Nathalie Perrier Bruno Agard Pierre Baptiste Jean-Marc Frayret André Langevin Robert Pellerin Diane Riopel Martin Trépanier.

Transcription

1 A Survey of Models and Algorthms for Emergency Response Logstcs n Electrc Dstrbuton Systems - Part I: Relablty Plannng wth Fault Consderatons Nathale Perrer Bruno Agard Perre Baptste Jean-Marc Frayret André Langevn Robert Pellern Dane Ropel Martn Trépaner February 2010 CIRRELT Bureaux de Montréal : Bureaux de Québec : Unversté de Montréal Unversté Laval C.P. 6128, succ. Centre-vlle 2325, de la Terrasse, bureau 2642 Montréal (Québec) Québec (Québec) Canada H3C 3J7 Canada G1V 0A6 Téléphone : Téléphone : Télécope : Télécope :

2 A Survey of Models and Algorthms for Emergency Response Logstcs n Electrc Dstrbuton Systems - Part I: Relablty Plannng Nathale Perrer, Bruno Agard, Perre Baptste, Jean-Marc Frayret, André Langevn *, Robert Pellern, Dane Ropel, Martn Trépaner Interunversty Research Centre on Enterprse Networks, Logstcs and Transportaton (CIRRELT), and Department of Mathematcs and Industral Engneerng, École Polytechnque de Montréal, P.O. Box 6079, Staton Centre-vlle, Montréal, Canada H3C 3A7 Abstract. Emergency response operatons n electrc dstrbuton systems nvolve a host of decson-makng problems at the relablty, performance montorng and evaluaton, and contngency plannng levels. Those operatons nclude fault dagnoss, fault locaton, fault solaton, restoraton, and repar. As the frst of a two-part survey, ths paper revews optmzaton models and soluton methodologes for relablty plannng problems wth fault consderatons related to electrc dstrbuton operatons. Contngency plannng problems of emergency dstrbuton response are dscussed n the second part. The present paper surveys research on determnng a dstrbuton substaton sngle-fault capacty, reallocatng excess load, confgurng dstrbuton systems, parttonng a geographcal area nto servce terrtores or dstrcts, and locatng materal stores and depots. Keywords. Electrc power dstrbuton, emergency response, operatons research. Acknowledgements. Ths work was supported by the Natural Scences and Engneerng Research Councl of Canada (NSERC). Ths support s gratefully acknowledged. Results and vews expressed n ths publcaton are the sole responsblty of the authors and do not necessarly reflect those of CIRRELT. Les résultats et opnons contenus dans cette publcaton ne reflètent pas nécessarement la poston du CIRRELT et n'engagent pas sa responsablté. * Correspondng author: Andre.Langevn@crrelt.ca Dépôt légal Bblothèque et Archves natonales du Québec, Bblothèque et Archves Canada, 2010 Copyrght Perrer, Agard, Baptste, Frayret, Langevn, Pellern, Ropel, Trépaner and CIRRELT, 2010

3 Introducton Plannng the operatons of emergency dstrbuton response nvolves a host of decson problems that can be modeled and solved usng operatons research methodologes. The mportance of these problems s obvous from the mpact of fault stuatons on customers and electrc utltes. Fault stuatons may cause n extrems states where servce s nterrupted n dstrbuton systems, thus reducng the qualty of servce and causng fnancal losses for electrc utltes. These losses are dffcult to quantfy monetarly but can be sgnfcant n specfc stuatons. For example, the snowstorms of January 2008 n the central-eastern-southern parts of Chna that brought down electrcty lnes and poles n several provnces affected nearly two thrds of Chna s total land and ncurred an estmated $10-bllon drect economc loss (Zhpng, 2008). As hghlghted by Ćurčć et al. (1996), electrc power generaton and transmsson system plannng has long been an deal feld for the development and applcatons of operatons research due to the complexty and challenges of the problems assocated wth those systems, the hgh nvestment, operatng and outage costs of almost any generaton or transmsson plant, as well as the huge number of customers that can be affected by possble outages n these systems. However, the lterature related to emergency dstrbuton response has experenced a slow growth. Ths stuaton s somewhat surprsng gven that dstrbuton systems account for up to 90% of all customer relablty problems largely due to the radal nature of most dstrbuton systems, the large number of elements nvolved, the sparsty of protecton devces and the proxmty of the dstrbuton system to end-use customers. In fact, the slow progress of operatons research n emergency dstrbuton response hghlghts the consderable dffculty of these problems. Problems faced by utlty dstrbuton planners are complex and ste specfc because of the dfference n characterstcs such as topologcal features of the network, operatonal capabltes and appled operatonal devces. Also, utlty dstrbuton planners have a mult-crtera envronment n whch they have to address problems n terms of conflctng crtera. For example, the fundamental tradeoff n restorng the electrc power n a localty s between mnmzng the length of the restoraton perod and maxmzng the number of customers wth a restored supply. A prevous survey by Khator and Leung (1997) suggests that most early contrbutons n power dstrbuton plannng were dealng wth smplfed models ether falng to address the ssue of equpment falure or accountng for t by merely factorng n a safety equpment capacty. In the last two decades however, a growng body of operatons research applcatons to emergency dstrbuton response has appeared n the lterature. The large number of components nvolved n dstrbuton systems, the complexty of dstrbuton networks, and the ever ncreasng capablty of utltes for CIRRELT

