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Procedía Engineering 89 (2014) 1176 - 1183

Procedía Engineering

www.elsevier.com/locate/procedia

16th Conference on Water Distribution System Analysis, WDSA 2014

Divide and Conquer Partitioning Techniques for Smart Water

Networks

A. Di Nardoa*, M. Di Natalea, G.F. Santonastasoa, V. Tzatchkov b, V.H. Alcocer

Yamanaka b

aDept. of Civil, Design, Structural and Environmental Engineering, via Roma 29, 81031, Aversa, Italy bUrban Hydraulics Department, Mexican Institute of Water Technology, Jiutepec, Mor., Mexico

Abstract

Water Network Partitioning (WNP), represents one of the best methodologies for applying water balance and pressure control of a water distribution system to reduce water leakage and improve water quality protection. Traditionally the WNP is based on empirical and 'trial and error' approaches used with hydraulic simulation, which are difficult to apply to large water networks. Recently some heuristic procedures, based on different techniques (graph theory, clustering, etc.) showed that is possible to find optimal solutions. The authors developed a software, called SWANP (Smart Water Network Partitioning) that allows finding automatically the optimal layout of District Meter Areas based on a multi-level algorithm. This paper compare SWANP with other procedures for WNP.

© 2014 Publishedby ElsevierLtd. This isanopen access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Peer-review under responsibility of the Organizing Committee of WDSA 2014

Keywords: Water network partitioning, District meter areas, graph theory, water leakage, smart water networks

1. Introduction

The development of new monitoring and control technologies and the recent growth of computational power used by simulation software have changed the traditional approach to analysis, design and management of Water Distribution Systems (WDS), from passive to smart actions, allowing the transformation of urban WDS in Smart WAter Networks (SWANs). This new approach, based on the use of control devices and Hydro informatics

* Corresponding author. Tel.: +39-081-5010202; fax: +39-081-5037370. E-mail address: armando.dinardo@unina2.it

1877-7058 © 2014 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Peer-review under responsibility of the Organizing Committee of WDSA 2014 doi: 10.1016/j.proeng.2014.11.247

techniques, allowed to division of water networks in subsystems implementing the paradigm of "divide and conquer" in SWANs [1]. This approach, defined also as Water Network Partitioning (WNP), allows improving the management of water distribution systems with reference to some important issues as water balance [2, 3], water pressure management [4] and water quality protection [5, 6]. WNP can be applied with greater effectiveness defining smaller permanent network subsystems, called District Meter Areas (DMAs) obtained by the insertion of gate valves and flow meters. The traditional criteria for the design of network DMAs are based on empirical suggestions (such as recommended number of properties or length of pipes per district, etc.) and on 'trial and error' approaches used with hydraulic simulation software [7]. Nevertheless these indications, the procedures are very difficult to apply to large water supply systems because the insertion of gate valves modifies the original network layout and may worsen the hydraulic performance of the water network. Indeed WNP changes the hydraulic performance [1] because it is in conflict with the traditional design criteria of pipe redundancy that markedly improves the network reliability, which can be significantly reduced by pipe closures, which are necessary to obtain permanent DMA.

In the last years different procedures have been proposed in literature (a detailed review is given in [1] and [8]) to finding automatically an optimal network partitioning based on different techniques, as for example breadth and depth first search [8, 9, 10, 11, 12, 13], multilevel partitioning [1, 4], community structure [14], clustering [9, 15] and multi-agent [16, 17, 18]; while for the selection of pipes on which to insert flow meters or boundary (or gate) valves based on iterative procedures [10, 14] or genetic algorithms [1, 18]. The effort of this preliminary study was to compare three procedures, based on different approaches, with another one, recently implemented by the authors in the SWANP software [19].

2. Water network partitioning

The procedures tested are based on different approaches starting from the representation of the water network as a simple weighted graph considering G = (V,E), where V is the set of n vertices (or nodes) and E is the set of m edges (or pipes). The aim of the partitioning phase of each procedure is to minimize the number Nec of edge-cuts (or links between the districts) with the constraint of the number of nodes that belong to each district. If the edges and vertices of the graph are weighted, the goal becomes to minimize the sum of associated weights on the edge-cuts and to balance the sum of node weights for each districts. The selection of edge cuts on which to insert boundary valves Nbv or flow meters Njm is not provided by each procedure because, in some cases, the gate valves are then inserted in all edge pipes defining a Water Network Sectorization (WNS), as defined in [12].

