The aim of this paper is to show that energy surplus indices, such as resilience index, besides providing a very good indirect measure of water distribution network reliability, also represent a valuable and effective indicator of network robustness under alternative network scenarios. It can thus be profitably used for network design under conditions of uncertain future demands. The methodology adopted consisted of: (a) multi-objective design optimization performed in order to minimize construction costs while maximizing the resilience index; and (b) retrospective performance assessment of the alternative solutions of the Pareto front obtained, under demand conditions far from those assumed during the design phase. Two case studies of different topological complexity were considered. Results showed that the resilience index, which is one of the most effective indirect indices of reliability, represents a very good measure of robustness as well.

INTRODUCTION

Climate change, population growth and increasing urbanization, make it not an easy task for technicians to define future water demand at water distribution network nodes. It is therefore very useful during the design phase to make use of effective indices characterizing the capability of the network to adequately perform even under demand conditions far from those assumed during the design phase (which are however the most critical working conditions that are reasonably possible to expect during the infrastructure technical life). This capability is known in the literature as robustness (Hashimoto et al. 1982a).

It is the authors' opinion that the robustness of a water distribution system design can be linked to its hydraulic reliability, which is classically defined as its capacity to fully satisfy users' demand in a given period of time (Hashimoto et al. 1982b).

Several specific performance indicators can be adopted for the direct estimation of service reliability (Gargano & Pianese 2000; Tanyimboh et al. 2001; Ciaponi 2009; Creaco & Franchini 2012) under critical operation scenarios such as segment isolation due to pipes bursts. The ratio of supplied water discharge to users' demand is a representative example of these performance indicators. For complex real networks, the evaluation of these performance indicators can unfortunately be a computationally heavy task since it entails the execution of numerous pressure-driven hydraulic simulations.

Various indirect indices of reliability, such as resilience (Todini 2000) and entropy (Tanyimboh & Templeman 2000), have been devised in recent years in order to limit the computational effort for network reliability assessment in the context of the design phase. These indices are conceived in such a way as to express the redundancy of the network under benchmark operation conditions.

In this context, Tanyimboh et al. (2011) and Greco et al. (2012) investigated which indirect index is more correlated to the network reliability, retrospectively estimated by direct performance indicators evaluation, and arrived at contrasting results: Tanymboh et al. indicate the entropy as the best indirect reliability measure while for Greco et al. resilience has to be preferred.

According to the authors’ experience (Creaco et al. 2014), resilience, which is linked to the energy redundancy of the network, is the more advisable indirect reliability index for both simple and complex networks while entropy, which is linked to the uncertainty characterizing the paths within the network that bring water to each node, proved not to be, in itself, a good and consistent indirect measure of network reliability. Similar results were achieved by Raad et al. (2010). More recently, Creaco et al. (2015) showed that a more complete estimation of network reliability can be obtained by combining the network resilience with the loop diameter uniformity.

Among the indirect indices of reliability, resilience index is believed by the authors to be a good surrogate index of design robustness as well. This paper is aimed at demonstrating the validity of the latter belief.

MATERIALS AND METHODS

The methodology used consists of two phases.

In the first phase, a multi-objective design optimization, aimed at minimizing network total cost while maximizing resilience index, is performed on a water distribution network employing the well-known Non-dominated Sorting Genetic Algorithm II (NSGA-II) multi-objective algorithm (Deb et al. 2002) and adopting, as decisional variables, the network pipe diameters. The results of the optimizations are Pareto fronts of optimal alternative solutions featuring increasing values of cost and resilience index.

The index of resilience, IR, is related to the hydraulic head surplus at network nodes compared to the minimum required heads to fully satisfy nodal demands under ordinary operating conditions. This head surplus represents an ‘energy reserve’ that can be dissipated under critical operational conditions, such as segment isolations or unforeseen nodal demand increase (which cause an increase in head losses), preventing or limiting water supply to users from being affected (Fortunato et al. 2012).

