Generalized Resilience and Failure Indices for Use with 3 Pressure-Driven Modeling and Leakage

: In 2000, the resilience and failure indices were introduced as a convenient and compact tool to express respectively water-dis-6 tribution network (WDN) surplus and deficit in satisfying users ’ demand, in terms of delivered power. In their original formulation, the 7 mentioned indices, originally thought as WDN design tools, were developed only considering the demand-driven modeling approach, which 8 would include pumps but not leakage. This paper extends the formulation of both indices and presents a generalized expression, more 9 convenient for use when dealing with pressure-driven modeling and capable of including the effect of leakage. Following the original concept, 10 the generalized indices were developed by calculating the power dissipated in the network as a function of the difference between the total 11 power inserted through source nodes and pumps and the net delivered power, whereas the leakage-related power is considered as a loss 12 similarly to the internally dissipated one. Applications to WDN analysis and design proved that using the new formulation in the presence of 13 leakage and pressure-dependent consumptions yields better description of the delivered power excess, compared to the original demand-14 driven formulation and to another pressure-driven formulation present in the scientific literature. DOI: 10.1061/(ASCE)WR.1943-5452 15 .0000656. © 2016 American Society of Civil Engineers.

).An overall mea-28 sure of reliability is then obtained by averaging the performance 29 indicators calculated in each category of critical scenarios (Ciaponi 30 2009).As an example, if the network reliability related to pipe 31 breakage is considered, the ratio of delivered water discharge to 32 users' demand has to be assessed for each possible pipe break 33 in the network, as done by Giustolisi et al. (2008a) and Creaco et al. 34 (2012).Apart from identifying the segment that includes the 35 generic broken pipe and the network part that remains connected 36 to the source following the segment isolation, this requires perfor-37 mance of one pressure-driven simulation for each network segment.

38
Though applying this procedure is not a heavy task in the analysis 39 of WDNs, it may turn out to be too cumbersome in the optimization context, when it has to be reiterated for each solution proposed by the optimizer.
To avoid using performance indicators in the optimization context, various researchers have tried to formulate compact indices of reliability, which require a single network simulation for being evaluated while being good surrogates for the more cumbersome performance indicators.In this context, the pressure head/energy related indices, such as the pressure surplus by Gessler and Walski (1985) and the resilience index by Todini (2000), aim to express the network reliability in terms of service pressure excess compared to the minimum desired value guaranteeing the full demand satisfaction.In particular, the resilience index by Todini (2000) is calculated taking as benchmark a network configuration that delivers network demands with the minimum desired pressure head value at all nodes.In particular, it is the ratio of the excess of power delivered to users, to the maximum power that can be dissipated in the network when satisfying the demand.Incidentally, the latter corresponds to the total power introduced into the network through the source nodes and eventually present pumps minus the minimum power necessary to satisfy the demand.Later, some authors (e.g., Prasad et  With the objective to design a WDN, the resilience index, as proposed by Todini (2000), was formulated in the context of demand-driven modeling (Todini and Pilati 1988), where nodal water discharges are assumed always equal to nodal demands and leakage cannot be accounted for but in an approximate way.Indeed, a generalization to the pressure-driven modeling (Germanopoulos 1985;Wagner et al. 1988 proposed by Saldarriaga et al. (2010).However, this approach, which does not fully follow the original concept, has the drawback of including leakage outflows in the numerator of the proposed resilience index.This may result in an undesired increase in the index as leakage grows, as if leakage were something good that needs to be recovered.
Following the original definition, this paper presents a generalized expression of the resilience index to pressure-driven modeling, which is able to incorporate both leakage and pressure dependent outflows to users in a robust and sound way.Along with the resilience index, the failure index, originally proposed by Todini (2000), which concerns the system supplied power under pressure deficit operating conditions, is also generalized as an extension to the negative values of the resilience index.In the following sections, first the methodology is described, reporting the pressure-driven modeling used for the calculations and the form of the generalized resilience and failure indices.The results that prove the applicability of the indices in both the analysis and the optimization contexts follows.

