## 2 Definition of Vulnerability Indices Based on the Proposed Network  
The growing demands and the structural complexity of power systems cause increases in the risk of cascading failures and blackouts in the power system [35]. These issues highlight the need for assessing vulnerable points in order to reduce the threats to society due to the involved sequence of cascading failures. In other words, the power system vulnerability can be introduced as an efficient tool to determine and measure the weakness and incidence of the lines or the other sections related to the power system with regards to the cascading events. Eventually, this paper tries to address two effective approaches to prevent large-scale blackouts from both the technical and cyber-security points of view. In this regard, there needs to be an appropriate modeling method to get the vulnerability assessment of the power grid. To this end, the case of evaluating the vulnerable sections of a power grid needs to be defined as disturbance-based indices comprising of several basic parameters computed with regards to two cases: (1) normal condition (2) operation of the power system under a simulated disturbance. Any perturbation including the transmission line outage or generation unit outage might affect the other small/large-scale sections of the power grid. Hence, assume a studied power grid with L line, B bus, and G generation unit in the first place. Then, the unexpected effects arising from _kth_ transmission line and _gth_ generator outages on the main sections of the grid, that is, buses, lines, and generation units can be calculated as follows:  
* Vulnerability indices of Bus This part is aimed to represent the disturbance impacts of disconnecting k transmission lines on the grid buses by using the definition of vulnerability indices as shown in Equations (1) and (2): \[B_{b}^{k}=\left(\frac{1-TV_{min,b}}{1-TV_{b,k}}\right)^{2}\times\left(\frac{( 1-TV_{normal,b})-(1-TV_{b,k})}{(1-TV_{normal,b})-(1-TV_{min,b})}\right)^{2}\] (1)  
\[VIB_{b}^{k}=\left\{ \begin{array}{c}1\to B_{b}^{k}>B_{b}^{basic}\\ 0\to B_{b}^{k}<B_{b}^{basic}\end{array} \right. \tag{2}\]  
where \(B_{b}^{k}\) is defined as the \(k\) line outage index of bus \(b\) and \(VIB_{b}^{k}\) indicates the bus index for the sake of disconnecting \(k\) line. It is needed to say that the variable \(TV_{b}\) is considered as the voltage deviation and \(1-TV_{b}\) shows the voltage of bus \(b\). Based on Equation (1), in the case that the outage of \(k\) line occurs, the voltage of bus \(b\) will be affected and altered from the voltage level of normal to critical, which is shown by \(1-TV_{b}^{k}\). This leads to a growing trend in the \(k\) line index (\(B_{b}^{k}\)) up to a basic level (\(B_{b}^{basic}\)). Keeping this issue in mind, the \(b\) bus index (\(VIB_{b}^{k}\)) exposed to the \(k\) line disconnection will be equal to 1, if \(k\) line index overtakes from the threshold. On the contrary, the \(VIB_{b}^{k}=0\) means that the \(k\) line index is still less than the basic level. In other words, the \(VIB_{b}^{k}=1\) shows that the \(k\) line outage can affect the voltage deviation of bus \(b\) and vice versa. * Vulnerability indices of line The same process related to the vulnerability indices of buses can be developed to assess the vulnerability of the grid lines under \(k\) line outage as follows: \[L_{l}^{k}=\left(\frac{P_{l}^{k}}{P_{l}^{max}}\right)^{2}\times\left(\frac{P_{ l}^{k}-P_{normal,l}^{k}}{P_{l}^{max}-P_{l}^{k}}\right)^{2}\] (3)\[VIL_{l}^{k}=\left\{ \begin{array}{c}1\to L_{l}^{k}>L_{l}^{\text{basic}}\\ 0\to L_{l}^{k}<L_{l}^{\text{basic}}\end{array} \right. \tag{4}\]  
As mentioned before, a blackout over the power grid may be launched through the cascading events. For instance, in the case that the outage of \(k\) line has occurred, the power flow of the other lines of the grid might increase up to a level that the loading of lines overtakes the maximum limit.