Reducing the Computational Effort of Stochastic Multi-Period DC Optimal Power Flow with Storage

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1 Redung the Computtonl Effort of Stohst Mult-Perod DC Optml Power Flow wth Storge Olver Mégel Görn Andersson Power Systems Lbortory ETH Zürh Zürh, Swtzerlnd {megel, Johnn L. Mtheu Eletrl nd Computer Engneerng Unversty of Mhgn Ann Arbor, USA Abstrt Due to the nrese of ntermttent renewble energy soures, t s beomng more mportnt to onsder renewble generton forest error when solvng the optml power flow (OPF) problem. The stohst OPF, whh uses multple forest senros, generlly leds to lower ost ompred to the stndrd determnst OPF, whh uses sngle forest. However, the stohst OPF s omputtonlly more demndng thn the determnst OPF. Both ost svngs nd omputton tmes further nrese when storge unts or rmp-onstrned genertors re nluded, s they requre solvng mult-perod OPF problem. Our ontrbuton s hybrd method pprohng the ost performne of the stohst OPF whle mntnng omputtonl burden lose to the determnst OPF. The method ombnes elements from both the stohst nd the determnst OPF, nd reles on Benders Cuts to nterfe them. Usng reedng horzon pproh over one yer, we fnd tht, bsed on eleven test ses, one verson of our hybrd method leds to t lest 70% of the ost mprovement of the stohst OPF, whle the omputton tme nrese s t most 40% of the stohst OPF tme nrese. Furthermore, the omputtonl dvntge of our method nreses wth the system sze. Two dfferent versons of the method llow fvorng of ether the omputtonl mprovement or the ost mprovement. We lso dentfy dretons for further mprovement. Fnlly, our method n be used for more generl problems n whh one wshes to ombne two models wth dfferent levels of omplexty. Index Terms Optml Power Flow, Stohst, Mult-Perod, Forest Senros, Storge, Benders Cuts. I. INTRODUCTION Due to nresng penetrtons of photovolts (PV) nd wnd, t s beomng more mportnt to onsder renewble generton forest error when solvng the optml power flow (OPF) problem. If renewble generton forest senros re vlble, one n solve the stohst OPF. When solvng mult-perod OPF wth nter-temporl onstrnts (due to storge dynms or rmp lmts) to dspth system subjet to forest error, the stohst OPF generlly leds to lower ost thn the determnst OPF, whh onsders only one renewble generton forest profle. Ths s beuse onsderng multple forest senros better represents unertnty, nd leds to more robust shedules tht requre lower use of pekng unts to ompenste the forest error [1]. However, the stohst OPF s omputtonlly more demndng thn the determnst OPF. Our gol s to develop method for solvng the OPF problem nludng storge tht strkes blne between the superor performne of the stohst OPF nd the lower omputtonl need of the determnst OPF. In dong so, we hve developed method tht pples more generlly to problems n whh we would lke to strke blne between the superor performne of one method nd the lower omputtonl need of nother method. We refer to the Mult-Perod OPF wth Storge s the MPS OPF. We fous on onvex (hene ontnuous) DC MPS OPF problems for systems tht nlude sgnfnt PV produton, s well s storge unts szed to ompenste the dy/nght PV yle. Hene, n optmzton horzon of two to fve dys s generlly enough to heve lmost optml dspth (for wnd domnted systems wth smlr storge ptes, the reommended horzon mght be slghtly dfferent). Ths s unlke systems wth lrge pumped hydro unts, whh requre horzon of one yer, sne the storge s lso used for the summer/wnter yle. However, solvng the problem for hndful of dys s lredy omputtonlly ntensve for lrge systems when usng multple forest senros. We solve the problem usng reedng horzon pproh, smlrly to [2]. We defne t s the tme step durton, here one hour, nd N s the number of tme steps of the optmzton horzon. The horzon (N t) orresponds to few dys. At eh tme step, we observe the PV output for the urrent tme step nd the forest for the rest of the optmzton horzon; solve the MPS OPF to shedule the onventonl plnts, storge unts, nd PV urtlment for the whole horzon; pply the dspth results orrespondng to the frst optmzton tme step; move forwrd n tme by one tme step nd repet the proess. Ths pproh llows us to use updted PV forest s soon s they beome vlble. Our settng s most smlr to U.S. hourly energy mrkets; we do not model dy-hed or ntrhour mrkets. A. Lterture Revew When solvng the MPS OPF problem, the dspth of the genertors nd storge unts t the frst tme step s ffeted by N. The lrger N the better the dspth, s the solver n foresee the onsequene of the desons further nto the future. Ths ws reognzed more thn 30 yers go [3] n

