# Leakage Control and Energy Consumption Optimization in the Water Distribution Network Based on Joint Scheduling of Pumps and Valves

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## Abstract

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## 1. Introduction

## 2. Model and Methodology

#### 2.1. Establishment of the Optimization Model

#### 2.1.1. Objective Function and Decision Variables

#### 2.1.2. Constraints

#### Water Supply Capacity Constraints for Each WTP

#### Speed Constraints of VSPs

#### Node Pressure Constraints

#### 2.1.3. The Calculation of Leakage

#### 2.1.4. The Simulation of PRV

#### 2.1.5. The Characteristics of VSP

#### 2.2. Genetic Algorithm for Optimization

## 3. Calculations and Results

^{3}, and the electricity price coefficient $\psi $ = 0.6 RMB/KWh. Motor efficiency is chosen as 90%. Considering that the solution space increases sharply as the number of PRV increases, the diameter multiplier interval is set to 0.1, with a total of 11 gears. Meanwhile, the node pressure constraint is difficult to satisfy when the pump speed closes to a lower limit (0.75 or 0.8). Thus, 0.85 to 1.0 is selected as the adjustment range of VSPs, with a total of 16 gears. The minimum node pressure ${H}_{i,min}$ = 28 m, the leakage constant ${C}_{L}={10}^{-5}$, and the leakage exponent coefficient $\gamma =1.18$.

## 4. Discussion

- This paper puts forward a new optimization model for controlling leakage and energy consumption by adjusting pumps and PRVs. The PRVs (location and setting) and VSPs (speed) in a small virtual network was optimized with the aim to reduce leakage and energy consumption. Regardless of the PRV cost, with the number of PRVs growing, the marginal effect of increased PRVs for the PRV-only strategy decreases quickly after incipient rising, while the value of the joint scheduling strategy grows all the time. It was found that joint scheduling not only had clear advantages in pressure control, but also reduced energy consumption (Figure 5, Figure 6 and Figure 7). Under the optimal strategy of joint scheduling, about 1148 ${\mathrm{m}}^{3}$ water (7% of the original consumption) and 722 $\mathrm{kWh}$ electric energy (25.4% of the original consumption) are saved per day. After considering the PRV cost, the actual cost savings began to decline after the number of PRVs reached a certain value.
- Regarding the stability and convergence of optimization results, even though there are some fluctuations in objective function values and changes in PRV locations, it is not clear (See Table 1 and Figure 3). Considering that the location selection is consistent to some degree (e.g., the same combination of 8, 9, 11, and 20 was chosen for the five trials when scheduling four PRVs, and 11, 20, and 30 were always chosen when scheduling four PRVs and VSPs), further refinement of the PRV setting and VSP speed can be redone after the position is selected. When comparing the PRV-only results with Nicolini et al. [11], the consistency of PRV locations for the multiple trials in this paper is better than theirs (identical PRV location combinations appear more times) when the PRV number is less than five (because the maximal PRV number in Nicolini et al. [11] is five).
- The PRVs and VSPs are optimized together under multiple working conditions, and the position and setting of PRVs are decision variables at the same time. This approach is more reasonable than methods that optimize the setting after fixing the positions first [12,13,18]. It can also meet the need to install new PRVs. Meanwhile, the encoding mode can be modified according to the number of working conditions, fixed speed pumps, and VSPs. Considering that fitting too many working conditions may lead to increased calculation difficulty, the maximum, minimum, and intermediate working conditions can be used to simulate when there are too many conditions, and others could be optimized after PRV locations are determined.
- The use of floating-point number coding is mainly based on the fact that it can greatly shorten the number of bits and can perform fast precision conversion and range adjustment, according to the requirements. However, as the number of PRVs, pumps, and working conditions increase, the code length of an individual increases, which results in a dramatic expansion of solution space. This problem makes the optimization in large networks very inefficient and hard [25]. To overcome it, strategies such as improving computer performance, pre-selection of potential PRV locations, and choosing an appropriate range of pump speed should be relied on. In the case network, the computing time varies from about 20 minutes to 360 minutes when the number of PRVs is one to seven, which seems to be accepted when compared with some time-consuming method [17] (due to the difference in model complexity, software, etc., the comparison may not have much meaning).
- The failure of direct combining optimal solutions of the PRV-only strategy and of the VSP-only strategy directly indicates that the joint effects of PRVs and VSPs must be considered. The joint scheduling strategy, which takes into account the interaction among all network components, has its advantages and necessity. Constraints on the water supply volume of each WTP are taken into account in this paper, such that the supplied volume of each WTP can maintain stability before and after optimization. Moreover, the proportion of costs contributed by leakage and by energy consumption may not be reasonable for real water distribution systems. In this case, within the network modified from previous researchers [11,21,22,23], the cost of energy consumption is too low when compared to the cost contributed by leakage, which seems to be unreal. However, it will not influence the application of this proposed method in other water networks.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Encoding mode and interpretation process. ${N}_{v}$ is the number of PRVs. ${N}_{p}$ is the number of VSPs. The pipe ID is where PRV installs. ${v}_{ij,k}$ is the diameter multiplier for condition $k$ (from 1 to $n$). ${\mathsf{\alpha}}_{{N}_{p},k}$ is the speed of the pump ${N}_{p}$ for condition $k$.

