Research ArticleModeling and Design of Single-Phase PV Inverter with MPPTAlgorithm Applied to the Boost Converter Using Back-SteppingControl in Standalone Mode
Omar Diouri ,1 Najia Es-Sbai,1 Fatima Errahimi,1 Ahmed Gaga,1 and Chakib Alaoui2
1Electrical Department, Laboratory of Renewable Energies and Intelligent Systems (LERSI), Faculty of Sciences and Technology,Sidi Mohamed Ben Abdellah University of Fez, Morocco2INSA, Euro-Méditerranée, EUROMED University, Fez, Morocco
Correspondence should be addressed to Omar Diouri; [email protected]
Received 3 March 2019; Revised 19 July 2019; Accepted 8 August 2019; Published 6 November 2019
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 proposedconverter uses a reference voltage that is generated by the Perturb and Observe (P&O) algorithm in order to extract themaximum power point (MPP) by responding accurately to varying atmospheric conditions. Another goal for using the boostconverter is to raise the voltage at the input of the inverter without using a transformer in this system, thus making the systemmore compact and less expensive. Secondly, the single-phase H-bridge inverter was controlled by using back-stepping control inorder to eliminate the error between the output voltage of the inverter and the desired value, even if there is acute load variationat the output of the inverter. The stability of the boost converter and H-bridge inverter was validated by using Lyapunov’sstability theory. Simulation results show that the proposed PV system with back-stepping controllers has a good extraction ofthe MPP with an efficiency of 99.93% and 1ms of response time. In addition, the sinusoidal form of the output voltage of theinverter is fixed to 220V and the total harmonic distortion of the output voltage was found to be less than 1%.
1. Introduction
In recent years, several researches were focused on how todecrease the environmental pollution on Earth by using cleansources of energy such as solar, wind, hydro, biomass, andbiogas [1]. These types of renewable energies are frequentlyapplied to distributed generation (DG) [2]. In 2014, theworld’s electricity consumption amounted to approximately20.7 trillion kilowatt-hours according to [3], having a netincrease of 7,323 terawatt-hours since 1980. Moreover, in2015, the World Bank estimated that 22.69% of the world’srural population were not able to access electricity, sincethe source of power is often located geographically far fromthe consumers, requiring expensive distribution of this elec-tricity by using transmission lines. According to the WorldBank, 8.26% of the worldwide-transmitted power in 2014 islost due to losses associated with transmission lines. One
way to circumvent these issues is to use distributed powergeneration units based on photovoltaic systems.
Currently, there are two types of PV systems: grid-connected and standalone [4]. In order to inject the DCenergy available at the output of photovoltaics into the grid,it is necessary to use converters as an interface [5, 6]. Thissystem is known as the grid-connected PV system. On theother hand, a standalone PV system consists of the transfor-mation of photovoltaic electricity to AC loads available at theconsumer’s sites. Power converters are necessary in order tomake interconnection between solar PV modules and ACloads. These power converters should accomplish two princi-pal functions: first, to ensure that the PV array always gener-ates the maximum power regardless of the variations of theatmospheric and load conditions. This is referred to as themaximum power point tracker or simply MPPT. The secondfunction is the conversion of the continuous voltage
HindawiInternational Journal of PhotoenergyVolume 2019, Article ID 7021578, 16 pageshttps://doi.org/10.1155/2019/7021578
generated by the PV array into the alternative voltage to beused by the AC loads. This AC output voltage should havethe 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 toattain these purposes.
In this paper, a converter with a two-stage topology isused. It consists of a boost converter and an H-bridgeinverter. The main objective of the first stage (boost con-verter) is allowing the PV array to generate the maximumpower using the MPPT technique [9]. There are several algo-rithms 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 MPPTtechniques: indirect MPP tracking like the fractional opencircuit voltage method [11], direct MPP tracking like theincremental conductance [12, 13], or the Perturb andObserve (P&O) method that is implemented in this work.The authors of [12] achieved a response time of 7ms in orderto track the suitable value of the power using the proposedmodified incremental conductance, and an efficiency of97.53%. In [14], the rise time of the back-stepping controlof MPPT and the integral back-stepping were 2.42ms and2.17ms, 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 mainobjectives are to achieve a lower response time and higherefficiency in the MPPT stage. The P&O algorithm takesadvantage of the fact that the P-V curve has a decreasingnature to the right of the MPP and an increasing nature tothe left of the MPP. The drawback of this algorithm is thatthe operating point is never stable and steady at the MPP.It is always oscillating around in the MPP region. This couldbe reduced using very small perturbation steps around theMPP. Another shortcoming is that there is no regulation ofthe output voltage of the DC-DC converter. This issue shouldbe taken into consideration as the authors mentioned in [15].By using the back-stepping control with the MPPT block, wecan generate the reference voltage to be tracked by the con-troller. In addition, the control for boost converter forcesthe PV array to provide the same voltage as the MPPT block.For nonlinear systems, the use of a suitable controller is nec-essary to stabilize the system in the point considered. There-fore, to test the robustness of this controller, we have forcedthis system to high changes of solar irradiance in a short timeand the results show that our proposed system tracks per-fectly the reference power using back-stepping.
