## 1 Introduction
SMC ensures current control for PFC when load is increasing up to 140%; however, the DC voltage presents variations of up to 13% around the equilibrium point and a delay of several grid cycles for voltage stabilization is observed. Basically, DC voltage was not considered in the sliding surface design; hence, the controller response is slow for DC bus regulation. In [26], the authors presented an analysis between integral and double integral SMC for AC current error. Their current sliding surface with an integral component presents low steady state error and high PF; nonetheless, the AC current has high \(THD_{i}\). The sliding surface with double integral reduces steady state error and \(THD_{i}\); however, a DC current component appears. For this sliding surface, DC voltage dynamic was not considered. [PERSON] et al. [27], [PERSON] and [PERSON] [28] implemented a hybrid PI-SMC for current error reduction. In this case, the DC voltage is controlled by the external PI loop and it does not present over peaks when the load or source are disturbed or when the reference voltage is changed. Nevertheless, a slow response with delay of seconds for both AC current and DC voltage is observed. [PERSON] and [PERSON] [15] implemented an AC/DC converter with hybrid PI-SMC for PFC. In this case, a genetic algorithm (GA) for obtaining the sliding coefficients was developed. AC current and DC voltage errors were considered in the sliding surface design; however, the DC voltage response presents oscillations and high overvoltagesin the presence of load disturbances since the SMC response was limited by the linear controller. In addition, \(THD_{i}\) increases due to the significant distortion produced in the AC current near zero crossing. [PERSON] et al. [23] implemented an AC/DC converter based on totem-pole topology with a hybrid PI-SMC controller. AC current and DC voltage errors were considered in the sliding surface design. An integral surface of AC current and DC voltage errors were also included, reducing the steady state errors. In this case, fast responses were reached when AC current changes were incorporated; nevertheless, an external limiter for protection of power switches was required. In addition, the DC voltage presented over peaks due to the linear component of the controller. [PERSON] and [PERSON] [18] presented an AC/DC converter with a hybrid PI-SMC controller and a PWM modulation for fixing switching frequency. The AC current and DC voltage with their corresponding integrative errors were considered. The DC voltage presented oscillations with over peaks of up to 11%, which were mitigated in several cycles of the source when the load changed, so that linear controller response was slower. Table 1 summarizes the key point of the literature review. According to the reviewed literature, the knowledge gap consists of developing a SMC controller for SBBC that simultaneously performs Power Factor Correction (PFC) and DC bus regulation without using cascade PI or PI-SMC controllers to achieve the best dynamic behavior that removes DC overvoltages and responds fast to DC voltage changes or DC load current perturbations. For this, we propose the use of SMC with a new sliding surface with three terms that improves the following aspects: (1) A normalized DC voltage error term removes DC voltage instability in start-up and large disturbances, since the controller gives priority to the current controller component increasing robustness [29]. (2) An AC current error term implements a PFC scheme and guarantees fast current stabilization during disturbances. (3) An integral sliding term for the AC current error increases the overall system stability and reduces the stable-state error in presence of load or source disturbances. It is worth mentioning that this paper presents a rigorous deduction of the sliding mode conditions giving specific details and explaining the phenomenon of current waveform distortion that is presented in zero crossings.