Defender–attacker–operator: Tri-level game-theoretic interdiction analysis of urban water distribution networks

Highlights

• Propose a tri-level defender–attacker–operator game-theoretic model to analyze the network interdiction problem of WDNs.

• Use hydraulic analysis to obtain the reliability index of WDNs as the equilibrium index of the dynamic game model.

• Pressure-driven model is employed to describe the actual water demand.

• A case study was conducted to evaluate the effectiveness of the proposed model.

Abstract

The aim of this paper is to advance the field of network interdiction analysis by introducing an application to the urban water distribution networks (WDNs), deploying protective resources against intentional attacks. The resource allocation problem for urban water supply systems is considered as a three-player (i.e., defender–attacker–operator) game, in which the attacker aims to maximize disruption impacts via interdicting water plants in the network, the defender aims to minimize the worst-case disruption impacts achieved by the attacker while the system operators fulfill the water demand in the residual urban water supply network. Considering the operating characteristics of the water supply network, we adopted the method of hydraulic analysis in the third level to obtain its reliability, and use this as the game equilibrium index of the first two levels. An effective modified variable neighborhood search method is devised to obtain the solution to the game. Finally, a case study was conducted based on the data of water supply network of a certain city in China to evaluate the effectiveness of protection resources against intentional attacks.

Introduction

Water distribution network, one of the most important parts of urban lifeline networks, plays important roles in maintaining urban social-economic operations and the livelihoods of urban residents. In recent years, urban lifeline network is increasing likely to become the target of intentional attacks, which has a serious potential impact on the public. Many studies demonstrate that terrorism is definitely one of the world’s most important issues from the late 20th century to the early 21st century. Attacks on the water distribution network will have multi-dimensional serious consequences. The direct consequences include blocking the supply of urban residential and industrial water, releasing the poisonous chlorine gas from water treatment plant, and the indirect consequences include making severe impacts on economy, livelihood and overall stability of the society, etc. Protecting the safety of urban water distribution networks against intentional attacks has become a hot issue in the engineering field [1], [2], [3], [4].

Focusing on the protection of urban water supply lifeline, scholars have done a lot of research [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. Michaud and Apostolakis [5] proposed a methodology for rating the components of water supply networks and assess the effects of the failure of each of the network components considering their capacity and repair time. For reliability assessment, Shuang et al. [7] evaluated the evolution of failure propagation time for a WDN experiencing cascading failures, and found the critical pipes which may reduce system reliability dramatically. Agathoklis et al. [10] studied the topological robustness and vulnerability assessment of WDNs based on network’s topology and its operating pressure.

Most of the above studies focus on WDN’s own attributes, such as topology design of pipe network, vulnerability analysis of components, or cascading failures of flow analysis. Moreover, these studies are aimed at situations such as earthquakes and other natural accidents, but there is little involved in the protection of WDNs under deliberate attacks. Although cyber–physical attacks [1], [3] have become a research hotspot in recent years, as far as infrastructure is concerned, the risks of terrorist attacks and air strikes still exist. The International Committee of the Red Cross (ICRC) [15] had reported that fighters were increasingly targeting water and sanitation facilities across the Middle East, exacerbating severe shortages for agriculture and households. In general, water distribution network is composed of multiple types of facilities, such as water treatments plants, water pump stations and supply pipelines [16]. These facilities are widely distributed in different areas. However, a city with limited protective resources generally has no ability to perfectly defend them all. With advances in drone strike technology and the high risk of terrorist attacks, it remains particularly important to strengthen the urban WDNs as part of its city lifeline to counter deliberate strikes. How to optimally allocate the limited protective resources in urban WDN with specific consideration of intentional attacks has become one of the most challenging problems for urban administration.

