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A heuristic design support methodologybased on graph theory for districtmetering of water supply networksArmando Di Nardo a & Michele Di Natale aa DIC, Department of Civil Engineering , Second University ofNaples , via Roma 29, 81031, Aversa, CE, ItalyPublished online: 29 Sep 2010.

To cite this article: Armando Di Nardo & Michele Di Natale (2011) A heuristic design supportmethodology based on graph theory for district metering of water supply networks, EngineeringOptimization, 43:2, 193-211, DOI: 10.1080/03052151003789858

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Engineering OptimizationVol. 43, No. 2, February 2011, 193–211

A heuristic design support methodology based on graph theoryfor district metering of water supply networks

Armando Di Nardo* and Michele Di Natale

DIC, Department of Civil Engineering, Second University of Naples, via Roma 29,81031 Aversa (CE), Italy

(Received 18 March 2009; final version received 04 March 2010 )

The management of existing water supply networks can be substantially improved by permanent waterdistrict metering (WDM) which is one of the most efficient techniques for water loss detection and pressuremanagement. However, WDM may compromise water system performance, since some pipes are usuallyclosed to delimit districts in order not to have too many metering stations, to decrease costs and simplifywater balance. This may reduce the reliability of the whole system and not guarantee the delivery of water atthe different network nodes. In practical applications, the design of district meter areas (DMAs) is generallybased on empirical approaches or on limited field experiences. In this work a design support methodology(DSM) is proposed, which helps to identify the position of flow meters and of boundary valves needed todefine permanent DMAs. The DSM is based on graph theory and is applied to a test case.

Keywords: water district metering; graph theory, water supply network; water leakage; resilience; DMA

1. Introduction

Water District Metering (WDM) is an important methodology for water Distribution System(WDS) management (AWWA 1999, 2003, Lambert and Hirner 2000). It consists of partitioninga water system into subsystems (called ‘districts’) delimited by boundary or control valves andby flow meters.

This methodology, developed in the UK (Wrc/WSA/WCA 1994) and already implemented inmany countries, aims at:

(a) simplifying network water balance calculation by monitoring night flows in each district inorder to detect unreported bursts and to enable leakage identification and location (usingacoustic methods, step tests, etc.);

(b) carrying out pressure management in order to reduce hydraulic head and water leakage;(c) improving water system management with district hydraulic data continuous monitoring in

order to prevent water shortage and to plan better maintenance operations.

*Corresponding author. Email: armando.dinardo@unina2.it

ISSN 0305-215X print/ISSN 1029-0273 online© 2011 Taylor & FrancisDOI: 10.1080/03052151003789858http://www.informaworld.com

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194 A. Di Nardo and M. Di Natale

As recommended in the literature (Water Industry Research Ltd 1999), the water balancetechnique requires the installation of flow meters at strategic points throughout the WDS. In thisway Minimum Night Flow (MNF) monitoring is simplified, can give precious information aboutdistrict water losses (Wrc/WSA/WCA 1994) and provides the basis for network operation andhelps to prioritise districts for leak detection and location activity. District MNF is monitored bycalculating the difference between the inflow and outflow in each district during hours of lowconsumption.

Flow meters may sometimes contain a pressure valve for water system pressure management.In these situations the division of the network in hydraulically independent districts (Alonso etal. 2000) allows a more effective pressure regulation since a different head level can be obtainedin every single network subsystem by installing boundary valves.

If WDM is carried out in the water supply network design phase, it is possible to plan differentdistricts in order to improve system management, pressure regulation and water leakage detectionwithout altering the network performance. When water systems are already in operation it is moredifficult to set the number and dimension of districts and, in many cases, WDM may even worsenconsiderably the hydraulic performance of water networks both in terms of water quality (creatingmore dead-end pipes and flow reversals) and of node heads (Wrc/WSA/WCA 1994).

Water district metering can be carried out in different ways and with various partitioning levels(Water Industry Research Ltd 1999): in particular, WDM can be ‘permanent’ or ‘temporary’,with reference to time duration, and can have different sizes as a function of its specific objective(network management, water balance, pressure regulation, etc.).

The best results in leakage monitoring can be achieved by defining a small permanent district,called District Meter Area (DMA). Depending on the characteristics of the network, DMAs maybe supplied by single or multiple feeds, may flow into adjacent DMAs or be self-contained.The fewer the flow meters the easier and cheaper the calculation of DMA water balance; thiscan be obtained by using boundary valves to reduce the number of pipe connections accordingto hydraulic constraints and performance. Indeed, for each district, there is a cost to purchase,install and maintain the equipment and, with an equal number of DMAs, it is better to have thelowest possible number of flow meters. The best condition would be to have one single inflowmeter for each district so as to make it easy for the operator to calculate the synchronous waterbalance.

DMA planning is simpler for branched networks and is much more complicated with loopednetworks that are typical of residential areas. Permanent WDM changes the original topologicallayout of water systems and reduces network water pressure, especially during peak demands.Though this is positive in terms of water loss reduction during night flow, it may lead to insufficientsupply to the customers during peak demand hours.

WDM changes the hydraulic behaviour since, in principle, it is in conflict with tradi-tional design criteria (Shamir and Howard 1968, Alperovits and Shamir 1977, Mays 2000)of looped water networks which allow water systems to be more reliable under mechanicaland hydraulic failure conditions (Mays 1996). The main design criterion is pipe redundancythat markedly improves network reliability which can be significantly reduced by pipe closurenecessary to obtain permanent WDM (that can also be generically described as ‘a permanentfailure’).

