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JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 6, ISSUE 2, APRIL 2011 17

Adaptive SA for Economic Dispatch with Multiple Fuel Options R. K. Santhi and S. Subramanian Abstract— The Economic Dispatch with Multiple Fuel Options (EDMFO) is one of the important optimization problems in a power system. The objective of economic dispatch problem is to determine the optimal combination of power outputs for all generating units, which minimizes the total fuel cost while satisfying load demand and operating constraints. This makes the economic dispatch problem with multiple fuel options a non-linear constrained optimization problem. The generating units, particularly those that are supplied with multi-fuel sources (coal, nature gas, or oil), lead to the problem of determining the most economic fuel to burn. The cost curve of such generator is become highly nonlinear, containing discontinuities due to valvepoint loadings, the cost function is more realistically denoted as a segmented piecewise quadratic function. This paper addresses the Adaptive Simulated Annealing (ASA) algorithm for the solution of realistic economic dispatch problem. The performance of the algorithm for the solution of EDMFO is evaluated by implementing the algorithm with 10-unit economic dispatch problem considering both multi-fuel effects and valve-point loadings. Index Terms—Adaptive simulated annealing, Economic dispatch, Multiple fuel options, Valve point effect.

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1 INTRODUCTION

T

HE objective of the Economic Dispatch (ED) problem of electric power generation, whose characteristics are complex and highly nonlinear, is to schedule the committed generating unit outputs so as to meet the required load demand at minimum operating cost while satisfying all unit and system equality and inequality constraints. Improvements in scheduling the unit outputs can lead to significant cost savings. In traditional ED problems, the cost function of each generator is approximately represented by a simple quadratic function and is solved using mathematical programming based on several optimization techniques, such as Dynamic Programming (DP), linear programming, homogenous linear programming, and nonlinear programming techniques. However, none of these methods may be able to provide an optimal solution, for they usually get stuck at a local optimum [1]. Recently, as an alternative to the conventional mathematical approaches, modern heuristic optimization techniques such as Simulated Annealing (SA), Genetic Algorithm (GA), Evolutionary Programming (EP) and Particle Swarm Optimization (PSO) have been given much attention by many researchers due to their ability to find an almost global optimal solution [2], [3], [4], [5]. In the traditional economic dispatch problem, the cost function for each generator has been approximately represented by a single quadratic function, and the valvepoint effects were ignored. The generating units, particularly those that are supplied with multi-fuel sources (coal, nature gas, or oil), lead to the problem of determining the most economic fuel to burn. As fossil fuel cost increases, it becomes more necessary to have a good model for the generation cost,

and the loss of accuracy by using traditional single quadratic cost functions should not be neglected. Therefore, more accurate representation is required. Since the cost curve of a generator is highly nonlinear, containing discontinuities due to valve-point loadings, the cost function is more realistically denoted as a segmented piecewise quadratic function. Such a problem has been solved using the Hierarchical Method (HM) of Lagrangian multipliers method to find the incremental fuel cost for subsystems comprising sets of units [6]. The solution searches for the optimal for various choices of fuel and generation range of the units iteratively. Hopfield Neural Network (HNN) and improved Adaptive HNN (AHNN) approach have been applied to solve economic dispatch problem with multiple fuel options [7], [8]. The HNN suffers with slow convergence rate and normally takes a large number of iterations. A Hybrid real coded GA [HGA] method has been presented for solving EDMFO problem [9]. An enhanced Lagrangian neural network has been applied to solve the economic load dispatch problems with piecewise quadratic cost functions [10]. In this method the convergence speeds are enhanced by employing by momentum technique and providing criteria for choosing the learning rate. Economic dispatch solutions with piecewise quadratic cost functions has been solved by using Improved Genetic Algorithm (IGA) [11]. The heuristic search techniques such as EP [12] and PSO [13] have also been applied to solve EDMFO. This paper presents an Adaptive SA (ASA) based solution methodology for solving economic dispatch with multiple fuel options.

