Modular multilevel converters (MMCs) have emerged as the most promising topology for high and medium voltage applications for the coming years. However, one particular negative characteristic of MMCs is the existence of circulating current, which contains a dc component and a series of lowfrequency evenorder ac harmonics. If not suppressed, these ac harmonics will distort the arm currents, increase the power loses, and cause higher current stresses on the semiconductor devices. Repetitive control (RC) is well known due to its distinctive capabilities in tracking periodic signals and eliminating periodic errors. In this paper, a novel circulating current control scheme base on RC is proposed to effectively track the dc component and to restrain the lowfrequency ac harmonics. The integrating function is inherently embedded in the RC controller. Therefore, the proposed circulating current control only parallels the RC controller with a proportional controller. Thus, conflicts between the RC controller and the traditional proportional integral (PI) controller can be avoided. The design methodologies of the RC controller and a stability analysis are also introduced. The validity of the proposed circulating current control approach has been verified by simulation and experimental results based on a threephase MMC downscaled prototype.
I. INTRODUCTION
In recent years, the demand for fully controllable voltage source converter (VSC) topologies is continuously increasing in the fields of electricity networks and motor drives. However, in highvoltage and mediumvoltage applications, to meet the requirements of higher voltage ratings and higher conversion efficiency, traditional twolevel VSC topologies are always replaced by multilevel converters, such as the diodeclamped multilevel converter, the flyingcapacitor converter (FC), the cascaded Hbridge converter (CHB), and the modular multilevel converter (MMC)
[1]
,
[2]
. Among them, the MMC is the most promising topology since it shows outstanding features such as lower power losses, reduced EMI noise, less semiconductor stresses, scalability, easy assembling, and nearly ideal sinusoidal shaped output waveforms. These advantages make the MMC very attractive for highvoltage applications, particularly in the HVDC sector
[3]

[8]
.
The circuit configuration of a threephase MMC is shown in
Fig. 1
. Each of its phases consists of two arms, an upper arm and a lower, which are connected through two buffer inductors. Each arm is formed by a series connection of N nominally identical submodules (SMs) and each SM contains a capacitor and either a halfbridge (HB) circuit or a fullbridge (FB) circuit
[9]
. Due to the series connection of the upper and lower arm SMs, the MMC shows a particular circulating current which circulates through both the upper and lower arms, and ideally, the circulating current should be a dc current. However, since an ac current flows through the SM capacitors, voltage fluctuations are unavoidable. These fluctuations will further impose on the buffer inductors and result in a series of lowfrequency harmonics in the circulating current
[10]
. It follows that these circulating current harmonics are unwanted and should be attenuated. If not, these harmonics will distort the arm currents, bring additional losses, and cause higher current stresses on the semiconductor devices.
Circuit configuration of the modular multilevel converter (MMC). (a) Topology structure. (b) Halfbridge SM. (c) Fullbridge SM.
The circulating current can be controlled by regulating the summation of the upperarm and the lowerarm voltages
[11]
. The existing circulating current harmonics suppression methods are generally based on a proportional integral (PI) controller, which can track the dc circulating current reference with no steadystate error
[12]
,
[13]
. However, this method cannot effectively attenuate the ac harmonics due to its deficient gain at nonzero frequencies. Since the most dominant harmonic frequency in the circulating current is the 2nd order, reference
[11]
presents another suppression scheme by forming a new PI controller under the secondfrequency negativesequence rotational frame. Nevertheless, this rotated controller needs extra calculation for the coordinate transformations and cannot apply to unbalanced threephase or singlephase systems. Further, multiple stationaryframe proportional resonant (PR) controllers have been adopted in
[14]
and
[15]
by attaching them to a traditional PI controller, and achieving an efficient gain by appropriately setting the resonant frequencies. Although this method is able to suppress the circulating current harmonics accurately, it is too complicated to design a series of multiple PR controllers since the circulating current contains too many harmonics such as the 2nd, 4th and 6th order harmonics.
