Comparisons of wake models profiles with LiDAR data measured downstream from an operating onshore wind turbine

Abstract

This study analyzes the performance of Park, Frandsen, and Larsen models to simulate wake development downstream from a wind turbine for freestream wind velocities ranging from 5 to 10 m/s. Analyses are performed in terms of normalized freestream velocity recovery for a longitudinal centerline downstream from the turbine and normalized wind velocity profiles for cross-sections located 500 and 700 m downstream from the wind turbine. Simulated results are compared with high resolution LiDAR data measured during operation of a North American wind farm. Comparisons of longitudinal profiles demonstrate that Larsen and Frandsen models provide the best agreement with measured data for the case of 5 m/s freestream wind velocity, whereas Park model performs best for the 6–9 m/s freestream wind velocity bins. Post-processing of measured data indicates asymmetry of wake profiles at the selected cross-sections. At these locations, Larsen model accurately predicts the west side of normalized velocity profiles, whereas Park and Frandsen models only predict the velocity recovery at the wake centerline.

Keywords

Analytical wake models wind turbine wakes wind turbine velocity recovery LiDAR measurements

Introduction

In recent years, wind energy has become one of the most promising renewable energy sources around the world due to the rapid expansion of wind energy conversion technologies (GWEC, 2019; IRENA, 2020). Even with the technological improvements occurred in the last decades, current wind turbine projects have demonstrated that specialized studies involving wind energy efficiency are still crucial to obtain the best performance from the technology (Edelenbosch et al., 2017; Lund, 2007). The assessment of certain wind properties, such as the wind distribution, shear stresses, and turbulence characteristics, and also the effect of wake developments downstream from the turbine, is necessary for a complete evaluation of the efficiency of power generation systems (Chehouri et al., 2015; Leung and Yang, 2012; Saidur et al., 2011).

Wind turbine operation generates wakes characterized by lowered momentum and enlarged turbulence levels downstream from the turbine. Such phenomena is induced by extraction of kinetic energy from the atmospheric flow, resulting in reduced mean velocity, and increased turbulence regions that are propagated downstream to the next wind turbine generator (WTG) (Kim et al., 2015; Mo et al., 2013; Tsalicoglou, 2012). The development and propagation of these recirculation regions can affect power output and increase the loading of downstream wind turbines in a wind farm. As a result, it becomes desirable to quantify the characteristics of the flow behind individual turbines.

An usual approach to address this problem is the use of analytical models that are able to calculate the velocity deficit and recovery along the wake region (Crespo et al., 1999; Renkema, 2007).In general, these models are based on simplified forms of mass and momentum conservation principles, and their main advantage relies on the low computational costs of simulations and on the simplicity for practical applications (Vicente, 2018).

A literature survey shows that some studies have investigated wake effects on wind turbine performance and recent ones reported that analytical wake models were able to reproduce wake velocity deficit and recovery accordingly. Barthelmie et al. (2006, 2007, 2009) reported that the average power losses detected by a single wind turbine due to wakes was approximately 10% and for multiple wind turbines displaced in arrows the power production losses could reach values of 20%. The authors concluded that the output energy reduction was likely to range from 5% to 8% of the annual energy yield depending on the wind farm configuration. Therefore, the understanding of power losses mechanisms by the action of wakes was crucial to improve wind farm design.

Jeon et al. (2015) performed simulation comparisons using analytical wake models (e.g. Jensen, Larsen, and Frandsen models). Simulations were developed considering single wake of a 2MW onshore wind turbine, located at Sungsan Wind Farm in South Korea, and the calculations were compared with field measured data. Jensen model showed the highest prediction accuracy of the velocity deficit estimated at the wake center for freestream wind velocity up to 8.5 m/s. When the wind velocity exceeded 8.5 m/s, Larsen model was the most accurate in predicting the width of the wake and its profile. The authors concluded that the applications of wake models for wake-effect assessment require previous analysis of the spacing between wind turbines, because the prediction accuracy of the wake models varied with the downstream distance condition.

A review study presented by Göçmen et al. (2016) analyzed the six widely used approaches of wake modeling developed at the Technical University of Denmark, including Jensen and Larsen analytical models. The models results were evaluated using data from the Sexbierum (onshore) and the Lillgrund (offshore) wind farms located at northern Netherlands and in the Swedish west coast, respectively. The benchmark evaluations demonstrated that the Jensen and Larsen models were adequate for large wind farm calculations, specially because these models are robust and computationally affordable. The results obtained by the models for the velocity recovery at the wake centerline were considered satisfactory for both onshore and offshore implementations, as long as the far wake region was considered and the atmospheric conditions were well defined.

