[Reddy
,
4(2): February, 2015] ISSN: 2277-9655
Scientific Journal Impact Factor: 3.449
(ISRA), Impact Factor: 2.114
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IJESRT
INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH
TECHNOLOGY
ECONOMIC LOAD DISPATCH PROBLEM WITH VALVE POINT EFFECT USING
FIREFLY ALGORITHM
Dinakara Prasad Reddy P*, M C V Suresh
ABSTRACT
This paper proposes application of Firefly algorithm for solving economic load dispatch problem with valve point
effect. Firefly algorithm is a metaheuristic population based method which was based on the flashing patterns and
behaviour of fireflies. The proposed approach has been examined and tested with the numerical results of economic
load dispatch problems with three and five generating units with valve point loading without considering prohibited
operating zones and ramp rate limits. The results of the proposed Firefly algorithm are compared with that of other
techniques such as lambda iteration and ABC.
KEYWORDS: Economic dispatch, Firefly algorithm, Artificial Bee Colony algorithm, valve point loading,
Mathematical modelling, PSO, GA.
INTRODUCTION
Economic load dispatch (ED) is an important task in
the power plants operation which aims to allocate
power generations to match load demand at minimal
possible cost while satisfying all the power units and
system constraints. The complexity of the problem is
due to the nonlinear and non-smooth characteristics of
the input-output curves of the generators, because of
valve-point effect, ramp rate limits and prohibited
operating zones. The mathematical programming
based optimization methods such as lambda iteration,
base point participation method, Gradient and
Newton’s methods can solve successfully t he convex
ED problems [1-4]. But unfortunately, these methods
are ineffective to handle the non convex ED problems
with non-differentiable characteristics due to high
complexity. Dynamic programming can solve such
type of problem, but it suffers from curse of
dimensionality. Hence for optimal solution this
problem needs a fast, robust and accurate solution
methodology. Now days heuristic search methods
such as simulated annealing (SA) [5]-[6], genetic
algorithm (GA) [7],[14-16], evolutionary
programming (EP) [8], particle swarm optimization
(PSO) [9], Biogeography based optimization [10],
chaotic ant swarm optimization [11] and firefly
algorithm [12] are employed to solve the ED problems
All the approaches have achieved success to a certain
extent.
This paper presents the application of proposed Firefly
algorithm [13] to economic load dispatch problem
with valve point loading.
ECONOMIC LOAD DISPATCH PROBLEM
The economic load dispatch problem is defined as to
minimize the total operating cost of a power system
while meeting the total load plus transmission losses
with in the generator limits. Mathematically, the
problem is defined as to minimize equation (1)
subjected to the energy balance equation given by (2)
and the inequality constraints given by equation (3).
𝐹𝑖(𝑃𝑖) = ∑(𝑎𝑖 𝑃𝑖2+ 𝑏𝑖𝑃𝑖+ 𝑐𝑖)
𝑁𝐺
𝑖=1 (1)
∑𝑃𝑖
𝑁𝐺
𝑖=1 = 𝑃𝐷+ 𝑃𝐿 (2)
𝑃𝑖𝑚𝑖𝑛 ≤ 𝑃𝑖≤ 𝑃𝑖𝑚𝑎𝑥 (i=1, 2, 3... NG) (3)
Where 𝑎𝑖, 𝑏𝑖 and 𝑐𝑖 are cost coefficients
𝑃𝐷 is load demand
𝑃𝑖 is real power generation
𝑃𝐿 is power transmission loss
NG is number of generators
One of the important, simple but approximate methods
of expressing transmission loss as a function of
generator powers is through B- coefficients. The
general form of the loss formula using B- coefficients
is 𝑃𝐿=∑ ∑ 𝑃𝑖𝐵𝑖𝑗𝑃𝑗
𝑁𝐺
𝑗=1
𝑁𝐺
𝑖=1 MW (4)
Where 𝑃𝑖, 𝑃𝑗 are real power injections at the ith, jth
buses 𝐵𝑖𝑗 are loss coefficients
The above loss formula (4) is known as George’s
formula.
In normal economic load dispatch problem the input –
output characteristics of a generator are approximated
using quadratic functions, under the assumption that
the incremental cost curves of the units are
monotonically increasing piecewise-linear functions.
