Control of a Doubly Fed Induction Machine
Wind Power Generator with Rotor Side
Matrix Converter
Ricardo Jorge Pires dos Santos
Abstract – A method for control of a doubly fed
induction machine (DFIM) in a variable speed
constant frequency (VSCF) wind generation system
is presented in this paper. It combines the
advantages of a recently developed direct power
control (DPC) algorithm, with a model reference
adaptive system (MRAS) method for sensorless
rotor positioning and a matrix converter.
Simulation results demonstrate that the controller
proposed has an excellent power control, at
constant frequency, as well as a perfect rotor sector
recognition using the MRAS observer.
Nomenclature
,
,
,
,
,
,
,
,
M
Ls
Lr
Stator active and reactive reference
powers.
Stator active and reactive Power.
Error in the active and reactive power,
respectively.
Stator and rotor voltage, respectively.
Stator and rotor current, respectively.
Angle between the stator flux and rotor
flux.
Stator and rotor frequency, respectively.
Stator flux, rotor flux and air-gap flux,
respectively.
Component of the rotor current along the
flux.
Fluxes in αβ domain.
Fluxes in dq domain.
Magnetizing inductance.
Stator resistance.
Stator inductance.
Rotor inductance.
PI proportional and integral parameters
respectively.
I – Introduction
The progressive depletion of fossil fuels, and concerns
regarding environmental changes, are acting as major
motivation for research and development of new sources
for electric power generation. Wind power generation is
becoming a viable and very attractive source of clean
and renewable energy. In a conventional wind power
generator, a constant speed mode of operation is used to
keep the output frequency identical to the grid. In this
paper, a VSCF system is proposed, since it has been
widely demonstrated that variable speed generation
systems have a higher energy capturing and converting
efficiency than the fixed speed generation. In such
system, the stator is directly connected to the grid, while
an electronic converter is connected to the rotor, to
control the active and reactive power, while maintaining
unity power factor and constant output frequency.
The conventional method used for independent control
of active and reactive powers is stator flux oriented
vector control, dependent on rotor position sensors.
However, the use of these sensors is undesirable, as it
causes the system to be highly dependent on the
accuracy of computation of the stator flux and the
accuracy of the rotor position, as well as drawbacks in
terms of maintenance, cost, cabling and robustness [1].
Several research groups have proposed alternative
methods that do not require rotor position information,
such as direct torque control (DTC), but this method has
been mostly used for cage rotor induction machines. In
this work, a method for direct and independent control
of active and reactive power, DPC, is presented, in
which the measurements are carried in one winding, the
stator, while the switching action is carried at another
winding, the rotor [2]. However, rotor positioning
information is needed, in order for the DPC technique to
work properly. The MRAS is a well known method for
the sensorless control of the squirrel cage induction
machine, and its application to the DFIM is recently
being analyzed. This system is used to estimate the
rotational speed of the rotor, and the position is derived
from that variable [1].
In this control system, a matrix converter is used to
regulate the exciting voltage applied to the rotor,
according to the principles of the integrated DPC/MRAS
system and a power factor control algorithm. This
method has the advantage of controlling the active and
reactive power as well as providing unity power factor at
constant output frequency, while processing only a
fraction of the total power being transferred by the
machine. The matrix converter control relies on the
space vector modulation and sliding mode control
methods [3], [4]. In figure 1, a basic model of the system
under study is presented, and the interaction between
each individual system mentioned above can be seen.
This paper starts with the introduction of some
relevant concepts, such as the DPC algorithm, the
MRAS observer and the matrix converter technology.
Then, simulation results are presented to validate the
performance of the system, and some major conclusions
are expressed in a last chapter.
1
DFIG
Grid
Filter
.
DPC
Power
Control
MRAS
P.Factor
Control
Figure 1. Control system basic model.
II - DPC Algorithm
It can be demonstrated, that active power handled by
the stator can be increased with an increase in the
angular separation between the rotor flux vector, and the
reactive power drawn by the stator can be increased by
reducing the magnitude of the rotor flux and vice versa
[2]. It is important, then, to understand the effect of
application of a voltage vector control to the amplitude
and magnitude of the rotor flux vector, which is the key
idea behind direct power control.
