solving the capacitor placement problem in a radial distribution system using tabu search approach

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1868 IEEE Transactions on Power Systems, Vol. 11, No. 4, November 1996

§olving the Capacitor Placement Problem in a Radial Distribution System Using Tabu Search Approach

Yann-Chang Huang Hong-Tzer Yang*, IEEE Member Ching-Lien Huang, IEEE Member

Department of Electrical Engineering National Cheng Kung University

Tainan, 701 TAIWAN

*Department of Electrical Engineering Chung Yuan Christian University

Chung-Li, 320 TAIWAN

Abstract - In this paper, the capacitor placement problem in a radial distribution system is formulated and solved by a Tabu Search (TS) based solution algorithm. The capacitor placement problem considers practical operating constraints of capacitors, load growth, capacity of the feeder and the upper and lower bound constraints of voltage at different load levels to minimize the investment cost of capacitors and system energy loss. A sensitivity analysis method is used to select the candidate installation locations of the capacitors to reduce the search space of this problem apriori. Comparison results of the TS method with the Simulated Annealing (SA) show that the proposed TS method can offer the nearly optimal solution to the capacitor placement problem within reasonable computing time. Keyword$: Capacitor Placement, Tabu Search, Combinatorial Optimization

I. INTRQDUCTIQN Capacitors are often installed in distribution system for

reactive power compensation to carry out power and energy loss reduction, voltage regulation, system security improve- ment, and system capacity release. Economic benefits of the capacitor depends mainly on where and how many capacities of the capacitor are installed and proper control schemes of the capacitors at different load levels in the distribution system.

A variety of methods have been devoted to solving the capacitor placement problem. In the early work, most of the re- searchers used conventional analytical method in conjunction with some heuristics [l]. Duran 121 considered the capacitor sizes as discrete variables and employed dynamic programming to solve the problem. Regarding both location and capacitor size as continuous variables, a gradient search based iterative procedure was proposed to deal with fixed and switched type capacitor installation problem. 133.

Chiang et al. [4] used the optimization technique, simulated annealing, to search the global optimum solution to the capacitor placement problem, which is formulated as a discrete combinatorial optimization problem in [5 ] . The simulated annealing method can provide the nearly global optimal

solution, but the associated computational burden is heavy. Sundhararajan and Pahwa [6] proposed the genetic algorithm approach to determine the optimal placement of capacitors. However, the setting of control parameters somewhat depends on experiences and remains as an open problem to be studied.

TS is a strategy for solving combinatorial optimization problem. The TS strategy has been applied in various fields, and the capability of the TS to obtain high quality solutions within reasonable computing time has been verified. The TS method is built upon a descent mechanism of a search process. The descent mechanism biases the search toward points with lower objective function values, while special features are added to avoid being trapped in the local minima [7,8].

Based on this mechanism, this paper has successfully applied the TS method to solve the capacitor placement problem. Problem description of the capacitor placement is first addressed. Then the general scheme of the TS method and its applications to the capacitor placement problem are presented. Finally, the numerical results of the TS method tested in a 69-node radial distribution system [5] are compared with those of the existing SA method.

11. PROBLEM DESCRIPTION The capacitor placement problem considered in this paper

is to determine the locations, types, number and sizes of capacitors to be installed in a radial distribution system, and the control schemes of the capacitor at Werent load levels. The objective is aimed to reduce the energy losses in the system and retain the voltage magnitudes of the system within pre- scribed maximum and minimum allowable values for different load levels while minimizing the total cost of the system.

Since the capacitor sizes as well as control schemes are treated as discrete decision variables, the capacitor placement problem is formulated as a combinatorial optimization problem with a non-differentiable objective function. For instance, as shown in Fig. 1, the investment cost of the capacitor is expressed as a non-differentiable step-like function of capacity.

The yearly load duration curve to be served by the distribution system is shown in Fig. 2. The period of the yearly load is divided into a number of intervals. During each interval, the load level is assumed constant and a cost value associated with per-unit energy loss is given. For an appropriate planning horizon (10 years considered m this paper), different load duration curves for each year are employed to take into

-account the load growth and maximum Capacity of the distribution lines.

36 WM 243-6 PWRS A paper recommended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for presentation at the 1996 IEEE/PES Winter Meeting, January 21- 25, 1996, Baltimore, MD. Manuscript submitted August 1, 1995; made available for printing November 29, 1995.

