# Service Pressure and Energy Consumption Mitigation-Oriented Partitioning of Closed Water Distribution Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Case Study

_{d}shown in Figure 2 is considered, in order to simulate the daily operation of the WDN in an extended simulation period.

## 3. Materials and Methods

- Partitioning (clustering and dividing) of the WDN;
- Optimization of the hourly settings of the pumping stations present in the WDN.

#### 3.1. WDN Partitioning

#### 3.1.1. Clustering

_{i}and α

_{i}are the three terms described above, and their respective weighing coefficients. The terms H

_{i}are calculated with the following relationships [13]:

_{b}and n

_{p}are the number of boundary pipes and the total number of pipes, respectively. U

_{i}is the property to be made uniform between the DMAs (for instance, DMA demand), while U

_{tot}is the sum of U

_{i}in all the M DMAs. With reference to the property to be made uniform inside each segment, e.g., ground elevation, u

_{i,j}is the property values in the j-th of the n(i) nodes of the i-th DMA. Then, $\overline{{u}_{i}}$ is the mean value of u

_{i,j}in the i-th DMA, while u

_{max}and u

_{min}are the maximum and minimum values of u

_{i,j}in the whole WDN, respectively.

- Construction of a configuration of the first attempt with the desired number M of clusters;
- Refinement of the configuration by means of an algorithm inspired by simulated annealing to search for a configuration with a better value of Q, the number M of clusters being the same.

_{i}are concerned [13], we recommend that a sensitivity analysis be performed by varying them within the range [0, 2], while making sure that their sum stays equal to 2. Based on the configurations obtained during the sensitivity analysis, the ultimate clustering solution can be chosen.

#### 3.1.2. Dividing

_{r}and failure I

_{f}indices, expressed as follows [25]:

**q**and

_{user}**d**are the vector of nodal outflows to users and user demands at demand nodes, respectively.

**H**and

**H**are the vectors of nodal heads and desired heads for full outflow at demand nodes. Namely, the generic element H

_{des}_{des}of

**H**can be calculated by summing the nodal ground elevation z and the desired pressure head h

_{des}_{des}.

**Q**and

_{0}**H**are the vectors of source outflows and heads, respectively. Finally,

_{0}**Q**and

_{p}**H**are the vectors of pump flows and heads, respectively. GRF is an index ranging from −1 to 1, in which the larger values are associated with high-pressure heads at demand nodes, guaranteeing full satisfaction of user demands. While the negative values of GRF accounted for in I

_{p}_{f}represent a power deficit in the WDN, the positive values of GRF accounted for in I

_{r}represent the surplus of power, compared to the minimum power necessary for meeting user demands with suitable service pressure. The GRF is close to 1 when small head losses are present in the WDN under peak demand conditions. In the present work, GRF variations are calculated to analyze the power dissipation effects produced by the closure of boundary pipes in the dividing phase of WDN partitioning.

_{f}of flow meters = number n

_{b}of boundary pipes). This configuration of flow meters is associated with the highest GRF = GRF

_{0}.

_{f}= n

_{b}− 1 and GRF= GRF

_{1}.

_{f}= n

_{b}− 2 and GRF = GRF

_{2}.

_{f}= n

_{b}− k and GRF = GRF

_{k}.

_{des}.

_{f}of flow meters installed as a function of the GRF, an effective and efficient approximation of the Pareto front in the trade-off between n

_{f}and GRF, to be minimized and maximized respectively, is obtained. The Pareto front can also be plotted in the trade-off between the number of closed isolation valves on boundary pipes N

_{civ}= n

_{b}− n

_{f}and GRF, for both to be maximized.

_{des}are encountered while reducing the number of flow meters from n

_{f}= n

_{b}down to n

_{f}= 0, the number N

_{c}of WDN configurations explored in the present iterative algorithm is:

_{b}= 50, the relationship (7) would yield N

_{c}= 1275 << ${n}_{b}^{2}$ = 2500.

#### 3.2. The Optimization of Pump Settings

_{i}and n

_{n}are the pressure head at the generic i-th demand node and the number of demand nodes in the WDN, respectively.

_{ps}pumping station is n

_{m,j}, the number of decision variables, i.e., the number n

_{set}of pump settings to optimize at each time instant, is the following, if a single setting is used for all pumps belonging to the same model in the generic station:

_{m,j}= 1 for each j), as in the case study network in Figure 1, the relationship (8) simply yields n

_{set}= n

_{ps}.

^{®}2023a environment [23], to approximate the optimum in this optimization problem. The hydraulic solver used for each solution proposed by the optimizer is the Epanet 2.2 toolkit developed by [26].

