*2.2. Existing MPC Methods and Corresponding AV2Rs*

For a two-level inverter, eight feasible voltage vectors (six active and two null voltage vectors) can be obtained according to the switching states, as presented in Figure 2a. In classical MPC method (termed as MPC1), the system can be simultaneously controlled by evaluating a cost function. With respect to the mentioned V2G inverter, the objective is to regulate the output current *i* = [*id*, *iq*] *<sup>T</sup>*. Hence, the cost function can be constructed as:

$$\mathbf{g} = \left\| \mathbf{i}^{ref} - \mathbf{i}^{k+1} \right\|^2 \tag{7}$$

Thereafter, eight feasible vectors are utilized to evaluate the cost function and the optimal vector is selected by minimizing the cost function. Nevertheless, since the switching state alternates between 0 and 1, the AV2R (highlighted with red) is insufficient and seven points are merely covered from the graphical subject of view, as observed in Figure 2b. As such, the applied vector is limited to these points. In this manner, the locus of applied vectors is necessarily discontinuous, and therefore, the control effect is also relatively limited.

**Figure 2.** Feasible voltage vectors and active voltage vector region (AV2R) of existing model predictive control (MPC) methods: (**a**) Feasible voltage vectors; (**b**) AV2R of MPC1; (**c**) AV2R of MPC2; (**d**) AV2R of MPC3.

To this end, an improved strategy using the DSVM method is studied in [23], termed as MPC2. Twelve virtual vectors in conjunction with eight actual vectors are evaluated during each sampling period. Hence, as illustrated in Figure 2c, the AV2R is extended to 19 points from seven points, which results in that the computation burden is increased extremely. Furthermore, a duty-ratio-based MPC is developed for a three-level inverter in [26], termed MPC3. Carrying out this algorithm in the two-level inverter, AV2R of six lines can be covered due to that the null vector is inserted as shown in Figure 2d. However, both of these two arts still suffer from the problem of an inadequate AV2R, and hence, the discontinuous locus of applied vectors.

Comparatively, the art proposed in [27] for a three-level converter is outperforming in terms of the AV2R. When this method is extended to the two-level inverter, the AV2R will reach the whole hexagon region. Nevertheless, all the actual vectors as well as the vector combinations of the six sectors should be assessed in the two-level inverter, which will introduce an exceptionally high computation burden.
