# Operation Rule Derivation of Hydropower Reservoirs by Support Vector Machine Based on Grey Relational Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Deterministic Reservoir Operation Optimization Model

#### 2.1. Objective Function

_{i,t}

_{,}, K

_{i,t}, H

_{i,t}and Q

_{i,t}are the power output, power production coefficient, water head, and turbine discharge of reservoir i at period t, respectively; ΔT

_{t}is the length of period t.

#### 2.2. Operation Constraints

_{min}(n,t) represents the minimum water level; Z

_{max}(n,t) denotes the maximum water level at time t of reservoir n.

^{out}(n,t) is the outflow at time t of reservoir n; ${Q}_{\mathrm{min}}^{\mathrm{out}}\left(n,t\right)$ is the minimum outflow; ${Q}_{\mathrm{max}}^{\mathrm{out}}\left(n,t\right)$ represents the maximum outflow at time t of reservoir n.

_{min}(n,t) is the minimum output at time t of reservoir n; N

_{max}(n,t) is the maximum output at time t of reservoir n;

#### 2.3. Optimization Method

## 3. Derivation Rule Method

#### Support Vector Machine (SVM)

_{i}, y

_{i})}, i = 1, 2, …, n. It is assumed that all the training data are fitted by linear functions without error under the ε:

_{i}, x

_{j}) is defined as a kernel function which represents a linear dot product of the nonlinear projection. SVM can efficiently solve complex problems with an appropriate kernel function. The common kernel functions are linear, sigmoid, radial basis function, and so on. We train SVM with each kernel function and choose the best one.

## 4. Using Grey Relational Analysis to Quantify the Influence of Input Vectors and Evolution Index

#### 4.1. Grey Relational Analysis

_{i}= (x

_{i}(1), x

_{i}(2),…, x

_{i}(n)), which can also be expressed as X

_{i}

^{0}= (x

_{i}

^{0}(1), x

_{i}

^{0}(2),…, x

_{i}

^{0}(n)), where x

_{i}

^{0}(k) = x

_{i}(k) − x

_{i}(1), k = 1, 2, …, n. The absolute grey relational grade and relative grey relational grade are expressed by Equations (14) and (15):

#### 4.2. Evolution Index

_{i}and $\overline{X}$ are the release in period i and average release during operational period obtained from the DP-POA model, respectively; ${X}_{i}^{\prime}$ and $\overline{{X}_{i}^{\prime}}$ are the release in period t and average release during operational period obtained from the SVM model, respectively; n is the number of data values.

#### 4.3. Rescaled Adjusted Partial Sums

_{k}is the mean value of the measured parameter in the year k; P

_{avg}is the average mean value in the period of observation; S

_{d}is the standard deviation of P

_{avg}; m = 1, …, n and n is the number of calculated years.

## 5. Case Study

#### 5.1. Correlation Analysis in the GRA

_{b}, expected output N

_{e}, initial water level last time Z

_{t−}

_{1}, initial water level this time Z

_{t}, inflow last time Q

_{in,t−}

_{1}, inflow this time Q

_{in,t}, water discharge last time Q

_{out,t−}

_{1}, the number of months T

_{m}, the period number of months T

_{x}(three periods in a month), water discharge last time in adjacent station ${Q}_{out,t-1}^{\prime}$, inflow last time in adjacent station ${Q}_{in,t-1}^{\prime}$, initial water level last time in adjacent station ${Z}_{t-1}^{\prime}$, and initial water level this time in adjacent station ${Z}_{t}^{\prime}$. The specific calculation results of grey relational grade are given in the Table 2. Figure 3 shows the comparison of different factors.

#### 5.2. Operating Rules Derivation and Results Discussion

## 6. Conclusions

- (1)
- The simulation results indicate that the significant correlation factor and potential correlation factors can improve the fitting accuracy. For correlation factor, the larger the grey relational grade is, the better the fitting accuracy will be.
- (2)
- Among the three GRA-SVM schemes, GRA-SVM-2 has the best fitting accuracy and the absolute error of hydropower generation are 2.57 and 0.42, respectively. Therefore, in practical application, as many related factors (comprehensive grey relational grade is more than 0.5) should be selected as possible.
- (3)
- The GRA quantifies the importance of each correlation factor, which may as the input of the SVM model. Among the water conservancy between Xi Luo-du and Xiang Jia-ba considering the backwater effect, the relevant data of adjacent hydropower station plays an important role to improve accuracy.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Location of the cascade reservoirs including the Xi Luo-du Reservoir and Xiang Jia-ba reservoir in the upper Yangtze River.

