Impact of probe volume and peak detection methods on lidar rotor effective wind speed and turbulence intensity estimations

Lidar simulation techniques are a suitable and increasingly reliable alternative for testing lidar measuring strategies and illustrating their response when combined with modelled wind fields. In this work, two simulation tools are combined to assess the uncertainty in the derivation of the rotor effective wind speed and the wind speed variance from a forward-looking nacelle-mounted continuous wave lidar wind speed estimations. These uncertainties are analysed for a variety of atmospheric turbulence levels and lidar measuring strategies. A synthetic turbulence generator is used to create the reference wind fields. Subsequently, a lidar simulator operated in a continuous-wave mode is used to scan the synthetic wind fields and perform a sensitivity analysis by comparing first- and second-order statistics against reference values. The lidar simulator is enhanced with three Doppler peak detection methods, namely the maximum, the median and the centroid, to extract radial wind speeds from the velocities found within the probe volume. The results show that probe volume and peak detection methods influence the uncertainty of the wind speed variance. The uncertainty in time-averaged and instantaneous rotor effective wind speed estimations is not sensitive to the lidar spatial averaging or peak detection methods investigated. Finally, we saw that the turbulence intensity influences the derived lidar quantities and is the main driver of the variations in rotor effective wind speed uncertainty estimations.


Introduction
As wind turbines become larger, hub height wind measurements result in insufficient information to properly describe the flow across the area swept by the blades [1].Forward-looking nacellemounted lidars can scan a volume in front of a wind turbine, providing information from larger areas (e.g., the area spanned by the rotor blades) compared to the fixed, point-wise measurements of anemometers.Despite well-known advantages over metmasts, lidar can only provide the alongbeam velocity component.Most exactly, the lidar measures all the velocities sensed along the so-called probe volume (PV) and this influences the estimated wind characteristics.
Lidars measure over a certain volume of air centred around the desired focus distance (f d ), where velocities at locations nearby contribute more to the estimated velocity, which is a sort of weighted estimate of the velocities within the probe volume [2].Despite this weighted value yields a good estimate of the fluid velocity [3], a bias can be found in the line-of-sight (LOS) velocities (v LOS ) when a velocity gradient is found within the PV [4].Moreover, an underestimation of the radial velocity variance occurs due to the spatial averaging from the PV.Also, scanning strategies affect the turbulence intensity (TI) [5].Therefore, the study of both PV effects and scanning patterns is crucial to improve lidar estimates of wind characteristics.
The rotor effective wind speed (REWS) can be defined as the average of the longitudinal component of the wind velocities along the rotor area of a wind turbine.It has the advantage of representing, better than a single measuring point would do, the equivalent mean velocity "seen" by the rotor blades, and is a key factor for wind turbine control [6].The potential of using lidars for deriving REWS has been explored in multiple studies [7][8][9][10].
Wind turbulence is closely related to atmospheric effects like vertical shear or buoyancy and can be described as the variations of the wind velocity around a mean value.This work is only based on the assessment of the variance in the u wind speed component, instead of TI to avoid contamination due to mean of the u component.Thus, assuming statistical stationarity [11], turbulence can be characterised by second-order moments (auto-and cross-covariances) of the wind components in 10-min wind speed samples (standard for wind energy applications [12]).Knowledge of TI is an important factor in site assessment and can be used for controller scheduling [13].
In this work, ViConDAR [14,15], for Virtual Constrained turbulence and liDAR measurements, has been used to scan wind fields and provide simulated lidar measurements.
ViConDAR is an open-source lidar simulator that allows mimicking forward-looking nacellemounted lidar measurements.It is coupled to external codes and outputs can be fed into turbulence generators to create constrained synthetic wind fields.It is also able to account for the dynamics of lidar motion to be used e.g., to analyse performance of floating lidars.A new module has been implemented in ViConDAR to improve simulated wind lidar measuring characteristics, accounting for a realistic spatial averaging method in the lidar along-beam direction and a Doppler spectra peak detection method.
The purpose of this paper is to assess the uncertainty observed in lidar-derived quantities, namely REWS and wind speed variance, due to the combined effect of the lidar spatial averaging and signal processing carried out to obtain the peak in the Doppler spectrum of radial velocities found within the PV.The sensitivity of the assessed wind characteristics is studied under combinations of various probe lengths, peak detection methods, TI levels (T I ref ) and scanning strategies.

