Transmission network optimal planning using the tabu search method

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  • ELSEVIER Electric Power Systems Research 42 (1997) 153-163

    ELECTRIC POUER SYSTErnS AESEFlRCH

    Transmission network optimal planning using the tabu search method

    Fushuan Wen, C.S. Chang * Depcwtment of Electricnl Engineering, Nationnl UrGersit?; of Singapore, 10 Kent Ridge Creswnt. Singuporr 119260, Singapore

    Received 8 November 1996

    Abstract

    A new method is presented in this paper to solve the single stage (horizon year) optimal planning problem for a transmission network, given future generation and load demands, and alternative types of lines available, subject to overload and right-of-way constraints. The problem is formulated as a 0 1 integer programming problem, and an efficient solving approach based upon the tabu search (7s) method is developed. TS has emerged as a new, highly efficient, search paradigm for quickly finding high quality solutions to combinatorial optimization problems. It is characterized by gathering knowledge during the search. and subsequently profiting from this knowledge. TS is inherently suitable for the transmission network optima1 planning problem, because a one-to-one mates is matched between the optimal planning procedure and the TS algorithm, which is then tuned for maximum algorithmic efficiency. In addition, a refined version of the TS method is utilized so as to find multiple optimal planning schemes for some planning problems. The research results for three test systems have verified the feasibility and efficiency of the developed TS based transmission network optimal planning method. 0 1997 Elsevier Science S.A.

    Keywords: Transmission network planning; Tabu search

    1. Introduction

    The transmission network optimal planning problem [l] is a nonlinear, large scale combinatorial optimiza- tion problem. The complication of the problem arises mainly from the large number of problem variables where a multitude of technical and economical con- straints are to be considered. The problem has been approached in many different ways, including using heuristic methods [2- 51. the traditional mathematical optimization methods such as linear programming [6,7], intcgcr programming [8], mixed integer programming [9- 121: dynamic programming [ 131, Benders decompo- sition [14- 161, the graph search method [17], and the hybrid method combining the heuristic methods and the mathematical optimization methods [ 181. The heuristic methods are based on intuitive analysis. They are relatively close to the way that engineers think. These approaches have the advantages of straightfor- wardness, flexibility, high speed of computation, and the easy involvement of personnel in decision making

    * Corresponding author. E-mail: eleccs@leonis.nus.sg

    0378.7796/97/%17,00 a 1997 Elsevier Science S.A. All rights reserved. PII SO378-7796(96)01200-X

    [I]. The main disadvantage of the heuristic methods is that they are not rigorous from the mathematical view- point. The traditional mathematical optimization meth- ods can take into account the interaction between variables, and are more rigorous in theory. However, because the number of network planning variables is very large and constraints are very complex, the tradi- tional mathematical optimization methods find it very difficult to solve large scale planning problems. Gener- ally, many simplifications must be conducted in using these approaches for solving practical planning prob- lems.

    With the development of artificial intelligence (AI) theory and techniques, some new Al-based approaches to the transmission network optimal planning problem have been proposed in recent years, such as the expert system based [19] and the artificial neural network (ANN) based [20] methods. The main advantage of the expert system based method lies in its ability to simu- late the experience of planning experts in a formal way. However, knowledge acquisition is always a very difficult task in applying this method. Moreover, maintenance of the large knowledge base is very

  • 154 F. Wen, C.S. Chang /Electric Power Systems Research 42 (1997) 153-163

    difficult. Research work in the application of ANN to the optimal planning of transmission networks is very preliminary, and much work remains to be done. The potential advantage of the ANN based method is its inherent parallel processing nature. In summary, al- though the transmission network optimal planning problem has been extensively studied, it is still not well solved.

