Control for a DC Microgrid for Photovoltaic–Wind Generation with a Solid Oxide Fuel Cell, Battery Storage, Dump Load (Aqua-Electrolyzer) and Three-Phase Four-Leg Inverter (4L4W)

Abstract

This paper proposes a nonlinear control strategy for a microgrid, comprising a PV generator, wind turbine, battery, solid oxide fuel cell (SOFC), electrolyzer, and a three-phase four-leg voltage source inverter (VSI) with an LC filter. The microgrid is designed to supply unbalanced AC loads while maintaining high power quality. To address chattering and enhance control precision, a super-twisting algorithm (STA) is integrated, outperforming traditional PI, IP, and classical SMC methods. The four-leg VSI enables independent control of each phase using a dual-loop strategy (inner voltage, outer current loop). Stability is ensured through Lyapunov-based analysis. Scalar PWM is used for inverter switching. The battery, SOFC, and electrolyzer are controlled using integral backstepping, while the SOFC and electrolyzer also use Lyapunov-based voltage control. A hybrid integral backstepping–STA strategy enhances PV performance; the wind turbine is managed via integral backstepping for power tracking. The system achieves voltage and current THD below 0.40%. An energy management algorithm maintains power balance under variable generation and load conditions. Simulation results confirm the control scheme’s robustness, stability, and dynamic performance.

1. Introduction

The integration of renewable energy sources (RESs) into microgrids introduces challenges due to their intermittent and nonlinear behavior. This paper studies a stand-alone DC microgrid depicted in Figure 1, composed of a PV generator, wind turbine (PMSG), lithium-ion battery, solid oxide fuel cell (SOFC), and an aqua-electrolyzer used as a dump load. Each component is interfaced with the DC bus via appropriate DC-DC converters: boost for PV and SOFC, buck-boost for the battery, and buck for the electrolyzer. The SOFC operates when the battery’s SOC falls below 20%, while the electrolyzer absorbs excess energy when the SOC exceeds 80%. The electrolyzer generates hydrogen, which can be stored for later uses as an input to the SOFC. In [1], a hybrid microgrid combining solar energy and wind turbines with doubly fed induction generators and fuel cells was proposed to streamline energy conversion processes. However, the design did not incorporate batteries to support fuel cell operations and power management. In [2], the MPC-based controller employs a nonlinear system model combined with ARIMA forecasting to anticipate environmental and load disturbances. However, the PV system continuously operates at its Maximum Power Point Tracking (MPPT) without taking into account the potential overcharging or undercharging conditions of the Battery Energy Storage (BES). In [3], a hybrid microgrid comprising fuel cells (FCs), an electrolyzer, and a Battery Bank Energy Storage System (BBESS) is presented, demonstrating confirmed performance and efficiency. However, to achieve optimal efficiency and power stability, the implementation of a three-level inverter is essential. In [4], a third energy source is essential. In this setup, a Supercapacitor Energy Storage System (SCESS) is used for power balancing, while a fuel cell and electrolyzer are integrated to improve energy quality [5].

Sliding mode control (SMC) is commonly utilized in photovoltaic (PV) system regulation because of its strong robustness and high control accuracy, particularly in environments with significant disturbances [6]. However, a major limitation of SMC is the chattering effect, which can reduce system reliability and hinder practical deployment. In response to this issue, researchers have investigated alternative control strategies. Backstepping control (BC) is a widely used nonlinear method developed to address the drawbacks of sliding mode control (SMC). To enhance system robustness, it is often combined with other techniques, including integration with SMC [7,8]. Model Predictive Control (MPC) has been applied to DC-DC boost converters [9,10], and Stochastic Fuzzy MPPT techniques have also been proposed [11]. While these methods deliver solid performance outcomes, they typically rely on comprehensive and precise information about PV system parameters such as ambient temperature and solar irradiance, which may not always be readily accessible or easy to measure accurately in real-world conditions.

The proposed integral backstepping–super twisting algorithm (IBSTA) combines robustness and fast convergence, making it suitable for systems with uncertainties and disturbances. Compared to Model Predictive Control (MPC), which excels in constraint handling but requires significant computational resources, IBSTA offers lower complexity and faster response, favoring real-time applications. Fuzzy Logic Control (FLC), while simple and adaptable to imprecise models, may lack strong stability guarantees in highly nonlinear systems. IBSTA thus provides a balanced alternative, offering both robustness and practical feasibility for embedded implementations.

Due to nonlinearity in the system components, advanced power electronic interfaces are essential. A four-leg, two-level voltage source inverter (VSI) is used to regulate output power quality and manage unbalanced loads [12]. These inverters are widely used in APF [13,14,15], UPS [16], DVR [17], UPQC [18], and stand-alone systems [19,20], and elimination of current leakage in PV applications [21].

In control design, various strategies have been explored for four-leg inverters, including the pole-placement method [22], sliding mode control (SMC) [23], deadbeat (DB) model-based control [24], MPC [25], and the resonant and repetitive control methods [26,27]. This paper proposes an integral backstepping control method based on Lyapunov theory for output voltage and current regulation. It ensures fast, robust dynamic performance with reduced complexity compared to sliding mode and hysteresis-based methods.

The control coordination between the multiple energy sources, the battery, and the dump load is illustrated in the energy management chart shown in Figure 2.

Results demonstrate clearly that the control strategy proposed in this paper provides the best solution, with good tracking and optimization performance, fast dynamic response, and stable static power output, even when weather conditions (irradiation and wind speed) are rapidly changing.

The proposed strategy is developed in the next sections. The control and quality energy performances are presented using Matlab/Simulink 2017a.

In this paper, we propose a robust nonlinear control scheme for a multi-source microgrid to maintain high power quality under unbalanced AC loads. To reduce chattering in conventional sliding mode control, we incorporate a super-twisting algorithm (STA) within a hybrid integral backstepping–STA (IBSTA) controller, improving PV power extraction and dynamic response. System stability is ensured via Lyapunov-based analysis, and voltage/current THD is maintained below 0.40%. Additionally, an adaptive energy management algorithm coordinates generation and storage under dynamic conditions.

The main contributions are as follows. We develop an integrated control framework combining integral backstepping, Lyapunov-based methods, and second-order sliding mode control through the STA. A four-leg voltage source inverter enables independent phase control, enhancing voltage quality under unbalanced loads. We also propose a real-time energy management strategy for coordinated operation of distributed energy resources and storage components.

The remainder of the paper is organized as follows. Section 2 presents the modeling of the DC microgrid components. Section 3 describes the case study and evaluates system performance through MATLAB/Simulink simulations. Finally, Section 4 provides the conclusions.

2. Modeling of DC Microgrid Components

2.1. Modeling of the PV Energy System in the DC Microgrid

This section focuses on the photovoltaic (PV) subsystem integrated into the DC microgrid depicted in Figure 1. The PV system consists of four identical arrays connected in parallel, with each array formed by eight modules arranged in series. Under standard irradiance conditions (G = 1000 W/m²), the PV module operates with a maximum power point voltage of approximately 30.1 V, a short-circuit current of 8.83 A, and an open-circuit voltage of 37.4 V, as illustrated in Figure 3 and Figure 4. To interface the PV arrays with the DC bus, a DC-DC boost converter is implemented. This converter serves dual purposes: (1) to maximize energy harvesting from the PV arrays through a maximum power point tracking (MPPT) algorithm, and (2) to elevate the lower PV voltage to the required DC bus level. The control strategy continuously monitors the PV voltage $V_{pv}$, PV current $I_{pv}$, and the output voltage of the boost converter $V_{dc}$ to regulate the switching and ensure optimal operation. The approach ensures stable DC bus voltage while maintaining high energy conversion.

The boost converter’s average state model is given by the following equations:

$$\left\{\begin{array}{c}{\displaystyle \frac{d{x}_0}{dt}}={\displaystyle \frac{1}{C_{pv}}}{i}_{pv}-{\displaystyle \frac{1}{C_{pv}}}{x}_1 \\ {\displaystyle \frac{d{x}_1}{dt}}={\displaystyle \frac{1}{L}}{x}_0-{\displaystyle \frac{(1-u_{pv})}{L}}{V}_{dc} \end{array}\right.$$

(1)

where:

$$x={\left[x_{0 }{x}_1\right]}^T={\left[V_{pv} I_L\right]}^T$$

(2)

Control of PV System: Integral Backstepping–STA Controller

The first variable that represents the tracking error is formulated as:

$$E_1=x_0-V_{pvref}$$

(3)

The integrator of the error is:

$${\gamma }_1={\int }_0^t{(x}_0-V_{pvref})dt$$

(4)

Based on system (1.2), the derivative of the tracking error is expressed as:

$$\dot{E_1}={\displaystyle \frac{1}{C_{pv}}}{i}_{pv}-{\displaystyle \frac{1}{C_{pv}}}{x}_1-\dot{{ V}_{pvref}}$$

(5)

The Lyapunov candidate function is defined as:

$$V_1={\displaystyle \frac{1}{2}}{E_1}^2+{\displaystyle \frac{1}{2}}{K_0{\gamma }_1}^2$$

(6)

where $K_0$ is a design parameter and must remain a positive constant.

This Lyapunov function represents a measure of system energy in terms of the PV voltage tracking error and its integral. Physically, it quantifies how far the system is from the desired voltage $V_{pvref}$. Ensuring that this function decreases over time implies that the error and its accumulation reduce steadily, leading the PV output voltage to converge toward the reference value.

The time derivative of ${ V}_1$ obtained from Equation (5) is:

$$\begin{array}{cc}{\dot{V}}_1& =\dot{E_1} E_1+K_0 E_1 {\gamma }_1\hfill \\ & =E_1({\displaystyle \frac{1}{C_{pv}}}{i}_{pv}-{\displaystyle \frac{1}{C_{pv}}}{x}_1-\dot{V_{pvref}+}{K}_0 {\gamma }_1)\hfill \end{array}$$

(7)

Judicious choice of the entity within the brackets permits:

$${\displaystyle \frac{1}{C_{pv}}}{i}_{pv}-{\displaystyle \frac{1}{C_{pv}}}{x}_1-\dot{V_{pvref}+}{K}_0 {\gamma }_1=-k_1 E_1$$

(8)

The derivative of ${ V}_1$ versus time becomes:

$${\dot{V}}_1=E_1({\displaystyle \frac{1}{C_{pv}}}{i}_{pv}-{\displaystyle \frac{1}{C_{pv}}}{x}_1-\dot{V_{pvref}+}{K}_0 {\gamma }_1)=-k_1 {E_1}^2$$

(9)

The condition ${\dot{V}}_1$, where, $k_1>0$, guarantees asymptotic stability of the voltage tracking error.

This has direct physical importance: it ensures the photovoltaic panel voltage will follow the reference set point despite internal dynamics, ensuring maximum power point operation or stable output behavior depending on the reference strategy.

