Artificial intelligent-based backstepping control of parallel-connected multi-machine wind generation system using a new fifteen-switch rectifier

Abstract

This academic work offers a new variable-speed wind power generation system (NVS-WPGS) that employs two 5-phase permanent magnet synchronous generators (PMSGs) controlled through a fifteen-switch rectifier (FSR) architecture for a system that’s integrated to the power grid. The two 5-phase PMSGs are linked to the DC bus through an FSR, which, in turn, connects the DC link to the grid via a three-phase inverter. To ameliorate the operational efficiency of the wind system under investigation, the backstepping control (BSC) approach was employed. Building upon this technique, the Adaptive Neuro-Fuzzy Inference System (ANFIS) was trained and implemented for controlling the wind system. The goal is to boost system performance and control. While BSC is a strong control method for reference signal tracking, its dependence on system parameters is a downside. This dependence typically degrades performance. On the other hand, ANFIS exhibits robustness to variations in system parameters and uncertainties. The efficaciousness of the suggested control methodology was evaluated via experimental investigations conducted using the Matlab/Simulink program. The simulations were performed to verify the system’s capacity to maintain controlled variables in accordance with desired references, even in the presence of fluctuations in wind velocity. Simulations exhibit that the suggested ANFIS-BSC control system accurately tracks controlled intended references. Furthermore, system performance improved. Speed overshoot was decreased by 100%. Additionally, the system’s efficiency was enhanced, reaching a level of 96.5%, beating the BSC approach’s 96%. These findings underscore the remarkable effectiveness of the ANFIS-BSC approach compared to BSC methodologies.

Keywords

Variable-speed wind power generation system (VS-WPGS)permanent magnet synchronous-generator (PMSG)fifteen-switch rectifier (FSR)backstepping control (BSC)adaptive neuro-fuzzy inference system (ANFIS)

Introduction

In recent years, the world has faced several economic and political crises, but the Ukrainian-Russian situation stands out. Oil and gas prices rose during this crisis due to the significant role of natural gas in electricity generation. Rising energy prices and the incapacity of numerous countries, mostly European, to overcome this crisis had a major influence on the energy industry. Despite their preference for alternative energy, European nations, the major consumers of Russian gas, were unprepared for this catastrophe. This energy crisis was the most crucial important turning point toward clean and renewable energy due to its economic harms. This turning point began a new era of fast development and expansion in alternative energy sources as governments worldwide sought to capitalize on their advantages. The US, China, and India are likely to be of the largest investors the most in alternative energy between 2022 and 2027. The EU REPOWEREU strategy aims to eliminate Russian fossil fuel use by 2027. International Energy Agency predictions show renewable energy sources will expand by 2400 GW more than expected by 2027 (World Economic Forum, 2023).

Figure 1.

Global power generation across various technologies in the years 2015, 2021, and 2027.

One of the most prominent, widely used, and rapidly developing energy sources in the modern period is wind power. The fact that wind energy has zero negative impacts on the environment (including air, water, and soil) is a major factor in its popularity. Not only does it not produce any waste or greenhouse emissions, but its cost has also witnessed a decline compared to previous years (Figure 2), primarily attributable to substantial advancements in the field of renewable energy (Benakcha et al., 2019; Soufi et al., 2016)

Figure 2.

Subdividing the overall quantity of electricity production by the generating technology.

One of the notable advantages of WPGS is their possibility of operating at either fixed or variable speeds. Extensive research has consistently demonstrated that the variable-speed operating system, specifically, is highly advantageous. The variable-speed system offers several benefits, including the ability to lower mechanical stress induced by varying wind conditions, the capacity to enhance energy production through the employment of maximum power point tracking (MPPT) strategies, and the capability to generate approximately 40% more energy compared to FS-WPGS operating under identical conditions. Consequently, the VS-WPGS is regarded as more appealing due to its ability to ensure system stability and reliability (Errami et al., 2015; Saidi et al., 2022).

Undoubtedly, this system has a selection of aspects that need meticulous consideration to get optimal outcomes that cater to the user’s requirements. Among these elements, the generator stands out as one of the most crucial and significant. Various types of generators have been utilized since the inception of these systems, including the double-feed induction generator, which has gained significant popularity in recent years, as well as the synchronous generator with field excitation, the induction generator with a squirrel cage, and the permanent magnet synchronous generator. Recently, PMSGs have become very popular in the world of wind energy production. This is due to their advantages: they contain a large number of poles, which allow them to operate at low speed; they can be used without a gearbox, which cuts costs; and they can produce more torque because they do not need an external excitation current, which means there are no rotor losses. Consequently, these characteristics enhance the assessment efficiency of this particular generator in comparison to other options for the wind power-producing system (Ayadi and Derbel, 2017; Errami et al., 2015; Nguyen et al., 2018; Raj and Kumar, 2023; Saidi et al., 2022; Soufi et al., 2016). Conversely, it has lately garnered significant attention in several research papers about the utilization of multi-phase generators and their incorporation into wind energy-generating systems, owing to their inherent benefits, such as (Peng et al., 2021):

Improved reliability and high fault tolerance.

Possibility of using low-power devices to attain low voltage generation and large power.

Possibility of working in the event of the failure of one or more phases of the generator or power transformer.

In this specific context, a multi-phase PMSG offers several advantages over a three-phase system, such as improved reliability, reduced capacitance, and an increased frequency of torque pulses. Therefore, a multi-phase PMSG is an exceptionally appropriate choice for harnessing wind power (Bounadja et al., 2019a, 2019b).

Regarding the study of WPGSs, it is imperative to address the issue of power electronics converters, which serve as an intelligent interface responsible for regulating voltage, frequency, and the injected energy into the power grid. These converters play a crucial role in maintaining control over the system’s operations under various circumstances (Blaabjerg et al., 2012). One of the distinctive options that have received great acclaim in this type of system is the use of PMSG with the back-to-back converter (BTBC) to meet the network’s energy needs (Shanmugam et al., 2017). PMSGs possess several notable attributes, including excellent controllability. The integration of BTBCs with PMSGs enables effective isolation between the generator and the network side, facilitating efficient management of two-way power flow. Additionally, this configuration allows for swift regulation of active and reactive power injection into the network (Kamel et al., 2020; Tahir et al., 2020).

Wind farm systems (WFS) typically comprise numerous autonomous wind turbine generators that are interconnected and function concurrently. Power electronics converters are considered to be among the most costly pieces of equipment in such systems (Errami et al., 2015). To minimize costs and achieve energy production at the most economical rates, various transformer types have been employed. Among the recent advancements in this domain is the utilization of a BTBC based on a fifteen-switch rectifier (FSR). The FSR is characterized as a combination of two five-phase rectifiers. This converter configuration offers the advantage of connecting two generators to a single rectifier simultaneously, obviating the requirement for two separate rectifiers. The implementation of this approach serves to reduce costs associated with transformer usage while ensuring efficient energy generation. This transformer is regarded as a novel approach in terms of both performance and cost. It is distinguished by reducing the number of switches, resulting in reduced switching losses without adding complexity to system control (Kamel et al., 2020).

