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Wind as a Stochastic Process

From the spectrum of the wind to the power extremum seeking loses, and the design that loses least, under the two IEC turbulence spectra.

The gradient that extremum seeking computes is noisy because the wind is. Its noise, the scatter it leaves in the gain, and the power that scatter costs all follow from one number: the wind’s spectral density at the dither frequency.

IEC 61400-1 gives two models of that density for the longitudinal wind, Kaimal and von Karman. They share the value at zero frequency (for each model’s own integral length scale) and Kolmogorov’s \(f^{-5/3}\) fall above the corner, but turn the corner differently, and with the standard’s lengths (\(340\) m and \(147\) m at the NREL 5 MW hub) they disagree by up to \(35\%\) at the dither frequencies that matter. The analysis is therefore worked out twice, on equal footing. The derivation up to the spectrum (Links A to C) is common to both decks; from Link D on, every number is computed and checked against simulation for that deck’s spectrum.

Presentations

Wind as a Stochastic Process: the Kaimal Model →

Wind as a Stochastic Process: the von Karman Model →

NREL 5 MW, \(8\) m/s, \(\mathrm{TI} = 10\%\), tracking time one day	Kaimal	von Karman
IEC length scale \(L\)	\(340\) m	\(147\) m
one period’s \(\sigma(\hat g)\) at \(T = 150\) s, predicted	\(5.21\times10^{-7}\)	\(5.43\times10^{-7}\)
best design	\(T = 60\) s, \(a = 9.4\%\) of \(u^\star\)	\(T = 60\) s, \(a = 10.0\%\) of \(u^\star\)
power lost by the best design	\(0.21\%\)	\(0.23\%\)
power lost by the published design (\(150\) s, \(13.6\%\))	\(0.38\%\)	\(0.39\%\)

The design rule, and its choice of a \(60\) s period, does not depend on which spectrum is used; the loss it achieves moves from \(0.21\%\) to \(0.23\%\). Both decks end with the same comparison section, including the shaping filter with which Lao et al. (ACC 2022) generate von Karman wind, and why the analysis here synthesizes from the spectrum directly instead.