With the three phase rotor winding orientation
presented in figure 2, and in order to make an
appropriate voltage vector selection, the phasor plane
can be subdivided in six different 60º sectors, 1,2,….,6,
with a voltage vector U1, U2,…,U6 corresponding to
each one of them, as can be seen in figure 3.
U3
Sector 3
U2
Sector 2
Sector 4
U4
U1
Sector 1
Sector 5
U5
Sector 6
U6
Figure 3. Phasor diagram of DPC Voltage Vectors.
Phase b
Phase a
Phase c
Figure 2. Three phase rotor winding spacial disposition.
In a generating mode of operation, and considering anti
clockwise direction of rotation of the flux vectors in the
rotor reference frame to be positive, the flux
is ahead
of , as can be seen in figure 3.
The application of voltage vectors U2 and U3 accelerates
in the positive direction, resulting in an increase in
angular separation between the two fluxes, consequently
leading to an increase in the active power generated by the
stator.
On the other hand, application of vector U5 and U6
would result in a decrease in angular separation between
the rotor and stator fluxes, leading to a decrease in the
active power generated by the stator.
The reactive powers are controlled through the
manipulation of the magnitude of the rotor flux
.
Application of voltage vectors U1, U2, and U6, results in
an increase in the magnitude of
, indicating that an
increased amount of reactive power is being fed from the
rotor side and consequently less power is being drawn
from the stator side. In an opposite direction, application
of vectors U3, U4, and U5, reduces the magnitude of ,
increasing the amount of reactive power being drawn from
the stator side, lowering the power factor. Assuming the
power drawn by the stator as being positive and the power
generated as being negative, and if the rotor flux is in the
sector k, where k=1,2,…6, it can be concluded, from the
above discussion, that [2]:
The application of vectors U(k+1) and U(k+2)
would result in a decrease in the stator active
power and vectors U(k-1), U(k-2) would result
in an increase in the stator active power.
2
Table 1. DPC algorithm voltage vectors.
Perr<=0
Perr>0
Qerr > 0
Qerr <=0
Qerr>0
Qerr <=0
Sector 1
U3
U2
U5
U6
Sector 2
U4
U3
U6
U1
The application of vectors U(k), U(k+1)
and U(k-1), decreases the reactive power
drawn from the stator side and U(k+2),
U(k-2), U(k+3) increases the reactive
power drawn from the stator side.
The central idea of the DPC algorithm is to use the
rotor side converter to select an appropriate voltage
vector, forcing the stator active and reactive powers
to follow the reference signal within the narrow band
permitted by the hysteresis controllers. To estimate
the active and reactive powers on the stator, and
assuming a balanced three phase three wire system,
only two stator currents and voltages need to be
measured. The two stator voltages can be expressed
as:
(1)
(2)
And for the currents:
Sector 3
U5
U4
U1
U2
Sector 4
U6
U5
U2
U3
Sector 5
U1
U6
U3
U4
Sector 6
U2
U1
U4
U5
This process is similar for both active and reactive
powers.
The control strategy that has been described can be
summarized, as is presented in the table 1. In this
table, the voltage vectors are organized according to
their effect in the active and reactive powers. If for
example it is convenient to decrease the active power
while simultaneously increasing the reactive power,
and assuming the rotor flux vector to be in sector 3,
then vector 5 should be applied.
III - MRAS Observer
The key idea behind MRAS theory is to create a
closed loop controller, where the output of the system
is compared to a desired response from a reference
model. The model parameters are continuously
updated based on this error. Applied to the sensorless
identification of the rotor flux vector sector, this
system has a voltage model and a current model [1].
The voltage model provides the reference flux, and
can be estimated through:
(3)
(8)
(4)
Finally, the powers can be calculated as:
(5)
(6)
These powers can be made to track any given
reference, staying in a narrow band defined by the
error that is assumed to be tolerable. This error is
defined by the difference between the estimated
powers (5), (6) and the reference. This concept is
illustrated in the figure 4.