0885-8950/96/$05.00 0 1996 IEEE

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225200

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56300

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1 I I I I I

A load level I L1 h L2J -

T I T2 T n Fig. 2 Load Duration Curve I

I n L a e

I A. Problem Formulation

The objective function of the problem can be expressed as follows to minimize the capacitor investment cost and system energy loss:

subject to Pn,,,,.(2, q') =O (power flow constraints) (2) vy"" I 14 I vy"= (voltage constraints) ( 3 )

q f = d = q P Z , j E { 1 , 2 , 3 ,..., L )

OldIqP i e {1 ,2 ,3 ,..., I )

q', : discrete variables, for fixed typed capacitor:

for switched type capacitor (4 )

(5) In above formulation, qo is the sizing vector whose com-

ponents are multiples of the standard size of one capacitor bank; q' is the control scheme vector at load level j whose components are discrete variables; C,(qo) represent the investment cost associated with the capacitor installed at location i, whose cost function, as shown in Fig. 1, is non-differentiable; Plowd is the power loss at load level] with a time duration T, and ked is different energy loss cost for each load level; xk=[Pk,, Q, , I V,l'], zk=[P,, Qk] represent state variable vectors of real and reactive powers Pb Q, as well as squared voltage magnitude lV,lz at branch k; L and I denote numbers of load levels and candidate locations to install the capacitors.

B. Power Flow Equations The real and reactive power flows at the receiving end of

branch k+Z, Pk+,, Qk+I, and the voltage magnitude at the send- ing end I Vkl, as shown in Fig. 3 , can be expressed by the following recursive set of equations [ 5 ] :

P ~ + I = pk -rk+l(P; + Q;)/Ivk12 -.P~,k+l

Qk+l = Q k - ~ k + i ( P i + Q i ) / l V k I ~ -QL.~+I -t QC,k+i

(7)

(8) In the distribution system power flow equations, several

1) at the substation, the voltage magnitude IVo1 is given, 2) at the end of the main feeder:

boundary conditions must be satisfied:

P&#= 0 (9) Q,= 0 (10)

Ph= 0 ( 1 1 ) Qh= 0 (12)

3) at the end of lateral k:

where n and rn are the numlber of nodes of the branches 0 and k, respectively.

The power losses in the network can be calculated as the sum of the i% loss in each branch, The total power losses can be calculated by

(13)

where r,,, is the resistance of the (j+l)th node of the ith branch, and b is the number of branches, m, is the number of nodes of the ith branch.

C. Sensitivity Analysis To consider the practical constraints, heuristic and engi-

neering judgments are first used to select the potential locations where capacitors can be installed. Then a sensitivity analysis approach is further incorporated into the algorithm to determine the candidate locations for placing the capacitor in the distribution system. A priori estimation of these candidate locations can help to reduce the search space of the optimization problem.

The sensitivity analysis IS a systematic procedure to select those locations which have: maximum impaict on the system real power losses with respect to the nodal reactive power. Equation (14) is used to evaluate the influences of nodal reactive power change on the power losses in the selection of candidate capacitor installation sites:

Fig. 3 One-line diagram of a distribution feeder

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111. Tasu SEARCH METHOD Fundamentals of TS are the use of flexible memory of

search history and thus guide the search process to surmount local optimal solutions. Basic features that are needed to im- plement the TS [7,8] are described briefly in this section.

A. Moves and Selection The first step of TS is to define a set of moves that may be

applied to a trial solution to produce a new one. Among all the neighboring solutions thus produced, TS seeks the one that improves most the objective function. In certain situa- tions, if there are no improving moves, a fact which means some local optimum exists, TS chooses the one that least de- grades the objective function.

In the capacitor placement problem, the process searching from a vector of trial solution to the other is called a move. A vector of trial solution denotes the candidate capacitor locations, the capacitor setting (control schemes) for different load levels, the installed capacity, the power losses, the voltage magnitude, and the objective function value corresponding to this move, and the frequency counter. The frequency counter is used to indicate total times that the solution has been visited. The solution structure is represented as follows:

structure solution {candidate locations, capacitor setting, installed capacity, power losses, voltage magnitude, objective function value, frequency counter)

B. Tabu List In order to avoid returning to the local optimum just vis-

ited, the reverse move that is detrimental to achieving the optimum solution must be forbidden. This is done by storing this move in a data structure, such as a finite length first-in-first- out structure, called tabu list. The elements of tabu list are called tabu moves. Due to the tabu moves, we can keep the search bias toward points with lower objective function values and escape from local optimum solution. As soon as a trial solution is generated, the trial solution is checked to be whether in this tabu list or not. If so, the corresponding search space is greatly reduced (e.g., one-third, called the reduction rate of search neighborhood) of the normal range.

