Reconstructing turbulent wind-fields using inverse-distance-weighting interpolation and measurements from a pulsed mounted-hub lidar

This study evaluates a numerical multi-beam pulsed lidar mounted on the hub of the NREL 5MW reference wind turbine using the HAWC2 v13.1 numerical sensor for synthetic lidar measurement generation. While initially designed for single-beam operations, it facilitates multi-beam configuration simulations. We conducted an analysis of full-rotor longitudinal wind speed reconstruction by combining inverse-distance-weighting with synthetic sensor data from HAWC2. Utilizing a Mann-generated turbulence box for wind input at U = 11.4 m/s, we examined three lidar configurations for efficacy. The hub-mounted lidar proved efficient in capturing the incoming flow towards the turbine, showing a 60% improvement in overall reconstruction accuracy across the plane compared to the baseline, where hublidar measurements are simple average across the rotor plane. The rotor average wind speed showed a 30% enhancement compared to the baseline. Crucially, the lidar configuration, which impacted the spatial distribution across the rotor plane, emerged as a pivotal factor for effective reconstruction. Proper configuration assessment is essential, especially given the implications of rotational sampling and its impact under various wind conditions, for optimal performance. The proposed method, combining inverse-distance weighting with hub-lidar data for high spatial resolution measurements across the rotor plane, shows significant potential for real-time windflow estimation and lidar-assisted control applications.


Introduction
Nacelle-mounted lidars are crucial for wind measurements in wind turbine applications, but their placement behind the rotor leads to blade blockage, significantly reducing data availability and increasing uncertainty in wind field reconstruction [1,2].
We propose a spinner-mounted multi-beam pulsed lidar (hereinafter referred to as hublidar), which enables spatial measurements across the rotor plane and may enhance wind-field estimation accuracy.
Previous research has explored the concept of spinner-mounted single-beam lidars.In [3], a single-beam continuous wave (CW) lidar (ZephIR) on a 2.3 MW turbine (Vestas NM80) successfully measured upwind wind and turbulence structures in real-time.In [4] multiple configuration scenarios for single-beam CW and pulsed lidars on the NREL 5MW turbine for feed-forward control were examined, finding optimal error minimization with a lidar scan radius of 75% and preview distances between 100 and 160 m.More recently, [5] implemented a spinner-mounted single-beam lidar, both CW and pulsed, to assess its advantages in feed-forward blade pitch control compared to a nacelle-mounted lidar, revealing greater control benefits with the spinner-based single-beam lidar, akin to a four-beam nacelle lidar in feed-forward control performance.
This study utilizes a multi-beam pulsed lidar on the hub of the NREL 5MW [6] with HAWC2 v13.1's hub-lidar numerical sensor [7].We focus on two main objectives: (1) reconstructing the longitudinal wind component across the full rotor plane using inverse-distance-weighting (IDW) interpolation with synthetic hub-lidar measurements from HAWC2 v13.1, and (2) evaluating the effect of three different hub-lidar configurations on our method's efficacy.
This paper is structured as follows: Section 2 details the specifications of the windfield turbulence simulation used as the reference case, the chosen lidar configurations, and a description of the implemented reconstruction method.Section 3 presents the results for wind-field reconstruction and rotor average wind speed estimation.A discussion on the study's assumptions is provided in Sec. 4. Finally, Sec. 5 covers the key conclusions.

Methodology
This section outlines the parameters for the reference turbulence box (Sec.2.1), details the generation of numerical hub-lidar measurements (Sec.2.2), and describes the wind-field reconstruction method using inverse distance weighting interpolation (Sec.2.3).

Mann-generated turbulence box
In this study, we use the Mann model [8] to simulate a turbulent wind for a wind speed of U = 11.4 m/s.The wind condition was defined using IEC-recommended [9] parameters for the Mann model (L=33.6 m, Γ=3.9), with the parameter αϵ 2/3 adjusted for a target turbulence intensity of 10%.
The turbulence box dimensions are 4096 × 64 × 64 (x × y × z), with a grid spatial resolution of 2.0625 m in the rotor plane, resulting in box dimensions of 132 m × 132 m (y × z).The longitudinal dimension extends to 6840 m over 600 s.The inflow wind profile was represented using a power-law model with a shear exponent of 0.2.This turbulence box serves as our reference case for extracting the synthetic lidar measurements.

