environmental and economic power dispatch of thermal generators using modified nsga-ii algorithm

Environmental and economic power dispatch of thermal generatorsusing modified NSGA-II algorithm

Rajkumar Muthuswamy1*,†, Mahadevan Krishnan2,Kannan Subramanian3 and Baskar Subramanian4

1National College of Engineering, Department of EEE, Tirunelveli 627151, Tamilnadu, India2PSNA College of Engineering & Technology, Department of EEE, Dindigul 624622, Tamilnadu, India

3Kalasalingam University, Department of EEE, Krishnankoil 626126, Tamilnadu, India4Thiagarajar College of Engineering, Department of EEE, Madurai 625015, Tamilnadu, India

SUMMARY

This paper presents the solution to the problem in fabricating Environmental and Economic Power Dispatch(EEPD) of thermal generators with valve-point loading effect and multiple prohibited operating zones(POZ). The valve-point effect introduces ripples in the input–output characteristics of generating units,and the existence of POZ breaks the operating region of a generating unit into isolated sub-regions, thusforms a nonconvex decision space. The EEPD problem becomes a nonsmooth optimization problembecause of these valve-point effect and POZ. Accuracy of the solution for a practical system is improved byconsidering the nonlinearities of valve-point loading effect and multiple POZ in the EEPD problem. Themulti-objective evolutionary algorithms, namely non-dominated sorting genetic algorithm-II (NSGA-II) andmodified NSGA-II (MNSGA-II) have been applied for solving the multi-objective nonlinear optimizationEEPD problem. To improve the uniform distribution of non-dominated solutions, dynamic crowding distanceis considered in the NSGA-II and developed MNSGA-II. These multi-objective evolutionary algorithms havebeen individually examined and applied to the standard IEEE 30-bus and IEEE 118-bus test systems. Real-coded genetic algorithm is used to generate reference Pareto-front, which is used to compare with the Paretofront obtained using NSGA-II and MNSGA-II. Numerical results reveal that MNSGA-II is effectively capablefor appreciable performance than NSGA-II to solve the different power system nonsmooth EEPD problem.Moreover, three different performance metrics such as convergence, diversity and Inverted GenerationalDistance are calculated for the evaluation of closeness of obtained Pareto fronts to the reference Pareto-front.In addition, an approach based on Technique for ordering Preferences by Similarity to Ideal Solution is appliedto extract best compromise solution from the obtained non-domination solutions. Copyright © 2014 JohnWiley & Sons, Ltd.

key words: environmental and economic power dispatch; valve-point effect; prohibited operating zones;multi-objective evolutionary algorithms; modified non-dominated sorting genetic algorithm-II; TOPSIS; non-dominated solutions

1. INTRODUCTION

The economic operation of electric power generation is the dispatch of generating units’ outputs withdifferent combinations that minimizes the total fuel cost while satisfying the system power balance andcapacity limit constraints [1]. Various conventional methods such as lambda-iteration method, basepoint and participation factor method, gradient method and Newton method are used to solveeconomic dispatch (ED) problem [2]. All these conventional methods consider the input–outputcharacteristics of thermal generators by linear function. However, in practice the input–output curveof modern generators is nonlinear due to the valve-point loading effect. The mathematical

*Correspondence to: M. Rajkumar, National College of Engineering, Department of EEE, Tirunelveli-627151,Tamilnadu, India.†E-mail: [email protected]

Copyright © 2014 John Wiley & Sons, Ltd.

INTERNATIONAL TRANSACTIONS ON ELECTRICAL ENERGY SYSTEMSInt. Trans. Electr. Energ. Syst. (2014)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/etep.1918

representation of ED problem with valve-point loading effect is performed by adding a rectifiedsinusoidal contribution to the conventional quadratic cost function. Conventional methods have failedto obtain global optimum solution for the nonlinear cost function [3]. Hence, the stochastic methodssuch as Genetic Algorithm (GA) [3], Evolutionary Programming (EP) [4], Improved EP [5], ParticleSwarm Optimization (PSO) [6], Differential Evolution (DE) [7] and Harmony Search Algorithm(HSA) [8] have been used to solve the ED problem with valve-point loading effect.Also practically, generator’s operating range is break into several disjoint sub-regions when

prohibited operating zones (POZ) are present, and conventional optimization methods cannot bedirectly applied to solve such a discontinuous cost function problem. Because these conventionalmethods are often converging to local optimum or diverge altogether, Lee and Breipohl haveproposed decomposition method, based on conventional Lagrange relaxation approach, for solvingthe problem of reserve constrained ED with POZ. However, a large number of decision spaces andmore execution time are the drawbacks of this method [9]. In order to reduce the decision spaces andexecution time, Fan and Mc Donald have implemented a practical approach for solving the EDproblem with POZ. However, this approach is not suitable if the problem having multiple POZconstraints for a generator [10]. Hence, stochastic methods such as GA [11,12], EP [13], improvedEP [14], hybrid EP with tabu search and quadratic programming [15], neural networks [16] and PSO [17]have been used to solve the ED problem with POZ. However, all these studies consider single-objectiveED problem only.Due to the recent increasing awareness of environmental protection, combined economic emission

