# Reducing Impacts of Contamination in Water Distribution Networks: A Combined Strategy Based on Network Partitioning and Installation of Water Quality Sensors

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## Abstract

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## 1. Introduction

- reduction in inauspicious consequences of contamination (both accidental and intentional) in terms of limitation of contaminated areas (direct action);
- optimal placement of quality sensors (indirect action).

## 2. Materials and Methods

#### 2.1. Procedure 1—WNP

**L**=

**D**−

**A**, in which

**D**is the diagonal matrix containing the node degree k

_{i}of each node, and

**A**is the adjacency matrix. In this matrix, the elements a

_{ij}= a

_{ji}= 1 if nodes n

_{i}and n

_{j}are connected by a pipe; otherwise, a

_{ij}= a

_{ji}= 0. Shi and Malik (2000) [42] demonstrated that through the first C smallest eigenvector of the normalized Laplacian matrix, the relaxed version of the min-cut problem can be solved. In fact, it corresponds to the minimization of the Rayleigh quotient. If C is the number of clusters in which the network must be divided, the first C smallest eigenvectors of the Laplacian matrix are considered and used to create a new matrix

**U**. A k-means algorithm is applied to the rows of

_{nxC}**U**for grouping the nodes of the network in C clusters. The main trick is to change the representation of the nodes in the eigenspace of the first C eigenvectors, which enhances the cluster-properties of the nodes in such a way that they can be trivially detected in the new representation. The spectral clustering algorithm proved to show a superior performance to other clustering procedures, in that the provided clustering layout features both a well-balanced cluster size and a minimum number of edge cuts [43]. The main spectral clustering steps in the case of a WDN are described by Di Nardo et al. (2018b) [44]. The graph of the WDN can be considered un-weighted (every connection between the nodes has the same importance, a

_{nxC}_{ij}= a

_{ji}= 1) or weighted (the value a

_{ij}= a

_{ji}can be related to pipe features, such as diameter d and length l). In the applications of this work, a

_{ij}and a

_{ji}were set at 1. The optimal number of clusters C (from a topological point of view) in which to subdivide the network is chosen as a function of the number n of nodes, according to the relationship C

_{opt}= n

^{0.28}[45]. The clustering phase provides the optimal cluster layout and, as a result, the edge-cut set, consisting of a group of N

_{ec}boundary pipes between clusters. In correspondence to each boundary pipe, the flow transfer between the adjacent clusters must always be known, if it is larger than zero, in order to make the dividing effective. Therefore, the choice must be made whether either a gate valve must be closed, or a flow meter must be installed in the generic boundary pipe. Following this choice, the sum of closed gate valves (as numerous as N

_{gv}) and installed flow meters (as numerous as N

_{fm}) must be equal to N

_{ec}. Closing gate valves has the effect of reducing the service pressure and, therefore, leakage in the WDN. However, if service pressure falls below the desired threshold value h

_{des}, this negatively impacts on WDN reliability. In this work, the trade-off between leakage and WDN reliability was explored through the bi-objective optimization, performed through the NSGAII genetic algorithm [46]. In this optimization, several decisional variables equal to N

_{ec}was considered, to encode, inside individual genes, gate valve closure (gene value equal to 1) or flow meter installation (gene value equal to 0) at boundary pipes. The first objective function f

_{1}to minimize was the daily leakage volume V

_{l}(m

^{3}):

_{1}= V

_{l}

_{l}is calculated as the sum of the temporal integral of the nodal leakage outflows, evaluated as a function of nodal pressure heads through the Tucciarelli et al. (1999) [47] formula.

_{2}relates to the global resilience failure index GRF index proposed by Creaco et al. (2016) [48] to represent the instantaneous power surplus/deficit conditions of the WDN. In fact, GRF is dimensionless and is the sum of the resilience (I

_{r}) and failure (I

_{f}) indices evaluated at the generic instant of WDN operation:

**d**and

**q**are the vectors of nodal demands

_{user}**d**(m

^{3}/s) and water discharges

**q**(m

_{user}^{3}/s) delivered to users, respectively, at WDN demanding nodes. In this work,

**q**was evaluated as a function of

_{user}**d**and pressure head h (m) at each node though the pressure-driven formula of Tanyimboh and Templeman (2010) [49], with calibration proposed by Ciaponi et al. (2014) [50].

