The 2018 International Power Electronics Conference
Comprehensive Analysis of Virtual Impedance-Based
Active Damping for LCL Resonance
in Grid-Connected Inverters
Teng Liu*, Zeng Liu, Jinjun Liu, Yiming Tu, Zipeng Liu
State Key Laboratory of Electrical Insulation and Power Equipment, Xi’an Jiaotong University
Xi’an, China
*E-mail: teng.liu@stu.xjtu.edu.cn
Abstract—LCL filters have been widely applied as the
interface between voltage source inverters (VSIs) and power
grid. However, inherent LCL resonance complicates the
design of current control or even threatens system stability.
Extensive approaches have been proposed to deal with the
resonance, among which the active damping (AD) methods
realized by the feedback of filter state variables are proved
to be effective and robust. In this paper, a comprehensive
analysis of such AD by utilizing its virtual circuit property
is presented. It is found that different filter state variables
can be selected to achieve the effective damping functions by
emulating proper virtual impedances (VIs). Meanwhile, the
connection of the VI with the LCL network is not restricted
by the selected state variable but mainly dependent on the
applied AD controller, which, in result, enables a thorough
investigation of such VI-based AD methods. Consequently,
all possible forms of the AD controller to realize different
connection types by utilizing different state variables are
derived. The optimal selection of the AD controller together
with the specific state variable is further analyzed, where
the obtained results can be used to guide the design of such
AD in practice. Finally, the correctness of the theoretical
analyses are verified by the experimental results.
Keywords—active damping, grid-connected inverters, LCL
resonance, virtual impedance.
I.
INTRODUCTION
With fast development of distributed power generation
systems (DPGSs), voltage source inverters (VSIs) have
been widely used for integrating DPGSs into power grid
[1]. To meet the grid codes, output filters should be
equipped with VSIs. Compared with traditional L filters,
LCL filters can provide better attenuation of switching
ripples with smaller volumes and lower costs [2]-[3].
However, inherent LCL resonance characteristic imposes
constraints on the current controller design, which may
limit the current control loop bandwidth or even cause an
unstable system [4].
To suppress the LCL resonance, extensive researches
have been conducted [5]-[15]. One direct way is the socalled passive damping, which is realized by inserting
passive resistors into the LCL filter network [5]. Even
This work was supported by the National Natural Science
Foundation of China under Grant 51437007.
©2018 IEEJ
2681
though it owns the merits of simple implementation and
good robustness, it may not be preferred due to the
inevitable extra power losses, especially in high-power
applications [6].
An alternative way is the active damping (AD) to
obtain a more efficient system. The existing AD methods
can be divided into two main categories. One type is
implemented by cascading a digital filter with the current
regulator aiming at filtering out the LCL resonant peak
[6]. However, system parameters should be well known
to design the digital filter, which also makes such kind of
AD methods quite sensitive to the variation of the system
parameters. Another type is realized by feeding back the
LCL filter state variables to form a dual-loop control [7][15]. It has been proved that different state variables,
such as the inverter-side current [7], the filter capacitor
current [8]-[12], the filter capacitor voltage [13], or the
grid-side current [14]-[15], can all be selected to realize
the effective damping performance. However, it should
be noticed that different state variables need different AD
controllers to achieve an effective damping performance.
For example, when the capacitor current or the inverterside current is fed back, the AD controller is usually
implemented by a proportional gain [7]-[9]. However,
when the feedback state variable is changed to the gridside current, the AD controller should be modified to a
first-order high-pass filter with negative sign accordingly
[14]-[15]. Therefore, it is worthwhile to figure out all the
possible forms of the AD controller when the feedback
state variable to achieve the AD is determined.
An intuitive way to obtain appropriate forms of the
AD controller is based on its equivalent virtual circuit. In
[8], it has been proved that the proportional capacitorcurrent-feedback AD is equivalent to adding a virtual
resistor in parallel with the filter capacitor when the time
delay effect is neglected. The formed virtual resistor can
thus dampen the LCL resonance effectively. When the
feedback variable is changed to the inverter-side current,
a proportional AD controller can also emulate a virtual
resistor in series with the inverter-side filter inductor [7].
