Electrical Power and Energy Systems 43 (2012) 607–616

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Reliability evaluation and enhancement of distribution systems in the presenceof distributed generation based on standby mode

Rajesh Arya a,⇑, S.C. Choube b, L.D. Arya c

a Electrical Engg. Deptt., Medi-Caps Institute of Science & Technology, Indore, M.P., Indiab Electrical & Electronics Engg. Deptt., UIT, Rajiv Gandhi Technological University, Bhopal, M.P., Indiac Electrical Engg. Deptt., Shri G.S. Institute of Technology & Science, Indore, M.P., India

a r t i c l e i n f o

Article history:Received 13 April 2010Received in revised form 8 April 2012Accepted 24 May 2012Available online 7 July 2012

Keywords:Distribution systemReliability indicesFailure rateRepair timeDifferential evolutionPSO

0142-0615/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2012.05.045

⇑ Corresponding author.E-mail address: aryarajesh@yahoo.com (R. Arya).

a b s t r a c t

This paper describes an analytical methodology for reliability evaluation and enhancement of distribu-tion system having distributed generation (DG). Standby mode of operation of DG has been consideredfor this purpose. Reliability of the composite distribution system has been optimized with respect to fail-ure rate and repair time of each section of the distribution system subject to constraints on (i) failurerates/ repair times and (ii) customer and energy based reliability indices i.e. SAFI, SAIDI, CAIDI and AFNS.The objective function includes (i) cost of modification for failure rates/repair times (ii) additional cost ofexpected energy supplied by DG. Differential evolution (DE), Particle swarm optimization (PSO) and co-ordinated aggregation based PSO (CAPSO) have been used to develop computational algorithms. Devel-oped algorithms have been implemented on a sample distribution system.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Distribution systems were not originally designed to accommo-date generation in itself. Hence increasing penetration level of DGis causing changes in the planning, operation and maintenance ofdistribution systems. Presence of DG in a distribution network af-fects network planning, operation and maintenance, auxiliary ser-vices, quality of service and regulatory aspects [1–4]. Ackerman etal. [5] have analyzed the term DG in detail. Usually in reference tothis present paper, DG will be considered as electricity generationsystems connected to distribution networks, characterized by theirsmall rating and located near connection points. Distribution sys-tem operators (DSOs) may have their own DG or may encouragelarge consumer to install DG which is owned and controlled bythese customers. Availability of such customer owned and con-trolled DG may not be high, but may be of great help in improvingthe reliability indices of the distribution network.

The reliability of a distribution system may be increased bymodifying failure rate and repair time of each section of the net-work. Such modifications may require additional investmentswhich in the presence of DG may be mitigated. This will result inannual savings. On the other hand cost per unit energy obtained

ll rights reserved.

form DG may be high. Further the traditional reliability indicescovered sustained interruption durations. The time necessary tostart up the DG should be taken in to account for the reliabilityevaluation of distribution system. If this time is sufficiently shortthe customers suffer a momentary interruption, while, if not, theysuffer a sustained interruption.

Various reliability optimization algorithms have been devel-oped by reducing failure rate and repair time of each section ofthe distribution systems. Sallam et al. [6] presented a methodologyfor determination of optimal reliability indices for distribution sys-tem using gradient projection method. Chang and Wu [7] devel-oped a methodology for obtaining optimal reliability design ofelectrical distribution system using a primal dual interior pointalgorithm. Chowdhury and Custer [8] developed a value based ap-proach for designing urban distribution system. Bhowmik et al. [9]described a distribution network planning strategy by consideringa nonlinear objective function with linear and non linear con-straints for radial distribution system. Su and Lii [10] obtainedoptimum failure rates and repair times for each section of distribu-tion system using modified genetic algorithms. Particle swarmoptimization along with decomposition has been used for reliabil-ity optimization of radial distribution systems by Arya et al. [11].Bae and Kim [12] developed methodology for evaluating reliabilityof distribution system having DG in operation mode. Bae et al. [13]developed an optimal operating strategy for distributed generationconsidering hourly reliability worth. Louit et al. [14] presented a

Load Point

λs, rs

λsw, s

λdg, rdg

Fig. 1b. Reliability network of the schematic diagram of Fig. 1a.

λeq, req

Fig. 1c. A single component equivalent of reliability network of Fig. 1b.

LP - iA

Distribution System

LP - iB

DG ~

Fig. 2a. Reliability modeling for ith load point having limited DG capacity available.

608 R. Arya et al. / Electrical Power and Energy Systems 43 (2012) 607–616

simple technique for obtaining optimal interval for major preventivemaintenance for distribution system without DG. Costa and Matos[15] described various modeling aspects for assessing the effect ofmicrogrids to the reliability of distribution network. Trebolle et al.[1] proposed a market mechanism, referred to as reliability optionsfor distributed generation (RODG), which provides distributionsystem operators with an alternative to the investment in new dis-tribution facilities.

In view of the above discussion this paper presents a new meth-odology for obtaining optimal failure rate and repair time of eachsection of a distribution system in the presence of distributed gen-eration which is specifically customer owned and controlled. It isassumed that utility has to pay additional price for each unitcharged. An objective function which reflects cost of modificationof failure rates, and repair rate and cost of energy supplied fromDG has been constructed and minimized subject to reliabilityconstraints.

