## 2 Influence of Wind Speed Variability on Wind Power
Dropping terms II and IV and re-arranging terms I and III, Equation (4) can be re-expressed as:  
\[\overline{P}=K\left[\overline{v}^{3}+3\overline{\widetilde{v}\overline{ \widetilde{v}(t)^{2}}}\right]=K\overline{v}^{3}\left[1+3I^{2}\right] \tag{5}\]  
where \(I=\frac{\sigma_{v}}{\overline{v}}\) is the turbulent intensity, or alternatively the coefficient of variation in wind speed. The presence of a second fluctuation term in Equation (5) requires a comparison of its strength relative to the mean term (\(\overline{v}^{3}\)). When the turbulent intensity is low (\(I\ll 1\)), wind conditions are close to steady, and the mean power follows the standard \(\overline{P}\sim\overline{v}^{3}\) relation. If the fluctuation magnitude is a significant fraction of mean speed (\(I\lesssim 1\)), then \(3I^{2}\sim 1\). Two scenarios must then be considered: one where the standard deviation of wind speed \(\sigma_{v}\) remains constant or decreases with increasing mean wind speed \(\overline{v}\), and the second where \(\sigma_{v}\) increases with \(\overline{v}\). A consideration of the two scenarios becomes important, because a constant \(\sigma_{v}\) leads to additive variability, whereas \(\sigma_{v}\) increasing with \(\overline{v}\) leads to multiplicative variability, which in turn has implications for the error. Our analysis of wind data presented below always shows an increase in \(\sigma_{v}\) with \(\overline{v}\), hence we discuss only the case of multiplicative variability. The standard deviation of power \(\sigma_{P}\equiv\sqrt{\overline{(P(t)-\overline{P})^{2}}}\) can also be computed as a function of \(\overline{v}\) and \(\sigma_{v}\). Defining instantaneous power as \(P(t)=\overline{P}+\widetilde{P}(t)\), where \(\widetilde{P}(t)\) is the fluctuation about the time-independent mean \(\overline{P}\), whose long time average is zero by definition (\(\overline{\widetilde{P}(t)}\equiv 0\)), and using Equations (3) and (4):  
\[\widetilde{P}(t)^{2}\equiv(P(t)-\overline{P})^{2}=K^{2}[(\overline{v}+ \widetilde{v}(t))^{3}-\overline{v}^{3}-3\overline{\widetilde{v}\overline{ \widetilde{v}(t)^{2}}}-\overline{\widetilde{v}(t)^{3}}]^{2} \tag{6}\]  
An algebraic expansion of Equation (6) followed by a long-time average of individual terms yields:  
\[ \begin{array}{ll}\overline{\widetilde{P}(t)^{2}}=&K^{2}[9\overline{v}^{4} \overline{\widetilde{v}(t)^{2}}+15\overline{v}^{2}\overline{\widetilde{v}(t)^ {4}}+\overline{\widetilde{v}(t)^{6}}+18\overline{v}^{3}\overline{\widetilde{v} (t)^{3}}\\ &+6\overline{v}\overline{\widetilde{v}(t)^{5}}-9\overline{v}^{2}(\overline{ \widetilde{v}(t)^{2}})^{2}-6\overline{v}\overline{\widetilde{v}(t)^{3}} \ \overline{\widetilde{v}(t)^{2}}-(\overline{\widetilde{v}(t)^{3}})^{2}]\end{array} \]  
Keeping only the leading order term and ignoring all higher order terms, we arrive at a first order approximation for the standard deviation of wind power:  
\[\sigma_{P}=\sqrt{\overline{\widetilde{P}(t)^{2}}}=3K\overline{v}^{2}\sqrt{ \overline{\widetilde{v}(t)^{2}}}=3K\overline{v}^{2}\sigma_{v} \tag{7}\]  
From Equation (7), we see that \(\sigma_{P}\) scales linearly with \(\sigma_{v}\), but is amplified by a factor of \(3K\overline{v}^{2}\). More importantly, we see that the magnitude of wind power fluctuations scales quadratically with mean wind speed (\(\sigma_{P}\sim\overline{v}^{2}\)). We note that the functional form for \(\sigma_{P}\) in Equation (7) is expected to apply between the cut-in and rated speeds. In Figure 1, we plot the power curve (\(\overline{P}\) vs. \(\overline{v}\), solid red circles) with the instantaneous power against speed (\(P(t)\) vs. \(v(t)\), solid grey circles) overlaid on top of the power curve for a 2.05 MW REPower MM92 turbine (cut-in speed: 3 m/s, rated speed: 11.2 m/s, cut-out speed: 24 m/s, rated power: 2050 kW) located at a wind farm operated by EverPower Wind Holdings in Howard,NY (Table 1).