dispatch (CEED) is proposed as an alternative to achieve simultaneously the minimization of fuel costand pollutant emission. Literature over the years shows several classical methods, which were used insolving the multi-objective CEED strategies. Gent and Lamont introduced the minimum emissiondispatch concept, where they developed a program for online steam unit dispatch that results in theminimizing of NOx emission [18]. Ramanathan has proposed an efficient weight estimation techniqueto add the emission constraints in the standard classical ED problem. However, this approach requiresmultiple runs to find a Pareto-optimal set [19]. Farag et al have proposed a linear programming basedoptimization method, in which the emission function is treated as a constraint. However, this approachhas a severe difficulty in obtaining the trade-off relations between cost and emission [20]. Nanda et alhave applied goal programming method for multi-objective CEED problem but this method requires apriori knowledge about the shape of the problem search space [21]. El-kieb et al have applied aLagrange relaxation-based algorithm to environmental constraints of ED problem [22]. However,these classical methods are frequently converging at local optimum solution. Later, the use of heuristicoptimization approaches such as GA [23], EP [24], modified price penalty factor [25] and DE [26] areused to solve the multi-objective constrained optimization problem.Recently, the multi-objective evolutionary algorithms (MOEAs) have been used to eliminate many diffi-

culties in the classical methods. As population of solutions is used in their search, multiple Pareto-optimalsolutions can be found in one single simulation run [27]. Abido has applied Non-dominated Sorting GA(NSGA) for solving the multi-objective CEED problem [28] but it suffers because of computationalcomplexity, non-elitist approach and the need of sharing parameter. An improved version of NSGAknown as NSAG-II uses elitism to create a diverse Pareto-optimal front [29,30]. Although in NSGA-II,crowding distance (CD) operator ensures diversity along the Pareto-optimal front, uniform diversityand lateral diversity are lost. A modified version of NSGA-II uses dynamic CD (DCD) and controlledelitism to overcome the drawbacks found in NSGA-II [31]. Most recently, the basic HSA is modifiedand named asmulti-objective HSA, which has been used for solving CEED problem [32]. Multi-objectiveDE (MODE) [33] algorithm and Enhanced MODE [34] algorithm have been proposed for solving CEEDproblem. However, all these studies [18–34] consider CEED problem without taking into account ofvalve-point effect and POZ constraint. Very few works have been reported for the CEED problem withthe consideration of valve-point effect.Gjorgiev and Cepin have proposed a novel multi-objective optimization method for solving the

dynamic combined economic-environmental power dispatch problem with valve-point effect. Themethod is integrated within a GA and applied on various thermal and hydrothermal power systems.The results are analyzed and the achieved optimal solutions are compared with other solutionsobtained by different techniques such as NSGA-II, MODE algorithm and Simulated Annealing based

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Copyright © 2014 John Wiley & Sons, Ltd. Int. Trans. Electr. Energ. Syst. (2014)DOI: 10.1002/etep

techniques [35]. Gjorgiev et al have developed a new model in the hydrothermal generation dispatchproblem, which is concerned with obtaining optimal power outputs of all generating units givensimultaneous minimization of the fuel cost and the release of gaseous pollutants in the environmentconcurrently with minimization generating units unavailability. An efficient multi-objective-basedGA is applied for optimization purposes [36]. Basu has applied fuzzy-based Simulated Annealingmethod [37] and MODE algorithm [38] for solving CEED problem with the consideration of valve-point effect. However, the effectiveness of these approaches is not validated. In this work, the resultsobtained are validated by calculating performance metrics of the NSGA-II and MNSGA-II. Further, tothe best of authors’ knowledge, POZ constraint has not been considered yet in the CEED problemreported in the literature.In this paper, Real Coded GA (RCGA), NSGA-II and MNSGA-II have been applied for three

different cases of CEED problems. In the first case, CEED problem is solved without considering bothvalve-point effect and POZ. In the second case, valve-point effect is considered alone with the costobjective, and POZ is considered as a constraint in the third case. In all the three cases, transmissionline losses are computed using Newton–Raphson load flow solutions. Three different performancemetrics such as diversity, convergence and Inverted Generational Distance (IGD) are evaluated, andTechnique for ordering Preferences by Similarity to Ideal Solution (TOPSIS) method is used to extractbest compromise solution (BCS) from the obtained Pareto-optimal solution.The rest of this paper is organized as follows: Section 2 explains the importance of valve-point effect

and POZ. Section 3 describes the CEED problem formulation. Implementation of MNSGA-II for theCEED problem is explained in Section 4. Section 5 describes various performance measures. TOPSISdecision approach to determine BCS is discussed in Section 6. The simulation results of various testcases are presented in Section 7, and finally, Section 8 concludes.

2. PRACTICAL ECONOMIC DISPATCH PROBLEM

Practically, the input–output characteristics of generating units are nonlinear because of valve-pointloading effect in the operation of thermal generators and the existent of POZ. In this section, the im-portance of valve-point effect and POZ are explained.

2.1. Valve-point effect

Large steam turbine generators would have a number of steam admission valves that are opened insequence to obtain ever-increasing output of the unit, which is resulted in nonsmooth input–outputcharacteristics. These “valve-points” are illustrated in Figure 1.The fuel costs of generators are usually approximated by second-order polynomial when the

traditional techniques are used. The assumptions make the problem easy to solve. However, the lossof accuracy induced by these approximations is not desirable. Ignoring the valve-point loading effects,some inaccuracy would result in the generation dispatch [3].

