## 2 Description of MCP Scenarios for Multiple Markets
### Building MCP Scenarios Based on the Copula Theory  
For any time period of \(t\), the predicted MCP of market \(i\) is denoted by \(\hat{\lambda}_{i,t}\), and the prediction error is \(\varepsilon_{i,t}\). The actual MCP \(\lambda_{i,t}\) can be described as a random variable of \(\hat{\lambda}_{i,t}\) and \(\varepsilon_{i,t}\). That is:  
\[\lambda_{i,t}=\hat{\lambda}_{i,t}+\varepsilon_{i,t}\left(\lambda_{i,t}^{\min }\leq\lambda_{i,t}\leq\lambda_{i,t}^{\max}\right) \tag{1}\]  
where, \(i=1,2,\cdots,I\) indicates the serial number of the market, and the total number of markets, \(I\); \(\lambda_{i,t}^{\max}\) and \(\lambda_{i,t}^{\min}\), indicate the upper and lower bounds for \(\lambda_{i,t}\), respectively. Due to the complex correlation between supply and demand of multiple markets, the MCPs of these markets can fluctuate greatly and are related to each other, thus the MCP of each market is not appropriate for separate description. In this paper, the Copula theory in econometrics is introduced to describe the correlation of MCPs in multiple markets, and a method for generating combined MCP scenarios (i.e., MCP scenarios for multi-markets) based on the Copula function is proposed. The Copula theory was firstly proposed by [PERSON] in 1959 to describe complex nonlinear correlations between random variables, which is now widely used in the financial sector as it has no restrictions on the marginal distribution of variables [26]. The Copula function can put the marginal distributions together to form a joint distribution, and its strict definition is given later by [PERSON] [27]. That is, there is a function \(C\) that makes:  
\[H(x_{1},x_{2},\cdots,x_{n})=C(H_{1}(x_{1}),H_{2}(x_{2}),\cdots,H_{n}(x_{n})) \tag{2}\]  
where, \(H\) is the joint distribution function of the random vectors \(x_{1},x_{2},\cdots,x_{n}\), and its marginal distribution is \(H_{1}(x_{1}),H_{2}(x_{2}),\cdots,H_{n}(x_{n})\), respectively. It has been proved that if all these marginal distributions are continuous, the Copula function \(C\) is unique. Defining \(H_{i}^{-1}(x_{i}),i\in\{1,2,\cdots,n\}\) as the inverse function of \(H_{i}(x_{i})\), the Copula function can be expressed as:  
\[C(u_{1},u_{2},\cdots,u_{n})=H\Big{(}H_{1}^{-1}(u_{1}),H_{2}^{-1}(u_{2}),\cdots,H_{n}^{-1}(u_{n})\Big{)} \tag{3}\]  
where, \(u_{i}(i=1,2,\cdots,n)\) is a variable obeying uniform distribution in \([0,1]\). Obviously, the Copula function \(C(u_{1},u_{2},\cdots,u_{n})\) is a joint distribution function constructed by \(n\) marginal distributions of uniform distribution in \([0,1]\). There are five commonly used Copula functions, including Gaussian-, t-, Gumbel-, Clayton- and Frank-Copula functions. Their expressions, characteristics, and density distribution plots are listed in Table 1.  
As mentioned above, the cascade hydropower plants in southwestern China mainly participate in two electricity markets, i.e., the provincial and external markets. Based on the Copula theory, the combined MCP scenarios were generated using the following steps:  
(1) Construct MCP marginal cumulative distribution functions for the provincial and external markets. As the MCP \(\lambda_{i,t}\) is related to its prediction value \(\hat{\lambda}_{i,t}\) and prediction error \(\varepsilon_{i,t}\), the distribution function for \(\varepsilon_{i,t}\) needs to be firstly determined. At the early stage of reform, the monthly MCP data were accumulated for a short time and the distribution of prediction error was not clear. Therefore, according to the central limit theorem, suppose that \(\varepsilon_{i,t}\sim N\big{(}0,\sigma_{i,t}^{2}\big{)}\), then \(\lambda_{i,t}\sim N\big{(}\hat{\lambda}_{i,t},\sigma_{i,t}^{2}\big{)}\), namely, \begin{table}
\begin{tabular}{c c c c} \hline \hline
\begin{tabular}{c} Distributed \\ asymmetrically, without \\ reflecting the tail \\ correlation \\ \end{tabular} & \\ \hline \end{tabular}
\end{table} 
Table 1: The general information of five commonly used Copula functions.