Approaches for predicting wind turbine hub-height turbulence metrics

Hub-height turbulence is essential for a variety of wind energy applications, ranging from wind plant siting to wind turbine control strategies. Because deploying hub-height meteorological towers can be a challenge, alternative ways to estimate hub-height turbulence are desired. In this paper, we assess to what degree hub-height turbulence can be estimated via other hub-height variables or ground-level atmospheric measurements in complex terrain, using observations from three meteorological towers at the Perdigão and WFIP2 field campaigns. We find a large variability across the three considered 5 towers when trying to model hub-height turbulence intensity (TI) and turbulence kinetic energy (TKE) from hub-height or near-surface measurements of either wind speed, TI, or TKE. Moreover, we find that based on the characteristics of the specific site, atmospheric stability and upwind fetch either determine a significant variability in hub-height turbulence or are not a main driver of the variability in hub-height TI and TKE. Our results highlight how hub-height turbulence is simultaneously sensitive to numerous different factors, so that no simple and universal relationship can be determined to vertically extrapolate turbulence 10 from near-surface measurements, or model it from other hub-height variables when considering univariate relationships. We suggest that a multivariate approach should instead be considered, possibly leveraging the capabilities of machine learning nonlinear algorithms. Copyright statement. This work was authored in part by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. Funding provided by the U.S. Depart15 ment of Energy Office of Energy Efficiency and Renewable Energy Wind Energy Technologies Office. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for U.S. Government purposes.

1 Introduction 20 As wind energy expands and becomes an increasingly large portion of energy portfolios, collecting data on wind speed and atmospheric conditions at turbine hub height is essential for the high performance of wind plants and their successful integration the sonic anemometer high-frequency measurements at the sites. We calculate TI as where σ U is the variance of the horizontal wind speed, and U is the mean horizontal wind speed, both calculated over 10-minute intervals to match the current wind energy industry standard (Emeis, 2018).
The TKE quantifies the mean kinetic energy per unit mass of air in turbulent flow and is defined as where we calculate the variances of the wind speed components over 30-minute intervals, a common choice to study atmospheric boundary layer processes (De Franceschi and Zardi, 2003;Babic et al., 2012).
To classify atmospheric stability, we calculate the Obukhov length, defined as where T v is the virtual temperature (K), u * =(u w 2 + v w 2 ) 1 4 is the friction velocity (m s −1 ), k = 0.4 is the von Kármán 65 constant, g = 9.8 is the acceleration due to gravity (m s −2 ), and w T v is the kinematic buoyancy flux (K m s −1 ). As for TKE, we use a 30-minute average period to apply the Reynolds decomposition. To simplify the atmospheric stability classification, we consider stable conditions for z/L > 0 m, and unstable conditions for z/L < 0 m. Other analyses of the stability conditions at these campaigns find few times with neutral stratification given the strong diurnal forcing (Menke et al., 2018;Bodini et al., 2019).

The Perdigão field campaign
The Perdigão field campaign was designed to observe the microscale flow across a valley between two nearly parallel ridges in central Portugal (Fernando et al., 2019). A multinational group of scientists carried out a massive data collection campaign from 1 May-15 June 2017. The site is largely covered in trees, which increases the surface roughness. The areas outside the ridges and valley are largely farmland and eucalyptus groves. Instrumentation at the site includes meteorological towers, scanning 75 and profiling lidars, sodars, radiosonde devices, and a host of other instruments. In our analysis, we consider observations from two 100-m meteorological towers: TSE04, located on the southwest ridge, and TSE09, located on the valley floor, as shown in the map in Figure 1. At each tower, we use data collected by sonic anemometers at 10 m above ground level and 80 m above ground level. The third 100-m tower at Perdigão was excluded from this analysis because of a data outage for a significant portion of the campaign at the 10-m sonic anemometer. We also discard from the analysis time periods where the 80 sonic anemometer observations are affected by the tower wake, which artificially enhance measured turbulence. To do so, we eliminate observations when the wind direction is ±30 • from the direction opposite to the tower boom (285-345 • for TSE04, 305-5 • for TSE09).
Because of the effect of the terrain, different wind regimes characterize the two towers considered here, as seen in the roses in Figure 2. Located on the SW ridge, TSE04 mostly experiences wind from the NE and SW quadrants, with the wind flowing the Perdigão campaign, the aim of the WFIP2 campaign was to better understand wind speed variability through complex terrain to improve numerical weather prediction models. Instrumentation was arranged in a series of nested arrays that range from the mesoscale, so that current numerical weather prediction models could be validated, to the microscale, with meteorological towers and remote sensing instruments deployed amid wind plants along the Columbia River Gorge (Wilczak et al., 2019).
The wind plant shown in Figure 3 is located in very hilly terrain, with turbines arranged on the slope or on top of hills. High 100 wind speeds are generated through the Columbia River Gorge by multiple influences, notably a pressure differential between the cool damp air at the coast, and the warmer drier air to the east of the Cascade Mountains (Sharp and Mass, 2004).
In our analysis, we consider sonic anemometer measurements collected at the so-called Physics Site, the most densely instrumented location at WFIP2 (Figure 3). The data span from 16 July 2016 to 17 March 2017. We use data from a sonic anemometer at 80 m above ground level on tower P12 to quantify hub-height wind speed and turbulence. We also use measure-105 ments collected from 10-m sonic anemometers at towers P04, P05, and P10, just a few kilometers west of P12. The median value of the measurements from the three 10-m sonics was used to characterize near-ground level quantities for this analysis.  At P12, we discard all observations when wind direction is between 45 • and 210 • , to filter out for wake effects caused by either the meteorological tower itself or the nearby wind turbines. No such corrections were needed for the 10-m towers, which are not in close proximity to wind turbines, and whose sonic anemometers were mounted directly on top of the towers, 110 therefore not affected by tower wakes. Roses showing wind speed, stability, TI, and TKE as a function of wind direction at WFIP2 are shown in Figure A1 in Appendix A.

