# Deriving Operating Rules of Hydropower Reservoirs Using Multi-Strategy Ensemble Henry Gas Solubility Optimization-Driven Support Vector Machine

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Optimal Operation Model of Hydropower Reservoirs

#### 2.1.1. Objectives

_{i,t}, k

_{i,t}, H

_{i,t}, and Q

_{i,t}are power output, efficiency coefficient, water head, and turbine discharge of reservoir i at period t, respectively; $\mathsf{\Delta}t$ is the operation interval.

#### 2.1.2. Constraints

_{i,t}, I

_{i,t}, ${Q}_{i,t}^{total}$, q

_{i,t}, and ${Q}_{i,t}^{s}$ are the storage volume, total inflow, total outflow, local inflow, and water spillage of reservoir i at period t, respectively; N

_{i}is the number of upstream reservoirs directly connected to the reservoir i; O

_{j,t}is the outflow from upstream reservoir j directly connected to the reservoir i at period t.

_{i,t}and ${Z}_{i,t}^{d}$ are the forebay water level and downstream water level of reservoir i at period t, respectively.

^{s}is the total required storage capacity of multiple reservoir system.

#### 2.2. Summary of Henry Gas Solubility Optimization (HGSO)

#### 2.3. The Proposed MVQIHGSO

#### 2.3.1. Multi-verse Optimizer (MVO)

#### 2.3.2. Quadratic Interpolation Strategy (QI)

#### 2.3.3. The Proposed Enhanced HGSO Algorithm

Algorithm 1. Detailed information of MVQIHGSO algorithm. |

Pseudocode of the MVQIHGSO 01: Inputs: the population size N; the maximum number of iteration MaxIt; the constant values including c _{1}, c_{2}, c_{3} in Equations (2) and (3), c_{4} and c_{5} in Equation (6), α, β, and ε in Equatios (4) and (5); the number of groups l02: Initialize the gas population within the lower and upper boundary 03: Divide the gas population into specific groups with the same Henry’s constant value 04: Evaluate the gas of each group in the population 05: Obtain the best gas of the whole population and the l best gases corresponding to l groups 06: for it from 1 to MaxIt do 07: Generate random number r within [0, 1] 08: if r smaller than 0.5 do 09: Update the gas position by basic HGSO 10: else do 11: Update the gas position by MVO 12: end if 13: Sort the updated gas population by HGSO and MVO 14: Updating the gas population from 1 to N-2 by QI strategy based on greedy law 15: Evaluate the fitness of the final updated gas population 16: Update the best gas position obtained so far and the l best gases corresponding to l groups 17: end for 18: Outputs: the best global gas position and the corresponding optimal fitness value |

#### 2.4. Support Vector Machine

_{i}is the input vector of sample point i; y

_{i}is the observation value corresponding to sample point i; $F({x}_{i},\widehat{w})$ denotes the predicted value of sample point i; $\epsilon $ is defined as a cube so that the loss is zero in the case of $F({x}_{i},\widehat{w})$ is within the tube, while the loss is the magnitude of the difference between $F({x}_{i},\widehat{w})$ and the radius $\epsilon $ of the tube [44]. In order to solve the optimization problem in term of the above primal loss function, the slack variables ${\xi}^{\ast}$ and $\xi $ are introduced to minimize:

**x**can be expressed as follows:

^{2}), root mean square error (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE). Different indices can validate the performances of the model in different dimensions. Specifically, R

^{2}is used to characterize the linear correlation between the observed data and its predicted data; RMSE is adopted to indicate the average degree of dispersion between the observed data and its predicted data; MAE and MAPE are applied to quantify the deviation of the predicted data from the observed data. The formulation of validation indices are as follows:

## 3. Experimental Evaluation and Discussion

#### 3.1. Statistical Results and Analysis

#### 3.2. Non-Parameter Test Results and Analysis

## 4. Case Study

#### 4.1. Study Region

#### 4.2. Data Description

#### 4.3. Results and Discussion

^{2}, RMSE, MAE, and MAPE values in the prediction of end water level during a period. The best model according to the highest values of R

