As part of the Crop/Wind-energy EXperiment 2011 (CWEX-11) (Rajewski et al. 2013), two vertically profiling Doppler wind lidars (Windcube V1, described in Courtney et al. 2008) were deployed within an operating wind farm in the agricultural fields of central Iowa, USA (Fig. 1). Historical data indicate that this region often experiences strong southerly winds (Fig. 2), and so the lidars were sited north and south of a turbine to intentionally sample turbine inflow and wakes during southerly flow. Except during a brief intercomparison period, one lidar (CU1) was located approximately 165 m south (\(2.2D\)) of a row of six modern multi-megawatt wind turbine generators (WTG) placed in a line running from west to east; the second lidar (CU2) was located 250 m north (\(3.4D\)) of the WTG row. In addition to the lidars, other equipment interrogated the effects of turbine wakes on the agricultural crops in the vicinity, including an array of two surface-flux stations south and north of the wind-turbine row, and an Integrated Surface Flux System (ISFS) south of the turbine row and an additional three ISFSs north of the turbines (NCAR ISFS 2012). Surface-flux data were recorded for the duration of the lidar operational period; these data are discussed in Rajewski et al. (2013). To focus on the turbine wakes specifically, only the lidar data are discussed here; future work will explore the impact of the wakes on surface quantities.
The WTG observed in this study is a GE 1.5 SLE, which has an 80-m hub height and 74-m rotor diameter extending from 43 to 117 m above ground level (a.g.l.). The turbine begins to rotate at a cut-in speed of \(3\,\text{ m } \text{ s }^{-1}\), below which no power is produced. Electrical power production reaches a maximum at \(14\,\text{ m } \text{ s }^{-1}\), the rated speed for the turbines. At speeds \(>\) the rated speed, power production remains constant with increasing wind speed. At the cut-out speed of \(25\,\text{ m } \text{ s }^{-1}\), the turbine ceases rotation.
The row of WTGs pictured in Fig. 1 is located at the southern end of a larger utility-sized wind farm at an elevation of approximately 335 m above sea level. The landscape surrounding the study site consists of soybean and corn agricultural fields, with corn as the primary crop surrounding the lidars and WTGs. Small farms and homesteads interrupt the upwind fetch, with the closest homesteads approximately 600 m to the north-west and south–south-east from the lidars. A few metres north of the turbines and running parallel to the row of wind turbines, a 10-m-wide gravel access road connects the WTGs. The lidar observational period began on 30 June and concluded on 16 August 2011. Approximate sunrise occurred between 0430 and 0515 local standard time (LST) while sunset ranged from 1915 to 2000 LST.
The lidar system records the radial velocity of boundary-layer aerosols at a half opening angle \(\phi \) (approximately 30\(^{\circ }\) from vertical) in each of the four cardinal directions once per second. Line-of-sight velocities at each of the lidar’s ten range gates are converted to zonal, meridional, and vertical wind speeds at each height assuming flow homogeneity throughout the volume scanned by the lidar. (The impact of flow inhomogeneities on the measurements is discussed in Sect. 2.1.) The lidars record horizontal and vertical components of wind speed every second for each specified height; these components are then averaged for a 2-min period to quantify horizontal and vertical components of the flow and the variances of those quantities. Any data that do not meet the carrier-to-noise ratio threshold of \(-\)22 dB are omitted from the recorded 2-min average. Wind shear, directional shear, horizontal turbulence intensity, vertical turbulence intensity, and a form of turbulent kinetic energy are calculated based on the 2-min data from the measured wind-speed components, wind direction, wind-speed variance, and measurement height. Periods of precipitation (as measured at the local ISFS stations) are omitted due to potential lidar signal contamination (Aitken et al. 2012).
