Accounting for Source Location on the Vulnerability Assessment of Water Distribution Network

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Accounting for Source Location on the Vulnerability Assessment of Water Distribution Network

Hieu Chi Phan, Ph.D.¹; Ashutosh Sutra Dhar, Ph.D., P.Eng.²; and Nang Duc Bui, Ph.D.³

Abstract:The vulnerability of water distribution networks (WDNs) to water-main breaks is used for prioritizing pipes in the network for maintenance planning. In this paper, a parameter of graph theory, algebraic connectivity (AC), is employed as a vulnerability metric for WDNs. The change in the magnitude of AC of the networks due to the removal of pipes was found to have a strong correlation with loss of robustness of the WDNs or the size of the network being isolated by the water-main breaks. However, because AC is a topographic measure of graphs, the effects of the location of water sources are not accounted for in ranking through the change in AC. A virtual network is proposed here to overcome the limitation and to move the centrality of the network to the pipes connecting to the water source. The resulting AC based ranking is found to have a correlation with hydraulic impact factor-based ranking of the pipes of WDNs.DOI:10.1061/(ASCE)IS.1943- 555X.0000620.© 2021 American Society of Civil Engineers.

Author keywords: Water distribution network (WDN); Vulnerability analysis; Algebraic connectivity (AC); Prioritizing pipes;

Consequence assessment; Virtual network.

Introduction

The water distribution system (WDS) is a major part of the munici- pal infrastructure system that comprises a complex network of in- terconnected pipes. Pipes in the WDS are subjected to deterioration with age, leading to water-main breaks. For rehabilitation of the deteriorating water mains in the water distribution network (WDN), it requires evaluating the importance level of each pipe component based on the effect of its break on the overall system. However, the task is very complex for the WDS with thousands of pipe compo- nents in the network. To this end, a risk assessment framework is proposed to identify the critical pipes by assessing risks based on scenarios of the likelihood of failure events and the consequence of the events. Water-main breaks can have various consequences, including: (1) the effect on the overall network performance (reduction of redundancy or isolation of a part), (2) economic impact as- sociated with water loss, rehabilitation cost, and the cost of any damage, and (3) the effects on public health, safety, and security.

The effect on the overall network performance is often expressed as the robustness of WDN. Researchers employed different matrics of graph theory to examine network connectivity as a measure of the performance of WDN (Meng et al. 2018; Pagano et al. 2018;

Yazdani and Jeffrey 2012). The metrics of the graph theory pri- marily provide the topological condition of WDN. A major limitation of the approach is that it does not account for the hydraulic condition of the network, including the direction of flow (Hwang and Lansey 2017). However, a detailed hydraulic based assessment of WDN suffers from different limitations, including uncertainty of the hydraulic model and the requirements of extensive computa- tionally prohibitive combinatorial simulations (Gheisi and Naser 2015;Ulusoy et al. 2018; Berardi et al. 2014; Diao et al. 2016).

The demand-driven or pressure-driven hydraulic simulations do not accurately replicate the behavior of WDNs (Gupta and Bhave 1994;Lee et al. 2016). As a result, there has been growing interest in using surrogate analytical measures for the performance assessment of WDNs (Ulusoy et al. 2018). Torres et al. (2017) examined the metrics of graph theory for characterizing the hydraulic and water quality performance of WDS. The graph metrics were found to show a strong correlation with WDS performance measures, including the system level hydraulic performance measured as the maximum hourly unit head loss. The head loss was found to decrease with increasing the graph metrics such as algebraic connectivity (AC). A higher value of AC corresponds to a well-connected network that yields a lower head loss.

The AC is a well-known parameter of graph theory that can be used for the assessment of the robustness of networks (Giudicianni et al. 2018;Pagano et al. 2018;Yazdani and Jeffrey 2010). In many cases, AC is found as an important metric for the topological characteristics of the networks. For example, Pandit and Crittenden (2012) revealed that the AC has the highest contribution to the network resilience compared with five other metrics of graph theory (i.e., graph diameter, characteristic path length, central-point dominance, critical ratio of defragmentation, n, and meshedness coefficient). Phan et al. (2017,2018,2019) observed that the AC yields different magnitudes for networks having the same number of nodes and links but different ways of connecting the nodes or for removal of the link(s) from the network. They took advantage of the change in the magnitudes of AC due to the removal of pipe segment(s) in the network to develop a procedure to identify the importance level of the segment(s). The method was successfully applied in ranking water mains based on the effect of their breaks

1Lecturer, Dept. of Civil and Industrial Engineering, Institute of Techniques for Special Engineering, Le Quy Don Technical Univ., 236 Hoang Quoc Viet, Hanoi 100000, Vietnam (corresponding author).

