LCL filters are widely used in threephase PWM converter for its advantages of small volume, low cost and inhibition of high frequency current harmonic. However, it is difficult to optimize its design because its parameters are mutually influenced while the value of each parameter for LCL filter has impacts on the converter's cost and size. In this paper, the target of optimization is to minimize the parameter values of LCL filter, and an optimization method for parameters of LCL filter of threephase PWM converter based on least square method is proposed. With this method, a quantitative calculation of the harmonic component of the converter’s side phase voltage is performed first, and then the quantitative relationship between phase voltage harmonics and grid phase current harmonics is analyzed. After that, the attenuation requirement of each harmonic is obtained by taking into account the requirements for each harmonic component of grid current. Then according to the optimization objective, the objective function with minimum harmonic attenuation deviation is established, and least squares method is adopted for threedimensional global searching of parameters for LCL filter. Thus, the designed harmonic attenuation curve approximates the minimum attenuation requirements, and the optimized LCL filter parameters are obtained. Finally, the effectiveness of the method is verified by the experiments.
1. Introduction
Because threephase voltagetype PWM converter has the advantage of high power factor, good dynamic response, and bidirectional power flowing, etc. it is widely used in gridconnected power generation, motor drive, control of battery charge and discharge and other fields
[1

4]
. Meanwhile, the threephase PWM converter with pulse width modulation (PWM) generates high frequency switching harmonics, so it requires highfrequency filter in its AC side. Compared to the traditional L and LC type filter, LCL filter enjoys the advantages of small size, low cost, and high attenuation effect on highfrequency current harmonic and so on, and thus it is more suitable for highpower applications. However, LCL filter has three parameters. Moreover, the parameters have a high degree of coupling and the mutual effects between them are difficult to analyze, and due to some other reasons, it is hard to design the parameters.
In recent years, scholars have done a great research work on the design and optimization of LCL filter parameters. In the parameter design, Marco Liserre has done a detailed theoretical analysis on the steps and constraints of parameters design for LCL filter
[5

6]
, and established a complete design approach for the first time. Based on that, the literature
[7

9]
got the containing range for the three parameters of LCL filter by studying the relationship between the changes of the filter’s each parameters and ripple rejection and resonant frequency, and then the appropriate parameters were determined after several attempts and repeated verifications, but the obtained LCL filter parameters are not optimal. In optimizing the filter parameters, the literature
[10]
considered the voltage harmonics generated by SVPWM method. Meanwhile, with maximum single harmonic attenuation of grid current as the optimization objective, the method for designing optimal LCL parameters by doing iterative computation based on harmonic voltage value was proposed in literature
[10]
. However, in the optimization process, the resonant frequency and inductance scale factor were known input, thus reducing the number of constraints, therefore, only a simple onedimensional local search was performed. Based on literature
[10]
, literature
[11]
took the minimum of total harmonic distortion (THD) as the target for optimization. Then, a THD estimation model was established through analysis and calculation of harmonic current value, and a more simplified gradient algorithm with multiple starting point was realized for LCL filter parameter optimization. However, the resonance frequency was a known input in the optimization process, thus reducing the constraints, and only a twodimensional local search was done. In literature
[12]
, in accordance with the requirements of filtering performance, a LCL filter parameter optimization method was proposed that the total amount of inductance was minimized, but in the design process, the power factor and the value ratio between harmonic current and voltage harmonic of the switching frequency were known and thus it is a onedimensional local search method.
In this paper, an optimization method for parameters of LCL filter of least square method based threephase PWM converter is proposed. First, through the analysis of working principle and characteristics of converter SVPWM modulation, peaks of each harmonic for side phase voltage of converter is quantitatively calculated based on Fourier series expansion method. Secondly, the quantitative relationship between phase voltage harmonics and gridconnected phase current harmonics is analyzed, and minimum requirements for the filter attenuation rate is obtained combining with requirements for each harmonic of grid currents. Thirdly, with the optimization target of minimizing the parameter values of LCL filter, the objective function with minimum harmonic attenuation deviation is established, and least squares method is adopted for threedimensional global searching of parameters for LCL filter, so that the actual attenuation curve is approaching the required minimum attenuation rate, resulting in optimized parameters value. Finally, the experimental platform is established and the validity of the method is verified.
