Intelligent backstepping control of power grid-connected wind power generation system using two five-phase PMSGs controlled through the fifteen-switch rectifier

Abstract

This scholarly paper offers a wind power generation system (WPGS) that utilizes a configuration of parallel five-phase permanent magnet synchronous generators (PMSGs). The control mechanism for this system is based on a fifteen-switch rectifier (FSR) topology, which is specifically designed for grid-connected applications. To enhance the control performance of the proposed wind system, an Adaptive Neuro-Fuzzy Inference System (ANFIS)-based Backstepping control (BSC) methodology is utilized for both the generators-side and grid-side converters. The suggested control strategy’s effectiveness is assessed by conducting simulations at varying wind velocities and comparing it to a BSC-based control system. The simulation findings demonstrate that the suggested control system consistently ensures precise tracking of the controlled variables. Furthermore, several aspects of the system’s performance show significant improvements. Specifically, the control method successfully reduces power ripples by 30.5%, DC bus voltage overshoot by 3%, and speed overshoot by 100%. Moreover, the proposed technique reduces the overall harmonic distortion of the network current by 44.44% compared to the BSC method. Additionally, the system’s efficiency is enhanced to 96.5%, surpassing the 96% achieved by the BSC method. Compared to good nonlinear control methods, the results indicate that the suggested strategy outperforms.

Introduction

Antecedents

The Ukrainian-Russian crisis served as a significant turning point that propelled the transition toward clean and renewable energy, primarily due to the subsequent rise in oil and gas prices. European nations, being major importers of Russian gas, faced significant challenges in coping with the crisis. Despite their previous focus on alternative energy sources, they struggled to mitigate the impact of the crisis. However, the crisis has accelerated the expansion and development of renewable energy, with notable investments being made by China, the United States, and India. The European Union’s REPowerEU initiative sets forth the objective of eliminating dependence on Russian fossil fuels by the year 2027 (Figure 1). The International Energy Agency anticipates that the growth of renewable energy will surpass expectations, resulting in an estimated increase of approximately 2400 GW. ¹

Figure 1.

Worldwide power production by different technologies in 2015, 2021, and 2027.

Wind energy is one of the world’s most significant, frequently utilized, and continuously developing forms of energy at the current time. This is because wind energy is an entirely clean kind of energy that does not have any adverse impacts on the environment, including the atmosphere, water, or soil. In addition to the fact that it does not produce any waste or greenhouse gases (Figure 2), it has also seen a reduction in price in comparison to the years before as a result of the significant advancements that have been made in the area of renewable energy.^2,3

Figure 2.

Division of total produced electricity capacity by the generating technology.

One of the best things about WPGS is that they can work at either a fixed or changeable speed. An extensive body of research has shown that the variable-speed operating system is the best choice. This is because it has many benefits, including the ability to be controlled to lower the mechanical stress caused by winds, the ability to increase the energy produced by using the MPPT strategy, and the fact that VSWPGS can produce about 40% more energy than FSWPGS under the same operating conditions. This is why VSWPGS is more attractive because it ensures a stable system and is more reliable.^4–6

There is no doubt that this system consists of a group of elements that must be chosen with great care to obtain the best possible results that meet the needs of the user, and among the most important and prominent of these elements is the generator. Since the emergence of these systems, several types of generators have been used, such as the double-feed induction generator, which has been very popular in the past years; the synchronous generator with field excitation; the induction generator with a squirrel cage; and the permanent magnet synchronous generator. Recently, the permanent magnet synchronous generator has been widely used in the field of wind energy generation due to its advantages as it contains a large number of poles, which provides work at a low speed, and its use without a gearbox, which contributes to reducing the cost of energy production while achieving greater torque as it does not need an external excitation current and the absence of losses in the rotary member. Therefore, these advantages give better evaluation efficiency for this type of generator compared to others as a distinct option for the wind power generation system.^2,4–9 On the other hand, it has recently become the subject of many studies on the use of multi-phase generators and their integration with wind energy generation systems due to their advantages, including ¹⁰ :

• Increased fault tolerance and improved dependability

• Potential of utilizing low-energy devices to accomplish reduced voltage production and high power.

• The potential for continued operation in the event of the failure of one or multiple phases of the generator or power transformer.

Within this particular framework, a multi-phase PMSG exhibits notable benefits compared to a three-phase system, including enhanced dependability, diminished capacitance, and a heightened frequency of torque pulses. Hence, a multi-phase PMSG is a very suitable option for wind power uses.^11,12

Motivation

Talking about wind energy generation systems cannot pass without talking about power electronics converters, which are considered an intelligent interface for regulating frequency and voltage, as well as the energy injected into the network and ensuring its control under different circumstances. ¹³ Back-to-back transformers with PMSG are considered a good option to meet network requirements, ¹⁴ as this type of generator has various features, most notably: good controllability, The possibility of using it with back-to-back converters (BTBC), which gives an effective separation between the network side and the generator, in addition to the possibility of managing the two-way power flow, as well as allowing the regulation of the injection of active and reactive power into the network in a rapid manner.^15,16

Wind systems (WS) usually consist of many individual wind turbine generators connected and operated simultaneously, and power electronics converters are among the most expensive equipment. ⁵ In order to reduce this cost In order to obtain energy at the lowest possible cost, several types of transformers were used, and among the latest of these methods is the BTBC based on a fifteen switch rectifier (FSR) which is defined as the combination of two five-phase rectifiers, This type of converter gives the possibility of connecting two generators to one rectifier at the same time without the need for two. This transformer is also considered a new strategy in terms of performance and cost, as it is characterized by reducing the number of switches thus reducing switching losses while not complicating system control. ¹⁶

Literature review

Wind systems often use classic BTB converters, which consist of a rectifier (MSC) responsible for controlling the generator’s rotational speed and an inverter (GSC) that controls the DC-bus voltage and the electricity pumped into the network. An effective improvement may be obtained, particularly for the generator side, by substituting the generator side transformer with a fifteen-switch rectifier (FSR) while preserving the respective roles of each side. Throughout the search for this topic, a few papers were found that examined the same research topic as this work, with the most similar topic being the FSR topic in Ref., ¹⁶ whereby the author proposed a new converter topology for the WS called fifteen-switch rectifier (FSR). The simulation outcomes showed that FSR has done well. It also has some advantages over the classical topology used in Ref., ⁵ such as reducing switching losses resulting from reducing the number of switches while taking into account control techniques and not complicating them. In Ref., ¹⁷ the author proposed using a nine-switch BTBC topology for use with multi-generator WS. The results show that this proposed topology has many advantages over traditional BTBCs. The most important of these advantages is that it cuts down on the number of switches in both converters. This reduces the lost energy from switching and achieves good separation between the generation stages and inverting. To keep the study’s scope small, the main focus was on the so-called nine-switch converter (NSC) topology. This is very similar to the subject of our research, as NSC has been widely used, especially in energy generation applications such as a DFIG-based wind generation system, ¹⁸ and in the field of electric vehicles to drive multi-phase motors. ¹⁹ Aside from these applications, many different approaches for the NSI to control two independent three-phase loads/machines or a single six-phase load have been mentioned throughout the scientific literature.^20–27 In Ref., ²⁸ the author offered the SMC method for dual load regulation using a NSI. The simulation results demonstrated the effectiveness of this architecture in managing double AC loads. other works^29–31 have explored the same problem but with a special emphasis on using various control systems based on model predictive control (MPC) for the NSI system. The findings from the experiments indicate that the direct MPC technique provides reliable control operation and efficient multi-phase current regulation.

Numerous nonlinear control approaches have been used in different literature on wind systems based on BTBC. These include input-output feedback linearization, ² backstepping control,^32,33 and various types of sliding mode control.^34–43 In Ref., ³⁴ the authors provide the implementation of high-order sliding mode control (HOSMC) and conduct a comparative analysis of its performance concerning first-order sliding mode control (FOSMC). The results indicate that HOSMC exhibits a quicker reaction. Although the phenomenon of chatter is reduced compared to FOSMC, it does not eliminate it completely, in addition to the presence of unwanted errors in tracking the desired value. Whereas in Ref., ³⁵ the author suggests using integral sliding mode control (ISMC) technology and conducting a comparative analysis with the standard PI controller. The simulation outcomes demonstrate the dynamic advantages of ISMC and its effective regulation of generator speed across various operating situations. However, it should be noted that an undesirably chattering phenomenon occurs, as in Ref., ³⁴ as does the existence of distortions in the grid current. In Ref., ³⁶ as in the previous two works,^34,35 the super-twisting sliding mode control (STSMC) strategy was implemented for wind turbines powered by PMSG. The approach that was presented exhibited both robustness and rapid reaction. Additionally, it efficiently addressed the problem of excessive chatter, which has been noted as a significant obstacle in previous research.^34,35 In the work presented in Ref., ³⁷ the authors implemented the PID-type terminal sliding mode control (PID-TSMC) system for the MS and GS converter-modified controllers of BTBCs. The control roles of the MSC and GSC were reversed, and the results were compared with the latest findings in the literature.^36,38–40 The approach being suggested exhibits exceptional performance and decreased reaction time in the presence of fluctuations in the DC-bus voltage while also demonstrating resilience against limited external disturbances. In Ref., ⁴¹ the author introduced a novel, robust control technique for sliding mode control utilizing the smooth continuous function methodology. This approach was developed to tackle the limitations associated with conventional sliding-mode control. The simulation findings demonstrated the superiority of the suggested approach compared to the classical method. The key advantages of the new approach can be summarized as the elimination of the chatter phenomenon and the enhancement of response time. In Ref., ⁴² as presented in the previously mentioned literature, the authors used a novel methodology known as Fuzzy Fractional Order Sliding Mode Control (FFOSMC) and evaluated its performance in comparison to numerous other studies cited in the literature.^35,43 The findings of this investigation demonstrated the superiority of the proposed strategy. In the findings presented in this paper and other studies, it is evident that the proposed method exhibits superior performance. A critical aspect of this superiority is its ability to mitigate the chattering phenomenon and quicken responses. Nevertheless, it demonstrates a weakness in steady-state error, an area where the research described in Ref. ³⁴ triumphs. Whereas in the study conducted by the author in Ref., ⁴⁴ a comparative analysis was performed to examine and contrast four prominent control strategies commonly utilized in the control of WPGS employing PMSG. The investigated control strategies include sliding mode control (SMC), backstepping control (BSC), model predictive control (MPC), and direct power control (DPC), which are widely recognized and frequently employed in the field. The simulation findings show predictive control outperforms conventional control methods in accuracy, simplicity, precision, reference point tracking, and injected current quality. It effectively addresses the limitations of traditional techniques like chattering in SMC, high-power ripples in DPC, and BSC’s complexity.

