Modelling the Wind Turbine Inflow with a Reduced Order Model based on SpinnerLidar Measurements

Preview measurements of the inflow by turbine-mounted lidar systems can be used to optimise wind turbine performance by increasing power production and alleviating structural loads. Here, we apply Proper Orthogonal Decomposition (POD) to the line-of-sight wind speed measurements of a SpinnerLidar obtained from a large eddy simulation of a wind turbine operating in a turbulent atmospheric boundary layer. The aim of this work was to identify the dominant POD modes to derive a reduced order representation of the turbine inflow without making strong assumptions about the flow field. This dimensional 5 reduction is a first step towards the development of a reduced order inflow model (ROM) that offers a trade-off between wind field reconstruction techniques requiring flow assumptions and more complex physics-based representations. We found that only a few modes are required to capture the dynamics of the wind field parameters commonly used for lidar assisted wind turbine control such as the effective wind speed, vertical shear, directional misalignment. A possible interpretation of the modes is presented by direct comparison with these wind field parameters. Evaluating six different metrics in the time and frequency 10 domains related to the spatial, frequency domain and energy quantities, we find that a 10 mode ROM could accurately describe most spatio-temporal variations in the inflow. The reduced order modelling was accomplished using the inherent volume averaging property of lidar devices that attenuates high frequency turbulence with lower importance for the overall turbine response thus allowing significant data compression. Based on the models inflow wind field reconstruction performance, this method has potential use for lidar-assisted control, loads validation and turbulence characterisation. 15

The wind data quality extracted by the lidar depends upon the quality and beam scanning strategy of the lidar device itself.
Modern lidar systems can perform very fast scanning measurements to capture the wind field with more detail. Such lidars represent an improvement over the multiple fixed-beam systems commonly found in commercial devices as they are instead outfitted with beam-steering mechanisms capable of moving and refocusing the laser beam to a predefined point or a scanning pattern in space (Mikkelsen et al., 2008). With such advanced devices available, it is possible to measure the inflow of a large 30 wind turbine with high spatial and temporal resolutions. Due to the trend toward larger rotors, local wind field variations affect the turbine dynamics more strongly, hence it is necessary to measure wind fields over the entire rotor area.
Due to lidar's inherent line-of-sight limitations, wind field reconstruction (WFR) methods are required to extract even relatively simple parameters such as effective wind speed, direction and shear. Two types of WFR models can be found in the literature, i.e. static and dynamic WFR methods (Borraccino et al., 2017). In the static approaches (Borraccino et al., 2017;Kapp, 2017), 35 wind fields are assumed to be stationary for a certain averaging period and spatial flow assumptions are made to determine relevant wind parameters. These models are adequate for power performance measurements as they well estimate the averaged wind characteristics. Kapp (2017) derived a five-parameter representation (i.e. the effective wind speed, and horizontal and vertical shear and directional parameters) from lidar measured wind fields by combining planar measurements at two different upstream distances. Taylor's Hypothesis (Taylor, 1938) is assumed between the measurement planes to resolve the ambiguity in 40 shear and direction. Borraccino et al. (2017) introduced a static method based on a model-fitting technique and a least-squares solver to robustly reproduce ten-minute averaged wind parameters. This model was validated using full-scale measurements performed with two commercially available lidar devices ( the Avent five-beam demonstrator and the ZephIR dual-mode lidar).
In the dynamic reconstruction methodologies, both spatial and temporal variations of the wind fields are taken into consideration. In Raach et al. (2014), a 3D model based dynamic WFR technique was presented by combining the static model presented 45 in Schlipf et al. (2012) with Taylor's frozen turbulence hypothesis. Towers and Jones (2016) described a dynamic reconstruction methodology extracting two-dimensional horizontal wind fields at hub height from a pulsed lidar system with two fixed beams using an unscented Kalman filter with a lower order dynamic model to fill in the non-measured 2D velocity components of the wind fields. Guillemin et al. (2018) presented means of extracting real time 3D wind field parameters using a recursive weighted least-squares method validated with simulated pulsed lidar measurements. Kidambi Sekar et al. (2018) investigated 50 the performance of LINCOM (a fast Navier-Stokes physics-based solver) to instantaneously reconstruct the local 3D velocity components from line-of-sight measurements of a SpinnerLidar on a spherical measurement plane upstream of the rotor.
Taylors hypothesis is assumed in the WFR methods proposed in Raach et al. (2014) and Kapp (2017): however, it does not hold in the turbine induction zone and complex inflow situations. Borraccino et al. (2017) used also a static method to provide tenminute wind statistics rendering the method unusable for control and loads validation. The 2D reconstruction method suggested 55 in Towers and Jones (2016) also assumes that measurements along all the beams are available simultaneously. Consequently, the performance of the models is limited for either reconstructing high frequency information (on the order of several seconds) or in situations where flow simplification assumptions are invalid, i.e. complex terrain or wind turbine wakes.
One alternative method is to accurately represent the inflow without making strong assumptions concerning the wind field.
With recent advancements in scanning lidar technology such as the DTU SpinnerLidar (Mikkelsen et al., 2013;Herges et al., 60 2 https://doi.org/10.5194/wes-2021-16 Preprint. Discussion started: 10 March 2021 c Author(s) 2021. CC BY 4.0 License. 2017), which provides inflow scans with unprecedented spatio-temporal resolutions, it is possible to apply reduced order modelling techniques to real-time lidar inflow measurements. Using such high resolution data directly as a control signal is not feasible. A crucial step towards a lidar-based turbine control is a reduction of the measurement data to a few key variables, which still capture the most important spatio-temporal flow variations in the wind.
There are multiple ways to extract dynamic flow field information in fluid dynamics. One straightforward approach involves decomposing the flow field into a collection of spatial modes of which the most dominant are used to create a reduced order model (ROM) representation of the flow. Proper Orthogonal Decomposition (POD) is such an dynamic approach. This technique describes a velocity field as a linear combination of modes containing spatial information about the flow and time varying weighing functions defining the evolution of the flow field in time (Berkooz et al., 1993;Holmes et al., 2012). Mathematically, the POD method calculates deterministic orthogonal basis functions for representing a spatio-temporal field. The decomposi-70 tion is unbiased because it does not look for prior information and the basis functions are obtained from the dataset itself, in contrast to other techniques. As the modes themselves are orthogonal, this method is suitable for constructing reduced order models by directly truncating higher modes or using Galerkin projection to capture dominant flow physics and to implement model based closed-loop flow control (Taira et al., 2017). The dominant structures obtained via POD decomposition are representative of the coherent motions in the wind flow (Holmes et al., 2012).

