We study the calibration of the Dynamic Wake Meandering (DWM) model using high-spatial- and high-temporal-resolution SpinnerLidar measurements of the wake field collected at the Scaled Wind Farm Technology (SWiFT) facility located in Lubbock, Texas, USA. We derive two-dimensional wake flow characteristics including wake deficit, wake turbulence, and wake meandering from the lidar observations under different atmospheric stability conditions, inflow wind speeds, and downstream distances up to five rotor diameters. We then apply Bayesian inference to obtain a probabilistic calibration of the DWM model, where the resulting joint distribution of parameters allows for both model implementation and uncertainty assessment. We validate the resulting fully resolved wake field predictions against the lidar measurements and discuss the most critical sources of uncertainty. The results indicate that the DWM model can accurately predict the mean wind velocity and turbulence fields in the far-wake region beyond four rotor diameters as long as properly calibrated parameters are used, and wake meandering time series are accurately replicated. We show that the current DWM model parameters in the IEC standard lead to conservative wake deficit predictions for ambient turbulence intensities above 12 % at the SWiFT site. Finally, we provide practical recommendations for reliable calibration procedures.

Wake effects are perceived as one of the largest sources of uncertainty in energy production and load estimates of onshore and offshore wind farms

Wind lidars have become popular for studying wind turbine wakes due to their higher spatial resolution and ease of installation compared to traditional anemometers mounted on meteorological masts

The Scaled Wind Farm Technology (SWiFT) experiment, conducted at Sandia National Laboratories between 2016 and 2017

Here, we analyse the SWiFT dataset aiming at calibrating and evaluating the DWM model. This model is recommended in the IEC 61400-1 standard

The underlying hypothesis of the DWM model is to consider the wake as a passive tracer of the large incoming turbulence structures. The so-called

The wake deficit formulation of the DWM model is mainly based on the work of

The fidelity of the simulated wake meandering dynamics also affects the accuracy of load predictions

Further, the added turbulence formulation in the DWM model accounts for additional mechanically generated turbulence caused by the wake shear and the breakdown of tip and root vortices. These contributions are modelled by a semi-empirical formulation that uses parameters, which were calibrated against CFD simulations

As described above, there is no consensus for the values of the DWM model parameters when studying load predictions at any given site. Also, and perhaps most importantly, we do not know the sources of uncertainty observed in previous studies that used the model

To address this issue, we estimate uncertainties in the calibration parameters of the DWM model by applying Bayesian inference

Derive wake flow features such as the two-dimensional velocity deficit and wake-added turbulence profiles as well as time series of the wake meandering in both lateral and vertical directions from the SpinnerLidar measurements under different inflow wind speeds and atmospheric-stability conditions.

Calibrate the DWM-model-based wake deficit and wake-added turbulence predictions using the SpinnerLidar-derived wake flow features and the Bayesian inference framework.

Propagate modelling uncertainties in fully resolved wake flow fields for robust predictions that take into account the calibrated uncertainties.

Conduct a sensitivity analysis to determine the most significant sources of uncertainty in simulated wake fields that are typically inputs to aeroelastic simulations.

This study contributes to the ongoing discussion regarding the accuracy of power and load predictions of wind turbines operating under wake situations

The work is organized as follows. Section

The DWM model resolves three main wake features: the quasi-steady velocity deficit, the wake-added turbulence, and the wake meandering. Each model component is described separately in the following subsections.

The quasi-steady velocity deficit component describes the wake expansion and recovery caused partly by the recovery of the rotor pressure field and partly by turbulence diffusion moving farther downstream of the rotor

In the

Based on classical mixing length theory,

The wake turbulence is composed of three turbulence sources and can be defined as follows

Here, the meandering model is confined to a single wake scenario, whereas multiple wake dynamics are described in

The SWiFT facility is a research site located in Lubbock, Texas, operated by Sandia National Laboratories

Figure

The SpinnerLidar is a research Doppler wind lidar developed at DTU based on a continuous-wave (CW) laser system

Once a scan was completed, the SpinnerLidar refocused at a different range, and this process took about 2 s

A schematic view of the SpinnerLidar's scanning pattern:

For extended periods of the campaign,

Figure

Inflow wind and operational conditions at the SWiFT site.

