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Center for Nonlinear Studies |
The power $P$ generated by a wind turbine depends upon the wind speed $v$ blowing past the turbine through the power relation $P \propto v^3$ derived a century ago by FW Lanchester (1915) and independently by A Betz (1920) and NE Joukowsky (1920). Naturally, generated wind power varies with wind speed, and indeed, the wind power fluctuation spectrum is generally interpreted as reflecting the Kolmogorov spectrum of atmospheric turbulence; both vary with frequency $f$ as $f^{-5/3}$. Whereas these fluctuations are smoothed when power output from geographically distributed wind farms is summed, aggregate wind power entering the electrical grid is still correlated up to almost a day. In this talk, I will explain the wind power fluctuation spectrum from the turbine through the grid scale and draw inferences with crucial implications for wind power. First, in violation of an underlying assumption of Kolmogorov theory (K41), wind power fluctuations for an individual turbine and farm result from the largest scales (integral scale $l_0$) of atmospheric turbulence influencing the small scales where turbines operate. Second, $l_0$ is several 100 km long and relates to the weather system, not the integral scale of the atmospheric boundary layer ($ \sim 100$ m). Consequently, the integral scale velocity fluctuations simultaneously couple turbines within and between wind farms within a distance $ l_0$, causing correlated power fluctuations at the electrical grid. Third, I will demonstrate with real data and argue that aggregate power entering the grid is smoothed until it approaches a theoretical limit with $f^{-7/3}$ spectrum derivable from K41, modulo an assumption that remains to be verified. Finally, using this spectral bound, I will deduce the time-dependent turbine power relation $P(t) \propto \overline{v}v(t)^2$ ($\overline{v}$ is mean wind speed, and $t$ represents time) and add a correction to the original time-independent power relation of Lanchester, Betz and Joukowsky. I will summarize engineering and policy implications of these results.
Host: Turab Lookman