2009 International Conference of Soft Computing and Pattern Recognition

Transcription

1 2009 International Conference of Soft Computing and Pattern Recognition Implementing Particle Swarm Optimization to Solve Economic Load Dispatch Problem Abolfazl Zaraki Control and instrumentation Engineering Dept-CIED Universiti Teknologi Malaysia, UTM Johor Bahru, Malaysia Abstract Economic Load Dispatch (ELD) is one of an important optimization tasks which provides an economic condition for a power systems. In this paper, Particle Swarm Optimization (PSO) as an effective and reliable evolutionary based approach has been proposed to solve the constraint economic load dispatch problem. The proposed method is able to determine, the output power generation for all of the power generation units, so that the total constraint cost function is minimized. In this paper, a piecewise quadratic function is used to show the fuel cost equation of each generation units, and the B-coefficient matrix is used to represent transmission losses. The feasibility of the proposed method to show the performance of this method to solve and manage a constraint problems is demonstrated in 4 power system test cases, consisting 3,6,15, and 40 generation units with neglected losses in two of the last cases. The obtained PSO results are compared with Genetic Algorithm (GA) and Quadratic Programming (QP) base approaches. These results prove that the proposed method is capable of getting higher quality solution including mathematical simplicity, fast convergence, and robustness to solve hard optimization problems. Keywords-Economic Load Dispatch (ELD); Particle Swarm Optimization (PSO); Quadratic Cost Function; Generation unit; transmission losses. I. INTRODUCTION Electrical power systems are capable to produce sufficient electrical power to feed a bounded and certain range of load demand. In all practical power systems, minimizing the total operation costs is very important. Thus, the ELD technique is applied for allocating power generation among the committed units, so that the total generation cost of the system and also transmission power losses are minimized, while satisfying all the constraints [1]. The input-output characteristic of generators in a power system are non linear, and there are some multiple local minimum points and a global minimum point in this curve, therefore, the characteristic of ELD problem (objective function) are multi model and highly non linear. The common mathematical practices to solve constraint optimization problems such as ELD problem are lambda iteration method, base point and participation factor method, gradient method, etc. These techniques require the incremental cost curves to be monotonically increasing. Mohd Fauzi Bin Othman Control and instrumentation Engineering Dept-CIED Universiti Teknologi Malaysia, UTM Johor Bahru, Malaysia fauzi@fke.utm.my To use these mathematical methods in optimization problems it is necessary to select a suitable initial starting point for their algorithms. The wrong initial starting point may cause the divergence or convergence of the algorithm to some local optimum points rather than the global point. [1, 3, 8, 15]. It has been found that Newton based algorithms will face problem in having large number of inequality constraints. Beside it have been approved short coming of Linear programming methods is associated with the piecewise linear cost approximation. On the other hand Non linear programming methods have also been applied to solve the convergence problem. Evolutionary Programming (EP) technique, evolutionary computation technique such as Genetic Algorithm (GA), Artificial Neural Network (ANN), Particle Swarm Optimization (PSO), etc, are some of the proposed methods to solve ELD problem for a power system [2, 4]. The proposed PSO method is composed of a set of particles called individuals, which are able to follow a certain algorithm to obtain the best solution for a optimization problem. These particles explore the search space with different velocities and positions. Each particle of swarm presents a potential solution for the optimization problem. The performance of individuals, evaluated by a fitness function (objective function). The particle swarm's algorithm is able to obtain local optimum points for multi variable optimization problems, in the multi dimension search space [5]. II. PROBLEM FORMULATION The main goal in this optimization problem is to obtain a particular set of points, including all outputs of the power generation units, such that all equality and inequality constraints are satisfied. In addition, the total cost function is minimized. In this paper, the equality and inequality constraints indicate the real power balance and limitation of power generation of each unit, respectively. Some of the other constraints including voltage level and security are assumed to be constant. Equation (1) denotes the total fuel cost for a power system which is an equal summation of all generation units cost functions, in a power system /09 $ IEEE DOI /SoCPaR