The four techniques compared in this paper are described synthetically below but, for a detailed description of each algorithm, it is evidently necessary to refer to the papers in which they have been published.

2.1. Multi-agent simulation

The Multi Agent Simulation (MAS) framework, proposed by Hajebi et al. [17], is a two-step algorithm to solve the partitioning of the WDS into DMAs. The first step uses a ¿-means graph-clustering algorithm [20] to partition the network geographically into a prefixed number of cluster or districts. The k-means algorithm is an unsupervised learning method for discovering cluster and cluster centers in a set of unlabeled data. It begins with choosing at random ¿-objects, which represent, at first run, the cluster centers; then it assigns all the remaining objects to the nearest cluster centers. The process is repeated until convergence or after a finite numbers of trials. The second step applies a multi-agent system negotiation [17] to adjust the boundary nodes. The negotiation considers the difference of the elevation of the boundary nodes with the neighboring clusters. If the elevation is closer to that of the other cluster than to its current one, the boundary node is assigned to that cluster and the negotiation starts again for the new network arrangement. All boundary nodes are affected in a random sequence. The process stops when the total number of changes is less than a predetermined threshold number of changes. As reported in [17], the MAS is a heuristic algorithm and the optimal results for different runs may be different. This procedure, once found the optimal water network partitioning, does not provide a specific algorithm for the selection of pipes on which to insert flow meters or boundary valves.

2.2. Hybrid graph partitioning

The Hybrid Graph Partitioning (HGP) proposed by Ferrari et al. [10] combines two graph theory algorithms: the Kernighan-Lin (KL) [21] and the depth first search [22]. The Kernighan-Lin is a bisection algorithm that divides a weighted graph with 2n nodes into two subsets, each of size n, minimizing the sum of the weights. Anyway, it can be extended to the more general case of k-way partitioning by recursive bisection. The Depth First Search algorithm that traverses a graph by visiting neighbor vertices is applied in order to verify the connectedness of the network each time. Once found the optimal water network partitioning, HGP uses iterative procedures for the selection of pipes to be closed.

2.3. Water spectral clusters

The core of the Water Spectral Clusters (WSC), proposed by Herrera et al. [15] is a semi-supervised clustering algorithm. The starting point is building the affinity matrix for the WDS by assuming water demand as node weights and pipes diameter as edge weights. Then, other dissimilarity matrices, using different inputs and constraints about the water supply system, can also be built. In particular, other possible inputs are: geographic information of nodes, distance between nodes and node elevation, while a constraint is that each DMA is supplied by one or more water sources. Once defined the matrices of input information, they are transformed into a kernel matrix [23], which represents a synthesis of the WDS information data. Thus, starting from the information arranged in the kernel matrix, it is possible to apply the spectral clustering algorithm with the minimization of a Min Cut objective function in order to partition the WDS into the required number of DMAs. Once the optimal water network partitioning is found, also this procedure does not provide a specific algorithm for the selection of pipes to be closed.

2.4. SWANP

The Smart WAter Network Partitioning (SWANP) software is based on a Multi Level Recursive Bisection (MLRB) algorithm [24]. The authors proposed an original procedure [1] adjusting the traditional phases of a MLRB: a) coarsening; b) partitioning; c) uncoarsening; d) refinement with swapping. The k-way partition is recursively solved by performing a sequence of 2-way partitions (or bisections). Once defined, for each recursive bisection, in one go, all the boundary pipes, the MLRB optimizes the edge-cuts (refinement) moving a vertex from one partition to another (swapping) in compliance with the goals (minimization of the edge-cuts or associated weights) and constraints (balancing of the nodes or associated weights). Once defined by the MLRB procedure the links between districts (or the set of boundary pipes), a special Genetic Algorithm (GA) allows to choose heuristically the location of flow meters and gate valves. In addition, SWANP includes some performance indices to compare different layouts and to provide to operators a Decision Support System for choosing the best layout in compliance with the goal of WNP.

3. Results

The comparison between each procedure and SWANP was carried out using the same water network and the same number of DMA proposed in the original paper of each author. Then, the graph-partitioning phase was achieved with SWANP using weights on pipes and nodes different from those used in the other procedures. Only pressure indices were computed to measure performance of WNP because no other information was provided for the case studies that would allow computing other indices of SWANP (e.g., resilience, resilience deviation, etc., as reported in [25]). Because the information of each procedure and case study is not complete, the operative strategy of this first attempt to compare different techniques is based on the following: the SWANP software was tested on the same network used by the other authors comparing with the results provided in their studies.

