Water Network Sectorization Based on Graph Theory and Energy Performance Indices

WATER NETWORK SECTORIZATION BASED ON GRAPH THEORY AND ENERGY 1

PERFORMANCE INDICES 2

Armando Di Nardo1, Michele Di Natale

2, Giovanni F. Santonastaso

3, Velitchko G. Tzatchkov

4, and 3

Victor H. Alcocer-Yamanaka5 4

Abstract 5

This paper proposes a new methodology for the optimal design of water network sectorization, which is 6

an essential technique for improving the management and security of multiple-source water supply 7

systems. In particular, the network sectorization problem under consideration concerns the definition of 8

isolated district meter areas, each of which is supplied by its own source (or sources) and is completely 9

disconnected from the rest of the water system through boundary valves or permanent pipe sectioning. 10

The proposed methodology uses graph theory principles and a heuristic procedure based on minimizing 11

the amount of dissipated power in the water network. The procedure has been tested on two existing 12

water distribution networks (WDNs, in Parete, Italy and San Luis Rio Colorado, Mexico) using 13

different performance indices. The simulation results, which confirmed the effectiveness of the 14

proposed methodology, surpass empirical trial-and-error approaches and offer water utilities a tool for 15

the design of multiple-source WDNs in isolated districts. 16

Keywords: water distribution network sectorization; graph theory; resilience; district metering areas; 17

performance indices. 18

19

INTRODUCTION 20

Currently, there is potential to change the traditional approach to the analysis, design, and management 21

of water supply networks from passive approaches to proactive, smart approaches that are based on the 22

development of new monitoring and control technologies and the recent, exponential growth in the 23

computational power of simulation software. The low cost and availability of new monitoring and 24

1 Department of Civil Engineering, Second University of Naples, Aversa, Italy, [email protected]

2 Department of Civil Engineering, Second University of Naples, Aversa, Italy, [email protected]

3 Department of Civil Engineering, Second University of Naples, Aversa, Italy, [email protected]

4 Urban Hydraulics Department, Mexican Institute of Water Technology, Jiutepec, Mexico, [email protected]

5 Urban Hydraulics Department, Mexican Institute of Water Technology, Jiutepec, Mexico, [email protected]

ManuscriptClick here to download Manuscript: Di_Nardo_Manuscript_Minor_revision2.docx

management devices that are controlled by a remote system is cause for optimism because such 25

systems can accelerate the upgrading of systems and the aligning of the water distribution systems to 26

other network services, such as electricity, gas, and the Internet. From this prospective, the possibility 27

of inserting remote control gate valves supports the development of water network partitioning (WNP) 28

for both district metering and sectorization. WNP can assist in modernizing water supply system 29

management, specifically for achieving water balance (Water Authorities Association and Water 30

Research Centre 1985), applying pressure control techniques (Alonso et al. 2000; Nicolini and Zovatto 31

2009; Di Nardo and Di Natale 2011), and protecting users from malicious attacks (Poulin et al. 2008; 32

Grayman et al. 2009; Di Nardo et al. 2012a). The partitioning of a water distribution network (WDN) 33

can be achieved by defining permanent districts, called district meter areas (DMAs), through the 34

insertion of boundary valves (or sectioning of existing pipes) and flow meters to create subsystems to 35

simplify water balance and identify water losses. The boundary (or gate) valves can be closed 36

permanently or controlled by a remote system. 37

The occurrence of new breaks can be identified by monitoring the inflows to DMAs and noting where 38

the minimum night flow (MNF) increases. This information enables a water utility to intervene and 39

repair leaks once the optimum level of leakage is exceeded. This methodology was initially developed 40

and applied in the United Kingdom (UK) and was then adopted in many other countries (Farley 2001; 41

AWWA 2003; Morrison 2007). A recent study (Fanner et al. 2007) demonstrated that WNP must 42

account for three key issues: the DMA has to meet design and fire flow requirements, it must be 43

metered in a practical and economical manner, and it must be designed to ensure that water quality is 44

maintained. Specifically, Sturm and Thornton (2005) and Fanner et al. (2007) identified the five 45

general design and planning criteria to meet these requirements for DMAs. First, closed pressure 46

reducing valves (PRVs) and check valves can be used as boundary valves to provide fire flow when 47

required. Second, boundary valves should be on smaller mains to minimize the effect of dead ends. 48

Additionally, where possible, include large customers near boundaries or dead ends to avoid water 49

stagnation and water quality problems. Furthermore, if the DMA cannot provide required fire flows or 50

minimum pressure for fire sprinkler systems when supplied through one feed, then it is necessary to 51

provide two feeds into the DMA, with one feed equipped with a meter and the other equipped with 52

PRVs that open up only for a fire flow event or if demand is too high for only one feed. Finally, the 53

DMA can be supplied through multiple metered feeds on a daily basis; the flow can be supplied 54

through a single meter on a temporary basis when data are required for analysis. 55

The current best management practices, endorsed by the International Water Association (IWA) 56

(Lambert 2002), World Health Organization (WHO) (Farley 2001), and American Water Works 57

Association (AWWA) (2003), for the control of real (physical) water losses can be summarized by the 58

following four actions: a) conduct pressure management, b) achieve active leakage control, c) improve 59

the speed and quality of leak repairs, and d) increase main replacement and rehabilitation. 60