ORCID: https://orcid.org/0000-0001-9603-486X. Email: phanchihieu@

lqdtu.edu.vn

2Associate Professor, Faculty of Engineering and Applied Science, Memorial Univ. of Newfoundland, 230 Elizabeth Ave., St. John’s, NL, Canada A1C 5S7. ORCID: https://orcid.org/0000-0001-5137-3921. Email:

[email protected]

3Associate Professor, Le Quy Don Technical Univ., 236 Hoang Quoc Viet, Hanoi 100000, Vietnam. ORCID: https://orcid.org/0000-0003-2767 -6003. Email: [email protected]

Note. This manuscript was submitted on March 12, 2020; approved on February 14, 2021; published online on June 9, 2021. Discussion period open until November 9, 2021; separate discussions must be submitted for individual papers. This paper is part of theJournal of Infrastructure Systems, © ASCE, ISSN 1076-0342.

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on network disconnection or reduction of redundancy (Phan et al. 2019). Note that although the AC provides the topological importance of each pipe in the WDN, its magnitude remains inde- pendent of the location of the water source. For example, the change of AC for removing a link will be the same regardless of whether it is connecting the network to the water source or not.

However, the pipe connecting the WDN to the water source is more important than the other pipes, which would subsequently change the importance levels (and the consequences of failure) for all other pipes in the network. Thus, the current approach of using AC is unable to account for the topological importance of a pipe or a network of pipes connecting the water source.

In this paper, a novel approach is developed to account for the effect of water source (i.e., tank) location in AC, a measure of the water-main breaks on the WDN performance (i.e., vulnerability).

The paper examines first the AC and other graph theory metrics through applications to various benchmark WDNs. Then, a virtual network approach is developed to account for the tank location in the AC for the pipes in an undirected network. The method is illustrated through application to some benchmark WDNs.

Finally, the performance of the developed method is compared with a hydraulic-based method. It is, however, recognized that the hydraulic-based method relies on the assumptions used in the hydraulic analysis and, therefore, may not represent the actual scenario of WDS.

Methodology

Algebraic Connectivity

The AC is a mathematical parameter derived from a network connectivity matrix. The WDN can be modeled as a graph of G¼GðV;PÞ, whereV= set ofnnodes, andP= set of m pipes.

An adjacent matrix A of G is used to describe the link between the nodes (network connectivity)

A¼a_ij ð1Þ

whereaij¼1if there is a link (pipe) between nodeiand nodej;

and a_ij¼0if there is no link (pipe) between node iand nodej.

The node-degree matrix is a diagonal matrix, which contains the information about the number of connections (node-degree) at each node, and is defined as:

D¼diagðd_iÞ ð2Þ di = number of connections (node-degree) of node i, where:

d_i¼Xⁿ

j¼1

a_ij ð3Þ

Then, the Laplacian matrix is given by Eq. (4):

L¼D−A ð4Þ The second smallest eigenvalue (λ2) of the Laplacian matrix is the AC. The magnitude of AC is different for networks having the same number of nodes but different connectivity, measuring the network connectivity.

Several other metrics of graph theory were also examined for the assessment of the robustness and redundancy of WDN (Yazdani and Jeffrey 2010). Some of them are appropriate for evaluating the overall network condition rather than the effects of specific interconnections between the nodes. For example, average node degreeandnode and link connectivityare the measures of robustness and redundancy of overall networks, which are defined as k¼2m=n, and p¼2m=ðn×ðn−1ÞÞ, respectively, where m is the number of links and n is the number of nodes. The AC has properties that could be used to relate to the effects of specific connections between the nodes. It also has a strong correlation with other metrics of graph theory. Table1shows a comparison of different measures of 18 benchmark WDNs available in the published literature with the layouts provided in Fig.1. It shows that the average node degree scatters around two and three with marginal changes. This narrow variation and the low average node degree cause a tendency that if the size of the network (the number of nodes, n) increases, the node connectivity decreases (Yazdani and Jeffrey 2010). The node and link connectivity are well-known for negative correlation with the AC (Giudicianni et al. 2018;

