## 3 Results and Discussion
Consequently, \(\overline{v}\) and \(\sigma_{v}\) retain residual correlations,  
Figure 3: (a) log–log scale: The standard deviation of wind speed fluctuations \(\sigma_{v}\) (red solid circles) versus mean wind speed \(\overline{v}\) and the standard deviation of wind power fluctuations \(\sigma_{P}\) (blue solid squares) versus mean wind power \(\overline{P}\) for the Howard data set. Power-law fits to the data provided \(\sigma_{v}=0.25\times\overline{v}^{0.73}\) (solid line) and \(3.9\times\overline{P}^{0.49}\) (dashed line); (b) Whereas \(\sigma_{v}\) vs. \(\overline{v}\) for Big Bear Lake (red solid circles) exhibits power-law scaling with a fit value of \(0.4\times\overline{v}^{0.6}\) (solid line); and (c) the Atacama data set (blue solid squares) reveals a monotonic increase in \(\sigma_{v}\) vs. \(\overline{v}\). Although we include a power-law fit \(\sigma_{v}=0.31\times\overline{v}^{0.68}\) (dashed line) for illustrative purposes, the scatter in the data does not permit one to place any confidence in the fit value of the exponent. which are revealed as systematic variation in \(\sigma_{v}\) relative to \(\overline{v}\) (the same applies to \(\sigma_{P}\) and \(\overline{P}\)). Such relative scaling between moments of a distribution is well known via Extended Self Similarity (ESS) scaling [27], and was fruitfully exploited to accurately estimate deviations from scalings predicted by the Kolmogorov theory of turbulence [28]. Our observed scalings between the mean and standard deviation of wind speed follow in the same spirit. One cannot escape the residual correlations--and hence the power-law relationship between \(\sigma_{v}\) and \(\overline{v}\)--unless the averaging time is increased from the IEC-stipulated 10 min to the signal correlation time of order 24 h. A 24 h averaging time for each point on the power curve calls for several months worth of data collection effort, and hence is clearly impractical. A more tractable--and still quantitatively defensible--approach is to incorporate the power-law scaling between \(\sigma_{v}\) and \(\overline{v}\) to self-consistently account for residual correlations in the power curve and its variability, as we show below in Equation (8). We can now substitute the scaling form \(\sigma_{v}=C\times\overline{v}^{\alpha}\) in Equations (5) and (7) to obtain:  
\[\overline{P} =K\overline{v}^{3}\left[1+\frac{\sigma_{v}^{2}}{\overline{v}^{2} }\right]=K\overline{v}^{3}[1+C^{2}\overline{v}^{2(\alpha-1)}] \tag{8a}\] \[\sigma_{P} =3K\overline{v}^{2}\sigma_{v}=3 CK\overline{v}^{(2+\alpha)} \tag{8b}\]  
Equation (8) therefore provides a minimal parameter description of \(\overline{P}\) and \(\sigma_{P}\) as a function of \(\overline{v}\) alone, thereby keeping our analysis in accord with the requirements of IEC standard 61400-12-1 [3]. Even when a scaling form for \(\sigma_{v}\) is unavailable (e.g., the Atacama data set), one can still exploit the monotonic variation in \(\sigma_{v}\) versus \(\overline{v}\) and numerically input \(\overline{v}\) and \(\sigma_{v}\) into Equation (7) and calculate \(\sigma_{P}\) to establish confidence intervals around \(\overline{P}\) for forecast projections. Failing this, the wind speed variability feeds into and amplifies wind power variability, and \(\sigma_{P}\) transforms into forecast uncertainty. Equation (8) applies between the turbine's cut-in and rated speeds. Furthermore, we emphasize the fact that Equation (8b) is only a leading order approximation for wind power variability between the cut-in and rated speeds. Between the rated and cut-out speeds, mean power decouples from mean speed, and the turbine acts to rectify wind power fluctuations, thereby suppressing any increase in power variability. When wind power fluctuations are considered as a whole between the cut-in and cut-out speeds, \(\sigma_{v}\) should follow a shallower scaling than would be suggested by Equation (8b).