Dual Approximate Dynamic Programming for Large Scale Hydro Valleys

Transcription

1 Dual Approxmae Dynamc Programmng for Large Scale Hydro Valleys Perre Carpener and Jean-Phlppe Chanceler 1 ENSTA ParsTech and ENPC ParsTech CMM Workshop, January Jon work wh J.-C. Alas, suppored by he FMJH Program Gaspard Monge for Opmzaon. P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 1 / 24

2 Movaon Elecrcy producon managemen for hydro valleys 1 year me horzon: compue each monh he Bellman funcons waer values ) sochasc framework: ran, marke prces large-scale valley: 5 dams and more We wsh o reman whn he scope of Dynamc Programmng. P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 2 / 24

3 How o push he curse of dmensonaly lms? Aggregaon mehods fas o run mehod requre some homogeney beween uns Sochasc Dual Dynamc Programmng SDDP) effcen mehod for hs knd of problems srong assumpons convexy, lneary) Dual Approxmae Dynamc Programmng DADP) spaal decomposon mehod complexy almos lnear n he number of dams approxmaon mehods n he sochasc framework Ths alk: presen numercal resuls for large-scale hydro valleys usng DADP, and comparson wh DP and SDDP. P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 3 / 24

4 Lecure oulne 1 Hydro valley modelng Opmzaon problem 2 3 P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 4 / 24

5 Hydro valley modelng Opmzaon problem 1 Hydro valley modelng Opmzaon problem 2 3 P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 5 / 24

6 Operang scheme Hydro valley modelng Opmzaon problem a 1 x 1 Dam 1 u 1 a 2 x 2 Dam 2 u 2 a 3 x 3 u 3 u : waer urbnaed by dam a me, x : waer volume of dam a me, a : waer nflow a dam a me, p : waer prce a dam a me, Dam 3 Randomness: w = a, p ), w = w 1,..., w N ). P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 6 / 24

7 Dynamcs and cos funcons Hydro valley modelng Opmzaon problem z a x Dam u s z +1 Dam dynamcs: x+1 = f x, u, w, z), = x u +a +z s, beng he ouflow of dam : z +1 = g x, u, w, z), z +1 { = u + max 0,x u +a +z x}. }{{} s We assume he Hazard-Decson nformaon srucure u s chosen once w s observed), so ha u u mn { u, x + a + z x }. Gan a me < T : L x, u, w, z ) = p u ɛu )2. Fnal gan a me T : K x T ) = a mn{0,x T x } 2. P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 7 / 24

8 Hydro valley modelng Opmzaon problem 1 Hydro valley modelng Opmzaon problem 2 3 P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 8 / 24

9 Sochasc opmzaon problem The global opmzaon problem reads: max E N X,U,Z) =1 T 1 Hydro valley modelng Opmzaon problem L X, U, W, Z ) + K X ) )) T, =0 subjec o: X +1 = f X, U, W, Z ),,, σ U ) σ W 0,..., W ),,, Z +1 = g X, U, W, Z ),,. Assumpon. Noses W 0,..., W T 1 are ndependen over me. P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 9 / 24

10 1 Hydro valley modelng Opmzaon problem 2 3 P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 10 / 24

11 Prce decomposon Dualze he couplng consrans Z +1 = g X, U, W, Z ). Noe ha he assocaed mulpler Λ +1 s a random varable. Mnmze he dual problem usng a graden-lke algorhm). z a A eraon k, he dualy erm: x Dam u s z +1 Dam + 1 Λ +1,k) Z +1 g X, U, W, Z ) ), s he dfference of wo erms: Λ +1,k) Z +1 dam +1, Λ +1,k) ) g dam. Dam by dam decomposon for he maxmzaon n X, U, Z) a Λ +1,k) fxed. P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 11 / 24

12 1 Hydro valley modelng Opmzaon problem 2 3 P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 12 / 24