Jamakovic and Uhlig 2007). Parameters of the networks shown in Table1also reveal that the value of AC decreases with the increase of the network size and has a positive correlation with node

Table 1.Graph theory metric of several benchmark networks

# Network

Number of nodes (n)

Number of pipes (m)

Average node degree (k)

Node and link connectivity (p)

Algebraic connectivity (AC)

1 East-Mersea 755 769 2.03709 0.00270 0.00020

2 Colorado Springs 1,786 1,994 2.23292 0.00125 0.00024

3 Kamasi 2,799 3,065 2.19007 0.00078 0.00009

4 Richmound 872 957 2.19495 0.00252 0.00006

5 Hanoi 32 34 2.12500 0.06855 0.06116

6 BakRyan 36 52 2.88889 0.08254 0.07860

7 New York Tunnel 20 21 2.10000 0.11053 0.11799

8 Jinlin 28 34 2.42860 0.08995 0.07667

9 GoYang 23 30 2.60870 0.11858 0.10741

10 Fossolo 37 58 3.13514 0.08709 0.21888

11 Pescara 74 98 2.64865 0.03628 0.00891

12 Modena 272 317 2.33088 0.00860 0.00908

13 Balerma Irrigation 447 454 2.03132 0.00455 0.00069

14 Mount Pearl 3,850 3,991 2.07325 0.00054 0.00002

15 Ozger 15 21 2.80000 0.20000 0.30837

16 Zhi Jiang 114 164 2.87719 0.02546 0.01244

17 KLM 936 1,269 2.71154 0.00290 0.00191

18 Exeter 1,893 2,418 2.55470 0.00135 0.00103

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and link connectivity (Fig.2). Generally, between two WDNs, the one with the larger size has more probability of having a lower AC.

A network with lower AC also indicates a larger maximum distance between any pair of nodes (i.e.,graph diameterdefined according to graph theory) (Godsil and Royle 2013).

The comparison with different networks in Table1reveals that the AC is well-correlated with some other metrics of graph theory in defining the robustness and redundancy of the network. Besides, it has other advantageous properties such as a decrease of its value with the decrease of redundancy of the network and an increase of Fig. 1.Layouts of 18 benchmark WDNs.

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Fig. 2.Correlation of AC with: (a) number of nodes; and (b) node and link connectivity.

Fig. 3.Effect of link breaks: (a) break leading to disconnection; and (b) change in the Laplacian matrix.

Fig. 4.Change of AC after a link break for: (a) the intact network; and

(b) isolated subnetworks. Fig. 5.Change in AC (ΔAC) leading to disconnection of WDN.

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its value when isolation of node or network occurs (Fiedler 1973;

Phan et al. 2018,2019). With these properties, the AC is more suitable than the other graph theory metrics in analyzing the vulnerability of the network.

Effects of Breaking Links on AC

A water-main break in a WDN can be idealized as the removal of a link from the network. The removal of a link from the network can

have two possible consequences. First, it can reduce the probability of having a connection between any two nodes in the network, which is termed as loss of redundancy. The loss of redundancy has a positive correlation with the AC because the calculated AC of a network is reduced with the removal of link(s).

Second, the removal of a link may disconnect/isolate a part of the network, splitting it into two separate networks (Fig. 3). In Fig.3(a), the removal of linkiseparates the network into two subnetworks A and B. If the subnetworks A and B have p and q nodes

Fig. 6.Effects of graph types on AC: (a) types of graph; and (b) effects on AC.

Fig. 7. Flowchart for the assessment of WDN with a tank: (a) identification of virtual network; and (b) ranking of pipes in the network.

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(total pþq nodes in the network), respectively, the Laplacian matrix takes the form as shown in Fig. 3(b). The second smallest eigenvalue of the Laplacian matrix, i.e., AC, is greater than the AC of the intake network (with pþq nodes). The AC after disconnection corresponds to the lower of ACs calculated for subnetwork A (AC_A) and subnetwork B (AC_B). Fig.4shows the ACs of a network and subnetworks resulting from separation due to the removal of a link. The figure shows that the AC of the network increased from 0.01241 to 0.01289 due to the disconnection. Thus, an increase of AC with the removal of a link indicates the isolation of a part of the network.

