![Fig. 1. Worldwide growth of wind energy installed capacity [1].](/figures/fig-1-worldwide-growth-of-wind-energy-installed-capacity-1-3pbxitz6.webp)
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High-Order Sliding-Mode Control of Variable-Speed
Wind Turbines
Brice Beltran, Tarek Ahmed-Ali, Mohamed Benbouzid
To cite this version:
Brice Beltran, Tarek Ahmed-Ali, Mohamed Benbouzid. High-Order Sliding-Mode Control of Variable-
Speed Wind Turbines. IEEE Transactions on Industrial Electronics, Institute of Electrical and Elec-
tronics Engineers, 2009, 56 (9), pp.3314-3321. �10.1109/TIE.2008.2006949�. �hal-00525191�
3314 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009
High-Order Sliding-Mode Control of
Variable-Speed Wind Turbines
Brice Beltran, Tarek Ahmed-Ali, and Mohamed El Hachemi Benbouzid, Senior Member, IEEE
Abstract—This paper deals with the power generation control
in variable-speed wind turbines. These systems have two oper-
ation regions which depend on wind turbine tip speed ratio.
A high-order sliding-mode control strategy is then proposed to
ensure stability in both operation regions and to impose the ideal
feedback control solution in spite of model uncertainties. This
control strategy presents attractive features such as robustness
to parametric uncertainties of the turbine. The proposed sliding-
mode control approach has been validated on a 1.5-MW three-
blade wind turbine using the National Renewable Energy
Laboratory wind turbine simulator FAST (Fatigue, Aerodynam-
ics, Structures, and Turbulence) code. Validation results show
that the proposed control strategy is effective in terms of power
regulation. Moreover, the sliding-mode approach is arranged so as
to produce no chattering in the generated torque that could lead to
increased mechanical stress because of strong torque variations.
Index Terms—High-order sliding mode, power generation
control, wind turbine.
NOMENCLATURE
v Wind speed (in meters per second).
ρ Air density (in kilograms per cubic meter).
R Rotor radius (in meters).
P
a
Aerodynamic power (in watts).
T
a
Aerodynamic torque (in newton–meters).
λ Tip speed ratio.
C
p
(λ) Power coefficient.
C
q
(λ) Torque coefficient.
ω
r
Rotor speed (in radians per seconds).
ω
g
Generator speed (in radians per seconds).
T
g
Generator electromagnetic torque (in newton–meters).
T
ls
Low-speed torque (in newton–meters).
T
hs
High-speed torque (in newton–meters).
K
g
Generator external damping (in newton–meters per
radian–second).
K
r
Rotor external damping (in newton–meters per
radian–second).
J
r
Rotor inertia (in kilogram–square meter).
J
g
Generator inertia (in kilogram–square meter).
J
t
Turbine total inertia (in kilogram–square meter).
K
t
Turbine total external damping (in newton–meters per
radian–second).
Manuscript received January 13, 2008; revised September 4, 2008. First
published October 31, 2008; current version published August 12, 2009.
The authors are with the Laboratoire Brestois de Mécanique et des Systèmes,
University of Brest, 29238 Brest Cedex 03, France (e-mail: brice.beltran@dga.
defense.gouv.fr; m.benbouzid@ieee.org).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2008.2006949
B
r
Rotor external stiffness (in newton–meters per
radian–second).
B
g
Generator external stiffness (in newton–meters per
radian–second).
B
t
Turbine total external stiffness (in newton–meters per
radian–second).
I. I
NTRODUCTION
W
IND ENERGY conversion is the fastest growing energy
source among the new power generation sources in the
world, and this trend should endure for some time. At the end
of 2006, the total U.S. wind energy capacity had grown to
11 603 MW or enough to provide the electrical energy needs
of more than 2.9 million American homes. Wind capacity
in the United States and in Europe has grown at a rate of
20%–30% per year over the past decade (Fig. 1). Despite this
rapid growth, wind currently provides less than 1% of total
electricity consumption in the United States. The vision of the
wind industry in the United States and in Europe is to increase
wind fraction of the electrical energy mix to more than 20%
within the next two decades [1].
