Modeling he Repair Process ofa Power Disribuion Sysem c. J. Zapaa, Member EEE, S. C. Silva, H. Gonzalez, O. L. Burbano and J. A. Hernandez Absrac-One remedial acion for improving service reliabiliy in a power disribuion sysem is o faser crew response o reduce ouage duraions. For planning aciviies, he applicaion of his kind of remedial acion requires he modeling of he repair process performed in he power disribuion sysem. Thus, his paper presens a mehodology which models he repair process performed in each service erriory of a power disribuion sysem using conceps of queuing heory and sochasic poin processes and assesses he repair process performance by means of a procedure of sequenial Monecarlo simulaion. Resuls of he applicaion of his mehodology o hree power disribuion sysems show: 1. The inpu and service of he repair process model are no necessarily homogeneous Poisson processes; hus he Mone Carlo simulaion mehod is required for he assessmen of he repair process performance. 2. The index ha beer reflec he repair process performance is he mean waiing ime. This mehodology gives a base for he opimal scheduling of he repair resources in accordance wih he failure process and he arges for reliabiliy indices. For planning aciviies, he applicaion of his kind of remedial acion requires he explici modeling of he repair process performed in he power disribuion sysem. For mainenance aciviies he power disribuion sysem is spli ino several zones or service erriories, each one assigned o a repair eam [3], as shown in Fig 1. ndex Terms-Mainenance, power disribuion reliabiliy, reliabiliy modeling.. NTRODUCTON DSTRBUTON reliabiliy is a field of grea ineres all around he world because he disribuion funcional zone of a power sysem conribues a leas wih 90% of he failures ha affecs service coninuiy. Thus, i has a grea poenial for he improvemen of sysem performance and savings [1]. There are several ways o improve he reliabiliy of a power disribuion sysem. R. E. Brown in [2] explains he following: special proecive schemes, auomaion, faser crew response, reduce failure raes and sysem reconfiguraion. Regarding faser crew response, his remedial acion can improve service reliabiliy because i reduces ouages duraions. C. J. Zapaa is a professor a Universidad Tecnol6gica de Pereira, Pereira, Colombia, and a PhD suden a Universidad de los Andes, Bogoa, Colombia. (e-mail: cjzapaa@up.edu.co). S. C. Silva is a suden a Universidad Tecnol6gica de Pereira, Pereira, Colombia, (e-mail: silvanas80(~yahoo.com). H.. Gonzalez is wih Empresa de Energia del Casanare S. A, Yopal, Colombia. He also is a graduae suden a Universidad de los Andes, Bogoa, Colombia, (e-mail: heyderivan2005@gmail.com). O. L. Burbano is a suden a Universidad Tecnol6gica de Pereira, Pereira, Colombia, (e-mail: olgabur@homail.com). 1. A. Hernandez is a graduae suden a Universidad de los Andes, Bogoa, Colombia, (e-mail: ja.hemandez913@;uniandes.edu.co). Fig. 1. s for mainenance in a power disribuion sysem The resources for repairs are he personnel, rucks, ools, spares, ec. available for his work. The way hese resources are organized, for example, he number of crews for each zone, is he logisics. The repair resources generae he repair process. The repair process is he sequence ofrepairs performed by crews in accordance wih he repair orders sen by he conrol cener, which eiher auomaically deecs componen failures or receives cusomer calls regarding service inerrupions. Thus, he repair process in each service erriory is a queuing sysem. The inpu o his sysem is he sequence of componen failures which produce service inerrupions ha have o be repaired by crews. The oupu of his sysem is he sequence ofservice resoraions performed by crews. The performance of he repair process is dependan on he qualiy and quaniy of repair resources and he logisics. These resources are limied and have o be carefully mached o follow he pace of componen failures in order o obain accepable ouage imes. Tradiional mehods for sudying he repair process of a power disribuion sysem do no model i as i really is. Thus, he subjec of his paper is heir modeling using conceps of queuing heory and sochasic poin processes. 978-1-4244-2218-0/08/$25.00 2008 EEE.
