# Modeling and Design of Single-Phase PV Inverter with MPPT Algorithm Applied to the Boost Converter Using Back-Stepping Control in Standalone Mode

Authors: Omar Diouri; Najia Es-Sbai; Fatima Errahimi; Ahmed Gaga; Chakib Alaoui
Journal: International Journal of Photoenergy
 (2019)

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## Abstract

We propose a high-performance and robust control of a transformerless, single-phase PV inverter in the standalone mode. First, modeling and design of a DC-DC boost converter using a nonlinear back-stepping control was presented. The proposed converter uses a reference voltage that is generated by the Perturb and Observe (P&O) algorithm in order to extract the maximum power point (MPP) by responding accurately to varying atmospheric conditions. Another goal for using the boost converter is to raise the voltage at the input of the inverter without using a transformer in this system, thus making the system more compact and less expensive. Secondly, the single-phase H-bridge inverter was controlled by using back-stepping control in order to eliminate the error between the output voltage of the inverter and the desired value, even if there is acute load variation at the output of the inverter. The stability of the boost converter and H-bridge inverter was validated by using Lyapunov’s stability theory. Simulation results show that the proposed PV system with back-stepping controllers has a good extraction of the MPP with an efficiency of 99.93% and 1 ms of response time. In addition, the sinusoidal form of the output voltage of the inverter is fixed to 220 V and the total harmonic distortion of the output voltage was found to be less than 1%.


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## Body

## 1. Introduction

In recent years, several researches were focused on how to decrease the environmental pollution on Earth by using clean sources of energy such as solar, wind, hydro, biomass, and biogas [1]. These types of renewable energies are frequently applied to distributed generation (DG) [2]. In 2014, the world’s electricity consumption amounted to approximately 20.7 trillion kilowatt-hours according to [3], having a net increase of 7,323 terawatt-hours since 1980. Moreover, in 2015, the World Bank estimated that 22.69% of the world’s rural population were not able to access electricity, since the source of power is often located geographically far from the consumers, requiring expensive distribution of this electricity by using transmission lines. According to the World Bank, 8.26% of the worldwide-transmitted power in 2014 is lost due to losses associated with transmission lines. One way to circumvent these issues is to use distributed power generation units based on photovoltaic systems.Currently, there are two types of PV systems: grid-connected and standalone [4]. In order to inject the DC energy available at the output of photovoltaics into the grid, it is necessary to use converters as an interface [5, 6]. This system is known as the grid-connected PV system. On the other hand, a standalone PV system consists of the transformation of photovoltaic electricity to AC loads available at the consumer’s sites. Power converters are necessary in order to make interconnection between solar PV modules and AC loads. These power converters should accomplish two principal functions: first, to ensure that the PV array always generates the maximum power regardless of the variations of the atmospheric and load conditions. This is referred to as the maximum power point tracker or simply MPPT. The second function is the conversion of the continuous voltage generated by the PV array into the alternative voltage to be used by the AC loads. This AC output voltage should have the same performance and parameters as the grid, namely, having a stable frequency, amplitude, and sinusoidal form. In [7, 8], an overview of many topologies was presented to attain these purposes.In this paper, a converter with a two-stage topology is used. It consists of a boost converter and an H-bridge inverter. The main objective of the first stage (boost converter) is allowing the PV array to generate the maximum power using the MPPT technique [9]. There are several algorithms used to track the MPP effectively; the authors of [10] showed that the back-stepping algorithm gives good results. Several publications explore two broad categories of MPPT techniques: indirect MPP tracking like the fractional open circuit voltage method [11], direct MPP tracking like the incremental conductance [12, 13], or the Perturb and Observe (P&O) method that is implemented in this work. The authors of [12] achieved a response time of 7 ms in order to track the suitable value of the power using the proposed modified incremental conductance, and an efficiency of 97.53%. In [14], the rise time of the back-stepping control of MPPT and the integral back-stepping were 2.42 ms and 2.17 ms, respectively. In [10], efficiencies of 96%, 96.5%, 98.2%, and 99.1% were obtained by using P&O algorithm, PI, neuro-fuzzy, and back-stepping, respectively. Our main objectives are to achieve a lower response time and higher efficiency in the MPPT stage. The P&O algorithm takes advantage of the fact that the P-V curve has a decreasing nature to the right of the MPP and an increasing nature to the left of the MPP. The drawback of this algorithm is that the operating point is never stable and steady at the MPP. It is always oscillating around in the MPP region. This could be reduced using very small perturbation steps around the MPP. Another shortcoming is that there is no regulation of the output voltage of the DC-DC converter. This issue should be taken into consideration as the authors mentioned in [15]. By using the back-stepping control with the MPPT block, we can generate the reference voltage to be tracked by the controller. In addition, the control for boost converter forces the PV array to provide the same voltage as the MPPT block. For nonlinear systems, the use of a suitable controller is necessary to stabilize the system in the point considered. Therefore, to test the robustness of this controller, we have forced this system to high changes of solar irradiance in a short time and the results show that our proposed system tracks perfectly the reference power using back-stepping.The second stage of the proposed solution consists of an inverter. In order to assure an efficient use of the DG units, especially designed inverters play the role of energy conversion and adaptation between the sources and the loads [16]. The conversion principle in these inverters is the use of a pulse width modulation (PWM) technique to offer a stable sinusoidal output voltage of 220 V AC to the load. Several inverters use power electronic switches such as MOSFET or IGBT in the output stage. The PWM technique makes these inverters suitable for all types of electrical appliances [16]. However, these inverters must have a low total harmonic distortion (THD), a fast transient response, and a high efficiency. Much attention has been paid to regulation of the PWM inverter in order to ensure a sinusoidal waveform voltage with low THD, unchangeable frequency, and fast dynamic response under different types of loads [17]. The most known methods of regulation are the proportional-integral-derivative (PID) control [18], sliding-mode control [19], linear control [20], Lyapunov control [21], linear resonant control [22], and passivity-based control [23]. Kalantar Zadeh et al. presented in [24] a comparison between three types of controllers: sliding mode, back-stepping, and fuzzy logic. As a conclusion, the back-stepping was found to be the best controller which provides a higher performance. Kolbasi and Seker [25] proposed a nonlinear controller for inverters by using a robust back-stepping. However, having more than two gains makes the controller harder to control.The objective of this study is to achieve a high-performance inverter having a fast dynamic response for quick reference tracking and a low THD for a purely sinusoidal voltage and that is more adaptive to different types of loads in the standalone mode. In order to reach these goals, we propose an inverter composed of two bridges of electronic switches concatenated with an LC filter circuit at the output. Our contributions in this stage of inverter is the regulation of the output voltage in different load values and in different values of solar irradiance with low THD and low response time. Our system is robust, since in the case of sudden change in weather conditions, such as a variation in temperature, sun irradiation, or both will cause the controller to quickly follow this variation. It is also robust in terms of regulation of the voltage since it maintains 220 V and a stable 50 Hz for any value of a varying load.The back-stepping control has attracted the attention of many researchers, thanks to its capability to stabilize nonlinear dynamical systems. To design these dynamical systems, an analysis of the stability is necessary. For nonlinear systems, it is more complicated to verify the stability of equilibrium than linear dynamical systems. For that, the Lyapunov function is used to regulate the stability [26]. In general, the major step to the use of the Lyapunov theory is in constructing a suitable Lyapunov function. Therefore, there is no specific technique for building Lyapunov functions for ordinary differential equations, and the construction of Lyapunov functions is known in numerous cases. One of the methods of designing nonlinear controllers is based on Aleksandr Lyapunov’s theory of stability of dynamical systems (Lyapunov, 1892) [27, 28]. Typically, the goal of the design is to find a negative function of the derivative of the Lyapunov candidate function. However, this task is rather complex for a large number of systems. Back-stepping is a design method developed by several authors including Petar V. Kokotovic (see [27]) and applied to certain classes of systems, which normalizes the design of the controller into a series of predefined steps. This strategy makes it possible to build progressively the expression of the command that can stabilize the system. The remainder of this paper is organized as follows: Section 2 talks about the overall description of the proposed single-phase PV inverter in the standalone mode. In Section 3, the dynamic model and back-stepping control design of both converters, the boost and the H-bridge, with a filter is presented. Finally, testing and simulation of the proposed system are shown by simulation results in Section 4. This paper ends with a conclusion.


