service restoration in distribution system using non-dominated sorting genetic algorithm

Electric Power Systems Research 76 (2006) 768–777

Service restoration in distribution system usingnon-dominated sorting genetic algorithm

Yogendra Kumar ∗, Biswarup Das 1, Jaydev SharmaDepartment of Electrical Engineering, Indian Institute of Technology, Roorkee 247667, India

Received 22 May 2005; received in revised form 21 September 2005; accepted 26 October 2005Available online 9 December 2005

Abstract

In this paper, a non-dominated sorting genetic algorithm-II (NSGA-II) based approach is presented for service restoration in power distributionsystems. In contrast to the conventional GA based methods, the proposed approach does not require weighting factors required for conversionof multi-objective function into an equivalent single objective function. Based on the simulation studies carried out in four different systems, theperformance of the proposed scheme has been found to be better than the performance of conventional GA technique based approaches. Moreover,by including the string representing the pre-fault configuration of the distribution system in the initial population, the speed of convergence ise©

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eywords: Genetic algorithm; Service restoration; Distribution system

. Introduction

Due to ever increasing demand of electric power, the size andomplexity of the modern day power systems are also increas-ng rapidly. Because of this enhanced size and complexity of theystems, the likelihood of occurrence of fault and size of the areaffected by faults have also enhanced significantly. The durationf fault and the size of the area affected by the fault directly affecthe safety and satisfaction of the customers, which, in turn, haveprofound effect on the revenue earned by the electricity supplyompany. As a result, fastest restoration of supply to the maxi-um possible out-of-service area is one of the most important

asks for any power supply company. After the service restora-ion also, from economic point of view, there should be minimumoss in the system. Now, the topology of the distribution systemhanges because of service restoration operation. However, dueo various reasons, such as ease of fault location, fault isolationnd protective device co-ordination, etc., power distribution sys-ems are often required to operate in a radial fashion. Hence, it

is important to maintain this radiality of the systems, even whenthe topology of the systems is undergoing changes during theservice restoration process. Moreover, because of the varyingtopology and the connected loads, the bus voltages and line cur-rents do also change during the service restoration process. Tomaintain the safety and security of different power system com-ponents (such as transformer and lines, etc.), it is important thatthe bus voltage and line current should not cross their respectiveoperational limits. Additionally, service restoration is essentiallyaccomplished by transferring the loads in the out-of-service areato the neighboring supporting feeders via ‘ON–OFF’ control ofdifferent switches in the distribution system. As the time taken bythe restoration process depends on the number of switching oper-ations, it follows from the above requirement that the numberof switching operations should be kept as minimum as possible.Also, the software run-time required by the service restorationalgorithm needs to be minimum for speedier solution.

From the above discussion, it is apparent that the servicerestoration task is a multi-objective, multi-constraint optimiza-tion problem. In the literature, different approaches have beensuggested to solve this complex problem. Both the heuristic rule

∗ Corresponding author. Tel.: +91 1332 855589; fax: +91 1332 273560.E-mail addresses: [email protected] (Y. Kumar),

[email protected] (B. Das), [email protected] (J. Sharma).1 Member, IEEE.

based approach [1–9] and expert system based approach [10–16]essentially attempt to capture the knowledge base of the oper-ators which they use to determine the switching sequences forsupply restoration under fault conditions. This knowledge base

378-7796/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.epsr.2005.10.008

Y. Kumar et al. / Electric Power Systems Research 76 (2006) 768–777 769

is typically stored in the form of ‘rules’, which are used by thesetwo approaches for arriving at the appropriate solution. However,acquisition of the knowledge base of the operators for this pur-pose is often a very difficult task. Mathematical programmingapproaches [17–19] have been proved to be computationallyvery costly for large systems. In fuzzy set approaches [20–22],out-of-service load, number of switching operation, bus volt-age, line current, loading of transformer, etc. are taken as fuzzyvariables and the solution is found on the basis of maximummembership function. But it also does not guarantee the optimalsolution. In Ref. [23], algorithm based on ‘interested tree’ hasbeen suggested. But for large system, the number of ‘interestedtree’ may be quite high, thereby making this approach compu-tationally unattractive.

To solve the above-mentioned limitations, GA based tech-niques have been proposed in the literature [24,25]. In thisapproach, initially the multi-objective optimization problem isconverted into a single objective optimization by using weight-ing factors, and subsequently GA is used to solve this singleobjective optimization problem. Now, the values of the weight-ing factors depend on the importance of the objective functionsas well as on the scaling of the objective functions and con-straints. Although the importance of the objective functionsdoes not generally vary from network to network, the valuesof the objective functions and constraints vary from network tonetwork. As a result, the scaling factors vary from network tonftnipf

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Objective functions:(1) Minimization of out-of-service area:

Min f1(x) =b1∑

i=1

Li −∑

i∈B

Li (1)

x is the network configuration undergoing service restora-tion, represented by status of switches, b1 the no. ofbuses in the network before fault, B the set of energizedbuses in the restored network and Li is the load on ithbus.

