Active Damping of Resonance Oscillations in LCL Filters Based on Virtual Flux and Virtual Resistor

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Page 1: Active Damping of Resonance Oscillations in LCL Filters Based on Virtual Flux and Virtual Resistor

Active Damping of Resonance Oscillations in LCL-Filters Based on Virtual Flux and Virtual Resistor

William Gullvik1, Lars Norum1, Roy Nilsen2

1Norwegian University of Science and Technology O. S. Bragstads plass 2A 7491 Trondheim, Norway

Tel.: +47 73594210, Fax: +47 73594279 2Wärtsila Norway AS

[email protected]@elkraft.ntnu.no

[email protected]

Keywords Active damping, Converter control, Sensorless control

Abstract This paper presents several methods for active damping of resonance oscillations in LCL-filters. All the methods are implemented using only estimated values of the capacitor voltages and currents. The sensorless control strategy used in this paper is based on the virtual flux concept and all voltages and currents with the exception of the dc bus voltage and the converter output currents are derived from the virtual flux. The presented theory is then verified experimentally in the lab.

Introduction Active front end converters are becoming more and more popular as the cost of power electronic converters and components are decreasing and the power ratings are increasing. They are turning into important parts of drive systems both on-shore and off-shore and they are important in the exploitation of renewable energy sources like for instance wind power and photovoltaic systems.

It is important to keep the cost of the system as low as possible and in some cases like on-board ships the volume should also be limited. The line-filter between the converter and the grid can be reduced by using an LCL-filter instead of an L-filter. The main drawback with this is that the LCL-filter will introduce a resonance frequency into the system. Harmonic components in the output voltage can lead to resonance oscillations and instability problems unless they are properly handled.

One way of reducing the resonance current is by adding a passive damping circuit to the filter. This damping circuit can be purely resistive, causing relatively high losses, or a more complex solutions consisting of a combination of resistors, capacitors and inductors [11][12].

A more attractive option is the use of active damping where the output voltage from the converter is used to damp out the resonance oscillations. Several methods for active damping have been presented in the literature, [1][2][5][6], but most of these are utilizing sensors for measuring the either the capacitor voltages or currents, only a few methods based on sensorless control have been presented [3][4][7]. The focus in this paper is on sensorless control and how to realize active damping without measuring anything but the output current from the converter and the dc bus voltage.

Active Damping of Resonance Oscillations in LCL-Filters Based on Virtual Flux and Virtual resistor GULLVIK William

EPE 2007 - Aalborg ISBN : 9789075815108 P.1

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The Virtual Flux Concept A typical LCL-filter is shown in Figure 1 where V1 is the grid side voltage, Vc is the voltage across the filter capacitor and V2 is the converter output voltage.

Figure 1: LCL-filter

The capacitor voltage, Vc, can be expressed by equation (1) whereas the grid side voltage,V1, can be expressed by equation (2).

2

2 2c

d iV V L

dt= − (1)

21 2 2

1 1 1 2 2 2c c

di dV d idV V L V L i C CL

dt dt dt dt= − = − − + (2)

The main problem with equation (2) is that it depends on the third derivative of the converter output current, a fast changing quantity containing a relatively large amount of ripple in addition to measurement and discretization noise. It is therefore desirable to express the equations in an alternative way. One such way is to use the virtual flux concept introduced by [10] for the VOC and later by [9] for the DPC. The converter side flux is given by equation (3) whereas the capacitor and grid side virtual flux is given by equation (4) and (5). Drift compensation of the virtual flux is done using the method described by Niemelä [8].

2 2V dtψ = (3)

2 2 2c cV dt L iψ ψ= = − (4)2

1 1 1 2 1 2c c

dV dt L i LC

dtψ ψ ψ= = − + (5)

The capacitor current can be estimated using equation (6).

2

2c c c

d di C V C

dt dtψ= = (6)

The control system based on the voltage oriented control (VOC) and virtual flux is shown in Figure 2.