Water district metering may cause an alteration of overall and localised hydraulic performanceof water systems; network districts should therefore be designed by experts and with the aid ofsimulation software.

Many technical and economic constraints have to be taken into account (topology, presence ofboundary valves, available budget, etc.) in order to define permanent DMAs that do not diminishthe hydraulic performance of the water distribution network. DMAs must also meet the needs ofplanners (areas with different pressure levels, compliance with levels of service required, etc.)

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Engineering Optimization 195

which are difficult to describe in mathematical terms. Therefore optimization methods basedon the definition of a precise multi-objective function cannot be easily applied. Actually WDMdepends essentially on the skills of the planner who must identify the best possible solution amongthe diverse district metering options available.

Several suggestions about DMA dimensions can be found in the technical literature: in WaterAuthorities Association and Water Research Centre (1985) a DMA has to include 1000–3000properties, while in Butler (2000) a permanent district has to contain 2500–12,500 inhabitantswith 5–30 km of water network and Water Industry Research Ltd (1999) recommends a numberof properties between 1000 (small DMA) and 3000 (medium DMA) and up to 5000 (large DMA).These indications are based on empirical considerations and sometimes on few test cases and maynot necessarily apply to large water supply systems.

In this article an original approach to define WDM for WDS already in operation is put forward.The methodology is based on techniques borrowed from graph theory (Berge 1958, Biggs et al.1986) that allow the analysis of network configurations (Kesavan and Chandrashekar 1972). Waternetwork configuration problems, in particular network topology and connectivity, are classifiedas ‘layout problems’ (Stanic et al. 1998, Goulter and Morgan 1985, Ostfeld 2005, Giustolisi et al.2008). There are not many articles on layout optimization in the literature and their focus ismainly on water system robustness and reliability (Wagner et al. 1988a, b), and not specificallyon WDM.

Several scientific works on layout problems use graph theory to analyse water network topo-logy (node plano-altimetric position), connectivity (the probability that a given demand node isconnected to a source), reachability (the probability that all nodes are connected to a source), inorder to investigate network vulnerability with respect to expected operations (valve regulations,pipe substitutions, etc.).

In Jacobs and Goulter (1989), graph theory is combined with integer goal programming inorder to optimize water supply network redundancy and improve system reliability by maximizingnetwork regularity. This goal was achieved by minimizing the sum of the deviations at each nodein terms of the number of links incident upon it.

In Ostfeld and Shamir (1996) the concept of a water network ‘backup subsystem’ is introduced;a subset of links of the system where a prescribed level of service is maintained when a failureoccurs. The backup subsystem is obtained using algorithms based on graph theory and topologicconsiderations. While in Ostfeld (2005) a frequency connectivity digraph matrix, that maintainsKirchoff’s Laws 1 and 2, is used.

In Savic et al. (1995) graph theory is used to partition the water network in ‘tree’ and ‘co-tree’ sets to obtain simpler network layouts and to better describe the optimization problem ofminimizing the heads by setting hydraulic regulation valves. More recently in Deuerlein (2008)a similar approach to water supply system control and management has been proposed, basedon the division of the network graph in two subsystems: a main graph (called ‘core’) consistingof looped pipes and a secondary graph (called ‘forest’) consisting of tree pipes; both graphs arelinked by connection elements called ‘bridges’. These methodologies allow simplified networkmodelling and have different views of the hydraulic system. More specifically in a simplifiedscheme of a water supply network, made of core, forest and bridge, or tree and co-tree sets, it ispossible to better understand the interactions between the different parts of the system such as,for example, their connection to network sources (reservoir, tank, pump, etc.) and, in some cases,to better define optimization problems.

As to the specific problem of optimal district partitioning of a water distribution system withseveral reservoirs, Tzatchkov et al. (2006) suggest an algorithm derived from graph theory toidentify independent supply sectors (or districts) of the network layout.

In this article a heuristic Design Support Methodology (DSM) is put forward to partition a watersupply system in DMAs, by resorting to graph theory principles that permit an analysis of the

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196 A. Di Nardo and M. Di Natale

minimum dissipated power paths computed from each reservoir to each node of water network.The DSM has been tested on an Italian real-life network, Villaricca, in the Province of Naples.The DSM has been assessed by using performance indices (statistical and hydraulic) to comparethe level of service of several district layouts. Specifically, the degeneration of a looped networkdue to water district metering and, as a consequence, the diminished capacity of the water systemto react and to overcome stress conditions, has been measured with resilience indices (Todini2000), based on energy approach and used, in this article, to compare different WDMs with theOriginal Network Layout (ONL).

2. Methodology

The number of possible WDMs of a large water supply network may be huge but many of themare not compatible with the hydraulic constraints; although the number of allowable configura-tions is limited, it is difficult to find them by following only mere empirical remarks from theliterature or approaches such as ‘trial and error’, even if used together with hydraulic simulationsoftware.

The methodology proposed is a procedure to rapidly identify a ‘cost-effective’ configurationof permanent districts. In particular, it helps designers to define DMAs by comparing severalpossible options. The approach is based on the identification of the pipes where boundary valvesand flow meters can be inserted by searching minimum dissipated power paths and using somegraph theory principles. In this way it is possible to rapidly find WDMs that are compatible withthe levels of service and the reliability of the original network layout (ONL).

‘Less important’ or ‘more redundant’ pipes are identified and closed to create DMAs.In this way the number of pipes where flow meters and boundary valves may be insertedis reduced.