The second stage of the proposed solution consists of aninverter. In order to assure an efficient use of the DG units,especially designed inverters play the role of energy conver-sion and adaptation between the sources and the loads [16].The conversion principle in these inverters is the use of apulse width modulation (PWM) technique to offer a stablesinusoidal output voltage of 220V AC to the load. Severalinverters use power electronic switches such as MOSFET orIGBT in the output stage. The PWM technique makes theseinverters suitable for all types of electrical appliances [16].However, these inverters must have a low total harmonic dis-tortion (THD), a fast transient response, and a high effi-
ciency. Much attention has been paid to regulation of thePWM inverter in order to ensure a sinusoidal waveformvoltage with low THD, unchangeable frequency, and fastdynamic response under different types of loads [17]. Themost known methods of regulation are the proportional-integral-derivative (PID) control [18], sliding-mode control[19], linear control [20], Lyapunov control [21], linear reso-nant control [22], and passivity-based control [23]. KalantarZadeh et al. presented in [24] a comparison between threetypes of controllers: sliding mode, back-stepping, and fuzzylogic. As a conclusion, the back-stepping was found to bethe best controller which provides a higher performance.Kolbasi and Seker [25] proposed a nonlinear controller forinverters by using a robust back-stepping. However, havingmore 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 forquick reference tracking and a low THD for a purely sinusoi-dal voltage and that is more adaptive to different types ofloads in the standalone mode. In order to reach these goals,we propose an inverter composed of two bridges of electronicswitches concatenated with an LC filter circuit at the output.Our contributions in this stage of inverter is the regulation ofthe output voltage in different load values and in differentvalues of solar irradiance with low THD and low responsetime. Our system is robust, since in the case of sudden changein weather conditions, such as a variation in temperature, sunirradiation, or both will cause the controller to quickly followthis variation. It is also robust in terms of regulation of thevoltage since it maintains 220V and a stable 50Hz for anyvalue of a varying load.
The back-stepping control has attracted the attention ofmany researchers, thanks to its capability to stabilize nonlin-ear dynamical systems. To design these dynamical systems,an analysis of the stability is necessary. For nonlinear sys-tems, it is more complicated to verify the stability of equilib-rium than linear dynamical systems. For that, the Lyapunovfunction is used to regulate the stability [26]. In general, themajor step to the use of the Lyapunov theory is in construct-ing a suitable Lyapunov function. Therefore, there is no spe-cific technique for building Lyapunov functions for ordinarydifferential equations, and the construction of Lyapunovfunctions is known in numerous cases. One of the methodsof designing nonlinear controllers is based on Aleksandr Lya-punov’s theory of stability of dynamical systems (Lyapunov,1892) [27, 28]. Typically, the goal of the design is to find anegative function of the derivative of the Lyapunov candidatefunction. However, this task is rather complex for a largenumber of systems. Back-stepping is a design method devel-oped by several authors including Petar V. Kokotovic (see[27]) and applied to certain classes of systems, which normal-izes the design of the controller into a series of predefinedsteps. This strategy makes it possible to build progressivelythe expression of the command that can stabilize the system.The remainder of this paper is organized as follows: Section 2talks about the overall description of the proposed single-phase PV inverter in the standalone mode. In Section 3, thedynamic model and back-stepping control design of bothconverters, the boost and the H-bridge, with a filter is
2 International Journal of Photoenergy
presented. Finally, testing and simulation of the proposedsystem are shown by simulation results in Section 4. Thispaper ends with a conclusion.
2. System’s Description
Figure 1 shows the block diagram of the proposed system. Itincludes two power electronic converters between the PVarray and the AC loads. Each converter is controlled by aback-stepping system, having the role of providing the max-imum power to the loads and ensuring a good conversionfrom DC to AC power. The first converter is a boost DC-DC that is used to track the maximum electrical energy gen-erated by the PV array, for different values of irradiance andtemperature, using a basic MPPT algorithm type Perturb andObserve (P&O). It generates the reference voltage to theback-stepping block in order to force the PV array to providethis voltage. The second converter is an H-bridge inverterwith LC filter having the role of converting continuous toalternative voltage with minimum harmonic distortion andgood stability in terms of amplitude and frequency in differ-ent values of resistive loads.
2.1. PV Array and P&O Algorithm. Photovoltaic energy isbased on the conversion photons into electricity usingsemiconductor materials. Several solar cells constitute thephotovoltaic generators; this solar cell is the basic elementthat can provide a few watts only. Therefore, a photovol-taic system uses solar panels which is an interconnectionof several solar cells in parallel and in series in order toincrease the current and the voltage, respectively. More-over, to obtain larger values of power for large electricalinstallations, the association of several solar panels in par-allel and/or in series is necessary and this association isreferred to as a PV array. In this work, the solar panelconsidered is the monocrystalline 245W and the totalpower of the PV array is 978W. The electrical characteris-tics of both the solar module and the PV array are listedin Table 1. The I-V and P-V curves associated with thePV array used for different values of solar irradiance andfixed temperature are shown in Figures 2 and 3.
The PV array generates different values of powerdepending on specific atmospheric conditions in terms ofsolar irradiance and temperature. However, there is onepoint of power that is considered as the maximum powerpoint (MPP). There is one MPP for each curve, considering
MPPT BSC1 BSC2
PV DC-DC boost
Boost converters
9 V 100 V
DC-AC inverter LC filter Loads
Vpv , ipv
Vpvref
U1 iLB VC2 U2 iLF UC , i0
Ucref
Figure 1: Block diagram of the proposed PV inverter system.
Table 1: Parameters of the PV array.
Typical electrical characteristics Value
Module data
Maximum power per module (Pmax) 244.62W
Cells per module (Ncell) 60
Open circuit voltage (Voc) 37.2 V
Short circuit current (Isc) 8.62A
Voltage at maximum power point (Vmpp) 30.2 V
Current at maximum power point (Impp) 8.1 A
Temperature coefficient of Voc -0.36901%/°C
Temperature coefficient of Isc 0.086995%/°C
Array data
Parallel strings 1
Series-connected modules per string 4
Maximum power of PV array 978W
Voltage at maximum power point of PV array 120.8V
Current at maximum power point of PV array 8.1 A
3International Journal of Photoenergy
that the shading is negligible. The PV array should generatethe maximum power using a specific algorithm to track thismaximum which is commonly called the maximum powerpoint tracking (MPPT). In this work, the P&O algorithmis applied to the PV array voltage, which would translateto an increase or decrease in power as shown in Figure 4.If a rise in voltage leads to a rise in power, this means thatthe operating point is the left of the MPP, and hence, fur-ther voltage perturbation is required towards the right toreach the MPP. Conversely, if a rise in voltage leads to adiminution in power, this means that the present operating
point is to the right of the MPP, and hence, further voltageperturbation is required towards the left to reach the MPP.In this way, the algorithm converges towards the MPP afterseveral perturbations.
