## 1 Introduction  
With the development of power grids, more and more problems occur, accompanied by convenience and benefits. While power grids and distribution networks are desired to simultaneously interact with a large quantity of loads, the increasing number of nonlinear loads connected to the grids has caused a negative effect on the quality of electric power supply such as low power factor, electrical harmonics, and so on. Instead of a passive filter, the active power filter (APF) is a new power electronic device used for its higher flexibility and better capability of suppressing harmonic currents.  
Due to the wide range of applications of APF in power systems over the past decades, many scholars have applied a variety of intelligent control methods to APF. [PERSON] et al. [1] developed a new oriented control model for shunt active power filter (SAPF)-load system, and designed an adaptive controller based on the new model. [PERSON] et al. [2] proposed a new robust adaptive controller based on shunt active power filters, to reduce high overshoot and large settling time in the DC link voltage. [PERSON] et al. [3], [PERSON] et al. [4], [PERSON] et al. [5], and [PERSON] et al. [6] investigated intelligent control methods such as an adaptive neural network controller and fuzzy neural controller for an active power filter to track command current to eliminate the harmonic current and improve the power quality of the power system. [PERSON] et al. [7] presented a SAPF configuration and showed advantages of harmonic distortion and power converter losses. [PERSON] et al. [8] presented a D-Q current control method using a phase lock loop to improve the power quality.  
Recently, fractional calculus as well as its application begins to attract more and more attention. In the last three decades, many engineers have applied fractional calculus and fractional differential equation theory with various applications [9; 10; 11; 12; 13; 14]. Backstepping control techniques are powerful tools for their systematic and recursive design methodology for nonlinear feedback control [15; 16; 17; 18].  
Adaptive fractional fuzzy sliding mode controls and adaptive fuzzy-neural fractional finite-time sliding controllers are developed for active power filters [19; 20; 21]. In the nonlinear systems, unknown nonlinearities can be approximated by intelligent methods such as fuzzy systems [22; 23; 24; 25; 26; 27; 28; 29] and neural networks [30; 31; 32; 33; 34; 35]. Intelligent control methods have been investigated for dynamic systems. Motivated by the above literature, in this paper, a fractional adaptive backstepping sliding mode control method combined with fuzzy controller for a three-phase APF is proposed. The main motivations are emphasized as follows:  
1. A backstepping control strategy is applied to the design of a fractional sliding mode adaptive fuzzy controller. We avoid establishing a precise mathematical model of active power filter by transforming the general circuit equation into an analogical cascade system where the backstepping approach can be implemented.
2. Based on the backstepping control design, this paper extends the conventional integer-order sliding surface to fractional ones for three-phase active power filter. That means the system can achieve an extra degree of freedom and there would be more parameters to be adjusted to improve total harmonic distortion (THD).
3. A fractional sliding mode controller ensures that the control system reaches the sliding surface while the adaptive control strategy and fuzzy controller are also combined together to approximate the unknown dynamic model term and identify adaptive parameters online.