Inflow wind field measurements from nacelle-based lidar systems offer great potential for different applications including turbine control, load
validation, and power performance measurements. On floating wind turbines nacelle-based lidar measurements are affected by the dynamic behavior of
the floating foundations. Therefore, the effects on lidar wind speed measurements induced by floater dynamics must be understood. In this work, we investigate the influence of floater motions on wind speed measurements from forward-looking nacelle-based lidar systems mounted on floating
offshore wind turbines (FOWTs) and suggest approaches for correcting motion-induced effects. We use an analytical model, employing the guide for the expression of uncertainty in measurements (GUM) methodology and a numerical lidar simulation for the quantification of uncertainties. It is found that the uncertainty of lidar wind speed estimates is mainly caused by the fore–aft motion of the lidar, resulting from the pitch displacement of the floater. Therefore, the uncertainty is heavily dependent on the amplitude and the frequency of the pitch motion. The bias of 10

With many countries worldwide having ambitious targets for FOWT installations and a pipeline of upcoming projects, the installed capacity of FOWT is
expected to increase exponentially in the coming decade. Forecasts expect the global installed capacity of floating wind to reach 16.5

For wind measurements in deep waters, typically met masts are not applicable due to high installation costs. As an alternative, floating lidar concepts
have been developed and are already used in industry projects and research applications. A comprehensive overview of the technology and challenges can
be found in

In

Another application of nacelle-based lidar systems is the use of different lidar-assisted wind turbine control strategies. These control strategies
aim to use knowledge about the approaching wind field to optimize the operation of the turbine. Investigated concepts include collective and
individual pitch control (see, e.g.,

Although different applications require wind speed measurements in different temporal resolutions (e.g., 1

While the abovementioned studies have mainly investigated the use of nacelle-based lidar systems for the measurement of inflow conditions of onshore or bottom-fixed offshore wind turbines, little experience exists for the use on FOWT. Since the floating dynamics of the FOWT causes translational and rotational displacement of nacelle-mounted lidars, it can be expected that these dynamics affect the measurements. Therefore, it is necessary to investigate motion-induced effects and evaluate the need for motion correction.

The quantification of uncertainties and correction of motion influence have in general already been approached by several authors for both floating
and fixed lidar systems. In

In

For nacelle-based lidar systems on FOWT,

The correction of motion influence in lidar measurements has been addressed by different studies.

In

With the present work, we aim to provide missing insights into the effect of floater dynamics on nacelle-based lidar inflow wind speed
measurements. Therefore, we systematically analyze the motion-induced effects for individual floater DOFs and provide methodologies for the correction
of these effects. In short, the objectives of this work are as follows:

to quantify floater motion-induced uncertainties and biases in nacelle-based lidar wind speed measurements on FOWT,

to introduce correction methods for motion-induced effects on lidar wind speed measurements on different timescales,

to assess the introduced correction methods for different floater characteristics and atmospheric conditions.

In Sect.

In Sect.

In Sect.

In Sect.

Utilization of analytical and numerical lidar measurement models.

The methodology covers the introduction of the numerical and analytical models for quantification of uncertainty and bias in lidar wind measurements
on floating wind turbines. A parametric study is used for the quantification of motion-induced effects. Based on the findings of the parametric study,
two approaches for the correction of time-averaged and instantaneous lidar wind measurements are introduced. An overview of the methodology used and
use of the analytical and numerical lidar measurement models is shown in Fig.

The data set analyzed for this study contains data from a forward-looking lidar mounted on the nacelle of the FLOATGEN (

The floating substructure is a barge-type floater employing BW Ideol's damping pool design. It is installed on the SEM-REV test site (

The measurement campaign employed a Wind Iris TC lidar (

The analytical uncertainty model aims to provide estimates of uncertainty for lidar LOS wind speed measurements and wind field characteristics reconstructed from LOS wind speed measurements. The uncertainty estimation according to the GUM methodology requires an analytical description of the measurement. For the case of nacelle-based lidar measurements on a FOWT, the model must contain the relevant dynamic behavior of the FOWT, a description of the wind field, and a model of the measurement itself.

Lidar pattern and coordinate systems for the analytical model.

