## 6 Conclusions  
This paper proposes a new sliding surface for controlling a SBBC which simultaneously incorporates PFC and DC bus regulation. The proposed sliding surface contains: (1) a normalized DC voltage error term; (2) an AC current error term; and (3) an integral of the AC current error term. The surface was validated using sliding mode conditions, simulations and experimental tests. The surface was implemented for PI-SMC and SMC, and an AHB was used for fixing the switching frequency and reducing \(THD_{i}\). Simulations were performed to compare the three controllers Cascade PI, PI-SMC and SMC in terms of their dynamic behavior. When applied a DC power change of 250 W (50%), it was found that SMC presented the best performance, since the DC voltage presents the lowest stabilization time (30 ms) and practically does not present overvoltage (0.1%) which guarantees the protection of sensitive loads. In addition, the AC current has the fastest time response (0.04 ms) and presents the best behavior in zero crossings which reduces \(THD_{i}\) (3.7%). Several simulations were also carried out concerning the behavior of the proposed sliding surface using SMC in start-up and without a pre-charge of the DC bus, forcing the system to work in adverse conditions. In this situation, the SMC responded adequately limiting DC overvoltages and stabilizing the AC current within the first half cycle despite the large start-up over current. In these simulations, the stability of the SBBC was evidenced when the SMC was implemented. Experiments were implemented for PI-SMC and SMC considering changes in the DC voltage set-point and in the DC load current. It was found that the SBBC responds better under changes and perturbations, even though the perturbations of the SMC were more severe than the ones of the PI-SMC. Finally, zero crossing comparison showed a better behavior for SMC working with AHB, which demonstrates the rapid response of SMC under instantaneous sign changes of the sliding surface. Conceptualization, J.R.O.-C., G.E.M.-R., N.M.-G. and J.B.C.-Q. ; Data curation, J.R.O.-C.; Formal analysis, J.R.O.-C., G.E.M.-R., N.M.-G. and J.B.C.-Q. ; Funding acquisition, N.M.-G. and J.M.L.-L.; Investigation, J.R.O.-C., G.E.M.-R. and N.M.-G.; Methodology, J.R.O.-C., G.E.M.-R., N.M.-G. and J.B.C.-Q. ; Project administration, N.M.-G.; Resources, J.R.O.-C. and G.E.M.-R.; Software, J.R.O.-C. and G.E.M.-R.; Supervision, N.M.-G. and J.M.L.-L.; Validation, J.R.O.-C. and G.E.M.-R.; Visualization, J.R.O.-C., N.M.-G. and J.M.L.-L.; Writing--original draft, J.R.O.-C.; and Writing--review and editing, G.E.M.-R., [PERSON]-G., [PERSON]-L. and J.B.C.-Q. All authors have read and agreed to the published version of the manuscript. This research was funded by the Colombia Scientific Program within the framework of the so-called Ecosistema Cientifico (Contract No. FP44842-218-2018). Not applicable. The data presented in this study are openly available in Github repository ([[https://github.com/joserca1989/appendix-paper-SMC](https://github.com/joserca1989/appendix-paper-SMC)]([https://github.com/joserca1989/appendix-paper-SMC](https://github.com/joserca1989/appendix-paper-SMC)), accessed on 19 January 2021). The authors thank to Corporacion Universitaria Minuto de Dios--UNIMINUTO for the support received in this research. The authors also acknowledge the support from the \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Criteria**} & \multicolumn{2}{c}{**Hybrid SMC-PI (Load 25\%)**} & \multicolumn{2}{c}{**Proposed SMC (Load 50\%)**} \\ \cline{2-5}  & **Simulation** & **Experimental** & **Simulation** & **Experimental** \\ \hline \(v_{o}\) Stabilization time & 1.6 s & 1.53 s & 0.03 s & 0 s \\ \(v_{o}\) Undervoltage & 15.5 v & 18 v & 0.4 v & 0 v \\ \(i_{s}\) Cycles to reach steady state & 24 & 27 & less than 1 & less than to 1 \\ \(THD_{i}\) & 4.25\% & 4.82\% & 3.7\% & 2.67\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental vs. simulation: test with load increase.