**1. Introduction**

Increasing demand for energy, decreasing conventional fossil-fuel energy sources, and environmental concerns are driving forces toward renewable energy sources [1]. Wind energy as a renewable source turns is becoming an important, promising source [2]. It has been the most growing source in terms of installed capacity. However, the dense researches that are executed in the wind area market create various wind energy topologies. One of the common configurations is the Doubly Fed Induction Generators (DFIGs) equipped with variable speed Wind Turbines (WTs) [3]. This configuration is widely adopted due to its significant merits, such as the independent control of active and reactive powers, low

Ibrahim, M.N.; Rezk, H.; Bouallègue, S. An Enhanced DC-Link Voltage Response for Wind-Driven Doubly Fed Induction Generator Using Adaptive Fuzzy Extended State Observer and Sliding Mode Control. *Mathematics* **2021**, *9*, 963. https:// doi.org/10.3390/math9090963

Academic Editor: Nicu Bizon

Received: 3 March 2021 Accepted: 23 April 2021 Published: 25 April 2021

**Citation:** Alhato, M.M.;

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**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/).

converter costs, and mechanical stress reduction [4]. The DFIG has rotor windings that are connected to the grid through power converters which are composed of the Rotor Side Converter (RSC) and the Grid Side Converter (GSC).

The classical control diagram of DFIG-based wind energy conversion systems is mainly constructed based on a vector control strategy in which the well-known Proportional-Integral (PI) regulators are commonly used [5]. At the GSC control part, the performance of the DC-link voltage dynamics depends on the proper tuning of the PI regulator gains and the system parameters, such as resistances and inductances [6]. Therefore, the performance of the DC-link voltage may degrade due to the deviation of the real system's parameters from its nominal values. Besides, the DFIGs are very sensible to grid disturbances because of the direct link of stator windings to the electrical net. Especially, the dip voltage action is not accepted in wind conversion systems since it leads to over-voltages and over-currents in the rotor windings. Moreover, such a phenomenon generates much oscillation in the DC-link voltage as well as in the stator active-reactive powers' dynamics, which leads to disconnect the WT and stop the power generation [7].

Hence, there is a need to develop a robust control strategy that can deal with parameters' mismatch, uncertainties, and external disturbances. To this end, the Sliding Mode Control (SMC) approaches were proposed. In the DFIG-based WT framework, a conventional SMC approach was adopted to regulate the DC-link voltage loop in [7,8]. However, the chattering issue stays a grave challenge when implementing the SMC method, which is highly undesirable in the DFIG systems. To tackle this problem, a new SMC law was developed to enhance the tracking performance and the robustness regarding operational uncertainties [9]. In [10], the fast exponential reaching law was performed and contrasted with the conventional one. In these studies, the obtained results indicated better performances in the DC-link voltage maintenance against parameters' variations and grid voltage disturbances in comparison with the classical PI controller. However, the DC-link voltage ripples introduced by the SMC strategies still result in poor performance.

To enhance the performance of the DC-link voltage dynamics in the presence of disturbances, feed-forward compensation methods were introduced. Several sensor-lessbased DC-link voltage control methods were proposed. However, the implementation of these methods was adopted only for the AC-DC converters. The load current was only considered as a constant DC-disturbance term as shown in [11–13]. On the contrary, the load current that represented the DC rotor had severe ripples due to the RSC control and switching behaviors of the IGBTs in the DFIG system. Therefore, the effect of load current on the dynamics of the DC-link voltage should be studied [14]. To this end, an Extended State Observers (ESO) combined with other controllers is proposed to improve the performance of the voltage control loop. The ESO is used to offer estimations of internal states and external disturbances with minimal data about the system. The ESO considers the lumped disturbances, i.e., parameters' mismatch and unmodeled dynamics, as an extended state, which is estimated and compensated in the control law.

Nevertheless, in the design of an ESO, the selection of the bandwidth parameter is a challenging task that affects the closed-loop system performance. Generally, the larger the ESO's bandwidth, the more accurate the estimation of states will be achieved. On the other hand, the increase in such a bandwidth may lead to noise vulnerability and loss of robustness. The design and tuning of an observer are a trade-off between the estimation performance and noise action [15]. Usually, the bandwidth of the ESO is selected to be about 5 to 15 times the DC-link voltage controller's bandwidth. This ensures that the estimated state dynamics have a fast-tracking performance when the actual state dynamics change. Also, to guarantee that the ESO does not affect the current controller's performance, the selected bandwidth of the ESO should prevent overlapping with the current controller's bandwidth [16]. To this end, the bandwidth of the designed ESO is selected to be between 1/200 and 15/100 of the switching frequency. However, to achieve a fast response of the ESO dynamics during the perturbation, this paper introduces a novel adaptive fuzzy method to tune the value of the ESO's bandwidth within the limited region. Fuzzy gainsscheduling mechanism is introduced to adjust the appropriate values of the observer's bandwidth and further better the performance of the DC-link voltage control scheme. The concept utilizes a fuzzy logic inference as adaptive supervisors to tune the bandwidth gain in real-time.

