SMC Algorithms in T-Type Bidirectional Power Grid Converter

Abstract

In this paper, the implementation of sliding mode control algorithms for the case of power grid current regulation in a T-type bidirectional inverter system connected via an LCL filter to the power grid is proposed and presented. A mathematical model of such a system has been proposed, which was then implemented in a simulation environment. The method of designing sliding controllers using the Lyapunov method to conduct a stability proof is presented. The article includes a comparative analysis of two sliding mode control algorithms: the classic one, which includes equivalent control, discontinuous part, and proportional reaching law, and the hybrid one, in which the discontinuous part and reaching law were modified.

1. Introduction

The requirements for the quality of control systems of devices cooperating with the electrical grid are constantly increasing. This is due, among other things, to the constant growth in popularity of renewable energy sources [1]. The increasing number of different devices connected to the power grid means that the variability of power grid load is increasing. The number of so-called microgrids, i.e., local electrical grids, often equipped with energy storage and renewable energy sources such as photovoltaic panels [1,2,3,4,5,6,7], is growing. These structures mainly cooperate with the power grid, but they can also work in an island mode, i.e., disconnected from the main power grid [1,2,3,4,8,9,10,11].

The inverter, being an active intermediary between the local and main power grid, should perform its work while ensuring the appropriate quality of the current received and delivered to the grid. Classic and known methods of inverter system regulation are based on linear PID (Proportional–Integral–Derivative) controllers [1,2,8,9,10]. These are simple systems and are characterized by low resistance to disturbances and load variability.

An interesting and developing issue is the field of sliding mode control (SMC). Sliding controllers belong to the category of nonlinear and robust control systems [12,13]. The algorithms of the aforementioned control methods allow the use of knowledge of the dynamics of the object to design the control law, but the most important role is played by the discontinuous part of the control, which is responsible for switching in the sliding motion, guaranteeing resistance to disturbances [12,13]. An additional signal, the so-called reaching law [13,14,15,16], may be responsible for efficiently bringing the system’s operating point to the sliding phase. The problem of sliding controllers may turn out to be so-called chattering, i.e., the phenomenon of oscillations of small amplitude and high frequency [12,13].

In this work, a model of a structure, which includes a T-type three-level inverter connected to the power grid through an LCL filter [4,5,6,7,17,18,19,20], was proposed. In the literature, it is possible to find several examples of implementation of various three-phase inverter control algorithms that ensure improvement of the quality of transmitted energy. A few can be mentioned, among others: a method based on a Fractional-Order Synergetic Controller using a shunt active power filter to reduce reactive power and the impact of harmonics [21], a method using classic SMC controllers containing a discontinuous part based on a hyperbolic tangent function for the NPP (Neutral Point Piloted) topology [22], as well as algorithms based on the Predictive Control Model using a Luenberger observer [23].

In the case of sliding mode control, it should be mentioned that many different methods of developing classical SMC algorithms have been proposed in the literature on the subject. These include, among others, Fractional-Order SMC with Synergetic Controllers, which allows us to increase the efficiency of the control system and reduce the settling time [24], the SMC algorithm with adaptation based on the Neural Network, which allows for estimating and reacting to uncertain system parameters and external disturbances [25], a method using TSMC (Terminal Sliding Mode Control) that allows for convergence with finite-time [26], Fuzzy SMC increasing the robustness of the system [27], SMC-ANFIS (Artificial Neuro Fuzzy System) algorithm offering a combination of Fuzzy SMC control features with Neural Networks [28] and Super Twisting SMC improving the quality of control and reducing the impact of chattering phenomena [29]. However, in this work, it was decided to use easily applicable methods, which would be based on analytical methods.

It is therefore necessary to look for solutions enabling the achievement of suitable control quality parameters while maintaining limited numerical possibilities. Therefore, the authors attempted to implement sliding control algorithms for current regulation in a three-level grid inverter. Sliding mode controllers were designed in a classic form (SMC) and a modified one (HSMC—hybrid sliding mode control), in which changes were made in the discontinuous control by introducing a variable amplitude and also adding a hybrid reaching law, which contains switching proportional and exponential parts [30,31,32,33]. The proposed modified mechanism of the sliding mode controller’s operation reduces the impact of the chattering phenomenon and improves the quality of control, thus allowing for better properties than classic regulators [31,32,33].

2. Methodology

This work focuses on the simulation studies of proposed sliding mode control systems, which perform the task of controlling the main grid current in the structure of an inverter connected via an LCL filter to the power grid. The sinusoidal signals present in the three-phase system have been subjected to dq0 transformations, which allow for the simplification of the control system and maintain frequency synchronization with the power grid [7,34,35,36]. The control algorithms presented in the further part of the work allow for full-range regulation of the d and q components of the grid current vector.

The converter used in the work is a bidirectional inverter in the T topology, which allows for obtaining three levels of output voltages [17,18,37]. Such a solution allows for reducing the impact of current ripples, which makes it possible to use higher switching frequencies than in two-level inverters. For this reason, the frequency in the implemented system is 50 kHz.

Figure 1 presents a simplified diagram of a bidirectional inverter in T topology, which cooperates with a DC voltage source through a capacitive filter. This source can, for example, constitute energy storage in a so-called microgrid with renewable energy devices [1,2,3,4,5,6,7].

There are many problems with this type of inverter application. One of them is that LCL filters tend to oscillate due to their lossless nature. This problem can be avoided by properly selecting the values of the filter elements in such a way that the resonant frequency ranges do not coincide with the scope of the adjustable harmonics and the switching frequency [38,39]. The real parameters of passive elements, such as filter and capacitance losses and their other parasitic parameters, are also of great importance for the scale of occurrence of undesirable phenomena. In grid inverter control systems, the role of one of the inductive elements is partially or completely played by the inductance of the power grid, which unfortunately makes filter design difficult because the nature of the grid depends on the power of the distribution transformers, the distance from them and the connected loads. The authors assumed that at the stage of simulation work, tests would be carried out for the widest possible range of changes in network parameters. The second way to achieve stability is to take this phenomenon into account in the control algorithm [40,41]. In the examined case, where the measured and regulated values are the currents of both coils and the capacitor voltage, there is a strong mechanism that dampens possible oscillations.

The main dynamic problem in the described system is the small short-circuit impedances of the power grid, which cause high current values to appear even with small voltage changes [19]. The state of the power grid largely depends on the loads connected to it, which is particularly evident when cooperating with highly variable renewable energy sources over time. Therefore, the control systems of inverters connected to the power grid are burdened with high requirements regarding control quality. Figure 2 shows a block diagram of the investigated system.

The power system consists of a power grid, an LCL filter, and a three-phase T-type inverter. The control system includes dq0 transformations and inverse dq0 transformations of measured currents and voltages and a sliding mode controller.

