WESWind Energy ScienceWESWind Energ. Sci.2366-7451Copernicus PublicationsGöttingen, Germany10.5194/wes-2-133-2017Turbulence characterization from a forward-looking nacelle lidarPeñaAlfredoaldi@dtu.dkhttps://orcid.org/0000-0002-7900-9651MannJakobhttps://orcid.org/0000-0002-6096-611XDimitrovNikolayDTU Wind Energy, Technical University of Denmark, Roskilde, DenmarkAlfredo Peña (aldi@dtu.dk)13March20172113315213December201619December201617February201721February2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://wes.copernicus.org/articles/2/133/2017/wes-2-133-2017.htmlThe full text article is available as a PDF file from https://wes.copernicus.org/articles/2/133/2017/wes-2-133-2017.pdf
We present two methods to characterize turbulence in the turbine inflow using
radial velocity measurements from nacelle-mounted lidars. The first uses a
model of the three-dimensional spectral velocity tensor combined with a model
of the spatial radial velocity averaging of the lidars, and the second uses
the ensemble-averaged Doppler radial velocity spectrum. With the former,
filtered turbulence estimates can be predicted, whereas the latter model-free
method allows us to estimate unfiltered turbulence measures. Two types of
forward-looking nacelle lidars are investigated: a pulsed system that uses a
five-beam configuration and a continuous-wave system that scans conically. For
both types of lidars, we show how the radial velocity spectra of the lidar
beams are influenced by turbulence characteristics, and how to extract the
velocity-tensor parameters that are useful to predict the loads on a turbine.
We also show how the velocity-component variances and co-variances can be
estimated from the radial-velocity unfiltered variances of the lidar beams.
We demonstrate the methods using measurements from an experiment conducted at
the Nørrekær Enge wind farm in northern Denmark, where both types of
lidars were installed on the nacelle of a wind turbine. Comparison of the
lidar-based along-wind unfiltered variances with those from a cup anemometer
installed on a meteorological mast close to the turbine shows a bias of just
2 %. The ratios of the unfiltered and filtered radial velocity variances
of the lidar beams to the cup-anemometer variances are well predicted by the
spectral model. However, other lidar-derived estimates of velocity-component
variances and co-variances do not agree with those from a sonic anemometer on
the mast, which we mostly attribute to the small cone angle of the lidar. The
velocity-tensor parameters derived from sonic-anemometer velocity spectra and
those derived from lidar radial velocity spectra agree well under both
near-neutral atmospheric stability and high wind-speed conditions, with
differences increasing with decreasing wind speed and increasing stability.
We also partly attribute these differences to the lidar beam configuration.
Introduction
Recently, lidars have been mounted on the nacelle of wind turbines to
investigate wake characteristics
and today are extensively used in a forward-looking (FL) mode to scan the
turbine inflow for many purposes. One of such is power-performance
measurements; FL nacelle lidars decrease the statistical uncertainty of the
measured power curve when compared to that based on mast measurements
. The statistical uncertainty associated with load
validation can potentially also be reduced . Another
important use of FL nacelle lidars is turbine control; they have the
potential to reduce loads and increase energy capture
. Irrespectively of the application,
FL nacelle lidars are primarily aimed to characterize the inflow in front of
the turbine. Inflow characterization has been performed using lidars of
different types and configurations for several years
. However, FL nacelle lidars
have the advantage of measuring the inflow in front of the turbines more
“effectively” than other types of lidars because they scan over the area in
front of the turbine and yaw with it. Therefore, they can potentially be used
for measuring the yaw misalignment of wind turbines . If
they become widely applied in the wind-energy industry, they could be used to
characterize wind resources in regions where measurements from meteorological
towers are scarce or non-existent.
Similar to ground-based lidars, there are two main types of FL nacelle
lidars, pulsed and continuous-wave (CW), which mainly differ, for the purpose
of turbulence estimation, on the measurement probe volume and the scanning
strategies (specific details are given later). As with any other Doppler
lidar, they only measure the radial velocity along the laser beam or
line-of-sight velocity. As their measurement probe volumes are generally
larger than those of cup and sonic anemometers, they might not be able to
measure small turbulent eddies, which leads to “filtered” turbulence
estimates; however, as they scan the atmosphere with laser beams in different
directions, there might be contributions (contamination) from different
velocity components that can lead, for some scanning configurations and under
certain turbulence conditions, to turbulence estimates that might be even
higher than those from cup or sonic anemometers. A detailed analysis on how
lidar-based turbulence estimates can be assessed, filtered, and contaminated
is presented in .
Here we use time series of radial velocity measurements from different beams
emitted by a FL nacelle lidar to estimate the turbulence parameters of the
three-dimensional spectral velocity tensor model by
(hereafter the Mann model). This model is chosen because it fits the
atmospheric-turbulence velocity spectra for different surface, wind, and
atmospheric-stability conditions within the first ≈ 100 m from the
ground well and is widely used to perform
aeroelastic simulations of wind turbines. The ultimate objective of this study
is to find out whether nacelle lidars can be used independently (i.e., without the
need of extra measurements, e.g., from instruments on meteorological masts) to
extract turbulence information from the inflow. Nacelle lidars can
potentially infer the inflow characteristics that actually impact the
turbines better than traditional nacelle or mast anemometry because they can
scan over an air volume, which is more representative of the flow entering
the rotor plane. We also use, when possible, information of the Doppler
radial velocity spectrum to estimate the “unfiltered” lidar beam variances
and, from those, we estimate the velocity-component variances and
co-variances .
This paper is organized as follows. In Sect. , we
introduce shortly the characteristics of the wind field, how this is
represented by the Mann model, and how to extract the turbulence
characteristics from velocity spectra. Section shows the
two types of nacelle lidars investigated here, Sect.
illustrates how to derive the radial velocity spectra from the different
lidar configurations and how these spectra are influenced by both the lidar
configuration and the turbulence characteristics of the Mann model, and
Sect. shows how to extract turbulence information from
the lidars' radial velocity spectra. Section
introduces the Nørrekær Enge wind farm and the measurements of the
experimental campaign. Section provides the details on
the way the measurements are analyzed, and Sect. shows the
comparison of turbulence characteristics extracted from nacelle-lidar
measurements and those from sonic- and cup-anemometer measurements. Finally,
we provide some discussion and conclusions in the last two sections.
Turbulence background
The wind field is described by a vector field u(x), where the
time argument is eliminated because Taylor's frozen turbulence hypothesis is
assumed and x is the position vector in space,
x=(x,y,z). The mean value of the homogeneous velocity field is
u(x)=(U,0,0), so the coordinate
x is in the mean wind direction. The wind field
can also be written as a Fourier integral,
u(x)=∫u(k)eik⋅xdk⇔u(k)=1(2π)3∫u(x)e-ik⋅xdx,
where k is the wave vector . The
ensemble average of the absolute squared Fourier coefficients is the spectral
tensor:
ui*(k)uj(k′)=Φij(k)δ(k-k′).
