# An Adaptive Model Predictive Voltage Control for LC-Filtered Voltage Source Inverters
## 5 Simulation and Experimental Results
Figure 5: Simulation results showing transient performance under step-change of the load from zero to 5 kW of the VSI, (**a**) the proposed adaptive MPC, (**b**) conventional MPC. Figure 6: Simulation results showing steady-state performance under nominal load (5 kW) and 75% error in the initial value of filter capacitance (\(\epsilon_{C}\) = 75%), (**a**) the proposed adaptive MPC, (**b**) conventional MPC. In Figure 8, the performance of the conventional and the proposed control methods in terms of the steady-state error and THD of the output voltage for a wide range of filter parameters uncertainties is investigated, which confirms the excellent robustness of the proposed control method over a wide range of parameters variations. In this figure, \(e_{L}\) and \(e_{C}\) are errors between the real values of the filter inductance and capacitance and the estimated ones in the control system:  
\[e_{L}=\left(\frac{L_{c}-L_{f}}{L_{f}}\right)\times 100,\ e_{C}=\left(\frac{C_{c}-C_{ f}}{C_{f}}\right)\times 100 \tag{13}\]  
Figure 8: Simulation results showing robustness analysis of the proposed adaptive MPC and conventional MPC against the filter parameters uncertainties, (**a**) voltage tracking error, (**b**) total harmonic distortion (THD\({}_{\text{no}}\)). Figure 7: Simulation results showing outputs of the proposed adaptive observer under nominal load (5 kW) and 75% error in the initial value of filter capacitance (\(e_{C}\) = 75%), (**a**) estimated capacitor voltages and voltage estimation errors (**b**) estimated inverter currents and current estimation errors, (**c,d**) estimated disturbance inputs and disturbance estimation errors. \(L\) and \(C\) are the filter inductance and capacitance, and subscripts \(f\) and \(c\) denote the output filter and control system parameters. Finally, to evaluate the testability of the proposed control method in practice, many tests have been carried out in the laboratory setup. Performance of the proposed adaptive MPC under different conditions, including steady-state operation under linear and nonlinear loads, and transient performance and load-current disturbance rejection are shown in Figure 2:0 The experimental results are the same as the simulated ones and confirm the proposed method's excellent performance in practice. In the end, it is worth noting that the proposed adaptive MPC improves system robustness and provides load-current-sensorless operation, however at the same time, it does not increase the complexity and computations of the conventional one considerably. Indeed, the conventional MPC employs the state-space model in (3) and measured signals to compensate for the control delay. In contrast, the proposed adaptive MPC replaces the open-loop estimator (3) with the closed-loop ones in (7) and (10). Therefore, as can be seen from (3), (7), and (10), the proposed estimator has eight more sums and four multiplications, which is negligible regarding the total calculations and existence of powerful processors. However, these extra computations remarkably increase the system performance and provide the proposed control method's sensorless operation, which increases the system reliability, and reduces the cost and size. Figure 2:1 Experimental results showing performance of the proposed adaptive MPC under different conditions, (**a**) and (**b**) steady-state performance under linear and nonlinear loads, and (**c**) transient performance under step-change of the load from zero to 5 kW.