MIP Reformulation for Max-min Problems in Two-stage Robust SCUC

Transcription

1 1 MIP Reforulaon for Max-n Probles n Two-sae Robus SCUC Honxn Ye, Meber, IEEE, Janhu Wan, Senor Meber, IEEE, Zuy L, Senor Meber, IEEE Absrac Wh ncreasn renewable peneraon n power syses, a lo of research effors have been focused on how o accoodae he unceranes fro renewables n he Secury- Consran Un Coen SCUC proble. One of he canddae approaches o handln unceranes s he wo-sae Robus SCUC RSCUC, whch enables syse o survve n any scenaro. The survvably s uaraneed by he soluon opaly of he ax-n proble n he second sae. However, as he non-convex ax-n proble s NP-hard, s dffcul o e he exac opal soluon n accepable e. In hs paper, we propose a new effcen forulaon whch recass he ax-n proble o a Mxed Ineer Proran MIP proble usn Bnary Expanson BE. The upper bound of he ap beween he new MIP proble and he ornal axn proble s derved. The ap, whch quanfes he soluon opaly of he ax-n proble, s conrollable. Two effecve acceleraon echnques are proposed o prove he perforance of he MIP proble by elnan nacve flow consrans and decoposn e-coupled uncerany bude consrans. Accordnly, he copuaon burden of solvn he ax-n proble s reduced reendously. The sulaon resuls for he IEEE 118-Bus syse valdae and deonsrae he effecveness of he new BE-based soluon approach o he wo-sae RSCUC and he acceleraon echnques. Index Ters robus SCUC, ax-n proble, renewables, bnary expanson, lne coneson I. INTRODUCTION The Renewable Enery Sources RES, such as wnd and solar, have low producon cos and are free of carbon esson. The peneraon level of RES keeps clbn n recen decades, whch helps lower he enery producon cos and proec he envronen. However, hey also pose ajor challenes o elecrcy arkes as he RES oupu canno be predced accuraely n day ahead. In he U.S. Dayahead Marke DAM, he Independen Syse Operaor ISO or Reonal Transsson Oranzaon RTO perfors he Secury-Consran Un Coen SCUC and Econoc Dspach ED o clear he arke [1], [2]. In he SCUC and ED proble, he ISO/RTO deernes he un coen UC and ED for he nex day wh lowes cos o supply he forecased load whle respecn a se of consrans. The syse-wde consrans ay nclude he load deand balance, ranssson capacy l, and reserve requreen. The unwse consrans are norally coposed of eneraon capacy Ths work s suppored n par by he U.S. Naonal Scence Foundaon Gran ECCS J. Wan s work s suppored by he U.S. Deparen of Enery DOE s Offce of Elecrcy Delvery and Enery Relably. H. Ye and Z. L are wh he Rober W. Galvn Cener for Elecrcy Innovaon a Illnos Insue of Technoloy, Chcao, IL 60616, USA. eal: hye9@hawk..edu; lzu@.edu. J. Wan s wh Aronne Naonal Laboraory, Aronne, IL, USA janhu.wan@anl.ov. ls, rapn rae ls, and nu on/off e ls [2], [3]. The unceranes fro RESs pose new challenes for he SCUC and ED probles n DAM, whch have araced exensve aenon n recen years [4] [13]. The wo-sae Robus SCUC RSCUC can accoodae any uncerany n he second sae accordn o he redspach process [8] [10]. I exacly ees he relably and secury requreen whch s he frs prory n power syse operaon. Alhouh he RSCUC s suded nensvely, has no been wdely appled n he real arkes. One of he reasons s ha he ax-n proble n he soluon approach s NPhard. I should be ephaszed ha he lares er of RSCUC, robusness, s dependen on he opaly of he soluon o he ax-n proble. The ax-n proble s ofen convered o a axzaon proble accordn o he dualy heory, whch nroduces blnear ers n he new objecve funcon. In he Exree Pon EP based soluon approach, EPs are explcly forulaed for unceranes o lnearze he blnear ers [8], [10]. However, he EP approach s dependen on he avalably of he closed for of he EPs. I s also nracable when he nuber of he EPs s lare. Researchers also eploy Ouer Approxaon OA [9] and Mounan Clbn MC [11], [12], [14] o heurscally solve he axn proble wh ood copuaon effcency. However, he lobal opaly s no uaraneed n hese approaches, whch ay lead o he loss of robusness. If KKT condons are adoped [15], [16], a aheacal prora wh equlbru consrans MPEC can be esablshed. Bu MPEC s noorously hard o solve n pracce. An alernave way s o avod he ax-n proble. The affne polcy AP as shown n [17] s successfully adoped n he leraure [16], [18] [20] wh beer copuaon perforance. However, replacn he full recourse sraey wh src AP wll furher deerorae he value of he robus soluon, whch s already crczed for over-conservas. In hs paper, we propose o solve he ax-n proble n an nnovave way. Insead of enueran he EPs of he uncerany se, he blnear ers n he objecve funcon s lnearzed based on he Bnary Expanson BE echnque. The conrbuons of hs paper are 1 The BE approach s proposed o recas he ax-n proble n he robus opzaon fraework no a xed-neer proran MIP proble. The BE soluon approach does no rely on he closed for EPs of he uncerany se. Hence, s sll applcable when s hard o forulae he EPs. In parcular, when sophscaed uncerany ses are consdered o releve he conservaveness, he BE approach shows

