Wind turbine wake models typically require approximations, such as wake superposition and deflection models, to accurately describe wake physics. However, capturing the phenomena of interest, such as the curled wake and interaction of multiple wakes, in wind power plant flows comes with an increased computational cost. To address this, we propose a new hybrid method that uses analytical solutions with an approximate form of the Reynolds-averaged Navier–Stokes equations to solve the time-averaged flow over a wind plant. We compare results from the solver to supervisory control and data acquisition data from the Lillgrund wind plant obtaining wake model predictions which are generally within 1 standard deviation of the mean power data. We perform simulations of flow over the Columbia River Gorge to demonstrate the capabilities of the model in complex terrain. We also apply the solver to a case with wake steering, which agreed well with large-eddy simulations. This new solver reduces the time – and therefore the related cost – it takes to simulate a steady-state wind plant flow (on the order of seconds using one core). Because the model is computationally efficient, it can also be used for different applications including wake steering for wind power plants and layout optimization.

In this work, we present an improved formulation of the curled wake model

Parabolic solvers for RANS equations are a promising tool for fast wind
farm flow solvers.

Wake steering is a promising wind plant control strategy used to maximize
the power output of a wind plant

The wake of a wind turbine in yaw has a unique shape known as the curled wake

The curled wake model uses a simplified version of the RANS equations to predict the wake of a wind turbine in yaw

The curled wake solver presented in this work focuses on reducing computational cost and capturing wake steering effects. This is done by solving only the streamwise component of the linearized RANS equations and parametrizing the effects of the spanwise and wall-normal components using semianalytical solutions.

We use the RANS equations to model the time-averaged flow field
through a wind plant. The continuity equation is

The velocity is decomposed into a background flow (capital letters) and a
wake deficit (

The background flow (

The time-averaged wake velocities are denoted by

The effect of turbulence in the RANS equations is described by the divergence of the Reynolds stress tensor. The streamwise component of the
divergence of the Reynolds stress for the background flow solution (Eq.

A mixing length model is used to represent the terms in Eq. (

The Reynolds stress model used in this study was selected because of its computational efficiency. Resolving the spatial variations in the eddy viscosity would require the solution of the full RANS momentum equations and additional transport equations for relevant parameters in the selected Reynolds stress model

Wakes are initialized according to the wind speed at the rotor location in the plane closest to where the turbine is. As the solution marches downstream and new wind turbines are encountered, a new wake deficit is added to the plane (

Equation (

This numerical equation is discretized using a forward-in-time centered-in-space method with the stability criteria shown in
Eq. (

Figure

Schematic of the computational strategy used to solve Eq. (

To better understand the low computational cost of the solver presented, we assess the number of floating point operations needed to obtain a solution to Eq. (

Scaling of the computational algorithm

The current solver and formulations can also be used in applications with complex terrain. The complex terrain geometry can be included by specifying the boundary condition (

Streamwise velocity contours showing a plane perpendicular to the predominant wind direction.

We use the model proposed to compare with two different cases. The first comparison is done using supervisory control and data acquisition (SCADA) data from the Lillgrund wind plant. Second, we compare the model to a series of LES for an array of turbines with different yaw combinations.

We use the model proposed to compute the flow field over the Lillgrund wind plant. Ten-minute average SCADA data are available for all turbines for different wind conditions. The SCADA data were organized by wind speed, turbulence intensity, and wind direction into bins with a width of 1 m s

Layout of the Lillgrund wind plant with wind direction used in each simulation.

Figure

Comparison of turbine power versus SCADA data for the Lillgrund wind plant for cases at three different wind directions (185, 215, 255

List of LES cases performed for comparison study.

We now compare the model to results from LES of wakes in steering conditions.
The simulations were performed using the Simulator fOr Wind Farm Applications (SOWFA) using an actuator disk model with rotation

Total power output for wind plant LES with wake steering compared with the model proposed.

Velocity at hub height normalized by average speed at hub height from the model proposed and from LES and power output for each turbine from the model proposed compared to results from LES. Simulations of a four-by-three turbine array. The bars in the LES power denote 1 standard deviation of the power. Turbine numbering is from bottom to top and left to right.

Same as Fig.

Figure

We select two representative cases and compare the power for all turbines and
velocity at hub height. Figures

Fast wind power plant flow solvers are much needed for wind plant controls and layout optimization. In this work, we presented a simplified and fast solver for wind turbine wakes based on the curled wake model presented in

Some of the limitations from the different approximations of the model include a turbulence model mixing length that only depends on the vertical coordinate, a linearized solution of the vortices from curl that do not decay, a missing near-wake formulation, and no pressure term in the equations. These approximations were done to reduce the computational cost. Future work will focus on addressing the limitations and, more specifically, comparing the model with RANS, improving the turbulence model without compromising computational cost, improving the near wake, implementing a vortex decay model, using the solver for yaw-angle optimizations in a wind plant, and improving code performance to increase speed. This solver will soon be incorporated into the FLORIS framework and will be freely available.

The turbulence model proposed uses a constant,

Comparison of turbine power versus SCADA data for the Lillgrund wind plant for cases at two different wind directions (185, 215

In this section, we show a different formulation for the development of the turbulence model. It is possible to invoke the eddy-viscosity hypothesis in the derivation for the base and wake deficit equations independently.
When doing this, we express the Reynolds stresses as

Here, we show a convergence study for the curled wake model based on one of the simulations for the Lillgrund wind farm in Sect.

Comparison of turbine power for all turbines using different number of grid points across the turbine diameter for the Lillgrund wind plant for cases at wind direction of 185

Code will be available upon request by contacting the correspondence author.

LAMT led the model development and wrote 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.

The authors would like to thank Paula Doubrawa for help in allocating funding for this work and Sheri Anstedt from NREL for their help in editing the manuscript. The authors would also like to acknowledge the comments from Paul van der Laan and the anonymous reviewer. Data were furnished to the authors under an agreement between the National Renewable Energy Laboratory, Siemens Gamesa Renewable Energy A/S, and Vattenfall. Data and results used herein do not reflect findings by Siemens Gamesa Renewable Energy A/S and Vattenfall.

A portion of the research was performed using computational resources sponsored by the US Department of Energy's Office of Energy Efficiency and Renewable Energy and located at the National Renewable Energy Laboratory. This work was authored by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the US Department of Energy (DOE) under contract no. DE-AC36-08GO28308. Funding provided by the US Department of Energy Office of Energy Efficiency and Renewable Energy Wind Energy Technologies Office.

This paper was edited by Katherine Dykes and reviewed by Paul van der Laan and one anonymous referee.