Nonlinear Self-Synchronizing Current Control for Single-Phase AC Inverters

Abstract

Grid-connected single-phase inverters require accurate phase detection for synchronization and power control. Traditionally, phase-locked loops (PLLs) are used to estimate grid parameters. This paper proposes a novel approach that determines the grid phase angle using only current feedback, eliminating the need for grid voltage measurements or cascaded control schemes. The proposed method integrates a phase angle observer with a current controller to regulate real and reactive power. Lyapunov stability analysis and hardware experiments validate the effectiveness of the approach.

1. Introduction

Modularity, high efficiency, and minimal to no emission of pollutants make renewable energy resources a favorable substitute to conventional power generation based on fossil fuels. The increasing prevalence of challenging loads, especially loads incorporating power electronics converters, raises concerns about the power quality, stability, and operation of the power system [1]. To operate in grid-connected mode, the voltage source inverter (VSI) is a necessary link between renewable units and the utility [2]. Nonlinear dynamic behaviors are exhibited in renewable energy systems tied to the grid due to the switching function and time-varying voltage and current input from the renewable resources. To effectively integrate renewable energy resources, there is a need to reevaluate the power system gradually shifting to an architecture that is centered on power inverter-based distributed generation (DG) and its control. The communication-less and decentralized control scheme known as Droop Control [3,4], commonly used for integrating variable renewable sources (VRSs), aids in comprehensive voltage and frequency control by imitating virtual impedance and virtual inertia [5,6]. Nevertheless, the dynamics of the entire AC system are less due to the renewable energy integration, as these sources lack a rotating mass to store kinetic energy [7]. As a result, mismatches between the amount generated and the amount consumed cannot always be balanced mechanically. To ensure system operation, renewable energy resources and loads should not be subjected to unwanted tripping [8]. Before connecting a distributed generation unit (renewable-interfaced inverter) to the power grid, a unit for synchronization must be utilized to ascertain the phase angle of the grid and frequency, preventing an out-of-phase connection and ensuring a safe connection between source and load. PLL-based synchronization has been incorporated for power quality enhancement [9]. Synchronous reference frame PLLs represent the predominant approach because they deliver apt results under stable conditions of the grid. However, they become highly susceptible to harmonic distortions and grid voltage imbalances [10,11,12]. An increase in the PLL bandwidth has been explored to enable quicker angle detection [13], but this approach is constrained due to the proximity of adjacent converters [14], fragile conditions of the grid associated with stiffness ratios [12,14], and extensive DG inverter penetration, resulting in grid instabilities and power oscillations at lower frequencies [15,16]. Numerous efforts to develop improved PLLs to manage these conditions have overlooked the effects of the coupling and interface between the network impedance of the system and the PLL, which influence the tuning of the PLL [13]. The aforementioned coupling can result in various instability problems when numerous inverters are integrated into the system. An analysis illustrating the risks associated with higher DG inverter penetration using PLLs is evident in California, USA. During a documented event, California faced a substantial generation loss, around 700 MW, as the PLL of the inverter sensed frequencies below 57 Hz, causing an immediate shutdown, although the minimum observed frequency merely decreased to 59.87 Hz. As indicated in [8], researchers must devise robust strategies even when there are numerous inverters on the system. In an alternative method, a PLL extracts information from the grid, and as an outer loop, it is combined with the current (primary) control method, resulting in a cascaded control system [17]. However, the sluggish dynamics of the PLL can cause latencies in detecting the frequency and angle of the grid. Each grid state is critical to controlling the current of the AC grid. In addition, the PLL of the primary control scheme increases the system’s complexity. When influenced by the Proportional-Integral (PI) controllers of PLLs connected to weak grids, small signals can disturb the phase information of the output. As a result, the coordinate system of the grid-tied inverter becomes misaligned with the control coordinate system, leading to the deterioration of the grid-tied inverter system via the current control [18].