Pressure-Driven Modeling Approach
Let us assume a generic network with n 0 source nodes with preassigned head and n 1 nodes with unknown head.Furthermore, let the network include n p pipes and n pumps pumps.In network resolution, vector H 0 (n 0 × 1) of preassigned heads (i.e., heads at the source nodes, reservoirs or tanks) and vector d (n 1 × 1) of demands at the n 1 unknown head nodes are generally known at each instant of network operation.In particular, in the case of a reservoir, the generic head H 0 is generally preassigned.In the case of a tank, instead, it is determined based on the series of tank inflows and outflows by applying the continuity equation to the tank.The generic nodal demand d is estimated by applying either the top-down or the bottom-up demand allocation approaches to the network users (Walski et al. 2003).Network resolution enables vectors Q (n p × 1), Q p (n pumps × 1) and H (n 1 × 1), associated with pipe water discharges, pump water discharges and unknown nodal heads respectively, to be calculated.This is accomplished by applying the following momentum and continuity equations to the n p network pipes and n pumps network pumps, and to the n 1 network nodes, respectively:) 5   can be accurately modeled.Vector q is calculated as

> < > :
175 where q user (n 1 × 1) and q leak (n 1 × 1) represent the outflow de-176 livered to the users and the leakage allocated to the nodes.The re-177 lationship between q user , d, and h takes on the following form: where matrix C user = diagonal matrix, whose generic element 179 C user ði; iÞ expresses the outflow/demand ratio q user =d for the users 180 at the ith node.According to the formulation by Wagner et al. (1988), which 182 finds its mathematical expression in Eq. ( 6), this ratio is equal 183 to 0 (i.e., nodal outflow q user ¼ 0) as long as the nodal pressure 184 head is lower than or equal to a threshold value h min .Starting from 185 h min , the ratio increases up to a value equal to 1, which means out-186 flow q user equal to users' demand, achieved when the nodal pres-187 sure head equals the threshold desired value h des .For nodal 188 pressure heads higher than h des , the ratio stays equal to 1, with 189 the users' demand being fully satisfied The exponent δ in Eq. ( 6) is generally set to 0.5.
where CLLðn p × 1Þ = vector whose ith element is equal to C L;i L i , The resilience (I rd ) and failure (I fd ) indices originally defined by 211 Todini (2000) through the demand-driven modeling approach take 212 on the following form, using the notation of this paper: Eq. ( 11) has the drawback of putting the vector of leakage outflows in the numerator, as if it were something good that would need to be recovered.
Hereinafter, the resilience and failure indices are generalized to the pressure-driven approach avoiding the flaw mentioned previously.
As a generalization of Todini (2000), the resilience index can be calculated as where user HÞ = actual amount of power dissipated in the network, through pipe resistances and leakage outflow, to supply the users; vector q user is evaluated through Eqs. ( 5) and ( 6), and Q 0 (n 0 × 1) = vector of water discharges leaving the source nodes (thus including leakages as well).
, the maximum power that would be dissipated in the network, under the theoretical condition of q user ¼ d and Following algebraic operations, the following relationship is obtained: Implicitly, it has to be underlined that Q T p H p in Eq. ( 13) also accounts for pumps working as turbines or turbines themselves installed in the WDN.In the case of pumps working as turbines and/ or turbines, the generic value Q p H p is negative, i.e., the device takes energy out of the WDN.In a similar way, for network tanks that receive water from the network, instead of releasing it, negative values of Q 0 H 0 would be obtained.
Network configurations for which q T user H < d T H des [i.e., they have negative numerator in Eq. ( 13)] are unsatisfactory in terms of power delivered to users.In fact, they have a deficit of power, rather than a surplus with respect to what is desired.Since the resilience index is meant to describe network redundancy, this index can be set to 0 for these networks.This results in the following relationship, which can be universally used for assessing the resilience index: Please note that q user , H, Q 0 and Q p have to be computed through a pressure-driven modeling approach.The function max in Eq. ( 14) is useful for getting a value of the resilience index I r equal to 0 in those network configurations that features a power deficit rather than a power surplus.Without this function, i.e., if I r were expressed like in Eq. ( 13), these configurations would feature illogical values of I r (sometimes even smaller than −1 or larger than 1), as will be shown in the applications.By inserting the max function, instead, the configurations with power deficit are assigned a null value of I r , while the entity of the power deficit is properly described through the failure index, whose definition follows.
Written as in Eq. ( 14), the resilience index always ranges from 0 to 1, for all the kinds of networks.The highest value of I r ¼ 1 is obtained for a theoretical network configuration with no leakage and energy dissipations along the pipes.
In a similar way, the failure index originally proposed by Todini (2000) can be generalized through the following relationship: Here, again, the quantities q user and H have to be computed through a pressure-driven modeling approach.The function min in Eq. ( 15) is useful for getting a value of the failure index I f equal to 0 in the network configurations that feature a power surplus rather than a power deficit and are better described through I r .
Written as in Eq. ( 15), the failure index always takes on values ranging from −1 to 0, for all the kinds of networks.In particular, values equal to 0 are obtained for network with no deficit of power, i.e., those that feature positive values of I r .Values lower than 0, instead, are obtained for networks with deficit of power, i.e., those that feature a value of I r ¼ 0. The lowest possible value I f ¼ − 1 is obtained when q user ¼ 0, with 0 being the zero vector, i.e., in a network supplying no water to all its users due to low service pressure conditions.
The main difference between the generalized resilience and failure indices [Eqs.( 14) and ( 15)], on the one hand, and the original ones [Eqs.( 9) and ( 10)], on the other hand, lies in the numerator of the former, where the vector q user of nodal outflows to users appears instead of the vector d nodal demands.Again, variables H and Q 0 in [Eqs.( 14) and ( 15)] are derived through pressure-driven modeling, whereas the corresponding ones in [Eqs.( 9) and (10)] are obtained through demand-driven modeling.
The continuity of I r and I f is shown in Fig. 1, as a function of power q T user H delivered to the WDN users.As formulated in this work, the indices are nonnegative and nonpositive respectively.
Furthermore, either index takes on values different from 0 if and only if the other is equal to 0. In light of this continuity, a generalized resilience/failure index (GRF) GRF ¼ I r þ I f can be used to give indications of the WDN power surplus/deficit.As a result of the definition of I r and I f , GRF equals I r , when the latter is larger than 0.
Otherwise, for network configurations under deficient power conditions for which I r ¼ 0, GRF is equal to the failure index I f , which always takes on nonpositive values.Index GFR can be profitably used in the optimization context, as will be shown hereinafter.

