Aalborg Universitet
Resonance Damping and Parameter Design Method for LCL-LC Filter Interfaced Grid-
Connected Photovoltaic Inverters
Li, Zipeng; Jiang, Aiting; Shen, Pan; Han, Yang; Guerrero, Josep M.
Published in:
Proceedings of 8th IEEE International Power Electronics and Motion Control Conference (IPEMC-ECCE Asia),
2016
DOI (link to publication from Publisher):
10.1109/IPEMC.2016.7512528
Publication date:
2016
Document Version
Early version, also known as pre-print
Link to publication from Aalborg University
Citation for published version (APA):
Li, Z., Jiang, A., Shen, P., Han, Y., & Guerrero, J. M. (2016). Resonance Damping and Parameter Design
Method for LCL-LC Filter Interfaced Grid-Connected Photovoltaic Inverters. In Proceedings of 8th IEEE
International Power Electronics and Motion Control Conference (IPEMC-ECCE Asia), 2016 IEEE Press.
https://doi.org/10.1109/IPEMC.2016.7512528
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners
and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
- Users may download and print one copy of any publication from the public portal for the purpose of private study or research.
- You may not further distribute the material or use it for any profit-making activity or commercial gain
- You may freely distribute the URL identifying the publication in the public portal -
Take down policy
If you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access to
the work immediately and investigate your claim.
Resonance Damping and Parameter Design Method
for LCL-LC Filter Interfaced Grid-Connected
Photovoltaic Inverters
Zipeng Li, Aiting Jiang, Pan Shen, Yang Han
Dept. of Power Electronics, School of Mechatronics Eng.,
University of Electronic Science and Technology of China
Chengdu 611731, China
E-mail: hanyang@uestc.edu.cn
Josep M. Guerrero
Department of Energy Technology
Aalborg University
Aalborg 9220, Denmark
E-mail: joz@et.aau.dk
Abstract—In order to attenuate PWM harmonics effectively
and reduce filter cost and volume, LCL-LC filter is proposed
using a combination of LCL filter and an LC series resonant
part. Compared with LCL filter, LCL-LC filter is characterized
with decreased total inductance and better switch-frequency
harmonics attenuation ability, but the resonant problem affects
the system stability remarkably. In this paper, active damping
based on the capacitor voltage feedback is proposed using the
concept of the equivalent virtual impedance in parallel with the
capacitor. With the consideration of system delay, this paper
presents a systematic design method for the LCL-LC filtered
grid-connected photovoltaic (PV) system. With this method,
controller parameters and the active damping feedback
coefficient are easily obtained by specifying the system stability
and dynamic performance indices, and it is more convenient to
optimize the system performance according to the predefined
satisfactory region. Finally, the simulation results are presented
to validate the proposed design method and control scheme.
Keywords—active damping; grid-connected photovoltaic (PV)
converter; LCL-LC filter; proportional resonant (PR) controller;
I.
I
NTRODUCTION
The increasing global energy consumption has greatly
accelerated the demand for renewable energy, such as wind
and solar energy, and the grid connected photovoltaic (PV)
power generation is growing with the tendency of increased
scale and capacity [1-3]. As far as grid-connected photovoltaic
(PV) inverters are concerned, the stability analysis and
dynamic interaction between PV plants and distribution
system are attracting a lot of attention. When the two-level
inverters are used, the output voltages are polluted with
considerable harmonic components, thus the injected grid
currents may contain high-frequency harmonics near the
switching frequency caused by the pulse-width modulation
(PWM) process. Therefore, a low-pass filter (typical LCL
filter) must be installed between each PV inverter and the
unity for attenuating the high frequency harmonics injected
into the point of common coupling (PCC) [4, 5].
The filter can be of different types [6]. Compared with the
first order L filter, the third-order LCL filter can meet the grid
code requirements with smaller size and cost, especially for
applications above several kilowatts. In [7], a new filter named
LCL-LC filter is proposed, which is a combination of an LCL
and an LC series resonant circuits and has the ability of
attenuating switching frequency harmonics with an attenuation
rate of -60 dB/decade. However, LCL-LC filter suffers from
the resonance problems, two resonant peaks are introduced by
LCL-LC filter which may cause system instability. Generally,
passive damping [5] or active damping [8, 9] methods are
adopted for damping the resonance of output filter in PV
inverter. Passive damping can be realized by adding passive
components in the system [10], a direct way to damp the
resonance of the LCL-LC filter is to place a damping resistor
in series with the filter capacitor [7]. In this way, two
resonance peaks are effectively damped, the harmonics
attenuation ability in the range of switching frequency is not
weaken, but the physical resistors will inevitably lead to
higher losses and decreases system efficiency [11].