4 operatng these networks all motvate the use of optmzaton technques at varous levels n the electrc dstrbuton utlty. Emergency response logstcs n electrc dstrbuton systems presents a varety of decson-makng problems that can be grouped nto a number of categores accordng to the plannng horzon whch s concerned (Zografos et al., 1998). The relablty plannng level nvolves strategc plannng decsons related to the desgn of more relable and robust dstrbuton networks n whch fault cases are taken nto account. The plannng horzon for relablty ssues s usually around fve years (Zografos et al., 1998). Decsons related to dstrbuton substaton capacty plannng, dstrbuton system confguraton and the establshment of servce centers and servce terrtores or dstrcts may be vewed as strategc. The performance montorng and evaluaton plannng level s related to short term plannng decsons, and generally nvolves the perodcal adustment of emergency response logstcs resources and the performance montorng of the emergency actons based on statstcal nformaton on the demand, supply and performance of the emergency response mechansm. Fnally, decsons related to real-tme management of the emergency response logstcs resources belong to the contngency plannng level. For example, the assgnment of servce calls to emergency response unts and the routng of emergency response unts could be termed real-tme. Ths paper s the frst of a two-part survey of optmzaton models and soluton algorthms for relablty and contngency plannng problems related to emergency response n electrc dstrbuton systems. The am of ths paper s to provde a comprehensve survey of optmzaton models and soluton methodologes for relablty plannng problems related to emergency dstrbuton operatons. These problems nclude determnng a dstrbuton substaton sngle-fault capacty, reallocatng excess load, confgurng dstrbuton systems, parttonng a geographcal area nto servce terrtores or dstrcts, and locatng materal stores and depots. The second part addresses fault dagnoss, fault locaton, fault solaton, emergency servce restoraton, repar vehcle routng, repar crew schedulng and crew assgnment models for emergency response n electrc dstrbuton systems (Perrer et al., 2010). Very lttle work has been accomplshed concernng the desgn of relable dstrbuton networks n the context of dstrbuton emergency response, whle contngency plannng problems have receved a lot of attenton n the lterature. Ths paper s organzed as follows. Secton 1 descrbes the operatng states of electrc dstrbuton systems and the relablty plannng problems wth fault consderatons related to electrc dstrbuton operatons. Models for the determnaton of a dstrbuton substaton sngle-fault capacty and the reallocaton of excess load are descrbed n Secton 2. Models that address the confguraton of relable dstrbuton networks wth fault consderatons are revewed n Secton 3. Secton 4 focuses on CIRRELT

5 parttonng a geographcal area nto servce terrtores or dstrcts for emergency dstrbuton operatons. Models dealng wth the locaton of resource and materal depots for emergency dstrbuton response are presented n Secton 5. Conclusons and future research paths n dstrbuton emergency response plannng are presented n the last secton. 1. Electrc dstrbuton systems Electrcty s produced and delvered to consumers through generaton, transmsson and dstrbuton systems. Generaton systems consst of generatng plants that produce electrcal energy from another form of energy such as fossl fuels, nuclear fuels or hydropower, and generaton substatons that connect generaton plants to transmsson lnes. Transmsson systems transport electrcty over long dstances from generaton substatons to substatons that serve subtransmsson or dstrbuton systems. As power s moved and dstrbuted from a few large generaton plants to a wdely dspersed consumer base, t s gradually moved down to lower voltage levels, where t s splt nto ever smaller parts, wth each part routed onto a dfferent path on lower capacty equpment. Usually, ths splttng of the power flow beng done smultaneously wth a reducton n voltage happens from three to fve tmes durng the course of power flow from generaton to consumer. Dstrbuton systems delver power from bulk power systems to retal customers. To do ths, dstrbuton substatons receve power from the transmsson grd and step down voltages wth power transformers. These transformers supply prmary dstrbuton systems made up of many dstrbuton feeders, typcally overhead dstrbuton lnes mounted on poles or underground bured or ducted cable sets that delver power from dstrbuton substatons to dstrbuton transformers. Passng through these transformers, power s lowered n voltage once agan, to the fnal utlzaton voltage and routed to the secondary system wthn very close proxmty to the consumer or drectly to the consumers. Usually, each transformer serves a small radal network of secondary and servce lnes of utlzaton voltage. These lead drectly to the meters of consumers n the mmedate vcnty. A dstrbuton substaton generally occupes an area of one acre or more on whch the necessary substaton equpment s located. Snce feeder routes must pass near every customer, each substaton uses multple feeders to cover an assgned servce terrtory. Normally, between two and 12 feeders emanate from any one substaton (Wlls et al., 2001). Feeders of a substaton that are not connected to other feeders are called ndependent feeders. These feeders supply power to solated load demands. Feeders that are lnked to the feeders of adacent substatons are called connected feeders. In emergency stuatons, connected feeders allow a substaton s load to be transferred to adacent substatons. A smplfed drawng of an overall electrc power system and ts generaton, transmsson CIRRELT

6 and dstrbuton subsystems s shown n Fgure 1. Here, the dstrbuton substaton supples four ndependent feeders to cover ts servce terrtory. Generaton plant Transmsson Subtransmsson Generaton substaton system Transmsson substaton system Dstrbuton transformer Dstrbuton substaton Prmary dstrbuton system Secondary dstrbuton system Substaton servce terrtory Fgure 1. A power system and ts subsystems Dstrbuton systems can be desgned as radal, loop, or network systems, dependng on how the dstrbuton feeders are arranged and nterconnected about a substaton. The radal system s characterzed by havng only one path between each consumer and a substaton. An alternatve to radal feeder desgn s a loop system consstng of a dstrbuton desgn wth two paths between the substatons and every consumer. Such systems are often called European because ths pattern s the preferred desgn of many European utltes (Wlls et al., 2001). Network systems have multple electrcal paths from the substaton to the consumer. Radal desgn s the most wdely used method of dstrbutng electrc power, accountng for over 99% of all dstrbuton constructon n North Amerca (Wlls et al., 2001). In fact, radal systems are much less costly than loop or network systems and are much smpler n plannng, desgn and operaton. However, radal systems are less relable than the other two alternatves because the electrcal power flows exclusvely away from the substaton and out to the consumer along a sngle path. Thus, f any element along ths path fals, a complete loss of power to the consumer results. Dstrbuton systems are generally descrbed through a graph, whose lnks represent feeders and whose nodes correspond to the substaton locatons, transformer nstallatons and load demand ponts. Most radal feeder systems are bult as networks, but operate radally by openng swtches at certan ponts throughout the physcal network, so that the resultng confguraton s electrcally radal and thus defnes a spannng tree. The followng secton contans a bref descrpton of dstrbuton operatng states of electrc dstrbuton systems. Relablty plannng problems related to emergency response n electrc dstrbuton systems, that have been addressed wth operatons research methodologes, are then CIRRELT