2 Opertng ost [10 6 Totl omputton tme [h], T Optmzton horzon length [tme steps], N Fgure 1. Opertng ost nd totl smulton tme for 1354-bus system wth storge, when vryng the optmzton horzon of the stohst OPF problem. more generl settng, nd nmed end effets. Fgure 1 shows how the opertng ost Z of power system depends on N, when smultng over ten dys nd usng the reedng horzon pproh. Ths fgure lso shows tht lrger N leds to hgher totl omputton tme, T, orrespondng to the CPU tme needed by the solver. The most usul wys to ddress the end effet n MPS OPF problems re to ) hoose N lrge enough [4]; ) ssume n d-ho vlue of the energy n the storge t the end of the horzon [5],.e. termnl vlue; or ) enfore tht the termnl storge energy level s the sme s the ntl energy level [6]. The frst opton n be omputtonlly ntensve (espelly when usng multple forest senros), nd the seond one s not esy to mplement sne the seleton of the d-ho termnl vlue s not trvl problem. The optml vlue should be the mrgnl vlue of the stored energy for the future tme steps. For hydro shedulng, ths vlue s often obtned by solvng longer term shedulng problem [7]. Fgure 2 shows, for smple system, the mrgnl vlue of stored energy (.e. the senstvty of the opertng ost to the storge level) t the frst optmzton tme step for dfferent ntl storge levels nd over one yer. We n see tht the vlue vres sgnfntly over the yer, refletng the ft tht dfferent plnts n be the mrgnl one. We lso see tht the vlue depends on the ntl storge level, ndtng tht termnl funton (of the energy level) would be more pproprte thn termnl vlue. Of ourse, omputng ths termnl funton would requre hvng n urte forest for the tme steps followng N. If these dt re vlble, one ould lso ntegrte them n the optmzton problem by extendng the horzon, t the ost of more omputtonlly demndng problem. The thrd opton bove s not optml f the optmzton horzon does not orrespond to the typl storge yle, for nstne yerly yle for lrge hydro storge. The uthors of [8] ddress the end effet whle keepng the omputtonl effort low by usng dfferent tme step durtons: fne tme grnulrty loser to delvery, nd orser grnulrty further nto the future. Whle ths method s useful when onsderng storge unts wth lrge energy/power rto, t s not prtulrly useful here, sne the opposte s true for the problems we fous on. B. Contrbuton Our method s bsed on the frst two solutons: We selet N lrge enough so tht the end effet n be negleted, but we seprte N nto two prts, smplfy the seond prt formulton, nd use nformton from the soluton of the Mrgnl vlue of ntl storge level: 10% ntl storge level: 50% ntl storge level: 90% Tme [hours] Fgure 2. Mrgnl vlue of stored energy t the frst optmzton tme step for dfferent ntl storge levels nd over one yer. seond prt to buld termnl ost funton for the frst prt. Hene, the seond prt problem nfluenes the frst prt problem smlrly to how long-term hydro shedulng problems nfluene short-term hydro shedulng problems. Our method dffers from [9] n tht we fous on smplfyng the seond prt problem nd tht we do not terte between the frst nd the seond prts. Our ontrbuton s twofold: Frst, for the problem presented here, we develop method tht leds to lower opertng osts thn the determnst MPS OPF, whle beng less omputtonlly demndng thn the stohst MPS OPF. Hene, wth the sme omputtonl power vlble, our method ould lso onsder more tme steps thn the stohst MPS OPF. The frst prt of the optmzton horzon s solved usng senro forests, whle the seond prt s solved usng sngle forest. We use Benders Cuts to lnk the two prts together. Hvng dfferent number of senros for eh prt s just one possble pplton. Therefore, the seond nd more generl ontrbuton s tht our method llows one to lnk hgher omplexty/ury models pturng tme steps loser to delvery to smplfed models pturng tme steps further hed. Hene, t n be used for dfferent prs of omplex ner-term / smplfed longer-term models, nludng prs tht nnot be redly ombned n sngle optmzton problem. Seton II defnes the problem nd provdes the determnst nd stohst MPS OPF formultons. Seton III desrbes our hybrd method, Seton IV detls our se studes, nd Seton V ther results. Seton VI onludes. II. PROBLEM DEFINITION Our fous s on mngng PV forest error, nd so we mke the ssumpton tht perfet lod forests re vlble over the N tme steps. We ssume tht PV forest senros re vlble nd, for smplty, we onsder them equprobble. We buld the sngle forest profle used n the determnst MPS OPF by vergng the forest senros, s n [1]. Followng the dsusson n Seton I-A, we hoose N lrge enough so tht we n omt termnl ost funton for the energy n the storge unts t the end of the optmzton horzon. Our setup orresponds to wt-nd-see problem [10] sne we mke dspth desons n rel tme, nd only fter knowng the relzton of the stohst nput (.e., the PV generton) for the urrent tme step. Sne forest error dereses s the look-hed tme dereses, t s resonble

