## 1. Introduction

Wind turbines convert energy from the freestream, resulting in a volume of disturbed flow behind the rotor characterized by reduced wind speed and increased turbulence. The velocity deficit (VD) in this wake region diminishes with distance, as faster-moving air outside is gradually entrained. Even so, a turbine in the lee of and proximate to another produces less power and experiences higher fatigue loads than it would otherwise. As utility-scale turbines rarely exist in isolation, detailed knowledge of the mean flow and turbulence structure inside wakes is necessary for correctly modeling both power production and turbine loading at modern wind farms; see, for example, Porté-Agel et al. (2011), Churchfield et al. (2012), and Fitch et al. (2012).

To this end, the Turbine Wake and Inflow Characterization Study (TWICS) was conducted in the spring of 2011 to quantify various wake features downstream of a multimegawatt turbine located at the National Wind Technology Center (NWTC), a research and development facility operated by the U.S. Department of Energy’s National Renewable Energy Laboratory (NREL) just south of Boulder, Colorado. Full-scale measurements of wake dynamics are hardly practical or even possible with conventional sensors, such as cup anemometers mounted on meteorological (met) masts. Accordingly, the high-resolution Doppler lidar (HRDL) developed by the Earth System Research Laboratory (ESRL) of the National Oceanic and Atmospheric Administration (NOAA) was employed to investigate wake characteristics under various atmospheric conditions. HRDL remotely senses line-of-sight (LOS) wind velocities *u*_{LOS} and has been used in several previous studies of boundary layer dynamics (Grund et al. 2001; Banta et al. 2002; Newsom and Banta 2003; Tucker et al. 2009; Pichugina et al. 2012). Note that complementary TWICS analysis is provided in Smalikho et al. (2013), in particular a study of the relationship between turbulent energy dissipation rate and wake length.

Largely because of the limited availability of field test data for model verification, wind farm wake modeling—and hence the optimization of wind turbine layouts—has suffered from an unacceptable degree of uncertainty to date, particularly in complex terrain (Barthelmie et al. 2010). The advent of innovative measurement techniques involving scanning remote sensors (Käsler et al. 2010; Bingöl et al. 2010; Trujillo et al. 2011; Clive et al. 2011; Hirth et al. 2012; Hirth and Schroeder 2013; Iungo et al. 2013) suggests the need for new approaches to identify and characterize wind turbine wakes. Here, we develop a set of quantitative procedures for determining various wake features—such as the velocity deficit, the size of the wake boundary, and the location of the wake centerline—that are designed to be broadly applicable to other remote sensor datasets and also to output from computational fluid dynamics (CFD) simulations of wind turbines. In what follows, section 2 provides a brief overview of the theory and empirical observation of wind turbine wake aerodynamics. In section 3, we describe our particular experimental setup and the postprocessing wake detection algorithms. Main results are presented in section 4, in which wake characteristics are categorized by ambient wind speed, atmospheric stability, and ambient turbulence. A summary and recommendations for future studies are offered in section 5.

## 2. Background and previous work

### a. Wind turbine wake aerodynamics

Studies of wind turbine wakes often distinguish between near and far wake regions, with the dividing line loosely taken to be a few rotor diameters (*D*) downstream (Vermeer et al. 2003). In the near wake, the velocity deficit profile in the transverse and vertical directions depends on the amount of lift produced along the span of the blade. Very little lift is generated at the blade root because of a suboptimal airfoil cross section and the connection to the hub, and also at the end of the blade because of tip vortices. Maximum lift, on the other hand, is generated near 75% blade span, and consequently the velocity deficit profile contains two local minima that correspond roughly to these points along the blades. Farther downwind, in the far wake, turbulent mixing results in the merging of the two troughs to form a single trough, which is approximately Gaussian in shape (Magnusson 1999).

### b. Velocity deficit

*x*, are summarized in Fig. 1 (Alfredsson and Dahlberg 1981; Baker and Walker 1984; Haines et al. 1986; Högström et al. 1988; Elliott and Barnard 1990; Magnusson and Smedman 1994; Barthelmie et al. 2003; Barthelmie et al. 2006; Hirth and Schroeder 2013). The fit to the data is made assuming that VD follows a power law (Högström et al. 1988; Frandsen et al. 2006), defined aswith VD

_{0}= 56% being the velocity deficit at

*x*= 1

*D*and

*n*= −0.57 being an exponent controlling the attenuation of the velocity deficit with downstream distance. Interestingly, the aggregate best-fit value

*n*= −0.57 nearly matches the value of the scaling exponent predicted by classical fluid mechanics similarity theory at infinite Reynolds number, which says the deficit should scale as (

*x*/

*D*)

^{−2/3}(Johansson et al. 2003).