2.2. Boost Converter. The second block after the PV array is abasic DC-DC converter of type boost that steps up the volt-age from low input voltage, coming from the PV array, intohigh output voltage, going to the input of the inverter. Theinput of the boost converter is connected to the PV array inorder to achieve the MPP in different atmospheric
0 50 100 150Voltage (V)
0
100
200
300
400
500
600
700
800
900
1000
1100Po
wer
(W)
1 kW/m2
0.9 kW/m2
0.6 kW/m2
0.2 kW/m2
Temperature: 25°C
0.7 kW/m2
Figure 2: P-V curves of a PV array with different values of sun irradiance.
0 50 100 150Voltage (V)
0
1
2
3
4
5
6
7
8
9
10
11
Curr
ent (
A)
Temperature: 25°C
1 kW/m2
0.9 kW/m2
0.6 kW/m2
0.2 kW/m2
0.7 kW/m2
Figure 3: I-V curves of a PV array for different values of sun irradiance.
4 International Journal of Photoenergy
conditions. Its output is connected to an H-bridge inverter inorder to obtain a higher voltage that can be supplied to theAC load without using a transformer. Figure 5 representsthe synoptic block of the DC-DC converter with its control-ler. It is made with a back-stepping module to track the PVreference voltage generated by the MPPT block as definedin the previous section. This controller can generate a suit-able duty cycle for controlling a power transistor of the boostconverter using a PWM generator.
2.3. H-Bridge Inverter. A PWM inverter, cascaded with anLC filter in the standalone mode with back-stepping con-troller, is modeled in Figure 6. This inverter system iscomposed of two essential parts: the electrical power partand the control unit of this system. The electrical powerpart is composed of
(i) an H-bridge converter which is typically composedof four electrical MOSFET transistors
(ii) an LC Filter that is necessary to obtain a sinusoidalwaveform with an appropriate frequency and havinga minimum distortion of the voltage at terminalloads
(iii) resistive loads, which represent the final consumer ofthis electricity
The second part of the system is the back-steppingcontroller who plays a significant role to achieve a highperformance of the inverter in the standalone mode. Itregulates the output voltage at terminal loads by usingthe control law of equation (43) developed in Section 3of this manuscript.
3. Dynamic Model and Back-SteppingControl Design
3.1. Boost Converter. The basic schematic of the boost con-verter studied is depicted in Figure 7.
ipv and Vpv are the photovoltaic current and the photo-voltaic voltage generated by the PV array, respectively. Vpvis the parameter that should be regulated to achieve theMPP. iLB and VC2 are the current in the inductor LB and
Figure 5: Block diagram of the first stage (boost converter).
LCfilter
BSC2
VC2
U2 iLF UC, i0
Ucref
DC-AC Loads
Figure 6: Block diagram of the second stage (H-bridge inverter).
5International Journal of Photoenergy
the output voltage of the boost converter, respectively. Theswitching frequency applied in the power electronic transis-tor has the value of 20 kHz. Applying the Kirchhoff theoremon the schematic of the boost as shown in Figure 7, (1) and(2) represent the dynamic model of the boost:
C1dVpvdt
= ipv − iLB, ð1Þ
LBdiLBdt
= Vpv − 1 − u1ð ÞVC2: ð2Þ
By choosing the voltage Vpv as the system state and thecontrol parameter u1 as the signal control of the boost con-verter, (1) and (2) can be rearranged as follows:
C1 _x1 = ipv − x2,LB _x2 = x1 − 1 − u1ð ÞVC2,
ð3Þ
where x is the state vector of the second-order system. x1 andx2 are the average value of Vpv and iLB, respectively. u1 is thecontrol law.
_x1 =1C1
ipv −1C1
x2,
_x2 =1LB
x1 −1 − u1ð ÞLB
VC2:
ð4Þ
The goal is to track the photovoltaic reference voltage inorder to produce the maximum power by the PV array usinga back-stepping control. The control law is generated basedon the stability theory of the Lyapunov dynamic systems.Therefore, e1 is the error and it is defined as
e1 = x1 − Vpvref ,
_e1 = _x1 − _Vpvref =1C1
ipv −1C1
x2 − _Vpvref :ð5Þ
The first Lyapunov function V1 is defined as
V1 =12 e
21, ð6Þ
so that its derivative is
_V1 = e1 _e1 = e11C1
ipv −1C1
x2 − _Vpvref
� �: ð7Þ
To get _V1 = −k1e21 < 0, it is necessary to have equation (8),where k1 is a positive constant:
1C1
ipv −1C1
x2 − _Vpvref = −k1e1: ð8Þ
The virtual control of the system is x∗2 that is equal to
x∗2 = ipv + C1k1e1 − C1 _Vpvref , ð9Þ
where the second error between the second state variable x2and its desired value x∗2 is defined as follows:
e2 = x2 − x∗2 ,x2 = x∗2 + e2:
ð10Þ
From the derivative of V1 in (7), the expression of deriv-ative of error e1 can be written as
_e1 =1C1
ipv −1C1
x∗2 + e2ð Þ − _Vpvref
= 1C1
ipv −1C1
x∗2 −1C1
e2 − _Vpvref
= 1C1
ipv −1C1
ipv + C1k1e1 − C1 _Vpvref� �
−1C1
e2 − _Vpvref :
ð11Þ
Therefore, the system equation of the two errors is
_e1 = −k1e1 −1C1
e2, ð12Þ
_e2 = _x2 − _x∗2 =1LB
x1 −1 − uð ÞLB
VC2 − _x∗2 : ð13Þ
T
iLBipv
Vpv
+
–
C1 C2 VC2
+
–
DiodeLB
Figure 7: Basic schematic of the boost converter.