Snapshot Simulation and Extended Period Simulation
When a single scenario is chosen as benchmark, assessment of the resilience and failure indices is easily done through Eqs. ( 14) and (15), respectively.In the case of extended period simulation, a single value can be calculated for either index at each network operation instant.Wherefore, the characterization of the whole operation period can be carried out by calculating, for either index, the temporal average or the minimum, median, and maximum values.The cumulative Weibull frequency of either index can also be estimated.

Case Studies
Applications concerned three different case studies-a synthetic case study (Fig. 2) and two real case studies of different complexity 317 (Figs. 3 and 4).Given that the focus of this paper is mainly the 324 The first case study is the simple network of Alperovits and 325 Shamir (1977), made up of n 1 ¼ 6 nodes with unknown head, n 0 ¼ 326 1 node with preassigned head, n p ¼ 8 pipes and no pumps (Fig. 2).

327
The network was analyzed in a snapshot scenario representative of  The results of the first part of the applications are reported in Fig. 6.
Figs. 6(a and b) show how the resilience index I r and the failure index I f , as defined in this work, evaluated through the pressuredriven modeling, are affected by leakage increase.In particular, when the ratio of leakage to the whole outflow varies from about 9% to about 32% as a result of C L variation, I r decreases from 0.28 to 0.00.Starting from a leakage percentage equal to 32%, the network has no power surplus and starts to have power deficit.
Therefore, for leakage percentages larger than 32%, I r remains equal to 0. As for the failure index, I f stays equal to 0 when leakage percentage varies from 9 to 32%.Then, it starts decreasing down to about −0.19 when leakage changes from 32 to 50%.Overall, In the context of the comparison of I r with I rs , it is believed that, in light of its realistic estimate of the power supplied to the network users and due to its clearly decreasing trend as a function of leakage, I r is closer to the original rationale used by Todini (2000) for defining the resilience index.In fact, the resilience index was conceived in order to yield indications of how much power reserve remains available in a certain network configuration, for facing eventual occurrence of critical scenarios still satisfying users' requests.Therefore, it is intuitive to think that an increase in leakage, and then in power dissipation through leakage, must always lead to a waste in the power reserve, and then to a decrease in the resilience index.This behavior is remarked in I r and is missed by I rs .In addition, the newly defined failure index I f helps in providing    .This happens because nodal heads are not bounded below in demand-driven network simulations (nodal outflows do not decrease with service pressure decreasing).In pressure-driven simulations, instead, pressure heads cannot go below h min , as nodal outflows are set equal to 0 for h < h min .The highest value achieved by I rd þ I fd is close to 0.9, which is much larger than the highest value of GRF in 7 Fig.7(a).This difference happens because, by including leakage in the numerator, I rd considers leakage outflows inside the good power delivered to the users.Therefore, I rd þ I fd is a wrong estimate of the real power delivered to the users.In order to be able to compare directly the optimization solutions (pressuredriven modeling) with the BO solutions (demand-driven modeling), the latter were reevaluated using the pressure-driven modeling.
In particular, for each BO solution, a pressure-driven network simulation where nodal outflows were calculated considering Eqs. ( 4)-( 8) was performed.The pressure driven-related I r and I f associated with each solution were then calculated through Eqs. ( 14) and (15), respectively.The reevaluated BO solutions were then plotted in terms of network cost and GRF ¼ I r þ I f in Fig.An analysis was then carried out to show that I r would take on illogical values for power deficient network configurations, if the numerator in Eq. ( 14) were not bounded below by 0. In Fig. 8(a), the I r value calculated through Eq. ( 13), which is unbounded below, was plotted against the network cost for the optimization so-  14) and (15), respectively, in an attempt to represent network configurations with power surplus and deficit in a separate way.
The remarks made previously also justify the introduction of the  1.

