A Repetitive Control Scheme for Harmonic Suppression ... - IEEE Xplore

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 30, NO. 1, JANUARY 2015

471

A Repetitive Control Scheme for Harmonic Suppression of Circulating Current in Modular Multilevel Converters Liqun He, Student Member, IEEE, Kai Zhang, Jian Xiong, and Shengfang Fan, Student Member, IEEE

Abstract—In a modular multilevel converter (MMC), the interaction between switching actions and fluctuating capacitor voltages of the submodules results in second- and other even-order harmonics in the circulating currents. These harmonic currents will introduce extra power loss, increase current stress of power devices, and even cause instability during transients. Traditional methods for circulating current harmonic suppression have problems such as limited harmonic rejection capability, limited application area, and complex implementation. This paper presents a plug-in repetitive control scheme to solve the problem. It combines the high dynamics of PI controller and good steady-state harmonic suppression of the repetitive controller, and minimizes the interference between the two controllers. It is suitable for multiple harmonic suppression, easy to implement, and applicable for both single-phase and threephase MMCs. Simulation and experimental results on a singlephase MMC inverter proved the validity of the proposed control method. Index Terms—Circulating current, harmonic suppression, modular multilevel converter (MMC), repetitive control.

NOMENCLATURE Vdc vx ix vcpj x vcn j x ipx in x vpx vn x

Dc source voltage of the MMC. Ac output voltage of phase x referred to midpoint of dc source (x = u, v, w. For a single-phase MMC, x = o.). Ac output current of phase x. Capacitor voltage of submodule j in upper arm of phase x. Capacitor voltage of submodule j in lower arm of phase x. Upper-arm current of phase x. Lower-arm current of phase x. Upper-arm voltage (total output voltage of submodules in upper arm) of phase x. Lower-arm voltage (total output voltage of submodules in lower arm) of phase x.

Manuscript received September 30, 2013; revised December 7, 2013; accepted January 23, 2014. Date of publication February 6, 2014; date of current version August 26, 2014. Recommended for publication by Associate Editor J. Mahseredjian. The authors are with the State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: [email protected]; kaizhang@ hust.edu.cn; [email protected]; shengfangfan@ hust.edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPEL.2014.2304978

vz x iz x Iz x iz x iz x ω1 L R C

0 1 k

Common-mode component of the upper- and lower-arm voltages of phase x. Circulating current of phase x. Dc component of iz x . Fundamental component of iz x . k-order harmonic of iz x . Fundamental angular frequency of an output ac voltage. Equivalent inductance in each arm. Equivalent resistance in each arm. Capacitance of each submodule.

I. INTRODUCTION ULTILEVEL converters have successfully made their way into industrial high-power applications [1]. Among the multilevel topology family, the modular multilevel converter (MMC) is attracting increasing interests for the advantages of modular structure, inherent redundancy, distributed and reliable dc capacitors, improved power quality, four-quadrant operation, freedom from multiple isolated dc sources, ease of expandability, etc. [2]–[4]. In a modular multilevel converter, the chief ripple component of the submodule (SM) capacitor voltage is the fundamental one. It can produce second-order harmonic in the output voltage of the SMs. The latter then causes second-order harmonic in the circulating current that flows through the dc source and the phase leg. Without proper control, the amplitude of the secondorder circulating current can be significant, and it can trigger a series of higher, even-order current harmonics. Detailed analysis of steady-state circulating current has been given in [5] and [6]. The harmonics in the circulating current increase power losses and reduce service life of power devices. More seriously, if left uncontrolled, they may cause instability during transients [7]. In [8] and [9], the authors present open-loop methods for circulating current control and harmonic suppression. A drawback of these methods is parameter sensitivity. In [10], a PI controller is used to suppress the circulating current harmonics. However, it cannot totally remove the harmonics due to its limited gains at those harmonic frequencies. In [11], a pair of PI controllers based on double line-frequency, negative-sequence rotating frame, are utilized to eliminate the second-order harmonic. Such rotating frames are difficult to define in singlephase systems. In addition, this method may need some modification under unbalanced three-phase conditions. Proportionalresonant (PR) controllers [12]–[14] are applicable to both singlephase and three-phase systems. However, implementation of PR