The classical Phase-Locked Loop (PLL) encounters problems under disturbed conditions of the grid due to the low-pass filter’s slow response and its reduced precision in the generation of reference current. An enhanced version of PLL, known as EPLL, has successfully resolved these issues, which are used for the fundamental extraction of current. However, the structure of EPLL faces challenges associated with the presence of DC components and performance degradation under transient conditions. This structure is improved, but it is more complex and necessitates increased sampling time for real-time processing. Consequently, many researchers have adopted PLL with an integrated design for the extraction of grid current in grid-tied renewable-based energy systems. This PLL structure offers better filtering due to a faster reference to current generation. The main challenge with such structures is proper management of the integrator due to its stability challenges [19]. Researchers have developed control schemes that operate without needing a synchronization system like PLLs to resolve possible instability concerns arising from the interaction between PLL schemes and the current control loop. In [20], the authors suggested a direct power controller that does not employ a PLL scheme for a single-phase grid-tied inverter system. Single-phase inverters may encounter heightened voltage imbalances that can impact the stability of the power supply, yet they are highly necessary for residential and commercial system applications. A PLL-free control scheme has been developed to achieve tracking of current without any information about the parameters of the grid. It utilizes a rotating reference frame, or γδ-frame, which is estimated for the system [21,22]. In this scheme, adaptive compensation parameters enable tracking of the current and accounting for the unknown parameters, i.e., frequency, voltage magnitude, and phase of the grid. This eliminates the necessity for any other additional measurement or any feedback systems like PLL.

The system that incorporates a PLL relies on voltage sensors, which are quite expensive and can induce DC offset and electrical noise into the system, necessitating compensation [23]. A PLL system operates as an independent feedback loop that is outside and separate from the primary or main current control system, establishing a structure like cascade control [24]. To ensure maximum stability and precision, the current control used needs to account for the error aspects of the PLL system. Investigators have highlighted probable concerns about the stability of the grid arising from PLL schemes in different conditions of the grid [25,26]. To address these problems of stability, new PLL schemes such as MCCF-PLL [27], MNSOGI-PLL [28], and Improved DSC-PLL [29] have been proposed. These schemes require carefully designed procedures [30]. Furthermore, PLLs incorporated in the outer loop of these control schemes introduce additional complications and transient interactions and can potentially destabilize the entire system [31].

These issues have spurred the development of control schemes that are not dependent on synchronization methods like PLL for single-phase as well as three-phase grid-tied inverter systems [32,33,34,35]. As an example, in [35], a nonlinear control scheme is proposed for single-phase grid-tied inverters that excludes the necessity of a PLL to coordinate the inverter with the grid. However, the measurements of the grid parameters are still essential and are utilized as part of the control input. To date, only a limited number of control schemes have been documented, predominantly reliant on PLL technology [36], which in turn presents several discussed challenges. This paper presents the control of a single-phase inverter that is linked to the grid, using an innovative nonlinear current control scheme. The objective of tracking the current without any information about the frequency, magnitude, and phase angle of the grid, or the use of a PLL, is achieved by the controller proposed in this paper. A Lyapunov analysis motivates the control scheme development and proves the control objective is met and the signals are bounded in the closed loop. A hardware experiment further validates the result.

2. System Modeling

Figure 1 shows a single-stage, grid-tied inverter system with an inductive filter. The following equations describe the system’s dynamics:

$$V_a\triangleq V_{dc}{D}_{ab},$$

(1)

where $V_{dc}$ and $D_{ab}$ are the respective input voltage and duty cycle of the inverter.

$$L{\displaystyle \frac{d{i}_a}{dt}}=-R{I}_a(t)+V_a(t)-v_g(t)$$

(2)

where $V_a\left(t\right)$ and $I_a\left(t\right)$ are the inverter output voltages and current. In Equation (2), the grid voltage can be described as follows:

$$v_g\left(t\right)\triangleq V_g\mathrm{c}\mathrm{o}\mathrm{s}(\theta \pm \varphi )$$

(3)

where $\theta \left(t\right)$ is the grid angle, $\varphi$ is grid offset and $V_g$ is the grid magnitude. Both $\theta \left(t\right)$ and $V_g$ are unknown parameters, and $\varphi$ = 0.