^{2}, and the lowest values of RMSE, MAE and MAPE are the proposed MVQIHGSO-SVM as the best optimum model by a value of R

^{2}= 0.998, RMSE = 0.340, MAE = 0.126, and MAPE = 0.021% for XLD and R

^{2}= 0.998, RMSE = 0.164, MAE = 0.075, and MAPE = 0.019% for XJB. This means that the proposed MVQIHGSO method can find the best hyperparameter combination to improve the ability of SVM with respect to operation rule derivation. The remaining models, from best to worst according to RMSE value, are Grid-SVM, SCA-SVM, PSO-SVM, and HGSO-SVM for XLD and Grid-SVM, SCA-SVM, PSO-SVM, and HGSO-SVM for XJB. The worst model for XLD and XJB of all hybrid models are HGSO-SVM. The performance of the proposed model is improved by 17.27% and 14.58% compared with HGSO in term of RMSE value for XLD and XJB, respectively. The good performance of the proposed model in predicting data can be attributed to the following two reasons: (1) SVM shows unique advantages in solving small sample and non-linear problems by using kernel functions; (2) the multiple strategies including MVO algorithm and QI strategy improve the optimization ability of the basic HGSO.

## 5. Conclusions

- (1)
- Multiple strategies are equipped into HGSO to improve its performance in exploration and exploitation. The multi-verse optimizer (MVO) is used to enhance the exploration capability of basic HGSO and help the inferior agent to escape from local optimal. Quadratic interpolation (QI) is used to improve the exploitation ability of HGSO. Finally, the exploration and exploitation are balanced by integrating the multiple strategies.
- (2)
- MVQIHGSO with multiple strategies is benchmarked by 23 classical benchmark functions. The results demonstrates that MVQIHGSO outperforms most of the well-known metaheuristic algorithms and has a superior efficacy compared to the competitors based on the convergence accuracy and speed.
- (3)
- MVQIHGSO-SVM model is used to derive operating rules of hydropower reservoirs. The XLD and XJB in the upper Yangtze River are selected as a case study. The results indicate that the proposed MVQIHGSO-SVM model can accurately obtain the joint operation rules of hydropower reservoirs. The total hydropower generation calculated by the proposed hybrid model is closer to the optimal operation result, and the hydropower generation increased the most compared to conventional scheduling, reaching 22.25 × 10
^{8}kWh, increasing by 1.15%.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Function | Dim | Range | f_{min} |
---|---|---|---|

${f}_{1}(x)={\displaystyle {\sum}_{i=1}^{n}{x}_{i}^{2}}$ | 30 | [−100, 100] | 0 |

${f}_{2}(x)={\displaystyle {\sum}_{i=1}^{n}\left|{x}_{i}^{}\right|}+{\prod}_{i=1}^{n}\left|{x}_{i}^{}\right|$ | 30 | [−10, 10] | 0 |

${f}_{3}(x)={{\displaystyle {\sum}_{i=1}^{n}\left({\displaystyle {\sum}_{j-1}^{i}{x}_{j}}\right)}}^{2}$ | 30 | [−100, 100] | 0 |

${f}_{4}(x)={\mathrm{max}}_{i}\left\{\left|{x}_{i}\right|,1\le i\le n\right\}$ | 30 | [−100, 100] | 0 |

${f}_{5}(x)={\displaystyle {\sum}_{i=1}^{n-1}\left[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right]}$ | 30 | [−30, 30] | 0 |

${f}_{6}(x)={\displaystyle {\sum}_{i=1}^{n}{\left(\left[{x}_{i}+0.5\right]\right)}^{2}}$ | 30 | [−100, 100] | 0 |

${f}_{7}(x)={\displaystyle {\sum}_{i=1}^{n}i{x}_{i}^{4}}+random\left[0,1\right)$ | 30 | [−1.28, 1.28] | 0 |

Function | Dim | Range | f_{min} |
---|---|---|---|

${f}_{8}(x)={\displaystyle {\sum}_{i=1}^{n}-{x}_{i}^{}}\mathrm{sin}(\sqrt{\left|{x}_{i}\right|})$ | 30 | [−500, 500] | −418.9829 × 5 |