Phan et al. (2018) took advantage of the properties of AC to identify the reduction of redundancy and network disconnection of WDS due to water-main breaks. For the reduction of redundancy, the amount of change of AC due to the removal of a link is calculated as the difference of the AC of the intake network from that of the network without a link. For example, the change in redundancy due to removing pipei is calculated as ΔACi¼ACi−ACintact. Because the AC_iis lower than AC_intact, AC_i−AC_intactis negative for the case of a reduction of redundancy. Because AC is related to the degree of connection between any two nodes, a higher decrease of AC implies that the link has higher importance on the network redundancy. Thus, based on the change in AC due to the removal of links, the pipes can be ranked in order of their importance (Phan et al. 2017,2018,2019).

For the case of disconnection, the AC of the network is taken as the minimum of the ACs of the subnetworks isolated by the removal of a link. Because the AC is higher for each of the separated networks (Fiedler 1973), the difference between the AC of the

network without the link and the intake network is positive. Thus, a positive change in AC is used to identify possible disconnection due to a water-main break (Phan et al. 2018). It is also observed that if a small part of the network is isolated from a large network, the change of AC is less. This is illustrated through application to a WDN in Fig.5(Phan et al. 2019).

Fig.5shows that each of four pipes marked as circle, square, star, and diamond can isolate a part of the network in shaded areas.

The area affected by the removal of the pipe marked as the circle is the largest, which is gradually less for the removal of pipes marked as square, star, and diamond, respectively. The amount of the change of AC (ΔAC) also decreases in the same order with the largest change for removing the pipe mark as the circle and the smallest change for removing the pipe marked as the diamond.

Thus, the magnitude ofΔAC can be used as a measure of the se- verity of disconnection (i.e., area affected) due to the removal of a pipe. The pipe break that causes more severe disconnection is more important to the network. Phan et al. (2018,2019) employed the characteristics of AC to rank water mains according to their importance. The AC-based technique was applied to assess a broader risk of the water distribution network using a fuzzy inference system in Phan et al. (2019).

However, if the water source (tank) is connected to the area affected by the pipe marked as diamond, failure of this pipe would have a more significant effect on the WDN because water would not be supplied to the rest of the network. This effect is not accounted for in the pipe ranking method employed in Phan et al. (2018,2019).

A virtual network approach is presented in this paper to account for the location of the water source in the pipe ranking usingΔAC.

Fig. 8. Networks with a single tank: (a) Hanoi; (b) Jinlin; (c) Zhi Jiang; and (d) Mount Pearl WDNs.

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Proposed Virtual Network Approach

The AC based approach was found to reasonably rank pipes based on their effect on the topological changes of the network (Phan et al. 2019), except for the effects of the tank location. Usually, the pipe connected to the tank, termed herein as theαpipe, is ranked as less important because their failure does not isolate a large network.

However, because these pipes supply water from the tank to the rest of the network, these are highly important. Similarly, pipes surrounding the α pipe supply water to the downstream levels and are also important. The importance level gradually decreases to- ward the downstream levels. To increase the importance level of theαpipes and their surrounding pipes, a virtual network is proposed to be added at the tank location, which would be connected by these pipes. Thus, theαpipes will be the bottlenecks connecting the original network and the virtual network, increasing their importance levels. The resulting augmented/adjusted network com- posed of the original network and a virtual network is then analyzed to rank the pipes.

However, the virtual network should have a significant effect on the topologic structure of the adjusted network to cause theαpipes to be the important bottlenecks. For a large network with thousands of nodes, adding a virtual subnetwork of several nodes and pipes may not be sufficient to become an important part of the adjusted

network. Therefore, it is necessary to identify the size and type of the virtual network to be added. In general, the addition of a virtual network would reduce the overall AC of the network. For computa- tional effectiveness, a virtual network with a limited number of nodes (to minimize computation time) but having a significant effect on the reduction of AC of the network is preferred. The size and type of the virtual network would typically depend on the characteristics of a particular network. A methodology is developed for selecting a suitable virtual network to increase the rank of theα pipes effectively.