2 By approximating the fuel cost for each generation unit( F P ), to a quadratic function, (2) can be obtained, thus the total cost function will be changed into the following equation 2 : Output power generation of unit i.,, : Fuel cost coefficients of unit i. The constraints considered in this paper include: Equality constraint: N : Total real power is demand : Total power losses. 3 By using the matrix form, the losses formula can be shown as in the following equation. 4 P is matrix of the output powers of units. B is square matrix of transmission coefficients. The method used in this paper for considering the transmission losses, has been developed by Kron and adopted by Kirchmayer, which is the loss coefficient method [6, 8, 13]. The output power of each generation unit is bounded between two limitations III. PARTICLE SWARM OPTIMIZATION The PSO algorithm which was first proposed by Kennedy and Eberhart has been inspired by the Social behavior of a simple system (flock of birds). This algorithm can be effectively useful in solving many non linear hard optimization problems [5]. Unlike the mathematical methods for solving optimization problems, this algorithm does not need any gradient information about objective or error function and it can obtain the best solution independently [7]. According to the PSO algorithm, a swarm of particles that have predefined restrictions starts to fly on the search space. The performance of each particle is evaluated by the value of the objective function and considering the minimization problem, in this case, the particle with lower value has more performance. The best experiences for each particle in iterations is stored in its memory and called personal best (Pbest). The best value of Pbests (less valus) in iterations determines the global best (Gbest). By using the concept of Pbest and Gbest the velocity of each particle is updated in (5) : Particle velocity at current iteration (k+1) Particle velocity at iteration k r1, r2: random number between [0, 1] c1, c2: acceleration constant After this, particles fly to a new position: : Current particle position at iteration k+1 Particle position at iteration k Particle velocity at iteration k+1 a) Inertia weight IV. INCREASE CONVERGENCE RATE 5 In attempting to increase the rate of convergence of the standard PSO algorithm to global optimum, the inertia weight is proposed in the velocity equation [10, 11].By using the new equation for velocity, according to this modification, the suggested particle velocity in (5) will be changed to: For 1,.,, denote the minimum and maximum output power generation of unit i W is the inertia weight. 61

3 Applying this factor in (5) causes some of the particle's velocity in the previous iteration to remain in the new iteration. In order to use the inertia weight in this paper, a descending linear function is used. The best range for changing this function value for the convergence and obtaining the best possible solution is between 0.9 and 0.4. Using the inertia weight in velocity equation enables the swarm to fly in larger area of the search space (W = 0.9) and at the end of the iterations, the search space will be smaller (W = 0.4).By using the inertia weight the chance to obtain a best solution for a optimization problem will be more. In general, a linear descending function for inertia weight equation is shown in the following equation [13, 18]. W : inertia weight factor W max : maximum value of weighting factor W min : minimum value of weighting factor Iter max : maximum number of iteration Iter: current number of iteration b) initial global best position One of the important things to increase the convergence rate is choosing a correct initial position for global minimum. In this paper, the initial global best is placed in a certain area of the search space. This location is determined regarding to the constraints. In general, the PSO flowchart for unconstraint optimization is shown below: Stopping criteria The iterative process will be stopped under the supervision of a change in the production costs value, as accuracy is desired or the maximum number of iteration is reached. In this paper, the PSO algorithm will be stopped when the maximum number of iteration is reached. In order to use this standard PSO algorithm, unconstraint cost function has been derived by Lagrange method. The unconstraint total cost function equation is shown below N CF P KDP L P K: Lagrange multiplier N 7 The maximum and minimum velocity are controlled during the iteration, by assigning the constant values to those particles that want to exceed the [V min, V max ] boundary. The used PSO parameters in this paper are shown in the following table. TABLE I. Pso Parameters Parameters Value V d(min), V d(max) - /2, /2 Population size 50 Max inertia weight 0.9 Min inertia weight 0.4 Initial velocity 0 Initial position random C1, C2 2 V. SETTING POPULATION SIZE If the population size is too small, then an insufficient number of particles are produced and the algorithm may not give the best possible solution since some of the best positions are missed. If the population size is too large, the algorithm is very slow and inefficient. As shown in the table above, the population size in this article is 50. [1] VI. RESULTS AND DISCUSSION In order to show the feasibility of the proposed PSO, four test cases including 3,6,15 and 40 thermal power plants are tested. For the first two cases, the transmission losses are considered and in the second two cases, they are neglected. The desired load demand for these methods are 150, 700, 2215, and in MW, respectively. Matrix B as losses coefficient matrix of power system to show the transmission line loss for an interconnection power system for test cases has been shown below. For all test cases, the cost function for a generator has been defined by the quadratic equation below Figure1. Standard PSO Flowchart 62

4 Test case 1: This test case is adopted from [6] and the cost function and transmission losses coefficient are as below TABLE II. Cost Function Coefficient, Test Case 1 Plant No ($/MW2) ($/MW) ($) (Mw) (MW) The transmission losses coefficient matrix is = The obtained output powers for each generator by three various methods for Pd=100MW are: TABLE III. Outputs Power Generation, Test Case 1 units QP GA PSO P P P Test case 2: This test case is adopted from [6] and the cost function and transmission losses coefficient are shown below: TABLE IV. Cost Function Coefficient, Test Case 2 Plant No ($/MW2) ($/MW) ($) (Mw) (MW) And the transmission losses coefficient matrix is 0.14 = The obtained output powers for each generator by three various methods for Pd=700MW are: TABLE V. Outputs Power Generation. Test Case 2 units QP GA PSO P P P P P P Test case 3: This test case is adopted from [16] and the cost function coefficients are available for reference. The obtained output powers (Neglected Losses) with neglected losses for each generator by three various methods for Pd=2215MW are: TABLE VI. Outputs Power Generation. Test Case 3 units QP GA PSO P P P P P P P P P P P P P P P Test case 4: This test case is adopted from [17] and the cost function coefficients are available in this reference. The obtained output powers (Neglected Losses) for each generator by three various methods for Pd=10500MW are shown below, and because of huge data, only some of the selected outputs have been demonstrated. TABLE VII. Outputs Power Generation. Test Case 4 Units QP GA PSO P P P P P P P P P P P The algorithm is used for the PSO and random initialization for particles, in order to obtain the final cost value for each test case. The program has been run three 63

