# Robust Control Examples Applied to a Wind Turbine Simulated Model
## 4 Simulations and Comparisons
### Stability Analysis  
The stability properties of the proposed control strategies have been checked by means of a Monte Carlo campaign based on the wind turbine benchmark. In fact, the Monte Carlo analysis represents the only method for estimating the efficacy of the developed control schemes when applied to the monitored process. It is worth noting that the works [31; 37] suggested an analytical demonstration of the stability of fuzzy and adaptive control schemes, whilst some practical issues were remarked in Sections 3.1 and 3.2 for the fuzzy approach and the adaptive strategy, respectively.  
All simulations were performed by considering noise signals modeled as Gaussian processes, according to the standard deviations reported in Table 5. Different wind sequences were generated by the wind turbine benchmark simulator. Moreover, the initial conditions of the dynamic models recalled in Section 2 (i.e., the drive-train, the generator/converter, and the pitch system) were changed randomly. Therefore, the random wind signal \(v(t)\), the parameters of Table 5, and the dynamic model initial conditions allowed obtaining different sequences of the wind turbine signals \(\beta(t)\), \(\tau_{g}(t)\), \(\lambda(t)\), \(\omega_{g}(t)\), and \(P_{g}(t)\) for each Monte Carlo simulation.  
As an example of a single Monte Carlo run, Figure 10 highlights that the main wind turbine model variables, such as the generator torque \(\tau_{g}(t)\), the tip-speed-ratio \(\lambda(t)\), and the generator power \(P_{g}(t)\) remain bounded around the reference values, proving the overall system stability in simulation, even in the presence of disturbance and uncertainty. These results refer to the case of full load operation with the data-driven fuzzy controllers.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Working Condition** & **Index** & **GKSV** & **EB** & **UDC** & **COK** & **GFM** & **Fuzzy** & **Adaptive** & **Baseline** \\ \hline \multirow{4}{*}{Partial Load} & FTR & 0.234 & 0.224 & 0.123 & 0.003 & 0.235 & 0.001 & 0.018 & 0.403 \\  & MTR & 0.343 & 0.333 & 0.232 & 0.029 & 0.532 & 0.003 & 0.001 & 0.596 \\  & CTR & 0.657 & 0.667 & 0.768 & 0.971 & 0.468 & 0.997 & 0.999 & 0.404 \\  & MTD (s) & 47.24 & 44.65 & 69.03 & 19.32 & 13.74 & 0.08 & 0.08 & 70.87 \\ \hline \multirow{4}{*}{Full Load} & FTR & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 & 0.003 \\  & MTR & 0.002 & 0.003 & 0.002 & 0.003 & 0.002 & 0.001 & 0.001 & 0.003 \\  & CTR & 0.978 & 0.977 & 0.987 & 0.977 & 0.982 & 0.999 & 0.999 & 0.865 \\  & MTD (s) & 0.03 & 0.03 & 0.04 & 0.32 & 0.05 & 0.02 & 0.01 & 0.89 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Comparison of the different control strategies applied to the wind turbine benchmark. GKSV, Support Vector machine based on a Gaussian Kernel; EB, Estimation-Based; UDC, Up-Down Counters; COK, Combined Observer and Kalman; GFM, General Fault Model; FTR, False Tracking Rate; MTR, Missed Tracking Rate; CTR, Correct Tracking Rate; MTD, Mean Tracking Delay.  
Figure 10: Sensitivity analysis in full load conditions with the data-driven fuzzy regulators.  
As further Monte Carlo example run, the results achieved with the model-based adaptive controller in full load are summarized in Figure 11.  
Figure 11 depicts the perturbed signals representing the generator speed \(\omega_{g}(t)\) and power \(P_{g}(t)\), with respect to their nominal values. Also in this case, the main wind turbine variables remain bounded around the reference values, thus assessing the overall system stability in simulation, even in the presence of modeling errors and disturbance.