Backstepping Control of an AC/DC/AC Converter in a Grid Connected
Wind Power System
YOUSSEF CHAOU1, SAID ZIANI2, HAFID BEN ACHOUR3, ABDELKARIM DAOUDIA1
YOUSSEF EL HASSOUANI3
1OTEA, Department of Physic,
Faculty of Science and Technology Errachidia,
Moulay Ismail University,
BP-509 Boutalamine Errachidia 52 000,
MOROCCO
2ENSAM, Mohammed V University in Rabat,
MOROCCO
3ENIM, Department of Physic,
Faculty of Science and Technology Errachidia,
Moulay Ismail University,
BP-509 Boutalamine Errachidia 52 000,
MOROCCO
Abstract: - The contribution of this paper is to study a nonlinear control approach known as the backstepping
control method to manage reactive and active power using new parametric values for the grid connected wind
power generation system. Specifically, the research focuses on a system containing a wind turbine system, a
permanent magnet synchronous generator (PMSG), and a grid connected via AC/DC/AC converter. A
comprehensive mathematical model for both the wind turbine system, PMSG, and the grid connected via
AC/DC/AC converter was developed for this purpose. The Simulation results of the grid-connected wind power
generation system are presented and analyzed to show the performance of the nonlinear backstepping control
method. The characteristics of the response to variations in generator speed and the power injected into the grid
show the highlight of the performance of the backstepping method, under varying wind conditions.
Key-Words: - Nonlinear Control, Backstepping Technique, AC/DC/AC Converter, Wind Power, Turbine-
PMSG systems, Reactive and Active Power.
Received: April 16, 2023. Revised: February 13, 2024. Accepted: March 21, 2024. Published: April 26, 2024.
1 Introduction
Wind power is a type of energy that taps into the
winds force to generate electricity. It stands as a
cost-plentiful energy source that can be captured
through wind turbines. These structures are typically
installed on structures. Utilize the wind's kinetic
energy to create electricity. The wind spins the
turbine blades connected to a generator that
transforms energy into power. This electrical power
can then be utilized to supply energy to households,
companies, and neighborhoods. Wind power offers
benefits notably being an eco-renewable energy
option. It doesn't emit air pollutants or greenhouse
gases playing a role, in reducing our reliance, on
fuels, [1]. Furthermore, the price of wind power is
getting more competitive compared to energy
sources, which is drawing interest, from
communities and businesses. In general wind power
can significantly contribute to the shift, towards a
dependable energy infrastructure. Presently the
primary generators utilized in wind power
generation include the fed generator and the
permanent magnet synchronous generator, [2].
Faced with the problems of wind and water power
production, the permanent magnet synchronous
generator has advantages such as no excitation
circuit, and low maintenance. Thanks to these
advantages, the use of the permanent magnet
synchronous generator makes the variable speed
wind power conversion systems more attractive than
the fixed speed ones because of the possibility of
extracting the optimal energy in different operating
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2024.19.13
Youssef Chaou, Said Ziani, Hafid Ben Achour,
Abdelkarim Daoudia, Youssef El Hassouani
E-ISSN: 2224-2856
127
Volume 19, 2024
conditions. In this paper, the authors propose a
modeling study and behavioral simulations of a
wind power generation system based on a
permanent magnet synchronous machine connected
to the grid by an AC/DC/AC converter, [3]. In a
wind power system connected to the grid, the
AC/DC/AC converter plays a role, in converting and
transmitting power. Its effectiveness directly
impacts the performance of the wind power setup.
To transfer the captured wind energy back to the
grid the three phases produced by the machine with
varying frequency and amplitude are rectified using
a back-to-back converter (AC/DC/AC). The AC
side connects, to the stator of PMSG while the
converter output (DC/AC) links directly to the grid
through a coil. This setup offers benefits. While
traditional PI controllers can manage this system
they may not always ensure performance in terms of
stability and disturbance control, [4], [5]. To address
this issue different advanced control techniques
have been suggested for managing and overseeing
the wind power generation setup, including input-
output linearization control, [6], sliding mode
control [7], backstepping control, [8], [9], [10], [11]
and DTC. Backstepping has emerged as a promising
alternative approach for controlling nonlinear
systems. It integrates the selection of Lyapunov
functions with control laws, enabling the
maintenance of global system stability at all times,
[12].
2 Modeling of the Systems Elements
2.1 Model of the Turbine and the Permanent
Magnet Synchronous Machine
The theoretical power delivered to the turbine can
be represented by equation (1), where denotes the
density of air, represents the circular area swept by
the turbine blades, is the pitch angle of the blades,
and indicates the velocity of the wind in meters
per second (m/s).