Pressure management of individual water sources is challenging for interconnected networks because 61

networks are affected by other water sources; it is therefore preferable to have a single feed. Active 62

leakage control involves MNF measurements and computation of performance indicators. MNF 63

measurements are essentially applicable only to isolated zones that correspond to small DMAs with 64

zooming or step-testing methodologies (Water Industry Research Ltd. 1999; Farley 2001; Di Nardo et 65

al. 2012b). In interconnected networks, performance indicators can only be computed on a global basis 66

(for the entire network), not individually (for each source). Furthermore, leak repairs, pipe replacement, 67

and pipe rehabilitation can be achieved faster and more simply in isolated zones because only one 68

source needs to be closed during maintenance, and this process does not cause alterations to the other 69

parts of the networks. Moreover, most cities, particularly medium and large cities, are fed by multiple 70

water sources. This network design is not the outcome of a unified planning and design process but 71

instead results from years of rapid responses to new and continually rising demands and the expansion 72

of existing WDNs. Consequently, water sources in many cities are hydraulically interconnected by the 73

city distribution network, and there is not a clear delineation of the zones supplied by each water 74

source. Although this situation can advantageous to the hydraulic redundancy of a water system, it 75

creates challenges for water quality management. The water quality of each water source varies, which 76

makes it difficult to predict and control water quality within a WDN with interconnected water sources. 77

In such cases, which are common in developing countries, the benefits of many pipe loops are less 78

important than the benefits obtained with the use of the “divide-and-conquer” paradigm for the water 79

supply network (Tzatchkov et al. 2006a, 2006b). In addition, WNP may reduce the risk of malicious or 80

accidental contamination of a water supply network (Poulin et al. 2008; Grayman et al. 2009), and 81

complete isolation of a DMA is more effective for network protection (Di Nardo et al. 2012a). 82

A good solution is to divide the WDN into isolated zones (sectors) such that each zone is fed by its 83

water source (or water sources); this process is referred to as sectorization. Sectorization is achieved by 84

closing gate valves in the network pipes that link the DMAs. Although the term “sectorization” is also 85

used as a synonym for “division of DMAs,” the type of water system partitioning is defined as water 86

network sectorization (WNS) in this paper to highlight the condition wherein each district in the system 87

is completely separated (or isolated) from all other districts and can thus be called an isolated DMA (i-88

DMA). WNS represents a more difficult challenge than other forms of WNP because there are a greater 89

number of boundary gate valves and the districts are not connected. Because water systems are 90

traditionally designed with many connections and loops (Mays 2000), the closure of gate valves has the 91

potential to degrade the hydraulic performance of water networks, if the system is not properly 92

designed. The reduction in the number of pipes through which water can travel may reduce water 93

pressure, particularly during peak demand, which can lead to diminished levels of service for users and 94

water system redundancy (Di Nardo and Di Natale 2011). 95

This paper shows that it is possible to obtain a sectorized network layout that is compatible with 96

hydraulic performance. Most water supply systems have been designed without optimization criteria, 97

instead using only the general idea of defining loops and inserting pipes in each street. Therefore, 98

although redundancy is a good aim, it is possible that some loops can be interrupted to yield the 99

benefits of sectorization. Moreover, relatively low-cost devices for automatic sectorization (remotely 100

controlled gate valves) can currently be found on the market, and their use ensures that the system can 101

be easily adjusted to accommodate specific unforeseen situations (e.g., breaks, maintenance, fire 102

protection) by restoring the loops, as also suggested by Sturm and Thornton (2005) and Fanner et al. 103

(2007). 104

DMAs and i-DMAs have traditionally been designed based on empirical guiding principles (such as the 105

maximum number of properties or total length of pipes in a DMA) (Water Authorities Association and 106

Water Research Centre 1985; WRC/WSA/WCA Engineering and Operations Committee 1994; Water 107

Industry Research Ltd. 1999; AWWA 2003) combined with trial-and-error procedures. For example, a 108

feasible solution is developed by choosing the pipes to be closed and repeatedly running a simulation 109

model of the WDN is run repeatedly until acceptable pressure and flow conditions are met. Such a 110

procedure lacks any rational basis; if a feasible solution is found, its quality compared to other feasible 111

solutions is unknown. Because there are a vast number of possible network sectorization schemes, even 112

in small networks (Di Nardo and Di Natale 2011), identifying the best option by trial-and-error 113

procedures is difficult. 114

The optimal definition of a district in a WDN is one of the “layout problems” of WDNs that is widely 115

discussed in the literature. It has two main classifications that differ from one another: topology and 116

connectivity (Goulter and Morgan 1985; Ostfeld 2005; Giustolisi et al. 2008a; Deuerlein 2008) and 117

reliability and security (Wagner et al. 1988a, b; Ostfeld and Salomons 2004). 118

Some techniques have been published for designing DMAs, such as techniques based on multi-agent 119

systems (Wooldridge 2002), spectral clustering techniques (Ng et al. 2011), graph theory principles 120

(Biggs et al. 1986), and graph partitioning (Chevalier and Safro 2009). With reference to multi-agent 121

systems, Izquierdo et al. (2011) recently proposed an original procedure to define the DMAs of a water 122

supply network in which each agent is a consumption node with a number of associated variables 123

