Near-wake effects of wind turbine models using the free-vortex wake have been studied extensively, but there is a lack of validation for such predictions in the mid to far wake. This paper presents a novel validation study using three free-vortex wake models of increasing complexity: an actuator disc, an actuator disc with rotation, and a lifting-line model. We emphasise the application for dynamic wind farm flow control optimisation with a focus on wake redirection using yaw misalignment. For this purpose, wake models should provide sufficiently accurate power predictions at a low computational expense to enable real-time control optimisation. Three sets of wind tunnel data are used for validation: flow measurements under steady yaw misalignment, time-resolved flow measurements for a step change in yaw, and turbine output measurements with yaw control and simulated wind direction variation. Results indicate that the actuator-disc model provides the best balance between computational cost and accuracy in power predictions for the mid to far wake, which is not significantly improved upon by the addition of rotation. In the near wake, the added complexity of the lifting-line model may provide value as it models blade loading and individual tip vortices. Altogether, this study provides important validation for further studies into optimisation of wake steering under time-varying conditions and suggests that the actuator-disc model is a suitable candidate for use in a model-predictive wind farm flow control framework.

The limited availability of offshore and onshore parcels for wind energy production means that large, densely spaced wind farms are commonly used. However, in these farms, wake effects can lead to a significant decrease in power production and an increase in fatigue loading. While farm topology is typically optimised to minimise aerodynamic interaction, it lacks flexibility for time-varying wind conditions

This paper focuses on the use of yaw misalignment for wake steering, where an intentional misalignment in the yaw angle with respect to the dominant wind direction is used to deflect the low-energy, turbulent wake behind the turbine. After demonstrations of effectiveness in both simulation and wind tunnel experiments, wake steering has been shown to yield power gains in wind farms for predefined yaw angle offsets under steady conditions in field studies

Wind turbine wake models are essential tools for developing and implementing wake steering control strategies. Accurate predictions of wake behaviour allow for optimisation of wind turbine controls for objectives such as power production and reduction in fatigue loading. Current control strategies are mostly based on lookup tables generated by steady-state optimisation with engineering wake models, such as those in the FLORIS toolbox

Instead of implementing the dynamics into steady-state models, physics-based approaches attempt to simplify first principles to reduce complexity while maintaining essential dynamics. Studies with large-eddy simulation have been successful in control optimisation

Free-vortex wake (FVW) methods are meshless methods, using Lagrangian elements to model flow dynamics based on the vorticity formulation of the Navier–Stokes equations

Several studies have applied FVW methods to model wind turbine wakes, such as a lattice method

Most FVW models focus on wake dynamics close to the rotor and, to the best of our knowledge, little validation has been done for the mid to far wake. We define the mid-wake from 1 to 4 D and the far wake beyond 4 D downstream from the rotor, where

The validity of these models for wake predictions under yaw misalignment is evaluated with three sets of experimental data from wind tunnel measurements: first, a set of data that consists of lidar measurements of wind turbine wakes under steady yaw misalignment

The contribution of this paper is twofold: (i) an analysis of model parameter choice and suitable levels of simplification of the turbine representation for modelling the wind turbine wake and (ii) a validation of free-vortex wake models for mid- to far-wake power predictions with wind tunnel data, in light of control optimisation for yaw control.

The remainder of this paper is structured as follows. Section

First, we construct the models of the wind turbine wake that are studied in this paper. The free-vortex methods and straight-line vortex filament definition are introduced in Sect.

