Energy and Power En gi neering, 2011, 3, 551-556
doi:10.4236/epe.2011.34068 Published Online September 2011 (
Copyright © 2011 SciRes. EPE
Modified Shuffled Frog Leaping Algorithm for Solving
Economic Load Dispatch Problem
Priyanka Roy1, Abhijit Chakrabarti2
1Electrical engineering Department, Techno India, Kolkata, India
2EE Department, Bengal Engineeri n g an d S ci e nce Univers i t y, Shibpur, Howrah, India
Received August 10, 2011; revised September 12, 2011; accept e d Septemb e r 24, 2011
In the recent restructured power system scenario and complex market strategy, operation at absolute mini-
mum cost is no longer the only criterion for dispatching electric power. The economic load dispatch (ELD)
problem which accounts for minimization of both generations cost and power loss is itself a multiple con-
flicting objective function problem. In this paper, a modified shuffled frog-leaping algorithm (MSFLA),
which is an improved version of memetic algorithm, is proposed for solving the ELD problem. It is a rela-
tively new evolutionary method where local search is applied during the evolutionary cycle. The idea of
memetic algorithm comes from memes, which unlike genes can adapt themselves. The performance of
MSFLA has been shown more efficient than traditional evolutionary algorithms for such type of ELD prob-
lem. The application and validity of the proposed algorithm are demonstrated for IEEE 30 bus test system as
well as a practical power network of 203 bus 264 lines 23 machines system.
Keywords: Economic Load Dispatch Modified Shuffled Frog Leaping Algorithm, Genetic Algorithm
1. Introduction
Economic load dispatch (ELD) is a familiar problem
pertaining to the allocation of the amount of power to be
generated by different units in the system on an optimum
economic basis. The generated power has to meet the
load demand and transmission losses. This implies that
the dispatch at the true minimum cost requires that we
take the network losses into account. Also for the secure
operation of the power system, the generators must dis-
patch in such a way so that the transmission capacity
limits are not exceeded.
Many researches are involved to tackle the ELD prob-
lem for significant economical benefit. Conventional
methods such as lamda iteration method, gradient based
method [1] are used to solve the ELD problem by
changing the fuel cost curve in a piecewise linear func-
tion or monotonically increasing function. These meth-
ods ignore the portion of incremental cost curve that are
not continuous or monotonically increasing. But in-
put-output characteristics of modern units are inherently
non-linear because of ramp rate limits, valve point load-
ings etc. So in classical method fuel cost curve is ap-
proximated according to their requirement but use of
such approximation may lead to huge loss of revenue
over the time. Dynamic programming, proposed in [2], is
a method to solve non-linear and discontinuous ELD
problem but with respect to system size, simulation time
is increased rapidly in this method. Other than classical
methods, different artificial intelligent based methods
have been successfully utilize d to compute ELD problem.
These methods are evolutionary programming [3], parti-
cle swarm optimization [4], tabu search [5], differential
evolution [6], biography based optimization [7], genetic
algorithm [8], artificial neural network [9], intelligent
water drop algorithm [10] etc. Each and every method
has its own disadvantages. Neural network suffers from
excessive iterations, resultin g huge calculation as well as
more processing time. Genetic algorithm has a disadvan-
tage of premature convergence, and due to that its per-
formance degrades and its search capability reduces. In
PSO, the algorithm progresses slowly and due to its in-
ability to adjust the velocity step size it may be difficult
to continue the search at a finer grain. For multi modal
function PSO sometimes fail to reach global optimal
point. DE has been found to yield better and faster solu-
tion, satisfying all the constraints, bo th for uni-modal and
multi-modal system by using its different crossover
strategies. However with increase in system complexity
and size, DE method is unable to map its entire unknown
variables together, in an efficient way.
Recently, a new meta-heuristic algorithm called Shuf-
fled Frog-Leaping Algorithm (SFLA) is introduced [11],
it aims to model and mimic the behavior of frogs search-
ing for food laid on stones randomly located in a pond. It
combines the advantages of the genetic-based memetic
algorithm (MA) and the social behavior-based Particle
Swarm Optimization (PSO) algorithm and has found
applications in areas such as optimizing bridge-deck re-
pairs [12], materialized views selection [13], bi-criteria
permutation flow shop scheduling problem [14], applica-
tion to reservoir flood control operation [15] and a
mixed-model assembly line sequencing problem [16].
This paper proposes a combined shuffled frog-leaping
algorithm (SFLA) and a genetic algorithm (GA) that
chooses genes (features) related to classification. It is
named as modified shuffled frog-leaping algorithm
(MSFLA) where two types of iterations (local and global
search) are simultaneously performed to get better opti-
mized value. In this method cross over operation has
been implemented in both global and local iterations.