Contribution

The primary contributions of this work may be succinctly described as follows:

• ✓ Design and implementation of the BSC approach for the BTBC based on FSR.

• ✓ Training an ANFIS controller by applying the concepts of BSC.

• ✓ Implementation of the trained ANFIS controller on the BTBC system.

• ✓ Comparative evaluation of the performance of the ANFIS controller and the BSC approach.

In this research work, we utilized both BSC technique and ANFIS for the control of the BTBC. The aim is to boost the system’s performance and achieve robust control. Despite BSC being widely regarded as one of the most powerful control techniques, particularly for reference signal tracking, ⁴⁴ its heavy reliance on system parameters is its primary drawback. ⁴⁵ This dependence often leads to degradation in performance. On the other hand, ANFIS exhibits robustness to variations in system parameters and uncertainties, where it can adaptively adjust its rules and fuzzy parameters, thereby accommodating parameter differences and uncertainties. ⁴⁶ To profit from the respective advantages of these two techniques, ANFIS was trained based on BSC. This topology not only boosts the system’s performance but also enhances response time, reduces overshoots, and achieves good tracking performance. By combining the strengths of BSC and ANFIS, we not only achieve improved control performance but also benefit from the robustness of ANFIS in handling parameter uncertainties.

Paper organization

The subsequent parts of this research project are structured as follows: Section II begins by providing a concise model of the wind turbine and the two PMSGs, along with general analyses of the FSR topology and its most important characteristics and advantages. Within this section, the FSR converter control utilizing the PWM block is extensively discussed. Section III presents the proposition and design of a BSC strategy, and the ANFIS was trained based on the BSC technique for controlling the BTBC. The simulation outcomes performed to validate the suggested control strategy were comprehensively analyzed in Section IV. Finally, Section V represents the concluding segment, summarizing the key outcomes and most important notes derived from this work.

WPGS modeling

Figure 1 displays the general shape of the WPGS based on two five-phase PMSG connected on three-phase grid. The setting of new back-to-back converters are made up of two converters, the first is a MSC, it is a five-phase FSR, and the second transformer is a GSC) it is classical three-phase inverter, and the link between them is a filter C.

Wind-turbine modeling

The below equation offers a mathematical equation of the aerodynamic energy produced by wind turbines^11,12:

$$P_V=\frac{1}{2}\rho A{C}_p(\lambda ,\beta )V_w^3$$

(1)

In order to calculate the energy generated by wind turbines, we depend on many significant coefficients, namely: $\rho$ represents the density of the air, A is the area that is swept by the blades ( $\mathrm{A}=\pi R^2$ ), β constitutes the pitch angle of the propeller blades, $V_w$ means the wind speed (m/s), and λ represents the tip velocity ratio.

Where $C_p$ is the wind turbine’s power conversion coefficient:

$$\{\begin{array}{c}{C}_p(\lambda ,\beta )=0.5176.(\frac{116}{{\lambda }_i}-0.4.\beta -5).\exp (-\frac{21}{{\lambda }_i})+0.0068.\lambda \hfill \\ \frac{1}{{\lambda }_i}=(\frac{1}{\lambda +0.08.\beta }-\frac{0.035}{{\beta }^3+1})\hfill \\ \lambda =\frac{{\mathrm{\Omega }}_{m}R}{V_w}\hfill \\ C_{p\_\max }(\lambda ,\beta )=\frac{16}{27}=0.59\hfill \\ {\lambda }_{\mathrm{opt}}=8.1\hfill \\ \beta =0\hfill \end{array}$$

(2)

Whereas: ${\mathrm{\Omega }}_m$ is the angular speed of the turbine and R represents the blades’ radius.

Modeling of the PMSG

The voltage, magnetic flux, and electromagnetic torque for every five-phase PMSG model in a rotating d-q-x-y frame are expressed by equations (3)–(5), which are written in the following order ¹⁶ :

$$\{\begin{array}{c}{V}_{\mathrm{dsj}}=R_{\mathrm{sj}}{I}_{\mathrm{dsj}}+\frac{d{\mathrm{\Psi }}_{\mathrm{dsj}}}{\mathrm{dt}}-w_{\mathrm{mj}}{\mathrm{\Psi }}_{\mathrm{qsj}}\hfill \\ V_{\mathrm{qsj}}=R_{\mathrm{sj}}{I}_{\mathrm{qsj}}+\frac{d{\mathrm{\Psi }}_q}{\mathrm{dt}}+w_{\mathrm{mj}}{\mathrm{\Psi }}_{\mathrm{dsj}}\hfill \\ V_{\mathrm{xsj}}=R_{\mathrm{sj}}{I}_{\mathrm{xsj}}+\frac{d_{\mathrm{\Psi }\mathrm{xsj}}}{\mathrm{dt}}\hfill \\ V_{\mathrm{ysj}}=R_{\mathrm{sj}}{I}_{\mathrm{ysj}}+\frac{d{\mathrm{\Psi }}_{\mathrm{ysj}}}{\mathrm{dt}}\hfill \end{array}$$

(3)

$$\{\begin{array}{c}{\mathrm{\Psi }}_{\mathrm{dsj}}=L_{\mathrm{dj}}{I}_{\mathrm{dsj}}\hfill \\ {\mathrm{\Psi }}_{\mathrm{qsj}}=L_{\mathrm{qj}}{I}_q+{\mathrm{\Psi }}_{\mathrm{fj}}\hfill \\ {\mathrm{\Psi }}_{\mathrm{xsj}}=L_{\mathrm{dj}}{I}_{\mathrm{xsj}}\hfill \\ {\mathrm{\Psi }}_{\mathrm{ysj}}=L_{\mathrm{qj}}{I}_{\mathrm{ysj}}\hfill \end{array}$$

(4)

$$T_{\mathrm{emj}}=\frac{5}{2}{P}_j[(L_{\mathrm{dj}}-L_{\mathrm{qj}})I_{\mathrm{dsj}}{I}_{\mathrm{qsj}}+{\mathrm{\Psi }}_f{I}_{\mathrm{qqj}}$$

(5)

Where: d, q, x, and y are the axes of the stator, while $V_{\mathrm{dsj}},V_{\mathrm{qsj}},V_{\mathrm{xsj}},\mathrm{and}{V}_{\mathrm{ysj}}$ are the stator voltages, with j = 1, 2. The stator currents on the d, q, x, and y axes are $I_{\mathrm{dsj}},I_{\mathrm{qsj}},I_{\mathrm{xsj}}\mathrm{and}{I}_{\mathrm{ysj}}$ , respectively. In the rotating frames, $L_{\mathrm{dj}}\mathrm{and}{L}_{\mathrm{qj}}$ denote the inductances in the rotating frames. The stator resistance is denoted as $R_{\mathrm{sj}}$ . The magnet flux is known as ${\mathrm{\Psi }}_{\mathrm{fj}}$ , and ${\mathrm{\Omega }}_{\mathrm{mj}}$ denotes the rotor’s electrical speed.

The wind-turbine system’s mechanical equation could be writing by the fundamental law of dynamic as ensue:

$$T_{\mathrm{mj}}=T_{\mathrm{emj}}+K_{\mathrm{fj}}{\mathrm{\Omega }}_{\mathrm{mj}}+J_j\frac{d{\mathrm{\Omega }}_{\mathrm{mj}}}{\mathrm{dt}}$$

(6)

While: $T_{\mathrm{mj}}$ is the wind turbine’s mechanical torque.

${\mathrm{T}}_{\mathrm{emj}}$ : is generator’s electromagnetic torque.