The dynamics of each of the three phases of the system shown in Figure 2 is described by a system of equations:

$$\left\{\begin{array}{c}\mathrm{L}\frac{\mathrm{d}{\mathrm{i}}_{\mathrm{k}}}{\mathrm{d}\mathrm{t}}=\mathrm{d}{\mathrm{U}}_{\mathrm{d}\mathrm{c}}-{\mathrm{u}}_{\mathrm{c}\mathrm{k}}\\ \\ {\mathrm{L}}_{\mathrm{g}}\frac{\mathrm{d}{\mathrm{i}}_{\mathrm{g}\mathrm{k}}}{\mathrm{d}\mathrm{t}}={\mathrm{u}}_{\mathrm{c}\mathrm{k}}-{\mathrm{U}}_{{\mathrm{g}}_{\mathrm{k}}}\\ \\ {\mathrm{C}}_{\mathrm{f}}\frac{\mathrm{d}{\mathrm{u}}_{\mathrm{C}\mathrm{k}}}{\mathrm{d}\mathrm{t}}={\mathrm{i}}_{\mathrm{k}}-{\mathrm{i}}_{\mathrm{g}\mathrm{k}}\end{array}\right.$$

(1)

where

–

$\mathrm{L}$—inductance of the filter on the output side of the inverter;

–

${\mathrm{i}}_{\mathrm{k}}$—output current of a given inverter phase;

–

$\mathrm{d}$—duty cycle of the voltage modulator;

–

${\mathrm{U}}_{\mathrm{d}\mathrm{c}}$—voltage on the capacitive filter on the energy storage side;

–

${\mathrm{u}}_{\mathrm{c}\mathrm{k}}$—voltage of a given phase on the capacitive filter on the power grid side;

–

${\mathrm{L}}_{\mathrm{g}}$—power grid inductance;

–

${\mathrm{i}}_{\mathrm{g}\mathrm{k}}$—power grid current of a given phase;

–

${\mathrm{U}}_{{\mathrm{g}}_{\mathrm{k}}}$—power grid voltage of a given phase;

–

${\mathrm{C}}_{\mathrm{f}}$—capacity of the filter on the power grid side;

–

$\mathrm{k}$—designation of one of the three phases a, b, and c.

According to the description in the introduction of this work, the system was transformed from three-phase to vector [20,34,35]:

$$\left\{\begin{array}{c}\mathrm{L}\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\left[\begin{array}{c}{\mathrm{i}}_{\mathrm{d}}\\ {\mathrm{i}}_{\mathrm{q}}\end{array}\right]={\mathrm{U}}_{\mathrm{d}\mathrm{c}}\left[\begin{array}{c}{\mathrm{d}}_{\mathrm{d}}\\ {\mathrm{d}}_{\mathrm{q}}\end{array}\right]-\left[\begin{array}{c}{\mathrm{u}}_{\mathrm{c}\mathrm{d}}\\ {\mathrm{u}}_{\mathrm{c}\mathrm{q}}\end{array}\right]-\mathrm{L}\mathsf{\omega }\left[\begin{array}{c}{-\mathrm{i}}_{\mathrm{q}}\\ {\mathrm{i}}_{\mathrm{d}}\end{array}\right]\\ \\ {\mathrm{L}}_{\mathrm{g}}\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\left[\begin{array}{c}{\mathrm{i}}_{\mathrm{g}\mathrm{d}}\\ {\mathrm{i}}_{\mathrm{g}\mathrm{q}}\end{array}\right]=\left[\begin{array}{c}{\mathrm{u}}_{\mathrm{c}\mathrm{d}}\\ {\mathrm{u}}_{\mathrm{c}\mathrm{q}}\end{array}\right]-\left[\begin{array}{c}{\mathrm{U}}_{\mathrm{g}\mathrm{d}}\\ {\mathrm{U}}_{\mathrm{g}\mathrm{q}}\end{array}\right]-{\mathrm{L}}_{\mathrm{g}}\mathsf{\omega }\left[\begin{array}{c}{-\mathrm{i}}_{\mathrm{g}\mathrm{q}}\\ {\mathrm{i}}_{\mathrm{g}\mathrm{d}}\end{array}\right]\\ \\ {\mathrm{C}}_{\mathrm{f}}\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\left[\begin{array}{c}{\mathrm{u}}_{\mathrm{c}\mathrm{d}}\\ {\mathrm{u}}_{\mathrm{c}\mathrm{q}}\end{array}\right]=\left[\begin{array}{c}{\mathrm{i}}_{\mathrm{d}}\\ {\mathrm{i}}_{\mathrm{q}}\end{array}\right]-\left[\begin{array}{c}{\mathrm{i}}_{\mathrm{g}\mathrm{d}}\\ {\mathrm{i}}_{\mathrm{g}\mathrm{q}}\end{array}\right]-{\mathrm{C}}_{\mathrm{f}}\mathsf{\omega }\left[\begin{array}{c}{-\mathrm{u}}_{\mathrm{c}\mathrm{q}}\\ {\mathrm{u}}_{\mathrm{c}\mathrm{d}}\end{array}\right]\end{array}\right.$$

(2)

where

–

${\mathrm{i}}_{\mathrm{d}}$, ${\mathrm{i}}_{\mathrm{q}}$—components of the output current vector of the inverter;

–

${\mathrm{d}}_{\mathrm{d}}$, ${\mathrm{d}}_{\mathrm{q}}$—components of the duty cycle of the voltage modulator;

–

${\mathrm{u}}_{\mathrm{c}\mathrm{d}}$, ${\mathrm{u}}_{\mathrm{c}\mathrm{q}}$—components of the voltage vector on the capacitive filter on the power grid side;

–

${\mathrm{i}}_{\mathrm{g}\mathrm{d}}$, ${\mathrm{i}}_{\mathrm{g}\mathrm{q}}$—components of the power grid current vector;

–

${\mathrm{U}}_{\mathrm{g}\mathrm{d}}$, ${\mathrm{U}}_{\mathrm{g}\mathrm{q}}$—components of the power grid voltage vector;

–

$\mathsf{\omega }$—pulsation.

For the simplification of further equations, the following was also assumed:

$$\left[\begin{array}{c}{\mathrm{v}}_{\mathrm{d}}\\ {\mathrm{v}}_{\mathrm{q}}\end{array}\right]={\mathrm{U}}_{\mathrm{d}\mathrm{c}}\left[\begin{array}{c}{\mathrm{d}}_{\mathrm{d}}\\ {\mathrm{d}}_{\mathrm{q}}\end{array}\right]$$

(3)

where

–

${\mathrm{v}}_{\mathrm{d}}$, ${\mathrm{v}}_{\mathrm{q}}$—components of the output voltage vector of the inverter.

The following state variables were adopted for the implementation of sliding control algorithms [20]:

$$\begin{gathered} \left[\begin{array}{c}{\mathrm{x}}_{1\mathrm{d}}\\ {\mathrm{x}}_{1\mathrm{q}}\end{array}\right]=\left[\begin{array}{c}{\mathrm{i}}_{\mathrm{d}}-{\mathrm{i}}_{{\mathrm{d}}_{\mathrm{S}\mathrm{P}}}\\ {\mathrm{i}}_{\mathrm{q}}-{\mathrm{i}}_{{\mathrm{q}}_{\mathrm{S}\mathrm{P}}}\end{array}\right] \\ \left[\begin{array}{c}{\mathrm{x}}_{2\mathrm{d}}\\ {\mathrm{x}}_{2\mathrm{q}}\end{array}\right]=\left[\begin{array}{c}{\mathrm{u}}_{\mathrm{c}\mathrm{d}}-{\mathrm{u}}_{{\mathrm{c}\mathrm{d}}_{\mathrm{S}\mathrm{P}}}\\ {\mathrm{u}}_{\mathrm{c}\mathrm{q}}-{\mathrm{u}}_{{\mathrm{c}\mathrm{q}}_{\mathrm{S}\mathrm{P}}}\end{array}\right] \\ \left[\begin{array}{c}{\mathrm{x}}_{3\mathrm{d}}\\ {\mathrm{x}}_{3\mathrm{q}}\end{array}\right]=\left[\begin{array}{c}{\mathrm{i}}_{\mathrm{g}\mathrm{d}}-{\mathrm{i}}_{{\mathrm{g}\mathrm{d}}_{\mathrm{S}\mathrm{P}}}\\ {\mathrm{i}}_{\mathrm{g}\mathrm{q}}-{\mathrm{i}}_{{\mathrm{g}\mathrm{q}}_{\mathrm{S}\mathrm{P}}}\end{array}\right] \end{gathered}$$

(4)

2.1. Standard Sliding Mode Control Concept

The basis for designing a sliding mode controller is the adoption of a certain sliding variable [12,20]. In the case where we represent three-phase inverter quantities in vector form, we define a sliding variable for both the d and q components:

$${\mathrm{s}}_{\mathrm{d}}={\mathsf{\alpha }}_1{\mathrm{x}}_{1\mathrm{d}}+{\mathsf{\alpha }}_2{\mathrm{x}}_{2\mathrm{d}}+{\mathsf{\alpha }}_3{\mathrm{x}}_{3\mathrm{d}}$$

(5)

$${\mathrm{s}}_{\mathrm{q}}={\mathsf{\alpha }}_1{\mathrm{x}}_{1\mathrm{q}}+{\mathsf{\alpha }}_2{\mathrm{x}}_{2\mathrm{q}}+{\mathsf{\alpha }}_3{\mathrm{x}}_{3\mathrm{q}}$$

(6)

where

–

${\mathrm{s}}_{\mathrm{d}}$, ${\mathrm{s}}_{\mathrm{q}}$—sliding variables;

–

${\mathsf{\alpha }}_1$, ${\mathsf{\alpha }}_2$, ${\mathsf{\alpha }}_3$—coefficients of the sliding variable.