**Figure 3.**The value of fsum without and with considering PRV cost: (

**a**) PRV-only scheduling without considering PRV cost; (

**b**) PRV-only scheduling considering PRV cost (100 RMB/day for each); (

**c**) Joint scheduling regardless of PRV cost; (

**d**) Joint scheduling considering PRV cost.

**Figure 4.**The PRV setting and VSP speeds when using four PRVs: (

**a**) PRV setting (diameter multipliers) for PRV-only scheduling, (

**b**) VSP speeds for VS-only scheduling, and (

**c**) settings of joint scheduling.

**Figure 5.**Pressure distribution before (

**a**) and after (

**b**) PRV, (

**c**) VSP or (

**d**) PRV and VSR joint scheduling. The number 1, 2, and 3 correspond to the low demand period condition, the intermediate demand period condition, and the peak demand period condition, respectively.

**Figure 6.**The distribution of nodal pressure before and after scheduling. The left is the cumulative probability distribution curve of the pressure. The right is the pie chart, (

**a**) original, (

**b**) PRV-only scheduling with four PRVs, (

**c**) VSP-only scheduling, and (

**d**) joint scheduling with four PRVs.

**Figure 7.**Average/maximum/minimum pressure and hourly cost under different optimization strategies: (

**a**) Original; (

**b**) PRV-only scheduling with four PRVs; (

**c**) VSP-only scheduling; (

**d**) joint scheduling with four PRVs.

**Figure 8.**The speed and outlet pressure of pump 39 under different optimization strategies: (

**a**) speed; (

**b**) outlet pressure.

Number of PRVs | Pipe-ID (PRV-Only Scheduling) | fsum1 (RMB/Day) | fsum2 (RMB/Day) | fsum (RMB/Day) | Pipe-ID (Joint Scheduling) | fsum1 (RMB/Day) | fsum2 (RMB/Day) | fsum (RMB/Day) |
---|---|---|---|---|---|---|---|---|

