9
Climatic sensorless maximum power point tracking in PV generation systems H. El Fadil n , F. Giri GREYC Laboratory, UMR CNRS, University of Caen Basse-Normandie, Caen, France article info Article history: Received 3 January 2010 Accepted 24 January 2011 Available online 23 March 2011 Keywords: Photovoltaic arrays dc–dc converters MPPT Sensorless control Adaptive control Uncertain reference trajectory abstract The problem of maximum power point tracking (MPPT) is addressed for photovoltaic (PV) arrays considered in a given panel position. The PV system includes a PV panel, a PWM boost power converter and a storing battery. Although the maximum power point (MPP) of PV generators varies with solar radiation and temperature, the MPPT is presently sought without resorting to solar radiation and temperature sensors in order to reduce the PV system cost. The proposed sensorless control solution is an adaptive nonlinear controller involving online estimation of uncertain parameters, i.e. those depending on radiation and temperature. The adaptive control problem at hand is not a standard one because parameter uncertainty affects, in addition to system dynamics, the output-reference trajectory (expressing the MPPT purpose). Therefore, the convergence of parameter estimates to their true values is necessary for MPPT achievement. It is formally shown, under mild assumptions, that the developed adaptive controller actually meets the MPPT objective. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction The end of cheap petroleum price era, together with increasing requirements for environmental preservation, makes it more than ever necessary to develop clean and sustainable power generation sources. In this respect, photovoltaic power generators have gained a great popularity in recent years, due to their increasing efficiency and decreasing costs. Indeed, PV systems produce electric power without harming the environment, by transform- ing into electricity a free inexhaustible source of energy, i.e. solar radiation. Furthermore, PV devices are now guaranteed to last longer than ever and manufacturer warranties go over 20 years. Also, governments encourage resorting to such energy solutions through significant tax credits. All these considerations assure a promising future for PV generation systems. The dependence of PV array power upon atmospheric conditions is commonly defined by power–voltage (P V) characteristics asso- ciated to PV arrays. Figs. 1 and 2 show these characteristics for the SM55 PV module and illustrate the corresponding MPP, denoted (V m , P m ). The characteristics show in particular that the array power depends nonlinearly on the array terminal voltage. Moreover, the MPP varies with the changing radiation and temperature, necessi- tating continuous adjustment of the array terminal voltage if maximum power is to be transferred. The problem of maximizing PV power transfer, for different loads, has been dealt with following various approaches. In Khouzam (1990) and Applebaum (1987), PV power maximization was attempted by adapting the PV module characteristics to particular loads. In Braunstein and Zinger (1981), the array configuration (parallel and series connections) was tuned so that the MPP matches the load demands. The point is that a given configuration is only optimal for specific atmospheric and load conditions. When these conditions change energy maximization is lost. Another solution of PV maximization, or MPPT, is that based on PV array load adjustment to meet the MPP all time (Hiyama & Kitabayashi, 1997; Hussein, Muta, Hoshino, & Osakada, 1995). In Chiang, Chang, and Yen (1998), a residential PV energy storage system including a dc–dc boost converter, represented by a small- signal linear model, was considered. The achievement of MPPT was attempted using a current-mode control. In Eakburanawat and Boonyaroonate (2006), a linear controller was proposed to ensure MPPT for a PV array controlled through a SEPIC dc–dc converter. The controller design was performed under the assumption that the temperature is the only varying parameter. In Leyva et al. (2006), stability analysis of a MPPT extremum-seeking based control system was made or a PV array supplying a dc–dc switching converter. The analysis did not take into account the converter dynamics, i.e. these were reduced to a steady-state gain. More recently, Fangrui, Shanxu, Fei, Bangyin, and Yong (2008) presented a modified variable step size incremental conductance algorithm seeking MPPT with speed and accuracy improvement. Again, the converter dynamics were neglected in the algorithm design. A common assumption in all previously mentioned works is that accurate measures of the solar radiation and ambient temperature are supposed to be available all time. It is technically possible to meet this requirement by equipping the PV panel with radiation and temperature sensors. The point is that this solution Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/conengprac Control Engineering Practice 0967-0661/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2011.01.010 n Corresponding author. E-mail address: [email protected] (H. El Fadil). Control Engineering Practice 19 (2011) 513–521

Climatic sensorless maximum power point tracking in PV generation systems

Embed Size (px)

Citation preview

Page 1: Climatic sensorless maximum power point tracking in PV generation systems

Control Engineering Practice 19 (2011) 513–521

Contents lists available at ScienceDirect

Control Engineering Practice

0967-06

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/conengprac

Climatic sensorless maximum power point tracking in PV generation systems

H. El Fadil n, F. Giri

GREYC Laboratory, UMR CNRS, University of Caen Basse-Normandie, Caen, France

a r t i c l e i n f o

Article history:

Received 3 January 2010

Accepted 24 January 2011Available online 23 March 2011

Keywords:

Photovoltaic arrays

dc–dc converters

MPPT

Sensorless control

Adaptive control

Uncertain reference trajectory

61/$ - see front matter & 2011 Elsevier Ltd. A

016/j.conengprac.2011.01.010

esponding author.

ail address: [email protected] (H. El Fad

a b s t r a c t

The problem of maximum power point tracking (MPPT) is addressed for photovoltaic (PV) arrays

considered in a given panel position. The PV system includes a PV panel, a PWM boost power converter

and a storing battery. Although the maximum power point (MPP) of PV generators varies with solar

radiation and temperature, the MPPT is presently sought without resorting to solar radiation and

temperature sensors in order to reduce the PV system cost. The proposed sensorless control solution is

an adaptive nonlinear controller involving online estimation of uncertain parameters, i.e. those

depending on radiation and temperature. The adaptive control problem at hand is not a standard

one because parameter uncertainty affects, in addition to system dynamics, the output-reference

trajectory (expressing the MPPT purpose). Therefore, the convergence of parameter estimates to their

true values is necessary for MPPT achievement. It is formally shown, under mild assumptions, that the

developed adaptive controller actually meets the MPPT objective.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The end of cheap petroleum price era, together with increasingrequirements for environmental preservation, makes it more thanever necessary to develop clean and sustainable power generationsources. In this respect, photovoltaic power generators havegained a great popularity in recent years, due to their increasingefficiency and decreasing costs. Indeed, PV systems produceelectric power without harming the environment, by transform-ing into electricity a free inexhaustible source of energy, i.e. solarradiation. Furthermore, PV devices are now guaranteed to lastlonger than ever and manufacturer warranties go over 20 years.Also, governments encourage resorting to such energy solutionsthrough significant tax credits. All these considerations assure apromising future for PV generation systems.

The dependence of PV array power upon atmospheric conditionsis commonly defined by power–voltage (P�V) characteristics asso-ciated to PV arrays. Figs. 1 and 2 show these characteristics for theSM55 PV module and illustrate the corresponding MPP, denoted(Vm, Pm). The characteristics show in particular that the array powerdepends nonlinearly on the array terminal voltage. Moreover, theMPP varies with the changing radiation and temperature, necessi-tating continuous adjustment of the array terminal voltage ifmaximum power is to be transferred. The problem of maximizingPV power transfer, for different loads, has been dealt with followingvarious approaches. In Khouzam (1990) and Applebaum (1987), PV

ll rights reserved.

il).

power maximization was attempted by adapting the PV modulecharacteristics to particular loads. In Braunstein and Zinger (1981),the array configuration (parallel and series connections) was tunedso that the MPP matches the load demands. The point is that a givenconfiguration is only optimal for specific atmospheric and loadconditions. When these conditions change energy maximization islost. Another solution of PV maximization, or MPPT, is that based onPV array load adjustment to meet the MPP all time (Hiyama &Kitabayashi, 1997; Hussein, Muta, Hoshino, & Osakada, 1995).In Chiang, Chang, and Yen (1998), a residential PV energy storagesystem including a dc–dc boost converter, represented by a small-signal linear model, was considered. The achievement of MPPT wasattempted using a current-mode control. In Eakburanawat andBoonyaroonate (2006), a linear controller was proposed to ensureMPPT for a PV array controlled through a SEPIC dc–dc converter. Thecontroller design was performed under the assumption that thetemperature is the only varying parameter. In Leyva et al. (2006),stability analysis of a MPPT extremum-seeking based control systemwas made or a PV array supplying a dc–dc switching converter. Theanalysis did not take into account the converter dynamics, i.e. thesewere reduced to a steady-state gain. More recently, Fangrui, Shanxu,Fei, Bangyin, and Yong (2008) presented a modified variable stepsize incremental conductance algorithm seeking MPPT with speedand accuracy improvement. Again, the converter dynamics wereneglected in the algorithm design.

A common assumption in all previously mentioned works isthat accurate measures of the solar radiation and ambienttemperature are supposed to be available all time. It is technicallypossible to meet this requirement by equipping the PV panel withradiation and temperature sensors. The point is that this solution

Page 2: Climatic sensorless maximum power point tracking in PV generation systems

0 5 10 15 20 25 300

10

20

30

40

50

60

Power-Voltage charaterestic (λ = 1000 W/m2 ; T var)

Pow

er P

p (W

)

5 4 3 2 1

Temperature T (K):

1. 288.152. 298.153. 318.154. 333.155. 363.15

Voltage Vp (volts)

MPP (Vm, Pm)

Fig. 1. P–V characteristics of the SM55 PV module with constant radiation and

varying temperature.

