Control of a Doubly-Fed Induction Wind Generator Under Unbalanced Grid
Voltage Conditions
Ted Brekken, Ned Mohan, Tore Undeland
UNIVERSITY OF MINNESOTA
200 Union St. S.E.
Minneapolis, MN 55455, USA
Tel.: +47 – 976 69 048
Fax: +47 – 735 94 279
E-Mail: tedbrekken04@fulbrightweb.org
URL: http://www.ece.umn.edu
Acknowledgements
I would like to thank my advisor, Ned Mohan. I would also like to thank Tore Undeland and the
Energiomforming department at the Norwegian University of Science and Technology for their support.
Keywords
«Doubly fed induction motor», «Vector control», «Windgenerator systems», «Active damping», «Power
quality»
Abstract
Wind energy is often installed in rural, remote areas characterized by weak, unbalanced power
transmission grids. In induction wind generators, unbalanced three-phase stator voltages cause a number
of problems, such as overheating, over-current, and stress on the mechanical components from torque
pulsations. Therefore, beyond a certain amount of unbalance, induction wind generators are switched out
of the network. This can further weaken the grid. In doubly-fed induction generators, control of rotor
currents allows for adjustable speed operation and reactive power control. In addition, it is possible to
control the rotor currents to correct for the problems caused by unbalanced stator voltages, including
torque pulsations and unbalanced stator currents. This paper presents a novel voltage mode controller
design for a doubly-fed induction generator that provides variable speed, reactive power control. Also,
under stator voltage unbalance conditions, the proposed control eliminates torque pulsations and draws
more balanced currents from the utility.
Introduction
Rural grids are prone to three-phase voltage unbalances that can cause many problems for induction
machines. The goal of the research is to develop a control method for doubly-fed induction wind
generators that addresses issues associated with weak rural grids. Commonly used control methods for
doubly-fed machines pay no special attention to these problems. The presented control method improves
the robustness and all-around performance of doubly-fed induction wind generators, allowing them to
operate in conditions in which they would normally be removed from the grid. A controller is designed
and tested in simulation and also tested on a 15 kW DSP-based hardware setup.
Symbols
• vsd , isd , isq : Stator d-axis voltage, stator d-axis current, stator q-axis current.
• vrd , vrq , ird , irq : Rotor d-axis voltage, rotor q-axis voltage , rotor d-axis current, rotor q-axis current.
•
•
•
•
ωmech , T , Qs : Rotor mechanical speed (rad/s), generator torque, stator reactive power.
Lm , p, n : Generator magnetizing inductance, number of generator poles, rotor/stator turns ratio.
J , B : System moment of inertia, frictional constant.
ωsyn , ωd : Synchronous speed, dq-axis frame speed.
All variables are SI units or per unit.
Induction generator unbalance effects
Wind energy generation equipment is most often installed in remote, rural areas. These remote areas
usually have weak grids, often with voltage unbalances and under/over-voltage conditions [1,2]. When
the stator phase voltages supplied by the grid are unbalanced, the torque produced by the induction
generator is not constant. Instead the torque has periodic pulsations at twice the grid frequency, which can
result in acoustic noise at low levels and at high levels can actually destroy the rotor shaft, gearbox, or
blade assembly [3]. Also, an induction generator connected to an unbalanced voltage will draw
unbalanced currents. These unbalanced currents tend to magnify the grid voltage unbalance in addition to
causing over current problems as the phase current can become quite large [4].
A special type of induction generator called a doubly-fed induction generator (DFIG), as shown in Figure
1, is becoming the most popular choice for wind turbines. The ability to control the rotor currents allows
for reactive power control and variable speed operation, so that a DFIG can operate at maximum
efficiency over a wide range of wind speeds. The goal of the research is to develop a control method that
uses the rotor control capabilities of a DFIG to deal with unbalanced stator voltages, while also providing
the standard reactive power and speed control. Grid side converter control is not considered, but is a good
candidate for future research.
stator
grid
DFIG
rotor
DC link
AC
DC
DC
AC
Fig. 1: Doubly-fed induction machine
Definition of unbalance
There are several different ways to define a three phase unbalance. For this paper, the voltage unbalance
factor (VUF) is defined as the negative sequence magnitude divided by the positive sequence magnitude
[4]. The advantage of this system is that it accounts for both magnitude and phase angle. The same
unbalance factor can be defined for three-phase current (IUF).
