Criticality Assessment of Pipes in Water Distribution Networks Based on the Minimum Pressure Criterion

Abstract

A new criticality indicator for Water Distribution Networks (WDNs) is presented. The new indicator is based on the minimum pressure (MP) model, which relies on the assumption that air can enter the pipes, e.g., when failure occurs in water scarcity scenarios, and maintain a minimum pressure equal to zero in the whole network. The proposed indicator properly integrates topological features, provided by structural hole theory, with the hydraulic constraints provided by the WDN steady-state solution, with a particular focus on pipes where occurring free surface flow leads to a serious reduction in the quality of the network service. The new indicator leads to a new criterion for the prioritized maintenance of pipes in existing networks, as well as for the design and planning of new ones, which is different from the one derived from other popular indicators. Three real-life WDNs are selected as test cases.

1. Introduction

Water Distribution Networks (WDNs) are the main assets required to guarantee water supply to customers both in urban and rural areas, according to given pressure and quality standards as defined by existing regulations. It is therefore essential to define proper computational tools aimed at evaluating the performance and reliability of the WDNs in regular operating conditions, as well as in more critical scenarios where water deficiency can affect customers for a certain period of time.

WDN reliability can be described as a parameter embedding the complex interactions existing among all the elements included in the networks, such as nodes, pipes, reservoirs, pumps, turbines, and isolation valves. Highly interconnected systems like WDNs are subject to significant performance reduction due to the malfunction of even a few elements, whereas the installation of new ones in key parts of the network can provide a significant improvement in their global reliability.

In order to account for all these considerations, several water performance indicators (WPIs) [1] have been developed in the past, encompassing both topological and hydraulic characteristics of the WDN, including physical parameters of the pipes in the network and cost/benefit information. Performance indicators can be integrated for the vulnerability assessments of WDNs: some studies [2] focus mainly on the identification of critical segments in the network, defined as those parts of the WDN that can be directly isolated by valve closure to repair a broken pipe [3] and corresponding to areas affected by water shortage; other research is based on the study of connectivity in WDN, after modeling networks through weighted graph and selecting appropriate measurements to determine the robustness of the network in the event of failures [4]. A significant number of studies employ graph theory to identify the most impactful and vulnerable parts of the network, with the aim of providing a preliminary screening of the WDN without the need for hydraulic calculations [5].

Some studies involve the evaluation of a network’s resilience to quantify the satisfaction rate of water supply demand [6] in the case of pipe failure, where valve positioning could protect the most critical segments and reduce the duration of failure. Integrated approaches were also developed to assess nodal vulnerability in a WDN combining topological, hydraulic, and water quality parameters and finally scheduling maintenance routines [7]. Authors such as Marlim et al. [8] developed different indicators comprehending supply shortage due to the failure of some elements in the WDN and pipe degradation, ultimately combining them into a single criticality indicator. Complex network theory was also considered by studies that classified WDNs as spatial networks in order to identify the most appropriate centrality metrics, as betweenness, closeness, and nodal degree, to assist in planning and management strategies [9].

A hydraulic method [10] uses reliability metrics such as Supply Shortage, Pressure Decline, Energy Loss per Unit Length, and Hydraulic Uniformity Index, obtained as output of WDN models. Although it overcomes the limitations of separate conventional metrics, the resulting metrics are a function of the hydraulic features of the network, without proper integration with topology-based metrics provided by graph theory.

An alternative approach [11], based on graph theory, identifies critical links without hydraulic simulations, using dynamic Demand Edge Betweenness Centrality and resistance of the pipe to quantify the Failure Magnitude. This is extremely computationally efficient, but its accuracy may decrease in large and complex networks, and it lacks topographic data (elevation) that could significantly impact the results.

A risk-based evaluation approach, presented in [12], classifies pipes by combining the probability of pipe failures, estimated using Random Forest analysis, and the consequence of failures, measured with the composite index Link Hydraulic Criticality, which incorporates the number of disconnected nodes and the flow conveyed by each pipe. A limitation of this approach is that the Link Hydraulic Criticality can overestimate the topological impact of branch pipes in the network, and the use of demand-driven (DD) analysis fails in detecting unsatisfied demand.

Literature analysis shows that there is a consistent amount of studies about performance and criticality indicators associating the probability of nodes and pipes failure only with the main topological features and metrics of the network, while other studies integrate these aforementioned data with the hydraulic performance of the WDNs, as the demand shortfall in the event of single or multiple pipe failure. Many of these approaches consider topological metrics and hydraulic features of the network as two separate entities. In reality, topological metrics can be strongly affected by variations in the computed flow regime of the network, as more appropriate metrics, such as the modified closeness centrality [13,14], show.

Moreover, several indicators tend to analyze only regular service conditions of the network, whereas real WDNs are often not able to guarantee the desired demands and pressures, specifically in countries affected by water scarcity for large periods of the year: this limits the validity of the rehabilitation planning suggested by these indicators, given that they are reliable only in the design scenarios assumed for their calculations. In different scenarios, with pressure deficiency conditions, these indicators need realistic hydraulic WDN calculations, which cannot be given by demand-driven modeling, still widely applied in criticality evaluation of water networks. A more detailed hydraulic analysis also implies, according to us, the simulation of possible air intrusion inside the pipes, as described in Tucciarelli et al. [15].

There are many reasons that could lead to air infiltration in water pipes: in aqueducts, air could infiltrate into the network through vents, aimed to free the pipes from possible air bubble clusters, when water pressure attains zero relative value. In urban distribution networks, water is usually delivered to tanks linked to apartments: when the elevation of the floating valve in the tank drops due to relative pressure drops, air enters the tank, and the pressure in the distribution pipe depends on the tank water level. A similar effect is provided by the floating valves of the flush boxes in the apartments.

Moreover, in distribution networks, air particles could enter the pipes through the leaks in pipes and junctions.

Finally, water scarcity scenarios and leakages could bring the water manager to reduce pressure in the WDN in order to limit water consumption, with an increased risk of air intrusion. The modeling of non-pressurized elements, realized by assuming shallow water uniform flow in all the parts of the pipes characterized by zero pressure, as well as a more detailed description and validation of the air intrusion hypothesis, can be found in the work by Tucciarelli et al. [15].

The zero minimum pressure (MP) assumption leads to a better estimation of the steady-state piezometric gradients and of the vertically averaged velocities, that in the case of zero relative pressure are, respectively, overestimated and underestimated by traditional hydraulic analysis; additionally, air infiltration in the network can lead in some pipes to free surface flow, characterized by higher velocities and a consequent turbidity increment, with a fast deterioration of the water quality [16,17,18].