75
For wind energy applications, POD have been used to develop and understand dynamic wake models (Bastine et al., 2015;Andersen et al., 2017). Saranyasoontorn and Manuel (2005) applied POD to study inflow wind fields based on stochastic wind field simulations. They concluded that the inflow of a wind turbine can be represented by a very small number of POD modes for the longitudinal wind component. This was confirmed by Kidambi Sekar and Kühn (2017) where the methodology was applied to full field SpinnerLidar measurements. A POD based reduced order model offers a trade-off between the simplicity 80 of traditional WFR methods requiring flow assumptions, and more complex solvers using the line-of-sight measurements as a boundary condition for solving the Navier-Stokes equations.
The objective of this paper is to introduce a POD based dynamic WFR methodology that does not require strong assumptions about the reconstructed wind field. We aim to identify the dominant inflow spatial modes that can be used to obtain a lower order model representation of the velocity field that is scanned with a spinner based lidar while retaining most of its information.

85
To this end, we apply the POD method to virtual SpinnerLidar inflow measurements of a wind field taken from a large eddy simulation (LES) to have a realistic representation of the atmospheric boundary layer flow. The POD modes of the inflow field are estimated and used to reconstruct a reduced order model of the original flow using kinetic energy as the filter. Our second aim is to assess the quality of the WFR method based on relevant information concerning the turbine inflow. The evaluated metrics include the effective wind speed, vertical shear and horizontal misalignment and turbulence spectra in the fixed and 90 rotating reference frames. Additionally, we aim to interpret the modes based on their relationship to the defined metrics. The efficiency of the resulting ROM in capturing the dominant dynamics of the inflow while allowing for data compression is evaluated.
The article is structured as follows. The measurement device and LES simulations are described in Subsection 2.1 and Subsection 2.2. Subsection 2.3 describes the POD theory. The application of the POD method and the results are described in Section 3 where the method is applied to the LES simulations. POD based reconstruction of the original velocity fields is introduced followed by a quantitative analysis of the reduced order model based on wind field metrics. The performance of these metrics are investigated and compared against a reference. Section 4 discusses the results and Section 5 presents the conclusions.

