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A solution to the optimal power flow using genetic algorithm

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A solution to the optimal power ﬂowusing genetic algorithm
M.S. Osman
a
, M.A. Abo-Sinna
b,*
, A.A. Mousa
b
a
High technological Institute, 10th Ramadan city, Egypt
b
Department of Basic Engineering Science, Faculty Of Engineering, Moenouﬁa University,Shebin El-Kom, Egypt
Abstract
Optimal power ﬂow (OPF) is one of the main functions of power generation oper-ation and control. It determines the optimal setting of generating units. It is therefore of great importance to solve this problem as quickly and accurately as possible. This paperpresents the solution of the OPF using genetic algorithm technique. This paper proposesa new methodology for solving OPF. This methodology is divided into two parts. Theﬁrst part employs the genetic algorithm (GA) to obtain a feasible solution subject todesired load convergence, while the other part employs GA to obtain the optimal so-lution. The main goal of this paper is to verify the viability of using genetic algorithm tosolve the OPF problem simultaneously composed by the load ﬂow and the economicdispatch problem. Six buses system are used to highlight the goodness of this solutiontechnique.
2003 Elsevier Inc. All rights reserved.
Keywords:
Nonlinear programming; Genetic algorithms; Load ﬂow; Economic dispatch
1. Introduction
The economic load dispatching (ELD) problem is one of key problems inpower system operation and planning. The ELD problem may be expressed byminimizing the fuel cost of generators units under constraints depending on
*
Corresponding author.
E-mail address:
mabosinna2000@yahoo.com (M.A. Abo-Sinna).0096-3003/$ - see front matter
2003 Elsevier Inc. All rights reserved.doi:10.1016/S0096-3003(03)00785-9Applied Mathematics and Computation 155 (2004) 391–405www.elsevier.com/locate/amc
load variations, the output of generators has to be changed to meet the balancebetween loads and generation therefore, the ELD is one of the fundamentalproblems in power system operation and planning. The primary objective of the economic dispatch (ED) problem is to ﬁnd a set of active powers deliveredby the committed generators to satisfy at any time the required demand subjectto unit technical limits at the production cost. It is therefore of great impor-tance to solve this problem as quickly and accurately as possible. Conventionaltechniques oﬀer good results but when the search space is nonlinear and hasdiscontinuities these techniques become diﬃcult to solve with a slow conver-gence ratio and not always seeking to the optimal solution. New numericalmethods are then needed to cope with these diﬃculties specially, those withhigh-speed search to the optimal and not being trapped in local minima. On theother hand, there has recently been a great deal of interest in promising geneticalgorithm (GA) [1,3–6] and its application to various disciplines includingpower system planning operation and control. Genetic algorithms are alsobeing applied to a wide range of optimization and learning problems in manydomains. Genetic algorithms lend themselves well to power system optimiza-tion since they are known to exhibit robustness, require to auxiliary informa-tion, and can oﬀer signiﬁcant advantages in a solution methodology andoptimization performance. In this paper we formulate optimal power ﬂow(OPF) problem in Section 2. Section 3 describes genetic algorithm techniques.In Section 4, we review some of constraint-handling techniques in genetic al-gorithm. In Section 5, we deﬁne the OPF problem variables. Section 6 describesthe proposed algorithm. In Section 7, sample six-bus system are solved and theresults are introduced in Section 8. Finally Section 9 describes the main fea-tures of the proposed algorithm.
2. Optimal power ﬂow formulation [11]
An OPF problem is generally formulated asMin
f
ð
x
Þ
s
:
t
:
g
i
ð
x
Þ ¼
0
;
i
¼
1
;
. . .
;
q
and
h
j
ð
x
Þ
6
0
;
j
¼
q
þ
1
;
. . .
;
m
:
The objective function for the OPF reﬂects the costs associated with generatingpower in the system. The quadratic cost model for generation of power will beutilized:
C
P
G
i
¼
a
i
þ
b
i
P
G
i
þ
c
i
P
2G
i
where
P
G
i
is the amount of generation in megawatts at generator
i
. The ob- jective function for the entire power system can then be written as the sum of the quadratic cost model at each generator.
392
M.S. Osman et al. / Appl. Math. Comput. 155 (2004) 391–405
f
ð
x
Þ ¼
C
t
¼
X
N
g
i
ð
a
i
þ
b
i
P
G
i
þ
c
i
P
2G
i
Þ
This objective function will minimize the total system costs, and does notnecessarily minimize the costs for a particular area within the power system,where
C
t
is the total generation cost;
a
i
,
b
i
,
c
i
, the cost function coeﬃcients of unit;
P
G
i
the real power generation of unit
i
;
N
g
, the total number of generationunits and
i
¼
1
;
2
;
. . .
;
N
g
.The OPF equality constraints [7,9,11] reﬂect the physics of the power systemas well as the desired voltage set points throughout the system. The physics of the power system are enforced through the power ﬂow equations which requirethat the net injection of real and reactive power at each bus sum to zero.Therefore
g
ð
x
Þ
is
D
P
p
¼
P
G
p
P
cp
X
N
B
q
¼
1
V
p
V
q
Y
pq
cos
ð
d
p
d
q
H
pq
Þ
D
Q
p
¼
Q
G
p
Q
cp
X
N
B
q
¼
1
V
p
V
q
Y
pq
sin
ð
d
p
d
q
H
pq
Þ
where
P
G
p
,
Q
G
p
are the real and reactive power generations at bus
P
;
P
cp
,
Q
cp
thereal and reactive power demands at bus
P
;
V
P
, the voltage magnitude at bus
P
;
V
q
, the voltage magnitude at bus
q
;
d
p
, the voltage angle at bus
p
;
d
q
; the voltageangle at bus
q
;
Y
Pq
, the admittance magnitude;
H
pq
, the admittance angle;
N
B
,the total number of buses;
P
¼
1
;
2
;
. . .