${\mathrm{J}}_{\mathrm{j}}$ : is the total inertia of the system (with: $\mathrm{J}=({\mathrm{J}}_{\mathrm{t}}/{\mathrm{G}}^2)+{\mathrm{J}}_{\mathrm{g}}$ and G = 1)

${\mathrm{K}}_{\mathrm{fj}}$ : is the factor of a viscous friction.

The power generated through each generator is able to be calculated using the formula that goes as follows

$$\{\begin{array}{c}{P}_{\mathrm{mj}}=T_{\mathrm{emj}}.{\mathrm{\Omega }}_{\mathrm{mj}}=\frac{5}{2}[V_{\mathrm{dsj}}.I_{\mathrm{dsj}}+V_{\mathrm{qsj}}.I_{\mathrm{qsj}}]\hfill \\ Q_{\mathrm{mj}}=\frac{5}{2}[V_{\mathrm{qsj}}.I_{\mathrm{dsj}}-V_{\mathrm{dsj}}.I_{\mathrm{qsj}}]\hfill \end{array}$$

(7)

Analysis of the FSR topology

Figure 3 depicts that the FSR comprises five legs, with each leg containing upper, middle, and bottom switches. The FSR can be viewed as a combination of two five-phase rectifiers, each consisting of a set of switches. Specifically, the upper rectifier is made up of the top and middle switches, while the lower rectifier is composed of the middle and bottom switches. ¹⁶

Figure 3.

WPGS’s general architecture.

The upper and lower switches are controlled using traditional PWM to create their respective signals. The gating signals for the middle switches, on the other hand, are formed by inputting the gating signals of both higher and lower switches into the XOR logical operator. The overall number of switches is decreased by 25% when compared to a normal dual five-phase rectifier design.

The function of switch $S_{\mathrm{ij}}$ , illustrated in Figure 3, can be defined using the switch function as follows:

$$S_{\mathrm{ij}}=\{\begin{array}{c}1,S_{\mathrm{ij}}\mathrm{closed}\hfill \\ 0,S_{\mathrm{ij}}\mathrm{open}\hfill \end{array};\mathrm{i}=\mathrm{U},\mathrm{M},\mathrm{L};\mathrm{j}=\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D},\mathrm{E}$$

(8)

We may express the restrictions as:

$$S_{\mathrm{Uj}}+S_{\mathrm{Mj}}+S_{\mathrm{Lj}}=2$$

(9)

Regarding the operational approach of this novel converter shown in Figure 3, it is possible to use three cases per leg. The switching states may be designated as 1, 0, and −1, as seen 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 fifteen-switch structure.

Switching states	$S_{\mathrm{Uj}}$	$S_{\mathrm{Mj}}$	$S_{\mathrm{Uj}}$	$V_u$	$V_l$
1	1	0	1	$V_{\mathrm{dc}}$	0
0	0	1	1	0	0
−1	1	1	0	$V_{\mathrm{dc}}$	$V_{\mathrm{dc}}$

Figure 4 depicts the suggested PWM technique for the FSR, which utilizes two synchronized PWM units from traditional five-leg rectifiers. Each PWM unit receives distinct voltage references, $V_{U\_\mathrm{ref}}$ and $V_{L\_\mathrm{ref}}$ , in order to produce 10 pulses corresponding to the top and lower outputs, successively. The XOR gate generates the middle switch pulses by processing the upper and lower gating signals.

Figure 4.

Proposed PWM for the FSR.

Grid model

The grid voltage in the Park frame (d, q) might be stated in the following way ⁴¹ :

$$\{\begin{array}{c}{V}_{\mathrm{gd}}=R_s{I}_{\mathrm{gd}}+L_d\frac{d{I}_{\mathrm{gd}}}{\mathrm{dt}}-w_g{L}_q{I}_{\mathrm{gq}}+V_{\mathrm{id}}\hfill \\ V_{\mathrm{gq}}=R_s{I}_{\mathrm{gq}}+L_q\frac{d{I}_{\mathrm{gq}}}{\mathrm{dt}}-w_g{L}_d{I}_{\mathrm{gd}}+V_{\mathrm{iq}}\hfill \end{array}$$

(10)

In the grid system, $I_{\mathrm{gd}}$ and $I_{\mathrm{gq}}$ represent the direct and quadrature components of the grid’s currents, while $V_{\mathrm{gd}}$ and $V_{\mathrm{gq}}$ 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 $V_{\mathrm{id}}$ and $V_{\mathrm{iq}}$ . 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.

Within the context of the d-q reference frame, the active and reactive powers are expressed by

$$\{\begin{array}{c}{P}_g=\frac{3}{2}(V_{\mathrm{gd}}.I_{\mathrm{gd}}+V_{\mathrm{gq}}.I_{\mathrm{gq}})\hfill \\ Q_g=\frac{3}{2}(V_{\mathrm{gq}}.I_{\mathrm{gd}}-V_{\mathrm{gd}}.I_{\mathrm{gq}})\hfill \end{array}$$

(11)

The control approach used for the GSC involves making $V_{\mathrm{qg}}=0$ and $V_{\mathrm{dg}}=\Vert V_g\Vert$ . This results in the power equations being modified as follows:

$$\{\begin{array}{c}{P}_g=\frac{3}{2}(V_{\mathrm{gd}}.I_{\mathrm{gd}})=\frac{3}{2}(\Vert V_g\Vert .I_{\mathrm{gd}})\hfill \\ Q_g=\frac{3}{2}(-V_{\mathrm{gd}}.I_{\mathrm{gq}})=-\frac{3}{2}(\Vert V_g\Vert .I_{\mathrm{gq}})\hfill \end{array}$$

(12)

It may be concluded that the active and reactive power flows can be controlled by $I_{\mathrm{dg}}$ and $I_{\mathrm{qg}}$ , respectively.

The DC-link provides a connection between the converter located on the generator side and the transformer located on the grid side. The dynamic demeanor of the DC-link voltage may be determined by considering power balancing and disregarding converter losses. ⁴⁷

$$P_{\mathrm{gen}}-P_g=\frac{1}{2}.C.\frac{{\mathrm{dV}}_{\mathrm{dc}}^2}{\mathrm{dt}}$$

(13)

PMSG-based WPGS control

MSC control

BSC technique

The backstepping control methodology is based on the representation of interconnected systems as Lyapunov-order 1 subsystems, leading to heightened resilience against disturbances and improved overall system stability. The approach of BSC employs a sequential process, wherein a virtual command is generated at each stage to ensure the system’s convergence toward its equilibrium state (Figure 5). The utilization of a Lyapunov function ensures stability at every synthesis step.^32,33 By applying the relationships described in equations (3)–(6), the state representation of the system can be expressed.

$$\{\begin{array}{c}\frac{d{I}_{\mathrm{dsj}}}{\mathrm{dt}}=\frac{-R_{\mathrm{sj}}}{L_{\mathrm{dj}}}{I}_{\mathrm{dsj}}+\frac{P_j{L}_{\mathrm{qj}}}{L_{\mathrm{dj}}}{I}_{\mathrm{qj}}{\mathrm{\Omega }}_{\mathrm{mj}}+\frac{1}{L_{\mathrm{dj}}}{V}_{\mathrm{dsj}}\hfill \\ \frac{d{I}_{\mathrm{qsj}}}{\mathrm{dt}}=\frac{-R_{\mathrm{sj}}}{L_{\mathrm{qj}}}{I}_{\mathrm{qsj}}-\frac{P_j{L}_{\mathrm{dj}}}{L_{\mathrm{qj}}}{I}_{\mathrm{dj}}{\mathrm{\Omega }}_{\mathrm{mj}}-\frac{P_j{\mathrm{\Psi }}_{\mathrm{fj}}}{L_{\mathrm{qj}}}{\mathrm{\Omega }}_{\mathrm{mj}}+\frac{1}{L_{\mathrm{qj}}}{V}_{\mathrm{qsj}}\hfill \\ \frac{d{I}_{\mathrm{xsj}}}{\mathrm{dt}}=\frac{-R_{\mathrm{sj}}}{L_{\mathrm{dj}}}{I}_{\mathrm{xsj}}+\frac{1}{L_{\mathrm{dj}}}{V}_{\mathrm{xsj}}\hfill \\ \frac{d{I}_{\mathrm{ysj}}}{\mathrm{dt}}=\frac{-R_{\mathrm{sj}}}{L_{\mathrm{qj}}}{I}_{\mathrm{ysj}}+\frac{1}{L_{\mathrm{qj}}}{V}_{\mathrm{ysj}}\hfill \\ \frac{d{\mathrm{\Omega }}_{\mathrm{mj}}}{\mathrm{dt}}=\frac{T_{\mathrm{mj}}}{J_j}-\frac{5}{2J_j}{P}_j(L_{\mathrm{dj}}-L_{\mathrm{qj}})I_{\mathrm{dsj}}{I}_{\mathrm{qsj}}-\frac{5}{2J}{P}_j{\mathrm{\Psi }}_{\mathrm{fj}}{I}_{\mathrm{qsj}}-\frac{K_{\mathrm{fj}}}{J_j}{\mathrm{\Omega }}_{\mathrm{mj}}\hfill \end{array}$$

(14)

Figure 5.

The general FSR control structure.

