Investigation of Active Damping Approaches for
PI-based Current Control of Grid-Connected PWM
Converters with LCL Filters
J. Dannehl*, F.W. Fuchs**
S. Hansen
P.B. Thøgersen
* Student Member, IEEE, ** Senior Member, IEEE
Inst. for Power Electronics and Electrical Drives,
Christian-Albrechts-University of Kiel
D-24143 Kiel, Germany
* dannehl@ieee.org, ** fwf@tf.uni-kiel.de
Member, IEEE
Danfoss Drives A/S
DK-6100 Graasten, Denmark
s.hansen@danfoss.com
Senior Member, IEEE
KK-Electronic A/S
DK-9220 Aalborg, Denmark
patho@kk-electronic.com
Abstract—This paper deals with different active damping
approaches for PI-based current control of grid-connected PWM
converters with LCL filters which are based on one additional
feedback. Filter capacitor current as well as voltage feedback
for purpose of resonance damping are analyzed and compared.
Basic studies in continuous Laplace domain show that either
proportional current feedback or derivative voltage feedback
yield resonance damping. Detailed investigations of these two
approaches in discrete z-domain taking into account the discrete
nature of control implementation, sampling, and PWM are carried out. Several ratios of LCL resonance frequency and control
frequency are considered. For high resonance frequencies only
current feedback stabilizes the system. For medium resonance
frequencies both approaches show good performance. For low
resonance frequencies stability gets worse even if voltage feedback
offers slighly better damping properties. Measurement results
validate the theoretical results.
I. I NTRODUCTION
Three-phase grid-connected PWM converters are often used
in regenerative energy systems and in adjustable speed drives
when regenerative braking is required [1]. Power regeneration,
adjustable power factor, controllable DC-link voltage, and
less line current harmonic distortion are the most important
advantages with respect to passive diode bridges. For reduction
of line current ripple LCL filters are often used as grid filters
[2]. Due to high efficiency and flexibility active resonance
damping solutions as part of the converter control attracted
much attention [3]–[8].
The converter control commonly consists of an outer DClink voltage PI control and inner current control loops. For
some applications with line current feedback the well-known
PI control in synchronous reference frame can be employed.
But the inner PI-based current control is not suitable for many
applications in dependence of the ratio of LCL-resonance
frequency and control frequency [9]. For lower ratios or
converter current feedback PI control leads to instability. For
these applications active resonance damping gets integrated
into the current control loop. Sensorless resonance damping
[10] as well as damping with an additional feedback are
presented in literature. In [3], [4], [10], and [11] resonance
is damped with additional feedback of the voltages across the
filter capacitors. In [12] and [13] the current through the filter
capacitors is used for resonance damping and in [7] different
solutions based on so-called virtual resistors are presented.
For certain system settings the resonance damping effect is
put into evidence and the control performance is analyzed.
Analyses with different system settings, especially different
ratios of resonance and control frequency, are often not carried
out. For this reason one damping approach presented in [7] is
analyzed in more detail in [14].
In this paper a more general analysis of different resonance damping solutions is carried out. Basic LCL-resonance
damping properties of different feedback states are studied.
Feedback of the current through or voltage across the LCL
filter capacitors with different feedback transfer functions are
taken into consideration and compared in continuous Laplace
domain. The results show how the different feedback signals
need to be fed back in order to achieve resonance damping. Detailed investigation of two most suitable damping approaches in discrete z-domain taking into account the discrete
nature of control implementation, sampling, and PWM follows
the general analyses. Different ratios of LCL filter resonance
and control frequency are analyzed. By that limitations of the
approaches based on one additional feedback of the current
through or voltage across the filter capacitors are shown.
First the system is described and mathematical model is
shown in section II and a control overview is given in III.
Basic study of damping properties is carried out in section
IV. Detailed investigation of filter capacitor current feedback
with different typical system settings is presented in section
V-A and filter capacitor voltage feedback in section V-B.
Measurement results are shown in section VI. Finally, a
conclusion is given.
