Backstepping Controller Design for Power Quality Improvement in a Two-Stage Grid-Connected Photovoltaic Systems with LCL Filter

In a grid-connected photovoltaic system, the quality of energy injected by the photovoltaic system into the grid is directly linked to the topology of the inverter used and to the efciency of its control technique. Tis paper addresses this problem for a two-level grid-connected photovoltaic inverter operating under low irradiance conditions. Te aim is to reduce the harmonic distortion on the electrical network and therefore improve its power quality. To achieve this goal, a control strategy was set up considering the nonlinearity of the dynamic system, and the high dimension of the system model. Tus, a nonlinear controller designed using the backstepping technique is proposed. Te efectiveness of this control strategy was evaluated by simulation in MATLAB/Simulink. Results show that the proposed control technique signifcantly improves the power quality of the grid-connected photovoltaic system by minimizing the current harmonic distortion rates in low irradiance conditions. Te current harmonic distortion rates obtained for solar irradiance of 1000 W/m 2 , 750W/m 2 , 500W/m 2 , and 250W/m 2 are 0.48%, 0.78%, 1.22%, and 2.16%, respectively. Te power factor is 0.988, and the DC bus voltage is maintained at its reference voltage of 600 V with a very low response time during the transient phases. A comparison of our simulation results with those found in the literature on other control techniques such as proportional-integral-derivative (PID) and synergetic controls shows the efciency, superiority, and satisfactory performance of the proposed control scheme to minimize harmonic distortion under low irradiance conditions. Te robustness and better dynamic performance of the proposed backstepping controller under varying irradiance conditions have also been shown.


Introduction
1.1.Background.Greenhouse gases are one of the most important sources of climate change.Tese greenhouse gases come mostly from the combustion of fossil fuels such as gas, oil, and coal.Tis combustion of fossil fuels produces an increase of up to 70% of greenhouse gases in the world, thereby causing global warming [1,2].One of the most urgent and inescapable solutions to this problem of global warming is to move to renewable energy sources by reducing the dependency of power system around the word on fossil fuels [3][4][5][6][7].Tis is why the energy transition towards renewable energy sources has received more attention in recent years, in various sectors due to their great potential in the reduction of greenhouse gas emissions [4].Te main renewable energy sources are wind, biomass, hydroelectricity, geothermal energy, biofuels, and solar energy [5][6][7][8].Advantages such as inexhaustibility, free of charge, and cleanliness make photovoltaic solar energy one of the most dominant energy sources, and the best alternative to other existing renewable energy sources in terms of electricity supply [3,9].State governmental measures in many countries around the world leading to the reduction in the cost of the solar photovoltaic module by more than 40% have contributed to a rapid spread of photovoltaic (PV) systems in these countries [10].A photovoltaic panel, also called photocell, enables the conversion of solar energy into direct electrical energy.Its production strongly depends on meteorological conditions especially, the irradiance and the ambient temperature [11,12].