**H**and

**H**are the vectors of nodal heads (m) at demanding nodes and sources, respectively.

_{0}**H**is the vector of desired nodal heads, which are the sum of nodal elevations and desired pressure heads h

_{des}_{des}(m). Finally,

**Q**is the vector of the water discharges leaving the sources. The GRF index has the advantage of being within range [−1, 1]. Higher values of GRF indicate higher power delivered to WDN users and, therefore, higher service pressure. With reference to WDN daily operation, the second objective function f

_{0}_{2}to maximize was calculated with the following relationship, as suggested by Creaco et al. (2016) [48]:

_{2}= median(GRF)

_{1}and f

_{2}can be calculated by applying a pressure-driven WDN solver (e.g., that of Creaco et al. 2016 [48]). They are mutually contrasting objectives: in fact, as the number of closed gates grows, f

_{1}, which has to be minimized, decreases. At the same time, f

_{2}, which has to be maximized, decreases as well due to the decreasing service pressure. This creates a trade-off between the two objectives, which takes the form of a Pareto front of optimal solutions, that is a group of solutions from which to select the final solution for the partitioning. To this end, additional criteria, such as the partitioning cost or demand satisfaction, can be adopted. In fact, the Pareto front of optimal solutions can be re-evaluated in terms of other functions, such as number N

_{fm}of flow meters and demand satisfaction rate I

_{ds}. In fact, N

_{fm}is a surrogate for the partitioning cost [34], whereas I

_{ds}represents the effectiveness of the service to WDN users. The latter index can be calculated as:

_{d}(m

^{3}) and w

_{tot}(m

^{3}) are the delivered water volume and the WDN demand, respectively. Variable w

_{d}can be calculated starting from the temporal integral of the water discharge q

_{user}delivered to the users at each node.

#### 2.2. Procedure 2—Optimal Sensor Placement

_{3}= N

_{sens}(number of installed sensors), as a surrogate for the installation cost for WDN monitoring, while the second objective function is:

_{4}is related to the contaminated population pop

_{r}before the first detection of the generic r-th contamination event. This corresponds to the sum of the inhabitants served by the contaminated nodes and can be evaluated using the EPANET quality solver [52], using an unreactive contaminant. The EPANET quality solver can be applied to the flow field obtained following procedure 1. If the r-th event is not detected, pop

_{r}includes all the nodes crossed by the contamination till the whole contaminant mass leaves the WDN. Though numerous objective functions can be used for the optimal installation of sensors, the population exposed to contamination was chosen as the objective function to minimize along with the number of sensors. This choice was made because, compared to other potential objective functions (such as detection likelihood and sensor redundancy), the population exposed to contamination represents more directly the impact of contamination, which is the most meaningful from the viewpoint of risk assessment and mitigation. The time interval Δt

_{react}for the activation of emergency operations is set to 0 hr hereinafter for simplifying purposes. This means that contamination is assumed to stop instantaneously after its detection. However, Δt

_{react}can be set to other values without loss of validity of the whole methodology. The function f

_{4}is therefore the average value pop of pop

_{r}. In the bi-objective optimization, functions f

_{3}and f

_{4}are minimized simultaneously through the NSGAII genetic algorithm [46]. In fact, the minimization of the former reduces the sensor cost while the minimization of the latter impacts positively on the system security. In the population individuals of NSGAII, the number of genes is equal to the number of network nodes where sensors can be installed. Each gene can take on the two possible values 0 and 1, which stand for absence and presence of the sensor in the node associated with the gene, respectively.

- Option 1, sensors can be installed at all nodes (typical greedy approach);
- Option 2, sensors can be installed only at the hydraulically upstream nodes of the boundary pipes;
- Option 3, sensors can be installed at the most central nodes of each district, identified through topological considerations;
- Option 4, sensors can be installed at the hydraulically upstream nodes of the boundary pipes and at the most central nodes of each district.

#### 2.3. Procedure 3—Comparison of Sensor Placement Solutions

_{4}in Equation (5), followed by functions f

_{5}, f

_{6}and f

_{7}reported in the following Equations (6)–(8), respectively.

_{5}is the detection likelihood (i.e., the probability of detection):

_{r}= 1 if contamination scenario r is detected, and zero otherwise; and S denotes the total number of the contamination scenarios considered.