Methodology
An investigation on the uncertainty in obtaining the REWS and the wind speed variance when scanning turbulence boxes using a lidar simulator has been carried out.The synthetic wind fields with desired characteristics are obtained by using the Mann model [16].Then, ViConDAR [14] has been used to scan the turbulence boxes and retrieve information of first-and second-order wind speed statistics, which will be used for the analysis.

The synthetic wind fields
The reference synthetic turbulent wind fields are created utilizing the Mann model, which is described by three parameters, namely αϵ 2/3 , L and Γ representing the turbulence energy dissipation rate, turbulence length scale and the flow anisotropy.The values used for these parameters are shown in Table 1 and correspond to T I ref of 2% and 18%.These TI levels are considered as representative for low and high turbulence intensities, correspondingly.The evolution of the inflow from the measuring points up to the rotor is not accounted for, thus we assume Taylor's frozen hypothesis.
To partially mitigate statistical dependence on single random seed turbulence realisations, 10 different randomly defined seeds are used to obtain the turbulence fields for each T I ref .As we aim to simulating two TI levels, this makes a total of 20 simulated wind fields.
L ] coordinate system.θ corresponds to the rotation angle around the y-axis of the lidar coordinate system, increasing clockwise from the xy-plane defined by the lidar coordinate system, ϕ corresponds to the rotation angle around the y-axis, increasing clockwise from the x-axis of the lidar coordinate system and f d is the focus distance set to be 100 m.
The properties of the turbulence boxes are summarised in Table 1.The turbulence box is used to define an euclidean right-handed coordinate system considered as our inertial coordinate system (ICS).Each point in the box grid, Figure 1, contains information about its positional vector p(x, y, z), and its wind velocity vector V (p) = (u, v, w), where u, v and w are the longitudinal (x direction), horizontal transversal (y direction) and vertical (z direction) wind velocity components.The discretisation in the direction of the mean wind, which coincides with the x-axis of the ICS is 0.05 s with a total duration of 600 s.The grid is made of 64 × 64 grid points with a discretisation in the y − z plane of 3 m, resulting in a square grid of 189 m centred at hub height (H hub ).The θ and ϕ angles are defined in the lidar coordinate system as the rotation angles over the y and z axes, commonly known as elevation and azimuth angles correspondingly.It is assumed a vertical gradient in the longitudinal wind velocity direction described by the power law with a shear exponent, α = 0.2.This value corresponds to moderate stable conditions and is the IEC recommended value [12] in the case of standard wind turbine classes.The dimensions of the turbulence boxes were conveniently chosen to fully cover the rotor of the wind turbine.For the analysis we chose the DTU 10MW Reference Wind Turbine, with a radius rotor of 89.15 m and 119 m hub height [17].