Brusca et al. (2018) performed wake velocity recovery simulations using Jensen, Larsen, and Frandsen models and compared with data measured in a wind tunnel experiment of a small-scale wind turbine. Different undisturbed wind velocities (5, 8, and 10 m/s) and turbine rotational velocities (1000, 2000, and 3000 RPM) were employed in the experiments. The final comparisons demonstrated that Larsen model was in good agreement with experimental data, whereas Jensen and Frandsen models were able to identify only mean and peak velocity deficit, respectively.

Because most of the analytical models used for practical wind engineering applications are based on several simplifying assumptions, comparisons with field data are usually required to evaluate the performance of simulations. Despite the increasing attention given to wake simulations in recent years, literature survey shows that relatively few investigators have performed detailed comparisons of analytical wake models results against field data. There is still a scarcity of researches devoted to comparisons of analytical wake simulations with field measured data performed for a large range of freestream velocity levels, for longitudinal and vertical comparison cross-sections, and using statistical indexes as tools for detailed error analyses. This gap is intended to be fulfilled as the main contribution of this work.

With this aspect in mind, the present work aims to analyze the performance of three analytical wake models by developing statistical comparisons with high resolution LiDAR data measured during operation of a North American wind farm. By suggestion of the company that owns the wind farm and provided the field data, three wake models are selected, namely Park, Frandsen, and Larsen models. These models are applied for wake simulations of a single wind turbine located in the first row of the wind farm. Simulations are performed to predict the velocity deficit and recovery for a longitudinal centerline and for transversal sections located downstream from the turbine. The accuracy of the comparisons is quantified by the maximum deviations and by the root mean square error (RMSE) of normalized freestream velocity recovery between measurements and simulations for a longitudinal centerline and for cross-sections located 500 and 700 m downstream from the wind turbine.

The remainder of this manuscript is organized as follows: section materials and methods provides an overall description of LiDAR and measurements conditions, the mathematical formulation of the selected wake models and a description of the simulations scenarios. Section results and discussions describes the comparisons of the simulations for longitudinal and cross-section profiles, discussing models characteristics and obtained results in detail, and presents RMSE analyses for all simulations. Finally, section conclusion presents the conclusions and recommendations for further work.

Materials and methods

The subject of this work involves simulations of a wind turbine currently operating in a North American wind farm. The site is characterized as a flat plateau 1500 m above sea level with annual average temperature of 10°C and prevailing neutral atmospheric conditions. Detailed description of site location, such as name of city, state or region, wind turbine characteristics and measurement equipment detailed specification are not allowed to be informed, following a confidentiality agreement established with the company that operates the wind farm and provided the geometrical and field data.

A backward-looking long range Galion LiDAR, which uses Doppler shift technology, was mounted inside a selected wind turbine positioned at the first row of the wind farm. A nacelle mounted deployment was preferred due to the retractable back door of the wind turbine generator at 80 m above ground, which enabled relatively easy installation and concurrent velocity measurements through the horizontal plane of the wake. A hub height meteorological mast was installed 75 m above ground and approximately 300 m west from the wind turbine in order to obtain concurrent measurements of wind direction (relative to the wind turbine yaw angle of 180°) and freestream wind velocity. The neighboring turbines were located 280 m east, 600 m west, and 1300 m downstream from the selected wind turbine.

A frequency analysis indicated that winds from the south and southwest directions were prevalent at the site, dominating approximately 65% of the sample occurrences, as shown in Figure 1. Temperature gradients between 2 and 75 m height measured at the meteorological station were in the range of ±2°C, indicating neutral atmospheric conditions during the entire measurement period (Stival, 2017).

Figure 1.

Frequency distribution of wind direction obtained for the sample measurements.

Single wake velocity measurements were performed using the backward-looking LiDAR for a longitudinal centerline downstream from the wind turbine at the hub height, and for cross-sections located 500 and 700 m downstream from the turbine, as shown schematically in Figure 2(a). The LiDAR azimuth orientation was aligned with the back edge of the nacelle deck, such that the 180° beam azimuth setting corresponds to a scan out the back of the nacelle perpendicular to the rotor plane. The Plane Position Indicator (PPI) scan mode was set to a constant elevation of 80 m and varying azimuth angles, such that data could be projected on a horizontal plane. For the cross-section measurements, a 81° width scan with ±3° azimuth increments was performed, centered on the 180° axis straight out behind of the wind turbine, as illustrated in Figure 2(b). The downstream distances were normalized by the rotor diameter, $D$ . The mean velocities were calculated and organized into 1 m/s bins of inflow wind velocities.

Figure 2.

(a) Longitudinal centerline and cross-sections employed for the LiDAR measurements downstream from the wind turbine and (b) plan view of the backward-looking LiDAR scan implemented for the cross-section measurements.

