Dynamic Modeling and Resilience for Power Distribution

Transcription

1 Dynamc Modeng and Resence for Power Dstrbuton Yun We and Chuany J Schoo of Eectrca and Computer Engneerng Georga Insttute of Technoogy Atanta, Georga Ema: yunwe@gatech.edu jchuany@gatech.edu Foyd Gavan, Stephen Couvon and George Oreana Entergy Servces, Inc. New Oreans, LA 753 Ema: fgavan@entergy.com scouv@entergy.com gorea@entergy.com Abstract Resence of power dstrbuton s pertnent to the energy grd under severe weather. Ths work deveops an anaytca formuaton for arge-scae faure and recovery of power dstrbuton nduced by severe weather. A focus s on ncorporatng pertnent characterstcs of topoogca network structures nto spata tempora modeng. Such characterstcs are new notatons as dynamc faure- and recovery-neghborhoods. The neghborhoods quantfy correated faures and recoveres due to topoogy and types of components n power dstrbuton. The resutng mode s a mut-scae non-statonary spata tempora random process. Dynamc resence s then defned based on the mode. Usng the mode and arge-scae rea data from Hurrcane Ike, unque characterstcs are dentfed: The faures foow the 8/2 rue where 74.3% of the tota faures resut from 2.7% of faure neghborhoods wth up to 72 components faed together. Thus the hurrcane caused a arge number of correated faures. Unke the faures, the recoveres foow 6/9 rue: 59.3% of recoveres resuted from 92.7% of a neghborhoods where ether one component aone or two together recovered. Thus about 6% recoveres were uncorreated and requred ndvdua restoratons. The faure and recovery processes are further studed through the resence metrc to dentfy the east resent regons and tme duratons. I. INTRODUCTION A key objectve of the smart grd s to mprove resence of the power grd to externa dsruptons from severe weather. Severe weather events such as hurrcanes and storms have been occurrng more frequenty n Amerca n recent years, each of whch resuted n a haf to severa mon customers wthout eectrcty for days [] [2]. Power dstrbuton was often mpacted the most, as a compound effect of severe weather and an out-dated nfrastructure. Dstrbuton networks e at the edge of the grd wth a arge number of components across a wde geographca span. Those components can be ether agng or not we-protected, and are thus susceptbe to severe weather. A fundamenta research ssue pertanng to ths rea probem s the resence of power dstrbuton to arge-scae externa dsruptons from severe weather [3]. Resence here corresponds to the abty of the grd to wthstand externa dsturbances and to recover rapdy from faures [4]. Emprca approaches have been used wdey n ndustry for weather nduced faures [5]. However, emprca approaches become nadequate for arge-scae weather-nduced dsruptons that occur frequenty n recent years [2]. Statc modes are deveoped for dentfyng varabes reated to faures of power dstrbuton [6] [7] [8]. However, dynamc modes and resence are needed for characterzng the tme-varyng nature of weather-nduced arge-scae faure and recovery [9]. Resence nvoves mutpe spata-tempora scaes. A sma spata scae s at the component-eve where faures and restoratons occur. A arge spata scae s at the subnetworkeve where resence s measured through aggregatng component faures and recoveres n a servce area. A sma tempora scae of submnutes s when topoogcay correated faures occur [], and when eectrcty s reganed through sefheang []. A arge tempora scae of mnutes and beyond s when faures and restoratons occur due to severe weather and manua repar respectvey. For resence to encompass the pertnent mut-scae characterstcs, a rgorous probem formuaton s necessary from bottom-up,.e., from modeng to defnng resence/vunerabty at the mutpe scaes. Ths work deveops such an approach by focusng on the foowng chaenges. The frst chaenge s stochastcty where faures and recoveres occur randomy and dynamcay. Faure stochastcty resuts from spata-tempora evouton of externa weather [9]. Recovery stochastcty resuts from envronmenta condtons of the aftermath of a severe weather event. In addton, faures and restoratons exhbt non-statonarty,.e., dfferent behavours at dfferent