0 5 10 15 20 25 300

10

20

30

40

50

60

70

Pow

er P

p (W

)

7

6

5

4

3

2

1

1. 1002. 2003. 4004. 6005. 8006. 10007. 1200

MPP

Power-voltage charaterestic (T = 45ºC; λ var)

Irradiance λ (W/m2):

Voltage Vp (volts)

Fig. 2. P–V characteristics of the SM55 PV module, with constant temperature and

varying radiation.

Sun

PVG

+ _ib

vb

ig

vg Battery Boost DC-DC

converter

Fig. 3. General diagram of the PV battery-charger system.

Vp

Ip

Iph

ID

Rsh

Rs

Fig. 4. PV module equivalent circuit.

H. El Fadil, F. Giri / Control Engineering Practice 19 (2011) 513–521514

becomes rapidly costly especially when a wide PV plant, involvingseveral hundreds of PV panels, is considered. As a matter of factthe sensing effort is costly and the cost rapidly grows with therequired accuracy and the involved number of sensors. Besides,the installation and periodic maintenance of these instrumentsare quite costly. In the present paper, we are seeking a relativelycheap and efficient solution to the problem of controlling PVsystems consisting of PV panels, PWM boost power convertersand storing batteries (Fig. 3). The aim is to ensure a satisfactoryMPPT, whatever the position of the PV panel. The low expenserequirement is met by developing a sensorless controller notnecessitating climatic variable sensors (i.e. solar radiation andtemperature). Specifically, a sensorless nonlinear adaptive con-troller is designed using the backstepping technique and thecontrol design is based on a nonlinear model describing thewhole system, including the boost converter, the PV arrays andthe battery. The proposed adaptive controller involves onlineestimation of the model uncertain parameters, namely thosedepending on radiation and temperature. Unlike standard

adaptive control problems where parametric uncertainty onlyaffects the system dynamics, the adaptive control problem understudy is characterized by the fact that parameter uncertainty alsoaffects the output-reference signal. Accordingly, the referencesignal parameters (together with those of the system dynamics)must be estimated online. Therefore, the purpose of perfectoutput-reference tracking (which is necessary to meet the MPPTobjective) cannot be achieved unless the parameter estimatesconverge to their true values, a requirement that is not necessaryis standard adaptive control problems (Krstic, Kanellakopoulos, &Kokotovic, 1995). It is well known (e.g. Ioannou & Fidan, 2006)that the parameter convergence requirement is guaranteed pro-vided the involved closed-loop signals are persistently exciting insome sense. Presently, a persistent excitation condition is for-mally defined and shown to be satisfied. Consequently, the MPPTobjective is shown to be achieved. Finally, it is worth to empha-size that, instead of photovoltaic array based systems, solarenergy can also be exploited using distributed solar collectors.Energy systems based on the last technology can also be con-trolled using adaptive control techniques, see e.g. Henriques, Gil,Cardoso, Carvalho, and Dourado (2010).

The present paper is organized as follows: in Section 2, the systemmodeling is presented; Section 3 is devoted to the controller designand closed-loop theoretical analysis; the controller tracking perfor-mances are illustrated through numerical simulations in Section 4. Aconclusion and a reference list end the paper.

2. System modeling

Fig. 3 shows the general structure of a PV battery-chargersystem. This consists of a photovoltaic generator (PVG), a boostdc–dc converter and a storing battery. The role of the powerconverter is to interface the PV array output to the battery and totrack the MPP of the PV array.

2.1. Photovoltaic generator model

Solar energy conversion into electrical power is naturallyperformed by solar cells. The equivalent circuit of a PV moduleis shown in Fig. 4 (Gow & Manning, 1999; Luque & Hegedus,2003; Tan, Kirschen, & Jenkins, 2004). The traditional Ip�Vp idealcharacteristics (i.e. Rs¼0, Rsh¼N) of a solar array are given by the

Page 3: Climatic sensorless maximum power point tracking in PV generation systems

H. El Fadil, F. Giri / Control Engineering Practice 19 (2011) 513–521 515

following equation:

Ip ¼ Iph�IofexpðAVpÞ�1g ð1aÞ

with

A¼q

NgKTð1bÞ

Io ¼ IorT

Tr

� �3

expqEGO

gK

1

Tr�

1

T

� �� �ð1cÞ

Iph ¼ ISCRþKIðT�TrÞ½ �l

1000ð1dÞ

where Iph is the photocurrent (current generated under a givenradiation), Io is the cell reverse saturation current, l is the solarradiation. The meaning and typical values of the remainingparameters in (1a)–(1d) can be found in places, see e.g. Enrique,Duran, Sidrach-de-Cardona, and Andujar (2007) and Chu andChen (2009). The PVG is composed of many strings of PV modulesin series, connected in parallel, in order to provide the desiredvalues of output voltage and current. This PVG exhibits a non-linear Ig�Vg characteristics given, approximately and ideally, bythe following equation:

Ig ¼ Iphg�IogfexpðAgVgÞ�1g ð2Þ

where Vg is the PVG voltage, Ig is the PVG current, Ag¼A/Ns is thePVG constant, Iphg¼NpIph is the photocurrent of the PVG, Iog¼NpIo

is the saturation current of the PVG, Ns is the number of PVconnected in series and Np is the number of parallel paths. The PVarray module considered in this paper is the SM55. This has 36series connected mono-crystalline cells. The array electricalcharacteristics are assembled in Table 1.

2.2. Boost dc–dc converter model

Fig. 5 shows the circuit of boost dc–dc converter used tointerface the PV array output to the battery. As the PV array is acurrent source nature, a capacitor is placed at the circuit input sothat classic rules of source association are complied with. Theconverter operates according to the well known Pulse WidthModulation (PWM) principle. Applying Kirchoff’s laws to the

Table 1Electrical specifications for the solar module SM55.

Maximum power, Pm (W) 55

Short circuit current, ISCR (A) 3.45

Open circuit voltage, Voc (V) 21.7

Voltage at max power point, Vm

(V)

17.4

Current at max power point, Im (A) 3.15

KI (A/K) 4�10�4

L

vg

ib

Eb

Rb

iL

uC

ig

Battery

Fig. 5. Boost dc–dc converter circuit loaded by a battery.

circuit of Fig. 5, one obtains the following instantaneous model:

diLdt¼�ð1�uÞ

½RbiLþEb�

1

Lvg ð3aÞ

dvg

dt¼�

1

CiLþ

1

Cig ð3bÞ

where iL denotes the inductor current, vg the capacitor voltage, igthe PVG current and u a switch position. Because it involves abinary control input u, this model is useful for circuit simulatordesign but not for controller design. For control design purpose, itis more convenient to consider the following averaged model (seee.g. Krein, Bentsman, Bass, & Lesieutre, 1990):

dx1

dt¼�ð1�mÞ ½Rbx1þEb�

1

Lx2 ð3cÞ

dx2

dt¼�

1

Cx1þ

1

Cig ð3dÞ

where x1, x2 and ig denote the averaged value, respectively, of thecurrent iL, the input voltage vg and current ig . The duty ratio m isthe average value of the binary control u; it takes values in thecontinuous interval [0 1]. The continuous signal m turns out to bethe input control signal of system (3c)–(3d). The problem at handis to act on this control signal is in order to achieve the MPPT.

2.3. Overall system model

Combining (2) and (3d), yields

dx2

dt¼�

1

Cx1þ

1

Cy1ðT,lÞ�

1

Cj1ðT,vgÞy2ðTÞ ð4aÞ

where

y1ðT ,lÞ ¼ Iphg ð4bÞ

y2ðTÞ ¼ Iog ð4cÞ

j1ðT,vgÞ ¼ expðAgðTÞvgÞ�1 ð4dÞ

In view of (1c)–(1d), the parameter y1 depends on both tempera-ture T and radiation l, while y2 only depends on T. Furthermore,in view of (1b), the function j1 depends on T. As the temperatureis generally slowly varying, the effect of variation of j1 in (2) isvery small compared to the effect of y1 and y2. Then, without lostof generality, the following approximation can be made:

j1ðT,vgÞ � exp½AgðTrÞvg ��1¼defj1ðvgÞ ð4eÞ

where Tr is a constant reference temperature. In view of (3c), (4a)and (4e), the following averaged nonlinear model of the PVbattery-charger system (including the dc–dc converter and thePV array) is obtained:

_x1 ¼�ð1�mÞ½Rbx1þEb�

1

Lx2 ð5aÞ

_x2 ¼�1

Cx1þ

1

Cy1ðT ,lÞ�

1

Cj1ðx2Þy2ðTÞ ð5bÞ

3. Adaptive controller design and analysis

In this section, we aim at designing a controller that would beable to ensure a perfect maximum power point tracking (what-ever the position of the PV panel). Specifically, the controller mustenforce the voltage x2 to track, as accurately as possible, theunknown (and slowly varying) voltage Vm. Note that Vm isunknown because it depends on the temperature T and solarvariation l which are not supposed to be accessible to measure-ment. As mentioned in the introduction, the use of additional

Page 4: Climatic sensorless maximum power point tracking in PV generation systems

H. El Fadil, F. Giri / Control Engineering Practice 19 (2011) 513–521516

sensors (for T and l) and their periodic maintenance has a price.The more accurate the sensors are the higher the price. Thebenefit of the proposed adaptive control solution, designed in thepresent section, is to save the extra cost related to climaticvariables sensing. That is, no measurements are necessary exceptfor electrical variables for which we know there are reliable andnot too costly (effect hall) sensors.

