A GraphTheoretic Framework for Assessing the Resilience of Sectorised Water Distribution Networks
Abstract
Water utilities face a challenge in maintaining a good quality of service under a wide range of operational management and failure conditions. Tools for assessing the resilience of water distribution networks are therefore essential for both operational and maintenance optimization. In this paper, a novel graphtheoretic approach for the assessment of resilience for large scale water distribution networks is presented. This is of great importance for the management of large scale water distribution systems, most models containing up to hundreds of thousands of pipes and nodes. The proposed framework is mainly based on quantifying the redundancy and capacity of all possible routes from demand nodes to their supply sources. This approach works well with large network sizes since it does not rely on precise hydraulic simulations, which require complex calibration processes and computation, while remaining meaningful from a physical and a topological point of view. The proposal is also tailored for the analysis of sectorised networks through a novel multiscale method for analysing connectivity, which is successfully tested in operational utility network models made of more than 100,000 nodes and 110,000 pipes.
Keywords
Water distribution networks Resilience Complex systems Graph theory Multiscale graph decomposition1 Introduction
Resilience can be defined as the ability of a system to maintain and adapt its operational performance in the face of failures and other adverse conditions (Laprie 2005; Strigini 2012). Depending on the application domain, different approaches are applied to assess the resilience of manmade systems. There is no single widely used definition for quantifying the resilience of water distribution networks (WDNs), a common method is to formulate the hydraulic resilience as a measure of the ability of a network to maintain supply under failure conditions. Todini (2000) proposed a resilience index based on the steady state flow analysis of WDNs and dissipated energy; consequently, the resilience of a water network was defined using a measure of the available surplus energy. Recent work extends this approach; for example, Raad et al. (2010) modelled the outcomes of this index proposing to work with alternative approaches to the original in Todini (2000); this was done to overcome computational difficulties in optimising a network design problem using Todini’s resilience index as an objective function. Therefore, by using parameters of physical significance in the supply as surrogates and not the real hydraulic details, di Nardo et al. (2013a) included these surrogate indices for optimizing the resilience of sectorised water distribution networks. Baños et al. (2011) proposed an extension to Todini (2000) by considering failures under various water demand scenarios. Other approaches based on a steady state hydraulic analysis include Prasad and Park (2004) who adapted Todini’s index by incorporating the effects of both surplus pressure and reliability of supply loops assessed by the variability in the diameter of pipes connected to the same node. They also applied the method to a multiobjective problem for the optimal design and rehabilitation of a water distribution network. Jayaram and Srinivasan (2008) used the same method from Prasad and Park (2004) for the analysis of networks with multiple sources. An alternative approach was proposed by Wright et al. (2015) who assess the resilience of networks with dynamically configurable topologies by estimating what percentage of operational demand can be supplied when a disruptive event occurs.
A common constraint of the above approaches is that the combination of possible failure scenarios grows exponentially as the network becomes bigger (Berardi et al. 2014) together with possible inconsistencies and uncertainties associated with hydraulic simulations (Gupta and Bhave 1994). Trifunović (2012) explores hydraulic properties of the network based on the statistical analysis of common parameters under normal operation and proposed them as indices to assess the resilience of a water network. However, these statistical parameters may not be valid under failure conditions (Giustolisi et al. 2008). Another inherent drawback is the dependence on well calibrated (accurate) hydraulic models for large networks. Beyond the implicit nonlinearity, the number of parameters involved in the hydraulic equations (Abraham and Stoianov 2015) and their large number of possible combinations introduce high complexity to the WDN calibration problem (an NPHard problem, which cannot be solved in polynomial time Takahashi et al. (2010).) For large WDNs, these have an expensive computational cost since, even for moderate size networks, the number of possible failure scenarios grows exponentially and are approached using either some linearising assumptions or the use of heuristics for failure simulations (Berardi et al. 2014).