On the basis of the previous considerations, a new criticality indicator is presented, able to incorporate the assumption of pipe permeability to air and zero minimum pressure, as proposed by Tucciarelli et al. [15]. The new indicator is developed by adapting the structural hole influence matrix [19] and by properly integrating the variation in flow rate and pressures in the WDN due to free-surface flow occurring in some parts of the network.

In the following, we will show the results of the proposed methodology applied to three real networks and how the new indicator is able to provide a different perspective for the planning of prioritized maintenance for pipes in a WDN, compared to both existing criticality indicators and to the use of a traditional head-driven model.

2. Materials and Methods

2.1. Structural Holes in Networks

The approach described in this paper is based on the structural hole theory, first introduced for social networks by Burt (1992) [20]. Given an undirected and unweighted network represented by a graph G, containing N nodes and M edges, the adjacency matrix A = [a_ij]_N×N, where i and j are the indices of two nodes, is defined as

$$a_{ij}=\left\{\begin{array}{c}1 \\ 0 \end{array}\right.{\displaystyle \genfrac{}{}{0pt}{}{\mathrm{i}\mathrm{f} \mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{s} i \mathrm{a}\mathrm{n}\mathrm{d} j \mathrm{s}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{e} \mathrm{a} \mathrm{c}\mathrm{o}\mathrm{m}\mathrm{m}\mathrm{o}\mathrm{n} \mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}}{\mathrm{i}\mathrm{f} \mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{s} i \mathrm{a}\mathrm{n}\mathrm{d} j \mathrm{d}\mathrm{o} \mathrm{n}\mathrm{o}\mathrm{t} \mathrm{s}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{e} \mathrm{a} \mathrm{c}\mathrm{o}\mathrm{m}\mathrm{m}\mathrm{o}\mathrm{n} \mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}}}$$

(1)

Nodes i and j are said to be connected if a_ij = 1. When two groups of nodes or two single nodes do not share a direct connection or an indirect redundancy relationship, then the link or relationship between them is defined as a structural hole [21]. In this context, nodes sharing a structural hole constitute a way to broker the flow of information and resources between the two sides of the hole [22,23]. With the aim of measuring the structural holes in the network, first define p_ij as

$$p_{ij}={\displaystyle \frac{a_{ij}}{\sum _{j\in \Gamma \left(i\right)}{a}_{ij}}}$$

(2)

where Γ(i) is the set of nodes connected to node i and p_ij represents the fraction of the total effort the node i has to make in order to maintain its connection with node j [13,19,24].

The definition of “effort” was originally given by Burt [17] with reference to social networks, where each node in the graph represents one person and the links with other people are represented by edges: in this context, each node i has to make an effort to maintain a connection with each one of its connected nodes. The total effort node i exerts is equal to the sum of all the p_ij efforts it has to exert to maintain each one of its connections, considering all nodes j ∈ Γ(i), and is always equal to one.

Therefore, when only a single node j is connected to node i, the total effort exerted by node i is directed only towards node j; instead, when more nodes are connected to node i, the total effort exerted by node i is equally divided among each of its connected nodes.

This is why, given the definitions of Equations (1) and (2), the p_ij effort that node i exerts to maintain connection with node j is also equal to the inverse of the number of the “active” connections (the physical pipes) of node i: this directly comes from the fact that every a_ij term, related to all nodes j connected to node i, is equal to one.

See, in Figure 1, an example of p_ij terms calculation for node 1 connected to three nodes (2, 3, 4), according to Equations (1) and (2).

Define the constraint coefficient of node i as

$$C_i={\sum }_{j\in \Gamma \left(i\right)}{\left(p_{ij}+{\sum }_k{p}_{ik}{p}_{ki}\right)}^2 k\ne i, j$$

(3)

where node k is a common neighbor node between nodes i and j. Low values of constraint coefficients are held by nodes sharing structural holes; these nodes connect separated areas of the network, do not have a common link to other nodes, and their influence on the spread of information is significant. Conversely, high constraint coefficient values are related to nodes sharing more than a common neighbor, and, consequently, they are usually part of a loop in the network.

In order to measure the influence the node location has on the other nodes in the network, the closeness centrality metric C_c can be considered as follows:

$$C_c(i)={\displaystyle \frac{N-1}{\sum _{j=1}^{N-1}{d}_{ij}}}$$

(4)

where N represents the total number of nodes in the network and d_ij is the length of the shortest path between nodes i and j. High values of closeness centrality for a node i entail its location in the network to be critical, whereas low values are related to peripheral nodes.

2.2. Structural Hole Influence Matrix

The structural hole concept was afterwards adapted by Zhu et al. [19] to develop a new methodology useful to identify key nodes in complex networks through the structural hole influence matrix. The combination of the adjacency matrix defined in Equation (1) and the closeness centrality of Equation (4) is employed to obtain the node impact factor matrix H_A, defined as follows:

$$H_A=\left[\begin{array}{cccc}1& a_{12}{C}_c\left(2\right)& \cdots & a_{1N}{C}_c\left(N\right)\\ a_{21}{C}_c\left(1\right)& 1& \cdots & a_{2N}{C}_c\left(N\right)\\ ⋮& ⋮& ⋮& ⋮\\ a_{N1}{C}_c\left(N\right)& a_{N2}{C}_c\left(2\right)& \cdots & 1\end{array}\right]$$

(5)

Each element H_A(i,j) = a_ij C_c(j) represents the impact factor of node j on node i and depends on two factors: the global influence factor, related to the location of node j in the whole network and expressed by C_c(j), and the local influence factor expressed by the adjacency term a_ij; each value of the matrix diagonal is defined equal to 1, due to the impact of each node on itself being 100%.

Combining the information provided by matrix (5) and the constraint coefficient of Equation (3), it is possible to define the structural hole influence matrix H_C:

$$H_C=\left[\begin{array}{cccc}{C}_1^{-1}& a_{12}{C}_c(2)C_2^{-1}& \cdots & a_{1N}{C}_c(N)C_N^{-1}\\ a_{21}{C}_c(1)C_1^{-1}& C_2^{-1}& \cdots & a_{2N}{C}_c(N)C_N^{-1}\\ ⋮& ⋮& ⋮& ⋮\\ a_{N1}{C}_c(N)C_1^{-1}& a_{N2}{C}_c(2)C_2^{-1}& \cdots & 1\end{array}\right]$$

(6)

where H_C(i,j) = a_ijC_c(j)C_j⁻¹ represents the fraction of influence that node j exerts on node i, and C_i⁻¹ is the inverse of the constraint coefficient of node i. The higher the closeness centrality of node j, the higher its influence on node i, while low values of constraint coefficient C_j increase H_C(i,j).