The control law used in this work is based on sliding mode control and consists of two major components [20]:

• Continuous part: equivalent control and reaching law;
• Discontinuous control.

The first component of the control law is equivalent control, which assures the system with stability in the sense of Lyapunov. Below are the Lyapunov functions for the d and q components, starting from the first component of the current vector:

$${\mathrm{V}}_{\mathrm{d}}=\frac{1}{2}{\mathrm{s}}_{\mathrm{d}}^2$$

(7)

where

–

${\mathrm{V}}_{\mathrm{d}}$—Lyapunov function for the d component of the current vector.

The form of the derivative of this function is as follows:

$$\dot{{\mathrm{V}}_{\mathrm{d}}}={\mathrm{s}}_{\mathrm{d}}\dot{{\mathrm{s}}_{\mathrm{d}}}$$

(8)

For the system described by Equation (2) to be characterized by stability in the sense of Lyapunov, the derivative presented in Equation (8) must be less than or equal to 0, so in the next step, we equate the derivative of the sliding variable $\dot{s_d}$ to zero:

$$\dot{{\mathrm{s}}_{\mathrm{d}}}={\mathsf{\alpha }}_1\dot{{\mathrm{x}}_{1\mathrm{d}}}+{\mathsf{\alpha }}_2\dot{{\mathrm{x}}_{2\mathrm{d}}}+{\mathsf{\alpha }}_3\dot{{\mathrm{x}}_{3\mathrm{d}}}=0$$

(9)

Substituting the derivatives of state variables from Equation (4), we obtain the following form:

$${\mathsf{\alpha }}_1(\dot{{\mathrm{i}}_{\mathrm{d}}}-\dot{{\mathrm{i}}_{{\mathrm{d}}_{\mathrm{S}\mathrm{P}}}})+{\mathsf{\alpha }}_2(\dot{{\mathrm{u}}_{\mathrm{c}\mathrm{d}}}-\dot{{\mathrm{u}}_{{\mathrm{c}\mathrm{d}}_{\mathrm{S}\mathrm{P}}}})+{\mathsf{\alpha }}_3(\dot{{\mathrm{i}}_{\mathrm{g}\mathrm{d}}}-\dot{{\mathrm{i}}_{{\mathrm{g}\mathrm{d}}_{\mathrm{S}\mathrm{P}}}})=0$$

(10)

Next, using the state Equation (2), we obtain the following:

$${\mathsf{\alpha }}_1\left(\frac{{\mathrm{d}}_{\mathrm{d}}{\mathrm{U}}_{\mathrm{d}\mathrm{c}}-{\mathrm{u}}_{\mathrm{c}\mathrm{d}}+\mathrm{L}\mathsf{\omega }{\mathrm{i}}_{\mathrm{q}}}{\mathrm{L}}-\dot{{\mathrm{i}}_{{\mathrm{d}}_{\mathrm{S}\mathrm{P}}}}\right)+{\mathsf{\alpha }}_2\left(\frac{{\mathrm{i}}_{\mathrm{d}}-{\mathrm{i}}_{\mathrm{g}\mathrm{d}}+{\mathrm{C}}_{\mathrm{f}}\mathsf{\omega }{\mathrm{u}}_{\mathrm{c}\mathrm{q}}}{{\mathrm{C}}_{\mathrm{f}}}-\dot{{\mathrm{u}}_{{\mathrm{c}\mathrm{d}}_{\mathrm{S}\mathrm{P}}}}\right)+{\mathsf{\alpha }}_3\left(\frac{{\mathrm{u}}_{\mathrm{c}\mathrm{d}}-{\mathrm{U}}_{\mathrm{g}\mathrm{d}}+{\mathrm{L}}_{\mathrm{g}}\mathsf{\omega }{\mathrm{i}}_{\mathrm{g}\mathrm{q}}}{{\mathrm{L}}_{\mathrm{g}}}-\dot{{\mathrm{i}}_{{\mathrm{g}\mathrm{d}}_{\mathrm{S}\mathrm{P}}}}\right)=0$$

(11)

Further transformation of Equation (11) leads to the definition of the output voltage signal of inverter $v_d,$ ensuring the stability in the sense of Lyapunov:

$$\frac{{\mathsf{\alpha }}_1{\mathrm{d}}_{\mathrm{d}}{\mathrm{U}}_{\mathrm{d}\mathrm{c}}}{\mathrm{L}}+{\mathsf{\alpha }}_1\left(\frac{-{\mathrm{u}}_{\mathrm{c}\mathrm{d}}+\mathrm{L}\mathsf{\omega }{\mathrm{i}}_{\mathrm{q}}}{\mathrm{L}}-\dot{{\mathrm{i}}_{{\mathrm{d}}_{\mathrm{S}\mathrm{P}}}}\right)+{\mathsf{\alpha }}_2\left(\frac{{\mathrm{i}}_{\mathrm{d}}-{\mathrm{i}}_{\mathrm{g}\mathrm{d}}+{\mathrm{C}}_{\mathrm{f}}\mathsf{\omega }{\mathrm{u}}_{\mathrm{c}\mathrm{q}}}{{\mathrm{C}}_{\mathrm{f}}}-\dot{{\mathrm{u}}_{{\mathrm{c}\mathrm{d}}_{\mathrm{S}\mathrm{P}}}}\right)+{\mathsf{\alpha }}_3\left(\frac{{\mathrm{u}}_{\mathrm{c}\mathrm{d}}-{\mathrm{U}}_{\mathrm{g}\mathrm{d}}+{\mathrm{L}}_{\mathrm{g}}\mathsf{\omega }{\mathrm{i}}_{\mathrm{g}\mathrm{q}}}{{\mathrm{L}}_{\mathrm{g}}}-\dot{{\mathrm{i}}_{{\mathrm{g}\mathrm{d}}_{\mathrm{S}\mathrm{P}}}}\right)=0$$

(12)

