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International Journal of Electrical & Computer Sciences IJECS-IJENS Vol:12 No:04 10

124103-04-3939-IJECS-IJENS August 2012 IJ ENS I J E N S

AbstractA photovoltaic system connected to the grid isconstituted by two main parts: a solar cell and a power converterDC-AC. The impedance adaptation between the two parties is atechnological problem that essentially means the transfer ofmaximum power of PV generator to the grid. In this paper, anew technique for nonlinear control of power converter DC-ACis presented. This converter is built on two stages: a DC-DC

stage and a DC-AC stage. The two blocks are bound by a DCvoltage intermediate bus. The control objective is threefold: i)control the input voltage of the converter to find the MPP(maximum power point), ii) regulating the DC bus voltage, iii)generate a sinusoidal current at industrial frequency that will beinjected into the network with a power factor near unity. Thesynthesis of controllers was performed using the technique ofbackstepping nonlinear control. A detailed analysis of thestability control system is presented. The performance ofcontrollers has been validated by numerical simulation inMATLAB / SIMULINK.

Index Term DC-AC converter, Backstepping, MPP,photovoltaic system, power factor.

I. INTRODUCTION

Solar radiation is the most shared energy resource on earthand the most abundant: The quantity of energy released by thesun (captured by the planet earth) for one hour may besufficient to meet global energy needs for one year.Part of thisradiation can be used to directly produce heat

Abdelhafid Ait Almahjoub is serving in

M.I.T. L'University Hassan 2 mohamedia-Casablanca, [email protected]

A.Ailane is serving inM.I.T. L'University Hassan 2 mohamedia-Casablanca, Morocco

M.Rachik is serving inM.I.T. L 'University Hassan 2 mohamedia-Casablanca, Morocco

A.Essadki is serving inD.GE ENSET University Mohamed 5 Rabat Morocco

J.Bouyaghroumni is serving inM.I.T. L 'University Hassan 2 mohamedia-Casablanca, Morocco

(solar thermal) or electricity, it is solar photovoltaics. This

production method does not require a distribution network.

Indeed it can generate electricity where it is consumed(Villages, isolated houses, relay communications, water

pumping, Shelters ...).

Some countries are introducing measures to encourage the

producing of electricity from solar energy. And within thisframework, the energy is purchased at an attractive price(price per kWh produced higher than the price per kWhconsumed and billed by the energy supplier).

The conversion of this energy to electrical energy is performedby through a photovoltaic cell (PV)[1].

The voltage generated may vary depending on the materialused for the manufacture of the cell. The combination ofseveral PV cells in series / parallel result in a photovoltaicgenerator (GPV) which has a current-voltage (I-V) non-linearwith a maximum power point.

The I-V characteristic of GPV depends on the level ofillumination (Figure 1) and the temperature of the cell (Figure2) and the aging of the assembly. Moreover, its operatingpoint depends directly on the load that it supplies. In order toextract in every moment the maximum power available at theterminals of the GPV, we introduce an adapting stage betweenthe generator and the load to couple the two elements asperfectly as possible.In this paper, only the adapting stage, which is performed by apower converter DC-AC, is investigated in order to implementcontrol strategies to improve the energy behavior of thesesystems.More investigations have been published on various topologies

of DC-AC converters [7] [8] [9] [10]. Among these

topologies, which is based on the cascade of DC-DCconverters and DC-AC has a great feature (simple

configuration). Figure 3 shows the schematic diagram of such

a system. It is constituted of the one part, of a bridge chopper

for converting the input DC voltage into a DC voltage higher,and secondly an inverter capable of generating, from this DC

voltage, an alternating current which is injected directly into

the power grid.

Non-linear Control of a Multi-loopDC-AC Power Converter Using in

Photovoltaic SystemConnected to the Grid

Abdelhafid Ait Almahjoub, A.Ailane, M.Rachik, A.Essadki, J.Bouyaghroumni

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Using means models, widely used in AC-DC convertersliterature, several control design techniques have beenproposed, such as the classic PID controller that uses small-signal linear models [11] [12]. In this work we propose tocontrol this type of converter by the nonlinear backsteppingapproach. The concept is to calculate an appropriate controllaw to guarantee the global asymptotic stability of the closedloop system [13] [14] [15] [16].