tme and ocatons [9]. Exstng approaches n power systems account for randomness of faures [2] [3] [5] but rarey the dynamcs nor spata-tempora non-statonarty. In our pror work, a spata-tempora nonstatonary stochastc mode s deveoped at the arge spata scae of ctes and tempora scae of mnutes [9]. However, network structures of power dstrbutons are not consdered. A recent work [4] provdes a rgorous formuaton for the mnmum number of PMU montors n dstrbuton networks, and motvates our formuaton. Ths work ncudes the effect of network structures through a new noton of dynamc neghborhoods. Dynamc neghborhoods characterze how weather-nduced faures are exacerbated by a network structures. The resutng mode s a spatatempora non-statonary random process that encompasses

2 topoogca network structures, dfferent network components, dynamc faure and recovery. The second chaenge s how to defne resence. Statc metrcs on resence have been wdey used [5]. These metrcs, however, do not characterze the dynamc nature of arge-scae faures nduced by severe weather. In addton, resence needs to ncude recovery from faures [6]. Ths work defnes the resence as a dynamc metrc, motvated by those from communcaton networks [7]. Importanty, the dynamc resence metrc s based on the spata tempora mode derved from bottom up. The metrc thus refects the mpact of dynamc network neghborhoods n addton to weather-nduced faures and recoveres. The non-statonary mode and the resence metrc are apped to a rea fe exampe of arge scae power faures durng Hurrcane Ike n 28. Rea data from an operatona network s used to earn parameters of faure and recovery processes as we as the resence metrc. Ths resuts n nsghts and understandng on the resence of operatona networks n Secton V. A. Probem Descrpton II. PROBLEM SETTING Consder a node that represents a network component such as a substaton, a transformer, or a nk as a feeder/power ne. Severe weather can nduce a faure drecty to a node. For exampe, foodng can cause a non-functona substaton and other equpment faures. Hgh wnds can cause faen poes or trees on power nes. Such weather-nduced faures often occur n mnutes resutng from evovng severe-weather condtons [9]. A weather-nduced faure, referred to as a faure n short, can resut n secondary faures through a network structure. A network structure conssts of a topoogy and dfferent types of components. For exampe, unbaanced currents from a faure can cause burned ne fuses as secondary faures. A faure upstream can aso resut n osses of eectrcty, but no damage, at nodes downstream n a dstrbuton tree. For exampe, ether a non-functona substaton or a broken nk cause a oss of power to downstream nodes. Those nodes wthout eectrcty servce are referred to as outages. Secondary faures and outages occur at a smaer tme scae of submnutes, as mpacted by a network structure. Dsruptons ncude faures, secondary faures and outages. Recovery occurs at two tme-scaes aso. Sef-recovery occurs n submnutes. Manua repars to damaged nodes occur at the tme scae of mnutes or onger [9]. When a faure- or an outage-node regans eectrcty suppy, downstream outage nodes regan the servce together. Hence, the mut-scae characterstcs need to be quantfed for dsrupton and recovery respectvey. B. Faure and Neghborhoods Let X (w) (t) be the state of node at tme t, where specfes both a network ocaton and a geo-ocaton of the node, n. n s the tota number of nodes n a network. t> 3 Fas at s Fas at t 6 Fas at u 7 s t u v w Recovers at w 5 Recovers at v 6 7 s t u v w V (f) 2 ( ) V (f) ( ) V (f) ( ) V (r) ( ) V (r) ( ) Fg.. Exampe of neghborhoods. s contnuous tme. For smpcty, a node takes two states: X (w) (t) =f node s n dsrupton. X (w) (t) =f node s n norma operaton. w specfes three scenaros: w = f for a faure nduced by exogenous weather, w = f for a secondary faure, and w = o for an outage. An outage or a secondary faure s nduced by a faure occurred at a network neghbor. A network neghbor here s a node at the upstream of a dstrbuton network wth a tree topoogy. Dsrupton A (w) (t) s a state transton from norma to dsrupton {X (w) (t t) =,X (w) (t) =}, whch occurs n (t