The steps of the heuristic design support methodology proposed are as follows (Figure 1):

(a) simulate WDS to carry out pipe flows qj and node heads hi ;(b) define adjacency A, incidence I , weight W matrices;(c) compute all shortest paths {p}sminand all path frequencies fj ;(d) define main network layout (MNL);(e) draw main graph GM ;(f) choose DMA number Nk and dimension;(g) insert Nf m flow meter;(h) close { v

l} DMA links with boundary valves;

(i) simulate WDS again;(l) compute performance indices (PI);

(m) evaluate PI and Nf m: if PI are not satisfactory return to step f) if Nf m is not satisfactory goto step n);

(n) delete one flow meter and return to step h);(o) choose WDM.

Specifically, starting from network model INPUT (with n nodes, identified by labels αi withi = 1, . . . , n, m pipes, identified by labels βj , with j = 1, . . . , m, node water demand distributionQi , with 1 = 1, . . . , n, source heads Hs , identified by labels σs , with s = 1, . . . , r reservoir, pipelength Lj and node elevations ei), pipe flow qj, node pressure hi and head loss �Hj for eachpipe can be calculated by a demand driven approach (Todini and Pilati 1988) with a hydraulicsimulation (step (a)).

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Engineering Optimization 197

satisfactory

PINOsatisfactory

Nfm

NOsatisfactory

(step a) WDS hydraulic

simulation

(step b) define A, I and

W matrices

(step c) compute Shortest

Path { }sp min

(step d) define Main

Network Layout (MNL)

(step f) choose DMA

number Nk and dimension

(step g) insert Nfm flow

meter

(step h) close DMA link

set {vl}

(step m) PI AND Nfm

(step o) chooseWDM

(step n) delete a flow

meter

(step i) new WDS hydraulic simulation

(step l) compute

performance indices PI

(step e) draw

Main Graph GM

Figure 1. DSM flow chart.

Next, step b is the definition of square adjacency matrix A of order n and incidence matrix I

of order n × m (Bondy and Murty 1976):

Axy ={

1 if node αx is linked to node αy (f or x �= y)

0 else(1)

Ixy =

⎧⎪⎨⎪⎩

+1 if nodes αx is start node of pipe βy

−1 if nodes αx is end node of pipe βy

0 else

(2)

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198 A. Di Nardo and M. Di Natale

and of sparse weight square matrix W of order n :

Wxy ={

wxy if (Axy = 1 and wxy > 0)

0 else(3)

with the following weight function computed between two generic network nodes αx and αy :

wjwxy = qj�Hxy = qj (Hαx− Hαy

) (4)

Then, compatibly with network layout ∀s, i it is possible to identify all node paths {pz}s,αi ={σs, . . . , αx, . . . , αi}s,αi , belonging to path set {p}s,αi = {ps,αi

1 , . . . , ps,αiz . . . ps,αi

np } with z =1..ns,αi

, that may be crossed by an infinitesimal flow dq from source s to the ith node in theworst operating condition (peak water demand). Then it is easy to identify, by means of the I

matrix, pipe set {pz}s,βj = {βs, . . . , βx, . . . , βj }s,βj corresponding to node set {pz}s,αi .Among all possible paths of set {pz}s,αi only one of them is the ‘minimum dissipated power

per unit weight path’ ps,αi

min = {σs, . . . , αx, . . . , αi}s,αi

min with (min∑

βj ∈{pmin}s,βj wj ) calculated forcorresponding path pipes. So a new set {p}smin = {ps,α1

min , . . . , ps,αx

min , . . . ps,αn

min }, composed of allminimum dissipated power paths n from reservoir s to each node αi, can be defined (step (c)).

This problem, known as the ‘shortest path search’, in the case of a WDS can be solved byresorting to graph theory to determine the minimum cost path between two nodes (s and i) ofa network directed graph G with n nodes, m links and a weight function wj that correlates acost �Hj to the j th pipe of the G (Liberatore and Sechi 2007) as in relation (3). Many shortestpath resolution algorithms (Skiena 1990) can be found in the literature; in this article the Dijkstraalgorithm has been chosen (Dijkstra 1959) that allows all minimum dissipated power paths {p}sminto be obtained easily if the weight matrix W is known (step (c)).

Then, using adjacency matrix A′, oriented according to flow direction, and defined as follows,

A′xy =

{1 if (Axy = 1 and wxy > 0)

0 else,(5)

it is also possible to draw the diagraph G of ONL by using different criteria of representation (DiBattista et al. 1999). In this methodology, a hierarchical approach has been chosen (Warfield 1977,Johnson 1977, Carpano 1980, Sugiyama et al. 1981) in which all network nodes are drawn indifferent layers with a distinct hierarchy of connection. In this approach a diagraph hierarchy caneasily be found and there is a correspondence between visual perception and network connectionanalysis. ‘Ancestor’ and ‘descendant’ nodes define a specific HL hierarchical level (Di Battista etal. 1999) thus defining simpler structures such as trees, loops, subsystems, groupings, etc., thatcan be needed during WDM design.

