EVERY year, seasonal hurricanes threaten coastal areas.

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1 1 Strategc Stockplng of Power System Supples for Dsaster Recovery Carleton Coffrn, Pascal Van Hentenryck, and Russell Bent Abstract Ths paper studes the Power System Stochastc Storage Problem (PSSSP), a novel applcaton n power restoraton whch conssts of decdng how to store power system components throughout a populated area to maxmze the amount of power served after dsaster restoraton. The paper proposes an exact mxed-nteger formulaton for the lnearzed DC power flow model and a general column-generaton approach. Both formulatons were evaluated expermentally on real-lfe benchmarks. The results show that the column-generaton algorthm produces near-optmal solutons quckly and produces orders of magntude speedups over the exact formulaton for large benchmarks. Moreover, both the exact and the column-generaton formulatons produce sgnfcant mprovements over a greedy approach and hence should yeld sgnfcant benefts n practce. I. BACKGROUND & MOTIVATION EVERY year, seasonal hurrcanes threaten coastal areas. The severty of hurrcane damage vares from year to year, but hurrcanes often cause power outages that have consderable mpacts on both qualty of lfe (e.g., crppled medcal servces) and economc welfare. Therefore consderable human and monetary resources are always spent to prepare for, and recover from, threatenng dsasters. At ths tme, polcy makers work together wth power system engneers to make the crtcal decsons relatng to how money and resources are allocated for preparaton and recovery of the power system. Unfortunately, due to the complex nature of electrcal power networks, these preparaton and recovery plans are lmted by the expertse and ntuton of power engneers. Ths paper s part of a larger effort to combne mathematcal optmzaton and dsaster-specfc predctons n order to plan and react to dsasters more effectvely. The overall goal s to explot the hgh-qualty predctons (e.g., ensembles of possble hurrcane tracks) produced by, say, the Natonal Hurrcane Center (NHC) of the Natonal Weather Servce and optmzaton technques to produce, n reasonable tme, robust plannng and response plans that sgnfcant outperform the practce n the feld. In partcular, ths paper consders a strategc plannng problem arsng n ths process: How to store power system repar components throughout a populated area to maxmze the amount of power served after dsaster restoraton. Ths Power System Stochastc Storage Problem (PSSSP) rases some fundamental computatonal ssues as t nvolves, not only dscrete storage decsons n a stochastc settng, but also the C. Coffrn and P. Van Hentenryck are wth the Department of Computer Scence, Brown Unversty, Provdence RI 02912, USA. R. Bent s wth Los Alamos Natonal Laboratory, Los Alamos NM modelng of the electrcal power network, whch s a complex physcal system. In partcular, smply evaluatng the benefts of a specfc storage confguraton for a sngle scenaro requres the solvng of a complex optmzaton model qute smlar to optmal transmsson swtchng models (e.g., [1]). The man contrbutons of ths paper s to present an exact mxed-nteger programmng (MIP) formulaton to the PSSSP (assumng a lnearzed DC power flow model) and a column-generaton algorthm that produces near-optmal solutons under tght tme constrants. The column-generaton algorthm s an teratve process whch alternates between generatng storage confguratons for each scenaro ndependently (the subproblem) and selectng the best confguratons for each scenaro globally (the Master problem). The two formulatons were evaluated on benchmarks produced by the Los Alamos Natonal Laboratory, usng the Unted States nfrastructure and dsaster scenaros generated by state-of-theart hurrcane smulaton tools smlar to those used by the Natonal Hurrcane Center. Expermental results ndcates that both formulatons provde sgnfcant benefts n recoverng from dsasters over greedy approaches (whch should already mprove over exstng practce n the feld). Moreover, the column-generaton algorthm s shown to scale well to largescale dsasters, producng orders of magntude mprovements over the exact MIP approach. Note also that our columngeneraton algorthm s ndependent of any specfc electrcal power smulaton tool. Ths s mportant snce the electrcal power ndustry has developed several tools for modelng the behavor (e.g. T2000, PSLF, Powerworld, PSS) and recognzes that there s not a sngle model for completely understandng the behavor of the electrcal power network. The rest of the paper s organzed as follows. Secton II postons ths research wth respect to related work n system recovery. Secton III presents a specfcaton of the PSSSP problem. Secton IV presents an exact MIP formulaton usng a lnearzed DC power flow model. Secton V presents the column-generaton algorthm for solvng PSSSPs. Secton VI presents greedy algorthms for PSSSPs amed at modelng current practce n the feld. Secton VII reports expermental results of the algorthms on some benchmark nstances to valdate the approach and Secton VIII concludes the paper. Note that, although the focus s on hurrcanes, the technques are largely dsaster ndependent and also apples to earthquakes. II. PREVIOUS WORK Power engneers have been studyng power system restoraton (PSR) for at least 30 years (see [2] for a comprehensve

