An Efficient Genetic Algorithm for Optimal Large- scale Power ...

0 downloads 0 Views 593KB Size Report
resolve the problem of real large-scale power distribution network planning. ... M. R. Irving is with Brunel Institute of Power Systems, Brunel University. Telephone: ..... Genetic Algorithms Laboratory, University of Illinois at Urbana-. Champaign ... [13] J. L. Bernal-Agustín, "Aplicación de algoritmos genéticos al diseño óptimo.
Paper accepted for presentation at 2003 IEEE Bologna Power Tech Conference, June 23th-26th, Bologna, Italy

An Efficient Genetic Algorithm for Optimal Largescale Power Distribution Network Planning F. Rivas-Dávalos and M. R. Irving

Abstract-- This paper presents a new efficient genetic algorithm for optimal large-scale power distribution network planning. The algorithm finds the best location and size of substations and feeders to minimize a cost function of the network, which represents investment (fixed cost) and operational costs (nonlinear variable costs). The main advantage of the algorithm over other genetic algorithms is its capability to overcome the problems of low heritability and topological infeasibility, resulting in reduced solution times. An effective representation of the candidate solutions was used and specialized genetic operators were introduced. The algorithm was tested on three networks and the results were compared with the results from other methods. From this comparison, we concluded that the proposed genetic algorithm is more efficient than several methods presented before and it is suitable to resolve the problem of real large-scale power distribution network planning. Index Terms— Genetic algorithms, optimization, power distribution planning.

A

I. INTRODUCTION

power distribution network is a network with substations (energy delivery nodes), feeders (branches) and customers (demand nodes). Planning these networks involve several tasks; one of these is to find the location and capacity of substations and feeders to minimize investment and operational costs. This task is a problem with hundreds of alternative solutions and the planners must find the optimum one. Several methods have been developed to deal with this problem [1]. Recently, methods based on genetic algorithms have been proposed to resolve problems with mathematical difficulties [2]-[8][10]. However, some of these methods have been applied on only small networks and others have long solution time. The main problem of these methods is their poor representation of the candidate solutions (encoding solution) and their genetic operators (crossover and mutation). In this paper we propose an efficient genetic algorithm for optimal large-scale power distribution networks planning, using an effective representation of the candidate solutions and specialized crossover and mutation operators.

F. Rivas-Dávalos is with Brunel Institute of Power Systems, Department of Electronic and Computer Engineering, Brunel University, Kingston Lane, Uxbridge, Middlesex UB8 3PH, London U.K.(e-mail: [email protected]). M. R. Irving is with Brunel Institute of Power Systems, Brunel University. Telephone: +44 1895 274000 (e-mail: [email protected]).

0-7803-7967-5/03/$17.00 ©2003 IEEE

The paper is organized as follow. In Section II, we present the power distribution network planning problem and the difficulties of a conventional implementation of genetic algorithms to this problem. The proposed genetic algorithm is presented in Section III. In Section IV, the algorithm is tested on three networks: a theoretical network [8], which is a complete graph with 10 nodes and 45 arcs, a 10-node power distribution network [5], and a large-scale network, based on a realistic network [4]. In Section V we present the conclusions. II. PROBLEM STATEMENT A. Power Distribution Network Planning Problem The problem of finding the location and size of substations and feeders, for a given demand level at the load locations, is formulated as minimizing the following cost function:

f cost =

Substs

∑ [CF

k

+ CVk * (

k =1

feeders

∑ [CF + CV * P l

l

l

SDk max 2 ) * Pcu k ] + SDk nom 2

]

(1)

l =1

where: CFk= CVk= SDkmax= SDknom= Pcuk= CFl= CVl= Pl=

Fixed cost of substation k. Variable cost coefficient of substation k. Maximum load of substation k. Nominal load of substation k. Copper losses coefficient of substation k. Fixed cost of feeder l. Variable cost coefficient of feeder l. Power flow carried through feeder l.