2 2 ore advanaes over he EP approach. 2 A closed-for upper bound of he opaly ap n he BE approach s derved. I can be obaned drecly whou solvn any opzaon proble. Furherore, he opaly ap can be adjused by chann he nuber of BE ers. Accordnly, he soluon qualy of he ax-n proble becoes observable and conrollable. 3 Two acceleraon echnques, blnear er reducon and decoposon, are appled o reduce he soluon e for he MIP proble n he BE approach. A eneralzed suffcen condons for nacve lne consrans n he ax-n proble are derved and used o reduce os of he blnear ers. By explorn he specal srucure of he ax-n proble, he ornal e-coupled axn proble s decoposed no saller e-decoupled ax-n probles. Whle he dea of he wo acceleraon echnques s no new, he cobned use of he echnques alon wh he BE approach s novel and akes uch ore effcen o solve he ax-n subproble, he os e-consun sep n he robus opzaon fraework. In hs paper, he arx s denoed wh bold uppercase leer, he vecor wh bold lowercase leer, and he scalar wh noral fon leer. The res of hs paper s oranzed as follows. In Secon II. he RSCUC fraework and he axn proble are nroduced. Then n Secon III, he BE based approach o solvn he ax-n proble s proposed and he closed for of he opaly ap s derved. In Secon IV, he wo acceleraon echnques o reduce he soluon e are presened. The case sudes are presened wh he IEEE 118-Bus syse n Secon V. Fnally, Secon VI concludes hs paper. A. Uncerany Se II. TWO-STAGE ROBUST SCUC The unceranes n he SCUC proble are anly due o he forecasn errors for renewable power oupu and load. In he RSCUC leraure [8], [9], hese unceranes are reaed as load perurbaon. The uncerany se s odeled as U := ɛ 1,, ɛ T R N d R N d : u ɛ u, 1 S ɛ h, 2 ɛ, Λ, 3 u, ɛ, Λ, 4 u, where N d and T are he nubers of unceran load njecons and scheduln perods, respecvely, and ɛ represens he uncerany vecor a e. ɛ, R s he enry n he vecor ɛ. Defne ɛ = [ɛ 1,, ɛ T ]. The U defned n hs paper cobnes he uncerany se n [9] [11]. Eq. 1 ples ha he unceranes are led n nervals. The eneral polyope 2 can nclude any poenal consrans. The snle-hour bude consran for unceranes s forulaed n 3. The e-coupled bude consran s forulaed n 4. The bude paraeer Λ and Λ are assued neers. By odeln consrans 2,3,4, he feasble reon of U can be reduced. Consequenly, he soluon of he RSCUC wll be less conservave. B. Two-sae Robus RSCUC Ths paper focuses on he soluon approach for he axn subproble n RSCUC, We frs brefly nroduce a selfconaned wo-sae RSCUC odel. The deals can be found n [10], [16]. The RSCUC s forulaed as and F := P n x,p F Cx, p 5 s.. Ax + Bp b 6 x, p : ɛ U, p such ha Dp + p + Eɛ h 7 F x + Gp + H p. 8 The objecve 5 s o fnd he UC and ED soluon wh he leas cos, whch can be unzed aans any realzaon of he uncerany predefned n U. Bnary varable vecor x denoes he UC varables. p R N GN T denoes he ED varables, respecvely. N G s he nuber of eneraors. A, B, D, E, F, G, and H are absrac arces for represenn consrans. Parcularly, E R Ns N dn T where N s s he nuber of rows n 7. Equaon 6 represens UC and nework consran for he base-case scenaro. p R N GN T sands for he eneraon adjusen for uncerany accoodaon. The re-dspach process respecs he syse-wde consrans n 7 as well as he un-wse consran 8. Slar o [10], he Colun Generaon CG n [15] s adoped o solve he RSCUC. The aser proble MP and he subproble SP are esablshed as follows. and MP s.. SP n x,p Rɛ := Cx, p Ax + Bp b Dp + p w h Eˆɛ w, w W F x + Gp + H p w, w W φ := ax n ɛ U s, p Rɛ 1 s s, p : s 0 D p s h Dp Eɛ H p F x Gp 9a 9b 10a 10b 10c 10d where W s he ndex se for uncerany pons ˆɛ whch are dynacally eneraed n SP durn he soluon procedure as descrbed n Alorh 1. The dealed forulaon of Rɛ can be found n Appendx A. In fac, MP s slar o he scenaro-based Sochasc SCUC SSCUC. I s noed ha boh MP and SP are NPhard probles [21], [22]. In parcular, s dffcul o e he exac opal soluon o he ax-n proble SP.