2 2 collecton of work). The goal of PSR research s to fnd fast and relable ways to restore a power system to ts normal operatonal state after a blackout event. PSR research has consdered not only steady-state behavor, n whch the flow of electrcty s modeled by physcal laws, but also dynamc behavor whch consders transent states occurrng durng the process of modfyng the power system state (e.g., when energzng components). Indeed, these short, but extreme, states may cause unexpected falures whch must also be consdered carefully [3]. Moreover, power systems are comprsed of many dfferent components (e.g., generators, transformers, and capactors) whch have some flexblty n ther operatonal parameters but may be constraned arbtrarly. For example, generators often have a set of dscrete generaton levels and transformers have a contnuous but narrow range of tap ratos. Restoraton algorthms must take these nto account. The PSR research communty has recognzed that global optmzaton s often mpractcal for such complex non-lnear systems and adopted two man solutons strateges. The frst strategy s to use doman-expert knowledge (.e., the power engneer ntuton) to gude an ncomplete search of the soluton space. These ncomplete search methods nclude knowledgebased and expert systems [4], [5], [6], [7] and local search [8], [9]. The second strategy s to approxmate the power system wth a lnear model and to try solvng the approxmate problem optmally [10], [1], [11]. Some work also hybrdze both strateges by desgnng expert systems that solve a seres of approxmate problems optmally [12], [13]. It s mportant to emphasze, however, that most PSR work assumes that all network components are operatonal and need to be reactvated (e.g., [3], [6]). The PSR focus s thus to determne the best order of restoraton and the best reconfguraton of the system components. Ths paper consders the stochastc stockplng of replacement parts n dsaster plannng,.e., How to stockple powersystem components n order to restore as much power as possble after a dsaster. The stochastc stockplng problem s only concerned wth the steady-state behavor of the network, but t ntroduces both a stochastc element and an nventory component to the restoraton of a power system. To the best of our knowledge, ths s the frst PSR applcaton to consder strategc stockplng decsons for a collecton of dsaster predctons. III. POWER SYSTEM STOCHASTIC STORAGE The PSSSP conssts of choosng whch power system repar components (e.g., components for reparng lnes, generators, capactors, and transformers) to stockple before a dsaster strkes and how those components are allocated to repar the dsaster damages. A dsaster s specfed as a set of scenaros, each of whch characterzed by a probablty and a set of damaged components. In practce, the repostory storage constrants may preclude a full restoraton of the electrcal system after a dsaster. Therefore, the goal s to select the tems to be restored n order to maxmze the amount of power served n each dsaster scenaro pror to external resources beng brought n. The prmary outputs of a PSSSPs are: Gven: Power Network: P Network Item: N Item Type: t Component Types: t T Volume: v t Storage Locatons: l L Capacty: c l Scenaro Data: s S Scenaro Probablty: p s Damaged Items: D s N Maxmze: p s flow s s S Subject To: Storage Capacty Output: Whch components to store at each locaton. Whch tems to repar n each scenaro. Fg. 1. The Power System Stochastc Storage Problem Specfcaton. 1) The amount of components to stockple before a dsaster (frst-stage varables); 2) For each scenaro, whch network tems to repar (second-stage varables). Fgure 1 summarzes the entre problem, whch we now descrbe n detal. The power network s represented as a graph P = V, E where V s a set of the network nodes (.e., buses, generators, and loads) and E s a set of the network edges (.e., lnes and transformers). Any of these tems may be damaged as a result of a dsaster and hence we group them together as a set N = V E of network tems. Each network tem has a type t that ndcates whch component (e.g. parts for lnes, generators, or buses) s requred to repar ths tem. Each component type t T n the network has a storage volume v t. The components can be stored n a set L of warehouses, each warehouse l L beng specfed by a storage capacty c l. The dsaster damages are specfed by a set S of dfferent dsaster scenaros, each scenaro s havng a probablty p s of occurrng. Each scenaro s also specfes ts set D s N of damaged tems. The objectve of the PSSSP s to maxmze the expected power flow over all the damage scenaros, where flow s denotes the power flow n scenaro s. In PSSSPs, the power flow s defned to be the amount of watts reachng the load ponts. The algorthms assume that the power flow for a network P and a set of damaged tems D s can be calculated by some smulaton or optmzaton model. Note also that the decsons on where to store the components can be tackled n a second step. Once storage quanttes are determned the locaton aspect of the problem becomes determnstc and can be modeled as a mult-dmensonal knapsack [14]. IV. THE LINEAR POWER MODEL APPROACH PSSSPs can be modeled as two-stage stochastc mxednteger programmng model provded that the power flow constrants for each scenaro can be expressed as lnear

