LCL-Filter Design for Grid-Connected Three-Phase
Inverter Using Space Vector PWM
SeungGyu Seo, Yongsoo Cho, and Kyo-Beum Lee
Department of Electrical and Computer Engineering
Ajou University
Suwon, Korea
handsome1705@ajou.ac.kr, marine_blue@ajou.ac.kr, kyl@ajou.ac.kr
Abstract—This paper proposes a LCL-filter design for gridconnected three-phase inverter using space vector pulse width
modulation (SVM). There is a variety of studies in progress.
However, the existing methods have an error because they are
not applicable in SVM. This paper presents the design method of
LCL-filter that optimized for SVM switching operation. The
LCL-filter design procedure performed step by step. Inverterside inductor was obtained by mathematical analysis of the ripple
components of the grid current according to the switching state.
Filter capacitor was chosen by consideration of the reactive
power absorption ratio. Grid-side inductor was designed by
ripple attenuation factor of the output current. The effectiveness
of the described design method of LCL-filter is verified by
simulation.
Keywords—LCL filter; 2-level inverter; Space vector pulse
width modulation (SVM)
I.
INTRODUCTION
Recently, there is growing interest in renewable energy
such as solar power and wind power in order to replace the
fossil energy and nuclear energy. These renewable energy
systems are used grid-connected inverter such as Fig. 1.
However, output currents of the inverter have not only the
fundamental wave but also an integral multiple of the
switching frequency [1]. Thus harmonic components are
limited by standards such as IEEE-519 and IEEE-1547 [2]-[3].
In order to satisfy the standard, L-filter has been used in gridconnected system. However, if the system capacity increases,
the capacity of L-filter is increased to reduce the harmonic
component. Increase of capacity leads to rise a size and price of
the filter. In addition, the large inductance is causing the
problem of reduction of the dynamic response speed and the
voltage drop.
On the other hand, LCL-filter can reduce the filter cost and
volume. Because the harmonic reduction effects of LCL-filter
is better than L-filter. Although the LCL-filter have these
advantages, the LCL-filter has a drawback that design process
is complicated [4]-[5]. Furthermore, if resonance occurs in the
output current due to the bad filter design, it is necessary to
compensate the resonance such as using an active damping [6].
Papers related to the LCL-filter design are focused on the
influence of the additional LC parameters [7]-[8]. However,
additional LC parameters are determined by the inverter-side
Fig. 1: Grid-connected three-phase inverter system
inductor L1. To accurately design the inverter-side inductor is
very important. In addition, a switching method of system is
important for design filter. The current ripple that decides an
inverter-side inductor is influenced by switching method. There
are many switching techniques for controlling a three-phase
grid-connected inverter, such as sinusoidal pulse width
modulation (SPWM), discontinuous pulse width modulation
(DPWM), and space vector pulse width modulation (SVM).
Therefore, optimized design method of LCL-filter is necessary
in accordance with a switching method.
This paper proposes the design method of LCL-filter
optimized SVM switching operation. The SVM method is
implemented using the offset injection method. It is possible to
design the inverter-side inductor through the analysis of output
currents, in the case of using only the L-filter. The grid-side
inductor can be designed through a current ripple attenuation
rate. Current ripple attenuation is defined as a ratio of the
inverter-side’s output current THD and the grid-side output
current THD. The filter capacitor is designed suitably in
consideration of the absorption of reactive power. In order to
confirm the performance of the designed LCL-filter, the
proposed filter design method is verified by the obtained results
through simulations.
II.
LCL-FILTER DESIGN
A. Three-phase Inverter System of SVM Method
The SVM method is the most widely used method for
switching a three-phase inverter [9]. In addition, the SVM
978-1-5090-1210-7/16/$31.00 ©2016 IEEE
Vdc
2
−
*
van
*
vbn
*
vcn
Ts
Vdc
2
vas*
Vdc
2
vbs*
vcs*
Sa
0
1
1
1
Sb
0
0
1
1
Sc
0
0
0
1
1
Ta
1
Tb
1
Tc
1
1
0
1
0
0
0
0
0
Fig. 3: effective time of the tree-phase pole voltage
voffset
−
Vdc
2
ωt
0
π
3
2π
3
π
4π
3
5π
3
2π
Fig. 2: Offset and reference voltage of SVM method
method is easily implemented through the offset injection
method. The reference phase voltage can be expressed as
vas* = M i
vbs* = M i
vcs* = M i
Vdc
3
Vdc
3
Vdc
3
sin(ωt )
sin(ωt − 2π / 3) .
(1)
sin(ωt + 2π / 3)
where v*as, v*bs, and v*cs are the reference phase voltages, Mi is
the modulation index and Vdc is the dc link voltage.
The offset voltage makes the absolute value of max
reference voltage and the minimum reference voltage equally.
As a result, injected offset voltage method can operate at SVM.
The injected offset voltage that allows the SVM operation can
be defined as
voffset
 M iVdc

