Nonlinear Robust Backstepping Control for Three-Phase …downloads.hindawi.com/journals/mpe/aip/3824628.pdf · probable mismatch between the Maximum Power Points ... of the PV module

Research ArticleNonlinear Robust Backstepping Control for Three-PhaseGrid-Connected PV Systems

Mohamed Habib Boujmil1 Afef Badis 2 andMohamed Nejib Mansouri2

1Higher Institute of the Technological Studies of Nabeul Nabeul Tunisia2Electronics and Microelectronics Laboratory (E120583E) The National Engineering School of Monastir (ENIM)University of Monastir Monastir Tunisia

Correspondence should be addressed to Afef Badis badisafefgmailcom

Received 30 January 2018 Revised 3 May 2018 Accepted 30 May 2018 Published 25 June 2018

Academic Editor Viktor Avrutin

Copyright copy 2018 MohamedHabibBoujmil et alThis is an open access article distributed under theCreativeCommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

This paper proposes a cascade control structure for three-phase grid-connected Photovoltaic (PV) systemsThe PV system consistsof a PV Generator DCDC converter a DC link a DCAC fully controlled inverter and the main grid For the control process anew control strategy using nonlinear Backstepping technique is developed This strategy comprises three targets namely DCDCconverter control tight control of the DC link voltage and delivering the desired output power to the active grid with unity powerfactor (PF) Moreover the control process relies mainly on the formulation of stability based on Lyapunov functions Maximizingthe energy reproduced from a solar power generation system is investigated as well by using the Perturb and Observe (PampO)algorithm The Energetic Macroscopic Representation (EMR) and its reverse Maximum Control Structure (MCS) are used toprovide respectively an instantaneous average model and a cascade control structureThe robust proposed control strategy adaptswell to the cascade control technique Simulations have been conducted using MatlabSimulink software in order to illustrate thevalidity and robustness of the proposed technique under different operating conditions namely abrupt changingweather conditionsudden parametric variations and voltage dips and when facing measurement uncertainties The problem of controlling the grid-connected PV system is addressed and dealt by using the nonlinear Backstepping control

1 Introduction

Solar Photovoltaic (PV) energy is a potential and envi-ronmentally friendly resource of energy which has becomewidely explored till date owing to its omnipresence availabil-ity free gas emission and reduced maintenance cost In factgases emitted from the combustion of conventional fuel havedramatic drawbacks on living organs human health and theOzone layer [1 2]

Moreover the PV output power is always changing withfast-changing weather conditions eg solar irradiance leveland temperature Thus adopting a Maximum Power PointTracking (MPPT) technique is imperative as there is a prob-able mismatch between the Maximum Power Points (MPP)of the PV module and the load characteristics Thanks to theMPPT the effectiveness of the energy extraction is definitelyimproved [3] ie the total number of required PV panelsis decreased which reduces the total cost [4] Perturb and

Observe (PampO) method is the most widely used algorithmit portrays a simplicity of use and easy implementation [5 6]

Therefore static converters (SCV) must be designed andcontrolled efficaciously to maximize the overall efficiency ofthe PV system in medium-voltage (MV) and low-voltage(LV) Usually the solar energy is exploited either for stand-alone systems or for grid-connected Photovoltaic (PV) sys-tems Several papers in the literature are targeting the issueof grid-connected PV Generator (PVG) [7 8] Differentconverter structures have been proposed andor studied inorder to interface renewable energy systems in both grid-connected and stand-alone application [9 10] In general theboosting and the inverting stages in a grid-connected PVsystem present the two stages of the power conversion systemfor optimum power transfer Because of the nonlinearity ofthe PV power systems and the unpredictable intrinsic andatmospheric changes the operating point is always varyingdue to the control unit (DCDC converter andor inverter)

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 3824628 13 pageshttpsdoiorg10115520183824628

2 Mathematical Problems in Engineering

and the parametric errors Thus the use of robust controllaws becomes a required and challenging task for ensuring thestabilization and the good tracking In fact there are alreadya lot of researches [11ndash15] targeting the conversion chain ofgrid-connected PV system However the proposed schemesin the latter studies require the detailed evaluations namely

(i) control the DCDC converter in order to force thePV array to operate at the Maximum Power Points(MPP)

(ii) maintain the DC link voltage at the desired constantvalue

(iii) deliver the desired output power to the active gridwith unity power factor (PF)

A considerable progress has been made over last decadein optimization techniques in order to perform the threefoldobjectives Among these methods classical PID controller isusually used in industry and literature because of its robust-ness low cost and ease of implementationThe conventionalcontrol strategies (PI regulator or state regulator) producegood results in linear systems [16] However for nonlinearsystems or with varying parameters those methods becomeinsufficient and unreliable especially when the performancerequirements of the system are rigorousThus it is imperativeto choose the adequate control strategy which is insensitiveto parameter variations perturbations and nonlinearityThispaper proposes a Backstepping based technique to design asuitable control system for the grid-connected PV systemThe designed control technique ensures the optimal energytransfer to the grid via sharing active and reactive power intothe grid regardless of the atmospheric changes and paramet-ric uncertainties The desired DC link voltage is maintainedas well by using Backstepping approach to control the staticboost converter based on Matrix Topology (MT) MT isknown to be themost industrially encountered inmost powerelectronics systems since it does not need energy storagecomponents Control Lyapunov functions are formulated atdifferent stages of the design process to evaluate the stabilityof the designed controller

The PV systemrsquos operation and the development of itsappropriate control techniques require as a compulsorya basic representation of the entire system by taking intoconsideration the integrality and the causality between everyone of systemrsquos components

The Energetic Macroscopic Representation (EMR) is agraphical portrayal of the system which has been createdin 2000 by the L2EP at the University of Lille in FranceIt is an extremely basic representation with a gathering ofstandards and elements [17 18] Since then EMR has beenused as a part of manymultidomainmultisources [19 20] forrepresentation andmodeling of different systems namely PVand wind energy conversion systems Electric Vehicles (EV)[21]

In this paper EMR and its reverse Maximum ControlStructure (MCS) are both used to provide an instantaneousaveragemodel and a control structure based on cascade loopsThe chosen structure gives promising dynamic performances

and limits each state variable to manage the system incomplete safety

In the present study the modeling of the Low-voltagegrid-connected PV system is developed in Section 2 Sec-tion 3 encompasses the design of the proposed controllerbased on cascade Backstepping strategy In Section 4 sim-ulation results of the proposed control system are discussedfor various operating conditions Comments supporting theperformance and robustness of the Backstepping controlapproach are given Finally in Section 5 the conclusion isdrawn and followed by references

2 Modeling of the Grid-ConnectedPhotovoltaic System

The following section presents the equations used for model-ing each component in the PV conversion chain The princi-ple mission of the chosen conversion system is to extract themaximum active power through a boost converter operatingwith a suitable MPPT and managing as well the activeand reactive power injected into the grid via the inverterThe system is comprised of simple subsystems (pictograms)related together These pictograms portray every element inthe system by functions as shown in Table 1 [20] The EMRfor all components are interconnected in order to frame theentire system EMR with respecting the integral causality andfollowing the action-reaction principle MCS which allowsthe control loop modeling is deduced by inversion of theEMR [20] The entire grid-connected PV system is shows inFigure 1 while the EMR of the entire system is depicted inFigure 2

The current electrical source (green oval) represents thePVGwhich is the association of series andor parallel PV cellsin order to raise the current voltage and power An idealPhotovoltaic cell is equivalent to a power source shunted by adiode as shown in Figure 3 [22]

The PV model is mathematically modeled using

119868119901V = 119868119901ℎ minus 119868119904 [exp( 119880119901V119873119904119899119870119861119879) minus 1] (1)

where 119868119901ℎ = 119873119901 times 119894119901ℎ is photocurrent Np parallel cells119868119901V = 119873119901 times 119894119901V is current supplied by Np parallel cells119868119904 = 119873119901 times 119894119904 is reverse saturation current ofNp 119889119894119900119889119890 119894119899 119901119886119903119886119897119897119890119897V119879 = 119899119870119861119879119902 is the thermodynamic potential119873119901 is number of cells in parallel119873119904 is number of cells in series119902 is 119890119897119890119888119905119903119900119899 119888ℎ119886119903119892119890(1 610minus19C)119870119861 is Boltzman Constant (1 3810minus23jK)119879 is junction temperature Kn is ideality factor of the picture of the solar cellincluding between 1 and 5 in practice

Mathematical Problems in Engineering 3

Table 1 Elements of EMR and of control

Sourceelement

Accumula-tion

element

Indirectinversion

Mono-physical

conversionelement

Mono-physicalcouplingelement

Directinversion

Multi-physical

conversionelement

Multi-physicalcouplingelement

Couplinginversion

Figure 1 The grid-connected PV system chain

PVG

Command

Photovoltaic Generator CL Filter Boostconverter

DC linkinverter

threendash phase Filter three_phasegrid

pvu

pvi Li

Limu

mi

mridc

u

dcu m_dqv r_dqi

r_123v

r_123i

MPPT

upv_ref iL_ref um_ref

REGm dq_REGm

Maximum control structure PVG side

Griddq

12

Command

-

-

dc_reacutefu

dcu

r_dqv

r_dq_refi

pvu

CPQ

Process

Control

Maximum control structure grid side

imr_reacutef

Figure 2 The EMR and its reverse MCS of the overall system


phipvu

pvi

Load

Figure 3 Equivalent circuit model of a solar PV cell

The PVG pictogram delivers a current iPV and receives byreaction with the system the DC bus voltage upv of the LCfilter The inductor and capacitor filters store energy Theyare represented by two accumulation elements whose statevariables are the current 119894119871 for the inductor and the voltage119906119901V for the capacitor DCDC and DCAC converters (basedon MT without energy storage) are represented by squarepictograms Both converters are modeled in average value(the switching functions are replaced by duty cycles) [21]

