International Journal of Engineering Trends and Technology (IJETT) – Volume 33 Number 6- March 2016
Sliding Mode Control for the Stability
Analysis of a Variable Speed Wind Turbine
Aleena k sunny#1, M.Nishanthi*2
#
PG student, Power Systems Engineering, Sri Krishna College of Technology, Coimbatore, TamilNadu, India.
*
Assistant Professor, EEE Department,Sri Krishna College of Technology, Coimbatore, TamilNadu, India.
Abstract - The variable speed wind turbine is a highly
nonlinear system. The design of controller for such an
unstable system is a challenging task. This paper deals with
the stability analysis of a variable speed wind turbine using
sliding mode control. Sliding mode control is a nonlinear
control technique featuring remarkable properties of
accuracy, robustness, and easy tuning and implementation.
Two methods of sliding mode control is compared here to
make the system stable. Sliding mode control systems are
designed to drive the system states onto a particular surface
in the state space, named sliding surface. Once the sliding
surface is reached, sliding mode control keeps the states on
the close neighbourhood of the sliding surface. Thus the
dynamic behaviour of the system may be tailored by the
particular choice of the sliding function and the closed loop
response becomes totally insensitive to some particular
uncertainties. The sliding mode control strategy improves
the overall stability of the variable speed wind turbine
despite of its model uncertainties and external disturbances.
The proposed strategy is applied and the results are
obtained using MATLAB.
Keywords – Variable Speed Wind Turbines (VSWT), Wind
Energy Conversion System (WECS), Sliding Mode Control
(SMC),State Space.
I. INTRODUCTION
In recent years wind energy is becoming
more popular among other renewable sources due to
more advanced technologies. Reduction in the cost of
wind energy requires more efficient technology which
is able to extract optimum power form the wind.
Nowadays, wind energy is one of the most popular
and fastest growing renewable sources among others.
In general WT are classified in to two typed i.e. fixed
speed WT and variable speed wind turbine (VSWT).
VSWT is having more advantage than fixed speed WT
because, it is possible to track the variation in wind
speed by controlling the shaft speed which ultimately
leads to optimal power generation So, control strategy
plays a major role on WT characteristics and transient
load to the network. In VSWT the operating regions
are classified in to two major categories i.e. below and
above rated wind speed. At below rated wind speed
the main objective of the controller (i.e. torque control)
is to optimize the wind energy capture by tracking the
wind speed. At above rated wind speed the major
objective of the controller (i.e. pitch control) is to
maintain the rated power of the WT. For extracting the
maximum power at below rated
ISSN: 2231-5381
wind speed the WT rotor speed should operate at
reference rotor speed which is derived from effective
wind speed.
Variable speed WTs is continuously
increasing their market share, since they can possible
to track the changes in wind speed by adapting shaft
speed and, thus, maintaining optimal power generation.
The more the variable speed WTs are investigated, the
more it becomes obvious that their behaviour is
significantly affected by the control strategy used.
Usually VS-WECS use aerodynamic controls in
combination with power electronics to regulate torque,
speed, and power. The aerodynamic control systems,
usually variable-pitch blades or trailing-edge devices,
are expensive and complex, especially when the
turbines are larger.
A convenient control law is proposed instead
of applying classical sliding mode approaches such as
the twisting or super twisting algorithms. The
objective of the sliding mode controller design is the
stabilisation of the variable speed wind turbine despite
of its model uncertainties and external disturbances.
The sliding mode control approach is recognized as
one of the efficient tools to design the robust
controllers for complex high order non-linear dynamic
plants operating under uncertainty conditions. The
major advantage of sliding mode control
low
sensitivity to parameter variations and disturbances
which eliminates the necessity of exact modelling.
In section II, a general introduction about the
wind energy conversion system is given. In section III
the sliding mode control technique is elaborated. The
steps involved in the design of a variable structure
control are also included. The modelling of a small
scale variable speed wind turbine is shown in section
IV. The known model of the plant is necessary for the
reaching law to synthesise the variable control law. In
section V, the variable structure controller is designed
and a sliding mode control law is implemented to
make the state variables to reach the desired steady
state value. A convenient control law is proposed
instead of applying classical sliding mode approaches
such as the twisting or super twisting algorithms. The
objective of the sliding mode controller design is the
stabilisation of the variable speed wind turbine. The
simulation results obtained using MATLAB software
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International Journal of Engineering Trends and Technology (IJETT) – Volume 33 Number 6- March 2016
package are shown in section VI. The results show the
convergence of the sliding variables to zero in a
specific time, ensuring the system to reach the steady
state values.
II. WIND ENERGY CONVERSION SYSTEM
Differential heating of the earth's surface by
the sun causes the movement of large air masses on
the surface of the earth, i.e., the wind. Wind energy
conversion systems convert the kinetic energy of the
wind into electricity or other forms of energy. Wind
power generation has experienced a tremendous
growth in the past decade, and has been recognized as
an environmentally friendly and economically
competitive means of electric power generation.
