## 4 Results and Discussion
### Case Study 1
#### 4.1.1 Mathematical Modelling of the Problem Formulation  
In this case, eight variables had been taken into consideration (i.e., the TMS of eight relays), eight constraints due to bounds on relay operating time (or bounds on the TMS of relays), and nine constraints due to coordination criteria. The duplex of the proposed problem will have 17 variables  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Fault Point**} & \multicolumn{6}{c}{**Relay**} \\ \cline{2-9}  & \multicolumn{2}{c}{**1**} & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline \multirow{3}{*}{A} & \(I_{\text{relay}}\) & 10 & 10 & 3.3 & - & - & - & - & 3.3 \\  & \(a_{\rho}\) & 2.971 & 2.971 & 5.749 & - & - & - & & 5.749 \\  & \(I_{\text{relay}}\) & 3.45 & 3.45 & 5.1 & 6.9 & - & - & - & - \\  & \(a_{\rho}\) & 5.584 & 5.584 & 4.227 & 3.551 & - & - & - & - \\  & \(I_{\text{relay}}\) & 2 & 2 & 10 & 4 & - & - & 4 & - \\  & \(a_{\rho}\) & 10.035 & 10.035 & 2.971 & 4.9804 & - & - & 4.9804 & - \\  & \(I_{\text{relay}}\) & 5 & 5 & 4 & 10 & - & - & - & 4.9804 \\  & \(a_{\rho}\) & 4.281 & 4.281 & 4.9804 & 2.971 & - & - & - & 4.9804 \\  & \(I_{\text{relay}}\) & 20 & 6 & 2 & - & 8 & - & - & 2 \\  & \(a_{\rho}\) & 2.267 & 3.837 & 10.035 & - & 3.297 & - & - & 10.035 \\  & \(I_{\text{relay}}\) & 6 & 20 & 2 & - & - & 8 & - & 2 \\  & \(a_{\rho}\) & 3.837 & 2.267 & 10.035 & - & - & 3.297 & - & 10.035 \\ \hline \hline \end{tabular}
\end{table}
Table 2: \(a_{\rho}\) constants and relay currents for Case 1.  
Figure 5: A multi-loop distribution system.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Fault Point**} & \multicolumn{1}{c}{**T. Fault Current**} & \multicolumn{1}{c}{**Primary Relay**} & \multicolumn{1}{c}{**Backup Relay**} \\ \hline A & 2330 & 1, 2, 8 & -, -, 3 \\ B & 1200 & 3, 4 & -, 1, 2 \\ C & 1400 & 3, 7 & -, 4 \\ D & 1400 & 4, 8 & 1, 2, 3 \\ E & 2800 & 1, 5 & -, 8 \\ F & 2800 & 2, 6 & -, 8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Total fault current and primary/backup relationships of relays for Case 1.  
and 8 constraints. In order to validate the efficiency of RTO, the CTI ratio of 0.6 s and minimum operating time (MOT) of the relay was taken as 0.1 s. The TMS of all the eight relays are \(x1\)-\(x8\).  
The problem for optimization can be demonstrated as:  
\[minz=28.975x_{1}+28.975x_{2}+37.736x_{3}+11.502x_{4}+3.297x_{5}+3.297x_{6}+4.980 7x_{7}+30.7994x_{8} \tag{15}\]  
The constraints owing to the MOT of the relays are:  
\[2.971x_{1} \geq 0.1 \tag{16}\] \[2.971x_{2} \geq 0.1\] (17) \[5.584x_{3} \geq 0.1\] (18) \[4.980x_{4} \geq 0.1\] (19) \[3.297x_{5} \geq 0.1\] (20) \[3.297x_{6} \geq 0.1\] (21) \[4.980x_{7} \geq 0.1\] (22) \[10.035x_{8} \geq 0.1 \tag{23}\]  
Hence, the constraints mentioned in Equations (18), (19), (22), and (23) violate the constraints of the minimum value of the (TMS). Hence, these constraints are reconstructed as:  
\[x_{3} \geq 0.025 \tag{24}\] \[x_{4} \geq 0.025\] (25) \[x_{7} \geq 0.025\] (26) \[x_{8} \geq 0.025 \tag{27}\]  
The constraints owing to the coordination of relays with CTI taken as 0.6 are:  
\[5.749x_{3}-5.749x_{8} \geq 0.6 \tag{28}\] \[5.584x_{1}-3.551x_{4} \geq 0.6\] (29) \[5.584x_{2}-3.551x_{4} \geq 0.6\] (30) \[4.980x_{4}-4.980x_{7} \geq 0.6\] (31) \[4.281x_{1}-2.971x_{4} \geq 0.6\] (32) \[4.980x_{3}-4.980x_{8} \geq 0.6\] (33) \[10.035x_{8}-3.297x_{5} \geq 0.6\] (34) \[10.035x_{8}-3.297x_{6} \geq 0.6 \tag{35}\]