The chosen surface:

$${ S}_{pv}=\left(i_{pv}-x_1-\dot{C_{pv}{ V}_{pvref}+}{C}_{pv}{K}_0 {\gamma }_1\right)+C_{pv}{k}_1 E_1$$

(10)

The derivative of the surface:

$$\dot{S_{pv}}=\dot{i_{pv}}-\dot{x_1}-C_{pv} \ddot{V_{pvref}}+C_{pv}{K}_0 E_1+C_{pv}{k}_1 \dot{E_1}$$

(11)

Integrating $\dot{x_1}$ from (1) in (11), $\dot{S_{pv}}$ becomes:

$$\dot{S_{pv}}=\dot{i_{pv}}-{\displaystyle \frac{1}{L}}{x}_0+{\displaystyle \frac{(1-u_{pv})}{L}}{V}_{dc}-C_{pv} \ddot{V_{pvref}}+C_{pv}{K}_0 E_1+C_{pv}{k}_1 \dot{E_1}$$

(12)

The second Lyapunov function considered is:

$$V_2={\displaystyle \frac{1}{2}}{E_1}^2+{\displaystyle \frac{1}{2}}{K_0{\gamma }_1}^2+{\displaystyle \frac{1}{2}}{S_{pv}}^2$$

(13)

The second Lyapunov function $V_2$ extends the stability analysis to include the sliding surface $S_{pv}$, which incorporates dynamic system variables and enhances the robustness of control. This reflects the total error of the system, including deviations in output voltage and their rates of change, which is essential for handling fast dynamics in PV systems.

The derivative of $V_2$ is:

$${\dot{V}}_2=\dot{E_1} E_1+K_0 E_1 {\gamma }_1+{ S}_{pv} \dot{S_{pv}}$$

(14)

Integrating $\dot{S_{pv}}$ from Equation (12) in Equation (14), ${\dot{V}}_2$ becomes:

$${\dot{V}}_2=\dot{E_1} E_1+K_0 E_1 {\gamma }_1+{ S}_{pv} ( \dot{i_{pv}}-{\displaystyle \frac{1}{L}}{x}_0+{\displaystyle \frac{\left(1-u_{pv}\right)}{L}}{V}_{dc}-C_{pv} \ddot{V_{pvref}}+C_{pv}{K}_0 {\gamma }_1+C_{pv}{k}_1 \dot{E_1})$$

(15)

The derivative ${\dot{V}}_2$ combines both tracking error dynamics and the system’s surface dynamics, allowing us to shape the closed-loop behavior. From a practical viewpoint, minimizing ${\dot{V}}_2$ leads to fast convergence and improved transient performance, which are critical for grid-connected PV applications subject to changing environmental conditions.

The surface law chosen in this case is:

$$\dot{S_{pv}}=-\rho {\left|S_{pv}\right|}^{{\displaystyle \frac{1}{2}}}sign\left(S_{pv}\right)-\sigma \int sign\left(S_{pv}\right)$$

(16)

This sliding mode law introduces robustness against model uncertainties and external disturbances by enforcing finite-time convergence to the sliding surface. Physically, this translates into the system rejecting small perturbations in irradiance or load without losing voltage regulation performance.

Equation (12) can be written as a nonlinear states space equation in the following form:

$$\dot{S_{pv}}=A\left(V_{pv}\right)+B\left(V_{pv}\right)u_{pv}$$

(17)

where:

$$\begin{array}{cc}A\left(V_{pv}\right)=& \dot{i_{pv}}-{\displaystyle \frac{1}{L}}{x}_0+{\displaystyle \frac{1}{L}}{V}_{dc}-{ C}_{pv} \ddot{V_{pvref}}+C_{pv}{K}_0 {\gamma }_1\hfill \\ & +C_{pv}{k}_1 \dot{E_1}\hfill \end{array}$$

(18)

$$B\left(V_{pv}\right)=-{\displaystyle \frac{1}{L}}{V}_{dc}$$

(19)

The equivalent control law $u_{pveq }$ is derived when $\dot{S_{pv}}=0$ [28]. Its expression can be obtained from Equation (17) as:

$$u_{pveq }=-{\displaystyle \frac{A\left(V_{pv}\right)}{B\left(V_{pv}\right)}}$$

(20)

Then the equivalent control law $u_{pveq }$ is equal to:

$$u_{pveq }=(\dot{i_{pv}}-{\displaystyle \frac{1}{L}}{x}_0+{\displaystyle \frac{1}{L}}{V}_{dc}-C_{pv} \ddot{V_{pvref}}+C_{pv}{K}_0 {\gamma }_1+C_{pv}{k}_1 \dot{E_1}) {\displaystyle \frac{L}{V_{dc}}}$$

(21)

where $x_0=V_{pv}$.

The discontinuous function $u_{pvn}$ is defined as follows:

$$u_{pvn }={\displaystyle \frac{u_{STA }}{B\left(V_{pv}\right)}}=-u_{STA }\left({\displaystyle \frac{L}{V_{dc}}}\right) .$$

(22)

where $u_{STA}$ is the STA term defined by [29].

$$u_{STA }=-\rho {\left|S_{pv}\right|}^{{\displaystyle \frac{1}{2}}}sign\left(S_{pv}\right)-\sigma \int sign\left(S_{pv}\right)$$

(23)

According to [30], the adaptive gains of the algorithm gains are calculated as follows:

$$\dot{\rho }=\left\{\begin{array}{c}\Omega \sqrt{{\displaystyle \frac{\theta }{2}} }sign\left(\left|S_{pv}\right|-\delta \right)\\ \varphi if {\alpha }_b \le {\alpha }_c\end{array}\right. \mathrm{i}\mathrm{f} {\mathsf{\alpha }}_{\mathrm{b}}>{\mathsf{\alpha }}_{\mathrm{c}}$$

(24)

$$\sigma =\partial {\mathsf{\alpha }}_{\mathrm{b}}$$

(25)

where $\Omega$, $\rho$, $\mathsf{\delta }$, $\partial$, θ, $\sigma$, φ, ${\mathsf{\alpha }}_{\mathrm{b}}$, and ${\mathsf{\alpha }}_{\mathrm{c}}$ are positive constants.

Then the expression for the control law $u_{pv }$ is determined by [28]:

$$u_{pv}=u_{pveq }+u_{pvn }$$

(26)

From Equations (21)–(23) the final expression of $u_{pv}$ is:

$$u_{pv}=(\dot{i_{pv}}-{\displaystyle \frac{1}{L}}{V}_{pv}+{\displaystyle \frac{1}{L}}{V}_{dc}-C_{pv} \ddot{V_{pvref}}+C_{pv}{K}_0 E_1+C_{pv}{k}_1 \dot{E_1}+\rho {\left|S_{pv}\right|}^{{\displaystyle \frac{1}{2}}}sign\left(S_{pv}\right)+\sigma \int sign\left(S_{pv}\right)){\displaystyle \frac{L}{V_{dc}}}$$

(27)

The final control law $u_{pv}$ combines equivalent and discontinuous components to achieve both accuracy and robustness. It is physically implementable through duty-cycle modulation of the DC-DC converter. The structure ensures that the PV voltage remains tightly regulated around the reference, even under rapid changes in solar input or load conditions.

Proof of the derivative of $V_2$ must be negative:

From Equation (13), $V_2$ is expressed as follows:

$$V_2={\displaystyle \frac{1}{2}}{E_1}^2+{\displaystyle \frac{1}{2}}{K_0{\gamma }_1}^2+{\displaystyle \frac{1}{2}}{S_{pv}}^2$$

This expression demonstrates that $V_2\ge 0$.

The derivative of $V_2$ is expressed as follows:

$${\dot{V}}_2={\dot{V}}_1+{ S}_{pv} \dot{S_{pv}}=\dot{E_1} E_1+K_0 E_1 {\gamma }_1+{ S}_{pv} \dot{S_{pv}}$$

Integrating the value of ${\dot{V}}_1$ from (8), ${\dot{V}}_2$ becomes:

$${\dot{V}}_2=-k_1 {E_1}^2+{ S}_{pv} \dot{S_{pv}}$$

(28)

From Equation (16), we integrate the expression of $\dot{S_{pv }}$ in the term

$$S_{pv} \dot{S_{pv}}:$$

$$\begin{array}{cc}\hfill { S}_{pv} \dot{S_{pv}}& ={ S}_{pv}(-\rho {\left|S_{pv}\right|}^{{\displaystyle \frac{1}{2}}}sign\left(S_{pv}\right)-\sigma \int sign\left(S_{pv}\right)) \hfill \\ & =-S_{pv}.sign(S_{pv})(\rho {\left|S_{pv}\right|}^{{\displaystyle \frac{1}{2}}}+\sigma t) ˂ 0\hfill \end{array}$$

(29)

Then, from Equations (26) and (27), ${\dot{V}}_2$ can be expressed:

$${\dot{V}}_2=-k_1 {E_1}^2-S_{pv}.sign(S_{pv})(\rho {\left|S_{pv}\right|}^{{\displaystyle \frac{1}{2}}}+\sigma t)$$

(30)

$$=(-k_1 {E_1}^2-\left|S_{pv}\right|.(\rho {\left|S_{pv}\right|}^{{\displaystyle \frac{1}{2}}}+\sigma t)\le 0$$

(31)

where $k_1,\rho$, and $\sigma$ are positive constants and $t$ is time. This assumption results in a negative derivative of the Lyapunov function $\dot{V_2}$, which guarantees that the error variables $(E_1,{\gamma }_1,S_{pv})$ will converge asymptotically to the origin. This, in turn, ensures that $x_0$ converges asymptotically to the origin $V_{pvref}$, thereby allowing the PV system to extract the maximum power from the array. The MPPT algorithm’s objective is to calculate the optimal output voltage $V_{pvref}$, ensuring that the system operates at its MPP. For this purpose, the Incremental Conductance (IncCond) algorithm is employed, and its steps are as follows:

$$\left\{\begin{array}{c}\begin{array}{c}{V}_{pvref}\left(t\right)=V_{pvref}\left(t-h\right)+k_{ink}sign\left({\displaystyle \frac{i_{pv}}{V_{pv}}}+{\displaystyle \frac{{∆I}_{pv}}{{∆V}_{pv}}}\right)\\ {∆i}_{pv}=i_{pv}\left(t\right)-i_{pv}\left(t-h\right)\\ {∆v}_{pv}=V_{pv}\left(t\right)-V_{pv}\left(t-h\right)\\ k_{ink}=k_3\left|∆V_{pv}\left(t\right)\right|\end{array} \\ \end{array}\right.$$

(32)

The control configuration of the DC-DC boost converter within the photovoltaic system is depicted in Figure 5.