The implementation of BSC on the generator may be broken down into two consecutive steps. The first stage offers instructions for the subsequent one. Furthermore, the state vectors and control vectors are chosen so that $\left[x\right]={\left[\begin{array}{ccccc}{I}_{\mathrm{dsj}}\hfill & I_{\mathrm{qsj}}\hfill & I_{\mathrm{xsj}}\hfill & I_{\mathrm{ysj}}\hfill & {\mathrm{\Omega }}_{\mathrm{mj}}\hfill \end{array}\right]}^T={\left[\begin{array}{ccccc}{x}_1\hfill & x_2\hfill & x_3\hfill & x_4\hfill & x_5\hfill \end{array}\right]}^T$ specifically refers to the state vectors, whereas $\left[u\right]={\left[\begin{array}{cccc}{V}_{\mathrm{dsj}}\hfill & V_{\mathrm{qsj}}\hfill & V_{\mathrm{xsj}}\hfill & V_{\mathrm{ysj}}\hfill \end{array}\right]}^T$ indicates the control vectors.

• ❖ Step1:

The tracking error’s state variable $x_5$ can be expressed using the below-mentioned equation:

$${\xi }_{\mathrm{\Omega }}={\mathrm{\Omega }}_{\mathrm{ref}\_j}-{\mathrm{\Omega }}_{\mathrm{mj}}$$

(15)

The speed’s error dynamic is expressed in the following manner:

$${\stackrel{·}{\xi }}_{\mathrm{\Omega }j}=\frac{{\stackrel{·}{d\xi }}_{\mathrm{\Omega }j}}{\mathrm{dt}}={\xi }_{\mathrm{ref}\_j}-{\stackrel{·}{\xi }}_{\mathrm{mj}}={\stackrel{·}{\mathrm{\Omega }}}_{\mathrm{ref}\_j}-\frac{1}{J_j}[T_{\mathrm{tj}}-\frac{5P_j}{2}(L_{\mathrm{dj}}-L_{\mathrm{qj}}).I_{\mathrm{dsj}}.I_{\mathrm{qsj}}-\frac{5P_j}{2}{\mathrm{\Psi }}_{\mathrm{fj}}.I_{\mathrm{qsj}}-k_{\mathrm{fj}}.{\mathrm{\Omega }}_{\mathrm{mj}}]$$

(16)

The primary aim of this first stage is to eliminate the error in speed tracking. We utilize the Lyapunov function, which is determined by the below relationship:

$$V_{\mathrm{\Omega }j}=\frac{1}{2}{\xi }_{\mathrm{\Omega }j}^2$$

(17)

The Lyapunov function’s time derivative is given as:

$${\stackrel{·}{V}}_{\mathrm{\Omega }j}={\xi }_{\mathrm{\Omega }j}.{\stackrel{·}{\xi }}_{\mathrm{\Omega }j}=-k_{\mathrm{\Omega }j}{\xi }_{\mathrm{\Omega }j}^2+\frac{{\xi }_{\mathrm{\Omega }j}}{J_j}[J_j.{\stackrel{·}{\mathrm{\Omega }}}_{\mathrm{re}{f}_j}-T_{\mathrm{tj}}+k_{\mathrm{fj}}.{\mathrm{\Omega }}_m+\frac{5P_j}{2}.{\mathrm{\Psi }}_{\mathrm{fj}}.I_{\mathrm{qsj}}+k_{\mathrm{\Omega }j}{J}_j{\xi }_{\mathrm{\Omega }j}]+\frac{5P_j}{2J_j}(L_{\mathrm{dj}}-L_{\mathrm{qj}}).I_{\mathrm{dsj}}.I_{\mathrm{qsj}}.{\xi }_{\mathrm{\Omega }j}$$

(18)

As part of the BSC design, the virtual inputs, referred to as stabilizing functions, are established by selecting the quadratic $I_{\mathrm{qsj}\_\mathrm{ref}}$ and direct $I_{\mathrm{dsj}\_\mathrm{ref}}$ stator current components.

$$\{\begin{array}{c}{I}_{\mathrm{dsj}\_\mathrm{ref}}=0\hfill \\ I_{\mathrm{qsj}\_\mathrm{ref}}=\frac{2}{5.P_j.{\mathrm{\Psi }}_{\mathrm{fj}}}[T_t-J_j.{\stackrel{·}{\mathrm{\Omega }}}_{\mathrm{ref}\_j}-k_{\mathrm{fj}}.{\mathrm{\Omega }}_{\mathrm{mj}}-k_{\mathrm{\Omega }j}.J_j.{\xi }_{\mathrm{\Omega }j}]\hfill \end{array}$$

(19)

By substituting equation (19) into the derivative expression ${\stackrel{·}{V}}_{\mathrm{\Omega }j}$ , we get the following outcome:

$${\stackrel{·}{V}}_{\mathrm{\Omega }j}=-k_{\mathrm{\Omega }j}.{\xi }_{\mathrm{\Omega }j}^2\le 0$$

(20)

• ❖ Step 2:

The system’s virtual inputs are $I_{\mathrm{dsj}},I_{\mathrm{qsj}},I_{\mathrm{xsj}},\mathrm{and}{I}_{\mathrm{ysj}}$ , while the genuine control inputs, $V_{\mathrm{dsj}},V_{\mathrm{qsj}},V_{\mathrm{xsj}},\mathrm{and}{V}_{\mathrm{ysj}}$ , are determined through calculations based on the differences in the stator currents.

The errors in stator current can be expressed in the following form:

$$\{\begin{array}{c}{\xi }_{\mathrm{dj}}=I_{\mathrm{dsj}\_\mathrm{ref}}-I_{\mathrm{dsj}}\hfill \\ \multicolumn{1}{c}{{\xi }_{\mathrm{qj}}=I_{\mathrm{qsj}\_\mathrm{ref}}-I_{\mathrm{qsj}}}\\ \multicolumn{1}{c}{{\xi }_{\mathrm{xj}}=I_{\mathrm{xsj}\_\mathrm{ref}}-I_{\mathrm{xsj}}}\\ \multicolumn{1}{c}{{\xi }_{\mathrm{yj}}=I_{\mathrm{ysj}\_\mathrm{ref}}-I_{\mathrm{ysj}}}\end{array}$$

(21)

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

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\xi }}_{\mathrm{\Omega }j}=\frac{d{\xi }_{\mathrm{\Omega }j}}{\mathrm{dt}}=\frac{1}{J_j}[-T_{\mathrm{tj}}+\frac{5P_j}{2}{\mathrm{\Psi }}_{\mathrm{fj}}.i_{\mathrm{qsj}}+\frac{5P_j}{2}(L_{\mathrm{dj}}-L_{\mathrm{qj}}).I_{\mathrm{dsj}}.I_{\mathrm{qsj}}+k_{\mathrm{fj}}.{\mathrm{\Omega }}_{\mathrm{mj}}]}\\ =\frac{1}{J_j}[-T_{\mathrm{tj}}+\frac{5P_j}{2}{\mathrm{\Psi }}_{\mathrm{fj}}.(I_{\mathrm{qsj}\_\mathrm{ref}}-{\xi }_{\mathrm{qj}})-\frac{5P_j}{2}(L_{\mathrm{dj}}-L_{\mathrm{qj}}).I_{\mathrm{qsj}}.{\xi }_{\mathrm{dj}}+k_{\mathrm{fj}}.{\mathrm{\Omega }}_{\mathrm{mj}}]\hfill \\ =\frac{1}{J_j}[-T_{\mathrm{tj}}+\frac{5P_j}{2}{\mathrm{\Psi }}_{\mathrm{fj}}\left[\frac{2}{5.P_j.{\mathrm{\Psi }}_{\mathrm{fj}}}[T_t-k_{\mathrm{fj}}.{\mathrm{\Omega }}_{\mathrm{mj}}-k_{\mathrm{\Omega }j}.J_j.{\xi }_{\mathrm{\Omega }j}]\right]-\frac{5P_j}{2}{\mathrm{\Psi }}_{\mathrm{fj}}.{\xi }_{\mathrm{qj}}-\frac{5P_j}{2}(L_{\mathrm{dj}}-L_{\mathrm{qj}}).I_{\mathrm{qsj}}.{\xi }_{\mathrm{dj}}+k_{\mathrm{fj}}.{\mathrm{\Omega }}_{\mathrm{mj}}]\hfill \\ =\frac{1}{J_j}[k_{\mathrm{\Omega }j}.J_j.{\xi }_{\mathrm{\Omega }j}-\frac{5.P_j}{2}{\mathrm{\Psi }}_{\mathrm{fj}}.{\xi }_{\mathrm{qj}}-\frac{5.P_j}{2}(L_{\mathrm{dj}}-L_{\mathrm{qj}})I_{\mathrm{qsj}}.{\xi }_{\mathrm{dj}}]\hfill \end{array}$$

(22)