M

0885-8993 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

472


controllers becomes difficult when multiple harmonics are to be eliminated. In [15], a repetitive-plus-PI control scheme is proposed. It is applicable to both single-phase and three-phase systems. Moreover, it can eliminate multiple harmonics in the circulating current with a single repetitive controller. However, the repetitive controller and the PI controller are paralleled in [15]. Such an arrangement imposes unnecessary limitation on the PI controller design and also complicates the repetitive controller design. This paper proposes a different repetitive-plus-PI control scheme. The improved plug-in configuration of the repetitive controller avoids the above problems while keeping all the advantageous features. From the power balancing point of view, the secon-order ripple power consumed by the load of a single-phase MMC inverter must equal the second-order ripple power provided by the dc source and SM capacitors. (This relationship also holds for each phase of a three-phase MMC inverter.) Since eliminating the second-order component in the circulating current effectively cuts out the supply of the second-order ripple power from the dc source, it is then a reasonable doubt that the SM capacitors will have to provide more second-order ripple power, which means severe second-order fluctuation of the SM capacitor voltage. A discussion is given in Section IV to clarify this issue, which reveals that the SM capacitors do not suffer from higher second-order voltage harmonic with the proposed control scheme. This paper is organized as follows. In Section II, a unified model of circulating current is built, which is applicable to both single-phase and three-phase MMCs. The entire circulating current control structure is presented and discussed as well. Section III presents the design procedure of the proposed repetitive controller. Sections IV and V validate the effectiveness of proposed control scheme by simulation and experimental results. Section VI concludes the paper.

Fig. 1.

Topology of a three-phase MMC (as an inverter).

Fig. 2. (a) Equivalent circuit of one-phase leg. (b) Single-phase equivalent circuit of circulating current.

II. MODELING AND ANALYSIS OF CIRCULATING CURRENT Substituting (3) into (2) yields

A. Mathematical Model of Circulating Current Fig. 1 is the topology of a three-phase MMC inverter. The total output voltages of the SMs in the upper (or lower) arm of each phase can be modeled as controlled voltage sources vpx (or vn x ), as shown in Fig. 2(a). The circulating current is defined as iz x = (ipx + in x )/2.

(1)

Differential equations of arm currents are given as follows: ⎧ Vdc dipx ⎪ ⎪ − vx − L − Ripx ⎨ vpx = 2 dt ⎪ ⎪ ⎩ vn x = Vdc + vx − L din x − Rin x 2 dt ⎧ ix ⎪ ⎨ ipx = iz x + 2 ⎪ ix ⎩ in x = iz x − . 2

(2)

(3)

2L

diz x + 2Riz x = Vdc − (vpx + vn x ). dt

(4)

Therefore, the control of iz x is realized by adjusting (vpx + vn x ), i.e., the common mode component of the arm voltages, defined as vz x . The equivalent circuit of iz x is shown in Fig. 2(b). According to [5] and [6], if the modulating signals for the SMs are purely sinusoidal (with dc offset), iz x should consist of a dc component Iz x 0 and even-order harmonics iz x k (k = 2, 4, 6. . .). Eliminating these harmonics is the purpose of this study. The dc component Iz x 0 in the circulating current represents the average power supplied by the dc source, which is to be consumed by the load and SM capacitors. This component can be used for total energy control, i.e., the control of average voltages of all the SM capacitors in one-phase leg. The fundamental component iz x 1 in the circulating current can transport energy between the upper- and lower-arm capacitors (unless it

HE et al.: REPETITIVE CONTROL SCHEME FOR HARMONIC SUPPRESSION OF CIRCULATING CURRENT IN MMC

Fig. 3.

473

Overall circulating current control of MMC (for phase x).

is orthogonal to the ac output voltage). Therefore, iz x 1 can be utilized in the differential energy control between the upper and lower arms [7], [16]. B. Control System for Circulating Current The overall control system for circulating current is shown in Fig. 3. The proposed control system consists of the basic proportional-integral (PI) controller and a repetitive controller, the detail of which will be discussed in the next section. For a three-phase MMC inverter, three such control systems are needed. The circulating current reference i∗z x consists of a dc component coming from total energy control, and a fundamental component coming from the differential energy control. Since the dc component i∗z x 0 in the current reference can be easily corrupted by the harmonics in the capacitor voltages, a moving average filter (MAF) is inserted in the i∗z x 0 path. The time span of the MAF is chosen as half the fundamental period, since the lowest order ripple in the summed capacitor voltages of one-phase leg is the second-order one. The output of the circulating current control system Δ vz x is halved and then added to the normal components of the arm voltage references ( V2d c − vx∗ and V2d c + vx∗ ) to form the final ∗ and vn∗ x ). vx∗ is the desired ac output voltage of references (vpx phase x: vx∗ = M

Vdc sin(ωt + ϕx ), M ∈ [0, 1]. 2

(5)

III. DESIGN AND ANALYSIS OF THE PROPOSED REPETITIVE CONTROLLER

Fig. 5.

