A Lyapunov Based Approach to Enchance Wind Turbine Stability

Indonesian Journal of Electrical Engineering and Informatics (IJEEI)
Vol. 5, No. 2, June 2017, pp. 145~154
ISSN: 2089-3272, DOI: 10.11591/ijeei.v5i2.311  145
Received Aug 19, 2016; Revised April 23, 2017; Accepted May 11, 2017
A Lyapunov Based Approach to Enchance Wind
Turbine Stability
A. Bennouk*, A. Nejmi, M. Ramzi
Mathematics and physics laboratory
Faculty of Sciences and Technics, Beni Mellal
Sultan Moulay Slimane University, Beni Mellal, Morocco
e-mail: bennouk.anasse@gmail.com
Abstract
This paper introduces a nonlinear control of a wind turbine based on a Double Feed Induction
Generator. The Rotor Side converter is controlled by using field oriented control and Backstepping strategy
to enhance the dynamic stability response. The Grid Side converter is controlled by a sliding mode. These
methods aim to increase dynamic system stability for variable wind speed. Hence, The Doubly Fed
Induction Generator (DFIG) is studied in order to illustrate its behavior in case of severe disturbance, and
its dynamic response in grid connected mode for variable speed wind operation. The model is presented
and simulated under Matlab/ Simulink.
Keywords: Double Fed Induction Generator, wind turbine, Rotor Side Converter, Lyapunov, Grid Side
Converter
1. Introduction
Today, the number of wind turbines connected to the grid is steadily increasing.
Variable speed wind turbines outperform constant speed turbines in aerodynamic efficiency
while also reducing stress on the mechanical apparatus so as to make it more productive and
cost-effective [1].
Double Fed induction generator (DFIGs) is becoming more used in modern wind power
generation systems due to their variable speed operation, four quadrant active and reactive
power regulation, low converter cost, and reduced power losses compared with Permanent
Magnet Synchronous generators (PMSG).
Classic control of grid connected to DFIG is usually based on VOC (Voltage Oriented
Control) or FOC (Flux Oriented Control), it decouples the d and q rotor currents in the
synchronous frame. Control of instantaneous active and reactive power is then achieved by
regulating the decoupled rotor currents, using proportional–integral (PI) parameters and
accurate machine parameters. The weakness is shown when the machine’s converters operate
beyond their linear limits [1].
Field Oriented Control (FOC) is improved in this paper by the use of Backstepping
strategy to control the Rotor Side inverter; this method has scored positive performances. This
kind of control is generally associated with Lyapunov functions in order to increase system
stability against variable wind fluctuations.
Moreover, Variable structure control or Sliding Mode Control (SMC) strategy is an
effective and a high-frequency switching control for nonlinear systems showing uncertainties. It
features simple implementation disturbance rejection, strong robustness and fast responses.
This method will be used in the Grid Side inverter control so as to enhance Udc stability and
then reach the grid parameters in term of frequency, Total Harmonics Distortion (THD) and
voltage unbalance [2].
This paper is organized as follows: the second section is due to deal with a wind energy
modeling under Matlab/Simulink. The third section illustrates the Field Oriented Control and
Backstepping linearization control for the rotor side converter. The fourth section introduces a
SMC destined to control of the Grid Side Converter performances. Finally, a simulation with its
inherent results will be presented.

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IJEEI Vol. 5, No. 1, March 2017 : 145 – 154
146
2. Wind Turbine Modeling
This section presents WECS based on a DFIG connected to the electric grid
Figure 1. Wind turbine based on DFIG
Figure 1 shows the scheme of electrical energy’s generation from the wind power on
the basis of DFIG. As it is demonstrated, the stator is connected directly to the grid whereas the
rotor is connected to the grid via a Back-to-back converter [2].
2.1. Wind Speed Model
Wind speed model contains four components [1]:
( ) ( ) ( ) ( ) ( ) (1)
where Vb is the base wind component (constant), Vr is the ramp wind component, Vn is the base
noise wind component and Vg is the gust wind, Vg is set to zero during simulation, all of them in
m/s. The model implementation of the wind speed in Matlab/Simulink is presented in Figure 2:
Figure 2. Wind speed model