In addition, circulating current control can also be realized by repetitive control (RC)
[16]
. RC is based on the internal model principle. This principal encompasses a signal generator for the periodical reference inside the control system to achieve zero steadystate tracking error
[17]
. The most outstanding merit of RC is its ability to eliminate all of the harmonics in a periodical signal with infinite gains at these harmonic frequencies. In
[16]
, the circulating current control of an MMC is performed by paralleling a traditional PI controller with a RC controller. Thus, the multiple circulating current harmonics can be well attenuated by the RC controller. However, due to the parallel arrangement, RC may conflict with the traditional PI controller. Therefore, it imposes complicated design limitations on the control parameters.
This paper proposes a new circulating current suppression method by only combining an RC controller with a proportional controller. It is demonstrated that the integration function is inherently embedded in the RC controller so that the traditional PI controller can be simplified into a easy proportional controller. Therefore, conflicts between the traditional PI controller and the RC controller can be avoided. The control parameters are easier to design, and the accuracy and robustness of the control system can also be improved. Moreover, the design methodologies of the RC controller are discussed in terms of the stability and convergence rate. In addition, voltage balancing control of the SM capacitors is also provided. Finally, simulations and experiments are performed to confirm the effectiveness of the proposed control method.
II. MATHEMATICAL MODEL OF THE CIRCULATING CURRENT AND REPETITIVE CONTROLLER
 A. Model of the Circulating Current Control in a MMC
As shown in
Fig. 2
, an equivalent circuit diagram of one phase of a MMC is used for analysis.
u_{oj}
is the output voltage of phase
j
(
j
∈ {
a
,
b
,
c
}),
i_{oj}
is the phase current, and
U_{dc}
is the dclink voltage.
u_{uj}
,
i_{uj}
and
u_{wj}
,
i_{wj}
represent the voltages and currents of the upper arm and lower arm, respectively. The following equations can be obtained by Kirchhoff’s voltage and current law:
where
R
is the parasitic resistance, and
L
is the arm inductance.
u_{Lj}
is the voltage across the resistors and inductors, and
i_{cj}
is the circulating current, which is given as (4).
Equivalent circuit of one phase of the MMC.
Referring to
[10]
, the circulating current always consists of a dc component and multiple evenorder harmonics:
where
I_{cj,0}
is the dc component,
I_{cj,k}
is the amplitude of the
k
th current harmonic,
ω
_{0}
is the fundamental angular frequency,
φ_{k}
is the phase angle, and
k
is an even integer. Specifically, the control objective of the circulating current control is to track the dc component while eliminating the harmonics.
The substitution of (4) into (3) leads to:
It is seen that the circulating current can be controlled by regulating the summation of the upperarm and the lowerarm voltages. Further, the transfer function of the circulating current model can be represented as:
 B. Repetitive Controller for the Circulating Current Control
Repetitive controllers (RCs) are widely used to provide precise tracking of periodical and highly complex reference signals. A general block diagram of an RC controller is shown in
Fig. 3
. The transfer function is:
where
e^{sT}
is a periodic time delay to produce the internal model of an ac signal
[17]
.
Q
(
s
) is a filter to guarantee stability, and
C
(
s
) is a compensator to make up the magnitude and phase of the control system.
Block diagram of a general repetitive controller (RC).
Since the circulating current of the MMC happens to be a complex signal associated with a series of current harmonics, the RC is a very suitable solution to control them. Applying an RC controller to circulating current control has been discussed in the literature
[16]
. Note that
[16]
simultaneously adopts two controllers, an RC controller and a traditional PI controller, in which the PI controller regulates the dc component of the circulating current, and the RC controller is used to attenuate unwanted ac harmonics. However, conflicts are inevitable between these two controllers. This causes complicated limitations in the controller design and degrades the performances of the RC controller.
