In this study, we use a 37-year time series of monthly mean wind speed and monthly total wind-energy production in the contiguous United States (CONUS). For wind speed, we use hourly horizontal wind components in the National Atmospheric and Space Administration's Modern-Era Retrospective Analysis for Research and Applications, Version 2 (MERRA-2), reanalysis data set (Gelaro et al., 2017; GMAO, 2015) from 1980 to 2016. We use these components to derive the monthly mean wind speed at 80 m above the surface, which represents hub height in this study, via the power law (Eq. 1) and the hypsometric equation (Eq. 2): uz2uz1=z2z1α,z2-z1=RdT‾ln⁡p2p1. In Eq. (1), uz1 and uz2 are the horizontal wind speeds, at heights z1 and z2, in which wind speeds are the square root of the sum of squared horizontal wind components, and α is the shear exponent. In Eq. (2), Rd is the dry air gas constant, T‾ is the average temperature between levels z1 and z2, and p1 and p2 are the atmospheric pressures at z1 and z2. In most grid cells, we use the MERRA-2 meteorological output at 10 and 50 m above the surface to calculate α, so as to extrapolate the wind speed at 80 m. In mountainous regions, the heights at 850 or 500 hPa may be closer to 80 than 10 m above the surface; in that case, we use data at the next available level of 850 or 500 hPa to derive the heights of that level and thus to extrapolate the wind speed at 80 m.

The horizontal resolution of the MERRA-2 is 0.5∘ in latitude (about 56 km) and 0.625∘ in longitude (about 53 km). The MERRA-2 reanalysis interpolates the data and the metadata at the exact output latitude and longitude; hence the wind speed, air density, and elevation refer to the grid points with the particular sets of latitude and longitude (Bosilovich et al., 2016). Thus, the longest distance between a wind farm and the closest MERRA-2 grid-cell center is about 39 km.

For energy-production data, we use the net monthly energy production of wind farms in megawatt hours (MWh) from the US Energy Information Administration (EIA) between 2003 and 2016. Each of the wind farms has a unique EIA identification number. After we leave out about 300 wind sites with incomplete or substantially zero production data, a total of 607 wind farms in the CONUS are selected for this analysis. For simplicity, the CONUS in this analysis is defined as the area bounded by 127∘ W, 65∘ W, 24∘ N, and 50∘ N, and geographically includes the 48 states in CONUS and Washington, D.C. (Fig. 1).

We select two sites from two different geographical regions with considerable wind-energy deployment, the southern Plains and the Pacific Northwest in the United States, to contrast the results of various variability metrics. Based on the site-specific regressions, we extend the monthly energy-production time series to 37 years (Fig. 3a and b) for the two sites. Both sites pass the R2 filter at 0.75 and the r filter at 0.8. Although the OR site is farther from the closest MERRA-2 grid point in a region with more complex terrain, the resultant R2 (0.87) and predicted–actual-energy Pearson's r (0.91) are larger than those of the TX site (0.79 and 0.81, respectively) (Table 2). The 37-year-average wind speed of about 7.6 m s-1 at the TX site is larger than that of the OR site at about 6.8 m s-1 (Table 2). Additionally, the 12-month-lag autocorrelations demonstrate that the annual cycle of monthly wind speeds of the TX site is stronger than that of the OR site, yet the autocorrelations of the sites, 0.53 and 0.32, are still lower than the CONUS median of 0.58 (Table 2).

None of the monthly and annual wind-speed distributions of the sites are perfectly Gaussian. According to the kurtosis, skewness, and YKI values of the monthly mean wind speeds (Table 2), the monthly wind-speed distribution at the OR site skews towards lower wind speeds with more and stronger extremes (Fig. 3c). The skewed distribution at the OR site leads to 71.2 % of the monthly wind speeds located within 1σ from the mean, compared to the classic Gaussian of 68.3 %. Nevertheless, although the TX site monthly wind-speed distribution is very close to symmetric with fewer outliers (Fig. 3d), which is supported by near-zero skewness and YKI (Table 2), only 64.6 % of monthly data fall within 1σ from its mean. For annual-mean wind speeds, the averaging with a 12-month time span at both sites reduces the ranges and thus leads to kurtosis close to -1 (Table 2). Although the skewness and YKI are close to 0 (Table 2), only 59.5 % and 56.8 % of the annual-mean wind speeds fall within 1σ from the means of the OR and TX sites, respectively.