A capacitor C is used for controlling the PV outputvoltage It is modeled by means of the following equation

1198771 997888rarr 119862119889119906119901V119889119905 + 119906119901V119877 = 119894119901V minus 119894119871 (2)

An inductor L is used to apply the source alternating ruleIt can be modeled by the following differential equation1198772 997888rarr 119871119889119894119871119889119905 + 119903119894119871 = 119906119901V minus 119906119898ℎ (3)

The aim of the following subsection consists in modeling ofthe DCDC boost converter with matrix based topology Infact the DCDC matrix based converter is a static devisewhich is initially proposed by Gyugi [23] in 1976 Sincethen most of the published researches have dealt with three-phase circuit topologies [24ndash26] The basic problem to beaddressed is to control the unregulated DC input voltage ofthe converter in order to reach the desired output voltageThe input of the boost converter consists of the PVG and theCL filter This input is equivalent to a current source Whilethe boost output is connected to the DC link voltage whichis equivalent to a voltage source The matrix based booststructure is depicted in Figure 4

The connection matrix is developed by

[119865] = [11989111 1198911211989121 11989122] (4)

and the conversion matrix is expressed by[119872] = [(11989111 minus 11989112)] (5)

In such a case according to [27] the conversion matrixis reduced to a scalar representing the unique conversionfunction 119898 So the modulated voltage 119906119898ℎ and current 119894119898ℎare shown below 1198771198981 997888rarr 119906119898 = 119898ℎ times 1199061198891198881198771198982 997888rarr 119894119898 = 119898ℎ times 119894119871 (6)

Figure 4 Operative system of the matrix based boost converter

Figure 5 The EMR and its reverse MCS of the PVG side

119898119892 is the command variable of the boost converterThe EMRof the PVG side is illustrated in Figure 5

For the grid side modeling a capacitor C1 is connected tothe DC link for the purpose of controlling the voltage appliedto the input of the three-phase inverter and maintaining itequal to a preset valueThe voltage acrossC1 can be describedby the following differential equation

1198773 997888rarr 1198621 119889119906119889119888119889119905 + 119906119889119888119877 = 119894119898 minus 119894119898119903 (7)

The input can be considered as a voltage source At theinverter output an inductive filter is used to connect theinverter to the three-phase active grid The grid and the filterall together are equivalent to a three-phase current source Athree-phase matrix converter topology shown in Figure 6 isused

Instead of using the abc-frame the Park transformationP() is used in order to transform currents and voltages totheir equivalent components in the dq reference frame to getconstant values which are easy to track

The dynamic model is transformed from abc-frame to dqreference frame and can be expressed by relations (7) to (15)of the grid side and we take the angular frequency 120596 = 2120587119891

The connection matrix of the inverter is developed by

[119865] = [11989111 11989112 1198911311989121 11989122 11989123] (8)


Figure 6 Operative system of the matrix based tree-phase inverter

and the conversion matrix [M] is expressed by[119872] = [(11989111 minus 11989113) (11989112 minus 11989113)] = [1198981 1198982] (9)

The simple voltages and currents modulated by theinverter in the park reference can be expressed by

1198771198983 997888rarr [V119898119889V119898119902

] = 1199061198891198882 [119898119889119898119902]1198771198984 997888rarr 119894119898119900 = 12 (119898119889119894119903119889 + 119898119902119894119903119902)(10)

1198774 997888rarr [V119898119889V119898119902

] = [119875 (120579)] [V1199031V1199032] (11)

1198775 997888rarr [119894119903119889119894119903119902] = [119875 (120579)] [11989411990311198941199032] (12)

where

[119875 (120579)] = radic23 [[[cos (120579) cos(120579 minus 21205873 )minus sin (120579) minus sin(120579 minus 21205873 )]]] (13)

1198776 997888rarr (1198711 119889119889119905 + 1199031)[119894119903119889119894119903119902]= [V119898119889

V119898119902] minus [V119903119889

V119903119902] + [ 0 1198711120596minus1198711120596 0 ][119894119903119889119894119903119902]

(14)

P and Q active and reactive powers respectively are com-puted using the conventional instantaneous power definitionin 119889119902 system [8] as shown in

1198777 997888rarr [119875119876] = [V119898119889 V119898119902V119898119902 minusV119898119889][119894119903119889119894119903119902] (15)

Figure 7 The EMR and its reverse MCS of the grid side

The representation of DC link is an accumulation elementThe grid with the filter are represented by source element andaccumulation element respectively as portrayed in Figure 7

where

(i) r1 L1 are the grid filter(ii) 119894119903119889 and 119894119903119902 are the 119889119902 components of the line current(iii) V119903119889 and V119903119902 are the 119889119902 components of the grid voltages(iv) V119898119889 and V119898119902 are the modulated voltages generated

at the front end of the converter and considered ascontrol laws

(v) 120596 is the angular velocity of the grid voltages

Based on the mathematical models of the PVG side in (1)ndash(6)and the grid side as described by (7)ndash(15) the nonlinearrobust Backstepping control process is elaborately discussedin the next section

3 Backstepping Based Control ofthe Three-Phase Grid-Connected PVG

The fundamental idea of Backstepping consists in conceiving(for each subsystem) a virtual control law A Lyapunovfunction which ensures the stability is developed in orderto exploit it later as a reference for the immediately super-posed subsystem until the accurate command system directlyinvolved in the static converter is obtainedMatrix convertersare used instead of conventional converters using energystorage components

31 Backstepping Based Cascade Control Sizing of the PVGSide One of themain targets of this work is tomake the PVGoperate at its operating point MPPT To do so it is necessaryto refer to the MCS of the PVG side [28] The voltage 119906119901V ischosen as an output and the control chain consists of twocascade controllers As well it is noticed that the controller119862119906 not only insures the control of the voltage 119906119901V but alsoprovides the current reference 119894119871 119903119890119891 (a virtual command) for


the controller 119862119894 which insures the control of the inductancecurrent 119894119871 and provides the real command 119898119892 of the boostconverterThe relations associatedwith the EMR on the PVGside can be redefined as follows

(1198781) 997904rArr 119889119906119901V119889119905 = 119901V = minus 1119877119862119906119901V + 1119862119894119901V minus 1119862119894119871(1198782) 997904rArr 119889119894119871119889119905 = 119894119871 = minus 119903119871119894119871 + 1119871119906119901V minus 1119871119906119898 (16)

while119906119901V and 119894119871 are the state variables and119898ℎ is the commandvariable of the boost converter As the subsystem is a second-order system the design is performed in two stages

Stage 1 (voltage loop across the PVG) The subsystem (S1)described via the relation (16) is considered where the voltage119906119901V is taken as an output and the state variable 119894119871 is treatedas a virtual control variable This first stage is dedicatedto identify the tracking error e1 which corresponds thedifference between the output PVvoltage119906119901V and its referenceupv ref obtained from the MPPT bloc1198901 = 119906119901V minus 119906119901V 119903119890119891 (17)

According to relations (16) and (17) the dynamic equa-tion of the error is deduced1198901 = 119901V minus 119901V 119903119890119891 = minus 1119877119862119906119901V + 1119862119894119901V minus 1119862119894119871 minus 119901V 119903119890119891 (18)

In this step the control Lyapunov function is chosen as1198811 = 1211989021 (19)

Its derivative can be written as follows1 = 1198901 1198901 = 1198901 (minus 1119877119862119906119901V + 1119862119894119901V minus 1119862119894119871 minus 119901V 119903119890119891) (20)

Themain objective to be reached consists inmaking the errore1 converge to zero and ensure the stability of 1198901 by taking1 lt 0 To do so 119894119871 is chosen as a stabilizing function (ie119894119871 = 119894119871 119903119890119891) Thus we set1198901 = minus11989611198901 119886V119890119888 1198961 gt 0 (21)

And relation (18) can be written as followsminus 1119877119862119906119901V + 1119862119894119901V minus 1119862119894119871 119903119890119891 minus 119901V 119903119890119891 = minus11989611198901 (22)

From relation (22) the virtual command is deduced such as119894119871 119903119890119891 = 119862(11989611198901 minus 1119877119862119906119901V + 1119862119894119901V minus 119901V 119903119890119891) (23)

Stage 2 (current loop 119894119871) In this stage the subsystem S2 of thecurrent loop iL is considered As it is seen before the designof this stage consists in forcing the current iL to follow itsreference 119894119871 119903119890119891 The tracking error e2 is defined by1198902 = 119894119871 minus 119894119871 119903119890119891 (24)

According to (16) and (24) the dynamic equations of the errore2 are deduced1198902 = 119894119871 minus 119894119871 119903119890119891 = minus 119903119871119894119871 + 1119871119906119901V minus 1119871119906119898 minus 119894119871 119903119890119891 (25)

The subsystem of the PVG side consists of two second-ordersubsystems (S1) and (S2) According to relations (18) (24)and (25) we obtain the error system (e1 e2)1198902 = 119894119871 minus 119894119871 119903119890119891 997904rArr 119894119871 = 1198902 + 119894119871 1199031198901198911198901 = minus 1119877119862119906119901V + 1119862119894119901V minus 1119862 (1198902 + 119894119871 119903119890119891) minus 119901V 119903119890119891

1198902 = minus 119903119871119894119871 + 1119871119906119901V minus 1119871119906119898 minus 119894119871 119903119890119891(26)

