## 4 Results and Discussion
### Case Study 2
#### 4.2.1 Mathematical Modelling of the Problem Formulation  
Table 4 demonstrates the line information of the given system. The primary/backup pair of relays for the system are shown in Table 5. For this delineation, four fault areas are taken, as shown in Figure 8. The CT proportions and plug setting are given in Table 6. For various fault areas, the current seen by relays and the \(a_{\rho}\) constant are given in Table 7.  
\begin{table}
\begin{tabular}{c c} \hline
**Line** & **Impedance (\(\Omega\))** \\ \hline
1 and 2 & 0.08 +/1 \\
2 and 3 & 0.08 +/1 \\
1 and 3 & 0.16 +/2 \\ \hline \end{tabular}
\end{table}
Table 4: Line data for Case 2.  
Figure 7: Optimized total operating time with the literature for Case 1.  
In this representation, the entire number of constraints that arise are 11. Six constraints are attributable to limits on relay working time, and five constraints are by reason of coordination criteria. The MOT of each relay is 0.1, and the CTI is considered as 0.3 s. The TMSs of all six relays are \(x_{1}\)-\(x_{6}\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Relay** & **CT Ratio** & **Plug Setting** \\ \hline
1 & 1000/1 & 1 \\
2 & 300/1 & 1 \\
3 & 1000/1 & 1 \\
4 & 600/1 & 1 \\
5 & 600/1 & 1 \\
6 & 600/1 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 6: CT ratios and plug settings of the relays for Case 2.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Fault Point**} & \multicolumn{5}{c}{**Relay**} \\ \cline{2-7}  & \multicolumn{2}{c}{**1**} & **2** & **3** & **4** & **5** & **6** \\ \hline A & \(\begin{array}{c}I_{\text{relay}}\\ a_{\rho}\end{array}\) & 6.579 & 3.13 & - & 1.565 & 1.565 & - \\  & \(\begin{array}{c}a_{\rho}\end{array}\) & 3.646 & 6.065 & - & 15.55 & 15.55 & - \\ B & \(\begin{array}{c}I_{\text{relay}}\\ a_{\rho}\end{array}\) & 2.193 & - & 2.193 & 2.193 & 2.193 & - \\  & \(\begin{array}{c}a_{\rho}\end{array}\) & 8.844 & - & 8.844 & 8.844 & - \\ C & \(\begin{array}{c}I_{\text{relay}}\\ a_{\rho}\end{array}\) & 1.096 & - & 1.096 & - & 5.482 & 1.827 \\  & \(\begin{array}{c}a_{\rho}\end{array}\) & 75.91 & - & 75.91 & - & 4.044 & 11.539 \\ D & \(\begin{array}{c}I_{\text{relay}}\\ a_{\rho}\end{array}\) & 1.644 & - & 1.644 & - & 2.741 & - \\  & \(\begin{array}{c}a_{\rho}\end{array}\) & 13.99 & - & 13.99 & - & 6.872 & - \\ \hline \hline \end{tabular}
\end{table}
Table 7: \(a_{\rho}\) constants and relay currents for Case 2.  
Figure 8: A single end-loop distribution system.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Fault Point** & **Primary Relay** & **Backup Relay** \\ \hline A & \(1,2\) & -, 4 \\ B & \(3,4\) & \(1,5\) \\ C & \(5,6\) & -, 3 \\ D & \(3,5\) & \(1,\) - \\ \hline \hline \end{tabular}
\end{table}
Table 5: Primary and backup relationships of the relays for Case 2.  
The problem for optimization can be demonstrated as:  
\[minz=102.404x_{1}+6.0651x_{2}+98.758x_{3}+24.403x_{4}+35.319x_{5}+11.539x_{6} \tag{36}\]  
The constraints owing to the MOT of the relays are:  
\[3.646x_{1}\geq 0.1 \tag{37}\]  
\[6.055x_{2}\geq 0.1 \tag{38}\]  
\[8.844x_{3}\geq 0.1 \tag{39}\]  
\[8.844x_{4}\geq 0.1 \tag{40}\]  
\[4.044x_{5}\geq 0.1 \tag{41}\]  
\[11.539x_{6}\geq 0.1 \tag{42}\]  
Hence, the constraints mentioned in Equations (38)-(42) violate the constraints of the minimum value of the TMS. Hence, these constraints are reconstructed as:  
\[x_{2}\geq 0.025 \tag{43}\]  
\[x_{3}\geq 0.025 \tag{44}\]  
\[x_{4}\geq 0.025 \tag{45}\]  
\[x_{5}\geq 0.025 \tag{46}\]  
\[x_{6}\geq 0.025 \tag{47}\]  
The constraints owing to the coordination of relays with the CTI taken as 0.3 are:  
\[15.55x_{4}-6.065x_{2}\geq 0.3 \tag{48}\]  
\[8.844x_{1}-8.844x_{3}\geq 0.3 \tag{49}\]  
\[8.844x_{5}-8.844x_{4}\geq 0.3 \tag{50}\]  
\[75.91x_{3}-11.53x_{6}\geq 0.3 \tag{51}\]  
\[13.998x_{1}-13.998x_{3}\geq 0.3 \tag{52}\]