Crespo et al. (1988) observed both numerically and experimentally that the point of maximum velocity deficit is located below the turbine axis because of tower shadow, shear of the incoming flow, and the presence of the ground. Other investigators have asserted that maximum velocity deficits occur below hub height (Elliott and Barnard 1990), at hub height (Kambezidis et al. 1990), and above hub height (Magnusson and Smedman 1994; Helmis et al. 1995); the velocity deficit profile has a maximum “near” hub height in Barthelmie et al. (2003). This discrepancy probably arises from differences in rotor tilt, stability conditions, and terrain among the experiments.

### c. Wake size and expansion rate

In addition to velocity deficit, optimal siting of individual turbines within wind farms hinges on accurately characterizing wake size and expansion rate. We use wake width (*w*) and height (*h*) in what follows to indicate the size of the wake in the lateral and vertical directions, respectively. In the near wake, the wake enlarges because of mass continuity, as the velocity deficit reaches a maximum at some nonzero distance behind the turbine. Farther downstream, turbulent mixing induces entrainment of the faster ambient flow, and the resulting momentum transfer causes the velocity deficit to decrease and the wake to expand. Here, the wake growth rate depends upon not just mechanical turbulence generated by the rotor but also upon atmospheric buoyant and shear-generated turbulence. Moreover, because of the presence of the ground, the growth rate of the wake in the vertical direction is less than that in the lateral direction.

*wake width*is subjective, and various definitions exist within the literature. Here, we take the wake width to be the 95% confidence interval of the Gaussian velocity deficit profile, which is analogous to the definition in Hansen et al. (2012) using power deficits. Wake width observations from previous field experiments can be seen in Fig. 2. Elliott and Barnard (1990) base their estimation of wake width on the wind speed profile in some cases, and on the turbulence intensity profile in others. The wake widths for Magnusson and Smedman (1994) and Trujillo et al. (2011) are inferred from figures in those papers using the 95% confidence interval criterion. The fit to the data is made assuming that

*w*follows a power law (Högström et al. 1988; Frandsen et al. 2006), defined aswith

*w*

_{0}= 1.3

*D*being the wake width at

*x*= 1

*D*and

*m*= 0.33 being an exponent controlling the expansion of the wake with downstream distance. Again, it is interesting to note that the aggregate best-fit value

*m*= 0.33 closely matches the value of the scaling exponent predicted by similarity theory at infinite Reynolds number, which says the boundary should scale as (

*x*/

*D*)

^{1/3}(Johansson et al. 2003).

## 3. Data and methods

Covered mostly with short grasses, the NWTC is located near the base of the foothills of the Rocky Mountains, to the east-southeast of Eldorado Canyon. During the winter and spring, the canyon funnels strong winds directly to the site, with a predominant wind direction of about 290° (Clifton and Lundquist 2012; Banta et al. 1995, 1996). The strong directionality of the flow justified the use of HRDL, which measures LOS velocities and remained at a fixed location during the experiment. Positioned 880 m at a bearing of approximately 310° from the turbine of interest, HRDL could best resolve winds blowing from this direction. In addition to HRDL, a Leosphere–NRG WINDCUBE version 1 vertically profiling lidar provided supplementary measurements of the turbine inflow conditions, and observations from an 80-m met tower were used to calculate the bulk Richardson number Ri_{B}, a metric of atmospheric stability. The layout of the instrumentation is depicted in Fig. 3, and specifications for HRDL and the wind turbine are given in Tables 1 and 2, respectively. Note that the turbine rotor diameter *D* = 101 m and the hub height *H* = 80 m. Nearly 100 h of wake measurements were collected over a period spanning from 5 April 2011 to 3 May 2011.

HRDL technical specifications.

Wind turbine technical specifications.

Wind resource characteristics for the site, as measured at 80 m by the WINDCUBE over the course of the field campaign, are shown in Fig. 4. The Weibull fit to the wind speed distribution (Justus et al. 1978) in Fig. 4a has scale parameter *λ* = 7.20 and shape parameter *k* = 1.46. The histogram and wind rose in Figs. 4b and 4c indicate that most of the strong winds came from the west-northwest, corresponding nearly to the line of sight from HRDL to the turbine; the fit to the wind direction data in Fig. 4b is a finite von Mises mixture distribution (Masseran et al. 2013) with three modes, at 7°, 175°, and 287°. In Fig. 4d, the turbulence intensity *I* is defined as the ratio of the horizontal wind speed standard deviation to the mean horizontal wind speed, taken over a 10-min interval. Turbulence intensity was calculated only for wind speeds greater than 4 m s^{−1}. The lognormal fit to the turbulence intensity distribution (Larsen 2001) has location parameter *μ* = −1.73 and scale parameter *σ* = 0.49. The mean wind speed and turbulence intensity during the course of the experiment were 6.52 m s^{−1} and 0.199, respectively. By comparison, Elliott et al. (2009) measured 80-m turbulence intensities between 0.09 and 0.13 at seven sites in the midwestern United States, where much of the nation’s wind power capacity is located.