## 2. System’s Description

Figure1 shows the block diagram of the proposed system. It includes two power electronic converters between the PV array and the AC loads. Each converter is controlled by a back-stepping system, having the role of providing the maximum power to the loads and ensuring a good conversion from DC to AC power. The first converter is a boost DC-DC that is used to track the maximum electrical energy generated by the PV array, for different values of irradiance and temperature, using a basic MPPT algorithm type Perturb and Observe (P&O). It generates the reference voltage to the back-stepping block in order to force the PV array to provide this voltage. The second converter is an H-bridge inverter with LC filter having the role of converting continuous to alternative voltage with minimum harmonic distortion and good stability in terms of amplitude and frequency in different values of resistive loads.Figure 1
Block diagram of the proposed PV inverter system.


### 2.1. PV Array and P&O Algorithm

Photovoltaic energy is based on the conversion photons into electricity using semiconductor materials. Several solar cells constitute the photovoltaic generators; this solar cell is the basic element that can provide a few watts only. Therefore, a photovoltaic system uses solar panels which is an interconnection of several solar cells in parallel and in series in order to increase the current and the voltage, respectively. Moreover, to obtain larger values of power for large electrical installations, the association of several solar panels in parallel and/or in series is necessary and this association is referred to as a PV array. In this work, the solar panel considered is the monocrystalline 245 W and the total power of the PV array is 978 W. The electrical characteristics of both the solar module and the PV array are listed in Table1. The I-V and P-V curves associated with the PV array used for different values of solar irradiance and fixed temperature are shown in Figures 2 and 3.Table 1
Parameters of the PV array.
Typical electrical characteristics
Value
Module data
Maximum power per module (Pmax)
244.62 W
Cells per module (Ncell)
60
Open circuit voltage (Voc)
37.2 V
Short circuit current (Isc)
8.62 A
Voltage at maximum power point (Vmpp)
30.2 V
Current at maximum power point (Impp)
8.1 A
Temperature coefficient ofVoc
-0.36901%/°C
Temperature coefficient ofIsc
0.086995%/°C
Array data
Parallel strings
1
Series-connected modules per string
4
Maximum power of PV array
978 W
Voltage at maximum power point of PV array
120.8 V
Current at maximum power point of PV array
8.1 AFigure 2
P-V curves of a PV array with different values of sun irradiance.Figure 3
I-V curves of a PV array for different values of sun irradiance.The PV array generates different values of power depending on specific atmospheric conditions in terms of solar irradiance and temperature. However, there is one point of power that is considered as the maximum power point (MPP). There is one MPP for each curve, considering that the shading is negligible. The PV array should generate the maximum power using a specific algorithm to track this maximum which is commonly called the maximum power point tracking (MPPT). In this work, the P&O algorithm is applied to the PV array voltage, which would translate to an increase or decrease in power as shown in Figure4. If a rise in voltage leads to a rise in power, this means that the operating point is the left of the MPP, and hence, further voltage perturbation is required towards the right to reach the MPP. Conversely, if a rise in voltage leads to a diminution in power, this means that the present operating point is to the right of the MPP, and hence, further voltage perturbation is required towards the left to reach the MPP. In this way, the algorithm converges towards the MPP after several perturbations.Figure 4
Flowchart of the P&O algorithm.


