# Optimal Regulation of the Cascade Gates Group Water Diversion Project in a Flow Adjustment Period

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. One Dimensional Hydrodynamic Model of the Cascade Gates Group

^{3}/s), x is the longitudinal distance of the canal along the mainstream direction (m), q is the side inflow (m

^{3}/s), α is the momentum correction coefficient, A is the discharge area (m

^{2}), g is the acceleration of gravity (m/s

^{2}), and S

_{f}is the friction ratio drop, which can be expressed by the following formula, n

_{c}represents Manning’s roughness coefficient of the water conveyance canal, and R is the hydraulic radius (m).

_{i}and Q

_{i+1}represent the flows (in m

^{3}/s) in front of and behind the gate, respectively, M is the comprehensive discharge coefficient, e is the gate opening, B

_{g}is the total overflow width of the gate, H is the weir head, h

_{s}is the water depth behind the gate, Z

_{i}is the water level of the control section in front of the gate, and Z

_{i}

_{+1}is the water level of the control section behind the gate.

#### 2.2. Multi-Objective Optimization Model

#### 2.3. Multi-Objective Optimal Regulation Model Coupled with the Hydrodynamic Process

#### 2.3.1. Objective Function

#### 2.3.2. Decision Variables and Constraints

^{th}control gate, m; and ${e}_{t,i}^{max}$ is the minimum allowable opening amplitude of a single adjustment at time t for the i

^{th}control gate, m.

## 3. Application

#### 3.1. Study Area

^{3}. The water is diverted from the head gate of the Taocha Channel in the Danjiangkou Reservoir to Tuancheng Lake in Beijing and the Waihuan River in Tianjin. Since the entire water supply line was opened in December 2014, the cumulative water delivery volume of the project has exceeded 25.5 billion m

^{3}. This greatly alleviates water shortage in the cities of the water-receiving area, and the water quality of the rivers and lakes along the line has been significantly improved.

#### 3.2. Model Validation

^{3}/s.

#### 3.3. Optimum Regulation of the Control Gates Group under Typical Operating Conditions

#### 3.3.1. Typical Working Condition Settings

^{3}/s. The discharge increased by 2 m

^{3}/s every 2 h during the flow adjustment period, with a total adjustment of 10 and 20 m

^{3}/s.

#### 3.3.2. Initial Condition

## 4. Results and Discussion

#### 4.1. Optimize the Control Results

^{3}/s could be adjusted, and the feedforward times were = 72, 48, 24, 12, 6, and 4 h, respectively. A group of optimal control schemes could be obtained by optimizing the control model, and the results are shown in Table 3. It can be observed that when the flow was 10 and 20 m

^{3}/s, the feedforward control times were more than 24 h (C1, C2, C7, C8), and the average water level deviation was less than 0.15 m. At this time, the larger the flow adjustment is, the more the gate becomes regulated, but the water level deviation was within the allowable range. When the current feedforward control time was less than 24 h (C3–C6, C9–C12), the average water level deviation was about 0.20 m, and it exceeded the allowable range at this time. To meet the demand of the downstream flow adjustment, more gates are needed to participate in the regulation. Thus, the number of gate regulations was greater than the longer feedforward control time.

#### 4.2. Response Process of the Water Level

**Figure 6.**Water level deviation when the flow adjustment was 10 m³/s and 20 m³/s. (

**a**–

**l**) shows the water level deviation results under working conditions C1–C12 respectively.

^{3}/s), the deviation is larger. The feedforward control time did not exceed 24 h. The shorter the feedforward control time, the greater the difference between the water levels before and after the flow adjustment. The downstream water level rises before the adjustment, and the downstream water level drops within a short time after the adjustment. The greater the flow adjustment, the greater the drop. After a period of time once the regulation is completed, the upstream water level drops slightly while the downstream water level rises.