Thus, in the estimated reference frame, the design of an observer for the grid parameters $\widehat{\theta }(t)$, ${\widehat{V}}_g(t)$, $\widehat{\mathsf{\omega }}(t)$ are needed to execute the $dq$ like transformations in the estimated reference frame [22]. Here, ω$\left(t\right)$ is grid frequency, ω$\left(t\right)$ = $\dot{\theta (}t)$. This creates a rotating orthogonal axis system referred to as $\gamma \delta$ frame. Following the transition in the $\gamma \delta$ frame [37], the system displayed in Equation (2) can be:

$$L\left[\begin{array}{c}{\dot{I}}_{\gamma }\\ {\dot{I}}_{\delta }\end{array}\right]=V_{dc}\left[\begin{array}{c}{D}_{\gamma }\\ D_{\delta }\end{array}\right]-\left[\begin{array}{cc}R& -\dot{\widehat{\theta }}L\\ \dot{\widehat{\theta }}L& R\end{array}\right]\left[\begin{array}{c}{I}_{\gamma }\\ I_{\delta }\end{array}\right]-V_g\left[\begin{array}{c}\cos \tilde{\theta }\\ \sin \tilde{\theta }\end{array}\right],$$

(4)

where $\tilde{\theta }\left(t\right)$ represents the error difference of the actual and observed phase angle of the grid.

3. Control System Design and Development

The primary aim of the controller and observer proposed is to design the duty cycle control signals $D_{\gamma }\left(t\right)\mathrm{a}\mathrm{n}\mathrm{d}{D}_{\delta }\left(t\right)$ of the above system without the information of the grid parameters: angle $\theta \left(t\right)$, frequency $\omega \left(t\right)$, and voltage $V_g\left(t\right)$ of the grid. In which, the $I_{\gamma }\left(t\right)$ and $I_{\delta }\left(t\right)$ can track their corresponding reference currents $I_{\gamma }^{*}\left(t\right)$ and $I_{\delta }^{*}\left(t\right)$, hence, $I_{\gamma }\left(t\right)\to I_{\gamma }^{*}\left(t\right)$ and $I_{\delta }\left(t\right)\to I_{\delta }^{*}\left(t\right)$ as $t\to \infty$. The following stability analysis will demonstrate that upon convergence of the grid angle observer (i.e., at $\tilde{\theta }\left(t\right)=0$), $I_{\gamma }\left(t\right)=I_d\left(t\right),I_{\delta }\left(t\right)=I_q\left(t\right)$. This implies that both active power and reactive power injected into the grid can be controlled.

Following are the assumptions established to support the development of this control:

Assumption 1.

Grid current; $I_a$(t) and $V_{dc}$(t) are measurable.

Assumption 2.

R and L which are nominal system parameters are known a priori.

Assumption 3.

The parameters of the grid (frequency of the grid ω, phase angle of the grid $\theta$, and magnitude of the grid voltage $V_g$) are unknown, and ω and $V_g$ are assumed to be constants.

