Comparison of modular analytical wake models to the Lillgrund wind plant

Wake losses represent one of the primary sources of lost energy (and therefore efficiency and capital) within any operating wind plant.^1–3 The scale of losses associated with wind turbine wake interaction can vary considerably from one plant to another plant, depending on the layout of turbines and the local atmospheric resource. Many engineering processes associated with wind plant operation, design, and control benefit from accurate and computationally efficient wind turbine wake models. Given the utility of analytical wake modeling, evaluating their ability to reproduce wind plant behavior is a necessary step toward their use in industrial settings.

Analytical wake models are frequently applied to engineering processes associated with wind plant operation, design, and control. Many current-generation wake models assume a Gaussian profile for the time-averaged velocity deficit in both the vertical and transverse directions. The Gaussian velocity deficit assumes an axisymmetric wake profile whose radius grows linearly in the streamwise direction.⁴ Linear growth of the velocity deficit moving downstream depends on turbulence intensity and is often defined to conserve mass. One limitation of the Gaussian velocity deficit model is that it is only self-similar in the far wake; thus, it is not a reliable representation of the velocity deficit near the turbine.

Subsequent research has led to corrections and modifications to the Gaussian model, which improve its performance in the near wake, including a scaling factor in the maximum velocity deficit that decays beyond the near wake⁵ or reframes the velocity deficit distribution in the near wake by allowing the exponent defining the Gaussian to vary.⁶ Both modifications are able reduce disparity between the model and canonical wake profiles in the near wake by making the deficit more rectangular near the turbine and resolve toward a Gaussian velocity deficit in the far wake.

In addition to modeling the wake deficit, increasing attention has been paid to models of turbulence intensity in wind plant flows. Nearly all velocity deficit models rely on the turbulence intensity incident on a given wind turbine in addition to the local rotor-effective wind speed and the wind turbine's thrust coefficient. The prevailing model⁷ assumes that wake-added turbulence can be approximated by a product of the induction factor and the freestream turbulence that decays moving downstream as is commonly seen in the literature.^8–10 An alternative wake-added turbulence model⁵ includes spatial variability in the vertical and transverse directions and captures the increased turbulence in the high-shear region at the edge of the wake seen in wind tunnel^9,11–13 and field experiments.^14,15

Engineering velocity deficit and wake-added turbulence models rely on a superposition scheme to be deployed in a wind plant setting; each model is defined only for a single wind turbine wake. The parameterized dependence on incident velocity, turbulence intensity, and turbine thrust coefficient should be sufficient to adapt wakes for wind turbines within wind plants, but they still need to be aggregated a posteriori. Recent work¹⁶ summarizes superposition schemes for wind turbine wake models and compares their performance against wind tunnel experiments and field observations with scanning lidar. Among the more commonly used wake superposition methods are the root-sum-squares approach,^17–19 the linear superposition,^20–22 and the maximum deficit method.²³ Each superposition model has been explored in the context of the interactions of two to three wind turbine wakes, but few results consider the superposition of many wakes, as is seen in the deep array. In the fully developed wind turbine array, power production along streamwise-aligned rows of turbines approaches an asymptotic level²⁴ and wind turbine wake statistics become quasi-periodic from row to row.^12,25,26

The current work explores the analytical wake models and compares results with operational data from the Lillgrund Wind Plant in order to assess their ability to accurately reproduce power production. The accuracy of the model configurations is compared by aggregating results by wind speed, the wind direction, and the number of wakes influencing a given wind turbine, highlighting conditions for which individual model components better represent operation conditions. Each model combines a velocity deficit, wake-added turbulence, and wake superposition model component as implemented in the FLOw Redirection and Induction in Steady State (FLORIS)²⁷ wind plant modeling framework from the National Renewable Energy Laboratory (NREL). A detailed review of the underlying theory for the wake model components is presented in Sec. II, followed by a summary of the procedures taken to ensure quality control (QC) in the observational data in Sec. III. Finally, results from each configuration of the wake model are compared with operational data in Sec. IV, followed by a discussion and outlook for wind plant performance modeling in Sec. V.

