## Abstract

The expansion of water distribution networks is nowadays essential to meet the pressing population growth in many cities worldwide. With the subsequent increase in water demand, the hydraulic behaviour of a water supply system can change dramatically due to the significant head losses in the pipes caused by the larger flows released by pumping systems and reservoirs to supply the higher water consumption of the network. Strategical studies are necessary to delimit risk regions where a demand increase may affect the system too negatively. To analyse expansion scenarios with the lowest risk of failure and damage for the supply network operations, this research studies hydraulic performance and connectivity under various demand increase scenarios using calculations of complex network metrics together with relevant hydraulic criteria. For these calculations, this research, developed in Python, uses, respectively, the NetworkX 2.5 and WNTR 0.3.0 packages. The C-Town network is employed as a case study, and demand increasing scenarios are implemented on 30 nodes along the peripheral regions of the network to simulate the growth of the cities. Then, these scenarios are evaluated using the TOPSIS methodology, thus determining the best and worst sectors to expand the capacity of the network.

## HIGHLIGHTS

Hydraulic analysis of water distribution networks as graphs with edges directed by flow, and weighted by flow and water travel time.

Use of hydraulic criteria to characterize the system under diverse demand increase scenarios.

Application of a multi-criteria analysis to rank expansion scenarios.

Identification of sectors with higher connectivity, robustness and hydraulic performance.

## INTRODUCTION

According to the World Health Organization (WHO) & the United Nations Children's Fund (UNICEF) (2019), in 2017, 29% of the world's population or, more specifically, 2.2 billion people did not have a safe supply of drinking water in their homes, and more than 1.4 billion needed to travel 30 min or more from their locations to some place where uncontaminated water could be collected.

Therefore, given that estimates point to an increase of 1.7 billion people in the world by the year 2050 and of 12.2% in the number of people living in urban areas (United Nations Population Fund 2022), it is understandable that the demand for drinking water will also grow in the coming years and the problem of lack of access to water will tend to worsen.

In this sense, an essential measure to meet the existing demand and fight the population's lack of access to drinking water is to expand the capacity of the water distribution networks (WDNs).

However, due to the water demand increase produced by any expansion, WDNs become undersized, thus leading to an increase in hydraulic headloss and, as a consequence, reducing the availability of pressure in the system, which increases the vulnerability of the network to failure (Huzsvár *et al.* 2021). Thus, WDNs have to be properly re-designed to provide water to the new population; as a result, their new structural connectivity patterns affect their reliability, resilience and efficiency (Pagano *et al.* 2019). One way to analyse water supply systems is to model them according to complex network theory, since complex systems can represent many real-world structures (energy, transport and water supply systems) and, in addition, offer metrics (indicators) that can be useful to explore the formation, topology, efficiency and vulnerability of WDNs (Yazdani & Jeffrey 2011; Meng *et al.* 2018; Simone *et al.* 2018).

The complex network theory has been widely applied to characterise and study the behaviour of many real systems, such as the relationships among individuals, internet interconnections, urban infrastructures (e.g., water supply systems, roads) and so on (Simone *et al.* 2018). These systems are defined as complex because, without applying mathematical theories, it is impossible to predict the collective behaviour of their components from the observation of their elements, which may have random behaviour (Da Mata 2020).

Complex network theory is a field based on mathematical abstractions called graphs. A graph is a set of nodes connected by edges that, depending on the network to be modelled, can be weighted, and directed, and can even be dynamic over time (Boccaletti *et al.* 2006). By making an analogy with established mathematical-computational models of water supply systems, in the abstraction to complex network theory, the nodes represent reservoirs, tanks, demands and simple junctions, while the edges are the pipes, valves and pumps (Castro-Gama *et al.* 2016).

In the literature, some works model WDNs as complex networks for water distribution system management. This is the case, among others, of district-metered area (DMA) design, topological analysis, investigation of complex network metrics, and analysis of resilience and vulnerability indices of water supply systems. To cite just a few, Yazdani & Jeffrey (2012) analysed network vulnerability through directed graphs weighted by the hydraulic capacity of the pipes; Giudiciani *et al.* (2018) investigated the influence of the topology in metrics based on network attributes such as connectivity and robustness, among others, for undirected and unweighted graphs; Di Nardo *et al.* (2018) applied graph theory for optimal DMA creation.

Another important issue to verify the performance of WDNs refers to their hydraulic indices. These indices allow for characterizing the variations of certain hydraulic parameters in specific simulation periods. They are also useful in decision-making when seeking to reduce the risks of operational disturbances and failures in a water supply system (Jeong & Kang 2018). Among several hydraulic quality indices, the literature uses pressure uniformity (*PU*) (Alhimiary & Alsuhaily 2007), weighted average water age above a given limit (*WA*) (Marchi *et al.* 2014) and hydraulic resilience (*R*) (Todini 2000; Jalal 2008).

Saldarriaga *et al.* (2019) used , and *R* as hydraulic criteria to evaluate the performance of DMAs design. Brentan *et al.* (2021) employed them as objectives of an optimization algorithm in a DMA design process. The authors developed a control valve allocation algorithm to identify the best scenarios for DMAs with the aid of multi-criteria analysis.