2. Calculation of PWM Voltage Harmonic
For better clarifying the design method proposed in this paper, threelevel converter is taken as the research object. The main circuit of threephase threelevel voltage source converter is shown in
Fig. 1
, where
u_{gx}
(
x
=
a
,
b
,
c
) represents each grid phase voltage, and
u_{x}
refers to output phase voltage of converter at the AC side, and
S
_{x1}
,
S
_{x2}
,
S
_{x3}
and
S
_{x4}
are the four switches on the bridge arm corresponding to
x
, and
L
is filter inductance of the converter bridge side, and
L_{g}
refers to filter inductance of gridside, and
C_{f}
is the filter capacitor, and
R_{d}
is the damping resistor.
Main circuit of the threephase threelevel gridconnected VSR with a LCL filter
 2.1 Analysis of SVPWM
Threelevel SVPWM has 27 voltage vectors which is composed of 3 zero vectors, 12 short vectors, 6 medium vectors and 6 long vectors. As shown in
Fig. 2
.
Voltage space vectors of SVPWM
Here,
U_{ref}
refers to reference voltage vector, and
r
is the rotation angle of the reference voltage vector.
In this paper, the threelevel SVPWM method of literature
[14]
is adopted. Take
U_{ref}
in
Fig. 2
as an example, which locates at the small section 5 of the big section I. Its small vectors are selected as 100 and 211, which is defined as
U
_{1}
=
U_{dc}
/3, and the action time is
T
_{1}
. Meanwhile, its medium vector is selected as 200, defined as
U
_{2}
=
U_{dc}
/ 2 ×
e
^{jπ/ 6}
, and the action time is
T
_{2}
. And its long vectors are selected as 210, defined as
U
_{3}
= 2
U_{dc}
/3, and the action time is
T
_{3}
. Then, the relationship among them is described in the following formula (2).
Here,
T_{s}
is the switching period, and
T
_{1}
+
T
_{2}
+
T
_{3}
=
T_{s}
. Meanwhile, by formula (2), the value of
T
_{1}
,
T
_{2}
and
T
_{3}
can be obtained, as shown in formula (3).
where the modulation ratio is
and the range of values for
m_{u}
is 0 ≤
m_{u}
≤ 1, and
U_{d}
, and the range of values for
m_{u}
is 0 ≤
m_{u}
≤ 1, and
U_{dc}
is the DC link voltage. Moreover, take phase a as an example, the voltage waveform of voltage
u_{ao}
with phase a relative to the capacitor neutral point o is analyzed. For the voltage
u_{ao}
in a switching cycle
T_{s}
, (1) when
S_{x}
= 1, it is 0 ; (2) when
S_{x}
= 2, it is +
u_{dc}
/2 , with a positive level duty cycle
d
_{u(k)}
^{+}
(0 ≤
d
_{u(k)}
^{+}
≤ 1); (3) when
S_{x}
= 0, it is −
u_{dc}
/2 with a negative level duty cycle
d
_{u(k)}
^{}
(1≤
d
_{u(k)}
^{}
≤ 0). Therefore, the duty cycle of
U_{ref}
in
Fig. 2
can be represented by formula (4).
According to the switch vector sequence
[14]
, the duty cycle
d_{u}
in each section of
Fig. 2
are shown in Appendix
Table 1
.
For the digital implementation of SVPWM, the rotation angle
r
of the voltage vector
U_{ref}
steps once per switching period. And the step angle is
d_{r}
=360°/
m_{f}
where
m_{f}
=
f_{sw}
/
f_{g}
is the carrier ratio, and
f_{sw}
is the switching frequency, and
f_{g}
is the grid frequency. Take −180° as the moment of 0 and + 180° as T, and then spread
u_{ao}
in a grid period.
As shown in
Fig. 3
, separate the switching period
T_{s}
into
m_{f}
intervals, where in the length of interval 1 and interval
m_{f}
+1 is
T_{s}
/2 while other intervals is
T_{s}
. Meanwhile, the rotation angle corresponding to interval
k
is
r_{k}
=(
k
1) ×360°/
m_{f}
.