${f}_{9}(x)={\displaystyle {\sum}_{i=1}^{n}\left[{x}_{i}^{2}-10\mathrm{cos}(2\pi {x}_{i})+10\right]}$ | 30 | [−5.12, 5.12] | 0 |

${f}_{10}(x)=-20\mathrm{exp}\u27ee-0.2\sqrt{\frac{1}{n}{\displaystyle {\sum}_{i=1}^{n}{x}_{i}^{2}}}\u27ef-\mathrm{exp}\u27ee\frac{1}{n}{\displaystyle {\sum}_{i=1}^{n}\mathrm{cos}(2\pi {x}_{i})}\u27ef+20+e$ | 30 | [−32, 32] | 0 |

${f}_{11}(x)=\frac{1}{4000}{\displaystyle {\sum}_{i=1}^{n}{x}_{i}^{2}}-{\prod}_{i=1}^{n}\mathrm{cos}\u27ee\frac{{x}_{i}}{\sqrt{i}}\u27ef+1$ | 30 | [−600, 600] | 0 |

$\begin{array}{l}{f}_{12}(x)=\frac{\pi}{n}\left\{10\mathrm{sin}(\pi {y}_{1})+{\displaystyle {\sum}_{i=1}^{n-1}{({y}_{i}-1)}^{2}\left[1+10{\mathrm{sin}}^{2}(\pi {y}_{i+1})\right]+{({y}_{n}-1)}^{2}}\right\}+{\displaystyle {\sum}_{i=1}^{n}u({x}_{i},10,100,4)}\\ {y}_{i}=1+\frac{{x}_{i}+1}{4}\\ u({x}_{i},a,k,m)=\left(\right)open="\{">\begin{array}{l}k{\left({x}_{i}-a\right)}^{m}{x}_{i}a\\ 0-a{x}_{i}a\\ k{\left(-{x}_{i}-a\right)}^{m}{x}_{i}-a\end{array}\end{array}$ | 30 | [−50, 50] | 0 |

${f}_{13}(x)=0.1\left\{{\mathrm{sin}}^{2}(3\pi x)+{\displaystyle {\sum}_{i=1}^{n}{({x}_{i}-1)}^{2}\left[1+{\mathrm{sin}}^{2}(3\pi {x}_{i}+1)\right]+{({x}_{n}-1)}^{2}\left[1+{\mathrm{sin}}^{2}(2\pi {x}_{n})\right]}\right\}+{\displaystyle {\sum}_{i=1}^{n}u({x}_{i},5,100,4)}$ | 30 | [−50, 50] | 0 |

Function | Dim | Range | f_{min} |
---|---|---|---|

${f}_{14}(x)={\u27ee\frac{1}{500}+{\displaystyle {\sum}_{j=1}^{25}\frac{1}{j+{\displaystyle {\sum}_{i=1}^{2}{({x}_{i}-{a}_{ij})}^{6}}}}\u27ef}^{-1}$ | 2 | [−65, 65] | 1 |

${f}_{15}(x)={{\displaystyle {\sum}_{i=1}^{11}\left[{a}_{i}-\frac{{x}_{1}({b}_{i}^{2}+{b}_{i}{x}_{2})}{{b}_{i}^{2}+{b}_{i}{x}_{3}+{x}_{4}}\right]}}^{2}$ | 4 | [−5, 5] | 0.00030 |

${f}_{16}(x)=4{x}_{1}^{2}-2.1{x}_{1}^{4}+\frac{1}{3}{x}_{1}^{6}+{x}_{1}{x}_{2}-4{x}_{2}^{2}+4{x}_{2}^{4}$ | 2 | [−5, 5] | −1.0316 |

${f}_{17}(x)=\u27ee{x}_{2}-\frac{5.1}{4{\pi}^{2}}{x}_{1}^{2}+\frac{5}{\pi}{x}_{1}-6\u27ef+10\u27ee1-\frac{1}{8\pi}\u27ef\mathrm{cos}{x}_{1}+10$ | 2 | [−5, 5] | 0.398 |