7 dscussed. A more detaled revew on dstrbuton systems, ther functon, components, characterstcs, and operatons s presented n the books by Brown (2002) and Wlls et al. (2001) Dstrbuton operatng states Dstrbuton systems must be contnually montored, adusted, expanded, mantaned and repared. These actvtes are collectvely referred to as dstrbuton operatons. As hghlghted by Gutérrez et al. (1987), dstrbuton system operatons can be grouped nto fve operatng states dependng on whether equpment outages and customer nterruptons occur or not: normal, alert, emergency, n extrems, and restoratve states. The system s referred to as the normal operatng state when all customers are adequately suppled wthn acceptable voltage tolerances, all components are operatng properly, the system s confgured n ts usual manner, and equpment loadng levels are wthn desgn lmts. The system s n the alert operatng state when the system s securty level s reduced, but the system s stll operated wthn allowable lmts (Lndenmeyer, 2000). Ćurčć et al. (1996) recognze the alert state as a pre-outage state where the operatng lmts are n eopardy. For example, a pre-outage state occurs when a pece of equpment tends to become overloaded and protecton devces could take t out of servce. Such a stuaton ntates the preventve actons requred to return the system to the normal state. A wde range of causes ncludng equpment falure, anmals, trees, severe weather and human error dsrupt normal operatng condtons and can lead to outages and nterruptons. In the emergency operatng state, also called the outage state (Ćurčć et al., 1996), the operatng lmts are volated due to a short crcut, called fault. For example, a feeder lne down, transformer out of servce, or a breaker that opens when t shouldn t. A fault occurrng on an overhead feeder component s called a feeder fault and a short crcut occurrng on a substaton component, a substaton fault. The pece of equpment out of servce cannot be returned to operaton before the cause of ts outage s cleared. If ths can be done quckly, the system can be taken back to the normal state. If not, the system frst enters n an n extrems operatng state where the operatng lmts are volated and servce s nterrupted for one or more customers. Then, a restoraton brngs the system nto the restoratve operatng state provdng the best possble servce wth the remanng peces of equpment. When the system s n the restoratve state, part of the system equpment s dsconnected n order to solate the faulted secton, causng customer servce nterruptons. Clearng the cause of outage enables the system to be returned to the normal state. Fgure 2 llustrates the possble transtons among the prevously defned operatng states. Ths popular state approach s also explaned by Morelato and CIRRELT

8 Montcell (1989) who defne the pre-outage, outage and n extrems states as a sngle operatng state, called emergency state. Servce nterruptons and operatng lmts volatons Servce nterruptons Normal Pre-outage Outage In extrems Restoratve Operatng lmts volatons Fgure 2. Operatng states of a dstrbuton system 1.2. Relablty plannng problems wth fault consderatons Ths secton descrbes relablty plannng problems of emergency dstrbuton response that have been addressed by operatons research technques. Relablty plannng problems nclude determnng dstrbuton substaton sngle-fault capacty, reallocatng excess load, confgurng dstrbuton systems, parttonng a geographcal area nto servce terrtores or dstrcts, and locatng materal stores and depots. Relablty plannng wth fault consderatons s usually addressed by determnng the maxmum load a substaton can handle and by reallocatng excess load demand to substatons so that certan loads can be restored after a fault occurs. Relablty plannng wth fault consderatons can also be addressed by locatng, routng, and szng potental new substatons, transformers, and feeders, by parttonng a geographcal area nto emergency repar dstrcts, and by locatng resource depots. The plannng of electrc dstrbuton relablty wth fault consderatons nvolves decsons related to the confguraton of more relable and robust dstrbuton networks so as to mantan power supply n fault stuatons (see Secton 3). At the most basc level, relablty strateges dealng wth faults address the determnaton of dstrbuton substaton sngle-fault capacty and the reallocaton of excess load demand to substatons. In many large electrc utltes, a substaton s capacty s determned based on the maxmum load t can handle durng emergences. One emergency polcy wdely used among large electrc utltes, called the sngle-fault polcy, allows a sngle transformer fault among the substatons of a servce area at any gven tme. More nvolved relablty strateges, n whch fault cases are consdered, concern the reconfguraton of the system by addton of new feeders, substaton transformers, or substatons. As hghlghted by Brown (2002), snce the maorty of customer nterruptons s due to sngle faults n radal systems, relablty plannng does not typcally consder the smultaneous falure of multple components. CIRRELT