General Integral Backstepping Control Framework:

The integral backstepping control (IBC) method is used for current regulation in the battery, wind, fuel cell, and dump load systems:

Let the error be defined as:

$$e=x_{ref}-x\mathrm{or} e=x-x_{ref}$$

(33)

Its derivative:

$$\dot{e}=\dot{x_{ref}}-\dot{x}\mathrm{or} \dot{e}=\dot{x}-\dot{x_{ref}}$$

(34)

The integral of the error:

$$\gamma ={\int }_0^t(x_{ref}-x)dt\mathrm{or} \gamma ={\int }_0^t(x-x_{ref})dt$$

(35)

The Lyapunov candidate function:

$$\mathrm{V}={\displaystyle \frac{1}{2}}{e}^2+{\displaystyle \frac{1}{2}}{(K}_1{\gamma )}^2$$

(36)

The derivative of the Lyapunov function is:

$$\dot{V}=\dot{e} e+K_1\gamma e$$

(37)

To ensure stability:

$$\dot{e}+K_1\gamma =-Ke$$

(38)

where $K_1$ and $K$ are positive constants.

This leads to a control law structure that is tailored to each subsystem’s dynamic model.

2.2. Modelling Battery in DC Microgrid

The DC-DC bidirectional buck-boost converter links the BESS [31,32,33] to the DC bus. The buck-boost converter includes two IGBT switches $Q_1$ and $Q_2$, an inductor $L_{bat}$ with its equivalent series resistance $R_{bat}$, and an output capacitor $C_{dc}$, and the operating voltage is around 220 V.

The switches are activated by binary signals sent to their gates. The converter, responsible for regulating the DC bus voltage, can operate in both charging and discharging modes, depending on the load and state of charge (SOC). In the discharging mode (Boost), with $Q_1$ on and $Q_2$ off, the battery supplies energy to the DC bus ($I_{batref}$ > 0). When $Q_1$ is off and $Q_2$ is on, the converter enters buck mode ($I_{batref}$ < 0), and the battery is charged.

The buck-boost converter’s average state model is given by the following equations:

$${\displaystyle \frac{d{I}_{bat}}{dt}}={\displaystyle \frac{V_{bat}{d}_b}{L_{bat}}}-\left(1-d_b\right){\displaystyle \frac{V_{dc}}{L_{bat}}}-{\displaystyle \frac{R_{bat}}{L_{bat}}}{I}_{bat}$$

(39)

$${\displaystyle \frac{d{V}_{dc}}{dt}}=\left({1-d}_b\right){\displaystyle \frac{I_{bat}}{C_{dc}}}-{\displaystyle \frac{I_{bato}}{C_{dc}}}$$

(40)

where $V_{dc}$, $I_{bat},$ $I_{bato}$, $C_{dc}$, and $d_b$ are average-state values of battery output voltage, battery current, the output current of the converter, the output capacitor, and the control signal.

2.2.1. Battery Current Control

Applying the IBC framework to the battery current control, where:

$$x=I_{bat}$$

$$x_{ref}=I_{batref}$$

The error chosen is:

$$e_{bat}=I_{bat}-I_{batref}$$

(41)

By combining Equation (39) with Equation (41), the derivative of ${ e}_{bat}$ can be expressed as:

$$\dot{e_{bat}}={\displaystyle \frac{V_{bat}{d}_b}{L_{bat}}}-\left(1-d_b\right){\displaystyle \frac{V_{dc}}{L_{bat}}}-{\displaystyle \frac{R_{bat}}{L_{bat}}}{I}_{bat}-\dot{I_{batref}}$$

(42)

From the latter equation and Equation (38) we can deduce:

$$d_b=\left(\dot{I_{batref}}+{\displaystyle \frac{V_{dc}}{L_{bat}}}+{\displaystyle \frac{R_{bat}}{L_{bat}}}{I}_{bat}\right){\displaystyle \frac{L_b}{V_{dc}+V_{bat}}}-{(K}_{bat}{e}_{bat}+K_{bat1}{\gamma }_{bat}){\displaystyle \frac{L_b}{V_{dc}+V_{bat}}}$$

(43)

where $K_1$ and $K$ are replaced successively by $K_{bat1}$ and $K_{bat}$.

2.2.2. DC Bus Voltage Regulation Based on STA Controller

The system model Equation (40) can be written as a nonlinear states space equation in the following form:

$$\dot{x}=a\left(x\right)+b\left(x\right)u_i$$

(44)

where $x=V_{dc}$, the DC bus voltage,

$$a\left(x\right)=-{\displaystyle \frac{I_{bato}}{C_{dc}}}$$

(45)

and:

$$b\left(x\right)=\left({1-d}_b\right){\displaystyle \frac{1}{C_{dc}}}$$

(46)

$b\left(x\right)\ne 0\forall t$, where t is time, and $u_i$ is the battery reference current $I_{batref}$ produced by the DC link voltage controller (STA controller).

We consider C, Km, KM, Um, and Q the arbitrary positive constants of the STA controller which are determined by considering the following condition of convergence given by [30].

$$\left|\dot{a}\left(x\right)\right|+\mathrm{U}\mathrm{m}\left|\dot{b}\left(x\right)\right|\le \mathrm{C};\mathrm{K}\mathrm{m}\le b\left(x\right)\le \mathrm{K}\mathrm{M};\left|{\displaystyle \frac{a\left(x\right)}{b\left(x\right)}}\right|<q\mathrm{M};0<q<1$$

The primary objective of employing the super-twisting algorithm (STA) controller is the creation of a second-order sliding mode on the surface $S_{Vdc}$ by cancelling $S_{Vdc}\left(Vdc\right)$ and its derivative $\dot{S_{Vdc}}\left(Vdc\right)$ in a limited time ($\dot{S_{Vdc}}\left(Vdc\right)=$ $S_{Vdc}\left(x\right)=0$). The STA controller is used in Equation (40) with a relative degree equal to one. The sliding surface is defined by:

$$S_{Vdc}\left(Vdc\right)=V_{dc}-V_{dcref}$$

(47)

In view of Equation (44), the first derivative of $S_{Vdc}\left(Vdc\right)$ is calculated as follows:

$$\dot{S_{Vdc}}=a\left(V_{dc}\right)+b\left(V_{dc}\right)u_i-\dot{V_{dcref}}$$

(48)

The parameter $V_{dcref}$ represents the reference voltage for the DC bus and is considered a constant value.

The control unit of a STA controller is constructed by two terms, a continuous sliding mode component ($u_{eq}$), and a discontinuous term ($u_n$). Therefore, the total control law is given by [28]:

$$u_i=u_{eq}+u_n$$

(49)

By imposing the condition $\dot{S_{Vdc}}=0$, the equivalent control ($u_{eq}$) is calculated [28]. Based on Equations (45), (46), and (48), its expression can be derived as follows:

$$u_{eq}=-{\displaystyle \frac{a\left(V_{dc}\right)}{b\left(V_{dc}\right)}}={\displaystyle \frac{I_{bato}}{\left({1-d}_b\right)}}$$

(50)

We define $u_n$ as follows:

$$u_n={\displaystyle \frac{u_{st}}{b\left(V_{dc}\right)}}$$

(51)

where,

$$u_{st}=-\alpha {\left|S_{Vdc}\right|}^{{\displaystyle \frac{1}{2}}}sign\left(S_{Vdc}\right)-\lambda \int sign\left(S_{Vdc}\right)$$

(52)

In this equation, $\alpha$ and $\lambda$ are the adaptive gains given by [30].

The following procedure is used to determine the adaptive gains of the algorithm:

$$\dot{\alpha }=\left\{\begin{array}{c}\omega \sqrt{{\displaystyle \frac{\phi }{2}} } sign\left(\left|S_{bat}\right|-\gamma \right)\\ \mu if {\alpha }_a \le {\alpha }_l\end{array}\right.\mathrm{ }\mathrm{i}\mathrm{f}\mathrm{ }{\mathsf{\alpha }}_{\mathrm{a}}\mathrm{ }>{\mathsf{\alpha }}_{\mathrm{l}}$$

(53)

$$\lambda =\tau \alpha$$

(54)

where, ${\mathsf{\alpha }}_{\mathrm{l}}$, $\mu$, $\omega$, τ, $\lambda$,$\gamma$, φ, and ${\alpha }_a$ are positive constants.

By compensating the term $u_i$ in Equation (49) with $I_{batref}$, its expression can be deduced as follows:

$$I_{batref}=({\displaystyle \frac{I_{bato}}{\left({1-d}_b\right)}})+(-\alpha {\left|S_{Vdc}\right|}^{{\displaystyle \frac{1}{2}}}sign(S_{Vdc})-\lambda \int sign\left(S_{Vdc}\right))C_{dc}$$

(55)

In the process of voltage control, it can be assumed that the battery current is controlled by the integral backstepping controller because the dynamic of the current controller is much faster than that of the voltage controller (see Figure 6 and Figure 7). In addition, $R_{bat}$ and $L_{bat}$ are very small; therefore, Equation (43) can be simplified as:

$$d_{bat}\approx {\displaystyle \frac{V_{dc}}{V_{dc}+V_{bat}}}$$

(56)

Based on Equation (56), the battery’s reference current can be expressed as:

$$I_{batref}=I_{bato}({\displaystyle \frac{V_{dc}{+V}_{bat}}{V_{bat}}})+(-\alpha {\left|S_{Vdc}\right|}^{{\displaystyle \frac{1}{2}}}sign(S_{Vdc})-\lambda \int sign(S_{Vdc}))C_{dc}({\displaystyle \frac{V_{dc}{+V}_{bat}}{V_{bat}}})$$

(57)

To ensure the stability of the Super-Twisting Algorithm (STA) controller, the Lyapunov function $V_b$ must have a negative time derivative for all $S_{Vdc}\ne 0$. Where,

$$V_b={\displaystyle \frac{1}{2}}{S_{Vdc}}^2$$

(58)

$$\dot{S_{Vdc}}=-\alpha {\left|S_{Vdc}\right|}^{{\displaystyle \frac{1}{2}}}sign(S_{Vdc})-\lambda \int sign\left(S_{Vdc}\right)\mathrm{d}\mathrm{t}$$

(59)

The time derivative of $V_b$ can be written as:

$$\dot{V_b}=S_{Vdc}\dot{S_{Vdc}}$$

(60)

$$S_{Vdc}\dot{S_{Vdc}}=-S_{Vdc}.sign(S_{Vdc})(\alpha {\left|S_{Vdc}\right|}^{{\displaystyle \frac{1}{2}}}+\lambda t) ˂ 0$$

(61)

Therefore, the stability of the STA controller is verified through the Lyapunov condition. As a result, the Lyapunov condition can consistently be satisfied, ensuring that the DC bus voltage converges to its constant reference value under all operating conditions.

The control strategy for a battery-connected DC-DC buck-boost converter is depicted in Figure 8.