One period spread waveforms of u_{ao}
To facilitate the unified computing, the zero level of
u_{ao}
is regarded as positive level duty time when
u_{ao}
outputs negative voltage pulse. Provided that
d
_{u(k)}
^{+}
is the positive level duty cycle of interval k and
t_{k}
is the transition time of voltage pulse
k^{th}
, the transition time of each voltage pulse can be obtained by formula (5).
 2.2 Calculation of voltage harmonic
For the voltage
u_{ao}
is a periodic function, the values of each harmonic can be calculated based on the principle of Fourier series expansion method, as shown in formula (6).
where
ω
_{1}
= 2
πf_{g}
is the grid angular frequency;
a_{0}
is the DC component and
h
is the harmonic number.
According to the transition time
t
_{0}
,
t
_{1}
, …
t
_{2mf +1}
, and substitute
u_{ao}
into integral formulas (6), the calculation of each voltage harmonic can be derived, as shown in formula (7)  (9).
For phase a, the relationship of
u_{ao}
,
u_{ap}
and
u_{po}
is represented as
u_{ao}
=
u_{ap}
+
u_{po}
, where
u_{po}
is the voltage of the load capacitance midpoint p relative to midpoint o of DC capacitor and
u_{po}
= (
u_{ao}
+
u_{bo}
+
u_{co}
) / 3. In terms of the relationship among each harmonic component of
u_{ao}
,
u_{ap}
and
u_{po}
[16], it can be known that the components of output phase voltage
u_{ap}
are composed of the fundamental wave and the rest components of
u_{ao}
excluded of carrier frequency multiplication and third sideband harmonics. Therefore, the values of
u_{aph}
at the
h
order of voltage
u_{ap}
can be gained by formula (10).
where
n
is the fundamental doubling harmonic and
m
is the carrier doubling harmonics.
Based on the above analysis, the calculation process of voltage harmonic
U_{aph}
for side output phase of converter based on threelevel VSC with SVPWM can be obtained, as shown in
Fig. 4
.
Flow chart of the harmonic calculation of phase voltage
With the calculation process shown in
Fig. 4
, each harmonic component of
U_{aph}
can be obtained.
3. Optimization of Filter Parameters
 3.1 Equivalent model of filter
Assuming that the converter keeps threephase equilibrium, and parameters of each phase are consistent, and the voltage source for the grid is ideal, then singlephase equivalent model of fundamental component and harmonic component based on the LCL filter are shown in
Fig. 5(a)
and
Fig. 5(b)
, respectively.
Singlephase equivalent circuit of LCL filter: (a) Singlephase equivalent model of fundamental component; (b) Singlephase equivalent model of harmonic component
In addition, the fundamental component of the output phase voltage at the AC side of converter is defined as
u
_{ap1}
(
t
) = │
U
_{ap1}
│ × cos(
ω
_{1}
t
+
θ
) and the fundamental component of grid phase current is defined as
i
_{g1}
(
t
) = 
I
_{g1}
 × cos(
ω
_{1}
t
+
β
) while that of the grid phase voltage is
u
_{g}
(
t
) = │
U
_{g}
│ × cos(
ω
_{1}
t
) . From
Fig. 5
, the relationship among fundamental component
u
_{ap1}
of output phase voltage at the AC side of the converter, the fundamental component of the grid phase current
i
_{g1}
and the grid phase voltage
u_{g}
in the sdomain can be represented at the following formula (11).
According to
Fig. 5
, the relationship between the
h
order harmonic component
U_{aph}
of output phase voltage at the AC side and the
h
order harmonic component
I_{gh}
of grid phase current in the sdomain is shown at the following formula (12).
 3.2 Attenuation curve of filter
IEEE519 and IEEE1547 are two standard protocols for current harmonics of grid converter, commonly used in distributed generation and renewable energy systems, whose spectrum standard of grid current harmonic is as shown in
Fig. 6
.
Spectrum standard of grid current harmonic
In
Fig. 6
,
I_{ghmax}
is the maximum value for
h
order grid current harmonics, and its standard value is the percentage of
h
order harmonics and the fundamental components
I
_{g1}
. Moreover, the standard also provides the number of even harmonics is 25% of that of odd harmonics.
In terms of
Fig. 6
, the maximum value for
h
order filter output current can be obtained, and combined with the
h
order harmonics
U_{aph}
of filter input voltage, the minimum requirements for attenuation rate
G_{hReq}
of
h
order harmonics for the filter can be known, and their relationship is as shown in formula (13).
