Environmental Engineering Reference
In-Depth Information
where
G
is the “gain” matrix. If the system consisting of (
A
,
B
) in Equation (14-3a) is “con-
trollable”, this feedback law can be used to place the poles of this system arbitrarily in the
complex plane]. This is what allows active damping to be added to normally low-damped
turbine components and thereby significantly reduce fatigue loads. Either pole placement or
linear quadratic regulation (LQR) can be used to calculate the gain matrix
G
[Kwakernaak
and Sivan 1972].
With LQR, a unique linear feedback control signal is found that will minimize the fol-
lowing quadratic cost function:
J
=
¥
0
D (
t
)
T
Q
D (
t
)+D (
t
)
T
R
D (
t
)
dt
(5)
ò
x
x
u
u
where, D
x
(
t
) represents the system states, D
u
(
t
) represents the control inputs,
Q
contains
weightings for the states, and
R
contains weightings for actuator usage. Fast state regulation
and low actuator use are competing objectives; therefore, the
Q
and
R
weightings allow a
trade-off of performance objectives with actuator use. Kwakernaak and Sivan [1972] pro-
vide more details regarding LQR.
To use full state feedback as the final control design, one would have to measure every
state contained in the linear model described by Equation (14-3). Most current commercial
wind turbines are not instrumented to the extent needed to measure all these states, especially
as the order of the model increases. Usually only a limited number of measured signals are
available for control, such as generator or rotor azimuth angle, rotor speed, tower-top accel-
eration, and blade-root bending moments. However, according to Kwakernaak and Sivan,
observability
allows us to estimate the states contained in the linear model based on just a
few turbine measurements.
Progress in Advanced Controls for Wind Turbines
Some very early work in developing advanced state-space controls for wind turbines
was performed by Mattson [1984] who used a state estimator in combination with LQR. He
described regulation of power for a fixed-speed machine using blade pitch, reporting good
results except in frequency intervals close to the natural frequency of the first drive train
torsional mode. This was due to the amplified effects of measurement noise caused by the
controller attempting to compensate for phase lag at this natural frequency. Liebst [1985]
developed an individual pitch control system for a wind turbine using LQR design to allevi-
ate blade loads caused by wind shear, gravity and tower deflection.
Important progress has been made in simplifying dynamic models as expressed in
Equation (14-3) for use in control design. Large multi-body dynamics codes [
e.g.
MSC.
ADAMS®] divide a complex structure into numerous rigid-body masses and then connect
these parts with springs and dampers. This approach leads to nonlinear dynamic models
with hundreds or thousands of
degrees of freedom
(DOF). The order of these models must
be greatly reduced to make them practical for control design. In addition, these nonlinear
models need to be linearized in order to apply linear control theory.
An example of tools for simulation and linear model generation is a nonlinear wind tur-
bine simulation tool called DUWECS that was developed at the Delft University of Technol-
ogy [Bongers 1994]. This code models the effects of blade and tower flexibility, nacelle yaw,
and rotor teeter motion, as well as drive train torsional flexibility. Blade and tower flexibility
are modeled using a rigid blade/tower hinge approach. This approach leads to models with
fewer DOF than those developed with either multi-body dynamics codes or finite element
methods. DUWECS tools not only simulate the entire nonlinear turbine but also generate a
linear model of a turbine for control design.
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