Capacitor current feedback active damping is extensively used in gridconnected converters with an
LCL
filter. However, systems tends to become unstable when the digital control delay is taken into account, especially in low switching frequencies. This paper discusses this issue by deriving a discrete model with a digital control delay and by presenting the stable region of an active damping loop from high to low switching frequencies. In order to overcome the disadvantage of capacitor current feedback active damping, this paper proposes a modified approach using grid current and converter current for feedback. This can expand the stable region and provide sufficient active damping whether in high or low switching frequencies. By applying the modified approach, the active damping loop can be simplified from fourthorder into secondorder, and the design of the grid current loop can be simplified. The modified approach can work well when the grid impedance varies. Both the active damping performance and the dynamic performance of the current loop are verified by simulations and experimental results.
I. INTRODUCTION
Nowadays, the number of distributed power generation systems (DPGS) connected to the grid is increasing rapidly. As a result, stricter standards for power quality have been issued
[1]
. The gridconnected converter plays a crucial part in injecting high quality power into the grid. Due to the
LCL
grid filters’ good filtering characteristic above resonance frequency, they are increasingly being used to mitigate PWM harmonics, especially in high power DPGSs whose switching frequency is limited by losses
[2]
. Unfortunately, the inherent resonance of the
LCL
filter can cause undesired oscillations or even instasbility. A simple method to damp this resonance is to add a passive resistor to the
LCL
filter
[3]
. However, this is not appropriate for high power DPGSs due to their power loss
[4]
.
Many active damping approaches have been proposed which are more flexible and efficient than the passive damping techniques
[5]

[8]
. Capacitor current feedback active damping has been extensively used because it is simple to realize among the various active damping approaches
[9]

[11]
. However, in a digitally controlled system, there is an inevitable digital control delay between the sampling instant and the PWM updating instant which is equivalent to applying a unit time delay
z
^{1}
in the control loop
[12]

[14]
. Few references have thoroughly discussed the stability and damping performance of capacitor current feedback active damping when the digital control delay is taken into account, especially at low switching frequencies. Reference
[15]
totally neglects the influence of digital control delay. As its switching frequency as high as 10 kHz, the influence is not obvious. Digital control delay is regarded as a firstorder inertia in reference
[16]
which is suitable for its application with a 5 kHz switching frequency. However, in high power systems, the switching frequency is limited by power losses and cannot be set very high. Furthermore, the above references design the active damping loop in the continuous time domain. However, this will lead to system instability and poor damping performance in low switching frequencies such as 2 kHz. Reference
[17]
demonstrates the stable region of active damping in the discrete time domain when the digital control delay is taken into account. However, it only presents the stable region in the 5 kHz switching frequency. When the switching frequency decreases, the system becomes inherently unstable no matter how the capacitor current feedback coefficient changes. Therefore, it is unsuitable to neglect the digital control delay at low switching frequencies when designing the active damping loop. This paper addresses this issue by theoretically presenting the stable regions of capacitor current feedback active damping in high, medium and low switching frequencies.
In high power applications, both the grid current and the converter current are sensed in the
LCL
filter system. The grid current is used in current loop control, and the converter current is used to protect IGBTs from overcurrent. The capacitor current can be calculated by the grid current minus the converter current. This means that the capacitor current feedback is equal to the grid current and converter current feedback with the same feedback coefficient which will lose one degree of control freedom. In order to increase the degree of control freedom and to expand the stable region, especially at low switching frequencies, this paper proposes a modified active damping approach with grid current and converter current feedback which can strengthen system stability and provide sufficient active damping. Moreover, the modified approach does not need any additional sensors. When the grid impedance varies, the modified approach can also work well. Simulation and experimental results based on a 300 kVA three phase prototype verify the theoretical conclusions presented in this paper.
II. MODELING OF THELCLFILTER
A diagram of a grid connected PWM converter with an
LCL
filter is shown in
Fig. 1
, where
i_{g}
,
i_{s}
,
u_{c}
,
i_{c}
and
u_{o}
represent the grid current, converter current, filter capacitor voltage, filter capacitor current and converter voltage, respectively.
Grid connected PWM converter with LCL filter.
The state space equation of an
LCL
filter in the continuous time domain yields as Equ. (1) by applying Kirchhoff laws
[18]
, where the state vector is defined as
x
(
t
) = [
i_{g}
(
t
)
i_{s}
(
t
)
u_{c}
(
t
)]
^{T}
.
