Mathematical Problems in Engineering

Volume 2015, Article ID 827816, 11 pages

http://dx.doi.org/10.1155/2015/827816

## A Cascade-Based Emergency Model for Water Distribution Network

^{1}Department of Construction Management, Beijing Jiaotong University, Beijing 100044, China^{2}Department of Construction Management, Dalian University of Technology, Dalian, Liaoning 116024, China

Received 24 November 2014; Revised 17 March 2015; Accepted 17 March 2015

Academic Editor: Carlo Cattani

Copyright © 2015 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

Water distribution network is important in the critical physical infrastructure systems. The paper studies the emergency resource strategies on water distribution network with the approach of complex network and cascading failures. The model of cascade-based emergency for water distribution network is built. The cascade-based model considers the network topology analysis and hydraulic analysis to provide a more realistic result. A load redistribution function with emergency recovery mechanisms is established. From the aspects of uniform distribution, node betweenness, and node pressure, six recovery strategies are given to reflect the network topology and the failure information, respectively. The recovery strategies are evaluated with the complex network indicators to describe the failure scale and failure velocity. The proposed method is applied by an illustrative example. The results showed that the recovery strategy considering the node pressure can enhance the network robustness effectively. Besides, this strategy can reduce the failure nodes and generate the least failure nodes per time.

#### 1. Introduction

Infrastructure system is a significant and complex system that supports the normal implementation of urban functions. Water supply system is one of the indispensable elements in infrastructure system. By the methods in complex network, a large number of the systems in the nature and society can be depicted in the form of network diagram, such as internet [1], power grid [2], and scientific citation network [3]. Hence, urban water supply system can be construed as a complex network with various physical edges. The performance of water distribution network (WDN) can be studied by the complex network theory [4].

The robustness of WDN refers to the ability to avoid the loss of functions and the ability to tolerate errors and failures after the whole network damage or the components damage. The study on the robustness is one of the typical issues in complex network [5]. Cascading failure, as a step-by-step failure process [6], turns out to be a topic of recent interest on robustness and network security [7]. A large amount of loads exist in WDN. The failure of a certain component (such as breakdown or attack) may redistribute the network load, which may further lead to the result that certain components suffer a failure caused by the exceeding of their bearing capacity. The failures of these components may be likely to result in a secondary failure and a chain reaction. After that, a large number of components and even the whole network may collapse, which cause severe damage to the network.

WDN is a geography-related distributed system. In correlation with geographic space, WDN is highly sensitive to natural disasters (e.g., earthquake, hurricane) or human attack (e.g., terrorist attack) [8]. In WDN, the failures of the critical components arising from the destruction may lead to people and property loss and even impact the economic and social development. With the study on the cascading failures of WDN, Adachi and Ellingwood [9] constructed the service function assessment model of the urban water distribution system under the impact of earthquake, with power system taken as the back-up measure. Sitzenfrei et al. [10] established a cascading risk map with GIS, which involves both hazard and cascade vulnerability. The applied research showed that the neglect of cascading events undervalues the risk in WDN. Yazdani and Jeffrey [11] argued the fact that WDN was a spatial organization network. The structure and vulnerability of the network were studied by complex network. Four benchmark water networks were measured with the network paths, cycles, efficiency, and connectivity. Yazdani et al. [12] studied the structure of Kumasi’s WDN and quantified its connectivity and redundancy. The relationship between these metrics and network robustness illustrated that the simple topological measure only depicted partial network structure without the ability to describe network property completely. The assessment on the robustness of WDN should be further considered with hydraulic attributes. Hawick [13] argued that, after a series of optimizations and designs, WDN developed into a highly complex network. Shuang et al. [14] established the node vulnerability analysis model of WDN based on cascading failures in combination with the network and hydraulic properties. It was proved that this model could identify the critical nodes of the system effectively under the cascading effect. Shuang et al. [15] simulated the reliability peak and persistent time of cascade propagation in WDN. The reliability assessment of WDN under cascading effect should introduce system uncertainties.

It is urgent for the government and society to cope with the disasters and bring down the losses resulting from disasters. It is proved that the approach to prevent and control the spread of disaster events is to explore the evolutionary mechanism of disaster and then bring forward an effective strategy. In the cascading failure model of WDN, if the load exceeds the bearing capacity, system components fail, which may further trigger the redistribution of network load and then the secondary failures. In actual life, however, there always exist emergency response mechanisms to reduce the loss of failures and restore the normal service function. Upon the failures of WDN, external emergency forces can intervene in the failing components to cope with emergencies and assist in their restoration.

The paper studies the emergency recovery strategies on WDN with the approach of complex network and cascading failures. The model of cascade-based emergency for WDN is built. The cascade-based model considers the network topology analysis and hydraulic analysis to provide a more realistic result. A load redistribution function with emergency recovery mechanisms is established. From the aspects of uniform distribution, node betweenness, and node pressure, six recovery strategies are given to reflect the network topology and the failure information, respectively. The recovery strategies are evaluated with the complex network indicators to describe the failure scale and failure velocity. The proposed method is applied by an illustrative example to analyze the impact on recovery with the six strategies.