${f}_{18}(x)=\left[1+{({x}_{1}+{x}_{2}+1)}^{2}(19-14{x}_{1}+3{x}_{1}^{2}-14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}^{2})\right]\times \left[30+{(2{x}_{1}-3{x}_{2})}^{2}\times (18-32{x}_{1}+12{x}_{1}^{2}+48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}^{2})\right]$ | 2 | [−2, 2] | 3 |

${f}_{19}(x)=-{\displaystyle {\sum}_{i=1}^{4}{c}_{i}}\mathrm{exp}\left(-{\displaystyle {\sum}_{j=1}^{3}{a}_{ij}{({x}_{j}-{p}_{ij})}^{2}}\right)$ | 3 | [1, 3] | −3.86 |

${f}_{20}(x)=-{\displaystyle {\sum}_{i=1}^{4}{c}_{i}}\mathrm{exp}\left(-{\displaystyle {\sum}_{j=1}^{6}{a}_{ij}{({x}_{j}-{p}_{ij})}^{2}}\right)$ | 6 | [0, 1] | −3.32 |

${f}_{21}(x)=-{{\displaystyle {\sum}_{i=1}^{5}\left[(X-{a}_{i}){(X-{a}_{i})}^{T}+{c}_{i}\right]}}^{-1}$ | 4 | [0, 10] | −10.1532 |

${f}_{22}(x)=-{{\displaystyle {\sum}_{i=1}^{7}\left[(X-{a}_{i}){(X-{a}_{i})}^{T}+{c}_{i}\right]}}^{-1}$ | 4 | [0, 10] | −10.4028 |

${f}_{23}(x)=-{{\displaystyle {\sum}_{i=1}^{10}\left[(X-{a}_{i}){(X-{a}_{i})}^{T}+{c}_{i}\right]}}^{-1}$ | 4 | [0, 10] | −10.5363 |

Functions | Indicator | MVQIHGSO | HGSO | PSO | DE | MVO | SCA | OBSCA | GWO | IGWO | |
---|---|---|---|---|---|---|---|---|---|---|---|

Unimodal | F1 | Mean | 0.00 | 1.71 × 10^{−71} | 2.33 × 10^{−40} | 5.29 × 10^{−12} | 1.79 × 10^{−1} | 2.74 × 10^{−3} | 1.11 × 10^{−27} | 2.48 × 10^{−70} | 2.57 × 10^{−71} |

Std | 9.34 × 10^{−71} | 0.00 | 1.04 × 10^{−39} | 2.22 × 10^{−12} | 5.80 × 10^{−2} | 7.88 × 10^{−3} | 4.22 × 10^{−27} | 7.10 × 10^{−70} | 6.07 × 10^{−71} | ||

F2 | Mean | 4.10 × 10^{−185} | 1.25 × 10^{−43} | 8.79 × 10^{−3} | 4.65 × 10^{−8} | 3.55 | 3.17 × 10^{−6} | 4.21 × 10^{−25} | 6.67 × 10^{−41} | 1.95 × 10^{−42} | |

Std | 6.83 × 10^{−43} | 0.00 | 3.15 × 10^{−2} | 1.23 × 10^{−8} | 1.81 × 10^{1} | 4.67 × 10^{−6} | 2.18 × 10^{−24} | 7.20 × 10^{−41} | 3.62 × 10^{−42} | ||

F3 | Mean | 0.00 | 1.04 × 10^{−69} | 1.09 × 10^{−1} | 2.41 × 10^{4} | 1.83 × 10^{1} | 2.10 × 10^{3} | 1.32 × 10^{−3} | 7.80 × 10^{−19} | 8.61 × 10^{−13} | |

Std | 5.66 × 10^{−69} | 0.00 | 1.28 × 10^{−1} | 2.77 × 10^{3} | 6.90 | 1.73 × 10^{3} | 5.17 × 10^{−3} | 3.92 × 10^{−18} | 4.38 × 10^{−12} | ||