The system matrix, input matrix and output matrix are shown as Equ. (2). The parasitic resistors of the inductors are not taken into account in the following derivation.
In a digital control system, a digital processor needs some time to make calculations
[12]
. Therefore, the calculated result of the present step is updated to output at the beginning of the next step which is shown in
Fig. 2
. As a result, there is an inevitable digital control delay between the sampling instant and the PWM updating instant which is equivalent to applying a unit time delay
z
^{1}
in the control loop. The digital control delay has a crucial influence on system stability and damping performance. It is necessary to take the digital control delay into account when designing an active damping loop, especially at low switching frequencies. In this paper, the sampling instant is located at the bottom and the summit of the carrier. Therefore, the sampling frequency is double the switching frequency.
Digital control delay in digital control system.
Fig. 3
shows the grid current control structure with the digital control delay. By active damping, the current loop can be controlled with a PI controller. The acitve damping is achieved by state feedback. Choosing different feedback states corresponds to different active damping approaches.
K
is the feedback matrix.
Control structure with digital control delay.
The discrete model is derived as Equ. (3), where the state vector is defined as
x
(
k
) = [
i_{g}
(
k
)
i_{s}
(
k
)
u_{c}
(
k
)]
^{T}
.
The discrete system matrix and the input matrix can be calculated with:
The systme matrix and the output matrix yield as follows:
Due to the digital control delay, there is a unit time delay
z
^{1}
between the reference voltage
u
(
k
) and the output voltage
u
_{o}
(
k
)
[19]
.
In order to exactly analyze the impact of the digital control delay on system stability and the design of the active damping loop, the state space equation taking the digital control delay into account is derived as Equ. (8), where the state vector is defined as
x
(
k
) = [
i_{g}
(
k
)
i_{s}
(
k
)
u_{c}
(
k
)
u_{c}
(
k1
)]
^{T}
[20]
.
Whereas:
with
e_{ij}
and
f_{i}
(
i
,
j
∈{1,2,3}) being entries in matrices
E
and
F
.
The transfer function of the active damping loop
I_{g}
(
z
)/
R
(
z
) can be calculated by equation Equ. (10), where
K
is the feedback matrix and
I
is the unit matrix. The following discussions are based on this transfer function.
III. STABILITY ANALYSIS OF THE CAPACITOR CURRENT FEEDBACK ACTIVE DAMPING
Based on the discrete model in section II, the influence of the digital control delay on stability and damping performance is discussed in this section. The grid current and converter current are both sensed. The capacitor current is calculated by the grid current minus the converter current. As a result, the feedback matrix is
K
= [
K_{ic}

K_{ic}
0 0] , where
K_{ic}
represents the capacitor current feedback coefficient. The capacitor voltage is not sensed so its feedback coefficient is 0. The transfer function of the active damping loop yields as Equ. (11) by substituting feedback matrix
K
into Equ. (10). This shows that an active damping loop with the digital control delay has four poles. Three of them move as the feedback coefficient changes. The other is located at
z
=1.
Fig. 4
shows the root loci of the capacitor current feedback active damping loop. This demonstrates the stable regions both while considering the digital control delay (dashed line) and while neglecting the digital control delay (solid line) in high (10 kHz), medium (5 kHz) and low (2 kHz) switching frequencies with parameters given in
Table I
.
Root loci of capacitor current feedback when considering digital control delay and neglecting digital control delay.
PARAMETER OF LCL FILTER
The active damping loop without the digital control delay also has four poles. Two of them move as the feedback coefficient changes. The others are located at
z
=1 and
z
=0, respectively. When the digital control delay is not taken into account, the system is always stable. However, a system with the digital control delay tends to become unstable, especially at low switching frequencies.
Fig. 4
(a) demonstrates the root loci of a high (10 kHz) switching frequency. The damping performance of the dashed line (considering the delay) is nearly the same as that of the solid line (neglecting the delay). Therefore, the influence of the digital control delay is not obvious in high switching frequencies.
Fig. 4
(b) demonstrates the root loci of a medium (5 kHz) switching frequency. The damping ratio of the dashed line (considering the delay) is much smaller than that of the solid line (neglecting the delay). It is easy for the system to become resonant or even unstable at a medium switching frequency when considering the digital control delay.
Fig. 4
(c) demonstrates the root loci of a low (2 kHz) switching frequency. The damping performance of the dashed line (considering the delay) is different from that of the solid line (neglecting the delay). A system with the digital control delay is inherently unstable no matter how the feedback coefficient changes at a low switching frequency.
