# Research on Sustainable Scheduling of Cascade Reservoirs Based on Improved Crow Search Algorithm

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Multi-Objective Sustainable Scheduling Model of Cascade Reservoirs

#### 2.1. Problem Description

^{3}. Lake Oroville reservoir (hereafter referred to as reservoir A) is the upstream reservoir of the cascade reservoir system; Thermolito Forebay reservoir (hereafter referred to as reservoir B) is a downstream reservoir; Thermolito Afterbay reservoir (hereafter referred to as reservoir C) is located at the end of reservoir B, and the length of the canal connecting reservoir B and reservoir C is only 2.8 km. Reservoir A, reservoir B and reservoir C jointly form the Oroville–Thermalito Complex.

#### 2.2. Model Establishment

## 3. PSO-CSA Algorithm Design

#### 3.1. Introduction to the Algorithm

- Multiple crow individuals form crow populations and live as populations.
- Each individual crow can remember where he or she hides their food.
- The crows in the crow population will find and steal each other’s food by tracking.
- When an individual crow being followed discovers that it is being followed, it takes steps to confuse the other party.
- Specifically, the specific process of the CSA algorithm is as follows:

#### 3.2. Improvement Strategy

#### 3.2.1. The Crow Flies at Variable Speed

#### 3.2.2. Raven Spiral Update Location Strategy

#### 3.2.3. Crow Memory Contraction Update Strategy

#### 3.3. Improve the Specific Solution Steps of the Algorithm

## 4. Results and Discussion

#### 4.1. Overview of Engineering Background

^{3}.

#### 4.2. Display of Simulation Results

#### 4.3. Discussion

^{3}, the demand of water supply area I is 49,724 $\times {10}^{4}$ m

^{3}, the average water supply guarantee rate is 95.33%, and the water deficit within one year is 2321.9 $\times {10}^{4}$ m

^{3}; reservoir B to water supply area II is 26,759.05 $\times {10}^{4}$ m

^{3}, the demand of water supply area I is 27,832 $\times {10}^{4}$ m

^{3}, the average water supply guarantee rate is 96.15%, and the water deficit within one year is 1072.95 $\times {10}^{4}$ m

^{3}. In addition, it is worth pointing out that the annual power generation of the PSO-CSA algorithm designed in this paper is 7.84–12.17% higher than that of the other three algorithms.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Topological structure diagram of cascade reservoir system in the Oroville–Thermalito Complex.

Symbol | Illustrate |
---|---|

$\Delta t$ | The duration of the unit period |

$T$ | scheduling period set, $T=\left\{1,2,\cdot \cdot \cdot ,t\right\}$ |

$N$ | Reservoir set, $N=\left\{1,2,\cdot \cdot \cdot ,n\right\}$ |

${L}_{n,t,U}$ | The upstream water level $n$ of the reservoir during the time period $t$ |

${L}_{n,t,D}$ | The downstream level $n$ of the reservoir during the time period $t$ |

${L}_{n,t,L}$ | Loss of head $n$ of reservoirs within time $t$ |

${Q}_{t,n,G}$ | The flow rate of generators in reservoir $n$ within the time period $t$ |

${Q}_{t,n,G\mathrm{max}}$ | The maximum flow rate of generators in reservoir $n$ in the time period $t$ |

${Q}_{t,P}^{n}$ | The discharge rate of the reservoir $n$ |

${Q}_{t,I}^{n}$ | The flow rate of the reservoir $n$ |

${\overline{Q}}_{n-1,n}$ | The three-year average runoff within the $n-1\to n$ river section |

${Q}_{t,R,I}^{n-1,n}$ | The water demand of industry within the time period $t$ of the $n-1\to n$ river section |

${Q}_{t,R,A}^{n-1,n}$ | The water demand of agriculture within the time period $t$ of the $n-1\to n$ river section |

${Q}_{t,R,D}^{n-1,n}$ | The water demand of residents during the time period $t$ of the $n-1\to n$ river section |

${Q}_{t,R,E}^{n-1,n}$ | The water demand of the ecological environment during the time period $t$ of the $n-1\to n$ river section |

${Q}_{t,P,I}^{n-1,n}$ | The amount of water provided by industry in the river section during the time period $t$ of the $n-1\to n$ river section |

${Q}_{t,P,A}^{n-1,n}$ | The amount of water provided by agriculture in the river section during the time period $t$ of the $n-1\to n$ river section |

${Q}_{t,P,D}^{n-1,n}$ | The amount of water provided by residents in the river section during the time period $t$ of the $n-1\to n$ river section |

${Q}_{t,P,E}^{n-1,n}$ | The amount of water provided of the ecological environment during the time period $t$ in the $n-1\to n$ river section |