Lidar Observations of Inhomogeneous Flow
In the CWEX domain, the lidar observations of flow into the turbine can be assumed to be homogeneous across the measurement volume. However, observations in the wake region likely incorporate inhomogeneous flow, and so the uncertainty of lidar velocity measurements in such flow must be assessed. The size of the volume sampled by the lidar varies with the height \(h\) of the measurement. For the Windcube v1 in the present campaign, \(h\) ranged from 40 to 220 m a.g.l. With a half-opening angle \(\phi \) of approximately 30\(^{\circ }\), the line-of-sight measurements at height \(h\) are collected over a horizontal extent of \(2h\) sin \(\phi \) (or approximately \(h\)). The effective probe length of the Windcube v1 is 18 m. Therefore, at an altitude of 40 m, the Windcube velocity measurements collected over a 4-s period represent a volume 40 m in the horizontal, 18 m in the vertical, and centered at 40 m above the surface. Similarly, at an altitude of 100 m, the Windcube velocity measurements collected over a 4-s period represent a volume 100 m in the horizontal, 18 m in the vertical, centered at 100 m above the surface.
At hub height (here, 80 m elevation), a turbine wake has a horizontal extent or width the size of the rotor disk (approximated here as 80 m); at either 60 or 100 m a.g.l., the wake width would be approximately 70 m. Very few measurements of wake expansion have been documented, but in the offshore wind farm Horns Rev, wakes have been observed to expand by 5–10\(^{\circ }\) as they move downwind (Barthelmie et al. 2010). With a 5\(^{\circ }\) wake expansion, the wake region at 80 m a.g.l. would expand to approximately 124 m wide by the time the wake reaches the downwind lidar located at \(3.4D\) (250 m) downwind, so that the wake will encompass the entire Windcube sampling volume at that elevation. Similarly, the wake at 100 m above the surface would expand to 114 m width at a location \(3.4D\) downwind of the turbine, and so the wake would again envelop the Windcube sampling volume. Therefore, it is reasonable to assume that for southerly flow, the downwind lidar samples turbine wake at all altitudes 100 m and below. The measurement volume of the downwind lidar may exceed that of the wake itself at 120 m a.g.l., and so information on the transverse component of the flow in the wake at the top of the rotor disk cannot be collected with certainty using the experimental design here. However, the estimates of the streamwise velocity would be collected within the wake. (We are hopeful that estimates of wake expansion will become more precise based on observational studies in the future, particularly studies using scanning Doppler lidar (Käsler et al. 2010; Iungo et al. 2013; Smalikho et al. 2013) or radar (Hirth and Schroeder 2013).)
It is useful to quantify wake characteristics throughout the rest of the rotor disk, 100 m and below, recognizing that the measurements are likely sampling inhomogeneous flow. Bingöl et al. (2008) quantified lidar measurement uncertainty due to inhomogeneous flow for the special case of the mean velocity field varying linearly across the measurement volume defined by the half-opening angle \(\phi \) (nominally 30\(^{\circ }\) for the Windcube v1). They found the lidar measurements of the zonal (\(u\)), meridional (\(v\)) and vertical (\(w\)) velocity components at measurement altitude \(h\) have an uncertainty given by a function only of the variation of the vertical velocity \(w\) as it varies in the zonal (\(x\)), meridional (\(y\)), and vertical (\(z\)) components:
$$\begin{aligned} u_\mathrm{lidar} (h)&= u(h)+h\frac{\partial w(h)}{\partial x},\end{aligned}$$
(1)
$$\begin{aligned} v_\mathrm{lidar} (h)&= v(h)+h \frac{\partial w(h)}{\partial y},\end{aligned}$$
(2)
$$\begin{aligned} w_\mathrm{lidar} (h)&= w(h)-\frac{h}{2\cos \left( \varphi \right) }\text{ tan }^{2}(\varphi )\frac{\partial w(h)}{\partial z}. \end{aligned}$$
(3)
