# Integrated Software Development and Case Studies for Optimal Operation of Cascade Reservoir within the Environmental Flow Constraints

^{1}

^{2}

^{3}

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## Abstract

**:**

^{3}/s, the maximum and the minimum are calculated by T-O and T-M, respectively. The power generation of cascade reservoir calculated by IWO is less than IIWO. The conclusions that IIWO has better convergence than IWO in solving cascade reservoir model, and the water volume of environmental flow has no obvious influence on cascade reservoir operation are drawn.

## 1. Introduction

## 2. Cascade Reservoir Optimal Operation Model under the Constraint of Environmental Flow

_{c}is the power generated by the cascade reservoir hydroelectric plants; Num is the number of cascade reservoir in the system; T is the number of periods; k

_{i}is the efficiency coefficient of the i-th hydroelectric plant; Q

_{e,i,t}is the power generation flow of the i-th hydroelectric plant at period t; h

_{i,t}is the effective head of the i-th hydroelectric plant at period t; Δt is the number of hours at period t; ${Q}_{in,t}^{i}$ and ${Q}_{r,t}^{i}$ are, respectively, the inflow and water release of the i-th reservoir at period t; ${V}_{t+1}^{i}$ and ${V}_{t}^{i}$ are, respectively, the water storage of the i-th reservoir at period t+1 and period t; ${Q}_{r,i,t}^{min}$ and ${Q}_{r,i,t}^{max}$ are, respectively, the minimum and the maximum water release of the i-th reservoir at period t; Q

_{EF,i,t}is the environmental flow of the i-th reservoir at period t; ${Q}_{other}^{min}$ is the minimum flow to meet the other water demand; ${Z}_{i,t}^{min}$ and ${Z}_{i,t}^{max}$ are, respectively, the minimum and the maximum water levels of the i-th reservoir at period t; and ${N}_{i,t}^{min}$ and ${N}_{i,t}^{max}$ are, respectively, the minimum power output and the installed capacity of the i-th hydroelectric plant at period t (MW).

## 3. Methodology

#### 3.1. Improved Minimum Monthly Average Runoff Method (IMMR)

_{ij}is the monthly average runoff in month j of the i-th year.

#### 3.1.1. Divide into Different Years

_{i}is the average runoff in the i-th year; and Q

_{a}is the multi-year average runoff.

#### 3.1.2. Division of Different Periods in One Year

_{j}is the percentage from mean of month j, and Q

_{j}is the multi-year average runoff of month j.

#### 3.1.3. IMMR for Environmental Flow

_{b}(b = 1, 2, 3) represent flood season, flat period, and dry season, respectively. Min(Q

_{ab}) is the minimum monthly average runoff in a-th year of b. n

_{b}is the number of years of b.

#### 3.2. Improved Invasive Weed Optimization Algorithm (IIWO)

#### 3.2.1. Invasive Weed Optimization Algorithm (IWO)

_{max}and F

_{min}are, respectively, the maximum and the minimum fitness values of the population; Seed

_{max}and Seed

_{min}are the maximum and minimum numbers of seeds that can be reproduced by weed, respectively; X

_{i}is the position of the i-th weed; X

_{i,s}is the position of the s-th seed reproduced by the i-th weed; $N\left(0,{\sigma}_{iter}^{2}\right)$ is a normal distribution with a zero mean and a standard deviation of σ

_{iter}, which is the standard deviation at the present time step; w is a nonlinear modulation index (w = 3); σ

_{ini}and σ

_{fin}are previously defined initial and final standard deviations (σ

_{ini}> σ

_{fin}); and iter

_{max}is the maximum number of iterations.

#### 3.2.2. Improvement from IWO to IIWO

#### Improvement of Spatial Dispersal Formula

#### Selection of Spatial Dispersal Rules

#### 3.3. Development of Integrated Software

#### 3.3.1. IIWO Convergence Test Module

_{1}and x

_{2}) in this function have a range of values [−10, 10]. Sin() is sine function.

_{1}and x

_{2}) in this function also have a range of values [−10, 10]. Cos() is cosine function. i is an integer.

_{i}) have a range of values [−5.12, 5.12]. Here, $\pi $ = 3.1415926.

_{ini}= 20 (the initial weeds population), P

_{fin}= 90 (the maximum weeds population), Seed

_{max}= 5, Seed

_{min}= 2, σ

_{ini}= 2, σ

_{fin}= 0.001, iter

_{max}= 60, and n = 3 (in Equation (15)).

#### 3.3.2. Environmental Flow Calculation Module

#### 3.3.3. Cascade Reservoir Operation Module

_{ini}= 20, P

_{fin}= 90, Seed

_{max}= 5, Seed

_{min}= 2, σ

_{in}

_{i}= 2, σ

_{fin}= 0.001, iter

_{ma}

_{x}= 60, w = 3 (in Equation (14)), and n = 3 (in Equation (15)). The environmental flow calculated by each method mentioned above is used as the constraint of water release to optimize both single reservoir and cascade reservoir operation, respectively.

## 4. Case Studies

^{2}. We consider here two cascade reservoirs in the upper reaches of Wujiang River, i.e., Hongjiadu Reservoir and Dongfeng Reservoir.