II. S YSTEM D ESCRIPTION
A three-phase IGBT voltage source converter is studied. It
is connected to the grid via LCL filter as shown in Fig. 1. In
addition to the DC link voltage and converter currents either
uL
Lfg
uCf
iCf
Cf
Current
PI-Control
i q*
+
u C*
+
uC
Resonance
Damping
Fig. 2. Cascaded control structure including resonance damping based on
additional feedback
UC
Ucf
Ucf
UC
the filter capacitor currents or filter capacitor voltages are
measured. The line voltages are measured for the purpose of
synchronizing the control with the line voltage. The converter
is loaded by an inverter-fed induction machine. In this paper
different filter capacitances are used in order to study different
ratios of resonance and control frequency.
Applying Kirchhoff’s laws yields the continuous model of the
LCL filter (uC : converter output voltage) [9]:
d dq
1 dq
dq
− jω · →
i dq
i
=
·
u
−
u
→L
→ Cf
L
dt → L
Lf g
d dq
1
i dq
i
=
· u→ dq − u→ dq − jω · →
(1)
Cf
C
C
dt → C
Lf c
d dq
1 dq
u→
=
· →
i −→
i dq − jω · u→ dq
L
C
Cf
dt Cf
Cf
The resonance frequency of the LCL filter can be calculated:
Lf g + Lf c
(2)
ωRes =
Cf Lf g Lf c
Because the outer DC-link voltage is outside the scope of
this paper, the reader is referred to [9] for the mathematical
description of the DC-link dynamics and its control. From (1)
cf
the transfer function GU
LCL (s) can be derived
1
1
UCf (s)
=
2
UC (s)
Lc Cf s2 + ωRes
(4)
III. C ONTROL OVERVIEW
The control structure is shown in Fig. 2. The DC-link
voltage is controlled with PI controller [9]. Active and reactive
converter currents are controlled in synchronous reference
frame with PI controllers. In this paper converter current
control is focused on. Similar analyses can be carried out with
line current control. PI controllers are tuned with symmetrical
optimum [15] whereas the L filter approximation of the LCL
filter is used [16]:
kp = −
Lf g + Lf c
3 · Tc
;
TI = 9 · Tc
Icf
K(s)
GC(s)
Fig. 3. Feedback of filter capacitor voltage (left) and filter capacitor current
(right).
With this PI tuning and backwards implementation [17] the
reference step response shows an overshoot of approx. 30%
and the reference is crossed after approx. 3-4 control periods.
Note that more stringent requirements with respect to overshoot could be fulfilled by different tuning. Additionally, a
feed forward decoupling is used [9]. For the LCL filter system
active resonance damping is required. Sensorless resonance
damping is possible [18], but it has been shown to be rather
sensitive to parameter variations [8]. Moreover, its application
is not possible for every kind of LCL filter design [9].
Additional feedback of system states offers improvements.
Various solutions can be found in literature. In this paper a
more generic study of approaches to damp resonance with
feedback of one additional system state is carried out.
IV. BASIC STUDY OF RESONANCE DAMPING WITH
DIFFERENT FEEDBACK STATES
(3)
and taking into account, that ICf (s) = s Cf UCf (s) holds,
gives
s
1
ICf (s) UCf (s)
·
=
2
2
UCf (s) UC (s)
Lc s + ωRes
K(s)
GC(s)
Icf
GLCL(s)
GLCL(s)
GIcf
LCL (s) =
Measurements
Gate Signals
Grid-connected PWM converter with LCL filter
cf
GU
LCL (s) =
System
PWM
iC
Control
Fig. 1.
i d*
UDC
PI-Control
Load
uDC
uL
UDC*
iload
iDC
Lfc
(5)
As the resonance is caused by the filter capacitors it is
reasonable to use the current through or voltage across them
for the purpose of resonance damping. It depends strongly
on the application and voltage as well as current level which
kind of sensor to chose. For higher power/current levels
(MW range, but still low voltage < 1000 V AC) current
sensors are much more expensive than voltage sensors. At
low power (few kW range) standard current sensors might
even be cheaper than voltage sensors with the same insulation
level. In the lowest power/current range, where the power
circuit typically is integrated on a PCB, other issues (such as
use of PCB area) might be important cost drivers. Moreover,
in some applications voltage sensors are used anyway for
grid synchronization [3]. The number of sensors can also
be reduced by estimation of feedback signals [19]. In this
paper resonance damping properties of both feedback states
are studied.
