Simulation of a Permanent Magnetic Synchronous
Machine Wind Turbine Using Interconnection and
Damping Assignment Passivity-Based Control
Allan Fagner Cupertino1,2, Heverton Augusto Pereira1,2, José Tarcísio de Resende2 and Selênio Rocha Silva1
1
Graduate Program in Electrical Engineering
Federal University of Minas Gerais
Av. Antônio Carlos 6627, 31270-901
Belo Horizonte, MG, Brazil
allan.cupertino@yahoo.com.br, selenios@cpdee.ufmg.br
2
Gerência de Especialistas em Sistemas Elétricos de Potência
Universidade Federal de Viçosa
Av. P. H. Rolfs s/nº, 36570-000
Viçosa, MG, Brazil
hevertonaugusto@yahoo.com.br, resende@ufv.br
Abstract— Nowadays the use of variable speed wind turbines is
preferred because more power can be extracted from the wind.
The doubly fed induction generators (DFIG) are strongly used
because their price is smaller. However, permanent magnetic
synchronous generators (PMSG) improve the performance of
the system because their application removes the gearbox,
reducing the maintenance costs. Moreover, the use of full
converters prevents the propagation of grid disturbances to the
machine. In this context, this work compares two control
techniques applied in a 10.5 kW wind turbine based on PMSG
connected to the distribution grid. The first technique studied is
the classical structure based on proportional-integral
controllers. The second technique consists in a new methodology
of nonlinear control called Interconnection and Damping
Assignment Passivity-based Control (IDA-PBC). Simulations
were developed to compare the behavior of the system during
symmetric voltage sags and wind speed variations. The
controllers
implemented
showed
robustness
against
disturbances and it was observed a better performance of the
IDA-PBC technique compared to the classical structure.
system enable the PMSG based wind turbine to operate with
higher efficiency, more reliability and lower noise [2], [3].
Index Terms—Wind Energy, Interconnection and Damping
Passivity-Based Control, Symmetric Voltage Sags, ProportionalIntegral Control, Permanent Magnetic Synchronous Machine.
This work presents the application of the passivity-based
control theory in the PMSM wind turbine. This technique has
been applied in DFIG based wind turbines [6] and PMSG
based wind turbines [7].
I.
INTRODUCTION
Electricity is a concern for society. The limitation of
reserves, environmental impact and raise fossil fuels prices
indicate that it is necessary to diversify the generation system.
These factors have motivated researches about renewable
sources. In this context, wind energy had growth considerably
during the last years and presents good forecasts [1].
There are many technologies of variable speed wind
turbines. Both the volume and the cost of PMSG based wind
turbine are higher than the DFIG based wind turbine [2].
However, the elimination of the gearbox and DC excitation
This work is supported by the Brazilian agencies CAPES, FAPEMIG
and CNPQ.
Moreover, low voltage ride through (LVRT) capability is a
DFIG drawback because the electromagnetic relationship
between the stator and the rotor is more complex than PMSG,
and the stator is connected directly to the grid [4], [5].
There are two topologies of full converters used in PMSG
wind turbines, which are shown in Figure 1. The topology
with PWM rectifier is preferred because this topology permits
a bidirectional power flow and the stator currents are
approximately sinusoidal.
The classic control technique used in back-to-back
converter is based on synchronous reference frame and uses
proportional-integral controllers. The grid side converter
(GSC) controls the DC link voltage and the reactive power
injected into the grid. The machine side converter (MSC)
controls the magnetization current and the angular speed of the
machine, doing the maximum power tracker of the turbine.
This technique will be compared with the traditional
strategy during symmetric voltage sags and wind speed
variations. The next section presents the modeling of all
components of the PMSM based wind turbine.
(a)
B. Traditional control structure
The GSC control was made in synchronous reference
frame. The synchronism technique used is a DSOGI-PLL,
proposed by [10]. A LCL filter is used in order to reduce the
harmonics generated by switching. The design of this
component is presented in [11].
(b)
Figure 1.Common topologies of full converters applied in PMSG based wind
turbine: (a) Diode rectifier + DC/DC converter + PWM inverter; (b) PWM
Back-to-back converter.
II.
MODELING OF THE SYSTEM
A. Wind Turbine
The wind turbine model used in this work is the proposed
by [8]. The mechanical power is given by:
=
1
2
( , )
(1)
Where ρ is the air density; V is the wind speed and A is the
area swept by the wind turbine blades;
( , ) is given by:
The power coefficient
( , ) = 0.22
116
.