Finally, given a node i, it is possible to evaluate its influence M_i by averaging the influence values H_C of its connected nodes with the constraint coefficient of node i itself:

$$\begin{gathered} M_i={\displaystyle \frac{1}{{nd}_i}}\left(H_C\left(i,1\right)+H_C\left(i,2\right)+\cdots +H_C\left(i,{nd}_i\right)\right)={\displaystyle \frac{1}{{nd}_i}}\sum _{j=1}^{{nd}_i}{H}_C\left(i,j\right)= \\ ={\displaystyle \frac{1}{{nd}_i}}\left(C_i^{-1}+{\sum }_{j=1.j\not\equiv i}^{{nd}_i}{a}_{ij}{C}_c(j)C_j^{-1}\right) \end{gathered}$$

(7)

Equation (7) provides an extensive approach to evaluate the influence of a node i by combining its constraint coefficient with both the global and the local influence of each connected node j, with respect to the nodal degree nd_i representing the number of nodes adjacent to node i.

2.3. Integrated Topological–Hydraulic Minimum Pressure Criticality Indicator

2.3.1. Criticality Indicator Derived from Minimum Pressure Approach and Structural Hole Theory

In the new proposed methodology, the influence that each node in a WDN exerts on the adjacent ones is derived by applying the structural hole influence matrix approach, described in Section 2.2, within the context of the minimum pressure (MP) model, as proposed by Tucciarelli et al. [15]. The MP model is based on the assumption of pipe permeability to the air and resulting minimum zero pressure inside the network, along with the possibility of free flow conditions in some of the pipes. The assumption of uniform free surface flow and of water–air phase separation inside these pipes allows for the solving of the MP problem by iteration of standard WDN solutions, where a restricted number of nodes is set with zero pressure. In each iteration, the zero-pressure nodes are treated as Dirichlet boundary conditions, with a flow rate resulting from the mass balance. This flow rate is split in node i into two components: the actual nodal flow Q_a,i and the complementary free surface flows occurring in the pipes linked to node i, with the elevation of the second node j being smaller than the elevation of node i.

The resulting solution is very robust, also because no new physical parameters are adopted for the model simulation. The same Manning coefficient of the Chezy equations is computed from the hypothesis that the minimum resistance in the pipe is the one occurring in fully pressurized conditions.

A short description of the model can be found in the Supplementary Material.

2.3.2. Computation of Weighted Adjacency Matrix by Means of Pipe Hydraulic Weights Calculation

The first step of the methodology is the development of a weighted adjacency matrix. In order to properly develop it, some preliminary considerations have to be made: we assume each node i in a WDN to first be characterized by topological features, including its position in the network and its nodal degree nd_i, as well as by hydraulic features such as the actual nodal flow Q_a,i and the actual mean velocity inside the connected pipes. Observe that the nodal flow Q_a,i depends on the hydraulic conditions occurring in the network, is estimated by proper numerical simulations, and it may differ from the nodal demand Q_d,i. This difference, occurring because of the nodal pressure reduction, is also estimated in the pressure-driven approach, but only the MP one takes into account the limitation due to the reduction in the head gradient and of the corresponding pipe flow rate.

With the aim of defining a dimensionless performance indicator and taking into account the previous observations, we introduce here the hydraulic pipe weight W_p, defined as

$$W_p=\left({\displaystyle \frac{Q_{d,i}}{Q_{net}}}+{\displaystyle \frac{Q_{d,i-}{Q}_{a,i}}{Q_{d,i}}}+{\displaystyle \frac{Q_{d,j}}{Q_{net}}}+{\displaystyle \frac{Q_{d,j-}{Q}_{a,j}}{Q_{d,j}}}+{\displaystyle \frac{q_p}{Q_{net}}}\right)\left({\displaystyle \frac{v_{p,ff}}{v_{p,sf}}}+1\right)$$

(8)

where v_p,sf is the mean velocity in the pressurized part of the pipe and v_p,ff is the uniform flow velocity along the zero pressure part of it, when free flow condition occurs, or the same v_p,sf otherwise. Given a pipe p connecting nodes i and j, Q_d,i and Q_d,j represent, respectively, the nodal demand in node i and j, while Q_a,i and Q_a,j represent the actual nodal flow occurring in the two nodes; Q_net is the total flow extracted from the network, obtained as sum of all the actual nodal flows Q_a,i, and q_p represents the flow rate in pipe p.

This formulation is aimed at giving higher weight to pipes where the nodal demands Q_d,i and Q_d,j are significant with respect to the total nodal demand Q_net, and also in the case when at least one of the two nodal demands in node i and j is not completely satisfied, due to the increasing difference Q_d,i − Q_a,i. The last factor (v_p,ff/v_p,sf + 1) is introduced to provide higher relevance to pipes where free surface flow occurs, for the reasons explained in the introduction. Observe that, when channel flow occurs in a pipe, the flow rate q_p computed by the minimum pressure approach is lower than the flow rate in the same pipe computed by the traditional head-driven model, but the factor v_p,ff/v_p,sf provides a much larger compensation.