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

$$\{\begin{array}{c}{\stackrel{·}{\xi }}_{\mathrm{dj}}={\stackrel{·}{I}}_{\mathrm{dsj}\_\mathrm{ref}}-{\stackrel{·}{I}}_{\mathrm{dsj}}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\xi }}_{\mathrm{qj}}={\stackrel{·}{I}}_{\mathrm{qsj}\_\mathrm{ref}}-{\stackrel{·}{I}}_{\mathrm{qsj}}}\\ \multicolumn{1}{c}{{\stackrel{·}{\xi }}_{\mathrm{xj}}={\stackrel{·}{I}}_{\mathrm{xsj}\_\mathrm{ref}}-{\stackrel{·}{I}}_{\mathrm{xsj}}}\\ \multicolumn{1}{c}{{\stackrel{·}{\xi }}_{\mathrm{yj}}={\stackrel{·}{I}}_{\mathrm{ysj}\_\mathrm{ref}}-{\stackrel{·}{I}}_{\mathrm{ysj}}}\end{array}$$

(23)

$${\stackrel{·}{\xi }}_{\mathrm{dj}}={\stackrel{·}{I}}_{\mathrm{dsj}\_\mathrm{ref}}-{\stackrel{·}{I}}_{\mathrm{dsj}}=\frac{R_{\mathrm{sj}}}{L_{\mathrm{dj}}}{I}_{\mathrm{dsj}}-\frac{P_j.L_{\mathrm{qj}}}{L_{\mathrm{dj}}}{I}_{\mathrm{qsj}}.{\mathrm{\Omega }}_{\mathrm{mj}}+\frac{1}{L_{\mathrm{dj}}}{V}_{\mathrm{dsj}}$$

(24)

$$\{\begin{array}{c}{\stackrel{·}{\xi }}_{\mathrm{qj}}={\stackrel{·}{I}}_{\mathrm{qs}{j}_{\mathrm{ref}}}-{\stackrel{·}{I}}_{\mathrm{qsj}}=\frac{2}{5.P_j.{\mathrm{\Psi }}_{\mathrm{fj}}}[-k_{\mathrm{fj}}.{\stackrel{·}{\mathrm{\Omega }}}_{\mathrm{mj}}-k_{\mathrm{\Omega }j}.J_j.{\stackrel{·}{\xi }}_{\mathrm{\Omega }j}]-(\frac{-R_{\mathrm{sj}}}{L_{\mathrm{qj}}}{I}_{\mathrm{qsj}}-\frac{P_j{L}_{\mathrm{dj}}}{L_{\mathrm{qj}}}{I}_{\mathrm{dj}}{\mathrm{\Omega }}_{\mathrm{mj}}-\frac{P_j{\mathrm{\Psi }}_{\mathrm{fj}}}{L_{\mathrm{qj}}}{\mathrm{\Omega }}_{\mathrm{mj}}+\frac{1}{L_{\mathrm{qj}}}{V}_{\mathrm{qsj}})\hfill \\ =\frac{2}{5.P_J.J_j{\mathrm{\Psi }}_{\mathrm{fj}}}\left((k_{\mathrm{\Omega }j}.J_j-k_{\mathrm{fj}})[T_{\mathrm{tj}}-\frac{5P_j}{2}(L_{\mathrm{dj}}-L_{\mathrm{qj}}).I_{\mathrm{dsj}}.I_{\mathrm{qsj}}-\frac{5P_j}{2}{\mathrm{\Psi }}_{\mathrm{fj}}.I_{\mathrm{qsj}}-k_{\mathrm{fj}}.{\mathrm{\Omega }}_{\mathrm{mj}}]\right)\hfill \\ +(\frac{R_{\mathrm{sj}}}{L_{\mathrm{qj}}}{I}_{\mathrm{qsj}}+\frac{P_j{L}_{\mathrm{dj}}}{L_{\mathrm{qj}}}{I}_{\mathrm{dj}}{\mathrm{\Omega }}_{\mathrm{mj}}+\frac{P_j{\mathrm{\Psi }}_{\mathrm{fj}}}{L_{\mathrm{qj}}}{\mathrm{\Omega }}_{\mathrm{mj}}-\frac{1}{L_{\mathrm{qj}}}{V}_{\mathrm{qsj}})\hfill \end{array}$$

(25)

$${\stackrel{·}{\xi }}_{\mathrm{xj}}={\stackrel{·}{I}}_{\mathrm{xsj}\_\mathrm{ref}}-{\stackrel{·}{I}}_{\mathrm{xsj}}=0-[-\frac{R_{\mathrm{sj}}}{L_{\mathrm{dj}}}{I}_{\mathrm{xsj}}+\frac{1}{L_{\mathrm{dj}}}{V}_{\mathrm{xsj}}]=\frac{R_{\mathrm{sj}}}{L_{\mathrm{dj}}}{I}_{\mathrm{xsj}}-\frac{1}{L_{\mathrm{dj}}}{V}_{\mathrm{xsj}}$$

(26)

$${\stackrel{·}{\xi }}_{\mathrm{yj}}={\stackrel{·}{I}}_{\mathrm{ysj}\_\mathrm{ref}}-{\stackrel{·}{I}}_{\mathrm{ysj}}=0-[-\frac{R_{\mathrm{sj}}}{L_{\mathrm{qj}}}{I}_{\mathrm{ysj}}+\frac{1}{L_{\mathrm{qj}}}{V}_{\mathrm{ysj}}]=\frac{R_{\mathrm{sj}}}{L_{\mathrm{qj}}}{I}_{\mathrm{ysj}}-\frac{1}{L_{\mathrm{qj}}}{V}_{\mathrm{ysj}}$$

(27)

An innovative Lyapunov function is suggested to ensure the stability of the whole system by employing the stator voltages as references while considering speed errors and stator current errors.

$$V_2=\frac{1}{2}({\xi }_{\mathrm{\Omega }j}^2+{\xi }_{\mathrm{dj}}^2+{\xi }_{\mathrm{qj}}^2+{\xi }_{\mathrm{xj}}^2+{\xi }_{\mathrm{yj}}^2)$$

(28)

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

$${\stackrel{·}{V}}_2=({\xi }_{\mathrm{\Omega }j}.{\stackrel{·}{\xi }}_{\mathrm{\Omega }j}+{\xi }_{\mathrm{dj}}.{\stackrel{·}{\xi }}_{\mathrm{dj}}+{\xi }_{\mathrm{qj}}.{\stackrel{·}{\xi }}_{\mathrm{qj}}+{\xi }_{\mathrm{xj}}.{\stackrel{·}{\xi }}_{\mathrm{xj}}+{\xi }_{\mathrm{yj}}.{\stackrel{·}{\xi }}_{\mathrm{yj}})$$

(29)

