Multiple Tabu Search Algorithm for Solving Economic Dispatch Problem

Saravuth Pothiya , Prinya Tantaswadi and Suwan Runggeratigul

School of Communication, Instrumentation & Control

Sirindhorn International Institute of Technology, Thammasat University, 131 Moo 5, Bangkadi, A. Muang, Pathumthani 12000, Thailand, Tel. (662) 5013505-20 ext. 1805, Fax (662) 5013505-20 ext 1801

E-mail: p_saravuth@hotmail.com, prinya@siit.tu.ac.th ,suwan@siit.tu.ac.th

Abstract In this paper proposed a novel method based on

simple Tabu Search Algorithm (TSA) to solves Economic Dispatch (ED) problem for thermal plants, which we call Multiple Tabu Search Algorithm (MTSA). This proposed method to improved TSA by addition multiple computation, local updating and global updating solutions to simple tabu search algorithm. The proposed method guarantees the near optimal solution and remarkably reduces the computation time. This proposed method is implemented with MATLAB program to solve economic dispatch problem with various case study. The results show that MTSA can converges to optimum solution faster than TSA and obtains the solution with high accuracy and less than Genetic Algorithm, Hopfield Neural Network and TSA. 1. Introduction

Economic dispatch (ED) problem is one of the most important problems in electric power system operation. Large-scale system, the problem is more complex and difficult to find out optimal solution because it is nonlinear function and it contains numbers of local optimal. The basic main purpose for solving economic dispatch problem is to schedule the outputs of the generating units so as to meet the system load at least fuel cost under various system and operating constraints. More recently, The researchers try to use other optimization methods for solve this problem i.e. Artificial Intelligence methods (AI) such as Simulate Annealing (SA)[1], Hopfield Neural Network (HNN) [2-3], Genetic Algorithm (GA)[4-8] and Tabu Search Algorithm (TSA) [10].

In this paper, we proposed a novel method based on simple Tabu Search Algorithm [12-13] which we call Multiple Tabu Search Algorithm (MTSA) to solve economic dispatch problem, which guarantees the near optimal solution and reducing calculation speed. In order to show the effectiveness of the proposed method, test results of two typical cases are shown for example.

Case 1 three generating unit with polynomial fuel cost functions without consider transmission line loss [2-5]. Case 2 thirteen generating unit with valve-point loading effect fuel cost functions [7]. Effectiveness of proposed method to solve these case

studies is compared with conventional methods such as GA, HNN and TSA.

The paper is organized as follows: Section 2 formulates the economic dispatch problem. Section 3 describes the simple Tabu Search Algorithm and a novel method is proposes. Section 4 computational results from the proposed method are compared with other methods. Lastly, conclusion is given in Section 5. 2. The Economic Dispatch Problem

The main objective of solving the economic dispatch problem in electric power system is to determine the optimum fuel cost operation of each generating unit. The economic dispatch problem can be modeled by

Minimize ∑1

)(n

iiiT PFF

== (1)

where TF is total fuel cost )( ii PF is fuel cost of generating unit i n is number of generating units in the system Generally, fuel cost of thermal generating unit will be in polynomial function.

2)( iiiiiii PcPbaPF ++= (2) where ia , ib and ic are cost coefficients of unit i, subject to following constraint: (i) Power balance constraint

Ln

ii PDP +=

=∑

1 (3)

where iP is output generation of unit i D is total load demand LP is total transmission line loss (ii) Generation limit constraint

max,min, ≤≤ iii PPP (4) where Pi, min is the minimum generation limit of unit i Pi, max is the maximum generation limit of unit i

3. Multiple Tabu Search Algorithm 3.1 Tabu Search Algorithm (TSA)

In this section, we first describe the working principle of a Tabu Search Algorithm (TSA). The TSA is a general heuristic optimization approach devised for finding optimal solution of optimization problems. This algorithm was developed by F. Glover [6,7] in 1989. It has a flexible memory to keep the information about the history search and employs it to create and explore the new solution in the search space. Afterwards, Bland and Dawson [8] presented TSA to solve the optimization problem. TSA is a powerful optimization approach that has been successfully applied to a number of combinatorial optimization problems. Basically, TSA is based on Hill-climbing method, however, it also allow moves out of a current solution that makes the objective function worse and then eventually achieve a better solution.

The two main components of TSA are the tabu list (TL) and the aspiration criterion (AC). The tabu list stores all tabu moves that are not permitted to be applied to the current solution. The tabu list record the move direction, frequency and recency. Aspiration criterion is employed to determine which move should be free in such case, that is, if a certain move criterion is satisfied, then this move is set to be allowable. The procedure of TSA can be described in Figure 1. 3.2 Algorithm of Multiple Tabu Search (MTSA)

Based on the described TSA, the Multiple Tabu Search Algorithm (MTSA) is proposed. The similarity between these approach is that they are able to move solution out of the local optimum. However, MTSA improves the disadvantage of TSA, ie. , imprudently initial solution and repeat iteration. Furthermore, MTSA’s performance is superior to that of TSA in the aspects of computation time and a c c u r a c y o f s o l u t i o n .

The procedure of MTSA is based on the procedure of TSA as given in the previous section, however, addition some procedures to increase the performance are needed. The addition procedures are:

1) Genera te severa l in i t i a l so lu t ions . 2) M u l t i p l e S e a r c h i n g p r o c e d u r e s . 3) C o m p a r e a n d e x c h a n g e s o l u t i o n 4) R e s t a r t i n i t i a l s o l u t i o n s . 5) U p d a t e t h e b e s t s o l u t i o n s

A. Generate several initial solutions.

The initial solution is important for the optimization methods. Because of the convergence to an optimal solution depends on the chosen initial solution. TSA is working with a single feasible initial solution in each iteration. But, MTSA work with several initial solutions, a number of solutions equal to a number of multiple searching.