Moreover, the difficulty of identifying the best and worst scenarios in WDN expansion, due to the multiple hydraulic criteria employed, is clear. In this research, the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) is employed to rank the solutions of the considered scenarios. TOPSIS was first introduced by Hwang & Yoon (1981). It is one of the most widely used decision-making techniques for multi-criteria settings to rank alternatives or scenarios from calculations that refer to the distances between a solution and the so-called positive and negative ideal solutions (Chen 2019; Yu & Pan 2021).

As some examples of relevant research involving water supply networks and applications of the TOPSIS method to rank solutions, we can cite the works of Lopes *et al.* (2012), in which the TOPSIS method was used in comparison with other techniques to determine the risk of failure of a water distribution network; Onu *et al.* (2017), where the TOPSIS method was implemented along with an adapted programming logic model to rank sustainable alternatives for a water supply system; and Brentan *et al.* (2019), where the TOPSIS method was applied to rank sets of solutions obtained after implementing optimization algorithms to sectorize a water distribution network.

As a negative factor, it is worth noting that the TOPSIS method is unable to solve ambiguous problems (unquantifiable information, incomplete information, unobtainable information, and partially uncertain facts) in decision-making processes (Chiu & Hsieh 2016). However, since this research is about indicators and parameters related to the metrics of complex networks and hydraulic criteria, which are quantifiable from applying equations, in this case, the method can be used to obtain the best- and worst-case scenarios for the expansion of water distribution networks.

Considering a WDN as a complex network and applying hydraulic performance indices to characterize the network state in certain periods, this research proposes a methodology for identifying increased demand scenarios with enhanced robustness, connectivity and hydraulic performance. In this regard, a graph analysis of the results corresponding to the complex network metrics is performed and, in the case of the hydraulic indicators, rankings are created according to the multi-criteria analysis method employed, to determine the expansion scenarios with the best hydraulic performance.

## METHODS

This section presents the elements used to elaborate the research. The methodology is subdivided into the following subsections: WDN as a graph, complex network metrics, hydraulic criteria and multi-criteria analysis based on TOPSIS.

### Water distribution network as a graph

The WDN is modelled as a graph with directed edges employing the Python programming language and using the packages: NetworkX 2.5, for creating, manipulating and studying the structure, dynamics and functions of complex networks (Hagberg *et al.* 2008); and Water Network Tool for Resilience (WNTR) (Klise *et al.* 2018), version 0.3.0, for interfacing the WDN data in EPANET 2.2 software (Rossman *et al.* 2020) with Python.

The Networkx 2.5 library is selected for the creation and analysis of complex networks, since it already presents most of the metrics and functions that constitute the theory of complex networks. The WNTR 0.3.0 library was employed for making the connection between the hydraulic modelling software for water distribution networks EPANET 2.2 and the Python programming language environment, and also for linking the characteristics and hydraulic simulation data of the water distribution network with the Networkx library functions.

*et al.*(2018), a WDN is considered as a planar vertex-edge set, where the edge-weighted graph is defined as:where

*N*is the set of nodes and, in this case, encompasses reservoirs, tanks and water consumption demand nodes, and

*w*is a weighting matrix assigned to each edge. In this work, two weights are considered: flow and water travel time.

The weighting of the edges by the flows – recorded at 1-h intervals – is done to obtain data on the volumes of water that are transported through the network pipes to the supply-demand nodes. Weighting of the edges by the water travel time is performed to analyse the water flow velocity in the pipes, quantify the supply time of the network nodes and characterize the quality of the water reaching the nodes (Sitzenfrei 2021).

### Complex network metrics

In this subsection, the complex network theory metrics employed to analyse the water supply system as a graph are presented.

- (a)Edge density (
*ρ*): according to Rodríguez-Alarcón & Lozano (2019), it is the ratio between the total number of edges (*m*) in the network and the maximum number of edges that can constitute the network, , where*n*is the total number of vertices in the network. Therefore, it is defined by:

This metric is useful in identifying how dense or sparse a network is. Denser networks present more edge clustering and, consequently, more alternative paths between vertices.

- (b)Average degree (
*Deg*_{ave}): according to Barabasi (2016), this metric represents the average number of connections between nodes of the complex network, and for directed networks is given bywhere is the average incoming degree of the network, that is, considering only the edges reaching the nodes; is the average outcoming degree of the network, which accounts for only the connections characterized by the edges leaving the nodes;*N*is the total number of nodes in the network and*l*is the total number of edges in the network. - (c)Normalized closeness centrality (CC
): this metric determines the importance of the nodes in the network according to their proximity or shortest path lengths to other nodes in the network (Zhu_{i}*et al.*2021). The normalized closeness centrality is given by the equation (Freeman 1978):where is the shortest path distance between*j*and*i*, and is the number of nodes reaching*i*.

Note that for directed graphs, there is an input closeness centrality , in which the inverse of the sum of the distances of the shortest possible paths from all other nodes in the network to node *i* is calculated. In this case, the outbound closeness centrality is calculated by the inverse of the sum of the distances of the shortest possible paths from node *i* to all other nodes in the network.