t, t] at node, t>, w = {f,f,o}. t > s suffcenty sma so that there s ony one faure, and one set of secondary faures or outages occurred n (t t, t]. Faure neghborhood V (f) (t) s a new noton of dynamc topoogy, consstng of the downstream nodes that are n norma operaton precedng faure at t t. That s, for any j V (f) (t), ether outage A (o) j (t) or secondary faure A (f ) j (t) occur wth faure. Hence, a faure neghborhood characterzes correated faures and outages. C. Recovery and Neghborhoods Recovery B (w) norma {X (w) j (t) s the state transton from dsrupton to (t t) =,X (w) j (t) =}, whch occurs n (t t, t] at node, t>, w = {f,f,o}. When w = f (or w = f ), faure s repared. When w = o, an upstream neghbor of node s repared. (t) s another new noton of dynamc topoogy, consstng of the downstream neghbors of node that are n outage at (t t, t) pror to the restoraton. t > s suffcenty sma so that there s one restoraton Recovery neghborhood V (r) and one set V (r) D. Exampe (t) of recoveres n (t t, t]. Fgure shows an exampe of dsrupton and recovery as we as the neghborhoods. Frst, node 2 fas at tme s, nducng secondary faure 3 and outage 4,.e., V (f) 2 (s) ={3, 4}. Then node fas at t > s, nducng outages to nodes n faure neghborhood V (f) (t) ={, 5, 6}. Then node fas at u>t, nducng outages to V (f) (u) ={7, 8}. Note that 8

3 node 5 s a downstream neghbor of both nodes and but ony beongs to faure neghborhood V (f) (t) of node by defnton. Hence faure neghborhoods are non-overappng for faures occurred at dfferent ocatons and tme. Faure s repared at v > u, restorng faure and outages n recovery neghborhood V (r) (v) = {5, 6, 7, 8}. Fnay, faure s repared at w>v, restorng faure and outages n V (r) (v) ={, 2, 4}. Secondary faure 3 remans to be restored. Ths exampe ustrates faure- and recoveryneghborhoods that are dynamcay changng due to evovng faures and reconfguraton n restoraton. III. NON-STATIONARY SPATIAL TEMPORAL PROCESSES A dynamc network envronment emerges from the above probem settng: Externa severe weather causes network nodes to fa. The faed nodes nduce secondary faures and outages at ther network neghbors. Faures/outages then recover together wth ther neghbors. Such dsrupton and recovery are modeed as non-statonary spata tempora random processes, wth dynamc neghborhoods at the component-eve, and aggregatons of components n a servce regon. A. Dsrupton Process Our modeng starts from the component-eve. n nodes n a tree topoogy form a spata tempora process, consstng of a network of random state transtons as bnary varabes (t)],i[b (w) (t)]}, for t>, n, w = {f,f,o}. I(A) s an ndcator functon. I(A) =f the event A occurs; {I[A (w) otherwse, I(A) =. Let N (d) (t) be the number of nodes that are dsrupted from eectrcty servce n (t t, t]. For a suffcenty sma t >, t s natura to assume that ony one weather-nduced faure occurs at node, and one set of reated secondary faures and outages n (t t, t]. Then N (d) (t) =I[A (f) outages n neghborhood V (f) (t)] + v (f) (t)i[a (f) (t)], () where v (f) (t)i[a (f) (t)] s a set of secondary faures and (t). v (f) (t) = V (f) (t) s the sze of the faure neghborhood at node and tme t. v (f) (t) characterzes correated dsruptons. The arger v (f) (t) s, the more correated dsruptons for the node at tme t. Defnton : Faure, outage and dsrupton rate. Faure rate of node at tme t s the expected number of state transtons from norma to (weather-nduced) faures per unt tme at node, whch s E λ (f) (t) = m t { I[A (f) t } (t)]. (2) Here E{ } represents expectaton. Outage rate that s nduced by faure at tme t s the expected number of state transtons from norma to outage per unt tme at faure neghborhood V (f) (t). For smpcty of notaton, the outage rate here ncudes secondary faures aso, where { } E v (f) λ (o) (t)i[a (f) (t)] (t) = m. (3) t t Dsrupton rate at node s λ (d) (t) = m t λ (d) t (d) E{ N (t)}, (t) =λ (f) (t)+λ (o) (t). (4) A dsrupton rate shows the mpact of severe weather and a network structure. A faure neghborhood shows expcty mpacts of topoogy and heterogeneous types of components. B. Recovery Process The number of nodes that are recovered n (t