The purpose of this study is to examine the extent to which the assumption of uncorrelated uncertainties – and, therefore, the combination of those uncertainties through a sum of squares approach – is accurate and appropriate for operational AEP calculations. Specifically, this study aims to identify potential correlations between AEP uncertainty components using data for over 470 wind plants. While in the analysis we focus on operational AEP calculations, we expect that the results from this analysis – namely, the potential identification of correlated uncertainty components – can be equally relevant for informing and improving preconstruction AEP methods.

In Sect. 2, we first describe the data sources used in this analysis (wind plant operational data and reanalysis products), the Monte Carlo approach to quantify single uncertainty components in operational AEP, and the approaches used to combine these uncertainty components. Section 3 presents the main results of our analysis in terms of uncertainty contributions and correlations among the different components. We conclude and suggest future work in Sect. 4.

Operational wind plant energy production data for this analysis are obtained from the publicly available Energy Information Administration (EIA) 923 database . This database provides reporting of monthly net energy production from all power plants in the United States, including wind plants. More than 670 unique wind plants are available from this data set.

Long-term wind speed data (needed to perform the long-term or windiness correction in an AEP estimate) are used from three reanalysis products over the period of January 1997 through December 2017:

Version 2 of the Modern-Era Retrospective analysis for Research and Applications (MERRA-2) . We specifically use the M2T1NXSLV data product which provides diagnostic wind speed at 50 m above ground level (a.g.l.), interpolated from the lowest model level output (on average about 32 m a.g.l.), using Monin Obukhov similarity theory

Please note that this product is provided in MERRA-2 directly and no further interpolation was performed.

The wind speed data are density corrected at their native time resolutions to correlate more strongly with wind plant power production (i.e., higher-density air in winter produces more power than lower-density air in summer, wind speed being the same): 2Udens,corr=Uρρmean1/3, where Udens,corr is the density-corrected wind speed, U is the wind speed, ρ is air density (calculated at the same height as wind speed), ρmean is the mean density over the entire period of record of the reanalysis product, and the exponent 1/3 is derived from the basic relationship between wind power and wind speed cubed . To calculate air density at the same height as wind speed, we first extrapolate the reported surface pressure to the wind speed measurement height, assuming hydrostatic equilibrium : 3p=psurfexp⁡gzRTavg, where p is the pressure at the wind speed measurement height, psurf is the surface pressure, g is the acceleration caused by gravity, z is the wind speed measurement height, R is the gas constant, and Tavg is the average temperature between the reported value at 2 m a.g.l. and that at the wind speed measurement height.

To lessen the impact of limited and/or poor-quality data on the results of our analysis, we filter for wind plants with a moderate-to-strong correlation with all three reanalysis products (R2>0.6). About 25 % of the EIA wind plants are discarded with this filter. We also impose a threshold of 8 months of wind plant data availability in order to investigate uncertainty as it relates to a low number of data points – but not so low as to make the use of a regression relationship questionable. A total of 472 wind plants are kept for the final analysis, and their locations are shown in Fig. . Because obtaining an accurate representation of wind data in complex terrain by reanalysis products is challenging , most of the selected wind plants are located in the Midwest and Southern Great Plains. Notably, no wind plants in California pass the filtering criteria because they are predominately located in areas with thermally driven wind regimes, such as Tehachapi Pass, where coarse-resolution reanalysis products are poor predictors of wind energy production.

The fundamental step in an AEP calculation involves a regression between density-corrected wind speed (here, from the reanalysis products) and energy production (here, from the EIA-923 database). To investigate whether a simple linear function can be assumed to express the relationship between density-corrected wind speed and wind plant energy production when considering monthly data, we show a scatterplot between MERRA-2 density-corrected monthly wind speed and monthly energy production across all 472 sites in Fig. . For each site, data have been normalized by the respective site mean. We show best-fits using a linear, quadratic, and cubic function and calculate the mean absolute error (MAE) of each fit.

We find that the difference between the normalized MAE values from the considered functions is less than 0.7 %. Therefore, the uncertainty connected with the choice of using a linear regression in the operational AEP methodology at a monthly time resolution appears minimal. Moreover, through conversations with wind industry professionals, we found that a linear regression based on monthly data is the standard industry approach when performing bankable

Results are accepted by banks, investors, and so on for use in financing, buying/selling, and acquiring wind plants.

Given the lack of existing guidelines for a standard approach for operational AEP calculations, we base our methodology on conversations with four major wind energy consultants who represent most of the operational market share in North America. These conversations overwhelmingly revealed the following characteristics for operational AEP analysis, and we follow the same approach in our analysis.

Wind speed data (measured or modeled) are density corrected at their native time resolution using Eq. ().

In the EIA-923 database, availability and curtailment data are not available. Therefore, in our analysis, we omit steps 4 and 9 of the list and only perform calculations on net energy data.

