SYNTHETIC INDICES OF ROBUSTNESS OF WATER DISTRIBUTION

NETWORKS

Armando Di Nardo, Roberto Greco and Giovanni Francesco Santonastaso

Dipartimento di Ingegneria Civile, Seconda Università di Napoli, Aversa, CE, (Italy)

roberto.greco@unina2.it

Abstract

Entropy and resilience have been evaluated as indices for measuring robustness of water distribution networks.

The effects on network performance, caused by the failure of one or two links, have been evaluated by means of

several indices for two medium sized water distribution networks serving two towns in Southern Italy. All the

possible network layouts obtained by suppressing one or two links have been studied, excluding only the cases in

which disconnection of some nodes from the remaining part of the network occurred. The hydraulic simulations,

carried out with a demand-driven approach by means of the EPANET2 software, have shown that, unlike

entropy, resilience may represent a useful index of network robustness with regard to link failures.

Keywords Entropy, resilience, robustness, performance indices, water supply network

1. INTRODUCTION

Several design procedures aiming at maximizing hydraulic network reliability, usually defined as its ability of

performing its required functions under stated conditions for a specified period of time, have been proposed in

the literature. A common shortcoming of most of them, besides the computational burden, is that they rely upon

the knowledge of the probability of the considered unfavourable events, such as pipe failures or critical demand

distributions. Indeed, few data sets exist allowing either to calculate the probability of mechanical components

unavailability or to calibrate the probability distribution of water demand. The extrapolation of the obtained

probabilities to networks others than those to which the available data originally refer seems questionable. Thus

far, in the studies about water distribution network reliability only some of the uncertainty sources about

hydraulic networks possible operating conditions have been accounted for: in most cases, amongst the possible

operating scenarios only the case of failure of network mechanical components (i.e., pipes, pumps, tanks, valves,

etc...) has been considered; only in few cases also the uncertainty about water demand spatial and temporal

distribution has been considered. Therefore, the deterministic concept of robustness, e.g. the capability of a

system of maintaining given performance levels in presence of unfavourable variations of operating conditions,

seems more useful than reliability for applications to real water distribution systems.

In this paper, the potential use of network entropy [1] and resilience index [2] as synthetic indices of network

robustness is investigated. Network entropy, which has been proposed by several authors as a surrogate for

network reliability [3], [4], [5], is related to looped network redundancy, thus inherently prone to measure

network robustness [6]. Resilience index has been introduced in order to allow taking into account network

robustness in a cost-based optimal network design procedure [2]. A first preliminary comparison between the

two indices was carried out from the authors in [7]. Entropy and resilience were computed for two cases, which

refer to real networks serving two towns close to the city of Naples, southern Italy, namely Villaricca and Parete.

2. METHODOLOGY

The robustness of the two networks has been studied by means of a demand driven hydraulic model carried out

by EPANET2 software [8], with reference to all the possible combinations of simultaneous failures of some

pipes, simulated as closures of links in the model. For all the obtained network configurations, the hydraulic

simulation has been carried out and the proposed robustness indices have been calculated. The relationships

between the two indices, as well as with other commonly used indices of network hydraulic performance, are

discussed. Specifically, the outlined procedure consists of the following steps: 1) Hydraulic simulation of the

network; 2) Evaluation of entropy and resilience indices; 3) Hydraulic simulation of all the possible network

configurations obtained by suppressing one link; 4) For each of the obtained configurations, evaluation of

hydraulic performance indices. Steps from 2 to 4 can be repeated by suppressing more links until the number of

considered network configurations becomes so large to make calculations infeasible. Such limit depends on the

initial number of links of the considered network: as it will be shown, for medium sized networks (between 100

and 200 links), more than two simultaneous failures can be hardly considered. The relationship between the

values of resilience and entropy for all the considered network configurations has been studied, in order to get

more insight about the information carried by each of the proposed robustness indices. Furthermore, the

suitability of such indices to quantify network robustness has been investigated by evaluating the relationship

between each of the proposed indices, calculated for a given network configuration, and the values of hydraulic

performance indices for all the possible configurations obtainable by depriving such network of one link.

