Methods of turbine wake modeling are being developed to more accurately account for spatially variant atmospheric conditions within wind farms. Most current wake modeling utilities are designed to apply a uniform flow field to the entire domain of a wind farm. When this method is used, the accuracy of power prediction and wind farm controls can be compromised depending on the flow-field characteristics of a particular area. In an effort to improve strategies of wind farm wake modeling and power prediction, FLOw Redirection and Induction in Steady State (FLORIS) was developed to implement sophisticated methods of atmospheric characterization and power output calculation. In this paper, we describe an adapted FLORIS model that features spatial heterogeneity in flow-field characterization. This model approximates an observed flow field by interpolating from a set of atmospheric measurements that represent local weather conditions. The objective of this method is to capture heterogeneous atmospheric effects caused by site-specific terrain features, without explicitly modeling the geometry of the wind farm terrain. The implemented adaptations were validated by comparing the simulated power predictions generated from FLORIS to the actual recorded wind farm output from the supervisory control and data acquisition (SCADA) recordings and large eddy simulations (LESs). When comparing the performance of the proposed heterogeneous model to homogeneous FLORIS simulations, the results show a 14.6 % decrease for mean absolute error (MAE) in wind farm power output predictions for cases using wind farm SCADA data and a 18.9 % decrease in LES case studies. The results of these studies also indicate that the efficacy of the proposed modeling techniques may vary with differing site-specific operational conditions. This work quantifies the accuracy of wind plant power predictions under heterogeneous flow conditions and establishes best practices for atmospheric surveying for wake modeling.

Low-fidelity wake modeling utilities such as FLOw Redirection and Induction in Steady State (FLORIS) are typically used for the estimation of wind farm power output or the implementation of wind farm controls that help improve the overall performance of a wind farm. This includes implementing real-time corrective strategies that aid in reducing stress-inducing loads on turbines

The consequences are particularly evident when observing the accuracy of power predictions for wind farms located within complex terrain or wind farms that are otherwise subject to spatially variant conditions in the atmosphere. Because these atmospheres are subject to dramatic changes in the velocity and direction of wind, it is difficult to anticipate how the resulting wakes will form and what kind of power output should be expected. In

It should be noted that there are existing wake models that incorporate elements of heterogeneous wake effects caused by varying atmospheric conditions. For example, one model presented in

The aforementioned models present many methods for approximating farm–flow interaction in heterogeneous conditions. As a contribution to this area of research, this article will present a modified version of FLORIS that features an advantageous capability in modeling wind farms with spatially variant weather conditions and complex terrain. This adapted version of FLORIS presents several novel developments within the scope of control-oriented wake modeling research: an interpolation algorithm is implemented, which allows the user to define a gradient of atmospheric characteristics across the flow field, based on several measurements within or adjacent to the wind farm; elements of spatially variant wind direction, wind speed, and turbulence intensity are integrated into wake calculations of the preexisting FLORIS model; and an additional method is introduced to minimize error in power prediction accuracy caused by high-turbulence intensity and wind speed variance.

The objective in developing this proposed model is to capture a more accurate representation of the effects of wind farm wake interactions within complex terrain without actually resolving any terrain geometry during simulation. This study aims to analyze the accuracy of power output predictions and wake modeling performance for the proposed wake model, through comparisons to large eddy simulation (LES) wind farm supervisory control and data acquisition (SCADA) records.

FLORIS

The operation and performance of a turbine is modeled with respect to the relationship between the thrust coefficient,

FLORIS provides an option to select particular models for wake velocity deficit and wake deflection separately to suit the user's performance needs. The variety in modeling capabilities reflects a range of trade-offs between computational efficiency and the number of detailed physics applications applied to calculations. If a model is more computationally expensive, it is likely to implement more sophisticated algorithms as well, in hopes of achieving a more accurate result. These models all have a different approach to modeling turbine wake interactions and offer different strengths and weaknesses in functionality. Most models can be classified as either a velocity deficit or a wake deflection calculation, but there are also the Gaussian and Curl models that incorporate both calculations and extend further into the overall FLORIS wake modeling structure and control tools. For the purposes of this article, only the Gaussian wake model will be explained in depth. See

The Gaussian wake model is comprised from a series of papers, including

The Gaussian model computes the streamwise velocity deficit at any point in a turbine's wake by using analytical formulations of Reynolds-averaged Navier–Stokes (RANS) equations to an assumed Gaussian wake profile. The Gaussian wake is based on the self-similarity theory used for free shear flows

