Transient Oscillation Suppression Method of Modular Multilevel DC Transformer

The modular multilevel DC transformer (MMDCT) is more likely to suffer from transient oscillation, which is caused by insufficient damping of the equivalent RLC circuit with smaller arm resistance and submodule capacitance and may threaten the system stability. To tackle this issue, this paper conducts an in-depth analysis of the oscillation generating mechanism and thereby proposes a transient oscillation suppression method by incorporating an oscillation suppression angle (OSA) between the upper and lower arm switching voltages, with which the direction of the circulating current could be adjusted during this overlapped period. However, the open loop control of OSA shows indistinctive improvement of the damping ratio so that a damping compensator for regulating the circulating current is utilized. Furthermore, the compensating gain is optimally designed to achieve the critical damping state based on the small-signal model viewing OSA as an input variable. Besides, the outer capacitor voltage control loop and the power control loop are incorporated to provide the circulating current reference and to track the load power demand, respectively. Extensive simulation results from a realistically designed MMDCT model in PSIM validate the feasibility and effectiveness of the proposed transient oscillation suppression method.


I. INTRODUCTION
As the concept of sustainable development is deeply rooted in people's minds, the development and utilization of renewable energy represented by hydropower, wind energy and solar energy has become a hot topic worldwide. However, due to the inherent intermittent nature of renewable energy resources, high penetration of renewable energy sources poses serious challenges to the stable operation of existing power systems. On the contrary, the DC power grid is considered to be a preferable way for renewable energy access in the future because it does not require reactive power compensation and frequency synchronization [1]. Therefore, as the The associate editor coordinating the review of this manuscript and approving it for publication was Xiaorong Xie . core component of the DC power grid, DC transformers have received more and more attention in recent years [2].
A typical DC transformer is the dual active bridge (DAB) structure [3], [4], which realizes the bidirectional power flow by properly applying voltage across the leakage inductance of the medium-frequency transformer. Furthermore, the conflict between the low voltage stress of commercial power semiconductor devices and high voltage rating of the power grid is preferably managed by using multilevel concepts. Due to the expansibility and flexibility of the modular multilevel converter (MMC) [5]- [9], a modular multilevel DC transformer (MMDCT) based on half bridge submodule (SM) is proposed in [10], [11].
A majority of papers [12]- [19] have cast attention to the common problem of the DC transformers, such as soft switching characteristics, current stress and efficiency optimization VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ to improve reliability and stability based on the square wave modulation scheme. Nevertheless, the issues unique to MMDCT are more worthy of in-depth study.
To be specific, the series connected structure of submodule capacitors, the arm inductor along with its parasitic resistor is liable to suffer from RLC resonance issue, which also occurs in some PWM controlled grid-connected circuits such as current/voltage source rectifier [21]- [25], indirect matrix converter [26] and voltage source motor drives [27], [28], with the LC filter incorporated to alleviate the harmonic distortion at the input side. Hence the associated transient oscillation will be excited in the LC filter by the outer transient disturbances and the grid voltage harmonics. The damping resistors physically existed within the LC elements does help to increase the damping ratio, but it is preferred to be avoided due to the unexpected power losses [28], especially in high power applications. On the other hand, the approach of using active damping to suppress the LC oscillation has been investigated and can be categorized into two types: virtual damping resistor [21], [22] and control signal shaping. To be specific, the virtual damping resistor is proposed in [21] aiming at equivalently increasing the resistor via control methods while the possible position of the virtual resistor significantly affects the implementation of the control algorithm. The other perspective to achieve active damping by modifying the control signal with compensators includes changing the modulation index in one cycle of the resonance frequency [23], voltage and current feedback control [24] and state feedback control by pole placement approach [27], etc. To synthesis the merits in both categories, a combination of the virtual resistor and control signal shaping is proposed in [23], [25], [26] to achieve better LC resonance mitigation. To sum up, these proposed techniques target at suppressing the input AC side LC oscillation and alleviating the existed harmonic distortion simultaneously.
When it comes to the modular structure such as MMC in high power applications, the damping resonance is distinct from that in AC-DC converters, since the inherent RLC series structure inclines to trigger the transient oscillation in the circulating current and the capacitor voltage at the DC-link side. Aside from that, the second-order circulating current component needs special attention in such topologies. An oscillation suppression method based on active damping is proposed in [29], [30] for MMC based on the small signal state-space model, with which the equivalent parasitic resistance in the arm inductor is increased and the damping ratio is correspondingly enhanced. Reference [31] deals with the RLC oscillation issue under the condition of variable frequency operation through closed-loop controlling the circulating current. Whereas, the quantitative relationship between the controller parameter and the damping property remains unclear.
However, the studies in [29]- [31] only focus on the oscillation suppression in a traditional PWM modulated MMC with sinusoidal modulation, where the modulation signal can be readily augmented. These strategies are inapplicable for the MMDCT scenario because the DC transformer typically adopts the square wave modulation for achieving higher DC voltage utilization and smaller lower frequency energy ripples [10], [11]. Therefore, because they do not need to buffer low-frequency power ripples, the specific characteristics of the RLC series configuration in MMDCT are more worthy of attention since the submodule capacitors in MMDCT are usually designed smaller than those in traditional MMCs [18], [19]. The smaller submodule capacitance value increases the possibility of oscillation, which seriously threatens the stability of the system. Moreover, the parasitic resistance in the inductance of the phase leg needs to be as small as possible to reduce the system loss, but it will in turn limit the decay rate of oscillation. A dual closed-loop control method in proposed in [32] to cope with the oscillation suppression in a full-bridge back-to-back MMDCT based on '−1/2' modulation. However, the capacitor voltages in each arm should be dynamically sorted to determine which submodule is inserted or bypassed to modify the arm voltage, and the design of the controller parameters is not provided.
Therefore, this paper first analyzes the causes and effects of the transient oscillation phenomenon in MMDCT. Then, the oscillation suppression angle (OSA) is introduced, and its working principle is studied in depth. In addition, the damping ratio of the system with or without OSA is compared by establishing a second-order state space averaging model, which shows the open loop control of OSA is insufficient to improve the system damping. In order to achieve an effective oscillation suppression, a closed-loop control algorithm is adopted, and the gain of the damping compensator is optimally designed to achieve the critical damping state according to the established small-signal model. Besides, the capacitor voltage closed-loop control and the output power control are both established to ensure the stable and safe operation of MMDCT. Comparative simulation results verify the correctness and effectiveness of the proposed transient oscillation suppression method.

