Electricity Network Reliability Optimization

Transcription

1 Electrcty Network Relablty Optmzaton Kavnesh Sngh Department of Engneerng Scence Unversty of Auckland New Zealand Abstract Electrcty dstrbuton networks are subject to random faults. On occurrence of a fault n the dstrbuton network, the crcut breaker on the faulty feeder trps, whch dsconnects supply to all customers connected to that feeder. Once the fault s located, the faulty component s solated and then repared. The cost of solatng and reparng faults depends on the way the network s confgured, and the sze and locaton of ts components, snce a major part of ths cost comes from compensatng customers f a fault n the network causes the nterrupton duraton to be greater than a gven threshold duraton. We descrbe a model developed n collaboraton wth a local lnes company for mnmzng the cost of relablty of dstrbuton networks by swtch reconfguraton. Three local search heurstcs to reconfgure radal dstrbuton networks are nvestgated. For each component, the tme between falures, the solaton duraton, and the repar duraton are modelled usng Webull dstrbutons. The model allows the company to reconfgure ther current radal dstrbuton networks to mnmze ther total expected cost of relablty, as well as provdng a tool for nvestgatng captal nvestment scenaros. Introducton The power dstrbuton network can be consdered as two sub networks. The hgh voltage sub transmsson network connects the Transpower natonal transmsson network at the grd ext ponts to zone substatons, at 0, or kv. Each substaton serves a partcular geographc area. At the substatons the voltages are further stepped down usng transformers to kv or 6.6kV []. The functon of the dstrbuton network s to delver electrcty from the zone substatons to customers. It ncludes a system of cables and overhead lnes known as feeders operatng manly at kv, wth some 6.6kV, whch dstrbute electrcty from the zone substatons to dstrbuton substatons. At the dstrbuton substatons the voltage s stepped down to 400V and delvered to customers ether drectly or through overhead lnes and cables. The dstrbuton system conssts of nterconnected radal crcuts orgnatng from zone substatons. Dstrbuton networks are normally operated n a radal confguraton for effectve coordnaton of ther protectve systems. More precsely, the radal crcuts are connected to the substatons va a crcut breaker that dsconnects the feeder n case of a fault. Supply can be restored from alternatve sources va the nterconnectons usng swtchng operatons.

2 . The Relablty of Supply The relablty of supply s governed by the frequency and the duraton of nterruptons to supply and the number of customers affected by nterrupton. On an occurrence of an electrcal fault, the correspondng feeder s crcut breaker opens at the substaton. Ths dsconnects the entre feeder from supply of electrcal energy. Consequently all customers on the faulted feeder experence an nterrupton n servce. In most cases the fault s detected by the operatons control systems or by trouble calls from customers, but n some cases a crew has to be dspatched to locate the fault. Once the fault s located, t s solated by separatng t from the rest of the network before restorng servce to the unfaulted secton. The faulty component can then be repared. Subsequently the dstrbuton system returns to ts normal operatng state.. The Cost of Relablty In general, the total cost of solatng and reparng faults depends on the way the network s confgured, and the sze and locaton of ts components. Increasng the expendture on mantenance or extra captal can lower ths cost by reducng the frequency and solaton tme of faults. For a set of components wth a gven frequency of fault occurrence, t s possble to make further reducton n cost by confgurng the swtches n the network. Dfferent confguratons wll not alter the type of faults that need to be repared but may reduce the number of customers affected due to a partcular fault. Ths wll decrease the cost of relablty, or more precsely the monetary compensaton by the dstrbuton company to ther customers f the nterrupton duraton due to solaton or repar of a faled component s greater than a specfc tme nterval called the threshold duraton.. Dstrbuton Network Reconfguraton The reconfguraton of dstrbuton networks changes a par of swtches to mantan the radal structure of the dstrbuton network. In more precse terms, durng reconfguraton, a normally open (NO) swtch s closed. Ths acton volates the radal network constrant by formng a loop. To redress the volaton, a normally closed (NC) swtch contaned n the loop formed, s opened. The result s a newly confgured radal network. The concept of reconfguraton s further explaned n secton. For a large power dstrbuton radal network, the number of possble confguratons s extremely large. To fnd an optmal confguraton wth mnmum total expected cost of relablty, a complete enumeraton of the possble confguratons would have to be done. Ths would consume extremely large amounts of tme (t s an NP-hard problem) and thus ths approach s very mpractcal. To overcome the tme consumng process of complete enumeraton, heurstcs are employed n ths model to reconfgure networks to get close to optmal solutons. Ths s further dscussed n secton. Modellng The Radal Power Dstrbuton Network