The spectral velocity tensor, Φij, is assumed to be described by the
Mann model, which, besides k, only contains three parameters: αε2/3, L, and Γ, where α is the spectral
Kolmogorov constant, ε the specific rate of destruction of
turbulent kinetic energy, L a length scale related to the size of the
turbulent eddies, and Γ a parameter describing the anisotropy of the
turbulence. From the spectral tensor, the one-point spectra are calculated by
Fij(k1)=∬Φij(k)dk2dk3,
and typically, the three auto-spectra of the u, v, and w components of
the wind velocity, F11, F22, and F33, respectively, together
with the one-point cross-spectrum, F13, are fitted simultaneously to
measured or theoretical spectra in order to obtain the Mann-model parameters
(hereafter referred to as Mann parameters). This procedure is described in
. In order to facilitate the fitting, a two-parameter
look-up table (LUT) with values of Fij(k1)=Fij(k1;αε2/3=1,L=1,Γ) is precomputed. The mathematical identity
Fij(k1;αε2/3,L,Γ)=L5/3αε2/3Fij(k1L;1,1,Γ)
is used to calculate the spectra for arbitrary values of k1, αε2/3, L, and Γ.
Nacelle lidars
Two types of FL nacelle lidars are investigated: a CW and a pulsed lidar. The
lidars are assumed to be mounted close to the center of the rotor with N
beams pointing in different directions (see Fig. ). For the
CW lidar studied here, the beams point on a cone with the symmetry axis
pointing upstream; Fig. 's left panel shows a configuration
with an arbitrary number of beams, N=13, of which 12 beams draw a conical
surface and 1 is perpendicular to the rotor plane. The half opening angle
of the cone is φ. The beams of the CW lidar are focused at some
distance, df. If other measurement planes are required,
refocusing of the laser beam is necessary. For the pulsed lidar studied here
(N=5), the beam directions also form a cone where four positions are within
the conical surface and one is perpendicular to the rotor plane
(Fig. , right panel).
The ith lidar beam points to the direction defined by the unit vector
ni (i=1,…,N). The unit vector can be expressed as
n=-cosφ,sinφcosθ,sinφsinθ, where θ is the angle between the y axis and n
projected onto the y–z plane. For the beam perpendicular to the rotor,
n=-1,0,0. If we assume that the lidars measure at a
point, instead of over a probe volume, and that the u, v, and
w components do not change over the scanned area, the radial velocity of
the lidar beams over the scanned circle can be estimated as
vr(θ)=-ucosφ+vsinφcosθ+wsinφsinθ.
Lidar radial velocity spectra and beam variances
and show an expression for the
spectra measured by a lidar beam
Fvr(k1)=ninj∬ϕ^(k⋅n)2Φij(k)dk2dk3,
where ϕ^ is the Fourier transform of the lidar weighting function
that considers the probe volume. For a CW lidar, this is typically
approximated by
ϕ^(k1)=exp(-|k|zR),
where zR is the Rayleigh length that
can be estimated as
zR=λdf2πrb2,
where λ is the laser wavelength and rb the beam radius at
the output lens. For the pulsed lidar
Geometry of the rotor and nacelle lidars. The x axis is in the
mean wind direction. The lidar beams point upwind in the directions
determined by the unit vectors ni. For the CW lidar (left frame) we
include a beam perpendicular to the rotor for comparison only. For the pulsed
lidar (right frame) we show a five-beam configuration where beam 0 is
perpendicular to the rotor. Beams perpendicular to the rotor are shown in
green (beam 0), top beams in blue and magenta (beams 1 and 2), bottom beams
in red and cyan (beams 3 and 4), and other beams in grey.
ϕ^(k1)=sinc2kzR/2.
In Eq. () zR is not the Rayleigh length as in
Eq. (), but rather half the length of a rectangular pulse
. Despite this discrepancy, we use the same symbol because
zR is the parameter that characterizes both weighting functions.
Notice that Fvr is not a function of df/L because
the turbulence is assumed homogeneous.
Examples of radial velocity spectra of the CW and pulsed lidars calculated
from Eq. () with a half opening angle of φ=15∘,
which are compared with the “ideal” sonic u spectrum, are shown in
Figs. and , respectively
(within the range of wave numbers that we are interested in, sonic anemometers
resolve the u spectrum well). As explained in , the
negative correlation between the vertical and horizontal velocities causes
the variance of the upward (and forward) pointing beam to generally be the
highest of all beams, while the variance is
generally the lowest for the downward
beam. The difference between the downward and upward pointing beam spectra is
smaller than the differences between u, v, and w spectra and
deteriorates with increasing zR/L. This ratio indicates the
amount of filtering of eddies due to the probe volume. We can also see that
for the pulsed lidar the radial spectra of the top beams (1 and 2) are above
the sonic u spectrum for zR/L=0.25, which is due to
contributions from different components of the spectral tensor. Similar
mechanisms can result in a middle beam radial velocity spectrum above the
top beam one, particularly for zR/L≥1.
Sonic and CW lidar velocity spectra from Eqs. ()
and (), corresponding to the beams shown in Fig.
(left panel). Values of zR/L are indicated, φ=15∘,
Γ=3, and αε2/3=0.1 m4/3 s-2.
Sonic and pulsed lidar velocity spectra from
Eqs. () and () corresponding to the beams
shown in Fig. (right panel). Values of zR/L are
indicated, φ=15∘, Γ=3, and αε2/3=0.1 m4/3 s-2.
Figure 's left panel shows the behavior of the ratio of the
lidar beam radial velocity variance, σvr2, to the
variance of the u component, σu2, for a number of zR/L
values and for both types of lidars based on the Mann model with Γ=3.
As expected from the results in Figs. and
, the ratio increases with decreasing
zR/L and for the 0/middle beam of the pulsed/CW lidars,
σvr2/σu2=1 at zR/L=0 as no averaging
due to probe volume occurs. Furthermore, both lidars' top beams variances can be
higher than σu2 for zR/L≈0. Another way to study
the contributions of the different velocity components to
σvr2 is shown in Fig. right panel.
There we illustrate the ratio of the variance of the other two components,
σv2 and σw2, as well as σu2 to
σvr2 as a function of the beam azimuthal position for
zR/L=0. With φ=15∘ and such turbulence
characteristics, we can only measure a portion of σv2 and
σw2, and σvr2≈σu2 at
θ≈11∘/169∘ (also if a middle beam is used, no
matter the turbulence characteristics). For the same turbulence
characteristics as those used in Fig. , if we use a lidar
with φ=60∘, σvr2<σu2 for all
azimuthal positions, whereas σvr2≈σv2 at
θ≈200∘/340∘ and
σvr2≈σw2 at
θ≈237∘/303∘ (not shown). It is also observed
that for the same zR/L value, the averaging by the CW lidar has a
stronger effect on the variance than the pulsed lidar.
Unfiltered lidar radial velocity variance
The unfiltered variance of
the lidar beams, σvr,unf2, can be estimated by using
the information of the instantaneous Doppler radial velocity spectrum.
Following the steps in or
, the ensemble-average Doppler spectrum of the
radial velocity 〈Svr〉 can be assumed to
be equal to the probability density function of vr, i.e., 〈Svr〉=p(vr). This is because the
average of vr along the beam does not change highly with radial
distance, as FL nacelle lidars use a small cone angle and so the velocity
gradient along the probe volume is negligible. Therefore,
σvr,unf2 can be estimated as the second central moment
of p(vr).
