# Robust Control Examples Applied to a Wind Turbine Simulated Model
## 4 Simulations and Comparisons
### Controller Performance Tests
With reference to the second adaptive design approach using adaptive solutions, the two outputs \(P_{g}(t)\) and \(\omega_{g}(t)\) of the wind turbine continuous-time nonlinear model of Equation (9) were approximated by two second-order time-varying discrete-time models of Equation (15) with two inputs and one output. Using these two LPV prototypes, the model-based approach for determining the adaptive controllers recalled in Section 3.2 was exploited and applied to the wind turbine benchmark of Section 2. Thus, according to Section 3.2, the parameters of the adaptive controllers were computed on-line. In particular, for each output, two second-order (\(n_{a}=n_{b}=2\)) time-varying prototypes were identified, and the adaptive regulator parameters in Equations (36) or (37) were computed analytically at each time step \(k\). Also in this case, with reference to the adaptive controller structure of Equations (36) or (37), the parameters of the adaptive controllers were tuned on-line via the Ziegler-Nichols rules, applied to the LPV models. This adaptive procedure is already implemented and available in [32; 33]. In this way, if both the model on-line parametric identification and the regulator recursive tuning procedure are exploited, the parameter adaptation mechanisms should lead to good control performances. The simulations with the adaptive regulators have been obtained in the same situation of the fuzzy controllers. In this case, three on-line regulators were exploited for the compensation of both the blade pitch angle \(\beta(t)\) and the generator torque \(\tau_{g}(t)\), in Region 1 and Region 2. The adaptive algorithm described above run with initial values for its parameters reported in Table 3.  
Also with reference to the model-based adaptive approach, Figure 8 depicts both the controlled outputs \(P_{g}\) and \(\omega_{g}\) in bold gray lines with respect to their reference values \(P_{r}\) and \(\omega_{nom}\), respectively, \begin{table}
\begin{tabular}{c c} \hline
**Recursive Algorithm Parameter** & **Value** \\ \hline \(\bar{\theta}(0)\) & \([0.1,\,0.15,\,0.20,\,0.25\,0.30,\,0.35]^{T}\) \\ \(\Sigma_{\hat{\psi}}(0)\) & \(10^{-1}\,I_{\gamma}\) \\ \(\delta\) & \(0.995\) \\ \hline \end{tabular}
\end{table} 
Table 3: Initialization parameters of the adaptive algorithm. Figure 7: Generator speed \(\omega_{g}\) and power \(P_{g}\) (bold gray line) with respect to their references (dashed black line) \(\omega_{nom}\) and \(P_{r}\) with the fuzzy controllers. in dashed black lines. As will be seen in the following, also for the case of the adaptive regulators, Figure 8 highlights that this approach leads to interesting performances. Note that the recursive scheme is able to react actively with respect to variations of the working conditions of the wind turbine system. In fact, the fuzzy method can be considered a passive (even if robust) control solution, since the controllers are identified to passively tolerate the disturbance acting on the system. On the other hand, the adaptive methodology is able to counteract any variation or disturbance effects of the controlled process, thus representing an active control solution. These definitions were provided for different frameworks, e.g., in [38; 39]. As further example, Figure 9 depicts the main wind turbine model variables in full load working conditions. In particular, Figure 9 depicts the wind signal \(v(t)\), the generator speed \(\omega_{g}\) and power \(P_{g}\), and the control input \(\beta\). In order to analyze the performance of the proposed adaptive strategy, Table 4 reports also the \(T_{d}\), \(T_{a}\), \(5\%\) and \(NSSE\) values computed for these controllers. According to the simulation results summarized in Table 4, good tracking capabilities of the suggested adaptive controllers seem to be reached, and they are better than both the fuzzy regulators and the baseline governor, recalled in Section 2.