Once all minimum energy paths {p}smin have been calculated for each single sth reservoir, it ispossible to compute (step (c)) the ‘path frequency’ for each single pipe counting how many timestj each pipe j is found in all s · n shortest paths obtained with the Dijkstra algorithm, as follows:

fj = tj

s • n(6)

In particular, pipes which are not included in any shortest path have fj = 0. Now it is easy tocompute a new oriented adjacency matrix A′′ deleting all pipes with path frequency (6) equal tozero and defined as follows:

A′′xy =

{1 if (A′

xy �= 0) AND (fy �= 0)

0 else(7)

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Engineering Optimization 199

So starting from A′′, a new network layout, called Main Network Layout (MNL), is defined(step (d)) which differs from ONL because it has not all pipes with fj = 0. By means of A′′matrix it is possible to draw the Main Graph GM (step (e)) of the network with a hierarchicalrepresentation that allows one to simplify the partitioning of the system by helping planners inchoosing the number and size of DMAs, as well as in finding the location of flow meters andboundary valves.

Three main remarks are at the basis of the criteria adopted in design support methodology tominimize the alterations of the performance of ONL. The first remark is that the higher the numberof pipes closed to define DMAs, the larger the decrease of pipe diameter availability of the watersupply system. The second one is that there are pipes in the network which can be defined as ‘lessimportant’ for the operation of the system, meaning that they are present in the lowest numberof total shortest paths. These pipes are characterized by a lower path frequency fj as defined inEquation (6). The third remark comes from the fact that, the main graph GM being made by all theminimum dissipated power paths which connect the reservoirs s to demand nodes i, the closureof one of its pipes forces the flow to change its path to reach the same demand node. Therefore,such an alternative path, although compatible with the network topology, does not occur on themain graph and will be inevitably characterized by higher power dissipation.

These remarks lead to the definition of two criteria to select the pipes where flow meters andboundary valves are to be inserted for the WDM of the network:

(a) to minimize pipe closures and in particular those which belong to the main graph, by insertingflow meters on it preferably;

(b) to reduce, if necessary, the number of flow meters by replacing them with boundary valvesto be inserted in the pipes showing the lowest path frequency and cost wj .

Therefore, once the number Nk and the size of districts (step (f)) have been defined according tothe analysis of the main graph and to technical-economic considerations, it is possible to identifyDMAs by inserting a certain number Nf m of flow meters (step (g)) on the pipes of GM . After thedefinition of the nodes of each district, univocally determined by the position of the flow meterson GM , it is easy – thanks to the incidence matrix I , (2) – to identify pipes j (with fj = 0)connecting districts where to insert Nbv boundary valves in a pipe set {vl} = {v1, . . . , vx, . . . vNbv

}in order to obtain the permanent partitioning (step (h)).

When the new configuration of the network is obtained, a new hydraulic test is performed (step(i)) and Performance Indices (PIs) are calculated (step (l)) in order to evaluate the WDM (step(m)). At this point, if PIs and Nf m are satisfactory for the planner the WDM is chosen (step (o)).If PI is not satisfactory, it is necessary to decrease the number Nk or the size of DMAs, goingback to step (f) since the closure of pipes has caused an excessive increase in head losses.

But if PIs are satisfactory, it is possible to reduce the number Nf m of flow meters by replacingthem, one by one, with boundary valves (step (n)) thus reducing costs and simplifying the districtwater balance. This operation can be carried out by removing, gradually, the flow meters insertedon the pipes of the main graph which show the lowest path frequencies and cost wj according tocriterion (b).

In order to illustrate the concepts and the possibilities offered by the approach proposed,this heuristic design support methodology has been tested on a real water system at Villaricca,Naples, Italy.

2.1. Performance indices

Based on the above descriptions, performance indices are required to compare all the partitioningoptions of the water supply network and provide the planner with the necessary information to

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200 A. Di Nardo and M. Di Natale

make the best choice. For this purpose several PI are used to test the behaviour of a single districtor the whole network:

(a) energetic,(b) statistical(c) hydraulic.

Energy indices

According to Todini (2000), network looped topology allows for redundancy, which assists inensuring that the system can overcome local failures. The redundancy decrease can be measuredwith a ‘resilience’ index which does not involve the statistical analysis of different types of uncer-tainty which are required for the definition of the reliability constraints. Water district meteringcan also be assimilated to permanent local failures that change the system layout by increasinginternal power dissipation and decreasing energetic redundancy; this effect is due to boundaryvalves that reduce network pipe diameter availability and remove some network loops. EachWDM will tend to increase the internal power dissipation and resilience indices can provide agood way to compare several system layouts.

Specific resilience and failure indices were used starting from power balance of a water network(Todini 2000), in the hypothesis of head loss �Hj due to friction computed by a uniform flowformula:

PA = PD + PN (8)

in which each single equation term is represented by:Available power (or total power):

PA = γ

r∑s=1

qsHs (9)

where qs and Hs are, respectively, the discharge and head relevant to each reservoir r and γ is thespecific weight of water.

Dissipated power (or internal power):

PD = γ

m∑j=1

qj�Hj (10)

where qj and �Hj are flow and head loss of each network pipe.Nodes power (or external power):

PN = γ

n∑i=1

QiHi (11)

where Qi and hi are, respectively, water demand and node head at each network node.Then, similarly to Todini (2000), two different indices are defined:Resilience index:

Ir = 1 − PD

PD max(12)

where PD max = PA − γ∑n

i=1 Qi/Hi is the maximum power necessary to satisfy the demandconstraints Qi and node head constraint H̄i = ei + h̄i with h̄i equal to the design pressure of the

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Engineering Optimization 201

ith node. Higher values of Ir (Equation 12) indicate better WDMs in the sense of lower values ofdissipated power and, consequentially, higher resilience.