From formula (13), it can be found that the calculation of actual attenuation curve for LCL filter is as shown in the following formula (14).
where
s
=
jhω
_{1}
when
h
=2, 3, 4…. If
U_{aph}
,
L
,
L_{g}
,
C_{f}
and
R_{d}
are known，the actual attenuation curve
G_{h}
of LCL filter can be calculated by formula (14).
 3.3 Filter parameters constraints
According to the literature
[5

9]
, the following LCL filter parameters constraints can be obtained:
1) Reactive power generated by capacitance is generally required to be no more than 5% of the system’s rated power
P_{n}
, as follows:
2) Considering the capacity of outputting active power by VSC under stable conditions, the total inductance
L_{T}
=
L
+
L_{g}
should satisfy the formula (16):
3) The resonant frequency is generally defined to be between 10 times of the fundamental frequency
f_{g}
and 0.5 times of the switching frequency
f_{sw}
, as shown in the following formula (17):
4) The design of damping resistor
R_{d}
needs to consider a compromise between the system damping and the loss of damping, and its value is generally 1/3 of capacitive reactance for filter capacitor
C_{f}
at a resonant frequency of
f_{res}
, as follows.
where
f_{res}
is the resonant frequency of the filter, and
U_{ph}
is the effective value of grid phase voltage, meanwhile,
U_{phm}
and
I_{phm}
are peak values of grid phase voltages and phase currents.
 3.4 Optimization of parameters based on least squares method
Least squares method is a mathematical optimization method, which finds the best matching function for a given set of data through minimizing the sum of the squares of the error. By the formula (14), it can be seen that the larger the value of the filter parameter, the smaller the value of
G_{h}
, i.e. the better the suppression effect of its current harmonics, but the greater the volume of the filter and the higher the cost at the same time. Therefore, the value of
G_{h}
can be increased as far as possible when the requirements of harmonic suppression are satisfied, in order to minimize the filter parameters. In this paper, the least squares method is applied to the optimization of filter parameters. With the goal of minimizing the parameters of LCL filter, and in terms of the constraints on filter parameters obtained by the formula (15)  (18), a threedimensional global search for
L
,
L_{g}
and
C_{f}
is performed, so that the actual attenuation curve constantly approaches the required minimum attenuation rate, and then the optimized filter parameters are found.
Considering that some margin is required in the practical application of the filter,
M
can be defined as the margin value of the filter attenuation curve, which ranges from 0 to 1. Meanwhile, the actual attenuation rate
G_{h}
of each harmonic and the required minimum attenuation rate
G_{hReq}
should satisfy the formula (19).
The sum of deviation between actual attenuation rate
G_{h}
of each harmonic and the minimum requirements of attenuation rate
G_{hReq}
can be calculated by formula (20).
In summary, the interrelationships of each part of the optimization design above is shown in
Fig.7
.
Interrelationship of each part of the optimization design method
Besides, the program flow chart of the filter parameters design is shown in
Fig. 8
and the design steps of the method for LCL filter parameters optimization proposed in the paper are as follows, where
L
*,
L_{g}
*,
C_{f}
* and
R_{d}
* are the optimal values of the filter parameters.
