DECOMPOSITION-COORDINATION METHOD FOR THE MANAGEMENT OF A CHAIN OF DAMS

DECOMPOSITION-COORDINATION METHOD FOR THE MANAGEMENT OF A CHAIN OF DAMS J-C. ALAIS, P. CARPENTIER, V. LECLÈRE Absrac. We sudy he managemen of a chan of dam hydroelecrc producon where we consder he expeced gan semmng from he producon as he creron o maxmze. However solvng drecly he problem by Dynamc Programmng approach can be numercally mpossble because of he so-called curse of dmenson.consequenly we wll use some decomposon-coordnaon mehod on hs problem. However f decomposon-coordnaon mehods are well known n a deermnsc seng, few resuls are avalable n a sochasc seng. We wll presen a smple problem wh hree dams, ha can be solved by dynamc programmng, and he Dual Approxmae Dynamc Programmng DADP decomposon mehod we are usng on hs problem. As we have he exac soluon of he problem, we can presen a orough sudy of he numercal properes of DADP. Hydroelecrcy s he man renewable energy n many counres. I provdes a clean no greenhouse gases emssons and fas-usable energy ha s cheap and subsuable for he hermal one. I s all he more mporan o ensure s proper use ha comes from a shared lmed ressource: he reservor waer. Ths s he dam hydroelecrc producon managemen purpose. Mos dams are nerconneced n a hydroelecrc valley, ha s he waer urbned by one dam s gong as an nflow n anoher one. Thus he dmenson of he problem s mulpled by he number of dams n he valley. As s well known, he curse of dmensonaly forbds o use Dynamc Programmng for hydraulc valley of more han 5 dams 1. Consequenly we have o use an approxmae algorhm, lke a prce decomposon mehod. However prces n a sochasc seng would be sochasc processes, and hus would be nracable. We approach he problem by replacng he prces by her condonal expecaon wh respec o some nformaon varable, and show he numercal properes of hs mehod. 1. Inroducon We are neresed n presenng a mehod of decomposon-coordnaon n a sochasc seng appled o he managemen of a chan of dams. There exss some mehods o address hs problem of hydroelecrc valley managemen lke Progressve-Hedgng see [6], Aggregaon see [8] or Sochasc Dual Dynamc Programmng SDDP, see [5] for he orgnal presenaon, and [7] for a recen analyss of he mehod. Mos of heses mehods are scenaro-ree based, whch mply some serous lmaons when you wan o oban a polcy. We are gong o work on hs problem by usng some decomposon-coordnaon mehods, as presened n [3]. However he exenson o a sochasc seng s no really smple see [2] for example. Recenly an algorhm has been proposed n [1] ha came from he prce-decomposon mehod, and we are applyng on he opmal managemen of a chan of dam problem. Dae: Augus 5, 2012. 1 Indeed wh our choce of dscrezaon we need abou 90 seconds o solve a 3 dam valley, hus a 5 dam valley would need abou 4 days of compung. 1 1161

2 J-C. ALAIS, P. CARPENTIER, V. LECLÈRE 1.1. Descrpon of he problem. We consder a chan of N dams where he ouflows of he dam are nflows for he dam +1. We consder ha all dams are conrolled by he same frm, and hus we wan o opmze he sum of he payoffs. We presen here he problem we are addressng, as shown n fgure 1. Fgure 1. The rver chan model 1.1.1. Dynamcs of he dam. Le me vary n {0,..., T }. For all dams {1,..., N}, he followng posve real valuaed random varables are defned on a probably space Ω, F, P: x, he sorage level of dam a he begnnng of perod [, + 1[, u he hydrourbne ouflows of dam durng [, + 1[, sae conrol w and p, he exernal nflows and he producon earnngs of dam durng [, + 1[. nose We assume ha he nose are random varables ha are muually and sep by sep ndependen. They are unformly dsrbued on a dscree se. The dynamcs of he reservor sorage level reads, for he frs dam of he chan : And for any oher dam > 1 we have x 1 +1 = f 1 x 1,u 1,w 1,0, = x 1 u 1 + w 1. x +1 = f x,u,w,z, = x u + w + z, where z = x 1 u 1 + w 1 + z 1 1 1162