Second harmonic disturbance
Using symmetrical components theory, a stator voltage unbalance can be seen as the addition of a negative
sequence to the stator voltage. The negative sequence rotates at 50 Hz in the opposite direction of the
positive sequence (assuming a 50 Hz synchronous frequency). From the perspective of the negative
sequence, the rotor is turning in the counter direction. Therefore the negative sequence sees a very large
slip, which causes a large amount of negative sequence current to flow. This causes unbalanced stator
currents, and can also cause an over-current condition. In addition, the torque and reactive power will
have a second harmonic (100 Hz) pulsation.
To understand the torque and reactive power second harmonics, it is useful to use space vectors in the
stationary, stator frame. In space vector theory, the positive sequence stator voltage space vector rotates
with constant amplitude in the positive direction, at the synchronous speed. The negative sequence stator
voltage space vector rotates at the synchronous speed with constant amplitude in the negative direction.
The same will also apply to the flux: the flux arising from the positive sequence voltage will rotate
synchronously in the positive direction, and the flux arising from the negative sequence voltage will rotate
synchronously in the negative direction. These two flux vectors will “meet” and add constructively and
destructively twice per revolution, which gives rise to the double synchronous frequency disturbance in
the torque and reactive power [4].
In a synchronously rotating dq reference frame, the d and q system components will nominally be dc
values, in steady state. However, in the presence of a stator voltage unbalance, the d and q components
will have a second harmonic in addition to the dc value. Therefore the entire dq frame control structure
will have the addition of this second harmonic. This in turn causes second harmonic pulsations in the
torque (active power) and reactive power.
Controller design
For the proposed control, stator voltage oriented dq vector control is used. This orientation can be called
“grid flux oriented” control [5]. In this scheme, the d-axis is aligned with the stator voltage space vector
(instantaneously). Therefore, vsq = 0 (steady state and transient) and ω d = ω syn (steady state only). The daxis is used to control torque and the q-axis is used to control reactive power. The control structure is
shown in Figures 2 and 3. The control topology is fairly standard except for the addition of the Cddr and
Cqdr controllers, which supplement the d and q-axis rotor voltage. (In this paper “dr” stands for
“disturbance rejection.”) These supplementary feedback controllers give the loops a very high gain at the
known disturbance frequency (100 Hz), allowing the controllers to compensate for the torque and reactive
power pulsations that arise when the stator voltage is unbalanced. Reducing the reactive power pulsation
dramatically improves the unbalance of the stator current.
*
ωmech
Cω mech
ird*
Cird
T
Fig. 2: d-axis control
Cddr
vrd '
vrd ,2
vrd
ird vrd ( s)
ird
ωmech ird ( s)
ωmech
Qs*
CQs
irq*
Cirq
Qs
Cqdr
vrq '
vrq
irq vrq ( s )
irq
Qs irq ( s )
Qs
vrq ,2
Fig. 3: q-axis control
From a symmetrical components point of view, compensating for torque and reactive power pulsations is
analogous to the controller injecting a negative sequence into the rotor circuit in such a manner as to
compensate for the negative sequence in the stator circuit. The net effect is to dramatically reduce the
torque pulsations, reactive power pulsations, and stator current unbalance under unbalanced grid voltage
conditions.
Design of Cird and Cirq
The inner loop ird and irq current controllers, Cird and Cirq , were designed by using linearized state
equations relating the voltage and current in the dq frame, as shown in Equation 1 [6]. This is difficult to
do symbolically, but easily solved numerically using Matlab® or an equivalent analysis program.
isdq 
 vsdq 
d isdq 
 i  = [ A]  i  + [ B ]  v 
dt  rdq 
 rdq 
 rdq 
(1)
From the linearized state equations, ird vrd (s) and irq vrq ( s) are determined. Cird and Cirq are simple PI
controllers each designed for a 100 Hz loop gain crossover frequency and 90 degrees of phase margin.
Doubly-fed machines naturally have a pair of poorly damped poles near the grid frequency. Fast inner
current loops have a fast response time, but also tend to push the poorly damped system poles toward the
right half plane [5]. Slow loops are more stable, but the controllers must be fast enough to handle the
rotor converter blanking time harmonics, which occur at the slip frequency sixth harmonic [7]. A loop
gain bandwidth of 100 Hz is found to be a good compromise.
