International Journal of Engineering Trends and Technology (IJETT) – Volume22 Number 4- April2015
Optimization of Economic Load Dispatch Thermal
Power Plant Using Differential Evolution Technique
Deepti Gupta #1, Rupali Parmar*2
#1 M.tech student, Electrical, MPCT, Gwalior, INDIA
*2 Asst prof, Electrical, MPCT, Gwalior, INDIA
Abstract— This paper presents an algorithm for solving optimal
power flow problem through the application of Differential
Evolution (DE). Economic Dispatch is the process of allocating
the required load demand between the available generation units
such that the cost of operation is minimized. There have been
many algorithms proposed for economic dispatch out of which a
Differential Evolution (DE) is discussed in this paper.
Differential Evolution also incorporates an efficient way of selfadapting mutation using small population. The effectiveness of
algorithm has been tested for a practical economic dispatch
problem on a test system having three generating units. Thermal
power generating firms needs to minimize fuel cost, for better
profit at the same time it should satisfy system load demand,
real, reactive power limit, voltage limit, power transmission limit
and other limitations. For generation cost minimization
Economic Load Dispatch (ELD) was developed. When cost is the
single objective, the power generation may pollute the
environment. Thermal electric power could not be generated
without pollution but this pollution can be reduced for the sake
of good and healthy atmospheric condition. The proposed
method is compared with Particle Swarm Optimization (PSO)
and its variants
Keywords— Differential Evolution, Economic Dispatch, Particle
Swarm Optimization, PSO variants
1.
Introduction
The Economic Dispatch Problems (EDPs) is to determine the
optimal combination of power outputs of all generating units
to minimize the total fuel cost while satisfying the load
demand and operational constraints [1]. Power utilities try to
achieve high operating efficiency to produce cheap electricity.
Competition exists in the electricity supply industry in
generation and in the marketing of electricity. The idea behind
economic dispatch problem in a power system is to determine
the optimal combination of power output for all generating
units which will minimize the total fuel cost while satisfying
load and operational constraints. The economic dispatch
problem is very complex to solve because of its massive
dimension, a non-linear objective
function and large
number of constraints. Various investigations on the ELD
have been undertaken till date. Suitable improvements in the
unit output scheduling can contribute to significant cost
savings. Also information about forming market clearing
prices is provided by it. To improve the quality of solution [2]
lot of researches have been done and various methods have
been evolved so far in the field of economic load dispatch.
Traditional methods or conventional methods like lambda
iteration method, mixed integer linear programming [3],
dynamic programming [4], linear programming [5], Lagrange
relaxation method [6], Newton-based techniques [7] and
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interior point methods [8], reported in the literature are used to
solve solving economic load dispatch problem employing
different objective functions but non-convex optimization
problem, which is a challenging and cannot be solved by the
traditional methods. In recent years, due to the power and
simplicity of evolutionary algorithms, many researchers use
these algorithms to solve ED problems, different heuristic
approaches have been proved to be effective with promising
performance, such as simulated annealing (SA) [9],
evolutionary programming (EP) [10], Ant colony optimization
[11],Genetic algorithm (GA) [12],particle swarm optimization
(PSO) [13-14]. EP is rather slow converging to a near
optimum for some problems. Simulated annealing is very time
consuming, and cannot be utilized easily to tune the control
parameters of the annealing schedule. Evolutionary
programming is difficult in defining effective memory
structures and strategies which are problem dependent.