2. Reliability modeling aspects with distributed generation

Distributed generation which is owned and controlled by cus-tomer or any other agency may be used as standby units for reli-ability enhancement at specific load points. This application issimilar to one used in transferring load from one substation to an-other by disconnecting feeders at their normal source (in the eventof failure of a section) end and closing the open point. Usually thisis done in the event of a failure causing loss of supply from thesource substation and it is not normal practice to parallel sources.Now in the changed scenario of deregulated environment, thetransfer of load at a load point may be partial or total accordingto the capacity available of DG. If DG capacity is sufficient to meetthe load at a load point in the event of failure of supply from sub-station then the situation at load point is represent as shown inFig. 1a.

Further it is to be understood that load is usually connectedfrom source via various distributor segments. By manual switchingthe load may be shifted to DG in the event of trouble from sourceside. The average time to carry out the switching operation is as-sessed as ‘s’ hours. The equivalent reliability diagram for the sys-tem is shown in Fig. 1b in which source side, and DG are directlyconnected in parallel but placed in series with important switchingelement with failure frequency ksw and restoration time ‘s’. Reli-ability network of Fig. 1b may reduced by the application of ser-ies–parallel approximate formulas [16]. The system of Fig. 1b isreduced to an equivalent network (keq, req) shown in Fig. 1c usingfollowing relations

keq ¼ kskdgðrs þ rdgÞ þ ksw ð1Þ

req ¼ks � kdg � rs � rdg þ ksw � sks � kdgðrs þ rdgÞ þ ksw

ð2Þ

where keq is the equivalent failure rate; req the equivalent interrup-tion duration; ks the represents total failure rate up to load pointfrom substation supply; rs the represents average interruption dura-tion from substation supply; kdg the failure rate of DG; rdg the aver-age outage duration of DG; and ksw the failure rate of manualswitch; and s is the service restoration time from DG.

Main Source DG

Load Point (LP)

λs, rs λdg, rdg

Fig. 1a. Schematic diagram of a load point fed from main source and as stand bysupply from DG.

Service interruption rate and average interruption/restorationtime are calculated for each load point by successive applicationsof series parallel reduction rules. This type of procedure is com-monly employed in distribution systems reliability studies.

A different reliability modeling will be required if the DG capac-ity available is insufficient to fulfill the maximum demand of allthe customers connected at a load point. In such situation the loadpoint is partitioned in two portions one of it is supplied from mainsource as well as from DG as stand by and the other part is suppliedfrom main source only. This reliability modeling aspect is shown in(see Fig. 2a). Partitioned load points are ‘i-A’ and ‘i-B’. In this casealternative supply is available for a fraction of customers con-nected at the load point. Figs. 2b and 2c shows reliability networkfor evaluating the indices at load point LP � iA and LP � iB. Rela-tions (1) and (2) along with series parallel reliability networkreduction formulas are used to evaluate the basic indices at LP � iAor LP � iB.

3. Problem formulation

Reliability indices may be improved for a radial distribution sys-tem by modifying failure rate and average repair time of each seg-ment of the system. Addition of distribution generation (DG) mayrequire additional cost per kWh to be paid to the private DG own-ers. In view of this the objective function is selected as follows

J ¼XNC

k¼1

ak

k2k

þXNC

K¼1

bK

rKþ ADCOSTðEENSO� EENSDÞ ð3Þ

In above kK is the failure rate of Kth section; rK the average repairtime of kth section; aK and bK are the cost coefficients; EENSO the

Source LP - iAλs, rs

Fig. 2b. Reliability network for evaluating indices at LP � iA.

LP - iB

λs, rs

λsw, s

λdg, rdg

Fig. 2c. Reliability network for evaluating indices at LP � iB.

R. Arya et al. / Electrical Power and Energy Systems 43 (2012) 607–616 609

expected energy not supplied without DG; EENSD the expected en-ergy not supplied with DG and ADCOST is the additional cost perkWh to be paid to DG owners.

The objective function constitutes of three parts i.e. (i) first partreflects cost of modifications of failure rates of each section. It in-cludes cost of preventive measures for reducing failure rate. Thequadratic form is based on the reliability growth model of Duane[17], (ii) second part gives cost of modifications in average repairtime of each section which is again based on a growth model of re-pair time. Larger is the repair time lesser is the cost of repair andvice versa, and (iii) third part represents additional cost to be paidto the owners of DG. This term in fact is the expected energy sup-plied (EENSO–EENSD) by DG multiplied by additional charge perkW h (ADCOST). Thus the objective function provides a balance be-tween additional cost incurred by the provision of DG and cost dueto modifications in failure rates and repair times. It is to be notedthat the operation of DG has been considered in standby modeand DG has been assumed to be privately owned. Hence power ob-tained from DG is costlier than that obtained from distributionsubstation. Other side one has to pay price for modifications in fail-ure rates and repair times to achieve desired values of reliabilityindices as given by relations (7)–(10). Hence a compromise hasto be achieved between failure rate and repair time modificationsand additive cost of energy obtained from DG sources. This, in fact,is achieved by minimizing the objective function given by relation(3). This framed objective function includes the cost of ‘modifica-tions’ and additional cost of energy purchased from DSO. Furthernote that as explained in Section 2, power cannot be taped imme-diately from DG, but with a time delay of ‘s’ (service restorationtime from DG) hours and this has been accommodated in modelingaspects. Another aspect which has been included that the demandmay be fulfilled even partially in emergency conditions. This hasbeen included in modeling aspects.

Expected energy not supplied (EENS) is calculated for all loadpoints as follows [18]

EENS ¼X

i

LiUSYS;i ð4Þ

where Li is average load at ith load point; USYS,i is average annualoutage duration at ith load point.

