# BPDM-GCN: Backup Path Design Method Based on Graph Convolutional Neural Network
## 3 Construction of Communication Network Model for WAMS  
### Communication Network System Model Construction  
In this study, an undirected graph \(G=(V,E)\) is employed to represent the communication network topology of the wide-area measurement system. In the context of network topology, the symbol \(V\) represents the set of switch nodes, \(V=\{v_{i}|i\in(1,\cdots,n)\}\), \(n\in N^{*}\). \(E\) represents the set of links between switch nodes in the network topology, \(E=\{e_{1,2},e_{1,3}\cdots e_{i,j}\},(i,j)\in N^{*}\). The link between switch nodes \(v_{i}\) and \(v_{j}\) is denoted as \(e_{i,j}\), and \(w_{e_{i,j}}\) represents the link weight of link \(e_{i,j}\). Define the switch connected to the PDC in the network topology as the destination node \(d\), the remaining nodes as the source node \(d\), and the phasor data transmission path from the source node \(d\) to the destination node \(d\) as \(path_{s,d}\).  
In compliance with the stringent maximum delay specifications of the wide-area measurement system communication network, it is imperative to uphold the service quality of the wide-area measurement system communication. This paper delineates the phase transmission delay within the communication network of the wide-area measurement system into two components, denoted as \(delay(path_{s,d})\) and \(delay(v_{i})\). In this context, \(delay(path_{s,d})\) represents the sum of all link delays \(delay(e_{i,j})\) in the phasor data transmission path \(path_{s,d}\). \(delay(v_{i})\) represents the delay consumed by the phasor data in the switch forwarding. The calculation process of the data transmission delay \(Delay(path_{s,d})\) between the phasor data from the source node \(s\) to the destination node \(d\) in the network topology is depicted in Equation (1).  
\[Delay(path_{s,d})=\sum_{e_{i,j}\in path_{s,d}}delay(e_{i,j})+\sum_{i\in path _{i,j}}delay(v_{i}) \tag{1}\]  
For the sake of argument, let us assume that link \(e_{i,j}\) is connected to port \(n\) of switch \(v_{j}\) through port \(m\) of switch \(v_{i}\). The bandwidth utilization rate \(Occupancy\big{(}e_{i,j}\big{)}\) of link \(e_{i,j}\) is determined by the number of bytes forwarded by port \(m\) of switch \(v_{i}\) and port \(n\) of switch \(v_{j}\) in a specified time period \(T\). The calculation process of bandwidth occupancy rate A is illustrated in Equation (2).  
\[Occupancy\big{(}e_{i,j}\big{)}=\frac{b_{i,m}(t)-b_{i,m}(t-T)+b_{j,n}(t)-b_{j,n }(t-T)}{T*Bandwith\big{(}e_{i,j}\big{)}} \tag{2}\]In Equation (2), \(b_{i,m}(t)\) and \(b_{i,m}(t-T)\) represent the number of bytes forwarded by port \(m\)of switch \(v_{i}\) at time \(t\) and time \(t-T\), respectively. The value of \(Bandwith\big{(}e_{i,j}\big{)}\) denotes the total bandwidth of link \(e_{i,j}\).  
In the communication network topology of the WAMS, there exist multiple paths for phasor data transmission between the source node \(s\) and the destination node \(d\). The bandwidth occupancy \(Occupancy(path_{s,d})\) of the phasor transmission path is contingent upon the bandwidth occupancy \(\max\big{[}Occupancy\big{(}e_{i,j}\big{)}\big{]}\) of link \(e_{i,j}\), as illustrated in Equation (3).  
\[Occupancy(path_{s,d})=\big{\{}\max\big{[}Occupancy\big{(}e_{i,j}\big{)} \big{]}\big{|}e_{i,j}\in path_{s,d}\big{\}} \tag{3}\]  
Due to the fact that the performance parameters such as data transmission delay \(Delay(path_{s,d})\), bandwidth utilization rate \(Occupancy(path_{s,d})\), and packet loss rate \(Loss(path_{s,d})\) associated with the phasor data transmission path \(path_{s,d}\) in the network topology have different units of measurement, it is necessary to establish a unified metric standard for parameter comparison. To this end, a standardization procedure is applied to the data transmission delay \(Delay(path_{s,d})\) by performing a min-max normalization as described in Equation (4).  
\[Delay^{\prime}(path_{s,d})=\frac{Delay(path_{s,d})-\min[Delay(path_{s,d})]}{ \max[Delay(path_{s,d})]-\min[Delay(path_{s,d})]} \tag{4}\]