# Monitoring System of Transmission Line in Mountainous Area Based on LPWAN
## 3 Model and Method
### Network Model Design  
In this section, the mathematical modeling of the transmission line architecture design is carried out, and a priority multi-objective optimization model of LoRa dynamic grouping is proposed. The optimal installation location and usage quantity of the cellular wireless transmission module under the specified delay conditions were found through the algorithm, and different networking modes can be realized under the constraint of different system energy consumption and delay transformation. In this paper, the network architecture diagram can be modeled as a data directed graph. The transmission tower near the substation can realize end-to-end data transmission directly through the LoRa transmission module. Therefore, the directional graph mainly considers the transmission tower that needs to transmit data through the cellular wireless transmission module. As shown in Figure 4, suppose that a transmission line contains \(N\) transmission line towers, where \(i\) represents the end-to-end data transmission link between transmission poles and towers, and \(j\) represents the data transmission link of towers through cellular wireless communication. Therefore, any data transmission link can be expressed as a vector (\(i\),\(j\)), and \(P\) is the set of all wireless transmission communication links, that is, all communication link vectors (\(i\),\(j\)) are included in \(P\), \(\forall(i,j)\in P\). The purpose of this model is to find a feasible communication link for each communication path used, minimize the delay caused by multiple links, and consider the energy constraint of each link. First, it is necessary to limit the delay generated in any communication link to be less than or equal to the maximum communication delay required, as shown in Equation (5):  
\[\sum_{(i,j)\in P}D_{i,j,k}M_{i,j,k}\leq D_{\max}\quad\forall k\in\mathrm{N} \tag{5}\]  
where \(D_{i,j,k}\) denote that the communication node \(k\) uses the data link (\(i\),\(j\)) as the generated delay; \(M_{i,j,k}\) are binary variables; if node \(k\) uses the data link (\(i\),\(j\)) as the communication link, \(M_{i,j,k}=1\); otherwise, it is 0; and Dmax represents the maximum transmission delay required by the system administrator. \[\sum_{i=1}^{N}\left(G_{i}-L_{i}\right)\leq 0 \tag{6}\]  
Equation (6) ensures that there are towers using cellular wireless transmission modules regardless of transmission mode. Where \(G_{i}\) and \(L_{i}\) are binary variables, if the \(i\) tower uses thee cellular wireless transmission module, then \(G_{i}=1\), otherwise it is 0; if tower \(i\) uses the LoRa transmission module, then \(L_{i}=1\), otherwise it is 0. \[M_{i,j,k}-O_{i,j}\leq 0\quad\forall(i,j)\in P,\quad\forall k\in N \tag{7}\]  
Figure 4: Network data transmission digraph. Equation (7) ensures that each communication link is reused and the link cost is calculated at most once, where \(O_{i,j}\) are binary variables. If the data transmission link (\(i\),\(j\)) is used, \(O_{i,j}=1\), otherwise it is 0. \[M_{i,j,k}\,L_{i}\,G_{i}\,O_{i,j}\in\{0\,1\}\quad\forall(i,j)\in P,\forall k\in N \tag{8}\]  
Equation (8) determines that the decision variables \(M_{i,j,k}\), \(G_{i}\), \(L_{i}\), and \(O_{i,j}\) are binary variables. The delay function of the network is composed of Equations (5)-(8). The binary variables in the equation are decision variables. \(O_{i,j}\) indicates that the link is used by at least one node. If node \(k\) chooses link (\(i\),\(j\)) as its transmission path, \(M_{i,j,k}=1\), and when there is a delay constraint, the frequency of the cellular wireless transmission access path \(j\) of transmission tower \(i\) is significantly higher than that of other transmission towers, then the cellular wireless transmission module must be installed on the tower.