3 3 Alorh 1 Colun Generaon Procedure o Solve P 1: W, w 1, φ +, defne feasbly olerance δ 2: whle φ δ do 3: Solve MP, oban opal ˆx, ˆp. 4: Solve SP wh x = ˆx, p = ˆp, e soluon φ,ˆɛ w 5: W W w, w w + 1 6: end whle III. A MIP REFORMULATION FOR MAX-MIN PROBLEM The ax-n proble SP s convered o a axzaon proble based on dualy heory [8], [9], [16]. The convered blnear axzaon proble BP s forulaed as BP φ = ax ɛ U,λ,µ λ h + λ Eɛ µ 11 s.. D λ + H µ = λ 1, µ 0, 13 where h := h Dp and := F x Gp. Due o he quadrac er λ Eɛ n he objecve funcon 11, BP s hard o solve. In hs paper, he Bnary Expanson BE s used o lnearze he blnear er λ Eɛ [23]. The basc dea of he BE s ornaed fro he converson fro a decal nuber o a bnary nuber. An neer n he decal nuber syse can be equvalenly convered o a nuber n he bnary nuber syse. Consder an enry λ n vecor λ. As λ [0, 1], 2 K λ K 2 k z,k, k =0 where K s an neer and z,k 0, 1, k. Hence, we e K λ 2 k z,k,, 14 k=0 z,k 0, 1,, k. 15 Wh he BE echnque, we have he follown heore o solve BP. Theore 1. Inroduce auxlary paraeer R q Ns, q R Ns, = n q E ɛ, q = ax E ɛ. 16 ɛ U ɛ U q The proble BP s approxaed as RP φ a K= ax ɛ U,λ,µ N s K 2 k ξ,k + λ q h µ k=0 s.. D λ + H µ = 0 K K 2 k z,k λ 2 k z,k + 2 K 17 k=0 k=0 q = Eɛ q 18 0 λ 1, µ 0 ξ,k q, ξ,k q z,k,, k z,k 0, 1,, k, where K s a ven neer. The proof of Theore 1 s ven n Appendx B. E denoes he h row n E. Theore 1 presens an alernave ehod o he soluon of he blnear proble. Noce a relaxed U n 16 s he box consran 1. In hs case, he closed fors of and q are avalable and all he prncples are sll applcable. q The closed fors are = E q u E+ u, q = 2E + u 2E u, 19 where u s he non-neave bound vecor of he uncerany. E and E + are E = E abse /2, E + = E + abse /2, where abs s a vecor whose eleens are he absolue values of he eleens n. The process of deernn of q and s as follows. 1: Forulae q arx E. 2: for = 1 o N s do 3: Generae E + and E 4: Ge q and q accordn o 19 5: end for We have he follown heore reardn he soluon qualy. Theore 2. The ap beween proble BP and RP follows N s φ φ a K 2 K q, 20 where φ s he opal value o BP and φ a K s he opal value o RP ven K. The proof of Theore 2 s ven n Appendx B. Theore 2 shows ha he soluon qualy, whch s represened by he ap, s observable and conrollable. The ap s observable as s always saller han 2 K N s q. The upper bound n Theore 2 can be deerned drecly by 19 whou solvn RP or BP. The ap s conrollable as can be adjused by he paraeer K. By ncreasn K, he ap can be lowered. Based on Theore 1 and 2, BP can be approxaely solved by RP wh a conrollable ap. Copared o he EP approach [8], he BE approach can also handle oher uncerany ses whose EPs are hard o odel or even do no exs. In fac, he MC and OA approaches can also approxaely solve BP [9], [11]. A ajor advanae of he EP approach s ha s soluon qualy s observable and conrollable whle he opaly of MC and OA approach s unknown. IV. ACCELERATION TECHNIQUES The lare nuber of rows n E leads o heavy copuaon burden n he MIP proble RP. In hs secon, wo effecve acceleraon echnques are proposed o reduce he copuaon burden. The frs acceleraon echnque, whch s also applcable o oher approaches, s derved based on a suffcen condon o reduce nacve nework consrans. The second echnque s proposed by explorn he specal srucure of he ax-n proble. We decopose he ornal e-coupled ax-n proble no saller e-decoupled probles a each hour.

4 4 A. Reducon of Blnear Ters The blnear er λ Eɛ s derved fro he load balance consrans and nework capacy ls. If he nuber of ranssson lnes s N L, hen here are 2 N L N T lne consrans odeled n he ax-n proble. Consequenly, hese syse-wde consrans lead o a lare nuber of neer varables, whch ncreases he proble sze of RP. Nex, we show ha only par of he wll rean afer he reducon of blnear ers based on he suffcen condons for nacve consrans. The sraey s o denfy he nacve lne consrans and elnae he fro SP. However, he denfcaon s ade ore coplcaed due o he slack varables n Rɛ. The nonneave s n 44 n Appendx A ay cause a lare nuber of lne consrans n SP o becoe non-redundan. In order o address hs ssue, consder R ɛ := p, s L +, s L : s L +, s L 0, p, + p, = d, + ɛ,, 21 45, 46, 47. Copared o Rɛ n Appendx A, 21 does no nclude slack varables n he load balance consran. Then a new subproble USP USP φ u := ax ɛ U n s, p R ɛ 1 s 22 can be forulaed. As here s no slack varable n 21, he follown suffcen condons are esablshed o denfy he nacve lne consrans based on [24]. Theore 3. Consder he ne power njecon p nj, [ p nj,, p nj,] on bus a e, and nroduce auxlary varable p nj+, = p nj, p nj, 0 p nj+, p nj+, = p nj, p nj,,,. 