Assuming homogeneous turbulence, once σvr,unf2 is computed, the scanning pattern can be used to extract the
velocity-component variances by taking the variance of vr in
Eq. (),
σvr,unf2(θ)=σu2cos2φ+σv2sin2φcos2θ+σw2sin2φsin2θ-2u′w′‾cosφsinφsinθ,
where u′w′‾ is the uw covariance, the primes denote
fluctuations, and the overbar a time average, and we ignore the terms where
u′v′‾ and v′w′‾ appear because these two are usually
small (u fluctuations are in the mean wind direction). In the case of
misalignment of the lidar beams with respect to the wind, because of either
misalignment of the turbine with the wind (yaw misalignment), lidar
misalignment with the turbine, or both, it is not difficult to derive an
expression for σvr,unf2 that accounts for the
misalignment angle β.
Turbulence characterization from nacelle-lidar measurements
The strategy is to calculate theoretical spectra (in the form of a LUT) that
include both the effect of pointing the lidar in the direction ni and of averaging. Then, the measured spectra are fitted to the LUT to get the
turbulence parameters. One can expect this procedure to be unsuccessful for
zR>L, i.e., if the lidar is averaging out most eddies as shown in
Figs. and .
The computational burden of creating a lidar-based LUT using
Eq. () is larger than in the standard case, i.e., using
Eq. (), because Fvr is not only a
function of the two parameters k1L and Γ but also of
zR/L, φ, and θ. Furthermore, lidar beam misalignment
can be an issue. Therefore, we need to add an extra dimension to the LUT
because such misalignment has a large effect on the lidar radial velocity
spectrum.
(Left) Variance of the lidar beams' radial velocity (divided by the
u component variance) of the 2/top, 3/bottom and 0/middle beams of the
pulsed/CW (solid/dashed lines) lidars as a function of zR/L.
(Right) Ratio of the variance of each of the velocity components to that of
lidar beam as function of azimuthal position for zR/L=0.
Turbulence characteristics are computed for φ=15∘ and
Γ=3.
Figure illustrates the effect of misalignment
(β=-2∘) on the pulsed lidar radial velocity spectra for a set
of Mann parameters. The effect of the relatively small misalignment is
noticeable; the spectrum of the beams that become more parallel to the wind
is clearly above that of those that become less parallel at the same height. For
this particular pulsed lidar configuration, misalignment can result in a
similar spectrum for the beams 0 (middle) and 1/2 (depending on the sign of
the misalignment).
Effect of lidar beam misalignment (with respect to the wind) on the
radial velocity spectra of a pulsed lidar for φ=15∘,
Γ=3, αε2/3=0.1 m4/3 s-2,
zR/L=0.5, and β=-2∘.
Site and measurementsSite
The Nørrekær Enge wind farm is located in the Himmerland region in
northern Jutland, Denmark, ≈ 300–400 m south-east of the waters of
Limfjorden (see Fig. ). It comprises 13 Siemens 2.3 MW-93 wind
turbines with hub height of 81.8 m and a rotor diameter D of 92.6 m. They
are aligned on a row at a direction 73.9∘ with the north. The
distance between the turbines is 487 m (5.2 D). A meteorological mast was
located at 101.2∘, at a distance of 232 m (2.5 D) from turbine number
4 (from left to right on the row). The wind farm is located over flat terrain
and the surface is characterized by a mix between croplands and grasslands,
and the fjord to the north. At ≈ 2 km southwest of turbine 4, the
terrain is no longer flat.
Measurements
The measurements here analyzed correspond to the period 27 October 2015 to
7 January 2016. There are three types of measurements: supervisory control
and data acquisition (SCADA) on turbine 4, FL nacelle-lidar measurements from
systems mounted on the nacelle of turbine 4, and meteorological mast
observations. Both lidars were pre-tilted down ≈ 0.30∘ so
that their axes pointed at hub height, at a position 2.5 D from the turbine
for maximum power–performance operating conditions, based on aeroelastic
simulations of the tower bending (A. Vignaroli, personal communication, 2016).
Turbine measurements
For this analysis we use the following SCADA 10 min means of turbine 4: yaw,
power, and turbine and grid status. The yaw and power signals provide
measurements of the position of the turbine and the converted power, and the
grid and turbine status signals show whether the turbine was grid-connected
(yes/no) and available (yes/no).
The Nørrekær Enge wind farm in northern Denmark on a digital
surface elevation model (UTM32 WGS84). The wind turbines are shown in
circles, that with the nacelle lidars in red and the mast in a triangle. The
sector used for the analysis is also indicated. The waters of Limfjorden are
shown in light blue.
Pulsed lidar
A five-beam Avent pulsed lidar (hereafter known as Avent) was mounted on the
nacelle of turbine 4. Ten different ranges were measured simultaneously per
beam position (49, 72, 95, 109, 121, 142, 165, 188, 235, and 281 m). The
beam configuration is exactly as that in the right panel of Fig.
(and we will use the same beam numbering), with φ=15.08∘ and
zR=24.75 m . The lidar accumulated radial
velocity spectra per beam position for 1 s before it moved to the next
beam position; thus, radial velocity time series can be analyzed at 0.2 Hz.
Each radial velocity estimate from the average Doppler spectrum was performed
by the instrument using a maximum-likelihood-estimator algorithm
.
Continuous-wave lidar
A ZephIR dual-mode CW lidar (hereafter known as ZephIR) was also mounted on
the nacelle of turbine 4. Five different ranges were considered (10, 30, 95,
120, and 235 m); for each range ≈ 50 azimuthal positions on the
circle formed with a cone with φ=15.05∘ were measured during
1 s; the system averaged Doppler radial velocity spectra within azimuthal
ranges of ≈ 7.38∘ to get an estimate of the radial velocity
per azimuth by computing the centroid of the average Doppler spectrum
. The system also kept a record of each average Doppler
radial velocity spectrum, which is used here to estimate the unfiltered
variance. The lidar characteristics λ=1.56×10-6 m and
rb=28 mm (M. Harris, personal communication, 2016) can be used to estimate
zR with Eq. (). Each range was sampled three times
before focusing to the next one; thus, radial velocities for the same range
and azimuthal position can be found every ≈ 18 s.
Mast measurements
We use measurements from cup anemometers (P2546A) at 80, 78, and 57 m
height, mounted on 3 m long booms 250∘ from the north; from a 3-D sonic
anemometer (CSAT3) at 76 m on a 2 m boom 190∘ from the north; and a
wind vane (Vector W200P) at 78 m on a 3 m boom 70∘ from the north,
all mounted on the meteorological mast. The mast is an equilateral triangular
lattice structure with a width of 0.4 m at 80 m.
Data analysisData selection and filtering
We analyze the time series of all data and their statistics in 10 min
periods. The total number of 10 min periods available for analysis is 9586.
The next steps are followed in the analysis:
We use the 10 min vane measurements to concentrate the analysis on a
wake-free sector covering the mast location (88.85–238.85∘) that
takes into account the obliquity of the wind farm row and a 15∘ wake
expansion (see Fig. ). A total of 5825 10 min measurements are available for analysis
where both lidars are also working (based on a 10 min status signal of both
lidars) and turbine 4 is grid-connected and available.
The availability of the Avent data is highest at the range 121 m
because this range is the closest to the focusing distance. Therefore, we
focus all our lidar-data analysis at this range, although the mast is at
232 m from turbine 4. Furthermore, when a carrier-to-noise (CNR) filter is
applied to the 5 s time series, the two lowest beams (3 and 4) return fewer
data than the others due to, among other things, obstruction from the blades (the
availability of beam 3 is lower than that of beam 4). A total of 3236 10 min periods
are available for analysis after filtering the 5 s Avent data so that for
each 10 min period there are a minimum of 110 samples for beams 0, 1, 2, and
4 with CNR >-22 dB.