3 3 constrants. The lnearzed DC model, sutably enhanced to capture that some tems may be swtched on or off, s a natural choce n ths settng, snce t was found to be reasonably successful n optmal transmsson swtchng [1] and network nterdcton [11]. Fgure 2 presents a two-stage stochastc mxed-nteger programmng model for solvng PSSSPs optmally (when usng a lnearzed DC power model). The frststage decson varable x t denotes the number of stockpled components of type t. Each second-stage s assocated wth a scenaro s. Varable y s specfes whether tem s workng, whle z s specfes whether tem s operatonal. Auxlary varable flow s denotes the power flow for scenaro s, Ps l the real power flow of lne, Ps v the real power flow of node, and θ s the phase angle of bus. The objectve functon (1) maxmzes the expected power flow across all the dsaster scenaros. Constrant (2) ensures that the stockpled components do not exceed the storage capacty. Each scenaro can only repar damaged tems usng the stockpled components. Constrant (3) ensures that each scenaro s uses no more than the stockpled components of type t. There may be more damaged tems of a certan type than the number of stockpled component of that type and the optmzaton model needs to choose whch ones to repar, f any. Ths s captured, for each scenaro s, usng a lnearzed DC power flow model (4 13), whch extends optmal transmsson swtchng [1] to buses, generators, and loads snce they may be damaged n a dsaster. Moreover, snce we are nterested n a best-case power flow analyss, we assume that generaton and load can be dspatched and shed contnuously. Constrants (4 7) capture the operatonal state of the network and specfy whether tem s workng and/or operatonal n scenaro s. An tem s operatonal only f all buses t s connected to are also operatonal. Constrant (4) specfes that all undamaged nodes and lnes are workng. Constrant (5 7) specfy whch buses (5), load and generators (6), and lnes (7) are operatonal. Constrant (8) computes the total power flow of scenaro s n terms of varables Ps v. The conservaton of energy s modeled n constrant (9). Constrant (10 11) specfy the bounds on the power produced/consumed/transmtted by generators, loads, and lnes. Observe that, when these tems are not operatonal, no power s consumed, produced, or transmtted. When a lne s non-operatonal, the effects of Krchhoff s laws must be gnored, whch s captured n the tradtonal power equatons through a bg M transformaton (12 13). In ths settng, M can be chosen as B π 3. Note also that the logcal constrants can be easly lnearzed n terms of the 0/1 varables. Ths MIP approach s appealng snce t solves PSSSPs optmally for a lnearzed DC model of power flow. In partcular, t provdes a sound bass to compare other approaches. However, t does not scale smoothly wth the sze of the dsasters and may be prohbtve computatonally n real-lfe stuatons. To remedy ths lmtaton, the rest of ths paper studes a columngeneraton approach. V. A COLUMN-GENERATION APPROACH When the optmal values of the frst-stage varables are known, the PSSSP reduces to solvng a restoraton problem for Let: D ts = { D s : t = t} V b = { N : t = bus} - the set of network busses = the set of generators connected to bus V l = the set of loads connected to bus L = the set of network lnes L = the from bus of lne L + = the to bus of lne LO b = the set of extng lnes from bus b LI b = the set of enterng lnes from bus b B = susceptance of lne ˆP l = transmsson capacty of lne = maxmum capacty or load of node V g ˆP v Varables: x t N - number of stockpled tems of type t y s {0, 1} - tem s workng n scenaro s z s {0, 1} - tem s operatonal n scenaro s flow s R + - served power for scenaro s Ps l ( ˆP l, ˆP l ) - power flow on lne Ps v (0, ˆP v ) - power flow on node θ s ( π, π ) phase angle on bus Maxmze: p s flow s (1) s S Subject To: v t x t c l (2) t T l y s x t t T, s S (3) D ts y s = 1 / D s s S (4) z s = y s V b s S (5) z s = y s y js j V b, V g j V j l, s S (6) z s = y s y L + s L L, s S (7) s flow s = Pjs v s S (8) j V l V b j V l Pjs v = Pjs v + Pjs l Pjs l V b, s (9) j LI j LO j V g 0 Ps v ˆP v z s V g j V j l, s S (10) ˆP s l z s Ps l ˆP s l z s L, s (11) Ps l B (θ L + s L s) + M ( zs) L, s S (12) Ps l B (θ L + s L s) M ( zs) L, s S (13) Fg. 2. The MIP Model for Power System Stochastc Storage Problems. each scenaro s,.e., to maxmze the power flow for scenaro s under the stored resources specfed by the frst-stage varables. The column-generaton approach takes the dual approach: It ams at combnng feasble solutons of each scenaro to obtan hgh-qualty values for the frst-stage varables. In ths settng, a confguraton s a tuple w = w 1,..., w k (T = {1,..., k}) where w t specfes the number of tems of type t beng stockpled. A confguraton s feasble f t satsfes the storage constrants,.e., v t w t c l. t T l L For each scenaro s, the optmal power flow of a feasble confguraton w s denoted by flow ws. Once a set of feasble confguratons s avalable, a mxed-nteger program (the Master problem) selects a set of confguratons, one per scenaro,

