Optimizing Infrastructure Design and Recovery Operations Under - PowerPoint PPT Presentation

Optimizing Infrastructure Design and Recovery Operations Under Stochastic Disruptions Siqian Shen Department of Industrial and Operations Engineering University of Michigan The 13th INFORMS Computing Society Conference January 07, 2013 Shen, U of Michigan Optimizing Designs and Operations of Interdependent Infrastructures 1 / 32

Outline Introduction Model 1: Optimal Design and Operations in a Single Network Problem Description and Formulation A Decomposition Framework Modifying Model 1 for Power Systems Model 2: Optimal Design and Interdependency Disconnections in Multiple Infrastructures An Exact Formulation Feasible solutions to Model 2 Lower bounds for SP q ( x )-Model 2 Computational Results Computing Model 1 Computing Model 2 Shen, U of Michigan Optimizing Designs and Operations of Interdependent Infrastructures 2 / 32

Critical Infrastructure Analysis: Literature Review ◮ Considered as networks with supply/demand/transshipment nodes, and service flows. ◮ Important to applications in energy, transportation, telecommunication, and many other areas. ◮ The literature includes ◮ system survivability under malicious attacks, nature disasters, or component failures (e.g., Brown et al. 2006, Murray et al. 2007, San Martin 2007). ◮ network design against deliberate attacks and the research of network interdiction (see, e.g., Cormican et al. 1998, Wood 1993). ◮ network vulnerability (e.g., Pinar et al. 2010) and cascading failures (e.g., Crucitti et al. 2004, Nedic et al. 2006). ◮ particular use in designing power grids (Faria Jr et al. 2005, Yao et al. 2007) and operations against blackouts (Alguacil et al. 2010). Shen, U of Michigan Optimizing Designs and Operations of Interdependent Infrastructures 3 / 32

Our Problems Combine phases of network design and operational planning, to minimize the expected costs of arc construction, flow operation, and service recovery under stochastic arc disruptions. Motivation: ◮ The forms of service recovery vary depending on disruption severity, system interdependency, and service priority. ◮ For small-scale failures, local repairing can be done immediately for fully restoring service. ◮ During large-scale and severe damages, disconnection operations are used to avoid cascading failures. Two stochastic model variants: ◮ Model 1 for repairing small-scale failures in a single network. ◮ Model 2 for avoiding large-scale cascading failures in multiple interdependent infrastructures. Shen, U of Michigan Optimizing Designs and Operations of Interdependent Infrastructures 4 / 32

A Single Network: Notation I Model 1 considers a single network with ◮ G ( N , A 0 ∪ A ): a directed connected graph with node set N = N + ∪ N = ∪ N − ◮ N + , N = , and N − : sets of supplies, intermediate transmissions, and demands. ◮ A 0 and A : the current existing arcs and potential arcs to be constructed ( A 0 = ∅ in this paper). Parameters: ◮ a ij , c ij , and d ij : flow capacity, construction cost, and unit flow cost of arc ( i , j ), ∀ ( i , j ) ∈ A . ◮ h i : unit generation cost of each supply node, ∀ i ∈ N + . ◮ S i : the maximum capacity of supply node i ∈ N + . ◮ D i : consumer’s demand at node i ∈ N − , with � i ∈N + S i ≥ � i ∈N − D i . Shen, U of Michigan Optimizing Designs and Operations of Interdependent Infrastructures 5 / 32

A Single Network: Notation II ◮ Q : a finite set of random disruption scenarios. ◮ I q ij ∈ { 0 , 1 } : an effect of a disruptive event on arc ( i , j ), ∀ ( i , j ) ∈ A , q ∈ Q , where I q ij = 0 if arc ( i , j ) fails, and 1 otherwise. ◮ b q ij : cost of repairing arc ( i , j ), ∀ ( i , j ) ∈ A , q ∈ Q with max q ∈ Q b q ij < c ij by assumption, ∀ ( i , j ) ∈ A . Decision Variables: ◮ x ij ∈ { 0 , 1 } : such that x ij = 1 if we construct arc ( i , j ), and 0 otherwise. ◮ y q ij ∈ { 0 , 1 } , such that y q ij = 1 if arc ( i , j ) is repaired in scenario q , and 0 otherwise. ◮ f q ij ≥ 0: the amount of flow on arc ( i , j ) in a repaired network, ∀ q ∈ Q . Shen, U of Michigan Optimizing Designs and Operations of Interdependent Infrastructures 6 / 32

Formulation of Model 1   1 � �  � h i g q � b q ij y q � d ij f q min: c ij x ij + i + ij + (1a) ij  | Q | ( i , j ) ∈A q ∈ Q i ∈N + ( i , j ) ∈A ( i , j ) ∈A � f q � f q ji − g q s.t. ij − i = 0 ∀ i ∈ N + , q ∈ Q (1b) j :( i , j ) ∈A j :( j , i ) ∈A � f q � f q ij − ji = − D i ∀ i ∈ N − , q ∈ Q (1c) j :( i , j ) ∈A j :( j , i ) ∈A � f q � f q ij − ji = 0 ∀ i ∈ N = , q ∈ Q (1d) j :( i , j ) ∈A j :( j , i ) ∈A y q ij ≤ x ij (1 − I q ij ) ∀ ( i , j ) ∈ A , q ∈ Q (1e) f q ij ≤ a ij ( I q ij x ij + y q ∀ ( i , j ) ∈ A , q ∈ Q ij ) (1f) 0 ≤ g q i ≤ S i ∀ i ∈ N + , q ∈ Q (1g) x ij ∈ { 0 , 1 } ∀ ( i , j ) ∈ A , y q ij ∈ { 0 , 1 } , and f q ij ≥ 0 ∀ ( i , j ) ∈ A , q ∈ Q . (1h) where ◮ Variables g q i in (1b) provide flow amount generated from supply nodes i ∈ N + . Shen, U of Michigan Optimizing Designs and Operations of Interdependent Infrastructures 7 / 32