(1)
The ratio of turbine speed to wind speed is
expressed by (2), where is the rotational speed of
the turbine, is the blade radius.
(2)
The power coefficient () has a theoretical
limit of 0.59 called the "Betz limit". This limit is
never reached in practice. This coefficient can be
estimated using (3).
(3)
With:
, , ,
, , and
The mechanical torque of the wind turbine
obtained from the mechanical power is expressed
by (4).
(4)
The mechanical equation of the system is
expressed by (5), where and have the moments
of inertia of the turbine and the generator,
respectively, and have the coefficients of the
viscous friction of the turbine and the generator,
respectively, is the rotational speed of the
generator, and the speed multiplier ratio.
(5)
With:
et
The electrical model of an PMSM in generator
operation is reproduced from the model of the
machine in motor operation model of the machine in
motor operation, by reversing the direction of the
currents and in the Park marks. The model of
the permanent magnet synchronous generator
PMSG thus obtained can be written in the following
form (6):
(6)
Note that for driving a PMSG by the
wind turbine where is the stator resistance,
and are the inductances in the (d, q) frame, and
are the stator currents, is the electrical velocity
of the and is the PMSG remanent flux.
2.2 Model of the Converters and the Electric
Grid
The system has a static AC/DC/AC converter with
the rectifier on the machine side and the inverter on
the grid side. The model of AC/DC/AC converter is
given by (7), where is the DC bus capacitance,
is the rectifier current and is the inverter
current.
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DOI: 10.37394/23203.2024.19.13
Youssef Chaou, Said Ziani, Hafid Ben Achour,
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(7)
The inverter model is presented by (8),
where, and are the three-phase voltages
at the output of the inverter and is the DC bus
voltage.
(8)
The analytical model of the power system is
presented by (9), where and are the
electromotive force of the system in the park
reference (d, q), Rg and Lg are the resistance and
inductance of the lines.
(9)
Whith: and
In addition, the active and reactive power
expressions can be calculated according to the
following equations:
(10)
3 System Control Studied by
Backstepping Technique
The equation (11) is the model of the global system
studied in the park reference (d, q) it can be written
as:
(11)
The equation (11) represents the dynamic model
of a nonlinear system whose general form (12) is the
following:
(12)
4 Backstepping Control Design
The backstepping controller is considered a very
useful tool when some states are controlled by other
states. This technique uses one state as a virtual
controller to another state since the system is in
triangular feedback form. It also overcomes the
problem of finding a Lyapunov control function as a
design tool. The design of backstepping control,
nonlinear systems or subsystems of the form (13).
(13)
Where:
We wish to make the output follow the
reference signal supposed to be known. The
system being of order n, the design is done in n
steps.
5 Designed of Backstepping Controller
The application of this approach on the permanent
magnet synchronous generator allows us to
determine the constituents of the control voltages of
the machine, by ensuring the machine, ensuring the
global stability by the Lyapunov theory.
The errors defined by the expressions:
(14)
The errors dynamics is given by:
(15)
5.1 Step 1: Control of
The current of the PMSG is always forced to be
zero, so the d-axis flux is zero and all coupling flux
is directed along the q-axis to obtain maximum
electromagnetic torque. To ensure the control of the
current, we adopt the following ''Lyaponov''
function:
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(16)
Making a derivative of this function (16), we will
have:
(17)
We choose:
(18)
Where is a positive scalar.
Then
and the backstepping
control law is designed as:
(19)
5.2 Step 2: Control of Rotor Speed
As the rotor speed is the main control variable, its
trajectory is defined as the reference value and the
control error as:
(20)
Define the second Lyapunov function as:
(21)
And
(22)
In order to obtain , we can choose:
(23)
Where is a positive scalar
Then
(24)
Considering that this leads to defining
the command necessary to determine the
voltage.