4 4 Let: W - the set of confguratons w t - the amount of components of type t n w W flow ws - the served power for w n scenaro s Varables: x t N - number of stockpled tems of type t y ws {0, 1} - 1 f confguraton w s used n s flow s R + - power flow for scenaro s Maxmze: p s flow s (1) s S Subject To: v t x t c l (2) t T l w t y ws x t s, t (3) w W y ws = 1 s (4) w W flow ws y ws = flow s s (5) Fg. 3. w W The The Master Problem for the PSSSP. maxmzng the expected power flow across all scenaros. Fgure 3 presents the MIP model. In the model, the objectve (1) specfes that the goal s to maxmze the expected flow. Constrant (2) enforces the storage requrements, whle constrant (3) lnks the number of components x t of type t used by scenaro s wth the confguraton varables y ws. Constrant (4) specfes that each scenaro uses exactly one confguraton. Constrant (5) computes the power flow of scenaro s usng the confguraton varables y ws. It s obvously mpractcal to generate all confguratons for all scenaros. As a result, we follow a column-generaton approach n whch confguratons are generated on demand to mprove the objectve of the Master problem. Recall that our goal s twofold: Frst, we am at desgnng an approach whch s largely ndependent of the power flow smulaton or optmzaton model. Second, we am at producng an optmzaton model whch scales to large dsasters. For these reasons, we generate confguratons based on technques nspred by onlne stochastc combnatoral optmzaton [15], [16]. A. The Column-Generaton Subproblem In our column-generaton approach, the confguratons are generated for each scenaro ndependently and n a systematc fashon. Fgure 4 descrbes a generc optmzaton model for generatng a confguraton. In the model, the storage and power flow constrants are abstracted for generalty. The storage constrants contan at least the storage constrants of the PSSSP but generally adds addtonal constrants to generate nterestng confguratons for the Master problem. The power flow constrants are abstracted to make the approach ndependent of the power flow model. The optmzaton model features a lexcographc objectve (1), seekng frst to maxmze the power flow and then to mnmze the number of repared components. The storage constrants (2) are expressed n terms Let: s - a scenaro T t = { N : t = t} - the set of nodes of type t Varables: r {0, 1} - 1 f tem s repared flow R + - served power Maxmze: (flow, r ) (1) Subject To: storageconstrant({t t } t T, {r } N ) (2) flowconstrants(p, D s, {r } N, flow) (3) Fg. 4. The Generc Confguraton-Generaton Model For a Scenaro. of the decson varables r that ndcates whether tem s repared: They wll be descrbed shortly. Constrant (3) encapsulates the power flow model whch computes the flow from the values of the decson varables. The column-generaton algorthm generates two fundamental types of confguratons for a gven scenaro. Each type takes nto account nformaton from the other scenaros and nstantates the generc model n Fgure 4 wth specfc storage constrants. a) Upward Confguratons: The ntuton behnd upward confguratons s as follows. Some storage decsons comng from other scenaros are fxed by a confguraton w and the goal s to generate as best a confguraton as possble for scenaro s gven these decsons,.e., a confguraton for scenaro s that maxmzes ts power flow gven the fact that the repars must nclude at least w t components of type t. More precsely, upwardconfguraton(w,s) denotes the soluton to the model n Fgure 4 where the storage constrant becomes v t max(w t, r ) c l. t T T t l b) Downward Confguratons: The ntuton behnd downward confguratons s as follows. Our goal s to gve an opportunty for a confguraton w comng from another scenaro to be selected n the Master problem, whle ensurng that all other scenaros generate a mutually compatble confguraton. As a result, we seek to maxmze the power flow for scenaro s, whle not exceedng the storage requrements of w. More precsely, downwardconfguraton(w,s) denotes the soluton to the model n Fgure 4 where the storage constrant becomes t T : T t r w t. Two specal cases of upward and downward confguratons are mportant for ntalzng the column-generaton process: the clarvoyant and the no-repar confguratons. c) Clarvoyant Confguratons: The clarvoyant confguraton for a scenaro s s smply the upward confguraton wth no storage constrant mposed: clarvoyant(s) = upwardconfguraton( 0,..., 0, s). Scenaros for whch ther clarvoyant confguraton s selected n the Master problem cannot be mproved.

