## 3 Wind System Control at Variable Wind Speeds from 3 [m/s] up to \(V_{MAX}\) [m/s]
### A Competitive Linear VRFT Controller for Nonlinear Control  
For comparison, we design a competitive PI controller algorithm based on the VRFT principle [44; 45; 46; 47; 48; 49] which was proposed for automated tuning based on (but not limited to) reference model specification. For data collection, we fix the wind speed value \(V=5\left[m/s\right]\) and use a pre-stabilizing feedback controller with transfer function \(-20-0.2/s\) applied to the closed-loop error \(e(t)=n_{ref}(t)-n(t)\). Here, \(11\leq n_{ref}\leq 20\ [\mathrm{rps}]\) is generated as piece-wise constant uniform random samples with periodic switching at \(3\left[s\right]\) additionally filtered with a lowpass second-order filter with function \(1/\left(s^{2}+2s+1\right)\). The control input \(f\) is saturated below at zero and it is further excited with a wide power spectrum uniform random noise with amplitude in \([-4;4]\) with period \(0.1\) [s]. The collected input \(f\) and controlled output \(n\) are recorded at a sampling period of \(T_{s}=0.05\ [\mathrm{s}]\) and are shown in Figure 3. The tuned PI controller is to be implemented in discrete time; thus, it is parameterized as \(C_{PI}^{VRFT}(q)=\left(p_{0}+p_{1}q^{-1}\right)/\left(1-q^{-1}\right)\). Here, \(q\) is the discrete-time single step forward operator, analogous to the \(z\) operator. The reference model for setting up the desired behavior from \(n_{ref}\) to \(n\) in closed loop is picked as \(M(s)=10^{2}/\left(s^{2}+2\cdot 0.9\cdot 10\cdot s+10^{2}\right)\) and discretized accordingly at \(T_{s}=0.05\ [\mathrm{s}]\), to reach \(M(q)\) discrete-time representation. Let the dataset of input-output samples be \(DS=\{f_{k},n_{k}\}\) with sample index \(1\leq k\leq N\), where in our case study, we measured \(N=4000\) sample points. A discrete-time VRFT prefilter is a common option for filtering the virtual feedback error defined as \(e_{k}^{v}=M^{-1}(q)n_{k}-n_{k}\) and also for filtering \(f_{k}\). This prefilter used in our case is set to \(0.8/\left(1-0.2q^{-1}\right)\) with sampled period \(T_{s}\). The loss function to be minimized is \(L_{VRFT}=1/N\sum_{k=1}^{N}\left(f_{k}-C_{PI}^{VRFT}(q)e_{k}^{v}\right)^{2}\). The resulting identified controller is \(C_{PI}^{VRFT}(q)=\left(-4.0534+0.6495q^{-1}\right)/\left(1-q^{-1}\right)\). Comparative control results obtained with \(C_{PI}(s)\) and \(C_{PI}^{VRFT}(s)\) (the latter being implemented in discrete time) are offered in Figure 4. Note that in the testing phase, \(V(t)=3+2.0071\times 10^{-3}t\), and it acts, as a parametric disturbance on the model \(P_{TV}\) but also imposes the setpoint as \(n_{ref}=n_{OPTIM}=3.8269\times V\). Initial conditions for the test scenario are set as \(f_{0}=0.953\ [\mathrm{Hz}],n_{0}=11.481\ [\mathrm{rps}]\). Figure 4: The closed-loop testing with \(C_{PI}(s)\) and \(C_{PI}^{FFT}(q)\) in the operating region OR1. Figure 3: The closed-loop collected data for VRFT PI controller design in the exploration phase. In the bottom subplot, the value of the additional resistance \(R_{ad}(n,f)\) from (13) affects both \(P_{GE}\) and \(I_{N}\) and it is plotted for informative purposes. We observe the following. Acceptable tracking control is obtained with respect to \(n_{ref}\) and relatively small steady-state errors are visible with respect to the ramp-like \(V(t)\). We stress that the underlying process is nonlinear time-varying due to the parameter \(V(t)\) acting as disturbance but also imposing the MPP operation via \(n_{OPTIM}\). Surprisingly, given the oscillatory nature of the transient response obtained with the VRFT-trained PI controller, it shows a reduced steady-state tracking error under time-varying \(V(t)\), when compared with the simple PI controller. Regarding the sensitivity of the design with respect to the parameter's choice, the VRFT controller seemed very sensitive to the choice of the reference model bandwidth.