 2 3
 M iVdc

2 3
=
 M iVdc

 2 3
 M iVdc

 2 3
2π
2π
π 
) + sin(ωt +
)), 0 ≤ ω t ≤
3
3
6 


2π π
π
(sin(ωt ) + sin(ωt − )), ≤ ω t ≤

3
6
2
 .(2)

π
2π
5π
(sin(ωt +
) + sin(ωt )), ≤ ω t ≤

3
2
6


2π
2π 5π
(sin(ωt − ) + sin(ωt +
)),
≤ωt ≤π
3
3
6

(sin(ωt −
*
*
*
, vbn
, and vcn
obtained
Finally, the reference pole voltage van
as follows :
Fig. 4: Three-phase inverter output phase voltage
*
van
= vas* + voffset
*
vbn
= vbs* + voffset
(3)
*
vcn
= vcs* + voffset
As shown in Fig. 2, the reference pole voltage is made by
the sum of reference phase voltage and offset voltage. For that
reason, a modulation index range extended.
B. Inverter-side Inductor Design
The design method of the inverter-side filter inductor Li can
be determined through a current ripple factor. The current
ripple factor RF is able to be expressed by the ratio of the rated
current in the system Irate and the current ripple Iripple. RF is
expressed as
RF =
I ripple
I rate
.
(4)
It is possible to obtain an inductance value of the filter by
using the equation (4). The ripple current flowing in Li is
determined on the basis of the level and the effective time of
the inverter output phase voltages. The effective time is
determined according to the switching operation. The effective
time can be confirmed by Fig. 3.
ea
ea
Fig. 6: Current ripple in the sector II, III
Fig. 5: Current ripple in the sector I, IV
Fig. 3 shows the effective time of each phase switches in a
one-period. The switching operation is determined through the
comparison of the magnitude between triangular carrier wave
and reference pole voltage. For that reason, effective time can
be calculated as
Ta =
Ts
2
*
 van

+ 0.5 

V
 dc

ea (ω t ) = M i
*

Ts  vbn
+ 0.5 

2  Vdc

*

T v
Tc = s  cn + 0.5  .
2  Vdc

(5)
Tb =
Ta, Tb, Tc is effective time of each phase switch and Ts is
control period.
The output phase voltage can be divided into four regions
in the half cycle. Fig. 4 shows that the waveform of the output
TABLE I.
Region
EFFECTIVE TIME OF EACH PHASE OUTPUT VOLTAGE
Output voltage
phase voltage and the average voltage are divided into four
regions. The voltage in inductor is represented by the
difference between the inverter output phase voltage and the
grid phase voltage. The grid phase voltage ea can be obtained
as follows:
Vdc
3
sin(ωt ) , 0 < ω t < π
(6)
As shown in Fig. 4, the output phase voltage of the inverter
has a five voltage level. The applied voltage in inductor varies
depending on the time. Therefore, it is possible to obtain the
current ripple by using applied voltage at inductor and effective
time.
The inductor current is changed according to the output
phase voltage. In addition, its magnitude is determined by
output phase voltage waveform. The output phase voltage is
divided in four regions. These four regions are symmetric each
other. Thus, it is possible to determine the RMS value of the
current ripple in the region I and II. The output phase voltages
(-2/3 Vdc, -1/3 Vdc, 0 Vdc, 1/3 Vdc, and 2/3 Vdc) have influence on
the effective time. The determined effective time is shown in
Table. 1.
Effective time
0Vdc
T0
Tb
I
1/3Vdc
T1
Ta−Tb
II
−1/3Vdc
0Vdc
2/3Vdc
T2
T0
T1
Tc−Ta
Tb
Ta−Tc
III
1/3Vdc
0Vdc
2/3Vdc
T2
T0
T1
Tc−Tb
Tc
Ta−Tb
IV
1/3Vdc
0Vdc
1/3Vdc
T2
T0
T1
Tb−Tc
Tc
Ta−Tc
−1/3Vdc
T2
Tb−Ta
Fig. 5 and 6 show the shape of the current ripple in
response to the inverter output phase voltage of each region. In
the half period of the region I and IV, the maximum value of
the current ripple is determined by inserting 0 Vdc during the T0,
(1/3) Vdc during the T1, and (−1/3)Vdc during the T2. The level
of the inverter output phase voltage which determines the
maximum value of the current ripple is (1/3) Vdc. It can be
confirmed through Fig. 5.
The maximum value of the current ripple in the region II
and III is similar with the region I and IV. It is determined by
inserting 0 Vdc during T0, (2/3) Vdc during T1, and (1/3) Vdc
during T2. The level of the inverter output phase voltage which
determines the maximum value of the current ripple is (2/3) Vdc
and (1/3) Vdc. However, (1/3) Vdc contribute to the maximum
value of the current ripple when it is greater than the ea. It can
be confirmed through Fig. 6.
Therefore, the maximum value of the current ripple in each
region is can be defined as
1 1
1