t, t] can be obtaned smary, N (r) where I[B (f) V (r) (t) =I[B (f) (t)] + v (r) (t)i[b (f) (t)], (5) (t)] s the state of recovery for faure. v (r) (t) = (t) s the sze of a recovery neghborhood V (r) (t). Recovery process s characterzed by the recovery rate defned as foows. Defnton 2: Recovery rate. The recovery rate for node and ts neghbors n V (r) E λ (r) (t) = m t (t) at tme t s { I[B (f) (t)]( + v (r) (t)) t }. (6) The recovery rate and neghborhoods are dynamc, showng a changng topoogy n restoraton. The tme-varyng rates and neghborhoods show the non-statonarty of dsrupton- and recovery-processes. C. Aggregaton at Subnetwork-Leve Now et N (w) (t) be the number of dsruptons n a subnetwork n a servce regon, E{N (w) (t)} = t λ (w) (τ) (τ)dτ, (7) where w = {f,f,o}. (τ) ndcates the ocaton of a dsrupton at tme τ. dτ s assumed to be sma so that at most one faure and one neghborhood of secondary faures/outages occur n (τ dτ, τ). The expected number of dsruptons E{N (d) (t)} occurred up to tme t s the sum of expected faures and outages (wth secondary faures), E{N (d) (t)} = E{N (f) (t)} + E{N (o) (t)}. (8) Let E{N (r) (t)} be the expected number of nodes whch recover n [,t] n a subnetwork, then E{N (r) (t)} = t IV. RESILIENCE λ (r) (τ) (τ)dτ. (9) The non-statonary spata tempora mode enabes a nove resence metrc for power dstrbuton. Before the metrc s defned, we frst characterze fast versus sow recovery based on concepts from nfant and agng mortaty [8]. Defnton 3: Infant (fast) and agng (sow) recovery: Let > be a threshod vaue. If a node remans n ether d

4 pm 9/2 2am 9/3 2pm 9/3 2am 9/4 (b) (a) Dsrupton Rate Faure Rate Faure Neghborhood Landfa 9/2 9/6 9/2 9/24 Tme Dsrupton Rate Recovery Rate Recovery Neghborhood Fg. 2. Emprca rate functons of the network: (a) Faure rate, dsrupton rate, and sze of faure neghborhood; (b) Recovery rate and sze of recovery neghborhood. faure or outage for ess than d duraton, the node has nfant recovery. Otherwse, the node has agng recovery. Usng threshod d, we can defne resence. Intutvey, resence measures network-wde performance from two aspects. One s for a power grd to wthstand externa dsruptons as much as possbe. The other s to rapdy restore eectrcty servce from faures. Hence, agng recovery s a compement of these two characterstcs [9]. Resence s then characterzed as one mnus agng recovery. Defnton 4: Resence: Consder a sub-network n regon Z wth m number of dsruptons. The resence of the subnetwork s, s(t, Z) = m t τ= w=f,o (τ) Z λ (w) (τ) (τ) Pr{D (w) (τ) (τ) >t τ + d } ) dτ. () The second term corresponds to the expected percentage of agng recoveres at tme t. The agng recoveres here correspond to dsruptons at tme t that woud not recover for at east addtona duraton d. For exampe, when w = f, λ (f) (τ) dτ s the expected number of dsruptons occurred n (τ dτ, τ]. Pr{D (w) (τ) (τ) >t τ + d } s the probabty for faures to ast a duraton onger than t τ + d. The product s the expected number of nodes that fa n (τ dτ, τ] and do not recover at tme t + d, whch s smpy the number of agng recoveres vewed at tme t. The ntegra adds up a agng recoveres n duraton [,t] and regon Z. Hence, s(t, Z) s the expected percentage of nodes n regon Z at tme t whch are ether n norma operaton or recover wthn addtona duraton d. The resence thus refects tempora evouton of a network n response to severe weather. V. HURRICANE IKE AND LARGE-SCALE REAL DATA The non-statonary spata tempora mode s now apped to studyng the mpact of a major hurrcane. Rea data from an Dsruptons Neghborhoods % 25.7% 2.7% 79.3% Mce Eephants 6am 9/2 2am 9/3 2am 9/4 Tme Mce Eephants Fg. 3. Hstogram of the dsruptons and faure neghborhoods over tme. Sze of eephants neghborhoods s more than 2. Recoveres Neghborhoods % 59.3% 7.3% 92.7% 9/2 9/6 9/2 9/24 Tme Mce Eephants Mce Eephants Fg. 4. Hstogram of the recoveres and recovery neghborhoods over tme. Sze of eephants neghborhoods s more than 2. operatona network s used to obtan emprca dsrupton and recovery rates and to understand mpacts of network structures. A. Rea Data and Processng Hurrcane Ike s one of the strongest hurrcanes that