The stator flux can also be calculated from the
stator current model, using the machine inductances,
the estimated rotor position
and the rotor and
stator currents.
(9)
The error between the reference flux
estimated flux , in α-β components is:
and the
(10)
Figure 4. Controlled active power variation.
The error in the active power is positive as long as
the estimated power keeps rising, until the limit
allowed by the hysteresis controllers is reached. At
that point, a change in the instantaneous switching
state is required, and afterwards the estimated power
begins to decrease, until the minimum value allowed
by the controllers, leading to the application of
another voltage vector.
Defined in this way, the error can be reduced by
adjusting the estimated rotor position . With the
above discussion in mind, a model of the MRAS
observer is presented in figure 5. The PI controller
drives the error defined in (10) to zero, and it´s output
is the estimated rotational speed, which by integration
provides the estimated rotor angle . The band-pass
filter in the voltage model is used to block the dc
components of the measured voltages and currents.
IV-MRAS: PI Controller Parameters
Estimation
3
+
\
Voltage Model
+
Current Model
∫∫
+
r
Figure 5. MRAS implementation model.
The error, in d-q components, is:
=
-
(11)
=-
+
-
(
=
(16)
The closed loop transfer function has to be defined,
to estimate the PI proportional parameter,
, and
integer parameter, :
The small signal model for the error is obtained as:
=
G (s) =
= 0) (12)
Assuming an orientation along the stator flux
(
= 0), and referring to a synchronous rotating
frame:
=
+
(13)
The d-q flux, derived from the current model, is not
a DC signal unless the estimated speed is equal to the
real speed, so replacing
in (13)
yields:
=
=
+
+
Finally, assuming
obtained as:
(17)
Assuming
=0, in order to isolate
in (17):
(18)
This transfer function has one simple pole, defined
as:
(14)
= 0,
and
(19)
are
Finally, for the PI parameters:
For T
=>
M
Δ
=10
(20)
(15)
Using expressions (12), (14) and (15), a smallsignal model, used for designing the PI controller can
be derived, as can be seen in figure 6.
(21)
V-MRAS: Stability Analysis.
In this section, the MRAS system stability is
analysed, through the inspection of the closed loop
transfer function:
+
-
=
(22)
Figure 6. MRAS small signal model.
The open loop transfer function, derived from
figure 6, is:
In this expression, M,
and
are constant.
Consequently, the system’s stability is only
dependent on the product
, which represents
4
the component of the rotor current along the flux, as
can be understood through the observation of figure 7
[5].
restrictions can be summarized in the following
expression:
This means that, at any instant, one and only one
switch can be ON in each of the matrix lines. With
these restrictions, only 27 states are allowed for the
control of the matrix converter. Applying the space
vector modulation method, these states can be
represented in vector form, as can be seen in table 2
[4], [6]:
Figure 7. Component of the rotor
current along the flux.
Table 2. Space vector modulation of possible commutation states
This component of the rotor current along the flux
can be computed as:
=
+
(23)
The condition that guaranties system stability is:
(24)
VI-Matrix Converter
The matrix converter is an AC-AC converter,
assembled from individual semiconductor switches
which act directly on the connections between the
generator and the load. For that reason it is
commonly called a direct converter. This converter
consists of nine bi-directional switches, arranged as a
3x3 matrix, as shown in figure 8, so that any input
phase can be connected to any output phase at any
time [4].
Figure 8: -Matrix converter topology.
The matrix converter topology being described can
be made to control the input currents and the output
voltages through an appropriate control strategy. A
space vector modulation (SVM) strategy is combined
with sliding mode control in this paper to control the
converter [3], [4].
Each individual switch has two possible states
(ON/OFF), and there are 9 switches, which
configures
possible
combinations.
