Quaderni di applicazione tecnica 10 - impianti fotovoltaici

7/29/2019 Quaderni di applicazione tecnica 10 - impianti fotovoltaici

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IJCSNS International Journal of Computer Science and Network Security, VOL.11 No.6, June 2011 105

Manuscript received June 5, 2011Manuscript revised June 20, 2011

Modelling and non linear control of a photovoltaic systemwith storage batteries: A bond graph approach

Zitouni.NP*P, Khiari.B*, Andoulsi.RP*P, Sellami.AP*P, Mami AP***PT,Hssen.A TP****P

* Laboratory of Photovoltaic and the Semiconductors in CRTEn, BP95 Hammam lif 2050 Tunisia,

** Research Unit: Networks and Machines Electric in INSAT Tunisia***Laboratory of automatic and control systems

**** Laboratory of Water and the Environment in CRTE

Abstract.This paper presents a bond graph modelling ofa photovoltaic source (PV). The insolation variation during

the day poses the problem of energy storage. For that, weused electrochemical batteries which present the best solution

by their good adaptation to photovoltaic source. A maximum

power point tracking (MPPT) device is located between PV

array and batteries to optimise the power transfer from the PV

array and batteries. A non linea control approach of a

photovoltaic system is introduced for the DC-DC converter.

The control strategy is based on state feedback input output

linearization, and the control law is determined from the

established bond-graph model.

Key words:bond graph, photovoltaic source, nonlinear control, storage

battery, DC/DC converter.

1. Introduction

In rural zones, conventional sources of energies arelimited, and photovoltaic energy (PV) becomes a

promising solution. In fact, many advantages arepresented by this energy like the availability on a largescale of planet area. A stand alone photovoltaic systemgenerally uses batteries to maintain supply when the solarenergy is not available [5]. In order to overcome theundesired effect of the PV output voltage, and to assureits maximum power point operating, its possible toinsert a DC-DC converter (MPPT) between the PV

generator and the batteries [3]. There is a thorough studyfound a MPPT control , In order to guide the generatortowards the optimum power point and to

exploit the maximum energy delivered by thephotovoltaic generator. Where [21] develop a PWMcontrol strategies, [20] use genetic assisted, multi-layer

perceptron neural network control,[18]and[19] use fuzzylogic control. In this paper, considering the nonlinearcharacteristic I-V of the PV system, a non linear (N.L)

control law is developed for this system . To exploit themodel bond graph of the photovoltaic system, Thecontrol law is determined from the established bond-

graph model [8][9]. In the first part of this paper, theB.G model for different elements of the system is

presented. In the second part, a theoretical study of thenonlinear control is presented, and the non linearapproach for the control based on the B.G model for thesystem and state feedback linearization is developed,Finally, different simulations are presented.

2. Bond graph modelling for the photovoltaic

source

2.1 The PV array modelThe PV array is basically formed by solar cells. Theequivalent circuit model of a solar cell consists of a

current generator ( phI ) and a diode (D ), serial ( sR )

and parallel ( shR ) resistances [2]. The equation (1) refers

to the PV array current. The equivalent circuit is given byfigure.1[2] [10].

sh

sp

T

psp

sphpR

IpRV

V

IRVIII

+

+= ]1)[exp( (1)

In literature, a model simplification is often done. This

simplification consists in neglecting resistances sR and

shR which facilitates considerably the exploitation of the

PV generator model by avoiding the implicit equations.The simplified equation of the photovoltaic courantintroduced by [2] is given as follows:


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IJCSNS International Journal of Computer Science and Network Security, VOL.11 No.6, June 2011106

Figure. 1. Equivalent schema of a real photovoltaic cell

]1)[exp( = TPSPhP VVIII (2)

Whereq

TKnV BT = R R is the thermodynamic potential,

PhI is the photocurrent of the PV cell proportional to

illumination, SI is the diode reverse saturation current, q

is the electron charge (1,6.10 P-19 Pc), bK is the Boltzman

constant given as flow: (1.38.10 P-23 Pj/k), T is thejunction temperature in Kelvin degree, n is the idealfactor of the photovoltaic cell. The term

]1)[exp( TPs VVI is the darkness diode current

noted DI .