The repetitive control scheme proposed in
[16]
is shown in
Fig. 4
, where a repetitive controller
RC
(
s
) is paralleled to a PI controller
PI
(
s
). The circulating current
i_{cj}
was then controlled to track its reference
. Thereby, the control plant seen from the repetitive controller
H'
(
s
) , can be derived as:
where
U_{RC}
(
s
) denotes the output of the repetitive controller
RC
(
s
). Hence,
H'
(
s
) needs to be carefully considered when designing the repetitive controller. Further, the openloop transfer function of the current control system can be obtained as:
Control structure in [16].
Fig. 5
and
Fig. 6
show a bode diagram of
H′
(
s
) and
G
(
s
), respectively. It should be noted that the gain of
H′
(
s
) attenuates below the break frequency of
PI
(
s
)
f_{break}
, while
PI
(
s
)=
K
_{P}
+
K
_{I}
/
s
and
f_{break}
=
K
_{I}
/
K
_{P}
.
H′
(
s
) should hold high gains at harmonic frequencies for good repetitive control. In addition, it can be seen from
Fig. 6
that the harmonics tracking capability of the repetitive control is decreased if
f_{break}
is above the 2
^{nd}
order frequency. Hence, in order to guarantee the validity of repetitive controller in
[16]
, there is a strict restriction on the PI controller so that
f_{break}
must be chosen below the 2
^{nd}
order frequency. Otherwise, conflicts between the RC and PI are inevitable.
Different plants of repetitive controllers in [16] and in this paper.
Openloop bode diagram of current control system with different f_{break} of PI controller.
III. PROPOSED CIRCULATING CURRENT CONTROL METHOD
This paper adopts a parallel structure, as shown in
Fig. 7
, for controlling the circulating current. A repetitive controller is paralleled to a proportional controller
P
(
s
) instead of a PI controller
PI
(
s
). Specifically, the proposed control scheme removes the
f_{break}
restriction, which exists in
[16]
. This is due to the fact that the integrator is inherently incorporated into the repetitive controller. Thus, conflicts are avoided and the designs of the RC and P controllers can be mutually independent.
Structure of the proposed circulating current control scheme.
 A. Inherent Integration Characteristic of Repetitive Control
With respect to the general repetitive controller shown in
Fig. 3
, and assuming that
Q
(
s
) and
C
(
s
) are both constant 1s, the ideal
RC
(
s
) can be written as:
where
ω
=2π/
T
. The following mathematical derivation can be obtained as:
where k is integer.
Substituting (12) into (11) results in:
Therefore, it is shown that
RC
(
s
) inherently consists of an integrator and a series of resonant controllers. Thus, the control gains at the resonant frequencies as well as the dc frequency are theoretically infinite. It is worth noting that the
RC
(
s
) intrinsically contains the integration function. Therefore, the integrator can be removed and the PI controller can be replaced by a simple P controller. The control plant seen from the repetitive controller now is improved as:
A bode diagram of
H
(
s
) is shown in
Fig. 5
. This indicates that the break point of the traditional PI controller is removed. Thus,
H
(
s
) can ensure a high gain at dc and harmonics frequencies. This is suitable for repetitive controllers.
Moreover, openloop bode diagrams of the circulating current control systems with different controllers are compared in
Fig. 8
. It can be observed that the P+RC structure is identical to the PI+RC structure (with same magnitude and phase). This shows that the integrator can be saved.
Comparison of system openloop bode diagrams with P, PI, (PI+RC), and (P+RC).
Furthermore, when compared to a traditional PI controller, the presented control scheme can achieve high gains at a series of resonant frequencies to effectively attenuate the ac harmonics of the circulating current.
 B. Design of Repetitive Controller
The design of the repetitive controller concentrates on the following considerations: improving both the stability of the circulating current control system and the response rate of the repetitive controller. The transfer function of the repetitive controller in the discrete domain is given as:
where
f_{sample}
is the sampling frequency,
f
_{0}
is the fundamental frequency (i.e., 50Hz), and
N_{RC}
=(
f_{sample}
/(2
f
_{0}
)) is the ratio of the sampling frequency to the 2
^{nd}
order harmonic frequency.