- (d)Average shortest path length (
*L*_{ave}): according to Zhang*et al.*(2021), it is the average length of the shortest path between all pairs of nodes in the network, and it is expressed by:in which, is the length of the shortest path from a node*i*to a node*j*, and*n*is the number of nodes in the network. Mao & Zhang (2017) corroborate that the average shortest path of a network measures the efficiency of the information traffic. - (e)Bridge density (
*D*_{br}): according to Wu*et al.*(2018), it is the ratio between the total number of bridges (*N*_{br}) and the total number of edges (*m*) of the complex network

Bridges are edges that connect large sets of nodes of the graph, which, if occasionally removed, can lead to the unavailability of a certain number of consumers and generate considerable financial and social losses. It is a metric that quantifies the robustness of the WDN. Higher values of bridge density determine a network that is more vulnerable to failure, due to fewer alternative paths, since more bridges in the network graph represent more sets of nodes connected by only one edge.

### Hydraulic criteria

This subsection presents the hydraulic criteria used to quantify the hydraulic performance and resilience of the WDN.

- (a)Pressure uniformity (PU): according to Alhimiary & Alsuhaily (2007) can be expressed by the equation:where is the pressure at junction
*i*at time step*t*; is the minimum operational pressure and is the average pressure in the network at time step*t*.

is a metric that accounts for the uniformity of the pressure distribution along the water supply system. This metric measures the difference among the pressures at nodes and the average and minimum pressures of the system, for a given time step. The higher this difference, the lower the uniformity pressure distribution in the system.

- (b)Weighted average water age above a given limit (WA): according to Marchi
*et al.*(2014), it is defined by the equation:where is given in hours; is the water age at junction*i*at time step*t*(excluding tanks and reservoirs); is the demand at junction*i*at time step*t*; is the water age limit (in hours) allowed by the standard and represents a binary variable, set to 1 if the water age is greater than or equal to the limit, and 0 otherwise.

quantifies how long, in general, it takes the system to distribute water to consumers above an established limit. Since quality deterioration and chlorine decay are strictly linked to travel time, this metric can be used to measure the quality of the supplied water.

- (c)Hydraulic resilience (
*R*): proposed by Todini (2000), this criterion measures the network's capability to withstand stress and failure conditions, and is defined by the equation:where is the number of demand nodes; is the number of reservoirs; is the number of pumps of the network; and are the demand and hydraulic head of the demand node*i*; and are, respectively, the flow and level of the reservoir/tank*r*; is the power of the pump*j*in the system;*γ*is the specific weight of water; and, finally, is the minimum hydraulic head required for supplying the system.

### Multi-criteria analysis based on TOPSIS

In this step, the TOPSIS multi-criteria method is used to rank the created scenarios to identify the best-performing hydraulic scenarios considering the different variations in the hydraulic parameters of the network after the respectively implemented expansions. The calculation and analysis steps that constitute the TOPSIS method, as recalled by Brentan *et al.* (2019), are the following.

Step 1: Calculation of the decision matrix (data input matrix) starting with the identification and characterization of the data to be used, where *i* corresponds to each alternative and *j* determines each evaluation criterion considered.

*i*, which represents how close solution

*i*is to the positive and negative ideal solutions, from the expression:Step 6: Creation of a ranking of solutions by ordering the previously calculated coefficients in descending order, where the closer is to 1, the better ranked is the solution, while the closer is to 0, the worse ranked is the solution, thus occupying the last positions of the ranking.

### Case study

The WDN used in this work is the C-Town network which, without expansion, consists of 429 pipes, 388 junction nodes, 7 tanks, 1 reservoir, 11 pumps and 5 valves. This network is selected because it has all the hydraulic components (tanks, valves and pumps) that urban water distribution networks have and therefore most closely simulates the hydraulic operation of a real network.

In this research, the network is sectorized into 5 DMAs, and 30 expansion scenarios are simulated in peripheral locations of the network by increasing the base demand at specific nodes. The expansion nodes are selected only in peripheral regions of the network to simulate new occupation of urban zones and, consequently, the increase of the urban network. The number of expansion nodes in total and per sector is defined according to the size of the respective districts that predominate in the extreme regions of the C-Town network.

*n*,

*n*being node ID number, and coloured DMAs.

In this study, for each of the 30 selected expansion nodes, four percentages of 0.1, 0.5, 5 and 10% of the total demand of the original network (approximately 270 L/s) are simulated, that is, 120 expansion scenarios are considered. These percentages of demand increase are selected because the C-Town network simulates the hydraulic operation of a real network of approximately 150,000 inhabitants and also because these percentage increases can possibly occur for a network with this population capacity.

## RESULTS AND DISCUSSION

In this section, all the results obtained in this research are described and, for better organization of the information, they are subdivided into the following subsections: results involving the complex network metrics and results concerning the hydraulic criteria.

### Results of the complex network metrics

From Figures 2 and 3, it can be noted that the larger the increase in demand, the more connected the network. This can be attributed to a larger number of nodes that tend to be supplied by the larger flows released by pumps, reservoirs and tanks to supply the increased demand. Furthermore, because of the same demand increase, by comparing the results only among the expansion nodes, it can be noted that some scenarios generate a larger number of edges in the graphs and, consequently, greater connectivity among the nodes in the network.

The metrics edge density (Equation (3)), and average degree (Equation (4)), in general, quantify the number of connections (edges) between nodes and, therefore, considering a graph with edges weighted by flow, higher values of these metrics determine more pipes occupied by water and, consequently, a more resilient supply system, since if a pipe breaks, the water will be more likely to reach the node through other paths.