## 4. Results

#### 4.1. Results of WDN Partitioning

#### 4.1.1. Results of Clustering

_{2}was kept equal to 0 in all combinations. Then, four combinations involving values of α

_{1}and α

_{3}ranging from 0.1 to 1.9, with sum equal to 2, were considered.

_{1}at the expense of α

_{3}tends to produce clustering solutions with progressively smaller values of H

_{1}(less numerous boundary pipes n

_{b}) and larger values of H

_{3}(higher heterogeneity of ground elevations inside each DMA) (see also Figure 4). Though not controlled in the refinement process due to α

_{2}= 0, an improvement in the uniformity of DMA demands (decrease in H

_{2}) is also obtained.

_{3}compared to the other solutions, one of the DMAs, i.e., the one in magenta, degenerates into a single node. Then, it was discarded in the subsequent analyses. As solutions 2 and 3 are very similar in terms of n

_{b}, H

_{1}and H

_{3}, only solution 3 was considered for the dividing, together with solution 4.

#### 4.1.2. Results of Dividing

_{civ}= n

_{b}− n

_{f}and GRF, shown in Figure 6a,b, respectively. In these calculations, the desired pressure head h

_{des}was considered equal to 20 m.

_{civ}) exist for values of N

_{civ}ranging from 0 to 38 (solution 3—Figure 6a) or 34 (solution 4—Figure 6b), as the closure of additional isolation valves on boundary pipes causes topological disconnections in the WDN. Expectedly, the two graphs show that GRF tends to decrease as N

_{civ}increases, as the closure of isolation valves always causes an increase in head losses and the lowering of service pressure in the WDN. However, the decrease in GRF is almost inappreciable until N

_{civ}= 30 for solution 3 (see Figure 6a) or N

_{civ}= 25 for solution 4 (see Figure 6b). To the right of these values, the decrease in GRF is appreciable, but still small. As nodal pressure heads never fall below h

_{des}, the extreme configurations to the right-hand side were selected as the ultimate partitioning corresponding to a similar value of GRF ≈ 0.57 and to values N

_{civ}= 38 and N

_{civ}= 34 for solutions 3 and 4, respectively. The choice of the configurations with the highest N

_{civ}is due to (i) the small decrease in GRF in comparison with the unpartitioned solution (GRF ≈ 0.65), and to (ii) the possibility of limiting the number of flow meters n

_{f}= n

_{b}− N

_{civ}. In fact, the number of flow meters is then n

_{f}= n

_{b}− N

_{civ}= 8 for both solutions 3 and 4, which is reasonable for a number M of DMAs equal to 7, to be added to the six flow meters already present at the exit of the pumping stations.

_{civ}is motivated by the need to limit partitioning costs, as the closure of an isolation valve (following its installation if the valve is absent) is much less expensive than the installation of a flow meter.

#### 4.2. Results of the Optimization of Pump Settings

_{min}) and average ($\overline{h}$) service pressure for solutions 3 and 4 on a typical day of operation, in comparison with the pattern obtained in the benchmark WDN configuration with no partitioning and no pump setting optimization. The analysis of this figure highlights a significant reduction in terms of service pressure. The comparison of solutions 3 and 4 points out that solution 3 enables better mitigation of $\overline{h}$, as a result of more uniform nodal ground elevations inside each DMA. However, the pattern of h

_{min}is constant at 20 m for both solutions, due to the minimum pressure constraint adopted in the optimization. Overall, solution 3 enables the reduction in the average and minimum service pressure by 48% and 55%, respectively. Solution 4, instead, enables a reduction in the average and minimum service pressure by 46% and 55%, respectively.

_{mean}and daily average total absorbed energy, with a solution of WDN partitioning into seven districts obtained via clustering based on the Girvan and Newman [27] algorithm, with no consideration of WDN altimetry. The dividing described in Section 3.1.2 was applied to this solution, resulting in n

_{f}= 9 of the n

_{b}= 28 boundary pipes to be equipped with the flow meter, in order to obtain a similar GRF to solutions 3 and 4 under peak demand conditions. Then, the optimization of pump settings was performed as explained in 3.2. Overall, Table 6 proves the solution based on the Girvan and Newman [27] algorithm to be less effective than solutions 3 and 4 in terms of the mitigation of service pressure and energy consumption. This attests to the benefits of performing the clustering while pursuing the uniformity of nodal ground elevations inside each DMA.