A campaign period of 6 weeks during September and October produced a database containing 6000 values, obtained for 10 minutes intervals. The measurement rate was chosen to obtain a 29-beam scan file in approximately 4 minutes to complete a scan cycle. The recovery rate of LiDAR data presented an average of 74% of the available data, which provided a representative sample to establish accurate statistics for the simulations comparisons. This was determined by a two-step process: a preliminary screening to identify the indicators of poor quality samples, and a subsequent quality control procedure based on the knowledge of the wind resource pattern and local weather conditions, where the outliers were removed. Preliminary analysis using a Windographer software (AWS Truepower, 2017) indicated mean inflow wind velocity of 5.1 m/s at the hub height, whereas the operational wind velocity range for the wind turbine generator varied from 4 to 25 m/s. Wind velocities below 10 m/s occurred approximately 93% of the time. For this reason, wind velocities from 5 to 10 m/s, divided into 1 m/s bins, were adopted as inflow boundary conditions for the simulations comparisons. During the field campaign all neighbor wind turbines were operating and no specific action was taken to guarantee minimum interference from wakes coming from other turbines. There was not any neighboring wind farms that could impact the data collection.

Three different wake models namely Park, Frandsen, and Larsen models are employed in this work, and the calculations are compared against LiDAR data for the selected inflow wind velocities (5–10 m/s). The chosen models are considered analytical models, conceived based on simplifying assumptions of mass and momentum conservation equations and largely applied by the wind industry. The main objective of these models is to estimate the flow velocity deficit and recovery, and the development of wakes diameter downstream from wind turbines.

Park model

The first wake model analyzed in this work is the Park model developed based on the study of Jensen (1983) and implemented in Wind Atlas Analysis and Application program (WAsP) (Mortensen et al., 1997). This is an analytical model that assumes an ideal axially symmetric flow and a top-hat shape for the velocity distribution on the wake cross-sections (Katic et al., 1987). For the longitudinal centerline, the model considers a gradually developing wake that is a linear function of the turbine downstream distance, $x$ . The wake diameter for a given downstream distance, $D_{w}$ , is calculated by:

D_{w} = D + 2 kx,

(1)

where $D$ is the rotor diameter and $k$ is an empirical decay constant that represents the dissipation process as the wake width increases. A common value of $k$ used for onshore simulations varies from 0.050 to 0.075 (Archer et al., 2018).

According to Park model the longitudinal velocity for a given position downstream from the turbine is given by:

\frac{u}{V_{0}} = 1 - 2 ω {(1 + \frac{2 kx}{D})}^{- 2},

(2)

where $u$ is the longitudinal component of the velocity vector downstream from the turbine, $V_{0}$ is the freestream velocity and $ω$ is an induction factor given by:

ω = \frac{1 - \sqrt{1 - C_{t}}}{2},

(3)

where $C_{t}$ is a thrust coefficient typically lower than 1.0, which is associated with the maximum thrust force resulting from the energy conversion in the equipment (Jensen, 1983).

Frandsen model

Frandsen model was developed by the application of mass and momentum conservation principles for a control volume enclosing the wind turbine generator (Frandsen, 1992; Frandsen et al., 2006). This model also assumes a top-hat shape for the velocity distribution on the wake cross-sections. The wake diameter for a given downstream position is calculated by:

D_{w} = D {(β^{\frac{n}{2}} + γ x)}^{\frac{1}{n}},

(4)

where $D_{w}$ is the wake diameter, $D$ is the rotor diameter, $x$ is the downstream distance, $n$ is the shape factor adopted equals 3, and $γ$ and $β$ are dimensionless parameters written as functions of the thrust coefficient according to:

γ = 1 - \sqrt{1 - C_{t}},

(5)

β = \frac{1 + \sqrt{1 - C_{t}}}{2 \sqrt{1 - C_{t}}} .

(6)

The longitudinal velocity for a given position downstream from the turbine can be written as:

u = \frac{V_{0}}{2} (1 \pm \sqrt{1 - 2 \frac{A}{A_{w}} C_{t}}),

(7)

where $u$ is the longitudinal component of velocity downstream from the turbine, $V_{0}$ is the freestream velocity, $A_{w}$ is the cross-sectional wake area associated with the diameter $D_{w}$ , and $A$ is the cross-sectional area at the beginning of the wake expansion.

Larsen model

Larsen model was developed based on analytical solutions of the Prandtl turbulent boundary layer equations, considering incompressible, stationary and axisymmetric flows, and assuming self similarity for the velocity cross-section distribution along the wake expansion (Larsen, 1988, 2009; Larsen et al., 2013). The model is able to calculate velocity distribution for selected cross-sections, as well as velocity recovery for a longitudinal centerline downstream from the turbine.