Figure 1. Incremental fuel cost curve for five-valve steam turbine unit.

ENVIRONMENTAL AND ECONOMIC POWER DISPATCH


2.2. Prohibited operating zones

The conventional ED problem, as a convex optimization problem and assumes that the whole of theunit operating range between the minimum and maximum generation limit is available for operation.In practical systems, the input–output curve of a generator may have POZ due to steam valve operationor vibration in shaft bearing. Because it is not easy to determine the POZ by actual performancetesting, the best economy is achieved by avoiding operation in areas that are in actual operation.The consideration of POZ creates a discontinuity in cost curve and is shown in Figure 2.

3. COMBINED ECONOMIC EMISSION DISPATCH PROBLEM FORMULATION

The CEED problem is a multi-objective optimization problem, which involves simultaneous optimiza-tion of two objective functions, that is, fuel cost and emission, while satisfying power balance and gen-eration capacity constraints. The formulation of CEED problem is as follows:

3.1. Objective functions

In this section, formulations of total fuel cost objective function and total emission objective functionare given.

3.1.1. Fuel cost. The fuel cost function of the thermal generators can be represented as quadratic equa-tion as follows [27]:

FT ¼ ∑N

i¼1ai þ biPi þ ciP

2i

� �(1)

whereFT is the total fuel cost in $/hai, bi and ci are the fuel cost coefficients of the ith generatorPi is real power output of the ith generatorN is the number of generators in the system.

3.1.2. Fuel cost function with valve-point loading effect. To model the effects of nonsmooth fuel costfunctions, a recurring rectified sinusoidal contribution is added to the second-order polynomialfunctions to represent the input–output equation as follows. The total fuel cost in terms of real poweroutput can be expressed as [3]

FT ¼ ∑N

i¼1ai þ biPi þ ciP

2i þ di sin ei P

mini � Pi

� �g jÞ��(2)

wheredi and ei are the valve-point coefficients of ith generatorPmini is the minimum power generation limit.

Figure 2. Cost function with prohibited operating zones.

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3.1.3. Emission. The total emission of atmospheric pollutants such as sulfur oxides (SOx) and nitrogenoxides (NOx) are given as a function of generator output, which is modeled as a direct sum of aquadratic function and an exponential term of the active power output of the generating units and isexpressed in the following form [29].

ET ¼ ∑N

i¼1αi þ βiPi þ γiP

2i þ ηi exp δiPið Þ� �

(3)

whereET is the total emission in ton/hαi, βi, γi, ηi, and δi are the emission coefficients of the ith generator in the system.

3.2. Constraints

There are three constraints considered for this CEED problem, that is, the generation capacity of eachgenerator, power balance of the total power system and POZ. Formulation of transmission line lossescalculation through load flow solution is also explained in this section.

3.2.1. Generator capacity constraint. The generated real power output of each generator should bewithin the lower and upper limits [31],

Pmini ≤ Pi ≤ Pmax

i ; i ¼ 1; 2; :::::;N (4)whereP mini is the minimum power generation limit of ith generator

P maxi is the maximum power generation limit of ith generator.

3.2.2. Power balance constraint. The total power generation must supply the total load demand andthe real power losses in the transmission lines [32]. It can be defined as

∑N

i¼1Pi � Pd � Ploss ¼ 0 (5)

wherePd is the total load demandPloss is the power loss in the transmission network.The Ploss can be calculated using Newton–Raphson load flow method, which gives all bus voltage

magnitudes and angles; it can be described as follows [32]:

Ploss ¼ ∑NL

k¼1gk U2

i þ U2j � 2UiUj cos θi � θj

� �h i(6)

wherei and j are the total number of buses (i≠ j)k is the kth network branch that connects bus i to bus jNL is the number of transmission linesUi and Uj are the voltage magnitudes at bus i and jgk is the transfer conductance between bus i and jθi and θj are the voltage angles at bus i and j, respectively.

3.2.3. Prohibited operating zones constraint. The feasible POZ of unit i can be described as follows [17]:

Pmini ≤ Pi ≤ PL

i;1

PUi;q�1 ≤ Pi ≤ PL

i;q; q ¼ 2; ::::Zi

PUi;Zi

≤ Pi ≤ Pmaxi

(7)

wherePLi;q, P

Ui;q are the lower and upper boundary of qth POZ of unit i

q is the index of POZZi is the number of POZ of unit i.



4. IMPLEMENTATION OFMODIFIED NON-DOMINATED SORTING GENETIC ALGORITHM-II

The NSGA-II is a fast and elitist MOEA, and it implements elitism for multi-objective search, using anelitism-preserving approach. Elitism enhances the convergence properties toward the true Pareto-optimalset. Crowded comparison operator is used for good spread of solutions in the obtained non-dominatedsolutions. The details of NSGA-II algorithm are given in [29]. Although the crowded comparison operatorensures diversity along the non-dominated front in NSGA-II, the uniform diversity is lost and hence leadsto slowing down of the search. To overcome this drawback of NSGA-II, a new diversity strategy calledDCD is introduced to improve uniform distribution of non-dominated solutions [31].