5 5 d) No-Repar Confguratons: The no-repar confguraton for a scenaro s s smply the downward confguraton wth no storage avalablty,.e., no-repar(s) = downwardconfguraton( 0,..., 0, s). The no-repar confguraton guarantees that each scenaro can select at least one confguraton n the Master problem. B. The Column-Generaton Algorthm Fgure 5 presents the complete column-generaton algorthm. Lnes 1 4 descrbe the ntalzaton process, whle lnes 7 12 specfy how to generate new confguratons. The overall algorthm termnates when the newly generated confguratons do not mprove the qualty of the Master problem. e) Intalzaton: The ntalzaton step generates the ntal set of confguratons. These contan the clarvoyant and the no-repar confguratons for each scenaro, as well as the downward confguratons obtaned from the expected clarvoyant,.e., the confguraton w e whose element t s defned by w e t = s S p s clarvoyant(s). f) The Colum-Generaton Process: At each teraton, the algorthm solves the Master problem. If the soluton has not mproved over the prevous teraton, the algorthm completes and returns ts confguraton,.e., the values of the decson varables x t descrbng how many elements of component type t must be stockpled. Otherwse, the algorthm consders each scenaro s n solaton. The key dea s to solve the Master problem wthout scenaro s, whch we call a restrcted Master problem, and to derve new confguratons from ts soluton w s. In partcular, for each scenaro s, the column-generaton algorthm generates 1) one upward confguraton for s (lne 9); 2) one downward confguraton for every other scenaro n S \ {s} (lne 12). The upward confguraton for s s smply the best possble confguraton gven the decsons n the restrcted Master problem,.e., upwardconfguraton(w s, s) The downward confguratons for the other scenaros are obtaned by selectng an exstng confguraton w +s for s whch, f added to the restrcted Master soluton, would volate the storage constrant: Downward confguratons of the form downwardconfguraton(w +s, j) are then computed for each scenaro j S\{s} n order to gve the Master an opportunty to select w +s. The confguraton w +s ams at beng desrable for s, whle takng nto account the requrement of the other scenaros. Intally, w +s s the clarvoyant soluton. In general, w +s s the confguraton n W whch, when scaled to satsfy the storage constrants, CONGIGURATIONGENERATION() 1 W {no-repar(s) s S} 2 W W clarvoyant(s) s S} 3 w e expectedclarvoyant(s) 4 W W downwardconfguraton(w e, s) s S} 5 whle The master objectve s ncreasng 6 do w m Master(S) 7 for s S 8 do w s Master(S \ {s}) 9 W W upwardconfguraton(w s, s) 10 w +s selectconfguraton(w s, s) 11 for j S \ {s} 12 do W W downwardconfguraton(w +s, j) 13 return w m Fg. 5. The Column-Generaton Algorthm for the PSSSP. maxmzes the power flow for scenaro s,.e., max w W subject to p flow w t T : w t > wt s t T : p w t wt s 0 p 1 g) Smulaton-Independent Optmzaton: The columngeneraton algorthm only reles on the power flow model n the subproblem of Fgure 4. Ths s essentally an optmal transmsson swtchng model wth lmts on the component types that can be repared. It can be solved n varous ways, makng the overall approach ndependent of the power flow model. VI. A GREEDY STORAGE MODEL Ths secton presents a basc greedy storage model whch emulates standard practce n storage procedures and provdes a baselne for evaluatng our optmzaton algorthms. However, to the best of our knowledge, the storage models used n practce are rather ad-hoc or based on past experence and the models presented n ths secton probably mprove exstng procedures. In fact, t s well-known that power companes often rely on spare parts from neghborng regons n postdsaster stuatons. The greedy storage algorthm generates a storage confguraton w h by computng frst a dstrbuton of the component types and then fllng the avalable storage capacty wth components to match ths dstrbuton. Once a scenaro s s revealed, the qualty of greedy storage can be evaluated by computng a downward confguraton downwardconfguraton(w h, s). The dstrbuton used to produce w h s based on the number of occurrences of the component types n the undamaged network. Ths metrc s meanngful because t models the assumpton that every component type s equally lkely to be damaged n a dsaster. The computaton for a dstrbuton

