Through numerical simulations and the analysis of field measurements, we investigate the dependence of the accuracy and uncertainty of turbulence estimations on the main features of the nacelle lidars' scanning strategy, i.e., the number of measurement points, the half-cone opening angle, the focus distance and the type of the lidar system. We assume homogeneous turbulence over the lidar scanning area in front of a Vestas V52 wind turbine. The Reynolds stresses are computed via a least-squares procedure that uses the radial velocity variances of each lidar beam without the need to reconstruct the wind components. The lidar-retrieved Reynolds stresses are compared with those from a sonic anemometer at turbine hub height. Our findings from the analysis of both simulations and measurements demonstrate that to estimate the six Reynolds stresses accurately, a nacelle lidar system with at least six beams is required. Further, one of the beams of this system should have a different opening angle. Adding one central beam improves the estimations of the velocity components' variances. Assuming the relations of the velocity components' variances as suggested in the IEC standard, all considered lidars can estimate the along-wind variance accurately using the least-squares procedure and the Doppler radial velocity spectra. Increasing the opening angle increases the accuracy and reduces the uncertainty on the transverse components, while enlarging the measurement distance has opposite effects. All in all, a six-beam continuous-wave lidar measuring at a close distance with a large opening angle provides the best estimations of all Reynolds stresses. This work gives insights on designing and utilizing nacelle lidars for inflow turbulence characterization.

Inflow turbulence characteristics are important for wind turbine load validation

Assuming statistical stationarity, turbulence can be represented by the variances and covariances of the wind field components

Compared to turbulence estimates from traditional anemometry, the accuracy and the uncertainty of lidar-derived turbulence characteristics can be
affected by not only the spatial and temporal resolutions intrinsic to the lidar systems and the characteristics of atmospheric turbulence but also
the lidar scanning strategies

Lidars measure the radial velocity (also known as the line-of-sight velocity) along the laser beam.

There are two main types of nacelle lidar systems, namely continuous-wave (CW) and pulsed. They mainly differ on the working principle and the way
they probe the atmosphere within their measurement volume. The probe volume of a CW system increases with the square of the focus distance, while the
probe volume of a pulsed system remains constant with measurement range

This work investigates the dependence of the accuracy and the uncertainty of the turbulence estimations on the main features of the nacelle lidars'
scanning strategy, i.e., the number of measurement positions within a full scan, the half-cone opening angle, the focus distance and the type of the
lidar system. We select eight scanning patterns, which are commonly known or widely used in the wind energy industry. Homogeneous frozen turbulence is
assumed throughout our analysis. The Reynolds stresses are estimated via a least-squares procedure using radial velocity variances instead of
computing from the reconstructed mean wind velocities. Estimates from a sonic anemometer at turbine hub height are used as reference. Compared to

This paper is organized as follows. Section

Assuming Taylor's frozen turbulence

We assume that the Mann model well describes the spatial structure of the turbulent flow. Besides

The unit vector

Definition of the coordinate system and beam angles for nacelle lidar modeling.

The radial velocity of a lidar can be written as the convolution of the weighting function

CW lidar

pulsed lidar

Variances calculated from the centroid-derived radial velocities are attenuated by the lidar probe volume, which acts like a low-pass filter to the
wind velocity fluctuations. Therefore, we refer to them as the “filtered” radial velocity variances. If we assume that the lidar probe volume can be
negligible and that

The unfiltered radial velocity variance can be derived by taking the variance of Eq. (

In practice, the unfiltered radial velocity variance

Assuming that all contributions of the radial velocity to the Doppler spectrum are because of turbulence,

The Reynolds stress tensor

To compute

The matrix

This can be written as

To solve the six Reynolds stresses from Eq. (

the lidar has at least six beams or measures at six different locations within one full scan;

the lidar beams have at least two different opening angles.

If a lidar has less than six beams, or the opening angles of all beams are identical and some of the six equations are linearly dependent, we have
fewer knowns than unknowns in Eq. (

All Reynolds stresses apart from

Turbulence is isotropic, i.e.,

The relations between velocity components' standard deviation

Selected lidar scanning patterns for numerical simulations. The SpinnerLidar (

We simulate lidar measurements on the nacelle of a wind turbine with a rotor diameter (

We simulate eight lidars with different scanning patterns, as shown in Fig.

Scanning trajectories of the 4-beam and the 50-beam lidars measuring at

Scanning strategies of the six-beam lidar with

We consider the lidar probe volume when we investigate the dependence of the Reynolds stress estimations on

Parameters for modeling the CW and pulsed lidar probe volume in numerical simulations.

Comparison of the modeled lidar probe volume for CW and pulsed lidars at three different focus distances.

The time lag between each measurement within a full scan is not considered, but it is assumed that measurements are taken at the same time. In the numerical
simulations neglecting lidar probe volume (see results in Sects.

During the period from 1 October 2020 to 30 April 2021, a SpinnerLidar was deployed on the nacelle of a Vestas V52 wind turbine at the DTU Risø campus in Roskilde, Denmark, measuring the flow in front of the turbine. The V52 wind turbine has a rotor diameter of 52

A digital surface elevation model (UTM32 WGS84) showing the Risø test site in Roskilde, Denmark. The height above the mean sea level is indicated by the color bar (in

The SpinnerLidar

The scanning trajectory of the SpinnerLidar in the measurement campaign.