    In recent years, there has been an enormous amount of interest in the applications of genetic algorithms (GAS), simulated annealing (SA) and tabu search (TS) for solving some difficult or poorly characterised opti- misation problems with a multi-modal or combinatorial nature. These methods are generally called modern heuristic techniques. Many successful applications of these methods in solving large scale practical problems have been reported recently. SA is powerful in obtain- ing the optimal solutions for large scale optimization problems and has been applied to solve the optimal planning problem of transmission networks [21], but its computational burden is very heavy. The computation speed of GAS is faster than that of SA. The preliminary application of GAS to the transmission network opti- mal planning problem has been reported recently [22], and good results obtained. TS [23-251 has emerged as a new, highly efficient, search paradigm for quickly finding high quality solutions to combinatorial optimi- sation problems. It is characterised by gathering knowl- edge during the search, and subsequently profiting from this knowledge. TS has been successfully applied to solve many large scale and complicated combinatorial optimisation problems in many areas including power systems [26-331, but its application to transmission network optimal planning has not yet been explored. Some previous research [32,33] has shown that the computational efficiency of TS is better than that of GAS. The main issue of this paper is to investigate its efficiency in solving the transmission network optimal planning problem, with an objective to find a more efficient method. First, the transmission network opti- mal planning problem is formulated as a O-l integer programming problem. Secondly, a TS based method is developed for solving this problem. The test results for three systems have verified the feasibility and efficiency of the developed TS based transmission network opti- mal planning method.

    The remainder of this paper is organized as follows. Section 2 presents the mathematical model of the trans- mission network optimal planning problem. Section 3 briefly describes the fundamental principle of the TS method and its application in the transmission network optimal planning problem. Three test systems are served for demonstrating the feasibility and efficiency of the developed TS based method in Section 4, and conclusions are presented in Section 5.

    2. The mathematical model of the transmission network optimal planning problem

    The transmission network optimal planning problem can be broadly categorised into static and dynamic planning problems. The static planning problem, which is dealt with in this paper, is also known as the single stage or horizon year planning problem. It involves determining the network connection scheme for a par- ticular generation and load pattern in a future horizon year, and does not deal with the timing problem of network connection schemes. For the single stage plan- ning problem, the available information consists of the initial network structure and parameters, the candidate lines (rights-of-way and parameters), the construction investment cost of each candidate line and the transmis- sion capacities (ratings) of each existing line and each candidate line. The objective of this problem is to determine the most economical planning scheme(s) to meet the load demand in the horizon year subject to the security or reliability constraints. The security or reli- ability constraints may consist of the overload con- straints in the basic planning network and the N - 1 security constraints. As the first attempt, we only take into account the overload constraints in this work, although the framework of the presented method does not limit the inclusion of the N- 1 security constraints [l]. Thus, the single stage (horizon year) transmission network optimal planning problem can be formulated as one of the two following problems:

    WlZ when Z > 1 (1.1) Mm f(s) = f C,s,+ W,G when Z = 1 (1.2)

    i=l

    where S is a N-dimension solution vector to the opti- mal planning problem, or in other words, S identifies a network planning scheme. N is the number of candidate lines. s, is the ith element of S, and si = 1 or 0 respec- tively denotes the ith candidate line added into the network planning scheme identified by S or not. Z is the number of isolated islands if the network planning scheme identified by S is a disjoint network, otherwise Z = 1. Z is a function of S. Based upon the topology structure of the initial network and the planning scheme identified by S, the connection relationship between the buses and branches (lines) can be obtained, and as a result Z can be determined very easily through graph search. C, is the construction investment cost of the ith candidate line. G is the total amount of over- loads in all overloaded lines of the planning network identified by S, so G is a function of S. G can be calculated by the DC power flow method or AC power flow method (such as NewtonRaphson power flow or

  • F. Wen, C.S. Chang /Electric Power Sptems Research 42 (1997) 153-163 155

    fast decoupled power flow) depending on the require- ment of accuracy and the availability of data. In this work, G is calculated using the DC power flow method [l]. W, and W, are positive penalty coefficients.