The concept of entropy of a water supply network has been derived from Shannon‟s information entropy [9], by

considering all the possible Np flow paths of water through the network, from source nodes to delivery nodes,

and assuming that the probability Pk that water flowing through a pipe belongs to k-th path might be expressed as

the ratio between path flow Qk and total network flow Q [10]. In such a way, the Np flow paths constitute a set of

mutually exclusive and completely exhaustive events, for which the entropy function may be written as [1]:

pN

kkk PlnPS

1

(1)

In this study, the calculation of S has been carried out making use of the equivalent recursive expression

proposed in [1]. For any given network topology, the above defined entropy is a measurement of the redundancy

of the paths available for water flow in the network, and becomes maximum when all the possible paths carry the

same flow Qk=Q/N, say when there are no „main‟ paths in the network. Such feature explains why entropy has

been proposed as a surrogate for network reliability, assuming that the more a network is redundant, the more is

reliable as reported in [3], [4] and [5]. However, such assumption should be handled with care, since there is no

direct relation between entropy and energy losses: paradoxically, two topologically identical networks, with

diameters and roughness coefficients changed in such a way that energy losses along all the links maintain their

mutual ratios, would share the same entropy in a demand-driven analysis, even in the case of an increase in

energy losses causing pressure deficit at some nodes; in the same case, a pressure-driven analysis would predict

also a reduction of delivered flows at some nodes, which would not necessarily produce a reduction of network

entropy, because both Qk and Q could decrease, unpredictably affecting the values of Pk in equation (1).

The concept of resilience introduced in [2] immediately resembles the above given definition of network

robustness. The proposed resilience index represents the fraction of the total available power which is not

dissipated in the network for delivering the design demands Q*D,j at nodes:

N,...,j

*j,D

*j,D

N,...,ji,Ri,R

N,...,j

*j,D

*j,D

N,...,jj

*j,D

MINOUTIN

MINOUT

*OUT

rHQHQ

HQHQ

PP

PPI

R 11

11 (2)

In equation (2), P*

OUT represents the power associated with the delivery of the design demands at the N nodes of

the network under the actual heads Hj; PMIN

OUT is the value assumed by the same quantity whereas the design

demands are delivered exactly under the minimum design heads, H*D,j; PIN

is the total power carried by the flows

QR,i entering the network through the NR nodes connected to reservoirs with head HR,i. The resilience index is in

the range [0, 1] if the design requirements are fulfilled, and represents the residual amount of available power,

which may allow the network to properly operate under stress conditions, such as the failure of one or more

links, and/or unpredicted demand concentrated peaks at some nodes. In this paper, the resilience index, which is

tested as network robustness index, has been calculated also for situations in which design requirements are not

fulfilled. In these cases, the resilience index may assume negative values. The capability of the proposed indices

to synthesize network robustness is tested by comparing them with some indices commonly used to evaluate the

hydraulic performance of water supply networks:

- mean head deficit,

with1

Q

QH

HN,...,j

*j,Dj,D

D

*jjj

*jj,D

*jjj,D

HH:jHHH

HH:jH 0 (3)

- mean head surplus,

with1

Q

QH

HN,...,j

*j,Dj,S

S

*jjj,S

*jj

*jjj,S

HH:jH

HH:jHHH

0 (4)

- hydraulic performance index,

with1

Q

Q

HPIN,...,j

*j,Dj

*jjj

*jjj

j*j

jjj

jjj

HH:j

HHz:jzH

zH

zH:j

1

0

21

(5)

In the above equations, zj represents the ground elevation at network nodes; HPI is a simplified version of the

hydraulic performance index proposed in [11].