### 2.2. Boost Converter

The second block after the PV array is a basic DC-DC converter of type boost that steps up the voltage from low input voltage, coming from the PV array, into high output voltage, going to the input of the inverter. The input of the boost converter is connected to the PV array in order to achieve the MPP in different atmospheric conditions. Its output is connected to an H-bridge inverter in order to obtain a higher voltage that can be supplied to the AC load without using a transformer. Figure5 represents the synoptic block of the DC-DC converter with its controller. It is made with a back-stepping module to track the PV reference voltage generated by the MPPT block as defined in the previous section. This controller can generate a suitable duty cycle for controlling a power transistor of the boost converter using a PWM generator.Figure 5
Block diagram of the first stage (boost converter).


### 2.3. H-Bridge Inverter

A PWM inverter, cascaded with an LC filter in the standalone mode with back-stepping controller, is modeled in Figure6. This inverter system is composed of two essential parts: the electrical power part and the control unit of this system. The electrical power part is composed of
(i)
an H-bridge converter which is typically composed of four electrical MOSFET transistors
(ii)
an LC Filter that is necessary to obtain a sinusoidal waveform with an appropriate frequency and having a minimum distortion of the voltage at terminal loads
(iii)
resistive loads, which represent the final consumer of this electricityFigure 6
Block diagram of the second stage (H-bridge inverter).The second part of the system is the back-stepping controller who plays a significant role to achieve a high performance of the inverter in the standalone mode. It regulates the output voltage at terminal loads by using the control law of equation (43) developed in Section 3 of this manuscript.


## 2.1. PV Array and P&O Algorithm

Photovoltaic energy is based on the conversion photons into electricity using semiconductor materials. Several solar cells constitute the photovoltaic generators; this solar cell is the basic element that can provide a few watts only. Therefore, a photovoltaic system uses solar panels which is an interconnection of several solar cells in parallel and in series in order to increase the current and the voltage, respectively. Moreover, to obtain larger values of power for large electrical installations, the association of several solar panels in parallel and/or in series is necessary and this association is referred to as a PV array. In this work, the solar panel considered is the monocrystalline 245 W and the total power of the PV array is 978 W. The electrical characteristics of both the solar module and the PV array are listed in Table1. The I-V and P-V curves associated with the PV array used for different values of solar irradiance and fixed temperature are shown in Figures 2 and 3.Table 1
Parameters of the PV array.
Typical electrical characteristics
Value
Module data
Maximum power per module (Pmax)
244.62 W
Cells per module (Ncell)
60
Open circuit voltage (Voc)
37.2 V
Short circuit current (Isc)
8.62 A
Voltage at maximum power point (Vmpp)
30.2 V
Current at maximum power point (Impp)
8.1 A
Temperature coefficient ofVoc
-0.36901%/°C
Temperature coefficient ofIsc
0.086995%/°C
Array data
Parallel strings
1
Series-connected modules per string
4
Maximum power of PV array
978 W
Voltage at maximum power point of PV array
120.8 V
Current at maximum power point of PV array
8.1 AFigure 2
P-V curves of a PV array with different values of sun irradiance.Figure 3
I-V curves of a PV array for different values of sun irradiance.The PV array generates different values of power depending on specific atmospheric conditions in terms of solar irradiance and temperature. However, there is one point of power that is considered as the maximum power point (MPP). There is one MPP for each curve, considering that the shading is negligible. The PV array should generate the maximum power using a specific algorithm to track this maximum which is commonly called the maximum power point tracking (MPPT). In this work, the P&O algorithm is applied to the PV array voltage, which would translate to an increase or decrease in power as shown in Figure4. If a rise in voltage leads to a rise in power, this means that the operating point is the left of the MPP, and hence, further voltage perturbation is required towards the right to reach the MPP. Conversely, if a rise in voltage leads to a diminution in power, this means that the present operating point is to the right of the MPP, and hence, further voltage perturbation is required towards the left to reach the MPP. In this way, the algorithm converges towards the MPP after several perturbations.Figure 4
Flowchart of the P&O algorithm.


## 2.2. Boost Converter

The second block after the PV array is a basic DC-DC converter of type boost that steps up the voltage from low input voltage, coming from the PV array, into high output voltage, going to the input of the inverter. The input of the boost converter is connected to the PV array in order to achieve the MPP in different atmospheric conditions. Its output is connected to an H-bridge inverter in order to obtain a higher voltage that can be supplied to the AC load without using a transformer. Figure5 represents the synoptic block of the DC-DC converter with its controller. It is made with a back-stepping module to track the PV reference voltage generated by the MPPT block as defined in the previous section. This controller can generate a suitable duty cycle for controlling a power transistor of the boost converter using a PWM generator.Figure 5
Block diagram of the first stage (boost converter).