6 International Journal of Photoenergy
Choosing a second Lyapunov function candidate V2,
V2 =V1 +12 e
22, ð14Þ
and its derivative is
_V2 = _V1 + e2 _e2 = e1 _e1 + e2 _e2: ð15Þ
Applying (12) and (13) in equation (15), the new expres-sion of the derivative of V2 is mentioned in
_V2 = e1 −k1e1 −1C1
e2
� �+ e2
1LB
x1 −1 − u1ð ÞLB
VC2 − _x∗2
� �
= −k1e21 + e2 −
1C1
e1 +1LB
x1 −1 − u1ð ÞLB
VC2 − _x∗2
� �:
ð16Þ
To get _V2 = −k1e21 − k2e22 < 0, where k1 and k2 are two
positive constants, it is necessary to have the following:
−1C1
e1 +1LB
x1 −1 − u1ð ÞLB
VC2 − _x∗2 = −k2e2: ð17Þ
The control law corresponding to “u1” for the boost con-verter is defined in
u1 = 1 − 1VC2
x1 − LB _x∗2 − LB
1C1
e1 − k2e2
� �� �: ð18Þ
u1 is the duty cycle that is the input of the PWM gen-erator issuing a suitable PWM signal to control the powertransistors in the boost converter. The error e1 tends tozero because the derivatives of V1 and V2 are negativefunctions.
3.2. Inverter with LC Filter. The single-phase inverter studiedis depicted in Figure 8.
VC2 is the DC voltage.UAB and UC are the output voltagebefore filtering and after filtering, respectively. iLF and i0are the current of the inductor LF and the current in theload, respectively. The switching frequency applied in elec-tronic switches has the value of 20 kHz, and it is signifi-cantly higher than the frequency of the system which is50Hz in order to obtain a good form of output voltageof the inverter. Therefore, voltages and currents arereplaced by their Root Mean Square (RMS) value. (19)and (20) represent the system model:
CdUC
dt= iLF − i0, ð19Þ
LFdiLFdt
=UAB −UC: ð20Þ
By choosing the voltage UC as the system state and thecontrol parameter u2 as the signal control of the inverter,(19) and (20) can be rearranged as
C _x1 = x2 − i0,LF _x2 =VC2u2 − x1,
ð21Þ
where x is the state vector of the second-order system. x1and x2 are the average values of UC and iLF, respectively.u2 is the control law.
_x1 =1Cx2 −
1Ci0,
_x2 =VC2L
u2 −1LF
x1:ð22Þ
The objective is to have the sinusoidal desired outputvoltage at load terminals by using a back-stepping control-ler in order to have a closed-loop regulation. The designtechnique of the back-stepping controller is based on thestability theory of the Lyapunov dynamic systems. Thissection aims to force the output voltage x1 to track thereference signal Ucref with the lowest THD and highrobustness. Therefore, e3 is the error and it is defined as
e3 =Ucref − x1: ð23Þ
The aim is to obtain e3 equal to zero. Taking thederivative of e3,
_e3 = _Ucref − _x1,
_e3 = _Ucref −1Cx2 +
1Ci0:
ð24Þ
By choosing the following Lyapunov candidate,
V1 =12 e
23, ð25Þ
so that its derivative is
_V1 = e3 _e3 = e3 _Ucref −1Cx2 +
1Ci0
� �, ð26Þ
+
–
T
T
T
T
LF
C
iLF i0
+
–
UCLoadUABVC2
Figure 8: Basic schematic of a single-phase H-bridge inverter.
7International Journal of Photoenergy
where the second error is defined as
e4 = α − x2, ð27Þ
x2 = α − e4, ð28Þso that its derivative is
_e4 = _α − _x2 = _α −VC2LF
u2 +1Lx1: ð29Þ
From (26) and (28),
_V1 = e3 _Ucref −1Cα −
1Ce4 +
1Ci0
� �: ð30Þ
To get _V1 < 0, choose α such that
α = C _Ucref +1Ci0 + k3e3
� �, ð31Þ
where k3 > 0, then
_V1 = e3 _Ucref − _Ucref −1Ci0 − k3e3 +
1Ce4 +
1Ci0
� �, ð32Þ
_V1 = e3 −k3e3 +1Ce4
� �, ð33Þ
_V1 = e3 _e3: ð34ÞTherefore,
_e3 = −k3e3 +1Ce4: ð35Þ
Choosing a second Lyapunov function candidate,
V2 =V1 +12 e
24, ð36Þ
and its derivative is
_V2 = _V1 + e4 _e4: ð37Þ
Basing on equations (33) and (37), we can get _V2:
_V2 = −k3e23 +
1Ce3e4 + e4 _e4, ð38Þ
_V2 = −k3e23 + e4
1Ce3 + _e4
� �: ð39Þ
As a result, from (29) and (39), the expression of _V2 is
_V2 = −k3e23 + e4
1Ce3 + _α −
VC2LF
u2 +1LF
x1
� �: ð40Þ
The derivative of α from equation (31) is
_α = C €Ucref +1C__i0 + k3 _e3
� �: ð41Þ
Therefore, the final expression of _V2 is
_V2 = −k3e23 + z2
1Ce3 + C €Ucref + __i0 + Ck3 _e3 −
VC2LF
u2 +1LF
x1
� �:
ð42Þ
To get _V2 < 0, choose the control law “u2” for theinverter as defined in equation (43), where k4 > 0:
u2 =LF
VC2
1Ce3 + C €Ucref + __i0 + Ck3 _e3 +
1LF
x1 + k4e4
� �:
ð43Þ
By applying this control law to the PWM inverter inthe standalone system, the error e3 tends to zero becausethe derivatives of V1 and V2 are negative functions.According to the Lyapunov theory, choose k3 > 0 and k4 > 0to ensure good stability of the back-stepping control forthe inverter.