The modular wake modeling framework is summarized in Fig. 1. Each column in the figure describes a model component (velocity deficit, wake-added turbulence, and wake superposition) and the associated parameters. All model components discussed in the current work are available in the FLORIS²⁷ software from the National Renewable Energy Laboratory (NREL). The current work is built from FLORIS v2.0.1, which includes all wake model components discussed. Key references for each wake model are provided in the figure although it should be noted that the versions available in FLORIS may differ slightly from those proposed in the original references.

Focus in the current work centers on the Gaussian wake model,⁴ and subsequent modifications.^5,6 All the models share a common Gaussian kernel and can be tuned to converge to nearly identical results in the far wake. Accordingly, legacy models^21,28,29 are not discussed in the context of the validation against performance data from the Lillgrund Wind Plant.

At their core, the velocity deficit in each model is represented with a Gaussian distribution,

where $ΔUU∞$ is the velocity deficit, defined as a change in bulk velocity (⁠$ΔU$⁠) with respect to some reference velocity (⁠$U∞$⁠) and the wake radius is defined as $r=y2+(z−H)2$⁠. y and z are the spanwise and vertical coordinates, respectively; y = 0 at the wind turbine hub and z = 0 at the ground (or sea) level; and H is the wind turbine hub height. x denotes the streamwise direction, and for wake models, it indicates the distance downstream of the wind turbine. The amplitude of the distribution is denoted by C(x), where x is the streamwise coordinate. Each model poses a slightly different formulation for the velocity deficit amplitude, C(x), but they all rely in some way on the operating thrust coefficient, C_T, of the modeled wind turbine and, thus, on the incident wind speed, as well as the local turbulence intensity, I. Two formulations discussed here^4,6 are based on previous research,¹⁷ defining the maximum velocity deficit as

In each case, the velocity deficit accounts for the expansion of the wake with the standard deviation of the distribution, σ, and decays moving downstream, described by a function $k*$⁠. Bastankhah and Porté-Agel⁴ frame the decay as $k*=∂σ/∂x$⁠, with ϵ defined as

and

Subsequent work by Niayifar and Porté-Agel²² posed the growth as an empirical formulation depending on the local turbulence intensity,

FLORIS defines wake growth according to the formulation in Eq. (5), with the additional allowance for growth to differ in the transverse and vertical directions. In most cases, wake growth is assumed to be axisymmetric, so that $k*=ky=kz=ka·I+kb$⁠. In FLORIS, k_a and k_b are parameters used to tune the Gaussian velocity deficit models to fit the wake width and height, if an axisymmetric wake is not assumed. The Gaussian model in FLORIS is only considered to be applicable in the far wake, which satisfies the condition

where $kα$ and $kβ$ are parameters tuned empirically.

Blondel and Cathelain⁶ pose a near-wake correction as

where $pNW=−1$ is a near-wake coefficient tuned to measurement data and $cNW$ is described using the induction factor $a=(1−1−CT)/2$ as

The system is closed with an expression for ϵ as the linear evolution of the standard deviation,

In Eqs. (8) and (9), a_s, b_s, and c_s are tuning coefficients fit to large eddy simulation data.

Ishihara and Qian⁵ propose an alternative correction using an additional term in the velocity deficit amplitude that only plays a role in the near wake,

The amplitude is reframed in terms of a set of parameters [κ₁, κ₂, and κ₃ in Eq. (10)] that each depends on C_T and I,

In purely Gaussian normal formulations, the power in the exponential term n = 2 [see Eq. (1)]; however, in the adaptation by Blondel and Cathelain,⁶ the velocity deficit distribution is given super-Gaussian properties by allowing the value of n to evolve as

where a_f, b_f, and c_f are the tuning parameters fit to the measurement data. To ensure consistency across the work by NREL, tuning coefficients for the models outlined in this section have been modified such that all models agree in the far wake.

Most modern literature on analytical wind turbine wake model development relies on a purely empirical wake-added turbulence model.⁷ The model relies on the thrust coefficient alone in the near wake (defined as $x/D<3$ for that case) and the thrust coefficient and the ambient turbulence in the far wake. Wake-added turbulence, $I+$⁠, is defined as

where the constant coefficients c_c, c_a, c_i, and c_d modulate the dependence on the scale (constant), axial induction factor, ambient turbulence intensity, and downstream distance, respectively.