## 2.3. H-Bridge Inverter

A PWM inverter, cascaded with an LC filter in the standalone mode with back-stepping controller, is modeled in Figure6. This inverter system is composed of two essential parts: the electrical power part and the control unit of this system. The electrical power part is composed of
(i)
an H-bridge converter which is typically composed of four electrical MOSFET transistors
(ii)
an LC Filter that is necessary to obtain a sinusoidal waveform with an appropriate frequency and having a minimum distortion of the voltage at terminal loads
(iii)
resistive loads, which represent the final consumer of this electricityFigure 6
Block diagram of the second stage (H-bridge inverter).The second part of the system is the back-stepping controller who plays a significant role to achieve a high performance of the inverter in the standalone mode. It regulates the output voltage at terminal loads by using the control law of equation (43) developed in Section 3 of this manuscript.


## 3. Dynamic Model and Back-Stepping Control Design

### 3.1. Boost Converter

The basic schematic of the boost converter studied is depicted in Figure7.Figure 7
Basic schematic of the boost converter.ipv and Vpv are the photovoltaic current and the photovoltaic voltage generated by the PV array, respectively. Vpv is the parameter that should be regulated to achieve the MPP. iLB and VC2 are the current in the inductor LB and the output voltage of the boost converter, respectively. The switching frequency applied in the power electronic transistor has the value of 20 kHz. Applying the Kirchhoff theorem on the schematic of the boost as shown in Figure 7, (1) and (2) represent the dynamic model of the boost:
(1)C1dVpvdt=ipv−iLB,(2)LBdiLBdt=Vpv−1−u1VC2.By choosing the voltageVpv as the system state and the control parameter u1 as the signal control of the boost converter, (1) and (2) can be rearranged as follows:
(3)C1ẋ1=ipv−x2,LBẋ2=x1−1−u1VC2,where x is the state vector of the second-order system. x1 and x2 are the average value of Vpv and iLB, respectively. u1 is the control law.
(4)ẋ1=1C1ipv−1C1x2,ẋ2=1LBx1−1−u1LBVC2.The goal is to track the photovoltaic reference voltage in order to produce the maximum power by the PV array using a back-stepping control. The control law is generated based on the stability theory of the Lyapunov dynamic systems. Therefore,e1 is the error and it is defined as
(5)e1=x1−Vpvref,ė1=ẋ1−V̇pvref=1C1ipv−1C1x2−V̇pvref.The first Lyapunov functionV1 is defined as
(6)V1=12e12,so that its derivative is
(7)V̇1=e1ė1=e11C1ipv−1C1x2−V̇pvref.To getV̇1=−k1e12<0, it is necessary to have equation (8), where k1 is a positive constant:
(8)1C1ipv−1C1x2−V̇pvref=−k1e1.The virtual control of the system isx2∗ that is equal to
(9)x2∗=ipv+C1k1e1−C1V̇pvref,where the second error between the second state variable x2 and its desired value x2∗ is defined as follows:
(10)e2=x2−x2∗,x2=x2∗+e2.From the derivative ofV1 in (7), the expression of derivative of error e1 can be written as
(11)ė1=1C1ipv−1C1x2∗+e2−V̇pvref=1C1ipv−1C1x2∗−1C1e2−V̇pvref=1C1ipv−1C1ipv+C1k1e1−C1V̇pvref−1C1e2−V̇pvref.Therefore, the system equation of the two errors is(12)ė1=−k1e1−1C1e2,(13)ė2=ẋ2−ẋ2∗=1LBx1−1−uLBVC2−ẋ2∗.Choosing a second Lyapunov function candidateV2,
(14)V2=V1+12e22,and its derivative is
(15)V̇2=V̇1+e2ė2=e1ė1+e2ė2.Applying (12) and (13) in equation (15), the new expression of the derivative of V2 is mentioned in
(16)V̇2=e1−k1e1−1C1e2+e21LBx1−1−u1LBVC2−ẋ2∗=−k1e12+e2−1C1e1+1LBx1−1−u1LBVC2−ẋ2∗.To getV̇2=−k1e12−k2e22<0, where k1 and k2 are two positive constants, it is necessary to have the following:
(17)−1C1e1+1LBx1−1−u1LBVC2−ẋ2∗=−k2e2.The control law corresponding to “u1” for the boost converter is defined in
(18)u1=1−1VC2x1−LBẋ2∗−LB1C1e1−k2e2.u1 is the duty cycle that is the input of the PWM generator issuing a suitable PWM signal to control the power transistors in the boost converter. The error e1 tends to zero because the derivatives of V1 and V2 are negative functions.