II. MATHEMATICAL MODEL AND TRANSIENT OSCILLATION GENERATION MECHANISM OF MMDCT A. MATHEMATICAL MODEL OF MMDCT
The typical MMDCT topology with medium frequency transformer isolation used for distribution network is depicted in Fig. 1. In the primary side of the transformer a single-phase MMC is used to directly access the high-level DC voltage while in the secondary side a simple and standard H-bridge converter is chosen to provide a low-level DC voltage port. The MMC and the H-bridge converter are connected via a medium frequency (MF) transformer, with the leakage inductor acts as the power exchange media between the primary and secondary side.
In the primary side, the MMC has two phase-legs and each phase-leg incorporates two strings of cascaded halfbridge submodules (SM), namely the MMC arm with N SMs, and a coupled arm inductor in between. The two ends of the phase-leg form the high-level voltage DC port while the midpoint of the inductor is the high frequency AC port connecting to the primary side of the MF transformer.
On the basis of the MMC operation principles [5], the input DC current is divided equally between two phase-legs and thus form the common-mode or the so-called circulating current that charges the SMs while the differential-mode current is driven identically from both ends of the upper and lower arms to the MF transformer that discharges the SMs and delivers the same amount of power to the secondary side of MMDCT. Taking the phase-leg a of the MMC system-level for instance, Applying Kirchhoff's voltage law to the upper loop (formed by the overall input DC link, the upper arm and the terminal voltage with respect to the neutral point) and the lower loop (formed by the overall input DC link, the lower arm and the terminal voltage with respect to the neutral point), respectively, yields where U dc and u aN represent the overall input DC voltage and the terminal voltage with respect to the neutral point, respectively; u p1 , u p2 , i p1 and i p2 represent the upper and lower arm voltages and currents, respectively; and L s , L m and R s represent the self-inductance, mutual inductance and the inherent parasitic resistance of the coupled arm inductor, respectively. For simplicity, the aforementioned system current variables can be represented by their common-mode and differential-mode components as where i cma and i dma represent the common-mode current and the differential-mode current, respectively and i dma is half of the leakage inductor current of the transformer, which is denoted by i lk . Adding and subtracting (1) and (2) then substituting from (3) and (4) gives Note the same conclusion applies for the phase-leg b and the primary side voltage of the MF transformer can be then calculated using the formula u ab = u aN − u bN .
On the other hand, the SM-level dynamics are simply described using Ohm's law of capacitor for both upper and lower arms, to give where u cp1j , i cp1j , u cp2j and i cp2j represent the voltages and charging currents of the jth SM (j = 1, 2, . . . N ) in the upper and lower arms, respectively.