Taking into account Equation (3), the $v_d$ signal can be rewritten as follows:

$${\mathrm{v}}_{\mathrm{d}}=-\frac{\mathrm{L}}{{\mathsf{\alpha }}_1}\left[{\mathsf{\alpha }}_1\left(\frac{-{\mathrm{u}}_{\mathrm{c}\mathrm{d}}+\mathrm{L}\mathsf{\omega }{\mathrm{i}}_{\mathrm{q}}}{\mathrm{L}}-\dot{{\mathrm{i}}_{{\mathrm{d}}_{\mathrm{S}\mathrm{P}}}}\right)+{\mathsf{\alpha }}_2\left(\frac{{\mathrm{i}}_{\mathrm{d}}-{\mathrm{i}}_{\mathrm{g}\mathrm{d}}+{\mathrm{C}}_{\mathrm{f}}\mathsf{\omega }{\mathrm{u}}_{\mathrm{c}\mathrm{q}}}{{\mathrm{C}}_{\mathrm{f}}}-\dot{{\mathrm{u}}_{{\mathrm{c}\mathrm{d}}_{\mathrm{S}\mathrm{P}}}}\right)+{\mathsf{\alpha }}_3\left(\frac{{\mathrm{u}}_{\mathrm{c}\mathrm{d}}-{\mathrm{U}}_{\mathrm{g}\mathrm{d}}+{\mathrm{L}}_{\mathrm{g}}\mathsf{\omega }{\mathrm{i}}_{\mathrm{g}\mathrm{q}}}{{\mathrm{L}}_{\mathrm{g}}}-\dot{{\mathrm{i}}_{{\mathrm{g}\mathrm{d}}_{\mathrm{S}\mathrm{P}}}}\right)\right]$$

(13)

The control signal in our situation is the component of the duty cycle vector $d_d$, which, in the case of equivalent control, was marked $d_{deq}$. Assuming the following:

$${\mathrm{i}}_{{\mathrm{g}\mathrm{d}}_{\mathrm{S}\mathrm{P}}}={\mathrm{I}}_{\mathrm{g}}$$

(14)

$${\mathrm{i}}_{{\mathrm{g}\mathrm{q}}_{\mathrm{S}\mathrm{P}}}=0$$

(15)

$${\mathrm{u}}_{{\mathrm{c}\mathrm{d}}_{\mathrm{S}\mathrm{P}}}={\mathrm{U}}_{\mathrm{g}\mathrm{d}}$$

(16)

$${\mathrm{u}}_{{\mathrm{c}\mathrm{q}}_{\mathrm{S}\mathrm{P}}}={\mathrm{U}}_{\mathrm{g}\mathrm{q}}+{\mathsf{\omega }\mathrm{L}}_{\mathrm{g}}{\mathrm{I}}_{\mathrm{g}}$$

(17)

$${\mathrm{i}}_{{\mathrm{d}}_{\mathrm{S}\mathrm{P}}}={\mathrm{C}}_{\mathrm{f}}\dot{{\mathrm{U}}_{\mathrm{g}\mathrm{d}}}+{\mathrm{I}}_{\mathrm{g}}-\mathsf{\omega }{\mathrm{C}}_{\mathrm{f}}{\mathrm{U}}_{\mathrm{g}\mathrm{q}}-{\mathsf{\omega }}^2{\mathrm{C}}_{\mathrm{f}}{\mathrm{L}}_{\mathrm{g}}{\mathrm{I}}_{\mathrm{g}}$$

(18)

$${\mathrm{i}}_{{\mathrm{q}}_{\mathrm{S}\mathrm{P}}}={\mathrm{C}}_{\mathrm{f}}\dot{{\mathrm{U}}_{\mathrm{g}\mathrm{q}}}+\mathsf{\omega }{\mathrm{C}}_{\mathrm{f}}{\mathrm{U}}_{\mathrm{g}\mathrm{d}}$$

(19)

$${\mathrm{I}}_{\mathrm{g}}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}=>\dot{{\mathrm{i}}_{{\mathrm{g}\mathrm{d}}_{\mathrm{S}\mathrm{P}}}}=\dot{{\mathrm{I}}_{\mathrm{g}}}=0$$

(20)

where

–

${\mathrm{I}}_{\mathrm{g}}$—the set value of the d component of the network current vector. We can determine the duty cycle signal for equivalent control of the d component:

$${\mathrm{d}}_{\mathrm{d}\mathrm{e}\mathrm{q}}=\frac{1}{{\mathrm{U}}_{\mathrm{d}\mathrm{c}}}\left[{\mathrm{u}}_{\mathrm{c}\mathrm{d}}-\mathrm{L}\mathsf{\omega }{\mathrm{i}}_{\mathrm{q}}+\dot{\mathrm{L}{\mathrm{i}}_{{\mathrm{d}}_{\mathrm{S}\mathrm{P}}}}-\frac{{\mathsf{\alpha }}_2\mathrm{L}}{{\mathsf{\alpha }}_1{\mathrm{C}}_{\mathrm{f}}}\left({\mathrm{x}}_{1\mathrm{d}}-{\mathrm{x}}_{3\mathrm{d}}+{\mathrm{C}}_{\mathrm{f}}\mathsf{\omega }{\mathrm{x}}_{2\mathrm{q}}\right)-\frac{{\mathsf{\alpha }}_3\mathrm{L}}{{\mathsf{\alpha }}_1{\mathrm{L}}_{\mathrm{g}}}\left({\mathrm{x}}_{2\mathrm{d}}+{\mathrm{L}}_{\mathrm{g}}\mathsf{\omega }{\mathrm{x}}_{3\mathrm{q}}\right)\right]$$

(21)

Similarly, the control signal $d_{qeq}$ for the q component was determined. The Lyapunov function for the second component of the current vector was adopted:

$${\mathrm{V}}_{\mathrm{q}}=\frac{1}{2}{\mathrm{s}}_{\mathrm{q}}^2$$

(22)

where

–

$V_q$—Lyapunov function for the q component of the current vector.

The derivative of this function is as follows:

$$\dot{{\mathrm{V}}_{\mathrm{q}}}=\frac{1}{2}{\mathrm{s}}_{\mathrm{q}}\dot{{\mathrm{s}}_{\mathrm{q}}}$$

(23)

We then compare the derivative of the sliding variable to zero:

$$\dot{{\mathrm{s}}_{\mathrm{q}}}={\mathsf{\alpha }}_1\dot{{\mathrm{x}}_{1\mathrm{q}}}+{\mathsf{\alpha }}_2\dot{{\mathrm{x}}_{2\mathrm{q}}}+{\mathsf{\alpha }}_3\dot{{\mathrm{x}}_{3\mathrm{q}}}=0$$

(24)

Analogously to Equations (10)–(13),we make a transformation:

$${\mathsf{\alpha }}_1(\dot{{\mathrm{i}}_{\mathrm{q}}}-\dot{{\mathrm{i}}_{{\mathrm{q}}_{\mathrm{S}\mathrm{P}}}})+{\mathsf{\alpha }}_2(\dot{{\mathrm{u}}_{\mathrm{c}\mathrm{q}}}-\dot{{\mathrm{u}}_{{\mathrm{c}\mathrm{q}}_{\mathrm{S}\mathrm{P}}}})+{\mathsf{\alpha }}_3(\dot{{\mathrm{i}}_{\mathrm{g}\mathrm{q}}}-\dot{{\mathrm{i}}_{{\mathrm{g}\mathrm{q}}_{\mathrm{S}\mathrm{P}}}})=0$$

(25)

$${\mathsf{\alpha }}_1(\frac{{\mathrm{d}}_{\mathrm{q}}{\mathrm{U}}_{\mathrm{d}\mathrm{c}}-{\mathrm{u}}_{\mathrm{c}\mathrm{q}}-\mathrm{L}\mathsf{\omega }{\mathrm{i}}_{\mathrm{d}}}{\mathrm{L}}-\dot{{\mathrm{i}}_{{\mathrm{q}}_{\mathrm{S}\mathrm{P}}}})+{\mathsf{\alpha }}_2(\frac{{\mathrm{i}}_{\mathrm{q}}-{\mathrm{i}}_{\mathrm{g}\mathrm{q}}-{\mathrm{C}}_{\mathrm{f}}\mathsf{\omega }{\mathrm{u}}_{\mathrm{c}\mathrm{d}}}{{\mathrm{C}}_{\mathrm{f}}}-\dot{{\mathrm{u}}_{{\mathrm{c}\mathrm{q}}_{\mathrm{S}\mathrm{P}}}})+{\mathsf{\alpha }}_3(\frac{{\mathrm{u}}_{\mathrm{c}\mathrm{q}}-{\mathrm{U}}_{\mathrm{g}\mathrm{q}}-{\mathrm{L}}_{\mathrm{g}}\mathsf{\omega }{\mathrm{i}}_{\mathrm{g}\mathrm{d}}}{{\mathrm{L}}_{\mathrm{g}}}-\dot{{\mathrm{i}}_{{\mathrm{g}\mathrm{q}}_{\mathrm{S}\mathrm{P}}}})=0$$