Scanning strategies and lidar measuring configuration
The lidar configuration considered has been adjusted for a set of lidar scanning patterns and Rayleigh distances (z R ).Lidar scanning patterns are depicted in Figure 2. In [11], authors stressed the importance of certain pattern characteristics to improve lidar velocity components' variance estimates.The patterns selected for the present work share part of this characteristics, and partly aim to check whether some particularities in the scanning geometries like including a central beam or combining different opening angles also influence REWS estimations.
The cross pattern combines several azimuth and opening angles with a maximum of ϕ = θ = 43.5 • in the x and y direction.This maximum coincides with the opening angle of the circular pattern, with a radius of 65 m at 100 m distance.The square pattern has a maximum opening angle of 31.3 • .All the patterns cover ∼ 75 % of the rotor disk depicted in the Figure 2.
The considered lidar measuring configurations are described in Table 2.The lidar is mounted on the nacelle at (y, z) = (0, 0) m, with no offset considered between the lidar and the rotor  center.The focus distance is located 100 m upstream the turbine rotor, in a plane perpendicular to the longitudinal lidar axis, and perfect alignment between the lidar and the mean wind direction is assumed.For this study, we have considered the time step between measurements within a single scan equal to zero, i.e. a time-synchronised multi-beam CW lidar configuration, probing the wind every 5 s at a fixed measurement plane.No noise sources have been considered in the calculations.

