## 4 Graph Theoretic Resiliency Framework  
### Preliminaries of the Resiliency Estimation Framework  
The \(PDR\) can be calculated at the PDC corresponding to each PMU. Based on the \(PDR\), the overall performance of the SCN can be evaluated. This is because \(PDR=1\) for the normal state, \(PDR\) is less than one for a partially failed state, and \(PDR=0\) for the fully failed state. Thus, we considered the \(PDR\) as a Figure of Merit (FoM) for resiliency estimation. Moreover, the average \(PDR\) value was considered for the overall system performance, as more than one PMU communicates synchrophasor data to a particular PDC. Thus, the average \(PDR\) at the PDC due to the corresponding PMUs can be determined as follows. If there are \(n\) PMUs connected to \(j^{th}\) PDC (say, \(PDC_{j}\)), then the average \(PDR\) at \(PDC_{j}\) is:  
\[PDR_{PDC_{j}}=\frac{\sum\limits_{i=1}^{n^{\prime}\leq n}PDR_{i}}{n^{\prime}} \tag{2}\]  
A resiliency curve using the average \(PDR\) as a figure of merit is shown in Figure 4 for different instances of time. As shown in Figure 4, the SCN is said to be in a normal state when all of its components work normally, resulting in an average \(PDR\) of 1. This can be observed up to time \(t_{1}\). Due to some external/internal factors, some components may fail, which will decrease the average \(PDR\), as shown beyond time \(t_{1}\). An SCN can tolerate a certain level of reduction in \(PDR\), but in order to perform satisfactorily, it must maintain a minimum value of \(PDR\), which is referred to as the _threshold_ value denoted as \(PDR_{th}\). Even during the interval \(t_{2}-t_{4}\) the \(PDR\) remains equal to or greater than the threshold value and, thus, the system performs satisfactorily. However, if the \(PDR\) becomes less than the threshold value, then the system disrupts and it enters the out-of-service state, i.e., the failed state. This is represented by the decaying exponential (in red colour) beyond time \(t_{2}\) in Figure 4. When the failed components of the system come back to the normal state, then the average \(PDR\) will improve, as depicted during \(t_{4}-t_{5}\). For time \(t>t_{5}\), all components recover to their original working state; thus, \(PDR\) again reaches the unity value. The overall operation of the SCN can be studied through the following phases: (1) degradation phase, (2) threshold-operated phase, and (3) recovery phase. 1. [label=(0)]
2. _Degradation phase:_ The phase during which the components of the SCN start failing partially, resulting in the degradation of the \(PDR\). However, the SCN is considered to still be operational, since \(PDR\geq PDR_{th}\). The degradation rate (\(\xi_{\text{deg}}\)) can be defined as \[\xi_{\text{deg}}=\frac{PDR_{t_{1}}-PDR_{t_{2}}}{t_{1}-t_{2}}\] (3)
3. _Degradation phase:_ The \(PDR\) is defined as \[\xi_{\text{deg}}=\frac{PDR_  
where \(PDR_{t_{1}}\) and \(PDR_{t_{2}}\) are the \(PDR\) received at times \(t_{1}\) and \(t_{2}\) respectively. Thus, if a node degrades at a rate of 0.075 s\({}^{-1}\), then it takes 13.34 s to degrade the \(PDR\) from 1 to 0.
2. _Threshold-operated phase:_ This is the state corresponding to the \(PDR\) value being close to the threshold value \(PDR=PDR_{th}\). During this phase, the SCN operates with minimum performance, since some packets may be lost.
3. _Recovery phase:_ When a partially failed component recovers from the partially failed state to the normal state, or even when a failed component is substituted by normal components, the \(PDR\) value increases such that \(\max\{PDR\}=1\) and \(PDR>PDR_{th}\). The recovery rate \((\xi_{rec})\) can be determined during the recovery phase \(t_{4}\) to \(t_{5}\) using Equation (4), such that \(\{4_{t},t_{5}\}\) \(\in\) recovery phase. \[\xi_{rec}=\frac{PDR_{t_{4}}-PDR_{t_{5}}}{t_{4}-t_{5}}\] (4) where \(PDR_{t_{4}}\) and \(PDR_{t_{5}}\) are the \(PDR\) received at times \(t_{4}\) and \(t_{5}\), respectively. Thus, if a node recovers at a rate of 0.075 s\({}^{-1}\), then it takes 13.34 s to restore the \(PDR\) from 0 to 1.