(26)

$${\frac{{\mathsf{\alpha }}_1{\mathrm{d}}_{\mathrm{q}}{\mathrm{U}}_{\mathrm{d}\mathrm{c}}}{\mathrm{L}}+\mathsf{\alpha }}_1(\frac{-{\mathrm{u}}_{\mathrm{c}\mathrm{q}}-\mathrm{L}\mathsf{\omega }{\mathrm{i}}_{\mathrm{d}}}{\mathrm{L}}-\dot{{\mathrm{i}}_{{\mathrm{q}}_{\mathrm{S}\mathrm{P}}}})+{\mathsf{\alpha }}_2(\frac{{\mathrm{i}}_{\mathrm{q}}-{\mathrm{i}}_{\mathrm{g}\mathrm{q}}-{\mathrm{C}}_{\mathrm{f}}\mathsf{\omega }{\mathrm{u}}_{\mathrm{c}\mathrm{d}}}{{\mathrm{C}}_{\mathrm{f}}}-\dot{{\mathrm{u}}_{{\mathrm{c}\mathrm{q}}_{\mathrm{S}\mathrm{P}}}})+{\mathsf{\alpha }}_3(\frac{{\mathrm{u}}_{\mathrm{c}\mathrm{q}}-{\mathrm{U}}_{\mathrm{g}\mathrm{q}}-{\mathrm{L}}_{\mathrm{g}}\mathsf{\omega }{\mathrm{i}}_{\mathrm{g}\mathrm{d}}}{{\mathrm{L}}_{\mathrm{g}}}-\dot{{\mathrm{i}}_{{\mathrm{g}\mathrm{q}}_{\mathrm{S}\mathrm{P}}}})=0$$

(27)

$${\mathrm{v}}_{\mathrm{q}}=-\frac{\mathrm{L}}{{\mathsf{\alpha }}_1}\left[{\mathsf{\alpha }}_1\left(\frac{-{\mathrm{u}}_{\mathrm{c}\mathrm{q}}-\mathrm{L}\mathsf{\omega }{\mathrm{i}}_{\mathrm{d}}}{\mathrm{L}}-\dot{{\mathrm{i}}_{{\mathrm{q}}_{\mathrm{S}\mathrm{P}}}}\right)+{\mathsf{\alpha }}_2\left(\frac{{\mathrm{i}}_{\mathrm{q}}-{\mathrm{i}}_{\mathrm{g}\mathrm{q}}-{\mathrm{C}}_{\mathrm{f}}\mathsf{\omega }{\mathrm{u}}_{\mathrm{c}\mathrm{d}}}{{\mathrm{C}}_{\mathrm{f}}}-\dot{{\mathrm{u}}_{{\mathrm{c}\mathrm{q}}_{\mathrm{S}\mathrm{P}}}}\right)+{\mathsf{\alpha }}_3\left(\frac{{\mathrm{u}}_{\mathrm{c}\mathrm{q}}-{\mathrm{U}}_{\mathrm{g}\mathrm{q}}-{\mathrm{L}}_{\mathrm{g}}\mathsf{\omega }{\mathrm{i}}_{\mathrm{g}\mathrm{d}}}{{\mathrm{L}}_{\mathrm{g}}}-\dot{{\mathrm{i}}_{{\mathrm{g}\mathrm{q}}_{\mathrm{S}\mathrm{P}}}}\right)\right]$$

(28)

Finally, assuming that

$$\mathsf{\omega }=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\Rightarrow \dot{\mathsf{\omega }}=0$$

(29)

we obtain the final form of the q component of the duty cycle vector for equivalent control:

$${\mathrm{d}}_{\mathrm{q}\mathrm{e}\mathrm{q}}=\frac{1}{{\mathrm{U}}_{\mathrm{d}\mathrm{c}}}\left[{\mathrm{u}}_{\mathrm{c}\mathrm{q}}+\mathrm{L}\mathsf{\omega }{\mathrm{i}}_{\mathrm{d}}+\dot{\mathrm{L}{\mathrm{i}}_{{\mathrm{q}}_{\mathrm{S}\mathrm{P}}}}+\frac{{\mathsf{\alpha }}_2\mathrm{L}}{{\mathsf{\alpha }}_1{\mathrm{C}}_{\mathrm{f}}}\left({\mathrm{x}}_{1\mathrm{d}}-{\mathrm{x}}_{3\mathrm{d}}+{\mathrm{C}}_{\mathrm{f}}\mathsf{\omega }{\mathrm{x}}_{2\mathrm{q}}\right)+\frac{{\mathsf{\alpha }}_3\mathrm{L}}{{\mathsf{\alpha }}_1{\mathrm{L}}_{\mathrm{g}}}\left({\mathrm{x}}_{2\mathrm{d}}+{\mathrm{L}}_{\mathrm{g}}\mathsf{\omega }{\mathrm{x}}_{3\mathrm{q}}\right)\right]$$

(30)

The second component of the continuous part of the control signal is the so-called reaching law, which ensures that the system reaches the sliding phase faster through the operating point than it would without this element. This part of the control also allows for limiting the amplitude of discontinuous control, thereby reducing the impact of so-called chattering on the system.

In this work, a proportional reaching law [30] was used. The control signals for the d and q components were named ${\mathrm{d}}_{\mathrm{R}\mathrm{L}\mathrm{d}}$ and ${\mathrm{d}}_{\mathrm{R}\mathrm{L}\mathrm{q}}$, respectively:

$${\mathrm{d}}_{\mathrm{R}\mathrm{L}\mathrm{d}}=-{\mathrm{k}}_{\mathrm{R}\mathrm{L}}{\mathrm{s}}_{\mathrm{d}}$$

(31)

$${\mathrm{d}}_{\mathrm{R}\mathrm{L}\mathrm{q}}=-{\mathrm{k}}_{\mathrm{R}\mathrm{L}}{\mathrm{s}}_{\mathrm{q}}$$

(32)

where

–

${\mathrm{k}}_{\mathrm{R}\mathrm{L}}$—reaching law control gain, ${\mathrm{k}}_{\mathrm{R}\mathrm{L}}>0$.

The basis of sliding mode control is the discontinuous part, which is responsible for the presence of sliding motion and causes the system to belong to the category of robust control systems [12]. This part is based on the use of the signum function:

$${\mathrm{d}}_{\mathrm{D}\mathrm{d}}=-{\mathrm{k}}_{\mathrm{D}}\mathrm{s}\mathrm{g}\mathrm{n}\left({\mathrm{s}}_{\mathrm{d}}\right)$$

(33)

$${\mathrm{d}}_{\mathrm{D}\mathrm{q}}=-{\mathrm{k}}_{\mathrm{D}}\mathrm{s}\mathrm{g}\mathrm{n}\left({\mathrm{s}}_{\mathrm{q}}\right)$$

(34)

where

–

${\mathrm{d}}_{\mathrm{D}\mathrm{d}}$, ${\mathrm{d}}_{\mathrm{D}\mathrm{q}}$—components of the discontinuous control signal;

–

${\mathrm{k}}_{\mathrm{D}}$—gain of discontinuous control, ${\mathrm{k}}_{\mathrm{D}}>0$.