IR is defined as: 
formula
1
where Pint is the hydraulic power dissipated by the water flowing through the network and Pint max is the maximum hydraulic power that can be dissipated within the system while the minimum heads at network nodes to supply the required demands are still met. Pint is given by the difference of the total hydraulic power that enters the network, Ptot, and the total hydraulic power provided to the users, Pext: 
formula
2
Ptot is in turn expressed by the following equation: 
formula
3
where Qk and Hk, respectively are the water flow entering the network and the hydraulic head at the k-th source node, nr is the number of source nodes and γ is the specific gravity of water.
The hydraulic power provided to the users is given by the equation: 
formula
4
wherein hi and qi are the actual head and supplied flow at the i-th node, respectively, and nn is the number of nodes of the network. During the design phase and under ordinary operating conditions, qi is equal to nodal demand, di.
Pint max is finally defined as: 
formula
5
where himin is minimum head at the i-th node to supply the required demand, di.

Under ordinary operating conditions, since constraints hihimin and qi = di are prescribed for all nodes during the design phase, it holds that PintPint max. IR can then only take on positive values and ranges within the interval [0, 1]: it cannot ever be strictly equal to 1 as this would imply absence of energy dissipations in the network.

In the second phase, in order to evaluate the extent to which the resilience index provides good representation of design robustness as cost grows, the performance of all the optimal solutions of the Pareto front is assessed a posteriori. In particular, following the approach proposed by Creaco & Franchini (2012), the demand satisfaction rate, S, representative of network robustness, is evaluated for each design configuration, under operational scenarios featuring different nodal demands from those assumed in the first phase. As a result of this, each design configuration, and then each configuration cost, is associated with demand satisfaction rate values assessed under various operation scenarios. In fact, this enables a relationship to be created between network cost and robustness. By analysing this relationship, water utility managers can make a more aware choice of the final design configuration, in the trade-off between the growth of their investment and the corresponding benefits in terms of network robustness.

The demand satisfaction rate, S, is defined as the ratio of total actual water flow supplied to users at network nodes, Q, to the total water demand, D: 
formula
6
where qi is the actual water flow supplied to users at node i, calculated via head-driven hydraulic simulation, and di is the water demand. The relationship between qi, di and pressure head hi at node i can be expressed as follows (Wagner et al. 1988): 
formula
7
where hmin,i is the minimum pressure head required to fully satisfy the nodal demands and h0,i is the minimum pressure head required to enable any nodal outflow; the exponent γ is generally set to 0.5.

CASE STUDIES

Two case studies were considered. The first one is the rather simple network of Tanyimboh et al. (2011), made up of 11 nodes with outflow, all with ground elevation of 0 m, and 17 pipes, all 1,000 m long with Hazen–Williams roughness coefficient equal to 130. This value is typical for plastic pipes, whose roughness does not vary significantly during the network useful life. The network features only one source node with 100 m pressure head and a global peak demand of 444.5 l/s (Figure 1). The minimum required pressure head for full demand satisfaction is 30 m.
Figure 1

Case study 1. Network of Tanyimboh et al. (2011).

Figure 1

Case study 1. Network of Tanyimboh et al. (2011).

The second case study is the distribution network serving the part of the city of Ferrara (Northern Italy) lying inside the medieval walls (Creaco & Franchini 2012). This network features 536 nodes with outflow and 825 pipes with a total length of about 90 km (Figure 2).
Figure 2

Case study 2. Network serving the part of Ferrara city, Italy, lying inside the medieval walls (Creaco & Franchini 2012).

Figure 2

Case study 2. Network serving the part of Ferrara city, Italy, lying inside the medieval walls (Creaco & Franchini 2012).

The whole network peak demand of 367 l/s is supplied by two source nodes. In the network layout adopted, all nodes have a ground elevation of 0 m, and both the source nodes have a hydraulic head of 30 m. The roughness coefficients considered within the design phase are those relative to old cast iron pipes. In particular, a Manning coefficient equal to 0.015 s/m1/3 was chosen in order to represent the condition of the cast iron pipes at the end of the useful life of the network. The minimum required pressure head for full demand satisfaction was set to 25 m.

Pipe diameters and unit costs considered for the design applications are reported in Table 1.

