Optimizing designs and operations of a single network or multiple interdependent infrastructures under stochastic arc disruption

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Abstract

In this paper, we consider an infrastructure as a network with supply, transshipment, and demand nodes. A subset of potential arcs can be constructed between node pairs for conveying service flows. The paper studies two optimization models under stochastic arc disruption. Model 1 focuses on a single network with small-scale failures, and repairs arcs for quick service restoration. Model 2 considers multiple interdependent infrastructures under large-scale disruptions, and mitigates cascading failures by selectively disconnecting failed components. We formulate both models as scenario-based stochastic mixed-integer programs, in which the first-stage problem builds arcs, and the second-stage problem optimizes recourse operations for restoring service or mitigating losses. The goal is to minimize the total cost of infrastructure design and recovery operations. We develop cutting-plane algorithms and several heuristic approaches for solving the two models. Model 1 is tested on an IEEE 118-bus system. Model 2 is tested on systems consisting of the 118-bus system, a 20-node network, and/or a 50-node network, with randomly generated interdependency sets in three different topological forms (i.e., chain, tree, and cycle). The computational results demonstrate that (i) decomposition and cutting-plane algorithms effectively solve Model 1, and (ii) heuristic approaches dramatically decrease the CPU time for Model 2, but yield worse bounds when cardinalities of interdependency sets increase. Future research includes developing special algorithms for optimizing Model 2 for complex multiple infrastructures with special topological forms of system interdependency.

Introduction

Infrastructure systems and their operations are essential and fundamental for the development of modern societies. Such systems are often considered as networks involving sets of supply, demand, and transshipment nodes. These infrastructures have been extensively studied in applications of energy, transportation, and telecommunication, to provide platforms for service delivery.

In particular, critical infrastructure problems investigate the survivability of systems under malicious attacks, nature disasters, or component failures (e.g., [6], [17], [24]). The related research has focused on deliberate attacks and network interdiction problems [8], [29], [23], [25], formulated as bilevel programs where a defender optimally allocates resources to minimize the maximum loss caused by an attacker. Meanwhile, another track of the research focuses on problems of network vulnerability (e.g., [20]) and cascading failures (e.g., [9], [18]). These problems are in particular of importance to power network design [11], [30] and operations against blackouts [3].

This paper considers infrastructure damages caused by uncertain weather conditions or node failures rather than malicious or antagonistic attacks. We combine network design and operational planning, to minimize the expected cost of arc construction, flow operation, and service recovery given stochastic arc disruption. However, recovery operations may vary among networks and applications, depending on disruption severity, system interdependency, and service priority. For small-scale failures, local repairing can be done immediately for fully restoring service. For instance, Verizon, a wireless company uses leased signal transmission towers to quickly recover a disrupted service [13]. In other cases, the damage may be severe and not repairable within a short amount of time. Recovery operations then include load shedding, redistributing flows, and islanding to avoid cascading failures (e.g., [10], [31]).

Corresponding to different assumptions of network and arc disruption, we develop two stochastic model variants, referred to as Models 1 and 2, for handling (i) small-scale non-deliberate disruptions in a single network and (ii) large-scale cascading failures in multiple interdependent infrastructures, respectively. In Model 1, failed connections (arcs) can be repaired or reconstructed. In Model 2, recovery operations are considered as load shedding and failure isolation via disconnections of system interdependency. We build both models as mixed-integer programming (MIP) formulations, and develop decomposition-based cutting-plane approaches for optimizing Model 1. For Model 2, we develop two heuristic approaches to obtain feasible solutions and upper bounds, and solve a relaxation of the MIP model to obtain a lower bound. Both Models 1 and 2 can be applied to a wide range of applications in managing power transmission networks and transportation systems. In particular, Model 2 can be used for optimizing islanding operations in micro-grids (e.g., [1], [12], [32]). Lee et al. [16] describe an application for which we can use Model 2 to design multiple interdependent systems in the New York City. The goal is to design optimal responding strategies for the Office of Emergency Management to prioritize tasks and coordinate responsible agencies. Here the coordinative operations include system disconnection and isolation, under the uncertainty of small-scale failures in subway, power-grid, and other infrastructures.

A class of Model 1 variants, referred to as network capacity expansion problems, have been well studied for various network design applications. Under the assumption of uncertain demand, decisions are made to design a network as well as to allocate extra capacities on a subset of selected arcs. The goal is to minimize the total cost of network construction, capacity expansion, together with the expected cost of product/service flow and demand loss penalty. Bienstock and Mattia [4] formulate a capacity expansion problem for power grids, and solve the corresponding mixed-integer program via a Branch-and-Cut algorithm. Ahmed et al. [2] propose a multi-stage stochastic integer programming approach for optimizing sequential decisions of capacity expansion given uncertain demand in supply chain management, and Ordóñez and Zhao [19] investigate a robust optimization model for general network-flow capacity expansion problems under demand uncertainty with unknown distributions. Riis and Andersen [21] and Riis and Lodahl [22] respectively study a multi-stage model and a bicriteria stochastic programming model for capacity expansion problems in telecommunication.

