Nondominated Sorting Genetic AlgorithmII (NSGAII) is applied for solving Combined Economic Emission Dispatch (CEED) problem with valvepoint loading of thermal generators. This CEED problem with valvepoint loading is a nonlinear, constrained multiobjective optimization problem, with power balance and generator capacity constraints. The valvepoint loading introduce ripples in the inputoutput characteristics of generating units and make the CEED problem as a nonsmooth optimization problem. To validate its effectiveness of NSGAII, two benchmark test systems, IEEE 30bus and IEEE 118bus systems are considered. To compare the Paretofront obtained using NSGAII, reference Paretofront is generated using multiple runs of Real Coded Genetic Algorithm (RCGA) with weighted sum of objectives. Comparison with other optimization techniques showed the superiority of the NSGAII approach and confirmed its potential for solving the CEED problem. Numerical results show that NSGAII algorithm can provide Paretofront in a single run with good diversity and convergence. An approach based on Technique for Ordering Preferences by Similarity to Ideal Solution (TOPSIS) is applied on nondominated solutions obtained to determine Best Compromise Solution (BCS).
1. Introduction
Economic Dispatch (ED) is the process of allocating generation levels, to the various generating units so that the system demand is fully met in the most economical way. The traditional ED has the objective of minimizing the fuel costs
[1]
. Operating at absolute minimum cost can no longer be the only criterion for dispatching electric power due to the increasing environmental pollution caused by the fossilfueled electric power plants. It forces the utilities to modify their design to reduce pollution and atmospheric emissions of the thermal power plants
[2]
. Hence, it is necessary to minimize both emission and cost. However, minimizing the emission and cost are usually two conflicting objectives. Thus, it is not possible to minimize both of them simultaneously and some form of conflicting resolution must be adopted to arrive at a solution
[3]
.
Several Economic Emission Dispatch (EED) strategies have appeared in the literature over the years. Lagrange relaxation method
[4]
, weighted sum method
[5]
, εconstrained algorithm
[6]
, Linear programming method,
[7]
, Goal programming technique
[8]
are used to solve the EED problem. However, these classical methods are highly sensitive and frequently converge at local optimum solution and computational time increases with the increase of the dimensionality of the problem.
Later, the use of heuristic optimization approaches such as Genetic Algorithm (GA)
[9]
, Evolutionary Programming (EP)
[10]
and Differential Evolution (DE)
[11]
are used to solve the multiobjective constrained optimization problem. Prabakar et al have applied modified price penalty factor method to Combined Economic Emission Dispatch (CEED) problem and converted into single objective problem
[12]
. Recently, the multiobjective evolutionary algorithms (MOEAs) are used to eliminate many difficulties in the classical methods
[13]
. Because, population of solutions is used in their search and multiple Paretooptimal solutions can be found in one single simulation run. Some of the popular MOEAs are Nondominated Sorting Genetic Algorithm (NSGA), Niched Pareto Genetic Algorithm, Strength Pareto Evolutionary Algorithm (SPEA), NSGAII, Pareto Archived Evolution Strategy etc.
[14]
. Abido has applied NSGA
[15]
, SPEA
[16]
and Multiobjective Particle Swarm Optimization (MOPSO)
[17]
approaches for solving the multiobjective CEED problem. NSGA suffers from computational complexity, nonelitist approach and the need to specify a sharing parameter. An improved version of NSGA known as NSGAII, which resolves CEED problems and uses elitism to create a diverse Paretooptimal front, has been subsequently presented
[18

21]
. In addition, TOPSIS method is employed to choose the BCS, which will be useful to the decision maker
[20
,
21]
. Wu
et al
proposed multiobjective DE (MODE) algorithm with elitist archive and crowding entropy based diversity measure to solve the environmental/economic power dispatch problem
[22]
. The premature convergence using MODE algorithm is overcome by enhanced MODE (EMODE) algorithm to solve EED problem by Youlin Lu
et al
[23]
. Though researchers have used several methods for solving single objective ED problem and multiobjective EED problem, but they do not considered valvepoint loading effect
[1