F4 | Mean | 2.06 × 10^{−72} | 1.51 × 10^{−183} | 1.09 × 10^{−1} | 2.00 | 6.33 × 10^{−1} | 1.36 × 10^{1} | 1.55 × 10^{−5} | 1.15 × 10^{−17} | 7.93 × 10^{−15} | |

Std | 1.13 × 10^{−71} | 0.00 | 7.10 × 10^{−2} | 2.44 × 10^{−1} | 2.77 × 10^{−1} | 9.40 | 2.88 × 10^{−5} | 1.69 × 10^{−17} | 6.16 × 10^{−15} | ||

F5 | Mean | 2.85 × 10^{1} | 2.80 × 10^{1} | 4.06 × 10^{1} | 4.65 × 10^{1} | 3.08 × 10^{2} | 5.54 × 10^{2} | 2.79 × 10^{1} | 2.65 × 10^{1} | 2.25 × 10^{1} | |

Std | 2.70 × 10^{−1} | 6.13 × 10^{−1} | 2.77 × 10^{1} | 2.31 × 10^{1} | 5.98 × 10^{2} | 2.18 × 10^{3} | 3.06 × 10^{−1} | 7.80 × 10^{−1} | 2.79 × 10^{−1} | ||

F6 | Mean | 1.74 × 10^{−1} | 3.44 | 3.63 × 10^{−23} | 4.92 × 10^{−12} | 1.66 × 10^{−1} | 4.29 | 4.10 | 4.13 × 10^{−1} | 1.01 × 10^{−5} | |

Std | 6.39 × 10^{−2} | 5.26 × 10^{−1} | 1.84 × 10^{−22} | 1.84 × 10^{−12} | 4.89 × 10^{−2} | 3.92 × 10^{−1} | 2.73 × 10^{−1} | 2.57 × 10^{−1} | 2.39 × 10^{−6} | ||

F7 | Mean | 6.76 × 10^{−5} | 7.92 × 10^{−4} | 9.32 × 10^{−3} | 2.59 × 10^{−2} | 1.22 × 10^{−2} | 2.35 × 10^{−2} | 1.47 × 10^{−3} | 4.72 × 10^{−4} | 8.64 × 10^{−4} | |

Std | 4.35 × 10^{−4} | 4.84 × 10^{−5} | 4.00 × 10^{−3} | 4.70 × 10^{−3} | 5.04 × 10^{−3} | 2.54 × 10^{−2} | 1.03 × 10^{−3} | 3.15 × 10^{−4} | 3.80 × 10^{−4} | ||

Multimodal | F8 | Mean | −1.02 × 10^{4} | −2.64 × 10^{5} | −6.68 × 10^{3} | −1.25 × 10^{4} | −8.18 × 10^{3} | −3.97 × 10^{3} | −4.08 × 10^{3} | −6.09 × 10^{3} | −9.62 × 10^{3} |

Std | 1.08 × 10^{3} | 6.38 × 10^{5} | 5.86 × 10^{2} | 8.39 × 10^{1} | 7.31 × 10^{2} | 2.69 × 10^{2} | 2.26 × 10^{2} | 7.63 × 10^{2} | 1.29 × 10^{3} | ||

F9 | Mean | 0.00 | 0.00 | 4.67 × 10^{1} | 6.20 × 10^{1} | 1.05 × 10^{2} | 1.29 × 10^{1} | 0.00 | 1.78 × 10^{−1} | 1.39 × 10^{1} | |

Std | 0.00 | 0.00 | 1.45 × 10^{1} | 5.96 | 3.31 × 10^{1} | 1.98 × 10^{1} | 0.00 | 8.08 × 10^{−1} | 6.96 | ||

F10 | Mean | 1.01 × 10^{−15} | 1.72 × 10^{−15} | 6.36 × 10^{−1} | 6.01 × 10^{−7} | 7.92 × 10^{−1} | 1.11 × 10^{1} | 1.09 × 10^{−1} | 1.26 × 10^{−14} | 9.06 × 10^{−15} | |

