Cf34498502

A.Hima Bindu, Dr. M. Damodar Reddy / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 4, Jul-Aug 2013, pp. 498-502
498 | P a g e
Economic Load Dispatch Using Cuckoo Search Algorithm
A.Hima Bindu1
, Dr. M. Damodar Reddy2
1
(M.Tech Student, Department of EEE, S.V.University, Tirupati)
2
(Associate Professor, Department of EEE, S.V.University, Tirupati)
ABSTRACT
Economic Load Dispatch problem (ELD)
is one of the most important optimization
problems in power system operation and
planning. This paper introduces a solution to ELD
problem using a new metaheuristic nature-
inspired algorithm called Cuckoo Search
Algorithm (CSA). The proposed approach has
been applied to various systems. This algorithm is
based on the obligate brood parasitic behavior of
some cuckoo species. The results proved the
efficiency of the proposed method when compared
with the other optimization algorithms.
Keywords - Cuckoo Search Algorithm (CSA),
Economic Load Dispatch (ELD)
I. Introduction
The economic load dispatch is an important
real time problem, in allocating the real power
demand among the generating units. The best
generation schedule for the generating plants to
supply the required demand, plus the transmission
loss with the minimum generation cost. Traditional
optimization techniques such as gradient method,
lambda iteration method, the linear programming
method and Newton‟s method are used to solve the
ELD problem with increasing cost function.
Economic dispatch methods require the
generator cost curve to be continuous. Hence the
operating cost function for each generator has been
approximately represented by the quadratic function.
It is of great importance to solve this problem quickly
and accurately as possible by considering all kind of
discontinuity in non-linear space. The conventional
methods include the base point and participation
factor, lambda-iteration method and gradient method
[1]. In these numerical methods, an essential
assumption is that the incremental cost curves of the
units are increasing.
The input-output characteristics of modern
units are highly non-linear. These non-linear
characteristics of generating units are due to
discontinuous prohibited operating zones, ramp rate
limits and cost functions, which are not smooth, thus
gives inaccurate results. Therefore more interests
have been focused on the application of Artificial
Intelligence (AI) technology for solution of these
problems. Some of AI methods are Genetic
Algorithm, simulated Annealing , Tabu Search,
Evolutionary Programming, Ant Colony
Optimization, Artificial Neural Networks,
Differential Evolution , Harmony search Algorithm,
Dynamic Programming and Particle Swarm
Optimization [2-11], have been developed and
applied successfully to small and large systems to
solve ELD problems in order to find better results.
Recently, a new metaheuristic search algorithm,
called Cuckoo Search (CS) [12-17], has been
developed by Yang and Deb. In this paper, Cuckoo
Search Algorithm has been used to solve the ELD
problem.
II. Economic Load Dispatch Formulation
2.1. Economic Dispatch
The main objective of ELD of electric
power generation is to schedule the generating units
so as to meet the load demand at minimum operating
cost while satisfying all the constraints. The
economic load dispatch problem including
transmission losses is expressed as
Minimize
1
( )
n
T i i
i
F F P

  (2.1)
Where TF is the total generation cost in RS/hr, „i‟ is
the number of generators, iP is the real power
generation of
th
i generator in MW, ( )i iF P is the
generation cost for iP . Subjected to equality and
inequality constraints. These constraints include:
2.1.1 Power Balancing Equation
For power balance condition, an equality
constraint should be satisfied. The total power
generation should be same as total load demand plus
the total line losses.
1
0
n
D L i
i
P P P

   (2.2)
Where DP is the total demand in MW, LP is the
transmission loss of the system in MW.
2.1.2 Generator Limits
The generation output of each generator
should lie between minimum and maximum limits.
The inequality constraints for each generator are
,min ,maxi i iP P P  (2.3)
Where ,miniP is the Minimum power output limit of
th
i generator in MW, ,maxiP is the Maximum power

499 | P a g e
output limit of
th
i generator in MW.
The generator cost function ( )i iF P is
usually expressed as a quadratic polynomial (cost-
curve equation):
2
( )i i i i i i iF P a P b P c   (2.4)
Where ia , ib and ic are fuel cost coefficients.
2.1.3 Transmission Losses
The Transmission losses are taken in to
account to achieve true economic load dispatch. It is
a function of unit generation. To calculate the
transmission losses two methods are there in general.
One is the penalty factors method and other is the B-
coefficients method. In the B-coefficients method,
network losses are expressed as a quadratic function:
1 1
n m
L i ij j
i j
P PB P
 