Challenges.
To ensure the extraction of the maximum power in a photovoltaic system under variable irradiance and temperature, a Maximum Power Point Tracking (MPPT) algorithm is implemented through a DC-DC converter [13,14].Te inverter can therefore convert this DC energy into AC form for AC loads [15].Photovoltaic systems can be mainly classifed into standalone and gridconnected photovoltaic systems [7,16].In the gridconnected PV systems, solar energy source is connected to the grid through inverters and these are used to fll the energy defcit of grids [15,16].During the coordination and operation of the grid-connected photovoltaic systems, undesirable efects on both the grid and end-user equipment can arise due to power quality disruptions due to the variations in meteorological conditions and the presence of nonlinear loads into the grid [17].One of the main and worst power quality disturbances is the injection of harmonic distortions of the current by the solar photovoltaic systems and nonlinear loads into the grid [9,18,19], especially in low irradiance conditions [20,21].Disturbances caused by these harmonic distortions of the current in the electrical grid are overheating of cables, breakdown of capacitors, increase in line losses, power factor degradation, vibration and acoustic noise from motors, malfunction of metering devices, and aging of lines [15,22].To address the aforementioned problems, various control strategies are proposed in the literature by many authors.
1.3.Literature Review.Much attention have been paid to inverter control techniques to ensure sinusoidal waveform of the current with low THD, fast dynamic response, and invariable frequency of the system under diferent types of loads [23].In their work, Yaïchi et al. have proposed the use of multilevel converters and thus allowing them to provide a multilevel wave with less total harmonic distortion depending on the number of switching cells [24].However, this method requires more switches than conventional converters and thus makes these converters expensive and bulky.In order to minimize the THD, Imarazene et al. have proposed techniques of harmonics selection in which harmful harmonics from the frequency spectrum are selected and removed [25].Te major drawback of this solution is the difculty to calculate the exact commutation angles.Tis leads to the complexity of the method and limits its feasibility.Mamane et al. have proposed an alternative to the abovementioned harmonics selection techniques.Teir technique consists of a reactive power compensation algorithm applied to parallel active flters [26].However, the use of this technique to improve of the power quality is often very bulky and expensive.Pesdjock et al. have proposed an improved variant of the associated classical synergetic control to reduce the THD of a grid-connected photovoltaic system [15].In their approach, these authors have modifed the traditional synergetic method by creating an intermediate virtual control.Sharma and Gali proposed a modifed hysteresis current controller switching scheme for multifunctional (MHCC) grid-tied photovoltaic inverters [27].Teir control technique consists of feeding an active power to the local grid during day time and acting as a shunt active power flter (SAPF) to mitigate reactive power, current harmonics, and switching frequency problems.Morey et al. have recently proposed a review of the latest grid-connected inverter control techniques and associated inverters confgurations [16].Te main control techniques for grid-connected inverters identifed by these authors include the synchronous reference frame (SRF) theory, the double synchronous reference frame (DSRF) theory, the stationary two-phase reference frame theory, a combined action of the proportional resonance and the harmonic compensator (PR + HC) controllers based on stationary twophase reference frame theory, and a combined action of a 3rd-order proportional resonance and proportional integral (PR + PI) controllers based on the theory of the stationary two-phase reference frame.Morey et al. have also discussed in depth control algorithm approaches based on synchronous reference frame (SRF) theory, IcosΦ, instantaneous symmetrical component theory, and instantaneous reactive power theory (IRPT), for controlling grid-connected PV inverters [28].In addition to the abovementioned control techniques, several other techniques exist among which PID control [29][30][31][32] is applied to LCL flter, sliding mode control (SMC) [33], linear control [34], linear resonant control [35], Lyapunov control [36], and passivity-based control [37].Tey are combined between them or with other modern control techniques in order to achieve better performances and power qualities of photovoltaic systems.Tus, Fanjip et al. [38] have applied the fuzzy logic control associated with the sliding mode control to a parallel active flter to eliminate harmonics distortion of the current in the same system as Pesdjock et al. [15].Finally, for the above PV system, Lakshmi and Hemamalini have proposed the design of a control decoupled from the system using a fractional PI controller to minimize the harmonic distortions of the current [39].Subsequently, Golzari et al. have proposed a direct predictive power control based on the Lyapunov function developed for the control of the three-phase grid-connected PV inverter [40].Tis control technique based on the discrete frstorder flter model aims to improve the tracking speed of the controller during rapid changes in solar irradiance.A nonlinear sliding mode controller called supertwisted integral sliding mode control has been proposed for a threephase grid-connected inverter [41,42].Tis controller has proven to be robust and maintains low THD in the presence 2 International Transactions on Electrical Energy Systems of grid impedance variation, flter parameter drift, and network harmonic distortion.Adding the supertwisting action helps to remove the chattering problem associated with the conventional sliding mode control strategy, and the integral action helps to improve the steady-state error of the grid current.A proportional integral (PI) control strategy associated with repetitive control for grid-connected photovoltaic inverters with a third-order