Further, in this paper, a Second Order Sliding Mode (SOSM) controller based on the well-known Super Twisting Algorithm (STA) is firstly developed to regulate the DC-link voltage. The total disturbance in the DC-link voltage dynamics is theoretically investigated and the proposed fuzzy tuned-ESO is adopted to estimate it. Thus, a combined control law consisting of SOSM controller and disturbance compensation through the fuzzy gainsscheduling based-ESO is elaborated for the DC-link voltage regulation loop. Moreover, this paper investigates the stability conditions using the Lyapunov theory of nonlinear systems to be used as operational constraints for the formulated optimization-based control problem consisting of the tuning of all effective parameters of the designed ESO-based SOSM controller. Such a hard and constrained optimization problem is solved thanks to advanced competitive global metaheuristics, such as Thermal Exchange Optimization (TEO) [17,18], Water Cycle Algorithm (WCA) [19], Particle Swarm Optimization (PSO) [20], Genetic Algorithm (GA) [21], Grasshopper Optimization Algorithm (GOA) [22] and Harmony Search Algorithm (HSA) [23]. The main advantages of the designed adaptive controller are: (1) it presents a fast time-domain response and high robustness of the closed-loop DC-link voltage loop under external disturbances, (2) it clearly reduces the chattering phenomenon that is introduced by the SMC strategies, and (3) it ensures high tracking performance through a selection of the optimal and adaptive gains for the proposed control method.

The rest of the paper is arranged as follows: An elaborated dynamical model of the GSC component is described in Section 2. Section 3 investigates the second-order sliding mode controller design based on constrained optimization methods. The established uncertainties and stability conditions in the sense of Lyapunov for nonlinear systems are served as operational constraints for the reformulated optimization problem. Section 4 discusses the design of the fuzzy tuned-ESO-based DC-link voltage control loop. Comparative simulation results, gained on a 1.5 MW DFIG, are investigated in Section 5. Finally, Section 6 states the conclusion and future works.

#### **2. Modeling of the DFIG Based Wind Energy Converter**

The GSC component is joined to the electrical net via an L or LCL filter. However, for a better harmonics reduction of the grid currents, architecture with an LCL filter is adopted [3,24]. The mathematical modeling of the GSC system is defined in the d-q synchronous reference frame as given by Equation (1). In these dynamics, *LT* states the sum of the converter and grid side inductances. Indeed, at the fundamental frequency, the LCL filter can be considered as an L-one with an inductance equal to the sum of the LCL-filter inductors [3]:

$$\begin{cases} L\_T \frac{di\_{d\xi}}{dt} = -R\_T i\_{d\xi} + \omega\_{\mathcal{S}} L\_T i\_{q\xi} + V\_{d\xi} - V\_{df} \\ L\_T \frac{di\_{q\xi}}{dt} = -R\_T i\_{q\xi} - \omega\_{\mathcal{S}} L\_T i\_{d\xi} + V\_{q\xi} - V\_{qf} \\ \frac{dV\_{d\xi}}{dt} = \frac{1}{C\_{dc}} \left( \frac{3}{2} \frac{V\_{d\xi}}{V\_{dc}} i\_{\xi d} - i\_{rdc} \right) \end{cases} \tag{1}$$

where *Cdc* is the capacitance of the DC-link circuit and *irdc* is the current between the DFIG rotor and the DC-link component.

#### **3. Design of Second-Order Sliding Mode Controller**

⎪⎪⎨

The main aim of the GSC control is to keep the DC-link voltage constant [5,7]. To perform this goal, Figure 1 presents the proposed control scheme. In such a design setup, the direct grid current is controlled thanks to the STA-based SOSM controller to maintain the DC-link voltage constant. The quadrature grid current can be used to regulate the reactive power's flow between the GSC and the grid.

**Figure 1.** General block diagram for the proposed control scheme method.