We then extract all radial velocities for all azimuthal positions of
the ZephIR for the range 120 m when no rain was detected by the instrument.
The azimuthal position of the ≈ 50 points over the scanned circle
changes after each revolution. A total of 2590 10 min periods are available for analysis in
which there are a minimum of 4500 radial velocities samples per 10 min
period at the 120 m range.
Finally, we extract the 1 Hz data of the sonic anemometer and cup
anemometer at 80 m, in which there are a minimum of 600 samples per 10 min
period. The final dataset thus contains 2273 10 min samples of concurrent
turbine–lidars–mast data.
Furthermore, each 10 min time series has been post-processed. For the Avent
data, we linearly detrend each radial velocity time series for each beam
before applying a despiking filter, where values above and below 3 standard
deviations from the mean are filtered out. The missing values are then filled
in using linear interpolation. The top left panel of Fig. shows an example of a 10 min time series of the Avent beams' radial
velocity. The solid lines show the final interpolated time series and the
markers show original radial velocities before post-processing.
An example of a 10 min time series of the radial velocity of
different beams for the Avent (top left) and the ZephIR (top right) lidar.
The radial velocities of the two lidars at all azimuthal positions are
illustrated in the bottom-left panel (see text for details). In the
bottom-right panel, we show a comparison of the 2273 10 min mean radial velocities of the Avent (beam 2) and ZephIR (bin 6) with the results of a
linear regression through the origin and coefficient of determination R2.
Measurements from beam 3 of the Avent are omitted due to low availability.
For the ZephIR data, we construct time series of radial velocities at
azimuthal positions similar to those of the Avent. Since the azimuthal
positions of the ZephIR change from revolution to revolution, we extract
radial velocities within azimuthal position bins of 7.2∘ on a fixed
frame of reference. Three of such bins, 43, 6, and 31, are “aligned” with
the Avent beams 1, 2, and 4, respectively. The time series per bin is then
threshold-filtered with a minimum radial velocity of 2 m s-1, and
detrended and despiked as with the Avent data.
The top right panel of Fig. shows the time series per
bin; we include four more bins (0, 12, 18, and 37) than those aligned with the
Avent beams and their positions can be inferred by color coding using the bottom left
panel of Fig. which shows the radial
velocities in a polar plot. In the top panels in Fig.
the effect of despiking is noticeable (the filtered time series are shown in
solid lines and the original are shown in markers), and in the bottom left panel of
Fig. , all the radial velocities
estimated from the Doppler spectrum within the 10 min period at the 120 m
range by the ZephIR are shown. Since the lidars were mounted behind the
rotor, the rotating blades sometimes interfered with the beam and the
estimated radial velocity became the projection of the radial velocity of
the blade onto the beam direction; the result is the figure of eight close to
zero radial velocity shown in the bottom left panel of Fig. . In this latter plot, we also include the radial velocities of the
three Avent beams that are aligned with the ZephIR bin positions. At these
three positions, both lidars show good agreement; a comparison of all 10 min
mean radial velocities estimated by the Avent and ZephIR for one of these
“aligned” positions, beam 2 and bin 6, respectively, is shown in the bottom right
panel of Fig. .
The top right panel of Fig. also shows that it is
possible to get more than one radial velocity value within the same azimuthal
bin (sometimes up to three values). Finally, the ZephIR's time series are
“completed” using linear interpolation.
Comparison of sonic and 80 m cup anemometer statistics: mean wind speed (left frame) and horizontal-wind variance (right frame). Each 10 min
sample is shown in grey markers, a 1 : 1 line is shown for guidance in
black, and the results of a linear regression through the origin and R2
are given.
For each 10 min period, the 1 Hz sonic and cup anemometer data are
detrended and despiked as with the lidar data, and mean and turbulence
statistics are computed. The sonic-anemometer wind-speed components are
rotated so that u is aligned with the mean wind. We estimate the friction
velocity, from the sonic wind speed and temperature fluctuations, as
u*=u′w′‾2+v′w′‾21/4,
and the Obukhov length estimated as
LO=-u*3κ(g/T‾)w′Θv′‾,
where κ is the von Kármán constant (≈ 0.4), g the
gravitational acceleration, T a reference temperature, and
Θv the virtual potential temperature. Spectra of all lidar
radial velocities, sonic-anemometer wind speed components and cup-anemometer
horizontal wind velocity are computed for each 10 min period. All 10 min
turbulence statistics and spectra from the sonic anemometer are also computed
on a 5 and a 18 s basis, mimicking the lidar sampling frequencies.
Sonic-anemometer measurements
When compared to the measurements from the 80 m cup anemometer, the
sonic-anemometer mean horizontal wind speeds are 2.6 % lower (see
Fig. left panel). This bias is higher than 0.6%,
which is the estimation that results from assuming, between the two
instruments' heights, the logarithmic wind profile
U=u*κlnzz0,
where z0 is the roughness length, ≈ 0.012 m, which is a typical
value of these surface conditions . When looking at
variances, the bias is 12 % (Fig. , right panel)
if we use the u component or the combined u and v components for
the estimation of the sonic-anemometer variance. The latter means that for
this site and at this height, the v variance has a low contribution to the
horizontal velocity variance (which is what a cup anemometer does
theoretically measure) and so we could assume the cup-anemometer variance to
give a good estimate of the u variance. On the other hand, the bias between
both instruments' variances cannot be explained simply; a 4 % bias is
expected, assuming the 2 % bias of the mean wind speed.
The behavior of the sonic-derived velocity spectra does not correspond well
with the notion of turbulence local isotropy within the inertial subrange,
where we expect the same spectral density for the v and w components and
the u component is 25 % lower .
Figure shows that within the inertial subrange
the ensemble-average sonic u spectrum (of all 10 min observed spectra) is
indeed ≈ 25 % lower than the v spectrum but so is the
w spectrum. Possible explanations for this are path-averaging errors and
transducer shadowing mainly attenuating the w spectrum measured by the
CSAT3 . Figure also
illustrates the fit to the three auto-spectra and cross-spectrum using the
Mann model (see Sect. ), which shows the expected
behavior within the inertial subrange (k1⪆0.03 m-1 for this
case). The fit is performed on the ensemble-average spectra that have been
logarithmically-averaged on the basis of the wavenumber (we will use such
logarithmically-averaged spectra when fitting Mann parameters). These
“average” Mann parameters (Γ=3.00, αε2/3=0.14 m4/3 s-2, and L=35.38 m) are similar
to those observed at a site with similar surface and turbulence
characteristics , but it should be noticed that these are
the average of spectra for a number of atmospheric and turbulence conditions
and that the Mann-model fitting procedure is normally performed over specific
wind speed, turbulence, or atmospheric-stability ranges.
Power spectrum for different velocity components. The solid lines
show the ensemble-average spectra of all 10 min sonic-anemometer spectra;
the markers, the k based logarithmically-average spectra of all 10 min
spectra; and the dotted lines, a fit to the spectra using the Mann model.
Ensemble-average spectrum of all 10 min Avent radial velocity
spectra (per beam), sonic-anemometer u spectrum, and 80 m cup-anemometer
spectrum. Original (left) and noise-filtered lidar radial velocity
spectra (right).