Genetic algorithm sometimes lacks a strong capacity of
producing better offspring and causes slow convergence near
global optimum, sometimes may be trapped into local
optimum. DE is a population based stochastic function
maximize relating to evolutionary computation, whose simple
yet powerful and straightforward features make it very
attractive for numerical optimization. DE uses a rather greedy
and less stochastic approach to problem solving than do
evolutionary algorithms (EA). DE combines simple
arithmetical operators with the classical operators of
recombination, mutation, and selection to evolve from a
randomly generated starting population to a final solution. DE
differs from conventional genetic algorithms in its use of
perturbing vectors, which are the difference between two
randomly chosen parameters vectors. This method is a
heuristic, population-based optimization method that uses a
population of points to search for global minimum of a
function over continuous search space. DE generates new
vectors of parameters by adding the weighted difference
between two vectors to a third one. If the result provides a
smaller objective function than a predetermined individual,
new result will replace the one with which it is compared. DE
has been used to solve different optimization problems
including power system planning, however, the main
difficulty with DE techniques, appears to be ensuring the
search diversity of population when the algorithm is
approaching the region of local and global optimum. The
improvement is mainly achieved through several mutation
methods by combining the fitness sharing scheme into the
solution process. The findings are supported by detailed
simulations based on mathematical function optimization
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International Journal of Engineering Trends and Technology (IJETT) – Volume22 Number 4- April2015
before solving a power system planning problem. The
objective of the economic dispatch problem (EDP) of electric
power generation, whose characteristics are complex and
highly nonlinear, is to schedule the committed generating unit
outputs so as to meet the required load demand at minimum
operating cost while satisfying all units and system equality
and inequality constraints. Chaos describes the complex
behavior of a nonlinear deterministic system. The application
of chaotic sequences instead of random sequences in DE is a
powerful strategy to diversify the DE population and improve
the DE‟s performance in preventing premature convergence to
local minima.
discontinuities due to valve-point loading in fossil fuel
burning plant. The valve-point loading effect has been
modelled as a recurring rectified sinusoidal function, such as
the one shown in equation 5 and Fig.1.
(5)
2. Economic Load Dispatch (ELD)
This section deals with the formulation of economic load
dispatch problem where the objective function has to be
minimized depending upon its equality and inequality
constraints. Transmission loss plays a major factor and affects
the optimum dispatch of generation. The objective of an ELD
problem is to find the optimal combination of power
generations that minimizes the total generation cost while
satisfying equality and inequality constraints [15][16].
2.1 Formulation of Classical ED Problem
The objective of the classical ELD is to minimize the total
fuel cost by adjusting the power output of each of the
generators connected to the grid. The total fuel cost is
modelled as the sum of the cost function of each generator.
The basic economic dispatch problem can be described
mathematically as a minimization of problem.
N
MinFT
Fi ( Pi )
(1)
i 1
Where Fi ( Pi ) is the fuel cost equation of the „i‟th plant. It is
the variation of fuel cost in $ with generated Power (MW).
2
Fi ( Pi )
(ai Pi
bi Pi
ci ) (2)
The total fuel cost to be minimized is subject to the following
constraints and equation (5) represents fuel cost including
valve point loading.
(3)
(4)
2.2 Fuel Characteristics with Valve
loading effect
Economic load dispatch (ELD) is considered one of the key
functions in electric power system operation. The economic
load dispatch problem is commonly formulated as an
optimization problem, with the aim of minimizing the total
generation cost of power system but still satisfying specified
constrains. The input-output characteristics (or cost functions)
of a generator are approximated using quadratic or piecewise
quadratic function, under the assumption that the incremental
cost curves of the units are monotonically increasing
piecewise-linear functions. However, real input-output
characteristics display higher-order nonlinearities and
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Fig.1 operating cost characteristics with valve
point loading
3. Optimization methodology based on DE approaches for the
economic dispatch problem
Differential Evolution is one of the most recent population
based stochastic evolutionary optimization techniques. Storn
and Price first proposed DE in 1995 [17-18] as a heuristic
method for minimizing non-linear and non-differentiable
continuous space functions. Differential Evolution is a simple
and efficient adaptive scheme for global optimization over
continuous spaces. In this section, we review the differential
evolution (DE) algorithm that was used for searching the
optimum solution of ELD problems. Differential evolution
solves real valued problems based on the principles of natural
evolution using a population P. the population size remains
constant throughout the optimization process Differential
Evolution (DE) is a parallel direct search method which
utilizes NP, D-dimensional parameter vectors, In DE, a
population of NP solution vectors is randomly created at the
start this population is successfully improved over G
generations by applying This optimization process is carried
out with three basic operations mutation, crossover and
selection operators, to reach an optimal solution [20]. The
main steps of the DE algorithm are given bellow:
The main steps of the DE algorithm are given bellow:
Initialization Mutation Crossover Evaluation Selection
Until (Termination criteria are met)
as a population for each generation G. NP does not change
during the minimization process. The initial vector population
is chosen randomly and should cover the entire parameter
space. As a rule, we will assume a uniform probability
distribution for all random decisions unless otherwise stated.