Lidar simulator and the new module
To better imitate lidar measurements ViConDAR has been extended with a new module accounting for a realistic approach to the lidar spatial averaging effect introduced by the PV.As mentioned, lidars do not measure in a single point but rather within a volume in the along-beam direction.The Rayleigh length (z R ) is commonly used to characterise the PV of a lidar device.For a CW lidar, z R corresponds to the half-width half-maximum of the weighting function of the probe volume [18].The Rayleigh length of a CW lidar increases with the square of the focus distance and it can be written as [19]: where s is the distance from the focus point along the beam direction.
The overall effect of the PV is that it filters out eddies smaller than the size of the PV.This results in an attenuation of the wind speed variance, thus in an underestimation of the TI that can be obtained from lidar measurements.
In real applications, the lidar yields, out of its frequency analyser, an average of an ensemble of frequency spectra where the dominant frequency corresponds to the Doppler shift, from which the v LOS is derived.To imitate this signal processing, we follow the approach in [20].There, the authors modelled CW lidar simulated measurements by the convolution of the v LOS sensed within the PV and the weighting function Ψ(s): where δ is the Dirac delta function, v are the different histogram velocity bins (here we use a resolution of 0.1 m/s), s is the distance from f d in the along-beam direction and v LOS (s) is the LOS wind velocity at a distance s from the focus point.
Due to the finite dimensions of the turbulence boxes, the weighting function (see Section 2.3) needs to be truncated to ensure that the lidar measurements lie within the limits established by the box grid.The truncation factor M was set to be six times the Rayleigh distance.
This method yields a velocity histogram over which the peak detection method is applied to find the most representative value of the actual v LOS .Figure 3 shows the histogram obtained when measuring an arbitrary grid point.Each point within the PV along the LOS yields a weighted velocity value which is added, based on the distance to f d , into the correspondent bin of the normalised histogram, Ŝ(v, t).ViConDAR has three commonly used peak detection methods implemented: the maximum method (maximum hereafter) simply yields the value corresponding to the velocity bin with highest occurrence (most common velocity); the median method yields the velocity where the cumulative sum of the velocity spectrum values reaches 0.5 (not applied in this work), and the centroid method (centroid hereafter) yields the velocity value corresponding to the geometric centre of the spectrum (arithmetic mean).
A simple analysis of the filtering effects of the different peak detection methods is shown in Figure 4.The analysis is performed for the variance of the reconstructed wind velocity at  an arbitrary location, using a Rayleigh distance of 25 m.The x-axis of the left-handed plot correspond to the 10 random seeds used and for the case of T I ref = 2%.The results of such analysis have been used as criteria for selecting the peak detection methods to be compared, choosing the extreme cases for our study i.e., the maximum and the centroid.In Figure 4 (right) we illustrate, as an example, the combined effect of the lidar spatial averaging and the centroid peak detection method over a u wind velocity component time series measured by the lidar.A lower variance than the reference is obtained in the post-processing of the retrived signal, leading to a underestimation of TI. reference velocity time series on a point-by-point basis.Instantaneous wind speed information can be useful if we want the turbine to react, in a relatively short time, to a specific atmospheric event that may affect the performance of the system.
where M is the number of k points measured by the lidar during the measuring period.The MAE of the wind speed variance is calculated as: where j is the number of seeds and σ 2 ref corresponds to the reference wind speed variance, obtained as the averaged variance over all the points in the grid.
The observed time-averaged REWS uncertainty is shown in Figure 5 for z R distances of 0, 5 and 25 m and T I ref of 2% and 18%.We found similar values and trends for the maximum and the centroid.Here, more than 75% of the uncertainty values are below 0.07 m/s −1 which corresponds to 0.47% of the simulated mean wind speed (see Table 1).We have not been able to identify any differentiating factor in REWS MAE variations attributable to variations in the PV length.However, the measurement patterns influenced the REWS estimations, meaning that attention should be paid to the selection of the pattern geometry when estimating REWS.The cross pattern was found to have the highest uncertainty values for both time-averaged and instantaneous REWS estimates in almost all measurement configurations and atmospheric scenarios evaluated, likely due to the combination of several opening and azimuth angles.We see a higher sensitivity of the maximum to changes in TI levels, suggesting that it is more sensitive to random velocity changes within the PV.This is confirmed by the results of the wind speed variance uncertainty (Figure 7) since the maximum filters out fewer variance, thus resulting into higher sensitivity to changes in the wind speed.
Figures 5 and 6 show that both peak detection methods have, for high TI levels, larger rotor effective wind speed MAE and RMSE standard deviations than for low TIs.However, these standard deviations are influenced neither by PV variations nor by the utilisation of different peak detection methods.This means that the precision of the uncertainty in the time-averaged and instantaneous REWS estimates is mainly influenced by TI and not by the PV length or the peak detection method.The same is true for the precision in the uncertainty in wind speed variance estimations (Figure 7).In Figure 6 we show the trends in the uncertainty of the instantaneous REWS values.For each lidar realisation (each time the lidar completes a pattern) the REWS is compared to the reference value.The maximum absolute difference due to PV variations occured for the centroid and is 0.14 m/s, which corresponds to 0.9% of the simulated mean wind velocity (see Table 1).These results, jointly with those shown in Figure 5, suggest that PV effects and the selection of the peak detection method has no influence over the estimation of the REWS uncertainty.The uncertainty in the estimation of instantaneous REWS values is visibly dependent on TI, a factor to be taken into account when applying turbine control techniques during high turbulence events.
Figure 7 shows the MAE of the wind velocity variance due to the effect of increasing PVs.As shown in Figure 4 (left) and confirmed here, the centroid filters out more variance than the maximum, which means less sensitivity to wind speed variations.The higher filtering level by the centroid can be explained by its definition as the arithmetic mean of the ensemble of wind speeds.The averaging effect leads to losses of possible singularities within the ensemble, thus to an underestimation of the variance.
The variance deficit is also correlated with the TI, having negligible differences between patterns and PV lengths for low TIs.The MAE of the wind velocity variance increases with increasing TI.For the TIs evaluated, the uncertainty in the estimation of the σ 2 lidar differ by at least two orders of magnitude.As an example, the MAE found when using the configuration: circular pattern, maximum and z R = 25 m, is 31% of that found for the centroid-which was the highest observed-under the same measurement conditions.The MAE analysis for T I ref = 2% (not visible in Figure 7) revealed similar trends as for T I ref = 18% at a lower scale, also increasing for increasing PV lengths.The precision in the estimations of the MAE wind speed variance remain approximately constant for each pattern along the different PVs and also for the different peak detection methods.