There are technical constraints to consider: power capacity limits of feeders; power supply limits of substations, voltage drop and radial configuration. Genetic algorithms have been widely used to resolve optimization problems [9]. A genetic algorithm consists of a population of data structures. Each data structure represents a possible solution (encoded solution) for the problem being optimized. These solutions are classified by an evaluation function, giving better value (fitness) to better solutions. The

population evolves over several generations to a population including the optimal solution (or suboptimal solutions) to the problem. This evolution is generated by two genetic operators: crossover, which creates new solutions (offspring) by combining parts from two solutions (parents), and mutation, which make changes in a single solution. A new population is created from the old population and the offspring population. This process is carried out in each iterations (generation) of the algorithm. The process is stopped by a predetermined stopping rule. B. Genetic Algorithm Implementation Problem Recently, power distribution network planning methods have been developed using GA [2]-[7]. In these references, some methods are presented with applications on small networks; meanwhile the others have a high solution time. The main causes of these limitations are the use of standard solution encoding techniques and genetic operators. The following problems have been observed [8][12]: ♦



Where best is the value of the solution with the minimum fitness, average is the average value of the population P and τ is a predetermined tolerance value. A direct encoding of a spanning tree was used to codify each solution of the problem. Fig. 1 shows an example of encoding solutions. Fig. 1a) represents a distribution network with one substation, four nodes and seven candidate feeders. Fig. 1b) is a possible solution, which is codified by an array containing the set of feeders forming the solution. In the formation of the initial population, only feasible solutions are created. First of all, the candidate feeders are randomly numbered. Then, to create each solution, an array is initialized to empty set and one feeder after the other is randomly selected and included in the array if it does not introduce a cycle with the feeders already included in the vector. This procedure terminates when a complete spanning tree (radial network) has been created.

Low heritability. A significant number of offspring generated by the crossover operator hardly have substructures of their parents. Topological unfeasibility. Many offspring do not represent a topologically valid solution.

To overcome such difficulties, we proposed a genetic algorithm for optimal large-scale power distribution network planning, using a direct encoding solution method and specialized genetic operators. The algorithm was developed based on an algorithm for the degree-constrained minimum spanning tree problem [11]. Fig. 1. An example of a direct encoding of a solution.

III. THE PROPOSED GENETIC ALGORITHM The structure of the proposed genetic algorithm is as follows: Procedure: Genetic Algorithm Begin t=0 Generate an initial population P(t) Evaluate P(t) While ((best-average) > τ) do begin Select parents Q(t) Recombine Q(t) to yield offspring population R(t) (CROSSOVER) Modify some offspring (MUTATION) Evaluate R(t) Select P(t+1) from P(t) and R(t) t=t+1 end end

The crossover operator to create offspring consists on two steps: In the first step, a set of feeders contained in both parents is selected to initialise the offspring. In the second step, feeders are randomly and successively selected from the rest of the feeders contained either in parent 1 or parent 2 (but not in both) to be included in the offspring (only feeders that do not introduce a cycle are included). If the offspring is not yet a spanning tree, other feeders from the candidate feeders not contained in the parents are chosen randomly until it is. Fig 2 shows an example of a crossover operation for two parent solutions of the network in Fig 1. Fig. 2c) is the offspring initialised with feeders contained in its both parents (feeders 3 and 5). Fig. 2d) illustrates that the feeders 1 (from parent 1) and 6 (from parent 2) were randomly selected and inserted into the offspring. This random selection can give feeders from only one of the parents. This crossover operator only produces feasible offspring solutions with substructures of their parents; resolving the problems of low heritability and topological unfeasibility.

A. A Theoretical Network[8] This network is a complete graph (every node has a direct connection with the rest of the nodes) with 10 nodes and 45 branches (with a space of 100 million possible solutions). The genetic parameter values used are the same ones used in [8]: population size = 60, Pc (crossover probability) = 0.8 and Pm (mutation probability) = 0.0. The proposed genetic algorithm gave the solution (Fig. 4) in less iterations than the algorithm proposed in [8].