DECOMPOSITION-COORDINATION METHOD FOR THE MANAGEMENT OF A CHAIN OF DAMS 3 s he waer nflows n dam comng from dam 1, s also he oal ouflows of dam 1. The bound consrans are:e x +1 x +1 x +1 and u u u, {0,..., T 1}. 2 Moreover we assume he Hazard-Decson nformaon srucure u s chosen once w s observed, so ha u u mn { u,x + w + z x }. 1.1.2. Objecve funcon. We are consderng he mulple sep managemen of a chan of dams, each dam produces elecrcy, wh an effcency coeffcen η, ha s sold a he same prce. Thus he hydroelecrc valley obeys he followng valorzaon 2 mechansm N T 1 =1 =0 p η u + εu 2 + K x T, 3 where K s a funcon valorzng he remanng waer a me T n he dam. The εu 2 erm s here o represen some non-lneary n he effcency of urbnes as well as numercally sablze he problem by makng srongly convex. As hs creron s random, we choose o mnmze he expeced cos Le F = { F }=0,...,T [ N E =1 T 1 =0 p η u + εu 2 }{{} =Lx,u,w,z be he flraon of pas noses: F = σ W 0,...,W, wh W = { W 1,...,W N }. Thus he sochasc opmzaon problem we are solvng reads ] +K x T. 4 subjec o: N T 1 mn E L,U,W,Z + K x T, 5a,U,Z =1 =0 +1 = f,u,w,z,,, Z +1 = g,u,w,z,,, 5b 5c as well as measurably consrans: U F,,. 5d 1.1.3. Some remarks abou he opmzaon problem. The noses W are ndependen over me, so ha he problem can be heorecally solved by Dynamc Programmng DP. The resulng opmal feedback laws a me depend on he curren saes of he dams: U = γ 1,...,,..., N,,. 2 As usual n opmzaon we choose o mnmze he oppose of he gan. 1163

4 J-C. ALAIS, P. CARPENTIER, V. LECLÈRE However DP s subjec o he curse of dmensonaly: he mehod s no numercally racable as soon as N 5. And hus we have o fnd anoher numercal soluon. Le s noe ha he couplng beween he dams arses only from Equaon 1 : Z +1 = g,u,w,z. And hs s he consran we wll dualze n order o use prce-decomposon mehod on. 2. Prce decomposon and Uzawa s algorhm The man dea of he prce decomposon of problem 5c s o see Z varable as shown n fgures 2 a and 2 b. as an ndependan a b Fgure 2. a: Whole Problem b: Decomposed Problem 2.1. Dualzaon of he couplng consran. We am a dualzng Consran 1 and a solvng he Problem 5 by usng he Uzawa algorhm: a eraon k, he assocaed mulpler s a fxed F -measurable random varable λ +1 k, and he erm under he expecaon nduced by dualy n he cos funcon s λ +1 k. Z +1 g,u,w,z, noe ha λ +1 k s relaed o. I can be decomposed as k.z +1 λ +1 : erm peranng o dam + 1. λ +1 k.g,u,w,z : erm peranng o dam. Fnally, he followng erm s added o he cos of dam λ k.z λ +1 k.g Consequenly he algorhm s done as follow : 1 we fx mulplers λ k for all and,,u,w,z. 1164

DECOMPOSITION-COORDINATION METHOD FOR THE MANAGEMENT OF A CHAIN OF DAMS 5 2 we have o solve N problems wh only one dam, 3 we updae he mulpler by a graden sep. 2.2. Opmzaon subproblem a eraon k. s: Consequenly opmzaon problem assocaed o dam a eraon k of he Uzawa algorhm mn E,U,Z subjec o: T 1 L,U,W,Z =0 + λ k.z λ +1 k.g,u,w,z + K xt, 6a +1 = f,u,w,z,, 6b and he measurably consrans: U F and Z F,. 6c Wh boundary condons: Z 1 0 and λ N+1 0. Ths problem s a one dmensonal dam problem and can be solved by DP or by any oher mehod. 3.1. DADP prncple. 3. Dual Approxmae Dynamc Programmng DADP The presence of he random varables λ k prevens us o use DP unless he propery =0,...,T 1 of ndependence of he λ over me s verfed, whch s no he case. The dea of DADP, as presened n [1] and [4] s o replace he known mulpler λ k by s condonal expecaon w.r.. a chosen nformaon varable, namely E λ k, or equvalenly o replace 5c by E Z g 1 1,U 1,W 1,Z 1. 7 Le s noe ha hs approxmaon s a relaxaon of he problem as he consran s loosened, and hus he sraeges ha we derve may no be admssble, even f he algorhm converges. Thus we sll have o consruc an admssble sraegy from he one we oban wh he DADP algorhm. In pracce, s a shor-memory process ha wll ener he sae varables of he subproblems. Possble choces for are: 1 cons: we deal wh he consran n expecaon, 2 = W 1 : we ncorporae he nose W 1 n Subproblem, 3 = f 1 : we mmc he dynamcs of 1. 1,W 1 We have choosen o explore he case where mmc 1. 3.2. Opmzaon subproblem n DADP. The condonal expecaon E λ k corresponds o a funcon ϕ k whch can be 1165