In Table 1, the first comparison refers to the MAS procedure, as proposed by Hajebi et al. [17], on the Net3 network. In the first four columns the characteristics of network partitioning are reported, respectively: the number of DMA k, the number of nodes belonging to each DMA, and the number of boundary valves Nbv and flow meters

Njm (with Nec= Nbv + Nfm). The comparison shows that the MAS procedure allowed to obtain, for that case study, a number of edge cuts Nec=5 significantly lower than the one obtained with SWANP (Nec=15) with node weights equal to "node elevation" and pipe weights equal to "water flow".

Table 1. Comparison between WNP obtained with MAS and SWANP for the Net3 original network

Network/WNP k DMA1 nodes Partitioning characteristics DMA2 nodes DMA3 nodes Nb, Njm Performance indices hmean [m] h^m] h^ [m]

Net3 - - - - - 41.73 4.19 92.45

MAS 3 38 24 30 3 2 41.87 4.21 92.54

SWANP 3 38 14 40 13 2 44.66 5.03 95.63

In the Fig. 1, the best water network partitioning with k=3 DMAs, obtained by the SWANP software, is reported indicating the location of flow meters and boundary valves.

Fig. 1. DMAs of Net3 network by SWANP.

Although the MAS procedure does not provide the selection of boundary pipes, in this work the heuristic algorithm proposed in [1] was used for this aim; in this way, fixing Nm=2 was found as the optimal number of flow meters and the alteration of hydraulic performance can be computed with pressure indices. In particular, because in this case the number of Nec obtained with MAS is small, it was possible to compute all combination of the location of Nbv and Nfm. The performance indices show an interesting result, as reported in the last three columns of Table 1. Although the number of edge cuts obtained with the MAS procedure is lower, it is expected that this solution would provide lower alteration of hydraulic performance than the solution obtained with the SWANP software, because the number of closed valves is higher fixing the same number of flow meters, all pressure indices computed with SWANP are higher and also higher than those of the original network. This singular result is certainly due to the presence of pump systems in the network that change the available power but also to the non-linearity of the problem and indicates that not only the minimization of edge cuts but also their location is a crucial aspect of water network partitioning. Anyway, comparing the results obtained by the MAS procedure with the original values of pressure indices of the network, it is evident that the MAS procedure is very effective and it is worth to test it on larger water networks, as already done for the SWANP software.

In the Table 2, the second comparison refers to the HGP procedure, proposed by Ferrari et al. [10] for a water network partitioning of Anytown network in k=2 DMAs. In this case, the SWANP software provides a number of edge cuts slightly lower than HGP (Nbv =9 vs Nbv =11) after the partitioning phase with only node weight equal to "water demand". Then, unlike the MAS procedure, the HGP procedure does include an iterative tool for selection of

the location of flow meters and boundary valves. In order to simplify the comparison, in this case, the heuristic procedure to select the location of boundary valves and flow meters did not use but the performance indices obtained using the same number Nfm=3 and location of flow meters and inserted all gate valves in the other edge cuts. As reported in Table 2, the solution obtained with the SWANP is clearly better than HGP in terms of hydraulic performance with pressure indices practically equal to the original network and with a value of hmin=30.02 m almost twice the one obtained with HGP (hmin=16.10 m). Anyway, the HGP procedure provided a good value of hmean=65.62 m compared with the value of original network (hmean=68.57 m); also in this case the pressure values were affected by the presence of pump systems.

Table 2. Comparison between WNP obtained with HGP and SWANP for Anytown original network Partitioning characteristics Performance indices

k DMA1 nodes DMA2 nodes Nbv Nfm hmean [m] hmin[m] h^ [m]

Anytown - - - - - 68.57 30.12 170.23

HGP 2 7 10 8 3 65.62 16.10 170.58

SWANP 2 8 9 6 3 70.00 30.02 170.41

In in this case, a lower value of Nec=9 obtained with SWANP was sufficient to obtain a better WNP, compared to the HGP procedure. It is reasonable to think that the use of heuristic optimization procedure in SWANP can find an even better solution. Finally, it is worth to test also the HGP procedure on larger water networks. In the Fig. 2, the best water network partitioning with k=2 DMA, obtained by the SWANP software, is reported indicating the location of flow meters and boundary valves.