2. TRADTONAL METHODS FOR STUDYNG THE REPAR PROCESS The repair process performed in a power disribuion sysem has been radiionally sudied in he following ways: A. By means ofsaisical analysis ofouage imes These kinds of sudies ake operaing daa of he power disribuion sysem and analyses he saisics ofouages imes by feeders, subsaions and geographical zones o give guidelines abou which zones of he sysem need improvemen on he repair process performance. Alhough hese kinds of sudies can include he modeling of he ouages imes using probabiliy disribuions or sochasic processes [4], [5], hey do no include an explici modeling ofhe repair process. As a service erriory can include pars of several feeders, hese kinds of sudies have o be exended o each service erriory because, in his case, a global analysis can be misleading. B. As par ofhe componen reliabiliy models This approach is exensively applied in power disribuion reliabiliy assessmens [6], nor maer he mehodology, cuses [7]-[8], analyical simulaion [9], Markov process [10] or Monecarlo simulaion [11]-[12]: he repair process is included as par of componen reliabiliy models, by means of he probabiliy disribuion ofimes o repair. This approach has he following disadvanages: No maer he probabiliy disribuion used, i is assumed repair resources are unlimied because every ime a componen fails a crew is available o repair i. So, he repair ime only depends on he paricular acions aken o fix each kind ofcomponen. As he repair process is represened by means of a probabiliy disribuion, i is assumed i is a saionary process i.e. he performance ofrepair eams is no affeced by inernal or exernal facors. However, in real life, he crew performance is affeced by exernal facors like weaher, raffic, ec. and also by inernal facors like, available ools, available skills, workload, ec. The endency ofhe repair process performed in he power disribuion sysem is los because he imes o repair are classified by componen ype and hus he chronological sequence in which hey occur is los. Mos mehods apply he n -1 loss of componen crieria wha i no a rue assumpion because a failure can occur independenly ifoher failures which occurred before have been repaired or no. Reliabiliy assessmens ofpower disribuion sysems only include high volage and medium volage componens. However in real life, repair eams also have o repair he low volage componens wha represens an imporan demand on repair resources. Moreover, reliabiliy surveys shows in some power disribuion sysems he low volage componens are he ones ha fail more frequenly [13]. A. Definiion Fig. 2. The concep of SPP. STOCHASTC PONT PROCESSES A sochasic poin process (Spp)l is a random process in which he number of evens N ha occur in a period of ime!::,. is couned, wih he condiion ha one and only one even can occur a every insan. Fig. 2 presens a picorial represenaion of a SPP where Xi denoes an iner-arrival inerval and i an arrival ime. f he dae when he observaion of he process sared is aken as reference,!::,. = - 0, only appears in he equaions ha describe he process. Even 1 Even 2 Even 3 j i Ai x J! L Xl ~ -00 0 11 ) ) i:jj.., - The mahemaical model of a SPP is defined by means of he inensiy funcion A(), which is he rae of change of he expeced number of evens. This parameer allows he calculaion of: The expeced number of evens: The variance: i E[N()] = A() = f~ A()d (1) VAR[N()] = A() The probabiliy ha k evens occur: P[N() = k] = ~[A() *e-a() for k = 0,1,2,,,' (3) k! B. The Concep oftendency The endency, defined as he change wih ime in he number of evens ha occur, is a very imporan feaure of a SPP. n Fig. 3 he following hree kinds of endency are depiced [14]: Posiive endency: Evens increase wih ime and inerarrival inervals decrease. A() is an increasing funcion. Zero endency: Evens ha occur and iner-arrival inervals do no show a paern of increase or decrease. A() is consan. Negaive endency: Evens decrease wih ime and inerarrival inervals increase. A() is a decreasing funcion. (2) 1 The Singular and plural of acronyms are spelled he same.
