Engineering wake models provide the invaluable advantage to predict wind turbine wakes, power capture, and, in turn, annual energy production for an entire wind farm with very low computational costs compared to higher-fidelity numerical tools. However, wake and power predictions obtained with engineering wake models can be insufficiently accurate for wind farm optimization problems due to the ad hoc tuning of the model parameters, which are typically strongly dependent on the characteristics of the site and power plant under investigation. In this paper, lidar measurements collected for individual turbine wakes evolving over a flat terrain are leveraged to perform optimal tuning of the parameters of four widely used engineering wake models. The average wake velocity fields, used as a reference for the optimization problem, are obtained through a cluster analysis of lidar measurements performed under a broad range of turbine operative conditions, namely rotor thrust coefficients, and incoming wind characteristics, namely turbulence intensity at hub height. The sensitivity analysis of the optimally tuned model parameters and the respective physical interpretation are presented. The performance of the optimally tuned engineering wake models is discussed, while the results suggest that the optimally tuned Bastankhah and Ainslie wake models provide very good predictions of wind turbine wakes. Specifically, the Bastankhah wake model should be tuned only for the far-wake region, namely where the wake velocity field can be well approximated with a Gaussian profile in the radial direction. In contrast, the Ainslie model provides the advantage of using as input an arbitrary near-wake velocity profile, which can be obtained through other wake models, higher-fidelity tools, or experimental data. The good prediction capabilities of the Ainslie model indicate that the mixing-length model is a simple yet efficient turbulence closure to capture effects of incoming wind and wake-generated turbulence on the wake downstream evolution and predictions of turbine power yield.

Wake interactions are responsible for significant power losses of wind farms

Wind turbine wakes present a structural paradigm where the flow responds to both turbine settings and incoming wind conditions: the former being associated with thrust coefficient affecting power production and wake velocity deficit

Engineering wake models have widely been used in the wind energy industry because they provide a good trade-off between fidelity, in terms of accuracy of the predicted flow and turbine power capture, and required computational costs. High-fidelity models, such as large-eddy simulation (LES), enable detailed characterization of the wake flow and dynamics, together with effects on wind turbine performance

For wind farm problems involving hundreds to thousands of simulations, engineering wake models represent suitable tools to achieve predictions of power capture from a wind turbine array in a limited amount of time

The pioneering work by

Other wake models have been developed without assuming the wake velocity distribution and expansion formula. For instance, Larsen used the RANS equations with the mixing-length model as turbulence closure

Field models (implicit models) use different methodologies to resolve the governing equations implicitly. Ainslie developed one of the classic field models and calculated the complete flow field numerically by solving the RANS equations with a turbulence closure based on the mixing-length assumption

Engineering wake models generally require parametric calibration. Light detection and ranging (lidar) measurements were used to calibrate and validate the wake growth rate of the Bastankhah wake model obtaining a good agreement between the model predictions and the experimental data

In this paper, we optimally tune parameters of four engineering wake models based on lidar measurements collected for a utility-scale wind farm. Firstly, the models and their parameters are examined and discussed. The optimization of the model parameters is carried out by minimizing the objective function defined by the percentage error between the average velocity field measured by the lidar and the respective one predicted by the engineering wake models. The optimal tuning of the model parameters is performed for various clusters of the lidar dataset based on the turbine thrust coefficient and incoming wind turbulence intensity at hub height. The optimally tuned parameters of the engineering wake models are then scrutinized through linear regression analysis. Limitations and advantages of the various models will be discussed based on the results obtained from the optimal calibration process.

The paper is organized as follows: in Sect.

A lidar experiment was carried out at a wind farm in northern Texas made of 39 2.3 MW wind turbines with rotor diameter,

Characterization of the test site:

The scanning pulsed Doppler wind lidar deployed for this experiment is a Windcube 200S manufactured by Leosphere, which emits a laser beam into the atmosphere and measures the radial wind speed, i.e., the velocity component parallel to the laser beam, from the Doppler frequency shift of the backscattered lidar signal. The lidar system operates in a spherical coordinate system and measures the radial velocity defined as the summation of three velocity components projected onto the laser beam direction. It features a typical scanning range of about 4 km with a range gate of 50 m, an accumulation time of 500 ms, and an accuracy of 0.5 m s

According to the wind farm layout and the prevalence of southerly wind directions (Fig.