3.1. Online estimation of the maximum power point (MPP)

The MPP (Vm, Pm) is reached when

@P

@Vg

����Vg ¼ Vm

¼ 0 with P¼ VgIg being the PV power:

That is, the controller must keep qP/qVg equal zero, whatever theradiation l and the temperature T, by acting on the duty cycle m.At the optimal point, one has

@P

@Vg¼ IgþVg

@Ig

@Vg¼ 0 ð6Þ

Using (2), (4b) and (4c), it follows from (6) that

y1�y2½ð1þVmAgðTrÞÞexpðAgðTrÞVmÞ�1� ¼ 0 ð7Þ

If y1 and y2 were precisely known, Eq. (7) would be used togenerate the optimal voltage Vm. The point is that y1 and y2 areunknown because they depend on the temperature and radiationwhich presently are supposed not to be accessible to measure-ment. Eq. (7) will then be replaced by its certainty equivalenceform, i.e.

y1�y2½ð1þVmAgðTrÞÞexpðAgðTrÞVmÞ�1� ¼ 0 ð8Þ

where y1 and y2 are online estimates of y1 and y2, respectively.The corresponding estimators will be designed in the nextsubsection, as a part of a nonlinear adaptive controller.

3.2. Adaptive controller design

Recall that the control objective is to enforce the voltage x2 totrack the optimal point Vm, despite the system parameter uncer-tainties. To this end, the backstepping design principles areinvoked (Krstic et al., 1995). As already mentioned, the way thistechnique is presently applied is not usual because the referencetrajectory, namely Vm, depends on the estimates (y1,y2) of theunknown system parameters (y1, y2). Therefore, the convergenceof the estimates to their true values is presently crucial for theachievement of the MPPT objective, unlike in standard adaptivecontrol problems (e.g. Krstic et al., 1995). There, perfect conver-gence is not necessary. Since system (5a)–(5b) is a second order,the controller design is performed in two steps (El Fadil, Giri, ElMagueri, & Chaoui, 2009).

Design Step 1: Introduce the following tracking error:

z1 ¼ x2�Vm ð9Þ

Achieving the tracking objective amounts to enforcing the error z1

to vanish. To this end, the dynamics of z1 have to be clearlydefined. Deriving (9), it follows from (5b) that

_z1 ¼�1

Cx1þ

1

Cy1�

1

Cj1ðx2Þy2�

_V m ð10Þ

In the above equation, the quantity �x1/C stands as a virtualcontrol variable. Let us consider the following Lyapunov function:

V1 ¼ 0:5z21þ0:5ð ~y

2

1=g1þ~y

2

2=g2Þ ð11Þ

where g140 and g240 are any real constants, called parameteradaptation gains. The time-derivative of V1 along the trajectory

of (10) is

_V 1 ¼ z1�x1

Cþw11y1þw12y2�

_V m

� ��~y1

g1

ð_y1�g1w11z1Þ

�~y2

g2

ð_y2�g2w12z1Þ ð12Þ

where we have defined the first two regressors

w11 ¼ 1=C, w12 ¼�j1ðx2Þ=C ð13Þ

We can eliminate ~y1 and ~y2 from _V 1 by considering the update

laws_y1 ¼ gt11 and

_y2 ¼ gt12, with

t11 ¼w11z1, t12 ¼w12z1 ð14Þ

Furthermore, z1 can be regulated to zero if �x1/C¼a1 where a1 isa stabilizing function defined by

a1 ¼�c1z1�w11y1�w12y2þ_V m ð15Þ

where c140 is a design parameter. Since �x1/C is not the actualcontrol input, one can only seek the convergence of the error �x1/

C�a1 to zero, and we do not use_y1 ¼ gt11 and

_y2 ¼ gt12 as anupdate laws. Instead we retain t11 and t12 as our first tuning

functions and tolerate the presence of ~y1 and ~y2 in _V 1. We thendefine the following second error variable:

z2 ¼�x1=C�a1 ð16Þ

The next step is to determine a variation law for the control signalm so that the set of errors z1 and z2 vanish asymptotically. But, letus first establish some useful equations. Eq. (10) becomes,using (16) and (15)

_z1 ¼�c1z1þz2þw11~y1þw12

~y2 ð17Þ

Also, derivative (12) of the Lyapunov function is rewritten

_V 1 ¼�c1z21þz1z2þ

~y1ðt11�_y1=g1Þþ

~y2ðt12�_y2=g2Þ ð18Þ

Design Step 2: Now, the objective is to enforce the errorvariables (z1, z2) to vanish. To this end, let us first determine thedynamics of z2. Deriving (16) with respect to time and using (5a),(15), (13), (4e) and (17), implies