Due to the uncertainty on the sonic-derived statistics, we will use the
cup-anemometer variance as a proxy for σu2. However, we will use the
sonic-based Mann parameters for comparison with the lidar-based Mann
parameters (and for estimations of σv,w2 and u′w′‾)
because it is the only reference we have for three-dimensional turbulence
measurements.
Undersampling and noise removal
Although the variances of a velocity time series sampled over a 10 min
period at a frequency fs of 0.2 or 0.06 Hz are not statistically
different from those estimated from 1 or 10 Hz records, aliasing and noise
might appear both in the sonic-anemometer and the lidar radial velocity
spectra. The left panel of Fig. shows the Avent
radial velocity spectrum that has been ensemble-averaged from all the 10 min
observed spectra for each of the beams. We conjecture that the increase in
the spectral densities at high frequencies is due to noise.
The right panel of Fig. shows the effect of a noise
filter, which is based on the method by , on the
ensemble-average Avent radial velocity spectra.
The noise filter seems to recover the shape of the Avent radial velocity
spectra. However, when tested on the 18 s sonic ensemble-average
u spectrum (not shown), the filter highly distorts the shape and the peak
of the spectrum. Therefore, we focus the spectra analysis on the measurements
performed at fs≥0.2 Hz, i.e., we exclude the ZephIR radial velocity
spectra for the analysis. For the results presented hereafter, the noise
filter is only applied to the Avent radial velocity spectra.
Figure also shows that for these
ensemble-averages, the spectral density of beam 2 is the highest, followed by
that of beams 0 and 1, and then that of beam 4. This behavior might be due to
three reasons: Excessive rolling of the Avent, so that beam 2 points higher
than beam 1; that the turbulence characteristics at the position of beam 2
are rather different than those at the position of beam 1; or that there is
yaw misalignment so that beam 2 points closer to the direction of the mean
wind compared to beam 1 (see Fig. ). Both ZephIR and
Avent have tilt and roll signals, and for the 10 min samples analyzed here
the maximum absolute 10 min mean tilt and roll are only 0.56 and
0.31∘, respectively. Also, the very flat terrain characteristics
should not have such an impact on the ensemble-average spectrum of two beams
that point at the same height, like beams 1 and 2 in this particular case.
So, the most plausible explanation is that beams 2 and 3 are more aligned with
the mean wind than beams 1 and 4.
In the right panel of Fig. , we can see that the
spectrum of beam 2 is slightly higher than that of the 80 m cup anemometer
and higher than that of the sonic anemometer (up to f≈0.04 and 0.07
Hz, respectively). Such behavior is expected for low zR/L values
(see Fig. top left panel) or under lidar
misalignment conditions (see Fig. ). We can also see
that the cup-anemometer spectrum is higher than that of the sonic anemometer,
as expected from the variance results in the right panel in Fig. .
Horizontal wind-speed reconstruction
For both lidars we need to reconstruct the horizontal wind speed at the
specific range of the lidars, which can later be used for spectral analysis
and for filtered along-wind variance estimates. We use a simplified version
of the linear-gradient model of ,
vr(θ)=-cosφu+Rddudzsinθ+vcosθsinφ,
where Rd is the radius of the disc formed by the scanning pattern
at the given range, to estimate u, v, and the vertical gradient of the
along-wind component, du/dz. In Eq. (),
we ignore w and other vertical and horizontal gradients of the wind
components because their contribution is small. For both lidars, the beams
selected in Sect. are used for the reconstruction,
which can be done on the time-series basis or the 10 min averages.
Figure shows the results of the lidar-based
reconstruction on all 10 min means compared to the 80 m cup anemometer and
between the lidars; for both lidars we show the horizontal wind-speed
magnitude but when using the mean wind speed we obtain the same results. The same
results, regarding the linear regression and R2 (not shown), as those
given in the left panel of Fig. are found when
comparing the radial velocity of beam 0 with the 80 m cup-anemometer wind speed on a 10 min basis.
Comparison of reconstructed and 80 m cup anemometer horizontal wind
speeds. (Left) Cup anemometer against Avent. (Right) Avent against ZephIR.
Each 10 min sample is shown in grey markers. A 1 : 1 line is shown for
guidance in black, and the results of a linear regression through the origin
and R2 are also given.
Ensemble-average Doppler radial velocity spectrum
The Doppler-spectrum analysis is performed over all the 2273 10 min periods
using the ZephIR data (the Doppler spectrum information is not available for
the Avent). While each of the 10 min radial velocity time series per bin
position is thresholded and despiked (see Sect. ), we
extract the normalized Doppler radial velocity spectrum for each of the
samples within that 10 min and bin position. We then sum all the normalized
Doppler spectra within the 10 min period and the resulting Doppler spectrum
is normalized to unit area before we estimate the variance in two ways: by
computing the second moment from the spectrum and by fitting a normal
distribution to the spectrum to extract its variance
.
Figure illustrates examples of ensemble-average
Doppler spectra for different 10 min periods for the positions of bins 0 and
31, where we intentionally show 10 min radial velocity distributions with
high and low mean values, and high and low variances, including double-peak
distributions (there are only a few of them). These few distributions give us
an idea of the variety of turbulence characteristics of the dataset.
Distributions with high and low radial velocities generally show high and low
variances, respectively, as expected. Particularly in the examples, there is
a 10 min period with very low variance for both bin positions with clear
larger radial velocities for bin 0 compared to those for bin 31, indicating
very high wind shear, which is normally associated with atmospheric stable
conditions. This is an early morning 10 min period in late October, in which
the sonic-derived LO value is 1.82 m, corresponding to extremely
stable conditions.
Examples of normalized Doppler radial velocity spectra measured over
five 10 min periods with the ZephIR at the positions of bin 0 (left) and bin
31 (right). The markers show the observed distributions and the solid lines show a
normal fit.
Results
The results are divided into five parts. In Sect. , we
illustrate the main turbulence characteristics of the site, which we use to
classify the data in a number of atmospheric-stability and wind-speed ranges.
In Sect. , we intercompare the ZephIR estimates of
variances and co-variances using the unfiltered lidar radial velocity
variances with the cup- and sonic-anemometer estimates.
Section shows the effect of the noise filter on
the Avent radial velocity variance for the atmospheric-stability and
wind-speed ranges. In Sect. , we explore the
effect of atmospheric stability on both the sonic and the lidar radial
turbulence spectra and intercompare the Mann parameters derived from both
types of spectra. Finally, in Sect. , we perform the same
exercise as in Sect. but on the basis of the
wind-speed ranges.