(elevation and demand are most important) that obtains different WNP scenarios. A spectral clustering 124

technique was proposed by Herrera et al. (2010) to partition a water supply network using dissimilarity 125

matrices (transformed into weighted kernel matrices) that are obtained from graphical and vector 126

information (pipes, demand nodes, and water constraints). 127

Referencing the use of graph theory principles, Ostfeld and Shamir (1996) introduced the concept of a 128

water network backup subsystem to define a subset of system links where a prescribed level of service 129

is maintained when failure occurs. Tzatchkov et al. (2006a) subsequently suggested an algorithm 130

derived from graph theory to identify independent supply sectors (or districts) of a network layout 131

based on the last-in-first-out (LIFO) stack approach. More recently, Giustolisi and Savic (2010) 132

described an algorithm for identifying the association between valves and isolated segments (or 133

sectors) based on the use of topological matrices of a network whose topology was modified to account 134

for the existence of the valve system; furthermore, a genetic algorithm (GA) was used to minimize the 135

number of isolation valves and the maximum total undeliverable demand. A heuristic design support 136

methodology (DSM) for partitioning a water supply system in DMAs was later proposed by Di Nardo 137

and Di Natale (2011). This DSM is based on graph theory and the use of energy indices; it allows for 138

the analysis of the minimum energy paths, which are computed from each reservoir to each node in a 139

water network, and it supports definition of the optimal districts. Finally, graph partitioning techniques 140

borrowed from informatics have been proposed by Sempewo et al. (2008) and Tzatchkov et al. (2012) 141

as tools for the optimal demarcation of water networks into zones based on balancing length, demand, 142

or flow within zones. More specifically, Di Nardo et. al. (2011) introduced an automatic methodology 143

for defining DMAs by integrating software with graph partitioning and hydraulic simulation and for 144

optimizing the definition of DMAs using an energetic approach. All of these approaches, with the 145

exception of that of Tzatchkov et al. (2006a), addressed the design of DMAs (i.e., WNP) but did not 146

explicitly address i-DMAs (i.e., WNS). This paper presents a novel approach for the automatic design 147

of i-DMAs that is based on graph theory principles coupled with a heuristic optimization technique for 148

the selection of pipes to close. It further minimizes an objective function developed for energy criteria. 149

The proposed approach has been applied to case study WDNs. 150

METHODOLOGY 151

The proposed methodology for WNS is illustrated in the flow chart in Figure 1. It is composed of the 152

following steps, which are illustrated using the example of a small hydraulic network (Figure 2a). 153

a) Define a simple (original) graph of the water network. 154

The first step consists of a) defining, from the adjacency matrix, a simple graph of the network 155

G=(V,E), where V is the set of n vertices (or nodes) and E is the set of m edges (or links). 156

b) Find the independent sectors using a depth first search (DFS) algorithm. This phase is based on 157

graph theory algorithms that are generally supposed to be more efficient than algorithms based on a 158

linear algebra topological matrix in terms of computation speed and storage requirements (Giustolisi et 159

al. 2008a; Giustolisi and Savic 2010). The graph theory algorithm that was used, known as depth first 160

search (DFS), was proposed by Tarjan (1972) and allows for the exploration of the connectivity of a 161

graph. The DFS algorithm begins at some node and explores as far as possible along each path (in 162

“depth”) until there are no more adjacent unvisited nodes; only then does it start a new path. This 163

algorithm is different from the breadth first search (BFS) algorithm (Pohl 1969), which starts at a root 164

node and explores all of the adjoining nodes (in “breadth”) until there are no more adjacent unvisited 165

nodes. Perelman and Ostfeld (2011), who applied the DFS algorithm to a water supply network, 166

proposed a procedure for a topological clustering of the nodes that can be utilized for different 167

purposes, such as for water security enhancements through sensor placements at clusters or for efficient 168

isolation of a contaminant intrusion. More specifically, Tzatchkov et al. (2006a) used the DFS 169

algorithm to identify the independent sectors of a WDN. The algorithm allowed for the identification of 170

all possible independent sectors (step b) starting from each source node in the network. The source 171

nodes correspond to root nodes. 172

The application of the DFS algorithm made it possible to identify a new graph structure of the network, 173

composed of trees and branches, called a DFS forest graph (Cormen et al. 1990). With regard to the 174

small network in Figure 2a, the DFS routine starts from nodes A and B (the two sources of the water 175

network or roots of the graph) and easily locates two trees with branches, as illustrated in Figures 2b 176

and 2c. 177

c) Identify the hierarchical level (HL) of the graph that corresponds to each source (common nodes 178

may exist). A hierarchical approach (Di Battista et al. 1999) was chosen to draw the tree graph in which 179

all network nodes were represented by different layers (levels) with a distinct hierarchy of connection. 180

In this approach, hierarchy can easily be identified, and there is a correspondence between visual 181

perception and network connection analysis; subsequently, “ancestor” and “descendant” nodes can be 182

defined (Di Battista et al. 1999) that assign a specific hierarchical level (HL). 183