In inhomogeneous flow, then, the uncertainty of velocity measurements is a function of the variation in vertical velocity across the horizontal extent (\(x\) or \(y\)) of the measurement volume. Note that, although variations in the horizontal velocities are permitted in this model, by continuity those terms may be replaced with variations in the vertical velocity as shown in Bingöl et al. (2008). The horizontal extent (\(x\) or \(y\)) of the measurement volume is, by virtue of \(\varphi \approx 30^{\circ }\), approximately \(h\), simplifying (1)–(3) after discretization of the partial derivatives to
$$\begin{aligned} u_\mathrm{lidar} (h)&= u(h)+\Delta w(h),\end{aligned}$$
(4)
$$\begin{aligned} v_\mathrm{lidar} (h)&= v(h)+\Delta w(h),\end{aligned}$$
(5)
$$\begin{aligned} w_\mathrm{lidar} (h)&= w(h)-\frac{h}{3\surd 3}\frac{\Delta w(h)}{\Delta z}, \end{aligned}$$
(6)
where \(\Delta z\) is the effective probe length of the Windcube, 18 m (Courtney et al. 2008). Therefore, uncertainties in the lidar estimates of the horizontal velocity components are on the order of the vertical velocity variation within the wake. Uncertainties in the vertical velocity measurements are a function of the magnitude of the vertical velocity and \(h/(3\surd 3 \Delta z)\) where the denominator \({\approx }94\,\text{ m }\). To quantify uncertainty of lidar measurements in inhomogeneous flow, then, it is important to quantify the variation of vertical velocity in a wake.
Using two Doppler lidars, Iungo et al. (2013) report observations of the horizontal and vertical velocity components in the wake of a 2.3 MW turbine, which is larger than the turbine studied here. They find “the mean vertical velocity is shown to be roughly negligible for all the tested downstream locations,” consistent with the modeling studies of Porte-Agel et al. (2011) and Wu and Porté-Agel (2011). Close inspection of the figures of Iungo et al. (2013) suggests a variation of vertical velocity in the wake of less than \(0.7\,\text{ m } \text{ s }^{-1}\). Therefore, according to Eqs. 4 and 5, we expect an error of \({<}1\,\text{ m } \text{ s }^{-1}\) in the horizontal wind-speed measurements within the wake reported herein, for measurements at heights 100 m and below. More detailed quantification of lidar uncertainty, perhaps using detailed computational fluid dynamics simulations, may be possible.
Lidar Intercomparison
The two lidars were co-located during the first two days of the 2011 observational campaign to quantify any bias in wind-speed measurement between the two lidar units. Both units were sited at the CU1 site (Fig. 1) separated by approximately 3 m. During the intercomparison, wind speeds were less than \(20\,\text{ m } \text{ s }^{-1}\) and wind direction was mostly from the south–south-west, unaffected by any turbines. 10-min averaged wind speeds from both units compare well (Fig. 3): the slope is near unity and the y-intercept of the ordinary least-squares best fit is close to zero. A high coefficient of determination, 0.998, shows a strong correlation between the data from each lidar. \(R^{2}\) values between the lidar units exceeded 0.997 when comparing wind speed and direction at all heights within the rotor disk. Both wind-speed and wind-direction data are highly correlated between the two lidars at all heights during the intercomparison. We conclude there was no detectable bias in wind speed between the two instruments. After the intercomparison period, the lidars were located at their respective locations shown in Fig. 1 to capture both upwind and wind-turbine wake data.
Wake Definition
During the CWEX observational campaign, wind direction was frequently (but not always) from the south (Fig. 2). Using the hub-height wind speed and wind direction as measured by the upwind lidar (CU1), we define periods when the northerly lidar (CU2) likely sampled the turbine wake. Requirements included a CU1 wind speed \({>}3\,\text{ m } \text{ s }^{-1}\) (to ensure turbine operation). A wind direction requirement based on the wake expansion of 5–10\(^{\circ }\) observed by Barthelmie et al. (2010) was calculated: wind directions between \(167^{\circ }\) and \(195^{\circ }\) should place a wake from turbine B3 (Fig. 1) directly over the lidar at CU2 so that all four beams from the lidar sample wake. Other wind directions produce wakes from other turbines; the analysis here incorporates only wakes from turbine B3, located directly between the two lidars.