Wind systems typically employ conventional BTBCs, which include a rectifier (MSC) that controls the rotational speed of the generator and an inverter (GSC) that manages the voltage of the DC link and the power sent into the grid. To enhance performance, especially on the generator side, one potential upgrade is to replace the generator side converter with a FSR while maintaining the respective functions of each side. During the search for this subject, a few publications were discovered that explored the same research issue as this study. The most closely similar topic was the FSR topic discussed in Kamel et al. (2020), where the author introduced a novel converter topology for the wind farm system known as the FSR. The simulation results demonstrated the commendable performance of the FSR. Additionally, it offers several benefits compared to the conventional architecture mentioned in Errami et al. (2015). These advantages include the reduction of switching losses by minimizing the number of switches, while also considering control mechanisms without adding unnecessary complexity. The author in Reusser et al. (2015) suggested using a nine-switch BTBC topology for multi-generator wind systems. The findings demonstrate that this suggested configuration has several benefits in comparison to conventional BTBCs. One crucial benefit of this is that it reduces the number of switches in both converters. This minimizes energy loss during switching and ensures effective isolation between the generating stages and the inverting process. In order to limit the study’s breadth, the primary emphasis was placed on the nine-switch converter (NSC) topology. The topic of our study closely aligns with this, since NSC has been extensively used, particularly in energy conversion applications such as a DFIG-based wind generating system (Wen et al., 2016) and an integrated motor drive and battery charger (Diab et al., 2016). Furthermore, several methods for the NSI to regulate two separate three-phase loads/machines or a single six-phase load have been discussed in the scientific literature (Abbache et al., 2021; Dos Santos et al., 2011; Goyal and Aware, 2017; Gulbudak et al., 2021; Kominami and Fujimoto, 2007a, 2007b; Oka and Matsuse, 2007; Reusser, 2016). The techniques largely covered include the incorporation of vector control and suitable pulse width modulation (PWM) techniques. Sliding mode control (SMC) and model predictive control (MPC) are two control techniques used for controlling load dynamics in the framework of NSI-driven, two-independent multiphase loads (Gokdag, 2022; Gulbudak and Gokdag, 2021, 2022; Gulbudak et al., 2023).

In various works on conventional BTBC-based wind systems, a plethora of nonlinear control techniques have been used. Backstepping control (Aziz et al., 2019; El Mourabit et al., 2020), input-output feedback linearization (Soufi et al., 2016), and different kinds of sliding mode control Nasiri et al., 2015a, 2015b; Mozayan et al., 2016; Zhuo et al., 2016; Nasiri et al., 2019; Mousa et al., 2020; Rhaili et al., 2020; Majout et al., 2022a; Mojtaba et al., 2022; Rhaili et al., 2022 are also examples of such techniques. The authors in Zhuo et al. (2016) provide the implementation of high-order sliding mode control (HOSMC) and do a comparative study of its performance with first-order sliding mode control (FOSMC). The findings suggest that HOSMC has a more rapid response. While the phenomenon of chatter is lesser in comparison to FOSMC, it is not entirely eradicated, with the existence of undesired inaccuracies in tracking the desired value. Whereas in Mousa et al. (2020), the author proposes the use of integral sliding mode control (ISMC) technology and does a comparison study with the conventional PI controller. The simulation results illustrate the dynamic features of ISMC and its efficient control of generator speed in different operating scenarios. Nevertheless, it is important to point out the occurrence of an undesirable chattering phenomenon, as in Zhuo et al. (2016), as well as the presence of distortions in the grid current. Similar to the previous two studies (Mousa et al., 2020; Nasiri et al., 2019; Zhuo et al., 2016) used the super-twisting sliding mode control (STSMC) approach for wind turbines driven by PMSG. The offered technique demonstrated both resilience and prompt responsiveness. Moreover, it effectively tackled the problem of excessive chatter, which has been identified as a key hindrance in prior studies (Mousa et al., 2020; Zhuo et al., 2016). In the work described in Mojtaba et al. (2022), the authors implemented the PID-type terminal sliding mode control (PID-TSMC) scheme for the machine-side and grid-side converter-modified controllers of BTBCs. By inverting the control functions of the MSC and GSC, the outcomes were compared to the most recent discoveries documented in the literature (Mozayan et al., 2016; Nasiri et al., 2015a, 2015b, 2019). The proposed methodology showcases outstanding performance and reduced reaction time when subjected to fluctuations in the DC-bus voltage. Furthermore, it exhibits resilience against limited external disturbances. In Majout et al. (2022a), the author instituted a new and strong control strategy for sliding mode control using the smooth continuous function methodology. This technique was devised to address the constraints associated with traditional sliding-mode control. The simulation results unequivocally showcased the superiority of the suggested methodology in comparison to the conventional method. The primary benefits of the new technique may be succinctly described as the eradication of the chatter phenomenon and the improvement of reaction time. In Rhaili et al. (2022), as illustrated in the previously stated literature, the authors used a new approach called Fuzzy Fractional Order Sliding Mode Control (FFOSMC) and assessed its effectiveness in relation to many other studies referenced in the literature (Mousa et al., 2020; Rhaili et al., 2020). The investigation’s results unequivocally established the superiority of the recommended technique. The data reported in this research, together with previous studies, clearly demonstrate that the suggested strategy displays higher performance. An essential element of this advantage is its capacity to alleviate the chattering phenomenon and accelerate replies. However, it highlights a weakness in the concept of steady-state error, which is well addressed in the study discussed in Mousa et al. (2020), which has shown clear superiority in this area. In Bounadja et al. (2024), the author conducted a comparative analysis between two robust control approaches, namely the Second-Order Super-Twisting Controller (SOSTC) and the Adaptive Third-Order Continuous Super-Twisting Control (ATOCSTC), for wind power generation systems. The findings of this work exhibit the evident superiority of ATOCSTC over SOSTC in several aspects. These include the reduction of power ripples, enhanced performance stability in diverse conditions, and improved resilience against parameter variations.

This work’s main contributions may be summarized as follows:

✓ The designing and implementation of the BSC technique for the FSR-based new BTBC.

✓ Utilizing BSC techniques to train an ANFIS controller.

✓ Implementing the ANFIS controller that has been trained for controlling the FSR.

✓ A comparative assessment of the performance between the ANFIS controller and the BSC approach.

Within the scope of this research project, we used both BSC technology and ANFIS to implement control over the FSR. Enhancing the functioning of the system and achieving strong control are the goals of this endeavor. Even though BSC is usually considered to be one of the most powerful control methods, especially for reference signal tracking (Majout et al., 2022b), the fact that it is so heavily dependent on system parameters is the fundamental disadvantage that it has (Salime et al., 2023). It is common for this reliance to result in a significant decline in performance. At the same time, the ANFIS demonstrates resilience to fluctuations in system parameters and uncertainties since it can adaptively adjust its rules and fuzzy parameters, effectively absorbing variations and uncertainties in parameters (Jang and Sun, 1995). To capitalize on the distinct benefits of these two methods, ANFIS was taught using BSC as a basis. This architecture not only improves the performance of the system but also increases reaction time, lowers overshoots, and provides accurate tracking performance. By integrating the advantages of BSC and ANFIS, we not only enhance control performance but also capitalize on the durability of ANFIS in managing parameter uncertainty.

The next parts of this research endeavor are organized as follows: In Section “WPGS model,” a succinct modeling of the wind turbine and the two PMSGs is presented, along with comprehensive analyses of the FSR topology and its key attributes and benefits. This section provides a comprehensive discussion of the control of the FSR converter using the PWM block. Section “ General synthesis of the control system” introduces the concept and plan of a BSC approach, and the ANFIS was trained using the BSC method to control the FSR. The outcomes of the simulation utilized to verify the suggested method of control have been carefully examined in Section “Results and discussion.” Section “Conclusion” serves as the last part of this study, providing a concise summary of the main results and significant observations.

WPGS model

Figure 3 illustrates the overall structure of the WPGS, which consists of two five-phase PMSGs connected to a three-phase grid. The configuration of the new back-to-back converters (NBTBC) consists of two converters. The first converter, known as the machine-side converter (MSC), is a five-phase FSR. The second converter, referred to as the grid-side converter (GSC), is a traditional three-phase inverter. These two converters are connected by a filter C.

Figure 3.

The overall structure of a WPGS.

