This paper presents a model to incorporate the secondary effects of wake steering in large arrays of turbines. Previous models have focused on the aerodynamic interaction of wake steering between two turbines. The model proposed in this paper builds on these models to include yaw-induced wake recovery and secondary steering seen in large arrays of turbines when wake steering is performed. Turbines operating in yaw-misaligned conditions generate counter-rotating vortices that entrain momentum and contribute to the deformation and deflection of the wake at downstream turbines. Rows of turbines can compound the effects of wake steering that benefit turbines far downstream. This model quantifies these effects and demonstrates that wake steering has greater potential to increase the performance of a wind farm due to these counter-rotating vortices especially for large rows of turbines. This is validated using numerous large-eddy simulations for three-turbine, five-turbine, and wind farm scenarios.

Wake steering is a type of wind farm control in which wind turbines in a wind farm operate with an intentional yaw misalignment to mitigate the effects of its wake on downstream turbines in order to increase overall combined wind farm energy production (

An early model of wake steering was provided in

Several recent papers proposed a new wake deficit and wake deflection model based on Gaussian self-similarity (

One of the main issues observed with the Gaussian model in FLORIS is that the model tends to under-predict gains in power downstream with respect to LES and field data. In addition,

This paper presents a hybrid wake model, which modifies the Gaussian model (

Vortices drive a process of added yaw-based wake recovery, which increases the gain from wake steering to match LES and field results.

The interaction of the counter-rotating vortices with the atmospheric boundary layer shear layer and wake rotation induces wake asymmetry naturally.

By modeling of the vortices, secondary-steering and related multi-turbine effects are included, which will be important for evaluating wake steering for large wind farms.

In this paper, we will introduce the analytical modifications made to the Gaussian model in Sect

This section briefly describes the Gaussian model used to describe the velocity deficit and the effects of wake steering in a wind farm. Figure

The proposed model, known as the Gauss–curl hybrid (GCH) model, builds upon the Gaussian model introduced in

Model setup that includes yaw-induced effects such as yaw-added recovery and secondary steering. The standard modeling for wake deflection is shown in gray, and the proposed deflection model in this paper is shown in red. These effects manifest through the spanwise and vertical velocities that are generated from yaw-misaligned turbines. These effects are described in Sect.

The wind turbine wake model used to characterize the velocity deficit behind a turbine in normal operation in a wind farm was introduced by several recent papers including

This wake model also computes added turbulence generated by turbine operation and ambient turbulence conditions. For example, if a turbine is operating at a higher thrust, this will cause the wake to recover faster. Conversely, if a turbine is operating at a lower thrust, this will cause the wake to recover slower. Conventional linear flow models have a single wake expansion parameter that does not change under various turbine operating conditions.

In addition to the velocity deficit, a wake deflection model is used to describe the flow behavior behind a yaw-misaligned turbine, which occurs when performing wake steering and is also implemented based on

The total deflection of the wake due to yaw misalignment is defined as

The spanwise and vertical velocity components are currently not computed in the Gaussian model, but they are critical components for modeling the effects of wake steering. These velocity components can be computed based on wake rotation and yaw misalignment as shown in

Wake rotation is included by modeling a Lamb–Oseen vortex, which makes sure that the vortex is not a singular point near the center of the rotor. The circulation strength for the wake rotation vortex is now

The vertical and spanwise velocities can then be computed using the strength of the vortex,

In addition to the wake rotation, when a turbine is operating in yaw-misaligned conditions, the turbine generates a collection of smaller counter-rotating vortices that are approximated as one pair of large counter-rotating vortices that are released at the top and the bottom of the rotor and generate additional spanwise and vertical velocity components that need to be accounted for in this approach (

As is done with wake rotation, the spanwise and vertical velocity components,

The spanwise and vertical velocities are combined using a linear combination at downstream turbines as is done in

Finally, the vortices generated by the turbines decay as they move downstream. The dissipation of these vortices is described in

The streamwise velocity and the wake deflection are influenced by the spanwise and vertical velocity components,

In

In particular,

The TKE is converted to a turbulence intensity through the following (

Three-turbine array in SOWFA where all turbines are aligned

In addition to added wake recovery, the model proposed in this paper is able to predict secondary steering that matches large-eddy simulations. The wake deflection model described in Sect.

Specifically, the vortices described in the previous section propagate far downstream, dissipate, and affect all turbines directly downstream of the turbine that generated the vortices. When they reach a downstream turbine, they impact the wake of the downstream turbine in a phenomenon called secondary steering (

Due to the presence of the effective yaw angle, downstream turbines generally do not have to yaw as much as upstream turbines to produce large gains. This phenomenon was observed in a wind tunnel study (

The addition of yaw-added recovery and secondary steering effects has increased the computational time by 3.5 times, e.g., a five-turbine case takes 0.007 s to run the GCH model compared to 0.002 s for the Gaussian model. However, the results in this paper indicate that when evaluating wake steering, GCH is necessary to include as the Gaussian model is not able to capture the compounding effects of wake steering.

First, the Gaussian model, GCH model, and SOWFA are compared in three-turbine array simulations. The three-turbine array demonstrates the benefits of the yaw-added recovery (YAR) effect as well as secondary steering (SS). The following plots show the contributions of YAR and SS compared with the Gaussian model and the full GCH model, which contains both YAR and SS. The YAR and SS models are computed by disabling the model produced in Sect.

