transmission network optimal planning using the tabu search method
TRANSCRIPT
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: [email protected]
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 TS’s 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 all the trial solutions thus produced, TS seeks the one that
156 F. Wen, C.S. ChanR /Electric Ponper Systems Research 42 (1997) 153-163
improves most of the objective function. In certain situations, if there are no improving moves, a fact which means some local optimum exists, TS chooses the one that least degrades the objective function. The most basic components of the tabu search are the moves, tabu list and aspiration level (criterion), which are briefly introduced below.
3.1. Moves
The search process of TS is implemented by the moves. A trial solution can be created by a move. Many kinds of moves are currently available [23-331, and the following two kinds of moves [27] are adopted in this work for solving the O-l integer programming problem of Eq. (1).
(a) Single move (denoted by nzi)
xtrial=xc-ntfUl (i= l,2,...,n) (3)
where ui is a vector with the same dimension as X, and its ith element is 1 (if the ith element of Furrent is 0) or - 1 (if the ith element of Furrent is 1) and all the other elements are zero.
(b) Exchange move (denoted by mu) Xrtrla’=Xcurrent+~i-~J (i,j=1,2,...,n, and i#j)
(4)
where u, is a vector with the same dimension as X, and its ith element is 1 (if the ith element of Furrent is 0) or - 1 (if the ith element of FUrrent is 1) and all the other elements are zero. u/ is a vector with the same dimen- sion as X, and its ith element is 1 (if the ith element of x CUrrent is 1) or - 1 (if the jth element and all the other elements are zero. exchange move can be implemented moves.
3.2. Tabu list
df XCUrre”t is 0)
Obviously, an by two single
In order to prevent from returning to the local optimum just visited, the reverse move that is detrimen- tal to achieving the optimum solution must be forbid- den [28]. This is done by storing this move in a tabu list. The elements of the tabu list are called tabu moves. The reverse moves are restricted from regions the search already explored. Due to the enforcement of tabu moves, the search process can escape from the local optimal solutions.
The dimension of the tabu list is called the tabu list size. Obviously, how to specify the tabu list size in the searching process plays an important role in the search for good solutions. In general, the tabu list size should grow with the size of the given problem, but how to specify the optimal tabu list size is still an open prob- lem. Up to now, the tabu list size has been determined
515 l&l
Fig. 1. 6-bus Garver network path schematic diagram
experimentally. In addition, how to manage the tabu list such as how long (how many iterations) a move can be retained in the tabu list is also an important prob- lem. Many methods to implement and manage the tabu list have been developed [23-331, and the methods used in this work are described below.
In this work, the tabu list is updated iteration by iteration [27]. At the end of each iteration, the new move is added to the tabu list, and an old move may be removed if it has been in the tabu list for T,,, itera- tions. T,,, is called the tabu tenure which plays an important role in finding good solutions. The detailed implementation is as follows.
(a) If a single move mi is selected in the kth iteration, then the following moves will be added to the tabu list:
-mi,mi, (j=1,2 ,..., n,j#i)
(b) If an exchange move m, is selected in the kth iteration, then the following moves will be added to the tabu list:
-mi,m,i (l-1,2 ,..., n, I#i)
m,,mjl (l-1,2 ,..., n, I#j)
Fig. 2. optimal planning network of the 6-bus Garver system
F. Wen, C.S. Chang /Electric Power Systems Research 42 (1997) 153-163 157
- Existing lines - -- Expandable lines
4’ ii
Fig. 3. The l&bus network path schematic diagram.
3.3. Aspiration level (criterion)
An important component of TS is the incorporation of an aspiration level (criterion) for any tabu move in the tabu list. The role of the aspiration level is to provide an added flexibility to choose good moves by allowing a tabu move to be overridden if its aspiration level is attained. This is because the tabu list may forbid certain worthy moves possibly leading to a better solu- tion than the best one found so far. An aspiration level is used to allow tabu moves to be released if they are judged to be worthy or interesting [28]. In other words, the aspiration level is to allow an ‘excellent’ tabu move to be selected if this level is attained. Many implemen- tation strategies for the aspiration level are available [23], and in this work the aspiration level is defined as [23,28]: if a tabu move from the current solution XCUTrent can reach a solution which is better than the best solution found so far, then the aspiration level for this tabu move is attained and can be overridden,
Fig. 4. The first optimal planning network of the l&bus system.