EENS is calculated using relation (4) without DG is termed asEENSO and that calculated with DG is termed as EENSD. Expectedenergy not supplied for all load points is calculated using relation(4). In this relation Li is load at ith load point and USYS,i is averageannual outage duration at ith load point and is calculated using fol-lowing relation for radial systems.

USYS;i ¼Xk2S

kkrk ð4aÞ

S denotes set of distribution segment in series upto kth loadpoint; kk is failure rate of kth segment and rk is average repair timefor kth segment

Now to calculate USYS,i accounting DG, the equivalent failurerate keq and equivalent repair time req are used as given by relations(1) and (2) in Section 2. Hence USYS,i accounting DG is evaluated asfollows

USYS;i ¼ keq � req ð4bÞ

Whereas relation (4a) is used to evaluate USYS,i and in turn used toevaluate EENSO (expected energy not supplied without DG).

Objective function (3) is minimized subject to following cus-tomer and energy based constraints [18]

(i) Bounds on the modification of failure rate and repair time ofeach section

kk;min 6 kk 6 kk;max ð5Þ

rk;min 6 rk 6 rk;max ð6Þ

(ii) Inequality constraint on system average interruption fre-quency index (SAIFI)

SAIFI 6 SAIFId ð7Þ

System average frequency index (SAIFI) is defined as follows

SAIFI ¼

Xi

kSYS;iNi

Xi

Ni

ð8Þ

(iii) Inequality constraint on system average interruption dura-tion index (SAIDI)

SAIDI 6 SAIDId ð9Þ

SAIDI is defined as

SAIDI ¼

XiUSYS;iNiP

Nið10Þ

(iv) Inequality constraint on customer average interruptionduration index (CAIDI)

CAIDI 6 CAIDId ð11Þ

CAIDI is evaluated as follows

CAIDI ¼

XiUSYS;i � NiXikSYS;iNi

ð12Þ

(v) Constraint on average energy not supplied (AENS) per cus-tomer also known as average system curtailment index(ASCI)

AENS 6 AENSd ð13Þ

Where AENS is given as follows

AENS ¼P

LiUSYS;iPNi

ð14Þ

In all above Ni is the number of customers at ith load point; Li aver-age load at ith load point; kSYS;i is average failure rate at ith loadpoint; rSYS,i is average interruption duration at ith load point; USYS,i

is average annual outage duration given as product of kSYS;i andrSYS,i.

SAIFId, CAIDId, SAIDId and AENSd are threshold values of the SAIFI,CAIDI, SAIDI and AENS respectively. In Eqs. (5) and (6) the upperlimits on failure rate (k) and repair time (r) are considered as cur-rent (present) values as provided by any system data. Lower limitson failure rate (kk,min) and repair time (rk,min) are the values whichcan be achieved in practice. In any case these values cannot bemade zero. These lower limits are obtained in practice using failure

610 R. Arya et al. / Electrical Power and Energy Systems 43 (2012) 607–616

and repair data analysis along with the associated costs. The proce-dure is similar to that of reliability growth model [17]. The thresh-old values of reliability indices i.e. SAIFId, SAIDId, CAIDId, and AENSd

are based on managerial/administrative decisions.The formulated reliability optimization problem has been

solved using robust population based optimization algorithmsknown as differential evolution (DE), particle swarm optimization(PSO) and one of its variant known as co-ordinated aggregationbased particle swarm optimization (CAPSO). The formulated prob-lem has been solved using these three optimization techniques soas to get confidence in the optimization results and applicability ofthese methods for solving such problem may be verified and com-parison of the result may be made. The main idea of the paper is toobtain the optimum values of failure rates and repair times of eachdistribution segment subject to satisfaction of threshold values ofreliability indices (SAIFI, SAIDI, CAIDI, and AENS) and incorporatingthe effects of distributed generation available in standby mode.The optimized values may be allocated as target values to preven-tive maintenance and corrective repair crew. Computational algo-rithms have been explained in next two sections.

4. Solution methodology using differential evolution (DE)

Storn and Price [19] developed a very simple population basedstochastic function optimizer known as deferential evolution (DE)technique. DE has been applied to solve various natures of engi-neering problems [20]. DE solves the optimization problem bysampling the objective function at multiple randomly chosen ini-tial points. Bounds on decision variables define the region fromwhich ‘M’ vectors in the initial population are chosen. DE generatesnew solution points in ‘D’ dimensional space that are perturbationsof existing points. DE perturbs vectors with the scaled difference oftwo randomly selected population vectors. Further DE adds thescaled random vector difference to a third selected population vec-tor (termed as base vector) to produce a mutated vector. A uniformcrossover is employed to produce a trial vector from target vectorand mutated vector. The DE based algorithm to solve the formu-lated problem is implemented in following steps.

Step-1. Initialization: Generate a population of size ‘M’. Eachvector consists of failure rates and repair times of each sectionof distribution system. The values of each component of vectorare obtained by sampling uniformly between lower and upperlimits as given by relations (5) and (6).

X0i ¼ ½k0

i ;�r0i �; i ¼ 1; . . . ;M

k0i ¼ ½k

0i1; k

0i2; . . . ; k0

i;NC �

�r0i ¼ ½r0

i1; r0i2; . . . ; r0

i;NC �

Step-2. Evaluate rSYS;i; kSYS;i and USYS,i at each load point with DGand without DG using series–parallel reduction formulas asexplained in Section 2.Step-3. Calculate reliability indices SAIFI, SAIDI, CAIDI, AENS,EENSO and EENSD as defined in the relations 8,10,12,14, and4 respectively.Step-4. Evaluate objective functions JðX0

i Þ for all vectors in thepopulation as given by relation (3).Step-5. Evaluate inequality constraints (7),(9),(11) and (13) foreach vector of population. If a vector satisfies these constraintscall it ‘F’ i.e feasible, otherwise flag it as ‘NF’ i.e. not feasible.Identify a vector X0

best i.e a vector feasible and is best from objec-tive function view point.Step-6. Select a target vector, i = 1.