output flter has been proposed by Li et al. [43].Te proposed control scheme also improves the harmonic suppression ability of the PV inverter and the system maintains better stability and fast dynamic response.Deželak et al. have proposed a comparison between the particle swarm optimization (PSO) and Ziegler-Nichols (ZN) tuning methods for controlling the inverter of the grid-connected PV system with a third-order output flter [44].Tese two approaches are generally used to determine the parameters of proportional integral (PI) controllers.Te PSO-optimized PI controller shows its superiority over the ZN optimization technique in terms of system stability and power quality.A fuzzy-PI and a fuzzy sliding mode controller for two-stage single-phase photovoltaic inverters connected to the grid through a third-order flter are also proposed by Zeb et al. [45].Teir controllers regulate the DC link voltage and improve the power quality of the system.One of the most used controllers for the grid-connected photovoltaic inverter is the backstepping controller.It can be used alone or associated with other controllers.Below are some relevant publications where this control strategy is used.Tus, a comparison between sliding mode, fuzzy logic, and backstepping controllers is presented by Zadeh et al. [46].Tey found that backstepping was the best controller with the highest performance.A robust and nonlinear backstepping controller for inverters is proposed by Kolbasi and Seker [47].However, this controller is hard to handle control because it contains more than two gains.Zouga et al. have proposed a backstepping controller based on the particle swarm optimization (PSO) algorithm for a threephase grid-connected solar system [48].Te authors use the backstepping technique based on the L-flter model to develop three cascade controllers.Te frst one aims to fx the DC voltage of the PV panels at the MPP (maximum power point).Te second controller is designed to inject a sinusoidal three-phase current into the grid through the control of an inverter.Te third controller is a proportional integral (PI) controller based on the particle swarm optimization (PSO) technique, which aims to regulate the voltage of the intermediate circuit to a constant reference value.Pervej et al. have proposed a nonlinear backstepping controller for three-phase grid-connected PV systems based on the direct power controller (DPC) approach [49].In their work, the internal current control loop is deleted; this simplifes the modeling of the grid-connected system by a frst-order flter as well as the controller design by improving the transient performance.In order to improve dynamic stability and power quality, and control the appropriate amount of active and reactive power to be injected into the grid, nonlinear adaptive backstepping controllers for grid-connected threephase PV systems through a frst-order and LC flters have been proposed by Roy et al. [50,51].A fltered nonlinear adaptive backstepping controller to connect a solar PV system to the three-phase grid through a frst-order flter has been proposed by Xu et al. [52].Tis control strategy allows the regulation of the DC link voltage of the PV system and of the current used to control the active or reactive power injected into the grid.Te technique also provides a solution to the intrinsic problem of diferential expansion and saturation in the backstepping control technique by using a control flter to eliminate the impact of time derivative and saturation from the controller.
Other works in which LCL flters are used to eliminate harmonics distortions in grid-connected photovoltaic systems have been presented by various authors.For example, a hybrid controller incorporating repetitive and stateful pole assignment control for a three-phase grid-connected inverter with LCL flter has been proposed by Liu et al. [53].A simplifed feedback linearization control of a three-phase PV inverter with an LCL flter has been proposed by Bao et al. [54].Teir work aims to increase the decoupling performance of the control system, and improve its dynamic performance and adaptability.An integrated design approach for LCL flters based on nonlinear inductors for gridconnected inverter applications has been proposed by Sgrò et al. [55].Teir work present a modifed design procedure for an LCL flter topology that takes into account the optimization of the current control loop in terms of robustness and dynamic performance.Based on the analysis of equations describing the behavior of the flter, a new methodology integrating the design of the flter inductances, a predictive controller and an active damping method have been developed.Li et al. have proposed an improved proportional resonant (PR) control strategy for a three-phase grid-connected inverter with LCL flter based on active damping [56].In this work, the authors frst compared and analyzed the proportional resonance technique and the quasiproportional resonance to the improved PR controller.In the second stage, the improved PR controller is compared with the traditional controller of the three-phase gridconnected inverter with LCL flter based on active damping.Te proposed improved current control strategy has good dynamic response.It can realize the current nonstatic error control of the grid-connected, as well as the decoupling control of active and reactive power when the load jumps.In addition, an adaptive harmonic compensation technique with a proportional resonance (PR) and PR integral (PRI) controller is proposed to minimize low-order harmonics [16].
From the above, frst-, second-, and third-order passive flters are regularly used in various control strategies of the grid-connected photovoltaic systems in order to reduce the total harmonic distortion (THD) of the current injected into the grid.However, as it can be seen in the literature cited above, the backstepping control strategy has already been applied to control the inverters of the grid-connected photovoltaic systems only for the L and LC flters.However, the frst-order flter has the disadvantage of a fairly large voltage drop across its terminals and a high weight in terms of size [57].Nevertheless, the LCL flter draws more attention from researchers due to its superior performance.