The basis of the vortex methods is the vorticity formulation of the Navier–Stokes equations. The FVW method is based on Lagrangian particles that advect downstream. These particles induce a velocity based on their associated circulation strength. The resultant flow velocity may be calculated at any position based on the freestream velocity and the sum of induced velocities. The vorticity formulation requires the assumption of inviscid and incompressible flow, although diffusion may be approximated. For a further description of the fundamentals, the reader is referred to aerodynamic literature, such as

The three-dimensional model formulations in this study are based on straight-line vortex filaments. The induced velocity

A Gaussian core with core size

Vortex filaments are convected over time according to the combination of the freestream velocity

Turbulence is not explicitly accounted for when using the FVW to construct models of wind turbine wakes. However, growth of the vortex core may be used to approximate the effects of turbulent and viscous diffusion as

The wind turbine models used for this study are the three-dimensional actuator disc as used by

The vortex filament structures and the direction of circulation for the three different free-vortex wind turbine representations under consideration:

The turbine thrust

An actuator-disc representation of a wind turbine is implemented with the free-vortex method and illustrated in Fig.

At fixed time intervals

Numerical parameters for the FVW models as used for this validation study.

The vorticity generated by the ADM is purely azimuthal as long as the turbine is yaw-aligned with the freestream wind direction. Under yaw misalignment, the vortex rings deform into the characteristic curled shape of the wake as a counter-rotating vortex pair is formed.

An extension of the ADM is the actuator disc model with rotation (ADMR). A root vortex is released along the centre line of the wake as shown in Fig.

Assuming again that the rotor is uniformly loaded, the thrust force is equally distributed over each of the blades,

The Joukowsky rotor model is a lifting-line model (LLM) that assumes uniform blade loading, forming a rotating horseshoe vortex system for each blade

An important aspect of the wind turbine models from Sect.

The streamwise spatial discretisation of the ADM is studied by constructing a cylindrical vortex tube with a length of 12 D from discretised vortex rings, approximating the ADM wake. The spacing between vortex rings is varied to study the effect on the wake deficit without the effects of temporal evolution. The number of rings is adjusted accordingly to maintain a constant wake length, and the circulation of the vortex filaments is adjusted to maintain the same distribution of total circulation. The velocity error

The convergence behaviour of the velocity deficit with an increasing number of rings is first order as is illustrated in Fig.

Relative error in the velocity deficit at the rotor plane

The variation in error over filament spacing within the wake, at

It is important to note that streamwise spatial discretisation is directly connected to the time discretisation and the computational complexity. The largest possible time step is such that a vortex ring is released at every time step. High spatial resolution is thus only possible for small time steps. Additionally, the large number of elements required to generate a wake of sufficient length with high streamwise resolution leads to large increases in the computational cost of the induced velocity calculation; the cost of the induced velocity evaluation increases quadratically with the number of vortex filaments. Small time steps and expensive induced velocity calculation both contribute to a significant increase in computational cost for a given prediction horizon. Therefore, a relatively large time step and coarse spatial resolution are chosen for the purpose of efficient optimisation of wind turbine controls.

The ADMR introduces a single extra vortex filament per time step compared to the ADM, which makes it about 1.1 times more expensive with the current numerical settings. The LLM requires 4 times as many filaments as the ADM for a wake of the same length, which makes a single time step 16 times more expensive. Accounting for the smaller time steps, simulating a given time with the LLM is theoretically about 140 times more expensive than the ADM.

A small benchmark is run on a regular laptop running Windows 10 on an i7-8650 CPU at 1

To put these numbers into perspective, evaluating a single wake in FLORIS with the cumulative-curl model

The time discretisation of the FVW is studied by examining convergence for a first- and second-order integration scheme. In order to perform this convergence experiment, it is necessary to decouple streamwise spatial discretisation and time discretisation. We reformulate the problem such that a number of sub-steps may be taken between releasing vortex rings.