23]
.
In general, discontinuity may also be observed in thermal power plants due to valvepoint loading. In reality, due to valvepoint effect, the cost function is nonsmooth and nonmonotonically increasing and conventional methods such as Lambdaiteration, Gradient method and Newton method have failed to obtain global optimum solution. Hence, stochastic methods such as GA
[24]
, EP
[25]
, Improved EP
[26]
, PSO
[27]
and DE
[28]
have been used to solve the ED problem with valvepoint loading effect by adding the rectified sinusoidal contribution to the conventional quadratic cost function.
Though researchers considered valvepoint effect in the single objective ED problem
[24

28]
, very few works are reported with the consideration of valvepoint effect for the CEED problem. Basu
[29]
analyzed the interactive fuzzy satisfying based simulated annealing technique for CEED problem with nonsmooth fuel cost and emission level functions. The major advantage of this method is obtaining a compromising solution in the presence of conflicting objectives. However, the longer execution time is the drawback of this method. Hemamalini
et al
have applied PSO algorithm to solve emission constrained ED problem with valvepoint loading effect. However, this formulation does require the knowledge of the relative importance of each objective and has a severe difficulty in getting the tradeoff relations between cost and emission
[30]
. MODE algorithm has been applied by Basu, for solving EED problems with valvepoint loading and extreme points obtained are compared with Partial DE, NSGAII and SPEA2 for different test systems. However, the selection of BCS from the estimated Paretooptimal set is not considered
[31]
. Also, the transmission line losses are calculated through B
_{mn}
coefficients
[29

31]
.
In this paper, NSGAII algorithm is used to solve CEED problem with valvepoint effect. Paretofront obtained by the NSGAII is compared with reference Paretofront found by RCGA. In addition, the transmission line losses are calculated through load flow solutions and BCS is obtained by TOPSIS method, which will be useful to the decision maker. The rest of this paper is organized as follows: Section 2 describes the CEED problem formulation. Implementation of NSGAII for the CEED problem is explained in Section 3. Section 4 incorporates TOPSIS decision approach to determine the BCS. The simulation results of various test cases are presented in Section 5 and Section 6 concludes.
2. Problem Formulation
The multiobjective CEED problem with its constraints is formulated as a nonlinear constrained problem as follows.
subject to power balance and generation capacity constraints, where,
F(P_{g}): Total fuel cost ($/hr),
E(P_{g}): Total emission (ton/hr).
 2.1 Objective functions
The fuel cost function or inputoutput characteristics of the generator may be obtained from design calculations or from heat rate tests. For large steam turbine generators, the inputoutput characteristics are not always smooth. Large steam turbine generators will have a number of steam admission valves that are opened in sequence to obtain everincreasing output of the unit. These “valvepoints” are illustrated in
Fig. 1
.
Ignoring the valvepoint loading effects, some inaccuracy would result in the generation dispatch. Therefore, the fuel costs of generators are usually approximated by secondorder polynomial when the traditional techniques are used.
The assumptions made the problem easier to solve. However, the loss of accuracy induced by these approximations is not desirable. To model the effects of nonsmooth fuel cost functions, a recurring rectified sinusoidal contribution is added to the second order polynomial functions to represent the inputoutput Eq. (2) as follows. The total fuel cost in terms of real power output can be expressed as
[24]
,
Incremental fuel cost curve of turbine unit
where,
F(P_{g}): Total fuel cost ($/hr),
a_{i}, b_{i}, c_{i}, d_{i}, e_{i} : Fuel cost coefficients of generator i,
P_{gi}: Power generated by generator i,
: Minimum power generation limit,
N : Number of generators.
The total emission of atmospheric pollutants such as Sulphur Oxides (SO
_{x}
) and Nitrogen Oxides (NO
_{x}
) from a fossilfired thermal generating unit depends upon the amount of power generated by each unit. For simplification, the total emission generated can be approximately modeled as a direct sum of a quadratic function and an exponential term of the active power output of the generating units and is expressed in the following form
[29]
.
where,
E(P_{g}): Total emission (ton/hr),
α_{i}, β_{i}, γ_{i}, η_{i}, δ_{i} : Emission coefficients of generator i.
 2.2 Constraints
Generation capacity constraint:
For stable operation, real power output of each generator is restricted by lower and upper limits as follows:
where,
: Minimum power generated,
: Maximum power generated.
Power balance constraint:
The total power generated must supply the total load demand and the transmission losses
[19]
.
where,
P_{d} : Total load demand,
P_{loss} : Transmission losses.
The real power loss Ploss can be calculated from Newton Raphson load flow solution, which gives all bus voltage magnitudes and angles; it can be described as follows
[22]
:
where,
i and j are the total number of buses (i ≠ j),
k is the k^{th} network branch that connects bus i to bus j,
N_{L} is the number of transmission lines,
V_{i }and V_{j} are the voltage magnitudes at bus i and j,
g_{k} is the transfer conductance between bus i and j,
θ_{i} and θ_{j} are the voltage angles at bus i and j respectively.
3. Implementation of NSGAII
The NSGAII algorithm and its computational flow are described in this section.
 3.1 NSGAII
NSGAII is a fast and elitist multiobjective evolutionary algorithm (MOEA) and implements elitism for multiobjective search, using an elitismpreserving approach. Elitism enhances the convergence properties towards the true Paretooptimal set. A parameterless diversity preserving mechanism is adopted. Diversity and spread of solutions are guaranteed without the use of sharing parameters. In this paper, NSGAII uses simulated binary crossover and polynomial mutation for solving the problem. The crowd comparison operator, guides the selection process towards a uniformly spread Pareto frontier
[18]
.
 3.2 Computational flow

