# Optimal Design of Intermittent Water Distribution Network Considering Network Resilience and Equity in Water Supply

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Pressure-Driven Analysis

- (1)
- Network modeling: Using EPANET, a hydraulic model of the WDN is created. The model contains data on the boundary conditions, nodal demands, pipe characteristics, and network topology. The network’s hydraulic behavior can be simulated using this model as a starting point.
- (2)
- Demand allocation: For realistic simulations, it is essential to accurately estimate demands at each node. Based on past trends, demographic estimates, and specific consumer data, the demands may be determined. To capture genuine demand profiles, seasonal fluctuations and nocturnal patterns are taken into account.
- (3)
- Pressure simulation: Various scenarios for the network’s behavior are simulated using the hydraulic model. The operating conditions, such as pump operation, valve settings, and tank levels, are taken into account during the simulation. Based on the water conservation and energy calculations, the simulation determines pressure heads and flow rates across the network.
- (4)
- Performance assessment: The network’s performance is then assessed using the simulation findings. It is possible to evaluate key performance indicators such as minimum and maximum pressures, pressure deficits, network resilience, uniformity coefficient, or equity and flow rates. These indicators aid in locating trouble spots, such as low-pressure locations, high-pressure areas, and probable water quality problems.

#### 2.2. Source Head Method

- 1.
- Run hydraulic analysis in EPANET (RunH) to determine the pressure head at the source (${H}_{s}$).
- 2.
- For each node in the network:
- a.
- Get the pressure head at the node.
- b.
- Calculate the head loss between the source and the node as: ${HL}_{j}$ = ${H}_{s}$ − ${H}_{j}$.
- c.
- Calculate the desired pressure head at the source for this node as:$${H}_{sj}^{des}=\mathit{elj}+{H}_{jmin}+{HL}_{j}+{H}_{j}^{des}\text{}s\text{}\left(\mathrm{where}\text{}\mathit{elj}\text{}\mathrm{is}\text{}\mathrm{the}\text{}\mathrm{elevation}\text{}\mathrm{at}\text{}\mathrm{node}\text{}j\right).$$

- 3.
- To calculate the minimum source pressure head at which discharge at each node is zero, for each node in the network:
- i.
- Set Q
_{j}(discharge) to 0. - ii.
- Run hydraulic analysis (RunH) to get the pressure head at the node.
- iii.
- Calculate the head loss between the source and the node when only Q
_{j}is zero, as: ${HL}_{j1}$ = ${H}_{s}$ − ${H}_{j}$. - iv.
- Reset Q
_{j}to its actual value. - v.
- Calculate the minimum pressure head at the source when only Q
_{j}is zero, as:$${H}_{sj}^{min1}=\mathit{elj}+{H}_{jmin}+{HL}_{j1}.$$

- 4.
- Sort the nodes in ascending order of ${H}_{sj}^{min1}$.
- 5.
- For each node in the sorted order:Set Q
_{j}(discharge) to 0.- a.
- Run hydraulic analysis (RunH) to get the pressure head at the node.
- b.
- Calculate the head loss between the source and the node when discharge at this node and also the nodes above this in the sorted list are zero, as: ${HL}_{j2}$ = ${H}_{s}$ − ${H}_{j}$.
- c.
- Calculate the minimum pressure head at the source when all discharge at this node and also the nodes above this are zero, as: ${H}_{sj}^{min1}$ = elj +${H}_{jmin}$ + ${HL}_{j2}$.

End of loop. - 6.
- After getting ${H}_{sj}^{min}$ and ${H}_{sj}^{des}$, for each node, calculate the corrected discharge and check the head (if ${H}_{ji\text{}}$ − ${h}_{ji\text{}}$> Limit, go for previous steps, otherwise, continue with step 6), calculate average supply ratio (SR), uniformity coefficient (CU).

#### 2.3. Performance Indicators

#### 2.4. Multi-Objective Evolutionary Optimization (MOEA)

#### 2.5. Analysis of Networks

#### 2.5.1. Network-1

#### 2.5.2. Network-2

#### 2.6. Multidimensional Pareto Analysis

- (a)
- Dominance matrix: The code constructs a dominance matrix where each element (i, j) represents whether solution i dominates solution j. This matrix captures the dominant relationships among solutions for multiple objectives.
- (b)
- Identifying Pareto solutions: By analyzing the dominance matrix, the code identifies solutions that are Pareto optimal, meaning they are not dominated by any other solutions for all the objectives.
- (c)
- Extracting Pareto front: The code extracts the Pareto front solutions from the original dataset based on the identified Pareto optimal solutions.

## 3. Results and Discussions

#### 3.1. Trade-off between the Objective Functions

#### 3.2. Multidimensional Pareto Analysis

## 4. Conclusions

- The approach accelerates the optimization of intermittent water distribution networks, considering into account factors such as equity, reliability, and cost. Enhancing equity has a significant influence on the overall reliability of the networks. Hence, conducting a comprehensive analysis that considers both aspects are crucial to understanding the mutual impact they have on each other.
- Two indices, network resilience (${I}_{n}$) and uniformity coefficient ($CU$), are taken as the performance indicators that improve the network reliability and equity in the distribution networks considered in the study. The results from the two networks show that increasing network resilience can also increase water supply equity. Optimizing the layout of the networks, especially in the case of large networks, may improve the equity in the water supply, without increasing the cost of the network.
- The application of NSGA-II to the benchmark networks from the literature to optimize the design of WDN is analyzed in the present study. Results show that network resilience can be increased up to 0.9 with minimal cost. As reliability approaches 1, cost of the network increases significantly with a marginal increment of reliability, which is uneconomical for the design.
- It is observed that the CU solution set obtained for the networks indicates increasing equity can be achieved to a certain extent (up to 0.85) with lesser increment in cost. Beyond the CU value of 0.85, the cost of networks increases, which indicates the effort needed in the selection of pipe, pump, and source tank to enhance the equity in supply with minimal cost.
- Overall, the present study considered equity and reliability as major factors for the design of WDNs along with cost, and results obtained from the application of NSGA-II conclude that equity and reliability of the networks are enhanced simultaneously to a reasonable extent with minimal cost.
- A multidimensional Pareto analysis carried out in the present study benefits the decision-makers to identify non-dominated solutions out of the Pareto front.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Tradeoff between objective functions for Network 1. (