Tables 1 and 2 show the five best- and worst-case scenarios obtained based on the results, respectively, of the network edge density and the average degree metrics, considering edges of the network weighted by the flow.

. | 0.1% . | 0.5% . | 5.0% . | 10.0% . | ||||
---|---|---|---|---|---|---|---|---|

Scenarios . | Node . | DMA . | Node . | DMA . | Node . | DMA . | Node . | DMA . |

Best | J25 | 2 | J377 | 1 | J74 | 5 | J379 | 1 |

J377 | 1 | J8 | 1 | J90 | 2 | J150 | 2 | |

J153 | 2 | J308 | 1 | J27 | 2 | J158 | 3 | |

J373 | 1 | J266 | 3 | J36 | 2 | J36 | 2 | |

J1058 | 1 | J27 | 2 | J379 | 1 | J90 | 2 | |

Worst | J144 | 1 | J504 | 2 | J504 | 2 | J266 | 3 |

J74 | 5 | J25 | 2 | J184 | 3 | J373 | 1 | |

J158 | 3 | J70 | 5 | J70 | 5 | J152 | 2 | |

J308 | 1 | J155 | 1 | J153 | 2 | J184 | 3 | |

J91 | 2 | J144 | 1 | J155 | 1 | J350 | 3 |

. | 0.1% . | 0.5% . | 5.0% . | 10.0% . | ||||
---|---|---|---|---|---|---|---|---|

Scenarios . | Node . | DMA . | Node . | DMA . | Node . | DMA . | Node . | DMA . |

Best | J25 | 2 | J377 | 1 | J74 | 5 | J379 | 1 |

J377 | 1 | J8 | 1 | J90 | 2 | J150 | 2 | |

J153 | 2 | J308 | 1 | J27 | 2 | J158 | 3 | |

J373 | 1 | J266 | 3 | J36 | 2 | J36 | 2 | |

J1058 | 1 | J27 | 2 | J379 | 1 | J90 | 2 | |

Worst | J144 | 1 | J504 | 2 | J504 | 2 | J266 | 3 |

J74 | 5 | J25 | 2 | J184 | 3 | J373 | 1 | |

J158 | 3 | J70 | 5 | J70 | 5 | J152 | 2 | |

J308 | 1 | J155 | 1 | J153 | 2 | J184 | 3 | |

J91 | 2 | J144 | 1 | J155 | 1 | J350 | 3 |

. | 0.1% . | 0.5% . | 5.0% . | 10.0% . | ||||
---|---|---|---|---|---|---|---|---|

Scenarios . | Node . | DMA . | Node . | DMA . | Node . | DMA . | Node . | DMA . |

Best | J25 | 2 | J377 | 1 | J74 | 5 | J150 | 2 |

J377 | 1 | J8 | 1 | J90 | 2 | J379 | 1 | |

J153 | 2 | J266 | 3 | J27 | 2 | J158 | 3 | |

J373 | 1 | J308 | 1 | J36 | 2 | J36 | 2 | |

J1058 | 1 | J27 | 2 | J379 | 1 | J90 | 2 | |

Worst | J150 | 2 | J504 | 2 | J266 | 3 | J266 | 3 |

J74 | 5 | J25 | 2 | J184 | 3 | J373 | 1 | |

J158 | 3 | J70 | 5 | J153 | 2 | J152 | 2 | |

J308 | 1 | J155 | 1 | J155 | 1 | J184 | 3 | |

J91 | 2 | J144 | 1 | J70 | 5 | J350 | 3 |

. | 0.1% . | 0.5% . | 5.0% . | 10.0% . | ||||
---|---|---|---|---|---|---|---|---|

Scenarios . | Node . | DMA . | Node . | DMA . | Node . | DMA . | Node . | DMA . |

Best | J25 | 2 | J377 | 1 | J74 | 5 | J150 | 2 |

J377 | 1 | J8 | 1 | J90 | 2 | J379 | 1 | |

J153 | 2 | J266 | 3 | J27 | 2 | J158 | 3 | |

J373 | 1 | J308 | 1 | J36 | 2 | J36 | 2 | |

J1058 | 1 | J27 | 2 | J379 | 1 | J90 | 2 | |

Worst | J150 | 2 | J504 | 2 | J266 | 3 | J266 | 3 |

J74 | 5 | J25 | 2 | J184 | 3 | J373 | 1 | |

J158 | 3 | J70 | 5 | J153 | 2 | J152 | 2 | |

J308 | 1 | J155 | 1 | J155 | 1 | J184 | 3 | |

J91 | 2 | J144 | 1 | J70 | 5 | J350 | 3 |

From these tables, for both metrics, the expansion nodes that stand out among the five best scenarios, for different percentage increases in demand, are: node J377, which is among the best for demand increases of 0.1 and 0.5%; and nodes J36 and J379, which stand out as being among the best for demand increases of 5.0 and 10.0%.

On the contrary, by analysing Table 1, it is identified that, for the case of applying the network edge density metric and according to the percentages of demand increase employed, the expansion nodes that stand out among the five worst-case scenarios are: node J144, which is present among the worst-case for demand increases of 0.1 and 0.5%; and node J184, which is present among the worst-case for 5.0 and 10.0% demand increases.