Initialization Identify initial Solution Create empty Tabu List Set Best Solution = Solution Define Terminated Conditions Done = False Repeat If value of Solution < value of Best solution Then Best Solution = Solution If Terminated condition which is not satisfied have been met Then Add Solution to Tabu List If Tabu List is full Then Delete oldest entry from Tabu list End Else Terminated condition is satisfied D o n e = T R U E End Else Find New Solution by some transformation on Solution If solution is not improved, New Solution was searched for a long time then Randomly generate New Solution If New solution is not in Tabu List then Solution = New Solution End End Until Done= TRUE

Figure 1. Principle of Tabu Search algorithm B. Multiple Searching procedures

In recently, the personal computer have high speed computational. For solving the large scale problem may be uses several computers run in parallel. It is call “parallel searching”. The idea of multiple searching, we use only one computer for run multiple procedures. C. Compare and exchange solution

During the search, the solution will be compare with other solutions which obtain by other procedures and exchange the best solution. D. Restart initial solutions

When the searching is struck on the local solution for a long time and procedure of TSA can not escape from local solution, we will restart initial solution for select the new current solution.

E. Update the best solutions After already searching for each iterations, we

will select the best solution of the each procedures and set this solution as the optimal solution.

4. Problem and Results

Case 1: Three power-generating units, total load demand is 850 MW, without consider transmission line loss. The test system shown in Figure 2, the data of each generating units shown in table 1 [2,5] Table 1 Data of generating unit for case study 1 Unit ai bi ci Pmin(MW) Pmax(MW)

1 561 7.92 0.0016 150 600 2 310 7.85 0.0019 100 400 3 78 7.97 0.0048 50 200

Transmissionnetwork System load

150 - 600 MW

100 - 400 MW

50 - 200 MW

Unit 1

Unit 3

Unit 2

Figure 2 Schematic diagram of case 1

A result examination in case 1 shown in table 2

compared with Hopfield Neural Network [2], Genetic [5] and Tabu Search. The result shows that MTSA ’s fuel cost function value is less than HNN, GA and almost equal with TSA, but MTSA can converge to solution faster. The Figure 3 shows convergence to solution of MTSA compared with TSA.

Figure 3 Convergence to solution of case study 1

Case 2: Thirteen power-generating units, total load demand is 2520 MW, without consider transmission line loss. But valve-point loading effect is clearly considered, fuel cost function given by equation (6). The data of each generating units shown in table 3.[7] Figure 4 Input-output curves (I/O curve) and valve-point loading effect shown in generating unit 1, 2 and 4 respectively.

|)](sin[|)( min,2

iiiiiiiiiii PPfePcPbaPF +++= (6) Table 2 Computational result of case study 1

P HNN GA TSA MTSA P1 392.9013 393.198 384.528 384.528 P2 334.3876 334.789 327.452 327.452 P3 122.1394 123.004 138.020 138.020 PT 849.4283 850.991 850.000 850.000 FT 8194.203 8194.360 8194.201 8194.201

Table 3 Data of generating unit for case study 2

P(MW) Unit Min Max ia ib ic ie if

1 0 680 550 8.1 0.00028 300 0.035 2 0 360 309 8.1 0.00056 200 0.042 3 0 360 307 8.1 0.00056 200 0.042 4 60 200 240 7.74 0.00324 150 0.063 5 60 200 240 7.74 0.00324 150 0.063 6 60 200 240 7.74 0.00324 150 0.063 7 60 200 240 7.74 0.00324 150 0.063 8 60 200 240 7.74 0.00324 150 0.063 9 60 200 240 7.74 0.00324 150 0.063 10 40 120 126 8.6 0.00284 100 0.084 11 40 120 126 8.6 0.00284 100 0.084 12 55 120 126 8.6 0.00284 100 0.084 13 55 120 126 8.6 0.00284 100 0.084

Figure 5 shows comparison in convergence of

TSA and MTSA. The figure shows that MTSA can converge to solution faster than TSA and it can escape from a local minimum to find out near global minimum efficiently. Simulation result for this case shown in table 4, compared with GA [7] and TSA. The fuel cost of MTSA is less than GA and TSA.

Figure 4 Input-output curves in case study 2

Figure 5 Convergence to solution in case 2

Table 4 Computational result of case 2

P(MW) GA TSA MTSA

P1 628.32 628.317 628.3182 P2 356.80 299.206 299.1962 P3 359.45 331.991 299.1950 P4 159.73 159.733 159.7310 P5 109.86 159.711 159.7310 P6 159.73 159.744 159.7322 P7 159.73 159.739 159.7330 P8 159.73 159.742 159.7320 P9 159.73 159.700 159.7280 P10 76.92 40.009 77.3974 P11 75 77.720 77.3974 P12 60 92.348 92.3974 P13 55.00 92.335 87.7111 PT 2520.000 2520.000 2520.000 FT 24400 24314.755 24169.956

In comparison of both cases, it can be seen that

when a large-scale problem, MTSA can clearly obtain solution better than other methods and converges to near global minimum with less search account. 5. Conclusion

This Multiple Tabu Search Algorithm shows the efficiency for search the optimum solution. It is high efficiency than other method such as GA and HNN. Computational result, MTSA convergences to the best solution and faster than TSA. Moreover, proposed method can be applied to solve other problems as well.

Acknowledgements Thank North Eastern University and King

Mongkut’s Institute of Technology Ladkrabang for work place and source of knowledge.

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