Recently, the equivalent virtual circuit property has also
been revealed when the grid-side current is fed back by a
first-order high-pass filter with negative sign, which is
The 2018 International Power Electronics Conference
equivalent to adding a virtual impedance (VI) in parallel
with the grid-side filter inductor. And this VI consists of
a series RL damper in parallel with a negative inductance
[15]. Therefore, the VI in different connections with LCL
filter has specific relationships with the AD controller.
To thoroughly reveal all the possible AD controllers
with different state variables for the effective damping,
this paper proposes a mathematical way to derive the
forms of the AD controller based on the virtual circuit
property. It is obtained that an arbitrary state variable can
be applied to emulate a VI in different connections with
LCL filter network through different AD controllers, and
the connection types will not be limited by the selected
state variable. For instance, the feedback of the filter
capacitor current can not only emulate a VI in series or
parallel with the capacitor, but can also realize possible
connections of VI with the inverter-side or grid-side filter
inductor through proper AD controllers. As a result, all
possible forms of the AD controller with different state
variables are derived. Further, the optimal selection of the
AD controller considering the availability of the sensors
for measuring the filter state variable is analyzed, where
the obtained conclusions can provide a guideline for the
design of such AD in practice. Finally, experimental
results validate the correctness of the theoretical analyses.
II.
kr s
s 2 + ω0 2
(a)
(b)
SYSTEM DESCRIPTION
Fig.1 shows the system structure of a three-phase gridconnected VSI system, where the inverter-side inductor
L1, the grid-side inductor L2, and the filter capacitor Cf
make up the LCL filters. Lg represents the grid impedance.
The dc-link voltage Vdc is assumed to be constant for
simplicity. vpcc is the point of common coupling (PCC)
voltage, whose phase angle θ is usually acquired by the
synchronous reference frame phase-locked loop (SRFPLL) for synchronizing the VSI with the grid. In practice,
the bandwidth of the SRF-PLL is always designed much
slower than that of the inner current control loop, which
means the effect of the SRF-PLL on the high frequency
resonance can be neglected [16].
For current regulation, either the inverter-side current
i1 or the grid-side current i2 can be controlled to inject the
demand power into the grid, where the control scheme
can be implemented either in the stationary αβ frame or
in the rotating dq frame. In this paper, the current control
in the αβ frame is chosen to avoid nonlinearity introduced
by the dq transformation. Herein, assuming a three-phase
balanced grid-connected VSI system, the per-phase block
diagrams of the current control with different AD, which
are shown in Fig. 2, can thus be used for the following
analyses. Gc(s) is the current regulator implemented with
a proportional-resonant (PR) controller expressed as
Gc ( s ) = k p +
Fig. 1. System structure of a three-phase grid-connected VSI.
(1)
where ω0 is the grid fundamental angular frequency. Gd(s)
is time delay effect existed in the digitally controlled VSI,
which includes both the computational and PWM delays.
2682
Fig. 2. Per-phase block diagrams of the current control with different
AD. (a) grid-side current control. (b) inverter-side current control.
When the synchronous sampling scheme is applied, Gd(s)
can be expressed as
Gd ( s ) = e−1.5Ts s
(2)
where Ts is the sampling period [8].
It is worthwhile to notice that the selection of the state
variable used for achieving the AD purpose is flexible
when i2 is fed back to control the injected power as
shown in Fig. 2(a). In this case, i1, i2, filter capacitor
current icf, or filter capacitor voltage vcf can all be adopted
to realize the effective damping performance. However,
when the regulated current is changed to i1, the same state
variable i1 may be the only choice to realize the AD
simultaneously, as the current or voltage sensors to detect
the other state variables may not be installed in most
practical applications for saving the cost. The relevant
current control diagram is shown in Fig. 2(b), where the
AD controller Gad(s) is usually implemented by a simple
proportional gain [7]. The proportional feedback of i1 is
equivalent to adding a virtual resistor in series with L1
when the time delay effect is neglected, which thus can
dampen the LCL resonance effectively.
According to the above analyses, this paper mainly
focuses on the situation where i2 is fed back to control the
injected power. It is known that different state variables
need different forms of Gad(s) to achieve the effective
damping performance. However, most of the existing
literatures presented Gad(s) directly without giving any
explanation, which may not be helpful for a deeper
insight of such AD methods. Hence, this paper derives all
The 2018 International Power Electronics Conference
possible forms of Gad(s) with the different state variables
based on the equivalent virtual circuit property, which
will be demonstrated in the following part.