Wind-turbine model

The following mathematical expression represents the energy produced by the wind turbine (Bounadja et al., 2019b):

P_{V} = \frac{1}{2} ρ A C_{p} (λ, β) V_{w}^{3}

(1)

In order to calculate the energy generated by wind turbines, we depend on many significant coefficients, namely: $ρ$ represents the density of the air, A is the area that is swept by the blades (A = π $R^{2}$ ), β represents the pitch angle of the propeller blades, $V_{w}$ means the wind speed (m/s), and λ represents the Tip Speed Ratio (TSR).

While $C_{p}$ is the power conversion coefficient of the wind turbine (Figure 4).

{\begin{matrix} C_{p} (λ, β) = 0.5176 . (\frac{116}{λ_{i}} - 0.4 . β - 5) . e x p (- \frac{21}{λ_{i}}) + 0.0068 . λ \\ \frac{1}{λ_{i}} = (\frac{1}{λ + 0.08 . β} - \frac{0.035}{β^{3} + 1}) \\ λ = \frac{Ω_{m} R}{V_{w}} \\ C_{p_m a x} (λ, β) = \frac{16}{27} = 0.59 \\ λ_{o p t} = 8.1 \\ β = 0 \end{matrix}

(2)

Figure 4.

Features of the power coefficient.

Whereas: $Ω_{m}$ is the angular speed of the turbine and R represents the blades’ radius (m).

PMSGs model

The following listed equations from (3) to (5) express the voltage, magnetic flux, and electromagnetic torque for each five-phase PMSG model in a rotating d-q-x-y frame (Kamel et al., 2020):

{\begin{matrix} V_{d s j} = R_{s j} I_{d s j} + \frac{d Ψ_{d s j}}{d t} - w_{m j} Ψ_{q s j} \\ V_{q s j} = R_{s j} I_{q s j} + \frac{d Ψ_{q}}{d t} + w_{m j} Ψ_{d s j} \\ V_{x s j} = R_{s j} I_{x s j} + \frac{d_{Ψ x s j}}{d t} \\ V_{y s j} = R_{s j} I_{y s j} + \frac{d Ψ_{y s j}}{d t} \end{matrix}

(3)

{\begin{matrix} Ψ_{d s j} = L_{d j} I_{d s j} \\ Ψ_{q s j} = L_{q j} I_{q} + Ψ_{f j} \\ Ψ_{x s j} = L_{d j} I_{x s j} \\ Ψ_{y s j} = L_{q j} I_{y s j} \end{matrix}

(4)

Where: d, q, x, and y are the axes of the stator, while $V_{d s j}, V_{q s j}, V_{x s j}, a n d V_{y s j}$ are the stator voltages, with j = 1, 2. The stator currents on the d, q, x, and y axes are $I_{d s j}, I_{q s j}, I_{x s j} a n d I_{y s j}$ , respectively. In the rotating frames, $L_{d j} a n d L_{q j}$ denote the inductances in the rotating frames (H). The stator resistance is denoted as $R_{s j}$ (Ω). The magnet flux is known as $Ψ_{f j}$ (Wb), and $Ω_{m j}$ denotes the rotor’s electrical speed (rad/s).

T_{e m j} = \frac{5}{2} P_{j} [(L_{d j} - L_{q j}) I_{d s j} I_{q s j} + Ψ_{f} I_{qqj}

(5)

The wind-turbine system’s mechanical equation can be written by the fundamental law of dynamics as follows:

T_{m j} = T_{e m j} + K_{f j} Ω_{m j} + J_{j} \frac{d Ω_{m j}}{d t}

(6)

While: $T_{m j}$ is the mechanical torque of the wind turbine (N.m).

$T_{emj}$ : is generator’s electromagnetic torque (N.m).

$J_{j}$ : is the total inertia of the system ( $Kg / m^{2}$ ).

$K_{fj}$ : is the factor of a viscous friction.

By using the following formula, it is possible to determine how much power each generator produces:

{\begin{matrix} P_{m j} = T_{e m j} . Ω_{m j} = \frac{5}{2} [V_{d s j} . I_{d s j} + V_{q s j} . I_{q s j}] \\ Q_{m j} = \frac{5}{2} [V_{q s j} . I_{d s j} - V_{d s j} . I_{q s j}] \end{matrix}

(7)

General analysis of the FSR structure

As can be seen in Figure 5(b), the FSR is made up of five legs, and each of those legs has both upper and middle switches as well as bottom switches. It is possible to say that the FSR is a combination of two five-phase rectifiers, each of which is made up of a set of switches, respectively. The upper and middle switches are the components that make up the upper rectifier, while the middle and lower switches are the components that make up the lower rectifier (Kamel et al., 2020).

Figure 5.

Two parallel five-phase rectifiers: (a) conventional rectifier (20-switch) and (b) 15-switch rectifier.

The signals for the upper and lower switches are generated by conventional pulse width modulation (PWM). Meanwhile, the XOR logical operator is used to construct the gating signals for the middle switches from the gating signals of the lower and upper switches, respectively. The total number of switches is also reduced by 25% compared to using two parallel conventional rectifiers (Figure 5(a)).

The function of switch $S_{i j}$ , illustrated in Figure 5(b), can be defined using the switch function as follows:

S_{i j} = {\begin{matrix} 1, t h e s w i t c h (S_{i j}) i s c l o s e d \\ 0, t h e s w i t c h (S_{i j}) i s o p e n \end{matrix}; Where : i = U, M, L, and : j = A, B, C, D, E

(8)

We may express the restrictions as:

S_{U j} + S_{M j} + S_{L j} = 2

(9)

Concerning the operational methodology of this novel converter illustrated in Figure 5(b), it is feasible to employ three scenarios per leg. The switching states can be denoted as 1, 0, and −1, as depicted in Table 1. In this table, j corresponds to legs A, B, C, D, or E, while U, M, and L correspond to the upper, middle, and lower switches, successively.

Table 1.

State of switching and voltage between the rectifier’s poles in a 15-switch structure.

Switching states	$S_{U j}$	$S_{M j}$	$S_{L j}$	$V_{u}$	$V_{l}$
1	1	0	1	$V_{d c}$	0
0	0	1	1	0	0
−1	1	1	0	$V_{d c}$	$V_{d c}$

The suggested PWM method for the FSR is shown in Figure 6. It uses two synced PWM units from conventional five-phase rectifiers. Each PWM unit receives two voltage references, $V_{U_r e f}$ and $V_{L_r e f}$ , so it can make 10 pulses that match the top and bottom outputs in a row. The XOR gate, which processes the upper and lower gating signals, produces the middle switch pulses.

Figure 6.

The suggested PWM block for the FSR.

Filter (R, L) and grid model

In the Park frame (d, q), the grid voltages could be written in the following way (Majout et al., 2022b):

{\begin{matrix} V_{g d} = R_{s} I_{g d} + L_{d} \frac{d I_{g d}}{d t} - w_{g} L_{q} I_{g q} + V_{i d} \\ V_{g q} = R_{s} I_{g q} + L_{q} \frac{d I_{g q}}{d t} - w_{g} L_{d} I_{g d} + V_{i q} \end{matrix}

(10)

In the grid system, $I_{g d}$ and $I_{g q}$ represent the direct and quadrature components of the grid’s currents, while $V_{g d}$ and $V_{g q}$ represent the d-q components of the grid’s voltage. On the other hand, the voltage vector of the GSC has two components in the d-q axis, which are $V_{i d}$ and $V_{i q}$ . In addition, $R_{g}$ and $L_{g}$ denote the resistance and inductance of the grid filter, respectively, while $w_{g}$ . signifies the angular frequency of the grid voltage.