Design of Cωmech
The outer speed and reactive power loops were designed in the normal cascade manner, having loop gain
crossover frequencies well below that of the inner current loops.
To design the speed controller, the ωmech ird ( s) transfer function must be determined. It becomes useful
to define magnetizing current.
isq + nirq = imq
(2)
isd + nird = imd
(3)
T=
p
p
p
Lm n(isq ird − isd irq ) = Lm n(imq ird − imd irq ) ≈ Lm nimq ird
2
2
2
(4)
The significant advantage of this form of the torque equation is that the magnetizing currents imd and imq
are predominantly determined by the stator side voltage, and therefore can be approximated to be constant.
The magnetizing current space vector will lag the stator voltage space vector by nearly 90 degrees. Since
the d-axis is chosen to be aligned with the stator voltage space vector, the majority of the magnetizing
current space vector will be reflected along the negative q-axis. Therefore imq is large negative, and imd is
very small. Therefore the torque can be approximated as having a proportional dependence on ird .
T
p
( s ) = Lm nimq
ird
2
(5)
Therefore,
p

Lm nimq 
2

ωmech ird ( s ) = 
( sJ + B )
(6)
A PI controller is used, and the loop gain bandwidth is designed for 1 Hz. A fast speed loop will be
quicker to respond to changes in the load, such as gusts of wind. This will cause the generator torque to
respond quickly, thereby causing the power into the grid to change rapidly, mirroring the wind gusts.
However, a slow speed loop will allow the generator to accelerated or decelerate as the wind changes, thus
using the rotational energy of the machine as a buffer to smooth the power into the grid [8].
Design of CQs
The reactive power outer loop is designed by relating stator reactive power, Qs , to irq . It is again helpful
to use the magnetizing current.
Qs = −vsd isq = −vsd (imq − nirq )
(7)
As explained above, the magnetizing current imq can be approximated to be constant. It then follows that
Qs
( s ) = nvsd
irq
(8)
A PI controller is used, and the loop gain bandwidth is designed for 1 Hz.
Design of disturbance rejection controllers Cddr and Cqdr
The Cddr and Cqdr controllers shown in Figures 2 and 3, respectively, come into effect only in the
presence of a stator voltage unbalance. They supplement the output of the nominal operation controllers,
Cird and Cirq , to remove the 100 Hz disturbance from the torque and reactive power. It is not enough to
simply increase the gain of Cird and Cirq . Removing the second harmonic from ird and irq will not
completely remove the second harmonic from the torque and reactive power.
The Cddr and Cqdr controllers are designed to have a large gain at the known disturbance frequency (twice
the synchronous frequency, 100 Hz) but also to have a negligible effect at all other frequencies. This is
done by using a high-Q, second order bandpass filter. The Cddr controller can be represented as having
two components: a bandpass filter feeding into a standard lead-lag controller.
Cddr = Cddr ,bp Cddr ,ll =
sω0 Q filt
( s + s ω0 Q filt + ω0 2 )
2
(9)
Cddr ,ll
The natural frequency of the filter, ω0 , is the second harmonic (100 Hz) and Q filt is set to some high value
(e.g. 100).
Figure 4 shows the inner loop of the d-axis with the Cddr controller.
ird*
Cird
T2* = 0
T ird ( s ) T
Cddr ,bp
Cddr ,ll
vrd '
vrd
ird vrd ( s )
ird
vrd ,2
T2
Fig. 4: ird loop with disturbance rejection controller
In this figure, T2 refers to the second torque harmonic (100 Hz). Stability of the augmented ird loop can
then be determined by using the Nyquist stability criterion for the augmented loop gain:
loop gain =


ird
T
( s )  Cird + Cddr ( s ) 
vrd
ird


(10)
The design of Cqdr follows the same procedure.
750 kW Simulation results
The proposed controller design was tested in simulation using Matlab/Simulink®. The generator model is
a 750 kW doubly-fed generator.
In Figure 5, the system is running under nominal conditions without the supplemental disturbance
rejection control loops. At t = 2 seconds, the grid becomes unbalanced with a stator voltage unbalance of
approximately 6.5 %. The unbalance was created by placing a resistance in series with the stator A-phase.
