S. Muthu Vijaya Pandian and K. Thanushkodi

Transcription

1 Solving Economic Load Dispatch Problem Considering Transmission Losses by a Hybrid EP-EPSO Algorithm for Solving both Smooth and Non-Smooth Cost Function S. Muthu Vijaya Pandian and K. Thanushkodi Abstract An Evolutionary Programming (EP) and Efficient Particle Swarm Optimization (EPSO) techniques are employed to solve Economic Dispatch (ED) problems including transmission losses in power system is presented in this paper. This paper is clearly justified with the results separately obtained for the above two techniques and also provided with the results by applying both the algorithms separately. With practical consideration, ED will have non-smooth cost functions with equality and inequality constraints that make the problem, a large-scale highly constrained nonlinear optimization problem. The proposed method expands the original PSO to handle a different approach for satisfying those constraints. In this paper, an Efficient Particle Swarm Optimization (EPSO) technique is employed so that optimized results are obtained, and by applying EP, faster convergence is obtained. To demonstrate the effectiveness of the proposed method it is being applied to test ED problems, one with smooth and other with non smooth cost functions considering valve-point loading effects. Comparison with other optimization and hybrid algorithm techniques showed the superiority of the proposed EP-EPSO approach and confirmed its potential for solving nonlinear economic load dispatch problems with losses. Index Terms Economic load dispatch, Evolutionary Programming, Efficient Particle swarm optimization, Valve point loading effect I. INTRODUCTION Economic Dispatch (ED) problem is one of the fundamental issues in power system operation. In essence, it is an optimization problem and its main objective is to reduce the total generation cost of units, while satisfying constraints.[1] Previous efforts on solving ED problems have employed various mathematical programming methods and optimization techniques excluding losses. Recently, Eberhart and Kennedy suggested a Particle Swarm Optimization (PSO) based on the analogy of swarm Dr.K.Thauskkodi is with Director of Akshaya College of Engineering and Technology, Coimbatore, Tamilnadu, , India. thanush_dr@rediffmail.com. Manuscript received October 30, S.Muthu Vijaya Pandian is with Department of Electrical and Electronics Engineering, V.L.B.Janakiammal College of Engineering and Technology, Coimbatore, Tamil Nadu, , India. (Phone : ); Fax : ( ajay_vijay@rediffmail.com). of bird and school of fish. In PSO, each individual makes its decision based on its own experience together with other individual s experiences. In artificial intelligence, an evolutionary algorithm (EA) is a subset of evolutionary computation, a generic population-based metaheuristic optimization algorithm. An EA uses some mechanisms inspired by biological evolution: reproduction, mutation, recombination, and selection. Candidate solutions to the optimization problem play the role of individuals in a population, and the fitness function determines the environment within which the solutions "live" (see also cost function). Evolution of the population then takes place after the repeated application of the above operators. Artificial evolution (AE) describes a process involving individual evolutionary algorithms; EAs are individual components that participate in an AE. The main advantages of the PSO algorithm are summarized as: simple concept, easy implementation, and computational efficiency when compared with mathematical algorithm and other heuristic optimization techniques[7]. The practical ED problems with valve-point loading effects are represented as a non smooth optimization problem with equality and inequality constraints. To solve this problem, many salient methods have been proposed such as dynamic programming, evolutionary programming, neural network approaches, and genetic algorithm. In this paper, an alternative approach is proposed to the non smooth ED problem using an Efficient PSO (EPSO), which focuses on the treatment of the equality and inequality constraints when modifying each individual s search. The equality constraint (i.e., the supply/demand balance) is easily satisfied by specifying a variable (i.e., a generator output) at random in each iteration as a slag generator whose value is determined by the difference between the total system demand (including losses) and the total generation excluding the slag generator. However, the inequality constraints in the next position of an individual produced by the PSO algorithm can violate the inequality constraints. In this case, the position of any individual violating the constraints is set to maximum or minimum depending on velocity evaluated. 560