The reduction of AC with the increase of nodes depends on the types of the network. The effect of increasing nodes for different types of networks, namely path graph, ring graph, and lattice graph, are examined in Fig. 6. Fig. 6 shows that the reduction of AC with the increase of nodes is the highest for the path graph (also for the ring graph) and the lowest for the lattice graph. The AC of the benchmark WDNs discussed earlier lies between the ACs of the lattice graph and the ring graph. Among the different networks, the path graph has the lowest AC. Thus, the path graph is expected to have the highest impact on the network with the addition of the lowest number of nodes. A path graph type of virtual network is therefore recommended for using to represent the water tank.

Fig. 9.Contour ofΔAC: (a) Hanoi; (b) Jinlin; (c) Zhi Jiang; and (d) Mount Pearl.

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Fig.7presents the framework proposed to implement the virtual network approach for ranking water mains considering the tank location, including identification of the suitable size of the virtual network. The intact network is first analyzed by removing each of the pipes at a time and calculating the change in the AC to rank the pipes based onΔAC. Then, the order of ranking of the pipe connected to the tank (αpipe) is examined. If the pipe has the highest ranking, no virtual network is required, and the ranking of the pipe obtained from the analysis is used for maintenance planning.

However, if the ranking of the pipe connected to the tanks is not the highest, a path graph type of virtual network is linked to the pipe.

The analysis of the network is performed with the addition of nodes to the virtual network until the pipe connected to the tank has the highest ranking.

Case Studies

The proposed method is demonstrated through application to different WDNs, such as: the Hanoi network (Fujiwara and Khang 1990), Jinlin network (Bi et al. 2015), Zhi Jiang network (Zheng et al. 2011), and the Mount Pearl network (Phan et al. 2019), shown in Fig.8. These networks cover various possible sizes of WDNs.

The Hanoi and Jinlin networks are small-sized networks with 32 and 28 nodes, respectively. The Zhi Jiang network is medium-sized

(114 nodes) and the Mount Pearl network (3,850 nodes) is relatively larger-sized. The Hanoi network contains large loops with three branches. The Jinlin network includes a single pipe connecting subnetworks. The Zhi Jiang network is a lattice type network.

The Mount Pearl network is separated into a tree-like subnetwork (upper part) and hierarchal loops (lower part) with different sizes and various near-end branches.

Fig.9plots the contours ofΔAC of the networks (intake network). The line thickness in the figure corresponds to the magnitude ofΔAC and hence the ranks of the pipes. The pipes could be ranked in numerical order starting from 1 (1 being the highest rank having the most significant impact) based on ΔAC (Phan et al.

2019). However, for simplification in presentation, the pipes with a range ofΔAC are placed in a group in Fig.9. Thus, each of blue, cyan, green, yellow, black, magenta, and red lines in the figure are for a range ofΔAC and are categorized as the pipes with high impact (redundancy group), moderate impact (redundancy group), low impact (redundancy group), insignificant impact, low impact (disconnection group), moderate impact (disconnection group), and low impact (disconnection group), respectively. As seen in the figure, the thick blue lines correspond to the highest negative ΔAC, failure of which would cause the most significant reduction of redundancy, whereas no part of the network will be isolated. On the other hand, the removal of thick red pipes in Fig.9causes the

Fig. 10.Ranking ofαandβpipes against the size of virtual network (path graph): (a) Hanoi; (b) Jinlin; (c) Zhi Jiang; and (d) Mount Pearl networks.

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highest positive ΔAC, indicating that the failure of these pipes would lead to the most significant disconnection of the networks.

However, none of the pipes having the highestjΔACjin the intact network is connected to the water source (tank) (termed herein as theβpipes). The pipes connected to the tank (αpipes) should have the most significant effect on the network, which is not revealed in the ranking of pipes shown in Fig.9. Therefore, virtual networks (path graph type) with varying numbers of nodes are added to each of the networks at the locations of the tank, and the pipes are re- ranked. The inclusion of a virtual network increases the importance levels of theαpipes and decreases the importance level of theβ pipes, which depends on the number of nodes added to the virtual network. Fig.10shows the change of ranking of theαandβpipes with the increase of nodes in the virtual networks.