13 DADP core dea The -h subproblem wres: max E U,Z,X T 1 =0 L X, U, W, Z ),k) + Λ Z Λ +1,k) g X, U, W, Z ) ) + K X T ) ), bu Λ,k) depends on he whole pas of noses W 0,..., W )... The core dea of DADP s o replace he consran Z +1 g X, U, W, Z ) = 0 by s condonal expecaon wh respec o Y : E Z +1 g X, U, W, Z ) Y ) = 0, where Y 0,..., Y T 1) s a well-chosen nformaon process. P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 13 / 24

14 DADP core dea The -h subproblem wres: max E U,Z,X T 1 =0 L X, U, W, Z ),k) + Λ Z Λ +1,k) g X, U, W, Z ) ) + K X T ) ), bu Λ,k) depends on he whole pas of noses W 0,..., W )... The core dea of DADP s o replace he consran Z +1 g X, U, W, Z ) = 0 by s condonal expecaon wh respec o Y : E Z +1 g X, U, W, Z ) Y ) = 0, where Y 0,..., Y T 1) s a well-chosen nformaon process. P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 13 / 24

15 Subproblems n DADP DADP hus consss of a consran relaxaon. Ths consran relaxaon s equvalen o replace he orgnal mulpler Λ,k) by s condonal expecaon E Λ,k) ) Y 1. The expresson of he -h subproblem becomes: max E U,Z,X T 1 L X, U, W, Z ),k) ) + E Λ Y 1 Z =0 E Λ +1,k) ) Y g X, U, W, Z ) ) + K X T ) ). If he process Y 1 follows a dynamcal equaon, DP apples. P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 14 / 24

16 A crude relaxaon: Y cse 1 The mulplers Λ,k) appear only n he subproblems by means of her expecaons E Λ,k) ), so ha each subproblem nvolves a 1-dmensonal sae varable. 2 For he graden algorhm, he coordnaon ask reduces o: E Λ,k+1) ),k) ) = E Λ + ρ E Z +1,k) g X,k), U,k), W, Z,k) ) ). 3 The consrans aken no accoun by DADP are n fac: E Z +1 g X, U, W, Z ) ) = 0. The DADP soluons do no sasfy he nal consrans: need o use an heursc mehod o regan admssbly. P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 15 / 24

17 How o regan admssble polces? We have compued N local Bellman funcons V a each me, each dependng on a sngle sae varable x... whereas we need one global Bellman funcon V dependng on he global sae x 1,..., x N) n order o desgn he decsons by usng one-sep Dynamc Programmng. Heursc proposal: V x 1,..., x N) = N =1 V x ), he one-sep DP problem o solve a me beng: N max L u 1,...,u N x, u, w, z ) + V ) +1 x +1, ) wh x +1 = f =1 x, u, w, z ) and z +1 = g x, u, w, z ). P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 16 / 24

18 How o regan admssble polces? We have compued N local Bellman funcons V a each me, each dependng on a sngle sae varable x... whereas we need one global Bellman funcon V dependng on he global sae x 1,..., x N) n order o desgn he decsons by usng one-sep Dynamc Programmng. Heursc proposal: V x 1,..., x N) = N =1 V x ), he one-sep DP problem o solve a me beng: N max L u 1,...,u N x, u, w, z ) + V ) +1 x +1, ) wh x +1 = f =1 x, u, w, z ) and z +1 = g x, u, w, z ). P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 16 / 24

19 How o regan admssble polces? We have compued N local Bellman funcons V a each me, each dependng on a sngle sae varable x... whereas we need one global Bellman funcon V dependng on he global sae x 1,..., x N) n order o desgn he decsons by usng one-sep Dynamc Programmng. Heursc proposal: V x 1,..., x N) = N =1 V x ), he one-sep DP problem o solve a me beng: N max L u 1,...,u N x, u, w, z ) + V ) +1 x +1, ) wh x +1 = f =1 x, u, w, z ) and z +1 = g x, u, w, z ). P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 16 / 24

20 1 Hydro valley modelng Opmzaon problem 2 3 P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 17 / 24

