In this paper, an active damping control scheme for LLCL filters based on the PR (proportionalresonant) regulator is proposed for gridconnected threelevel Ttype PWM converter systems. The PR controller gives an infinite gain at the resonance frequency. As a result, the oscillation can be suppressed at that frequency. In order to improve the stability of the system in the case of grid impedance variations, online grid impedance estimation is applied. Simulation and experimental results have verified the effectiveness of the proposed scheme for threephase Ttype AC/DC PWM converters.
I. INTRODUCTION
In recent years, voltage source converters (VSC) have been widely employed for gridconnected applications with the advantage of bidirectional power flow and its ability to control the voltage and power factor. In particular, multilevel converters such as Ttype threelevel inverters have been used due to their higher performance and efficiency when compared with the twolevel converters
[1]

[3]
. To suppress switching harmonics, a large size filter inductor is needed in the VSC. However, this deteriorates the system dynamic performance and increases the cost of the inductor in high power applications.
Due to the advantages in terms of cost and dynamics, LCL filters are preferred over L filters for gridconnected PWM converters since smaller inductors can be used in the LCL filters. However, there are drawbacks of LCL filters such as the resonance phenomena and the complex control algorithm
[4]

[7]
. To suppress the filter resonance, a number of passive and active damping methods have been proposed
[8]

[16]
. The passive damping methods are simple and reliable, but power loss is an issue especially for high power applications
[8]

[10]
. On the other hand, the active damping schemes do not increase power losses, but the control algorithm is modified to suppress resonance
[11]

[17]
. In
[12]
,
[13]
, an active damping method based on a virtual resistor has been proposed. However, an extra capacitor current sensor is required. In
[14]
, a genetic algorithm has been applied to tune the notch filter for the active damping of LCL filters. Although no extra sensor is needed, the optimization algorithm is complicated. The PR regulator has been discussed for LCL filters in
[15]
,
[16]
. However, the grid impedance variation has not been investigated which is an issue in the case of a variable grid impedance.
To reduce the inductor size in LCL filters, a new structure for the LLCL filters of singlephase VSCs has been proposed
[18]
. To suppress the resonance of the LLCL filters, the passive damping method has been employed, where an additional resistor was used
[9]

[11]
.
This paper is an extended version of
[16]
, where an active damping method based on the PR control has been proposed for the LLCL filters of threelevel Ttype converters. To suppress the resonance in the case of variable grid impedances, the online grid impedance estimation method is applied. In addition, the system stability is analyzed to show the validity of the proposed control algorithm. The effectiveness of the proposed method is verified by simulation and experimental results obtained for a threelevel Ttype converter.
II. SYSTEM DESCRIPTION
 A. ThreeLevel Ttype AC/DC PWM Converters
A threelevel Ttype AC/DC PWM converter is shown in
Fig. 1
. It is connected to the grid through LLCL filters. The grid is modeled as a sinusoidal voltage source with variable line impedances. A resistive load is connected to the DC output terminal of the converter. For the PWM converter, the voltage equations are expressed in the synchronous reference frame as
[19]
:
Gridconnected threelevel Ttype converters with LLCL filters.
where:

ede,eqe:dqaxis grid voltages

ide,iqe:dqaxis converter currents

vde,vqe:dqaxis converter voltages

ω: angular frequency of the grid voltage

R: sum of the resistances in the filter inductors (R1+R2)

L: sum of grid and converterside inductors (L1+L2)
By aligning the qaxis of the synchronous reference frame to the grid voltage,
e_{de}
= 0. The DClink voltage dynamics can be expressed as:
where:

vdc: DClink voltage

iDC: converterside DC current

iL: load current
where the losses in the filter and the converter are neglected.
 B. LLCL Filters
In LLCL filters, the transfer functions of the grid and converter currents to the converter voltage are expressed as:
where:
From (4) and (5), the resonance frequency in the LLCL filters is given by:
The frequency responses of the grid and converter currents to the converter voltage are shown in
Fig. 2
(a) and (b), respectively. It can be seen that the impedance of the LLCL filters is very low near the switching frequency due to the series inductor in the capacitor branch.
Bode plots of transfer functions in LCL and LLCL filters. (a) G_{1}(s) = I_{g}(s)/V_{i}(s). (b) G_{2}(s) = I_{c}(s)/V_{i}(s).
 C. Parameter Design of LLCL Filters
In the design of the filter, some limitations on the parameter values should be considered as follows
[6]
,
[8]
:

1) The converterside inductorL1is determined based on the allowable current ripple of the converter.

Where:

irpMax: maximum ripple component in the rated current

fsw: switching frequency

The gridside inductorL2is designed according to the IEEE 5191992 standard recommendations, where harmonic currents higher than the 35thorder component should be less than 0.3% of the fundamental.

2) The capacitor valueCis determined based on the reactive power absorbed under the rated condition, where the upper limit is given by:

where:

Vg: fundamental voltage of the grid

ω: fundamental frequency in radian per second

3) The series inductorLfis designed depending on the capacitance in the LLCL filters. For the zero impedance at the switching frequency,

4) The resonance frequency is in the range between ten times the fundamental frequency and a half of the switching frequency
III. PROPOSED ACTIVE DAMPING CONTROL
Fig. 3
shows the control block diagram of the gridconnected Ttype converter with the LLCL filters, where the PR control is applied for active damping.
Control block diagram of the gridconnected PWM converter with LLCL filters.
 A. Active Damping Control
The PR controller is operated in the
abc
reference frame, where the steadystate error is eliminated at the specified resonance frequency. The transfer function of the PR controller is given as
where:

Kpr: proportional gain

Kr: resonant gain

ωres: resonance frequency
The magnitude and phase characteristics of the openloop transfer function for the PR controller with respect to the different resonant gains are illustrated in
Fig. 4
. It can be seen that the higher resonant gains can eliminate the steadystate error. However, this leads to a wider bandwidth.
Bode plots of the resonant controller for different gains (K_{r}).
 B. Grid Impedance Estimation
To suppress the resonance effectively, the grid impedance needs to be known. In this study, an online estimation of the grid impedance is applied to find the total inductance on the grid side of the LLCL filters
[20]
. When a voltage component of the specified frequency is added to the PWM voltage reference, a harmonic current component is incurred at the same frequency. Next, the Discrete Fourier analysis for the specific injected harmonic component is applied as:
where:

N: number of the samples per fundamental period

g(n) :input signal (voltage or current) at the sampling
point n

: complex Fourier vector of the hthharmonic of the input signal

Fhr: real part of

Fhi: imaginary part of
Then, the harmonic current and voltage components are obtained from (12). From these the grid impedance at the specified harmonic frequency is calculated as:
where:

Rgrid: grid resistance at the injected harmonic hx.

Lgrid: grid inductance at the injected harmonic hx.
The grid impedance in terms of the fundamental frequency is obtained by:
Fig. 5
illustrates variations of the filter resonance frequency according to variations of the grid impedance. By increasing the grid inductance from
L_{grid}
= 0 to 0.5mH the filter resonance frequency,
f_{res}
, varies from 1.9 kHz to 1.48 kHz.
Bode plots of transfer functions of LLCL filters for different values of grid inductance.
IV. STABILITY ANALYSIS
To determine the feasibility of the proposed active damping method, the system stability is analyzed. The overall current control scheme is depicted in
Fig. 6
. Since the PI current controller (
G_{PI}
(
s
) =
K_{p}
+
K_{i}
/
s
) is used in the synchronous reference frame, its equivalent form for the transfer function,
G_{PR}
_{_f}
(
s
), in the stationary reference frame is given by:
The proposed closedloop current control in stationary reference frame.
where:

Kp1: proportional gain

Kr1: resonant control gain
A BPF (bandpass filter) is needed for the PR controller to damp the filter resonance, where the cutoff frequency is equal to the resonance frequency of the LLCL filters. The transfer function of the BPF is expressed as:
where:

ωb: center frequency

Q: quality factor

B: bandwidth
Therefore, the transfer function of the current controller,
G_{c}
(
s
), is the sum of
G_{PR}
_{_f}
(
s
) in (15) and the multiplication of
G_{PR}
_{_r}
(
s
) and
G_{BPF}
(
s
) in (11) and (16) respectively, which can be expressed as:
The stability is evaluated in the Zdomain, where one sampling delay of the digital processing and onehalf a sampling delay of the PWM converters are involved in the PWM block, which is expressed as:
Then, the openloop transfer function of the current control system is expressed as:
To analyze the frequency response of the proposed current control system, the closedloop transfer function in (20) is investigated.
The root loci for the proposed current control system are illustrated in
Fig. 7
. In
Fig. 7
(a), the root loci is plotted at the switching frequency of
f_{sw}
= 4
kHz
. The control system is stable since all of the poles are placed inside the unit cycle. As the switching frequency is increased, the stability may be degraded. However, by tuning the controller gains, the system can still be stable at
f_{sw}
= 6
kHz
and
f_{sw}
= 8
kHz
, as illustrated in
Fig. 7
(b) and (c), respectively. A further increase of the switching frequency may cause instability since the system poles are placed outside the unit cycle.
Root loci of the proposed current control system. (a) f_{sw}= 4 kHz. (b) f_{sw}= 6 kHz. (c) f_{sw}= 8 kHz.
The frequency response of (19) is illustrated in
Fig. 8
where the controller gains are
K_{p}
= 9,
K_{i}
= 3000,
K_{pr}
= 6,
K_{ir}
= 50 and the bandwidth of the controller is approximately 500 Hz, which results from the definition of
GM
= −3
dB
in the closedloop control system. A bandpassfiler (BPF) with a cutoff frequency of 1.9 kHz and a bandwidth of 200 Hz are employed to extract the resonant components in the grid currents. In order to guarantee stability, a phase margin of
PM
> 45° and a gain margin
GM
> 6
dB
are generally desirable
[13]
,
[21]
.
Fig. 8
shows that the gain and phase margins of the openloop current control system are 13 dB and 46°, respectively.
Bode plot of the openloop current control system.
Fig. 9
shows the step responses of the closedloop system in (20), where a resonance phenomenon does not appear.
Step response of the closedloop current control system.
V. SIMULATION RESULTS
A simulation is performed for the threelevel Ttype PWM converter. The simulation parameters are listed in
Table I
. The different parameters of the filters are shown in
Table II
.
PWMCONVERTER PARAMETERS
FILTER PARAMETERS
Fig. 10
illustrates the performance of the grid impedance estimation. The estimated and real values of the grid inductance are shown in (a), where the estimation error is less than 5%. In (b), the magnitude of the injected harmonic and its frequency are shown. The distortion in the grid current, due to the harmonic injection, appears as shown in (c). In (d), the resonance frequency of the LLCL filters for different grid impedances is illustrated, where the resonance frequency varies from 1.9 kHz in the case of zero grid impedance to 1.48 kHz in the case of a grid impedance of 0.5 mH.
Grid impedance estimation. (a) Grid inductance. (b) Harmonic voltage. (c) Grid current. (d) Resonance frequency.
Fig. 11
shows the grid and converter currents and the FFT spectra in the case I (L
_{1}
=1.2 mH, L
_{2}
= 0.8 mH) of LCL filters. The magnitude of the dominant harmonic component in the grid current is lower than 0.1 A (0.18% of the fundamental component). The total harmonic distortion (THD) factors are listed in
Table III
. For this Case I the THD is 2.91%.
Grid and converter currents with FFT spectra in case I of LCL filters: L_{1} =1.2 mH, L_{2} =0.8 mH.
THD AND SIDEBAND HARMONIC MAGNITUDE OF THE GRID CURRENT IN DIFFERENT CASES
