Water Distribution Network Reliability Assessment beyond the Resilience Index ^†

Abstract

Water distribution network design must consider cost and reliability, with reliability being complex to assess, involving mechanical, hydraulic, and water quality aspects. Current metrics like flow entropy and resilience index have known flaws. This paper presents a new index, addressing the known weaknesses of the original resilience index and including critical network features in reliability assessment. The new proposed index introduces a novel pressure surplus threshold, setting more realistic pressure limits in operational management.

1. Introduction

Water distribution networks (WDNs) are crucial urban infrastructure, ensuring continuous access to drinking water. However, they face challenges like climate change and aging, highlighting the need for resilience—the ability to maintain functionality during disruptions. Current resilience indices have limitations, such as neglecting network redundancy and diameter uniformity. Building on prior work [1] this paper introduces an alternative resilience index formulation, considering a broader range of factors. It explores how these factors influence WDN resilience and applies the new index to three network versions to assess its real-world effectiveness.

2. Discussion

This chapter delves into the various influences of WDN design and operational regimes on resilience. It deepens how these aspects impact the network’s ability to withstand and recover from adverse events, ensuring uninterrupted water supply. Through a comprehensive analysis, authors investigate their implications on the overall functionality and responsiveness of the network in emergency situations.

• Redundancy in Network Topology: Looped networks excel in resilience due to alternative paths, unlike tree-like configurations that limit resilience. Introducing a redundancy coefficient offers a quantitative measure to evaluate redundancy.
• Uniformity of Diameters: Uniform pipe diameters enhance resilience by providing effective alternative paths, while heterogeneous distributions may compromise it. Evaluating a coefficient for diameter uniformity is crucial for resilience.
• Network Operating Pressure Regime: Higher pressure networks exhibit resilience, but reliance on surplus pressure may overlook diminishing returns and increase failure risks. Integrating diminishing returns into resilience indices ensures optimal resilience.
• Node Location: Assumptions about uniform node contributions overlook the varying impact of pipe breaks based on proximity to the tank. Incorporating a weight based on the flow supplying each node improves resilience assessment accuracy.
• Node Demand: Traditional indices overlook nodes without demand, neglecting their potential resilience significance. Considering every node enhances resilience assessment comprehensiveness.

3. Materials and Methods

The assessment of network resilience hinges on key factors: topology, pipe connection uniformity, and node significance. Authors propose a novel resilience index formulation incorporating these factors. A weighted resilience index introduces two coefficients—topological and uniformity—to modify junction weights. It integrates pressure surplus for all junctions, not just those with non-zero demand. The topological coefficient penalizes junctions with fewer connections, while the uniformity coefficient favors networks with similar diameter pipes. Calculations avoid graph theory complexity, using only counts of pipes supplying each node and their diameters.

3.1. Importance of Each Junction

The junctions through which a greater flow passes (upstream part of the network) are more important for the network functioning, and therefore their resilience (or lack of it) is more impactful overall. The presence of a measure that puts more weight on the junctions through which more flow passes allows to better account for the areas near the tanks or main pipelines. A junction located in the peripheral area of the network has a marginal impact compared to the water mains near a tank. This is accomplished by using $Q_{in}^i$ (flow entering junction i) in the assessment of each node contribution for the network resilience.

3.2. Topological Coefficient

The topological coefficient reduces the contribution of the junctions for which there is a single entering pipe. It is assumed that such junctions contribute less to network resilience. The topological coefficient-K^T (1) is a multiplicative coefficient that can assume values between 0.5 and 1.5 and can be estimated as follows:

$$K_j^T=0.5+{\displaystyle \frac{N_{in}^j-1}{N_{in}^j}}$$

(1)

where $N_{in}^j$ is Total number of pipes entering junction $j$.

3.3. Uniformity Coefficient

In a WDN, the connection redundancy does not ensure resilience by itself. The uniformity coefficient is based on the assumption that the pipes converging into a junction are effectively redundant, and therefore resilient, the more their diameters are similar.

The uniformity coefficient is a multiplicative coefficient that rewards the uniformity of the diameters and penalizes situations in which the diameters of the incoming pipes are very different. According to (2) it will assume values between 0.5 and 1. The uniformity coefficient-K^U (2) can be assessed as follows:

$$\mathrm{If} N_{in}^j=1\Rightarrow K_j^U=0.5; \mathrm{Otherwise} K_j^U={\displaystyle \frac{\sum _{i=1}^{N_{in}^j}{\left({D_{in}^j}_i\right)}^2}{N_{in}^j\ast {\left(MAX\left({D_{in}^j}_i\right)\right)}^2}}$$

(2)

where

• $N_{in}^j$: Total number of pipes entering junction $j$;
• ${D_{in}^j}_i$: Diameter of the $i^{\mathrm{t}\mathrm{h}}$ pipe that enters junction $j$.

The uniformity coefficient only takes into consideration the pipes entering the junctions, and not the diameter, but its square, because the pipe section is proportional to the square of the diameter.

