A. B. M. Nasiruzzaman,  M. G. Rabbani, M. R. I. Sheikh,  M. S. Anower, Shubhashis K. Shil, Md. Ashraful

Eastern University Digital Library

International Conference on Electronics, Computer and Communication (ICECC 2008) 
University of Rajshahi, Bangladesh 
 
ISBN 984-300-002131-3 
 
A Genetic Algorithm Based Approach for Solving Optimal Power Flow Problem  
 
A. B. M. Nasiruzzaman1, M. G. Rabbani2, M. R. I. Sheikh3, M. S. Anower4, Shubhashis K. Shil5, Md. Ashraful 
Haque6 
1-2,4EEE & 5-6CSE  Department, Rajshahi University of Engineering & Technology, 6204, Bangladesh 
3Kitami Institute of Technology, 165 Koen-cho, Kitami, Hokkaido, 090-8507, Japan 
E-mail: nasir_zaman_eee@yahoo.com1 
 
 
Abstract: The objective of this paper is to evolve 
simple and effective methods for the optimal power 
flow (OPF) problem in thermal units, which are 
capable of obtaining optimal power generations for 
a large-scale system. In optimization, GA has some 
well-known advantages and disadvantages. The 
classical (derivative based) method is applied as 
economic dispatch problem with transmission 
capacity constraints in literature. In this study, a 
similar investigation for economic dispatch is 
considered using simple Genetic Algorithm. The 
performance of results obtained with both of the 
methods is nearly same but from the viewpoint of 
calculation time GA is much superior and its 
effectiveness increases as the constraints of the 
problem are increased.  
Key Words: Genetic Algorithm, Optimal Power 
Flow. 
 
1.     INTRODUCTION 
In a practical power system, the power plants are not 
located at the same distance from the centre of loads 
and their fuel costs are different. Also under normal 
operating conditions, the generation capacity is more 
than the total load demand and the losses. Thus, there 
are many options for scheduling generation. In an 
interconnected power system the objective is to find 
power scheduling of each power plant in such a way 
as to minimize the operating cost. This means that 
the generator’s power is allowed to vary within 
certain limits so as to meet a particular load demand 
with minimum fuel cost. This is called the optimal 
power flow problem. The OPF is used to optimize 
the power flow solution of large scale power system. 
This is done by minimizing selected objective 
function while maintaining an acceptable system 
performance in terms of generator capability limits 
and the output of compensating devices. The 
objective functions, also, known as cost functions, 
may represent economic costs, system security, or 
other objectives.  
 
The OPF has been studied by many researchers and 
methods have been presented in several classical 
papers [2], [3], and [4], and in well established 
textbooks [5], and [6]. These classical methods 
require specific assumptions on the objective 
function such as continuity, differentiability. 
Moreover, these methods are slowly convergent.  
Genetic Algorithms (GA) are general search 
technique based on the mechanism of natural 
selection and natural genetics. In GA, genetic strings 
represent search points. The search process uses 
probabilistic transition rules instead of deterministic 
ones as in traditional optimization method. The 
search is conducted over a population of solutions. 
Moreover, no specific assumptions on the objective 
function, such as continuity, differentiability, etc. are 
required in GA.    
Genetic Algorithm is different from traditional or 
conventional methods in four ways: 
1. GA work with a coding of the parameter 
set, not the parameters themselves. 
2. GA search from a population of points, not 
a single point. 
3. GA use payoff (objective function) 
information, not derivatives or other auxiliary 
knowledge. 
4. GA use probabilistic transition rules, not 
deterministic rules. 
In several papers [1] the optimal power flow problem 
is solved using GA considering a power plant 
composed of six generators.  
In our paper, analysis is done for any number of 
buses with any number of generators.  
The OPF problem has a nonlinear objective function 
having many variables and constraints. In this paper 
GA has been used to solve the OPF problem as it 
completely replaces classical mathematical 
programming methods and also it is easy to 
implement as an optimization procedure.       
In view of the above, the main thrust of the research 
work presented in this paper is to propose a new but 
effective method for solving OPF using GA. The 
main objectives of the research work presented are as 
follows: 
 
1.   To present a systematic approach for the design 
of a new method for solving optimal power flow 
problem using GA. 
2.  To compare the performance of the GA based 
method with the conventionally derivative based 
methods [1], [2], [4], and [5]. 
 