Solving ELD problem using MSFLA technique is new
and the application and validity of the proposed algo-
rithm are demonstrated for IEEE 30 bus test system as
well as for a practical power network of 203 bus 264
lines 23 machines eastern India grid system. It has been
observed that compared to GA and common traditional
method, MSFLA based ELD solutions yield better re-
sults from economic point of view.
A brief description and mathematical formulation of
ELD problem has been discussed in the following sec-
tion. The concept of SFLA is discussed in section III
while the respective algorithm and parameter setting of
MSFLA has been provided in section IV. Simulation
studies are discussed in section V and conclusion is
drawn in section VI.
2. Nomenclature
In the analytical model following symbols have been
m and n: Number of buses
B: Loss coefficients for active power
: Power factor angles of bus load
: Phase angles of bus voltages
P: Real power demands
P: Real power outputs
P: Real loss .
Suffix i stands for ith bus while suffix j stands for jth
bus. The variables have been expressed in p.u. while the
angles have been expressed in degree
R: Series resistance of lines
3. Economic Load Dispatch
The aim of ELD is to optimize the cost function sub-
jected to linear and non-linear equality and inequality
constraints. The cost function
of an N-bus
power system having NG number of fossil fuel units is
given by
total iii
unit of cost/hr,
The active loss is conventionally expressed using
B-coefficient (or loss coefficient) matrix and can be rep-
resented as [17],
00 0
iGi GiijGj
 
For a system of N-plants, the loss coefficients are
given by [1 7]:
cos cos
00 11
 PDj
P and
3.1. Power Balance
The total generating power has to be equal to the sum of
load demand and transmissio n-line loss:
, (4)
where D is total load, L is transmission loss and C is
generated power.
The transmission loss can be represented by the B-
coefficient method as described in Equations (2) and (3).
3.2. Maximum and Minimum Li mits of Power
The generation power of each generator has some limits
and it can be expressed as
min max
4. Shuffled Frog Leaping Algorithm
The shuffled frog-leaping algorithm (SFLA) is a heuris-
tic search algorithms. It attempts to balance between a
Copyright © 2011 SciRes. EPE
P. ROY ET AL.553
wide scan of a large solution space and also a deep
search of promising location for a global optimum which
can not be solved by traditional optimization techniques.
It combines the benefits of a gene-based memetic algo-
rithm (MA) and social behavior-based particle swarm
optimization (PSO). MA is a gene-based optimization
algorithm similar to a GA. In a GA, chromosomes are
represented as a string consisting of a set of elements
called “genes.” Chromosomes in MA are represented by
elements, called “memes.” MA and GA differ in one
aspect, i.e. MA implements a local search before cross-
over or mutations to determine offspring. After the local
search, new offspring that obtains better results than
original offspring, replaces original offspring and thus
the evolutionary process continue. PSO is an evolution-
ary algorithm in which individual solutions are called
“particle” (analogous to the GA chromosome), but PSO
does not apply crossover and mutation to construct a new
particle. Each particle changes its position and velocity
based on the individual particle’s optimal solution and
the corporate optimal solution until a global optimal so-
lution is found.
The SFLA is derived from a virtual population of
frogs in which individual frogs are equivalent to the GA
chromosomes and represent a set of solutions. Each frog
is distributed to a different subset of the whole popula-
tion, called a memeplex. An independent local search is
conducted for each frog memplex and is called meme-
plex evolution. After a defined number of memetic evo-
lutionary steps, frogs are shuffled among memeplexes
enabling frogs to interchange messages among different
memplexes. This ensures that they move to an optimal
position similar to particles in PSO. Local search and
shuffling continue until defined convergence criteria are
SFLA have demonstrated effectiveness in a number of
global optimizatio n problems which are difficult to solve
using other method viz. intelligent water drop technique
[10].The detail steps involved in SFLA is given as under.
4.1. Initial Population
An initial population of P frogs is created randomly for a
S-dimensional problem. A frog i is represented by S
ii i
Ffff (6)
4.2. Sorting and Distribution
Frogs are sorted in descending order based on their
fitness values. The entire population is then divided into
m memeplexes, each containing n frogs (i.e., P= m × n ).
The first frog is distributed to the first memeplex, the
second frog to the second, th e m frog to the m memeplex,
and the m – 1 frog to the first m emeplex and so on.
