xW
yW zW
Figure 3.4: Snapshot of the time variant vector field as a general description of wind. Wind speed vectors for aero-elastic simulations are only assigned to the rotor plane, but need to be assigned also
to other locations to simulate lidar systems.
3.2 Wind Modeling 35
xI
yI zI
αh
αv xW
yW zW
Figure 3.5:Orientation of the wind coordinate system (subscriptW) in the inertial coordinate system (subscriptI). Rotation order is defined as azimuth → elevation (αh →αv).
components [ui,W vi,W wi,W]T in a point i from the wind to the inertial coordinate system is then calculated with the rotation matrix TIW by
ui,I
vi,I
wi,I
=
cos(αh) −sin(αh) 0 sin(αh) cos(αh) 0
0 0 1
| {z }
Tazimuth
cos(αv) 0 sin(αv)
0 1 0
−sin(αv) 0 cos(αv)
| {z }
Televation
| {z }
TIW
ui,W
vi,W
wi,W
. (3.12)
The transformation from the inertial to the wind coordinate system is done by
ui,W
vi,W wi,W
=TWI
ui,I
vi,I wi,I
with TWI =TIW−1 =Televation−1 Tazimuth−1 . (3.13)
In the majority of this work, the wind turbine is assumed to be perfectly aligned with the mean wind direction. This implies that both inflow angles are zero and the wind coordinate system coincides with the inertial coordinate system. However, for the wind reconstruction in Chapter 4, the three-dimensional wind flowing towards the wind turbine is considered to slowly change its direction. In the next subsections, several wind models in the inertial coordinate system are presented.
3.2.2 Wind Models for Aero-Elastic and Lidar Simulations
As described in Chapter 2, time-variant wind fields defined on a two-dimensional grid are, in general, sufficient to be used for aero-elastic simulations, because only the aerodynamic forces and moments in the rotor plane need to be calculated. These vector fields [u2D v2D w2D]T are usually aligned with the inertial coordinate system and assign a three-dimensional wind vector [ui,W vi,W wi,W]T to each point i with lateral and horizontal coordinate yi,W and zi,W at each time ti by interpolation. This leads to the following definition:
Definition (Wind model for aero-elastic simulation). The wind model for aero-elastic simulation assigns a wind speed vector[ui,I vi,I wi,I]Tto each pointiwith coordinates[0yi,I zi,I]T at time ti, with a given vector field [u2D v2D w2D]T, as described by
ui,I vi,I wi,I
=
ui,W vi,W wi,W
with
ui,W vi,W wi,W
=
u2D(yi,W, zi,W, ti) v2D(yi,W, zi,W, ti) w2D(yi,W, zi,W, ti)
and
"
yi,W
zi,W
#
=
"
yi,I
zi,I
#
. (3.14)
However, for the simulation of lidar systems measuring in front of the rotor plane, time-variant wind fields defined for coordinates outside the rotor plane (xi,I 6= 0) are necessary, see Fig-ure 3.5.
In this work, the method introduced by [1] is used to obtain wind speed vectors for coordinates away from the rotor. Taylor’s Frozen Turbulence Hypothesis [66] assumes that the turbulence remains unchanged while traveling with the mean wind speed ¯u. Hence, the wind at time ti
in the downwind location [xi,W yi,W zi,W]T passed by the point [0 yi,W zi,W]T at the rotor disc earlier at time tR,i = ti− xi,Wu¯ . The advantage of this method is that the same wind field can be used for the aero-elastic simulation as well as for the lidar simulations. For ¯u, the constant mean wind speed at hub height from the wind field can be applied. Thus, the following wind model is defined:
Definition (Wind model for lidar simulation). The wind model for lidar simulation assigns a wind speed vector [ui,I vi,I wi,I]T to each point i with coordinates [xi,I yi,I zi,I]T at time ti, with a given wind vector field [u2D v2D w2D]T, with a mean wind speedu, and with given inflow¯ angles αh and αv, as described by
ui,I
vi,I
wi,I
=TIW(αh, αv)
ui,W
vi,W
wi,W
with
ui,W
vi,W
wi,W
=
u2D(yi,W, zi,W, tR,i) v2D(yi,W, zi,W, tR,i) w2D(yi,W, zi,W, tR,i)
,
tR,i =ti− xi,W
¯
u , and
xi,W yi,W zi,W
=TWI(αh, αv)
xi,I yi,I zi,I
. (3.15)
3.2 Wind Modeling 37
xI
yI zI
xW
yW zW
αh αv
δvzW
δhyW v0
Figure 3.6: Three-dimensional inhomogeneous flow model: the wind vector field is parameterized by a rotor effective wind speed v0, a linear horizontal and vertical shears δh and δv, and the horizontal
and the vertical inflow anglesαh and αv.