Error Signal Development: To Aid in the Development of the Control, the Current Tracking Errors Are Defined in Equation (5)

$$\left[\begin{array}{c}{\tilde{\mathrm{I}}}_{\mathsf{\gamma }}\\ {\tilde{\mathrm{I}}}_{\mathsf{\delta }}\end{array}\right]\triangleq \left[\begin{array}{c}{\mathrm{I}}_{\mathsf{\gamma }}^{\mathrm{*}}\\ {\mathrm{I}}_{\mathsf{\delta }}^{\mathrm{*}}\end{array}\right]-\left[\begin{array}{c}{\mathrm{I}}_{\mathsf{\gamma }}\\ {\mathrm{I}}_{\mathsf{\delta }}\end{array}\right].$$

(5)

The estimation error of the phase angle, frequency of the grid and magnitude of the grid are given in Equations (6), (7) and (8), respectively,

$$\tilde{\mathsf{\omega }}\triangleq \mathsf{\omega }-\widehat{\mathsf{\omega }},$$

(6)

$$\tilde{\mathsf{\theta }}\triangleq \mathsf{\theta }-\widehat{\mathsf{\theta }},$$

(7)

$${\tilde{V}}_g\triangleq v_g-{\widehat{V}}_g.$$

(8)

According to the sub-segment of Lyapunov stability, the inverter control signals are shown below in Equation (9), where $k_{1,}{k}_2\in {\mathbb{R}}^{+}$ are the control gains and are positive

$$\left[\begin{array}{c}{\mathrm{D}}_{\mathsf{\gamma }}\\ {\mathrm{D}}_{\mathsf{\delta }}\end{array}\right]\triangleq {\displaystyle \frac{1}{{\mathrm{V}}_{\mathrm{d}\mathrm{c}}}}\left(L\left[\begin{array}{c}{\dot{\mathrm{I}}}_{\mathsf{\gamma }}^{\mathrm{*}}\\ {\dot{\mathrm{I}}}_{\mathsf{\delta }}^{\mathrm{*}}\end{array}\right]+\left[\begin{array}{cc}\mathrm{R}& -\dot{\widehat{\mathsf{\theta }}}\mathrm{L}\\ \dot{\widehat{\mathsf{\theta }}}\mathrm{L}& \mathrm{R}\end{array}\right]\left[\begin{array}{c}{\mathrm{I}}_{\mathsf{\gamma }}\\ {\mathrm{I}}_{\mathsf{\delta }}\end{array}\right]+\left[\begin{array}{c}{\mathrm{v}}_{\mathrm{g}}\\ 0\end{array}\right]+\left[\begin{array}{cc}{k}_1& 0\\ 0& k_2\end{array}\right]\left[\begin{array}{c}{\tilde{\mathrm{I}}}_{\mathsf{\gamma }}\\ {\tilde{\mathrm{I}}}_{\mathsf{\delta }}\end{array}\right]\right).$$

(9)

The closed-loop aspects of the current error are as shown in Equation (10)

$$L\left[\begin{array}{c}{\dot{\tilde{\mathrm{I}}}}_{\mathsf{\gamma }}\\ {\dot{\tilde{\mathrm{I}}}}_{\mathsf{\delta }}\end{array}\right]=-\left[\begin{array}{cc}{\mathrm{k}}_1& 0\\ 0& {\mathrm{k}}_2\end{array}\right]\left[\begin{array}{c}{\tilde{\mathrm{I}}}_{\mathsf{\gamma }}\\ {\tilde{\mathrm{I}}}_{\mathsf{\delta }}\end{array}\right]+\left[\begin{array}{c}{\tilde{V}}_g\\ {\mathrm{V}}_g\sin \tilde{\mathsf{\theta }}\end{array}\right].$$

(10)

The grid angle observer’s update law is shown in Equation (11)

$$\dot{\widehat{\theta }}\triangleq \widehat{\omega }+{\tilde{I}}_{\delta }+V_g\mathit{sin}\tilde{\theta }.$$

(11)

The realizable from of the estimators in Equation (11) is given in Equation (12):

$$\widehat{\mathsf{\theta }}\triangleq \mathrm{L}{\tilde{\mathrm{I}}}_{\mathsf{\delta }}+\int (\widehat{\mathsf{\omega }}+\left({\mathrm{k}}_2+1\right){\tilde{\mathrm{I}}}_{\mathsf{\delta }})\mathrm{d}\mathsf{\sigma }.$$

(12)