Synthetic lidar measurements
In HAWC2 version 13.1 [7], a novel sensor simulating a hub-mounted single-beam wind lidar with a pulsed laser system, known as hub-lidar, has been implemented.The hub-lidar rotates with the rotor and is constrained to the azimuthal angle of blade 1 body in the HAWC2 wind turbine model [10].
We utilized this sensor to generate synthetic lidar measurements from a wind-field created as outlined in Sec.2.1, conducting aeroelastic simulations on a rigid tower without tilt.This will neglect the effect of the tilt angle and the relative tower moment on the line-of-sight measurements from the hub-lidar.
The sensor computes line-of-sight (LOS) velocities, V LOS , at each scanning location by converting the u, v, and w components of the synthetic turbulent field into the LOS coordinate system.To mimic the lidar's probe volume, the sensor employs the weighting function described in [11].
Users can simulate various lidar configurations by incorporating multiple beams with distinct parameters in the HAWC2 output channels.Additionally, the sampling frequency can be adjusted by post-processing the HAWC2 output results.
In this study, we explored three six-beam lidar configurations, with each beam sampled consecutively at 200 ms intervals without any switching delay.The six-beam hub-lidar settings in HAWC2 included range lengths from 50 to 300 m at each 10 m (26 ranges), a range-gate length of ∆P = 38.4m, a beam width of ∆L = 24.75m, normalized integration volume halfwidth of dv = 3.0, and integration points equal to N lidar = 200.These parameters were chosen based on a convergence study of the HAWC2 hub-lidar sensor detailed in [10].
The half-cone and azimuthal angles for each configuration are specified in Table 1, with the initial azimuthal angle determined by the starting position of blade 1, measured clockwise from downwind perspective, as detailed in [10].These configuration were determined via a preliminary investigation into good beam settings, not detailed here for brevity.
Table 1.Lidar configurations: Half-cone (θ i ) and azimuthal (ϕ i ) angles Beam Number HAWC2 generates outputs for each measurement, including the location (x, y, z), nominal and volume-averaged LOS, and free wind components u, v, w.
The u lidar velocity is obtained by projecting V LOS onto the longitudinal axis, disregarding the v-and w-components, and is defined as u lidar = V LOS / cos(θ), where θ represents the half-cone angle of each beam.To ensure accuracy, measurements are filtered to fit within the dimensions of the turbulence box.
Figure 1 presents the selected lidar configurations in two perspectives: a rotating frame showing beams in their initial positions on the spinner, and a non-rotating frame demonstrating beam placements after a complete scan, considering a rotational speed of ω = 12.1 rpm for the NREL 5MW reference wind turbine at t = 380 s.The appearance of the non-rotating frame changes with time due to variations in rotational speed.Although configurations 1 and 2 appear similar in the rotating frame, their primary distinction is found in the sequence of beam scanning, attributed to differences in their azimuthal angle definitions.

Inverse-distance-weighting interpolation
IDW interpolation is a technique used to estimate values at unmeasured locations using known values from nearby points.It is based on the concept that a known point's influence diminishes with increasing distance, meaning points closer to the target location have a larger impact on the interpolated value.
Let's define d(x, x i ) as the distance between the target value u(x) (wind speed value that we want to estimate) located at x and a measurement point u i located at x i .In this method, if the target value is located at the same location of a measurements point (x = x i ), then our target value u(x) will be equal to the wind speed at the measurement point, u i .
Mathematically, IDW is expressed as: if d(x, x i ) ̸ = 0 for all i, or equal to u i if d(x, x i ) = 0 for some i, where u(x) is the estimated value at location x, u i are the known values at surrounding points, w i are the weights assigned Figure 1.For the study's selected lidar configurations, the rotating frame is depicted on the left, illustrating beam positions in a fixed plane, while the non-rotating frame on the right shows beam positions after one complete scan at ω = 12.1 rpm for the NREL 5MW reference wind turbine, starting at t = 380 s.It is important to note that the key distinction between configurations 1 and 2 lies in the sequence of azimuthal angles, thereby affecting the order in which the lidar beams conduct scanning.
to each known value, and n is the number of known points used in the estimation.The weight w i is calculated as: where p is a positive exponent that controls the rate at which the weight decreases with distance.A higher value of p results in a steeper decrease in influence with distance.In this study, we chose p = 3, after it showed better performance in a preliminary evaluation.
To calculate longitudinal wind speeds across the rotor plane (YZ) at each time step t, we adopt Taylor's frozen hypothesis [12].For the IDW reconstruction, we select hub-lidar measurements from a specific longitudinal range, including all measurements within this span.This range is determined by the duration of a number of complete six-beam lidar scan, which is t lidar = 1.2 s, given that each beam scans at 5 Hz, equating to 0.2 s per scan.
An illustration of data selection for a specific time step, say t target = 200 s, considering two full scans (n scans = 2) and a wind speed of U = 11.4 m/s, is provided in Fig. 2. Here, the YZ plane reconstruction occurs at x target = 2280 m (calculated as U × t target ), with selected hub-lidar measurements contained within a defined cuboid.This cuboid extends (in the xaxis) from x initial = 2266.32m (calculated as U × (t target − t lidar 2 × n scans )) to x end = 2293.68m (U × (t target + t lidar 2 × n scans )).Measurements within this cuboid (highlighted in blue) are utilized in the IDW reconstruction.
In this study, we evaluated from one to five full-scans (1.2 to 6 s), corresponding to longitudinal ranges of length, x = x initial − x end , from 13.6 m up to 68.4 m.It's important to note that x target represents a longitudinal distance across the entire length of the turbulence box where the reconstruction is executed.In a real-time case scenario, the nearest distance from the wind turbine where the reconstruction can be performed would be approximately 70 m, determined by the hub-lidar's minimum measurement range and the number of scans under consideration.
Using selected data (all measurement points situated across the selected range), IDW calculates longitudinal wind speeds at x target , weighting available V LOS / cos(θ) values.This results in longitudinal wind speeds at each grid point on the YZ plane (N y = 64, N z = 64).
This process is replicated at every time step to produce reconstructed turbulence boxes for the longitudinal component.The finalized longitudinal dimension of these turbulence boxes is N t = 3413, after excluding the initial 80 s and final 20 s from the aeroelastic HAWC2 simulation.This exclusion ensures that lidar measurements are encompassed within the generated box, culminating in a 500 s turbulence box.
For comparison, a baseline is established by averaging lidar measurements across the YZ plane at each t target time step, considering a range for one full scan (13.68 m).This is to mimic the standard practice of estimating the rotor-average wind speed from lidar measurements.