B. TRANSIENT OSCILLATION GENERATION MECHANISM OF MMDCT
The common-mode equivalent circuit model of MMDCT is shown in Fig. 2(a), where the SM capacitors are in series with VOLUME 8, 2020 the arm inductors, so it is susceptible to LC oscillations, viz., the capacitor will charge the arm inductor and generate the magnetic energy stored in the inductor coil. After a period of time, the arm inductor will charge the capacitor in a similar manner in order to convert the magnetic energy back into electrical energy. Over and over again, if there is no loss in the series circuit, the energy exchange between the two passive devices will cause continuous and constant oscillation, as shown by the dotted line in Fig. 2(b). However, in reality, the loss in the parasitic resistor of the arm inductor cannot be ignored, so the oscillation amplitude will decay with time, as shown by the solid line in Fig. 2(b). But since the loss is relatively small in a well-designed system, the oscillation attenuation is very slow. That is, the system has insufficient damping. In this case, if the DC voltage or the transmission power command is varied, a significant oscillation of the circulating current and the capacitor voltage inevitably occurs during the transition time.
In order to perform quantitative analysis to verify the oscillation mechanism, the state space equation should be established by selecting the sum of 2N submodule capacitor voltages u ca_sum and the circulating current i cma as the corresponding state variables (Assuming the submodule capacitor voltages are well balanced, and thus u cp1j and u cp2j are represented by u ca_sum /2N ). Given that the switching ripples of the state variables in a switching period is small (i.e., small ripple approximation [33]), their averaged values during a switching period are used instead as the actual state variables, to give where T s is the switching period and <u ca_sum > T s and <i cma > T s represent the averaged value of u ca_sum (t) and i cma (t) in each switching period in the phase leg a, respectively.
Based on the traditional 50% square wave modulation [18], the relationship between the system-level and the SM-level voltages and currents are able to be established, as depicted in Fig. 3.

