An Efficient Simulated Annealing Algorithm for Economic Load Dispatch Problems

This paper presents an efficient simulated annealing (SA) algorithm with a single decision variable to solve the economic load dispatch (ELD) problems. The philosophy involves the introduction of a new decision variable through a prudent mathematical transformation of the relation between the decision variable and the optimal generations. The objectives of ELD problems in electric power generation is to programmed the devoted generating unit outputs so as to meet the mandatory load demand at lowest amount operating cost while satisfying all units and system equality and inequality constraints. Global optimization approaches is inspired by annealing process of thermodynamics. The proposed SA algorithm presented here is applied to two case studies, which analyze power systems having three, and six generating units. The results determined by the proposed approach are compared to those found by conventional quadratic programming (QP) and genetic algorithm (GA).


Introduction
Economic load dispatch (ELD) is one of the most important problems in power system operation and planning.The main objective of ELD is to determine the optimal combination of power outputs of all generating units to meet the required load demand at minimum operating cost while satisfying system equality and inequality constraints [1].In this problem, fuel cost of generation is represented as cost curves and overall calculation minimizes the operating cost by finding a point where total output of generators equals to total system load that must be delivered plus losses.In the traditional ELD problem, the cost function for each generator has been approximately represented by a single quadratic function and is solved using mathematical programming based on the optimization techniques such as lambda-iteration method, gradient-based method, etc. [2].
Classical methods like Newton-based and gradient methods cannot perform very well for problems having highly nonlinear characteristics with large number of constraints and many local optimum solutions.Dynamic programming is one of the approached to solve non-linear and discontinuous ELD problem, but it suffers from problem of curse of dimensionality or local optimality [3].Methods based on artificial intelligence techniques, such as artificial neural  [4][5][6].However, neural network-based approaches may suffer from excessive numerical iterations, resulting in huge calculations.Heuristic search techniques, such as evolutionary programming [7], particle swarm optimization [8], genetic algorithms [9][10][11], differential evolution [12], tabu search [13] and ant colony optimization [14] have also been successfully applied to ELD problems.
Simulated annealing (SA) algorithm is a promising heuristic algorithm for handling the combinatorial optimization problems.It has been theoretically proved that SA algorithm converges to the optimal solution.Another strong feature of SA algorithm is that a complicated mathematical model is not required and the problem constraints can be easily incorporated.In power systems, SA has been applied to a number of power system optimization problems with impressive successes [15,16].
In this paper, the proposed SA is discussed to solve the ELD problem by considering the linear equality and inequality constraints for the three units and six units system and the results were compared with conventional quadratic programming and GA.The algorithm described in this paper is capable of obtaining optimal solutions efficiently.

Research Method 2.1. Economic Load Dispatch Formulation
The objective of an ELD problem is to find the optimal combination of power generations that minimizes the total generation cost while satisfying an equality constraint and inequality constraints.The fuel cost curve for any unit is assumed to be approximated by segments of quadratic functions of the active power output from the generating units.For a given power system network, the problem may be described as optimization (minimization) of total fuel cost as defined by (1) under a set of operating constraints.
where T F is total fuel cost of generation in the system ($/hr), a i , b i , and c i are the cost coefficient of the i th generator, P i is the power generated by the i th unit and N is the number of generators.
The cost is minimized subjected to the following power balance and generator capacity constraints.

Power Balance Constraint:
where P D is the total power demand and P Loss is total transmission losses.
The transmission loss P Loss can be calculated by using B matrix technique and is defined by (3) as, where B ij 's are the elements of loss coefficient matrix B. The conditions for optimality can be obtained by using Lagrangian multipliers method and Kuhn Tucker conditions as follows [1]: The following steps are followed to solve the economic load dispatch problem with the constraints: Step-1: Allocate lower limit of each plant as generation, evaluate the transmission loss and incremental loss coefficients and update the demand.
Step-2: Apply quadratic programming to determine the allocation new i P of each plant.If the generation hits the limit, it should be fixed to that limit and the remaining plants only should be considered for next iteration.
Step-3: Check for the convergence: where  is the tolerance.Repeat until the convergence criteria is meet.
A brief description about the quadratic programming method is presented in the next section.