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Compared to the frst-or second-order flter, it ofers signifcant attenuation of the high-frequency components of harmonic distortions on the one hand, and cost savings in terms of reduced overall weight and component size on the other [58].LCL flters are also used in grid-connected PV systems to get rid of interactive resonances [59].In addition, the dynamic model of the LCL flter in the dq synchronous reference frame is nonlinear, high order (6th order), complex, and its implementation through conventional control techniques is very difcult and inaccurate.

Scopes and Motivation.
In view of the aforementioned literature, there is a need for an efcient control technique able to control dynamic systems with high-order variable parameters.Signifcant attention has recently given to the backstepping approach as one of the most efective approaches to control such systems.Te backstepping control technique is a recursive design procedure developed for designing stabilizing controls for a special class of nonlinear dynamical systems built from subsystems.It designing procedure links the choice of a control Lyapunov function with the design of a feedback controller and guarantees global asymptotic stability of strict feedback systems [60].Tis control method has two main advantages.On one hand, it has the ability to decompose a complex high-order system into several subsystems of frst order and on the other hand, for simplifying the control design procedure, it introduces virtual control signals in each subsystem [61].

Contributions.
Te main contribution of this work is to systematically control a high-order diferential equation system of a three-phase grid-connected solar photovoltaic system with a LCL output flter (third-order flter) operating under fast sunlight conditions.Tis is done by using the ability of the backstepping control technique to stabilize at each step any nonlinear dynamic subsystems of frst-order leading to the overall stability of the system [8].
Tis control scheme will contribute (i) to improve the power quality of the grid-connected solar photovoltaic system operating in the variable and low irradiance conditions by injecting into the grid the highest possible sinusoidal waveform that minimizes the rate of harmonic distortion (ii) to improve the overall stability of the solar photovoltaic system in closed loop (iii) to improve the robustness of the system under variable and low irradiance conditions Tus, this paper deals with the control of a two-level voltage inverter using a nonlinear backstepping controller intended for a three-phase grid-connected photovoltaic system.Te proposed backstepping controller uses the nonlinear dynamic model of an LCL flter in the dq synchronous reference frame to generate the reference voltage (Vref ) of the two-level voltage inverter.Te main rule of this control technique is the reactive power compensation and the reduction of the harmonic distortions generated by the inverter.
1.6.Organization of the Paper.Te rest of the research article is structured as follows in the next sections.Section 2 concern the description and modeling of a grid-connected PV system.Tis section ends with the design of the nonlinear backstepping controller.Numerical simulations are used in Section 3 to illustrate the efectiveness of the controller in minimizing the harmonic distortions rate and improving the power factor of the grid-connected PV system.Te article ends with a conclusion in Section 4.