3 A radal network s equvalent to a spannng tree. Fgure. shows a dagram of a radal dstrbuton network. Key Substaton Crcut Breaker Normally Closed Swtch Normally Open Swtch Transformer Power Lne Fgure. A radal dstrbuton network The radal network confguraton s modelled as a graph, where: The edges represent swtches The vertces represent collectons of power systems gear contanng no swtches, whch we shall henceforth refer to as a component. Each component comprses power systems gear such as power lnes and cables, transformers, power poles, fuses etc. The swtches (labelled accordng to ther locaton, functon and status) represent the NC swtches, NO swtches and Crcut Breakers. Each component conssts of all the power systems gear between adjacent swtches. When clumpng power systems gear nto a component, t s assumed that upon falure of any power systems gear wthn the component, all the power systems gear represented by that component and all the consumers connected to the component would be dsconnected durng solaton and repar.. Fault Effect Overvew When there s a fault n the radal network, the closest crcut breaker upstream of the fault trps and dsconnects the entre network downstream of the crcut breaker. It s assumed that the network remans dsconnected untl the fault s solated. Then power s restored to all the possble areas of the nterrupted network by openng and closng swtches. All the components n the solated area of the network reman dsconnected untl the faulty component s repared. In general most power dstrbuton companes have to compensate ther customers f the nterrupton duraton due to solaton or repar of a faled component s greater than the threshold duraton. In our model, the solated area may consst of a sngle component n whch the power systems gear has faled, or nclude several components that are drectly or ndrectly lnked to the faled component

4 whle havng no other swtches connectng them to the rest of the network thus restrctng ther restoraton.. Formulaton of The Stochastc Model The tme between falures, the solaton duraton, and the repar duraton nvolve uncertanty. Therefore these duratons need to be modelled usng a stochastc model. We approxmate these stochastc duratons usng the Webull dstrbuton. The falure frequency for a component s the number of tmes the component fals over a perod of tme. A falure for a component can be descrbed as transtonng from an operatng (Up) state to an solaton state. The probablty of ths transton occurrng s approxmated usng a Markov process. It can be seen that the component state probabltes are ndependent of the embedded Markov Chan statonary state probabltes. Usng the component state probabltes, the state frequency Fr(j) for component j s Fr( j) =, ( µ + µ + µ ) u where µ s the Webull mean duraton and subscrpts u, r and represent the component Up, Isolaton and Repar states respectvely. It can be seen that the state frequency s the same for all component states. In computng the expected cost of relablty, the probabltes for the solaton or repar duraton beng greater than the threshold duraton need to be determned. We assume that durng solaton and repar of a faled component n a power dstrbuton network, no other component fals. Ths assumpton s justfable when falures are ndependent and occur wth small probablty. Let: I = Isolaton Duraton, R = Repar Duraton, Q = Threshold Duraton. We assume that I, R and Q each have a Webull dstrbuton. The Webull Dstrbuton s defned by the followng probablty densty functon and cumulatve dstrbuton functon: f () t β = t β η β e β t η, () t r β t η F = e. Probablty that the solaton duraton s greater than the threshold duraton s: Pr( I > Q) = F ( Q) β Q η = e. Probablty that the solaton duraton s less than the threshold duraton, but the solaton + repar duraton s greater than the threshold duraton s: Pr( I < Q I I + R > Q). The dstrbuton of I+R s gven by a convoluton ntegral: Pr( I < Q I I + R > Q) = f I Q 0 () t f ( y) dy dt y= Q t r

