Control Design of Fractional Multivariable Grey Model-Based Fast Terminal Attractor for High Efficiency Pure Sine Wave Inverters in Electric Vehicles

Abstract

In this paper, a fast and efficient control method is proposed for a pure sine wave inverter used in an electric vehicle system, which can provide better performance under transient and steady-state conditions. The proposed control technique consists of a fast terminal attractor (FTA) and a fractional multivariable grey model (FMGM). The FTA with finite time convergence offers a faster convergence rate of the system state and a singularity-free solution. However, if the uncertain system boundaries are overestimated or underestimated, chatter/steady-state errors can occur during the FTA, which can lead to significant harmonic distortion at the output of the pure sine wave inverter. A computationally efficient FMGM is incorporated into the FTA to solve the chatter/steady-state error problem when an uncertain estimate of the system boundary cannot be satisfied. Simulation results show that the proposed control technique exhibits low total harmonic distortion. Experimental results of a prototype pure sine wave inverter are presented to support the results of the simulation and mathematical analysis. Since the proposed pure sine wave inverter outperforms the classical TA (terminal attractor)-controlled pure sine wave inverter in terms of convergence speed, computational efficiency, and harmonic distortion elimination, this paper will serve as a useful reference for electric vehicle systems.

1. Introduction

Pure sine wave inverters are not only receiving increasing attention but are also widely used in electric vehicles [1,2]. The requirements for high-performance pure sine wave inverter systems are generally based on the following standards: (1) Linear or nonlinear loads (especially rectifier loads) with low total harmonic distortion (THD) output voltage waveforms. (2) Rapid instantaneous response during sudden load changes. (3) The steady-state error should be limited to the smallest possible range. In order to meet these standards, several existing control techniques have been considered in the literature, such as deadbeat control, fast Fourier transform technique, and iterative control. Deadbeat control technology with fast dynamic response can be used; however, deadbeat control technology depends on the accuracy of the parameters [3]. Both fast Fourier transform (FFT) and iterative control can overcome the effects of uncertain internal parameters and external disturbances. However, these control techniques are either difficult to implement or computationally complex [4,5]. Sliding mode control (SMC) has attracted attention due to its insensitivity to parameter changes and suppression of external interference. The control of pure sine wave inverters generally adopts SMC. A fixed switching frequency sliding mode is suggested for single-phase unipolar inverters, employing a traditional sliding surface that causes large voltage sag when faced with step load changing [6]. To address the incomplete system dynamics of a grid-connected inverter, a sliding surface based on multi-resonance is built. While this approach improves both steady-state and transient performance, it requires considerable calculation time [7]. The enhanced sliding mode controller capable of mitigating uncertain disturbances has been developed for voltage regulation in inverter-based islanded microgrids, but it comes with complicated designs [8]. An integral sliding mode approach has been suggested for the robust voltage tracking control of three-phase inverters. While this method can effectively reduce steady-state errors, it is important to note that chattering may still arise unless a smoothing function is incorporated [9]. The aforementioned SMC methods are subject to challenges related to long-term convergence.

In recent years, the so-called fast terminal attractor (FTA) has been realized by a series of interesting SMC controllers, and it has been widely used in various fields [10,11]. The FTA can be driven to converge to the equilibrium state of the system in a finite time, while still maintaining the robustness of classical SMC. This type of sliding mode control significantly enhances both the dynamic and steady-state responses of the controlled system. However, if the boundary is inaccurately estimated due to significant uncertainties, it can lead to chattering or steady-state errors. The chattering is an undesirable phenomenon characterized by high-frequency oscillations. This sinuous movement along the sliding surface results in poor control accuracy, and excessive thermal dissipation in the power electronics transistors. Additionally, the chattering can cause highly distorted outputs from the pure sine wave inverter and decreased efficiency, potentially triggering unmodeled high-frequency dynamics that destabilize the system. Prior investigations have examined the utilization of high gain feedback to mitigate interruptions arising from uncertain bounds. Such a high gain value within the controlled system facilitates ease of execution; however, it also results in steady-state inaccuracies [12]. Numerous efforts have been made to develop adaptive and observer techniques for evaluating the level of uncertainty within systems, particularly in the context of power converter systems. Although these methodologies have successfully diminished the extent of chattering, the conditions for long-term convergence of the system state have become increasingly significant [13,14]. The predictive methodology has the potential to improve the chattering phenomenon; however, it is important to note that this approach is constrained by the necessity for a considerable amount of execution time [15]. The grey prediction was first proposed by Dr. Deng in 1982 and has been successfully applied in many engineering fields to effectively solve the prediction problem of uncertain systems. For dynamic systems, the grey prediction technique is used to portray and analyze the future trend of sequence numbers based on past and present data. The grey model can be constructed from the output signals of the system and a small amount of sampling data; however, it reduces the prediction accuracy on highly volatile data [16,17]. To improve the prediction accuracy, this paper introduces the fractional multivariable grey model (FMGM), which provides a more accurate prediction [18,19]. Therefore, in this paper, it will be a good idea to combine the FTA with the FMGM compensator for pure sine wave inverters. It can be seen that the proposed control technique produces a low THD, transient fast closed-loop pure sine wave inverter. A comparison of the proposed control technique with some other methods is presented in