The quadratic function of Lyapunov is applied in (27)

1198812 (1198901 1198902) = 1198811 + 1211989022 = 1211989021 + 1211989022 (27)

whose derivative is written below after an elementary calcu-lation2 (1198901 1198902)= 1198901(minus 1119877119862119906119901V + 1119862119894119901V minus 1119862119894119871 119903119890119891 minus 119901V 119903119890119891)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

minus11989611198901+ 1198902(minus 119903119871 119894119871 + 1119871119906119901V minus 1119871119906119898 minus 119894119871 119903119890119891 minus 11198621198901)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟minus11989621198902

(28)

K2 being a positive constant is defined to guarantee thenegativity of V2 Besides for this stage it is essential to makethe error e2 converge to zero in these conditions the choiceof the real command119898ℎ 119903119890119892 becomes evident Using (16) and(28) we obtain119898ℎ 119903119890119892 = 119871119906119889119888 (11989621198902 minus 119903119871119894119871 + 1119871119906119901V minus 119894119871 119903119890119891 minus 11198621198901) (29)

Thanks to this choice the derivative of the control Lyapunovfunction is reduced to2 = minus11989621198902 minus 11989611198901 (30)2 can be negative definite (2 lt 0) or semidefinite 2 le 0)which proves the asymptotic stability towards the origin ofthe subsystem (S2)

32 Backstepping Based Cascade Control Sizing on the GridSide Referring to the MCS of the grid side in Figure 7 it isnoticed that according to the control chain highlighted ingreen the DC link voltage 119906119889119888 is considered as an outputThe controller C4 not only ensures the voltage control 119906119889119888but also provides the reference current 119894119898119900 119903119890119891 Afterwardsthe latter reference current is used by the block ldquoControlP-Qrdquo in order to determine the reference currents 119868119903 119889119902 ofthe multivariable controller C3 which in turn monitors the


currents injected into the grid With this intention it mustprovide the main control commands (119898119892119889 119903119890119892 and119898119892119902 119903119890119892) tobe applied to the three-phase inverter

The DC voltage controller is exploited to produce thereference current Its target consists on keeping the voltageconstant on theDC sideThe current loop is considered as theinner loop while the DC voltage loop presents the outer loopThe DC link voltage is controlled by means of the converterside DC current as follows

(1198783) 997904rArr 119889119906119889119888119889119905 = 119889119888 = minus 111987711198621 119906119889119888 + 11198621 119894119898ℎ minus 11198621 119894119898119900 (31)

While the current controllers are used to achieve the trackingof the grid currents 119868119903 119889119902 the control laws are obtained in (32)as follows(1198784)997904rArr

119889119894119903119889119889119905 = 119894119903119889 = 11198711 (minus1199031119894119903119889 + V119898119889 minus V119903119889 + 1198711120596119894119903119902)119889119894119903119902119889119905 = 119894119903119902 = 11198711 (minus1199031119894119903119902 + V119898119902 minus V119903119902 minus 1198711120596119894119903119889) (32)

(i) 119898119892119889 119903119890119892 and 119898119892119902 119903119890119892 are the three-phase invertercommand variables

As the grid side subsystem is a second-order system its designis performed in two stages

Stage 1 (DC link voltage loop) The subsystem (S3) describedby relation (31) is considered and the DC link voltage isdefined as an output while the current imo is treated as avirtual command This step consists in identifying the errore3 as the difference between the DC link voltage 119906119889119888 and itsreference 119906119889119888 119903119890119891 such as1198903 = 119906119889119888 minus 119906119889119888 119903119890119891 (33)

The same structure is kept and the Backstepping cascadecontrol is done according to the same process established onthe PVG side The relation of the virtual command imo ref iswritten by

119894119898119900 119903119890119891 = 1198621 (11989631198903 minus 111987711198621 119906119889119888 + 11198621 119894119898ℎ minus 119889119888 119903119890119891) (34)

Stage 2 (grid current loop) According to the MCS in Figures4 and 8 the currents 119894119903119889 and 119894119903119902 are considered as outputs andthe voltages V119898119889 and V119898119902 are treated as virtual commandsAs a multivariable controller C4 is used the dq-axis currenttracking errors are respectively identified as the differencebetween current 119894119903119889 and 119894119903119902 and their references such as119890119889 = 119894119903119889 minus 119894119903119889 119903119890119891 997904rArr 119894119903119889 = 119890119889 + 119894119903119889 119903119890119891119890119902 = 119894119903119902 minus 119894119903119902 119903119890119891 997904rArr 119894119903119902 = 119890119902 + 119894119903119902 119903119890119891 (35)

Figure 8 Equivalent scheme of the grid side Photovoltaic installa-tion in the dq frame

According to (32) and (35) we obtain the dynamic equationsof the errors 119890119889 and 119890119902

119890119889 = 11198711 (minus1199031119894119903119889 + V119898119889 minus V119903119889 + 1198711120596119894119903119902) minus 119894119903119889 119903119890119891119890119902 = 11198711 (minus1199031119894119903119902 + V119898119902 minus V119903119902 minus 1198711120596119894119903119889) minus 119894119903119902 119903119890119891 (36)

Taking into account relations (10) (35) and (36) and thedynamic equation of the e3 error we obtain after an elemen-tary calculation the grid side subsystem relation formed by(S3) and (S4) within the errors space (e3 ed eq)

1198903 = minus 111987711198621 119906119889119888 + 11198621 119894119898ℎ minus 11198621 12 (119898119889119894119903119889 + 119898119902119894119903119902)minus 119889119888 119903119890119891 (37)

Then 119894119903119889 and 119894119903119902 of relation (37) are replaced by their expres-sions obtained in the relation (35) we obtain the followingrelation1198903= minus11989631198903⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞minus 111987711198621 119906119889119888 + 11198621 119894119898ℎ minus 11198621 12(119898119889119894119903119889 119903119890119891+119898119902119894119903119902 119903119890119891)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119894119898119900 119903119890119891

minus 119889119888 119903119890119891minus 121198621 (119898119889119890119889 + 119898119902119890119902)

(38)

the latter equation can be written in a simpler form

1198903 = minus11989631198903 minus 121198621 (119898119889119890119889 + 119898119902119890119902) (39)

The design of this step consists in forcing the current 119894119903119889 and119894119903119902 injected to the grid to follow their references namely toforce the errors 119890119889 and 119890119902 to converge towards zero In thiscase the quadratic function of Lyapunov V3 is increased bytwo terms 1198814 = 1198813 + 119881119889 + 119881119902 = 1211989023 + 121198902119889 + 121198902119902 (40)


when (40) is derived we obtain4 = 1198903 1198903 + 119890119889 119890119889 + 119890119902 119890119902 (41)

By introducing the dynamic equation of the error e3 and theexpression (36) in the relation (41) and after a little long butelementary calculation (41) becomes as shown below

4 = minus119896311989023 + 119890119889( 11198711 (minus119903119894119903119889 + V119898119889 minus V119903119889 + 1198711120596119894119903119902) minus 119890321198621119906119889119888 V119898119889 119903119890119891 minus 119894119903119889 119903119890119891⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟minus119896119889119890119889

)+ 119890119902( 11198711 (minus119903119894119903119902 + V119898119902 minus V119903119902 minus 1198711120596119894119903119889) minus 119890321198621119906119889119888 V119898119902 119903119890119891 minus 119894119903119902 119903119890119891⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

minus119896119902119890119902

)(42)

A wise choice of the tensions V119898119889 and V119898119902 would make119889 and 119902 respectively negative and would ensure the sta-bility at the origin of the subsystem on the grid side In thiscontext V119898119889 and V119898119902 are considered as virtual commandswith their references V119898119889 119903119890119891 and V119898119902 119903119890119891 respectively

V119898119889 119903119890119891 = 2119862121198621119906119889119888 minus 1198903 (minus119896119889119890119889+ 11198711 (119903119894119903119889 + V119903119889 minus 1198711120596119894119903119902) + 119894119903119889 119903119890119891)V119898119902 119903119890119891 = 2119862121198621119906119889119888 minus 1198903 (minus119896119902119890119902+ 11198711 (119903119894119903119902 + V119903119902 + 1198711120596119894119903119889) + 119894119903119902 119903119890119891)

(43)

The main target of this stage is to control the grid currentswhich are needed to determine the control commands119898119889 119903119890119892and 119898119902 119903119890119892 to be applied at the entrance of the three-phaseinverter

119898119889 119903119890119892 = V119898119889 119903119890119891119906119889119888119898119902 119903119890119892 = V119898119902 119903119890119891119906119889119888 (44)

4 Results of Simulation

The instantaneous average model of the overall systemis developed under the software package MatlabSimulinkenvironment The results of simulation are carried out usingthe following conditions

119875 = 1119896119882119862 = 220120583119865119877 = 100119896Ω

119871 = 231198981198671198621 = 50001205831198651198771 = 10119896Ω1199031 = 00002Ω1198711 = 1119898119867119880119903 = 380119881119891 = 50119867119911(45)

and with the following control parameter values which areacquired using trial and error for the purpose of satisfying thealreadymentioned theoretical conditions during the previoussection 119896119906119901V = 76119896119894119871 = 104119896119906119889119888 = 2000119896119889 = 119896119902 = 105

(46)

The performances of the designed nonlinear controllerwill be evaluated by simulation on a three-phase low-voltagegrid-connected PV system under different operating scenar-ios