III.
G p _ m ( s) =
Gad ( s ) =
A. Derivation of Gad(s) With Different State Variables
To derive possible forms of Gad(s) with different state
variables intuitively, Fig. 2(a) is equivalently transformed
as shown in Fig. 3, where Gp(s) is the transfer function
from the output voltage of the VSI vo(s) to i2(s), and Gt(s)
represents the transfer function from i2(s) to the certain
state variable fed back for realizing the AD. Herein, Gp(s)
can be derived based on Fig. 2(a), which is expressed as
i2 ( s )
1
=
vo ( s ) s ( L1 L2 C f s 2 + L1 + L2 )
(5)
1
1
1
⋅(
−
)
Gd ( s ) ⋅ Gt ( s ) G p _ m ( s ) G p ( s )
(6)
Further, if i2 is fed back to realize the AD purpose,
which makes Gt(s) = 1, the form of Gad(s) can finally be
obtained as
Gad ( s ) =
(3)
Fig. 3 can be further transformed to Fig. 4 to reveal the
equivalent virtual circuit property introduced by the AD.
Based on Fig. 4, it is clear that the components inside the
dashed box can be regarded as the modified virtual power
plant which is also represented by the transfer function
from vo(s) to i2(s). Hence, the characteristic of the virtual
power plant Gp_m(s), which is modified by the AD, can be
derived as
G p _ m ( s) =
s C f L1 L2 Z v s + L1 L2 s + ( L1 + L2 ) Z v 
Based on (3), (4) and (5), the form of Gad(s) to realize
such Zv(s) in parallel with Cf can be derived as
DERIVATION AND OPTIMAL SELECTION OF Gad(s)
WITH DIFFERENT STATE VARIABLES
G p (s) =
Zv
2
G p ( s)
i2 ( s )
(4)
=
vo ( s ) 1 + G p ( s ) ⋅ Gad ( s ) ⋅ Gd ( s ) ⋅ Gt ( s )
Meanwhile, Gp_m(s) can also be obtained based on the
introduced equivalent virtual circuit of the LCL filter
network. In this paper, implementing a virtual impedance
Zv(s) in parallel with Cf is taken as an example to
demonstrate the derivation process of Gad(s). The relevant
virtual circuit of the LCL filter network is shown in Fig. 5,
based on which Gp_m(s) can be obtained as
L1 L2 s 2
Zv
(7)
where Gd(s) = 1 is assumed. Even though the time delay
will change the final form of the VI, and may also have
negative effect on the damping performance due to the
resulted negative virtual resistor in high frequency range
[8], this assumption is still acceptable to derive the basic
form of Gad(s), as the time delay effect can be partly
compensated by different delay compensation techniques
[8], [17]. In this paper, a lead-lag compensator proposed
in [17] is directly cascaded with Gad(s) in the final control
scheme to reduce the time delay effect.
From the above derivation process, it can be found that,
even though the feedback state variable for the AD is the
grid-side current, the formed virtual impedance can still
be in parallel with Cf, which means the connection type
of Zv(s) with LCL filter network will not be limited by the
selected state variable.
Consequently, all the possible forms of Gad(s) with
different state variables can be derived by following the
similar process. The obtained results are listed in Table I.
It is clear that the connection type of Zv(s) is mainly
dependent on the specific form of Gad(s) rather than the
selected state variable. With a proper Gad(s), an arbitrary
state variable can be fed back to emulate Zv(s) in any
connection with the LCL filter for achieving the effective
damping performance.
B. Optimal Selection of Gad(s)
After obtaining possible forms of Gad(s) with different
state variables, the optimal selection of Gad(s) should be
discussed. As suppressing the LCL resonance is the main
topic of this paper, Zv(s) is realized as a virtual resistor,
i.e., Zv(s) = Rv. Other possible Zv(s), such as a virtual
inductor or a more complicated VI for different purposes,
will be investigated in the further work.
The first criterion for the optimal selection of Gad(s) is
the complexity to implement Gad(s). The comparisons are
made among different Gad(s) with the same state variable.