According to Larsen model the wake radius for a given position downstream from the turbine, $R_{w}$ , is written as:

R_{w} = {(\frac{35}{2 π})}^{3 / 10} {(3 c_{1}^{2})}^{1 / 5} {(C_{t} A (x + x_{0}))}^{1 / 3},

(8)

where $A$ is the cross-sectional area at the beginning of the wake expansion, $C_{t}$ is the thrust coefficient, $x$ is the downstream distance, and $x_{0}$ is the normalized position of the wind turbine generator.

The term $c_{1}$ is a constant associated with the Prandtl characteristic mixing length, written as:

c_{1} = {(\frac{D}{2})}^{5 / 2} {(\frac{105}{2 π})}^{- 1 / 2} {(C_{t} A x_{0})}^{- 5 / 6},

(9)

and the normalized position of the wind turbine generator is given by:

x_{0} = \frac{9.5 D}{{(\frac{2 R_{9.5}}{D_{e}})}^{3} - 1},

(10)

where $R_{9.5}$ is the wake radius at a downstream distance of 9.5 rotor diameters from the hub and $D_{e}$ is the effective rotor diameter.

The parameter $R_{9.5}$ is calculated as follows:

R_{9.5} = 0.5 [R_{nb} + \min (H, R_{nb})],

(11)

in which $R_{nb}$ is defined by:

R_{nb} = \max (1.08 D, 1.08 D + 21.7 D (I_{a} - 0.05)),

(12)

where $I_{a}$ represents the ambient turbulence intensity and $H$ is the hub height.

The effective rotor diameter is calculated according to:

D_{e} = D \sqrt{\frac{1 + \sqrt{1 - C_{t}}}{2 \sqrt{1 - C_{t}}}},

(13)

where $C_{t}$ is the thrust coefficient typically lower than unit.

According to Larsen model the wind velocity deficit inside the wake is calculated as follows:

\begin{matrix} V_{0} - u = \frac{- V_{0}}{9} {(C_{t} A {(x + x_{0})}^{- 2})}^{1 / 3} \\ {r^{\frac{3}{2}} {(3 c_{1}^{2} C_{t} A (x + x_{0}))}^{- 1 / 2} - {(\frac{35}{2 π})}^{3 / 10} {(3 c_{1}^{2})}^{- 1 / 5}}^{2}, \end{matrix}

(14)

where $V_{0}$ is the freestream velocity, $u$ is the wind velocity in the wake and $V_{0} - u$ is the velocity deficit in the wake.

Simulation scenarios

The simulations are performed for inflow velocities ranging from 5 to 10 m/s, divided into 1 m/s intervals, and the obtained results are compared with LiDAR measured data for longitudinal and cross-section profiles. The selected simulation scenarios were defined based on the main characteristics of measured data, which consisted of prevailing winds coming from south and southwest directions (about 65% of the sample occurrences, as shown in Figure 1), mean wind velocity of 5.1 m/s and wind velocities below 10 m/s occurring approximately 93% of the time. The turbulence intensity condition was not determined during the data collection.

LiDAR measurements were statistically processed and the median values associated with inflow wind velocities from 5 to 10 m/s, divided into 1 m/s bins, were retained for comparisons. The velocity probability distribution allowed to determine the standard deviations for each point of measurement, which is used as a preliminary evaluation in the comparisons. The data scattering is more pronounced in the far wake regions, particularly for distances downstream from the 9D section, due to the access of smaller samples associated with the reduced sensitivity of the LiDAR Doppler shift system to capture the return pulses over long distances. The final errors are quantified by the maximum deviations and by the root mean square error (RMSE) of normalized inflow velocity recovery between LiDAR measurements and models simulations, for each inflow wind velocity bin.

Results and discussions

Comparisons of longitudinal profiles

For the longitudinal profiles, the downstream distances are normalized by the rotor diameter, $D$ . The results are plotted in $x - y$ graphical format, where the vertical axis represents the normalized freestream velocity recovery in the wake centerline, $U = u / V_{0}$ , and the horizontal axis stands for the normalized longitudinal distances in terms of the rotor diameter. Comparisons are performed for a maximum downstream distance of 12 $D$ in the longitudinal direction.

Figure 3(a) to (f) show the longitudinal velocity recovery in the wake centerline for inflow wind velocities ranging from 5 to 10 m/s, respectively. The solid dots stand for median values of the LiDAR measured data, and the opened symbols indicate the wake models results. The vertical bars show measured data scattering, where each bar corresponds to ±1 standard deviation of the bin sample probability distribution.

Figure 3.

Normalized velocity recovery for longitudinal distances downstream from the wind turbine in the wake centerline. Comparisons for inflow wind velocities of: (a) 5 m/s, (b) 6 m/s, (c) 7 m/s, (d) 8 m/s, (e) 9 m/s, and (f) 10 m/s.