Case 1 (control under sudden irradiation variations) Theirradiation can quickly change by environmental condi-tions Four irradiation steps were simulated and irradiancegoes from 1000Wm2 to 800Wm2 then it steps down to600Wm2 and finally steps up to 900Wm2 Under constanttemperature when a step change of irradiance happens theability ofMPP tracking is demonstrated for the PampOmethodAs a result the power delivered into the gridwill be optimizedsince the maximum PV current and voltage are extractedFigures 9 and 10 show the results obtained with the PampOmethod They show the PampO good tracking in case of fast-changing conditions


(a)

0

05

1

15

2

25

3

35

4

45

5PV

Cur

rent

(A)

50 100 150 200 250 3000PV Voltage (V)

1000 WＧ2

900 WＧ2

800 WＧ2

600 WＧ2

Figure 9 I-V curve with PampO

(b)

50 100 150 200 250 3000PV Voltage (V)

0

200

400

600

800

1000

1200

PV P

ower

(W)

Figure 10 P-V curve with PampO

Furthermore the perfect follow-up of the PV voltage andits reference is depicted in Figure 11

The reference DC link voltage is fixed at 700V TheDCDC boost converter is used to maintain the DC linkvoltage at the desired value through the DC link capacitorIn fact it is noticed from Figure 12 that the changing solarirradiation has no significant effect on the DC link voltageand it remains at 700V

In Figures 13 and 14 the current loop controller isvalidated since the measured grid currents are trackingperfectly their references and vary significantly whenever a

Upv

230240250260270280290300310

PV V

olta

ge (V

)

235240245250

1 15 25 305 2Time (seconds)

５ＪＰＬ

14

145

15

155

16

165

17

Figure 11 Upv voltage reference and measured

(b)

2

7007000002

7007000002

699

6992

6994

6996

6998

700

7002

7004

7006

7008

701

Vdc b

us (V

)

Udc５dcＬ

1 15 25 305 2TTime (seconds)

Figure 12 DC link voltage

variation of solar irradiation happens A detailed view on thecontroller performance during irradiation change (at t=2 sec)is provided by the zoom of Figure 14 From the two figures itcan be recognized that the current controller response is veryfast and is reaching its reference in a brief span

Case 2 (control under voltage dips withoutwith currentamplitude limitation) Voltage dips are considered as oneof the most challenging problems in grid-connected PVsystems The proposed control strategy have to ensure agood voltage dip immunity In fact due to a voltage dipat t=008 sec grid voltages have decreased by 50 andmaintain this level for 006 sec Figure 15 highlights theimpact of the grid voltage dip on the currents injected intothe grid without amplitude limitationThe grid currents have


(a)

ir1ir2

ir3

2

minus1

0

1

1000WＧ2900WＧ2800WＧ2

600WＧ2

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

Curr

ents

ir1 i

r2 an

d ir3

(A)


Figure 13 Grid currents under irradiation variation

(b)

2 200006

199 2 201 202198Time (seconds)

minus3

minus2

minus1

0

1

2

3

4

5

Grid

Cur

rent

ir1

(A)

ir1ＣＬ1Ｌ

Figure 14 Grid current ir1 measured and its reference

increased by 50 accordingly in order to mitigate the powerloss caused by voltage dips Furthermore one of the aimsof the Backstepping control strategy is to keep the DC linkvoltage stable independently of the power variation Figure 16is validating the DC link voltage controller that provides agood tracking of themeasured dc link voltage to its reference

(a)

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

8

Grid

Cur

rent

s ir1

ir2

and

ir3 (A

)

014008Time (seconds)

ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ

Figure 15The voltage dip effect on grid currents without limitationof their amplitudes

699999

700

699999

700

DC

link

volta

ge (V

)

(b)


UdcUdc ref

Figure 16 The voltage dip effect on DC link voltage

At t=008 sec and t=014 sec a transient phenomenon can bediscerned due to voltage dip The DC controller is trying tokeep the DC link at the same constant value and reduce veryfast the error


(a)

minus3

minus2

minus1

0

1

2

3

4

5

6G

rid cu

rren

ts ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ

Figure 17The voltage dip effect on grid currents with limitation oftheir amplitudes

(b)


700

702

704

706

708

710

DC

link

volta

ge (V

)

udcudcＬ


Voltage dip has no significant impact on grid currentswith amplitude limitation as depicted in Figure 17 The gridcurrents remain unchanged while the DC link voltage inFigure 18 has gradually increased during the voltage dip toreach a steady-state with a new value (7087V)

4

5

6

7

8

9

10

11

T1=L

1r1

002 004 006 008 01 012 014 016 018 020Time (seconds)

Figure 19 Time constant of the grid line

(a)

minus3

minus2

minus1

0

1

2

3

Grid

curr

ents

ir1 i

r2 an

d ir3

(A)


Figure 20 The effect of 50 line resistance sudden decrease on thegrid currents

Case 3 (control sensitivity to parameter variations) Theparameters of the PV system can suddenly vary As a resultit is necessary to evaluate the robustness and reliability of theBackstepping control strategy

Specifically a reduction of 50 in resistance r1 leads to anincrease of 100 of the time constant (T1=L1r1) as shown inFigure 19 The parameter change is performed at t=008 sec

The results of simulation given in Figure 20 prove thereliability of the Backstepping control to sudden parametricvariations In fact it shows capability to deliver the desiredoutput power to the grid with unity power factor in otherwords it keeps the output current in phase with the gridvoltage Grid currents remain unchangedwhen the resistancevalue varies The simulation results obtained confirm theexcellent performance and the robustness of the control byBackstepping

Case 4 (control under measurement uncertainties) Underconstant temperature (T=25∘C) the PV system is subjected


Figure 21 Measurement uncertainties random error insertion inthe current control loop

(a)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ

Figure 22 Measurement uncertainty effect on grid currents noisewith an amplitude of +-10 of grid current amplitude

to measurement uncertainty by inserting a noise called alsoa random error as an external disturbance at the entrance ofthe Backstepping control bloc of the current control loop asportrayed in Figure 21 First the noise is of amplitude of+-10 and then +-20 of the grid current amplitude

Besides the solar irradiation level steps down from1000Wm2 to 800Wm2 curves at t=008 sec Figures 22and 23 show the robustness of the Backstepping control tomeasurement uncertainties It is worth noting that at thepresence of noise grid currents vary slightly and the unitypower factor is always maintained

5 Conclusion

A robust nonlinear Backstepping Controller is proposed inthis paper for a grid-connected PV system The EMR andthe MCS are proposed in order to provide respectively theinstantaneous average model and a cascade control struc-ture For superior tracking efficiency a PampO based MPPTalgorithm is employed to extract maximum power from PV

(b)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ


panelsThe control strategy is designed in order to control allcascade loops in the conversion chain The system responsesare performed when fast-changing solar irradiation voltagedip parametric variations and measurement uncertaintiesare experienced The Backstepping control addresses all thealready quoted challenges and the problem of controlling athree-phase grid-connected PV system is addressed Simula-tions have been conducted using MatlabSimulink validatingthe functionality robustness and simplicity of the algorithm

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] M A Green ldquoPhotovoltaic principlesrdquo Physica E Low-dimen-sional Systems and Nanostructures vol 14 no 1-2 pp 11ndash172002

[2] G Xu P Moulema L Ge H Song and W Yu A UnifiedFramework for Secured Energy Resource Management in SmartGrid Smart Grid CRC Press 2016

[3] W Xiao andW G Dunford ldquoA modified adaptive hill climbingMPPT method for photovoltaic power systemsrdquo in Proceedingsof the 35th Annual Power Electronics Specialists Conference(PESC rsquo04) vol 3 pp 1957ndash1963 IEEE June 2004


[4] D Hohm and M Ropp ldquoComparative study of maximumpower point tracking algorithms using an experimental pro-grammable maximum power point tracking test bedrdquo inProceedings of the 28th IEEE Photovoltaic Specialists Conferencepp 1699ndash1702 IEEE Anchorage AK USA 2000

[5] A Badis M H Boujmil and M N Mansouri ldquoA comparativestudy on maximum power point tracking techniques of pho-tovoltaic systemsrdquo International Journal of Energy Optimizationand Engineering vol 7 no 1 pp 66ndash85 2018

[6] A Badis M N Mansouri and A Sakly ldquoPSO and GA-based maximum power point tracking for partially shadedphotovoltaic systemsrdquo in Proceedings of the 7th InternationalRenewable Energy Congress IREC 2016 March 2016

[7] M H Boujmil M N Mansouri andM F Mimouni ldquoCompar-ative study of the cascade state ad-justment applied to a threephase low volt-age grid connected to a pv systemsrdquo STA20092009

[8] G A Raducu Control of Grid Side Inverter in a B2B Configu-ration for WT Applications [Master thesis] Aalborg University2008

[9] M Amirabadi ldquoAnalog control of AC link universal powerconverters The key to very high frequency AC link conversionsystemsrdquo in Proceedings of the 30th Annual IEEE Applied PowerElectronics Conference and Exposition APEC 2015 pp 2301ndash2308 March 2015

[10] M Basavaraj and P Nagabhushan ldquoSolar based high frequencyAC link inverterrdquo International Research Journal of Engineeringand Technology (IRJET) vol 3 no 5 pp 828ndash834 2016

[11] B N Alajmi K H Ahmed S J Finney and B W WilliamsldquoFuzzy-logic-control approach of a modified hill-climbingmethod for maximum power point in microgrid standalonephotovoltaic systemrdquo IEEE Transactions on Power Electronicsvol 26 no 4 pp 1022ndash1030 2011

[12] T L Kottas Y S Boutalis and A D Karlis ldquoNew maximumpower point tracker for PV arrays using fuzzy controller in closecooperation with fuzzy cognitive networksrdquo IEEE Transactionson Energy Conversion vol 21 no 3 pp 793ndash803 2006

