Method for Minimum-Loss Reconfiguration of Distribution System by Tabu Search

YUJI MISHIMA,1 KOICHI NARA,1 TAIJI SATOH,1 TAKAMITSU ITO,2 and HIROTOSHI KANEDA2

1Ibaraki University, Japan2Meidensha Corp., Japan

SUMMARY

This paper proposes a loss minimum reconfigurationmethod by tabu search for an open-loop radial distributionsystem with distributed generators. The problem is to findthe optimal normal open sectionalizing switch positionswhich minimize the total distribution line losses subjectedto the line/transformer capacity constraints and voltageconstraint. Generally, the problem is mathematically for-mulated as a complex combinatorial optimization problemor mixed integer programming problem, and is solved byusing mathematical programming method, heuristic algo-rithm, intelligent method, and so on. However, a satisfac-tory algorithm for power companies has not yet beenattained in both computational burden and solution accu-racy. Thus, in this paper, the authors propose a method tosolve the above problem by using tabu search (TS). Reversepower flow caused by distributed generators can be in-cluded in the solution algorithm. TS is one of the meta-heu-ristic algorithms, and sometimes has been evaluated to bebetter than genetic algorithm (GA) or simulated annealing(SA) from viewpoints of both computational speed andsolution accuracy. In order to evaluate the validity andefficiency of the algorithm, several numerical examples areshown in this paper. © 2005 Wiley Periodicals, Inc. ElectrEng Jpn, 152(2): 18–25, 2005; Published online in WileyInterScience (www.interscience.wil ey. com ) . DOI10.1002/eej.20086

Key words: distribution system; loss minimiza-tion; load transfer; distributed generators; tabu search.

1. Introduction

Most power distribution systems in Japan are radialnetworks with an open loop, which are referred to asnormally open-loop networks. The distribution lines aredivided into multiple load sections by section switches. Theline loss strongly depends on the configuration of normallyopen section switches. Thus, the open switch positions mustbe optimized to reduce line losses. Various solutions to thisproblem have already been proposed [1–14]. In addition,with the recent trend toward connecting fuel cells, microgas turbines, and other small distributed generators to dis-tribution networks, the reverse power flow produced bysuch distributed generators must be considered when solv-ing the above problem. In this study, the authors propose amethod based on tabu search [15], a kind of meta-heuristics,to solve the problem of loss minimization for distributionsystems involving distributed generators. Solutions to theproblem of distribution loss minimization with regard todistributed generators are proposed in Refs. 11–13. Themethod proposed in Ref. 11 treats the distribution line loadsas a concentrated load. The authors have developed amethod of dealing with loads distributed equally in sec-tions, and have applied it to optimal configuration of dis-tributed generators aiming at minimization of distributionloss [12]. Similarly, in Ref. 13, the load is assumed to beequally distributed and tabu search is applied to distributionloss minimization, using the (n – 1) criterion for distributedgenerators.

This paper proposes a method to speed up calcula-tions even though distributed generators are connectedarbitrarily to a distribution system; in this method, neighborsolutions of tabu search are used.

The problem of loss minimization in a distributionsystem including distributed generators is formulated inSection 2, and Section 3 presents the proposed methodbased on tabu search. In Section 4, numerical examplesusing a medium-scale system model involving about 100

© 2005 Wiley Periodicals, Inc.

Electrical Engineering in Japan, Vol. 152, No. 2, 2005Translated from Denki Gakkai Ronbunshi, Vol. 123-B, No. 10, October 2003, pp. 1149–1155

Contract grant sponsor: Supported in part by a JSPS Grant-in-Aid (C-2-1365-0299).

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sections are given to verify the effectiveness and speed ofthe proposed method. The results obtained in this study arethen summarized.