A. Control System Configuration For its better transient performance, the PI control is kept as a preliminary measure of harmonic suppression. The block diagram and the Bode plot of the PI controlled circulating current loop are as shown in Fig. 4. The transfer function of the PI controller is: Kp s + K i PI(s) = . (6) s According to (3), the transfer function of plant G(s) is G(s) =

1 . 2Ls + 2R

Fig. 4. (a) Block diagram and (b) Bode plot of the PI controlled circulating current loop.

(7)

Control structure proposed in [15].

In [15], a repetitive controller is paralleled with the PI controller to improve harmonic suppression. The resulted control structure is shown in Fig. 5. Such a parallel configuration of a repetitive controller and the existing high-dynamic controller (the PI controller in this case) can also be found in [17]–[20]. In this paper, however, a different structure as shown in Fig. 6 is adopted. An important consideration behind a parallel structure is perhaps the transient performance. It seems that by paralleling the two controllers instead of cascading them, the quick response of the PI controller will not be affected by the slow repetitive controller. But the seemingly “cascaded” structure in Fig. 6 can also

474


Fig. 6.

Proposed plug-in repetitive control structure.

Fig. 7.

Equivalent form of Fig. 6.

avoid this. With the feedforward path of current reference, the control structure in Fig. 6 is equivalent to the classical “plug-in” structure [21]–[24] as shown in Fig. 7. It is easier to see from Fig. 7 that the plug-in repetitive controller is more separated from the PI controlled circulating current loop. Therefore, the PI controller can be designed independently, and the dynamics of the whole system will not be hold back by the repetitive controller. The plug-in repetitive controller only deals with the residual, repetitive error left by the PI controller. The proposed control structure also provides a more friendly “plant” for the repetitive controller. In Fig. 6 or Fig. 7, the plant (i.e., all the dynamics from yr p to iz x ) as seen from the repetitive controller is the PI-controlled circulating current loop, with the transfer function P (s) =

PI(s)G(s) . 1 + PI(s)G(s)

(8)

The frequency characteristics of P (s) has already been given in Fig. 4(b). It exhibits unity gain from zero frequency up to the break frequency, and then a monotonically decreasing gain after the break frequency. This is a frequency characteristic that is most desirable for the repetitive controller design [23]. With the parallel structure, the plant of the repetitive control is P (s) =

G(s) . 1 + PI(s)G(s)

(9)

The difference between (9) and (8) is the transfer function of the PI controller. Shown in Fig. 8 are the equivalent block diagram and the Bode plot of P (s). The gain curve of P (s) becomes a trapezoid. The constant-gain region (where it is convenient for the repetitive controller to achieve good command following and disturbance rejection) greatly narrows. As indicated in [23], slope sections in the gain curve pose potential threats for stability of the repetitive control system. With a gain curve like Fig. 8(b), the design of the repetitive controller becomes difficult, since the designer now faces two instead of only one such slop section. It also should be noted that in order for the fundamental frequency to remain in the flat-top region of the gain curve, the break frequency fP I of the PI controller has to be kept well

Fig. 8. (a) Block diagram and (b) Bode plot of the plant of RP controller for parallel control structure.