    If the network planning scheme identified by S is a disjoint network, then use Eq. (1.1) to calculate f(S), otherwise use Eq. (1.2) to calculate f(S). The value of f(S) reflects the goodness of the planning scheme iden- tified by S. The smaller f(S) is the better the planning scheme identified by S. Obviously, it is an essential requirement that the optimal network structure be a connected network, i.e. no isolated islands exist. In order to ensure that the minimization off(S) reflect the objective of optimal network planning, W, should be much bigger than W,. This means that any connected planning network scheme is preferred to any disjoint planning network scheme. W, is used to penalise the planning scheme with overloaded lines, and it should be large enough. W, and W, can be specified based on experience. In the following three numerical examples, we specify W, = lo5 and W, = 500.

    The remaining problem is how to find S by utilizing Eqs. (1.1) and (1.2), and the given data such as the initial network structure and parameters, the candidate lines (rights-of-way and parameters), the construction investment cost, and the generation and load pattern in a future horizon year, or in other words, how to find the value of S which minimizesf(S). Obviously, this is a O-l integer programming problem. In the following section, we will introduce a TS method to solve this problem. The motivation for adopting the TS for solv- ing this problem lies in its ability to find the global optimal solution efficiently.

    The developed transmission network optimal plan- ning method includes two main steps:

    (a) To input the system data, including the topologi- cal structure and parameters of the existing transmis- sion network, the candidate lines (rights-of-way and parameters), the construction investment cost of each candidate line, and the generation and load pattern in a future horizon year.

    (b) To conduct the planning procedure with the following: 1. To formulate the 0- 1 integer programming model

    as presented above for the transmission network optimal planning problem.

    2. To produce an initial planning scheme (an initial or current solution), S*, by assigning all its elements to be 1. This means that all candidate lines are added into the network as the initial solution or the initial planning scheme. This initial planning scheme re- sults in a dummy network which is a strongly connected, very secure, highly redundant and uneco- nomic scheme. In this way, the occurrences and evaluations of disjoint network planning schemes during the TSs solving process can be avoided to a

    great extent, and as a result high solution efficiency can be attained.

    3. To apply a TS, which is presented in Section 3, to search for better schemes progressively until the prespecified maximum permitted iteration number has been reached. For each hypothesis scheme S produced during this procedure, we calculate 2 first so as to know if the scheme corresponds to a disjoint network. If so (i.e., Z > l), use Eq. (1.1) to computef(S), otherwise (i.e., Z = 1) use Eq. (1.2) to compute f(S).

    3. Tabu search [32] and its application to the optimal planning of a transmission network

    The tabu search approach described below is im- proved from the one described in Ref. [32].

    With its roots going back to the late 1960s and early 1970s the tabu search was proposed in its present form a few years ago by Glover [23-251. It has now become an established optimization approach that is rapidly spreading to many new fields. For example, successful applications of TS have been reported recently in solv- ing some power system problems, such as hydro-ther- ma1 scheduling [26], multi-level reactive source planning [27], capacitor placement in radial distribution systems [28] and thermal unit maintenance scheduling [29], the optimal determination of tie lines in distribution sys- tems [30], unit commitment [31], fault section estima- tion [32] and alarm processing [33]. Together with SA and GAS, TS has been singled out by the Committee on the Next Decade of Operations Research [23] as ex- tremely promising for the future treatment of practical applications.

    TS is a restricted neighbourhood search technique, and its computational flowchart is an iterative process [23]. The fundamental idea of TS is the use of the flexible memory of search history which thus guides the search process to surmount local optimal solutions. To describe the workings of TS, we consider a combinato- rial optimization problem in the following form:

    Minimize H(X) (2)

    where X is a vector of dimension n, and its elements are integers. H(X) is the objective function (cost or penalty function), and can be linear or nonlinear. The first step of TS is to produce an initial (current) solution Xcurrent either randomly or using an existing (heuristic) method for the given problem. The second step is to define a set of moves that may be applied to the current solution to produce a set of trial solutions. As an example, the move can take the form of Pia = Xcurrent + AX. Here, AX is a vector with the same dimension as X. In fact, a move produces a neighbourhood search. Among...

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