Looking at Table 2, it can be seen that, in the case of using the average degree metric, the expansion nodes that stand out among the five worst scenarios, for the different percentage increases in demand, are the J184 and J266 nodes, which are among the worst for 5.0 and 10.0% increases in demand. It is also worth mentioning that, for the application of both metrics, the J373 expansion node presents the five worst scenarios, namely for 1.0, 2.0 and 10.0% increases in demand.

In general, the results represented in Tables 1 and 2 are virtually the same for some percentages of demand, due to both metrics quantifying the number of edges (connections) between the nodes of the network graph. Therefore, the more connections between the nodes of the network graph, the higher the edge density and average degree values will be. On the other hand, the fewer connections between the nodes of the network graph, the lower the edge density and average degree values will be. This context can be justified by Equations (3) and (4).

Furthermore, it should be noted that only the bounds of demand increase percentages are considered in Figures 4 and 5, which highlights the difference in the closeness centrality values for the different weightings of edges.

Observing Figure 4, it is noted that the larger the demand increase applied to the network, the lower, in general, the normalized closeness centrality of the graph with edges weighted by the flow. The main reason is that the length of the shortest paths is inversely proportional to the closeness centrality and, in this way, the larger the flows, the longer the lengths of the edges corresponding to the shortest paths due to the weighting process of the graph and, consequently, the lower the closeness centrality of the scenarios’ graphs.

From Figure 4, it is also observed that, for a 0.1% increase in demand, there is a greater variation among the input centrality data, with lower values for the scenarios corresponding to expansion nodes J36, J74, J91, J155, J184, J240, J266 and J504. These respective scenarios can be characterized as more vulnerable to failure than the others because, according to Equation (5), the lower the proximity centrality, the fewer nodes are connected or the lower flows pass along the pipes.

Comparing Figures 4 and 5, while is smaller for the case of graphs with edges weighted by flow due to the larger volumes of water travelling through the pipes, is larger for the graphs with edges weighted by water travel time, because the larger the flows, the higher the flow velocities and, consequently, the shortest the water travel times.

Figure 6 shows that the larger the increase in demand, the longer the average lengths of the shortest flow-weighted edge paths, and the shorter the average lengths of the graphs with water travel time-weighted edges.

Also, regarding the use of complex network metrics, in this work, we have applied the bridges density metric to network graphs with flow-weighted edges to identify scenarios with the least number of bridges or, equivalently, with more alternative paths between nodes and, consequently, more robust layouts.

From Figure 7, it can be seen that the larger the increase in demand at the expansion nodes, in general, the lower the bridges density values of the respective implemented scenarios. Consequently, the scenarios become more robust because, due to the larger flows in the network, more weighted edges are added to the graphs and thus more connections are generated between the nodes in the graph.

Considering the application of the bridges density metric, it can also be seen in Figure 7 that some expansion nodes obtained a lower bridges density for the same demand increase and, consequently, they generated networks with more alternative connections. Table 3 presents the five best and worst expansion scenarios considering the hourly averages of the bridge density metric to all demand increase scenarios, over the entire simulation time.

. | 0.1% . | 0.5% . | 5.0% . | 10.0% . | ||||
---|---|---|---|---|---|---|---|---|

Scenarios . | Node . | DMA . | Node . | DMA . | Node . | DMA . | Node . | DMA . |

Best | J32 | 2 | J334 | 1 | J32 | 2 | J350 | 3 |

J70 | 5 | J27 | 2 | J379 | 1 | J70 | 5 | |

J150 | 2 | J36 | 2 | J158 | 3 | J155 | 1 | |

J27 | 2 | J158 | 3 | J36 | 2 | J145 | 3 | |

J308 | 1 | J32 | 2 | J240 | 2 | J308 | 1 | |

Worst | J350 | 3 | J184 | 3 | J70 | 5 | J191 | 1 |

J91 | 2 | J350 | 3 | J145 | 3 | J158 | 3 | |

J504 | 2 | J144 | 1 | J266 | 3 | J265 | 3 | |

J373 | 1 | J266 | 3 | J184 | 3 | J1058 | 1 | |

J1058 | 1 | J145 | 3 | J144 | 1 | J36 | 2 |

. | 0.1% . | 0.5% . | 5.0% . | 10.0% . | ||||
---|---|---|---|---|---|---|---|---|

Scenarios . | Node . | DMA . | Node . | DMA . | Node . | DMA . | Node . | DMA . |

Best | J32 | 2 | J334 | 1 | J32 | 2 | J350 | 3 |

J70 | 5 | J27 | 2 | J379 | 1 | J70 | 5 | |

J150 | 2 | J36 | 2 | J158 | 3 | J155 | 1 | |

J27 | 2 | J158 | 3 | J36 | 2 | J145 | 3 | |

J308 | 1 | J32 | 2 | J240 | 2 | J308 | 1 | |

Worst | J350 | 3 | J184 | 3 | J70 | 5 | J191 | 1 |

J91 | 2 | J350 | 3 | J145 | 3 | J158 | 3 | |

J504 | 2 | J144 | 1 | J266 | 3 | J265 | 3 | |

J373 | 1 | J266 | 3 | J184 | 3 | J1058 | 1 | |

J1058 | 1 | J145 | 3 | J144 | 1 | J36 | 2 |

Analysing Table 3, it can be seen that, for the case of applying the bridges density metric, the expansion nodes that stand out among the five best scenarios, for the percentage demand increases considered, are: the J27, which is among the best for demand increases of 0.1 and 0.5%; the J32, which is among the best for demand increases of 0.1, 0.5 and 5.0%; J70 and J308, which provide some of the best scenarios, for 0.1 and 10.0% demand increases.

