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(1. INTRODUCTION
As a result of their advantages, multilevel inverters have received much attention in industrial applications (Maamar et al., 2021). Compared to traditional inverters, the use of Multilevel Inverters (MLIs) in power circuits has many benefits, such as the reduction of Total Harmonic Distortion (THD), the reduction of static switch stress, the reduction of power losses and filter volume, and the enhancement of power factor with the increase of the voltage level (Chauca Llusca et al., 2014). According to Akagi (2017), there are three standard varieties of MLIs: the FCMLI, Flying Capacitor Multilevel Inverter, the CHBMLI, Cascaded Half-Bridge Multilevel Inverter with separate DC, Direct Current sources, and the NPCMLI, Neutral Point Clamped Multilevel Inverter. The CHBMLI is the most commonly operated in industrial systems because of its high voltage capability, high reliability, high power, and modularity (Oumaymah et al., 2020; Toubal Maamar et al., 2020). Generating multiple output voltage levels from the conventional topologies of MLIs is possible but can require many DC sources, capacitors, diodes, and switching devices, which then results in cost increases and the complexity of control (Toubal et al., 2020). To address the drawbacks of conventional MLIs, the amount of switching components and devices has been the subject of numerous research. In the literature, the authors Albakhait and Abdulkareem (2022) propose an H-bridge inverter with a new switch configuration model, producing five output voltage levels. Similarly, two H-bridges are cascade connected to obtain nine levels at the output of an inverter. Under normal and defective conditions, to balance the voltage of the capacitors, a new T-type inverter structure with a fewer number of devices was presented in the paper (Kumar et al., 2022). Moreover, Selvakumar and Patel (2021) propose an inverter with fewer switches that produces a high number of levels. Furthermore, Toubal Maamar et al. (2021) present different inverter topologies that can optimize the efficiency of inverters by using appropriate control.
To achieve a good dynamic performance under various loads, many studies have been conducted in recent years to investigate the closed-loop control of multilevel inverters. In 2021, Odeh et al. proposed a single carrier-based pulse width modulation (PWM) scheme for a cascaded H-bridge multilevel inverter; its operational concept is to adapt the sinusoidal modulation waveform to a range of single triangular carrier signals to generate the desired output waveform pattern for PWM. Sakthisudhursun et al. (2016) propose the control of a three-level five-phase inverter by simplified Space Vector Pulse Width Modulation (SVPWM). Oumaymah et al. (2020a) present the SVPWM technique for a five-level NPC inverter to reduce the total harmonic distortion and increase the power factor. Hossam-Eldin et al. (2020) developed a wind power conversion system from back-to-back multilevel converter systems using third harmonic and sinusoidal pulse width modulation approaches. In Jiang et al. (2019a), a new PWM strategy for controlling the neutral point voltage based on carriers is shown, allowing under any modulation index and power factor to reach the neutral point voltage balance. Non-linear control strategies, such as the backstepping method, are worthy of study if we consider that the inverter connected to the grid is a nonlinear system of order three (Medina Sánchez & Naranjo Cevallos, 2021; Oumaymah et al., 2021; Pozo & Pacas, 2013). Manai et al. (2020) proposed a control strategy based on the backstepping method associated with a hysteresis current controller to produce the inverter control signals that automatically balance the capacitor voltage.
On the other hand, the inverters cannot be directly connected to grids because of harmonics. Filters, such as LCL, LC, and L, are generally used to address this situation. The filters are employed to improve the quality of the currents fed into the grid by attenuating the output harmonics of the inverter (Huang et al., 2020). A simple filter, the L filter attenuates the frequencies by 20 dB/decade (Zhang et al., 2019). On the other hand, there is the LC filter. It is a second-order filter with better efficiency than the L filter. LC filter provides 40 dB/decade of attenuation (González et al., 2014). LCL is a third-order filter that offers good attenuation 60 dB/decade at frequencies above the cut-off (Jiang et al., 2019). It also improves inverter-grid decoupling than other types of passive filter (Villanueva et al., 2020). The utilization of the LCL filter is challenging due to its high-order nature, which entails intricate analysis and mathematical description of the system.