1) THE VOLTAGE RELATIONSHIP
Taking the switching signal S p1j in the upper arm as an example, when S p1j is in the 'ON' state the arm output voltage equals to the SM capacitor voltage while when S p1j is 'OFF' the arm output voltage is 0. The same conclusion applies for the other arms. Therefore, the averaged upper and lower arm voltages <u p1 > T s and <u p2 > T s can be calculated as 2) THE CURRENT RELATIONSHIP Similarly, when S p1j is in the 'ON' state, the SM capacitor current i cp1j equals to the arm current i 1 , which can be decomposed into a common-mode current i cma and half a leakage inductor current i lk /2. In contrast, when S p1j is in the 'OFF' state, i cp1j is zero. Note that since the leakage inductor current i lk is an AC term varying significantly over a switching period, it is inapplicable for the small ripple approximation. Instead, this current may be averaged by P/2u ab , where P denotes the transmitted power in MMDCT and u ab represents the primary side voltage of the transformer, respectively. To further simplify the analysis, this averaged current is approximately seen as a load disturbance, which does not rely on the system state variables, to develop a linear equation. Therefore, u ab approximately equals to -U dc when S p1j = 1, and then the capacitor charging current can be calculated as Substituting (11) and (12) into (5), (7) and (8) yields By solving the state space matrix in Laplace transformation form in (13), the characteristic roots are readily derived as In general, the system parameters determine that R 2 s /(L s + L m ) 2 is always smaller than N /(L s + L m )C, hence the system characteristic roots s 1,2 are a pair of conjugate complex roots with negative real parts, viz.
where α, ω d , ξ , ω n represent the damped oscillation decay rate, the damped oscillation frequency, the damping ratio and the natural oscillation frequency of the MMDCT system, respectively, and are all positive values. From (14) and (15) it is evident that the oscillation phenomenon is associated with the system parameters R s , L s , L m , C and N , where the damping ratio ξ along with the natural oscillation frequency ω n can be calculated as   4 illustrates how the damping ratio and the oscillation frequency are dependent on the system parameters. When the submodule capacitance C is fixed at 940uF, the damping ratio increases with the increase of the parasitic resistance and the decrease of the arm inductance [see Fig. 4(a)]. It can be seen from Fig. 4(a) that only if the parasitic resistance reaches 1 and the arm inductance is smaller than 1mH can the system reaches a satisfactory damping state. Furthermore, when the parasitic resistance is fixed (for example with a typically value of 0.1 ), the damping ratio will be increased with a smaller arm inductance and larger submodule capacitance [see Fig. 4(b)], but the damping ratio is limited within 0.2 even if the capacitor reaches an unrealistically large value. On the other hand, as depicted in Fig. 4(c) and (d), the oscillation frequency is inversely proportional to the arm inductance and the submodule capacitance while independent of the parasitic resistance.
To sum up, the nature of the MMDCT parameter values indicate the system is liable to suffer from the damped oscillation. Moreover, increasing the damping ratio via changing the physical parameters in MMDCT system is not feasible in practice for the following three reasons: 1) R s should be designed as small as possible to reduce losses; 2) C does not need to buffer the low frequency energy as a traditional MMC and thus can be selected with a small value to save costs; 3) L s VOLUME 8, 2020 should be larger than a certain value to suppress the possible surge current in the phase leg.
Therefore, it is preferable to increase the system damping by introducing additional control variables under a given system parameter configuration, and thereby an effective control method is to be proposed and detailed in the next section.

III. INCORPORATING AN OSCILLATION SUPPRESSION ANGLE AND ITS EFFECT ON THE SYSTEM DAMPING
An oscillation suppression method is proposed in this section aimed at eliminating the transient oscillation in the circulating current as well as in the capacitor voltages. This method is achieved by incorporating a phase shift angle, which is hereinafter renamed as the oscillation suppression angle (OSA) δ d , at each edge of the square modulation signals.
A. OPERATING PRINCIPLE OF OSA Fig. 5(a) and (b) illustrates how the circulating current can be controlled upward and downward through incorporating a positive OSA and a negative OSA, respectively. Their corresponding waveforms are presented with the solid lines in comparison with the waveforms using the conventional 50% square wave modulation with dotted lines. Note the small inter-SM phase shift angles [18] for reducing dv/dt stress across the transformer are neglected here for brevity.
When δ d is positive [see Fig. 5(a)], during the rising edge the driving signals for the upper arm SMs is delayed by δ d while during the falling edge is led by δ d as compared to the traditional 50% square wave modulation. The same principle applies for the lower arm, and thus a zero voltage zone lasting for 2δ d is generated in u p1 + u p2 . Therefore, the arm inductor voltage u La equals to U dc in this 2δ d interval that gives rise to the increase in the circulating current.
On the contrary, when a negative δ d is incorporated, the arm inductor voltage equals to U dc − U ca_sum , which is negative, during the 2δ d period, and thereby the circulating current is forced decreasing during this period. Note U ca_sum is the value of <u ca_sum > T s .
To sum up, the incorporation of OSA produces an additional square wave voltage across the arm inductor and its parasitic resistor, which can be controlled positive or negative through choosing a correct polarity of δ d . As a result, the circulating current is able to be appropriately controlled upward or downward every half switching cycle to suppress the transient oscillation caused by the insufficient damping. Yet, it is desirable to provide quantitative relationship between the enhanced damping ratio and the incorporated OSA.