below the fundamental frequency. This is clearly an unwelcome limit imposed on PI controller design, which may affect the optimization of PI parameters. B. Design of the Proposed Repetitive Controller Fig. 6 is redrawn in z-domain and with more detail in Fig. 9, which shows the inner structure of the repetitive controller. For convenience, the disturbance has been equivalently moved to the output side of G(z). The design process of the repetitive controller is based on the parameters listed in Table I. The sampling frequency fs of the control system equals the carrier frequency (which is also the equivalent switching frequency of the MMC); therefore, fs = 4 kHz. Since odd-order harmonics may also arise due to imperfections in practical applications, and the current reference contains fundamental component, the base frequency of the repetitive controller is chosen as the fundamental frequency (50 Hz) of the MMC. Therefore, there are Ns = 80 error samplings within one repetitive control cycle. Note that although phase disposition PWM (PDPWM) is employed here, other PWM techniques, e.g., carrier phase-shifted PWM (CPSPWM), can also be used with the proposed control method, provided that the equivalent switching frequency is the same. As the core of the repetitive controller, the modified internal model 1/(1 − Q(z)z N s ) integrates the error on a cycle basis. The Q(z) filter, which enhances robustness by slightly attenuating the integration action, can be a constant close to 1, or a low-pass filter. Q(z) = (z + 4 + z −1 )/4 is adopted here. It is a low-pass filter with zero-phase shift at low frequencies. The time delay unit z −N s postpones control action by one repeating period so that the time advance unit z k (for phase compensation purpose) as well as the noncausal Q(z) filter can be realized.


Fig. 9.

475

Detailed block diagram of the proposed repetitive control scheme for circulating current. TABLE I PARAMETERS OF THE SINGLE-PHASE MMC UNDER STUDY

Fig. 10.

Bode plot of S(z)P (z) and z −k .

The characteristic equation of the repetitive control system is z N s − [Q(z) − Kr S(z)z k P (z)] = 0.

In the repetitive compensator Kr S(z)z k , the filter S(z) provides magnitude compensation for the plant P (z). Since the low-frequency gain of P (z) is already a constant 0 dB, S(z) is chosen as a simple secon-order low-pass filter with natural frequency ωn = 10ω1 and damping ratio ζ = 1. It provides a steeper descending slope of the gain curve at high frequencies, which is helpful to system stability. Time advance unit z k performs phase compensation for S(z)P (z). Fig. 10 shows that z 4 can perfectly cancel out the phase delay of S(z)P (z) up to10ω1 . Repetitive gain Kr (0 < Kr ≤ 1) ultimately determines the amplitude of controller output yr p . A smaller Kr brings larger stability margin but slower error convergence speed. The transfer function from the error signal ierr (z) to reference i∗z x (z) in Fig. 9 is:

ierr (z) =

[1 − P (z)][z N s − Q(z)] i∗ (z) z N s − [Q(z) − Kr S(z)z k P (z)] z x −

[1 − P (z)][z N s − Q(z)] iz x k (z). z N s − [Q(z) − Kr S(z)z k P (z)]

(10)

(11)

The necessary and sufficient condition for system stability is that Ns roots of (11) are inside the unity circle centered at the origin of the z-plane. To simplify the design process, a sufficient condition for system stability can be derived by small gain theorem [25]: Q(ej ω t ) − Kr S(ej ω t )ej k ω t P (ej ω t ) < 1, ω ∈ [0, π/Ts ] (12) Define H(ej ω t ) = Q(ej ω t ) − Kr S(ej ω t )ej k ω t P (ej ω t ). Equation (12) means that the end of vector H(ej ω t ) should never exceed the unity circle. Smaller magnitude of H(ej ω t ) indicates larger stability margin, faster error convergence, and better steady-state harmonic suppression [23]. Fig. 11(a) shows the locus of vector H(ej ω t )(ω ∈ [0, π/Ts ]) with Kr = 1. The entire locus remains well within the unity circle. A similar Nyquist plot based on parameters of [15] is given in Fig. 11(b) for comparison. The initial section of the locus is found to be dangerously close to the stability boundary. This is because the low-frequency gains of P’(s) are significantly below 0 dB in the parallel control structure. If the control structure is changed to the proposed plug-in style shown in Fig. 6 or Fig. 7 while all the control parameters are kept unchanged (except for the repetitive gain being reduced Kp times to account for the different loop gains of these two structures), the Nyquist plot will become the one shown in Fig. 11(c). The stability margin is improved, and magnitudes of H(ej ω t ) at 50 and 100 Hz are

476


Fig. 11. Nyquist plot of H (ej ω t ) for (a) proposed plug-in repetitive control structure, (b) parallel repetitive control structure presented in [15], and (c) the same RP controller of [15] but with proposed plug-in structure.

Fig. 12. Comparison of open-loop gains |G(ej ω t )|, |PI(ej ω t )G(ej ω t )|, and |G r p (ej ω t )P (ej ω t )|.

