# Risk Analysis for Short-Term Operation of the Power Generation in Cascade Reservoirs Considering Multivariate Reservoir Inflow Forecast Errors

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Multivariate Inflow Forecast Errors

_{1}, B

_{2}, …, B

_{n}) and k reservoirs. There are T forecast periods and M sets of forecast inflow.

^{1}, X

^{2}, …, X

^{J}), then the following forecast error matrix can be established under M sets of historical runoff data, such that:

_{n}under one set of runoff data is expressed as (Y

^{1}, Y

^{2}, …, Y

^{J}), and its error matrix can be expressed as ${\left[{y}_{n,i}^{j}\right]}_{M\times J}$ where n, $n\in (1,2,\cdots ,N)$ is the tributary from which runoff is derived.

_{1}are ${\left[{x}_{i}^{j}\right]}_{M\times J}$ and ${\left[{y}_{1,i}^{j}\right]}_{M\times J}$, respectively. The inflow forecast errors of the second reservoir are then a combination of the error matrices of A and B

_{1}, which have M × M combinations in total. Correspondingly, the forecast errors of reservoir k should be the combination of the error matrices of A, B

_{1}, B

_{2}, …, B

_{n}, with ${M}^{(n+1)}$ combinations in total.

#### 2.2. Stochastic Simulation of Multivariate Inflow Forecast Errors

^{j}, its probability density function can be obtained by using the GMM, such that:

_{n}is:

^{2}). The D

^{2}value is calculated as follows:

## 3. Risk Analysis for Short-Term Operation of the Power Generation in Cascade Reservoirs

#### 3.1. Short-Term Operation of the Power Generation Model of Cascade Reservoirs

#### 3.1.1. Maximum Power Generation Capacity Model of Cascade Reservoirs

_{t}is the total output of the system in the T period; Δt is the length of the period; K is the output coefficient; ${Q}_{i}^{t}$ is the generation reference flow of i power station in the t period; and ${H}_{i}^{t}$ is the generating head of i power station in the t period.

#### 3.1.2. Minimum Energy Consumption Model of Cascade Reservoirs

#### 3.1.3. Restrictions

#### 3.2. Risk Analysis for Short-Term Operation of the Power Generation in the Cascade Reservoirs

#### 3.3. Calculation Steps for Short-Term Operation of the Power Generation Risk Rate of Cascade Reservoirs

## 4. Case Study

#### 4.1. Analysis of Forecast Error Characters in Different Forecast Periods

#### 4.2. Analysis of the Joint Distribution Function

^{2}value of the t-Copula was 0.1658. The fitting result is good. Hence, the t-Copula function was selected to fit the forecast errors of the Jinxi Reservoir inflow and the Guandi Reservoir interval inflow in each forecast period.

#### 4.3. Risk Analysis for Short-Term Operation of the Power Generation

## 5. Discussion

- From the analysis of this paper, it can be found that the simulated result is close to the measured data, and the average value of simulated accuracy is 97.52%. The average value of simulated accuracy for reference [25] is 97.87%. Compared with reference [25], the simulated accuracy is similar. The GMM-Copula in this study also exhibited a satisfactory performance that was consistent with those reported by Ji et al. [24]. These indicate that the methodology proposed in this study can effectively describe the statistical characteristics of the inflow forecast error series and provides a reference value for short-term operation of the power generation in large cascade reservoirs.
- From the aspect of risk rate, the hydropower generation plan usually takes the forecast runoff as the input data directly, without considering the inflow forecast errors, which leads to the risk and failure of the generation plan. We also can see there is little difference in the risk rate value of insufficient output among the three representative days. The predetermined output we select is the target output obtained with the forecast runoff process as the input, while the calculated output is obtained with the simulated runoff process based on the forecast error as the input. Because the forecast error distribution function we consider is the same, the risk rate value is similar. The risk rate value obtained in this study can be a useful reference for the decision-making. Therefore, this paper not only considers the risk of hydropower generation caused by inflow forecast errors but also analyzes and discusses the seasonal change of risk of hydropower generation in different periods including the dry period and wet period in detail. Compared with reference [26] which also analyzes the risk of short-term operation of the power generation of reservoirs, the simulated forecast errors considering the correlation between each forecast period are closer to the actual process, and the risk rate value in our study can be more reasonable. When planning hydropower generation, it is helpful to reduce the risk rate of power generation by adding the prediction value and the simulation prediction errors.
- This approach provides some guidance for hydropower station operations. Since the characteristics of the inflow forecast error vary seasonally, the risk rate of hydropower generation operation also varies. Therefore, examination of the risk for hydropower generation operation of cascade reservoirs under inflow uncertainty for different runoff periods for a comprehensive analysis will be the focus in the next study.

