## 5 Simulation and Discussion  
In this section, the feasibility of the proposed fractional backstepping sliding mode adaptive fuzzy controller is verified at the platform of MATLAB/SIMULINK with SimPower Toolbox. Comparison between the controller with fractional modules and the controller with integer orders is also given to show the advantage and effectiveness of the proposed scheme. The membership we choose is \(\mu=\exp[-(x+4-(i-1)*1.6)^{2}]\), \(i=1,\ldots,6\) as shown in Figure 3.  
In the fractional backstepping sliding mode adaptive fuzzy controller, \(\lambda_{1}=1,\lambda_{2}=0.1,\lambda_{3}=0.000001\), \(\theta_{fk}=\left[ \begin{array}{cccc}\theta_{hk1}&\theta_{hk2}&\theta_{hk3}& \theta_{hk4}&\theta_{hk5}&\theta_{hk6}\end{array} \right]^{T}\), where \(k=1,2,3\), and adaptive gain \(r=100,000\), \(c=100,000\). The main parameters of the APF system are shown in Table 1.  
When the electronic system starts to work at 0 s, the source current of A phase is shown in Figure 4, which is seriously affected by the harmonic current. At the time t = 0.04 s, the switch is closed and the proposed APF controller begins to work, the source current in Figure 5 tends to a steady state after a half cycle which is about 0.01 s. \begin{table}
\begin{tabular}{c c} \hline \hline Supply voltage and frequency & \(V_{s1}=V_{s2}=V_{s3}=220\) V, \(f=50\) Hz \\ \hline Switching frequency & \(f_{sw}=10\) KHz \\ \hline The non-linear load & \(R=10\) \(\Omega\), \(L=2\) mH \\ \hline Active power filter parameters & \(L=10\) mH, \(R=100\)\(\Omega\), \(C=100\) uF, \(v_{dref}=850\) V \\ \hline PI controller & \(k_{p}=0.03\), \(k_{i}=0.01\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters. Figure 3: The member function degree of \(x\). When the fractional order is set with different values, the effects of tracking the instruction current obtained are not exactly the same. Figure 6 shows the specific waveforms of the instruction current and compensation current with different values of fractional order. It can be observed that if \(\alpha\) is too small (\(\alpha=0.1\)), the compensation current can track the instruction current reluctantly with considerable error which cannot meet the tracking standard. While \(\alpha=0.5\), tracking quality is significantly improved, however, the effect is not excellent for its relative larger tracking error compared with the result when we choose \(\alpha=0.9\). Logically, we choose \(\alpha=0.9\) in order to achieve better effects. Correspondingly, the compensation current tracking error while \(\alpha=0.9\) is shown in Figure 7, further illustrating the advantage of the proposed fractional backstepping sliding mode adaptive fuzzy controller in tracking instruction current. The values of total harmonic distortion (THD) before and after the proposed APF controller is implanted into the system are 24.71% and 1.5%, respectively. The values of the THD index decrease from the high value of 24.71% to the low value of 1.5% in 0.06 s, which is far less than the harmonic standard of IEEE of 5% [36], meaning that the designed controller has a good effect in decreasing distortion. Figure 8 shows the changing tendencies adaptive parameter \(\theta_{f}\). Obviously, the parameters converge to stable values. In order to verify the good robustness of the proposed controller in the presence of load changes, we add the loads in a ladder-type increase. Specifically speaking, we add the same loads to the system at the time 0.1 s and 0.2 s to see the performance of the controlled system. THD in 0.16 s and 0.26 s is 1.39% and 1.83%, respectively, still under 5%, which verifies the strong robustness of the system. Moreover, we adopt the Propotional-Intergral (PI) controller at the DC side to acquire a stable DC capacitor voltage. In Figure 9, it can be seen the DC capacitor voltage can tend to be stable regardless of the changes of the applied load. Figure 4: The previous source current of A phase. Figure 5: The improved source current of A phase.