## 2 Methods
### Two-Stage Risk-Benefit Hedging Calculation Considering Forecast Uncertainty
#### 2.5.2 Calculation of Downstream Flood Risk Rate Considering Forecast Information  
The input runoff information plays a decisive role in the reservoir regulation decision in the facing stage, and its prediction accuracy is high compared with the residual stage prediction error, which is small, so it can be regarded as a deterministic forecast. Therefore,  
Figure 5: Diagram of the relationship between water volume storage in advance and added value of benefit in stage 1.  
in order to ensure that there is no flood control risk, the amount of water discharged from the reservoir must be smaller than the downstream safe amount of water. \[R_{1}^{b}=P(Q_{1}^{b}\geq Q_{1,max}^{b})=0 \tag{4}\]  
In the residual stage, the forecast information of inflow runoff within the future rolling period \(\Delta T\) should be considered and the impact of its prediction error on the downstream flood control risk should not be ignored. The water level must be controlled at the lower limit of the control domain of flood limit water level at the end of the prediction period:  
\[V_{2}=V(Z_{d}^{-})=V_{1}+(Q_{2}^{a}+\theta_{2}^{a})-D_{2}=V_{0}+Q_{1}^{a}-D_{1 }+(Q_{2}^{a}+\theta_{2}^{a})-D_{2} \tag{5}\]  
Therefore, the uncertainty of reservoir discharge decision \(D_{2}\) is caused by the error of the inflow runoff forecast. The expected discharge \(\overset{-}{D_{2}}\) in stage 2 is expressed as follows:  
\[D_{2}=\overline{D}_{2}+\theta_{2}^{a}=(W_{1}+\overline{Q}_{2}^{a})+\theta_{2 }^{a} \tag{6}\]  
where \(W_{1}\) is the pre-storage water of stage 1 and \(W_{1}=V_{1}-V(Z_{d}^{-})\). Therefore, when \(\theta_{2}^{a}>0\), that is, the predicted inflow is less than the actual inflow, and the actual discharge of the reservoir is greater than the expected discharge \(D_{2}>\overset{-}{D_{2}}\), the downstream needs to bear certain flood control risks. In \(\theta_{2}^{a}<0\), the predicted inflow is relatively large and the actual discharge of the reservoir is smaller than the expected discharge, that is, \(D_{2}<\overset{-}{D_{2}}\). In this case, the downstream flood control safety can be guaranteed. The probability of the combined inflow (including the expected discharge and interval forecast inflow) exceeding the safe flow in stage 2 is used to define the downstream flood control risk rate, as shown in Equation (7):  
\[R_{2}^{b}=P(Q_{2}^{b}>Q_{2,max}^{b})=\int_{Q_{2,max}^{b}}^{+\infty}h(Q_{2}^{b}) dQ_{2}^{b} \tag{7}\]  
According to the evolution equation of river flood (excluding the propagation time of river flood), the combined discharge volume of downstream control stations is the sum of the reservoir discharge volume and forecast error, assuming that the forecast error follows the standard normal distribution, and the downstream flood control risk rate with forecast error as the independent variable is calculated as follows:  
\[R_{2}^{b}=P(\theta_{2}^{b}>Q_{2,max}^{b}-Q_{2}^{b})=\int_{Q_{2,max}^{b}-Q_{2}^{ b}}^{+\infty}h(\theta_{2}^{b})d\theta_{2}^{b}=\ P(\theta_{2}^{b}>\sigma)=\int_{\sigma}^{+ \infty}h(\theta_{2}^{b})d\theta_{2}^{b} \tag{8}\]  
where \(h(Q_{2}^{b})\) represents the probability density function of combined incoming water \(Q_{2}^{b}\) of the downstream control station in the second stage; \(h(\theta_{2}^{b})\) represents the probability density function of the incoming water forecast error \(\theta_{2}^{b}\) of the downstream control station in the second stage; and \(Q_{2,max}^{b}-Q_{2}^{b}\) is the flood control safety value, that is, the difference between the safe overflow water and the expected inflow of the downstream reservoir, denoted by \(\sigma\):  
\[\sigma=Q_{2,max}^{b}-\overline{Q}_{2}^{b} \tag{9}\]  
The flood control safety value \(\sigma\) is the safety range set aside in advance for the forecast uncertainty. The larger the value, the larger the forecast error that can be accepted by the downstream.