IV. MODIFIED ACTIVE DAMPING APPROACH
As discusses in section III, the capacitor current feedback is equal to the grid current and converter current feedback with the same feedback coefficient which loses one degree of control freedom. This is unsuitable at a low switching frequency. In order to increase the degree of control freedom, this paper sets different feedback coefficients for the grid current and converter current which strengthens the system stability. Moreover, the modified approach does not need additional sensors. Therefure, the feedback matrix is
K
= [
k_{ig} k_{is}
0
k_{u}
] . Where,
k_{ig}
and
k_{is}
represent the feedback coefficients of the grid current and the converter current, respectively. The capacitor voltage is not sensed so its feedback coefficient is 0.
k_{u}
represents the feedback coefficient of the reference voltage
u
(
k1
).
The transfer function of the active damping loop
I
_{g}
(
z
)/
R
(
z
) is derived by substituting feedback matrix
K
and the parameters of
TABLE I
into Equ. (10). The characteristic equation of the active damping loop is shown as Equ. (12).
This shows that the active damping loop has four poles. Assume that the desired poles are
p_{1}
,
p_{2}
,
p_{3}
,
p_{4}
, where
p_{3}
,
p_{4}
represent the pair of conjugate complex poles. Therefore, the desired characteristic equation is derived as Equ. (13).
Where:
Expressions for the feedback coefficients in terms of the desired pole locations are obtained by comparing Equ. (12) with Equ. (13). The equations are derived as Equ. (15).
In linear algebra, the solvable condition of the equation
A
‧
x
=
b
is that
Rank
(
A

b
)=
Rank
(
A
)=
number of variable
[21]
. In Equ. (15),
Rank
(
A
)=
number of variable
=
3
. Therefore, the solvable condition of Equ. (15) is
Rank
(
A

b
)=
3
. Use Gaussian Elimination to transform the augmented matrix
A

b
to an upper triangular matrix as Equ. (16).
When the fourth row of the augmented matrix
A

b
is equal to 0,
Rank
(
A

b
)=
3
. Therefore, the solvable condition of Equ. (15) is derived as Equ. (17).
The desired poles must satisfy Equ. (17). In order to derive the root loci of the active damping loop in the discrete domain, set the dominant pole
p_{1}
=
0.9
, and the nondominant pole
p_{2}
=
0.1
, and assume the pair of conjugate complex poles to be
p_{3}
=
α
+
j
‧
β
and
p_{4}
=
α

j
‧
β
. Substituting
p_{1}
,
p_{2}
,
p_{3}
,
p_{4}
into Equ. (17) yields the equation of the conjugate complex poles as Equ. (18).
The root loci of the active damping loop, comparing modified approach with the conventional capacitor current feedback active damping, is shown in
Fig. 5
.
Root loci of active damping loop comparing modified approach with conventional capacitor current feedback.
Fig. 5
(a) shows the root loci of a medium (5 kHz) switching frequency. The modified approach (solid line) can provide a larger damping ratio than the conventional capacitor current feedback active damping (dashed line) in a medium switching frequency.
Fig. 5
(b) shows the root loci of a low (2 kHz) switching frequency. The conventional capacitor current feedback active damping (dashed line) is unstable. However, the modified approach (solid line) can overcome this issue at a low switching frequency.
When compared with the conventional capacitor current feedback, the modified approach can strengthen stability and provide sufficient damping performance, especially in low switching frequencies. Choosing proper locations of the desired poles
p_{1}
,
p_{2}
,
p_{3}
,
p_{4}
and substituting them into Equ. (15), the feedback matrix
K
can be solved out.
Fig. 6
shows a bode diagram of the active damping loop, comparing the modified approach with the conventional capacitor current feedback approach, at a 2 kHz switching frequency. It demonstrates that the conventional capacitor current feedback (marked line) causes resonance. By applying the modified approach (solid line), the resonance can be damped.
Bode diagram of active damping loop comparing modified approach with conventional capacitor current feedback in 2 kHz switching frequency.
By applying the modified approach, the pair of conjugate complex poles
p_{3}
,
p_{4}
can be placed away from the dominant pole
p_{1}
. Therefore, the conjugate complex poles have little influence on the current loop in the low frequency band. The cut off frequency of the current loop is always set to the low frequency band so that it is suitable to neglect the conjugate complex poles when designing the current loop
[22]
.
By neglecting the pair of conjugate complex poles, the active damping loop is simplified from fourthorder into secondorder. This will simplify the design of the current loop.
