Resilience enhancing expansion strategies for water distribution systems: A network theory approach

Abstract

Planners and engineers attempting to improve the resilience of water distribution systems face numerous challenges regarding the allocation and placement of redundancy so as to reduce the likelihood and impact of asset failures and take into consideration the growing demand for clean water, now and into the future. Water distribution systems may be represented as networks of multiple nodes (e.g. reservoirs, storage tanks and hydraulic junctions) interconnected by physical links (e.g. pipes) where the connectivity patterns of this network affects its reliability, efficiency and robustness to failures. In this paper we employ the link-node representation of water infrastructures and exploit a wide range of advanced and emerging network theory metrics and measurements to study the building blocks of the systems and quantify properties such as redundancy and fault tolerance, in order to establish relationships between structural features and performance of water distribution systems. We study the water distribution network of a growing city from a developing country and explore network expansion strategies that are aimed to secure and promote structural invulnerability, subject to design and budget constraints.

Introduction

Increasing water stress in many developing countries both endangers lives and restricts economic growth. Although levels of access to a piped water distribution are increasing at a global level, many of the developing world’s cities are served by old and poorly maintained networks with consequential problems in terms of both water quality and supply reliability (UNDP, 2006). The problem is compounded by the increasing frequency and scale of extreme natural events (CRED, 2009) and man-made catastrophes that cause major flow disruptions. Like many other infrastructural systems, water distribution systems consist of multiple interconnected components, whose individual or simultaneous failure may have adverse consequences in terms of disruption to water services. Therefore, improving system reliability and reducing system susceptibility to damage and perturbation are prime concerns for system engineers and utility managers responsible for the design, operation and protection of the Water Distribution Networks (WDN). As the cities of developing countries expand (often very rapidly) through urban migration, the expansion of existing WDNs provides an opportunity to enhance service resilience through the incorporation of strategic network redundancy.

Assessing WDN susceptibility to damage requires defining system performance indicators such as failures, reliability, resilience and vulnerability, as well as characterizing reductions in network structural qualities such as redundancy and optimal connectivity. Failures in water distribution systems mainly fall into two closely related groups (Ostfeld and Shamir, 1993): (i) mechanical failure of system components, and (ii) hydraulic failures in meeting consumer demand. The reliability of a water distribution system is usually defined as the probability of non-failure over a given period of time. The analytical evaluation of the reliability of large WDNs is deemed extremely cumbersome and difficult due to the high dependency of the involved procedures on data from individual system components and their failure modes, and the intricacies of impacts on system performance by the set of all possible subsystem failures (Mays, 2000, p18.2).

Hashimoto et al. (1982) were among the first to propose the use of indices of resilience (the speed of recovery from failure) and vulnerability (the extent of failure) for the assessment of water resource system performance. Later, Todini (2000), provided an index of resilience related to “the intrinsic capability of the system to overcome failures”, which to some extent addressed the problem of accounting for reliability. In this work we use a rather conceptual (but perhaps broader) definition of system resilience provided by Bruneau et al. (2003) characterized by the four infrastructural qualities of robustness, redundancy, resourcefulness and rapidity, which largely incorporate the notions of risk (likelihood and impacts of failures), reliability, recovery, and system tolerance at both pre- and post-failure stages. System vulnerability is consequently regarded as the antonym of resilience and hence we use the words resilience and invulnerability interchangeably. These characteristics viewed individually or together, can be expected to reduce the likelihood and impact of service failures and the total service disruption during emergency conditions (e.g. pipe breakage, fire demand and pump outage).

A comprehensive assessment of WDN resilience would entail an analysis and modeling of system performance or failure data during both normal and abnormal operational circumstances, the incorporation of any alternative supply options, as well as disruption response options. Because the data needed to support such an analysis are typically non-existent in a developing world context, this work adopts a restricted yet useful approach by studying the relationship between WDN structure (in particular, redundancy and optimal connectivity) and overall vulnerability. Such a structural analysis, using graph theory as the underlying analytical tool, provides an intuitively robust model of the water distribution system which leads to further explorations of the interplay between system layout and performance, efficiency and vulnerability. Moreover, it will demonstrated that performing such a limited assessment of water system resilience by looking at the structure of the network brings additional advantages such as low computational overheads and data requirements.

Analysis of the vulnerability of complex networks using graph theory techniques is concerned with the study and quantification of network building blocks (e.g. loops, connectivity) and the identification of structural weaknesses, critical locations and the impacts of failures of the core components and highly connected nodes or hubs (Albert et al., 2000). This is typically carried out using analytical techniques which assess the impacts of random failures or successive targeted attacks on the efficiency and performance of the network (Crucitti et al., 2005). Notable studies concerned with the structural analysis and assessment of the vulnerability of complex networks in other sectors have addressed the resilience of urban transport networks (Zio and Sansavini, 2007, Masucci et al., 2009), the vulnerability of power grids (Bompard et al., 2009, Crucitti et al., 2005 and Holmgren, 2006) and the World Wide Web (Albert et al., 2000). Furthermore, other methodologies have been proposed that analyze cascading failures in such networks, along with the traditional path-based methods in the analysis of infrastructure network reliability (Dueñas-Osorio and Vemuru, 2009).