The spanwise and vertical dependence was introduced to a wake-added turbulence model in effort to generate a more physically representative distribution of $I+$⁠.⁵ Noting that wind turbine wakes typically exhibit elevated levels of $I+$ at the edges of the wake, the new model has a double-Gaussian distribution for $I+$ that depends on the wake radius. The formulation that they implement is analogous to that of the velocity deficit in Eq. (10) with tuning coefficients, κ₄, κ₅, and κ₆ [that have definitions noted in Eq. (11)],

The coefficients k₁ and k₂ denote the radial dependence of the Gaussian functions,

The last term in Eq. (14) is introduced to account for the relatively weak $I+$ in the lower part of the wake,

The presence and interaction of velocity deficits from multiple individual wakes requires that some means of superposition be implemented. Some of the earliest work on the wake superposition by Lissaman²⁰ treated the velocity deficit as a passive flow component, analogous to the transport of a passive scalar, as a justification for combining velocity deficits linearly,

for each turbine wake i up to the total number of wakes being considered, N. The velocity deficit, $ΔUi(x)=(1−Ui(x)/U∞,i)$⁠, is the velocity deficit, defined as the local average velocity field, $Ui(x)$⁠, divided by the local inflow velocity $U∞,i$⁠. A linear superposition scheme accounts for the continuous decrease in momentum with additional wakes and has been used extensively in subsequent research.^21,22

Other research^17–19 proposed the superposition of energy deficits, rather than velocity or momentum, leading to a root-sum-squares formulation. In the sum-of-squares method, the total field of velocity deficits contributed by all turbines is

Given that wind plant flows are high-Reynolds number flows, they are dominated by convective terms in the Navier–Stokes equations, and thus, they are highly non-linear. This observation could imply that no simple superposition method should be expected to accurately represent the complex interactions of multiple wind turbine wakes and has led to several other wake combination schemes. The third method available in the FLORIS framework and implemented in the current work takes the maximum velocity deficit in any region of the flow:²³ 

Regardless of the superposition scheme implemented, the working hypothesis is that with a physically representative velocity deficit and wake-added turbulence model, the atmospheric flow within a wind plant will be adapted to consider the energy extraction by the wind turbines. In this way, the flow regime typical of the deep array is expected to emerge with a sufficient level of wake interactions.^24,25

A principal goal of the current validation effort is to quantify FLORIS's ability to accurately model the power production of a large wind plant where strong wake interactions are present throughout. The default wake model configuration in FLORIS implements a Gaussian wake model,³⁰ a wake-added turbulence model,⁷ and the sum-of-squares wake superposition scheme. FLORIS currently includes other wake models—several of which have an explicit near-wake formulation—which are to be quantitatively compared with the default configuration. The current design of FLORIS splits a wake model into constituent components, each of which can be selected and combined with others in a modular sense. Model configurations tested in the current work are summarized in Table I.

Each configuration is labeled with an acronym that describes the single model component that has been changed from the default (def) configuration. Configurations changing the velocity deficit model are vmb and vmi (for the Blondel and Ishihara velocity models, respectively), changing the turbulence model are tmi and tmc (for the Ishihara and Crespo models, respectively), and changing the combination model are cmf and cmm (for the freestream-linear and maximum superposition models, respectively). The key difference between the def and tmc configurations (both have Gaussian velocity deficit, Crespo–Hernandez turbulence, and SOSFS combination models) is the tuning of the turbulence model parameters. Recent work of King et al.¹⁰ implemented a version of the Crespo and Hernandez turbulence model that, in conjunction with physical mechanisms introduced in the curled wake model,³¹ reduces disparity from similar wake statistics from high-fidelity models. With the release of FLORIS v2.0.1,²⁷ default parameters have changed, as reported in Table II.

The Lillgrund Wind Plant is located 10 km off the coast of southern Sweden in the Kattegat Strait. It contains 48 Siemens SWT-2.3–93 wind turbines and has a rated nameplate capacity of 110 MW. Each wind turbine has a rotor diameter of D = 93 m and a hub height of H = 65 m. The layout of Lillgrund is shown in Fig. 2, where each turbine location is denoted with a blue marker; a met mast, located at the southwest corner of the wind plant, is indicated with an open marker. Wind directions for which the met mast is considered free from the influence of upstream obstructions are shown by the green shaded region. Unobstructed wind directions are calculated based on the proximity of the met mast to the wind turbines following International Electrotechnical Commission (IEC) guidance.³² One key feature of Lillgrund that makes it an interesting wake modeling case study is that the turbines are relatively closely spaced, as near as $3.3D$⁠. This leads to strong wake interactions within the wind plant and a rapid onset of deep array flow conditions.