The expression of the active and reactive powers within the d-q reference frame is as follows:

{\begin{matrix} P_{g} = \frac{3}{2} (V_{g d} . I_{g d} + V_{g q} . I_{g q}) \\ Q_{g} = \frac{3}{2} (V_{g q} . I_{g d} - V_{g d} . I_{g q}) \end{matrix}

(11)

The control strategy utilized for the GSC includes making $V_{q g} = 0$ and $V_{d g} = | | V_{g} | |$ . Consequently, the power equations undergo modification in the following manner:

{\begin{matrix} P_{g} = \frac{3}{2} (V_{g d} . I_{g d}) = \frac{3}{2} (| | V_{g} | | . I_{g d}) \\ Q_{g} = \frac{3}{2} (- V_{g d} . I_{g q}) = - \frac{3}{2} (| | V_{g} | | . I_{g q}) \end{matrix}

(12)

The regulation of active power flow may be achieved via $I_{d g}$ , whereas $I_{q g}$ , is responsible for controlling reactive power flow.

The DC-link offers a connection between the converter situated on the generator side and the converter situated on the grid side. The DC-link voltage’s dynamic behavior is feasibly defined by taking into account power balancing and neglecting losses in the converter (Benchagra et al., 2012).

P_{g e n} - P_{g} = \frac{1}{2} . C . \frac{{d V}_{d c}^{2}}{d t}

(13)

General synthesis of the control system

MSC control

Backstepping control strategy

The backstepping control approach is based on representing the interconnected systems as Lyapunov-order 1 subsystems. This results in heightened resilience to disruptions and improved general steadiness in the system. The backstepping control employs a multistep method, generating a virtual command at each step to ensure the system’s convergence to its equilibrium state. The Lyapunov function ensures the stability of each synthesis step (Aziz et al., 2019; El Mourabit et al., 2020). Based on the relationships outlined in equations (3)–(6), the system’s state form could be represented as:

{\begin{matrix} \frac{d I_{d s j}}{d t} = \frac{- R_{s j}}{L_{d j}} I_{d s j} + \frac{P_{j} L_{q j}}{L_{d j}} I_{q j} Ω_{m j} + \frac{1}{L_{d j}} V_{d s j} \\ \frac{d I_{q s j}}{d t} = \frac{- R_{s j}}{L_{q j}} I_{q s j} - \frac{P_{j} L_{d j}}{L_{q j}} I_{d j} Ω_{m j} - \frac{P_{j} Ψ_{f j}}{L_{q j}} Ω_{m j} + \frac{1}{L_{q j}} V_{q s j} \\ \frac{d I_{x s j}}{d t} = \frac{- R_{s j}}{L_{d j}} I_{x s j} + \frac{1}{L_{d j}} V_{x s j} \\ \frac{d I_{y s j}}{d t} = \frac{- R_{s j}}{L_{q j}} I_{y s j} + \frac{1}{L_{q j}} V_{y s j} \\ \frac{d Ω_{m j}}{d t} = \frac{T_{m j}}{J_{j}} - \frac{5}{2 J_{j}} P_{j} (L_{d j} - L_{q j}) I_{d s j} I_{q s j} - \frac{5}{2 J} P_{j} Ψ_{f j} I_{q s j} - \frac{K_{f j}}{J_{j}} Ω_{m j} \end{matrix}

(14)

The process of setting up backstepping control on the grid side may be divided into two sequential steps. The first phase provides instructions for the next one. Furthermore, the state vectors and control vectors are chosen so that $[x] = {[\begin{matrix} I_{d s j} I_{q s j} I_{x s j} I_{y s j} Ω_{m j} \end{matrix}]}^{T} = {[\begin{matrix} x_{1} x_{2} x_{3} x_{4} x_{5} \end{matrix}]}^{T}$ specifically refers to the state vectors, whereas $[u] = {[\begin{matrix} V_{d s j} V_{q s j} V_{x s j} V_{y s j} \end{matrix}]}^{T}$ indicates the control vectors (Figure 7).

Figure 7.

The overall control structure of the FSR.

Step1

The state variable $x_{5}$ of the tracking error is expressed through the formula that follows:

ξ_{Ω} = Ω_{r e f_j} - Ω_{m j}

(15)

The speed’s error dynamic might be expressed as follows:

{\overset{\cdot}{ξ}}_{Ω j} = \frac{d ξ_{Ω j}}{d t} = ξ_{r e f_j} - {\overset{\cdot}{ξ}}_{m j} = {\overset{\cdot}{Ω}}_{r e f_j} - \frac{1}{J_{j}} [T_{t j} - \frac{5 P_{j}}{2} (L_{d j} - L_{q j}) . I_{d s j} . I_{q s j} - \frac{5 P_{j}}{2} Ψ_{f j} . I_{q s j} - k_{f j} . Ω_{m j}]

(16)

The primary aim of this initial phase is to minimize the error ratio in accurately tracking the reference speed. To achieve this, the Lyapunov function is employed, which is mathematically defined as follows:

V_{Ω j} = \frac{1}{2} ξ_{Ω j}^{2}

(17)

The Lyapunov function’s time derivative is given as:

{\overset{\cdot}{V}}_{Ω j} = ξ_{Ω j} . {\overset{\cdot}{ξ}}_{Ω j} = - k_{Ω j} ξ_{Ω j}^{2} + \frac{ξ_{Ω j}}{J_{j}} [J_{j} . {\overset{\cdot}{Ω}}_{re f_{j}} - T_{t j} + k_{f j} . Ω_{m} + \frac{5 P_{j}}{2} . Ψ_{f j} . I_{q s j} + k_{Ω j} J_{j} ξ_{Ω j}] + \frac{5 P_{j}}{2 J_{j}} (L_{d j} - L_{q j}) . I_{d s j} . I_{q s j} . ξ_{Ω j}

(18)

As part of the backstepping control design, the virtual inputs, referred to as stabilizing functions, are established by selecting the quadratic $I_{q s j_r e f}$ and direct $I_{d s j_r e f}$ stator current components.

{\begin{matrix} i_{d s j_r e f} = 0 \\ i_{q s j_r e f} = \frac{2}{5 . P_{j} . Ψ_{f j}} [T_{t} - J_{j} . {\overset{\cdot}{Ω}}_{r e f_j} - k_{f j} . Ω_{m j} - k_{Ω j} . J_{j} . ξ_{Ω j}] \end{matrix}

(19)

We get the following result when we substitute equation (19) into the derivative formula ${\overset{\cdot}{V}}_{Ω j}$ (equation (18)):

{\overset{\cdot}{V}}_{Ω j} = - k_{Ω j} . ξ_{Ω j}^{2} \leq 0

(20)

$k_{Ω j}$ is constant with positive values, and $ξ_{Ω j}^{2} \geq 0$ .

Step 2

While $I_{d s j}, I_{q s j}, I_{x s j}, a n d I_{y s j}$ , represent the system’s virtual inputs, $V_{d s j}, V_{q s j}, V_{x s j}, a n d V_{y s j}$ represent the actual control inputs, which are calculated through the variations in the stator currents.

The form that could be used to express errors in stator current is as follows:

{\begin{matrix} ξ_{d j} = I_{d s j_r e f} - I_{d s j} \\ ξ_{q j} = I_{q s j_r e f} - I_{q s j} \\ ξ_{x j} = I_{x s j_r e f} - I_{x s j} \\ ξ_{y j} = I_{y s j_r e f} - I_{y s j} \end{matrix}

(21)

By utilizing equations (19) and (21), we can update the formula for speed dynamics to a new form.

\begin{matrix} {\overset{\cdot}{ξ}}_{Ω j} = \frac{d ξ_{Ω j}}{d t} = \frac{1}{J_{j}} [- T_{t j} + \frac{5 P_{j}}{2} Ψ_{f j} . i_{q s j} + \frac{5 P_{j}}{2} (L_{d j} - L_{q j}) . I_{d s j} . I_{q s j} + k_{f j} . Ω_{m j}] \\ = \frac{1}{J_{j}} [- T_{t j} + \frac{5 P_{j}}{2} Ψ_{f j} . (I_{q s j_r e f} - ξ_{q j}) - \frac{5 P_{j}}{2} (L_{d j} - L_{q j}) . I_{q s j} . ξ_{d j} + k_{f j} . Ω_{m j}] \\ = \frac{1}{J_{j}} [- T_{t j} + \frac{5 P_{j}}{2} Ψ_{f j} [\frac{2}{5 . P_{j} . Ψ_{f j}} [T_{t} - k_{f j} . Ω_{m j} - k_{Ω j} . J_{j} . ξ_{Ω j}]] - \frac{5 P_{j}}{2} Ψ_{f j} . ξ_{q j} - \frac{5 P_{j}}{2} (L_{d j} - L_{q j}) . I_{q s j} . ξ_{d j} + k_{f j} . Ω_{m j}] \\ {\overset{\cdot}{ξ}}_{Ω j} = \frac{1}{J_{j}} [k_{Ω j} . J_{j} . ξ_{Ω j} - \frac{5 . P_{j}}{2} Ψ_{f j} . ξ_{q j} - \frac{5 . P_{j}}{2} (L_{d j} - L_{q j}) I_{q s j} . ξ_{d j}] \end{matrix}

(22)

Formulae (14) and (25) might be used to express the dynamics of stator current errors.