In Eq. (1), it is assumed that in a ‘n’ bus power system,the buses are numbered from 1 to n and in the pre-faultcase, all the buses in the network are energized. Therefore,‘b1’ is equal to ‘n’. However, in the post-fault scenario, allthe buses would not be necessarily energized. Hence, ‘B’would contain only the energized buses. For example, in afive bus system, b1 = 5 and if, in the post-fault case, bus 3cannot be energized, then B = (1, 2, 4 and 5).

(2) Minimization of no. of switch operation:

Min f2(x) =s∑

i=1

|swi − xi| (2)

s is the no. of switches in the network, swi the status of ith

etwork which, in turn, causes the variation of weighting factorsor different networks. Hence, for every network, the weights areo be tuned. In the fuzzy-GA approach [26], the multi-objectiveature of the service restoration problem is retained by represent-ng each objective function by a suitable fuzzy set. However, thearameters of the fuzzy sets need to be tuned for every networkor satisfactory performance.

To alleviate the above problems, in this work, a non-ominated sorting genetic algorithm-II (NSGA-II) [27] basedpproach is proposed for solving the service restoration prob-em. In this technique, the multi-objective nature of the serviceestoration problem is retained without the need of any tunableeights or parameters. As a result, the proposed methodology

s generalized enough to be applicable to any power distribu-ion network. To improve the performance, in this method, thelite-preserving operator, which favors the elites of a populationy giving them the opportunity to be directly carried over to theext generation, is used. Rudplph [28] has proved that GAs con-erge to the global optimal solution in the presence of elitism.long with convergence, it is also desired that GA maintains aood spread of solutions in the obtained set of solutions (callediversity). The diversity, in this method, is achieved with the helpf the crowded tournament selection operator (CTSO) that doesot require any tuning parameter. CTSO used in this paper hasower computational complexity and hence the run time is saved.

. Problem formulation

The objective functions and the constraints of the serviceestoration problem are described below:

switch just after fault (decided by the protective devices)and xi is the status of ith switch in the restored network.

(3) Minimization of losses:

Min f3(x) = losses in x (3)

Constraints:(1) Radial network structure should be maintained.(2) Bus voltages should be within acceptable range.

vmin < vi < vmax (4)

vmin is the minimum acceptable bus voltage, vi the voltageat ith bus and vmax is the maximum acceptable bus voltage.

(3) Line currents should be within acceptable range

imin < ii < imax (5)

imin is the minimum acceptable line current, ii the currentin the ith line and imax is the maximum acceptable linecurrent.

In this work, it is assumed that the preferential knowl-edge of the objective function considered herein is known.In this paper, from the point of view of customer’s satis-faction, the first objective function (i.e. minimization ofout-of-service area) is kept on first preference. To improvethe customer’s satisfaction and reliability of service, therestoration time should be minimum and hence the secondobjective (i.e. minimization of number of switch opera-tions) is kept on the second preference. Finally, the objec-tive function of minimization of losses has been kept onthe third preference.

770 Y. Kumar et al. / Electric Power Systems Research 76 (2006) 768–777

3. Non-dominated sorting genetic algorithm-II

Non-dominated sorting genetic algorithm is essentially amodified form of conventional GA. Like conventional GA, italso uses selection, crossover and mutation operator to createmating pool and offspring population. The step-by-step proce-dure of NSGA-II for one generation is described here for readyreference. However, the following terminologies, which are cen-tral to the concept of NSGA-II, are described in detail in Refs.[27,29], and hence their definitions are not repeated here: (a)domination, (b) constrained-domination, (c) crowded tourna-ment selection operator and (d) crowding distance. The basicalgorithm of NSGA-II is as follows:

Step 1: Initially a random parent population Po of size N iscreated (i.e. N is the number of strings or solutions inPo). The length of each string is LS (i.e. LS is the numberof bits in each string).

Step 2: Create offspring population Qo of size N by applyingusual GA operators (i.e. selection, crossover and muta-tion) on Po.

Step 3: Assign Pt = Po and Qt = Qo, where Pt and Qt denotethe parent and offspring population at any general ‘tth’generation, respectively.

Step 4: Create a combined population Rt = Pt ∪Qt. Thus, thesize of R is 2N.

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included non-dominated front are discarded from Pt+1.To choose the solutions to be discarded, initially thesolutions of the last included non-dominated front aresorted according to their crowding distances and sub-sequently, the solutions having least (n−N) crowdingdistances are discarded from Pt+1.