566
The results of the analysis are plotted in the graphs in Fig. 9.In the GRF distribution tends to be very asymmetric with tail towards the lower values, symmetric and very asymmetric with tail towards the higher values, in the scenarios of the three categories defined previously, respectively.However, in the scenario of the second category, the variance is smaller than in the others.By summing up these results, all the scenarios feature most of the GRF values close to the median, which can then be considered as the most representative value of GRF to represent synthetically the power conditions in the network.

Third Case Study
As for the third case study, the leakage percentages obtained as a function of the network age are reported in Table 2. Figs. 10     (a and b) report, for the third case study, the same kind of results as Figs.9(a and b), respectively.The main difference between the results of the two case studies lies in the fact that, in the third case study, GRF is always positive during the day in all the seven scenarios.Furthermore, according to the three categories of scenario defined previously, all the scenarios of the third case study belong to the first category, with distribution of GRF values very asymmetric with tail towards the lower values.The fact that the network in the third case study is never under power deficit conditions is due to suitable performance indicators (Gargano 20 and Pianese 2000; Tanyimboh et al. 2001; Ciaponi 2009; Creaco 21 and Franchini 2012).These indicators relate the water discharges 22 delivered to network users, to their demands under critical opera-23 tional scenarios, which occur due to either mechanical (pipe break-24 age, pump failure, power outages, control valve failure, etc.) or 25 hydraulic (such as changes in demand or in pressure head, aging 26 of pipes, inadequate pipe sizing, insufficient pumping capacity, in-27 sufficient storage capability) failure al. 2003; Raad et al. 2010; Pandit and Crittenden 2012; Cimellaro et al. 2015) proposed other definitions of the resilience index.In particular, Prasad et al. (2003) and Raad et al. (2010) incorporated into the original index by Todini (2000) the uniformity of pipe diameters and water discharges respectively, in order to obtain a better representation of network reliability.However, these modified versions have the drawback of corrupting the original physical meaning of the resilience index.To preserve this physical meaning while taking account of the uniformity of pipe sizes in the WDN, Creaco et al. (2015) proposed using an additional loop diameter uniformity index, along with the original resilience index, in the optimization context.Their calculations proved that dealing with resilience index and loop diameter uniformity as separate objective functions helps in obtaining a more comprehensive representation of the network reliability.Nevertheless, despite positive correlation with reliability (Atkinson et al. 2014), the resilience index is a global index and, as such, can only give an overall and the network operating conditions.In fact, the local demand shortfalls that occur under the usual operating conditions or critical scenarios, such as those associated with segment isolation or open hydrant(s), cannot be detected through the resilience index and require the burdensome calculation of performance indicators.