4.1. Dynamic crowding distance

The NSGA-II uses CD measure in population maintenance to remove the excess individuals found inthe non-dominated set when the number of non-dominated solutions exceeds the population size. Theindividuals having lower value of CD are preferred over individuals with higher value of CD in theremoval process. Individual’s CD can be calculated as follows:

CDi ¼ 1Nobj

∑Nobj

g¼1f giþ1 � f gi�1

�� (8)

whereNobj is the number of objectives,f giþ1 is the gth objective of the i+ 1th individual,f gi�1 is the gth objective of the i-1th individual.

The major drawback of CD is the lack of uniform diversity in the obtained non-dominated solutions.If normal CD is applied, some of the individuals helping to maintain uniform spread are removed [30].To overcome this problem, DCD method is suggested in [31]. The individuals’ CD is calculated

only once during the process of population maintenance but the individuals’ DCD is varying dynam-ically during the process of population maintenance. In the DCD approach, one individual with lowestDCD value during every time is removed, and DCD is recalculated for the remaining individuals. Theindividuals’ DCD are calculated as follows:

DCDi ¼ CDi

log 1Vari

� � (9)

whereCDi is calculated by Equation (8),

Vari ¼ 1Nobj

∑Nobj

g¼1f giþ1 � f gi�1

�� CDi

� �2(10)

Vari is the variance of CDs of individuals, which are neighbors of the ith individual. Vari can giveinformation about the difference in the variations of CD for different objectives. Therefore, ifDCD is used in population maintenance, individuals in the non-dominated set will have morechance to get retained.

4.2. Computational flow

The algorithm can be described by the following steps:

Step 1: Choose population size N, crossover and mutation probability, crossover and mutationindex, maximum number of iterations and control variable limits.

Step 2: Initially, generate a random parent population of size N within control variable limits. Setthe iteration count i= 0.

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Step 3: For each individual in the population, run power flow algorithm using Newton–Raphsonmethod and evaluate the objective functions and constraint violations.

Step 4: Create offspring population of size N by using the crowded tournament selection, SimulatedBinary Crossover and polynomial mutation.

Step 5: Combine the parent population and offspring population, which makes the size of combinedpopulation is 2N.

Step 6: Perform non-dominated sorting to the combined population and identify different fronts.Step 7: If the size of non-dominated set M is greater than the population size N, then remove

M–N individuals from non-dominated set by using DCD-based strategy; elsewhere, goto Step 4.

Step 8: The process can be stopped after a fixed number of iterations. If the criterion is not satisfiedthen the procedure is repeated from Step 3 after creating the new population from the parentpopulation. The new population obtained under the MNSGA-II will, in general, be morediverse than that obtained by using NSGA-II approach.

5. PERFORMANCE METRICS

To evaluate the performances ofmulti-objective optimization algorithms, somemeasures of performancesare essential. The existing performance metrics can be classified into three classes: metrics forconvergence (γ), metrics for diversity (Δ) and metrics for both convergence and diversity. These metricsare helpful for evaluating the closeness of the obtained Pareto-front to the reference Pareto-front and forevaluating diversity among non-dominated solutions [27].

5.1. Convergence metric or distance metric (γ)

This convergence metric (γ) evaluates average distance between non-dominated solutions found andthe actual Pareto-optimal front, as follows:

γ ¼∑N

i¼1di

N(11)

where di is the distance between non-dominated solutions found and the actual Pareto-optimal front,and N is the number of solutions in the front. The smaller this metric value, the better is theconvergence toward the Pareto-optimal front [29].

5.2. Spread metric or diversity metric (Δ)

This diversity metric (Δ) measures the extent of spread achieved among the obtained solutions. First, thecalculation of the Euclidean distance di between consecutive solutions in the obtained non-dominated setand then calculates the average d of these distances. Thereafter, from the obtained set of non-dominatedsolutions, calculate the extreme solutions and then use the following metric to calculate the non-uniformity in the distribution:

Δ ¼df þ dl þ ∑

N�1

i¼1di � d��

df þ dl þ N � 1ð Þd (12)

where df and dl are the Euclidean distances between the extreme solutions and the boundarysolutions of the obtained non-dominated set. The parameter d is the average of all distances di,i = 1, 2… (N-1), assuming that there are N solutions on the best non-dominated front and (N-1)consecutive distances. According to this metric, if an algorithm finds a smaller Δ value is ableto find a better diverse set of non-dominated solutions [29].



5.3. Inverted Generational Distance

The IGD is designed for both convergence and diversity. IGD is calculated as shown in the following:

IGD ¼ ∑ v∈P�d v;Pð ÞP�j j (13)

whereP* is a set of uniformly distributed points in true Pareto-front,P is the non-dominated solutions obtained by MOEAs,d (v,P) is the minimum Euclidean distance between v and the point in P.A value of IGD equal to 0 indicates that P should be close to P* [27].