The final control law for the d and q components of the network current vector is obtained by summing the previously mentioned individual control signals:

$${\mathrm{d}}_{\mathrm{d}}={\mathrm{d}}_{\mathrm{d}\mathrm{e}\mathrm{q}}+{\mathrm{d}}_{\mathrm{R}\mathrm{L}\mathrm{d}}+{\mathrm{d}}_{\mathrm{D}\mathrm{d}}$$

(35)

$${\mathrm{d}}_{\mathrm{q}}={\mathrm{d}}_{\mathrm{q}\mathrm{e}\mathrm{q}}+{\mathrm{d}}_{\mathrm{R}\mathrm{L}\mathrm{q}}+{\mathrm{d}}_{\mathrm{D}\mathrm{q}}$$

(36)

Using Equations (21) and (30), we obtain the following dependencies:

$$\dot{{\mathrm{V}}_{\mathrm{d}}}\le 0$$

(37)

$$\dot{{\mathrm{V}}_{\mathrm{q}}}\le 0$$

(38)

Therefore, the Lyapunov stability of the system has been proven. Taking into account the total control signals from (35) and (36), apart from the stability point, we obtain the following inequalities for ${\mathrm{s}}_{\mathrm{d}}\ne 0$ and ${\mathrm{s}}_{\mathrm{q}}\ne 0$:

$$\dot{{\mathrm{V}}_{\mathrm{d}}}={\mathrm{s}}_{\mathrm{d}}\left(-{\mathrm{k}}_{\mathrm{D}}\mathrm{s}\mathrm{g}\mathrm{n}\left({\mathrm{s}}_{\mathrm{d}}\right)-{\mathrm{k}}_{\mathrm{R}\mathrm{L}}{\mathrm{s}}_{\mathrm{d}}\right)<0$$

(39)

$$\dot{{\mathrm{V}}_{\mathrm{q}}}={\mathrm{s}}_{\mathrm{q}}(-{\mathrm{k}}_{\mathrm{D}}\mathrm{s}\mathrm{g}\mathrm{n}\left({\mathrm{s}}_{\mathrm{q}}\right)-{\mathrm{k}}_{\mathrm{R}\mathrm{L}}{\mathrm{s}}_{\mathrm{q}})<0$$

(40)

Using inequalities (39) and (40), asymptotic Lyapunov stability for both components of the power grid current vector has been proven.

In both cases presented using inequalities (39) and (40), the derivative of the given Lyapunov function will be negative for the positive value of the sliding variable, as well as for its negative value.

Unfortunately, this control is characterized by oscillations, etc., which is why the authors propose a modified version of the control.

2.2. Modified Sliding Mode Control Concept

The solution proposed in this work is the application of the so-called hybrid reaching law and linking this part of the control with discontinuous control by using a common parameter [21].

The hybrid reaching law includes two switching control laws:

• Proportional reaching law [14,15];
• Exponential reaching law [14,15,16,33].

The switching of these two signals depends on the current value of the sliding variable. The aim is to shorten the reaching phase time and then enable better cooperation of discontinuous control with reaching law. The first goal will be achieved by the exponential part, while the second will be by the proportional part.

The applied reaching law is described by the following equations:

$${\mathrm{d}}_{\mathrm{R}\mathrm{L}\mathrm{H}\mathrm{d}}=-{\mathrm{k}}_{\mathrm{R}\mathrm{L}}\mathrm{H}\left({\mathrm{s}}_{\mathrm{d}},\mathsf{\beta }\right)$$

(41)

$${\mathrm{d}}_{\mathrm{R}\mathrm{L}\mathrm{H}\mathrm{q}}=-{\mathrm{k}}_{\mathrm{R}\mathrm{L}}\mathrm{H}\left({\mathrm{s}}_{\mathrm{q}},\mathsf{\beta }\right)$$

(42)

where

–

${\mathrm{d}}_{\mathrm{R}\mathrm{L}\mathrm{H}\mathrm{d}}$, ${\mathrm{d}}_{\mathrm{R}\mathrm{L}\mathrm{H}\mathrm{q}}$—modified reaching law controls;

–

$\mathrm{H}({\mathrm{s}}_{\mathrm{d}},\mathsf{\beta })$, $\mathrm{H}({\mathrm{s}}_{\mathrm{q}},\mathsf{\beta })$—functions implementing hybrid reaching law control;

–

$\mathsf{\beta }$—a parameter that is the exponent of power affecting the control gain, $\mathsf{\beta }\in [0,1]$.

The function $\mathrm{H}(\mathrm{s},\mathsf{\beta })$ for both components of the vector (d and q) takes the following form depending on the values of sliding variables:

• For $\left|s\right|>1:$

$$\mathrm{H}\left(\mathrm{s},\mathsf{\beta }\right)={\left|\mathrm{s}\right|}^{1+\mathsf{\beta }}\mathrm{s}\mathrm{g}\mathrm{n}\left(\mathrm{s}\right)$$

(43)

• For $\left|s\right|<1:$

$$\mathrm{H}\left(\mathrm{s},\mathsf{\beta }\right)=\mathrm{s}$$

(44)

Equation (44) presents the classical, proportional implementation of reaching law control, while Equation (43) includes exponential control dependent on the current value of the sliding variable and the parameter, which is the exponent.

The next step is the previously mentioned change in discontinuous control, which will be linked with the hybrid reaching law by the parameter $\beta$. This link allows for an analytical selection of gains of individual components of the control law. The higher the value of the $\beta$ parameter, the smaller the gain of the discontinuous control part and the greater the gain of the reaching law. Appropriate tuning of this value allows for maintaining the system robustness guaranteed by this part of the control while reducing the impact of chattering. In the case of reaching law, the $\beta$ parameter makes it possible to shorten the reaching phase by applying greater gain until the point that describes the current state of the system approaches the sliding hyperplane. Therefore, the modified discontinuous control takes the following form:

$${\mathrm{d}}_{\mathrm{D}\mathrm{H}\mathrm{d}}=-{\mathrm{k}}_{\mathrm{D}}{\left|\mathrm{s}\right|}^{1-\mathsf{\beta }}\mathrm{s}\mathrm{g}\mathrm{n}\left({\mathrm{s}}_{\mathrm{d}}\right)$$

(45)

$${\mathrm{d}}_{\mathrm{D}\mathrm{H}\mathrm{q}}=-{\mathrm{k}}_{\mathrm{D}}{\left|\mathrm{s}\right|}^{1-\mathsf{\beta }}\mathrm{s}\mathrm{g}\mathrm{n}\left({\mathrm{s}}_{\mathrm{q}}\right)$$

(46)

where

–

${\mathrm{d}}_{\mathrm{D}\mathrm{H}\mathrm{d}}$, ${\mathrm{d}}_{\mathrm{D}\mathrm{H}\mathrm{q}}$—modified discontinuous controls.

The above-proposed modified forms of discontinuous controls are characterized by a variable amplitude, which depends on the signal of sliding variables. This modification should eliminate the impact of chattering on the system, which will result in an improvement in the quality of regulation.