Step1: Initially, a random parent population of sizeNis created.

Step2: The population is sorted based on the nondomination. Each population is assigned a rank equal to its nondomination level or front number (1 is the best level, 2 is the next best level, and so on). Calculate the crowding distance (CD) of populations in each nondomination level and sort populations in descending order of CD.

Step3: Select two individuals at random. Compare their front number and CD. Select the better one and copy it to the mating pool.

Step4:The Simulated Binary Crossover (SBX) and polynomial mutation have been used to create offspring population of sizeN.

Step5: Combine the parent population and child population.

Step6:The combined population is sorted according to nondomination and crowding distance. Since all parent and offspring population members are included, elitism is ensured.

Step7: The process can be stopped after a fixed number of iterations. If the criterion is not satisfied then the procedure is repeated from Step 3 after creating the new population from the parent population[18].
Computational flow of NSGAII
4. TOPSIS Method
In general, the result of MOEAs is a set of nondominated front. From the best obtained Paretofront, it is usually required to select one solution for implementation. A multi attribute decision making (MADM) approach is adopted to rank the obtained NSGAII solutions and the BCS is calculated in a deterministic environment with a single decision maker. From the decision maker’s perspective, the choice of a solution from all Paretooptimal solutions is called a posteriori approach and it requires a higher level decision making approach, which is to determine the best solution among a finite set of Paretooptimal solutions with respect to all relevant attributes. In this paper, MADM technique based on TOPSIS is employed in posterior evaluation of Paretooptimal solutions to choose the best one among them. The concept of TOPSIS is described as: In the absence of a natural course of action for overall summary measure and ranking, the most preferred alternative should not only have the shortest distance from the positive ideal solution, but also have the longest distance from the negative ideal solution. Almost all MADM methods require predetermined information on the relative importance of the attributes, which is usually given by a set of normalized weights. The weights of two objectives are calculated by Shannon’s entropy method. The entropy method is based on information theory, which assigns a small weight to an attribute if it has similar attribute values across alternatives, because such attribute does not help in differentiating alternatives
[20
,
21]
.
The classical MADM model is described as follows:
Let
R
=
R_{ij}, i
=1, 2....
n
(no. of Paretooptimal solutions),
j
= 1, 2.....
m
(no. of objectives) is the
n
×
m
decision matrix, where
R_{ij}
, is the performance rating of alternative
X_{j}
(Paretooptimal solutions) with respect to attribute
A_{i}
(objective function values).
To determine objective weights by the entropy measure, the decision matrix needs to be normalized for each objective
A_{j}
as
As a consequence, a normalized decision matrix representing the relative performance of the alternatives is obtained as
The amount of decision information contained in Eq. (8) and emitted from each attribute
A_{j}
(
j
= 1, 2, ....
m
) can thus be measured by the entropy value
e_{j}
as
The degree of divergence
d_{j}
of the average intrinsic information contained by each attribute
A_{j}
(
_{j}
= 1, 2, ....
m
) can be calculated as
The objective weight for each attribute
A_{j}
(
j
= 1,2, ....
m
) is thus given by
The weighted normalized value
v_{ij}
is calculated as
After determining performance ratings of the alternatives and objective weights of the attributes, the next step is to aggregate them to produce an overall performance index for each alternative. This aggregation process is based on the positive ideal solution (
A
^{+}
) and the negative ideal solution (
A
^{}
), which are defined, respectively by
Separation between alternatives can be measured by the
n
dimensional Euclidean distance. The separation of each alternative from the ideal solution is given as
Similarly, the separation from the negative ideal solution is given as
The relative closeness to the ideal solution of alternative
X_{j}
