# Assessing the Impact of Partitioning on Optimal Installation of Control Valves for Leakage Minimization in WDNs

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## Abstract

**:**

## 1. Introduction

- Waste of energy used to pump and treat water that does not reach customers.
- Potential deterioration of small breaks to pipe bursts.
- Potential intrusion of pollutants through pipe breaks when negative pressure occurs.

- Monitoring and control of consumption and leakage in the network.
- Implementation of pressure management.
- Identification of pipe bursts.
- Protection of the network from contamination events.
- Management practices in intermittent WDNs.
- Placement of sensors for the identification of contamination events.

## 2. Materials and Methods

#### 2.1. Optimization of Control Valves

#### 2.1.1. Optimization of Valve Settings

_{Δt}of temporal steps, the solution at each time step enables estimation of nodal heads and water discharges at demanding nodes and links, respectively, starting from nodal demand and source head values. In this context, the presence of N

_{val}control valves can be simulated by suitably modifying the resistance of the N

_{val}valve-fitted pipes [14,15]. To this end, the resistance of the generic valve fitted pipe and at the generic time step can be divided by a valve setting coefficient ranging between 0 and 1, corresponding to fully closed and fully open valve, respectively. If the final aim is to attenuate leakage, the N

_{Δt}× N

_{val}valve settings can be optimized to minimize the daily leakage volume ${W}_{L}$, expressed as follows:

_{L,i}the leakage volume at the generic time step of the day, obtained as the sum of leakage from demanding nodes, evaluated as a function of service pressure using the same formulation as [4], multiplied by the time step.

#### 2.1.2. Optimization of Control Valve Locations

_{val}of control valves and the daily leakage volume W

_{L}in the WDN are simultaneously minimized. This algorithm carries out the deterministic exploration of the research space of possible combinations of control valves in the WDN by adding one control valve at each step. This results in the significant reduction in combinations in comparison with the total enumeration. Despite the inherent simplifications, the algorithm can yield well-performing solutions with a small computational overhead, especially when the effects of installed control valves do not interfere. The effectiveness of this algorithm was analyzed and compared with that of a multi-objective genetic algorithm well established in the scientific literature in the work [16].

_{max}of installable control valves and the n

_{p}candidate locations for control valve installation in the WDN must be fixed. In most cases, n

_{p}is equal to the total number of pipes. However, in some cases, some pipes must be excluded from the list of candidates, e.g., those belonging to the connection line between a pump station and a tank. The operation of this algorithm can then be summarized as follows.

_{L}.

_{p}potential locations in the WDN. For each location, a control valve is simulated in the WDN model and its settings in the typical day of operation are optimized to minimize the daily leakage volume W

_{L}, as explained in Section 2.1.1. The performance of the various locations is compared in terms of W

_{L}, enabling identification of that with the lowest value. Therefore, at the end of step 1, the optimal configuration with one present control valve is obtained.

_{p}− 1 candidate locations. Therefore, n

_{p}− 1 combinations of two control valves, the first valve of which has been found in step 1, are considered. For each combination, valve settings are optimized to minimize W

_{L}, as explained in Section 2.1.1. At the end of step 2, the W

_{L}performance of the n

_{p}− 1 combinations are compared, enabling identification of the optimal combination with the lowest W

_{L}. Therefore, at the end of step 2, the optimal configuration with two present control valves is obtained.

_{val}, the N

_{val}-th optimal location to be added to the N

_{val}− 1 optimal locations identified in the previous steps is searched for, to minimize W

_{L}. Therefore, at the generic step N

_{val}, the optimal configuration with N

_{val}control valves is obtained.

_{max}steps, the Pareto front of optimal solutions can be easily derived, by plotting the W

_{L}values obtained as a function of N

_{val}.

#### 2.2. WDN Partitioning Based on Minimum Transport

_{i}and Q

_{i}are the length and water discharge of the generic pipe, respectively.

_{1}demanding nodes, which is guaranteed by enforcing continuity equation such as the following:

_{j}and n

_{p,j}are the demand at the generic j-th demanding node and the number of pipes connected to it, respectively.

