# Dynamic Load Redistribution of Power CPS Based on Comprehensive Index of Coupling Node Pairs
## 2 The Comprehensive Characteristics of the Coupling Node Pair
### Node Pair Utilization Index  
Coupled networks in power systems have proven to be more vulnerable than single-layer networks [14].  
As shown in Figure 2, on the power side, the ratio of the power transmitted on the line to the upper limit of its capacity is defined as the utilization rate of the transmission line [15]. The power nodes are connected by transmission lines, so the node utilization rate is defined, which is the ratio of the power flowing through the node to the capacity of the node. The higher the utilization, the easier it is for the node to approach the limit value when the load increases. Therefore, the power node utilization rate can be defined as:  
\[U_{pi}=\frac{\sum{(\sqrt{PF_{li}^{2}+QF_{li}^{2}}+\sqrt{PT_{li}^{2}+QT_{li}^{2} })}}{2C_{pi}} \tag{4}\]  
where \(li\) is the adjacent line of node \(i\). \(PF_{li}\) and \(QF_{li}\) are the active and reactive power at the outflow end of line \(li\), respectively. \(PT_{li}\) and \(QT_{li}\) are the active and reactive power at the receiving end of the line \(li\), respectively.  
For a power-side node, the one delivering power to the node is called the parent node, and the one receiving power from the node is called the child node, which is shown in Figure 3 below.  
Figure 3: Parent and child nodes of a power node.  
Figure 2: Utilization model for coupled node pairs.  
The more children nodes a node has, the more risk will be passed on to that node, making it more vulnerable, and conversely, the more parent nodes it has, the more risk it can be to help share. Therefore, the vulnerability indicator of the power side can be defined as:  
\[\eta_{pi}=U_{pi}\cdot e^{k_{io}-k_{ii}} \tag{5}\]  
The cyber side adopts a variety of technologies for defense, such as firewalls, identity authentication, VPN, intrusion detection, isolation, encryption, and other means [16]. Therefore, the threat degree formula for the cyber-side coupling node can be defined as follows:  
\[\eta_{ci}=f_{util}I_{dis}k_{i}\cdot\frac{C_{i}}{C_{ci}} \tag{6}\]  
Among them, \(f_{util}\) represents the vulnerability utilization of node \(i\) on the cyber side; \(I_{dis}\) indicates the degree of vulnerability distribution; \(k_{i}\) is the degree of node \(i\). The security zone where the vulnerability is located is used as the main basis to assess the vulnerability utilization and vulnerability distribution impact degree. After a comprehensive statistical measurement [16]. it can be seen that the threatened degree \(f_{util}\cdot I_{dis}\) of vulnerabilities in each security zone is shown in Table 1:  
Therefore, the comprehensive vulnerability indexes of the coupling node pair based on utilization can be established as follows:  
\[\eta_{vi}=\rho(\gamma_{cp}\eta^{\prime}_{pi}+(1-\gamma_{cp})\eta^{\prime}_{ci}) \tag{7}\]  
Further, \(\rho\) is the probability of inter-network failure (\(0<\rho<1\)), characterizing the odds of risk spreading across space, and \(\gamma_{cp}\) is the vulnerability weight coefficient, and \(\eta^{\prime}_{pi}\) and \(\eta^{\prime}_{ci}\) are normalized subnet node vulnerability metrics, respectively.