# Fault Detection Algorithms in Medium-Voltage Direct-Current (MVDC) Grids
## 3 Study Case and Results
### Link Faults
#### 3.2.2. Undervoltage Algorithm  
Figure 10a shows a pole-to-pole fault in the 0% of link L24. On the other hand, Figure 10b shows a pole-to-ground fault in the middle of link L34.  
High-resistance faults introduce an additional impedance into the fault circuit, which can reduce the flow of the fault current and slow down the rate at which the voltage decreases. This slower voltage decay results in a longer time for the voltage to reach the threshold required to trigger the undervoltage protection mechanisms.  
Conversely, pole-to-pole faults typically present a lower impedance path for the fault current, allowing a more rapid voltage drop and therefore a shorter time to trigger undervoltage protection. As a result of the analysis, a limitation of the undervoltage algorithm for high-resistance faults has been found. It is 8 \(\Omega\) for pole-to-pole faults and 3 \(\Omega\) for pole-to-ground faults. For higher fault resistances, the algorithm starts losing stability, as in Figure 10 and Tables 9 and 10, for 20 \(\Omega\) fault resistance. It starts detecting faster faults at the opposite end than at the end closer to the relay.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Fault at Link L24**} & \multicolumn{8}{c}{**Detection Time at R24 (ms)**} & \multicolumn{8}{c}{**Detection Time at R42 (ms)**} \\ \cline{2-11}  & \multicolumn{3}{c}{**P2P**} & \multicolumn{3}{c}{**P2G+**} & \multicolumn{3}{c}{**P2P**} & \multicolumn{3}{c}{**P2G+**} \\ \cline{2-11}  & **0\%** & **50\%** & **100\%** & **0\%** & **50\%** & **100\%** & **100\%** & **50\%** & **0\%** & **100\%** & **50\%** & **0\%** \\ \hline \multirow{3}{*}{OC} & R = 0.01 \(\Omega\) & 0.180 & 0.180 & 0.180 & 0.160 & 0.180 & 0.180 & 1.800 & 1.710 & 1.625 & 7.625 & 3.475 & 2.760 \\ \cline{2-11}  & R = 2 \(\Omega\) & 0.185 & 0.185 & 0.180 & 0.165 & 0.180 & 0.180 & 1.835 & 1.740 & 1.650 & 7.695 & 7.700 & 3.260 \\ \cline{2-11}  & R = 20 \(\Omega\) & 0.185 & 0.190 & 0.200 & 0.205 & 0.210 & 0.225 & 2.235 & 2.120 & 1.985 & 8.740 & 8.740 & 8.690 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Fault at link L24, detection times at R24 and R42, for the OC algorithm.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Fault at Link L34**} & \multicolumn{8}{c}{**Detection Time at R34 (ms)**} & \multicolumn{8}{c}{**Detection Time at R43 (ms)**} \\ \cline{2-11}  & \multicolumn{3}{c}{**P2P**} & \multicolumn{3}{c}{**P2G+**} & \multicolumn{3}{c}{**P2P**} & \multicolumn{3}{c}{**P2G+**} \\ \cline{2-11}  & **0\%** & **50\%** & **100\%** & **0\%** & **50\%** & **100\%** & **100\%** & **50\%** & **0\%** & **100\%** & **50\%** & **0\%** \\ \hline \multirow{3}{*}{OC} & R = 0.01 \(\Omega\) & 1.240 & 1.435 & 1.645 & 1.695 & 2.010 & 2.515 & 0.950 & 0.810 & 0.665 & 1.050 & 0.905 & 0.700 \\ \cline{2-11}  & R = 2 \(\Omega\) & 1.260 & 1.455 & 1.665 & 1.750 & 2.095 & 8.390 & 0.970 & 0.820 & 0.680 & 1.090 & 0.935 & 0.735 \\ \cline{2-11}  & R = 20 \(\Omega\) & 1.390 & 1.620 & 1.855 & 2.465 & - & - & 1.100 & 0.930 & 0.795 & 1.460 & 1.215 & 1.000 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Fault at link L34, detection times at R34 and R43, for the OC algorithm.  
Figure 9: DC currents: (**a**) pole-to-pole fault L34 link 50%, and (**b**) pole-to-ground fault L24 link 100%.