Gaussian FLOWERS: Wind-rose-based analytical integration of Gaussian wake model for extremely fast AEP estimation

In commercial wind farms, turbines are typically placed six to ten rotor diameters apart (Stevens , 2017). This relatively close proximity means there is an insufficient distance for the turbulent structures in their wakes to fully diffuse and for the induced velocity deficits to fully recover before being transported into downwind turbines. These wake interactions lead to a 10%–20% discrepancy in energy production between the leading turbines and the farm average, as observed in the generation data of utility-scale farms (Sorensen and Nielsen, 2006; Barthelmie , 2010).

Reducing the negative effects of these wake interactions is the primary challenge of the wind farm layout optimization (WFLO) problem. The problem is recognized as computationally complex, nonlinear, and nonconvex (Herbert-Acero , 2014; González , 2014). Nevertheless, studies have shown that annual energy production (AEP) can be considerably increased through tailoring the layout of turbines to the expected wind conditions (Feng and Shen, 2015; Kirchner-Bossi and Porté-Agel, 2018; and Thomas , 2019). Numerous methods have been proposed to find solutions to the WFLO problem. Azlan (2021) provide a comprehensive review. A common feature among the methods is the repeated [on the order of thousands (Porte-Agel , 2020)] calculation of farm AEP to evaluate candidate layouts.

The use of high-fidelity computational fluid dynamics (CFD) within layout optimization studies is currently impractical owing to the high computational cost of each simulation, the large number of simulations required to cover every bin, and the large number of iterations required for a satisfactory solution (Shakoor , 2016; Parada , 2017). Instead, computationally inexpensive engineering wake models are often used, which simplify the modeling of individual turbine wakes by considering only the infinite-time-average (steady-state) flow field. Using simplified equations based on the conservation of key flow quantities such as mass and momentum and assuming a discrete “top-hat” velocity deficit profile, Jensen (1983) and Frandsen (2006) each derived widely used engineering wake models. Bastankhah and Porté-Agel (2014) replaced the discrete velocity profile with a more realistic Gaussian profile. Compared to the Jensen model, the steady-state velocity field predicted by the Gaussian model is in closer agreement with experimental (Bastankhah and Porté-Agel, 2017), LiDAR (Zhan , 2020; Kaldellis , 2021), and numerical (Bastankhah and Porté-Agel, 2014) findings. When used for AEP prediction, the Gaussian model shows good agreement with historical power generation data (Doekemeijer , 2022).

To create a wind farm model, the individual wakes are combined using a superposition approach. Lissaman (1979) suggested linear superposition of velocity deficits. Later, Katic (1987) suggested linear superposition of energy deficits. Both are used extensively in the literature and in industry (Porte-Agel , 2020). More recently, physics-based approaches such as those suggested by Zong and Porté-Agel (2020) and Bastankhah (2021) have demonstrated reduced error in AEP prediction, albeit with increased complexity.

Despite the use of engineering wake models, calculating AEP remains a significant computational cost; Kirchner-Bossi and Porté-Agel (2018) reported the optimization of an 80-turbine farm using the Gaussian wake model required a computation time of about 16 CPU days. An alternative approach by LoCascio (2022) introduced a solution for AEP based on the analytical integration of the Jensen engineering wake model across wind directions. When used to predict AEP, the FLOWERS model reduced computation time by an order of magnitude compared to the numerical integration approach. However, one identified drawback of FLOWERS was a consistent discrepancy in AEP estimates on the order of 10%.

This paper builds on the FLOWERS approach to estimating AEP with an analytical integral formulation, but a Gaussian wake velocity deficit model rather than the Jensen model is used, along with an alternative approach to handling the nonlinear terms involved in the integration. There are two objectives of this study: (1) to present the derivation of the Gaussian-FLOWERS model and justify its modeling assumptions, and (2) to compare the accuracy and performance of the Gaussian-FLOWERS predictions against the (Jensen) FLOWERS model and conventional AEP models based on numerical integration. This paper is organized as follows. The derivation of the Gaussian-FLOWERS model is detailed in Sec. II. In Sec. III, the FLOWERS model and reference models are defined. Section IV details the evaluation of the Gaussian-FLOWERS model compared to two other engineering models. Section V outlines the study's takeaways.

Assuming x and y, respectively, denote east and north directions, $θ′=0$ corresponds to a (westerly) wind bearing $θWB$ of 270° (with the turbine wakes aligned with the positive x axis). A schematic of the coordinate system used is shown in Fig. 2(b).