However, since the converter is supplied by a voltage
source, the input phases must never be shorted, and
due to the inductive nature of the load, the DFIG,
none of the output phases can be left open. These
State Vector
1
2
-
-
3
4
5
6
-
-
7
+1
8
-1
-
-
9
10
+2
-2
-
-
11
+3
-
-
-
-
- +
- +
+
+
- +
+
+
- +
-
12
-3
13
+4
14
-4
-
-
15
16
+5
-5
-
-
17
+6
-
-
-
-
18
-6
19
+7
20
-7
21
+8
-
-
-
22
-8
-
23
24
+9
-9
-
25
26
27
0
0
0
-
-
-
To avoid further complexity, the vectors
numbered from 1 to 6 and the zero vectors are not
used. The voltage and current vectors can be
represented in the α-β plane, as shown in figures 9
and 10.
4, 5,
0
1, 2,
7, 8,
Figure 9: - Generic voltage vectors representation.
5
2, 5,
a)
3, 6,
b)
0
When
the error.
When
the error.
=>
, to decrease
=>
, to decrease
To implement this control system, hysteresis
controllers are used to maintain the error within
tolerable limits (typically 2 Δ), using the command
law (28):
1, 4,
(28)
Figure 10: - Generic current vectors representation.
The vectors represented in these two figures, are
fixed in space, but their magnitudes can change,
reflecting the variations in the input voltages, in the
case of voltage vectors or the output currents in the
case of current vectors. At any given time and in each
one of the directions shown, there are three vectors,
all with different magnitudes, and three in the
opposite direction in the α-β plane. For control
purposes, and because a fast response of the system is
highly desirable, only the two with bigger magnitudes
are used [6], [10].
Along with SVM, sliding mode control is used in
this work. This is a non-linear control technique,
particularly fit to be used in power converters, in
which the application of a high-frequency switching
control between two discontinuity surfaces allows for
the state space trajectory of the system to slide
through the discontinuity surface [3].
Hysteresis
Band
Figure 11. Space state trajectory.
In sliding mode control, the application of an
infinite frequency switching control would allow the
state space trajectory to exactly follow the reference
signal. But as conventional semiconductors are
frequency limited, it is impossible to exactly follow
the reference signal, so there is an error defined as:
VII - DFIG Control with Rotor Side Matrix
Converter
With the previous chapter’s theoretic considerations
in mind, it is now possible to focus in the key idea
behind this work, the control of a DFIG, whose stator
is directly connected to the power grid, while acexcited by a matrix converter in the rotor side (Fig.
1). To achieve this objective, the converter output
(rotor side) voltages and input currents (grid side)
must be controlled. The output voltages are imposed
by the DPC algorithm, which generates voltage
vectors depending on the estimated stator powers and
according to the rotor flux position information,
provided by the MRAS observer. Following this
method, active and reactive power control can be
achieved. On the other side, input currents must be
controlled to guaranty grid side unity power factor.
The active and relative power control can be
achieved by establishing a relation between the 6
DPC method voltage vectors, with the 18 voltage
vectors resulting from the space vector modulation
technique. It must be kept in mind, though, that these
18 voltage vectors have fluctuating magnitudes,
depending on the input voltages relative position [6].
Considering that at a certain instant in time they have
the following α-β plane representation (Fig. 12):
-6
Δ
(26)
Using this control law, the system will be immune
to perturbations and independent of circuit
parameters, or operation point. The high frequency
control applied, has to be alternately positive and
negative, to guaranty a near zero dynamic error Δ,
and so, condition (27) has to be followed [7], [8].
(27)
Consequently, it can be understood that:
-8
+5
-7
+4
(25)
A control law is presented (26), in which Δ is a
dynamic error:
+9
+3
-2
+1
-1
+2 -3
+7 -4
+8
-9
-5
+6
Figure 12. Output Voltage Vectors.
The selection of the U3 voltage vector (Fig. 3),
according to the DPC method, implies that, to
properly control the active and reactive powers, a
decrease in the α component and an increase in the β
component of the voltages is mandatory. Then,
assuming the matrix converter voltage vectors
6
Park
Transformation
d,q  αβ
+
Concordia
Transformation
αβ  a,b,c
,
-
DPC
algorithm
,
Grid
,
Current
commutation
Function
Vector
Selection
+
,
Vector
,
Matrix
Converter
DFIG
Vector
MRAS
Observer
Stator Power
Estimation
Figure 13. Control System Implementation Model.