For example, when i2 is selected to realize the AD, three
forms of Gad(s), which make the formed virtual resistor in
series with L1, in series with L2 or in parallel with Cf, are
much simpler than the other three forms of Gad(s). When
the state variable is changed to icf, a proportional gain
L1/(CfRv) is the simplest form of Gad(s). Consequently, all
the simpler forms of Gad(s) are marked in red in Table I.
Fig. 3. Equivalent control block diagram of Fig. 2(a).
Fig. 4. Equivalent block diagram transformation to reveal the virtual
circuit property of the AD.
Fig. 5. Equivalent virtual circuit of the LCL filter network when Zv(s)
is in parallel with Cf.
2683
The 2018 International Power Electronics Conference
TABLE I
DIFFERENT FORMS OF Gad(s) WITH DIFFERENT STATE VARIABLES USED FOR ACTIVE DAMPING
Variable used for the active damping
Virtual impedance
connection type
Grid-side current i2
In series with L1
(1 + L2 C f s ) ⋅ Z v
(1 +
2
2
In parallel with L1
Capacitor current icf
−
2
(1 + L2 C f s ) ⋅ L1 s
) ⋅ Zv
L2 C f s 2
2
−
2
−
L1 L2 C f s + L2 C f Z v
2
2
In parallel with L2
In series with Cf
In parallel with Cf
−
2
(1 + L1C f s ) ⋅ L2 s
L2 s + Z v
−
2
f
C L1 L2 Z v s
L1 L2 s
Zv
−
(1 + L1C f s ) ⋅ L2
L2 C f s + C f Z v
4
C f Zv s + 1
2
2
2
−
C f L1 Z v s
−
L1
2
(1 + L1C f s ) ⋅ L2 s
2
f
C L1 L2 Z v s
(1 + L2 C f s 2 ) ⋅ (C f Z v s + 1)
2
L1 L2 s + L2 Z v
L2
s+
1
L2 s
) ⋅ Zv
2
−
(1 + L1C f s ) ⋅ sL2
L2 s + Z v
2
4
−
C f L1 Z v s
3
C f Zv s + 1
2
L1 s
2
Zv
(1 + L2 C f s ) ⋅ Z v
It can be concluded that a simple proportional gain can be
regarded as an optimal selection of Gad(s) when icf or i1 is
selected to realize the AD. However, if the state variable
is changed to vcf or i2, there exist three possible selections
of Gad(s) which can meet the requirement of simplicity.
To further compare these three Gad(s), the frequency
responses of Gp_m(s) with different connections of the
virtual resistor are plotted in Fig. 6 respectively. It is seen
that Gad(s) to form a virtual resistor in parallel with Cf can
easily achieve the satisfied damping performance. For the
other two connections, even though the formed Rv can
dampen the resonant peak, it will cause a flat and small
magnitude gain in the low frequency range, which may
lead to a relatively low current control bandwidth when
the PR controller is adopted as the current regulator.
Therefore, a virtual resistor connected in parallel with Cf
is much preferred among all the connection types.
However, it should also be noticed that, when Rv is in
) ⋅ Zv
(1 + L2 C f s ) ⋅ sL1
L1C f
2
(1 + L2 C f s 2 ) ⋅ ( L2 s + Z v )
L1 L2 s
C f Zv
(
2
2
−
2
C f Zv s + 1
−
L1 s + Z v
1 + L2 C f s
1
L2 s
2
L21 s 2
(1 + L1C f s ) ⋅ Z v
2
L2 C f s
(C f s +
2
(1 + L1C f s ) ⋅ Z v
2
(1 + L1C f s ) ⋅ Z v
In series with L2
Capacitor voltage vcf
Zv
(1 + L2 C f s ) ⋅ L1
2
L1 s + Z v
1
Inverter-side current i1
parallel with Cf, Gad(s) will contain the derivative term if
the selected state variable is vcf or i2. And this derivative
term is not practically feasible due to the possible noise
amplification. Fortunately, this limitation can be solved
by approximating the derivative term with a non-ideal
generalized integrator [18].