476
International Conference on Electronics, Computer and Communication (ICECC 2008) 
University of Rajshahi, Bangladesh 
 
ISBN 984-300-002131-3 
 
2.    MATHEMATICAL MODEL 
Mathematical models are of fundamental importance 
in understanding any physical system. This section 
focuses on the modeling of OPF problem and its 
formulation. The factors influencing power 
generation at minimum cost are operating 
efficiencies of generators, fuel cost, and transmission 
losses. The most efficient generator in the system 
does not guarantee minimum cost as it may be 
located in an area where fuel cost is high. Also if the 
plant is located far from the load center, transmission 
losses may be considerably higher and hence the 
plant may be overly uneconomical. Hence the 
problem is to determine the generation of different 
plants such that the total operating cost is minimum. 
In all practical cases, the fuel cost of generator i with 
power output iP can be represented as a quadratic 
function of real power generation 
2
iiiiii PPC γβα ++=  
where, ,, ii βα and iγ represent unit cost 
coefficients. 
The economic dispatch problem is to find the real 
power generation for each plant such that the 
objective function (i.e., total production cost) as 
defined by the equation 
∑
=
++=
n
i
iiiiit PPC
1
2γβα    
is minimum subject to the inequality constraints 
given by  
(max)(min) iii PPP ≤≤    gni ,......,2,1=  
where (min)iP and (max)iP are the minimum and 
maximum generating limits respectively for plant i, 
subject to the constraint that generation should equal 
total demand plus losses, i.e., 
∑
=
+=
gn
i
LDi PPP
1
 
where tC is the total production cost, iC is the 
production cost of the i-th plant, iP is the generation 
of i-th plant, DP is the total load demand, and gn is 
the total number of displaceable generating plants 
and PL is the total transmission loss as a quadratic 
function of the generator power outputs given by 
Kron’s loss formula  
∑∑∑
== =
++=
gg g n
i
ii
n
i
n
j
jijiL BPBPBPP
1
000
1 1
 
The coefficients ijB are called loss coefficients. 
3.    GENETIC ALGORITHM 
 
Genetic algorithms are computerized search and 
optimization algorithms based on the mechanics of 
natural genetics and natural selection. They combine 
the survival of the fittest among string structures with 
a structured yet randomized information exchange to 
form search algorithm with some of the innovative 
flair of human search. In every generation, a new set 
of artificial creatures (strings) is created using bits 
and pieces of the fittest of the old; an occasional new 
part is tried for good measure. While randomized, 
genetic algorithms are no simple random walk. They 
efficiently exploit historical information to speculate 
on new search points with expected improved 
performance [7], and [8]. 
 
3.1 General Structure of GA 
The sequential steps for searching optimal solution 
of PSS parameters using GA is shown in Fig.1. 
 
 
Fig. 1  Computational flow chart. 
 
3.2 Design Methodology 
The design problem can be formulated as the 
following optimization problem:  
 
1.  Minimize the objective function 
2
iiiiii PPC γβα ++=  subjected to 
constraint maxmin PPP i ≤≤ . Here the fitness 
477
International Conference on Electronics, Computer and Communication (ICECC 2008) 
University of Rajshahi, Bangladesh 
 
ISBN 984-300-002131-3 
 
function )0(ii CCf −= is taken and it is to be 
minimized. 
2. An initial population of 256 individuals is 
generated each of 8 bits with the following 
equations. So, N=8. 
(a) First the accuracy with following 
parameters is chosen: 
12 −
−= N
LU XXAccuracy ; 
where maxPX
U = and minPX L = .  
(b) Chromosome/Individual value                       
= Accuracy*Si where Si=0,1,2,…,255. 
3.  From this initial population of 256, 128 
individuals are taken randomly for every 
generation. 
4. Roulette Wheel Method of reproduction is used.   
5. Single cross-site of crossover is used with 
crossover rate 0.5 and mutation rate 0.05. 
6.  As a terminating condition number of      
generation is chosen to be 100. 
 