Step 7: Create the offspring population Qt+1 by application ofCTSO, crossover and mutation operator on Pt+1. InCTSO, the winner (better) solution is selected by com-paring two solutions based on their rank and crowdingdistance. The solution having lower rank is declaredwinner. If two solutions have same rank, the solu-tion having higher crowding distance is declared win-ner. Now, to create offspring, two solutions are pickedup randomly from the parents’ population, and subse-quently the winner of these two solutions is collected.This process is repeated till the number of solutions col-lected is lesser than size of population. After collectingrequired number of solutions, crossover and mutationoperators are applied on collected solutions.

Step 8: Test for convergence. If the algorithm has convergedthen stop and report the results. Else, t← (t + 1),Pt←Pt+1, Qt←Qt+1 and go back to step 4.

4. Implementation of NSGA-II in service restorationproblem

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tep 5: Perform non-dominated sorting on Rt. Non-dominatedsorting divides the population in different fronts. Thesolutions in Rt, which do not constrained-dominate eachother but constrained-dominate all the other solutionsof Rt, are kept in the first front or best front (calledset F1). Among the solutions not in F = F1, the solu-tions which do not constrained-dominate each other butconstrained-dominate all the other solutions, are keptin the second front (called set F2). Similarly, amongthe solutions not belonging to F = F1 ∪F2, the solu-tions which do not constrained-dominate each other butconstrained-dominate all the other solutions, are kept inthe third front (called set F3). This process is repeateduntil there is no solution in Rt without having its ownfront. Subsequently, these generated fronts are assignedtheir corresponding ranks. Thus, F1 is assigned rank 1,F2 is assigned rank 2 and so on.

tep 6: To create Pt+1, i.e. the parent population in the nextor ‘(t + 1)th’ generation, the following procedure isadopted. Initially, the solutions belonging to the set F1are considered. If size of F1 is smaller than N, then allthe solutions in F1 are included in Pt+1. The remainingsolutions in Pt+1 are filled up from the rest of the non-dominated fronts in order of their ranks. Thus, if afterincluding all the solutions in F1, the size of Pt+1 (let it bedenoted by ‘n’) is less than N, the solutions belonging toF2 are included in Pt+1. If the size of Pt+1 is still less thanN, the solutions belonging to F3 are included in Pt+1.This process is repeated till the total number of solu-tions (i.e. n) in Pt+1 is greater than N. To make the sizeof Pt+1 exactly equal to N, (n−N) solutions from the last

In this work, before NSGA-II is implemented for solving theervice restoration problem, the original distribution network isapped to a graph involving nodes and branches. The nodes of

he graph represent the various zones of the original distributionystem. A zone is defined by a partial network of the distributionystem, which does not contain any switch. The branches of theraph represent the switches (either ‘ON’ or ‘OFF’) of the orig-nal distribution system (but they do not contain any feeder ofhe original network). Inside any zone (i.e. the partial network,hich also contains the system feeders), the structure is radial

nd all the relevant network data like load data, feeder data etc.re known from the given distribution system data. Hence, if thetatuses of the switches are such that the zones are connected inmeshed fashion, then the original network would also operate

n a meshed fashion. On the other hand, if the statuses of thewitches connect the various zones in a radial fashion, then theriginal network would also operate in a radial manner. As aesult, the structure, whether radial or meshed, of the originalistribution system network is completely determined by thetructure of the graph, i.e. if the structure of the graph is radial,he original distribution system network also operates radially,hereas if the structure of the graph is meshed, the original distri-ution system also operates in a meshed fashion. Moreover, theonfiguration of the original network is completely determinedy the status of the switches. Fig. 1 shows a simple distributionetwork and its corresponding graph is shown in Fig. 2. It iso be noted that if each bus is separated from other bus(es) bywitch(es), then each zone would contain only one bus and restf the algorithm would follow. Now, various issues of NSGA-IIre discussed below.


Fig. 1. Network configuration of system-1 before fault.

Fig. 2. System-1 divided in zones.

4.1. String representation

As the configuration of a network is represented by status ofall the switches in the network, the string in the service restora-tion problem represents the status of all the switches in thesystem. The length of each string (i.e. the number of bits ina string) is equal to the number of switches in the system. In thiswork, the binary coding system has been adopted. Thus, the sta-tus of the ‘closed’ and ‘open’ switch in the system is representedby the binary digit ‘1’ and ‘0’, respectively. For instance, in thenetwork shown in Fig. 2, switches s1, s2, s3, s5, s7, s8 and s9are closed and rest are open. Thus, the corresponding string forthis configuration is s = [1 1 1 0 1 0 1 1 1 0], where the indices ofthis string represent the switch number.