The grid frequency estimator is in Equation (13), where $k_{\omega }\in {\mathbb{R}}^{+}$ is a positive estimator gain

$$\dot{\widehat{\omega }}=k_{\omega }{V}_g\mathit{sin}\tilde{\theta }$$

(13)

Its realizable form is as follows:

$$\widehat{\omega }\triangleq k_{\omega }\left(L{\tilde{I}}_{\delta }+k_2{\int }_{t_0}^t{\tilde{I}}_{\delta }\mathrm{d}\mathsf{\sigma }\right)$$

(14)

and the grid magnitude observer is shown in Equation (13), where $k_v\in {\mathbb{R}}^{+}$

$${\dot{\widehat{\mathrm{V}}}}_g\triangleq {\mathrm{k}}_{\mathrm{v}}{\tilde{\mathrm{I}}}_{\mathsf{\gamma }}.$$

(15)

Stability Analysis:

Theorem 1.

The error performance of closed loop system guarantees that, the signals of error described in Equations (5)–(8) are controlled as:

$${\tilde{I}}_{\gamma }\left(t\right),{\tilde{I}}_{\delta }\left(t\right),\tilde{\theta }\left(t\right)\to 0\mathrm{as}t\to \infty .$$

(16)

Proof of Theorem 1.

To evaluate the system’s stability, a Lyapunov candidate function is selected as shown below with respect to time or $V\left(t\right).$ □

$$V\triangleq {\displaystyle \frac{1}{2}}L{{\tilde{I}}_{\gamma }}^2+{\displaystyle \frac{1}{2}}L{{\tilde{I}}_{\delta }}^2+V_g\left(1-\mathit{cos}\tilde{\theta }\right)+{\displaystyle \frac{1}{2k_{\omega }}}{\tilde{\omega }}^2+{\displaystyle \frac{1}{2k_V}}{\tilde{V}}_g^2.$$

(17)

Assumption 4.

The above $V\left(t\right)$ in Equation (17) is positive definite in the local region $\tilde{\theta }\left(t\right)\in (-2\pi ,2\pi )$. Assuming that $\tilde{\theta }$(t) is bound in a way that the effective domain of $\tilde{\theta }$(t) is $\tilde{\theta }\left(t\right)\in \left[-\pi ,\pi \right)$. This function is determined to be globally positive definite.

The derivative of the Lyapunov function (17) with respect to time is obtained as:

$$\dot{\mathrm{V}}=\mathrm{L}{\tilde{\mathrm{I}}}_{\mathsf{\gamma }}{\dot{\tilde{\mathrm{I}}}}_{\mathsf{\gamma }}+\mathrm{L}{\tilde{\mathrm{I}}}_{\mathsf{\delta }}{\dot{\tilde{\mathrm{I}}}}_{\mathsf{\delta }}+{\mathrm{V}}_{\mathrm{g}}\dot{\tilde{\mathsf{\theta }}}\sin \tilde{\mathsf{\theta }}+{\displaystyle \frac{1}{{\mathrm{k}}_{\mathsf{\omega }}}}\tilde{\mathsf{\omega }}\dot{\tilde{\mathsf{\omega }}}+{\displaystyle \frac{1}{{\mathrm{k}}_{\mathrm{V}}}}{\tilde{\mathrm{V}}}_{\mathrm{g}}{\dot{\tilde{\mathrm{V}}}}_{\mathrm{g}}.$$

(18)

By replacing the error dynamics of closed loop from Equation (10) and the time derivative of Equations (6)–(8) into Equation (18), and utilizing A3, $\dot{V}$(t) can be obtained as:

$$\dot{V}={\tilde{I}}_{\gamma }\left(-k_1{\tilde{I}}_{\gamma }+{\tilde{V}}_g\right)+{\tilde{I}}_{\delta }\left(-k_2{\tilde{I}}_{\delta }+V_g\sin \tilde{\theta }\right)+V_g\sin \tilde{\theta }\left(\omega -\dot{\widehat{\theta }}\right)-{\displaystyle \frac{1}{k_{\omega }}}\tilde{\omega }\dot{\widehat{\omega }}-{\displaystyle \frac{1}{k_V}}{\tilde{V}}_g{\dot{\widehat{V}}}_g.$$

