Model‐independent periodic stability analysis of wind turbines

Abstract

In this work, a new method is proposed for the stability analysis of wind turbines. The method uses input–output time histories obtained by conducting virtual excitation experiments with a suitable wind turbine simulation model. Next, a single‐input/single‐output periodic reduced model is identified from the recorded response and used for a stability analysis conducted according to the Floquet theory. Since only input–output sequences are used, the approach is model independent in the sense that it is applicable to wind turbine simulation models of arbitrary complexity. The use of the Floquet theory reveals a much richer picture than the one obtained by widespread classical approaches based on the use of the multi‐blade coordinate transformation of Coleman. In fact, it is shown here that, for each principal mode computed by the classical approach, there are in reality infinite super‐harmonics of varying strength fanning out from the principal one at multiples of the rotor speed. The relative strength of each harmonic in a fan provides for a way of measuring how periodically one specific fan of modes behaves. The notion of super‐harmonics allows one to justify the presence of peaks in the response spectra, peaks that cannot be explained by the classical time‐invariant analysis. The Campbell diagram, i.e., the plot of system frequencies vs. rotor speed, is in this work enriched by the presence of the super‐harmonics, revealing a much more complex pattern of possible resonant conditions with the per‐rev excitations than normally assumed. Copyright © 2014 John Wiley & Sons, Ltd.

Modal dynamics of structures with bladed isotropic rotors and its complexity for two-bladed rotors

“…This arbitrary s/rev shift between the periodicity of the eigenvector and the frequency of the eigenvalue is identical to the frequency indeterminacy in Floquet analysis due to the logarithm of the complex Floquet multipliers (Skjoldan and Hansen, 2009;Bottasso and Cacciola, 2015). The advantage of Floquet analysis is that there are only 2N D solutions and the frequency indeterminacy can be chosen arbitrarily for each of them; here the concept of participation factors introduced by Bottasso and Cacciola (2015) is helpful.…”

“…For anisotropic three-bladed rotors, Floquet theory or Hill's method is needed to obtain an eigenvalue problem which leads to the periodic eigenvectors of the principal eigenvalue solutions (Skjoldan and Hansen, 2009;Skjoldan, 2009;Skjoldan and Hansen, 2009;Bottasso and Cacciola, 2015). To handle the frequency indeterminacy of the periodic eigenvalue solutions from these methods, Skjoldan and Hansen (2009) suggest to select the principal solution such that the harmonic components on the ground-fixed degrees of freedom are minimized.…”

“…To handle the frequency indeterminacy of the periodic eigenvalue solutions from these methods, Skjoldan and Hansen (2009) suggest to select the principal solution such that the harmonic components on the ground-fixed degrees of freedom are minimized. Bottasso and Cacciola (2015) introduce the concept of modal participation factors in which the norm of the individual harmonic components of a periodic eigenvector determines how much the particular component contributes to the response of the particular mode. They also introduce the concept of periodic Campbell diagrams to plot the frequencies of these harmonic components along with the principal frequency.…”

“…Early studies (Warmbrodt and Friedmann, 1980;Wendell, 1982;Kirchgäßner, 1984) have used Floquet theory to investigate the aeroelastic stability of two-bladed rotors without focusing on their modal dynamics. However, Kirchgäßner (1984) introduces the concept of dominant eigenfrequencies for the harmonic components of the Floquet solutions with the largest magnitude in the corresponding eigenvector, and he plotted the frequencies of all harmonic components with magnitudes larger than a threshold relative to the dominant component in a periodic Campbell diagram similar to Bottasso and Cacciola (2015). A later study by Stol et al (2002) considers the dynamic stability of a teetered two-bladed turbine using Floquet theory on a model with up to seven degrees of freedom.…”

Aeroelastic stability of idling wind turbines

“…Stability analysis in the case of yaw misalign-ment requires the application of Floquet theory (Skjoldan and Hansen, 2009;Bottasso and Cacciola, 2015). This is because in yawed flows, periodicity on aerodynamic loads introduced as a result of non-axisymmetric inflow conditions cannot be eliminated through the application of the Coleman multiblade transformation.…”