Description of the Grid-Connected Photovoltaic System.
Te three-phase two-stage grid-connected solar photovoltaic system that is presented in this article is adapted from the work in [15,62,63].Its general diagram which consists of fve groups of blocks is presented in Figure 1.Te frst block group is composed of a photovoltaic power source (PV block) connected to a voltage step-up chopper (DC-DC block).Tis frst group of blocks also contains a maximum power point tracking controller (MPPT block) associated with a PI regulator (PI block), making it possible to improve the performance of the PV block via the DC-DC block.Te second block comprises a capacitor C dc used to flter the supply voltage of the two-level inverter (DC-AC block).Te inverter is driven by a vector pulse width modulation signal (SVPWM block) and then injects into the grid a three-phase alternating current.Te third group is made up of a thirdorder flter which flters the harmonic distortions of the current injected into the grid by the inverter of the PV system.Te fourth group contains a nonlinear load composed of a graëtz bridge (AC-DC block) supplying a load (P2, Q2) and a linear load (P1, Q1) connected to the network.Te ffth group consists only of the block describing the proposed backstepping technique.

Mathematical Modeling of PV Systems. According to
Kirchhof's law relating to currents applied to the electrical circuit of Figure 1, the following relationship is obtained: where V dc is the voltage across the flter capacitor C dc .i out and i dc are, respectively, DC-DC boost converter output current and inverter input current.
According to the law of conservation of energy and neglecting the power losses of the inverter, the power balance relationship between DC side, i.e., the input of the inverter and AC side of the inverter, i.e., the output can be given by the following relationship [64,65]: International Transactions on Electrical Energy Systems where v gd , v gq , i gd , and i gq are, respectively, the grid voltages and currents in the dq synchronous reference frame.Te average value of v gq is equal to zero in steady state.Ten, by replacing (2) in (1), the dynamic of the DC link voltage can be expressed as follows: Te single-phase model of the LCL flter in the reference frame (abc) is represented in Figure 2.
In Figure 2, R i and L i are, respectively, the flter resistance and the inductance on the inverter side; R g and L g are, respectively, the flter resistance and inductance on the grid side; C 2 and R d are capacitance and damping resistance of the flter, respectively; ω being the pulsation of the grid.
Te mathematical model by phase of the flter in the state space is given by the following equation: In applying the Park transform to the system of equation ( 4), the dynamic mathematical model of the LCL flter in Park's dq synchronous rotating frame can be described as follows:   International Transactions on Electrical Energy Systems where By substituting equations (3) in ( 5), the grid-connected photovoltaic system can be described by the fnal dynamic mathematical model of the LCL flter as follows: 2.3.Backstepping Control Design.Tis section is devoted to the development of a backstepping controller for gridconnected inverters with LCL-type third-order flters at the output.Te dynamic system of equations in equation ( 7) is of order seven and has seven state vectors.Te design of the regulator will also be done in seven steps.
Step 1. Stabilization of the frst subsystem (frst equation) Considering the following frst subsystem to be stabilized, Te frst regulation error between the measurement x 1 � V dc and its reference x 1d � V dcref is Te derivative of the regulation error becomes