Turbulence characteristics
Figure shows the overall turbulence characteristics
of the site based on cup- and sonic-anemometer observations, using the 2273
10 min concurrent data. In the left frame, we illustrate the behavior of the
turbulence intensity, σU/U, with wind speed, using the 80 m
cup-anemometer measurements; wind speeds are in the range
≈ 5–23 m s-1 with low σU/U values within the low
wind-speed range and σU/U increasing with wind speed. In the right
frame, we illustrate the behavior of the dimensionless wind shear,
ϕm=κz/u*∂U/∂z, with dimensionless
atmospheric stability, z/LO; we use the cup-anemometer wind-speed
measurements at 78 and 56 m to estimate ∂U/∂z
(≈ΔU/Δz) and the sonic-derived u* to compute
LO and ϕm. Figure. , in the right
panel, shows that the atmosphere during the analyzed period is mostly stable
(z/LO>0) and that, as expected, ϕm increases with
increasing z/LO. Such atmospheric conditions explain the low
σU/U values for low wind speeds. In the left panel of Fig. , we include a prediction of σU/U, using
Eq. () with σU=2.5u* and zo=0.012 m, which
fairly agrees with the data for high wind speeds only, as expected. In
Fig. (right panel), we include, for comparison only,
the prediction ϕm=1+4.7z/LO from surface-layer theory
that is offset with the data because ϕm, and so
z/LO, is estimated at a mean height of z=67 m with only two
wind-speed observations that were 22 m apart, whereas the turbulence estimates are
from the sonic anemometer at 76 m.
(Left) Turbulence intensity σU/U as a function of mean wind
speed U from the 80 m cup-anemometer observations. (Right) Dimensionless
wind shear ϕm as a function of dimensionless stability, z/LO,
based on the sonic- and cup-anemometer observations. The grey markers show
2273 10 min concurrent samples and the solid lines are theoretical
predictions (see text for details).
Based on the observed turbulence characteristics and knowing that we need to
average a number of 10 min spectra to be able to robustly extract the Mann
parameters , we classify the concurrent data into 10
classes as illustrated in Table , ensuring that
there are a close to 100 10 min samples per class as a minimum. From the
atmospheric-stability classes, we can see that the data comprise mainly
stable conditions, with stability 1 being the only close-to-neutral class
(〈z/LO〉=0.0625, with z=67 m). The more stable the
atmospheric conditions the lower the wind speed and the friction velocity, as
expected. Most of the data range within the stability 2 class (〈z/LO〉=0.1489), i.e., most of the observations are nearly
stable. From the wind-speed classes, we observe most of the data within a
high-speed range (11–13 m s-1) and, similarly to the stability
classification, the lower the wind speed the more stable the atmosphere
(except for the speed 1 and 2 classes), and so the lower the friction
velocity. Interestingly, for the speed 1 and 2 classes
z/LÕ=0.5084 and 0.7196, respectively (where
̃ indicates the median value), which are higher values than the
mean dimensionless stability of the most stable class (stability 5). We use
the median for the speed classes since the LO values highly
fluctuate within those speed ranges.
Atmospheric-stability and wind-speed classes and ranges based on the
cup- and sonic-anemometers' observations (see text for details). The
ensemble-average values of the dimensionless stability, wind speed, and
friction velocity per range are also provided. For the speed ranges we use
the median of the dimensionless stability. z=67 m is here the mean height
used for the dimensionless atmospheric-stability estimates.
Classz/L0No. of 10 min samples〈z/LO〉〈U〉 (m s-1)〈u*〉 (m s-1)stability 1-0.1–0.12250.062512.750.68stability 20.1–0.26290.148912.540.61stability 30.2–0.33500.243511.340.48stability 40.3–0.42250.347510.710.42stability 50.4–0.51530.445710.020.35classU (m s-1)no. of 10 min samples〈U〉 (m s-1)z/LÕ〈u*〉 (m s-1)speed 15–7936.650.50840.21speed 27–95167.980.71960.23speed 39–1150610.070.36840.37speed 411–1374111.940.21330.52speed 513–1527813.820.14020.64Unfiltered lidar turbulence
Based on the ZephIR configuration (φ=15.05∘), we are able to
predict all variances' ratios σvr2/σu,v,w2, using the Mann model with a given Γ parameter for the unfiltered lidar
radial velocity variances, i.e., using Eq. () with
zR/L=0. This is a procedure similar to the one we use for the
results in Fig. (right panel).
Figure shows a comparison of the ZephIR
“unfiltered” radial velocity variances (for bins 0 and 31) with the
cup-anemometer variances for all the 2273 10 min data, together with the
Mann-model prediction, using Γ=3. We present variance estimations that
are computed from the normal distribution fit to the average normalized
Doppler spectrum, instead of those calculating the second moment from the
spectrum, since the latter method is more sensitive to “spurious” data that
appear far from the area where most radial velocities are concentrated. This
is particularly seen for the lower bins (18 and 31) and might be due to
non-filtered blade-obstructed data, noise, or sudden jumps in the radial
velocity within the 10 min period.
Comparison of the 80 m cup-anemometer and the unfiltered ZephIR
radial velocity variances for bins 0 (left) and 31 (right). We show a 1 : 1
line for guidance and the predictions of the Mann model using Γ=3.
Results of a linear regression through the origin and R2 are also given.
For bin 31, σvr2>11 m2 s-2 for two 10 min
periods.
As expected, based on the results in Fig. , the top (bin
0) and a lower beam (bin 31) show a higher and lower variance, respectively,
than that of the “u” component (in quotation marks because we use the
cup-anemometer measurements). The Mann-model-based results slightly
underpredict the ratio σvr2/σu2 for these two
beams compared to the raw data. Reducing the value of Γ or accounting
for misalignment improves the predictions; e.g., with Γ=2.5 and
β=0∘ the Mann-model results predict biases of 11 and
-13 % for bins 0 and 31, respectively (not shown). It is important to
mention that the original (filtered) radial velocity variances for these
two bins are 13 and 31 % lower than the cup-anemometer measurements (not
shown) with slightly higher R2 values, 0.785 and 0.798, respectively.
Furthermore, we can also estimate σu,v,w2 and u′w′‾ for
each 10 min period through a least-squares fit of
Eq. () that does not depend on the Mann
parameters but assumes homogeneous turbulence within the scanned volume, using
the unfiltered radial velocity variances. Figure (left
panel) shows the estimate of σu2 based on the unfiltered radial
velocity variances of all bins without accounting for misalignment compared
to σcup2. The lidar-variance estimate is only 2 %
larger than the cup-anemometer value and the R2 value is higher than that
of any other comparison between cup-anemometer and lidar beam radial velocity
variances (filtered or not).
Comparison of the 80 m cup anemometer and the unfiltered (left) and
filtered (right) u variances from the ZephIR estimated under the assumption
of homogeneous turbulence within the measurement volume (see text for
details). We show a 1 : 1 line for guidance. Results of a linear regression
through the origin and R2 are also given.
In Fig. (right panel) we show a similar comparison to the
plot in the left panel but for the “filtered” u variance, which was
computed by reconstructing the u and v components, as described in
Sect. , using the ZephIR measurements on the seven
bins, but from the 18 s radial velocity measurements. The comparison with
the filtered values shows poor agreement with a 50 % underestimation of
the variance by the ZephIR. However, reconstructed u velocities from the
18 s radial velocities and averaged within 10 min periods compare well with
the reconstructed values from the 10 min means; the mean bias is 0 % and
R2=0.999 (not shown).
We also compare the lidar-derived σv,w2 and
u′w′‾ values with the sonic-anemometer estimates; the biases are
very high and R2 values are very low (not shown). This is not surprising
given the weight of the σv,w2 and u′w′‾ terms in
Eq. () when using low φ values. With this
lidar configuration, the reconstruction of the v component from, for example, Eq. (14) is not sound either; the yaw misalignment based on both
the Avent and ZephIR reconstructed u and v components shows poor
agreement when compared to the difference between the wind-vane and the
turbine-yaw 10 min signals.