{\begin{matrix} {\overset{\cdot}{ξ}}_{d j} = {\overset{\cdot}{I}}_{d s j_r e f} - {\overset{\cdot}{I}}_{d s j} \\ {\overset{\cdot}{ξ}}_{q j} = {\overset{\cdot}{I}}_{q s j_r e f} - {\overset{\cdot}{I}}_{q s j} \\ {\overset{\cdot}{ξ}}_{x j} = {\overset{\cdot}{I}}_{x s j_r e f} - {\overset{\cdot}{I}}_{x s j} \\ {\overset{\cdot}{ξ}}_{y j} = {\overset{\cdot}{I}}_{y s j_r e f} - {\overset{\cdot}{I}}_{y s j} \end{matrix}

(23)

{\overset{\cdot}{ξ}}_{d j} = {\overset{\cdot}{I}}_{d s j_r e f} - {\overset{\cdot}{I}}_{d s j} = \frac{R_{s j}}{L_{d j}} I_{d s j} - \frac{P_{j} . L_{q j}}{L_{d j}} I_{q s j} . Ω_{m j} + \frac{1}{L_{d j}} V_{d s j}

(24)

{\begin{matrix} {\overset{\cdot}{ξ}}_{q j} = {\overset{\cdot}{I}}_{qs j_{r e f}} - {\overset{\cdot}{I}}_{q s j} = \frac{2}{5 . P_{j} . Ψ_{f j}} [- k_{f j} . {\overset{\cdot}{Ω}}_{m j} - k_{Ω j} . J_{j} . {\overset{\cdot}{ξ}}_{Ω j}] - (\frac{- R_{s j}}{L_{q j}} I_{q s j} - \frac{P_{j} L_{d j}}{L_{q j}} I_{d j} Ω_{m j} - \frac{P_{j} Ψ_{f j}}{L_{q j}} Ω_{m j} + \frac{1}{L_{q j}} V_{q s j}) \\ = \frac{2}{5 . P_{J} . J_{j} Ψ_{f j}} ((k_{Ω j} . J_{j} - k_{f j}) [T_{t j} - \frac{5 P_{j}}{2} (L_{d j} - L_{q j}) . I_{d s j} . I_{q s j} - \frac{5 P_{j}}{2} Ψ_{f j} . I_{q s j} - k_{f j} . Ω_{m j}]) \\ + (\frac{R_{s j}}{L_{q j}} I_{q s j} + \frac{P_{j} L_{d j}}{L_{q j}} I_{d j} Ω_{m j} + \frac{P_{j} Ψ_{f j}}{L_{q j}} Ω_{m j} - \frac{1}{L_{q j}} V_{q s j}) \end{matrix}

(25)

{\overset{\cdot}{ξ}}_{x j} = {\overset{\cdot}{I}}_{x s j_r e f} - {\overset{\cdot}{I}}_{x s j} = 0 - [- \frac{R_{s j}}{L_{d j}} I_{x s j} + \frac{1}{L_{d j}} V_{x s j}] = \frac{R_{s j}}{L_{d j}} I_{x s j} - \frac{1}{L_{d j}} V_{x s j}

(26)

{\overset{\cdot}{ξ}}_{y j} = {\overset{\cdot}{I}}_{y s j_r e f} - {\overset{\cdot}{I}}_{y s j} = 0 - [- \frac{R_{s j}}{L_{q j}} I_{y s j} + \frac{1}{L_{q j}} V_{y s j}] = \frac{R_{s j}}{L_{q j}} I_{y s j} - \frac{1}{L_{q j}} V_{y s j}

(27)

A novel Lyapunov function is suggested to guarantee the stability of the whole system by using the stator voltages as references, taking into account the speed errors and stator current errors.

V_{2} = \frac{1}{2} (ξ_{Ω j}^{2} + ξ_{d j}^{2} + ξ_{q j}^{2} + ξ_{x j}^{2} + ξ_{y j}^{2})

(28)

The new Lyapunov function’s time derivative is given as:

{\overset{\cdot}{V}}_{2} = (ξ_{Ω j} . {\overset{\cdot}{ξ}}_{Ω j} + ξ_{d j} . {\overset{\cdot}{ξ}}_{d j} + ξ_{q j} . {\overset{\cdot}{ξ}}_{q j} + ξ_{x j} . {\overset{\cdot}{ξ}}_{x j} + ξ_{y j} . {\overset{\cdot}{ξ}}_{y j})

(29)

\begin{matrix} {\overset{\cdot}{V}}_{2} = - k_{Ω J} . ξ_{Ω J}^{2} - k_{d j} . ξ_{d j}^{2} - k_{q j} . ξ_{q j}^{2} - k_{x j} . ξ_{x j}^{2} - k_{y j} . ξ_{y j}^{2} + ξ_{Ω j} [- \frac{5 P_{j}}{2 J_{j}} Ψ_{f j} . ξ_{q j} - \frac{5 P_{j}}{2 J_{j}} (L_{d j} - L_{q j}) I_{q s j} . ξ_{d j}] \\ + ξ_{d j} [\frac{R_{s j}}{L_{d j}} I_{d s j} - \frac{P_{j} . L_{q j}}{L_{d j}} I_{q s j} . Ω_{m j} + \frac{1}{L_{d j}} V_{d s j} + k_{d j} . ξ_{d j}] \\ + ξ_{q j} [\frac{2}{5 . P_{J} . J_{j} Ψ_{f j}} ((k_{Ω j} . J_{j} - k_{f j}) [T_{t j} - \frac{5 P_{j}}{2} (L_{d j} - L_{q j}) . I_{d s j} . I_{q s j} - \frac{5 P_{j}}{2} Ψ_{f j} . I_{q s j} - k_{f j} . Ω_{m j}]) \\ + \frac{R_{s j}}{L_{q j}} I_{q s j} + \frac{P_{j} L_{d j}}{L_{q j}} I_{d j} Ω_{m j} + \frac{P_{j} Ψ_{f j}}{L_{q j}} Ω_{m j} - \frac{1}{L_{q j}} V_{q s j} + k_{q j} . ξ_{q j}] + ξ_{x j} [\frac{R_{s j}}{L_{d j}} I_{x s j} - \frac{1}{L_{d j}} V_{x s j} + k_{x j} . ξ_{x j}] \\ + ξ_{q j} [\frac{R_{s j}}{L_{q j}} I_{y s j} - \frac{1}{L_{q j}} V_{y s j} + k_{y j} . ξ_{y j}] \\ {\overset{\cdot}{V}}_{2} = - k_{Ω J} . ξ_{Ω J}^{2} - k_{d j} . ξ_{d j}^{2} - k_{q j} . ξ_{q j}^{2} - k_{x j} . ξ_{x j}^{2} - k_{y j} . ξ_{y j}^{2} + ξ_{d j} \\ [\frac{R_{s j}}{L_{d j}} I_{d s j} - \frac{P_{j} . L_{q j}}{L_{d j}} I_{q s j} . Ω_{m j} + \frac{1}{L_{d j}} V_{d s j} + k_{d j} . ξ_{d j} - \frac{5 P_{j}}{2 J_{j}} (L_{d j} - L_{q j}) I_{q s j} . ξ_{Ω j}] + \\ + ξ_{q j} [\frac{2}{5 . P_{J} . J_{j} Ψ_{f j}} ((k_{Ω j} . J_{j} - k_{f j}) [T_{t j} - \frac{5 P_{j}}{2} (L_{d j} - L_{q j}) . I_{d s j} . I_{q s j} - \frac{5 P_{j}}{2} Ψ_{f j} . I_{q s j} - k_{f j} . Ω_{m j}]) \\ + \frac{R_{s j}}{L_{q j}} I_{q s j} + \frac{P_{j} L_{d j}}{L_{q j}} I_{d j} Ω_{m j} + \frac{P_{j} Ψ_{f j}}{L_{q j}} Ω_{m j} - \frac{1}{L_{q j}} V_{q s j} + k_{q j} . ξ_{q j} - \frac{5 P_{j}}{2 J_{j}} Ψ_{f j} . ξ_{q j}] + ξ_{x j} [\frac{R_{s j}}{L_{d j}} I_{x s j} - \frac{1}{L_{d j}} V_{x s j} + k_{x j} . ξ_{x j}] \\ + ξ_{q j} [\frac{R_{s j}}{L_{q j}} I_{y s j} - \frac{1}{L_{q j}} V_{y s j} + k_{y j} . ξ_{y j}] \end{matrix}