Equation (7) represents the midpoint rule numerical integration of wind farm power production over the wind direction bins (King , 2020).

A number of changes to Eq. (8) are needed to make the integral tractable. These simplifications will be discussed in the following, and when needed, more details on the validity of these simplifications are provided for an interested reader in the  Appendixes A and B.

For a given wind direction $θ′$⁠, turbines may experience lower local wind velocities due to upstream wakes, which could lead to downstream turbines operating with different power coefficients. However, for simplicity, we base the power coefficient on the undisturbed farm inflow $U0$ instead of on local flow conditions $Uw,n$⁠, so $Cp(Uw,n)$ in Eq. (8) is replaced with $Cp(U0(θ′))$⁠.

Because the universal thrust coefficient is constant and does not need to be found in turn from the furthest upwind turbine, the condition on the superposition sum is changed from $m<n$ to $n≠m$⁠.

In Eq. (15), $mod(a,b)$ is the modulo function, returning the remainder of the division of a by b. The function $m(θ)$ wraps angle $θ$ into the range $[−π$⁠, $π]$⁠. It is required to maintain the $2π$ periodicity of the wake model that is otherwise lost by using a small angle approximation.  Appendix B demonstrates that this small angle approximation has a negligible impact on accuracy.

AEP predictions based on the Gaussian-FLOWERS formulation are compared with two other engineering wake models, summarized below.

This model, developed by Bay (2023) and based on the cumulative model introduced in Bastankhah (2021), uses a Gaussian-shaped wake velocity deficit model, accounts for the effect of wake-added turbulence in the wake recovery rate, and uses a physics-based cumulative method for the superposition of individual wakes. The added complexity of the cumulative curl model enables better predictive accuracy (Bay , 2023), and it is currently the default wake model in the FLORIS wake modeling library (NREL, 2023). A single grid point was used to sample the wind velocity across the rotor, although FLORIS supports a two-dimensional grid of points across the rotor area. The turbulence intensity was set to $6.86%$⁠; default values were used for the remaining parameters. We would like to clarify that by comparing our model's performance with this model, we do not intend to imply that this so-called “reference” model is equivalent to the “truth.” It is important to remember that the cumulative curl model is also an engineering low-fidelity model. However, this model has already been validated against high-fidelity simulations and historical data of offshore wind farms (Bay , 2023) and is commonly used for AEP estimations and farm layout optimizations. Our goal here is to evaluate how our model compares with this state-of-the-art engineering wake model, and whether it can provide comparable AEP predictions at much lower computational costs.

The original FLOWERS model developed by LoCascio (2022) is referred to as the Jensen-FLOWERS model. This approach uses analytical integration to predict AEP but uses the top-hat wake velocity deficit model suggested by Jensen (1983). This approach uses an alternative method of linearizing the integral, where power generation is based on the weight-averaged wake velocity. Further detail is given in  Appendix A. A value of 0.05 was used for the Jensen wake growth parameter, which is a commonly used value for offshore cases (Barthelmie , 2007). Both the Jensen-FLOWERS and Gaussian-FLOWERS models are similarly implemented in vectorized Python to facilitate direct comparisons of performance.

Twelve wind roses for use in the evaluation were generated using the WIND Toolkit dataset. The WIND Toolkit hosts publicly available annual wind forecasts at 126,000 likely wind plant sites across the continental United States (Draxl , 2015). The locations of the sites selected, which include both onshore and offshore cases, are shown in Fig. 3. The wind rose at each site is shown in Fig. 4. The wind data were binned using $360×1°$ bins and a single average wind speed per directional bin. The wind roses had minimum, average, and maximum mean wind speeds, of 7.54, 9.20, and 11.23 ms⁻¹, respectively (calculated from $∑IU0,ifi)$⁠. Non-uniform wind farm layouts, discussed in Sec. IV, were generated by using the Poisson disk sampling algorithm outlined in Bridson (2007) to fill a square boundary. This method produces a nonuniform set of points with a specified minimum separation.

We now examine the AEP prediction accuracy and performance of Gaussian-FLOWERS for three illustrative examples using a single nonuniform layout and three different wind roses.

The nonuniform layout consists of 50 turbines positioned within a $42D×42D$ square boundary with a minimum separation of 5.1D and mean nearest neighbor distance of 6.3D, where D is the turbine rotor diameter. This is an equivalent farm density to a 7D spaced rectangular grid layout (with 49 turbines placed within the same boundary).