2. Distribution Loss Minimization Problem withRegard to Distributed Generators

2.1 Definition and formulation of problem

This paper deals with a normally open-loop distribu-tion system involving distributed generators. In particular,the problem consists of finding the configuration of nor-mally-open switches that minimizes distribution loss whilesatisfying the given constraints. This problem is formulatedas a combinatorial optimization problem by means of Eqs.(1) to (6). As regards the validity of the assumptions listedbelow, the loads in actual power networks can only beapproximately estimated, which inevitably results in largeerrors: thus, the assumptions are not expected to signifi-cantly reduce accuracy.

Assumptions

1. The load in every section is a constant-current,known load.

2. The loads in all sections are distributed equally.3. The load power factor is 1.0.4. The lines have only resistance and reactance.5. The installation points and outputs of the distrib-

uted generators are known.6. The distributed generators are handled as constant-

current sources.

Objective function

Constraints Line capacity constraint

Transformer capacity constraint

Voltage upper/lower limit constraint

Power supply constraint

Feeder section connection constraint

Here Lossjt is the line loss in section j at time t; xij is avariable that takes a value of 1 when section j correspondsto feeder i, and a value of 0 otherwise; Ijt is the load insection j at time t; bik is the line capacity at the k-th controlpoint of feeder i; Jik is the set of sections from section k tothe feeder’s terminal (including k); bs is the capacity oftransformer s; Ti is the set of sections connected to trans-former s; Ji is the set of sections connected to feeder i; zj isthe impedance of section j; uqt = Iqt (q ≠ l), = Iqt/2 (q = l);Tij is the set of sections from the root of feeder i to sectionj; Vi is the voltage at the root of feeder i; Vij__, Vij

__ are the lower

and upper voltage limits in section j of feeder i; IDGmt is theoutput of distributed generators m at time t; and Dj is the setof distributed generators from section j to the feeder’sterminal (including j).

Objective function (1) is designed to minimize thetotal loss in all sections (Lossjt) during the period underconsideration. The line loss in section j at time t depends onthe reverse flow caused by the distributed generators [12],and its calculation algorithm is explained in Section 2.2.Equations (2) and (3) give the constraints on the currentcapacity of the lines and transformers, respectively, and theconstraints on the upper and lower voltage limits are givenby Eq. (4). Equation (5) states that a feeder is connected tosection j, and power must be supplied. Finally, Eq. (6) statesthat every section is connected to a feeder, and that thesections must be connected continuously from the begin-ning (source side to end of distribution line).

2.2 Distribution loss in presence of distributedgenerators

Figure 1 shows the current diagram when distributedgenerators are connected to a distribution line. In particular,

(1)

(2)

(3)

(4)

(5)

(6)

Fig. 1. Distribution line with distributed generators.

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the diagram plots the current flowing through the line onthe vertical axis, and the distance from the transformer onthe horizontal axis; the connection of distributed generatorsis shown by circles. The shaded portions show the outputcurrent of the distributed generators, which means addinga negative current.

As shown in Fig. 1, the differential coefficient of thecurrent flowing through the line has a discontinuity everytime the load section changes or a distributed generator isconnected. Therefore, assuming equally distributed loads,the distribution loss must be calculated for every continuousinterval. For this reason, the section h with distributedgenerators connected is divided into subsections hi.

Below we show the calculation of the distributionloss in the subsections depending on the capacities of thedistributed generators (magnitudes of reverse power flow).

(1) Case of no influence by distributed generators Ifthere is no distributed generator connected from section jthrough its feeder’s terminal, that is, if Eq. (7) is true, theline loss in section j can be found as in Eq. (8) [4], with Ijt

and Ijtl representing, respectively, the load current and tran-sit current in section j at time t (here rj is the line resistanceof section j):

(2) Case of distributed generators connected but noreverse power flow As shown in Fig. 2, if at time t, distrib-uted generators are connected from section j to the feeder’sterminal but their total output is small compared to thepassing current Ijtl without generators, that is, when Eq. (9)is true, the line loss in section j can be calculated by Eq.(10) [12]:

(3) Case of distributed generators connected withreverse power flow When a reverse power flow occurs insection j at time t and then ceases within the section asshown in Fig. 3, that is, when Eq. (11) is true, the line lossin section j at time t can be calculated by Eq. (12) [12]:

Here lj is the percentage of section j in which the reversepower flow disappears, as shown in Fig. 3.