Fig. 6
demonstrates that dashed line (neglecting the conjugate complex poles) is nearly the same as the solid line (considering the conjugate complex poles) in the low frequency band.
Variations of the grid impedance in weak grids has an impact on active damping performance
[23]
. Therefore, it is necessary to do a sensitivity analysis when the grid impedance is taken into account.
Fig. 7
shows the root loci of the active damping loop when the grid impedance varies from 0 μH to 225 μH which is equal to 15% pu. The conjugate complex poles move towards the circle center, and the other two poles just move a little on the real axis.
Root loci of active damping loop when grid impedance varies from 0 μH to 225 μH (15% pu).
Fig. 8
shows a bode diagram of the active damping loop comparing no grid impedance with a 15% pu grid impedance. It demonstrates that the bode diagram with no grid impedance (solid line) is nearly the same the one with a 15% pu grid impedance (marked line) in the low frequency band. Moreover, the damping performance of the 15% pu grid impedance is better than that of the no grid impedance. This also demonstrates that the dashed line (neglecting the conjugate complex poles) is nearly the same as with the marked line (15% pu grid impedance). Therefore, it can be seen that the modified active damping method works well when the grid impedance varies.
Bode diagram of active damping loop comparing no grid impedance with 15% pu grid impedance.
V. SIMULATION AND EXPERIMENT RESULTS
The analysis is verified with MATLAB and a 300 kVA three phase prototype. The prototype is shown in
Fig. 9
. The system parameters are given in
Table II
.
Prototype of 300 kVA PWM converter.
SYSTEM PARAMETER
The active damping performance is verified by reactive current reference steps from 0 to 500A. The conventional capacitor current feedback simulation results are unstable at a 2 kHz switching frequency.
Fig. 10
shows the results of the modified approach at a 2 kHz switching frequency.
Fig. 10
(a) shows the simulation results, and
Fig. 10
(b) shows the experiment results. It can be seen that the system is stable and that the resonance is well damped. The total harmonic distortion (THD) of the grid current is 2.27% for the simulation and 2.52% for the experiment. The experimental results match the simulation results well. This demonstrates that the modified approach can strengthen the system stability and provide sufficient damping performance.
Reactive current step from 0 to 500A in 2 kHz switching frequency.
Grid current waveforms are obtained by an oscilloscope whose sampling frequency is 500 kHz. In order to see the dynamic performance of
Fig. 10
, the grid current is transformed from the three phase stationary frame into the two phase rotary frame by MATLAB.
Fig. 11
shows the step response of the reactive current component.
Fig. 11
(a) shows the simulation results, and
Fig. 11
(b) shows the experiment results. There is a onestep delay between the references and the results because of the digital control delay. The experimental results match the simulation results well. The rising time and overshoot are acceptable. This demonstrates that modified approach has good dynamic performance.
Step respond of reactive current component from 0 to500A.
The test of a DCload step from noload to fullload is performed with a 2Ω resistor which is parallel with a DC link capacitor.
Fig. 12
shows the results of the modified approach at a 2 kHz switching frequency.
Fig. 12
(a) shows the simulation results, and
Fig. 12
(b) shows the experimental results. The DC bus is 750V and the grid current is 600A in the steady state.
DCload step from noload to fullload in 2 kHz switching frequency.
This demonstrates that the system is stable and that the resonance is well damped. The dynamic respond is fast. The experimental results match the simulation results well.
In order to verify the modified approach when grid impedance is taken into account, a simulation of reactive current stepping from 0 to 500A is done, with the grid impedance set as 225 μH.
Fig. 13
shows the simulation results. The resonance is well damped and the dynamic performance is acceptable. This shows that the modified approach still works well when grid impedance is taken into account.
Reactive current step from 0 to 500A when grid impedance equals 225 μH.
VI. CONCLUSION
A modified active damping approach for grid connected PWM converters with a
LCL
filter has been proposed and analyzed. A discrete model considering the digital control delay has been derived. The feedback coefficients of the modified approach have been yielded in terms of the desired pole locations. The stable regions of the conventional capacitor current feedback active damping and the modified active damping approach have been discussed. The root loci and bode diagram of the active damping loop have been plotted when the grid impedance varies.
This paper shows that it is acceptable for the capacitor current feedback active damping to neglect the digital control delay at a high switching frequency but that systems will become inherent unstable at a low switching frequency. Therefore, it is unsuitable to neglect the digital control delay at a low switching frequency when the designing active damping loop. The modified approach can strengthen system stability and provide sufficient damping performance. By applying the modified approach, the active damping loop can be simplified from fourthorder into secondorder and the design of the current loop can be simplified. The modified approach can work well when the grid impedance varies. The theoretical results are verified by simulation and experiment results.