$$\begin{array}{l}{\stackrel{.}{V}}_2=-k_{\mathrm{\Omega }J}.{\xi }_{\mathrm{\Omega }J}^2-k_{dj}.{\xi }_{dj}^2-k_{qj}.{\xi }_{qj}^2-k_{xj}.{\xi }_{xj}^2-k_{yj}.{\xi }_{yj}^2\\ +{\xi }_{\mathrm{\Omega }j}[-\frac{5P_j}{2J_j}{\mathrm{\Psi }}_{fj}.{\xi }_{qj}-\frac{5P_j}{2J_j}(L_{dj}-L_{qj})I_{qsj}.{\xi }_{dj}]+{\xi }_{dj}\left[\frac{R_{sj}}{L_{dj}}{I}_{dsj}-\frac{P_j.L_{qj}}{L_{dj}}{I}_{qsj}.{\mathrm{\Omega }}_{mj}+\frac{1}{L_{dj}}{V}_{dsj}+k_{dj}.{\xi }_{dj}\right]\\ +{\xi }_{qj}[\frac{2}{5.P_J.J_j{\mathrm{\Psi }}_{fj}}(\left(k_{\mathrm{\Omega }j}.J_j-k_{fj}\right)[T_{tj}-\frac{5P_j}{2}\left(L_{dj}-L_{qj}\right).I_{dsj}.I_{qsj}-\frac{5P_j}{2}{\mathrm{\Psi }}_{fj}.I_{qsj}-k_{fj}.{\mathrm{\Omega }}_{mj}])\\ +\frac{R_{sj}}{L_{qj}}{I}_{qsj}+\frac{P_j{L}_{dj}}{L_{qj}}{I}_{dj}{\mathrm{\Omega }}_{mj}+\frac{P_j{\mathrm{\Psi }}_{fj}}{L_{qj}}{\mathrm{\Omega }}_{mj}-\frac{1}{L_{qj}}{V}_{qsj}+k_{qj}.{\xi }_{qj}]+{\xi }_{xj}[\frac{R_{sj}}{L_{dj}}{I}_{xsj}-\frac{1}{L_{dj}}{V}_{xsj}+k_{xj}.{\xi }_{xj}]\\ +{\xi }_{qj}\left[\frac{R_{sj}}{L_{qj}}{I}_{ysj}-\frac{1}{L_{qj}}{V}_{ysj}+k_{yj}.{\xi }_{yj}\right]\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2=-k_{\mathrm{\Omega }J}.{\xi }_{\mathrm{\Omega }J}^2-k_{\mathrm{dj}}.{\xi }_{\mathrm{dj}}^2-k_{\mathrm{qj}}.{\xi }_{\mathrm{qj}}^2-k_{\mathrm{xj}}.{\xi }_{\mathrm{xj}}^2-k_{\mathrm{yj}}.{\xi }_{\mathrm{yj}}^2}\\ +{\xi }_{\mathrm{dj}}[\frac{R_{\mathrm{sj}}}{L_{\mathrm{dj}}}{I}_{\mathrm{dsj}}-\frac{P_j.L_{\mathrm{qj}}}{L_{\mathrm{dj}}}{I}_{\mathrm{qsj}}.{\mathrm{\Omega }}_{\mathrm{mj}}+\frac{1}{L_{\mathrm{dj}}}{V}_{\mathrm{dsj}}+k_{\mathrm{dj}}.{\xi }_{\mathrm{dj}}-\frac{5P_j}{2J_j}(L_{\mathrm{dj}}-L_{\mathrm{qj}})I_{\mathrm{qsj}}.{\xi }_{\mathrm{\Omega }j}]\hfill \\ +{\xi }_{\mathrm{qj}}[\frac{2}{5.P_J.J_j{\mathrm{\Psi }}_{\mathrm{fj}}}\left((k_{\mathrm{\Omega }j}.J_j-k_{\mathrm{fj}})[T_{\mathrm{tj}}-\frac{5P_j}{2}(L_{\mathrm{dj}}-L_{\mathrm{qj}}).I_{\mathrm{dsj}}.I_{\mathrm{qsj}}-\frac{5P_j}{2}{\mathrm{\Psi }}_{\mathrm{fj}}.I_{\mathrm{qsj}}-k_{\mathrm{fj}}.{\mathrm{\Omega }}_{\mathrm{mj}}]\right)\hfill \\ +\frac{R_{\mathrm{sj}}}{L_{\mathrm{qj}}}{I}_{\mathrm{qsj}}+\frac{P_j{L}_{\mathrm{dj}}}{L_{\mathrm{qj}}}{I}_{\mathrm{dj}}{\mathrm{\Omega }}_{\mathrm{mj}}+\frac{P_j{\mathrm{\Psi }}_{\mathrm{fj}}}{L_{\mathrm{qj}}}{\mathrm{\Omega }}_{\mathrm{mj}}-\frac{1}{L_{\mathrm{qj}}}{V}_{\mathrm{qsj}}+k_{\mathrm{qj}}.{\xi }_{\mathrm{qj}}-\frac{5P_j}{2J_j}{\mathrm{\Psi }}_{\mathrm{fj}}.{\xi }_{\mathrm{qj}}]+{\xi }_{\mathrm{xj}}[\frac{R_{\mathrm{sj}}}{L_{\mathrm{dj}}}{I}_{\mathrm{xsj}}-\frac{1}{L_{\mathrm{dj}}}{V}_{\mathrm{xsj}}+k_{\mathrm{xj}}.{\xi }_{\mathrm{xj}}]\hfill \\ +{\xi }_{\mathrm{qj}}[\frac{R_{\mathrm{sj}}}{L_{\mathrm{qj}}}{I}_{\mathrm{ysj}}-\frac{1}{L_{\mathrm{qj}}}{V}_{\mathrm{ysj}}+k_{\mathrm{yj}}.{\xi }_{\mathrm{yj}}]\hfill \end{array}$$

(30)

$k_{\mathrm{dj}},k_{\mathrm{qj}},k_{\mathrm{xj}},\mathrm{and}{k}_{\mathrm{yj}}$ are constants with positive values.

To ensure the negativity of the derivative of the Lyapunov function and achieve system stability, the stator reference voltages are applied in the following manner:

$$V_{\mathrm{dsj}\_\mathrm{ref}}=R_{\mathrm{sj}}.I_{\mathrm{dsj}}-P_j.L_{\mathrm{dj}}.{\mathrm{\Omega }}_{\mathrm{mj}}.I_{\mathrm{qsj}}-\frac{5P_j.L_{\mathrm{dj}}}{2J_j}(L_{\mathrm{dj}}-L_{\mathrm{qj}})I_{\mathrm{qsj}}.{\xi }_{\mathrm{\Omega }j}+k_{\mathrm{dj}}.L_{\mathrm{dj}}.{\xi }_{\mathrm{dj}}$$

(31)

$$\begin{array}{c}\multicolumn{1}{c}{V_{\mathrm{qsj}\_\mathrm{ref}}=\frac{2}{5.P_J.J_j{\mathrm{\Psi }}_{\mathrm{fj}}}\left((k_{\mathrm{\Omega }j}.J_j-k_{\mathrm{fj}})[T_{\mathrm{tj}}-\frac{5P_j}{2}(L_{\mathrm{dj}}-L_{\mathrm{qj}}).I_{\mathrm{dsj}}.I_{\mathrm{qsj}}-\frac{5P_j}{2}{\mathrm{\Psi }}_{\mathrm{fj}}.I_{\mathrm{qsj}}-k_{\mathrm{fj}}.{\mathrm{\Omega }}_{\mathrm{mj}}]\right)}\\ +\frac{R_{\mathrm{sj}}}{L_{\mathrm{qj}}}{I}_{\mathrm{qsj}}+\frac{P_j{L}_{\mathrm{dj}}}{L_{\mathrm{qj}}}{I}_{\mathrm{dj}}{\mathrm{\Omega }}_{\mathrm{mj}}+\frac{P_j{\mathrm{\Psi }}_{\mathrm{fj}}}{L_{\mathrm{qj}}}{\mathrm{\Omega }}_{\mathrm{mj}}-\frac{1}{L_{\mathrm{qj}}}{V}_{\mathrm{qsj}}+k_{\mathrm{qj}}.{\xi }_{\mathrm{qj}}\hfill \end{array}$$

(32)

$$V_{\mathrm{xsj}\_\mathrm{ref}}=R_{\mathrm{sj}}.I_{\mathrm{xsj}}+k_{\mathrm{xj}}.L_{\mathrm{dj}}.{\xi }_{\mathrm{xj}}$$

(33)

$$V_{\mathrm{ysj}\_\mathrm{ref}}=R_{\mathrm{sj}}.I_{\mathrm{ysj}}+k_{\mathrm{yj}}.L_{\mathrm{yqj}}.{\xi }_{\mathrm{yj}}$$

(34)

In simple terms, the behavior of the Lyapunov function may be summarized in the following way:

$${\stackrel{·}{V}}_2=-k_{\mathrm{\Omega }J}.{\xi }_{\mathrm{\Omega }J}^2-k_{\mathrm{dj}}.{\xi }_{\mathrm{dj}}^2-k_{\mathrm{qj}}.{\xi }_{\mathrm{qj}}^2-k_{\mathrm{xj}}.{\xi }_{\mathrm{xj}}^2-k_{\mathrm{yj}}.{\xi }_{\mathrm{yj}}^2\le 0$$

(35)

ANFIS technology

Description

ANFIS is an acronym for Adaptive Neuro-Fuzzy Inference System. This is a hybrid machine learning approach that combines the capabilities of neural networks with fuzzy logic systems. ANFIS is mostly used for modeling and control applications across many domains, including nonlinear control. The system employs a fuzzy inference technique to establish input-output correlations and adjusts the system’s parameters via the use of a neural network^48,49

Figure 6 explains the ANFIS design. Regardless of the neural network component, it is identical to the fuzzy inference system. The ANFIS structure is comprised of five levels, which are delineated in the following order^50,51:

• The first layer, known as the input layer, consists of input variables. Each node inside this layer analyzes triangular, trapezoidal or Gaussian membership functions. The most optimal function is determined by the one that yields the lowest training error. The input variables are then sent to the subsequent layer.

• Layer 2, known as the minimum selection layer, is in charge of determining the minimum value among the inputs provided. It performs the task of choosing the input with the minimum value among the available inputs.

• Layer 3, known as the normalization layer, is responsible for normalizing every input concerning the other inputs. The output of node $i^{\mathrm{th}}$ in this layer is computed by dividing the input $i^{\mathrm{th}}$ by the aggregate of all the other inputs. The normalization procedure guarantees that the contribution of each input is suitably adjusted for the overall input values.

• In Layer 4, the value produced by the $i^{\mathrm{th}}$ node in this layer can be computed through a linear function that considers both the output of the $i^{\mathrm{th}}$ node in the third layer and the input signals of the ANFIS.

• Layer 5, designated as the summation layer, performs the task of aggregating all the incoming signals. The ANFIS architecture can be tuned to automatic tuning through the utilization of a least-square estimation method for output membership functions and a backpropagation algorithm for both output and input membership functions.

Figure 6.

The ANFIS’s overall architecture.

ANFIS training

The Matlab/Simulink model was used to train ANFIS based on the nonlinear control technique (BSC) to control the WPGS to improve the system performance, improve the quality of the produced power, and reduce tracking errors. The stages of training this controller can be described through the following steps^48,52–54:

The first step in the ANFIS training process contains the collection of training data, which is important for determining the inputs and outputs of the system. In this stage, the inputs are defined as speed error $\left({\xi }_{\mathrm{\Omega }j}\right)$ and current error $({\xi }_{\mathrm{dj}},{\xi }_{\mathrm{qj}},{\xi }_{\mathrm{xj}},{\xi }_{\mathrm{yj}})$ , representing the difference between the actual values and the desired values. On the other hand, the outputs are composed of the reference value of current $I_{\mathrm{qsj}\_\mathrm{ref}}$ (output of the speed controller) and the control signals (reference voltages ${\mathrm{V}}_{\mathrm{dsj}\_\mathrm{ref}}$ , ${\mathrm{V}}_{\mathrm{qsj}\_\mathrm{ref}},{\mathrm{V}}_{\mathrm{xsj}\_\mathrm{ref}}$ , and ${\mathrm{V}}_{\mathrm{ysj}\_\mathrm{ref}}$ ,). Properly gathering and defining this training data is essential for training the ANFIS controller effectively.