2.2. Bond-graph modelling of the photovoltaicsource

The PV generator has a non-linear current-voltage (I-V) characteristic. This one is characterized by an optimaloperating point corresponding to the maximum powerdelivered by the PV generator. To simulate (I-V)characteristic, we use the bond-graph model representedin figure.2.

Figure. 2. Equivalent scheme of simplified PV array model (a) and itsBond graph model (b)

In order to test the theoretical model, we hadcompared the experimental (I-V) characteristic of the PV

array to the corresponded of BG simulated one. The twocurves are shown in figure.3.

0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

16

18

Panel voltage Up(V)

PanelcurrentIp(A)

0 5 10 15 20 25 30 35 40

0

2

4

6

8

10

12

14

16

Ca ract? istique? perimentale I=f(V)

Ip(A)

Vp(V)

Figure. 3. Simulated (a) and experimental (b) (I-V) characteristic of thePV array.

2.3. Bond graph modelling of the battery and chargeregulator

The storage device (load acid batteries) is difficult tocharacterise and several models are used in literature[2][3][5]. Nevertheless, some of these models require

(a)

(b)

(a)

(b)


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0 time 16.5

A

A

A

30

10

Ub (V)

B

B

B

50

-20

Ip (A)

C

C

C

70

-30

Ib (A)

D

D

D

50

0

Up(V)

E

E

E

1

0

reg-cont

na59 - exp13

D

D

C

C

B

B

AA

information on battery parameters, which are not easilyobtained. The photovoltaic generators present a limitedand fluctuating energy source. For that, batteries

allowing the storage of energy at illumination period areused. The accumulators are characterized by the state of

charge Q , the f.e.m )(QVa and internal resistance aR .There are several alternatives of battery modelling. Weretain that presents an electric model relating to the lead-acid batteries. For an ideal model, the battery is

represented by a simple voltage source bE in series with

a resistance bR (with bR is the internal resistance of the

battery) and a storage capacity bC to model the effect of

charge and discharge of the battery [3].

The model parameters are deduced from the current-voltage characteristics during the battery charge anddischarge. figure.4 depicts the equivalent electric schemaof the energy components. The corresponding bond-graph model is represented in figure.5. The bond-graphmodel simulations are illustrated by figure.6 [10][11].

Where bU is the battery voltage, bI represents the

battery current, PV is PV panel voltage, PI is the panel

current, and chI is the load current.

In this work we modelled a charge regulator, which isassimilated to two switches in the PV generator circuitdisposed on both sides of the battery. The charge

programme establishes operates the switches accordingto the battery charge state and the energy availability(solar illumination).

Figure. 4. Complete electric diagram of the energy components

The order of the switchers must hold in account thesense of battery current (charge or discharge phase). Forthe correct operation of the battery, the regulatordeveloped maintains the battery terminal voltage betweentwo limits. The switch model is developed in [10] [22].

Figure. 5. Bond-graph model of the energy components

In the bond-graph model, the battery is modelled by

an effort source bESe = in series with a resistance

brR : and a capacity bCC: , supplied with aphotovoltaic source (already presented). The charge ofthe battery from PV generator and the discharge towardsa receptor is controlled by a charge regulator. The last isassimilated for two switches, put in series on both sidesof the battery, allowing to limit the over charge and thedischarge of the battery. The switches are modelled by

modulated transformerMTF in serial with resistive

element onRR : . These elements are controlled by aprogram pre-established depending on the battery stateand the PV source. The simulation of the BG model is

presented by figure.6.

Figure. 6. Simulation of the Bond-graph model of the energycomponents

From simulations of figure.6, we have distinguishedthe charge and discharge phases of the battery between

two limits ( bdU and bcU ). The battery voltage is the

indicator of state of charge. According to this state theregulator commutates between two phases, where the

RchRRDIR h VRp

IRch

CRp

rRb

CRb

ERb

Chargeregulator

IR

Sf : IR h 0 : URp 1 : IR 0 : URb 1: IRch R : RRch

1 : IR R : rR

Se : ERb

C : CR

C : CR

R: RonR1 R: RonR2

R:RRD

Switchcontrol


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battery current change the sign panel current joints zero

and the panel voltage )( pV towards the panel open

circuit value)( po

V .

