Active Damping of LCL-Filter Resonance based
on Virtual Resistor for PWM Rectifiers –
Stability Analysis with Different Filter
Parameters
Christian Wessels, Jörg Dannehl, student member, IEEE and Friedrich W. Fuchs, senior member, IEEE
Christian-Albrechts-University of Kiel
Insitute of Power Electronics and Electrical Drives
Kaiserstr. 2, 24143 Kiel, Germany
Abstract—This publication presents the investigation of
active damping of resonance oscillations with virtual
resistor for grid-connected PWM rectifiers with LCL-filter
for different filter parameters. Using the voltage-oriented PI
current control with converter current feedback, additional
active damping of the filter resonance is necessary for stable
operation. In the literature different methods are proposed
that differ in number of sensors and complexity of control
algorithms. If higher damping of the switching ripple
current is required LCL-filters with lower resonance
frequencies can be used. Resulting low ratios between
resonance frequency and control frequency challenge the
control with respect to damping of resonance. Moreover,
some active damping methods are not suitable for these
filter settings.
Here the active damping concept based on virtual resistor is
analyzed concerning stability for two significant filter
configurations. It turns out that it is applicable for
configurations with higher resonance frequency, whereas
systems lower resonance frequencies can poorly be damped.
Additionally the method exhibits the advantage of simple
implementation but the disadvantage of additional current
sensors. Theoretical analyses and of the selected method
with time-discrete implementation are shown in this paper.
Theory is verified by experimental results.
I.
INTRODUCTION
PWM rectifiers are applied where bidirectional flow of
energy by converters is needed, e.g. in variable speed
drives with regenerative braking or regenerative energy
systems. As advantages they offer an adjustable power
factor and emit less current harmonic distortion,
compared to passive diode rectifiers.
To damp the switching harmonics, grid side filters are
used to connect the PWM rectifier with the grid. LCLfilters are more interesting, because they are more costeffective compared to simple L-filters as smaller
inductors can be used to achieve the same damping
effect. To overcome the disadvantage of resonance
oscillations of LCL-filters, damping of the filter
resonance is necessary. Simple passive damping with
resistor in series to the LCL-filter-capacitor [1] creates
additional power losses and decreases the system
performance. Thus active damping by modifying the
control algorithm is preferred because of no additional
power losses and more flexibility. Different active
damping methods are presented in literature. In [2-4] the
converter current control with additional feedback of the
voltages across the filter capacitors is shown. In [5] the
line current control with additional feedback of the
current through the filter capacitors is presented. In [6,7]
the converter current control using only the converter side
current sensors is shown. An overview about different
multiloop approaches can be found in [8-10]. In [11-13]
the control is designed by using the complete state
information.
The control methods differ in several criteria like
number of sensors, complexity of control algorithm and
robustness against parameter variations for certain LCLfilter configurations. In [14] limitations of control with
converter current feedback and additional active damping
with notch-filter as presented in [6] are shown. If an
LCL-filter with a low resonance frequency is chosen for
the purpose of high damping of switching harmonics, the
design of the active damping gets very difficult and a
poor robustness is obtained. A method utilizable for a
large set of system parameters is desirable.
In this paper the applicability of the voltage-oriented PI
control with active damping based on virtual resistor
concept [9] is presented including the analysis of stability
for two significant filter settings. The analysis is verified
by means of measurement results.
The publication is organized as follows: in section II
the system description and modeling is shown. Section III
describes the control structure and in section IV the
virtual resistor concept including theoretical analyses is
shown. Measurement results are presented and analysed
in section V. Finally, a conclusion closes this publication.