The dimension of the tabu list is called tabu list size. The choice of tabu list size is critical. If the size is too large, appealing moves may be forbidden and higher quality solutions thus can not be explored. While if this value is too small, cycling may occur in the search process and the process often returns to the solution just visited. Empirically, tabu list sizes should grow with the size of the problem. In this paper, we use a dynamic link list to serve as the tabu list, whose size is 15 to record the solution states just visited. This means the move will be kept tabu for a duration of 15 moves. The older tabu move is released from the address of the tabu element. The new tabu move is attached to the tabu list by allocating the memory and giving a erasing pointer to it. C. Aspiration Criterion

Since the tabu list may forbid certain worthy or interesting

moves possibly leading to a better solution than the best one found so far. An aspiration criterion is used to allow tabu moves to be released if they are judged to be worth or interesting. In other words, the aspiration criterion is to allow "exce- llent" tabu moves to be selected if the aspiration level is attained.

In the paper, an aspiration criterion takes the form: if a tabu move (tabu-move) from the current trial solution S, can reach a solution (&+tabu-move) better than the best so far solution 9, obtained up to iteration k then this tabu-move can be attractive or available to be released i.e., if

f(S,+tabu-move) ) then the tabu restriction will be overridden, and (S,+tabu-move) is viewed as the next trial solution. H e r e p ) is the objective function to be minimized.

D. Intensificution and Diversijicution

In order to obtain optimal solution, TS use intensification and diversification techniques. The former means intensification of the search in the neighborhood of the sub-optimal solution, the latter means diversification of the search to so far unexplored regions of the solution space. If intensification is missing, the search becomes an iterated random sampling; if diversification is missing, the process may be trapped in a sub-optimal region.

The frequency counter in the solution structure is used in the intensification and &versification techniques. The frequency counter denotes the times the solution (or the move) having been visited throughout the solution process.

For example, according to the size of neighborhood to be searched in the capacitor placement problem, a threshold (named the frequency counter threshold) is set at three, that is to say, if the solution has been visited three or more times, it will be heavily penalized and thus loose its attractiveness. Search will, therefore, be directed to so far unexplored region of the solution space to diversify the search process. When the frequency counter is less than three times, intenslfyrng the search in the neighborhood of the sub-optimal region is possible. In this manner, the tabu search is expected to be able to find the optimal solution to the combinatorial optimization problem.

IV. SOLUTION ALGORITHM FOR CAPACITOR PLACEMENT In this section, a step by step TS based solution algorithm

for the capacitor placement problem is presented. Step 1: Input system data. Input system configuration, network data and parameter setting (e.g., lower and upper bounds on operating voltage, tabu list size, etc.). Step 2: Conduct sensitivity analysis. We select those nodes which have maximum impact on the system real power losses with respect to the nodal reactive power as the candidate locations to install the capacitors. Step 3: Generate an initial feasible solution state.

1) Randomly select a solution state from the solution space. 2) For each load level, execute the distribution power flow to

check feasibility. If any constraint is violated, go to I ) ;

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otherwise proceed to next step. Step 4: Perform Tabu search procedure. 1) Select best move as next move direction from the move

2) Check feasibility as in Step 3-2). If not feasible, go to 1);

Step 5: Update the best solution state. If the move is not tabu and lead to another solution state better than so far visited, or if the move is tabu but aspiration criterion is attained, then update the best solution state. Step 6: Sethelease tabu move. Record the executed move as tabu move in the tabu list, or release the tabu move if the aspiration criterion is attained. Step 7: Check the stopping criterion. If the change of the objective function value of successive best so far solution is less than a given value, then stopping criterion is satisfied and go to next step; otherwise, return to Step 4, and continue the tabu search procedure. Step 8: Output the optimal solution state. Output of the solution algorithm includes the optimal locations, types, number and sizes of capacitors to be installed and the optimal control schemes for different load levels.