21 Three case sudes dam 1 dam 1 dam 1 dam 2 dam 2 dam 2 dam 3 Dscrezaon T 12 X 41 U 6 W 10 3Dams Valley dam 3 dam 4 4Dams Valley dam 3 dam 4 dam 5 5Dams Valley P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 18 / 24

22 Resuls Valley 3Dams 4Dams 5Dams DP CPU me DP value Table: Resuls obaned by DP Valley 3Dams 4Dams 5Dams DADP CPU me DADP value Gap wh DP 3.2% 2.0% 1.1% Dual value Table: Resuls obaned by DADP Expecaon Resuls obaned usng a 4 cores 8 hreads InelRCore 7 based compuer. P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 19 / 24

23 Resuls Valley 3Dams 4Dams 5Dams DP CPU me DP value SDDP d value SDDP d CPU me Table: Resuls obaned by DP and SDDP d Valley 3Dams 4Dams 5Dams DADP CPU me DADP value Gap wh DP 3.2% 2.0% 1.1% Dual value Table: Resuls obaned by DADP Expecaon Resuls obaned usng a 4 cores 8 hreads InelRCore 7 based compuer. P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 19 / 24

24 Resuls Valley 3Dams 4Dams 5Dams DP CPU me DP value SDDP d value SDDP d CPU me Table: Resuls obaned by DP and SDDP d Valley 3Dams 4Dams 5Dams DADP CPU me DADP value Gap wh DP 3.2% 2.0% 1.1% Dual value Table: Resuls obaned by DADP Expecaon Resuls obaned usng a 4 cores 8 hreads InelRCore 7 based compuer. P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 19 / 24

25 1 Hydro valley modelng Opmzaon problem 2 3 P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 20 / 24

26 Three valleys Gnoure Izour Soulcem Bor dam 1 Auza Mareges dam 2 dam 3 Sabar Agle dam 4 Dscrezaon T 12, W 10 fne grds for X and U Vcdessos Valley Chasang Sabler Dordogne Valley dam 5 dam 6 Soop Valley P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 21 / 24

27 Resuls Valley Vcdessos Dordogne Soop SDDP d CPU me SDDP d value Table: Resuls obaned by SDDP d Valley Vcdessos Dordogne Soop DADP CPU me DADP value Gap wh SDDP d +0.2% 2.4% 2.8% Dual value Table: Resuls obaned by DADP Expecaon P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 22 / 24

28 Resuls Valley Vcdessos Dordogne Soop SDDP d CPU me SDDP d value Table: Resuls obaned by SDDP d Valley Vcdessos Dordogne Soop DADP CPU me DADP value Gap wh SDDP d +0.2% 2.4% 2.8% Dual value Table: Resuls obaned by DADP Expecaon P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 22 / 24

29 CPU me comparson P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 23 / 24

30 Conclusons and perspecves Conclusons for DADP Fas numercal convergence of he mehod. Near-opmal resuls even when usng a crude relaxaon. Mehod ha can be used for very large valleys General perspecves Apply o more complex opologes smar grds). Connecon wh oher decomposon mehods. Theorecal sudy. P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 24 / 24

31 P. Carpener e G. Cohen. Décomposon-coordnaon en opmsaon déermnse e sochasque. En préparaon, Sprnger, P. Grardeau. Résoluon de grands problèmes en opmsaon sochasque dynamque. Thèse de docora, Unversé Pars-Es, J.-C. Alas. Rsque e opmsaon pour le managemen d énerges. Thèse de docora, Unversé Pars-Es, V. Leclère. Conrbuons aux méhodes de décomposon en opmsaon sochasque. Thèse de docora, Unversé Pars-Es, K. Bary, P. Carpener, G. Cohen and P. Grardeau, Prce decomposon n large-scale sochasc opmal conrol. arxv, mah.oc , P. Carpener & J.-P. Chanceler DADP appled o large scale hydro valleys CMM Workshop 24 / 24