II. SYSTEM DESCRIPTION
The investigated system is shown in Fig. 1. The PWM
rectifier is connected with the grid via LCL-filter. The
DC side of the rectifier consists of the DC capacitor and
is connected to a load. The system parameters are given
in Tab. I. Here, two LCL-filter configurations with
different resonance frequencies are used. Choosing a
higher filter capacitor yields to higher damping of the
Tab. I: System parameters for analysis
Symbol
Quantity
Value
vL
Nominal line voltage
230 V (rms)
iL
Nominal line current
15 A (rms)
Lfg
Grid-side filter inductance
2,0 mH
Rfg
Resistance of grid-side filter inductor
30 m
Lfc
Converter-side filter inductance
3,0 mH
Rfc
30 m
Cf
Resistance of converter-side filter
inductor
Filter capacitance
fc
Switching/Control frequency
8 F /
48 F
4 kHz
The transfer function of the converter is:
Fig. 1: Grid-connected PWM rectifier with LCL-Filter
switching harmonics, but reduces the resonance
frequency of the filter as it can be seen in the Bode
diagram in Fig. 2. For the purpose of feedback the DC
link voltage as well as the converter and line currents are
measured.
The line voltage is measured for synchronizing the
control with the grid frequency.
Here the space vector notation is used. The three-phase
values are transformed into stationary reference frame
and further, using the line voltage vector, into rotating dq
coordinates in order to perform the voltage-orientedcontrol. From control point of view it is advantageous to
control DC values since PI controllers can achieve
reference tracking without steady state errors. As
disadvantage the coordinate transformation leads to
current dynamics coupling. Modeling the LCL-filter in a
dq-reference frame gives
dq
di
dq
dq
dq
L fg L  v L - v Cf - ( R fg  jL fg ) i L
dt
dq
d v Cf
dq
dq
dq
Cf
 i L - i C  jC f v Cf
dt
dq
di
dq
dq
dq
L fc C  v Cf - v C - ( R fc  jL fc ) i C
dt
For the control design the delays caused by the PWM,
sampling and computation are taken into account by
modeling the converter as one sample delay with a time
constant of one switching period (
).
G PWM ( s ) 
vC
1

*
vC sTc  1
(3)
For the control loops, PI controllers with proportional
gain k and time constant Ti are used, which are modeled
as shown in equation (4).
GPI ( s)  k
sTi  1
sTi
(4)
Due to the discrete nature of the control algorithm
implementation the stability analyses in this paper are
performed time discretely as well in the Z-domain. All
other transfer functions are discretized with the zeroorder-hold method [15] with a sampling frequency equal
to the control frequency. Couplings between current
components are neglected for stability analysis as a
decoupling network is used [14].
(1)
The copper losses of the inductors are taken into
account by Rfg and Rfl.
Neglecting the losses of the converter and of the filter,
the power balance between grid side and DC side gives
vDC iDC = 3/2 vLd iLd . The dynamics of the DC link
voltage can be expressed by:
C DC
dv DC
dt
 i DC  i Load 
3 i Ld v Ld
 i Load (2)
2 v DC
Fig. 2: Bode diagram of transfer functions (converter output voltage to
line current) of LCL-line-filters with different resonance frequencies.
III. CONTROL STRUCTURE
In this paper the voltage-oriented PI control [1,16] with
converter current feedback and additional resonance
damping is used to control the PWM rectifier with LCLfilter. The cascaded control structure is shown in Fig. 3.
For the control design the converter is modeled as
shown in (3). The PI controller parameter (k I, TI) are
tuned as described in [14]
k I  k I ,opt 
 Lf
2Tc
; TI  a I Tc ; aI  3
2
(8)
Additionally,
anti-windup
mechanisms
are
implemented to avoid the arising problems in case of
limitation of the current and voltage references. Grid
synchronization is done with a PLL algorithm.
IV. VIRTUAL RESISTOR CONCEPT
Fig. 3: Overview of complete control structure
(γL: line voltage phase angle)
To regulate the DC-voltage of the outer control loop to
its constant value PI controllers are used. To design the PI
controller parameters (kDC,TDC), the inner control loop is
modeled as a first order delay element with the delay time
of Tinner = 4 Tc. and the controller is tuned with the
symmetrical optimum [17]:
k DC 
*
2 VDC
C DC
2
; TDC  a DC
Tinner ; a DC  3 (5)
3  a DC Tinner v Ld
The inner current control is performed in rotating dqcoordinates with PI controllers as well.
In the low frequency range the LCL-filter behaves
similar to the L-filter as shown in [1]. In this frequency
range the control will mostly act. Therefore a designer
needs to model the system in the rotating frame of the Lfilter-based active rectifier for the control and consider
the transfer function of the overall filter with damping for
stability and dynamic purposes [1]. The approximation as
L-filter will be used for the control design in this paper.
