## 4 Results and Discussion  
The threshold current or peak make current and the short circuit breaking current can be calculated by finding the RMS symmetrical current \(i^{\prime\prime}k\); the peak make current for a single radial current is calculated using the peak factor \(k\). \[i_{p}=k\sqrt{2i^{\prime\prime}k}, \tag{7}\]  
where \(k=1.02+0.98x10^{-3}\left(\frac{X}{R}\right)\). \[Peakfactor=\sqrt{2}\left[1+sin\left\{tan^{-1}(\frac{X}{R})\right\}exp\left\{ \frac{-[\frac{\pi}{2}+tan^{-1}(\frac{X}{R})]}{\frac{X}{R}}\right\}\right], \tag{8}\]  
where the \(\frac{X}{R}\) is the system ratio at the fault point. The RMS current is given by \(I_{r}ms\) total at \(\frac{1}{2}\) cycle (KA),  
\[I_{r}ms=\sqrt{I+2 exp\left[-\frac{\pi}{2}\right]}XI_{r}ms, \tag{9}\]  
at \(\frac{1}{2}\) cycle(KA) and \(\frac{X}{R}=\left(\frac{XP}{R_{r}^{2}}\right)\times\frac{f}{f_{c}}\) for single phase to Earth faults. \(f\) is the normal frequency and \(f_{c}\) is the equivalent frequency. The model generated the threshold current when the fault resistance was set at 0.01 \(\Omega\), 50 \(\Omega\) and 100 \(\Omega\). The response times of the three different relays were also analysed, as seen in Tables 4-6. Figure 2: Instantaneous overcurrent relay block model. \begin{table}
\begin{tabular}{c c c} \hline
**Curve Types** & **K** & \(\kappa\) \\ \hline Normal Inverse Curve & 0.140 & 0.020 \\ \hline Very Inverse Curve & 13.5 & 1 \\ \hline Extremely Inverse Curve & 80 & 2 \\ \hline Long-time Standard Curve & 120 & 1 \\ \hline \end{tabular}
\end{table} 
Table 3: IDMT curve constant. The tripping time varied as the fault resistance changed from 0.01 \(\Omega\) to 100 \(\Omega\) at different fault conditions. In Table 4, when the fault resistance was at 0.01 \(\Omega\), the trip time was 0.01 s at the three relays and all the fault conditions. At the same time, it was quite different when the fault resistance changed to 50 \(\Omega\). The trip time was reduced to zero seconds at a double line to ground and a single line to ground fault at relays 1 and 3 with a variation in relay 2 of about 0.91 s as seen in Table 5. The time multiplier setting (TMS) for each of the relays was set at 1 s, 2 s and 5 s, and the error after tripping was about 0.9 s, which is minimal, as seen in Table 6 (0.98 s, 0.89 s and 0.96 s) for the three phases to ground fault. This can be seen in Figure3 A-C, where the TMS was set to 5 s, 2 s and 1 s, respectively. A backup relay was added to the network for optimal system performance to prevent feedback faults. In instances where relay 1 failed to operate, it sent the signal to relays 2 and 3 for better coordination and protection. \begin{table}
\begin{tabular}{c c c c} \hline
**Fault Types** & **Fault Resistance = 0.01 \(\Omega\)** & \\ \hline Three-phase to ground fault & Relay 1 & Relay 2 & Relay 3 \\ \hline Threshold Current (kA) & 259 & 8.524 & 2.826 \\ \hline TMS (Seconds) & 2.991 & 0.1421 & 0.04709 \\ \hline Trip Time (Seconds) & 0.01 & 0.01 & 0.01 \\ \hline Double phase to ground fault & Relay 1 & Relay 2 & Relay 3 \\ \hline Threshold Current (kA) & 358.9 & 13.25 & 4.424 \\ \hline TMS (Seconds) & 2.991 & 0.2208 & 0.07373 \\ \hline Trip Time (Seconds) & 0.01 & 0.01 & 0.01 \\ \hline Single line to ground fault & Relay 1 & Relay 2 & Relay 3 \\ \hline Threshold Current (kA) & 358.8 & 16.43 & 5.478 \\ \hline TMS (Seconds) & 2.99 & 0.2739 & 0.0913 \\ \hline Trip Time (Seconds) & 0.01 & 0.01 & 0.01 \\ \hline \end{tabular}
\end{table} 
Table 4: Threshold current at fault resistance of 0.01 \(\Omega\).