B. CONTROLLABLE SYSTEM DAMPING RATIO WITH OSA
By incorporating the OSA at both edges of the switching signal, the actual duty cycle of each SM changes from 50% to (π −2δ d )/2π. As a result, the relationship between the systemlevel and SM-level voltages and currents, as per Fig. 6, Then, the averaged state space equation of (13) is rewritten by substituting (17) to into (5), (7) and (8), to give Similarly as Section II.B, this state-space equation can be solved via calculating the characteristic roots of the state matrix, and then the oscillation decay rate and oscillation frequency are attained as Furthermore, the damping ratio ξ δ is enhanced due to the introduction of δ d , which is expressed as It can be seen from (20) that the damping ratio is controllable owing to the incorporated OSA δ d . Fig. 7 shows this relationship when R s is fixed at 100m . Overserving from Fig. 7 reveals that the MMDCT system is extremely underdamped when R s is small, with the damping ratio smaller than 0.04. By increasing δ d , only a small growth can be observed in ξ δ initially, whereas a surge in ξ δ occurs when δ d reaches around 1.5 rad.
However, since δ d is the angle that adjusts the charging/discharging current and maintains the capacitor voltage levels, it is very small in practice. As a result, the introduction of δ d only provides a possibility to balance the stored power of the system, and it does not assist sufficiently in improving the damping ratio if used in an open loop control manner. Therefore, an active damping method in a closed loop control manner should be implemented to further enhance the system damping, as detailed in the next section. VOLUME 8, 2020

IV. FURTHER ENHANCING THE SYSTEM DAMPING WITH CLOSED LOOP CONTROL A. SMALL SIGNAL MODEL AND TRANSFER FUNCTION
In order to further improve the MMDCT system damping and accelerate the oscillation attenuation rate, a closed loop control method for suppressing oscillation is required. Since the mathematical model is nonlinear as per (18) when the OSA δ d is seen as an input variable, a small signal model that describes the linear relationship between the input signal δ d and the state variable (seen as the output variable) needs to be developed.
Firstly, the input variable OSA δ d and the system state variables are perturbed (seen as a DC term plus a small-signal AC term) as below: Then substituting (21) into (18) gives Note in (22) the differential-mode current <i dma > T s is alternatively described by P/<u ca_sum > T s to improve the accuracy of the model, since the transmitted power is regulated in a closed-loop form according to the load current variations via the phase shift ratio between the primary and secondary side square wave signals, and thus the current becomes dependent on the capacitor voltage.
The DC term in (22) is the quiescent operating point that matches the steady-state analysis, which is written as follows: It can be seen from (23) that the quiescent OSA value d is able to be used to adjust the quiescent capacitor voltage U ca_sum to an expected value under a given system working power P. Since the parasitic resistance R s is relatively small and the reference of U ca_sum is 2U dc , d is also very small in practice.
Secondly, by neglecting the DC term and the second-order AC term, Eq. (22) becomes (24) Finally, by transforming (24) into the Laplace form, the control-to-output transfer function can be then derived as where M and O are specified as