Based on the above overview, we find that the multilevel inverters with reduced switches and components are effective in several areas. Still, their popularity for the LCL-Grid system is limited. Additionally, the most commonly used inverters in the literature for implementing the LCL-grid system are the NPC inverter in three-phase systems and the H-bridge inverter in single-phase systems. The PID, Proportional-Integral-Derivative, and the PI, Proportional-Integral controllers are commonly employed in practical applications. The majority of industrial applications, namely in motor control, utilize these controllers, with a usage rate above 90% (De la Cruz & Camacho, 2015; Yandun et al., 2018). Moreover, we noted that the nonlinear Backstepping approach based on the Lyapunov stability rule is an interesting method for controlling engineering systems, especially nonlinear systems, where this approach has many advantages and benefits such as robustness and high-quality performances in dynamic systems control.
This original paper introduces and designs a new power circuit composed of: a novel five-level inverter, which utilizes fewer switching components to supply power to an LCL filter and the grid. Then, design an appropriate LCL filter to achieve the synchronization operation between the inverter and the grid; to attain high-quality grid current with the decrease of high degree harmonics. Lastly, the backstepping control is applied to generate a trigger pulse corresponding to the adequate current-voltage synchronization through the Lyapunov stability rule and grid current control.
This paper is structured in the following manner: the second section delineates the structure of the proposed system and explains the design of the backstepping control law. The discussion of the findings may be found in Section III. The paper comes to a close with Section IV.
2. METHODOLOGY
Figure 1 shows the proposed grid-connected inverter scheme.
The proposed system is composed of two fundamental parts: the electrical power part and the control part. The electrical power part consists of: an emerging five-level inverter topology, an LCL Filter, and an electrical grid.
The control part consists of: the Backstepping Control (BSC) method, and the pulse generator that utilizes the high-frequency modulation approach to generate gating pulses SPWM for the five-level inverter.
Figure 1 The proposed scheme of the grid-connected five-level inverter with LCL filter and the nonlinear backstepping controller
2.1 The emerging five-level inverter topology
A five-level inverter that is typically composed of six switches (S 1, S 2, S 3, S 4, S 5, S 6) and two DC power supplies (V 1=V 2=V dc ) (Maamar et al., 2020). Table 1 indicates the possible switching states and the output voltages (u i ) of the discussed emerging five-level inverter.
2.2 The LCL filter design
The filter is required to obtain a sinusoidal waveform (Elamri et al., 2022). Table 2 indicates the system parameters.
The LCL-Filter parameters are calculated with the following system of Equations (1) (Elamri et al., 2022):
Where:
ω sω is the angular switching-frequency (ω sω =2πf sω ), ω is the angular fundamental-frequency (ω=2πf), ω 0 is the angular resonance-frequency (ω 0 =2πf 0 ).
2.3 The backstepping control (BSC) method
Neglecting the damping resistance, the Equations (2) of the inverter connected to the grid with LCL filter are as follows:
Where:
u i represents the voltages produced by the five-level inverter, i i represents the current flowing through the inverter, i g represents the current flowing through the grid, i c represents the current flowing through the capacitor, u g represents the voltage of the grid, and u c represents the voltage of the capacitor.
By selecting the voltage as the system state and the control parameter as the inverter signal control, Equation (2) can be rearranged as follows:
Where:
The principle is to produce the desired sinusoidal output voltage, to obtain a closed loop regulation, by applying a Backstepping controller. This control consists in subdividing the complex nonlinear system into several subsystems.
In this study, the system will be divided into 3 steps. A suitable virtual loop is chosen in each subsystem, and then a virtual control is introduced, utilizing its error to form the Lyapunov function, which provides the strength and global stability of the asymptotic tracking error (Wang and Wai, 2020). Therefore, the following tracking error is defined on the current:
Where: x 1 * is the reference signal.
The temporal derivative of the reference signal is determined using the Equation (3.1) by:
Consider the following Lyapunov function:
This is how you find the time derivative of the Lyapunov function:
The Lyapunov function time derivative must be negative in order to keep the subsystem stable. So, the Lyapunov function time derivative can be found by:
Where: C 1 is a real positive parameter of the controller design.