Here, we investigate the variability in the wake flow characteristics under varying stability and inflow wind speed conditions. We classify each 10 min sonically derived statistic into atmospheric-stability classes defined by ranges of the dimensionless stability parameter (

The statistics of the inflow wind parameters are presented separately in Tables

Dataset from

Similar to Table 1 but for

Similar to Table 1 but for

As lidars only measure the line-of-sight (LOS) velocity (

Considering the small elevation angles and the typical low values of

To perform comparisons with predicted velocity deficits from the DWM model, we aim at isolating the contribution of the wake deficit from that of the vertical wind shear in lidar measurements. As defined in

Figure

Ensemble-average velocity deficit profiles in the MFoR measured at 2, 3, 4, and 5

Turbulence measures derived from lidar radial velocity measurements are “filtered” because of their relatively large probe volume

Examples of 10 min ensemble-averaged Doppler spectra obtained at three fixed locations across the scanned area – a wake centre, a wake edge, and a wake-free position – are shown in Fig.

Examples of normalized Doppler LOS velocity spectra measured over a 10 min period at 2.5

To characterize the spatial distribution of the wake turbulence within the scanned area, we derive

Figure

Two-dimensional spatial distribution of the horizontal wind velocity variance (

The calibration of the wake deficit and wake-added turbulence components are conducted in the MFoR using a Bayesian inference framework. We describe the Bayesian model in Sect.

The basis of the Bayesian inference is to estimate the probability distribution of the model parameters based on available observations. Let

The outcome of the calibration is a joint probability distribution of the inferred model parameters. From this joint PDF, we can estimate the posterior PDF of any wake feature simulated by the DWM model, i.e. the wake deficit and wake-added turbulence profiles in the MFoR or the fully resolved wakes in the FFoR, among others, which we denote by

We use lidar-derived wake deficit profiles in the MFoR collected during

The set of observable variables comprises

We select uniform prior distributions on model parameters

Joint and marginal posterior PDFs of

We propagate the uncertainties in

The lidar-estimated deficits are approximately Gaussian under stable to near-neutral conditions, whereas the Gaussian shape is lost under more unstable conditions. This may result from errors in the wake tracking procedure due to the larger meandering amplitudes and also due to the presence of large-scale turbulence structures in the inflow

The DWM-model-predicted deficit profiles with parameters specified by their posterior distributions are in good agreement with the lidar observations for distances beyond 4

It can be observed that the uncertainties in

We provide comparisons between measured and predicted wake deficit profiles using the calibration from this work as well as those reported in early studies in Fig.

By considering

Comparison between measured and predicted ensemble-average spanwise velocity deficit profiles resolved in the MFoR at hub height and obtained at 2, 3, 4, and 5

Ensemble-average spanwise velocity deficit profiles computed in the MFoR at hub height obtained at 5

The wake-added turbulence model (Eq.

Due to these two assumptions, we propose an improved semi-empirical formulation of the wake-added turbulence scaling factor

DWM-model-predicted flow characteristics.

The calibration parameters in Eq. (

As a first step, we fit the “original” analytical formulation of

To infer the posterior PDFs of

The resulting posterior PDFs of the parameters follow a normal distribution (not shown) with statistical properties tabulated in Table

Wake-added turbulence predictions. Panels

Here, we investigate the relationship between the inflow turbulence fluctuations and the lidar-tracked wake positions to characterize the large-scale eddies responsible for the meandering. The analysis is carried out by comparing the spectra of the lidar-tracked meandering time series, which are derived by means of the tracking algorithm in Eq. (

Results of the spectral analysis for both lateral and vertical meandering are shown in Fig.

Figure

Normalized ensemble-average power spectral density (PSD) of the lateral and vertical wake meandering tracked by the SpinnerLidar (red) and that derived using the meandering model (DWM

The validation of the DWM model is performed by resolving wake fields in the FFoR; thus the simulated wakes include the combined effects of the velocity deficit, added turbulence, and wake meandering dynamics in both lateral and vertical directions. This analysis is carried out using data from

The SpinnerLidar measurements collected during

We derive the two-dimensional spatial distribution of the mean wind speed in the wake region (

Mean (

Correlation coefficients between model parameters estimated using Bayesian inference.

We compute

A good agreement between measurements and predictions is found. The vertical profile of the mean wind speed exhibits a single-peak shape resulting from the combined effects of the inflow vertical shear (modelled by a power law) and the wake-induced Gaussian-like deficit shape. The wake turbulence in the lateral direction exhibits a double-peak shape with larger values near the locations associated with strong velocity gradients that are further enhanced by the wake meandering. Enhanced turbulence levels in proximity of the upper wake region are observed from lidar measurements. The deviations between

The wake simulation uncertainties shown in Fig.