In case a preliminary solution is available, the initial
population might be generated by adding normally distributed
random deviations to the nominal solution. DE generates new
parameter vectors by adding the weighted difference between
two population vectors to a third vector. Let this operation be
called mutation. The mutated vector‟s parameters are then
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International Journal of Engineering Trends and Technology (IJETT) – Volume22 Number 4- April2015
mixed with the parameters of another predetermined vector,
the target vector, to yield the so-called trial vector. Parameter
mixing is often referred to as “crossover” in the EScommunity and will be explained later in more detail. If the
trial vector yields a lower cost function value than the target
vector, the trial vector replaces the target vector in the
following generation. This last operation is called selection.
Each population vector has to serve once as the target vector
so that NP competitions take place in one generation. More
specifically DE‟s basic strategy can be described as follows:
3.1 . Initialization
The first step in the DE optimization process is to create an
initial population of candidate solutions by assigning random
values to each decision parameter of each individual of the
population. Such values must lie inside the feasible bounds of
the decision variable.
3.2.Mutation
After the population is initialized, the mutation operator
creates mutant vectors by perturbing a randomly selected
vector xa with the difference of two other randomly selected
vectors xb and xc, Where xa, xb and xc are randomly chosen
vectors among the NP population, and a ≠ b ≠ c. xa, xb and xc
are selected anew for each parent vector. The scaling constant
F is an algorithm control parameter used to adjust the
perturbation size in the mutation operator and improve
algorithm
convergence.
For
each
target
vector
a mutant vector is generated
using a differential mutation operation according to following
equation.
(2)
Where F is a real and constant factor, commonly known as
scaling factor or amplification factor, is a positive real
number, typically less than 1.0, that controls the amplification
of the differential variation
with
random
indexes
constant
which has to be determined by the user
. Is a randomly chosen index
ensures that
which
gets at least one parameter from
.
3.4.Selection
The selection operator forms the population by choosing
between the trial vectors and their predecessors (target vectors)
those individuals that present a better fitness or are more
optimal. This optimization process is repeated for several
generations allowing individuals to improve their fitness as
they explore the solution space in search of optimal values.DE
has three essential control parameters: the scaling factor (F),
the crossover constant (CR) and the population size (NP). The
scaling factor is a value in the range [0, 2] that controls the
amount of perturbation in the mutation process. The crossover
constant is a value in the range [0, 1] that controls the
diversity of the population. The population size determines the
number of individuals in the population and provides the
algorithm enough diversity to search the solution space.
To decide whether or not it should become a member of
generation G+1 the trial vector
is compared to the
target vector
using the greedy criterion. If vector
yields a smaller cost function value than
then
is set to
otherwise, the old value
is
retained.
4. Results and Discussion
The proposed DE-based approach has been developed and
implemented using the MATLAB software. In order to
investigate we experimented with 2 standard test cases. They
are 10 unit systems without VPL effect and, 10 unit systems
with VPL effect systems with a varying percentage of the
maximum power as demand .All the simulations were carried
out using MATLAB 9.0.1 on core i3, 2nd generation
processor having 2.4 GHz. With 4 GB RAM on 64-bit
operating system.
Case Study 1: 10 unit system with VPL
3.3.Crossover
DE uses binomial crossover operation this crossover constant
CR is an algorithm parameter that controls the diversity of the
population The crossover operator creates the trial vectors,
which are used in the selection process. In order to increase
the diversity of the perturbed parameter vectors, crossover is
introduced. To this end, the trial vector:
(3)
(4)
In (4)
is the
evaluation of a uniform random
number generator with outcome
. CR is the crossover
The system is taken from [22]. The test system consists of 10
generating units with valve point loading effect with total load
demand of 1036MW.The fuel-cost characteristics are given in
Table A1 in the Appendix.