9 Relablty plans wth fault consderatons can also help to partton a geographcal area nto emergency repar dstrcts. Gven the large dspersed geographc extent of most emergency dstrbuton operatons, a utlty generally parttons ts servce area nto subareas, called dstrcts. All dstrcts are treated smultaneously by separate crews to facltate the organzaton of the emergency repar operatons and thus reduce the duraton of electrc power nterruptons. The dstrct desgn problem conssts of parttonng a large servce area nto non-overlappng small dstrcts accordng to several crtera such as contguty, sze and workload. The contguty crteron requres that dstrcts do not nclude dstnct parts separated by other dstrcts. Also, to balance the level of servce offered to the customers across dstrcts, they are often approxmately the same sze and are balanced n workload,.e. dstrcts are assgned equvalent emergency repar resources. Fnally, relablty plans wth fault consderatons can help to locate resource depots. A resource depot s a place where resources for restorng the electrc power n a localty are stored. These resources nclude repar crews, vehcles, poles and transformers. The depots may be dfferent,.e. the types of the resources and the amount of each type of resources n each depot may be dfferent. The resource depot locaton problem conssts of smultaneously selectng the proper stes to allocate dfferent depots wth resource capactes, and determnng the amounts of the resources shpped from the depots to varous geographcally scattered locatons or customers n order to satsfy the demands of the customers, whle mnmzng the total transportaton cost for the power restoraton. Depot locaton problems n the context of dstrbuton emergency response are generally formulated as network locaton problems, n whch facltes can be located only on the nodes or lnks of the network. 2. Dstrbuton substaton sngle-fault capacty and load reallocaton models Under the sngle-fault polcy, the load of the servce area must be satsfed f falure occurs to ether the largest transformer at the substaton beng evaluated or one of ts adacent substaton s largest transformer (not necessarly the largest transformer of all the adacent substatons). Leung et al. (1995) proposed a lnear programmng formulaton for the problem of determnng a substaton s sngle-fault capacty. Let S be the set of the substatons wthn the servce area (ncludng substaton k, the substaton beng evaluated). The sum of the capactes of a substaton s transformers s referred to as the normal substaton capacty,.e., the load a substaton can handle under normal condtons. When a transformer fals at a substaton, load of the faled transformer can be temporarly transferred to the remanng n-servce transformers operatng at an above 100% emergency rate for a short perod of tme untl the transformer s repared or a moble transformer s n place. The sum of the capactes of the substaton s n-servce transformers operatng under emergency rates s called the emergency CIRRELT

10 substaton capacty. For every substaton S, let UD be a nonnegatve real varable representng the unsatsfed demand of substaton, and defne NC, EC, LD and FC as the normal capacty of substaton, the emergency capacty of substaton, the load demand at substaton and the feeder transfer capacty of substaton, respectvely. The transfer capacty of a substaton, gven ts load demand satsfed, s the excess feeder capacty of the substaton. The transfer capacty of a substaton lmts the amount of load that can be permanently reallocated to the substaton as well as the amount of power that can be temporarly transferred from ths substaton to ts adacent substatons. Let S S be the set of the adacent substatons to substaton, S. For every substaton S and for every adacent substaton S, let P be a nonnegatve real varable representng the amount of power transferred from substaton to substaton va connected feeders when substaton s under emergency, and let AFC represent the aggregate capacty of the feeders connectng substaton to substaton. Defne also M as the total power transfer lmt. We present here the equvalent nonlnear verson of the Leung et al. (1995) formulaton for the substaton sngle-fault capacty problem (we elmnate the new varable and the addtonal lnear constrants ntroduced by Leung et al. (1995) n the equvalent lnear program). Maxmze mn EC k + α k Pk ; NC k Pk (1 + δ ) UD (2.1) Sk Sk Sk subect to S α P + EC LD UD ( S k ) (2.2) mn EC k + α k Pk ; NC k Pk LDk UDk (2.3) Sk Sk P FC LD ( S, S ) (2.4) { AFC AFC } P mn, ( S, S ) (2.5) P NC LD ( S \ {k}, S k ) (2.6) S α P M ( S) (2.7) P 0 ( S, S ) (2.8) UD 0 ( S) (2.9) CIRRELT

11 The obectve functon (2.1) maxmzes the load that substaton k can handle under the sngle-fault polcy. When the largest transformer of substaton k fals, the sngle-fault capacty of substaton k corresponds to the frst term whch s the sum of ts emergency load capacty and the power t receves va feeder from adacent substatons. The emergency load capacty of the substaton s the load the substaton can handle wth ts remanng n-servce transformers operatng above 100% of ther rated capacty. The parameter α k s a dscountng factor to take nto account voltage drop n feeders. Voltage drops n dstrbuton systems are permtted to reduce system demand. When the largest transformer of an adacent substaton to substaton k fals, the sngle-fault capacty of substaton k corresponds to the second term whch s the remanng capacty of the substaton, after supplyng power to the adacent substaton. The mnmum of the two capactes s the maxmum load the substaton can handle under the sngle-fault polcy. The parameter δ s a very small value to gve a penalty for havng unsatsfed demand. Constrants (2.2) and (2.3) guarantee that the load demand of the servce area s satsfed f a transformer fault occurs. Constrants (2.2) state that, for each adacent substaton to k, the total power receved from ts adacent substatons plus ts emergency load capacty must be greater than ts load demand mnus ts unsatsfed demand. Constrant (2.3) state that that the sngle-fault capacty of substaton k must be greater than ts load demand mnus ts unsatsfed demand. Constrants (2.4) (2.6) assure that the power transfer lmts mposed by dstrbuton capactes, substaton normal capactes, and forecasted load demands are respected. The dstrbuton capacty of a substaton s the sum of the capactes of the feeders suppled by the substaton. Constrants (2.4) requre that the power transferred from one substaton to another adacent substaton durng emergency depends on the transfer capacty of the supply substaton. Constrants (2.5) assure that the power suppled from a substaton to another adacent substaton s less than the aggregate capacty of the feeders connectng the two substatons. Constrants (2.4) and (2.5) assume that the load for a substaton can be redstrbuted among ts transformers durng emergency. However, f load redstrbuton wthn a substaton s not possble, then these constrants must be replaced by the followng constrants. P FL mn AFC FL, ( S, S ) (2.10) α For every substaton S and for every adacent substaton S, let FL be the load on the feeders connectng to. Constrants (2.10) assure that the power transfer lmts mposed by ether the connectng feeder s transfer capacty or the load on ts neghborng staton s feeders are respected. Constrants (2.6) ensure that the power transferred from an adacent substaton of substaton k to one CIRRELT

