**4. Simulation and Interpretation**

In order to confirm the authenticity and advantage of the proposed control technique, the system has been implemented, validated and realized using the Matlab/Simulink package. In the simulation, the attitude of each control and its performance, such as the PI controller, fuzzy logic controller and backstepping controller, are analyzed in order to verify the e fficiency of the active hybrid HAPF filter used to control the proposed backstepping, to compensate for harmonics and improve the quality of electrical power. The mathematical calculation of the parameters of the backstepping controller is complex, so these are carefully chosen to achieve the desired objective. The other parameters will be given as follows *Vs*1 = *Vs*2 = *Vs*3 = 220 V, the passive filter parameters are as follows *L* = 0.01 H, *C* = 150 μF, the reference voltage of the DC bus is equal to 620 V, the pollutant load is a three-phase diode rectifier, its output an inductance of 0.003 H, in series with a resistance of 18 Ω, and the energy storage capacity is chosen from 2000 μF. The Simulink model realized is illustrated in Figure 6.

The Matlab/Simulink package was used to realize the Simulink model of the system with the proposed approach. The latter is composed by a three-phase source connected to a rectifier bridge used as a non-linear load supplying a load of type RL, a block of the *p*-*q* method for identifying reference currents from the load currents, a static power converter in series with a passive filter connected in parallel with the load, a linked control block using the backstepping controller for voltage regulation *Vdc* to the capacitor terminals *C* with the identification block. More than one block linked to the identification block allows the regulation of the injected current and transfers the control pulses to the converter for the semiconductors. The model is illustrated in Figure 6.

Figures 7 and 8 show the charging current (for clarity, only one phase is exposed) and its harmonic spectrum. It is clearly proven that there is a significant distortion of the charging current, and that the total harmonic distortion (THD) is proportionally high (29.52%).

**Figure 6.** Realized Simulink model.

**Figure 8.** FFT load current analysis.

Figures 9–11 present the source current spectrum, after the HAPF compensation using the PI controller, fuzzy logic controller and the proposed backstepping controller, respectively. We notice that the THDs are reduced to 2.53%, 1.81% and 1.37%, all within the standard IEEE harmonics limits of 5%, but the backstepping is significantly reduced.

**Figure 9.** FFT analysis of the source current after filtering using the PI controller.

**Figure 10.** FFT source current analysis after filtering with fuzzy logic controller.

**Figure 11.** FFT source current analysis after filtering with backstepping controller.

Figures 12–14 illustrate the source current after filtering. It is observed that the source current with the backstepping controller is clearly sinusoidal. Also for the PI controller, and the fuzzy logic controller, the source current is almost sinusoidal but including disturbances.

**Figure 12.** Source current after filtering with the PI controller.

**Figure 13.** Source current after filtering with the Fuzzy logic controller.

**Figure 14.** Source current after filtering with the backstepping controller.

Figures 15–17 present the reference current and the compensation current. It is observed that the compensation current is coincided with the reference current, and can accurately follow the reference current using the proposed backstepping controller, whereas in the case of the PI controller and the fuzzy logic controller there is a small error between the two currents. In general, this indicates that the proposed control technique can guarantee the exact monitoring of the reference current.

**Figure 15.** Reference and compensation current used by the PI controller.

**Figure 16.** Reference and compensation current used by the Fuzzy logic controller.

**Figure 17.** Reference and compensation current of the backstepping controller.

Figures 18–20 indicate the DC bus voltage *Vdc* followed the variation of its reference with the backstepping controller at a better speed. The system is stabilized at the time *t* = 0.06 s, and we notice a good accuracy, but the result with the PI linear controller contains an error between the DC bus voltage *Vdc* and its reference which varies from 600 V to 620 V (as a ripple in the transient regime) and during the delay as shown in Figure 18. It is noted that *Vdc* does not follow its reference variation in the transient regime. Also, a high response time is found. The system is only stabilized at the time

*t* = 0.135 s, which translates into a poor speed and consequently a degradation of the performance of the HAPF. Concerning the fuzzy non-linear logic controller, the system is stabilized at the time *t* = 0.0755 s. The delay of the voltage *Vdc* deviates from its reference and presents a response time at 5% lower. These make the system slower, and it contains oscillations in the permanent regime, consequently the system degrades the accuracy. The controller's earnings are elected by test to achieve satisfactory performance. It should be noted that the THD with backstepping is significantly lower than the PI control, and fuzzy logic. We can say that the backstepping command has better control performance in terms of oscillations and response time compared to the PI and fuzzy logic controller. The response of the HAPF can be improved by using the proposed control method that achieves the desired performance.

**Figure 18.** DC bus voltage *Vdc* used with PI controller.

**Figure 19.** DC bus voltage *Vdc* used with Fuzzy logic controller.

**Figure 20.** DC bus voltage *Vdc* used with backstepping controller.
