Scalar- and vector wind speeds, with a Doppler Beam Swinging LiDAR

This post deals with a topic relevant for wind data users: the differences between scalar- and vector averages of the horizontal wind speeds using cups/sonic and LiDARs. I will be using measurements from:

• Sonics and Doppler Beam Swinging LiDAR (Leosphere WindCube) from the XPIA campaign (Lundquist et al, 2017);
• Cups and Doppler Beam Swinging LiDAR (Leosphere WindCube) from the FINO3 met mast (publicly available).

I looked into this for two main reasons:

1. The release, earlier this year, of a statement from Leosphere, indicating they would for now on change their retrieval algorithm from scalar to a mix of scalar- and vector averages;
2. A validation (publicly-available) of a Leosphere WindCube at a site in Texas, showing large deviations between LiDAR and cup measurements (Kirincich, 2019, also on eo-winds.net library).

Both of 1. and 2. indicate that, in some conditions, scalar estimates with WindCube LiDARs overestimate the wind speed compared with cup anemometers (that are scalar measurements): see for instance the plot below, showing large deviations for small wind speeds at this site in Texas:

If you recall my previous post on LiDAR validations, the literature cited therein, as well as the dozens of validation reports on eo-winds.net library, this may seem strange: such large deviations are not reported for those flat-terrain, Northern-European sites, and in particular offshore.

I was myself involved (a bit) in a research project, long time ago: the “Bankable LiDAR” project in DONG Energy. LiDARs were tested against cup anemometer and sonics at the Høvsøre test site in Denmark, using both vector- and scalar methods; see Section 2.2 of (Courtney, 2014). Aaaand…. the test (at the coastal location Høvsøre) did not reveal any particular bias which should have been accounted for.

As the site in Texas is also (very) flat terrain, with no orography-induced large scale turbulence, I started to wonder…

Previous works, and in particular studies such as (Albers, 2018) and (Leosphere, 2019), do indicate that when the flow is “complex” (i.e. when the assumption of flow homogeinity across the LiDAR measurement volume isn’t verified) such deviations are expected; but then again: the site in Texas didn’t look that complex, so…

Actually, if large scale flow motions are present in flat terrain, they are likley induced either by convection (during the day), or stable oscillations (at night). So a likely culprit for these differences betwen LiDARs and cups is, like often, atm stability. But then: what about offshore conditions? Offshore sites have their share of very unstable (think: open convection cells, Danish North Sea), or very stable (minute-scale gravity waves): so are the biases also present offshore?

This post proposes to address these questions using the following method:

1. Briefly explain what scalar- and vector wind speeds are, for point measurements;
2. Recall how these are calculated from WindCube measurements;
3. Compare LiDAR vector- and scalar estimates against scalar estimates from anemometers (sonics, cups);

1) Vector vs scalar averaging for point measurements (i.e. not for LiDARs)

See below. Let us imagine a steady state flow where the wind blows from South to North (left): there is only one wind speed value. But, if now the wind blows 50% of the time to the North East, and 50% of the time to the North West, with equal speed, of course the vector average of these two vectors will be smaller than the scalar average (middle). For a more realistic flow, in the boundary layer (turbulent), the wind direction will oscillate and lead to the vector average being always smaller or equal to the scalar average (right).

This is discussed for instance in (Clive, 2008), which shows that the ratio between vector- and the scalar average decreases when the standard deviation of the wind direction increases; see below. As a side note, for offshore sites, values are approximately [0;5] degrees.

While cups and sonic anemometers are both “point” measurements, they not measure the same quantities. Cup anemometer measure (to some approximation) the total horizontal instantaneous wind speed, thereby providing scalar average values. To compute vector averages with cup anemometer data, a wind direction measurement (from a wind vane) is required. On the other hand, sonic anemometers compute instantaneous u, v and w (vertical) components, allowing for both scalar- and vector averages to be computed with one instrument.

The differences between scalar- and vector averages, in particular in relation to wind turbine power curve measurements have long been discussed. As discussed in (Pedersen, 2002), the differences between wind speed averaging methods (including also the vertical fluctuations) were (for up to 15% turbulence intensity) small compared with the differences observed between different types of cup anemometers (some being more or less sensitive to turbulence); therefore, and since wind turbines control system aim at “following” the wind direction, it was chosen to use scalar wind speeds from cup anemometers as reference for power curve testing. Yet, it is worth noting that the mean wind speed used as input to Blade Element Momentum (BEM) codes, or any computational model actually, are (longitudinal) vector wind speeds, not the scalar.

2) Vector vs scalar averaging for WindCube measurements

To start with, and if you are not already familiar with wind the different type of wind LiDARs and their measurement principles, see this nice report: “Remote Sensing for Wind Energy (Peña et al., 2013)“. The tl;dr is as follows:

• LiDARs are measurements instruments equipped with lasers;
• Laser beams are emitted from the instrument and into the atmosphere;
• The light is reflected by small aerosols flowing at the speed of the wind;
• The instrument measures the shift in frequency between the emitted- and reflected signals;
• The shift in frequency is proportional to the speed of the aerosols (Doppler effect) in the direction the laser beam (the radial wind speed);
• By shooting at different places over a relatively short period of time, the instrument reconstructs a wind vector from the radial wind speeds.

See below for a Leosphere WindCube:

The radial wind speeds measured by the LiDAR are instantaneous wind speeds. For doing Wind Resource Analysis, or for measuring the performance of a wind turbine, the mean wind speed over longer period is needed, (typically: 10 minutes). Two main methods are usually considered for computing a mean wind speed value from a set of radial wind speeds, they both rely on the estimation of the two horizontal wind speed components u and v, see how the equations looks like for the Leosphere LiDAR below: u and v are functions of the radial wind speeds, and of the angle of the beams theta.