12 of ts adacent substatons (other than k) durng emergency condton s less than ts normal capacty mnus ts load. The lmt on the total power transferred to any substaton under emergency from all ts adacent substatons s respected va constrant set (2.7). Model (2.1)-(2.9) s repeated for each substaton n the servce area. In the same paper, Leung et al. (1995) addressed the reallocaton of excess load demand to substatons so that certan loads can be restored after a fault occurs. When a transformer fals at a substaton, the adacent substatons can temporarly meet part of the faled substaton s demand load by transferrng power to t va connected feeders. However, when a substaton s forecasted load demand exceeds the maxmum load of the substaton, utlty planners can permanently reallocate the excess load to adacent substatons wthout necesstatng new captal nvestments such as buldng feeders, purchasng transformers, or constructng new substatons. We now descrbed a lnear programmng formulaton proposed by Leung et al. (1995) for the reallocaton of excess load wthn a network of substatons. For every substaton S and for every adacent substaton S, let R be a nonnegatve real varable representng the amount of load reallocated from substaton to adacent substaton, and let V represent the maxmum load that can be reallocated wth respect to the voltage ratngs of the feeders from substaton toward substaton. Mnmze S S R (2.11) subect to S ( R R ) EC + α P LD + ( S) (2.12) S ( R R ) NC P LD + ( S, S ) (2.13) S P FC LD + R k Rk ( S, S ) (2.14) k S k S { AFC AFC } P mn, ( S, S ) (2.15) S α P M ( S) (2.16) { V V } R mn, ( S, S ) (2.17) P, R 0 ( S, S ) (2.18) CIRRELT

13 The obectve functon (2.11) mnmzes the total load reallocated to adacent substatons. Constrants (2.12) and (2.13) ensure that load demand requrements are met after load reallocaton under the sngle-fault emergency stuaton when a substaton or an adacent substaton s under emergency, respectvely. Constrants (2.14)-(2.16) ensure that feeder capacty lmts are respected. Constrants (2.14) requre that the total load allocated to every substaton does not exceed ts total feeder capacty. Constrants (2.15) and (2.16) are dentcal to ther respectve counterparts (2.5) and (2.7) of the model (2.1)-(2.9). Voltage ratng lmts of the feeders connectng all pars of adacent substatons are respected va constrants (2.17). Leung et al. used the MPS mathematcal programmng package to solve the two models (2.1)-(2.9) and (2.11)-(2.18) wth a set of data from the substaton network of the Fort Myers Dstrct of Florda contanng 12 substatons connected va feeders. The authors concluded that there exsts consderable synergstc behavor n an electrc dstrbuton system. For example, addng capacty to the substaton under emergency mght not be the most economcal relablty strategy, whle addng capacty to a substaton can provde relef to ts multple adacent substatons. 3. System confguraton models wth fault consderatons More nvolved relablty strateges, n whch fault cases are consdered, concern the reconfguraton of the system by addton of new feeders, substaton transformers, or substatons Feeder confguraton models Feeder confguraton s a mult-year dstrbuton plannng process whch prescrbes the least cost feeder confguraton for a network of substatons over the plannng horzon. The followng lnear MIP model, proposed by Sarada et al. (1995), provdes an expanson plan whch determnes the perod-toperod nstallaton tmes and locatons of new feeders. The model also concurrently determnes load reallocatons and power transfer decsons and ensures that all loads n the network be met under the sngle-fault polcy. The obectve s to mnmze the cost of new feeders, whle satsfyng load demand requrements, voltage ratngs, and equpment loadng lmts. For every substaton S, defne VR, NF and NTR as the ncrease n feeder capacty of substaton obtaned by addng a feeder, the number of exstng feeders at substaton, and the number of transformers avalable at substaton, respectvely. For every substaton S and for every adacent substaton k S, let d k be the dstance from substaton k to the uncton of feeders from substaton. Let T be the set of tme perods, expressed n years. For every substaton S and for every tme perod t T, defne LD as the forecast load demand of substaton for tme perod t and T m as the percentage change, due to load growth, n the load at substaton between the tme perod m 1 to m, m < t. Let D be the set of new load demand CIRRELT

14 locatons. Fgure 3, taken from Sarada et al. (1995), llustrates connected exstng feeders (sold lnes) and potental feeders (dotted lnes) between substatons and k, and the new load demand locaton at the uncton of potental feeders. k Fgure 3. Exstng feeders, potental feeders and new load locaton For every new load demand locaton D and for every tme perod t T, defne L t as the forecast load demand at new load locaton for tme perod t. For every new load demand locaton D, for every substaton S and for every tme perod t T, let y t be a bnary varable equal to 1 f and only f a new ndependent or connected feeder s to be nstalled from substaton toward new load locaton n tme perod t, and let NLT t be a nonnegatve real varable representng the amount of load at new locaton assgned to substaton n tme perod t. Thus, the load at a new load locaton may be splt up between two or more substatons. For every substaton S, for every adacent substaton k S and for every tme perod t T, let x kt be a bnary varable equal to 1 f and only f a new connected feeder s to be nstalled from substaton k toward substaton n tme perod t, let P kt be a nonnegatve real varable representng the amount of power transferred from substaton k to substaton va connected feeders when substaton s under emergency n tme perod t, let R kt be a nonnegatve real varable representng the amount of load reallocated from substaton k to adacent substaton n tme perod t and let CF kt be the cost of addng a feeder of unt length from substaton k toward substaton n perod t. For every new load demand locaton D, for every substaton S, for every adacent substaton k S and for every tme perod t T, let NFC kt be a bnary varable equal to 1 f and only an ncrease n connectng feeder capacty occurs n tme perod t when both substatons and k are connected va new load locaton. Fnally, defne numf as the maxmum number of feeders per transformer n any tme perod (both exstng and new) at a substaton and PR as the penalty cost for reallocatng a unt of load. All other operatonal parameters are defned as n Secton 2. The formulaton s gven next. CIRRELT

