Og2423252330

K. Sudhakara Reddy, Dr. M. Damodar Reddy/ International Journal of Engineering Research
and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2325-2330
Economic Load Dispatch Using Firefly Algorithm

K. Sudhakara Reddy1, Dr. M. Damodar Reddy2
1-(M.tech Student, Department of EEE, SV University, Tirupati
2- (Associate professor, Department of EEE, SV University, Tirupati)

ABSTRACT
Economic Load Dispatch (ELD) problem problem solving algorithms. In the Firefly
in power systems has been solved by various algorithm[14][15][16], the objective function of a
optimization methods in the recent years, for given optimization problem is associated with the
efficient and reliable power production. This flashing light or light intensity which helps the
paper introduces a solution to ELD problem using swarm of fireflies to move to brighter and more
a new metaheuristic nature-inspired algorithm attractive locations in order to obtain efficient
called Firefly Algorithm (FFA). The proposed optimal solutions.
approach has been applied to various test systems. In this research paper we will show how the
The results proved the efficiency and robustness firefly algorithm can be used to solve the economic
of the proposed method when compared with the load dispatch optimization problem. A brief
other optimization algorithms. description and mathematical formulation of ELD
problem has been discussed in the following section.
Keywords - Economic Load Dispatch, Firefly The concept of Firefly Algorithm is discussed in
Algorithm. section 3. The respective algorithm and parameter
setting of FFA has been provided in section 4.
1.Introduction Simulation studies are discussed in section 5 and
Economic Load Dispatch (ELD) seeks the conclusion is drawn in section 6.
best generation schedule for the generating plants to
supply the required demand plus transmission loss 2. Formulation of the Economic Load
with the minimum generation cost. Significant Dispatch Problem
economical benefits can be achieved by finding a 2.1. Economic Dispatch
better solution to the ELD problem. So, a lot of The objective of economic load dispatch of
researches have been done in this area. Previously a electric power generation is to schedule the
number of calculus-based approaches including committed generating unit outputs so as to meet the
Lagrangian Multiplier method [1] have been applied load demand at minimum operating cost while
to solve ELD problems. These methods require satisfying all units and operational constraints of the
incremental cost curves to be monotonically power system.
increasing/piece-wise linear in nature. But the input- The economic dispatch problem is a
output characteristics of modern generating units are constrained optimization problem and it can be
highly non-linear in nature, so some approximation is mathematically expressed as follows:
required to meet the requirements of classical
dispatch algorithms. Therefore more interests have
been focused on the application of artificial (2.1)
intelligence (AI) technology for solution of these Where, FT : total generation cost (Rs/hr)
problems. Several AI methods, such as Genetic n : number of generators
Algorithm[2] Artificial Neural Networks[3], Pn : real power generation of n th
Simulated Annealing[4], Tabu Search[5], generator (MW)
Evolutionary Programming[6], Particle Swarm Fn(Pn) : generation cost for Pn
Optimization[7], Ant Colony Optimization[8], Subject to a number of power systems
Differential Evolution[9], Harmony search network equality and inequality constraints. These
Algorithm[10], Dynamic Programming[11], Bio- constraints include:
geography based optimization[12], Intelligent water
drop Algorithm[13] have been developed and applied 2.2. System Active Power Balance
successfully to small and large systems to solve ELD For power balance, an equality constraint
problems in order to find much better results. Very should be satisfied. The total power generated should
recently, in the study of social insects behavior, be the same as total load demand plus the total line
computer scientists have found a source of inspiration losses
for the design and implementation of optimization
algorithms. Particularly, the study of fireflies’
behavior turned out to be very attractive to develop

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Where, PD : total system demand (MW) The firefly algorithm has three particular
PL: transmission loss of the system idealized rules which are based on some of the major
(MW) flashing characteristics of real fireflies.
The characteristics are as follows:
2.3. Generation Limits i) All fireflies are unisex and they will move towards
Generation output of each generator should more attractive and brighter ones regardless their sex.
be laid between maximum and minimum limits. The ii) The degree of attractiveness of a firefly is
corresponding inequality constraints for each proportional to its brightness which decreases as the
generator are distance from the other firefly increases. This is due
to the fact that the air absorbs light. If there is not a
brighter or more attractive firefly than a particular
(2.3) one, it will then move randomly.
Where, Pn,min : minimum power output limit iii) The brightness or light intensity of a firefly is
of nth generator (MW) determined by the value of the objective function of a
Pn,max : maximum power output limit of nth given problem. For maximization problems, the light
generator (MW) intensity is proportional to the value of the objective
The generation cost function Fn(Pn) is usually function.
expressed as a quadratic polynomial:
3.2. Attractiveness
In the firefly algorithm, the form of
attractiveness function of a firefly is given by the
(2.4) following monotonically decreasing function:
Where, an, bn and cn are fuel cost
coefficients. with m≥1
2.4. Network Losses (3.1)
Since the power stations are usually spread Where, r is the distance between any two
out geographically, the transmission network losses fireflies,
must be taken into account to achieve true economic β0 is the initial attractiveness at r
dispatch. Network loss is a function of unit =0, and
generation. To calculate network losses, two methods γ is an absorption coefficient
are in general use. One is the penalty factors method which controls the decrease of the light intensity.
and the other is the B coefficients method. The latter
is commonly used by the power utility industry. In 3.3. Distance
the B coefficients method, network losses are The distance between any two fireflies i and j at
expressed as a quadratic function: positions xi and xj respectively can be defined as :