_z2 ¼1

LCð1�mÞðRbx1þEbÞþc� €V mþw21

~y1þw22~y2þw11

_y1þw12_y2

ð19Þ

with

c¼�1

LCx2�c2

1z1þc1z2þAg

C2y2ð1þj1ðx2ÞÞðx1�y1þf1ðx2Þy2Þ ð20aÞ

where w21 and w22 represent the second regressor functionsdefined as follows:

w21 ¼w11j2ðx2,y2Þ ð20bÞ

w22 ¼w12j2ðx2,y2Þ ð20cÞ

with

j2ðx2,y2Þ ¼ c1�Ag

C1þj1ðx2Þ

y2 ð20dÞ

In (19), the actual control input arises for the first time. One isfinally in a position to make a convenient choice of the parameterupdate law and the control law to stabilize the whole system withstate vector is (z1, z2, ~y1, ~y2). Consider the augmented Lyapunovfunction candidate

V ¼ V1þ1

2z2

2 ¼1

2z2

1þ1

2z2

2þ1

2g1

~y2

1þ1

2g2

~y2

2 ð21Þ

Page 5: Climatic sensorless maximum power point tracking in PV generation systems

PV battery

charger system

Controller (25)

and adaptive laws

(23a-b)

Digital resolution of (8)

T λ

Duty ratio

µ

ig

vg

V1θ

Perturbations

Fig. 6. A nonlinear adaptive control block-diagram for the MPPT of PV systems.

Table 2Parameters of controlled system.

Eb 24 V

Rb 0.65 OL 1 mH

C 4700 mF

PWM frequency 50 kHz

40

60

PV Power P (W)

10

15

PV Voltage vg (V)

H. El Fadil, F. Giri / Control Engineering Practice 19 (2011) 513–521 517

Deriving V with respect to time gives, using (18) and (19)

_V ¼ _V 1þz2 _z2

¼�c1z21þz2 z1þ

1

LCð1�mÞðRbx1þEbÞþc� €V þw11

_y1þw12

_y2

� �

þ ~y1�_y1

g1

þt11þw21z2

0@

1Aþ ~y2

�_y2

g2

þt12þw22z2

0@

1A ð22Þ

To eliminate ~y1 and ~y2 from _V , the following update laws areconsidered:

_y1 ¼ g1t21 ð23aÞ

_y2 ¼ g2t22 ð23bÞ

with

t21 ¼ t11þw21z2 ¼w11z1þw21z2 ð24aÞ

t22 ¼ t12þw22z2 ¼w12z1þw22z2 ð24bÞ

The control signal m is chosen to let the term in z2 (betweenbrackets on the right side of (22)) equal to �c2z2. Doing so, oneobtains

m¼ 1þLC

Rbx1þEbðz1þc2z2þc� €V mþw11g1t21þw12g2t22Þ ð25Þ

where c240 is a design parameter. The adaptive controller thusdesigned includes the parameter adaptive law (23a)–(23b) andthe adaptive control law (25). The resulting closed-loop system isanalyzed in the following Theorem.

Theorem 1. (Main result). Consider the closed-loop system consist-

ing of the controlled system (5a)–(5b) and the adaptive controller

including the control law (25) and the parameter adaptive

laws (23a)–(23b). Then, one has the following properties:

205

(1) The tracking error z1¼x2�Vm converges to zero, whatever the

initial condition z1(0).

00

(2) Suppose the following persistent excitation condition is fulfilled:

0 0.1 0.2 0.3 0.40 0.1 0.2 0.3 0.4

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8Duty ratio μ

time (s)0 0.1 0.2 0.3 0.4

290

300

310

320

330

340Temperature T (K)

time (s)

Fig. 7. MPPT achievement in presence of temperature changes.

lim inft-1

Z t

0WðtÞT WðtÞdt40 ð26Þ

Then, ~y converges to zero. It follows that the MPPT objective is

accurately achieved.

Proof. See Appendix. &

The above theorem shows that the MPPT objective is actuallyachieved provided the persistent excitation assumption (26) isfulfilled. It is checked using several simulations that such arequirement is not an issue (Section 4.1). In standard adaptivecontrol problems (e.g. those dealt with in Krstic et al. (1995))persistent excitation is not required because the output-referencetrajectory does not depend on unknown parameters. Then, theconvergence of the parameter estimates to their true values is notnecessary for perfect output-reference convergence.