This means that the q component of the input
current must be null, in order to achieve unity power
factor. Using sliding mode control, the error between
this current and the defined reference (
=0) is
defined by:
(30)
1.2
1
0.8
0.6
0.4
[p.u.]
(29)
bidirectional power flow, unity power factor control
and input/output characteristics are evaluated in this
chapter.
Fig.14, 15 shows the system’s transient response
due to a step change in active and reactive power
command, from the initial
=0 p.u., to
=0.9
p.u. at t=0.04s, while
is maintained at 0.2 p.u.
Active Power [p.u.]
representation of figure 12, it is clear that vector -6,
+5, +4 should be applied to the rotor to obtain the
same effect in the αβ components described above.
On the other hand, it can be demonstrated that the
unity power factor can be controlled, if the following
condition is respected [6]:
0.2
The input current can then be made to track the
given reference, using hysteresis controllers, with the
following commutation function [8], [9]:
0
-0.2
0
0.04
0.06
0.08
0.1
0.12 0.14
0.16
0.18
0.2
Figure 14. Active power transient response.
0.28
0.26
0.24
0.22
0.2
0.18
0.16
0.14
[p.u.]
Reactive Power [p.u.]
The
function can only assume two logic levels,
“1” and “-1”. When (31) is at level “1”, meaning that
reference signal
is superior to the current, the
system is expected to apply a vector with positive q
component, decreasing the error, and otherwise, when
is at level “-1”, a vector with negative q
component should be applied.
These two strategies, the DPC voltage control and
the unity power factor control have to be combined,
when selecting an appropriate vector to control the
machine. The DPC voltage control imposes the
matrix converter output voltages, while the input
currents are defined by a power factor control
method. This means, that both the machine side and
the power grid side, have to be taken under
consideration at any instant in time.
0.02
t [s]
(31)
0.12
0.1
0
0.04
0.08
0.12
0.16
0.2
t[s]
Figure 15. Reactive power transient response.
VIII - Simulation Results
A simulations study is carried out to verify the
above control strategy. The power control,
7
0.3
1
0.2
0.5
0
-0.5
0.1
0
-0.1
[p.u.]
Input Current [p.u.]
Output voltage[p.u.]
1.5
-0.2
-1
-0.3
-1.5
0.06 0.0620.0640.0660.0680.070.0720.0740.0760.0780.08
0
0.02 0.04
0.06 0.08
0.5
0.4
0.4
0.3
0.2
0.1
0.05
0.1
-0.4
-0.6
-0.8
0
0.15
0.05
0.1
0.15
t [s]
Figure 20. Active and reactive powers.
1
0.8
0.6
0.4
0.2
0
-0.2
Input
-0.4
0.6
0.4
0.2
0
-0.2
-0.4
[pu]
voltage and current
1
[p.u.]
Input voltage and current [p.u.]
0.2
-0.2
0.8
-1
0
0.18
0
Figure 17. Active and reactive powers.
-0.8
0.16
0.2
t [s]
-0.6
0.14
[pu]
Active and Reactive power [pu]
0.6
[p.u.]
Active and Reactive power [p.u.]
0.6
-0.1
0
0.12
Figure 19. Input currents
Figure 16. Output voltage
0
0.1
t [s]
t [s]
-0.6
-0.8
0.05
0.1
0.15
t [s]
-1
0
0.05
0.1
0.15
t [s]
Figure 18. Converter input voltage and current.
Figure 21. Converter input voltage and current.
As can be seen, the system exhibits excellent active
and reactive power control, with a fast response to the
active power reference step. The responses in the
active and reactive power are perfectly decoupled. In
the same simulation, Fig. 16 shows the output
voltage, with a characteristic waveform, due to the
high-frequency semiconductors commutation. The
input currents presented in Fig. 19, exhibit some
distortion, due to the fact that in this work, the use of
null vectors was discarded. This leads to an excessive
commutation frequency which causes current
oscillation and noise.
