Reinforcement of gas transmission networks with MIP constraints and - PowerPoint PPT Presentation

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Reinforcement of gas transmission networks with MIP constraints and uncertain demands 1 . Babonneau 1 , 2 and Jean-Philippe Vial 1 F 1 Ordecsys, scientific consulting, Geneva, Switzerland 2 Swiss Federal Institute of Technology, Lausanne, Switzerland ISMP 2015 - Pittsburgh 1 Research supported by the Qatar National Research Grant 6 1035 5 126.

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Operations and design of gas transmission networks 1 Robust optimization to deal with uncertain demands 2 Numerical experiments (preliminary) 3

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Gas transmission networks Objective of this research: Extend the continuous and deterministic formualtion of ( F . Babonneau, Y. Nesterov and J.-P . Vial. Design and operations of gas transmission networks. Operations Research, Operations Research, 60(1):34-47, 2012 ) to uncertain demands, fixed investment costs, and commercial diameters.

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Operations of gas transmission networks Find a flow x and a system of pressures p such that l a β x a | x a | 2 − p j 2 , a = ( i , j ) , a ∈ E p = p i (1a) D a 5 ¯ φ ≤ Ax ≤ φ (1b) 2 − p j 2 ) x a ≤ 0 , x a ( p i ≥ 0 , a = ( i , j ) , a ∈ E c (1c) 2 − p j 2 ) x a ≥ 0 , x a ( p i ≥ 0 , a = ( i , j ) , a ∈ E r (1d) ¯ p i ≤ p i ≤ p i , i ∈ V . (1e) ⇒ Nonlinear and non convex set of inequalities. We rely on a two-step procedure of find a feasible pair of flows and pressures

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) First step: Finding feasible flows The first problem computes flows min E ( x ) − � f , x � − � d , Ax � (2a) x φ ≤ Ax ≤ ¯ φ (2b) x a ≥ 0 , a ∈ E c ∪ E r . (2c) The function E ( x ) = � a ∈ V E a ( x a ) is separable and is defined by | x a | 3 β E a ( x a ) = (3) l a , a ∈ E p D 5 3 a

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Second step: Finding compatible pressures The second one computes compatible pressures. Given x ∗ , find a system of pressures p and an action vector f such that f + A T ( p ) 2 E ′ ( x ∗ ) = (4a) f a = 0 , a ∈ E p (4b) p 2 i − p 2 0 , if x ∗ ≤ a > 0 , a = ( i , j ) , a ∈ E c (4c) j p 2 i − p 2 0 , if x ∗ ≥ a > 0 , a = ( i , j ) , a ∈ E r (4d) j p i ≤ p i ≤ ¯ p i , i ∈ V . (4e) By applying the simple change of variable P i = p 2 i , system (4) is linear in f and P .

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Reinforcement problem Let E n to be the set of arcs on which reinforcement takes place and ¯ C be an upper bound on the total investment cost. The investment problem can be stated as � � min min E a ( x a ; D a ) + E a ( x a ) (5a) x ( D a , a ∈ E n ) a ∈ E n a ∈ E p I ( D ) ≤ ¯ C (5b) φ ≤ Ax ≤ ¯ φ (5c) x a ≥ 0 , a ∈ E c ∪ E r . (5d) Assumption Data analysis shows that the investment cost can be approximated by I ( D ) = l × ( k 1 × D 2 . 5 + k 2 ) , (6) where l is the length of the arc and D ≤ ¯ D.

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Convex and continuous formulation of reinforcement problem Let us perform the change of variable y a = D 2 . 5 , a ∈ E n , a | x a | 3 | x a | 3 β β � � � � min min l a + l a (7a) y 2 D 5 3 3 x ( y a , a ∈ E n ) a a a ∈ E n a ∈ E p 1 × y a ≤ ¯ � l a × k a C (7b) a ∈ E n φ ≤ Ax ≤ ¯ φ (7c) x a ≥ 0 , a ∈ E c ∪ E r . (7d) The function | x | 3 / y 2 is jointly convex in x and y .

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Extension to integer considerations Let k be the set of commercial diameters.   | x a | 3 β  �  + � min l a E ( x a ) y 2 x , y , z 3 a a ∈ E n a ∈ E p 1 D 5 / 2 2 ) ≤ ¯ � � z ak l a ( k a + k a C k a ∈ E n k z ak D 5 / 2 � y a = , k = 1 , . . . , K k k � (8) z ak ≤ 1 , k = 1 , . . . , K k φ ≤ Ax ≤ ¯ φ x a ≥ 0 , a ∈ E c ∪ E r z ak ∈ [ 0 , 1 ] a ∈ E n , k = 1 , . . . , K .