Quadratic Programming Method
A linearly constrained optimization problem with a quadratic objective function is called a quadratic programming (QP) [17].Due to its numerous applications; quadratic programming is often viewed as a discipline in and of itself.Quadratic programming is an efficient optimization technique to trace the global minimum if the objective function is quadratic and the constraints are linear.Quadratic programming is used recursively from the lowest incremental cost regions to highest incremental cost region to find the optimum allocation.Once the limits are obtained and the data are rearranged in such a manner that the incremental cost limits of all the plants are in ascending order.
The general quadratic programming can be written as: where c is an n-dimensional row vector describing the coefficients of the linear terms in the objective function, and Q is an (n × n) symmetric matrix describing the coefficients of the quadratic terms.If a constant term exists it is dropped from the model.As in linear programming, the decision variables are denoted by the n-dimensional column vector x, and the constraints are defined by an (m× n) A matrix and an m-dimensional column vector b of righthand-side coefficients.We assume that a feasible solution exists and that the constraint region is bounded.When the objective function f(x) is strictly convex for all feasible points the problem If there are only equality constraints, then the QP can be solved by a linear system.Otherwise, a variety of methods for solving the QP are commonly used, namely; interior point, active set, conjugate gradient, extensions of the simplex algorithm etc.The direction search algorithm is minor variation of quadratic programming for discontinuous search space.For every demand the following search mechanism is followed between lower and upper limits of those particular plants.For meeting any demand the algorithm is explained in the following steps: (i) Assume all the plants are operating at lowest incremental cost limits.(ii) Substitute and make the objective function quadratic and make the constraints linear by omitting the higher order terms.(iv) Solve the ELD problem using quadratic programming recursively to find the allocation and incremental cost for each plant within limits of that plant.(v) If there is no limit violation for any plant for that particular piece, then it is a local solution.(vi) If for any allocation for a plant, it is violating the limit.It should be fixed to that limit and the remaining plants only should be considered for next iteration.(vii) Repeat steps 2, 3, and 4 till a solution is achieved within a specified tolerance.

Simulated Annealing Algorithm 2.3.1. Overview
Simulated annealing is an optimization technique that simulates the physical annealing process in the field of combinatorial optimization.Annealing is the physical process of heating up a solid until it melts, followed by slow cooling it down by decreasing the temperature of the environment in steps.At each step, the temperature is maintained constant for a period of time sufficient for the solid to reach thermal equilibrium.At any temperature T, the thermal equilibrium state is characterized by the Boltzmann distribution.This distribution gives the probability of the solid being in a state i with energy E i at temperature T as where k is a constant.
Metropolis et al. [18] proposed a Monte Carlo method to simulate the process of reaching thermal equilibrium at a fixed value of the temperature T. In this method, a randomly generated perturbation of the current configuration of the solid is applied so that a trial configuration is obtained.Let E c and E t denote the energy level of the current and trial configurations, respectively.If c t E E  ; then a lower energy level has been reached, and the trial configuration is accepted and becomes the current configuration.On the other hand, if the trial configuration is accepted as current configuration with probability proportional to exp(∆E/T), ∆E = E t -E c .The process continues until the thermal equilibrium is achieved after a large number of perturbations, where the probability of a configuration approaches Boltzmann distribution [19,20].
By gradually decreasing the temperature T and repeating Metropolis simulation, new lower energy levels become achievable.As T approaches zero least energy configurations will have a positive probability of occurring.The flowchart of simulated annealing (SA) is shown in Figure 1.The general algorithm of SA can be described in steps as follows: Step 1: Set the initial value of C p0 and randomly generate an initial solution x initial and calculate its objective function.Set this solution as the current solution as well as the best solution, i.e. x initial = x current = x best Step 2: Randomly generate an n 1 of trial solutions in the neighborhood of the current solution.
Step 3: Check the acceptance criterion of these trial solutions and calculate the acceptance ratio.If acceptance ratio is close to 1 go to Step 4; else set C p0 = α.C p0 ; α > 1; and go back to Step 2.
Step 4: Set the chain counter k = 0.
Step 5: Generate a trial solution x trial .If x trial satisfies the acceptance criterion set x current = x trial , J(x current ) = J(x trial ) and go to Step 6; else go to Step 6.
Step 6: Check the equilibrium condition.If it is satisfied go to Step 7; else go to Step 5.
Step 7: Check the stopping criteria.If one of them is satisfied then stop; else set k = k + 1 and C p = μ.C p ; μ < 1; and go back to Step 5.