4.2. Generation of initial strings

As discussed in step 1 of Section 3, generally the initial pop-ulation Po is generated randomly. This is the simplest method,in which no knowledge about the network is required. How-ever, if the pre-fault network is well behaved (i.e. all loads are

properly served, various constraints are satisfied, etc.), it is rea-sonable to expect that the final solution would be found near theoriginal configuration. In that case, if one of the randomly gen-erated strings is replaced by the string representing the original,pre-fault configuration (PFC), then the spread of the solutionin the initial population reaches closer to the optimal solutionand hence the chance of reaching the final solution in shortertime enhances. Even if the optimal solution is quite far fromthe original configuration, keeping the original configuration inPo gives at least a good spread of solutions (called diversity),which is advantageous to NSGA-II [27]. Moreover, to keep thefaulted zone always isolated, if any of the elements (i.e. bits)corresponding to the switches around the faulted zone is ‘1’, itis made ‘0’ before proceeding further. Also, if any of these ran-domly generated strings, the bit corresponding to the root switch(i.e. the switch directly connected to the substation) is ‘0’, it isimmediately made ‘1’ (if the root switch is ‘OFF’, the substationis isolated, which means no power can be supplied to the rest ofthe distribution system).

As the bit corresponding to the root switch must always bemaintained as ‘1’ to ensure supply to the rest of the system,another alternative methodology for generation of initial stringscan also be adopted. In this alternative methodology, the bitcorresponding to the root switch would always be fixed as ‘1’ anda string of ‘n− 1’ bits (where ‘n’ is the number of switches in thesystem) would be generated randomly. However, it is to be notedtabaocparNaioi

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hat for evaluation of all the objective functions and constraints,ll the ‘n’ bits would be necessary. Therefore, a little extra bit ofook keeping is necessary to concatenate the ‘root switch bit’nd the string comprising of ‘n− 1’ bits before evaluation of thebjective functions and constraints. On the other hand, no suchoncatenation is necessary in the first approach (described in therevious paragraph). However, it is to be observed that there isvery minor difference between these two approaches and as a

esult, there will be virtually no difference of performance of theSGA-II algorithm using either of these two approaches. As the

uthors felt more comfortable in the first approach, in this workt has been adopted (frankly second option can also be used byther researchers with equal ease, if they feel more comfortablen it, with no change of NSGA-II performance).

.3. Radiality checking

To check the radiality of the system, a breadth-first-traversalf the network, which starts from the root switch and proceedsowards the downstream side of the distribution system, is com-

enced. The switch at which the traversal starts (i.e. the rootwitch) is called the first level switch. If any switch under con-ideration is closed, the zones connected at the downstream sidef this switch can be reached and hence is marked ‘visited’. Onhe other hand, if this switch is open, the zones connected at theownstream side cannot be reached and these zones are markedunvisited’. After all the first level switches are considered, zonesarked ‘visited’ are admitted in a list L. The switches connected

t downstream side of the zones currently admitted in list L arealled second level switches. After considering the second levelwitches in same manner as just described, the list L is updated.


The switches connected at the downstream side of the zones mostrecently admitted in the list L are called ‘third level switches’.This process is repeated till there is no switch left in the nextlevel (i.e. all the switches are covered). The illustration of thisabove procedure is given below for the network shown in Fig. 2,in which s1 is the root switch.

First level switch: s1.Zones visited, connected with first level switch: z1.Second level switches: s2 and s3.Zones visited, connected with second level switches: z2 andz4.Third level switches: s4, s5, s7 and s8.Zones visited, connected with third level switches: z3, z5 andz7.Fourth level switches: s6, s9 and s10.Zones visited, connected with fourth level switches: z6.Fifth level switches: nil, i.e. all zones have been visited.

During this traversal, if any of the zones is ‘visited’ morethan once, the presence of a loop is detected. To maintain theradiality of the system, the switch currently under considerationis made ‘OFF’ immediately (i.e. the corresponding bit in thestring is changed from ‘1’ to ‘0’). After the network traversalis complete (i.e. all the network switches are considered), allthe zones marked ‘visited’ are put in a list EZ called “existingzes

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4.6. Partial restoration

During restoration process, if no such string in population isgenerated which can restore whole out-of-service area withoutviolating any constraint, the string which can restore out-of-service area partially without violating any constraint, is selectedand kept in a better front. Subsequently, the NSGA-II algorithmas explained in Section 3 is followed. Thus, in this proposedalgorithm, there is no need to increase the length of the stringsas suggested in Ref. [26], which is advantageous as the speed ofexecution of GA reduces when the string length is increased.

4.7. Front formation

Following step 5 of Section 3, the combination of parentand offspring population having length 2N is divided in variousranked non-dominated fronts. Because of the front formationfrom the combination of parent and offspring population, chanceis given to the current best solution in parents to compete with theoffspring solutions. If no better solution is generated in offspring,the current best solution in parent becomes winner again. In thisway, the elitism is maintained and due to the presence of elitism,convergence is improved [28].