2.3. Wind Energy Conversion System Modeling for DC Microgrid [34]

In continuous conduction mode (CCM), a DC-DC boost converter is employed to regulate the rotational speed of the turbine-generator assembly by functioning as an apparent load for the generator. The converter includes an input inductor $L_w$ with an associated series resistance $R_w$, a diode D, output capacitor $C_{dc}$, and IGBT switch Sb. The system’s dynamic behavior can be represented by the average-state model, which is formulated through the following set of differential equations:

$${\displaystyle \frac{d{V}_{dc}}{dt}}=-{\displaystyle \frac{i_{wo}}{C_{dc}}}+\left(1-u_w\right){\displaystyle \frac{i_w}{C_{dc}}}$$

(62)

$${\displaystyle \frac{d{i}_{\mathrm{w}}}{dt}}={\displaystyle \frac{V_{inw}}{L_w}}-{\displaystyle \frac{R_w}{L_w}}\mathrm{ }{i}_w-\left(1-\mathrm{ }{u}_w\right){\displaystyle \frac{V_{dc}}{L_w}}\mathrm{ }$$

(63)

The average-state quantities of the converter are denoted as follows: $V_{dc}$ for the voltage across the DC bus, $V_{inw}$ for the input voltage, $i_w$ for the inductor current, $u_w$ for the control signal, and $i_{wo}$ for the output current of the boost converter.

Wind Energy Conversion System Control Strategy

Applying the IBC framework to the wind energy conversion current control, where:

$$x=i_w$$

$$x_{ref}=i_{wref}$$

The chosen error is:

$$e_w=i_{wref}-i_w$$

(64)

Using the system dynamics from Equations (62) and (63), and applying the same IBC methodology, we derive the control input:

$$u_w=1+{\displaystyle \frac{d{i}_{wref}}{dt}}{\displaystyle \frac{L_w}{V_{dc}}}-{\displaystyle \frac{V_{inw}}{V_{dc}}}+{\displaystyle \frac{R_w}{V_{dc}}}{i}_w+{\displaystyle \frac{K_w{e}_w{L}_w}{V_{dc}}}+{\displaystyle \frac{{\gamma }_w{L}_w}{V_{dc}}}$$

(65)

where $K_1$ and $K$ are replaced successively by 1 and $K_W$.

The MPPT technique is employed to compute the desired reference current.

The reference torque was determined by estimating the generator speed (${\omega }_m$), as given by [34]:

$$T_{mref}=K_{opt}{{ \omega }_m}^2$$

(66)

$$K_{opt}=1.67\times {10}^{-3}\mathrm{N}\mathrm{m}/\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$$

(67)

Based on the wind turbine characteristics, this value is considered constant. The corresponding reference current is extracted using the measured input voltage to the rectifier voltage ($V_{inw}$), according to:

$$i_{wref}={\displaystyle \frac{T_{mref}{\omega }_m}{V_{inw}}}$$

(68)

The characteristics of the wind turbine in case 1:$i_w$ and its reference are depicted in Figure 9, the mechanical torque $T_m$, reference torque $T_{mref}$, and generator electromagnetic torque $T_e$ are depicted in Figure 10, the power coefficient Cp is depicted in Figure 11, the angular speed ${\omega }_m$ (rad/s) is depicted in Figure 12, the tip speed ratio λ is depicted in Figure 13 and the wind speed $V_w$ (m/s) is depicted in Figure 14.

The regulation approach for the DC-DC boost converter integrated in wind power system is presented in Figure 15.

2.4. Modeling Approach for SOFC System in DC Microgrid [35,36]

A solid oxide fuel cell (SOFC) model (

Table 1. SOFC parameters.

Parameter	Value
Absolute temperature (T)	1273 K
Universal gas constant (R)	8314 J/(kmol K)
Faraday’s constant (F)	96,487 C/kmol
Ideal standard potential (${\mathrm{E}}_0$)	1.18 V
Number of cells in series in stack (${\mathrm{N}}_0$)	325
Constant, ${\mathrm{K}}_{\mathrm{r}}={\mathrm{N}}_0/4\mathrm{F}$	$0.842\times {10}^{-6} \mathrm{k}\mathrm{m}\mathrm{o}\mathrm{l}/\left(\mathrm{s}\mathrm{A}\right)$
Valve molar constant for hydrogen (${\mathrm{K}}_{{\mathrm{H}}_2}$)	$2.81\times {10}^{-4} \mathrm{k}\mathrm{m}\mathrm{o}\mathrm{l}/(\mathrm{s} \mathrm{a}\mathrm{t}\mathrm{m})$
Valve molar constant for oxygen (${\mathrm{K}}_{{\mathrm{O}}_2}$)	$2.52\times {10}^{-3} \mathrm{k}\mathrm{m}\mathrm{o}\mathrm{l}/(\mathrm{s} \mathrm{a}\mathrm{t}\mathrm{m})$
Response time for hydrogen flow (${\mathrm{T}}_{{\mathrm{H}}_2}$)	$26.1 \mathrm{s}$
Response time for water flow (${\mathrm{T}}_{{\mathrm{H}}_2\mathrm{O}}$)	$78.3 \mathrm{s}$
Response time for oxygen flow (${\mathrm{T}}_{{\mathrm{O}}_2}$)	$2.91 \mathrm{s}$
Ohmic loss/cell (r)	$32.813\times {10}^{-8} \mathsf{\Omega }$

) is developed based on references [35,36]. The SOFC operates at a voltage of 430 V, which is stepped up to the reference DC voltage ($V_{dcref}$) of 880 V using a boost converter. For MATLAB simulation, the stack output voltage is modeled and connected to a controlled voltage source, as illustrated in Figure 16. The dynamic SOFC model is linked to the DC bus via the boost converter.

The SOFC system is designed to meet the peak load requirements in scenarios where wind and solar power are unavailable, and the battery’s state of charge (SOC) drops to 20%. The sizing of the SOFC unit is determined based on the system’s peak load demand, which is 9.690 kW, as illustrated in Figure 17. To ensure optimal utilization and account for a safety margin, the SOFC is oversized by 20%, resulting in a final capacity of 11.628 kW.

The SOFC is interfaced with the DC microgrid through a DC-DC boost converter. The converter configuration includes an input inductor $L_{FC}$ with associated series resistance $R_{FC}$, a diode D, an output capacitor $C_{dc}$, and a controllable IGBT switch $S_C$. The dynamic behavior of the converter is represented using an average-state space model to facilitate control system design and performance analysis. The primary average-state variables in this model include the output current from the converter $I_{FCo}$, the inductor current $I_{FC }$, the DC bus voltage $V_{dc}$, the control signal $u_{FC }$, and the input voltage to the converter, denoted as $V_{inFC}$.

The average-state space model of the boost converter is expressed according to:

$${\displaystyle \frac{d{I}_{FC}}{dt}}={\displaystyle \frac{V_{inFC}}{L_{FC}}}-{\displaystyle \frac{R_{FC}}{L_{FC}}} I_{FC}-\left(1- u_{FC}\right){\displaystyle \frac{V_{dc}}{L_{FC}}}$$

(69)

$${\displaystyle \frac{d{V}_{dc}}{dt}}=-{\displaystyle \frac{I_{FCo}}{C_{dc}}}+\left(1-u_{FC}\right){\displaystyle \frac{I_{FC}}{C_{dc}}}$$

(70)

2.4.1. Fuel Cell Current Control

Here:

$$x=I_{FC}$$

$$x_{ref}=I_{FCref}$$

The current tracking error is defined as follows:

$$e_{FC}=I_{FCref}- I_{FC}$$

(71)

Using the system dynamics from Equations (69) and (70), and applying the same IBC methodology, we derive the control input:

$$u_{FC}=1+{\displaystyle \frac{d{I}_{FCref}}{dt}} {\displaystyle \frac{L_{FC}}{V_{dc}}}-{\displaystyle \frac{V_{inFC}}{V_{dc}}}+{\displaystyle \frac{R_{FC}}{V_{dc}}} I_{FC}+{\displaystyle \frac{K_{FC }. e_{FC} . L_{FC} }{V_{dc}}}+{\displaystyle \frac{{\gamma }_{FC}{L}_{FC}}{V_{dc}}}$$

(72)

where $K_1$ and $K$ are replaced successively by 1 and $K_{FC}$.

2.4.2. DC Bus Voltage Control Strategy

To ensure that the DC bus voltage remains constant, a controller based on Lyapunov stability theory is employed. The error signal $e_1$ is defined as the difference between the desired DC voltage and the measured DC voltage. This error can be formulated as follows:

$$e_{VFC}=V_{dcref}-V_{dc}$$

(73)

The Lyapunov function V is chosen as follows:

$$V={\displaystyle \frac{1}{2}} C_{dc}{{(e}_{VFC}(t))}^2+{\displaystyle \frac{1}{2}} K_{FC I}({{\int }_{{\mathit{t}}_0}^{\mathit{t}}{e}_{VFC}(t)dt)}^2$$

(74)

where $K_{FC I}$ denotes the integral gain of the DC bus voltage controller. Consequently, the derivative of the Lyapunov function becomes:

$$\dot{V}=C_{dc}{e}_{VFC}\left(t\right)\dot{e_{VFC}}+K_{FC I} .e_{VFC}\left(t\right).{\int }_{t_0}^t{e}_{VFC}\left(t\right)dt$$

(75)

After replacing the derivative of $V_{dc}$ from Equation (73) in Equation (75), the time derivative of the Lyapunov function V is obtained as:

$$\begin{gathered} \dot{V}=-C_{dc}{e}_{VFC}\left(t\right)\left(-{\displaystyle \frac{I_{FCo}}{C_{dc}}}+\left(1-u_{FC}\right){\displaystyle \frac{I_{FC}}{C_{dc}}}\right) \\ +K_{FC I} .e_{VFC}(t){\int }_{t_0}^t{e}_{VFC}\left(t\right)dt \end{gathered}$$

(76)

The current control dynamics are significantly faster than those of the voltage control loop (see Figure 18 and Figure 19, primarily because the current controller is based on an integral backstepping approach while the voltage controller is designed using a Lyapunov-based method. Since the resistances, $R_{SOFC}$ and $L_{FC}$, are very small, Equation (72) can be expressed as:

$$u_{FC}\approx 1-{\displaystyle \frac{V_{inFC}}{V_{dc}}} .$$

(77)

To guarantee that the voltage control error converges to zero, the derivative of V must be negative, leading to the following condition:

$$\begin{array}{c}-C_{dc}\left(-{\displaystyle \frac{I_{FCo}}{C_{dc}}}+\left(1-u_{FC}\right){\displaystyle \frac{I_{FC}}{C_{dc}}}\right)+K_{FC I}. {\int }_{t_0}^t{e}_{VFC}\left(t\right)dt\hfill \\ =-K_{pFC}{e}_{VFC}\left(t\right)\hfill \end{array}$$

(78)

In this case, $K_{pFC}$ represents a positive constant for the proportional gain. By inserting the value of $u_{FC}$ from Equation (77) in Equation (78), $I_{FCref}$ can be formulated as:

$$I_{FCref}={(I}_{FCo}+K_{FC l}.{\int }_{t_0}^t{e}_{VFC}\left(t\right)dt+K_{pFC}{e}_{VFC}\left(t\right) ){\displaystyle \frac{V_{dc}}{V_{inFC}}}$$

(79)

By incorporating Equation (79) into Equation (78) and considering the condition specified in Equation (77), the resulting expression for the derivative V is obtained via:

$$\dot{V}=-K_{pFC}{e_{VFC}\left(t\right)}^2\le 0$$

(80)

As a result, the Lyapunov stability criterion can consistently be fulfilled, guaranteeing the convergence of the DC bus voltage to its steady-state reference under all circumstances.