The instantaneous horizontal wind speeds values are then:

$$V_{h,scalar} = \overline{V_{h}} = \overline{\sqrt{u^{2} + v^{2}}}$$

From these (scalar) values, the so-called scalar-averaged wind speed is derived:

$$V_{h,scalar} = \overline{V_{h}} = \overline{\sqrt{u^{2} + v^{2}}}$$

But, there is another option: computing first the mean u and v components individually, and then adding the two mean vector components together. This is the vector-averaged wind speed:

$$V_{h,vector} = \sqrt{\overline{u}^{2} + \overline{v}^{2}}$$

According to the documentation from (Leosphere, 2020) and the slides of the accompagnying webinar, both of these two methods (scalar and vector), differ from the real horizontal wind speed for large values of standard deviations of the wind direction:

Just, some information is still pending (at least I could not find it):

• Is it the standard deviation of the cup + vane, or that of the LiDAR?
• Where have these validations been carried out, against what type of cups, and how?

I will elaborate on these questions at the end of this post.

3) Comparisons onshore from the XPIA campaign (Boulder, CO)

As mentioned earlier,I have used the XPIA data; in part from the A2e database (mast), and in part from https://breeze.colorado.edu/ftp/ (LiDAR). I have computed both scalar- and vector averages from the (mast-effect free) sonic anemometer time series on the mast, and from the LiDAR data.

The results of the comparison between vector and scalar wind speeds (10-minute samples) are shown below for two (approximate) stability classes, first for the sonics, and then for the LiDAR data. In these figures, the black line shows the relationship:

$$ \frac{WS_{vec}}{WS_{sca}} = \frac{\sin\sqrt{3}\sigma}{ \sqrt{3}\sigma}$$

From these figures, we see that the ratio between scalar- and vector averages follow indeed the simple model (gaussian turbulence) for small to medium values of standard deviations. We also note that for unstable conditions (day), the relationship for the LiDAR data differ depending if the sonics- or the LiDAR standard deviation is used. In effect, as for the standard deviation of wind speed, the LiDAR wind direction standard deviation is overestimated when large scale turbulence is present; see below.

Therefore (see the plots below), when comparing the mean wind speeds values between LiDAR (vector and scalar) and sonic (scalar), we find that:

• During the night (i.e. for, most likely, stable conditions, small values of wind direction standard deviation): both vector- and scalar LiDAR averages show the same distributions or relative deviations compared with the sonics scalar;
• During the day (i.e. for, most likely, unstable conditions, large values of wind direction standard deviation): the LiDAR scalar wind speeds are significantly larger than the vector values. For the LiDAR vector wind speeds, the distribution of relative differences shows a larger proportion of small values, yet the average of the distribution does not seem to change compared with the night.

The analysis of the dependency of relative differences to the wind direction standard deviation, see below for 50 and 100 mASL, shows that:

• For similar values of standard deviations, the stable- and unstable subsets show similar values;
• The differences increase with the standard deviation for the LiDAR scalar averages;
• The differences seem to be constant with the standard deviation for the LiDAR vector averages, and only decrease for large values of standard deviation (about 10°, as displayed on the webinar) – but for unstable conditions only.

4) Comparisons offshore at the FINO3 met mast (German North Sea)

Offshore, the main problem for making such comparisons are the more proeminent mast effects. As explained in multiple papers about the FINO masts, the large structure heavily disturbs the flow, affecting both the cups (slowdown and speed-ups) and the LiDAR (beams in the wake of the mast). See below how the LiDAR- and top anemometer wind speed compare: there are only few wind directional bins where one can expect a good correspondence!

With this in mind, see the comparisons below, between LiDAR scalar- and vector wind speeds, cup anemometer wind speeds, and LiDAR and cup wind direction standard deviations. The obukhov length has been estimated using ERA5.

As for the mast in Boulder, we see that for unstable conditions, the LiDAR overestimate the wind direction standard deviation. Yet, these values remain small, and the differences between LiDAR scalar- and vector averages remain very small.

5) Conclusions, and remaining questions

From this short analysis, my only conclusion is that for offshore sites with small turbulence, we don’t need to worry too much: while there are small differences between LiDAR scalar- and vector averages, they remain an order of magnitude smaller than the other biases we need to account for.

However, for onshore sites, and in particular for flat terrain, small wind speeds and highly convective situations, I think that more analysis is needed. We may be able to tell that, for large values of wind direction standard deviation:

• The LiDAR scalar wind speeds overestimate the sonics (and cup I guess) scalar wind speed, regardless of the stability conditions;
• The LiDAR vector wind speeds understimate the sonics/scalar wind speeds for unstable conditions.

Some things remain unclear:

• In Leosphere’s analysis: is the wind direction standard deviation that of the cup? or the LiDAR?
• Same as above, but for the algorithm.
• For large value of standard deviation, is the LiDAR vector wind speed underestimating the cup scalar wind speed also in stable conditions? I guess such large values occur very little in stable conditions, but still it may be good to know.
• Are the validation reports for the DWG classification study available to third parties? The very large sensitivities observed at all elevations at the DNV mast in Hamburg seem questionable, as well as the ones close to 4 m at GF1 and GF2.

That’s all for now for these high-level tech. notes. Comments and questions are welcome.

Rémi

[update 2021-11-08]: corrected typo in yaxis of subplots(2,2,4) showing vec_lidar/sca_sonics vs std(wd).