(19)

The update laws from Equations (11) and (13) and the grid magnitude estimator from Equation (15) are substituted into Equation (19); the below succeeding simplified negative semi-definite form of $\dot{V}\left(t\right)$ can be obtained as:

$$\dot{V}=-k_1{\tilde{I}}_{\gamma }^2-k_2{\tilde{I}}_{\delta }^2-V_g^2{\sin }^2\tilde{\theta }.$$

(20)

From $V\left(t\right)$ and $\dot{V}\left(t\right),$ it can be concluded that ${\tilde{I}}_{\gamma },{\tilde{I}}_{\delta },\tilde{\theta },\tilde{\omega },{\tilde{V}}_g$ are bounded, hence ${\tilde{I}}_{\gamma },{\tilde{I}}_{\delta },\tilde{\theta },\tilde{\omega },{\tilde{V}}_g\in {\mathcal{L}}_{\infty }$. From the closed-loop current error equation in (10) and the preceding conclusions, we can say that ${\dot{\tilde{I}}}_{\gamma },{\dot{\tilde{I}}}_{\delta }\in {\mathcal{L}}_{\infty }$. Since $\tilde{\omega }\in {\mathcal{L}}_{\infty }$ and by using Assumption 3, it is evident that $\widehat{\omega }\in {\mathcal{L}}_{\infty }$. As ${\tilde{I}}_{\delta },\widehat{\omega },\tilde{\theta }\in {\mathcal{L}}_{\infty }$ and from Equation (14) we can determine that $\dot{\widehat{\theta }}\in {\mathcal{L}}_{\infty }$. As $\omega ,\dot{\widehat{\theta }}\in {\mathcal{L}}_{\infty }$ and based on the derivative of Equation (7), it can be stated that $\dot{\tilde{\theta }}\in {\mathcal{L}}_{\infty }$. As ${\tilde{I}}_{\gamma },{\tilde{I}}_{\delta },\tilde{\theta }\in {\mathcal{L}}_{\infty }$ and ${\dot{\tilde{I}}}_{\gamma },{\dot{\tilde{I}}}_{\delta },\dot{\tilde{\theta }}\in {\mathcal{L}}_{\infty }$ Barbalat’s Lemma [38] is utilized to demonstrate that ${\tilde{I}}_{\gamma }\left(t\right),{\tilde{I}}_{\delta }\left(t\right),\tilde{\theta }\left(t\right)\to 0$ as t→∞.

4. Hardware Results

To verify the controller proposed in the paper, we utilized the PLECS RT Box-1 Power Hardware in Loop (P-HIL) to control an experimental single-phase inverter, with the controller activated at t = 0.05 s. The experimental setup of the single-phase grid-tied system is illustrated in Figure 2. A laboratory-based programmable DC power source provided the input to the grid-tied system. The single-phase grid-following inverter was then used to manage the AC-side current. We are using VARIAC and the utility grid at a 60 Hz fundamental frequency. This AC source is connected in parallel to a 48 Ω power resistor and is sized to ensure that the power drawn by the resistor is greater than the power supplied by the inverter. This setup guarantees that no current is exported to the AC supply source.

An inductor filter was employed with the inverter to eliminate harmonics in the AC waveform. Future work could explore a grid-forming inverter control, which would necessitate changes to the output filter. The output of the single-phase inverter, equipped with an inductive filter, was fed to a resistive load connected in parallel with a single-phase AC grid. PWM output for the inverter was transmitted to each respective component through the RT Box.