_{i}, the pipe water discharges occurring under daily average or peak demand conditions. In the starting condition, the topological positive direction is set in each pipe in such a way that all the Q

_{i}values are nonnegative. If peak values of nodal demands are obtained by applying multiplicative coefficients to daily average values, considering either daily average or peak demand leads to the same topological positive directions for the starting condition. Furthermore, the results of the linear programming under peak demand conditions are simply proportional to those obtained under daily average demand conditions.

_{0}and n

_{l}are the number of sources and geometric loops present in the WDN configuration, respectively, the solution of the linear programming problem yields n

_{l}+ n

_{0}− 1 values equal to 0 and n

_{t}− (n

_{l}+ n

_{0}− 1) values larger than 0. The number n

_{l}+ n

_{0}− 1 is equal to the number of loops, including both geometric loops and source interconnection paths. Indeed, it represents the maximum number of pipes that can be removed while guaranteeing that all demanding nodes remain connected to one source. The removal of the n

_{l}+ n

_{0}− 1 pipes transforms a looped network configuration into a system of branched networks, each of which is fed by a single source. Each branched network can be considered an independent partition. As a result, the minimization of T enables clustering the nodes of the WDN into a number of partitions equal to the number n

_{0}of sources. However, the partitions with all pipes removed cannot be selected as the ultimate solution since their branched structure would guarantee a too low level of redundance. To make up for this drawback, the pipes removed that do not belong to any source interconnection paths can be re-introduced. The other removed pipes represent, instead, the boundary pipes between the partitions.

- It is a topological procedure that considers explicitly neither altimetric aspects, which may impact on service pressure in resulting partitions, nor practical engineering criteria, such as the uniformity of partitions in terms of total demand or other variables.
- In this basic formulation, it cannot be applied when the desired number of partitions is different from the number of sources.

## 3. Applications

#### 3.1. Case Studies

_{1}= 32 with unknown head and n

_{0}= 2 source nodes with fixed head, i.e., nodes 33 and 34) and n

_{p}= 41 pipes. The daily average demand of all WDN users is around 18.5 L/s with nodal demands at demanding nodes ranging from 0.1156 to 1.156 L/s. This WDN features a quite large variability in terms of ground elevations, ranging between 394.8 m above sea level and 465 m above sea level for the demanding nodes. As for leakage, the ratio of daily leakage volume (W

_{L}= 1243 m

^{3}) to total daily consumption (user consumption + leakage) volume (2841 m

^{3}) in the WDN was modelled to be equal to 44%. The main features of the WDN nodes and pipes were derived from the referenced works [15,23]. The daily patterns for the nodal demand multiplicative coefficient C and for the head H

_{0}at the sources are shown in twelve 2 h-long time slots in Figure 2.

_{1}= 272 demanding nodes, n

_{0}= 4 sources with fixed head, and n

_{p}= 317 pipes. In the work of Bragalli et al. [24], the peak demand of about 407 L/s is considered and no pressure-dependent leakage is implemented. In the present work, the yearly average demand was obtained by halving the peak demand. In the context of the average demand, a typical day of operation was considered with three representative time slots, associated with values of the multiplying demand coefficient C equal to 0.7, 1.0, and 1.3, respectively. Nodal emitters were set to obtain a leakage percentage equal to 15%. While this case study has been used in various works [24] for the application of WDN design algorithms, a redundant configuration of diameters in comparison with the minimum cost was considered in the present work, to make this case study feasible partitioning and pressure regulation. The variability in ground elevation at demanding nodes in the second case study, from 30.39 to 74.5 m above sea level, is smaller than in the first case study.

#### 3.2. Results for the WDN of Santa Maria di Licodia

#### 3.2.1. Analysis of Service Pressure in the Unpartitioned WDN

_{des}value for full demand satisfaction can be assumed equal to 15 m. Therefore, as is evident from Figure 3, there is a very large excess of service pressure compared to the desired value h

_{des}for full demand satisfaction.