23 Denoe p v = p nj,, and fl, v = N d exss an neer j [1, N d ] so ha j 1 j =1 Γ l, p nj,. If here p nj+ n, p v p nj+ n 24 n=1 n=1 j 1 Γ l,n Γ l,j p nj+ n, + Γ l,j p v + fl, v F l 25 n=1 Γ l,1 Γ l,2 Γ l,nd 26 holds, hen he flow consran for lne l wh a capacy of F l n he posve drecon a s nacve. n s he bus wh he n h lares shf facor Γ l,n for lne l. The proof of Theore 3 s shown n Appendx C. Theore 3 s a eneralzed and exended work nspred by [25]. The an dfferences are ha Theore 3 s esablshed for axn probles consdern unceranes and can be appled o oher b-level probles. In [25], he search for he upper/lower bound of he power flow s fnshed when he su of he eneraor capaces s larer han he consan load deand. Alorh 2 Solve P wh Elnaon Technques 1: W, w 1, φ +, defne feasbly olerance δ 2: whle φ δ do 3: Solve MP, oban opal ˆx, ˆp. 4: Updae arx E by applyn condons n Theore 3 5: Solve RP wh x = ˆx, p = ˆp, e soluon φ, ˆɛ w 6: W W w, w w + 1 7: end whle In he ax-n proble, he load deand s no consan anyore. Insead, s a varable. In Theore 3, a key ask s o fnd he h enouh lower and upper bounds for ne power njecon. Forunaely, ven p, he closed fors of p nj, and p nj, can be obaned accordn o rapn ls and eneraon capaces ha l p and he box consran for ɛ. They are p nj, = pnj, = G G p, + r, x, p d, + u,,, p, +, x, p d, u,,, r where r, x, p and, x, p are he upper bound and lower r bound of p,. They are he funcons of ven UC and ED as well as un rapn raes and eneraon capaces. By applyn Theore 3, arx E can be snfcanly reduced. Consequenly, unnecessary bnary decson varables and consrans n RP are also elnaed. Once he shf facors are ordered, he copuaon coplexy s ON d n Theore 3. The new procedure o solve P s presened n Alorh 2. The expense of replacn R wh R s ha he dual varables for load balance consrans becoe unbounded. However, n Alorh 2, 0 λ 1 s sll used n RP. The follown proposon shows ha wll no aler he convered soluon o P. Proposon 1. The opal value φ u o USP s an upper bound of he opal value φ o SP or BP,.e. φ u φ. Gven he opal x, p o P, hen φ u = φ, and opal soluon λ, µ o BP s also he opal dual varables o he nner-level nzaon proble n USP. The proof of Proposon 1 s ven n Appendx E. Proposon 1 shows ha f he procedure n Alorh 2 has no convered a eraon w, ˆɛ w ay no be he wors pon. Forunaely, ˆɛ w s uaraneed o be he wors pon f he procedure converes a eraon w. Therefore, he robusness s also uaraneed. Accordnly, 25 can be replaced wh j 1 Γ l,n Γ l,j p nj+ n,+γ l,1 δ+γ l,j p v n=1 +f v l, F l 27 n Theore 3 when feasbly olerance δ, whch s a erc for he soluon robusness, s consdered. The BE and nacve consran denfcaon conrbue o he perforance proveen oeher. In he proposed BE approach, he nacve consran denfcaon helps elnae

5 5 he blnear ers whose opal values are zero. Consequenly, he nuber of bnary varables and relaed consrans are draacally reduced n RP, and he perforance s proved. B. Decoposon of Te-coupled Bude Consrans In hs subsecon, a odel dependen decoposon echnque s proposed by explorn he specal srucure of SP. As x, p s ven n he aser proble, he re-dspach varable p and slack varable s are e decoupled. The only e-coupled varable s ɛ due o he consrans 4. In he follown, a echnque s proposed o handle he ecoupled bude consrans. We llusrae he echnque wh SP, whch s also applcable n Alorh 2. Consder an uncerany se a hour V v := ɛ R N d : 1, 2, ɛ, v. 28 u, Then a new ax-n proble a hour can be forulaed as φ v := R ɛ := ax n ɛ V v s 1 s 29a,ˆp R ɛ s, p : s 0 29b D p s h D p E ɛ 29c H p F x G p 29d where subscrp denoes he paraeers or varables assocaed wh hour. I s observed ha bude paraeer v s used n he above ax-n proble. We have he follown proposon reardn v and he soluon o SP. Proposon 2. Consder he proble IP : φ IP :=ax φ v 30 v, s.. v Λ, v Λ, v Z +, 31 φ IP = φ holds. Denoe he opal soluon o IP as v,. Gven v, he soluon ɛ v o proble 29a s a sub-vecor of opal soluon ɛ o proble SP a. The proof of Proposon 2 s ven n Appendx 2. Wh he values φ v obaned by solvn 29a, IP can be cased as an MIP proble or sply an ordern proble, whch can be solved effcenly. In fac, all he subprobles 29a wh dfferen v a varous e nervals can be solved n parallel. Moreover, we also desn an effcen sequenal alorh o solve he proble IP. I s presened n Alorh 3. Is copuaon coplexy s analyzed by counn he ax-n subprobles as follows. Theore 4. There are a os 2T + Λ Λ ax-n proble 29a o be solved. Alorh 3 Alorh o Solve Proble IP 1: φ IP 0 2: for = 1 o T do 3: Solve 29a wh v = Λ 4: φ IP φ IP + φ v 5: end for 6: whle v > Λ do 7: for = 1 o T do 8: f φ 1 0 and φ v 1 s no avalable hen 9: Calculae φ v 1 by solvn 29a 10: = φ v φ v 1 11: end f 12: end for 13: v v 1, where s he ndex of he nu one wh v 1 n 1,, T 14: φ IP φ IP 15: end whle Load Deand MW 6,000 5,000 4, Hour F. 1. Load Deand Profle n 24 Hours V. CASE STUDY The proposed soluon approach o solve he ax-n proble s esed n he IEEE 118-Bus syse hp://oor.ece.. edu/daa/roscuc 118.xls. The experens are perfored on PC wh Inel @3.40GHz. Gurob s ulzed o solve he MIP proble [26]. The MIPGap s se as and TeL s se as 400 seconds for Gurob. The syse consss of 54 uns and 186 ranssson lnes. The load profle s llusraed n F. 1. I s assued ha unceranes exs a 50 buses. u,, an enry n u, s se as 10% of he load a bus. Three roups of case sudes are perfored. 