We can also estimate σu2 through a least-squares fit of
Eq. () but using the unfiltered radial velocity
variances of the horizontal bins (12 and 37) only and the comparison with
σcup2 shows similar results (bias of 3 % and
R2=0.842). This indicates, firstly, the small but positive effect of
adding the top and lower beams' variances, and secondly, that the
contributions of other velocity components are not that significant for the
estimation of σu2 with the actual lidar configuration. Accounting
for misalignment does not improve the variance comparison (the bias increases
from 2 to 7 %).
Effect of the noise filter on the lidar variances
We also classify the 10 min 80 m cup anemometer variances and Avent radial
velocity spectra into the classes given in Table ,
ensemble-average the spectra within each class, and compute the variance of
each ensemble-average spectrum. The comparison of such variances, for each
Avent beam, is illustrated in Fig. (raw). We
also show a similar comparison but for the noise-filtered Avent radial
velocity ensemble-average spectra. Furthermore, we include the prediction
σvr2/σu2 based on the Avent lidar configuration
using the Mann model with fixed Mann parameters (same as those found in
Sect. using the ensemble-average sonic-anemometer
velocity spectra).
Comparison of the 80 m cup anemometer with the Avent radial
velocity variances for different beams for the 10 turbulence classes (filled
circles) in Table . Raw and noise-filtered data are
shown as well as a 1 : 1 line (for guidance) and the prediction of the
Avent filtered radial velocity variance based on the Mann model, using
Γ=3.00, L=35.38 m, and β=0∘.
When the noise filter is applied, the ratio σvr2/σcup2 is well predicted by the
Mann model. The largest difference is observed for beam 4 but this is because
the noise filter highly reduces the variance for one particular class only.
For beams 1 and 2, the Mann model predicts the same
σvr2/σcup2 value as here we do not
take into account lidar misalignment.
Effect of atmospheric stability on turbulence spectra
The ensemble-average sonic and Avent radial velocity spectra are used
separately to extract two independent sets of Mann parameters for each of the
atmospheric stability classes in Table by fitting
the sonic- and lidar-based LUTs computed through the use of
Eqs. () and ().
Figure shows the results of the two stability
classes farthest apart (stabilities 1 and 5).
Normalized power spectra of the different velocity components based
on the sonic-anemometer observations (left) and of the Avent radial velocity
for different beams (right). The top panels show the results for the first
stability range (stability 1) and the bottom panels for the last stability
range (stability 5).
For the stability 1 class, the Mann model agrees well with the sonic velocity
spectra and for stability 5 the differences between the model and the
sonic-anemometer observations are larger, as expected, since the Mann model
was developed for near-neutral atmospheric conditions. Both the sonic
observations and the Mann model show the spectral peaks to move to higher
wave numbers with increasing stability because the size of the turbulence
eddies decreases with stability in agreement with the study of
. The lidar radial velocity spectra also show similar
features to the sonic-based spectra: higher normalized spectral densities for
the most stable compared to the close to neutral class and spectral peaks
that move to higher wave numbers with increasing stability. The former
feature might be due to the way we normalize the spectra: we make use of the
76 m u* value instead of one close to the ground where surface-layer
scaling is more valid, particularly for stable conditions. The agreement of
the Mann-model-based spectra also deteriorates with stability but the
lidar-based LUT seems to follow the behavior of the radial
velocity spectra for these two classes fairly well.
In Fig. , we show the results of the Mann parameters
extracted from the ensemble-average sonic and lidar radial velocity spectra
for all atmospheric stability classes. There is a slight decrease in Γ with
stability (based on the sonic-anemometer data) and the lidar-based value
closely follows the sonic-based one, with best agreement at the highest
stability range. The sonic-based αε2/3–parameter slightly
decreases with stability and, for the near-neutral stability class, the
lidar-based value is close to the sonic-based one. A similar feature is found
for the L parameter; both types of data show a very close value for
near-neutral conditions and the sonic-based value slightly decreases with
stability as expected. The increasing differences between the sonic- and the
lidar-based αε2/3 and L parameters with stability are
interconnected. We cannot expect to measure eddies below the size of the
lidar probe volume, which in this case means that we are not able to
accurately estimate the length scale when L⪅zR. This
occurs already at the stability 3 class. These two Mann parameters are,
in practical terms, scaling factors in the velocity spectra as seen from
Eq. (), and so an underestimation of L generally
leads to an overestimation of αε2/3 when fitting the
lidar-based LUT.
Mann parameters for a number of atmospheric-stability conditions
(see Table ) derived from sonic anemometer and
lidar radial velocity spectra.
We also have to notice that when using this type of lidar configuration, we
are extracting turbulence information from the radial velocity spectra of
beams, whose spectral densities are rather close (since all beams measure a
close to u spectrum), whereas in the case of the sonic-anemometer
observations we use three auto-spectra and a cross-spectrum that are
relatively far apart in terms of spectral densities. This issue is discussed
further in Sect. .
Effect of wind speed on turbulence spectra
We now perform a similar procedure as that in
Sect. but for each of the wind-speed classes
in Table , and the results of the two wind-speed
classes most far apart (speeds 1 and 5) are shown in
Fig. . For the speed 1 class, the Mann model does not
agree with the sonic velocity spectra as well as it does when compared to the
speed 5 class, as expected, since the atmospheric conditions are closer to
neutral for the latter class. Both the sonic-anemometer observations and the
Mann model show spectral peaks that move to lower wave numbers with
increasing wind speed because of the combined effect of stability and wind
speed; the larger the turbulent eddies, the higher the wind speed and the
lower the stability.
Similar to Fig. , but here the top panels
show the results for the first wind-speed range (speed 1) and the bottom
panels for the last wind-speed range (speed 5).
The lidar radial velocity spectra also show similar features to the
sonic-based spectra; lower normalized spectral densities for the high-wind
compared to the low-wind class and spectral peaks that move to lower wave
numbers with increasing wind speed. The agreement of the Mann-model-based
spectra deteriorates with decreasing wind speed, but the lidar-based LUT also
seems to follow the behavior of the radial velocity spectra for
these two classes fairly well (similarly as it does when comparing spectra for the range
of stability classes).
In Fig. , we show the results of the Mann parameters but
for the wind-speed classes. Based on the sonic-anemometer data, Γ, is
rather constant with wind speed, a behavior already observed by
for the same height and the lidar-based value agrees well
with the sonic-based one for all wind-speed classes, particularly the two low
wind-speed classes. Similar to the results from the atmospheric-stability
classes, the differences between the sonic- and the lidar-based
αε2/3 and L parameters are larger than those for
Γ, but for these wind-speed classes the αε2/3 parameter does not differ largely under the classes
where L differs the most, i.e., speed classes 1 and 2, where the average
conditions are very stable. Turbulence characteristics under these two
classes are similar and L is higher within speed 1 compared to the
speed 2 class. The highest differences in the estimations of L are also
found for those classes in which L⪅zR (speed classes
1–3).
Mann parameters for a number of wind-speed ranges (see
Table ) derived from sonic anemometer and lidar
radial velocity spectra.
(Left panel) Pulsed lidar radial velocity spectra for different
beams. (Right panel) Contributions of the spectral velocity tensor components
to the lidar radial velocity spectrum for beams 2 (solid lines) and 3 (dashed
lines). Values of β=0∘, φ=60∘, Γ=3,
αε2/3=0.1 m4/3 s-2, and zR/L=0.5
are used for the computation.