The wake width in the

The findings of

In the scope of this study, it is important to note that the introduction of spatial heterogeneity in initial wind conditions (which is a key principle in the proposed model) violates the original assumption of no pressure gradient for the derivation of the Gaussian wake model. Although this limits the model's ability to conserve key principles that govern the physical dynamics of fluid flow, the results of this study show that the measured improvements in model accuracy outweigh the consequences of incomplete conservation. In

The Gaussian model also implements methods proposed by

To implement the effects of shear,

The Gaussian model was designed to avoid the inaccuracies caused by neglecting the effects of turbulence intensity by implementing methods introduced by

The turbulence intensity,

Based on the original definition proposed in

As noted earlier, the Gaussian wake model was developed under the assumption of flat terrain. Since the heterogeneous model was specifically designed to best benefit wind farms located in complex terrain, it is important to know the consequences of violating this assumption. In

Previously, FLORIS derived the initial wind speed, wind direction, and turbulence intensity by using one value to represent the entire flow-field domain. In this article, we describe the modifications to FLORIS to accommodate heterogeneous flows. This section will explain the methods used to calculate wakes based on the gradient of values observed in the undisturbed flow field without wake effects. The motivation behind this development was to create a more detailed characterization of the initial state of the atmosphere, which leads to improvements in the power predictions of a wind farm.

To implement heterogeneity in FLORIS, an interpolation is performed based on several input values assigned to spatially varying coordinates inside or adjacent to the wind farm (see Fig.

A diagram representing the processes performed during the initialization of the heterogeneous FLORIS model.

A visual depiction of the methods used to interpolate and define atmospheric characterization values at specific points within the input coordinates.

The process begins with implementing a piecewise linear interpolation method for all points within the region defined by the input coordinates. First, Delaunay triangulation is performed using the Quickhull algorithm discussed in

The next step in determining the interpolated values is to use the established triangular elements to perform barycentric interpolation. During this step, the barycentric coordinates of each point of interest are determined relative to the triangular element in which it resides. Based on each set of barycentric coordinates, the interpolated result is calculated using a weighted average of the values defined at the triangle's vertices

Linear barycentric interpolation was chosen be implemented for this step because it is relatively efficient in computation and can be easily implemented without requiring any input parameters other than the locations and values of wind measurements. Although, it must be noted that the accuracy of the interpolated values is dependent on the quality of input measurements provided, the complexity of the terrain geometry, and the weather patterns observed in the physical wind farm.

The extrapolation process implements a nearest-neighbor interpolant to calculate all remaining unknown values. Using the recently interpolated point values in addition to the original input values, this method operates by selecting a single value at the nearest location to the point being extrapolated and assigning this nearest value to the extrapolated point. A visualization of this calculation is depicted in Fig.

The extrapolation process used to define the remaining values for characterization of the initial state of fluid flow.

The nearest-neighbor extrapolation method was chosen because it defines a feasible relationship between input measurements and does not attempt to extrapolate using a formula derived from a curve-fitting or trend-predictive algorithm. Many other extrapolation methods attempt to predict a rate of change outward of the interpolation domain by implementing a function that approximates a predicted progression of extrapolated values. For example, it was found that the analytic continuation of radial basis functions (RBFs) and fitted polynomial splines outside of the initial domain often produced a non-feasible output that did not respect the physical limitations of the atmospheric characteristic being extrapolated. Although it was speculated that these methods could likely be adjusted with tuning factors to fit extrapolated data within feasible bounds, efforts to do this were not explored in this study. Instead, the nearest-neighbor algorithm was chosen to simplify implementation of realistic extrapolation within the model.

When solving for the interpolated and extrapolated values for turbulence intensity and wind speed, values are easily computed because they are defined by values on a non-cyclical scale. Because wind direction is represented using angles in degrees, the interpolation and extrapolation methods must be circular. The issue of interpolating circular data was addressed by simply computing the interpolation twice for each angle of wind direction,

It should be noted that the vertical (

Before FLORIS performs any calculations for velocity deficit in wakes, it first assigns an initial value of wind speed (

Visualizations of two planes showing the FLORIS flow field during a simulation with heterogeneous wind speed.

Similar to wind speed, an interpolation of wind direction is initially established across the flow-field grid through the methods of interpolation discussed in Sect.