4. Simulation and Results
Figure 9 shows the circuit schematic of the PV inverter in thestandalone mode simulated in the Simulink platform. At thebeginning of the simulation, the resistor R1 is the only con-sumer of the solar energy generated by the PV array. After0.65 seconds, another resistive load R2 is added in parallelfor an additional 0.1 sec. At 0.75 sec, R2 is disconnected fromthe load and only R1 remains connected. Parameters of thePV module used in this study are listed in Table 1. The over-all system is tested and validated with parameters of the sys-tem which are listed in Table 2.
The type of PWM block used in Simulink for both theboost converter and the inverter is the PWM generatorDC-DC and single-phase half-bridge (2 pulses), respectively.The minimum and maximum values of the input PWM gen-erator for the inverter, which is the control law u2, are -1 and1, respectively.
The internal design blocs of back-stepping controlnumbers 1 and 2 are shown, respectively, in Figures 10 and11. The control law equation can be presented in the formof blocks.
The initial value of irradiance is set to 600W/m2; aftereach 0.2 s, it is changed to the following values: 200W/m2,700W/m2, 1000W/m2, and 900W/m2 in order to haveinstantaneous step values of irradiance in a short time fortesting the capability of the controller to track the suitablevalue of power generated by the PV array. During this simu-lation, the temperature is kept at 25°C. Figure 12 shows thesun irradiance profile that is applied to the PV array withthe aim of simulating the proposed system in different condi-tions and to examine the dynamic response of the two back-stepping controllers.
8 International Journal of Photoenergy
As shown in Figure 13, the MPPT algorithm block gener-ates successfully the corresponding peak voltage Vpvref to beused later by the back-stepping controller. At the beginningof simulation, the PV array voltage Vpv started at 0V valueand it attained the initial value of the Vpvref which is 121Vwith 2V of ripples in the transitory regime that lasted10ms. Until 0.2 s, the ripples of Vpv are 0.6V in the steadystate. At 0.2 s, there is a change of solar irradiance from600 to 200W/m2 which causes a slight decrease in refer-
ence voltage from 121V to 118V; also, the value of thePV array voltage Vpv successfully tracks the Vpvref . It canbe seen from Figure 13 that the PV array voltage Vpv is chan-ged due to the variation of solar irradiance and it follows thereference voltage Vpvref with some disturbances in the transi-tory regime due to the low performance of using the basicP&O algorithm.
The variation of irradiance causes little variation in thePV voltage and high variation in the PV current. Figure 14shows that the current generated by PV array ipv is variatedwith a big jump between two values against the voltage whichvaries with a small difference. At the start, the current valueipv is set at 5.2A and it became stable after 15ms from thebeginning at 4.85A in the steady state. At 0.4 s, the irradianceis changed from 200 to 700W/m2 which causes the rapidincrease in current value with some ripples in the transitoryregime that lasted only 6ms; the new current value is5.68A at 700W/m2.
Similarly, Figure 15 shows the photovoltaic power of thePV array during 1 s according to the solar irradiance profilechosen in Figure 12. At the beginning, the sun irradiance isset at 600W/m2 that means that the reference power gener-ated by the PV array is 589W. The power value is 0W, andthe back-stepping controller starts executing its role whichmakes the PV array generate the power for different atmo-spheric conditions. The transient phase contains little ripplesof 20W, and this controller can create a suitable control tothe boost converter to track the reference power. For this typeof power supply application, the overshoot of our MPPT sys-tem is around 40W, in the transient phase, in which the valuecan be considered as negligible compared to the total powerof the system which is 1 kW. The transient phase contains lit-tle ripples of 20W, and this controller can create a suitablecontrol to the boost converter to track the reference power.After 7ms from starting the simulation, the power generatedby the PV array is 588W. At 0.2 s, the irradiance changed itsvalue and it became 200W/m2 and automatically the power
Table 2: Parameters of the system elements.
Parameter Value
Ucref (RMS) 220V
FPWM 20 kHz
f 50Hz
R1 100ΩR2 100ΩOffset (MPPT bloc) 0.0005V
C 47e − 6 FC1 100e − 6 FC2 100e − 6 FLF 4:7e − 3HLB 3e − 3HT 25°C
Figure 9: Schematic of the proposed PV inverter system.
9International Journal of Photoenergy
Vpv
Vpvref
3
1
2
5
ipv
iLB
VC24
Derivative1
G3
G2G1+–
-K-
G4
Constant
1
Divide
Derivative3 G6
G5
Product
1
1Constant1U1
+–
++
+–
+
×
–
+–
–
+
–
-K-
-K-
-K-
-K-
-K-
×÷
Δ𝜇
Δt
Δ𝜇
Δt
Figure 10: Internal blocs of back-stepping control 1.
i0
iLF
UC
1
3
4
2Ucref
Integrator1 Integrator
Derivative2
G5
G4
G8
G11G10
G7
G6
G9 U2
1
G3
G2
G1
Δ𝜇
Δt
1
S
1
S
-K-
-K-
-K-
-K-
-K-
-K-
-K-
-K-
-K-
-K-
-K-
+
++
–
+
–+
–
+
+
+
+
+
+
+
Figure 11: Internal blocs of back-stepping control 2.
0200
300
400
500
Irra
dian
ce (W
/m2 )
600
700
900
800
1000
0.1Time (s)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 12: Sun irradiance profile.
10 International Journal of Photoenergy
0
20
40
60V
olta
ge (V
)
80
100
120
140
0 0.1
0 0.01
0.195
118
120
122
0.2 0.205
1120.395 0.4 0.405 0.41
0.6118120122124126128
0.65 0.7 0.75 0.8
Vpvref
Vpv
114116118120122
0.21
0.02115
120
125
130
Time (s)0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 13: PV array voltage with its reference.
0 0.1Time (s)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11
2
3
4
5
6
Curr
ent (
A)
7 0 0.45.55.65.75.85.9
0.402 0.404 0.406 0.408 0.41
7.40.65 0.7
7.67.8
88.2
4.6
4.8
5
5.2
0.005 0.01 0.015 0.02
8
9
ipv
Figure 14: PV array current.
