## 3 Proposed Discontinuous Model Predictive Control with Zero Vector Selection
where \(\mathbf{v}_{a}^{*}(k+1)\), \(\mathbf{v}_{b}^{*}(k+1)\), and \(\mathbf{v}_{c}^{*}(k+1)\) denote the inverter output phase voltages at the next sampling point, and \(\mathbf{i}_{ao}^{*}(k+1)\), \(\mathbf{i}_{ho}^{*}(k+1)\), and \(\mathbf{i}_{co}^{*}(k+1)\) denote the inverter output phase currents at the next sampling point, respectively. The discontinuous switching intervals are determined based on the voltage and current magnitudes. The maximum and minimum reference voltages calculated using Equation (8) are calculated as follows:  
\[ \begin{array}{l}\mathbf{v}_{max}^{*}(k+1)=max\ \big{(}\mathbf{v}_{a}^{*}(k+1),\mathbf{v}_{ b}^{*}(k+1),\mathbf{v}_{c}^{*}(k+1)\big{)}\\ \mathbf{v}_{min}^{*}(k+1)=min\ \big{(}\mathbf{v}_{a}^{*}(k+1),\mathbf{v}_{b}^{*}(k+1),\mathbf{v}_{ c}^{*}(k+1)\big{)}\end{array}  \tag{9}\]  
where \(\mathbf{v}_{max}^{*}(k+1)\) denotes the maximum voltage, and \(\mathbf{v}_{min}^{*}(k+1)\) denotes the minimum voltage, respectively. Subsequently, the currents flowing in the phases where these two voltages occur are identified and designated as the maximum and minimum currents, respectively. In particular, the current in the phase where \(\mathbf{v}_{max}^{*}(k+1)\) occurs is assigned as \(\mathbf{i}_{max}^{*}(k+1)\), whereas the current in the phase where \(\mathbf{v}_{min}^{*}(k+1)\) occurs is assigned as \(\mathbf{i}_{min}^{*}(k+1)\). If the absolute value of \(\mathbf{i}_{max}^{*}(k+1)\) exceeds that of \(\mathbf{i}_{min}^{*}(k+1)\), the offset voltage polarity becomes positive; otherwise, it becomes negative. The offset voltage is calculated as follows:  
\[ \begin{array}{l}\mathbf{v}_{offset}(k+1)=\left\{\begin{array}{ll}\frac{V_{dc}}{ 2}-\mathbf{v}_{max}^{*}(k+1),&\text{if }|\mathbf{i}_{max}^{*}(k+1)|>|\mathbf{i}_{min}^{*}(k+1)|\\ -\frac{V_{dc}}{2}-\mathbf{v}_{min}^{*}(k+1),&\text{if }|\mathbf{i}_{min}^{*}(k+1)|>|\mathbf{i}_{max}^{*}(k+1)| \end{array}\right.\end{array}  \tag{10}\]  
where \(\mathbf{v}_{offset}(k+1)\) denotes the offset voltage. When its polarity is positive (+), it is defined as a P-type offset voltage, and when it is negative (-), it is defined as an N-type offset voltage. The calculated offset voltage is added to the future reference voltage to clamp the phase with a high current to either a positive or negative DC link. Figure 5 depicts the calculation process of the offset voltage. Figure 5: Flowchart for offset voltage calculation. In the proposed method, \(V_{7}(111)\) is selected when the offset voltage is P-type, and \(V_{0}(000)\) is selected when it is N-type. Figure 6 depicts the selection of the zero-voltage vectors based on the offset voltage polarity. This selection of the zero-voltage vectors affects the switching losses and alters the switching states of the considered MVVVs. For the MVVV, \(V_{m1}\)\(\sim\)\(V_{m6}\), the zero-voltage vectors, are essential for generation. Therefore, they are classified as P-type and N-type multi-vectors based on the offset voltage polarity. Since \(V_{m7}\)\(\sim\)\(V_{m12}\) is generated by synthesizing only the active vectors, it is a neutral vector that is unaffected by polarity. Table 1 lists the switching states of the inverters used in the proposed method.