## 2 Problem Definition
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(G_{A}\) & Active distribution networks (ADNs). \\ \(G_{P}\) & The power networks of the ADN. \\ \(G_{I}\) & The communication networks of the ADN. \\ \(V_{P}\) & The nodes set of the PN. \\ \(V_{I}\) & The nodes set of the CN. \\ \(NoN_{P}\) & The number of the nodes in the PN. \\ \(NoN_{I}\) & The number of the nodes in the CN. \\ \(V_{f\_initial}\) & The unintentional random initial failure nodes set of the PN. \\ \(\infty\) & The steady-state of the ADN after the CCF. \\ \(G_{P1}(\infty)\), \(G_{P2}(\infty)\),... & The connected clusters in the PN of the ADN after the CCF stops. \\ \(\in S_{P}(V_{f\_initial},\infty)\) & The connected clusters containing generators in the PN of the ADN after the CCF stops. \\ \(G_{Pg1}(\infty)\), \(G_{Pg2}(\infty)\),... & The connected clusters containing generators in the PN of the ADN after the CCF stops. \\ \(\in S_{Ig}(V_{f\_initial},\infty)\) & The connected clusters in the CN of the ADN after the CCF stops. \\ \(G_{I1}(\infty)\), \(G_{I2}(\infty)\),... & The connected clusters in the CN are interdependent with the connected clusters \\ \(G_{Iq}(\infty)\), \(G_{Iq2}(\infty)\),... & The connected clusters in the CN are interdependent with the connected clusters \\ \(\in S_{Ig}(V_{f\_initial},\infty)\) & containing generators in the PN of the ADN after the CCF stops. \\ \(Num\) & The number of experiments. \\ \(N\) & The number of iterative steps in the propagation of the CCF. \\ \(R_{P}(N)R_{I}(N)\) & The expected proportion of remaining nodes in the PN/CN at step \(N\) after the CCF occurs. \\ \(R_{P}(N)R_{Iz}(\infty)\) & The expectation of the quotient between the number of the remaining nodes in the clusters that containing generators and the number of the remaining nodes of the whole PN at step \(N\) after the CCF occurs. \\ \(\Theta_{P}\) & The probability that a single node in the PN fails randomly. \\ \(Ed_{SP}(n)Ed_{SI}(n)\) & The edge set (excluding the edges connecting the PN and the CN) of a node \(n\) in the PN/CN. \\ \(<\)\(K_{2}\)\(>\) & The average control fan in of a node in the PN. \\ \(<\)\(O_{1}\)\(>\) & The average power fan in of a node in the CN. \\ \(B_{ctrl}\) & The set of the paths involved in an effective control \(Econ_{I}\). \\ \(T_{PL}(C)T_{IL}(e)\) & Load threshold of an edge \(e\) in the PN/CN. \\ \(Run(n)^{-}Run(n)\) & A node or an edge \(n\) runs normally/ abnormally. \\ \(\alpha_{P}/\alpha_{I}\) & A tolerance parameter of a node \(n\) in the PN/CN which represents the ratio of the maximum capacity to nominal capacity of the node \(n\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation used in this paper. The final expected proportion of normal operational nodes was an index reflecting the robustness of the ADN against the CCF. The problem to be solved in this paper was to calculate the relationship between the final expected proportion of normal operational nodes \(R_{PF}(\infty)/R_{IF}(\infty)\) in the PN and CN of the ADN and distribution of generators, unintentional random initial failure, interdependence between the sub-networks, network topology, and load redistribution, when the CCF stops. According to the notations and relationship in Table 1, it can also be seen that the final expected proportion \(R_{PF}(\infty)/R_{IF}(\infty)\) of normal operational nodes is shown in Equation (1). \[\left\{ \begin{array}{l}R_{PF}(\infty)\approx\frac{1}{Num}\sum\limits_{Num} \frac{\sum\limits_{s\in S_{Ig}(V_{s,initial},\infty)}^{s\in S_{Ig}(V_{s,initial },\infty)}|s|}{NON_{I}}\\ R_{IF}(\infty)\approx\frac{1}{Num}\sum\limits_{Num}\frac{\sum\limits_{s\in S_{ Ig}(V_{s,initial},\infty)}^{s\in S_{Ig}(V_{s,initial},\infty)}|s|}{NON_{I}}\end{array} \right.. \tag{1}\]