A node set st is a network subgraph that can be associated with a specific category of graph, called a 184

tree graph, where any two vertices in the graph are connected by only one path. With regard to the 185

example in Figure 2a, with s=2, the two tree graphs, 1t and 2t , had 11 hierarchical levels (step c), 186

as illustrated by Figures 2b and 2c. 187

d) Obtain the independent and common node sets for each HL of the graph. In Figure 2b, starting from 188

the first level of tree graph 1t (corresponding to source node A), the algorithm examines whether there 189

are nodes common to the other tree graph, 2t (corresponding to source node B) for each HL. Figure 3 190

shows that until HL 4, no node from set 14

1 tt

belongs to set 2

4

2 tt

; therefore, the 4

1t

is 191

fixed as a subset of i-DMA1, and the 4

2t

is fixed as a subset of i-DMA2. Starting from HL 5, there are 192

nodes that belong to sets 4

111 ttt

and 4

222 ttt

; thus, it was necessary to choose which 193

i-DMA was better for assigning these nodes. Until this step of the methodology, hydraulic simulation 194

of the network was unnecessary, and the graph was treated as an undirected graph. 195

e) Design the required i-DMA limiting gate valves using a heuristic procedure based on the 196

minimization of dissipated power and a GA. This step, based on hydraulic simulation and an 197

optimization procedure, can be divided into two substeps as follows: 198

e1) Perform hydraulic analysis. A heuristic procedure was applied to design the required i-DMAs 199

using the output from the hydraulic analysis. The heuristic procedure defined a new subset: 200

21 ttC

(1) 201

that was divided into two subsets, {C1} and {C2}, with 021 CC , to obtain a water subsystem 202

supplied by only one source through the insertion of gate valves in the links (pipes) between the nodes 203

that belong to subsets {C1} and {C2}. In the example in Figure 2, the nodes belonging to i-DMA1 were 204

14

1 Ct , and the nodes belonging to i-DMA2 were 2

4

2 Ct . 205

e2) Achieve node swapping between common node sets such that the amount of dissipated power is 206

minimized. 207

To locate the two subsets {C1} and {C2} (step e2), a special technique of node swapping (Kernighan 208

and Lin 1970; Fiduccia and Mattheyses 1982) was developed by the authors through the 209

implementation of a suitable GA (Goldberg 1989). This GA determines the optimal layout for the i-210

DMAs by inserting valves in the pipes through node swapping, that is, moving some nodes belonging 211

to subsets {C1} and {C2} from one subset to another while maintaining compliance with a specific 212

objective function (OF). The equation that defines the OF was chosen by following the applicable 213

results obtained through an energy approach, introduced by Di Nardo and Di Natale (2011), for 214

selecting pipe closures that minimize the dissipated power of the water network. WNS changes the 215

system layout by increasing head loss and internal power dissipation but decreasing “diameter 216

availability” (i.e., the number of pipes through which water can travel) and energy redundancy; this 217

effect is caused by the closed valves, which reduce the number of network pipes and remove some 218

network loops. Therefore, each WNS, obtained by inserting gate valves and thus closing some pipes, 219

will increase the dissipated power and reduce the energy resilience of the water system. For this reason, 220

network resilience can be a useful way to compare different system layouts by their dissipated power. 221

Prior to node swapping, it was necessary to perform a hydraulic analysis that assigned a flow and head 222

loss to each pipe used to compute the dissipated power (Di Nardo and Di Natale 2011). Pressure driven 223

analysis (PDA) (Giustolisi et al. 2008; Giustolisi et al. 2011) was used for a network with a given node 224

water demand distribution Qi, with i=1..n; source heads Hs, with s=1..r; reservoirs; pipe lengths Lj, and 225

node elevations zi. The analysis provided the pipe flows qj, with j=1..m;, node heads Hi, and head loss 226

ΔHj for each pipe. PDA is an appropriate approach for analyzing WNS designs because hydraulic 227

performance may be affected, which could cause the water pressure to fall below the design pressure. 228

The power balance of a water network (Di Nardo and Di Natale 2011) can be defined as 229

NDA PPP (2) 230

where

r

s

ssA HqγP1

is the available power (or total power), qs and Hs are the discharge and head 231

relevant to each reservoir, respectively, and γ is the specific weight of water.

m

j

jjD HqγP1

is the 232

dissipated power (or internal power), where qj and ΔHj are the flow and head loss for each network 233

pipe, respectively, and

n

i

iiN HQγP1

is the node power (or external power), where Qi and Hi are the 234

water demand and head at each network node, respectively. 235

In this manner, the chosen objective function was the sum of the dissipated power in all ms=(m-Nbv) 236

pipes of the sectorized network (where Nbv is the number of links in which boundary valves are 237

inserted) as follows: 238

sm

j

jj HqγOF1

min (3) 239

The minimization of (3) was conducted with a GA by employing the Genetic Toolbox of MATLAB©

240

(Mathworks, Inc., Natick, MA, USA). 241

The decision variables in this minimization consisted of assigning each node in the subset {C} to either 242

{C1} or {C2}, and the constraints were that both i-DMA1 and i-DMA2 had to be “connected 243

subgraphs”. 244

Therefore, for each source s, every individual in the GA was composed of a sequence of chromosomes 245

whose length was equal to the number of nodes that belonged to subset {C}. 246

Each chromosome i (decision variable) assumed the value 0 (zero) if the i-th node belonged to {C1} 247

and was thus assigned to i-DMA1; alternatively, it assumed the value 1 (one) if it belonged to {C2} and 248

was thus assigned to i-DMA2. Then, the optimization procedure had to check whether i-DMA1 and i-249