FLORIS results for the three-turbine case shown for the GCH model with the centerline of the wake computed for GCH (red) and the Gaussian model (blue), where the first turbine is yawed

FLORIS results for the three-turbine case shown for the GCH model with the centerline of the wake computed for GCH (red) and the Gaussian model (blue), where the first turbine is yawed

The three-turbine scenario was simulated at 8 m/s with 6 % and 10 % turbulence intensities and spaced 7

Comparison of changes in power when sweeping the angle of the second turbine, i.e., Turbine 2, when the angle of the first turbine, i.e., Turbine 1, is set to +20

Comparison of changes in power when sweeping the angle of the second turbine, i.e., Turbine 2, when the angle of the first turbine, i.e., Turbine 1, is set to +20

Next, several simulations were run at each turbulence intensity where the first turbine was yawed

The Gaussian model is not able to capture the secondary effects of wake recovery and secondary steering. The Gaussian model is able to capture gains in low-turbulence (6 %) conditions; see Fig.

Next, five turbines were simulated in SOWFA, the Gaussian model, and the GCH model for different combinations of yaw angles, starting with all aligned, the first turbine yawed 25

Flow field of five turbines using GCH. The resulting centerline behind each turbine is shown for GCH in red and the Gaussian model in blue. The optimized yaw angles are based on the values shown in Table

Absolute power values for each turbine (excluding the upstream Turbine 1) in the five-turbine array for a wind speed of 8 m/s and low turbulence, i.e., 6 % turbulence intensity. Total turbine power is shown in the rightmost plot. The

Figure

Figure

Power gains for each turbine (excluding the upstream turbine) in the five-turbine array for a wind speed of 8 m/s and low turbulence, i.e., 6 % turbulence intensity. Total power gain is shown in the rightmost plot. The

Absolute power values for each turbine (excluding the upstream turbine) in the five-turbine array for a wind speed of 8 m/s and high turbulence, i.e., 10 % turbulence intensity. Total turbine power is shown in the rightmost plot. The

Power gains for each turbine (excluding the upstream turbine) in the five-turbine array for a wind speed of 8 m/s and high turbulence, i.e., 10 % turbulence intensity. Total power gain is shown in the rightmost plot. The

Five-turbine results for low- and high-turbulence conditions using SOWFA, the Gaussian model, and the GCH model. The bold font indicates the largest power gains identified in SOWFA, which also correspond to yaw angles identified by the GCH model.

Engineering wake models in FLORIS are often used to determine optimal set points for wake steering and assess the performance of these set points. The results of optimizing the Gaussian and GCH models are compared in this section. Specifically, the Gaussian and GCH models were optimized individually for the five-turbine case under low- and high-turbulence conditions. These yaw angles from each optimization were simulated in SOWFA. The power predicted in SOWFA, the Gaussian model, and the GCH model, for each set of yaw angles, are compared in Table

Flow field results from SOWFA where the wind direction is 270

In both low- and high-turbulence cases, the GCH optimized yaw angles produced higher power gains in SOWFA compared with the Gaussian model and also outperformed simply operating all turbines (except the last turbine) at a maximum yaw angle of 25

Finally, a full wind farm analysis was performed to quantify the potential of wake steering when effects such as yaw-added recovery and secondary steering are included. For this analysis, we used a 38-turbine wind farm used as in

Flow fields of the optimized Gaussian model

Flow field of the optimized GCH model

Wind farm results for low- and high-turbulence conditions for SOWFA, the Gaussian model, and the GCH model. The bold font indicates the largest power gains identified by SOWFA, which also correspond to the yaw angles identified by the GCH model.

Optimizations were performed with the Gaussian model and the GCH model for low (6 %) and high (10 %) turbulence conditions. Flow fields are shown in Fig.

The results of the optimization are shown in Table

Lastly, a full optimization over a wind rose was run for the wind farm in low- and high-turbulence conditions. The wind rose is shown in Fig.

Wind rose used to compute the AEP gains from wake steering.

Wind farm AEP results for low and high turbulence intensity (TI).

This paper introduces an analytical model that better captures the secondary effects of wake steering in a large wind farm. These secondary effects include yaw-added wake recovery as well as secondary wake steering that significantly boosts the impact of wake steering. The results of this model were compared with LES for 3- and 5-turbine arrays as well as a 38-turbine wind farm. The model compared well with results from LES and outperformed the Gaussian model in most cases. Furthermore, this paper demonstrated the possible gains in a large wind farm when considering these large-scale flow structures. Controllers can be developed in the future to manipulate these flow structures to significantly improve the performance of a wind farm.

The code is publicly available and can be accessed on

Please reach out to the authors for access to the data.

JK led the model development and led the writing of the article. PF ran all of the LES cases, led the analysis, and contributed significantly to the text of the article. All authors provided input to this paper.

The authors declare that they have no conflict of interest.

The views expressed in the article do not necessarily represent the views of the DOE or the US Government. The US Government retains and the publisher, by accepting the article for publication, acknowledges that the US Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for US Government purposes.

This work was authored (in part) by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under contract no. DE-AC36-08GO28308. Funding was provided by the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy Wind Energy Technologies Office.

This research has been supported by the Department of Energy (contract no. DE-AC36-08GO28308).

This paper was edited by Carlo L. Bottasso and reviewed by Bart M. Doekemeijer and one anonymous referee.