B. DESIGN OF DAMPING COMPENSATOR
A damping compensator using a proportional controller that is effective in improving the system damping ratio is depicted in Fig. 8. Through solving the characteristic equation 1 + k pi G iδ (s) = 0, the characteristic roots can be calculated as where M and O are specified as It can be seen from (28) that by incorporating a positive gain k pi the oscillation amplitude decays faster and the oscillation frequency becomes lower as compared to the open loop control system with fixed d . As a result, the damping ratio ξ k further updates to Combining (23) with (29), the variation of ξ k depending on that of k pi and δ d is illustrated in Fig. 9. It is clear from The specific performance of the damped oscillation with different damping compensator gains k pi at a fixed d (selected as 0.002rad) is listed in Table 1. It can be seen from the table that the oscillation decay rate along with the damping ratio increases significantly as expected with a small increase in k pi while the oscillation frequency keeps decreasing with the increase of k pi . When k pi reaches 0.029 the MMDCT system reaches the critical damping state with the oscillation frequency decays to zero. Note the critical damping state with lower bandwidth is preferable in this case rather than the industrially suggested damping ratio of 0.707.
Theoretical calculation of the precise k pi is by setting ξ k = 1 in (29), to give It is worth noting that the optimal k pi is dependent on the system steady-state operating point, which may vary under different load/power conditions, and thus the damping compensator gain should be adjusted in a self-adaptive manner to guarantee a satisfactory dynamic response, particularly under large load step up/down scenarios. Fig. 10 provides a comparison of the circulating current dynamic response without and with utilizing damping compensation methods. Fig. 10(a) depicts the circulating current waveform without using any damping compensation strategy in which a significant oscillation occurs during the transient period. When the open loop damping compensation method with a fixed OSA is incorporated, the damping ratio remains the same response [see Fig. 10(b)] which is in consistent with  the conclusion in Section III. In contrast, the damping compensation with closed loop control of OSA presents significant improvement in the dynamic response [see Fig. 10  Additionally, the SM capacitor voltage control using a PI regulator is typically required to provide a circulating current VOLUME 8, 2020 reference, as illustrated in Fig. 11, as well as to balance the input and output power. The controller parameters k pu and k iu are set according to offline design using MATLAB, where the current-to-voltage transfer function G ui (s) is derived from (24), as below

C. STABILITY ANALYSIS
The stability of the inner current loop and the outer voltage loop is verified in Fig. 12(a) and Fig. 12(b), respectively. In Fig. 12(a), a pair of conjugate complex roots with negative real part appear in the MMDCT system without using any damping compensation method, in which case the frequency is 369rad/s and the damping ratio is 0.0248. In contrast, the roots evolve into two distinct negative real values with the damping ratio compensated to 1 (i.e., with k pi calculated in (30)). Note that there is a positive zero in both cases, which may adversely affect the dynamic process of the system.
The stability of the overall dual loop control system is shown in Fig. 12(b), where the resonant peak reaches as high as 49.7dB at the frequency of 369rad/s. With adjusting the PI regulator, the resonance is completely eliminated with a phase margin of 75 deg. This makes sure the overall voltage control loop reaches a satisfactory damping state, and no oscillation will appear in the capacitor voltages during transient period.

D. THE OVERALL CLOSED LOOP CONTROL SYSTEM OF MMDCT
The overall control diagram of the MMDCT system is given in Fig. 13. Apart from the dual loop control in the primary side, an outer phase shift angle undertakes the role of adjusting the active power transfer between the primary and secondary side of MMDCT, in order to fulfill the assumption made in (22) that the transmitted power is the disturbance relying solely on the load while the differential-mode current depends on the capacitor voltages. Hence the output voltage is closed-loop controlled with a PI regulator to track the load power demand.