DMA2 remained connected subgraphs, that is, whether there was a path from any point to any other 250

point inside them. In this paper, this step was performed by a DFS algorithm (optimization constraint). 251

One-hundred generations were performed with a population that was composed of 20 individuals and a 252

crossover percentage of Pcross=0.8. Next, by repeating the procedure illustrated in the flow chart in 253

Figure 1 (from step (d) to step (e)) for each of the other r-1 sources, the remaining i-DMAs (or 254

districts) could be identified. 255

In general, it is possible to divide the network into i-DMAs only by a heuristic approach, although there 256

are a vast number of possible combinations. The DFS algorithm allowed for the identification of the set 257

{C}, which significantly simplified the search for suboptimal solutions. The GA (or another heuristic 258

optimization procedure) was applied only to the nodes belonging to set {C}, which greatly reduced the 259

domain of possible solutions by automatically eliminating a large number of inadequate solutions that 260

may have been included when using a purely heuristic algorithm, such as i-DMAs that include nodes 261

that are disconnected from sources. 262

f) Compute the performance indices of the new (sectorized) water system and compare them to the PIs 263

of the original system. Finally, in step f), performance indices (PIs) were computed to evaluate the 264

expected alteration of network hydraulic behavior due to the sectorization; specifically, the following 265

three categories of PIs were used to assess different sectorization layouts: 266

f1) resilience indices. Prasad and Park (2004) proposed the concept of network resilience, which 267

combines the effects of surplus power and reliable loops. Specifically, the surplus power at node i is 268

given by , where

**

iii hzH and *

ih is the design pressure for the i-th node; a 269

loop is considered reliable if the pipes incident with a node do not vary widely in diameter. Thus, the 270

uniformity at node i is given as 271

ip

ip

nip

n

j

j

iddn

d

C

,

,

,...,max 1,

1

(4) 272

where np,i is the number of pipes incident with the node i and dj is the diameter of the incident pipe.

273

In this manner, the following index is defined: 274

- Resilience network index (Prasad and Park 2004): 275

max

1

,

D

n

i

isi

rnP

PCγ

I (5) 276

Higher values of Irn indicate better WNSs due to the higher values of available power surplus, a more 277

uniform incident pipe distribution, and thus higher network resilience. To compare different layouts of 278

the network, a new index was proposed in this work as follows: 279

- Resilience network deviation index: 280

1001*

n

nrnd

I

II 1001

*

rn

rnrnd

I

II (6) 281

where *

rnI is the resilience network index of the WNS layout. This index immediately shows the 282

resilience network percentage deviation between the WNS and original water network (OWN), with 283

higher values of Irnd indicating a worse WNS.

284

f2) pressure indices. Energy indices refer to the entire water network, whereas WNS also affects 285

individual i-DMAs; therefore, other district-specific indices were employed. These types of indices, 286

such as the mean hmean, maximum hmax, minimum hmin, and standard deviation hsd, are traditionally used 287

to measure node pressure deviation and aid in summarizing the most important information about the 288

level of service of a water system. 289

f3) flow deficit index. This type of index was computed in the PDA approach in a manner similar to the 290

total unsupplied nodal demand (Giustolisi et al. 2008c) index; in the demand driven analysis (DDA) 291

approach (Todini and Pilati 1988), this index is always equal to 1.00 because 292

i

ia

iiia

iiia

n

i

i

n

i

ii

fd

Q

QαQQ

αQQ

Q

Qα

I ,

,

,

1

1

0

1

(7) 293

where Qa,i represents the actual nodal demand delivered in the PDA approach. 294

CASE STUDIES 295

The methodology was applied to two case studies of real WDNs: a) Parete, a small network in Italy (Di 296

Nardo and Di Natale 2012), and b) San Luis Rio Colorado, a large network in Mexico (Tzatchkov et al. 297

2006b). 298

Parete, with a population of 10,800, is located in a densely populated area south of the province of 299

Caserta (Italy). Water consumption can be characterized as exclusively residential with predominantly 300

three- to four-story houses that were built in the 1970s and 1980s. The network is supplied by two 301

sources. 302

San Luis Rio Colorado is a Mexican city located in the northern part of the state of Sonora, which is 303

near the Mexico-United States (US) border. As of 2010, the city had a population of 178,376 and a 304

total of 48,400 connections, of which 45,850 were residential, 2,445 were commercial, and 105 were 305

industrial. The distribution network was approximately 50 years old and was composed of 60-mm to 306

500-mm asbestos cement and plastic (polyvinyl chloride, PVC) pipes. This case study is a clear 307

example of why a municipality should, under similar conditions, sectorize its network into i-DMAs. 308

The water supply sources consisted of 18 deep water wells, which were fully interconnected by the 309

distribution network at the beginning of the sectorization project and which did not have water tanks. 310

