## 3 Problem Formulation  
The aim of our work is to present an optimization framework for the optimal number and position of the IoT-based fault indicator in power distribution systems in rural areas. The objective function of this optimization problem is given by  
\[TC=IC+FIC \tag{1}\]  
where \(TC\) represents the total cost, which is composed of two main parts: interruption cost (\(IC\)) and cost for installing and maintaining the fault indicator (\(FIC\)). Therefore, the goal of fault indicator optimization is to minimize the objective function. The interruption cost (\(IC\)) refers to the economic loss caused by the power-supply interruption once the fault occurs. The power loads connected to the faulted section and the downstream sound sections would lose their power supply during the fault seeking and fault resection process. Hence, the \(IC\) consists of \(IC_{local}\) and \(IC_{others}\). \(IC_{local}\) represents the economic loss caused by the load interruption connected to the faulted section. \(IC_{others}\) represents the economic loss caused by the load interruption connected to the downstream sound sections. Thus, the \(IC\) is given by  
\[IC=IC_{local}+IC_{others} \tag{2}\]  
In addition, the \(IC_{local}\) caused by the load interruption connected to the faulted section is given by  
\[IC_{local}=\sum_{l=1}^{n}\left[\left(T_{s\text{e}\text{s}\text{e}\text{-}l}+T _{r\text{e}\text{s}\text{e}\text{-}l}\right)\cdot\lambda\cdot\text{Price} \cdot Load_{l}\right] \tag{3}\]  
where \(T_{s\text{e}\text{-}l}\) is the time for fault seeking of faulted section \(l\), \(T_{r\text{e}\text{s}\text{e}\text{-}t}\) is the time for the fault resection, \(\lambda\) is failure probability of distribution line consisted of all sections, \(Price\) is electricity price per kWh, \(Load_{l}\) represents the amounts of loads connected to the faulted section \(l\), \(n\) represents the number of sections in the distribution lines. For the faulted section \(l\), the \(IC_{others}\) caused by the load interruption connected to the other downstream sound sections is given by  
\[IC_{others}=\sum_{k=l+1}^{n}\left[\left(T_{s\text{e}\text{s}\text{e}\text{-}l }+T_{r\text{e}\text{s}\text{e}\text{-}l}\right)\cdot\lambda\cdot\text{Price} \cdot Load_{k}\right] \tag{4}\]  
where \(Load_{k}\) represents the amounts of loads connected to the downstream sound sections, \(k=l+1,l+2\), \(\cdots,n\). Among them, \(T_{resect}\), \(\lambda\), _Price_, and the amount of power loads connected to each section are all fixed value, \(T_{seek\text{-}l}\) is a variable value, which can be calculated by Equation (5). \[T_{seek\text{-}l}=\frac{1}{V}\cdot\sum_{a=1}^{sl}l_{a} \tag{5}\]  
where \(V\) represents the speed of patrolling the line, \(l_{a}\) is the length of subsection \(a\) in the faulted section \(l\), \(sl\) represents the total number of the subsections in the faulted section \(l\). Hence, \(\sum\limits_{a=1}^{sl}l_{a}\) is the total length of all subsections in the faulted section \(l\). It is calculated from the upstream subsection where the last fault indicator detects an over-current status to the downstream subsection where the first fault indicator detects an under-current status and includes the subsections where no fault indicator is installed between them. It should be noted that the fault may occur at any point in the faulted section. We should patrol the faulted section from the beginning to the end, which is identified by two adjacent fault indicators. When the fault point is further located, we do not need to patrol the remaining subsections in the faulted section. Therefore, the actual distance of the section being patrolled will be shorter than the whole length of the faulted section. However, we cannot predict the particular fault position in the faulted section. Hence, we can only use the whole length of all subsections in the faulted section to calculate the fault seeking time. In this way, we get the maximum distance of each subsection in the faulted section needed to be patrolled, which is described as \(l_{a}\).