 ¼ 580.7 Р6 ¼ 61.4685 F6 ¼ 622.5Р7 ¼ 124.2220 F7 ¼ 406.0 Р7 ¼ 98.6272 F7 ¼ 117.1 Р7 ¼ 91.1493 F7 ¼ 670.7 Р7 ¼ 123.7407 F7 ¼ 934.8Р8 ¼ 174.9636 F8 ¼ 1911.9 Р8 ¼ 138.0825 F8 ¼ 1828.9 Р8 ¼ 154.7463 F8 ¼ 516.6 Р8 ¼ 141.0835 F8 ¼ 1230.8Р9 ¼ 40.0000 F9 ¼ 3000.0 Р9 ¼ 197.4452 F9 ¼ 2934.9 Р9 ¼ 112.7697 F9 ¼ 2789.9 Р9 ¼ 159.4427 F9 ¼ 2786.5Р10 ¼ 279.9255 F10 ¼ 2012.6 Р10 ¼ 137.8533 F10 ¼ 1742.6 Р10 ¼ 222.0743 F10 ¼ 1985.8 Р10 ¼ 190.3753 F10 ¼ 2540.7

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geneticalgorithm

-II,Energy





1,131378 $ and emission decreases to 563,327 lb for the case ofemission dispatch. In case of economic emission dispatch by usingNSGA-II, cost is 1,096,467 $ which is more than 1,072,842 $ and lessthan 1,131,378 $ and emission is 599,093 lb which is less than686,620 lb and more than 563,327 lb.

It is seen from Tables 2, 4 and 5 that results obtained fromeconomic dispatch, emission dispatch and economic emissiondispatch are a bit different from case 1 to case 3 and case 4.

For both case 3 and case 4, optimum economic dispatch, op-timum emission dispatch and optimum economic emissiondispatch are not achieved. But this may well compensate for thepenalty that might be otherwise imposed for not maintaining thefuel contract.

5.2. Test system 2

This test system considers ten coal-burning generatingunits which remain on line for a 3-week period. The data oftest system 1 is doubled to obtain a ten unit system. Load de-mand and fuel delivered during the scheduling period are alsodoubled.

Here, three cases are considered. In case 1 economic dispatch,emission dispatch and economic emission dispatch are donewithout considering fuel constraints. In case 2 economic fueldispatch, emission fuel dispatch and economic emission fueldispatch are done considering fuel constraints when all the unitshave sufficient coal. Case 3 is purposely structured to show theinteraction of the fuel deliveries and different types of dispatch ofthe generating units when there is fuel shortage at unit 2. Table 6shows specification for this test system.

In order to explore the extreme points of the trade-off surface,fuel cost and emission objectives are minimized individually byusing real coded genetic algorithm (RCGA) for all the three cases. Incase of RCGA, the population size, maximum number of iterations,crossover and mutation probabilities have been selected as 100,100, 0.9 and 0.2, respectively for all the three cases of this testsystem.

NSGA-II has been applied to optimize both cost and emissionobjectives simultaneously for all the three cases. For comparison,SPEA 2 has been applied to solve the problem. In case of NSGA-IIand SPEA 2, the population size, maximum number of iterations,crossover and mutation probabilities have been selected as 20,50, 0.9 and 0.2, respectively for all the three cases of this testsystem.

Table 7 shows the dispatch solutions for case 1. Fig. 12 depictscost and emission convergence obtained from case 1. The distri-bution of 20 nondominated solutions obtained in the last iterationof proposed NSGA-II and SPEA 2 for case 1 is shown in Fig. 13. It isseen from Table 7 that in case of economic dispatch, cost is2,115,312 $ and emission is 1,702,439 lb. But cost increases to2,254,749 $ and emission decreases to 1,122,981 lb for the case ofemission dispatch. In case of economic emission dispatch by usingNSGA-II, cost is 2,180,098 $ which is more than 2,115,312 $ and lessthan 2,254,749 $ and emission is 1,278,305 lb which is less than1,702,439 lb and more than 1,122,981 lb.