Raw atmospheric data used include high-frequency (20-Hz) observations of wind speed, u, and wind direction, θ, reported by the met mast between February 2009 and December 2010. Wind speed and direction data were binned to a temporal resolution of 10 min, from which the mean and standard deviations were calculated. Turbulence intensity is estimated as the ratio of the retained 10-min statistics for wind speed as $I=σu/u$⁠. Wind speed and turbulence intensity roses are shown in Fig. 3. Additional quality-control steps for the data include omitting any period when instrumentation is not correctly reporting (e.g., data are missing or report a single, fixed value) from further consideration. Wind speed and wind direction observations are collected using a cup anemometer and wind vane at 65 and 61 m above the sea level, respectively.

Stationary atmospheric conditions are identified and quantified using the total variation method.³³ Continuous data of 60 min are collected into blocks; used to estimate a covariance matrix; and reduced to a scalar-valued metric, $V$⁠, which describes the variability of the atmospheric data. Processing data through the total variation algorithm provides a distribution of values that can be used to organize observations according to statistical stationarity. The distribution of observations according to their total variation is shown in Fig. 4. Data whose time stamps occur in periods that pass the quality control (QC) on total variation are used to filter the supervisory control and data acquisition (SCADA) data. In the current work, any time period when the total variation is less than that of the 75th percentile, $V75%$⁠, is considered for further quality control. Any time period greater than $V75%$ either is too noisy to be of much use for steady-state wake model validation or contains transient atmospheric dynamics. The selection of the 75th percentile is somewhat arbitrary and was chosen to eliminate observations only when the atmospheric variable exhibits the most transient behavior, while retaining as much data as possible.

The atmospheric conditions used to drive the FLORIS simulations are derived from reports by the met mast. Each set of conditions is defined by a bin of wind speed (width 1 m/s, centered on integer values of wind speeds), wind direction (width of $5°$⁠), and turbulence intensity (width of 2%). Only conditions with a minimum of 30 10-min observations were included for validation against FLORIS. Additionally, only conditions for which the met mast located at the southwest corner of the wind plant was not waked were considered, resulting in a total of 432 cases.

The full set of SCADA data for Lillgrund comprises operational data from all 48 wind turbines downsampled to the industry standard 10-min bins. Each turbine reports channels including status condition flags, power production, nacelle position, generator speed, and a host of other SCADA channels. Only mean data are available for each 10-min bin, channel, and turbine.

Met mast data are taken to be a more accurate representation of the atmospheric conditions although reliable records are available only for wind directions between $110°$ and $310°$⁠, following guidance from IEC,³² shown by the green region in Fig. 2. Given that the met mast is located southwest of the wind plant, atmospheric conditions represented by the anemometer and wind vane do not necessarily reflect those seen by wind turbines on the other side of the plant. This is especially evident in the power reported by the wind turbines, shown in Fig. 5(a), which contrasts power curves using different wind speed estimates. In the figure, points in blue are located according to the power reported by turbine A01 (on the east end of Lillgrund) and the wind speed reported by the met mast. The deviation from the expected power curve results from using a wind speed that is not reflective of that incident on A01 during many of the 10-min observations. In contrast, the points in green show the same power production by A01 plotted against the wind speed observed using a nacelle-mounted anemometer. Even without correcting the nacelle anemometer for flow disturbance, the power curve is in much clearer focus.

SCADA data are quality controlled to allow only time periods when the reported power was within 25% or 300 kW of the nominal power curve (shown in red in Fig. 5). The nominal power curve was defined as the average power by unwaked wind turbines (B08, C08, and D08) near the met mast. Time periods when wind turbine power production was less than 10 kW were eliminated. Anytime when more than a single turbine was not reporting correctly, either according to the status flags in the SCADA data or because of detected curtailment, was eliminated as unreliable for the purposes of model validation. Similar quality control data have been obtained in the previous wind plant analysis.^34,35 In more recent work, the quality control procedures are quite conservative, also requiring constant C_T, in addition to the procedures used in the current work.