  (2.5)
Where ijB is a constant which is called B-
coefficients or loss coefficients.
III. Cuckoo Search
3.1. Cuckoo Behavior
Cuckoos are the fascinating birds, not only
because of their beautiful sounds they make, but also
because of their reproduction strategy. Some species
such as the ani and Guira cuckoos lay their eggs in
nests, though they may remove other‟s eggs to
increase the hatching probability of their eggs. A
number of species engage the brood parasitism by
laying their eggs in the nest of other host birds [12].
There are three basic types of brood parasitism: intra-
specific brood parasitism, co-operative breeding, and
nest take-over. Some host birds can engage direct
conflict with the intruding cuckoos. If a host bird
discovers the eggs are not their own, they will either
throw these alien eggs away or simply abandon its
nest and build a new nest elsewhere. Some cuckoo
species such as the New World brood-parasitic
Tapera have evolved in such a way that female
parasitic cuckoos are often very specialized in the
mimicry in color a pattern of the eggs of a few
chosen host species. This reduces the probability of
their reproductivity [13].
3.2 Levy Flights
In nature, animals search for food in a
random or quasi-random manner. In general, the
foraging path of an animal is electively a random
walk because the next move is based on the current
location/state and the transition probability to the
next location [14].
3.3 Cuckoo Search
In describing our new Cuckoo Search, we
now use the following three idealized rules:
i. Each cuckoo lays one egg at a time, and
dump its egg in randomly chosen nest;
ii. The best nests with high quality of eggs
will carry over the next generations;
iii. The number of available host nests is
fixed, and the egg laid by a cuckoo is
discovered by the host bird with a
probability
aP ε [0. 1].
In this case, the host bird can either throw
the egg away or abandon the nest, and build a
completely new nest. For simplicity, this last
assumption can be approximated by the fraction aP
of the n nests are replaced by new nests.
For maximization problem, the quality or
fitness of a solution can simply be proportional to the
value of the objective function. Other forms of fitness
function in genetic algorithms [15]. For simplicity,
we can use the following simple representations that
each egg in a nest represents a solution, and a cuckoo
egg represents a new solution. When generating new
solutions
( 1)t
x 
for, say, a Cuckoo i, a Levy flight is
performed
( 1) ( )
( )t t
i ix x Levy 
   (3.1)
where 0  is the step size which should be
related to the scales of the problem of interests. In
most cases, we can use 1  [16]. The product 
means entry wise multiplications. This entry wise
product is similar to those used in PSO.
The Levy flight essentially provides a
random walk while the random step length is drawn
from a Levy distribution
,Levy u t 
 (1 3)  ( 3.2)
which has an infinite variance with an infinite mean.
Levy walk around the best solution obtained so far,
this will speed up the local search [17].
IV. Cuckoo Search Algorithm
Step 1: Read the system data which consists of fuel
cost coefficients, minimum and maximum power
limits of all generating units, power demand and B-
coefficients.
Step 2: Initialize the parameters and constants of
Cuckoo Algorithm. They are “non”, Pa, beta,
itermax.
Step 3: Generate “non” number of nests randomly
between min and max
Step 4: Set iteration count to 1.
Step 5: Calculate the fitness value corresponding to
“non” number of cuckoos.
Step 6: Obtain the best fitness value Gbest by
comparing all the fitness values and also obtain the
best nests corresponding to the best fitness value
Gbest.
Step 7: Determine sigma value using the following

500 | P a g e
equation:
1/
( 1)/2
(1 )sin( / 2)
*[(1 ) / 2] *2
u


 

  
   
  
  
Step 8: Find newnest by using stepsize between the
min and max limits.
Step 9: Find the fitness value, if tfitness > fitness
value then send the nest values to newnest. Next
update Gbest by comparing fitness values.
Step 10: New solution by Random Walk
In this if random value > pa then find the stepsize1
between any two nests. Then find newnest1, where
newnesst1 = nest + stepsize1, the newnest1 must be
with in the limits. Again update the Gbest by
comparing fitness values. If this condition violates
then go to step 5 and repeat this procedure.
Step 11: Finally Gbest gives the optimal solution of
the Economic Load Dispatch problem and the results
are printed.
V. Search Results
The proposed Cuckoo Search Algorithm has
been implemented successfully to solve the ELD
problem of units. Cuckoo method is tested with three
generating unit system and six generating unit
system. The problem is solved by Lambda iterative
and the Cuckoo Search methods optimize the
problem.
The economic load dispatch problem
requires the loss coefficient matrix for calculating
transmission line losses and also satisfying the power
balancing equation and also the operational
constraints. The Cuckoo program is written in
MATLAB software package.
5.1 Three-Unit System
The loss coefficient matrix or B-coefficient
matrix of three unit system are given below in table
5.1. Economic Load Dispatch for three unit system is
solved by Lambda iteration method and Cuckoo
search method. The results of Cuckoo search method
are given in below table 5.2. Comparison of Lambda
iteration method and Cuckoo search method results
are tabulated in below table 5.3.
The loss coefficient matrix of 3-Unit System
0.000071 0.000030 0.000025
0.000030 0.000069 0.000032
0.000025 0.000032 0.000080
ijB
 