6. TECHNIQUE FOR ORDERING PREFERENCES BY SIMILARITY TO IDEAL SOLUTIONMETHOD

In general, the result of MOEAs is a set of non-dominated front. From the best obtained Pareto-front, itis usually required to select one solution for implementation. A multi-attribute decision-making(MADM) approach is adopted to rank the obtained NSGA-II solutions. The BCS is calculated in adeterministic environment with a single decision maker. From the decision maker’s perspective, thechoice of a solution from all Pareto-optimal solutions is called a posterior approach, and it requiresa higher level decision-making approach, which is to determine the best solution among a finite setof Pareto-optimal solutions with respect to all relevant attributes. In this paper, MADM techniquebased on TOPSIS is employed in posterior evaluation of Pareto-optimal solutions to choose the bestone among them. The concept of TOPSIS is described as follows: In the absence of a natural courseof action for overall summary measure and ranking, the most preferred alternative should not onlyhave the shortest distance from the positive ideal solution but also have the longest distance fromthe negative ideal solution. Almost all MADM methods require predetermined information on therelative importance of the attributes, which is usually given by a set of normalized weights. Theweights of two objectives are calculated by Shannon’s entropy method. The entropy method is basedon information theory, which assigns a smaller weight to an attribute if it has similar attribute valuesacross alternatives, because such attribute does not help in differentiating alternatives [31].

Table I. Power generation schedule with power loss, extreme solution and best compromise solution forIEEE 30-bus system (case 1).

CEED without valve-point effect and POZ – IEEE 30-bus system

Power generation /losses (MW)

RCGA NSGA-II MNSGA-II

Cost ($/h)Emission(ton/h) Cost ($/h)

Emission(ton/h) Cost ($/h)

Emission(ton/h)

P1 11.38 41.10 11.60 40.00 11.55 40.74P2 30.60 46.30 30.52 46.51 31.70 44.85P3 59.85 54.41 59.73 54.11 54.44 55.33P4 98.23 39.00 98.21 39.22 94.51 40.85P5 51.29 54.41 51.22 53.63 54.92 54.78P6 35.52 51.50 35.60 53.00 39.66 50.19Total generation 286.87 286.72 286.88 286.47 286.78 286.74Losses 3.47 3.32 3.48 3.07 3.38 3.34Cost ($/h) 608.1247 645.7245 608.1291 644.9107 608.1257 643.2614Emission (ton/h) 0.2200 0.1942 0.2200 0.1942 0.2167 0.1942Best compromisesolution

— — 630.67 0.1949 631.39 0.1948

Execution time(seconds)

3251.8 86.281 87.093

Extreme solutions for cost and emission are bold in the table entries.CEED, combined economic emission dispatch; POZ, prohibited operating zones; RCGA, real-coded genetic algorithm; NSGA-II,non-dominated sorting genetic algorithm-II; MNSGA-II, modified NSGA-II.

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7. SIMULATION RESULTS AND DISCUSSION

The RCGA, NSGA-II and MSGA-II are implemented in MATLAB version 7.11 and executed on aPentium-IV Intel (R) Core(TM) i3-2310M CPU operating at 2.10GHz speed with 4GB RAM.

7.1. Description of the test systems

The standard IEEE 30-bus system is composed of six generating units with a demand of 283.4MWand IEEE 118-bus system consists of 19 generating units with a demand of 3668MW are taken to

Table II. Power generation schedule with power loss, extreme solution and best compromise solution forIEEE 30-bus system (case 2).

CEED with valve-point effect – IEEE 30-bus system





Emission(ton/h)

P1 11.71 40.04 11.12 37.49 11.42 40.22P2 30.81 45.96 30.67 45.88 30.68 46.23P3 59.03 56.16 59.67 53.24 59.56 53.85P4 98.24 39.94 97.95 41.24 98.25 39.28P5 51.07 53.14 51.59 55.83 51.22 54.39P6 35.92 50.91 35.79 52.38 35.65 51.93Total Generation 286.78 286.15 286.79 286.06 286.78 285.90Losses 3.38 2.75 3.39 2.66 3.38 2.50Cost ($/h) 611.0551 648.5538 611.0684 646.1838 611.0552 648.9837Emission (ton/h) 0.2210 0.1942 0.2200 0.1942 0.2200 0.1942Best compromisesolution

— — 634.9 0.1948 634.9 0.1947


3458.2 89.109 87.422

Extreme solutions for cost and emission are bold in the table entries.CEED, combined economic emission dispatch; RCGA, real-coded genetic algorithm; NSGA-II, non-dominated sorting geneticalgorithm-II; MNSGA-II, modified NSGA-II.

Table III. Power generation schedule with power loss, extreme solution and best compromise solution forIEEE 30-bus system (case 3).

CEED with POZ – IEEE 30-bus system





Emission(ton/h)

P1 11.52 40.75 10.16 38.73 11.53 39.53P2 30.59 45.81 40.01 46.56 30.63 46.31P3 59.99 54.53 57.13 54.91 60.02 54.89P4 98.32 40.74 96.49 39.65 98.26 39.39P5 51.45 54.43 48.69 55.05 51.44 54.77P6 35.00 51.01 34.39 51.78 34.99 51.79Total Generation 286.87 287.27 286.87 286.68 286.87 286.68Losses 3.47 3.87 3.47 3.28 3.47 3.28Cost ($/h) 608.128 645.109 609.330 643.989 608.128 644.553Emission (ton/h) 0.2201 0.1942 0.2183 0.1942 0.2198 0.1942Best compromisesolution

— — 633.9 0.1952 632.6 0.1944


5402.75 90.36 91.593




verify the effectiveness of RCGA, NSGA-II and MNSGA-II. The detailed fuel cost coefficients,emission coefficients, the minimum power limits and the maximum power limits are taken from[5,25,30]. Valve-point coefficients for IEEE 30-bus system are taken from [5] and appropriately assumedfor IEEE 118-bus system. POZ data for both the test systems are given in the Appendix. The bus data andthe line data are taken from [39]. MATPOWER software is used for power flow calculations [39].