In contrast, as shown in Table 3, it can also be observed that the expansion nodes that stand out among the five worst-case scenarios for the percentage demand increases employed are: the J184, which is among the worst for 0.5 and 5.0% demand increases; the J144, which gives one of the worst scenarios for 0.5 and 5.0% demand increases; and the J1058, also one of the worst expansion nodes for 0.1 and 10.0% demand increase scenarios.

We may conclude that some expansion scenarios of higher edge density and average degree do not have lower bridges density because more edges in the graph mean more connectivity and not necessarily fewer bridges.

### Results of the hydraulic criteria

In addition to complex network metrics, in this research, some hydraulic criteria were also applied to analyse the behaviour of the C-Town network model, specifically, pressure uniformity (PU), weighted average water age above the set limit (WA) and hydraulic resilience (*R*). Tables 4–7 present the five best and worst scenarios for the considered demand increases of 0.1, 0.5, 5.0 and 10.0%, according to the results of , and *R*, used for calculating the closeness coefficient of the TOPSIS method.

. | Expansion node . | DMA . | . | . | . | . | Ranking . |
---|---|---|---|---|---|---|---|

Best solutions | J27 | 2 | 496.9705 | 4.5640 | 0.4048 | 0.9951 | 1st |

J240 | 2 | 497.0459 | 4.5636 | 0.4048 | 0.9941 | 2nd | |

J153 | 2 | 496.9480 | 4.5657 | 0.4048 | 0.9930 | 3rd | |

J25 | 2 | 496.9705 | 4.5648 | 0.4047 | 0.9919 | 4th | |

J91 | 2 | 496.9580 | 4.5660 | 0.4048 | 0.9918 | 5th | |

Worst solutions | J158 | 3 | 496.9052 | 4.8912 | 0.4030 | 0.0138 | 26th |

J266 | 3 | 496.9091 | 4.8918 | 0.4029 | 0.0128 | 27th | |

J184 | 3 | 496.9052 | 4.8904 | 0.4028 | 0.0126 | 28th | |

J145 | 3 | 496.9051 | 4.8915 | 0.4028 | 0.0120 | 29th | |

J265 | 3 | 496.9083 | 4.8918 | 0.4028 | 0.0120 | 30th |

. | Expansion node . | DMA . | . | . | . | . | Ranking . |
---|---|---|---|---|---|---|---|

Best solutions | J27 | 2 | 496.9705 | 4.5640 | 0.4048 | 0.9951 | 1st |

J240 | 2 | 497.0459 | 4.5636 | 0.4048 | 0.9941 | 2nd | |

J153 | 2 | 496.9480 | 4.5657 | 0.4048 | 0.9930 | 3rd | |

J25 | 2 | 496.9705 | 4.5648 | 0.4047 | 0.9919 | 4th | |

J91 | 2 | 496.9580 | 4.5660 | 0.4048 | 0.9918 | 5th | |

Worst solutions | J158 | 3 | 496.9052 | 4.8912 | 0.4030 | 0.0138 | 26th |

J266 | 3 | 496.9091 | 4.8918 | 0.4029 | 0.0128 | 27th | |

J184 | 3 | 496.9052 | 4.8904 | 0.4028 | 0.0126 | 28th | |

J145 | 3 | 496.9051 | 4.8915 | 0.4028 | 0.0120 | 29th | |

J265 | 3 | 496.9083 | 4.8918 | 0.4028 | 0.0120 | 30th |

. | Expansion node . | DMA . | . | . | . | . | Ranking . |
---|---|---|---|---|---|---|---|

Best solutions | J240 | 2 | 497.6727 | 4.4671 | 0.4029 | 0.9191 | 1st |

J153 | 2 | 497.1879 | 4.4927 | 0.4033 | 0.9040 | 2nd | |

J27 | 2 | 497.2899 | 4.4884 | 0.4029 | 0.9035 | 3rd | |

J91 | 2 | 497.2557 | 4.4932 | 0.4033 | 0.9031 | 4th | |

J90 | 2 | 497.3052 | 4.4924 | 0.4029 | 0.8970 | 5th | |

Worst solutions | J373 | 1 | 497.0960 | 4.8248 | 0.4026 | 0.0817 | 26th |

J308 | 1 | 496.9941 | 4.8271 | 0.4026 | 0.0779 | 27th | |

J1058 | 1 | 496.8281 | 4.8373 | 0.4026 | 0.0627 | 28th | |

J8 | 1 | 496.8577 | 4.8373 | 0.4021 | 0.0525 | 29th | |

J334 | 1 | 497.0478 | 4.8404 | 0.4018 | 0.0386 | 30th |

. | Expansion node . | DMA . | . | . | . | . | Ranking . |
---|---|---|---|---|---|---|---|

