A turbine's power curve is zero below a cut-in speed of

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Earth, Climate & Energy Physics · Energy resources and conversion

Energy from Physics, Ch Natural energy flows: rotors, turbines, waves, channels and hot rock

Problem

A turbine's power curve is zero below a cut-in speed of 3 m s−13\,\mathrm{m\,s^{-1}}, rises as Prated(v/11 m s−1)3P_{\rm rated}(v/11\,\mathrm{m\,s^{-1}})^3 up to a rated speed of 11 m s−111\,\mathrm{m\,s^{-1}}, stays at PratedP_{\rm rated} up to a cut-out speed of 25 m s−125\,\mathrm{m\,s^{-1}}, and is zero above that. The site's wind follows a Rayleigh distribution (kW=2k_{\rm W}=2) with mean speed 7.0 m s−17.0\,\mathrm{m\,s^{-1}}; take Γ(1.5)=0.886\Gamma(1.5)=0.886. ((a)) Find the scale speed cWc_{\rm W}. ((b)) Find the fraction of the year the turbine spends at rated power. ((c)) Estimate the rest of the capacity factor by splitting 33 to 11 m s−111\,\mathrm{m\,s^{-1}} into four bins of 2 m s−12\,\mathrm{m\,s^{-1}}, each weighted by the power at its mid-speed, and give the capacity factor before losses.

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