0 PRV | - | 0 | 0 | 0 | - | 0 | 0 | 0 |

Trial 1 | ||||||||

1 PRV | 11 | 1848 | 42 | 1890 | 11 | 3061 | 389 | 3450 |

2 PRV | 8, 11 | 2255 | 60 | 2315 | 9, 11 | 3178 | 361 | 3539 |

3 PRV | 8, 9, 11 | 2600 | 68 | 2668 | 10, 11, 20 | 3248 | 415 | 3663 |

4 PRV | 8, 9, 11, 20 | 2753 | 68 | 2821 | 1, 11, 20, 31 | 3375 | 407 | 3782 |

5 PRV | 8, 9, 11, 20, 31 | 2776 | 67 | 2843 | 3, 11, 20, 23, 29 | 3325 | 426 | 3751 |

6 PRV | 8, 9, 11, 20, 23, 27 | 2773 | 74 | 2847 | 9, 11, 20, 22, 29, 31 | 3440 | 398 | 3838 |

7 PRV | 5, 8, 9, 10, 11, 21, 29 | 2779 | 75 | 2854 | 1, 8, 9, 11, 20, 26, 31 | 3471 | 395 | 3866 |

Trial 2 | ||||||||

1 PRV | 11 | 1848 | 42 | 1890 | 11 | 3056 | 409 | 3465 |

2 PRV | 5, 11 | 2346 | 76 | 2422 | 11, 13 | 3160 | 392 | 3552 |

3 PRV | 8, 9, 11 | 2600 | 68 | 2668 | 11, 13, 20 | 3263 | 395 | 3658 |

4 PRV | 8, 9, 11, 20 | 2742 | 72 | 2814 | 1, 11, 20, 31 | 3445 | 432 | 3877 |

5 PRV | 8, 9, 11, 20, 25 | 2631 | 66 | 2697 | 9, 11, 14, 20, 29 | 3427 | 408 | 3835 |

6 PRV | 1, 8, 9, 11, 20, 25 | 2797 | 59 | 2856 | 9, 10, 11, 13, 20, 29 | 3489 | 422 | 3911 |

7 PRV | 6, 8, 9, 11, 20, 24, 36 | 2825 | 59 | 2884 | 9, 10, 11, 20, 24, 29, 31 | 3479 | 441 | 3920 |

Trial 3 | ||||||||

1 PRV | 11 | 1848 | 42 | 1890 | 11 | 3056 | 409 | 3465 |

2 PRV | 8, 11 | 2255 | 58 | 2313 | 11, 13 | 3214 | 418 | 3632 |

3 PRV | 1, 5, 11 | 2564 | 73 | 2637 | 11, 20, 25 | 3263 | 430 | 3693 |

4 PRV | 8, 9, 11, 20 | 2755 | 74 | 2829 | 9, 11, 20, 31 | 3372 | 394 | 3766 |

5 PRV | 1, 6, 8, 11, 20 | 2766 | 78 | 2844 | 9, 10, 11, 20, 31 | 3354 | 417 | 3771 |

6 PRV | 5, 9, 11, 13, 19, 25 | 2709 | 67 | 2776 | 9, 11, 13, 20, 23, 25 | 3378 | 411 | 3789 |

7 PRV | 8, 9, 10, 11, 20, 23, 34 | 2771 | 67 | 2838 | 9, 10, 11, 20, 23, 26, 31 | 3393 | 426 | 3819 |

Trial 4 | ||||||||

1 PRV | 11 | 1848 | 42 | 1890 | 11 | 3111 | 411 | 3522 |

2 PRV | 5, 11 | 2346 | 55 | 2401 | 11, 20 | 3131 | 406 | 3537 |

3 PRV | 8, 9, 11 | 2522 | 71 | 2593 | 3, 11, 20 | 3245 | 433 | 3678 |

4 PRV | 8, 9, 11, 20 | 2755 | 74 | 2829 | 11, 20, 22, 31 | 3261 | 420 | 3681 |

5 PRV | 8, 9, 11, 20, 24 | 2779 | 72 | 2851 | 8, 11, 20, 26, 31 | 3320 | 429 | 3749 |

6 PRV | 8, 9, 11, 20, 23, 26 | 2750 | 67 | 2817 | 9, 11, 20, 22, 29, 31 | 3419 | 433 | 3852 |

7 PRV | 2, 8, 9, 10, 11, 20, 27 | 2799 | 64 | 2863 | 1, 8, 9, 11, 20, 23, 29 | 3580 | 445 | 4025 |

Trial 5 | ||||||||

1 PRV | 11 | 1848 | 42 | 1890 | 11 | 3077 | 389 | 3466 |