The dispatch solutions for case 2 are summarized in Table 8.Cost and emission convergence obtained from case 2 are shownin Fig. 14. The distribution of 20 nondominated solutions ob-tained in the last iteration of proposed NSGA-II and SPEA 2 forcase 2 is depicted in Fig. 15. It is seen from Table 8 that undereconomic dispatch, cost is 2,115,339 $ and emission is1,705,350 lb. But in case of emission dispatch, cost increases to2,255,937 $ and emission decreases to 1,125,132 lb. In case ofeconomic emission dispatch by using NSGA-II, cost is 2,186,266 $which is more than 2,115,339 $ and less than 2,255,937 $ and


emission is 1,275,961 lb which is less than 1,705,350 lb and morethan 1,125,132 lb.

It is seen from Tables 7 and 8 that results obtained from eco-nomic dispatch, emission dispatch and economic emission dispatchare very close to each other both for case 1 i.e. without consideringfuel constraints and case 2 i.e. when all the units have sufficientcoal.

The dispatch solutions for case 3 are shown in Table 9. Fig. 16shows cost and emission convergence obtained from case 3. Thedistribution of 20 nondominated solutions obtained in the lastiteration of proposed NSGA-II and SPEA 2 for case 3 is depicted inFig. 17. It is seen from Table 9 that in case of economic dispatch, costis 2,129,740 $ and emission is 1,689,524 lb. But cost increases to2,259,380 $ and emission decreases to 1,151,955 lb for the case ofemission dispatch. In case of economic emission dispatch by usingNSGA-II, cost is 2,198,294 $ which is more than 2,129,740 $ and lessthan 2,259,380 $ and emission is 1,302,447 lb which is less than1,689,524 lb and more than 1,151,955 lb.

It is seen from Tables 7 and 9 that results obtained from eco-nomic dispatch, emission dispatch and economic emission dispatchare a bit different from case 1 to case 3.

For case 3, optimum economic dispatch, optimum emissiondispatch and optimum economic emission dispatch are notachieved. But this may well compensate for the penalty that



1 ¼ 2000, V02 ¼ 500, V0

3 ¼ 5000, V04 ¼ 8000, V0

5 ¼ 8000, V06 ¼ 2000, V0

7 ¼ 5000, V08 ¼ 5000, V0

9 ¼ 8000, V010 ¼ 8000.


Generation(MW)








Objective

1 Р1 ¼ 64.3155 F1 ¼ 0 Cost ¼2,129,740 $

Р1 ¼ 74.3780 F1 ¼ 385.3 Cost ¼2,259,380 $

Р1 ¼ 72.1573 F1 ¼ 674.8 Cost ¼2,198,294 $

Р1 ¼ 45.3799 F1 ¼ 0 Cost ¼ 2,197,458 $Р2 ¼ 80.8564 F2 ¼ 996.7 Р2 ¼ 76.2347 F2 ¼ 996.8 Р2 ¼ 87.7233 F2 ¼ 963.4 Р2 ¼ 69.2353 F2 ¼ 847.8Р3 ¼ 175.0000 F3 ¼ 2000.0 Emission ¼

