Sensitivity analysis for the outages of nuclear power plants Kengy - PowerPoint PPT Presentation

Introduction Study of the reference problem Sensitivity analysis Sensitivity analysis for the outages of nuclear power plants Kengy Barty † , Fr´ eric Bonnans ∗ , and Laurent Pfeiffer ∗ ed´ ∗ INRIA Saclay and CMAP, Ecole Polytechnique, † EDF R&D February 2012

Introduction Study of the reference problem Sensitivity analysis Introduction Study of a two-level problem: optimization of the dates of the outages of nuclear power plants optimization of the production of electricity. Our approach: 1 fix a schedule τ 2 optimize the production of electricity: V ( τ ) 3 perform a sensitivity analysis: compute V ′ ( τ ) 4 improve the schedule.

Introduction Study of the reference problem Sensitivity analysis Introduction Study of a two-level problem: optimization of the dates of the outages of nuclear power plants optimization of the production of electricity. Our approach: 1 fix a schedule τ 2 optimize the production of electricity: V ( τ ) 3 perform a sensitivity analysis: compute V ′ ( τ ) 4 improve the schedule.

Introduction Study of the reference problem Sensitivity analysis 1 Study of the reference problem Model Pontryagin’s principle Structure of optimal controls 2 Sensitivity analysis Abstract theorem Toy example Application to the outages

Introduction Study of the reference problem Sensitivity analysis 1 Study of the reference problem Model Pontryagin’s principle Structure of optimal controls 2 Sensitivity analysis Abstract theorem Toy example Application to the outages

Introduction Study of the reference problem Sensitivity analysis Model General notations: S the set of plants, of cardinal n T the horizon of the problem d t the demand of electricity Control variables: u i t the rate of production of plant i t ≤ u i the bound on production 0 ≤ u i i ∈ S [0 , u i ] U = �

Introduction Study of the reference problem Sensitivity analysis State variables: s i t the level of fuel of plant i [ τ i b , τ i e ] the dates of the outages a i the rate of refuelling Dynamic: � s i − u i e ] + a i 1 t ∈ [ τ i ˙ t = t 1 t / ∈ [ τ i b ,τ i b ,τ i e ] s i s i , 0 0 = State constraints: � s i b = 0 τ i s i T ≥ 0

Introduction Study of the reference problem Sensitivity analysis Optimization criterion: � T � � � u i min d t − d t + φ ( s T ) , c t 0 i ∈ W ( t ) where: c and φ and strongly convex and smooth and φ is decreasing W ( t ) is the set of working plants at time t . Functional spaces: u ∈ L ∞ (0 , T ; R n ) s ∈ W 1 , ∞ (0 , T ; R n )

Introduction Study of the reference problem Sensitivity analysis Pontryagin’s principle The Hamiltonian is independent on the state! � u i � p i � � � � − u i 1 t / e ] + a i 1 t ∈ [ τ i H ( t , u , p ) = c d t − + . ∈ [ τ i b ,τ i b ,τ i e ] i ∈ W ( t ) i ∈ S Proposition If ( u , s ) is optimal, there exists a costate t �→ p ( t ) such that 1 Each coordinate p i takes only two values over time, p i 0 on [0 , τ i b ] and p i T on [ τ i b , T ] such that p i T = D s i φ ( s T ) if s i p T ≤ D s i φ ( s T ) and T = 0 . 2 For almost all t in [0 , T ] , H ( t , u t , p t ) ≤ H ( t , v , p t ) , ∀ v ∈ U .

Introduction Study of the reference problem Sensitivity analysis Pontryagin’s principle The Hamiltonian is independent on the state! � u i � p i � � � � − u i 1 t / e ] + a i 1 t ∈ [ τ i H ( t , u , p ) = c d t − + . ∈ [ τ i b ,τ i b ,τ i e ] i ∈ W ( t ) i ∈ S Proposition If ( u , s ) is optimal, there exists a costate t �→ p ( t ) such that 1 Each coordinate p i takes only two values over time, p i 0 on [0 , τ i b ] and p i T on [ τ i b , T ] such that p i T = D s i φ ( s T ) if s i p T ≤ D s i φ ( s T ) and T = 0 . 2 For almost all t in [0 , T ] , H ( t , u t , p t ) ≤ H ( t , v , p t ) , ∀ v ∈ U .

Introduction Study of the reference problem Sensitivity analysis Stucture of optimal controls Each stock of fuel i has two marginal prices associated: − p i − p i 0 ≥ 0 and T ≥ 0 . At time t , the Hamiltonian is the sum of � i ∈ W ( t ) u i � the integral cost: c d t − � i ∈ S − p i t u i . the cost associated to fuel: � Moreover, the costate induces an ordering of the plants. If t > − p j − p i t , then plant i is used only if plant j produces at its maximum rate.

Introduction Study of the reference problem Sensitivity analysis Stucture of optimal controls Each stock of fuel i has two marginal prices associated: − p i − p i 0 ≥ 0 and T ≥ 0 . At time t , the Hamiltonian is the sum of � i ∈ W ( t ) u i � the integral cost: c d t − � i ∈ S − p i t u i . the cost associated to fuel: � Moreover, the costate induces an ordering of the plants. If t > − p j − p i t , then plant i is used only if plant j produces at its maximum rate.