${\xi}_{t,L}^{n-1,n}$ | Water scarcity indicator function during the time period $t$ within a river section $n-1\to n$ |

${\xi}_{t,L,E}^{n-1,n}$ | Water shortage index function of ecological environment during the time period $t$ in the river section $n-1\to n$ |

${\delta}_{n}$ | The output coefficient of the generator of the reservoir $n$ |

Month | 1 | 2 | 3 | 4 | 5 | 6 | Total | |

Reservoir | Purpose | $\mathbf{Water}\mathbf{Demand}(\mathbf{Unit}:\times {\mathbf{10}}^{\mathbf{4}}{\mathbf{m}}^{\mathbf{3}})$ | ||||||

A | Industry | 1000 | 1012 | 1224 | 1200 | 1200 | 1500 | |

Agriculture | 509 | 500 | 2400 | 3972 | 4675 | 5825 | ||

Domestic | 85 | 90 | 90 | 100 | 115 | 115 | ||

Ecology | 75 | 88 | 90 | 115 | 167 | 199 | ||

B | Industry | 750 | 780 | 795 | 795 | 740 | 800 | |

Agriculture | 240 | 250 | 709 | 771 | 895 | 950 | ||

Domestic | 600 | 590 | 580 | 625 | 707 | 794 | ||

Ecology | 133 | 135 | 139 | 180 | 188 | 195 | ||

Month | 7 | 8 | 9 | 10 | 11 | 12 | ||

Reservoir | Purpose | Water Demand (Unit:$\mathbf{\times}{\mathbf{10}}^{\mathbf{4}}$m^{3}) | ||||||

A | Industry | 1400 | 1500 | 1224 | 1200 | 1000 | 1344 | 14,804 |

Agriculture | 5990 | 4614 | 1200 | 1150 | 504 | 300 | 31,641 | |

Domestic | 120 | 140 | 140 | 122 | 95 | 95 | 1307 | |

Ecology | 204 | 207 | 200 | 188 | 152 | 107 | 1792 | |

B | Industry | 820 | 950 | 890 | 825 | 825 | 755 | 9725 |

Agriculture | 945 | 900 | 864 | 752 | 400 | 295 | 7972 | |

Domestic | 790 | 787 | 746 | 690 | 605 | 610 | 8124 | |

Ecology | 199 | 195 | 180 | 172 | 150 | 145 | 2011 |

**Table 3.**The optimal value, the worst value and the average value of the fitness function of the four algorithms in the process of running 30 times.

Algorithm | CSA | PSO | GA | PSO-CSA |
---|---|---|---|---|

Index | Numerical Value | |||

The optimal value | 16.06 | 16.29 | 16.72 | 15.00 |

The worst value | 16.55 | 16.50 | 17.04 | 15.74 |

The average value | 16.36 | 16.45 | 16.81 | 15.48 |

Month | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

Reservoir | $\mathbf{Water}\mathbf{Demand}(\mathbf{Unit}:\times {10}^{4}{\mathbf{m}}^{3})$ | |||||

A | 1522.7 | 1613.4 | 3559.4 | 5216.5 | 6014.5 | 7534.7 |

B | 1591.5 | 1733.2 | 2172.0 | 2208.1 | 2420.8 | 2505.0 |

Month | 7 | 8 | 9 | 10 | 11 | 12 |

Reservoir | Water Demand (Unit:$\times {\mathbf{10}}^{\mathbf{4}}$m^{3}) | |||||

A | 7063.2 | 6110.1 | 2643.3 | 2621.8 | 1732.8 | 1769.7 |

B | 2640.0 | 2791.45 | 2614.7 | 2386.9 | 1972.3 | 1723.1 |

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## Share and Cite

**MDPI and ACS Style**

Liu, X.; Lu, J.; Zou, C.; Deng, B.; Liu, L.; Yan, S.
Research on Sustainable Scheduling of Cascade Reservoirs Based on Improved Crow Search Algorithm. *Water* **2023**, *15*, 578.
https://doi.org/10.3390/w15030578

**AMA Style**

Liu X, Lu J, Zou C, Deng B, Liu L, Yan S.
Research on Sustainable Scheduling of Cascade Reservoirs Based on Improved Crow Search Algorithm. *Water*. 2023; 15(3):578.
https://doi.org/10.3390/w15030578

**Chicago/Turabian Style**

Liu, Xiaoshan, Jinyou Lu, Chaowang Zou, Bo Deng, Lina Liu, and Shaofeng Yan.
2023. "Research on Sustainable Scheduling of Cascade Reservoirs Based on Improved Crow Search Algorithm" *Water* 15, no. 3: 578.
https://doi.org/10.3390/w15030578