The design phase of the Fuzzy Inference System (FIS): During this stage, the fuzzy sets and membership functions for every variable are established. This involves setting the appropriate membership functions and defining their forms based on the features and characteristics of each variable. The determination of membership function shapes is influenced by various factors, including the data distribution, the desired level of granularity, and the linguistic understanding of the variables.

The Generate phase of Fuzzy Rules: To generate accurate and meaningful fuzzy inference results, it is necessary to establish a set of fuzzy rules that describe the relationship between the inputs and outputs. This is accomplished by analyzing the patterns and relationships within the data. The algorithm identifies clusters and generates fuzzy rules that capture the underlying structure and behavior of the system. The (trimf) membership function was chosen for the ANFIS model due to its superior performance in comparison to other alternatives (Table 2), which is a critical point to emphasize. Additionally, the optimal number of membership functions is determined by selecting a specific number of MFs (Figure 7).

Table 2.

Available MFs.

Function name	Function symbol
Triangular mf	trimf
Trapezoidal mf	trapmf
Generalized bell curve mf	gbellmf
Gaussian curve mf	gaussmf
Two-sided Gaussian curve mf	Gauss2mf
PI-shaped curve mf	pimf
Difference of two sigmoid mf	dsigmf
Product of two sigmoid mf	psigmf

Figure 7.

ANFIS model training interface.

The training phase of the ANFIS model: During this stage, specific algorithms, such as the backpropagation algorithm and the hybrid learning algorithm are utilized to repetitively tune the parameters of the model. This process aims to reduce the disparity between the anticipated outputs produced by the ANFIS model and the actual outputs acquired from the training data. By fine-tuning the model’s parameters using these training algorithms, the ANFIS model becomes more precise and capable of effectively capturing the essential patterns and relationships within the data. The objective is to achieve the optimum possible level of concurrence between the predicted outputs and the real outputs.

GSC control

BSC technology

The grid-side converter (GSC) controls the distribution network’s electrical energy flow. The DC bus voltage and power factor stability are the goals of this regulation.^45,47

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

$$\{\begin{array}{c}\frac{d{I}_{\mathrm{gd}}}{\mathrm{dt}}=\frac{1}{L_g}{V}_{\mathrm{id}}-\frac{R_g}{L_g}{I}_{\mathrm{gd}}+W_g{I}_{\mathrm{gq}}-\frac{1}{L_g}{V}_{\mathrm{gd}}\hfill \\ \frac{d{I}_{\mathrm{gq}}}{\mathrm{dt}}=\frac{1}{L_g}{V}_{\mathrm{iq}}-\frac{R_g}{L_g}{I}_{\mathrm{gq}}-W_g{I}_{\mathrm{gd}}-\frac{1}{L_g}{V}_{\mathrm{gq}}\hfill \\ \frac{{\mathrm{dV}}_{\mathrm{dc}}^2}{\mathrm{dt}}=\frac{2}{C}(-\frac{3}{2}{V}_{\mathrm{gd}}.I_{\mathrm{gd}}+P_{\mathrm{gen}})\hfill \end{array}$$

(36)

The implementation of backstepping control on the grid side may be broken down into two consecutive steps. The first stage offers instructions for the subsequent one. Furthermore, the state vectors and control vectors are chosen so that $\left[x\right]={\left[\begin{array}{ccc}{I}_{\mathrm{gd}}\hfill & I_{\mathrm{gq}}\hfill & V_{\mathrm{dc}}^2\hfill \end{array}\right]}^T={\left[\begin{array}{ccc}{x}_1\hfill & x_2\hfill & x_3\hfill \end{array}\right]}^T$ specifically refers to the state vectors, whereas $\left[u\right]={\left[\begin{array}{cc}{V}_{\mathrm{id}}\hfill & V_{\mathrm{iq}}\hfill \end{array}\right]}^T$ indicates the control vectors.

• Step 1:

The tracking error’s state variable $x_3$ can be expressed using the below-mentioned equation:

$$e_{\mathrm{dc}}=V_{\mathrm{dc}\_\mathrm{ref}}^2-V_{\mathrm{dc}}^2$$

(37)

Due to the fact that our objective necessitates that the error converges to zero, we are able to fulfill it by selecting $I_{\mathrm{gd}}$ as a virtual control variable in the equation presented above and utilizing it to control the error $e_{\mathrm{dc}}$ . By using the Lyapunov function, which may be defined as:

$$V_1=\frac{1}{2}{e}_{\mathrm{dc}}^2$$

(38)

The Lyapunov function’s time derivative is given as:

$${\stackrel{·}{V}}_1=e_{\mathrm{dc}}.{\stackrel{·}{e}}_{\mathrm{dc}}=e_{\mathrm{dc}}({\stackrel{·}{V}}_{\mathrm{dc}}^2-\left[\frac{2}{C}(-\frac{3}{2}{V}_{\mathrm{gd}}.I_{\mathrm{gd}}+P_{\mathrm{gen}})\right])$$

(39)

As part of the BSC design, the virtual input, referred to as the stabilizing function, is established by selecting the grid’s current $I_{\mathrm{gd}}$ components.

$$I_{\mathrm{gd}\_\mathrm{ref}}=\frac{1}{V_{\mathrm{gd}}}(\frac{2}{3}{P}_{\mathrm{gen}}-\frac{C}{3}[{\stackrel{·}{V}}_{\mathrm{dc}}^2+k_{\mathrm{dc}}.e_{\mathrm{dc}}])$$

(40)

By substituting equation (40) into the derivative expression ${\stackrel{·}{V}}_1$ , we get the following outcome:

$${\stackrel{·}{V}}_1=-k_{\mathrm{dc}}.e_{\mathrm{dc}}^2\le 0$$

(41)

• Step 2:

The system’s virtual inputs are $I_{\mathrm{gd}}\mathrm{and}{I}_{\mathrm{gq}}$ , while the genuine control inputs, $V_{\mathrm{gd}}\mathrm{and}{V}_{\mathrm{gq}}$ , are determined through calculations based on the differences in the stator currents.

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

$$\{\begin{array}{c}{e}_d=I_{\mathrm{gd}\_\mathrm{ref}}-I_{\mathrm{gd}}\hfill \\ \multicolumn{1}{c}{e_q=I_{\mathrm{gq}\_\mathrm{ref}}-I_{\mathrm{gq}}}\end{array}$$

(42)

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

$${\stackrel{·}{e}}_{\mathrm{dc}}=-k_{\mathrm{dc}}.e_{\mathrm{dc}}^2-\frac{3}{C}{V}_{\mathrm{gd}}.e_d$$

(43)

Moreover, the dynamic equation for the error signals $e_d\mathrm{and}{e}_q$ may be determined in the following manner:

$$\{\begin{array}{c}{\stackrel{·}{e}}_d={\stackrel{·}{I}}_{\mathrm{gd}\_\mathrm{ref}}-(\frac{1}{L_g}{V}_{\mathrm{id}}-\frac{R_g}{L_g}{I}_{\mathrm{gd}}+W_g{I}_{\mathrm{gq}}-\frac{1}{L_g}{V}_{\mathrm{gd}})\hfill \\ {\stackrel{·}{e}}_q={\stackrel{·}{I}}_{\mathrm{gq}\_\mathrm{ref}}-(\frac{1}{L_g}{V}_{\mathrm{iq}}-\frac{R_g}{L_g}{I}_{\mathrm{gq}}-W_g{I}_{\mathrm{gd}}-\frac{1}{L_g}{V}_{\mathrm{gq}})\hfill \end{array}$$

(44)

An innovative Lyapunov function is suggested to ensure the stability of the whole system by employing the voltages $V_{\mathrm{id}}\mathrm{and}{V}_{\mathrm{iq}}$ as references while considering DC-bus voltage error and grid current errors.

$$V_2=\frac{1}{2}(e_{\mathrm{dc}}^2+e_d^2+e_q^2)$$

(45)

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

$${\stackrel{·}{V}}_2=e_{\mathrm{dc}}.{\stackrel{·}{e}}_{\mathrm{dc}}+e_d.{\stackrel{·}{e}}_d+e_q.{\stackrel{·}{e}}_q$$

(46)

So:

$$\begin{array}{c}{\stackrel{·}{V}}_2=-k_{\mathrm{dc}}.e_{\mathrm{dc}}^2-k_d.e_d^2-k_q.e_q^2+e_d\hfill \\ [{\stackrel{·}{I}}_{\mathrm{gd}\_\mathrm{ref}}-(\frac{1}{L_g}{V}_{\mathrm{id}}-\frac{R_g}{L_g}{I}_{\mathrm{gd}}+W_g{I}_{\mathrm{gq}}-\frac{1}{L_g}{V}_{\mathrm{gd}})+k_d.e_d-\frac{3}{C}{V}_{\mathrm{gd}}.e_{\mathrm{dc}}]\hfill \\ +e_q[{\stackrel{·}{I}}_{\mathrm{gq}\_\mathrm{ref}}-(\frac{1}{L_g}{V}_{\mathrm{iq}}-\frac{R_g}{L_g}{I}_{\mathrm{gq}}-W_g{I}_{\mathrm{gd}}-\frac{1}{L_g}{V}_{\mathrm{gq}})+k_q.e_q]\hfill \end{array}$$

(47)

$k_d\mathrm{and}{k}_q$ are constants with positive values.