This paper assesses the resilience of WDNs from a topological perspective where properties such as network configuration and redundancy in connectivity are taken into account together with physicalbased flow properties. This assessment is based on a computationally efficient implementation of the Kshortest paths algorithm (Eppstein 1998). It measures in a statistically robust way how every WDN demand node would be affected by disruptions in supply. This is done by analysing alternative paths between demand nodes and water sources (i.e. tanks and reservoirs). In addition, the paths are weighted by the hydraulic attributes of the supply routes to make the measures physically meaningful. Graphtheoretic indices have been applied to assess the resilience of water networks and used also as a surrogate measure in design problems (Yazdani et al. 2011). These surrogate indices are measures that correlate to physical/hydraulic based indices of resilience but are not necessarily based on physical indices whose parameters are either unavailable or are difficult to compute. These indices can be broadly classified into “statistical” and “spectral” measures (GutiérrezPérez et al. 2013). Among the statistical indices, the meshedness coefficient (Buhl et al. 2006) provides an estimation of the topological redundancy of a network. Centralpoint dominance (Freeman 1977), gives a measurement of the network vulnerability to failures corresponding to central locations. Flow Entropy (Raad et al. 2010) measures the strength of supply to a node both in terms of the number of connections and their similarity. The socalled demandadjusted entropic degree in Yazdani and Jeffrey (2012) is another alternative that uses demand on nodes and volume capacity on edges to compute a weighted entropic degree. More recently, (Liu et al. 2014) have extended the flow entropy measure to also consider the impact of pipe diameter on reliability.
The work also extends the proposed adaptation of Kshortest paths algorithm for the analysis of WDNs divided into sectors or District Metered Areas (DMAs) by utilizing multiscale network decomposition (Albert and Barabási 2002; Lee and Maggioni 2011). Demand nodes of a given DMA are aggregated into a sectornode, where a new graph (DMAgraph) is defined with vertices representing DMA areas and edges that abstract the sectortosector connectivity (Izquierdo et al. 2011). A mapping function is used to establish a relationship between the network graph of demand nodes and the DMAgraph, where every DMAvertex (or sectornode) has new features such as number of customers, total demand, and an average pressure aggregated from information of individual nodes of the DMA. The physical connections between sectors, which consist of pipes and boundary valves, form the edges of this DMAgraph and define the level of its connectivity. Resilience assessments carried out on individual demand nodes in the DMAs are also linked to the corresponding sectornodes, which are then used for a systemlevel resilience analysis. Considering both the network graph of demand nodes and a DMAgraph, a multiscale analysis of resilience is proposed.
2 A Graph Theory Perspective on the Definition of WDN Resilience
The network connectivity of a WDN can be modelled as a nearlyplanar mathematical graph^{1}, G = (V, E), where V (vertices) corresponds to n nodes and E (edges) corresponds to m pipes of the water system. This graph also has the particular characteristics that every node or edge should normally have at least one path of edges connecting it to a source node (tanks and/or reservoirs).
The weighted KSP algorithm returns infinity for Eq. 2 if there is no path of water distribution connecting the given node with a supply source. The contribution of such a source to the resilience index for a node in Eq. 1 will be zero. Consequently, both the resistance of paths to sources and redundancy in supply are explicitly taken into account by the resilience index.
In Eq. 2, the flow resistance over K paths is averaged since using the shortest path only may not be sufficient to represent the connectivity and indirectly resilience. By averaging over the complete set of routes (i.e. K → 2^{ n }, where n is a very big number for operational networks (Eppstein 1998)), the true connectivity of a node to a source is measured. However, the computational complexity of this measure grows exponentially (\(\mathcal {O}(2^{n} n)\) for the proposed index) and it quickly becomes infeasible for medium to large networks. Averaging over the Kshortest paths is a computationally feasible approximation that takes into account a sufficiently large number of paths in a robust statistical method.
2.1 Validation of the Resilience Assessment Method
A previously published network CTown (Ostfeld et al. 2012) is used as a validation casestudy for the graphtheoretic method proposed. This is a medium size network that consists of 333 nodes, 429 pipes, 4 valves, 5 pump stations, 7 elevated tanks, and 1 reservoir. A graphtheoretical analysis of resilience in WDNs (Yazdani et al. 2011) identifies a node of special importance as those with high betweenness centrality index, which measures the relative number of shortest paths from all vertices to all others that pass through a node. Nodes with betweenness centrality values larger than 2 times the standard deviation (SD) from the network average computed over all nodes will be denominated as high betweenness nodes. This work considers “critical transfer nodes” to those of topological importance with respect to the flow distribution with a WDN. Thus, critical transfer nodes are, a priori, those belonging to the main trunk and those with high betweenness.
Using K paths has two main advantages. Firstly, it emphasises the need to consider multiple paths of transport that contribute to the resilient operation of a network and it guarantees a robust statistical estimation since multiple paths are used to capture the average connectivity and capacity instead of the unique shortest path connecting two nodes. Secondly, the sum in Eq. 1 converges for relatively low values of K to obtain a good approximation of the analysis over all possible routes.