Std | 1.53 × 10^{−15} | 6.49 × 10^{−16} | 7.67 × 10^{−1} | 1.10 × 10^{−7} | 7.47 × 10^{−1} | 9.72 | 5.12 × 10^{−1} | 2.97 × 10^{−15} | 2.31 × 10^{−15} | ||

F11 | Mean | 0.00 | 0.00 | 1.66 × 10^{−2} | 7.82 × 10^{−11} | 4.49 × 10^{−1} | 1.72 × 10^{−1} | 4.67 × 10^{−11} | 4.53 × 10^{−4} | 1.89 × 10^{−3} | |

Std | 0.00 | 0.00 | 2.13 × 10^{−2} | 1.51 × 10^{−10} | 8.69 × 10^{−2} | 2.35 × 10^{−1} | 2.56 × 10^{−10} | 2.48 × 10^{−3} | 4.56 × 10^{−3} | ||

F12 | Mean | 7.73 × 10^{−4} | 3.43 × 10^{−1} | 4.49 × 10^{−2} | 6.66 × 10^{−13} | 8.71 × 10^{−1} | 1.17 | 4.48 × 10^{−1} | 3.01 × 10^{−2} | 7.46 × 10^{−7} | |

Std | 3.65 × 10^{−4} | 1.18 × 10^{−1} | 7.04 × 10^{−2} | 3.91 × 10^{−13} | 8.25 × 10^{−1} | 1.89 | 9.21 × 10^{−2} | 2.30 × 10^{−2} | 2.44 × 10^{−7} | ||

F13 | Mean | 2.00 × 10^{−2} | 2.49 | 2.07 × 10^{−2} | 3.01 × 10^{−12} | 3.78 × 10^{−2} | 3.20 | 2.28 | 3.05 × 10^{−1} | 1.63 × 10^{−2} | |

Std | 8.67 × 10^{−3} | 3.23 × 10^{−1} | 3.65 × 10^{−2} | 1.72 × 10^{−12} | 1.87 × 10^{−2} | 1.52 | 1.47 × 10^{−1} | 2.03 × 10^{−1} | 3.70 × 10^{−2} | ||

Fixed-dimension multimodal | F14 | Mean | 9.98 × 10^{−1} | 1.14 | 2.58 | 9.98 × 10^{−1} | 9.98 × 10^{−1} | 1.33 | 1.20 | 3.22 | 9.98 × 10^{−1} |

Std | 1.61 × 10^{−12} | 2.87 × 10^{−1} | 2.01 | 0.00 | 6.12 × 10^{−12} | 7.52 × 10^{−1} | 6.05 × 10^{−1} | 3.54 | 4.12 × 10^{−17} | ||

F15 | Mean | 3.08 × 10^{−4} | 3.53 × 10^{−4} | 3.84 × 10^{−4} | 6.49 × 10^{−4} | 3.33 × 10^{−3} | 9.72 × 10^{−4} | 7.45 × 10^{−4} | 4.35 × 10^{−3} | 3.34 × 10^{−4} | |

Std | 4.16 × 10^{−8} | 4.13 × 10^{−5} | 2.95 × 10^{−4} | 9.37 × 10^{−5} | 6.80 × 10^{−3} | 4.47 × 10^{−4} | 1.14 × 10^{−4} | 8.14 × 10^{−3} | 1.43 × 10^{−4} | ||

F16 | Mean | −1.03 | −1.03 | −1.03 | −1.03 | −1.03 | −1.03 | -1.03 | −1.03 | −1.03 | |

Std | 5.34 × 10^{−12} | 2.10 × 10^{−5} | 6.78 × 10^{−16} | 6.78 × 10^{−16} | 4.39 × 10^{−8} | 1.50 × 10^{−5} | 7.18 × 10^{−7} | 2.78 × 10^{−9} | 6.78 × 10^{−16} | ||

F17 | Mean | 0.398 | 0.399 | 0.398 | 0.398 | 0.398 | 0.399 | 0.398 | 0.398 | 0.398 | |

Std | 1.49 × 10^{−10} | 9.22 × 10^{−4} | 0.00 | 0.00 | 7.35 × 10^{−8} | 6.37 × 10^{−4} | 2.89 × 10^{−4} | 3.25 × 10^{−7} | 0.00 | ||