Methods
To obtain a realistic wind field dataset to investigate the inflow to a wind turbine and create a reduced order model representation 100 we employ virtual SpinnerLidar data derived from high fidelity LES. The SpinnerLidar specifications and working principles are presented in Subsection 2.1. The LES providing the wind field data are explained in Subsection 2.2 along with the Lidar Simulator (LiXim). In Subsection 2.3, the ROM is introduced based on the POD methodology while Subsection 2.4 describes the method used to extract and compare wind field parameters.

105
The SpinnerLidar (Mikkelsen et al., 2013;Herges et al., 2017) is a modified ZephIR 300 continuous-wave Doppler lidar with a 2D scan head developed by the Technical University of Denmark (Fig. 1). This lidar can perform performing 2D measurements on a spherical surface of the radial line-of-sight wind speeds in front of a wind turbine with very high spatial and temporal resolutions. The device consists of two rotating prisms deviating the lidar's focused beam by an angle of 15 • while rotating at a fixed ratio of 7 to 13. The resulting scan pattern movement creates a fast rosette trajectory covering a large area with a 110 quasi-homogeneous spatial resolution. The lidar is capable of providing 2D wind field scans at a temporal sampling rate of 1 Hz with a variable focal distance from 10 m to 150 m (albeit with a constant opening angle of 30 • ). It can be set to sample a maximum of 500 radial line-of-sight measurements points distributed over each completed scan trajectory (Fig. 2). We note that the lidar can only provide wind speed measurements along the beam direction (so called the line-of-sight (v los ) 115 wind speeds): usually referred to as the 'cyclops dilemma' in literature (Schlipf et al., 2011). The line-of-sight v los speed is expressed as a projection of the three wind speed components on the line-of-sight, as described by Harris et al. (2007) and (1) Here, α is the azimuth and δ is the elevation angle of the horizontal and vertical direction of the focused laser beam, respectively, The probe length is considered to be twice the half width at half maximum (Γ) which is the distance from the focal point at which the backscatter spectrum is reduced to half its peak power and depends quadratically on the focal distance (Fig. 3). The

Large Eddy Simulations (LES) and the Lidar Simulator (LiXim)
The wind field information for the analysis was generated using LES. LES models are capable of accurately resolving the turbulent kinetic energy in the atmosphere making them a suitable candidate for simulating realistic inflow conditions while 140 also providing a full 3D reference wind field for comparison and quality assessment. The LES data were obtained from simulations with the Parallelised Large Eddy Simulation Model (PALM). The PALM code is widely used for atmospheric boundary layer studies and works by solving the filtered, incompressible, non-hydrostatic Navier-Stokes equations. Further details of the model are available in Raasch and Schroter (2001) and Maronga et al. (2015). PALM employs the Schumann volume averaging approach and uses central differences to discretise the non-hydrostatic and incompressible Boussinesq approximation   To extract the SpinnerLidar measurements from the LES wind field, we used the integrated lidar simulation toolbox LiXim (Lidar Scanner Simulator) developed by Trabucchi (2020). The SpinnerLidar is simulated in the LES wind field using the LiXim simulator measuring at a focal distance of 126 m with a temporal resolution of 1 Hz for a total duration of 3700 s. The decomposition procedure is to sample the data, calculate the auto-covariance matrix, and solve the corresponding eigenvalue problem which can then be used to construct an orthogonal basis. Several references describing the general POD methodology 180 can be found in Berkooz et al. (1993), Sirovich (1987) and Holmes et al. (2012). The practical implementation of this method for application to SpinnerLidar measurements is discussed here. The line-of-sight measurements of the SpinnerLidar v los (X, t) Where n is the number of snapshots and N is the total number of grid points in each snapshot. For a line-of-sight velocity field 185 V (X, t), ||V (X, t)|| 2 is related to its turbulent kinetic energy. It is common to subtract the mean value to obtain the fluctuating where φ j (X) are called the spatial POD modes optimal with respect to the flow turbulent kinetic energy and Z j (t) are the time evolving POD weighing coefficients. This solution is obtained by solving the eigenvalue problem of the covariance matrix resulting in a set of eigenvectors φ j denoted as POD modes and a set of corresponding eigenvalues λ j which can be ordered as The flow field can now be denoted as a linear combination of N uncorrelated spatial modes:

195
where the j th weighing coefficient is obtained as Z j (t) contains the temporal gains determining the development of the POD modes in time. A reduced order representation of the flow fieldV (X, t) = V (X, t) +V (X, t) can be obtained by sorting and assembling the modes by decreasing magnitude of the eigenvalues and by truncating the higher modes and retaining only M < N modes where For such a reduced order approximation, the POD modes of Eq. 5 are optimal with respect to the turbulent kinetic energy in the flow. Hence, they are a set of optimal modes with the least mean square error given by The spatial modes φ j (X) contain information about coherent structures as the POD method can be seen as an energy filter 205 that unravels the large spatial turbulent structures. However, it must be noted that these structures might not be actual physical structures present in the flow field, but merely a result of the mathematical operation. An important property of this method is that the POD modes are orthonormal and their temporal gains are uncorrelated (Holmes et al., 2012). With large sizes of the flow matrix being common in fluid flow problems, the dimension of the covariance matrix R becomes quite high, thereby making the application of classical POD very time consuming. To avoid this, the method of snapshots proposed in Sirovich

210
(1987) is used whereby the temporal covariance matrix R = V T V is solved to obtain the same dominant spatial POD modes.
Due to the reduced computational and memory resources needed, the snapshot method is the most commonly used method for obtaining POD modes from flow data.
We can now apply the snapshot POD methodology to the line-of-sight velocity field v los (X, t) obtained via SpinnerLidar measurements, truncate the higher modes and create a reduced order model of the flow. The order M of the reduced model in 215 Eq. (8) is crucial. Improper selection of M might lead to a dimensional reduction that is either very large or very small and important flow field data may be lost. We aim to identify dominant inflow POD modes that will preserve the inflow's dynamic characteristics while providing a significant dimensional reduction.

Quantifying the accuracy of the Reduced Order Model
The main objective of any inflow model is to capture the inflow, along with its variations that significantly impact the wind 220 turbine. Thus, the quality of ROM reconstruction should be evaluated with respect to wind field parameters that directly affect the turbine itself. Typical parameters required to implement a standard individual pitch controller are the rotor effective longitudinal wind speed u eff , horizontal misalignment δ h and vertical shear s v . To extract the wind field parameters from the SpinnerLidar line-of-sight measurements, we use a methodology based on Kapp (2017). As two measurement planes are needed to resolve the ambiguity between the shear and direction parameters, a simplified 3-parameter model consisting of the 225 rotor effective wind speed u eff , horizontal misalignment δ h and vertical shear s v is obtained by omitting the horizontal shear s h and the upflow δ v . The effective wind speed quantity u eff is related directly to the turbine's dynamic response and power output. This is the main variable for selecting the operational condition of the turbine and an input for collective pitch control.
The vertical shear s v is important for individual pitch control algorithms, whereby the controller compensates for vertical shear present in the ABL by pitching the blades individually. The horizontal misalignment δ h is an important parameter for 230 determining the turbine's yaw setting. To quantify the reconstruction error associated with these wind field parameters, we write: where Y (t) is the set of rotor-averaged wind field parameters u eff , δ h , s v calculated by applying a 3-parameter functionŶ to the line-of-sight velocities. Similarly, the wind field parameters for the reduced order model with M modes can be represented 235 as: The quality of the reconstructed wind field v M los (X, t) and the corresponding wind field parameter Y M (t) with M modes can be calculated by comparison to the original wind field parameters Y (t). To evaluate the reconstruction efficiency, two different error parameters are introduced as defined by Bastine et al. (2015). The error associated with the reconstruction of a wind field 240 parameter Y (t) is defined as: where std and dyn represent the standard and dynamic error respectively. These values respectively quantify the mean error and the error associated with the fluctuations Y (t) − Y (t) in the wind field. The definition of the dynamic error is motivated by the fluctuation of wind field parameters having has the largest effect on the loading and the fatigue on the turbine in contrast 245 to the mean wind field parameters. To investigate the relationship between the POD temporal gains Z j (t) and the wind field parameter time series, the Pearson correlation coefficient ρ is used: Here, cov is the covariance of Z j (t) and Y (t) and σ are the respective standard deviations.