Table 1. Comparison of the proposed control technique with some other methods.

Methods	Characteristics	Advantages	Disadvantages
Proposed control technique	Strong insensitivity	Finite state convergence time, Fast calculation, and no singularities	Slightly higher hardware realization cost
Deadbeat control technology [3]	Constant switching frequency	Fast dynamic response, and short settling time.	Dependent on the accuracy of the parameters
Fast fourier transform analysis [4]	Overcoming disturbances effectively	Processing large amounts of data	Computational complexity
Iterative learning control [5]	Attenuating repeating disturbances	Improvement of tracking accuracy	Long time calculation

.

The research purpose and contribution delineated in this paper can be encapsulated as follows: (i) The proposed FTA facilitates a finite-time, singularity-free convergence to equilibrium during the sliding phase while enhancing the speed at which the state behavior approaches the sliding surface in the reaching phase. (ii) The FMGM demonstrates improved accuracy in predicting state points, even in the presence of system uncertainties, effectively addressing chatter and steady-state errors. This leads to a rapid and stable convergence of state trajectories to the equilibrium point. (iii) The proposed pure sine wave inverter is capable of delivering quick dynamic responses and stable steady-state performance across a range of load conditions. The structure of the paper is delineated as follows: Section 1 examines the limitations associated with classical sliding mode control and conventional grey prediction, offering recommendations aimed at enhancing the performance of the inverter. Section 2 provides a comprehensive analysis of the dynamic modeling of pure sine wave inverters, while Section 3 details the proposed control technique design. In Section 4, both simulation and experimental results are presented to validate the efficacy of the proposed control technique. Section 5 presents a comprehensive discussion and analysis. Section 6 wraps up the paper with concluding remarks.

2. System Modeling

As illustrated in Figure 1, the pure sine wave inverter utilizes a bridge configuration made up of four transistor switches connected by inductive-capacitive (LC) low-pass filters, allowing it to produce a regulated alternating current (AC) voltage. Generally, the switching frequency of the pure sine wave inverter is significantly higher than the fundamental frequency of the AC output, which means that the dynamic characteristics of the inverter can often be disregarded, treating it as a simple proportional gain. Based on this assumption, the inverter’s dynamic behavior can be effectively modeled using the LC low-pass filter associated with a load. The load may consist of either a resistor-based linear load or a dynamically changing step load, while the inductive-capacitive parameters are subject to change. This variability is essential for evaluating the performance response of the pure sinusoidal inverter.

Using Kirchhoff’s voltage law (KVL) and Kirchhoff’s current law (KCL), and defining $e_1=v_o-v_{ref}$ and ${\dot{e}}_1=e_2={\dot{v}}_o-{\dot{v}}_{ref}$, the error dynamic state space equation can be obtained as

$$\{\begin{array}{l}{\dot{e}}_1=e_2\\ {\dot{e}}_2=a_1{e}_1+a_2{e}_2+bu+f\end{array},$$

(1)

where a₁ = −1/LC, a₂ = −1/RC, $b=k_{pwm}/LC$, $k_{pwm}$ is the equivalent gain of the inverter and $f=-a_1{v}_{ref}-a_2{\dot{v}}_{ref}-{\ddot{v}}_{ref}$ denotes the system uncertainty. By using the proposed control technique, the output voltage $v_o$ of the pure sine wave inverter can be forced to follow the sinusoidal reference voltage $v_{ref}$. From (1), it can be seen that the control signal $u$ must be designed to converge $e_1$ and $e_2$ to zero. Therefore, since the FMGM is used to reduce chatter, the FTA can drive the system tracking error to converge to zero in a finite amount of time, thus ensuring closed-loop stability and producing higher performance.

