Dams management problem DADP in a nutshell Numerical experiments Dual Approximate Dynamic Programming for Large Scale Hydro Valleys P. Carpentier, J-P. Chancelier, V. Lecl` ere, F. Pacaud supported by the FMJH Program Gaspard Monge for Optimization. ENSTA ParisTech and ENPC ParisTech, France SOWG DADP applied to large scale hydro valleys Sept. 20, 2017 1 / 27
Dams management problem DADP in a nutshell Numerical experiments Motivation Electricity production management for hydro valleys 1 year time horizon : compute each month the “values of water” (Bellman functions) stochastic framework : rain, market prices large-scale valley : 5 dams and more We wish to remain within the scope of Dynamic Programming. SOWG DADP applied to large scale hydro valleys Sept. 20, 2017 2 / 27
Dams management problem DADP in a nutshell Numerical experiments How to push the curse of dimensionality limits? Aggregation methods fast to run method require some homogeneity between units Stochastic Dual Dynamic Programming (SDDP) efficient method for this kind of problems strong assumptions (convexity, linearity) Dual Approximate Dynamic Programming (DADP) spatial decomposition method complexity almost linear in the number of dams approximation methods in the stochastic framework This talk : present numerical results for large-scale hydro valleys using DADP, and comparison with DP and SDDP. SOWG DADP applied to large scale hydro valleys Sept. 20, 2017 3 / 27
Dams management problem DADP in a nutshell Numerical experiments Lecture outline Dams management problem 1 Hydro valley modeling Optimization problem DADP in a nutshell 2 Spatial decomposition Constraint weakening Numerical experiments 3 Academic examples More realistic examples SOWG DADP applied to large scale hydro valleys Sept. 20, 2017 4 / 27
Dams management problem Hydro valley modeling DADP in a nutshell Optimization problem Numerical experiments Dams management problem 1 Hydro valley modeling Optimization problem DADP in a nutshell 2 Spatial decomposition Constraint weakening Numerical experiments 3 Academic examples More realistic examples SOWG DADP applied to large scale hydro valleys Sept. 20, 2017 5 / 27
Dams management problem Hydro valley modeling DADP in a nutshell Optimization problem Numerical experiments Operating scheme a 1 t u 1 x 1 t a 2 t t Dam 1 u 2 x 2 a 3 t t t Dam 2 u 3 x 3 t t Dam 3 u i t : water turbinated by dam i at time t , x i t : water volume of dam i at time t , a i t : water inflow at dam i at time t , p i t : water price at dam i at time t , w t = ( w 1 Randomness: w i t = ( a i t , p i t , . . . , w N t ) , t ). SOWG DADP applied to large scale hydro valleys Sept. 20, 2017 6 / 27
Dams management problem Hydro valley modeling DADP in a nutshell Optimization problem Numerical experiments Dynamics and cost functions Dam dynamics: zi ai t t x i t +1 = f i t ( x i t , u i t , w i t , z i t ) , si = x i t − u i t + a i t + z i t − s i t , t ui xi z i +1 being the outflow of dam i : t t t zi +1 ⊕ z i +1 = g i t ( x i t , u i t , w i t , z i Dam i t ) , t t � t − x i � = u i 0 , x i t − u i t + a i t + z i t + max . � �� � s i t We assume the Hazard-Decision information structure ( u i t is chosen � t − x i � t is observed), so that u i ≤ u i once w i u i , x i t + a i t + z i t ≤ min . L i t ( x i t , u i t , w i t , z i Gain at time t < T : t ) = p i t ) 2 . t u i t − ǫ ( u i K i � � x i = − a i min { 0 , x i Final gain at time T : x i } 2 . T − � T SOWG DADP applied to large scale hydro valleys Sept. 20, 2017 7 / 27
Dams management problem Hydro valley modeling DADP in a nutshell Optimization problem Numerical experiments Dams management problem 1 Hydro valley modeling Optimization problem DADP in a nutshell 2 Spatial decomposition Constraint weakening Numerical experiments 3 Academic examples More realistic examples SOWG DADP applied to large scale hydro valleys Sept. 20, 2017 8 / 27
Dams management problem Hydro valley modeling DADP in a nutshell Optimization problem Numerical experiments Stochastic optimization problem The global optimization problem reads: � N ��� � T − 1 � � � � + K i � L i X i t , U i t , W i t , Z i X i max ( X , U , Z ) E , t t T i =1 t =0 subject to: X i t +1 = f i t ( X i t , U i t , W i t , Z i t ) , ∀ i , ∀ t , � � � � U i ⊂ σ W 0 , . . . , W t ∀ i , ∀ t , σ , t Z i +1 = g i t ( X i t , U i t , W i t , Z i t ) , ∀ i , ∀ t . t Assumption. Noises W 0 , . . . , W T − 1 are independent over time . SOWG DADP applied to large scale hydro valleys Sept. 20, 2017 9 / 27