7.2. Parameter settings

The parameter settings of NSGA-II andMNSGA-II for solving CEED problem are as follows: In general,six times of the number of decision variables are set as population size. For IEEE 30-bus system, the pop-ulation size and iteration are fixed at 40 and 200, respectively. For IEEE 118-bus system, the populationsize and maximum iteration number are fixed at 100 and 500, respectively. The crossover probability (Pc)is selected between 0.8 and 0.95, in steps of 0.01. It is found that, after experimentation, Pc=0.85 pro-duces best results. Other parameters such as mutation probability (Pm), crossover index (ηc) and mutationindex (ηm) are selected as 1/n (where n-number of variables), 5 and 15, respectively.

7.3. IEEE 30-bus and IEEE 118-bus test systems

To demonstrate the effectiveness of RCGA, NSGA-II and MNSGA-II, three different cases haveconsidered as follows:

Table IV. Power generation schedule with power loss, extreme solution and best compromise solution forIEEE 118-bus system (case 1).

CEED without valve-point effect and POZ – IEEE 118-bus system

Powergeneration /losses (MW)




Emission(ton/h)

P1 673.52 310.33 640.18 314.74 611.297 330.361P2 73.064 425.52 54.086 396.63 62.239 436.532P3 76.074 89.992 83.849 82.989 89.071 89.833P4 299.99 299.99 285.21 299.49 298.658 299.491P5 40.00 399.99 40.495 395.52 40.072 394.789P6 1.0202 1.666 1.449 8.837 5.328 5.682P7 17.653 18.706 12.749 22.667 9.500 8.788P8 30.001 239.99 30.032 236.20 31.862 239.601P9 10.398 49.989 48.723 49.480 41.875 40.019P10 138.69 199.95 151.96 199.66 195.868 197.999P11 199.99 199.99 190.139 198.92 197.051 194.059P12 399.89 291.42 394.83 340.86 395.269 303.409P13 399.950 399.83 397.633 386.81 398.515 385.217P14 599.999 167.68 590.541 174.59 582.462 157.579P15 3.685 1.645 3.992 4.895 3.022 3.788P16 690.625 307.753 672.044 303.70 658.537 310.376P17 228.31 292.74 240.91 276.89 243.364 291.661P18 5.3177 49.988 36.108 38.283 10.799 46.155P19 8.303 26.841 6.9604 39.801 5.704 39.741Total generation 3896.5 3774.0 3881.9 3771.0 3880.5 3775.1Losses 228.5 106.0 213.9 103.0 212.5 107.1Cost ($/h) 11 509.778 18 227.583 11 577.500 17 993.470 11 552. 081 18 311.064Emission (ton/h) 14.9726 5.4876 14.1910 5. 4950 13.7960 5. 5466Bestcompromisesolution

— — 16 369.9 5.7001 16 193.0 5.7339


31 806.157 838.171 825.875


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Case 1 CEED problem without valve-point effect and POZ constraint,Case 2 CEED problem with valve-point effect andCase 3 CEED problem with POZ.

Transmission line losses are computed for all the three cases. IEEE 30-bus system with a demand of283.4MW and IEEE 118-bus system with a demand of 3668MW are employed to find the effectivenessof the approaches.Extreme solutions obtained out of 10 trial runs for cost and emission using NSGA-II and MNSGA-II

for IEEE 30-bus system are reported in Tables I–III and that for IEEE 118-bus system are reported inTables IV–VI. Extreme solutions for cost and emission are obtained using RCGA also reported in Ta-ble I–VI. From Tables I–VI, it can be concluded that the MNSGA-II is capable of providing better ex-treme solutions than the NSGA-II. Referring to Tables I–VI, it is found that extreme solutions of cases2 and 3 are better in accuracy compared with those of case 1 for the two IEEE test systems. Becauseignoring the valve-point effect and POZ constraint in the CEED problem shows the inaccurate value incase 1. Execution time is lesser using NSGA-II and MNSGA-II compared with RCGA method, thuscomputationally more efficient than RCGA method.Best Pareto-fronts obtained using RCGA, NSGA-II and MNSGA-II for cases 1–3 with respect to

IEEE 30-bus system are shown in Figures 3–5, respectively. Similarly, Figures 6–8, respectively

Table V. Power generation schedule with power loss, extreme solution and best compromise solution forIEEE 118-bus system (case 2).

CEED with valve-point effect – IEEE 118-bus system





Emission(ton/h)