For the original Hanoi network [Fig.10(a)], theαpipe is ranked 32 out of total 34 pipes without the virtual network, implying that its effect is insignificant in the network. With the increase of nodes in the virtual network, the importance levels of theαpipe increases (the rank becomes less) and of theβpipe decreases (the rank becomes high). The rank of theαpipe is 2nd for a virtual network containing five nodes. The rank of the pipe is the first for a virtual network of 11 nodes. A virtual network of 11 nodes is therefore used for ranking of the pipe in the Hanoi network.

Similarly, the rank of the αpipe in the Jinlin, Zhi Jiang, and Mount Pearl networks reach the first position with virtual networks having at least six nodes, 15 nodes, and 400 nodes, respectively.

Note that the number of nodes to be chosen for the virtual network usually is higher for larger WDNs. Because of the large size of the Mount Pearl network, a 100-node interval is chosen to increase the

Fig. 11.Ranking based onΔAC for adjusted networks: (a) Hanoi; (b) Jinlin; (c) Zhi Jiang; and (d) Mount Pearl network.

Fig. 12.Hydraulic importance factors for: (a) Hanoi; (b) Jinlin; (c) Zhi Jiang; and (d) Mount Pearl.

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size of the virtual network for each step of the analysis. However, there is no general rule for choosing the exact size of the virtual network, which can depend on both the topologic character and location of the tank in the network. Thus, the minimum size of the virtual network is determined through iteration.

The contours ofΔAC for the pipes in the networks adjusted with the virtual networks are shown in Fig.11. It shows that the importance levels of the pipes have been changed from those in

Fig.9(for the original networks). The pipes connecting to the tank have the highest importance level (thick red) with positiveΔAC (indicating disconnection).

Comparison with a Hydraulic Method

To compare the ranking of the pipe obtained using the proposed method with those from the hydraulic importance factor (HIF)

Fig. 13.Ranking of AC versus HIF approaches for: (a) Hanoi; (b) Jinlin; (c) Zhi Jiang; and (d) Mount Pearl.

Table 2.Pearson correlation coefficient of AC-based and HIF-based ranking

# Network

Number of nodes (n)

Number of links (m)

Algebraic connectivity (AC)

Minimum nodes required for virtual network (n_min-vir)

Pearson correlation factor (ρ)

1 Hanoi 32 34 0.06116 11 0.7391

2 Jinlin 28 34 0.07667 6 0.7022

3 Zhi Jiang 114 164 0.01244 15 0.9228

4 Mount Pearl 3,850 3,991 0.00002 400 0.6961

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method (Yoo et al. 2014), hydraulic analysis of the networks was performed using EPAnet version 2.0 software. Yoo et al. (2014) employed the calculated water flows of an intake network and defined an HIF, after (Jun et al. 2008), as [Eq. (5)]:

HIFi¼ðQ_iþQ_i;segÞ

Q_sum ð5Þ

whereQ_iandQ_i_;_seg= flows in theith pipe and in the pipes within the affected segment, respectively, if theith pipe breaks. Q_sum = summation of flow in all pipes within the WDN. HIF_ireflects the hydraulic importance of theith pipe, where a higher HIF_i value indicates higher importance of the pipe. Although the breaking of a pipe may change the hydraulics of the network, the water flow obtained from the intake network is employed in the ranking using this approach. Thus, the ranking obtained from the HIF method is not considered as the benchmark but used here to see how the results from this method compare with those from the proposed method.

The contours of the HIF for the Hanoi, Jinlin, Zhi Jiang, and Mount Pearl WDNs are plotted in Fig.12. It reveals that the pipes connecting to the tank (αpipes) have the highest HIFs, which are consistent with those obtained using the method proposed in this paper (ΔAC approach). The next important pipes are located around the pipes connected to the tank. The pipe rankings obtained by using the proposed method and the HIF method are compared in Fig.13. It shows that the pipe rankings by theΔAC approach and the HIF approach are scattered around the1∶1line, implying that correlation exists between the ranking obtained using the two methods. However, a perfect correlation between the results is not expected due to the inherent difference between the methods. A large scatter is observed for the Mount Pearl network, which is more complex. The Pearson correlation coefficients between the two ranking methods are calculated, as shown in Table2, which are greater than 0.7. For the Zhi Jian network, it is as high as 0.92.