The influence of filter capacitor current or voltage feedback on
the system poles is studied separately. The analysis is carried
KuCf
KuCf
KiCf
KiCf
KuCf
KiCf
Fig. 4. Impact of different continuous feedback characteristics on the system pole-zero locations in continuous s-plane (PWM neglected): Filter capacitor
voltage feedback (top) and filter capacitor current feedback (bottom).
out in continuous Laplace domain. For this purpose delays
caused by PWM and computation as well as discretization
effects are neglected at this stage. More detailed analyses are
presented later in section V. The loops illustrated in Fig. 2 are
analyzed whereas different feedback transfer functions K(s)
are taken into consideration, namely proportional, derivative,
and integrative feedback. The closed-loop transfer function
Gc (s) can be calculated:
GLCL (s)
(6)
Gc (s) =
K(s) + GLCL (s)
whereas for UCf feedback (3) is used and (4) for ICf
feedback.
In this paper feedback on the voltage references is considered. Note, that feedback on the current references instead of
voltage references is possible as well. This can be beneficial
with respect to current overshoot during transients [6]. In [7]
and [14] it is used within the so-called concept of virtual
resistor. In [6] it is used as part of deadbeat state space control.
Similar analyses including the PI-controller in the loop give
the required feedback types. As this is beyond the scope of
this paper results are not shown here.
A. Filter capacitor voltage feedback
Evaluations of (6) with (3) for various feedback types are
shown in Tab. I. For proportional feedback the transfer func-
tion of a resonator is obtained whereas the natural resonance
frequency ωRes is shifted in dependence of K. As can be seen
in Fig. 4 the complex conjugated poles are shifted along the
imaginary axis in dependence of the proportional gain. Thus
no resonance damping is achieved by proportional feedback
of the filter capacitor voltage.
Derivative feedback gives a second order delay transfer function. The natural resonance frequency ωRes is kept but a nonzero damping factor is achieved which is the term related to s.
The complex conjugated poles are shifted into the left s-plane
in dependence of the derivative gain. Thus resonance damping
is achieved by derivative feedback of the filter capacitor
voltage without frequency shift.
From Fig. 4 it can be seen that the system poles are shifted
into the right s-half plane for integrative feedback type. Thus
the system gets unstable for integrative feedback.
It can be concluded that derivative filter capacitor voltage
feedback is required for resonance damping. Approaches in literature using the filter capacitor voltage for resonance damping
basically follow this result. In [3] and [10] the voltage is fed
back with a lead element that exhibits high pass characteristics.
In [4] a high pass filter is directly selected from the start.
TABLE I
C LOSED - LOOP TRANSFER FUNCTIONS WITH UCf AND ICf FEEDBACK FOR DIFFERENT FEEDBACK TYPES
Feedback type
Gc (s) for UCf feedback
Gc (s) for ICf feedback
K(s) = K (P-type)
1
1
2
Lc Cf s2 + (ωRes
+ K/(Lc Cf ))
1
s
2
Lc s2 + s (KiCf /(Lc ) +ωRes
damping
K(s) = K s (D-type)
1
1
2
Lc Cf s2 + s K/(Lc Cf ) +ωRes
1
Lc + KiCf
s2
+
s
Lc
Lc + KiCf
2
ωRes
damping
K(s) =
K
(I-type)
s
1
s
2
Lc Cf s3 + sωRes
+ K/(Lc Cf )
1
s
Lc s2 + KiCf /Lc + ω 2
Res
IC*(z)
+
B. Filter capacitor current feedback
Based on the result of the previous section it can be expected
that proportional ICf feedback yields resonance damping, as
ICf (s) = s Cf UCf (s) holds. The pole zero maps in Fig. 4
and the transfer functions in Tab. I confirm that expectation.
Approaches in literature using the filter capacitor current for
resonance damping, follow this result. In [12] [13] proportional
feedback is used.
-
-
UC*(z)
UC(z)
z-1
IC(z)
LCL
K(z)
resonance
damping
ICf(z)
or
UCf(z)
Fig. 5. Detailed current control structure: PI control with resonance damping
by feedback of ICf or UCf .