#### 3.2.2. Application of the Sequential Addition Algorithm to the Unpartitioned WDN

_{des}or at the minimum daily pressure head value in the case of pressure excess or deficit, respectively, in comparison with h

_{des}. The results of the sequential addition algorithm up to N

_{max}= 10 valves are shown in Table 1, showing that significant leakage reductions (by 17%) can be obtained with a single control valve installation in pipe 27. The results improve sensibly up to N

_{val}= 4 valves in pipes 27, 7, 3, and 14, for which the leakage reduction compared to the no-control scenario adds up to about 40%. The Pareto front of optimal solutions in the trade-off between N

_{val}and W

_{L}is reported in Figure 4, showing expectedly decreasing values of W

_{L}as N

_{val}increases.

#### 3.2.3. WDN Partitioning

_{l}= 9 loops, i.e., 8 geometric loops + 1 source interconnection path, the minimization of the transport function led to the removal of 9 pipes, namely, pipes 9, 13, 16, 18, 22, 26, 32, 34, and 39, for opening the loops. As expected, the resulting WDN configuration was a system of two branched networks, each of which was fed by a single source (Figure 5). In fact, the minimization of the transport function enabled clustering the nodes of the WDN into two partitions. To restore a suitable level of reliability in terms of number of closed loops, which help water supply in scenarios of mechanical failure in the WDN, while keeping two separate partitions, all the removed pipes except for pipes 18 and 26 were re-introduced. In fact, pipes 9, 13, 16, 22, 32, 34, and 39 do not contribute to WDN partitioning, while pipes 18 and 26 do. These two pipes were then considered the boundary pipes between the left and right partitions of the WDN (see Figure 6).

_{des}= 15 m, attesting to the feasibility of the physical separation.

_{L}reduction from 1243 to 1176 m

^{3}.

#### 3.2.4. Application of the Sequential Addition Algorithm to the Partitioned WDN

_{L}= 1176 m

^{3}also in the partitioned WDN. The valve locations considered in the sequential addition were slightly different from the case of the unpartitioned WDN, due to the flow variations induced by the partitioning. These differences arose starting from the fourth valve installed in the WDN, after the first three valves were installed in pipes 27, 7, and 3 in both cases. While the sequential addition suggested pipe 14 as the location for the fourth valve in the unpartitioned WDN, it suggested pipe 25 after partitioning. In fact, the optimal 4-valve solution obtained in the unpartitioned WDN, including valve locations 27, 7, 3, and 14, yielded a suboptimal leakage volume W

_{L}= 780 m

^{3}in the partitioned WDN, in comparison with the solution 27, 7, 3, and 25, which features W

_{L}= 730 m

^{3}. The choice of pipe 25 instead of pipe 14 as the fourth valve location is motivated as follows. First, the node downstream of pipe 25, i.e., node 16, features very large values of service pressure and leakage, due to its small ground elevation. However, the abatement of service pressure at this node was discouraged in the unpartitioned WDN, since it would have required installation of two control valves, i.e., at pipes 25 and 26, respectively. Therefore, the sequential addition algorithm preferred to choose another location for the fourth valve in the unpartitioned WDN. In the case of the partitioned WDN, instead, the disconnection of pipe 26 from node 16 due to the partitioning made pipe 25 branched, and, therefore, a suitable site for control valve installation and pressure regulation.

_{val}of control valves being equal, the partitioning was observed to yield benefits in terms of W

_{L}reduction. These benefits were estimated to be ranging from about 1% to about 8% (last column in Table 2). This is the result of the lower service pressure existing in the partitioned WDN, starting from the initial scenario with no control valves installed.

_{val}-W

_{L}between unpartitioned and partitioned WDN is shown in Figure 4, highlighting lower values of W

_{L}in the partitioned WDN for each value of N

_{val}.