3 Fig. 3. Tendency on SPP C. Types ofspp Posiiw ZeD Negaive A SPP wihou endency is saionary or imehomogeneous. Homogeneiy means iner-arrival inervals are independen and idenically disribued. So, evens ha occur are independen. The opposie is rue for a SPP wih endency. The endency allows he basic classificaion of SPP shown in Fig. 4. No endency: SaDwy, homgeneous -+ plocesses EXPonemw(HPP)} Renewal plocesses -+Gamma. 1 (RP) Weibull l() = - Dc. E(~ Nan Wih endenc:y: Homogerl OUS PowerLaw 1() = 1 ~P-l Nan-saim.aJ.y: ---... Po' ---...., --.-- l5san --.. Alemaing li) =1 + Q Sl...(l) + b... non-ldnnge:reou:s plocesses "4~:J plocesses (NHPP) Ec. Fig. 4. A basic classificaion of SPP Saionary SPP are called renewal processes (RP) preceded by he name of he iner-arrival inervals disribuion. The inensiy funcion of a RP is he inverse ofhe expeced value of iner-arrival inervals. The mos famous RP is he exponenial one, commonly called Homogeneous Poisson Process (HPP). Non saionary SPP are in general called Non Homogeneous Poisson Processes (NHPP). D. Procedurefor he Selecion of a SPP Model[14], [15] The procedure for fiing a SPP model o a sample daa aken from a random poin phenomenon is as follows: 1. Deermine if here is a endency in he arrival imes sample by means of he Laplace es or graphic mehods [16]. 2. f here is evidence of a endency, selec a NHPP model, esimae is parameers and apply a goodness of fi es. A problem wih NHPP is ha mehods for parameer esimaion and goodness offi are specific for each kind of model and for some ofhem are no developed ye. 3. f here is no evidence of a endency, apply an independence es o iner-arrival inervals such as he scaer diagram or he correlaion plo [17]. 4. f iner-arrival inervals are independen, fi a probabiliy disribuion using he radiional mehods for parameer esimaion and goodness of fi. n his case, a RP model is obained. E. The Power Law Process While here are many NHPP models, he approach here is o use he Power Law Process () developed by L. Crow in 1974 [18] because: is an acceped model o represen he failure process of repairable componens [19]. There are mehods for parameer esimaion and goodness of fi [19], [20], [21], [22]. can represen a process wih or wihou endency can represen he HPP. The inensiy funcion ofhis process is: Where: A: Scale parameer greaer han zero. p: Shape parameer greaer han zero. The shape parameer conrols he endency ofhe model in he following way: f p> 1: Posiive endency f ~ < 1: Negaive endency f p=1: F. How o Generae Samplesfrom SPP Models 1) Renewal Processes 1. Le o = 0. 2. Generae a uniform random number U i (4) Zero endency. n his case, he represens he HPP. 3. Ge an iner-arrival inerval Xi = F- 1 (U i ) using he probabiliy disribuion funcion of he iner-arrival inervals. 4. The arrival ime is i = i - 1 + Xi' 5. Go o sep 2 unil he sopping rule is reached: a given number ofevens or a sample period T. 2) Non homogeneous Poisson Processes [17] 1. Generae a sequence of n arrival imes from an HPP wih inensiy funcion A= 1.0 which covers sample period T. These imes are called 1 ', 2 ',. ",n'. 2. Find he inverse funcion of he mean cumulaive number ofevens ofhe NHPP under sudy ( A- ). 3. Calculae he arrival imes of he NHPP as i = A- (i') As poined ou by Law and Kelon [17], he applicaion of his algorihm depends on how easy he inversion of A is. n he case of he recursive equaion is: (5)
4 V. MODELNG OF THE REPAR PROCESS The repair process of each zone (service erriory) of a power disribuion sysem is modeled as he queuing sysem shown in Fig. 5. failures process '-- Crew 1... Repairs.~ Crew2 -.-LLLL Crewm Fig. 5. Queuing model ofhe repair process in a service erriory of a power disribuion sysem For his queuing sysem he following is defined: Cliens: Failures which produce service inerrupions and have o be repaired by crews Resources: The number of crews in he zone. A crew corresponds o a server in queuing heory erminology. Capaciy: nfinie, because all he failures considered here have o be repaired. Queuing discipline: Firs Come - Firs Served (FCFS) npu process: The zone failure process. is he superposiion of he failure processes of he componens locaed in he zone. Only failures which produce service inerrupions and have o be repaired by a crew are considered. This process has an failure inensiy A F (). Service process: The SPP ha represens he equivalen capaciy of all crews assigned o he zone in he form of a repair inensiy A R (). Oupu process: The SPP of he repairs performed by crews. These repairs are relaed o service resoraions. The oupu process is he resul of he ineracion beween he inpu and he service processes. Using Kendall's noaion [23] his queuing sysem is described as follows: G/G/m/oo/FCFS The firs and second "G" indicae ha boh he inpu and service processes are general SPP (RP or NHPP). m, 00, and FCFS indicae, respecively, he number of crews, he sysem capaciy and he queuing discipline. The raffic inensiy index ac) is defined as [23]: Alhough ac) is dimensionless, i is measured in Erlangs. A raffic inensiy of 1.0 Erlang means one failure uses or occupies he repair resources 100% of he ime. Traffic inensiy higher han 1.0 means he failures arrives faser han repairs can be performed. Thus, ac) have o be less or equal o 1.0 in order o have a sable queuing sysem. (6) V. ASSESSMENT OF THE REPAR PROCESS PERFORMANCE A. Obaining he zonefailure process From operaing records, obain a sample of arrival imes of hose componen failures which caused service inerrupions and were repaired by