Environmental Engineering Reference
In-Depth Information
Proof
We now present the Lyapunov function
V
¼
1
2
e
2
:
ð
15
:
6
Þ
Then, based on Eq. (
15.3
), the time derivative along the trajectory of the system
yields
V
¼
e e
¼
e
ð
ae
K
a
sgn
ð
e
ÞÞ ¼
ae
2
K
a
j
e
j
\0
:
ð
15
:
7
Þ
Thus, V is globally positive definite and radially unbounded, while the time
derivative of the Lyapunov-candidate-function is globally negative definite; so
the equilibrium is proven to be globally asymptotically stable. Moreover, finite
time stability can be proven. Equation (
15.7
) can be written as
p
2
p
V
V
K
a
j
e
j¼
K
a
:
p
2
p
V
V
þ
K
a
is negative semidefinite and Theorem 1 in [
4
] can be
applied to conclude that the origin is a finite time stable equilibrium. Furthermore,
from [
4
], the settling time function t
s
is described as
Thus,
1
p ð
V
Þ
1
=
2
;
t
s
2
K
a
and using Eq. (
15.6
) leads to
t
s
e
K
a
:
h
The proposed simple nonlinear torque control, see Eq. (
15.5
), does not require
information from the turbine total external damping or the turbine total inertia.
This control only requires the generator speed and electrical power of the WT.
Thus, our proposed controller requires few WT parameters and it does not need to
filter the generator speed measurement.
15.5.3 Pitch Control
To assist the torque controller with regulating the wind turbine electric power
output, while avoiding significant loads and maintaining the rotor speed within
acceptable limits, a collective pitch controller is added to the rotor speed tracking
error. The pitch angle controller is a gain scheduling PI-controller with the gen-
erator speed as input and the pitch servo set-point as output. That is,