Fig. 14. Capacitor voltages of all SMs in one phase leg (simulation results). (a) With PI controller only. (b) With MAF + PI controller. (c) With MAF + PI controller +proposed repetitive controller.

Fig. 13. Upper arm current ip , lower arm current in , and circulating current iz (simulation results). (a) With PI controller only. (b) With MAF + PI controller. (c) With MAF + PI controller +proposed repetitive controller.

reduced, indicating better command-following and disturbancerejection at these frequencies. The performance of the parallel structure can definitely be improved with more sophisticated repetitive controller design [17], [19]. But this seems an unnecessary effort since a better control structure could have prevented most of the difficulties in the first place.


477

Fig. 15. Phasor diagram of second-order pulsating power when circulating currents are separately controlled by (a) PI controller only, (b) MAF + PI controller, and (c) MAF + PI controller + proposed repetitive controller.

C. Harmonic Rejection Ability The transfer function of a repetitive controller in Fig. 9 is Kr S(z)z k yr p (z) = . (13) i∗z x (z) z N s − Q(z) The open-loop gains of original plant G(ej ω t ), PI control system |PI(ej ω t )G(ej ω t )|, and repetitive control system |Gr p (ej ω t )P (ej ω t )| are compared in Fig. 12. Fig. 12 indicates that the repetitive controller provides much higher gains at fundamental frequency and its multiples than the PI controller. Therefore, the repetitive controller can achieve much better command following and harmonic suppressing capability. If Q(z) = 1, and the disturbance as well as reference are purely repetitive, i.e., z N s i∗z x = i∗z x , z N s iz x k = iz x k , then (10) becomes Gr p (z) =

z N s ierr (z) = H(z)ierr (z)

(14)

Equation (14) reveals that after each control cycle (i.e., fundamental period), the magnitude of the error reduces to |H(ej k ω 1 t )| times its previous value. For the chief harmonic component iz x 2 , |H(ej 2ω 1 t )| = −13.8 dB, indicating that the second-order harmonic in the circulating current can be suppressed within 3∼4 fundamental periods by the repetitive controller. Due to the dynamics of the MMC (mainly the SM capacitors’ voltages) in practical operation, the “purely repetitive” assumption does not hold exactly; therefore, the actual error convergence will be slower. IV. SIMULATION RESULTS A simulation model of a single-phase MMC inverter is established in MATLAB/Simulink to verify the proposed control scheme. The parameters are already given in Table I. Three cases are simulated: 1) iz is controlled by the PI controller only; 2) the moving average filter (MAF) is added; and 3) the proposed repetitive controller is also added. The arm currents and circulating currents in the steady state are shown in Fig. 13, which indicates that the moving average filter provides positive but limited effect on harmonic suppression, while

the proposed repetitive controller can eliminate nearly all the harmonics in the circulating current. The peak values of arm currents in the above three cases are respectively 11.7, 10.6, and 9.2 A. The reduced peak current is beneficial for the power devices. This advantage becomes more important for high-power applications. Now that the circulating current is rid of the second-order content, the dc source of the MMC no longer supplies the secondorder ripple power consumed by the single-phase load. The latter then has to be supplied solely by the SM capacitors. To investigate whether the second-order ripple power is increased for the SM capacitors, Fig. 14 shows the capacitor voltages of the eight SMs, in which vcp1 ∼vcp4 are capacitor voltages of SM1 ∼SM4 in the upper arm and vcn 1 ∼vcn 4 are capacitor voltages of SM1 ∼SM4 in the lower arm. Since voltage-balancing control keeps these voltages almost identical, the FFT analysis is only done for vcp1 . It is interesting to see that among the three situations, there is no significant change in the second-order harmonic of the capacitor voltages, indicating no significant change of second-order ripple power. The following is an analysis of the reason. Define the second-order ripple powers of the dc source, the load, and the SM capacitors as Pdc 2 , Po 2 and PC 2 , respectively. These three powers should balance out if the ripple power of the buffer inductors is neglected. Based on the simulation data, the phasor diagram of Pdc 2 , Po 2 , and PC 2 in each situation is given in Fig. 15. The phase of the output voltage vx is taken as the phase reference. In these three powers, Po 2 is determined by the load and therefore is almost the same, while other two are quite dependent on the control method. In Fig. 15(a), amplitude of Pdc 2 is significant due to the significant second-order harmonic in the circulating current [as shown in Fig. 13(a)]. Fig. 15(a) also shows that Pdc 2 and PC 2 are so out of phase that the sum of their amplitudes far exceeds the need of the load (Po 2 ). In Fig. 15(b), the phase angle between Pdc 2 and PC 2 narrows a bit. Consequently, Pdc 2 becomes smaller. In Fig. 15(c), Pdc 2 is eliminated and Po 2 is totally supplied by PC 2 . It then becomes clear that since Pdc 2 counteracts PC 2 in the first two cases, its elimination does not result in an increase of PC 2 .