(4) Case of distributed generators connected withreverse power flow in all sections When reverse power flowoccurs in the whole section j at time t as shown in Fig. 4,that is, when Eq. (13) is true, the line loss in section j at timet can be calculated by Eq. (14) [12]:

Fig. 2. No reverse power flow exists (with DG).

(7)

(8)

Fig. 3. Reverse power flow exists.

(9)

(10)

(11)

(12)

Fig. 4. Reverse power flow exists throughout section.

(13)

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3. Solution Method of Distribution LossMinimization Using Tabu Search

3.1 Algorithm outline

Since the problem in the above is a combinatorialoptimization problem of finding the switch configurationof a normally-open loop distribution network that mini-mizes the distribution loss, the tabu search is employed tosolve the problem. The general flowchart is shown in Fig.5. In the figure, the objective function of the initial solutionis calculated. Then the objective function is calculated forall neighbors of the current solution, and the solution offer-ing the best value of objective function is chosen. Theprocedure is repeated until a preset number of iterations isreached.

3.2 Objective function of tabu search

Tabu search is a full-area search method in which atabu list is created to prevent repetitive search for the same

solutions, and the neighborhood search progresses withprohibition of movement in the directions specified in thetabu list [15]. When tabu search is applied to the problemin question, constraints (5) and (6) can be satisfied bycreating neighbor solutions that do not violate the con-straints. However, as regards constraints (2) to (4), it isdifficult to create nonviolating neighbor solutions, or to addviolating solutions to the tabu list. Thus, we attempt to relaxthe constraints and incorporate them into the objectivefunction. The objective function of the tabu search is de-fined in Eq. (15). In this equation, the first term representsthe distribution line loss, the second and third terms repre-sent the constraint penalties on the current capacity of,respectively, the lines and transformers, and the fourth termrepresents the penalty regarding the upper and lower volt-age limits.

Objective function of tabu search

Here α1, α2, α3 are weight coefficients.

3.3 Definition of neighbor solution

Since the accuracy and processing time depend onhow the neighbor solutions are created, the following twotypes of neighborhoods are defined:

(1) Method 1. In the closed circuit obtained when oneof the currently open switches is closed, the neighborhoodis defined as the network configuration obtained by openingthe neighbor switches that can be operated [14]. For exam-ple, in Fig. 6, when switch (2) is closed, the availableswitches are (1), (2), (3), and (4). In this case, neighborsolutions are obtained by opening the neighbor switches of(2), that is, (1) or (3).

(2) Method 2. When one of the currently openedswitches is closed, the neighborhood is defined as thenetwork configuration obtained by opening the switches inthe section that has the smallest current in the closed circuit.In other words, as shown in Fig. 7, when a switch is closed,

(14)

Fig. 5. General flowchart of solution algorithm.

(15)

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the loop current ∆I flowing from a toward b can be calcu-lated in the following way by using Thevenin’s theorem:

Here Va and Vb are the voltages at points a and b before theswitch is closed, and Zloop is the impedance of the closedloop created by closing the switch (= Za + Zb), with Za andZb denoting the impedances from, respectively, a and b tothe transformer bus. In addition, Zs is the impedance of theclosed switch, which is small enough to assume that Zs = 0.

Denoting by Ia and Ib the transit currents in the sectionbefore the switch is closed, the transit currents change as inEq. (17) when the switch is closed, and the smallest currentcan be found from Eq. (18):

Here Imin is the smallest transit current in the closed loopformed by closing the switch, A is the set of sections frompoint a through the transformer bus, and B is the set ofsections from point b through the transformer bus.