The lidar measurements were typically performed by using a range gate of 50 m, elevation angle of

About 10 000 PPI lidar scans of isolated wind turbine wakes have been processed to provide the nondimensional average velocity fields used for this study

By leveraging the average velocity field of wakes measured with a scanning Doppler wind lidar for different atmospheric-stability regimes and rotor thrust coefficients, we perform optimal tuning of four widely used engineering wake models, namely the Jensen model

For this work, it is noteworthy that the thrust coefficient of the turbine rotor can be estimated through the lidar measurements,

Additionally, the rotor thrust coefficient, which is referred to as

For the Jensen wake model, mass conservation is applied for a control volume located immediately downstream of a turbine rotor, while an explicit formula is derived to predict the wake velocity field by using only two parameters as input, namely the rotor thrust coefficient,

In

For the optimization of the parameters of the Jensen model, the thrust coefficient,

Optimal tuning of the Jensen wake model:

In analogy with the model proposed in (

The optimization of the parameters for the Jensen model also produces estimates of the rotor thrust coefficient,

Linear regression of the thrust coefficient obtained from the optimal tuning of the wake models,

Optimally tuned parameters from Bastankhah wake model:

For the Bastankhah wake model, a Gaussian profile is used to describe the wake velocity field in the transverse direction at a given downstream location. This Gaussian velocity profile is then used to solve the mass and momentum budgets as for jets evolving in a boundary layer

The optimized wake expansion factor of the Bastankhah wake model,

The offset of the standard deviation of the velocity profile,

Figure

It is noteworthy that

Similarly to

For the Larsen wake model, the RANS equations are simplified by neglecting gradients with a smaller order of magnitude considering the boundary layer approximation and dropping the viscous term due to the high Reynolds numbers involved for applications to utility-scale wind turbines

In the case study shown in Fig.

The optimally tuned parameters for the Larsen wake model are reported in Fig.

Optimally tuned parameters for the Larsen wake model:

Regarding the wake recovery rate reported in Fig.

Similarly to the Larsen wake model, the Ainslie wake model is derived from the RANS equations for incompressible flows

For the Ainslie wake model, the wake width,

Summarizing, the inputs of the Ainslie model are the thrust coefficient,

Optimal tuning of the parameters for the Ainslie wake model:

Normalized velocity for the cluster with

Since the two model parameters

Once the parameters of the four considered engineering wake models have been optimally tuned based on the mean velocity fields retrieved from the lidar measurements, and their trends as functions of the normalized incoming wind speed at hub height,

Velocity profiles predicted from four wake models and compared with lidar data for

The velocity field predicted through the Bastankhah wake model looks very similar to the mean velocity field measured by the lidar, especially in the far wake, indicating that the velocity profiles in the radial direction can be modeled with a good level of accuracy through a Gaussian function, which is the underlying assumption of the Bastankhah wake model. A larger wake velocity deficit with respect to the reference lidar data is observed in the near wake for the model predictions. This feature of the Bastankhah wake model can be better understood through the velocity profiles in the radial direction reported for various downstream locations and incoming-turbulence intensity in Fig.

Predictions of the near-wake velocity field are improved for the Larsen wake model (Fig.

Percentage error, PE (Eq.

Accuracy in the wake-flow predictions obtained through the optimally tuned engineering wake models is quantified through the average percentage error, PE (Eq.

In Fig.

Finally, a sensitivity analysis of the optimally tuned model parameters as a function of

Linear regression between inputs from lidar cluster and output parameters from the wake models. Bold values indicate high correlation between the respective parameters.

As mentioned above, the optimal tuning of the Jensen, Bastankhah, and Larsen wake models generates an estimate of the thrust coefficient of the turbine rotor,

Regarding the model parameters representing the wake turbulent diffusion, the slope obtained through the linear regression of these parameters with the incoming-turbulence intensity, TI, is between 0.86 and 1.12 (

Low-computational costs and easy implementation are key factors for the wide application of engineering wake models in wind energy for both industrial and academic pursuits. However, it is challenging to tune parameters of these engineering wake models to achieve satisfactory accuracy for predictions of wakes and power capture required for the design and control of wind farms. Furthermore, this calibration can be even more challenging when wake models are used for a broad range of atmospheric-stability regimes or in the presence of flow distortions induced by the site topography.