Wind-field reconstruction
To assess the accuracy of IDW-reconstructed wind-fields against reference fields, we calculated the Root Mean Square Error (RMSE) as = ( 1/n n i (y i − ŷi ) 2 ), comparing the reconstructed (y i ) to the reference field (ŷ i ), with n = 3413 representing the grid size longitudinally over 500 s.
Table 2 outlines the RMSE for the longitudinal component of the wind-field across varying ranges (or number of scans), for both IDW reconstructions and baseline cases.This RMSE spans the entire reconstructed turbulence box (3413 × 64 × 64).
The optimal case for each configuration varies.For instance, configuration 1 yields the best results at three scans, showing a 60% improvement over the baseline.For configuration 2, the highest performance is at four scans, marking a 57% enhancement compared to the baseline.Configuration 3's optimal result is at five scans, achieving a 43% improvement relative to the baseline.Additionally, it's evident that configuration 1 achieves the best performance, while configuration 3 shows the least effectiveness among the chosen setups.Notably, less optimal configurations benefit from a higher number of scans, enhancing performance.In contrast, for an optimal configuration, extending the range length beyond a certain point begins to diminish the method's overall effectiveness.
Lastly, the benefit of adding an extra scan (range) for reconstruction is marginal.For configuration 1, an improvement of 4% is observed when increasing from one to three scans.In configuration 2, the enhancement is 3% moving from one to four scans.For configuration 3, the improvement is 4% when expanding from one to five scans.
Figure 3 illustrates the IDW reconstruction across the YZ plane for a specific time step (t = 315.95s), showcasing the best performance ranges (number of full-scans) for each of the three evaluated configurations.The sequence, from left to right, includes available hublidar measurements for reconstruction, followed by the original and reconstructed longitudinal wind speed at the selected time step, and concludes with a comparison of the reference and reconstructed cases, by calculating the difference at each grid point.
The configuration's influence is evident, as the spatial distribution of measurements directly affects method performance.In configuration three (Fig. 3 c), despite having a similar number of points, the beams' overlapping leads to a limited distribution across the rotor plane, consequently reducing the reconstruction's effectiveness.
The comparison reveals that the most significant differences in interpolation occur in areas where no lidar measurement points are available within specific grids of the YZ plane.Additionally, it's important to note that even though the configurations were estimated for a constant rotational speed, since the wind speed selected represents the rated wind speed, the wind turbine is operating at different regimes.This variation in wind speed leads to fluctuations in the rotational speed, and therefore to variations in the spatial distribution for the lidar measurements.

Rotor average wind speed
To evaluate the rotor average wind speed (RAWS), the aeroelastic response of the NREL 5MW reference wind turbine in HAWC2 was evaluated.
In HAWC2, the RAWS is calculated as the rotor average free wind speed excluding tower top motion.In this study, only the longitudinal wind speed was included for the RAWS calculation.
Figure 4 illustrates the RMSE for the RAWS over a 400-second period, showing that all RAWS estimates from the hub-lidar outperformed the baseline, with improvements of 30%, 33%, and 28% for configurations one, two, and three, respectively, compared to their baselines.
This mirrors trends seen in Sec.3.1, where the lowest overall RMSE occurs at three, four, and five scans for configurations one, two, and three, respectively.The optimal reconstruction was achieved by configuration one at three scans, with a RM SE = 0.589 [m/s].Root-mean-square error (RMSE) for the RAWS, calculated between each reconstructed case with the reference one.The selected cases are the best-performing range for each configuration.Additionally, the baseline cases are also presented.