250
The performance of the POD based reduced order model is tested for the virtual SpinnerLidar in the LES wind field. The simulations are used as a benchmark for comparing and quantifying the accuracy of the inflow model based on different reconstruction metrics such as the convergence of POD modes, eigenvalue distributions, velocity field reconstruction, wind field parameterisation and turbulent spectra in the fixed and rotational frames of reference.

255
The first test of the performance of the wind field reconstruction model is to check for convergence of the POD modes.
Typically, the dominant high energy POD modes associated with the large-scale turbulent structures are well converged. To test the convergence of POD modes, velocity data over various numbers of timesteps are taken and the change of POD modes with time is examined. Ideally, the process is said to converge if the eigenvalues and POD modes shapes remain constant as the numerical solution progresses. Figure 5 illustrates the eigenvalues associated with the first six POD modes and their 260 corresponding rates of change with respect to the number of samples. The magnitude of the first six eigenvalues vary up to a dataset size of n = 1000 samples and start to converge around n = 3500 samples due to the temporal correlations in the wind field. The eigenvalues rate of change of the falls below 5% for the first 6 modes calculated with more than 3000 samples. does not produce any significant changes in the shape of the modes. As the dominant modes obtained for the dataset are well converged, the use of POD to extract large scale structures is considered to be justified (Hekmati et al., 2011). Further on, we apply the POD method to the total sample duration of 3700 s.

Application of the POD methodology to SpinnerLidar measurements
The eigenvalues and eigenvectors associated with the covariance matrix for the line-of-sight velocity field are calculated based The normalised magnitudes of the eigenvalues λ j and the fraction of energy associated with each mode of the v los measurements are shown in Fig. 7. The first POD mode contains 57.7% of the total measured energy while the second, third, and fourth equivalent to that in the wind field due to volume averaging induced turbulence attenuation of the line-of-sight measurement.
This may be surprising since in a turbulent flow, energy is distributed over the different scales and its representation might usually require an enormous number of POD modes. However, a lidar system acts as a low-pass filter for small scale turbulence due to its volume averaging property. Hence, the small scale turbulence is filtered out and an accurate representation of the remaining SpinnerLidar measured total TKE can be recovered with very few modes. the horizontal and vertical directions respectively. Modes 5 and 6 have quadrupole distributions while modes 7 and 8 show weak hexapole distributions. These observations exhibit weak statistical isotropy commonly observed in turbulent flows. As expected, the higher mode patterns become increasingly complex compared with the first few modes. It should be noted that the shape of the POD modes will differ based on the inflow conditions due to variable behaviour of the ABL under different 290 stratification conditions.

Reconstruction of the velocity field
The extracted POD modes are used to reconstruct the velocity field based on Eq. (8) by choosing increasing values of M (the number of modes used for reconstruction). A snapshot of the velocity field at an arbitrary time t = 256 s is reconstructed and illustrated in Fig. 9 for different number of modes along with its corresponding planar velocity reconstruction error in 295 comparison to full lidar measurements. As expected, a more detailed wind field reconstruction is achieved with an increasing number of modes. For the reconstruction with M = 1 (first mode alone), only the mean wind speed of the measurements is obtained, as indicated by a constant velocity distribution over the whole measurement plane which is also supported by the M = 1 error plot with respect to the full lidar measurements. The addition of more modes into Eq. (8) adds more localised wind field information as the smaller wind field fluctuations which are contributed by the higher modes are taken into account.

300
The velocity field reconstruction with ten modes shows close agreement with the untruncated full lidar measurements with reconstruction error effectively reducing to zero throughout the measurement plane. The high recovery of kinetic energy in the first few modes as discussed in Subsection 3.2 indicates that only a few modes are required to create a meaningful reduced order model capturing all spatial variations in the wind field.   The dynamic error for u eff drops below 3% when at least the first three modes are added. The dynamic error of s v reduces to 10% with the addition of the first four modes. The dynamic error of δ h declines below 10% with the addition of the first seven modes. Both the standard and dynamic error fall below 0.5% for all wind field parameters by including 10 modes in the reconstruction. The discontinuous behaviour shown by the standard and dynamic error occurring at identical mode numbers indicates a relationship between the particular mode and the corresponding wind field parameter.

