## 6 Methodology
### Case (1): Constant Output Voltage Control
#### 6.1.1 Turn-Off Angle Control  
In this case, the PID is implemented to generate the turn-off angle at a constant turn-on angle. The optimal PID parameters obtained based on multiple runs using the PSO technique for a discrete system at a sampling time of 1 usec are \(k_{p}\) = 0.338, \(k_{i}\) = 73.25, and \(k_{d}\) = 0.0042. The tuning of the PID controller, in this case, is highlighted in Table 2. The output current responses at two reference voltage values of 100 V and 180 V are shown in Figure 6a,b, respectively. The excitation current for Vref = 180 V has a peak value more than the excitation current for Vref = 100 V because the turn-off angle for the 180 V is more than the turn-off angle for 100 V, while the turn-on angle is fixed at 0\({}^{\circ}\).  
Figure 4: Magnetization curve for SRG model.  
The phase current has two intervals: the excitation and the generation intervals. The peak value of the phase current occurs during the generation interval. The different generated turn-off angles at different initial turn-on angles and reference output voltages are shown in Figure 7a. It is shown that increasing the initial turn-on angle affects the generated turn-off angle, where, at an initial turn-on angle of 20\({}^{\circ}\), the turn-off angle is at  
\begin{table}
\begin{tabular}{c c c c} \hline  & \(\mathbf{k_{p}}\) & \(\mathbf{k_{i}}\) & \(\mathbf{k_{d}}\) \\ \hline PS1 & 0.382976 & 72.97717 & 0.004077 \\ \hline PS2 & 0.388455 & 74.30645 & 0.004164 \\ \hline PS3 & 0.388088 & 73.30799 & 0.004163 \\ \hline PS4 & 1.405067 & 176.1952 & 0.004016 \\ \hline PS5 & 0.047432 & 100.0567 & 0.003584 \\ \hline PS6 & 0.390093 & 73.22360 & 0.004168 \\ \hline PS7 & 0.701417 & 97.90111 & 0.004426 \\ \hline PS8 & 0.341268 & 47.79083 & 0.004200 \\ \hline PS9 & 0.409198 & 102.9344 & 0.003757 \\ \hline PS10 & **0.338800** & **73.25463** & **0.004200** \\ \hline \end{tabular}
\end{table}
Table 2: Tuning of PID controller in turn off.  
Figure 5: (**a**) Addressed case studies and (**b**) the SRG system block diagram.  
its maximum and equal to 45\({}^{\circ}\) for V\({}_{\text{ref}}\) = 200 V. However, the maximum turn-off angle is 37.5\({}^{\circ}\) at an initial turn-on angle of 25\({}^{\circ}\) for V\({}_{\text{ref}}\) = 100 V. The peak value of the phase current at different reference output voltages and different initial turn-on angles are illustrated in Figure 7b, which shows that the peak value is constant at a high turn-on angle.  
The output voltage response at the turn-off angle control and zero initial turn-on angle, \(\theta_{\text{on}}\) = 0.0\({}^{\circ}\), is shown in Figure 8a, which confirms that the output response has a constant steady-state settling time of approximately 0.03 s. The maximum peak overshoot changes from 28% for V\({}_{\text{ref}}\) = 50 V to 10% for V\({}_{\text{ref}}\) = 100 V. The controlled turn-off angle response is simulated in Figure 8b.  
The designed controlled system has the advantage of generating different maximum output voltages at different turn-on angles, as shown in Figure 9a, or at different maximum excitation currents, as shown in Figure 9b. The control system can generate a maximum voltage at different fixed turn-on angles and different maximum reference excitation currents. The output response is over-damped and has a steady-state settling time of 0.06 s.  
Figure 6: Current response at (**a**) V\({}_{\text{ref}}\) = 100 V, (**b**) V\({}_{\text{ref}}\) = 180 V.  
Figure 7: The effect of the initial turn-on angle on (**a**) the generated turn-off angle and (**b**) the maximum generated current at different reference output voltages.