*3.4. AV2R Analysis and Summary of the Proposed MPC*

Consequently, the proposed method can enhance the AV2R and improves the steady-state performance consequently. Note that (*d*<sup>0</sup> \* , *d*<sup>1</sup> \* , *d*<sup>2</sup> \* ) is calculated by solving the optimization problem with constraints shown in (10), which ensure (*d*<sup>0</sup> \* , *d*<sup>1</sup> \* , *d*<sup>2</sup> \* ) is a feasible solution. Hence, the proposed method can expand the AV2R to the region of the whole hexagon theoretically, as presented in Figure 5. Compared to three methods of MPC1-3, since all vectors within the hexagon can be generated, the proposed method yields a sufficient AV2R to create the continuous locus of applied vectors and hence leads to a better steady-state performance.

**Figure 5.** AV2R of the proposed MPC method.

To sum up, the implementation of the Lagrange multipliers method to determine the duration of the null vector and two active vectors is the core idea of this paper. The process of the proposed algorithm steps can be listed as follows.

*Step 1.* Measurement of grid-voltage and output currents. Then, to calculate θ, *id*, *iq*, *ed* and *eq* using phase-locked-loop (PLL) and Park's transformation.

*Step 2.* Determination of two active voltage vectors using MPC1, as discussed in Section 3.1.

*Step 3.* Duty ratio calculation for null and active vectors, as discussed in Section 3.2.

*Step 4.* PWM generation based on the SVM concept, as discussed in Section 3.3.