The four selected variability methods yield similar resultant monthly variabilities that are close to the respective CONUS medians based on the 37-year monthly data. For variabilities of monthly wind speeds, the differences between the two sites are slight because the comparison among the results of the four metrics is inconclusive (Table 2): the monthly variabilities are not far from the national medians (Table 2). However, results from the normalized spread metrics (RCoVs, range divided by trimean, and CoV) using the 37-year and the observed energy production illustrate that the OR site generates more variable wind power than the TX site (Table 2). The magnitudes of the variabilities between the 37-year and the actual monthly energy production are also comparable, and the discrepancies between them are larger at the TX site than the OR site. Nonetheless, the predicted and the observed monthly energy production of the two sites demonstrate similar variability characteristics overall.

Moreover, when we apply the four selected methods to the annual-mean data, the metrics describe IAV exactly. For both variables, wind speed and energy generation, nearly all metrics illustrate that the OR site has stronger IAV than the TX site, except for using σ to quantify energy-production IAV (Table 2). Echoing the results of the monthly data mentioned previously, the use of normalized metrics suggests the energy production at the OR site varies more than that at the TX site, intermonthly and interannually. Note that all the IAVs are smaller than the variabilities calculated using monthly data (Table 2), because the annual averaging collapses variations in the data.

Additionally, the magnitudes of energy variabilities and IAVs are also nearly or more than twice as large as those of wind speed (Table 2). The reason is the nature of the power curve: wind-power generation is a function of wind speed cubed at wind speeds below rated. Therefore, small wind-speed variations propagate into large energy-production fluctuations that are discernible in monthly and yearly data.

Matching the wind-speed and energy variabilities over 37 years at each r-filtered site, RCoV, as a statistically robust and resistant metric, yields the highest Pearson's r (0.86) among the four highlighted methods as well as all the variability metrics evaluated (Fig. 4 and Table B1). A perfect variability measure would link wind-speed and wind-power variations closely together with a correlation of unity, and so RCoV, with the highest Pearson's r, is the best of all. On the one hand, a strong correlation between the wind-speed RCoV and the energy-production RCoV implies that the high wind-speed variability at a wind farm translates to high energy-generation variability, and vice versa (Fig. 4a). For instance, the moderate 37-year wind-speed RCoVs of the OR and TX sites indicate modest fluctuations in energy production between months (Fig. 4a). On the other hand, a nonresistant method, range divided by trimean, leads to a lower r (0.64) and suggests the OR site has variable wind speed and energy production (Fig. 4b). For the other two nonrobust and nonresistant methods, the CoV results in a modest r (0.70) with a similar scatter as the RCoV (Fig. 4c); the σ, not normalized by an average metric, does not relate wind-speed and energy variabilities effectively (Fig. 4d). The positions of the two wind farms relative to the rest of the sites in Fig. 4 illustrate that the TX site experiences average variabilities in wind resource and energy production, whereas the OR site has above-average energy-generation variability. Overall, the four methods lead to different representations of energy variability at the OR site.

By increasing the years included in the variability calculations using monthly data, the resultant correlations of most metrics vary less, the correlations gradually converge to their 37-year values, and their asymptote periods vary. The 37-year Pearson's r values from the four selected metrics between wind-speed and energy-production variabilities in Fig. 4 transform into the 37-year marks in Fig. 5, and we use a 5 % threshold of normalized deviation to determine the asymptote periods. Particularly, the r's from RCoV and CoV (Fig. 5a and c) reach their respective asymptotes steadily with longer length of data, whereas the r's from range divided by trimean do not (Fig. 5b). The 37-year correlation using σ is weak and thus the method is not actually useful: while the r's approach the 37-year benchmark (Fig. 5d), this correlation value is so low (0.2) as to be ineffective. Paired with a high long-term r, the asymptote period of a metric indicates the appropriate time span of wind-speed data required to represent the variability of wind-energy production. For example, the resultant r's using RCoV approach a high value after just 3 years, meaning one needs 3 years of wind-speed data to estimate the wind-speed variability so as to adequately infer the energy-production variability of a certain or potential wind farm via RCoV.

