#### Abstract

A backstepping-based adaptive controller with neural compensator is designed for harmonic suppression in a three-phase active power filter (APF). The fundamental rule of backstepping method is to take some state variables as “virtual controls” and then design intermediate controller. An adaptive neural controller using radial basis function (RBF) is derived to estimate the APF system nonlinearity and strengthen the current’s tracking property and power grid quality. Simulations studies indicate the proposed backstepping-based adaptive neural controller has good current tracking behavior and increased power quality.

#### 1. Introduction

The harmonic distortion of power grid quality is becoming a serious issue with the increasing nonlinear loads in electrical equipment and power systems. Shunt active power filters (SAPF) become the main harmonic treatment way because they can effectively reduce current harmonic distortion and reactive power.

The active power filter (APF) is an intelligent harmonic control device, which detects harmonics and injects compensation current into the grid to improve power quality. There are some current controllers, i.e., hysteresis current control, triangular wave modulation control, and space vector modulation control. With the introduction of the smart grid concept, the control algorithms can achieve more accurate harmonic current tracking control effects. Yue et al. [1] designed a predictive double loop controller in order to increase the robustness and adaptive property of an active filter. Swain et al. [2] improved the robustness and stability of APF with a novel sliding controller. Intelligent control methods were developed in [3–5] to remove the current with harmonics and increase the power grid quality. Carpinelli et al. [6] adopted a multiobjective method in a multiconverter distribution system. Tareen et al. [7] decreased the power switches and improved the properties of grid-connected inverters by reducing the cost, weight, and size.

In the nonlinear systems, unknown nonlinearities can be approximated by intelligent methods such as neural networks [8] and fuzzy systems [9]. Liu et al. [10] derived an adaptive fuzzy controller with output feedback for full state constrained nonlinear systems. Wu et al. [11] proposed mixed fuzzy/boundary schemes for nonlinear parabolic PDE systems. Li et al. [12] developed a novel adaptive neural strategy with prescribed property for nonlinear systems with switched and interconnected uncertainties. Peng et al. [13] designed a novel dynamic surface controller with neural network based on a predictor for nonlinear system with uncertainty. Wang et al. [14–16] developed neural and fuzzy controllers for nonlinear systems with unknown nonlinearities. Xu et al. [17] derived and applied composite neural control strategy to hypersonic flight dynamics. Chen et al. [18] designed a novel intelligent sliding controller with dynamic structure using fuzzy neural controller. Wai et al. [19] designed a fuzzy-neural sliding controller to deal with the chattering.

Backstepping controller can obtain the targets of tracking and stabilization because it is a controller with recursive property, dividing a full system into lower order systems. It can relax the matching condition in the strict feedback system and avoid cancelling the useful existing nonlinearities. The fundamental idea of backstepping is to recursively derive a controller and step back from the subsystem progressively, ensuring stability for each step, until getting to the final step. Thus, adaptive control is combined with neural control and backstepping approach for dynamic systems [20, 21].

Adaptive neural controller was put forward to treat the harmonics in APF and improve the power grid quality in [22–24]. In this work, an adaptive neural backstepping scheme is designed to guarantee the current tracking and improve the system robustness. The innovative points can be listed as follows:

A backstepping method is incorporated with the adaptive neural control to obtain the desired harmonic suppression related to the current in power grid system. The adaptive neural backstepping controller is used to compensate the nonlinear loads and increase the current tracking property and total harmonic distortion (THD) index.

This control method is realized by neural controller without known accurate model of APF, making the controller simpler and easier to be achieved, strengthening the power supply quality. A robust current compensation controller is added to solve the nonzero issue with respect to the approximation errors existing in the neural system.

#### 2. System Description

The schematic diagram of a three-phase shunt APF discussed in this paper is depicted in Figure 1. The main components are nonlinear loads, source and PWM generator, control system, and harmonic current detector module. The control system aimed to stabilize DC link voltage according to a basic value and track the instruction current so as to generate the compensating current to decrease the distortion current caused by nonlinear loads. In Figure 1, , , and represent the voltages in the grid, , , and represent the power currents, , , and represent the loading currents, , , and represent the voltages in the public joint points, , , and represent the compensating current, is the capacitor in the DC side, is the voltage in the capacitance C, is the current in the capacitance C, is the inductance in the AC side, and denotes the equivalent resistance.

The model of an APF system is derived in the next procedure. The circuit relationshipsare obtained by Kirchhoff rules [3], where is the voltage between and .

Assuming the balanced AC supply voltage, and summing equations in (1), considering the absence of the zero-sequence yield

To indicate the IGBT working status, switch function is defined aswhere .