Resilience deviation index:

Ird =(

1 − I ∗r

Ir

)100 =

(P ∗

D − PD

PD max − PD

)100 (13)

where I ∗r = P ∗

D/PD max and P ∗D is Dissipated Power of WDM network layout. This index shows

immediately the resilience percentage deviation between WDM and ONL: naturally, higher valuesof (13) mean worst WDM.

Statistical indices

Energy indices refer to the entire water network but WDM also affects the individual DMAs,therefore other district indices have been used: district performance statistical indices that computeMean, Minimum, Maximum and SQM values of hydraulic head for each district nodes. Statisticalindices allow the most important information about nodal pressures traditionally used to measurethe level of service of WDS to be summarized.

Hydraulic indices

Finally district performance hydraulic indices have also been developed in order to evaluate thebehaviour of WDM with reference to district design pressure, as follows:

Root of mean squares (RMS) of pressure deviations:

RMSPDk =√∑nk

i=1 (hki − h̄i)2

Nk

(14)

where hki is water pressure in the ith node of the kth DMA; h̄i is design pressure in the same node;nk is the number of the kth district nodes. RMSPDk index measures node pressure deviation, bothpositive and negative, from district design pressure at each node: low values of this index show asmall alteration of DMA pressure distribution, otherwise high values indicate a strong effect ofwater district metering on network original status.

Mean pressure surplus:

MPSk =∑n0k

i=1 IS

nSk

(15)

where

IS ={

0 ∀ i : h̄i > hi

hi − h̄i ∀ i : h̄i ≤ hi

(16)

and nSk is the node number with h̄i ≤ hi in the kth district. This index gives a measure of nodesurplus: the higher the values, the higher the DMA pressure surplus and vice versa.

Mean pressure deviations:

MPDk =∑n0k

i=1 ID

nDk

(17)

where

ID ={

0 ∀ i : hki > h̄i

hki − h̄k ∀ i : hki ≤ h̄i

(18)

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202 A. Di Nardo and M. Di Natale

and nDk is the node number with hki ≤ h̄k in the kth district. This index gives a measure of nodepressure deviation: clearly the lower the negative values, the smaller the DMA pressure deviationand vice versa.

Mean frequency of pressure deviations:

MFPDkx =∑nxk

i=1 IDFx

nxk

(19)

where

IDFx ={

0 ∀ i : hki > (h̄i − x)

1 ∀ i : hki ≤ (h̄i − x)(20)

and nxk is the node number of the kth district in which pressure is x metres less than node designpressure h̄i . MFPDkx index measures the frequency of pressure deviation from prefixed pressurevalues (h̄i − x) in order to have a more specific knowledge of pressure deviation distribution. Inthe article the value of x has been assumed to be 0, 2 and 5 m, in order to test pressure deviationrange; in general, low values of this index show good system performance but, more specifically,high MFPDk0 values can also indicate very small deviations from district design pressure whilelow MFPDk10 indicates large pressure deviation in DMAs.

3. Case study

Each single step of DSM application is illustrated in the following case study ofVillaricca network(Figure 2) with 30,000 inhabitants, 387 × 104 m2 of flat surface, in a densely populated area Northof Naples (Italy), already studied by the authors in Di Nardo et al. (2008).

TheVillaricca network is supplied from three different sources, identified in Figure 2 with nodesσ1 = 197, σ2 = 198 and σ3 = 199. The kind of water consumption in Villaricca is exclusivelyresidential with a prevalence of houses, built in 1970s and 1980s, with four to five floors. Therefore,the same design pressure for all nodes and equal to h̄i = h̄ = 30 m ∀i = 1 . . . n has been assumed.

Following the DSM flow chart reported in Figure 1, the first step (a) is a hydraulic simula-tion of the Villaricca network model, reported in Table 1 and available on the author web site(http://docenti.unina2.it/dinatale), carried on using a free software EPANET (Rossman, 2000) inorder to calculate unknown pipe flows qj and node heads hi . Hydraulic simulation is carried outduring peak water demand because permanent WDM effects on hydraulic performance are clearlyworse in this operating condition.

Figure 2. Villaricca network hydraulic model.

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Engineering Optimization 203

Table 1. Villaricca network model characteristics.

Nodes (n) 199Pipes (m) 249Pipe total length 36,018 mPipe material Steel and cast ironReservoir Node 197 and 198: 151 m; node 199: 150 mRoughness (Manning) 0.009; 0.016; 0.011; 0.09Total inflow (peak demand) 209.29 l/s

Table 2. Villaricca network shortest paths from reservoir σ1 = 197 to all αi nodes.

Shortest path set Shortest path Minimum cost Label αi of shortest path nodes

{p}197min p

197,α1min

∑βj ∈{pmin}s,α1

wj σ1, . . . , α1

p197,α2min 4,419 197, 129, 159, 126, 8, 7, 6, 2, 4, 1

p197,α3min 4,416 197, 129, 159, 126, 8, 7, 6, 2

p197,α4min 4,421 197, 129, 159, 126, 8, 7, 6, 2, 4, 3

p197,α5min 4,418 197, 129, 159, 126, 8, 7, 6, 2, 4

.

.

.... . . .

p197,αxmin

∑βj ∈{pmin}s,αx

wj σ1, . . . , αx

.

.

.... . . .

p197,αnmin

∑βj ∈{pmin}s,αn

wj σ1, . . . , αn

Then (step (b)) adjacency A, incidence I , weight W and oriented adjacency A′ matrices aredefined and, subsequently (step (c)) all shortest paths {p}smin from each reservoir to all networknodes are computed by Dijkstra algorithm. For example, in Table 2 a few minimum paths ofVillaricca network, from the reservoir σ1 = 197 to ith node, are represented. Once all path fre-quencies fj have been calculated (step (c)) it is possible to define new MNL (step (e)) computingmatrix A′′ (7); specifically the Villaricca network pipes with fj = 0 are 38.