### a. High-resolution Doppler lidar

The signature instrument in TWICS was HRDL, a pulsed coherent Doppler lidar with 30-m range resolution and 5 cm s^{−1} velocity precision, depending on atmospheric and operating conditions. Following the terminology used in radar display, sweeping the azimuth angle of the beam while holding the elevation angle fixed is known as a plan position indicator (PPI) scan. In contrast, a range–height indicator (RHI) scan involves sweeping the elevation angle while holding the azimuth angle fixed. At low elevation angles, PPI scans yield close approximations of horizontal wind speed near the surface, while RHI scans provide vertical cross sections of the radial wind field. By employing a well-collimated beam, HRDL does not suffer from antenna sidelobe contamination, allowing wind profile measurements to be taken very close to the surface and to other obstacles (Grund et al. 2001).

These scanning techniques were used to examine the structure of the wake behind the turbine. Sample pseudocolor plots of HRDL-measured LOS velocity for both PPI and RHI scans, in which wakes appear as regions of cooler colors, are shown in Figs. 5 and 6, respectively. During the study, a PPI scan typically involved holding the elevation angle constant at a value between 3° and 4°, while sweeping the azimuth through an angle of about 30°, centered on the turbine, at a rate of roughly 1.5° s^{−1}. Normally lasting approximately 20 s, a full sector sweep usually included measurements at about 30 discrete azimuth angles. By contrast, a typical RHI scan involved holding the azimuth angle constant at a value within ±3° from the line connecting HRDL to the turbine while sweeping the elevation angle at a rate slightly less than 1° s^{−1} from 0° to between 10° and 15°. Because of the geometry of ground-based remote sensing, the altitude at which PPI scans sample the flow increases with range, such that measurements are eventually taken above the wake at some distance far behind the turbine, depending on the elevation angle. Moreover, RHI scans may not always intersect the wake, especially at longer range gates, because of wake meandering and variations in wind direction and turbine yaw. The scans were, however, designed to capture as much of the wake as possible, given these geometric constraints.

In addition to measuring properties of the wake, HRDL was well positioned to quantify the variability of the inflow to the turbine. We define the spatial turbulence intensity *I*_{space} as the ratio of the wind speed standard deviation to the mean wind speed, calculated over a given region of space, which we take here to be the annular sector of each scan with radii spanning from 4*D* to 2*D* upwind of the turbine. These distances are chosen to correspond to the standard range for measuring freestream winds in the determination of wind turbine power performance (IEC 2005). Figure 7a shows an example of the LOS velocity measurements within this region, using the same PPI scan as the one depicted in Fig. 5. To estimate *I*_{space} in the case of the PPI scans, we assume horizontal homogeneity of the wind field over the sensed area, with uniform wind speed *u* and direction *φ*, as is typical for wind lidar; see, for example, Frehlich et al. (2006). When the LOS velocity signal is displayed as a function of azimuth angle *θ*, a plot as the one shown in Fig. 7b is obtained, since *u*_{LOS} = *u* cos(*θ* − *φ*). A weighted nonlinear regression scheme (Box et al. 2005) is used to fit this equation to the data, in which each weight is equal to the reciprocal of the variance of the measurement. Velocity precision is estimated using the Cramér–Rao lower bound (Rye and Hardesty 1993), because HRDL operates close to this theoretical limit in both high and low signal-to-noise ratio (SNR) conditions (Grund et al. 2001). The spatial turbulence intensity is then calculated by dividing the root-mean-square deviation (RMSD) of the fit by the modeled wind speed *u*. In this particular example, RMSD = 5.24 m s^{−1} and *u* = 15.4 m s^{−1}, such that *I*_{space} = 34%. To be clear, the fit here is intended to establish an average wind speed within the inflow region, so that the variation about that average may be quantified, analogous to the way turbulence intensity is normally calculated over a temporal interval. In addition to indicating the turbulence within the flow, the amplitude of the spatial wind fluctuations is an important factor in determining the minimum velocity deficit that can be captured by the wake detection algorithms described in section 3d.

*δ*is the beam elevation angle,

*k*= 0.4 is the von Kármán constant,

*z*

_{0}is the roughness length (Stull 1988), and

*d*is an offset factor to account for the complex terrain. The vertical coordinate

*z*is measured with respect to the ground elevation at HRDL, where

*z*= 0. From the wind speed profiles measured by the met tower,

*z*

_{0}was empirically determined to be 0.01 m for west-northwesterly winds at the site. After a series of trial and error, we rejected the inclusion of stability corrections to the logarithmic profile because they were found to be particularly ill suited for describing the flow at the NWTC. The incorporation of the parameter

*d*essentially allows the model to control the height at which the wind speed goes to zero, since the flow often did not neatly conform to the terrain. As shown in the elevation profile of Fig. 9, the slope along the line of sight from HRDL to the turbine is quite modest, but afterward there exists a sequence of steep gullies. To be clear, the parameter

*d*is traditionally included to account for obstacles, such as trees or buildings, to the flow, but this is not the sense in which it is used here. With

*k*and

*z*

_{0}as fixed constants and

*d*as adjustable parameters, nonlinear regression is used to acquire the best fit to the measured wind speed profile, an example of which is given in Fig. 8b. In the case of the RHI scans, the spatial turbulence intensity is calculated by dividing the RMSD of the fit by the modeled wind speed at hub height, which also corresponded roughly to the average measurement height of the scans. For the example in Fig. 8, RMSD = 1.58 m s

^{−1}and

*u*= 8.24 m s

^{−1}at hub height, such that

*I*

_{space}= 19%.