[13] J Li and H Wang ldquoMaximum power point tracking of pho-tovoltaic generation based on the fuzzy control methodrdquo inProceedings of the 1st International Conference on SustainablePower Generation and Supply (SUPERGEN rsquo09) pp 1ndash6 Nan-jing China April 2009

[14] D Menniti A Pinnarelli and G Brusco ldquoImplementation ofa novel fuzzy-logic based MPPT for grid-connected photo-voltaic generation systemrdquo in Proceedings of the 2011 IEEE PESTrondheimPowerTechThePower of Technology for a SustainableSociety POWERTECH 2011 June 2011

[15] B Yang W Li Y Zhao and X He ldquoDesign and analysis of agrid-connected photovoltaic power systemrdquo IEEE Transactionson Power Electronics vol 25 no 4 pp 992ndash1000 2010

[16] M H Boujmil M N Mansouri and M F Mimouni ldquoOptimalcontrol of PVG interconnected to a three-phase lv networkrdquoJTEA2010 2010

[17] K S Agbli D Hissel M-C Pera and I Doumbia ldquoEnergeticMacroscopic Representation (EMR) new approach for multi-physics energetic flows modellingrdquo in Proceedings of the 8thPower Plant and Power SystemControl Symposium PPPSC 2012pp 723ndash728 September 2012

[18] D Chrenko M Pera and D Hissel ldquoFuel cell system modelingand control with energetic macroscopic representationrdquo inProceedings of the 2007 IEEE International Symposium onIndustrial Electronics pp 169ndash174 Vigo Spain June 2007

[19] L Horrein A Bouscayrol Y Cheng and M E Fassi ldquoMul-tiphysical modeling and description of a permanent magnetsynchronous machine using energetic macroscopic representa-tion for EVHEV applicationsrdquo in Proceedings of the 2013 15thEuropeanConference onPower Electronics andApplications EPE2013 September 2013

[20] W Lhomme P Delarue F Giraud B Lemaire-Semail and ABouscayrol ldquoSimulation of a photovoltaic conversion systemusing energetic macroscopic representationrdquo in Proceedings ofthe 15th International Power Electronics and Motion ControlConference and Exposition EPE-PEMC 2012 ECCE Europe pDS3e76 September 2012

[21] J S Martınez D Hissel M-C Pera and M Amiet ldquoPracticalcontrol structure and energy management of a testbed hybridelectric vehiclerdquo IEEETransactions onVehicular Technology vol60 no 9 pp 4139ndash4152 2011

[22] I Tyukhov H Rezk and P Vasant ldquoModern optimizationalgorithms and applications in solar photovoltaic engineeringrdquoSustaining Power Resources through Energy Optimization andEngineering pp 390ndash445 2016

[23] L Gyugi and B R Pelly Static Power Frequency ChangersTheory Design and Applications Wiley-Interscience New YorkNY USA 1976

[24] J Oyama T Hihuchi E Yamada T Koga and T Lipo ldquoNewcontrol strategies for matrix converterrdquo in Proceedings of the20th Annual IEEE Power Electronics Specialists ConferencePESC rsquo89 Record vol 1 pp 360ndash367 1989

[25] T Sobczyk ldquoNumerical study of control strategies for frequencyconversion with a matrix converterrdquo in Proceedings of the PowerElectronics and Motion Control pp 497ndash502 Warsaw Poland1994

[26] J G Cho and G H Cho ldquoSoft-switched matrix converter forhigh frequency direct AC-to-AC power conversionrdquo Interna-tional Journal of Electronics vol 72 no 4 pp 669ndash680 1992

[27] J-P Hautier and J-P Caron Convertisseurs Statiques Method-ologie Causale de Modelisation et de Commande EditionsTechnip 1999

[28] M R Barakat B Tala-Ighil H Chaoui H gualous Y Slamaniand D Hissel ldquoEnergetic macroscopic representation of marinecurrent turbine system with loss minimization controlrdquo IEEETransactions on Sustainable Energy 2017

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and the parametric errors Thus the use of robust controllaws becomes a required and challenging task for ensuring thestabilization and the good tracking In fact there are alreadya lot of researches [11ndash15] targeting the conversion chain ofgrid-connected PV system However the proposed schemesin the latter studies require the detailed evaluations namely

(i) control the DCDC converter in order to force thePV array to operate at the Maximum Power Points(MPP)

(ii) maintain the DC link voltage at the desired constantvalue

(iii) deliver the desired output power to the active gridwith unity power factor (PF)

A considerable progress has been made over last decadein optimization techniques in order to perform the threefoldobjectives Among these methods classical PID controller isusually used in industry and literature because of its robust-ness low cost and ease of implementationThe conventionalcontrol strategies (PI regulator or state regulator) producegood results in linear systems [16] However for nonlinearsystems or with varying parameters those methods becomeinsufficient and unreliable especially when the performancerequirements of the system are rigorousThus it is imperativeto choose the adequate control strategy which is insensitiveto parameter variations perturbations and nonlinearityThispaper proposes a Backstepping based technique to design asuitable control system for the grid-connected PV systemThe designed control technique ensures the optimal energytransfer to the grid via sharing active and reactive power intothe grid regardless of the atmospheric changes and paramet-ric uncertainties The desired DC link voltage is maintainedas well by using Backstepping approach to control the staticboost converter based on Matrix Topology (MT) MT isknown to be themost industrially encountered inmost powerelectronics systems since it does not need energy storagecomponents Control Lyapunov functions are formulated atdifferent stages of the design process to evaluate the stabilityof the designed controller

The PV systemrsquos operation and the development of itsappropriate control techniques require as a compulsorya basic representation of the entire system by taking intoconsideration the integrality and the causality between everyone of systemrsquos components

The Energetic Macroscopic Representation (EMR) is agraphical portrayal of the system which has been createdin 2000 by the L2EP at the University of Lille in FranceIt is an extremely basic representation with a gathering ofstandards and elements [17 18] Since then EMR has beenused as a part of manymultidomainmultisources [19 20] forrepresentation andmodeling of different systems namely PVand wind energy conversion systems Electric Vehicles (EV)[21]

In this paper EMR and its reverse Maximum ControlStructure (MCS) are both used to provide an instantaneousaveragemodel and a control structure based on cascade loopsThe chosen structure gives promising dynamic performances

and limits each state variable to manage the system incomplete safety

In the present study the modeling of the Low-voltagegrid-connected PV system is developed in Section 2 Sec-tion 3 encompasses the design of the proposed controllerbased on cascade Backstepping strategy In Section 4 sim-ulation results of the proposed control system are discussedfor various operating conditions Comments supporting theperformance and robustness of the Backstepping controlapproach are given Finally in Section 5 the conclusion isdrawn and followed by references

2 Modeling of the Grid-ConnectedPhotovoltaic System

The following section presents the equations used for model-ing each component in the PV conversion chain The princi-ple mission of the chosen conversion system is to extract themaximum active power through a boost converter operatingwith a suitable MPPT and managing as well the activeand reactive power injected into the grid via the inverterThe system is comprised of simple subsystems (pictograms)related together These pictograms portray every element inthe system by functions as shown in Table 1 [20] The EMRfor all components are interconnected in order to frame theentire system EMR with respecting the integral causality andfollowing the action-reaction principle MCS which allowsthe control loop modeling is deduced by inversion of theEMR [20] The entire grid-connected PV system is shows inFigure 1 while the EMR of the entire system is depicted inFigure 2

The current electrical source (green oval) represents thePVGwhich is the association of series andor parallel PV cellsin order to raise the current voltage and power An idealPhotovoltaic cell is equivalent to a power source shunted by adiode as shown in Figure 3 [22]

The PV model is mathematically modeled using

119868119901V = 119868119901ℎ minus 119868119904 [exp( 119880119901V119873119904119899119870119861119879) minus 1] (1)

where 119868119901ℎ = 119873119901 times 119894119901ℎ is photocurrent Np parallel cells119868119901V = 119873119901 times 119894119901V is current supplied by Np parallel cells119868119904 = 119873119901 times 119894119904 is reverse saturation current ofNp 119889119894119900119889119890 119894119899 119901119886119903119886119897119897119890119897V119879 = 119899119870119861119879119902 is the thermodynamic potential119873119901 is number of cells in parallel119873119904 is number of cells in series119902 is 119890119897119890119888119905119903119900119899 119888ℎ119886119903119892119890(1 610minus19C)119870119861 is Boltzman Constant (1 3810minus23jK)119879 is junction temperature Kn is ideality factor of the picture of the solar cellincluding between 1 and 5 in practice



Sourceelement

Accumula-tion

element

Indirectinversion

Mono-physical

conversionelement


Directinversion

Multi-physical

conversionelement


Couplinginversion


PVG

Command


DC linkinverter


pvu

pvi Li

Limu

mi

mridc

u

dcu m_dqv r_dqi

r_123v

r_123i

MPPT


REGm dq_REGm


Griddq

12

Command

-

-

dc_reacutefu

dcu

r_dqv

r_dq_refi

pvu

CPQ

Process

Control


imr_reacutef



phipvu

pvi

Load




1198771 997888rarr 119862119889119906119901V119889119905 + 119906119901V119877 = 119894119901V minus 119894119871 (2)




[119865] = [11989111 1198911211989121 11989122] (4)







1198773 997888rarr 1198621 119889119906119889119888119889119905 + 119906119889119888119877 = 119894119898 minus 119894119898119903 (7)





[119865] = [11989111 11989112 1198911311989121 11989122 11989123] (8)