478


Fig. 17.

Fig. 16. board.

Overview of the test setup.

(a) Submodules of the MMC inverter. (b) DSP + FPGA control

The above simulation results are for a single-phase MMC inverter. In a three-phase MMC inverter, the circulating current of each phase can be controlled independently. Therefore, the proposed control scheme can be directly applied to three-phase scenario.

Fig. 18. Upper arm current ip , lower arm current in , and circulating current iz (experimental results) (a) With PI controller alone. (b) With proposed repetitive control scheme.

V. EXPERIMENTAL RESULTS Experiments are carried out on a single-phase MMC test setup, of which the main parameters have been summarized in Table I. Fig. 16 shows the submodules and the control board of the MMC inverter. The control system is implemented with a TMS320F2812 DSP from Texas Instruments. An EP1C12Q240I7 FPGA from Altera then receives the modulating signals from the DSP and performs the pulse-width modulation. Fig. 17 is an overview of the test setup. A. Steady-State Performance Fig. 18(a) shows the arm currents ip , in , and circulating current iz in the steady state when the PI controller is used alone. Fig. 18(b) shows the same currents when the proposed repetitive controller is in effect. Fig. 19 gives FFT analysis of the circulating current in these two situations. It can be seen that harmonics are sufficiently suppressed by the repetitive controller. The peak value of arm currents in Fig. 18(a) is 11.6 A. It is reduced to 9.2 A in Fig. 18(b). Shown in Fig. 20 are waveforms of the capacitor voltages, output voltage, and load current with the proposed control scheme. Fig. 21 compares the capacitor voltages vcp1 and vcn 1 with and without the proposed repetitive control scheme. The maximum voltage as well as the peak-to-peak ripple voltage sees no

Fig. 19. FFT analysis of circulating current. (a) With PI controller alone. (b) With proposed repetitive control scheme.

Fig. 20. Experimental waveforms of (a) all SM capacitor voltages, and (b) ac output voltage v o and current io (with proposed control scheme).


Fig. 22.

479

Startup process of proposed control scheme (At t = t1 , RP controller starts to work).

Fig. 23. Output voltage/current, capacitor voltages, and circulating current of the MMC inverter during 50% load step change (experimental results). (a) With PI controller alone. (b) With proposed control scheme.

significant increase after the proposed repetitive control scheme is applied.

B. Dynamic Performance Fig. 22 shows the dynamic error convergence of the proposed repetitive control scheme. Before t = t1 , circulating current is controlled by the PI controller alone, and residual second-order harmonic in the circulating current can be clearly seen. At t =

t1 , the proposed repetitive control scheme starts to work. The harmonic content is suppressed in about 8 fundamental periods. Fig. 23 compares the transient processes of 50% load change with the PI controller alone and with the proposed control scheme. It can be seen that the dynamic response of the MMC inverter has not been affected by the plug-in repetitive controller. This is consistent with the conclusion in Section III-A, which states that the transient performance is determined by the PI controller while the repetitive controller only deals with the residual error so as to improve steady-state harmonic suppression.

480


[10] [11] [12]

Fig. 21. SM capacitor voltages of upper arm and lower arm. (a) With PI controller alone. (b) With proposed repetitive control scheme.

VI. CONCLUSION The second-order as well as other higher order harmonics in circulating current brings extra power losses and may affect stable operation of the MMC. This paper proposed a “PI + Repetitive” control scheme to suppress these harmonics in the circulating current. It greatly improves the harmonic suppression of the conventional PI controller. It is applicable to both single-phase and three-phase systems, and is able to eliminate multiple harmonics with a single controller. Compared with another “PI + Repetitive” control scheme in which the two controllers are paralleled, the control structure proposed in this paper results in a more friendly plant for the repetitive controller, and poses no design limit on the PI controller. Simulation and experiments are made on a single-phase MMC inverter. The results show good harmonic suppression of the proposed control scheme, and indicate that the plug-in repetitive controller does not affect the transient performance of the PI controller. The results also show that after the second-order harmonic in the circulating current is cut out, there is no significant increase of second-order ripple in the SM capacitor voltages.