## 5. Discussion

- The modelled service pressure remained above the threshold of 20 m for full demand satisfaction.
- The rate of leakage in the real WDN is currently small.
- The number of nodes in the WDN layout is high, meaning that demand is always modelled close to its associated user along the pipe.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**Pareto fronts of solutions in the trade-off between N

_{civ}and GRF, obtained in the dividing of solutions (

**a**) 3 and (

**b**) 4.

**Figure 7.**Daily pattern of minimum and mean pressure in the WDN in the benchmark unpartitioned layout and in partitioned solutions 3 and 4.

**Table 1.**Characteristics of the pumping stations in terms of number of pumps and maximum water discharge Q

_{max}and head H

_{max}.

Station ID | Station Name | Number of Pumps | Q_{max} (L/s) | H_{max} (m) |
---|---|---|---|---|

1 | Villalunga | 1 | 50 | 80 |

2 | Nord | 4 | 144 | 47 |

3 | Libertà | 1 | 38 | 85 |

4 | Borgo Ticino | 1 | 38 | 85 |

5 | Mirabello | 1 | 50 | 80 |

6 | Est | 4 | 144 | 47 |

**Table 2.**Solutions and weights in the clustering. Characteristics of the solutions in terms of number n

_{b}of boundary pipes and modularity contributions H.

Solution | Weights | n_{b} (-) | H_{1} (-) | H_{2} (-) | H_{3} (-) |
---|---|---|---|---|---|

1 | α_{1} = 0.1; α_{2} = 0.0; α_{3} = 1.9 | 53 | 0.0112 | 0.2351 | 0.0315 |

2 | α_{1} = 0.5; α_{2} = 0.0; α_{3} = 1.5 | 50 | 0.0105 | 0.1668 | 0.0468 |

3 | α_{1} = 1.0; α_{2} = 0.0; α_{3} = 1.0 | 46 | 0.0097 | 0.1643 | 0.0478 |

4 | α_{1} = 1.5; α_{2} = 0.0; α_{3} = 0.5 | 42 | 0.0089 | 0.1642 | 0.0494 |

Time (h) | Setting in Station 1 | Setting in Station 2 | Setting in Station 3 | Setting in Station 4 | Setting in Station 5 | Setting in Station 6 |
---|---|---|---|---|---|---|

0–1 | 0.648 | 0.607 | 0.754 | 0.770 | 0.717 | 0.571 |

1–2 | 0.646 | 0.607 | 0.760 | 0.776 | 0.708 | 0.569 |

2–3 | 0.648 | 0.606 | 0.756 | 0.774 | 0.715 | 0.569 |

3–4 | 0.648 | 0.606 | 0.756 | 0.774 | 0.715 | 0.569 |

4–5 | 0.646 | 0.615 | 0.750 | 0.764 | 0.780 | 0.569 |

5–6 | 0.707 | 0.729 | 0.678 | 0.710 | 0.680 | 0.728 |

6–7 | 0.880 | 0.893 | 0.943 | 0.665 | 0.643 | 0.909 |

7–8 | 0.884 | 0.939 | 0.890 | 0.790 | 0.673 | 0.943 |

8–9 | 0.880 | 0.893 | 0.943 | 0.665 | 0.643 | 0.909 |

9–10 | 0.769 | 0.774 | 0.786 | 0.656 | 0.740 | 0.798 |

10–11 | 0.769 | 0.774 | 0.786 | 0.656 | 0.740 | 0.798 |

11–12 | 0.777 | 0.795 | 0.806 | 0.725 | 0.888 | 0.831 |

12–13 | 0.717 | 0.751 | 0.738 | 0.663 | 0.725 | 0.765 |

13–14 | 0.707 | 0.729 | 0.678 | 0.710 | 0.680 | 0.728 |

14–15 | 0.672 | 0.667 | 0.702 | 0.702 | 0.663 | 0.631 |

15–16 | 0.699 | 0.686 | 0.761 | 0.779 | 0.658 | 0.648 |

16–17 | 0.714 | 0.707 | 0.646 | 0.721 | 0.671 | 0.703 |

17–18 | 0.717 | 0.751 | 0.738 | 0.663 | 0.725 | 0.765 |

18–19 | 0.694 | 0.716 | 0.729 | 0.717 | 0.708 | 0.706 |

19–20 | 0.653 | 0.659 | 0.734 | 0.734 | 0.638 | 0.610 |

20–21 | 0.652 | 0.647 | 0.730 | 0.740 | 0.702 | 0.597 |

21–22 | 0.636 | 0.619 | 0.748 | 0.756 | 0.791 | 0.572 |

22–23 | 0.640 | 0.613 | 0.758 | 0.769 | 0.736 | 0.571 |

23–24 | 0.649 | 0.605 | 0.730 | 0.733 | 0.719 | 0.571 |

Time (h) | Setting in Station 1 | Setting in Station 2 | Setting in Station 3 | Setting in Station 4 | Setting in Station 5 | Setting in Station 6 |
---|---|---|---|---|---|---|