To ensure the negativity of the derivative of the Lyapunov function and achieve system stability, the stator reference voltages are applied in the following manner:

$$V_{\mathrm{id}\_\mathrm{ref}}=R_g.I_{\mathrm{gd}}-W_g.L_g.I_{\mathrm{qg}}+V_{\mathrm{gd}}-\frac{3}{C}{V}_{\mathrm{gd}}.L_g.e_{\mathrm{dc}}+k_d.L_g.e_d$$

(48)

$$V_{\mathrm{iq}\_\mathrm{ref}}=R_g.I_{\mathrm{gq}}+W_g.L_g.I_{\mathrm{dg}}+V_{\mathrm{gq}}+k_q.L_g.e_q$$

(49)

Ultimately, the behavior of the Lyapunov function might be described as outlined below:

$${\stackrel{·}{V}}_2=-k_{\mathrm{dc}}.e_{\mathrm{dc}}^2-k_d.e_d^2-k_q.e_q^2\le 0$$

(50)

ANFIS control technique

The process of training the ANFIS for grid-side converter control based on BSC involved following similar steps to those used for generator and machine-side converter control. Nevertheless, there are differences in terms of the inputs and outputs of the system.

Three variables were specified for the inputs. The first input indicates the error in the DC-bus voltage ( $e_{\mathrm{dc}}$ ), indicating the difference between the actual DC-bus voltage and the required reference value. The remaining two inputs represent the network current errors ( $e_d\mathrm{and}{e}_q)$ , which signify the discrepancies between the actual grid currents and their reference values. These inputs are essential for detecting and analyzing the differences and variations inside the system.

Regarding the outputs, there are three outputs. The first output is the reference value of the current $I_{\mathrm{gd}\_\mathrm{ref}}$ , which represents the output of the DC-link voltage controller. The two other outputs are the control signals, which are the reference voltages $V_{\mathrm{id}\_\mathrm{ref}}$ and $V_{\mathrm{iq}\_\mathrm{ref}}$ .

Using these inputs and outputs to train the ANFIS model based on the BSC method, the system can learn and adapt to reduce the differences between what it should do and what it does. The ANFIS model can make accurate control signals and reference values thanks to this training process. This makes sure that the grid-side converter is controlled efficiently and reliably.

Results and discussion

To assess the efficaciousness of control strategies, as explained in this scholarly paper, and to demonstrate their efficiency, the described approaches were tested through experiments under various wind speed profiles, which can be seen in Figures 5 and 8. Simulation experiments were conducted using the MATLAB/Simulink program. The numerical simulations utilized a reactive power reference of $Q_g=0$ to regulate the GSC. For the MSC control, the currents $I_{\mathrm{dsj}},I_{\mathrm{xsj}},\mathrm{and}{I}_{\mathrm{ysj}}$ were set to zero. In addition, the DC-bus’s reference voltage is set at 2500V; while the grid frequency stays constant at 50 Hz. Table 3 presents the complete set of specifications for the wind turbine, including the 5-phase PMSGs, DC-link, and grid.

Figure 8.

The general GSC control structure.

Table 3.

The wind turbine, PMSGs, Grid, and DC-link 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	$V_{\mathrm{dc}}$	2500 V
Stator resistance	$R_s$	0.821 mΩ	Specific density of air	$\rho$	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	${\lambda }_{\mathrm{opt}}$	8.1	Grid frequency	$F_g$	50 Hz
Pole number	P	26	Optimal power coefficient	$C_{p-\max }$	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	${\mathrm{\Psi }}_f$	13.0282 Wb				Grid voltage	$V_g$	690 V

MSC outcomes

The performance assessment of the generation system is carried out under different wind velocities, with an average velocity of 10.5 m/s, as depicted in Figure 9(a). The study is conducted at a rotating angle of 0°. The utilization of the ANFIS method produces virtually ideal values with a minimal error rate for both TSR (λ) and $C_p$ , measuring 8.1 and 0.48, respectively, as shown in Figure 9(b) and (c). In addition, Figure 9(d) illustrates the mechanical power produced by the generators, whereas Figure 9(e) presents a comparative assessment of the total torque of the generators and the ideal torque. Remarkably, a strong match is observed between them, indicating good tracking of the ANFIS to the reference value.

Figure 9.

Simulation outcomes of the two PMSG wind systems controlled by ANFIS-BSC: (a) wind speed profile, (b) power coefficients, (c) tip speed ratio, (d) mechanical powers, (e) total electromagnetic torque.

Figure 10 displays the mechanical speed of the generators in comparison to the reference speeds. Both control approaches demonstrate strong performance in accurately following the reference values. The superiority of the ANFIS method over BSC is evident, especially in closer observation where ANFIS shows a stronger adherence to the desired reference values.

Figure 10.

Comparative analysis between the ANFIS and BSC controllers: (a) the speed of the first generator and (b) the speed of the second generator.

The following Tables 4 to 7 summarizes the highlights and key findings that are demonstrated in Figure 10.

Table 4.

Overshoot and response time of the speed generators.

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

Table 5.

Overshoot and response time of the DC-link voltage.

Controller	Overshoot ratio (%)	Response time (ms)
BSC	16.6	85
ANFIS	13.6	78

Table 6.

Active/reactive power ripples.

Controller	Active power ripples (W)	Reactive power ripples (VAR)
BSC	360,000	42,500
ANFIS	250,000	38,500
Reduction ratio	30.50%	9,42%

Table 7.

Efficiency, grid power factor, and grid current THD.

Controller	Efficiency (%)	Grid current THD (%)	Power factor	Grid current frequency (Hz)
BSC	96	1.89	1	50
ANFIS	96.5	1.05	1	50

GSC results

Figure 11 illustrates a comparative examination of the outcomes on the grid side, specifically comparing the ANFIS and the BSC controller. The key findings of this study include an analysis of the DC-bus voltage, injected powers, power factor, grid frequency, a detailed examination of grid voltage and current waveforms, a closer examination of single-phase grid current, and an evaluation of the THD ratio of grid current. Figure 11(a) and (b) demonstrate that the DC-link voltage stays consistent, irrespective of variations in wind speed. The ANFIS has a clear advantage over BSC in terms of both the rate of overshoot and the time it takes to respond. The ANFIS technique exhibits improved performance by decreasing overshooting and delivering quicker response times in comparison to the BSC approach. Figure 11(c) and (d) illustrate the active and reactive power that is being injected into the network, respectively. The active power exhibits responsiveness to changes in wind speed while maintaining accurate alignment with the reference value. Nevertheless, it is important to mention that the ANFIS demonstrates a somewhat higher level of effectiveness in reducing ripples when compared to the BSC technique. In contrast, the reactive power maintains close adherence to its reference value of zero in order to attain a power factor of unity, as seen by Figure 11(e) and (g). Figure 11(i) displays the THD ratio, demonstrating that both control strategies produce excellent outcomes. However, ANFIS exhibits higher performance compared to the BSC.

Figure 11.

Comparative study of GS results between the ANFIS and BSC: (a, b) DC-bus voltage; (c) active power injected into the grid; (d) grid reactive power; (e) power factor; (f) grid frequency; (g) closer examination of grid current and voltage; (h) closer observation of grid current; and (i) the overall harmonic distortion of grid current.

The following Tables 4 to 6 summarizes the highlights and key findings that are demonstrated in Figure 11.

To evaluate the efficiency of the control systems under investigation, the theoretical output power is compared to the actual power being injected into the network (Figure 12), as defined in the following:

$${\eta }_{\mathrm{sys}}=\frac{{\int }_0^t{P}_g}{{\int }_0^t{P}_{\mathrm{th}}}\times 100\mathrm{\%}$$

(51)

Where: ${\eta }_{\mathrm{sys}}$ is the system’s efficiency $P_g$ is the power transmitted to the grid, and $P_{\mathrm{th}}$ is the extracted power.

Figure 12.

Overall efficiency of the system.

Conclusion

This article discusses the implementation of BSC and ANFIS control technologies for a grid-connected wind system. This wind system is based on two five-phase PMSGs that are controlled by a FSR. The ANFIS control was trained using BSC, which is based on Lyapunov theory, to guarantee system stability. This control method is well-known for its intrinsic benefits, including simplicity, high accuracy, and stability. Comprehensive simulations are conducted to evaluate the efficaciousness of the suggested control techniques and compare their outcomes with each other.

The outcomes of the simulations displayed that both techniques achieved separate control between the PMSGs. They also showed several enhancements in both the rapid responses and the steady-state effectiveness. These improvements encompass rapid response, precision, 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.