2 II. FORMULATION OF ECONOMIC DIAPATCH PROBLEM A. ED Problem with Smooth Cost Functions Economic load dispatch (ELD) pertains to optimum generation scheduling of available generators in an interconnected power system to minimize the cost of generation subject to relevant system constraints. Cost equations are obtained from the heat rate characteristics of the generating machine. Smooth costs functions are linear, differentiable and convex functions. The most simplified cost function of each generator can be represented as a quadratic function as given in whose solution can be obtained by the conventional mathematical methods [2] : where C Fj aj bj cj Fj aj bj cj C= F j P j (1) F j P j =a j +b j P j +c j P j 2 total generation cost cost function of generator j cost coefficients of generator j cost function of generator j cost coefficients of generator j While minimizing the total generation cost, the total generation should be equal to the total system demand plus the transmission network loss. The transmission loss is given by the equation, (2) PL= BojPj (3) where B oj is the loss co-efficient matrix. The equality constraint for the ED problem can be given by, Pj = D + PL (4) where D is the total demand needed by the load or consumer. The generation output of each unit should be between its minimum and maximum limits. That is, the following inequality constraint for each generator should be satisfied Pjmin<Pj<Pjmax (5) where P jmin, P jmax are the minimum and maximum output of individual generators. B. ED Problem with Non-smooth Cost Functions In reality, the objective function of an ED problem has non differentiable points according to valve-point effects. Therefore, the objective function should be composed of a set of non-smooth cost functions. In this paper, one case of non-smooth cost function is considered i.e. the valvepoint loading problem where the objective function is generally described as the superposition of sinusoidal functions and quadratic functions [7]. C. Non-smooth Cost Function with Valve-Point Effects The generator with multi-valve steam turbines has very 561 different input-output curve compared with the smooth cost function[6]. Typically, the valve point results in, as each steam valve starts to open, the ripples like in to take account for the valve-point effects, sinusoidal functions are added to the quadratic cost functions as follows: FjPj=aj+bjPj+cjPj2+ejxsin(fjx(Pjmin-Pj)) (6) III. EVOLUTIONARY PROGRAMMING EP is a near global search stochastic optimization method starting from multiple points, which placed emphasis on the behavioural linkage between the parents and their offsprings, rather than seeking to emulate specific operators as observed in nature to find a solution. However EP takes a long computation time to find a solution and sometime EP suffers from the convergence problem. On the other hand, EPSO is a gradient based optimization method starting from a single point and using gradient information to obtain a solution. The solution obtained from EPSO is a local optimal solution. In order to obtain a high quality solution, the first part, EP is applied to obtain a near global solution. After the specified termination criteria for EP is reached, EPSO is applied in the second part by using the solution from EP as an initial starting point and searches by using a gradient information to obtain the final optimal solution [2]. A. Evolutionary Programming Subproblem 1) Representation For m,particles in the swarm and n generators of the systems, the array of control variable vectors (S) can be shown as P 11 P 12.. P 1m P 21 P 22.. P 2m P n1 P n2. P nm 2) Initialization To begin the population of chromosomes is uniform randomly initialized within the operation range of the generator [5]. 3) Fitness Evaluation The active power generations at all the buses except the first bus in all intervals are control variable, which are itself constrained. Equality constraint can be handled as P P P ij Dj ij i= 2 n = 4) Creation of Offspring A new population of solutions is produced from the existing population by adding a guassian random number with zero mean and pre-defined standard deviation as follows:

3 Pit = Pit + N(0,σit2) (7) Where σ it can be calculated from the following equation: σit = * fs/fmax(pitmax- Pitmin) (8) Where ϐ is a scaling factor, which can be tuned during the process of search for optimum. 5) Selection And Competition The selection technique used here is the stochastic tournament method. The 2P individuals compete with each other for selection. A weight value ωs is assigned to each individual as follows p ωs = ωj (9) j= 1 ωj = 1 if fs<fr ωj =0 otherwise In biological DNA systems, the basic units are the adenine (A), thymine (T), guanine (G) and cytosine (C) nucleotides that join the helical strands. In genetic algorithms, the basic unit is called a symbol. The nature of symbols depends on the particular genetic algorithm. In gene expression programming, the symbols consist of functions, variables and constants. Symbols for variables and constants are called terminals, because they have no arguments. An ordered set of symbols form a gene, and an ordered set of genes form a chromosome. In GEP programs, genes typically have 4 to 20 symbols, and chromosomes are typically built from 2 to 10 genes; chromosomes may consist of only a single gene. The DNA strand for a mammal typically contains about 5x10 9 nucleotides. B. Gene Expression Programming Gene Expression Programming is a procedure that mimics biological evolution to create a computer program to model some phenomenon. Gene expression programming can be used to create many different types of models including decision trees, neural networks and polynomial constructs. The type of gene expression programming implemented in DTREG is Symbolic Regression so named because it creates a symbolic mathematical or logical function [4]. Fig.1 Biological evolution of EP DTREG provides a full implementation of the Gene Expression Programming algorithm developed by Cândida Ferreira. Here are some of the features of DTREG s implementation : Continuous and categorical target variables Automatic handling of categorical predictor variables A large library of functions that you can select for inclusion in the model Mathematical and logical (AND, OR, NOT, etc.) function generation Choice of many fitness functions Both static linking functions and evolving homeotic genes Fixed and random constants Nonlinear regression to optimize constants Parsimony pressure to optimize the size of functions Automatic algebraic simplification of the combined function Several forms of validation including crossvalidation and hold-out C. Expression Trees and Karva The key to GEP s ability to quickly mutate valid expressions is the way it encodes symbols in genes. This notation is called the Karva Language. Expressions encoded using Karva are called K-expressions. Consider the simple mathematical expression: a*b+c (10) This can be encoded as an expression tree of the form Fig 2: Karva-expressions encoding model 1 An expression tree is an excellent way to represent an expression in a computer, because the tree can be arbitrarily complex, and expression trees can be evaluated quickly. To convert an expression tree to the Karva notation, start at the left-most symbol in the top line of the tree and scan symbols left-to-right and top-to-bottom. Each time a symbol is encountered, add it to the K-expression in leftto-right order. When there are no more symbols on a line, advance to the left end of the following line. Using this method, the tree shown above is converted to the K- expression: +*cab (11) Note that + is the first symbol found on the first line, at the end of that line scanning begins on the second line and finds * followed by c. It then starts with the third line 562

4 and finds a and b. As a second example, consider the expression a*b+sqrt(c*d) (12) The corresponding expression tree is Fig 3: Karva-expressions encoding model 2 Where Q represents square root. This can be translated to the K-expression +*Qab*cd (13) The process of converting an expression tree to a K- expression can be carried out quickly by a computer. A reverse process can quickly convert a K-expression back to an expression tree. D. Implementation of PSO for ED Problems The PSO algorithm searches in parallel using a group of individuals, in a physical dimensional search space, the position and velocity of individual i are represented as the vectors Xi = (xib.xin) and Vi = (vib.vin) ) respectively, be the position of the individual i and its neighbors best position so far. Using the information, the updated velocity of individual i is modified under the following equation in the PSO algorithm[3]: V i k+1 =ωv i k +c 1 rand1 x (P besti k - x i k )+c 2 rand 2 x (G best k x i k ) Where k+1 V i velocity of individual of iteration at k ω weight parameter C 1, C 2 acceleration factors rand 1 rand 2 random numbers between 0 and 1. k X i Position of individual i at iteration k Pbest best position of group throughout iteration k Each individual moves from the current position to the next one by the using the following equation: X ik+1 =X ik +X ik+1 The search mechanism of the PSO using the modified velocity and position of the individual i based on (7) and (8) is illustrated in fig (4) Fig. 4 Search mechanism of PSO IV. EFFICIENT PSO FOR ED PROBLEMS In this section, a new approach to implement the PSO algorithm will be described while solving the ED problems considering losses [3]. The main process of the efficient PSO algorithm can be summarized as follows: Step1) Initialization of a group at random while satisfying constraints. Step2) Velocity and position updates while satisfying constraints Step3) Update of Pbest and Gbest. Step4) Calculate transmission losses for the obtained Pbest and Gbest Step5) Increment the demand with the transmission losses Step6) Go to Step 2 until satisfying stopping criteria. In the subsequent sections, the detailed implementation strategies of the EPSO are described. A. Initialization of Individuals In the initialization process, a set of individuals (i.e., generation outputs) is created at random. Therefore, individual i position at iteration 0 can be represented as the vector of n is the number of generators.[3] The velocity of individual i is given by corresponds to the generation update quantity covering all generators. The following procedure is suggested for satisfying constraints for each individual in the group: Step1) Set j=1, i=1 element (i.e., generator) of an individual i. Step2) Select the jth element of the individual i. Step3) Create the value of the element (i.e., generation output) at random satisfying its inequality constraint. Step4) If j=n-1 then go to step 5; otherwise j=j+1 and go to Step 2. Step5) The value of the last element of an individual is determined by subtracting P from the total demand Step6) If i=no of individuals go to step 7; otherwise put i=i+1 and go to step 2. Step7) Stop the initialization process. After creating the initial position of each individual, the velocity of each individual is also created at random. The following strategy is used in creating the initial velocity: (P min - )-P ij 0 <v ij 0 <P max - -P ij 0 Where e is a small positive real number (14) The velocity of element j of individual i is generated at random within the boundary [8]. B. Velocity Update To modify the position of each individual, it is necessary to calculate the velocity of each individual in the next stage, which is obtained from (7). In this velocity updating process, the values of parameters such as w, c1 and c2 should be determined in advance. The weighting function is defined as follows 563 w=w max -(w max -w min /iter max )*iter (15)