The major components of a typical wind
energy conversion system include a wind turbine,
generator, interconnection apparatus and control
systems. Wind turbines can be classified into the
vertical axis type and the horizontal axis type. Most
modern wind turbines use a horizontal axis
configuration with two or three blades, operating
either down-wind or up-wind. A wind turbine can be
designed for a constant speed or variable speed
operation.
Variable speed wind turbines can produce
8% to 15% more energy output as compared to their
constant speed counterparts, however, they necessitate
power electronic converters to provide a fixed
frequency and fixed voltage power to their loads.
Most turbine manufacturers have opted for reduction
gears between the low speed turbine rotor and the high
speed
three-phase
generators.
Direct
drive
configuration, where a generator is coupled to the
rotor of a wind turbine directly, offers high reliability,
low maintenance, and possibly low cost for certain
turbines.
For small to medium power wind turbines,
permanent magnet generators and squirrel cage
induction generators are often used because of their
reliability and cost advantages. Induction generators,
permanent magnet synchronous generators and wound
field synchronous generators are currently used in
various high power wind turbines. Interconnection
apparatuses are devices to achieve power control, soft
start and interconnection functions. Very often power
electronic converters are used as such devices.
III.SLIDING MODE CONTROL
The Sliding Mode Control (SMC) strategy
became the principle operation mode in so called
variable structure control (VSC) and its structure
changes with current value of its state [8, 14].
Variable structure systems consist of a set of
continuous subsystems with a proper switching logic
ISSN: 2231-5381
and, as a result, control actions are discontinuous
functions of sys-tem state, disturbances (if they are
accessible for measurement), and reference inputs. In
the course of the entire history of control theory,
intensity
of
discontinuous
control
systems
investigation has been maintained at a high enough
level. In particular, at the first stage, on-off or bangbang regulators are ranked highly due to ease of
implementation and efficiency of control hardware.
Also the system switches infinitely many times in a
single time instant [16].
Consider a plant described as,
(1)
(2)
Where x is an n-vector, u is a scalar, and A and B are
of appropriate dimensions. The design of the variable
structure control is recognised to have the following
steps:
Determination of the switching function s(x)
such that the sliding mode on the switching
plane is constantly stable. i.e., s(x) = 0.
Determination of a control law such that a
reaching condition must be pleased. i.e.,
(3)
This infers that any state, starting from any
initial state, will move toward the switching plane on
the switching surface and attain the steady state in
finite time.
The physical meaning of above statement is as follows:
Design a switching surface s(x) = 0 to
represent a desired system dynamics, which
is of lower order than the given plant.
Design a variable structure control u(x, t)
such that any state x outside the switching
surface is driven to reach the surface in finite
time. On the switching surface, the sliding
mode takes place, following the desired
system dynamics. In this way, the overall
VSC system is globally asymptotically
stable.
As a result, the closed loop control system is
globally stable. The response of such a system in
general consists of three modes, namely, the reaching
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International Journal of Engineering Trends and Technology (IJETT) – Volume 33 Number 6- March 2016
mode (RM), sliding mode (SM), and steady-state
mode (SS), as shown in Figure.1 (for a second order
system).
Instead of first establishing an analytical
expression of a reaching condition and then designing
a control law to meet the situation, here a different but
much more convenient method is adopted, called the
reaching law approach [13, 12].This reaching law
unswervingly dictates the dynamics of the switching
function s(x) = 0 and then, a VSC control law is
created from the reaching law with recognized model
of the plant and known constraints of perturbations.
where,
Jtot is the total inertia of the system [Kg/m2].
ΩG is the angular velocity of the generator [rad/s].
N is the gear ratio.
Tr is the force provided by the rotor [Nm].
KG is the generator’s torque constant [Nm/A].
I is the current through the generator coil [A].
B is the viscous friction coefficient for the system
[Nm/(rad/s)].
The electrical equation for the generator is given by,
(6)
Where,
UL is the voltage provided by the load [V].
RG is the terminal resistance of the generator [Ω].
The servomechanism can be modeled by a first order
linear differential equation as,
(7)
Fig.1 Trajectories of a VSC system
A convenient reaching law for the VSC of a plant is
given by,
where,
β is the pitch duty cycle.
The state, which is wanted to control, together with
the input and output of the system, has to be
recognized, in order to derive a state model. They are
listed below:
States: The angular velocity, ΩG and the pitch duty
cycle, β.
Input:
Voltage on the generator terminals, U L and
(4)
pitch duty cycle, βref.
Output: The angular velocity, ΩG.
IV. MODELLING OF A SMALL SCALE
VARIABLE SPEED WIND TURBINE
The state space model of the system is derived using
(5) and (6) combined with (7).