3. Control Design

The sliding function can be expressed as

$$\sigma =e_1+{\displaystyle \frac{1}{\alpha }}{e}_2^{\beta },$$

(2)

where $\alpha >0$ and $1<\beta <2$. Then, it is suggested that the presented sliding mode reach the law below:

$$\dot{\sigma }=-{\kappa }_1{\left|\sigma \right|}^{{\tau }_1}\mathrm{gd}(\sigma /\epsilon )-{\kappa }_2{\left|\sigma \right|}^{{\tau }_2}\mathrm{gd}(\sigma /\epsilon )-{\kappa }_3{\left|\sigma \right|}^{{\tau }_3}\sigma ,$$

(3)

where ${\kappa }_1>0$, ${\kappa }_2>0$, ${\kappa }_3>0$, $1<{\tau }_1$, ${\tau }_2>0$, ${\tau }_3>0$, $\epsilon >0$, and $\mathrm{gd}(\sigma /\epsilon )$ is a saturation function based on the Gudermannian function.

According to (1)–(3), the FTA control law $u$ can be derived as

$$u=-b^{-1}[a_1{e}_1+a_2{e}_2+{\displaystyle \frac{\alpha }{\beta }}{e}_2^{2-\beta }+{\kappa }_1{\left|\sigma \right|}^{{\tau }_1}\mathrm{gd}(\sigma /\epsilon )+{\kappa }_2{\left|\sigma \right|}^{{\tau }_2}\mathrm{gd}(\sigma /\epsilon )+{\kappa }_3{\left|\sigma \right|}^{{\tau }_3}\sigma ].$$

(4)

Proof. 

Assuming the following candidate for the Lyapunov function

$$V=0.5{\sigma }^2,$$

(5)

The time derivative $V$ can be calculated in the following manner:

$$\dot{V}=\sigma \dot{\sigma }=\sigma {\left(e_1+{\displaystyle \frac{1}{\alpha }}{e}_2^{\beta }\right)}^{\prime }\le -\sigma \cdot ({\kappa }_1{\left|\sigma \right|}^{{\tau }_1}\mathrm{gd}(\sigma /\epsilon )+{\kappa }_2{\left|\sigma \right|}^{{\tau }_2}\mathrm{gd}(\sigma /\epsilon )+{\kappa }_3{\left|\sigma \right|}^{{\tau }_3}\sigma ).$$

(6)

From Equation (6), we can deduce that both $\sigma$ and $e_2$ parameters are non-zero, indicating that $\dot{V}$ will be negative, which is consistent with the principles of Lyapunov’s stability theorem. The robust and resilient FTA feedback system is anticipated to reach the equilibrium region swiftly and within a limited timeframe. □

However, it is important to note that significant internal changes in plant parameters or external load disturbances to the pure sine wave inverter may lead to persistent high-frequency chattering or steady-state errors. This could result in inaccuracies in the tracking trajectory during both transient and steady-state conditions. To mitigate chattering and steady-state errors, it is essential to develop the FMGM utilizing the appropriate procedures as follows [20,21]:

Step 1: Define ${\chi }_1^{(0)}=\{x_1^{(0)}(1), x_1^{(0)}(2), \dots , x_1^{(0)}(m)\}$, and then increasing $\rho$ order to the sequence ${\chi }_j^{(0)}$ ($j=2, 3, \dots , \ell$) obtains the following sequence:

$${\chi }_j^{(\rho )}=\{x_j^{(\rho )}(1), x_j^{(\rho )}(2), \dots , x_j^{(\rho )}(m)\},$$

(7)

where $x_j^{(\rho )}(n)={\displaystyle \sum _{i=1}^n\frac{\Gamma (\rho +n-i)}{\Gamma (n-i+1)\Gamma (\rho )!}{x}_j^{(0)}(i)}$, $n=1, 2, \dots , m$, and $\Gamma (\cdot )$ stands for a Gamma function.

Step 2: A first-order differential equation for the multivariable grey model can be built as

$${\displaystyle \frac{d{x}^{(\rho )}}{dt}}+A{x}^{(\rho )}=B,$$

(8)

where $A$ and $B$ stand for the development and driving coefficients, respectively.