P1 657.562 299.243 619.244 309.409 647.701 318.552P2 56.724 429.010 95.494 413.208 52.964 406.683P3 83.989 83.298 89.436 84.303 88.715 80.825P4 295.475 283.872 299.840 284.142 295.434 297.839P5 40.363 399.735 40.392 395.502 41.977 396.482P6 3.9975 9.876 2.917 6.534 7.013 8.393P7 17.0116 22.160 10.568 22.851 16.575 22.857P8 30.1108 236.782 33.217 238.812 30.898 232.410P9 46.0110 44.705 33.631 49.865 47.872 43.542P10 152.7939 199.219 198.876 178.827 171.924 199.396P11 199.624 199.488 191.022 194.309 192.813 198.899P12 399.867 362.944 386.319 320.015 399.759 330.965P13 384.559 371.465 395.338 389.456 399.464 390.694P14 597.156 168.441 548.729 159.185 581.203 177.041P15 2.939 1.8848 1.0965 3.028 1.181 4.802P16 651.217 282.917 661.368 347.362 602.693 316.599P17 239.035 296.966 252.201 288.186 271.819 263.657P18 23.039 49.159 5.973 48.763 6.766 47.787P19 5.507 31.699 4.800 37.942 6.165 34.532Total generation 3886.981 3772.864 3870.462 3771.699 3862.936 3771.955Losses 218.981 104.864 202.462 103.699 194.936 103.955Cost ($/h) 12 039.249 18 795.778 12 043.882 18 533.628 11 944.047 18 482.067Emission (ton/h) 14.284 5.563 13.277 5.539 13.593 5.5210Bestcompromisesolution

— — 16 902.76 5.8756 16 863.85 5.7747


31 938.32 867.342 870.523

Extreme solutions for cost and emission are bold in the table entries.CEED, combined economic emission dispatch; RCGA, real-coded genetic algorithm; NSGA-II, non-dominated sorting geneticalgorithm-II; MNSGA-II, modified NSGA-II.



represent best Pareto-fronts obtained for cases 1–3 using RCGA, NSGA-II and MNSGA-II in IEEE118-bus system. Multiple Pareto-optimal solutions are obtained from single run using NSGA-IIand MNSGA-II, whereas RCGA produces the Pareto front in multiple runs. From the obtained

Table VI. Power generation schedule with power loss, extreme solution and best compromise solution forIEEE 118-bus system (case 3).

CEED with POZ – IEEE 118-bus system





Emission(ton/h)

P1 825.009 300.004 652.0570 311.3760 661.1104 316.7928P2 150.009 450.003 300.5014 386.7490 158.5739 452.1963P3 80.999 89.999 66.8183 89.7640 68.7743 68.5363P4 229.987 249.999 232.6436 174.6248 299.3478 246.6635P5 40.006 388.018 101.1264 348.1243 41.2964 347.0228P6 1.003 9.999 8.7802 9.8013 8.7574 9.3865P7 22.987 22.996 18.6192 22.9905 6.2725 8.5190P8 30.003 239.996 30.6023 234.4095 31.9419 238.1627P9 39.999 49.999 32.0140 39.6480 39.2816 49.8023P10 132.354 199.999 122.0069 199.9880 196.5573 199.9692P11 199.999 199.993 88.3731 199.4457 159.6604 199.1819P12 324.883 250.414 322.5816 350.6196 391.2183 279.9553P13 399.999 372.413 383.3625 399.9634 399.6819 399.6730P14 562.952 149.545 555.7721 220.4216 433.5458 157.9624P15 1.000 4.994 2.9170 3.8987 1.3063 1.7803P16 549.973 450.001 662.4273 451.0764 652.2405 451.4738P17 224.889 255.329 217.1451 289.0008 285.5039 282.6564P18 5.000 49.999 8.0953 5.2316 15.1694 45.1264P19 4.006 39.995 33.0911 31.4340 7.0886 19.0417Total generation 3825.1 3773.7 3838.9344 3768.5672 3857.3286 3773.9026Losses 157.1 105.7 170.9344 100.5672 189.3286 105.9026Cost ($/h) 11 655.856 18 240.610 12 862.488 16 910.675 11 904.102 17 541.703Emission (ton/h) 13.0098 5.7111 13.0403 6.3309 11.9724 6.0496Bestcompromisesolution

— — 16 238.20 6.4512 16 052.14 6.2936


63 638.3 1781.69 1762.45

Extreme solutions for cost and emission are bold in the table entries.CEED, combined economic emission dispatch; POZ, prohibited operating zones; RCGA, real-coded genetic algorithm; NSGA-II, non-dominated sorting genetic algorithm-II; MNSGA-II, modified NSGA-II.

Figure 3. Pareto-front of real-coded genetic algorithm (RCGA), non-dominated sorting genetic algorithm-II(NSGA-II) and modified NSGA-II (MNSGA-II) for IEEE 30-bus system (case 1).

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Figure 6. Pareto-front of real-coded genetic algorithm (RCGA), non-dominated sorting genetic algorithm-II(NSGA-II) and modified NSGA-II (MNSGA-II) for IEEE 118-bus system (case1).



Pareto-optimal front using NSGA-II and MNSGA-II, BCS is extracted using TOPSIS method andis also shown in Figures 3–8 whose values are reported in Tables I–VI. BCS represents the op-erating point, which the power system operator may consider it for practical use. There arediscontinuities in the Pareto-front shown in Figures 5 and 8, because of the presence of POZconstraint in CEED problem of the two test systems.



Table VII. Statistical results of performance measures (case1) – IEEE 30-bus system.

Measure Algorithm Best Worst Mean Standard deviation

γ NSGA-II 0.3091 0.4043 0.3553 0.0417MNSGA-II 0.3328 0.4218 0.3718 0.0378

Δ NSGA-II 0.3777 0.5638 0.4676 0.0765MNSGA-II 0.2486 0.3961 0.3370 0.0545

IGD NSGA-II 0.0024 0.0029 0.0026 0.0002MNSGA-II 0.0025 0.0029 0.0028 0.0001

IGD, Inverted Generational Distance; NSGA-II, non-dominated sorting genetic algorithm-II; MNSGA-II, modified NSGA-II.