A depiction of the initial processes before the calculation of wakes. Panel

Using the rotated turbine map shown in Fig.

Depiction of the process performed in FLORIS to align the flow-field grid with the location and wind direction of turbine T6 (as defined in Fig.

Next, to calculate the velocity deficit caused by each turbine's wake, all of the grid points in the flow field are rotated to replicate the effects of changing wind direction. These rotated grid points represent the redirection of the flow in response to changing wind direction within the flow field (see Fig.

Visualizations of FLORIS calculating velocity deficit turbine T6 in conditions of heterogeneous wind direction. The velocity deficit is calculated using the grid points in the fully rotated position

As discussed in Sect.

Visualization of a flow field with heterogeneous wind direction. Turbine rotors are indicated by black lines.

A second visualization of a flow field with more complex heterogeneous wind direction. Wind input measurements are indicated using diamond markers, and turbine rotors are shown with black lines.

Visualization of a flow field with heterogeneous turbulence intensity. The turbines that experience higher turbulence intensity show a faster rate of wake recovery, and vice versa. Turbine rotors are indicated by black lines.

The grid point spacing in the

To further exemplify the applications of this functionality, Fig.

It is important to consider that this model was not designed to calculate the effects of changes in wind direction that are extremely dynamic. A change in wind direction that is too drastic will cause grid points in the rotated flow-field grid (the red points shown in Fig.

Although it may be possible for the wind direction within a wind farm to change this drastically, these conditions often involve multiple adjacent domains of flow that are separated by a boundary, which are difficult to represent in this model. These weather conditions are also most often observed in instances of lower wind speeds and therefore can be considered not as lucrative in regards to power production. Plans for future developments to FLORIS involve designing a more inclusive model that is capable of mitigating issues concerning rapid changes in wind direction.

The geographic distribution of turbulence intensity is established for the initial state of the flow field through the interpolation methods discussed in Sect.

It is important to note that in the interest of conserving computational efficiency, calculations for evaluating the rate of wake expansion and recovery are only dependent on the updated turbulence intensity at the location of the turbine creating the wake.

In addition to the heterogeneous features, developments were also made to reduce inaccuracies in power output predictions caused by turbulent operating conditions. As mentioned in Sect.

Specifically, this approach adjusts the power output with respect to the level of turbulence intensity at a turbine. The adjusted power is calculated by using distribution of the wind speed fluctuations at the turbine, based on calculations that consider the original wind speed and the standard deviation in wind speed. The first step in this algorithm is to create a normalized probability density function,

The value of the power coefficient,

Adjusted power curve for the NREL 5 MW reference turbine for different turbulence intensities. The dashed lines denote the cut-in, rated, and cutout wind speeds and also represent the boundaries of the first, second, and third regions, respectively.

The following expression may be used to calculate the final value of adjusted power output,

In future work, this turbulence-correction model could be improved by implementing a similar consideration of the thrust coefficient,

It is important to note that similar models have been developed that incorporate methods of turbulence re-normalization based on machine-learned or empirically derived data

Two validation studies are presented to analyze the effectiveness of the adapted FLORIS wake model. In Sect.

This section presents a validation study that evaluates the power prediction accuracy of the proposed heterogeneous FLORIS model in comparison to large eddy simulations of a 38-turbine array, calculated using NREL's tool Simulator fOr Wind Farm Applications (SOWFA)

Twelve test cases were evaluated for this study; each was simulated using different wind directions, varying from 10 to 340

Horizontal cut planes of the FLORIS simulations with wind direction at 270

To analyze the effects of wake losses for the simulated wind farm, additional heterogeneous and homogeneous FLORIS simulations were conducted excluding all FLORIS wake calculations. Figure

Total wind farm power output calculated using the homogeneous and heterogeneous FLORIS models, with and without wake losses. The results of the SOWFA simulation are also plotted for reference in black.

The absolute error of the total wind farm power output was calculated for each model to analyze power prediction accuracy at every wind direction evaluated in the case study. The results of these calculations are shown in Fig.

Absolute error in total farm power output calculated using the homogeneous and heterogeneous FLORIS models, with and without wake losses.

The mean absolute error (MAE) of power predictions at individual turbines in the wind farm was also calculated using Eq. (

Mean absolute error of individual turbine power output predictions calculated using the homogeneous and heterogeneous FLORIS models, with and without wake losses.