Active Damping of Resonance Oscillations in LCL-Filters Based on Virtual Flux and Virtual resistor GULLVIK William

EPE 2007 - Aalborg ISBN : 9789075815108 P.2

Page 3: Active Damping of Resonance Oscillations in LCL Filters Based on Virtual Flux and Virtual Resistor

dqαβ abc

αβ

abcαβdq

αβ

θ

Figure 2: Control system using VOC and virtual flux

Active Damping I. Using lead-lag compensator

A method for active damping using a lead-lag compensator was introduced in [1], [2], and a sensorless solution of the same method was simulated in [3]. The lead compensator used to add positive phase to the system is shown in equation (7). The compensator has to be tuned for the systems resonance frequency.

( ) 11

dd

d

T sL s k

T sα+

=+

(7)

dqαβ abc

αβ

abcαβdq

αβ

θ

,c dv

,c qv,corr dv

,corr qv

Figure 3: Control system using VOC and a lead-lag compensator for active damping.

The control system with virtual flux and active damping using lead-lag compensators is shown in Figure 3. The estimated capacitor voltages are coming from the virtual flux model and the two correction voltages from the lead-lag element are added to the output from the two current regulators.

Active Damping of Resonance Oscillations in LCL-Filters Based on Virtual Flux and Virtual resistor GULLVIK William

EPE 2007 - Aalborg ISBN : 9789075815108 P.3

Page 4: Active Damping of Resonance Oscillations in LCL Filters Based on Virtual Flux and Virtual Resistor

The effect of the lead-lag compensator can be seen by comparing the two bode plots in Figure 4 and Figure 5. The first bode plot shows the open loop response of the current control loop without the lead-lag compensator whereas the latter shows the response with the compensator.

-100

-50

0

50

100M

agni

tude

(dB

)

100

101

102

103

104

105

106

-90

-45

0

45

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Figure 4: Open loop bode plot of the current control loop without active damping

-60

-40

-20

0

20

40

60

Mag

nitu

de (

dB)

100

101

102

103

104

105

106

-180

-135

-90

-45

0

45

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Figure 5: Open loop bode plot of the current control loop with lead-lag compensator

II. Using a virtual resistor

This method is similar to the one presented by [2] but uses estimated values instead of measurements. The objective of the method is to emulate the damping resistor used in passive damping through modifications of the control loop.

The transfer function from the converter output voltage V2 to the capacitor voltage Vc in a pure LCL-filter without damping element is given by equation (8) when the resistances found in the filter inductors are ignored. Whereas the transfer function leading to the capacitor current, Ic, is given by equation (9).

Active Damping of Resonance Oscillations in LCL-Filters Based on Virtual Flux and Virtual resistor GULLVIK William

EPE 2007 - Aalborg ISBN : 9789075815108 P.4

Page 5: Active Damping of Resonance Oscillations in LCL Filters Based on Virtual Flux and Virtual Resistor

( ) 11 2

2 1 2 1 2

cV LH s

V s L L C L L= =

+ +(8)

( )2c

c

IH s

VsC= = (9)

When passive damping is achieved by placing a resistor in series with the filter capacitor the transfer functions found in equations (8) and (9) are changed in to those given by equations (10) and (11), where V’c is the voltage across the series connected filter capacitor and damping resistor.

( ) ( )( )( )

13 2

2 1 2 1 2

1'1

c L sRCVH s

V s L L C L L sRC+

= =+ + +

(10)

( )4 ' 1c

c

I sCH s

V sRC= =

+(11)

The goal is to modify the control structure so that the transfer functions for the non-damped system are changed into that of the passively damped system without physically adding a resistor in series with the capacitor. The block diagram in Figure 6 shows a modified control structure where the capacitor current multiplied with a constant k is subtracted from the converter output voltage, V2.