Before computing the weighted adjacency matrix, pipe weights W_p must be normalized, with respect to the maximum W_p,max, in the whole network as

$$W_{p,n}={\displaystyle \frac{W_p}{W_{p,max}}}$$

(9)

At this point, a weighted adjacency matrix can be computed through pipe weights as follows:

$$a_{w,ij}=\left\{\begin{array}{c}{W}_{p,n} \\ 0 \end{array}\right.{\displaystyle \genfrac{}{}{0pt}{}{\mathrm{i}\mathrm{f} \mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e} i \mathrm{a}\mathrm{n}\mathrm{d} j \mathrm{a}\mathrm{r}\mathrm{e} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d}}{\mathrm{i}\mathrm{f} \mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e} i \mathrm{a}\mathrm{n}\mathrm{d} j \mathrm{a}\mathrm{r}\mathrm{e} \mathrm{n}\mathrm{o}\mathrm{t} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d}}}$$

(10)

Some authors [13,25] propose different hydraulic pipe weight W_p formulations for the construction of the weighted adjacency matrix. We chose to normalize W_p values before defining the weighted adjacency matrix because in this way, W_p,n values always belong to the range [0, 1]. Indeed, if we instead applied in Equation (7) the W_p values without normalization, in place of the a_ij terms, this would lead to a variation in the lone term $\sum _{j=1.j\not\equiv i}^{{nd}_i}{a}_{ij}{C}_c(j)C_j^{-1}$, representing the influence of all adjacent nodes j, without a corresponding impact on the influence of node i itself (given by C_i⁻¹), which is independent from the hydraulic pipe weight W_p. This possible variation, due only to a variation in the a_ij range, could lead, in Equation (7), to two completely different orders of magnitude, respectively, for the influence of node i (C_i⁻¹) and for the influence of connected nodes j ($\sum _{j=1.j\not\equiv \mathrm{i}}^{{nd}_i}{a}_{ij}{C}_c(j)C_j^{-1}$), depending on the actual W_p values. This implies that one of these two terms could be negligible with respect to the other one. On the contrary, normalization according to Equation (9) guarantees values within the range [0, 1], regardless of the specific W_p formulation.

We chose to compute the constraint coefficient C_i, for each node i, on the only basis of the surrounding topology, as provided by the a_ij values of the unweighted network (Equation (1)), and not by the a_w,ij ones (Equation (10)), because all hydraulic characteristics are already taken into account by the W_p,n formulation.

See, in Figure 2, the W_p,n computed in a simple network with only one reservoir (node 16) and 16 nodes. In each node, Q_d,i = 2.5 L/s is set, and all the pipes share the same geometry and material; a needle valve is located in pipe 7, so that the corresponding flow rate leaving the reservoir is equal to 31 L/s, and a water leakage affects node 6 (more details in Supplementary Material).

It can be observed in Figure 2 that the maximum W_p,n value, with the exception of pipe 1, which is located next to the reservoir, is attained by pipes 8 and 9 that are in the 2nd and 3rd position in W_p,n ranking. This is due to the unsatisfied nodal demands in their nodes and to the free surface flow occurring in almost the entire length of these pipes; pipe 7 has W_p,n = 0.58, mainly due to its proximity to pipes 8 and 9. Pipe 2, which is located downstream of pipe 1, has W_p,n = 0.52 and is in the 5th position, while pipe 3 is in the 6th position, due to its link with three pipes to which it delivers water. This simple test shows clearly how the formulation of Equation (8) is able to represent the physics of the problem, giving greater weight to pipes with high flow rate but also to pipes connecting nodes whose demands are not satisfied.

2.3.3. Minimum Pressure Criticality Indicator

After computation of the pipe weights, it is possible to compute the criticality of a node i according to Equation (7); however, although a_w,ij terms can now be computed through Equation (10), the original formulation of Equation (7) only depends on topological characteristics and needs to be adjusted on the basis of the actual hydraulic dynamics. The first point is the number of nodes nd_i directly connected to node i in Equation (7): this parameter is inversely proportional to the criticality M_i, because the more the connections shared by node i with other nodes, the more paths are available to reach node i and the lower is its criticality M_i; nonetheless, in a directed network, where the edges are oriented according to the flow direction, not every node can be reached by a single node. This implies that each node i has a set of reachable nodes n_ℜ(i) and a set of nodes nd_WF(i) that can reach node i, depending on the flow direction in all pipes of the WDN. For this reason, it is more appropriate to consider, among all the nodes j adjacent to node i, only the nd_WF(i) nodes where fluxes are directed towards node i.

A similar comment can be made about the closeness centrality C_c of Equation (4), previously defined upon the assumption that each node can be reached from every connected node i: this is no longer true in a directed network, where a better modified version of closeness centrality [13,14] is given by

$$C_{WF\left(i\right)}={\displaystyle \frac{n_{\mathfrak{R}\left(i\right)}-1}{N-1}}{\displaystyle \frac{n_{\mathfrak{R}\left(i\right)}-1}{{\sum }_{j=1}^{n_{\mathfrak{R}\left(i\right)}-1}\left({d_{Ri}+d}_{ij}\right)}}$$

(11)

where C_WF(i) depends both on the shortest path d_ij (between node i and all its reachable nodes j) and the shortest path d_Ri (between the closest reservoir R and node i), assuming only reservoirs to be source nodes. This implies that key nodes near the reservoir will have a larger set of reachable nodes n_ℜ(i) and higher C_WF(i) values.

In Figure 3, the C_WF calculation for node 6 is shown:

On the basis of all the previous comments and of the original Equation (7), we propose here the Minimum Pressure Criticality Indicator (MPCI):

$${MPCI}_i={\displaystyle \frac{1}{{(nd}_{WF\left(i\right)}+1)}}\left(C_{\left(i\right)}^{-1}{C}_{WF\left(i\right)}+{\sum }_{j=1,j\not\equiv i}^{{nd}_i}{W}_{p,n(i,j)}{C}_{WF\left(j\right)}{C}_{\left(j\right)}^{-1}\right)$$

(12)

This criticality indicator, integrated by the MP approach, incorporates both topological and hydraulic factors and computes a node criticality on the basis of the topological characteristics of itself and of all the adjacent nodes and connected pipes, as well as of the hydraulic dynamics of the whole network. Observe that the modified closeness centrality C_WF affects in Equation (12) both node i and all nodes j in order to account for their respective sets of reachable nodes n_ℜ. We introduced the term (nd_WF(i) + 1) to avoid a zero value for the reservoirs with nd_WF(i) = 0, when fluxes are all directed outwards from them.

The normalized version can be expressed as follows:

$${MPCI}_{i,n}={\displaystyle \frac{{MPCI}_i}{{MPCI}_{i,max}}}$$

(13)

Finally, given a pipe p connecting nodes i and j, its criticality indicator MPCI_p can be computed as the mean MPCI value between the two nodes:

$${MPCI}_p={\displaystyle \frac{{MPCI}_{i,n}+{MPCI}_{j,n}}{2}}$$

(14)

We present in Figure 4 a synthetic flowchart of the MPCI_p calculation and in the Supplementary Material a detailed step-by-step calculation of MPCI_p for the network of Figure 2.