However, measurements in front of the turbine reveal that the wind field evolves when moving from the point of measurement to the rotor plane [67]. There are several approaches to include the wind evolution in the wind fields used for simulations: in [20, 68] wind component signals are generated at the lidar measurement position using the wind evolution model of [56] in addition to the two-dimensional wind fields used for aero-elastic simulations. Even more realistic three-dimensional time-variant wind fields can be obtained from Computational Fluid Dynamics (CFD) simulations. The advantage of these wind fields is that effects, such as the induction zone, can be included by coupling CFD simulations to aero-elastic simulations [69]. However, these coupled simulations are still computationally very extensive.
3.2.3 Reduced Wind Model for Wind Field Reconstruction
Usually, the following homogeneous flow model is used by commercial lidar systems:
Definition (Homogeneous flow model). The homogeneous flow model assigns the same wind speed vector [uI vI wI]T to each point i:
ui,I vi,I wi,I
=
uI vI wI
. (3.16)
However, this model has some shortcomings as explained later in Chapter 4. Here, an inho-mogeneous flow model is proposed that neglects the lateral and vertical wind components vW and wW in the W-system. The longitudinal wind componentuW is parameterized by the rotor effective wind speed v0, the linear horizontal shear δh, the linear vertical shear δv, the horizon-tal inflow angle αh, and the vertical inflow angle αv (see Figure 3.6). These parameters are denoted in this work as wind characteristics. In the static case, the wind field and the wind characteristics are assumed to be constant for a given period of time:
Definition (Static inhomogeneous flow model). The static inhomogeneous flow model assigns a wind speed vector [ui,I vi,I wi,I]T to each pointi with coordinates [xi,I yi,I zi,I]T, with given wind characteristics v0, δh, δv, αh, and αv, as described by:
ui,I
vi,I wi,I
=TIW(αh, αv)
ui,W
vi,W wi,W
with
ui,W
vi,W wi,W
=
v0+δhyi,W+δvzi,W
0 0
and
xi,W yi,W
zi,W
=TWI(αh, αv)
xi,I yi,I
zi,I
. (3.17)
For lidar-assisted control, it is also beneficial to take the temporal relationship of the wind propagating along the mean wind direction into account. Here, the rotor effective wind speed v0(t) and the shears δv(t) and δh(t) are assumed to be time-dependent and Taylor’s Frozen Turbulence Hypothesis is applied. This is similar to model (3.15), which is used for the lidar simulation. With the time-dependent wind characteristics, the following model is defined:
Definition (Dynamic inhomogeneous flow model). The dynamic inhomogeneous flow model assigns a wind speed vector[ui,I vi,I wi,I]T to each pointi with coordinates[xi,I yi,I zi,I]T at time ti, with given time-dependent wind characteristics v0(t), δh(t), and δv(t), with given scalar wind characteristics αh and αv, and with a given mean wind speed u, as described by:¯
ui,I
vi,I
wi,I
=TIW(αh, αv)
ui,W
vi,W
wi,W
with
ui,W
vi,W
wi,W
=
v0(tR,i) +δh(tR,i)yi,W+δv(tR,i)zi,W
0 0
tR,i=ti− xi,W
¯
u , and
xi,W
yi,W
zi,W
=TWI(αh, αv)
xi,I
yi,I
zi,I
. (3.18)
The idea behind the wind characteristics is to have signals representing the effect of the wind field on the rotor of a simplified wind turbine. A wind field with the same rotor effective wind
3.2 Wind Modeling 39 speed v0 should generate the same aerodynamic torque. In the following subsection, a linear unweighted and a squared weighted case are presented, which will both be used in this work.
3.2.4 Wind Characteristics from Wind Fields
In the linear and unweighted case, it is assumed that the rotor effective wind speed is an average of the wind flowing through the rotor disk. This can be expressed by the integral of the longitudinal wind component over the rotor area with rotor radius R divided by the rotor area:
v0 = 1 πR2
Z2π 0
ZR 0
uI rdrdφ, (3.19)
with the polar coordinates r=p
yI2 +zI2 and φ = arctan(yI/zI).
