# The Setpoint Curve as a Tool for the Energy and Cost Optimization of Pumping Systems in Water Networks

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Setpoint Curve

#### 2.2. Objective Functions

#### 2.3. Optimization Algorithms

## 3. Case Studies and Results

#### 3.1. TF Network

^{3}; ${\eta}_{i,N17}$ = 65%, $T{T}_{i,N17}$ = 0.20 €/m

^{3}; y ${\eta}_{i,N18}$= 60%,$T{T}_{i,N18}$ = 0.30 €/m

^{3}). The network has 17 nodes and a total of 24 pipes. The average daily flow is 100 L/s. The information on all the nodes as well as of the pipes is presented in Table 1 and Table 2. The roughness of all the pipes is 0.1 mm. The minimum pressure required in the network is ${H}_{min}$ = 20 mWC. The demand curve of the network, as well as the electricity rates, are presented in Table 3 and Figure 3.

#### 3.2. Richmond Network

## 4. Discussion

## 5. Conclusions

- If the WDN have tanks, the critical node pressure is dependent on the tank level. Consequently, the setpoint curves present oscillations with respect to the required pumping heads because it is not possible to maintain a constant pressure at the critical node of the network. However, it is possible to know the maximum savings that can be obtained as long as it is possible to adjust the curve of each pumping system to the corresponding setpoint curve, hence the usefulness when determining if a system is over- or under-dimensioned.
- The use of the setpoint curve allows for evaluating whether raising the storage tanks represents a significant economic saving, although this aspect requires a deeper analysis.
- Given that the setpoint curve depends to a large extent on the pressure at the critical node, it is possible to combine the methodology with sectorization processes with the intention of better managing the pressure of the network and leading to greater savings in energy costs.
- The studied concept can also serve as a methodology for sizing tanks that correspond to pumping stations capable of adapting to optimal setpoint curves and that further minimize operating costs.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbols | |

Q | Demand flow |

H | Head pressure |

Hpe_{i,n} | Pressure head of the reservoir |

Hnc_{i,n} | Pressure head of the critical node |

He_{i} | Pumping head of the station represented by the reservoir |

H_{i,s} | Pumping head of each station (s) for the simulation period (i) |

Q_{e,i} | Flow supplied by the reservoir |

Q_{i,s} | Flow to be supplied by each pumping station (s) during the analysis period (i) |

Qd_{i} | Flow demanded during a simulation period (i) |

Qmax_{i,s} | Maximum flow supplied by each pumping station (s) during the analysis period (i) |

Qmin_{i,s} | Minimum flow supplied by each pumping station (s) during the analysis period (i) |

Qmd | Average daily demand flow |

H_{min} | Minimum required pressure |

l_{in,ta} | Initial level of the tank |

l_{fin,ta} | Final level of the tank |

Nta | Total numbers of tanks |

η_{(i,s)} | Performance of the station (s) during the period (i) |

t_{i} | Pumping time that corresponds to the duration of the simulation period (i) |

Tf_{i,s} | Electricity rate assigned to the station (s) for the period (i) |

Abbreviations | |

SC | Setpoint Curve |

TT | Treatment rate |

FD | Demand factor |

WDN | Water Distribution Network |

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**Figure 10.**Setpoint curves (SC) and characteristic curves of the pumps (CB) for the pumping stations of the Richmond network.