1 We valdae he effecveness of he proposed BE approach n handln a roup of uncerany ses. 2 We copare he copuaon perforance and soluon qualy of he proposed BE approach wh hose of he EP approach [8], [10] and MC approach [11], [12]. 3 We deonsrae he perforance of he proposed acceleraon echnques. A. Effecveness Valdaon of BE Approach The IEEE 118-bus syse s coposed of 3 zones. The bus se n each zone s denoed as Z n. In hs subsecon, we assue he oal unceranes n a zone respecs L z n, Z n ɛ, U z n,, n,. 32

6 6 Cos $ TABLE I OPTIMALITY GAPS V.S. INCREASING K K Gap MW VoWors MW AveraeCPUTeRP s α = 1, Λ = 30, Λ = Λ a Operaon Cos UCs h Λ b UC Hours F. 2. Operaon Cos and UCs v.s. Λ α = 1, Λ = Λ, K = 20 Cos $ α a Operaon Cos UCs h α b UC Hours F. 3. Operaon Cos and UCs v.s. α Λ = 30, Λ = 360, K = 20 TABLE II BE V.S. MC AND EP Approach BaseCos $ VoWors MW ToalCPUTe s BE 1,987, MC 1,987, EP 1,987, ,653 α = 1, Λ = 30, Λ = 600, K = 20 For splcy, L z n, and U z n, are paraeerzed wh α,.e. L z n, = α Z n u, and U z n, = α Z n u,. If α = 1, hen 32 becoes nacve. I s noed ha consran 2 s eneral enouh o odel oher lnear consrans. The procedure presened n Alorh 2 converes n 8 eraons. The sulaon resuls wh dfferen bude paraeers Λ are shown n F. 2. The curve for he base-case cos s drawn wh respec o dfferen Λ n F. 2a. I s observed ha he base-case cos s also low when Λ s sall. Wh he ncrease of Λ, he base case cos s also rapn up. The UC hours, he su of all coed un hours, are depced n F. 2b. When Λ = 20, around 720 UC hours are scheduled. The lares UC hours around 748 happens when Λ s 50. I ndcaes ha when he bude paraeer s lare, ore reserves are requred o anan he soluon robusness. F. 3 presens he sulaons wh dfferen α. Paraeer α reflecs he zonal confdence level for he uncerany. The e-coupled bude paraeer Λ s also se o 360. I can be observed ha he base-case cos ncreases onooncally wh α. When α = 0.2, he cos s he lowes around $1,982,000. When α = 0.8, he cos ncreases o around $1,986,000. The UC hours depced n F. 3b show a slar rend wh α. In fac, when he paraeer Λ or α ncreases, he uncerany se U s also enlared. The experens show ha a larer uncerany se leads o a ore conservave soluon o RSCUC n ers of he base-case coss and UC hours. Noe ha addn sroner consrans based on he hsorcal daa n U s preferred n RSCUC o reduce he conservaveness. The BE approach does no rely on he EPs of U o add consrans, whch s one advanae of he BE approach over he EP approach. Table I shows how he opaly ap s conrolled by K based on Theore 2. The upper bound of he ap, Gap, decreases wh K. The copuaon burden AveraeCPUTeRP, whch s he averae CPU e of solvn RP when he reducon echnque s appled, norally ncreases wh K n Gurob. Colun VoWors shows he consran volaon n he wors-case scenaro. I can be observed ha he acual volaon s uch saller han he upper bound of he ap. Consran Volaon MW MC BE BE Hour F. 4. Consran Volaon n he Wors Scenaro α = 1 B. Coparson wh Oher Approaches In hs par, he soluon qualy and perforance of he BE approach are copared wh hose of he EP approach and MC approach [11], [12]. The opaly ap of he axn proble s n fac also he soluon robusness of he soluon o RSCUC. The advanae of he MC approach s ha only LP probles are solved o fnd he soluon o he ax-n proble. However, he soluon s only locally opal. In conras, he EP approach and he BE approach can fnd he lobally opal soluon wh a known ap. We se he feasbly olerance or he robusness olerance δ = 0.01 and he paraeer used o conrol he opaly of he BE approach K = 20. Two cases are sulaed for he hree soluon approaches, wh he acceleraon echnques n Secon IV-A appled n all hree approaches. 1 α = 1. Consran 32 becoes nacve. All hree soluon approaches are esed. 2 α = 0.5. I becoes ore coplcaed for he EP approach o handle he uncerany se. For splcy, only he BE approach and MC approach are esed. 1 α = 1: Table II shows he experen resuls. BaseCos shows he base-case cos. I can be observed ha he opal value obaned by he BE or EP approach s around $214 hher han ha obaned by he MC approach. In ers of he base-case cos, sees ha he soluon fro he MC approach s beer han hose fro he BE or EP approach. However, he fac s ha soe wors pons are ssed n he MC approach, as shown n colun VoWors, whch represens he consran volaon n he wors-case scenaro.

