## 4 Case Studies to Validate the Self-Tuning Algorithm
\begin{table}
\begin{tabular}{c c c} \hline \(\rho\) & \(\Delta T_{m}\) & **Control Performance** \\ \hline \(\rho=0.0001\) (constant) & \(\Delta T_{m}\leq 16\%\) & Stable system and very good control performances (settling time: 0.2 s) \\ \hline \(\rho=0.001\) (constant) & \(\Delta T_{m}\leq 16\%\) & Stable system and good control performances (settling time: 0.5 s) \\ \hline \(\rho=0.01\) (constant) & \(\Delta T_{m}\leq 16\%\) & Stable system and satisfactory control performances (settling time: over 1 s) \\ \hline \(\rho=0.1\) (constant) & \(\Delta T_{m}\leq 18\%\) & Stable system but poor control performances (settling time: over 5 s) \\ \hline \(\rho\in\{0.0001,\ 0.1\}\) (variable) & \(\Delta T_{m}\leq 24\%\) & Stable system and very good control performances (settling time: under 0.5 s) \\ \hline \end{tabular}
\end{table} 
Table 1: The case studies—results for maximum allowed values of mechanical torque variation (\(\Delta T_{m}\)). Figure 3: _Cont._  
The results are changed by considering an increase by an order of the control penalty factor (\(\rho=0.1\), see Figure 3d-f). For the same maximum allowed disturbance as in the previous cases (\(\Delta T_{m}=16\%\)), the control performance is weaker, but the system remains stable (see Figure 3d). Although a significant attenuation of the oscillations can be noticed, the settling time exceeds 2 s, and therefore the control system responds slower. However, in this case, for higher disturbances (\(\Delta T_{m}=18\%\), \(\Delta T_{m}=19\%\)), the major change consists in the fact that the system remains stable (see Figure 3e,f). This stabilization was not possible for the previous lower values of the control penalty. For \(\Delta T_{m}=18\%\), the settling time increases over 5 seconds and the control system response could still be considered acceptable (Figure 3e). For \(\Delta T_{m}=19\%\) the control system remains stable, but an unacceptably long transient regime (over 6 s) and a too long settling time (over 10 s) can be noticed (Figure 3f). Similar results are obtained for a disturbance over this value. The conclusion is that a high value of \(\rho\) can stabilize the control system, even when it is highly disturbed, but the control performance is far too weak to be considered acceptable (too long settling time and oscillating regime). It should be noted that in all performed tests, the overshoot is relatively constant (slightly over 4 volts) due to the constraint imposed by the connection to a power system (which tries to maintain the generator terminal voltage at the constant value of power system). Therefore, a comparative analysis of the extreme cases of control penalty setting (\(\rho=0.0001\) and \(\rho=0.1\)) leads to the following conclusions (which are the starting point for the design of the proposed self-tuning algorithm):  
* A small value of \(\rho\) ensures a very good performance, but only for small disturbances;
* A high value of \(\rho\) ensures a system stabilization even for higher disturbances, but with the cost of an unacceptable control performance degradation. Taking into consideration the previous results, the proposed tuning strategy uses the following rules by implementing an algorithm based on a sharp hysteresis curve (Figure 1):  
* In the stationary regime, when the output error is under a certain threshold \(\mathrm{e}_{2}\), the control penalty factor \(\rho\) is set to a low value (\(\rho_{0}=0.0001\)) that ensures the best control performance. * When a high disturbance occurs and, as a consequence, the output error passes over the threshold \(\mathrm{e}_{2}\) (see Figure 1), the control penalty factor is set to a high value \(\rho_{1}=0.1\), in order to stabilize the system through a strong penalty of control. Such a high value of \(\rho\) can ensure the stability of the control system, even when it is highly disturbed.