5 Q = 0 βr Q t ηr f () t e dt. Substtutng the probablty densty functon f (t), gves Q β βr t Q t + β β r η η Pr( I < Q I I + R > Q) = t e dt. () β η 0. Computng the Cost of Relablty Usng the falure frequences and the duraton dstrbutons derved above, t s smple to compute the total expected cost of relablty (COR) for a gven radal confguraton. Let: R C () = expected cost of relablty due to falure at component ; C I () = cost of compensatng customers when the tme to solate a fault at component s greater than the threshold duraton; C IR () = cost of compensatng customers when the tme to solate a fault at component s less than the threshold duraton, but the solaton plus repar tme s greater than the threshold duraton; Fr() = frequency of falures at component ; P I () = Pr( I > Q) = probablty that the tme to solate a fault at component s greater than the threshold duraton; P IR () = Pr( I < Q I I + R > Q) = probablty that the tme to solate a fault at component s less than the threshold duraton, but the solaton plus repar tme s greater than the threshold duraton. Then the total expected COR s computed as: N ( ) = Fr( ) [ C ( ) P ( ) + C ( ) P ( )] N R = = C I I IR IR where N s the total number of components. The calculaton of the expected COR s straghtforward except n the case of the term P IR () whch s the convoluton ntegral (). In practce we compute ths numercally usng Smpson s rule..0 Radal Dstrbuton Network Reconfguraton A power dstrbuton company wants to confgure ther dstrbuton network to mnmse the expected cost of relablty. To determne the confguraton that gves the mnmum expected cost of relablty; all possble radal confguratons have to be evaluated to determne the best confguraton. The computatonal tme taken to completely enumerate all possble confguratons for a large network would be very long. Hence a complete enumeraton approach s mpractcal, and therefore we use a local search approach.

6 . Local Search Heurstcs Local search heurstcs are employed to reconfgure radal networks to gve a local mnmum for the total expected cost of relablty. The confguraton that gves a local mnmum could be a confguraton that gves a global mnmum, but ths s not guaranteed. A local search soluton s not guaranteed to be a global mnmum, because a local search heurstc does not explore all possble confguratons; consequently a possble confguraton not evaluated could be the global mnmum. A local search heurstc s a neghbourhood search algorthm that explores neghbourng solutons (radal network confguratons). The solutons explored depend on the neghbourhood rule mplemented and how the neghbourhood structure s defned. The neghbourhood of a partcular radal network confguraton s the set of all radal network confguratons that can be obtaned from the partcular radal network confguraton usng the operatons set out by the neghbourhood rule. A further characterstc of a local search heurstc s ts descent rule. The descent rule determnes the neghbourng soluton that s to be accepted out of all the possble neghbourng solutons defned n the neghbourhood structure. The followng s some notaton used to descrbe the local search neghbourhood structures and the rules used n ths model. An Enterng Swtch (ES) s a NO swtch that we select to become a NC swtch. A Leavng Swtch (LS) s a NC swtch that we select to become a NO swtch. A general neghbourhood search algorthm that can be used to reconfgure a radal network s as follows. An ES s selected for the current radal network confguraton. Closng ths swtch forms a loop (cycle n the spannng tree) n the network that volates the necessary radal constrant. To restore the network back to a radal confguraton, a LS has to be chosen from the loop formed. Observe that the choce of potental LS s depends on the ES chosen, though we suppress ths dependence n the notaton. As an example, consder the radally confgured network n Fgure. called RN-0. RN Loop Enterng Swtch s 7 Closed swtches 4, 7 & 8 forms a loop Fgure. The ntal stage of reconfguraton where swtch 7 s selected as the ES from RN-0. Ths forms a loop that volates the radal network

