## 3 Identification of Critical Nodes base on Controllability Theories of Complex Networks  
### Controllability Index  
For a power system, let \(G\) be a directed network as defined in the previous section with \(N\) nodes. Based on the controllability of complex networks, the controllability index can be defined as follows:  
\[\lambda_{i}=\left(N^{D}_{i}-N^{D}_{orig}\right)+\frac{\sum_{j\in\Omega}\left( N^{D}_{j}-N^{D}_{orig}\right)}{K_{i}} \tag{2}\]  
where \(N^{D}_{orig}\) is the minimum number of driver nodes of the original network, \(N^{D}_{i}\) is the minimum number of driver nodes of the network after the node \(i\) is in fault, \(\Omega\) is adjacency nodes of node \(i\) and \(K_{i}\) is the degree of node \(i\), \(N^{D}_{j}\) is the minimum number of driver nodes of the network after the node \(j\) is in fault.  
The physical significance of expression (2) is as follows: a node may be considered critical if when it is in fault there needs to be an increase in \(N_{D}\) to maintain full control of the power system, which can be quantified using \(N^{D}_{j}-N^{D}_{orig}\). Furthermore, if the small numbers of nodes which are considered of importance in the system fault simultaneously, then it can create two serious problems. One is that the minimum number of driver nodes can rapidly increase, which means the power systems need to increase additional input signals to maintain full control of the power networks. The other one is that the driver nodes' distribution may be changed greatly, which means it leads to a significant driver nodes shift. Thus, the difficulty of fully controlling the power systems will be increased by these two aspects. They may lead to further failures and eventual cascades if the power systems cannot be readjusted in time. Although the power systems can be readjusted, it is very difficult to control the power systems as well. However, the importance of every node is not only related to itself, but also can be related to its adjacency nodes. For example, whether a node is a driver node can be changed if one of its adjacency nodes is in fault. In this paper, the mean value of \(N^{D}_{j}-N^{D}_{orig}\) is used to quantify this idea. In addition, \(K_{i}\) can reflect the nodes' structural characteristics of the network.  
Equation (2) can be normalized for its relative values by dividing the maximum values. Then the controllability index can be further defined as:  
\[\overline{\lambda}_{i}=\frac{\lambda_{i}}{\max\lambda} \tag{3}\]The nodes will be ranked with the controllability index after calculating Equation (3), and the nodes are regarded as critical if they have higher values of the controllability index.  
Remark 1: It is well known that the state of power systems can be restricted into a certain range by certain control measures, if it is controllable. Therefore, when faults happen, if a power system is controllable, the state of the power system can be restricted into permitted range by adding external control inputs, so that the blackout risk can be effectively mitigated. Conversely, if a power system is uncontrollable, some serious problems may also happen, e.g., voltage collapse [19,20]. Therefore, whether the power system is controllable or not can have an important influence on the security of the power system. Furthermore, [PERSON]'s structural controllability theorem verified that a physical system is controllable if it is structurally controllable [26]. Thus, the concept of structural controllability analysis can also be used to define the security of the power system.