A WDN can be described as a spatially organized system of multiple interconnected components arranged in a non-trivial and rather complex configuration. Such a system may be represented by a mathematical graph as a collection of nodes (e.g. junctions such as pipe intersections, reservoirs and consumers) and links (e.g. pipes and sometimes valves) which define the relationship between such nodes. In the literature, the problem of reliable design and (capacity) expansion of water distribution systems has been mostly addressed in the context of heuristic methods based on trade-off between reliability and cost by using stochastic simulation-based or mult-iobjective optimization techniques (Simpson et al., 1994, Savic and Walters, 1997, Todini, 2000, Todini, 2009, Prasad and Park, 2004, Agrawal et al., 2007, Tanyimboh and Kalungi, 2008). However, modeling WDNs by using network theory measurements has remained a relatively less explored area, with most of the previous works concerned with reliability or aggregation/skeletonization problems (Jacobs and Goulter, 1988, Walters and Lohbeck, 1993, Yang et al., 1996, Ostfeld, 2005, Giustolisi et al., 2008, Perelman and Ostfeld, 2011 and references therein). In general, graph models do not account for the importance and location of the isolation valves in a WDN reliability analysis (Walski, 1993), and alternative representations based on the segments (i.e. the portions of water distribution systems that can be isolated by valves) are required to make the use of link-node representations more realistic. Apart from such limitations and some other necessary simplifications that will be made to make modeling viable, the vulnerability and robustness of WDN may still be systematically exposed to analysis by graph theory and complex network measurements (Yazdani and Jeffrey, 2011a, Yazdani and Jeffrey, In Press).

Generally speaking, reliability analysis is a probabilistic approach quantifying the likelihood of failure, while structural robustness (invulnerability) analysis is a deterministic approach to quantifying susceptibility of a network structure to damage by using graph properties (Boesch et al., 2009). Studying the structure of water distribution systems may offer a simple framework for assessing network robustness against single or multiple component failures and providing support for decisions on assessment of water distribution system resilience. To this end, this work should be regarded as a contribution falling under the second viewpoint (i.e. the study of structural robustness and interplay between system structure and function) which complements but does not replace general reliability analysis.

Among notable contributions based on the use of graph models are work by Jacobs and Goulter, 1988, Jacobs and Goulter, 1989 who showed that the most invulnerable-to-failure design water distribution system is a regular graph with equal number of links incident to each node, while the inverse relationship is not necessarily true due to the existence of bridges (links whose removal disconnects the network) and articulation points (nodes whose removal along with the removal of their incident links disconnects the network). Kessler et al. (1990) used graph theory to develop a methodology for least cost design of invulnerable WDNs by incorporating reliability in the design of the network. Furthermore, Ostfeld and Shamir (1996) and Ostfeld (2005) utilized graphs to study the selection of one-level system redundancy “backups” in a WDN undergoing failures. A recent contribution uses topological clustering of water distribution systems for the purpose of connectivity analysis under various conditions including failure scenarios (Perelman and Ostfeld, 2011). In the work described below, by employing the graph representation of WDNs and quantifying the structure of such a graph, we demonstrate that useful information can be obtained about levels of network redundancy and optimal connectivity.

In such a representation, the redundancy (i.e. the existence of the alternative supply paths) is to some extent captured by the loops in the network designs, while structural vulnerability or lack of robustness is observed as graph cut-sets whose failure (i.e. the removal of their constituent components from the graph) may result in the disconnection of a large number of nodes (e.g. consumers or distribution branches) from water supply sources. Using graph models and network measurements enables a systematic quantification of the structure of WDNs even for large complex systems. This, in turn, allows for a comparison of connectivity patterns for alternative design, which can be used to improve system robustness and resilience by avoiding critical locations and network bottlenecks. This is, in particular, extremely important to water system engineers and urban planners contemplating sustainable strategies for WDNs within a wider context of optimally-designed next-generation infrastructures.

This paper is organized as follows. First, we briefly introduce some graph theory metrics and measurements and explain how they might be interpreted and used to capture qualities such as redundancy, optimal connectivity and structural robustness. Second, we study the structure of the water distribution network of Kumasi (Ghana’s second largest city), quantify its connectivity and redundancy and explore the relationship of these metrics to the network’s structural invulnerability and robustness. We then examine a number of scenarios for the expansion of this network to serve a growing part of the city (based on current development plans), and investigate different investment options to improve structural invulnerability subject to limited financial resources. We make heuristics-based recommendations on how to achieve budget-constrained network expansions supported by the undertaken measurements. Finally, we discuss the scope, usefulness and limitations of the proposed methodology in dealing with the resilience of real water distribution systems and briefly look at future directions for this work.