Conclusions and outlook
We conducted a simulation-based work to assess the uncertainty of the REWS and the wind velocity variance obtained from wind velocity estimations of a forward-looking nacelle-mounted CW lidar under a variety of z R distances, Doppler spectra peak detection methods and atmospheric turbulence levels.Instantaneous REWS uncertainties were evaluated using the MAE metric and time-averaged REWS uncertainties were evaluated using the RMSE.The uncertainty of the wind speed variance is assessed using the MAE.The simulations combined Mann turbulence boxes to create the wind fields and ViConDAR [14] to simulate the lidar measurements.Comparison of the results against actual lidar data is necessary to validate the methodology.
From the results we can say that the REWS uncertatinty is not sensitive to the spatial averaging or the peak detection method.The precision in the estimation of the uncertainty of the time-average and instantaneous values of the REWS is mainly driven by TI.In contrast, both the selection of the peak detection method and the effects of spatial averaging influenced the estimation of the wind speed variance.This effect typically increases with PV lengths and is significantly relevant for the centroid.This is especially important in the case of CW lidars, whose PV increases with the square of focus distance, which makes this type of technology unsuitable for long-distance wind measurements.We see that the precision in the estimations of the wind speed variance uncertainty is also influenced by TI.This precision decreases for higher levels of TI.
Morevover, the uncertainty of instantaneous wind speeds increased with PV lengths and opening angles when measuring vertically heterogeneous flows with P V ̸ = 0 m (not shown here).This effect is more pronounced as TI and wind speed gradients increase within the PV.
As a comparison between the measurement strategies evaluated, we can conclude that, if the objective were to measure the time-averaged REWS, the combination of the maximum and the square pattern would show the lowest uncertainty for longer PVs.If the objective were to minimise the uncertainty in the instantaneous REWS, the combination of the centroid and the square or circular patterns would show the lowest uncertainty.If the purpose of our measurements were to minimise the variance deficit, the maximum should be used to recover the peak frequency of the Doppler spectrum, regardless of the measurement pattern used.
Although other sources of uncertainty are well known, such as inaccuracies in the lidar focusing process or calibration errors, the mathematical framework used does not yet allow this type of lidar drawbacks to be addressed.Future work should therefore focus on the combination of such uncertainty sources and the study of their relative contribution to the final uncertainty.This could help identify the main drivers of uncertainty in lidar estimates, enabling effective alignment of efforts towards improving lidar technology, which in turn would increase confidence and acceptance by the wind industry, and eventually lead to its ultimate consolidation as the all-in-one wind measuring device.

Figure 1 .
Figure1.Angles convention, inertial [x I ,y I ,z I ] and lidar [x L ,y L ,z L ] coordinate system.θ corresponds to the rotation angle around the y-axis of the lidar coordinate system, increasing clockwise from the xy-plane defined by the lidar coordinate system, ϕ corresponds to the rotation angle around the y-axis, increasing clockwise from the x-axis of the lidar coordinate system and f d is the focus distance set to be 100 m.

Figure 2 .
Figure 2. Illustration of the 3 scanning patterns used for comparison.They are set to cover ∼ 75% of the wind turbine rotor.The solid line corresponds to the 89.15 m rotor radius of DTU's 10 MW wind turbine.

Figure 3 .
Figure 3. Ilustration of the velocity spectrum obtained due to the PV spatial averaging effect.Vertical lines in the histogram show the velocity value at the centroid (red), maximum (green) and lidar focus point (black).The z R is set to 25 m and the alpha exponent is set to 0.2.

Figure 4 .
Figure 4. Analysis of filtering effects of the different peak detection methods on the variance measured at an arbitrary location for the 10 seeds used (left) and combined effect of the lidar spatial averaging and the centroid peak detection method over the u-component of the wind velocity (right).

Figure 5 .
Figure 5.Comparison between the REWS mean absolute error (MAE) calculated for the centroid (left) and the maximum (right) for the different scanning patterns.Error bars represent one standard deviation.

Figure 6 .
Figure 6.Comparison of the REWS root mean squared error (RMSE) for the centroid (left) and the maximum (right) for the different measuring strategies.Error bars represent one standard deviation.

Figure 7 .
Figure 7. σ 2 MAE comparison for the centroid (left) and the maximum (right) for the different measuring strategies.Error bars represent one standard deviation.