## 5 Semi-Active Structural Control
### Semi-Active Controller Based on the Frequency of the Vibrations  
It is standard practice to tune TMDs to a specific vibration mode, typically the first mode, as it often induces the largest amplitude response. Adaptive control, however, offers the capability to tune a TMD at multiple frequencies. To achieve this, the TMD must be configured with optimal parameters to mitigate the system response across various vibration modes. In order to explore this approach, a frequency-domain analysis of the system response is necessary. The simplified model as described in Section 3 provides the theoretical framework. It includes an external force that simulates wave effects, allowing for variations in the tower's oscillations. The external force applied is a sinusoidal signal with a magnitude of 90,000 kN, featuring frequencies aligned with the inherent frequencies of the system, as detailed in the subsequent lines. To evaluate the FOWT response in the frequency domain and identify the vibration modes, the power spectral density (PSD) must be determined. By analyzing the PSD, the frequencies at which power peaks occur can be identified, revealing the system's dominant vibration modes. This estimation process involves calculating the Fourier transform of the signal's autocorrelation function, which, in this case, corresponds to the movement of the wind turbine tower top. For non-periodic signals, it is common to assume periodicity over specified time intervals. The Fourier transform is then computed for these intervals, yielding the discrete Fourier spectrum of the signal, represented by the Discrete Fourier Transform (DFT). The DFT is defined as follows:  
\[Y(k)=\sum_{j=1}^{n}X(j)W_{n}^{(j-1)(k-1)} \tag{13}\]  
where  
\[W_{n}=e^{(-2\pi i)/n} \tag{14}\]  
is one of \"\(n\)\" roots of unity and the autocorrelation function is the correlation of the signal with a delayed copy of itself as a function of delay. Based on this analysis, two power peaks were identified: one at 0.08 Hz and another at 0.54 Hz, corresponding to the first and second vibration modes of the tower-platform system, respectively. These results are consistent with those reported in reference studies such as [11]. The passive TMD used for structural control was tuned based on a free decay test, which matched the first natural mode at 0.08 Hz. In this case, the optimal values obtained for the TMD spring stiffness and damping coefficient were 4868 N/m and 2419 Ns/m, respectively. While this optimized passive TMD effectively reduced the system response at the specific frequency of 0.08 Hz, it did not mitigate the response at the second mode frequency. The proposed semi-active control strategy addresses this limitation by dynamically adapting the TMD to the prevailing frequency, which may be influenced by wave-induced \begin{table}
\begin{tabular}{c c c} \hline \hline
**Section** & \(k_{\text{TMD}}\) **(N/m)** & \(d_{\text{TMD}}\) **(Ns/m)** \\ \hline
3.1 & 4420 & 568 \\
3.2 & 2217 & 5348 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: TMD values obtained for each sub-section of Section 3.  
forces at a different frequency. The current frequency is identified through the Fast Fourier Transform (FFT). The scheme for optimizing the frequency-based semi-active TMD is presented in Figure 7, using the same color code as in Figure 5. In this configuration, the scheduler generates a set of frequency wave scenarios, and the wind turbine (WT) structure is tested under wave excitation at these primary frequencies. The scheduler begins by assigning a value for the wave frequency, \(f_{waves}\). It then resets the Genetic Algorithm (GA) and initiates the optimization process. The GA generates a pair of tuning parameters for the TMD \(\left(k_{STMD},\ d_{gTMD}\right)\), which are applied as inputs to the TMD. After the simulation completes, the standard deviation of the TTD is calculated and used as the fitness function for the GA.