_{val}-W

_{L}for both the unpartitioned and partitioned WDNs (Figure 4). This attests to the fact that, up to N

_{val}= 4, the addition of a control valve is effectively paid back in terms of leakage reduction. For both the unpartitioned and partitioned WDNs, the valve settings expressed in terms of pressure head at the downstream node are quite constant in the day and very close to h

_{des}= 15 m, due to the altimetry of the urban center. Most nodes in the unpartitioned WDN have slightly lower pressure heads than those in the partitioned WDN. The only evident exception is the situation of node 16, for which a large difference exists between the pressure head in the unpartitioned WDN (around 65 m) and in the partitioned WDN (around 15 m). This is the result of what was highlighted about the optimal addition of the fourth control valve in unpartitioned and partitioned WDN. However, this significant difference of service pressure at node 16 enables lower daily leakage volume to be obtained for the partitioned WDN (730 vs. 751 m

^{3}). Overall, as expected, lower pressure heads are observed in the presence of four control valves than in the absence of control valves (compare Figure 9 with Figure 8).

^{3}/s in scenario 1, with overshooting and undershooting present mainly at nighttime and daytime, respectively. The partitioning causes the lowering of the pattern to the average value of 0.0136 m

^{3}/s (compare scenario 3 with scenario 1). The distance of the pattern in scenario 3 from the pattern in scenario 1 is larger at daytime than at nighttime. In the scenarios with control valves, i.e., scenarios 2 and 4, the patterns are much flatter than in the no control scenarios 1 and 3. In fact, they feature very small fluctuations around their average values of 0.0087 and 0.0084 m

^{3}/s, respectively, resulting from poorly variable service pressure conditions in the day.

#### 3.3. Results for the WDN of Modena

#### 3.3.1. Analysis of Service Pressure in the Unpartitioned WDN

_{des}value for full demand satisfaction is assumed equal to 20 m. Therefore, as is evident from Figure 12, there is a very large excess of service pressure compared to the desired value h

_{des}for full demand satisfaction.

#### 3.3.2. Application of the Sequential Addition Algorithm to the Unpartitioned WDN

_{des}= 20 m as the minimum pressure head constraint at the generic demanding node. The Pareto front of optimal solutions obtained in the trade-off between N

_{val}and W

_{L}is reported in Figure 13, showing expectedly decreasing values of W

_{L}as N

_{val}increases. Like for the first case study, the decrease is significant up to N

_{val}= 4, for which a reduction in W

_{L}by about 38% is observed in comparison with the no control scenario.

#### 3.3.3. WDN Partitioning

_{l}= 49 loops, i.e., 46 geometric loops + 3 source interconnection paths, the minimization of the transport function led to the removal of 49 pipes, resulting in a system of 4 branched partitions each of which fed by a single source. To restore a suitable level of reliability in terms of loops while keeping four separate partitions, 31 of the removed pipes were re-introduced. The other 18 pipes were then considered the boundary pipes between the partitions of the WDN (see Figure 15). Like in the first case study, the physical separation of the partitions was verified to be feasible and sustainable in terms of service pressure. The physical separation was obtained by closing the isolation valves at one end of each pipe. The feasibility check of the partitioning is carried out in Figure 16, showing the comparison of pressure head values between the unpartitioned and partitioned WDN. Globally, this Figure shows that the physical separation of the four partitions causes pressure decreases at some nodes and pressure increases at others. Though the number of nodes with pressure decreases prevails, no inacceptable decreases were observed considering h

_{des}= 20 m, attesting to the feasibility of the physical separation.

_{L}reduction from 3101 to 3100 m

^{3}.