To maintain the largest possible sample size for any given atmospheric condition, missing points were replaced using a K-nearest neighbor regression algorithm.³⁶ During replacement, a single missing value was estimated to be the average of the most similar reports from the wind SCADA record that, otherwise, passed all quality control measures. To reduce errors and uncertainty introduced during the data replacement step, only time periods with a single missing value were considered for replacement.

The performance of each wake model configuration is determined by the errors between the observational data and model results. The absolute error for any case represents the difference between the FLORIS estimate and the median of the reported SCADA observations for the respective atmospheric condition,

and has the units of kW. Relative error,

normalizes absolute error by the expected power production for the range of wind speeds in region 2 of the wind turbines.

Figure 6 shows absolute and relative error estimates for the default FLORIS model configuration (def). Each point in the polar error distributions is defined by a unique combination of wind speed and wind direction bins, averaged over the bins of turbulence intensity, I. Peak $εabs$ is associated with higher wind speeds, following the wind speed–power cube law. Accordingly, for low wind speeds, the def case shows small values of $εabs$ for all wind directions. Relative error (center panel) is normalized by observed power and shifts the focus to more moderate wind speeds. The number of observations within each point in the following polar plots is shown in the rightmost panel of Fig. 6.

Figure 7 shows polar plots of the absolute and relative errors for each model configuration in the first and third columns. The second and fourth columns reflect the change in either $εabs$ or $εrel$ from that reported for the def case. Consistently, absolute error figures tend to highlight higher wind speeds, where a small change in predicted behavior could result in a large absolute discrepancy between the observed and modeled power production. For lower wind speeds, the inner region of each plot shows smaller absolute errors, expected because the baseline for comparison is simply lower.

Each model configuration examined in the current work exhibits the greatest absolute error in the same regions of the atmospheric condition space, although to different degrees. Regardless of the particular wake model configuration, the highest absolute error is associated with wind directions coming from the west-northwest and from the southwest. Wind from the northwest corresponds to the extremes where the met mast is free of upstream flow disturbances. The consistency across model configurations could indicate that the absolute error in these wind directions arises in part from uncertainty in the characterization of flow conditions or that conditions reported by the met mast do not accurately reflect those experienced by the wind plant as a whole. From the southwest, high $εabs$ appears to arise from the wake models themselves. Wind directions around $220°$ and $255°$ correspond to directions for which the wind plant has the close spacing and for which the most wakes interact. The vmb and tmi cases, both of which have near wake corrections, show the greatest reduction in $εabs$⁠. Several model configurations (vmi, cmf, and cmm) highlight the layout directions of Lillgrund, most evident in the plots showing the change in $εabs$ in the second column.

The third column of Fig. 7 shows the distribution of $εrel$ by the wind speed and direction and more evenly highlights errors across the wind speeds in Region 2 of SWT 2.3–93. Evident in the figure, the highest relative error is associated with all wind plant alignment directions, again illustrated by vmi, cmf, and cmm. Interestingly, these configurations serve to decrease relative prediction error for the nonaligned wind directions, where wakes do not directly impinge on downstream machines.

Changing the velocity deficit component of the wake model has little effect on $εrel$ compared to the default configuration (def). For the Blondel velocity model, $εrel$ is slightly decreased (∼5%–10%) for winds from the southwest, but is unchanged elsewhere. In the case of the tmi model configuration, relative error is slightly increased for a few cases with high wind speeds and wind directions around $120°$ and decreased everywhere else.

Quantifying the changes in model performance using the def model configuration as a baseline is accomplished by tabulating the results presented in the rightmost column of Fig. 7 into histograms, shown in Fig. 8. Each histogram shows the distribution of changes in $εrel$ from the def case. Histograms are not normalized and present the total number of observations with a given change in $εrel$⁠. Positive values in any of the histograms indicate an increase in relative error over the performance of def, and negative values indicate a reduction in error. The alternate superposition models (cmf and cmm) and the Ishihara and Qian velocity model (tmi) make the most consistent improvement to modeling the SCADA data from Lillgrund. Noted in the above polar scatter plots, observations increasing error (to the right of the vertical dashed line) are aligned with the wind plant layout, and reductions in error are noted for wind directions that do not lead to direct wake impingement.