4.8. Selection of N strings

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ones”. It is to be noted that only the “existing zones” are actuallynergized zones (i.e. each of these zones is connected to theubstation via some combination of ‘closed’ switches).

.4. String evaluation

After checking the radiality, all strings give radial configura-ion, and their corresponding existing zones are recorded. Withhe help of the existing zones, existing buses and existing linesre found. With the knowledge of the existing buses, the out-of-ervice area in terms of the disconnected load is calculated usingq. (1). Using Eq. (2), the number of switch operations is calcu-

ated. To calculate system loss, bus voltage violations and lineurrent violations, ac load flow (using backward/forward sweepethod) [30] study is conducted after deleting the non-existing

ines and buses from and adding the existing lines and buses tohe input data file.

.5. String operation

To generate the offspring population, single point crossoverethod is used. Moreover, the mutation operator is applied ran-

omly in any string. After the offspring population is created,he radiality of all offspring configurations is checked. If any ofhe offspring configurations is found to be non-radial, it is madeadial following the procedure described in Section 4.3. Subse-uently, with the help of CTSO, mating pool is created for theext generation. This operator maintains: (i) convergence as theolution having better front is selected and (ii) diversity as theolution having higher crowding distance within the same fronts selected.

N strings for parent population of the next generation fromt (which has 2N strings) of current generation are selected fol-

owing the procedure described in step 6 of Section 3.

.9. Convergence

To check for convergence, at each generation, the candi-ate configurations in parent Pt and offspring Qt are comparedfter they are made radial (following the procedure described inection 4.3). If both populations are same, the convergence isonsidered to be achieved, otherwise not.

.10. Selection of final solution after convergence

After the convergence is achieved, the best solutions are con-ained in the first front of Rt. If the first front has only oneolution, then obviously it is the final solution of the serviceestoration problem. On the other hand, if the first front hasore than one solutions, the final solution is chosen follow-

ng a M stage (where M is the number of objective functions)rocedure which uses the knowledge of preference of differentbjective functions. In the first stage, those solutions (from therst front of Rt) are picked up which have minimum value of theost preferred objective function and put them in a set FS. If FS

ontains only one solution, this is declared to be the final solu-ion. If not, then in the second stage, those solutions from FS areicked up which have minimum value of the second preferredbjective function and FS is updated with these solutions (i.e.S now contains the solutions obtained after the second stage).gain, if FS now contains only one solution, it is declared toe the final solution. If not, the above procedure is repeated. In


general, at any ‘mth’ stage, those solutions from FS are pickedup which have minimum value for the ‘mth’ preferred objectivefunction and the set FS is updated with these solutions. If FScontains only one solution at any stage, this is declared to bethe final solution and the algorithm terminates. Otherwise thealgorithm proceeds for the next stage. If, even after all the Mobjective functions are considered, FS still contains more thanone solution, any solution in FS is equally good and hence canbe declared as the final solution.

4.11. Steps of algorithm for service restoration

Step 1: The information available to the algorithm are: (i) sys-tem data, (ii) PFC and (iii) post-fault configuration.

Step 2: Generate Po following the procedure described in Sec-tion 4.2.

Step 3: Check the radiality of the solutions in Po and modifythem, if necessary, following the procedure describedin Section 4.3.

Step 4: Evaluate the strings in Po as described in Section 4.4and assign Pt = Po.

Step 5: Generate the offspring population Qo as described inSection 4.5. Moreover, in all the solutions of Qo, thefaulted zone is isolated and the root switch is alwaysmade ‘closed’ as described in Section 4.2.

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2

3

4. No. of switches: 75, no. of buses: 173, nominal voltage:33 kV, no. of load buses: 88 (P = 169.4, Q = 16.4).

5. Simulation results and discussion

For testing the validity of the proposed technique, it has beenapplied to all the four test systems described in Section 4.12.For each of these systems, two different versions of the pro-posed algorithm have been considered. In the first version, noneof the randomly generated strings in Po has been replaced bythe string representing the PFC. Henceforth, this case wouldbe termed as “NSGA method-1”. In the second version, one ofthe randomly generated strings in Po has been replaced by thestring representing the PFC, and henceforth, this case would betermed as “NSGA with PFC”. To compare the performance ofthe proposed NSGA methodologies to some of the algorithmsalready published in the literature, the similar two versions (i.e.one with PFC and another without it) of the algorithm proposedin Ref. [26] have also been applied to the same service restora-tion problems considered in this work. For any particular system,the values of the crossover probability, mutation probability andthe population size are kept same for all these four methods.However, their values are different for different systems. Thevalues are given in Table 1. Due to space limitation, the resultscorresponding to the smallest system (i.e. the 10 switch system)as

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Step 6: Evaluate the strings in Qo as described in Section 4.4and assign Qt = Qo.