The largest time step considered is

Figure

Relative error in the position of the vortex filaments for varying time discretisation, comparing the first-order explicit Euler method and the second-order explicit Heun method in simulating the wake of a yaw-misaligned rotor. The reference solution for time step

For the numerical parameters presented here, the methods converge as expected. The chosen time step

The convergence of azimuthal discretisation is tested by varying the number of elements in the vortex rings. A simulation with a yaw misalignment of 30

Figure

Relative error in the cross-stream rotor-averaged velocity profile at 5 D downstream for the ADM under a yaw misalignment of

The time discretisation of the LLM is chosen such that it achieves the same azimuthal resolution, which is for

The choice of vortex core size

Illustration of vortex particle/filament trajectories for varying initial core sizes. Larger core sizes produce more stable, i.e. less unstable, trajectories. The results in this paper are produced for an initial core size

Variation in the vortex core size has very little influence on the initial wake depth. However, the wake recovery can be tuned with vortex growth, implemented with Eq. (

Introducing vortex core growth using the Lamb–Oseen model allows for tuning of the diffusion to approximately match turbulent mixing in the wake. Slices at

Following the study of numerical model parameters, it is essential to validate the wake flow and power predictions of the FVW model for yaw control optimisation. This section first presents the available data from the three wind tunnel experiments in Sect.

Three sets of experimental data are used in this paper for the model validation study. The first is a set of steady-state flow measurements for the wind turbine wake under yaw misalignment

The wind direction

All experiments used the MoWiTO 0.6 turbine with a rotor diameter

The first set of experimental data (labelled WTA) was recorded in the wind tunnel at ForWind, University of Oldenburg, and has been published in

The data contain measurements of wakes for yaw misalignment angles

The second set of experimental data (labelled WTB) was recorded in the Open Jet Facility (OJF) at the TU Delft, with the same set-up as used in

A PIV set-up with four cameras was used to measure the flow velocity at downstream distances

The third set of experimental data (labelled WTC) was again recorded in the wind tunnel at ForWind, University of Oldenburg

The upstream turbine was stationary and yaw-controlled to achieve wake steering. The second turbine was placed 2.66 D downstream with an increased hub height of 0.16 D. The downstream turbine is mounted on an

The yaw set points for the upstream turbine were stored in a lookup table and applied differently for each control experiment. The experiments are labelled BW30, BW60, BW120, BW300, and BW600 based on the length of the wind direction averaging window used in the controller, with shorter windows leading to more frequent yaw variations. Figure

Time series data from WTC showing yaw misalignment on the upstream turbine and the wind direction variation for the BW30 experiment. The changes in wind direction are achieved by translation of the downstream turbine on an

The performance measures in this study reflect the purpose of this model. It is oriented towards control for power maximisation, and therefore the predictive qualities for wake deflection and downstream aerodynamic power availability are important aspects to measure.

Wake deflection is determined according to the wake centre position, which is defined as the cross-stream position where aerodynamic power available for a virtual rotor at hub height is minimal. The potential power follows quite directly from the measured or simulated flow field and is directly related to downstream turbine performance as available aerodynamic power

For statistical analysis, the fits of the power predictions are evaluated with the variance accounted for (VAF),

A total comparison of power at different yaw angles is performed by binning the results according to wind direction and yaw angle bins and calculating the mean and standard deviation of the power in each bin. The analysis is the same for both model and experimental data, thus allowing equivalent comparison.

A visual, qualitative comparison of the available flow measurements provides a general overview of the strengths and weaknesses of each of the models. These are provided for steady-state measurements from WTA and WTB. The cross-stream planes of the flow are considered to be more important than hub height planes because of the three-dimensional nature of wind turbine wakes under yaw misalignment.

A quantitative analysis of the steady-state wake deflection from WTA is performed by analysis of the flow cross-sections for the cross-stream position where potential power is minimal. This is considered to be the wake centre and a measure for predictive power for wake deflection under steady-state conditions.

The WTB experiment is replicated with all three FVW models to analyse the temporal dynamics of the model at high time resolution. The time series of potential power production provide insight into the propagation of yaw effects downstream through the wake.

