## 3 Proposed Discontinuous Model Predictive Control with Zero Vector Selection  
We propose a discontinuous MPC (D-MPC) that minimizes the switching losses by setting discontinuous intervals based on the current magnitude. We improve the current error and THD by using the multi-vector approach presented by [PERSON] and [PERSON]'s research [35] while reducing switching losses through D-MPC [39; 40]. The zero-voltage vectors, \(\mathbf{V}_{7}\) and \(\mathbf{V}_{0}\), are selected based on the polarity of the offset voltage calculated for discontinuous switching, thereby suppressing unnecessary switching. Figure 4 depicts a Block diagram of the proposed D-MPC. Switching losses are proportional to the magnitude of the load current flowing through the inverter leg during switching. The proposed D-MPC applies discontinuous switching by determining sections with the highest current and voltage magnitudes. By adding an offset voltage to the future reference voltage, the voltage in the maximum current sections is fixed at \(+\mathrm{V_{dc}}/2\) or \(-\mathrm{V_{dc}}/2\). The switching losses are reduced by clamping the phase with the maximum current to either a positive or a negative DC link. Additionally, the proposed method selects an optimal zero-voltage vector by utilizing the offset voltage polarity. Although the zero-voltage vectors, \(\mathbf{V}_{7}\) and \(\mathbf{V}_{0}\), share the same magnitude of zero, their switching states vary as (111) and (000), respectively. The improper selection of zero-voltage vectors can cause unnecessary switching, thereby increasing the losses or creating nonoptimal discontinuous intervals. Therefore, the appropriate selection of zero-voltage switching states is crucial for reducing the switching losses. The offset voltage for clamping is calculated based on the voltage and current magnitudes. When the polarity of the calculated offset voltage is positive (+), V7(111) is used as the zero-voltage vector; whereas, when it is negative (\(-\)), V0(000) is used. Selecting the appropriate zero-voltage vectors according to the offset voltage polarity helps in maintaining the switching state of the already clamped phase, thereby preventing additional switching losses. This selection of zero-voltage vectors affects both the switching losses and MVVV generation. Zero-voltage vectors are essential for generating MVVVs. Since different zero-voltage vectors are used based on the offset voltage polarity, the multi-virtual vectors are also distinguished based on the polarity. The vectors generated via this process form the candidate voltage group. The optimal voltage vector that minimizes the difference from the calculated reference voltage is determined using a cost function. The desired load current is tracked by applying the switching state corresponding to the inverter. In summary, the proposed D-MPC presents a balanced performance by reducing the current error and distortion while improving the overall power conversion efficiency. The proposed method constructs a voltage-based cost function. The output phase voltage of the inverter at the next sampling point, calculated based on Equation (6) and the inverse dynamic model, is expressed as follows:  
\[ \begin{array}{l}\mathbf{v}_{a}^{*}(k+1)=\frac{1}{T_{s}}\left\{L_{o}\mathbf{i}_{a0}^{* }(k+1)+(R_{o}T_{s}-L_{o})\mathbf{i}_{a0}(k)+\mathbf{e}_{a}(k)T_{s}\right\}\\ \mathbf{v}_{b}^{*}(k+1)=\frac{1}{T_{s}}\left\{L_{o}\mathbf{i}_{b0}^{*}(k+1)+(R_{o}T_{ s}-L_{o})\mathbf{i}_{b0}(k)+\mathbf{e}_{b}(k)T_{s}\right\}\\ \mathbf{v}_{c}^{*}(k+1)=\frac{1}{T_{s}}\left\{L_{o}\mathbf{i}_{c0}^{*}(k+1)+(R_{o}T_{ s}-L_{o})\mathbf{i}_{c0}(k)+\mathbf{e}_{c}(k)T_{s}\right\}\\ \end{array}  \tag{8}\]  
Figure 4: Block diagram of the proposed discontinuous model predictive control (D-MPC).