1,689,524 lbР3 ¼ 154.1125 F3 ¼ 1182.0 Emission ¼

1,151,955 lbР3 ¼ 149.0880 F3 ¼ 1498.6 Emission ¼

1,302,447 lbР3 ¼ 164.3586 F3 ¼ 1621.0 Emission ¼

1,327,848 lbР4 ¼ 72.0319 F4 ¼ 2943.5 Р4 ¼ 215.8619 F4 ¼ 2177.5 Р4 ¼ 133.5417 F4 ¼ 1642.3 Р4 ¼ 119.4722 F4 ¼ 2830.8Р5 ¼ 299.8533 F5 ¼ 2997.6 Р5 ¼ 166.2925 F5 ¼ 1091.6 Р5 ¼ 247.9039 F5 ¼ 2074.3 Р5 ¼ 300.0000 F5 ¼ 1076.9Р6 ¼ 52.7862 F6 ¼ 0 Р6 ¼ 75.0000 F6 ¼ 887.3 Р6 ¼ 75.0000 F6 ¼ 114.2 Р6 ¼ 73.4129 F6 ¼ 1000.0Р7 ¼ 125.0000 F7 ¼ 0 Р7 ¼ 109.5850 F7 ¼ 963.7 Р7 ¼ 125.0000 F7 ¼ 846.0 Р7 ¼ 101.4922 F7 ¼ 98.0Р8 ¼ 175.0000 F8 ¼ 1999.7 Р8 ¼ 156.0613 F8 ¼ 1323.4 Р8 ¼ 122.1048 F8 ¼ 654.7 Р8 ¼ 165.4602 F8 ¼ 1697.5Р9 ¼ 60.0350 F9 ¼ 2451.8 Р9 ¼ 215.8320 F9 ¼ 2361.9 Р9 ¼ 172.2926 F9 ¼ 2606.4 Р9 ¼ 129.9990 F9 ¼ 3000.0Р10 ¼ 295.1217 F10 ¼ 610.6 Р10 ¼ 156.6420 F10 ¼ 2630.5 Р10 ¼ 215.1885 F10 ¼ 2925.3 Р10 ¼ 231.1897 F10 ¼ 1827.9

2 Р1 ¼ 75.0000 F1 ¼ 1000.0 Р1 ¼ 73.1680 F1 ¼ 639.4 Р1 ¼ 61.7026 F1 ¼ 48.8 Р1 ¼ 70.4240 F1 ¼ 117.1Р2 ¼ 76.1953 F2 ¼ 1000.0 Р2 ¼ 80.1141 F2 ¼ 999.8 Р2 ¼ 66.0329 F2 ¼ 998.2 Р2 ¼ 53.8459 F2 ¼ 954.6Р3 ¼ 175.0000 F3 ¼ 2000.0 Р3 ¼ 157.6614 F3 ¼ 17.1 Р3 ¼ 169.1237 F3 ¼ 1318.5 Р3 ¼ 170.6224 F3 ¼ 1884.2Р4 ¼ 123.3936 F4 ¼ 3000.0 Р4 ¼ 243.9557 F4 ¼ 3000.0 Р4 ¼ 217.0270 F4 ¼ 2009.5 Р4 ¼ 239.0137 F4 ¼ 2567.7Р5 ¼ 299.9087 F5 ¼ 2167.2 Р5 ¼ 239.7262 F5 ¼ 2643.7 Р5 ¼ 282.1252 F5 ¼ 2520.8 Р5 ¼ 250.2077 F5 ¼ 2664.3Р6 ¼ 75.0000 F6 ¼ 1000.0 Р6 ¼ 74.9193 F6 ¼ 763.2 Р6 ¼ 60.4821 F6 ¼ 996.5 Р6 ¼ 74.8781 F6 ¼ 626.6Р7 ¼ 125.0000 F7 ¼ 0 Р7 ¼ 121.5405 F7 ¼ 227.1 Р7 ¼ 120.0362 F7 ¼ 0 Р7 ¼ 114.6962 F7 ¼ 20.2Р8 ¼ 175.0000 F8 ¼ 0 Р8 ¼ 167.8783 F8 ¼ 1702.9 Р8 ¼ 169.0304 F8 ¼ 1332.5 Р8 ¼ 163.6059 F8 ¼ 1807.6Р9 ¼ 175.8587 F9 ¼ 3000.0 Р9 ¼ 225.8689 F9 ¼ 2112.2 Р9 ¼ 176.4017 F9 ¼ 2764.7 Р9 ¼ 226.1345 F9 ¼ 2346.7Р10 ¼ 299.6437 F10 ¼ 832.8 Р10 ¼ 215.1677 F10 ¼ 1894.7 Р10 ¼ 278.0382 F10 ¼ 2010.5 Р10 ¼ 236.5717 F10 ¼ 1011.0

