The MMC main circuit topology is shown in Figure 1A. The three-phase structure is the same, with a total of six bridge arms. Each phase can be divided into upper and lower bridge arms, and each bridge arm is composed of n sub-modules with the same structure cascaded with the bridge arm inductor L. In Figure 1A, U _dc is the DC output voltage; u _Px and _uNx are the output voltages of the upper and lower bridge arms of the x phase, respectively. Phase x = a,b,c; i _Px and i _Nx represent the current flowing through the upper and lower bridge arms of the phase x, respectively; u _x is the output voltage of the MMC AC port; u _sx and i _x are AC-side phase voltage and phase current, respectively; L _s is the filter inductor for the energy interaction between the AC power supply and the MMC.

In this paper, MMC uses a half-bridge sub-module, and the topology is shown in Figure 1B. When running, according to different modulation and sub-module equalization sequencing algorithms, it can work in three states: input, cut-off and latching. When it is running stably, it only works in the state of input and cut-off. The current i _SM flowing path and its port voltage u _SM are shown in Table 1. 1 and 2 are the outlet ports of the sub-module; in the input state, u _SM = U _c ; in the cut-off state, u _SM = 0.

MMC single-phase equivalent circuit is shown in Figure 2. Ignore the effects of AC power and AC-side filter inductors. Taking a single-phase as an example, the KVL equation is established for the upper and lower bridge arms (Tan et al., 2021): U d c 2 − u P x − L d i P x d t − u x = 0 U d c 2 − u N x − L d i N x d t + u x = 0 (1)

The MMC three-phase structure is symmetrical, and the DC-side current is evenly distributed among the three phases. The capacitor voltage of the sub-module of the phase unit fluctuates, resulting in the voltage difference among the phase units, forming a double frequency circulation. Therefore, the interphase circulation icir in this paper includes the DC circulation and the interphase double frequency circulation that are evenly distributed among the three phases. Thereby, the current equations of the single-phase upper and lower bridge arm nodes are established: i P x = i x 2 + i c i r i N x = − i x 2 + i c i r (2)

The circulation current in the bridge arm current intensifies the fluctuation of the capacitor voltage of the sub-module, resulting in a decrease in the accuracy of the DC output voltage and an increase in the system loss (Bahrani et al., 2016). Therefore, it is necessary to introduce circulation suppression when designing the deadbeat predictive current control system applied to MMC.

By discretizing formula (1), the formula for the output voltage of the upper and lower bridge arms of the MMC in the kth control period (Zhang et al., 2021) is obtained: u P x k = U d c k 2 − u x k − L i P x * k + 1 − i P x k T s u N x k = U d c k 2 + u x k − L i N x * k + 1 − i N x k T s (3) where U _dc (k) and u _x (k) are the DC-side voltage and the AC-port output voltage in the kth control period, respectively; u _Px (k) and u _Nx (k) are the sum of the voltages of all submodule capacitors of the control cascaded into the upper and lower bridge arms of the MMC main circuit in the kth control period; i _Px (k) and i _Nx (k) are the currents flowing through the upper and lower bridge arms, respectively; i ^* _Px (k+1) and i ^* _Nx (k+1) are the upper and lower bridge arm currents at the beginning of the k+1 control period, that is, the reference values for the prediction of the upper and lower bridge arm currents. According to formula (2), formula (4) can be obtained: i P x * k + 1 = i P x _ r e f = i x _ r e f 2 + i c i r _ r e f i N x * k + 1 = i N x _ r e f = − i x _ r e f 2 + i c i r _ r e f (4) where i _{x_ref} is the AC current reference value; i _{cir_ref} is the reference value for interphase circulation to suppress the system circulation and reduce the voltage fluctuation of the capacitor of the sub-module. The interphase circulation reference value i _{cir_ref} can be obtained by PI control of the difference between capacitor voltage reference value U _{c_ref} of the sub-module and its average value U _{c_aver} . i c i r _ r e f = U c _ r e f − U c _ a v e r × k p + k i s (5) where kp and ki are proportional and integral coefficients, respectively.

Formula (4) is substituted into formula (3) and formula (5) to construct a deadbeat predictive current control system, as shown in Figure 3. The MMC deadbeat predictive current control system is divided into three parts: voltage outer loop (circulation current suppression), current inner loop (deadbeat predictive current control), carrier phase-shift modulation and sequencing algorithm. The reference value of the interphase circulation is obtained by the voltage outer loop i _{cir_ref} and the reference value of the AC current is substituted together with the reference value of the AC current i _{x_ref} into formula (4) to get the reference value of the output current of the upper and lower arms of each phase i _{Px_ref} and i _{Nx_ref} . The reference values u _Px (k) and u _Nx (k) of the voltage of the upper and lower bridge arms of each phase in the kth control period can be obtained by substituting i _{Px_ref} and i _{Nx_ref} into formula (3). The sub-modules that should be put in the upper and lower bridge arms of each phase are obtained by the carrier phase-shifting modulation and sequencing algorithm, and the trigger pulse of the sub-module of each phase is generated.