The three correlation coefficients (Pearson's r, Spearman's rs, and Kendall's τ) yield consistent results among all variability metrics tested; hence we primarily present the results using Pearson's r here. Table 3 summarizes the 37-year correlations (r, rs, and τ), between the wind-speed variabilities and the energy-production variabilities using the r-filtered data, and the respective asymptote periods of the methods. The r and τ of RCoV are the largest (0.86 and 0.67, respectively) among all variability metrics, and the associate asymptote periods are also relatively short (2 to 3 years) (Table 3). Another normalized, robust, and resistant spread metric, interquartile range (IQR) divided by median, results in the highest rs, and the rs of RCoV is the second largest (Table 3). More importantly, the asymptote periods of RCoV are the smallest of all, regardless of the choice of correlation coefficient. In other words, fewer years of data are necessary to calculate RCoV to effectively relate wind-speed and energy variabilities than any other metric. Overall, when a spread metric yields strong correlations between variabilities of wind speed and energy generation, the correlation metrics agree with each other (Table 3). Therefore, the results in this paper focus on Pearson's r, which is a commonly used correlation coefficient.

In addition to the spread metrics, other distribution diagnostics also yield strong correlations between the 37-year monthly wind speed and energy production. For example, kurtosis and skewness result in r and rs above 0.9. Because we determine the asymptote periods based on normalized deviations, when the 37-year correlation benchmark of a metric is high, the respective asymptote period tends to be shorter. Therefore, only 1 year of monthly data is required to compute kurtosis and skewness adequately, except for using rs in kurtosis, where those rs's of the smaller number of years are low (Table 3). Moreover, the symmetry and the shape of the energy-production distribution can be characterized using wind-speed data, given the moderately strong correlations of YKI and the Weibull shape parameter (Table 3).

Additionally, we also perform the same correlation and asymptote analyses on the data from changing the R2 and r filter thresholds as well as the data with random error, and RCoV again yields the strongest correlations and the shortest asymptote periods among all methods. We adjust the R2 and r requirements in the linear-regression process, thus changing the filtered sample sizes. On the one hand, reducing the R2 threshold to 0.6 and the r threshold to 0.7 increases the respective sample sizes to 461 and 306 wind farms, but weakens the correlations between wind-speed and energy variabilities for all methods (Table B3). On the other hand, increasing the R2 threshold to 0.85 and the r threshold to 0.9 strengthens the wind-speed–energy correlations of all the metrics and shrinks the sample sizes to 212 and 83 wind farms, respectively (Table B3). Modifying the filtering thresholds leads to different r's yet similar asymptote periods among all metrics. Moreover, we also test the vigorousness of our findings by introducing an error term, randomized based on the standard error, in predicting the 37-year energy production. The error term adds uncertainty to resemble the reality of noisy wind-speed and power-production data. We introduce the error term to the predicted energy production for each of the 349 wind farms that pass the original R2 threshold of 0.75. This approach weakens the correlations and lengthens the asymptote periods for most metrics (Table B3). Overall, according to the results from the R2–r threshold and the random error tests, RCoV yields the highest r's among all methods, and its asymptote periods remain reasonably short.