In the meantime, considering , thus (1) is expressed as

Denote the switching state as

Equation (5) shows the relation between and ; then, (5) with the consideration of eight permissible IGBT switching states generates

Then, (4) can be written in simplified form as

Define two state variables

Differentiating and with respect to time yields

Considering the external disturbances, the model of the APF system is rewritten aswhere ), ), , is an unknown, bounded external disturbance satisfying ,

#### 3. Adaptive Backstepping Control

The design of backstepping method consists of two steps. Firstly, a “virtual” controller is designed. Secondly, the real backstepping controller is derived. The detailed procedure with respect to the backstepping method is introduced as follows.

*Step 1. *The ideal current is denoted as with continuous second-order derivatives. The tracking error isThen,The virtual control is designed aswhere .

Define current tracking error asWe choose the first Lyapunov function candidate asDifferentiating (16) yieldsIf , then . So we need to take the second step.

*Step 2. *From (15), we can getThe second Lyapunov function is selected as and the derivative of isTo make , backstepping controller is proposed aswhere .

Then,Based on Lyapunov stability theory, the asymptotic stability is ensured.

#### 4. Adaptive Neural Backstepping Controller

Figure 2 is a block diagram of a three-layer RBF neural network structure, which mainly includes input layer, hidden layer, and output layer. The hidden layer maps the signal from the input space to a higher-dimensional space, and the output layer performs a weighted summation operation to generate an RBF network output value.

RBF network can approach any nonlinear function over a compact set with arbitrary precision. The block diagram of adaptive neural backstepping system is designed in Figure 3.

Since in (11) is unknown, a RBF neural estimator is used to approach . Because there is minimum approximation error in the neural netwro ksystem, in order to guarantee the stability of the closed-loop system, a compensation controller is added to the controller (21). The detailed reason why the compensation controller is incorporated in the control will be discussed in the next derivation steps. As shown in Figure 3, based on (21), the new controller is proposed aswhere In (27), radial basis , is the base width of the node, is the centric vector of the node , denotes the weight of the neural structure:where is a positive constant.

Define optimal parameterwhere is an assemble for .

Define minimum approximation error:where .

Define the third Lyapunov function candidate aswhere .

The derivative of is calculated aswhere .

Substituting (25) into (29) yields

If we choose , thenThis implies that is negative semidefinite, and , , and are bounded signals. From Barbalat’s Lemma [25], that is, if a scalar function is uniformly continuous such that exists and is finite, then . Then, we can conclude , .

#### 5. Simulation Discussion

Simulation studies using Matlab/Simulink and SimPower Toolbox are conducted to verify the performance of the proposed controller. The APF controller starts to work at 0.05 s, and load shock is introduced at 0.12 s. Table 1 shows the simulation parameters adopted in the APF control system: The parameters of APF: inductance on the AC side is selected ; capacitor voltage on the DC side is .

Figures 4 and 5 show the source current and harmonics spectrum, giving that A phase current contains harmonic whose THD value is 24.71%. Figure 6 is the source current harmonic graph at the beginning. After 0.05 s, APF begins to work to make the current close to sine wave in no more than 0.01 s. The load shock is added at 0.12. Figures 4, 7, and 9 plot the source current and harmonics spectrum using backstepping method, and Figures 5, 8, and 10 show the source current and harmonics spectrum using the proposed method. The source current is close to sine waveform after adaptive neural backstepping compensation even with load shock. The THD values are decreased to 2.96% and 3.81% with the backstepping method, whereas the THD values are decreased to 1.63% and 2.08% with the proposed method. The proposed method has better compensation property in the presence of the load shock than the backstepping control.

Figure 11 is the current compensation using the designed method where the tracking performance is well in the presence of nonlinear load shock. For the voltage control, we add the loads in a ladder-type increase. Specifically speaking, we add the same loads to the system at the time 0.1s and 0.2s to see the performance of the controlled system. PI controller is used in the voltage control. DC capacitor voltage is shown in Figure 12, indicating that it is in the range of the reference voltage, and tends to be steady state quickly under the load shock.

#### 6. Conclusion

An adaptive NN backstepping controller is designed for the harmonic suppression of a three-phase APF. A RBF NN controller is used to adaptively estimate and compensate the system nonlinearities, enhancing the robust performance. A compensation control is added to the controller to guarantee the stability. Simulation studies demonstrated that the proposed control strategy can reduce THD values effectively, improving the electric quality.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The authors thank the anonymous reviewers for their useful comments that improved the quality of the paper. This work is supported by National Science Foundation of China under Grant no. 61873085 and Natural Science Foundation of Jiangsu Province under Grant no. BK20171198. The Fundamental Research Funds for the Central Universities under Grant no. 2017B 20014.