The MMC typically employs a double-closed-loop PI control, as shown in Figure 4. P _ref, V _dcref, Q _ref, U _smref, i _dref, i _qref, u _abcref are the references of active power, reactive power, DC voltage and the PCC AC voltage, d-axis current, q-axis current, modulation of arm voltage, respectively; and P, Q, V _dc, U _sm, i _sabc, u _sd, u _sq, ω _ref, are the output active power, reactive power, DC voltage, PCC AC voltage, PCC AC current, d-axis voltage, q-axis voltage, angular frequency, respectively; k _p1, k _i1, k _p2, k _i2, are out-loop and inner-loop proportional and integral coefficients.

The powers are calculated in the dq-frame as: P = 1.5 u s d i s d + u s q i s q Q = 1.5 u s q i s d − u s d i s q (9)

In the dq reference frame, (Equation 9) expressed as: L e q d d t i s d i s q = − R e q i s d i s q + u d i f f d u d i f f q − u s d u s q + ω L e q − ω L e q i s d i s q (10)

R _eq and L _eq in (Equation 10) are expressed in R e q = R g + 1 2 R a r m L e q = L g + 1 2 L a r m ,

In (Equation 10) expressed Laplace domain as: R + L s i s d s = − u s d s + u d i f f d s + ω L i s d s R + L s i s q s = − u s q s + u d i f f q s + ω L i s q s (11)

Transfer functions for output variables i _sd, i _sq and voltages V _d and V _q expressed as: i s d s V d s = 1 R e q + L e q s i s q s V q s = 1 R e q + L e q s (12) Where V d s = − u s d s + u d i f f d s + ω L i s d s V q s = − u s q s + u d i f f q s + ω L i s q s

By Equations 11, 12, the input-output relationship (Figure 5) of d-axis current, q-axis current expressed as:

According to the negative feedback theory, the current reference value i_dref and i_qref can be well tracked by PI control, similarly the active power and reactive power can be well tracked and controlled.

Traditional MMC double-PI control typically uses fixed reference values for its targets, making it incapable of actively adjusting to variations in grid voltage and frequency. The core concept of grid-forming control is to modify the converter control mode, enabling it to exhibit rotational inertia and autonomously provide active or reactive power support to the grid. The mathematical equations governing the grid-forming virtual synchronous generator (VSG) control for the MMC are presented as follows. P * − P = J ω 0 d ω d t + D p ω 0 ω − ω 0 (13) E = E n + k q Q * − Q + k v U * − U (14) L v d i j r e f d t + R v i j r e f = e j r e f − u j (15)

Equations 13–15 describe the rotor motion equation, the reactive power-voltage control equation, and the electrical equation of the VSG body used for virtual synchronous control.

In these equations, P* and P are the reference power and actual active power, respectively. J and D _p are the VSG control parameters refer to virtual rotational inertia and damping coefficient respectively, ω₀and ω refer to the rated velocity and actual electrical angular velocity, respectively, Q* and Q are the reference power and actual active power, respectively. E _n is the no-load electromotive force in the synchronous generator. R _v and L _v are the virtual resistance and inductance of VSG, respectively. i _jref is the three-phase current reference value of VSG output. e _jref is the internal potential reference value. u _j is the three-phase voltage of the MMC grid connection point.

The control block diagram for the VSG in the MMC is shown in Figure 6. The stator voltage is determined using P-f control and Q-V control, with virtual impedance R _v and L _v incorporated into the electrical section of the VSG model. The current command value i _jref, which satisfies the stator voltage equation, is computed through the virtual impedance loop. The d-q current references, i _dref and i _qref, are then used as inputs to the current inner loop in Figure 4, completing the simulation of the stator voltage equation in VSG control.

By adjusting the values of J and D _p, different active power response characteristics of the MMC can be achieved. The dynamic response of the output power for the MMC, with different J and D _p coefficients, under VSG control is shown in Figures 7, 8.

As illustrated in Figures 7, 8, increasing the virtual inertia J results in a larger amplitude and a longer response time to the P/Q reference signals. Conversely, increasing the virtual damping coefficient D _p leads to a smaller amplitude and a shorter response time. In practice, D _p is generally determined by grid standards, and the dynamic performance of the MMC is primarily adjusted by modifying J.

Model predictive control (MPC) is an advanced control method for power electronics, based on predicting the next AC grid-side current by establishing a discrete mathematical model of the converter. It then performs a traversal calculation of all possible switching states of the power electronics, ultimately achieving optimal control within each control period.

Unlike traditional control methods, MPC does not rely on a PI control loop, making it structurally simpler, requiring no complex parameter tuning. MPC offers fast tracking responses, minimal overshoot, and superior dynamic performance (Zheng, 2020). In this paper, MPC is applied to the virtual synchronous control of the MMC, and a corresponding cost function is designed to address circulating currents and DC voltage fluctuations, as illustrated in Figure 9.

The power control outer loop of grid-forming control is implemented as MPC-VSG, where the traditional PI current loop is replaced by the MPC model predictive controller. In this setup, the current reference i _jref of the VSG output (as shown in Figure 6) serves as the input reference for the model predictive control. The discrete mathematical Equations 7, 8, of the MMC are combined to construct the cost function for MPC.