2.4 The bond graph modeling of DC/DC converter

The DC/DC converter model

When the PV generator is directly connected to theload, the system will operate at the intersection of the

VI curve and load line, which can be far from themaximum power point MPP. The converter consists of

power components and control circuit. A control

algorithm permits to assure maximum output power. TheMPPT (Maximum power point tracking) controller usedin our case is based on a boost converter. The equivalentschema of DC/DC converter is presented by figure.7.

Figure. 7 Schema of the buck converter

3.1 The bond graph modeling of DC/DC converter

This method supposes the ideal switches and givesindirectly the average models appropriate to the variousstatic inverters which permit to deduce easily the bond-graph model. For the case of the boost converter, theaverage values of electric parameter are given byequation (3) and (4) [2][6][22].

=+=T

TIdtti

Tdtti

TI

210 11).(

1).(

1 (3)

10 222).(1).(1 Vdttv

Tdttv

TV

T

T

=+= (4)

The BG model corresponding to the DC/DC converterdeveloped by [2][22] is illustrated in the figure.7.The complete BG model of the photovoltaic source isobtained by associating the models developed previously.The global model and the bond-graph model arerespectively represented by the figure.9 and figure.10.

Figure. 8. The bond-graph model of boost converter

Figure. 9. Complete electric schema of the solar energy supply

Figure. 10. Global bond-graph model of photovoltaic system

3. Non linear control law

The main objective is to charge batteries, it consist tomaximize the power absorbed by the batteries [7][8]. Thecontrol law can be determined by an analytical method ordirectly from the B.G model. In this paper we will exploitthe B.G model to determine the control law, this methodincludes five stage [8][9]. In what follows we will

present a study of the input output linearization controllaw


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The control of a broad class of non-linear systems canhave a linear input-output behaviour by choosing anonlinear control law per state return. We point out the

theory of this control for the monovariable case [12] [5].We consider the monovariable model with single output:

( ) ( )( )xhy

uxgxfx

=

+=&(5)

WherenRx , mRu , (.)f et (.)g are two fields

of vectors ofn

R

By deriving y we obtains:

=y& LRfR h+ LRg Rh u uhLhLy gf +=& (6)

Where hLf et LRg Rh hLg respectively indicate the Lie

derivation of h in the direction of vectors fields f

and g .

)(,)(1

xfx

hfdhxhL j

n

j j

f =

>====<

Ifn

g RxxhL ,0)( , The control low has thefollowing form:

vxxu )()( += (8)

Where (.) and (.) are two function of nR This law is given by:

)(1

vhLhL

u fg

+=

(9)

Thus leading to the linear system:

vy =& If 0)( xhLg , we derives (5) to have:

uhLLhLy fgf )(2 +=&& (10)

With )(2 hLLL fff = and )( hLLhLL fgfg =

As previously, If 0hLL fg ,n

Rx the control

law:

( )vLhLL

u ffg

+= 21

(11)

In a way more general, if r is the smallest entirety such

as 02 hLL fg for 2,...,0 = ri

Andnr

fg RxxhLL ,0)(1 ,

then the control law is written:

( )vhLhLL

ur

fr

fg

+=1

1

(12)

Giving:

vyr =)(

The number of derivation of the output necessary toreveal the input is known as relative degree of the model,extension of the definition of the relative degree intolinear[14].The diagram block of a control system by linearization

input-output by return of state is as follows:

Figure. 11 : diagram block of a controller process by linearizationinput-output by return of state

Once the linearization is achieved, several controlmethod can be carried out such as the placement of pole,the reference model .... etcBy using the input-output linearization by state return,the dynamic of a nonlinear system are broken up partlyexternal (input-output) and an internal part(unobservable). Since the external part makes up of alinear relation between y and v (or per equivalence the

canonical form of commandability between y and u ), it

is easy to conceive input v , where the output has the

Input-output

linearizati

Externallinear

control

Processus Output

R

vx

u y


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desired behaviour. It remains to know if the internsdynamic well will behave (if the internal states willremain limited). The interns stability for nonlinear

systems was treated in various work [15][16]. Theessential definitions and results are as follows:

Dfinition 1 :

System (5) have a relative degree rif:

)()()( 2 xhLLxhLLxhL rfgfgg== K = 0 and

)(2 xhLL rfg 0 xRPnP.