Flow chart of the LCL filter parameters design
Step 1
: The system’s rated power
P_{n}
, the switching frequency
f_{sw}
, the DC bus voltage
U_{dc}
, grid phase voltage
U_{ph}
and frequency
f_{g}
are known, and then the ranges of filter parameter constraints are calculated as 0<
L
<
L_{Tmax}
, 0<
L_{g}
<
L_{Tmax}
and 0<
C_{f }
<
C_{fmax}
according to formula (15)  (17);
Step 2
: given a set of
L
,
L_{g}
and
C_{f}
within the range got in Step 1, then if the parameters are successfully defined, use formula (18) to calculate the damping resistor
R_{d}
and go to step 3, otherwise go to step 8;
Step 3
: the current peak value 
I
_{g1}
 and power factor angle
β
are known, and thus the peak value and phase angle of the converter output phase voltage at the AC side can be obtained by formula (11);
Step 4
: the modulation ratio
m_{u}
can be obtained by the peak value of the output phase voltage, and then
U_{aph}
can be obtained by
Fig. 5
for peak value of each harmonic of phase voltage;
Step 5
: the maximum value
I_{ghmax}
of peak value for each harmonic of grid current can be obtained from
Fig. 7
, and then the required minimum attenuation rate
G_{hReq}
can be calculated by the formula (13) ;
Step 6
: given the margin value of filter attenuation curves as
M
, then the actual attenuation curve of LCL filter can be obtained by substituting
L
,
L_{g}
,
C_{f}
and
R_{d}
into formula (14), and it can be seen whether it satisfies formula (19), so return to step 2 if not satisfied, otherwise go to step 5;
Step 7
: calculate by formula (20) and determine whether it satisfies
G_{Err}
<
G_{Err}
^{*}
, and if it satisfies, then update the value of optimal filter parameters as
G_{Err}
^{*}
=
G_{Err}
,
L
*=
L
,
L_{g}
*=
L_{g}
,
C_{f}
*=
C_{f}
and
R_{d}
*=
C_{d}
while back to step 2;
Step 8
: stop searching, and output the values of optimal filter parameters as
L
*=
L
,
L_{g}
*=
L_{g}
,
C_{f}
*=
C_{f}
and
R_{d}
*=
C_{d}
 3.5 A practical design example
The design parameters of system are as follows: rated power
P_{n}
is 10 kVA, and DC voltage
U_{dc}
is 700 V, and the percentage between effective value of grid phase voltage
U_{ph}
and frequency
f_{g}
are 220V/50Hz, and the switching frequency
f_{sw}
is 9kHz.
The peak value of fundamental wave for phase voltage
U
_{g1}
and modulation ratio
m_{u}
is 1when the power factor of the converter, and the peak value of grid current
I
_{g1}
is 21A, and the phase
β
is zero. To verify the effects of the design method in extreme cases, the margin value
M
of filter attenuation curve is 1.
The filter parameters constraints can be obtained by (15)(16):
C_{max}
=14uF and
LT_{max}
=10mH. Then, the threedimensional searching scope can be 0.1mH
L
10mH, 0.1mH
L_{g}
10mH and 0.2uF
C_{f}
10uF. Given the value of capacity
C_{f}
and searching
L
and
L_{g}
, a set of values of L and L can be obtained while the sum of harmonic deviation
G_{Err}
is minimal. And by repeating the same searching process, the parameters of
L
,
L_{g}
, and
G_{Err}
with various
C_{f}
are shown in
Fig. 9
.
Values of L and L_{g} and the minimal values of G_{Err} with various values of C_{f}
In
Fig. 9
, it can be found that the parameters of
G_{Err}
,
L
and
L
+
L_{g}
converge to the minimum point while
C_{f}
is 3.1uF, which have been marked in the figure. Therefore, the values of
L
,
L_{g}
and
C_{f}
at the minimum point, as shown in
Table 1
, are the optimal parameters obtained by the proposed method.
LCL filter parameters before & after optimization
LCL filter parameters before & after optimization
The traditional method proposed in literature
[5]
and the new proposed method are respectively adopted to design LCL filter parameters, as it shown in
Table 1
.
From
Table 1
, it can be clearly seen that the parameters using the traditional method are bigger than those using new method.
The required minimum attenuation rate
G_{hReq}
, the attenuation rate of filters
G_{hOld}
by the traditional method and the attenuation rate of filters
G_{hNew}
by the method proposed in this paper are described as corresponding curves and compared, as shown in
Fig. 10
.
Attenuation curve of LCL filter
From the attenuation curve of threelevel LCL filter shown in
Fig. 10
, the attenuation curve of filters with the traditional method
G_{hOld}
shows downward deviation and is relatively far from the required minimum attenuation rate
G_{hReq}
. The attenuation curves of filters with the method proposed in this paper
G_{hNew}
is closer to
G_{hReq}
than
G_{hOld}
, and in terms of
G_{hNew}
<
G_{hOld}
, it can be known that the filters meet the suppression requirements of each current harmonic.
4. Experimental Verification
An experimental platform according to the system parameters given above in section 3.5 is built. The comparative analysis of two different sets of filter parameters in
Table 1
for the experimental platform are made as follows.
The calculated values for harmonic spectrum of grid current with the traditional method and the method proposed in this paper are shown in
Fig. 11
.