7 7 TABLE III BE V.S. MC Approach BaseCos $ VoWors MW ToalCPUTe s BE 1,983, MC 1,983, α = 0.5, Λ = 30, Λ = 600, K = 20 TABLE IV ACCELERATION PERFORMANCE Decoposon Reducon ToalCPUTe s Obj $ No No NaN NaN Yes No 4,799 1,983,909 No Yes 197 1,983,879 Yes Yes 61 1,983,823 α = 0.5, Λ = 30, Λ = 600, K = 20 Consran Volaon MW MC BE Hour Load Deand MW 6,000 5,000 4,000 F. 5. Consran Volaon n he Wors Scenaro α = Hour Wner Suer The value n VoWors s obaned by solvn anoher axn proble usn he EP approach ven he UC and ED soluons fro he BE and MC approach. I s observed ha he soluon fro he MC approach s no robus. In he worscase scenaro, he consran volaon s 35MW. The dealed values for he hourly consran volaons are depced n F. 4. F. 4 shows ha he consran volaons occur a Hour 9, 10, 13, 17, 19, 20, and 21. The lares volaon 10MW happens a Hour 21. The oal soluon e for RSCUC n colun ToalCPUTe verfes he copuaonal advanae of he MC approach. The EP approach has he lares copuaon burden alhouh s soluon s exac. I should be poned ou ha he copuaon burden of he EP approach s relaed o he nuber of he EPs. When here are only 30 buses wh unceranes n he syse, he EP approach can fnd he soluon n less han 200 seconds. In he EP approach, Theore 3 elnaes he redundan consrans, bu he nuber of he bnary varables reans he sae afer he elnaon. In he BE approach, Theore 3 reduces he bnary varables and elnaes he relaed consrans. The proposed BE approach shows a ood rade-off beween he robusness and he copuaon burden accordn o he daa n Table II. 2 α = 0.5: Table III and F. 5 depc he sulaon resuls. Copared wh he uncerany se when α = 1, he sze of he uncerany se n hs par s reduced. Therefore, he base-case coss n Table III are lower han hose n Table II. I can also be observed n Table III ha he basecase cos obaned fro he BE approach s hher han ha fro he MC approach, alhouh he dfference $379 = 1, 983, 823 1, 983, 444 s relavely sall. However, he daa n VoWors shows ha he BE approach s uch beer han he MC approach n ers of soluon robusness. In he worscase scenaro, he oal consran volaon of he MC approach s abou 62MW. I s observed ha he consran volaons are lare a Hour 10-13, and 19-21, when he load deands are hh. The daa n Table II and Table III show ha he soluon robusnesses can be snfcanly dfferen even f he operaon coss are close. Ths ndcaes he crcaly of he soluon F. 6. Addonal Load Deand Profles n 24 Hours opaly of he ax-n proble. The fac ha he soluon fro he MC approach fals o survve n he wors-case scenaro s also conssen wh he analyss n he frs pararaph n hs subsecon. C. Perforance of Reforulaon and Acceleraon In hs par, we presen he perforance of he proposed reforulaon and he wo acceleraon echnques nroduced n Secon IV. For he IEEE 118-bus syse, here are 8, 928 = nework consrans and 24 equaly consrans for load balance. When K s se as 20, we need o odel 179, 040 bnary varable z n he ornal RP shown n Secon III. The decoposon echnque n Secon IV-B breaks he snle e-coupled lare ax-n proble no a seres of saller probles, one for each e nerval. Theore 3 can reove abou 95% nacve nework consrans n he nner nzaon proble of he ax-n proble. Table IV shows he dfferen copuaon perforances when we apply he acceleraon echnques. Colun Toal- CPUTe presens he oal CPU e when he procedure s convered. Colun Obj shows he base-case operaon cos. There are sall cos dfferences, whch are whn $100. I can be observed ha no soluon s found whn he ven e l f we do no apply any acceleraon echnques. Specfcally, he MIP solver canno e any feasble soluon o proble RP whn 400 seconds. Wh he decoposon echnque only, he soluon s found n 4,799 seconds afer solvn ulple MIP probles. Wh he reducon echnque only, he soluon e s reduced snfcanly o 197 seconds. The bes perforance a soluon e of 61 seconds s acheved whle applyn boh acceleraon echnques. I s observed ha n he ax-n proble, he nework consran s he an facor leadn o copuaon coplexy. The copuaonal coplexy s also dependen on he load profles. Hence, addonal sulaons are perfored wh wner and suer load profles fro hps://

8 8 Decoposon Reducon TABLE VI NUMERICAL INFORMATION FOR THE MIP MODELS BE BE Presolved EP Presolved # of Cons. # of Var. # of Bn. # of Cons. # of Var. # of Bn. # of Cons. # of Var. # of Bn. No No 389, , , , , ,816 9,744 19,222 2,400 Yes No 16,215 15,904 7,420 14,401 14,398 6, No Yes 10,803 20,937 2,100 7,324 7,279 2,019 9,276 10,026 2,400 Yes Yes 606 1, # of Cons. : Nuber of Consrans; # of Var. : Nuber of Connuous Varables and Bnary Varables; # of Bn. : Nuber of Bnary Varables. TABLE V AVERAGE CPU TIME OF SOLVING ONE MAX-MIN PROBLEM Decoposon Reducon Wner Profle Suer Profle BE s EP s BE s EP s No No Yes No No Yes Yes Yes α = 1, Λ = 30, Λ = 720, K = 20 washnon.edu/research/psca/rs/rs96/table-04.x. The load curves are llusraed n F. 6. The TeL for Gurob s chaned o 3600 seconds. Table V presens he averae CPU e of solvn one ax-n proble. Theore 3 can sll elnae over 95% lne consrans. The soluon process enerally converes afer up o 5 eraons usn Alorh 2, and n each eraon eher one ul-perod ax-n proble or ulple snle-perod