Fig. 2. An example of a crossover operation for two parent solutions of the network in Fig. 1.

The mutation operator is described as follows (see Fig. 3): In a first step, a candidate feeder currently not in the offspring is randomly chosen and inserted in the offspring (Fig. 3a)). A cycle will be formed with this action so a random choice among the branches in the cycle is then made (excluding the new branch inserted), and the chosen one is removed from the offspring (Fig. 3b)).

Fig. 4. Evolution of the best, worst and average value solution from the proposed genetic algorithm tested on a complete graph with 10 nodes.

B. A 10-node Power Distribution Network[5] This network is a power distribution network with eight demand nodes, two candidate substations and fourteen candidate feeders. The test consisted on finding the optimum location of substations and feeders. The genetic parameter values used are the same ones used in [5]: population size = 10, Pc = 0.8 and Pm = 0.1. The proposed genetic algorithm gave the solution (Fig. 5) in less iterations than the algorithm presented in [5]. Fig. 3. An example of a mutation operation for the offspring in Fig. 2.

In the initial population process and crossover operation, a union-find algorithm was used to test efficiently whether two nodes are connected by a cycle [14]. In the mutation operation, it is required to find the feeders which conform a cycle when a feeder is inserted in the offspring. To do this task efficiently, an adjacency list representation of the tree is created temporarily and a depth-first search algorithm is applied [14]. IV. COMPUTATIONAL RESULTS The proposed solution algorithm was tested on three networks. The tests were done using a PC compatible 400 MHz Pentium III with 128 Mb of RAM, Windows NT and a Visual C++ compiler.

Fig. 5. Evolution of the best, worst and average value solution from the proposed genetic algorithm tested on a 10-node power distribution network.

C. A Large-scale Network[4] The proposed genetic algorithm was tested on a large-scale network based on a realistic network presented in [4]. This network has 201 nodes with 43 existing feeders and 184 proposed new feeders. The proposed genetic algorithm converged in 47 generations (about 5 minutes) (Fig. 6) using the following parameter values: population size = 200, Pc = 0.8 and Pm = 0.07. The genetic algorithm described in [4] was tested on a network with 63 nodes and 110 feeders and gave the solution in 1411 seconds (about 23 minutes)[13] using these parameter values: population size = 200, Pc = 0.3 and Pm = 0.02.

networks. Three of these were presented in this paper. From the results we concluded that: ♦ ♦



The direct representation is an effective representation of solutions for power distribution network planning. The offspring generated by the crossover operator consist mostly of substructures of their parents. (high heritability). All of offspring generated represent a topological valid solution (topological feasibility).

Therefore, the proposed genetic algorithm is suitable to resolve the problem of large-scale power distribution networks planning and it is more efficient than several previous methods. VI. ACKNOWLEDGMENT The authors gratefully acknowledge the financial assistance given to F. Rivas-Dávalos by the Consejo Nacional de Ciencia y Tecnología (National Council for Science and Technology), México. F. Rivas-Dávalos would like to thank Instituto Tecnológico de Morelia (Morelia Institute of Technology), México for granting him study leave to carry out PhD studies at Brunel University, England, UK. VII. REFERENCES

Fig. 6. Evolution of the best, worst and average value solution from the proposed genetic algorithm tested on a large-scale power distribution network.

The authors compared the proposed algorithm with the algorithms presented in [5] and [8] in terms of iteration because of lack of further information in those references. However, it can been said that the algorithm in [5], as well as in [4], uses the standard solution encoding technique and genetic operators and the results in section IV.C shows that the proposed encoding technique and genetic operators make our genetic algorithm more efficient. In reference [8], the authors presented a new recombination method. This method and the proposed method basically are similar in small computational effort and topological feasibility. V. CONCLUSIONS The existing power distribution network planning methods based on genetic algorithms have the disadvantage of limited problem size and high solution time. The main causes of this weakness are the standard representation of the solutions and the use of standard genetic operators. We proposed a new efficient genetic algorithm for large-scale power distribution network planning, using a direct encoding solution method and specialized genetic operators. The algorithm was developed based on an algorithm for the degree-constrained minimum spanning tree problem [11]. The proposed genetic algorithm has been tested on several