At t = 3 seconds, the disturbance rejection controllers are switched in. The figure shows that the
disturbance rejection controllers significantly reduce the torque and reactive power pulsations, while also
decreasing the unbalance factor of the stator current.
Also, several steady state simulations were run, with the generator at synchronous speed and rated power.
The 100 Hz torque pulsation magnitude and stator current unbalance factor are plotted versus stator
voltage unbalance factor. The results are shown in Figure 6. The linear interpolation lines in Figure 6
show a reduction of 3.4/0.33 = 103 for the torque, and a reduction of 2.7/0.99 = 2.7 for the stator current
unbalance over a range of stator voltage unbalance from 0 to 0.1.
Generator Torque
-0.8
Generator Torque 100 Hz Magnitude
0.25
-0.85
-0.9
0.2
torque (per unit)
torque (per unit)
-0.95
-1
-1.05
-1.1
0.15
0.1
-1.15
0.05
-1.2
-1.25
-1.3
1
1.5
2
2.5
3
3.5
4
4.5
0
5
1
1.5
2
Generator Reactive Power
0.2
2.5
3
3.5
4
4.5
5
time (seconds)
time (seconds)
0.2
Generator Stator Reactive Power 100 Hz Magnitude
0.18
0.15
reactive power (per unit)
reactive power (per unit)
0.16
0.1
0.05
0
-0.05
-0.1
0.14
0.12
0.1
0.08
0.06
0.04
-0.15
-0.2
0.02
1
1.5
2
2.5
3
3.5
4
4.5
0
5
1
1.5
2
time (seconds)
2.5
3
3.5
4
4.5
5
time (seconds)
Stator Current Magnitude
Stator Current Unbalance Factor (IUF)
0.2
isa
isb
isc
1.15
0.18
0.16
1.1
unbalance factor
current (per unit)
0.14
1.05
1
0.95
0.12
0.1
0.08
0.06
0.9
0.04
0.85
0.8
0.02
1
1.5
2
2.5
3
3.5
4
4.5
0
5
1
1.5
2
time (seconds)
2.5
3
3.5
4
4.5
5
time (seconds)
Fig. 5: 750 kW generator transient simulation with approx. 0.065 stator voltage unbalance factor
Generator Torque 100 Hz Magnitude
Stator Current Unbalance Factor (IUF)
standard
with dr
0.25
0.2
unbalance factor
0.2
torque (per unit)
standard
with dr
0.25
y=3.4e+000*x-0.01
0.15
0.1
0.05
0.15
y=2.7e+000*x-0.01
0.1
0.05
y=9.9e-001*x-0.00
y=3.3e-002*x-0.00
0
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
stator voltage unbalance factor (VUF)
Fig. 6: 750 kW generator steady state simulations
0
0.02
0.03
0.04
0.05
0.06
0.07
0.08
stator voltage unbalance factor (VUF)
0.09
0.1
15 kW Simulation and hardware results
The proposed control was tested on a 15 kW doubly-fed induction generator controlled with the rapid
prototyping DSP package dSpace®, as shown in Figure 7. The stator voltage unbalance was created by
placing a high-power variable resistance in series with the stator A-phase.
In Figure 8, the system is running in steady state with a 5 % stator voltage unbalance. At t = 0, the
disturbance rejection controllers are activated. The figures on the right hand side also have the behaviour
as predicted by simulation. Simulation results were obtained using a Matlab/Simulink® model of the
control and 15 kW DFIG. The simulation results were not included on the left hand side plots for clarity.
The results show that the disturbance rejection controllers greatly reduce the second harmonic torque
pulsations, reactive power pulsations, and stator current unbalance.
230V 3ph AC
15 kW DFIG
15 kW DC
inverter
220V DC
DSP
220V DC
inverter
dSpace
PC
Fig. 7: 15 kW doubly-fed test setup
The hardware results closely match the simulation results, except in a few cases. The actual 100 Hz
torque magnitude is larger than predicted by simulation. This is likely partially due to 100 Hz noise
coming from measurements and external sources, not necessarily from an unbalance. The torque shown
for hardware results is calculated from measured currents. Therefore noise and offset errors in the current
measurements will be present in the calculated torque. Also the measured stator current unbalance is
greater than predicted by the simulation. Some of this error is likely due to offset errors in the current
measurement. The unbalance calculation is sensitive, and even small offset errors of 1 % can have a large
impact on the calculated unbalance factors.
