# Water Distribution Network Optimization Model with Reliability Considerations in Water Flow (Debit)

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## Abstract

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## 1. Introduction

## 2. Problem Description

#### 2.1. Water Distribution Network Optimization Model Components

#### 2.1.1. Hydraulic Network Components and Sets Required in Model Development

- Vertices (a subset of $N$): ($N$ as a set of vertexes and the set of natural numbers may not be confused);Other node sets can represent groundwater, desalinization, wastewater treatment plants, etc.
- Aqueducts (Arcs) (a subset of $A$):The mathematical model of system planning/design that pays attention to reliability on constraints such as this requires the optimization problem to adopt probabilistic rules on the constraints so that the optimization model that is formed becomes an optimization model problem with probabilistic constraints or opportunity constraints. A water distribution network optimization model, in its planning and application design, is inseparable from one that seeks to satisfy demands in its services. Therefore, a water distribution network optimization model is proposed by adopting probabilistic rules of several constraints that can increase service satisfaction. This water distribution network optimization model is called the chance-constrained model. In its application, we attempt to display several possible performance sets of a water distribution network system as different uncertain parameters.The general form of the opportunity constraint model can be written as:

#### 2.1.2. Pressure Drop

- $\u2206P$ is the pressure drop across the pipe.
- $f$ is the Darcy friction factor, which depends on the roughness of the pipe surface and the Reynolds number (a dimensionless quantity describing the flow regime).
- $L$ is the length of the pipe.
- $D$ is the diameter of the pipe.
- $\rho $ is the density of the fluid (water in this case).
- $V$ is the velocity of the fluid within the pipe.

#### 2.2. Deterministic Chance-Constrained Forms

#### 2.3. Joint Chance-Constrained Form

- Chance-constrained models (individual or joint);
- Assumptions of random vectors (whether continuous or discrete, or independent);
- Types of stochastic inequalities (linear, convex, right-hand random constraints).

## 3. Results and Discussion

#### 3.1. The Algorithm

- Stage 1.

- Stage 2.

#### 3.2. Example

^{3}/s, and the roughness coefficient is set at 100. For all $i\in N$, the minimum allowable pressure head value, denoted by ${ph}_{\mathrm{min}}\left(i\right)$, is 40 m, while the maximum allowable value, denoted by ${ph}_{\mathrm{max}}\left(i\right)$, is 121 eval(i) meters. Table 2 describes the network’s structure and pipe lengths. In terms of the diameters of the pipes we have available, we have thirteen distinct options to choose from. Table 1 displays the $(D\left(e,r\right),C\left(e,r\right))$ value for all $r=1,\dots ,{r}_{e}$ for each $e\in E$. Elevation and demand data for each junction are shown in Table 1, beside the reservoir.

#### 3.3. Solution Found by the MINLP Model

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The challenge of the network is solved by employing arcs whose widths are directly related to the sizes of the circles they connect. Unnumbered pipes have a 0.06 m internal diameter.

$\mathit{i}$ | $\mathit{e}\mathit{l}\mathit{e}\mathit{v}\left(\mathit{i}\right)$ | $\mathit{d}\mathit{e}\mathit{m}\left(\mathit{i}\right)$ |
---|---|---|

1 | 65.15 | 0.00049 |

2 | 64.40 | 0.00104 |

3 | 63.35 | 0.00102 |

4 | 62.50 | 0.00081 |

5 | 61.24 | 0.00063 |

6 | 65.40 | 0.00079 |

7 | 67.90 | 0.00026 |

8 | 66.50 | 0.00058 |

9 | 66.00 | 0.00054 |

10 | 64.17 | 0.00111 |

11 | 63.70 | 0.00175 |

12 | 62.64 | 0.00091 |

13 | 61.90 | 0.00116 |

14 | 62.60 | 0.00054 |

15 | 63.50 | 0.00110 |

16 | 64.30 | 0.00121 |

17 | 65.50 | 0.00127 |

18 | 64.10 | 0.00202 |

19 | 62.90 | 0.00188 |

20 | 62.83 | 0.00093 |

21 | 62.80 | 0.00096 |

22 | 63.90 | 0.00097 |

23 | 64.20 | 0.00086 |

24 | 67.50 | 0.00067 |

25 | 64.40 | 0.00077 |

26 | 63.40 | 0.00169 |

27 | 63.90 | 0.00142 |

28 | 65.65 | 0.00030 |

29 | 64.50 | 0.00062 |

30 | 64.10 | 0.00054 |

31 | 64.40 | 0.00090 |

32 | 64.20 | 0.00103 |

33 | 64.60 | 0.00077 |

34 | 64.70 | 0.00074 |

35 | 65.43 | 0.00116 |

36 | 65.90 | 0.00047 |

$\mathit{r}$ | $\mathcal{D}\left(\mathit{e},\mathit{r}\right)$ | $\mathcal{C}\left(\mathit{e},\mathit{r}\right)$ |
---|---|---|

1 | 0.060 | 19.8 |

2 | 0.080 | 24.5 |

3 | 0.100 | 27.2 |

4 | 0.125 | 37.0 |

5 | 0.150 | 39.4 |

6 | 0.200 | 54.4 |

7 | 0.250 | 72.9 |

8 | 0.300 | 90.7 |

9 | 0.350 | 119.5 |

10 | 0.400 | 139.1 |

11 | 0.450 | 164.4 |

12 | 0.500 | 186.0 |

13 | 0.600 | 241.3 |

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**MDPI and ACS Style**

Mawengkang, H.; Syahputra, M.R.; Sutarman, S.; Weber, G.W.
Water Distribution Network Optimization Model with Reliability Considerations in Water Flow (Debit). *Water* **2023**, *15*, 3119.
https://doi.org/10.3390/w15173119

**AMA Style**

Mawengkang H, Syahputra MR, Sutarman S, Weber GW.
Water Distribution Network Optimization Model with Reliability Considerations in Water Flow (Debit). *Water*. 2023; 15(17):3119.
https://doi.org/10.3390/w15173119

**Chicago/Turabian Style**

Mawengkang, Herman, Muhammad Romi Syahputra, Sutarman Sutarman, and Gerhard Wilhelm Weber.
2023. "Water Distribution Network Optimization Model with Reliability Considerations in Water Flow (Debit)" *Water* 15, no. 17: 3119.
https://doi.org/10.3390/w15173119