3.4. New Formulation for the Resilience Index

The next equation shows the suggested new formulation for the resilience index—$I_r$:

$$I_r={\displaystyle \frac{{\sum }_{i=1}^n{Q}_{in}^i{·\mathsf{\Delta }h}_i}{{\sum }_{i=1}^n{Q}_{in}^i{·\mathsf{\Delta }h}_i^{max}}}$$

(3)

$${\mathsf{\Delta }h}_i=Max\left(0; Min\left(h_i-h_i^{min}; h_i^{+}\right)\right) {\mathsf{\Delta }h}_i^{max}=Min\left(h_i^{max}-h_i^{min}; h_i^{+}\right)$$

(4)

where $n$—number of network’s nodes; $Q_{in}^i$—flow entering junction I; $h_i$ piezometric head (or pressure) of the $i$^th node; $h_i^{min}$—minimum piezometric head (or pressure) for the $i$^th node; $h_i-h_i^{min}$—piezometric head (or pressure) surplus for the $i$^th node; $h_i^{+}$—piezometric head (or pressure) surplus threshold for the $i$^th node; $h_i^{max}$—maximum piezometric head (or pressure) for the $i$^th node.

In the presented formulation, the minimum piezometric head/pressure represents the target head/pressure value, also used in Todini’s resilience index formulation [2]. This value may be provided by regulations or inferred from the characteristics of the end-users supplied by the node to which it refers. The maximum piezometric head/pressure represents the value that the head/pressure should not exceed in the network’s operation to avoid damaging the infrastructure. The piezometric head/pressure surplus threshold represents a value beyond which the resilience of the node does not benefit from further increases in head/pressure.

3.5. Weighted Resilience Index

The topological and uniformity coefficients are two dimensionless multiplicative coefficients that can be integrated into different formulas for the assessment of resilience indices. The coefficients are calculated for each junction and multiply the numerator of (3). As they are formulated, in general, a resilience index that integrates these coefficients ($I_r^w$—weighted resilience index) should be lower than the original one:

$$I_r^w={\displaystyle \frac{{\sum }_{i=1}^n{\left(K_i^T{K}_i^U\right)·Q}_{in}^i{·\mathsf{\Delta }h}_i}{{\sum }_{i=1}^n{Q}_{in}^i{·\mathsf{\Delta }h}_i^{max}}}$$

(5)

4. Results

4.1. Case Study

This study utilizes variants of the Villa Rosa WDN, a section of the Northwest System in Tampa, Florida, USA, from a previous work [1]. Three network variants are analyzed:

• Normal Network: Current configuration serving Villa Rosa.
• Looped Network: 25 pipes strategically added to maximize loops while maintaining consistent diameters.
• Tree-like Network: 45 pipes removed to create a completely tree-like configuration, with each junction supplied by a single upstream pipe.

4.2. Simulation Settings

The WaterNetGen software [3] has been modified to compute the new formulation proposed for the resilience index, both in its simple (3) and weighted versions (5).

For the three variants, equal values of minimum (30 m) and maximum (60 m) pressure were used across the entire network to simplify the assessment of the proposed index’s sensitivity. For all the networks analyzed, the pressure surplus remains near 20 m, with all node pressures below the maximum pressure value chosen.

4.3. Indices Assessment

The analysis conducted on the three networks involves four different values for the pressure surplus threshold (h⁺): 15, 20, 22.5, and 25 m (

Table 1. Results of the three networks obtained with h⁺ = 15, 20, 22.5 and 25 m.

	Results for h+ = 15 m			Results for h+ = 20 m
Metric	Tree	Normal	Looped	Tree	Normal	Looped
%Nodes 1	100.00%	100.00%	100.00%	88.34%	97.55%	100.00%
Todini [2]	0.8813	0.8948	0.9179	0.8813	0.8948	0.9179
${\mathit{I}}_{\mathit{r}}$	1	1	1	0.995	0.999	1
$K_{avg}^T$$/K_{avg}^U$	0.5/0.5	0.638/0.632	0.698/0.636	0.5/0.5	0.638/0.632	0.698/0.636
${\mathit{I}}_{\mathit{r}}^{\mathit{w}}$	0.25	0.305	0.388	0.249	0.305	0.388
	Results for h+ = 22.5 m			Results for h+ = 25 m
Metric	Tree	Normal	Looped	Tree	Normal	Looped
%Nodes	41.72%	41.72%	80.98%	0.61%	0.61%	3.07%
Todini [2]	0.8813	0.8948	0.9179	0.8813	0.8948	0.9179
${\mathit{I}}_{\mathit{r}}$	0.97	0.98	0.995	0.901	0.913	0.94
$K_{avg}^T$$/K_{avg}^U$	0.5/0.5	0.638/0.632	0.698/0.636	0.5/0.5	0.638/0.632	0.698/0.636
${\mathit{I}}_{\mathit{r}}^{\mathit{w}}$	0.242	0.299	0.386	0.225	0.278	0.363

¹ Percentage of nodes that exceed the h⁺ value.

).

5. Conclusions

Assessing the resilience of WDNs is intricate and encompasses factors beyond those discussed here, such as tank redundancy, pumping stations, pressure-reducing valves (PRVs), and District Metered Areas (DMAs), and their analysis and integration into resilience assessments remain ongoing. The new formulations proposed for the resilience index have proven effective in considering multiple factors historically overlooked by classical resilience index formulations. This result has been achieved while maintaining a simple structure for the indices, ensuring their easy usability.