### 3.2. Inverter with LC Filter

The single-phase inverter studied is depicted in Figure8.Figure 8
Basic schematic of a single-phase H-bridge inverter.VC2 is the DC voltage.UAB and UC are the output voltage before filtering and after filtering, respectively. iLF and i0 are the current of the inductor LF and the current in the load, respectively. The switching frequency applied in electronic switches has the value of 20 kHz, and it is significantly higher than the frequency of the system which is 50 Hz in order to obtain a good form of output voltage of the inverter. Therefore, voltages and currents are replaced by their Root Mean Square (RMS) value. (19) and (20) represent the system model:
(19)CdUCdt=iLF−i0,(20)LFdiLFdt=UAB−UC.By choosing the voltageUC as the system state and the control parameter u2 as the signal control of the inverter, (19) and (20) can be rearranged as
(21)Cẋ1=x2−i0,LFẋ2=VC2u2−x1,where x is the state vector of the second-order system. x1 and x2 are the average values of UC and iLF, respectively. u2 is the control law.
(22)ẋ1=1Cx2−1Ci0,ẋ2=VC2Lu2−1LFx1.The objective is to have the sinusoidal desired output voltage at load terminals by using a back-stepping controller in order to have a closed-loop regulation. The design technique of the back-stepping controller is based on the stability theory of the Lyapunov dynamic systems. This section aims to force the output voltagex1 to track the reference signal Ucref with the lowest THD and high robustness. Therefore, e3 is the error and it is defined as
(23)e3=Ucref−x1.The aim is to obtaine3 equal to zero. Taking the derivative of e3,
(24)ė3=U̇cref−ẋ1,ė3=U̇cref−1Cx2+1Ci0.By choosing the following Lyapunov candidate,(25)V1=12e32,so that its derivative is
(26)V̇1=e3ė3=e3U̇cref−1Cx2+1Ci0,where the second error is defined as
(27)e4=α−x2,(28)x2=α−e4,so that its derivative is
(29)ė4=α̇−ẋ2=α̇−VC2LFu2+1Lx1.From (26) and (28),
(30)V̇1=e3U̇cref−1Cα−1Ce4+1Ci0.To getV̇1<0, choose α such that
(31)α=CU̇cref+1Ci0+k3e3,where k3>0, then
(32)V̇1=e3U̇cref−U̇cref−1Ci0−k3e3+1Ce4+1Ci0,(33)V̇1=e3−k3e3+1Ce4,(34)V̇1=e3ė3.Therefore,(35)ė3=−k3e3+1Ce4.Choosing a second Lyapunov function candidate,(36)V2=V1+12e42,and its derivative is
(37)V̇2=V̇1+e4ė4.Basing on equations (33) and (37), we can get V̇2:
(38)V̇2=−k3e32+1Ce3e4+e4ė4,(39)V̇2=−k3e32+e41Ce3+ė4.As a result, from (29) and (39), the expression of V̇2 is
(40)V̇2=−k3e32+e41Ce3+α̇−VC2LFu2+1LFx1.The derivative ofα from equation (31) is
(41)α̇=CU¨cref+1Ci̇̇0+k3ė3.Therefore, the final expression ofV̇2 is
(42)V̇2=−k3e32+z21Ce3+CU¨cref+i̇̇0+Ck3ė3−VC2LFu2+1LFx1.To getV̇2<0, choose the control law “u2” for the inverter as defined in equation (43), where k4>0:
(43)u2=LFVC21Ce3+CU¨cref+i̇̇0+Ck3ė3+1LFx1+k4e4.By applying this control law to the PWM inverter in the standalone system, the errore3 tends to zero because the derivatives of V1 and V2 are negative functions. According to the Lyapunov theory, choose k3>0 and k4>0 to ensure good stability of the back-stepping control for the inverter.


## 3.1. Boost Converter

The basic schematic of the boost converter studied is depicted in Figure7.Figure 7
Basic schematic of the boost converter.ipv and Vpv are the photovoltaic current and the photovoltaic voltage generated by the PV array, respectively. Vpv is the parameter that should be regulated to achieve the MPP. iLB and VC2 are the current in the inductor LB and the output voltage of the boost converter, respectively. The switching frequency applied in the power electronic transistor has the value of 20 kHz. Applying the Kirchhoff theorem on the schematic of the boost as shown in Figure 7, (1) and (2) represent the dynamic model of the boost:
(1)C1dVpvdt=ipv−iLB,(2)LBdiLBdt=Vpv−1−u1VC2.By choosing the voltageVpv as the system state and the control parameter u1 as the signal control of the boost converter, (1) and (2) can be rearranged as follows:
(3)C1ẋ1=ipv−x2,LBẋ2=x1−1−u1VC2,where x is the state vector of the second-order system. x1 and x2 are the average value of Vpv and iLB, respectively. u1 is the control law.
(4)ẋ1=1C1ipv−1C1x2,ẋ2=1LBx1−1−u1LBVC2.The goal is to track the photovoltaic reference voltage in order to produce the maximum power by the PV array using a back-stepping control. The control law is generated based on the stability theory of the Lyapunov dynamic systems. Therefore,e1 is the error and it is defined as
(5)e1=x1−Vpvref,ė1=ẋ1−V̇pvref=1C1ipv−1C1x2−V̇pvref.The first Lyapunov functionV1 is defined as
(6)V1=12e12,so that its derivative is
(7)V̇1=e1ė1=e11C1ipv−1C1x2−V̇pvref.To getV̇1=−k1e12<0, it is necessary to have equation (8), where k1 is a positive constant:
(8)1C1ipv−1C1x2−V̇pvref=−k1e1.The virtual control of the system isx2∗ that is equal to
(9)x2∗=ipv+C1k1e1−C1V̇pvref,where the second error between the second state variable x2 and its desired value x2∗ is defined as follows:
(10)e2=x2−x2∗,x2=x2∗+e2.From the derivative ofV1 in (7), the expression of derivative of error e1 can be written as
(11)ė1=1C1ipv−1C1x2∗+e2−V̇pvref=1C1ipv−1C1x2∗−1C1e2−V̇pvref=1C1ipv−1C1ipv+C1k1e1−C1V̇pvref−1C1e2−V̇pvref.Therefore, the system equation of the two errors is(12)ė1=−k1e1−1C1e2,(13)ė2=ẋ2−ẋ2∗=1LBx1−1−uLBVC2−ẋ2∗.Choosing a second Lyapunov function candidateV2,
(14)V2=V1+12e22,and its derivative is
(15)V̇2=V̇1+e2ė2=e1ė1+e2ė2.Applying (12) and (13) in equation (15), the new expression of the derivative of V2 is mentioned in
(16)V̇2=e1−k1e1−1C1e2+e21LBx1−1−u1LBVC2−ẋ2∗=−k1e12+e2−1C1e1+1LBx1−1−u1LBVC2−ẋ2∗.To getV̇2=−k1e12−k2e22<0, where k1 and k2 are two positive constants, it is necessary to have the following:
(17)−1C1e1+1LBx1−1−u1LBVC2−ẋ2∗=−k2e2.The control law corresponding to “u1” for the boost converter is defined in
(18)u1=1−1VC2x1−LBẋ2∗−LB1C1e1−k2e2.u1 is the duty cycle that is the input of the PWM generator issuing a suitable PWM signal to control the power transistors in the boost converter. The error e1 tends to zero because the derivatives of V1 and V2 are negative functions.