Another simulation was performed, in which the
bidirectional power flow in the matrix converter was
evaluated. Fig 17 and 18, show the active and
reactive powers, and the input (grid-side) voltage and
current respectively, when the DFIG is operating in
sub synchronous mode. The input voltage and current
are approximately in phase with each other
, and the reactive power is nearly zero, indicating that
unity power factor can be achieved. In this situation,
the rotor is being supplied with active power from the
grid side, through the matrix converter. On the other
hand, Fig. 20 and 21 show the active and reactive
power as well as the input voltages and currents, with
the machine operating in super synchronous mode. In
this case, the opposite phase relation between the
input voltage and current indicates that the rotor is
supplying active power to the grid. The reversing of
the input current between sub synchronous and super
synchronous mode of operation, demonstrates the
matrix converter bidirectional power flow capacity.
Another simulation was performed on the MRAS
observer, and the results are presented in Fig. 22 and
23. The first simulations done using this method
encountered some stability problems, and so a robust
system was tested, validating the condition expressed
8
6
0.035
0.03
5
0.02
3
0.015
[p.u.]
Error
4
r
Sector
0.025
0.01
0.005
2
0
1
0
0.05
0.1
0.15
-0.005
0
0.2
t [s]
0.01
0.02
0.03
0.04
0.05
0.06 0.07
0.08
0.09
0.1
t [s]
Figure 22. MRAS sector identification.
Figure 23. MRAS error
in (24). Perfect rotor flux sector identification was
achieved (Fig. 22), and a rapid error convergence to
zero is observed (Fig. 23).
IX – Conclusion
The simulation study on the proposed method, has
clearly demonstrated that sensorless power control,
through the application of an appropriate alternated
excitation to the rotor, using a matrix converter is a
viable solution to a DFIG based wind power
generator control. The DPC algorithm allows
independent active and reactive power control, and is
insensitive to the parameters of the machine, since it
depends only on the currents and voltages measured
in the stator. The MRAS observer exhibits an
excellent tracking performance under simulation,
allowing sensorless power control. The matrix
converter has bidirectional power flow capacity,
implementing
the
DPC
algorithm
while
simultaneously controlling the power factor and
maintaining a constant output frequency. Input and
output waveforms show an acceptable distortion
level, which could be significantly decreased if null
vectors were used.
Aparício, L.; Esteves, P.; “Conversor Matricial:
Aplicação em Aproveitamentos Eólicos com Máquina
de Indução Duplamente Alimentada”, Trabalho Final de
Curso da Licenciatura em Engenharia Electrotécnica e
de Computadores, Instituto Superior Técnico, Setembro
de 2007.
[9] Huber, L.; Borojevic, D., Burany, N.; “Analysis, design
and implementation of the space-vector modulator for
forced-commutated cycloconvertors”, IEE ProceedingsB, Vol. 139, No. 2, March 1992.
[10] Holmes, D.; Lipo, T.; "Implementation of a Controlled
Rectifier Using AC-AC Matrix Converter Theory";
IEEE Transactions on Power Electronics, Vol. 7, No 1,
January 1992, pp. 240-250.
[8]
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[1]
[2]
[3]
[4]
[5]
[6]
[7]
Cárdenas,R.; Peña, R., Asher,G., Clare, J., Cartes, J.,”
MRAS Observer for Doubly Fed Induction Machines”,
IEEE Transactions on Energy Conversion, Vol. 19, No.
2, June 2004.
Datta, R. ;Ranganathan, V.;”Direct Power Control of
Grid-Connected Wound Rotor Induction Machine
Without Rotor Position Sensors”, IEEE Transactions on
Power Electronics, Vol. 16, No. 3, May 2001.
Huber, L.; Borojevic, D.;, “Space Vector Modulated
Three-phase to Three-phase Matrix Converter with
Input Power Factor Correction”, IEEE Transactions on
Industry
Applications,
Vol.
31,
No
6,
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