Depending on disturbances in the AC grid frequency or voltage, or input changes in the active power command (P _ref or V _dc) or reactive power command (Q _ref or U_sm), the three-phase current reference i _sjref (t + T_s) (j = a, b, c) is calculated using Equations 13–15.

The first cost function of MPC-VSG is built. J j = i j r e f t + T s − i s j t + T s (16)

Combined with (Equation 7), the upper and lower bridge arms voltage difference value e j t + T s = u n j k + 1 − u p j k + 1 corresponding to the minimum J j can be denoted as e j * t + T s , whose value range is V d c N − N 2 , − N − 1 2 , . . . , 0 . . . , − N − 1 2 , N 2 , and V _dc is the DC bus voltage. According to the combined Equations 5, 6, the number of upper and lower bridge arms can be calculated, as n n j * and n p j * .

Equation 8 highlights the presence of circulating currents within the MMC, which, while circulating only inside the MMC, do not affect the external AC or DC sides. However, these currents contribute to increased power device losses and must be suppressed. In this work, we address this issue by controlling the circulating currents through adjustments to the additional voltage v d i f f applied simultaneously by Equations 17, 18: i z j k + 1 − i z j k T s = − i z j k + 1 R a r m L a r m − u n j k + 1 + v d i f f + u p j k + 1 + v d i f f 2 L a r m + V d c k + 1 2 L a r m (17) v d i f f j = V d c / N − m , − m − 1 , . . . 0 , m − 1 , m (18)

The second cost function of MPC-VSG is obtained. J d i f f j = i d c t + T s / 3 − i z j t + T s (19) where i d c is the DC side current. When J d i f f j is the lowest value, the corresponding v d i f f value is calculated and m denoted as v d i f f j * t + T s and m * . The upper bridge arm submodules number under the MPC-VSG control strategy is obtained by combining cost function (Equations 16, 19). N p j = n p j * + m * (20)

The lower number under the MPC-VSG control strategy is obtained. N n j = n n j * + m * (21)

Submodule voltage balancing is achieved using the nearest-level approximation method. By combining Equations 20, 21, the submodule sequence for each arm is determined, and the corresponding control pulses are applied to the MMC.

Three control objectives are incorporated into the proposed Level MPC these are AC current, circulating current, and strict fault current limit. control objectives are achieved through the cost function J outlined in Equation 22. J = λ 1 + ξ i j r e f t + T s − i s j t + T s + λ 2 i d c t + T s / 3 − i z j t + T s (22) ξ = ∞ , i f i s j ≥ I max 0 , i f i s j < I max (23)

Within the cost function, λ 1 λ 2 introduces the weighting factor, in order to tracking of reference and regulate the circulating current to its desired value, ξ is additional penalty weighting factor to the fault current strict limit in Equation 24.

To mitigate transient overcurrents, the proposed method limits the amplitude of the reference current under the d-q framework while maintaining the current phase. This ensures that power supply quality is preserved during fault conditions to the greatest extent possible. The strategy is described as follows:

where I m a x denotes the maximum allowable current of the converter. i s d q _ ⁡ lim * denotes the saturated d-q current. i s q * , i s d * , i s d _ ⁡ lim * , and i s q _ ⁡ lim * denote the reference current of axis d and q before and after the limiting respectively. i s d q _ ⁡ lim * = i s d q *   i s d 2 + i s q 2 < I max i s d _ ⁡ lim * = I max ⁡ cos arctan ⁡ 2 i s q * i s d *   i s d 2 + i s q 2 ≥ I max i s q _ ⁡ lim * = I max ⁡ sin arctan ⁡ 2 i s q * i s d *   i s d 2 + i s q 2 ≥ I max (24)

The basic principles of model predictive control refer to Figure 10.

The active power step simulation of the MMC converter is conducted under a grid-forming control strategy. The simulation parameters for the MMC converter and system are provided in Table 3. The number of submodules, denoted as “m,” used to suppress circulating current is set to 1. The time-domain simulation results are presented in Figure 11. Before t = 3 s, MMC operates statically at −5 MW, with a circulating current of approximately 20 A. At t = 3 s, the active power jumps to 5 MW, with the circulating current remaining around 20 A. During this transition, the capacitor voltage fluctuates by about 6% due to the circulating current. The MMC converter operates stably throughout the step process, fulfilling the specification requirements.

To fully compare the dynamic response characteristics of the inertia support with the proposed control strategy, the simulation experiment under sudden change of load is carried out. During normal operation, the synchronous generator is 20 MVA, the load is 20 MW, and MMC-HVDC provided 5 MW power to the load. At 3 s, the 7 MW load is suddenly put into, with the frequency transient response characteristics under the two controls observed. As shown in Figure 12, at t = 3 s, based on PI-VSG control, the grid frequency f drops to 49.25 Hz. However, based on MPC-VSG control, the frequency occurs 49.29 Hz, which reduced by 0.04 Hz.

When the system frequency decreases, both strategies can achieve the rapid response adjustment of active power. Compared with PI-VSG control, MMC under MPC-VSG control has a faster power tracking speed.