   
  
5.2 Six-Unit System
The loss coefficient matrix or B-coefficient
matrix of six unit system are given below in table 5.4.
Economic Load Dispatch for six unit system is
solved by Lambda iteration method and Cuckoo
search algorithm method. The results of Cuckoo
search method are given in below table 5.5.
Comparison of Lambda iteration method and Cuckoo
search method results are tabulated in below table
5.6.
The loss coefficient matrix of 6-Unit System
0.000014 0.000017 0.000015 0.000019 0.000026 0.000022
0.000017 0.000060 0.000013 0.000016 0.000015 0.000020
0.000015 0.000013 0.000065 0.000017 0.000024 0.000019
0.000019 0.000016 0.000017 0.000072 0.000030 0.000025
0.00
ijB 
0026 0.000015 0.000024 0.000030 0.000069 0.000032
0.000022 0.000020 0.000019 0.000025 0.000032 0.000085
 
 
 
 
 
 
 
 
 
VI. Conclusion
Economic Load Dispatch problem is solved
by using Lambda iteration and Cuckoo search
methods. The Results of Lambda iteration method
and Cuckoo search algorithm are compared for three
generating unit and six generating unit systems. The
program is written in MATLAB software package.
The Economic Load Dispatch has to meet the load
demand at minimum operating cost while satisfying
the constraints of power system of all units.
Table 5.1 Cost of power generation and power limits
of 3-Uint System
Table 5.2 Cuckoo Search Results for 3-Unit system
Units an bn cn Pn,min Pn,max
1 1243.5311 38.30553 0.03546 35 210
2 1658.5696 36.32782 002111 130 325
3 1356.6592 38.27041 0.01799 125 315
SI.No
Power
Demand
(MW)
λ 1P (MW) 2P (MW) 3P (MW) lossP
(MW)
Fuel Cost
(Rs/hr)
1 350 44.4387 70.3012 156.267 129.208 5.77698 18564.5
2 400 45.4762 82.0784 174.994 150.496 7.56813 20812.3
3 450 46.5291 93.9374 193.814 171.862 9.61271 23112.4
4 500 47.5977 105.88 212.728 193.306 11.9144 25465.5
5 550 48.6824 117.907 231.738 214.831 14.4769 27872.4
6 600 49.7836 130.021 250.846 236.437 17.3040 30334.0
7 650 50.9017 142.223 270.053 258.124 20.3997 32851.0

501 | P a g e
Table 5.3 Comparison results of Lambda iteration
method and Cuckoo search method for 3-Unit System
SI.
No
Power
Demand
(MW)
Fuel Cost (Rs/hr)
Lambda
Iteration
Method
Cuckoo
Search
Algorithm
1 350 18570.7 18564.5
2 400 20817.4 20812.3
3 450 23146.8 23112.4
4 500 25495.2 25465.5
5 550 27899.3 27872.4
6 600 30359.3 30334.0
7 650 32875.0 32851.0
8 700 35446.3 35424.4
Table 5.4 Cost of power generation and power limits
of 6-Uint System
Unit an bn cn Pn,min Pn,max
1 756.79886 38.53 0.15240 10 125
2 451.32513 46.15916 0.10587 10 150
3 1049.9977 40.39655 0.02803 35 225
4 1242.5311 38.30443 0.03546 35 210
5 1658.5696 36.32782 0.02111 130 325
6 1356.6592 38.27041 0.01799 125 315
Table 5.5 Cuckoo Search Results for 6-Unit system
Table 5.6 Comparison results of Lambda iteration method and Cuckoo search method for 6-Unit system
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8 950 49747.4 49682.1

502 | P a g e
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