15 Mnmze ( ktd k x kt ) + ( CFtd yt ) + ( PR R kt ) S k S t T CF (3.1) D S t T S k S t T subect to EC + k S P kt LD + TR + D D NLT t ( S, t T) (3.2) NC P LD + TR + NLT ( S, k S, t T) (3.3) k kt t { V V } R mn, ( S, k S, t T) (3.4) k k NLT t VR t m= 1 y NLT t = S m L t ( D, S, t T) (3.5) ( D, t T) (3.6) Pk EFC ( S, k S, t T) (3.7) P k AFC k + VR t k m= 1 x km + VR t k D m= 1 NFC km ( S, k S, t T) (3.8) P k AFC k + VR t m= 1 x km + VR t D m= 1 NFC km ( S, k S, t T) (3.9) 2 NFC y + y 1+ NFC ( D, S, k S, t T) (3.10) kt kt t t kt km kt 2 NFC y + y 1+ NFC ( D, S, k S, t, m T, m < t) (3.11) kt m kt kt 2 NFC y + y 1+ NFC ( D, S, k S, t, m T, m < t) (3.12) k S kt P M ( S, t T) (3.13) kt NF t + xkm + y m= 1 k S D m numf NTR { 0,1} ( S, t T) (3.14) x, y, NFC ( D, S, k S, t T) (3.15) kt t kt R, P, NLT 0 ( D, S, k S, t T) (3.16) kt kt t where, for every substaton S and for every tme perod t T, CIRRELT

16 TR = t R km m= 1 k S k S R km + t T R km km 1 m= 2 k S k S T m R km 1 and EFC t t = FC VR xkm VR ym LD NLT m= 1 k S m= 1 D D t + TR are two ntermedary nonnegatve real varables representng the total load reallocated to substaton n tme perod t and the excess feeder capacty of substaton n tme perod t, respectvely. The total load reallocated to a substaton n a gven tme perod s the dfference between the sum of the loads t receves from ts adacent substatons and the loads t reallocates to them up to that tme perod. The excess feeder capacty of a substaton s the dfference between ts total feeder capacty and ts net load. The obectve functon (3.1) mnmzes the total costs of new feeders. The frst term corresponds to the total nstallaton cost of new connected feeders. The second term s the total nstallaton cost of feeders along new routes requred to meet loads at new locatons. The fnal term s the penalty cost assgned to load reallocatons to prevent unnecessary redstrbuton of loads. Constrant sets (3.2)-(3.4) are smlar to ther respectve counterparts (2.12), (2.13) and (2.17) of the model (2.11)-(2.18). Constrant set (3.5) mposes a lmt on the amount of load, at each new locaton, that can be assgned to an adacent substaton n a gven tme perod. Ths lmt s equal to the capacty of the new feeder nstalled n ths tme perod or earler from the adacent substaton toward the new load locaton. Constrant set (3.6) ensures that the load demand at each new locaton s satsfed for each tme perod. Constrant sets (3.7) and (3.8)-(3.9), smlar to ther respectve counterparts (2.14) and (2.15) of the model (2.11)-(2.18), mpose upper bounds on the power transferred by each substaton to an adacent substaton under emergency for each tme perod. Constrant sets (3.10)-(3.12) lnk new feeder locatons and feeder capacty ncrease. They ensure an ncrease n nterconnectng feeder capacty between each par of adacent substatons only when new feeders are nstalled from both substatons va a new load locaton. Constrant set (3.13) s smlar to ts counterpart (2.16) of the model (2.11)- (2.18). Fnally, a lmt on the total number of feeders, both exstng and new, at every substaton for each tme perod s mposed by constrant set (3.14). Agan, the model was appled to the Fort Myers Dstrct of Florda contanng 12 substatons, for a plannng horzon of two tme perods, and solved by the branch-and-bound algorthm of the MPS package. As mentoned by Sarada et al. (1995), the problem sze may be reduced by explotng the spatal nature of the problem. Theoretcally, each substaton can be connected to all other substatons n the dstrbuton network va new feeders. However, gven the hgh cost of nstallng new feeders over long dstances, t s practcal to consder new feeders only to the substatons n the vcnty. Ths CIRRELT

17 reduces the sze of the problem consderably. The authors also emphaszes that the model can be extended to nclude decsons related to expanson of the substaton capactes such as upgradng exstng transformers or addng new transformers Substaton transformer confguraton models The sngle-fault polcy requres that a substaton capacty be planned at the transformer level. Hence, when the addton of feeders does not ensure that load demand requrements are met, transformer confguraton must be consdered. For a network of substatons, Leung et al. (1996) proposed a 0-1 lnear programmng model to dentfy the optmal transformer confguraton. The soluton of the model prescrbes the optmal transformer relocaton as well as purchase optons. The obectve s to mnmze transformer capacty requrement and transformer procurement cost whch ncludes purchase cost, transport cost, dsassembly cost, and nstallaton cost, whle satsfyng voltage drop lmts and equpment overloadng lmts under the sngle-fault polcy. Let I 1 and I 2 be two sets of transformer destnatons. The set I 1 denotes substatons where new transformers may be added or substatons for whch transformers may be upgraded. The set I 2 denotes transformer storage locatons. Snce transformers n storage can be used n other dstrcts, the decson of movng a transformer to a storage locaton mplctly nvolves relocatng the transformer to another servce area for subsequent use. For every destnaton substaton I 1, defne NC, EC, MT and M as the normal capacty of substaton, the emergency load capacty of substaton, the maxmum number of transformers that substaton can take on, consderng physcal or other constrants, and the maxmum amount of power that can be receved by substaton durng emergency condtons, respectvely. The lmt on the total power receved by a substaton depends on the polcy of the electrc utlty as well as the practcalty of the substaton. For every destnaton substaton I 1, let N be the set of substatons adacent to substaton. For every destnaton substaton I 1 and for every adacent substaton n N, let P n be a nonnegatve real varable representng the amount of power transferred from substaton n to adacent substaton va connected feeders when substaton s under emergency, and defne FC n, LD n and AFC n as the feeder transfer capacty of substaton n, the load demand at substaton n, and the aggregate capacty of the feeders connectng substaton n to substaton, respectvely. Let J be the set of transformer sources (vendors, transformer storage locatons or substatons where transformers may be removed or downgraded). For every source of transformers J, let K be the set of transformers n source. For every transformer source J and for every transformer k K, defne c k as the capacty of transformer k of source. For every transformer destnaton I 1 I 2, for every transformer source J and for every transformer k K, let x k be a CIRRELT