BIO
Zhiqiang Wan was born in Jiangxi, China, in 1990. He received his B.S. degree from the Harbin Institute of Technology, Harbin, China, in 2012. He is presently working towards his M.S. degree in the School of Electrical and Electronics Engineering, Huazhong University of Science and Technology (HUST), Wuhan, China. His current research interests include PWM rectifiers, LCL filters and digital control techniques.
Jian Xiong was born in Hubei Province, China. He received his B.S. degree from the East China Shipbuilding Institute, Zhenjiang, China, in 1993, and his M.S. and Ph.D. degrees from the Huazhong University of Science and Technology (HUST), Wuhan, China, in 1996 and 1999, respectively. He joined HUST as a Lecturer in 1999, and was promoted to an Associate Professor in 2003. His current research interests include uninterruptible power systems, ac drives, switchmode rectifiers, STATCOMs, and their related control techniques.
Ji Lei was born in Jiangxi Province, China, in 1991. He received his B.S. degree in Electrical Engineering and Automation from the Hefei University of Technology, Hefei, China, in 2013. He is presently working towards his M.S. degree in Electrical and Electronics Engineering at the Huazhong University of Science and Technology (HUST), Wuhan, China. His current researches include digital control technologies and threephase unbalance compensation.
Chen Chen was born in Jiangsu Province, China, in 1991. He received his B.S. degree from Jiangsu University, Zhenjiang, China, in 2012. He is presently working towards his Ph.D. degree in the School of Electrical and Electronics Engineering, Huazhong University of Science and Technology (HUST), Wuhan, China. His current research interests include digital control techniques, the design and control of power electronics systems, high power factor rectifiers and modular multilevel converters.
Kai Zhang was born in Henan Province, China. He received his B.S., M.S., and Ph.D. degrees from the Huazhong University of Science and Technology (HUST), Wuhan, China, in 1993, 1996, and 2001, respectively. He joined HUST as an Assistant Lecturer, in 1996. He was a Visiting Scholar at the University of New Brunswick, Saint John, NB, Canada, from 2004 to 2005. He was promoted to a Full Professor in 2006. He is the author of more than 40 technical papers. His current research interests include uninterruptible power systems, railway traction drives, modular multilevel converters, and electromagnetic compatibility techniques for power electronic systems.
Blaabjerg F.
,
Teodorescu R.
,
Liserre M.
,
Timbus A. V.
2006
“Overview of control and grid synchronization for distributed power generation systems,”
IEEE Trans. Ind. Electron.
53
(5)
1398 
1409
DOI : 10.1109/TIE.2006.881997
Jalili K.
,
Bernet S.
2009
“Design ofLCLfilters of activefrontend twolevel voltagesource converters,”
IEEE Trans. Ind. Electron.
56
(5)
1674 
1689
DOI : 10.1109/TIE.2008.2011251
Wang T. C. Y.
,
Zhihong Y.
,
Gautam S.
,
Xiaoming Y.
“Output filter design for a gridinterconnected threephase inverter,”
in Power Electronics Specialist Conference, 2003. PESC '03. 2003 IEEE 34th Annual
2003
Vol. 2
779 
784
Gabe I. J.
,
Montagner V. F.
,
Pinheiro H.
2009
“Design and implementation of a robust current controller for VSI connected to the grid through anLCLfilter,”
IEEE Trans. Power Electron.
24
(6)
1444 
1452
DOI : 10.1109/TPEL.2009.2016097
Wessels C.
,
Dannehl J.
,
Fuchs F. W.
“Active damping of LCLfilter resonance based on virtual resistor for PWM rectifiers ; stability analysis with different filter parameters,”
in Power Electronics Specialists Conference, 2008. PESC 2008. IEEE
2008
3532 
3538
Kukkola J.
,
Hinkkanen M.
2014
“Observerbased statespace current control for a threephase gridconnected converter equipped with anLCLfilter,”
IEEE Trans. Ind. Appl.
50
(4)
2700 
2709
DOI : 10.1109/TIA.2013.2295461
Liserre M.
,
Aquila A. D.
,
Blaabjerg F.
2004
“Genetic algorithmbased design of the active damping for an LCLfilter threephase active rectifier,”
IEEE Trans. Power Electron.