with respect to
A
^{+}
is defined as
Since
and
, then clearly,
C_{ j}
∈ [0,1].
Choose an alternative with maximum
C_{j}
or rank alternatives according to
C_{j}
in descending order. It is clear that an alternative
X_{j}
is closer to
A
^{+}
than to
A
^{–}
as
C_{j}
approaches 1
[20
,
21]
.
5. Simulation Results and Discussion
The RCGA and NSGAII algorithms are coded in MATLAB version 7.11 on a PC with PentiumIV Intel (R) Core(TM) i32310M CPU operating at 2.10 GHz speed with 4 GB RAM.
 5.1 Description of the test systems
The standard IEEE 30bus system consists of six generating units with a demand of 283.4 MW and IEEE 118bus system is composed of 19 generating units with a demand of 3668 MW are taken as test systems, to verify the effectiveness of NSGAII. The detailed fuel cost coefficients, emission coefficients, the lower and the upper power limits are taken from
[12
,
26
and
30]
. The bus data and the line data are taken from
[32]
. MATPOWER software is used for power flow calculations
[32]
.
 5.2 Parameter settings
The parameter settings of NSGAII for solving CEED problem is as follows: In general, the population size of six times the number of decision variables is considered. For IEEE 30bus system, the population size and iteration are set as 40 and 200 respectively. For IEEE 118bus system, the population size and maximum iteration number are set as 100 and 500 respectively. The crossover probability (
P_{c}
) is varied between 0.8 to 0.9 and other parameters such as mutation probability (
P_{m}
), crossover index (
η_{c}
) and mutation index (
η_{m}
) are selected as 1/
n
(where nnumber of variables), 5 and 15 respectively
[14]
.
 5.3 Generation of reference Paretofront
To compare the performance of NSGAII, a reference Paretofront obtained by using multiple runs of Real Coded Genetic Algorithm (RCGA) with weighted sum approach is considered. In reference Paretofront generation, CEED problem is treated as single objective optimization problem by linear combination of objectives as follows:
where,
w is a weighing factor and the sum of weighting factor must be 1.
f_{1} is the cost objective and f_{2} is the emission objective.
To get 50 nondominated solutions, the algorithm is applied 50 times with varying weight factors as a uniform random number varying between 0 and 1 in each trial. Different population sizes and iteration numbers are selected depending upon the number of decision variables
[20]
.
 5.4 IEEE 30bus system
In this case, the NSGAII and RCGA have been applied to solve CEED problem for the standard IEEE 30bus system. The single line diagram of this system is given in
[19]
. The power system is interconnected by 41 transmission lines and the total system demand for the 21 load buses are 283.4 MW. Extreme solutions for cost and emission are obtained out of ten trial runs using NSGAII for IEEE 30bus system for the problem of CEED without valvepoint effect and are reported in
Tables 1
and
2
respectively and for the problem of CEED with valvepoint effect are reported in
Tables 3
and
4
respectively. Extreme solutions for cost and emission are obtained using RCGA also reported in
Table 1
to
Table 4
. Referring to
Table 1
, PSO algorithm as reported in the literature
[30]
, able to give better results compared to other methods but it needs weight factors to convert multiobjective problem into single objective algorithm and also multiple runs are required to obtain the Paretooptimal solutions. From the
Tables 2
to
4
, it can be concluded that, the NSGAII is capable of providing better results compared to other methods for the CEED problem and it can be observed that the inaccuracy in the resulting dispatch when the valvepoint loading effects are ignored. The execution time is also short using NSGAII, thus computationally more efficient than RCGA. Best Paretofront obtained in the problems of CEED with valvepoint effect and without valvepoint effect using NSGAII and RCGA are shown in
Fig. 3
. The NSGAII produces the Paretooptimal front in a single simulation run and it is clear that the solutions are diverse and well distributed. Furthermore, the Paretofront generated using NSGAII and multiple runs Paretofront obtained using RCGA are almost identical.
Comparative result of Extreme solution for cost  IEEE 30bus system in the case of CEED without Valvepoint effect
