Active Damping of LCL-Filter Resonance based on Virtual Resistor for ...

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Abstract—This publication presents the investigation of active damping of resonance oscillations with virtual resistor for grid-connected PWM rectifiers with LCL- ...
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]

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