Therefore, the active damping method is a preferred
solution for the residential PV inverters with a higher
reliability and reduced power loss. In this paper, a systematic
parameters design method for LCL-LC filtered grid-connected
photovoltaic (PV) system using capacitor voltage feedback
scheme is proposed. By using this method, the controller
parameters and active damping feedback coefficient can be
easily obtained by selecting the reasonable range of system
stability and dynamic performance indices, hence it is
convenient to optimize the system performance according to
the satisfactory region. Further, the active damping scheme for
the LCL-LC grid-connected PV system based on the capacitor
voltage feedback is found to have a strong robustness against
the grid impedance variation and the grid voltage fluctuation
conditions.
This paper is organized as follows. In section II, the system
model and the passive damping methods for LCL-LC filter are
presented. On this basis, the active damping method based on
capacitor current feedback of the LCL-LC is presented in
section III. In section IV, a systematic controller design
method for LCL-LC filtered grid-connected photovoltaic (PV)
978-1-5090-1210-7/16/$31.00 ©2016 IEEE
This work was supported in part by the National Natural Science Foundation
of China under Grant 51307015, and in part by the Open Research Subject o
f
Sichuan Province Key Lab of Power Electronics Energy-Saving Technologies
& Equipment under Grant szjj2015-067, and in part by the Open Research
Subject of Artificial Intelligence Key Laboratory of Sichuan Province unde
r
Grant 2015RZJ02, and in part by the Fundamental Research Funds of Central
Universities of China under Grant ZYGX2015J087.
2016 IEEE 8th International Power Electronics and Motion Control Conference (IPEMC-ECCE Asia)
978-1-5090-1210-7/16/$31.00 ©2016 IEEE
system is proposed, the design example is also given. In
Section V, the simulation results validate the proposed control
scheme and the parameter design method. Finally, Section VI
concludes this paper.
II. S
YSTEM
C
ONFIGURATION AND
P
ASSIVE
D
AMPING
M
ETHODS
FOR THE
LCL-LC
F
ILTER
-
BASED
PV
I
NVERTERS
Fig. 1(a) shows the main circuit of a two-stage LCL-LC
filtered grid-connected PV system. The boost DC-DC
connected to the PV panels step up the voltage of the DC bus
to a proper level for the PV inverter. The H-bridge DC-AC
inverter produces PWM sinusoidal current injected into the
grid based on the Maximum Power Point Tracking (MPPT)
algorithms [12]. The voltages at the PCC are sampled for
Phase Locked Loop (PLL), the injected grid current i
g
is
sensed for current control and the capacitor current u
c
is
sensed for active damping.
And the corresponding current control circuit is shown in
Fig. 1(b), K
pwm
is the transfer function of PWM inverter, which
is defined as V
dc
/V
tri
, where V
dc
is amplitude of the input dc
voltage and V
tri
is the amplitude of the triangular carrier [13].
G
d
(s) represents the system delay, G
c
(s) is the transfer function
of the outer loop controller. The proportional resonant (PR)
regulator is introduced to eliminate the static error at the
fundamental frequency. To simplify the analysis, the PR
controller at fundamental frequency is given by:
22
0
2
()
2
c
cpr
c
s
Gs K K
ss
ω
ωω
=+
++
(1)
where parameters K
p
and K
r
are the proportional and resonant
gain, respectively. ω
0
is the fundamental angular frequency
and ω
c
is the bandwidth at -3 dB cutoff frequency. Generally,
K
p
determines the dynamic performance of the controller and
affect the stability of the system, while K
r
determines the gain
at the selected frequency and controls the bandwidth around it.
Fig. 1. Topology of a two-stage LCL-LC filtered grid-connected PV
system (a) Main circuit (b) Control circuit
Fig. 2. Schematic diagrams of LCL-LC filter.