To sum up, if the capacitor current sensor is available,
Gad(s) realized only by a simple proportional gain can be
regarded as the first choice to obtain a satisfied damping
performance. However, in practice, the capacitor current
sensor is usually unavailable for the purpose of saving the
cost. At this time, the current sensor for i2 or the voltage
sensor for vcf may be applied, where a virtual resistor in
parallel with Cf is preferred to dampen the LCL resonant
peak. If there is only the current sensor for i1 available,
the optimal selection of Gad(s) may be a proportional gain
due to the simplest implementation.
IV.
CASE STUDY OF THE GRID-SIDE CURRENT
FEEDBACK ACTIVE DAMPING
As many existing literatures have already presented
the AD realized by the feedback of icf or i1, this paper
only presents the grid-side current i2 for realizing the AD
to demonstrate the correctness of the theoretical analyses.
It should be mentioned that choosing i2 as the feedback
variable can avoid the extra sensors, which thus can save
the system cost.
A. Grid-side Current Feedback Active Damping
The per-phase block diagram of the grid-side current
control with the grid-side current feedback AD is shown
in Fig. 7. It is seen that the grid-side current i2 is fed back
for both injected power regulation and active damping
purposes. Herein, the time delay Gd(s) is considered to
make the analysis more close to the real case.
Based on Table I, Gad(s) for obtaining a virtual resistor
Fig. 6. Frequency responses of Gp_m(s) with different connections of
the virtual resistor.
2684
The 2018 International Power Electronics Conference
where λ is defined as the compensation coefficient. A
smaller λ means a better compensation capability. Kc is
used to attenuate the high frequency noise amplification,
whose value should be smaller than 1.
B. Damping Performance Evaluation
To illustrate the damping performance of this grid-side
current feedback AD, the current control loop gain T(s)
can be derived based on Fig. 7, which is expressed as
Fig. 7. Per-phase block diagram of the grid-side current control with
the grid-side current feedback active damping.
T ( s ) = Gc ( s ) ⋅
in parallel with Cf can be expressed as
L1 L2 s 2
Rv
(8)
where Rv is the emulated virtual resistor. Its value can be
determined by the desired damping coefficient.
As mentioned above, the derivative term s2 in (8) can
not be directly realized due to the severe high frequency
noise amplification. To solve this issue, a second-order
transfer function proposed in [19] is adopted to replace s2
term. Consequently, the final form of Gad_x(s) can be
expressed as
Gad _ x ( s ) =
ω 2 ⋅ s2
L1 L2
⋅ 2 c
Rv s + ζ s + ωc2
1 + Gad ( s ) ⋅ Gcomp ( s ) ⋅ Gd ( s ) ⋅ G p ( s )
(11)
From (11), it is clear that T(s) without any damping
can be obtained by setting Gad(s) = 0. As a result, the
frequency responses of T(s) without and with the gridside current feedback AD can be plotted in Fig. 9, where
the main circuit parameters are listed in Table II.
Based on Fig. 9, it is seen that the frequency responses
of T(s) without any damping exist a resonant peak at the
LCL resonance frequency, which will cause a negative
180° phase falling. According to the stability criterion, the
system will be unstable. When the proposed grid-side
current feedback AD is applied, this resonant peak can be
suppressed effectively leading to a stable system.
Besides, it can also be seen that the frequency responses
of T(s) with the AD is similar with those of T(s) with an
ideal virtual resistor in parallel with Cf, which thus proves
the correctness of the theoretical analyses.
Fig. 8. Frequency responses of Gad(s) in (8) and Gad_x(s) in (9).
Gad ( s ) =
Gd ( s ) ⋅ G p ( s )
TABLE II
MAIN CIRCUIT PARAMETERS
Item
Symbol
DC-link voltage
Grid voltage (l-g, rms)
Switching frequency
Sampling frequency
Inverter-side inductor
Grid-side inductor
Filter capacitor
Vdc
vs
fsw
fs
L1
L2
Cf
Value
400V
120V
10kHz
10kHz
3.5mH
1.75mH
15μF
(9)
where ζ is used for damping to avoid the infinite gain at
ωc. The larger value of ωc makes Gad_x(s) more coincident
with Gad(s). However, a larger magnitude gain at the high
frequency range will be accompanied. Besides, the larger
value of ζ ensures the flatter peak at ωc. In this paper, ωc
= 3×104 rad/s and ζ = 104 are chosen. The frequency
responses of Gad(s) in (8) and Gad_x(s) in (9) can thus be
plotted in Fig. 8, respectively.