6 6 Benchmark N S max s S ( D s ) BM BM BM BM TABLE I FEATURES OF THE PSSSP BENCHMARKS. P r proceeds as follows. When all of the components have a unform sze v, the quantty of component type s ( l L c l P r())/v. When the component types have dfferent szes, the storage confguraton s the soluton of the optmzaton problem mn t T subject to wh t T wh P r() wt h v t t T l L 0 w h. c l VII. BENCHMARKS & RESULTS Ths secton reports the expermental results of the proposed models. It starts by descrbng the benchmarks and the algorthms. It then presents the qualty results and the effcency results. It concludes by reportng a varety of statstcs on the behavor of the column-generaton algorthms. A. Benchmarks The benchmarks were produced by Los Alamos Natonal Laboratory and are based on the electrcal nfrastructure of the Unted States. The dsaster scenaros were generated by state-of-the-art hurrcane smulaton tools smlar to those used by the Natonal Hurrcane Center [17], [18]. Ther szes are presented n Table I whch gves the number of network tems, the number of scenaros, and the sze of the largest damage produced by the scenaros. For nstance, BM5 consders a network wth 1789 tems and 4 dsaster scenaros. The worst dsaster damages 255 tems n the network. Each of these benchmarks s evaluated for a large varety of storage capactes (.e., dfferent values of c l ). Ths makes t possble to evaluate the behavor of the algorthms under a wde varety of crcumstances, as well as to study the tradeoff between storage (or budget) and the qualty of the restoraton. Overall, more than 70 confguratons of each algorthm were studed n these experments. B. Implementaton of the Algorthms The optmzaton algorthms were mplemented n the COMET system [19], [20], [21] and the experments were run on Intel Xeon CPU 2.80GHz machnes runnng 64-bt Lnux Deban. The experments use the standard lnearzed DC power flow equatons for the subproblems. To quantfy ts benefts compared to the practce n the feld, the column-generaton algorthm s compared to the greedy storage procedures presented earler. The global qualty of the column-generaton % Power Restored (scale MW) Stochastc Storage Power Flow Storage Capacty Clarvoyant Optmal MIP* Column Generaton Greedy Fg. 6. Soluton Qualty for the PSSSP Algorthms on Benchmark 4. algorthm s evaluated by comparson to the optmal MIP model presented n Secton IV and to the clarvoyant soluton,.e., the soluton obtaned when the actual dsaster scenaro s known n advance. Benchmarks 1 and 3 are small enough that the MIP models can be solved to optmalty. However, benchmarks 4 and 5 present large damage scenaros wth over 100 damaged components and optmal solutons to the MIP models may not be obtaned n a reasonable amount of tme. The followng tme lmts are then mposed: 18 hours of the global MIP, 2 hours for the evaluaton of each greedy and clarvoyant scenaro, and 15 mnutes for column-generaton sub-problems. C. Qualty of the Algorthms Fgure 6 depcts the qualty results for Benchmark 4,.e., the percentage of the backout restored by the varous algorthms for a gven storage capacty. The fgure shows the results for the greedy procedure, the column-generaton algorthm, the MIP model, and the expected clarvoyant soluton. The results ndcate that the MIP model and the columngeneraton algorthm produce sgnfcant mprovements over the greedy approach, reducng the blackout by almost 20% n the best case. The mprovements are especally sgnfcant for low storage capactes, whch seems to be the case n practce as mentoned earler. Remarkably, the column-generaton algorthm outperforms the qualty of the MIP model, whch always reaches ts tme lmt. On ths set of benchmarks, the MIP model and the column-generaton algorthm are close to the clarvoyant soluton, ndcatng that the stochastcty s reasonable n PSSSPs. Ths mples that our algorthms produces hgh-qualty solutons for the scenaros. Space constrants prevent us from ncludng smlar graphs for all benchmark classes. Instead, we present an aggregaton of these results for each benchmark class, averagng the qualty