Step-7. Select two random vectors XðkÞp and XðkÞq from the popu-lation, where p – q – i – best.Step-8. Generate a mutated vector using following relation

V ðKÞi ¼ XðKÞbest þ rðXðKÞp � XðKÞq Þ ð15Þ

r is scale factor usually lies in the range [0,1] and V ðKÞi is mutantvector.

Step-9. If any component of mutant vector i.e. V ðKÞi violates thebounds on decision variables then bounce back mechanism [20]is applied. The bounce back method replaces violated elementby the new element whose value lies between the base param-eters (selected as Xbest) value and bounds being violated asfollows

v ðkÞij ¼xbest;j þ randðxj;min � xbest;jÞ; if v ðKÞij 6 xj;min

xbest;j þ randðxj;max � xbest;jÞ; if v ðKÞij P xj;max

8<:

ð16Þ

rand is random digit between [0,1].Step-10. Trial vector tðkÞi is obtained using uniform cross over asfollows

tðkÞij ¼v ðkÞij ; if randj 6 Cr; or j ¼ jrand

xðkÞij ; otherwise

8<: ð17Þ

Cr is a crossover probability control parameter selected in therange [0,1], Cr is user defined value which controls the numberof decision variables which are copied from the mutant. randj

is random digit in the range [0,1].Step-11. performs reliability evaluation for trail vector tðKÞi andobtain SAIFI, SAIDI, CAIDI, and AENS.Step-12. If all the inequality constraints i.e. 7,9,11, and 13 aresatisfied than call the trial vector as ‘F’ otherwise flag it as ‘NF’.Step-13. Obtain EENSO and EENSD as given in relation (4).Step-14. Evaluate objective function (3) for trail vector i.e. JðtðkÞi ÞLampinen’s constraint handling technique is used to select orreject trial vector in new population [21]. Trail vector tðkÞi isselected in new population under following conditions.

(i) tðKÞi is feasible and JðtðKÞi Þ 6 JðXðKÞi Þ.(ii) tðKÞi is ‘F’ and XðKÞi is ‘NF’.

(iii) tðKÞi and XðKÞi both ‘NF’, but tðKÞi does not violate any con-straint more than XðKÞi . Otherwise XðKÞi is retained innew population.

Step-16. Select another target vector i = i + 1.Step-17. If i 6M, repeat from Step-7.Step-18. Increase generation count, k = k + 1.Step-19. If k 6 kmax, repeat from step-6, otherwise stop. Wherekmax is maximum number of generations specified.

It is stressed that DE algorithm may even be stopped before exe-cution of maximum number of generation, if improvement inobjective function is not observed in a pre-specified generations.

5. Solution methodology using particle swarm optimization

Particle swarm optimization was originally proposed by Ken-nedy and Eberhart [22]. The methodology is population basedand applied to solve the formulated problem. The implementationof PSO for solving the reliability optimization problem is in follow-ing steps.

Step-1. Population of size ‘M’ is generated where each particlecontains failure rates and repair times of each section sampleduniformly within bounds governed by relation (5) and (6).Selected particles are assumed to be feasible i.e. these satisfy

R. Arya et al. / Electrical Power and Energy Systems 43 (2012) 607–616 611

inequality constraints 7,9,11, and 13. The generated particlesare represented as

Xð0Þi ¼ ½kð0Þi1 ; k

ð0Þi2 ; . . . ; kð0Þi;NC ; r

ð0Þi1 ; r

ð0Þi2 . . . ; rð0ÞiNC �; i ¼ 1; . . . ;M

Step-2. This step consist of initialization of velocities of eachparticle selected in step-1. The velocity of each element ‘j’ ofparticle ‘i’ is generated at random within the boundary givenas follows

kj;min � k0ij 6 q0

ij 6 kj;max � koij

rj;min � r0ij 6 q0

ij 6 rj;max � roij

ð18Þ

Step-3. The initial best position, Pð0Þbest;i, of particle ‘i’ is set as itsinitial position and initial best position of group, Gð0Þbest , is deter-mined as the position of an individual with minimum value ofobjective function as obtained using relation (3).Step-4. Set iteration count k = 1.Step-5. Velocity of each individual ðqðkÞi Þ is modified using fol-lowing relation

qðkÞi ¼ w � qðk�1Þi þ c1 � rand1ðPðk�1Þ

best;i � Xðk�1Þi Þ

þ c2 � rand2ðGðk�1Þbest � Xðk�1Þ

i Þ ð19Þ

where Pðk�1Þbest;i is the best previous position ith particle; Gðk�1Þ

best is theglobal best position among all the participation in the swarm fromobjective function view point; rand1, rand2 are random digits in therange [0,1]; w is inertia weight and c1, c2 are acceleration constantsA large inertia weight (w) facilitates global exploration and a smal-ler inertia weight tends to facilitate local exploration to fine tunethe current search area. Therefore the inertia weight ‘w’ is animportant parameter for convergence behavior of PSO. A suitablevalue of inertia weight usually provides balance between globaland local exploration abilities and consequently results in a betteroptimum solution. In view of this it has been suggested that inertiaweight w is varied as follows [23]

w ¼ wmax �wmax �wmin

kmax� k

where kmax is the maximum number of iteration specified and k de-notes current iteration. wmax and wmin denotes maximum and min-imum values of inertia weight.