We finally want to point out that the assumption of pipe permeability to air is central to the MP numerical model but not to the proposed indicator: MPCI equations were formulated in order to integrate the results provided by MP approach, especially in water scarcity scenarios where free surface flow could occur, but its formulations can be nonetheless used with traditional demand-driven (EPANET) and head-driven numerical models. We compare in the following section both results obtained using MP and the traditional head-driven approach, demonstrating how MPCI is able to provide accurate and physically based results, compared to other criticality indicators, even when only the head-driven approach is used. We will also show that, when MPCI is used to compute results in water scarcity scenarios, differences between the results of MP and of more traditional numerical models become more remarkable and can lead to very different points of view for a prioritized maintenance of pipes.

3. Results and Discussion

3.1. North Marin WDN

In order to test the validity of the proposed indicator, the W_p,n and MPCI_i,n values of all the nodes of three real-life WDNs have been computed. The first case is the EPANET Net3 system [26], based on the North Marin WDN, which is one of the most used networks for validation available in the literature; all related data can be found in the online database (https://uknowledge.uky.edu/wdsrd/ (accessed on 11 November 2024)). We analyzed the Net3 configuration at 12:00 A.M., as shown in Figure 5. The network has 95 nodes, four reservoirs (nodes 92, 93, 94, and 95), 115 pipes, and one pump.

A water scarcity scenario was simulated with the HD approach, where three reservoirs (93, 94, and 95) were removed from the WDN and a needle valve with a 94.3% opening degree was installed downstream of reservoir 92. We compared the results (shown in Figure 6) obtained using the proposed W_p,n and MPCI_i,n formulations with the results obtained with the W_p formulation and Hydraulic Connectiveness Criticality (HCC) indicator, based on structural hole theory, as proposed by Marlim et al. [13] for the WDN criticality assessment (more details in Supplementary Material).

In

Table 1. Computed parameters for North Marin WDN.

Node ID	C(i)	CWF(i)	MPCI(i)	Node ID	C(i)	CWF(i)	MPCI(i)
1	0.500	0.001	0.004	47	0.333	0.011	0.102
2	0.333	0.002	0.012	48	0.333	0.013	0.126
3	0.333	0.004	0.033	49	0.500	0.013	0.066
4	1.000	0.000	0.000	50	1.000	0.000	0.037
5	0.333	0.002	0.012	51	0.250	0.019	0.121
6	0.333	0.003	0.022	52	0.333	0.015	0.077
7	1.000	0.000	0.000	53	0.500	0.016	0.080
8	0.333	0.001	0.006	54	0.500	0.019	0.093
9	0.333	0.000	0.002	55	0.333	0.013	0.065
10	1.000	0.000	0.000	56	0.500	0.012	0.065
11	0.333	0.007	0.053	57	0.250	0.016	0.107
12	0.500	0.008	0.039	58	0.333	0.014	0.069
13	0.500	0.001	0.007	59	0.500	0.020	0.096
14	0.333	0.003	0.020	60	0.333	0.022	0.164
15	0.333	0.002	0.014	61	0.334	0.021	0.158
16	0.333	0.017	0.132	62	0.500	0.000	0.003
17	0.333	0.010	0.049	63	0.333	0.070	0.830
18	0.333	0.011	0.052	64	1.000	0.000	0.000
19	0.333	0.023	0.115	65	0.500	0.001	0.005
20	0.333	0.024	0.118	66	0.333	0.002	0.013
21	0.333	0.017	0.084	67	1.000	0.000	0.000
22	0.333	0.025	0.325	68	1.000	0.000	0.001
23	0.250	0.000	0.003	69	0.500	0.072	0.749
24	1.000	0.000	0.000	70	1.000	0.000	0.000
25	0.500	0.024	0.116	71	1.000	0.000	0.000
26	0.333	0.026	0.191	72	1.000	0.000	0.023
27	0.500	0.000	0.000	73	1.000	0.000	0.000
28	0.500	0.013	0.114	74	0.500	0.001	0.005
29	0.333	0.014	0.068	75	0.333	0.002	0.010
30	0.500	0.001	0.004	76	0.500	0.004	0.021
31	1.000	0.000	0.000	77	0.500	0.003	0.015
32	1.000	0.000	0.000	78	0.333	0.006	0.067
33	0.333	0.016	0.203	79	0.333	0.007	0.125
34	0.250	0.023	0.323	80	0.500	0.003	0.017
35	0.333	0.031	0.349	81	1.000	0.000	0.000
36	0.500	0.032	0.293	82	0.320	0.067	0.861
37	0.500	0.001	0.005	83	0.320	0.033	0.198
38	0.333	0.028	0.338	84	0.333	0.016	0.174
39	0.333	0.007	0.056	85	0.500	0.069	0.823
40	0.500	0.005	0.028	86	0.334	0.023	0.120
41	0.500	0.033	0.375	87	0.458	0.020	0.071
42	0.500	0.008	0.121	88	0.320	0.057	0.725
43	0.500	0.009	0.045	89	0.333	0.030	0.235
44	0.333	0.012	0.088	90	0.333	0.001	0.004
45	0.500	0.008	0.042	91	1.000	0.000	0.000
46	0.333	0.009	0.048	92	1.000	0.068	1.000

, the computed values of C_(i), C_WF(i), and MPCI_(i) are shown. Related color maps are also available in the Supplementary Material.

In Figure 6, pipes are classified in six criticality groups, according to each different indicator (e.g., considering the WDN with 112 pipes, “1–10%” refers to the first 11 (about 10%) pipes with the highest values of the computed indicator). In Figure 6a,c, results are shown, respectively, for W_p,n and MPCI_p, while in Figure 6b,d, results obtained with, respectively, W_p and HCC formulations [13], are shown:

We can observe in Figure 6a,b that pipes with the highest W_p,n values (between top 1% and top 10%) are the ones carrying most of the flow in the network and are located downstream of the reservoir and in the central branch of the WDN, while pipes located in peripherals areas generally have low positions in W_p,n ranking.

Pipes 112 and 110, even if located near the reservoir, have a W_p value of zero (Figure 6b) due to zero nodal demand in each node of these pipes. Instead, with the new W_p,n formulation (Figure 6a), the same pipes rank in the first criticality group because, even though nodal demands are zero, the flow rate in these pipes is the highest among all pipes. Equation (8) correctly captures this criticality, which is also consistent with the topology of the WDN: in fact, a disconnection of these two pipes from the network would lead to hydraulic isolation of all the pipes downstream of the reservoir.