Δimax1 (ωt ) =  Vdc − ea (ω t )  ⋅ (Ta − Tb ) − ea (ωt ) ⋅ Tb
L3
L

Δimax 2 (ωt ) =
+
Inverter side current
2ms / div
(a)
12
1

 Vdc − ea (ω t )  ⋅ (Ta − Tc ) − ea (ωt ) ⋅ Tb (7)
L3
L

Current ripple
2ms / div
2
1   1
 1

ω
−
+  Vdc − ea (ω t )   ⋅ (Tc − Tb ) .
V
e
t
(
)
dc
a


2L   3
 3
 

(b)
The maximum value of the determined current ripple has a
symmetrical structure. As shown in Fig. 7, it shows the current
ripple and zoomed current ripple. The zoomed current ripple
looks like a triangular wave of high frequency. Therefore, the
absolute value of the current ripple is required to determine the
RMS value of the current ripple which is approximated to
triangular waveform with high frequency. For that reason, it is
possible to calculate the RMS value of the current ripple when
integrate the RMS value of the triangular wave [10]. The RMS
value of the current ripple can be expressed as
I ripple =
π

2  π6 2
2
2
 0 Δimax1 (θ )dθ + π Δimax 2 (θ )dθ  .
3π 
6

5µ s / div
(c)
Fig. 7: Output current waveform using a L-filter in one period: (a)
inverter-side current, (b) current ripple, (c) zoomed current ripple
I rate =
(8)
Li =
1.719 ⋅ Ts ⋅ Vdc
L( Mi )
.
⋅
4
π3
Li ⋅10
(9)
3 2Zb
=
M iVdc
.
2 6 f n Lb
(10)
The Li can be obtained from (4), (9), and (10). The Li can
be indicated as
Substituting (6) and (7) into (8), the following equation is
are derived:
I ripple =
M iVdc
3.43788 6 ⋅ Ts ⋅ f n ⋅ Lb
L( Mi )
⋅
.
π
RF ⋅ Mi ⋅104
(11)
The modulation index Mi and base inductance Lb are
expressed as
Equation (7) is expressed by the Li, Vdc, Ts, Mi, and L(Mi).

3
4
2
 −23.13Vdc ⋅ Mi + 8.15Vdc ⋅ Mi + 17.56Vdc ⋅ Mi 


2
2
2
 − 0.784 Mi Vdc ( 4 − 6.9282 Mi + 3 Mi )



2
2
 − 0.738Mi Vdc ( 2 − 6.9282 Mi + 6 Mi )



 + 0.1995Mi Vdc 2 ( 4 − 6.9282 Mi + 3 Mi 2 )





2
2
2
where: L( Mi) =  + 4.136Mi Vdc ( 2 − 6.9282 Mi + 6 Mi )
.


