## 2 Description of MCP Scenarios for Multiple Markets
### Reducing MCP Scenarios  
Generally, the results will be more accurate with more simulated scenarios, but the computation scale and time will increase accordingly. To guarantee the result accuracy and simultaneously strike a balance with the computing workload, the number of scenarios needed to be properly determined. Clustering is a commonly used scenario reduction method, but the number of reserved clusters is difficult to determine. In this paper, the hierarchical clustering method combined with inconsistent values is introduced to reduce the MCP scenarios, and the silhouette value was used to measure the rationality of clustering results. The process of reducing scenarios is detailed below:  
(1) Take each MCP scenario as a cluster and calculate the distance between every two clusters. Then, the absolute distance is adopted. (2) Merge the nearest two clusters into a new cluster and calculate the distance between the new cluster and other clusters. (3) Repeat steps (1) and (2) until all MCP scenarios are merged together. (4) Calculate the inconsistent values after each merging and express them as \(\{\theta_{k},k=1,2,\cdots,S-1\}\), and  
\[\theta_{k}=\frac{d_{k}-\overline{d}_{k}}{\sigma_{k}} \tag{8}\]where, \(\theta_{k}\) is the inconsistent value after the \(k\)th merging. \(S\) is the number of total original scenarios, \(d_{k}\) is the distance between the two clusters of the \(k\)th merging, \(\overline{d}_{k}\) is the mean value of the distances calculated in the \(k\)th merging, and \(\sigma_{k}\) is the standard deviation of the distances calculated in the \(k\)th merging. The inconsistent value was calculated by the inconsistent function in MATLAB. Each inconsistent value was a measure of separation between the two merged clusters, compared to the separation between subclusters merged within those clusters. If only two original MCP scenarios are merged in the \(k\)th merging, the inconsistent value is 0. Defining \(\Delta\theta_{k}=\theta_{k}-\theta_{k-1}\), the larger the \(\Delta\theta_{k}\) was, the worse clustering result of the \(k\)th merging was, and the better the clustering result of the \(k-1\)th merging was. Generally, the larger the number of reserved clusters, the more the reserved scenarios remain consistent with the original MCP scenarios, and the computation time will increase accordingly. The final number of reserved clusters could be determined by reference to the variation of inconsistent values \(\{\theta_{k},k=1,2,\cdots,S-1\}\). (5) The silhouette value proposed by [PERSON] [28] is often used to evaluate the result of the cluster analysis, which can be calculated by Equation (9):  
\[\beta_{i}=\frac{b_{i}-a_{i}}{\max(a_{i},b_{i})} \tag{9}\]  
where, \(\beta_{i}\) is the silhouette value of scenario \(i\), \(i=1,2,\cdots,S\), \(r\) is the cluster index that scenario \(i\) belongs to, \(a_{i}\) is the average distance between scenario \(i\) and other scenarios in the cluster \(r\), and \(b_{i}=\min\)\(\left(b_{1,1},\cdots,b_{i,l_{1}},\cdots,b_{i,R}\right)\), \(b_{i,l}\) is the average distance between scenario \(i\) and all scenarios in the cluster \(l\), \(l\
eq r\). The range of \(\beta_{i}\) is \([-1,1]\), and the larger the \(\beta_{i}\), the better the clustering result of scenario \(i\). (6) Define the number of reserved clusters as \(S^{\prime}\). For cluster \(s(1\leq s\leq S^{\prime})\), the representative scenario is \(\xi^{s}\), and can be calculated as below:  
\[\left\{ \begin{array}{l}\Omega_{s}=\left\{\xi^{s,1},\cdots,\xi^{s,m},\cdots, \xi^{s,M}\right\}\\ \xi^{s,m}=\left\{\lambda_{1,t}^{s,m},\cdots,\lambda_{i,t}^{s,m},\cdots,\lambda _{N,t}^{s,m}\right\}\\ \lambda_{i,t}^{s}=\frac{1}{M}\sum\limits_{m=1}^{M}\lambda_{i,t}^{s,m}\\ \xi^{s}=\left\{\lambda_{1,t}^{s},\cdots,\lambda_{i,t}^{s},\cdots,\lambda_{N,t }^{s}\right\}\end{array} \right.