Case 1.A-Effect of population size 50, 100,200 and 500 in 10generating units with VPL
The cost coefficients and other data of the 10 generating
unit system [1] and PD is 1036 MW, DE achieved quite
effective result. A parameter tuning was done to find
optimal values of trails 100, C.R. 0.9, F 0.9, NP 500,
itermax 500, and Different Population 50, 100, 500 sizes
for the practical ELD problem using DE. It can be
observed from Table 1. Fig 1, Fig 2, Fig 3 and fig 4 show
the convergence characteristics of the 10-generating units
with VPL effect.
Table 1. Effect of Population Size on 10-Unit System
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International Journal of Engineering Trends and Technology (IJETT) – Volume22 Number 4- April2015
Population 50, 100, 500 sizes for the practical ELD
problem using DE. It can be observed from Table 2. Fig
5, Fig 6, and fig 7 show the PD effect of the 10-generating
units with VPL effect. Table 2 shows that load increase so
cost is also increase and SD increase.
Table 2 power demand effect in 10 unit system with VPL
Fig 1 .Effect of population size 50 and trials 100 (10 unit system with VPL)
Fig 5. 10 Unit Systems with VPL Power Demand Effect (PD 800MW and
Trials 100)
Fig 2 .Effect of population size 100 and trials 100 (10 unit system with VPL)
Fig 6. 10 Unit Systems with VPL Power Demand Effect (PD 1200MW and
Trials 100)
Fig 3 .Effect of population size 200 and trials 100 (10 unit system with VPL)
Fig 7. 10 Unit Systems with VPL Power Demand Effect (PD 1500MW and
Trials 100)
Fig 4 .Effect of population size 500 and trials 100
(10 unit system with VPL)
Case 1- (B)-Effect of power demand 800 MW, 1200 MW,
and 1500 MW in 10-generating units with VPL
In this case different power demand 800 MW, 1200 MW,
and 1500 MW in 10 unit systems with VPLeffect.
Parameter tuning was done to find optimal values of trails
100, C.R. 0.9, F 0.9, NP 400, itermax 500, and Different
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Case 2-(C)-Comparison of Results for 10 unit system
In this case Parameter tuning was done to find optimal
values of trails 100, C.R. 0.9, F 0.9, NP 500, itermax
1000 and power load demand (PD) 1036 MW for the
practical ELD problem using DE.DE achieved quite
effective result. Results obtained from proposed DE
algorithm have been compared with PSO, CPSO,
WIPSO and MRPSO [22], with valve Point Loading
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International Journal of Engineering Trends and Technology (IJETT) – Volume22 Number 4- April2015
Effects. The best results of all evolutionary methods
are shown in Table 3.and fig 8 shows that the proposed
DE algorithm provided better results compared to other
reported evolutionary techniques.
Fig 8. Best result of DE in 10 unit system with VPL
Table 3 Compare of Different Technique with DE in 10 units system
with VPL
Fig 9 .Effect of population size 50 and trials 100 (10 unit system without VPL
Fig 10 .Effect of population size 100 and trials 100 (10 unit system without
VPL)
Fig 11 .Effect of population size 200 and trials 100 (10 unit system without
VPL)
Case Study 2: 10 unit system with VPL
The system is taken from [22]. The test system consists of 10
generating units with valve point loading effect with total load
demand of 1036MW.The fuel-cost characteristics are given in
Table A2 in the Appendix.
Case 2-(A)-Effect of population size 50, 100,200 and 500 in
10-generating units without VPL
The cost coefficients and other data of the 10 generating
unit system [1] and PD is 1036 MW, DE achieved quite
effective result. A parameter tuning was done to find
optimal values of trails 100, C.R. 0.9, F 0.9, NP 500,
itermax 500, and Different Population 50, 100, 500 sizes
for the practical ELD problem using DE. It can be
observed from Table 4. Fig 9, Fig 10, Fig 11 and fig 12
show the convergence characteristics of the 10-generating
units without VPL effect.
Table 4. Effect of Population Size on 10-Unit System without VPL
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Fig 12 .Effect of population size 500 and trials 100 (10 unit system without
VPL)
Case 2-(B)-Effect of power demand 800 MW, 1200 MW, and
1500 MW in 10-generating units without VPL
In this case different power demand 800 MW, 1200 MW,
and 1500 MW in 10 unit systems without VPL effect.