4.3. Memeplex Evolution
Within each memeplex, frogs with the best and the worst
fitness are identified as Xb and Xw, and the frog with the
global best fitness is identified as Xg separately. To im-
prove upon the worst solution, an equation similar to
PSO is used to update the worst solution, e.g., Equations
(7) and (8):
Change in frog position
X (7)
New position Xw = current position ( Xw + Di) (8)
max max
where rand() is a random number between 0 and 1 and
Dmax is the maximum change allowed in a frog’s posi-
tion. If this process produces a better solution, it replaces
the worst frog. If Equations (7) and (8) do not improve
the worst solution, Xb of Equation (7) is changed to Xg
and adapted to Equatio n (9).
Change in frog position
1rand .
DXX (9)
If Equations (7) and (9) do not improve the worst so-
lution, then a new solution is randomly generated to re-
place that worst frog.
4.4. Shuffling
After a defined number of memeplex evolution steps, all
frogs of memeplexes are collected, and sorted in de-
scending order based on their fitness. Step 2 divides frogs
into different memeplexes again and then step 3 is per-
4.5. Terminal Condition
If a global solution or a fixed iteration number is reached,
the algorithm stops.
5. Programming Parameter and Algorithm
In MSFLA programming a number of parameters need to
be adjusted to compute best optimal value of the vari-
ables i.e., population size, number of memplexes, and
number of global and local iteration.
1) Population size: it is a number of set of variables,
defined as a total number of frogs. In this simulation
procedure, population size has been taken as 100. In-
Copyright © 2011 SciRes. EPE
crease in number of population means good accuracy but
it will lead to more propagation delay. After running the
program with different number of population size, it has
been observed that for this optimization problem, typi-
cally a population size of 100 is most suited for optimiz-
ing both processi ng time and value.
2) Number of memplexes: In this programming, num-
ber of memplexes is fixed at 10. As population size and
number of memplexes are user input, the given input of
number of memplexes is such that there exists a certain
number of frogs (population size/total number. of mem-
plexes) in each memplexes.
3) Number of global iteration: In this type of iteration,
the cross-over between best frog & worst frog is done
taking the whole population. One global iteration con-
sists of local iterations as many as number of memplexes
present. It is taken 10 here. Maximizing the number of
global iteration gives more accurate results but it takes
more time to process.
4) Number of local iteration: In this type of iteration,
the cross-over between best frog & worst frog is don e in
every single memplexes. Number of local iterations are
taken as 20 here. Maximizing the number of local itera-
tions also gives more accuracy but it gives more delay.
All the MSFLA parameters value discussed above is
for IEEE 30 bus test system.
Modified Shuffled Frog Leaping Algorithm
Step 1: start
Step 2: population size (n), no. of memeplexes (m),
number of local search within each memeple xes and num-
ber of global search are given as inputs.
Step 3: generate population of frogs (F) randomly from
the given data.
Step 4: evaluate fitness of F.
Step 5: sort F in descending order.
Step 6: cross-over between worst frog (Fw) and best
frog (Fb) is done to get two new offsprings.
Step 7: replace Fb and Fw with two best frogs (ac-
cording to their fitness) from four frogs(two parents and
two offsprings).
Step 8: partition F into m memeplexes such that each
memeplexes gets (F/m) frogs.
Step 9: find Fb and Fw from each memeplexes and do
cross-over between them.
Step 10: get two new offspring from them and replace
Fb and Fw with two best frogs(according to their fitness)
from four frogs.
Step 11: check whether number of local search is com-
pleted or not, if not then go to step 9.
Step 12: if local search is completed then check whe-
ther number of global search is completed or not, if not
then go to step 5.
Step 13: if global iteration is completed then get the
best solution (best fitn ess) from F.
Step 14: en d.
It has been observed from the above algorithm that
proposed MSFLA performs two simultaneous crossover,
i.e., global (Step 6 and 7) and local (Step 9 and 10)
search to produce new offspring which gives better result
compared to SFLA.
6. Simulation
To examine the validity of MSFLA model for the ELD
problem, IEEE 30 bus test system and a practical power
network of 203 bus 264 lines 23 machines system have
been considered. The result of proposed MSFLA model
has been compared with GA based ELD result and
classical iteration method. A reasonable B-loss coeffi-
cient matrix of the system has been employed to calcu-
late transmission loss. The detail calculation part of ELD
problem is concentrated on IEEE 30 bus test system
followed by the power scheduling of practical system.
The test system and production units’ properties are given
in Tables 1 and 2 for IEEE 30 bus system.
Table 3 shows different parameters of ELD schedul-
ing of IEEE 30 bus test system with three computational
techniques. It has been shown that compared to GA and
classical method, MSFLA technique in respect to cost
and power scheduling is better than the other two me-
thods, though the compu tational time is more in MSFLA
rather than GA. In MSFLA, cross over operation is per-
formed in local as well as global iteration time where as
in GA there is only one cross over operation. With re-
spect to transmission loss, MSFLA computation yields
Table 1. Test system properties.