5 times and the average results are demonstrated. The results for all the test cases with various load demands are then shown below. Also by using the loss formula coefficients, the transmission loss has been calculated and shown below. TABLE VIII. # Test Case Comparison Results of Three Various Methods Cost($/h) T.losses PLosses(MW) QP GA PSO 1 3 units Pd=150 Yes units Pd=700 Yes units Pd=2215 No units Pd=10500 No A short comparison between the results of total fuel cost values in terms of ($/h) have been done. This diagram includes the result of three various methods for different test cases with various load demands. Figure 2. Cost Fuction Comparision for test cases VII. CONCLUSION The problem of economic load dispatch with some constraints has been investigated in this paper. The ELD problem has also been solved by quadratic programming, genetic algorithm, and particle swarm optimization methods for 4 various test cases and load demands, and all the results obtained from three various methods have been compared. The simulation results have shown that the proposed method is capable in obtaining better minimization results, although the difference is small, it will be huge in a big power system. Also, there is no mathematical complexity in the proposed algorithm since it has a simpler structure than other proposed methods. VIII. ACKNOWLEDGMENTS The author wishes to thank University Technology Malaysia (UTM) for facilitating the supporting equipments and also for supporting the overall process. IX. REFERENCES [1]. Vanaja, B.; Hemamalini, S.; Simon, S.P, Artificial Immune based Economic Load Dispatch with valve-point effect, TENCON , TENCON2008. IEEE Region 10Conference Nov Page(s):1-5. [2]. Fukuyama, Y.; Ueki, Y, An application of artificial neural network to dynamic economic load dispatching, Neural Networks to Power Systems, 1991., Proceedings of the First International Forum on Applications of July 1991 Page(s): Digital Object Identifier /ANN [3]. Ling, S.H.; Lu, H.H.C.; Chan, K.Y.; Ki, S.K, Economic load dispatch: A new hybrid particle swarm optimization approach, Universities9-12Dec.2007 Page(s):1 8 Digital Object Identifier /AUPEC [4]. Pancholi, R.K.; Swarup, K.S, Particle swarm optimization for security constrained economic dispatch, Intelligent Sensing and Information Processing, Proceedings of International Conference on 2004 Page(s):7-12 Digital Object Identifier /ICISIP [5]. Andries P.engelbrecht, Computational Intelligence, John Wiley & sons, New York, [6]. Hadi Saadat, Power System Analysis, McGraw-hill companies, Inc, [7]. Mekhamer, S.F.; Moustafa, Y.G.; EI-Sherif, N.; Mansour, M.M, A modified particle swarm optimizer applied to the solution of the economic dispatch problem, Electrical, Electronic and Computer Engineering, ICEEC ' International Conference on, 5-7 Sept Page(s): [8]. J.Woods and B.F Wollenberg, power generation, operation, and control, John Wiley & sons, [9]. Nadia Nedjah, Luiza de Macedo Mourelle, Swarm Intelligent System, Spring Berlin Heidelberg New York, [10]. Y.shi, R.Eberhart, A modified Particle Swarm Optimizer, proc. IEEE int. Conf. on Evolutionary Computation.PP 69-73, [11]. Y.shi, R.Eberhart, Parameter Selection in Particle Swarm Optimization, In : portovw, Saravanan N, waagen D and Eiben AE(eds) Evolutionary Programming VII, PP ,1998. [12]. Gomes, M.H.R.; Saraiva, J.T, Active / reactive dispatch in competitive environment, Power Tech, 2005 IEEE Russia June 2005 Page(s):1-7 Digital Object Identifier /PTC [13]. F.vanden Berg, An Analaysis of Particle Swarm Optimizer, PHD thesis,department of computer science,university of Pretoria, South Africa, [14]. Zdenek-Dostal, Optimal Quadratic Programming Algorithms, Springer,1 edition (March 5, 2009). 64

6 [15]. Nidul Sinha, Chakrabarthi.R and Chattopadhyay.P.K, Evolutionary Programming Techniques for Economic Load Dispatch, IEEE Transactions on evolutionary computation, Vol. 7, 2003, pp [16]. Z.L. Gain, Particle Swarm Optimization to solve the Economic Dispatch Considering the Generator Constraints, IEEE Trans. on Power System, Vol. 18,No.3,pp , August [17]. P.-H. Chen and H.-C Chang, Large-scale economic dispatch by genetic algorithm, IEEE Trans. Power Syst., vol.10, pp , Nov [18]. M. Sudhakaran, P. Ajay - D - Vimal Raj and T.G. Palanivelu, Application of Particle Swarm Optimization for Economic Load Dispatch Problems, Intelligent Systems Applications to Power Systems, ISAP International Conference on 5-8 Nov Page(s):1-7 65