Assuming the d- and q-current dynamics decoupled (2)
gives the following dynamics:
Applying the PI control structure with converter
current feedback without additional active damping
yields the root loci shown in Fig. 6 for LCL filter with
high resonance frequency and Fig. 8 for low resonance
frequency. Marked are the system poles for optimal
proportional gain kI=kI,opt. For both sets of parameters the
system becomes unstable because the resonant poles fall
outside the unity circle.
Active damping with the virtual resistor concept as
presented in [9] is based on the idea, that resonance
oscillations in a network can be damped by connecting a
real resistor in series to the filter capacitor. By modifying
the control algorithm similar behavior can be achieved
without using a real resistor. Thus, no additional power
losses are generated.
The one phase equivalent-circuit of a passively
damped LCL-filter by means of additional resistor R is
shown in Fig. 4.
Fig. 4: One phase equivalent circuit of LCL-filter
di
L f Ld  v Ld - vCd - R f iLd
dt
diLq
Lf
 -vCq - R f iLq
dt
(6)
(s)
The line voltage is treated as disturbance and therefore
not taken into consideration during the control design
process. Therefore the same parameters can be used for
the d- and q-current controller. The control design and
analysis will be performed for the d-axis only.
Transforming the first line of (6) into the Laplace domain
yields the first order behavior:
GL ( s) 
 1/ R f
I Ld

VCd s ( L f / R f )  1
The block diagram of the LCL-filter follows from
equation (9)
(7)
(9)
and is shown in Fig. 5 (upper). Rearranging the block
diagram yields the one shown in Fig. 5 (lower).
It can be seen that in order to emulate a real resistor R in
series to Cf an additional damping term (sCfRv) has to be
added to the converter current reference.
The new system behaves like a network with damping
resistor, but instead of a real resistor, additional current
sensors and differentiation are needed.
It should be noted that, instead of using measured
values, capacitor current estimation is known from [10]
but complicates the control algorithm.
By discretizing the differentiation with backwardsdifference-approximation
[15] with
as
the system sampling time the converter reference current
becomes:
(10)
A differentiator may cause noise problems in the
control because it will amplify high-frequency signals
[9], but neither in simulation nor experimental results
problems were noticed in this work.
In the laboratory the filter-capacitor-current icf is
calculated as difference of the converter current iC and the
line current iL.
As already mentioned, some active damping methods
are not applicable for line filter configurations with a low
resonance frequency. To investigate the applicability of
the presented virtual resistor concept, the analyses are
done for two significant filter settings with different
parameters, one with a high resonance frequency and the
other with a low resonance frequency. Fig. 7 shows the
pole zero map of the closed current control loop for high
resonance frequency with different virtual resistor values
varying from Rv=0  to Rv=15 and constant
proportional gain kI,opt.
Fig. 6: Root locus without virtual resistor for LCL-filter with high
resonance frequency (fres=1,6 kHz)
Fig. 8: Root locus without virtual resistor for LCL-filter with low
resonance frequency (fres=660 Hz)
Fig. 7: Pole zero map with virtual resistor
(Rv=0…15insteps) for LCL-filter with high resonance frequency
(fres=1,6 kHz)
Fig. 9: Pole zero map with virtual resistor
(Rv=-6…6insteps) for LCL-filter with low resonance frequency
(fres=660 Hz)
Fig. 5: Block diagram of LCL-Filter with real damping resistor in series
to capacitance (upper) and rearrangement to virtual damping resistor
(lower)
For zero virtual resistance the active damping is
ineffective and the same system pole configuration as
marked in Fig. 6 without active damping can be seen. For
increasing values of virtual resistor the resonant poles are
attracted into the inner unity circle and the system gets
stabilized. A virtual resistance of Rv=10 yields
effective active damping. Further increase of R v leads to
instability again, because the system pole on the
imaginary axis move outside the unity circle.
Fig. 9 shows the same pole zero map for low resonance
frequency filter parameters and different virtual
resistances from Rv= -6  to Rv=6 .