7 7 Benchmark Clarvoyant MIP Model Column Generaton Greedy BM1 100% 99.1% 98.5% 73.4% BM3 100% 97.4% 96.6% 71.8% BM4 100% 91.4% 97.5% 88.3% BM5 100% 77.5% 85.4% 69.3% TABLE II EXPERIMENTAL RESULTS ON AGGREGATED SOLUTION QUALITY. Benchmark Clarvoyant MIP Model Column Generaton Greedy BM1 100% 97.9% 96.3% 66.0% BM3 100% 92.8% 92.8% 21.9% BM4 100% 62.7% 75.0% 34.2% BM5 100% 70.7% 81.5% 11.4% TABLE III AGGREGATED SOLUTION QUALITY ON SMALL STORAGE CAPCITIES. over 20 storage capactes for each benchmark. The results are presented n Table II, whch depcts the average relatve gap of each algorthm from the clarvoyant soluton. The columngeneraton approach brngs substantal benefts over the greedy approach n smaller benchmarks and over both the greedy and MIP approach on larger ones, demonstratng ts scalablty. Table III gves another perspectve as t only aggregates results for small storage capactes whch often correspond to reallfe stuatons. The benefts of the column-generaton approach over the greedy algorthm are dramatc and the gap over the MIP s even more sgnfcant on the larger benchmarks. seconds (log scale) Stochastc Storage Runtme Storage Capacty Clarvoyant Optmal MIP* Column Generaton Greedy Fg. 7. Expermental Results for the PSSSP Algorthm on Benchmark 4. Benchmark Clarvoyant MIP Model Column Generaton Greedy BM BM BM BM TABLE IV PSSSP BENCHMARK RUNTIME (SECONDS) D. Performance of the Algorthms Fgure 7 depcts the performance results for Benchmark 4 under 20 dfferent storage capactes. It depcts the runtme of the MIP Model and the column-generaton algorthm n seconds, usng a log scale gven the sgnfcant performance dfference between the algorthms. The results show substantal mprovements n performance for the column-generaton algorthm over the MIP model. Ths s especally the case for low to medum storage capactes, whch typcally model the realty n the feld. For these storage confguratons, the column-generaton algorthm runs from 2 to 100 tmes faster than the MIP model (despte ts tme lmt) and s about 3 tmes faster n average. In summary, on large nstances, the column-generaton approach mproves both the qualty and performance of the MIP models. Table IV presents the average runtme results for all benchmarks. It also gves the tme to evaluate the greedy and expected clarvoyant problems as a bass for comparson. These results are qute nterestng n the sense that the MIP model behaves very well for reasonably small dsasters, but does not scale well for larger ones. Snce our goal s to tackle dsaster plannng and restoraton at the state level, they pont out to a fundamental lmtaton of a pure MIP approach. It s also nterestng to observe that the column-generaton algorthm takes about 4 tmes as much as tme as evaluatng the greedy power flow. Ths s qute remarkable and ndcates that the column-generaton s lkely to scale well to even larger problem szes. Overall, these results ndcate that the column-generaton algorthm produces nearoptmal solutons under tght tme constrants and represents an appeallng approach to PSSSPs. E. Behavor of the Column-Generaton Algorthm It s useful to study the behavor of the column-generaton algorthm n order to understand ts performance. Table VI reports, n average, the number of confguratons generated, as well as the number of downward and upward confguratons n the fnal soluton. The results ndcate that the algorthm only needs a small number of confguratons to produce a hghqualty soluton. Recall that there are 18 scenaros n BM3 and BM4, whch means that the algorthm generates n average less than 7 confguratons per scenaro. The types of confguratons are relatvely well balanced, although there are clearly more upward confguratons n average. Ths s partly caused by the varance n the scenaro damages. As the storage capacty ncrease more of the small damage scenaros can select ther clarvoyant soluton. Fgure 8 detals these results for BM4. VIII. CONCLUSION Ths paper studed a novel problem n power system restoraton: the Power System Stochastc Storage Problem (PSSSP). The objectve n PSSSPs s to decde how to stockple components n order to recover from blackouts as best as possble after a dsaster. PSSSPs are complex stochastc optmzaton problems, combnng power flow smulators, dscrete storage decsons, dscrete repar decsons gven the storage decsons, and a collecton of scenaros descrbng the potental effects of the dsaster. The paper proposed an exact mxed-nteger formulaton and a column-generaton approach. The columngeneraton subproblem generates confguratons from each scenaro ndependently, takng nto account storage decsons