Fig. 5. The second optimal planning network of the IS-bus system.
3.4. A framework for solving the transmission network optimal planning problem using tabu search
The initial configuration of the future (horizon year) network is frequently disconnected (for example, due to the construction of new generation facilities or the occurrences of new load centers) especially for the long term transmission network planning problem. In this case, the computational efficiency of the TS method depends heavily on the initial solution, because the TS method is a neighbourhood search method. If the initial solution represents a weak and disjointed network, then the occurrences and evaluations of disjoint planning network schemes may involve much computational ef- fort during the TS’s computational procedure. In order to prevent this situation to the greatest extent from occurrence and to enhance the computational effi- ciency, we can first add all candidate lines into the initial network, and take this as the initial solution (or initial planning scheme). A general algorithm of the TS based transmission network optimal planning, which takes into account this problem, can be described as follows:
(a) Produce the initial (current) solution S* by as- signing all its elements to be 1, and set the iteration counter k = 0. Set the best solution vector Sbest = S*. (b) If k is equal to a prespecified maximum permitted iteration number K,,,, then output Sb”“’ as the final result and stop. Otherwise, set k = k + 1, and go to Step (c). (c) Select a trial solution Stria’ from the neighbour- hood of S* by applying the two kinds of moves as defined in Section 3.1 and compute the correspond- ing f(Strial). If the network identified by Stria’ is a disjoint one (i.e., 2 > l), then use Eq. (1.1) to calcu- late f(Strial), otherwise use Eq. (1.2) to calculate f(Sttia'). Repeat this process until the specified neigh- bourhood sampling number, SP,,,, has been reached. (d) If Sbest is not better than the best trial solution which has the smallest objective function value, then
158 F. Wen, C.S. Chang/EIectric Power Systems Research 42 (1997) 153-163
Fig. 6. The optimal transmission plan for the modified New England 39-bus test system.
assign this best trial solution to Sbest. Otherwise, go to Step (e). (e) S* is updated to the best trial solution which has the smallest objective function value as evaluated in Step (c) and the corresponding move is not in the tabu list or its aspiration level is attained. Then, add the move to the tabu list, and go to Step (b). If the best trial solution corresponds to a tabu move and its aspiration level is not attained, then check the next best trial solution, and repeat this step.
3.5. A method to find multiple optimal solutions by tabu search
The tabu search technique as stated above is power- ful in finding the global or near global optimal solution of an optimization problem. But it is shown from the simulation results that the TS can usually find only one global or near global optimal solution. For the trans- mission network optimal planning problem, multiple solutions may exist, especially for large scale power systems with many candidate lines. In order to find all reasonable solutions, the TS must be modified. The modification is as follows. At the first iteration, we put the initial solution in a specially designed array. In each of the follow-up iterations, we check if the best trial solution(s) in the current iteration is better than the
solution(s) stored in the array. If yes, we use the best trial solution(s) in the current iteration to replace the record of the array. If the best trial solution(s) in the current iteration is as good as the solution(s) stored in the array and they are not the same solutions, then we put the best solution(s) in the current iteration into the array, thus the array is expanded. Otherwise, we do not change the record of the array. Please note that the array only contains those solutions which are found to be the best up to the current iteration, and only a copy can be stored in the array for each of the best solutions. Thus, at the end of the TS operation, the array will contain all the different best solutions found during the operation. If the parameters of TS (i.e., K,,,, SP,,, and r,,,) are properly specified, the multiple global optimal solutions can be found by this way. This has been verified by the following numerical examples.