7 To restore the network to radal confguraton, a leavng swtch (other than the ES 7) n the loop has to be opened. Suppose swtch 4 s chosen to be the LS. Fgure. shows the next step of reconfguraton resultng n a new radally confgured network RN-. RN Loop The LS s swtch 4 The new radally confgured network Fgure. The second stage of reconfguraton where a swtch from the loop s nomnated as the LS to gve radal network RN- In the followng sectons we descrbe three heurstcs: the Adjacent Neghbourhood Steepest Descent Search, the Cycle Neghbourhood Steepest Descent Search and the Cycle Neghbourhood Next Descent Search... Adjacent Neghbourhood Steepest Descent Search In the Adjacent Neghbourhood (AN) search, the neghbourhood structure of the current radal network s the set of all radal networks that can be obtaned by the operaton of closng a NO swtch to form a cycle and openng ether one of the two NC swtches n the cycle adjacent to the NO swtch, so as to produce a radal network. The descent rule utlzed n the local search heurstc for the AN structure s the steepest descent rule. The AN steepest descent rule takes the followng approach. For any ncumbent radal network, we evaluate the total expected cost of relablty (COR) of all solutons n the adjacent neghbourhood, and select that whch has lowest total expected COR, (f that gves an mprovement) to be the ncumbent. These searches are done repeatedly untl there s no soluton found whch gves a lower total expected COR than the ncumbent soluton. At ths pont the ncumbent soluton s declared to be the local mnmum... Cycle Neghbourhood Steepest Descent Search In the Cycle Neghbourhood (CN) search, the neghbourhood structure of the current radal network s the set of all radal networks that can be obtaned by the operaton of closng a NO swtch (from the set of possble ES s) to form a cycle (set of possble LS s), and then selectng a LS from the set of possble LS s, so as to produce a radal network. The descent rule utlzed n the local search heurstc for the CN structure s the steepest descent rule. The CN steepest descent rule takes the followng approach. At

8 each search by the CN steepest descent local search heurstc, each ES from the set of ES s s consdered. Subsequently for each ES, the total expected COR s determned for all the radal networks formed when consderng the correspondng LS s one-by-one. Out of all the confguratons explored n a sngle search, the ES and LS combnaton that gves the mnmum total expected COR s accepted as the ncumbent soluton. The ncumbent soluton s stored and then another steepest descent local search starts wth the radal network confguraton gven by the ncumbent soluton as the ntal radal network. These searches are done repeatedly untl there s no soluton found whch gves a lower total expected COR than the ncumbent soluton. At ths pont the ncumbent soluton s declared to be the local mnmum... Cycle Neghbourhood Next Descent Search The Cycle Neghbourhood Next Descent local search has exactly the same neghbourhood structure defnton as that of the CN steepest descent local search. The dfference s n the descent rule employed. Gven a neghbourhood rule wth a CN structure, a local search heurstc wth the Next Descent rule evaluates neghbourng solutons (radal confguratons) and accepts the frst soluton found whch gves a lower total expected COR then the ncumbent soluton. The next descent rule s also referred to as the Frst Improvement rule. The CN next descent rule takes the followng approach. Frst an ES s selected along wth a LS. Ths forms a new radally confgured network for whch the total expected COR s computed. If the total expected COR s less than that for the ncumbent (ntally set to be the total expected COR of the ntal network), then we accept the new radally confgured network as the ncumbent soluton. Otherwse we consder the next LS from the set of LS s to form a new radal confguraton. If all LS s from the set have been consdered, we then consder the next ES from the set of ES s and thus a LS from ts correspondng set of potental LS s, to form a new radal confguraton. A sngle search of the CN Next Descent local search conssts of the aforementoned steps. Every search starts wth the most recently accepted radal network confguraton (.e. the ncumbent soluton) as the ntal radal network from whch the sets of ES s and LS s are determned n accordance wth the defned neghbourhood structure. These searches are done repeatedly untl there s no soluton found whch gves a lower total expected COR than the ncumbent soluton. At ths pont the ncumbent soluton s declared to be the local mnmum.. Modellng Network Load Capacty Whle t s mportant for a power dstrbuton company to have ther radal networks confgured for mnmum total expected cost of relablty, t s also essental that the reconfgured radal networks do not exceed the load capactes of the network components. When reconfgurng the radal network, each reconfguraton has to be checked to ensure that the load capacty s not exceeded at any part of the network. One approach s to carry out a load flow analyss, usng power systems prncples to calculate the power flows and voltages of the power system under normal operaton condtons. We take a smpler approach to model the load capactes of the network. We assume that the clumped power system gear represented by a node n the radal network has the capacty to supply all the customers connected to t and all the power system components n successor nodes. We specfy a load capacty for each swtch, and check