crews. is recommended he sample covers a leas one yearof sysem operaion. is imporan o remember ha: No all service inerrupions are solved by crews; some of hem are solved by means of a reconnecion performed by a circui breaker or recloser. Low volage componens also cause service inerrupions which in mos ofhe cases have o be repaired by crews. Apply o he failure arrival imes sample he procedure for selecing an SPP model. n accordance o he endency on he resuling zone failure process i can be concluded he following: Zero endency: The populaion of componens locaed in he zone are in heir useful life. This is, heir reliabiliy is no improving neiher deerioraing. Posiive endency: The populaion of componens locaed in he zone shows aging. Negaive endency: The populaion of componens locaed in he zone shows reliabiliy improvemen. This kind of modeling implies repairs are minimal [24], i.e. hey only reurn he componens o he operaing sae wihou improving or deerioraing heir reliabiliy condiion. B. Obaining he zone service process For each failure ha caused a service inerrupion and was repaired by means a crew acion obain he ime o repair (r ). Time o repair includes: he ransporaion ime o he place where cusomers are wihou service, ime o find he failed componen, ime o fix he failed componen and reconnecion ime. A r does no include he waiing ime (w) he period while he crew receives he repair order and is free o go o repair he failure. The waiing ime is resul of he congesion on he repair process, he fac ha when a crew receives a repair order i can be busy repairing a failure ha occurred before. Apply o he sample of imes o repair he procedure for selecing an SPP model. n accordance o he endency on he resuling service process i can be concluded he following: Zero endency: The crew performance is no increasing no decreasing Posiive endency: The crew performance is increasing because as ime evolves repairs ake less ime o be performed Negaive endency: The crew performance is decreasing because as ime evolves repair ake more ime o be performed.
5 C. Assessing he repair process performance The repair process is observed arificially for a period T of one or more years by means of a sequenial Mone Carlo simulaion procedure [25]. A simulaion consiss of N ieraions or arificial observaions of he repair process performance during T. n each ieraion, he sequence of componen failures and repairs is generaed using he inpu and service processes. Fig. 6 shows he ineracion beween he failure and service processes for a zone wih one crew or one equivalen crew. Failures 1; h h J; 1; f all crews are busy when failure i arrives, his failure has o wai unil some crew finishes a repair j and fixes i (Congesion). r; = r j + r; f a crew is free when failure i arrives, he repair for his failure sars immediaely (No congesion) r; = f; + r; (9 ) 6. Calculae he ouage duraion (8 ) o d; = r; - f; (10) ~~ ~ l-+---+-~ ~ l ~ '11.. 7. Calculae he repair waiing ime: w; = od; - r; (1 1) Repairs 8. For T or is sub-periods (monh, semeser, ec.) compue he mean waiing ime (mw), he mean ouage duraion ( mod ) and he congesion ( C ) defined as: C = m w / m o d * 1 00 0A, (1 2) Fig. 6. Calculaion ofouage duraions Every ime a failure arrives, i is assigned o a crew ha performs a repair j in a ime lj. The arrival imes of and j are if and lj, respecively. Each ieraion produces a sample of number of failures ( nf), imes o repair (r), ouage duraion (od) and waiing imes ( w ). Two sopping rules are used for he simulaion: A fixed number of ieraions or he coefficien of variaion of a load poin index. D. eraion Procedure 12 ~ ~,..--. ~ od. od~ 1. Generae he inpu process for a period T using he zone failure SPP model and algorihms ofsecion.f. 2. Generae he service process. This is, for each failure J; generae a lj using he zone service SPP model and algorihms of secion.f. 3. Compue he mean raffic inensiy ma() 4. The arrival ime ofhe firs repair is: r l = fl + r l (7 ) 5. The arrival ime of he nex repair is deermined in he following way: V. EXAMPLES Tradiional queuing analyses assume he inpu and service processes are Markovian (Boh HPP) or semi-markovian (One HPP and he oher a RP). However, for he repair process ofa power disribuion sysems i is no known which SPP models can represen his processes. Thus, daa of hree Colombian power disribuion sysems was gahered in order o know hese models and o apply he proposed mehodology. Table shows general descripion of he sudied sysems. For each sysem an assessmen of he repair process performance was carried ou for T = 1.0 year wih simulaions of 150 ieraions. Tables o V shows resuls. Confidence level ofinpu and service process models is 95%. These resuls show: For Pereira and Paso sysems he inpu and repair processes are non-saionary wih posiive endency. This means alhough he reliabiliy of he componens is deerioraing, he repair process is adjusing o follow he increasing paern of failures arrivals. For Casanare sysem he inpu and repair processes are saionary bu hey do no correspond o he HPP. The performance of he repair process is direcly conneced wih he size (area) of he service erriory. For Pereira and Casanare sysems he wors indexes corresponds o zones (service erriories) wih highes areas. The effec of remedial acions proposed o reduce ouage duraions can be esed wih his mehodology. A low congesion or raffic inensiy does no mean a low waiing ime or consequenly a low ouage duraion. Resuls of mean ouage ime corresponds o hose values observed during operaion ofhe sudied sysems.