The mean sequence $G_1^{(\rho )}$ can be obtained from consecutive neighbors of ${\chi }_1^{(\rho )}$ as follows:

$$G_1^{(\rho )}=\{g_1^{(\rho )}(2), g_1^{(\rho )}(3), \dots , g_1^{(\rho )}(m)\}.$$

(9)

and

$$g_1^{(\rho )}(n)={\displaystyle \frac{1}{2}}\times (x_1^{(\rho )}(n)+x_1^{(\rho )}(n-1)) , n=2, 3, \dots , m.$$

(10)

Thus, a discrete sequence of the (8) is expressed as

$$x_1^{(\rho -1)}(n)+A{g}_1^{(\rho )}(n)={\displaystyle \sum _{j=2}^{\ell }{B}_j{x}_j^{(\rho )}(n)}+w_1(n-1)+w_2, n=2, 3, \dots , m,$$

(11)

where $w_1$ represents the linear modified item, while $w_2$ denotes the term associated with the grey effect.

Solving for $A$, $w_1$, $w_2$, $B_2$, …, and $B_m$ by the ordinary least squares method yields

$$Z={({\Lambda }^T\Lambda )}^{-1}{\Lambda }^{T}Y,$$

(12)

where $Z=\left[\begin{array}{c}{B}_2\\ ⋮\\ B_m\\ A\\ w_1\\ w_2\end{array}\right]$, $\Lambda =\left[\begin{array}{ccccccc}{x}_2^{(\rho )}(2)& x_3^{(\rho )}(2)& \cdots & x_{\ell }^{(\rho )}(2)& -g_1^{(\rho )}(2)& 1& 1\\ x_2^{(\rho )}(3)& x_3^{(\rho )}(3)& \cdots & x_{\ell }^{(\rho )}(3)& -g_1^{(\rho )}(3)& 2& 1\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ x_2^{(\rho )}(m)& x_3^{(\rho )}(m)& \cdots & x_{\ell }^{(\rho )}(m)& -g_1^{(\rho )}(m)& m-1& 1\end{array}\right]$, and $Y=\left[\begin{array}{c}{x}_1^{(\rho -1)}(2)\\ x_1^{(\rho -1)}(3)\\ \cdot \\ \begin{array}{l}\cdot \\ \cdot \end{array}\\ x_1^{(\rho -1)}(m)\end{array}\right]$.

Step 3: The predicted output can be computed as follows:

$${\widehat{x}}_1^{(0)}(n)={\left({\widehat{x}}_1^{(\rho )}\right)}^{-\rho }(n)$$

(13)

where $n=2, 3, \dots , m$.

Lastly, an improved grey prediction ($u_{igp}$) component has been incorporated into the control methodology outlined in (4) to mitigate chattering effects:

$$u_{igp}(k)=\{\begin{array}{c}0\\ \Omega \widehat{\sigma }(k)sat(\sigma (k)\widehat{\sigma }(k))\end{array}\begin{array}{c}, \left|\widehat{\sigma }(k)\right|<\kappa \\ , \left|\widehat{\sigma }(k)\right|\ge \kappa \end{array}$$

(14)

where $\Omega$ designates a constant, $\widehat{\sigma }(k)$ refers to the prediction of $\sigma (k)$, $sat(\cdot )$ is a saturation continuous function, and $\kappa$ marks the border of the system. The latest summary of findings indicates that the proposed control technique successfully reduces the chattering of the state variable motion trajectory before it converges to the origin, while a continuous function is being utilized. When the constant $\epsilon$ is considerably large, it is expected that the slope of the function will diminish, leading to a smoother curve. In contrast, a smaller constant $\epsilon$ results in a function slope that more closely resembles the signum function. As demonstrated in (3), it is evident that when parameter $\dot{\sigma }=0$ is considered, the system can achieve stabilization at the equilibrium point. In the case of parameter $\left|\sigma \right|>>0$, the switching gain associated with the power term facilitates a rapid convergence of the state variable towards the sliding surface, thereby significantly enhancing the system’s approach speed. Conversely, as parameter $\left|\sigma \right|$ approaches zero, the power term mechanism results in a reduction in the system’s switching gain when in proximity to the sliding surface, which effectively mitigates the chattering. Furthermore, the third term of $\dot{\sigma }$ can be utilized to enhance the convergence rate of the system when it is situated at a considerable distance from the sliding surface. The proposed control technique embodies a hybrid approach that combines FMGM with FTA, making it suitable for application in pure sine wave inverters. Within this framework, the FTA effectively reduces the state error to zero, while the FMGM provides precise predictions of state points, thereby minimizing the likelihood of chattering effects.

