Reinforcement of gas transmission networks with MIP constraints and uncertain demands 1

Transcription

1 Reinforcement of gas transmission networks with MIP constraints and uncertain demands 1 F. Babonneau 1,2 and Jean-Philippe Vial 1 1 Ordecsys, scientific consulting, Geneva, Switzerland 2 Swiss Federal Institute of Technology, Lausanne, Switzerland ISMP Pittsburgh 1 Research supported by the Qatar National Research Grant

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

3 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.

4 Operations of gas transmission networks Find a flow x and a system of pressures p such that l aβ xa xa D a 5 = p i 2 p j 2, a = (i, j), a E p (1a) φ Ax φ (1b) (p i 2 p j 2 )x a 0, x a 0, a = (i, j), a E c (1c) (p i 2 p j 2 )x a 0, x a 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

5 First step: Finding feasible flows The first problem computes flows min x E(x) f, x d, Ax (2a) φ Ax φ x a 0, a E c E r. The function E(x) = a V Ea(x a) is separable and is defined by (2b) (2c) E a(x a) = β x a 3 l a Da 5 3, a Ep (3)

6 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 j 0, if x a > 0, a = (i, j), a E c (4c) p 2 i p 2 j 0, if x a > 0, a = (i, j), a E r (4d) 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.

7 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 x a(x a; D a) + E a(x a) (5a) (D a, a E n) a E n a E p I(D) C φ Ax φ x a 0, a E c E r. (5b) (5c) (5d) Assumption Data analysis shows that the investment cost can be approximated by I(D) = l (k 1 D k 2 ), (6) where l is the length of the arc and D D.

8 Convex and continuous formulation of reinforcement problem Let us perform the change of variable y a = D 2.5 a, a E n, min x min (y a, a E n) ( β l a 3 a E n x a 3 y 2 a l a k1 a ya C (7b) a E n φ Ax φ x a 0, a E c E r. The function x 3 /y 2 is jointly convex in x and y. ) + β x a 3 l a 3 D 5 a E p a (7a) (7c) (7d)

9 Extension to integer considerations Let k be the set of commercial diameters. min β x a 3 l a x,y,z 3 y 2 + E(x a) a E n a a E p z ak l a(k1 a D5/2 k + k2 a ) C a E n k y a = z ak D 5/2 k, k = 1,..., K k z ak 1, k = 1,..., K k φ Ax φ (8) x a 0, a E c E r z ak [0, 1] a E n, k = 1,..., K.

10 1 Operations and design of gas transmission networks 2 Robust optimization to deal with uncertain demands 3 Numerical experiments (preliminary)

11 Relaxed formulation of reinforcement problem By partial dualization, the problem is max min min α 0 x (y a, a E n) with the inner minimization problem { C a(x a) = min y a l a β 3 E(x; y) + E(x) + α(i(y) C) φ Ax φ x a 0, a E c E r. (9a) (9b) (9c) } x a 3 ya 2 + αl ak1 a ya, a E n. (10)

12 Solving the inner minimization problem Theorem (Babonneau, Nesterov, Vial) C a(x a) is convex and is given by C a(x a) = l aβ 1/3( 3αk ) 1 2/3 xa 2 The optimal diameter in problem (11) is D a = ( 2β 3αk a 1 ) 2 15 x a 2 5

13 Simplified formulation of the reinforcement problem For a given α min x ( 3αk l aβ 1/3 a 1 2 a E n ) 2/3 xa + β x a 3 l a 3 D 5 a E p a (11a) 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)

14 Uncertainty model We consider the demand parameter at delivery nodes i as uncertain such that φ i = φ n i + ξ i ˆφ i where φ n i = φ i is the nominal demand, ˆφ 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.

15 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 a + i V d ξ i ν i a, a E p. In that formulation, the new decision variables are the coefficients ν 0 a R and νi a R.

16 Uncertain formulation with robust constraints We can now replace x and φ by their definition. H(α) = min t,ν ( 3αk a 1 2 a E n νa 0 + ) 2/3ta + β ta 3 l a 3 D 5 a E p a (12a) ξ i νa i t a a E n ξ Ξ (12b) i V d A i (νa 0 + ξ i νa i ) φn i + ξ i ˆφi, i V d i V d ξ Ξ (12c) φ i A i (νa 0 + ξ i νa i ) φ i, i V d i / V d ξ Ξ (12d) νa 0 + ξ i νa i 0, a E c E r i V d ξ Ξ. (12e)

17 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 is ā T x + (P T x) T ξ b, for all ξ Ξ = {ξ ξ 2 k}, ā T x + k P T x 2 b, with an associated satisfaction probability of (1 exp( k2 2.5 ))

18 1 Operations and design of gas transmission networks 2 Robust optimization to deal with uncertain demands 3 Numerical experiments (preliminary)

19 Belgian instance

20 Belgian instance #Nodes Label φ φ p p 1 Zeebrugge Dudzele Brugge Zomergem Loenhout Antwerpen Gent Voeren Berneau Liège Warnand Namur Anderlues Péronnes Mons Blagneries Wanze Sinsin Arlon Pétange We increase the bounds of the demands and the supplies with a factor 1.3 to make the existing design under-sized. We assume demand variability of 5%. Mosek conic MIP optimizer (beta version)

21 Impact of fixed costs on reinforcement Formulation #Arc No fixed costs Fixed costs No fixed costs Fixed costs No fixed costs Fixed costs Costs CPU

22 Impact of commercial diameters on reinforcement Investment budget with fixed costs #Arc Continuous Commercial Continuous Commercial Continuous Commercial CPU

23 Robust investment solutions for α = 10 Arc Probability satisfaction # (O,D) Existing Determinist 10%( k =0.5) 33% (k =1) 80% (k =2) 1 (1,2) (1,2) (2,3) (2,3) (3,4) (5,6) (6,7) (7,4) (4,14) (8,9) (8,9) (9,10) (9,10) (10,11) (10,11) (11,12) (12,13) (13,14) (14,15) (15,16) (11,17) (17,18) (18,19) (19,20)

24 Next steps Validation procedure : generate a set of demand scenarios and find compatible pressures. Compute a quality of service. Combine all features. Improve compressor modelling Thanks!!!