Step-6. position of each particle, XðK�1Þi , is updated using follow-

ing relation

XðkÞi ¼ Xðk�1Þi þ qðkÞi ð20Þ

Step-7. If any component of updated particles violates thebounds given by 5,6 then it is set to limiting value.Step-8. Perform reliability evaluation for each updated particleand evaluate SAIFI, SAIDI, CAIDI, AENS, EENSO and EENSD.Step-9. Evaluate for each updated particle inequality con-straints 7,9,11, and 13. If all these constraints are satisfiedthe updated particle is retained in updated population. Ifany of the constraints is not satisfied a fly-back mechanism[24] is adopted for handling the inequality constraints. In thisfly back mechanism the idea is to maintain a feasible popula-tion by flying back such updated particle to its previous posi-tion. Since the population is initialized in the feasible regionflying back to its previous position will guarantee the solutionto be feasible. Flying back to its previous position when anupdated particle violates the constraints will allow a newsearch closer to boundary.Step-10. update PðkÞbest;i and GðkÞbest as follows

PðkÞbest;i ¼XðkÞi ifJðXðkÞi Þ < JðPðk�1Þ

best;i Þ

Pðk�1Þbest;i otherwise

8<: ð21Þ

GðKÞbest at this current iteration is set as the best value in terms ofobjective function among all PðkÞbest;i.

Step-11. Increase iteration count k = k + 1

If k 6 kmax repeat from Step-5, otherwise stop. PSO is a heuristicsearch procedure and as such no analytical convergence criterionexists, except that PSO is terminated when a pre-specified maxi-mum number of iterations are executed. The maximum numberof iteration may be specified based on computational experience[23]. Further the convergence may be observed by plotting a graphbetween JðGðKÞbestÞ and number of iterations.

6. Solution methodology using co-ordinated aggregation basedparticle swarm optimization (CAPSO)

CAPSO is an improved variant of conventional PSO [25]. Eachparticle moves considering only the positions of particles with bet-ter achievements than its own with the exception of the best par-ticle which moves randomly. The coordinated aggregation may beconsidered as a type of active aggregation where particles are at-tracted only by places with the most food. Hence CAPSO differsfrom PSO described in previous section only in updating the veloc-ity of each particle (Step-5 in previous section). Velocity of eachparticle except the best in swarm is updated using followingrelation.

qðkÞi ¼ w � qðk�1Þi þ

XrjAFðk�1Þ

ij Xðk�1Þj � Xðk�1Þ

i

h ið22Þ

rj denotes random digit between [0,1] AFðk�1Þij is known as achieve-

ment factor, which is the ratio of difference between achievement(objective function) of particle and better achievements by parti-cle-j to the sum of all these differences. Expression for AFðk�1Þ

ij is gi-ven as follows.

AFðk�1Þij ¼

JðXðK�1Þi Þ � JðXðK�1Þ

j ÞX

l2TJðXK�1

i Þ � JðXðK�!Þl Þ

h i ð23Þ

T represents the set of particles-j with better achievement in termsif objective function.

Relation (22) is used to update the velocities of all particles ex-cept the best particle in the swarm. The velocity of best particle inthe swarm is updated using a random co-ordinator calculated be-tween its position and the position of a randomly chosen particle inthe swarm as follows.

qðkÞp ¼ w � qðk�1Þp þ r Xðk�1Þ

m � Xðk�1Þp

h ið24Þ

Xðk�1Þm is randomly selected particle, Xðk�1Þ

p is the best particle in theswarm and ‘r’ is random digit between [0,�1]. The updating of bestparticle of group behaves in a crazy way and helps the swarm to es-cape from local minima. Remaining steps are same as explained inprevious section. CAPSO has better convergence characteristics andrequire only inertia weight parameter to start the updatingprocedure.

7. Results and discussions

The algorithm developed in this paper has been implementedon a sample radial distribution system as shown in Fig. 3 [11].The system has seven load points in addition to a source sub-sta-tion and seven distributor segments. The system has been modifieddue to presence of DG at load points LP-3, LP-6 and LP-8. In the ab-sence of loss of supply at these nodes continuity is maintained withthe help of these distributed sources. Further it is assumed thatDG-6 present at LP-6 is sufficient to feed the need of only 50% con-sumers connected at the load point and distributed generator DG-3

LP-2

LP-3LP-5

LP-7

LP-8

#1 # 2

#3

# 4

# 5

#6

# 7

1

DG-8

Source

DG-3

DG-6

LP-6

~

~

~

Fig. 3. Radial distribution system with DG at selected load points.

Table 2Average load and number of customer at load points.

Load point LP-K 2 3 4 5 6 7 8

Average load, LK, kW 1000 700 400 500 300 200 150Number of customer 200 150 100 150 100 250 50

Table 3Cost coefficients ak and bk for calculating objective function (J).

Sr. No. 1 2 3 4 5 6 7

ak (Rs.) 246 40 81 250 32 4.5 5.0bk (Rs.�102) 100.00 72.90 72.00 200.00 180.00 38.40 93.60

612 R. Arya et al. / Electrical Power and Energy Systems 43 (2012) 607–616

and DG-8 connected at load points LP-3 and LP-8 are having suffi-cient capacities to meet demand of all the consumers connected atthese load points. Table 1 gives current failure rates ðk0

j Þ, repairtime ðr0

j Þ and bounds on these quantities for each section. Due topresence of distributed energy resources modifications on bothsides of decision variables have been considered. Records ofgrowth model of these variables allow to estimate bounds on thesevariables. Table 2 provides average loads and number of customersat each load point. Table 3 provides cost coefficient ak, bK as re-quired to evaluate objective function (J). The ADCOST is selectedas Rs.1.5 per kW h.