2.5. Load-Side Control of the Voltage Source Inverter (VSI)

As shown in Figure 20, the configuration of the four-leg inverter is followed by an output low-pass LC filter. Each phase of the inverter is connected to the load via an LC low-pass filter. To enable the zero-sequence component of the load current to flow, the fourth leg, marked as “n”, is linked to the neutral point of the load. This setup ensures that the four-leg inverter is capable of supporting highly unbalanced loads while delivering excellent performance under nonlinear load conditions, offering two distinct advantages. To mitigate switching of the current ripple and enhance load voltage quality, an inductor is placed in series with the fourth leg, similar to the inductors used in the other legs. This addition reduces the current rating of the switches in the fourth leg and also limits the fault current during short-circuit conditions. The four-leg inverter model in the abc reference frame is provided by Equations (81)–(84).

The topology of the three-phase four-leg inverter is depicted in Figure 20.

$${\displaystyle \frac{d{i}_{Lx}}{dt}}=({\displaystyle \frac{1}{L_f}}\mathrm{ }\left(u_{xn}-u_x\right)+{\displaystyle \frac{L_n}{L_f}}\mathrm{ }{\displaystyle \frac{{di}_n}{dt}})$$

(81)

$${\displaystyle \frac{d{u}_x}{dt}}=({\displaystyle \frac{1}{C_f}}\mathrm{ }\left(i_{Lx}-i_x\right))$$

(82)

$$\left(u_{xn}=V_{dc}\times \mathrm{ }{d}_{xn}\mathrm{ }\right)$$

(83)

$$i_a+\mathrm{ }{i}_b+\mathrm{ }{i}_c+\mathrm{ }{i}_n=0$$

(84)

In the above and below formulas, x = a, b, c; $u_x$ is the load voltage; $u_{xn}$ is the output voltage of the inverter; $i_{Lx}$ is the inductor current; $i_x$ is the load current; and dxn is the duty cycle of the phase voltage. The control strategy selected for the four-leg inverter with an inductor involves the use of integral backstepping control. The primary goal of the integral backstepping controller is to generate a reference current, and then to create a reference voltage for each phase in the (a, b, c) frame.

2.5.1. Generation of the Reference Currents

Integral backstepping control:

The error chosen here is $={\epsilon }_{xi}$.

$${\epsilon }_{xi}=u_{xref}-u_x$$

(85)

Then:

$${\epsilon }_{ai}=u_{aref}-u_a$$

(86)

$${\epsilon }_{bi}=u_{bref}-u_b$$

(87)

$${\epsilon }_{ci}=u_{cref}-u_c$$

(88)

The derivative of the error is:

$$\dot{{\epsilon }_{xi}}={\displaystyle \frac{d{u}_{xref} }{dt}}- {\displaystyle \frac{d{u}_x }{dt}}$$

(89)

The following Lyapunov function is considered:

$$V_{xi}={\displaystyle \frac{1}{2}}{{\mathrm{ԑ}}_{xi}}^2+{\displaystyle \frac{1}{2}}{{\gamma }_{xi}}^2$$

(90)

where

$${\gamma }_{xi}={\int }_{t0}^t{\epsilon }_{xi}dt$$

(91)

The Lyapunov function derivative is determined as follows:

$${\dot{V}}_{xi}=\dot{{\mathrm{ԑ}}_{xi}}\mathrm{ }{\mathrm{ԑ}}_{xi}+{\mathrm{ԑ}}_{xi} {\gamma }_{xi}$$

(92)

To ensure $u_x$ converges to $u_{xref}$, the time derivative of $V_{xi}$ should be negative semi-definite, i.e.,:

$${\dot{V}}_{xi}\le 0$$

(93)

Then we can deduce:

$$\dot{{\mathrm{ԑ}}_{xi}}+{\gamma }_{xi}=-K_{x1}{\mathrm{ԑ}}_{xi}$$

(94)

where $K_{x1}$ is a positive constant.

Proof of Equation (93):

From Equation (92), the Lyapunov function time derivative is expressed as:

$${\dot{V}}_{xi}=\dot{{\mathrm{ԑ}}_{xi}}{\mathrm{ԑ}}_{xi}+{\mathrm{ԑ}}_{xi} {\gamma }_{xi}$$

then

$${\dot{V}}_{xi}=\mathrm{ }{\mathrm{ԑ}}_{xi}(\dot{{\mathrm{ԑ}}_{xi}}+{\gamma }_{xi})$$

(95)

Integrating Equation (94) into Equation (95) yields the following expression:

$${\dot{V}}_{xi}=\mathrm{ }-K_{x1}{{\mathrm{ԑ}}_{xi}}^2<0$$

(96)

From Equation (96), we can conclude that the function $V_{xi}$ decreases steadily and its derivative is negative semi-definite.

Consequently, the Lyapunov stability criterion remains fulfilled, and the measured voltage reliably tracks its reference across all conditions.

From Equations (82), (89), (91), and (94) we can deduce:

$${\displaystyle \frac{d{u}_{xref} }{dt}}-\left({\displaystyle \frac{1}{C_f}}\mathrm{ }\left(i_{Lx}-i_x\right)\right)+{\gamma }_{xi}=-K_{\mathrm{x}1}{\mathrm{ԑ}}_{xi}$$

(97)

From Equation (97), $i_{Lxref}$ can be determined as follows:

$$i_{Lxref}=i_x+C_f(K_{x1}{\mathrm{ԑ}}_{xi}+{\gamma }_{xi}+{\displaystyle \frac{d{u}_{xref} }{dt}})$$

(98)

Then the three reference currents in the (a, b, c) frame are:

$$i_{Laref}=i_a+C_f(K_{a1}{\mathrm{ԑ}}_{ai}+{\gamma }_{ai}+{\displaystyle \frac{d{u}_{aref} }{dt}})$$

(99)

$$i_{Lbref}=i_b+C_f(K_{b1}{\mathrm{ԑ}}_{bi}+{\gamma }_{bi}+{\displaystyle \frac{d{u}_{bref} }{dt}})$$

(100)

$$i_{Lcref}=i_c+C_f(K_{c1}{\mathrm{ԑ}}_{ci}+{\gamma }_{ci}+{\displaystyle \frac{d{u}_{cref} }{dt}})$$

(101)

2.5.2. Development of Reference Voltages for the Three-Phase System in the (a, b, c) Frame

The error chosen here is $={\epsilon }_{xi1}$.

$${\epsilon }_{xi1}=i_{Lxref}-i_{Lx}$$

(102)

Then:

$${\epsilon }_{ai1}=i_{Laref}-i_{La}$$

(103)

$${\epsilon }_{bi1}=i_{Lbref}-i_{Lb}$$

(104)

$${\epsilon }_{ci1}=i_{Lcref}-i_{Lc}$$

(105)

The derivative of the error is:

$$\dot{{\epsilon }_{xi1}}={\displaystyle \frac{d{i}_{Lxref} }{dt}}- {\displaystyle \frac{d{i}_{Lx} }{dt}}$$

(106)

The Lyapunov function adopted is defined as follows:

$$V_{xi1}={\displaystyle \frac{1}{2}}{{\mathrm{ԑ}}_{xi1}}^2+{\displaystyle \frac{1}{2}}{{\gamma }_{xi1}}^2$$

(107)

where

$${\gamma }_{xi1}={\int }_{t0}^t{\epsilon }_{xi1}dt$$

(108)

The time derivative of the Lyapunov function is derived as:

$${\dot{V}}_{xi1}=\dot{{\mathrm{ԑ}}_{xi1}} {\mathrm{ԑ}}_{xi1}+ {\mathrm{ԑ}}_{xi1} {\gamma }_{xi1}$$

(109)

To ensure $i_x$ converges to $i_{xref}$, the time derivative of $V_{xi1}$ is required to be negative semi-definite, i.e.,:

$${\dot{V}}_{xi1}\le 0$$

(110)

Then, from Equations (109) and (110), we can deduce:

$$\dot{{\mathrm{ԑ}}_{xii}}+{\gamma }_{xi1}=-K_{x2}{\mathrm{ԑ}}_{xi1}$$

(111)

where $K_{x2}$ is a positive constant.

Proof of Equation (110):

From Equation (109), the time derivative of the Lyapunov function is derived as:

$${\dot{V}}_{xi1}=\dot{{\mathrm{ԑ}}_{xi1}}{\mathrm{ԑ}}_{xi1}+{\mathrm{ԑ}}_{xi1} {\gamma }_{xi1}$$

then

$${\dot{V}}_{xi1} =\mathrm{ }{\mathrm{ԑ}}_{xi1}(\dot{{\mathrm{ԑ}}_{xi1}}+ {\gamma }_{xi1})$$

(112)

By integrating Equation (111) into Equation (112), the inequation presented below can be derived:

$${\dot{V}}_{xi1}=\mathrm{ }-K_{x2}{{\mathrm{ԑ}}_{xi1}}^2<0$$

(113)

From Equation (113), we can conclude that the Lyapunov function $V_{xi1}$ steadily decreases over time and its derivative is negative semi-definite. Hence, the Lyapunov stability criterion can consistently be fulfilled, and the measured current reaches its reference value in all scenarios.

From Equations (81) and (111), we can deduce:

$${\displaystyle \frac{d{i}_{Lxref} }{dt}}-\left({\displaystyle \frac{1}{L_f}}\mathrm{ }\left(u_{xn}-u_x\right)+{\displaystyle \frac{L_n}{L_f}}\mathrm{ }{\displaystyle \frac{{di}_n}{dt}}\right)+{\gamma }_{xi1}=-K_{\mathrm{x}2}{\mathrm{ԑ}}_{xi1}$$

(114)

From Equation (114), the reference voltage $u_{xn}$ can be determined as follows:

$$u_{xnref}=u_x+L_f\left(K_2{\mathrm{ԑ}}_{xi1}+{\gamma }_{xi1}+{\displaystyle \frac{d{i}_{Lxref} }{dt}}\right)-L_n{\displaystyle \frac{{di}_n}{dt}}$$

(115)

Then the three reference voltages in the (a, b, c) frame are:

$$u_{anref}=u_a+L_f\left(K_2{\mathrm{ԑ}}_{ai1}+{\gamma }_{ai1}+{\displaystyle \frac{d{i}_{Laref} }{dt}}\right)-L_n{\displaystyle \frac{{di}_n}{dt}}$$

(116)

$$u_{bnref}=u_b+L_f\left(K_2{\mathrm{ԑ}}_{bi1}+{\gamma }_{bi1}+{\displaystyle \frac{d{i}_{Lbref} }{dt}}\right)-L_n{\displaystyle \frac{{di}_n}{dt}}$$

(117)

$$u_{cnref}=u_c+L_f\left(K_2{\mathrm{ԑ}}_{ci1}+{\gamma }_{ci1}+{\displaystyle \frac{d{i}_{Lcref} }{dt}}\right)-L_n{\displaystyle \frac{{di}_n}{dt}}$$

(118)

Stability Verification:

To perform the stability analysis, the positive definite Lyapunov function ${ V}_{total}$ is employed:

$$\begin{array}{cc}{V}_{total}=& {\sum }_{x=a}^{x=c}{V}_{xi}+{\sum }_{x=a}^{x=c}{V}_{xi1}={\sum }_{x=a}^{x=c}{\displaystyle \frac{1}{2}}{{\mathrm{ԑ}}_{xi}}^2+{\sum }_{x=a}^{x=c}{\displaystyle \frac{1}{2}}{{\gamma }_{xi}}^2\hfill \\ & +{\sum }_{x=a}^{x=c}{\displaystyle \frac{1}{2}}{{\mathrm{ԑ}}_{xi1}}^2+{\sum }_{x=a}^{x=c}{\displaystyle \frac{1}{2}}{{\gamma }_{xi1}}^2.\hfill \end{array}$$

(119)

The derivative of ${ V}_{total}$ must be negative semi-definite for the system to be asymptotically stable.