Harnessing wind energy for electric power generation is an
area of research interest, and nowadays, the emphasis is given
to the cost-effective utilization of this energy aiming at quality
and reliability in the electricity delivery [2]. During the last two
decades, wind turbine sizes have been developed from 20 kW
to 2 MW, while even larger wind turbines are being designed.
Moreover, a lot of different concepts have been developed and
tested [3].
In fact, variable-speed wind turbines (VSWTs) are continu-
ously increasing their market share, since it is possible to track
the changes in wind speed by adapting shaft speed and thus
maintaining optimal power generation. The more VSWTs are
investigated, the more it becomes obvious that their behavior
is significantly affected by the control strategy used. Typically,
VSWTs use aerodynamic controls in combination with power
electronics to regulate torque, speed, and power. The aerody-
namic control systems, usually variable-pitch blades or trailing-
edge devices, are expensive and complex, particularly when
the turbines are larger. This situation provides a motivation to
consider alternative control approaches.
The main control objective of VSWT is power efficiency
maximization. To reach this goal, the turbine tip speed ratio
should be maintained at its optimum value despite wind vari-
ations. Nevertheless, control is not always aimed at capturing
as much energy as possible. In fact, in above rated wind speed,
the captured power needs to be limited. Although there are
0278-0046/$26.00 © 2009 IEEE
Authorized licensed use limited to: Mohamed Benbouzid. Downloaded on August 25, 2009 at 08:29 from IEEE Xplore. Restrictions apply.
BELTRAN et al.: HIGH-ORDER SLIDING-MODE CONTROL OF VARIABLE-SPEED WIND TURBINES 3315
Fig. 1. Worldwide growth of wind energy installed capacity [1].
Fig. 2. VSWT global scheme.
both mechanical and electrical constraints, the more severe
ones are commonly on the generator and the converter. Hence,
regulation of the power produced by the generator (i.e., the
output power) is usually intended, and this is the main objective
of this paper.
II. W
IND TURBINE MODELING
The global scheme for VSWT is given by Fig. 2. The system
modeling is inspired from the study in [4] and [5]. Moreover, a
fixed pitch VSWT, which is considered in this paper, could be
schematically represented by Fig. 3.
The aerodynamic power P
a
captured by the wind turbine is
given by
P
a
=
1
2
πρR
2
C
p
(λ)v
3
(1)
where C
p
represents the wind turbine power conversion effi-
ciency. It is a function of the tip speed ratio λ as well as the
blade pitch angle β in a pitch-controlled wind turbine. λ is
defined as the ratio of the tip speed of the turbine blades to
wind speed and is given by
λ =
Rω
r
v
. (2)
The C
p
−λ characteristics, for different values of the pitch
angle β, are shown in Fig. 4. This figure indicates that there is
one specific λ at which the turbine is most efficient.
Normally, a VSWT follows the C
p max
to capture the max-
imum power up to the rated speed by varying the rotor speed
to keep the system at λ
opt
. Then, it operates at the rated
power with power regulation during high wind periods by active
control of the blade pitch angle or passive regulation based on
aerodynamic stall [6].
The rotor power (aerodynamic power) is also defined by
P
a
= ω
r
T
a
. (3)
Moreover
C
q
(λ)=
C
p
(λ)
λ
. (4)
Fig. 3. Wind turbine drive train dynamics.
It comes then that the aerodynamic torque is given by
T
a
=
1
2
πρR
3
C
q
(λ)v
2
(5)
and the optimum torque by
T
opt
= k
opt
ω
2
, with k
opt
=
1
2λ
2
opt
ρπR
5
C
q max
where λ
opt
is the optimal tip speed ratio.