(30)

$k_{d j}, k_{q j}, k_{x j}, a n d k_{y j}$ are constants with positive values.

To guarantee that the derivative of the Lyapunov function is negative and the system is stable, the stator reference voltages are implemented as follows:

V_{d s j_r e f} = R_{s j} . I_{d s j} - P_{j} . L_{d j} . Ω_{m j} . I_{q s j} - \frac{5 P_{j} . L_{d j}}{2 J_{j}} (L_{d j} - L_{q j}) I_{q s j} . ξ_{Ω j} + k_{d j} . L_{d j} . ξ_{d j}

(31)

\begin{matrix} V_{q s j_r e f} = \frac{2}{5 . P_{J} . J_{j} Ψ_{f j}} ((k_{Ω j} . J_{j} - k_{f j}) [T_{t j} - \frac{5 P_{j}}{2} (L_{d j} - L_{q j}) . I_{d s j} . I_{q s j} - \frac{5 P_{j}}{2} Ψ_{f j} . I_{q s j} - k_{f j} . Ω_{m j}]) \\ + \frac{R_{s j}}{L_{q j}} I_{q s j} + \frac{P_{j} L_{d j}}{L_{q j}} I_{d j} Ω_{m j} + \frac{P_{j} Ψ_{f j}}{L_{q j}} Ω_{m j} - \frac{1}{L_{q j}} V_{q s j} + k_{q j} . ξ_{q j} \end{matrix}

(32)

V_{x s j_r e f} = R_{s j} . I_{x s j} + k_{x j} . L_{d j} . ξ_{x j}

(33)

V_{y s j_r e f} = R_{s j} . I_{y s j} + k_{y j} . L_{y q j} . ξ_{y j}

(34)

Lyapunov function behavior might be simplified in the following way:

{\overset{\cdot}{V}}_{2} = - k_{Ω J} . ξ_{Ω J}^{2} - k_{d j} . ξ_{d j}^{2} - k_{q j} . ξ_{q j}^{2} - k_{x j} . ξ_{x j}^{2} - k_{y j} . ξ_{y j}^{2} \leq 0

(35)

ANFIS control strategy

Description

ANFIS indicates the Adaptive Neuro-Fuzzy Inference System. This strategy utilizes a combination of neural networks and fuzzy logic systems, resulting in a hybrid machine-learning method. ANFIS is primarily used for modeling and control applications in several fields, including nonlinear control. The system uses a fuzzy inference approach to build connections between inputs and outputs, and it adjusts the system’s parameters by using a neural network (Al-Majidi et al., 2019; Basappa and Viswanathan, 2022).

Figure 8 depicts the overall architecture of the Adaptive Neuro-Fuzzy Inference System (ANFIS). It is important to mention that, while the ANFIS design has a neural network component; it nevertheless adheres to the structure of a fuzzy inference system. The ANFIS design has five layers, which are successively identified as follows (Areed et al., 2010; Miloudi et al., 2007):

Figure 8.

The general architecture of the ANFIS.

The first layer, which is referred to as the input layer, is made up of variables that are entered. Within this layer, every node performs an analysis of membership functions that are either triangular, trapezoidal, or Gaussian. The one that results in the least amount of training error is the one that is considered to be the most optimal function. The variables that were entered are then sent to the layer that comes after it.

Layer 2, sometimes referred to as the minimum selection layer, is responsible for identifying the lowest value among the given inputs. It accomplishes the function of selecting the input with the smallest value from the options that are offered.

Layer 3, commonly known as the normalization layer, is accountable for normalizing each input relative to the other inputs. The computation of the output of node $i^{t h}$ in this layer involves dividing the input $i^{t h}$ by the sum of all the other inputs. The normalization technique ensures that the impact of each input is appropriately calibrated based on the total input values.

Layer 4: The output of the i1 node in this layer can be determined using a linear function that takes into account the output of the i1 node in the third layer and the input signals of the ANFIS.

The summation layer, which is Layer 5, is responsible for combining all of the incoming signals. Automatic tuning of the ANFIS architecture is achieved by applying a backpropagation algorithm to both the input and output membership functions and a least-square estimation approach to the output membership functions.

Training of the suggestion ANFIS

The Matlab/Simulink program was utilized to train an ANFIS based on the BSC approach. The purpose was to regulate the wind power generating system, aiming to enhance system performance, increase power quality, and minimize tracking errors. The process of training this controller may be described in the following stages (Al-Majidi et al., 2019; Jang, 1993; Jang et al., 1997; Kumpati and Kannan, 1990):

The first stage of the ANFIS training process involves gathering training data, which is essential to identifying the inputs and outputs of the system. At this level, the inputs are specified as the speed error $(ξ_{Ω j})$ and the current errors $(ξ_{d j}, ξ_{qj}, ξ_{x j}, ξ_{y j})$ , which indicate the disparity between the actual values and the intended values. On the flip side, the outputs consist of the reference value of the current $I_{q s j_r e f}$ (result from the speed controller) and the control signals (reference voltages $V_{dsj_ref}$ , $V_{qsj_ref}, V_{xsj_ref}$ , and $V_{ysj_ref}$ ,) Accurately collecting and specifying this training data is crucial for efficiently training the ANFIS controller

The Fuzzy Inference System (FIS) design phase: At this step, the fuzzy sets and membership functions for each variable are determined. This entails establishing suitable membership functions and determining their forms depending on the attributes and qualities of each variable. The identification of membership function shapes is impacted by several aspects, such as the distribution of the data, the required degree of detail, and the linguistic comprehension of the variables.

The Generate phase of Fuzzy Rules involves creating a collection of precise and meaningful fuzzy rules that depict a relationship between the inputs and outputs to get reliable fuzzy inference results. This is achieved by examining the patterns and correlations within the data. The algorithm determines clusters and produces fuzzy rules that represent the fundamental structure and behavior of the system.

The training phase of the ANFIS model: At this step, certain techniques, like the backpropagation algorithm and the hybrid learning algorithm, are used to iteratively adjust the parameters of the model. The objective of this method is to minimize the discrepancy between the predicted outputs generated by the ANFIS model and the actual outputs obtained from the training data. Through the process of adjusting the model’s parameters using these training methods, the ANFIS model improves its precision and ability to accurately capture the fundamental patterns and relationships present in the data. The goal is to get the highest degree of concurrence between the projected outputs and the actual outputs.