3 to ssume perfet rel-tme forest. Rel power systems re operted by solvng both here-nd-now problems (suh s dy-hed shedulng) nd wt-nd-see problems (suh s dspth of regulton serves) [1]. We restrt ourselves to systems whh hve suffent rel-tme flexblty, for exmple, from pekng unts or PV urtlment. Hene, we onsder problems tht re lwys fesble. For smplty, throughout ths pper, we wrte equtons ssumng onventonl genertor, PV genertor, storge unt, nd lod re onneted t eh bus. For ny spef bus, vrbles orrespondng to resoures tht do not exst t tht bus re set to zero. The ndex k K = [1,...,N] denotes the tme steps of the optmzton horzon, I the buses, l L the lnes, nd s S the dfferent forest senros. We use for the set rdnlty. Vrbles n bold typefe represent olumn vetors over I. A onventonl genertor s defned by ts mxmum power P, ts symmetrl rmpng lmt R, nd ts lner nd qudrt opertng ost oeffents g,1, g,2. A storge unt s defned by ts power lmt A (on both hrge nd dshrge), ts energy lmt E, ts hrgng nd dshrgng effenes η,ηd, nd ts lner nd qudrt opertng ost oeffents,1,,2, whh re ppled to both hrgng nd dshrgng power. These oeffents represent for nstne degrdton proess. The onventonl genertor ost funton nd the storge ost funton per tme step nd for eh unt re f g (g k)= ( g k g,1 +(g k ) 2 g,2 f ( k, d k) = ( ( k + d k ) t, (1) ) 2 ),2 t, (2) ),1 + ( k+ d k where g k s the power produton of the onventonl genertor t bus t tme step k, nd k, d k the hrgng nd dshrgng power of the storge unt t bus t tme step k. We use YALMIP [11] to represent our set of equtons nd onstrnts, nd CPLEX to solve the optmzton problems. We next ntrodue the determnst nd stohst MPS OPF formultons, nd our hybrd method s desrbed n Seton III. In the reedng horzon pproh used here, eh of the three methods s used to obtn the dspth t the frst tme step of the optmzton horzon. A. Determnst Mult-Perod Optml Power Flow Wth Storge The determnst MPS OPF problem mnmzes f g (g k)+f ( k, d k) subjet to: k K I (3) 0 g k P,k (4) R g k g k 1 R,k (5) 0 k, d k A,k (6) 0 e k E,k (7) e k+1 =e k + ( kη d k/η d ) t,k (8) 0 h pv k ppv k,k (9) p nj k =g k+p pv k hpv k +d k k d k,k (10) p nj k =Bθ k k (11) Pl b θlk lne /x l Pl b l,k (12) θ 1,k =0 k (13) g 0 =g prev (14) e 1 =e n (15) where (4) defnes the lmts on g k, (5) the rmpng lmts on g k, (6) the lmts on k nd d k, (7) the lmts on the storge unts energy level e k, (8) the dynml equton of the storge unts, (9) the lmts on PV urtlment where h pv k s the urtlment nd ppv k the relzed or expeted PV generton, nd (10) the nodl blne equton where p nj k s the power njeton t bus, nd d k the lod demnd. Equton (11) represents the DC power flow equton, B the nodl dmttne mtrx, nd θ k the bus voltge ngle vetor. Equton (12) represents the lne power lmt where θ lne lk s the voltge ngle dfferene between the two buses onneted through lne l t tme step k, x l the lne seres retne, nd Pl b ts symmetrl power lmt; (13) fxes the slk bus voltge ngle; nd (14)-(15) defne the ntl system stte, bsed on the prevous set ponts of the onventonl genertors g prev nd on the ntl energy levels e n Seton II, p pv 1. As ndted n orresponds to the rel PV output, whle ppv k 1 orresponds to forest vlues. The deson vrble vetor for the determnst problem s X det = [x det 1 ;x det 2 ;...;x det N ], where x det k = [g k; k ;d k ;hpv k ]. B. Stohst Mult-Perod Optml Power Flow Wth Storge The stohst MPS OPF problem mnmzes 1 f g S (g ks)+f ( ks, d ks) s S k K I (16) subjet to modfed versons of (4)-(15) tht pply for ll s nd where re ll the optmzton vrbles re ddtonlly ndexed over s, s well s to the followng onstrnts: g 11 =g 1s, s 1 (17) 11= 1s, s 1 (18) d 11= d 1s, s 1 (19) h pv 11 =hpv 1s, s 1 (20) whh ensure tht the sme ton s hosen t the frst tme step for ll senros sne ths s the ton ppled to the (rel) system. Note tht g 0s nd e 1s re lredy onstrned by the modfed versons of (14)-(15), nd tht p pv 1s s the sme for ll senros sne t orresponds to the tul vlue. The vlues of p pv (k 1)s orrespond to the dfferent forest senros. The deson vrble vetor for the stohst problem s X sto, nd s S tmes lrger thn X det, sne ll the deson vrbles re replted for eh senro.