Analysis of hub-height quantities
Hub-height TI and TKE are directly calculated using wind speed components, so the first relationship we consider is how they vary as a function of hub-height wind speed itself. We first analyze the wind speed distributions at each tower ( Figure 4), where 115 Weibull distributions have been fitted to each case. TSE09 has the the lowest average wind speed given its location in the valley, whereas wind speed at the WFIP2 tower has a much wider distribution, with high wind speed occurring more frequently than at Perdigão.
Hub-height wind speeds influence hub-height TI and TKE. Figures 5 and 6 show how hub-height TI and TKE vary as a function of hub-height wind speed at the three considered towers. Because of the large number of raw data points involved, in 120 this and many of the subsequent plots in this paper we adopt a binning approach. In each plot, we bin individual measurements based on the magnitude of the x-axis quantity, so that there are an equal number of raw data points represented by each point shown in the plots. The x and y coordinates of each point shown in the plots are calculated as the median of all the raw data points in each bin. A total of 100 points is shown for each relationship. Furthermore, we use a polynomial regression to determine the line of best fit for each set of points. When looking at such a general relationship between binned data, a defined 125 relationship between hub-height wind speed and TI emerge, but still with a large variability among different sites. In general, hub-height TI displays an inverse relationship with speed-higher wind speed values display lower values of TI-as has been seen at other locations (Emeis, 2018). At TSE04, this relationship levels out at wind speeds higher than about 7 m s −1 . On the other hand, the relationship at TSE09 is closer to being linear, likely because high wind speed rarely occurs at this location. On the other hand, the WFIP2 data show a remarkable agreement with the TSE04 tower. Also, the WFIP2 data set contains more 130 measurements at high wind speeds, where the TI values asymptote as wind speed increases.
On the other hand, hub-height TKE and hub-height wind speed have a direct relationship at all towers ( Figure 6), again with a large variability among the three locations. TSE09 generally displays higher values of TKE compared to TSE04, which is consistent with the TKE roses in Figure 2. WFIP2 has lower TKE values at all wind speeds, but demonstrates a highly consistent relationship.