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In this frst subsystem, x 2 � i gd is considered as a virtual command.If α 2 is the form that x 2 must have to stabilize the frst subsystem, then the regulation error will be Substituting x 2 � z 2 + α 2 in the derivative (10) gives Let us choose a Lyapunov function V 1 in quadratic form, in order to determine the frst virtual command α 2 so that Its derivative becomes To determine the virtual command α 2 which ensures the negativity of the derivative _ V 1 of the Lyapunov function V 1 , the following equality is stated from equation ( 14): where k 1 is a design constant, with k 1 > 0.
By replacing α 2 from equations ( 16) in ( 14), the following derivative of the Lyapunov function _ V 1 is obtained: From equation (17), it is noted that if z 2 � 0, then, 1 < 0, which means that z 1 ⟶ 0. Te convergence of z 2 will be obtained at the next step.
Step 2. Stabilization of the second subsystem (2nd equation) Let the second subsystem be stabilized, and it will be as follows: Te second regulation error between the measurement x 2 � i gd and its reference α 2 � i gdref which is the virtual control obtained in Step 1: Te derivative of the regulation error becomes In this second subsystem, x 4 is considered as a virtual command.If α 4 is the form that x 4 must have to stabilize the second subsystem, then the regulation error will be Substituting x 4 � z 4 + α 4 in the derivative (20) leads to Let us choose an extended Lyapunov function V 2 , in order to determine the second virtual command α 4 so that Its derivative becomes Replacing _ V 1 and _ z 2 , respectively, from equations ( 20) and ( 22) into (24) gives To determine the virtual command α 4 which ensures the negativity of the derivative _ V 2 of the extended Lyapunov function V 2 , the following equality is stated from equation (25): where k 2 is a design constant, with k 2 > 0.
By replacing α 4 of equations ( 27) in (25), the following derivative of the Lyapunov function _ V 2 is obtained: From equation (28), it is noted that if < 0, which means that z 1 ⟶ 0 and z 2 ⟶ 0. Te convergence of z 4 will be obtained in the third step.
By following the same method as in Steps 1 and 2, the third to the seventh subsystems can be successively stabilized.Table 1 gives the Lyapunov function used at each step, International Transactions on Electrical Energy Systems obtained virtual commands, and the flter input voltages (v id , v iq ) in the dq synchronous reference frame.Tese two components of the flter input voltage are necessary to build the SVPWM needed to drive the electronic switches (IGBT) of the inverter.Constants k 3 , k 4 , k 5 , k 6 , and k 7 are design constants defned during the stabilization of the third to the seventh subsystems.Te seven steps of the stabilization of the system can be implemented through the block diagram of Figure 3. Abbreviations used in this paper are listed in Table 2.

Results and Discussion
To evaluate the performance of the designed backstepping controller applied to the two-level inverter based on the LCL flter model, numerical simulations were carried out in the MATLAB/Simulink platform using the same test conditions as those of the proportional integral (PI) controller proposed in [66] and synergic control in [15,63].Te parameters of the PV generator used in this work are listed in Table 3. Te overall system is tested and validated with the system parameters listed in Table 4.
To generate drive pulses, the space vector PWM (SVPWM) was used.Te initial irradiance value is fxed at 1000 W/m 2 ; after every 0.5 s, it changes, respectively, to the following values: 500 W/m 2 , 750 W/m 2 , and 250 W/m 2 to have irradiance variations in a short time and to test the ability of the controller to follow the appropriate value of power generated by the PV generator.Tese changes in the irradiance are shown in Figure 4. Tis solar irradiance profle is applied to the photovoltaic array with the aim of simulating the proposed system under diferent conditions and examining the dynamic response of the backstepping controller.Troughout the simulation, the temperature is maintained at 25 °C.Te instantaneous active power in steady state is approximately 50.57kW at this solar irradiance.At t � 1 s, the solar irradiance passes at 750 W/m 2 .Te oscillations decrease more during the transient phase which lasts tr � 28.5 ms, and the instantaneous active power in the steady state is about 75.57kW.Finally, at t � 1.5 s, the solar irradiance passes at 250 W/m 2 .Te oscillations increase further during a response time of 15.36 ms.Ten the instantaneous active power in steady state is around 25.56 kW at this solar irradiance.Tus, it can be seen that the active power injected into the grid by the PV system is the maximum one expected at each irradiance value.So, the proposed control scheme allows to extract the maximum power from the PV panels.

Reactive Power.
Te total reactive power injected or transmitted into the electrical grid for 2 s according to the solar irradiance profle of Figure 4 is illustrated in Figure 6.It can be seen that this power is almost constant over time.In other words, the reactive power transmitted by the PV system is very low and oscillates around almost zero constantly over time regardless of the irradiance value.

Power Factor.
Te power factor of the studied gridconnected photovoltaic system for the given irradiance profle is illustrated in Figure 7.It is equal to 0.988 during the Lyapunov function used from step 1 to step 6 of the backstepping control scheme (V) ω Pulsation of the grid (rad/s) Components of state vector of the system Components of the system reference state vector      International Transactions on Electrical Energy Systems in the steady state.Tus, it is noted that the DC bus voltage perfectly follows its set point value whatever the value of the irradiance is.