Meanwhile, a significant number of publications has been found for studying network flow problems under arc-disruption uncertainty, which are closely related to the research in this paper. To name a few, Sorokin et al. [27] consider the fixed charge network flow problem under uncertain arc failures, and seek a robust optimal flow assignment to restrict potential losses using Conditional Value-at-Risk. Janjarassuk and Linderoth [14] study a stochastic network interdiction problem where the successful destruction of an arc is a Bernoulli random variable, and the objective is to minimize the maximum expected flow of the adversary. The authors run a computational-grid-based parallel algorithm for optimizing a mixed-integer programming formulation of the problem. In this paper, we optimize both network design and flow assignment under random arc disruption. Model 1 is formulated as a two-stage stochastic integer program where decisions of network design appear at the first stage, and the arc repairing and network flows are recourse decisions corresponding to specific scenarios. With regard to Model 2, few papers have been found to have similar problem settings. Cavdaroglu et al. [7] formulate service restoration and task scheduling in interdependent systems. The resulting formulation is a large-scale mixed-binary-integer program and is directly solve by off-the-shelf solvers.

The remainder of the paper is organized as follows. Section 2 formulates Model 1 for designing a single network and develops several cutting-plane algorithms. Section 3 formulates Model 2 for designing multiple interdependent infrastructures, and uses heuristic approaches for quickly finding feasible solutions and objective bounds. Section 4 tests both models on a set of randomly generated single or multiple network systems with diverse parameter settings. We demonstrate the computational efficacy of cutting-plane approaches in Model 1 and compare heuristic results with MIP solutions in Model 2. Section 5 concludes the paper and proposes future research directions.

Section snippets

Notation and problem formulation

Let G(N,A0A) denote a directed connected graph with node set N=N+N=N-, where N+,N=, and N- respectively represent sets of supplies (generators), intermediate transmissions (transformers), and demands (consumers). Assume that N+N-=N+N-=N-N-=. Arc sets A0 and A respectively represent the current existing arcs and potential arcs to be constructed, with both A0 and AN×N. Without loss of generality, we assume that A0= for all problems discussed in this paper. For all (i,j)A, denote aij, cij

An exact formulation

Model 2 optimizes the design of multiple infrastructures, being interdependent and possessing a risk of cascading failures. With the same assumptions of first-stage operations, we revise our second-stage recourse operations to mitigate demand losses caused by large-scale cascading failures. Rather than repair arcs, Model 2 emphasizes on two major responses after certain arcs are randomly destroyed: (i) load shedding at demand nodes, and (ii) isolating failures by disconnecting interdependent

Computational results

We first test Model 1 on an IEEE 118-bus system based on the US Midwest power grid. We compare Benders cuts (22) and cuts (25) (referred to as “BAC cuts”), and also test a hybrid cut-generation scheme by systemically mixing Benders and BAC cuts. (Details of how to execute the hybrid scheme are described later.) We present average CPU time for different cutting-plane approaches, compared with directly solving the monolithic MIP model of Model 1 in default CPLEX. Section 4.1 provides details of

Conclusions

This paper investigates problems of critical infrastructure design and recovery under stochastic arc disruption. We consider both small-scale failures for a single network, and large-scale cascading failures for multiple interdependent infrastructures. An MIP model is respectively proposed for each case. In Model 1, we first require all demand being fully satisfied, and revise the model by adding Big-M constraints to enforce the Kirchhoff's Voltage Law in power transmission networks. We develop

Acknowledgments

The author is grateful to anonymous referees and the editor for their constructive comments. The author also gratefully acknowledges the support of IBM Smarter Planet Innovation Faculty Award.

References (32)

  • G. Brown et al.

    Defending critical infrastructure

    Interfaces

    (2006)
  • B. Cavdaroglu et al.

    Integrating restoration and scheduling decisions for disrupted interdependent infrastructure systems

    Annals of Operations Research

    (2012)
  • K.J. Cormican et al.

    Stochastic network interdiction

    Operations Research

    (1998)
  • P. Crucitti et al.

    Model for cascading failures in complex networks

    Physical Review E

    (2004)
  • A. Delgadillo et al.

    Analysis of electric grid interdiction with line switching

    IEEE Transactions on Power Systems

    (2010)
  • H. Faria et al.

    Power transmission network design by greedy randomized adaptive path relinking

    IEEE Transactions on Power Systems

    (2005)
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