 
4. COMPARATIVE STUDY OF GA BASED 
SOLUTION WITH CONVENTIONAL 
SOLUTION 
To evaluate the effectiveness of the proposed GA 
based OPF problem solution examples from [5] are 
taken as reference, the performance of the GA is 
examined under different conditions 
 
1. Economic dispatch neglecting losses and    
generator limits (Example 7.4).  
2. Economic dispatch neglecting losses and including 
generator limits (Example 7.6). 
3.  Economic dispatch including losses and 
generator limits (Example 7.7). 
Comparative results are given in the following 
tables: (Table 1, Table 2, and Table 3) 
 
Table 1:  Comparison of classical and genetic based 
optimal dispatch of generation problem neglecting 
losses and generator limits. 
 
 Classical Genetic 
P1(MW) 400.0000 398.4313 
P2(MW) 250.0000 250.9803 
P3(MW) 150.0000 150.5882 
Cost ($/h) 6682.20 6682.51 
Time(sec) 50.765 0.9677 
 
 
Table 2:  Comparison of classical and genetic based 
optimal dispatch of generation problem neglecting 
losses and including generator limits. 
 
 Classical Genetic 
P1(MW) 450.0000 447.3529 
P2(MW) 325.0000 325.0000 
P3(MW) 200.0000 202.6470 
Cost ($/h) 8236.25 8237.66 
Time(sec) 105.765 1.0103 
 
Table 3:  Comparison of classical and genetic based 
optimal dispatch of generation problem including 
losses and generator limits. 
 
 Classical Genetic 
P1(MW) 35.0907 31.1764 
P2(MW) 64.1317 67.0588 
P3(MW) 52.4767 50.0000 
Cost ($/h) 1592.65 1566.45 
Time(sec) 250.765 1.723 
 
From the tables it is evident that GA based solutions 
are faster than the traditional derivative based 
methods and the computational time differences are 
much more prominent as the constraints are 
increased. 
 
5. CONCLUSION 
A new approach for solving the challenging optimal 
load flow problem is presented. The model for 
solving the mentioned problem is formulated and 
solved at various conditions. The results are 
compared with the results obtained using traditional 
methods. The results are quite similar except that GA 
is much faster than the traditional ones. This 
indicates that the GA approach is quite promising 
and deserves serious attention because of its 
probabilistic transition rules, not deterministic ones.   
References 
[1]  A. Demirören and H.L. Zeynelgil, “The 
Investigation  of Environmental/Economic 
Dispatch using a Genetic Algorithm”, proc. of 
Modelling, Identification, and Control, MIC-
2003. 
[2] R.C. Burchett, H. H. Happ, and D. R. 
Vierath,  “Quadratically Convergent Optimal 
Power Flow,” IEEE Transactions PAS-103, no. 
10,  pp. 3267–3275, Nov. 1984. 
[3] R.C. Burchett, H. H. Happ, and K. A. 
Wirgao, “Large Scale Optimal Power Flow” 
IEEE Transactions PAS-103, no. 10  pp. 3722–
3731, Oct. 1982. 
478
International Conference on Electronics, Computer and Communication (ICECC 2008) 
University of Rajshahi, Bangladesh 
 
ISBN 984-300-002131-3 
 
[4] J. F. Dopaz, O. A. Klitin, G. W. Stagg, and 
M. Watson “An optimization technique for real 
and reactive power allocation” proc. of  IEEE, 
Vol. 55, no. 11, pp1877-1885, 1967. 
[5] W. D. Stevenson, Jr, “Elements of Power 
System Analysis”, McGraw-Hill Book 
Company, 2002. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
[6] Hadi Saadat, “Power System Analysis”, Tata 
McGraw-Hill Publishing Company Limited, 
2002. 
[7] David E. Goldberg, “Genetic Algorithms in 
Search, Optimization and Machine Learning”, 
Pearson Education, 2002. 
[8] S.Rajasekharan, S. A. Vijayalekshmi Pai,              
“Neural Networks, Fuzzy Logic & Genetic 
Algorithms”, Prentice Hall of India, 2003. 
 
 
 
 
 
 
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A genetic algorithm based approach for solving optimal power flow problem

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