## 3 Optimization Model  
In this paper, the cascade hydropower plants are assumed as one generator in the electricity market and cleared by the MCP. The annual market is considered and each time period represents one month. The operation objective was to maximize the market benefits. Considering the different MCP scenarios in the future period, the objective function can be defined as Equation (12). \[\max B=\sum\limits_{t=1}^{T}\left[\sum\limits_{s=1}^{S^{\prime}}\left(\xi_{1,t}^{s}\lambda_{1,t}^{s}+\xi_{2,t}^{s}\lambda_{2,t}^{s}\right)\times P_{r,t} (\xi_{s})\right] \tag{12}\]where, \(e_{i,t}^{s}\) is the supply energy for market \(i\) in the time period \(t\) under the representative scenario \(\xi^{s}\), \(S^{\prime}{}_{t}\) is the number of reserved scenarios in period \(t\). The hydropower generation cost mainly involved the constant cost with little influence on the optimization results, and so it was not considered in the optimization model. For the cascade hydropower plants in southwestern China, the provincial and external markets are considered, \(i=1\) for the provincial market and \(i=2\) for the external market. To get the proposed model more applicable to the scheduling in practice, the energy ratio for different electricity markets and the minimum supply energy for the provincial market, during the whole scheduling period, were also included in the constraints. The constraints involved in the model were as follows, in which the hydraulic constraints also can be seen in [8; 9; 13]. (1) Water balance equation:  
\[\left\{ \begin{array}{l}V_{m,t+1}=V_{m,t}+(Q_{m,t}-q_{m,t}-d_{m,t})\Delta T_{t }\\ Q_{m,t}=q_{m-1,t}+d_{m-1,t}+I_{m,t}\end{array} \right. \tag{13}\]  
where, \(V_{m,t}\) and \(V_{m,t+1}\) are the storage of reservoir \(m\) at the beginning and end of the time period \(t\), respectively, m\({}^{3}\); \(Q_{m,t}\) is the total inflow of reservoir \(m\) in the period \(t\), \(q_{m,t}\), \(d_{m,t}\), and \(I_{m,t}\) are the power release, non-power release, and natural inflow of reservoir \(m\) in the time period \(t\), respectively, m\({}^{3}\)/s, \(m=1,2,\cdots,M\) is the reservoir index, \(m-1\) is the index of the upstream reservoir of reservoir \(m\), and \(\Delta T_{t}\) is the number of seconds in the period \(t\). (2) Initial and end reservoir level constraints:  
\[Z_{m,1}=Z^{\prime}{}_{m,1}\forall m\in M \tag{14}\]  
\[Z_{m,T}=Z^{\prime}{}_{m,T}\forall m\in M \tag{15}\]  
where, \(Z^{\prime}{}_{m,1}\) and \(Z^{\prime}{}_{m,T}\) are the initial and end reservoir levels of planning horizon, respectively, which are specified before the optimization process, \(m\). In the case study, \(Z^{\prime}{}_{m,1}\) and \(Z^{\prime}{}_{m,T}\) were specified based on the historical records for practical operation. (3) Outflow constraint:  
\[\underline{q}_{m,t}\leq q_{m,t}\leq\overline{q}_{m,t} \tag{16}\]  
where, \(\underline{q}_{m,t}\) and \(\overline{q}_{m,t}\) are the minimum and maximum outflow of plant \(m\) in the time period \(t\), m\({}^{3}\)/s. (4) Power generation constraint:  
\[\underline{p}_{m,t}\leq p_{m,t}\leq\overline{p}_{m,t} \tag{17}\]  
where, \(\underline{p}_{m,t}\) and \(\overline{p}_{m,t}\) are the minimum and maximum power generation of plant \(m\) in the time period \(t\), MW. (5) Storage boundary constraint:  
\[\underline{V}_{m,t}\leq V_{m,t}\leq\overline{V}_{m,t} \tag{18}\]  
where, \(\underline{V}_{m,t}\) and \(\overline{V}_{m,t}\) are the minimum and maximum storage for reservoir \(m\) at the beginning of the time period \(t\), respectively, m\({}^{3}\). (6) Market-to-market energy ratio constraint:  
\[\left\{ \begin{array}{l}\frac{E_{1}}{E_{2}}=\theta\\ E_{1}=\sum\limits_{t=1}^{T}e_{1,t}\\ E_{2}=\sum\limits_{t=1}^{T}e_{2,t}\end{array} \right.