4 III. HYBRID MULTI-PERIOD OPTIMAL POWER FLOW WITH STORAGE Our hybrd method dvdes the optmzton horzon nto the frst-prt problem (k K 1 = [1,...,N 1]) nd the seondprt problem (k K 2 = [N,...,N]). The frst-prt problem s solved usng multple forest senros (s the stohst MPS OPF), nd the seond-prt problem s solved usng sngle forest (s the determnst MPS OPF). We do so beuse the ddtonl nformton provded by the forest senros (ompred to sngle forest) s most vluble lose to delvery, nd smplfyng the seond-prt nto determnst problem redues the omputtonl burden n exhnge for only modest loss of performne. We solve the two prts s dfferent optmzton problems, strtng wth the seond prt. In order to lnk the two problems together, we ntrodue U N (e N ), termnl ost funton for the frst-prt problem tht depends on the storge energy levels t tme step N. The funton U N (e N ) s obtned by solvng the seond-prt problem. It s lower pproxmton of V N (e N ), the true expeted ost-to-go over the seond-prt horzon. The onept of dvdng mult-perod problem nto dfferent temporl subproblems nd usng stte-dependent funton to lnk these subproblems omes from (stohst) dul dynm progrmmng, often used for lrge hydro shedulng problems [9]. The frst-prt problem mnmzes { } 1 { f g S (g ks)+f( ks, d ks)} +UN (e N s) s S k K 1 I (21) subjet to the stohst versons of (4)-(15), nd to (17)-(20). Note tht N 1<<N, so we need termnl ost funton for the frst-prt problem, unlke n (3) or (16). We dsuss nludng genertor nter-temporl rmp onstrnts n U N n Seton III-D. To buld U N (e N ) we must frst solve severl nstnes of the seond-prt problem for dfferent vlues of e N, referred to s trl ponts nd desgnted by e j N, wth j J. These trl ponts defne the ntl energy levels n the seondprt problem, nd re smlr to e n n the stohst or determnst MPS OPF. The seond-prt problem mnmzes f g (g k)+f ( k, d k) (22) k K 2 I s.t. (4) (13), g N 1 = g x N 1 (23) e N = e j N (24) where gn x 1 wll be desrbed n Seton III-A. The seondprt problem orresponds to the determnst MPS OPF, exept t uses dfferent set of tme steps nd (23)-(24) nsted of (14)-(15). We denote by z j N the objetve vlue of (22) nd by λj N the Lgrnge multpler orrespondng to (24), whh tells us how muh z j N would hnge wth respet to hnge n ej N. Therefore, sne (22) s onvex, z j N nd λj N gve us lner pproxmton (lower bound) of V N (e N ), urte n the V N' (e N' ), true ost-to-go funton 1 Cut bsed on trl pont e =100 N' 2 Cut bsed on trl pont e =500 N' 3 Cut bsed on trl pont e =900 N' U (e ), ost-to-go pproxmton N' N' Storge energy level t tme step N' [MWh], e N' Fgure 3. Illustrtve exmple of the ost-to-go pproxmton from N untl N, for sngle storge unt (dm(e N )=1, so the bold typefe n be omtted). neghborhood of e j N. Ths pproxmton n be nterpreted s Benders Cut [9]. A set of uts n be used to buld lower pproxmton of V N (e N ) over domn, nd hene form U N (e N ), s shown n Fg. 3 for smple exmple. Eh ut s obtned by solvng (22) for dfferent trl pont. We n reformulte the frst-prt problem (21) to mke explt the role of the uts, speflly, the problem mnmzes { 1 S s S k K 1 I { f g (g ks)+f ( ks, d ks)} +un s subjet to stohst versons of (4)-(15), (17)-(20), nd } (25) λ j N T[ en s e j N ] +z j N u N s j, (26) where u N s s the pproxmte ost-to-go from N to N for senro s, nd (26) defnes set of lower bounds for u N s s funtons of e N s, prmetrzed by the uts. Sne we solve mnmzton problem, (26) together wthu N s n the objetve funton represent U N (e N s), whh s peewse lner onvex pproxmton, but modeled here s the mxmum of J ffne funtons; hene, we solve the problem wthout nteger vrbles. Note tht the frst-prt problem orresponds to the stohst MPS OPF, exept t uses dfferent set of tme steps nd nludes both u N s nd (26). The key feture of the hybrd method s tht U N (e N ) s obtned by solvng J nstnes of determnst MPS OPF. The pproh s bsed on the mplt ssumpton tht the verge forest over the senros (.e. the determnst forest) s generlly suffent to obtn the generl shpe of the ost-to-go funton strtng N tme steps n the future. A. Cut plement To obtn stsftory ury for U N (e N ) whle keepng the number of uts low (nd so the number of nstnes of the seond-prt problem), the trl ponts should be n the neghborhood of e N tht the system wll vst. Therefore, before solvng the seond-prt problem, we solve n explorton problem to dentfy the neghborhood subspe. The explorton problem s extly the determnst MPS OPF, nd ts only use s to generte ) the energy levels t tme step N gven the determnst formulton nd denoted e x N nd ) the onventonl genertor dspth t N 1 gven the determnst formulton nd denoted g x N 1.

5 Energy Level, storge #3 [GWh] Energy Level, storge #2 [GWh] Energy Level, storge #1 [GWh] Fgure 4. Exmple of trl pont plement, for three storge unts. The green pont orresponds to e x N nd to e 1 N, used n both the Sngle-Cut nd the Mult-Cut versons. The ornge ponts re used only n the Mult-Cut verson, nd orrespond to e 2 N to e 9 N. The lengths of the edges orrespond to 40% of the orrespondng storge pty, s the ponts re loted t e x N ±0.2E. We then use e x N to generte the trl ponts. Here, we show two smplst wys to generte them; more dvned tehnques re the subjet of future reserh. The frst s lled the Sngle-Cut verson nd retes only one trl pont: e 1 N =ex N ( J =1), ledng to sngle ut n (26). The seond s the Mult-Cut verson whh leds to J =2 Y +1, where Y s the number of storge unts. In ddton to e 1 N =ex N, ths verson genertes two ddtonl levels for eh storge unt: e x N 0.2E nd e x N +0.2E. By enumertng ll possble ombntons of these two ddtonl levels per storge unt, we obtn olleton of 2 Y trl ponts. Ths olleton forms hyperbox n the storge spe round e x N, nd defnes the ponts e 2 N to +1 e2y N, s shown n Fg. 4. Note tht both versons provde n urte pproxmton of V N (e N ) n only subspe of the storge spe. B. Summry of the Hybrd Method At eh tme step, our hybrd method requres solvng three problems n the followng order: One explorton problem, usng the determnst formulton nd N tme steps, to obtn e x N nd gx N 1 ; J nstnes of the seond-prt problem, usng the determnst formulton nd N N +1 tme steps, to buld U N (e N ); One frst-prt problem, usng the stohst formulton (s well s u N s nd (26)) nd N 1 tme steps, to determne the dspth t k=1. The method n ether be seen s wy to redue the omplexty of stohst MPS OPF wth horzon N, or s wy to provde termnl ost funton for stohst MPS OPF wth horzon N 1. The method of usng Benders Cuts to nterfe the frst- nd seond-prt problems s generl, nd ould be ppled to vrety of problems n whh there re benefts to onsderng omplex, urte models n the frst prt nd smpler, less urte models n the seond prt. C. Computtonl Complexty All the problems onsdered n ths pper re qudrt progrmmng problems, wth polynoml tme omplexty O((ΩM) w ), where Ω s the number of tme steps, M the problem sze per tme step nd per senro (bsed on the numbers of optmzton vrbles nd onstrnts), nd w>1 polynoml ftor [12]. The omplextes of the dfferent problems re gven n Tble I. The hybrd method wll be fster thn the stohst MPS OPF f the totl omputton Problem TABLE I. determnst MPS OPF explorton problem J seond-prt problems frst-prt problem stohst MPS OPF PROBLEM COMPLEXITY. Computtonl Complexty O(...) (N M) w (N M) w J ((N N +1)M) w ( S (N 1) M) w ( S N M) w tme of the explorton problem, the J seond-prt problems, nd the frst-prt problem s smller thn the stohst MPS OPF omputton tme. Speflly, our method wll be fster for lrge S, smll J, nd smll (N 1)/N. The sze of ths rto leds to trde-off: the smller t s, the fster our method, but the worst the results, s we wll see n the seond se study. Our method wll never be fster thn the determnst MPS OPF, sne the explorton phse lone requres s muh effort thn the determnst MPS OPF. D. Rmp Constrnt nd Cost-to-Go Approxmton Our method uses onlyes the system