The distributions of spatial turbulence intensity for the PPI and RHI scans over the duration of the experiment are shown in Fig. 10. The RHI scans feature relatively lower spatial turbulence according to the definition given here—note that the distribution peaks around *I*_{space} = 30% for the PPI scans, and *I*_{space} = 10% for the RHI scans. In the following discussion, low (high) spatial turbulence intensity is defined as being less (greater) than 30% for the PPI scans, whereas *I*_{space} = 10% is taken to be the value separating these two categories for the RHI scans.

### b. WINDCUBE lidar

Positioned about 300 m to the west-northwest of the turbine, the WINDCUBE collected wind speed and direction profiles from 40 up to 200 m above ground level (AGL), depending on atmospheric conditions (Cariou 2011; Aitken et al. 2012). Technical specifications are given in Table 3. WINDCUBE measurements were used to seed and check the accuracy of the wake detection algorithms described in section 3d.

WINDCUBE technical specifications.

### c. Meteorological tower

The met tower is located about 1 km to the west-northwest of the turbine, as shown in Fig. 3. Wind speed and direction data were collected at 2, 5, 10, 20, 50, and 80 m AGL on the met tower using Met One SS-201 cup anemometers and Met One SD-201 wind vanes, respectively. Wind speeds were measured with an accuracy of the greater of 2% of reading or 0.5 m s^{−1}, while wind directions were measured with an accuracy of 3.6°. Additionally, air temperature was measured with an accuracy of 0.1°C using Met One T200A platinum resistance thermometers at 2, 50, and 80 m AGL. Measurements of dewpoint and barometric pressure were also taken at 2 m AGL. At each measurement height, data were collected at 1 Hz and then stored as 10-min averages (Johnson and Kelley 2000). The met tower data and instrumentation documentation are publicly available and can be downloaded (from http://www.nrel.gov/midc/nwtc_m2/).

*g*is gravitational acceleration,

*θ*

_{υ}is the virtual potential temperature difference across a layer of thickness Δ

*z*, and Δ

*u*is the change in horizontal wind speed across that same layer (Stull 1988). Atmospheric stability was determined using calculations of Ri

_{B}between 2 and 80 m AGL, with Ri

_{B}< −0.03, |Ri

_{B}| < 0.03, and Ri

_{B}> 0.03 indicating unstable, neutral, and stable conditions, respectively (see Fig. 11 for the distribution of Ri

_{B}during the experiment). Given the accuracy of the temperature and wind speed sensors, we estimate that, on average, the uncertainty in Ri

_{B}is about 0.02. The stability classifications were chosen to divide the amount of data roughly into thirds and to avoid overlap between the unstable and stable categories arising from the uncertainty in Ri

_{B}. The 2–80-m layer—as opposed to the 2–50- or 50–80-m layers—was chosen to maximize the Δ

*θ*

_{υ}measurement (and therefore minimize the uncertainty in Ri

_{B}) and because measurements of conditions near the surface are important for characterizing atmospheric stability.

### d. HRDL data processing

HRDL LOS velocity measurements were disregarded when the corresponding SNR was less than −15 dB. In addition, for each scan, measurements that did not lie within three standard deviations of the median were identified as outliers and removed from the analysis (W. A. Brewer 2011, personal communication). Most outliers were the result of hard target strikes—such as the laser beam hitting the wind turbine tower or rotor blades—or signal drop-off. Expanding upon the work of Bingöl et al. (2010) and Trujillo et al. (2011), a set of statistical models (Aster et al. 2013) is developed below to extract various wake attributes from the HRDL measurements using weighted nonlinear regression, in which observation weights are specified using the Cramér–Rao lower bound, as discussed previously.

#### 1) One-dimensional PPI algorithm

As depicted in Fig. 12, we define a coordinate system, centered at HRDL, with longitudinal axis *r*, *θ*), with *θ* > 0 (*θ* < 0) for clockwise (counterclockwise) rotations.