Note: The u _{Px_ref} and u _{Nx_ref} in Figure 3 are u _Px (k) and u _Nx (k) in formula (3)

Normally, we want to complete the sampling, computational control, and output of each physical quantity of the system at the same time. However, in the actual control system, there are delays in the process of system sampling, calculation control, and loading duty cycle, which cannot be completed at the same time, so the influence of time delay should be considered when designing the control system (Wang et al., 2015a). Figure 4 is a schematic diagram of the time delay of considering system sampling, calculating control, loading duty cycle, and generating target signals. i _Px and i _Nx were used as the control variables, u _Px and u _Nx were the controlled variables, and T _s was approximately equivalent to the carrier period (i.e., one beat). At the time of kT _s , start sampling i _Px and i _Nx ; During the control period from kT _s to (k+1)T _s , the sampling of i _Px and i _Nx is completed, and the calculation of u _Px and u _Nx is completed. At the time of (k+1)T _s , start loading duty cycle; During the control period from (k+1) T _s to (k+2)T _s , the duty cycle conversion is completed and the voltages of the upper and lower bridge arms of the MMC, u _Px and u _Nx , are output. At the time of (k+2)T _s , the target current (MMC upper and lower arm currents) i _Px and i _Nx can be obtained.

Through the above analysis, it can be seen that there is a delay of one beat between the output voltage u _Px and u _Nx , that is, the delay of one control period T _s from the start time of current sampling to the start time of loading duty cycle. There is a delay of two beats between the output current i _Px and i _Nx , that is, the delay is 2T _s (two beats) between the start time of current sampling and the time when the target current signal is obtained, resulting in a lag in the control effect.

Previous studies have shown that the delay of the control system will reduce the output accuracy, affect the stability, and reduce the follow-up performance (Zhang et al., 2017). Therefore, the delay of the output voltage/current is compensated by the control period at the same time to achieve a good control effect.

In order to compensate for the influence of the delay on the control effect, the output voltage of the AC port is predicted one beat in advance, and the output current of the upper and lower bridge arms is predicted two beats in advance, so the control period plus one can be processed for formula (3) to obtain (Zhang et al., 2021): u P x k + 1 = U d c k + 1 2 − u x k + 1 − L i P x * k + 2 − i P x k + 1 T s u N x k + 1 = U d c k + 1 2 + u x k + 1 − L i N x * k + 2 − i N x k + 1 T s (6)

When the MMC is running stably, the DC voltage fluctuation is relatively small, and it can be considered that: U d c k = U d c k + 1 = U d c _ r e f (7)

From formula (3) and formula (6), it can be seen that the output voltage u _x of the AC port is a linear relationship with the predicted currents i ^* _Px and i ^* _Nx of the upper and lower bridge arms, and the control period plus one. Since T _s is approximately equivalent to the carrier period, and its value is small, it can be considered that the increments of u _x in each control period are equal, and the first-order forward difference method is used to predict and downtime u _x , and it can be obtained (Abdel-Rady Ibrahim Mohamed and El-Saadany, 2007): u x k + 1 − u x k = u x k − u x k − 1 (8)

The predicted reference values of ux in the k+1 control period are: u x k + 1 = 2 u x k − u x k − 1 (9)

First, substituting formula (9) into formula (6); Secondly, i ^* _Px (k+1) and i ^* _Nx (k+1) in formula (3) are replaced by i _Px (k+1) and i _Nx (k+1), and the expressions i _Px (k+1) and i _Nx (k+1) are obtained by formula (3) and substituted into formula (6). Finally, in combination with formula (7), formula (10) is obtained: u P x k + 1 = U d c _ r e f 2 − u x k − L i P x * k + 2 − i P x k T s ⏟ Conventional Deadbeat Current Predictive Control + U d c _ r e f 2 − u x k − u P x k ⏟ Arm Inductance Voltage Correction Term − u x k − u x k − 1 ⏟ AC port output voltage correction term u N x k + 1 = U d c _ r e f 2 + u x k − L i N x * k + 2 − i N x k T s ⏟ Conventional Deadbeat Current Predictive Control + U d c _ r e f 2 + u x k − u N x k ⏟ Arm Inductance Voltage Correction Term + u x k − u x k − 1 ⏟ AC port output voltage correction term (10)

From formula (10), it can be seen that the mechanism of two-beat deadbeat predictive current control is to add the correction terms of bridge arm inductor voltage and AC-port output voltage on the basis of the traditional deadbeat predictive current control, so as to increase the speed and accuracy of the control system. The traditional deadbeat predictive current control can realize the fast tracking of the bridge arm current, so that the system can run stably. In the actual circuit, the bridge arm inductor current will cause the bridge arm inductor value to change, and the bridge arm inductor voltage correction term can be used to correct this change; Similarly, the AC-port output voltage correction term can be used to correct changes in the grid-side voltage.