6 J-C. ALAIS, P. CARPENTIER, V. LECLÈRE pre-compued by a leas-sqare fng on some known rajecores for example. Consder he choce: = f 1 1,W. Subproblem wres: mne 1 T 1 L,U,W,Z + ϕ k.z ϕ +1 =0 subjec o measurably consrans are omed: k +1 +1 = f,u,w,z, +1.g,U,W,Z +K x T, 8a 8b = f 1 1 1,W, 8c = f +1 1,W. The sae s a 3-dmensonal vecor, consequenly Dynamc Programmng can be used o solve he sub-problem. 3.3. dealed algorhm. We gve here a formal presenaon of he algorhm. Frs he nalzaon of he algorhm should be done as follow We fx some random parcles ha s some rajecores of he nose W l [0,T ] ha wll be used hroughou he algorhm. We nalze λ 0 as deermnsc well chosen consans zero by defaul, and ϕ 0 as consan funcons. We defne 0 := 0 A beer sarng pon for λ could be found from he opmal soluon on he mean scenaro for example. Then a he begnnng of eraon k we should have defned A nose varable ξ. A varable of nformaon k whch should be an unconrolled process = f 1,ξ. A funcon ϕ k such ha ϕ ky E λ k k = y For each we solve 8d T mn E[ L,U,Z,U =0,W + ϕ k.z ϕ +1 k +1.g +1 = f,u,z,w +1 = f k,ξ +1 +1 = f +1 k +1,ξ +1,U,W ],Z U F Z F 1166

DECOMPOSITION-COORDINATION METHOD FOR THE MANAGEMENT OF A CHAIN OF DAMS 7 Ths gves us some opmal feedback laws γ k,, +1,W,ξ,ξ +1 η k,, +1,W,ξ,ξ +1 U Z ha are used wh,l k, U,l k, Z,l k,,l U,l k = γ k k, k, +1 Z,l k = η k k, k, +1,l k, o compue k, +1 k,w,l k,w,l,ξ,l,ξ +1,l,ξ,l,ξ +1,l and And fnally we can,l k +1 = f,l +1 +1,l +1,l k, U,l k, Z,l k = f k,l k,ξ,l k = f +1 k +1,l k,ξ +1,l k,w,l Updae of he prces rajecores: λ +1,l k := Z +1,l k+1 := λ +1,l k g,l k, U,l wh,l Defne a new nformaon dynamcs f k+1 Smulae,l k+1. Make a regresson of λ,l whch ermnae sep k. k + ρ k,l k,. k,w,l k+1 on,l k+1 o oban, Z,l k. ϕ k+1y E λ k+1 k+1 = y. 3.4. Heursc o consruc an admssble soluon. Once he algorhm has converged we have some feedbacks laws ha mus verfy he consran 7 E Z g 1 1,U 1,W 1,Z 1 1. s no verfed. Con- whch means ha he mechancal consran Z = g 1,U 1,W 1,Z 1 sequenly one has o defne an heursc o urn hs sraeges no an admssble one. As we have choosen such ha should mmcs 1 we can consruc an approxmae Bellman s value funcon for he global problem as he sum of he Bellman s value funcon of each subproblem where s replaced by. Consequenly we oban a global admssble sraegy by dong a one me sep opmzaon of he global problem. More precsely f we wre V,, +1 he Bellman s funcon obaned for subproblem when he DADP algorhm has converged, we defne Ṽ x = N =1 V, 1,, wh he convenon ha x 0 = 0. Then he conrol we choose a me when he chan of dams s n he sae x s he opmal soluon of mn u N =1 L,U,W,Z +Ṽ f x,u,w. 9 1167