00
100
200
300
400Pow
er (W
)
500
600
7000.005
540
560
580600
0.01 0.015
0.4680682684686688
0.45
0.6940
960
980
1000
0.65 0.7 0.75
Ppv
0.8
0.5 0.55 0.6
0.2191
191.5
192
0.25 0.3 0.35 0.4
800
900
1000
0.1Time (s)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 15: PV array power.
11International Journal of Photoenergy
will decrease; therefore, the new value of the power generatedby the PV array is 191.6W. At 0.4 s, the power value is 687Wand it corresponds to 700W/m2 of sun irradiance. For1000W/m2, the power increases and it becomes 978.2W, inthe last part or the last value of irradiance which is900W/m2. At 0.8 s, the sun irradiance changes from 1000to 900W/m2 which causes diminution in power. Figure 16shows a zoom of Ppv between 0.8 and 0.80006 s; therefore,the response time to attain the exact value of the power isaround 1ms. Figure 17 gives an idea about the maximumpower at 900W/m2 that is 881.2W. Referring to Figures 16and 17, the efficiency of the MPPT system is 99.93%. More-over, the MPP is successfully tracked by the controller whichverifies the high robustness and performance of the back-stepping control to generate the maximum power to the load.
Figure 18 shows the performance of the inverter with theproposed back-stepping controller. This figure representstwo waveforms which are the output voltage of the inverterat the terminal load UC and the sinusoidal signal referenceUcref . At the beginning, the irradiance value was 600W/m2
and the output voltage of the boost converter increased untilit became greater than 311V which is the nominal value ofthe input inverter.
Figure 19 presents a zoom of the output voltage of theinverter between 0 and 0.1 s to show that the inverter followsperfectly the reference voltage using the back-stepping con-trol. For our inverter, as mentioned in Figure 19, the responsetime to have a good form of the output voltage for electricalloads is 30ms at the beginning in which the power generatedby the PV array is 589W. Moreover, in this time, the RMSvalue of the output voltage of the inverter is less than thenorm 230V. Therefore, this transient phase has no effect onpower quality in electrical loads. There is distortion of outputvoltage UC until 40ms. This is due to insufficient voltage inthe input of the inverter and not to the back-stepping control.After 40ms from the beginning, the input voltage of theinverter becomes greater than 311V and the output voltageof inverter UC started to track the reference voltage Ucref .Moreover, with the input voltage of the inverter greater than311V, the response time of this controller is less than 15msthat ensures the high robustness of this controller. At 0.2 s,the solar irradiance became 200W/m2 which means lesspower generated by the PV array. Therefore, the boost con-verter was not able to provide a suitable voltage to theinverter and the voltage was less than 311V. For this reason,a voltage drop appeared in the output voltage of the inverter
0.8860
880
900Po
wer
(w)
920
940
960
980
Time (s)0.80001 0.80002 0.80003 0.80004 0.80005 0.80006
X: 0.8001Y: 880.6
Ppv
Figure 16: Zoom area of the PV array power.
X: 120.5Y: 881.2
00
100200300400500600700800900
10001100
50
Voltage (V)
0.9 kW/m2
150100
Pow
er (w
)
Figure 17: P-V characteristic of the PV array for 900W/m2.
12 International Journal of Photoenergy
as show in Figure 18 between 0.2 and 0.4 s. After 0.4 s, thesolar irradiance increased, and automatically, the powerincreased. Therefore, there is no issue with voltage dropdespite the addition of a second resistive load.
Figure 20 shows the current curve i0 of the AC loads. Thisfigure represents the pure sinusoidal waveform of the currentconsumed by loads. Between 0.2 and 0.4 s, there is a currentdrop due to the insufficient current produced by the PVarray. At 0.65 s, there are two loads in parallel. It can be seenfrom Figure 20 that the current increased to 6.2A as a max-imum value; after 0.75 s, it decreased to 3.1A as a maximumvalue. The controller of the second part of the proposedinverter is robust, and it shows the robustness and high per-formance of the chosen controller.
Figure 21 shows the frequency analysis of the inverter’s out-put voltage. The fundamental is about 310.7V at a frequency of50Hz, and the THD is about 0.78%. From Figures 18–21, wecan conclude that the inverter studied with the proposed
back-stepping control in the standalone mode has a low har-monic distortion, high conversion efficiency, strong controlperformance, and a high quality of sinusoidal waveform.
The efficiency of the MPPT system and the efficiency ofthe inverter are mentioned in Table 3 in different values ofsolar irradiation. As shown for different solar irradiance levels,our first controller of the MPPT system presents a high per-centage of efficiency and a good manner to track the powercompared with other works as mentioned in [12, 14]. Forthe second stage which is the inverter, it also presents goodand high efficiency, only that there is a certain limit of solarirradiance wherein the inverter cannot generate a good wave-form of voltage to loads. As shown, the efficiency in 200W/m2
is bad because the input power of the inverter is less than thethreshold power that must be generated by the PV array.
Table 4 shows the performance comparison of MPPTalgorithms between our system and other works. The 1msof the response time and the 99.93% of efficiency
0.01Time (s)
0 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1–400
–300
–200
–100Volta
ge (V
)
0
100
200
300
400
UC
Ucref
Figure 19: Output voltage inverter after filtering with its reference (zoom area).
0.10–400
–300
–200
–100Volta
ge (V
)
0
100
200
300
400
Time (s)0.6
UC
Ucref
0.2 0.3 0.4 0.5 0.7 0.8 0.9 1
Figure 18: Output voltage inverter after filtering with its reference.
13International Journal of Photoenergy
demonstrate the high performance and robustness of P&Owith back-stepping algorithm of our MPPT system.