From the complete set of 12 wind roses shown in Fig. 4, sites 9, 1, and 6 are selected for examination, chosen to cover the widest range of scenarios in terms of the performance of the Gaussian-FLOWERS model. Compared to site 9 for which the model provides the most accurate predictions, sites 1 and 6 feature more directionally concentrated winds and progressively higher average wind speeds (of 7.6, 8.5, and 10.6 ms⁻¹, respectively), which may explain the less satisfactory performance of the model in these cases. Among the twelve sites shown in Fig. 4, site 1 is the median case in terms of Gaussian-FLOWERS accuracy, while site 6 is the worst case. For these three different sites, the wind rose, farm layout, mean velocity field, farm AEP, and computational costs are presented in the left, middle, and right columns of Fig. 5.

The mean wind velocity field in Fig. 5 illustrates the impact of turbine wakes within wind farms. Turbines on the farm boundary operate undisturbed for a wide range of wind directions, while central turbines frequently experience reduced wind velocities due to upstream wakes. This results in notable variances in energy production between turbines. The largest reduction in wind speed occurs opposite the most frequent wind directions. When the wind direction is highly concentrated, the effects of turbine wakes are similarly localized. For example, for site 6, which features strongly westerly winds, the greatest reduction in wind speed is observed immediately to the east of the turbines. Figure 5 reports that for these three sites, Gaussian-FLOWERS predicts the AEP over 2 orders of magnitude faster than the cumulative curl model, with a small trade-off in the accuracy of $0.1%−7.4%$⁠. This is an improvement on the Jensen-FLOWERS approach which achieves comparable improvement in computation time but with a larger trade-off in accuracy of $7.6%−11.5%$⁠. The reported computation times were obtained using a single thread of an AMD Ryzen 7 3700X processor, running with 16 GB of DDR4 RAM.

The AEP of turbines that are more frequently waked are predicted with less accuracy than the leading turbines as shown in the figure. This is because simplifications 1, 2, and 4 discussed in Sec. II D 2 are less valid when there are large wake velocity deficits at the turbine locations. Although the percentage error is higher for these turbines, their contribution to AEP in absolute terms is small, so the overall accuracy is not impacted as greatly. Site 6 frequently contains wind speeds above the rated speed of the turbine, where $(dCT/dU0)$ and $(dCP/dU0)$ are large. Assuming a constant thrust and a power coefficient based on $U0$ (not $Uw)$ is less valid, and as a result, AEP prediction accuracy diverges.

Predictions of Gaussian-FLOWERS presented in Sec. IV A were based on using the maximum number of Fourier terms (i.e., 181 terms for a wind rose with 360 bins). We now demonstrate the ability to significantly decrease the computational cost of Gaussian-FLOWERS. This is achieved by reducing the number of terms in the Fourier series representation of the wind rose, as shown in Fig. 6 for the three example cases described in Sec. IV A.

For site 1, reducing the number of Fourier terms from 181 to 10 reduces computation time by almost 18 times (from 100 times faster to 1750 times faster) but accuracy decreases by a negligible $0.03%$⁠. This speedup and sensitivity to Fourier modes is consistent with the work of LoCascio (2022). This is enabled by the ability of the Fourier series to compactly and efficiently represent the information in the wind rose. The most important information about the wind rose is contained within the lowest frequency (higher amplitude) terms, which allows the removal of the higher-frequency terms without greatly impacting accuracy.

We now examine the AEP prediction accuracy of Gaussian-FLOWERS more generally using 120 test cases generated by pairing the complete set of 12 wind roses with 10 nonuniform layouts. The number of turbines in each layout was sampled from a uniform distribution between 5 and 60 turbines, and the minimum spacing for each layout was sampled from a uniform distribution between 3.7D and 7D. This variation in density is equivalent to a regular rectangular layout with spacing between 5D and 9D, respectively. The layouts generated are shown in Fig. 7. The aggregated results for the 120 cases are shown in Table I. Over the 120 cases, Gaussian-FLOWERS predicts AEP more reliably and with greater accuracy than Jensen-FLOWERS. The Gaussian-FLOWERS approach has a lower median error of $−1.59%$ and standard deviation of $3.30%$ compared to an error of $−6.09%$ with a deviation of $14.17%$ of the Jensen-FLOWERS approach.