Results aggregated across all atmospheric conditions and for all wind turbines in the wind plant can provide only a high-level picture of the ability of the various wake model configurations to replicate wind plant performance. To identify the specific context in which any particular model component reduces error around power estimates, results need to be sliced around variables of interest. In Figs. 9(a), 9(b), and 9(d), solid lines indicate the root mean square (RMS) error for each case, aggregated around a particular quantity of interest. The shaded regions indicate one standard deviation above or below the RMS value.

In Fig. 9(a), RMS relative errors are aggregated according to wind speed. It is notable in the figure that each model configuration shares the same performance curve—all the models share the same general trend of low absolute error for low wind speeds and increasing error as wind speeds tend toward Region 2.5. The benefit of considering a relative metric for performance error is also evident in the result aggregated by wind speed. Although the RMS trends for every model configuration are quite similar, $εrel$ clearly distinguishes between certain groups of configurations. Specifically, the configurations using the maximum velocity deficit superposition model (cmf), the maximum deficit model (cmm), or the Ishihara and Qian velocity model (vmi) clearly exhibit a lower relative error for wind speeds between 6 and 12 m/s; however, it is also evident in the figure that these two model configurations have a larger standard deviation, indicating that although they perform better on average, a good deal of variability remains for any given wind speed. Improvements by the superposition models indicate that the sum-of-squares approach to combining velocity deficits is not sufficiently representative of the complex flow dynamics that go into wake interaction.

The panel on the right side of Fig. 9(a) indicates the change in error for a given model configuration from def. Again, the solid line indicates the RMS value, and the shaded region indicates one standard deviation above or below. For each model configuration, a change in error is calculated as the test configuration minus the default configuration; thus, a negative delta in error represents a reduction in error compared to the default configuration, and positive values indicate an increase in error. It is worth noting that when wind speeds are greater than 12 m/s, many of the trends observed in Region 2 reverse; the cmf, cmm, and tmi cases show an increase in $εrel$ from the def case, and the vmb case shows slight improvements.

In Fig. 9(b), results are aggregated by the wind direction reported by the met mast. As discussed in the quality control section, only wind directions for which the met mast is unwaked are considered. Dashed lines in Fig. 9(b) indicate wind directions aligned with the arrangement of Lillgrund, $θ∈[120°,180°,222°,255°,300°]$⁠. Aggregating error by the wind direction helps to provide some context for the present results. Although they perform better overall, the cmf, cmm, and tmi model configurations clearly perform poorly when the wind direction is aligned with the wind plant layout. When the wind direction is such that wakes directly impinge on downstream machines, the cmm and tmi models tend to increase relative error. When the wind comes from an oblique angle, however— between the dashed vertical lines—the cmm and tmi models are able to reduce prediction error by as much as 35%. These cases include wind directions that lead to partial wake interaction and those that lead to wakes going between downstream wind turbines. Interestingly, the vmb configuration, which represents the least overall improvement over the def, is able to consistently reduce $εrel$ for wind directions aligned with the wind plant layout. The near-wake formulation in the vmb case is thought to be the cause of the error reduction for wind directions leading to direct wake impingement, significant in this study due to the close spacing within Lillgrund.

Aggregating by turbulence intensity, as in Fig. 9(c), highlights some of the limitations on the wake model configurations. The greatest reductions in $εrel$ are shown for the lowest values of turbulence intensity for all model configurations. Stably stratified atmospheric conditions with very low turbulence are expected to be fairly common in offshore settings,³⁷ such as is the case for Lillgrund. In instances of low turbulence intensity, the wake models shown here should provide an accurate picture of offshore wind plant performance. For cases with high turbulence intensity, however, model configurations that, otherwise, perform well actually increase $εrel$ by as much as 25%.

The final mean of error aggregation undertaken in the current work is to tabulate RMS prediction error according to the number of wakes that impact the performance of a given wind turbine. The IEC standards³² prescribe a method for estimating the influence of an upstream wind turbine or obstruction on a given measurement location, used here to determine the number of wakes influencing each wind turbine. Tabulating error according to the number of wakes considered for a given wind turbine helps to identify the sources of uncertainty in a given prediction.