By varying the virtual resistor value the system poles
can be moved, but they are not attracted into the inner
unity circle, thus the system stability cannot be increased.
Stable system operation can only be achieved by
lowering the proportional gain of the current control,
which reduces the system bandwidth, or by using other
active damping methods like state space controllers,
which is not presented here.
V. EXPERIMENTAL RESULTS
To verify the theoretical analysis measurement results
are taken at a test bench of the system as shown in Fig. 1.
The control algorithm is implemented on a dSPACE DS
1006 board. The self-built 22 kW-PWM rectifier is
loaded by an inverter-fed 4-pole induction motor. The
effective DC link capacitance is 4450 F.
Fig. 10 shows the line and converter currents for the
LCL-filter with high resonance frequency (Cf=8F)
without active damping (upper) and with active damping
by virtual resistor (lower). The effectiveness of the virtual
resistor becomes clear as the resonance is well damped.
The current spectra with active damping in Fig. 12 and
without active damping in Fig. 13 also illustrate a good
resonance damping. Figure 16 shows the currents during
activation of active damping. Obviously, the resonance
oscillations are damped fast and effectively by the virtual
resistor.
Fig. 10: Measured converter (ch 2) and line currents (ch 4) without
(upper) and with virtual resistor (lower) for LCL-filter with high
resonance frequency (fres=1,6 kHz) (20A/div)
Fig. 11: Measured converter (ch 2) and line currents (ch 4) without
(upper) and with virtual resistor (lower) for LCL-filter with low
resonance frequency (fres=660 Hz) (20A/div)
Fig. 12: Measured converter (upper) and line (lower) current spectra
without virtual resistor for LCL-filter with high resonance frequency
(fres=1,6 kHz)
Fig. 14: Measured converter (upper) and line (lower) current spectra
without virtual resistor for LCL-filter with low resonance frequency
(fres=660 Hz)
Fig. 13: Measured converter (upper) and line (lower) current spectra
with virtual resistor (Rv=15) for LCL-filter with high resonance
frequency (fres=1,6 kHz)
Fig. 15: Measured converter (upper) and line (lower) current spectra
with virtual resistor (Rv=-2) for LCL-filter with low resonance
frequency (fres=660 Hz)
Theoretically, no stable operation is possible with the
LCL-filter with low resonance frequency (Cf = 48F) as
the system poles are outside the unity circle for all PI
gains (see root locus in Fig. 8). But due to additional
natural damping of the filter elements which is not
modeled in this work operation with reduced PI gain is
possible.
Fig. 11 shows measurement results obtained with the
LCL-filter with low resonance frequency (Cf = 48F) and
reduced PI gain (kI=0,8 kI,opt). Fig. 14 and 15 show the
spectra. The resonance oscillations are mainly visible in
the converter current and can be damped by a small
virtual resistor. Higher virtual resistor values leads to
instability again very easily. The system is close to the
stability limit and small increase of R v or kI lead to
instability making the converter tripping. Line current is
more distorted by low grid harmonics (5th and 7th). Due to
the low PI gain the low frequency distortions are damped
worse.
Fig. 16: Measured converter (ch 2) and line currents (ch 4) during
activation of virtual resistor (ch 3) for LCL-filter with high resonance
frequency (fres=1,6 kHz) (20A/div)
VI. CONCLUSION
In this paper the voltage-oriented PI control with
converter current feedback and additional resonance
damping is used to control the PWM rectifier with grid
side LCL-filter. The active damping method based on the
virtual resistor concept is analysed for two significant
settings of line filter parameters.
This method offers the advantage of simple
implementation but the disadvantage of additional needed
current sensors. To show the instability without
additional active damping the system is analyzed in root
locus. The tuning procedure of the virtual resistor and its
result on the control performance is presented in the pole
zero map. The performance of the investigated control
system is verified by measurements at a test drive.
From theoretical analysis and experimental results it
becomes clear, that active damping with virtual resistor
damps resonance effectively only for high resonance
frequency LCL-filter. For LCL-filter with low resonance
frequency the system stability can only be achieved by
lowering the proportional gain of the current controller,
which reduces the system bandwidth or by using more
complex control algorithms like state space control,
which is not considered here.
