Author : Ganga Reddy Tankasala

International Journal of Scientific & Engineering Research Volume 3, Issue 1, January-2012

ISSN 2229-5518

Download Full Paper : PDF**Abstract— **This paper deals with optimization of fuel cost of coal fired generators of a modern power sytem. The conventional method of solving economic load dispatch (ELD) uses Newton Raphson, Gauss and Gauss Siedel techniques whose time of computation increases exponentially with the size. Inorder to overcome the dreawbacks of conventional methods, Artificial Intelligent (AI) techniques likes like Genetic Algorithm (GA), Nueral Networks (NN), Artificial Immune systems (AIS) and Fuzzy Logics etc… are used. One such AI technique used is Artificial Bee Colony optimization (ABC) inspired from the foraging behaviour of bees. The ABC is applied for ELD and compared with the other AI techniques. The results show that ABC promises global minimum of the solution while others may land in local minimum.

**Index Terms— **Artificial bee colony, Artificial intelligent techniques, Economic load dispatch, Genetic Algorithm, Power systems

**1 INTRODUCTION ** Artificial Bee Colony optimization algorithms are formulated on the basis of natural foraging behaviour of honey bees. ABC was first developed by Dr.Korba. Some artificial ideas are added to construct a robust ABC .Very unlike to classical search and optimization methods ABC starts its search with a random set of solutions (Colony size), instead of single solution just like GA. Each population member is then evaluated for the given objective function and is assigned fitness.The best fits are entertained for next generation while the others are discarded and compensated by a new set of random solutions in each generation. The only stopping criterion is the completion of maximum no of cycles or generations. At the end of cycles the solutions with best fit is the desired solution.

Economic Load Dispatch (ELD) is one of the important opti-mization problems in modern Energy Management Systems (EMS). ELD determines the optimal real power settings of generating units in order to minimize total fuel cost of thermal plants. Various mathematical programming methods and optimization techniques have previously been applied for solution of ELD. These include Lambda iteration method, participation factors method and gradient methods. ELD problems in practice are usually hard for traditional mathematical programming methodologies because of the equality and inequality constraints.

ABC is applied for solution of ELD.A generating unit based encoding scheme is used, however when applied to large size systems, the number of maximum iterations or generations has to be increased proportinally.The solution time grows approximately linearly with problem size rather than geometrically.

**2 ECONOMIC LOAD DISPATCH**2.1 Problem Formulation

The objective of Economic Load Dispatch (ELD) for power system consisting of coal fired thermal generating units is to find the optimal combination of power generations that minimizes the total fuel cost for generation while satisfying the specified equality and inequality constraints. The fuel cost function of the generator is modeled as a quadratic function of generator active powers (P). The minimization function ‘A’ can be obtained as sum of the fuel costs Fi of all the generating units.

Min A = ∑ Fi ∀ i ϵ (1, 2,3…, NG) (1)

Subjected to

∑ PGi = PD + Ploss ∀ i ϵ (1, 2,3…, NG) (2)

PGimin ≤ PGi ≤ PGimax ∀ i ϵ (1, 2, 3…, NG) (3)

The fuel cost of generating unit is given by

Fi = (ai + bi Pi + ciPi2 ) (4)

Where ai, bi, ci are cost coefficients of generating unit i, Pi or PGi is real power generation of unit ‘i’. PD is the total demand and Ploss represents the transmission losses. PGimin and PGimax are the minimum and maximum generation lim-its of ith unit.

This is a constrained optimization problem that may be solved using calculus methods that involve Legrange function. The necessary condition for the minimization of fuel cost is that the incremental cost rates of all the units be equal to some undetermined value Lambda (λ). Along with the above condition, the equality constraint, the sum of the power outputs must be equal to the combined power demand and losses. If transmission system losses are neglected, the equality constraint becomes, the sum of the power outputs must be equal to the total power demand by the load. Also, the power output of each unit must be with in its generation range.

2.2 Transmission system Losses

Since always transmission losses are involved with a network, in order to achieve exact ELD, transmission system losses must be taken into account.Using B-coefficients method, the network losses are expressed as a quadratic function of unit generations as

Ploss =ΣΣPi Bij Pj ∀ i,j ∈ [1,2,3, … NG] (5)

In (5) Bij are called as B-coefficients or loss coefficients which are constant under certain assumed conditions. The above loss formula is known as the George’s formula.

**3 ARTIFICIAL BEE COLONY FORAGING BEHAVIOUR**To find the optimal decision variables, to optimize an objective function and to satisfy the constraints, the variables are bounded to the limits. Eqn. (6) gives a function defined to [1] take care of variable bounds.

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