# A Heuristic Algorithm for Locating Line-to-Line Faults in Photovoltaic Systems
## 3 Proposed Method
### Heuristic Algorithm for Line-to-Line Fault Location
#### 3.2.2 Fault Location Determination
Case 3: Multiple module groups experience intra-string short-circuit faults, as shown in Figure 10. The fault occurs between module groups \(V_{1,6}\) and \(V_{1,7}\). In accordance with the \begin{table}
\begin{tabular}{c c c c} \hline  & & **Stage 1** & \\ \hline String \# & \(k_{i}\) & \(d_{i}\) & \(L_{i}\) \\ \(i=1\) & 2 & \(-1\) & 2, 6 \\ \(i=2\) & 0 & 0 & null \\ \hline \multicolumn{4}{c}{**Stage 2**} \\ \hline String \# & \(k_{i}\) & \(d_{i}\) & \(L_{i}\) \\ \(i=1\) & 2 & \(-1\) & 2, 6 \\ \(i=2\) & 0 & 0 & null \\ \hline \multicolumn{4}{c}{**Modified**} \\ \hline String \# & \(k_{i}\) & \(d_{i}\) & \(L_{i}\) \\ \(i=1\) & 2 & \(-1\) & 1, 7 \\ \(i=2\) & 0 & 0 & null \\ \hline \end{tabular}
\end{table} 
Table 3: Case 2 fault location process. Figure 9: Case 2 fault. In accordance with the fault marking process proposed in this paper, correctly marking the fault location requires knowledge of the relative magnitudes between A and B, as well as between C and D. This relationship can be determined through comparison. Using the fault structure in Figure 11 as an example, the array currents are labeled as shown in Figure 12.  
In accordance with [PERSON]'s Voltage Law, the sum of voltages in any closed loop is zero, resulting in the following equations:  
\[\left\{ \begin{array}{l}8\times A=11\times C\\ 13\times B=10\times D\end{array} \right. \tag{4}\]  
In accordance with the principle of proportionality, the following relationships are established:  
\[\left\{ \begin{array}{l}A>C\\ B<D\end{array} \right. \tag{5}\]  
Based on the I-V characteristic curve of the PV module, it is understood that higher voltage corresponds to lower current, so:  
\[\left\{ \begin{array}{l}I_{A}<I_{C}\\ I_{B}>I_{D}\end{array} \right. \tag{6}\]  
In accordance with [PERSON]'s Current Law, the current flowing into any node is always equal to the current flowing out, implying that \(I_{fault}\) must be greater than 0. Therefore:  
\[\left\{ \begin{array}{l}I_{A}<I_{B}\\ I_{C}>I_{D}\end{array} \right. \tag{7}\]  
This inference is also confirmed by simulation, as shown in Figure 13. In Figure 13a, the simulation conditions are set to normal operating cell temperature (NOCT), with irradiance G at 800 W/m\({}^{2}\) and module temperature T at 45 \({}^{\circ}\)C. When the array reaches maximum power, the currents in Figure 13b exhibit the relationships described in Equations (6) and (7). Although \(I_{fault}\) approaches 0 A at lower voltages, a closer inspection reveals that it remains greater than 0 A at all times. Then, based on the I-V characteristic curve of the PV module, it can be concluded that:  
\[\left\{ \begin{array}{l}A>B\\ C<D\end{array} \right. \tag{8}\]  
With the known voltage relationships between A and B, and C and D, the fault marking process is applied to the fault structure in Figure 11. In String 1, the fault points 7, 3, and 4 are sequentially marked, with the final recorded voltage direction as 1. In String 2, the fault  
Figure 12: Cross-string fault current. points 1, 3, and 4 are marked in sequence, with the final recorded voltage direction as \(-1\). The results are shown in Table 5, Stage 1. At the beginning of the fault determination process, the marked results are organized and sorted, as shown in Table 5, Stage 2. At this point, \(d_{1}=1\) and \(d_{2}=-1\), so s is set to 1. Upon inspection, the last marked point in the low-potential string \(L_{2}\) is not 6, and the first marked point in the high-potential string \(L_{1}\) is not 2, indicating that the fault is located in the middle of both strings. Therefore, the fault location is identified as the first fault point in the high-potential string and the last fault point in the low-potential string. In this case, the fault is determined to be a short circuit between module group 3 in String 1 and module group 4 in String 2. Case 5: A cross-string short-circuit fault occurs between two strings, with the low-potential end located in the last module group of the string, as shown in Figure 14. The fault occurs between module groups \(V_{1,6}\) and \(V_{2,7}\).