Discussion
Our study introduces a straightforward IDW interpolation method using hub-mounted lidar measurements for full-rotor flow estimation.However, some assumptions made under this study merit further discussion.
In our initial study, the LOS wind speed is scaled by cos(θ), omitting v and w components.This approach may need adjustments for more detailed analyses.Additionally, errors in LOS measurements due to a flexible tower and tilt could impact the accuracy of longitudinal wind speed estimates used in reconstruction.These elements will be examined in subsequent phases of our research.
The concept of perfectly known shear wind speed profiles is overly idealistic.While lidar measurements allow reasonable shear profile estimation, real-world scenarios with more complex shear patterns may pose greater challenges.Future phases of this study will involve performance evaluation using actual hub-lidar data to ensure comprehensive analysis of this phenomenon.
Furthermore, the assumption of a constant wind-flow rate may not hold true in practice (Taylor's frozen hypothesis).Additional research is required to verify the hypothesis's applicability, especially considering factors like rotor induction.
Finally, the method's dependency on measurement spatial distribution presents challenges for turbines with variable rotor speeds.One potential solution could be selecting scans based on data availability to fill in measurement gaps.

Conclusions
Our research focuses on reconstructing longitudinal wind speed across the full-rotor plane at each time step, employing inverse distance weighting (IDW) interpolation with data from an innovative hub-mounted lidar that overcomes blade blockage issues.
We used synthetic measurements from HAWC2 v13.1's numerical hub-lidar sensor, simulating aeroelastic responses with a Mann-generated turbulence box [8] at 11.4 m/s wind speed and 10% turbulence intensity.Three lidar configurations were assessed to gauge the IDW method's effectiveness, alongside a baseline case determined by calculating rotor average wind speed within data a 13.68 m range.
When comparing full-rotor reconstruction, the study revealed that our method significantly enhances accuracy compared to the baseline, with the best configuration yielding a 60% improvement.For rotor average wind speed, we observed a 30% improvement over the baseline, achieving an RMSE of 0.589 m/s.
We observed the hub-lidar configuration's substantial impact on both baseline and IDW reconstruction performance, emphasizing the need for spatially distributed measurements across the rotor plane.Poor configurations led to error increases up to 42%, underscoring the disparity between the worst (configuration 3) and best (configuration 1) reconstructions.
Expanding the measurement range (i.e.including high number of scans in the reconstruction) yielded modest improvements of about 3%, benefiting configurations with fewer data points.However, for data-rich configurations, extending the range eventually results in diminishing returns.
The hub-mounted lidar demonstrates significant promise for real-time wind-flow estimation, with the proposed method averaging a computation time of 0.15 s per time step on a standard personal computer.Its high spatial resolution provides detailed measurements across the rotor plane, facilitating real-time wind-field reconstruction when paired with an appropriate reconstruction technique.This capability is vital for the development of a flow-aware controller, aligning with the objectives of the CONTINUE project.
Ongoing research is exploring further techniques, such as Proper Orthogonal Decomposition (POD) coupled with Inverse Distance Weighting (IDW), to effectively bridge data gaps.

Figure 2 .
Figure 2. Illustration of hub-lidar measurement selection based on the number of scan considered.The hub-lidar measurements inside the limits of the blue cuboid are the ones considered for the reconstruction with inverse-distance-weighting interpolation.The length of the cuboid will be determined by the number of scans selected, in this case n scans = 2, for a time step t target = 200 s.

Figure 3 .
Figure 3. Wind-field reconstruction comparison at t = 315.95s, at slice 2157, for (a) Configuration 1 with three scans, (b) Configuration 2 with four scans, and (c) Configuration 3 with five scans.Sequence from left to right: Available hub-lidar measurements for reconstruction, reference and reconstructed wind-fields, and the differences between them across the YZ plane.

Figure 4 .
Figure 4.Root-mean-square error (RMSE) for the RAWS, calculated between each reconstructed case with the reference one.The selected cases are the best-performing range for each configuration.Additionally, the baseline cases are also presented.

Table 2 .
Root-mean-square error (RMSE) [m/s] for the longitudinal wind-field reconstruction across varying ranges (or number of scans), for three different lidar configurations.