4. Test results
We have used three systems to test the developed TS based transmission network optimal planning method. The first system is the well known Garver 6-bus system [6,22], the second system is a 18-bus system from [I], and the third system is a modified 39-bus New England test system from [IO]. Please note that the convergence
F. Wen, C.S. Chang /Electric Power Systems Research 42 (1997) 153-163 159
0000000000000 ddddddddddddd
Trmwr-mmO-Nmd-mw ------NNN~mNN
F.TR 2 *Ob000mN00000 Piddtiddeitiddddd
0000000000000 ddddddddddddd
-Nmd-v3w~mmO-Nm M-F.-
criterion adopted in this work is that a prespecified maximum permitted iteration number, K,,,,,, has been reached.
The network path schematic diagram of the first system is shown in Fig. 1 (taken from Ref. [22]). The dashed lines in Fig. 1 denote the candidate rights-of- way, and the symbol ‘xk’ associated with each right-of- way identifies the maximum number of candidate lines (k) which can be added in this right-of-way. The system parameters such as the generation and load pattern, line parameters (resistance, reactance and transmission capacity) of the existing and the candidate lines, and the length of each candidate line can be found in many publications such as [6,22]. Assuming that the construc- tion investment is proportional to the line length [6,22], we can use the line length to replace the cost C, in Eq. (1.2). From Fig. 1, it is known that there are 21 candidate lines. Using the developed TS based method, the well known optimal planning scheme as shown in Fig. 2 can be obtained very easily. The optimal plan- ning scheme as shown in Fig. 2 can be obtained by setting the parameters in TS as follows:
K max = 50, SP,,,,, = 22, T,,,,, = 20
The computing time for planning this system is about 14 s on a 486 microcomputer.
The network path schematic diagram of the second system as shown in Fig. 3 is taken from Ref. [l] (P.390). This is a simplified actual system in the western part of China. The system parameters such as the generation and load pattern, line parameters (resistance, reactance and transmission capacity) of the existing and the can- didate lines, and the length of each candidate line can be found in [1,22]. Assuming that the construction investment is proportional to the line length [1,22], we can use the line length to replace the cost Ci in Eq. (1.2). From Fig. 3 and the system data as given in [1,22], it is known that there are 36 candidate lines. Using the developed TS based method, two optimal planning schemes as shown in Figs. 4 and 5 can be obtained. This shows that the TS based method can find multiple optimal planning schemes in a single run. The method presented in [l] only obtained the first optimal planning scheme as shown in Fig. 4. The optimal planning schemes as shown in Figs. 4 and 5 can be obtained by setting the parameters in TS as follows:
K max = 50, SP,,, = 36, T,,,,, = 30
The computing time for planning this system is about 32 s on a 486 microcomputer.
The network path schematic diagram of the third system as shown in Fig. 6 is taken from Ref. [lo]. In this figure, the phantom lines represent old rights-of- way that can be reinforced, and the dashed lines denote new rights-of-way that can be equipped. The system parameters are taken from [lo], and listed in Tables
Tabl
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0.00
08
0.01
56
1.02
5 19
.23
F. Wen, C.S. Chang /Electric Ponaer Systems Research 42 (1997) 153-163 161
Table 3 The modified 39-bus New England system: right-of-way planning set (from [IO])
Line R X Transmission capacity (I-sting) Alternative number Candidate lines Construction investment cost
From to
2 3 0.001480 0.014800 6.08 1” 0 0.0
0.000987 0.009867 9.12 2 1 11.25 0.000740 0.007400 12.16 3b 2 16.88 0.000592 0.005920 15.20 4 3 28.13 0.000493 0.004933 IS.24 5 4 33.76
6 23 0.003440 0.034400 2.62 I” 0 0.0 0.002293 0.022933 3.92 2 1 26.15 0.001720 0.017200 5.24 3h 2 39.23
15 24 0.000820 0.008200 10.98 1” 0 0.0 0.000547 0.005467 16.47 2 1 6.23 0.000410 0.004100 21.96 3b 2 9.35
0.000328 0.003280 27.45 4 3 15.59 0.000273 0.002733 32.94 5 4 18.70
23 24 0.004220 0.042200 2.13 Id 0 0.0