4. Simulation and Experimental Results

Table 2. Parameters of the pure sine wave inverter.

Parameters	Value
DC-link voltage ($V_{dc-link}$)	200 V
AC output voltage ($v_o$)	110 Vrms
Power of inverter	500 W
Frequency of AC output-voltage	60 Hz
Filter inductor ($L$)	1 mH
Filter capacitor ($C$)	20 μF
Resistive load ($R$)	12 ohm
Switching frequency	36 kHz

presents the specifications for the pure sine wave inverter. Both simulated and experimental results were conducted on a 500 W pure sine wave inverter to demonstrate the effectiveness of the proposed control technique. The proposed control technique uses a digital implementation based on dSPACE (dSPACE GmbH, Paderborn, Germany) digital signal processor (DSP). The control environment of dSPACE DSP can be combined with MATLAB (version 6.1, MathWorks Inc., Natick, MA, USA)/Simulink (version 4.1, MathWorks Inc., Natick, MA, USA) for fast modeling to validate the control algorithm. Although dSPACE DSP increases the cost of the proposed inverter, this is an advantage for researchers who would like to quickly verify the effectiveness of the proposed algorithm. In the future, the dSPACE DSP can be substituted by other digital control devices, such as a field-programmable gate array (FPGA) or a peripheral interface controller (PIC), to reduce the cost of the proposed pure sine wave inverter.

4.1. Simulation Results

The simulated output voltage waveforms produced by the pure sine wave inverter utilizing the proposed control technique as well as the conventional TA (terminal attractor) under full load situation are reported in Figure 2 as well as Figure 3, correspondingly. The output voltage becomes a true sine form, rendering a stabilized AC without distortion. Since the load remains a linear condition of the purely resistive property with no dynamic variability, this all points to the fact that the output of a pure sine wave inverter experiences consistent response behavior.

Figure 4 displays the output voltage difference between the proposed control technique and the conventional TA for full loads. Both the output voltage trajectories nearly coincide, showing a commendable sinusoidal steady state since the load is a linear resistor.

With the intention of checking the steady-state effect behavior of the pure sine wave inverter beneath the variability of the inductive-capacitive filter values, the simulated output voltage waves of the proposed control technique as well as the conventional TA are rendered in Figure 5 as well as in Figure 6, separately. The proposed pure sine wave inverter output voltage throughout the steady state period incurs a very low total harmonic distortion, which can be deemed as a pure sine shape; nonetheless, the conventional TA’s pure sine wave inverter output voltage shape suffers from warped oscillations in steady state, having a gross total harmonic distortion.

Figure 7 displays the difference in the simulated output voltage and current in response to variations in the filter parameters, where the output voltage from the conventional TA exhibits a deviated steady-state behavior away from the sinusoidal trajectory.

A pure sine wave inverter used with the proposed control technique demonstrates the simulated output voltage waveform at a triggering angle of ninety degrees (starting from no load to full load) as a transient step response according to Figure 8. At the onset of the triggering angle, although the voltage droops momentarily, the decrement is mild and it immediately rebounds to the maximized amplitude of the target alternating current voltage. Following conventional TA, Figure 9 depicts the simulated output voltage waveform with a transient load step alteration (from no load to full load) at ninety triggering angles for a pure sine wave inverter. The conventional TA compensation lacks significant improvement in terms of dynamical following impact, which can be monitored by the relatively big voltage droop coupled with the considerably longer transient retrieval period.

Figure 10 illustrates a comparison of the simulated output voltage difference between the proposed control technique and the conventional TA during a step load changing from no load to full load. The output voltage utilizing the proposed control technique demonstrates a recovery from a minor voltage dip, reaching approximately 10.612 volts, back to the nominal voltage level. In contrast, the output voltage associated with the conventional TA experiences a significant voltage dip of 56.633 volts before returning to the nominal voltage.