Then, thanks to MATLAB Bioinformatics Toolbox© (by The MathWorks), by adopting thehierarchical criterion to represent nodes (Siek et al. 2002), the oriented graph G (Figure 3) ofONL and the oriented main graph GM (Figure 4) of MNL are drawn (step (e)). In particular,Figure 3 also shows the HL = 31 hierarchical levels into which nodes are divided.

By comparing the two figures, it is possible to see that, unlike the graph G (Figure 3), the maingraph GM (Figure 4) is open or tree-like which makes the district design easier.

By comparing the two graphs in Figures 3 and 4 with the traditional layout of the water networkin Figure 2 it is easy to realize that this visualization is more effective to design a WDM. Inparticular, the main graph GM (Figure 4) allows an immediate visualization of the ‘most important’pipes of the water system which generally correspond to those with a lower hierarchical level HL

and a higher path frequency. Furthermore, the GM allows an easier visualization of the several nodegroups of the network which can give useful information to choose the number and size of districts.

Therefore, based on the analysis of the main graph, it has been decided to partition the networkinto Nk = 5 DMAs (step (f)) by inserting Nf m = 6 flow meters (step (g)) which, along with theflow meters traditionally installed downstream of the three reservoirs, will allow to make districtwater balances in order to evaluate the flow losses in accordance with the sizes of DMAs suggested

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204 A. Di Nardo and M. Di Natale

Figure 3. Villaricca network graph G with indication of: 31 HL and five DMAs.

in the literature. At this point, by knowing the individual nodes belonging to each DMA, it is easyto identify all connection pipes j where to insert the Nbv = 9 boundary valves (step (h)) in orderto obtain the five DMAs univocally identifying the WDM A1, shown in Figures 3 and 4. Thesingle pipes j (with the relevant fj ), where flow meters and boundary valves have been inserted,are reported in the first row of Table 3, together with the total number of nodes of each DMA.

A new simulation of the water network has then been performed (step (i)) and PIs have beencalculated (step (l)), as defined in Section 2.1. The results are reported in the first row of Tables4, 5 and 6 corresponding to WDM A1.

Subsequently, since PIs can be considered as satisfactory (step (m)), it has been decided toreduce the number of flow meters in district A1 with 5 DMAs (step (n)) by identifying a newlayout called A2, with Nf m = 5, created by inserting a boundary valve in the pipe βj = 140 ofGM with the lowest fj = 1, according to criterion (b) to try and close those pipes with the lowest

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Engineering Optimization 205

Figure 4. Villaricca network main graph GM with indication of five DMAs and three flow meters.

path frequencies and cost wj . As a consequence, the number of flow meters, as shown in thesecond raw of Table 3, decreased from Nf m = 5 to Nf m = 4 and total boundary valves increasedfrom Nbv = 9 to Nbv = 10.

Then, a new hydraulic simulation has been performed and the PIs reported in the second rawof Tables 4, 5 and 6 have been calculated again. Step (n) is repeated several times, creating theWDM A3 and A4 with the gradual reduction of the flow meters in the pipes having increasinglyhigher fj values. The process stops when, by replacing another flow meter with a boundary valve,a whole district would be disconnected by making the water supply from reservoirs impossible.

The results of simulations, for each WDM, are reported in Tables 4, 5 and 6, as a function ofthe sum F of all the path frequencies fj of pipes βj belonging to the set {vl} of the Nbv boundaryvalves closed to define districts:

F =∑

βj ∈{vl}fj (21)

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206 A. Di Nardo and M. Di Natale

Table 3. Water district metering layouts with five DMAs.

WDM DMA boundary pipes DMA number nodes

Boundary valves in Boundary valves outNk ID Nf m Nbv Flow meter in GM GM (with relative fj ) GM (with fj = 0) 1 2 3 4 5

A1 6 9 140, 178, 142, 25, 107, 111 – 50 22 23 69 32A2 5 10 178, 142, 25, 107, 111 140(1) 15, 21, 26, 45, 78,

5 A3 4 11 142, 25, 107, 111 140 (1), 178 (15) 92, 133, 150, 198A4 3 12 25, 107, 111 140 (1), 178 (15), 142 (36)

Table 4. Energy indices of Villaricca network WDMs and ONL.

Power (kWatt) Energy indices

WDM ID PA PN PD Ir Ird F

ONL 324.20 304.02 20.18 0.369 – –A1 324.20 303.07 21.13 0.340 7.92 0A2 324.20 303.02 21.18 0.338 8.33 1A3 324.20 302.89 21.31 0.334 9.37 16A4 324.20 302.74 21.46 0.329 10.66 52

Table 5. Statistical indices of Villaricca network WDMs and ONL.