Reliability modeling aspects are described in Section 2. kSYS; rSYS

and USYS, for load points LP-2, LP-4, LP-5 and LP-7 (these do nothave DG) are evaluated using following relation.

kSYS;2 ¼ k1

rSYS;2 ¼ r1

USYS;2 ¼ k1r1

kSYS;4 ¼ k1 þ k2 þ k3

Table 1System data for sample radial distribution system.

Distribution section k0j ð1=yrÞ kj;minð1=yrÞ k

1 0.4 0.2 02 0.2 0.05 03 0.3 0.1 04 0.5 0.1 05 0.2 0.15 06 0.1 0.05 07 0.1 0.05 0

USYS;4 ¼ k1r1 þ k2r2 þ k3r3

rSYS4 ¼USYS;4

kSYS;4

kSYS;5 ¼ k1 þ k4

USYS;5 ¼ k1r1 þ k4r4

rSYS;5 ¼USYS;5

kSYS;5

kSYS;7 ¼ k1 þ k2 þ k6

USYS;7 ¼ k1r1 þ k2r2 þ k6r6

rSYS;7 ¼USYS;7

kSYS;7

Since distribution generations are present at LP-3, LP-6, and LP-8,the calculation of basic reliability indices will be different. The reli-ability network to evaluate basic indices for LP-3 is shown in Fig. 4as per modeling aspects described in Section 2.

kS3 ¼ ðk1 þ k2Þ

rS3 ¼k1r1 þ k2r2

ðk1 þ k2Þ

kSYS;3 and rSYS,3 with DG are obtained using relation (1) and (2) asfollows

kSYS;3 ¼ kS3kd3ðrs3 þ rd3Þ þ kSW3

rSYS;3 ¼ks3 � kd3ðrs3 þ rd3Þ þ kSW3 � s3

kSYS;3

In above kd3; rd3 are failure rate and average down time of DG-3 kSW3

failure rate of manual switch of DG-3 s3 average service restorationtime of DG-3Similarly reliability diagram for LP-8 is shown in Fig. 5;basic reliability indices kSYS;8; rSYS;8 and USYS,8 are evaluated using fol-lowing relations.

j;maxð1=yrÞ r0j ðhrsÞ rjmin(hrs) rjmax(hrs)

.6 10 6 15.0

.3 9 6 14.0

.5 12 4 18.0

.8 20 8 30.0

.3 15 7 22

.15 8 6 12

.15 12 6 18

LP - 3

λs3, rs,3

λsw3, s3

λd3, rd3

Fig. 4. Reliability network for LP-3.

LP - 8

λs8, rs,8

λsw8, s8

λd8, rd8

Fig. 5. Reliability network for LP-8.

Table 4Failure rate and average down time for distributed generation.

Sr.No.

DG FailureRatekdð1=yrÞ

Downtime rd

(h)

Failure rate ofmanual switch ksw

(1/yr)

Servicerestorationtime s (h)

1 DG-3

1.50 15.78 0.5 1.5

2 DG-6

1.75 18.68 0.5 1.5

3 DG-8

2.00 20.61 0.5 1.5

Table 6Initial population generated for PSO and CAPSO.

Decision variables #1 #2 #3 #4 #5

k1 0.3040 0.2253 0.2208 0.2248 0.2339k2 0.0719 0.0623 0.0674 0.1946 0.0539k3 0.1392 0.2257 0.2834 0.4197 0.2940k4 0.1522 0.2511 0.2043 0.1499 0.2087k5 0.1980 0.2519 0.2638 0.2388 0.2219k6 0.0941 0.1041 0.0944 0.1384 0.1231k7 0.0928 0.0943 0.0886 0.0654 0.0955r1 6.2027 7.4363 7.5631 6.0613 7.7577r2 7.4233 11.0731 8.5684 11.8895 10.0196r3 5.1253 7.5651 4.2047 6.1739 10.8947r4 22.0984 14.5564 14.1669 11.9885 8.4468r5 15.4493 8.9905 17.8474 8.7557 13.3241r6 9.5325 7.6173 8.6555 11.1945 6.2040r7 6.7674 8.3229 10.6841 6.3030 11.0793Objective function,

J � 1042.7322 2.2322 2.2402 2.0557 2.6592

Table 7Control parameter for DE, PSO and CAPSO.

S. No. Parameters Values of parameters

1 Population size, (DE) 10, 202 Step size (a) 0.73 Cross over rate (Cr) 0.34 Maximum generation specified kmax 3005 c1 1.06 c2 1.07 wmax 18 wmin 0.19 Maximum iteration in PSO/CAPSO 300

10 Population size (PSO/CAPSO) 5, 10, 20

R. Arya et al. / Electrical Power and Energy Systems 43 (2012) 607–616 613

kS8 ¼ k1 þ k2 þ k6 þ k8

rS8 ¼k1r1 þ k2r2 þ k6r6 þ k8r8

kS8

kSYS;8 ¼ kS8 � kd8 þ kd8ðrS8 þ rd8Þ þ kSW8

rSYS;8 ¼kS8 � kd8ðrS8 þ rd8Þ þ kSW8 � s8

kSYS;8

kd8; rd8 are failure rate and average interruption duration of DG-8.kSW8; s8 are failure rate of manual switch and service restoration

time for DG-8.DG capacity available is sufficient to satisfy the requirement of

only 50% consumers connected at LP-6. The reliability networkfor LP-6 is shown in Fig. 4. LP-6 is partitioned in two separatenodes i.e. 6A and 6B. 50% consumer may be supplied by either

Table 5Initial population generated for DE.