Calculated value of harmonic spectrum of grid current: (a) The traditional method; (b) The method proposed in this paper
To verify the results of the above design, a threelevel threephase PWM converter prototype with TMS320F 28335 as the control center is established, and the YO KOGAWA power analyzer (WT1800) is adopted to measure the harmonic spectrum of grid current. With the traditional method, the parameter
L
is 4.0mH@21A,
L_{g}
is 2.0mH@21A, and
C_{f}
is 6uF when Voltech transformer tester (AT3600) is used for inductance measurement under the condition of DC bias as 21A. However, with the method proposed in this paper, L is 1.6mH@21A,
L_{g}
is 1.3mH @21A and
C_{f}
is 2.2uF under the same condition. And the measured values for harmonic spectrum of grid current is shown in
Fig. 12
.
Measured value of harmonic spectrum of grid current: (a) The traditional method; (b) The method proposed in this paper
From the grid current harmonics shown in
Fig. 12 (a)
and
Fig. 12 (a)
, the value of each current harmonic obtained by the traditional method is smaller, that is, its filtering effect is better, but from
Table 1
it can be seen that its filter parameters are too large. Meanwhile, from
Table 1
it can be seen that with the method proposed in this paper this method, the obtained value of each current harmonic is larger but can meet the requirements given by IEEE519, and the obtained filter parameters
L
,
L_{g}
and
C_{f}
are smaller, and the total value of inductance and capacitance value are decreased by about half.
From
Fig. 12
, it can be found that the current harmonic orders in the resonant frequency, the first, second and third of switching frequency and its sideband are the key points. Therefore, a comparison between the calculated value and the experimental value for grid current harmonic with the traditional method and the new method proposed in this paper are shown in
Table 2
.
Contrast of current harmonic before and after LCL filter parameters optimization
Contrast of current harmonic before and after LCL filter parameters optimization
From
Table 2
, the value of current harmonics obtained by the method proposed in this paper is closer to standard value than traditional methods, which is especially closer to the standard value of 0.075% when the frequency is resonant frequency and one time switching frequency. In addition, it can be seen from
Fig. 11
,
Fig. 12
and
Table 2
that the calculated and experimental values of grid current harmonic have high consistency at the resonance frequency
f_{res}
and opening frequency
f_{sw}
, 2
f_{sw}
and 4
f_{sw}
, indicating the effectiveness of the calculation method for current harmonics proposed in this paper.
LCL filter is used to suppress high harmonics, but due to the presence of dead zone for PWM in practical application, loworder harmonics are existed in actual grid current, which are mainly composed of 5, 7, 11 and 13order harmonics. However, harmonic compensation can be realized for the loworder harmonics by means of control, and thus the validity of this method is not affected.
5. Conclusion
In the paper, an optimization method for parameters of least square method based LCL filter is proposed. The required minimum attenuation rate is obtained by combining the calculated voltage harmonics generated by PWM and standard of each grid current harmonic. Then, with the goal of minimizing parameter values for LCL filter, a threedimensional global search for LCL filter parameters is performed, making the actual attenuation curve approximating the required minimum attenuation rate. Based on the constraint scopes for LCL filter parameters of the traditional method, this paper searches for the optimal filter parameters by optimizing the algorithm, which has addressed the problem of the traditional method, that is, it requires repeated trial and simulation of parameters. Finally, the effectiveness of this method has verified by the experimental results.
BIO
Hong Zheng He received Master degree in control science and engineering from Sichuan University. His research interests are renewable energy system control and power electronics and motion control.
Zhengfeng Liang He received B.S degree in Automation Engineering from university of Electronic Science and Technology of China. Currently, he is a M.S student in Automation Engineering, Electronic Science and Technology University, China. His research interests are power electronics, filtering and storage converters.
Mengshu Li She received B.S degree in Automation Engineering from University of Electronic Science and Technology of China. Currently, she is a M.S student in Automation Engineering, Electronic Science and Technology University, China. Her research interests are power electronics, filtering and storage converters.
Kai Li He received his B.S, M.S and PhD degree in Automation Engineering from University of Electronic Science and Technology of China, China, in 2006, 2009 and 2014. Currently, he is a Lecturer Professor in University of Electronic Science and Technology of China. His research interests include multilevel inverters, storage converters and microgrid.
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