Parameter tuning was done to find optimal values of trails
100, C.R. 0.9, F 0.9, NP 400, itermax 500, and trail 100
for the practical ELD problem using DE. It can be
observed from Table 5. Fig 13, Fig 14, and fig 15 show
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International Journal of Engineering Trends and Technology (IJETT) – Volume22 Number 4- April2015
the PD effect of the 10-generating units without VPL
effect. Result shows that load increase so cost is also
increase with SD increase.
are shown in Table 3.and fig 8 shows that the proposed
DE algorithm provided better results compared to other
reported evolutionary techniques.
Table 5 power demand effect in 10 unit system without VPL
Fig 16. Best result of DE in 10 unit system with VPL
Table 6 Compare of Different Technique with DE in 10
units system without VPL
Fig 13. 10 Unit Systems without VPL Power Demand Effect (PD 800MW
and Trials 100)
Fig 14. 10 Unit Systems without VPL Power Demand Effect (PD 1200MW
and Trials 100)
Fig 5. 15 Unit Systems without VPL Power Demand Effect (PD 1500MW
and Trials 100)
Case 2-C-Comparison of Results for 10 unit system
without VPL
In this case Parameter tuning was done to find optimal
values of trails 100, C.R. 0.9, F 0.9, NP 1000, itermax
1000 and power load demand (PD) 1036 MW for the
practical ELD problem using DE.DE achieved quite
effective result. Results obtained from proposed DE
algorithm have been compared with PSO, CPSO,
WIPSO and MRPSO [22], with valve Point Loading
Effects. The best results of all evolutionary methods
ISSN: 2231-5381
Conclusion
In this paper, the proposed Differential Evolution algorithm
has been presented to optimize conventional quadratic and
non-convex fuel cost functions. The effectiveness of the
proposed method has demonstrated through 10 bus system.
The results obtained for test systems using proposed method
are compared with existing methods. The differential
evolution algorithm has been successfully implemented to
solve ED problems with the generator constraints as linear
equality and inequality constraints the algorithm is
implemented for 10 units system. The observations reveal that
the results obtained using proposed method is close to the
existing method. Also, it is clear that the iterative process
begins with best population and convergence rate is fast for
proposed method. Hence, the computation time has been less
for proposed method in comparison with PSO variants method.
So it is clear that the proposed algorithm has the ability to find
the better quality solution, computational efficiency, less CPU
time and minimum S.D. per iteration when compared to other
methods such as PSO and PSO variant.
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REFRENCES
Page 158
International Journal of Engineering Trends and Technology (IJETT) – Volume22 Number 4- April2015
[1] Leandro dos Santos Coelho a,*, Chu-Sheng Lee b, “Solving economic load
dispatch problems in power systems using chaotic and Gaussian particle
swarm optimization approaches” Electrical Power and Energy Systems 30
(2008) 297–307.
[2] Devendra Bisen 1 , Hari Mohan Dubey 1 , Manjaree Pandit 1 and B. K.
Panigrahi2, “Solution of Large Scale Economic Load Dispatch Problem using
Quadratic Programming and GAMS: A Comparative Analysis”, Journal of
Information and Computing Science Vol. 7, No. 3, 2012, pp. 200-211
[3] M. Gar CW, Aganagic JG,Tony Meding Jose B & Reeves S, “Experience
with mixed integer linear programming based approach on short term
hydrothermal scheduling”
,‖ IEEE transaction on power system
2001;16(4),pp.743-749.
[4] Shi CC, Chun HC, Fomg IK & Lah PB., “Hydroelectric generation
scheduling with an effective differential dynamic programming algorithm”,
IEEE transaction on power system 1990,5(3),pp.737-743
[5] K.Ng and G.Shelbe, “ Direct load control a profit-based load
management using linear programming ”, IEEE transaction on power
system,vol.13,no.2,1998,pp.688-694.
[6] Erion Finardi C, silva Edson LD, & Laudia sagastizabal CV., “Solving
the unit commitment problem of hydropower plants via Lagrangian relaxation
and sequential quadratic programming”,Computaional & Applied
Mathematics 2005,24(3).
[7] D.I. sun, B.Ashley,B.Brewer,A.Hughes and W.F. Tinney, “Optimal
power flow by Newton Aproach”, IEEE transaction on power system,
vol.103,1984,pp.2864-2880.