ax-n probles should be solved. In eneral, he CPU e has he sae rend n he wner and suer load profles. Table V shows ha s he os effcen o solve he snle-perod ax-n proble wh he reducon echnque. In he BE approach, he averae copuaon e s less han one second. Wh he reducon echnque, he BE approach can also solve he ul-perod ax-n proble whn 14 seconds on averae. I s observed ha applyn decoposon echnque leads o slhly loner averae soluon e n each eraon wh he wner profle,.e = > The boo lne s ha provdes an opon for he hh-perforance parallel copun, whch s especally poran when he sze of he NP-hard proble s lare. In he EP approach, he MIP solver ofen sops when he e l s reached. The EP approach has uch heaver copuaonal burden han he BE approach accordn o Table V. In he wner case, he averae CPU e s seconds for solvn a snle-perod ax-n proble n he EP approach. I s 434 es ha n he BE approach. In suary, he daa n Table V suess ha he BE approach becoes powerful wh he acceleraon echnques. I also ndcaes ha he acceleraon echnques are ore useful n he BE approach. Table VI shows he nuercal nforaon for he ypcal MIP odels n he wo approaches. The Gurob MIP solver can reduce he proble sze by presolvn he MIP proble. The nforaon of he presolved odel s also lsed n he colun BE Presolved and EP Presolved. I can be observed ha 178,080 varables are ornally odeled n he BE approach. I s noed ha 178,080 s saller han 179, 040 = The reason s ha he shf facors for one lne are all zeros wh respec o he buses wh unceranes. In he presolved odel, he nuber of bnary varables s reduced by 15, 264 = 178, , 816 and he nuber of consrans s reduced by 43, 186 = 389, , 974. The decoposon echnque alone can reduce he nuber of bnary varables o 4.17%.e. 4.17%=6,784/162,816. The reducon echnque alone can reduce he nuber of bnary varables o 1.24%.e. 1.24%=2,019/162,816. Boh decoposon and reducon echnques oeher can reduce he nuber of bnary varables o 0.094%.e %=153/162,816. I can be observed ha boh acceleraon echnques can snfcanly reduce he sze of he MIP odel n he BE approach. In conras, he reducon echnque n he EP approach can only reduce he nubers of consrans and connuous varables. The nuber of bnary varables reans 2,400 when he reducon echnque s appled n he EP approach. The perforance of he MIP solver s no only relaed o he proble sze, bu also o he odel self. As shown n he las row of Table VI, he MIP odels n he BE approach and he EP approach have slar szes.e. around 400 consrans and 500 varables when he wo acceleraon echnques are appled. However, he MIP solver has uch beer copuaonal perforance n he BE approach han ha n he EP approach accordn o Table V. If only he decoposon echnque s appled, he presolved MIP odel has 14,401 consrans and 6,784 bnary varables n he BE approach, whch s uch larer han he MIP odel wh 406 consrans and 100 bnary varables n he EP approach. Bu he averae soluon e n he BE approach s sll uch less han ha n he EP approach accordn o Table V. These observaons ndcae ha he MIP solver has beer copuaonal perforance for he odels n he BE approach. I suess ha he cun plane echnque and oher echnques n he odern MIP solver are ore effecve for he proposed odel. VI. CONCLUSION In hs paper, a novel BE approach o solve he ax-n proble s proposed n he RSCUC fraework. The nonconvex ax-n proble s reforulaed as a MIP proble. The new reforulaon s no dependen on he closed fors for he exree pons n uncerany se. The soluon qualy of he ax-n proble s observable and conrollable n he

9 9 proposed BE approach. To reduce he copuaon burden, wo effecve acceleraon echnques are also proposed n hs paper. The effecveness and advanaes are deonsraed n he case sudy wh IEEE 118-Bus syse. The NP-hard ax-n proble n he robus SCUC fraework s noorously hard o solve. The lobal opaly of he ax-n proble s he os essenal par n RSCUC because represens he soluon robusness. The BE approach proposed n hs paper shows he ood perforance n he opaly as well he copuaon e. Furherore, s applcable when oher uncerany ses are adoped. I wll be useful n he daa-drven RSCUC whch res o overcoe he conservaveness ssue. If RSCUC s used for arke clearn, he conrollable and observable soluon qualy n he proposed BE approach has poran plcaons, as dfferen soluons ay lead o varous prces. I s an poran fuure work. I s also possble o furher prove he copuaon perforance wh a hybrd soluon approach cobned wh BE and MC/OA. The BE approach can be appled o check he lobal opaly and enerae he new wors pon afer he RSCUC soluon s convered wh he MC/OA approach. The soluon fro he BE approach can also serve as a sarn pon n he MC/OA approach. Anoher fuure work s o es he proposed BE approach on lare-scale syses. Based on he resuls of a prelnary sulaon sudy wh he ISO-scale Polsh syse [27], he proposed BE approach s sll racable for solvn he axn proble. We also plan o work wh several ISOs n he U.S. o es he proposed BE approach usn he real daa of her syses. n y,z,i,p F s. APPENDIX A DETAILED FORMULATION OF P AND Rɛ Cp,, I, 33 p, = d,, 34 F l Γ l, p, d, F l, l, 35 I, p n G p, I, p ax,, 36 p, p, 1 r u 1 y, + p n y,,, 37 p, + p, 1 r d 1 z, + p n z,,, 38 nu on/off e l, 39 where p,, I,, y,, and z, are un dspach, un saus, sarup ndcaor, and shudown ndcaor, respecvely. r u and r d are un rapn up and rapn down raes, respecvely. In hs paper, x denoes he bnary varable vecor ncludn I,, y,, and z,, and p denoes he dspach vecor. 