The LCL-LC filter is composed of a traditional LCL filter
and a LC series resonant circuit as shown in Fig. 2. The
transfer function from inverter output voltage u
i
to injected
grid current i
g
is given as
2
53
12 1 2 1 2
1
()
()()
rr
LCL LC
rr f
sLC
Gs
s
LL LCC s D D s L L
−
+
=
++++
(2)
where L
1
is the converter-side inductor, L
2
is the grid-side
inductor. C
f
is the filter capacitor, L
r
is the resonant inductor
and C
r
is the resonant capacitor, and the terms in the
denominator are defined as:
112
()
rf
D
LL C C=+
,
212
()
rr
D
LC L L=+
(3)
The resonant frequency of the L
f
-C
f
series resonant circuit is
at the switching frequency f
sw
, thus the value of L
f
is much
smaller than that of L
1
or L
2
. The main idea of designing the
LCL-LC filter is to decompose the LCL-LC filter into a
traditional LCL part and a LC series branch part. The
parameters of the three-phase inverter system and the designed
LCL-LC filter parameters are listed in Table I.
The f
r1
and f
r2
are the frequency where the two resonant
peaks exist and they can be denoted as:
TABLE
I
P
ARAMETERS OF
T
HE
T
HREE
-P
HASE
I
NVERTER
Parameter Symbol Va lu es
Line Voltage E 110 Vrms
Rated Power P 2.2 kW
Grid frequency f
0
50 Hz
Switching frequency f
s
w
10 kHz
Sampling frequency f
s
10 kHz
Modulation index m 0.9
Dc-link voltage U
dc
650 V
Inverter-side inductor L
1
0.4 mH
Grid-side inductor L
2
0.2 mH
Capacitor C
f
12 uF
LC circuit inductor L
r
21 uH
LC circuit capacitor C
r
8 uF
First resonant frequency f
r
1
3.08 kHz
Second resonant frequency f
r
2
13 kHz
Grid-side inductor L
2
0.2 mH
12 345
1
6
12 345
2
6
r
r
D
DDDD
f
D
D
DDDD
f
D
+− ++
=
++ ++
=
(4)
where the parameters D
3
~D
6
are denoted as:
22 2
312
22 2
412
512 12
612
()
()
2()()
2
rf
rr
rr r f
rf
DLLCC
DLCLL
DLLLCLLCC
DLLCC
=+
=+
=+−
=
(5)
To damp the resonance, the simplest way is to place a
passive resistor in series or parallel with the inductor or
capacitor of the LCL-LC filter. Generally speaking, when the
damping resistor is in series with L
1
or L
2
, the resonance at f
r1
can be fully attenuated, but the resonance at f
r2
cannot be fully
eliminated, the damping effect of the resonance at f
r2
increases
with a larger damping resistor, but the gain at low frequency is
also reduced.
2016 IEEE 8th International Power Electronics and Motion Control Conference (IPEMC-ECCE Asia)
III.
A
CTIVGE
D
AMPING
M
ETHOD FOR
LCL-LC
Compared with passive damping methods, active damping
strategy is more flexible and efficient, which adopts virtual
resistor to eliminate power loss and can be easily incorporated
into the existing control algorithm. Similar to traditional LCL
filter, there are various feedback variables feasible for
damping the resonance and stabilize the system, i.e., LC series
branch voltage u
c
, capacitor current i
c1
and i
c2
, and inverter
side current i
1
. And the current control diagrams are shown as
Fig. 3.
Generally, the digital controlled system, contains the
computation delay, sampler continuous approximation, and
PWM delay. In the continuous domain, one-sample
computation delay is expressed as
()
s
s
T
d
Gs e
−⋅
=
(6)
The zero-order hold (ZOH) introduces an extra delay of
0.5T
s
, which results in a total delay of 1.5T
s
or G
d
(s)=e
-sλTs
with λ= 1.5.
When the delay is considered, frequency f
R
of capacitor
current or LC series branch current feedback is smaller than
that of capacitor voltage feedback. Therefore, in terms of
closed-loop bandwidth, the proportional capacitor voltage
feedback is preferred. However, a negative proportional gain
K
uc
is needed to stabilize the system.
Fig.3. Current control using active damping method with (a) capacitor voltage
u
c
, (b) capacitor current i
c1
and (c) LC circuit current i
c2
feedback
IV.
C
ONTROLLER
D
ESIGN
M
ETHOD FOR
LCL-LC
F
ILTERED
G
RID
-C
ONNECTED
I
NVERTER
The control block diagrams of LCL-LC type grid-connected
PV system with capacitor voltage feedback is shown in Fig.
