## 4 Robustness Analysis of the ADN against the CCF
### Relationship Analysis of Robustness and Tolerance Parameters  
With the increase of the value of parameter reflecting the nodes thresholds under random attack strategy, the survivability of the single PN increased gradually against cascading failures, as shown in Figure 4. Therefore it was necessary to analyze the nodes thresholds influence on the robustness of the ADN. These nodes thresholds analyses provided conditions for preventing the CCF. Assuming that the degree (it equals the sum of fan in and fan out) of a node \(n_{Pi}\) in the PN is \(d_{P}\). Considering the power fluctuation of the distributed generators and the initial load of the node \(n_{Pi}\) at step 0 is shown in Equation (10), which is modified from the conclusions in the literature [23]. \[\langle load_{P}\rangle\ =\ \sum_{d_{P}\ =\ 1}^{\infty}F_{P}(d_{P})\times Prob _{P}(d_{P})+g \tag{10}\]where the load \(F_{P}(d_{P})\) of the node \(n_{Pi}\) in the PN is the function of its degree \(d_{P}\). According to the Central Limit Law, this paper used the conclusions in literature [13] to model the randomness of the output power of distributed generators. So, \(\xi_{Pi}\) is a random variable obeying Gaussian distribution. The expected load of the node \(n_{Pi}\) in the PN is shown in Equation (11). \[\langle load_{P}\rangle\ =\ \sum_{d_{P}=\ 1}^{\infty}F_{P}(d_{P})\times Prob_{P}(d _{P})+g \tag{11}\]  
where \(g\) represents the expected output power by the distributed generators at the node \(n_{Pi}\), if the node \(n_{Pi}\) belongs to the set of generation nodes \(V_{P1}\), \(g\) is not equal to zero, otherwise, it is zero. \(Prob_{P}(d_{P})\) is the probability that the degree of the node \(n_{Pi}\) is \(d_{P}\). The threshold of a node \(n_{Pi}\) in the PN at step \(N\) is equal to \(T_{PP}(n_{Pi}N)=\alpha_{P}\times F_{P}(d_{P})\). Where \(\alpha_{P}\) is a tolerance parameter of the node \(n_{Pi}\) and satisfies \(\alpha_{P}\geq 1\). The expected value of the threshold of the node \(n_{Pi}\) in the PN is shown in Equation (12). \[\langle Thres_{P}\rangle\ =\ \langle\alpha_{P}\rangle\times\sum_{d_{P}=\ 1}^{\infty}F_{P}(d_{P})\times Prob_{P}(d_{P}) \tag{12}\]  
Similarly, assuming that the degree of a node \(n_{Pi}\) in the CN is \(d_{I}\), then the initial load (data packets) of node \(n_{Pi}\) at step \(0\) after the CCF occur is shown in Equation (13). \[Load_{I}(n_{Pi},0)\ =\ F_{I}(d_{I}) \tag{13}\]  
The expected load of the node \(n_{Pi}\) in the CN is shown in Equation (14). \[\langle load_{I}\rangle\ =\ \sum_{d_{I}=\ 1}^{\infty}F_{I}(d_{I})\times Prob_{I}( d_{I}) \tag{14}\]  
where \(Prob_{I}(d_{I})\) is the probability that the degree of the node \(n_{Pi}\) is \(d_{I}\). The threshold of a node \(n_{Pi}\) in the CN is equal to \(T_{IP}(n_{Pi}N)=\alpha_{I}\times F_{I}(d_{I})\). Where \(\alpha_{I}\) is a tolerance parameter of the node \(n_{Pi}\) and satisfies \(\alpha_{I}\geq 1\). The expected value of the threshold of the node \(n_{Pi}\) in the CN is shown in Equation (15). \[\langle Thres_{I}\rangle\ =\ \langle\alpha_{I}\rangle\times\sum_{d_{I}=\ 1}^{\infty}F_{I}(d_{I})\times Prob_{I}(d_{P}) \tag{15}\]  
Figure 4: The UCTE (Union for the Co-ordination of Transmission of Electricity) summer case data 2002 curve using MATCASC (cascading line outages analyse tool for power grids). It is assumed that the unintentional random initial failure rate of nodes in the PN and CN are \(\Theta_{P}\) and \(\Theta_{I}\) respectively. The expected proportion of the remaining normal operational nodes in the PN is \((1-\Theta_{P})\times f_{P}(1-\Theta_{P})\). When the node fails, the load of the original node will be redistributed to the remaining nodes in the PN, and the total redistributed load in the PN is \(NON_{P}\times(1-(1-\Theta_{P})\times f_{P}(1-\Theta_{P}))\times<load_{P}>\). Correspondingly, the total redistributed load (data packets) in the CN is \(NON_{I}\times(1-(1-\Theta_{I})\times f_{I}(1-\Theta_{I}))\times<load_{I}>\).