Fig. 2. DMAs of Anytown network by SWANP.

In Table 3, the last comparison between SWANP and the WSC procedure, proposed by Herrera et al. [15] for the real water network of Celaya in Mexico, is reported. It is worth to highlight that the WSC procedure [15] is proposed specifically for Water Network Sectorization (WNS) and not for water network partitioning. In other terms, the goal of the procedure is to find isolated district (or i-DMA, as defined in [12, 26]), each one supplied from one or more sources, inserting gate valves in each edge cuts. Then the partitioning algorithm should find a solution that, at same time, minimizes the number of Nec and the alteration of hydraulic performance of each i-DMA. Thus, no algorithm is required for the selection of the location of flow meters. In this case, reported in Table 3, the SWANP software provided a number of edge cuts significantly lower than WSC obtained with no weights on pipes and nodes: Nec =34 by the WSC procedure and Nec =14 by the SWANP software. In this case, the comparison is not

very effective because the location of gate valves in the Celaya network provided from Herrera et al. [15] is not completely clear. Anyway, in order to provide a useful hint to further comparison, the same location of the Nm=4 flow meters, and the same number k=3 of DMA were selected by SWANP, as illustrated in the Fig. 3.

Table 3. Comparison between WNP obtained with WSC and SWANP for Celaya original network

Network/WNP k DMA1 nodes Partitioning characteristics DMA2 nodes DMA3 nodes Nb, Njm Performance indices hmean [m] h^m] h^ [m]

Celaya - - - - - 10.55 4.86 14.70

WSC 3 122 84 127 34 - - - -

SWANP 3 114 110 115 10 4 9.88 2.89 14.22

The performance indices reported in Table 3 show a slight alteration of hmean=9.88 m and hmax=14.22 m compared to the values of original network (hmean=10.55 m and hraa*=14.70 m) but a more significant worsening of value of hmin from 4.86 m (that shows a bad hydraulic performance also of the original network) to 2.89 m (limited to only 20 nodes of the whole network where pressure is under 4 m).

Fig. 3. DMAs of Celaya network by SWANP.

As explained in [27], if the aim of Herrera et al. [15] was only to design a WNS, the simulation results obtained with SWANP are not comparable with their layout, because the number of nodes and the flow to assign to each source in a WNS depend on network topology and hydraulic characteristics. If the algorithms tries to match the balance constraint, as happens in a WNP, hydraulic performance can be significantly worsen. Indeed this constraint can be useful in a WNP because each DMA can be balanced in terms of nodes (or district flow) as it is not assigned to only one source but is connected with other districts, while it can be even negative in a WNS because it can generate layouts incompatible with the level of service [27].

4. Conclusion

In the last decade, some procedures for water network partitioning and sectorization have been proposed in the literature to overcome the empirical approaches followed by the operators to design district meter areas (DMAs) and isolated district meter areas (i-DMAs). In this study, a preliminary comparison between three of these procedures and a software developed by the authors was achieved showing their effectiveness but also some critical issues as the following:

• the algorithms have to be tested on larger water networks; the authors suggest to identify a benchmark network with many loops, similar to a real water system, with a low value of resilience [28];

• the procedures should provide more details on each of their steps in order to allow a comparison; at this moment, it is practically impossible to compare each procedure on the same network because some information that is required is not available (e.g., in one case the model used, in other case the design pressure or details about the algorithm, etc.);

• the papers should provide a clear definition of their "divide and conquer" aims, possibly using the classification in Water Network Partitioning (WNS) and Water Network Sectorization (WNS);

• performance indices to compare the multi-objective optimization problem should be defined; in each of the tested procedures this aspect was completely neglected.

With reference to the specific results obtained in this study, the SWANP software showed better results with respect to the HGP and WSC procedures in terms of both hydraulic performance and minimization of the number of edge cuts, while the MAS procedure provided a better result only in terms of edge cuts. Then the SWANP software is arranged as a decision support system providing to operators partitioning and sectorisation layouts that can be compared with the use of different performance indices.

Finally, it is worth to highlight that the research on the topic of water network partitioning can be considered advanced because the automatic procedures show a good ability to find the best solution, although the previous issues should be overcome in order to transfer the results to the water market.

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