## Appendix A Derivation of CDF  
The real power flow on line \(k\) connected between bus \(i\) and bus \(j\) is formulated as [30]:  
\[P_{ij}=|V_{i}||V_{j}||Y_{ij}|cos(\theta_{ij}-\delta_{i}+\delta_{j})-V_{i}^{2}Y_{ ij}\ cos\ \theta_{ij}. \tag{10}\]  
Applying the approximation of a Taylor series expansion and ignoring the effects of the remaining higher-order terms, Equation (10) gives  
\[\Delta P_{ij}=\frac{\partial P_{ij}}{\partial\delta_{i}}\Delta\delta_{i}+\frac {\partial P_{ij}}{\partial\delta_{j}}\Delta\delta_{j}+\frac{\partial P_{ij}} {\partial V_{i}}\Delta V_{i}+\frac{\partial P_{ij}}{\partial\delta_{j}}. \tag{11}\]  
Equation (11) can be rewritten as  
\[\Delta P_{ij}=a_{ij}\Delta\delta_{i}+b_{ij}\Delta\delta_{j}+c_{ij}\Delta V_{i }+d_{ij}\Delta V_{j}. \tag{12}\]  
The coefficients in Equation (12) are formulated from the partial derivatives of the real power injection corresponding to the variables \(\delta\) and \(V\) and given as  
\[a_{ij}=V_{i}V_{j}Y_{ij}\ sin(\theta_{ij}+\delta_{j}-\delta_{i}), \tag{13}\]  
\[b_{ij}=-V_{i}V_{j}Y_{ij}\ sin(\theta_{ij}+\delta_{j}-\delta_{i}), \tag{14}\]  
\[c_{ij}=V_{j}Y_{ij}\ cos(\theta_{ij}+\delta_{j}-\delta_{i})-2V_{i}Y_{ij}\ cos\ \theta_{ij}, \tag{15}\]  
\[d_{ij}=V_{i}Y_{ij}\ cos(\theta_{ij}+\delta_{j}-\delta_{i}). \tag{16}\]  
The Newton-Raphson Jacobian relationship is considered in determining the CDFs as  
\[\left[\begin{array}{c}\Delta P\\ \Delta Q\end{array}\right]=\left[J\right]\left[\begin{array}{c}\Delta\delta \\ \Delta V\end{array}\right]=\left[\begin{array}{cc}J_{11}&J_{12}\\ J_{21}&J_{22}\end{array}\right]\left[\begin{array}{c}\Delta\delta\\ \Delta V\end{array}\right] \tag{17}\]  
Taking the coupling between \(\Delta P\)-\(\Delta\delta\) and \(\Delta Q\)-\(\Delta\delta\) into consideration and assuming that the reactive power flows are constant (i.e., \(\Delta Q=0\)), the power injections variations can be expressed by  
\[\Delta P=J_{11}\Delta\delta+J_{12}\Delta V, \tag{18}\]  
\[0=J_{21}\Delta\delta+J_{22}\Delta V. \tag{19}\]  
Equations (18) and (19) then become  
\[\Delta P=J_{11}\Delta\delta-J_{12}J_{22}^{-1}J_{21}\Delta\delta=J_{r}\Delta\delta. \tag{20}\]From Equation (A11), the value of the voltage angle variation is given by  
\[\Delta\delta=[J_{r}]^{-1}\Delta P.\] (A12)  
From Equation (A10), the voltage variation with respect to the variations in power can be written as  
\[\Delta V=J_{22}^{-1}J_{21}\Delta\delta=J_{22}^{-1}J_{21}[J_{r}]^{-1}\Delta P.\] (A13)  
Equations (A12) and (A13) are formulated as  
\[\Delta\hat{\epsilon}_{i}=\sum_{l=1}^{n}m_{il}\Delta P_{l}\quad i=1,\,2,\,...,n, \quad i\
eq s,\] (A14)  
\[\Delta V_{i}=\sum_{l=1}^{n}m_{ilv}\Delta P_{l}\quad i=1,\,2,\,...,n,\quad i\
eq s.\] (A15)  
Substituting Equations (A14) and (A15) into Equation (8), the variation in real power flow becomes  
\[\Delta P_{ij}=a_{ij}\sum_{l=1}^{n}m_{il}\Delta P_{l}+b_{ij}\sum_{l=1}^{n}m_{jl} \Delta P_{l}+c_{ij}\sum_{l=1}^{n}m_{jlv}\Delta P_{l}+d_{ij}\sum_{l=1}^{n}m_{ jlv}\Delta P_{l}.\] (A16)  
Equation (A16) can be rewritten as  
\[\begin{array}{ll}\Delta P_{ij}=&(a_{ij}m_{i1}+b_{ij}m_{j1}+c_{ij}m_{i1v}+d_ {ij}m_{j1v})\Delta P_{1}\\ &+...(a_{ij}m_{in}+b_{ij}m_{jn}+c_{ij}m_{inv}+d_{ij}m_{inv})\Delta P_{n}\end{array}.\] (A17)  
Thus, the CDF corresponding to both the \(n\)th bus and the line \(ij\), connecting buses \(i\) and \(j\), can be obtained as  
\[\Delta P_{ij}=CDF_{1}^{ij}\Delta P_{1}+CDF_{2}^{ij}\Delta P_{2}+...+CDF_{n}^{ ij}\Delta P_{n},\] (A18)  
\[CDF_{n}^{ij}=a_{ij}m_{in}+b_{ij}m_{jn}+c_{ij}m_{inv}+d_{ij}m_{inv}.\] (A19)