Offshore wind farms are more commonly installed in wind farm clusters, where wind farm interaction can lead to energy losses; hence, there is a need for numerical models that can properly simulate wind farm interaction. This work proposes a Reynolds-averaged Navier–Stokes (RANS) method to efficiently simulate the effect of neighboring wind farms on wind farm power and annual energy production. First, a novel steady-state atmospheric inflow is proposed and tested for the application of RANS simulations of large wind farms. Second, a RANS-based wind farm parameterization is introduced, the actuator wind farm (AWF) model, which represents the wind farm as a forest canopy and allows to use of coarser grids compared to modeling all wind turbines as actuator disks (ADs). When the horizontal resolution of the RANS-AWF model is increased, the model results approach the results of the RANS-AD model. A double wind farm case is simulated with RANS to show that replacing an upstream wind farm with an AWF model only causes a deviation of less than 1 % in terms of the wind farm power of the downstream wind farm. Most importantly, a reduction in CPU hours of 75.1 % is achieved, provided that the AWF inputs are known, namely, wind farm thrust and power coefficients. The reduction in CPU hours is further reduced when all wind farms are represented by AWF models, namely, 92.3 % and 99.9 % for the double wind farm case and for a wind farm cluster case consisting of three wind farms, respectively. If the wind farm thrust and power coefficient inputs are derived from RANS-AD simulations, then the CPU time reduction is still 82.7 % for the wind farm cluster case. For the double wind farm case, the RANS models predict different wind speed flow fields compared to output from simulations performed with the mesoscale Weather Research and Forecasting model, but the models are in agreement with the inflow wind speed of the downstream wind farm. The RANS-AD-AWF model is also validated with measurements in terms of wind farm wake shape; the model captures the trend of the measurements for a wide range of wind directions, although the measurements indicate more pronounced wind farm wake shapes for certain wind directions.

The growth of offshore wind energy has led to wind farm clustering, where wind farm interaction is unavoidable. Recently, the Danish government released a report with plans for a new 10 GW offshore wind farm cluster situated around an artificial energy island hub in the North Sea

Low-fidelity engineering wind turbine wake models

It is not trivial to verify the wind farm parameterizations in mesoscale models because, e.g., in the WRF model the wind farm parameterizations are implemented within 1D planetary boundary layer (PBL) schemes

Wind farm (cluster) modeling with RANS is challenging when an inflow model with an atmospheric boundary layer (ABL) height is employed together with a large horizontal domain in the order of 200 km and more

The wind farm cluster validation case and the corresponding supervisory control and data acquisition (SCADA) measurements are discussed in Sect.

Figure

Wind farm cluster site.

Wind farm cluster definition.

This study uses 3 years of SCADA data from Dudgeon, from 1 January 2018 to 1 January 2021. For each turbine, the data are first averaged to 10 min periods, and then each period is kept only if it does not contain any missing data, non-production or curtailment periods, low power values (

In parallel, ERA5 data interpolated from the nearest ERA5 grid points to the wind farm location, for the same 3-year period, are used to estimate the Obukhov length

The wind farm simulations are performed with PyWakeEllipSys v3.0

In this work, different RANS flow domains are used to model wind farm clusters containing single, double, and triple wind farms. A sketch of the flow domain types of the different applied actuator methods is depicted in Fig.

Sketch of different RANS domain types. Cyan and magenta boxes contain refined cells to resolve the wind turbine and farm wakes. Blue disks and boxes represent AD and AWF models, respectively.

RANS grid and boundary conditions (BCs) for the wind farm cluster validation case. Horizontal

Figure

Figure

The RANS-AD wind farm simulation setup is similar to the one used in a previous work

The AWF model represents each wind farm as a single entity, and its effect on the flow is modeled by a distributed thrust force:

One could employ additional terms in the turbulence model equations to account for the effect of under resolving the wind farm layout in the AWF model when using large horizontal cells

The drag force of the AWF model

AWF horizontal drag force distribution integrated over the height for Race Bank (RB)

Equation (

For each wind farm modeled as an AWF,

RANS-AD simulations are performed to calculate the total wind farm thrust force for a range of wind speed and directions (

RANS-AWF simulations are performed using the same flow cases as step 1a in order to calculate

RANS-AD-AWF or RANS-AWF cluster simulations are performed.