The model (5) have a relative degree r if eachn

Rx the output requires r derivations before revealing the

control in (6).

If a system has a relative degree r, there is easy to check

that eachn

Rx 0

there is a vicinity 0u of 0x suchas the change of co-ordinates for the state defined by:

)(

)(

122

111

xhLzT

xhzT

f==

==(13)

M

)(11

xhLzTr

frr

==

With 0)()(1 =xgxdT for nri ....1+= is adiffeomorphism

If we considerT

nr TTZ )....( 12 += ,the equation (5) canbe written in the following normal canonical form:

1211 zz =&

M

rr zz 111&&

= (14)

),(

),(),(

212

2112111

zzz

zzgzzfz r

=

+=

&

&

11zy = (15)

In (14) ),( 211 zzf represent )(xhLr

f and ),( 211 zzg

represent )(1 xhLrf . Now if 0=x is a balance point of

the system below ( 0)0(0)( == handof ), then the

dynamics (16) are the zeros dynamics.

),0( 22 zz =& (16)

These dynamics are the dynamic ones madeunobservable by the state return. Indeed the lawlinearization by state return is the nonlinear equivalentof the poles zero placement in loop closed of the model,in order to make them unobservable [15].It is noted that the subspace:

{ } { }0:)0()(: 1010 ==== zUxhLxhLUxL rff LCan be invariant while choosing:

)),((),(

1211

211

vzzfzzg

u += (17)

The dynamics (16) are the dynamics on this space.The system (5) is known as minimal of phase if thedynamics of the zeros are asymptotic stable as indicates itthe following definition:

Dfinition 2 : [16]

A nonlinear system is known as asymptotically phaseminimal if the dynamics of the zeros is asymptoticallystable.The dynamic interns associated with the linearizationinput-output, corresponding to the last equations: normalform, generally depend on the output. For that wedefines an intrinsic property of the nonlinear systemconsider the system internal dynamics when the entrymakes it possible to maintain the output in zeros. Tosuppose the null output implies that all its derivatives are

null. For that the dynamic interns correspond to themodel where the zeros dynamic describe the movements

on a surface 0M of rn dimension defined by

01 =Z [14].

4. Determination of the control law directly on Bond

Graph

The control law can be determined directly from the B.Gmodel, this method includes five stage [8][9].


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The photovoltaic state equation can be put in thefollowing form (5).

(18)

Stage 1: Bond-graph model with standard causality

The minimal dynamic way or output-input way ( )pv, ,having only one integrator. The number of this integrator

gives the relative output-input degree: 1=r .

Figure.12. Bond-graph model with the minimal dynamic way

Stage 2: development of the reverses model

The method is based on the causal inversion input-outputfor the B.G of the system; it consists to deduce the staticand dynamics input-output control law directly on bond-graph model. For the design of the control system, it isoften useful to determine the input from the desiredoutput. This is obtained by the inversion of the systemdynamics where the input is replaced by the output,which corresponds for a control of the opposite system.To determine the opposite model, the bond graphtechnique was extended by the introduction of theconcept of bi causality [13].

Figure.13. The opposite bond-graph model

The currents3

f et6

f became the output of the

transformerWith:

6

31

f

fm ==

(19)

From this equation we can deduce the control law :U

6

3

f

fU ==

=

=

63

36

ff

ee

(20)

Stage 3: Deduction of the minimal dynamic equation

We deduce a mathematical model reverses calledminimal dynamic equation. The equation is obtainedwhile reading the bond graph model with oppositecausality.The cyclic ratio of the boost converter is the input of thecontrol law U .

The state vector isT

cbcpch qqpx ][=

The output is the voltage of the generator: pvy =

225

46 ,, fdt

dvcq

R

ef

l

pf

p

Dch

ch ==== & (21)

According to(20) and (21) we obtain:

+

=

ph

b

cb

cp

ch

ch

pDch

bpchb

cb

cp

ch

I

E

q

q

p

l

cRl

cclr

q

q

p

00

10

01

001

01

1

&

&

&


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ch

chTpsph

ch

ch

pTtpsph

p

lycvvII

p

lU

vcvvIIf

&

&

=

=

)1)(exp[

]1)[exp(3(22)