335
The huge reductions in the standard and dynamic errors can be explained by the plots of correlation between the time series of the POD time coefficients Z j (t) and the wind field parameters X(t) as visualised in Fig. 11 (c). Confirming the hypothesis based on the time series reconstruction of the rotor effective velocity (Fig. 10), the first mode is highly correlated with spatial fluctuations in the wind field with |ρ(Z 1 , u eff )| = 0.99. The yaw misalignment (δ h ) is related to the third mode with |ρ(Z 3 , δ h )| = 0.99. The vertical shear (s v ) is related to the second and the fourth modes |ρ(Z 2 , s v )| = 0.72 and 340 |ρ(Z 4 , s v )| = 0.91. The argument of the relationship between the wind field parameters and the modes is also strengthened by the shapes of the POD modes (Fig. 8): the third and fourth mode exhibit horizontal and vertical structures while the first mode resembles the v los variations.

Evaluation of the ROM in the frequency domain
The dynamic loading induced on the turbine blades is significantly determined by the fluctuations of the longitudinal wind 345 velocity impinging on the blades and how quickly they rotate. As the blades move through the turbulent wind field, they perform a so-called 'rotational sampling' of the turbulent structures, which differ from the velocities observed at a stationary point (Kristensen and Frandsen, 1982). To investigate this effect we calculate the autospectral density of the line-of-sight wind speed of the reduced order model for the stationary hub centre and rotating reference frames for a radial position of 90% on the first blade. The ROM calculated spectra are evaluated with respect to the turbulent spectra directly determined by the lidar 350 measurements and the reference LES (Fig. 12).
The spectra were calculated via Welch's modified periodogram method with a Hanning window and 300 s data segments and a 50% overlap between segments. For the hub centre point visualised in Fig. 12 (a), the spectra of the different reduced order models exhibit very similar behaviours. The lidar measurements for all cases show a remarkable drop-off from the -5/3 Kolmogorov slope at 0.03 Hz, evident of the low-pass filtering effect of the lidar. The spectrum calculated for the reduced order 355 model with the first mode underpredicts the energy content by one order of magnitude while the addition of more modes pushes the spectrum upwards toward the full lidar measurements. The spectra of the reduced order model with ten modes and the full lidar measurements show no difference, indicating that with the first ten modes, almost all energy is recovered in comparison to the full lidar measurements. However, completely different behaviour is observed when examining the rotationally sampled spectra of the projected longitudinal wind speed (ω = 11.88 rpm) shown in Fig. 12 (b). The light blue line shows the spectrum