## 6. Conclusions

- GMM-Copula model was more suitable to simulate the inflow errors in different forecast periods. By comparing the mean values, variance and variation coefficients of the simulated and the actual inflow forecast errors, the accuracy of the joint simulation was greater. Thus, the proposed approach provides a novel means of simulating inflow forecast errors with multivariate combinations.
- Through the analysis of power generation risk during the non-flood season, it was determined that the risk rates of wasted water (3.50%) and beyond-or-below-limit water levels (2.02%) were the highest on wet days. The risk rate of insufficient output was the highest on dry days, which offers new insights into the short-term operation of the power generation of the Jinguan hydropower stations.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Cascade reservoirs in a basin. In order to make the calculation easy to be understood, the interval branch is simplified as one.

**Figure 4.**Probability density function curves of inflow forecast errors; (

**a**) Jinxi Reservoir inflow; (

**b**) Guandi Reservoir interval inflow.

**Figure 5.**Probability density function curves of inflow forecast errors in different forecast periods; (

**a**) Jinxi Reservoir inflow; (

**b**) Guandi Reservoir interval inflow.

Hydropower Station | Installed Capacity MW | Firm Power MW | Designed Annual Energy Output ×10 ^{8} kW·h |
---|---|---|---|

Jinxi | 3600 | 1086 | 166.20 |

Jindong | 4800 | 1443 | 237.60 |

Guandi | 2400 | 709.80 | 110.16 |

**Table 2.**Fitted Gaussian mixture model (GMM) parameters of Jinxi Reservoir inflow in different forecast periods.

Forecast Period | K | Weight (α) | Mean Value (u) | Variance (σ^{2}) |
---|---|---|---|---|

6 h | 2 | α_{1} = 0.9577, α_{2} = 0.0423 | u_{1} = −1.0001, u_{2} = 18.8304 | σ_{1}^{2} = 32.5671, σ_{2}^{2} = 28.6849 |

12 h | 2 | α_{1} = 0.4928, α_{2} = 0.5072 | u_{1} = −3.0380, u_{2} = −0.8373 | σ_{1}^{2} = 112.2997, σ_{2}^{2} =29.8075 |

18 h | 2 | α_{1} = 0.6912, α_{2} = 0.3088 | u_{1} = 1.6888, u_{2} = −2.5080 | σ_{1}^{2} = 50.7381, σ_{2}^{2} = 36.5538 |

24 h | 2 | α_{1} = 0.6404, α_{2} = 0.3596 | u_{1} = −2.8084, u_{2} = 3.6344 | σ_{1}^{2} = 47.6983, σ_{2}^{2} = 78.0387 |

Forecast Period | K | Weight (α) | Mean Value (u) | Variance (σ^{2}) |
---|---|---|---|---|

6 h | 2 | α_{1} = 0.8264, α_{2} = 0.1736 | u_{1} = −0.2519, u_{2} = 0.5758 | σ_{1}^{2} = 10.8288, σ_{2}^{2} = 0.3925 |

12 h | 2 | α_{1} = 0.2239, α_{2} = 0.7761 | u_{1} = −0.2722, u_{2} = −0.5653 | σ_{1}^{2} = 50.2290, σ_{2}^{2} = 7.8424 |

18 h | 2 | α_{1} = 0.6840, α_{2} = 0.3160 | u_{1} = −0.4602, u_{2} = 1.7341 | σ_{1}^{2} = 15.5119, σ_{2}^{2} = 50.1798 |

24 h | 2 | α_{1} = 0.4690, α_{2} = 0.5310 | u_{1} = −0.5955, u_{2} = −0.1940 | σ_{1}^{2} = 69.3037, σ_{2}^{2} = 12.8721 |

x_{(6)} | x_{(12)} | x_{(18)} | x_{(24)} | y_{(6)} | y_{(12)} | y_{(18)} | y_{(24)} | ||
---|---|---|---|---|---|---|---|---|---|

x_{(6)} | Correlation coefficient | 1.000 | 0.312 | 0.123 | 0.230 | −0.022 | 0.032 | −0.030 | 0.020 |

p value of bilateral significance test | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

x_{(12)} | Correlation coefficient | 0.312 | 1.000 | 0.160 | 0.226 | 0.001 | 0.001 | 0.003 | −0.016 |

p value of bilateral significance test | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | |

x_{(18)} | Correlation coefficient | 0.123 | 0.160 | 1.000 | 0.282 | 0.021 | 0.003 | −0.040 | 0.001 |

p value of bilateral significance test | 0 | 0 | - | 0 | 0 | 0 | 0 | 0 | |

x_{(24)} | Correlation coefficient | 0.230 | 0.226 | 0.282 | 1.000 | −0.004 | −0.005 | −0.011 | 0.008 |

p value of bilateral significance test | 0 | 0 | 0 | - | 0 | 0 | 0 | 0 | |

y_{(6)} | Correlation coefficient | −0.022 | 0.001 | 0.021 | −0.004 | 1.000 | −0.025 | 0.004 | 0.034 |

p value of bilateral significance test | 0 | 0 | 0 | 0 | - | 0 | 0 | 0 | |

y_{12)} | Correlation coefficient | 0.032 | 0.001 | 0.003 | −0.005 | −0.025 | 1.000 | 0.006 | 0.023 |

p value of bilateral significance test | 0 | 0 | 0 | 0 | 0 | - | 0 | 0 | |

y_{(18)} | Correlation coefficient | −0.030 | 0.003 | −0.040 | −0.011 | 0.004 | 0.006 | 1.000 | 0.001 |

p value of bilateral significance test | 0 | 0 | 0 | 0 | 0 | 0 | - | 0 | |

y_{(24)} | Correlation coefficient | 0.020 | −0.016 | 0.001 | 0.008 | 0.034 | 0.023 | 0.001 | 1.000 |

p value of bilateral significance test | 0 | 0 | 0 | 0 | 0 | 0 | 0 | - |

Forecast Period | Inflow | Mean Value | Variation Coefficient | Variance | |||
---|---|---|---|---|---|---|---|

Simulated | Measured | Simulated | Measured | Simulated | Measured | ||

6 h | Jinxi Reservoir inflow | 0.315 | 0.309 | 30.573 | 31.456 | 96.504 | 94.453 |

Guandi Reservoir interval inflow | −0.034 | −0.033 | −114.425 | −116.529 | 21.141 | 20.659 | |

12 h | Jinxi Reservoir inflow | −0.646 | −0.669 | −14.778 | −15.179 | 104.810 | 103.239 |

Guandi Reservoir interval inflow | −0.031 | −0.030 | −195.442 | −196.682 | 33.812 | 34.822 | |

18 h | Jinxi Reservoir inflow | −3.733 | −3.626 | −3.291 | −3.420 | 149.759 | 153.795 |

Guandi Reservoir interval inflow | 0.493 | 0.508 | 15.009 | 14.582 | 55.801 | 54.902 | |

24 h | Jinxi Reservoir inflow | −4.229 | −4.133 | −3.602 | −3.495 | 210.848 | 208.739 |

Guandi Reservoir interval inflow | 0.642 | 0.661 | 14.050 | 13.544 | 79.093 | 80.054 |

Typical Day | Risk Rate of Insufficient Output/% | Risk Rate of Wasted Water/% | Risk Rate of Beyond-or-Below-Limit Water Level/% |
---|---|---|---|

Wet day | 1.48 | 2.02 | 3.50 |

Normal day | 1.56 | 0.63 | 2.19 |

Dry day | 1.80 | 0.52 | 2.32 |

Mean value | 1.61 | 1.06 | 2.67 |

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## Share and Cite

**MDPI and ACS Style**

Wu, Y.; Wang, L.; Wang, Y.; Zhang, Y.; Wu, J.; Ma, Q.; Liang, X.; He, B.
Risk Analysis for Short-Term Operation of the Power Generation in Cascade Reservoirs Considering Multivariate Reservoir Inflow Forecast Errors. *Sustainability* **2021**, *13*, 3689.
https://doi.org/10.3390/su13073689

**AMA Style**

Wu Y, Wang L, Wang Y, Zhang Y, Wu J, Ma Q, Liang X, He B.
Risk Analysis for Short-Term Operation of the Power Generation in Cascade Reservoirs Considering Multivariate Reservoir Inflow Forecast Errors. *Sustainability*. 2021; 13(7):3689.
https://doi.org/10.3390/su13073689

**Chicago/Turabian Style**

Wu, Yueqiu, Liping Wang, Yi Wang, Yanke Zhang, Jiajie Wu, Qiumei Ma, Xiaoqing Liang, and Bin He.
2021. "Risk Analysis for Short-Term Operation of the Power Generation in Cascade Reservoirs Considering Multivariate Reservoir Inflow Forecast Errors" *Sustainability* 13, no. 7: 3689.
https://doi.org/10.3390/su13073689