Stage 4: development of control law

From the dynamic equation minimal, we replace

)1()( =ry r& by a new control input v ; the input-outputrelation system (with feed-back) becomes linear

according to the equation: vyy r == & The linearization static law is:

vp

lcIcvqI

p

lU

ch

chphTcs

ch

ch += ])1)exp([( (23)

The feedback linearization control law is inserted inPV voltage control loop in order to reach the optimalvoltage. This objective can be achieved by a simple

control law )(: yykvv d =

In this case, the control transformed the system intointegrating block, a proportional P regulator issuitable.We can make the measurable sizes:

vI

cIvvI

IU

b

phTps

b

+= ])1)exp([(1

(24)

The diagram of the system is given by the followingfigure.

Figure.14. Diagram of the system with control law

This control law made two states unobservable

( chch pq , ) from the new control v . According to thenonlinear control theory, this variety represents thedynamics of the zeros of the system whose study ofinternal stability is essential

Stage 5: study of stability

The dynamics of the zeros is obtained by

imposing 0=y , all bonds attached of 0 junction iscancelled, consequently the power is cancelled. We applythe second method of lyapunov to the bond-graph toanalyze the stability of dynamics of the zeros. We canchoose the sum of the energies stored in the elements of

storage Iet C like function candidate of lyapunov [9].

))()

(2

1

)()()(

22

1 1

b

cb

ch

ch

n

i

n

j

jjnii

c

q

l

pV

qpVtEi j

i

+=

+== = =

+

(25)

The derivative of this energy represents the differencebetween the power delivered by the source and theconsumption by the dissipative elements.

0])([)()(

)()()(

22

1 1211


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and acV ). The battery voltage is the indicator of charge

state of the battery. According to this state the regulatorcommutates between two phases, where the battery

current change the sign and panel current joins zero. Thecontrol law stabilise the system on the desired output

power and the output follows the input.

Figure 15. The BG model with feedback linearization input outputcontrol

5. Conclusion

The pilot unit, the object of our study, consists of aphotovoltaic generator, storage battery with chargeregulator, Maximum power point tracking (MPPT) andthe loads. With these varied elements, the modelling ofthis pilot unit becomes complex. For that, we proposed a

unified approach of modelling based on graphictechnique known as bond-graph. This technique issystematic and has a sufficient flexibility to be able tointroduce various components into the system and toestablish a nonlinear control law.In this paper an electric model of the photovoltaic sourceis presented and the complete BG model for differentelements of the system is elaborated. This model wasused to design a state feedback linearization input outputcontrol law. A non linear approach has been proposed tocontrol the loading of a battery and exploit the maximumof energy delivered by the PV panel. In this paper wehave exploit the B.G model to determine the control law,

(this method includes five stages). Reliable simulationresults are presented to demonstrate the validity of the

proposed control approach.

0 2 4 6 8 10 12 14 16

28

30

32

Up

and

Upref(V)

0 2 4 6 8 10 12 14 160

0.5

1

controllaw

u

0 2 4 6 8 10 12 14 1610

20

30

Ub(V)

0 2 4 6 8 10 12 14 165

10

15

Times (s)

Ib(V)

Figure.16. Simulation of model BG of the system with state feedbacklinearization control

0 2 4 6 8 10 12 14 16

20

30

40

Up,

Ub,

Uo

p

(V)

0 2 4 6 8 10 12 14 160

0.5

1

charge

control

0 2 4 6 8 10 12 14 16

0

10

20

Ip(A)

0 2 4 6 8 10 12 14 16-20

-10

0

10

Times(s)

Ib(A)

Ub(V)

Up(V)

Upref(V)

Figure. 17. Simulation of model BG of the system with chargeregulator and state feedback linearization input-output control

law.


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for averaged modelling of power electronic converters.

Proceeding of the SCS International Conference on Bond-graph Modeling (ICBGM 97). Phoenix, USA, Vol 29, N1, 201-206.

[2] R.Andoulsi, A. Mami, G. Dauphin-Tanguy, M. annabi,Modelling and simulation by Bond Graph technique of aDC motor fed from a photovoltaic source via MPPT boostconverter. Proceeding of CSSC99, pp 4181-4187.