Best solutions | J240 | 2 | 497.6727 | 4.4671 | 0.4029 | 0.9191 | 1st |

J153 | 2 | 497.1879 | 4.4927 | 0.4033 | 0.9040 | 2nd | |

J27 | 2 | 497.2899 | 4.4884 | 0.4029 | 0.9035 | 3rd | |

J91 | 2 | 497.2557 | 4.4932 | 0.4033 | 0.9031 | 4th | |

J90 | 2 | 497.3052 | 4.4924 | 0.4029 | 0.8970 | 5th | |

Worst solutions | J373 | 1 | 497.0960 | 4.8248 | 0.4026 | 0.0817 | 26th |

J308 | 1 | 496.9941 | 4.8271 | 0.4026 | 0.0779 | 27th | |

J1058 | 1 | 496.8281 | 4.8373 | 0.4026 | 0.0627 | 28th | |

J8 | 1 | 496.8577 | 4.8373 | 0.4021 | 0.0525 | 29th | |

J334 | 1 | 497.0478 | 4.8404 | 0.4018 | 0.0386 | 30th |

. | Expansion node . | DMA . | . | . | . | . | Ranking . |
---|---|---|---|---|---|---|---|

Best solutions | J350 | 3 | 493.2814 | 4.0202 | 0.4201 | 0.9932 | 1st |

J266 | 3 | 493.2540 | 4.0159 | 0.4182 | 0.9750 | 2nd | |

J265 | 3 | 493.1894 | 4.0148 | 0.4145 | 0.9267 | 3rd | |

J91 | 2 | 495.2969 | 4.0357 | 0.4061 | 0.8147 | 4th | |

J240 | 2 | 500.2542 | 4.0228 | 0.4066 | 0.8133 | 5th | |

Worst solutions | J334 | 1 | 497.6796 | 4.1693 | 0.3774 | 0.4339 | 26th |

J70 | 5 | 498.0677 | 4.4907 | 0.3971 | 0.4209 | 27th | |

J152 | 2 | 493.7058 | 4.0343 | 0.3605 | 0.4206 | 28th | |

J32 | 2 | 497.0604 | 4.4782 | 0.3887 | 0.3448 | 29th | |

J36 | 2 | 497.0853 | 4.4841 | 0.3870 | 0.3250 | 30th |

. | Expansion node . | DMA . | . | . | . | . | Ranking . |
---|---|---|---|---|---|---|---|

Best solutions | J350 | 3 | 493.2814 | 4.0202 | 0.4201 | 0.9932 | 1st |

J266 | 3 | 493.2540 | 4.0159 | 0.4182 | 0.9750 | 2nd | |

J265 | 3 | 493.1894 | 4.0148 | 0.4145 | 0.9267 | 3rd | |

J91 | 2 | 495.2969 | 4.0357 | 0.4061 | 0.8147 | 4th | |

J240 | 2 | 500.2542 | 4.0228 | 0.4066 | 0.8133 | 5th | |

Worst solutions | J334 | 1 | 497.6796 | 4.1693 | 0.3774 | 0.4339 | 26th |

J70 | 5 | 498.0677 | 4.4907 | 0.3971 | 0.4209 | 27th | |

J152 | 2 | 493.7058 | 4.0343 | 0.3605 | 0.4206 | 28th | |

J32 | 2 | 497.0604 | 4.4782 | 0.3887 | 0.3448 | 29th | |

J36 | 2 | 497.0853 | 4.4841 | 0.3870 | 0.3250 | 30th |

. | Expansion node . | DMA . | . | . | . | . | Ranking . |
---|---|---|---|---|---|---|---|

Best solutions | J158 | 3 | 491.8816 | 3.1362 | 0.4478 | 0.9839 | 1st |

J350 | 3 | 492.1709 | 3.2545 | 0.4509 | 0.9443 | 2nd | |

J145 | 3 | 491.8694 | 3.1108 | 0.4353 | 0.9436 | 3rd | |

J266 | 3 | 492.0957 | 3.2618 | 0.4475 | 0.9398 | 4th | |

J265 | 3 | 491.7857 | 3.2603 | 0.4393 | 0.9276 | 5th | |

Worst solutions | J379 | 1 | 494.7395 | 3.5598 | 0.3032 | 0.4383 | 26th |

J1058 | 1 | 490.2083 | 3.6282 | 0.2857 | 0.3706 | 27th | |

J504 | 2 | 506.1459 | 3.1514 | 0.2143 | 0.3030 | 28th | |

J153 | 2 | 494.9171 | 3.1993 | 0.2205 | 0.3015 | 29th | |

J152 | 2 | 496.5184 | 3.2054 | 0.1957 | 0.2723 | 30th |

. | Expansion node . | DMA . | . | . | . | . | Ranking . |
---|---|---|---|---|---|---|---|