[13]

[14] [15]

[16]

[17] [18]

[19]

[20] [21] [22]

REFERENCES [1] S. Kouro, M. Malinowski, K. Gopakumar, J. Pou, L. G. Franquelo, B. Wu, J. Rodriguez, M. A. Pérez, and J. I. Leon, “Recent advances and industrial applications of multilevel converters,” IEEE Trans. Ind. Electron., vol. 57, no. 8, pp. 2553–2580, Aug. 2010. [2] A. Lesnicar and R. Marquardt, “An innovative modular multilevel converter topology suitable for a wide power range,” in Proc. IEEE Power Tech Conf., 2003, pp. 1–3. [3] M. Glinka and R. Marquardt, “A new AC/AC multilevel converter family,” IEEE Trans. Ind. Electron., vol. 52, no. 3, pp. 662–669, Jun. 2005. [4] H. Akagi, “Classification, terminology, and application of the modular multilevel cascade converter (MMCC),” IEEE Trans. Power Electron., vol. 26, no. 11, pp. 3119–3130, Nov. 2011. [5] Q. Song, W. Liu, X. Li, H. Rao, S. Xu, and L. Li, “A steady-state analysis method for a modular multilevel converter,” IEEE Trans. Power Electron., vol. 28, no. 8, pp. 3702–3713, Aug. 2013. [6] K. Ilves, A. Antonopoulos, S. Norrga, and H.-P. Nee, “Steady-state analysis of interaction between harmonic components of arm and line quantities of modular multilevel converters,” IEEE Trans. Power Electron., vol. 27, no. 1, pp. 57–68, Jan. 2012. ¨ [7] A. Antonopoulos, L. Angquist, and H.-P. Nee, “On dynamics and voltage control of the modular multilevel converter,” in Proc. Eur. Conf. Power Electron. Appl., Barcelona, Spain, Sep. 8–10, 2009, pp. 1–10. [8] L. Angquist, A. Antonopoulos, D. Siemaszko, K. Ilves, M. Vasiladiotis, and H. Nee, “Inner control of modular multilevel converters—An approach using open-loop estimation of stored energy,” in Proc. Int. Power Eng. Conf., Sapporo, Japan, Jun. 21–24, 2010, pp. 1579–1585. [9] K. Ilves, A. Antonopoulos, L. Harnefors, S. Norrga, and H.-P. Nee, “Circulating current control in modular multilevel converters with fundamental

[23] [24] [25]