F18 | Mean | 3.00 | 3.00 | 3.00 | 3.00 | 3.00 | 3.00 | 3.00 | 3.00 | 3.00 | |

Std | 2.86 × 10^{−10} | 1.13 × 10^{−5} | 1.50 × 10^{−15} | 1.90 × 10^{−15} | 6.23 × 10^{−7} | 1.51 × 10^{−5} | 4.30 × 10^{−6} | 3.90 × 10^{−6} | 7.14 × 10^{−16} | ||

F19 | Mean | −3.86 | −3.86 | −3.86 | −3.86 | −3.86 | −3.86 | −3.86 | −3.86 | −3.86 | |

Std | 1.53 × 10^{−9} | 2.17 × 10^{−3} | 2.71 × 10^{−15} | 2.71 × 10^{−15} | 2.67 × 10^{−7} | 2.89 × 10^{−3} | 1.76 × 10^{−3} | 1.30 × 10^{−3} | 2.71 × 10^{−15} | ||

F20 | Mean | −3.32 | −3.12 | −3.29 | −3.31 | −3.29 | −2.96 | −3.16 | −3.29 | −3.31 | |

Std | 3.02 × 10^{−2} | 7.03 × 10^{−2} | 5.54 × 10^{−2} | 1.38 × 10^{−15} | 5.56 × 10^{−2} | 3.10 × 10^{−1} | 4.12 × 10^{−2} | 5.20 × 10^{−2} | 3.02 × 10^{−2} | ||

F21 | Mean | −10.2 | −4.89 | −6.15 | −9.96 | −7.28 | −4.28 | −9.30 | −9.31 | −9.82 | |

Std | 1.93 | 8.91 × 10^{−2} | 3.44 | 1.68 × 10^{−3} | 2.79 | 2.16 | 1.07 × 10^{−1} | 1.92 | 1.28 | ||

F22 | Mean | −10.4 | −4.91 | −9.06 | −10.0 | −8.58 | −4.05 | −10.2 | −10.2 | −10.4 | |

Std | 1.35 | 1.15 × 10^{−1} | 2.77 | 5.07 × 10^{−8} | 2.90 | 2.26 | 1.44 × 10^{−1} | 9.63 × 10^{−1} | 7.32 × 10^{−9} | ||

F23 | Mean | −10.5 | −4.96 | −8.11 | −10.5 | −9.56 | −4.97 | −10.4 | −10.4 | −10.5 | |

Std | 2.59 × 10^{−5} | 1.02 × 10^{−1} | 3.55 | 1.49 × 10^{−13} | 2.25 | 1.70 | 1.04 × 10^{−1} | 9.79 × 10^{−1} | 1.09 × 10^{−14} |

**Table 5.**Statistical results of the proposed MVQIHGSO and other state-of-art algorithms from Friedman test.

Algorithms | Friedman Ranks | Final Ranks |
---|---|---|

MVQIHGSO | 2.543 | 1 |

HGSO | 4.608 | 3 |

PSO | 5.652 | 7 |

DE | 4.608 | 4 |

MVO | 6.456 | 8 |

SCA | 7.804 | 9 |

OBSCA | 5.260 | 6 |

GWO | 4.695 | 5 |

IGWO | 3.369 | 2 |

**Table 6.**Statistical results of the proposed MVQIHGSO and other state-of-art algorithms from Wilcoxon test ($p\ge 0.05$).