18 bnary varable equal to 1 f and only f transformer k of source s allocated to destnaton, and defne pc k as the procurement cost of movng transformer k from source to destnaton. Dependng on the correspondng source and destnaton of a transformer, the procurement cost may nclude purchase cost, transportaton cost, dsassembly cost, nstallaton cost, savngs of subsequently utlzng n another dstrct a transformer moved to a storage locaton, cost of usng a storage unt, etc. The formulaton for the confguraton of substaton transformers under the sngle-fault polcy can be stated as follows: Mnmze c k xk + δ pc x (3.17) I1 J k K I1 I2 J k K k k subect to NC = c x ( I 1 ) (3.18) J k K k k n P n FC LD ( I 1, n N ) (3.19) { AFC AFC } n n n n P mn, ( I 1, n N ) (3.20) EC + α P LD ( I 1 ) (3.21) n N n n EC NC P LD ( I 1, n N ) (3.22) n ( NC c x ) k k β ( I 1, J, k K ) (3.23) n N J k K α P M ( I 1 ) (3.24) n I 1 n x 1 ( J, k K ) (3.25) k x k MT { 0,1} ( I 1 ) (3.26) x ( I 1 I 2, J, k K ) (3.27) k P 0 ( I 1, n N ) (3.28) n where δ s the opportunty cost per unt capacty. The obectve functon (3.17) mnmzes the sum of opportunty cost and procurement cost. Constrants (3.18) defne the normal capacty of every substaton. Constrants (3.19)-(3.22) are very smlar to ther respectve counterparts (2.4), (2.5), (2.2) and (2.6) of the model (2.1)-(2.9). Note that for stuatons where a substaton s load can not be CIRRELT

19 redstrbuted among ts transformers durng emergency, constrants (3.19) and (3.20) must be replaced by the followng constrants. P n FLn mn AFC n FLn, ( I 1, n N ) (3.29) α n Constrants (3.21) and (3.22) assure that load demand requrements are satsfed when a fault occurs to ether the largest transformer of each destnaton substaton or the largest transformer of each substaton adacent to a destnaton substaton, respectvely. The parameter α n s a dscountng factor to take nto account voltage drop n feeders connectng substaton n to substaton. Constrants (3.23) state that the emergency capacty of each destnaton substaton must be computed from the worst scenaro,.e., fault of the substaton s largest transformer. The parameter β s the emergency rate of the remanng substaton s n-servce transformers operatng under emergency condtons. The maxmum amount of power that each destnaton substaton can receve durng emergency condtons s respected va constrant set (3.24). Constrants (3.25) requre each transformer to be assgned to at most one destnaton. Fnally, constrants (3.26) mpose a lmt on the number of transformers wthn each destnaton substaton. Computatonal tests usng MPS mathematcal programmng package were performed on data from the substaton network of the Fort Myers Dstrct of Florda contanng 12 substatons connected va feeders and between one and three transformers for each substaton. The model was also used to analyze a varety of scenaros for extensons to the basc model, ncludng maxmzaton of sngle-fault capacty and allocaton of transformers over a mult-perod horzon Compound feeder, transformer and substaton confguraton models As hghlghted by Khator and Leung (1995), the nstallaton of new feeders s closely lnked to the addton or upgradng of transformers. However, these nterdependent problems are most often solved separately. Typcally, when a substaton s forecast load demand exceeds ts sngle-fault capacty and reallocaton of load s not possble due to nsuffcent dstrbuton capacty, the cheapest alternatve s to frst nstall new feeders. When the substaton s load can not stll be met, ts capacty may then be ncreased by ether replacng the exstng unts wth transformers of hgher capacty or addng transformers to the substaton. Should that fal to overcome the capacty shortage, the last and most expansve alternatve s to buld a new substaton. Obvously, ths sequental approach may lead to suboptmal decsons. CIRRELT

20 Nara et al. (1994) proposed a lnear mxed nteger programmng formulaton for the combned feeder, transformer and substaton confguraton problem n whch faults are taken nto consderaton. The formulaton, whch s based on a prevous mult-perod expanson plannng model developed by Nara et al. (1991), ncorporates the radalty constrant, load demand requrements, lne voltage relatons, voltage drop lmts and loadng lmts. The obectve s to mnmze nstallaton costs. Fgure 4, adapted from Nara et al. (1994), provdes an example of a dstrbuton network descrbed through a graph wth one source node (transmsson substaton), three substaton nodes, sx transformer nodes and 11 load demand nodes. The lnks between nodes represent the electrcal connecton of nodes. For example, a lnk between two load ponts represents an exstng or potental feeder. New nstallaton faclty canddates nclude substatons, transformers and feeders. S 3 S 2 S 1 Fgure 4. Example of a dstrbuton network. potental feeder exstng feeder Let I and J be the sets of nodes and lnks, respectvely. For each lnk J, let w be a bnary varable equal to 1 f and only f lnk s nstalled, and let also c be the nstallaton cost of lnk (c = 0 for exstng lnks). It should be notced that the cost of a lnk between a source node and a substaton node corresponds to the cost of buldng the substaton. Also, the cost of nstallng a feeder between a substaton node and a transformer node corresponds to the cost of nstallng the transformer. Fnally, the cost of a lnk between a transformer node and a load pont corresponds to the fxed cost of nstallng a feeder secton between the two nodes, whereas the cost of a lnk between two load ponts corresponds to the varable feeder cost. For each lnk J, defne LC and R as the loadng capacty CIRRELT