0–1 | 0.648 | 0.606 | 0.756 | 0.769 | 0.716 | 0.571 |

1–2 | 0.658 | 0.604 | 0.760 | 0.778 | 0.713 | 0.569 |

2–3 | 0.649 | 0.605 | 0.760 | 0.776 | 0.721 | 0.569 |

3–4 | 0.649 | 0.605 | 0.760 | 0.776 | 0.721 | 0.569 |

4–5 | 0.642 | 0.615 | 0.750 | 0.758 | 0.773 | 0.571 |

5–6 | 0.829 | 0.730 | 0.765 | 0.763 | 0.763 | 0.729 |

6–7 | 0.873 | 0.890 | 0.886 | 0.860 | 0.786 | 0.979 |

7–8 | 0.944 | 0.938 | 0.976 | 0.985 | 0.744 | 0.999 |

8–9 | 0.873 | 0.890 | 0.886 | 0.860 | 0.786 | 0.979 |

9–10 | 0.766 | 0.792 | 0.832 | 0.811 | 0.639 | 0.820 |

10–11 | 0.766 | 0.792 | 0.832 | 0.811 | 0.639 | 0.820 |

11–12 | 0.754 | 0.823 | 0.697 | 0.723 | 0.649 | 0.922 |

12–13 | 0.735 | 0.763 | 0.839 | 0.745 | 0.659 | 0.776 |

13–14 | 0.829 | 0.730 | 0.765 | 0.763 | 0.763 | 0.729 |

14–15 | 0.662 | 0.667 | 0.705 | 0.715 | 0.765 | 0.607 |

15–16 | 0.670 | 0.692 | 0.739 | 0.803 | 0.665 | 0.623 |

16–17 | 0.742 | 0.713 | 0.824 | 0.849 | 0.683 | 0.648 |

17–18 | 0.735 | 0.763 | 0.839 | 0.745 | 0.659 | 0.776 |

18–19 | 0.696 | 0.727 | 0.815 | 0.890 | 0.639 | 0.669 |

19–20 | 0.677 | 0.656 | 0.804 | 0.798 | 0.815 | 0.592 |

20–21 | 0.674 | 0.651 | 0.732 | 0.740 | 0.715 | 0.601 |

21–22 | 0.637 | 0.623 | 0.768 | 0.764 | 0.791 | 0.572 |

22–23 | 0.645 | 0.612 | 0.758 | 0.764 | 0.742 | 0.571 |

23–24 | 0.648 | 0.606 | 0.756 | 0.769 | 0.716 | 0.571 |

**Table 5.**Mean power (KW) absorbed in each pumping station and in the whole WDN, in the benchmark unpartitioned layout and in partitioned solutions 3 and 4.

Station | Benchmark | Solution 3 | Solution 4 |
---|---|---|---|

Station 1 | 29.62 | 11.40 | 12.10 |

Station 2 | 138.95 | 66.17 | 67.29 |

Station 3 | 19.71 | 9.00 | 9.92 |

Station 4 | 19.25 | 7.49 | 9.84 |

Station 5 | 30.22 | 13.45 | 12.24 |

Station 6 | 145.64 | 67.62 | 71.69 |

Total | 383.40 | 175.13 | 183.08 |

**Table 6.**Comparison of solution 3, solution 4 and the solution based on clustering [27] in terms of daily average mean pressure h

_{mean}and total mean absorbed energy.

Results | Solution 3 | Solution 4 | Solution Based on Clustering [27] |
---|---|---|---|

h_{mean} (m) | 28.88 | 30.09 | 30.83 |

Total Energy (KW) | 175.13 | 183.08 | 213.21 |

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## Share and Cite

**MDPI and ACS Style**

Creaco, E.; Giudicianni, C.; Tosco, A.
Service Pressure and Energy Consumption Mitigation-Oriented Partitioning of Closed Water Distribution Networks. *Water* **2023**, *15*, 3218.
https://doi.org/10.3390/w15183218

**AMA Style**

Creaco E, Giudicianni C, Tosco A.
Service Pressure and Energy Consumption Mitigation-Oriented Partitioning of Closed Water Distribution Networks. *Water*. 2023; 15(18):3218.
https://doi.org/10.3390/w15183218

**Chicago/Turabian Style**

Creaco, Enrico, Carlo Giudicianni, and Alessandro Tosco.
2023. "Service Pressure and Energy Consumption Mitigation-Oriented Partitioning of Closed Water Distribution Networks" *Water* 15, no. 18: 3218.
https://doi.org/10.3390/w15183218