− 0.4 − 5
(2)
The factor is calculated by (3) and is dependent on two
variables: The factor λ which is the tip-speed-ratio, given by
equation (4) and which is the pitch angle.
1
=
1
0.035
−
+ 0.08
+1
A relation between the pitch angle and the electric power
can be obtained if it is considered:
≈ −
+
; [9]
 The electrical dynamic is faster than the mechanical
dynamic and the losses are negligible ( ≈ );
It is used a compensator ( ) = −(K
closed loop transfer function is:
( )
=
∗( )
,
+
,
+
,
+1
+
,
+
,
,
Adjusting the gains of the compensator, it can be obtained
a response without overshoot.
Figure 2.Pitch control scheme.
+ ∙
−
∙ ∙
+
=
∙
+ ∙
+
∙ ∙
+
The modeling of the GSC control loops is presented in
[12]. The complete control structure of the GSC is presented
in Figure 3 (a). The outer loops, slower, control the reactive
power (and consequently, the power factor) and the DC bus
voltage. These loops use proportional-integral controllers with
anti-windup action.
The inner loops, faster, control the injected current and use
proportional controllers with feed-forward actions. The
dynamic of the machine connected to the MSC in rotor
synchronous reference frame is [13]:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
Where Ψ = λ
= −
−
+
= −
−
−
=
=
+
−
+ pL ω i
(7)
+
+
and Ψ = pL ω i .
For the MSC it is used a
rotor-orientated control. In
this orientation, the direct axis current represents the
magnetization axis and the quadrature current represents the
torque current. Externally of the quadrature current loop it is
implemented a loop of angular speed whose reference is
dependent on the wind (principle of the maximum power
tracker). This reference is calculated by:
), and the
(5)
∙
Where
and
represents the inverter output voltage,
and
the grid voltage, and the injected current, is
the sum of inductances and is the sum of resistances of the
filter.
(4)
The pitch angle β control compares the electric power
generated with the rated value, Figure 2. It was used a
simplified model for the servomechanism, that considers a
first order constant τ and some saturations in the angle.
=
(6)
(3)
=

In the fundamental frequency ω , the LCL filter can be
approximated for a L filter. The dynamic of the GSC in
synchronous reference frame is:
=
Where
turbine.
(8)
is the optimum tip speed ratio of the wind
The reference of quadrature axis is zero because it is not
implemented a flux control loop in this work. The complete
control structure of MSC is presented in Figure 3 (b).
⎧
⎪
⎪
⎪
⎪
⎪
+
+
=−
−
−
+
=−
+
+
−
=−
−
+
−
(12)
⎨
⎪
⎪
⎪
⎪
⎪
⎩
(a)
=−
= −
−
=
−
−
−
+
−
−
The PCH model is obtained considering the state vector
] and
= [
the
] . The storage energy function is:
inputs = [
( )=
(b)
( )
Figure 3.Traditional control structure of the back-to-back converter.
C. IDA-PBC control of the back-to-back converter
Traditionally, the passivity-based control is applied in
systems which has an Euler-Lagrange model and stabilizes the
response by minimizing an energy function associated with
the storage system. The technique IDA-PBC (Interconnection
and Damping Assignment Passivity-based Control) extends
the conventional ideas of PBC for a larger class of systems
called PCH (Port-Controlled Hamiltonian) which considers
the modeling of the total energy of the system [14], [7]. The
canonical form of a PCH system is given by [15]:
( )
= [ ( ) − ( )]
( )
=
+
(9)
( )
Where x ∈ R is the state vector; and represent the inputs
( )>0
and outputs of the system, respectively; ( ) =
represents the dissipative terms. The interconnection structure
is represented by ( ) and the matrix J(x) = − ( ). H (x)
is the storage energy function of the system.
In order to obtain the PCH model of the PMSM based
wind turbine it is considered that the LCL filter can be
approximated by a L filter and the terminal voltage of the
generator is, in average, imposed by the MSC. In this
situation:
=
;
=
(10)
Where
and
are the direct and quadrature axis
modulation of the MSC. The dynamic of the DC bus voltage,
considering that the machine operates like motor, can be
written like (11).
= −
+
−
+
The complete model of the system is:
(11)
Where
1
2
⟹
= [
=
]
[
=
(13)
].