4. Illustration of the adaptive controller and simpler versions

4.1. Adaptive controller performances

The theoretical performances described by Theorem 1 of theadaptive controller, including the control law (25) and theparameter adaptive laws (23a)–(23b), designed in Section 3, arenow illustrated by simulation. The experimental setup, describedby Fig. 6, is simulated using MATLAB. The characteristics of thecontrolled system are gathered in Table 2. It is worth noting that

the controlled system is simulated using its instantaneous model.The averaged model (5a)–(5b) is in effect used only in thecontroller design. The control design parameters are given thefollowing values which proved to be convenient: c1¼240,c2¼3�103, g1¼8�10�4 and g2¼5�10�12. The resulting closed-loop control performances are illustrated by Fig. 7–10.

4.1.1. Temperature variation effect

Fig. 7 illustrates the adaptive controller behavior when facingtemperature changes. Specifically, the temperature T variesbetween 298.15 and 333.15 K (i.e. between 25 1C and 60 1C),while the radiation l is constant equal to 1000 W/m2. It is seenthat the controller keeps the whole system at the optimal

Page 6: Climatic sensorless maximum power point tracking in PV generation systems

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.452.3

2.4

2.5

∧θ1 (A)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.4502468

x 10-5∧θ2 (A)

Fig. 8. Estimates of unknown parameters in presence of temperature variations.

0 0.1 0.2 0.3 0.40

1020304050

PV Power P (W)

0 0.1 0.2 0.3 0.40

5

10

15

PV Voltage vg (V)

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8Duty ratio μ

time (s)0 0.1 0.2 0.3 0.4

400

600

800

1000

Radiation λ (W/m2)

time (s)

Fig. 9. MPPT achievement in presence of radiation changes.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

1

2

3

∧θ1 (A)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.5

1

x 10-4∧θ2 (A)

Fig. 10. Estimates of unknown parameters in presence of radiation variations.

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

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

time (s)

x 109

Fig. 11. Variation of the smallest eigenvalue of the matrix ð1=tÞR t

0 WðtÞT WðtÞdt.

H. El Fadil, F. Giri / Control Engineering Practice 19 (2011) 513–521518

operation conditions. Indeed, the captured PV power P achievesthe values 44.08 or 54.1 W which correspond (on the powercurves of Fig. 1) to the maximum points associated to thetemperatures 333.15 and 298.15 K, respectively. Fig. 8 showsthe estimates y1 and y2 of the unknown parameters y1 and y2.It is seen that both estimates matches well their true valuesconfirming Theorem 1.

4.1.2. Radiation variation effect

Fig. 9 illustrates the MPPT achievement in presence of radiationchanges. Specifically, the radiation varies between 400 and 1000 W/m2, meanwhile the temperature is kept constant, equal to 318.15 K(i.e. 45 1C). The figure shows that the captured PV power variesbetween 16.3 and 48.06 W. These values correspond (see Fig. 2) tothe maximum points on the curves associated to the consideredradiations. Again, it is observed (see Fig. 10) that the estimates of theunknown parameters do converge to their true values matchingwell the changing operating conditions. Such a good parameteradaptation is due the fulfilment of the persistent excitation

condition (26). Fig. 11 shows that the smallest eigenvalue of thematrix ð1=tÞ

R t0 WðtÞT WðtÞdt remains all time above zero, despite

the changing conditions. Condition (26) is thus checked.

4.2. Benefit of accounting for the PV nonlinearity

The adaptive regulator, composed of (25) and (23a)–(23b), isdesigned using the nonlinear model (5a)–(5b) where j1(x2) repre-sents the PV nonlinearity. The question is whether the adaptivecontroller performances are preserved if the function j1(x2) isreplaced (in the regulator design) by a linear approximation?

To answer this question, the two nominal conditions areconsidered

T ¼ T0 ¼ 298:15 K, l¼ l0 ¼ 1000 W=m2, P¼ P0 ¼ 44:08 W,

Vg ¼ vgo ¼ 16 V ð27Þ

T ¼ T0 ¼ 318:15 K, l¼ l0 ¼ 400 W=m2, P¼ P0 ¼ 16:66 W,Vg ¼ vgo ¼ 13 V ð28Þ

Page 7: Climatic sensorless maximum power point tracking in PV generation systems

H. El Fadil, F. Giri / Control Engineering Practice 19 (2011) 513–521 519

Performing a first-order Taylor series expansion of j1(vg) aroundvgo, one obtains from (4e)

j1ðvgÞ �j1 linearizedðvgÞ ¼def

exp½AgðTrÞvgo�þAgexp½AgðTrÞvgo�ðvg�vgoÞ�1

ð29Þ

The two linear approximations, corresponding to the nominalconditions (27) and (28), are successively used, instead of the truefunction j1(vg), in the design procedure of Section 3. Twosimplified adaptive controllers are thus obtained and appliedsuccessively to the true (instantaneous model of the) systeminvolving the true PV nonlinearity j1(vg). The simulated experi-ment illustrated by Fig. 12 (resp. Fig. 13) involves a temperature(resp. radiation) deviation (from its nominal value) within the

0 0.1 0.2 0.3 0.40

20

40

60

PV Power P(W)

X: 0.2045Y: 42.2

0 0.1 0.2 0.3 0.40

5

10

15

20

PV Voltage Vg (V)

0 0.1 0.2 0.3 0.4

0

0.5

1

Duty ratio μ

time (s)0 0.1 0.2 0.3 0.4

300

310

320

330

Temperature T (K)

time (s)

Fig. 12. Performances of the simplified adaptive controller involving the linear

approximation (29) corresponding to nominal conditions (27).