 − 2.214 Mi 3 Vdc 2 ( 2 − 6.9282 Mi + 6 Mi 2 )



 + 0.2877 Mi 3 Vdc 2 4 − 6.9282 Mi + 3 Mi 2

(
)




2
2
 + 0.372 Vdc ( 4 − 6.9282 Mi + 3 Mi )



 − 0.526 Vdc 2 ( 4 − 6.9282 Mi + 3 Mi 2 )



The rated current of the system is determined by
2 En
.
Vdc
(12)
En2
.
2π f n Pn
(13)
Mi =
Lb =
Table. 2: System parameters
Parameter
Rated output power
Grid line-to-line voltage
DC-link voltage
Grid frequency
Switching frequency
Target current THDi
Target current THDg
Li
Cf
Lg
Value
3 kW
220 Vrms
400 V
60 Hz
7.8 kHz
12 %
2.6 %
1.4 mH
4.4 μF
709 μH
Set a initial
parameter
Calculate a base value
Select a design goal
Inductor-side current THD
Grid-side current THD
Design a Li
Design a Lg
Design a Cf
fn < fres< ½ fsw
Check a filter
performance
Fig. 9: Current waveform of the LCL-filter in simulation: (a)
inverter-side current, (b) grid-side current, (c) FFT analysis of
inverter-side current, and (d) FFT analysis of grid-side current
Design LCLfilter
E. Resonant Frequency
The resonance frequency fres is determined by the designed
filter parameters. It can be expressed by
Fig. 8: LCL-filter design diagram
C. Filter Capacitor Design
The based capacitance Cb at rating condition and the filter
capacitance Cf can be defined as
Cb =
1
.
2π f n Z b
C f = xCb .
(14)
(15)
where En is the grid line voltage, Pn is the three-phase power
and fn is the line frequency.
When the absorption rate of reactive power is too large,
because of increasing the current passing the inverter-side
inductor, loss is increased. On the other hand, when absorption
rate of reactive power is too small, additional inductor is
required in inverter-side inductor and grid-side inductor.
Therefore, it is general to set an appropriate value within 5 %
[11].
D. Grid-side Inductor Design
The Lg is determined between the basic current ripple
reduction ratio r and Li. The value of Lg is determined as
Lg = rLi .
(16)
The r can be determined by using through the relationship
between the inverter-side current ripple ii and the grid-side
current ig ripple [11].
f res =
1
2π
Li + Lg
Li Lg C f
.
(17)
The LCL-filter has a resonant component, unlike the L-filter.
In order to avoid resonance in the resonance frequency band,
the resonance frequency is limited between 10 times of the
line frequency and 1/2 of the switching frequency fsw [12].
F. Systematic Filter Design
Fig. 8 is the algorithm for designing the LCL-filter. The
first step is to set an initial parameter such as Pn, fn, fsw, Vdc,
and ea. In the next step, calculation of a base impedance, base
inductance, and base capacitance is conducted. The LCL-filter
parameter is obtained by using (11), (15) and (16) with
selected target THD. After that, the resonance frequency is
confirmed. When the resonance frequency has a problem in
this procedure, system repeat the previous step. Lastly, system
check a filter performance.
III. SIMULATION RESULTS
The performance of the designed LCL-filter verified
through PSIM simulation. Table. 2 shows the system
parameters. In order to design the target THD of the inverterside current to 12%, the inverter-side filter inductor is
determined 1.4 mH by equation (11). The filter capacitance can
be designed to 4.4 μF when setting the reactive power
absorption rate to 2.67 %. In order to achieve the THD 2.6 %,
the grid-side inductor can be designed to 709 μH.
As shown in Fig. 9(a), the THD of ii is confirmed to 12.18
%. Finally, the output current of the designed LCL-filter ig is
described in Fig. 9(b) as THD of ig is 2.41 %. Therefore, the
proposed design method is confirmed the accuracy with an
error 0.19 %. Fig. 9(c) and (d) show the FFT analysis
waveform of the current in each part. Harmonics are generated
at integer multiples of the switching frequency. Additionally, ig
was confirmed that the harmonics is substantially suppressed.
IV. CONCLUSION
This paper proposed the LCL-filter design for gridconnected three-phase inverter by using the SVM. Design
method of the inverter-side inductor is shown mathematically
in order to improve the accuracy of the filter design through the
analysis of the current ripple. The proposed LCL-filter design
method is verified by showing an improved result between
actual current THD and target current THD. The simulation
results demonstrate the effectiveness of the proposed design
method.
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