We employ a two equation turbulence model in the form of a

We employ an analytically prescribed potential temperature profile

The effect of the prescribed temperature is represented by a buoyancy source term in the

Atmospheric inflow model precursor results employing a prescribed temperature profile compared to a neutral ASL solution and results from the WRF model. Results are shown in terms of wind speed

Summary of input and derived parameters for the ABL inflow model.

Since we lack measurements of the freestream, we need an alternative source to determine the input parameters for the inflow model. The New European Wind Atlas (NEWA) database

In the present work, we only use one inflow profile for all simulations, for simplicity and also when the inflow wind speed cases are not equal to 8 m s

The new inflow model still includes a low eddy viscosity region above the ABL height, similar to the ABL model based on

Depiction of the WRF model domains (orange frames): outermost parent domain (9 km grid spacing) with two nested inner domains (3 and 1 km grid spacing, respectively). The location of the Dudgeon, Sheringham Shoal, and Race Bank wind farms are indicated in blue, green, and red, respectively.

In the previous section, output from a mesoscale long run using the WRF model was used to derive input parameters for the RANS inflow model. Another set of mesoscale simulations using the WRF model was performed on the Dudgeon wind farm area (including the Dudgeon, Sheringham Shoal, and Race Bank wind farms) but using an implementation of the WRF model version 3.7.1, in which the explicit wake parameterization (EWP) is included

The TurbOPark model from

The TurbOPark wake model is implemented in DTU Wind's open-source wind farm simulation tool PyWake

TurbOPark, as implemented in PyWake, is used with two different setups: one reflects the original model setup from

The horizontal drag distribution in the AWF model represents the wind turbine density per cell. If large horizontal cells are used in the AWF model, e.g., in the order of the wind turbine interspacing in a wind farm, then the AWF grid orientation with respect to the wind farm layout can have a large influence on the wind farm wake shape when counting or binning the number wind turbines per cell. We refer to this method as the binning method. One can solve the issue by using a superposition of two-dimensional Gaussian functions for the wind turbines to represent the number of wind turbines per cell. To illustrate the difference between the binning and the Gaussian methods, a square wind farm with

The effect of different horizontal drag force distribution methods for a coarse AWF model (

The effect of different horizontal drag force distribution methods for a coarse AWF model (

The effect of different horizontal spacing in the AWF model for a square wind farm. Contours of integrated canopy density

The results shown in Fig.

The effect of the horizontal resolution in the AWF grid

Difference between AD and AWF models in terms of streamwise velocity and turbulence intensity for a square wind farm layout with 8 D spacing. AWF models differ in horizontal spacing (

Same as Fig.

The difference between the AD and the AWF simulation results are depicted in Fig.

Double wind farm case: effect of Sheringham Shoal (ShS) on Dudgeon (Du) for a wind direction of

The effect of Sheringham Shoal on Dudgeon for a wind direction of

The contour plots in Fig.

Double wind farm case: effect of Sheringham Shoal on Dudgeon in terms of wind farm power loss

There is a maximum of 0.9 % difference between the RANS-AD and RANS-AD-AWF models in terms of the equivalent wind speed extracted from the first row of wind turbines (with respect to the freestream wind speed). Note that it is not possible to perform this exercise with the RANS-AWF simulations, since both wind farms are represented by an AWF model. This difference in wind speed is smaller along the transect, and all three RANS models predict a wake magnitude in the range of the WRF model results, as shown in Fig.

Grid size and computational effort of RANS wind farm (cluster) simulations including Dudgeon (Du), Sheringham Shoal (ShS), and Race Bank (RB). Percentage between brackets reflects the reduction when using AWF models with respect to only using ADs. CPU and run hours are listed per flow case.

The wind farm power loss of Dudgeon due the presence of Sheringham Shoal is depicted in Fig.