A Decomposition Framework Decompose Model 1 into two stages with binary variables x at the first stage, and | Q | independent subproblems at the second stage. ◮ A relaxed master problem: c ij x ij + 1 � � min: η q | Q | ( i , j ) ∈A q ∈ Q s.t. L q ( η q , x ) ≥ 0 ∀ q ∈ Q η q ≥ η q x ij ∈ { 0 , 1 } ∀ ( i , j ) ∈ A , ∀ q ∈ Q . ◮ Given a solution x , subproblem SP q ( x ) -Model 1 is � h i g q � b q ij y q � d ij f q η q = min: i + ij + ij i ∈N + ( i , j ) ∈A ( i , j ) ∈A s.t. (1b)–(1g), y q ij ∈ { 0 , 1 } , and f q ij ≥ 0 ∀ ( i , j ) ∈ A . Shen, U of Michigan Optimizing Designs and Operations of Interdependent Infrastructures 8 / 32

Cutting Plane Generations I Generate L q ( η q , x ) ≥ 0 as LP-based Benders Cuts: ◮ Relax y q ≥ 0 in SP q ( x )-Model 1, and let ˜ λ q α q ij , and ˜ β q i , ˜ ij be optimal dual solutions associated with (1b)–(1d), (1e) and (1f), respectively. ◮ Given that SP q ( x )-Model 1 has a feasible solution, � � � (1 − I q α q ij + a ij I q ij ˜ β q � ˜ λ q � λ q ˜ η q ≥ − ij )˜ x ij − i S i − i D i ij i ∈N + i ∈N − ( i , j ) ∈A (2) is valid for all q ∈ Q . ◮ Proof: Weak duality theorem. Shen, U of Michigan Optimizing Designs and Operations of Interdependent Infrastructures 9 / 32

Cutting Plane Generations II Combine Benders cuts with Laporte-Louveaux (LL) inequalities to enforce convergence: X 1 as the set of arcs { ( i , j ) ∈ A : ˆ x , denote ˆ ◮ Given ˆ x ij = 1 } X 0 as the set of arcs { ( i , j ) ∈ A : ˆ and ˆ x ij = 0 } . ◮ Suppose that the current ˆ x is not optimal. ◮ Because at least one x variable will change its current value in next iteration, � � (1 − x ij ) + x ij ≥ 1 . (3) ( i , j ) ∈ ˆ ( i , j ) ∈ ˆ X 1 X 0 is valid to MP. Shen, U of Michigan Optimizing Designs and Operations of Interdependent Infrastructures 10 / 32

Modifying Model 1 for Power Systems Apply Model 1 for optimizing design and service restoration in power transmission networks. ◮ Let θ i and θ j be voltages at locations i and j , and f ij be the electricity flow between i and j . ◮ The Kirchhoff’s Voltage Law: θ i − θ j = R ij f ij , where R ij is the reactance between locations i and j (a DC flow model). ◮ Add two constraints to SP q ( x )-Model 1: θ q i − θ q j ≥ R ij f q ij + M + ( I q ij x ij + y q ij − 1) ∀ ( i , j ) ∈ A (4) θ q i − θ q j ≤ R ij f q ij − M − ( I q ij x ij + y q ij − 1) ∀ ( i , j ) ∈ A , (5) where both M + and M − are sufficiently large numbers. Shen, U of Michigan Optimizing Designs and Operations of Interdependent Infrastructures 11 / 32

A Penalty-based Subproblem Relaxation ◮ Develop valid cuts by allowing unsatisfied demands at nodes i ∈ N − . ◮ This variant refers to the “load shedding” operation in practice, in which the goal is to minimize costs of arc construction, repair, and the penalties incurred by unmet demands. ◮ u q i ≥ 0: unsatisfied demands at nodes i ∈ N − . ◮ The new Model 1 imposes a penalty p q i for each unit of unsatisfied demand, and formulate R-SP q ( x ) -Model 1 : � h i g q � p q i u q � b q ij y q � d ij f q min: i + i + ij + ij i ∈N + i ∈N − ( i , j ) ∈A ( i , j ) ∈A s.t. (1b), (1d)–(1g), (4), (5) � f q ji − u q − i = − D i ∀ i ∈ N − j :( j , i ) ∈A u q i ≥ 0 , ∀ i ∈ N − , y q ij ∈ { 0 , 1 } , and f q ij ≥ 0 , ∀ ( i , j ) ∈ A . Shen, U of Michigan Optimizing Designs and Operations of Interdependent Infrastructures 12 / 32

Valid Inequalities Through Branch-and-Cut I x , in subproblem q , branch on arc sets A + ⊆ A and ◮ Given ˆ A − ⊆ A \ A + , such that y q ij = 1, ∀ ( i , j ) ∈ A + , and y q ij = 0, ∀ ( i , j ) ∈ A − . ij -values for all arcs ( i , j ) ∈ A + � A − after ◮ To ensure binary y q branching, add y q ∀ ( i , j ) ∈ A + ij ≥ 1 (6) − y q ∀ ( i , j ) ∈ A − . ij ≥ 0 (7) to subproblems and compute η q . Shen, U of Michigan Optimizing Designs and Operations of Interdependent Infrastructures 13 / 32