Best solutions | J158 | 3 | 491.8816 | 3.1362 | 0.4478 | 0.9839 | 1st |

J350 | 3 | 492.1709 | 3.2545 | 0.4509 | 0.9443 | 2nd | |

J145 | 3 | 491.8694 | 3.1108 | 0.4353 | 0.9436 | 3rd | |

J266 | 3 | 492.0957 | 3.2618 | 0.4475 | 0.9398 | 4th | |

J265 | 3 | 491.7857 | 3.2603 | 0.4393 | 0.9276 | 5th | |

Worst solutions | J379 | 1 | 494.7395 | 3.5598 | 0.3032 | 0.4383 | 26th |

J1058 | 1 | 490.2083 | 3.6282 | 0.2857 | 0.3706 | 27th | |

J504 | 2 | 506.1459 | 3.1514 | 0.2143 | 0.3030 | 28th | |

J153 | 2 | 494.9171 | 3.1993 | 0.2205 | 0.3015 | 29th | |

J152 | 2 | 496.5184 | 3.2054 | 0.1957 | 0.2723 | 30th |

Looking at Tables 4–7, it can be seen that, in general, and are reduced with the increase in supply-demand and this is due to the larger flows released by pumps, reservoir and tanks to supply the consumption increase, which develop higher hydraulic head losses at the nodes and flow velocities in the pipes. Furthermore, analysing the tables, it can be seen that the larger the increase in demand, the slightly larger the resilience at some nodes. This can be attributed to the operational changes in the pumping systems, which can generate greater hydraulic heads in the DMAs.

Thus, analysing the maps in Figure 8 and Tables 4 and 5, it can be seen that initially, for a demand increase of 0.1 and 0.5%, all of the best scenarios were located in DMA #2, something that can be attributed to the shorter travel time of water to supply the nodes (WA) and the larger hydraulic resilience indices (*R*) of the scenarios in this sector of the network. However, according to Tables 4 and 5, for demand increases of 0.1 and 0.5%, the pressure uniformity (PU) of the top scenarios presents large values and does not differ much from the worst scenarios.

One has to note that seeks to standardize the pressure distribution throughout the water supply system, based on an indicator value that represents the differences between the pressures at the network nodes and the average and minimum pressures of the system, for a certain time stage.

Thus, the larger the pressure uniformity value, the greater the variation of the system pressure in relation to the average and minimum pressures and, consequently, the greater the risk of supply failures, since very high pressures characterize an energy overload in the system, which can even lead to the rupture of pipes, and pressures below the minimum can make it difficult to supply water to consumers.

However, for larger demand increases, as shown in Tables 6 and 7, the best scenarios are predominantly identified in DMA #3, due to its lower variations, higher *R* values and lower indexes. In this case, the is a metric that quantifies, in general, how long it takes, above the established limit, for the water distributed through the pipes to supply the consumption demand nodes.

Therefore, the larger the weighted average age value of water above the established limit, the worse the efficiency of the system, since longer supply times can favour the proliferation and deposition of organic material, and the decay of chlorine in the water in the pipes, thus reducing the water quality supplied to the system. On the other hand, high levels of *R* determine lower vulnerability of the water supply system to failures and stress conditions.

Observing Tables 4–7, together with the maps in Figure 8, it can be seen that the nodes in DMA #2, J91 and J240 stand out for the demand increases of 0.1, 0.5 and 5.0%, as these are the scenarios that generate shorter water supply times to consumption nodes and higher hydraulic resilience in the WDN. The other scenarios that obtained good results for higher percentages of demand increase (5.0 and 10.0%) correspond to nodes J350, J266 and J265 in DMA #3.

## CONCLUSIONS AND RECOMMENDATIONS

In this research, we have tried to validate the modelling of a water supply network from the perspective of the complex network theory. The edges of the graphs of the water network are weighted by hydraulic parameters and, after applying metrics of complex networks, it has been possible to characterize the correct correlation between water travel time and flow from the shortest path metric.

In addition, from the results of complex network metrics and the hydraulic performance rankings for expansion scenarios, it is possible to identify scenarios of larger connectivity, more robustness and better quality of supply, as is the case of expansion nodes J25 and J153 for demand increases of 0.1%; J27 for demand increases of 0.5% and J377 for demand increases of 5 and 10% (as seen in the complete ranking of scenarios). On the opposite side, observing the complete ranking and considering the results of complex network metrics and hydraulic criteria, the worst scenarios are J155 and J350 for demand increases of 0.1 and 0.5%; and J144 for demand increases of 5 and 10%.

As the objective of this research was reached, it is concluded that hydraulic parameters can be synergistically involved with complex network metrics through suitable weighting to obtain hydraulically efficient options for the expansion of WDNs. The developed mathematical and hydraulic analysis of the expansion framework in a WDN proves to be a powerful tool to understand the impacts of increased demand.

For future works, it is suggested to carry out studies that address the metrics of complex networks and hydraulic criteria before and after the implementation of engineering measures (replacement of pipes and pumps, addition of tanks) for the rehabilitation of the minimum pressures of the system after demand expansions. It is also considered relevant to the coupling of water distribution network expansion proposed in this paper with water consumption demand forecast models in cities, for a more practical and precise analysis of better expansion scenarios in a real supply network. In addition, for future studies, it is recommended to carry out additional research comparing the modelling of water distribution networks as different types of graphs (directed, undirected, weighted and unweighted) to understand more precisely the benefits and limitations of using the theory of complex networks in water supply systems.

## DATA AVAILABILITY STATEMENT

Data cannot be made publicly available; readers should contact the corresponding author for details.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*Performance Measurement of Water Distribution Systems (WDS): A Critical and Constructive Appraisal of the State-of-the-Art*

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