V. SIMULATION VERIFICATION
The feasibility and effectiveness of the proposed transient oscillation suppression method has been investigated in PSIM, with the system configuration and parameters listed in Table 2. In order to better compare the effects without and with using the proposed damping compensation method, the underdamped oscillation is designed to be more significant with the parasitic resistance of the phase leg set to a small value (R s = 10m ). In order to perform the simulation in a more practical way, the quasi-two level modulation is adopted here to reduce dv/dt. Furthermore, results using the saturated medium-frequency transformer are compared to that from a traditional linear transformer.

A. TRANSIENT RESPONSE RESULTS WITHOUT USING THE PROPOSED CONTROL METHOD
The transient response of the system, including the submodule capacitor voltages u cij , the circulating current i cma , the arm inductor voltage u La and the output side voltage u o , without using any damping compensation method is illustrated in Fig. 14, where a load power step increase from 112.5kW to 225kW is triggered at t = 0.4 s. All these variables present significant oscillation phenomenon, demonstrating that the original MMDCT system is underdamped as explained in Section II. Note the oscillation frequency is around 729 rad/s that is consistent with the theoretical analysis in (16), verifying the correctness of the quantitative analysis of the oscillation mechanism. Since the decay rate of the transient oscillation correlates to the product of ξ and ω n , the oscillation will disappear after a long time if there is no load change. However, since the disturbance is inevitable in the MMDCT system, the MMDCT system is likely to oscillate frequently, threatening the safe operation of the system. Moreover, the load transient at t = 0.4s triggers much more serious oscillations in the aforementioned variables with higher resonance peak value. In summary, both steadystate response and transient response are undesirable without using any damping compensation method in MMDCT system, indicating the necessity of incorporating the proposed control method to suppress the transient oscillation in a second-order system.

B. TRANSIENT RESPONSE RESULTS WITH CLOSED LOOP CONTROL OF OSA
The transient response of i cma and δ d during load step up/down using a single circulating current control loop are depicted in Fig. 15. When the load power steps down [refer to Fig. 15] from 225kW to 112.5kW at t = 0.6s, the circulating current reference decreases suddenly, and then the actual circulating current is larger than the new target value during the transient process. As a result, δ d is adjusted negative to produce an overlapped gating signals [refer to S p1j , S p2j in Fig. 15], whose duty ratio increases to (π −δ d )/2π , because of which the circulating current will decrease in the 2δ d period to track the new reference. Eventually, i cma will be stable around the new reference after several switching periods and δ d will return to its stable operation point as well.
On the other hand, when the load power steps up [refer to Fig. 16] from 112.5kW to 225kW at t = 0.6s, the actual circulating current will be smaller than the new reference, that is, δ d will be controlled positive to provide a 2δ d zone for i cma to increase until it reaches the new reference, with the duty ratio of S p1j and S p2j being decreased to (π − δ d )/2π. Note that there is relatively larger ripple current in i cma during the start-up transient procedure, which is on account of the larger difference between the actual current and the reference and thus δ d is adjusted larger to provide a relatively longer rising/falling time in the circulating current. With i cma approaching the steady-state value, the value of δ d turns back smaller, giving rise to smaller ripple in the current.
Note that the operation mechanism of OSA is to modify the duty ratio of the 180 • phase shifted signals in each switching pair (S p1j and S p2j ), while the inner submodule phase shift angel θ in the quasi-two level modulation changes the phase shift angle among N submodules per arm with the 180 • phase shift characteristic maintained between S p1j and S p2j . In other words, the incorporation of θ will not affect the feasibility of OSA, while it only affects the moment when each switching pair acts on the circulating current.