1(a). To simplify the analysis, the grid current is expressed as
12
()
()
(s) ( ) ( )
1() 1()
() ()
b
L
gref g
LL
gg
Gs
Gs
iisvs
Gs Gs
is i s
=−
++
=+
(7)
where the loop gain G
L
(s) can be expressed as
2
53
12 7 8
() () (1 )
(s)
cd PWM rr
L
rr f
GsGsK LCs
G
s
LLLCC s D sD
+
=
++
(8)
And the parameters in the denominator of (8) and G
b
(s) in (7)
can be expressed as
712 2
8212
11
21 1 11
()
()
()
()
rr ucd PWM
uc d PWM
L
b
LL L
DDDLCLKGsK
DKGsK LLL
ZZ
Gs
ZZ Z ZZ
=++
=++
+
=
++
(9)
where
1
1
()
()()
()
()
cf uc cr Lr
cf cr Lr uc cf cr Lr
uc
uc d PWM
ZZ Z Z
Z
Z
ZZZZZZ
sL
Zs
KG sK
+
=
++ + +
=
(10)
The influence of the filter capacitor can be ignored when
calculating the magnitude of the loop gain lower or equal to f
c
.
Thus G
L
(s) and G
2
(s) can be approximated as:
212
() ()
(s) ( )
(() )
cd PWM
L
uc d PWM
GsG sK
GTs
s
KG sK L L L
≈=
++
(11)
()
2
111
2
322 2
21 12 12 1
() ()
(s) ( )
(()() )
uc d PWM
b
uc d PWM
sL K s G s K sL sL
GTs
s
LL s K sG sK LL LL L
++
≈=
+++
(12)
The steady-state error of the grid-connected inverter
includes the amplitude error E
A
and the phase error δ. The PR
controller can eliminate the static error at the fundamental
frequency, thus the phase error δ can be minimized.
Considering T(s)
≫
1 and |G
d
(j2πf
0
)|≈1, the amplitude of the
grid current i
g
at the fundamental frequency f
0
can be
expressed as
20
00
0
(2 )
(2 ) (2 ) ()
(2 )
gref g
Tj f
ij f i j f vs
Tj f
π
ππ
π
≈−
(13)
and the amplitude error E
A
can be expressed as
0
0
(2 )
(1)
1
(2 )
gref
guc
A
ref c ref PWM
ij f i
VK
E
iGjfIK
π
π
−
+
=≈⋅
(14)
where I
ref
and V
g
are the root mean square (RMS) values.
At the fundamental frequency f
o
, the PR controller can be
approximated as:
0
(2 )
cpr
Gj f K K
π
≈+
(15)
Substituting (15) into (14) yields
_
(1)
guc
rEA p
Aref PWM
VK
K
K
EI K
+
=−
(16)
where K
r_EA
is the critical value of the integral gain K
r
constrained by E
A
and K
uc
. To ensure the system dynamic
performance, the crossover frequency f
c
is always set to be a
value greater than 10 times over the fundamental frequency f
0
.
The magnitude of the PR controller can be approximated to K
p
at the frequencies equal or higher than f
c
, the magnitude
frequency response of the system is zero at f
c
. Therefore,
considering G
c
(j2πf
c
) ≈ K
p
and |T(j2πf
c
)|
=
1, the proportional
gain K
p
can be written as
212
2( )
cucPWM
P
PWM
f
KK L L L
K
K
π
++
≈
(17)
2016 IEEE 8th International Power Electronics and Motion Control Conference (IPEMC-ECCE Asia)
From (17), it can be inferred that K
p
is proportional to f
c
,
and a larger K
p
ensures a faster dynamic response. When the
expected crossover frequency f
c
and K
uc
are selected, the
proportional gain K
P
can be determined.
As for LCL-LC filter, the phase curve crosses over -180°
only at the second resonance frequency f
r2
. Therefore, the gain
at f
r2
needs to be adjusted below 0 dB to ensure GM >0, and
the resonant peak at f
r2
needs to be effectively damped by
tuning the active damping coefficient. Besides, the magnitude
curve crosses over 0 dB at the cross over frequency f
c
, the
phase margin PM should also be higher than 0dB to ensure
system stability.
As discussed earlier, the gain margin GM can be expressed
as
2
GM 20lg ( 2 )
Lr
Gj f
π
=−
(18)
At the second resonance frequency f
r2
, the influence of the
filter capacitor is not negligible, considering G
c
(j2πf
r2
) ≈ K
p
,
substituting G
L
(s) into (18), the boundaries of K
uc
constrained
by GM can be derived as (19), as shown at the bottom of this
page.