#### 4.3. Response Process of the Volume Capacity of the Channel Pool

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 7.**Storage capacity when the flow adjustments are 10 m³/s and 20 m³/s. (

**a**–

**l**) shows the volume capacity variation results under working conditions C1–C12 respectively.

Control Gate No. | Stake No. | Gate Bottom Elevation (m) | Single Hole Width (m) | Number of Holes |
---|---|---|---|---|

G48 | 970 + 379 | 70.403 | 6.6 | 3 |

G49 | 980 + 116 | 67.787 | 6.0 | 3 |

G50 | 1002 + 169 | 66.721 | 6.0 | 3 |

G51 | 1017 + 385 | 65.344 | 6.0 | 3 |

G52 | 1036 + 963 | 64.554 | 5.5 | 3 |

G53 | 1046 + 196 | 63.785 | 5.5 | 3 |

G54 | 1071 + 847 | 65.151 | 7.0 | 3 |

G55 | 1085 + 024 | 61.643 | 6.0 | 3 |

G56 | 1112 + 074 | 60.588 | 7.8 | 2 |

G57 | 1121 + 840 | 60.783 | 5.0 | 3 |

G58 | 1136 + 825 | 57.492 | 5.0 | 2 |

G59 | 1157 + 652 | 55.764 | 5.5 | 2 |

G60 | 1172 + 353 | 55.596 | 5.4 | 2 |

G61 | 1197 + 669 | 55.974 | — | 2 |

Diversion Gate No. | Stake No. | Diversion Gate No. | Stake No. |
---|---|---|---|

F63 | 983 + 866 | F69 | 1079 + 569 |

F64 | 1007 + 496 | F70 | 1104 + 313 |

F65 | 1030 + 769 | F71 | 1117 + 631 |

F66 | 1036 + 023 | F72 | 1156 + 414 |

F67 | 1061 + 371 | F73 | 1180 + 707 |

F68 | 1070 + 370 | F74 | 1195 + 724 |

F86 | 1121 + 720 |

Exit Gate No. | Stake No. | Exit Gate No. | Stake No. |
---|---|---|---|

T42 | 977 + 801 | T48 | 1096 + 976 |

T43 | 993 + 346 | T49 | 1110 + 179 |

T44 | 1015 + 126 | T50 | 1135 + 088 |

T45 | 1044 + 822 | T51 | 1157 + 002 |

T46 | 1077 + 350 | T52 | 1184 + 713 |

T47 | 1084 + 675 | T53 | 1197 + 636 |

Channel Pool No. | Entrance | Outlet | Entrance Bottom Elevation (m) | Outlet Bottom Elevation (m) | Bottom Width (m) | Slope Coefficient | Design Roughness |
---|---|---|---|---|---|---|---|

C1 | G48 | G49 | 70.403 | 69.821 | 10.0–22.2 | 2.5–3.0 | 0.014 |

C2 | G49 | G50 | 69.987 | 69.129 | 18.0–23.5 | 2.0–3.0 | 0.014 |

C3 | G50 | G51 | 68.879 | 68.360 | 18.0–21.5 | 2.5–3.0 | 0.014 |

C4 | G51 | G52 | 67.574 | 66.513 | 18.0–22.0 | 2.5–3.0 | 0.014 |

C5 | G52 | G53 | 66.284 | 65.998 | 15.0–16.5 | 3.0 | 0.014 |

C6 | G53 | G54 | 65.985 | 65.151 | 18.7–23.0 | 2.0–2.5 | 0.014 |

C7 | G54 | G55 | 65.151 | 64.264 | 18.5–23.0 | 2.0–2.5 | 0.014 |

C8 | G55 | G56 | 64.140 | 61.525 | 15.0–23.0 | 1.0–2.5 | 0.014 |

C9 | G56 | G57 | 61.143 | 60.783 | 18.5–22.5 | 0.75–2.5 | 0.014 |

C10 | G57 | G58 | 60.679 | 59.877 | 14.5–20.0 | 1.0–2.5 | 0.014 |

C11 | G58 | G59 | 59.767 | 58.649 | 7.5–12.5 | 0.8–2.5 | 0.014 |

C12 | G59 | G60 | 58.542 | 57.788 | 7.5–12.1 | 0.75–2.5 | 0.014 |

C13 | G60 | G61 | 57.696 | 55.974 | 7.5–13.0 | 0.75–2.5 | 0.014 |

Target | Gate Regulation Times | Mean Water Level Deviation (m) | ||
---|---|---|---|---|