2 PRV | 5, 11 | 2348 | 71 | 2419 | 11, 13 | 3157 | 394 | 3551 |

3 PRV | 8, 9, 11 | 2600 | 73 | 2673 | 10, 11, 20 | 3310 | 432 | 3742 |

4 PRV | 8, 9, 11, 20 | 2760 | 69 | 2829 | 11, 20, 25, 31 | 3287 | 431 | 3718 |

5 PRV | 8, 9, 11, 20, 22 | 2766 | 67 | 2833 | 4, 9, 11, 20, 29 | 3401 | 402 | 3803 |

6 PRV | 5, 8, 9, 11, 20, 22 | 2680 | 63 | 2743 | 9, 11, 12, 13, 19, 20 | 3393 | 414 | 3807 |

7 PRV | 1, 5, 8, 9, 11, 12, 24 | 2776 | 73 | 2849 | 1, 10, 11, 14, 20, 29, 31 | 3489 | 450 | 3939 |

fsum1 (RMB/Day) | fsum2 (RMB/Day) | fsum (RMB/Day) | |
---|---|---|---|

Trial 1 | 2553 | 467 | 3020 |

Trial 2 | 2543 | 469 | 3012 |

Trial 3 | 2545 | 467 | 3012 |

Trial 4 | 2551 | 465 | 3016 |

Trial 5 | 2553 | 465 | 3018 |

Scheduling | Total Consumption of Water (m^{3}/Day) | Leakage (m^{3}/Day) | Water Savings (m^{3}/Day) | Water Savings/Original Total Consumption (%) | Cost Contributed by Leakage (RMB/Day) | Reduction (%) | Total Consumption of Electric Energy (kWh/Day) | Electric Energy Savings (kWh/Day) | Cost Contributed by Energy Consumption (RMB/Day) | Reduction (%) | Total Cost (RMB/Day)(without PRVs Cost) | Reduction (%) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Original | 16,297 | 3337 | - | - | 10,010 | - | 2837 | - | 1702 | - | 11,712 | - |

PRV-Only | 15,377 | 2417 | 920 | 5.6 | 7250 | 27.6 | 2722 | 115 | 1633 | 4.1 | 8883 | 24.2 |

VSP-Only | 15,446 | 2486 | 851 | 5.2 | 7457 | 25.5 | 2059 | 778 | 1235 | 27.4 | 8692 | 25.8 |

Joint | 15,149 | 2189 | 1148 | 7.0 | 6566 | 34.4 | 2115 | 722 | 1269 | 25.4 | 7835 | 33.1 |

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## Share and Cite

**MDPI and ACS Style**

Shao, Y.; Yu, Y.; Yu, T.; Chu, S.; Liu, X.
Leakage Control and Energy Consumption Optimization in the Water Distribution Network Based on Joint Scheduling of Pumps and Valves. *Energies* **2019**, *12*, 2969.
https://doi.org/10.3390/en12152969

**AMA Style**

Shao Y, Yu Y, Yu T, Chu S, Liu X.
Leakage Control and Energy Consumption Optimization in the Water Distribution Network Based on Joint Scheduling of Pumps and Valves. *Energies*. 2019; 12(15):2969.
https://doi.org/10.3390/en12152969

**Chicago/Turabian Style**

Shao, Yu, Yanxi Yu, Tingchao Yu, Shipeng Chu, and Xiaowei Liu.
2019. "Leakage Control and Energy Consumption Optimization in the Water Distribution Network Based on Joint Scheduling of Pumps and Valves" *Energies* 12, no. 15: 2969.
https://doi.org/10.3390/en12152969