C. THE OVERALL TRANSIENT RESPONSE RESULTS WITH THE PROPOSED CONTROL METHOD
A load power step-up transient response from P = 112.5kW to P = 225kW and a step-down transient response from P = 225kW to P = 112.5kW are both provided to verify the effectiveness of the proposed closed-loop oscillation suppression control method, as depicted in Fig. 17 and Fig. 18, respectively. Both steady-state and transient response are satisfactory without evident oscillations, thus the damping characteristics and the stability of MMDCT system get improved sufficiently. This is achieved with the optimal controller gain designed to compensate the system to the critical damping state.
Hence, the pulse voltage with the duty ratio of δ d /π is generated across the arm inductor with the closed-loop control of the circulating current, and accordingly the transformer primary side voltage augments to three level where the zero level lasts for 2δ d . To be specific, if the actual circulating current is larger than the desired value, a negative perturbation will be overlaid on the steady state operating point of d to reduce the increment of i cma as compared to the previous switching period and vice versa, back and forth, the circulating current  will be adjusted around the reference and the oscillation in i cma will be completely suppressed as well as that in u cpij , u La and u o , without imposing any burden into the system losses.
In addition, the outer capacitor voltage control provides the reference of the circulating current and eliminates the steady state error in capacitor voltages effectively. Besides, the output voltage is well maintained to the reference of 750V via the output power closed loop control with only 4% voltage drop during transient process. Thereby, a desirable stability state of a MMDCT system is achieved with critical damping ratio though sacrificing the dynamic response to some extent. Besides, the outer phase shift angle between the primary and secondary side increases (relatively small due to the input DC link voltage is large) under the circumstance of the load power transient to adjust the transmitted power through the transformer, as depicted in Fig. 19. Since the quasi-two level modulation is adopted to limit dv/dt where the inter SM phase shift angle θ = 0.1%π, the overlapped zone in the N switching signal pairs are generated by OSA and phase shifted sequentially by θ, hence the primary side voltage u ab has 2N + 1 levels during the falling and rising edges, as depicted in Fig. 20. Nevertheless, it only lasts for a short period of (2 * OSA + (N − 1)θ) since OSA is around 1%π during the transient process and zero during the steady state and θ is also small enough. Therefore, the introduced OSA and θ will not contribute to the saturation of the mediumfrequency transformer in MMDCT.
In order to investigate the validity of the proposed transient oscillation suppression method using a saturable transformer, simulation waveforms are compared in Fig. 21 and Fig. 22. When the piecewise linear relationship between the flux Φ and magnetizing current i m satisfies (3.75 * 10 −4 Wb, 0.6A)(3 * 10 −3 Wb, 60A), the transformer will be saturated since the maximum flux Φ max determined by Φ − i m curve equals to 0.003Wb (much smaller than Φ m = 0.04Wb solved in equation U = 4.44fNΦ m , where U is the effective value of U dc ), hence the primary side transformer current i leakage keeps increasing with a sharply increased slope introduced, as shown in the zoomed view of the red circle of Fig. 21. When Φ −i m curve satisfies (0.0375Wb, 0.6A)(0.3Wb, 60A), the maximum flux Φ max equals to 0.3Wb and the transformer will not suffer from saturation so the primary side current in transformer is stable, as shown in Fig. 22. Note that when the transformer is saturated, the circulating current shows divergence while the oscillation in the circulating current is  suppressed with the proposed transient oscillation suppression control method. When the transformer is not saturated, the circulating current is stable with no transient oscillation phenomenon. Hence, the circulating current is well regulated without any oscillation no matter the transformer is saturated or not, and the proposed transient oscillation method is of validity.

VI. CONCLUSION
The study in this paper shows that MMDCT has a lowfrequency oscillation problem due to the small value of the submodule capacitance, which is more likely to occur than the traditional MMC. By introducing an OSA designed to adjust the direction of change in the circulating current, it is able to improve the system damping and suppress the transient oscillation. Through the small-signal linearization of the established second-order state space model, the transfer function of OSA-to-current is derived, and the improved damping ratio of the MMDCT system with closed-loop control of OSA is achieved. Simulation results from an industrial-level MMDCT model in PSIM validate the effectiveness of the proposed OSA along with the damping compensator control approach.