Introduction Study of the reference problem Sensitivity analysis Total production Demand � � � # # � � � � � � � � � The bounds � � / � # depend on: � �, W(t), and p(t).

Introduction Study of the reference problem Sensitivity analysis If some plants share the same costate, then the optimal controls are not unique. In our model, the total production is unique. The costate has to be considered as a dual variable, characterized by ( p 0 , p T ). It is not necessarily unique.

Introduction Study of the reference problem Sensitivity analysis 1 Study of the reference problem Model Pontryagin’s principle Structure of optimal controls 2 Sensitivity analysis Abstract theorem Toy example Application to the outages

Introduction Study of the reference problem Sensitivity analysis Abstract theorem Consider the abstract family of optimization problems P y V ( y ) = min x f ( x , y ) , s.t. g ( x , y ) ≤ 0 , and its Lagrangian L ( x , y , λ ) = f ( x , y ) + � λ, g ( x , y ) � . The functions f and g are continuously differentiable. For a reference value y 0 , suppose that P y 0 is convex and denote by S ( y 0 ), the set of solutions of P y 0 Λ( y 0 ), the set of Lagrange multipliers.

Introduction Study of the reference problem Sensitivity analysis Abstract theorem Consider the abstract family of optimization problems P y V ( y ) = min x f ( x , y ) , s.t. g ( x , y ) ≤ 0 , and its Lagrangian L ( x , y , λ ) = f ( x , y ) + � λ, g ( x , y ) � . The functions f and g are continuously differentiable. For a reference value y 0 , suppose that P y 0 is convex and denote by S ( y 0 ), the set of solutions of P y 0 Λ( y 0 ), the set of Lagrange multipliers.

Introduction Study of the reference problem Sensitivity analysis Theorem Suppose that 1 problem P y 0 has solutions 2 problem P y 0 is qualified, at all the solutions 3 for all sequence y n → y 0 , P y n has a solution x n such that ( x n ) n has a limit point x in S ( y 0 ) Then, V is directionally differentiable at y 0 in all direction h and V ′ ( y 0 , h ) = inf sup D y L ( x , λ, y 0 ) h . x ∈ S ( y 0 ) λ ∈ Λ( y 0 ) Our goal: applying this result to V ( τ b , τ e ).

Introduction Study of the reference problem Sensitivity analysis Theorem Suppose that 1 problem P y 0 has solutions 2 problem P y 0 is qualified, at all the solutions 3 for all sequence y n → y 0 , P y n has a solution x n such that ( x n ) n has a limit point x in S ( y 0 ) Then, V is directionally differentiable at y 0 in all direction h and V ′ ( y 0 , h ) = inf sup D y L ( x , λ, y 0 ) h . x ∈ S ( y 0 ) λ ∈ Λ( y 0 ) Our goal: applying this result to V ( τ b , τ e ).

Introduction Study of the reference problem Sensitivity analysis An example of a directionally differentiable function: f : x ∈ R �→ | x | . At 0, we have: � if h ≥ 0 , h f ′ (0 , h ) = − h if h ≤ 0 .

Introduction Study of the reference problem Sensitivity analysis Toy example We consider a simplified problem with parameter τ . � τ � 1 V ( τ ) = min c 1 ( t , x t , u t ) d t + c 2 ( t , x t , u t ) d t 0 τ  x t = ˙ f 1 ( t , x t , u t ) , t ∈ [0 , τ ] ,   s.t. x t = ˙ f 2 ( t , x t , u t ) , t ∈ [ τ, 1] ,  x 0 . x 0 =  In this framework, impossible to apply the general result and compute V ′ ( τ )!

Introduction Study of the reference problem Sensitivity analysis A piecewise affine change of variables θ τ enables us to fix the date of the perturbation. � � � � � ��� 1 � � 0 0 � � 1

Introduction Study of the reference problem Sensitivity analysis We obtain: � τ 0 � 1 θ τ ˙ s c 1 ( θ τ θ τ ˙ s c 2 ( θ τ V ( τ ) = min s , x s , u s ) d s + s , x s , u s ) d s , 0 τ 0 = ˙  θ τ s f 1 ( θ τ x s ˙ s , x s , u s ) , s ∈ [0 , τ 0 ] ,   = ˙ θ τ s f 2 ( θ τ s.t. ˙ s , x s , u s ) , s ∈ [ τ 0 , 1] , x s  = x 0 . x 0  The Lagrangian is � τ 0 θ τ ˙ s H 1 ( θ τ L ( x , u , τ, p )= s , x s , u s , p s ) d s 0 � 1 � 1 ˙ θ τ s H 2 ( θ τ + s , x s , u s , p s ) d s − p s ˙ x s d s , τ 0 0 where H 1 ( t , x , u , p ) = c 1 ( t , x , u ) + � p , f 1 ( t , x , u ) � .