V. D ETAILED ANALYSIS OF OVERALL DIGITAL CURRENT
TABLE II
S YSTEM PARAMETERS
CONTROL SYSTEM
For the following detailed study a typical system setting
of a medium power drive is selected, see Tab. II. In order to
cover a typical range of LCL filter parameters three different
LCL filter settings are studied. The choice of filter elements is
a trade-off between switching harmonics attenuation, reactive
power consumption, relative short circuit voltage drop, grid
decoupling, filter losses as well as costs and size of filter
elements. Research on LCL filter design is documented in
[16], [20]. From control and resonance damping points of view
the ratio of resonance frequency to control frequency is most
important. Therefore the control is analyzed in detail for three
characteristic ratios. In this paper the control frequency is kept
constant and the resonance frequency is varied by changing the
filter capacitance. The inductances are not changed. Table III
shows the LCL filter parameter settings selected for the following analyses. Settings are classified into high, medium, and
low resonance frequency. The results can easily be transferred
to other LCL filter settings and switching frequencies.
From the previous basic study it is clear that proportional
filter capacitor current feedback and derivative filter capacitor
voltage feedback gives resonance damping. Its discrete realization on a microprocessor-based system gives some additional
challenges that are to be considered. Moreover, the overall
current control system needs to be studied as active resonance
damping directly influences the current dynamic. The model
used for system analysis is shown in Fig. 5. Taking the
sampling process into account leads to the analysis in discrete
z-domain. The continuous LCL filter model is discretized by
zero order hold [17]. Moreover parasitic series resistances of
+
PI
Symbol
Quantity
Value
UL
ω
UDC
P
fc
CDC
Line voltage (phase-to-phase, rms)
Line angular frequency
DC-link voltage
Nominal power (grid-side)
Switching/ control frequency
DC link capacitance
400 V
2 π 50 Hz
700 V
25 kW
2.5 / 5kHz
2710 μF
TABLE III
LCL FILTER SETTINGS AND CORRESPONDING RESONANCE FREQUENCIES
Setting
I
II
III
Cf /μF
(p.u.)
16.3
(3.25 %)
32.6
(6.5 %)
97.8
(19.5 %)
Lf g /mH (p.u.)
Rf g /mΩ (p.u.)
0.75 (3.7 %)
50 (0.79 %)
0.75 (3.7 %)
50 (0.79 %)
0.75 (3.7 %)
50 (0.79 %)
Lf c /mH (p.u.)
Rf c /mΩ (p.u.)
2.0 (9.9 %)
60 (0.95 %)
2.0 (9.9 %)
60 (0.95 %)
2.0 (9.9 %)
60 (0.95 %)
fRes /
kHz
1.69
(high)
1.19
(medium)
0.69
(low)
Lf c : Grid-side filter inductance, Rf g : Resistance of grid-side filter
inductor, Lf c : Converter-side filter inductance, Rf c : Resistance of
converter-side filter inductor, Cf : Filter capacitance
the inductors are taken into account but parallel resistances
modeling core losses are neglected. It is worth noting that in
practice they give some natural resonance damping [9]. The
coupling terms in (1) are neglected and the analysis is done for
one axis only. The fundamental PI controller in its backwards
implementation [17] and a one sample period delay due to
the computation and PWM are included. Discrete control
implementation of resonance damping K(z) is considered.
KiCf = 0
KiCf = 0
KiCf = -5
KiCf = 0
KiCf = -4
Fig. 6. ICf feedback: Impact of feedback gain KiCf on the pole zero map for LCL filters with different resonance frequencies: fRes = 1.69 kHz (left),
fRes = 1.19 kHz (middle), fRes = 0.69 kHz (right).
KuCf = 0
KuCf = 0
KuCf = - 1,5
KuCf = 0
KuCf = - 3
Fig. 7. UCf feedback: Impact of feedback gain KuCf on the pole zero map for LCL filters with different resonance frequencies: fRes = 1.69 kHz (left),
fRes = 1.19 kHz (middle), fRes = 0.69 kHz (right).
Step response, (low fRes)
*
From: iC
To: C
i
1.4
1.2
Amplitude
1
0.8
0.6
KuCf/Cf = −1.5 /F
0.4
0.2
0
0
Fig. 8.
ICf feedback: Impact of feedback
gain KiCf on the step response for LCL filters
with different resonance frequencies: fRes =
1.69 kHz (top) and fRes = 1.19 kHz (bottom).