The results of the 120 cases, ordered by mean wind speed, number of turbines, and turbine spacing, are shown in Fig. 8 from left to right, respectively. A large discrepancy in accuracy between the two approaches is observed when the mean wind speed exceeds 9.5 ms⁻¹. The Jensen-FLOWERS model tends to overestimate AEP, while the Gaussian-FLOWERS model slightly underestimates. This is due to the differing methods of including the power coefficient. In Jensen-FLOWERS, the power coefficient is constant across all wind directions, based on the average wind velocity at each turbine. In Gaussian-FLOWERS, the power coefficient is variable across wind direction and is based on conditions at the farthest upwind turbine. Allowing the power coefficient to vary with directions results in more accurate power prediction when the turbine operates in Region III, which frequently occurs when the average wind speed approaches the rated speed of the turbine (11 ms⁻¹ for the IEA 10MW turbine used in this study).

Increasing the number of turbines in the farm or reducing the turbine spacing decreases AEP prediction accuracy. Either action results in increased wake velocity deficits at the turbine locations due to an increased number of overlapping wakes or a reduced distance for the wake to recover, respectively. This reduces the validity of simplifications 1, 2, and 4, which assume a small wake velocity deficit. A well-optimized layout avoids large wake velocity deficits for directions that have the largest contribution to AEP. Therefore, we expect an improvement in the accuracy of AEP prediction will be observed for well-optimized layouts relative to the random layouts presented here. The dependence of the error on the wind rose and layout characteristics will be examined in more detail in  Appendix B.

In this paper, we derived a new engineering AEP model called Gaussian-FLOWERS. This is an analytical formulation for AEP of a wind farm using a Gaussian wake velocity deficit model, following the FLOWERS approach introduced by LoCascio (2022). The errors introduced by our modeling assumptions, which were necessary to permit a tractable analytical integral, were characterized, and their impact on overall accuracy was shown to be small through a series of case studies. The developed model was evaluated against a current state-of-the-art engineering model and showed greater than 2 orders of magnitude performance increase with a small trade-off in accuracy. We showed the performance can further be improved by truncating the number of terms in the Fourier series, with negligible impact on accuracy. Compared to Jensen-FLOWERS, Gaussian-FLOWERS predicts farm AEP with greater accuracy, $−1.59%$ compared to $−6.09%$⁠, and with less variation, $3.30%$ compared to $14.17%$⁠.

Moreover, the analytical approach may provide a convenient differentiable expression if gradient-based optimization methods are used for farm layout optimization.

In general, we have demonstrated that the simplifications made in Sec. II D 2 do not typically lead to large errors in AEP estimation for most wind farms, and those simplifications were found to be necessary in order to develop an analytical formulation for the AEP. However, in certain operating conditions, these simplifications may lead to an elevated error. Future research may further explore the possibility of improving the modeling framework in the following areas:

• Average wind velocity binning. Modeling can be improved by describing the range of wind velocity variation within each bin rather than using an average velocity for each wind direction. For example, an alternative approach could be to integrate over a Weibull distribution whose parameters vary with direction.

• Power coefficient. The power coefficient for each direction is set by the farthest upwind (unwaked) turbine. The possibility of using the local wind velocity can be explored.

• Thrust coefficient. The thrust coefficient is constant over all directions and turbine positions. Using the local wind velocity found in turn from the farthest upwind turbine can improve model predictions.

• Nonlinear cross terms. To linearize the integral, we neglect the “cross” terms in the expansion of $(1−∑ΔUU0)3$⁠. This leads to error, and in the most extreme cases, nonphysical power generation.

• 3D effects. We sample the wind velocity at a single point (the turbine hub centre), neglecting wind shear and variation of velocity across the swept area of the rotor.

• Superposition method. Linear superposition is used within the analytical framework for simplicity. More recently, physics-based approaches have shown increased accuracy.

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 provided by the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy Wind Energy Technologies Office. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. 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 U.S. Government purposes.

The authors have no conflicts to disclose.

Caidan Whittaker: Conceptualization (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal). Michael J. LoCascio: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Luis A. Martinez-Tossas: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Writing – review & editing (equal). Christopher J. Bay: Conceptualization (equal); Formal analysis (equal); Writing – review & editing (equal). Majid Bastankhah: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

This simplification is necessary, as the number of cross terms grows combinatorially with the number of turbines—including these terms would negate the performance advantage of the analytical method.