The PM of the system can be expressed as
o
2
PM 180 ( )
c
L
s
jf
Gs
π
=
=+∠
(20)
To calculate the phase margin, using G
c
(j2πf
c
)
≈
K
p
+2K
r
is
more accurate. Substituting s = j2πf
c
into G
L
(s) in (20), the
PM can be rewritten as
3
0
2
4
PM 180 arctan
4
PPrr
rc rcrr
K
KLC
K
KLC
ππ
ωπω
−
=− +
−
(21)
According to (21), K
r
can be expressed as
3
o2
4
tan(PM+180 )( 4 )
PPrr
r
ccrr
KKLC
K
LC
ππ
ωπω
−
=
−
(22)
Substituting (16) and (17) into (22), the proportional
damper K
uc
constrained by PM and E
A
can be derived as
12
_PM
2
() 2 ( )
(2())
EA PWM d g PM EA c
uc
gPM EA cPWM d
GK Gs VG G fL L
K
VG G fK G sL
π
π
−+ +
=
−
(23)
where the G
PM
and G
EA
is related to the specified phase
margin PM and the amplitude error E
A
, respectively, which is
shown in (24)
02
3
tan(PM+180 )( 4 )
(4 )
P
Mccrr
EA A ref r r
GLC
GEI LC
ωπω
ππ
=−
=−
(24)
From the above analysis, it can be seen that if the upper and
lower boundaries of GM and PM, and E
A
are specified, the
satisfactory region constrained by K
uc
and f
c
can be obtained
from (19) and (23). From Fig. 4, the crossover frequency f
c
and the proportional damper K
uc
can be easily selected. Then,
with the possible value of f
c
and K
uc
, the proportional gain K
P
can be calculated from (17), the relationship between K
p
, K
uc
and f
c
is plotted in Fig. 5. Generally, a larger f
c
is always
selected to ensure good dynamic response, thus the area for
satisfactory range of K
p
are depicted in the shaded area in Fig.
5. Finally, with the obtained K
p
and K
uc
, the integral gain K
r
can be calculated from (16), and the relationship between K
r
,
K
uc
and K
p
is plotted in Fig. 6. As can be seen, a bigger K
r
is
required to reduce the amplitude error E
A
, and the area for
satisfactory range of K
r
is depicted in the shaded area with
better dynamic and less steady-state error.
To verify the design method, an example is given. If the
upper and lower boundaries of PM and GM is chosen as 30° ≤
PM ≤ 60° and 3 dB ≤ GM ≤ 12 dB, respectively, which ensure
good dynamic performance and stability margin. Besides, the
magnitude error E
A
is expected to be less than 0.4%. Based on
(19) and (23), the satisfactory region constrained by K
uc
and f
c
can be obtained. From Fig. 4, point A is selected with f
c
=
2150 Hz and K
uc
=-0.28. With (17), the calculated
proportional gain K
p
is 0.49. Substituting K
uc
, K
p
and E
A
into
(16), K
r
=40 can be obtained. Considering the previous steps,
the designed parameters can be defined as:
0.49, 40, 0.28
pruc
KKK===−
(25)
Based on the designed parameters, the system bode diagram
with compensation is shown in Fig 7. The gain margin (GM)
is 5.21 dB, the phase margin (PM) is 43° and the crossover
frequency f
c
is 2.15 kHz. Therefore, both the dynamic
performance and stability of the LCL-LC filter interfaced PV
system are achieved.
Fig. 4. Region of f
c
and K
uc
constrained by GM PM and E
A
.
Fig. 5. Relationship between K
p
, K
uc
and f
c
.
GM
55 22 33
20
212 122 212212
_GM
GM
22 33
20
22 2 22 2
32 2 ( )(4 1)10 8 ( ) 2 ( )
2()(14)108 ()2 ()
rfrr c rrr r r
uc
c PWM d r r r r r r PWM d r PWM d
f
CCLLL f L L f LC f G G f L L
K
f
KGsL fLC fLCKGsL fKGsL
πππ ππ
πππ π
++ −− ++ +
=
−+ −
(19)
2016 IEEE 8th International Power Electronics and Motion Control Conference (IPEMC-ECCE Asia)
