# Dynamic Load Redistribution of Power CPS Based on Comprehensive Index of Coupling Node Pairs
## 3 Power CPS Risk Propagation Considering Load Redistribution
### Subnetwork Load Reallocation
#### 3.2.1 Cyber Network Load Reallocation  
The best routing of the cyber network itself may not be optimal for the power system, so optimal routing in favor of the power system should be established to ensure the accessibility of critical information flows [22]:  
1. The first round of redistribution on the information side When multiple nodes fail at t-time, the nodes with relatively high importance in the front row are prioritized for load redistribution. The specific allocation steps are as follows: The first stage of load redistribution of a single failed node \(v_{i}\) is carried out, and the full load of the \(v_{i}\) node is distributed to all neighbor nodes. For the information-side failure node \(v_{i}\), the load is zeroed, that is, \(L_{i}=0\), the relevant row and column elements are zeroed in the coupling adjacency matrix, and the neighbor node set of \(v_{i}\) is collected and recorded while removing the line connected to the \(v_{i}\) node. For multiple neighbor nodes of node \(v_{i}\), their comprehensive vulnerability is sorted according to at the \(t+1\) moment, one of \(v_{i}\)'s neighbor nodes \(v_{j}\) will be allocated to a certain proportion of the load, then \(v_{i}\) reassigned to \(v_{j}\). The load increment is: \[L_{ij}(t+1)=L_{i}(t)\cdot\frac{\eta(v_{j})}{\sum\limits_{n\in\Gamma_{i}}\eta( v_{n})}\] (16) where \(L_{ij}(t+1)\) is the load reallocated to node \(j\) at the moment of \(t+1\), and \(L_{i}(t)\) is the load before node \(i\) fails; \(\eta(v_{j})\) is the comprehensive vulnerability index of node \(j\), and \(\frac{\eta(v_{j})}{\sum\limits_{n\in\Gamma_{i}}\eta(v_{n})}\) is the weight assigned to the load, following the node parameters after each round of redistribution; \(\Gamma_{i}\) is the set of neighbor nodes of node \(i\). If multiple neighbor nodes of node \(j\) fail, then all failed neighbor nodes may redistribute load to \(j\), and the load increment on node \(j\) at the \(t+1\) moment is: \[\Delta L_{j}(t+1)=\sum\limits_{i\in\Gamma_{j}}L_{ij}(t+1)\] (17) where \(\Gamma_{j}\) is the set of failed neighbor nodes for node \(j\). The total load on node \(j\) is: \[L_{j}(t+1)=\Delta L_{j}(t+1)+L_{j}(t)\] (18) where \(L_{j}(t)\) is the load of node \(j\) at \(t\)-time. 2. The second round of redistribution on the information side The first round of redistribution ends with updating the loads and metrics for all changed nodes. At this time, the status of the remaining nodes of the information network is divided into three cases: 1. Normal state, can operate normally; 2. In the overload state, but not exceeding the capacity, it can maintain normal operation in the short term, but it needs to be processed as soon as possible; 3. Exceeding the maximum capacity and being in a failed state requires a new round of load redistribution. No processing is required when the node is in the first case. Repeat the actions in the first round when the third case occurs. In the second case, the overload of the information node will cause the throughput of the node to mismatch, and the system will gradually become congested. During this time, the excess portion of the load can be redistributed again.  
The steps for the second round of load redistribution are:  
Step 1: Collect nodes that are in an overloaded state and newly failed with their load.  
Step 2: Redistribute the excess load of the overloaded nodes, and redistribute the entire load of the nodes that exceed the capacity according to the method of the first round.  
Step 3: Calculate the load amount of all nodes after this round of load redistribution, if the case of no node overload is satisfied (1), no operation is required, and the cascade failure ends; if (2) or (3), return to step 1.  
Constraints satisfied: \(L_{j}(t+1)\leq C_{cj}\). \(C_{cj}\) is the capacity of node \(j\). When \(L_{j}(t+1)>C_{cj}\), it means that the load of node \(j\) exceeds its capacity, and the cyber node \(j\) fails, causing a new load redistribution.