From Equations (96) and (113),$\dot{{ V}_{total}}$ is calculated as:

$$\dot{{ V}_{total}}=\sum _{x=a}^{x=c}-K_{x1}{{\mathrm{ԑ}}_{xi1}}^2+\sum _{x=a}^{x=c}-K_{x2}{{\mathrm{ԑ}}_{xi1}}^2<0$$

(120)

By using the Lyapunov stability theory, the stability of the closed-loop control system is ensured. After determination of the reference voltages, we will determine the offset. The fourth leg of the four-leg inverter not only supplies a neutral wire but also controls the zero-sequence voltage ($u_z)$. In the scalar PWM approach, the zero-sequence voltage ($u_z$) is added to the reference voltage ($u_{xnref}$), yielding the following new reference voltages:

$$u_{anref}+u_z={u_{anref}}^{*}$$

(121)

$$u_{bnref}+u_z={u_{bnref}}^{*}$$

(122)

$$u_{cnref}+u_z={u_{cnref}}^{*}$$

(123)

In addition, the zero-sequence voltage is assigned as the reference voltage for the inverter’s fourth leg.

The following equation describes the zero-sequence voltage [37,38]:

$$u_z=V_{dc}\left(0.5-\tau \right)-\left(1-\tau \right)u_{MAX}-\tau u_{MIN}$$

(124)

In this context, $u_{MAX}$ and $u_{MIN}$ denote the highest and lowest values among the three reference-phase voltages during each sampling period The distribution ratio $\tau$ determines the duration assigned to the zero-voltage vectors within each switching cycle. This period, known as the zero-vector timing, occurs when all upper switches of the voltage source inverter (VSI) are either simultaneously ON or OFF. For proper operation, $\tau$ must lie within the interval [0, 1] [38].

The voltage controller and current controller of the three-phase four-leg inverter are depicted in Figure 21.

The topological scheme of scalar PWM is depicted in Figure 22.

2.6. Modeling of Dump Load System [39]

The components of the buck converter are listed as follows: $C_{bk}$ is the capacitor, $L_{bk}$ the inductor, and the switching devices are Q and D linked to the electrolyzer with a voltage $V_e$. $I_{Lbk}$ is the inductor current, $V_{dc}$ is the source voltage.

The average-state space model of the buck converter is expressed according to:

$${\displaystyle \frac{d{I}_{Lbk}}{dt}}=u_{bk}{\displaystyle \frac{V_{dc}}{L_{bk}}}-{\displaystyle \frac{V_c}{L_{bk}}}$$

(125)

$${\displaystyle \frac{d{V}_C}{dt}}={\displaystyle \frac{I_{Lbk}}{C_{bk}}}-{\displaystyle \frac{I_{obk}}{C_{bk}}}$$

(126)

2.6.1. Dump Load System Current Control

Here:

$$x=I_{Lbk}$$

$$x_{ref}=I_{Lbkref}$$

The current tracking error is defined as follows:

$$e_{bk}=I_{Lbkref}-\mathrm{ }{I}_{Lbk}$$

(127)

Using the system dynamics from Equations (125) and (126), and applying the same IBC methodology, we derive the control input:

$$u_{bk}=({\displaystyle \frac{d{I}_{Lbkref}}{dt}}+{\displaystyle \frac{V_c}{L_{bk}}}+ {\gamma }_{bk}+K_{bk}{e}_{bk}){\displaystyle \frac{L_{bk}}{V_{dc}}}$$

(128)

2.6.2. Electrolyzer Voltage Regulation Based on Lyapunov-Based Function Controller

To regulate the DC bus voltage at a constant level, a control approach based on Lyapunov theory is introduced. The voltage tracking error, denoted by $e_b$, quantifies the deviation between the desired reference voltage $V_{eref}$ and the measured voltage output of the buck converter $V_e$. This tracking error at any given time ttt is defined as:

$$e_b=V_{eref}- V_e$$

(129)

The Lyapunov expression used to assess the buck converter’s stability, given by $V_{buck}$, is expressed as:

$$V_{buck}={\displaystyle \frac{1}{2}} C_{bk}.{\left(e_b\left(t\right)\right)}^2+{\displaystyle \frac{1}{2}} K_{Ibk}.{({\int }_{t0}^t{e}_b\left(t\right) dt)}^2$$

(130)

The derivative of $V_{buck}$ becomes:

$$\begin{array}{cc}\dot{V_{buck}}=& C_{bk} .e_b\left(t\right) . \dot{e_b\left(t\right)}\hfill \\ & +K_{Ik} .e_b\left(t\right) .\left({\int }_{t0}^t{e}_b\left(t\right) dt\right) .\hfill \end{array}$$

(131)

$$\begin{array}{cc}\dot{V_{buck}}=& {-C}_{bk} .e_b\left(t\right) . \dot{V_C}\hfill \\ & +K_{Ibk} .e_b\left(t\right) .\left({\int }_{t0}^t{e}_b\left(t\right) dt\right)\hfill \end{array}$$

$$\dot{V_{buck}}=e_b\left(t\right) .( {-C}_{bk} . \dot{V_C}+K_{Ibk} .({\int }_{t0}^t{e}_b\left(t\right) dt))$$

(132)

When we replace the derivative of the buck converter voltage $\dot{V_C}$ from Equation (126), we get:

$$\begin{gathered} \dot{V_{buck}}=e_b\left(t\right) .( {- C}_{bk} .( {\displaystyle \frac{I_{Lbk}}{C_{bk}}}- {\displaystyle \frac{I_{obk}}{C_{bk}}}) \\ +K_{Ibk} .({\int }_{t0}^t{e}_b\left(t\right) dt)) \end{gathered}$$

(133)

Equation (133) can be simplified in the following manner:

$$\dot{V_{buck}}=e_b\left(t\right) .( - (I_{Lbk}-I_{obk}) +K_{Ibk} .({\int }_{t0}^t{e}_b\left(t\right) dt))$$

(134)

To achieve asymptotic convergence of the voltage error, it is necessary that the derivative of $V_{buck}$ remains negative, which leads to the condition below:

$$- \left(I_{Lbk}-I_{obk}\right) +K_{Ibk} .\left({\int }_{t0}^t{e}_b\left(t\right) dt\right)=-K_{pbk}.e_b\left(t\right)$$

(135)

where $K_{pbk}$ is a constant positive proportional gain.

Then, from Equation (135), $I_{Lbkref}$ can be calculated as follows:

$$I_{Lbkref}=I_{obk} +K_{Ibk} .\left({\int }_{t0}^t{e}_b\left(t\right) dt\right)+K_{pbk} .e_b\left(t\right)$$

(136)

The buck converter controller is depicted in Figure 23.

The non linear load is depicted in Figure 24.

3. Case Study

Case 1: The inverter loading a three-phase balanced load.

Steady-State and Dynamic Performance Evaluation:

In the initial simulation scenario, a four-leg inverter is used to supply a balanced three-phase resistive load, where each phase has a resistance of $R=10 \mathsf{\Omega }$. The objective is to assess the effectiveness of the proposed integral backstepping controller under steady-state conditions. Key metrics include voltage harmonic distortion, voltage regulation accuracy, and neutral current behavior. The simulation results, shown in Figure 25, Figure 26 and Figure 27, demonstrate that the inverter produces high-quality output voltages with extremely low total harmonic distortion. Specifically, THD values were measured at 0.09% for phase a, and 0.08% for phases b and c—well below industry-accepted thresholds. Additionally, the steady-state voltage error $e_v(\%)$, represented in Figure 28, remains under 0.2%, indicating excellent voltage control performance. Because the system operates with a balanced load, the net current through the neutral conductor is minimal. This is confirmed in Figure 29, which shows that the neutral current is effectively negligible, validating the current symmetry across all three phases.

Current Control and Transient Response:

The controller’s ability to handle fast transients is illustrated in Figure 30, where the line currents rapidly track their respective reference signals and settle within 0.005 s. This fast dynamic response highlights both the precision and robustness of the inverter control architecture.

Battery Management and Power Balance:

The system also incorporates a battery energy storage system to maintain the power balance between generation and consumption. The battery control relies on an integral backstepping approach to regulate current and a sliding mode-based (STA) controller to stabilize the DC-link voltage. As shown in Figure 31, neither the solid oxide fuel cell (SOFC) nor the dump load engages during this period, since the battery’s state of charge (SOC) stays within its safe operating range (between 0.2 and 0.8). Moreover, Figure 32 reveals that the DC-link voltage exhibits minimal overshoot of less than 1 V and very low ripple, emphasizing the control system’s stability and effectiveness under varying conditions.

PV and Wind System Performance:

The photovoltaic (PV) subsystem’s voltage control performance is highlighted in Figure 33 and Figure 34, where the PV input voltage quickly and accurately follows its reference, achieving convergence in under 0.005 s. This performance remains consistent even with changes in irradiance levels, as illustrated in Figure 35. Regarding the wind energy system, a detailed evaluation is provided in Section 2.3, supported by figures that include measurements of electromagnetic torque, rotor speed of the PMSG, inductor current (measured and reference), power coefficient, and tip speed ratio. These results confirm that the wind energy conversion system is successfully regulated by the proposed integral backstepping controller, keeping the turbine operation within the optimal range defined by the MPPT strategy. This ensures efficient energy harvesting from wind resources while maintaining system stability. The load current is depicted in Figure 36, the load voltage is depicted in Figure 37, the currents $i_{Laref}$ and $i_{La}$ are depicted in Figure 38, the currents $i_{Lbref}$ and $i_{Lb}$ are depicted in Figure 39, the currents $i_{Lcref}$, $i_{Lc}$ are depicted in Figure 40.

Case 2: The inverter loading a single-phase load.

