23-02-2013, 02:44 PM
Solving economic load dispatch problems in power systems using chaotic and Gaussian particle swarm optimization approaches
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Abstract
The objective of the Economic Dispatch Problems (EDPs) of electric power generation is to schedule the committed generating units
outputs so as to meet the required load demand at minimum operating cost while satisfying all units and system equality and inequality
constraints. Recently, global optimization approaches inspired by swarm intelligence and evolutionary computation approaches have proven
to be a potential alternative for the optimization of difficult EDPs. Particle swarm optimization (PSO) is a population-based stochastic
algorithm driven by the simulation of a social psychological metaphor instead of the survival of the fittest individual. Inspired by the
swarm intelligence and probabilities theories, this work presents the use of combining of PSO, Gaussian probability distribution functions
and/or chaotic sequences. In this context, this paper proposes improved PSO approaches for solving EDPs that takes into account nonlinear
generator features such as ramp-rate limits and prohibited operating zones in the power system operation. The PSO and its variants
are validated for two test systems consisting of 15 and 20 thermal generation units.
Introduction
The Economic Dispatch Problems (EDPs) is to determine
the optimal combination of power outputs of all generating
units to minimize the total fuel cost while satisfying
the load demand and operational constraints [1].
In a liberalized electricity market, the optimization of
economic dispatch is of economic value to the network
operator. The economic dispatch is a relevant procedure
in the operation of a power system. Over the past years,
many optimization methods have been proposed in the literature.
A spectrum of the advances in economic dispatch
is well discussed in [2–28]. When compared with the conventional
(classical) techniques [4–13].
Formulation of an EDP with generator constraints
The EDP is to find the optimal combination of power
generation that minimizes the total fuel cost while at thermal
power units satisfying the total demand subjected to
the operating constraints of a power system with a defined
interval (typically 1 h). The essential operation constraints
are the power balance constraint, where the total generated
power must be equals to the load demands plus the transmission
losses on the electrical network, and the power
limit constraints, where individual generator units must
be operated within their specified range.
Optimization methodology based on PSO for the EDP
Social insect societies are distributed systems, which
despite the simplicity of their individuals, present a highly
structured social organization. As a result of this organization,
insect societies can accomplish complex tasks that, in
some cases, far exceed the individual capabilities of a single
insect, as ants for example. The field of swarm intelligence
is an emerging research area that presents features of selforganization
and cooperation principles among group
members bio-inspired on social insect societies. Swarm
intelligence is inspired by nature, based on the fact that
the live animals of a group contribute with their individual
experiences to the group, rendering it stronger to face other
groups. The most familiar representatives of swarm intelligence
in optimization problems are the food-searching
behavior of ant colonies [45], particle swarm optimization
[46], artificial immune systems [47], and bacterial foraging
[48].
The proposal of PSO algorithm was put forward by several
scientists who developed bio-inspired computational
simulations of the movement of organisms such as flocks
of birds and schools of fish. Such simulations were heavily
based on manipulating the distances between particles, i.e.,
the synchrony of the behavior of the swarm was seen as an
effort to keep an optimal distance between them. In the
next subsection, the fundamentals and implementation
details about the PSO are described.
New PSO approaches based on Gaussian distribution
and/or chaotic sequences
Coelho and Krohling [40] proposed the use of a truncated
Gaussian and Cauchy probability distribution to generate
random numbers for the velocity updating equation of
PSO. In this paper, new approaches to PSO are proposed
which are based on Gaussian probability distribution linked
with chaotic sequences. Firstly, random numbers are generated
using the Gaussian probability distribution and/or
chaotic sequences in the interval [1, 1], and then mapped
to the interval [0, 1]. The use of chaotic sequences in PSO
could be useful to escape from local optima, while the
Gaussian distribution could provide a faster convergence
in local search.
An essential feature of chaotic systems is that small
changes in the parameters or the starting values for the
data lead to vastly different future behaviors, such as stable
fixed points, periodic oscillations, bifurcations, and ergodicity.
These behaviors can be analyzed based on the meaning
of Lyapunov exponents and the attractor theory
[51,52].
Conclusion
This paper has demonstrated the feasibility of employing
modified PSO approaches for efficient solving of EDPs
with generator constraints. PSO is an effective optimization
method that belongs to the category of evolutionary methods.
Its development is based on the observations of social
behavior of animals such as bird flocking, fish schooling,
and swarm theory. Like evolutionary algorithms, PSO
technique conducts search using a population of particles,
corresponding to individuals. Each particle represents a
candidate solution to the problem at hand.
In relation to the procedure involved in solving the
EDP, the simulation results achieved by PSO(4) and
PSO(3) to the case studies I and II, respectively, were better
than the presented results in literature. The results of these
simulations with modified PSO approaches are very
encouraging and represent an important contribution to
PSO algorithm setups.