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A Lyapunov Based Approach to Enchance Wind… (A. Bennouk)
147
The kinetic energy in this level is given by:
(2)
where m is the air mass, ρ is the air density, is the wind speed and S is the covered surface of
the turbine.
2.2. Wind Turbine Modeling
Wind turbine is applied to convert the wind energy into mechanical torque. The
mechanical torque of the turbine can be calculated from mechanical power at the turbine
extracted from wind power. Then, the power coefficient of the turbine (Cp) is used. The power
coefficient is function of pitch angle (β) and tip speed (λ). The power coefficient maximum of
(Cp) is known as the limit of Betz [2].
The power coefficient is given by:
( ) (3)
where
and
C1, C2, C3, C4, C5 and C6 are constants given by the turbine constructor (C1 = 0.516, C2
= 116, C3 = 0.4, C4 = 5.1, C5= 21 and C6 =0.0068), R is the rotor radius and ωm is the rotor
speed generator. The power coefficient is a nonlinear function of the tip speed ratio λ and the
blade pitch angle β (in degrees).If the swept area of the blade and the air density are constant,
the value of Cp is a function of λ and it is maximum at the particular λ optimum. Hence, to fully
utilize the wind energy, λ should be maintained at λopt, which is determined from the blade
design, Cp is defined as [3]:
and
(4)
where Pm is the mechanical output power of the turbine.
The Rotor Side converter is used to extract the MPPT (Maximum Power Point
Tracking), in this purpose the DFIG angular speed reference is calculated permanently with an
approach to follow the wind speed fluctuation, the wind turbine control generates the DFIG
speed reference signal, performed by the RSC controller in DFIG control level. This reference
signal is determined from the predefined characteristic P-ω, based on filtered measured
generator speed. Whereas the Grid Side Converter is used to control the DC link voltage and
guarantees unity power factor in the rotor branch. The transmission of the reactive power from
DFIG to the grid is thus only through the stator, a conventional vector with Lyapunov approach
will be used in this command algorithm to enhance system stability.
3. Rotor Side Converter control
The standard vector control is used to control the RSC. In order to ensure stability when
applying this control strategy mechanism, the stator flux magnitude must remain constant. To
this purpose, Lyapunov’s approach is used in order to enhance the system’s stability [2].
The electrical energy conversion is described by the following equation according to a
d-q frame :

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(5)
and
(6)
Electromagnetic torque is expressed by:
( ) (7)
V is the voltage, Rs and Rr are respectively the stator and rotor resistance. Ls,Lr and Lm are
stator, rotor and mutual inductance between stator and rotor θs, and θr present angles of stator
and rotor frames.
The aim of this approach is to catch the MPPT and to stabilize active power through
regulating the electromagnetic torque depending basically on irq and ird, the stator flux is, thus,
oriented to d axis in order to simplify the equations [5], [6]:
and (8)
The new system inputs are respectively d-q rotor current and angular speed expressed
by the following equations:
( ) (9)
( )

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It is clear from the dynamic model above, the nonlinearity, because of the coupling
between the d-q rotor currents and the speed, a variable change is initiated so as to introduce
the Lyapunov function:
(10)
is the angular speed, is the reference of , and are respectively the d and q axis
rotor current, , and are their references.
Lyupunov function is chosen as :
(11)
̇ ̇ ( ̇ ) (12)
The derivative of the complete Lyupunov function is negative defined, if the quantities
between parentheses in equation (12) are equal to zero. iqref is given by:
( ) (13)
̇ ̇ ̇
and
̇ ̇ ̇
V 2 and V3 are defined respectively as :
(14)
Lyapunov function is defined as:
( ) (15)
̇ ( ̇ ̇ ̇ ) ( ̇ ̇ )
In order to fulfill Lyapunov’s conditions, we integrate the following equations [2]:
( ̇ ̇ ̇ ) (16)
and
( ̇ ̇ ) (17)
We conclude after calculating direct and quadrature voltage that :
( ) (18)

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( )
4. Grid Side Converter control
When the grid voltage changes suddenly, a control approach is adopted in the GSC to
keep the DC voltage constant and to assure a zero rotor reactive power [7], [8], [9].
First, the system equations are defined as follows:
(19)
Ud, Uq :d-q components of converter voltage
vd,vq : d-q grid voltage components
id, iq :d-q current components
We suppose that the grid voltage is aligned to the d-axis. We may define the state
system as [10]:
[
̇
] [ ] [ ] [ ] (20)
(21)
(22)
In order to keep the system stability, the sliding system should be:
̇̇ ̇̇
Then we deduce the Ud and Uq :
( ) (23)
( ) ( ) ( ̇̇ )
5. Simulation and results
The test wind profile with full field turbulence is generated through the use of Wind
turbine block as presented in Figure 1. This block shows the hub height wind speed profile. In
general, wind speed consists of two components, mean wind speed and turbulence component.
The simulation is realized in order to illustrate Grid parameters stability, Grid voltage unbalance,
THD and Frequency.
The wind turbine and DFIG parameters are illustrated in Table 1 [11]:

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Table 1. DFIG parameters
Parameter Symbol Value
Active Power P 2MW
Stator resistance Rs 0.001518 Ω
Stator inductance Ls 0.059906 H
Rotor inductance Lr 0.082060 H
Rotor resistance Rr 0.002087 Ω
Mutual inductance Lm 2.4 H
Pole pairs N 2
Inertia J 17.23Kg.m²
Gear box ng 5.065
Simulation and results are presented first with a constant wind profile, then during a
variable wind speed to illustrate the used approach robustness based on Lyapunov theory:
Figure 3. Wind speed Figure 4. Rotor angular speed
Figure 5. Power Coefficient (Cp) Figure 6. Vdc
Figure 7. Active Power

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It’s obvious that for a constant wind speed,the wind power captured and deliverd to the
grid had the same shape as the wind, active power is kept constant and equal to 1.26MW for a
wind speed of 9.9m/s, the MPPT approach is well achieved and the power coeffiscient is
maintained at 0.47. The DFIG speed is the image of the wind speed, it is following properly its
reference. Lyapunov apporach used to control GSC shows also good performances and the
Udc voltage was equal to its reference .
We present here simulations result following variable wind speed :
Figure 8. Variable wind speed Figure 9. Electromagnetic torque
Figure 10. Rotor angular speed Figure 11. Vdc voltage
Figure 12. Total Harmonics Distorsion (THD) Figure 13. Frequency
Figure 14. Grid voltage Figure 15. Grid Current

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Figure 16. Active power Figure 17. Reactive power
The dynamic responses of the DFIG generation system under an intermittent wind
action are shown in Figure 8. It is clear that when a mean wind speed changes, the
electromagnetic torque varies. We may deduce then that when the wind speed increases
11m/s, the electromagnetic field is set at 1pu. Consequently, the delivered power is more or
equal to 2MW. Conversely, when the wind speed is inferior, the electromagnetic and active
power generated decreases proportionally. The DFIG speed is the image of the wind speed, it is
following properly its reference
The DC link voltage also varies according to the wind speed fluctuations, but the
variation is not obvious as shown in Figure 11. The DC voltage is generally kept at its reference.
The Total Harmonic Distortion presented in Figure 12 had demonstrated good
performances and did not exceed 5%, the assigned threshold of national grid code. The
frequency, as shown in Figure 13, did not also exceed the threshold of 1.2 pu. The grid side
voltage had shown also good performances apart from some a slight distortion caused by
dynamic response and wind speed fluctuations. The balance voltage remained inferior to 1%,
which is the fixed threshold of National Grid Code [2].
The active power variations illustrated in Figure.16 depend and follow the wind speed
variation which has demonstrated also good performances. The active power reaches its
highest values when the wind speed exceeds11m/s.
The reactive power had shown a good performance as presented in Figure.17 except in
200s, when we observe that the reactive power exceeds 600 Kvar following a quick fluctuation
of Udc link voltage.
6. Conclusion
With the growing level of penetration scored by wind-origin power production into the
general power system, many national codes have been applied so as to ensure stability to the
electric grid. Many simulations and complete models are needed to be established prior to any
connection between wind farm and electric grid [2].
A nonlinear control of variable speed WT is, thus, proposed. The main aim here is to
maximize the energy capture from the wind while reducing Grid parameters deviations.
Also, and with the aim of illustrating from one side the system stability and to what
extend it matches the national grid code, we have identified and presented the THD, frequency
and voltage unbalance in the Point of Common Coupling (PCC), as these parameters had
shown when they were under their limit during the simulation time.
Stability had been improved by using FOC and Lyupunov conditions in the Generator
Side Converter. In the Grid Side Converter, SMC control was applied. The different PID
controllers had been turned so as to obtain good performances. The use of SMC associated to
Lyapunov had also demonstrated good performances in terms of Grid parameters stability.

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References
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