In order to reduce the dependence of the predicted current of the bridge arm on the inductor of the bridge arm and further improve the accuracy of the control system, in this paper, the Newton interpolation method is used to predict the output current of the bridge arm. There is a series of unequal independent variables (z_0,z_1,z₂…,z _n , where m≠n,z _m ≠z _n ) and their function f(z). f (z _n )-f (z _m )/(z _n -z _m ) becomes the first-order difference quotient of the function f(z) at the point z _m and z _n , which is denoted as f [z _n ,z _m ]; f [z₁,z₂,…,z _k ]-f [z₀,z ₂,…,z _k−1]/(z _k -z₀) is the k-order difference quotient.

According to the definition of the difference quotient (Zhang et al., 2017), the Newton interpolation first-order polynomial is obtained: f 1 z = f z 0 + f z 0 , z 1 z − z 0 (11)

The quadratic polynomial is: f 2 z = f z 0 + f z 0 , z 1 z − z 0 + f z 0 , z 1 , z 2 z − z 0 z − z 1 (12)

Therefore, the nth degree polynomial of f(z) is: f n z = f z 0 + f z 0 , z 1 z − z 0 + f z 0 , z 1 , z 2 z − z 0 z − z 1 + ⋯ + f [ z 0 , z 1 , ⋯ , z n ] z − z 0 ⋯ z − z n − 1 (13)

According to formula (4), the reference value of the bridge arm current prediction is related to the AC current reference value and the circulation reference value, and the circulation reference value can be obtained by the voltage outer loop, so the prediction of the bridge arm current can be converted into the prediction of the AC current reference value. Therefore, the reference values of AC current prediction at (k+1)T _s and (k+2)T _s can be predicted according to formula (13) and the sampling values of AC current at (k-2)T _s , (k-1)T _s and kT _s respectively.

Newton interpolation quadratic polynomial is used to establish the AC current prediction expression, shown as follows: i x _ r e f t k + 1 = i x t k − 2 + i x t k − 2 , t k − 1 t k + 1 − t k − 2 + i x t k − 2 , t k − 1 , t k t k + 1 − t k − 2 t k + 1 − t k − 1 = i x t k − 2 − 3 i x t k − 1 + 3 i x t k (14)

The expression for the prediction of the AC current established by Newton interpolation cubic polynomial is: i x _ r e f t k + 2 = i x t k − 2 + i x t k − 2 , t k − 1 t k + 2 − t k − 2 + i x t k − 2 , t k − 1 , t k t k + 2 − t k − 2 t k + 2 − t k − 1 + i x t k − 2 , t k − 1 , t k , t k + 1 t k + 2 − t k − 2 t k + 2 − t k − 1 t k + 2 − t k = − i x t k − 2 + 4 i x t k − 1 − 6 i x t k + 4 i x t k + 1 (15)

Substituting the AC current value at time t _k+1 predicted by formula (14) into formula (15), formula (16) can be obtained: i x _ r e f t k + 2 = 3 i x t k − 2 − 8 i x t k − 1 + 6 i x t k (16)

Substituting i _{x_ref} (t _k+2) with the i _{x_ref} in formula (4), formula (17) can be obtained: i P x * k + 2 = i P x _ r e f = i x _ r e f 2 + i c i r _ r e f i N x * k + 2 = i N x _ r e f = − i x _ r e f 2 + i c i r _ r e f (17)

Substituting formula (17) into formula (10) constructs an improved two-beat deadbeat synchronous predictive current control system based on Newton interpolation method, and the structure is shown in Figure 6.

The improved two-beat deadbeat synchronous predictive current control system is applied to MMC, and the DC voltage and current waveforms during stable operation are shown in Figure 7. The two-beat deadbeat predictive current control system is applied to MMC, and Figure 8 shows the DC voltage and current waveforms during stable operation.

As can be seen from the comparison of Figure 7 and Figure 8, both control systems can stabilize the DC voltage at 10 kV and the DC current at 100 A. The two-beat deadbeat synchronous predictive current control is improved, the rise time is 0.0369s, and the steady-state error is about +0.13% and −0.08% at about 3s. In order to stabilize the DC voltage at 10 kV, the inductor value of the bridge arm needs to be set from 3.23mH to 3.29 mH in the simulated environment with the same parameters. The two-beat deadbeat predictive current control has a rise time of 0.0375s and a steady-state error of about +0.12% and −0.04% at about 3s. Table 3 shows the comparison of the above parameters.