Finally, the dynamic experiment in WTC is fully replicated with the ADM free-vortex wake model. The upstream turbine is set to the specified yaw angle as recorded in the experiment data and operated under a constant thrust coefficient. The downstream turbine performance is evaluated from the rotor-averaged velocity over a rotor disc at the downstream turbine position. This position varies over time as the turbine is translated to track the specified wind direction from the experiment. For both upstream and downstream turbines, the rotor-averaged velocity is recorded. The downstream velocity is increased by

The wake model provides an estimate of the velocity, but the experimental data record generator power. A simple turbine model is specified to account for inertial dynamics,

This section presents the core results of this paper, the comparison of the model predictions with wind tunnel data for validation. Yaw misalignment under steady conditions is discussed in Sect.

A visual comparison of cross-stream wake velocity profiles is provided in Fig.

A comparison of normalised streamwise velocity for wind turbines under yaw misalignment. The experimental data from WTA are compared to model results with the ADM, ADMR, and LLM with slices at

For yaw-aligned flow (

Both positive and negative yaw misalignment angles result in the generation of a counter-rotating vortex pair and subsequently a curled wake shape, which becomes apparent from 3 D downstream. The ADM produces wakes that are symmetric between positive and negative yaw misalignment, as expected. The inclusion of the root vortex in the ADMR models some of the asymmetry in wake shape that is also present in the experimental data. The LLM produces a similar asymmetric deformation of the wake.

A large deformation of the wake is visible from 5 D onwards for the wake under yaw misalignment. In the FVW models, this leads to stretching of the vortex filaments and unstable wake structures. The wake of the LLM breaks down in particular beyond 5 D downstream because of the large number of vortex filaments in close proximity. The ADMR still demonstrates similar stability in wake structure as the ADM at 7 D downstream, although resemblance of the wind tunnel data is reduced.

Finally, some of the details of the experimental data are not represented in the FVW models. The effect of the wake from the turbine tower on power predictions is assumed to be minor, and therefore it is not considered in the FVW models, although it is present in the wind tunnel measurements. Furthermore, the models assume the rotor to be uniformly loaded, which is not the case in the experiment. The inclusion of non-uniform rotor loading would increase the model complexity but might improve the modelling of wake deflection

The displacement of the wake centre is evaluated according to the cross-stream position where available aerodynamic power is minimal. Figure

Wake centre deflection over downstream distance calculated from the flow slices illustrated in Fig.

The ADM appears to have the best fit to the experimental data over the measured downstream distances. The ADMR and LLM show a similar deflection profile, which only shows good agreement with the experimental data up to

Experiment WTB recorded wake development at downstream distances

Flow slices showing streamwise velocity for yaw misalignment

The good quality of the fit follows expectations because the wind tunnel has uniform inflow and a low turbulence intensity.

More importantly, this experiment can provide insight into the wake dynamics for changes in yaw misalignment with a high temporal resolution. The rotor-averaged wind speed for a virtual rotor is evaluated based on the PIV snapshots and is shown in Fig.

The realisation of the

The value of VAF and NMAE for the three dynamic model simulations is listed in Table

Fit quality of the time series of potential power estimates from the replication of the WTB experiment, listing VAF (V) and NMAE (N). The experimental data and ADM estimate, both using steady-state assumptions and dynamic simulation, are illustrated in Fig.

The dynamic ADM and ADMR perform to a similar level of accuracy in this mid-wake region. They are marginally outperformed by the LLM, which is considerably more computationally expensive. These results support the findings from Sect.

The replication of the yaw step experiment with the ADM in Fig.

The WTC data set provides many performance measurements for varying yaw angle and wind direction. It is replicated only with the ADM, as it appears to perform similarly to the ADMR, and the LLM yields only minor improvements at a computational expense, which is prohibitive for control optimisation in the mid to far wake.

The replication of the experiment with the BW30 controller settings with the ADM yields a power estimate for which the time series is shown in Fig.

Power predictions with the ADM are compared with experimental data from WTC for the BW30 control setting. The time series fit VAF is

The upstream turbine power is estimated with a NMAE of

Fit quality of the time series of generator power estimates from the replication of the WTC experiment with the ADM, listing VAF (V) and NMAE (N). The experimental data and power estimate for the BW30 control strategy are illustrated in Fig.