0.002813 0.028133 3.19 2 1 32.09 0.002110 0.021100 4.26 3b 2 48.13 0.001688 0.016880 5.32 4 3 80.22 0.001407 0.014067 6.39 5 4 96.26
3 15 0.0 0.0 0.0 0 0 0.0 0.005600 0.05600 1.61 1 1 21.29 0.002800 0.02800 3.21 2 2 31.93 0.001867 0.018667 4.82 3 3 53.22
0.001400 0.014000 6.43 4b 4 63.87 0.001120 0.011200 8.04 5 5 85.16 0.000933 0.009333 9.64 6 6 95.80
3 25 0.0 0.0 0.0 0 0 0.0 0.003920 0.039200 2.30 1 14.90 0.001960 0.019600 4.59
:b 2 22.35
0.001307 0.013067 6.89 3 3 37.26 0.000980 0.009800 9.18 4 4 44.71
0.000784 0.007840 11.48 5 5 59.61 0.000653 0.006533 13.78 6 6 67.06
a Existing lines. b The optimal planning scheme.
l-3. The meaning of the symbol ‘n,jn,’ associated with each right-of-way as shown in Fig. 6 is the same as in [IO], i.e., n, and n, respectively identify the alternative scheme sequence numbers in the existing initial network and the optimal planning network obtained by the TS based method. From the system data given in Tables l-3, it is known that there are 26 candidate lines. Using the developed TS based method, the optimal planning scheme as shown in Fig. 6 (identified by the number n2 in ‘n,/n,’ associated with each right-of-way) and Table 3 (identified by ‘b’ in the ‘Alternative num- ber’ column) can be obtained. This optimal planning scheme is different from, and more economical than, the one given in [lo]. The construction investment cost for this optimal planning scheme is 199.8 1, while, for the one given in [lo], it is 256. Please note the difference between this work and the work presented in [lo]. The DC power flow is utilized in this work, while a different
linear power flow model is used in [lo]. The optimal planning scheme as shown in Fig. 6 and Table 3 can be obtained by setting the parameters in TS as follows:
Km,, = 50, SP,,,=26, T,,,,,=20
The computing time for planning this system is about 47 s on a 486 microcomputer.
Based upon the test results of these three systems, it can be concluded that the proposed TS based transmis- sion network optimal planning method is efficient and has potential for practical applications.
5. Conclusions
In this paper, a new approach to the optimal plan- ning of transmission network is developed using the
162 F. Wen, C.S. Chang/Electric Power Systems Research 42 (1997) 153-163
tabu search technique. At first, based upon the DC power-flow model, the problem is formulated as a O-l integer programming problem. Then, a new method is proposed for solving the problem using the tabu search technique.
The test results on three sample systems have shown that the developed TS based method is feasible and efficient, and has potential for practical applications. The main advantages of the presented TS based method include the ease of its implementation and a high computational efficiency. In our experience, the parameters in TS, such as the maximum permitted iteration number K,,,, the neighbourhood sampling number SP,,, and the tabu tenure T,,,, can easily be tuned in the solution process. As a first attempt, the N - 1 security constraints have not yet been included in the formulation in this work. However, the framework of this developed method does not limit the inclusion of these constraints. Future work will include the N- 1 security constraints in the formulation and to test the developed TS based approach for actual power systems.
Appendix A. Nomenclature
the number of candidate lines a solution to the transmission network opti- mal planning in vector form with N elements the construction investment cost of the ith candidate line the number of isolated islands in the plan- ning network identified by S the total amount of overloads in all over- loaded lines in the planning network iden- tified by S penalty coefficients the trial solution vector to an optimization problem the current solution vector a single move an exchange move the best solution vector for the transmission network optimal planning problem the maximum permitted iteration number for the tabu search the tabu tenure the neighborhood sampling number in each iteration of the tabu search
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