To explore the dynamic response and voltage regulation capabilities of the proposed control strategy under non-ideal conditions, the inverter is tested with an unbalanced load. Specifically, a single-phase resistive load of ($R_a=15\mathsf{\Omega })$ is connected to phase a, while phases b and c remain open. This test case introduces significant phase asymmetry and serves to verify the controller’s capacity to maintain voltage stability, suppress harmonics, and ensure system balance.

Voltage Quality and Harmonic Distortion:

Despite the unbalanced loading, the output voltage waveform in phase a remains highly sinusoidal. As illustrated in Figure 41 the total harmonic distortion (THD) of the phase a voltage is only 0.09%. This low value, even under asymmetrical load conditions, indicates that the control algorithm maintains high voltage quality and actively suppresses harmonic content introduced by the imbalance. Performance like this is typically difficult to achieve in conventional three-leg inverters without complex compensating techniques.

Voltage Regulation and Tracking Accuracy:

The ability of the controller to regulate voltage is further evidenced in Figure 42 which presents the steady-state load voltage error $e_v(\%)$. The voltage deviation remains below 0.3%, confirming that the proposed controller can maintain output voltage levels very close to the desired reference, even when only one phase is loaded. This accurate tracking demonstrates the robustness of the integral backstepping control method and its effectiveness in decoupling and regulating individual phase voltages.

DC-Link Voltage Stability:

The behavior of the DC-link voltage under unbalanced conditions is a critical indicator of overall system stability. In Figure 43, the input DC-link voltage remains tightly regulated around 880 V. The transient response is notably fast, with the voltage settling quickly after initial disturbances and exhibiting minimal overshoot. Importantly, the voltage ripple does not increase significantly during the unbalanced loading scenario, which highlights a key advantage of the four-leg inverter: the presence of the fourth leg allows neutral current circulation, preventing unbalanced currents from affecting the DC side. In contrast to three-leg configurations—where unbalanced loads often lead to DC-link voltage fluctuations—the four-leg design effectively isolates the DC-link from such disturbances. This structural benefit enhances the overall power quality and reliability of the system in real-world applications where load symmetry cannot be guaranteed.

Voltage Decoupling and Neutral Current Management:

The voltage waveforms across all three phases are shown in Figure 44, where each phase voltage maintains its reference profile without mutual interference. The unbalanced load applied to phase a does not affect the waveforms of phases b and c, indicating that the controller successfully decouples the phase voltages and regulates them independently. This level of control precision is essential for multi-phase systems operating in dynamic and unbalanced environments.

The phase and neutral current behavior is analyzed in Figure 45 and Figure 46. The current in phase a and the neutral line are nearly identical in magnitude but opposite in phase (approximately 180° out of phase). This behavior confirms that the fourth leg effectively provides a return path for the unbalanced current, ensuring that the inverter can safely and accurately handle non-zero neutral currents without degrading performance.

Case 3: The inverter loading a three-phase nonlinear load.

As nonlinear loads become increasingly prevalent in residential, industrial, and renewable-integrated systems—especially those using power electronics such as rectifiers, motor drives, or switched-mode power supplies—the ability of stand-alone inverters to manage such conditions is critical. These loads typically inject substantial harmonic currents into the power supply, degrading voltage quality and potentially affecting both the inverter and connected sensitive equipment. In this context, the four-leg inverter topology, combined with the proposed integral backstepping control strategy, is rigorously tested against three representative nonlinear load scenarios: resistive, inductive DC, and capacitive DC loads. These load types introduce progressively more distortion, offering a comprehensive evaluation of the inverter’s harmonic mitigation and dynamic voltage control capabilities.

Impact of Load Type on Current Waveforms:

Simulation results reveal that, with the nonlinear presented in Section 2.3, the inverter current remains close to sinusoidal. This is attributed to the linear nature of the voltage–current relationship in resistive elements, even when subjected to switching events or waveform distortions. Consequently, the inverter output voltage also retains a clean sinusoidal profile with negligible deviation. However, when the load involves inductive or capacitive DC characteristics, the situation changes significantly. These elements introduce nonlinear dynamics, such as current lags (in inductive cases) and sharp charging/discharging behavior (in capacitive cases), leading to pronounced waveform distortion. Among all test conditions, the capacitive DC load imposes the most severe current distortion, manifesting as sharp, pulse-like currents due to the sudden energy absorption behavior of the capacitor. These current spikes represent high-frequency harmonics that can destabilize conventional inverter systems.

Inverter Response and Harmonic Mitigation:

Despite these harsh conditions, the inverter successfully preserves voltage waveform quality. Figure 47 illustrates that even with a heavily distorted current drawn by the capacitive load, the output voltage remains nearly sinusoidal. This outcome highlights the controller’s effectiveness in decoupling the output voltage from the current distortion, an essential feature for maintaining power quality in real-world nonlinear load environments.

Furthermore, the measured THD values of the output voltages across all phases, as shown in Figure 48, Figure 49 and Figure 50 are impressively low. Even in the worst-case harmonic scenario, the THD remains under 0.36%, which is far superior to the 8% upper limit defined by the IEC 62040-3 standard for stand-alone power systems [40]. The steady-state voltage error $e_v$ is less than 0.7% Figure 51. This performance reinforces the inverter’s suitability for deployment in critical or sensitive environments, such as off-grid systems, remote energy stations, or small-scale industrial plants.

Controller Contribution and Decoupling Benefits:

The results affirm the role of the integral backstepping controller in dynamically adjusting inverter switching actions to suppress voltage distortion. The ability to maintain precise output control, independent of load behavior, is further enhanced by the four-leg inverter architecture, which isolates and compensates for neutral current paths—something that three-leg systems inherently lack.

As observed in Figure 52 the neutral wire carries non-zero current, confirming the presence of zero-sequence components generated by the nonlinear load. The fourth leg ensures that this current is safely and accurately processed, preventing imbalance in the output phase voltages.

Stability of the DC-Link Voltage:

A key element in ensuring system-wide performance is the stability of the DC-link voltage, which acts as the intermediary energy buffer between generation sources (e.g., PV, wind, battery) and the AC load. Figure 53 confirms that the DC-link voltage remains centered around 880 V, with minimal ripple despite the high harmonic stress. This stability is achieved through a hybrid control strategy involving integral backstepping for smooth current regulation from the battery and electrolyzer; STA (super-twisting algorithm) control to control the DC-link voltage; and a Lyapunov function-based adaptive controller to control the voltage $V_e$ of the electrolyzer.

These coordinated efforts ensure that even under nonlinear and highly variable loading conditions, the inverter operates within its desired voltage margin, minimizing stress on components and improving overall reliability.

The results of this study confirm that the proposed system offers a resilient solution for real-world applications involving nonlinear or unbalanced loads. The combination of low voltage distortion under nonlinear conditions, fast and stable DC-link control, effective neutral current handling, and compliance with international harmonic standards makes this system an excellent applicant for stand-alone power systems, particularly in renewable energy microgrids, electric vehicle charging infrastructure, and rural electrification projects where load profiles are unpredictable.

Currents under nonlinear load is depicted in Figure 54.

Case 4: The inverter loading under 100% load changes.

The comprehensive evaluation of the proposed system demonstrated its superior performance in managing dynamic load variations and maintaining system stability under a wide range of operating conditions. The test results provide valuable insights into the robustness of the proposed control method, especially under scenarios involving fluctuations in solar irradiance (Figure 55), wind conditions (described in Section 2.3), and load changes, all of which are common in real-world microgrid applications.

System Response to Load Variations:

In the load variation tests, the proposed control method was able to maintain load voltages within an acceptable range, even when subjected to significant load changes. The 100% load change scenario provided a challenging test case for the system, with both balanced and unbalanced resistive loads being applied in succession. A balanced three-phase resistive load (15 Ω per phase) was applied from 0.75 s to 1.5 s, and an unbalanced resistive load with values of 17 Ω, 15 Ω, and 20 Ω for phases a, b, and c, respectively, was applied from 2.0 s to 2.5 s. During the balanced load step, the system demonstrated its ability to recover from a maximum voltage drop of 23.3 V in phase “c” (Figure 56) within 1 ms, restoring the load voltage with a minimal variation of 8.5%. Similarly, under the unbalanced load condition, the maximum voltage drop of 35 V in phase “b” (Figure 57) was efficiently corrected, with the load voltage restored within 1 ms and a maximum voltage deviation of 13%. The voltage of the load reaches its reference in less than 0.005 s (Figure 58). These rapid recovery and tracking times are critical for ensuring the stability and reliability of the microgrid, especially in stand-alone applications where voltage deviations beyond certain limits can lead to equipment damage or inefficient operation. As mentioned, the IEC62040-3 international standard permits a voltage deviation of up to ±30% for stand-alone applications, but the proposed system outperforms this specification, achieving voltage recovery times faster than the prescribed 5 ms, and maintaining voltage deviations well within the acceptable range.

Power Quality and Harmonic Distortion:

The total harmonic distortion (THD) analysis is another crucial aspect of the system’s performance, as excessive harmonic distortion can degrade the quality of the electrical supply, reduce the efficiency of connected loads, and potentially damage sensitive equipment. The THD of the three-phase load voltages remained below 0.09% (Figure 59, Figure 60, Figure 61, Figure 62, Figure 63 and Figure 64), and the value of the steady-state error $e_v$ (%) was lower than 0.2% (Figure 65), well within the limits set by international standards for power quality. This indicates that the proposed inverter control effectively minimizes harmonic distortion, ensuring that the power supplied to the load is of high quality. The ability to maintain low THD is particularly important in systems that rely on sensitive electronic equipment, where voltage waveform distortion can lead to malfunction or inefficiency. The results further validate that the proposed control strategy can provide clean, stable power even under varying load conditions.

Dynamic Power Management with Energy Storage and SOFC:

Another key feature of the system is its ability to maintain the power balance between generation and load demand through the integration of an energy storage system (battery) and a solid oxide fuel cell (SOFC) (Section 2.4.1 Various powers). The battery and SOFC work in tandem to ensure that power demand is met even when the renewable generation (solar and wind) fluctuates. The SOFC begins supplying power when the battery’s state of charge (SOC) drops below a threshold of 0.2 (Figure 66), ensuring a continuous and reliable power supply to the load. This feature highlights the system’s ability to handle transient periods where renewable generation may not be sufficient to meet demand, offering a resilient solution for microgrid applications that require a constant power supply. The battery and SOFC also play a crucial role in reducing the reliance on external grid connections, which is particularly beneficial in remote or off-grid locations.

DC-Link Voltage Regulation and Controller Performance:

The input DC-link voltage was maintained at a stable value of approximately 880 V, with minimal ripple (see Section 2.4.2 (Proof)). This stable DC-link voltage is a key factor in the efficient operation of the inverter, as any significant fluctuations could impact the inverter’s performance and lead to inefficiencies. The use of advanced control methods, including the integral backstepping controller for current regulation and the STA and Lyapunov-based controllers for DC-link voltage management, ensured the robustness and accuracy of the system’s voltage regulation. Furthermore, the response times of the current controller and voltage controller were notably fast, with the current controller showing faster dynamics than the voltage controller, as shown in (Section 2.4.2).