3 Р1 ¼ 45.9335 F1 ¼ 1000.0 Р1 ¼ 73.5911 F1 ¼ 1000.0 Р1 ¼ 41.5623 F1 ¼ 725.5 Р1 ¼ 41.9914 F1 ¼ 779.4Р2 ¼ 60.1562 F2 ¼ 1000.0 Р2 ¼ 61.2059 F2 ¼ 1000.0 Р2 ¼ 55.5549 F2 ¼ 928.2 Р2 ¼ 65.8343 F2 ¼ 984.1Р3 ¼ 173.3245 F3 ¼ 1999.8 Р3 ¼ 143.2503 F3 ¼ 2000.0 Р3 ¼ 142.2694 F3 ¼ 1385.7 Р3 ¼ 171.5112 F3 ¼ 1915.5Р4 ¼ 43.7764 F4 ¼ 417.5 Р4 ¼ 181.8214 F4 ¼ 2649.2 Р4 ¼ 177.5282 F4 ¼ 2702.3 Р4 ¼ 124.0273 F4 ¼ 2279.5Р5 ¼ 299.9995 F5 ¼ 2999.9 Р5 ¼ 147.5333 F5 ¼ 531.5 Р5 ¼ 242.4880 F5 ¼ 2086.2 Р5 ¼ 178.0187 F5 ¼ 1130.7Р6 ¼ 44.6254 F6 ¼ 999.9 Р6 ¼ 75.0000 F6 ¼ 1000.0 Р6 ¼ 57.3152 F6 ¼ 415.5 Р6 ¼ 65.7265 F6 ¼ 503.4Р7 ¼ 121.8623 F7 ¼ 942.6 Р7 ¼ 111.4974 F7 ¼ 153.0 Р7 ¼ 104.3221 F7 ¼ 418.1 Р7 ¼ 116.9062 F7 ¼ 200.5Р8 ¼ 174.8939 F8 ¼ 2000.0 Р8 ¼ 144.9976 F8 ¼ 1548.4 Р8 ¼ 129.6762 F8 ¼ 1459.3 Р8 ¼ 166.2208 F8 ¼ 1516.3Р9 ¼ 40.0043 F9 ¼ 3.2 Р9 ¼ 202.6489 F9 ¼ 2574.7 Р9 ¼ 149.7492 F9 ¼ 2019.2 Р9 ¼ 150.2429 F9 ¼ 3000.0Р10 ¼ 295.4239 F10 ¼ 2637.0 Р10 ¼ 158.4541 F10 ¼ 1543.3 Р10 ¼ 199.5345 F10 ¼ 1860.1 Р10 ¼ 219.5206 F10 ¼ 1690.6

M.Basu

/Energy

xxx(2014)

1e16

14Pleasecite

thisarticle

inpress

as:Basu

M,Fuel

constrainedeconom

icem

issiondispatch

usingnondom

inatedsorting

geneticalgorithm

-II,Energy





might be otherwise imposed for not maintaining the fuelcontract.

6. Conclusion

This paper examines the usefulness of nondominated sortinggenetic algorithm-II for solving fuel constrained economic emissiondispatch problem of thermal generating units. The results showthat fuel consumption can be adequately controlled to satisfyconstraints imposed by suppliers using the proposed method.Optimum economic emission dispatch is not achieved always, butthis is generally much less than the penalty that could be imposedfor violating the fuel system constraints.