The comparisons for the 5 m/s inflow wind velocity (Figure 3(a)) shows that Larsen model obtained the most accurate overall representation of the LiDAR data profile among the three models. The largest discrepancies are observed at a distance of 3D downstream from the hub, where simulated values exceeded the measured velocity recovery by approximately 12%. The results of Park model are also satisfactory, even though a maximum deviation of 38% is observed in the beginning of the wake development. Frandsen model is the only one that overestimates the velocity recovery in the first half of the wake development (until the distance of 6D), presenting maximum deviation of approximately 29% at 3D distance. Further downstream, for longitudinal distances greater than 6D, all models converge to good agreement with measured data.

For the cases of 6 and 7 m/s inflow velocities (Figure 3(b) and (c), respectively), the Park model results present excellent agreement with LiDAR data. Most of the simulated values are positioned within the range of the measured data ±1 standard deviation. The most accurate points of the comparisons occurred at the distance of 7D, reaching only 0.2% and 0.4% deviation for the 6 and 7 m/s cases, respectively. Larsen model results also follow the measured data profile, with maximum deviations of about 18%. Frandsen model again overestimates the velocity recovery on the first half of the wake development, reaching maximum deviations of 29% and 23% for the 6 and 7 m/s cases, respectively.

For the cases of 8, 9, and 10 m/s inflow velocities (Figure 3(d)–(f), respectively) all simulations overestimate the velocity recovery when compared with measured data, specially for downstream distances lower than 8 D. Despite the similarity among models results, Park model performs slightly better than the others for the 8 m/s case, presenting maximum deviations lower than 10%, as well as for the 9 m/s case, where the maximum deviation reaches 19% at the 3 D downstream distance. For downstream distances larger than 8 D, Larsen and Frandsen models results improve their agreement with measured data. As the inflow velocities increased, the behavior of the three models seems to be quite similar. In particular, the models results are practically coincident for the inflow velocity levels of 9 and 10 m/s.

Comparisons of cross-section profiles

The comparisons on the transversal direction are performed for cross-sections located at 500 and 700 m downstream from the wind turbine, corresponding to the distances of 5 and 7 $D$ , respectively. For each cross-section, profiles of normalized velocity are calculated for a radius of 200 m around the wake centerline. A total of twenty-one points of measurements, uniformly distributed at each 20 m intervals across the wake diameter at the hub height, are adopted for both cross-sections.

For all freestream wind velocities, the measured wake cross-section profiles are clearly asymmetrical. Despite the fact that freestream wind velocity and direction interference is not desirable, this situation is susceptible to occur during turbine operation, specially in the cases that no proper action is taken to guarantee minimum interference from wakes coming from neighbor turbines in the wind farm (Chen et al., 2018; Lefebvre and Jones, 2020). As the analytical wake models employed in this work are not designed to predict asymmetric cross-section distributions, some level of disagreement between simulation results and field measurements is expected.

Figure 4(a) to (f) show the normalized wind velocity distributions for a cross-section located 500 m downstream from the turbine, considering inflow wind velocities from 5 to 10 m/s, divided into 1 m/s bins. The solid dots stand for median values of the LiDAR measured data, and the lines indicate wake models results. The comparisons for the 5 m/s inflow wind velocity (Figure 4(a)) show that Larsen model obtained excellent agreement with measure data on the west side of the wake. A maximum deviation of 35% occurs on the east side of the wake, where asymmetry of measured data is pronounced. Park model is able to predict the minimum normalized velocity recovery of 0.72 occurred at the centerline and Frandsen model fails to simulate this value, overestimating the normalized velocity to 0.9 at this location.

Figure 4.

Normalized velocity distribution for a cross-section located 500 m downstream from the wind turbine. Comparisons for inflow wind velocities of: (a) 5 m/s, (b) 6 m/s, (c) 7 m/s, (d) 8 m/s, (e) 9 m/s, and (f) 10 m/s.

For the cases of 6 and 7 m/s inflow velocities (Figure 4(b) and (c), respectively) Larsen model is again effective to predict the west side distribution of the measured data profile. On the east side of the wake, maximum deviations of approximately 26% and 28% are observed for the 6 and 7 m/s cases, respectively.

Park model is more accurate than Frandsen model on the comparisons of centerline velocity recovery. For the case of 6 m/s, Park model obtained a velocity recovery of 0.76, whereas Frandsen model obtained 0.86, corresponding to deviations of 8% and 22% in relation to the measured data, respectively. For the case of 7 m/s, Park model obtained velocity recovery of 0.79, whereas Frandsen model obtained 0.83, corresponding to deviations of 5% and 10%, respectively.