It is important to mention that some countries have specific standards regarding the magnitude of harmonic current distortion. The harmonic currents are usually generated by nonlinear loads, such as rectifiers, fluorescent lamps, variable frequency drives, and switching power supplies. The IEEE (Institute of Electrical and Electronics Engineers) 519-1992 [22] specifies the current total harmonic distortion (THD) required by a limit of less than 5%. The simulated output voltage and current waveforms of the pure sine wave inverter with the proposed control technique are shown in Figure 11 under nonlinear rectifier load conditions. The good steady-state accuracy is clearly observed, showing that the voltage and current total harmonic distortions are 1.352% and 3.124%, respectively. The simulated output voltage and current waveforms of a conventional TA-controlled pure sine wave inverter subjected to a nonlinear rectifier load are presented in Figure 12. The total harmonic distortions of the voltage and current are up to 27.785% and 36.173%, respectively, which indicate a heavily distorted sinusoidal output.

4.2. Experimental Results

The experimental output voltage waveforms for the pure sine wave inverter, comparing the proposed control technique with the conventional TA under full load conditions, are illustrated in Figure 13 and Figure 14. Both controllers demonstrate AC voltage waveforms that are remarkably undistorted.

Figure 15 presents the output voltage difference between the proposed control technique and the conventional TA for full loads. Because the pure sine wave inverter is connected to a linear resistive load, it shows an almost distortion-free sinusoidal waveform for both output voltages.

The output voltage waveform for the inverter using the proposed control technique under filter parameters varying is shown in Figure 16. This waveform maintains a nearly sinusoidal shape without distortion, resulting in a total harmonic distortion (THD) of just 0.040%, which is well within the industry standard. In contrast, Figure 17 depicts the output voltage waveform for the conventional TA-controlled inverter under filter parameters varying, which exhibits a significantly higher THD of 6.137%, indicating a non-sinusoidal distortion.

To thoroughly evaluate the performance of the pure sine wave inverter, it is essential to operate it under strictly varying filter parameters. Figure 18 presents a comparison of the simulated output voltage difference in the context of filter parameters varying for the proposed control technique and the conventional TA. The proposed control technique generates a high-quality AC sine wave, whereas the conventional TA results in oscillations and distortions in the output voltage.

Figure 19 and Figure 20 present the experimental voltage waveforms for the proposed control technique and the conventional TA during transient step loads (with a ninety-degree transition from no load to full load). Notably, the proposed control technique effectively reduces voltage droop and recovers to the reference voltage more quickly than the conventional TA.

The comparative analysis of the experimental output voltage difference for the proposed control technique and the conventional TA during the step load changing is depicted in Figure 21. The voltage drop observed in the conventional TA is approximately five times greater than that of the proposed control technique, indicating a considerably inferior transient response.

Overall, the proposed control technique significantly improves the steady-state response of the pure sine wave inverter when subjected to highly filter parameters varying, compared to the conventional TA.

Table 3. Percent of total harmonic distortion (THD) and voltage drop for experimental AC output-voltage.

Proposed Control Technique
Simulations	Resistive loading	Step load changing	Filter parameters varying
	%THD	Voltage droop	%THD
	0.025%	10.612 volts	0.040%
Conventional TA
Simulations	Resistive loading	Step load changing	Filter parameters varying
	%THD	Voltage droop	%THD
	0.031%	56.133 volts	5.854%
Proposed control technique
Experiments	Resistive loading	Step load changing	Filter parameters varying
	%THD	Voltage droop	%THD
	0.027%	10.785 volts	0.042%
Conventional TA
Experiments	Resistive loading	Step load changing	Filter parameters varying
	%THD	Voltage droop	%THD
	0.039%	58.741 volts	6.137%

summarizes the simulated and experimental THD percentages and output voltage droop values for various operation conditions. Therefore, the proposed control technique has proved to outperform the conventional TA against purely resistive loads, transient step load variability, as well as an inductive-capacitive filter with changing parameters, providing higher dynamical as well as steady state representations.

