## 3 Voltage Source Inverter Model
Figure 4 illustrates the eigenvalues that define the bandwidth of the open-loop system. Although the model is stable, it exhibits resonance points near 4500 rad/s and 5000 rad/s. Therefore, the control design must ensure robustness to effectively mitigate these oscillations at the resonant frequency. Moreover, it is essential to impose current constraints considering the nominal current of the VSI's power semiconductors and ensure a feasible modulation voltage for safe operation. The voltage magnitude constraints at the converter output, which serves as the input to the LC filter, along with the system's current constraints, are used to formulate the MPC optimization problem. The MPC formulation requires that the constraints be expressed as linear inequalities using the QP approach [15; 21]. However, controlling the VSI in \(dq\) coordinates introduces specific challenges. To overcome these issues, this work adopts the approximations proposed in [29], using polygonal restrictions to the modulated voltage and current restrictions. To ensure the operation of the current within the nominal range, it is crucial to limit the output current of the converter (\(\mathbf{i}_{\mathrm{dq}}\)) within a circular boundary of radius \(r=i_{\mathrm{n}}\), where \(i_{\mathrm{n}}\) is the nominal current of the inverter. As discussed previously, these constraints must be reformulated as linear inequalities. In this study, this is achieved by approximating the circle with a regular dodecagon, as depicted in Figure 5. The current generation capacity accounts for 95.5% of the total circumference, with the remainder sacrificed due to the approximation. This strategy ensures rapid and robust control and maintains safe operation within the limits of the semiconductor. However, it involves a trade-off between the current restriction design's precision and the optimization solver's computational cost. The current constraints \(\mathbf{i}_{\mathrm{dq}}\) shown in Figure 5 can be expressed as inequality constraints by utilizing trigonometric identities, as follows:  
\[\mathbf{l}_{\mathrm{x}}\leq\mathbf{J}\mathbf{x}_{\mathrm{k}}\leq\mathbf{u}_{ \mathrm{x}}, \tag{8}\]  
where the vectors and matrices are given by the following:  
\[\mathbf{l}_{\mathrm{x}}=i_{\mathrm{n}} \begin{bmatrix}3.73\\ 1.36\\ 1.00\\ 1.00\\ 1.36\\ 3.73\end{bmatrix} ,\quad\mathbf{J}= \begin{bmatrix}3.73&1.00&0.00&0.00\\ 1.00&1.00&0.00&0.00\\ 0.27&1.00&0.00&0.00\\ -0.27&1.00&0.00&0.00\\ -1.00&1.00&0.00&0.00\\ -3.73&1.00&0.00&0.00\end{bmatrix} \quad\mathbf{u}_{\mathrm{x}}=-\mathbf{l}_{ \mathrm{x}}. \tag{9}\]  
Figure 4: Frequency response of the open loop VSI. In (8), the current constraints are represented using a constant matrix (**J**) and a vector parameterized by \(i_{\text{n}}\), allowing generalization for various VSI generation capacities. The voltage actuation, on the other hand, is achieved using space vector pulse width modulation (SV-PWM). This technique is widely employed in power electronics and is readily implemented in digital signal processors. The objective of the SV-PWM modulator is to generate a pulsed voltage with the desired average value, ensuring constant switching frequency and optimal utilization of the DC bus voltage. SV-PWM achieves approximately 15% greater utilization of the DC bus voltage compared to sinusoidal PWM techniques [30]. When the Clarke transformation is applied to all modulated inverter voltages, \(v_{\text{sa}\beta,t}=v_{\text{sa},t}+jv_{\text{s}\beta,t}\), these voltages are constrained within the hexagon defined by the six active vectors of the inverter, as illustrated in Figure 6. These constraints can be formulated as constant linear inequalities in the \(\alpha\beta\) frame. However, the control of the inverter with an LC filter is performed in the \(dq\) frame, requiring the rotation of these constraints. This results in \(\theta\)-dependent, time-varying constraints [29], which can be expressed in the \(dq\) frame but involve a high computational cost.