Some of the well pumps were equipped with variable speed drives that allowed the pumps to follow 311

water demand variation and stopped them when water demand was very low. The extent of the areas 312

supplied by each well was unknown, and it was suspected that some well pumps were frequently 313

stopped, not because of low demand but because of higher hydraulic heads at other wells. Thus, MNF 314

measurements and pressure management were worthless, and computation of performance indicators 315

was only possible for the entire network. The indicators were not able to reveal the problematic (high 316

water loss) areas of the network. The division of the city WDN into i-DMAs was the only way to 317

accomplish all of the following necessary actions: optimization of the pump operation, water balance 318

analysis for each source, pressure management, and improved water quality control. The main 319

characteristics of the hydraulic models for the two networks are reported in Table 1. The water 320

networks were modeled by WDNetXL (Giustolisi et al. 2008b) in a PDA. The hydraulic simulations 321

were specifically conducted for peak water demand in the summer because permanent WNS effects on 322

hydraulic performance were deemed more important under this operating condition. 323

The Parete network. The Parete network had a low original resilience network index of Irn=0.33 that 324

was computed with a design pressure of h*=25 m for each node, indicating a “low availability” (Greco 325

et al. 2012) of the water system to be partitioned. In other words, it would be difficult to change its 326

original layout with the insertion of valves without a significant decrease in hydraulic performance. 327

Robustness describes the ability of a system to maintain given performance levels in the presence of 328

unfavorable variations in operating conditions (e.g., closure or burst of a pipe). It was possible to say 329

that the Parete network had a low robustness and low availability for sectioning because resilience can 330

be used as a surrogate measure for robustness (Greco et al. 2012), 331

For this case study, the challenge was to develop a WNS scheme that did not significantly affect 332

hydraulic performance with only the use of traditional approaches based on empirical suggestions and 333

simulation techniques. 334

As expected, the WNS design of the Parete network, which was achieved with the proposed 335

methodology, isolated two i-DMAs; each was supplied by one reservoir. The corresponding simulation 336

results are reported in Tables 2 and 3, which provide the power balance and performance indices for the 337

entire OWN and WNS and for each i-DMA. The global results, as shown in Table 2, confirm the 338

effectiveness of the proposed methodology, with suitable values for resilience deviation indices 339

(Irnd=7.59%) that indicate a low alteration of hydraulic performance of the OWN. The sectorization was 340

achieved by inserting Nbv=6 boundary valves, which allowed for complete isolation of each i-DMA. 341

This result is widely compatible with the hydraulic performance of the network and the level of service 342

for users, as confirmed by the pressure indices reported in Table 3; all mean, maximum, and minimum 343

values were very close to the corresponding original values for the entire network and each district. 344

The flow deficit index showed a delivered flow that was almost equal to the design demand (Ifd=0.999 345

in both the OWN and WNS). This result was expected based on the implications of the PDA approach 346

for a resulting water pressure that was slightly below the design pressure (25 m) at a few network 347

nodes, as can be deduced from the values of hmin in Table 3. In contrast, the simulation results show a 348

slight improvement of mean and minimum pressure due to the different network layout and to a 349

reduction of Ifd. 350

Finally, in Figure 2, the two i-DMAs (represented with boundary lines) of the Parete WNS, which were 351

obtained with the proposed methodology, are shown. Valves were inserted in all pipes that intersected 352

the boundary lines of the DMAs. 353

San Luis Rio Colorado network. Unlike the Parete network, the original San Luis Rio Colorado 354

network had a low resilience network index of Irn=0.47 because the network had great diameter 355

variability. Resilience was computed with a design pressure of h*=15 m based on local design 356

standards. The simulation results are reported in Tables 4, 5, and 6. The proposed methodology isolated 357

10 i-DMAs, each of which was supplied by one or two water wells, as illustrated in Figure 5. Table 4 358

shows a very low alteration of Irnd of 9.39% despite the insertion of Nbv=168 boundary valves. 359

In Table 5, the computed pressure indices reveal that the results are compatible with the level of service 360

for users in all of the i-DMAs, as the pressure in the nodes of all i-DMAs are close to their original 361

values. The mean and maximum pressure values, as reported in Table 5, are suitable for each i-DMA, 362

which confirms that the proposed methodology accommodated a change in the original layout of the 363

water system without significantly affecting its performance. 364

The values of the available power supplied by each well before and after the water network 365

sectorization are listed in Table 6, along with the corresponding i-DMAs. All isolated districts were 366

supplied by a pair of water wells, except for i-DMA8 and i-DMA9. The percentage deviations were 367

between +8.75% and -5.01%, which indicates quality performance for all water wells. 368

This hydraulic performance was confirmed by the values of the obtained flow deficit index that 369

indicated that flows delivered in each i-DMA were equal to the design demand of Ifd=1.0, given that 370

water pressure values were above the design pressure (15 m) at all network nodes, for both OWN and 371

WNS, as can be deduced from Table 5. 372

Finally, the sectorization obtained by the proposed methodology was compared to the sectorization 373

previously obtained for the San Luis Rio Colorado network by the traditional trial-and-error procedure 374

(Tzatchkov et al. 2006b). A better WNS was obtained in this study, with a significant improvement in 375

resilience indices (Irn=0.43 versus the previous Irn= 0.32), which was certainly related to a lower 376

number of pipe closures (Nbv=168 versus Nbv=170) and the minimization of dissipated power achieved 377

by the proposed methodology. The WNS1 achieved by Tzatchkov et al. (2006b) did not satisfy the 378

minimum pressure constraint of hmin=11.42, and thus, in the PDA approach, the node water demand 379

was also not fulfilled, with Ifd=0.997, whereas WNS2 satisfied all indices with hmin=16.33 and Ifd=1.00. 380