Figure 9(d) shows $εrel$ as a function of the number of wakes influencing each turbine, analogous to the plots showing the errors aggregated by the wind speed and wind direction. For wind turbines encountering zero or one wake from upstream, all models perform very similarly. The next region includes wind turbines experiencing between two and six wakes from upstream machines. This represents the region in the wind plant that could be considered transitional—that is, the power production has not reached its asymptotic level. In the transitional region of the wind, the greatest reduction in model error is derived from the velocity deficit models that include an explicit near-wake formulation. In these regions, the vmb case offers a more accurate representation of the flow physics and then the def configuration. The improvement offered by the vmb case in the transitional region likely arises from two factors: turbines are spaced close enough to one another to still be in the near wake and wind turbine wakes in the transitional region do not have the same morphology as the fully developed region. It is also clear that in the transitional region, wake combination and wake turbulence models do not offer any benefit to wind plant performance estimates; rather, they tend to increase error by as much as $εrel=20%$ (⁠$εabs≈100$ kW).

The final region of a wind plant—where wind turbines experience the influence of more than six upstream devices—is considered the deep array. In the deep array, we see a reversal of the trend noted previously; velocity deficit models lead to an increase in $εrel$ as high as 20%, whereas wake combination and turbulence models tend to decrease $εrel$ by as much as 50%. The distinction of performance by all model configurations indicates that different flow physics dominates the fidelity of a simulation. The prediction of power produced by unwaked turbines depends largely on the characterization of atmospheric conditions and the generator model (the c_p and c_t curves and the generator efficiency). Velocity deficit models play a large role in the accuracy of the power predictions in the transitional region. This is the region in which a small change in velocity can lead to a relatively large change in power production. In the deep array, power production tends toward an asymptotic value in a time-averaged sense, and the quality of the FLORIS prediction seems to benefit greatly by reconsidering the turbulence model and the method by which velocity deficits are superimposed. Elevated levels of turbulence in the deep array region drive faster wake expansion, leading to more uniform velocity on the rotor and shortening the near wake region. For this reason, velocity deficit models with near wake formulations do not improve model accuracy over the def case.

Results aggregated by atmospheric condition and the number of wakes influencing each turbine aim to categorize the performance of FLORIS model configurations in a broad sense. With that said, aggregated results by necessity cannot provide a fully granular picture of wind plant prediction for every wind turbine and for every atmospheric condition. The following figures show the performance of model configurations against the collection of observations from SCADA that conform to particular atmospheric conditions. Each plot in Fig. 10 represents the total observations from the SCADA data corresponding to a specific set of atmospheric conditions. From top are the atmospheric conditions that to the minimum and maximum $εabs$ and $εrel$ for the default FLORIS configuration as well as the case with a maximum number of observations. The gray box plots indicate the distribution of SCADA observations for each wind turbine (gray diamonds are statistical outliers), and the colored markers indicate the predicted power output of each wind turbine by each model configuration. Finally, the purple vertical bands in the background indicate the number of wakes influencing each wind turbine.

Unsurprisingly, the minimum absolute error case is one for which the wind speed is relatively low at 5 m/s. In this case, it is very difficult to distinguish the performance of the different model configurations, with the exception of the freestream linear with the superposition model. In contrast, the maximum absolute error keys clearly demarcate model configuration performance, with the exception of the freestream linear superposition model (CFM). It is evident in the figure that all the model configurations do fairly well with unwaked wind turbines; however, including the influence of only a few upstream wakes quickly leads to sizeable prediction errors by all models. For this particular condition (WS = 11.0 m/s, WD = $265°$⁠, and I = 3%), the def model configuration is outperformed by all other models.

The following cases represent the minimum and maximum relative error for the default configuration (third and fourth subfigures from the top, respectively). For the minimum $εrel$ case, the wind speed is fairly high (12.0 m/s) and I has a moderate value (5%). At the high end of Region 2, the reported power production (and thus the denominator in $εrel$⁠) is relatively large, reducing the relative error. Turbulence intensity is also squarely in the range for which many of the models are defined. The maximum relative error case, in contrast, has extremely low turbulence intensity. In the FLORIS framework, I allows us to determine how quickly wind turbine wakes dissipate. Figure 10 shows that the Ishihara and Qian turbulence model can more accurately predict wind plant performance for low I cases.

Figure 11 communicates similar information as in the boxplots but offers information about the spatial distribution of the model error instead of statistical distributions for each turbine. In each subfigure, the colored dots communicate the relative error of the default model configuration for each atmospheric condition. Purple dots in the background indicate the number of upstream wakes influencing each turbine.