Q
Li in the ith pipe, to be allocated to either end node, can be cal-193 culated as 194where C L;i , L i , and h a;i = leakage coefficient, the length and the 195 average pressure head in the generic pipe, respectively; n leak = leak-196 age exponent, which generally takes on values within the range 197 [0.5, 1.5] (Van Zyl and Cassa 2014).Vector q leak of leakage 198 allocated to the unknown head nodes can be expressed in the fol-199 lowing compact vector form, derived from Creaco and Pezzinga 200 (2015b, a):

202 and h (n 1 × 1 )(n 1 × 1 )
and h 0 (n 0 × 1) = vector of pressure heads in the 203 unknown and fixed head nodes, respectively.Incidentally, these 204 vectors can be obtained from H and H 0 , by subtracting z 205 and z 0 (n 1 × 1), vectors of ground elevations for the un-206 known and fixed head nodes, respectively.In Eq. (8), the division 207 by 2 and the exponent n leak apply to each element of the matrices.208 Resilience and Failure Indices in the Pressure-Driven 209 Modeling Approach 210

213
Whereas the failure index has never been generalized to the 214 pressure-driven modeling, a generalization of the resilience index 215 was proposed by Saldarriaga et al. (2010) in the presence of leakage 216 and in the absence of pumps.In particular, using the notation of this 217 paper, the Saldarriaga et al. (2010) resilience index I rs (where s 218 stands for Saldarriaga) takes on the following form: 318assessment of the benefits of the new extended resilience index for-319 mulation, water demand is assumed perfectly known and the usual 320 network operation with no failure is considered in all the case stud-321 ies.Furthermore, when network design is performed, it is done in 322 one step, without considering the phasing of construction in time 323 (seeCreaco et al. 2014).

1 Fig. 1 .Fig. 2 .
328 the peak demand.As in the original paper, a Hazen Williams rough-329 ness coefficient equal to 130 m 0.37 s −1 was used for all network 330 pipes.The data relative to the preassigned head at the source node, 331 nodal demands and pipe lengths can be found in the original paper.332 In the present work, values of h min and h des equal to 5 and 30 m 333 respectively were considered for the calculations.The leakage ex-334 ponent n leak in Eq. (8) was set to 1.18, as was done by Pezzinga and 335 Pititto (2005).This value lies in the range [0.5, 1.5] of typical val-336 ues and is mainly associated with the presence of longitudinal 337 cracks in plastic pipes (Van Zyl and Cassa 2014).The choice of 338 such a simple network as first case study is motivated by the ne-339 cessity of facilitating the analysis of the results.This was done in 340 light of the focus of the paper, which is to present expressions for 341 assessing the resilience and failure indices in the pressure-driven 342 modeling approach.343A first application was carried out to show how the resilience 344 and failure indices proposed in this paper vary when leakage 345 percentage changes.This was done by initially considering, for ex-346 plicative purposes, a network configuration with uniform pipe 347 diameters equal to 457.2 mm.To obtain different leakage outflows, 348 pipe leak coefficients C L;i were modified uniformly in the network.349 In particular, the network leakage coefficient C L was set to various 350 values within the range ½5 × 10 −8 ; 1 × 10 −6 m 0.82 s −1 .Though the 351 wide range adopted for C L extends beyond the usual values of real 352 water loss, it helps, on the one hand, in fully describing the unac-353 counted-for water, which also includes apparent losses such as 354 theft, meter underregistration, unmetered users, flushing, firefight-355 ing.On the other hand, it enables clearly analyzing how the newly F1:Ranges of validity for the resilience and failure indices in terms F1First case study: network of Alperovits and Shamir (1977); F2:2 node IDs close to the nodes; pipe IDs inside the brackets [] (adapted F2:3 from Alperovits and Shamir 1977) application, the two-objective design of the net-359 work was carried out to minimize the total network cost (sum 360 of pipe costs) and maximize GRF.As in the original problem presented by Alperovits and Shamir (1977), the pipe sizes were considered the decisional variables for the design.In this work, the same pipe costs per unit length as a function of the pipe diameter, as those defined by Alperovits and Shamir (1977) without any cost unit, were used.In the context of the two-objective network design, an optimization was carried out using the NSGAII algorithm (Deb et al. 2002) and considering a uniform value C L ¼ 5 × 10 −8 m 0.82 s −1 in network pipes.The second case study of the paper concerned a network featuring n 0 ¼ 1 source node, n 1 ¼ 70 nodes with outflow and n p ¼ 95 pipes (Fig. 3).The network, which represents the water distribution system serving a town in northern Italy, features a total end-to-end length of about 14 km.The pipe and nodal characteristics of this network are reported in the work by Creaco et al. (2012).Unlike the latter work, in which the demands were allocated along the network pipes, in the present work they are allocated to the network nodes with unknown head.The whole network peak demand is 15 L=s, including about 20% of leakage.In the calculations, the pipe resistance was modeled through the Manning formula.Like in the first case study, the leakage exponent n leak in Eq. (8) was set to 1.18.The network performance in terms of I r and I f (i.e., GRF) was analyzed in seven extended period scenarios, each of which aimed at representing the day of peak demand.The scenarios, aimed at representing various ages in the network, differed in the values of the network leakage coefficient C L and of the pipe Manning roughness coefficient.As Table 1 shows, the scenarios, associated with growing network ages from 0 to 60 years, featured C L values ranging