THD AND SIDEBAND HARMONIC MAGNITUDE OF THE GRID CURRENT IN DIFFERENT CASES
Case II of the LCL filters (L
_{1}
=1.2 mH, L
_{2}
= 0.35 mH) is illustrated in
Fig. 12
, where the dominant harmonic magnitude in the grid current is lower than 0.3 A (0.56 % of the fundamental current) with a THD of 5.12%. It is obvious that Case II of the LCL filters does not meet the requirement of the IEEE standard 5191992.
Grid and converter currents with FFT spectra in case II of LCL filters: L_{1} =1.2 mH, L_{2} =0.35 mH.
Case III of the LLCL filters (L
_{1}
=1.2 mH, L
_{2}
= 0.35 mH, and L
_{f}
= 0.08 mH) is shown in
Fig. 13
, where the magnitude of the dominant harmonic in the grid current is lower than 0.92 A (0.17 % of the fundamental component) with a THD of 2.64% (
Table III
).
Grid and converter currents with FFT spectra in case III of LLCL filters: L_{1} =1.2 mH, L_{2} =0.35 mH, L_{f} =0.08 mH.
VI. EXPERIMENTAL RESULTS
To verify the proposed damping control, experiments have been conducted on a 3kW converter system. A 32b DSP chip (TMS320F28335) is used for the main controller. The parameters of the threelevel Ttype converter are listed in
Table. IV
. The PWM frequency of the converter is 5 kHz and the dead time is set to 2 µs. For the IGBT module (4MBI300VG120R50), two gating drivers have been employed such as VLA51301R for the midpoint IGBTs and VLA54201 for the leg IGBTs. For the reducedscale system of the simulation, the LLCL filters are redesigned by the same procedure described in section II (
Table. V
).
Fig. 14
shows the threelevel Ttype PWM converter used for the tests.
Threelevel Ttype PWM converter.
PWMCONVERTER PARAMETERS
FILTER PARAMETERS
The grid and converter currents and the FFT spectra are shown in
Fig. 15
, for Case I of the LCL filters (L
_{1}
=2 mH, L
_{2}
= 1 mH) where the magnitude of the dominant harmonic component in the grid current is lower than 0.021 A (0.2% of the fundamental component) with a THD of 3.72%.
Grid and converter currents with FFT spectra in the case I of LCL filters (L_{1}=2 mH, L_{2}=1 mH).
Fig. 16
illustrates the same waveforms as in
Fig. 15
, in Case II of the LCL filters (L
_{1}
=2 mH, L
_{2}
= 0.4 mH), where the dominant harmonic magnitude in the grid current is lower than 0.052 A (0.46 % of the fundamental current) with a THD of 4.04%. It is obvious that Case II of the LCL filters does not meet the requirement of the IEEE standard. In Case III of the LLCL filters (L
_{1}
=2 mH, L
_{2}
= 0.4 mH, and L
_{f}
= 0.1 mH), as shown in
Fig. 17
, the magnitude of the dominant harmonic in the grid current is lower than 0.015 A (0.13 % of the fundamental component) with a THD of 2.47% (
Table VI
).
Grid and converter currents with FFT spectra in the case II of LCL filters: L_{1}=2 mH, L_{2}=0.4 mH.
THD AND SIDEBAND HARMONICS MAGNITUDE OF THE GRID CURRENT IN DIFFERENT CASES
THD AND SIDEBAND HARMONICS MAGNITUDE OF THE GRID CURRENT IN DIFFERENT CASES
Grid and converter currents with FFT spectra in case III of LLCL filters: L_{1}=2 mH, L_{2}=0.4 mH, L_{f}=0.1 mH.
Fig. 18
shows the grid phase currents and FFT spectra in the case of different grid impedances. Even though the grid impedance is changed from 0.1 mH to 0.2 mH, the PR controller can suppress the resonance effectively with the estimated grid impedance. In addition, a THD of about 2.4% in the grid currents is within the acceptable level.
Grid currents and FFT spectra with LLCL filters. (a) L_{grid} = 0.1 mH. (b) L_{grid} = 0.2 mH.
Fig. 19
shows the converter performance in the case of a resistive load application, where the grid current and the qaxis converter current are shown in (a) and (b), respectively. The DClink voltage is fluctuated at the point where the converter qaxis current changes abruptly, where the undershoot in the DClink voltage is lower than 10% as shown in (c). The converter phase current is illustrated in (d). As can be seen, it is more distorted than the grid current.