The complete control law is presented in the following form:

$${\mathrm{d}}_{\mathrm{d}}={\mathrm{d}}_{\mathrm{d}\mathrm{e}\mathrm{q}}+{\mathrm{d}}_{\mathrm{R}\mathrm{L}\mathrm{H}\mathrm{d}}+{\mathrm{d}}_{\mathrm{D}\mathrm{H}\mathrm{d}}$$

(47)

$${\mathrm{d}}_{\mathrm{q}}={\mathrm{d}}_{\mathrm{q}\mathrm{e}\mathrm{q}}+{\mathrm{d}}_{\mathrm{R}\mathrm{L}\mathrm{H}\mathrm{q}}+{\mathrm{d}}_{\mathrm{D}\mathrm{H}\mathrm{q}}$$

(48)

The two parts of the control, which have been modified, still provide negative definiteness of the derivatives of both Lyapunov functions (39) and (40), so the system remains asymptotically stable in the sense of Lyapunov for ${\mathrm{s}}_{\mathrm{d}}\ne 0$ and ${\mathrm{s}}_{\mathrm{q}}\ne 0$:

$$\dot{{\mathrm{V}}_{\mathrm{d}}}={\mathrm{s}}_{\mathrm{d}}\left(-{\mathrm{k}}_{\mathrm{D}}{\left|\mathrm{s}\right|}^{1-\mathsf{\beta }}\mathrm{s}\mathrm{g}\mathrm{n}\left({\mathrm{s}}_{\mathrm{d}}\right)-{\mathrm{k}}_{\mathrm{R}\mathrm{L}}\mathrm{H}({\mathrm{s}}_{\mathrm{d}},\mathsf{\beta })\right)<0$$

(49)

$$\dot{{\mathrm{V}}_{\mathrm{q}}}={\mathrm{s}}_{\mathrm{q}}\left(-{\mathrm{k}}_{\mathrm{D}}{\left|\mathrm{s}\right|}^{1-\mathsf{\beta }}\mathrm{s}\mathrm{g}\mathrm{n}\left({\mathrm{s}}_{\mathrm{q}}\right)-{\mathrm{k}}_{\mathrm{R}\mathrm{L}}\mathrm{H}\left({\mathrm{s}}_{\mathrm{q}},\mathsf{\beta }\right)\right)<0$$

(50)

The sign dependence of the given derivatives of the Lyapunov function presented using inequalities (49) and (50) is the same as in the case described in Equations (39) and (40).

The modified control laws have been implemented in a simulation environment to re-examine the system. The results are presented and described in Section 3.

2.3. Simulation Model of T-Type Bidirectional Power Grid Converter

The discussed model of a bidirectional T-type inverter with an LCL filter and the proposed control system has been implemented in the MATLAB (version R2021a) Simulink environment for the purpose of conducting a simulation examination. The simulation model is presented in Figure 3.

The most important submodules of this model include a signal generator for the set values, a grid current control system, dq0 transforms for voltages and currents, an inverse dq0 transform for the set voltages for the modulator, a modulator, an inverter with an LCL filter, a power grid, and a phase synchronization module.

Table 1. System parameters used for simulation.

Name of the Parameter	Value
Simulation time step	100 ns
Modulator frequency	50 kHz
DC source voltage	350 V
Filter inductance (inverter output side)	1.4 mH
Filter capacitance	0.2 mF
Filter inductance (load side)	0.127 mH

presents the parameters describing the components used throughout the system.

3. Results

The aim was to examine the system of network current vector control in the discussed structure of the inverter with a filter connected to the power grid. Five cases were tested for classic SMC and its modified version: one static case and four dynamic cases, including step changes of grid current vector d and q components. In order to evaluate the results, the following measures of control performance were collected: integral squared error (ISE), integral absolute error (IAE), overshoot value, settling time, and maximum and minimum values. The time duration of each test is equal to 0.08 s.

3.1. Simulation Results of SMC and Modified SMC—Static Case

The first simulation test pertains to a case in which the set signal for the d component of the grid current vector is constant and is equal to 10 A. The initial value is the same. For the q component, these values are zero. The waveforms of current grid vector component signals were collected and are presented in Figure 4.

One can see that in the case of the classic sliding mode control algorithm (a), the system is stable but, at the same time, characterized by the presence of regular low-amplitude oscillations and noises. This is due to the constant amplitude of discontinuous control. The gain of this part of the control law should be greater than the total gain of all disturbances acting on the system so that the system is robust and remains in sliding motion [42]. Reducing the constant amplitude can cause the system to lose total robustness and cause the operating point to not move in a sliding motion at all times. However, the variable amplitude of discontinuous control gain can be used when implementing hybrid reaching law. The use of the proposed modified sliding mode control method allows for a significant reduction in the impact of the above-mentioned oscillations for both the d and q components of the grid current vector. This is shown in Figure 4b. Additionally, the impact of current ripples caused by voltage drops on transistors when phase currents pass through 0 has been reduced.

Figure 5 presents the control measures collected for the static case. Part (a) of this graphic contains the integral squared error and integral absolute error values for both components of the grid current vector. The presented numerical values were extracted based on Figure 4.

The use of a modified algorithm (HSMC) allows us to obtain approximately twice better results (52% reduction) of the IAE value of the d component than in the case of the classical algorithm (SMC). The control measure values for the q component are significantly lower for the hybrid version of sliding mode control, so the use of this algorithm allows us to effectively limit the reactive power of the system (92% reduction in IAE and 99% reduction in ISE). The additional measures presented in Figure 5b allow us to conclude that the use of the classic version of sliding mode control in the discussed problem is subject to a much greater spread of values caused by oscillations and noise than in the case of the proposed algorithm (for d component: 73% reduction in maximum and 79% reduction in minimum; for q component: 94% reduction in maximum and 95% reduction in minimum).

3.2. Simulation Results of SMC and Modified SMC—Dynamic Cases

The next test case concerns dynamic scenarios, i.e., those characterized by a change in the set signal during the operation of the system. The following situations were tested:

• Change in ${\mathrm{i}}_{\mathrm{d}}$ from 0 to 10 A, ${\mathrm{i}}_{\mathrm{q}}$ equal to 0 A;
• Change in ${\mathrm{i}}_{\mathrm{d}}$ from 10 to −10 A, ${\mathrm{i}}_{\mathrm{q}}$ equal to 10 A;
• ${\mathrm{i}}_{\mathrm{d}}$ equal to 0 A, ${\mathrm{i}}_{\mathrm{q}}$ change from 0 to 10 A;
• ${\mathrm{i}}_{\mathrm{d}}$ equal to 10 A, ${\mathrm{i}}_{\mathrm{q}}$ change from 10 to −10 A.

Figure 6 shows the waveforms of the current vector components. Part (a) presents the classic control method and (b) the modified one.

Again, there is less oscillation and noise present in the signals for modified solution.

Figure 7 shows the control measures for the test performed—(a) includes IAE and ISE for two current components, while (b) concerns the adjustment time. The presented numerical values were extracted from Figure 6.

For the hybrid algorithm, this indicator is slightly larger, which may be at the expense of reducing oscillations to a great extent (13% higher value of settling time). The indicators presented in (a) indicate that better quality regulation is provided by HSMC (IAE: for d component reduced by 23% and for q component reduced by 76%).

In the second dynamic test, the case was examined when the set signal of the grid current vector component passed through the value 0. The obtained waveforms are presented in Figure 8.

This test was performed to check whether such action does not cause additional negative consequences. Figure 8a,b show that such an influence cannot be observed.

Figure 9 shows the control measures for the test performed—(a) includes IAE and ISE for two current components, while (b) concerns the adjustment time. The presented numerical values were extracted based on Figure 8.

The results are similar to the previous test, but this time, the settling time is shorter for the results of the modified algorithm (10% reduction). As in the previous test cases, control measures for the q component of the current vector are lower for the hybrid sliding mode control version (IAE: for d component reduced by 42% and for q component reduced by 79%).

The third dynamic case is analogous to the first one, but the change concerns ${\mathrm{i}}_{\mathrm{q}}$. The obtained waveforms are presented in Figure 10.

In this scenario, we do not observe overshoot, and its absence is visible in the shown graphics.

Figure 11 shows the control measures for the test performed—(a) includes IAE and ISE for two current components, while (b) concerns the adjustment time. The presented numerical values were extracted from Figure 10.

Based on the results shown in Figure 11, we can once again confirm the improvement in the quality of regulation thanks to the hybrid algorithm (IAE: for d component reduced by 59% and for q component reduced by 51%), this time at a visible cost in the form of settling time (50% higher settling time value).