Contrast | Actual Regulation | Optimal Regulation | Actual Regulation | Optimal Regulation |

G48 | 2 | 3 | 0.043 | 0.042 |

G49 | 4 | 7 | 0.027 | 0.059 |

G50 | 4 | 4 | 0.051 | 0.035 |

G51 | 5 | 4 | 0.082 | 0.060 |

G52 | 6 | 7 | 0.113 | 0.019 |

G53 | 6 | 5 | 0.077 | 0.019 |

G54 | 5 | 4 | 0.087 | 0.038 |

G55 | 6 | 4 | 0.093 | 0.042 |

G56 | 6 | 7 | 0.053 | 0.086 |

G57 | 4 | 3 | 0.059 | 0.085 |

G58 | 9 | 2 | 0.025 | 0.095 |

G59 | 10 | 4 | 0.023 | 0.098 |

G60 | 10 | 5 | 0.018 | 0.099 |

Total/Average | 77 | 59 | 0.058 | 0.060 |

Feedforward Time (h) | Working Condition | Flow (m³/s) | Upstream Boundary (Water Level) (m) | Downstream Boundary (Initial Discharge) (m³/s) |
---|---|---|---|---|

72 | Condition 1 (C1) | 10 | 76.49 | 23.26 |

Condition 7 (C7) | 20 | |||

48 | Condition 2 (C2) | 10 | ||

Condition 8 (C8) | 20 | |||

24 | Condition 3 (C3) | 10 | ||

Condition 9 (C9) | 20 | |||

12 | Condition 4 (C4) | 10 | ||

Condition 10 (C10) | 20 | |||

6 | Condition 5 (C5) | 10 | ||

Condition 11 (C11) | 20 | |||

4 | Condition 6 (C6) | 10 | ||

Condition 12 (C12) | 20 |

Working Condition | Average Water Level Deviation (m) | Gate Regulation Times |
---|---|---|

C1 | 0.125 | 67 |

C2 | 0.101 | 71 |

C3 | 0.144 | 73 |

C4 | 0.176 | 66 |

C5 | 0.160 | 72 |

C6 | 0.183 | 73 |

C7 | 0.104 | 83 |

C8 | 0.085 | 86 |

C9 | 0.183 | 92 |

C10 | 0.202 | 86 |

C11 | 0.206 | 82 |

C12 | 0.204 | 90 |

Working Condition | Total Volume Difference of the System (×10^{4} m³) | Working Condition | Total Volume Difference of the System (×10^{4} m³) |
---|---|---|---|

C1 | 173.5460 | C7 | 220.9510 |

C2 | 95.7520 | C8 | 183.1010 |

C3 | 115.0820 | C9 | 50.3860 |

C4 | 33.4700 | C10 | 30.3700 |

C5 | 61.2320 | C11 | 24.8800 |

C6 | 49.0320 | C12 | 29.4670 |

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**MDPI and ACS Style**

Zhu, J.; Zhang, Z.; Lei, X.; Yue, X.; Xiang, X.; Wang, H.; Ye, M. Optimal Regulation of the Cascade Gates Group Water Diversion Project in a Flow Adjustment Period. *Water* **2021**, *13*, 2825.
https://doi.org/10.3390/w13202825

**AMA Style**

Zhu J, Zhang Z, Lei X, Yue X, Xiang X, Wang H, Ye M. Optimal Regulation of the Cascade Gates Group Water Diversion Project in a Flow Adjustment Period. *Water*. 2021; 13(20):2825.
https://doi.org/10.3390/w13202825

**Chicago/Turabian Style**

Zhu, Jie, Zhao Zhang, Xiaohui Lei, Xia Yue, Xiaohua Xiang, Hao Wang, and Mao Ye. 2021. "Optimal Regulation of the Cascade Gates Group Water Diversion Project in a Flow Adjustment Period" *Water* 13, no. 20: 2825.
https://doi.org/10.3390/w13202825