9 that ths s not volated when a confguraton s chosen. In the current verson of our model no load capacty check s carred out for the resultng network confguraton durng repar, although ths feature can be added wthout any essental changes. 4.0 Results and Conclusons Usng the models and concepts developed n the project, a Vsual Basc spreadsheetbased tool called the Network Relablty Optmzer (NeRO) has been developed. Once all the data defnng the components and the swtches have been entered nto NeRO, t can be executed. The program uses the graph defnton to randomly generate a radally confgured network. Randomly generated radal networks that meet the load feasblty crtera, allows each local search heurstc to have dfferent startng confguratons, thus ncreasng the chance that a global mnmum wll be found. Usng the randomly generated radal network as an ntal confguraton, the program subsequently computes the total expected COR and stores the network as the ncumbent soluton. Fnally the program starts the local search heurstc. 4. Case Study of Takann7 Feeder A case study was carred out on the Takann7 feeder. Ths feeder s part of VECTOR Ltd dstrbuton network and has a poor performance hstory. VECTOR Ltd s nvestng n a power restoraton lnk (PRL) at the end of the Takann7 feeder from a neghbourng dstrbuton company. Usng NeRO, the total expected cost of relablty was compared for the current confguraton wth and wthout the PRL. Analyss wth the model ndcated a sgnfcant decrease n the total expected cost of relablty, hence supportng the nvestment decson. Further, the feeder was optmally confgured wth the PRL nvestment, to gve an even lower total expected cost of relablty. 4. Performance Comparson of the Local Search Heurstcs To analyse the performance of the local search heurstcs developed n ths project, a large network was used. The network had components and 6 swtches. All the data that was requred to defne a complete network was randomly generated. Fgure 4. shows how each local search heurstc would perform for 00 runs gven that every soluton accepted at each run would be no worse than the soluton found n the precedng run. From ths perspectve, the plot for each local search heurstc shows ts convergence characterstcs. It can be concluded that the Adjacent Neghbourhood Steepest Descent local search heurstc s the best performng heurstc as t fnds a local mnmum that s no worse than that found by the Cycle Neghbourhood Steepest Descent and the Cycle Neghbourhood Next Descent local search heurstcs, n the shortest tme perod.

10 Comparson of the Local Search Heurstcs (00 Sorted Results) Total Expected Cost of Relablty ($000/month) NeRO enables VECTOR to reconfgure ther current radal dstrbuton networks to mnmse ther total expected cost of relablty. Addtonally NeRO can be used as a plannng and desgn tool to nvestgate captal nvestment scenaros. It can be enhanced to carry out a full load flow analyss usng power systems concepts and provdes a good foundaton for development of programs to mnmse power loss, and optmse capacty usage n conjuncton wth mnmsng the cost of relablty. Acknowledgments The author would lke to acknowledge the contrbutons of Assocate Professor Andrew Phlpott and Dr. Murray Smth of the Unversty of Auckland, and Peter Rchards and Ashok Parsotam of VECTOR Lmted. References Adjacent Neghbourhood Steepest Descent Cycle Neghbourhood Next Descent Cycle Neghbourhood Steepest Descent Average Tme Per Local Mnmum 8., 8.8 and.8 seconds Local Search Run Number Fgure 4. Comparson of the convergence characterstcs of local search heurstcs [] Asset Management Plan (000/00). VECTOR Lmted. Auckland, NZ.