GSC control

Backstepping control strategy

The GSC is responsible for controlling the flow of electrical energy across the distribution network. These regulations are designed to achieve the aims of maintaining power factor stability and DC bus voltage (Benchagra et al., 2012; Salime et al., 2023).

Based on the relationships (10) and (13), the state form of this system can be written as follows:

{\begin{matrix} \frac{d I_{g d}}{d t} = \frac{1}{L_{g}} V_{i d} - \frac{R_{g}}{L_{g}} I_{g d} + W_{g} I_{g q} - \frac{1}{L_{g}} V_{g d} \\ \frac{d I_{g q}}{d t} = \frac{1}{L_{g}} V_{i q} - \frac{R_{g}}{L_{g}} I_{g q} - W_{g} I_{g d} - \frac{1}{L_{g}} V_{g q} \\ \frac{{d V}_{d c}^{2}}{d t} = \frac{2}{C} (- \frac{3}{2} V_{g d} . I_{g d} + P_{g e n}) \end{matrix}

(36)

The process of setting up backstepping control on the grid side might be divided into two sequential steps. The first phase provides instructions for the next one. Furthermore, the state vectors and control vectors are chosen so that $[x] = {[\begin{matrix} I_{g d} I_{g q} V_{d c}^{2} \end{matrix}]}^{T} = {[\begin{matrix} x_{1} x_{2} x_{3} \end{matrix}]}^{T}$ specifically refers to the state vectors, whereas $[u] = {[\begin{matrix} V_{i d} V_{i q} \end{matrix}]}^{T}$ indicates the control vectors (Figure 9).

Figure 9.

The overall BSC structure of GSC.

Step 1

The tracking error’s state variable $x_{3}$ can be expressed using the below-mentioned equation:

e_{d c} = V_{d c_r e f}^{2} - V_{d c}^{2}

(37)

To achieve our aim of reducing the error to zero, we may do this by designating $I_{g d}$ as a virtual control variable in the above equation and using it to regulate the error $e_{d c}$ . Through utilizing the Lyapunov function, which can be expressed as:

V_{1} = \frac{1}{2} e_{d c}^{2}

(38)

The Lyapunov function’s time derivative is given as:

{\overset{\cdot}{V}}_{1} = e_{d c} . {\overset{\cdot}{e}}_{d c} = e_{d c} ({\overset{\cdot}{V}}_{d c}^{2} - [\frac{2}{C} (- \frac{3}{2} V_{g d} . I_{g d} + P_{g e n})])

(39)

Within the framework of the backstepping control design, the stabilizing function, known as the virtual input, is determined by choosing the $I_{g d}$ components of the grid’s current.

I_{g d_r e f} = \frac{1}{V_{g d}} (\frac{2}{3} P_{g e n} - \frac{C}{3} [{\overset{\cdot}{V}}_{d c}^{2} + k_{d c} . e_{d c}])

(40)

By substituting equation (40) into the derivative expression ${\overset{\cdot}{V}}_{1}$ , we get the following outcome:

{\overset{\cdot}{V}}_{1} = - k_{d c} . e_{d c}^{2} \leq 0

(41)

Step 2

While $I_{g d} a n d I_{g q}$ , represent the system’s virtual inputs, $V_{i d} a n d V_{i q}$ , represent the actual control inputs, which are calculated using the variations in the grid currents.

The errors in grid currents can be expressed in the following form:

{\begin{matrix} e_{d} = I_{g d_r e f} - I_{g d} \\ e_{q} = I_{g q_r e f} - I_{g q} \end{matrix}

(42)

Therefore, the equation for the error can be written as follows:

{\overset{\cdot}{e}}_{d c} = - k_{d c} . e_{d c}^{2} - \frac{3}{C} V_{g d} . e_{d}

(43)

Additionally, the equation that specifies the dynamic behavior of the error signals $e_{d} a n d e_{q}$ might be established as follows:

{\begin{matrix} {\overset{\cdot}{e}}_{d} = {\overset{\cdot}{I}}_{g d_r e f} - (\frac{1}{L_{g}} V_{i d} - \frac{R_{g}}{L_{g}} I_{g d} + W_{g} I_{g q} - \frac{1}{L_{g}} V_{g d}) \\ {\overset{\cdot}{e}}_{q} = {\overset{\cdot}{I}}_{g q_r e f} - (\frac{1}{L_{g}} V_{i q} - \frac{R_{g}}{L_{g}} I_{g q} - W_{g} I_{g d} - \frac{1}{L_{g}} V_{g q}) \end{matrix}

(44)

A novel Lyapunov function is suggested to guarantee the stability of the whole system by using the voltages $V_{i d} a n d V_{i q}$ as references, taking into account the errors in DC-bus voltage and grid currents.

V_{2} = \frac{1}{2} (e_{d c}^{2} + e_{d}^{2} + e_{q}^{2})

(45)

The novel Lyapunov function’s time derivative is given as:

{\overset{\cdot}{V}}_{2} = e_{d c} . {\overset{\cdot}{e}}_{d c} + e_{d} . {\overset{\cdot}{e}}_{d} + e_{q} . {\overset{\cdot}{e}}_{q}

(46)

So:

\begin{matrix} {\overset{\cdot}{V}}_{2} = - k_{d c} . e_{d c}^{2} - k_{d} . e_{d}^{2} - k_{q} . e_{q}^{2} + e_{d} [{\overset{\cdot}{I}}_{g d_r e f} - (\frac{1}{L_{g}} V_{i d} - \frac{R_{g}}{L_{g}} I_{g d} + W_{g} I_{g q} - \frac{1}{L_{g}} V_{g d}) + k_{d} . e_{d} - \frac{3}{C} V_{g d} . e_{d c}] \\ + e_{q} [{\overset{\cdot}{I}}_{g q_r e f} - (\frac{1}{L_{g}} V_{i q} - \frac{R_{g}}{L_{g}} I_{g q} - W_{g} I_{g d} - \frac{1}{L_{g}} V_{g q}) + k_{q} . e_{q}] \end{matrix}

(47)

$k_{d} a n d k_{q}$ are constants with positive values.

To achieve system stability and ensure that the derivative of the Lyapunov function is negative, the stator reference voltages are applied in the following particular way:

V_{i d_r e f} = R_{g} . I_{g d} - W_{g} . L_{g} . I_{q g} + V_{g d} - \frac{3}{C} V_{g d} . L_{g} . e_{d c} + k_{d} . L_{g} . e_{d}

(48)

V_{i q_r e f} = R_{g} . I_{g q} + W_{g} . L_{g} . I_{d g} + V_{g q} + k_{q} . L_{g} . e_{q}

(49)

In the end, the behavior of the Lyapunov function could be expressed in the following way:

{\overset{\cdot}{V}}_{2} = - k_{d c} . e_{d c}^{2} - k_{d} . e_{d}^{2} - k_{q} . e_{q}^{2} \leq 0

(50)

Results and discussion

In order to evaluate the efficacy of the control tactics outlined in this academic article and to showcase their efficiency, the provided methods were subjected to experimentation across different wind speed profiles, as seen in Figure 10(a). The MATLAB/Simulink application was used to perform simulation experiments. The numerical simulations used a reactive power reference at $Q_{g} = 0$ to control the GSC. The currents $I_{d s j}, I_{x s j}, a n d I_{y s j}$ were adjusted to a value of zero for the MSC control. Furthermore, the reference voltage of the DC-link is established at 2500 V, while the grid frequency remains consistent at 50 Hz. Table 2 displays the comprehensive parameters for the wind turbine, including the 5-phase PMSGs, DC-link, and grid.