3. CASE STUDY

The suitability of entropy and resilience as robustness indices has been tested for two medium-sized hydraulic

networks, serving two towns near Naples (Italy), already used respectively in [7] and [12]: Villaricca, with

30000, inhabitants and Parete, with 11000 inhabitants. The layouts of the two networks are given in Figure 1.

a) Villaricca b) Parete

Figure 1. Sketch of the layouts of the two networks: a) Villaricca and b) Parete

In Table 1 the numbers of network layouts obtained by considering all possible combinations of simultaneous

pipes failures are reported; the numbers of configurations producing the disconnection of nodes from the rest of

the network are also given. These layouts have been excluded from the calculation set because hydraulic head at

such nodes could not be evaluated. Figure 2 shows that no correlation exists between entropy and resilience for

both networks: indeed, correlation coefficients result -0.297 and -0.398 for Villaricca and Parete, respectively.

4.5

5

5.5

6

6.5

-8 -7 -6 -5 -4 -3 -2 -1 0 1I r

S

5.5

6

6.5

7

7.5

8

8.5

9

-4 -3 -2 -1 0 1I r

S

a) Villaricca b) Parete

Figure 2. Relationship between resilience, Ir, and entropy, S, for all the considered network configurations: a)

network of Villaricca and b) network of Parete

Table 1. Comparison of performance indices for different number of DMAs

Simulation characteristics Villaricca Parete

Number of studied network layouts 31125 39903

Number of network layouts with disconnected nodes 11714 44

Network entropy, S 5.217 5.771

Network resilience, Ir 0.512 0.351

For the hydraulic network of Villaricca, the distribution of the values of the performance indices HD, HS and HPI

of all the network layouts, derived from a configuration with a single link failure (indicated in the following as

mother network configuration) by suppressing one more link (indicated in the following as son network

configurations), are plotted in Figures 3a and 4a, in form of box and whiskers plots [13], respectively against

resilience and entropy of the mother network.

a) Villaricca b) Parete

Figure 2. Box plots of the distribution of hydraulic performance indices of all the network configurations with

two link failures deriving from a mother network configuration with one link failure vs. resilience of the mother

network: a) mean hydraulic head deficit; b) mean hydraulic head surplus; c) hydraulic performance index

The subscript of the indices indicates the number of simultaneous link failures of the considered network layouts.

The five lines of the box and whiskers plots represent, respectively, the 10%, 25%, 50%, 75% and 90%

quantiles. Figures 3b and 4b show the same diagrams for the network of Parete.

a) Villaricca b) Parete

Figure 4. Box plots of the distribution of hydraulic performance indices of all the network configurations with

two link failures deriving from a mother network configuration with one link failure vs. entropy of the mother

network: a) mean hydraulic head deficit; b) mean hydraulic head surplus; c) hydraulic performance index

In all cases the distributions of the values of the various indices are such that 50%, 75% and 90% quantiles are

nearly identical, so that the relevant lines in box plots collapse into each other; in most cases, the same happens

also for the 25% quantile, so that the only whisker clearly distinguishable from the others is the 10% quantile.

These features indicate that most of the network configurations deriving from a common mother configuration

assume the highest values of all the considered performance indices, while the few cases for which the indices

indicate lower performances correspond to configurations in which the failure of links which are particularly

crucial due to network topology have been considered. It looks clear that the 10% quantile of the distributions of

the performance indices of Villaricca result in all cases lower than for Parete, indicating that, in this case, the

mother networks are less robust, presumably because of their less redundantly looped layout (cfr. Figure 1). This

feature is probably also the cause of the very high number of network layouts with disconnected nodes reported

in Table 1.

The diagrams show that the entropy of the mother network does not allow to predict the performance of the son

networks, whatever is the performance index used. High values of HPI2 and HD2 are achieved regardless the

entropy S1 of the mother network, while the values of HS2 result chaotically spread. Conversely, all the

considered performance indices of the son networks show a strong dependence on the resilience of the mother

network, indicating bad performances of the son networks when the resilience of the mother network is negative.