The dynamic behavior of the FOWT is modeled considering four DOFs, namely the rotational displacement in yaw, pitch, and roll direction, as well the
heave displacement of the floater (see Fig.

The model parameters defining the dynamic behavior of the FOWT are summarized in Table

Considered FOWT dynamics parameters.

In reality, the LOS measurement of a lidar is influenced by various atmospheric and technical parameters. For the analytical calculation of
motion-induced uncertainties, a simplified model of the atmosphere and the lidar measurement is introduced. The horizontal wind speed is modeled using
a power-law profile given by

For the consideration of the dynamic LOS position due to floater motion, two coordinate systems are introduced. Following the notation of

The specific beam geometry used in this study is given in Sect.

Finally, the LOS velocity is mathematically described by a projection of the local wind vector

The dynamic measurement height

Considering the vertical wind profile and the wind direction

Following the recommendation of

The analytical model for the estimation of uncertainty and biases introduced in Sect.

Similar to the analytical model, the positions of lidar focus points after rotational displacement are obtained by multiplication of the LOS vectors
by a rotation matrix

The LOS measurements of the individual beams are modeled as a projection of the wind vector

Lidar pattern and coordinate systems for the numerical model.

Synthetic turbulent wind fields with desired characteristics are created using a turbulence generator. In this work the open-source turbulence
generator TurbSim (

Analysis of measurement campaign data.

Figure

Additionally, the standard deviation within each 10

Since this velocity component is superimposed on the measurement, it can be expected that the pitch motion and related translational velocities of the nacelle will cause significant fluctuations in the lidar measurements. The effect of mean pitch angles and related shift of measurement positions is expected to have a smaller influence on the measurement for this floater type.

This is confirmed by the time series example in Fig.

Finally, the power spectral density (PSD) of the inclination pitch signal and one LOS velocity signal as well as the reconstructed

In this section the findings from uncertainty and bias estimation of lidar measurements under motion influence, as defined in
Appendix

This configuration represents a fixed-beam lidar system with four beams, arranged in a rectangular pattern. The opening angle (angle to the center line)
of all four beams is set to

Modeled LOS velocities and reconstructed

In the first analysis, the influence of displacement in individual DOFs on the measured LOS velocities is examined with the help of the analytical
model. Figure

For a vertical shear exponent of

Modeled LOS velocities as a function of translational nacelle velocities in

For the pitch DOF, the upper and lower beam LOS velocities have different characteristics. Therefore, the reconstructed wind speed is
fluctuating. It is also important to note that the relationship between reconstructed wind speed and pitch angle is nonlinear. This nonlinear
relationship can cause bias in averaged lidar wind speed estimates. Heave displacement in combination with a nonlinear vertical wind shear profile
causes fluctuation in the reconstructed wind speed due to changing measurement elevation. As a result, the reconstructed

Figure

This analysis shows that the most relevant DOF for the influence of nacelle-based lidar wind measurements is pitch motion. Under the presence of a
vertical wind shear profile, the rotational pitch displacement causes a significant variation in reconstructed wind speed. This variation is
nonlinear as a function of pitch displacement, indicating that the displacement could introduce bias in the measurement. The rotational pitch motion
also causes translational velocities of the nacelle in the

Uncertainty estimation of

In this section, we quantify motion-induced uncertainties in reconstructed lidar wind speed estimates as a function of dynamic input parameters using
the analytical and numerical models. For the analytical model, the wind field is only defined by the inflow wind speed of 10

Dynamics parameters for uncertainty quantification.

The resulting uncertainty of the reconstructed

In the next step, the influence of the frequency of rotational pitch movement was investigated. Figure

Verification of the analytical results is done employing the numerical model. Turbulent synthetic wind fields used in the simulations are generated
for a mean wind speed of 10

Wind field parameters.

Comparison between analytical and numerical uncertainty and MAE estimation. Panels

Figure

In general, the pattern of the numerical results follows the estimation of the analytical uncertainty model. The MAE is heavily dependent on the pitch
amplitude and period, which is determining the magnitude of the fore–aft motion of the lidar system. The shear exponent and mean pitch angle variation
show no significant effect on the MAE. Quantitatively, both models show similar magnitudes of uncertainty and MAE values. It can be seen that there is a linear relationship between uncertainty in the

This analysis shows that the uncertainty in reconstructed

In this section, we quantify the motion-induced bias in reconstructed lidar wind speed estimates as a function of dynamic input parameters using the analytical and numerical models. The input wind fields for the numerical model and the dynamics parameter space are the same as previously used for the uncertainty quantification.