#### 3.3.4. Application of the Sequential Addition Algorithm to the Partitioned WDN

_{L}(N

_{val}) is shown in the graph in Figure 13 in comparison with the front obtained in the unpartitioned WDN. Like in the first case study, all the dots of the Pareto front of the partitioned WDN are slightly below those of the unpartitioned WDN, highlighting lower leakage volumes by up to about 7%, the number of installed control valves being the same. To have better insight into this aspect, the optimal location of four control valves in the partitioned WDN can be analyzed (see Figure 17) and compared with the four-valve scenario in the unpartitioned WDN (Figure 14). As Figure 14 and Figure 17 show, the valve locations obtained in the case of the partitioned WDN (pipes 314, 315, 316, and 317) are different from the case of the unpartitioned WDN, due to the flow variations induced by the partitioning, though being all close to the WDN sources. Furthermore, the downstream pressure settings at the control valves are equal to 22.93, 22.50, 25.04, and 21.58 m, with an average value of 23.01 m. The settings are smaller than those equal to 25.72, 29.28, 23.13 and 25.72 m (average value of 25.97 m), obtained for the control valves installed in pipes 52, 134, 314, and 316, respectively, in the unpartitioned WDN. This proves that the partitioning improves regulation of service pressure in the WDN. Therefore, leakage volume is lower in the partitioned WDN (W

_{L}= 1796.61 m

^{3}) than the unpartitioned WDN (W

_{L}= 1931.91 m

^{3}). The lower leakage volume is consistent with the results in Figure 18, globally pointing out lower pressure heads for the partitioned WDN.

## 4. Discussion

- Analysis of service pressure in the unpartitioned WDN.
- Optimal location of control valves in the unpartitioned WDN.
- WDN partitioning in the absence of control valves.
- Optimal location of control valves in the partitioned WDN.

- When involving physical separation between partitions, WDN partitioning can result per se in the slight lowering in service pressure and, therefore, in leakage attenuation.
- Due to variations in flow distribution, the valve locations optimally selected in a partitioned WDN may differ from those in the unpartitioned WDN.
- The number of optimally installed being the same, the partitioned WDN enables achievement of better leakage reduction performance than the unpartitioned WDN.
- In both the unpartitioned and partitioned WDNs, the installation of control valves makes the daily pattern of leakage outflows flatter, by reducing the variability of service pressure in the day.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Water distribution networks (WDNs) of (

**a**) Santa Maria di Licodia and (

**b**) Modena. Arrows indicate flow direction after the preliminary analysis.

**Figure 2.**WDN of Santa Maria di Licodia. Daily patterns of demand multiplicative coefficient C and head H

_{0}at sources.

**Figure 3.**WDN of Santa Maria di Licodia. Cumulative frequency F(−) of maximum and minimum daily pressure heads h (m), in comparison with the minimum desired value h

_{des}.

**Figure 4.**WDN of Santa Maria di Licodia. Pareto fronts of optimal solution in the trade-off between N

_{val}and W

_{L}.

**Figure 5.**WDN of Santa Maria di Licodia. WDN with loops opened based on the results of the optimization.

**Figure 8.**WDN of Santa Maria di Licodia. In the absence of control valves, comparison of daily maximum (

**a**) and minimum (

**b**) pressure heads h (m) between the unpartitioned and partitioned WDN.

**Figure 9.**WDN of Santa Maria di Licodia. In the case of four control valves, comparison of daily maximum (

**a**) and minimum (

**b**) pressure heads h (m) between the unpartitioned and partitioned WDN.

**Figure 10.**WDN of Santa Maria di Licodia. Location of control valves in the unpartitioned and partitioned WDNs.

**Figure 11.**WDN of Santa Maria di Licodia. Daily pattern of leakage outflows in no control and control scenarios, for both the unpartitioned and partitioned WDN.

**Figure 12.**WDN of Modena. Cumulative frequency F(−) of daily maximum and minimum pressure heads h (m), in comparison with the minimum desired value h

_{des}.

**Figure 13.**WDN of Modena. Pareto fronts of optimal solution in the trade-off between N

_{val}and W

_{L}.

**Figure 14.**WDN of Modena. In the case of four control valves, positions of the valves in the unpartitioned WDN.

**Figure 16.**WDN of Modena. In the absence of control valves, comparison of daily maximum (

**a**) and minimum (

**b**) pressure heads h (m) between the unpartitioned and partitioned WDN.

**Figure 18.**WDN of Modena. In the presence of four control valves, comparison of daily maximum (

**a**) and minimum (

**b**) pressure heads h (m) between the unpartitioned and partitioned WDN.

**Table 1.**Water distribution network (WDN) of Santa Maria di Licodia. Results of the sequential addition of control valves in the unpartitioned WDN. Locations and leakage volumes.