Further, the transfer function from the error
i_{err}
to the circulating current reference
in
Fig. 7
can be derived as:
The characteristic equation of the circulating current control system is:
A necessary condition for system stability is that all of the
N_{RC}
characteristic roots of (17) are inside the unit circle of the zplane. Thus, the following constraint must be satisfied to guarantee stability:
In order to ensure the stability of the circulating current control system,
Q
(
z
) and
C
(
z
) should be designed appropriately to satisfy (18). When
Q
(
z
)=1, the RC controller (15) can be rewritten as:
A polezero map of (19) is shown in
Fig. 9
. The poles, namely the roots of
RC
(
z
) all seat at the unit circle. Hence, it is marginally stable for
RC
(
z
), which threatens the stability of the circulating current control system. On the other hand,
Fig. 9
(a) shows that when
Q
(
z
) is selected as 0.95, all of the poles move inside the unit circle, whereas it impairs the control gain of
RC
(
z
) at lowfrequency harmonics. Thereby, the desired
RC
(
z
) for harmonic suppression is that the lowfrequency poles (i.e., 2
^{nd}
, 4
^{th}
, and 6
^{th}
harmonics) are set at the unit circle to get high control gains, while the highfrequency poles should be located inside the circle to guarantee stability. Therefore, this paper designs
Q
(
z
) as a movingaverage filter (MAF) (z
^{1}
+2+z)/4, which has zero phaseshift and attenuates rapidly at high frequencies. A polezero map of this
RC
(
z
) is included in
Fig. 9
(b). It is worth noting that the MAF only moves the high frequency poles into the unit circle without affecting the lowfrequency poles of
RC
(
z
).
Polezero maps of RC. (a). Polezero map when Q(z)=0.95. (b). Polezero map when Q(z)=(z^{1}+2+z)/4.
Moreover, the compensator
C
(
z
) is designed as:
where
K_{RC}
is a proportional coefficient and
L
is a positive integer.
z^{L}
is a phaselead element to compensate the computation delay of the DSP in real implementation. The compensated phase angle produced by
z^{L}
for a signal with a frequency of
f
is:
In addition,
K_{RC}
determines the control strength of the RC controller, a larger
K_{RC}
brings a higher convergence rate
D
and a lower steadystate error. Thus,
K_{RC}
should be selected so that it is large enough to achieve a high convergence rate. However, the system will be unstable if the oversized
K_{RC}
leads to
D
>1. Finally, the designed repetitive controller in the discrete domain is shown in
Fig. 10
.
Designed repetitive controller in discrete domain.
 C. Complete Control Structure for the MMC
The complete control strategy for the MMC is depicted in
Fig. 11
, including the proposed circulating current controller, the overall dc voltage regulation, the SM capacitor voltage balancing control, and the phaseshifted carrier PWM generator. The overall dc voltage regulation ensures that the energy stored in one phase of the MMC remains stabilized by regulating the dc component of the circulating current. The dcbus voltage
U_{dc}
, is used as a command and compared with one half of the sum of the SM capacitor voltages in phase
j
, that is:
where
U_{cap_j}
(
i
) is the capacitor voltage of the
i
th SM (
i
=1~2
N
) in phase
j
. A PI controller is then adopted to mitigate this voltage difference and to generate the reference of the dc circulating current
, that is:
where
K
_{p1}
and
K
_{i1}
are the control parameters.
Proposed current control structure for MMC.
Then, the proposed circulating current controller forces the circulating current to follow
, while it attenuates the unwanted harmonics.
K
_{p2}
is the gain of the proportional converter. Thus, the references for the upper arm and the lower arm voltages can be obtained by combining the output of the circulating current controller
u_{ref_cir}
with the ac voltage reference
u_{ref_oj}
:
Additionally, from a practical point of view, the MMC SMs can never be identical because parametric differences are unavoidable, such as the component parameter variations, unequal power distribution, and the inconsistent transmission delay of the driving signals. Further, it should be noted that all of these parametric differences will ultimately lead to a capacitor voltage unbalance among the SMs. As shown in
Fig. 9
, the SM capacitor voltage balancing control is realized by adding an adjustment to the reference signal of each SM, which multiplies the capacitor voltage error with the circulating current, that is:
where
u_{ref_bal}
(
i
) represents the balance adjustment of the
i
th SM, and
K
_{p3}
denotes the balancing gain. This equation indicates that for SMs with voltages lower than the average voltage, the product of
u_{ref_bal}
(
i
) and the circulating current
i_{cj}
will form a positive power transfer to charge these SMs. On the other hand, a negative power transfer will be generated to discharge the SMs when their voltages are higher than the average voltage.
