6 In-Operation Wind Turbine Modal Analysis via LPV-VAR Modeling 55 d(A1,1) 2 4 6 d(A2,1) 2 4 6 d(A3,1) 2 4 6 d(A4,1) 2 4 6 d(A5,1) 2 4 6 d(A6,1) 2 4 6 d(A7,1) 2 4 6 d(A8,1) 2 4 6 d(A9,1) 2 4 6 d(A10,1) 2 4 6 d(A1,2) 2 4 6 d(A2,2) 2 4 6 d(A3,2) 2 4 6 d(A4,2) 2 4 6 d(A5,2) 2 4 6 d(A6,2) 2 4 6 d(A7,2) 2 4 6 d(A8,2) 2 4 6 d(A9,2) 2 4 6 d(A10,2) 6 2 4 6 6 2 4 6 6 2 4 6 6 2 4 6 6 6 6 2 4 6 6 2 4 6 6 2 4 6 6 2 4 6 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 6 2 4 2 4 6 2 4 6 -2 -1 0 1 2 3 log10 d(Ai,j) Fig. 6.4 Mahalanobis distance of each one of the model parameters to zero where .M/p;q denotes the .p; q/th entry of the matrix M, while O†vecf‚g is the estimated covariance matrix of the vectorized parameter matrices. This distance function attempts to determine the proximity of each parameter of the LPV-VAR model to zero. Thus, one could consider removing a parameter matrix if its entries are sufficiently close to zero. Figure 6.4 shows the results of this analysis in the estimated parameters of the LPV-VAR.10/2 model of the blade’s vibrational response. The resulting distances indicate that despite few individual entries reaching values close to zero, all matrices feature sufficiently large distance components, thus demonstrating the necessity of a complete representation. 6.3.3 Model Based Analysis In this section, the dynamic characteristics of the vibration response of the wind turbine blade are analyzed with respect to the identified LPV-VAR.10/2 model. The “frozen” PSDs of the wind turbine blade vibration at each one of the sensor locations are shown in Fig. 6.5. Notice that only the elements in the diagonal of the “frozen” PSD matrix are shown in the plot. The plot demonstrates the presence of different modal components, some of them with clear evidence of amplitude and frequency modulation. Additionally, it is evident that the power of the vibration is much higher at farther positions on the blade, while the relative power of some modes is also dependent on the sensor location. This latter characteristic may be associated with the mode shapes as shall be analyzed next. Figure 6.6a demonstrates the “frozen” natural frequencies obtained from the LPV-VAR model. From the complete set of “frozen” natural frequencies, only those with associated “frozen” damping lower than 25% are selected for calculation of their respective mode shapes, as shown in Fig. 6.6b. According to the results obtained from eigenanalysis based on a physical model of the wind turbine shown in [13], the first two flapwise bending modes should lie in the vicinity of 0.7 and 2 Hz spectral lines. Both modes may be correlated with modes M3 and M8 obtained from the LPV-VAR model. As can be seen in both the “frozen” PSDs and in the “frozen” natural frequencies, these modes demonstrate visible amplitude modulations and less evident (but still important) frequency modulations. These results confirm earlier findings obtained from single-channel simulations, which have been reported in previous contributions [5]. A significant added benefit of the proposed formulation is the inference of the mode shapes, which result in agreement with those calculated from the physical model, as reported in [13] and also displayed with the mode shapes of modes M3 and M8 in Fig. 6.6b. 6.4 Conclusion This work proposes a method for the identification of non-stationary/time-dependent dynamics based on multiple acceleration response measurements by means of the postulated Linear Parameter Varying Vector AR model. Key elements of this formulation are the estimation of the parameter matrices and the covariance matrix of the parameter estimates, as well as the calculation of “frozen” modal quantities from identified models. The method is exemplified on a simulated case study of a wind turbine blade under regular operating conditions.
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