## 3.2. Inverter with LC Filter

The single-phase inverter studied is depicted in Figure8.Figure 8
Basic schematic of a single-phase H-bridge inverter.VC2 is the DC voltage.UAB and UC are the output voltage before filtering and after filtering, respectively. iLF and i0 are the current of the inductor LF and the current in the load, respectively. The switching frequency applied in electronic switches has the value of 20 kHz, and it is significantly higher than the frequency of the system which is 50 Hz in order to obtain a good form of output voltage of the inverter. Therefore, voltages and currents are replaced by their Root Mean Square (RMS) value. (19) and (20) represent the system model:
(19)CdUCdt=iLF−i0,(20)LFdiLFdt=UAB−UC.By choosing the voltageUC as the system state and the control parameter u2 as the signal control of the inverter, (19) and (20) can be rearranged as
(21)Cẋ1=x2−i0,LFẋ2=VC2u2−x1,where x is the state vector of the second-order system. x1 and x2 are the average values of UC and iLF, respectively. u2 is the control law.
(22)ẋ1=1Cx2−1Ci0,ẋ2=VC2Lu2−1LFx1.The objective is to have the sinusoidal desired output voltage at load terminals by using a back-stepping controller in order to have a closed-loop regulation. The design technique of the back-stepping controller is based on the stability theory of the Lyapunov dynamic systems. This section aims to force the output voltagex1 to track the reference signal Ucref with the lowest THD and high robustness. Therefore, e3 is the error and it is defined as
(23)e3=Ucref−x1.The aim is to obtaine3 equal to zero. Taking the derivative of e3,
(24)ė3=U̇cref−ẋ1,ė3=U̇cref−1Cx2+1Ci0.By choosing the following Lyapunov candidate,(25)V1=12e32,so that its derivative is
(26)V̇1=e3ė3=e3U̇cref−1Cx2+1Ci0,where the second error is defined as
(27)e4=α−x2,(28)x2=α−e4,so that its derivative is
(29)ė4=α̇−ẋ2=α̇−VC2LFu2+1Lx1.From (26) and (28),
(30)V̇1=e3U̇cref−1Cα−1Ce4+1Ci0.To getV̇1<0, choose α such that
(31)α=CU̇cref+1Ci0+k3e3,where k3>0, then
(32)V̇1=e3U̇cref−U̇cref−1Ci0−k3e3+1Ce4+1Ci0,(33)V̇1=e3−k3e3+1Ce4,(34)V̇1=e3ė3.Therefore,(35)ė3=−k3e3+1Ce4.Choosing a second Lyapunov function candidate,(36)V2=V1+12e42,and its derivative is
(37)V̇2=V̇1+e4ė4.Basing on equations (33) and (37), we can get V̇2:
(38)V̇2=−k3e32+1Ce3e4+e4ė4,(39)V̇2=−k3e32+e41Ce3+ė4.As a result, from (29) and (39), the expression of V̇2 is
(40)V̇2=−k3e32+e41Ce3+α̇−VC2LFu2+1LFx1.The derivative ofα from equation (31) is
(41)α̇=CU¨cref+1Ci̇̇0+k3ė3.Therefore, the final expression ofV̇2 is
(42)V̇2=−k3e32+z21Ce3+CU¨cref+i̇̇0+Ck3ė3−VC2LFu2+1LFx1.To getV̇2<0, choose the control law “u2” for the inverter as defined in equation (43), where k4>0:
(43)u2=LFVC21Ce3+CU¨cref+i̇̇0+Ck3ė3+1LFx1+k4e4.By applying this control law to the PWM inverter in the standalone system, the errore3 tends to zero because the derivatives of V1 and V2 are negative functions. According to the Lyapunov theory, choose k3>0 and k4>0 to ensure good stability of the back-stepping control for the inverter.