2 PROBLEM FORMULATION

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• R.K. Santhi, Professor of Electrical Engineering, Annamalai University, Annamalainagar-608002, Tamil Nafu, India. • S.Subramanian, Professor of Electrical Engineering, Annamalai University, Annamalainagar-608002, Tamilnadu, India.

A piecewise quadratic function is used to represent the inputoutput curve of a generator with multiple fuel option. A generator with k fuel options the cost curve is divided into k discrete regions between lower and upper bounds. In reality, the

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objective function of economic dispatch problem has nondifferential points according to valve-point loadings and multiple fuels. Therefore, the objective function should be composed of a set of non-smooth cost functions. In this chapter, two cases of cost functions are considered. First is the case with the multiple fuels problem where the objective function is expressed as the piecewise quadratic cost function. The other is the case with both valve-point effects and multiple fuels for the economic dispatch operation, where the objective function is represented as a set of piecewise superposition of sinusoidal functions and quadratic functions. The objective function of economic dispatch problem with is given by (1) Minimize FT = ∑ Fi (Pi )

valve-point effects and multiple fuel options makes the complication to find global optimum solution. The minimization of the generation cost is subjected to the following constraints: (i) Power balance constraint

2.1 Economic Dispatch Problem with Multiple Fuels A piecewise quadratic function is used to represent the inputoutput curve of a generator with multiple fuel option. A generator with k fuel options the cost curve is divided into k discrete regions between lower and upper bounds. The economic dispatch problem with piecewise quadratic function is defined as ai1 Pi 2 + bi1 Pi + ci1 , fuel 1, Pi min ≤ Pi ≤ Pi1  a P 2 + bi 2 Pi + ci 2 , fuel 2, Pi1 < Pi ≤ Pi 2 Fi ( Pi ) =  i 2 i M  2 max aik Pi + bik Pi + cik , fuel k , Pik −1 < Pi ≤ Pi

where Pimin and Pimax are the minimum and maximum power outputs of the i th unit.

$/h (2) where Fi(Pi) is the fuel cost function of ith unit, Pi is the power output of ith unit, and aik, bik, and cik are cost coefficients of the ith unit using the fuel type k, ei and fi are the fuel cost coefficients of generator i reflecting valve point effects.

2.2 Economic Dispatch Problem with Valve-Point Effects and Multiple Fuels The accurate and practical modeling of PED problem should include the valve-point effects and multiple fuel options in the problem formulation. Therefore, the total cost function should combine multiple fuel options and valve-point loading effects [11] and is formulated as follows:

a P2 + b P + c + e sin( f ( Pmin − P )) , i1 i i1 i1 i1 i1 i1  i1 i  fuel1, Pimin ≤ Pi ≤ Pi1  a P2 + b P + c + e sin( f ( Pmin − P )) , i2 i i2 i2 i2 i2 i2  i2 i  Fi ( Pi ) =  fuel2, Pi1 < Pi ≤ Pi2 M  a P2 + b P + c + e sin( f ( Pmin − P )) , ik i ik ik ik ik ik  ik i  fuelk , Pik−1 < Pi ≤ Pimax   $/h (3) The practical economic dispatch problem including

n

∑P = P i

(4)

D

i =1

where is PD the total system demand in MW. (ii) Generating capacity constraint

Pi min ≤ Pi ≤ Pi max

(5)

3 ADAPTIVE SA APPROACH TO EDMFO SA is a stochastic algorithm, requires a number of parameters to specify a cooling schedule. Most of the existing SA based ED approaches require time consuming trial and error testing procedure to obtain optimal cooling schedule. A solution strategy involving SA that adaptively adjusts the cooling schedule has been developed to obtain a robust solution for EDMFO is presented in this section.