The other matrices are:
( )=
0
[
]
(14)
0 0 0
⎡0 0 0⎤
⎢
⎥
( )= ⎢ 0 0 0 ⎥
⎢−1 0 0 ⎥
⎢ 0 −1 0 ⎥
⎣ 0 0 −1⎦
0
⎡
⎢ 0
( )= ⎢ 0
⎢ 0
⎢−
⎣−
Where
=
0
0
0
0
−
( +
and
0
0
0
0
0
−
−
)
0
0
−
0
=
.
(15)
− (
0
0
+
0
0
0
0
)⎤
⎥
⎥
⎥
⎥
⎦
(16)
The design of the passivity-based controller consists in
change the dissipative and the interconnection structure and
adds damping in closed loop. Virtual resistances are emulated
by the controller, reducing the storage energy at the operation
point [16]. The stability proof and the deduction of the control
law are obtained by the propositions 1 and 2.
Proposition 1 (stability) [15]: Suppose a system that has a
PCH model given by equation (9), an equilibrium point
= − , and a closed loop storage energy
function ( ) > 0. It supposes e
satisfying:
J (x) = J(x) + J = −J (x)
(17)
( )= ( )+
(18)
=
( )>0
If the equation can be transformed to the dynamic error
equation:
= [ ( )−
( )]
( )
The system is asymptotically stable.
(19)
Proof:
( )
⎧
⎪
( )
=
( )
=
[ ( )−
( )]
( )
(20)
Because J (x) is skew-symmetric matrix, then:
( )
= −
( )
( )
( )
<0
(21)
So, by Lyapunov’s stability theorem, the system is
asymptotically stable at the equilibrium point x .
Proposition 2 (Control law) [15]: Suppose a system that
has a PCH model given by equation (9). If
and
satisfying (17) and (18), and:
( +
)= ( )+ ( )
( )=
(22)
( )=
,
(23)
The system (9) can be written in the form (19) if the
relation (24) is satisfied:
=[
+ ( )]
+
− ( )
+ ( )
(24)
Proof:
Define =
results in (25).
−
=
, then
+
. Substituting into (9),
⎨
⎪
⎩
= −
= −
= −
−
+ (
−
−
)
+
+
)
− ( −
= −
+
−
(
)
+
(30)
−
The references of current are calculated using
proportional-integral controllers like in the traditional strategy.
The variables r and r are parameters of the controller.
D. LVRT capability and protection circuit
The LVRT capability is an important requisite of a wind
farm. Many countries have standards regulating the operation
of wind farms during voltage sags. During voltage sags, the
generated power P cannot be injected into the grid due to
inverter current limitation.
Generally, it is used a protection chopper that dissipates
the exceeding energy. In this work, when the DC bus voltage
is equal to 10 % of overvoltage, the chopper is activated by a
hysteresis control.
E. Simulations
It was simulated in Matlab/Simulink one 10.5 kW wind
turbine connected to the distribution grid whose voltage level
is 220 V. The parameters of the simulated system are
presented in TABLE I. The parameters of the generator are
presented in [17].
TABLE I. PARAMETERS OF THE SIMULATED SYSTEM.
Wind Turbine parameters
12
Rated wind speed
=[ ( +
)− ]
( +
)+ ( +
) −
(25)
Besides, using the equation (9) it is possible to obtain (26).
= [ ( )− ]
(26)
+ ( )
/
6
Blade diameter
214.5
Rated angular speed
Servomechanism delay time
0.25
Rate limitation of pitch angle
5 °/
LCL Filter parameters
Equation (26) can be rewritten in the form (27).
= [ ( )−
( )
]
(27)
+
Where K is given by (28).
= −[
−
− ( )]
=[
+
+ ( )]
4.7
Second inductance of the filter
0.26
Capacitance of the filter
23 μ
Damper resistance
6.6
Back-to-Back converter parameters
+ ( )
+ g(x)u + g(x )u
(28)
If = 0, equation (27) is equivalent to (19). This
condition can be rewritten as (29).
(x)
First inductance of the filter
− ( )
+ ( )
(29)
In this situation, the control law is obtained solving the
equation (29). In this work, it was defined
null and
[
0 0]. It is easy to see that
=
these matrices are according to (17) and (18). Accomplishing
some algebraic manipulations, it is possible to obtain the
equations that calculate de modulation index for both
converters:
Switching frequency
5
Dc bus capacitance
3.06
DC bus voltage
500
DC Chopper resistance
24
Back-to-Back converter parameters
Switching frequency
5
Dc bus capacitance
3.06
500
DC bus voltage
DC Chopper resistance
24
Controller parameters
Proportional gain of the pitch controller
Integral gain of the pitch controller
Proportional gain of the GSC current
controller
0.006°/
0.003°/(
15.4 Ω
)
overcurrent limit. The detail shows that the PI based control
presents an oscillatory behavior that does not exist in the IDAPBC response.
