**1. Introduction**

Nowadays, renewable energy sources (RES) such as photovoltaic (PV) solar systems, wind turbines, and others are integrated into conventional power systems to avoid the high cost of constructing new or expanded facilities [1]. The final stage of the integration of PV systems consists of DC-AC inverters. Special consideration for inverter topologies and controls is required to preserve the network stability and to achieve acceptable dynamic performance of the voltage and frequency [2]. Different controllers for micro-grid inverters during grid-connected and islanded operation modes have been investigated in previous studies [3,4]. Corresponding to behavior and operating conditions of the electrical grid, the controllers of the inverter system can be classified as linear, non-linear, robust, adaptive, predictive, and intelligent controllers [5]. Various types of linear controllers for microgrid inverters including classical controllers, Proportional Resonant (PR) controllers, and Linear Quadratic Gaussian (LQG) controllers were reported [6–8]. Non-linear controllers for grid-connected inverter systems (GCIS) such as sliding mode, feedback linearization, and hysteresis controllers have been proposed in [9–13]. In [14], a current-control is proposed for voltage-source inverters using the H∞ robust control technique. Additionally, adaptive control techniques and model predictive controllers for grid-connected and standalone inverters were reported in [15–18]. In all studies referred to, the proposed non-linear controllers showed better performance when compared with linear controllers'

**Citation:** Shadoul, M.; Yousef, H.; Al Abri, R.; Al-Hinai, A. Adaptive Fuzzy Approximation Control of PV Grid-Connected Inverters. *Energies* **2021**, *14*, 942. https://doi.org/ 10.3390/en14040942

Academic Editor: Oscar Barambones

Received: 18 January 2021 Accepted: 5 February 2021 Published: 11 February 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

performance. The main drawback of nonlinear control methods is that they rely on the system mathematical model and system parameters availability.

Intelligent control systems including neural network controllers, repetitive controllers, fuzzy logic controllers (FLCs), and autonomous controllers are introduced for nonlinear systems control. The advantage of intelligent controllers is that they do not rely on the system mathematical model and they can handle many nonlinear and uncertain systems. In [19], a discrete-time repetitive controller (RC) was proposed to improve the output current and to overcome the drawback of using a linear PI controller in the presence of non-linearities in the system components. Type-1 and Type-2 FLCs have been widely used in various applications and have achieved remarkable success in managing higher levels of uncertainty [20–25]. FLC applications also include intelligent control for marine applications, traction diesel engines, robotic control, internet bandwidth control, industrial system controllers, power managemen<sup>t</sup> and electrical control, aircraft control evolutionary computing, and DC-DC converters. Moreover, type-2 fuzzy logic has also proven successful in clinical diagnosis, differential diagnosis, and nursing evaluation in the health field [26].

For GCIS, different FLCs were presented in [27–29]. In [28], the real time testing for FLC for three phase grid-connected inverters to control the voltage and the current was presented. The results demonstrated FLC ability to generate high-quality PV power while maintaining the power factor of unity. A grid side inverter system control using a simple FLC that works well for grid interconnected variable speed wind generators was proposed in [29]. In another work, type-2 FLC (T2FLC) was implemented to control a DC-DC buck converter [30]. For PV systems, interval T2FLC (IT2FLC) based on maximum power point tracking (MPPT) method was proposed in [31]. In addition, the work in [32] implemented a T2FLC as a MPPT to handle the rules' uncertainties during high weather condition variations. The proposed MPPT based on the FLC showed a faster response in the transient response and a stable steady state. A further IT2FLC was developed for single phase grid connected PV systems in [33], where IT2FLC was used as MPPT algorithm. Simulation results showed that the proposed IT2FLC-based MPPT controller has a fast transient response.

In [34], an FLC for inverters in PV applications was presented; the work discussed several factors and challenges and provided guidelines for developing capable and effective inverter control systems.

Moreover, a fuzzy neural network controller based on the Takagi–Sugeno–Kang type approach presented to control the active and reactive power of three-phase grid-connected PV systems during grid faults was reported in [35,36].

Furthermore, to overcome the disadvantages of conventional controllers for uncertain nonlinear systems, adaptive fuzzy control (AFC) techniques were proposed to control uncertain nonlinear systems [37,38]. Due to its advantage in handling complex uncertain nonlinear systems, researchers have used the AFC techniques in different applications [39–43]. The AFC technique was applied for induction motor control in [39], the optimal power conversion control for standalone wind energy conversion systems in [40], permanent magne<sup>t</sup> synchronous motor control, and fuzzy fault-tolerant switched systems in [41,42]. To the best of the authors' knowledge, there is no reported work available that describe the application of AFC to the GCIS. This motivates the authors to propose an AFC scheme that exploits the concept of the multi-input multi-output (MIMO) feedback linearization and the approximation capability of fuzzy systems. PV GCIS are highly nonlinear and uncertain systems due to the intermittent nature of the PVand the inverter pulse widthmodulation (PWM) technique. Without fast-acting inverter control, these nonlinearities and uncertainties lead to power quality, output harmonics, voltage regulation, losses, and system implementation issues. The proposed AFC, based on the method of feedback linearization, is developed to solve these nonlinearities and uncertainties due to the method's ability to manage complex nonlinear control systems without the need for a mathematical model. The fuzzy system's capability to approximate unknown parameters of the GCIS for different operation cases will be used to design the controller. The objectives of the

proposed AFC for GCIS are to control both the power factor and the dc voltage. The quality of the designed controller will be tested to validate its effectiveness in achieving the control objectives for different simulation cases. Moreover, to evaluate the efficiency of the proposed controller, a comparison between its performance, the PI controller, and a Takagi–Sugeno–Kang-type probabilistic fuzzy neural network controller (TSKPFNN) performances was conducted. The main contribution of the paper can be summarized in the following:


The rest of the paper is organized in five sections. The MIMO of GCIS and the feedback linearization are presented in Section 2. Section 3 gives the design of an AFC for a general MIMO. Based on the analysis presented in Section 3, the proposed AFC for GCIS is explained in Section 4. Simulation results are presented in Section 5 and conclusions are drawn in Section 6.

#### **2. Grid-Connected Inverter System (GCIS)**