5 Where w max w min final weight initial weight iter max maximum number of iteration iter current iteration number C. Position Modification Considering Constraints The position of each individual is modified by (8). The resulting position of an individual is not always guaranteed to satisfy the inequality constraints due to over/under velocity [4]. If any element of an individual violates its inequality constraint due to over/under speed then the position of the individual is fixed to its maximum or minimum operating point. Therefore, this can be formulated as follows: (16) To resolve the equality constraint problem without intervening the dynamic process inherent in the PSO algorithm, we propose the following procedures: Step1) Set j=1, i=1. Let the present iteration be k. Step2) Select the jth element (i.e., generator) of an individual i. Step3) Modify the value of element j using (7), (8), and (11).And satisfy inequality constraint. Step4) If j=n-1 then go to Step 5, otherwise j=j+ 1and go to Step 2. Step5) The value of the last element of an individual is obtained by subtracting P ij 0 from the total system demand. Step6) If i=no. of individuals then go to step 7; otherwise i=i+1 and go to Step2 Step7) Stop the modification procedure D. Update of Pbest and Gbest The Pbest of each individual at iteration k+1 is updated as follows: P k+1 k+1 k+1 k besti =X i if TC i < TC i P k+1 besti =P k k+1 k besti if TC i > TC i (17) (18) Where TC i object function evaluated at the position of the individual i. Additionally, Gbest at iteration k+1 is set as the best evaluated position among Pbest k+1 V. SIMULATED RESULT ANALYSES A. ED Problem with Non Smooth Cost Functions with Valve point effect TABLE 1: INPUT DATA FOR 40 UNIT SYSTEM Generator P jmin P jmax a i b i c i e i f i

6 TABLE 2: HYBRID RESULTS FOR (NN-EPSO) 40 UNIT SYSTEMS Unit Output (MW)

7 Fig 4: Convergence plot for 40 unit systems in (NN-EPSO) Method B. Simulated result sfor forty Units hybrid EP-EPSO TABLE 3: OUTPUT RESULTS FOR FORTY UNITS HYBRID (EP-EPSO ) METHOD Generator Output (MW) Demand = MW Losses = MW Optimal cost = $/hr Elapsed Time = sec The above table 2 given the output results for forty unit systems hybrid Neural network and Efficient particle swarm optimization method [8]

8 Demand = MW Losses = MW Optimal cost = $/hr Elapsed Time = sec TOTAL COST ($/hr) 1.32 x ECONOMIC LOAD DISPATCH USING HYBRID EP-EPSO ITERATIONS Fig 5: Convergence plot for 40 unit systems in (EP-EPSO) Algorithm TABLE 4: COMPARISON BETWEEN HYBRID EP-EPSO AND NEURO-EPSO METHOD Generator Output Of Hybrid Output Of Hybrid EP- NEURO-EPSO (MW) EPSO (MW)