In order to easily exhibit the excellence and to show theperformance of the proposed back-stepping in the standa-lone mode, the results of the PV inverter with other works[17, 29–33] are summarized in Table 5. As shown in this com-parison, our proposed system has good and high performancecontrol 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 ofenergy from photovoltaic modules to the use of this energyby electric charges. It is a dual function or double role betweentracking the maximum power and regulating the voltage tohave a sinusoidal waveform at the output of the inverter. How-ever, most works deal only with the DC to AC conversionwithout MPPT [34]. With a very low THD and a veryadvanced nonlinear controller, our system is the most practi-cal for the realization of photovoltaic inverters in the standa-lone mode.
5. Conclusion
A robust control scheme combined with a high performancePV inverter system has been presented in this paper. Simula-tion results show that the two stages of converter successfullyrespond to the two principal objectives which are, firstly, theextraction of the maximum power from the PV array with anefficiency of 99.93% and 1ms of response time to show the
Figure 21: Frequency analysis of the output voltage inverter.
Table 3: Efficiency of two stages of the whole system under differentsolar 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 96.88
14 International Journal of Photoenergy
fast dynamic response of our MPPT algorithm using theboost converter in order to avoid the use of the transformer.Secondly, we obtained a stable sinusoidal waveform of theoutput voltage of the inverter which is 220V with 0.78% ofthe THD and fixed 50Hz frequency. This performance isobtained by using a nonlinear back-stepping control whichcan quickly track the reference by having the error convergeto zero. This PV inverter system delivers the high qualitysinusoidal power to the AC load as long as the solar radiationis higher than 300W/m2.
Data Availability
There is no underlying data in the research article.
Disclosure
The authors would to thank the organizers of the Interna-tional Conference on Wireless Technologies, Embeddedand Intelligent Systems (WITS 2019) which was organizedon April 3-4, 2019, in Fez, Morocco. All accepted papersin this conference were published on IEEE Xplore; amongthose items accepted was our article titled “Control of Sin-gle Phase Inverter Using Back-Stepping in Stand-AloneMode” which was the basis to accomplish other work thatis presented in this paper.
Conflicts of Interest
The authors declare that there is no conflict of interestregarding the publication of this paper.
References
[1] B. R. Lin and J. Y. Dong, “New zero-voltage switching DC–DCconverter for renewable energy conversion systems,” IETPower Electronics, vol. 5, no. 4, pp. 393–400, 2012.
[2] S. Sharma and B. Singh, “Control of permanent magnet syn-chronous generator-based stand-alone wind energy conver-sion system,” IET Power Electronics, vol. 5, no. 8, pp. 1519–1526, 2012.
[3] Statista website, “World power consumption,” https://www.statista.com/statistics/280704/world-power-consumption/.
[4] M. Morales-Caporal, J. Rangel-Magdaleno, H. Peregrina-Bar-reto, and R. Morales-Caporal, “FPGA-in-the-loop simulationof a grid-connected photovoltaic system by using a predictivecontrol,” Electrical Engineering, vol. 100, no. 3, pp. 1327–1337, 2018.
[5] Y. Yang and F. Blaabjerg, “Overview of single-phase grid-connected photovoltaic systems,” in Renewable Energy Devicesand Systems with Simulations in MATLAB® and ANSYS®, F.Blaabjerg and D. M. Ionel, Eds., pp. 41–66, CRC Press, BocaRaton, FL, USA, 2017.
Table 4: Performance comparison of MPPT algorithms.
[6] A. H. Mollah, P. G. K. Panda, and P. P. KSaha, “Single phasegrid-connected inverter for photovoltaic system with maxi-mum power point tracking,” International Journal ofAdvanced Research in Electrical, Electronics and Instrumenta-tion Engineering, vol. 4, no. 2, pp. 648–655, 2015.
[7] L. Hassaine, Power Converters and Control of Grid-ConnectedPhotovoltaic Systems, Springer International Publishing,Cham, Switzerland, 2016.
[8] Z. Zeng, H. Yang, R. Zhao, and C. Cheng, “Topologies andcontrol strategies of multi-functional grid-connected invertersfor power quality enhancement: a comprehensive review,”Renewable and Sustainable Energy Reviews, vol. 24, pp. 223–270, 2013.
[9] R. Ayop and C. W. Tan, “Design of boost converter based onmaximum power point resistance for photovoltaic applica-tions,” Solar Energy, vol. 160, pp. 322–335, 2018.
[10] A. D. Martin and J. R. Vazquez, “MPPT algorithms compari-son in PV systems: P&O, PI, neuro-fuzzy and backsteppingcontrols,” in 2015 IEEE International Conference on IndustrialTechnology (ICIT), pp. 2841–2847, Seville, Spain, 2015.
[11] J. Ahmad, “A fractional open circuit voltage based maximumpower point tracker for photovoltaic arrays,” in 2010 2ndInternational Conference on Software Technology and Engi-neering, pp. 247–250, San Juan, PR, USA, 2010.
[12] S.Motahhir, A. El Hammoumi, and A. El Ghzizal, “Photovoltaicsystem with quantitative comparative between an improvedMPPT and existing INC and P&O methods under fast varyingof solar irradiation,” Energy Reports, vol. 4, pp. 341–350, 2018.
[13] Y. Cheddadi, F. Errahimi, and N. Es-sbai, “Design and verifica-tion of photovoltaic MPPT algorithm as an automotive-basedembedded software,” Solar Energy, vol. 171, pp. 414–425, 2018.
[14] M. Arsalan, R. Iftikhar, I. Ahmad, A. Hasan, K. Sabahat, andA. Javeria, “MPPT for photovoltaic system using nonlinearbackstepping controller with integral action,” Solar Energy,vol. 170, pp. 192–200, 2018.
[15] M. Kaouane, A. Boukhelifa, and A. Cheriti, “Regulated outputvoltage double switch Buck-Boost converter for photovoltaicenergy application,” International Journal of Hydrogen Energy,vol. 41, no. 45, pp. 20847–20857, 2016.
