# Fault Detection Algorithms in Medium-Voltage Direct-Current (MVDC) Grids
## 3 Study Case and Results
### Bus Faults
#### 3.1.5 Differential Algorithm  
During normal operating conditions, the differential current should be zero, as in Figure 7,b. In this case, as the simulated faults are located at bus 2, following Equation (8), the sum of the currents of relays R21 and R24 is compared with the total current of the bus. Figure 7 simulates a pole-to-pole fault. Figure 7 simulates a pole-to-ground fault. The sum of the currents of R21 and R24 and the total current of the bus are equal but with the opposite direction in a steady state. When the fault occurs for t = 6.5 s, this condition changes and a differential current appears. Table 5 shows the detection time of the algorithm for the different fault type and fault resistance values.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Fault at Bus 2**} & \multicolumn{3}{c}{**Detection Time at R21 (ms)**} & \multicolumn{3}{c}{**Detection Time at R24 (ms)**} \\ \cline{2-6}  & **P2P** & **P2G+** & **P2P** & **P2G+** \\ \hline \multirow{3}{*}{ROCOV} & R = 0.01 \(\Omega\) & 0.010 & 0.010 & 0.010 & 0.010 \\ \cline{2-6}  & R = 2 \(\Omega\) & 0.010 & 0.010 & 0.010 & 0.010 \\ \cline{1-1} \cline{2-6}  & R = 20 \(\Omega\) & 0.010 & 0.010 & 0.010 & 0.010 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Fault at bus 2, detection times at R21 and R24, for the ROCOV algorithm.  
Figure 6: Fault at bus 2, ROCOV at R21 and R24: (**a**) pole-to-pole fault, and (**b**) pole-to-ground fault.  
In Figure 7b, it can be observed that at one point the differential current changes its polarity. The reason why this occurs is that, when a pole-to-ground fault occurs, that part of the fault current circulates through the ground. As a result, the measured currents in the positive pole and the negative pole are not going to be equal.  
In this case, the fault is in the positive pole to ground, and the resulting differential current is negative. As a certain percentage of the fault current is circulating from the positive pole to ground, the remaining current in the positive pole will have a lower value than the current in the negative pole. For this scenario, the predefined threshold value is \(\text{THR}_{\text{DIFF}}=10\) A, in absolute values. The 10 A threshold for differential current protection is chosen based on the system's operating current range and fault detection requirements. It ensures sensitivity to internal faults while minimizing false alarms from normal operational fluctuations. This threshold is high enough to avoid responding to transient current variations but low enough to detect a wide range of faults. Additionally, it aligns with the expected fault characteristics, allowing for effective fault isolation without causing false fault detections. This threshold strikes a balance between fast fault detection and system reliability.  
The differential algorithm fulfills the protection requirements for both pole-to-pole and pole-to-ground faults. During bus faults, the differential algorithm has no time delay, because there is no distance between the measurement relays and the fault.