The wind farm cluster consisting of the three wind farms Dudgeon, Sheringham Shoal, and Race Bank is simulated with RANS-AD-AWF, where Dudgeon is represented by ADs, and the other two wind farms are modeled with the AWF model. A range of wind directions between 190 and 300

Measured and simulated effect of the upstream wind farms on the southern wind turbine row of Dudgeon in terms of wake shape for different wind directions

Same as Fig.

Figures

The RANS results of the western row in Fig.

The main motivation for the AWF model is the reduction in computational effort compared to using ADs. Table

The RANS-AD grids for multiple wind farms could be reduced by using more complex grid topologies, where each wind farm is situated in a refined region instead of using a single large refined area that includes all wind farms.
However, it is more challenging to generate such a grid topology compared to box-type domains. Furthermore, it should be noted that the simulations are performed on an in-house shared computer cluster. Hence, the listed CPU hours and wall clock time in Table

When the investigated wind farm cluster is simulated with Dudgeon as ADs and the other wind farms as AWF models, 76 % of cells can be saved compared to modeling the wind farm cluster with only ADs due to a reduction in horizontal spacing outside the Dudgeon wind farm area, as listed in Table

A RANS-based wind farm parameterization, the AWF model, is proposed. It uses a wind farm thrust force as a momentum sink similar to a forest canopy model. The AWF model can be used as an obstacle model to a downstream wind farm of interest represented by ADs, or it can be used to estimate wind farm power when the downstream wind farm is also modeled as an AWF. The AWF model simulates wind farm interaction by a calibrated controller of wind farm thrust and power as a function of the wind farm volume-averaged wind direction and wind speed.

When the horizontal spacing in the AWF model is refined, the wind farm flow approaches the results from a RANS-AD wind farm simulation. This is achieved by calibrating the thrust force magnitude with precursor RANS-AD wind farm simulations and employing a horizontal thrust force distribution in the form of wind turbine density using a superposition of the wind turbine coordinates represented by two-dimensional Gaussian functions. The verification study showed that the Gaussian superposition method solves the problem of artificial wind farm wake effects that can occur when the number of wind turbines are binned for large horizontal cells, as current wind farm parameterizations implemented in numerical weather models, such as the WRF model, do.

A new atmospheric inflow model is introduced that is potentially more suited for wind farm cluster simulations because it does not rely on an ABL height set by a global turbulence length-scale limiter that can result in nonphysical wind farm wakes. The proposed inflow model relies on a prescribed analytical temperature profile including an inversion height and inversion strength, while a temperature equation is not solved for in order to maintain a steady-state inflow model. The model is shown to be dependent on three non-dimensional numbers. The new inflow model does not (and is not expected to) solve the problem of numerical instabilities related to the low eddy viscosity region above the ABL in combination with large horizontal domains associated with wind farm clusters. The problem is mitigated in the present work by using a relatively low domain height, which can introduce additional numerical blockage, thereby negatively influencing the predictions. An alternative solution needs to be found in the future to allow the more preferred, taller domain heights. In addition, the new ABL model requires more validation with measurements.

The proposed RANS-AWF and inflow models are employed to simulate two neighboring wind farms and a cluster consisting of three wind farms, where either one of the wind farms is modeled by ADs, and the remaining wind farms are represented by AWF models (RANS-AD-AWF) or all wind farm are AWF models (RANS-AWF). The results for the double wind farm case are compared with TurbOPark, WRF, and RANS-AD simulations (for the latter, all wind turbines are modeled by ADs) using the wind speed derived from the front row turbines of the downstream wind farm and the horizontal wind speed extracted from a transect 1.4 km upstream of the wind farm farthest downstream. The latter is performed because the chosen resolution for the WRF model simulations was not sufficient to resolve the front row wind turbine wind speed. While the overall horizontal wind speed at the wind farms in the WRF model simulations is quite different with respect to the RANS results, the horizontal wind speed at the transect from the WRF results compares well with those from the RANS-AD, RANS-AD-AWF, and RANS-AWF models. In addition, the front row wind speed in the RANS-AD-AWF only deviated by 0.9 % compared to the RANS-AD results with respect to the freestream. The original formulation of TurbOPark shows stronger wind farm wake effects compared to the other models, but its result in terms of wind farm wake shape compared well with all models. This indicates that a comparison in terms of wind farm wake shape should not be the only type of validation. A revised TurbOPark setup, where the ground model is switched off and a larger wake expansion coefficient of 0.06 is used, predicts much better results compared to the higher-fidelity models. Unfortunately, the RANS-AD-AWF simulations of the wind farm cluster could only be validated with the shape of the front row wind speed because the SCADA measurements of this row are used to both measure the wake of the upstream wind farm and determine the freestream conditions due to a lack of concurrent freestream measurements. The trends of the RANS-AD-AWF simulation results of the upstream wind farm wake shapes compare reasonably well with the results from the SCADA, although the measured shapes indicate stronger wind farm wake effects, possibly due to near-stable conditions that were not filtered out from the SCADA in order to maintain a large enough data set. More validation of both the AWF model and prescribed temperature inflow model is required. In addition, we need SCADA with concurrent inflow measurements in order to validate the magnitude of wind farm wakes and its impact on neighboring wind farms.