stte. However, due to rmpng onstrnts, the prevous onventonl genertor dspth lso defnes the system stte, s shown by (14). Hene, to be omplete, the method should lso onsder ths vrble when buldng the ost-to-go pproxmton, whh would then beome U N (e N,g N 1). In tht se, the Mult-Cut verson would onsder 2 Y+Ygen +1 uts, where Y gen s the number of onventonl genertors. Nevertheless, for the power system models we nlyzed, we found tht the mplton of the rmp onstrnt N tme steps nto the future ws mnor, so (23) s suffent. We refer to the seond se study for dsusson on tht subjet. IV. DEFINITIONS OF CASE STUDIES The frst se study ms to determne how the system sze (numbers of buses, genertors, nd storge unts) nfluenes the performne of the dfferent methods, nd when the hybrd method s of nterest. For dfferent system szes (.e. ses), we pply the reedng horzon pproh nd ompre the solver omputton tme nd the yerly ost obtned when usng the determnst MPS OPF (Seton II-A), the stohst MPS OPF (II-B), nd the Sngle-Cut or Mult-Cut versons of the hybrd method (III). Eh se s smulted over one yer, nd they ll hve the sme lod profles, relzed PV produton, PV forest, nd totl onventonl genertor pty. Addtonlly, ther totl onventonl genertor pty brekdown n terms of bse plnts (low opertng ost, low rmpng pblty), mdmert plnts, nd pekng plnts (hgh opertng ost, hgh rmpng pblty) s dentl. In yerly energy terms, the PV produton (not onsderng urtlment) orresponds to 50% of the lod onsumpton, nd s produed by sngle plnt. We defne three dfferent grd szes, whh lso relte to the number of onventonl genertors: Smll: Copper plte (one-bus) system, where (11)-(13) re repled by I pnj k =0, wth three onventonl plnts; Medum: Sx-bus system, wth four onventonl plnts;

6 TABLE II. Plnt tegory MARGINAL COSTS OF THE DIFFERENT PLANT CATEGORIES. Conventonl, bse lod plnts 5-10 Conventonl, md-mert plnts Conventonl, pekng plnts Storge unts (one-wy ost, losses not nluded) 8-9 Mrgnl ost [e/mwh] Lrge: 30-bus system, wth fve onventonl plnts. The totl storge energy pty lwys orresponds to bout seven hours of verge lod onsumpton, but for eh grd sze we desgn versons wth dfferent numbers of storge unts: one to three for the smll system, two to four for the medum one, nd one to fve for the lrge one. Ths leds to totl of 11 ses. We set N=72 nd N =9. The lod profles re sled Swss nd Germn lod profles (ENTSO-E dt). The PV forest nd generton profles re omputed usng PV model [13] nd wether forest/mesurement from the Swss wether serve. However, we dded Brownn moton nd Gussn whte nose to nrese the normlzed root men squre forest error to 10% over the frst twelve hours of the forest. Ths ws done to hghlght the bltes of the dfferent methods to hndle the forest error. The forests re updted every 12 hours nd onssts of 16 senros. The ost oeffents hve been hosen to hghlght the performne of the dfferent dspthes, nd not to represent ny spef tehnology. The mrgnl ost rnges for the dfferent plnt tegores re gven n Tble II, nd we ssume one-wy storge effeny of 88 to 92%. Fnlly, to ensure omprble results, we enfore yerly yl behvor for the storge level. Our smultons strt wth storge stte of hrge of 10%, on Jnury 1st t 00:00, nd we wnt to ensure the sme level on Jnury 1st t 00:00 of the followng yer. For ths, whenever the end-of-the-yer tme step s n sght of the optmzer (.e wthn the optmzton horzon), we dd n ddtonl onstrnt mposng the stte of hrge t tht spef tme step. The seond se study nlyzes the senstvty of the results to N. It uses the medum grd wth two storge unts, desrbed n the frst se study. All the dt nd ssumptons of the frst se study pply, exept tht we vry N from 3 to 24, for both the Sngle-Cut nd the Mult-Cut versons. The results ndte to wht extent the ssumptons of ) usng the determnst forest to buld U N (e N ) nd ) negletng the onventonl genertors prevous dspth n ths funton re ffeted by the length of the frst-prt problem. Both se studes re