March 26-29
Comparison Between Linear and Nonlinear Control
Strategies for Variable Speed Wind Turbine Power
Capture Optimization
Boubekeur Boukhezzar
Automatic Control Department, Supélec,
Plateau du Moulon, 3, rue Joliot-Curie,
91192 Gif-sur-Yvette cedex, FRANCE
E-mail: boubekeur.boukhezzar@supelec.fr
Houria Siguerdidjane
Automatic Control Department, Supélec,
Plateau du Moulon, 3, rue Joliot-Curie,
91192 Gif-sur-Yvette cedex, FRANCE
E-mail: houria.siguerdidjane@supelec.fr
Copyright © 2009 MC2D & MITI
Abstract: The purpo of this work is to compare some linear and nonlinear control strategies, with the aim of benefiting as well as possible of wind energy conversion systems. Below rated wind speed, the main control objective is to perform an optimal wind power capture while avoiding strong loads on the drive train shafts. To explicitly take into consideration the low speed shaft flexibility, a two-mass nonlinear model of the wind turbine is ud for controllers synthesis. After adapting a LQG controller bad on the linearized model, nonlinear controllers bad on a wind speed estimator are developed. They take into account the nonlinear dynamic aspect of the wind turbine and the turbulent nature of the wind. The controllers are validated upon an aeroelastic wind turbine simulator for a realistic wind speed profile. The study shows that nonlinear control strategies bring more performance in the exploitation of wind energy conversion systems.
Keywords: LQG control, nonlinear control, two-mass model, variable speed wind turbines.
1.Introduction
In perspective of wind energy electric power production increasing, it is necessary to optimize wind energy conversion systems (WECS) exploitation. Efficient production tools and methods are then necessary. Even if they are less implemented and more complicated to be controlled, variable speed wind turbines (VSWT) show many advantages compared to fixed speed wind turbines [1], [2]. Their annual production exceeds the fixed speed one by 5 to 10 %. For this kind of turbines, it is shown that the action of the control system can have a major impact on the loads experienced by the turbine [3], [4]. The design of the controller must take into account the effect on loads, and at least ensure that excessive loads will not result from the control action [5], [6]. Many of wind turbines control systems are bad on linear models. This is due to veral reasons. On the one hand, there are generally a simple analytical solutions to many control problems (LQR, pole placement, Kalman-Fltering). On
Figure 1. two-mass wind turbine model characteristics.
Figure 2. LQG controller bad on augmented state x a
the other hand, it is easier to implement such controllers in practical applications. Until now, the major part of implemented wind turbine controllers are then bad on linearized models [7], [8], [9], [10].
Wind turbine controller objectives depend on the operating area [11]. For low wind speeds, it is more important to optimize wind power capture while it is recommended to limit power production and rotor speed above the rated wind speed. The objective of this paper is to design a controller, for pow
er capture optimization, that takes into consideration the nonlinear nature of the wind turbine behavior, the flexibility of the drive-train shaft and the turbulent nature of the wind. The aspects are considered in previous works, but not simultaneously. Linear controllers bad on the two-mass wind turbine model were propod in [4], [12] and [13]. We have also propod nonlinear controllers with wind speed estimators in [14] using a one-mass model. In this work, we propo to extend this method using a two-mass model of the drive train. This is motivated by the fact that the one mass-model cannot report the flexibility of the low speed shaft. However, this flexibility induces a flexible resonant and non-resonant modes that can cau system oscillations.
This paper is organized as fellows: In Section II, after a brief prentation of the aerodynamic model, the two-mass model is given in state-space form. A backgroud of WT control objectives in low speed area is given in Section
III. LQG controller is then deduced in Section
IV from the linearized model in the aim of satisfying a compromi between wind power capture optimization and drive train load reduction. To overcome the limited performance of the linear controller, nonlinear controllers bad on the two-mass nonlinear wind turbine model is prented in Section V.
The controllers are coupled with a wind speed estimator. In Section VI, the developed
controllers are validated upon an aeroelastic wind turbine simulator for realistic high-turbulence wind speed profile under some constraints (input perturbation and measurement noi). The simulations results show better performance for the nonlinear controller regarding wind power capture while standing in acceptable control loads and drive train torque transient loads.
2.WIND TURBINE MODELLING
A two-mass model is ud to describe the wind turbine dynamics. This one is commonly ud
in the literature [13], [15], [16]. Its scheme is illustrated in Fig. 1(a).
The u of a two-mass model for controllers synthesis is motivated by the fact that the control law deduced from this model are more general and can be applied for wind turbines of different sizes. Particularly, this controllers are more adapted for high-flexibility wind turbines
that cannot be properly modelled with a one mass model [14]. In fact, it is also shown in [17]
that the two-mass model can report flexible modes in the drive train model that cannot be highlighted with the one mass model. The aerodynamic power captured by the rotor is
given by
(1)
The power coefficient depends on the blade
pitch angle
and the tip speed ratio ¸ defined
as,
(2)
the aerodynamic torque is then
(3)
infrastructure中文where
(4)
is the torque coefficient.