The temporal derivative of the error can be determined by comparing between Equations (7) and (8), as follows:
By identifying between Equations (5) and (9), we find:
In accordance with the Backstepping method and to ensure the current stability, the virtual command x 2 * is determined by the following equation:
Since x 2 * is not the actual order input, we determine a second tracking error:
Using Equations (11), (12) and applying some mathematical manipulations, the equation (5) can be expressed as:
By putting Equation (13) into Equation (7), the temporal derivative of Lyapunov function can be expressed as:
Using Equations (3.2) and (12) and applying some mathematical manipulations, the temporal derivative of the second tracking error can be expressed as:
The virtual input x 3 is exploited for the stability of the voltage loop. Then, we examine the function of the Lyapunov candidate:
By Equation (14), it is determined the Lyapunov candidate's time derivative, as:
To keep the subsystem stable, the Lyapunov candidate's time derivative must be negative. This means that:
Where: C 2 > 0. C 2 is a real positive parameter of the controller design.
Since x 3 * is not the actual order input, we determine a second tracking error:
Using Equations (15), (18), (19) and applying some mathematical manipulations, the temporal derivative of the second tracking error is determined by:
By putting Equation (20) into Equation (17), the time derivative of the Lyapunov candidate presented in Equation (17) is determined by the following expression:
Using Equations (3.3), (18), (19), the time derivative of e 3 can be calculated as follows:
To stabilize the whole system, we must make a suitable choice of the real control signal u, for all errors (e 1, e 2, e 3) converge to zero. As such, the next step is to put a model of the Lyapunov augmented candidate:
The time derivative of the Lyapunov augmented candidate is determined using Equations (21) and (23) by:
Combining Equations (22) and (24), one gets the following control law:
Where u * is the actual input. C 3 is a real positive parameter of the controller design.
In the case (u = u *), the temporal derivative of e 3 given in Equation (22) and the time derivative of the Lyapunov augmented candidate given in Equation (24) are written as:
According to Equation (21), a negative definition is available, implying that the equilibrium (e 1, e 2, e 3) = (0, 0, 0) is generally stable. As a result, all tracking errors are exponentially fading. This completes the proof of proposal (Idrissi et al., 2018).
To account for the effects of sampling in real applications, we have extended our analysis to a discrete-time model. The discretization of the Backstepping controller was performed using the Euler method, with sampling period T S . The discretized equations 11, 18 and 25 are given by:
Discrete Lyapunov functions can be calculated as follows:
Where discrete stability analysis is written as follows:
Where O(Ts²) represents higher-order terms introduced by discretization, which become negligible for sufficiently small sampling periods and are thus omitted in the first-order stability analysis.
To stabilize the whole system, we must take:
A discrete Lyapunov stability analysis shows that stability is maintained as long as the sampling period T S satisfies the stability condition given in Equation (30). This condition ensures that the difference of the Lyapunov function ΔV 3[k] remains negative, thus ensuring the convergence of tracking errors to zero.
It is important to note that the choice of sampling period affects not only the stability of the system but also its dynamic response. A sampling period that is too long can lead to performance degradation, while one that is too small can unnecessarily increase the computational load on the microcontroller. In our case, the value of T S ≤ 0.1 ms offers a good compromise between performance and implementation feasibility for the five-level grid-connected inverter system with LCL-filter.
3. RESULTS AND DISCUSSION
MATLAB and Simulink tools are used to test the theory study and design of the system. This part talks about what the current and voltage patterns showed. Two system performance tests show that the proposed nonlinear backstepping approach works well for controlling the current in a five-level inverter that is connected to the grid and has an LCL filter. The system operating parameters are given in Table 2.
3.1 System responses with constant set-point
The system reactions are compared with a constant set-point (reference) to see how well the suggested nonlinear backstepping controller works for controlling the current of a five-level inverter that is connected to the grid and has an LCL filter. The constant set-point value equal to "Current (10 A)".
Figure 2 illustrates the grid-current (i g ) waveform of the system using a constant set-point. The illustration demonstrates how the proposed control method can make the system grid-current (i g ) follow the reference model (i gref ). The correct selection of the controller parameters, which are in charge of enhancing the current transient performance in terms of rising time, overshoot, and settling time, justifies this. Figure 3 shows the curves of the inverter output-voltage (u i ) and the grid-current (i g ) at a constant set-point. As can be seen in the figure, the grid-current (i g ) has a pure sine-wave form because it was filtered via LCL-filter. Moreover, the figure indicates that the grid-current (i g ) is in phase with the inverter output-voltage (u i ).