Further, normalized and simple spread metrics yield different relative wind-speed variabilities between wind sites. On the one hand, the correlations coefficients between 37-year monthly mean wind-speed RCoV and CoV, two spread metrics that are normalized by average metrics, are nearly unity (Fig. 6a). The comparison between two simple spread metrics, MAD and σ, results in correlation coefficients close to 1 also (Fig. 6d). The relative positions of the OR site highlight the differences between Fig. 6a and d: compared to other wind farms, the OR site has moderate wind-speed RCoV and CoV, but small MAD and σ. Compared to Fig. 6a, the lower rs and τ in Fig. 6d illustrate that MAD and σ can misrepresent the relative wind-speed variabilities of a wind site. On the other hand, the results between a normalized spread metric (RCoV and CoV) and the respective simple spread metric (MAD and σ), which is also the numerator of the normalized spread metric, lead to weaker correlations (Fig. 6b and c). The r, rs, and τ between 37-year monthly wind-speed RCoV and σ are 0.684, 0.738, and 0.579, respectively (not shown). The wind sites with slower average wind speeds and thus disproportionately larger normalized spread results cause the deviations from perfect correlations in Fig. 6b and c. Therefore, normalized spread metrics, which account for the differences in wind-speed magnitude, become advantageous over simple spread metrics in comparing variabilities of wind sites. Note that we demonstrate similar comparisons between wind-speed spread metrics via annual-mean data in Fig. A2 (Appendix A).

Meanwhile, using annual-mean data to compute IAVs can lead to misleading interpretations. Scatterplots of the 37-year wind-speed and energy IAVs similar to Fig. 4 are illustrated in Fig. A1, via the same 195 r-filtered sites. The correlations via yearly averages are generally weaker except for a few metrics, including range divided by mean, which yields the largest r of all (Table B4). However, the 37-year correlations do not adequately represent the long-term values (Table B4), so even though the resultant asymptote periods are longer than those using monthly data, the asymptote analysis method is unsuitable for annual data. Moreover, using annual averages greatly limits the sample size at each site even with 37 years of hourly wind-speed data. Statistically, a smaller sample leads to a smaller spread of that distribution. Accordingly, with few years of data, small spreads in annual-mean wind speeds result in a tight cluster of IAVs among all the wind farms. Therefore, the compact collection of wind-speed and energy-production IAVs causes strong correlations, solely because of the small number of annual averages used in the IAV calculation. Thus, the correlations via annual means demonstrate a downward trend with increasing length of data, regardless of the variability metrics chosen (Fig. 7). Although the correlations approach the 37-year values, the weakening correlations with more years included in the IAV calculations imply that using less data is preferred in connecting the two IAVs. Note that the spread cannot be computed with one data point and hence the correlations between wind-speed IAVs and energy IAVs do not exist with a single year of data (Fig. 7). Overall, the asymptote analysis causes deceptive results, and, given the nature of the annual data, we cannot determine the sufficient length of data to effectively link the IAVs of wind speed and energy production. In other words, relating wind-speed IAV and energy-generation IAV with annual-mean data is flawed.

Now that we have established that RCoV is a powerful and accurate way to relate wind-speed and energy-generation variations, we assess the required amount of data to calculate the RCoV of wind speed. We compute the site-specific RCoVs using different spans of monthly mean wind speeds, including the OR and the TX sites (Fig. 8). The variations of RCoVs decrease as more years are included in the calculations, and for each location we use the 37-year wind-speed RCoV as the long-term benchmark. For example, the 37-year wind-speed RCoV of 0.082 at the OR site means that the median among the absolute deviations from the median is 8.2 % of the median monthly mean wind speed (Fig. 8a and Table 2). We determine the 37-year σ's as 10 % and 5 % of the 37-year RCoV, and we apply the χ2 approach at 90 % and 95 % confidence levels, respectively, to derive the convergence years, or the minimum length of wind-speed data required to calculate RCoV effectively. The convergence years of the OR and TX sites are 12 and 25 years with a 90 % confidence, and 20 and 31 years with a 95 % confidence, respectively (Table B5). In other words, for the OR site, one needs 12 years of monthly mean wind speeds to compute RCoV with a 90 % confidence that the resultant RCoV is within a 10 % deviation from the 37-year RCoV.