For a discrete-dimensional wind field, Equation (3.19) is simplified by a mean over all nR
longitudinal wind components within the rotor disk (Riemann sum):
v0 = 1 nR
nR
X
i=1
ui,I. (3.20)
In the squared and weighted case, it is assumed that the impact of the longitudinal wind component uI depends on the contribution to the aerodynamic torque:
v0 = vu uu uu uu uu ut
Z2π 0
ZR 0
∂cP
∂r (r)u2Irdrdφ Z2π
0
ZR 0
∂cP
∂r (r)rdrdφ
. (3.21)
The stationary spanwise variation of power extraction ∂c∂rP(r) with ZR
0
∂cP
∂r (r) dr =cP,max≤cP,Betz with cP,Betz = ZR
0
32
27R2rdr= 16
27 (3.22)
can be obtained by modeling tip and root losses following [70] and [41]. Figure 3.7 shows the used weighting function with cP,max and rmin from Table C.2. This curve is an approximation of the aerodynamics and will differ in aero-elastic simulations from the actual values. However, it still covers the effect of tip and root losses compared to a simple average over the rotor disc.
For a discrete wind field, Equation (3.21) is simplified to a weighted sum of all longitudinal
r/R[m/m]
R∂cP/∂r[-]
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1 1.2
Figure 3.7:Used span-wise variation of power extraction in the presence of tip and root losses (black) and Betz-optimal curve without losses (gray).
wind components within the rotor disc:
v0 =
nR
X
i=1
cP,i ui,I nR
X
i=1
cP,i
, (3.23)
with cP,i = ∂c∂rP(ri) evaluated at the distance ri =q
yi,2I +zi,2I from the hub.
A linear vertical shear δv is chosen for this work instead of a logarithmic or exponential shear, which are often used for the description of averaged vertical wind profiles. One advantage of using linear vertical shear is that it can be directly used as a disturbance input for the reduced wind turbine model. Another advantage is that fitting a wind field with a normal turbulence level to a linear shear is more robust compared to logarithmic or exponential shear models.
In the linear and unweighted case, the shears can be extracted from a two-dimensional wind field using a linear fit. The equations for longitudinal wind components in (3.17) can be arranged in the following form for all nR points inside the rotor disk and for each time ti:
u1,I
: unR,I
| {z }
m
=
1 y1,I z1,I
: : :
1 ynR,I znR,I
| {z }
A
v0
δh
δv
| {z }
s
. (3.24)
The wanted vectorscan then be obtained by applying the least squares method given that the matrix A represents the linear coefficients and vector mthe longitudinal wind components:
mins km−Ask2. (3.25)
For the rotor effective wind v0, Equation (3.25) is equivalent to (3.20) due to the linearity.
3.2 Wind Modeling 41
32 28 30 26 24 22 20 16 18
14 12 10 8 6
θ [deg]
Ma[MNm]
0 5 10 15 20 25 30
2 4 6 8
Figure 3.8:Contour lines of rotor effective wind speeds at Ω = 12.1 rpm for the NREL 5 MW turbine.
3.2.5 Wind Characteristics from Turbine Data
The rotor effective wind speedv0 can be also obtained from turbine data by an estimator similar to the one presented in [71]. Here, the reduced model from Section 3.1.2 is further simplified by neglecting the tower motion. Then, with the equations for the drive train dynamics (3.3a), the aerodynamic torque (3.9a), and the tip speed ratio (3.10), one can obtain the following model
J˙Ω +MLSS=Ma (3.26a)
Ma = 1
2ρπR3cP(λ, θ)
λ v02 (3.26b)
λ= ΩR
v0 . (3.26c)
Here, MLSS is the low-speed shaft torque, which can be either measured directly or calculated from the electrical power (3.7c) or the generator torque:
MLSS = MG
iGB = Pel
ηelΩ. (3.27)
With measured data of Ω and MLSS, the aerodynamic torque Ma can be calculated using Equation (3.26a). Due to numerical issues, Equation (3.26b) is reorganized into:
λ3 = 1
2ρπR5cP(λ, θ)
Ma Ω2. (3.28)
Because of the λ-dependency of cP, an explicit expression cannot be found. The equation is solved with a set of Ma, Ω and θ, and a three-dimensional look-up table v0R(Ma,Ω, θ) is generated (see Figure 3.8), which can then be used to get a time series of v0R by a three-dimensional interpolation. An adjustment is necessary ifcPwas determined with an air density different from the measured mean value. This estimator is used for post-processing. More sophisticated estimators such as a Kalman filter [72] might be better suited for online purposes.
xI
yI zI
xL
yL zL
ΨL ΘL
ΦL
xL,I yL,I zL,I
Figure 3.9:Orientation of the lidar coordinate system (subscriptL) in the inertial coordinate system (subscriptI): Origin of the L-system within theI-system is [xL,I yL,I zL,I]T and rotation order from
L toI is defined as yaw→ pitch→ roll (ΨL→ΘL→ΦL).