V. NUMERICAL RESULTS The proposed TS based solution methodology for capacitor

placement has been implemented using Turbo-C language and executed at a Personal Computer PC 586-90. The test system is a 69-node radial distribution system which includes one main feeder and seven laterals [ 5 ] , as shown in Fig. 4. The system and load data can be found in [ 5 ] . The control parameters of TS were set at 15 of the tabu list size, 1/3 of the reduction rate of search neighborhood, and 3 of the frequency counter threshold.

In this paper, we consider ten-year planning horizon. For the first three years, a yearly load growth rate of 9.55% was assumed. After that, since the peak load has reached the maximum capacity of the feeder, 5,000 kW, the load was assumed constant till the end of the planning horizon. The investment cost of the capacitors included installation cost, purchase cost of the capacitor and the associated protection equipment (arrester and fuse cutout). Timer and oil switch were also included in the cost of switched type capacitors. The investment costs for fixed typed capacitor were NT$56,300/bank and NT$74,900/bank for switched type capacitor with one bank of 300 kvar. The load level and energy loss cost data used are given in Table I.

For practical installation space consideration, maximum capacity of the installed capacitor is of four banks (1,200 kvar). The capacities are regarded as discrete variables and as multiples of a standard bank (300 kvar). In this paper maximum number of locations to be installed was limited to five. After sensitivity analysis, we selected ten candidate locations to install the capacitors, i.e., node 7, 10, 11, 20, 36, 37, 38, 48, 50 and 53 in the test system of Fig. 4.

Based on the results of the sensitivity analysis, two cases are studied for fixed type and mixed fixedswitched type

set based on the objective function evaluated,

otherwise proceed to next step. 15 16 17 18 I9 20 21 22 23

tt. \ \rii” I - h M H 42 43 444546 474849 5051 5:’ 53 n

27282930 31 32 33 34

ffl

I Fig. 4 The diagram of a 69-bus test systcem I

TABLE I DATA ON LOAD AND ENERGY LOSS Con

Load Level Time Intervals Energy Loss Cost (first vear’l (hours) mr$flcwh)

Level L, L, L3 T, T, T, ke, ke, ke3 Value 1 0.8 0.5 1.000 6.760 1.000 0.7 1.78 2.95

capacitor placement problems, respectively. The results obtained from the proposed ‘TS method were compared to the existing SA approach in the aspects of total system cost and computing time.

Case I

In Case 1, we present the test results for Ihe fixed type capacitor placement problem of the studied system using the solution algorithm proposed in Sec. IV. The computing time of the proposed methodology is 13 sec. in contrast to 16 sec. of the SA for the same optimal solution achieved. The energy loss cost, investment cost, and total system cost for different number of placement locat ion are shown in Table I1 and Fig. 5 . The optimum number of placement location is equal to 3 (Table I1 and Fig. 5). The optimal locations and sizes are in nodes 20, 50 and 53 with 300 kvar (one bank) installed in node 20, with 600 kvar (two banks) in node 50, and with 300 kvar (one bank) in node 53. As noted, the optimal control schemes are same as the capacity installed. The test results are summarized in Tables I11 and IV.

The total real power losses of ten years arid voltage magnitude profile of the test system witWwithouit fixed capacitor placement are given in Table IV. To investigate the robust- ness of the TS method, 1100 random initial solutions were

TABLE 11 SYSTEM COST FOR DIFFERENTNUMBER OF PLACE WENTLCCATTON

IN CASE 1 No. of Energy Loss Investment Total System

Location Cost ( N T $ ) Cost (NT$) - Cost (NT$) 0 38,824,378 0 38,824,378 1 29,718,2162 168,900 29,887,862 2 27,750,792 225,200 27,975,992 3 26,623,125 225,200 26,848,325 4 26,610,2.70 337,800 26,948,070 5 26,632,194 394,100 27,026,294

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total system cost energy loss cost investment cost

- energy loss cost + investment cost

WT$) OJT$)

yo- total system cost

2sE+7 t / 1 1.OE+5

2.OE+7 O.OE+O 0 1 2 3 4 s

number of location Fig. 5 Total system cost for different number of

capacitor placement location in case 1

TABLE ID OPIIMAL SOLUTION IN CASE 1

Optimal Control Setting (kvar) Optimal Location light load medium load peak load Sizes (kvar)