8 8 Benchmark Clarvoyant MIP Model Column Generaton Greedy BM BM BM BM TABLE V BENCHMARK RUNTIME ON SMALL STORAGE CAPACITIES (SECONDS) Benchmark Confguratons Downward Confg. Upward Confg. BM BM BM BM TABLE VI THE BEHAVIOR OF THE COLUMN GENERATION FOR PSSSPS. column count Stochastc Storage Runtme Downward Upward for other scenaros. A subset of these storage confguratons are then selected n the Master problem, to produce the global storage decsons. The algorthms were evaluated on benchmarks produced by the Los Alamos Natonal Laboratory, usng the electrcal power nfrastructure of the Unted States. The dsaster scenaros were generated by state-ofthe-art hurrcane smulaton tools smlar to those used by the Natonal Hurrcane Center. Expermental results show that the column-generaton algorthm produces near-optmal solutons and produces orders of magntude speedups over the exact formulaton for large benchmarks. Moreover, both the exact and the column-generaton formulatons produce sgnfcant mprovements over greedy approaches and hence should yeld sgnfcant benefts n practce. The results also seem to ndcate that the column-generaton algorthm should ncely scale to even larger dsasters, gven the small number of confguratons necessary to reach a near-optmal soluton. As a result, the column-generaton algorthm should provde a practcal tool for decson makers n the strategc plannng phase just before a dsaster strkes. REFERENCES [1] E. Fsher, R. O Nell, and M. Ferrs, Optmal transmsson swtchng, Power Systems, IEEE Transactons on, vol. 23, no. 3, pp , aug [2] M. Adb, Power System Restoraton(Methodologes & Implementaton Strateges), [3] M. Adb and L. Fnk, Power system restoraton plannng, Power Systems, IEEE Transactons on, vol. 9, no. 1, pp , feb [4] T. Sakaguch and K. Matsumoto, Development of a knowledge based system for power system restoraton, Power Apparatus and Systems, IEEE Transactons on, vol. 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Upfal, Onlne Stochastc Optmzaton under Tme Constrants, Annals of Operatons Research (Specal Issue on Stochastc Programmng), vol. 177, pp , [17] Fema hazus overvew, [18] D. A. Reed, Electrc utlty dstrbuton analyss for extreme wnds, Journal of Wnd Engneerng and Industral Aerodynamcs, vol. 96, no. 1, pp , [19] Comet 2.1 user manual, dynadec nc [20] L. Mchel, A. See, and P. Van Hentenryck, Transparent Parallelzaton of Constrant Programmng, INFORMS Journal on Computng, vol. 21, no. 3, pp , [21] P. Van Hentenryck, Constrant-Based Local Search. Cambrdge, Mass.: The MIT Press, Carleton Coffrn receved a B.Sc. n Computer Scence and a B.F.A. n Theatrcal Desgn from the Unversty of Connectcut, Storrs, CT and a M.S. from Brown Unversty, Provdence, RI. He s currently a Ph.D. canddate n the department of computer scence at Brown Unversty. Pascal Van Hentenryck s a Professor of Computer Scence at Brown Unversty. He s a fellow of the Assocaton for the Advancement of Artfcal Intellgence, the recpent of the 2002 ICS INFORMS award, the 2006 ACP Award, a honorary degree from the Unversty of Louvan, and the Phlp J. Bray award for teachng excellence. He s the author of fve MIT Press books and numerous publcatons n the premer journals n artfcal ntellgence, numercal analyss, operatons research, and programmng languages. Most of ths research n optmzaton software systems has been commercalzed and s wdely used n academa and ndustry. Russell Bent receved hs PhD n Computer Scence from Brown Unversty n 2005 and s currently a staff scentst at Los Alamos Natonal Laboratory n the energy and nfrastructure analyss group servng as a capablty research and development task leader. Russell s responsble for the development of LogSms, a project for resource management, plannng, and dstrbuton for dsasters. He s also co-prncpal nvestgator for a project optmzaton and control theory for Smart Grd technologes. Russell s publcatons nclude determnstc optmzaton, optmzaton under uncertanty, nfrastructure modelng and smulaton, constrant programmng, algorthms, and smulaton. In the last fve years, Russell has publshed 1 book and over 25 artcles n peer revewed journals and conferences n artfcal ntellgence and operatons research.