5. Discussion and Analysis

All simulation and experimental findings of the proposed pure sine wave inverter are in accordance with the required level of performance for both percent of total harmonic distortion as well as voltage droop. To be precise, IEEE (Institute of Electrical and Electronic Engineers) standard 519-2014 [23] advises that there should be a less than eight percent maximal voltage total harmonic distortion target from systems operating at minimally one thousand volts or lower. IEEE standard 1159-2019 [24] limits voltage drop of 0.1 to 0.9 pu (per unit) measured as the root-mean-square value of the voltage or current at the power source frequency during a period of time ranging from one-half cycle to one minute. A further issue to be explored is the LC filter for pure sine wave inverters, whose design can be followed depending on the recommendations of literature works [25,26]. Let us start with the following assets: 1 to 0.8 lagging for linear loading, crest factor ranging from 1.414 to 5.0 for nonlinear rectified loading, total harmonic distortion percentage lower than eight percent for linear as well as nonlinear loading, and the modulation index of 0.8 for rated resistive loading. The aim of decreasing the volume of the filter leads to the decision of a moderately large switching frequency, normally ranging from 20 kHz to 40 kHz regarding metal-oxide-semiconductor field-effect transistor (MOSFET) switches. Next, there is a factor to be considered corresponding to the cut-off frequency of the LC filter. A lower value of this factor means more decay as well as less augmentation of the switching as well as fundamental frequencies separately; it is possible to find minimizing values for this factor, while 0.95 or lower modulation value would be desirable. It is also relevant to choose a factor dependent on the switching frequency as well as the inductor ripple current, where it is advisable to have the inductor’s ripple current around twenty to forty percent. Once these recommended factors have been taken into account, the values of L and C can be evaluated.

The other attractive aspect consists in the fact that some greater than second-order sliding-mode control technologies have been exploited to smooth the control signal, thereby limiting chattering as well as enlarging the reliability of the controlled system, for instance, the third-order sliding mode control as well as the full-order sliding mode control. A microgrid inverter control structure on the basis of adaptive third-order sliding mode as well as optimal droop control is suggested. Amongst them, the third-order sliding mode controller works well to minimize chattering, leading to an effective suppression of voltage as well as loading perturbations in microgrid inverters, independent of whether the inverter runs in grid-tie or islanded modes, or transitioning from grid-tie to islanded modes [27]. There is a full-order terminal sliding mode control strategy presented for the virtual synchronous generator-based inverter. The mentioned full-order terminal sliding mode is capable of diminishing the chattering trouble incurred from high-speed switching so that enhancement of the ruggedness as well as the tracking rate in the system becomes achievable. In this case, the generator can deliver stabilized energy into the grid or sustain steady-state performance despite the islanding situation [28].

Recent publications on grey prediction have already strengthened the quality of forecasting by associating wavelet transformed signal analysis methods as well as artificial intelligence learning with neural networks; such a subject is examined to be beneficial in forecasting systems containing uncertainties. A modified wavelet transform-based grey Markov chain is modeled considering a variety of energy types. The discrete wavelet transform is adopted to remove noise, and an augmented grey model replaces the classic model, while fuzzy logic and meteorological approach are integrated as part of the Markov chain. The resulting forecasts of both the generation and conservation of energy resources have been tested for applicability as well as efficacy [29]. Based on wavelet transformation as well as gray back-propagation neural network, a hybrid version of the model for forecasting the depth foundation pit deflection has been ameliorated. Using wavelet noise removal gives accurate data on the deflection of the pits; the gray model is utilized to forecast the deformation of the foundation pit, while the forecast data serves as the incoming sample data for the back-propagation neural network. A strong level of precision with good robustness could be achieved with the hybridization model [30].

In prospective research efforts, the proposed control technique can be utilized with various kinds of inverters, including bidirectional T-type multilevel inverters designed for electric vehicles, switched-capacitor-inductor-based inverters, and three-phase transformerless photovoltaic inverters. Furthermore, the application areas of the obtained results can be effectively utilized not only for electric transport but also in generating equipment based on renewable energy sources, such as solar photovoltaic power generation, wind power generation, and solid oxide fuel cell power generation.

6. Conclusions

In this paper, the proposed control technique is demonstrated on a pure sine wave inverter controlled by a digital signal processor. Compared to the conventional TA, the FTA has a systematically fast convergence time, which provides better response under transient and steady-state conditions. In addition, the FMGM helps to eliminate the chatter that occurs in the FTA when the load becomes a highly nonlinear environment. The proposed pure sine wave inverter can provide high-quality AC output voltage when combined with FTA and FMGM. Both simulation and experimental findings indicate that the controlled pure sine wave inverter achieves low total harmonic distortion, rapid transient response, elimination of chatter, and minimal steady-state error across various operational load scenarios. The effectiveness of the proposed control technique is substantiated through theoretical analysis, simulation, and experimental results. In the future, the possible research applications of the proposed methodology involve high step-up DC-DC converters with maximum power point tracking (MPPT) for photovoltaic systems, bidirectional DC-DC converters for energy storage systems, AC-DC-AC converters for electric vehicles.