For the plots in Fig. 11 showing average relative error, results from the default model configuration were grouped by the wind direction indicated in each figure. The formulation of $εrel$ already normalizes results by wind speed, and so the results shown are effectively averaged over only the turbulence intensity. Consistently, the wake model configurations tested in the current work show less sensitivity to turbulence intensity than they do for the wind speed and wind direction. Thus, the plots in Fig. 11 are able to combine the validation results aggregated by wind speed, turbulence intensity, and the number of wakes influencing the performance of any given wind turbine for selected wind directions. Figures 10 and 11 together communicate the benefits of wake model components for different regions of the wind plant. Consider that alternate velocity deficit models reduce the underprediction of power production in the transitional region of the wind plant and alternate turbulence and superposition models mitigate overprediction in the deep array.

Reducing the analysis of wake model configuration performance into a single value necessarily excludes correlation of model error to atmospheric conditions. Table III offers a high-level metric for the performance of each model configuration in terms of the error metric considered (absolute or relative) and the inflow model (uniform or heterogeneous) used to drive the simulation. At a glance, all the models perform similarly, with notable improvements introduced by changing the wake superposition scheme. Only moderate changes are introduced considering alternate velocity deficit or turbulence models.

The FLORIS²⁷ wind plant modeling tool provides a framework for composing wake models in a modular sense, flexibly combining velocity deficit, wake-added turbulence, superposition, and deflection models. These model components were tested against wind plant operational data to quantify their ability to reproduce wind plant behavior in a range of atmospheric conditions, represented with wind speed, wind direction, and turbulence intensity. Regardless of the particular configuration of the wake model, the highest absolute errors are generally associated with wind speeds, arising from the cubic relationship between wind speed and power production.

Disparity between modeled and observed wind plant performance in a relative sense effectively normalizes the validation results for wind speed. The relative error highlights that changing the wake-added turbulence model and wake superposition scheme can increase model accuracy for flows that are not aligned with the wind plant layout. This is especially evident in the Ishihara and Qian⁵ turbulence model and the maximum deficit superposition model discussed by Gunn et al.;²³ however, these models increase the disparity between observational and model data when the wind direction is aligned with the rows of wind turbines. This suggests that direct wake impingement requires a different treatment from the partial wake overlap or the superposition of wakes that merge laterally.

The distribution of relative error aggregated by wind speed highlights that the linear and maximum wake combination models and the Ishihara and Qian⁵ velocity model all reduce the relative error in Region 2 × 10%–15% from the default wake model configuration. However, these benefits are only seen in the deep array setting, where six or more wakes influence the performance of a given wind turbine. In the deep array, power production tends toward an asymptotic value, and the model predictions benefit greatly by reconsidering the turbulence model and the method by which velocity deficits are superimposed. In the transitional region, where fewer wakes need to be considered, the velocity model of Blondel and Cathelain⁶ is better able to accurately capture wind turbine power production.

The Lillgrund Wind Plant represents an interesting and valuable case study for wake model validation. Its highly structured layout leads to regular and significant wake interaction, and the relatively close wind turbine spacing makes the transition toward deep array behavior clear. These benefits serve to highlight the different capabilities of the wake model components to reproduce the flow physics—and therefore the wind plant performance—for a wide range of atmospheric conditions; however, the unique layout of Lillgrund makes the results presented in the current work difficult to generalize to other wind plants. Results presented in the current work serve to establish the baseline performance of the FLORIS framework for wake modeling and wind plant performance modeling. Testing model components individually provides a snapshot for the benefits of selecting each model and identifies regions and conditions where they improve the representation of the complex interaction between the wind plant and the atmospheric flow field. A future study will consider optimizing the tuning parameters in each wake model against a collection of wind plants with distinct atmospheric resources, layouts, and wind turbine models and should provide a set of parameter values that consistently reduce model error.

This work was authored 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 this 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 this 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. 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.

The data that support the findings of this study are available from Siemens Gamesa Renewable Energy A/S and Vattenfall. Restrictions apply to the availability of these data, which were used under license for this study. Data are available from the authors upon reasonable request and with the permission of Siemens Gamesa Renewable Energy A/S and Vattenfall.