Dams management problem Spatial decomposition DADP in a nutshell Constraint weakening Numerical experiments Dams management problem 1 Hydro valley modeling Optimization problem DADP in a nutshell 2 Spatial decomposition Constraint weakening Numerical experiments 3 Academic examples More realistic examples SOWG DADP applied to large scale hydro valleys Sept. 20, 2017 10 / 27
Dams management problem Spatial decomposition DADP in a nutshell Constraint weakening Numerical experiments Price decomposition Dualize the coupling constraints Z i +1 = g i t ( X i t , U i t , W i t , Z i t ). t Note that the associated multiplier Λ i +1 is a random variable. t Minimize the dual problem (using a gradient-like algorithm). zi ai t t At iteration k , the duality term: � � si Λ i +1 , ( k ) Z i +1 − g i t ( X i t , U i t , W i t , Z i t ) · t , t t ui xi t t is the difference of two terms: zi +1 Dam i t Λ i +1 , ( k ) · Z i +1 � dam i +1, t t Λ i +1 , ( k ) · g i � � � dam i . t t · · · Dam i + 1 Dam by dam decomposition for the maximization in ( X , U , Z ) at Λ i +1 , ( k ) fixed. t SOWG DADP applied to large scale hydro valleys Sept. 20, 2017 11 / 27
Dams management problem Spatial decomposition DADP in a nutshell Constraint weakening Numerical experiments Dams management problem 1 Hydro valley modeling Optimization problem DADP in a nutshell 2 Spatial decomposition Constraint weakening Numerical experiments 3 Academic examples More realistic examples SOWG DADP applied to large scale hydro valleys Sept. 20, 2017 12 / 27
Dams management problem Spatial decomposition DADP in a nutshell Constraint weakening Numerical experiments DADP core idea The i -th subproblem writes: � T − 1 � � � � + Λ i , ( k ) L i X i t , U i t , W i t , Z i · Z i max U i , Z i , X i E t t t t t =0 �� �� � + K i � − Λ i +1 , ( k ) X i t , U i t , W i t , Z i X i · g i , t t t T but Λ i , ( k ) depends on the whole past of noises ( W 0 , . . . , W t ) . . . t The core idea of DADP is to replace the constraint Z i +1 t ( X i t , U i t , W i t , Z i − g i t ) = 0 by its t conditional expectation with respect to Y i t : � � � Z i +1 t ( X i t , U i t , W i t , Z i � Y i − g i t ) = 0 , E t t where ( Y i 0 , . . . , Y i T − 1 ) is a “well-chosen” information process. SOWG DADP applied to large scale hydro valleys Sept. 20, 2017 13 / 27
Dams management problem Spatial decomposition DADP in a nutshell Constraint weakening Numerical experiments DADP core idea The i -th subproblem writes: � T − 1 � � � � + Λ i , ( k ) L i X i t , U i t , W i t , Z i · Z i max U i , Z i , X i E t t t t t =0 �� �� � + K i � − Λ i +1 , ( k ) X i t , U i t , W i t , Z i X i · g i , t t t T but Λ i , ( k ) depends on the whole past of noises ( W 0 , . . . , W t ) . . . t The core idea of DADP is to replace the constraint Z i +1 t ( X i t , U i t , W i t , Z i − g i t ) = 0 by its t conditional expectation with respect to Y i t : � � � Z i +1 t ( X i t , U i t , W i t , Z i � Y i − g i t ) = 0 , E t t where ( Y i 0 , . . . , Y i T − 1 ) is a “well-chosen” information process. SOWG DADP applied to large scale hydro valleys Sept. 20, 2017 13 / 27
Dams management problem Spatial decomposition DADP in a nutshell Constraint weakening Numerical experiments Subproblems in DADP DADP thus consists of a constraint relaxation. This constraint relaxation is equivalent to replace the original � � � . multiplier Λ i , ( k ) Λ i , ( k ) � Y i − 1 by its conditional expectation E t t t The expression of the i -th subproblem becomes: � T − 1 � � � � � � � Λ i , ( k ) L i X i t , U i t , W i t , Z i � Y i − 1 · Z i max + E U i , Z i , X i E t t t t t t =0 �� � � � � Λ i +1 , ( k ) � Y i X i t , U i t , W i t , Z i · g i − E t t t t �� + K i � X i . T If each process Y i follows a dynamical equation, DP applies. SOWG DADP applied to large scale hydro valleys Sept. 20, 2017 14 / 27
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