In this paper, we have considered four widely used engineering wake models, namely the Jensen model, Bastankhah model, Larsen model, and Ainslie model. The tuning parameters of these engineering wake models have been optimally calibrated by minimizing the mean percentage error between the wake flow predicted through the models and the mean velocity fields measured through a scanning Doppler wind lidar deployed at an onshore wind farm in northern Texas. Statistics of the wake velocity field are obtained through a cluster analysis based on the incoming-turbulence intensity at hub height and the normalized hub-height wind speed. The results of the optimal-tuning procedure have shown that the thrust coefficient obtained through this numerical approach is in very good agreement with the values obtained by applying the mass and streamwise momentum budgets on the mean lidar data by neglecting pressure gradients and turbulent stresses. Furthermore, the model parameters representing the wake turbulent expansion and recovery are roughly linearly proportional to the incoming wind turbulence intensity at hub height.

This study has shown that the Jensen model has a lower yet comparable accuracy than the remaining three wake models, which is mainly connected with the simplistic top-hat assumption used for modeling the wake velocity deficit. However, a good estimate of the mean kinetic energy within the wake as a function of the downstream location is achieved through the Jensen wake model. The Larsen wake model has generally shown better accuracy than the Jensen model yet lower than for the Bastankhah and Ainslie wake models. This feature seems to be the effect of a slightly more complex formulation of the model, leading to the presence of parameters that are not easy to tune through a data-driven approach, as for this work.

The Bastankhah wake model has shown great accuracy in wake predictions upon the optimal tuning of the model parameters for a broad range of incoming-turbulence intensity and incoming wind speed at hub height, namely thrust coefficient of the turbine rotor. The main assumption of the Bastankhah wake model consists of modeling the wake velocity profile in the radial direction through a Gaussian function. Therefore, significant differences between the predictions and the lidar data have been observed in the near wake and/or for relatively low incoming-turbulence intensity for which the velocity profiles might not be axisymmetric and may differ from a Gaussian-like profile. Therefore, we recommend using the Bastankhah wake model only for downstream locations and wind conditions for which the Pearson correlation coefficient between the actual velocity field and the Gaussian model is expected to be higher than 0.99.

Finally, the Ainslie wake model has shown great accuracy, indicating that the mixing-length model for the RANS equations is a simple yet efficient turbulence closure model to capture the effects of incoming turbulence and wake-generated turbulence on wake downstream evolution and recovery. The Ainslie wake model provides a great advantage to use as input the velocity profile at a specific streamwise location. This input can be obtained through experiments, numerical simulations, or other models.

The optimal tuning of the considered wake models has enabled us to significantly reduce the mean percentage error in the predictions of the wake velocity field. For certain clusters of the lidar dataset, the mean percentage error has been 4 times smaller than for the respective baseline wake prediction obtained by using standard parameter values available from the literature. Considering that the wind farm under investigation is characterized by a typical layout, flat terrain, and typical daily cycle of the atmospheric stability for onshore wind farms, we expect that similar improvements in wake-prediction accuracy can be generally achieved for wind farms with similar characteristics by using the reported optimally tuned model parameters.

For the Larsen wake model, the coefficient representing the wake turbulent diffusion is

For the terms in the second-order contribution of the Larsen model solution, they are defined as

The authors noticed that for roughly identical predictions in streamwise velocity component from the Larsen and Ainslie wake models, the radial velocity predicted from the former is 1 order of magnitude larger than that for the latter while having the opposite sign. Subsequently, we calculated the divergence in cylindrical coordinate and nonconservative form, (

Assessment of the Larsen wake model

The Ainslie wake model consists of two governing equations, the continuity and momentum budgets, which are solved through the boundary layer approximation

The lidar dataset used is publicly available to download at

The code for the optimal tuning of the models is available at

This research has been funded by a grant from the National Science Foundation CBET Fluid Dynamics (award number 1705837). This material is based upon work supported by the National Science Foundation under grant IIP-1362022 (collaborative research: I/UCRC for Wind Energy, Science, Technology, and Research) and from the WindSTAR I/UCRC members Aquanis, Inc.; EDP Renewables; Bachmann Electronic Corp.; GE Energy; Huntsman; Hexion; Leeward Asset Management, LLC; Pattern Energy; and TPI Composites. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the sponsors.

The authors declare that they have no conflict of interest.

This research has been supported by the National Science Foundation, Directorate for Engineering (grant no. 1705837), and the National Science Foundation, Directorate for Engineering (grant no. IIP-1362022).

This paper was edited by Julie Lundquist and reviewed by two anonymous referees.