The rotor-side inertia
like to do 和like doing的区别dynamics is given by
(5)
the low-speed shaft torque acts as a breaking
torque on the rotor
(6)
The generator inertia is driven by the high-speed shaft and braked by the electromagnetic
torque
(7)
If we assume an ideal gearbox with a ratio ,
one have
(8)
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deducing time derivative from (6) and using (7) and (8) leads to the following dynamical
system
(9)
with
(10)
and
3. Control Objectives
One can distinguish two operating areas of a variable speed wind turbine: below and above the rated wind speed. Below the nominal power, the main control
objectives are:
1) Maximize wind power capture.
2) Reduce loads submitted by the drive train
shaft.
The power coefficient curve has a unique maximum that corresponds to an optimal
wind energy capture (Fig. 1(b)).
(11)
where (12)
Conquently, in partial load operating mode, in order to maximize wind power extraction, the
blade pitch angle is fixed to its optimal value
and with the aim of maintaining at its
optimal value, the rotor speed must be
adjusted to track the optimal reference given by
(13)
4. LQG Control
Given the diversity of available methods, it is more common to synthesize wind turbines controllers from the linearized model for an operating point corresponding to the mean wind speed. The LQG controller propod in this ction is inspired from [17] and adapted to the two-mass model we have propod. For this, we will first linearize the WT model. A. Model Linearization
As shown in (3), the nonlinear nature of WT dynamical model comes from the aerodynamic
torque.
depends on the rotor speed , the
blade pitch angle and the wind speed that is a
random highly fluctuating uncommendable input. Linearizing the aerodynamic torque for a
given operating point leads to the following
expression
注会综合阶段难吗
(14)
where
,
and are constant coefficients. We define then new state variables corresponding to the variations around the operating point
(15)
In order to achieve a compromi between aerodynamic power capture optimization and control load
s reduction, linear quadratic controller is ud with a Kalman filter to
minimize the criterion .
For low wind speeds, the blade pitch angle is
fixed
. The WT model is then a SISO system where the generator torque constitutes
the input and the rotor speed the output. The LQG block scheme, bad on extended linear reprentation (19) is shown in Fig. 2.
The control input is obtained by a linear
state feedback upon the estimated state via a
constant gain resulting from the solution of the algebraic Riccati equation
The aerodynamic torque expression is then (16)
In this ca, the linearized state space model is
(22)
(17)
with
(23)
with
the control input is then,
The state-space model matrices are given in
(17).
(24)
5. Nonlinear State Feedback Control
Assimilating the wind speed to a linear filtered non-correlated white noi, one can
小学英语阅读100篇write
(18) In order to overcome the LQG performance, the nonlinear dynamical aspect of the two-mass
model must be taken into consideration. A
nonlinear control strategy is then adopted bad
on a wind speed estimator. We will only prent
the control parts in this work, for details about
the estimator, the reader is referred to [18].
where
and depend on the mean wind speed, ground characteristics and wind speed
assumed turbulence [17]. is a unity variance
non-correlated white-noi.
A. Nonlinear Static State Feedback Control
Finally, the whole state-space reprentation of
the linearized model is
Starting from
expression
(25)
(19)
jea
one may deduce the cond time derivative of the rotor speed
B. LQG Controller Synthesis
(26)
The wind power capture optimization and load torque fluctuation reduction objectives can be simultaneously taken into account by inimizing
the criterion [17]
it is also possible to extract
from (9)
(27)
(20)
an internal exception occured
Replacing (25) and (27) in (26), it comes out
and are the weighting factors. It is shown in [17] that the minimization of (20) is
equivalent to minimize the following criterion
(28)
(21)
Let be the tracking error defined as
where
(29)
We want to impo a cond order dynamics to defined as
and
(30)
(31)
and
are chon such that is
Hurwitz. Replacing
given by (25) and given by (28), and substituting the state variable by their estimates, we conclude to the
expression of
(32)
with
B. Nonlinear Dynamic State Feedback Control The static state feedback controller is unable to deal with control disturbances. In order to reject the effect of an additive constant control perturbation, a third order error tracking
dynamics is impod
(33)
,hzx
and are chon such that the
多少钱英语
polynomial
is Hurwitz.
The time derivative of is obtained from (28) .
From (25) for
and (27) for
, we reach
as
inaccordance
(34)
with
Substituting this expression in (32) as well as
and given respectively by (25) and (28), and replacing all the variable by their estimates, the
control dynamics is thus
(35)
The coefficients Ci are detailed in (35). The nonlinear dynamic state feedback controller with estimator scheme is shown in Fig. 3. In order to achieve a compromi between wind power capture optimization and control loads reduction, we have adopted the following principles:
• The choice of a tracking dynamics that allows to fulfill the mean tendency of the wind speed, over a given time interval, while avoiding the local high-speed variations due to the turbulence.
•
The filtering of the control torque
by a low-pass filter to smooth the control input, we relieve in that way the drive train from strong efforts and fast transients.