The actual hardware of the inverter circuits included four digital outputs from the digital board, connected to the RT Box. The actual and the reference grid current are shown in Figure 3. It is observable that the current of the grid successfully follows the reference currents. The proposed scheme illustrates the synchronization between the grid and the inverter before inserting current when the connection is expected to occur at t = 0.05 s.

The component ratings are detailed in

Table 1. Ratings for each component.

NO.	COMPONENTS
1	Prog. DC Power Supply 300 V/5.2 A 1.56 kW	DC Supply (XLN30052)	186 [V]
2	1 ɸ Inverter	IGBT (STGWA20H65DFB2)	650 [V], 20 [A]
3	Filter	L	12 × ${10}^3$ [H]
4	R Load	R	48 [Ω]
5	AC Source (GRID)	$v_g\left(t\right)$	140 [Vrms], 60 [Hz]
6	Switching Frequency	$f_{sw}$	25 Khz
7	Current control Gains	$k_1$	45
		$k_2$	6
		$k_v$	12.5
		$k_w$	30

, listing specifications for the DC power supply, IGBT-based inverter, inductive filter, resistive load, AC grid voltage, and switching frequency. Additional waveforms, including grid phase angle tracking, grid frequency estimation, and grid voltage estimation, further validate the controller’s performance. The experimental findings confirm that the proposed scheme effectively controls the inverter’s current without requiring prior grid information or a phase-locked loop (PLL), ensuring self-synchronization between the inverter and the grid.

Table 1 summarizes the ratings of each component used in the Single-Phase Grid tied power system.

Figure 4 shows the current error in $\gamma \delta$ frame. It is clear from this figure that a large current error is experienced at t = 0.05 [sec.] when the scheme is initiated. In less than 0.1 [sec.] the current error is converging to zero, hence validating Theorem 1, hence the grid angle has been determined, and the closed loop system is under control. Figure 5 represents control signal in $\gamma \delta$ frame, respectively. The control signals are responsible for regulating the inverter’s switching behavior to maintain the desired power flow. The signals are bound and are not saturated. Effective control signals indicate a well-tuned system capable of maintaining stable operation under different grid conditions.

Figure 6a shows the grid angle observer tracks the phase angle of the grid. A PLL is used here to display the grid angle, but this PLL is not used in the scheme. Figure 6b represents the estimated grid frequency, where the estimated voltage of the grid can be seen in Figure 6c. The waveforms from the PLL algorithm are presented here solely for comparative analysis. It is evident that the method provides excellent dynamic performance, as well as validates Theorem 1. Figure 5 and Figure 7 demonstrate the scheme produces a bounded and realizable control signal. A smooth, bounded control signal in this figure indicates that the proposed method effectively regulates the inverter’s operation without instability or excessive oscillations.

5. Conclusions

A scheme has been presented that controls the inverter current in a single-phase AC system without requiring any information of grid information ($\theta \left(t\right)$, $\omega \left(t\right)$, $V_g\left(t\right)$) or the use of any cascade control scheme. Additionally, excluding the need for any synchronization unit or PLL, self-synchronization between the grid and the inverter used in the grid has been guaranteed with the scheme proposed in the paper. The scheme adaptively estimates the angle, frequency, and magnitude of the grid. A Lyapunov closed-loop system analysis has validated the controller and estimation scheme. Experimental results further validate the control scheme. Experimental results validate the approach, demonstrating accurate phase tracking and stable operation under grid-connected conditions.

In future work, we will take two paths. The first will complete a more in-depth study to consider a sensitivity analysis for this approach to consider a wide range of parameter variations. This study will consider optimal control gain sections, as well as sensitivity to non-ideal parameter values. Other trials will consider grid abnormalities and their effects on our proposed scheme. The second track of our future work will extend the proposed self-synchronizing control approach to grid-forming inverters. This will involve developing a robust control strategy to ensure stable voltage and frequency regulation, enabling seamless operation in microgrid and islanded scenarios.