20 300 300 300 300 50 600 600 600 600 53 300 300 300 300

TABLE IV SYSTEM WITH /WITHOUT FIXED TYPE CAPACITOR PLACEMENT FOR

THREE LOADLEVELS IN CASE 1 Real Power Losses (kW) Voltage (P.u.)

light load medium load peak load Vmin Vmax Case -- Without 825.71 2,265.68 3,723.36 0.9014 1.0 capacitor

capacitor With 568.73 1,536.21 2,623.76 0.9523 1.0

Total System Cost (NT$) 2.70E+7

2.69E+7

2.68E+7

2.67E+7

2.66E+7 0 25 50 7s 100

Note: Min. Cost: 26,848,325 Max. Cost: 26,866,501 Average Cost. 26,850,589

Fig. 6 Distribution of final solutions of 100 m in case 1

given. The total system cost distribution of final solutions of the 100 runs is shown in Fig. 6, where the statistics of the solutions are listed as well.

Case 2

In this Case, the test results for the mixed fixedswitched type capacitor placement in the test system are presented.

TABLE V SYSTEM COSTFOR DIFFERENTNUMBER OFPLACEMENTL~CATTON

IN CASE 2 No. of Energy Loss Investment Total System

Location cost (NT$) C&t (NT$) Cost @IT$) 0 38,824,378 0 38,824,378 1 29,224,117 225,200 29,449,317 2 27,057,930 412,200 27,470,130 3 25,958,263 430,800 26,389,063 4 25,945,210 449,900 26,395,110 5 25,934,994 506,200 26,44 1,194

total system cost energy loss cost investment cost

~ energy loss cost WT$) 4.OE+7 -8- investmentoost 1 6.OE+5

WT$)

5.OE+5 3.5E+7

4.OE+5

3.OE+7 3.OE+5

2.OE+5

-O-- total system cos

2.5E+7 1 .OE+S

2.OE+7 O.OE+O 0 1 2 3 4 5

number of location Fig 7 Total system cost for different number o f

capacitor placement location in case 2

Planning results of the TS method are exactly the same as those of the SA. The computing time for the TS is 23 sec. and that for the SA is 28 sec. The energy loss cost, investment cost, and total system cost for different number of placement location are shown in Table V and Fig. 7. The optimum number of placement location, as shown in the Table V and Fig. 7, is also equal to 3. The optimal sizes and locations of capacitors are 600 kvar (two banks, switched type) installed in node 11, 900 kvar (three banks, switched type) installed in node 50, and 300 kvar (one bank, fixed type) installed in node 53. Ta- ble VI shows the optimal control schemes and sizes of the

TABLE VI O ~ M A L SOLLITION IN CASE 2

Optimal Control Setting (kvar) Optimal Location light load medium load peak load Sizes (kvar)

11 300 600 600 600 50 300 600 900 900 53 300 300 300 300

TABLE VI1 SYSTEM WITH /WITHOUT MIXED TYPE CAPACITORPLACEMENTFOR

THREE LOAD LEVELS IN CASE 2 Real Power Losses (kW) Voltage (P.u.)

light load medium load peak load Vmin Vmax Case

Without 825.71 2,265.68 3,723.36 0.9014 1.0 capacitor

With 560.73 1,522.48 2,456.29 0.9584 1.0 canacitor

Total System Cost (NT$) 2 66E+7

2.65E+7

2.64E+7

2.63E+7

2.62E+7 0 25 50 75 100

Note: Min. Cost: 26,389,063 Max. Cost: 26,417,998 Average Cost: 26,391,850

Fig. 8 Distribution of final solutions of 100 runs in case 2

installed capacitors. Given in Table VI1 are total real power losses of 10 years and voltage magnitude of the test system with/without mixed capacitor placement are . The total system cost distribution of final solutions of the 100 runs with different random initial solutions is shown in Fig. 8.

From the test results, we have the following observations: In each case, minimum and maximum voltages are within allowable range (0.95-1.05 P.u.) for all different load levels . Voltage and energy losses have been greatly improved with capacitor placement, and mixed fixedswitched type capacitor placement is better than the fixed type capacitor placement. Total system costs (including energy loss cost and investment cost) have been greatly reduced with mixed fixedswitched type capacitor placement when compared with that of the fixed type (refer to Tables V and 11).