CONCLUSIONS 381

WNS, which refers to the division of a network into i-DMAs, represents an important technique for 382

improving multiple-source water network management. However, designing WNS for large water 383

distribution systems is a very complex task. A few relevant methodologies have been proposed for the 384

design of districts that are compatible with hydraulic performance, but these methods mainly address 385

DMAs. In this paper, a new methodology for the design of i-DMAs was proposed and applied to two 386

case studies of existing city WDNs: Parete (Italy) and San Luis Rio Colorado (Mexico), where 387

sectorization projects are in progress. For these case studies, optimal isolated districts were designed to 388

be supplied exclusively by one or two water sources and disconnected from other sectors through gate 389

valves or by sectioning existing pipes. The proposed methodology was based on graph theory 390

principles, a DFS technique for searching independent branches of the network, and energy 391

considerations to minimize dissipated power by employing a specially developed GA for node 392

swapping. Using performance indices, a comparison of the simulation results with otherwise obtained 393

sectorization layouts confirmed the effectiveness of the methodology in defining a WNS that was 394

compatible with the level of service for users and that required negligible alterations to hydraulic 395

performance with regard to network resilience and supplied node water demand. The simulation results 396

showed that it is possible to find sectorization layouts that are compatible with network resilience and 397

robustness and that also have almost the same level of fire protection as the original network, as 398

demonstrated by only slight alterations to the minimum pressure in each sector. The use of gate valves 399

controlled by a remote system can ensure a rapid recovery of the original redundancy, in preserving the 400

capacity to open pipe links to face specific situations (i.e., breaks and maintenance). Furthermore, the 401

proposed methodology to sectorize a water network (or to define i-DMAs) can also be used to partition 402

(or to identify DMAs), leaving open some pipes between districts. Clearly, the sectorization represents 403

the most difficult challenge for WNP design. If a WNS is good, a corresponding WNP, with some gate 404

valves opened, will certainly be better. 405

Finally, the methodology can be applied to large water systems. It offers water utilities a design tool 406

that is based on performance indices and exceeds the empirical trial-and-error approaches that are 407

traditionally used for WNS. 408

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523

Notation 524

525

Ci Pipe uniformity at node i

C Set of nodes subject to swapping

1C Subset of C supplied exclusively from source 1

2C Subset of C supplied exclusively from source 2

dj Diameter of the incident pipe [mm]

E Set of edges

G Graph of the network

Hi Node heads [m] *

iH Design pressure for the i-th node [m]

HL Hierarchical level

Hs Head at source s [m]

i Node index, chromosome

Ifd Flow deficit index

Irn Resilience network index

Resilience network index of the sectorized layout

Irnd Resilience network deviation index

j Pipe index

Lj Length of pipe j s [m]

m Number of edges

ms=m - Nbv Number of links in which boundary valves are not inserted

n Number of vertices

Nbv Number of links in which boundary valves are inserted

np,i Number of pipes incident with node i

OF Objective function

PA Available power [W]

Pcross Crossover percentage

PD Dissipated power [W]

PDmax Maximum power necessary to satisfy the demand Qi and node head constraints [W] *

DP Dissipated power computed for the WNS layout [W]

PN Node power [W]

Ps,i Surplus power at node i

Qa,i Actual demand delivered at node i in the PDA approach [m3 sec

-1]

Qi Water demand at node i [m3 sec

-1]

qj Flow in pipe j [m3 sec

-1]

qs Discharge of each reservoir [m3 sec

-1]

r Number of reservoirs (sources)

s Source index

st Network tree subgraph stemming from source s

HL

st

Set of nodes in st belonging to level HL

*

rnI

HL

sss ttt

Set of nodes in st excluding nodes belonging to level HL

V Set of vertices

zi Elevation of node i [m]

αi Reduction factor

Specific weight of water [N m-3

]

ΔHj Head loss in pipe j [m]

526

Table 1. The hydraulic characteristics of two networks

Network characteristics Hydraulic network

Parete San Luis Rio Colorado

Number of nodes, n 182 1890

Number of links, m 282 2681

Number of reservoirs, r 2 18

Number of pumps, p 18

Hydraulic head of reservoirs

or geodetic height of water level in

wells [m]

110.0 -2,00; -8.87; -6.45; -2.85; -9.38;

-0.75; -4.10; -7.23; 0.05; 0.62;

-3.19; -3.80; 3.55; 2.43; -7.32;

-3.71; 1.85; 3.73

Total pipe length, LTOT [km] 32.7 599.06

Minimum ground elevation, zMIN [m] 53.1 0.00

Maximum ground elevation zMAX [m] 78.6 40.11

Pipe materials cast iron asbestos cement and PVC

Pipe diameters [mm] 60; 80; 100; 110; 125; 150;

200

60; 62.5; 75; 100; 150; 200;