(25)
5.3 Step 3: Control of
In the following, the assurance of stability and
convergence of the component to the
reference, leads us to choose the following
''Lyapunov'' function:
(26)
Then
To obtain , we can choose
(27)
Where is a positive scalar
(28)
We deduce the final backstepping control law
is designed as:
(29)
5.4 Step 4: Control of
In the following, the assurance of stability and
convergence of the component to the reference
, leads us to choose the following ''Lyaponov''
function:
(30)
Making a derivative of this function (30), we will
have:
(31)
We choose:
(32)
Where is a positive scalar
Then
and the backstepping
control law is designed as:
(33)
5.5 Step 5: Control of
In the following, the assurance of stability and
convergence of the component to the reference
, leads us to choose the following ''Lyaponov''
function:
Define the second Lyapunov function as:
(34)
And (35)
To obtain , we can choose
(36)
Where is a positive scalar
Then
(37)
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The backstepping control law is designed as:
(38)
5.6 Step 6: Control of
The reactive power is always forced to be zero
(Q=0), to ensure the quality of the energy injected
into the electrical network and also to determine the
reference current . In view of ensuring the
control of the current , we adopt the following
''Lyaponov'' function:
(39)
Then
To obtain , we can choose:
(40)
Where is a positive scalar
(41)
We deduce the final backstepping control law
is designed as:
(42)
Finally we define from the backstepping
control, the reference variables ,
and necessary for the control
respectively of the PMSG and the AC/DC/AC
converter, while requiring stability of the cascaded
subsystems to ensure an asymptotic stability of the
overall system. The Figure 1 represents the global
nonlinear control scheme of a grid-connected wind
turbine equipped with a PMSG and an AC/DC/AC
converter, managed by a robust nonlinear control
strategy called backstepping, which allows us to
follow dynamic references and stabilize the system
against disturbances.
Fig. 1: Schematic representation of the backstepping
control of the Turbine-PMSG and AC/DC/AC
converter
6 Simulation Results and Discussion
6.1 Simulations Results
The adopted control is based on the Backstepping
method applied to a GMSM, whose model is
nonlinear and multivariable, the Table 1 shows the
parameter values used to test the system through
numerical simulation.
Table 1. The parameters of all systems
4
0.6
0.0014
0.0028
0.2
0.02
0.0014
0.39
1.22 [kg /m3]
10
G
6
0.40
0.025
C
0.0042
6.2 Results and Discussion
The simulation results are obtained for the random
wind speed closest to the evolution of the real wind
are implemented in Figure 2, to adapt it to the slow
dynamics of the system studied, And to prove
control robustness Backstepping on the grid-
connected wind energy conversion system. DC bus
voltage regulation is shown in Figure 3. To control
the voltage , the inverter injects surplus current
into the network or vice versa, to discharge/charge
the capacitor until (=), at the same time as
the inverter transmits the power used to the network.
So, according to Figure 3, the DC bus voltage is
well controlled at its 350V setpoint after a response
time of 0.02s. After 0.02s, it is the permanent
regime. Generally, the DC bus voltage is fixed,
even in the presence of load variations and network
conditions.The Figure 6 and Figure 7 show the
results of (,) control in the Park reference frame
(d,q). The current signals in the Park frame, and
, are presented about their references in Figure 4
and Figure 5. Through these currents, we can
control the active power and reactive power . In
Figure 6 and Figure 7, we observe that the active
power injected by the inverter into the grid closely
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follows the reference Pref. Consequently, the
current perfectly tracks its reference . The
same applies to the current , which oscillates
around its reference and has an average value
close to zero. The reactive power follows the
reference , which is set to zero, to maintain a
unity power factor in the system. Figure 8 illustrates
the phase difference between the current and the
voltage on the grid-side converter, which is
equal to zero. They also have the same frequency of
50Hz, which is the grid frequency.
Fig. 2: Random wind speed profile
Fig. 3: The DC link voltage
Fig. 4: The grid direct currants ( )
Fig. 5: The grid quadratur currants (,)
Fig. 6: The active power injected into the grid
Fig. 7: The reactive power injected into the grid
Fig. 8: The phase difference between the current
i_ga and the voltage V_gaon the grid-side converter
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The research work addressed in this study can be
applied in the field of renewable energies and can
even be extended and combined with other tools
such as wavelets and artificial intelligence, as
illustrated in references, [13], [14].
7 Conclusions
In our study, we thoroughly examined the use of
backstepping control in managing a grid connected
wind power generation system that utilizes a
permanent magnet synchronous generator (PMSG)
connected to the grid through an AC/DC/AC
converter. We conducted simulations based on
varying wind speeds to reflect real-world
conditions. The results demonstrate the capability of
the system to optimize power extraction, from wind
sources maintain DC bus voltage, and manage the
exchange of reactive and active power, with the
grid effectively. The objective of this research was
to develop a robust and efficient control scheme that
can ensure stable operation, seamless grid
synchronization, and optimal power transfer in grid
connected wind power systems.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- YOUSSEF CHAOU: Methodology, Software and
Writing - original draft
- SAID ZIANI: Visualization and Validation
- HAFID BEN ACHOUR: Visualization
- ABDELKARIM DAOUDIA: Supervision
- YOUSSEF EL HASSOUANI: Supervision
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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DOI: 10.37394/23203.2024.19.13
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