5.2.1 Determination of Lidar Auto-spectrum
Ignoring the filtering effect of a lidar and assuming Taylor’s Frozen Turbulence Hypothesis [66]
to be fully valid (no wind evolution), the lidar estimate of the rotor effective wind speed is considered equal to the longitudinal wind velocity uH at the hub:
v0L =uH. (5.4)
Therefore, the spectrum of a perfect staring lidar system is simply modeled with the longitudinal Kaimal auto-spectrum (2.19) by
SLL =F{uH}F∗{uH}=SHH=Sii,u= 4σ2uLu¯u
1 + 6fL¯uu(5/3). (5.5)
5.2.2 Determination of Rotor Auto-spectrum
With the definition of the rotor effective wind speed (3.19), the rotor averaged spectrum SRR can be derived by
SRR =F
1 πR2
Z2π 0
ZR 0
uI rdrdφ
F∗
1 πR2
Z2π 0
ZR 0
uI rdrdφ
= 1
(πR2)2 Z2π
0
ZR 0
Z2π 0
ZR 0
F{ui,I}F∗{uj,I}
| {z }
Sij,u
ri rjdridφidrjdφj. (5.6)
To obtain the cross-spectrum Sij,u of the longitudinal wind speed component between a point i and j, the coherence of the longitudinal velocity component (2.21) between two points with distancerij is rewritten as
γij,ur = exp (−κ rij) with κ= 12s f
¯ u
2
+0.12 Lu
2
. (5.7)
Here, κ is the frequency-dependent lateral decay parameter. With the definition of coherence γij,ur2 = |Sij,u|2
Sii,uSjj,u (5.8)
and the assumption that there is no phase shift between the longitudinal wind speed component in point i and j and with Sii,u =Sjj,u, the cross-spectrum is obtained by
Sij,u =|Sij,u|=γij,urSii,u= exp (−κ rij)Sii,u. (5.9)
5.2 Correlation of a Perfect Staring Lidar System 85 With (5.9), the explicit solution of (5.6) can be found as
SRR=2Sii,u
(Rκ)3
L1(2Rκ)−I1(2Rκ)− 2 π +Rκ(−2L−2(2Rκ) + 2I2(2Rκ) + 1)
, (5.10)
where L are modified Struve functions1 and I are modified Bessel functions of the first kind2. Details of this calculation can be found in Appendix C.4.
5.2.3 Determination of Cross-spectrum
Similar to (5.6), the cross-spectrum between the staring lidar system and the rotor is
SRL =F
1 πR2
Z2π 0
ZR 0
uI rdrdφ
F∗{uH}
= 1
πR2 Z2π
0
ZR 0
SHj,u rjdrjdφj, (5.11)
whereSHj,u is the cross-spectrum of the longitudinal wind component between the hub and the point j. Assuming that the lidar signal uI is time-shifted to the rotor, the cross-spectrum can be replaced similar to (5.9) by the coherence and auto-spectrum and then (5.11) is solved by
SRL = 2Sii,u
R2κ2
1− Rκ+ 1 exp(Rκ)
. (5.12)
5.2.4 Determination of Coherence and Transfer Function
Finally, the transfer function GRL and the squared coherence γRL2 can be calculated based on the cross- and auto-spectra of (5.5), (5.10), and (5.12) using (5.2) and (5.1).
With the definition of the wavenumber
k= 2πf
¯
u , (5.13)
the coherence and transfer function are independent of the mean wind speed ¯u. Figure 5.2 shows both for the NREL 5 MW reference turbine.
The model is compared to the coherence and transfer function obtained from a time domain simulation. As illustrated in Figure 5.1, a wind field needs to be generated and scanned to
1Seewww.mathworld.wolfram.com/ModifiedStruveFunction.html.
2Seewww.mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html.
k [rad/m]
|GRL|[-]
k [rad/m]
γ2 RL[-]
10−3 10−2 10−1 100
10−3 10−2 10−1 100 0
0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Figure 5.2:Coherence (left) and transfer function (right) between a perfect staring lidar and a turbine with a rotor diameter of 126 m. From model (black) and from estimation (dark gray) with ±σ(ˆγRL2 )
and ±σ(|GˆRL|) confidence bounds (light gray).
extract the signals for the rotor effective wind speed v0 and its lidar estimate v0L. Here, a large TurbSim wind field with a time step of 0.25 s based on the Kaimal turbulence model with nR = 777 points distributed equally over the rotor area with 4 m separation is used (for details see Table C.3). A time length of 8192 s is chosen to obtain a good estimation of the correlation.