DMA DMA

WDM ID Index 1 2 3 4 5 Index 1 2 3 4 5

ONL Mean 35.27 36.38 41.56 37.18 33.66 Min 25.12 32.95 35.68 31.90 29.77A1 36.95 37.04 40.73 35.80 31.93 27.31 34.29 34.72 30.69 25.98A2 36.93 37.01 42.24 35.48 31.65 27.28 34.38 36.38 30.42 25.71A3 36.96 37.01 42.24 35.02 32.17 27.31 34.38 36.38 30.53 26.05A4 37.56 36.92 42.23 34.74 31.68 27.96 34.34 36.37 30.17 25.56

ONL Max 44.29 43.61 45.94 43.22 36.40 SQM 4.06 2.87 2.61 2.47 1.89A1 44.33 43.58 45.31 42.31 34.92 3.91 2.39 2.70 2.52 2.10A2 44.33 43.60 46.53 41.89 34.64 3.92 2.44 2.57 2.49 2.10A3 44.33 43.60 46.53 41.80 35.37 3.91 2.44 2.57 2.54 2.27A4 44.35 43.59 46.52 41.58 34.87 3.75 2.47 2.57 2.57 2.26

First of all, in Table 4, it is worth noticing that the original network layout (ONL) shows arather low resilience index Ir = 0.369 which confirms the low redundancy of the system with adesign pressure h̄ = 30 m. It can also be observed that WDMs showing the lowest Ird values alsocorrespond to those with the highest F values. In particular, WDM A1 shows a value F = 0 sinceit was created without closing any pipes on GM ; in fact, in this case, the only pipes closed byboundary valves are not included in any shortest path and they all have, obviously, fj = 0.

The efficacy of the approach proposed can be evaluated by observing that the four WDMs (A1,A2, A3 and A4) show low deviations from the resilience index, below 11.00%, with a negligibleincrease in dissipated power PD and equal to 1280 kW at the maximum. Such small deviationsare compatible with the performances of the system as confirmed by statistical (Table 5) andhydraulic (Table 6) indices.

In more detail, in order to analyse performances in terms of node hydraulic head, the statisticalindices of Villaricca network indicate good results in each DMA of all WDM A1, A2, A3 and A4.The district statistical indices show how the WDM alters the distribution of network pressures ina non-homogeneous way. In fact, Table 5 shows, for example, an increase in the mean pressure in

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Engineering Optimization 207

Table 6. Hydraulic indices of Villaricca network WDMs and ONL.

DMA DMA

WDM ID Index 1 2 3 4 5 Index 1 2 3 4 5

NDM RMSPD 6.66 7.00 11.85 7.59 4.12 MPS 6.09 6.38 11.56 7.18 3.79A1 7.98 7.43 11.07 6.33 2.85 7.47 7.04 10.73 5.80 2.54A2 7.96 7.42 12.50 6.02 2.67 7.45 7.01 12.24 5.48 2.46A3 7.98 7.42 12.50 5.62 3.14 7.47 7.01 12.24 5.02 2.82A4 8.44 7.35 12.49 5.39 2.81 7.75 6.92 12.23 4.74 2.78

NDM MPD −2.14 0.00 0.00 0.00 −0.23 MFPD0 10.00 0.00 0.00 0.00 3.13A1 −1.16 0.00 0.00 0.00 −1.37 6.00 0.00 0.00 0.00 15.63A2 −1.22 0.00 0.00 0.00 −1.23 6.00 0.00 0.00 0.00 21.88A3 −1.14 0.00 0.00 0.00 −1.33 6.00 0.00 0.00 0.00 15.63A4 −2.04 0.00 0.00 0.00 −1.11 2.00 0.00 0.00 0.00 28.13

NDM MFPD2 4.00 0.00 0.00 0.00 0.00 MFPD5 0.00 0.00 0.00 0.00 0.00A1 2.00 0.00 0.00 0.00 3.13 0.00 0.00 0.00 0.00 0.00A2 2.00 0.00 0.00 0.00 3.13 0.00 0.00 0.00 0.00 0.00A3 2.00 0.00 0.00 0.00 3.13 0.00 0.00 0.00 0.00 0.00A4 2.00 0.00 0.00 0.00 3.13 0.00 0.00 0.00 0.00 0.00

DMAs 1 and 2 and a decrease in DMAs 3, 4 and 5; the same occurs for minimum and maximumpressures with relatively reduced SQM. Such behaviour of the system is due both to the layoutchanges made with WDMs and the position of the three reservoirs which may, in some cases,increase the pressure of a district to the detriment of others. It is therefore advisable to use indicesother than the resilience one which, alone, cannot show the effect of the WDM on individualDMAs.

Also hydraulic indices, reported in Table 6, confirm the good performances of WDM obtainedwith DSM. The RSMPDk (14) index shows that, vis-à-vis the design pressure h̄, there is a slightworsening of the performance of DMA 4, whereas the worsening is stronger for DMA 5 in allfour WDMs A. In DMAs 1, 2 and 3, as already pointed out with statistical indices, there is aslight improvement that results into a decrease in the deviation, measured by MPDk (17), in thosenodes having pressure lower than the design value, and into an increase of the surplus measuredby MPSk (15), in those nodes having pressure higher than h̄.

It is possible to evaluate further the analysis of the performance of single districts by usingthe MFPDkx (19) index, which measures the mean frequency of the deviation of single nodes.In Table 6, the MFPDk0, calculated for x = 0, shows a remarkable increase in the number ofnodes with pressure lower than design value in DMA 5, whereas it shows a slight decrease in

Figure 5. Traditional representation of Villaricca WDM A4.

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208A

.DiN

ardoand

M.D

iNatale

Table 7. Examples of others water district metering layouts.