H1(s)

H2(s)

-

k

V2(s) Vc(s)

Ic(s)

Figure 6: Control loop with capacitor current feedback

The transfer function for the modified system is shown in equation (12).

( ) 15 2

1 2 1 1 2

LH s

s L L C skLC L L=

+ + +(12)

Replacing k with equation (13) makes the denominators of equation (10) and (12) equal. 1 2

1

L Lk RL+= (13)

The RC component found in the numerator of equation (10) is much smaller than one making this term negligible.

The open loop bode plot of the current control loop with the virtual resistor is shown in Figure 7, looking at the resonance frequency shows that the resonance peak has been attenuated.

III. Using a virtual resistor working on the high frequency component of the current

In this work an alternative virtual resistor method is suggested. This method is based on the circuit for passive damping of LCL-filters shown in Figure 8 ([11]). The purpose of the circuit is to limit the losses created by the damping resistor by inserting an inductor in parallel with the damping resistor. If

Active Damping of Resonance Oscillations in LCL-Filters Based on Virtual Flux and Virtual resistor GULLVIK William

EPE 2007 - Aalborg ISBN : 9789075815108 P.5

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the size of the inductor is properly chosen most of the fundamental frequency current will pass through it whereas the resonance current will pass through the damping resistor.

-100

-50

0

50

100

Mag

nitu

de (

dB)

100

101

102

103

104

105

106

-135

-90

-45

0

45

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Figure 7: Open loop bode plot of the current control loop with virtual resistor

Figure 8: Passive damping using a resistor in parallel with an inductor

Analyzing the passive damping method shown in Figure 8 gives the following transfer function between the converter output voltage V2(s) and the voltage across the capacitor and damping elements, V’c(s):

( ) ( )( )( )

21

6 3 2 22 1 2 1 2 1 2

' f fc

f f f

L R sL s RL CVH s

V s L L L C s L L RC L L R sL s RL C

+ += =

+ + + + +(14)

The transfer function for the capacitor current is given by equation (15). Whereas the frequency response of the capacitor current can be expressed as in equation (16).

( ) ( )( )

2

7 2'c

c

I s s LC sRCH s

V s s RLC sL R+

= =+ +

(15)

( )( )

2 2

7 2 4 2 2 22 2 2

2 2 2

21

CH

R C R LCR C

L R

ωω

ω ωωω

=+

+ −+

(16)

Active Damping of Resonance Oscillations in LCL-Filters Based on Virtual Flux and Virtual resistor GULLVIK William

EPE 2007 - Aalborg ISBN : 9789075815108 P.6

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At low frequencies the frequency response approaches that of the non-damped system as shown in equation (17), meaning that the nominal frequency current is passing through the inductor instead of the damping resistor ( L is small compared to R).

( )7H Cω ω≈ (17)

For high frequencies the situation changes and the transfer function is now approaching that of the resistor damped system, meaning that most of the current is now flowing through the resistor as illustrated by equation (18) which shows the approximated magnitude of the transfer function H7 for high frequencies. This is the same as the magnitude of transfer function for the resistor damped LCL filter, H4, shown in equation (11).

( )7

2 2

2 2 21C

HCR

ωω

ω≈

+(18)

The capacitor currents can be estimated using equation (6), which is the second derivative of the virtual flux across the capacitors or the first derivative of the capacitor voltage.

When the capacitor currents are estimated from the capacitor voltage, it is possible to remove the fundamental frequency component without using a high-pass filter and thereby preventing phase lag and distortion of the remaining high-frequency components. Starting with equation (19) for the capacitor currents in the two phase system, and the knowledge of the voltage components shown in equation (20).

'c c

di C V

dt= (19)

( ) ( ), , cos sinc c c c cV V jV V jVα β θ θ= + = + (20)

Combining the two equations above results in an expression for the capacitor currents that doesn’t contain any derivatives as shown in equation (21). The expression is only valid for the fundamental frequency component and is not the true capacitor current.