Number of buses 30
Number of generator units 6
Number of branches 43
Number of tie lines 6
Table 2. Production units’ properties.
Cost Co-efficient
No Pmax (MW)Pmin (MW)
1 145.5 120.5 0.074 1.08325
2 70.6 50.6 0.089 1.03324
3 35.6 20.4 0.089 1.03322
4 50 30 0.074 1.08321
5 25.9 10.8 0.089 1.03323
6 25.9 10.8 0.053 1.17 29
Copyright © 2011 SciRes. EPE
P. ROY ET AL.555
Table 3. Computation of best outputs of 6 units systems
using different method.
Item (p.u. values) CLASSICAL
P1 1.3848 1.385 1.391
P2 0.5756 0.576 0.533
P3 0.2456 0.246 0.237
P4 0.35 0.35 0.368
P5 0.179 0.178 0.159
power of 6
P6 0.1689 0.169 0.144
Total power (pu) 2.9039 2.904 2.832
Active loss (pu) 0.065 0.071 0.06
Cost of power ( INR/hr) 147.2998 147.3 147.220
Processing time (Sec) 0.035 0.008 0.009
lower loss compared to classical method and GA.
In Table 4, another simulation result has been shown
where a practical power system having 203 buses, 264
lines, 23 machines of eastern grid of India is simulated
using MSFLA, GA and classical method.
It has been observed from Table 4 that in a large
system MSFLA based ELD scheduling still shows better
result compared to GA and classical method. The total
power generated using MSFLA being lesser in com-
parison to compute power by other two methods, it is
most logical that MSFLA technique provides a tool for
reduction in total cost of generated power.
The advantages of MSFLA over the other methods are
discussed below:
1) In MSFLA, two types of iterations (local and global
search) are simultaneously performed to get better op-
timized value whereas in GA there is only global search.
MSFLA has the ability to reach global minima in a con-
sistent manner with better convergence characteristic. In
case of classical method only global search is performed.
2) In case of memory usage and time complexity, it has
been observed from table III that MSFLA based solution
takes more CPU time compared to GA based solution as
two simultaneous iterations (global and local) are com-
puted in MSFLA whereas GA based programming has
been performed with single iteration. But in case of mini-
mizing generation cost which is the main objective of
ELD problem, MSFLA shows better result compared to
GA and any other traditional method. More over trans-
mission losses are also reduced using this method.
7. Conclusions
In this paper economic load dispatch problem has been
Table 4. Computation of best outputs of 23 units system
using different method.
Power Generation
Of each unit
(p.u. values)
1 5.4 5.48226 5.38
2 5.4 5.49353 5.4
3 3.8 3.48685 3.332
4 5.4 5.50104 5.456
5 3.8 3.49652 3.75
6 2.4 2.40876 2.5
7 0.9 0.89 0.8
8 0.46 0.40835 0.456
9 1.8 1.85684 1.79
10 2 2.18023 2.05
11 1.8 1.88354 1.889
12 2.7 2.83909 2.56
13 0.243 0.241 0.251
14 0.108 0.1 0.1005
15 0.54 0.51197 0.495
16 6 6.09554 6.01
17 0.5 0.54414 0.49
18 1.5 1.55 1.64
19 6 6.08307 6.07
20 0.204 0.2 0.23
21 0.2 0.2 0.21
22 0.22 0.2 0.2
23 0.4 0.39 0.41
Total Power (pu) 51.775 52.0427351.4695
Active loss (pu) 1.4452 1.4 1.387
Processing time (sec) 1.2 0.8 1.1
Cost of power (INR/hr)1318.1 1317.89 1316.78
formulated as a multi objective problem to optimize fuel
cost as well as to minimize system loss. This problem is
solved using MSFLA, GA and classical lamda iteration
method. Using IEEE 30 bus test system as a standard
system and a practical power system of 203 bus 264 lines
23 machines, it has been observed that MSFLA method
is more efficient than other programming techniques.
The good performance of MSFLA in ELD problem
discussed in this paper provides some evidence that
Copyright © 2011 SciRes. EPE
Copyright © 2011 SciRes. EPE
MSFLA theory can be successfully applied to various
practical power system optimization problems in the
8. Acknowledgements
The authors would like to thank Mr. Pritam Roy, Com-
puter science department, Govt. College of ceramic tech-
nology, for his relevant support related to the MSFLA
programmi ng.
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