To quantify the intermonthly variability of wind speed at a wind farm, RCoV requires 10 years of monthly wind-speed records with a 90 % confidence. In general, the σ's of wind-speed RCoVs across the CONUS decrease with more years included in the RCoV calculation (Fig. 9a). For each grid point, the sample size of RCoV also becomes smaller, from 37 RCoVs of 1 year of data to 1 RCoV of 37 years of data, and hence the σ of RCoV decreases as the length of the analysis period of wind speed increases (Fig. 9a). With the σ's of RCoVs across 37 years, we determine the convergence years via the χ2 method. For a certain confidence level, the cumulative fraction of the CONUS grid cells that exceed the associated threshold of χ2-derived confidence intervals increases with the length of data (Fig. 9b). Among all of the MERRA-2 grid cells in the CONUS, the median convergence year is 10 years and the associated MAD is 3 years at a 90 % confidence level (Fig. 9b and Table B5). In other words, to assess the wind-speed variability via RCoV with a maximum of 10 % error from the long-term value and a 90 % confidence, one needs 10±3 years of monthly mean wind-speed records.

Moreover, raising the confidence level extends the minimum length of wind-speed data to compute RCoV. At the 95 % confidence level, the median convergence year is 20 years, and 2.5 % of grid points in the CONUS require more than 37 years of monthly mean data to calculate RCoV (Fig. 9b and Table B5). Additionally, using yearly mean wind speeds instead of monthly data to calculate RCoV requires much longer time to reach convergence. At a 95 % confidence, 33 years of annual-mean data is the average required length, and half of the CONUS grid points have convergence years of more than 37 years (Fig. 9b and Table B5). We also perform the same analysis on CoV and σ of wind speeds (Table B5). Although CoV and σ need fewer years to attain convergence, these nonrobust and nonresistant methods yield worse correlations between wind-speed and energy-production variabilities than RCoV, and hence we focus on demonstrating the RCoV results.

Spatial distributions of wind-speed RCoVs across the CONUS identify locations with reliable wind resources. Based on the site-specific convergence years at a 90 % confidence level (Fig. 10a), we calculate the RCoVs with monthly mean wind speeds of the particular time spans at each grid point and normalize with the CONUS median (Fig. 10b). Regions requiring long wind-speed records are irregularly scattered across the continent, such as the Northeast, the Dakotas, and Texas. The mountainous states generally illustrate high RCoVs, including the Appalachians and the Rockies. Given the strong correlations between the wind-speed RCoV and energy-production RCoV, Fig. 10b offers a realistic estimation of the general spatial pattern of the variability in wind-energy production as well. Note that, qualitatively, Fig. 10b is similar to the maps of wind-speed variability in Fig. 13a of Gunturu and Schlosser (2012) and in Fig. 3 in Hamlington et al. (2015), which also illustrate the variability of wind resources in the CONUS. In addition, using a 10-year fixed length of wind-speed data for all CONUS grid points to compute RCoV results in a nearly identical spatial distribution to the pattern in Fig. 10b.

Further, an ideal location for wind farms should exhibit ample wind speeds with low variability. We combine the spatial variations of the normalized RCoV and the long-term wind resource (Fig. 10b and c), and we differentiate regions according to the CONUS median RCoV and wind speed (Fig. 10d). Favorable candidates for wind farm developments have above-average wind speeds and below-average variabilities, such as the Plains, parts of the upper Midwest, spots in the Columbia River region, and pockets nears the coasts of the Carolinas; poor places for wind power with weak winds and strong variabilities include the Appalachians and most of the Northeast.

The convergence years in some CONUS grid points are beyond 37 years when we increase the confidence level from 90 % to 95 % (Fig. 9b and Table B5), and those grid points do not demonstrate any geographical pattern as in Fig. 10a. Additionally, when using RCoV to represent IAV, the spatial patterns of required data lengths and the resultant normalized RCoVs for annual data are notably different from the monthly mean results, and geographical features seem to be irrelevant (Fig. A3). Furthermore, the categorical features of CoV resemble those of RCoV for onshore wind resources in the CONUS, whereas using σ results in notably distinct classifications of CONUS wind resources (Figs. 10d and A4).