The ranking of the pipes leading to disconnection (blue points) matches well with the ranking of the pipes based on HIF.

Networks with Multiple Tanks

The method discussed above has been applied to two water distribution networks having two water tanks each. One of the networks is relatively small (called herein Rural WDN), having 381 nodes and 475 links/pipes[Fig.14(a)]. The AC of this network is calculated as6.334×10⁻³. The removal of most of the pipes was found to reduce AC, indicating loss of redundancy (redundancy loss group), as shown in the ranking map [Fig.14(b)] developed based on the magnitude ofΔAC. Only a few pipes can lead to the disconnection of some parts of the network. Fig.14(b)shows that the high ranked pipes are scattered all over the network. A virtual network can be added to the location of each water tank to have higher ranks for the pipes connecting to the tanks. Fig.14(c)shows the ranking obtained using a virtual network of 100 nodes (path graph) at the location of each tank. The virtual network has significantly changed the ranks of the pipe in the WDN, with high-rank links around the water tanks followed by the links ranked medium and less important levels, respectively.

The other network, known as the Exeter WDN, has 1,893 nodes and 2,416 pipes with two tanks (Tank 1 and Tank 2) located close to each other [Fig.15(a)]. The ranks of the pipe based onΔAC are shown in Fig.15(b). It shows the high-rank pipes scattered in several areas. The pipes connected to Tank 1 and Tank 2 are ranked 404th and 1644th, respectively [Fig.15(b)]. The ranks of the pipes are changed to 1st and 4th, respectively, for adding a virtual network of 500 nodes at the location of each tank [Fig.15(c)]. Ranks for all other pipes are also changed accordingly. The effect of virtual networks of different sizes is also investigated considering a network of 500 nodes at the location of Tank 1 and of 50 nodes at the location of Tank 2 [Fig.15(d)]. The pipes connected to Tank 2 are not the most critical in Fig.15(d), indicating that the size of the virtual network plays a significant role. Whereas the number of

Fig. 14.Ranking of a small network with two water tanks for: (a) the rural WDN; (b) pipe ranking in the original network; and (c) pipe ranking in the modified network.

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nodes of the virtual network is chosen here arbitrarily to examine the effects, selecting the size of virtual networks for complex WDNs requires further development.

Conclusion

The properties of the AC are employed for ranking pipes in WDNs based on the impact of water- main breaks on the network. A water- main break (removal of a pipe from the network) causing the big- gest change in AC (positive or negative) of the network is found to have the highest impact on the network. A negative change in AC indicates a reduction of redundancy and robustness. A positive change in AC indicates that a part of the network will be separated due to the water-main break. A larger positive change in AC implies that a larger part of the network will be isolated. Thus, changes in the magnitudes of AC of WDN due to the removal of links can be used for prioritizing the pipes in the network. However, the

conventional AC is completely a topological measure and, therefore, unable to account for the effect of the location of the water source. The pipes in the vicinity of water sources are expected to be more important because these pipes supply water to the downstream pipes. In this paper, a virtual network method is proposed to account for the location of water sources in ranking the water mains using AC.

A virtual network is proposed at the tank location, which increases the priority level of the pipes connecting to the water source. For virtual network, path type, ring type, and lattice type networks are examined. Among them, the path type network is found to have the most significant effect on the network with minimum numbers of nodes. The number of nodes in the virtual network should be large enough to have an effect on changing the ranks of the pipe connecting to the water source. In general, a larger WDN would require a larger size of a virtual network. A framework is proposed for the determination of the size of the virtual network and the ranking of the pipes in the network.

Fig. 15.Ranking of a large network with two tanks for: (a) the Exeter WDN; (b) pipe ranking in the original network; (c) pipe ranking with the same-sized virtual network; and (d) pipe ranking with different sizes of virtual networks.

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The virtual network method can also be applied for WDNs with multiple water sources, as demonstrated through application to the networks with two tanks. However, further research is required for the selection of the size of virtual networks with multiple tanks.

Data Availability Statement

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Acknowledgments

Comments from the anonymous reviewers significantly improved the quality of the article and are gratefully acknowledged.

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