*u*and direction

*φ*, an angle with the same conventions as

*θ*. Because the angle between the HRDL LOS and the vector describing the flow field at a given point is |

*θ*−

*φ*|, the HRDL-measured LOS velocity is the actual wind speed modified by the function cos(

*θ*−

*φ*), which can be rewritten asThe wake itself is modeled as either a single- or symmetric double-Gaussian function subtracted from the uniform background flow, in an attempt to account for the difference between the shape of the VD profile in the near and far wakes. For each sweep of the beam, and at each range gate

*r*, three models were fit to the LOS velocity data to identify the wake, if any: a wake-free model,a single-Gaussian wake model,and a double-Gaussian wake model,In both Eqs. (7) and (8),

*a*denotes the amplitude of the Gaussian and

*s*

_{w}is a parameter controlling the width of the wake. In Eq. (7),

*y*

_{c}denotes the location of the wake center, whereas

*y*

_{l}and

*y*

_{r}are used to distinguish the locations of the left and right local minima in Eq. (8), respectively. For a given range gate

*r*,

*F*test was used to determine the simplest model (i.e., the model with the least number of parameters) to fit the data, in which the threshold

*p*value was set to 0.05. If the

*p*value was less than 0.05, then the simpler model was rejected and the more complicated model was deemed to fit the data significantly better (Kleinbaum et al. 2007).

*u*

_{wake}=

*u*−

*a*, so the velocity deficit is given byIn addition, the single-Gaussian wake width is defined to be the size of the 95% confidence interval of the velocity deficit profile,as mentioned in section 2c. On the other hand, for the double-Gaussian wake, VD is calculated by taking

*u*

_{wake}to be the minimum modeled wind speed inside the wake, while the wake width is calculated as

Using the subscript 0 to denote initial coefficient estimates, seeding for the regression algorithm proceeded for each range gate *r* as follows: *φ*_{0} was set to correspond to the median WINDCUBE-measured wind direction at hub height (80 m) over the duration of the beam sweep (~20 s), *u*_{0} to the median HRDL-measured LOS velocity for the range gate of interest, *a*_{0} to the difference between *u*_{0} and the minimum HRDL-measured LOS velocity, *y*_{c0} to the *y* coordinate of the minimum HRDL-measured LOS velocity (in the double-Gaussian case *y*_{l0} and *y*_{r0} were set to the *y* coordinates of the two smallest HRDL-measured LOS velocities), and *s*_{w0} to 0.25D (a value corresponding to a wake width of one rotor diameter).

An example model fit at a distance of 2D behind the turbine is shown in Fig. 13, in which the parameters were estimated as *φ* = −26.6°, *u* = 14.5 m s^{−1}, *a* = 7.2 m s^{−1}, *y*_{c} = −0.99*D*, and *s*_{w} = 0.42, such that VD = 49.6% and *w* = 1.7*D*. Furthermore, the ability of the one-dimensional PPI algorithm to capture the meandering of the wake centerline is demonstrated in the top panel of Fig. 14, in which the black line indicates the estimates of *y*_{c} for all range gates included in the scan.

#### 2) Two-dimensional PPI algorithm

*u*and direction

*φ*, as before. Letting

*d*

_{HRDL}= 880

*m*= 8.8

*D*be the distance between HRDL and the wind turbine, we define a new unprimed coordinate system related to the primed coordinate system of the previous section through the following transformation:Note that the unprimed coordinate system is centered at the turbine and rotated from the primed system by the angle

*φ*, such that, ignoring yaw misalignment and wake meandering, the

*x*axis is aligned with the wake centerline (see Fig. 12). As before, the wake is represented as having a Gaussian profile, with the velocity deficit and width following the power laws given bywhere VD is written as a decimal and not a percentage, and VD

_{0},

*s*

_{0},

*n*, and

*m*are parameters. Similar to the one-dimensional PPI case, two models were fit to the LOS velocity data to identify the wake, if any: a wake-free model,and a wake model,Note that

*x*and

*y*, since the primed coordinates are related to the unprimed coordinates via the inverse of Eq. (13), written asso that

*x*and

*y*are the independent variables in Eq. (17). As before, an extra sum-of-squares

*F*test was used to determine the most appropriate model to fit the data, with the threshold

*p*value = 0.05.

As in the one-dimensional PPI model, for each beam sweep, *φ*_{0} was set such that the wake centerline was aligned with the median WINDCUBE-measured wind direction at 80 m during the duration of the sweep, and *u*_{0} was set to the median HRDL-measured LOS velocity. The other initial coefficient estimates were VD_{00} = 0.56, *s*_{00} = 0.33*D*, *n*_{0} = −0.57, and *m*_{0} = 0.33, values corresponding to the fits of previous wake measurements from section 2. An example model fit is shown in Fig. 14b, in which the parameters were estimated as *φ* = −28.3°, *u* = 15.6 m s^{−1}, VD_{0} = 0.81, *s*_{0} = 0.37*D*, *n* = −0.79, and *m* = 0.37. While of course not capturing any wake meandering, the model reasonably estimates the average ambient wind speed as well as the gross features of the wake.

