## 4 Robustness Analysis of the ADN against the CCF
### Robustness Analysis
#### 4.1.1 Distribution of Generators and Robustness of the ADN  
The resulting-iteration-equation-sets are concluded in (4a,b), where \(R_{I}(0)=\)1 and \(R_{IF}(0)=\)1. The iteration Equation (4a) calculates the value of \(R_{P}(2N+1)\) using the value of \(R_{IF}(2N)\), and the iteration Equation (4b) calculates the value of \(R_{I}(2N)\) using the value of \(R_{PF}(2N-1)\). As \(N\rightarrow\infty\), the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Domain** & **Type** & & **Normal Operation Conditions** \\ \hline \multirow{4}{*}{PN} & Edge & (1) & The load flowing through the edge \(e_{P}\) does not exceed its threshold \(T_{PI}(e_{P})\) at step \(N\). i.e., \\  & Edge & & \((l\)_{Iand}(e_{P},N)\leq T_{PI}(e_{P}))\wedge(e_{P}\in E_{P})\) \\  & \(e_{P}\) & (2) & The nodes on both sides of the edge \(e_{P}\) are in normal operation. i.e., \\  & & & \(\left(e_{P}\ =\ (n_{P_{X}}\in V_{P},n_{P_{Y}}\in V_{P})\right)\rightarrow( run(n_{P_{X}})\wedge run(n_{P_{Y}}))\) \\ \hline \multirow{4}{*}{CN} & & (1) & There is at least one normal operating node \(n_{P}\) in the PN that provides power to this node \(n_{I}\). i.e., \\  & & & \(3n_{P}\). \(\left((e\ =\ (n_{P}\in V_{P},n_{I}\in V_{I}))\right)\to run(n_{P})\) \\  & Node & & \(n_{I}\). the CN does not exceed its threshold at step \(N\). i.e., \\  & \(n_{I}\) & & \((l\)_{Iand}(n_{I},N),n_{I}\in V_{I})\leq T_{IP}(n_{I})\) \\  & & & \((3)\). The node \(n_{I}\) is part of at least one path in the set \(B_{tr}\). i.e., \\  & & & \(3n_{I4}\). \(path(n_{I},n_{I})\in B_{P}\) \\ \hline \multirow{4}{*}{\
\begin{tabular}{c} Edge \\ \(e_{P}\) \\ \end{tabular} } & (1) & The load (data packets) flowing through the edge \(e_{P}\) does not exceed its threshold \(T_{IL}(e_{P})\) at step \(N\). i.e., \\  & Edge & & \((l\)_{Iand}(e_{P},N)\leq T_{IL}(e_{P}))\wedge(e_{P}\in E_{I})\) \\  & \(e_{P}\) & (2) & The nodes on both sides of edge \(e_{P}\) are in normal operation. i.e., \\  & & & \(\left(e_{P}\ =\ (n_{IL}\in V_{I},n_{I}\in V_{I})\right)\rightarrow( run(n_{IL})\wedge run(n_{I}))\) \\ \hline \multirow{2}{*}{
\begin{tabular}{c} The interdependence \\ between the PN and \\ the CN \\ \end{tabular} } & Edge & (1) & The nodes on both sides of the edge \(e\in(E_{PI}\cup E_{IP})\) are in normal operation. i.e., \\  & \(e\) & \(((e\ =\ (n_{IP_{X}}\in V_{P},n_{ILx}\in V_{I}))\vee(e\ =\ (n_{ILx}\in V_{I},n_{PP_{X}}\in V_{P}))) \rightarrow( run(n_{IP_{X}})\wedge run(n_{ILx}))\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._iteration Equation (4a,b) calculate the sequences \(R_{I}(0)\), \(R_{P}(1)\), \(R_{I}(2)\)..., \(R_{P}(\infty)\), \(R_{I}(\infty)\) and \(R_{IF}(0)\), \(R_{PF}(1)\), \(R_{IF}(2)\)..., \(R_{PF}(\infty)\), \(R_{IF}(\infty)\). \[\left\{ \begin{array}{l}R_{P}(2N+1)\!=\ (1\!-\!\theta_{P})\times\left[1-(1-R_{IF}(2 N))^{\langle K_{2}\rangle}\right]\\ R_{PF}(2N+1)\ =\ R_{P}(2N+1)\times f_{P}(R_{P}(2N+1))\end{array} \right.,N\ =\ 0,1,2,\ldots \tag{4a}\] \[\left\{ \begin{array}{l}R_{I}(2N)\ =\ 1\!-\!\left[1-R_{PF}(2N-1)\right]^{ \langle 0_{1}\rangle}\\ R_{IF}(2N)\ =\ R_{I}(2N)\times f_{I}(R_{I}(2N))\end{array} \right.,N\ =\ 1,2,\ldots \tag{4b}\]  
Figure 3: Venn diagram for computing \(R_{PF}(N)/R_{IF}(N)\) in the first three iterations. When the CCF stops, the ADN enters the steady state and then satisfies the equations \(R_{P}(2N-1)=R_{P}(2N)=R_{P}(2N+1)=\lambda\) and \(R_{I}(2N-1)=R_{I}(2N)=R_{I}(2N+1)=\gamma\). According to Equation (4a,b), a set of steady-state nonlinear equations is obtained after the replacement as above, shown in Equation (5). \[\left\{ \begin{array}{l}\gamma\ =\ 1-[1-\lambda\times f_{P}(\lambda)]^{\langle \Omega_{1}\rangle}\\ \lambda=(1-\theta_{P})\times\left[1-(1-\gamma\ast f_{I}(\gamma))^{\langle K_{ 2}\rangle}\right]\end{array} \right. \tag{5}\]  
At steady state, the final expected proportions \(R_{PF}(\infty)\) and \(R_{IF}(\infty)\) of remaining normal operational nodes in the PN and CN are shown in Equation (6). \[\left\{