## 4 Wind System Control at Variable Wind Speeds from \(V_{max}\) [m/s] up to 21 [m/s]
### Testing the Controllers Designed for OR2  
The test scenario for the competitive controllers designed for the region OR2 is captured by a wind speed variation \(V(t)=21-11e^{-0.01t}\), initial conditions at time \(t=0\) are \(V_{0}=10\) [m/s] (close to \(V_{MAX}\) but rounded to a smaller integer value, without generality loss), \(n_{0}=38.269\) [rps], \(\beta_{0}=0\) [deg] and \(P_{TV0}=2.4423\) [MW]. For time intervals up to several tens of seconds, the variation of \(V(t)\) can be considered ramp-like, as will be apparent from the plots. For robustness, the poles of the closed loops of the linearized system with the double integrator controller are allocated, resulting in \(p_{1}=-5,\ p_{0}=-20\). For the sliding mode controller, the main parameter is chosen as \(K=10^{6}\), and for chattering reduction, the sign function in the controller switching is smoothed as \(sgn(g)=\tanh\ (10g)\). For MPAC application, we choose \(N=3\); the blade pitch angle derivative was saturated within thresholds \(\beta_{min}=-40,\ \beta_{max}=40\), and the control effort on the pitch angle derivative was set to \(\lambda=1\). An interior-point barrier optimizer was employed. A sample period of \(T_{s}=0.05\) [s] was used, as with the VRFT discrete-time controller. The control results measured with all proposed controllers are visible in Figure 5. Fast transient responses are obtained with a slight advantage for the sliding mode controller. In the disturbance rejection, the linear double integrator rejects the disturbance with steady-state tracking error close to zero, similarly to the sliding mode controller. This is surprising,as the double integrator is linear; however, it displays sufficient robustness. The MPAC controller displays decreasing steady-state tracking error at a slower pace. The control in operating region OR2 is deemed feasible and performant with the proposed controllers. That is, reliable operation follows by regulating \(P_{TV}(t)\) to the desired value \(P_{TV}^{*}(t)=1.5\) [MW]. Chattering with \(\beta(t)\) is not seen due to its small value; however, its indirect effect is observed with the sliding mode controller, even though the computed \(u(t)\) is integrated once to arrive at \(\beta(t)\). The results were tested on the entire wind speed variation domain [10; 21] [m/s] and they hold the same control behavior. Due to the fast-transient responses, only the first 50 s are presented in Figure 5. Due to its simplicity and ability to simultaneously reject disturbance and track the reference, the double integrator may be the best tradeoff. However, testing the robustness in extreme conditions under additional operational constraints may leave a significant space for either the sliding-mode controller or for the MPAC one. Regarding the sensitivity of the compared controllers with respect to the parameter's choice, we disclose the following. The linear double integrator provided the baseline performance, with two main criteria, in this order: (1) fast transient response and (2) null steady-state tracking error with respect to the ramp-like wind speed. For the sliding mode controller, we expected to outperform the linear double integrator; however, we could not make the transient response much faster, since the chattering increased. Notice from the theoretical design that the chattering will be present under any circumstance, since we chose to not compensate explicitly for the uncertainty; thus, we leave it to the switching component to deal with it. This way, the resulting controller has much simpler expression and does not need to evaluate the uncertainty factor \(\sigma\) at each integration time step, so the computational complexity is heavily reduced. With the chattering result, as seen in the plot, the performance was considered good enough, with zero-on-average steady-state tracking error. For the MPAC controller, although the targeted performance was better than with the linear double integrator, we notice its apparent lower performance when compared with the double integrator.