Fig. 1. I=F(V) T =25C depending on irradiation

Fig. 2. I=F(V) depending on the junction temperature

Fig. 3. DC-AC converter

II. MATHEMATICAL MODEL OF THE CONVERTER

The converter shown in Figure 3 consists of a chopper andan inverter using switches such as IGBTs. These transistorsare controlled by the principle, widely known in the literature,the pulse width modulation (PWM). Thus, the arms ((k1, k2),(k3, k4)) and the switch k5 we associate respectively the binary

switching functions 1 and 2 such that:

1 4 2 31

1 4 2 3

1 if k and k are ON and k and k are OFF

1 if k and k are OFF and k and k are ON

52

5

1 if k is ON

1 if k is OFF

The cyclical and binary signals 1 and 2 are the control

inputs of the AC-AC. They vary from one period to another,and their variations can determine the trajectories of the statevariables of the converter such as the currents in the inductorsand the voltages across the capacitors.

The development of the model switched system is based on theapplication of Kirchhoff's laws. Thus, we obtain

1out

o o out rs dc

diL r i v v

dt (1a)

21

1

2dc

out in

dvC i i

dt

(1b)

21

2in

i pv i in dc

diL v r i v

dt

(1c)

pvinpv

i iidt

dvC (1d)

The above model proves to be unsuitable for the development

of continuous control laws since it involves, as input variables,the binary signals 1 and 2 . To overcome this

inconvenience, the equivalent average model is used. Thus,we obtain the model

1 1 1 2o o resL x r x v u x (2a)

22 1 1 3

1

2

uCx u x x

(2b)

iinipv

vpv

vres

iout

vdc

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23 3 4 2

1

2i i

uL x r x x x

(2c)

4 3i pvC x x i (2d)

where 1x , 2x , 3x , 4x , 1u , 2u and pvi represent, respectively,

the means values, over a period of cutting, of variables outi ,

dcv , ini , pvv , 1 , 2 and pvi .

Note that the mathematical model (2) is nonlinear because ofthe products involving the state variables and input signals.

The non-standard form of this model leads us to choosetechniques taking into account the nonlinearities such asapproach backsteeping.

III. CONTROLLER DESIGN

A. The aim of controller

The converter control strategy must be developed to: i)achieve the MPPT (Maximum Power Point Tracking) side ofthe PV cell by setting the operating point current / voltage, ii)

ensure the setting of the DC bus voltage iii) manage thetransfer of power from the network side, such that theincoming average active power equals outgoing active power(around losses). Moreover, the quality of injected current hasto be the best possible.

The controller synthesis will be performed in three steps.First, an output current inner loop (regulator 1) is designed toinject a sinusoidal current in the network and have a powerfactor close to unity. In the second step, an input current innerloop (regulator 2) is designed to have a DC voltage at theinput whose amplitude is determined by the reference signal(command received from the system controlling MPPT). Inthe end an outer voltage loop (regulator 3) is built-up to

generate the reference signal, which will be used by thecontroller 1, to regulate the DC bus voltage at its desired

reference value 2 DCx =v

.

The proposed control system will have the structure shown inFigure 4. Both controllers 1 and 2 will be synthesized by atechnique using Backstepping approach and the third will bedone by a simple proportional-integral corrector.

Fig. 4. Les boucles de rgulations du convertisseur DC-AC

B. Output current inner loop design (regulator 1)

The PFC objective means that the converter output currentshould be sinusoidal and in phase with the grid supplyvoltage. It amounts to ensuring current harmonics rejection.We therefore seek a regulator that enforces the current x1 to

track a reference signal of the form *1 inx = v when IR .

The block diagram in Figure 5 shows the basic structure of thecontrol loop of the output current.Taking into account the equations (2a) the relative degree ofthe system in relation with the variable x1 is 1, thebackstepping synthesis of the regulator will be done in onestep.

Fig. 5. Output current inner loop

Consider the tracking error z1 defined by

)( *111 xxLz o

its dynamics is given by:

)( *111 xxLz o (3)

Let us use the Lyapunov candidate function211 5.0 zV

As its derivative with respect to time, is given by

111 zzV

the choice 111 zkz (4)

Where k1 is a positive constant synthesis, leads to a Lyapunov

candidate function whose dynamics is negative definite.So 2111 zkV

Therefore global asymptotic stability is achieved and z1tends exponentially to 0.Using (2a) , (3) and (4) we will have :

*1 1 1 1 2 1o res ok z r x v u x L x

x2 will initially be equal to 2E because the capacitor (C) of

the DC bus is automatically loaded by the DC-AC converterthat acts as a rectifier diode (D1, D2, D3 and D2) at systemstartup if we delay the switch control (k1 , k2, k3, k4).Solving the previous equation with respect to u1 led to thebackstepping control law as follows:

*1 1 1 1 1

2

1( )o res ou k z r x v L x

x (5)

Proposition 1.Consider the DC-AC converter of Figure 3 which is

described by the model means (2). If the first derivative of isavailable, then the control law (5) guarantees globalasymptotic stability of the error signal z1.