Besides, as the time delay Gd(s) may have negative
influence on the introduced virtual resistor, a secondorder lead-lag compensator proposed in [17] should be
added in cascaded with Gad_x(s) to compensate the delay
effect as shown in Fig. 7. Its expression is given as
Gcomp ( s ) = K c ⋅
(1 + 0.75Ts s ) 2
0 < λ < 0.75 K c ≤ 1 (10)
(1 + λTs s ) 2
Fig. 9. Frequency responses of T(s) without and with the proposed gridside current feedback AD.
2685
The 2018 International Power Electronics Conference
measured phase-a PCC voltage and three-phase grid-side
currents are shown in Fig. 10. It is clear that the grid-side
currents are severely oscillated indicating an unstable
system, which is consistent with the analyses in Fig. 9.
Then, the proposed grid-side current feedback AD is
applied. The measured steady-state waveforms are shown
in Fig. 11(a), where the system can be well stabilized.
Meanwhile, a satisfied dynamic performance is obtained
by setting the current magnitude reference from 5A to
10A, whose measured waveforms are shown in Fig. 11(b).
Consequently, the proposed grid-side current feedback
active damping is proved to be effective to suppress the
LCL resonance.
V.
Fig. 10. The measured phase-a PCC voltage and three-phase grid-side
currents when the VSI is operated without any damping.
CONCLUSIONS
This paper presented a comprehensive analysis of the
AD implemented by the feedback of the LCL filter state
variables. It was found that different state variables can
be adopted to achieve the effective damping function
by emulating a proper VI. Besides, the connection type
of VI with the LCL filter network is not constrained by
the selected state variable but mainly dependent on the
specific form of the AD controller. Therefore, all possible
forms of the AD controller to realize the VI in different
connections with the LCL filter network under different
state variables are mathematically derived. Further, the
optimal selection of the AD controller is discussed by
considering the complexity of the controller’s realization
as well as the availability of the measuring sensors. The
obtained conclusions may guide the design of such AD
methods. Finally, a case study focusing on one particular
AD, which is realized by the feedback of the grid-side
current to emulate a virtual resistor in parallel with the
filter capacitor, was presented. The damping performance
evaluation in the frequency domain together with the
experimental results have verified the correctness of the
above theoretical analyses.
(a)
REFERENCES
[1]
[2]
[3]
(b)
Fig. 11. The measured phase-a PCC voltage and three-phase grid-side
currents when the VSI is operated with the proposed grid-side current
feedback AD. (a) steady-state waveforms, (b) dynamic waveforms.
[4]
C. Experimental Results
To verify the effectiveness of such grid-side current
feedback AD, a laboratory setup, where the main circuit
parameters listed in Table II are applied, is built up. The
structure is exactly the same with that shown in Fig. 1.
The ac power grid is emulated by a Chroma Regenerative
Grid Simulator 61860.
Firstly, the VSI is operated without any damping. The
2686
[5]
[6]
[7]
F. Blaabjerg, R. Teodorescu, M. Liserre, and A. V. Timbus,
“Overview of control and grid synchronization for distributed
power generation systems,” IEEE Trans. Ind. Electron., vol. 53,
no. 5, pp. 1398–1409, Oct. 2006.
M. Liserre, F. Blaabjerg, and S. Hansen, “Design and control of an
LCL-filtered-based three-phase active rectifiers,” IEEE Trans. Ind.
Appl., vol. 41, no. 5, pp. 1281-1291, Sept./Oct. 2005.
Y. Tang, P. C. Loh, P. Wang, F. H. Choo, F. Gao, and F.
Blaabjerg, “Generalized design of high performance shunt active
power filter with output LCL filter,” IEEE Trans. Ind. Electron.,
vol. 59, no. 3, pp. 1443-1452, Mar. 2012.
K. Jalili and S. Bernet, “Design of LCL filters of active-front-end
two-level voltage-source converters,” IEEE Trans. Ind. Electron.,
vol. 56, no. 5, pp. 1674-1689, May 2009.