The measurements used for the analysis are from the wind sectors, which are relatively aligned with the mast-turbine direction (i.e.,
the 10

The SpinnerLidar measurements are post-processed to remove the signals reflected by the wind turbine blades, the telescope lens (the beam can hit the
lens perpendicularly) or other hard targets. Such a procedure filters out some measurements close to the middle of the pattern. To compensate for the
nacelle movement, we rotate the system-reported beam scanning coordinates using the 10

Selected lidar scanning patterns (in red) from the gridded SpinnerLidar scans (in light gray), which are at the focus distance of 62

Selected grid cells for the six-beam lidar with three different levels of the half-cone opening angle. The central grid coincides in the three cases. The gridded SpinnerLidar scans are shown in light gray.

To imitate lidars with different scanning strategies, we select SpinnerLidar measurements at certain grid cells to estimate the Reynolds stresses, as
marked in red in Fig.

In this section, we show comparisons of the Reynolds stresses computed from the considered lidars against those from the sonic anemometer at turbine
hub height in bar plots. In the plots, markers correspond to the means of the estimations from 100 turbulence fields, and the error bars are

Reynolds stresses derived from the sonic anemometer and lidars, which have more than six beams and measure at a single distance.

We show in Fig.

Results in Fig.

Figure

Reynolds stresses derived from the virtual sonic anemometer, as well as the four- and five-beam lidars measuring at a single plane and multiple (multi) planes from 100 simulated wind fields. The lidars' probe volumes are neglected.

In case the nacelle lidar has fewer than six beams, not all six Reynolds stresses can be solved from Eq. (

Relative error (%) of the mean values of the lidar-derived along-wind variance to the one from the sonic anemometer. The lidars' probe volumes are neglected in the simulations. Results from the simulations are computed using measurements at a single plane (same set up as Fig.

Both simulation and measurement results show, as a general trend, that lidar-derived

Table

The along-wind variance derived from all considered lidars using the LSP-

Dependence of the Reynolds stress estimations on the increasing half-cone opening angle

Dependence of the Reynolds stress estimations on the increasing focus distance

The results shown in this section include the averaging effect of the lidar probe volume. In Fig.

We study the dependence of the Reynolds stress estimations on the increasing focus distance

Results shown in Fig.

As shown in Fig.

In this work, we characterize turbulence in front of a small wind turbine at

We show from both simulations and measurements that all six Reynolds stress components can be estimated accurately when using a nacelle multi-beam
lidar. Although the spectral turbulence model used here (the Mann model), which is the basis of our simulated turbulence fields, assumes two of these
components to be zero, namely

This study investigated the dependence of the Reynolds stress estimations on different number of beams, half-cone opening angles, focus distances, single or multiple measurement planes, and different types of the Doppler wind nacelle lidars using both numerical simulations and measurements. The considered lidar scanning patterns included the staring lidar (single beam); the 2-, 4-, 5-, 6-, 50-, and 51-beam lidars; and the SpinnerLidar, which reports 400 radial velocities with one scan. We assumed a homogeneous inflow turbulence (both for the simulations and measurements) and Taylor's frozen turbulence (for the simulations). The lidar-retrieved turbulence estimations were compared with those from a sonic anemometer at turbine hub height. Analysis of both numerical simulations and measurements showed that to estimate all the six Reynolds stresses accurately, a nacelle lidar system with at least six beams is required. Also, one of the beams of this system should have a different opening angle. Adding one central beam improves the estimations of the velocity components' variances. Measuring at multiple planes with the same beam orientations only reduces the uncertainty but not the bias in the reconstruction, if Taylor's frozen turbulence hypothesis is applied. All considered lidars can estimate the along-wind variance accurately by using the least-squares procedure and the assumption that the relations of the velocity components' variances are as suggested in the IEC standard. Also, the Doppler radial velocity spectra are needed for the accurate estimations. For both CW and pulsed lidars, increasing the opening angle reduces both the error and uncertainty of the estimations, while increasing the focus distance has opposite effects. In short, from all tested scanning strategies, a six-beam CW lidar measuring at a close distance with a large opening angle gives the best estimations of all Reynolds stresses. The optimum value of the opening angle depends on the Reynolds stress term of interest and also the wind turbines' size. Further studies or experiments are needed to study the best opening angle of the six-beam lidar for different applications.

In this work, the single-point turbulence statistics are estimated using the least-squares procedure, which assumes homogeneity over the lidar scanning
area. Wind turbines nowadays are often operating inside a wind farm or have large spans over the swept area. The assumption of homogeneous turbulence
can be violated under those conditions. Therefore, further studies on the optimized lidar scanning strategy for turbulence estimation should consider
the inhomogeneity of the inflow. Additionally, the proposed nacelle lidar scanning strategies can be used to study the wind evolution, study the spatial
correlations of turbulence and estimate multi-point statistics, which better characterize the inflow that interacts with the turbine than the hub
height ones. The wind field reconstruction of the inhomogeneous wind fields can benefit from constrained simulations, which incorporate lidar
measurements into three-dimensional turbulence wind fields. Future works could also consider the non-Gaussianity of turbulence

Measurements from the SpinnerLidar are not publicly available due to a non-disclosure agreement between the authors and the provider of the data. Simulated nacelle lidar measurements are available upon requests.

All authors participated in the conceptualization and design of the work. WF and AS performed numerical simulations of nacelle lidars without probe volume. AS extended the simulations with lidar probe volume. WF conducted the analysis of field measurements and drafted the manuscript. AP and JM supported the whole analysis. All authors reviewed and edited the manuscript.

At least one of the (co-)authors is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The campaign was conducted as a part of the LIdar-assisted COntrol for REliability IMprovement (LICOREIM) project at DTU Wind Energy.

This research has been supported by the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement (LIKE – Lidar Knowledge Europe, H2020-MSCA-ITN-2019 (grant no. 858358)).

This paper was edited by Sukanta Basu and reviewed by two anonymous referees.