Compared Algorithms | Unimodal Functions | Multimodal Functions | Fixed-Dimension Functions |
---|---|---|---|

MVQIHGSO vs. HGSO | 2.5940 × 10^{−8} | 0.1012 | 7.4567 × 10^{−4} |

MVQIHGSO vs. PSO | 1.4838 × 10^{−12} | 4.1333 × 10^{−10} | 0.8573 |

MVQIHGSO vs. DE | 6.6342 × 10^{−19} | 2.5731 × 10^{−4} | 0.0916 |

MVQIHGSO vs. MVO | 1.7618 × 10^{−32} | 5.2700 × 10^{−31} | 0.0273 |

MVQIHGSO vs. SCA | 9.6394 × 10^{−27} | 6.4376 × 10^{−25} | 3.2725 × 10^{−5} |

MVQIHGSO vs. OBSCA | 1.0240 × 10^{−9} | 4.1657 × 10^{−4} | 0.0227 |

MVQIHGSO vs. GWO | 5.5859 × 10^{−7} | 0.0072 | 0.0654 |

MVQIHGSO vs. IGWO | 7.5243 × 10^{−5} | 0.0011 | 0.0402 |

Characteristics | Hydropower Stations | Units | |
---|---|---|---|

Xiluodu | Xiangjiaba | ||

Completion date | 2013 | 2012 | - |

Watershed area | 0.45 | 0.45 | million km^{2} |

Dead water level | 540 | 370 | m |

Flood control limited water level | 560 | 370 | m |

Normal water level | 600 | 380 | m |

Regulated storage | 6.46 | 0.903 | billion m^{3} |

The minimum release | 1200 | 1200 | m^{3}/s |

The minimum output | 1000 | 1000 | MW |

Installed capacity | 12,600 | 6000 | MW |

Efficiency coefficient | 8.8 | 8.8 | - |

Reservoirs | Models | R^{2} | RMSE | MAE | MAPE |
---|---|---|---|---|---|

XLD | MVQIHGSO-SVM | 0.998 | 0.340 | 0.126 | 0.021% |

HGSO-SVM | 0.997 | 0.411 | 0.151 | 0.025% | |

SCA-SVM | 0.997 | 0.405 | 0.149 | 0.025% | |

PSO-SVM | 0.998 | 0.405 | 0.150 | 0.025% | |

Grid-SVM | 0.998 | 0.357 | 0.151 | 0.025% | |

XJB | MVQIHGSO-SVM | 0.998 | 0.164 | 0.075 | 0.019% |

HGSO-SVM | 0.997 | 0.192 | 0.098 | 0.026% | |

SCA-SVM | 0.997 | 0.189 | 0.092 | 0.025% | |

PSO-SVM | 0.996 | 0.192 | 0.096 | 0.025% | |

Grid-SVM | 0.997 | 0.187 | 0.086 | 0.023% |

**Table 9.**Total hydropower generation based on observed and predicted data as well as optimal operation.

Reservoirs | Hydropower Generation (TWh) | ||||||
---|---|---|---|---|---|---|---|

Observed | MVQIHGSO-SVM | HGSO-SVM | SCA-SVM | PSO-SVM | Grid-SVM | Optimization | |

XLD | 129.82 | 131.30 | 131.03 | 131.03 | 131.03 | 131.12 | 131.41 |

XJB | 64.03 | 64.78 | 64.56 | 64.56 | 64.56 | 64.61 | 65.00 |

Total | 193.85 | 196.08 | 195.59 | 195.58 | 195.58 | 195.73 | 196.41 |

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## Share and Cite

**MDPI and ACS Style**

Qiu, H.; Hu, T.; Zhang, S.; Xiao, Y.
Deriving Operating Rules of Hydropower Reservoirs Using Multi-Strategy Ensemble Henry Gas Solubility Optimization-Driven Support Vector Machine. *Water* **2023**, *15*, 437.
https://doi.org/10.3390/w15030437

**AMA Style**

Qiu H, Hu T, Zhang S, Xiao Y.
Deriving Operating Rules of Hydropower Reservoirs Using Multi-Strategy Ensemble Henry Gas Solubility Optimization-Driven Support Vector Machine. *Water*. 2023; 15(3):437.
https://doi.org/10.3390/w15030437

**Chicago/Turabian Style**

Qiu, Hongya, Ting Hu, Song Zhang, and Yangfan Xiao.
2023. "Deriving Operating Rules of Hydropower Reservoirs Using Multi-Strategy Ensemble Henry Gas Solubility Optimization-Driven Support Vector Machine" *Water* 15, no. 3: 437.
https://doi.org/10.3390/w15030437