## Appendix A  
The power forecast error probability density function of wind farm represented by the GMM is expressed as:  
\[f^{(i)}(\Delta P_{W}^{(i)})=\sum_{k=1}^{m}c_{k}^{(i)}g(u_{k}^{(i)},\sigma_{k} ^{(i)},\Delta P_{W}^{(i)}) \tag{1}\]  
where, \(m\) denotes the number of single Gaussian distributions in the GMM, and \(c_{k}^{(i)},u_{k}^{(i)},\sigma_{k}^{(i)}\) denote the weight coefficient, mean and standard deviation, respectively, of the \(k\)th single Gaussian distribution in wind farm \(i\). The total uncertain power of the system is expressed as:  
\[P_{\text{un}}=\Delta p_{W1}+\Delta p_{W2}+\cdots+\Delta p_{WN} \tag{2}\]  
where, \(N\) is the number of wind farms. Compared to the forecast error of wind power output power, the forecast error of load is small, so the forecast error of the load is ignored. The change of active power flow of branch \(s-t\) caused by the change of node active power output \(\Delta p\) is \(\Delta P_{s-t}\) in power system power flow analysis. \[\Delta P_{s-t,j}=G_{s-t,j}\;\Delta P_{W,j} \tag{3}\]  
where, \(G_{s-t,j}\) is the distribution factor of generator output power transfer of node \(j\) for line \(s-t\). Considering that multiple nodes are connected to the wind farm, the influence of all wind turbines on a branch should be considered. The linear relationship between the power error of the line \(s-t\) and all wind farm errors is expressed as:  
\[\Delta P_{s-t}=\beta^{(1)}\Delta p^{(1)}+\beta^{(2)}\Delta p^{(2)}+\cdots+ \beta^{(N_{W})}\Delta p^{(N_{W})} \tag{4}\]  
where, \(\beta^{(1)}\beta^{(2)}\cdots\beta^{(N_{W})}\) are constant coefficients, which can be obtained by the wind turbine output power transfer distribution factor matrix \(G\). Several properties of probability calculation are introduced. Suppose that \(x_{1}\) and \(x_{2}\) are two independent random variables, and their probability density functions are \(f^{(x_{1})}(\cdot)\) and \(f^{(x_{2})}(\cdot)\), respectively. \(y\) is the linear function of the random variables \(x_{1}\) and \(x_{2}\), and its probability density function is \(f^{(y)}(\cdot)\). (1) If y = \(hx_{1}\) and \(h\) is a constant, then \(f^{(y)}(y)=f^{(x_{1})}(y/h)/h\). (2) If y = \(x_{1}+x_{2}\), then \(f^{(y)}(y)=f^{(x_{1})}(y)\otimes f^{(x_{2})}(y)\). If \(x_{1}\) and \(x_{2}\) are subject to the normal distribution and \(y=x_{1}+x_{2}\), then \(y\) is still subject to the normal distribution, the expected value of \(y\) is the sum of the expected values of \(x_{1}\) and \(x_{2}\), and the variance of \(y\) is the sum of the variance of \(x_{1}\) and \(x_{2}\). According to the above properties, the probability density function of line power \(P^{(s-t)}\) can be deduced as follows:  
\[f^{(s-t)}(p^{(s-t)})=\sum\limits_{k=1}^{3}\sum\limits_{m=1}^{3}\cdots\sum \limits_{n=1}^{3}c^{(s-t)}g(\bar{u}^{(s-t)},\sigma^{(s-t)},p^{(s-t)}) \tag{11}\]  
where  
\[c^{(s-t)}=c_{k}^{(1)}c_{m}^{(2)}\cdots c_{n}^{(N_{w})}\]  
\[\bar{u}^{(s-t)}=\bar{p}^{s-t}+\beta_{st}^{(1)}\bar{\mu}_{1}^{(1)}+\beta_{st}^{( 2)}\bar{\mu}_{k}^{(2)}+\cdots+\beta_{st}^{(N_{w})}\bar{\mu}_{k}^{(N_{w})} \tag{12}\]  
\[\left(\sigma^{(s-t)}\right)^{2}=(\beta_{st}^{(1)}\sigma_{k}^{(1)})^{2}+(\beta_ {st}^{(2)}{\sigma_{m}^{(2)}}^{2})^{2}+\cdots+(\beta_{st}^{(N_{w})}\sigma_{n}^{ (N_{w})})^{2}\]  
where, \(\bar{p}^{s-t}\) is the expected value of the line power, which can be calculated by DC power flow. The cumulative distribution function of \(\bar{p}^{s-t}\) is derived as follows:  
\[ \begin{array}{l}F^{(s-t)}(p^{(s-t)})=f^{p^{(s-t)}}_{-\infty}f^{(s-t)}(x)dx\\ =\sum\limits_{k=1}^{3}\sum\limits_{m=1}^{3}\cdots\sum\limits_{n=1}^{3}c^{(s-t) }\int_{-\infty}^{p^{(s-t)}_{-\infty}}g(\bar{u}^{(s-t)},\sigma^{(s-t)},x)dx\\ =\sum\limits_{k=1}^{3}\sum\limits_{m=1}^{3}\cdots\sum\limits_{n=1}^{3}c^{(s-t )}\int_{-\infty}^{\frac{(s-t)}{\sigma^{(s-t)}}}g(0,1,x)dx\\ =\sum\limits_{k=1}^{3}\sum\limits_{m=1}^{3}\cdots\sum\limits_{n=1}^{3}c^{(s-t )}G(\frac{p^{(s-t)}-\bar{u}^{(s-t)}}{\sigma^{(s-t)}})\end{array}