F3: 1 Fig. 3 . 1 Fig. 4 . 4 .
Second case study: reference network of a town in northern Italy; node IDs close to the nodes; pipe IDs inside the brackets []; in Node 71, F3:2 supply of a district F4:Third case study: reference network of a city in northern Italy 5×10 −9 m 0.82 s −1 (estimated value for the real network) to 389 1.58 × 10 −8 m 0.82 s −1 .In each scenario, the Manning coefficient 390 value of each pipe was obtained starting from the real one, reported 391 by Creaco et al. (2012), by adding the quantity 0.00015× 392 ðnetwork ageÞ) up to a maximum value of 0.015 m −1=3 s.This 393 was done to account for pipe deterioration as time goes by.The 394 trend of the hourly demand coefficient reported in Fig. 5 was 395 assumed valid in all the scenarios.396 The third case study of the paper concerned a network featuring 397 n 0 ¼ 2 source nodes, n 1 ¼ 536 nodes with outflow and n p ¼ 825 398 pipes (Fig. 4).The network, which represents the water distribution 399 system serving a part of a city in northern Italy (Creaco and 400 Franchini 2012), features a total end-to-end length of about 90 401 km.In this network layout, all nodes have a ground elevation of 402 0 m a.s.l., and the two source nodes have heads at 30 m a.s.l.403 The whole network peak demand is 367 L=s, including about 404 20% of leakage.In the calculations, the pipe resistance was mod-405 eled through the Manning formula.Like in the other case studies, 406 the leakage exponent n leak in Eq. (8) was set to 1.18.Like in the 407 second case study, the network performance in terms of I r and I f 408 (i.e., GRF) was analyzed in seven extended period scenarios, rep-409 resenting the peak daily demand at various network ages.The differ-410 ences between the scenarios in terms of leakage coefficient C L and 411 pipe roughness, both assumed uniform over the network, are shown 412 in Table 2.The same trend of hourly demand coefficient (Fig. 5) as 413 the second case study was also used for the third case study.414 Results 415 First Case Study-Network Simulation 416 As far as the first case study is concerned, the leakage coefficient 417 C L variation within the range ½5 × 10 −8 ; 1 × 10 −6 m 0.82 s −1 produced leakage percentage rates within the range [9-50%].

Figs. 6 (
Figs. 6(a and b) give a numerical proof of the continuity of I r and I f , which was shown in Fig. 1 in a qualitative way.The behavior of I r and I f is because the nodal pressure heads and delivered powers decrease with the leakage outflow, and then the whole outflow Q 0 [Eq.(14)] increasing.A comparison was then made between the resilience index I r defined hereinbefore and that defined by Saldarriaga et al. (2010) [I rs in Eq. (11)].As Fig. 6(a) shows, I rs takes on values within the range [0.7, 0.8].These values are much larger than those of I r , within the range [0, 0.3].This happens because leakage appears in the numerator of I rs .Therefore, the latter index provides an unrealistic estimate of the power supply delivered to the users.Furthermore, I rs stays almost constant, not being strongly influenced by leakage increase.In particular, it increases a little bit up to a leakage percentage of about 25%.Then, after a plateau, it slightly decreases.The existence of these two trends depends on the fact that up to a leakage percentage of about 25%, the outflows to the users stay almost equal to the demands [see Fig. 6(c)] in Fig. 6 reporting performance indicator mean (q user =d) related to the satisfaction of the users' demand).When leakage percentage further increases, the outflow to the users starts decreasing, because of some nodal pressure heads h being lower than h des , and this results in a decrease in I rs .