250; 300; 350; 400; 450; 500

Peak demand, Q [m3/s] 0.110 1.735

Design pressure head, h* [m] 25 15

TableClick here to download Table: Di_Nardo_Tables_Minor_revision.docx

Table 2. Power and Energy Indices of Parete

Layout

Power (kWatt) Resilience Indices Nbv

PA PN PD Irn Irnd

- % -

OWN 120.73 104.14 16.59 0.33 - -

iDMA1 54.66 47.91 6.75 0.31 7.59 6

iDMA2 66.19 55.64 10.55

Table 3. Pressure Indices of Parete

Layout hmean hmin hmax hsd

OWN 31.40 21.61 50.53 5.67

WNS 31.66 23.67 49.92 4.31

iDMA1 29.93 23.67 38.89 2.90

iDMA2 35.08 27.88 49.92 4.61

Table 4. Power and Energy Indices of S.L. Rio Colorado

Layout

Power (kWatt) Resilience Indices Nbv

PA PN PD Irn Irnd

% -

OWN 1118.62 1076.77 41.85 0.47 - -

1 99.74 98.28 1.47

0.43 9.39 168

2 128.90 125.98 2.91

3 58.13 54.74 3.39

4 97.10 87.25 9.85

5 117.94 114.41 3.53

6 150.53 142.04 8.49

7 131.08 127.37 3.71

8 62.09 60.32 1.76

9 103.56 99.93 3.63

10 170.68 158.63 12.06

Table 5. Statistical Indices of S.L. Rio Colorado

Layout hmean hmin hmax hsd

OWN 28.87 20.97 62.23 4.28

WNS 29.21 16.33 69.28 6.93

1 32.96 23.97 46.23 5.68

2 33.29 29.15 69.28 2.80

3 20.89 16.33 32.73 3.30

4 23.76 18.89 37.75 5.00

5 34.55 29.35 48.01 3.62

6 34.88 28.81 43.36 3.96

7 18.74 16.51 51.54 2.54

8 20.17 19.03 21.48 0.54

9 25.72 23.98 29.00 1.14

10 34.34 31.56 40.16 1.91

Table 6. Available Power of each water well before and after WNS

Water

well iDMA

PA [KWatt] Water

well iDMA

PA [KWatt]

OWN WNS OWN WNS

1 2 61.33 61.83 10 1 53.54 52.84

2 9 104.69 103.56 11 7 89.22 86.39

3 6 77.24 76.87 12 7 44.25 44.69

4 10 88.51 90.33 13 5 30.07 30.14

5 8 59.67 62.09 14 4 40.34 42.30

6 4 56.30 54.80 15 10 76.67 80.35

7 1 51.40 46.90 16 3 40.35 41.93

8 6 77.09 73.66 17 2 63.87 67.07

9 5 88.13 87.80 18 3 15.97 16.20

Figure_1Click here to download high resolution image

Figure_2Click here to download high resolution image

Figure_3Click here to download high resolution image

Figure_4Click here to download high resolution image

Figure_5Click here to download high resolution image

Figure 1 – Flow chart of proposed methodology

Figure 2 – a) Example network, b) DFS tree graph , c) DFS tree graph

Figure 3 – Network Sets for swapping phase

Figure 4 – Water Network Sectorization design of Parete with two i-DMAs. Bold lines show district

limits.

Figure 5 – Water Network Sectorization design of San Luis Rio Colorado with ten i-DMAs. Bold

lines show district limits.

1t 2t

Figure Caption List

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3

13

31

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500

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6771 subtototal

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TablesWord

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Word

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2 158 2 315

3 158 3 158

4 158 4 158

5 158 5 315

6 158 67 7

8 89 9

10 10

11 11

12 12

13 13

14 14

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2681 17

7269 18

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9950 20 and 21

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WATER NETWORK SECTORIZATION BASED ON GRAPH THEORY AND ENERGY PERFORMANCE INDICES

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Date Issued:October 24, 2012

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This certificate may be verified at www.journalexperts.com/certificate. This document certifies that the manuscript listed above was edited for proper Englishlanguage, grammar, punctuation, spelling, and overall style by one or more of the highly qualified native English speaking editors at American JournalExperts. Neither the research content nor the authors' intentions were altered in any way during the editing process. Documents receiving this certificationshould be English-ready for publication; however, the author has the ability to accept or reject our suggestions and changes. To verify the final AJE editedversion, please visit our verification page. If you have any questions or concerns about this edited document, please contact American Journal Experts [email protected].

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punctuation, spelling, and overall style by one or more of the highly qualified native

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Manuscript title:ASCE Journal of Water Resources P lanning and Management Paper

Authors:Armando Di Nardo, Michele Di Natale, G iovanni F. Santonastaso, Velitchko G. Tzatchkov and Victor H. Alcocer-Yamanaka

Date Issued:February 7, 2013

Certificate Verification Key:81EA-A665-430B-B6AE-046D

This certificate may be verified at www.journalexperts.com/certificate. This document certifies that the manuscript listed above was edited for proper Englishlanguage, grammar, punctuation, spelling, and overall style by one or more of the highly qualified native English speaking editors at American JournalExperts. Neither the research content nor the authors' intentions were altered in any way during the editing process. Documents receiving this certificationshould be English-ready for publication; however, the author has the ability to accept or reject our suggestions and changes. To verify the final AJE editedversion, please visit our verification page. If you have any questions or concerns about this edited document, please contact American Journal Experts [email protected].

American Journal Experts provides a range of editing, translation and manuscript services for researchers and publishers around the world. Our top-quality PhD editors are all native Englishspeakers from America's top universities. Our editors come from nearly every research field and possess the highest qualifications to edit research manuscripts written by non-native Englishspeakers. For more information about our company, services and partner discounts, please visit www.journalexperts.com.