## 2 Methods
### Two-Stage Risk-Benefit Hedging Calculation Considering Forecast Uncertainty
#### 2.5.2 Calculation of Downstream Flood Risk Rate Considering Forecast Information
For example, when the safe flow volume of the control station is 50 \(\times\) 10\({}^{6}\) m\({}^{3}\) and the expected combined inflow is 30 \(\times\) 10\({}^{6}\) m\({}^{3}\), the downstream flood control safety value is 20 \(\times\) 10\({}^{6}\) m\({}^{3}\); so long as the prediction error does not exceed B, the downstream flood control safety can be guaranteed. In conclusion, the flood control risk \(R_{2}^{b}\) at the downstream of the reservoir is the probability that the combined incoming water \(Q_{2}^{b}\) exceeds the downstream safe flow volume \(Q_{2,\text{max}}^{b}\). If the prediction accuracy is high, in order to ensure the flood control safety of the reservoir design, the sum of the reservoir discharge and the interval inflow must not exceed the downstream safe flow (\(Q_{2}^{b}\leq Q_{2,\text{max}}^{b}\)). However, if the forecast error is large, it is necessary to analyze the effect of the forecast error on flood control risk. When the combined inflow \(Q_{2}^{b}\) increases, the downstream flood control risk \(R_{2}^{b}\) increases; meanwhile, when the foresight period increases, the response risk also increases, as shown in Figure 6.  
Within a dispatch period, it is necessary to ensure that the residual stage (stage 2) can discharge all of the pre-storage water in the decision stage (stage 1), then the downstream flood control safety value is as follows:  
\[\sigma=Q_{2,max}^{b}-\overline{Q}_{2}^{a}-\overline{Q}_{2}^{ab}-W_{1} \tag{10}\]  
According to the above equation, in order to increase the profit and benefit of the reservoir, the water storage \(W_{1}\) in the decision-making stage (stage 1) will increase and the flood control safety value \(\delta\) in the second stage will decrease, which will cause the flood control risk rate to increase. On the contrary, \(W_{1}\) decreases and \(\delta\) increases at the same time. It can be seen that the reservoir profit benefit and flood control risk are contradictory and competitive with each other, and meet the following relationship:  
\[W_{1}+\delta=Q_{2,\text{max}}^{b}-\overline{Q}_{2}^{a}-\overline{Q}_{2}^{ab} =Q_{2,\text{max}}^{b}-\overline{Q}_{2} \tag{11}\]  
where \(\overline{Q_{2}}\) represents the sum of the predicted downstream interval and the incoming water from the reservoir and \(Q_{2,\text{max}}^{b}-\overline{Q}_{2}^{a}-\overline{Q}_{2}^{ab}\) is the difference between the predicted incoming water and the downstream safe flow and is called residual flood control capacity (\(RF\)):  
\[R_{1}^{b}=P(Q_{1}^{b}\geq Q_{1,max}^{b})=0 \tag{4}\]0  
It can be seen from Equation (12) that the meaning of \(RF\) is the maximum amount of water discharged from the reservoir in the second stage when the prediction error is 0. \[R_{1}^{b}=P(Q_{1}^{b}\geq Q_{1,max}^{b})=0 \tag{4}\]1 Diagram of the relationship between combined inflow and downstream flood control risk rate in stage 2. When \(RF>0\), the reservoir in the first stage can realize more water storage, so as to increase the benefit, or to ensure the flood control safety in the second stage, so as to reduce water storage. In other words, when the forecast is too large or too small, the tradeoff between the two objectives of reservoir flood control and benefit is the division of the remaining flood control capacity RF between the first stage pre storage water \(W_{1}\) and the second stage flood control safety value \(\delta\). In a word, the risk benefit value of the reservoir in the reserving stage and decision-making stage is as follows:  
\[ \begin{array}{l}G_{1}(W_{1})=R_{1}^{b}+B_{1}^{-}=(1-\frac{W_{1}}{W_{N}})^{m} \\ G_{2}(\delta)=R_{2}^{b}+B_{2}^{-}=\int_{\sigma}^{+\infty}h(\theta_{2}^{b})d \theta_{2}^{b}\end{array}  \tag{13}\]  
Equation (14) calculates the first-order partial derivatives \(G_{1}^{\prime}(W_{1})\) and \(G_{2}^{\prime}(\delta)\).