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Interestingly, hub-height TI and TKE, when compared directly, show a larger divergence in trends between the three towers ( Figure 7). Both TSE04 and TSE09 have lines of best fit showing a proportional relationship between TI and TKE, as is expected from the way those parameters are calculated in Section 2. However, the WFIP2 data show a different trend, with the line of best fit trending slightly downward because of several high values of TI at low values of TKE, likely caused by the faster horizontal winds at this location. If these points were excluded, WFIP2 would also show a proportional relationship between 140 TI and TKE.
Although first-order relationships to predict hub-height turbulence metrics, such as the ones shown in Figures 5 and 6, might seem simple and well-defined (at least when considering binned data), a host of atmospheric and topographic variables have a strong influence on both TI and TKE. In fact, atmospheric turbulence is affected by a variety of conditions, including the stability condition of the atmosphere, the wind direction, and the interaction between the wind flow and upwind terrain. As all these factors can greatly affect TI and TKE, analyzing different conditions separately will lead to a more accurate analysis of the TI and TKE data sets. In the remainder of the analysis, we dive deeper into such multivariate variability to provide more specific results. This analysis emphasizes the challenge of using surface variables to predict hub-height quantities.
To guide this additional analysis, we first assess the impact of atmospheric stability on the distributions of TI and TKE at hub height. Histograms of TI and TKE at the TSE04, TSE09, and WFIP2 towers (Figure 8)  Wind direction regimes can also impact atmospheric turbulence, because the flow interacts with different roughness and vegetation based on the upwind fetch. Figure 9 shows histograms of hub-height TI and TKE for the dominant wind direction 160 regimes (based on the wind roses in Figure 2) at the two Perdigão towers. At TSE04, we consider northeasterly (between 34 • Figure 6. Hub-height TKE as a function of hub-height wind speed at the Perdigão and WFIP2 towers. between 5 and 10 m 2 s −2 compared to the ridge tower. At the WFIP2 Physics Site, winds are dominantly from the west, and a significant portion of the rest of the wind direction range was discarded from the analysis to avoid contamination caused by wake effects, as shown in Figure A1. As a consequence of this reduced variability, we will not investigate the TI and TKE 170 variability as a function of wind direction at WFIP2.
This preliminary analysis shows how clear differences emerge when TI and TKE are segregated by stability condition and wind direction. In the following sections, we dive deeper into how these conditions affect the ability to estimate hub-height turbulent quantities from ground-level measurements.

Predicting hub-height TI from surface-based quantities
As mentioned in the Introduction, hub-height TI is essential for a variety of wind-energy-related tasks, ranging from siting and turbine selection to wake steering and power forecasting. However, hub-height measurements can be difficult to obtain, whereas ground-level measurements are more readily available. Here we test the ability of ground-level atmospheric measurements to predict hub-height TI at the Perdigão and WFIP2 towers.

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The close relationship between hub-height TI and hub-height wind speed at the TSE04 and WFIP2 in Figure 5 points to the potential predictive power of this metric. We test whether this still holds when using ground-level wind speed instead in Figure   10. As noted, we find reduced variability between the Perdigão tower TSE04 and WFIP2, while TSE09 predicts consistently higher values of TI at 80 m due to its location in the valley. Even for TSE04 and WFIP2, which show a wide range of nearsurface wind speeds, the resulting relationship with hub-height TI is close to a constant horizontal line for the majority of the 185 wind speed range, therefore limiting the practical applicability of these relationships to predict hub-height TI from a specific value of near-surface wind speed.
Because ground-level wind speed cannot be used as a strong predictor of hub-height turbulence, there may be a better extrapolation performance when considering the same variable at both ground level and hub height. Figure 11 shows hub-height TI as a function of ground-level TI for the three considered towers. Each site displays a different relationship between hub-height 190 TI and surface-based TI. The WFIP2 data appear in a tightly gathered cluster of near-surface TI data centered around 0.2, while the TSE09 near-surface TI data are more loosely grouped and centered around 0.5, and TSE04 data are spread more evenly across a wider range of near-surface TI. These differences in compactness and locations of data groupings drive differences in the best-fit lines for each case. Although useful for visualizing an overall trend for the relationship between ground-level and hub-height TI, the binning process applied in making these plots might obscure other important trends that could be used to 195 estimate hub-height TI more accurately, if segregating the data based on atmospheric stability or wind direction.
We start by considering the impact of atmospheric stability, which we already found to have a significant impact on the distribution of hub-height turbulence properties (histograms in Figure 8). When we segregate TI data at both hub height and near the ground by stability condition (Figure 12  ground-level TI. This separation is not at all evident in TSE09, where the stable and unstable cases exist as overlapping clouds, indicating that stability is not the governing factor for TI in the Perdigão valley, producing no clear trends. At WFIP2, the unstable condition displays higher values of hub-height TI overall, but the two sets of points overlap at low values of TI, and the trend for the stable condition is downward, unlike the other lines that all trend upward. Therefore, when considering the impact of atmospheric stability, there is no common trend between the three cases, and no general relationship to vertically 205 extrapolate TI emerges. Next, we consider how the relationship between ground-level and hub-height TI varies when considering different wind direction regimes at Perdigão (Figure 13). In fact, these two 100-m meteorological towers at Perdigão experience different  We quantify this variability in terms of the different terrain roughness and vegetation height, which affect the wind flow for the 215 considered wind direction regimes (Table 1). These terrain data sets are obtained from a lidar terrain survey at 20-m resolution described in Santos et al. (2017); Costa et al. (2019). For each tower and each considered wind direction sector, we calculate the mean and standard deviation of each terrain segment within a 5-km radius sector centered on the location of the tower.
We find that at TSE04, both upwind terrain roughness and vegetation height show larger values (both in terms of their means and standard deviations) when the wind is coming from the SW compared to the NE. This variability is consistent with the 220 large differences seen in Figure 13, so that the wind flow at TSE04 is much more turbulent when the wind has interacted with a highly irregular terrain. On the other hand, up-valley and down-valley flow at TSE09 show similar surface roughness and vegetation height, which is consistent with the little variability seen in Figure 13. As a reminder, WFIP2 was not included in this analysis because the vast majority of the data share the same westerly wind regime. Figure 11. Hub-height TI as a function of ground-level TI at the Perdigão and WFIP2 towers. Finally, we consider whether ground-level TKE might be a better predictor of hub-height TI (Figure 14). Once again, each tower displays a different relationship, with the data largely forming loose clouds rather than clear trend lines, even when considering binned data. This loose relationship demonstrates the low potential effectiveness for ground-level TKE to predict TI aloft. As was done for ground-level TI, we segregated the relationships between hub-height TI and surface-based TKE by stability condition and wind direction. We found no novel results, and included the corresponding plots in the appendix 230 ( Figures A2 and A4).