(2.5) (3.2)
Where Bmn are constants called B where xi,k is the kth component of the spatial
coefficients or loss coefficients coordinate xi of the ith firefly and d is the number of
dimensions.
3. The Firefly Algorithm
3.1. Description 3.4. Movement
The Firefly Algorithm is a metaheuristic, The movement of a firefly i which is
nature-inspired optimization algorithm which is attracted by a more attractive i.e., brighter firefly j is
based on the social flashing behavior of fireflies. It is given by :
based on the swarm behavior such as fish, insects or
bird schooling in nature. Although the firefly
algorithm has many similarities with other algorithms (3.3)
which are based on the so-called swarm intelligence, Where the first term is the current position
such as the famous Particle Swarm Optimization of a firefly, the second term is used for considering a
(PSO), Artificial Bee Colony optimization (ABC) firefly’s attractiveness to light intensity seen by
and Bacterial Foraging (BFA) algorithms, it is indeed adjacent fireflies and the third term is used for the
much simpler both in concept and implementation. random movement of a firefly in case there are no
Its main advantage is that it uses mainly real random brighter ones. The coefficient α is a randomization
numbers, and it is based on the global communication parameter determined by the problem of interest,
among the swarming particles called as fireflies. rand is a random number generator uniformly
distributed in the space [0,1].

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4. Algorithm A reasonable loss coefficient matrix of
Step 1: Read the system data such as cost power system network was employed to draw the
coefficients, minimum and maximum power transmission line loss and satisfy the transmission
limits of all generator units, power demand capacity constraints. The program is written in
and B-coefficients. MATLAB software package.
Step 2: Initialize the parameters and constants of
Firefly Algorithm. They are noff, αmax, αmin, 5.1 Three-Unit System
β0, γmin, γmax and itermax (maximum number The generator cost coefficients, generation
of iterations). limits and B-coefficient matrix of three unit system
Step 3: Generate noff number of fireflies (xi) are given below. Economic Load Dispatch solution
randomly between λmin and λmax . for three unit system is solved using conventional
Step 4: Set iteration count to 1. Lambda iteration method and Firefly algorithm
Step 5: Calculate the fitness values corresponding to method. Test results of Firefly method are given in
noff number of fireflies. table 5.2. Comparison of test results of Lambda-
Step 6: Obtain the best fitness value GbestFV by iteration method and Firefly algorithm are shown in
comparing all the fitness values and also table 5.3.
obtain the best firefly values GbestFF
corresponding to the best fitness value Table 5.1 Cost coefficients and power limits of 3-
GbestFV. Unit system.
Step 7: Determine alpha(α) value of current
Unit iteration using the following equation: an bn cn Pn,min Pn,max
α (iter)= αmax -(( αmax - αmin) (current
iteration number )/ itermax) 1 1243.5311 38.30553 0.03546 35 210
Step 8: Determine the rij values of each firefly using
the following equation: 2 1658.5696 36.32782 0.02111 130 325
rij= GbestFV -FV
rij is obtained by finding the difference 3 1356.6592 38.27041 0.01799 125 315
between the best fitness value GbestFV
(GbestFV is the best fitness value i.e., jth
firefly) and fitness value FV of the ith firefly. The loss coefficient matrix of 3-Unit system
Step 9: New xi values are calculated for all the
Bij=
fireflies using the following equation:

Table 5.2 Test results of Firefly Algorithm for 3-Unit
(4.1) System
Where, β0 is the initial attractiveness Pow
γ is the absorption co-efficient Sl
er
P1 P2 P3 Ploss
Fuel
rij is the difference between the best fitness value Dem Cost
. (M (M (M (M
GbestFV and fitness value FV of the ith firefly. and λ (Rs/
N W) W) W) W)
α (iter) is the randomization parameter ( In this o.
(M hr)
present work, α (iter) value is varied between 0.2 W)
and 0.01) rand is the random number between 0 and 44.4 70.3 156. 129. 5.77 185
1 350
1. 387 012 267 208 698 64.5
In this present work, x → λ 45.4 82.0 174. 150. 7.56 208
2 400
Step 10: Iteration count is incremented and if 762 784 994 496 813 12.3
iteration count is not reached maximum then 46.5 93.9 193. 171. 9.61 231
3 450
go to step 5 291 374 814 862 271 12.4
Step 11: GbestFF gives the optimal solution of the 47.5 105. 212. 193. 11.9 254
Economic Load Dispatch problem and the 4 500
977 88 728 306 144 65.5
results are printed. 48.6 117. 231. 214. 14.4 278
5 550
824 907 738 831 769 72.4
5. Simulation Results 49.7 130. 250. 236. 17.3 303
The effectiveness of the proposed firefly 6 600
836 021 846 437 040 34.0
algorithm is tested with three and six generating unit 50.9 142. 270. 258. 20.3 328
systems. Firstly, the problem is solved by 7 650
017 223 053 124 997 51.0
conventional Lambda iterative method and then 52.0 154. 289. 279. 23.7 354
Firefly algorithm optimization method is used to 8 700
371 514 360 894 680 24.4
solve the problem.

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Table 5.3 Comparison of test results of Lambda- Bij=
iteration method
and Firefly Algorithm for 3-Unit System
Fule Cost (Rs/hr)
Lambda
Sl. Power Firefly
iteration
No. Demand(MW) Algorithm
method
1 350 18570.7 18564.5

2 400 20817.4 20812.3 Table 5.5 Test results of Firefly method for 6-Unit
System
3 450 23146.8 23112.4
Po
4 500 25495.2 25465.5
w F
5 550 27899.3 27872.4 er ue
S D Pl l
P1 P2 P3 P4 P5 P6
6 600 30359.3 30334.0 l e oss C
( ( ( ( ( (
. m ( os
M M M M M M
7 650 32875.0 32851.0 N an λ M t
W W W W W W
o d W (R
) ) ) ) ) )
8 700 35446.3 35424.4 . ( ) s/
M hr
W )
)
5.2 Six-Unit System 47 23 95 10 20 18 14 32
The generator cost coefficients, generation 60 .3 .8 .6 0. 2. 1. .2 09
limits and B-coefficient matrix of six unit system are 1 10
0 41 60 38 70 83 19 37 4.
given below. Economic Load Dispatch solution for 9 3 9 8 2 8 4 7
six unit system is solved using conventional Lambda 48 26 10 10 21 19 16 34
iteration method and Firefly Algorithm method. Test 65 .1 .0 7. 9. 6. 6. .7 48
results of Firefly method are given in table 5.5. 2 10
0 73 67 26 66 77 95 28 2.
Comparison of test results of Lambda-iteration 1 9 4 8 5 4 1 6
method and Firefly Algorithm are shown in table 5.6.
49 28 11 11 23 21 19 36
70 .0 .2 8. 8. 0. 2. .4 91
3 10
0 14 90 95 67 76 74 31 2.
Table 5.4 Cost coefficients and power limits of 6-
6 8 8 5 3 5 9 2
Unit system
49 30 11 13 12 24 22 22 39
Unit an Bn Cn Pn,min Pn,max 75 .8 .4 .2 0. 7. 4. 8. .3 38
4
0 45 75 26 44 51 46 18 10 4.
1 756.79886 38.53 0.15240 10 125 1 6 5 6 5 6 2 8 0
2 451.32513 46.15916 0.10587 10 150 50 32 14 14 13 25 24 25 41
80 .6 .5 .4 1. 6. 7. 3. .3 89
5
3 1049.9977 40.39655 0.02803 35 225 0 61 86 84 54 04 66 00 31 6.
3 3 3 8 5 4 9 2 9
4 1243.5311 38.30553 0.03546 35 210 51 34 17 15 14 27 25 44
28
85 .4 .7 .7 2. 4. 0. 7. 45
6 .5
5 1658.5696 36.32782 0.02111 130 325 0 83 10 67 70 61 89 85 0.
56
9 2 5 8 4 7 9 3
6 1356.6592 38.27041 0.01799 125 315 52 36 21 15 27 31 47
16 28
90 .3 .8 .0 3. 2. .9 04
7 3. 4.
0 16 48 77 22 73 88 5.
The loss co-efficient matrix of 6-Unit system 93 17
2 1 5 7 7 1 3
53 38 24 17 16 29 35 49
28
95 .1 .9 .4 5. 1. 7. .6 68
8 7.
0 58 99 14 21 88 48 29 2.
64
5 8 5 4 2 1 5 1

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