Instantaneous
Current in the Grid.Te instantaneous current of phase a during the variation of the irradiance is presented in Figure 9.At t � 0 s, the solar irradiance applied to the photovoltaic generator is G � 1000 W/m 2 , the maximum intensity of the current injected into the grid is about 200 A. At t � 0.5 s, the solar irradiance applied to the photovoltaic generator is G � 500 W/m 2 , the grid receives a current with a maximum intensity of about 100 A. At t � 1 s, the solar irradiance applied to the photovoltaic generator is G � 750 W/m 2 , the instantaneous current in the grid has a maximum value of about 149 A. Finally, at t � 1.5 s, the solar irradiance applied to the photovoltaic generator is    12 International Transactions on Electrical Energy Systems G � 250 W/m 2 and the maximum intensity of the current injected into the grid is about 56 A. Globally, the maximum value of the current in that phase is proportional to the irradiance value.Tis is directly linked to the behavior of the current provided by the PV modules.

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In Figure 10(c), the frequency analysis of the current injected in the grid by the phase number a of the inverter is given when the solar irradiance is 1000 W/m 2 .Its fundamental is around 232.1 A at a frequency of 50 Hz, and the current THD is about 0.48%.Figure 11(c) presents the same frequency analysis when the solar irradiance is 750 W/m 2 .Te fundamental is around 169.9 A at 50 Hz, and the THD is about 0.78%.Figure 12(c) shows the frequency analysis when the solar irradiance is 500 W/m 2 .Te fundamental is around 107.6 A at 50 Hz, and the THD is around 1.22%.Finally, Figure 13(c) shows the frequency analysis when the solar irradiance is 250 W/m 2 .Its fundamental is about 47.19 A at 50 Hz, and the THD is about 2.16%.Tis leads to the conclusion that the backstepping control based on an LCL flter proposed for the control of the grid-connected inverter in PV system allows to have low current harmonic distortion rates in the grid whatever the solar irradiance.Terefore, a high power quality is obtained in the considered PV power system.[15] Improved synergetic control 1.02 1.10 1.25 2.32 Wai et al. [23] Backstepping control with LC flter 1.29 ---Diouri et al. [8] Backstepping control with LC flter 0.78 ---Xu et al. [64] Backstepping control 1.62 ---Xu et al. [64] Adaptive fuzzy sliding mode command-fltered backstepping (AFSCB) 0.53 ---Roy and Mahmud [65] Nonlinear robust adaptive 2.12 ---Roy and Mahmud [65] Backstepping controller (NRABC) 2.20 ---Roy et al. [50] Nonlinear backstepping control  [23]) and for inverter in standalone PV system (0.78% [8]), proving that the output voltage of the studied grid-connected inverter supplied by the PV system is lightly afected by harmonic distortions.
To highlight the interest of this work, a comparison of the performances of the proposed backstepping controller based on a LCL flter during the variation of the solar irradiance with those in the literature was also carried out.Te average values of the current THD collected during various operating tests are presented in Table 5. Te efciency of the inverter operating with the proposed backstepping controller was evaluated for diferent values of solar irradiance and given in Table 6.Analysis of the results in these tables shows that the THD values obtained with the proposed backstepping controller are always within the accepted standards whatever the solar irradiance variations are.Tese quantitative and qualitative results show that the proposed control approach has better performance in terms of reducing harmonic distortions of the currents compared to other approaches in the literature [15,63,66] and even in terms of efciency.