Bias of

Figure

Non-zero mean pitch angles result, in general, in upwards- or downwards-shifted focus points. For negative mean pitch angles (upwards-shifted focus points) and the assumption of a power-law wind profile, this results in a positive bias in the LOS measurements. Depending on the magnitude of vertical shear this effect exceeds the negative effect from the pitch motion and leads to an overestimation of wind speed. Consequently, the overestimation is most pronounced for high negative mean pitch and low pitch amplitudes.

For the relationship between vertical shear, pitch amplitude, and resulting bias, it can be seen that the calculated bias is not 0, even in the
presence of zero pitch amplitude. This bias is not introduced by any dynamics but is a result of the definition of the reference value used for bias
calculation. As detailed in Appendix

Figure

Comparison of bias wind speed estimation from the analytical and numerical model. Fixed parameters:

The analytical model results are verified using the numerical model. Figure

This analysis shows that the bias in reconstructed

The results of the numerical and analytical uncertainty and bias estimation have identified the main dynamic effects influencing the measurements of
nacelle-based lidars on floating wind turbines. Motion-induced uncertainties in measurement time series are mainly influenced by the following:

Translational velocity components caused by the fore–aft movement of the nacelle due to floater pitch motion cause fluctuation in horizontal wind speed estimates.

Motion-induced biases in time-averaged wind speed measurements are mainly influenced by the following:

Rotational oscillations of the beam direction due to floater pitch motion cause an underestimation of horizontal wind speed components.

Negative mean pitch angles of the FOWT are causing upwards-shifted focus points, resulting in an overestimation of wind speed.

It is important to note that these effects have different orders of magnitude and are relevant at different timescales. Translational velocities
cause fluctuations in the measurement time series, which can be of the order of 1

Depending on the intended use of the measurement, a correction of these effects can be necessary. Power performance measurement of wind turbines is a
key application of lidar wind measurements. According to IEC 61400-12-1, the performance measurements should be performed by measuring the
10

The use of lidar inflow measurements for turbine load validation has been investigated by several studies using different methods. In

Another application of nacelle-based lidar systems on FOWT is the use of lidar wind speed measurements for turbine pitch control. Here the wind speed
information of the inflow wind field is used as an input to an additional control loop that aims to compensate for changes in wind speed, e.g., through
gusts, by changing the rotor blade pitch angles in order to maintain the rotor speed. This control approach can significantly reduce platform motions
and variations in rotor speed due to disturbance in the form of wind gusts (see, e.g.,

The abovementioned use cases show that motion correction of measurements from nacelle-based lidar systems is necessary, while different timescales have to be considered. In this work, we suggest a method for the correction of fore–aft motion-induced fluctuation based on frequency filtering and a simple model-based correction approach of the turbine mean pitch employing the analytical model.

As shown in Sect.

Instead of correcting lidar measurement time series based on the instantaneous turbine tilt angles, we suggest correcting the reconstructed wind speed
with a model-based approach. The analytical model introduced in Sect.

In this way a four-dimensional look-up table with correction values

Parameter space correction look-up table.

The remaining model parameters are set to fixed values of

Assuming the mean pitch angle and the mean pitch amplitude are known for each 10

The approach avoids the need for synchronized motion time series data, which might not be available in all cases. Inclination sensor signals might be noisy or not accurate enough due to the influence of nacelle acceleration on the sensor. Thus, the suggested approach is easy to implement for practical applications. However, it relies on the availability and accuracy of floater dynamics statistics. Inaccuracies could occur in transient conditions, where mean floater dynamics do not represent the actual floater dynamics sufficiently.

In this study, we investigate the use of frequency filters for the correction of motion-induced fluctuations. We suggest the application of a
frequency filter on the time series of reconstructed

For frequency filtering, we use a bandstop filter characterized by three parameters. The stopband frequency range is given by a

The filter parameters applied in Sect.