In particular, for the case study of Parete, the mean head surplus is less than 1.0m until the mother resilience Ir1

is negative, and shows a rapid linear growth in the range of positive Ir1; similarly, the values of HS2 of Villaricca

network slowly grow until the mother resilience is smaller than 0.2, and then show a rapid linear growth for

larger Ir1. The trends of HD2 and HPI2 result similar for both the case studies, showing a constant improvement of

son networks performances with mother network resilience Ir1, until it reaches the value of 0.2, beyond which the

at least 90% of the son networks still provide high performances.

4. CONCLUSIONS

The effectiveness of entropy and resilience indices for measuring the robustness of water distribution networks

with regards to link failures has been studied. The obtained results show that no correlation can be found

between such indices. Indeed, network configurations with high resilience result more robust with respect to link

failures: indeed, the distributions of the values assumed by the performance indices, for all the possible network

configurations obtained by suppressing one link starting from a common network configuration, indicate that the

higher the resilience, the more the performance level is maintained in case of link failures. Conversely, no

relationship has been found between the entropy of a network configuration and the values assumed by the

performance indices for the configurations obtained by depriving the starting configuration of one link.

Therefore, while a high value of resilience indicates a high robustness of the network with regards to link

failures, network entropy, often proposed as a surrogate of network reliability in network design procedures,

does not provide useful information about the capability of the network to assure good performances in case of

link failures.

References

[1] T.T. Tanyimboh and A.B. Templeman, “Calculating Maximum Entropy Flows in Networks”, J. Oper. Res.

Soc., vol. 44, no. 4, 1993, pp. 383-396.

[2] E. Todini, “Looped water distribution networks design using a resilience index based heuristic approach”,

Urban Water, vol. 2, no. 2, 2000, pp. 115-122.

[3] T.T. Tanyimboh and A.B. Templeman, “A quantified assessment of the relationship between the reliability

and entropy of water distribution systems”, Eng. Opt., vol. 33, no. 2, 2000, pp.179-199.

[4] T.T. Tanyimboh and C. Sheahan, “A maximum entropy based approach to the layout optimization of water

distribution systems”, Civ. Eng. and Env. Syst., vol. 19, no. 3, 2002, pp. 223-253.

[5] Y. Setiadi, T.T. Tanyimboh and A.B. Templeman, “Modelling errors, entropy and the hydraulic reliability

of water distribution systems”, Adv. Eng. Softw., vol. 36, no. 11-12, 2005, pp.780-788.

[6] W.K. Ang and P.W. Jowitt, “Some observations on energy loss and network entropy in water distribution

networks”, Eng. Opt., vol. 35, no. 4, 2003, 375-389.

[7] A. Di Nardo, M. Di Natale, R. Greco and G.F. Santonastaso, “Resilience and Entropy Indices for Water

Supply Network Sectorization in District Meter Areas”, in Proc. 9th International Conference on

Hydroinformatics, Tianjin, Cina: Chemical Industry Press, 2010, pp. 2365-2373.

[8] L.A. Rossman. EPANET2 Users Manual. Cincinnati (OH): US E.P.A., 2000.

[9] C.E. Shannon, “A mathematical theory of communication”, Bell System Tech. J., vol. 27, no. 3, 1948,

pp.379-423.

[10] K. Awumah, I. Goulter, and S.K. Bhatt, “Assessment of reliability in water distribution networks using

entropy-based measures”, Stochastic Hydrology and Hydraulics, vol. 4, no. 4, 1990, pp. 325-336.

[11] R. Gargano and D. Pianese, “Reliability as Tool for Hydraulic Network Planning”, J. Hydr. Engrg., vol.

126, no. 5, 2000, pp.354-364.

[12] A. Di Nardo and M. Di Natale, “A design support methodology for district metering of water supply

networks”, in Proc. of the Annual Int. Symposium on Water Distribution Systems Analysis WDSA, Tucson,

Arizona, US, 2010.

[13] J.W. Tukey, Exploratory Data Analysis, Reading (MA): Addison-Wesley, 1977.