## 2 Definition of Vulnerability Indices Based on the Proposed Network
For this reason, the security control system of the power grid has to make the needed edicts to disconnect the other lines, getting to a steady-state operation. In this regard, vulnerability assessment is provided by Equation (3) in which \(L_{l}^{k}\) is defined as the \(k\) line outage index of the grid lines formulated based on the normal and fault conditions. Focusing on Equation (4), the \(VIL_{l}^{k}=1\) means that the \(k\) line outage will affect the power flow of line \(l\) and vice versa. * Vulnerability indices of generator  
It can be inferred that the generation units of the grid may be affected when a \(k\) line outage happens. Hence, there needs to be defined the index of \(g\) generator (\(VIG_{g},VIQ_{g}\)) as the disturbance effects arising from disconnecting \(k\) line. It is worth mentioning that the index of \(g\) generator comprises of active and reactive power indexes (\(VIG_{g}\), \(VIQ_{g}\)) depending on the generated active/reactive powers pertaining to the \(g\) generator. Accordingly, Equations (5)-(8) indicate the formulation related to the indexes of \(g\) generator, which are dependent on the k line outage index of g generator (\(Q_{g}^{k},G_{g}^{k}\)). It is important to say that the argument mentioned in previous sections about vulnerability indices can be developed for finding the critical points of the generation units. \[G_{g}^{k}=\left(\frac{P_{g}^{k}-P_{normal,g}^{k}}{P_{S}^{max}-P_{g}^{min}} \right)^{2}\times max\left[\left(\frac{P_{g}^{min}}{P_{g}^{k}}\right)^{2}, \left(\frac{P_{g}^{k}}{P_{S}^{max}}\right)^{2}\right] \tag{5}\]  
\[Q_{g}^{k}=\left(\frac{Q_{g}^{k}-Q_{normal,g}^{k}}{Q_{g}^{max}-Q_{g}^{min}} \right)^{2}\times max\left[\left(\frac{Q_{g}^{min}}{Q_{g}^{k}}\right)^{2}, \left(\frac{Q_{g}^{k}}{Q_{g}^{max}}\right)^{2}\right] \tag{6}\]  
\[VIG_{l}^{k}=\left\{ \begin{array}{c}1\to G_{g}^{k}>G_{g}^{basic}\\ 0\to G_{g}^{k}<G_{g}^{basic}\end{array} \right. \tag{7}\]  
\[VIQ_{l}^{k}=\left\{ \begin{array}{c}1\to Q_{g}^{k}>Q_{g}^{basic}\\ 0\to Q_{g}^{k}<Q_{g}^{basic}\end{array} \right. \tag{8}\]  
As shown in (9), \(\overset{\rightarrow}{LO}\) is defined as a logical matrix including the indexes of generators and lines with a domination of L-to-R (B + L + G). The row/column subscripts of \(\overset{\rightarrow}{LO}\) are assigned to the grid assets, that is, bus, line, and generators and the disconnected lines, respectively. \[\overset{\rightarrow}{LO}=\left[ \begin{array}{c}\overset{\rightarrow}{VIB }\\ \overset{\rightarrow}{VIL}\\ \overset{\rightarrow}{VIQ,VIG}\end{array} \right] \tag{9}\]  
Similar to the process related to line outage, the generator outages can be simulated instead of the line outage and the vulnerability indices pertaining to the bus, line, and gen eration unit indexes are calculated. By doing so, the vulnerability matrix arising from the \(g\) generator outage index is provided by Equation (10). \[\overset{\rightarrow}{G\O}=\ \ \left[ \begin{array}{c}\overset{g:1,2,.....,G}{ \longmapsto}\\ \overset{\rightarrow}{VIB}\\ \overset{\rightarrow}{VIL}\\ \overset{\rightarrow}{VIQ,\overset{\rightarrow}{VIG}}\end{array} \right] \tag{10}\]  
The network line outage index (\(NLO\)) can be computed as the ratio of disturbance effects caused by launching the line outage as indicated in Equation (11) in which parameter \(R\) is comprised of elements from a number of networks such as bus, line, and generation unit (\(R=L+B+G\)). In the same way, the network generator outage index (\(NGO\)) is formulated by (12), each element of which expresses the vulnerability value considering the g generator outage. \[\overset{\rightarrow}{NLO}=\frac{\overset{\rightarrow}{LO}}{R} \tag{11}\]  
\[\overset{\rightarrow}{NGO}=\frac{\overset{\rightarrow}{GO}}{R} \tag{12}\]  
Finally, the objective function of the vulnerability index of the power grid comprises both the network line/generator outage indexes. Hence, Equation (13) is provided as the main goal of this paper which should be minimized. \[NV=\sum_{R}NLO+NGO \tag{13}\]