ment to drw generl, qulttve onlusons. A rel power system s fr more omplex thn the exmples onsdered here, nd the solver nd the mhne t runs on ply key role. V. RESULTS For the 11 ses onsdered n the frst se study, the stohst dspth leds on verge to yerly opertng ost 0.50% smller thn the determnst dspth. However, the verge omputtonl effort, n term of CPU tme, s 15 tmes Yerly opertng ost [ Determnst OPF Hybrd Method, Sngle-Cut Hybrd Method, Mult-Cut Stohst OPF Z=73%, T=18% Z=77%, T=71% Totl omputton tme [s], T Fgure 5. Comprson of the dspth methods, smulted over yer for the medum grd wth three storge unts (one se of the frst se study). lrger. Ths rto nreses wth the problem sze (t rehes 20 for the lrge grd wth fve storge unts) nd so the stohst MPS OPF would be hllengng to mplement on rel power system models wth hundreds of buses. We report our hybrd method s performne ompred to the trdtonl methods, nd defne the reltve ost mprovement δz nd the reltve CPU tme nrese δt : δz= Zdet Z hyb Z det Z sto ; δt=t hyb T det T sto T det, where the exponents det, sto, nd hyb orrespond to the determnst dspth, the stohst dspth, nd our hybrd method, respetvely. For both ndtors, vlue of 0% orresponds to the determnst MPS OPF, nd 100% to the stohst MPS OPF. Hene, lrge δz nd smll δt would ndte tht the hybrd method performs well. The performne of the methods s shown n Fg. 5 for the medum grd wth three storge unts. We see tht both the Sngle-Cut nd the Mult-Cut hybrd versons led to ost lower thn the determnst dspth, but hgher ost thn the stohst one, nd tht the omputton tme s hgher thn for the determnst dspth, but lower thn for the stohst dspth, s expeted. For ths se, the Mult-Cut verson results n slghtly lower ost thn the Sngle-Cut one. The full results of the frst se study re shown n Tble III, nd hghlght the behvor of the hybrd method wth respet to the system sze. Frst, we see tht the Sngle-Cut verson leds to t lest 70% of the ost mprovement of the stohst MPS OPF, whle the omputton tme nrese s t most 40% of the omputton tme nrese of the stohst MPS OPF. We lso see tht ths reltve omputton tme nrese s smller for lrger systems (n terms of both grd sze nd number of storge unts), ndtng tht the Sngle-Cut verson s wellsuted for lrge systems. As lredy ndted n Fg. 5, we see tht the mrgnl beneft of the Mult-Cut verson over the Sngle-Cut verson s rther smll. Whle the Mult-Cut verson leds to lower or equl ost n ten out of 11 ses, the verge ddtonl mprovement ompred to the Sngle-Cut verson s only 2%. On the other hnd, the verge ddtonl CPU tme nrese ompred to the Sngle Cut verson s 61%. We lso see tht the Mult-Cut verson sles bdly wth the number of storge unts. In ses shown n red, the Mult-Cut verson s more omputtonlly demndng nd leds to hgher ost thn the stohst MPS OPF, nd s therefore not of nterest. These ses re hrterzed by lrge number of storge

7 Yerly opertng ost [10 6 TABLE III. RESULTS OF THE FIRST CASE STUDY. δz: RATIO OF THE COST IMPROVEMENT OVER THE STOCHASTIC MPS OPF COST IMPROVEMENT; δt: RATIO OF THE CPU TIME INCREASE OVER THE STOCHASTIC MPS OPF CPU TIME INCREASE. Grd Sze Smll Medum Lrge Storge Number Sngle-Cut Mult-Cut δz [%] δt [%] δz [%] δt [%] Determnst OPF Hybrd, Sngle-Cut. Prmeter: N' Hybrd, Mult-Cut. Prmeter: N' Stohst OPF Totl omputton tme [s], T Fgure 6. Senstvty nlyss on N, for the medum grd sze wth two storge unts (seond se study). N ={3,6,...,21,24}. unts reltve to the system sze, nd hene lrge number of uts. The results show tht there s stll on verge n dded vlue (n term of ost) of usng more thn one ut to buld the ost-to-go pproxmton, but tht too mny uts mke the Mult-Cut verson prohbtvely slow. The results of the seond se study re shown n Fg. 6. Inresng N leds to lower ost but hgher omputton tme, nd the dependeny on N s fr from lner. A vlue of three leds to hgher ost thn the determnst se, for both the Sngle- nd the Mult-Cut versons. Inresng N to sx leds to sgnfnt