Step 7: Follow steps 4–7 of NSGA-II as described in Section3 to obtain Pt+1 and Qt+1.

Step 8: In all the solutions of Qt+1, the faulted zone is iso-lated and the root switch is always made ‘closed’ asdescribed in Section 4.2.

Step 9: Check the radiality of the solutions in Qt+1 and modifythem, if necessary, following the procedure describedin Section 4.3.

tep 10: Check for convergence as described in Section 4.9. Ifthe algorithm has converged, find the final solution asdiscussed in Section 4.10. Otherwise go to step 11.

tep 11: Evaluate the strings in Qt+1 as described in Section 4.4.tep 12: Update Pt←Pt+1, Qt←Qt+1 and go back to step 7.

.12. Test systems

The proposed methodology has been applied on four differentistribution systems. All test systems are of different size. Fig. 1hows diagram of one (10-switch system) of the test systems.ollowing are the parameters of the test systems. The notationsand Q denote the total active and reactive load in the system

n MW and MVAR, respectively.

. No. of switches: 10, no. of buses: 13, nominal voltage: 11 kV,no. of load buses: 12 (P = 26.5, Q = 8.6).

. No. of switches: 14, no. of buses: 10, nominal voltage:13.8 kV, no. of load buses: 9 (P = 5.6, Q = 4.0).

. No. of switches: 37, no. of buses: 32, nominal voltage:12.66 kV, no. of load buses: 31 (P = 3.7, Q = 2.3).

re only illustrated in detail in this paper, whereas, for otherystems, the summary of the important results are provided.

In the system shown in Fig. 1, it has been assumed that theault has taken place in zone z4 between bus no. 4 and 8, andence switch s3 is tripped to isolate the fault. As a result, theower supply to the loads in bus no. 8–13 is interrupted. Now, asus nos. 8 and 9 are in the faulted zone z4, the supply to the loadsf these two buses cannot be restored till the faulty component isepaired. However, the loads on the remaining unsupplied busesi.e. bus no. 10–13) can be restored as they belong to the healthyones z5, z6 and z7.

Fig. 3 shows the iteration wise variation of all three objec-ive functions when the service restoration problem is solvedy “NSGA method-1”. At first two iterations, the values of thehree objective functions do not change. At third iteration, thealues of second and third objective function reduce while thatf the first objective function still remains the same. Hence, theolution at third iteration is better than the solution of secondteration. At fourth and fifth iteration, there is no change in thealue of any objective function. At sixth iteration the values ofhe second and third objective function increase while that ofhe first objective function decreases. As the first objective func-ion is of topmost (i.e. first) preference, the solution obtained at

able 1A parameters for different system

10 Switchsystem

14 Switchsystem

37 Switchsystem

75 Switchsystem

opulation size 4 5 6 8rossover probability 0.6 0.6 0.68 0.7utation probability 0.04 0.04 0.04 0.03


Fig. 3. Iteration wise variation of the objective functions using NSGA.

sixth iteration is better than that obtained at the previous iter-ation. After sixth iteration, there is no change in the value ofany of the objective functions and the algorithm converges ateighth iteration. Thus, it can be said that the optimal solution isobtained at sixth iteration taking 0.2474 s of elapsed time (ET),whereas the algorithm converges after eighth iteration taking0.33 s of ET. It is to be noted that the ET has been calculatedby using the ‘tic’ and ‘toc’ commands of MATLAB [31]. When“NSGA with PFC” is applied for this same problem, it is foundthat the topmost preferred objective function is minimized at thefirst iteration itself (i.e. the optimal solution is achieved), whichtakes 0.11 s of ET and the algorithm converges in three itera-tions (taking 0.22 s of ET). The final, restored network is shownin Fig. 4.

The performances of the algorithm of [26] (without PFC)and NSGA-II are shown in Fig. 5. It is observed from Fig. 5that with the algorithm [26], the optimal solution is found at10th iteration (which takes 3.9882 s of ET to achieve) and thealgorithm finally converges at 34th iteration, taking 13.56 s ofET. When the algorithm of Ref. [26] is executed by includingPFC in Po, the corresponding values for convergence time andtime for best solution are 7.69 and 2.98 s, respectively. It is to benoted that in Fig. 5, there is only one fitness function as in [26],the multi-objective problem is converted into a single objectivefunction optimization problem before the application of GA.Thus, apparently, the proposed NSGA based algorithm achievest

Fig. 5. Objective function vs. iteration.