1198771198983 997888rarr [V119898119889V119898119902

] = 1199061198891198882 [119898119889119898119902]1198771198984 997888rarr 119894119898119900 = 12 (119898119889119894119903119889 + 119898119902119894119903119902)(10)

1198774 997888rarr [V119898119889V119898119902

] = [119875 (120579)] [V1199031V1199032] (11)

1198775 997888rarr [119894119903119889119894119903119902] = [119875 (120579)] [11989411990311198941199032] (12)

where


1198776 997888rarr (1198711 119889119889119905 + 1199031)[119894119903119889119894119903119902]= [V119898119889

V119898119902] minus [V119903119889

V119903119902] + [ 0 1198711120596minus1198711120596 0 ][119894119903119889119894119903119902]

(14)


1198777 997888rarr [119875119876] = [V119898119889 V119898119902V119898119902 minusV119898119889][119894119903119889119894119903119902] (15)



where
























1198812 (1198901 1198902) = 1198811 + 1211989022 = 1211989021 + 1211989022 (27)



(28)














119894119898119900 119903119890119891 = 1198621 (11989631198903 minus 111987711198621 119906119889119888 + 11198621 119894119898ℎ minus 119889119888 119903119890119891) (34)








119894119898119900 119903119890119891

minus 119889119888 119903119890119891minus 121198621 (119898119889119890119889 + 119898119902119890119902)

(38)


1198903 = minus11989631198903 minus 121198621 (119898119889119890119889 + 119898119902119890119902) (39)







minus119896119902119890119902

)(42)



(43)


119898119889 119903119890119892 = V119898119889 119903119890119891119906119889119888119898119902 119903119890119892 = V119898119902 119903119890119891119906119889119888 (44)



119875 = 1119896119882119862 = 220120583119865119877 = 100119896Ω

119871 = 231198981198671198621 = 50001205831198651198771 = 10119896Ω1199031 = 00002Ω1198711 = 1119898119867119880119903 = 380119881119891 = 50119867119911(45)


(46)




(a)

0

05

1

15

2

25

3

35

4

45

5PV

Cur

rent

(A)

50 100 150 200 250 3000PV Voltage (V)

1000 WＧ2

900 WＧ2

800 WＧ2

600 WＧ2


(b)

50 100 150 200 250 3000PV Voltage (V)

0

200

400

600

800

1000

1200

PV P

ower

(W)





Upv

230240250260270280290300310

PV V

olta

ge (V

)

235240245250


５ＪＰＬ

14

145

15

155

16

165

17


(b)

2

7007000002

7007000002

699

6992

6994

6996

6998

700

7002

7004

7006

7008

701

Vdc b

us (V

)

Udc５dcＬ






(a)

ir1ir2

ir3

2

minus1

0

1

1000WＧ2900WＧ2800WＧ2

600WＧ2

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

Curr

ents

ir1 i

r2 an

d ir3

(A)



(b)

2 200006

199 2 201 202198Time (seconds)

minus3

minus2

minus1

0

1

2

3

4

5

Grid

Cur

rent

ir1

(A)

ir1ＣＬ1Ｌ



(a)

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

8

Grid

Cur

rent

s ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


699999

700

699999

700

DC

link

volta

ge (V

)

(b)


UdcUdc ref




(a)

minus3

minus2

minus1

0

1

2

3

4

5

6G

rid cu

rren

ts ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


(b)


700

702

704

706

708

710

DC

link

volta

ge (V

)

udcudcＬ



4

5

6

7

8

9

10

11

T1=L

1r1

002 004 006 008 01 012 014 016 018 020Time (seconds)


(a)

minus3

minus2

minus1

0

1

2

3

Grid

curr

ents

ir1 i

r2 an

d ir3

(A)









(a)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ




5 Conclusion


(b)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ



Data Availability




References





































Journal of












Journal of







Volume 2018






Volume 2018










Sourceelement

Accumula-tion

element

Indirectinversion

Mono-physical

conversionelement


Directinversion

Multi-physical

conversionelement


Couplinginversion


PVG

Command


DC linkinverter


pvu

pvi Li

Limu

mi

mridc

u

dcu m_dqv r_dqi

r_123v

r_123i

MPPT


REGm dq_REGm


Griddq

12

Command

-

-

dc_reacutefu

dcu

r_dqv

r_dq_refi

pvu

CPQ

Process

Control


imr_reacutef



phipvu

pvi

Load




1198771 997888rarr 119862119889119906119901V119889119905 + 119906119901V119877 = 119894119901V minus 119894119871 (2)




[119865] = [11989111 1198911211989121 11989122] (4)







1198773 997888rarr 1198621 119889119906119889119888119889119905 + 119906119889119888119877 = 119894119898 minus 119894119898119903 (7)





[119865] = [11989111 11989112 1198911311989121 11989122 11989123] (8)





1198771198983 997888rarr [V119898119889V119898119902

] = 1199061198891198882 [119898119889119898119902]1198771198984 997888rarr 119894119898119900 = 12 (119898119889119894119903119889 + 119898119902119894119903119902)(10)

1198774 997888rarr [V119898119889V119898119902

] = [119875 (120579)] [V1199031V1199032] (11)

1198775 997888rarr [119894119903119889119894119903119902] = [119875 (120579)] [11989411990311198941199032] (12)

where


1198776 997888rarr (1198711 119889119889119905 + 1199031)[119894119903119889119894119903119902]= [V119898119889

V119898119902] minus [V119903119889

V119903119902] + [ 0 1198711120596minus1198711120596 0 ][119894119903119889119894119903119902]

(14)


1198777 997888rarr [119875119876] = [V119898119889 V119898119902V119898119902 minusV119898119889][119894119903119889119894119903119902] (15)



where
























1198812 (1198901 1198902) = 1198811 + 1211989022 = 1211989021 + 1211989022 (27)



(28)














119894119898119900 119903119890119891 = 1198621 (11989631198903 minus 111987711198621 119906119889119888 + 11198621 119894119898ℎ minus 119889119888 119903119890119891) (34)








119894119898119900 119903119890119891

minus 119889119888 119903119890119891minus 121198621 (119898119889119890119889 + 119898119902119890119902)

(38)


1198903 = minus11989631198903 minus 121198621 (119898119889119890119889 + 119898119902119890119902) (39)







minus119896119902119890119902

)(42)



(43)


119898119889 119903119890119892 = V119898119889 119903119890119891119906119889119888119898119902 119903119890119892 = V119898119902 119903119890119891119906119889119888 (44)



119875 = 1119896119882119862 = 220120583119865119877 = 100119896Ω

119871 = 231198981198671198621 = 50001205831198651198771 = 10119896Ω1199031 = 00002Ω1198711 = 1119898119867119880119903 = 380119881119891 = 50119867119911(45)


(46)




(a)

0

05

1

15

2

25

3

35

4

45

5PV

Cur

rent

(A)

50 100 150 200 250 3000PV Voltage (V)

1000 WＧ2

900 WＧ2

800 WＧ2

600 WＧ2


(b)

50 100 150 200 250 3000PV Voltage (V)

0

200

400

600

800

1000

1200

PV P

ower

(W)





Upv

230240250260270280290300310

PV V

olta

ge (V

)

235240245250


５ＪＰＬ

14

145

15

155

16

165

17


(b)

2

7007000002

7007000002

699

6992

6994

6996

6998

700

7002

7004

7006

7008

701

Vdc b

us (V

)

Udc５dcＬ






(a)

ir1ir2

ir3

2

minus1

0

1

1000WＧ2900WＧ2800WＧ2

600WＧ2

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

Curr

ents

ir1 i

r2 an

d ir3

(A)



(b)

2 200006

199 2 201 202198Time (seconds)

minus3

minus2

minus1

0

1

2

3

4

5

Grid

Cur

rent

ir1

(A)

ir1ＣＬ1Ｌ



(a)

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

8

Grid

Cur

rent

s ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


699999

700

699999

700

DC

link

volta

ge (V

)

(b)


UdcUdc ref




(a)

minus3

minus2

minus1

0

1

2

3

4

5

6G

rid cu

rren

ts ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


(b)


700

702

704

706

708

710

DC

link

volta

ge (V

)

udcudcＬ



4

5

6

7

8

9

10

11

T1=L

1r1

002 004 006 008 01 012 014 016 018 020Time (seconds)


(a)

minus3

minus2

minus1

0

1

2

3

Grid

curr

ents

ir1 i

r2 an

d ir3

(A)









(a)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ




5 Conclusion


(b)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ



Data Availability




References





































Journal of












Journal of







Volume 2018






Volume 2018









phipvu

pvi

Load




1198771 997888rarr 119862119889119906119901V119889119905 + 119906119901V119877 = 119894119901V minus 119894119871 (2)




[119865] = [11989111 1198911211989121 11989122] (4)







1198773 997888rarr 1198621 119889119906119889119888119889119905 + 119906119889119888119877 = 119894119898 minus 119894119898119903 (7)





[119865] = [11989111 11989112 1198911311989121 11989122 11989123] (8)





1198771198983 997888rarr [V119898119889V119898119902

] = 1199061198891198882 [119898119889119898119902]1198771198984 997888rarr 119894119898119900 = 12 (119898119889119894119903119889 + 119898119902119894119903119902)(10)

1198774 997888rarr [V119898119889V119898119902

] = [119875 (120579)] [V1199031V1199032] (11)

1198775 997888rarr [119894119903119889119894119903119902] = [119875 (120579)] [11989411990311198941199032] (12)

where


1198776 997888rarr (1198711 119889119889119905 + 1199031)[119894119903119889119894119903119902]= [V119898119889