6 TABLE GENERAL DATA OF STUDED SYSTEMS Sysem Pereira Paso Casan.ue Region MunicipaliyofPereira MunicipaliyofSan Juan de Paso Deparlmerd ofcasan.ue Uiliy Empresa de Energia de Pereira S. A. Cenrales Eledricas de Narifio S. A. Empresa de Energia. del Casan.ue S. A Area [)em"] 702 1181 44640 UJban populaion 371239 317:377 20ffi52 lwral populaion 72315 70241 94401 Service erriories 3 1 3 Crews per mne 3 2 2 Noes: 1. ncolombia. a deparlmerd is a group ofmurlicipaliies 2. Thecapial ofdeparl:men mcasanare is Yopal ciy (nhabiards: 90218 mban and 166]4!Ural) 3. Populaion in given ininhabiards 4. Source for populadn daa: Colombian ce!lsus ofyear :;:n05 (www.dane.gov.co) 5. Casanare Deparlmerd is rnain:v grassland pladu wih vej:ydifflcw ransjlj: condiiors during rainy season which las a leas 6 monhs. A. Resulsfor Pereira Sysem 2 TABLE PERERA SYSTEM -NPUT AND SERVCE PROCESSES npu plocess Service Process [Failureslhou:r] [Repairshour] A. p = 0.1471 ~p =1.0539 A., = 0.3765 ~, =1.0291 PiP A p =0.0363 ~,. =1.1584 A... = 0.4218 fl... =1.0560 3 A. p = 0.0715 /lap =1.1254 A p =0.2313 ;3p =1.1133 Noe: These models were buil Wlh daaof ~ar 2J05 TABLE PERERA SYSTEM -REPAR PROCESS PERFORMANCE mr mod n'jw C ma(f) [Houn] [Hours] [Hours] rh] ro] 1 2.08 4.13 2.05 49.63 49.11 2 1.53 2.00 0.47 23.jJ 21.74 3 1.72 2.79 1.07 38.35 34.53 B. Resulsfor Casanare Sysem 2 TABLE V CASANARE SYSTEM - NPUT AND REPAR PROCESSES npu plocess Service Process [Failureshour] [Repairslhour] Weibull RP A... =0.1814 Weinill. RP A, =0.2132 ap =0.3133..e... =0.1547 a, =0.5150 J" =0.6611 Weibull RP A.,. =0.1971 WeibulJ. RP A. o =0.3086 a p = 0.4492..e,. =0.6285 a, = 0.5179 ~, =0.1001 Weibull RP A... =0.2175 Weibull RP A... =0.3125 3 a p =0.3911 fi... =0.1154 a... = 0.6f68 fj lr = 0.5712 Noes: 1. These nod.els were buil wih daa ofyem2004-ljl)5 2. The Weibull densiy flmchn is defmed a5 () = a.j3p-l eiqj(- ajp) TABLE V CASANARE SYSTEM - REPAR PROCESS PERFORMANCE mr mod mw C ma(f) [Hours] [H0Ul'S] [Hou.rsJ ro] ro] 1 3.66 19.72 16.GS 81.44 68.0 2 3.24 18.22 14.98 82.22 65.0 3 3.19 23.41 20.22 86.37 70.0 C. Resulsfor Paso Sysem TABLE V PASTO SYSTEM - NPUT AND SERVCE PROCESSES npu plocess Service Process [Failu.reslhour] [Repairshour] 1... =0.5589 ~p =1.0464 A R =1.6853 JJ, =1.0 189 Noe: These models were buil Wlh daaof ~ar 2J1)S TABLE V PASTO SYSTEM - REPAR PROCESS PERFORMANCE mr mod n'lw C ma() [lainues] [Minues] [Minues] r/6] r...] 1 32.34 53.49 21.15 39.54 37.10 V. CONCLUSONS 1. The repair process performed in each service erriory of a power disribuion sysem is a queuing sysem. Thus, i has o be modelled using queuing models, no as par of componen reliabiliy models, he radiional approach applied in reliabiliy assessmens. Also, i is no realisic o apply he deerminisic crieria n -1 for he sysem reliabiliy assessmens because a failure can occur independenly if he previous failure has been or no repaired. 2. As shown in he examples, he inpu and service of he repair process of a power disribuion sysem are no necessarily HPP; hey can be RP or NHPP, and for his reason, he sysem reliabiliy assessmen has o be performed by means of a sequenial Mone Carlo simulaion. The approach presened here, ha considers saionary and non-saionary SPP for he failure and he