Discussion
A reduced order model of the wind turbine inflow was created using POD based on the inflow data from a 3700 s long interval in a large eddy simulation. The main goal of this work was to apply POD to inflow wind fields measured by a turbine-mounted 375 SpinnerLidar and to identify the most energetic and dominating modes which can be used to create a reduced order inflow model. By doing so, the application of low dimensional modelling methods to the temporal dynamics of the POD coefficients is possible. As our goal was to obtain a reduced order representation of the inflow, metrics related to the entire wind field were chosen. Our method provides a trade-off between conventional wind field reconstructions that require strong wind field assumptions and time-consuming physics-based solvers. Even though this study could benefit from full 3D wind fields, POD To evaluate the performance of the reduced order model, we define the first metric as the POD mode convergence. The convergence of the eigenvalues and mode shapes (Figs. 5 and 6) indicate that a dataset with around 3500 to 4000 samples, corresponding to approximately 1 hour for the SpinnerLidar sampled at 1 Hz, should be able to provide converged results and deterministic modes. Due to the relatively fast computation (30 s on a laptop), the analysis can be performed on the fly using a 390 moving window to obtain POD modes which correspond to variable inflow conditions that a turbine experiences in operation.
Using converged POD modes, it is feasible to develop reduced order inflow models capable of capturing the essential inflow features relevant to the turbine response.
The second metric is the cumulative energy distribution across the modes which is useful for determining the truncation point of the reduced order model. The results indicate that the first few modes contain the most energy in the wind field. In the LES 395 data, the first 10 modes contribute 96.6% of the total energy measured by the lidar while with 100 modes, 99.95% of the total energy is recovered (Fig. 7). The first mode contributes more than half of the total measured energy in the wind field by being highly correlated to the mean inflow wind speed. This is because the longitudinal velocity component, which dominates the line-of-sight measurement, is strongly correlated over the various spatial scales, as reported in Saranyasoontorn and Manuel (2005) and Kidambi Sekar and Kühn (2017). The sharp slope of the eigenvalue distribution also indicates that most energy 400 is concentrated in large scale structures. Most of the measured energy is concentrated in the first few modes as the energy associated with the small scale turbulence is effectively low-pass filtered due to the inherent volume averaging property of the lidar. This effect also nullifies the apparent disadvantage of the POD methodology whereby a large number of modes are generally required to capture small turbulent structures. Hence, the method utilizes the lidar's inherent limitation to its advantage.
Note that the lidar does not capture the total kinetic energy in the wind field as some of it is lost due to (1)  With the truncation point at 10 modes determined initially based on the eigenvalue distribution, we define the third metric as the data compression achieved due to the truncation itself. As the ROM eliminates modes that do not contribute significantly 410 to the flow, only the most energetic modes are taken for reconstruction. From the LES, data compression rates of 99.5% are achieved while retaining approximately 90% of the measured TKE with a reduced order representation of 10 modes. This is a significant data reduction that is beneficial for developing real-time turbine control algorithms by compressing the inflow information into just a few signals.
The fourth metric is a qualitative one related to the reconstruction of the velocity field from the ROM. With a single mode, the 415 reconstruction only captures the variation of the rotor effective wind speed in the measurement plane. Taking more modes into account introduces more spatial variation in the velocity field (Fig. 9). With ten modes, we obtain very good visual agreement with full lidar measurements. The model outperforms the standard three-parameter WFR of the velocity field which is unable to resolve the localised spatial structures.
control which are also evaluated as metrics. The time series of the reconstructions with different number of modes are presented in Fig. 10 for the LES case. The low frequency characteristics of the three commonly used wind field parameters (rotor effective wind speed, vertical shear and horizontal misalignment) can be captured by the first few modes alone. The high frequency fluctuations in the wind field parameters time series were reconstructed with high accuracy by taking the first ten POD modes.
To quantify the reconstruction accuracy of these three parameters, two definitions of standard std and dynamic dyn errors 425 are introduced. The behaviour of these errors for the simulated lidar measurements is shown in Fig. 11, which indicates their change in std and dyn as a function of number of modes. As expected, the addition of modes to the reconstruction decreases the error and causes both std and dyn to decline almost to zero when the first ten modes are taken into account. Interestingly, the addition of certain modes to the reduced order model decreases the values of the standard and dynamic errors substantially, leading to a better representation of the wind field. For instance, the addition of the second and fourth modes reduces the stan-430 dard and dynamic error associated with vertical shear quite substantially. The same behaviour is seen for the third mode, whose addition to the reconstruction reduces the estimation error in the horizontal misalignment. This suggests that certain modes play a major role in the definition of specific wind parameters while the specifics of the relationship itself will depend on the variability in the inflow conditions. The relationship between the modes and the three wind field parameters was quantified by calculating the total correlation between the time evolution of the modes and the wind field parameter itself as seen in Fig. 11.