Table 1

Pipe diameters, d, and unit costs, c, adopted during the design phase for both case studies

d [mm]c [€/m]
45 185 
60 203 
80 227 
100 231 
150 272 
200 299 
250 328 
300 360 
350 399 
400 439 
d [mm]c [€/m]
45 185 
60 203 
80 227 
100 231 
150 272 
200 299 
250 328 
300 360 
350 399 
400 439 

RESULTS AND DISCUSSION

The multi-objective design optimization was applied to both case studies, using a population of 200 and 1,000 individuals in the first and second case study, respectively. The total number of generations was set at 200 and 5,000 in the first and second case study, respectively. The choice of the number of individuals and generations was made following a preliminary investigation, whose results are not herein reported and which was aimed at analysing the trade-off between computation burden and accuracy of the results. Inside the algorithm NSGA-II, integer numbers were used for encoding pipe diameters. In the generation of the offspring population, 90% of the individuals were generated through uniform cross-over; the remaining 10% were drawn from the parent population, with their genes undergoing mutation in 1% of the cases.

The Pareto fronts of optimal solution, IR versus network cost, reported in Figures 3 and 4 were finally obtained. After the optimization was carried out for both case studies, the alternative optimal network configurations were retrospectively evaluated in terms of demand satisfaction rate, as described in the ‘Materials and methods' section. For the retrospective analysis, two scenarios were considered, in which nodal demands were increased by 50% and 100%, respectively, in comparison with the design scenario.
Figure 3

Case study 1. On the left, Pareto front of optimal solutions in the cost C – resilience IR space. On the right, solutions of the front re-assessed in terms of the ratio of the water discharge Q supplied to the users to the network demand D.

Figure 3

Case study 1. On the left, Pareto front of optimal solutions in the cost C – resilience IR space. On the right, solutions of the front re-assessed in terms of the ratio of the water discharge Q supplied to the users to the network demand D.

Figure 4

Case study 2. On the left, Pareto front of optimal solutions in the cost C – resilience IR space. On the right, solutions of the front re-assessed in terms of the ratio of the water discharge Q supplied to the users to the network demand D.

Figure 4

Case study 2. On the left, Pareto front of optimal solutions in the cost C – resilience IR space. On the right, solutions of the front re-assessed in terms of the ratio of the water discharge Q supplied to the users to the network demand D.

The results of the analysis for the first and the second case study are presented in Figures 3 and 4, respectively. From these graphics, it is possible to remark that resilience maximization yields network design configurations robust enough to guarantee values of demand satisfaction rate S sufficiently high even for nodal demands 100% larger than the original ones. As a result of this, the resilience index can be considered to be a good measure of design robustness.

In particular, starting from the minimum cost solutions of the Pareto fronts and keeping on the left of the characteristic ‘knee’, it is possible to obtain a large increase in S with relatively small additional investments. This means that it is always profitable to choose solutions that are slightly more expensive than the minimum cost solutions, since the small increase in the investment is paid back by an increase in the network robustness.

In the first case study, solutions with resilience index just above 0.60 guarantee demand satisfaction levels larger than 85%, in the most critical scenario considered. In the second case study, for a resilience index equal to 0.6 and for an increase in nodal demands by 100%, S is already close to 95%, and then is larger than the S value observed in the first case study in correspondence to the same resilience value. This difference is most probably due to the higher topological complexity and redundancy of the network of Ferrara, which makes design configurations intrinsically more resilient and robust compared to the simple network of the first case study.

At this stage, a remark has to be made about the possibility of extending the methodology described to tackle more complex problems than that considered here, in which only network diameters are optimized. In more complex problems, additional aspects, such as those related to the tank filling/emptying processes and to water quality (e.g. flow velocities, detention times, and chlorine concentration), could also be taken into account. For the extension of the methodology, extended period simulations could be preferred to snapshot simulations and other decisional variables, such as tank sizes, could also be considered.

CONCLUSIONS

In this a paper, a two steps methodology was set up in order to prove that the index of resilience, which is generally used as a surrogate of network reliability, also represents an effective indicator of the network robustness, which is the network's capability to fulfil users' demands under demand conditions different from those assumed in the design phase.

The first step of the methodology consisted in a network design multi-objective optimization performed in a reference benchmark demand scenario and aimed at minimizing costs and maximizing resilience. The subsequent second step entailed retrospectively assessing the solutions of the optimal Pareto front in other demand scenarios, in terms of demand satisfaction rate.

Results pointed out that optimal solutions featuring increasing resilience also feature an increasing demand satisfaction rate value, thus indicating a positive correlation between the resilience index and the network robustness.

The methodology can be successfully used by water utility managers in the framework of network design, in order to understand the extent to which increasing investments, in comparison with the minimum cost that enables demand to be met, leads to profitable benefits in terms of network robustness.

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