7 service processes, is very differen from radiional queuing modeling, which assumes ha hese processes are Markovian (HPP) or semi-markovian (One HPP and he oher a RP). 3. The index ha beer reflecs he performance ofhe repair process is he waiing ime. A low congesion or raffic inensiy does no necessarily mean a low waiing ime or consequenly a low ouage duraion. 4. The proposed mehodology explicily evaluaes he performance of he repair process performed in a power disribuion sysem and gives an analyical base for he opimal scheduling of he repair resources in accordance wih he failure process generaed by he componens and he arges for reliabiliy indices. V. ACKNOWLEDGMENT The auhors graefully acknowledge he conribuion of Fernando Valencia (Empresa de Energia de Pereira S. A.) and Raul Oriz (Cenrales Elecricas de Narifio S. A.) for supplying he informaion for his sudy. X. REFERENCES [1] Kjolle G, Rolfsegn L, Dahl E, "The economic aspec of reliabiliy in disribuion sysem planning", EEE Trans. Power Delivery, Vol. 5, No. 2,1990. [2] Brown R. E, Elecric Power Disribuion Reliabiliy, Marcel Dekker, 2002. [3] Zografos K. G, Doulgeris C, Tsoumpas P, "An inegraed framework for managing emergency-response logisics: he case of he elecric uiliy companies", EEE Transacions on Engineering Managemen, vol. 45, May, 1998. [4] Chow M. Y, Taylor L. S, Chow M. S, "Time of ouage resoraion analysis in disribuion sysems", EEE Transacions on Power Delivery, Vol. 11, No.3, 1996. [5] Balijepalli N, Subrahmanyan S. Venkaa, Chrisie R. D, "Modeling and analysis of disribuion reliabiliy indices", EEE Transacions on Power Delivery, Vol. 19, No.4, Ocober 2004. [6] Zapaa C. J, Silva S. C, Burbano O. L, "Repair models of power disribuion componens", EEE Transmission & Disribuion Lain America, 2008. [7] EEE, Power Sysem Reliabiliy Evaluaion, Tuorial Course 82 EHO 195-8-PWR, 1982. [8] EEE Recommended Pracice for he Design ofreliable ndusrial and Commercial Power Sysems, EEE Sandard 493,1997. [9] Kjolle G, Sand K, "RELRAD - An analyical approach for disribuion sysem reliabiliy assessmen", EEE Transacions on Power Delivery, Vol. 7, No.2, 1992. [10] Billinon R, Allan R. N, Reliabiliy Evaluaion ofengineering Sysems: Conceps and Techniques, Plenum Press, 1992. [11] Billinon R, Allan R. N, Reliabiliy Evaluaion of Power Sysems, Plenum Press, 1996. [12] Billinon, R., Wang Peng, "Teaching disribuion sysem reliabiliy evaluaion using Mone Carlo simulaion", EEE Transacions on Power Sysems, Vol. 14, No.2, 1999. [13] Zapaa C. J, Monealegre P. A, "Technical problems on he qualiy of elecriciy service", Revisa Mundo Elecrico, No. 68, 2007. (n Spanish). [14] Ascher H, Feingold H, Repairable sysems reliabiliy: Modeling, inference, misconcepions and heir causes, Marcel Dekker, 1984. [15] Ascher H. E, Hansen C. K, "Spurious exponenially observed when incorrecly fiing a disribuion o nonsaionary daa", EEE Transacions on Reliabiliy, Vol. 47, No.4, December 1998. [16] Wang P, Coi D. W, "Repairable sysems reliabiliy rend es and evaluaion", EEE Annual Reliabiliy and Mainainabiliy Symposium, 2005. [17] Law A. M, Kelon D. W, Simulaion Modeling and Analysis, Mc-Graw Hill, 2000. [18] Crow L. H, "Evaluaing he reliabiliy of repairable sysems", EEE Annual Reliabiliy and Mainainabiliy Symposium, 1990. [19] EC Power law model - Goodness-of-fi es and esimaion mehods, EC Sandard 61710, 2000. [20] Park W. J, Kim Y. G, "Goodness-of-fi ess for he power law process", EEE Transacions on Reliabiliy, Vol. 41, No.1, March 1992. [21] Park W. J, Kim Y. G, "More goodness-of-fi ess for he power law process", EEE Transacions on Reliabiliy, Vol. 43, No.2, June 1994. [22] Klefsjo B, Kumar U, "Goodness-of-fi ess for he power law process based on he TTT plo", EEE Transacions on Reliabiliy, Vol. 41, No. 4, December 1992. [23] Chee H. N, Queuing Modeling Fundamenals, Wiley, 1996. [24] Rigdon S. E, Basu A. P, Saisical Mehods for he Reliabiliy of Repairable Ssysems, Wiley, 2000. [25] CGRE Task Force, "Sequenial Probabilisic Mehods for Power Sysem Operaion and Planning", Elecra, No. 179, pp. 69-97, Aug. 1998. [26] Zapaa C. J, Burbano O. L, Silva S. C, "Assessing he performace ofhe repair process of a power disribuion sysem", Revisa Scienia e Thecnica, No.37, Universidad Tecno16gica de Pereira, 2007. Available: hp://www.up.edu.co. (in Spanish). [27] Gonzalez H., "Sudy of he repair process of a power disribuion sysem by means of sochasic poin processes", Universidad de los Andes, 2007. [28] Hernandez J. A, "Sudy of he repair process of he power disribuion sysem in he San Juan de Paso Municipaliy", Universidad de los Andes, 2008. X. BOGRAPHES Carlos J. Zapaa (S'1993, AM' 1997, M'2004) obained his BScEE from he Universidad Tecno16gica de Pereira, Pereira, Colombia, in 1991 and his MScEE from he Universidad de Los Andes, Bogoa, Colombia, in 1996. From 1991 o 2001 he worked for Concol S. A, Bogoa, Colombia, where he paricipaed in 41 projecs of power sysem sudies, elecrical designs and sofware developmen. Since 2001, he has worked as professor a he Universidad Tecno16gica de Pereira. Currenly, he is working owards his PhD a he Universidad de Los Andes, Bogoa, Colombia. Silvana C. Silva obained his BScEE from he Universidad Tecno16gica de Pereira, Pereira, Colombia, in 2007. During years 2005 and 2006 she worked in he research projec "Mehodologies for reliabiliy assessmens in power disribuion companies". Heyder. Gonzalez obained his BScEE from he Universidad ndusrial de Sanander, Bucaramanga, Colombia, in 1999. For four years he worked in several public services governmenal offices in he Deparameno del Casanare, Colombia. Since 2003, he works for Empresa de Energia del Casanare S. A, Yopal, Colombia, as operaions manager. Currenly, he is a graduae suden a Universidad de los Andes, Bogoa, Colombia. Olga L. Burbano obained his BScEE from he Universidad Tecno16gica de Pereira, Pereira, Colombia, in 2007. During years 2005 and 2006 she worked in he research projec "Mehodologies for reliabiliy assessmens in power disribuion companies". She worked for UP Servicios, Pereira, Colombia from June 2006 unil July 2007. Since Augus 2007 she works for Cenrales Elecricas de Narino S. A, Paso, Colombia. Julio A. Hernandez obained his BScEE from he Universidad Nacional de Colombia, Bogoa, Colombia, in 2004. During years 2005 and 2006 he worked for Cenrales Elecricas de Narino S. A, Paso, Colombia, as engineer in projecs ofrural power disribuion. Currenly, he works for AREVA T&D Company, Bogoa, Colombia, as design engineer and he is a graduae suden a Universidad de los Andes, Bogoa, Colombia.