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C. input current inner loop design (regulator 2)

The block diagram of figure 6 shows the control loop of theinput voltage. It calculates the control law of the second stageof the power converter DC-DC to regulate the output voltagex4 to its desired reference value x4

*.

Fig. 6. Input voltage inner loop

Taking into account the equations (2c) and (2d) the relativedegree of the system, by respect to the variable x4, is 2, the

backstepping synthesis of the regulator will realized in two

stages.

We define an error variable*

4 4 4( )iz C x x

the dynamics of the error z4 is given by*

4 4 4( )iz C x x (7)

So with (2e)*

4 3 4pv iz x i C x (8)

Consider the following candidate Lyapunov function2

4 40.5V z

its derivative with respect to time is given by

4 4 4V z z

The choice 4 4 4z k z (9)

Where k4 is a positive constant synthesis, leads to a

Lyapunov candidate function whose dynamics is negativedefinite.using (8) and (9) we will have

*4 4 3 4pv ik z x i C x (10)

if we choose x3 as virtual control input we deduce thestabilizing function

* *3 4 4 4pv ix k z i C x

(11)As x3 is not the control input, a new error variable z3 betweenthe virtual control x3and its desired value x3

* is introduced:*

3 3 3z x x (12)

The dynamics of the error z3 is given by:*3 3 3z x x (13)

Using (8) , (9), (11) and (12), we will have :

4 3 3 3z z k z (14a)2

4 4 4 3 4V k z z z

(14b)

Now consider the dynamics of the error z3 taking into account(2c), so:

*23 3 4 2 3

11

2i

i i i

r uz x x x x

L L L

(15)

We see appear for the first time, the true signal of the controlnoted u2.The objective now is to stabilize the system (z4, z3),for this we take as a Lyapunov candidate function thefollowing function:

2 2

3 4 3

1 1

2 2V z z

Its dynamics is given by: 3 4 4 3 3V z z z z .

Using (14a) 23 4 4 3 4 3( )V k z z z z

The choice 4 3 3 3z z k z (16)

Where k3 is a positive constant synthesis, which guaranteesthe negativity of the dynamics of the Lyapunov candidatefunction, because:

2 23 4 4 3 3V k z k z

0

the equations (15) and (16) lead to:

*23 4 2 3 3 3 4

11

2i

i i i

r ux x x x k z z

L L L

We deduce then the following control law:

*2 3 4 3 3 3 4

2

21 ( )i i i iu r x x L x k L z L z

x (17)

Proposition 2.Consider the DC-AC converter of Figure 3 which is

described by the model means (2). The control law (17)guarantees the global asymptotic stability of error signals z3and z4. Moreover the dynamics of these errors is described bythe following model:

4 4 4

3 3 3

1

1

z k z

z k z

D. outer voltage loop design (regulator 3)

The block diagram in Figure 7 shows the structure of theregulator of DC bus voltage.

Fig. 7. Structure for regulating the DC bus voltage

The aim of the outer loop is to generate a tuning law for the

ratio in such a way that the DC bus voltage x2 be regulatedto a given reference value x2

*. The first step in designing is toestablish the relation (model) between the ratio (controlinput) and the DC voltage x2.

Hypothesis:H1

The inner loop input current (regulator 1) and the inner loopoutput current (regulator 2) are supposed to have fast

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dynamics compared to of the outer loop of DC bus voltage(regulator 3).

H2The voltage drops across the different coils and their parasiticresistances are assumed negligible compared to the input andoutput voltages.Based on the hypotheses H1 and H2, the control laws (5) and

(17) reduce to:

12

1resu v

x (18a)

2 42

21u x

x (18b)

If we substitute u1 and u2 given by expressions (18) in (2b)the model becomes:

2 2 1 4 3

1( )resx x xv x x

C (19)

setting 22xy , its dynamics becomes:

2

E ( )y p tC (20)

Where2

3 4

E 2( ) cos(2 )p t t x x

C C

Note that the model (20) can be seen as an integratorperturbed by the signal )(tp . The controller design is based on

the average model as follows:

)(tpky (21)

with

T

ydtT

y

0

1

2Ek

C

Tdt

T 0

1

3 4

2( )

0

Tp t x x dt

CT

Where T is the period of the network.The control model (21) has been verified by simulation. So,the step response results in a signal y (the square of the outputvoltage) ramp type perturbed by undulations of low amplitude.