Once the tuning process provides an estimate of the K value, the proposal proceeds to compute the inverse of the resistance to supply, g(K), for the chosen K. In Fig. 1b g(30) is plotted for all individual nodes. The same plot also represents the values of g(30) for central nodes (maintrunk nodes and those with a betweenness centrality of at least 2 times the SD of the network average; they are plotted with red triangles). Most of the critical nodes have high g(K) compared to the average node g(K) value. In general, nodes of higher betweenness centrality are those that will appear more frequently in any shortest path, including paths to sources, and so will tend to have higher resilience as defined by Eq. 1.
2.2 Comparison of KShortest Paths Index (I _{ G T }) with Alternative Approaches
Resilience indices computed when a percentage of pipes are removed
Index  0 % removed  2.5 % removed  5 % removed 

I _{ r }  0.4476  0.3697  0.2561 
I _{ G T }  0.6306  0.5711  0.4166 
I _{ C B }  0.0715  0.0620  0.0550 
I _{ m }  0.0597  0.0508  0.0419 
I _{ S }  0.0369  0.0355  0.0351 
As shown in Fig. 2b, graphtheoretic indices show consistency with the hydraulic index^{2}. The index I _{ G T } changes in a proportional way to variations in the condition of the water distribution network. The entropic degree, I _{ S }, is the measure which provides the poorest estimation of the deterioration in resilience as more pipes are removed, similarly to the results presented by (Greco et al. 2012). Figure 2b also shows that the meshedness index, I _{ m }, has a narrow distribution. This is because meshedness is a measure that depends only on the relative number of nodes and pipes, without taking into account their physical or hydraulic properties.
3 Assessing Resilience for Large Scale Water Networks
The network topology of a WDN includes both treelike (usually hierarchical and lower redundant networks, i.e. having an equivalent proportion of nodes and links connecting them) and well meshed structures, where there usually exists several paths connecting two nodes and so a greater number of links compared to the number of nodes. These two type of structures can be interconnected by links forming graphbridges, which transport water from the transmission mains to these distribution areas. One way to analyse largescale WDNs is by partitioning the network into different components (assets: valves, pumps, pipes, and meters, among others). Alternatively, a WDN can be divided into subnetworks in order to make the analysis simpler than working with the whole system model. This is of interest especially in the case of largescale WDNs, which can contain hundreds of thousands of nodes and pipes, where consequently the classical methods of analysis are not efficient because of the large dimensions.
Many WDNs have been divided into sectors (DMAs) mainly for leakage management (Tabesh et al. 2009; Gomes et al. 2013; Xin et al. 2014). For this and a variety of other reasons (Herrera 2011), WDNs are divided into sectors nowadays and researches have focused their attention on different sectorisation approaches. The most studied works on water network division are based on variations of graph clustering (di Nardo and di Natale 2011; Perelman and Ostfeld 2011), spectral clustering (Herrera et al. 2010; Candelieri et al. 2014; Herrera et al. 2012), community detection (Diao et al. 2012), multiagent systems (Izquierdo et al. 2011; Hajebi et al. 2013), breadth and depth first search (Deuerlein 2008; di Nardo et al. 2014) or multilevel partitioning (di Nardo et al. 2013b). The work of Mandel et al. (2015) is the first to propose a sectorisation approach for largescale water networks. For these previously sectorised networks, the resilience index in Eq. 1 measures the level of connectivity of each demand node through various supply roots. A DMA division of the network automatically allows the use of two different scales of this resilience measure, having approaches of resilience for every individual node or measures summarised by sector. This provides new ways for analysing and managing the system and allows network configurations that have a major impact on resilience to be easily identified; for example multifeed, singlefeed, cascading DMAs.
3.1 Multiscale Resilience Analysis
When high variability exists, this constitutive rule computes the DMA resilience index over the whole set of demand nodes in the DMA and identifies the existence of low resilient nodes. The two Eqs. 5 and 7 assist operators to identify areas for a further investigation where a threshold of demand nodes or critical customers might not have the required level of resilience.
3.2 Multiscale Representation of a WDN
By considering all links hydraulically connecting DMAs, the edges of the new graph are formed by taking into account the open/closed status of valves, and consequently links which can provide additional operational resilience are identified. The weights of the new graph edges are proportional to the aggregated volumetric properties of individual pipes and valves that connect sectornodes. Sectornodes can also be used to visualize average levels of demand, pressure, elevation, etc. and the sectornode size can inherit topological characteristics of DMAs (eg. number of nodes or customers).