The case of changing the set signal of the q component of the network current vector is different from changing the d component. There is no overshoot, and the difference in control measure values for both algorithms is not multiple for the changed component, as was the case with changing the previous component. Still, the results using the modified algorithm are much better than those of the classic solution, but at the visible cost of settling time.

Figure 12a,b show the waveforms for the case of a dynamic change in ${\mathrm{i}}_{\mathrm{q}}$ with the signal value crossing 0.

Figure 13 shows the control measures for the test performed—(a) includes IAE and ISE for two current components, while (b) concerns the adjustment time. The presented numerical values were based on data from Figure 12.

The results visible in Figure 13, as in the previous test scenario, indicate lower values of control measures for HSMC (IAE: for d component reduced by 53% and for q component reduced by 40%) at the expense of settling time (43% higher settling time value).

For convenience of simulation data analysis of experiments presented in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13, we also present these data of investigated parameters in

Table 2. Control measures for static simulation case.

Cases	IAE_d	IAE_q	ISE_d	ISE_q
Static case (SMC)	0.0042	0.0072	0.0004	0.0011
Static case (HSMC)	0.0021	0.0005	0.0001	0.000005

,

Table 3. Maximum and minimum values of grid current component values for static simulation case.

Cases	${\mathbf{i}}_{\mathit{d}}$ Maximum [A]	${\mathbf{i}}_{\mathit{d}}$ Minimum [A]	${\mathbf{i}}_{\mathbf{q}}$ Maximum [A]	${\mathbf{i}}_{\mathbf{q}}$ Minimum [A]
Static case (SMC)	0.2247	−0.2039	0.3326	−0.4055
Static case (HSMC)	0.0599	−0.0411	0.0199	−0.0184

,

Table 4. Control measures for dynamic simulation cases.

Cases	IAE_d	IAE_q	ISE_d	ISE_q
${\mathrm{i}}_{\mathrm{d}}$: from 0 to 10 A (SMC)	0.0066	0.0079	0.0079	0.0015
${\mathrm{i}}_{\mathrm{d}}$: from 0 to 10 A (HSMC)	0.0051	0.0019	0.0112	0.0006
${\mathrm{i}}_{\mathrm{d}}$: from 10 to −10 A (SMC)	0.0118	0.0156	0.0199	0.0069
${\mathrm{i}}_{\mathrm{d}}$: from 10 to −10 A (HSMC)	0.0068	0.0032	0.0177	0.0026
${\mathrm{i}}_{\mathrm{q}}$: from 0 to 10 A (SMC)	0.0059	0.0165	0.0007	0.0263
${\mathrm{i}}_{\mathrm{q}}$: from 0 to 10 A (HSMC)	0.0024	0.0081	0.0002	0.0225
${\mathrm{i}}_{\mathrm{q}}$: from 10 to −10 A (SMC)	0.0068	0.0259	0.0010	0.0956
${\mathrm{i}}_{\mathrm{q}}$: from 10 to −10 A (HSMC)	0.0032	0.0153	0.0004	0.0824

and

Table 5. Settling time values for dynamic simulation cases.

Cases	${\mathbf{i}}_{\mathbf{d}}$ Settling Time [s]	${\mathbf{i}}_{\mathbf{q}}$ Settling Time [s]
${\mathrm{i}}_{\mathrm{d}}$: from 0 to 10 A (SMC)	0.0042	-
${\mathrm{i}}_{\mathrm{d}}$: from 0 to 10 A (HSMC)	0.0047	-
${\mathrm{i}}_{\mathrm{d}}$: from 10 to −10 A (SMC)	0.0067	-
${\mathrm{i}}_{\mathrm{d}}$: from 10 to −10 A (HSMC)	0.0060	-
${\mathrm{i}}_{\mathrm{q}}$: from 0 to 10 A (SMC)	-	0.0046
${\mathrm{i}}_{\mathrm{q}}$: from 0 to 10 A (HSMC)	-	0.0069
${\mathrm{i}}_{\mathrm{q}}$: from 10 to −10 A (SMC)	-	0.0057
${\mathrm{i}}_{\mathrm{q}}$: from 10 to −10 A (HSMC)	-	0.0081

.

Table 2 contains control measures for the static case: integral squared error and integral absolute error values for the two components of the network current vector.

Table 3 contains the maximum and minimum values of the mentioned components.

Table 4 contains control measures for dynamic cases: integral squared error and integral absolute error values for the two components of the network current vector.

Table 5 refers to the settlement time values for dynamic cases.

4. Conclusions

This paper discusses the issue of controlling the current of a three-phase power grid to which an inverter with an LCL filter is connected. The inverter, implemented in T topology, allows for achieving three voltage levels, which enables the use of a higher switching frequency by reducing the influence of current ripple. The control of the inverter’s gates is carried out by duty cycle signals that go to the modulator. To simplify the implementation of the discussed control task, a dq0 transformation was used, which utilizes Clarke’s and Park’s transformations. As a result, instead of three sinusoidal signals, the system operates on two constant components of the current vector.

The first proposed control system was a system based on sliding mode control. The stability of the system was examined using the Lyapunov function, and asymptotic stability in the sense of Lyapunov was obtained. The proposed control system was tested in the MATLAB Simulink environment. The control algorithm fulfilled its task, the system was stable, and the set values were achieved quickly; however, the current waveforms were characterized by non-decaying oscillations of high frequency and low amplitude.

The second control system was based on the previously discussed structure. Two parts of the control law were modified: reaching law and discontinuous control. A so-called hybrid reaching law was introduced. Both changed parts were linked by one parameter, which allows for the analytical selection of parameters for reaching law. The results obtained by introducing these changes into the simulation environment showed that the measured quality of regulation was significantly improved, and the presence of oscillations in current signals was greatly reduced.

It is worth mentioning once again the exemplary results obtained using HSMC: for the static case, IAE was reduced by 52% for the d and 92% for the q component, which means a significant reduction in reactive power in the system. For one of the cases of a step change in the d component, it was possible to reduce the IAE by 23% for the d and 76% for the q component. For the scenario in which the step change concerns the q component, the results were 53% and 40% better, respectively. These data confirm that the modified SMC algorithm allows us to obtain better results than classical SMC controllers in the case of the discussed system. The proposed method is feasible to implement in typical microcontroller-based devices used in power electronics, especially in PV inverters, EV chargers, and energy storage devices connected to the grid.

The final noteworthy aspect pertains to comparing our results with other published works. In the study presented in [19], the authors investigated a control system for a three-phase inverter with an LCL filter, employing a PI controller. In this case, low-amplitude oscillations and regular current overshoots can be observed in network current waveforms. Both of these problems were successfully mitigated using our HSMC approach. Another work, ref. [20], faced challenges related to oscillations in the errors of both components of the current vector, likely due to chattering phenomena. Our hybrid approach effectively reduced these oscillations. In another study involving a T-type inverter, in [21], researchers implemented a PI (Proportional–Integral) controller combined with the FOSC (Fractional-Order Synergetic Controller) method. This approach effectively reduced unwanted current pulsations. However, it did not fully account for system dynamics or ensure robustness. Furthermore, ref. [5] demonstrated an NPC (Neutral Point Clamped) inverter with an LCL filter. The results indicated positive network current control quality, stability, and high-quality signal reception. Notably, the authors employed a QPR (quasi-proportional resonant) controller with a hybrid active damping control strategy mechanism, so the structure of the inverter system and the control algorithms are much more complicated than those used in our research. Although the presented method enhanced robustness to some extent, it did not guarantee robustness in the meaning of nonlinear robust control systems that can be provided by SMC or other VSC (Variable Structure Controller) or even by adaptation mechanism in some cases.