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
ACKNOLEDGMENT
This work has partly been financed by European Social
Fund / Innovation Fund Schleswig-Holstein and carried
out as part of CE wind, competence centre wind energy
Schleswig Holstein
[12]
[13]
REFERENCES
[1]
[2]
[3]
M. Liserre, F. Blaabjerg, and S. Hansen, “Design and control
of an LCL-filter-based three-phase active rectifier ,” IEEE
Trans. on Industry Applications, vol. 41, no. 5, pp. 1281-91,
2005.
V. Blasko and V. Kaura, “A novel control to actively damp
resonance in input LC filter of a three-phase voltage source
converter,” IEEE Trans. on Industry Applications, vol. 33,
no. 2, pp. 542-550, 1997.
M. Liserre, A. Dell’Aquila, and F. Blaabjerg, “Stability
improvement of an LCL-filter based three-phase active
rectifier,” in Proc. Power Electronics Specialist Conference,
vol.3, pp. 1195-1201, 2002.
[14]
[15]
[16]
[17]
M. Malinowski, M.P. Kazmierkowski, W. Szczygiel, and S.
Bernet, “Simple Sensorless Active Damping Solution for
three-phase PWM Rectifier with LCL Filter,” Proc. IEEE
Industrial Electronics Conference, pp. 987-991,2005.
E. Twining and D.G. Holmes, “Grid Current Regulation of a
Three-Phase Voltage Source Inverter with an LCL Input
Filter,”
IEEE Trans. on Power Electronics, vol. 18, no. 3, 2003.
M. Liserre, R. Teodorescu, and F. Blaabjerg, “Stability of
Photovoltaic and Wind Turbine Grid-Connected Inverters for
a Large Set of Grid Impedance Values,” IEEE Trans. on
Power Electronics, vol. 21, no. 1, pp. 263- 272, 2006.
J. Dannehl, F.W. Fuchs, and S. Hansen, “PWM Rectifier
with LCL-Filter using different Current Control Structures,”
Proc. European Conference on Power Electronics and
Applications,
CD-ROM, 2007.#
P.C. Loh and D.G. Holmes, “Analysis of multiloop control
strategies for LC/CL/LCL-filtered voltage-source and
current-source inverters,” IEEE Trans. on Industry
Applications, vol. 41, no. 5, pp. 644- 654, 2005.
P.A. Dahono, “A control method to damp oscillation in the
input LC-filter,” in Proc. Power Electronics Specialist
Conference, vol. 4, pp. 1630–5, 2002.
W. Gullvik, L. Norum, R. Nilsen, „Active Damping of
Resonance Oscillations in LCL-Filters Based on Virtual Flux
and Virtual Resistor”, Proc. European Conference on Power
Electronics and Applications,
CD-ROM, 2007.
M. Bojrup, P. Karlsson, M. Alakula, and L. Gertmar, “A
multiple rotating integrator controller for active filters,”
Proc. European Conference on Power Electronics and
Applications, CD-ROM, 1999.
F.A. Magueed and J. Svensson, “Control of VSC connected
to the grid through LCL-filter to achieve balanced currents,”
Proc. IEEE Industry Applications Society Annual Meeting,
vol. 1, pp. 572-8, 2005.
E. Wu and P.W. Lehn, “Digital current control of a voltage
source converter with active damping of LCL resonance,”
IEEE Trans. on Power Electronics, vol. 21, no. 5, pp. 13641373, 2006.
J. Dannehl, C. Wessels, F.W. Fuchs, “Limitations of
Voltage-Oriented PI Current Control of Grid-Connected
PWM Rectifiers with LCL-Filter”, IEEE Trans. on
Industrial Electronics (submitted), 2008
K.J. Aaström and B. Wittenmark, Computer-controlled
systems: theory and design, Prentice Hall, 1997.
M.P. Kazmierkowski, R. Krishnan, and F. Blaabjerg,
Control in Power Electronics: Selected Problems, Academic
Press, 2002.
D. Schröder, Elektrische Antriebe 2, Regelung von
Antriebssystemen, Springer, 2001.