Conclusion
In this work, a control strategy that could efciently control a three-phase two-level grid-connected inverter of a photovoltaic system was developed, with the aim of improving the power quality of the electrical grid.Two of the most important constraints of this inverter control system were taken into account, namely, (i) the nonlinearity of the dynamics of the system, and (ii) the high dimension of the system model.For these reasons, a nonlinear controller was proposed.It basically consists of the control law synthesized using the backstepping technique.A theoretical analysis, using Lyapunov's stability theory at each stage of the technique, proves the overall stability of the system.Te results of the simulation show that the total harmonic distortion rates of the current are 0.48%, 0.78%, 1.22%, and 2.16%, respectively, for solar irradiance of 1000 W/m 2 , 750 W/m 2 , 500 W/m 2 , and 250 W/m 2 ; the DC bus voltage is maintained at 600 V, and the power factor is 0.988 for the same irradiance values.Tus, the proposed controller meets its main control objective which was to improve the quality of energy.Te simulation results further underline the robustness of the controller with regards to rapid changes in solar irradiance.An experimental study of the proposed backstepping command and an analysis of the cost savings would be done later.

3. 1 . 1 .
Total Power Injected into the Grid.Te total power injected into the electrical grid for 2 s according to the solar irradiance profle of Figure 4 is shown in Figure 5.At t � 0 s, when the solar irradiance is 1000 W/m 2 , the instantaneous active power in steady state is approximately 100 kW after a response time tr � 15.05 ms.Moreover, at t � 0.5 s, the solar irradiance is 500 W/m 2 , slight oscillations are observed during the transient phase which lasts about tr � 19.82 ms.

iq
Final control law on the direct and quadratic axis i i , i g Current on the inverter side and on the grid side10 International Transactions on Electrical Energy Systems entire period of time when the solar irradiance varies; proving that the reactive power is almost zero.It is nevertheless observed a large and rapid drop in the power factor at t � 1.5 s when the solar irradiance drops from 750 W/m 2 to 250 W/m 2 .3.2.DC BusVoltage.Te DC bus voltage V dc (curve in red) and its reference (curve in blue) V dcref � 600 V are illustrated in Figure 8.At t � 0 s, the solar irradiance G � 1000 W/m 2 is applied to the photovoltaic generator, a response time tr � 22.32 ms is observed during the transient phase, after which the DC bus voltage returns to its reference value of 600 V in steady state.Moreover, at t � 0.5 s, the solar irradiance G � 500 W/m 2 is applied to the photovoltaic generator, a response time tr � 8.6 ms is observed during the transient phase, after which the DC bus voltage returns to the reference value of 600 V in steady state.At t � 1 s, the solar irradiance increases to the value G � 750 W/m 2 , a response time tr � 29 ms is observed during the transient phase, after which the DC bus voltage returns to its reference value in the steady state.Finally, at t � 1.5 s, the solar irradiance decreases to the value G � 250 W/m 2 , a response time tr � 26.05 ms is observed during the transient phase, after which the DC bus voltage also returns to its reference value

Figure 6 :
Figure 6: Reactive power transmitted into the grid by the PV system according to variations in irradiance.

Figure 7 :
Figure 7: Power factor according to variations in irradiance.

Figure 8 :
Figure 8: DC bus voltage according to variations in irradiance.

Figure 9 :Figure 10 :
Figure 9: Instantaneous current in phase number a of the grid according to the variation in irradiance.

Table 1 :
Lyapunov function used at each stabilization steps and the obtained virtual commands and the flter input voltages.

Table 4 :
Power system parameters.

Table 5 :
Comparative table of THD i a (%) for diferent solar irradiance levels.

Table 6 :
[21]ency of inverter under diferent solar irradiation values.Nevertheless, this power quality degrades as the value of the solar irradiance becomes lower and lower as discussed by Tchofo Houdji et al.[21].In the meantime, the frequency analysis of the voltage shown in Figures10(d)-13(d) presents a fundamental of 565.6 V at a frequency of 50 Hz and voltage THD of 0.24%, 0.41%, 0.46%, and 0.51%, respectively, for irradiance values of 1000 W/m 2 , 750 W/m 2 , 500 W/m 2 , and 250 W/m 2 .Tese values of the voltage THD are less that those obtained in the literature with other backstepping controllers for grid-connected inverter (1.90%