In Method 1, two neighbor solutions are created forone open switch, and when there are N open switches, thenumber of neighbor solutions for the current solution maybe as large as 2N. On the other hand, in Method 2, themaximum number is N, which promises a faster search. Inaddition, in both cases, the processing time is reduced by

skipping objective function calculation for neighbor solu-tions for which the variation of the objective functionremains unchanged.

3.4 Tabu move and aspiration criterion

Here “tabu move” applies to the reopening of aclosed switch during a period determined by the tabulength. The aspiration criterion is the best value of theobjective function obtained so far.

4. Numerical Examples

4.1 Distribution system model

The proposed method was verified by simulations onthe distribution system model shown in Fig. 8, including103 sections, 112 switches, and 10 open switches [4]. In thediagram, switches are marked by ×, and switches in theinitial state are marked by ⊗. FD denotes the feeder number,and the number stands for the electric current (A) at thefeeder root. The parameters employed in the tabu search arelisted in Table 1. The tabu length and weight coefficientsα1, α2, α3 in Eq. (15) were determined by trial and error.

4.2 Results and discussion

Because of space limitations, this paper shows threenumerical examples:

Case 1: distributed generators are not connected.Case 2: one distributed generator is connected.Case 3: multiple distributed generators are connected.

The solutions found by the proposed method for every caseare presented in Figs. 9 to 11, and the objective functionsand processing time are given in Table 2. In the diagrams,the dotted-line circles represent switches that are open in

Fig. 6. Definition of neighborhood for Method 1.

(16)

Fig. 7. Current occurring when open switch is closed. Fig. 8. Distribution system model.

(17)

(18)

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the initial solution, and the solid-line circles representswitches that are open in the obtained solution. Blackcircles show locations at which distributed generators areconnected, and heavy lines represent sections in whichreverse power flow occurs. The figures next to the blackcircles denote the output current (A) of the distributedgenerators. For both methods, the tabu search was restrictedto 100 iterations, and solutions were obtained in both cases,as is evident from Table 2. However, Method 2 required lesscalculation time.

The validity of the results was checked by compari-son with results obtained for Case 1 by using the geneticalgorithm (GA) and simulated annealing (SA) proposed inRef. 4, as shown in Table 3. The open switch configurationobtained by the GA is presented in Fig. 12. The solutionobtained by SA is identical to that found by the proposedmethod. TS in Table 3 denotes the results obtained byMethod 2 (the same as the results for Case 1 calculated byMethod 2 in Table 2). Thus, the proposed Method 2 basedon tabu search provides a better objective function than GA,while obtaining a solution comparable to that of SA butmore rapidly.

Table 1. Parameters of tabu search

Fig. 9. Solution obtained for Case 1 (Methods 1, 2).

Fig. 10. Solution obtained for Case 2 (Methods 1, 2).

Fig. 11. Solution obtained for Case 3 (Methods 1, 2).

Table 2. Calculation results

Table 3. Calculation results by GA and SA

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5. Conclusions

This paper deals with normally-open loop distribu-tion systems involving distributed generators. In particular,a method of configuring normally-open switches so as tominimize distribution losses in normal operation is pro-posed. The proposed method can find the minimum-lossconfiguration even when reverse power flow is produceddue to distributed generators. The method has been verifiedby several simulations on a medium-scale distribution sys-tem model.

Acknowledgments

The authors express their gratitude to Miss A. Gojo,a master’s student at Ibaraki University, for her help withthe calculations. This study was supported in part by a JSPSGrant-in-Aid (C-2-1365-0299).

REFERENCES

1. Liu CC, Lee SJ, Venkata SS. An expert system opera-tional aid for restoration and loss reduction of distri-bution systems. IEEE Trans PWRS 1988;3:619–626.

2. Civanlar S, Grainger JJ, Yin H, Lee SSH. Distributionfeeder reconfiguration for loss reduction. IEEE TransPWRD 1988;3:1217–1223.