Decision variables #1 #2 #3 #4

k1 0.2585 0.3914 0.2320 0.4874k2 0.0636 0.1299 0.2093 0.1163k3 0.2208 0.4862 0.4127 0.4514k4 0.2217 0.5069 0.7234 0.5934k5 0.2749 0.2705 0.2323 0.2777k6 0.0767 0.0506 0.0939 0.1408k7 0.1001 0.0890 0.0711 0.0996r1 7.5019 10.4011 8.6868 6.7307r2 11.5107 12.0494 8.4746 11.8140r3 5.7112 11.9502 14.4646 5.4722r4 8.8760 27.5932 8.5542 13.7090r5 14.1625 19.2267 7.2507 17.1514r6 11.8015 8.3456 10.3021 10.9863r7 16.3618 8.8771 8.4427 14.5380Objective function, J � 104 2.2411 1.0170 0.9647 0.8031F/NF* F NF NF NF

NF denotes not feasible solution.* F denotes feasible solution.

distribution network or DG and remaining are supplied by DG-6in case of loss of supply from source node. The basic reliability indi-ces are evaluated for LP-6A using following formula.

kSYS;6A ¼ k1 þ k4 þ k5

USYS;6A ¼ k1r1 þ k4r4 þ k5r5

rSYS;6A ¼USYS;6A

kSYS;6A

Basic reliability indices at LP-6B is evaluated using following rela-tions as given by relations (1) and (2)

#5 #6 #7 #8 #9 #10

0.4799 0.3470 0.2565 0.3955 0.3186 0.29850.1278 0.1009 0.0955 0.2051 0.0523 0.08440.1356 0.2651 0.3874 0.1332 0.1849 0.23360.3538 0.6560 0.3548 0.4777 0.4668 0.45350.2767 0.1829 0.1969 0.2419 0.2273 0.20820.1374 0.1409 0.1448 0.0984 0.0501 0.06200.1354 0.0945 0.1019 0.0840 0.0543 0.09926.4992 12.8142 13.3613 12.2991 11.7836 12.61517.8369 10.7832 6.5648 10.3990 12.5356 11.042711.7145 6.5088 7.8263 8.4846 17.3843 14.833129.5521 27.6815 23.9893 11.3934 11.4329 24.929912.6668 20.6401 13.8181 16.8318 13.2455 21.65279.5162 7.0240 7.7196 10.3243 6.7416 9.181812.5851 6.1168 11.6384 12.6372 10.2956 14.47491.2098 1.1411 1.3185 1.3519 2.5975 1.4889NF NF NF NF NF NF

Source #1 #4 #5 LP – 6A

#1 #4 #5 LP – 6B

DG - 6 ~

Fig. 6. Reliability network for indices evaluation at LP-6.

Table 8Optimized values of failure rates and repair times as obtained by DE, PSO and CAPSO.

Variables Magnitudes asobtained by DE

Magnitudes asobtained by PSO

Magnitudes asobtained by CAPSO

k1 (yr) 0.2331 0.2471 0.2245k2 (yr) 0.1789 0.1357 0.1743k3 (yr) 0.3104 0.2932 0.3449k4 (yr) 0.3430 0.3685 0.3731k5 (yr) 0.2845 0.2998 0.2719k6 (yr) 0.0920 0.0952 0.0901k7 (yr) 0.1483 0.1487 0.1239r1 (h) 6.0033 6.0023 6.4414r2 (h) 6.0767 6.0012 6.7191r3 (h) 6.1595 13.2311 8.6474r4 (h) 8.0100 8.0029 8.1497r5 (h) 7.0643 7.0685 7.2750r6 (h) 9.2649 11.3771 6.1672r7 (h) 8.3064 6.0047 6.7933Objective

function(J) � 104

1.1701 1.1836 1.1841

Table 9CPU Time for convergence required for DE, PSO, and CAPSO Techniques on Intelcore2duo processor, 2.9 GHz.

S. No Techniques Population size CPU time (ms)

1 DE 10 12520 250

2 PSO 05 7810 14120 281

3 CAPSO 05 6210 14120 250

614 R. Arya et al. / Electrical Power and Energy Systems 43 (2012) 607–616

kSYS;6B ¼ kSYS;6A � kd6ðrSYS;6A þ rd6Þ þ kSW6

rSYS;6B ¼kSYS;6A � kd6 � rSYS;6A � rd6 þ kSW6 � s6

kSYS;6B

In above kd6; rd6 are failure rate and average down time of DG-6. ksw;6

is failure rate of manual switch of DG-6. s6 denotes average servicerestoration time of DG-6.

The basic indices evaluated as above are used to evaluateSAIFI, SAIDI, CAIDI and AENS as given by relation 8,10,12, and 14

Table 8aPopulation as obtained at the converged iteration along with values of objective function.