occurred n 28. Ike resuted n more than two mon customers wthout eectrcty n Lousana and Texas [9]. A major utty provder coected data on component faures and outages. Our data set conssts of 24 sampes (faures or outages) that occurred from 7 a.m. September 2th to 4 a.m. September 4th, durng whch Hurrcane Ike made the andfa. Each sampe conssts of occurrence tme, duraton and ocaton (attude and ongtude) of a dsrupton for a component n a dstrbuton network. The accuracy for tme t s one mnute. A faure neghborhood ncudes sampes whose faure/ outage occurrences fa wthn a mnute. There are 24 faure neghborhoods of szes 72. The remanng 26 faures occurred ndvduay wth mnutes apart. Smary, sampes wth recovery occurrences wthn a mnute are n a recovery neghborhood. There are 24 recovery neghborhoods and 824 ndvdua recoveres. B. Emprca Non-Statonary Processes We now study the emprca non-statonary processes, the mpact of topoogca network structures, and the resence usng the rea data. ) Emprca Dsrupton Process: We estmate emprca faure rate ˆλ (f) (t) and dsrupton rate ˆλ (d) (t) by aggregatng over the components. The dsrupton rate takes nto account of faure neghborhoods. The faure rate s cacuated by aggregatng dsruptons occurred n a mnute as one faure.

5 Resence h 9/2 9/6 9/2 9/24 Tme d =h d =2h d =24h Fg. 5. Resence of the entre power dstrbuton. A ( smpe movng average s used [2], where ˆλ (w) (t) = 2τ ˆN (w) (t + τ) ˆN (w) (t τ)), wth τ = hour, w = {d, f}. Fgure 2(a) shows the dsrupton and faure rates respectvey. The rates and the faure neghborhoods are tme-varyng, showng the non-statonarty of the dsrupton process. The faure rate ncreased to the peak vaue of 5 new faures per hour around the andfa. The dsrupton rate exhbted a smar behavor but had a arger peak vaue of 45 new dsruptons per hour around the andfa. The much arger dsrupton rate refects the mpact of dynamc network structure: There were a arge number of correated dsruptons durng the hurrcane. Ths s further shown n Fgure 2(a) where arge faure neghborhoods occurred many durng the ntense hurrcane perod, wth as many as 72 dsruptons n one neghborhood. Hence, the network components and topoogy were mpacted by the hurrcane dfferenty durng ts evouton. 2) Emprca Recovery Process: Recovery rate ˆλ (r) (t) and the sze of recovery neghborhood v (r) (t) (t) are estmated smary and shown n Fgure 2(b). Two bursts of recoveres emerge. The frst s nfant recovery that occurred aong wth major faures wthn sx hours after the andfa. The second s agng recovery that occurred about 7.7 hours after. The recovery rate and the sze of the recovery-neghborhoods vary wth tme, showng the non-statonarty of the recovery process. C. Neghborhoods: Impact of Network Structures Dynamc faure neghborhoods are ndcatve of the mpacts of topoogca network structures and the hurrcane. The arge faure neghborhoods around the andfa shown n Fgure 2(a) ndcate that the hurrcane nduced a arge number of correated dsruptons. In contrast, faures that occurred ndvduay happened many before and after the hurrcane. Ths suggests that correated faures/outages occurred at the sma tme scae of submnutes s a pertnent characterstc of the hurrcane-nduced dsruptons. Usng the anaogy of eephant and mce fows n computer communcaton [2], we refer arge neghborhoods as eephants, and sma neghborhoods as mce. An 8/2 rue emerges for the dsrupton process: Eephants faure neghborhoods of sze more than 2 contrbute to 74.3% of tota Lattude Lattude Lattude (a) Reducton n Resence Longtude Longtude.5..5 (a2) Eephant Faure Neghborhood Longtude pm, 9/ (a3) Mce Faure Neghborhood 5 5 Lattude Lattude Lattude (b) Reducton n Resence Longtude Longtude (b2) Eephant Faure Neghborhood Longtude 3 am, 9/ (b3) Mce Faure Neghborhood Fg. 6. Two snapshots of the resence over the geographca area of the power dstrbutons: (a) Reducton n percentage (of tota number of dsruptons) for resence; (b) Number of neghborhoods for eephants faures; (c) Number of neghborhoods for mce faures. dsruptons as shown n Fgure 3. However, the eephants faure neghborhoods amount to ony 2.7% of the tota faure neghborhoods. Ths mpes that the