The curves for the grid voltage (u g ) and grid current (i g ) at a fixed set point are shown in Figure 4. As appears in the illustration, the grid-current (i g ) and grid-voltage (u g ) are in phase, allowing the grid and the inverter to be synchronized.
Figure 5 depicts the FFT analysis of the inverter output-voltage (u i ) and the Total Harmonic Distortion (THD) value. The FFT analysis shows that the THD voltage of the five-level inverter is approximately 0.21 %. Figure 6 shows the FFT analysis of the grid-current (i g ) and the THD value. The THD of the grid-current is approximately 0.1 %.
THD analysis is a crucial parameter to consider in power system design. THD values below 5% are considered excellent and comply with international standards, as recommended (Maamar et al., 2021). The achieved low THD values demonstrate the effectiveness of the LCL filter and the control method.
3.2 System responses with variable set-point
In order to assess the effectiveness of the suggested nonlinear backstepping controller system for current control of a grid-connected 5-level converter with LCL filter, the system reactions when subjected to a variable set-point (reference) are analyzed and compared. Figure 7 illustrates the curve of the variable set-point, where the set value changes from "Current (5 A) to (15 A)" at "Time (t=40 ms)"; then, changes from "Current (15 A) to (10 A) at "Time (t = 70 ms)".
Figure 8 illustrates the grid-current (i g ) waveform of the system with a variable set-point. The figure shows a good tracking performance without oscillations in response set-point changes. By forcing the system grid-current (i g ) to adhere to the reference model (i gref ), the system demonstrates excellent tracking capability. In this instance, the tracking error is approaching zero and the control parameters are converging to a constant value. The proper selection of the backstepping controller parameters based on the Lyapunov function justifies the high performances of the system replies.
Figure 9 depicts the curves of the inverter output-voltage (u i ) and the grid-current (i g ) using a variable set-point. As can be seen in the figure, the grid-current (i g ) is in phase with the inverter output-voltage (u i ). The system performance is satisfactory, there is a good tracking accuracy of the reference, and it does not exhibit oscillations in response to set-point changes, which means that the system is stable. In addition, the system accurately tracks the desired reference without significant error or deviation over time.
Figure 3 Curves of the inverter output-voltage (u i ) and the grid-current (i g ) at constant set-point
Figure 9 Curves of the inverter output-voltage (u i ) and grid-current (i g ) with the variable set-point
The results were achieved using the straightforward backstepping control technique. The emerging topology of the five-level inverter used in this study requires fewer switches and drives compared to the conventional H-bridge five-level inverter, which typically uses eight switches and drives (Toubal Maamar et al., 2021a). As a result, the implementation cost of the circuit was reduced.
In control systems, the effectiveness or performance can be assessed through several key metrics and criteria, as stability, tracking accuracy, ease of implementation, and low THD. By evaluating our control system against these criteria, we can confirm its compliance with the desired performance standards, making it suitable for its intended application.
4. CONCLUSION
The objective of this study was to understand the effects of nonlinear backstepping controller for controlling the current of a five-level inverter connected to the grid and has an LCL-filter. The proposed system has been described step by step, and the simulation tests have been verified the performance of the system. According to the presented results, THDs of the grid-current and the inverter output-voltage are less than 5 % and this value matches to the international standards. Moreover, the simulations revealed a precise grid regulation, stability of system, and being able to follow reference signals while ensuring a rapid response time to sudden changes. The present work also demonstrates that the backstepping method is a purely mathematical problem based on Lyapunov function for searching the stability of the system; this high-performance method requires a good knowledge of mathematic development and analysis.
As an extension to the work, it would be interesting to assess the effects of system experimental implementation. On the other hand, to explore new research topics, we are paving the way to study other related systems such as: the extension of system to three-phase converter topologies, integration with more advanced approaches of grid scheduling, integration of renewable energy such as PV solar, wind, or hybrid systems like PV-Wind, which have garnered significant attention due to their sustainability.
