## 4 Simulation Results
### Sliding Mode Control Behavior  
This section presents the stability of SMC, forcing the system to work in adverse conditions consisting on the start-up without a pre-charge of the DC bus or under large perturbations. The response of the DC voltage in SBBC start-up is presented in Figure 11. Figure 11a shows the DC voltage without DC-bus pre-charge. Note that \(v_{o}\) does not present instability or over peaks and it increases until reaching the stabilization point (400 V) in 1.52 s. The stabilization time can be modified by means of the sliding coefficient depending on the desired priority order, increasing or decreasing the response time for DC voltage or AC current. Figure 11b presents the behavior of the normalized DC voltage error (\(S_{1}\)); initially, voltage error is bounded by its minimum value \((-1)\). Therefore, \(S_{1}\) at start-up according to the sliding coefficient value. Then, \(S_{1}\) rapidly reaches the convergence point (stabilization point) and keeps sliding around \(S_{1}=0\). Normalization of output voltage error allows reducing the output voltage impact during large disturbances, giving priority to current control and avoiding non-minimum phase behavior [29]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Grid voltage \((v_{s}=v_{in})\) & 120 Vrms \\ Grid frequency \((f)\) & 60 Hz \\ DC bus capacitor \((\mathrm{C})\) & 2.2 mF \\ DC bus voltage \((V_{o}=V_{ref})\) & 400 V \\ Inductors \((L_{1}=L_{2}=L)\) & 2.2 mH \\ Switching frequency \((f_{sw})\) & 40 kHz \\ Rated power \((P_{o})\) & 500 W \\ Rated Load \((R)\) & 320 \(\Omega\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. Figure 10: DC voltage transient response when load increases from 250 W to 500 W, comparing the performance of (**a**) PI control; (**b**) PI-SMC; and (**c**) SMC. Figure 9: AC current performance of: (**a**) PI control; (**b**) PI-SMC; and (**c**) the proposed SMC. The behavior of AC current in SBBC in start-up is presented in Figure 12. In Figure 12a, a transient with an over peak near 15 times the desired value is observed; nevertheless, the over peak is only presented for the first half cycle of the wave and the AC current rapidly stabilizes in a sinusoidal wave in the second grid cycle while zero crossing deformation is gradually reduced in four cycles. Figure 12b presents the behavior of AC current error (\(S_{2}\)). Initially, \(S_{2}\) also has a couple of large transient oscillations, and then the magnitude of oscillations is significantly reduced, being only evident for the first zero crossings. \(S_{2}\) reaches the convergence zone oscillating near the equilibrium point in the adaptive hysteresis band. The main function of \(S_{2}\) consists on properly following the AC current reference according to load requirements which permits avoiding the non-minimum phase behavior; in addition, another function of \(S_{2}\) consists on performing PFC. The behavior of \(S_{3}\) in start-up is presented in Figure 13. \(S_{3}\) is reset each electric grid cycle to set the cumulative error as zero. \(S_{3}\) begins in zero and immediately decreases near \(-1.2\) for the first peak; then, it increases again and reaches the convergence zone around \(S_{3}=0\). \(S_{3}\) provides robustness under disturbances; its effect is more evident during start-up reducing oscillations before reaching the steady state. To illustrate the impact of \(S_{3}\) during start-up, Figure 14 shows the input current without using \(S_{3}\). Comparing Figures 12a and 14, it can be seen that current overshoot and stabilization time are reduced when \(S_{3}\) is included in the sliding surface. The stabilization of DC voltage and AC current and the convergence to zero of their errors (including integral) in SBBC start-up demonstrate the stability of the proposed sliding surface outside the operation zone (hitting condition and Lyapunov criteria). Figure 11: DC voltage behavior in start-up: (**a**) DC voltage; and (**b**) DC voltage error \(S_{1}\).