majorty of dsruptons are correated and nduced by eephant faures. Unke the dsruptons, recovery neghborhoods foow the 6/9 rue: The mce recovery neghborhoods of sze 2 or ess take up 92.7% of a recovery neghborhoods, amountng to 59.3% of recoveres, as shown n Fgure 4. Ths suggests that around 6% recoveres were uncorreated, and thus requred ndvdua restoratons. D. Resence: Identfyng Vunerabty We now obtan the emprca resence n terms of agng recovery usng rea data. ) Vunerabe Tme: In genera, threshod d can be determned through faure and recovery rates (see [9] for detas). Here n Fgure 2(b), the emprca recovery rate ceary shows that nfant recovery occurred aong wth the majorty of the faures, and s for the faures that asted ess than 2 hours. After the nfant recovery, there s a deay of 7.7 hours before agng recovery occurred. Therefore, the threshod s d = 2 hours. The resence s cacuated usng d = 2 n Equaton. As shown n Fgure 5, the resence decreased from the norma operatons aong wth the faure occurrences, and reached a maxmum reducton of 8.7% of tota dsruptons. The tme at the mnmum resence s 32 hours snce the nta faure occurrence. Ths was the most vunerabe tme when the nfant recovery aready ended and the agng recovery was yet to begn. The mnmum resence ndcates that 8.7% of tota 4 3 2

6 dsruptons needed at east another d = 2 hours to recover. Ths s consstent to the resence curve that t took up to 4 days for a dsruptons to recover. What f threshod d s chosen dfferenty? If d =s chosen, the nfant recovery woud be ncorrecty consdered as a part of non-resence. The resutng resence s thus overy pessmstc wth a maxmum reducton of 83.7% rather than 8.7% n Fgure 5. On the other hand, f d = 24 s chosen, the threshod fasey excudes parts of agng recovery, resutng n overy optmstc resence. Hence, dentfyng an optma threshod s mportant and sha be consdered n the future work. 2) Vunerabe Areas: The resence metrc can aso be used to dentfy vunerabe areas n a servce regon. Fgure 6(a) and (b) show the percentage reducton of the resence at two tme epoches: 4 hours before the andfa and the tme of the mnmum resence. The regons wth more than 5% and 6% reducton of resence appear as vunerabe areas for the two tme epoches. Fgure 6(a2) and (b2) show the number of eephant faure neghborhoods at the two tmes respectvey. The vunerabe areas concde wth the regons that have a arge number of eephant faure neghborhoods. Ths s consstent wth the fndng n Secton V-C that eephant faure neghborhoods contrbute to the majorty of the dsruptons and thus a sgnfcant deducton of the resence. Mce faure neghborhoods, however, are not couped wth the vunerabe areas. These fndngs pose nterestng drectons for more detaed study at the component eve. VI. CONCLUSION Ths work deveops a spata-tempora non-statonary random process to mode arge-scae dsruptons of power dstrbuton nduced by severe weather. The mode combnes nonstatonary faure- and recovery-random processes wth network structures. Dynamc faure- and recovery-neghborhoods are defned to characterze a topoogca network structure. The neghborhoods characterze correated faures and recoveres wthn networks. Dynamc dsrupton- and recovery-rates are used as smpe quanttes for faure- and recovery processes at both component- and subnetwork-eve. A resence metrc resutng from the mode then characterzes the evouton of faure and recovery. Rea data from an operatona network durng Hurrcane Ike s used to study the resence and the mpact of dynamc neghborhoods. An 8/2 rue emerges for faures, showng that hurrcane-nduced power-dsruptons are mosty correated due to network structures. In contrast, recoveres occur many n sma patches, and thus requred ndvdua restoratons. These fndngs revea dspartes between arge-scae faures and recoveres processes, suggestng more n-depth studes at the component eve for dentfyng vunerabtes and mprovng resence. [2] W. N. Bryan, Emergency Stuaton Reports: Hurrcane Sandy, Offce of Eectrcty Devery and Energy Reabty of U.S. Department of Energy, Hurrcane Sandy Stuaton Report 2, November 22. [3] G. A. Pagan and M. 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