R. N. Beres, X. Wang, F. Blaabjerg, C. L. Bak, and M. Liserre,
“Comparative evaluation of passive damping topologies for
parallel grid-connected converters with LCL filters,” in IEEE 2014
International Power Electron. Conf., 2014, pp. 3320-337.
J. Dannehl, M. Liserre, and F. Fuchs, “Filter-based active damping
of voltage source converters with LCL filter,” IEEE Trans. Ind.
Electron., vol. 58, no. 8, pp. 3623-3633, Aug. 2011.
J. Xu, S. Xie, C. Kan, and D. Ji, “An improved inverter-side
current feedback control for grid-connected inverters with LCL
filters,” in Proc. IEEE ICPE-ECCE Asia, Jul. 2015, pp. 984-989.
The 2018 International Power Electronics Conference
[8]
[9]
[10]
[11]
[12]
[13]
D. Pan, X. Ruan, C. Bao, W. Li, and X. Wang, “Capacitorcurrent-feedback active damping with reduced computation delay
for improving robustness of LCL-type grid-connected inverter,”
IEEE Trans. Power Electron., vol. 29, no. 7, pp. 3414-3427, Jul.
2014.
D. Pan, X. Ruan, C. Bao, W. Li, and X. Wang, "Optimized
controller design for LCL-type grid-connected inverter to achieve
high robustness against grid-impedance variation," IEEE Trans.
Ind. Electron., vol. 62, no. 3, pp. 1537–1547, Mar. 2015.
X. Li, X. W, Y. Geng, X. Yuan, C. Xia and X. Zhang, “Wide
damping region for LCL-type grid-connected inverter with an
improved capacitor-current-feedback method,” IEEE Trans.
Power Electron., vol. 30, no. 9, pp. 5247-5259, Sep. 2015.
X. Wang, F. Blaabjerg, and P. C. Loh, “Virtual RC damping of
LCL-filtered voltage source converters with extended selective
harmonic compensation,” IEEE Trans. Power Electron., vol. 62,
no. 3, pp. 1537-1547, Mar. 2015.
J. Dannehl, F. Fuchs, S. Hansen, and P. B. Thøgersen,
“Investigation of active damping approaches for PI-based current
control of grid-connected pulse width modulation converters with
LCL filters,” IEEE Trans. Ind. Appl., vol. 46, no. 4, pp. 15091517, Jul. 2010.
M. Malinowski and S. Bernet, “A simple voltage sensorless active
damping scheme for three-phase PWM converters with an LCL
2687
[14]
[15]
[16]
[17]
[18]
[19]
filter,” IEEE Trans. Ind. Electron., vol. 55, no. 4, pp. 1876-1880,
Apr. 2008.
J. Xu, S. Xie, and T. Tang, “Active damping-based control for
grid-connected LCL-filtered inverter with injected grid current
feedback only,” IEEE Trans. Ind. Electron., vol. 61, no. 9, pp.
4746-4758, Sep. 2014.
X. Wang, F. Blaabjerg, and P. C. Loh, “Grid-current-feedback
active damping for LCL resonance in grid-connected voltagesource converters,” IEEE Trans. Power Electron., vol. 31, no. 1,
pp. 213-223, Jan. 2016.
S. G. Parker, B. P. McGrath, and D. G. Holmes, “Regions of
active damping control for LCL filters,” IEEE Trans. Ind. Appl.,
vol. 50, no. 1, pp. 424-432, Jan./Feb. 2014.
T. Liu, Z. Liu, J. J. Liu, and Y. Tu, “An improved capacitorcurrent-feedback active damping for LCL resonance in gridconnected inverters,” in Proc. IEEE IFEEC-ECCE Asia, Jul. 2017,
pp. 2111-2116.
Z. Xin, X. Wang, P. C. Loh, and F. Blaabjerg, “Realization of
digital differentiator using generalized integrator for power
converters,” IEEE Trans. Power Electron., vol. 30, no. 12, pp.
6520-6523, Dec. 2015.
T. Liu, Z. Liu, J. J. Liu, and Y. Tu, “Virtual impedance-based
active damping for LCL resonance in grid-connected voltage
source inverters with grid current feedback,” in Proc. 2016 IEEE
ECCE, Oct. 2016, pp. 1-8.