Fig. 5 .
Hourly demand coefficient used for the calculations in the sec-F5:2 ond and third case studies

1 Fig. 6 .F7: 1 Fig. 7 .
second part of the applications, the results of the opti-471 mization is reported in the graphs in Fig.7.Fig.7(a) shows that, as 472 expected, the Pareto front obtained in the optimization is limited 473 between values of −1 and 1 of objective function GRF ¼ I r þ 474 I f .Fig. 7(a) also shows the leakage percentage rate on the second 475 vertical axis.This rate equals 100% in the optimization solution 476 featuring GRF ¼ − 1, in which all nodal outflow is leakage 477 and there is no flow delivered to users.As the network cost grows, 478 it rapidly decreases up to a minimum value.Then, it grows rapidly 479 again and stabilizes around a value close to 11%, which is also the 480 average percentage rate over the optimization solutions.For the 481 sake of comparison, a benchmark optimization (BO) where 482 the resilience and failure indices were evaluated following the de-483 mand-driven approach [Todini 2000 Eqs.(9) and (10)] was carried 484 out.In BO, nodal demands were then considered to be pressure 485 independent.Compared to the users' demands in the optimization, 486 the demands in BO were increased by 12%, in order to account, in 487 all the BO solutions, for a constant (pressure independent) leakage 488 rate of 11%, equal to the average leakage percentage over the op-489 timization solutions.In fact, in the demand-driven approach (where 490 q ¼ d), leakage cannot be expressed as a function of nodal pressure 491 heads and it has to be fixed a priori in an approximate way.The 492 Pareto front obtained in BO was reported in Fig. 7(b).Unlike 493 the Pareto fronts of the optimization, which only include values 494 of GRF ¼ I r þ I f larger than or equal to −1, the Pareto front of 495 BO does not have a lower boundary (though the graph is bounded F6:First case study: as a function of leakage percentage compared F6:2 to the whole outflow, trends of (a) resilience index, as was defined by F6:3 Saldarriaga et al. (2010) (I rs ) and as is defined in this work (I r ); (b) fail-F6:4 ure index I f as is defined in this work; (c) performance indicator mean F6:5 (q user =d) First 10 case study: (a) Pareto front obtained in the optimization F7:2 and leakage percentage rate in the various solutions; (b) Pareto front F7:3 obtained in the benchmark optimization (BO); (c) comparison between F7:4 the optimization solutions and reevaluated BO solutions 7(c), along with the optimization solutions.The comparison in this graph highlights the fact that the solutions of the benchmark optimization BO are dominated by the optimization solutions up to a network cost value close to 600 8 ,000.Furthermore, using the demand-driven approach and including leakage in an approximate way leads to wrong assessment of the minimum cost solution which guarantees h > h des at all nodes (cost ¼ 541,000 from the benchmark optimization BO, instead of cost ¼ 464,000 9 from the pressure-driven optimization).Fig. 7(c) shows that the GRF values of the optimization and BO solutions are closer (almost coincident) when the optimization solutions feature a leakage rate close to 11% [see Fig. 7(a)], which is equal to the constant pressure-independent leakage rate assumed in BO.More evident differences appear when the optimization solutions feature leakage percentage rates far from 11%.In general, for any value of leakage rate chosen for BO, the curve of reevaluated BO solutions is close to the optimization Pareto front in correspondence to the optimization solutions featuring a close leakage rate.Discrepancies occur in correspondence to the optimization solutions featuring a different leakage rate from that assumed in the BO optimization.In fact, it is not possible to pick a single value of leakage rate that enables the curve of reevaluated BO solutions to be close to the optimization Pareto front over the whole Pareto front length.

11 ,
lutions.This figure clearly shows that, without the lower boundary, I r takes on illogical values smaller than −1 and larger than 1, for power deficient network configurations (i.e., network costs lower than 400 000).Unlike GRF [see Fig.7(a)], the unbounded I r is not a monotonic function of the network cost in the region of the power deficient network configurations.Fig. 8(b) shows the values of the unbounded I f , obtained neglecting the min function in Eq. (15), for the optimization solutions.Unlike the unbounded I r , the unbounded I f has a regular monotonic trend.However, it features smaller (positive) variations (within the range [0, 0.11]) than GRF [see Fig. 7(a)] and this fact prevents it from properly differentiating the solutions with power surplus.The loss of physical meaning for the unbounded I r and the small positive variations in the unbounded I f corroborate the definition of I r and I f through Eqs. ( the second case study is concerned, the leakage percent-564 age and the average value of the pipe Manning roughness coeffi-565 cient obtained as a function of network age are reported in Table