Comparative result of Extreme solution for cost  IEEE 30bus system in the case of CEED without Valvepoint effect
Comparative result of Extreme solution for emission  IEEE 30bus system in the case of CEED without Valvepoint effect
Comparative result of Extreme solution for emission  IEEE 30bus system in the case of CEED without Valvepoint effect
Comparative result of Extreme solution for cost of IEEE 30bus system in the case of CEED with Valvepoint effect
Comparative result of Extreme solution for cost of IEEE 30bus system in the case of CEED with Valvepoint effect
Comparative result of Extreme solution for emission of IEEE 30bus system in the case of CEED with Valvepoint effect
Comparative result of Extreme solution for emission of IEEE 30bus system in the case of CEED with Valvepoint effect
Extreme solution for cost  IEEE 118bus system in the case of CEED without Valvepoint effect
Extreme solution for cost  IEEE 118bus system in the case of CEED without Valvepoint effect
 5.5 IEEE 118bus system
Simulations are conducted on the standard IEEE 118bus, 19 generatorstest system applying RCGA and NSGAII. Two different cases are considered for the study. In the first case, valvepoint effect is not considered but in the second case the cost function is modeled as a quadratic function summed with a sine term to include valvepoint effect. In both cases, transmission line losses are included for the load demand of 3668 MW. Extreme solutions for cost and emission are obtained out of ten trial runs using NSGAII for IEEE 118bus system in the problem of CEED without valvepoint effect are reported in
Tables 5
and
6
respectively and in the problem of CEED with valvepoint effect are reported in
Tables 7
and
8
respectively. Extreme solutions for cost and emission are obtained using RCGA also reported in
Tables 5
to
8
and it can be observed that the inaccuracy in the resulting dispatch when the valvepoint loading effects are ignored. Execution time of the NSGAII is very less, thus computationally more efficient than RCGA. Best Paretofront obtained in the cases of CEED with valvepoint effect and without valvepoint effect using NSGAII and RCGA are shown in
Fig. 4
. The NSGAII produces the Paretooptimal front in a single simulation run but RCGA produces the Paretofront in multiple runs. Furthermore, the Paretofront generated using NSGAII and multiple runs Paretofront obtained using RCGA are almost identical.
Extreme solution for emission  IEEE 118bus system in the case of CEED without Valvepoint effect
Extreme solution for emission  IEEE 118bus system in the case of CEED without Valvepoint effect
Extreme solution for cost  IEEE 118bus system in the case of CEED with Valvepoint effect
Extreme solution for cost  IEEE 118bus system in the case of CEED with Valvepoint effect
Extreme solution for emission  IEEE 118bus system in the case of CEED with Valvepoint effect
Extreme solution for emission  IEEE 118bus system in the case of CEED with Valvepoint effect
Best compromise solution for IEEE 30bus and IEEE 118bus system in the case of CEED with Valvepoint effect
Best compromise solution for IEEE 30bus and IEEE 118bus system in the case of CEED with Valvepoint effect
Reference Paretofront using RCGA and best obtained Paretofront using NSGAII for IEEE 30 bus system
Reference Paretofront using RCGA and best obtained Paretofront using NSGAII for IEEE 118 bus system
 5.6 Best Compromise Solution (BCS)
From the estimated Paretooptimal set, it is usually required to choose one of them for implementation. Moreover, the choice of one solution over the other requires additional knowledge about the CEED problem. MADM technique is commonly used to evaluate the Paretooptimal solutions and choose the best one
Table 1
Comparative result of Extreme solution for cost – IEEE 30 bus system in the case of CEED without Valvepoint effect among them. A large number of methods have been developed for solving multiple attribute problems. The concept of TOPSIS from the decision maker’s perspective, the choice of a solution from all Paretooptimal solutions, which is to determine the best solution among a finite set of Paretooptimal solutions with respect to all relevant attributes is used in this work. This procedure has been applied to NSGAII results and the BCS is arrived.
Table 9
gives the BCS value of IEEE 30bus and IEEE 118bus systems. It can be noticed that, NSGAII is capable of providing better results in the BCS compared to PSO algorithm for IEEE 30bus system.