Predicting hub-height TKE from surface-based quantities
TKE is commonly used in atmospheric science, dispersion modeling, drone piloting, and forecasting applications, and therefore estimating hub-height TKE from ground-level measurements would be beneficial to a variety of applications. The relationship between hub-height TKE and ground-level wind speed ( Figure 15) shows a similar positive, near-linear relation as seen when 235 considering hub-height wind speed ( Figure 6). However, in Figure 15 the trend lines for each tower diverge even more, with WFIP2 predicting consistently low TKE values, and TSE09 predicting higher TKE values, but with data only available for low wind speeds. Once again, the specific characteristics of each site have a strong impact on the variability of TKE, limiting the predictive capability of ground-level wind speed.
We anticipate that using ground-level TKE to estimate hub-height TKE would produce the most straightforward relationship, 240 because they are the same quantity measured at two heights. Figure 16 shows that for each tower studied, hub-height TKE has a strong near-linear relationship with ground-level TKE. However, the slope of the trend lines varies depending on the site.
At TSE09 in the valley, surface-based measurements seen in Figure 16 correspond to hub-height measurements of TKE with slopes higher than a 1:1 line. On the other hand, the trends at TSE04 and WFIP2 have slopes lower than a 1:1 line, allowing for little universal predictive power for this metric across the three towers.  Finally, the relationship between hub-height TKE and surface-based TI is explored in Figure 17. While the best-fit lines for the two towers at Perdigão have similar slopes, the WFIP2 tower shows a near-horizontal line. Once again, overall, using TI at ground-level to predict hub-height TKE would not produce accurate results that can be generalized across different sites.

Conclusion
The ability to use measurements taken at ground level of wind speed, wind direction, TI, TKE, and atmospheric stability to es-250 timate hub-height quantities would be extremely useful in all applications where hub-height measurements are crucial, ranging from wind turbine control to power forecasting and-in a broader perspective-pollutant dispersion and drone flight forecasting. Meteorological towers that are tall enough to directly measure hub-height turbulence are both expensive and difficult to construct, especially considering the ever-increasing size of commercial wind turbines, while near-surface measurements are much more readily available. Despite this intense motivation to find accurate approaches for using ground-level measurements 255 to estimate turbulence conditions at hub height, few models have been proposed to date for such application, each with their inherent uncertainty and limitations. The challenge is even larger for sites in complex terrain.
We analyzed atmospheric data at 10 m and 80 m above ground level from three meteorological towers at two sites in complex terrain: the TSE04 and TSE09 towers from the Perdigão field campaign, and a third tower from the WFIP2 field campaign.  First, we investigated the relationships between hub-height TI and TKE as a function of other hub-height quantities. Although satisfactory site-specific relationships emerge when considering bin-averaged data, the raw data actually present a large scatter, which makes such relationships murkier, as seen in Figure A3. In addition, a large variability among the different sites emerges, even when comparing the TSE04 and TSE09 towers at Perdigão, which are only located a few kilometers apart. The limited skill in predicting hub-height turbulence from a single other atmospheric variable is also confirmed when considering ground-level measurements as predictors. We find large across-site variability when trying to predict hub-height TI and TKE from ground-265 level wind speed, TI, or TKE. Also, when segregating the data by atmospheric stability, a more nuanced picture emerges.
Although unstable conditions are connected to stronger turbulence at WFIP2 and the ridge-top TSE04, atmospheric stability is not a main driver of hub-height turbulence regimes when considering the TSE09 valley tower. Finally, we find that the impact of different wind direction regimes on hub-height turbulence is again highly site-specific, arising from the terrain roughness and vegetation upwind.