[8] X.Yan & V.H. Quintana, “Improving an interior point based OPF by
dynamic adjustments of step sizes and tolerances”, IEEE transaction on power
system , vol.14,no.2,1999,pp.709-717.
[9] Kamlesh Kumar Vishwakarma1, Hari Mohan Dubey2* , Manjaree Pandit3
and B.K. Panigrahi4 “Simulated annealing approach for solving economic
load dispatch problems with valve point loading effects”, International
Journal of Engineering, Science and Technology Vol. 4, No. 4, 2012, pp. 6072.
[10] K.P. wong & J. yuryevich, “Evolutionary based algorithm for
environmentally constraints economic dispatch”,IEEE transaction on power
system , vol.13,no.2,1998,pp.301-306.
[11] Gaurav Prasad Dixit, Hari Mohan Dubey, Manjaree Pandit,
B.K.panigrahi, “Economic Load Dispatch using Arifitial Bee Colony
Optimization”, International conference on Advanced Computing,
Communication and Networks,2011.
[12] Bishnu Sahu, Avipsa Lall, “Economic Load Dispatch in Power System
using Genetic Algorithm”, International Journal of Computer Applications
(0975 – 8887)
Volume 67– No.7, April 2013.
[13] Charanjeet Madan 1, Bhuvnesh Khokhar 2, Rohini Sharma 3, “Economic
Load Dispatch Using AI Technique”International Journal of Advanced
Research in Electrical, Electronics and Instrumentation Engineering Vol. 2,
Issue 7, July 2013.
[14] Shubham Tiwari1, Ankit Kumar2, G.S Chaurasia3, G.S Sirohi4
“
Economic
Load
Dispatch
Usingmparticle
Swarm
Optimization”international journal of application or innovation in engineering
& management(IJAIEM), Volume 2, Issue 4, April 2013
[15] R.Subramanian, K.Thanushkodi , “An Efficient Novel TANAN‟s
Algorithm for Solving Economic Load Dispatch Problems”, International
Journal of Soft Computing and Engineering (IJSCE) ISSN: 2231-2307,
Volume-3, Issue-3, July 2013.
[16] Emmanuel Dartey Manteaw*, Dr. Nicodemus Abungu Odero**,
“Combined Economic and Emission Dispatch Solution Using ABC_PSO
Hybrid Algorithm with Valve Point Loading Effect”, International Journal of
Scientific and Research Publications, Volume 2, Issue 12, December 2012 2
ISSN 2250-3153.
[17] G. Nageswara Reddy, S. S. Dash, S. Sivanagaraju, Ch. V. Suresh,
“Economic Load Dispatch using Imperialistic Competitive Algorithm: An
Effect of Control Variables”, International Journal of Engineering and
Advanced Technology (IJEAT) ISSN: 2249 – 8958, Volume-3 Issue-6,
August 2014.
[18] Surekha P, Dr.S.Sumathi, “Solving Economic Load Dispatch problems
using Differential Evolution with Opposition Based Learning” Wseas
Transactions On Information Science And Applications, Issue 1, Volume 9,
January 2012
ISSN: 2231-5381
[19]C.Kumar, T.Alwarsamy, “Solution of Economic Dispatch Problem using
Differential Evolution Algorithm”, International Journal of Soft Computing
and Engineering (IJSCE) ISSN: 2231-2307, Volume-1, Issue-6, January 2012.
[20] Sunil Kumar Soni, Vijay Bhuria, “Multi-objective Emission constrained
Economic Power Dispatch Using Differential Evolution”, International
Journal of Engineering and Innovative Technology (IJEIT) Volume 2, Issue 1,
July 2012.
[21] Abazar Shamekhi1, “An Improved Differential Evolution Optimization
Algorithm”
IJRRAS
15
(2)
●
May
2013
/Volumes/Vol15Issue2/IJRRAS_15_2_01.
[22] Nagendra Singh1, Yogendra Kumar2, “Constrained Economic Load
Dispatch Using Evolutionary Technique ”, Asian Journal of Technology &
Management Research [ISSN: 2249 –0892] Vol. 03 – Issue: 02 (Jul - Dec
2013)
Table -Appendix A1
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Table -Appendix A2
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