120 A. Hatstatt and K. E. Tatsis 0 5 10 15 20 25 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 Frequency [Hz] Power Spectral Densityh m2 s2Hzi Original Time Series Generated Time Series 0 15 30 45 60 −0.2 0 0.2 Time [s] ¨q2(t) [m/s 2] Original Time Series 0 15 30 45 60 −0.2 0 0.2 Time [s] ¨q2(t) [m/s 2] Generated Time Series Fig. 3 Comparison of the spectra (left) between the original and the generate time series (right) of the second mode in FA direction 200 400 600 800 10−3 10−2 10−1 100 Cycle counts Amplitude [m/s2] Original Time Series Generated Time Series 200 400 600 800 10−3 10−2 10−1 100 Cycle counts Amplitude [m/s2] Original Time Series Generated Time Series Fig. 4 Comparison of the modal strain cycle counts for the first (left) and second (right) mode in the FA direction frequencies (eigenfrequencies of the system and multiples of rotational frequency). These two particular characteristics of the physical problem allow the surrogate model to replicate the real-world complex dynamics in a satisfactory manner with a very limited number of hyperparameters. One could argue that more complex models could replicate the dynamics in a better way, which is undeniably the case, however this would overlook the final goal of the surrogate. The final target being to find a mapping between scarce SCADA data and the hyperparameters of the surrogate model, simpler models are preferred, as fewer parameters usually exhibit higher interpretability and predictabilty. Moreover, the presented approach allows to find analytically the hyperparameters of the model, avoiding the need for optimization algorithms, which, based on the implemented cases, seem to induce a lot more variability in the identified parameters. Lastly, the quantity of interest is neither the spectra of the obtained signals nor the pseudo time series directly, but their predictive capabilities for structural analysis. A very critical example is the quantification of fatigue accumulation, and the applicability of the generated pseudo time series for this purpose is investigated in the remaining of this section. Fatigue damage is related to the strain cycles history in the tower basis. Following the methodology proposed in [14], the strain history is retrieved from the modal contributions q(t), by initially integrating the modal accelerations in the Laplace domain [15] and subsequently transforming the model displacements to strains through the deformation matrix of the Finite Element model of the system. Once the strain time series are obtained, the number of cycles for each strain level is counted, which is herein performed using the rainflow counting algorithm. The obtained results are presented in Figs. 4, where a satisfactory agreement between the generated and the original time series is achieved.
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