ID | Elev (m) | Demand (L/s) | ID | Elev (m) | Demand (L/s) |
---|---|---|---|---|---|

N1 | 8 | 5 | N10 | 7 | 5 |

N2 | 8 | 4 | N11 | 7 | 10 |

N3 | 5 | 3 | N12 | 5 | 5 |

N4 | 8 | 4 | N13 | 4 | 2 |

N5 | 4 | 3 | N14 | 3 | 10 |

N6 | 2 | 8 | N15 | 3 | 15 |

N7 | 5 | 7 | N16 | 4 | 0 |

N8 | 6 | 10 | N17 | 0 | 0 |

N9 | 2 | 9 | N18 | 0 | 0 |

Node 1 | Node 2 | Lenght (m) | Diam. (mm) | Node 1 | Node 2 | Lenght (m) | Diam. (mm) |
---|---|---|---|---|---|---|---|

N1 | N2 | 200 | 150 | N11 | N7 | 300 | 80 |

N2 | N3 | 150 | 100 | N11 | N4 | 250 | 150 |

N3 | N4 | 150 | 100 | N8 | N12 | 250 | 80 |

N4 | N1 | 200 | 200 | N5 | N13 | 100 | 60 |

N5 | N6 | 200 | 60 | N3 | N12 | 98 | 60 |

N7 | N8 | 400 | 80 | N3 | N14 | 300 | 80 |

N6 | N7 | 300 | 60 | N14 | N15 | 500 | 80 |

N8 | N5 | 300 | 80 | N2 | N15 | 400 | 100 |

N8 | N4 | 250 | 150 | N16 | N10 | 125 | 100 |

N7 | N9 | 300 | 100 | N12 | N13 | 52 | 60 |

N10 | N11 | 300 | 100 | N17 | N12 | 1 | 2000 |

N9 | N16 | 125 | 100 | N14 | N18 | 1 | 1000 |

Time (h) | 1–5 | 6–7 | 8 | 9 | 10 | 11 | 12 | 13–14 | 15 | 16 | 17 | 18–20 | 21 | 22 | 23 | 24 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Demand factor (DF) | 0.4 | 0.7 | 1.0 | 1.2 | 0.7 | 0.7 | 1.7 | 2.0 | 1.7 | 1.0 | 0.8 | 1.1 | 1.5 | 1.5 | 1.1 | 0.4 |

N16, N17, N18 (Є/kWh) | 0.0672 | 0.1094 | 0.2768 | 0.1094 | 0.0672 |

ID Source | Maximum Flow (L/s) | Performance (%) |
---|---|---|

(1A, 2A) | 100.00 | 75 |

(3A) | 70.00 | 77 |

(4B) | 111.50 | 72 |

(5C) | 6.11 | 71 |

(6D) | 13.89 | 58 |

(7F) | 6.00 | 54 |

ID Tanks | Diameter (mm) | Initial Level (m) | Min Level (m) | Max Level (m) |
---|---|---|---|---|

A | 23.5 | 2.050 | 1.02 | 3.37 |

B | 15.4 | 2.030 | 2.03 | 3.65 |

C | 6.6 | 0.500 | 0.50 | 2.00 |

D | 11.8 | 1.100 | 1.10 | 2.11 |

E | 8.0 | 1.992 | 0.20 | 2.69 |

F | 3.6 | 1.293 | 0.19 | 2.19 |

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**MDPI and ACS Style**

León-Celi, C.F.; Iglesias-Rey, P.L.; Martínez-Solano, F.J.; Mora-Melia, D.
The Setpoint Curve as a Tool for the Energy and Cost Optimization of Pumping Systems in Water Networks. *Water* **2022**, *14*, 2426.
https://doi.org/10.3390/w14152426

**AMA Style**

León-Celi CF, Iglesias-Rey PL, Martínez-Solano FJ, Mora-Melia D.
The Setpoint Curve as a Tool for the Energy and Cost Optimization of Pumping Systems in Water Networks. *Water*. 2022; 14(15):2426.
https://doi.org/10.3390/w14152426

**Chicago/Turabian Style**

León-Celi, Christian F., Pedro L. Iglesias-Rey, Francisco Javier Martínez-Solano, and Daniel Mora-Melia.
2022. "The Setpoint Curve as a Tool for the Energy and Cost Optimization of Pumping Systems in Water Networks" *Water* 14, no. 15: 2426.
https://doi.org/10.3390/w14152426