[1]

M. Vaziri, K. Tomsovic, and T. Gönen, "Distribution expansion problem revisited part 1: Categorical analysis and future directions, " IASTED Power and Energy Systems Proceedings. Paper No. 319-155, September 2000, Marbella, Spain pp. 283-290. [2] V. Miranda, J. V. Ranito, and L. M. Proenca, "Genetic algorithms in optimal multistage distribution network planning, " IEEE Trans. Power Systems, vol. 9, pp. 1927-1933, Nov. 1994. [3] E. –C. Yeh, S. S. Venkata, and Z. Sumic, "Improved distribution system planning using computational evolution, " IEEE Trans. Power Systems, vol. 11, pp. 668-674, May 1996. [4] I. J. Ramírez-Rosado and J. L. Bernal-Agustín, "Genetic algorithms applied to the design of large power distribution systems, " IEEE Trans. Power Systems, vol. 13, pp. 696-703, May 1998. [5] W. M. Lin, Y. S. Su, and M. T. Tsay, "Genetic algorithm for optimal distribution system planning, " in Proc. 1998 Power System Technology International Conf., pp. 241-245. [6] M. Filipec, D. Skrlec, and S. Krajcar, "New genetic algorithm for link distribution system design, " in Proc. 2000 Mediterranean Electrotechnical Conf., pp. 887-890. [7] P. M. S. Carvalho, L. A. F. M. Ferreira, F. G. Lobo, and M. F. Barruncho, "Distribution network expansion planning under uncertainty: A hedging algorithm in an evolutionary approach, " IEEE Trans. Power Delivery, vol. 15, pp. 412-416, Jan. 2000. [8] P. M. S. Carvalho, L. A. F. M. Ferreira, and M. F. Barruncho, "On spanning-tree recombination in Evolutionary large-scale network problems – Application to electrical distribution planning, " IEEE Trans. Evolutionary Computation, vol. 5, pp. 623-630, Dec. 2001. [9] D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning. New York: Addison Wesley, 1980. [10] Y. H. Song and M. R. Irving, "Optimisation techniques for electrical power systems part 2 Heuristic optimisation methods, " Power Engineering Journal, pp. 151-160, Jun. 2001. [11] G. R. Raidl, "An efficient evolutionary algorithm for the degreeconstrained minimum spanning tree problem, " in Proc. 2000 Evolutionary Computation Congress, pp. 104-111. [12] J. Gottlieb, B. A. Julstrom, G. R. Raidl, and F. Rothlauf, "Prufer numbers: A poor representation of spanning trees for evolutionary search, " Illinois Genetic Algorithms Laboratory, University of Illinois at UrbanaChampaign, Urbana, IL, IlliGAL Report No. 2001001, Jan. 2001.

[13] J. L. Bernal-Agustín, "Aplicación de algoritmos genéticos al diseño óptimo de sistemas de distribución de energía eléctrica, " Ph.D. dissertation, Dept. of Electrical Engineering, University of Zaragoza, Spain, 1998. Available: http://www.jeo.org/emo/ [14] R. Sedgewick, Algorithms in C++. Addison Wesley, 1992.

VIII. BIOGRAPHIES Francisco Rivas-Dávalos received both the B.Eng. and M.Sc. degrees in Electrical Engineering from Instituto Tecnológico de Morelia, México in 1994, and 2000 respectively. He is currently a Ph.D. student at the Brunel Institute of Power Systems, Brunel University, U.K. His current research interests lie in power distribution systems planning, and operation. Malcolm Irving received a B.Eng. degree in Electrical Engineering from the University of Sheffield, U.K., in 1974, and the degree of Ph.D. in Control Engineering, also from the University of Sheffield in 1977. He has been researching in power systems for over 25 years and is presently Professor of Power Systems and Director of the Brunel Institute of Power Systems, Brunel University, U.K.