## 5. Results and Discussion

#### 5.1. IIWO Convergence Test

#### 5.2. Environmental Flow Calculation

#### 5.3. Single Reservoir and Cascade Reservoir Optimal Operation

#### 5.3.1. Operation Results in Different Years

#### 5.3.2. Rolling Correction of Cascade Reservoir Operation Curves

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 4.**The sites of Hongjiadu and Dongfeng on Wujiang River and the location of Wujiang river basin in China.

**Figure 5.**Monthly runoff data of Wujiang River Basin and its monthly runoff forecast results of 2019: (

**a**) Inflow of Hongjiadu; (

**b**) interval inflow between Hongjiadu and Dongfeng; (

**c**) the initial forecast monthly runoff data of 2019 and its revised results.

**Figure 6.**Convergences of IIWO to Schaffer, Shubert, and Rastrigrin: (

**a**) Convergence of IIWO to Schaffer; (

**b**) convergence of IIWO to Shubert; (

**c**) convergence of IIWO to Rastrigrin.

**Figure 7.**Environmental flow of Hongjiadu and Dongfeng: (

**a**) Environmental flow of Hongjiadu; (

**b**) environmental flow of Dongfeng.

**Figure 8.**Impacts of different environmental flows on reservoir operation (using the annual environmental flow and annual output as statistical data): (

**a**) Impacts of different environmental flows on single-reservoir operation (Hongjiadu); (

**b**) impacts of different environmental flows on cascade reservoir operation (Hongjiadu and Dongfeng).

**Figure 9.**Rolling correction of Hongjiadu and Dongfeng reservoirs dispatch curves: (

**a**) Rolling correction of Hongjiadu operation curve; (

**b**) rolling correction of Dongfeng operation curve.

Percentage from Mean | High Flow Year (Flood Season) | Normal Flow Year (Flat Period) | Low Flow Year (Dry Season) |
---|---|---|---|

µ | µ > 20% | −20% < µ ≤ 20% | µ ≤ −20% |

µ_{j} | µ_{j} > 20% | −20% < µ_{j} ≤ 20% | µ_{j} ≤ −20% |

Name | Formula | Sketch |
---|---|---|

Schaffer | $f(x)=0.5+\frac{{\left(sin\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}\right)}^{2}-0.5}{{\left[1+0.001\left({x}_{1}^{2}+{x}_{2}^{2}\right)\right]}^{2}}$ | |

Shubert | $f(x)=\left\{{\displaystyle \sum}_{i=1}^{5}icos\left[\left(i+1\right){x}_{1}+i\right]\right\}\cdot \left\{{\displaystyle \sum}_{i=1}^{5}icos\left[\left(i+1\right){x}_{2}+i\right]\right\}$ | |

Rastrigrin | $f(x)={\displaystyle \sum}_{i=1}^{D}\left[{x}_{i}^{2}-10cos(2\pi {x}_{i})+10\right]$ |

Flow Condition | October-March | April-September |
---|---|---|

Outstanding | 40% average annual flow | 60% average annual flow |

Excellent | 30% average annual flow | 50% average annual flow |

Good | 20% average annual flow | 40% average annual flow |

Fair or degrading | 10% average annual flow | 30% average annual flow |

Poor or minimum | 10% average annual flow | 10% average annual flow |

Name | Hongjiadu | Dongfeng | Location Diagram of the Two Reservoirs |
---|---|---|---|

Normal water level (m) | 1140 | 970 | |

Flood control level (m) | 1138 | 970 | |

Dead water level (m) | 1076 | 936 | |

Guaranteed output (MW) | 159.1 | 100 | |

Installed capacity (MW) | 600 | 695 | |

Efficiency coefficient | 8.4 | 8.35 |

Reservoir | IMMR | MMR | Q90 | Q95 | T-O | T-E | T-G | T-F | T-M |
---|---|---|---|---|---|---|---|---|---|

Hongjiadu | 495 | 491 | 837 | 764 | 888 | 710 | 533 | 355 | 178 |

Dongfeng | 1376 | 1360 | 2031 | 1877 | 1988 | 1590 | 1193 | 795 | 398 |

Total | 1871 | 1851 | 2868 | 2641 | 2876 | 2300 | 1726 | 1150 | 576 |

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**MDPI and ACS Style**

Wu, C.; Fang, G.; Liao, T.; Huang, X.; Qu, B.
Integrated Software Development and Case Studies for Optimal Operation of Cascade Reservoir within the Environmental Flow Constraints. *Sustainability* **2020**, *12*, 4064.
https://doi.org/10.3390/su12104064

**AMA Style**

Wu C, Fang G, Liao T, Huang X, Qu B.
Integrated Software Development and Case Studies for Optimal Operation of Cascade Reservoir within the Environmental Flow Constraints. *Sustainability*. 2020; 12(10):4064.
https://doi.org/10.3390/su12104064

**Chicago/Turabian Style**

Wu, Chengjun, Guohua Fang, Tao Liao, Xianfeng Huang, and Bo Qu.
2020. "Integrated Software Development and Case Studies for Optimal Operation of Cascade Reservoir within the Environmental Flow Constraints" *Sustainability* 12, no. 10: 4064.
https://doi.org/10.3390/su12104064