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Clearly, the results of our analysis underscore the sensitivity of hub-height turbulence to numerous different factors that simultaneously contribute to the variability of hub-height turbulence, so that no simple and universal relationships can be derived when considering a univariate approach. Given the complexity of the desired relationship, and the inherent nonlinearity of turbulent processes, machine learning could be leveraged to successfully model hub-height TI and TKE from a set of other atmospheric variables. Recently, machine learning approaches have been successfully developed and applied to vertically 275 extrapolate wind speed (Bodini and Optis, 2020b,a;Optis et al., 2021), both onshore and offshore, and to better parameterize surface fluxes (Kosovic et al., 2020). Future work could expand these recently developed approaches to predict hub-height turbulence instead, such that assessments of wake steering potential would not need to assume constant values of turbulence Bensason et al. (2021). In addition, our analysis could be replicated in more simple terrain (Takle et al., 2019) or offshore, to assess whether relationships to derive hub-height turbulence can more easily be derived in less complex topography.

Appendix A: Additional analysis for predicting hub-height TI
In this study, parameters from the WFIP2 field campaign were not separated by wind direction as was done with the Perdigão towers. As seen in Figure A1, the dominant wind direction is from the west. Measurements from 45-210 • were discarded due 285 to wake effects from the tower or nearby wind turbines.
When segregated by stability condition, Figure 14 shows that stable conditions consistently display lower values of TI at each TKE level. Even though TKE at ground level will not likely lead to an accurate prediction of TI at hub height, the differences caused by stability further demonstrate the inability of TKE data at ground level to accurately predict TI aloft ( Figure A2). Figure A1. Roses of wind speed, atmospheric stability (expressed in terms of z/L), TI, and TKE at WFIP2, using data from the 80-m sonic anemometers.
At both TSE04 and TSE09, when segregated by wind direction, there is little distinction between the two dominant wind 290 directions at each tower, as seen in Figure A4. For these two cases, segregating by wind direction does not improve the predictive ability of ground-level TKE.
Appendix B: Additional analysis for predicting hub-height TKE Hub-height TKE as a function of ground-level TKE segregated by stability condition is shown in B1. Each tower displays widely varying trends between stable and unstable cases. At TSE09 there is little distinction, while at TSE04 and WFIP2 the 295 fit lines of the stable and unstable cases depart dramatically. Overall, no universally valid relationship for using ground-level TKE and atmospheric stability to predict hub-height TKE is found.  When considering the relationship between hub-height TKE and ground-level TKE when segregated by wind direction at Perdigão ( Figure B2), TSE04 shows a greater distinction between wind directions than TSE09, likely because there was a greater difference in roughness and vegetation height in the SW and NE flow directions at TSE04, than between the NW and 300 SE directions at TSE09 in the valley.
Hub-height TKE as a function of ground-level TI segregated by stability condition can be seen in Figure B3. Stable conditions consistently predict lower values of hub-height TKE; however, the trend lines vary significantly across the sites, not providing useful information regarding the relationship.
Finally, hub-height TKE as a function of ground-level TI segregated by wind direction is shown in Figure B4   Perdigão field campaign. In particular, we are grateful for the human and logistic support Felicity Townsend provided to our research group in the field. We thank Jose Laginha Palma for providing the vegetation data used in the analysis. We express appreciation to NCAR/EOL, the Danish Technical University and INEGI for their herculean efforts to construct and maintain the datastreams from the networks of towers at Perdigão. We thank the WFIP2 teams from the National Oceanic and Atmospheric Administration, the Pacific Northwest National Laboratory, and Notre Dame for their efforts in constructing and maintaining the physics site at WFIP2. We also thank Dr. Mike Optis for the 315 initial conversations to envision the analysis. Support for JKL and HGL was partially provided by the National Science Foundation awards AGS-1565498 and AGS-1554055. Support for NB was provided by the DOE Wind Energy Technologies office under the AWAKEN project.