#### 3) RHI algorithm

*r*denotes the HRDL range gate, while

*δ*specifies the HRDL elevation angle. Similar to the one-dimensional PPI case, the wake is modeled as either a single- or symmetric double-Gaussian function subtracted from the background flow, which is taken to follow the logarithmic profile of Eq. (3). For each sweep of the beam, and at each

*r*, three models were fit to the LOS velocity data to determine the structure of the wake, if any: a wake free model,a single-Gaussian wake model,and a double-Gaussian wake model,Similar to the PPI models,

*a*denotes the amplitude of the Gaussian and the parameter

*s*

_{h}controls the height of the curve. In the single-Gaussian case,

*z*

_{c}denotes the vertical location of the wake center, whereas

*z*

_{l}and

*z*

_{u}are used to distinguish the locations of the lower and upper local minima for the double-Gaussian case, respectively. Here,

*z*is the independent variable, whereas all other variables appearing on the right-hand sides of Eqs. (19)–(21) are parameters, with the exception of

*k*and

*z*

_{0}, which are fixed. An extra sum-of-squares

*F*test was used to determine the simplest model to fit the data, again with the threshold

*p*value = 0.05.

For each beam sweep, the initial estimate for *d*_{0} was set equal to −10 m, which corresponds roughly to the elevation of the terrain at the turbine relative to HRDL. Furthermore, *a*_{0} was set to the difference between the median and minimum HRDL-measured LOS velocities, *z*_{c0} to *H* = 0.8*D* = 80 m (in the double-Gaussian case, *z*_{l0} and *z*_{u0} were set to *H* − 0.25*D* = 0.55*D* = 55 m and *H* + 0.25*D* = 1.05*D* = 105 m, respectively), and *s*_{h0} to 0.25*D* = 25 m.

*u*

_{wake}=

*u*

_{ambient}−

*a*, so the velocity deficit is given byIn addition, the single-Gaussian wake height is defined to be the size of the 95% confidence interval of the velocity deficit profile, usingOn the other hand, for the double-Gaussian wake, the velocity deficit is calculated by taking

*u*

_{wake}to be the minimum modeled wind speed inside the wake, while the wake height is defined asThe effective vertical location of the wake center in the double-Gaussian case is taken to be

*z*

_{c}= (

*z*

_{l}+

*z*

_{u})/2.

An example model fit at a distance 1*D* behind the turbine is shown in Fig. 16, in which the parameters were estimated as ^{−1}, *d* = 35.9 m, *a* = 8.10 m s^{−1}, *z*_{c} = 40.4 m = 0.505*H*, and *s*_{h} = 38.4 m, such that VD = 70% and *h* = 154 m = 1.54*D*. Note that *z*_{c} is normalized by *H*, since the maximum velocity deficit is expected to occur near hub height, and *h* is normalized by *D*, to facilitate comparison to the wake width.

#### 4) Model acceptance criteria

Using the above-mentioned algorithms, wake parameters were determined for every beam sweep in the HRDL dataset. In addition to the extra sum-of-squares *F* test used to find the best-fit models, some models were rejected for having unusual parameter estimates. If, for a particular model fit, any parameter or confidence interval fell outside three standard deviations of the respective median value, then the fit was deemed an outlier and disregarded. Moreover, model fits with unphysical parameter estimates were also eliminated from the analysis. Specifically, in both the one-dimensional PPI and RHI algorithms, a wake is necessarily characterized by 0 < *a* < *u*_{ambient} because 0% < VD < 100% by definition. Because of unusually large ambient variability, undetected hard target strikes, or signal dropout, the algorithms could occasionally determine *a* to be outside of the valid range; such unphysical model fits were excluded from consideration.

We note here that the probability of detecting a wake is a complex function of the amplitude of the velocity deficit, the spatial turbulence, the wind direction, and the extent of the wake boundary. A thorough Monte Carlo simulation would be needed to properly quantify how each parameter influences wake detection. While falling outside our intended scope here, this would likely be an important area of study for future research.

## 4. Results

### a. Velocity deficit profile

A total of 5971 (3926) beam sweeps were evaluated using the one-dimensional PPI (RHI) algorithms. Figures 17a and 17b show, as a function of downwind distance, the percentage of scans for which a physical wake was modeled with sufficient goodness of fit—as determined by the *F* test—by these respective algorithms. (As an aside, we note here that the two-dimensional PPI algorithm detected a wake in about 20% of the scans.) In both cases, the number of detected wakes decreased with downwind distance, as the velocity deficit scaled more with the variability in the ambient flow. The scans, moreover, eventually sampled the flow outside the wake at longer range gates due to the geometrical constraints mentioned earlier, so wakes were detected on average out to about *x* = 7D, although longer wakes were certainly identified on an individual basis. Surprisingly, the PPI algorithm detected far more single-Gaussian than double-Gaussian wakes even at short distances just behind the turbine, suggesting that the azimuth scan rate was slightly too fast to resolve the double-Gaussian structure of the near wake in many cases. This deficiency could perhaps be corrected in a future study by slowing the PPI scan azimuth rate, such that more measurements are taken in a single beam sweep. Because the beam azimuth had to be more or less aligned with the wake centerline for an RHI scan to capture a meaningful portion of the wake, the RHI algorithm detected relatively fewer overall wakes than the PPI algorithm. With finer angular resolution, however, the RHI scans seem to do a better job of detecting the double-Gaussian shape in the near wake. In general, the ratio of single- to double-Gaussian wakes was measured to generally increase with downwind distance, as turbulent mixing caused the two troughs in the initial velocity profile to later merge into a single trough. We do not observe a well-defined demarcation of the near and far wakes, and a small number of double-Gaussian wakes are detected at longer range gates. Again, we expect that the shape of the velocity deficit profile in both the near and far wakes would be better resolved by slowing the lidar scan rate.