In the comparisons for the cases of 8 and 9 m/s inflow wind velocities (Figure 4(d) and (e), respectively) Larsen model profile follows closely the measured data on the west side of the wake. On the eastern side of the wake, maximum deviations of approximately 30% are observed for both 8 and 9 m/s cases. Park and Frandsen models predict normalized velocity recovery between 0.88 and 0.90 at the wake centerline. For the case of 10 m/s inflow velocity (Figure 4(f)), Larsen model is able to capture the west side of the wake, whereas Park and Frandsen models obtain similar results, presenting normalized velocity recovery of 0.76 at the wake centerline.

Figure 5(a) to (f) show the normalized wind velocity distributions for a cross-section located 700 m downstream from the turbine, considering inflow wind velocities from 5 to 10 m/s, divided into 1 m/s bins. The comparisons for the 5–9 m/s inflow wind velocities (Figure 5(a)–(e)) show that Larsen model is generally able to capture the measured data profiles on the west side of the wakes. On the east side of the wakes, simulated and measured profiles do not agree due to the asymmetric distribution of the measured data, presenting maximum deviations ranging from 19% for the case of 7 m/s to 38% for the case of 8 m/s.

Figure 5.

Normalized velocity distribution for a cross-section located 700 m downstream from the wind turbine. Comparisons for inflow wind velocities of: (a) 5 m/s, (b) 6 m/s, (c) 7 m/s, (d) 8 m/s, (e) 9 m/s, and (f) 10 m/s.

Park and Frandsen models results are significantly different only for the cases of 5 and 6 m/s inflow wind velocities (Figure 5(a) and (b)). Frandsen model provides a better representation of the overall profile, whereas both models show some level of disagreement when comparing velocity recovery at the wind turbine centerline. For the case of 5 m/s, Park model obtained a velocity recovery of 0.79, whereas Frandsen model obtained 0.90, corresponding to deviations of 2% and 9% in relation to the measured data, respectively. For the case of 6 m/s, Park model obtained velocity recovery of 0.82, whereas Frandsen model obtained 0.88, corresponding to deviations of 1% and 8%, respectively.

For the cases of inflow wind velocities from 7 to 9 m/s (Figure 5(c) to (e)), Park and Larsen models produce similar results, obtaining good agreement with measured data for velocity recovery at the wake centerline. The simulated velocity recovery at the wake centerline are approximately 0.85, 0.85, and 0.87 for the cases of 7, 8, and 9 m/s, corresponding to deviations of 0%, 3%, and 8%, respectively. For the case of 10 m/s inflow wind velocity (Figure 5(f)), the measured data statistics could not be processed accordingly due to the small sample of recovered data.

Models discussion and RMSE analysis

Except for the 5 m/s inflow wind velocity case, the comparisons of longitudinal profiles indicate small differences among the wake models behavior. This similarity occurs because the models attempt to represent mass and momentum conservation principles using similar simplifying assumptions, such as incompressible fluid, stationary, and unperturbed flow conditions. The velocity formulations are all functions of the turbine downstream distance and depend upon trust coefficients and wake diameter developments. Frandsen and Park models approaches for the wake diameter calculation are equivalent, even though Park model employs an empirical decay constant instead of a trust coefficient in the formulation, so in that case the wake diameter estimate do not depend on the freestream wind velocities. Larsen model calculates an effective wake diameter that is associated with a thrust coefficient.

From the three selected models, Larsen is the only one able to predict non-uniform velocity cross-section distribution. It is interesting to observe that, in spite of the limitation assumption of axisymmetric velocity distribution, Larsen model profiles are in quite good agreement with measured data on the western half of the wakes. This suggests that Larsen model can be useful to partially estimate asymmetrical wake cross-section profiles, which are susceptible to occur by influences of wakes coming from neighboring wind turbines. Park and Frandsen models assume uniform velocity cross-section distribution (top-hat velocity distribution) and for this reason these models are suitable mainly to estimate wake diameter development and velocity recovery values at the turbine centerline.

In order to assess the models results and to quantify the accuracy of the simulations when compared to measured data, a root mean square error index (RMSE) is computed according to:

RMSE = \sqrt{\frac{1}{N} Σ_{i = 1}^{N} {(U_{s} - U_{o})}^{2}},

(15)

where $N$ is the number of data points employed in the comparisons of longitudinal and cross-section profiles, $U_{s}$ is the simulated normalized velocity and $U_{o}$ is the observed normalized velocity. This evaluation aims to obtain a statistical measure of the wake models performance when compared with LiDAR data for longitudinal and cross-section distributions.