Finally, phaseshifted triangular carriers are adopted for comparison with the references of the SMs, which produces the gating signals for the IGBTs in each of the SMs.
IV. SIMULATION
A simulation model of a threephase MMC is established in MATLAB/Simulink software to verify the validity of the proposed repetitive control method. The parameters of the simulation model are listed in
Table I
. The simulation results under traditional PI control and under the proportional repetitive control are both provided.
Fig. 12
(a) shows the simulation waveform when the traditional PI controller is adopted. It can be seen that the arm currents are clearly distorted and that significant circulating current harmonics exist. This can be confirmed by the harmonic spectrums in
Fig. 12
(b). On the other hand, when the designed RC controller is adopted, as can be observed in
Fig. 13
, almost all of the current harmonics are eliminated, which results in a smooth dc circulating current. Moreover,
Fig. 14
indicates that the SM capacitor voltages are well balanced with the proposed SM balancing control method.
PARAMETERS FOR SIMULATION
PARAMETERS FOR SIMULATION
Currents and FFT analysis of phase A in steadystate with PI. (a). Upper arm current i_{uA}, lower arm current i_{wA}, circulating current icA. (b) FFT analysis of i_{cA}.
Currents and FFT analysis of phase A in steadystate with RC. (a). Upper arm current i_{uA}, lower arm current _{iwA}, circulating current icA. (b) FFT analysis of i_{cA}.
Capacitor voltages of upperarm and lowerarm SMs with RC.
V. EXPERIMENTAL RESULTS
To further verify the effectiveness of the proposed circulating current control method, a downscaled threephase MMC prototype with three SMs per arm has been built in the laboratory, as shown in
Fig. 15
. The circuit parameters are listed in
Table II
. The proposed circulating current controller is implemented in a TMS320F28335 floatingpoint DSP, and an EP3C25Q240C8 FPGA is adopted to perform the PWM modulation and to communicate with all of the SMs. The control parameters for the proposed circulating current controller are designed as follows: the current sample frequency is
f_{sample}
=10kHz,
N_{RC}
=100,
K_{RC}
=7.8V/A, and the proportional gain is
K
_{P2}
=31.2V/A. Moreover,
L
is selected as 3 in this experiment. This shows the good compensation of the computation delays.
Photograph of the prototype.
PARAMETERS IN EXPERIMENTS
PARAMETERS IN EXPERIMENTS
The experimental results are shown in
Figs. 16

20
. The arm currents, output current, circulating current, and capacitor voltages under traditional PI control are shown in
Fig. 16
. On the other hand, the proposed circulating current control method with the RC is shown in
Fig. 17
. As further depicted in
Fig. 18
(a), it is clear that under the traditional PI control method, the 2
^{nd}
, 4
^{th}
and 6
^{th}
order harmonics are included in the circulating current, due to the limited gain of the PI controller at these ac frequencies. On the other hand,
Fig. 18
(b) indicates that the magnitudes of the circulating current harmonics are dramatically reduced as a result of the infinite gain of the repetitive controller.
Experimental results with traditional PI control method: (a). Currents of upper and lower arms i_{uA}, i_{wA}, output current i_{oA}. (b). Circlating current i_{cA}, capacitor voltage of an upper arm SM U_{cap_uA}, and a lower arm SM U_{cap_wA}.
Experimental results with proposed circulating current control method: (a). Currents of upper and lower arms i_{uA}, i_{wA}, output current i_{oA}. (b). Circlating current i_{cA}, capacitor voltage of an upper arm SM U_{cap_uA}, and a lower arm SM U_{cap_wA}.