_{6}is the detection time. For each detected contamination scenario, the sensor detection time corresponds to the elapsed time from the start of the contamination event, to the first identified presence of a nonzero contaminant concentration. If t

_{j}is the time of the first detection (referred to the j-th sensor location), the detection time (t

_{d}) for the solution for each contamination event, is the minimum among all present sensors t

_{d}= min(t

_{j}); the characteristic detection time of the solution is defined as the mean of all t

_{d}for the contamination scenarios detected by at least one sensor:

_{7}is the sensor redundancy. In a generic scenario, the variable Red corresponds to the number of sensors (including the first) that detect the contamination within 30 minutes from the first detection; the redundancy Red of the solution is defined as the mean of all the values of redundancy Red for all the considered contamination scenarios:

## 3. Case Study

_{des}= 19 m was assumed for the demanding nodes, coming from the sum of the maximum building height in the town, which is 9 m in Parete, and 10 m, as prescribed by the Italian guidelines. Reference was made to the day of maximum consumption in the year when the total nodal demand ranges from 7.6 L/s at nighttime to 77.2 L/s in the morning and midday peaks, with an average value of 36.3 L/s. The leakage volume of the networks in the day of maximum consumption adds up to 930 m

^{3}(about 23% of the total outflow from the sources). The number of users connected to each WDN node was derived based on its average nodal demand.

## 4. Results and Discussions

_{opt}= n

^{0.28}proposed by Giudicianni et al. (2018) [45] to calculate the optimal number of clusters yields C

_{opt}= 4.3 for this WDN. The number of nodes for each DMA are DMA

_{1}= 20, DMA

_{2}= 35, DMA

_{3}= 39, DMA

_{4}= 41 and DMA

_{5}= 49, with N

_{ec}= 21. For the dividing, the optimization through NSGAII yielded the Pareto front reported in Figure 1a, showing, as expected, growing values of median(GRF) with V

_{l}growing. In fact, both variables are growing functions of the service pressure in the WDN. Figure 1b,c report the number N

_{fm}of flow meters and the demand satisfaction rate I

_{ds}, respectively, re-evaluated from the Pareto front and plotted against V

_{l}. Globally, Figure 1b highlights that the higher values of N

_{fm}tend to be associated with the high values of V

_{l}. This is because V

_{l}tends to grow when fewer gate valves are closed (and then more numerous flow meters are installed) at the boundary pipes. Finally, Figure 1c shows that I

_{ds}tend to grow with V

_{l}increasing, since both variables are increasing functions of the service pressure.

_{fm}(= 8), highest number of closed valves N

_{gv}(= 13), which ensures I

_{ds}= 100%, was finally chosen. An important remark to be made is that among the several advantages of the WNP, the adopted partitioning solution enables also reducing leakage, from 930 m

^{3}(for the un-partitioned layout) to 895 m

^{3}(partitioned solution with 13 gate valves closed and 8 flow meters installed). This corresponds to a 3.7% leakage reduction without negatively affecting I

_{ds}and GRF. In fact, for this solution median(GRF) is equal to 0.32, very close to the value of 0.36 for the un-partitioned network. The layout of the partitioned layout is reported in Figure 2. The optimal sensor placement is then carried out. The following assumptions were made for the construction of the set S of contamination events considered in the optimization:

- all the 182 demanding nodes were considered to be potential locations for contaminant injection;
- 24 possible contamination times in the day (hour 0, 1, 2, …, 22, 23);
- single value of the mass injection rate equal to 350 g/min;
- single value of the injection duration equal to 60 min.

- Var1Op1: Optimal sensor placement on the partitioned WDN allowing sensor installation on all nodes (182 potential locations);
- Var1Opt2: Optimal sensor placement on the partitioned WDN allowing sensor installation only on the nodes hydraulically upstream from the flowmeter fitted boundary pipes (8 potential locations);
- Var1Opt3: Optimal sensor placement on the partitioned WDN allowing sensor installation only on the most central nodes of each district (15 potential locations, i.e., three locations for each district, which feature a much higher betweenness centrality value than the other nodes);
- Var1Opt4: Optimal sensor placement on the partitioned WDN allowing sensor installation on the nodes hydraulically upstream from the flowmeter fitted boundary pipes and on the most central nodes of each district (23 scenarios).