According to Fig. 3, the aerodynamic torque T
a
will drive the
wind turbine at the speed ω
r
. The low-speed torque T
ls
actsasa
braking torque on the rotor. The generator is driven by the high-
speed torque T
hs
and braked by the generator electromagnetic
torque T
g
. Through the gearbox, the rotor speed is increased by
the gearbox ratio n
g
to obtain the generator speed ω
g
while the
low-speed torque is augmented.
The rotor dynamics, together with the generator inertia, are
characterized by the following differential equations:
J
r
˙ω
r
= T
a
− K
r
ω
r
− B
r
θ
r
− T
ls
J
g
˙ω
g
− T
hs
− K
g
ω
g
− B
g
θ
g
− T
g
.
(6)
The gearbox ratio is defined as
n
g
=
ω
g
ω
r
=
T
ls
T
hs
. (7)
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3316 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009
Fig. 4. Wind turbine power and torque coefficients [7].
It comes then that
J
t
˙ω
r
= T
a
− K
t
ω
r
− B
t
θ
r
− T
g
(8)
where
⎧
⎪
⎪
⎨
⎪
⎪
⎩
J
t
= J
r
+ n
2
g
J
g
K
t
= K
r
+ n
2
g
K
g
B
t
= B
r
+ n
2
g
B
g
T
g
= n
g
T
em
.
(9)
Since the external stiffness B
t
is very low, it can be neglected
(the combined inertia of the generator and the rotor is dominat-
ing). This leads to represent the drive train as single lumped
mass for control purposes. We will then use the following
simplified model for control purposes:
J
t
˙ω
r
= T
a
− K
t
ω
r
− T
g
. (10)
The generated power will finally be given by
P
g
= T
g
ω
r
. (11)
III. R
OBUST CONTROL DESIGN
A. Problem Formulation
Wind turbines are designed to produce electrical energy as
cheaply as possible. Therefore, they are generally designed so
that they yield maximum output at wind speeds around 15 m/s.
In case of stronger winds, it is necessary to waste part of the
excess energy of the wind in order to avoid damaging the wind
turbine. All wind turbines are therefore designed with some sort
of power control. This standard control law keeps the turbine
operating at the peak of its C
p
curve
T
g
= kω
2
, with k =
1
2
πρR
5
C
p max
λ
3
opt
.
There are two significant problems with this standard
control. The first one is that there is no accurate way to
determine k, particularly since blade aerodynamics can change
significantly over time. The second one concerns the fact that
the term J
t
˙ω
r
+ K
t
ω
r
is neglected, which means that the
captured power is supposed to be equal to T
g
ω
t
. It is obvious
that in many cases, and particularly for turbulent winds, this
assumption will not be realistic. Moreover, even when it is
assumed that k can be accurately determined via simulation
or experiments, wind speed fluctuations force the turbine to
operate off the peak of its C
p
curve much of the time. Indeed,
tight tracing the maximum C
p
would lead to high mechanical
stress and transfer aerodynamic fluctuations into the power
system. This, however, will result in less energy capture.
The proposed control strategy will therefore solve the second
problem. Indeed, the proposed solution to the problem of wind
turbine maximum power point tracking (MPPT) control strat-
egy relies on the estimation of the aerodynamic torque using a
high-order sliding-mode observer [8], [9]. This estimate is then
used to derive a high-order sliding-mode controller that ensures
T
opt
tracking in finite time.
B. Aerodynamic Torque Observer
In order to estimate the aerodynamic torque, we will use the
supertwisting algorithm [9]. This algorithm has been developed
for systems with relative degree 1 to avoid the chattering
phenomena. The control law comprises two continuous terms
that, again, do not depend upon the first time derivative of the
sliding variable. The discontinuity appears only in the control
input time derivative.
