## 3 Problem Formulation
Then the total distance of the faulted section needed to be patrolled is described as \(\sum\limits_{a=1}^{sl}l_{a}\). Therefore, the total interruption cost (\(IC\)) can be expressed by  
\[IC=\sum\limits_{l=1}^{n}\left(T_{seek\text{-}l}+T_{resect}\right)\cdot\lambda \cdot Price\cdot(Load_{l}+\sum\limits_{k=l+1}^{n}Load_{k})] \tag{6}\]  
In addition, in order to consider the influence of communication attenuation, the probability of error in information transmission (\(p\)) is added, which can be expressed by  
\[p=\sqrt{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j}\right)^{2}}\ /d \tag{7}\]  
where \(\left(x_{i},y_{i}\right)\) is the position of the acquisition unit, \(\left(x_{j},y_{j}\right)\) is the position of the collection unit, \(d\) is effective communication distance of LoRa. As we all know, if the distance between the acquisition unit and the collection unit is farther, the probability of transmission error will be higher. In fact, there is a nonlinear relationship between the probability of error and distance. However, it is difficult for us to obtain the optimal configuration results under the complex nonlinear function. Therefore, we simplify it to a linear model in our proposed model, which is within a distance of 10 km. We also study the impact of the relationship between probability of error and distance on the optimal configuration scheme in 4.4, which shows that the simplified linear relationship between probability of error and distance is appropriate to be used in the optimal configuration scheme of fault indicators. Equation (7) is obviously a function of the position of the acquisition unit and the collection unit. Therefore, the optimal placement of the acquisition unit and the collection unit helps to keep the communication quality of the collection unit above a certain predetermined level. Moreover, the interruption cost will be influenced by changing the value of \(p\). If the distance is too long, there will be an error in communication, and the fault indicator will not work properly. As a result, we need to expand the domain of patrolling if we want to locate the fault accurately. Therefore, the length of line that needed to be patrolled will increase and the probability of its increase is \(p\)We assume that the original length of line that we need to patrol is \(l_{original-a}\), and the increased length of line that we need to patrol is \(l_{addition-a}\), so the interruption cost will be changed to be expressed by  
\[ \begin{array}{l}IC=\sum\limits_{l=1}^{n}\left[(\sum\limits_{a=1}^{sl}\frac{l_ {original-a}}{V}+T_{recect})\cdot\lambda\cdot Price\cdot(Load_{l}+\sum\limits_{k= l+1}^{n}Load_{k})\right.\\ \left.+(p\cdot\sum\limits_{a=1}^{sl}\frac{l_{addition-a}}{V}+T_{recect})\cdot \lambda\cdot Price\cdot(Load_{l}+\sum\limits_{k=l+1}^{n}Load_{k})\right]\end{array}  \tag{8}\]  
In order to satisfy the constraints, we assume that \(l_{original-a}\approx l_{addition-a}=l_{a}\), so the interruption cost will be expressed by  
\[IC=\sum\limits_{l=1}^{n}\left[(\sum\limits_{a=1}^{sl}\frac{l_{a}}{V}\cdot(1+p) +T_{recect})\cdot\lambda\cdot Price\cdot(Load_{l}+\sum\limits_{k=l+1}^{n}Load_ {k})\right] \tag{9}\]  
According to Equation (5), the interruption cost will be expressed by  
\[IC=\sum\limits_{l=1}^{n}\left[(T_{\text{seek-}l}\cdot(1+p)+T_{recect})\cdot \lambda\cdot Price\cdot(Load_{l}+\sum\limits_{k=l+1}^{n}Load_{k})\right] \tag{10}\]  
The second part of the objective function is the cost of fault indicator based on LoRa and NB-IoT, which includes the cost of acquisition unit and collection unit.