For the protection of infrastructure [17], [18], [19], [20], [21], [22], [23] under intentional attacks, most scholars adopt the game theory to conduct the research [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], which is a universal method, additionally considering the allocation of protection resources and the construction of recovery on this basis. In those game theory model [24], [26], [27], most scholars use abstract methods such as supply–demand models and network flow models etc., when modeling infrastructure networks. Nafiseh et al. [24] proposed a tri-level protection–interdiction–restoration model to enhance the resilience of a system of interdependent infrastructure networks under hazard of intelligent attack. In their model, they generally depicted each infrastructure network, i.e., water, gas, and power networks, consisting of a set of nodes (including supply, demand and transshipment nodes). Ouyang et al. [25] proposed a tri-level defender–attacker–defender model to mitigate electric power system vulnerability to worst-case spatially localized attacks. Some scholars [14], [28], [29], [30], [42] use network flow methods to model infrastructure networks. Jin et al. [29] introduced a game-theoretic approach for enhancing urban transit networks’ robustness to intentional disruptions via optimally allocating protection resources. Based on the models of network flow, capacity and demand, Roberto et al. [14] presented a probabilistic procedure to assess network reliability and resilience. Some scholars [31], [32], [33], [34], [43], [44], [45], [46], [47] use the complex network theory to model the infrastructure network in the game model. Xu et al. [31] studied the impact of recovery resources allocation approaches on the recovery of scale-free networks under the constraint of a fixed amount of total resources. Paola et al. [32] introduced a game-theoretic approach for allocating protection resources among the components of a shortest-path network so as to maximize its robustness to external disruptions.

Above all, in these models for optimizing resources, scholars often simplify infrastructure to, such as supply and demand models, network flow models, and complex network models. This can help the optimization calculation of the model. However, different infrastructure networks have different network flow characteristics. Therefore, many previous studies are flawed in the ability to reflect the characteristics of a specific infrastructure network. For example, the power network involves power flow calculation, the natural gas network involves steady-state simulation, and the WDNs requires hydraulic analysis. Especially for WDNs, the computation cost of optimization is high due to the time-consuming nonlinear hydraulic analysis. And few scholars have studied the game-theoretic model of deliberate attacks on WDNs based on the hydraulic analysis.

Hence, we address this important missing area of literature by proposing a game-theoretic model combined with hydraulic analysis to analyze the network interdiction problem of urban water supply lifeline. To against intentional attack, the allocation of protective resources in urban WDN involves a strategic game among the attacker, the defender, and the operator. Specifically, the defender allocates protective resources to maximize the system’s robustness to external disruptions, while the attacker responds correspondingly to generate greatest disruption via interdicting certain components (e.g., water plants) of WDN. From the view of operating function, the water supply capacity can well reflect the robustness of the WDNs. Therefore, the goal of the game is to focus on water supply capacity, while taking into account the secondary disasters or resilience funds of the water plant. In the first level, the system defender allocates protective resources among candidate water plants with the objective of minimizing the maximal potential disruption impacts that can be achieved by attackers. In the second level, the attacker responds correspondingly and interdicts water plants in order to reduce the water delivery capacity of the WDNs, making it unable to meet the water pressure or water demand requirements. Genuinely, the defender cannot protect all units, but to protect a few units due to resource constraints. When the defender decides to protect several of these units, the corresponding defense resources will be consumed, and the attacker is the same. We abstract and simplify the attack and defense resources, which is a universal approach, and will not lose generality. Through the two actions of defender and attacker, the third level models is the water delivery system that aims to give priority to users’ water supply demands under the current situation. We not simply abstract WDNs into supply–demand pair network model or complex network model, but combines hydraulic analysis to obtain the equilibrium solution of game model.

Since this kind of problem is NP-complete [48] and requires substantial computational effort, we have designed an effective heuristic algorithm based on the modified neighborhood search algorithm. A case study of a water supply network in a city in China demonstrates the effectiveness of investing protective resources. The contribution of this study lies in the following aspects:

• Propose a tri-level defender–attacker–operator game-theoreticmodel to analyze the network interdiction problem of WDNs.

• Use hydraulic analysis to obtain the reliability index of WDNs as the equilibrium index of the dynamic game model.

• Demonstrate the practical significance of the proposed method by a case study based on a simplified WDN of a Chinese city.

The remainder of this paper is organized as follows. A tri-level game-theoretic programming model is developed in Section 2 and a modified variable neighborhood search solution method is proposed in Section 3. Section 4 carries out a case study based on the water supply network data of a certain city in China. Finally, conclusions are drawn in Section 5.