We evaluate the correction approaches using the numerical lidar simulation framework ViConDAR. Lidar measurements are simulated by coupling an
aeroelastic simulation of specific FOWT models to ViConDAR. In this way, the dynamics input response to wind and wave conditions is used as an input
to the lidar simulation. Details on the coupling approach can be found in

The first FOWT model employed for the simulation of FOWT dynamics is the Windcrete floater design concept (

Wind field parameters.

Mean errors of

Figure

Parameter space wave conditions, where

The Windcrete FOWT model shows a different behavior. The mean error of the

We first test the filtering approach using the lidar simulation framework ViConDAR for a random realization of a synthetic turbulent wind field with prescribed floater dynamics.

Figure

Time series and PSD of simulated lidar wind speed estimate with and without frequency correction for wave case 3 (see Table

In Fig.

MAE of

Figure

ME of

For the given wind and wave conditions the Windcrete floater shows very small pitch excitation, resulting in small translational velocities of the nacelle and no significant peak in the frequency spectrum. In this case, the MAE is not affected by the frequency filtering approach. On the contrary, the MAE of the filtered lidar wind speed estimates is slightly higher compared to the uncorrected value because the applied frequency acts on the measured wind field spectrum itself. Thus, not only motion-induced frequency components are filtered. MAE values are not significantly influenced by wave conditions because of the small dependency between floater pitch response and different wave conditions.

Besides MAE, the second metric used to evaluate the performance of frequency filtering is ME. While for FLOATGEN, filtering of the pitch frequency
does not introduce additional bias, for Windcrete bias of up to 0.2

The motion influence on nacelle-based lidar measurements was investigated with two different models. The introduced analytical model for the estimation of uncertainties and bias introduces several novelties and benefits compared to already-existing lidar simulation and uncertainty quantification frameworks.

First, it specifically addresses nacelle-based lidar systems on floating wind turbines for which very limited academic and industry experience exists,
and uncertainty quantification is crucial for the application in various use cases. The advantage of this model over other, already available
simulation tools lies mainly in its simplicity. The assumption of a power-law wind profile with no representation of turbulence does not require the
generation of synthetic turbulent wind fields. The representation of floater dynamics in terms of frequency and amplitude parameters of the individual
DOF avoids the need for numerical aeroelastic turbine-floater simulations. Thus, it allows an efficient estimation of motion-induced uncertainties
and biases based on basic design parameters of the FOWT concept and the lidar configuration. In this way, expensive computational numerical
simulation can be avoided, while still considering the most relevant effects of floater motion on the measurement. Based on these estimations and the
intended use of the lidar measurements, decisions about correction approaches can be facilitated. As shown in Sect.

The numerical model is more sophisticated and considers several characteristics which are neglected by the analytical model. In particular, it uses synthetic turbulent wind fields to account for the turbulent nature of real wind fields. Additionally, the probe-volume-averaging effect of lidar measurements is considered. The lidar setup is represented in a more realistic way, considering the temporal relation between the individual LOS measurements. A comparison of the results from the analytical and numerical models shows good agreement between both models. This indicates that the combined effect of turbulence, probe volume averaging, and time relation between the LOS measurements is small. The individual influence of these characteristics cannot be derived from our results, since no sensitivity study was conducted. However, comparing the results of the analytical and numerical model, we find good agreements between the two models, which in general gives confidence in the quantification of uncertainties. Based on a parametric study it was found that the most influential floater DOF for nacelle-based lidar measurement is the pitch displacement, leading to different effects relevant to different timescales. For time-averaged measurements, the pitch motion in combination with a vertical shear wind profile lead to an underestimation of wind speed. For FOWT configurations, operating at a non-zero mean pitch angle, shifted measurement positions in combination with a vertical shear profile lead to an overestimation of wind speed. Instantaneous wind speed measurements are mostly influenced by translational velocities of the nacelle, which are also caused by the rotational pitch movement of the floater. Two different approaches for the correction of these effects were introduced and tested using the numerical lidar simulation framework. The model-based bias correction approach is using bias estimates from the analytical model to calculate correction values as a function of pitch amplitude, frequency, and present shear conditions.