This fast response time is essential for maintaining system stability during load transients and minimizing voltage fluctuations.

Inverter Performance and Efficiency:

The inverter performance was evaluated by examining the input DC power (P_DC) and the output AC power (P_AC), as shown in Figure 67. The results indicate that the inverter operates with negligible losses, reinforcing the system’s overall efficiency. In particular, the ability of the inverter to quickly adjust to variations in the input voltage from the PV system—reaching its reference value in less than 0.005 s—further underscores the system’s rapid response to changing conditions (Figure 68 and Figure 69).

This fast response time is vital in maintaining stable and efficient operation in microgrids, where the power generation from renewable sources like solar and wind can be intermittent. The inverter’s ability to quickly adapt to changes in input power allows it to seamlessly integrate with the renewable energy sources while ensuring that the power quality and system stability are maintained.

Current Control and Zero-Sequence Components:

The fourth wire current (Figure 70, Figure 71 and Figure 72), which indicates the presence of zero-sequence components, was non-zero only when the inverter was supplying an unbalanced load. This behavior is consistent with the expected operation of the system, as unbalanced loads generate zero-sequence currents, which the system must handle appropriately. In contrast, when the load was balanced or no load was present, the fourth wire current remained zero. Figure 73, Figure 74 and Figure 75 reveal that the line currents follow their references and reach their references in less than 0.005 s. In a 100% descending step load, the current in all three phases was zero and the load voltage tolerances were not considerable (Figure 76 and Figure 77). The load voltage under unbalanced load and no load is depicted in Figure 78, the load current under balanced and no load is depicted in Figure 79, the current of 100% load step change under no load and unbalanced load is depicted in Figure 80, the current $i_{La}$ and its reference $i_{Laref}$ under balanced load and no load is depicted in Figure 81 and Figure 82, the Current $i_{La}$ and its reference $i_{Laref}$ under no load and unbalanced load is depicted in Figure 83, the current $i_{Lb}$ and its reference $i_{Lbref}$ under no load and balanced load is depicted in Figure 84, the current $i_{Lb}$ and its reference $i_{Lbref}$ under no load and unbalanced load is depicted in Figure 85, the current $i_{Lc}$ and its reference $i_{Lcref}$ under no load and unbalanced load is depicted in Figure 86, the voltage 100% load step change under unbalanced load and no load is depicted in Figure 87. This further validates the system’s ability to detect and manage unbalanced load conditions, a common challenge in microgrid systems.

Case 5: The inverter loading under 100% load changes and reference voltages variation.

In certain specialized applications, there is a need to supply loads with different voltage levels, particularly in stand-alone renewable energy-based distributed generation systems. This can include scenarios where loads require different voltage amplitudes, or where there is a need for unbalanced three-phase voltages. A four-leg inverter is an ideal solution for such applications, as it is capable of generating three-phase voltages with unbalanced characteristics, meeting these specific needs. This flexibility in voltage generation makes the four-leg inverter an excellent choice for voltage stabilization and balancing applications, especially in systems with varying load demands or renewable energy generation sources.

Unbalanced Voltage Generation and Control System Performance:

As demonstrated in the simulation results shown in Figure 88 and Figure 89, the four-leg inverter is capable of generating unbalanced three-phase voltages, which is essential for certain applications where loads require unbalanced or differing voltage amplitudes. The simulation results clearly confirm that the inverter can handle these unbalanced conditions effectively and precisely, showcasing the inverter’s versatility in providing both balanced and unbalanced load voltages. A key part of evaluating the system’s performance involves assessing the inverter’s ability to follow reference voltage tracking during step changes in reference load voltages. In the test, at t = 0.4 s to 0.85 s, phase “c” did not change, while phase “b” and phase “a” experienced 234.5 and 155.6 volts, respectively; step changes in reference balanced voltages were applied at different times, including a 50% falling step change from 0.95 s to 1.3 s and a rising step change from 1.3 s to the end of the simulation. In addition, various load changes were applied, including a 100% balanced resistive load step change between 0.75 s and 1.5 s, a 100% unbalanced resistive load step change between 1.666 s and 2.166 s, and a 100% nonlinear load step change between 2.35334 s and 2.8666 s. The results of these tests, as shown in Figure 90, indicate that the inverter successfully tracked the reference voltages in less than 0.005 s, achieving high performance in voltage regulation. For example, when phase “b” and phase “a” experienced voltage changes of 234.5 V and 155.6 V, respectively, during a voltage reference change, the inverter was able to bring the voltages back to their reference values quickly and precisely.

Battery Control and Power Balancing:

As seen in Figure 91, the battery plays a crucial role in maintaining the power balance between the production and the demand of the load. In the test scenario, when the state of charge (SOC) Figure 92 of the battery reached 0.8, the dump load was activated to help balance the system. The battery also effectively followed the load variations between 0.4 s and 0.85 s, adjusting for the unbalanced load caused by the difference in phase voltages at the same time. This action was vital in ensuring that the inverter was able to maintain power quality and stability despite the fluctuations in the reference voltages. The performance of the battery and dump load controllers was further validated by observing the small overshoot of less than 3 V and minimal ripple in the input DC-link voltage, as shown in Figure 93. This is a clear indication that the proposed controllers (integral backstepping for the battery current and STA for the DC-link voltage, and integral backstepping and PI for the dump load) are highly effective in ensuring smooth operation of the system.

Response to Voltage Reference Step Changes:

The dynamic response of the system to step changes in reference voltages was evaluated in terms of the load voltage’s ability to return to the reference value. As shown in Figure 89, the maximum voltage drop of about 23 V occurred for phase “c” when the reference voltage changed. The system quickly restored the voltage within a very short time interval, typically less than 1 ms, demonstrating its excellent voltage recovery capability. A similar observation can be made in Figure 94, where a maximum voltage drop of 27 V occurred in phases “c” and “b” when the load changed from no load to a 100% balanced load. The control system was able to retrieve the voltage within a very short time, maintaining high stability. Additionally, the system’s ability to handle the transition from unbalanced to balanced loads was also assessed. As shown in Figure 95, when the load changed from no load to a 100% unbalanced load, the system experienced a maximum voltage drop of about 35 V for phases “c” and “b”. However, even under these conditions, the inverter successfully restored the voltages in less than 1 ms, showcasing its ability to rapidly adapt to varying load conditions.

Compliance with IEC Standards:

According to the IEC62040-3 international standard, a maximum of ±30% voltage deviation is allowed for stand-alone applications during load changes, provided that these deviations occur within a 5 ms window. The proposed control system, however, demonstrated that it can retrieve the load voltages in less than 1 ms, with a maximum voltage deviation of 35 V. This exceeds the specifications set by the standard, confirming the system’s rapid recovery capabilities and its suitability for real-time applications requiring quick voltage adjustments.

Power Quality and THD:

The THD% of the three-phase load voltages were analyzed and presented in Figure 96, Figure 97, Figure 98, Figure 99, Figure 100, Figure 101 and Figure 102, where the value of $e_v$ (%) was found to be lower than 0.2% (Figure 103). A THD value of less than 0.40% is well below the IEEE 519 limits for all voltage levels, indicating a high-quality power system with minimal harmonic distortion, which is a testament to the effectiveness of the proposed control strategy. The control system effectively minimized any harmonic distortions, ensuring that the load voltages remained clean and stable, which is especially important for sensitive equipment that could be damaged by high harmonic levels.

Current Control and Zero-Sequence Components:

The current of the fourth wire, which represents the zero-sequence component of the load current, was shown to be non-zero when the inverter was supplying unbalanced loads or when the reference voltage was variable Figure 104. As expected, this zero-sequence current was zero when the load was balanced or when there was no load, confirming that the system properly identifies and manages unbalanced conditions. Figure 105, Figure 106, Figure 107, Figure 108, Figure 109, Figure 110, Figure 111, Figure 112, Figure 113, Figure 114 and Figure 115 reveal that the line currents follow their references and attain their references in less than 0.005 s which confirms the good performance and the robustness of the proposed control of the inverter (integral backstepping controller). This feature adds flexibility and robustness to the inverter’s design, enabling it to work effectively under a wide range of conditions.

Inverter and Dump Load Control:

The performance of the inverter and dump load controllers was further validated through their fast response times in following their respective reference values. As shown in Figure 116 and Figure 117, the inductor current of the buck converter $I_{Lbkref}$ attained its reference value in less than 0.001 s, and the measured load voltage attained the reference voltage value in less than 0.01 s (Figure 93), indicating the high-speed operation of the inverter’s current controller and confirming the excellent performance and robustness of the dump load controller, which employed an integral backstepping controller for current regulation and a Lyapunov-based controller for electrolyzer voltage regulation.

The irradiation is depicted in Figure 118 and the current of the load is depicted in Figure 119 and the wind conditions (described in Section 2.3).

4. Conclusions

This work presents a robust and comprehensive control strategy for voltage regulation in DC microgrids, validated through extensive simulation studies. The proposed decentralized control scheme exhibits excellent performance under both steady-state and transient conditions, effectively regulating voltage even in the presence of linear, nonlinear, and unbalanced load profiles. The system consistently maintains sinusoidal and symmetrical voltage waveforms, with fast dynamic response and low overshoot, thereby ensuring high power quality and operational reliability. The decentralized nature of the control architecture, which relies solely on local measurements, eliminates the need for communication between distributed units, enhancing the system’s scalability, modularity, and fault resilience. The theoretical stability of the microgrid is rigorously demonstrated using Lyapunov-based methods, reinforcing the reliability of the proposed control design. A tailored control approach is employed for each component of the microgrid:

The battery system is managed using an integral backstepping current controller coupled with a super-twisting algorithm (STA) for voltage regulation.

The SOFC employs an integral backstepping current controller along with a Lyapunov-based voltage controller.

A hybrid nonlinear controller combining integral backstepping and STA is implemented for the PV generator.

The wind energy system is governed by an integral backstepping controller.

Control of the dump load (aqua-electrolyzer) involves a multi-level strategy: integral backstepping for inductor current control, a PI controller for regulating the DC-link voltage, and a Lyapunov-based approach for managing the electrolyzer voltage.

The load-side voltage source inverter (VSI) is controlled using an integral backstepping method.

Simulation results validate the effectiveness and robustness of the proposed controllers across various operating conditions. The system exhibits strong voltage regulation performance and stable behavior under dynamic and complex loading scenarios. Furthermore, the control strategy’s performance under unbalanced single-phase load conditions highlights the capability of the four-leg inverter configuration. The system successfully maintains voltage waveform quality, ensures accurate voltage tracking, stabilizes the DC-link voltage, mitigates neutral current disturbances, and achieves phase decoupling. These attributes confirm the suitability of the proposed control framework for practical deployment in decentralized energy systems, particularly in applications characterized by variable or unpredictable loads—such as residential microgrids, renewable energy sources, and industrial settings with significant single-phase equipment presence.