( ) ( )( ) ( ) ( )( )' cos sin sin cosc c c c c

di C V jV C V jV

dtθ θ θ θ= + = − + (21)

The two individual current components are shown in equations (22) and (23). , ,'c ci V Cα β= − ⋅ (22)

, ,'c ci V Cβ α= ⋅ (23)

The two current components, ',ci α and '

,ci β , are then subtracted from the estimated capacitor current found from equation (6). After the subtraction only the high frequency components of the capacitor currents will remain.

The high frequency components are then multiplied with the factor k from equation (13) and added to the converter output voltage.

Experimental Results

The system was tested in the lab on a 10 kW converter using a Xilinx Virtex II Pro with an internal PowerPC core for the control system. The LCL-filter used 720 μH inductors for both the grid and the converter side and a 60 μF filter capacitor was used. The system was rated for 230 V line-line voltages and 25 A current; the switching frequency was 5 kHz. The dc bus voltage reference was held constant at 1 pu or 376 V for all the experiments.

Active Damping of Resonance Oscillations in LCL-Filters Based on Virtual Flux and Virtual resistor GULLVIK William

EPE 2007 - Aalborg ISBN : 9789075815108 P.7

Page 8: Active Damping of Resonance Oscillations in LCL Filters Based on Virtual Flux and Virtual Resistor

I. Lead-lag compensator

When the system was running at nominal load without active damping the grid side current was as shown in Figure 9. The THD of the current was 3.3 % and the resonance oscillations were clearly visible.

Enabling the lead-lag compensator resulted in the current shown in Figure 10. The resonance oscillations were no longer visible and the THD was reduced to 1.6%. The efficiency of the lead-lag compensator can be seen from Figure 11 showing the output current from the converter when the lead-lag compensator was enabled at time 0.0s. As seen from the figure the resonance oscillations were damped out within 7 ms.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-50

-40

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-20

-10

0

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20

30

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50

Time [s]

Cur

rent

[A

]

ia

ib

Figure 9: Grid side current without active damping

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-40

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40

Time [s]

Cur

rent

[A

]

ia

ib

Figure 10: Grid side current with lead-lag compensator

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05-1.5

-1

-0.5

0

0.5

1

1.5

Time [s]

Cur

rent

[pu]

i2,α

i2,β

Figure 11: The inverter output currents when active damping is enabled at time 0.0 s

Active Damping of Resonance Oscillations in LCL-Filters Based on Virtual Flux and Virtual resistor GULLVIK William

EPE 2007 - Aalborg ISBN : 9789075815108 P.8

Page 9: Active Damping of Resonance Oscillations in LCL Filters Based on Virtual Flux and Virtual Resistor

II. Virtual resistor

The result of using a virtual resistor is shown in Figure 12. As with the lead-lag compensator it is not possible to see any significant resonance oscillations in the grid side current at nominal load.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-40

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0

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Time [s]

Cur

rent

[A]

iaib

Figure 12: Grid side current with virtual resistor

III. Virtual resistor working on the high frequency component of the current

The method was implemented and tested in the lab under the same conditions as the other two methods. The experimental results are shown in Figure 13. As with the previous two methods all the significant resonance harmonics are removed from the grid side current proving that the method works.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-40

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0

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Time [s]

Cur

rent

[A

]

ia

ib

Figure 13: Grid side current with modified virtual resistor working only on the resonance current

Conclusion

Three methods for active damping of the resonance oscillations in LCL-filters based on sensorless control have been presented in this paper. All three methods were implemented without any feedback from AC voltage sensors or measured capacitor currents but were relying on the estimates coming from the virtual flux model. The only measurements used by the virtual flux model were two of the phase currents out from the converter and the DC bus voltage.