VI. CONCLUSIONS In this paper, we have proposed a TS based solution meth-

odology to determine ( i ) the installed locations of capacitors, ( i i ) the types, number and sizes of capacitor, and (iii) the control schemes of these capacitors at different load levels. A sensitivity analysis based method was used to a priori select the candidate locations to install the capacitors and reduce the solution space of the problem.

The effectiveness of the TS method to solve the combinatorial optimization problem of capacitor placement has been demonstrated through the numerical examples. In our experiences, control parameters of the TS, e.g., the tabu list size, the reduction rate of search neighborhood, and the frequency counter threshold, are easily tuned in the solution process. Comparing the results of TS with those of SA reveals that the proposed solution methodology can offer .the nearly optimal solution to the capacitor placement problem within less computing time. Future possible practical applications of the proposed TS based method for the capacitor placement problem and other combinatorial optimization problems in the power systems are thus encouraged.

VII. ACKNOWLEDGMENTS

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The authors are grateful to Mr. T.C. Liang, senior engi- neer of Tainan area T&D system of Taiwan Power Company, for his valuable discussions; and suggestions of the work presented in this paper. Financial supports from the National Sci- ence Council, TAIWAN, R.O.C. are also acknowledged

VIIIL REFERENCES M. Kaplan, "Optimization of number, location, size, control type, and control setting of shunt capacitors on radial distribution feeder," IEEE Trans. on Power Apparatus and Systems, vol. 103, No. 9, pp, 2659-2663, September 1984. H. Duran, "Optimum number, location, and size of shunt capacitors in radial distribution feeder: A dynamic programming Ap- proach," IEEE Trans. on Power Apparatus and Systems, vol. 87,

J.J. Grainger, S. Civanlar, and K.N. Clinard, "Optimal voltage dependent continuous-time control of reactive power on primary feeders," IEEE Trans. ofit Power Apparatus and Systems, vol. 103, No. 9, pp. 2714-2722, September 1984. H.D. Chiang, J.C. Wang, 0. Cockings and H.D. Shin, "Optimal capacitor placements in diistribution systems: Part I and Part 11," IEEE Trans. on Power Lklively, vol. 5, No. 2, pp. 634- 649, January 1990. M.E. Baran and F.F. Wu, ''Optimal capacitor placement on radial distribution system," I'EEE Trans. on Power Delivery, vol. 4,

S. Sundhararajan and A. Fahwa, "Optimal selection of capacitors for radial distribution sysl.ems using a genetic algorithm," IEEE PESSummer Meeting, paper no. 93 SM 4994,1993. F. Glover, "Tabu search - Part I," ORSA J. Cornput., vol. 1, pp.

F. Glover, "Tabu search - Part II," ORSA J. Cornput., vol. 2, pp.

NO. 9, pp. 1769-1774, J~tnl~ary 1983.

NO. 1, pp. 725-734, J ~ ~ w Y 1989.

190-206, 1989.

4-32,1990.

Hong-Tzer Yang received the B.19. and M.S. degrees in electrical engineering from National Cheng-Kung Univasity, Tainan, Taiwan iin 1982 and 1984, respectively. He received his Ph.D. dlegree in electrical engineering from National Tsing-Hua University, Hsin-Chu, Taiwan in 1989. He was a technical superin- tendent of Chung Shan Institute of Science and Technology from 1989 to 1995. He is now an associate professor of' Electrical Enginee'ring Department at Chung Yuan Christian University. His present research interests are neural networks and expert system applications in power systems. Dr. Yang is a member of Phi Tau Phi, and the IEEE PES and CSS.

Yann-Chang Huang received the B.S.E.E. degree from National Taiwan Insti- tute of Technology, Taipei, Taiwan, in 1993. He is currently working toward his Ph.D. degree at E.E. Department of National Cheng-Kung University, Tainan, Taiwan. His research interests are on the planning and automation of power transmission and distribution systenls.

Ching-Lien Huang received B.S. idegree in electrical engineering fiom National Cheng-Kung University, Tainan, 'Taiwan, in 1957, and M.S.E.E. degree from Osaka University, Osaka, Japan, in 1973. Since 1964, he has been with the De- partment of Electrical Engineering, National Cheng-Kung University, where he is now a professor. His major research interests are on high voltage engineering, power system switching surge and protection, and power system planning.

solving the capacitor placement problem in a radial distribution system using tabu search approach

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