### b. Velocity deficit attenuation

In Figs. 18–20, velocity deficit is plotted as a function of downwind distance, as determined by the one-dimensional PPI, two-dimensional PPI, and RHI algorithms, respectively. Here, the shaded error bars in each plot encompass both 1) inherent interindividual variability in the parameter estimates, since lidar scans were taken under a range of wind speeds, turbulence levels, and stability conditions; and 2) error in the estimation procedure itself. This stipulation is true of the error bars in later figures, as well. In general, the deficit was found to decrease, as faster-moving air was entrained within the wake, as expected. Notably, significant velocity deficits were still apparent even as far as (6−7)*D* behind the turbine. The RHI scans measured a somewhat lower velocity deficit than the PPI scans, presumably because the RHI scans did not transect the central part of the wake as often.

Despite significant overlap between subgroups when categorizing the results by ambient conditions, it is perhaps instructive to qualitatively examine the median results. As seen in Figs. 18–20b, wind speed—as measured at 80 m by the WINDCUBE—had the most pronounced effect on controlling the magnitude of the velocity deficit, especially in the near wake, with differences of 10%–20% between region 2 (below rated power; 4 < *u* < 12 m s^{−1}) and region 3 (at rated power; *u* > 12 m s^{−1}) of the power curve. The initial velocity deficit is a function of the amount of momentum extracted by the turbine from the ambient flow, and therefore of the turbine thrust coefficient, which is usually nearly constant for wind speeds below rated and decreases with wind speed above rated (Emeis 2013). In the case of turbulence (Figs. 18–20c), there are two competing influences: higher turbulence levels should 1) cause the velocity deficit to recover more quickly but 2) preclude the detection of relatively small velocity deficits. These two effects seem to more or less cancel out. The influence of stability on the rate of velocity deficit attenuation (Figs. 18–20d) is similarly difficult to discern from the results here, likely because of the physical setup at the site: typically, flow originating at a glacier passes down a mountain over a forest and then it spends just a few minutes of transit time over some rolling grassland before landing at the NWTC. The effect of local stability on the flow is probably rather weak in this situation.

Simply to check the accuracy of the ambient wind direction and speed estimates by the two-dimensional PPI algorithm (and perhaps to inspire confidence in the estimated wake parameters), the corresponding WINDCUBE measurements at hub height are compared in Fig. 21. Here, an individual dot represents the measurements for a single beam sweep. The abscissa is the average WINDCUBE measurement over the duration of the sweep (~20 s), while the ordinate is the corresponding parameter, as estimated by the algorithm, for that sweep. Given the algorithm’s assumption of flow uniformity and, moreover, the spatial separation between the different regions of flow sampled by HRDL and the WINDCUBE, the parameter estimates agree quite well with the WINDCUBE measurements: the correlation coefficient for wind direction (speed) is 0.90 (0.79).

### c. Wake boundary expansion

Figures 22–24 illustrate the growth of the wake boundary with downwind distance, as determined by the one-dimensional PPI, two-dimensional PPI, and RHI algorithms, respectively. Because of the presence of the ground, the expansion of the wake in the vertical direction was very slight and certainly much less pronounced than in the horizontal. Median wake width observations agree well with previous experiments. It is interesting to note that the wake width prediction by the industry-standard Park model (Barthelmie et al. 2006)—using the onshore value for the wake decay constant *k* = 0.075—is not in good agreement with our median results nor those of previous field experiments. Although wind speed did not seem to exhibit much influence on the expansion of the wake, higher turbulence levels and unstable conditions modestly increased the rate at which the wake width expanded. On the other hand, turbulence and stability did not have as much impact on the wake height, because the wake was not able to expand much at all in the vertical direction.

### d. Vertical wake structure

Figure 25 shows the estimate of the terrain offset by the RHI algorithm decreasing with distance behind the turbine and following the average overall slope of the terrain. The elevation of the actual terrain was slightly underestimated by the model; from a series of simulated RHI scans, we found that the model underestimated (overestimated) the elevation of the ground when the wind shear was low (high). Here, the terrain elevation was underestimated on average because the median wind shear exponent, as measured by the WINDCUBE, was just 0.085.