Appendix-1

Simulated Binary Crossover (SBX) operator

The procedure of computing child populations c1 and c2 fromtwo parent populations y1 and y2 under SBX operator as follows:

1. Create a random number u between 0 and 1.2. Find a parameter g using a polynomial probability distribution

as follows:


g ¼(ðuεÞ1=ðqc þ 1Þ; if u � 1=ε

ð1=ð2� uεÞÞ1=ðqc þ 1Þ; otherwise

where ε ¼ 2� f�ðqcþ1Þ and f is calculated as follows:

f ¼ 1þ 2y2 � y1

min½ðy1 � ylÞ; ðyu � y2Þ�

Here, the parameter y is assumed to vary in [yl,yu]. Here, theparameter qc is the distribution index for SBX and can take any non-negative value. A small value of qc allows the creation of childpopulations far away from parents and a large value restricts onlynear-parent populations to be created as child populations.

3. The intermediate populations are calculated as follows:

cp1 ¼ 0:5½ðy1 þ y2Þ � gðjy2 � y1jÞ�

cp2 ¼ 0:5½ðy1 þ y2Þ þ gðjy2 � y1jÞ�Each variable is chosen with a probability pc and the above SBX

operator is applied variable-by-variable.

Polynomial Mutation operator

A polynomial probability distribution is used to create a childpopulation in the vicinity of a parent population under the muta-tion operator. The following procedure is used:

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

8>>>><h2uþð1�2uÞð1�cÞðqmþ1Þi 1

ðqmþ1Þ �1; if u�0:5
x¼>>>>:1�
h2ð1�uÞþ2ðu�0:5Þð1�cÞðqmþ1Þi 1

ðqmþ1Þ; otherwise

where c ¼ min½ðcp�ylÞ;ðyu�cpÞ�ðyu�ylÞ

The parameter qm is the distribution index for mutation andtakes any non-negative value.

3. Calculate the mutated child as follows:

c1 ¼ cp1 þ xðyu � ylÞ

c2 ¼ cp2 þ xðyu � ylÞThe perturbance in the population can be adjusted by varying qm

and pm with generations as given below:

qm ¼ qmmin þ iter

pm ¼ 1nþ iteritermax

�1� 1

n

�

where qmmin is the user defined minimum value for qm,pm is theprobability of mutation, and n is the number of decision variables.


Table A.2Load demand and fuel delivered during scheduling period

Interval Duration (h) Load demand РD (MW) Fuel delivered FD(ton)

1 168 700 70002 168 800 70003 168 650 7000

Appen

dix-2

TableA.1Gen

erator

characteristics

Unit

Рmin

iMW

Рmax

iMW

a i$/h

b i$/MW

hc i$/(M

W)2h

d i$/h

e irad/M

Wailb/h

bilb/M

Wh

gilb/(MW

)2h

silb/h

q i1/MW

hiton/h

d iton/M

Wh

miton/(MW

)2h

l iton/h

rirad/M

WFm

ini ton

Fmax

i ton

Vmin

iton

Vmax

iton

120

7525

2.0

0.00

8010

0.01

280

�0.805

0.01

800.00

80.07

350.83

612

0.06

6889

0.00

0267

561.0

0.15

010

000

1000

02

2012

560

1.8

0.00

3020

0.01

050

�0.555

0.01

500.00

90.06

552.00

669

0.06

0200

0.00

0100

331.5

0.10

010

000

1000

03

3017

510

02.1

0.00

1230

0.00

970

�0.955

0.01

150.01

00.05

043.34

448

0.07

0230

0.00

0040

132.0

0.09

020

000

2000

04

4025

012

02.2

0.00

4040

0.00

845

�0.600

0.00

800.01

50.03

404.01

338

0.07

3578

0.00

0133

782.5

0.07

030

000

3000

05

5030

040

1.8

0.00

1550

0.00

730

�0.555

0.01

200.01

70.02

851.33

779

0.06

0200

0.00

0050

173.0

0.05

030

000

3000

0



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fuel constrained economic emission dispatch using nondominated sorting genetic algorithm-ii

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