Mathematical Problems in Engineering

Volume 2016, Article ID 1431457, 10 pages

http://dx.doi.org/10.1155/2016/1431457

## Serviceability Assessment for Cascading Failures in Water Distribution Network under Seismic Scenario

^{1}Department of Construction Management, School of Economics and Management, Beijing Jiaotong University, Beijing 100044, China^{2}Beijing Center for Industrial Security and Development Research, Beijing 100044, China^{3}School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China

Received 29 November 2015; Revised 23 March 2016; Accepted 31 March 2016

Academic Editor: Stefano de Miranda

Copyright © 2016 Qing Shuang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The stability of water service is a hot point in industrial production, public safety, and academic research. The paper establishes a service evaluation model for the water distribution network (WDN). The serviceability is measured in three aspects: (1) the functionality of structural components under disaster environment; (2) the recognition of cascading failure process; and (3) the calculation of system reliability. The node and edge failures in WDN are interrelated under seismic excitations. The cascading failure process is provided with the balance of water supply and demand. The matrix-based system reliability (MSR) method is used to represent the system events and calculate the nonfailure probability. An example is used to illustrate the proposed method. The cascading failure processes with different node failures are simulated. The serviceability is analyzed. The critical node can be identified. The result shows that the aged network has a greater influence on the system service under seismic scenario. The maintenance could improve the antidisaster ability of WDN. Priority should be given to controlling the time between the initial failure and the first secondary failure, for taking postdisaster emergency measures within this time period can largely cut down the spread of cascade effect in the whole WDN.

#### 1. Introduction

Water distribution network (WDN) is a basic component in civil infrastructure systems. Its stability and reliability are important to ensure industrial production and public safety. Nowadays, WDN has developed into a large-scale network with thousands of pipes and nodes [1]. In such circumstance, how to design, improve, monitor, and repair the components efficiently in WDN becomes a critical problem in risk and reliability analysis. However, such analyses are often challenging due to complex network topology [2], components interdependency [3], and hydraulic calculation. Component failures may lead to the cascade effects and secondary failures under seismic scenario [4, 5]. This cascade reaction will prolong the repair cycle and lead to economic losses [6]. Therefore, it is necessary to quantify the effects of such cascading failures, to develop a system reliability evaluation method under natural hazards and to further analyze the serviceability of the WDN.

Studies on the WDN reliability under seismic scenario have been attracting extensive attention. Shinozuka et al. [7] analyzed the WDN of Memphis and evaluated the consumer demand under seismic scenario. Hwang et al. [8] analyzed the damage of pipes and the soil liquefaction under seismic scenario by using GIS technology. The serviceability was simulated by using Monte Carlo method. Regarding the power system as the backup of WDN, Adachi and Ellingwood [9] made serviceability analysis on the interaction influence of the power system and WDN under seismic scenario. Brink et al. [10] evaluated and compared the WDN emergency measures of Los Angeles Hydropower Board under seismic scenario. However, the studies above are concentrated on the reliability and serviceability of pipe network under seismic scenario, instead of considering the effect of cascade.

The identification of critical nodes is an important aspect of the system design and antidisaster ability of urban infrastructure [11]. Research shows that the failure of the critical nodes or edges may trigger the disastrous consequences, such as widespread avalanche and complete collapse [12, 13]. The research is focused on cascading failure of complex network, which is to find out the critical nodes of the network. Furthermore, an important step in WDN evaluation under seismic scenario is to identify the nodes which influence the serviceability seriously [14, 15]. Hence, to WDN, analyzing the serviceability of the system and identifying the critical nodes are crucial in guaranteeing the urban safety.

To analyze the serviceability in WDN, technologies from complex network and system reliability can be used. The serviceability can be measured in three perspectives: functionality of structural components under disaster environment; recognition of cascading failure process; and reliability calculation and further evaluation of the system serviceability.

This paper studies the serviceability of WDN in cascading failure caused by seismic action. Three factors, that is, the seismic attenuation, cascading failure, and reliability of water supply, are taken into consideration. The cascading failure process is provided with the balancing of water supply and demand. The node and edge failures in WDN are interrelated under seismic excitations. The matrix-based system reliability (MSR) method is used to represent the system events and calculate the nonfailure probability. The influence of serviceability is evaluated with system reliability and cascading failure process. This method is applied to WDN. On the consideration of the antiseismic reliability of the single component and the whole system, the paper analyzes the influence of the network tolerance parameter on system serviceability. The method adopted in the paper can help the decision-makers to identify the critical nodes, resist the possible widespread network failure under disasters, and improve the WDN serviceability.

#### 2. Simulation of the Cascading Failure Process in WDN

##### 2.1. Cascading Failure Model

The load distribution on the network is determined by many factors. The load can be the material, information, and energy [12]. In many entity networks, the load is transmitted along the edges based on the strategy of shortest path. The nodal capacity is the maximum load that a node can bear. As the goal of system optimization is to maximize the system operation effects with the minimum cost, it can be assumed that there exists a proportional relation between the bearing capacity and the initial load:where is the node capacity; is the node load; is the tolerance parameter, . is a tunable parameter [16]. It gives a way to control the strength of the capacity.

When the load is out of the capacity range, the node loses its functions and triggers the flow redistribution. The flow redistribution may lead to new failure nodes. This step-by-step process is a cascading failure [17–19]. Time is used to describe the cascading failure step. In the paper, is used for an algorithmic step. indicates no failure in WDN. describes the initial failure. show the cascading failure process.

##### 2.2. Cascading Failure Modelling in WDN

Whether a node in WDN can provide sufficient pressure and flow to the customers is the basic condition to judge the system operation. The service pressure is defined as the load. The nodal pressure is neither too high nor too low. In this scenario, a water node is operational if it can operate effectively and it functions as intended. It required that the nodal pressure is neither less than the design lowest nor higher than its capacity. The highest capacity is calculated by (1). After the initial node is attacked, if there are other nodes’ loads out of its capacity range, new failure nodes are generated. The end condition is that no other node loses its function.

In order to measure the cascading failure in WDN after a particular node failed, it is assumed that the consequence of a failure node is completely damaged. Its adjacent edges quit from the system after the attack. Therefore, the failure is equivalent to deleting the node and its adjacent edges from the water network, which means the network’s topology changed. With changes in the WDN topology structure, the water flow redistributed. Pressure of the operational nodes is calculated according to the laws of conservation of mass and energy. The Newton-Raphson method and node equations are involved to simulate the pressure throughout the network. Figure 1 shows the flowchart of cascading failure simulation in WDN.