435
Very clear relationships between certain wind field parameters and the time evolution of the particular modes are seen from their correlation. While these relationships will change between datasets, it is significant that only the lower modes are strongly related to the three wind field parameters. This could be exploited and the corresponding modes could be chosen with respect to a particular application by focusing on certain parameters based on either the inflow or the turbine response.
The efficiency of the POD reconstruction in the frequency domain was also investigated by calculating the turbulence spectra 440 in the fixed and the rotating reference frames for different reduced order model reconstructions (Fig. 12). For the stationary hub height spectra, the lower order models show very similar behaviour across the whole frequency range with the SpinnerLidar measurements deviating from the Kolmogorov slope at the probe length induced drop-off frequency of 0.03 Hz. The sampling in the rotational frame of reference differs compared to the fixed frame as the large spatial coverage of the SpinnerLidar captures the periodic fluctuations in the sampled wind speed regardless of the volume averaging effect (Kidambi Sekar et al.,445 2020). In the rotational spectra, the reconstruction with the first mode is incapable of predicting the magnitudes of the 1P and 2P loads in contrast to the ROM with three or more modes. As more modes are added to the reconstruction, more spatial variation in the wind field will be captured, leading to a better representation of the eddy slicing effect from which the fluctuating blade loads could be modelled.
Creating an inflow representation based on the POD methodology offers certain advantages over existing WFR methods. This 450 method does not require strong assumptions about the wind field unlike Kapp (2017) and Raach et al. (2014) and can calculate the full wind field information instantaneously if suitable POD modes are available. This makes it attractive for wind turbine control. The location of the upstream scan distance is a very important topic of research as it is important to model the turbine's induction slowdown and wind evolution to make optimal use of turbine-mounted lidar data. An investigation by Mann et al. fected by the presence of the rotor until they are very close to the rotor plane (≤ 0.5D). Hence, the first POD modes and their corresponding time evolution (which are related to these large scale structures) can be considered to be unaffected in the induction zone. This could be especially exploited to perform inflow measurements for wind turbines with rotor diameters larger than the maximum scanning area of the SpinnerLidar. The large scale structures related to the wind inflow parameters can be reconstructed with high accuracy even when measurements are performed closer to the rotor. This analysis can be extended 460 to include different atmospheric conditions (wind speeds, turbulence intensity, shear, stability, transient gusts) and operating conditions (rotor axis tilt, dynamic yaw misalignment, full and partial wake impingement).
To improve the robustness of the results, additional analysis should be carried out using simulations and data from full field experiments for a range of inflow conditions. As previously mentioned, the WFR's quality depends directly upon the lidar data quality and is thus subject to inaccuracies caused by the device limitations itself. These shortcomings which are inherent device 465 properties must be investigated in detail based on its potential lidar based application. In this study, the metrics for quantifying the accuracy of the model were chosen based on the inflow wind field itself. To further investigate the relationship between lidar measured wind fields and turbine dynamics, a detailed evaluation of the POD model can be performed by choosing quantities that describe the turbine's response. In further investigations, it would be beneficial to choose metrics from the turbine response to quantify the ROM performance as suggested in Saranyasoontorn and Manuel (2005) and Bastine et al. (2018).

Conclusions
Turbine-mounted lidar measurements can be used to derive information about the inflow to the wind turbine which can subsequently be used for turbine control, loads validation or turbulence characterisation. As lidar capabilities improve due to improved hardware and larger datasets, it is crucial to reduce the measurement data to a few variables that can still capture the spatio-temporal dynamics relevant for describing the wind field. Such models will offer a trade-off between the simple 475 wind field reconstruction methods that require certain wind field assumptions on the one hand, and the complex CFD-based reconstructions on the other. Here, we have suggested a Reduced Order Model (ROM) for turbine-mounted SpinnerLidar measurements of the turbine inflow, based on Proper Orthogonal Decomposition (POD). The inflow model was tested with virtual SpinnerLidar measurements performed in an LES wind field. Well defined inflow modes are obtained and around 90% of the turbulent kinetic energy measured by the SpinnerLidar in the line-of-sight direction in the wind field is captured with just the 480 first ten POD modes. This very strong dimensional reduction indicates that the development of simplified inflow models is possible. Our method provides a way to capture most of the spatio-temporal flow information with just ten modes leading to a data reduction of 99% in comparison to the full lidar measurements for our investigated case. The velocity fields reconstructed with these dominant modes agree well with the full lidar measurements providing a method for extracting local spatial structures in the inflow. We demonstrated that certain modes are closely related to common wind field parameters such as the rotor 485 effective wind speed, vertical shear and horizontal misalignment even though the exact relationship will differ on a case by case basis. The data reduction was possible due to the volume averaging effect of the lidar, filtering out smaller turbulent structures deemed of lower importance for the overall turbine response, thereby taking advantage of one of the lidar's limitations. This method provides more information than classical wind field reconstruction methods: for instance, it captures the rotationally sampled wind field and the associated first and second harmonics (1P, 2P), which dominate the dynamic blade loading quite 490 well. The inflow model introduced here seems to be applicable to other scanning lidar systems capable of scanning the wind field with a sufficiently high spatio-temporal resolution. The method is also scalable, with respect to the evolution of more powerful lidar systems capable of even higher spatio-temporal resolution scans or better optical systems resulting in smaller probe lengths. Based on the results in this paper, our method is considered to have potential uses for lidar-based wind turbine control, loads validation and turbulence studies.

495
Data availability. The data of the LES simulations can be made available on request.
Author contributions. APK designed the research, developed the methodology, performed the LES simulations, performed the data analysis and wrote the paper. MvD developed the analysis toolbox for the SpinnerLidar together with APK, implemented the three-parameter WFR method, and provided thorough reviews of the manuscript. AR and MK contributed with intensive discussions on the scientific content and reviewed the manuscript thoroughly. MK supervised the research.