## 4. Simulation and Results

Figure9 shows the circuit schematic of the PV inverter in the standalone mode simulated in the Simulink platform. At the beginning of the simulation, the resistor R1 is the only consumer of the solar energy generated by the PV array. After 0.65 seconds, another resistive load R2 is added in parallel for an additional 0.1 sec. At 0.75 sec, R2 is disconnected from the load and only R1 remains connected. Parameters of the PV module used in this study are listed in Table 1. The overall system is tested and validated with parameters of the system which are listed in Table 2.Figure 9
Schematic of the proposed PV inverter system.Table 2
Parameters of the system elements.
Parameter
Value
Ucref (RMS)
220 V
FPWM
20 kHz
f
50 Hz
R1
100Ω
R2
100Ω
Offset (MPPT bloc)
0.0005 V
C
47e−6F
C1
100e−6F
C2
100e−6F
LF
4.7e−3H
LB
3e−3H
T
25°C
k1
9000
k2
9000
k3
20000
k4
30000
Sample time
1e−6s
Switch on
0.65–0.75 sThe type of PWM block used in Simulink for both the boost converter and the inverter is the PWM generator DC-DC and single-phase half-bridge (2 pulses), respectively. The minimum and maximum values of the input PWM generator for the inverter, which is the control lawu2, are -1 and 1, respectively.The internal design blocs of back-stepping control numbers 1 and 2 are shown, respectively, in Figures10 and 11. The control law equation can be presented in the form of blocks.Figure 10
Internal blocs of back-stepping control 1.Figure 11
Internal blocs of back-stepping control 2.The initial value of irradiance is set to 600 W/m2; after each 0.2 s, it is changed to the following values: 200 W/m2, 700 W/m2, 1000 W/m2, and 900 W/m2 in order to have instantaneous step values of irradiance in a short time for testing the capability of the controller to track the suitable value of power generated by the PV array. During this simulation, the temperature is kept at 25°C. Figure 12 shows the sun irradiance profile that is applied to the PV array with the aim of simulating the proposed system in different conditions and to examine the dynamic response of the two back-stepping controllers.Figure 12
Sun irradiance profile.As shown in Figure13, the MPPT algorithm block generates successfully the corresponding peak voltage Vpvref to be used later by the back-stepping controller. At the beginning of simulation, the PV array voltage Vpv started at 0 V value and it attained the initial value of the Vpvref which is 121 V with 2 V of ripples in the transitory regime that lasted 10 ms. Until 0.2 s, the ripples of Vpv are 0.6 V in the steady state. At 0.2 s, there is a change of solar irradiance from 600 to 200 W/m2 which causes a slight decrease in reference voltage from 121 V to 118 V; also, the value of the PV array voltage Vpv successfully tracks the Vpvref. It can be seen from Figure 13 that the PV array voltage Vpv is changed due to the variation of solar irradiance and it follows the reference voltage Vpvref with some disturbances in the transitory regime due to the low performance of using the basic P&O algorithm.Figure 13
PV array voltage with its reference.The variation of irradiance causes little variation in the PV voltage and high variation in the PV current. Figure14 shows that the current generated by PV array ipv is variated with a big jump between two values against the voltage which varies with a small difference. At the start, the current value ipv is set at 5.2 A and it became stable after 15 ms from the beginning at 4.85 A in the steady state. At 0.4 s, the irradiance is changed from 200 to 700 W/m2 which causes the rapid increase in current value with some ripples in the transitory regime that lasted only 6 ms; the new current value is 5.68 A at 700 W/m2.Figure 14
PV array current.Similarly, Figure15 shows the photovoltaic power of the PV array during 1 s according to the solar irradiance profile chosen in Figure 12. At the beginning, the sun irradiance is set at 600 W/m2 that means that the reference power generated by the PV array is 589 W. The power value is 0 W, and the back-stepping controller starts executing its role which makes the PV array generate the power for different atmospheric conditions. The transient phase contains little ripples of 20 W, and this controller can create a suitable control to the boost converter to track the reference power. For this type of power supply application, the overshoot of our MPPT system is around 40 W, in the transient phase, in which the value can be considered as negligible compared to the total power of the system which is 1 kW. The transient phase contains little ripples of 20 W, and this controller can create a suitable control to the boost converter to track the reference power. After 7 ms from starting the simulation, the power generated by the PV array is 588 W. At 0.2 s, the irradiance changed its value and it became 200 W/m2 and automatically the power will decrease; therefore, the new value of the power generated by the PV array is 191.6 W. At 0.4 s, the power value is 687 W and it corresponds to 700 W/m2 of sun irradiance. For 1000 W/m2, the power increases and it becomes 978.2 W, in the last part or the last value of irradiance which is 900 W/m2. At 0.8 s, the sun irradiance changes from 1000 to 900 W/m2 which causes diminution in power. Figure 16 shows a zoom of Ppv between 0.8 and 0.80006 s; therefore, the response time to attain the exact value of the power is around 1 ms. Figure 17 gives an idea about the maximum power at 900 W/m2 that is 881.2 W. Referring to Figures 16 and 17, the efficiency of the MPPT system is 99.93%. Moreover, the MPP is successfully tracked by the controller which verifies the high robustness and performance of the back-stepping control to generate the maximum power to the load.Figure 15
PV array power.Figure 16
Zoom area of the PV array power.Figure 17
P-V characteristic of the PV array for 900 W/m2.Figure18 shows the performance of the inverter with the proposed back-stepping controller. This figure represents two waveforms which are the output voltage of the inverter at the terminal load UC and the sinusoidal signal reference Ucref. At the beginning, the irradiance value was 600 W/m2 and the output voltage of the boost converter increased until it became greater than 311 V which is the nominal value of the input inverter.Figure 18