In comparison to Fig.

Horizontal planes of two different FLORIS simulations, taken at the same time-step iteration.

This section summarizes a validation study presenting comparisons of FLORIS power predictions to SCADA-recorded power outputs from an observed operational wind farm. A large, utility-scale wind farm located within mountainous terrain was chosen for this study because it is often subject to unpredictable and dramatic shifts in weather conditions. More information regarding the physical layout and characteristics of this wind farm can be found in Appendix

FLORIS simulations were performed using heterogeneous inputs of wind direction, turbulence intensity, and wind speed, which were taken from the wind farm's SCADA records. These inputs include four wind measurement values for each atmospheric characteristic, derived from meteorological (MET) tower measurements placed in various locations throughout the wind farm. Similar simulations were performed using an identical FLORIS model but with a singular homogeneous input for wind speed, wind direction, and turbulence intensity. These homogeneous inputs were derived by evaluating the average of the five heterogeneous input values at each time step. The resulting power output of all simulations was recorded with the inclusion of the turbulence correction and without. All cases were simulated using data averaged at time steps of 10 min over a range of 2 months.

In the following discussion, the results from all FLORIS simulations are presented and analyzed to determine the accuracy of power predictions from each test case. Figure

Although these visualizations do not give direct estimates of power prediction, they are helpful in translating the input measurements into a form that characterizes the general behavior of wind farm dynamics for the interpretation of the observer. The cut plane visualization is helpful in performing qualitative analysis of turbine wake interactions and is more useful when displaying the estimated weather conditions characteristic of each location in the flow field, which is improved in the heterogeneous model.

When comparing the performance of the simulations, the calculated power output was tabulated and compared for accuracy. In Fig.

Power output calculated by FLORIS for homogeneous (red), heterogeneous without the turbulence correction (blue), and heterogeneous with the turbulence correction (green), compared with SCADA data shown in black. Each shaded region represents the difference between predictions of power output and the measured power output from SCADA data.

In Fig.

It is possible for wake models to overpredict the power output of some turbines, and underpredict others, in a way that produces a total wind farm power estimate that seems accurate but is not using reliable and precise methods of calculation. To verify that the recent additions to FLORIS have improved the power-predicting capabilities, it must be confirmed that the new model produces a consistently accurate estimate with respect to each iteration in the time series and each turbine within the wind farm individually. To prove this model's consistency in accuracy, the normalized absolute error was calculated at each turbine at each iteration of the time series for this same day. The sum of the absolute error at all turbines within the wind farm is calculated for each simulation model at each time iteration. To calculate the sum of absolute error (SAE) for all turbines, the following formula was applied to each time iteration of the simulation.

Sum of the normalized absolute error at each turbine in the wind farm, computed at each time step.

The trends observed in Fig.

To ensure these same trends of accuracy persist over the entire 2-month period, the percent error of the total wind farm power output was calculated at each time-step iteration using the following equation.

Percent error of all three FLORIS models, plotted for comparison within varying ranges of wind speeds.

Although the plots for the wind speed domains vary slightly in distribution, it is clear that each histogram exemplifies a trend toward accuracy in simulations that incorporate heterogeneity and turbulence-correction calculations. It is important to note that only the data points shown in the percent-error range of each histogram were used to calculate the respective binned averages. The outliers were omitted because they tend to skew the presentation of the data set in a way that obscures the actual trend of data.

The mean absolute percent error (MAPE) values of all time-step iterations are also reported in Table

Mean absolute percent error in total wind farm power output for all FLORIS models, tabulated for comparison within varying ranges of wind speeds.

When comparing the MAPE values in Table

Although MAPE is an informative metric for analyzing the average percent error relative to a specific power output range, methods that use unweighted averaging are sometimes misleading in the analysis of overall power prediction accuracy. The relative error during time-step iterations with lower power output can seem large, even when the absolute error is insignificant in comparison to the magnitude of total farm output.

A more comprehensive representation of relative model accuracy is presented in the following table, where the mean absolute error (MAE) is evaluated for total wind farm output. This was calculated by evaluating the absolute error at each time step and then taking the mean of these error values. This calculation is performed using Eq. (

By taking an average of absolute errors instead of relative errors, MAE is a more effective metric in representing the overall accuracy of total wind farm power prediction. The resulting MAE values are shown in Table

Mean absolute error in total wind farm power output for all FLORIS models, tabulated for comparison within varying ranges of wind speeds. Total rated wind farm output was scaled to 100 MW for reference.