To quantify the impact of the single uncertainty components on the long-term operational AEP estimate obtained using the methodology described in the previous section, we implement a Monte Carlo approach. In general, a Monte Carlo method involves the randomized sampling of inputs to, or calculations within, a method which, when repeated many times, results in a distribution of possible outcomes from which uncertainty can be deduced. This is usually calculated as the standard deviation or the coefficient of variation (i.e., standard deviation normalized by mean) of the resulting distribution . Monte Carlo methods have been used in different applications for uncertainty quantification within the wind energy industry, ranging from the prediction of extreme wind speed events , to offshore fatigue design , and to the economic analysis of the benefits of wind energy projects . Here, we apply this approach to derive a distribution of long-term operational AEP values from which the uncertainty can be calculated. Using a Monte Carlo approach provides a direct estimate of AEP uncertainty by sampling the relevant parameters connected to the various uncertainty components. By contrast, traditional approaches to assessing uncertainty are often less direct. For example, wind resource interannual variability is often calculated and then converted to AEP uncertainty through an “energy / velocity” (EV) ratio estimated from the wind and energy data. A Monte Carlo approach avoids this intermediate ratio and any uncertainty or error associated with it.

In our analysis, we separately consider five operational-based uncertainty components so that only the sampling of one parameter is performed in each Monte Carlo configuration. The following uncertainty components are included in our proposed Monte Carlo methodology for long-term operational AEP.

Revenue meter measurement error. To incorporate this uncertainty component in the Monte Carlo simulation, we sample monthly revenue meter data from a synthesized normal distribution centered on the reported value and a 0.5% imposed standard deviation. In fact, a value of 0.5 % is consistent with what is typically assumed in the wind energy community as revenue meter uncertainty .

Once the contribution from each uncertainty component to the long-term operational AEP uncertainty has been quantified, the different components need to be combined to obtain the total AEP uncertainty. As stated in the Introduction, it is common practice for wind energy consultants to assume that all uncertainty components are uncorrelated and combine them using Eq. () to obtain σtot,uncorr. To test the validity of this assumption, we apply Eq. () in which each of the five considered uncertainty components, σi, is quantified as the coefficient of variation of the corresponding operational AEP distribution obtained by running the Monte Carlo simulation with a single sampling performed. We note that the same values of σtot,uncorr would be obtained by running the Monte Carlo simulation with, at each iteration, all of the five samplings performed independently of each other.

We contrast the total AEP uncertainty calculated assuming uncorrelated components with what we obtain by taking into account these correlations in the calculation. Following the guidance in , we combine the various uncertainty components and calculate the total long-term operational AEP uncertainty for each wind plant as follows: 7σtot,corr=∑i=1Nσi2+2∑i=1N-1∑j=i+1NRijσiσj, where, in our analysis, N equals 5 and Rij is the correlation coefficient between each pair of uncertainty components calculated from the results obtained for all 472 wind plants considered in the analysis.

The comparison between σtot,uncorr and σtot,corr will give insights into the error arising from ignoring the correlations existing between the various uncertainty components.

Distributions of each uncertainty component, expressed in terms of the percent coefficient of variation of the resulting AEP distributions, across all 472 wind plants are shown in Fig. . Uncertainty connected to wind resource IAV is found to contribute the most (average 4.1 % across all wind plants). The uncertainty in the linear regression model has the second-largest contribution (1.5 %), followed by the uncertainty of the reference wind speed data (0.8 %; here, of the reanalysis products) and revenue meter data (here, imposed at 0.5 %). The long-term windiness correction has the smallest uncertainty component (0.4 %). Therefore, the number of years used for the long-term windiness correction does not have a large impact on the overall uncertainty in operational AEP, at least for the sampled range of 10–20 years. Using as few as 10 years seems sufficient to give stability to the long-term AEP estimate, and adding additional years does not provide a significant reduction in the uncertainty connected with the long-term estimate. As already mentioned in Sect. 2, these results are obtained for wind plants in mostly simple terrain and with a moderate-to-strong correlation between reanalysis wind resource and wind energy production and, therefore, with an overall low operational AEP uncertainty. We acknowledge that the inclusion of wind plants with a weaker correlation with the reanalysis products would modify the relative contribution of the various uncertainty components (e.g., the importance of the regression uncertainty would increase).

To be able to assess the validity of the uncorrelated assumption when combining different uncertainty components, we assess potential correlations between uncertainty components by analyzing the Pearson's correlation coefficients, Rij (needed in Eq.  to calculate σtot,corr), from each pair of AEP uncertainty components across the 472 wind plants, and we summarize the results in the correlation matrix in Fig. .

To assess which of the obtained correlations have statistical significance, we calculate the p value associated with the 10 correlation coefficients. The test reveals that for three pairs of uncertainty components, the probability of finding the observed not-zero correlation coefficients if the actual correlation coefficient were, in fact, zero (p value) is less than 10-5. Therefore, the following three correlations have strong statistical significance.

The wind resource IAV and the long-term windiness correction uncertainties are moderately correlated (R=0.49, p=1.9×10-29).