The vertical location of the wake center relative to the base of the turbine, as determined by the RHI algorithm, is shown in Fig. 26. As expected, the maximum velocity deficit occurred very close to hub height immediately behind the turbine. Similar to the observation in Trujillo et al. (2011), we measured an upward shift in the center of the wake deficit with downwind distance, which we attribute to a superposition of the effects on the flow by the tilt of the rotor and the grade of the terrain. With the upward rotor tilt being 6° and the average downward slope of the ground being less than 1°, the result was a net upward shift of the wake centerline.

## 5. Summary and conclusions

A set of statistical models has been developed for the characterization of wind turbine wakes from scanning remote sensor measurements, with particular reference here to the TWICS experiment performed at the National Wind Technology Center near Boulder, Colorado, using long-range Doppler lidar. The models are parameterized to discern the dependence of various wake features—such as the velocity deficit, the lateral and vertical dimensions of the wake, and the horizontal and vertical locations of the wake centerline—on variations in atmospheric conditions. TWICS observations, which were classified according to ambient wind speed, turbulence, and atmospheric stability, agree well with those from previous field experiments. Initially about 50%−60% immediately behind the turbine, the velocity deficit decreased with downwind distance to a value of 15%−25% at *x* = 6.5*D*. The wake also expanded as it moved downwind of the turbine, albeit less so in the vertical direction because of the presence of the ground: initially the same size as the rotor immediately behind the turbine, the extent of the wake swelled to 2.7*D* in the horizontal at *x* = 6.5*D*, but only to 1.2*D* in the vertical at the same distance. Although the ambient wind speed was not seen to significantly influence the expansion of the wake, the difference in velocity deficit between regions 2 and 3 of the power curve was found to be 10%−20% on average, depending on downwind distance.

Improved understanding of wake formation and propagation is essential for optimizing wind turbine layouts, and in turn, power production and profits at wind farms. Although turbines are conventionally spaced at about (6−10)*D* along the prevailing wind direction and (1.5−3)*D* along the crosswind direction (Ammara et al. 2002), computational studies based on large-eddy simulation have found that optimal turbine spacing—in terms of power production and overall cost—may be considerably higher, on the order of 15*D* (Meyers and Meneveau 2012). Still, comparison with data from field tests is necessary to verify and improve such CFD models. Accordingly, in a future study, we plan to validate CFD-based numerical methods nested within the Weather Research and Forecasting Model (WRF) using the experimental data from TWICS.

Whereas the approach developed here can be generally applied to extract wake characteristics from both scanning remote sensor datasets and CFD model output, we would like to emphasize that the specific quantitative results presented in this paper are unique to the turbine, site, and inflow conditions considered herein. To expand upon these results and those of previous studies, additional wake experiments will need to be conducted for a variety of locations and turbine models. In particular, long-range Doppler lidar units would ideally be mounted on the nacelles of utility-scale turbines in future field tests, as was the ZephIR (maximum range = 200 m) on a 95-kW machine in Bingöl et al. (2010) and Trujillo et al. (2011). Such an arrangement is advantageous because PPI scans can sample the wake at a zero elevation angle and, moreover, RHI scans would more often transect the wake centerline because the lidar yaws, along with the turbine, into the direction of the ambient flow. Preferably, these experiments would be conducted in low-turbulence environments, where the assumption of ambient flow uniformity would apply and where velocity deficits in the wake would scale with ambient variability farther behind the turbine. In addition, the lidar scan rate should likely be reduced to better resolve the shape of the velocity deficit profile and therefore to determine a clearer distinction between the near and far wakes. The procedure would ideally incorporate dual Doppler lidar to map the vector, and not LOS, velocities in the flow field (Newsom et al. 2005). Future researchers may also wish to include additional features in the statistical wake models, such as the speed up around the edge of the wake boundary resulting from flow blockage by the rotor (Bingöl et al. 2010; Rajewski et al. 2013) and/or stability corrections to the logarithmic vertical wind speed profile (Stull 1988). The basic procedure developed here is well suited to quantify wake characteristics from experiments that incorporate these suggested modifications.

## Acknowledgments

It is our distinct pleasure to thank Alan Brewer, Raul Alvarez, and Scott Sandberg at NOAA for their efforts in collecting, processing, and interpreting the HRDL data. Much credit goes to Kelley Hestmark, a recent graduate of the University of Colorado Boulder, for her assistance with the initial data analysis. We are also immensely grateful for the help of Michael Ritzwoller and Balaji Rajagopalan at the University of Colorado Boulder, who both graciously offered indispensable advice regarding parameter estimation and inverse problems. Many thanks also to Andrew Clifton at NREL for his encouragement and valuable suggestions, to Neil Kelley for helping design the experiment, and to Jeffrey Mirocha at Lawrence Livermore National Laboratory for technical management of the project. We greatly appreciate the staff of the turbine manufacturer for its facilitation of observational periods. This work benefited greatly from anonymous reviewer comments and was sponsored by the U.S. Department of Energy’s Wind and Hydropower Technologies program, under the direction of the Office of Energy Efficiency and Renewable Energy.

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