The approximation is based on three assumptions: (i) the linear term dominates the binomial expansion so the quadratic term can be approximated without a large impact on accuracy; (ii) the wakes are narrow, so they infrequently overlap; and (iii) the wakes decay quickly, making the product of overlapping wakes negligible.

While this is a limitation of the Gaussian-FLOWERS approach, we argue it is unlikely to drastically affect AEP prediction accuracy in practice. For directions that have the largest contributions to AEP, and therefore the largest impact on AEP prediction accuracy, highly waked turbines are avoided by maximizing streamwise spacing and avoiding aligning wakes onto downstream turbines; for the other directions, accuracy is reduced over the narrow range of directions with aligned wakes, but these directions do not have as great a contribution to AEP.

In  Appendix B, we further discuss this simplification along with other simplifications for more realistic wind farms.

Assuming the error contributions are independent, the sum of the individual errors should equal the total error in Gaussian-FLOWERS (using the numerical solution as a reference). Note that simplification 3, the Fourier series representation of the wind roses, introduces negligible error and is therefore not discussed.

We now examine the effect of varying the average inflow velocity, $U0$⁠, on the accuracy of Gaussian-FLOWERS. A single nonuniform layout and three wind roses with increasing wind velocities are used. The nonuniform layout consists of 50 turbines positioned within a $42D×42D$ square boundary with a minimum separation of 5.1D. The directional frequency of the wind roses follows a von Mises distribution $M(θ,μ,κ)$⁠, with mean direction $μ=0$ (westerly) and concentration parameter $κ=10$⁠. Uniform wind speeds are applied across all directions, with corresponding results depicted for three different wind speeds of 6, 10, and 12 ms⁻¹ in the left, middle, and right columns of Fig. 11, respectively.

As the wind speed is increased, the error in Gaussian-FLOWERS varies from a positive value of $+0.1%$ to a negative value of $−10.4%$⁠.

The power coefficient simplification results in a positive error at 6 ms⁻¹, as $dCpdU0>0$ around this speed. At 10 ms⁻¹, the error is negligible as $dCpdU0≈0$⁠, and at 12 ms⁻¹, the error is negative since $dCpdU0<0$ around this speed.

The error from the thrust coefficient simplification is non-negligible at 12 ms⁻¹ because turbines no longer operate within the constant thrust coefficient region (Region III).

The error due to neglecting the cross terms diminishes at higher wind speeds because the wakes are narrow, so they overlap less frequently. The figure also shows that the small angle approximation (abbreviated as Sml Angle in the figure) discussed in the main report as simplification 5 introduces negligible error in all cases.

We now examine the effect of varying the density of the farm layout on the accuracy of Gaussian-FLOWERS. A single wind rose and three layouts are used. The directional frequency of the wind rose follows a von Mises distribution $M(θ,μ,κ)$⁠, with mean direction $μ=0$ (westerly), concentration parameter $κ=10$⁠, and a uniform wind speed of 10 ms⁻¹ across all directions. The three nonuniform layouts consist of approximately 50 turbines positioned within a square area with side lengths of 54, 42, 32D. This is an equivalent spacing to a 9, 7, 5.3D regular rectangular grid layout, respectively. Corresponding results are depicted in the left, middle, and right columns of Fig. 12, respectively.

As the farm density is increased, the error in Gaussian-FLOWERS increases from $−0.8%$ to $−5.8%$⁠. The primary source of error is the cross-terms simplification (i.e. simplification 4). With increased density, the distance for wakes to recover reduces, so the assumption that the product of overlapping wakes is negligible is less valid.

We now investigate the effect of increasing the farm size on the accuracy of Gaussian-FLOWERS.

For simplicity, the directional frequency of the wind rose is determined according to a von Mises distribution $M(θ,μ,κ)$ with a mean direction $μ=0$ and concentration parameter $κ=10$⁠. A uniform wind speed across all directions of 10 ms⁻¹ was used. The three nonuniform layouts consist of 27, 48, 72 turbines positioned within a square area with side lengths of 32, 42, 54D, respectively. As shown in Fig. 13, with an increase in the number of turbines, the error in Gaussian-FLOWERS ranges from $−1.1%$ to $−2.6%$⁠. The primary source of error is again the cross-term simplification. With increased farm size, the number of overlapping wakes deep within the farm increases, so the assumption that the product of overlapping wakes is negligible is less valid.