[3] Bogdan S.Borouy, Zied M. Salameh: Dynamic reponse ofa stand-Alone wind energy Conversion System with

battery energy storage to a wind Gust. IEEE transaction onEnergy Conversion. Vol 12.N1,marsh1997.

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stand-Alone wind energy conversion system with batterystorage to a wind gurst, IEEE Transaction on Energyconversion vo1. 12, issue 1 Marsh 1997, 73-75.

[6] V. Mahout,On the robustness of the flatcontrol, ISC3,juin 2003.

[7] H. SiraRamirez and Silva Navarro G.,Regulation andtraking for the overage boost converter circuit: ageneralized proportional integral approach, Int. Journal ofcontrol, vol. 75, pp. 988-1001, 2002.

[8] Bertrand, J.C. Sueur, G. Dauphin-Tangy. Bond Graph forModelling and Control Structural Analysis Tools for thedesign input output Decoupling State Feedbacks.Proceding ICBGM1997/Simul.Series 29:1.

[9] Junco. Sergio, Linarisation exacte entre-sortie et stabilit

de la dynamique des zeros directement sur bon graph dessystmes non linaires. Actes de CIFA 2000, Premireconfrence Internationale Francophone dautomatique, 4-7

juillet 2000.Lille, France.[10]Junco. Sergio, Lyapunovs second method and feed-back

stabilization directly on Bond Graph, Proceeding ofICGBM01, 2001, pp 86-98.

[11]Zitouni.N, Andoulsi.R, Sellami.A, Mhiri.R, Bond graphmodelling of a UV water Disinfection pilot unit fed by a

photovoltaic source. JTEA'2006, 12-14 may 2006 Hammamet, Tunisie.

[12]P.J. Gawthrop, D.J. Wagg, S.A. Neild, Bond graph basedcontrol and substructuring, Simulation Modelling Practiceand Theory 17 (2009) pp 211227.

[13]N. Yadaiah, N. Venkata Ramana, Linearisation of multi-

machine power system: Modeling and control A survey,Electrical Power and Energy Systems 29 (2007) 297311

[14]Roger F. Ngwompo, Peter J. Gawthrop, Bond graph-basedsimulation of non-linear inverse systems using physical

performance specifications, Journal of the FranklinInstitute 336 (1999) 1225}1247

[15]J. J. E. Slotine W.Li, Applied nonlinear control, Printice-hall international editions. Englewood Cliffs, New Jersey1991.

[16] C Kravaris, Input/output linearization: A nonlinear analogof placing poles at process zeros. AICHE Journal, Vol 34,

N11, November 1988, pp 1803-1812.

[17] A. Isidori, A tool for semi-global stabilizationof uncertainnon-minimum-phase nonlinear systems via output feed-

back. IEEE trasaction on automatic control, Vol 45.N 10October 2000, pp 1817-1827.

[18]I.H. Atlas and A.M.Sharaf, Anouvel maximum powerfuzzy logic controller for photovoltaic solar energysystems, Renewable energy 2007.

[19]Nopporn Patcharaprakiti, Suttichai.P andYosanai.S,Maximum power point traking usingadaptative fuzzy logic control for grid-connected

photovoltaic system, Renewable energy 30 (2005), pp1771-1788.

[20]R.Akkaya, A.A.Kulaksiz and . Aydogdu, DSPimplementation of a PV system with GA-MLP-NN basedMPPT controller supplying BLDC motor drive, EnergyConversion and Management 48 (2007), pp 210-218.

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MPPT and PWM control strategies, IEEE MELECOM2006.[22]D.Mezghanni, R. Andoulsi, A. Mami, G. Dauphin-Tanguy,

Bond graph modelling of a photovoltaic system feedingan induction motor-pump Simulation Modelling Practice andTheory 15 (2007) 12241238.

Naoufel,Zitouni is born in Tunis ; Tunisia 1978. He obtainedthe engineer diploma of engineer and master diploma inIndustrial Informatics and Automatics from (ESSTT) Tunisiarespectively in 2002 and 2004. He is a thesis student. His fieldof interest concerns the photovoltaic power, treatment water,

bond-graph modelling.Zitouni Naoufel, FST,Post address: campus universitaire, el Manar II, Tunis, email:

[email protected]

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Quaderni di applicazione tecnica 10 - impianti fotovoltaici