## 4 Case Studies to Validate the Self-Tuning Algorithm
By analyzing the results of both performed cases (mechanical torque variation and electrical resistive load), it can be concluded that \(\rho=0.0001\) is the single constant value of the control penalty factor that simultaneously ensures the best control performance and the control system stability. Unfortunately, for such a constant set value, the maximum variations of the two considered disturbances (for which the system can still be stabilized) are \(\Delta T_{m}=16\%\) and \(\Delta R=7\%\). These two values are considered as benchmarks for assessing the effectiveness of the proposed control strategy, (in terms of system stabilization for a wider value range of disturbances). For larger disturbances, the system can operate stably only using the proposed solution. Overall, the final conclusion for both types of disturbance (mechanical torque variation and electrical load/unload) is that the proposed control strategy extends the variation range of disturbances by over 50%. For a mechanical torque variation (which can be measured), the solution proposed in [17] offers a better result, allowing a maximum variation of mechanical torque up to 33%, compared with 24% in this case. As mentioned, without any tuning strategy of the control penalty (so for a constant penalty), the maximum allowed variation is only 16%. However, the main advantage of the solution proposed in this paper is that it can also treat the case of unmeasurable perturbation (such as electrical load/unload). Two additionally case studies are conducted to analyze the performances of the proposed control strategy under action of strong random disturbances, in order to prove the stability and robustness of such a control system. In the first case (Figure 8), a long sequence of random variations of mechanical torque is considered acting on the system. The controlled output (terminal voltage (Figure 8a)) is stable, proving the control system robustness and good performances of the proposed strategy. Both small and large disturbance can be noticed in Figure 8b, and Figure 8c shows the activation cycle of the tuning mechanism that sets the control penalty factor. Similarly, for the second case (Figure 9), a long sequence of random electrical load/unload is considered as acting on the controlled process (Figure 9b). A stable system output can be noticed in Figure 9a under control of the proposed tuning mechanism which adjusts the penalty factor \(\rho\) (Figure 9c). Again, the performed test proves the stability and robustness of the control system, also for this type of disturbance. Thus, even under stress conditions produced by various random disturbances, the performed case studies demonstrate the validity of the proposed self-tuning mechanism of control penalty factor, improving the performances of such a minimum-variance controller by significantly extending the value range of disturbances (acting on the controlled process) for which the control system remains stable and operational. Figure 8: Robustness test—mechanical torque variation: (**a**) controlled output (terminal voltage); (**b**) mechanical torque variation (disturbance); (**c**) variable setting of control penalty factor. Figure 9: Robustness test—electrical load/unload: (**a**) controlled output (terminal voltage); (**b**) electrical resistance variation (disturbance); (**c**) variable setting of control penalty factor. A comparative study between the proposed self-tuning control strategy (involving a variable control penalty), a classical PID controller (tuned for a fixed operating point), an adaptive PID control system and, of course, the classical minimum-variance control law (with a constant control penalty) was conducted in the following. First, we consider a classical PID controller. As already mentioned in Chapter 1, an induction generator (integrated into a wind energy conversion system) is a nonlinear process, described by a seventh order nonlinear model ([PERSON]'s equations) [20; 21; 22; 23; 24].