Comparisons between SpinnerLidar (SL)-measured and DWM-model-predicted spatial distribution of the mean and standard deviation velocity computed in the FFoR and obtained at 5

To evaluate the performance of the DWM model, we calculate two flow metrics that are relevant in aeroelastic simulations and compare them to relative measured quantities: the rotor-effective wind speed (

Further, a sensitivity analysis indicates that the variations in

Comparison of the SpinnerLidar-measured (SL) and DWM-model-predicted rotor-effective wind speeds

The SpinnerLidar measurements show that the wake recovers faster under unstable compared to stable conditions primarily due to the high turbulence levels of the former

Although the SWiFT campaign provides a comprehensive dataset, including a wide range of inflow wind conditions and detailed two-dimensional lidar measurements of the wake field, the calibration parameters obtained from the 192 kW turbine with 32 m diameter should be further evaluated for multi-megawatt turbines with larger rotors. While this study cannot explicitly demonstrate the transferability of the obtained results for modern-size turbines, it highlights the need for datasets that include observations of the wake deficit profiles under varying stability conditions, inflow wind speeds, and downstream distances for the reliable and robust calibration of engineering wake models. Here, we find that the velocity deficit's recovery rate for increasing turbulence is the main difference among calibrations reported in the literature (see Fig.

We also demonstrate that uncertainties in calibration parameters (e.g. COV

We recommend conducting DWM model calibrations using two-dimensional high-temporal- and high-spatial-resolution measurements of the wake field. Such resolution can be achieved by research-based nacelle lidars today, which allow the resolution of wake flow features including the wake deficit and wake-added turbulence profiles in the MFoR for model calibration. When using power production data for such calibrations (the IEC standard values are based on this approach), we are unable to distinguish between uncertainties from inaccurate wake deficit predictions and those from erroneous wake meandering representations. The latter play an important role in the accuracy of the fully resolved wake fields, which are inputs to aeroelastic simulations.

As wind turbines are typically spaced 5

Characterizing wake turbulence using lidars is challenging due to the limited sampling frequency and probe volume effects

We analysed high-spatial- and high-temporal-resolution SpinnerLidar measurements of the wake field collected at the SWiFT facility and derived wake features such as the wake deficit, wake-added turbulence, and wake meandering under varying atmospheric-stability conditions, inflow wind speeds, and downstream distances. The SpinnerLidar-estimated wake characteristics computed in the MFoR were used to determine uncertainties in the DWM model parameters using Bayesian inference. The uncertainties in model parameters were propagated to predict fully resolved wake flow fields in the FFoR. This approach allowed us to quantify uncertainties in the DWM-model-simulated wake fields and to investigate the sensitivity of the model parameters to flow features that primarily affect power and load predictions.

The SpinnerLidar-derived wake deficit profiles revealed the strong impact of atmospheric stability on wake evolution. In particular, we observed the faster recovery of the deficit under unstable compared to stable regimes as higher turbulence intensities characterized the former. These effects were accurately reproduced by the eddy viscosity term of the DWM model with the inferred parameters for distances beyond 4

We proposed and verified an improved semi-empirical formulation of the wake-added turbulence model that captured the effects of the atmospheric shear and the ambient turbulence on the spatial re-distribution of the wake turbulence observed at 2.5

The underlying hypothesis of the DWM model (i.e. wakes are advected passively by the large eddies in the incoming wind field) was verified by means of the SpinnerLidar-tracked meandering time series. The spectral analysis indicated that large eddies associated with sizes larger than 2

In a future study, we will quantify uncertainties in power and load predictions based on the proposed calibration at two different sites, the SWiFT facility and the Nørrekær Enge wind farm in Denmark

Figure

Time series of the wake meandering in the lateral direction observed by the SpinnerLidar (red markers) and those derived from the meandering model of Eq. (

Sensitivity analysis is conducted to identify the most important parameters affecting the accuracy of wake simulations. Here, we only investigate the uncertainties in the model parameters, which are listed in Table

The Sobol indices computed from wake simulations at five rotor diameters behind the rotor are illustrated in Fig.

Variance decomposition (Sobol indices) for the rotor-effective wind speed

The SpinnerLidar and meteorological mast data can be requested from the authors.

DC, ND, AP, and TH participated in the conception and design of the work. DC conducted the data analysis and wrote the draft manuscript. ND and AP supported the overall analysis and critically revised the manuscript. TH contributed with the acquisition of the dataset and critically revised the manuscript.

The authors declare that they have no conflict of interest.

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

Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the US Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525.

This paper was edited by Sandrine Aubrun and reviewed by two anonymous referees.