In addition to the points above, a large unbalance can drive the machine to operate in regions outside of
normal operation. Therefore second-order effects that are neglected in simulation, such as saturation, slot
effects, iron losses, and winding harmonics can become significant [9]. This may account for some of the
differences in simulation and hardware.
In addition to transient analysis, a series of steady state analysis with a swept stator voltage unbalance are
shown in Figures 9 and 10. Figure 9 is the simulation result, and Figure 10 is the hardware result. The
generator is at synchronous speed and rated power. Table I shows the reduction in torque pulsations and
stator current unbalance.
Table I: Torque and stator current unbalance reduction for the 15 kW DFIG
Simulation
Hardware
Torque
6.8/0.59 = 11.5
9.3/0.32 = 29.1
Stator Current Unbalance
6.1/0.82 = 7.4
7.1/1.3 = 5.5
Generator Torque
Generator Torque 100 Hz Magnitude
0.5
-0.4
hardware
simulation
0.45
0.4
-0.6
0.35
torque (per unit)
torque (per unit)
-0.8
-1
-1.2
0.3
0.25
0.2
0.15
-1.4
0.1
0.05
-1.6
0
0.5
1
1.5
2
2.5
0
3
0
0.5
1
1.5
2
3
time (seconds)
Generator Reactive Power
Generator Stator Reactive Power 100 Hz Magnitude
hardware
simulation
0.25
0.25
0.2
0.15
reactive power (per unit)
reactive power (per unit)
2.5
time (seconds)
0.1
0.05
0
-0.05
-0.1
-0.15
0.2
0.15
0.1
0.05
-0.2
-0.25
0
0.5
1
1.5
2
2.5
0
3
0
0.5
1
Stator Current Magnitude
2
2.5
3
Stator Current Unbalance Factor (IUF)
hardware
simulation
isa
isb
isc
1.1
0.25
1
unbalance factor
current (per unit)
1.5
time (seconds)
time (seconds)
0.9
0.8
0.7
0.2
0.15
0.1
0.05
0.6
0
0.5
1
1.5
2
2.5
0
3
0
0.5
1
1.5
2
2.5
3
time (seconds)
time (seconds)
Fig. 8: 15 kW hardware and simulation transient results with approx. 0.05 stator voltage unbalance factor
Torque 100 Hz Component
0.4
Stator Current Unbalance Factor (IUF)
standard
with dr
0.35
0.25
unbalance factor
torque (per unit)
0.3
y=6.8e+000*x-0.00
0.25
0.2
0.15
y=6.1e+000*x-0.00
0.2
0.15
0.1
0.1
y=5.9e-001*x-0.00
0.05
0
standard
with dr
0.3
0
0.01
0.02
0.03
0.04
0.05
y=8.2e-001*x-0.00
0.05
0.06
stator voltage unbalance factor (VUF)
Fig. 9: 15 kW simulation steady state results
0
0
0.01
0.02
0.03
0.04
stator voltage unbalance factor (VUF)
0.05
0.06
Torque 100 Hz Component
0.4
0.35
0.25
unbalance factor
torque (per unit)
standard
with dr
0.3
0.3
0.25
y=9.3e+000*x+0.01
0.2
0.15
y=7.1e+000*x-0.01
0.2
0.15
y=1.3e+000*x+0.02
0.1
0.1
y=3.2e-001*x+0.02
0.05
0.05
0
Stator Current Unbalance Factor (IUF)
0.35
standard
with dr
0
0.01
0.02
0.03
0.04
0.05
0.06
stator voltage unbalance factor (VUF)
0
0
0.01
0.02
0.03
0.04
0.05
0.06
stator voltage unbalance factor (VUF)
Fig. 10: 15 kW hardware steady state results
Conclusion
A control methodology for the operation of doubly-fed induction generators connected to an unbalanced
stator voltage is presented. Simulation and hardware testing have shown that the control is simple to
implement and very effective at compensating for the torque pulsations, reactive power pulsations, and
unbalanced stator current that normally occur when the stator voltage is unbalanced. This greatly reduces
the wear on the mechanical components, such as the shaft, gear box, and blade assembly, while also
improving the quality of the power fed into the grid. The presented control topology allows the generator
to tolerate a much larger stator voltage unbalance than is acceptable with standard controllers.
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