Another significant difference between Figure 6a,b is with respect to pipes 33 and 34, where nodal demand in one of their nodes, despite being relatively low, is not satisfied at all; again, this is properly taken into account by Equation (8) and thus W_p,n is able to provide detailed information about pipes where nodal demand is at least partially not satisfied.

The MPCI_p and HCC rankings are shown in Figure 6c,d, respectively. Unlike W_p,n, which takes into account only hydraulic parameters, MPCI_p and HCC integrate topological and hydraulic characteristics, and the results provide evidence of that.

In Figure 6c, pipes 33 and 34 show moderate criticality values, albeit located near pipes with high positions in the MPCI_p ranking. This can be explained by the relatively low fraction of the total water flow carried by them to the top right area of the WDN. The major fraction is carried instead by pipes 20 and 35, which are the only two connected to both the central branch of the WDN and the top right region: as expected, MPCI_p rises to the first criticality group in spite of W_p,n ranking (Figure 6a) in the third group (21–30%).

MPCI_p and HCC (Figure 6d) criticality rankings are considerably different, especially for pipes 112 and 110, which have the lowest ranking according to HCC formulation (Figure 6d) as a result of the zero nodal demands of their nodes. This criticality also affects pipes downstream of pipe 112 (pipes 17, 18, 19) and lowers their criticality ranking. All these pipes, instead, according to the MPCI_p ranking (Figure 6c), are in the first criticality group as they carry a large portion of flow, and again, this is consistent with the topology of the WDN. Considerable differences can also be observed in pipes 24, 26, and 29 (top right region in Figure 6c,d) that are located in a peripheral region of the WDN: while their HCC ranking is high (Figure 6c), because of their distance from water source and the low number of connected pipes, MPCI_p ranking is low, because it gives higher criticality to pipes (like 20 and 35) required by pipes 24, 26, and 29 to remain connected to water sources.

With the aim of quantifying the differences shown in Figure 6, Spearman’s rank correlation coefficient was calculated, and the results are shown in

Table 2. Spearman’s correlation coefficient r_s for Net3.

Wp,n-Wp (Figure 6a,b)	Wp,n-MPCIp (Figure 6a–c)	Wp-HCC (Figure 6b–d)	MPCIp-HCC (Figure 6c,d)
0.536	0.477	0.386	−0.446

:

While a good correlation (r_s = 0.536) exists between W_p,n and W_p (Marlim et al. [13]), mainly due to similar computed pipe flows, and also between W_p,n and MPCI_p, instead, MPCI_p and HCC show completely different trends (r_s = −0.44), in compliance with criticality rankings depicted in Figure 6c,d, because of the nodes with zero nodal demands as described above.

In order to give a perspective about how the computed results could lead to a signification variation in water quality, simulations of Net3 were carried out considering a realistic 24 h demand pattern, provided in the Supplementary Material. Results indicate that, within 24 h, the velocities in 106 pipes of the 112 pipes are subject to an increase of up to 203%. Following Braga et al. [27], We assume that a significant daily velocity variation can lead to systematic detachment of material accumulated over the pipe walls, especially when shear stress on the walls increases significantly. This effect could become more remarkable when a WDN operates for a larger period of time (e.g., during a water scarcity scenario) with several pipes with zero flow rate: when the network returns to regular operating conditions, this would lead to the detachment of all the material accumulated on pipe walls during the previous water scarcity period.

From this first test, it is clear that MPCI_p accurately integrates hydraulic and topological features of the WDN; however, even W_p,n alone provides significant insights, especially for free surface flow pipes, which can lead to a significant degradation of the quality of the delivered water.

3.2. Campofelice Di Roccella WDN

The second case study is the new WDN designed for the “Campofelice di Roccella” residential area of Sicily, in Italy, shown in Figure 7: the network is modeled with one reservoir, 128 nodes, and 177 pipes, and in the design scenario, the flow rate leaving the reservoir is Q_supply = 39.2 L/s. In this case, results obtained with MP and HD approaches for the hydraulic computations have been compared (for both the proposed W_p,n and MPCI_p), in order to highlight the relevant effect of the properly adopted hydraulic solution. In the analyzed water scarcity scenario, a reduced water supply was considered, with a corresponding flow rate Q_supply = 30.4 L/s (about 75% of the design water supply) obtained through a needle valve located downstream of the reservoir and a water leakage occurring in each internal node, with a total leakage Q_leakage = 5.2 L/s (see Supplementary Material for network data).

In Figure 7a,b, the results provided, respectively, by the MP and HD models, are shown.

In this scenario, remarkable differences appear between the two models: while the HD model (Figure 7b) provides zero pipes with no flow rate, the MP model (Figure 7a) provides 13 pipes with free surface flow and 36 pipes with no flow rate, which represent about 20% of the total number of pipes in the WDN. These results directly impact W_p,n and MPCI_p ranking for both models, as shown in Figure 8:

The most critical pipes for W_p,n and MPCI_p rankings are, as expected, the ones near the reservoir and in the central area of the WDN, followed in the ranking by their adjacent pipes.

In particular, W_p,n rankings do not differ significantly between MP (Figure 8a) and HD (Figure 8b) models in key regions of the network, due to the high number of pipes where nodal demands of their nodes are not partially or completely satisfied: this shows clearly how the W_p formulation of Equation (8) is able to identify critical pipes from an hydraulic point of view and it is not strongly affected by the adopted hydraulic model, although pipes where free surface flow occurs generally have an higher ranking with MP model (Figure 8a) than with the HD one (Figure 8b). Observe that, due to water leakage in nodes, q_p increases in some pipes and W_p,n increases as well: this is consistent because criticality of these pipes has to take into account the effect of leakage, and, as a consequence, only the actual nodal flow, excluding leakages, is considered in Q_a,i and Q_a,j of Equation (8).

However, the results provided by MPCI_p offer a very different perspective on the criticality ranking: in fact, according to the HD model (Figure 8d), there is an entire group of pipes, located in the central WDN area (identified by the dotted box in Figure 8d), with a rank between the third and fifth group of criticality. This is because the HD model is unable to estimate the zero flow rate in this area, computed instead by the MP model, as shown before in Figure 7a.

According to the HD model, these pipes have a major relevance because they carry the major fraction of the flow to the upper part of the WDN. The pipes in the external areas of the network are instead ranked in the least critical group (49, 48, 86, 101, 100, and 119 on the left in Figure 8d).