With the AWF model, one can simulate large wind farm clusters with RANS; the wind farm cluster validation case showed a reduction of 92.3 % and 99.9 % in CPU hours when two of three wind farms or all wind farms are represented by AWF models instead of using ADs, respectively, when the input wind farm thrust and power coefficients are known. If the wind farm power and thrust coefficients are calculated from a RANS-AD simulation of each wind farm, as performed in the present work, then the reduction in computational effort is 82.7 % when the wind farm cluster validation case is solely modeled by AWF models. Simpler and faster models that can generate the AWF input are currently investigated in a follow-up work

The AWF model can be used to further develop wind farm parameterizations in the WRF; this can be achieved by implementing the two-dimensional Gaussian superposition method of the wind turbine density and by using a wind farm drag coefficient that is dependent on the wind direction instead of using the wind turbine thrust curves. Idealized WRF simulations could be used to compare the resulting wind farm wakes with RANS-AWF simulation results.

The current implementation of the AWF model does not include any sources in the turbulence transport equations because the employed horizontal spacing of 2 D turned out to be sufficiently fine. If larger horizontal cell sizes are desired, one could investigate the use of additional turbulence-related source terms.

In a previous work

Similarity of prescribed temperature inflow model for

A new atmospheric inflow model is proposed in Sect.

Summary of input and derived parameters for the ABL inflow models, both using

Table

The four inflow profiles listed in Table

Atmospheric inflow model precursor results employing a global turbulence length-scale limiter

Contours of the streamwise velocity at the reference height of the validation case, simulated with RANS-AD-AWF, for a wind direction of 235

Wake magnitude of the validation case for a wind direction of 235

An overview of the numerical setup of the original and revised TurbOPark model, as implemented in PyWake

Summary of the original and revised TurbOPark setup in PyWake.

An overview of the numerical setup of the WRF simulations, including wind farms, is provided in Table

Summary table of WRF model configurations including initial and boundary conditions.

The numerical results are generated with proprietary software, although the data presented can be made available by contacting the corresponding author.

MPVDL drafted the article and produced the figures. MPVDL and OGS proposed the idea of a wind farm model with a drag coefficient as a function of wind direction. MPVDL proposed representing the wind turbine positions in the AWF model as two-dimensional Gaussian functions. AMF suggested using a standard deviation that is twice the horizontal AWF grid size and proposed the use of Eq. (

The contact author has declared that none of the authors has any competing interests.

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

The first author was inspired by Andrey Sogachev to model a wind farm as a forest canopy. Furthermore, the authors would like to thank Mads Mølgaard Pedersen for his support with the implementation of TurbOPark in PyWake. We also gratefully acknowledge the computational and data resources provided on the Sophia HPC Cluster at the Technical University of Denmark,

This is work has been co-financed by Equinor ASA. Oscar García-Santiago was supported by the European Union's Horizon 2020 research and innovation program under grant agreement no. 861291 as part of the Train2Wind (

This paper was edited by Cristina Archer and reviewed by Gonzalo Pablo Navarro Diaz and two anonymous referees.