5.3 Ω
Gain
of the IDA-PBC
20 Ω
Gain
of the IDA-PBC
200 Ω
Proportional gain of the DC bus voltage
controller
1.89 Ω
Integral gain of the DC bus voltage controller
33.63 (Ω s)
Proportional gain of the reactive power
controller
0.0002
33.63 (
Integral gain of the reactive power controller
Proportional gain of the generator speed
controller
155.4
Integral gain of the generator speed controller
3662.3
)
The first simulation implemented compares the two
presented control techniques during symmetric voltage sags.
The profile of voltage simulated is the worst case in the
Brazilian standards which is shown in Figure 4. In this case
the inverter works with unitary power factor and the wind
speed is maintained in the rated value.
During the voltage sag, the GSC current reach its limit,
and the protection chopper dissipates the exceeding power. In
this situation, the DC bus voltage oscillates between 525 and
550 V (the limits of the hysteresis control) like is shown in
Figure 7. When the most critical part of the sag finishes, the
inverter can inject all generated power and the DC bus voltage
returns to the rated value.
Figure 8 presents the generator angular speed that
maintains almost constant due to the isolation proportioned by
the back-to-back converter. The detail shows that the IDAPBC technique reduces the overshoot and the oscillation
during the sag.
40
30
25
The second simulation compares the two control
techniques during variations in the wind speed. The profile
simulated is presented in Figure 5. In this case, the PCC
voltage is 1
and the power factor is unitary.
40
20
35
15
10
30
5
25
0
2.5
200
3
3
3.02
3.04
3.06
3.5
4
4.5
time(s)
5
3.08
3.1
5.5
6
6.5
Figure 6. Current RMS injected by the GSC during symmetric voltage sag.
150
Vpcc (V)
PI
IDA-PBC
35
Ipcc(A)
Proportional gain of the MSC current
controller
PI
100
vdc (V)
540
50
520
500
2.5
0
0
2
4
6
8
3
3.5
4
4.5
time (s)
IDA-PBC
5
5.5
6
6.5
3
3.5
4
4.5
time (s)
5
5.5
6
6.5
10
time (s)
Figure 4. Symmetric voltage sag simulated.
vdc (V)
540
25
520
500
20
Vw (m/s)
2.5
15
Figure 7. DC bus voltage during symmetric voltage sag.
10
23
PI
IDA-PBC
22.8
5
22.6
22.4
0
10
20
30
40
time (s)
50
60
70
Figure 5. Wind speed variation simulated.
III.
RESULTS
A. Symetric voltage sags analysis
The RMS current injected into the grid during the voltage
sag is presented in Figure 6. At the interval 3 to 3.5 seconds
the voltage reduces to 0.2
and the GSC works in the
g (rad/s)
0
22.5
22.2
22.48
22
21.8
22.46
21.6
22.44
21.4
22.42
21.2
3
21
2.5
3
3.5
3.5
4
4.5
time (s)
4
5
5.5
6
6.5
Figure 8. Generator angular speed during symmetric voltage sag.
Finally, Figure 9 and Figure 10 present the behavior of the
active and reactive power. The details show that the IDA-PBC
reduces the overshoot in the active and reactive power
responses.
12
10
8
6
4
P pcc (kW)
9
2
2.8
8
0
7
10
20
2.7
6
5
30
40
time (s)
50
60
(a)
PI
IDA-PBC
2.5
3
3.5
4
3
3.2
4.5
time(s)
5
3.4
5.5
6
6.5
Figure 9. Active power injected by the GSC during symmetric voltage sag.
Qpcc (kVAr)
0.2
3
70
0.3
2.6
4
2.5
PI
IDA-PBC
10
Ppcc (kW)
PI
IDA-PBC
11
the rated value, the pitch controller increases the angle in
order to reduce the power coefficient. Both controls showed
the same behavior, Figure 14.