**Figure 2.**Comparation of output process between conventional operating and deterministic optimization results.

Reservoirs | Xi Luo-Du | Xiang Jia-Ba |
---|---|---|

Flood control water level (m) | 560 | 370 |

Normal pool level (m) | 600 | 380 |

Dead water level (m) | 540 | 370 |

Flood storage (billion m^{3}) | 46.50 | 9.03 |

Regulating storage (billion m^{3}) | 64.60 | 9.03 |

Regulation ability | Seasonal | Seasonal |

Power Capacity (MW) | 12,600 | 6400 |

Firm output (MW) | 3850 | 2009 |

Reservoirs | Xi Luo-Du | Xiang Jia-Ba | ||||
---|---|---|---|---|---|---|

Grey Relational Grade | Absolute | Relative | Comprehensive | Absolute | Relative | Comprehensive |

Z_{b} | 0.61 | 0.50 | 0.56 | 0.83 | 0.50 | 0.67 |

N_{e} | 0.57 | 0.50 | 0.53 | 0.50 | 0.50 | 0.50 |

Z_{t−}_{1} | 0.96 | 0.51 | 0.73 | 0.50 | 0.50 | 0.50 |

Z_{t} | 0.96 | 0.51 | 0.74 | 0.50 | 0.50 | 0.50 |

Q_{in,t−}_{1} | 0.73 | 0.52 | 0.63 | 0.72 | 0.56 | 0.64 |

Q_{in,t} | 0.73 | 0.52 | 0.63 | 0.77 | 0.66 | 0.72 |

Q_{out,t−}_{1} | 0.79 | 0.51 | 0.65 | 0.78 | 0.51 | 0.64 |

T_{m} | 0.94 | 0.51 | 0.72 | 0.96 | 0.50 | 0.73 |

T_{x} | 0.52 | 0.50 | 0.51 | 0.96 | 0.50 | 0.73 |

${{Q}^{\prime}}_{out,t-1}$ | 0.78 | 0.51 | 0.65 | 0.77 | 0.51 | 0.64 |

${{Q}^{\prime}}_{in,t-1}$ | 0.78 | 0.89 | 0.84 | 0.77 | 0.66 | 0.72 |

${Z}_{t-1}{}^{\prime}$ | 0.50 | 0.50 | 0.50 | 0.94 | 0.52 | 0.73 |

${Z}_{t}{}^{\prime}$ | 0.50 | 0.50 | 0.50 | 0.82 | 0.52 | 0.67 |

Reservoirs | Xi Luo-Du | Xiang Jia-Ba | ||||
---|---|---|---|---|---|---|

Methods | GRA-SVM-1 | GRA-SVM-2 | GRA-SVM-3 | GRA-SVM-1 | GRA-SVM-2 | GRA-SVM-3 |

R | 0.918 | 0.961 | 0.953 | 0.990 | 0.994 | 0.992 |

RMSE | 134.63 | 94.33 | 103.73 | 26.51 | 21.32 | 23.42 |

MAE | 87.87 | 57.54 | 60.94 | 20.18 | 15.73 | 18.75 |

Methods | Conventional Operation | DP-POA | GRA-SVM-1 | GRA-SVM-2 | GRA-SVM-3 | |||
---|---|---|---|---|---|---|---|---|

Assessment Index | Generation (10 ^{8} kWh) | Generation (10 ^{8} kWh) | Generation (10 ^{8} kWh) | Gap (%) | Generation (10 ^{8} kWh) | Gap (%) | Generation (10 ^{8} kWh) | Gap (%) |

Xi Luo-du | 1115.46 | 1230.66 | 1183.43 | 3.84 | 1199.10 | 2.57 | 1192.19 | 3.13 |

Xiang Jia-ba | 578.36 | 638.37 | 626.87 | 1.80 | 635.68 | 0.42 | 626.93 | 1.79 |

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**MDPI and ACS Style**

Zhu, Y.; Zhou, J.; Qiu, H.; Li, J.; Zhang, Q. Operation Rule Derivation of Hydropower Reservoirs by Support Vector Machine Based on Grey Relational Analysis. *Water* **2021**, *13*, 2518.
https://doi.org/10.3390/w13182518

**AMA Style**

Zhu Y, Zhou J, Qiu H, Li J, Zhang Q. Operation Rule Derivation of Hydropower Reservoirs by Support Vector Machine Based on Grey Relational Analysis. *Water*. 2021; 13(18):2518.
https://doi.org/10.3390/w13182518

**Chicago/Turabian Style**

Zhu, Yuxin, Jianzhong Zhou, Hongya Qiu, Juncong Li, and Qianyi Zhang. 2021. "Operation Rule Derivation of Hydropower Reservoirs by Support Vector Machine Based on Grey Relational Analysis" *Water* 13, no. 18: 2518.
https://doi.org/10.3390/w13182518