Fig. 9. Impact of the discretization method on
the frequency characteristic of the approximated
differentiators in comparison to the continuous
one.
0.002
0.004
0.006
Time (sec)
0.008
0.01
Fig. 10. UCf feedback: Impact of feedback
gain KuCf on the step response for LCL filters
with different resonance frequencies: fRes =
1.19 kHz (top) and fRes = 0.69 kHz (bottom).
A. Filter capacitor current feedback
Proportional ICf feedback is suitable for resonance damping, that is: K(z) = KiCf . Fig. 6 shows pole zero maps
for high, medium, and low resonance frequencies. It can be
seen that resonance poles can be damped into the unity circle
with proper tuning of feedback gain KiCf only for high and
medium resonance frequency. With low resonance frequency
the interactions with the low frequency poles increase and
resonance damping becomes impossible. Fig. 8 show the
step responses for high and medium resonance frequency,
respectively. It can be seen that with proper tuning of KiCf
resonance damping is obtained.
Line currents
40
40
20
20
iLd/A
iCd/A
Converter currents
0
0
Reference Current
Sampled Current
−20
10
20
−20
30
40
40
20
20
iLq/A
iCq/A
0
0
10
0
10
0
10
20
−20
30
30
Sampled Current
t/ms
20
30
t/ms
Fig. 11. PI control without feedback of ICf and UCf : measured
step response of dq converter current (left) and line current (right) for
medium resonance frequency.
Line currents
40
40
20
20
iLd/A
iCd/A
Converter currents
0
0
Reference Current
Sampled Current
−20
10
20
−20
30
40
40
20
20
iLq/A
iCq/A
0
0
Sampled Current
0
10
0
10
20
30
0
Reference Current
Sampled Current
−20
0
10
20
−20
30
Sampled Current
t/ms
20
30
t/ms
Fig. 12. ICf feedback: measured step response of dq converter current
(left) and line current (right) for medium resonance frequency.
Line currents
40
40
20
20
iLd/A
iCd/A
Converter currents
0
0
Reference Current
Sampled Current
−20
0
10
20
−20
30
40
40
20
20
iLq/A
iCq/A
Basic considerations in chapter IV show that derivative
feedback is necessary for resonance damping. Basically, it
is similar to proportional feedback of the filter capacitor
current, as ICf (s) = s Cf UCf (s) holds. Theoretically, there
is no difference in continuous domain but due to the discrete
implementation of the differentiation there are differences even
from a system theoretical point of view.
There are different possible discretization methods [17]. The
frequency behavior of differentiators discretized with Forward
difference, Backward difference, Tustin approximation and a
continuous differentiator are shown in Fig. 9. Even though
Tustin gives a perfect match in the phase, the deviations in
magnitude close to the Nyquist frequency are high. Root loci
that are not shown in this paper indicate that the system gets
unstable in any case if the Tustin approximation is used. With
respect to the magnitude, forward and backward discretization
yield better matches than Tustin. However, deviations in the
phase occur, ±90◦ at the Nyquist frequency. As the forward
discretization gives an implicit control algorithm backwards
discretization is used here: K(z) = KuCf · Cf (1 − z −1 )/Tc .
Fig. 7 shows pole zero maps for different resonance frequencies. It can be seen that high resonance frequency poles cannot
be damped. This is due to the discrete implementation of
the differentiator. Fig. 9 shows that with increasing frequency
the characteristic of the backward differentiator changes from
differentiator to proportional gain, that is constant magnitude
and zero phase. As proportional UCf -feedback is not suitable
for resonance damping (see Fig. 4) no damping is obtained.
For medium resonance frequency damping works well. As the
interactions with low frequency poles becomes higher for low
resonance frequencies stability gets worse in that case. Fig.
10 shows the step responses for medium and low resonance
frequency. Proper tuning yields resonance damping. However,
for low resonance frequency the resonance oscillations decay
slower. Higher feedback gain KuCf cannot improve damping,
but instead poles are pushed outside again, as can be seen in
Fig. 7 (right).
Compared with ICf feedback the interactions of the resonance poles with the dominant low frequency pole pair are in
general higher.