DMA boundary pipesWDM DMA number nodes Ird

Boundary BoundaryNk ID Nf m Nbv Flow meter in GM valves in GM valves out GM 1 2 3 4 5 6 7 [%] F

B1 9 14 140, 115, 178, 142, 67, 107, 111, 25, 106 – 14, 15, 21, 26, 13.53 0B2 8 15 115, 178, 142, 67, 107, 111, 25, 106 140 45, 78, 89, 92, 13.89 1

7 B3 7 16 178, 142, 67, 107, 111, 25, 106 140, 115 133, 150, 179, 32 18 32 33 36 23 22 15.68 8B4 6 17 142, 67, 107, 111, 25, 106 140, 115, 178 198, 215, 19.89 23B5 5 18 67, 107, 111, 25, 106 140, 115, 178, 142 222 21.49 59

C1 8 12 115, 178, 142, 25, 140, 107, 106, 111 – 15, 21, 26, 45, 13.21 0C2 7 13 178, 142, 25, 140, 107, 106, 111 115 78, 92, 133, 15.08 76 50 32 33 36 23 22 –C3 6 14 142, 25, 140, 107, 106, 111 115, 178 150, 179, 19.39 22C4 5 15 25, 140, 107, 106, 111 115, 178, 142 198, 215, 222 20.88 58

D1 6 10 207, 107, 111, 140, 176, 141 – 8.35 015, 26, 28, 30, 45,D2 5 11 207, 107, 111, 140, 176 140 8.75 1

5 78, 85, 92, 123, 48 22 23 54 49 – –D3 4 12 207, 107, 111, 140 140, 176 8.76 15215D4 3 13 207, 107, 111 140, 176, 141 11.65 53

E1 5 8 140, 178, 142, 25, 107 – 7.25 015, 92, 26, 45,E2 4 9 178, 142, 25, 107 140 8.23 1

4 133, 198, 21, 50 45 69 32 – – –E3 3 10 142, 25, 107 140, 178 9.27 16150

E4 2 11 25, 107 140, 178, 142 10.55 52

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Engineering Optimization 209

DMA 1. On the contrary, the same index calculated for x = 2 shows that in DMA 5 there arevery few nodes, and even no nodes for x = 5, below pressure h̄. This result confirms the efficacyof DSM since it is clear that a decrease in pressure by a few meters only is acceptable for theplanner. The results achieved for the WDM A in five districts show that the DSM gives the plannerseveral possible solutions with a lower and lower resilience deviation index Ird but with fewerdistricts. A possible choice could be made on WDMA A4 which, although it shows a slightlyhigher deviation for Ird , equal to 10.66, compared with the other three A1, A2 and A3, needs alow number of flow meters, equal to Nf m = 3 (with Nbv = 12), which would simplify the waterbalance and reduce installation and maintenance costs. This choice would be reasonable whenevaluating district statistical and hydraulic indices of WDM A4 which can be considered, in anyway, satisfactory. In Figure 5, the WDM A4 was reported directly on the traditional representationof the hydraulic model of the Villaricca network.

Finally, it is worth stressing that there is another way to choose the number and size of dis-tricts according to the various plan requirements, existing constraints, but also the different PIvalues which could be unsatisfactory after a first selection of districts. The planner’s discretionaryjudgement is a key factor in the partitioning process of a water distribution system. Therefore,Table 7 shows other four WDMs of Villaricca network, created by choosing a different numberof districts Nk and, with reference to WDM D with Nk = 5, by indentifying a different size ofdistricts. This decision aims at showing the robustness of the methodology proposed. WDM B,C, D and E, equal to a number of districts Nk = 7, Nk = 6, Nk = 5 and Nk = 4 respectively,were also created thanks to the DSM by following the same procedure shown in the flow chart(Figure 1) adopted for the WDM A. For a brief comparison of the results, Table 7 shows the indexIrd and the value of F . First of all, it is possible to see from the table how most district partitionsshow acceptable Ird values, ranging between 7.25% and 21.49%. Also in these cases, with anequal total number of closed pipes Nbv , the WDMs with higher Ird have a higher F value.

Furthermore, this comparison shows that WDMs with more districts (WDM B and WDM C)have a greater negative effect on the network performance since, as shown in the fourth columnof Table 7, it is necessary to insert a higher number Nbv of boundary valves. The same happensfor the WDM D with Nk = 5 which, compared with WDM A, shows a worse Ir since, with anequal number of DMAs, it was created by inserting a higher number of boundary valves in spiteof criterion (a).

This result is also confirmed by better Ird values, shown in Table 7, with reference to WDM E1at 4 DMAs, which was created with an even lower number of boundary valves than WDM A1.

4. Conclusions

This article proposes a heuristic design support methodology to define a permanent water districtmetering of a water supply network compatible with system hydraulic performances. The DSMis based on graph theory principles and, specifically, on the identification of minimum dissipatedpower paths, called shortest paths and computed by the Dijkstra algorithm.

The methodology, starting from the shortest paths, first defines the main network layout andthen draws the main graph GM that has specific features to simplify WDM design helping watersystem operators.

The results obtained by testing the heuristic procedure on the network of Villaricca show thatDSM quickly identifies WDMs with satisfactory performance indices and satisfies all designconstraints.

The methodology proposed is robust because, even for large networks that have many possibleWDMs, DSM allows good DMA layouts to be rapidly defined with resilience index val-ues close to original network layout. Furthermore DSM is very flexible because it provides

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210 A. Di Nardo and M. Di Natale

WDM planners with some alternative DMA layouts with different numbers of flow meters andoperational costs.

DSM may provide a valid technical support to water utilities in WDM design overcomingempirical approaches and is a flexible tool applicable to networks of any size using commerciallyavailable software (e.g. EPANET and MATLAB).

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