**a**) Tradeoff between cost and uniformity coefficient. (

**b**) Tradeoff between cost and reliability. (

**c**) Tradeoff between reliability and uniformity coefficient.

**Figure 4.**Tradeoff between objective functions for Network 2. (

**a**) Tradeoff between cost and uniformity coefficient. (

**b**) Tradeoff between cost and reliability. (

**c**) Tradeoff between reliability and uniformity coefficient.

S. No | Pipe Diameter (in mm) | Total Cost of Pipe (in INR per meter) |
---|---|---|

1 | 100 | 560 |

2 | 150 | 900 |

3 | 200 | 1303 |

4 | 250 | 1757 |

5 | 300 | 2267 |

6 | 350 | 2848 |

7 | 400 | 3485 |

8 | 450 | 4220 |

9 | 500 | 4820 |

10 | 550 | 5795 |

11 | 600 | 6794 |

12 | 650 | 7352 |

13 | 700 | 8050 |

14 | 750 | 9280 |

15 | 800 | 10,692 |

Parameter | Value |
---|---|

Size of population | 200 |

Probability of crossover | 0.6 |

Probability of mutation | 1/(number of real variables) |

Mutation distribution index | 15 |

Crossover distribution index | 25 |

Node ID | Elevation, m | Demand in Meter Cube per Hour (m^{3}/h) |
---|---|---|

1 | 210 | Tank |

2 | 180 | 100 |

3 | 190 | 100 |

4 | 185 | 120 |

-5 | 180 | 270 |

6 | 195 | 330 |

7 | 190 | 200 |

Link ID | Start Node | End Node | Length, m | Roughness Coefficient |
---|---|---|---|---|

1 | 1 | 2 | 1000 | 100 |

2 | 2 | 3 | 1000 | 100 |

3 | 2 | 4 | 1000 | 100 |

4 | 4 | 5 | 1000 | 100 |

5 | 4 | 6 | 1000 | 100 |

6 | 6 | 7 | 1000 | 100 |

7 | 3 | 5 | 1000 | 100 |

8 | 5 | 7 | 1000 | 100 |

Node ID | Elevation (In m) | Demand in Meter Cube per Day (m^{3}/day) |
---|---|---|

1 | 180 | Tank |

2 | 178 | 600 |

3 | 179 | 1000 |

4 | 180 | 900 |

5 | 181 | 1200 |

6 | 183 | 900 |

7 | 182 | 800 |

8 | 181 | 800 |

9 | 180 | 1200 |

10 | 182 | 1200 |

11 | 181 | 600 |

12 | 181 | 800 |

13 | 183 | 1200 |

14 | 184 | 800 |

15 | 179 | 800 |

16 | 180 | 600 |

17 | 181 | 900 |

Link ID | Start Node | End Node | Length, m | Roughness Coefficient |
---|---|---|---|---|

1 | 1 | 2 | 1400 | 100 |

2 | 2 | 3 | 1700 | 100 |

3 | 3 | 4 | 1000 | 100 |

4 | 4 | 5 | 900 | 100 |

5 | 5 | 6 | 1350 | 100 |

6 | 2 | 7 | 900 | 100 |

7 | 7 | 8 | 1100 | 100 |

8 | 8 | 5 | 1400 | 100 |

9 | 5 | 9 | 900 | 100 |

10 | 5 | 10 | 1000 | 100 |

11 | 6 | 10 | 1200 | 100 |

12 | 1 | 11 | 1100 | 100 |

13 | 11 | 12 | 800 | 100 |

14 | 12 | 13 | 1400 | 100 |

15 | 13 | 9 | 800 | 100 |

16 | 10 | 14 | 1100 | 100 |

17 | 11 | 15 | 1200 | 100 |

18 | 15 | 16 | 800 | 100 |

19 | 16 | 13 | 900 | 100 |

20 | 16 | 17 | 1400 | 100 |

21 | 17 | 14 | 1200 | 100 |

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**MDPI and ACS Style**

Ramani, K.; Rudraswamy, G.K.; Umamahesh, N.V.
Optimal Design of Intermittent Water Distribution Network Considering Network Resilience and Equity in Water Supply. *Water* **2023**, *15*, 3265.
https://doi.org/10.3390/w15183265

**AMA Style**

Ramani K, Rudraswamy GK, Umamahesh NV.
Optimal Design of Intermittent Water Distribution Network Considering Network Resilience and Equity in Water Supply. *Water*. 2023; 15(18):3265.
https://doi.org/10.3390/w15183265

**Chicago/Turabian Style**

Ramani, Katineni, G. K. Rudraswamy, and Nanduri V. Umamahesh.
2023. "Optimal Design of Intermittent Water Distribution Network Considering Network Resilience and Equity in Water Supply" *Water* 15, no. 18: 3265.
https://doi.org/10.3390/w15183265