20
0
Reference Current
Sampled Current
−20
B. Filter capacitor voltage feedback
Sampled Current
0
0
Sampled Current
0
10
0
10
20
30
0
Reference Current
Sampled Current
−20
0
10
20
t/ms
30
−20
Sampled Current
20
30
t/ms
Fig. 13. UCf feedback: measured step response of dq converter current
(left) and line current (right) for medium resonance frequency.
Line Current
40
40
40
30
30
30
0
0
L
L
20
i /%
50
10
20
10
1
2
3
4
5
6
0
0
7
20
10
1
2
3
4
5
6
0
0
7
4
kHz
50
40
40
30
30
30
10
C
C
20
i /%
50
20
10
2
3
Converter Current
40
1
2
kHz
Converter Current
50
0
0
1
kHz
Converter Current
i /%
iC / %
Line Current
50
i /%
iL / %
Line Current
50
3
4
5
6
7
0
0
5
6
7
5
6
7
20
10
1
2
3
kHz
4
5
6
7
0
0
1
2
kHz
3
4
kHz
Fig. 14. Measured steady state spectra of phase line currents (top) and converter currents (bottom) for medium resonance frequency without any additional
feedback (left), ICf feedback (middle), and UCf feedback (right) (sampling frequency: 100 kHz).
VI. M EASUREMENT R ESULTS
In order to validate the theoretical analysis measurements
have been carried out on a laboratory test set-up with a
self-built back-to-back converter feeding a 22-kW induction
motor. See Table II for most important system settings. The
control algorithm is implemented on a dSPACE DS 1006 board
and excecuted twice per switching period. Unless otherwise
stated, tests are carried out with motor speed of 1000 rpm.
The load torque generated by a DC machine is adjusted in
order to obtain a constant active power of 10 KW on the
grid side. Current reference steps are applied to the reactive
current component in order to test the dynamic behaviour. In
steady state reactive power of 10 kVA is supplied to the grid.
Harmonic compensators on the grid side as shown in [21] are
not used here.
Due to parasitic damping the LCL resonance poles are
naturally damped [9]. In order to clearly excite the LCL
resonance also in steady state the PI gain is slightly increased
by 20 %. The response of the PI controlled system without
feedback of any filter capacitor signal is shown in Fig. 11 and
the steady state current spectra are shown in Fig. 14 (left).
The currents are much distorted by the system resonance at
1.2 kHz. However, low frequency distortions due to the 5th
and 7th line voltage harmonics are also visible.
The results obtained with the resonance damping approaches are shown in Figs. 12 and 14 (middle) for ICf
feedback and in Figs. 13 and 14 (right) for UCf feedback,
respectively. A fast dynamic can be seen in the step responses.
There are more oscillations visible in the line side current and
there is coupling between the d and q axes. The resonance
is well damped, but the currents are still distorted by the
low frequency line voltage harmonics. Note that harmonic
compensators could reduce these distortions.
VII. C ONCLUSION
Basic study of resonance damping properties show that
either proportional feedback of the currents through the LCL
filter capacitors or derivative feedback of the voltages across
them is required. A detailed analysis with detailed discrete
models and with different system settings show limitations
in damping capability of these two approaches. For different
ratios of resonance frequency and control frequency their
behaviour is different. For high resonance frequencies only
current feedback can stabilize the system. For medium resonance frequencies both approaches show good performance.
For low resonance frequencies stability gets worse even though
voltage feedback offers slighly better damping properties. The
results in this paper are partly obtained from analyses with
several typical system settings. However, they can easily be
transferred to other particular settings with other parameters.
For future work it can be considered to achieve stability
of systems with even low LCL resonance frequencies by
feedback of both, current through and voltage across the filter
capacitors. Roughly speaking, the proportional voltage feedback is used for shifting the frequency into the frequency range
in which the current feedback achieves resonance damping.
But as the complete LCL filter state vector needs to be known
anyhow, state space control approaches can be applied. In
contrast to manual tuning of two damping parameters, pole
placement design offers a straight-forward design procedure
and the possibility of specifying desired closed-loop pole zero
locations. Estimation of certain system states could reduce the
sensor effort. For example the derivation of the filter capacitor
voltage is a simple way of filter capacitor current estimation.
ACKNOWLEDGMENT
This work has been funded by German Research Foundation
(DFG) and industry.
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