#### 2. The Cascade-Based Emergency Model of WDN

##### 2.1. The Topology Structure

Prior to the simulated computation of WDN, there is a need to store the graphic information in a certain way and then establish the model for the WDN. Characterized by the network-like topological structure, WDN can be analyzed by the graph theory. Since the water in pipe flows along a certain direction, WDN is a directed graph [16]. Water reservoirs, consumers, and tanks can be abstracted as nodes. Pipes, pumps, and valves can be represented as edges. The neighboring nodes are connected by pipes. The incidence matrix is used to describe the relationship between the nodes and pipes in the network. The number of rows is equivalent to the nodes, and the number of columns is equivalent to the pipes, respectively. in the matrix can be expressed as

##### 2.2. Parameters

###### 2.2.1. Load

Load is a significant physical parameter in the cascading failure model. The dynamic load changes lead to the cascading failures of the critical infrastructures such as power grid, water distribution system, gas supply network, traffic network, and communication network. Load also exists on the components such as nodes and pipes in WDN. Both the excessively large and small loads result in the flow changes, which may further trigger a series of cascading failures. After attack, the WDN redistributes the water pressure and flow in the network according to the topological structure and hydraulic changes. The complex network distributes the network load according to betweenness. To further study the cascade propagation of WDN, the node service pressure in the normal operation is used as the initial load [14].

###### 2.2.2. Capacity

Capacity is the load that a node can bear. If the load exceeds its bearing capacity, components suffer from failures. These failures cause the flow redistribution, leading to secondary failures. With the increase in operation time and the expansion in urban size, WDN has been further expanded based on the original construction size. Meanwhile, water demand changes randomly with the requirements of urban population and industry. As a result of these, the node pressures are no longer the initial value but they will be left in a changing state. Therefore, for the water distribution system that keeps running for a long time, there is a need to take account of the changes in node pressure and measure the robustness of WDN by its new constraints. The node minimum capacity is defined as the acceptable minimum water pressure . The node maximum capacity is defined as follows:where is the service pressure of node . The service pressure can be obtained in normal operation. It ensures that all the water demand can be satisfied. The node maximum capacity refers to the upper limit of the nodes constrained by cost or aging. In (2), expresses the node tolerance, which implements the control over the intensity of initial load and load distribution. has evaluated the extra pressure that a nodes in WDN can bear. has presented a way to assess the performance of WDN from system perspective. The greater the value of is, the huger the difference between the components will be; that is, the load distribution will be more nonuniform.

The research focuses on the dynamic changes in WDN. The network remains stable in the initial stage. The load of every node is smaller than its capacity. After the failure occurs, the load on the failure node is distributed to its neighboring nodes. If the neighboring nodes are unable to process the extra load (i.e., the load exceeds its capacity), the neighboring nodes fail and then result in cascading failures. Equation (3) describes the failure propagation process. If the redistributed node pressure is within its capacity, the node can still supply water; otherwise, the node is recognized as a new avalanched node. The connected pipes are closed to avoid risk expansion. Consider

###### 2.2.3. Actual Demand

The pressure-driven simulation method is based on the laws of conservation of mass and energy to determine pressure and flow distribution. The pressure-driven simulation method is effective in describing WDN which can prevent the negative water pressure under failure conditions. With the help of pressure-driven strategy, Wagner’s model suggests that there is a relationship between actual flow and node pressure [17]. This function is widely applied with the high ability in calculating node demands [18, 19].

In Wagner’s model, the demand supplied equals the required one as a customer controls a faucet when the system capacity is not exceeded [20]. The demand becomes the maximum allowed by the actual pressure in pressure-deficient condition. In addition, the node pressure should be neither too low nor too high. Abnormally high pressures may cause aging pipes to burst and lose service functions. Therefore, each node pressure must be controlled between the maximum pressure and the minimum pressure. When , the abnormally high-pressure condition occurs. To avoid risk expansion, it is assumed that the section where the failure demand node covered is isolated from the rest of the network. The failure node is removed out of the network and its connected pipes are closed. Hence, based on Wagner’s model, the relationship between actual demand and node pressure is revealed in combination with the minimum and maximum pressures. Considerwhere is the actual demand of node at time . is the required demand of node at time . The required demand is the full demand at the node. It can be obtained as the WDN performing normally. is the pressure of node calculated at time . is the minimum pressure of node . is the service pressure of node . is the maximum pressure of node .

There is still a lack of mature and universal hydraulic analysis software that can support the pressure-driven simulation method. In consideration of this, the robustness of WDN under the condition of cascading failures is figured out in combination with EPANET 2 and the actual demand function.

##### 2.3. The Cascading Failure Model

###### 2.3.1. EPANET 2

EPANET 2 is used to run the hydraulic simulation. EPANET 2 [21] is open source software developed by the United States Environmental Protection Agency (EPA). EPANET 2 can simulate the water hydraulic and water quality of WDN in a certain period of time. The gradient algorithm proposed by Todini and Pilati [22] is combined in EPANET 2 to find a solution for steady flow equations. EPANET 2 has been widely applied because of its high computing speed and satisfactory simulation effect. In terms of academic study, EPANET 2 is nearly deemed as the standard computing engine due to the accurate computing result and good compatibility [19, 23, 24].

The numerical simulation of WDN is implemented by calling Toolkit from EPANET 2 through MATLAB under the failure condition. EPANET 2 has provided the users with Toolkits, which is implemented by the way of dynamic link library (DLL) technology. It allows developers to customize computing engine of EPANET 2 according to their specific demands. The “.inp” output file of EPANET 2 has presented a targeted development approach that can be embedded in the optimal design or parameter analysis to increase the function of aided analysis. The process of MATLAB calling EPANET 2 Toolkits with pressure-driven strategy is shown in Figure 1.