The MP model, in contrast (Figure 8c), is able to detect free surface flow pipes and, in particular, pipes with zero flow rate, and this leads to substantially different results: a significant variation in criticality ranking affects almost all the pipes in the central region of the WDN (dotted box in Figure 8c), that rank in the least-criticality group even though, from just an hydraulic point of view, they are quite critical (as attested by high W_p,n ranking in Figure 8a). These pipes are, in fact, no more essential in the considered scenario, because a large part of the entering flow is redirected to a different part of the network, specifically to pipes on the left side (49, 48, 86, 101, 100, and 119). All these pipes now show greater relevance, and they are essential for the water supply of the upper part of the network that, according to the MP model, cannot be supplied by the central WDN region anymore.

This relevant difference in criticality ranking is caused by several adjacent pipes in the central region with zero flow rate: the nodes connecting these pipes will have a zero C_WF value (Equation (11)), because their number n_ℜ of reachable nodes is equal to zero and, as a consequence, their MPCI_i decreases dramatically or even drops to zero. This also proves that C_WF(i) is necessary in Equation (12) because otherwise, if the influence of node i were only given by C_(i)⁻¹ (and not by C_(i)⁻¹C_WF(i)), then node i would always have a much greater influence than all the other connected nodes (given by $\sum _{j=1,j\not\equiv i}^{{nd}_i}{W}_{p,n(i,j)}{C}_{WF\left(j\right)}{C}_{\left(j\right)}^{-1}$) due to the higher order or magnitude of C_(i)⁻¹.

A final consideration concerns pipe 27, which, despite being in the first criticality group in the W_p,n ranking (Figure 8a), is in the last group in the MPCI_p ranking (Figure 8c): this major variation is the effect of zero or low flow rate in all the pipes connected to pipe 27. This is correctly taken into account by the MPCI_p formulation (Equation (12)); again, as mentioned before, a low MPCI_p ranking suggests that pipe 27’s role is not fundamental for the WDN in the analyzed scenario.

Spearman’s coefficient results are provided in

Table 3. Spearman’s correlation coefficient r_s for Campofelice di Roccella WDN.

Wp,n (MP)-Wp,n (HD) (Figure 8a,b)	Wp,n (MP)-MPCIp (MP) (Figure 8a–c)	Wp,n (HD)-MPCIp (HD) (Figure 8b–d)	MPCIp (MP)-MPCIp (HD) (Figure 8c,d)
0.703	0.478	0.903	0.651

:

Results confirm what is shown in Figure 8; in particular, a strong correlation (r_s = 0.903) can be observed between the hydraulic weight W_p,n and MPCI_p. The reason is that the HD approach is unable to determine free surface flow pipes, unlike the MP approach, and this leads to a smaller correlation (r_s = 0.478).

3.3. Marchi Rural WDN

A third test case is the Marchi Rural water distribution network in Australia, based on a real irrigation system, which can be classified as a transmission dense-loop network, unlike the first two analyzed WDNs. The network has two reservoirs, 380 nodes, and 477 pipes; data can be found at the online database (https://uknowledge.uky.edu/wdsrd/ (accessed on 7 February 2025)).

In this test, only a regular scenario, where all nodal demands are satisfied, was analyzed, and the HD approach was used for the hydraulic model.

In Figure 9a, a comparison between the hydraulic weight computed according to Marlim et al. [13] and according to the new formulation of Equation (9) can be observed:

Major differences are evident between Figure 9a,b, and this is mainly due to the large number of nodes with zero nodal demand. In fact, when both nodes of one pipe have zero demand, the computed pipe hydraulic weight (Marlim et al. [13]) will always be zero, regardless of the pipe flow, which instead could be very high, especially near the two reservoirs (R1 and R2) and in critical pipes connecting separated areas of the WDN. Because of this, in Figure 9a, the real hydraulic dynamics of the network cannot be observed, and the most critical pipes (depicted in red in Figure 9a) are always the ones located in the most peripheral areas of the WDN, irrespective of every other parameter.

Formulation of Equation (9) instead correctly captures the hydraulic dynamics and, as expected, the most critical pipes are those located near the reservoirs, while medium criticality (between the top 21–30% and top 31–40% W_p,n) is assigned to pipes acting as a bridge between different regions of the network.

This test clearly shows that the new formulation performs accurately even in scenarios where only a few nodes have higher-than-zero demand; finally, even though only the HD approach was applied, the results are consistent with water dynamics, and, again, this proves that the new hydraulic weight does not necessarily need to always rely on the MP approach.

Finally, the computed Spearman’s coefficient r_s = 0.323, in compliance with the large divergence between Figure 9a,b.

3.4. Sensitivity Analysis of Normalization Method

With the aim of determining whether the choice to normalize both hydraulic weight and criticality indicator values using the maximum value of the corresponding parameter, as Equations (9)–(13) provide, could significantly impact the final MPCI_p output values (Equation (14)), a sensitivity analysis was performed, computing W_p,n (and similarly MPCI_i) at first using the L1 norm, and successively using the L2 norm, as described by Caldarola & Maiolo [28]; the equations, respectively, are presented here:

$$W_{p,n}={\displaystyle \frac{W_p}{\sum _{p=1}^M{W}_p}}$$

(15)

$$W_{p,n}={\displaystyle \frac{W_p}{\sqrt{\sum _{p=1}^M{W}_p^2}}}$$

(16)

where M is the total number of pipes. In order to show the differences between the resulting criticality rankings, Spearman’s correlation coefficient r_s was calculated for the three test cases. While, as expected, the W_p,n criticality rankings computed with L1 norm (Equation (15)) and L2 norm (Equation (16)) are identical to the ones computed with the chosen maximum norm (Equation (9)), given that in all three normalization methods all hydraulic weights divide the same scalar value (which implies r_s = 1), MPCI_p formulation is a more complex combination of topological and hydraulic factors, and the corresponding results are shown in the following

Table 4. Spearman’s correlation coefficient r_s for the three test cases considering different normalization methods for MPCI_p criticality rankings.

North Marin WDN		Campofelice di Roccella WDN		Marchi Rural WDN
Maximum Norm-L1 Norm	Maximum Norm-L2 Norm	Maximum Norm-L1 Norm	Maximum Norm-L2 Norm	Maximum Norm-L1 Norm	Maximum Norm-L2 Norm
0.984	0.994	0.980	0.994	0.999	0.999

:

As can be observed, an extremely strong correlation (r_s > 0.98) between the different criticality rankings emerges comparing the results obtained with the maximum norm, both to the ones provided by the L1 norm and by the L2 norm. Results indicate that the overall criticality ranking remains almost unvaried when choosing a different normalization method, implying that the equilibrium between topological and hydraulic terms in Equation (12) is not altered sufficiently by the selected normalization to vary the final MPCI_p.