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Table VIII. Statistical results of performance measures (case2) – IEEE 30-bus system.


γ NSGA-II 0.2880 0.3609 0.3236 0.0218MNSGA-II 0.2700 0.3845 0.3314 0.0370

Δ NSGA-II 0.3156 0.4206 0.3768 0.0347MNSGA-II 0.2812 0.3968 0.3352 0.0395



Table IX. Statistical results of performance measures (case3) – IEEE 30-bus system.


γ NSGA-II 0.0032 0.0047 0.0037 0.000518MNSGA-II 0.0036 0.0040 0.0038 0.000167

Δ NSGA-II 0.4579 0.5421 0.4998 0.0338MNSGA-II 0.3676 0.5302 0.4440 0.0739



Table X. Statistical results of performance measures (case1) – IEEE 118-bus system.


γ NSGA-II 0.0222 0.0243 0.0234 0.0008MNSGA-II 0.0238 0.0241 0.0240 0.0001

Δ NSGA-II 0.3994 0.5021 0.4403 0.0420MNSGA-II 0.1845 0.2569 0.2324 0.0316



Table XI. Statistical results of performance measures (case2) – IEEE 118-bus system.


γ NSGA-II 0.0148 0.0928 0.0556 0.0286MNSGA-II 0.0112 0.0628 0.0286 0.0199

Δ NSGA-II 0.3384 0.5224 0.4401 0.0561MNSGA-II 0.2105 0.3467 0.2574 0.0375



Table XII. Statistical results of performance measures (case3) – IEEE 118-bus system.


γ NSGA-II 0.0472 0.1106 0.0805 0.0231MNSGA-II 0.0272 0.0835 0.0439 0.0231

Δ NSGA-II 0.6617 0.8238 0.7294 0.0665MNSGA-II 0.3862 0.7201 0.5559 0.1493





The statistical analysis such as best, worst, mean and standard deviation results of multi-objectiveperformance metrics for IEEE 30-bus system are reported in Tables VII–IX, and for IEEE 118-bussystem are reported in Tables X–XII for cases 1–3. It can be seen that, for most of the performancemetrics, values obtained by MNSGA-II are better than NSGA-II.

8. CONCLUSION

In this paper, RCGA, NSGA-II and MNSGA-II are applied to solve CEED problem with three differenttest cases. In the first case, the problem is solved without considering both valve-point effect and POZconstraint. In the second case, valve-point loading is considered, and in the third case, the CEED problemis solved taking into account of POZ constraint. The performance of NSGA-II and MNSGA-II arevalidated on the standard IEEE 30-bus and IEEE 118-bus systems and are compared with the generatedreference Pareto-front by RCGA approach. Pareto-optimal solutions are obtained in single simulation runby NSGA-II and MNSGA-II and thus computationally more efficient than RCGA method. Pareto-frontobtained by MNSGA-II shows significant improvement in the uniform distribution of non-dominatedsolutions compared with NSGA-II. The performance of NSGA-II and MNSGA-II are compared withrespect to various statistical performance measures such as convergence metric, diversity metricand inverted generational distance metric. By using the statistical performance measures, it can beconcluded that the MNSGA-II is better with respect to most of the multi-objective performancemetrics values. The MADM procedure is followed for choosing the BCS from the obtainedPareto-optimal solutions based on TOPSIS.

ACKNOWLEDGEMENTS

The authors are thankful to the managements of the National College of Engineering, Tirunelveli, PSNA Collegeof Engineering and Technology, Dindigul, Kalasalingam University, Krishnankoil and Thiagarajar College ofEngineering, Madurai for the continuous encouragements and facilities provided to carry out this research.

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APPENDIX

Table A. Prohibited operating zones (POZ) data for IEEE 30-bus system.

Generator number POZ1 POZ2 POZ3

1 [15–20] [25–30] [45–50]2 [5–10] [20–25] [35–40]3 [45–50] [65–75] [95–100]4 [20–30] [60–65] [105–110]5 [20–30] [40–45] [80–85]6 [10–15] [20–25] [55–60]



http://www.pserc.cornell.edu/matpower

http://www.pserc.cornell.edu/matpower

Table B. Prohibited operating zones (POZ) data for IEEE 118-bus system.

Generator number POZ1 POZ2 POZ3

1 [200–300] [550–650] [725–825]2 [125–150] [250–300] [425–450]3 [20–30] [45–55] [70–80]4 [50–100] [175–200] [250–275]5 [50–100] [125–150] [350–375]6 [0–0] [0–0] [0–0]7 [4–6] [10–12] [10–12]8 [60–80] [120–140] [190–200]9 [10–20] [25–30] [40–45]10 [40–60] [80–100] [150–160]11 [50–60] [90–110] [160–180]12 [120–170] [200–225] [325–350]13 [100–150] [175–200] [300–325]14 [170–220] [350–400] [450–550]15 [0–0] [0–0] [0–0]16 [110–210] [350–450] [550–650]17 [50–100] [125–150] [225–250]18 [10–15] [25–30] [25–30]19 [10–15] [20–25] [20–25]

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environmental and economic power dispatch of thermal generators using modified nsga-ii algorithm

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