The method based on lead-lag compensators was the easiest one to realize in the lab since it does not need any estimates of the capacitor currents. The method was also found to be reliable and it reacted relatively fast when activated, one drawback with this method is that it has to be tuned to the resonance frequency of the LCL-filter and the grid. Changes in the grid may therefore make it less efficient if this change is not compensated for. The other two methods utilize the estimated capacitor currents and gave good results when these estimates were reliable.

Active Damping of Resonance Oscillations in LCL-Filters Based on Virtual Flux and Virtual resistor GULLVIK William

EPE 2007 - Aalborg ISBN : 9789075815108 P.9

Page 10: Active Damping of Resonance Oscillations in LCL Filters Based on Virtual Flux and Virtual Resistor

The lead-lag compensator gave better results than the two virtual resistor based methods at low loading conditions whereas all three methods preformed equally good at higher loads.

References

[1] V. Blasko, V. Kaura, “A novel control to actively damp resonance in input LC filter of a three phase voltage source converter”, Applied Power Electronics Conference and Exposition, APEC 1996. Conference Proceedings. 11th Annual, 3-7 March 1996, Pages: 545-551 vol. 2.

[2] Dahono, Pekik Argo “A Control Method for DC-DC Converters That Has an LCL Output Filter Based on New Virtual Capacitor and Resistor Concepts”. 35th Annual IEEE Power Electronics Specialists Conference, Aachen 2004, Germany, Pages: 36-42 vol. 1

[3] Malinowski, M et al. “Sensorless Operation of Active Damping Methods for Three-Phase PWM Converters”. IEEE ISIE 2005, June 20-23, Dubrovnik, Croatia

[4] Mariusz Malinowski, Marian P. Kazmierkowski, Wojciech Szczygiel, Steffen Bernet, “Simple Sensorless Active Damping Solution for Three-Phase PWM Rectifier with LCL Filter”. IECON 2005, 31st Annual Conference of IEEE, 6-10 Nov. 2005, pages: 987-991.

[5] E. Twinings, D. G. Holmes, “Grid Current Regulation of a Three-phase Voltage Source Inverter with an LCL Input Filter”. PESC 2002, IEEE 33rd Annual, Volume 3, Pages: 1189-1194.

[6] Olve Mo, Magnar Hernes, Kjell Ljøkelsøy “Active damping of oscillations in LC-filter for line connected, current controlled, PWM voltage source converters”. EPE 2003

[7] M. Liserre, A. Dell’Aquila, F. Blaabjerg, ”Genetic algorithm-based design of the active damping for an LCL-filter three-phase active rectifier”, IEEE Transactions on Power Electronics, Issue 1, Jan. 2004, Vol. 19, pages: 76-86.

[8] Niemelä, Markku. “Position sensorless electrically excited synchronous motor drive for industrial use based on direct flux linkage and torque control”. PhD thesis, Lappeenranta University of Technology, Finland 1999

[9] M. Malinowski, M. P. Kazmierkowski, S. Hansen, F. Blaabjerg, G.D. Marques, “Virtual flux based Direct Power Control of three-phase PWM rectifier”, IEEE Trans. on Ind. Applications, vol 37, no. 4, pp. 1019-1027, Jul/Aug 2001.

[10]P. J. M. Smidt, J. L. Duarte, ”An unity power factor convertor without current measurements”, EPE’95, Sevilla, pp. 3.275-3.280, 1995

[11]Timothy CY Wang, Zhihong Ye, Gautam Sinha, Xiaoming Yuan, “Output Filter Design for a Grid-interconnected Three-Phase Inverter”. PESC 2003, IEEE 34th Annual, Volume 2, 15-19 June 2003, Pages: 779-784.

[12]Pradeep V, Amol Kolwalker, Ralph Teichmann, “Optimized filter design for IEEE 519 compliant grid connected inverters”. IICPE 2004

Active Damping of Resonance Oscillations in LCL-Filters Based on Virtual Flux and Virtual resistor GULLVIK William

EPE 2007 - Aalborg ISBN : 9789075815108 P.10