In excess of 4,000 10-min time periods are available for analysis; more than 600 time periods (6,000 min) meet the criteria based on wind speed, wind direction, and precipitation for a wind-turbine wake detected by the lidar. Approximately half of the data presented here (55 h) were collected between the hours of 0700 and 1900 LST, providing an even distribution of daytime and nighttime conditions.
Quantities Observed
From the 2-min estimates of the two horizontal components (\(u, v\)) and the vertical component (\(w\)) of wind velocity, as well as the variances of these quantities over the 2-min period, several useful quantities can be calculated. 10-min averages of these quantities are presented in the figures below.
Previous investigations have observed enhanced turbulence intensity in the wake. Turbulence intensity is calculated from the variances (\(\sigma ^{2}\)) of the \(u\) and \(v\) components of the flow as in Eq. 7,
$$\begin{aligned} I=\frac{\sqrt{\sigma _u^2 +\sigma _v^2 }}{U}, \end{aligned}$$
(7)
where \(U\) is the mean horizontal wind speed (Stull 1988) at the level at which the velocities are observed. (Note that some investigators focusing on wind-tunnel studies, such as Chamorro and Porté-Agel (2009), normalize turbulence intensity with hub-height wind speed rather than the wind speed at the altitude of the measurement.) In the wind energy industry, turbulence intensity is usually calculated over a 10-min period, although it is likely that other averaging times are more appropriate for capturing all the energetic length scales of turbulent fluctuations (Mahrt 1998).
These variances may also be used to calculate a quantity approximating turbulent kinetic energy (TKE), a measure of turbulence in the atmosphere (Eq. 8). TKE incorporates both horizontal and the vertical components of flow variability (Stull 1988), and so an estimate of lidar TKE (here \(E)\) can be calculated as,
$$\begin{aligned} E=0.5\left( {\sigma _u^2 +\sigma _v^2 +\sigma _w^2 } \right) . \end{aligned}$$
(8)
In a detailed comparison of tower-based sonic anemometry to ground-based lidar, Sathe et al. (2011) suggest that ground-based lidar systems fail to accurately calculate horizontal and vertical variances as compared to sonic anemometry. Without tower data for comparison here, we do not suggest that the lidar can directly calculate TKE as it would be measured by a sonic anemometer. Rather, we provide a comparison between two similar lidar systems as they quantify fluctuations in the wind as represented by \(E\).
The wind profile power law (Eq. 9) compares wind-speed measurements between two heights \(z_{i}\) and \(z_{i+1}\). The power law is used in the wind energy industry not because it is an accurate portrayal of the true wind profile, but because the resulting coefficient \(\alpha \) captures information about wind shear that is easily comparable between multiple geographic locations (Schwartz and Elliott 2006),
$$\begin{aligned} \frac{U_{i+1} }{U_i }=\left( {\frac{z_{i+1} }{z_i }} \right) ^{\alpha }. \end{aligned}$$
(9)
In neutral stability, \(\alpha = 1/7\) is commonly used for approximations (Brower 2012). This assumption does not hold true for strongly stable or strongly unstable atmospheric conditions or under high wind shear conditions (Walter et al. 2009; Wharton and Lundquist 2012a, b, among others). Large positive values of \(\alpha \) indicate a large increase of wind speed with height; negative values indicate a decrease of wind speed with height. Changes of wind direction with height are not captured by the power-law coefficient. As discussed in the references above, the use of \(\alpha \) to quantify wind shear is not optimal, as it implicitly assumes a logarithmic wind profile that may only be expected in the surface layer or under neutral stratification. Herein it is used only to facilitate comparisons with previous work. Wherever possible, we recommend quantification of the complete wind-speed and wind-direction profiles rather than analysis of only a power-law coefficient \(\alpha \) calculated between two levels.