For the testing case using prescribed dynamics based on amplitude and frequency parameters, the correction approach yields very accurate
results. Here, it should be noted that the analytical and numerical model use the same dynamics inputs. In reality, the modeled floater dynamics
of the analytical model might not exactly represent the real floater dynamics. Also, the parameters determining the correction value for every
10

For higher pitch periods, translational velocities of the nacelle caused by floater pitch motions are not increasing the MEA significantly. Thus, the MEA is not predominantly determined by the pitch motion of the floater. Filtering of these frequencies yields increased MAEs, since relevant parts of the wind field spectrum are filtered out. Here, it should also be mentioned that the pitch frequency is assumed to be exactly known for each simulation run and not changing over time, which does not accurately represent the behavior of FOWTs under varying environmental conditions.

The testing cases using the simulated dynamics from FLOATGEN and Windcrete FOWT confirm the findings of the idealized conditions. For FLOATGEN, which
has a relatively low natural pitch period of around 11

In this study, we analyzed motion-induced effects on lidar measurements from forward-looking nacelle-mounted lidars on FOWT. For this analysis, we introduced a new analytical model for the estimation of uncertainty and bias in lidar-estimated wind speed. FOWT dynamics are modeled using amplitudes and periods of floater DOF in yaw, pitch, roll, and heave directions. The deterministic wind field is modeled by a simple power-law profile. Further, we applied the GUM methodology to derive combined uncertainties in the LOS measurements and in the reconstructed WFC. To verify the model outputs we compared the results to uncertainties derived with the numerical lidar simulation framework ViConDAR. This lidar simulation follows a more detailed modeling approach, in particular taking into account turbulent wind fields.

Results of a parametric study showed that the uncertainty of lidar-estimated wind speed estimates is mainly caused by the fore–aft motion of the lidar
resulting from the pitch displacement of the floater. Therefore, the uncertainty is heavily dependent on the amplitude and the frequency of the pitch motion. The estimated bias in 10

Further, we introduced two approaches for the correction of motion-induced effects. We used the analytical model to derive a look-up table of
correction values for 10

Following the GUM methodology for expression of uncertainty (

Accordingly the total uncertainty in the LOS measurements, induced by floater dynamics, is given by

In this section, we derive uncertainties in the reconstructed wind field characteristics based on previously derived LOS uncertainties. Since the
lidar is only able to measure the wind speed in the line-of-sight direction, wind field characteristics including the horizontal wind speed components
need to be reconstructed from the LOS measurements. It is not possible to reconstruct the three-dimensional wind vector from a single LOS measurement
unambiguously. Different approaches to this problem like the velocity–azimuth display technique introduced by

For the analytical derivation of uncertainty, we employ a simple wind field reconstruction algorithm that assumes the

Thus, the reconstructed wind speed component

Again, the uncertainty of the reconstructed horizontal wind speed components can be estimated by combining the standard of uncertainties of the input
quantities by following the GUM methodology. The combined uncertainty of

The correlation coefficients between the LOS measurements of the individual beams are needed as a parameter for the calculation of uncertainty of
reconstructed wind field characteristics. They are influenced by the changing LOS directions and the position of focus points in space and the assumed
wind field. Thus, correlation coefficients are dependent on the set of dynamic input parameters and the phasing between the single DOF. In the model,
the correlation coefficients are calculated for the present set of model input parameters. This is done by evaluating Eq. (

The lidar measurement model introduced in Sect.

Equation (

Figure

The rotation matrix is given by

The analytical lidar uncertainty estimation tool is available on

MG developed the analytical model for the uncertainty estimation and the software implementation. MG conducted the simulation studies and drafted the paper. VP contributed to conceptualization, code review, and review of the paper. JG and PWC contributed to discussions and reviewed the paper.

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.

This study has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Actions (grant no. 860879). Lidar measurement data were kindly provided by the VAMOS project, funded by the German Federal Ministry for Economic Affairs and Climate Action (BMWK) (grant no. 03EE2004A).

This research has been supported by the H2020 Marie Skłodowska-Curie Actions (grant no. 860879). This open-access publication was funded by the University of Stuttgart.

This paper was edited by Jakob Mann and reviewed by three anonymous referees.