The system (21) can be stabilized using a simple PI controllerthat has the transfer function given by

kksC ip1

)( .

The block diagram of the controlled system is shown in Figure8.

Fig. 8. Equivalent loop voltage

In closed loop, the output signal y depends on the referencesignal y* and the disturbing signal p (t) by the equation:

)().()().()( * sPsGsYsFsY (22)

With:

20

2

0

1

21

1)(

ss

ssF

and

20

2

0

2

21

)(

ss

ssG

Where:

ip kkk /1 , ik/12 , ik0 ,

i

p

k

kk

2

1

The transfer relation (22) shows that the proposed controller

guarantees a perfect pursuit *lim( ) 0t

y y

and a rejection

disturbance since G (s) contains a derivator effect.The Bode diagram of Figure 9 shows the frequency responseof the transfer function F (s).

100

101

102

103

-90

-60

-30

0

P

hase

(deg)

BodeDiagram

Frequency (rad/sec)

-30

-20

-10

0

M

agnitude

(dB)

Fig. 9. Bode diagram of F (s)

IV. SIMULATIONRESULTS

The performance of the proposed controller has beenvalidated by simulation in the environment MATLAB /SIMULINK. Table 1 shows all the parameters of thecontrolled system.

TABLE ISimulation parameters

Item valuegrid E 230 2V

100 rad/sDC -DC converter Li 20mH

ri 30mCi 2000F

DC-AC converter Lo 20mHro 30mC 4000F

Switching frequency fPWM 10Khz

Figures 10 to 15 gives the simulation results when thereference of the DC bus voltage (VDC) is an echelon going

from 500V to 600V and the reference voltage input (Vpv) is set

at 10V and then passed to 12V.

yC(s)y*

-k 1 /s

p(t)

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Figure 10 shows that the output current of the converter is

sinusoidal, and in Figure 11, we note that the output current issinusoidal and in phase with the grid voltage, this shows that

the power factor correction is perfectly realized.

Figure 12 shows the evolution of the converter input voltage,in particular it is noted that this voltage follows the reference

voltage.

Figure 13 shows that the DC bus voltage follows perfectly (onaverage) its reference.Finally in Figures 14 and 15, are presented the controlsignals 1 and 2 , it is clear that they are bounded.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-100

-50

0

50

Fig. 11. Courant iout

0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3-6

-4

-2

0

2

4

6

ioutvrs

Fig. 12. Courant iout et tension vrs (vrs/60)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

vpv

vpvref

Fig. 13. Tension lentre vpv

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1200

400

600

800

vdc

vdcref

Fig. 14. Tension du bus continu vdc et sa rfrence

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

Fig. 15. La loi de commande 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

Fig. 16. La loi de commande

2

V. CONCLUSIONS

In this paper, we presented a new control technique using

the backstepping approach to control DC-AC converters used

in photovoltaic connected to the grid. The control objective isthreefold i) injecting a perfectly sinusoidal current in the grid

and power factor correction ii) regulating the input voltage iii)

regulating the DC bus voltage between the chopper stage and

the inverter stage of the DC-AC single phase used. Thestudied system is described by a representation of nonlinear

state average of order 4. The synthesis of the regulator was

achieved by having recourse to advanced tools of nonlinearcontrol such as stability in the sense of Lyapunov. It wasshown by simulation that the proposed controller guarantees

the desired tracking performance and stability.

ACKNOWLEDGEMENTS

This work is supported by faculty Ben Msik of University

Hassan 2, L.T. Elkhaouarizmy, Moroccan school scienceengineer (EMSI), the network of systems theory of Morocco

REFERENCES

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[4] C. Alonso, M. F. Shraif, A. Martinez, Brevet CNRS, US 2005099166,Power converter control for automatic maximum power point tracking2005.

[5] C. CABAL, Optimisation nergtique de ltage dadaptation lectroniqueddi la conversion photovoltaque.Thse Universit Paul Sabatier,

Toulouse 2009.[6] M. Ahmed, A. Hably, S. Bacha Grid-connected Kite Generator System:

Electrical Variables Control with MPPT, 37th Annual Conference of the

IEEE Industrial Electronics Society (IECON 2011)[7] K. Noppadol, W. Theerayod, S. Phaophak, FPGA Implementation ofMPPT using variable step-size P&O algorithm for PV applications,Communication and Information Technologies, ISCIT06, IEEEInternational Symposium On, pp. 212- 215, Sept. 2006.

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