V119898119902] minus [V119903119889

V119903119902] + [ 0 1198711120596minus1198711120596 0 ][119894119903119889119894119903119902]

(14)


1198777 997888rarr [119875119876] = [V119898119889 V119898119902V119898119902 minusV119898119889][119894119903119889119894119903119902] (15)



where
























1198812 (1198901 1198902) = 1198811 + 1211989022 = 1211989021 + 1211989022 (27)



(28)














119894119898119900 119903119890119891 = 1198621 (11989631198903 minus 111987711198621 119906119889119888 + 11198621 119894119898ℎ minus 119889119888 119903119890119891) (34)








119894119898119900 119903119890119891

minus 119889119888 119903119890119891minus 121198621 (119898119889119890119889 + 119898119902119890119902)

(38)


1198903 = minus11989631198903 minus 121198621 (119898119889119890119889 + 119898119902119890119902) (39)







minus119896119902119890119902

)(42)



(43)


119898119889 119903119890119892 = V119898119889 119903119890119891119906119889119888119898119902 119903119890119892 = V119898119902 119903119890119891119906119889119888 (44)



119875 = 1119896119882119862 = 220120583119865119877 = 100119896Ω

119871 = 231198981198671198621 = 50001205831198651198771 = 10119896Ω1199031 = 00002Ω1198711 = 1119898119867119880119903 = 380119881119891 = 50119867119911(45)


(46)




(a)

0

05

1

15

2

25

3

35

4

45

5PV

Cur

rent

(A)

50 100 150 200 250 3000PV Voltage (V)

1000 WＧ2

900 WＧ2

800 WＧ2

600 WＧ2


(b)

50 100 150 200 250 3000PV Voltage (V)

0

200

400

600

800

1000

1200

PV P

ower

(W)





Upv

230240250260270280290300310

PV V

olta

ge (V

)

235240245250


５ＪＰＬ

14

145

15

155

16

165

17


(b)

2

7007000002

7007000002

699

6992

6994

6996

6998

700

7002

7004

7006

7008

701

Vdc b

us (V

)

Udc５dcＬ






(a)

ir1ir2

ir3

2

minus1

0

1

1000WＧ2900WＧ2800WＧ2

600WＧ2

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

Curr

ents

ir1 i

r2 an

d ir3

(A)



(b)

2 200006

199 2 201 202198Time (seconds)

minus3

minus2

minus1

0

1

2

3

4

5

Grid

Cur

rent

ir1

(A)

ir1ＣＬ1Ｌ



(a)

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

8

Grid

Cur

rent

s ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


699999

700

699999

700

DC

link

volta

ge (V

)

(b)


UdcUdc ref




(a)

minus3

minus2

minus1

0

1

2

3

4

5

6G

rid cu

rren

ts ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


(b)


700

702

704

706

708

710

DC

link

volta

ge (V

)

udcudcＬ



4

5

6

7

8

9

10

11

T1=L

1r1

002 004 006 008 01 012 014 016 018 020Time (seconds)


(a)

minus3

minus2

minus1

0

1

2

3

Grid

curr

ents

ir1 i

r2 an

d ir3

(A)









(a)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ




5 Conclusion


(b)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ



Data Availability




References





































Journal of












Journal of







Volume 2018






Volume 2018












1198771198983 997888rarr [V119898119889V119898119902

] = 1199061198891198882 [119898119889119898119902]1198771198984 997888rarr 119894119898119900 = 12 (119898119889119894119903119889 + 119898119902119894119903119902)(10)

1198774 997888rarr [V119898119889V119898119902

] = [119875 (120579)] [V1199031V1199032] (11)

1198775 997888rarr [119894119903119889119894119903119902] = [119875 (120579)] [11989411990311198941199032] (12)

where


1198776 997888rarr (1198711 119889119889119905 + 1199031)[119894119903119889119894119903119902]= [V119898119889

V119898119902] minus [V119903119889

V119903119902] + [ 0 1198711120596minus1198711120596 0 ][119894119903119889119894119903119902]

(14)


1198777 997888rarr [119875119876] = [V119898119889 V119898119902V119898119902 minusV119898119889][119894119903119889119894119903119902] (15)



where
























1198812 (1198901 1198902) = 1198811 + 1211989022 = 1211989021 + 1211989022 (27)



(28)














119894119898119900 119903119890119891 = 1198621 (11989631198903 minus 111987711198621 119906119889119888 + 11198621 119894119898ℎ minus 119889119888 119903119890119891) (34)








119894119898119900 119903119890119891

minus 119889119888 119903119890119891minus 121198621 (119898119889119890119889 + 119898119902119890119902)

(38)


1198903 = minus11989631198903 minus 121198621 (119898119889119890119889 + 119898119902119890119902) (39)







minus119896119902119890119902

)(42)



(43)


119898119889 119903119890119892 = V119898119889 119903119890119891119906119889119888119898119902 119903119890119892 = V119898119902 119903119890119891119906119889119888 (44)



119875 = 1119896119882119862 = 220120583119865119877 = 100119896Ω

119871 = 231198981198671198621 = 50001205831198651198771 = 10119896Ω1199031 = 00002Ω1198711 = 1119898119867119880119903 = 380119881119891 = 50119867119911(45)


(46)




(a)

0

05

1

15

2

25

3

35

4

45

5PV

Cur

rent

(A)

50 100 150 200 250 3000PV Voltage (V)

1000 WＧ2

900 WＧ2

800 WＧ2

600 WＧ2


(b)

50 100 150 200 250 3000PV Voltage (V)

0

200

400

600

800

1000

1200

PV P

ower

(W)





Upv

230240250260270280290300310

PV V

olta

ge (V

)

235240245250


５ＪＰＬ

14

145

15

155

16

165

17


(b)

2

7007000002

7007000002

699

6992

6994

6996

6998

700

7002

7004

7006

7008

701

Vdc b

us (V

)

Udc５dcＬ






(a)

ir1ir2

ir3

2

minus1

0

1

1000WＧ2900WＧ2800WＧ2

600WＧ2

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

Curr

ents

ir1 i

r2 an

d ir3

(A)



(b)

2 200006

199 2 201 202198Time (seconds)

minus3

minus2

minus1

0

1

2

3

4

5

Grid

Cur

rent

ir1

(A)

ir1ＣＬ1Ｌ



(a)

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

8

Grid

Cur

rent

s ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


699999

700

699999

700

DC

link

volta

ge (V

)

(b)


UdcUdc ref




(a)

minus3

minus2

minus1

0

1

2

3

4

5

6G

rid cu

rren

ts ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


(b)


700

702

704

706

708

710

DC

link

volta

ge (V

)

udcudcＬ



4

5

6

7

8

9

10

11

T1=L

1r1

002 004 006 008 01 012 014 016 018 020Time (seconds)


(a)

minus3

minus2

minus1

0

1

2

3

Grid

curr

ents

ir1 i

r2 an

d ir3

(A)









(a)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ




5 Conclusion


(b)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ



Data Availability




References





































Journal of












Journal of







Volume 2018






Volume 2018
























1198812 (1198901 1198902) = 1198811 + 1211989022 = 1211989021 + 1211989022 (27)



(28)














119894119898119900 119903119890119891 = 1198621 (11989631198903 minus 111987711198621 119906119889119888 + 11198621 119894119898ℎ minus 119889119888 119903119890119891) (34)








119894119898119900 119903119890119891

minus 119889119888 119903119890119891minus 121198621 (119898119889119890119889 + 119898119902119890119902)

(38)


1198903 = minus11989631198903 minus 121198621 (119898119889119890119889 + 119898119902119890119902) (39)







minus119896119902119890119902

)(42)



(43)


119898119889 119903119890119892 = V119898119889 119903119890119891119906119889119888119898119902 119903119890119892 = V119898119902 119903119890119891119906119889119888 (44)



119875 = 1119896119882119862 = 220120583119865119877 = 100119896Ω

119871 = 231198981198671198621 = 50001205831198651198771 = 10119896Ω1199031 = 00002Ω1198711 = 1119898119867119880119903 = 380119881119891 = 50119867119911(45)


(46)




(a)

0

05

1

15

2

25

3

35

4

45

5PV

Cur

rent

(A)

50 100 150 200 250 3000PV Voltage (V)

1000 WＧ2

900 WＧ2

800 WＧ2

600 WＧ2


(b)

50 100 150 200 250 3000PV Voltage (V)

0

200

400

600

800

1000

1200

PV P

ower

(W)





Upv

230240250260270280290300310

PV V

olta

ge (V

)

235240245250


５ＪＰＬ

14

145

15

155

16

165

17


(b)

2

7007000002

7007000002

699

6992

6994

6996

6998

700

7002

7004

7006

7008

701

Vdc b

us (V

)

Udc５dcＬ






(a)

ir1ir2

ir3

2

minus1

0

1

1000WＧ2900WＧ2800WＧ2

600WＧ2

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

Curr

ents

ir1 i

r2 an

d ir3

(A)



(b)

2 200006

199 2 201 202198Time (seconds)

minus3

minus2

minus1

0

1

2

3

4

5

Grid

Cur

rent

ir1

(A)

ir1ＣＬ1Ｌ



(a)

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

8

Grid

Cur

rent

s ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


699999

700

699999

700

DC

link

volta

ge (V

)

(b)


UdcUdc ref




(a)

minus3

minus2

minus1

0

1

2

3

4

5

6G

rid cu

rren

ts ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


(b)


700

702

704

706

708

710

DC

link

volta

ge (V

)

udcudcＬ



4

5

6

7

8

9

10

11

T1=L

1r1

002 004 006 008 01 012 014 016 018 020Time (seconds)