mprovement. Any nrese fter tht leds to smll ost reduton, for both versons. Ths lerly ndtes tht the hybrd method s senstve to the hoe of N, t lest up to threshold. Ths threshold depends on the rto of the storge pty over the verge demnd, nd on the rmp onstrnts of the onventonl plnts. Note tht even the Sngle-Cut verson wth N =24 s fster thn the the Mult-Cut verson for ny vlue of N. However, ths relton depends on the system sze, nd on the rnge ofn. The Mult- Cut verson generlly leds to slghtly lower ost, s n the frst se study. Bsed on the results of these two se studes, we would reommend usng the Sngle-Cut verson, s the smll ddtonl ost mprovement of the Mult-Cut verson omes wth rther lrge ddtonl omputtonl burden. However, t should be noted tht the ddtonl trl pont plement t e x N ±0.2E n the Mult-Cut verson s rbtrry. A dfferent hyperbox sze, or better wy to ple the uts, mght led to lower ost wth smlr or lower number of uts (nd hene smlr or lower omputton tme). Prelmnry results show tht the performne of the Mult-Cut verson s senstve to the ut plement, whh s the subjet of future reserh. VI. CONCLUSION We hve presented hybrd MPS OPF method tht nludes stohst prt nd determnst prt. Bsed on 11 test systems, we showed tht the Sngle-Cut verson of the hybrd method heves muh of the ost mprovement of the stohst MPS OPF, wth omputtonl burden lose to the determnst MPS OPF. We hghlghted tht the length of the frst-prt problem s key prmeter ffetng method performne. Ths method ould lso be used to desgn nd solve other type of mult-perod hybrd problems tht ontn both omputton-ntensve hgh-performne prt nd lower omputton lower-performne prt. We lso showed tht the omputtonl performne of the Mult-Cut verson of the hybrd method s senstve to the number or storge unts reltve to the system sze. Ths s beuse the number of Benders Cuts n tht verson grows exponentlly wth the number of storge unts. Therefore, further reserh wll fous on developng trl pont plement tehnques tht mxmze the usefulness of multple uts. Extensons to nteger vrbles or AC OPF wll lso be onsdered. REFERENCES [1] A. Tuohy, P. Mebom, E. Denny, nd M. O Mlley, Unt ommtment for systems wth sgnfnt wnd penetrton, Power Systems, IEEE Trnstons on, vol. 24, no. 2, pp , [2] A. Sturt nd G. Strb, Effent stohst shedulng for smulton of wnd-ntegrted power systems, Power Systems, IEEE Trnstons on, vol. 27, no. 1, pp , [3] R. C. Grnold, Model buldng tehnques for the orreton of end effets n multstge onvex progrms, Opertons Reserh, vol. 31, no. 3, pp , [4] A. Gbsh nd P. L, Atve-retve optml power flow n dstrbuton networks wth embedded generton nd bttery storge, Power Systems, IEEE Trnstons on, vol. 27, no. 4, pp , [5] E. Pérez, H. Beltrn, N. Apro, nd P. Rodríguez, Predtve power ontrol for pv plnts wth energy storge, Sustnble Energy, IEEE Trnstons on, vol. 4, no. 2, pp , [6] S. Ah, T. Green, nd N. Shh, Effets of optmsed plug-n hybrd vehle hrgng strteges on eletr dstrbuton network losses, n Trnsmsson nd Dstrbuton Conferene nd Exposton, IEEE PES, [7] G. W. Chng, M. Agng, J. G. Wght, J. Medn, T. Burton, S. Reeves, nd M. Chrstofords, Experenes wth mxed nteger lner progrmmng bsed pprohes on short-term hydro shedulng, Power Systems, IEEE Trnstons on, vol. 16, no. 4, pp , [8] S. Deml, A. Ulbg, T. Borshe, nd G. Andersson, The role of ggregton n power system smulton, n IEEE PES PowerTeh 2015, Endhoven, Netherlnds, [9] M. V. F. Perer, Optml stohst opertons shedulng of lrge hydroeletr systems, Interntonl Journl of Eletrl Power & Energy Systems, vol. 11, pp , [10] R.-B. Wets, Stohst progrmmng models: Wt-nd-see versus herend-now, n Deson Mkng Under Unertnty. Sprnger, [11] J. Löfberg, YALMIP : A toolbox for modelng nd optmzton n MATLAB, n Proeedngs of the CACSD Conferene, Tpe, Twn, [Onlne]. Avlble: [12] S. A. Vvss, Complexty theory: Qudrt progrmmng, n Enyloped of Optmzton. Sprnger US, 2009, pp [13] Snd Ntonl Lbortores. (2014) PV LIB Toolbox.