34 represens he eneraon/load balance consran; 35 represens he power flow consran; 36 represens he nu and axu capacy consrans; represen he rapn up and rapn down consrans, respecvely. Consrans are rewren as Ax + Bp b. The feasble se F s defned as F := y, z, I, p : ɛ U, p, p, + p, = d, + ɛ,, 40 F l Γ p, + p, d, ɛ, G F l, l, 41 p p, r, x, p,, 42 r,x, where r, x, p and, x, p are he upper bound and lower r bound of p,. Consrans are rewren as Consran 42 s rewren as Dp + p + Eɛ h. F x + Gp + H p. Rɛ s forulaed as Rɛ := p, s D +, s D, s L +, s L : 43 s D +, s D, s L +, s L 0 p, + p, = p nj, = G d, + ɛ, + s D+ s D, 44 p, + p, d, ɛ,,, 45 F l Γ l, p nj, + s L+ l s L l F l, l, 46 p p, r, x, p,, 47 r,x, where 44 denoes he eneraon/load balance consrans, 45 defnes he ne power njecon p nj,, 46 denoes he lne capacy consrans, 47 denoes he ls of he un eneraon adjusen p, as a funcon of ven UC and ED as well as un rapn raes and eneraon capaces. APPENDIX B PROOF OF THEOREMS 1 AND 2 λ Eɛ s approxaed as N s λ Eɛ = λ q + = λ q + λ q q N s K 2 k z,k q + λ q k=0 Denoe ξ,k := z,k q. We have q, f z,k = 1, 48 ξ,k = 0, f z,k = ɛ s bounded accordn o he defnon of U. Hence, Eɛ s also bounded. Wh 18, we also have 0 q q. Therefore, consrans 4849 are equvalen o he follown consrans ξ,k 0, ξ,k q q + q z,k 50 ξ,k q z,k, ξ,k q 51

10 10 Furherore, as RP s a axzaon proble and he coeffcens of ξ,k are posve, he consrans 50 are redundan. The feasble reon of λ and µ n RP rean he sae, alhouh 17 s nroduced o lnearze he blnear er n he objecve funcon. Then, we only need o consder he dfference of he objecve funcons. Ns K q λ Eɛ 2 k z,k q + λ k=0 N s N s K = λ q 2 k z,k q k=0 N s K = λ 2 k z,k q 2 K q N s k=0 The frs equaly follows 18. Wh he upper bound of BE approxaon λ K k=0 2 k z,k 2 K, he nequaly follows. APPENDIX C PROOF OF THEOREM 3 The flow consran for lne l n he posve drecon a s nacve f lne capacy F l s reaer han ax Γ l, p nj, : Wh he nroducon of p nj+, proble s reforulaed as ax Γ l, p nj+, + f v l, : p nj, = 0, p nj, p nj, p nj,. := p nj, p nj,, he above p nj+, = p v, 0 p nj+, p nj+, 52 If 52 s feasble, here us exs an neer j sasfyn 24. If he shf facors are ordered as 26, he opal soluon o 52 s p nj+ p nj+ n,, n = 1,, j 1 n, = p v j 1 n=1 pnj+ n,, n = j 0, n = j + 1,, N d, and he opal value s j 1 n=1 j 1 j 1 Γ l,n p nj+ n, + Γ l,j p v n=1 p nj+ n, = Γ l,n Γ l,j p nj+ n, + Γ l,j p v + f v n=1 Therefore, Theore 3 s proved. l,. + f v l, APPENDIX D PROOF OF THEOREM 4 Fro sep 2 o sep 5 n Alorh 3, here are T axn subprobles. In he frs eraon fro sep 6 o sep 15, here are T ax-n subprobles. Afer he frs eraon fro sep 6 o sep 15, here s only one ax-n subproble, and v decreases by 1 each e. The axal volaon afer sep 5 s Λ Λ. The whle loop ends unl v = Λ afer Λ Λ eraons. Hence, he oal nuber of axn subproble s 2T + Λ Λ. In fac, f φ v = 0, hen φ v 1 us be 0. Hence, n hs case, here s no need o solve anoher ax-n subproble o e he value of φ v 1. APPENDIX E PROOF OF PROPOSITION 1 The ax-n proble USP can also be reforulaed as a blnear proran proble accordn o he dualy heory. Denoe as UBP. Consder λ ud as he dual varable for 21 and λ d for 44. On a sde noe, 44 s expressed as wo nequaly consrans n 10c. Then he only dfference beween BP and UBP s ha 1 λ d 1, 53 whle λ ud s unbounded. Hence, φ u φ follows. When he opal soluon x, p o P s found, we have φ = φ u = 0. I eans s D + = s D = 0. Therefore, addn consran 1 λ ud 1, n UBP does no aler he opal soluon. APPENDIX F PROOF OF PROPOSITION 2 I s observed ha V 1 v 1,, V T v T : v Λ, v Λ, v Z + = U. On he oher hand, ven x, ɛ, he nner proble n SP s e-decoupled. Therefore, he feasble reon of IP and ha of SP are he sae. Hence, φ IP = φ and soluon ɛ v s a sub-vecor of ɛ. REFERENCES [1] X. Guan, P. B. Luh, H. Yan, and J. Aalf, An opzaon-based ehod for un coen, Inernaonal Journal of Elecrcal Power & Enery Syses, vol. 14, no. 1, pp. 9 17, [2] M. Shahdehpour, H. Yan, and Z. L, Marke Operaons n Elecrc Power Syses: Forecasn, Scheduln, and Rsk Manaeen, 1s ed. Wley-IEEE Press, [3] Z. L and M. Shahdehpour, Secury-consraned un coen for sulaneous clearn of enery and ancllary servces arkes, IEEE Trans. Power Sys., vol. 20, no. 2, pp , [4] Ineraon of wnd no syse dspach, New York ISO, Tech. Rep., [5] Ineraon of renewable resources, Calforna ISO, Tech. Rep., [6] L. Wu, M. Shahdehpour, and T. L, Sochasc secury-consraned un coen, IEEE Trans. Power Sys., vol. 22, no. 2, pp , [7] J. Wan, A. J. Conejo, C. Wan, and J. Yan, Sar rds, renewable enery neraon, and clae chane aon fuure elecrc enery syses, Appled Enery, vol. 96, pp. 1 3, [8] R. Jan, J. Wan, and Y. Guan, Robus un coen wh wnd power and puped sorae hydro, IEEE Trans. Power Syses, vol. 27, no. 2, pp , 2012.

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