07-11-2012, 12:37 PM
A Genetic Algorithm for Solving the Optimal Power Flow Problem
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Abstract
This paper presents solution of optimal power flow problem of large
distribution systems via a simple genetic algorithm. The objective is to
minimize the fuel cost and keep the power outputs of generators, bus voltages,
shunt capacitors/reactors and transformers tap-setting in their secure limits.
CPU times can be reduced by decomposing the optimization constraints to
active constraints manipulated directly by the genetic algorithm, and passive
constraints maintained in their soft limits using a conventional constraint load
flow. The IEEE 30-bus system has been studied to show the effectiveness of
the algorithm.
INTRODUCTION
The optimal power flow has been frequently solved using classical optimization
methods. The OPF has been usually considered as the minimization of an objective function
representing the generation cost and/or the transmission loss. The constraints involved are the physical laws governing the power generation-transmission systems and the operating
limitations of the equipment.
Effective optimal power flow is limited by (i) the high dimensionality of power systems and
(ii) the incomplete domain dependent knowledge of power system engineers. The first
limitation is addressed by numerical optimization procedures based on successive
linearization using the first and the second derivatives of objective functions and their
constraints as the search directions or by linear programming solutions to imprecise models
[1-4]. The advantages of such methods are in their mathematical underpinnings, but
disadvantages exist also in the sensitivity to problem formulation, algorithm selection and
usually converge to a local minimum [5]. The second limitation, incomplete domain
knowledge, precludes also the reliable use of expert systems where rule completeness is not
possible.
Objective Function
The most commonly used objective in the OPF problem formulation is the
minimisation of the total cost of real power generation. The individual costs of each
generating unit are assumed to be function, only, of active power generation and are
represented by quadratic curves of second order. The objective function for the entire power
system can then be written as the sum of the quadratic cost model at each generator.
Types of equality constraints
While minimizing the cost function, its necessary to make sure that the generation still
supplies the load demands plus losses in transmission lines. Usually the power flow
equations are used as equality constraints.
Description of Genetic Algorithms
The genetic algorithms are part of the evolutionary algorithms family, which are
computational models, inspired in the Nature. Genetic algorithms are powerful stochastic
search algorithms based on the mechanism of natural selection and natural genetics. GAs
works with a population of binary string, searching many peaks in parallel. By employing
genetic operators, they exchange information between the peaks, hence reducing the
possibility of ending at a local optimum. GAs are more flexible than most search methods
because they require only information concerning the quality of the solution produced by each
parameter set (objective function values) and not lake many optimization methods which
require derivative information, or worse yet, complete knowledge of the problem structure
and parameters.
There are some difference between GAs and traditional searching algorithms [8][9].
They cold be summarized as follows:
• The algorithms work with a population of string, searching many peaks in parallel, as
opposed to a single point.
• GAs work directly with strings of characters representing the parameters set not the
parameters themselves.
• GAs use probabilistic transition rules instead of deterministic rules.
• GAs use objective function information instead of derivatives or others auxiliary knowledge.
• GAs have the potential to find solutions in many different areas of the search space
simultaneously.
GA Applied to optimal power flow
A simple Genetic Algorithm is an iterative procedure, which maintains a constant size
population P of candidate solutions. During each iteration step (generation) three genetic
operators (reproduction, crossover, and mutation) are performing to generate new populations
(offspring), and the chromosomes of the new populations are evaluated via the value of the
fitness witch is related to cost function. Based on these genetic operators and the evaluations,
the better new populations of candidate solution are formed.
Chromosome coding and decoding
GAs works with a population of binary string, not the parameters themselves. For
simplicity and convenience, binary coding is used in this paper. With the binary coding
method, the active generation power set of 9-bus test system (Pg1,Pg2 and Pg3) would be
coded as binary string of O’s and 1’ with length B1, B2, and B3 (may be different),
respectively. Each parameter Pgi have upper bound Ui and lower bound Li .The choice of B1,
B2, and B3 for the parameters is concerned with the resolution specified by the designer in the
search space.