A multiscale network decomposition to assess the resilience of water systems is approached by studying their performance and network structure at various granularity levels and by analysing topological and physical information inherited from single demand nodes to sectors. Other important details that are obtained in the analysis include the number and location of multifeed DMAs, the number of valves within each DMA, and the pressure differential between neighbouring DMAs. In addition, the sectornode graph is used to identify which part of the network topology should require further investigation to improve resilience.
4 Experimental Study
Two operational WDNs were used to analyse the proposed methodology for assessing the resilience of sectorised water distribution networks. Firstly, a medium size WDN divided into 3 DMAs is investigated. A resilience assessment is carried out for each DMA. The second casestudy corresponds to a large water supply zone with high population density in England, UK. In both cases, certain sensitive information about the networks is withheld in order to maintain confidentiality.
4.1 CaseStudy A
Summary of the DMA configuration and resilience indices for WDNA
Sector ID  # nodes  # pipes  I _{ G T }(D M A) 

DMA1  1626  1770  0.0654 
DMA2  2416  2638  0.0869 
DMA3  778  826  0.0465 
TOTAL  4820  4032  0.0663 
In order to compute the proposed graphtheoretic index, I _{ G T } of Eq. 1, a sufficient number of shortest paths K need to be used. The analysis outlined in Fig. 1 was carried out on different nodes and it showed that K = 30 as sufficient for WDNA. In Fig. 4b, we plot the values of g(30) for each node of the WDN and for demand nodes with high betweenness centrality (labelled as critical nodes and also marked in red in Fig. 4a). For this network, analysis in Fig. 4b shows that the measure g(K) of the most critical nodes exceeds the average for the network.
From Table 2, it can be concluded that DMA3 is the sector with lowest resilience given the index I _{ G T }(D M A _{3}), while I _{ G T }(D M A _{2}) index is higher than the other two. DMA2 is directly connected with the only source that supplies water to the whole network. Thus, topologically and hydraulically, DMA2 is the most resilient sector of this network.
4.2 CaseStudy B
Summary of the DMA configuration of the top 5 WDNB sectors
Sector ID  # nodes  # pipes  Multifeed  I _{ G T }(D M A)  #CNs 

DMA29  2003  2081  Yes  0.0124  118 
DMA184  1119  1242  No  0.0179  32 
DMA85  784  834  Yes  0.0194  25 
DMA101  2106  2181  No  0.0212  60 
DMA208  2081  2249  Yes  0.0268  114 
As shown in Table 3, where the DMAs are ranked by resilience, 3 of the 5 DMAs contain more than 2,000 nodes, while only one of them has a size less than 1,000 nodes. Taking into account the number of critical transfer nodes within each sector (column #CNs of Table 3), it is shown that DMA29, DMA101, and DMA208 are sectors with larger number of transfer nodes. These DMAs also have a varying level of resilience; for example DMA29 has the smallest resilience index and DMA208 has the highest.
5 Conclusions

An inclusion of physical attributes of the water distribution network within graph theoretic indices gives the resilience analysis a hydraulically relevant and easy to quantify measure of how well connected a node is to the available water sources.

Computations for the proposed index scale quasilinearly as a function of the number of nodes (\(\sim \mathcal {O}(m+n\log (n)+kn)\)). This makes the measure feasible for application to large scale water networks, where the classical critical link hydraulic based analysis of network resilience would scale combinatorially.

Constitutive rules and functions facilitate the inclusion of empirical knowledge.
The proposed graphtheoretic framework is complimented by a novel multiscale decomposition method that converts the original WDN layout to DMAbased graphs. The developed method allows us to work with a WDN at two different levels of abstraction: single demand node and DMA. This enhances the results obtained for assessing the resilience of a WDN, especially in the case of large scale networks.
Footnotes
 1.
A planar graph is one that can be drawn on a plane without having edges intersecting other than at a node mutually incident with them. WDN graphs are not strictly planar due to the existence of some possible crossovers but are almost planar.
 2.
Please note that the proposed index does not replace/estimate hydraulically simulated indices. The presented analysis validates the consistency of results as failure conditions change.
Notes
Acknowledgments
This work has been performed with the support of the NEC  Imperial research project on Big Data Technologies for Smart Water Networks.
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