Figure 9A is the improved two-beat deadbeat synchronous predictive current control of three-phase AC voltage/current, and the three-phase AC phase voltage peak is 3266 V; the phase current peak is 224.2A, the phase current lag phase voltage is 827 μs, and the power factor cos (14.89°)≈0.9664 is improved. Figure 9B is a two-beat deadbeat predictive current control of three-phase AC voltage/current, and the three-phase AC phase voltage peak is 3266 V; the phase current peak is 228.2A, the phase current lag phase voltage is 965 μs, and the power factor cos (17.37°) ≈ 0.9543.

Figure 10A shows the total harmonic distortion rate of the AC current in phase A of the improved two-beat deadbeat synchronous predictive current control based on Newton interpolation method. 10 power frequency periods are selected for measurement starting from 3s, and the total harmonic distortion rate is 4.86%. As shown in Figures 10B, 10 power frequency periods are selected for measurement at the same time starting from 3s, and the total harmonic distortion rate is 7.74%. From formula (6) and formula (8), it can be seen that the two-beat deadbeat predictive current control system only uses the first-order forward difference method to predict the output voltage of the AC port for one control period, but formula (6) contains i ^* _Px (k+2) and i ^* _Nx (k+2), so it is necessary to predict the AC current i _x for two control periods, therefore, this paper uses Newton interpolation quadratic and cubic interpolation polynomials to predict the AC current for two periods to form i ^* _Px (k+2) and i ^* _Nx (k+2). Under the same system simulation parameters, the inductor of the bridge arm is reduced by 0.06 mH, which reduces the dependence of the predicted current of the bridge arm on the inductor of the bridge arm and improves the accuracy.

Table 4 shows the comparison of the THD data of phase-A AC current measured by 10 power frequency periods from 1 to 8s, and Figure 11 shows the data reconstruction curve of Table 4. From the comparison of the data in Table 4, it can be seen that the improved two-beat deadbeat synchronous predictive current control for three-phase AC current THD fluctuates between 4.7% and 4.9%, and the two-beat deadbeat predictive current control for three-phase AC current THD fluctuates between 7.7% and 7.8%.

As shown in Figure 12A, with the improved two-beat deadbeat synchronous predictive current control, the system circulation is gradually decreasing. In the range of 1–8s, the positive peak value of phase A circulation attenuates from 58.24 A to 48.28A, and the circulation inhibition rate is 17.10%. The positive peak value of phase B circulation attenuates from 56.04 A to 46.94A, and the circulation inhibition rate is 16.24%. The positive peak value of phase C circulation attenuates from 61.31 A to 46.49A, and the circulation inhibition rate is 24.17%. As shown in Figure 12B, with the two-beat deadbeat predictive current control, the three-phase circulation is also gradually decreasing. In the range of 1–8s, the positive peak of phase A circulation attenuates from 61.61 A to 42.98A, and the circulation inhibition rate is 30.24%. The positive peak value of phase B circulation attenuates from 58.49 A to 42.59A, and the circulation inhibition rate is 27.18%. The positive peak value of phase C circulation attenuates from 60.14 A to 43.75A, and the circulation inhibition rate is 27.25%. From the above analysis, it can be seen that the circulation of the system is well suppressed by the two control systems, and the effectiveness of the control of the voltage outer loop (circulation suppression) is verified.

Figure 13 shows the capacitor voltage and fluctuation of the MMC Phase-A upper bridge arm sub-modules. Phase A of the improved two-beat deadbeat synchronous predictive current control is shown in Figure 13A. At 1s, the sub-module capacitor voltage fluctuation rate is 2.5%. At 8s, the sub-module capacitor voltage fluctuation rate is 2.9%. The capacitor voltage of phase-A upper bridge arm sub-module of the two-beat deadbeat predictive current control is shown in Figure 13C. At 1s, the sub-module capacitor voltage fluctuation rate is 2.5%. At 8s, the sub-module capacitor voltage fluctuation rate is 2.9%. The analysis shows that the capacitor voltage fluctuation rate of the MMC sub-module meets the requirement of ±10% for both control systems. The effectiveness of the control of the voltage outer loop (circulation current suppression) is indirectly verified.

In order to verify the dynamic performance of the two control systems, the transient working conditions are simulated. As can be seen from Figures 14A, B, the improved two-beat deadbeat synchronous predictive current control achieves the peak value of three-phase AC current of 224.2 A before 1s. At 1s, the AC current burst is set, and the output current is tracked to the predicted reference value after 427 μs After entering the steady state, the peak value of the three-phase AC current is 271.5A, and it can be seen from Figures 15A, B that the peak value of the three-phase AC current is 227.8 A before 1s. At 1s, the AC current burst is set, the output current is tracked to the predicted reference value after 1.267 ms. After entering the steady state, the peak value of the three-phase AC current is 273.5 A. Therefore, the method proposed in this paper has better current fast tracking ability than the two-beat deadbeat predictive current control.