Figure 10.

Findings of simulations for the two 5-phase PMSG wind systems that ANFIS-BSC controls: (a) speed profile of winds, (b) power coefficients, (c) TSR, (d) mechanical powers, and (e) overall electromagnetic torque.

Table 2.

The PMSGs, wind turbine, DC-link and grid parameter.

Generator			Wind turbine			DC-link and grid
Parameters	Symbol	Value	Parameters	Symbol	Value	Parameters	Symbol	Value
Power generator	$P_{n}$	2 MW	Radius of the turbine blade	R	39 m	DC-link voltage	$V_{d c}$	2500 V
Stator resistance	$R_{s}$	0.821 mΩ	Specific density of air	$ρ$	1.225 kg/ $m^{3}$	Capacitor of the DC-link	C	0.038 F
Stator dq-axis inductance	$L_{d}, L_{q}$	1.573 mH	Tip-speed ratio	$λ_{o p t}$	8.1	Grid frequency	$F_{g}$	50 Hz
Pole number	P	26	Optimal power coefficient	$C_{p - m a x}$	0.48	Grid resistance	$R_{g}$	0.01 Ω
Moment of inertia	J	6300 kg/ $m^{2}$				Grid inductance	$L_{g}$	0.001 H
Flux linkage	$Ψ_{f}$	13.0282 Wb				Gridvoltage	$V_{g}$	690 V

MSC results

The evaluation of the generating system’s performance is conducted at various wind speeds, with a median velocity of 10.5 m/s, as seen in Figure 10(a). The investigation is done at a fixed angle of 0°. The implementation of the ANFIS approach yields very accurate results with a negligible margin of error for both TSR (λ) and $C_{p}$ , measuring 8.1 and 0.48, respectively, as seen in Figure 10(b) and (c). Furthermore, Figure 10(d) depicts the mechanical power generated by the generators, while Figure 10(e) provides a comparative evaluation of the total torque of the generators and the ideal torque. A notable correlation is detected between them, demonstrating accurate tracking of the ANFIS to the reference value.

Figure 11 illustrates the mechanical speed of the generators as compared to the reference velocities. Both control systems exhibit good performance in correctly following the reference values. The ANFIS approach has a clear advantage over BSC, particularly when closely examined; since it exhibits better adherence to the required reference values.

Figure 11.

A comparison of the ANFIS and BSC controllers concerning (a) the first generator’s speed and (b) the second generator’s speed.

The most significant aspects of the outcomes demonstrated in Figure 11 are summed up in Table 3.

Table 3.

Overshoot and response time of the speed generators.

Controller	Overshoot ratio (%)	Response time (ms)
BSC	3.15	58.5
ANFIS	0	44.5

GSC results

Figure 12 presents an analysis of the outcomes observed on the grid side, which are controlled by the BSC technique. The study encompasses various parameters, including the DC-link voltage, injected powers, power factor, grid frequency, grid voltage and current waveforms, single-phase grid current, and the THD ratio of grid current. Figure 12(a) depicts the stability of the DC-link voltage, which remains consistent regardless of fluctuations in wind speed. Figure 12(b) illustrate the active and reactive power injected into the network, respectively. The active power demonstrates responsiveness to changes in wind speed while maintaining precise alignment with the reference value. Meanwhile, the reactive power closely adheres to its reference value of zero to achieve a unity power factor, as evidenced by Figure 12(c) and (e). Figure 12(d) presents the grid current frequency, indicating that it remains constant at 50 Hz despite variations in wind speed. The THD ratio, shown in Figure 12(h), demonstrates the good outcomes produced by the control strategy.

Figure 12.

The GS results: (a) DC-link voltage, (b) powers injected into the grid, (c) power factor, (d) grid frequency, (e) grid current & voltage, (f) closer observation of grid current and voltage, (g) closer examination of grid current, and (h) grid-current harmonics spectrum.

The subsequent Table 4 provides a summary of the characteristics and the important notes of the outcomes illustrated in Figure 12.

Table 4.

The main outcomes of the grid side.

Controller	Overshoot ratio (%) of DC-bus	DC-bus voltage ripples (V)	Grid current THD (%)	Power factor	Grid current frequency (Hz)
BSC	16.6	2.3	1.89	1	50

Test stop one of the generators

This section analyzes the system’s response when one of the generators stops. The second generator is halted at t = 2.8 seconds, as seen in Figures 13 and 14, and remains inactive for 1.7 seconds. The generator is restarted at t = 4.5 seconds. This test examines a potential issue inside the system to verify the resilience of the control system while operating under multiple circumstances.

Figure 13.

The MS results test: (a) Torque of first subsystem, (b) Torque of second subsystem, and (c) First PMSG’s speed.

Figure 14.

The GS results test: (a) powers injected into the grid, (b) DC-link voltage, (c) grid current & voltage, (d) closer observation of grid current and voltage, and (e) grid-current harmonics spectrum.

Figure 13 demonstrates the MS outcomes: Figure 13(a) and (b) illustrates the electromagnetic torque of the generators and the turbine torque under normal operating conditions, as well as during a disturbance. It is evident from the figure that the torque of the first PMSG remains unchanged even when a fault occurs in the second PMSG, albeit with a slight overshoot during the fault event. Moreover, as shown in Figure 13(c), the speed of the first generator remains at the desired value with a very small percentage of error.

Figure 14 depicts the test outcomes for the network side, with Figure 14(a) illustrating the energy injected into the network. It is evident from the figure that there is a noticeable lack of injected energy at the moment of the fault, particularly in terms of active energy. However, despite this, the generation system continues to supply energy to the network without encountering any detrimental issues. Figure 14(b) displays the DC-link voltage; it clearly shows stability in the continuous voltage and tracking of the reference value. Notably, there is an overshoot during the occurrence of the fault, as well as the moment the system returns to its normal state. Figure 14(c) and (d) provide a close examination of the network current and voltage for one phase, respectively. Figure 14(e) further presents the percentage of THD in the network current, revealing a slight increase compared to the system’s normal state.

In order to assess the efficiency of the control systems under investigation (Figure 15), the theoretical output power is compared to the actual power injected into the grid (Mousa et al., 2020), as defined in:

η_{s y s} = \frac{\int_{0}^{t} P_{g}}{\int_{0}^{t} P_{t h}} \times 100 %

Where $η_{s y s}$ is the system efficiency, $P_{g}$ is the injected power into grid, and $P_{t h}$ is the theoretical extracted power.

Figure 15.

The overall efficiency of the WPGS.

Conclusion

This paper examines the use of Backstepping control (BSC) and adaptive neuro-fuzzy inference system (ANFIS) control technologies in a wind system that is linked to the power grid. The wind system employs two five-phase permanent magnet synchronous generators (PMSGs), which are controlled by a fifteen-switch rectifier (FSR). The ANFIS control was trained using BSC, a method grounded in Lyapunov theory, to ensure the stability of the system. The fundamental characteristics of this control system are well known, including simplicity, exceptional precision, and steadfastness. Extensive simulations are conducted to assess the efficacy of the proposed control mechanisms and compare their results.

The results of the simulations showed that both techniques achieved separate control of the two PMSGs. They also showed many improvements in both the dynamic responses and the steady-state performance. These improvements encompass speed response, accuracy, and ripple reduction. Notably, the ANFIS control approach has exhibited several good advancements when compared to the BSC technique. Key enhancements include faster rising time, excellent speed tracking without overshoots, and fewer ripples in injected power into the grid.

The objective of incorporating the fifteen-switch converter (FSR) into the wind system is to optimize its efficiency by lowering the overall number of switches compared to the conventional system, which is based on two parallel rectifiers, thus reducing losses associated with switching while keeping the control of the system simple. The results indicate that the suggested topology is very successful in providing outstanding performance under both steady-state and transient circumstances.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Abderrahim SAKOUCHI

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