0.1
0
-0.1
-0.2
1.2
10
1
1
0.8
0.6
Qpcc (kVAr)
0.4
0
0.2
60
22
-0.2
2.8
3
3.2
3.4
3.5
4
4.5
time (s)
5
5.5
6
6.5
B. Wind speed variation analysis
The second simulation analyzes the performance of the
controllers during variations in the wind speed. It is used the
wind speed profile of Figure 5.
The DC bus voltage is presented in Figure 11. It can be
observed an increase in the ripple when the wind speed is
large, due to the increase of power processed by the converter.
g (rad/s)
3
PI
IDA-PBC
18
PI
IDA-PBC
Figure 10. Reactive power injected by the GSC during symmetric voltage
sag.
16
5.625
14
5.62
12
5.615
5.61
10
5.605
8
5.6
6
10
20
16
30
40
time (s)
18
50
20
60
70
Figure 13. Generator angular speed during wind speed variation.
30
20
PI
(º)
510
505
10
PI
IDA-PBC
500
495
490
70
20
0
-0.2
vdc (V)
50
0.2
0.4
-0.4
2.5
30
40
time (s)
(b)
Figure 12. Active (a) and reactive (b) power injected by the GSC during
wind speed variation.
0.8
0.6
20
0
10
20
30
40
time (s)
IDA-PBC
50
60
70
510
10
20
30
40
time (s)
50
60
70
Figure 14. Pitch angle during wind speed variation.
vdc (V)
505
500
IV.
CONCLUSIONS
495
490
10
20
30
40
time (s)
50
60
70
Figure 11. DC bus voltage during wind speed variation.
The active and the reactive power injected are presented in
Figure 12. When the wind speed is smaller than the rated
value, the angular speed control maximizes the power
extraction.
Figure 13 shows that the generator angular speed changes
with the wind. Again, it can be observed the larger oscillation
caused by PI technique. When the wind speed is larger than
This work compared two control techniques for a smallscale PMSG based wind turbine: The traditional technique
based on PI controllers and a non-linear technique known as
IDA-PBC.
During symmetric voltage sags, it was observed a better
response of the IDA-PBC controller. This technique reduces
the oscillations in the injected current and overshoots in the
power during transients. During wind speed variations it was
observed a similar response for the two techniques because
this disturbance is less intense than voltage sags.
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BIOGRAPHIES
Allan Fagner Cupertino received the B.S. degree in
electrical engineering from the Federal University of
Viçosa (UFV), Viçosa, Brazil, in 2013. He is
integrant of GESEP, where developed works about
power electronics applied in renewable energy
systems. Currently he is Master Student from Federal
University of Minas Gerais (UFMG), Belo Horizonte,
Brazil.
His research interests include solar
photovoltaic, wind energy, control applied in power
electronics and grid integration of dispersed generation systems.
Heverton Augusto Pereira received the B.S. degree
in electrical engineering from the Federal University
of Viçosa (UFV), Viçosa, Brazil, in 2007, the M.S.
degree in electrical engineering from the State
University of Campinas (UNICAMP), Campinas,
Brazil, in 2009. Currently he is Ph.D. student from
the Federal University of Minas Gerais (UFMG),
Belo Horizonte, Brazil. Since 2009 he has been with
the Department of Electric Engineering, UFV, Brazil.
His research interests are wind power, solar energy and power quality.
José Tarcísio de Resende received the M.S. degree
in electrical engineering from the Federal University
of Itajubá (UNIFEI), Itajubá, Brazil, in 1994 and the
Ph.D. degree in electrical engineering from the
Federal University of Uberlândia (UFU), Uberlândia,
Brazil, in 1999. Since 2004, he has been with the
Department of Electrical Department of Electric
Engineering, UFV, Brazil. His research interests
include modeling of electric machines,power systems
and renewable energy.
Selênio Rocha Silva received the B.S. degree and the
M.S. degree in electrical engineering from the
Federal University of Minas Gerais (UFMG), Belo
Horizonte, Brazil, in 1980 and 1984, respectively,
and the Ph.D. degree in electrical engineering from
the Federal University of Paraíba, currently Federal
University of Campina Grande (UFCG), Campina
Grande, Brazil, in 1988. Since 1982, he has been with
the Department of Electrical Engineering, UFMG,
where, in 1995, he became a Full Professor. His main research interests
include ac motor drives, power quality, variable-speed generators for
wind turbines and grid integration of DG.
Prof. Silva is a member of the Brazilian Power Electronics Association, the
Brazilian Automatic Control Association, and the IEEE Industry Application
Society.