Decision Variables #1 #2 #3 #4

k1 0.2215 0.2195 0.2324 0.2231k2 0.1501 0.1600 0.1616 0.1646k3 0.3951 0.3716 0.3264 0.3733k4 0.3893 0.3933 0.3632 0.3843k5 0.2642 0.2672 0.3000 0.2658k6 0.0992 0.0978 0.0936 0.0872k7 0.1460 0.1472 0.1497 0.1369r1 6.1616 6.4672 6.0052 6.6341r2 6.0016 6.0016 6.0174 6.0015r3 5.5218 5.3490 5.0580 6.0633r4 8.0227 8.0153 8.0009 8.0070r5 7.1219 7.0767 7.0000 7.1101r6 6.9751 7.3910 6.0009 6.6675r7 7.4315 7.1725 17.9415 10.2842Objective function, J � 104 1.1796 1.1786 1.1769 1.1836F/NF* F F F F

NF denotes not feasible solution.* F denotes feasible solution.

respectively. Similarly EENSO and EENSD are also evaluated usingrelations 4,4a,4b.

Table 4 gives failure rate, average down time for distributedgenerations present at load points 3, 6 and 8. It also provides fail-ure rate (ksw) and service restoration time (s) for manual switch ofthese distributed generations. In evaluating EENSO (Without DG)the DG at location 3, 6, and 8 load points are not considered. Theproblem as formulated in Section 3 has been solved using DE,PSO and CAPSO based algorithm as explained in Sections 4–6.Table 5 gives initial population generated for DE. Table 6 presentsthe initial population generated and used in PSO and CAPSO algo-rithms. These populations were generated with an inbuilt randomnumber generator program. Table 7 gives the final control param-eter selected for DE, PSO and CAPSO (see Fig. 6).

Table 8 gives optimum failure rate and repair time of each sec-tion as obtained by these three methods, along with cost function.It is observed that results obtained by these three methods are inclose agreement. Further in DE based optimization all the membersof population become feasible and cluster around the optimumsolution at the end of convergence. This has been an importantcharacteristics of the solutions obtained by DE and has beenpointed out in Ref. [20]. This has been also observed by us inour computational procedure at the end. This has been put inTable 8a. Table 8a shows all solutions (population) along withone optimal solution with associated values of objective functions.This table validates the mentioned characteristics. Table 9 givesCPU time required to solve the optimization problem by threemethods with various population sizes. It was observed that PSOconverges with even five population size whereas DE does not con-verge with five population size. Solution with DE converges withten population size. But as population size is increased CPU timewith PSO increases from 78 ms to 281 ms with population size 5

#5 #6 #7 #8 #9 #10

0.2331 0.2312 0.2200 0.2247 0.2328 0.23330.1789 0.1619 0.1639 0.1683 0.1576 0.15750.3104 0.3286 0.3381 0.3320 0.3283 0.32960.3430 0.3683 0.3928 0.3749 0.3702 0.36280.2845 0.3000 0.2492 0.2922 0.3000 0.30000.0920 0.0920 0.1065 0.0890 0.0914 0.09560.1483 0.1495 0.1490 0.1448 0.1500 0.15006.0033 6.0006 6.4915 6.0185 6.0033 6.00966.0767 6.0013 6.2205 6.2610 6.0095 6.00046.1595 6.8840 5.1022 5.0691 7.7476 5.84328.0100 8.0004 8.0366 8.0115 8.0009 8.00127.0643 7.0001 7.1857 7.0374 7.0000 7.00109.2649 6.0009 6.9686 6.5093 6.0000 6.00148.3064 17.9447 6.8025 8.1343 17.9345 17.95071.1701 1.1768 1.1818 1.1757 1.1774 1.1772F F F F F F

Table 10Current and optimized reliability indices for radial distribution system.

S. No. Index Current values Optimized values Threshold values

DE PSO CAPSO

1 SAIFI interruptions/customer 0.6610 0.4997 0.5000 0.5000 0.50002 SAIDI h/customer 7.2671 3.0731 3.3297 3.2324 4.00003 CAIDI h/customer interruption 10.9933 6.1495 6.6595 6.4648 8.00004 AENS kW h/customer 22.2334 9.0454 9.9999 9.8024 10.0000

Fig. 7. Variation of best value of objective function (J) with population size 5 in PSO,CAPSO and population size 10 in DE.

Fig. 8. Variation of best value of objective function (J) with population size 10 inPSO, CAPSO and DE.

Fig. 9. Variation of best value of objective function (J) with population size 20 inPSO, CAPSO and DE.

R. Arya et al. / Electrical Power and Energy Systems 43 (2012) 607–616 615

and 20 respectively. The same is observed with DE and CAPSO. Forpopulation size 20 the CPU time required by DE is less than that re-quired by PSO but equals to that required by CAPSO. In all the threemethods CPU time increases as population size increases. Table 10gives reliability indices as obtained using current value of decisionvariables as well as those obtained using optimized variables bythree methods. Figs. 7–9 shows convergence curves as obtainedby three methods for different population sizes.

8. Conclusions

This paper has presented a methodology for reliability design ofa radial distribution system in the presence of distributed genera-tion. An optimization problem has been formulated which obtainsoptimum failure rates and repair times of each section subject tosatisfaction of inequality constraints based on energy and cus-tomer based reliability indices. Operating philosophy of DG hasbeen assumed in standby mode and cost of purchased energy ismore than that supplied from distribution substation. Formulatedproblem has been solved using two most robust and popular pop-ulation based algorithm, i.e. DE and PSO. The solution has also beenobtained using CAPSO, one of the variant of PSO. The results ob-tained have been compared. PSO requires less CPU time (popula-tion size 5) even with less population size than used in DE(population size 10). It was also observed that CPU time requiredfor convergence increases with population size for all the threetechniques. It was also observed that for population size twentyCPU time required by DE is less than that required by PSO.

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