However, for any solution technique of any optimizationproblem, not only the speed of solution but also the qualityof solution is equally important. To compare the quality of thesolutions obtained by these four techniques, the “equivalent fit-ness value (EFV)” of these four methods is calculated using theexpression of fitness function EFV = w1f1 + w2f2 + w3f3 +w4c1 + w5c2 [26], where w1, w2, w3, w4 and w5 are the weight-ing factor of f1, f2, f3, c1 and c2, respectively, while c1 and c2are voltage and current constraint violation, respectively. It isto be noted that for any particular test system, the values ofthese weights are kept same for all these four methods. How-ever, for different test systems, the weights are different. Now,after “NSGA method-1” is converged at eighth iteration, thevalues of the three objective functions are substituted into theexpression of EFV to calculate the EFV of this method. Simi-larly, the EFV of “NSGA with PFC” has also been calculated.Of course, the EFV of algorithm of Ref. [26] is directly com-puted by the algorithm itself. It has been found that for thecase described above, the EFVs of all the four algorithms aresame and equal to 10. Hence, it can be said that among allthese four methods, “NSGA with PFC” is best for this case as itachieves the optimal solution in least computation time. More-over, it can also be observed that “NSGA method-1” is betterthan the algorithm of [26] as it achieves the same solution in lesscomputation time.

To test the performance of the proposed methodology further,adsgciliessoha

he optimum solution faster than the algorithm of Ref. [26].

Fig. 4. Network configuration of system-1 after restoration.

large number of simulation studies have been carried out forifferent single and multiple fault cases in the other three testystems also and for each case, the EFV, time taken for conver-ence and the time needed to reach the best solution have beenalculated by all the four algorithms. As it is not possible toncorporate all the numerical results in this paper due to spaceimitation, some of the representative results are shown below tollustrate the performance of the proposed scheme. Towards thisnd, a set of single fault conditions considered in the 75 switchystem are shown in Table 2 and the results for this fault set arehown in Tables 3–5. Moreover, to investigate the performancef the proposed technique for multiple faults, simulation resultsave also been carried out for various cases of multiple faultsnd results for a representative set of multiple faults as described


Table 2Various single fault conditions considered in 75 switch system

Case 1 Zone 18, between bus 38 and 40Case 2 Zone 53, between bus 104 and 105Case 3 Zone 59, between bus 104 and 105Case 4 Zone 38, between bus 69 and 73

Table 3EFV obtained after convergence for fault set in Table 2

Case 1 Case 2 Case 3 Case 4

NSGA method-1 0.752 9.29279 3.549083 18.80863NSGA with PFC 0.752 9.29279 3.549083 18.80863Technique of Ref. [26] 0.752 10.52665 3.549083 18.90863Ref. [26] with PFC 0.752 10.52665 3.549083 18.90863

Table 4Time taken to reach convergence for fault set in Table 2



Table 5Time taken to reach the best solution for fault set in Table 2



in Table 6 are given in Tables 7–9, respectively. It is to be notedthat the unit of time in Tables 4, 5, 8 and 9 is in seconds.

From these results, it can be observed that the inclusion ofPFC in Po does not have any effect on the value of EFV attainedby NSGA, whereas, for the technique of [26], inclusion of PFCin Po sometimes helps to achieve better result (e.g. case 2 inTable 7). Also, the proposed NSGA technique (both the versions)always attains the minimum EFV, whereas, the same cannot bealways guaranteed with algorithm of Ref. [26] (e.g. case 2 inTable 3 and case 3 in Table 7). However, the inclusion of PFC in

Table 6A set of representative multiple faults

Case Description

1 37 switch system, faulted zones 9, 25 between buses 8 and 9 and25 and 26, respectively

2 37 switch system, faulted zones 6, 27, 31 between buses 5 and 6,27 and 28 and 31 and 32, respectively

3 75 switch system, faulted zones 59, 38, 15 between buses 1 and130, 1 and 69 and 24 and 25, respectively

4 75 switch system, faulted zones 59, 18, 15, 52 between buses 1 and130, 38 and 40, 24 and 25 and 104 and 105, respectively

Table 7EFV obtained after convergence for multiple faults



Table 8Time taken to reach convergence for multiple faults



Po reduces the time taken by any technique (NSGA or Ref. [26])to reach the convergence/best solution considerably compared tothe versions in which PFC is not included in Po. Moreover, theproposed ‘NSGA with PFC’ technique always takes the leastamount of time to reach convergence/best solution among allthese four techniques. Thus, among all these four techniques,the method ‘NSGA with PFC’ can be considered to give thebest performance with respect to both accuracy and speed.