_{sens}> 20. In this heuristic algorithm, for each NSGAII solution violating N

_{sens}= 20, a random integer number within the range (1, 20) is generated, representing the target number of sensors for that solution. Then, starting from the initial value of N

_{sens}, the least effective sensors in terms of pop are removed one by one to reach the target. Though increasing the computation time for each NSGAII generation by about 30 times, this algorithm proved to solve the issue of slow convergence. This heuristic algorithm was not applied to the optimizations Var1Opt2, Var1Opt3 and Var1Opt4. This made the NSGAII optimizations in the two latter applications much lighter from the computational viewpoint.

_{sens}increases up to 20. However, for high values of N

_{sens}, the additional benefit of a further sensor installed in the network tends to decrease, as already pointed out by Tinelli et al. (2017) [56]. In the present work, N

_{sens}= 6 appears to be the threshold of benefit for the installation of an additional sensor, slightly to right of the knee of the Pareto fronts (which lies around N

_{sens}= 3).

_{sens}≤ 6, pop for the un-partitioned WDN (Var0Opt1) is always higher than pop for the Var1Opt1 for all the number N

_{sens}of sensors installed in the network. The minimum value of pop = 462 is for Var1Opt1. Var1Opt2 (sensors allowed only upstream from boundary pipes), Var1Opt3 (sensors allowed only on topologically central nodes in DMAs) and Var1Opt4 (sensors allowed upstream from boundary pipes and on topologically central nodes in DMAs) give similar results to Var1Opt1 up to N

_{sens}= 2. For N

_{sens}> 2, Var1Opt2 and Var1Op3 degenerate while the good performance of Var1Opt4 persists. This is evidence that constraining sensor installation only upstream from boundary pipes or on topologically central nodes may lead to remarkably sub-optimal solutions. However, the combination of locations upstream from the boundary pipes and of topologically central nodes offers a good set of potential locations in the problem of optimal sensor placements. Figure 3b–d report the results of the reprocessing of the optimal solutions in terms of detection likelihood, detection time, and redundancy as a function of N

_{sens}. Along with Figure 3a, they give indications on the effectiveness of the solutions obtained in the NSGAII runs. Globally, the Var1 solutions obtained on the partitioned graph, especially Var1Opt1, Var1Op2, and Var1Opt4, tend to perform better in terms of pop, detection time, and sensor redundancy. Conversely, they feature slightly worse values in terms of detection likelihood. This may be because the optimization was carried out considering pop as objective function, which is slightly contrasted with detection likelihood [56]. In fact, the former variable mainly contributes to the system’s early warning capacity whereas the latter contributes to the system safety. As for Figure 3, it must be remarked that the curves in Figure 3a are Pareto fronts while those in the other Figure 3b–d are obtained by reprocessing the optimal solutions in terms of other assessment criteria. Since these curves are not Pareto fronts, they are not strictly monotonous. Figure 4 shows the sensor placement solutions obtained for N

_{sens}= 6 with three optimizations (Var0Opt1, Var1Opt1, and Var1Opt4). In this context, it must be noted that the Var1Opt4 solution has three of the six sensors placed close to flowmeters (the other three sensors are in the most central nodes according to the betweenness centrality). This solution yields managerial and economic benefits, due to the closeness of some sensors to installed flow meters and due to the possibility of sharing some electronical components for data acquisition, sharing, and transmission. Summing up, the Var1Opt4 solution represents a quasi-optimal solution in the explored trade-off between pop and N

_{sens}, while offering significant potentials for improved management. Another advantage compared to the Var0Opt1 and Var1Opt1 solutions with N

_{sens}= 6 is that it was obtained at a much lower computation cost (about 1/30), due to the research space reduction mentioned above for Options 2–4. Overall, the advantages in terms of computational lightness during the optimization as well as the possibility of inspecting and maintaining sensors in proximity to flow meters make solutions obtained in Opt4 preferable from the water utilities’ viewpoint. The results highlighted that nodes close to flow meters used for the monitoring of DMAs, which must always be easily accessible sites, represent good sensor locations for WDN monitoring from contaminations, when they are inserted into an optimization framework that also includes topologically central nodes inside DMAs. As for the optimal positions of the sensors in Var1Opt1 (partitioned network and all nodes as potential candidates) and Var1Opt4 (partitioned network and sensor installations restricted to entry points and central nodes in DMAs), it must be remarked that many locations are similar in the two cases (see Figure 4). This corroborates the fact that entry points and central nodes in DMAs are good candidate locations in the present case study.