1 Fig. 8 .
567 particular, Fig. 9(a) shows how GRF varies during the day in 568 the various scenarios, which are representative of network aging 569 and then feature growing values of pipe resistance (from the actual 570 values up to 0.015 m −1=3 s) and of leakage percentage (from the 571 actual 29-57% of the total outflow).Overall, the daily trend of 572 GRF tends to lower, as the network gets older.For network ages 573 larger than 20 (i.e., in Scenarios 4-7), GRF even takes on negative 574 values (I r ¼ 0 and I f < 0), representative of power deficit.In all the 575 scenarios, the values of GRF are always positive at nighttime since 576 the network is always able to deliver sufficient power (I r > 0 and 577 I f ¼ 0) in this part of the day, which features low nodal demands.578 Furthermore, at nighttime GRF always takes on values close to 0 579 because of the low values of the power delivered to the users com-580 pared to the power leaving the source node (which also includes the 581 power dissipated through leakage).Three different categories of 582 scenario can be distinguished in Fig. 9(a).The first category in-583 cludes Scenarios 1 and 2, in which GRF tends to be larger in 584 the day than at nighttime.The second includes only Scenario 3, 585 in which GRF is always close to 0 throughout the day.Finally, 586 the third includes Scenarios 4-7, in which GRF tends to be lower 587 in the day than at nighttime.The reason for the different behaviors 588 of GRF lies in the fact that, in each scenario of the first category, the 589 network power redundancy globally prevails over the network F8:(a) I r and (b) I f calculated 12 neglecting the max and min function F8:2 in Eqs.(14) and (15), as a function of network cost for optimization F8terms of size and duration.In the scenario of the second category, the duration and size of network redundancy and deficit compensate for each other.Finally, in each scenario of the third category, the network deficit prevails over the network redundancy.Fig. 9(b) reports, for the each scenario, the box plot of the GRF values in the day.The analysis of the box plots shows that 617 the larger redundancy of the network in terms of loops and pipe 618 sizes compared to the network of the second case study.619 Conclusions 620 In this paper the resilience and failure indices, originally proposed 621 by Todini (2000) using demand-driven modeling, were extended to 622 pressure-driven modeling also accounting for leakage.This was 623 done by defining a new generalized resilience/failure, following 624 the original definition in terms of available and delivered power.625 Besides deriving pipe water discharges and nodal heads through 626 pressure-driven modeling, the new formulation requires power loss 627 due to leakage to be excluded from the power delivered to satisfy 628 users demand.Applications to the WDN analysis showed that, 629 thanks to the formulation adopted, the indices describe properly 630 the variations in the power delivered to the users, as the ratio of 631 leakage to whole network outflow changes.The generalized indices 632 proved to be more sensitive to leakage variations than the pressure-633 driven resilience formulation proposed by Saldarriaga et al. (2010).634 Statistics on the index values can also be useful to analyze the net-635 work operation in an extended period simulation and the median 636 appears to be the most representative value of GRF to describe 637 synthetically the WDN power conditions.Applications to the mul-638 tiobjective design in the presence of pressure-dependent outflow 639 proved that considering the generalized indices yields benefits.640 In fact, network configurations are obtained, which dominate, in 641 terms of cost and delivered power, those obtained using the indices

F9: 1 Fig. 9 .
Second case study: (a) index GRF ¼ I r þ I f during the day in F9:2 the various scenarios; (b) for each scenario, box plot of the GRF values F9:3 during the day F10:1 Fig. 10.Third case study: (a) index GRF ¼ I r þ I f during the day in F10:2 the various scenarios; (b) for each scenario, box plot of the GRF values F10demand-driven modeling, where pres-643 sure dependent outflows are considered in an approximate way.644 Summing up, in light of the current tendency to prefer the pres-645 sure-driven approach to the demand-driven one in the modeling of 646 WDNs, it is expected that the generalized indices may replace the 647 original ones in most applications.

Table 1 .
Second Case Study: Network Age, Leakage Coefficient C L , C L ðm 0.82 s −1 Þ

Table 2 .
Third Case Study: Network Age, Leakage Coefficient C L , Leakage Percentage and Pipe Manning Coefficient Associated with