Mean absolute error in individual turbine power output for all FLORIS models, tabulated for comparison within varying ranges of wind speeds. Total rated wind farm output was scaled to 100 MW for reference.

Lastly, values of MAE were also calculated to represent the accuracy of the model at each individual turbine within the wind farm. Using this metric, Table

To analyze the influence of wake effects in this study, an identical set of simulations were performed excluding FLORIS wake loss calculations, and the results for MAE at the overall farm and individual turbine levels are reported in Tables

As noted in Sect.

This article introduces a method to include heterogeneous flow fields into the FLORIS simulation tool, as well as a turbulence correction to the power reported at each turbine. To analyze the developed model's improvements in accuracy, several FLORIS simulations with and without these changes were compared to large eddy simulations and SCADA data from a utility-scale wind farm. The results of the FLORIS simulations indicate that these two modifications improve power predictions of the wind farm at the turbine and wind farm level. The increased accuracy of this model's power prediction capabilities shows that this method is more precise in predicting farm–flow interaction in heterogeneous and turbulent environments, which previous versions of FLORIS were not able to simulate.

Overall, the heterogeneous and turbulence-intensity correction modifications presented in this article showed a positive effect on the accuracy of FLORIS capabilities. This improved model provides a more detailed quantitative and qualitative analysis of wind farm flow, including the demonstration of heterogeneous flow in cut-plane velocity plots and improved accuracy in power prediction at individual turbines as well as total wind farm power output. Comparing FLORIS power predictions to LES, the heterogeneous FLORIS model showed an 18.9 % decrease in mean absolute error (MAE) for total wind farm power output and a 19.5 % decrease in MAE for individual turbine power predictions compared to the homogeneous FLORIS model. In comparisons to SCADA data, FLORIS simulations that implemented the heterogeneous flow model showed a 14.6 % decrease in MAE for wind farm power output predictions compared to homogeneous model simulations. With the use of the proposed turbulence-intensity correction method in addition to the heterogeneous model, the MAE in farm power output predictions showed a 31.42 % MAE decrease compared to the homogeneous model.

These modifications to FLORIS have outlined a framework for a wake model that features atmospheric heterogeneity and turbulence-intensity corrections to the power curve and provides a platform for further developments in this area of research. In agreement with this study, the findings of

Further studies relating to the effectiveness of this model when applied to wind farm controls would be very beneficial in determining future developments to these algorithms. Additionally, more extensive investigations should be considered to evaluate the efficacy of the proposed model in a wider variety of operational conditions, particularly those with lower wind speeds and extreme variations in wind direction. Other future work will investigate alternative interpolation methods for the flow field that consider the wind farm terrain map, capabilities for simulating more dynamic changes in wind direction, implementing enforcement of momentum conservation, and optimizing the model's computational efficiency.

Map of a selected section of the observed wind farm discussed in Sect.

This table lists several key attributes that characterize the nature of the terrain and turbine layout of the observed wind farm discussed in Sect.

Absolute error of total wind farm power output predictions for four different FLORIS models compared to LES simulations, as discussed in Sect.

Mean absolute error of individual turbine power outputs for four different FLORIS models compared to LES simulations, as discussed in Sect.

Analysis of wake influence in the observed wind farm discussed in Sect.

Analysis of wake influence in the observed wind farm discussed in Sect.

FLORIS

The data used for the 32-turbine validation case discussed in Sect. 4.1 can be made available upon request, by contacting Jennifer King at jennifer.king@nrel.gov. SCADA data and other farm-specific information for the commercial wind farm discussed in Sect. 4.2 are not publicly available due to proprietary privacy restrictions.

AF was the primary author of this paper and main developer for the proposed additions to FLORIS code, with direct guidance from JK and CD. CD and JK initiated and organized the necessary resources for this research project and collaborated with AF in performing simulations and data analysis. RM, NH, CB, PF, and ES aided in the documentation, refinement, and integration of the added subroutines to ensure continuity with preexisting FLORIS framework. All co-authors contributed to a thorough internal review of 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.

Data were furnished to the authors under an agreement between the National Renewable Energy Laboratory and Avangrid. Data and results used herein do not reflect findings by Avangrid.

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 Sandrine Aubrun and reviewed by two anonymous referees.