## 2 Influence of Wind Speed Variability on Wind Power  
The starting point of our analysis is the instantaneous kinetic energy flux \(p(t)\) in wind with air density \(\rho\), blowing past a hypothetical turbine of cross-sectional area \(A\):  
\[p(t)=\frac{1}{2}\rho A\
u(t)^{3} \tag{1}\]  
Here \(v(t)\) is the time-varying streamwise component of velocity blowing past the turbine. We ignore directional fluctuations merely in the interest of keeping our arguments accessible (as also recently explained by [PERSON] [8]), but the subsequent analysis can be easily extended to include directional fluctuations [4]. Performing Reynolds decomposition on the time-varying velocity \(v(t)=\overline{v}+\widehat{v}(t)\) into a time-independent mean speed \(\overline{v}\) and a time-varying fluctuation \(\widehat{v}(t)\) whose long-time average \(\overline{\widehat{v}(t)}\equiv 0\) by definition, and substituting it in Equation (1), we obtain:  
\[p(t)=\frac{1}{2}\rho A[\overline{v}+\widehat{v}(t)]^{3}=\frac{1}{2}\rho A[ \overline{v}^{3}+3\overline{v}^{2}\widehat{v}(t)+3\overline{v}\widehat{v}(t )^{2}+\widehat{v}(t)^{3}] \tag{2}\]  
Owing to the turbine experiencing a drag force, it does not extract all the energy available in the wind. Indeed, the power \(P(t)\) generated by the turbine is given by:  
\[P(t)=K[\overline{v}^{3}+3\overline{v}^{2}\widehat{v}(t)+3\overline{v}\widehat {v}(t)^{2}+\widehat{v}(t)^{3}] \tag{3}\]  
where \(K\leq\left(\frac{16}{27}\right)\left(\frac{1}{2}\right)\rho A\) and \(\left(\frac{16}{27}\right)\) is the efficiency factor representing the theoretical upper bound for power conversion. We now take the long-time average of Equation (3) to obtain the mean power extracted by the turbine, which upon performing an appropriate time-average of each term on the right-hand side (RHS) yields:  
\[\overline{P}=K[\underbrace{\overline{v}^{3}}_{\text{Term I}}+\underbrace{3 \overline{v}^{2}\overline{\widehat{v}(t)}}_{\text{Term II}}+\underbrace{3 \overline{v}\overline{\widehat{v}(t)^{2}}}_{\text{Term III}}+\underbrace{ \widehat{v}(t)^{3}}_{\text{Term IV}}] \tag{4}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Name** & **Location Coordinates** & **Elevation** & **Terrain** & **Duration** & **Sampling Rate** \\ \hline Howard, New York & 42.339693\({}^{\circ}\), \(-\)77.569523\({}^{\circ}\) & 605 m & farm land & 20 Days & 0.2 Hz \\ Big Bear Lake, California & 34.25836\({}^{\circ}\), \(-\)116.92125\({}^{\circ}\) & 2085 m & forested & 12 Days & 1 Hz \\ Atacama, Chile & \(-\)23.01667\({}^{\circ}\), \(-\)67.75\({}^{\circ}\) & 5080 m & desert & 13 Days & 1 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Details of data sets analysed in the present study. Term I in RHS of Equation (4) represents the turbine's mean power output under steady (time-independent) wind speed conditions. Term II is zero, since \(\overline{\overline{v}(t)}\equiv 0\) by definition. \(\widetilde{v}(t)^{3}\) can take both positive and negative values, owing to its being an odd function of time \(t\). Consequently, Term IV on the RHS of Equation (4) takes a very small but non-zero value, and will be neglected in the foregoing analysis. Term III captures wind speed variability, and \(\widetilde{v}(t)^{2}\), being quadratic (an even function in time \(t\)), is always positive. Indeed, \(\overline{\widetilde{v}(t)^{2}}\equiv\overline{(v(t)-\overline{v})^{2}} \equiv\sigma_{v}^{2}\), the quantity of specific interest to our present analysis. As a point of comparison, a 20 day long-time average for the Howard dataset (Table 1) yielded the values \(\overline{v}^{3}=299.85\) m\({}^{3}\)/s\({}^{3}\) for term I, \(3\overline{\widetilde{v}(t)^{2}}=192.26\) m\({}^{3}\)/s\({}^{3}\) for term III, and \(\overline{v(\widetilde{t})^{3}}=23.06\) m\({}^{3}\)/s\({}^{3}\) for term IV. We are therefore justified in keeping term III (64% of term I) and dropping term IV (only 8% of term I).