## Appendix B Derivation of Mapping \(\mathbf{f_{P}}\) and \(\mathbf{f_{I}}\)  
Firstly, it is assumed the degree of the node \(n_{Pi}\) is \(d_{P}\), and the number of neighbor nodes set \(\textit{Neigh}_{P}(n_{Pi})\) of a node \(n_{Pi}\) in the PN of the ADN is \(D_{P1}=d_{P}-O_{2}\). In order to calculate the conditional probabilities \(\textit{Probp}(\textit{Eetr}_{3}/[\textit{Neigh}_{P}(n_{Pi})])\), then the probability needed is shown in Equation (A1). \[\textit{Probp}_{P}\left(\frac{\textit{Eetr}_{n}}{D_{P2}}=\frac{d_{P}}{p}\right) =\textit{Probp}\left(d_{P}^{\prime}\times\left(\sum\limits_{d_{P}=1}^{\infty} \frac{(F_{P}(d_{P})+\xi)\times\textit{Wp}(n_{Pi})\cdot n_{Pi})}{\sum\limits_{ np\in\textit{Neigh}_{P}(n_{Pi})}W_{P}(n_{Pi},n_{Pi})}\times\textit{Probp}_{P}(d_{P}) \right)+F_{P}(d_{P})+\xi_{i}>\alpha_{P}\times F_{P}(d_{P})\right)\] (A1)  
where \(\left(\sum\limits_{d_{P}=1}^{\infty}\frac{(F_{P}(d_{P})+\xi)\times\textit{Wp} (n_{Pi})\cdot n_{Pi})}{\sum\limits_{np\in\textit{Neigh}_{P}(n_{Pi})}W_{P}(n_{Pi },n_{Pi})}\times\textit{Probp}_{P}(d_{P})\right)\) is the expected load redistributed to the node \(n_{Pi}\) after the failure of a neighbor node \(n_{Pi}\) connected to the node \(n_{Pi}\), and then there is Equation (A2). \[\textit{Probp}_{P}\left(\frac{\textit{Eetr}_{3}}{D_{P2}}=\frac{d_{P}^{\prime} }{p}\right) =\left\{ \begin{array}{l}1,d_{P}^{\prime}>\frac{(n_{P}-1)\times F _{P}(d_{P})-\xi_{i}}{\left(\sum\limits_{d_{P}=1}^{\infty}\frac{(F_{P}(d_{P})+ \xi)\times\textit{Wp}(n_{Pi})\cdot n_{Pi})}{\sum\limits_{np\in\textit{Neigh} _{P}(n_{Pi})}\times\textit{Probp}_{P}(d_{P})}\right)}\\ 0,\text{other}\end{array} \right.\] (A2)  
After simplification, it is obtained in Equation (A10). \[\mathit{Prob}_{P}(\mathit{E}\mathit{t}\mathit{t}_{\mathit{P}4})\ =\ (1-\theta_{P})\times \Bigg{(}1-\mathit{Prob}_{P}(\frac{\mathit{E}\mathit{t}\mathit{t}^{\prime}_{ \mathit{P}2}}{\mathit{E}\mathit{t}\mathit{t}_{\mathit{P}1}})^{d_{P}-O_{2}} \Bigg{)}\times\Bigg{(}1-\mathit{Prob}_{P}(\frac{\mathit{E}\mathit{t}\mathit{t} \mathit{t}_{\mathit{P}3}}{D_{P1}\ =\ d_{P}-O_{2}})\Bigg{)}.\] (A10)  
Thus, the expected probability that any node belongs to the connected clusters containing generators is shown in Equation (A11). \[\sum\limits_{d_{P}\ =\ \max\{0,O_{2}\}}^{\infty}\mathit{Prob}_{P}(d_{P})\times \mathit{Prob}_{P}(\mathit{E}\mathit{t}\mathit{t}_{\mathit{P}4}).\] (A11)  
Therefore, considering the power fluctuations (intermittent or random) of distributed generators, it is defined \(f_{P}(1-\Theta_{P})\) as the expected probability of the connected clusters containing generators divided by \(1-\Theta_{P}\), then it is as shown in Equation (A12). \[f_{P}(1-\theta_{P})\ =\ \frac{\sum\limits_{d_{P}\ =\ \max\{0,O_{2}\}}^{\infty} \mathit{Prob}_{P}(d_{P})\times\mathit{Prob}_{P}(\mathit{E}\mathit{t}\mathit{ t}_{\mathit{P}4})}{1-\theta_{P}}.\] (A12)  
The above formula is further equal to Equation (A13). \[f_{P}(1-\theta_{P})\ =\ \sum\limits_{d_{P}\ =\ \max\{0,O_{2}\}}^{\infty} \mathit{Prob}_{P}(d_{P})\times\Bigg{(}1-\mathit{Prob}_{P}(\frac{\mathit{E} \mathit{t}\mathit{t}^{\prime}_{\mathit{P}2}}{\mathit{E}\mathit{t}\mathit{t} _{\mathit{P}1}})^{d_{P}-O_{2}}\Bigg{)}\times\Big{(}1-\mathit{Prob}_{P}(\frac{ \mathit{E}\mathit{t}\mathit{t}_{\mathit{P}3}}{D_{P1}\ =\ d_{P}-O_{2}})\Big{)}.\] (A13)  
In addition, \(\mathit{Prob}_{P}(\mathit{E}\mathit{t}^{\prime}\mathit{t}_{\mathit{P}2}/ \mathit{E}\mathit{t}\mathit{t}_{\mathit{P}1})\) is the probability that there is a particular neighbor node \(n_{P}\) of a node \(n_{P1}\), and this node \(n_{P1}\) is not connected to the connected clusters containing generators in subsequent neighbor nodes except this node \(n_{P1}\). Thus, the conclusion is drawn as shown in Equation (A14).