19
(1)
76 
86
DOI : 10.1109/TPEL.2003.820540
Wang X.
,
Blaabjerg F.
,
Loh P.
“Gridcurrentfeedback active damping for LCL resonance in gridconnected voltage source converters,”
IEEE Trans. Power Electron.
to be published
Chenlei B.
,
Xinbo R.
,
Xuehua W.
,
Weiwei L.
,
Donghua P.
,
Kailei W.
“Design of injected grid current regulator and capacitorcurrentfeedback activedamping for LCLtype gridconnected inverter,”
in Energy Conversion Congress and Exposition (ECCE), 2012 IEEE
2012
579 
586
Wagner M.
,
Barth T.
,
Ditmanson C.
,
Alvarez R.
,
Bernet S.
“Discretetime optimal active damping ofLCLresonance in grid connected converters by proportional capacitor current feedback,”
in Energy Conversion Congress and Exposition (ECCE), 2013 IEEE
2013
721 
727
Sun W.
,
Wu X.
,
Dai P.
,
Zhou J.
“An over view of damping methods for threephase PWM rectifier,”
in Industrial Technology, 2008. ICIT 2008. IEEE International Conference on
2008
1 
5
Sanbao Z.
,
Czarkowski D.
2007
“Modeling and digital control of a phasecontrolled seriesparallel resonant converter,”
IEEE Trans. Ind. Electron.
54
(2)
707 
715
DOI : 10.1109/TIE.2007.891766
HuuPhuc T.
,
Rahman M. F.
,
Grantham C.
“Time delay compensation for a DSPbased currentsource converter using observerpredictor controller,”
in Power Electronics and Drive Systems, 2007. PEDS '07. 7th International Conference on
Bangkok
2007
1091 
1096
Nussbaumer T.
,
Heldwein M. L.
,
Guanghai G.
,
Round S. D.
,
Kolar J. W.
2008
“Comparison of prediction techniques to compensate time delays caused by digital control of a threephase bucktype PWM rectifier system,”
IEEE Trans. Ind. Electron.
55
(2)
791 
799
DOI : 10.1109/TIE.2007.909061
Yi T.
,
Poh C. L.
,
Peng W.
,
Fook H. C.
,
Feng G.
,
Blaabjerg F.
2012
“Generalized design of high performance shunt active power filter with outputLCLfilter,”
IEEE Trans. Ind. Electron.
59
(3)
1443152 
2012
Guohong Z.
,
Rasmussen T. W.
,
Lin M.
,
Teodorescu R.
“Design and control of LCLfilter with active damping for active power filter,”
in Industrial Electronics (ISIE), 2010 IEEE International Symposium on
2010
2557 
2562
Parker S.
,
McGrath B.
,
Holmes G.
2014
“Regions of active damping control for LCL filters,”
IEEE Trans. Ind. Appl.
50
(1)
424 
432
DOI : 10.1109/TIA.2013.2266892
Dannehl J.
,
Wessels C.
,
Fuchs F. W.
2009
“Limitations of voltageoriented PI current control of gridconnected PWM rectifiers withLCLfilters,”
IEEE Trans. Ind. Electron.
56
(2)
380 
388
DOI : 10.1109/TIE.2008.2008774
Lehn P. W.
,
Irvani M. R.
1999
“Discrete time modeling and control of the voltage source converter for improved disturbance rejection,”
IEEE Trans. Power Electron.
14
(6)
1028 
1036
DOI : 10.1109/63.803396
Dannehl J.
,
Fuchs F. W.
,
Th X. F.
,
Gersen P. B.
2010
“PI state space current control of gridconnected PWM converters withLCLfilters,”
IEEE Trans. Power Electron.
25
(9)
2320 
2330
DOI : 10.1109/TPEL.2010.2047408
Otto B.
2008
Linear Algebra with Applications
4th ed.
Prentice Hall Publishers
USA
Chap. 1
Guoqiao S.
,
Dehong X.
,
Luping C.
,
Xuancai Z.
2008
“An improved control strategy for gridconnected voltage source inverters with anLCLfilter,”
IEEE Trans. Power Electron.
23
(4)
1899 
1906
DOI : 10.1109/TPEL.2008.924602
Wang X.
,
Blaabjerg F.
,
Loh P.
2015
“Virtual RC damping of LCLfiltered voltage source converters with extended selective harmonic compensation,”
IEEE Trans. Power Electron.
30
(9)
4726 
4737
DOI : 10.1109/TPEL.2014.2361853