Proposed SA
In all the existing SA algorithm based approaches for solving ELD problems, the real power generation of all generating units are considered as the decision variables that makes the size of the problem vary large, slow down the speed of these algorithms and hence not suitable for systems having larger number of generating units.In the proposed approach, the penalty factor λ of the classical λ -iteration is considered as the only decision variable irrespective of the number of generating units.The real power of all the generating plants are considered as the problem dependant variables and expressed as a function of λ.The real power generations are computed using (5) for each λ value obtained during the SA iterations.
The lower and upper limits of the decision variable-λ depend on the minimum and maximum power demands that the system can supply.The first step in obtaining these values is to compute the lower and upper incremental cost values by substituting the respective to real power limits in ( 5) for all the plants as The next step is choosing the lowest and highest incremental cost value, obtained from (11), as the limits for λ.The SA searches for the optimal solution by minimizing a cost function.In the proposed formulation, the net fuel cost of all the generating plant is considered as the cost function.However, a penalty term is included in the cost function to handle the explicit power balance constraint.The penalty term increases the cost function for infeasible solutions.The cost function is therefore built as a blend of fuel cost function and the power balance constraint through the use of a penalty factor as The number of decision variables in this formulation is always one, whereas the existing SA based approaches require the generation of all the plants as the variables.This reduction in decision variables will reduce the overall computational burden and improves the convergence rate.The algorithm of the proposed SA for solving the ELD problem is outlined.1. Read the input data of the ELD problem 2. Set k = 0 3. Choose initial temperature T t , cooling coefficient α, number of iterations for each temperature N t and maximum number of iterations N max .4. Choose a random start point λ 0 within the specified range 5. Repeat the following: a. Select a random point λ k from the neighbourhood of λ 0 within the specified range b.Solve (5) for P i while imposing the limits given by (4) c.Calculate If converged, stop and print the ELD corresponding to the λ 0 .Otherwise, set k = k + 1 6.Reduce the temperature by the factor α and go to step 5.

Results and Analysis
To verify the feasibility and effectiveness of the proposed SA algorithm, two different power systems were tested consisting of three and six generating units [21,22].Results of the proposed SA algorithm are compared with conventional quadratic programming (QP) and genetic algorithm (GA) methods.A reasonable B-loss coefficients matrix of power system network has been employed to calculate the transmission losses.The software has been written in the MATLAB-7 language.

Case 1: 3-Generating Units
In this case, a simple power system consist of three-generating units is used to demonstrate how the work of the proposed approach.Characteristics of thermal units are given in Table 1, followed by coefficient matrix B ij losses.For the system loads of 275 MW, 300 MW, 350 MW, and 400 MW, the conventional QP method is applied and results obtained are shown in Table 2.The optimal scheduling of generators obtained by the proposed SA algorithm for three unit systems is shown in Table 3.The comparison of results between conventional QP method and the proposed SA method are shown in Table 4.The comparison of results shows that the proposed SA algorithm is better than conventional QP method for each loading and it is very reliable in the aspect of solution quality.Case 2: 6-Generating Units In this case, a standard of six-generating units (IEEE 30 bus test systems) is used to demonstrate how the work of the proposed approach, as shown in Figure 2. Characteristics of thermal units are given in Table 5, followed by coefficient matrix B ij losses.
The simulation results using the proposed SA algorithm are shown in Table 6 and Table 7 respectively for the load variation of 700 MW and 800 MW.The simulation results show that the generation outputs of each unit obtained were smaller than those of the genetic algorithm (GA), which is taken from [22].Further, as a result, there was some reduction of the total generation cost and transmission losses.

Conclusion
In this paper, an efficient simulated annealing (SA) algorithm with a single decision variable has been successfully introduced to obtain the optimum solution of economic load dispatch problem.The SA method has been tested on two test cases consisting of 3generating units and 6-generating units systems and the results are compared to those of the conventional quadratic programming method and the GA method.Test results have shown that the proposed method can provide better solution than above mentioned methods.

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ISSN: 1693-6930 TELKOMNIKA Vol.11, No. 1, March 2013 : 37 -46 40 has a unique local minimum which is also the global minimum.A sufficient condition to guarantee strictly convexity is for Q to be positive definite.

TELKOMNIKA,
Vol. 11, No. 1, March 2013 : 37 -46 42 Else select a random number  in the range [0otherwise discard the trial point e.Check convergence by comparing the number of iterations k with N max

2.1.2. The Generator Capacity Constraint:
, min and P i, max are the minimum and maximum power output of the i th unit.

Table 1 .
Generating unit capacity and coefficients

Table 2 .
Economic dispatch for 3-generating units using QP method

Table 3 .
Economic dispatch for 3-generating units using the proposed SA

Table 4 .
Comparison of results between QP and the proposed SA for 3-generating units

Table 5 .
Generating unit capacity and coefficients