The mechanical model describes what forces
and torques affect the differential mechanical parts of
the system. Figure.2 shows how the different parts are
linked together.
The equation can be solved with respect to I and is
given by,
(8)
Inserting the above equation into the mechanical
differential equation for the generator, it becomes
(9)
Fig.2 Mechanical model of wind turbine
i.e.,
The system modeling is stimulated from the
study [4].The mechanical equation for the generator is
given by,
(10)
(5)
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International Journal of Engineering Trends and Technology (IJETT) – Volume 33 Number 6- March 2016
Tr is linearized around a specific operating point, as it
is a nonlinear function of together wind speed, rotor
speed and pitch angle. The linearization is done with
respect to ΩG and β, thus resulting in the gradient at
the operating point as,
In order to verify whether this model is
realistic, the poles are calculated and both are seem to
be negative. The motor and gear parameters have been
determined from the datasheet as follows:
TABLE I Motor and gear parameters
(11)
i.e.,
PARAMETERS
VALUE
Generator torque constant, KG
38.2*10-3Nm/A
Viscous force coefficient, B
42.4*106
Nm/(rad/s)
Gear ratio, N
1/11
Speed coefficient, Br
0.773*10-3
Terminal resistance, RG
7.19Ω
Total inertia of system, Jtot
199*10-6Kgm2
Pitch coefficient, Kβ
38.4
Time constant, τ
0.02s
(12)
Replacing Tr with (10), we get
(13)
Using the following matrix notation,
V. DESIGN OF VARIABLE STRUCTURE
CONTROLLER
(14)
Consider a plant,
(15)
(18)
The system can now be represented by the set of
matrices as,
Where x is an n-vector, u is a scalar, and A and B are
of appropriate dimensions.
Consider a linear switching function as,
(19)
Then the linear switching plane is given by,
(20)
On substituting equation in equation, we get
(16)
(21)
Solving for u(t), an equivalent control is given by
(17)
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(22)
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International Journal of Engineering Trends and Technology (IJETT) – Volume 33 Number 6- March 2016
VI.
SIMULATION AND RESULTS
System response with an initial speed of 8m/s
8
To prove the effectiveness of the SMC
technique in VSWT, a small scale variable speed wind
turbine system is used and the parameters of that
system are,
7
6
---------
Wind Speed
5
;
;
Gao’s methodology
Barwtodis methodology
4
3
2
1
;
0
;
-1
0
20
40
60
80
100
120
Time
Fig.4 Sliding variable, Ω Vs Time
The initial condition assumed is
.
System response with an initial pitch angle of 9
9
With the above values, the system is simulated
and the results obtained are given in Figure 3, Figure 4
and Figure 5.
In Figure 3, the sliding surface is shown. As
it can be noticed, s(t) reach steady state irrespective of
the initial perturbations. In Figures 4 and 5, the state
variables, viz., the speed and the pitch angle of the
wind turbine are shown. It is shown from the figures
that the two variables reaches smoothly to steady state
without any overshoot.Also we can see that the
stability can be attained faster in Gao’s methodology
than in Barwtodis methodology.
8
7
Pitch angle
6
----------
Gao’s methodology
Barwtodis methodology
5
4
3
2
1
0
-1
0
20
40
60
80
Time
Fig.5 Sliding variable, β Vs Time
System response with an initial speed of 8m/s
8
7
The
---------
Wind Speed
5
control effort is calculated as
. The following Table 2
summarizes the performance parameters of the
controller considering without model uncertainties and
external disturbances.
Gao’s methodology
Barwtodis methodology
6
4
3
2
TABLE II Performance parameters of the controller
1
0
-1
SYSTEM STATE
0
20
40
60
80
100
120
Time
Fig.3 Evolution of the sliding surface, s(t)
CASE
Without model
uncertainties
and external
disturbances
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CONTROL
EFFORT
MAXIMUM
PEAK
OVERSHOOT
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International Journal of Engineering Trends and Technology (IJETT) – Volume 33 Number 6- March 2016
[3]
VII. CONCLUSION
This paper dealt with the problem of
instability of the variable speed wind turbines despite
of model uncertainties and external disturbances. The
sliding mode control strategy implemented in this
paper has its special characteristics such as it ensures
stability and robustness despite model uncertainties
and external disturbances. Thus, this paper has
presented the theory and design technique for variable
structure control of a small scale variable speed wind
turbine. The results obtained clearly shows that the
system reach the steady state in finite time irrespective
of the initial perturbations in Gao’s methodology than
in Barwtodis methodology. Although the controller
proposed here can stabilize a small scale variable
speed wind turbine, it is important to implement them
in real time system.
ACKNOWLEDGEMENT
The authors would like to thank the Principal,
the Hod and to all faculty members of EEE
Department, Friends who have render their valuable
help in completing this paper successful.
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