FFT anaysis of experimental i_{cA}. (a). FFT anaysis of experimental i_{cA} with PI. (b). FFT anaysis of experimental i_{cA} with RC.
Experimental results when proposed circulating current control method is enabel at t=0.49s: (a). Currents of upper and lower arms i_{uA}, i_{wA}, output current i_{oA}. (b). Circlating current i_{cA}, capacitor voltage of an upper arm SM U_{cap_uA}, and a lower arm SM U_{cap_wA}.
Experimental results with the proposed circulating current control method with a load step change at t=0.28s: (a). Currents of upper and lower arms i_{uA}, i_{wA}, output current i_{oA}. (b). Circlating current i_{cA}, capacitor voltage of an upper arm SM U_{cap_uA}, and a lower arm SM U_{cap_wA}.
Fig. 19
gives the dynamic performances when the proposed circulating current controller replaces the PI control at t=0.49s. It can be seen that the circulating current harmonics are suppressed quickly after the proposed control is put into action. In the meantime, the capacitor voltages are kept well balanced without any deviations. In addition, the dynamic performances with a load step change are also tested, as shown in
Fig. 20
. Although overshoot occurs due to the time delay of the repetitive controller, the circulating current gradually converges within about 0.2s.
VI. CONCLUSIONS
This paper presents a novel circulating current control method to track the dc component while suppressing the lowfrequency ac harmonics. It is shown that the integrating function is inherently integrated in the RC controller, so that only a proportional controller is paralleled with the RC controller. As a result, conflicts between the RC controller and the traditional PI controller can be avoided, and the RC controller and the proportional controller can be designed independently. The accuracy and robustness of the control system are then improved. Based on the proposed circulating current control method, the complete control structure for the MMC is presented, with emphasis on the SM capacitor voltage balancing. Finally, simulation results are provided and experiments based on a threephase MMC prototype are performed. Both the simulation and experimental results confirm the validity of the proposed circulating current control method. It can be concluded that the proposed circulating current controller is a flexible and effective control solution for MMCs.
Acknowledgements
This work was supported by National Natural Science Foundation of China (51237002) and by grants from the Power Electronics Science and Education Development Program of Delta Environmental and Educational Foundation.
BIO
Binbin Li received his B.S. and M.S. degrees in Electrical Engineering from the Harbin Institute of Technology (HIT), Harbin, China, in 2010 and 2012, respectively, where he is currently working toward his Ph.D. degree. His current research interests include highpower electronics, multilevel converters, control algorithms, and PWM techniques.
Dandan Xu was born in Xinjiang, China, in 1990. She received her B.S. degree in Electrical Engineering from the Harbin Institute of Technology (HIT), Harbin, China, in 2013, where she is currently working toward her M.S. degree. Her current research interests include modular multilevel converters.
Dianguo Xu received his B.S. degree in Control Engineering from the Harbin Engineering University, Harbin, China, in 1982, and his M.S. and Ph.D. degrees in Electrical Engineering from the Harbin Institute of Technology (HIT), Harbin, China, in 1984 and 1989 respectively. In 1984, he joined the Department of Electrical Engineering, HIT, as an Assistant Professor. Since 1994, he has been a Professor in the Department of Electrical Engineering, HIT. He was the Dean of the School of Electrical Engineering and Automation, HIT, from 2000 to 2010. He is now the Assistant President of HIT. His current research interests include renewable energy generation technologies, multiterminal HVDC systems based on VSCs, power quality mitigation, speed sensorless vector controlled motor drives, and high performance PMSM servo systems. He has published over 600 technical papers. Dr. Xu is a Senior Member of the IEEE, and an Associate Editor for the IEEE Transactions on Industrial Electronics. He is serving as the Chairman of the IEEE Harbin Section, the Director of the Lighting Power Supply Committee of the CPSS, Vicedirector of the Electric Automation Committee of the CAA, the Electrical Control System and Equipment Committee of the CES, and the Power Electronics Committee of the CES.
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