;
N
B
and
q
¼
1
;
2
;
. . .
;
N
B
.The inequality constraints of the OPF reﬂect the limits on physical devices inthe power system as well as the limits created to ensure system security.Physical devices that require enforcement of limits include generators, tapchanging transformers, and phase shifting transformers.
P
Gmin
6
P
G
p
6
P
Gmax
;
Q
Gmin
6
Q
G
p
6
Q
Gmax
;
V
min
6
V
P
6
V
max
;
d
min
6
d
p
6
d
max
3. Genetic algorithm (GA)
GA, invented by Holland [6] in the early 1970s, is a stochastic global searchmethod that mimics the metaphor of natural biological evaluation. GAs op-erates on a population of candidate solutions encoded to ﬁnite bit string calledchromosome. In order to obtain optimality, each chromosome exchanges in-formation by using operators borrowed from natural genetic to produce thebetter solution. Fig. 1 shows outline of GAs for optimization problems. TheGAs diﬀer from other optimization and search procedures in four ways:
M.S. Osman et al. / Appl. Math. Comput. 155 (2004) 391–405
393
(1) GAs work with a coding of the parameter set, not the parameters them-selves. Therefore GAs can easily handle the integer or discrete variables.(2) GAs search from a population of points, not a single point. Therefore GAscan provide a globally optimal solutions.(3) GAs use only objective function information, not derivatives or other aux-iliary knowledge. Therefore GAs can deal with the non-smooth, non-continuous and non-diﬀerentiable functions which are actually existed ina practical optimization problem.(4) GAs use probabilistic transition rules, not deterministic rules.
4. Review of constraint-handling techniques [3–6]
One of the major components of any evolutionary system is the evaluationfunction. Evaluation functions are used for assign a quality measure for in-dividuals in a population. Whereas evolutionary computation techniques as-sume the existence of an (eﬃcient) evaluation function for feasible individuals,there is no uniform methodology for handling (i.e., evaluating) unfeasible ones.The simplest approach, incorporated by evaluation strategies and the versionof evolutionary programming (for numerical optimization problems), is toreject unfeasible solutions. But several other methods for handling unfeasibleindividuals have emerged recently. We review such methods (using a domain of nonlinear programming problems). For an excellent full review of constraint-handling techniques in genetic algorithm, the reader is referred to [4].
4.1. Methods based on penalty functions
The penalty function method is widely used in the mathematical program-ming literature. It essentially adds to the objective function some terms whichpunish a solution that is not feasible.
Fig. 1. Outline of GAs for optimization problems.394
M.S. Osman et al. / Appl. Math. Comput. 155 (2004) 391–405
A general mathematical problem is deﬁned as follows:Max
x
f
ð
x
Þ
s
:
t
:
x
¼ ð
x
1
;
x
2
;
. . .
;
x
n
Þ 2
X
R
n
g
i
ð
x
Þ
6
0
;
i
¼
1
;
2
;
. . .
;
k h
i
ð
x
Þ ¼
0
;
i
¼
k
þ
1
;
. . .
;
m
Let
F
R
n
be the feasible set for the above problem. Applying the idea of penalty functions, the above constrained optimization problem can be trans-formed into an unconstrained optimization problem. The objective function of the unconstrained optimization problem, which will be used as the ﬁtnessfunction in the associated genetic algorithm designed to solve the initial con-strained problem, has the following format:eval
ð
x
Þ ¼
f
ð
x
Þ
;
if
x
2
F
f
ð
x
Þ þ
penalty
ð
x
Þ
;
otherwise
where penalty (
x
) is zero, if no violation occurs, and is positive, otherwise.Usually, the penalty function is based on the distance of the solution form thefeasible region
F
, or on the eﬀort to ‘‘repair’’ the solution, i.e., to force it into
F
.The former case is the most popular one; in many methods a set of functions
f
j
ð
1
6
j
6
m
Þ
is used to construct the penalty, where the function
f
j
ð
x
Þ
measuresthe violation of the
j
th constraint in the following way:
f
j
ð
x
Þ ¼
max
f
0
;
g
j
ð
x
Þg
if 1
6
j
6
k
j
h
j
ð
x
Þj
if
k
þ
1
6
j
6
m
How the penalty function is designed and applied to infeasible solutions maydiﬀer in important details across problems.
4.1.1. Static penalty function
The static penalty function assumes that for every constraint we establish afamily of intervals which determine an appropriate penalty coeﬃcient
R
ij
[4]. Itworks as follows: (1) for each constraint, create several (
l
) levels of violation(these levels measure the degree of violation, e.g., slightly or heavily); (2) foreach level of violation and for each constraint, create a penalty coeﬃcient
R
ij
<
0 (
i
¼
1
;
2
;
. . .
;
l
,
j
¼
1
;
. . .
;
m
); higher degree of violation requires heavierpunishment (i.e., larger
R
ij
). The evaluation function has the following struc-ture:eval
ð
x
Þ ¼
f
ð
x
Þ þ
X
m j
¼
1
R
ij
f
2
j
ð
x
Þ
M.S. Osman et al. / Appl. Math. Comput. 155 (2004) 391–405
395

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