(a)

minus3

minus2

minus1

0

1

2

3

Grid

curr

ents

ir1 i

r2 an

d ir3

(A)









(a)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ




5 Conclusion


(b)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ



Data Availability




References





































Journal of












Journal of







Volume 2018






Volume 2018


















119894119898119900 119903119890119891 = 1198621 (11989631198903 minus 111987711198621 119906119889119888 + 11198621 119894119898ℎ minus 119889119888 119903119890119891) (34)








119894119898119900 119903119890119891

minus 119889119888 119903119890119891minus 121198621 (119898119889119890119889 + 119898119902119890119902)

(38)


1198903 = minus11989631198903 minus 121198621 (119898119889119890119889 + 119898119902119890119902) (39)







minus119896119902119890119902

)(42)



(43)


119898119889 119903119890119892 = V119898119889 119903119890119891119906119889119888119898119902 119903119890119892 = V119898119902 119903119890119891119906119889119888 (44)



119875 = 1119896119882119862 = 220120583119865119877 = 100119896Ω

119871 = 231198981198671198621 = 50001205831198651198771 = 10119896Ω1199031 = 00002Ω1198711 = 1119898119867119880119903 = 380119881119891 = 50119867119911(45)


(46)




(a)

0

05

1

15

2

25

3

35

4

45

5PV

Cur

rent

(A)

50 100 150 200 250 3000PV Voltage (V)

1000 WＧ2

900 WＧ2

800 WＧ2

600 WＧ2


(b)

50 100 150 200 250 3000PV Voltage (V)

0

200

400

600

800

1000

1200

PV P

ower

(W)





Upv

230240250260270280290300310

PV V

olta

ge (V

)

235240245250


５ＪＰＬ

14

145

15

155

16

165

17


(b)

2

7007000002

7007000002

699

6992

6994

6996

6998

700

7002

7004

7006

7008

701

Vdc b

us (V

)

Udc５dcＬ






(a)

ir1ir2

ir3

2

minus1

0

1

1000WＧ2900WＧ2800WＧ2

600WＧ2

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

Curr

ents

ir1 i

r2 an

d ir3

(A)



(b)

2 200006

199 2 201 202198Time (seconds)

minus3

minus2

minus1

0

1

2

3

4

5

Grid

Cur

rent

ir1

(A)

ir1ＣＬ1Ｌ



(a)

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

8

Grid

Cur

rent

s ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


699999

700

699999

700

DC

link

volta

ge (V

)

(b)


UdcUdc ref




(a)

minus3

minus2

minus1

0

1

2

3

4

5

6G

rid cu

rren

ts ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


(b)


700

702

704

706

708

710

DC

link

volta

ge (V

)

udcudcＬ



4

5

6

7

8

9

10

11

T1=L

1r1

002 004 006 008 01 012 014 016 018 020Time (seconds)


(a)

minus3

minus2

minus1

0

1

2

3

Grid

curr

ents

ir1 i

r2 an

d ir3

(A)









(a)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ




5 Conclusion


(b)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ



Data Availability




References





































Journal of












Journal of







Volume 2018






Volume 2018













minus119896119902119890119902

)(42)



(43)


119898119889 119903119890119892 = V119898119889 119903119890119891119906119889119888119898119902 119903119890119892 = V119898119902 119903119890119891119906119889119888 (44)



119875 = 1119896119882119862 = 220120583119865119877 = 100119896Ω

119871 = 231198981198671198621 = 50001205831198651198771 = 10119896Ω1199031 = 00002Ω1198711 = 1119898119867119880119903 = 380119881119891 = 50119867119911(45)


(46)




(a)

0

05

1

15

2

25

3

35

4

45

5PV

Cur

rent

(A)

50 100 150 200 250 3000PV Voltage (V)

1000 WＧ2

900 WＧ2

800 WＧ2

600 WＧ2


(b)

50 100 150 200 250 3000PV Voltage (V)

0

200

400

600

800

1000

1200

PV P

ower

(W)





Upv

230240250260270280290300310

PV V

olta

ge (V

)

235240245250


５ＪＰＬ

14

145

15

155

16

165

17


(b)

2

7007000002

7007000002

699

6992

6994

6996

6998

700

7002

7004

7006

7008

701

Vdc b

us (V

)

Udc５dcＬ






(a)

ir1ir2

ir3

2

minus1

0

1

1000WＧ2900WＧ2800WＧ2

600WＧ2

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

Curr

ents

ir1 i

r2 an

d ir3

(A)



(b)

2 200006

199 2 201 202198Time (seconds)

minus3

minus2

minus1

0

1

2

3

4

5

Grid

Cur

rent

ir1

(A)

ir1ＣＬ1Ｌ



(a)

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

8

Grid

Cur

rent

s ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


699999

700

699999

700

DC

link

volta

ge (V

)

(b)


UdcUdc ref




(a)

minus3

minus2

minus1

0

1

2

3

4

5

6G

rid cu

rren

ts ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


(b)


700

702

704

706

708

710

DC

link

volta

ge (V

)

udcudcＬ



4

5

6

7

8

9

10

11

T1=L

1r1

002 004 006 008 01 012 014 016 018 020Time (seconds)


(a)

minus3

minus2

minus1

0

1

2

3

Grid

curr

ents

ir1 i

r2 an

d ir3

(A)









(a)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ




5 Conclusion


(b)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ



Data Availability




References





































Journal of












Journal of







Volume 2018






Volume 2018









(a)

0

05

1

15

2

25

3

35

4

45

5PV

Cur

rent

(A)

50 100 150 200 250 3000PV Voltage (V)

1000 WＧ2

900 WＧ2

800 WＧ2

600 WＧ2


(b)

50 100 150 200 250 3000PV Voltage (V)

0

200

400

600

800

1000

1200

PV P

ower

(W)





Upv

230240250260270280290300310

PV V

olta

ge (V

)

235240245250


５ＪＰＬ

14

145

15

155

16

165

17


(b)

2

7007000002

7007000002

699

6992

6994

6996

6998

700

7002

7004

7006

7008

701

Vdc b

us (V

)

Udc５dcＬ






(a)

ir1ir2

ir3

2

minus1

0

1

1000WＧ2900WＧ2800WＧ2

600WＧ2

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

Curr

ents

ir1 i

r2 an

d ir3

(A)



(b)

2 200006

199 2 201 202198Time (seconds)

minus3

minus2

minus1

0

1

2

3

4

5

Grid

Cur

rent

ir1

(A)

ir1ＣＬ1Ｌ



(a)

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

8

Grid

Cur

rent

s ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


699999

700

699999

700

DC

link

volta

ge (V

)

(b)


UdcUdc ref




(a)

minus3

minus2

minus1

0

1

2

3

4

5

6G

rid cu

rren

ts ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


(b)


700

702

704

706

708

710

DC

link

volta

ge (V

)

udcudcＬ



4

5

6

7

8

9

10

11

T1=L

1r1

002 004 006 008 01 012 014 016 018 020Time (seconds)


(a)

minus3

minus2

minus1

0

1

2

3

Grid

curr

ents

ir1 i

r2 an

d ir3

(A)









(a)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ




5 Conclusion


(b)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ



Data Availability




References





































Journal of












Journal of







Volume 2018






Volume 2018









(a)

ir1ir2

ir3

2

minus1

0

1

1000WＧ2900WＧ2800WＧ2

600WＧ2

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

Curr

ents

ir1 i

r2 an

d ir3

(A)



(b)

2 200006

199 2 201 202198Time (seconds)

minus3

minus2

minus1

0

1

2

3

4

5

Grid

Cur

rent

ir1

(A)

ir1ＣＬ1Ｌ



(a)

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

6

7

8

Grid

Cur

rent

s ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


699999

700

699999

700

DC

link

volta

ge (V

)

(b)


UdcUdc ref




(a)

minus3

minus2

minus1

0

1

2

3

4

5

6G

rid cu

rren

ts ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


(b)


700

702

704

706

708

710

DC

link

volta

ge (V

)

udcudcＬ



4

5

6

7

8

9

10

11

T1=L

1r1

002 004 006 008 01 012 014 016 018 020Time (seconds)


(a)

minus3

minus2

minus1

0

1

2

3

Grid

curr

ents

ir1 i

r2 an

d ir3

(A)









(a)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ




5 Conclusion


(b)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ



Data Availability




References





































Journal of












Journal of







Volume 2018






Volume 2018









(a)

minus3

minus2

minus1

0

1

2

3

4

5

6G

rid cu

rren

ts ir1

ir2

and

ir3 (A

)


ir1ir2

ir3

ＣＬ1Ｌ

ＣＬ2Ｌ

ＣＬ3Ｌ


(b)


700

702

704

706

708

710

DC

link

volta

ge (V

)

udcudcＬ



4

5

6

7

8

9

10

11

T1=L

1r1

002 004 006 008 01 012 014 016 018 020Time (seconds)


(a)

minus3

minus2

minus1

0

1

2

3

Grid

curr

ents

ir1 i

r2 an

d ir3

(A)









(a)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ




5 Conclusion


(b)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ



Data Availability




References





































Journal of












Journal of







Volume 2018






Volume 2018










(a)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ




5 Conclusion


(b)

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

Grid

Cur

rent

(A)

002 004 006 008 01 012 014 0160Time (seconds)

ir1ＣＬ1Ｌ



Data Availability




References





































Journal of












Journal of







Volume 2018






Volume 2018









































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

Nonlinear Robust Backstepping Control for Three-Phase …downloads.hindawi.com/journals/mpe/aip/3824628.pdf · probable mismatch between the Maximum Power Points ... of the PV module