Although the proposed NSGA technique gives superior per-formance compared to the traditional GA technique, severalrelated issues still need to be addressed. Firstly, in this work,it has been assumed that all the loads are of equal importance.However, in any distribution system, there are always someloads, which are of highest priority (e.g. hospital). In the eventof partial load restoration, supply must first be restored to thehighest priority customers and this fact should be reflected in thefinal solution of the service restoration problem. Secondly, in thispaper, corresponding to the objective function of ‘minimizationof restoration time’, it has been assumed that all the restora-tion time associated with all the switches installed in the systemare same. However, in a distribution grid, generally two typesof switches (i.e. manual and automated) are used. The restora-tion times associated with these two types of switches are alsodifferent. As a result, the objective function corresponding to‘minimization of restoration time’ must also accommodate thesetwo different restoration times. Thirdly, from Tables 4, 5, 8 and 9ib(H

TT

NNTR

t is observed that in all the cases, the time needed to reach theest solution (TBS) is less than the time needed for convergenceTC) and in some cases; TBS is considerably smaller than TC.ence, in all the cases, the software continues the execution

able 9ime taken to reach the best solution for multiple faults


SGA method-1 10.7255 10.1637 3671.291 1271.2SGA with PFC 0.9183 0.8480 227.5240 22.05echnique of Ref. [26] 6.1137 6.0656 250.6744 1072.990ef. [26] with PFC 2.8192 2.0820 586.3299 289.51


even after it reaches the best solution, which in turn, increasesthe time for service restoration. Hence, for enhancing the speedof service restoration further, this gap between TBS and TC mustbe reduced (ideally should be made zero). The work is going onpresently to address these three important issues and the resultswould be reported in due time in future.

The fourth issue regarding the application of NSGA-II toservice restoration problem is related to some exceptional cases,which might occur in some rare occasions. As already describedin Section 4.10, the solution in the first front with lowest valueof “out-of-service” area would be chosen as the final solution.However, if in some exceptional case, this solution containsvery high values of objective functions f2 and f3, and then frompractical point of view, it may not be an acceptable solution.Indeed, it would be desirable to prevent this solution from beingincluded in the first front. In the proposed technique, this caneasily be achieved by imposing two more constraints in the ser-vice restoration problem. The first constraint would specify amaximum allowable limit of number of switching operation andthe second constraint would specify a maximum allowable limitof losses in the system. Thus, in this case, total number of con-straints would be five (instead of three as described in Section2). For any solution having exceptionally high value of either f2or f3, at least one of its constraints (out of five) would be violatedand therefore, it would be constraint-dominated by some othersolution. As a result, following the description of step 5 of Sec-tfnpo

6

gdb(tbrGbc

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[5] Y.Y. Hsu, et al., Distribution system service restoration using a heuristicapproach, IEEE Trans. Power Deliv. 7 (2) (1992) 734–740.

[6] S. Toune, et al., Comparative study of modern heuristic algorithms toservice restoration in distribution systems, IEEE Trans. Power Deliv. 17(1) (2002) 173–181.

[7] V.S. Devi, G. Anandalingam, Optimal restoration of power supply inlarge distribution systems in developing countries, IEEE Trans. PowerDeliv. 10 (1) (1995) 430–438.

[8] K.N. Miu, H.D. Chiang, B. Yuan, G. Darling, Fast service restoration forlarge scale distribution systems with priority customers and constraints,IEEE Trans. Power Syst. 13 (3) (1998) 789–795.

[9] K.N. Miu, H.D. Chiang, R.J. McNulty, Multi-tier service restorationthrough network reconfiguration and capacitor control for large scaleradial distribution networks, IEEE Trans. Power Syst. 15 (3) (2000)1001–1007.

[10] C.C. Liu, S.J. Lee, S.S. Venkata, An expert system operational aid forrestoration and loss reduction of distribution systems, IEEE Trans. PowerSyst. 3 (2) (1988) 619–626.

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[14] H. Kim, Y. Ko, K.H. Jung, Algorithm of transferring the load of thefaulted substation transformer using the best first search method, IEEETrans. Power Deliv. 7 (3) (1992) 1434–1442.

[15] S.S. Ahmed, S. Rabbi, R. Mostafa, A.R. Bhuiya, Development of an

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ion 3, this solution would automatically be relegated to lowerronts. However, for the case studies carried out in this paper,o such exceptional cases has been observed and hence, in thisaper, these two additional constraints have not been imposedn the service restoration problem.

. Conclusion

In this paper, an algorithm based on NSGA-II has been sug-ested for solving the service restoration problem in poweristribution systems. The advantage of the proposed NSGA-IIased technique is that it does not require any weighting factoras needed in a conventional GA based technique). The proposedechnique has been found to be superior to the conventional GAased technique in the sense that it always attains the best serviceestoration solution in lesser time (compared to the conventionalA). Also, it has been found that by including PFC of the distri-ution system in the initial population, the speed of convergencean be increased significantly.

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service restoration in distribution system using non-dominated sorting genetic algorithm

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