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Operations and design of gas transmission networks 1 Robust optimization to deal with uncertain demands 2 Numerical experiments (preliminary) 3

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Relaxed formulation of reinforcement problem By partial dualization, the problem is E ( x ; y ) + E ( x ) + α ( I ( y ) − ¯ C ) max α ≥ 0 min min (9a) x ( y a , a ∈ E n ) φ ≤ Ax ≤ ¯ φ (9b) x a ≥ 0 , a ∈ E c ∪ E r . (9c) with the inner minimization problem � � | x a | 3 β + α l a k a C a ( x a ) = min l a 1 y a , a ∈ E n . (10) y 2 3 y a a

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Solving the inner minimization problem Theorem (Babonneau, Nesterov, Vial) C a ( x a ) is convex and is given by l a β 1 / 3 � 3 α k 1 � 2 / 3 C a ( x a ) = | x a | 2 The optimal diameter in problem (11) is � 2 � 15 2 β D ∗ 2 a = | x a | 5 3 α k a 1

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Simplified formulation of the reinforcement problem For a given α l a β 1 / 3 � 3 α k a | x a | 3 β � 2 / 3 � 1 � min | x a | + l a (11a) D 5 x 2 3 a a ∈ E n a ∈ E p A i x ≥ φ i , i ∈ V d (11b) φ i ≤ A i x ≤ ¯ φ i , i / ∈ V d (11c) x a ≥ 0 , a ∈ E c ∪ E r . (11d)

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Uncertainty model We consider the demand parameter at delivery nodes i as uncertain such that φ i = φ n i + ξ i ˆ φ i i = φ i is the nominal demand, ˆ where φ n φ n i = γφ i is the demand dispersion and ξ i is a random factor with support [ − 1 , 1 ] . The problem of reinforcement gas transmission networks is now a two-stage problem with recourse . In the first stage, reinforcement investment is selected and in the second stage the decision concerns the flow (and the activity of compressor and regulator stations) to satisfy observed demands.

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Affine decision rules Given the demand model, we can define a decision rule as a function from the space of demands realizations to the space of recourse flow decisions in order to capture the fact that flows can be adjusted to fit observed demands. We propose affine decision rules (ADR) x a = ν 0 � ξ i ν i a + a , ∀ a ∈ E p . i ∈ V d In that formulation, the new decision variables are the coefficients ν 0 a ∈ R and ν i a ∈ R .

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Uncertain formulation with robust constraints We can now replace x and φ by their definition. � 3 α k a t 3 β � 2 / 3 � 1 � a H ( α ) = min t a + l a (12a) D 5 2 3 t ,ν a a ∈ E n a ∈ E p | ν 0 � ξ i ν i a + a | ≤ t a a ∈ E n ∀ ξ ∈ Ξ (12b) i ∈ V d A i ( ν 0 � ξ i ν i a ) ≥ φ n i + ξ i ˆ a + φ i , i ∈ V d ∀ ξ ∈ Ξ (12c) i ∈ V d φ i ≤ A i ( ν 0 � ξ i ν i a ) ≤ ¯ a + φ i , i / ∈ V d ∀ ξ ∈ Ξ (12d) i ∈ V d ν 0 � ξ i ν i a + a ≥ 0 , a ∈ E c ∪ E r ∀ ξ ∈ Ξ . (12e) i ∈ V d

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Applying robust optimization We can now state the robust equivalent of the robust constraint. Theorem (Ben-Tal, El Ghaoui, Nemirovski) Let ξ i , i = 1 , . . . , m be independent random variables with values in interval [ − 1 , 1 ] and with average zero: E ( ξ i ) = 0 , the robust equivalent of the constraint a T x + ( P T x ) T ξ ≤ b , for all ξ ∈ Ξ = { ξ | || ξ || 2 ≤ k } , ¯ is a T x + k || P T x || 2 ≤ b , ¯ with an associated satisfaction probability of ( 1 − exp ( − k 2 2 . 5 ))

Operations and design of gas transmission networks Robust optimization to deal with uncertain demands Numerical experiments (preliminary) Operations and design of gas transmission networks 1 Robust optimization to deal with uncertain demands 2 Numerical experiments (preliminary) 3