9 [5] N.Sinha, R. Chakrabarti and P.K.chattopadhyay, Evolutionary Programming Techniques for Economic Load Dispatch, IEEE Trans. Evol. Comput,vol 7, pp.83-94, Feb [6] J.Nanda,A.Sachan,.L.Pradhan M.L.Kothari,,A.Koteswara Rao, Application of Artificial Neural Network to Economic Load Dispatch, IEEE Trans. Power Systems, vol 22, [7] B.H.Chowdhury and S.Rahman, A Review of Recent Advances in Economic Dispatch, IEEE Trans. Power Systems, vol 5,no.4, pp , Nov [8] S.Muthu Vijaya Pandian and Dr.K.Thanushkodi, A Hybrid (Neural Network Efficient Particle Swarm Optimization (NN- EPSO) for Economic Dispatch Problems Considering Transmission Losses with Non-smooth Cost Functions International Journal of Computer Science and Electrical Engineering (IJCEE) no.3, vol1 June 2009,pg TABLE 5 COMPARISION OF TOTAL PRODUCTION COST AND SIMULATION TIME AMONG NN-EPSO AND EP-EPSO Method Cost $/hr Time(s) NN EPSO EP NN-EPSO EP-EPSO EP-SQP VI. CONCLUSION In this paper, a new methodology for solving nonsmooth ED problem including valve point loading using EP combined with EPSO. The proposed algorithm consists of two parts. The first part employs the property of EP, which can provide a near global search region at the beginning. When the specified termination criteria of EP is reached, the local search EPSO is applied to tune the control variables in order to obtain the final optimal solution. It is clear from the Table 5 mean cost value and simulation obtained by EP-EPSO is comparatively less compared to all other methods. Simulation results demonstrate that the proposed method can give a cheaper total production cost than those obtained from EPSO, EP, NN-EPSO and EP-SQP. The resultant EP-EPSO has been suggested as a powerful optimization tool for non-convex ED problem. S.Muthu Vijaya Pandian was born in Karaikudi, Tamil Nadu, India on the 19th of June He received his M.B.A from Madurai Kamaraj University, Madurai,Tamil Nadu, India, in He then received his M.E in Power Systems from Government College of Technology, Coimbatore, Tamil Nadu, India., in 2004 and his currently pursuing his Ph.D in Anna university, Chennai, India. He is currently Lecturer at the Department of Electrical and Electronics Engineering,V.L.B.Janakiammal College of Engineering and Technology, Coimbatore, Tamil Nadu, India. He has published Three International Journal, One International conference and his current research interests include areas of Power Systems. Dr.K.Thanushkodi was born in Theni District, Tamil Nadu, India in He received the B.E degree in Electrical and Electronics Engineering and the M.sc.(Engg) degree from Madras University, Chennai, India in 1972 and 1974, respectively, and the Ph.D degree in Electrical and Electronics Engineering from Bharathiar University, Coimbatore, India, in He served as thirty three years of teaching experience and guided 4 Ph.D. Now he guiding 12 Research scholars in Anna University, Chennai and 30 Research scholar in Anna University, Coimbatore, India. He published 75 papers in International, National Journals and Conferences. He is currently Director of Akshaya College of Enggineering and Technology, Coimbatore and also Syndicate member Anna University,Chennai, India. His research include computer modeling and simulation, computer networking and power systems.. REFERENCES [1] A.J. Wood and B.F.Wollenberg, Power Generation,Operation and Control, New York: John Wiley & Sons, Inc., [2] Pathom Attaviriyanupap and Hiroyuki Kita, A Hybrid EP and SQP for Dynamic Economic Dispatch with Non-smooth Fuel Cost Function IEEE Transactions on Power Systems, vol 17,no.2, May [3] J.B.Park, K.S.Lee, K.Y.Lee, A Patticle Swarm Optimization for Economic Dispatch with Non-smooth Cost Function, IEEE Trans. Power Systems, vol 20, Feb [4] H.T.Yang, P.C.Yang, and C.L.Huang, Evolutionary Programming Based Economic Dispatch for Units with Non- Smooth Fuel Cost Functions, IEEE Trans. Power Systems, vol 11, no.1, pp ,feb