switching frequency,” in Proc. Int. Power Electron. Motion Control Conf., Harbin, China, Jun. 2–5, 2012, pp. 249–256. M. Hagiwara and H. Akagi, “Control and experiment of pulsewidthmodulated modular multilevel converters,” IEEE Trans. Power Electron., vol. 24, no. 7, pp. 1737–1746, Jul. 2009. Q. Tu, Z. Xu, and L. Xu, “Reduced switching-frequency modulation and circulating current suppression for modular multilevel PWM Converters,” IEEE Trans. Power Del., vol. 26, no. 3, pp. 2009–2017, Jul. 2011. X. She and A. Huang, “Circulating current control of double-star choppercell modular multilevel converter for HVDC system,” in Proc. Annu. Conf. ´ IEEE Ind. Electron. Society, ETS Montréal, QC, Canada, Oct. 25–28, 2012, pp. 1234–1239. X. She, A. Huang, X. Ni, and R. Burgos, “AC circulating currents suppression in modular multilevel converter,” in Proc. Annu. Conf. IEEE ´ Ind. Electron. Society, ETS Montréal, QC, Canada, Oct. 25–28, 2012, pp. 190–194. Z. Li, P. Wang, Z. Chu, H. Zhu, Y. Luo, and Y. Li, “An inner current suppressing method for modular multilevel converter,” IEEE Trans. Power Electron., vol. 28, no. 11, pp. 4873–4879, Nov. 2013. M. Zhang, L. Huang, W. Yao, and Z. Lu, “Circulating harmonic current elimination of a CPS-PWM-based modular multilevel converter with a plug-in repetitive controller,” IEEE Trans. Power Electron., vol. 29, no. 4, pp. 2083–2097, Apr. 2014. S. Fan, K. Zhang, J. Xiong, and Y. Xue, “Control structure, improved modulation and voltage balancing method for modular multilevel converters,” in Proc. IEEE Energy Convers. Congr. Expo., Denver, CO, USA, Sep. 16–20, 2013, pp. 4000–4007. P. Mattavelli, L. Tubiana, and M. Zigliotto, “Torque-ripple reduction in PM synchronous motor drives using repetitive current control,” IEEE Trans. Power Electron., vol. 20, no. 6, pp. 1423–1431, Nov. 2005. X. H. Wu, S. K. Panda, and J. X. Xu, “Design of a plug-in repetitive control scheme for eliminating supply-side current harmonics of threephase PWM boost rectifiers under generalized supply voltage conditions,” IEEE Trans. Power Electron., vol. 25, no. 7, pp. 1800–1810, Jul. 2010. D. Chen, J. Zhang, and Z. Qianu, “Research on fast transient and 6n±1 harmonics suppressing repetitive control scheme for three-phase gridconnected inverters,” IET Power Electron., vol. 6, no. 3, pp. 601–610, Mar. 2013. Y. Cho and J. Lai, “Digital plug-in repetitive controller for single-phase bridgeless PFC converters,” IEEE Trans. Power Electron., vol. 28, no. 1, pp. 165–175, Jan. 2013. C. Cosner, G. Anwar, and M. Tomizuka, “Plug in repetitive control for industrial robotic manipulators,” in Proc. IEEE Int. Conf. Robot. Autom., May 13–18, 1990, pp. 1970–1975. C. Kempf, W. Messner, M. Tomizuka, and R. Horowitz, “Comparison of four discrete-time repetitive control algorithms,” IEEE Control Syst. Mag., vol. 13, no. 6, pp. 48–54, Dec. 1993. K. Zhang, Y. Kang, J. Xiong, and J. Chen, “Direct repetitive control of SPWM inverter for UPS purpose,” IEEE Trans. Power Electron., vol. 18, no. 3, pp. 784–792, May 2003. Z. Keliang and D. Wang, “Digital repetitive controlled three-phase PWM rectifier,” IEEE Trans. Power Electron., vol. 18, no. 1, pp. 309–316, Jan. 2003. G. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback Control of Dynamic Systems. Reading, MA: Addison-Wesley, 1991.

Liqun He (S’12) was born in Hubei, China, in 1989. She received the B.E. degree from the Huazhong University of Science and Technology, Wuhan, China, in 2010. She is currently working toward the Ph.D. degree at the School of Electrical and Electronic Engineering. Her research interests include design and control of power electronics systems, high-power factor rectifiers, and modular multilevel converters.


Kai Zhang was born in Henan Province, China. He received the B.E., M.E., and Ph.D. degrees from the Huazhong University of Science and Technology (HUST), Wuhan, China, in 1993, 1996, and 2001, respectively. He joined HUST as an Assistant Lecturer in 1996. He was a Visiting Scholar at the University of New Brunswick, Fredericton, NB, Canada, during 2004– 2005. He was promoted to a Full Professor in 2006. He is an author of more than 40 technical papers. His research interests include uninterruptible power system, railway traction drives, modular multilevel converters, and electromagnetic compatibility techniques for power electronic systems.

Jian Xiong was born in Hubei Province, China. He received the B.E. degree from East China Shipbuilding Institute, Zhenjiang, China, in 1993, and the M.E. and Ph.D. degrees from the Huazhong University of Science and Technology (HUST), Wuhan, China, in 1996 and 1999, respectively. He joined HUST as a Lecturer in 1999, where he was promoted to an Associate Professor in 2003. His research interests include uninterruptible power system, ac drives, switch-mode rectifiers, STATCOM, and the related control techniques.

481

Shengfang Fan (S’12) was born in Hubei, China, in 1985. He received the B.E. degree from the Huazhong University of Science and Technology, Wuhan, China, in 2009. He is currently working toward the Ph.D. degree at the School of Electrical and Electronic Engineering. From December 2011 to November 2012, he was an Intern in Corporate Technology, Siemens Corporation in Princeton, NJ, USA. His research interests include design and control of power electronics systems, ac drives, high-power factor rectifiers, and modular multilevel converters.