8 J-C. ALAIS, P. CARPENTIER, V. LECLÈRE Le us noe ha hs problem s global on he chan, bu only done on one me-sep, and hus numercally racable. 4. Numercal Resuls In order o make some neresng sudy of hs mehod we have choosen a problem wh hree dams.e :N = 3, n order o be able o solve explcely he problem by dynamc programmng. Thus we can compare he soluon of DADP algorhm wh he exac soluon. Moreover we have done some sascal sudes on he opmal soluon n order o choose wsely. 4.1. Numercal parameers of he problem. The characerscs of he sudy are: {mn, max} bounds on = {0, 80} hm 3,, ; me seps number T = 12 one sep a monh over a year; {mn, max} bounds on U = {0, 40} hm 3 monh 1,,. The sochasc unverse s fne. The nose processes are whe and unformly dsrbued and he nflows a he hree dams reservors are correlaed. The smulaon s based on 500 nflows scenaros. Fgure 3 represens sx nflows scenaros a dam 1, dam 2 and dam 3 and Fgure 4 represens he prce scenaro. We se η 1 = η 2 = η 3. 4.2. Opmal soluon. We solve he problem by usng he dynamc programmng algorhm. expeced oal gan = 1.470 10 6 e; The Fgure 5, par a, shows sx represenave sorage level rajecores n hm 3, over 12 monhs ha we obaned by he negraon of he dynamc programmng-compued sraegy. 4.3. DADP soluon. We solve he problem by usng he DADP algorhm. The mulplers processes are esmaed by her expeced values. The scenaros whch are used o run he Uzawa algorhm are dfferen from hose whch are used o smulae he compued sraegy. The expeced values of he mulplers converge Fgure 6 and he resuls are: expeced oal gan = 1.405 10 6 e; eraons number = abou 3000. The approxmaon ha we make by esmang he mulplers as her expeced values leads o a loss of abou 1%. Ths s all he more promsng ha we use he smples nformaon varable. The smulaed sorage level rajecores appear que smlar n Fgure 5 o he opmal ones. Of course, hey are no exacly he same and we can see some sgnfcan dfferences bu her global aspecs correspond. I s hen neresng o noce, hanks o Fgure 5, he fac ha he sorage levels a dam 2 and dam 3 are lkely o be hgher wh he opmal sraegy han wh he approxmaed one; whereas does no a dam 1. Ths s due o he msesmaon of he couplng beween he reservors whch s relave o he combnaon of he approxmaon of he mulplers by her expeced values and he use of a heursc o make he sraegy admssble. The spaal correlaon beween he nflows nose random varables and he sharng of a common prce beween he dams may explan he fac ha he subopmal gan say prey close o he opmal one, however. By he way, we observe n Fgure 7 ha he sraeges aren far from beng almos surely he same for dam 1 and for dam 2 as he prces are sgnfcanly neresng or no. We see ndeed, ha he DP-compued and he DADP-compued sraeges que correspond a me 3, as he prce s he hghes, and a me 8, as s he lowes. Moreover, we see ha he sraeges are almos surely wh respec o he 500 scenaros he same for dam 3 from me 1 1168

DECOMPOSITION-COORDINATION METHOD FOR THE MANAGEMENT OF A CHAIN OF DAMS 9 Fgure 3. Sx nflows scenaros Fgure 4. Deermnsc prce rajecory o me 6. Ths s explaned by he abundance of waer a dam 3 durng {1,..., 6} whch leads o he opmal sraegy U 3 τ = u τ, τ {1,..., 6}. 1169

10 J-C. ALAIS, P. CARPENTIER, V. LECLÈRE Fgure 5. Sx sorage level rajecores Fgure 6. Convergence of he mulplers expeced values along 3000 eraons 5. Concluson The fronal approach by dynamc programmng o a problem lke he opmzaon of an hydroelecrc valley s no numercally racable because of he so-called curse of dmensonaly. 1170

DECOMPOSITION-COORDINATION METHOD FOR THE MANAGEMENT OF A CHAIN OF DAMS 11 Fgure 7. Dfferences n sock and conrols on he 500 smulaon scenaros ω Decomposon-coordnaon approach can no be drecly appled alogeher as he probablsc srucure mples ha each sub-problem would be as complcaed as he orgnal one. Thus DADP appears as a way of dong a prce-decomposon approach by replacng he mulpler λ by s condonal expecaon. Once he soluon of he relaed problem s found we use an heursc o oban an admssble sraegy from he one gven by DADP algorhm. Numercal resuls are que encouragng, and sascal sudes on he opmum λ gve some good nsghs n order o choose a good nformaon varable. 1171

12 J-C. ALAIS, P. CARPENTIER, V. LECLÈRE References [1] Bary, K. and Carpener, P. and Cohen, G. and Grardeau, P. Prce decomposon n large-scale sochasc opmal conrol. submed. [2] Carpener, P. and Cohen, G. and Culol, J.-C. Sochasc opmal conrol and decomposon-coordnaon mehods. Recen Developmens n Opmzaon. Lecure Noes n Economcs and Mahemacal Sysems, 1995. [3] Cohen, G. Auxlary problem prncple and decomposon of opmzaon problems. J. Opmz. Theory & Appl., 32:277 305, 1980. [4] Grardeau, P. Résoluon de grands problèmes en opmsaon sochasque dynamque e synhèse de los de commande. PhD hess, 2010. [5] MVF Perera and L. Pno. Mul-sage sochasc opmzaon appled o energy plannng. Mahemacal Programmng, 521:359 375, 1991. [6] Rockafellar, R.T. and Wes, R.J.B. Scenaros and polcy aggregaon n opmzaon under uncerany. Mahemacs of operaons research, pages 119 147, 1991. [7] A. Shapro. Analyss of sochasc dual dynamc programmng mehod. European Journal of Operaonal Research, 2091:63 72, 2011. [8] A. Turgeon. Opmal operaon of mul-reservor power sysems wh sochasc nflows. Waer Ressource Research, 1980. 1172