Converter control performance with LLCL filters. (L_{1} =2 mH, L_{2} = 0.4 mH, and L_{f} = 0.10 mH). (a) Grid phase current. (b) Converter qaxis current. (c) DClink voltage. (d) Converter phase current.
Fig. 20
shows the performance of the converter in the case of unbalanced grid voltages, where one phasevoltage of the grid is decreased by 20% as shown in (a). The grid phase current in the case of resistive load variations is shown in (b), where the switching ripples meet the IEEE standard of 0.3%. The fluctuations in the DClink voltage do not exceed 10 % as shown in (c), which is acceptable under this condition.
Converter control performance with LLCL filters in the case of 20% drop in grid voltage. (L_{1} =2 mH, L_{2} = 0.4 mH, and L_{f} = 0.1 mH). (a) Grid threephase voltages. (b) Grid phase current (c) DClink voltage.
VII. CONCLUSION
In this paper, an active damping method using a PR controller has been proposed for the LLCL filters connected to the threelevel Ttype PWM converter. In addition, the grid impedance has been estimated online to compensate for the resonance frequency of the filter in the case of a variable grid impedance. The damping effect of the PR controller on LLCL filters has been verified by simulation and experimental results. In the LLCL filters, the gridside inductor has been reduced by 60%, when compared with the LCL filters, from 0.8 mH to 0.35 mH in the simulation and from 1 mH to 0.4 mH in the experiment. In this case, the THD and dominant harmonic component of the grid current have met the IEEE standard. It has been shown that the control performance for the LLCL filters is satisfactory both in the transient state and under grid voltage unbalances.
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2012R1A1A4A01015362).
BIO
Payam Alemi was born in Tabriz, Iran, in 1982. He received his B.S. degree from the University of Tabriz, Tabriz, Iran, in 2005, and his M.S. degree from the Science and Research Branch, Islamic Azad University, Tehran, Iran, in 2008, and Ph.D. degree in Electrical Engineering, from Yeungnam University, Gyeongsan, Korea, in 2014. His current research interests include the control of multilevel power converters, power loss analysis for converters, LCL filters and machine drives.
SeonYeong Jeong was born in 1988. She received her B.S. and M.S. degrees in Electrical Engineering from Yeungnam University, Gyeongsan, Korea, in 2011 and 2014, respectively. Her current research interests include converter control and power quality.
DongChoon Lee received his B.S., M.S., and Ph.D. degrees in Electrical Engineering from Seoul National University, Seoul, Korea, in 1985, 1987, and 1993, respectively. He was a Research Engineer with Daewoo Heavy Industry, Korea, from 1987 to 1988. Since 1994, he has been a faculty member in the Department of Electrical Engineering, Yeungnam University, Gyeongsan, Korea. As a Visiting Scholar, he joined the Power Quality Laboratory, Texas A&M University, College Station, TX, USA, in 1998; the Electrical Drive Center, University of Nottingham, Nottingham, UK, in 2001; the Wisconsin Electric Machines and Power Electronic Consortium, University of Wisconsin, Madison, WI, USA, in 2004; and the FREEDM Systems Center, North Carolina State University, Raleigh, NC, USA, from September 2011 to August 2012. His current research interests include ac machine drives, the control of power converters, wind power generation, and power quality. Professor Lee is currently the EditorinChief for the Journal of Power Electronics of the Korean Institute of Power Electronics.
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