Discussion
It is important to notice that some of the differences between turbulence
statistics estimated from the sonic-, cup-anemometer, and lidars'
measurements are not only due to the way they probe the atmosphere but also
because the lidar measurements are affected by optical and instrumental noise
(and by the blades, hard targets, and fog, among other factors), the cup- and
sonic-anemometers are inherently affected by flow distortion from the mast
structure and by the instrument itself, which we do not take into account, and
that there are differences in the heights of the measurements. For example,
the axes of the lidars pointed close to hub height when the wind turbine was
operating, and the 80 m cup and sonic anemometer are 1.8 and 5.8 m below
hub height, respectively. Also, the mast is 111 m from the range that we use
to extract the lidar measurements when the wind is directly from the mast to
the turbine. Wind speeds, variances, and velocity spectra from the mast and
the lidars' selected range are expected to be comparable due to the
topographic conditions of the site for the selected wind directions, but not
equal. Further details regarding how cup anemometers, sonic anemometers and
lidars measure turbulence are provided in ,
, and , respectively.
We assume turbulence to be homogeneous within the lidar scanning area, both
when extracting the Mann parameters and when studying the unfiltered
turbulence. This is a rather simplistic assumption as shown in the study by
, in which the Mann parameters are extracted from
sonic-anemometer measurements at different heights. However, we expect that
such an assumption results in turbulence parameters that are more
representative of the turbine operation as they are estimated from
measurements over a larger area.
In Sects. and , we show
normalized power spectra for the two most “extreme” classes in order to
understand the spectra behavior for the changing atmospheric and wind-speed
conditions; spectra results for the other classes are not shown, but lie in
between, as illustrated from the derived Mann parameters in
Figs. and . In
Sect. , we mention that part of the problem of
extracting the Mann parameters from the current lidar measurements is the
small difference between the beams' radial velocity spectra, all being
relatively close to the u spectrum. The Mann model needs more than
one-component spectra to fit the LUT to measurements/simulations, otherwise
the Mann parameters are ill-determined.
We find very good agreement between the along-wind variance estimate of the
ZephIR (when using the ensemble-average Doppler radial velocity spectrum) and
the cup-anemometer measurement, but for the other velocity-component variances
and co-variances, when compared to those from the sonic anemometer, the
biases are too large. But, can we improve such estimates, e.g., increasing the
cone angle φ? On the one hand, one can make the theoretical exercise of
predicting σu,v,w2 and u′w′‾ from the Mann model (with
a given set of Mann parameters). In parallel, we can use Eq. ()
with zR=0 to estimate the unfiltered σvr2
for different beams and use Eq. () to estimate
σu,v,w2 and u′w′‾ from the unfiltered beam variances.
If we compare the former predicted with the latter estimated variances, e.g.,
using a four-beam lidar (θ=0, 90, 180, and 270∘), with
φ=15∘, the result for the u-variance is a 2 % bias,
whereas the v and w variances show biases larger than 50 %. The
result for the v and w components improves when increasing φ; the
biases for both components' variances are below 20 % for
φ=60∘ but the bias deteriorates for the u variance with
increasing φ. If a central beam is added and we are able to extract
the unfiltered variance of this beam, i.e., σu2, the comparisons are
unbiased for all velocity components (no matter the value of φ).
On the other hand, using a lidar with φ=60∘ increases the
relative differences between the radial velocity spectra densities of the
beams, e.g., with the current Avent configuration as it uses a central beam.
Figure shows that with such a cone angle, the
central beam spectrum peaks close to the u spectrum peak, and the lower
beams peak at ≈ 20 % of the u spectrum peak (with
φ=15∘, the lower beams peak at ≈ 75 % of that of
the central beam spectrum). This is mainly due to the large negative
contribution of Φuw for the lower beams, as shown in
Fig. right panel. The difference between the u and
w spectra is ≈ 60 % only.
It is also important to highlight to the reader that wind turbine loads and power
performance are directly impacted by turbulence, in particular σu2.
The latter affects the turbine's power output differently, depending on the
wind speed . The Mann parameters add value for
understanding the behavior of loads but are not critical .
In this study we demonstrate that σu2 can be estimated by FL nacelle
lidars, and current research demonstrates that lidar-based σu2
values reduce the gap between loads and power measurements, as well as simulations.
It is difficult to compare our results with those from previous work on lidar
turbulence measurements . First, with a FL
lidar we are able to point the beam in a direction close to the mean wind,
whereas most lidars use beams pointing closer to the vertical wind component.
Second, we do not need to reconstruct wind components to estimate variances,
but the radial velocity spectrum and variance for each of the beams can be
directly computed; this allows us to create a LUT useful to extract the Mann
parameters.
Conclusions
We characterize turbulence using measurements from two types of
forward-looking nacelle lidars that were mounted on the nacelle of a wind
turbine. We compare such characteristics with those from sonic- and
cup-anemometer measurements on a mast, which is 111 m from the lidar
measurement range when the turbine and mast are aligned with the wind (thus
this distance increases for other wind directions). By using information of
the 10 min ensemble average Doppler radial velocity spectrum, we are able to
estimate 10 min unfiltered radial-velocity variances of the beams of a CW
lidar. These unfiltered beam variances are well predicted by the Mann model.
Assuming homogeneous turbulence within the lidar scanned area,
σu,v,w2 and u′w′‾ are estimated from the unfiltered
beam variances; comparison with the 10 min cup-anemometer variances reveals
a 2 % bias for the u variance, whereas the biases are very high for the
other velocity components.
We divide the 10 min time series and the sonic-anemometer and lidar beam
radial velocity spectra into atmospheric-stability and wind-speed classes
based on the mast measurements. Most of the conditions are stable and
relatively windy. We observe that the pulsed lidar beam variances are
affected by noise as clearly seen in the lidar radial velocity spectra.
Therefore, we noise filter the lidar beam spectra, and the resulting variances
show very good agreement with the prediction using the Mann and spatial
averaging models.
We also extract the Mann parameters from sonic-anemometer and lidar beam
radial velocity spectra and intercompare them for each of the classes. Under
high wind and near-neutral atmospheric conditions the agreement is good, and
the differences increase with higher stability and lower wind speed, where
the Mann model also has limitations fitting the sonic-anemometer velocity
spectra. This is partly because increasing stability and decreasing wind
speed results in turbulence length scales comparable to or lower than the
length of the lidar probe volume. We suggest to improve lidar-based
Mann-parameter estimations by increasing the lidars' cone angle, always keeping
a central beam, which will also aid in the estimations of the non-wind-aligned velocity variances and covariances, although the flow
homogeneity assumption becomes less valid.
Turbine data are not publicly available because there is a non-disclosure agreement between the partners in the UniTTe project. Lidar and mast data can be requested from Rozenn Wagner at DTU Wind Energy (rozn@dtu.dk).
The authors declare that they have no conflict of
interest.
Acknowledgements
Funding from Innovation Fund Denmark, grant number 1205-00024B, to the UniTTe
project (http://www.unitte.dk) is acknowledged. We are grateful for the continuous technical support by Antoine Borraccino and Andrea
Vignaroli, DTU Wind Energy. Edited by: S.
Aubrun Reviewed by: W. Bierbooms and one anonymous referee
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