Table 1 summarizes the maximum deviations obtained from the longitudinal profile comparisons and shows the normalized velocity RMSE between wake models results and measured data for a longitudinal profile downstream from the wind turbine. Simulations using Larsen model provides the most accurate results for the 5 m/s inflow wind velocity case, presenting normalized velocity RMSE of 0.0448 and maximum local deviation of 12%. Park model provides the best agreement with measured data for the inflow wind velocity levels of 6, 7, 8, and 9 m/s, obtaining maximum local deviations of 13%, 8%, 10%, and 19%, respectively. As far as the RMSE analysis is concerned, even though the three models demonstrate overall similar behavior, Park model provide slightly better results, presenting normalized velocity RMSE of 0.0309, 0.0343, 0.0381, and 0.0673, respectively. For the case of 10 m/s inflow wind velocity, the models simulations are quite similar obtaining maximum local deviations of 18%–20% and RMSE on the order of 0.11–0.12.

Table 1.

Maximum deviations and RMSE between wake models results and measured data for a centerline longitudinal profile located downstream from the wind turbine.

Inflow wind velocity (m/s)	Park		Frandsen		Larsen
Inflow wind velocity (m/s)	RMSE	Max Dev. (%)	RMSE	Max Dev. (%)	RMSE	Max Dev. (%)
5	0.0904	38	0.1067	29	0.0448	12
6	0.0309	13	0.0957	29	0.0594	18
7	0.0343	8	0.0736	23	0.0538	14
8	0.0381	10	0.0718	21	0.0599	17
9	0.0673	19	0.0938	29	0.0855	25
10	0.1267	18	0.1133	19	0.1268	20

Table 2 shows the simulated wake diameters and the normalized velocity RMSE between wake models results and measured data for a cross-section located 500 m downstream from the wind turbine. The results of Larsen model are in good agreement with the west side of the measured wake profiles for all inflow wind velocity cases. However, due to the large asymmetry of the east side of the measured profiles, the normalized velocity RMSE of Larsen model becomes similar to the other two models, varying from about 0.10 to 0.15. Park model predicts a wake diameter of 162.38 m for all inflow wind velocities. Frandsen model predicts diameters of 200.08, 177.00, 169.66, 166.57, 162.74, and 154.59 m, whereas Larsen model calculates 298.27, 295.11, 294.45, 294.22, 293.96, and 293.53 m for the 5–10 m/s inflow wind velocity cases, respectively.

Table 2.

Simulated wake diameter and RMSE between wake models results and measured data for a cross-section profile located 500 m downstream from the wind turbine.

Inflow wind velocity (m/s)	Park		Frandsen		Larsen
Inflow wind velocity (m/s)	RMSE	$D_{w}$ (m)	RMSE	$D_{w}$ (m)	RMSE	$D_{w}$ (m)
5	0.1366	162.3827	0.1377	200.0800	0.1347	298.2794
6	0.1063	162.3827	0.1192	177.0085	0.1171	295.1144
7	0.0981	162.3827	0.1019	169.6642	0.1063	294.4546
8	0.1223	162.3827	0.1267	166.5779	0,1338	294,2224
9	0.1427	162.3827	0.1470	162.7410	0,1591	293,9673
10	0.1296	162.3827	0.1386	154.5902	0.1388	293.5318

Table 3 shows the simulated wake diameters and the normalized velocity RMSE between wake models results and measured data for a cross-section located 700 m downstream from the wind turbine. Simulation using Larsen model provides RMSE values of 0.1117, 0.1119, 0.0959, 0.14553, and 0.1339 for the 5–9 m/s inflow wind velocity cases, respectively. Park and Frandsen models provide results on the same order of magnitude as Larsen model for all inflow wind velocities. For the 10 m/s inflow velocity case, measured data statistics could not be processed correctly due to excessive data scattering.

Table 3.

Simulated wake diameter and RMSE between wake models results and measured data for a cross-section profile located 700 m downstream from the wind turbine.

Inflow wind velocity (m/s)	Park		Frandsen		Larsen
Inflow wind velocity (m/s)	RMSE	$D_{w}$ (m)	RMSE	$D_{w}$ (m)	RMSE	$D_{w}$ (m)
5	0.1311	187.7358	0.1127	211.9316	0.1117	328.5282
6	0.1150	187.7358	0.1171	189.3017	0.1119	327.1127
7	0.0987	187.7358	0.0982	181.7483	0.0959	326.8199
8	0.1461	187.7358	0.1465	178.5137	0.1455	326.7170
9	0.1341	187.7358	0.1340	174.4423	0.1339	326.6041
10	0.1300	187.7358	0.1322	165.6147	0.1168	326.4117

Simulations using Park model provide wake diameter of 187.73 m. Frandsen model predicts 211.93, 189.30, 181.74, 178.51, 174.44, and 165.61 m, whereas Larsen model predicts wake diameters of 328.52, 327.11, 326.81, 326.71, 326.60, and 326.41 m for the 5–10 m/s inflow velocity cases, respectively. It can be observed that wake diameters predicted by Larsen model is generally 75% larger than the one predicted by Park model for this cross-section. However, it is difficult to establish the accuracy of these results compared to field measurements due to the lack of symmetry toward the east side of the measured profiles.