Best Compromise Solution using TOPSIS method for IEEE 30bus system
Best Compromise Solution using TOPSIS method for IEEE 118bus system
Figs. 5
and
6
show the position of BCS on Paretofront obtained using NSGAII for the CEED problem of with valvepoint effect in IEEE 30bus and IEEE 118bus system respectively.
6. Conclusion
In this paper, RCGA and NSGAII have been applied to solve the CEED problem with complexities of valvepoint loading effect and transmission line losses. The problem has been formulated as multiobjective optimization problem with competing fuel cost and emission objectives. By means of stochastically searching multiple points at one time and considering trial solutions of successive iterations, the NSGAII avoids entrapping in local optimal solutions than conventional methods. The NSGAII was tested for the IEEE 30bus and IEEE 118bus systems and compared with the generated reference Paretofront by RCGA. In all the cases, valvepoint loading effect is included. Simulation results reveal that the NSGAII can identify the Paretooptimal front with a good diversity for the CEED problems of with and without valvepoint effect. Moreover, the solutions are obtained in a single simulation run with less computational time. The MADM procedure is followed for choosing the BCS from the obtained Paretooptimal solutions based on TOPSIS.
BIO
M. Rajkumar was born in Tirunelveli, Tamilnadu, India, on August 1975. He received the B.E. degree in Electrical and Electronics Engineering from National Engineering College, Kovilpatti, Tamilnadu, India, in 1999 and M.E. degree in Power Systems from Arulmigu Kalasalingam College of Engineering Krishnankoil, Tamilnadu, India, in 2004. He has presented various papers in the National and International conferences. His current research interests include Power system optimization and evolutionary computation technique. He is currently an Associate Professor in the department of Electrical & Electronics Engineering, National College of Engineering, Maruthakulam, Tirunelveli, 627 151, Tamilnadu, India. He is a member of IET and life member of ISTE.
K. Mahadevan was born in Thirumangalam, Tamilnadu, India. He graduated in Electrical and Electronics Engineering in 1993 and Post graduated in Industrial Engineering in 1997 and PhD in 2006 from Madurai Kamaraj University, Tamilnadu, India. His fields of interest are Power System Operation and Control and Evolutionary Computation. Currently, he is Professor of Electrical & Electronics Engineering, PSNA College of Engineering & Technology, Dindigul, Tamilnadu, India.
S. Kannan received his B.E., M.E., and Ph.D Degrees from Madurai Kamaraj University, Tamilnadu, India in 1991, 1998 and 2005 respectively. His research interests include Power System Deregulation and Evolutionary Computation. He was a visiting scholar in Iowa State University, USA (October 2006September 2007) supported by the Department of Science and Technology, Government of India with BOYSCAST Fellowship. He is Professor and Head of Electrical and Electronics Engineering Department, Kalasalingam University, Krishnankoil, Tamilnadu, India, where he has been since July 2000. He is a Sr. Member of IEEE, Fellow of IE (I), Sr. Member in CSI, Fellow in IETE, Life member SSI and Life member of ISTE.
S. Baskar received the B.E., and the PhD Degrees from Madurai Kamaraj University, Madurai, Tamilnadu, India, in 1991 and 2001 respectively and the M.E., degree from Anna University, India, in 1993. He did his postdoc research in Evolutionary Optimization at NTU, Singapore. His research interests include the development of new Evolutionary Algorithm and applications to engineering optimization problems. He is the reviewer for IEEE Transactions on Evolutionary Computation. He has published over 50 papers in journals in the area of Evolutionary Optimization and applications. He is Professor in the department of Electrical & Electronics Engineering, Thiagarajar College of Engineering, Madurai, Tamilnadu, India. He is a Sr. Member of IEEE, Fellow of Institution of Engineers (India) and Life member of the Indian Society for Technical Education. He was the recipient of the Young Scientists BOYSCAST Fellowship during 20032004 supported by the Department of science and Technology, Government of India.
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