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. LCL-

filters are more interesting, because they are more cost-

effective 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 LCL-

filter 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

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-oriented-

control. 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

Cfcfc

dq

C

dq

Cf

dq

C

fc

dq

Cff

dq

C

dq

L

dq

Cf

f

dq

Lfgfg

dq

Cf

dq

L

dq

Lfg

iLjRvvdt

idL

vCj iidt

vdC

iLjRvvdt

idL

)(--

-

)(--

(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:

Load

DC

LdLdLoadDC

DCDC i

v

viii

dt

dvC

2

3 (2)

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 ( ).

Tab. I: System parameters for analysis

The transfer function of the converter is:

1

1)(

*

cC

CPWM

sTv

vsG

(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).

i

iPI

sT

sTksG

1)(

(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 zero-

order-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].

Fig. 2: Bode diagram of transfer functions (converter output voltage to

line current) of LCL-line-filters with different resonance frequencies.

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 Resistance of converter-side filter

inductor 30 m

Cf Filter capacitance 8 F /

48 F

fc Switching/Control frequency 4 kHz

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 LCL-

filter. The cascaded control structure is shown in Fig. 3.

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]:

3 ; ;3

2 2*

DCinnerDCDC

LdinnerDC

DCDCDC aTaT

vTa

CVk

(5)

The inner current control is performed in rotating dq-

coordinates 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 L-

filter-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:

LqfCq

Lqf

LdfCdLdLd

f

iRvdt

diL

iRvvdt

diL

--

--

(6)

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:

1)/(

/1)(

ff

f

Cd

LdL

RLs

R

V

IsG (7)

For the control design the converter is modeled as

shown in (3). The PI controller parameter (kI, TI) are

tuned as described in [14]

3; ;2

2

,

IcII

c

f

optII aTaTT

Lkk

(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

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

The block diagram of the LCL-filter follows from

equation (9)

(s) (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.

Fig. 5: Block diagram of LCL-Filter with real damping resistor in series

to capacitance (upper) and rearrangement to virtual damping resistor (lower)

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.

Fig. 6: Root locus without virtual resistor for LCL-filter with high

resonance frequency (fres=1,6 kHz)

Fig. 7: Pole zero map with virtual resistor

(Rv=0…15insteps) for LCL-filter with high resonance frequency

(fres=1,6 kHz)

By discretizing the differentiation with backwards-

difference-approximation [15] with as

the system sampling time the converter reference current

becomes:

(10)

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. 8: Root locus without virtual resistor for LCL-filter with low

resonance frequency (fres=660 Hz)

Fig. 9: Pole zero map with virtual resistor

(Rv=-6…6insteps) for LCL-filter with low resonance frequency

(fres=660 Hz)

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 Rv 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.

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)

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=8F)

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. 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. 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)

Theoretically, no stable operation is possible with the

LCL-filter with low resonance frequency (Cf = 48F) 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 = 48F) 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 Rv 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. 14: Measured converter (upper) and line (lower) current spectra without virtual resistor for LCL-filter with low resonance frequency

(fres=660 Hz)

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)

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.

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

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