Comparison of the obtained results with recent studies available in the literature demonstrates two main similarities: the ability of the models to predict wake-centered velocity deficit varies with the inflow wind velocity conditions, and the prediction accuracy of the wake models varies with the downstream distance location. This is corroborated with the results of the present work, which shows that differences among the models results are more pronounced for lower velocity levels (e.g. 5 and 6 m/s), and particularly for shorter downstream distances. For long downstream distances, the three models tend to predict similar velocity recovery for the longitudinal direction.

Physically, the obtained results show that the model with less simplifying assumptions is more successful in simulating wake characteristics in detail. In that respect, Park and Frandsen models are successful mainly to predict the velocity deficit measured at the wake centerline. Frandsen model presents slightly better performance for the near wake, whereas Park model performs best for intermediate downstream distances. Nevertheless, it is not possible to establish an individual model which presents best performance for all wake conditions.

Larsen model shows a good agreement with the measurements not only at the wake centerline, but also for the cross-section profiles. The comparisons performed in the present work show that Larsen model is able to reproduce wake profiles with a good level of agreement, even when compared with asymmetric measured data.

Conclusion

This work performed comparisons of Larsen, Park, and Frandsen wake models profiles against field data obtained by a LiDAR equipment for a wind turbine positioned on the first row of a wind farm. A longitudinal centerline at 80 m hub height and two cross-sections located 500 and 700 m downstream from the wind turbine were employed for comparisons. The field data consisted of velocity measurements obtained for a wind turbine generator currently operating in a North American wind farm. Profiles of normalized velocity were compared and discussed.

Based on the wind velocity and direction at the site, freestream wind velocities of 5–10 m/s, divided into 1 m/s bins, were selected as simulations inflow boundary conditions. The accuracy of simulation results was quantified by the maximum normalized velocity deviations and by the normalized velocity root mean square errors (RMSE) between simulated results and measured data.

Comparisons of longitudinal profiles demonstrated that Larsen model was the most accurate for the 5 m/s inflow wind velocity bin, presenting normalized velocity RMSE of 0.0448. For the range of 6–9 m/s inflow wind velocity bins, all models showed an overall good agreement with measured data. Larsen and Park models showed slightly better performance for the near wake. Park model RMSE evaluations were slightly lower than the others, ranging from 0.0309 to 0.0673. For the 10 m/s inflow wind velocity bin, the three models performed similarly.

Regarding the cross-section comparisons, the measured data presented asymmetrical distributions probably due to interference of wind diversion coming from a neighboring wind turbine. Nevertheless, Larsen model results were in good agreement with the western half of wake profiles for both cross-section located 500 and 700 m downstream from the turbine. The asymmetry of measured data conducted to approximately uniform normalized velocity RMSE evaluations among the models, all resulting on the range of 0.09–0.16.

Park and Frandsen models were in good agreement with measured data to predict the velocity recovery at the cross-section centerline. In addition, both models provided wake diameters on the same order of magnitude. Larsen model indicated wake diameters considerably larger than the former two models. However, it was difficult to perform comparisons with field measurements to determine the accuracy of the simulated wake diameters due to the lack of symmetry toward the east side of the measured profiles (it was impossible to visually define the measured wake diameters).

Finally, the significance of this work relies on the detailed comparisons performed among the three selected analytical wake models. These comprehensive comparisons allowed to conclude that the accuracy of the models varies with the freestream wind conditions and with the downstream distances, confirming similar wake models behavior observed by previous works. It is not possible to determine unequivocally the superior performance of a particular model. However, this study showed that Larsen model is suitable to simulate wake cross-section profiles even when compared with asymmetric measured data.

For the future, it is suggested to implement other measurement campaigns during different periods of the year in order to investigate the interference on the cross-section wake profiles caused by neighbor wind turbines. Further modeling work will concentrate on developing relationships between wakes velocity deficit/recovery and power production of downstream turbines to estimate potential wake effects on power generation reduction for different wind farm configurations. Additional future activity will concern evaluation of the three studied wake models against more sophisticated computational fluid dynamics models, aiming to categorize the main advantages of using simplified analytical wake models for wind turbine applications.

Footnotes

Acknowledgements

Leandro Lemes Stival gratefully acknowledge the financial support received during his master degree from the Brazilian Coordination for the Improvement of Higher Education Personnel (CAPES). The authors thank the support of the Graduate Program in Water Resources and Environmental Engineering from the Federal University of Paraná (PPGERHA-UFPR).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Leandro José Lemes Stival

Fernando Oliveira de Andrade

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