3. Shirmohammadi D, Hong HW. Reconfiguration ofelectric distribution networks for resistive line lossesreduction. IEEE Trans PWRD 1989;4:1492–1498.

4. Nara K, Shiose A, Kitagawa M, Ishihara T. Imple-mentation of genetic algorithm for distribution sys-tems loss minimum re-configuration. IEEE TransPWRS 1992;7:1044–1051.

5. Zhou Q, Shirmohammadi D, Liu WHE. Distributionfeeder reconfiguration for operation cost reduction.IEEE Trans PWRS 1997;12:730–735.

6. Borozan V, Rajakovic N. Application assessments ofdistribution network minimum loss reconfiguration.IEEE Trans PWRD 1997;12:1786–1792.

7. Lin WM, Chin HC. A new approach for distributionfeeder reconfiguration for loss reduction and servicerestoration. IEEE Trans PWRD 1998;13:870–875.

8. McDermott TE, Drezga I, Broadwater RP. A heuristicnonlinear constructive method for distribution sys-tem reconfiguration. IEEE Trans PWRS 1999;14:478–483.

9. Morton AB, Mareels IMY. An efficient brute-forcesolution to the network reconfiguration problem.IEEE Trans PWRD 2000;15:996–1000.

10. Jeon YJ, Kim JC, Kim JO, Shin JR, Lee KY. Anefficient simulated annealing algorithm for networkreconfiguration in large-scale distribution systems.IEEE Trans PWRD 2002;17:1070–1078.

11. Söder L. Estimation of reduction electrical distribu-tion losses depending on dispersed small scale energyproduction. Proc 13th Power Systems ComputationConference, p 1229–1234, 1996.

12. Nara K, Hayashi Y, Deng B, Ikeda K, Ashizawa T.Optimal allocation of dispersed generators for lossminimization. Trans IEE Japan 2000;120-B:672–677. (in Japanese)

13. Hayashi Y, Matsuki J, Takano H. Determinationmethod for loss minimum distribution system con-figuration considering disconnection of installed dis-persed generators. Trans IEE Japan 2002;122-B:1376–1383. (in Japanese)

14. Nara K, Mishima Y, Gojyo A, Ito T, Kaneda H. Lossminimum reconfiguration of distribution system bytabu search. Proc IEEE PES T&D Conference andExhibition 2002 Asia Pacific, Vol. 1, p 232–236.

15. Glover F, Laguna M. Tabu search. Kluwer Academic;1997.

Fig. 12. Resulting solution for GA.

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AUTHORS (from left to right)

Yuji Mishima (member) completed the first stage of his doctorate (electrical and electronic engineering) at AkitaUniversity in 1996, completed the second stage at Hokkaido University in 1998, and joined the faculty of Ibaraki University asa research associate. His research interests are planning and optimization of power systems and distribution networks. He holdsa Ph.D. degree, and is a member of ORSJ.

Koichi Nara (member) completed his M.E. program (electrical engineering) at Hokkaido University in 1970 and joinedMitsubishi Electric. He later served as a faculty member at Kitami Institute of Technology and at Hiroshima University, andhas been a professor at Ibaraki University since 1992. His research interests are planning and optimization of power systemsand distribution networks. He holds a D.Eng. degree, and is a member of ORSJ, IPSJ, and IEEE.

Taiji Satoh (member) completed his doctorate at Hiroshima University in 1985. After serving on the faculty there and atYamaguchi University, he has been an associate professor at Ibaraki University since 2002. His research interests are planningand optimization of power systems and distribution networks. He holds a D.Eng. degree, and is a member of ORSJ, IEICE,SICE, and IPSJ.

Takamitsu Ito (member) completed his M.E. program at Niigata University in 1997 and joined Meidensha Corp. He hasbeen engaged in analysis of power systems.

Hirotoshi Kaneda (member) graduated from Waseda University (electrical engineering) in 1994 and joined MeidenshaCorp. He has been engaged in research on wind power generation.

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