Figure

Besides possible differences between the model turbines themselves, this increase in power is likely due to a combination of two effects. First, the inflow has a shear profile with a power-law exponent of

The impact of wake steering through yaw misalignment on downstream power production is clouded in Fig.

Wind turbine power curve for the modelled yaw misalignment distribution, showing the mean and standard deviation of power in

The power-yaw curve of the upstream turbine is slightly asymmetric in the wind tunnel experiment due to the operation under sheared inflow conditions. This asymmetric power profile is not represented in the ADM because of the model symmetry and uniform inflow. The downstream expected power production matches well given the model simplicity. The benefit of wake redirection is slightly underestimated for large misalignment angles.

The important aspect of the ADM estimate of power for these two turbines is that the trends are captured well. In the end, what matters for control optimisation of wake steering is accuracy in representing the optimal operating point more than exactness in the predicted power. The presented data indicate that there is a considerable correspondence between the model and experiment, but model implementation in a control strategy will have to point out whether that is sufficient. Additional error integration and state estimation could always be included if required, such as for tracking a power reference.

Three free-vortex wake wind turbine models (ADM, ADMR, LLM) are presented in this work for the prediction of wake dynamics under yaw misalignment for control optimisation. The two highlights in this work are (i) a study of parameter sensitivity and convergence and (ii) a comparison with three sets of experimental data from wind tunnel measurements in order to validate the power predictions in the far wake.

The parameter and convergence study indicated that the best results for mid- to far-wake predictions are achieved with ADM, i.e. when the wake model has minimum complexity. The addition of rotation does improve qualitative agreement of flow fields with experimental data but does not necessarily improve power predictions under yaw misalignment. The LLM may generate a slightly more accurate response, but the computational cost is prohibitive for use in online control optimisation for wake redirection.

The comparison with experimental data illustrates to what extent the FVW models can provide predictions for available power when utilising wake steering control. Even under the assumptions of uniform inflow and uniform rotor loading, there is considerable agreement with experimental data in terms of steady-state wake deflection, dynamic response to yaw change, and power estimates with yaw control and wind direction variation. However, implementation in a control framework will have to point out whether the accuracy is sufficient for the intended purpose of yaw control for power maximisation.

In conclusion, the ADM appears to be a suitable candidate for efficiently predicting the dynamic mid- to far-wake effects of wake steering, a range from approximately 1 to 7 D. As such, it could play a central role in the development of novel model-based strategies for wind farm flow control. These new controllers could further improve wind farm energy yield as more accurate wake dynamics are included in control optimisation for wake redirection.

For near-wake stability and rotor-plane effects, the LLM has added value as it models individual blades and tip vortices. Further downstream, large wake deformations under yaw misalignment limit the usefulness of the vortex filament approach. A transition to vortex particles or engineering wake models may be a suitable option to continue wake predictions further downstream.

Model code supporting the work in this paper is available at

MJvdB: conceptualisation, methodology, software, validation, investigation, writing (original draft), visualisation. DDT: conceptualisation, writing (review and editing). PH: investigation (experiments WTA and WTC), writing (review and editing). DvdH: investigation (experiment WTB), writing (review and editing). BS: writing (review and editing), supervision. JWvW: writing (review and editing), conceptualisation, resources, funding acquisition.

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 made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This work is part of the robust closed-loop wake steering for large densely spaced wind farms research programme with project number 17512, which is (partly) financed by the Dutch Research Council (NWO). The experimental data (labelled WTA and WTC) are acquired in the scope of the CompactWind II research project (ref. no. 0325492H) and are funded by the Federal Ministry for Economic Affairs and Energy according to a resolution by the German Federal Parliament.

This paper was edited by Johan Meyers and reviewed by two anonymous referees.