Output voltage inverter after filtering with its reference.Figure19 presents a zoom of the output voltage of the inverter between 0 and 0.1 s to show that the inverter follows perfectly the reference voltage using the back-stepping control. For our inverter, as mentioned in Figure 19, the response time to have a good form of the output voltage for electrical loads is 30 ms at the beginning in which the power generated by the PV array is 589 W. Moreover, in this time, the RMS value of the output voltage of the inverter is less than the norm 230 V. Therefore, this transient phase has no effect on power quality in electrical loads. There is distortion of output voltage UC until 40 ms. This is due to insufficient voltage in the input of the inverter and not to the back-stepping control. After 40 ms from the beginning, the input voltage of the inverter becomes greater than 311 V and the output voltage of inverter UC started to track the reference voltage Ucref. Moreover, with the input voltage of the inverter greater than 311 V, the response time of this controller is less than 15 ms that ensures the high robustness of this controller. At 0.2 s, the solar irradiance became 200 W/m2 which means less power generated by the PV array. Therefore, the boost converter was not able to provide a suitable voltage to the inverter and the voltage was less than 311 V. For this reason, a voltage drop appeared in the output voltage of the inverter as show in Figure 18 between 0.2 and 0.4 s. After 0.4 s, the solar irradiance increased, and automatically, the power increased. Therefore, there is no issue with voltage drop despite the addition of a second resistive load.Figure 19
Output voltage inverter after filtering with its reference (zoom area).Figure20 shows the current curve i0 of the AC loads. This figure represents the pure sinusoidal waveform of the current consumed by loads. Between 0.2 and 0.4 s, there is a current drop due to the insufficient current produced by the PV array. At 0.65 s, there are two loads in parallel. It can be seen from Figure 20 that the current increased to 6.2 A as a maximum value; after 0.75 s, it decreased to 3.1 A as a maximum value. The controller of the second part of the proposed inverter is robust, and it shows the robustness and high performance of the chosen controller.Figure 20
Current consumed by the loads.Figure21 shows the frequency analysis of the inverter’s output voltage. The fundamental is about 310.7 V at a frequency of 50 Hz, and the THD is about 0.78%. From Figures 18–21, we can conclude that the inverter studied with the proposed back-stepping control in the standalone mode has a low harmonic distortion, high conversion efficiency, strong control performance, and a high quality of sinusoidal waveform.Figure 21
Frequency analysis of the output voltage inverter.The efficiency of the MPPT system and the efficiency of the inverter are mentioned in Table3 in different values of solar irradiation. As shown for different solar irradiance levels, our first controller of the MPPT system presents a high percentage of efficiency and a good manner to track the power compared with other works as mentioned in [12, 14]. For the second stage which is the inverter, it also presents good and high efficiency, only that there is a certain limit of solar irradiance wherein the inverter cannot generate a good waveform of voltage to loads. As shown, the efficiency in 200 W/m2 is bad because the input power of the inverter is less than the threshold power that must be generated by the PV array.Table 3
Efficiency of two stages of the whole system under different solar irradiation values.
Solar irradiation (W/m2)
Efficiency of MPPT (%)
Efficiency of inverter (%)
200
99.68
71.00
600
99.83
95.40
700
99.92
95.82
900
99.93
96.80
1000
99.96
96.88Table4 shows the performance comparison of MPPT algorithms between our system and other works. The 1 ms of the response time and the 99.93% of efficiency demonstrate the high performance and robustness of P&O with back-stepping algorithm of our MPPT system.Table 4
Performance comparison of MPPT algorithms.
Ref.
Algorithm type
Response time
Efficiency
[10]
P&O algorithm
—
96%
PI
—
96.5%
Neuro-fuzzy
—
98.2%
Back-stepping
—
99.1%
[12]
Proposed modified incremental conductance
7 ms
97.53%
[14]
Integral back-stepping
2.17 ms
—
Our study
P&O with back-stepping
1 ms
99.93%In order to easily exhibit the excellence and to show the performance of the proposed back-stepping in the standalone mode, the results of the PV inverter with other works [17, 29–33] are summarized in Table 5. As shown in this comparison, our proposed system has good and high performance control to extract the maximum power generated by the PV array and to regulate the output voltage of the inverter. Firstly, the function performed by our system is a conversion of energy from photovoltaic modules to the use of this energy by electric charges. It is a dual function or double role between tracking the maximum power and regulating the voltage to have a sinusoidal waveform at the output of the inverter. However, most works deal only with the DC to AC conversion without MPPT [34]. With a very low THD and a very advanced nonlinear controller, our system is the most practical for the realization of photovoltaic inverters in the standalone mode.Table 5
Quality comparisons of the whole system with other proposal work.
Ref
Function
Standalone/grid connected
MPPT
Transformer
Powered from
THD
Control type
System robustness
[29]
DC to AC
Standalone
No
Yes
Battery source
THD voltage 2%
Multilevel PWM control
Fair
[30]
MPPT+DC to AC
Standalone
Yes
Yes
PV source
—
Proposed MPPT
Fair
[31]
MPPT+DC to AC
Standalone
Yes
No
PV source
THD voltage 2.7%
SPWM
Poor
[17]
DC to AC
Standalone+grid connected
No
No
Battery source
THD voltage 1.29%
Back-stepping
Favorable
[32]
DC to AC
Standalone
No
No
Battery source
—
Direct quadrature and droop Lyapunov
Fair
[33]
DC to AC
Grid connected
No
No
Renewable energy source
Current THD 4.5%
Direct quadrature
Fair
Our study
MPPT+DC to AC
Standalone
Yes
No
PV source
THD voltage 0.78%
Back-stepping
Favorable


## 5. Conclusion

A robust control scheme combined with a high performance PV inverter system has been presented in this paper. Simulation results show that the two stages of converter successfully respond to the two principal objectives which are, firstly, the extraction of the maximum power from the PV array with an efficiency of 99.93% and 1 ms of response time to show the fast dynamic response of our MPPT algorithm using the boost converter in order to avoid the use of the transformer. Secondly, we obtained a stable sinusoidal waveform of the output voltage of the inverter which is 220 V with 0.78% of the THD and fixed 50 Hz frequency. This performance is obtained by using a nonlinear back-stepping control which can quickly track the reference by having the error converge to zero. This PV inverter system delivers the high quality sinusoidal power to the AC load as long as the solar radiation is higher than 300 W/m2.



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*Source: 7021578-2019-11-06.xml*