The rotor effective wind speedv0 is obtained by using the average of all longitudinal wind speed components (3.20). Its lidar estimate v0L in this case is equal to the longitudinal wind speed at hub height.
The process of estimating the coherence and the transfer function from time signals is quite complex and is here briefly explained to show the limitations and adjusting possibilities. First, the two signals are divided into nC = 32 data chunks of the same length (here 1024 data points). Then, the coherence and the transfer function are estimated using Welch’s averaged periodogram method3:
ˆ
γRL,biased2 =
nC
X
i=1
V0L,iV0,i∗
2
nC
X
i=1
|V0,i|2
nC
X
i=1
|V0L,i|2
(5.14a)
GˆRL =
nC
X
i=1
V0L,iV0,i∗
nC
X
i=1
|V0L,i|2
, (5.14b)
3Done using no overlap and a rectangular window withmscohere and tfestimate of The MathWorks Inc., Matlab R2013b, Natick, USA (2013).
5.2 Correlation of a Perfect Staring Lidar System 87 where V0,i and V0L,i are the FFTs of the signals v0 and v0L from the data chunk i. The bias of the coherence is then corrected using the approach proposed in [87].
ˆ
γRL2 = ˆγRL,biased2 − 1
nC(1−γRL2 )2
. (5.15)
Although the spectra are averaged over several chunks, the random error in both estimates is still visible. Based on [86], the standard deviation of the random error can be calculated by
σ(ˆγRL2 ) =
s2γRL2 (1−γRL2 )2
nC (5.16a)
σ(|GˆRL|) =
s(1−γRL2 )|GRL|2
2γRL2 nC . (5.16b)
The estimates and the corresponding ±σ-confidence bounds are shown in Figure 5.2. The agreement not only validates the analytic approach, but also demonstrates the advantage: with the proposed model the correlation between lidar measurements and the rotor effective wind speed can be directly calculated without the inaccuracy involved in the spectra estimation process. By dividing the signals in more data chunks, the bias and the random error can be decreased. However, there will be less data points per chunk and thereby the frequency resolution will degrade (here the sampling frequency isfs= 5121 Hz). It has to be mentioned that usually the segments are weighted with cosine windows (e.g., von Hann or Hamming functions) to decrease the spectral leakage (smearing of spectral content due to the finite length of the FFT) and overlapped to recover the information lost by the windowing.
5.2.5 Comparison to Semi-analytical Approach
These calculations above show that a fully analytic approach is already very complicated for the simplest lidar set-up when the integration has to be done “by hand”. However, to evaluate several more realistic lidar set-ups, a simpler and automated approach is important. For this purpose, a semi-analytic approach is developed to avoid the integration necessary for the deriva-tion of the correladeriva-tion model. Here, the basic idea is to discretize and linearize the equaderiva-tions where possible before applying the Fourier transformation. Using the linearity property of the Fourier transform, all spectra can be finally calculated by sums and products of wind spectra and coherences.
The determination of the auto-spectrumSRR and the cross-spectrum SRL can be simplified by
nR= 777 nR= 193
nR= 45 nR = 9
nR= 1
nR=∞ nR= 777 nR= 193 nR= 45 nR= 9 nR= 1
k [rad/m]
SRR[(m/s)2 /Hz]
10−3 10−2 10−1 100
10−4 10−2 100 102
Figure 5.3:Auto-spectrum of the rotor effective wind speed (rotor diameter 126 m) with the analytical model (black) and several semi-analytical models (shades of gray).
using the discrete definition of the rotor effective wind speed (3.20):
SRR=F ( 1
nR nR
X
i=1
ui,I )
F∗ ( 1
nR nR
X
i=1
ui,I )
= Sii,u n2R
nR
X
i=1 nR
X
j=1
γij,ur
SRL =F ( 1
nR
nR
X
i=1
ui,I
)
F∗{uH} = Sii,u
nR
nR
X
j=1
γHj,ur. (5.17)
The transfer function GRL and the squared coherence γRL2 can be calculated with (5.17) in a straightforward manner and the differences compared to using (5.10) and (5.12) decrease with increasing nR. In Figure 5.3, Cartesian grids with grid resolutions of 64, 32, 16, 8, and 4 m are used, resulting in averages over nR = 1, 9, 45, 193, and 777 grid points within the rotor disc.
The figure shows that the semi-analytic spectra are approaching the analytic one (nR = ∞) as the number of grid points increases. The point of divergence from the analytical model to a trend parallel to the single-point-spectrum (nR = 1) can be roughly doubled by halving the distance between the grid points.
The advantage of the semi-analytic model is that a weighting considering tip and root losses (3.23) can be easily implemented, only slightly increasing the complexity of the model.