The first correlation noted earlier (wind resource IAV and long-term windiness correction) is explained simply by the fact that both uncertainty components are driven by wind resource variability. At a site with large wind variability, IAV will be large by definition and so will the uncertainty introduced by different lengths of time series used for the long-term AEP calculation.

The correlation between linear regression and reference wind speed data uncertainties can be justified given the dependence of both these uncertainty components on the number of data points used in the regression between energy production data and concurrent wind speed data (Fig. ).

Both the slope and intercept errors (Eqs.  and ), on which the linear regression uncertainty depends (as described in Sect. 2.3), are inversely proportional to the number of data points so that when a regression is performed on only a few data points, its uncertainty increases. This dependence is exemplified in Fig. , in which we have compared the sampling sets of regression lines for two stations in the EIA data set; for these two cases, the standard errors of regression slope and intercept for the station with 8 data points (on the right) are 30–50 times larger than what is found for the station with 90 data points (on the left).

The number of data points used for the regression also has an impact on the reference wind speed data uncertainty. In fact, a short period of record of a wind plant's operation can lead to different interpretations from the reference wind resource data sets used as to whether that short period of record was above, equal to, or below the long-term average resource. Over a longer period of record, these potential discrepancies between different wind resource data sets (in our case, reanalysis products) tend to average out, leading, therefore, to a reduced uncertainty. We illustrate this phenomenon by exploring the long-term trend of the reanalysis products for the wind plant with one of the highest reported reference wind speed data uncertainties (EIA ID 60502 reported a 3.7 % reference wind speed data uncertainty). Figure  shows the result. The period of record for wind plant operation (shown by a shaded blue area in Fig. ) was only 12 months. As shown in the figure, the various reanalysis products have very different interpretations of the wind resource in the short period of record relative to the long term (ERA-I: 4 % above average; MERRA-2: 1 % below average; NCEP-2: 1 % above average). Consequently, the use of each reanalysis product will lead to different magnitudes (both positive and negative) in the long-term windiness corrections, leading to high uncertainty in the resulting operational AEP calculation. By increasing the period of record (i.e., increasing the number of data points used in the regression), such discrepancies tend to average out. This is illustrated in Fig. , where we show how the period of record to long-term wind speed ratio varies as we extend the period of record by increasing the number of months while keeping December 2017 as the fixed ending time. For short periods of record, there is considerable deviation of this ratio among the different reanalysis products (i.e., the reference wind speed data uncertainty is high). As the length of the period of record increases, this ratio tends to converge to 1.0, and the spread between the three reanalysis products decreases (i.e., the reference wind speed data uncertainty is low).

Finally, the (weak) negative correlation between linear regression and wind resource IAV uncertainties is linked to the fact that they respond differently to the R2 coefficient between the reanalysis wind speed and the energy production data (Fig. ). Predictably, the linear regression uncertainty is inversely proportional to the coefficient of determination because a stronger correlation between wind and energy production will lead to a reduced uncertainty of the regression between the two variables. On the other hand, wind resource IAV uncertainty shows a positive correlation with the regression R2 coefficient. This dependence can be explained because both quantities are positively correlated with the total variance of wind speed or, equivalently, produced energy. Figure  shows the relationship between IAV uncertainty and the total sum of squares, SStot,WS, of reanalysis wind speed (here, using MERRA-2 monthly data), which is proportional to the variance of the data: 8SStot,WS=∑i(WSi-WS‾)2. A positive correlation between IAV uncertainty and SStot,WS emerges. At the same time, the linear regression R2 coefficient also depends on the variance of the produced energy (and, equivalently, of wind speed) as it is defined as follows: 9R2=1-SSresSStot, where SSres is the total sum of the residuals from the linear regression. Equation () shows that when the total sum of squares SStot increases, so does R2, thus confirming the positive correlation between R2 and the variance in the data.

After having revealed the correlations existing between different AEP uncertainty components and having explained their sources, we can compare the total operational AEP uncertainty calculated when allowing for these correlations (Eq. ) with the total uncertainty calculated with the uncorrelated assumption using the conventional sum of squares approach (Eq. ). Figure  shows the results of this comparison for the 472 wind plants considered as a scatterplot and also as a histogram of the difference σtot,corr-σtot,uncorr. A weak bias can be observed with a mean value of +0.1% in uncertainty difference (and differences up to 0.5 % for specific wind plants). In other words, if correlations between the different uncertainty components are ignored in the calculation method, the whole operational AEP uncertainty is then, on average, slightly underestimated.

This difference can be explained by comparing the contributions Rijσi‾σj‾ from the various uncertainty pairs in Eq. () averaged over the 472 considered wind plants. Figure a shows the mean magnitude (across all wind plants) of these contributions for all of the considered uncertainty pairs. The negative correlation between IAV and linear regression has the largest single impact because this correlation involves the two largest uncertainty components (Fig. ). However, the sum of the contributions from all of the positive correlations exceeds the sum of the contribution from the negatively correlated components (Fig. b), thus resulting in the overall average increase in total operational AEP uncertainty when the correlations are taken into account in the calculation.