3.1 Adaptive SA The SA is extensively used to solve ED problem [2]. This approach finds the optimal solution using point-by-point iteration rather than a search over a population of individuals. Though this method is simple to formulate, require lower memory requirement than that of GA, they suffer from huge computational burden and end up with consuming exhaustively large execution times due to the improper choice of cooling schedule. The performance can be improved by adaptively adjusting the cooling schedule during the iterative process. The adaptive cooling schedule reduces the number of iterations and thus improves the computational speed. In ED problem, the real power generation of all the committed generating units is considered as the decision variables. PG1

PG2

PG3

---

PGk

---

PGng

3.1.1 Adaptive Cooling Schedule Cooling schedule determines functional form of the change in temperature required in SA. It is used at the probability step in SA and therefore, it governs the move. A wise choice of annealing schedule can save computational time and can improve the quality of solution. A sincere effort is required to choose an optimal cooling schedule for a problem. The cooling schedule of SA has been based on the analogy with physical annealing. Therefore, initial

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temperature has been set high enough to accept all transitions, which means heating up substances till all the molecules are randomly arranged in liquid. It reduces the temperature by a constant cooling factor. Finally the temperature becomes very small and it does not search any smaller energy level. This proportional cooling schedule does not lead to equilibrium at low temperature. Therefore, there is a need for a small number of transitions to be sufficient to reach the thermal equilibrium. An adaptive cooling schedule adjusts the rate of temperature decrease based on the past history of the run and tries to keep the annealing temperature close to the equilibrium as well as reducing the number of transitions to reach equilibrium. It is given by

1 Tk 3 Tt +1 = Tt − M k σ 2 ( Tt )

(6)

Where, σ2= variance of the objective function at equilibrium

Mk =

F max + Tt ln( 1 + δ ) σ 2 ( Tt ) ln( 1 + δ )

Tt

(7)

Fmax = estimated maximum value of the objective function σ = a small real number

3.1.2 Constraint Handling Mechanism The objective of the EDMFO is to minimise total fuel cost, while satisfying unit and system constraints. The unit minimum and maximum loading limits are taken care of by enforcing these limits while generating values for the problem variables. But the power balance constraint is handled through a penalty function approach. Penalty terms are incorporated in the cost function and are set to increase the cost depending on the magnitude of the violation. The cost function is therefore formed by augmenting the EDMFO objective function, and the power balance constraint, using a penalty factor σ as Minimise ng

ng

i =1

i =1

COST = ∑ {Fi ( PGi )+ hi Ei ( PGi )} + η ∑ PGi − PD − PLOSS (8) A larger value in the range of several hundreds has been chosen for σ. During the search process, if the constraint is not satisfied for a random SA solution point, the mismatch is multiplied by the large σ value and is reflected in the augmented objective function. This will consequently increase the cost function value and the SA will accept this solution point based on the probability P (T).

3.2 Implementation of ASA for EDMFO The implementation of adaptive simulated annealing for economic dispatch problem with valve-point effects and multiple fuel options is detailed in this section. In economic dispatch problem including multiple fuel options, it is desired to operate all the committed units to meet the total demand PD at minimum cost with optimal choice of fuel for each unit. Step (i) The ASA algorithm consists of exploring the solution space starting from a randomly selected solution and generating a new one by perturbing it. At generation G = 1, a

candidate solution is generated randomly within the limits. These initial individual values are chosen at random from within user-defined bounds. The structure of an individual for economic dispatch problem is composed of set of elements (i.e., generation output of units). An array of control variable of randomly generated candidate solution can be represented as

PG = [(P1 P2 P3 L Pn )]

(6)

Step (ii) A candidate solution is evaluated using the fitness function of the problem to minimize the fuel cost function. The power balance constraint is augmented with the objective to form a generalized fitness function fk as given below n  n  f k = ∑ Fi ( Pi ) + µ ∑ Pi − PD  i =1  i =1 

2

(7)

where µ is penalty parameter. The penalty term reflects the violation of the equality constraint and assigns a high cost of penalty function to candidate point far from feasible region. The upper and lower generation limit of generating unit is violated then it can fixed in the bound range by forcing it to lower or upper limit. The fuel cost of individual generating unit is calculated by identifying the fuel options for individual units for candidate solution and by using the respective fuel cost coefficients. Step (iii) Generate a new candidate solution by perturbing the previous solution. If the fitness function of this new solution is equal to current fitness function accept this solution as the current solution to generate another new candidate solution. Step (iv) If the fitness of neighborhood solution is greater than the current fitness function, determine the probability function Pacc and check the following condition is satisfied or not Pacc