[16] A. Mohd, E. Ortjohann, W. Sinsukthavorn, M. Lingemann,N. Hamsic, and D. Morton, “Inverter-based distributed gener-ation control using droop/isochronous load sharing,” IFACProceedings Volumes, vol. 42, no. 9, pp. 368–373, 2009.
[17] R.-J. Wai, C.-Y. Lin, H.-N. Huang, and W.-C. Wu, “Design ofbackstepping control for high-performance inverter withstand-alone and grid-connected power-supply modes,” IETPower Electronics, vol. 6, no. 4, pp. 752–762, 2013.
[18] O. Diouri, F. Errahimi, and N. Es-Sbai, “Regulation of the out-put voltage of an inverter in case of load variation,” IOP Con-ference Series: Materials Science and Engineering, vol. 353,article 012021, 2018.
[19] L. Schirone, F. Celani, and M. Macellari, “Discrete-time con-trol for DC–AC converters based on sliding mode design,”IET Power Electronics, vol. 5, no. 6, pp. 833–840, 2012.
[20] D.-C. Lee and G.-M. Lee, “Linear control of inverter outputvoltage in overmodulation,” IEEE Transactions on IndustrialElectronics, vol. 44, no. 4, pp. 590–592, 1997.
[21] I. S. Kim and M. J. Youn, “Variable-structure observer forsolar-array current estimation in a photovoltaic power-generation system,” IEE Proceedings - Electric Power Applica-tions, vol. 152, no. 4, pp. 953–959, 2005.
[22] A. Hasanzadeh, C. S. Edrington, H. Mokhtari,B. Maghsoudlou, and F. Fleming, “Multi-loop linear resonantvoltage source inverter controller design for distorted loadsusing the linear quadratic regulator method,” IET Power Elec-tronics, vol. 5, no. 6, pp. 841–851, 2012.
[23] H. Komurcugil, “Steady-state analysis and passivity-basedcontrol of single-phase PWM current-source inverters,” IEEETransactions on Industrial Electronics, vol. 57, no. 3,pp. 1026–1030, 2010.
[24] A. K. Zadeh, L. I. Kashkooli, and S. A. Mirzaee, “Designing apower inverter and comparing back-stepping, sliding-modeand fuzzy controllers for a single-phase inverter in an emergencypower supply,” Ciência e Natura, vol. 37, pp. 175–181, 2015.
[25] E. Kolbasi and M. Seker, “Nonlinear robust backstepping con-trol method approach for single phase inverter,” in 2016 21stInternational Conference on Methods and Models in Automa-tion and Robotics (MMAR), pp. 954–958, Miedzyzdroje,Poland, 2016.
[26] Z. Ji, W.Wu, Y. Feng, and G. Zhang, “Constructing the Lyapu-nov function through solving positive dimensional polynomialsystem,” Journal of Applied Mathematics, vol. 2013, Article ID859578, 5 pages, 2013.
[27] A. Vyas Ojha and A. Khandelwal, “Control of non-linear sys-tem using backstepping,” International Journal of Research inEngineering and Technology, vol. 4, no. 5, pp. 606–610, 2015.
[28] L. Ammeh, H. E. Fadil, A. Yahya, K. Ouhaddach, F. Giri, andT. Ahmed-Ali, “A nonlinear backstepping controller forinverters used in microgrids,” IFAC-PapersOnLine, vol. 50,no. 1, pp. 7032–7037, 2017.
[29] F.-S. Kang, S.-J. Park, S. E. Cho, C.-U. Kim, and T. Ise, “Mul-tilevel PWM inverters suitable for the use of stand-alone pho-tovoltaic power systems,” IEEE Transactions on EnergyConversion, vol. 20, no. 4, pp. 906–915, 2005.
[30] H. A. Sher, A. A. Rizvi, K. E. Addoweesh, and K. Al-Haddad,“A single-stage stand-alone photovoltaic energy system withhigh tracking efficiency,” IEEE Transactions on SustainableEnergy, vol. 8, no. 2, pp. 755–762, 2017.
[31] P. K. Peter and V. Agarwal, “Photovoltaic module-integratedstand-alone single-stage switched capacitor inverter with max-imum power point tracking,” IEEE Transactions on PowerElectronics, vol. 32, no. 5, pp. 3571–3584, 2017.
[32] S. K. Hosseini, M. Mehrasa, S. Taheri, M. Rezanejad,E. Pouresmaeil, and J. P. S. Catalao, “A control technique foroperation of single-phase converters in stand-alone operatingmode,” in 2016 IEEE Electrical Power and Energy Conference(EPEC), pp. 1–6, Ottawa, ON, Canada, 2016.
[33] M. Mehrasa, M. Rezanejhad, E. Pouresmaeil, J. P. S. Catalao,and S. Zabihi, “Analysis and control of single-phase convertersfor integration of small-scaled renewable energy sources intothe power grid,” in 2016 7th Power Electronics and Drive Sys-tems Technologies Conference (PEDSTC), pp. 384–389, Tehran,Iran, 2016.
[34] O. Diouri, N. Es-Sbai, F. Errahimi, A. Gaga, and C. Alaoui,“Control of single phase inverter using back-stepping instand-alone mode,” in 2019 International Conference on Wire-less Technologies, Embedded and Intelligent Systems (WITS),pp. 1–6, Fez, Morocco, 2019.
16 International Journal of Photoenergy
TribologyAdvances in
Hindawiwww.hindawi.com Volume 2018
Hindawiwww.hindawi.com Volume 2018
International Journal ofInternational Journal ofPhotoenergy
Hindawiwww.hindawi.com Volume 2018
Journal of
Chemistry
Hindawiwww.hindawi.com Volume 2018
Advances inPhysical Chemistry
Hindawiwww.hindawi.com
Analytical Methods in Chemistry
Journal of
Volume 2018
Bioinorganic Chemistry and ApplicationsHindawiwww.hindawi.com Volume 2018
