Environmental Engineering Reference
In-Depth Information
4.2 Wind Turbine Modeling
The energy captured by a wind turbine is a function of the rotor radius R, the wind
speed V, the rotor speed X
r
and the pitch angle b. More precisely, the rotor power
can be expressed as
P
r
ð
V
;
b
;
X
r
Þ¼
pqR
2
2
C
P
ð
k
;
b
Þ
V
3
;
ð
4
:
1
Þ
where q is the air density and k
¼
X
r
R
=
V is the tip-speed ratio. The efficiency of
the energy capture is characterized by the power coefficient C
P
ðÞ
. Figure
4.1
shows the power coefficient of the 5 MW NREL wind turbine benchmark reported
in [
6
].
The rotor torque results from dividing the captured power by the rotational
speed:
T
r
ð
V
;
b
;
X
r
Þ¼
P
r
ð
V
;
b
;
X
r
Þ=
X
r
:
ð
4
:
2
Þ
Modern wind turbines are complex mechanical systems exhibiting coupled
translational and rotational movements. This complex dynamic behavior is in
general well-captured by aeroelastic simulation codes such as the FAST (Fatigue,
Aerodynamics, Structures, and Turbulence) code developed by the National
Renewable Energy Laboratory (NREL) [
7
]. However, these models are not suit-
able for control design purposes. Simpler models including only a few oscillation
modes are enough to design a control law. Here, for the sake of clarity, a two-mass
model capturing just the first drive-train mode is used, whereas the unmodeled
dynamics will be covered by additive uncertainty. The dynamical equations
describing this model are
H
¼
X
r
X
g
=
N
g
;
X
r
¼
T
r
T
sh
;
ð
4
:
3
Þ
J
t
X
g
¼
T
sh
=
N
g
T
g
;
J
g
where the state variables are the torsion angle H, the rotor speed X
r
and the
generator speed X
g
. The model variables T
g
and T
sh
= K
s
H + B
s
(X
r
- X
g
) are
the generator and shaft torques, respectively. The model parameters are the inertia
J
t
combining the hub and the blades, the generator inertia J
g
, the gear box ratio N
g
,
and the shaft stiffness K
s
and friction B
s
coefficients. A representation of this two-
mass model can be observed in Fig.
4.2
.
In variable-speed wind turbines, the electrical generator is interfaced by a full
or partial power converter that controls the generator torque T
g
and decouples the
rotational speed from the electrical grid. Since the power converter and generator
dynamics are much faster than the mechanical subsystem, it can be assumed for