21 and mpedance of lnk. Impedance s one mportant feeder characterstc. Impedances are seres resstances and reactances that determne ohmc losses and voltage drops. Resstance s nfluenced by feeder materal, conductor temperature and current waveform frequency. Reactance s prmarly determned by constructon geometry, wth compact desgns havng a smaller reactance than desgns wth large phase conductor separaton. Let T be the set of predetermned fault cases. For each lnk J and for each fault t T, let y be a bnary varable equal to 1 f and only f lnk s used n fault case t, and let also x + and x be the forward and nverse drecton power flows n lnk n fault case t, respectvely, as shown n Fgure 5. For each node I and for each fault case t T, let v t be a nonnegatve real varable representng the voltage at node n fault case t. For each load demand node I and for each fault case t T, defne d t as the load demand at node n fault case t (d t = 0 f load pont s an element of the faulted secton n fault case t). Let I D I be the set of load demand nodes. For each load demand node I D, let J be the set of lnks ncdent to load pont. For each load demand node I D, for each lnk J and for each fault case t T, defne the bnary constant a t equal to 1 f and only f load pont can be suppled va ncdent lnk n fault case t (a t = 0 f lnk s an element of the faulted secton n fault case t). Fnally, defne n as the number of nodes, ncludng the source node and v as the allowable voltage drop. v t x + x k v kt Fgure 5. Feeder between two nodes. The formulaton s gven next. subect to J Mnmze J c (3.30) w y w ( J, t T) (3.31) y = n 1 and the graph s connected (t T) (3.32) x x + My My ( J, t T) (3.33) ( J, t T) (3.34) CIRRELT

22 J a t ( x x ) = d ( I D, t T) (3.35) ( v v ) R ( x x ) + s (1 y ) s (1 y ) = 0 ( J, t T) (3.36) t kt t x x + LC LC ( J, t T) (3.37) ( J, t T) (3.38) v t v ( I, t T) (3.39) { 0,1} w ( J) (3.40) { 0,1} y ( J, t T) (3.41) + + x, x, s, s 0 ( J, t T) (3.42) v 0 ( I, t T) (3.43) t The obectve functon (3.30) mnmzes the sum of the nstallaton costs for all the predetermned fault cases. Constrant set (3.31) guarantee that each lnk can be used as a part of a post-fault confguraton of the dstrbuton feeder n each fault case only f ths lnk s nstalled. Constrant set (3.32) assures that a radal confguraton,.e. a spannng tree, s defned for each fault case. The lnkng constrant sets (3.33) and (3.34) ensure that the forward or nverse power flow of a gven lnk n a gven fault case s postve f the lnk s used n ths fault case. M s a suffcently large postve number. Constrant set (3.35) requres that the load demand s satsfed for each load pont n each fault case. For each lnk J and for each fault t T, the dfference (x + x ) denotes the power flow n lnk n fault case t. For every lnk where an exstng or potental feeder exsts as shown n Fgure 5, constrant set (3.36) must be satsfed. If a lnk s used, then ths set assures that the lne voltage drop relaton holds. Otherwse, the voltage dfference between two nodes can be absorbed by ether s + or s accordng to the sgn of the voltage dfference. The varables s + and s are two nonnegatve real slack varables. Constrant sets (3.37)-(3.38) assure that the power flow lmts mposed by current capactes are respected n each fault case. Constrant set (3.39) mposes a lower voltage bound on the voltage at every node for each fault case. Ths model s solved wth a three-phase composte heurstc. Gven a set of nstalled facltes, the frst phase constructs, for each fault case, an ntal tree confguraton whch satsfes all the constrants except current capacty lmts and voltage drop constrants. Then, n the second phase, the ntal set of tree confguratons s made feasble by applyng fve procedures successvely. In the frst procedure, a CIRRELT

23 sequence of lnk exchanges s performed for each fault case. A lnk exchange frst constructs a cycle by addng one lnk n an ntal tree confguraton, and then removes another lnk along the cycle to form a new tree confguraton. A canddate lnk, to add or to delete for the exchange, s chosen so as to mnmze both nstallaton cost and constrant volatons. However, when capacty lmts are respected, the canddate lnk s chosen so as to both mnmze nstallaton cost and maxmze voltage. It should be noted that maxmzng voltage mplctly means to reduce operatng cost (losses) as much as possble. The search ends when no canddate lnk can mprove the tree confguratons. The second procedure stll attempts to elmnate constrant volatons by applyng lnk exchanges for each fault case. However, tree confguratons volatng current capacty lmts and voltage drop constrants are allowed durng the search process. The thrd, fourth and ffth procedures try to elmnate constrant volatons by addng one, two or even all canddate facltes to the exstng system, respectvely. To do ths, lnk exchanges are ndependently carred out for the fault cases for each procedure. When a procedure termnates, f no constrant volatons exst, the last phase attempts to reduce the nstallaton cost by removng unnecessary nstalled facltes or by replacng costly facltes wth cheaper ones, provded that these lnk exchange operatons do not cause any constrant volatons. The fve mprovement procedures and the last phase are appled to several ntal sets of tree confguratons and the best system confguraton plan s selected. The authors also proposed a smplfed verson of the heurstc where a feasble tree confguraton s determned for each fault case ndependently. Results on two problems nvolvng 59 nodes, 69 lnks and 6 fault cases ndcated that the three-phase composte heurstc allows nstallaton cost savngs of up to 73.39% over the tree confguratons produced by the smplfed method wth computng tmes less than 8 mnutes Compound substaton capacty, load reallocaton and system confguraton models Largely due to the nature of the sngle-fault polcy, there are strong nteractons between the determnaton of substaton load capacty, the permanent reallocaton of excess load, the nstallaton of new feeders, and the addton of substaton transformers. In an effort to ntegrate these closely nterrelated decsons nto a sngle decson scheme, Khator and Leung (1995) proposed a heurstc approach for the combned problem of substaton sngle-fault capacty plannng, load reallocaton, feeder confguraton and transformer confguraton over a mult-year plannng horzon. The approach, whch s based on the three models proposed by Leung et al. (1995), Sarada et al. (1995) and Leung et al. (1996), also ntegrates a substaton transformer confguraton model wth no fault consderaton developed by Leung and Khator (1995). Fgure 6, taken from Khator and Leung (1995), depcts the CIRRELT