## 5 Case Study  
### Study System  
The proposed model was applied to the cascade hydropower plants of the Wu River in the Guizhou Province, China. Seven hydropower plants have been built on the mainstream of the Wu River, with a total installed capacity of 8345 MW. Figure 2 shows the distribution chart of the Wu River cascade, with reservoir characteristics in Table 2. Four hydropower plants have long-term regulation abilities in the studied cascade, and they are Hongjiadu, Dongfeng, Wujiangdu, and Goupitan. The storage capacities of the other three reservoirs were much smaller, and they were addressed as runoff hydropower plants in the long-term operation. Thus, the cascade power decisions were distributed through the operation of these four large reservoirs. In this paper, the long-term provincial and external electricity markets are considered, assuming that the energy ratio between the markets is 3:1. The estimated electricity MCP and the standard deviation of estimated error are treated as known and given in Table 3. The monthly local inflow data in 2016 were used for simulation and shown in Figure 3. In the later text, market 1 represents the provincial electricity market and market 2 represents the external electricity market. All the basic data of the study system was collected from the dispatching center of the Wu River basin.  
Figure 1: The flow chart for calculation.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Plant Name**} & **Installed** & **Normal** & \multirow{2}{*}{**Dead Level**} & \multirow{2}{*}{**Regulation**} & \multicolumn{2}{c}{**Maximum**} & **Generation** \\  & **Capacity** & **Level** & **(m)** & **Coefficient** & **Power Release** & **Efficiency** \\  & **(MW)** & **(m)** & **(m)** & **Coefficient** & **(m\({}^{3}\)/s)** & **(kW/(m\({}^{3}\)/s)/m)** \\ \hline Hongjiadu & **600** & **1140** & **1076** & **0.72** & **490.5** & **8.50** \\ Dongfeng & **695** & **970** & **936** & **0.05** & **632.1** & **8.35** \\ Suofengying & **600** & **837** & **822** & – & 994.5 & 8.50 \\ Wujiangdu & 1280 & 760 & 720 & 0.09 & 1087.0 & 8.30 \\ Goupitan & 3000 & 630 & 590 & 0.14 & 1909.0 & 8.50 \\ Silin & 1050 & 440 & 431 & – & 1772.8 & 8.50 \\ Shatuo & 1120 & 365 & 353.5 & – & 1969.2 & 8.50 \\ \hline \hline \end{tabular} Note: The regulation coefficient could be calculated from the division of the beneficial storage by the average inflow water volume.  
\end{table}
Table 2: Characteristics of the studied hydropower plants on the Wu River.  
Figure 3: The monthly local inflow of each reservoir in 2016.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Month**} & \multicolumn{2}{c}{**Market 1**} & \multicolumn{2}{c}{**Market 2**} & \multirow{2}{*}{**Month**} & \multicolumn{2}{c}{**Market 1**} & \multicolumn{2}{c}{**Market 2**} \\ \cline{2-3} \cline{5-10}  & \(\hat{\lambda}\) & \(\sigma\) & \(\hat{\lambda}\) & \(\sigma\) & & \(\hat{\lambda}\) & \(\sigma\) & \(\hat{\lambda}\) & \(\sigma\) \\ \hline Jan. & 311.56 & 15.58 & 303.30 & 15.17 & Jul. & 265.11 & 13.26 & 298.42 & 14.92 \\ Feb. & 325.32 & 16.27 & 322.68 & 16.13 & Aug. & 266.79 & 13.34 & 300.76 & 15.04 \\ Mar. & 316.65 & 15.83 & 300.83 & 15.04 & Sep. & 246.27 & 12.31 & 246.43 & 12.32 \\ Apr. & 293.86 & 14.69 & 300.93 & 15.05 & Oct. & 255.16 & 12.76 & 290.43 & 14.52 \\ May & 287.12 & 14.36 & 283.43 & 14.17 & Nov. & 288.58 & 14.43 & 261.76 & 13.09 \\ Jun. & 283.23 & 14.16 & 279.26 & 13.96 & Dec. & 316.87 & 15.84 & 258.71 & 12.94 \\ \hline \hline \end{tabular} Note: \(\hat{\lambda}\) indicates the estimated electricity MCP; \(\sigma\) indicates the standard deviation of the estimated error.  
\end{table}
Table 3: The parameters of market clearing price (MCP) for market 1, 2 in a different month (yuan/MWh).  
Figure 2: The distribution chart of the Wu River cascade.