The chosen maximum norm of Equations (9) and (13) was made in order to guarantee, as already explained in Section 2.3.2, the same order or magnitude for all terms in Equation (12) and also to guarantee that there is always a single pipe with the maximum hydraulic weight value of 1, implying that the building of the weighted adjacency matrix of Equation (10) is coherent with the original definition of the adjacency matrix of Equation (1).

The high correlation obtained proves that the criticality ranking provided by MPCI is quite robust and independent of the specific normalization method chosen.

3.5. Global Enhancement of Network Performance via Localized Interventions Guided by MCPI

With the purpose of providing an idea of how acting on the most important pipes, suggested by MPCI, could lead to an overall enhancement of the network, we first calculated the resilience index I_r, proposed by Todini [24], in the Campofelice di Roccella WDN analyzed in design conditions, where all nodal demands are satisfied, and successively in improved conditions where the diameters of a number of selected pipes were increased.

In Figure 10, the MPCI_p criticality ranking, computed in design conditions, is shown:

Initially, the computed resilience index I_r = 0.206, showing that the network is able to satisfy every nodal demand and guarantee the requested pressures, but also suggesting that the surplus energy available in the WDN is not particularly significant.

According to the MPCI_p ranking, the pipes characterized by the highest MPCI_p values, belonging to the top 10%, are the 17 pipes depicted in red in Figure 10, with a total length of roughly 1024 m. A second simulation was run, in which each of the 17 pipes in the first criticality group was replaced by a corresponding pipe with a 50 mm increased diameter. As a consequence of these improvements, the nodal heads available in each node increased, and the corresponding resilience index rose to I_r = 0.244, showing that the surplus energy owned by the WDN was enhanced significantly.

On the other hand, a third simulation was run considering only pipes belonging to the last criticality group (between 51 and 100% top criticality) in Figure 10, and choosing an adequate number of pipes to replace, so that their total length is close enough to the total length (1024 m) of the 17 pipes previously chosen, given that the cost of maintenance usually depends on the lengths of the pipes to be replaced. In this case, a total of 16 pipes were considered, and each of their diameters was increased by 50 mm: the results lead to a final value of I_r = 0.209, clearly indicating that, even though the same total pipe length was improved, the WDN was not substantially strengthened by this maintenance.

It is therefore clear how MPCI_p is able to provide useful indications that water managers could follow in order to perform prioritized maintenance by selecting the most critical pipes to reinforce. Results of the test cases described above are summarized in

Table 5. Todini’s resilience index I_r for Campofelice di Roccella WDN. Percentage variations compared to design conditions are indicated in parentheses.

Ir (Design Conditions)	Ir (50 mm Increased Diameters in ~1024 m of Pipes Belonging to First Criticality Group of Figure 10)	Ir (50 mm Increased Diameters in ~1024 m of Pipes Belonging to Last Criticality Group of Figure 10)
0.206	0.244 (+18%)	0.209 (+1.75%)

:

3.6. Discussion and Limitations of the Proposed Approach

Even though MPCI represents a holistic approach that encompasses topological and hydraulic factors, limitations could arise in networks where hydraulic factors may overestimate the criticality of some pipes, especially when several nodal demands are not guaranteed in different and adjacent regions of the network. Although the topological terms involved in MPCI calculation should mitigate the impact of excessively critical pipes, other tests would be useful to ascertain the influence of hydraulic terms in these conditions; additionally, it would be useful to analyze MPCI results calculated when a different pipe hydraulic weight formulation (which, at the same time, is able to include the information about free surface flow, as Equation (8) provides) is used, in order to more accurately establish the effect of the computed hydraulic criticality.

Furthermore, in the proposed methodology, the constraint coefficient of Equation (3) was computed relying only on topological terms provided by the unweighted adjacency matrix: while this choice was made to avoid an improper combination of topological and hydraulic terms, it would be worthwhile to find out if a different choice for constraint coefficient calculation methodology could influence the final MPCI values.

It also has to be considered that the modified closeness centrality of Equation (11) usually represents the most computationally expensive term to calculate, although overall MPCI calculation time, for mid-sized network as those shown in this paper, still remains under 10 s; however, if MPCI were implemented in optimization algorithms for WDNs, in the case of really complex networks with more than 1000 nodes, the total computation time could arise significantly. Therefore, a different and simpler version of closeness centrality could be used to determine its influence and possibly reduce the required computational burden.

4. Conclusions

A new indicator for WDN criticality assessment is developed, integrating the minimum pressure (MP) approach, which relies on the assumption that the minimum pressure inside the network always remains greater than or equal to zero. The new MPCI also relies on the new defined hydraulic pipe weight W_p, which embeds both nodal and link parameters, according to the structural hole theory. In the three analyzed test cases, numerical results show that MPCI can usually provide a reliable estimate of the most critical pipes even using a standard head-driven (HD) approach for hydraulic modeling. In the case of the water scarcity scenario, the use of the MP modeling approach leads to very different outcomes with respect to the classical head-driven approach, and this results in a different prioritized maintenance schedule for pipes.

Finally, observe that well-designed WDNs never attain, with the design nodal flows, negative pressures. In this case, the MP approach provides the same results as the head-driven model. On the other hand, the assumption of failure of one or more network components could lead to significantly different outcomes for the proposed criticality indicator between the MP model and other WDN models.

The computed results call for further improvements of the proposed methodology and for further tests of the MPCI, especially in more complex networks with multiple sources and valves, by analyzing the impact that valves’ isolation has on the WDN hydraulic dynamics and the effect that the resulting variations in the flow distribution have on the proposed indicator. MPCI could also be incorporated in water quality studies to determine the relationship between water quality and the velocity variations during long time periods, particularly in the event of a network segment’s isolation, which could lead to unexpected flow direction change in pipes. Moreover, the presented criticality indicator relies on the nodal and pipe flows computed by the chosen hydraulic model, and, therefore, its validity could be strengthened by comparisons with real experimental data on WDN, which could lead to a different criticality map and, consequently, to alternative maintenance scheduling. Finally, the proposed indicator could be implemented as an objective function in the planning of new WDNs to optimize their layout and to increase their reliability.