## 5. Conclusions

- almost identical performance in terms of WDN monitoring, compared to the option of considering all nodes as potential locations;
- money savings thanks to the possibility of sharing some electronical components for data acquisition, sharing, and transmission;
- easiness of access to the sensors for maintenance;
- reduction in the search space and, therefore, in the computational complexity in the optimizations for optimal sensor placement;
- easier identification of the area from which the contamination starts with the subsequent possibility of isolating the district, assuring a higher resilience of the system to the spreading of the contamination.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Dividing phase considering the clustered graph of the Parete WDN (Variant 1). Pareto front of optimal solutions in the trade-off between daily median (GRF) index and leakage volume V

_{l}(

**a**), re-evaluated solutions in terms of number of flow meters N

_{fm}(

**b**), and of demand satisfaction rate I

_{ds}(

**c**). In all graphs, the selected solution is highlighted with a grey vertical line.

**Figure 3.**For the original un-partitioned WDN (Var0Opt1), reported as benchmark, and for the partitioned WDN (Var1Opt1, Var1Opt2, Var1Opt3 and Var1Opt4), Pareto front of optimal sensor placement solutions in the trade-off between N

_{sens}and contaminated population pop (

**a**), re-evaluated solutions in terms of N

_{sens}and detection likelihood P

_{s}(

**b**), N

_{sens}and detection time T

_{mean}(

**c**), and N

_{sens}and redundancy Red (

**d**).

**Figure 4.**Optimal location of 6 sensors in (

**a**) original un-partitioned WDN (Var0Opt1), (

**b**) partitioned WDN (Var1Opt1), and (

**c**) partitioned WDN (Var1Opt4).

Option | Variant 0 (Un-Partitioned) | Variant 1 (Partitioned) |
---|---|---|

1 (all nodes) | Var0Opt1 | Var1Opt1 |

2 (only boundary nodes) | - | Var1Opt2 |

3 (only central nodes) | - | Var1Opt3 |

4 (boundary nodes + central nodes) | - | Var1Opt4 |

**Table 2.**Simulation results in terms of exposed population from the four optimizations for sensor placement in the Parete WDN, considering N

_{sens}up to 6.

N_{sens} | Var0Opt1 | Var1Opt1 | Var1Opt2 | Var1Opt3 | Var1Opt4 |
---|---|---|---|---|---|

0 | 2806 | 2458 | 2458 | 2458 | 2458 |

1 | 1438 | 1274 | 1274 | 1274 | 1274 |

2 | 982 | 919 | 953 | 974 | 953 |

3 | 789 | 648 | 741 | 679 | 653 |

4 | 667 | 559 | 638 | 598 | 569 |

5 | 589 | 500 | 572 | 561 | 515 |

6 | 514 | 462 | 564 | 548 | 472 |

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## Share and Cite

**MDPI and ACS Style**

Ciaponi, C.; Creaco, E.; Di Nardo, A.; Di Natale, M.; Giudicianni, C.; Musmarra, D.; Santonastaso, G.F.
Reducing Impacts of Contamination in Water Distribution Networks: A Combined Strategy Based on Network Partitioning and Installation of Water Quality Sensors. *Water* **2019**, *11*, 1315.
https://doi.org/10.3390/w11061315

**AMA Style**

Ciaponi C, Creaco E, Di Nardo A, Di Natale M, Giudicianni C, Musmarra D, Santonastaso GF.
Reducing Impacts of Contamination in Water Distribution Networks: A Combined Strategy Based on Network Partitioning and Installation of Water Quality Sensors. *Water*. 2019; 11(6):1315.
https://doi.org/10.3390/w11061315

**Chicago/Turabian Style**

Ciaponi, Carlo, Enrico Creaco, Armando Di Nardo, Michele Di Natale, Carlo Giudicianni, Dino Musmarra, and Giovanni Francesco Santonastaso.
2019. "Reducing Impacts of Contamination in Water Distribution Networks: A Combined Strategy Based on Network Partitioning and Installation of Water Quality Sensors" *Water* 11, no. 6: 1315.
https://doi.org/10.3390/w11061315