Lidar simulations for the design of wake measurement campaigns 15
Figure 2.1: Sketch of the radial direction of the laser beam defined by the azimuth and elevation anglesχ andζ, respectively. VR denotes the radial component of the wind velocity vectorV.
Inverse projection
If the wind directionVdir is known, a simple approach to calculate the magnitude of the horizontal wind vector |VH|assumes no vertical wind (w= 0) to rewrite Eq. 2.1 as
VR=|VH|cos (∆α) cos (ζ) (2.2) where ∆α = Vdir−χ is the difference between the wind direction and azimuth angle. This formulation of the radial wind component can be directly inverted to calculate the horizontal wind speed:
|VH|= VR
cos (∆α) cos (ζ). (2.3)
Doppler beam swinging (DBS) method
The DBS technique (Werner, 2005) can be applied to lidar measurements as the ones exemplified in Fig. 1.5a. In the basic form, the lidar sequentially scan three radial directions: one oriented vertically and two at a fixed elevation angle (for instanceζ =60 ) with a mutual offset of 90°
between their azimuth angles. The corresponding radial wind speed components can be written as VRV = w
VRN = usin (ζ) +wcos (ζ) VRE = vsin (ζ) +wcos (ζ)
. (2.4)
considering, for instance, the vertical beam forVRV and the azimuth angle corresponding to the North and East directions forVRN andVRE, respectively.
Working on these equations, it is possible to obtain the two horizontal components of the wind vector:
u = VRN−Vsin (ζ)RV cos (ζ)
v = VRE−Vsin (ζ)RV cos (ζ) . (2.5)
If additionally the radial wind speed componentsVRS andVRW with the azimuth angle of the remaining South and West directions, respectively, it is possible to write the horizontal components
16 2.2. WIND SPEED AND WAKE PROFILES FROM LIDAR MEASUREMENTS
of the wind vector as
u = V2 sin (ζ)RN−VRS v = VRE2 sin (ζ−VRW)
w = VRN+VRE4 cos (ζ+VRS)+VRW
. (2.6)
This approach is commonly implemented in lidar by cycling the azimuth angle over the four cardinal directions under the assumption that the wind field is horizontally homogeneous and does not vary during the measurement along the different directions.
Visual azimuth display (VAD)
To apply the VAD method (Werner, 2005), the lidar measures along a 360° azimuthal scan with closely separated radial directions as drafted in Fig. 1.5b. The corresponding radial wind component varies as a sinusoidal function of the azimuth angle:
VR=aV AD+bV ADcos (χ−χmax) (2.7) where the offset aV AD, amplitude bV AD and phase shift χmax. The parameter of VR can be estimated by means of a least-square-fit to the radial measurements of a complete azimuthal scan.
Under the assumption that the wind field is horizontally homogeneous and does not vary during the measurement along the different directions, the results can be related to the wind vector components by
u = −bV ADsin (χmax)/cos (ζ) v = −bV ADcos (χmax)/cos (ζ) w = −aV AD/sin (ζ)
. (2.8)
Dual-Doppler techniques
If two or more synchronised lidars measure simultaneously at the same target position and the corresponding radial directions are linearly independent, the temporal and spatial averaging implied by the DBS and VAD methods could be avoided applying the matrix inversion method: If three radial wind speed componentsVRi and the corresponding radial directions are collected into the vectorVRT = [VR1VR2VR3] and into the matrixMeR= [eR1eR2eR3], respectively, it is possible to write and solve a linear system to find the components of the wind vector:
VT = [u v w] =VRT
·M−1eR (2.9)
This system has a solution only if the matrixMeR is invertible, which means that the three radial directions need to be linearly independent. Dual-Doppler algorithms (Drechsel et al., 2010) could be applied to relax this limitation. Similar approaches were applied for instance in the works by Cherukuru et al., (2017) and van Dooren et al., (2016) to resolve the wake wind field inside an offshore wind farm from azimuthal scans (plan position indicator, PPI) at low and zero elevation, respectively. Dual-Doppler measurements are very convenient also in complex terrain where wind field inhomogeneity due to the site topography adds up to the one of the wake. For example, Wildmann et al., (2018) analysed how a wake runs over a ridge using the dual-Doppler method and coplanar vertical scans (range height indicator, RHI) of the wake.
Lidar simulations for the design of wake measurement campaigns 17
2.2.2 Wake characterisation
Procedures applied to identify the overall properties of the wind velocity deficit at a downstream section of the wake are commonly indicated as wake tracking methods. Different approaches have been applied to lidar measurements during the last years. Some representative examples are described in the following1; the reader interested in an evaluation of common wake tracking approaches is invited to consult Doubrawa et al., (2017) and Vollmer et al., (2016).
Direct fit of Gaussian surfaces
This method was first introduced by Trujillo et al., (2011) for lidar wake measurements from the nacelle of a wind turbine. The goal was to estimate the centre position of the wake at a fast repetition rate and to qualitatively describe the corresponding steady wake. The measurements covered a two-dimensional cross-section of the wake. First, the wind deficit was isolated from the wake flow subtracting an estimation of the shear profile from the radial wind speed measurements.
Then a Gaussian surface was fit to the wind deficit in order to track the centre of the position of the wake.
Fit of projected Gaussian profiles
To fully characterise the wake and its meandering path (centre position, width and amplitude of the wind speed deficit) within a PPI or RHI scan by means of ground-based lidars, Aitken et al., (2014) modelled the wind deficit measured in the wake at each range-gate with a one-dimensional Gaussian profile. Slightly different models were applied to take into consideration (i) the scanner strategy, (ii) measurements in the free-flow, (iii) measurements in the near-wake or (iv) measurements in the
far-wake.
The free-flow model dealt only with the projection of the wind vector on the radial direction of the measurements; in PPI measurements the elevation was disregarded; in the RHI measurements, the vertical profile of the wind speed was incorporated into the wind model as a logarithmic profile.
A double or a single Gaussian profile was added to the free-flow wind model in the near- or far-wake region, respectively.
To characterise the average development of wind speed deficit in the wake (i.e. wake direction, recovery and expansion rates), Aitken et al., (2014) fitted the model of PPI measurements in the far-wake region considering for each sweep all the range-gates at once. First, they rotated the model into the rotor frame of reference; then, they imposed a power law to describe the expansion and recovery of the wind deficit.
Centroid methods
The centroid method can be applied to detect the centre of the wake deficit at a certain downstream cross-section and to study the meandering path of the wake. Herges et al., (2018) applied this
1Please note that, depending on the scanning pattern and on the availability of the measurements, interpolation of the data might be required before starting the analysis of the wake. These interpolation methods are out of the scope of this section.