Let us consider the following observer based on the super-
twisting algorithm
˙
ˆω = x − K
t
ω −
T
g
J
t
− A
1
|ˆω − ω|
1
2
sgn(ˆω − ω)
˙x = −A
2
sgn(ˆω − ω)
(12)
and the observation error e
ω
=ˆω − ω; thus, we have
˙e
ω
= −
T
a
J
t
+ x − A
1
|e
ω
|
1
2
sgn(e
ω
)
¨e
ω
= −
˙
T
a
J
t
+˙u with u = x − A
1
|e
ω
|
1
2
sgn(e
ω
).
According to the supertwisting algorithm, the gains A
1
and
A
2
are chosen as
A
1
> Φ
1
A
2
2
≥
4Φ
1
(A
1
+Φ
1
)
(A
1
−Φ
1
)
(13)
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BELTRAN et al.: HIGH-ORDER SLIDING-MODE CONTROL OF VARIABLE-SPEED WIND TURBINES 3317
Fig. 5. Optimal efficiency loci depicting the different regions of turbine control.
where Φ
1
is a positive constant which satisfies |
˙
T
a
/J
t
| < Φ
1
.
Thus, we will guarantee the convergence of e
ω
and ˙e
ω
to0in
a finite time t
f
, and from this, we deduce an estimation of the
aerodynamic torque defined by
T
a
= J
t
x ∀ t>t
f
. (14)
C. Proposed Control Strategy
To effectively extract wind power while at the same time
maintaining safe operation, the wind turbine should be driven
according to the following three fundamental operating regions
associated with wind speed, maximum allowable rotor speed,
and rated power [10], [11]. The three distinct regions are shown
in Fig. 5, where v
r max
is the wind speed at which the maximum
allowable rotor speed is reached, while v
cut−off
is the furling
wind speed at which the turbine needs to be shut down for
protection.
In practice, there are two possible regions of turbine op-
eration, namely, the high- and low-speed regions. High-speed
operation (III) is frequently bounded by the speed limit of the
machine. Conversely, regulation in the low-speed region (II) is
usually not restricted by speed constraints. However, the system
has nonlinear nonminimum phase dynamics in this region.
This adverse behavior is an obstacle to perform the regulation
task [10].
A common practice in addressing the control problem of
wind turbines is to use a linearization approach. However,
due to the stochastic operating conditions and the inevitable
uncertainties inherent in the system, such control methods come
at the price of poor system performance and low reliability
[4], [12]. Hence the need for nonlinear and robust control to
take into account these control problems [13]. In this context,
sliding-mode control seems to be an interesting approach.
Indeed, it is one of the effective nonlinear robust control
approaches since it provides system dynamics with an in-
variant property to uncertainties once the system dynamics
are controlled in the sliding mode [14]. Moreover, it is easy
to implement. For wind turbine control, sliding mode should
provide a suitable compromise between conversion efficiency
and torque oscillation smoothing [15]–[23].
The proposed control strategy combines a second-order
sliding-mode observer with a second-order sliding-mode con-
troller. This strategy does not use the wind velocity and
avoids the chattering phenomena. With this approach, effective
improvements are brought regarding a previously proposed
sliding-mode control strategy [23]. It should be noted that this
strategy requires only a rotation speed sensor.
The control objective is to optimize the capture of wind
energy by tracking the optimal torque T
opt
.
For that purpose, let us consider the tracking error
e
T
= T
opt
− T
a
(15)
where T
a
is deduced from the observer. Then, we will have
˙e
T
=2k
opt
ω(T
a
− K
t
ω − T
g
) −
˙
T
a
.
If we define the functions F and G as follows:
F =2k
opt
ω
G =2k
opt
ω(T
a
− K
t
ω) −
˙
T
a
thus, we have ¨e
T
= −F
˙
T
g
+
˙
G.
Now, let us consider the following high-order sliding-mode
controller based on the supertwisting algorithm [9]:
T
g
= y + B
1
|e
T
|
1
2
sgn(e
T
)
˙y =+B
2
sgn(e
T
)
(16)
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