N_{val} | Valve Locations on Unpartitioned WDN | W_{L} (m^{3}) on Unpartitioned WDN |
---|---|---|

0 | – | 1243 |

1 | 27 | 1029 |

2 | 27, 7 | 885 |

3 | 27, 7, 3 | 805 |

4 | 27, 7, 3, 14 | 751 |

5 | 27, 7, 3, 14, 33 | 725 |

6 | 27, 7, 3, 14, 33, 4 | 708 |

7 | 27, 7, 3, 14, 33, 4, 2 | 692 |

8 | 27, 7, 3, 14, 33, 4, 2, 41 | 680 |

9 | 27, 7, 3, 14, 33, 4, 2, 41, 6 | 670 |

10 | 27, 7, 3, 14, 33, 4, 2, 41, 6, 30 | 659 |

**Table 2.**WDN of Santa Maria di Licodia. Results of the sequential addition of control valves in the partitioned WDN. Locations, leakage volumes, and benefits of the partitioning.

N_{val} | Valve Locations on Partitioned WDN | W_{L} (m^{3}) on Partitioned WDN | W_{L} (m^{3}) on Unpartitioned WDN | Benefits (%) of Partitioning |
---|---|---|---|---|

0 | - | 1176 | 1243 | 5.42 |

1 | 27 | 978 | 1029 | 4.94 |

2 | 27, 7 | 867 | 885 | 2.00 |

3 | 27, 7, 3 | 795 | 805 | 1.24 |

4 | 27, 7, 3, 25 | 730 | 751 | 2.82 |

5 | 27, 7, 3, 25, 26 | 699 | 725 | 3.54 |

6 | 27, 7, 3, 25, 26, 2 | 676 | 708 | 4.56 |

7 | 27, 7, 3, 25, 26, 2, 33 | 654 | 692 | 5.45 |

8 | 27, 7, 3, 25, 26, 2, 33, 4 | 637 | 680 | 6.37 |

9 | 27, 7, 3, 25, 26, 2, 33, 24 | 619 | 670 | 7.58 |

10 | 27, 7, 3, 25, 26, 2, 33, 24, 17 | 607 | 659 | 7.85 |

**Table 3.**WDN of Modena. Leakage outflow rates (m

^{3}/s) in the three time slots of the day in different scenarios.

Time Slot | Scenario 1 Unpartitioned WDN, no Control | Scenario 2 Unpartitioned WDN, 4 Valves | Scenario 3 Partitioned WDN, no Control | Scenario 4 Partitioned WDN, 4 Valves |
---|---|---|---|---|

1 | 0.0363 | 0.0233 | 0.0365 | 0.0207 |

2 | 0.0359 | 0.0218 | 0.0359 | 0.0208 |

3 | 0.0354 | 0.0220 | 0.0352 | 0.0209 |

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**MDPI and ACS Style**

Creaco, E.; Castagnolo, D.; Pezzinga, G.
Assessing the Impact of Partitioning on Optimal Installation of Control Valves for Leakage Minimization in WDNs. *Water* **2021**, *13*, 1003.
https://doi.org/10.3390/w13071003

**AMA Style**

Creaco E, Castagnolo D, Pezzinga G.
Assessing the Impact of Partitioning on Optimal Installation of Control Valves for Leakage Minimization in WDNs. *Water*. 2021; 13(7):1003.
https://doi.org/10.3390/w13071003

**Chicago/Turabian Style**

Creaco, Enrico, Dario Castagnolo, and Giuseppe Pezzinga.
2021. "Assessing the Impact of Partitioning on Optimal Installation of Control Valves for Leakage Minimization in WDNs" *Water* 13, no. 7: 1003.
https://doi.org/10.3390/w13071003