0 0.1 0.2 0.3 0.40

20

40X: 0.2645Y: 33.81

PV Power P (W)

0 0.1 0.2 0.3 0.40

5

10

15

20

PV Voltage Vg (V)

0 0.1 0.2 0.3 0.4

0

0.5

1

Duty ratio μ

time (s)0 0.1 0.2 0.3 0.4

400

600

800

1000Radiation λ (W/m2)

time (s)

Fig. 13. Performances of the simplified adaptive controller involving the linear

approximation (29) corresponding to nominal conditions (28).

subinterval [0.15 s 0.3 s]. The figures show that the performancesof the simplified adaptive controller are quite comparable to thoseof the complete controller (defined by (25) and (23a)–(23b)) as longas the operation conditions are such that j1(x2) is accuratelyapproximated by j1linearized(x2) which is the case in the intervals[0 s 0.15 s] and [0.3 s 0.45 s]. In particular, the MPPT objective is thereaccurately achieved. However, when the above approximation doesnot hold, which is the case during the subinterval [0.15 s 0.3 s], theperformances of the simplified adaptive controller deteriorate. Actu-ally, it is observed in Fig. 12 that the extracted power is 42.2 W whilethe MPP is44.08 W. That is the loss is nearly 4.26%. Similarly, Fig. 13shows that the extracted power is 33.81 W while the MPP is48.78 W.That is the loss is nearly 30.69%.

4.3. Limit of linear control

As already pointed out, parameter adaptation (on Iph and Io) isresorted to achieve a sensorless MPPT. Specifically, no sensors areneeded for the measurement of the temperature T and radiation l.As a matter of fact, simpler control solutions exist if these sensorswere available (Chiang et al., 1998; Houssamo, Locment, &Sechilariu, 2010). For simplicity, a linear PI regulator Kp(1+1/Tis)is presently considered within the simulated experimental setupof Fig. 14. The PI regulator parameters have been tuned based onseveral system step responses. The following parameter valuesproved to be convenient: Kp¼0.02, Ti¼1.6 ms. As emphasized bythe simulated setup (Fig. 14), the temperature and radiationmeasures are supposed to be noised. Here, the noise is simply aband limited random signal. As a matter of fact, the controlperformances depend on the noise characteristics (mean andvariance). Figs. 15 and 16 illustrate the resulting control perfor-mances in presence of measurement noises with the mean 5% andvariance 5% of nominal values. It is observed in Fig. 15 that the PIcontrol performances are not as satisfactory as those of theproposed adaptive controller (illustrated in Section 4.1). Indeed,during the subinterval [0.15 s 0.3 s], the extracted power is nearly37.34 W while the MPP is 44.08 W. That is the lack is nearly15.29% of the MPP. Similarly, it is observed in Fig. 16 that theMPPT is not accurately achieved. Indeed the extracted power isnearly 44.15 W while the MPP is 48.78 W. That is the lack isnearly 9.49%.

The PI control strategy failure to achieve the MPPT becomeseven more problematic in presence of higher noise characteristics.Table 3 and Fig. 17 show the variation of the MPPT error (in % ofMPP value) in function of the noise mean (in % of its nominalvalue). Clearly, the MPPT error increases rapidly with thenoise mean.

In summary, the performances of the simple nonadaptivecontrol solution depend on the measurement quality which in

PV battery charger system

PI regulator

Digital resolution of (7)

Climate variables T λ

Duty ratio µ

vg

Vm Calculation of �1and �2 using

(4b-c) and (1c-d)

Sensors

Tmes

Noised measures

�mes

�1

�2

Fig. 14. Sensor based control strategy involving PI regulator.

Page 8: Climatic sensorless maximum power point tracking in PV generation systems

0 0.1 0.2 0.3 0.40

20

40

PV Power P (W)

X: 0.2146Y: 44.15

0 0.1 0.2 0.3 0.40

10

20

PV Voltage Vg (V)

0 0.1 0.2 0.3 0.4

0

0.2

0.4

0.60.8

Duty ratio μ

time (s)0 0.1 0.2 0.3 0.4

400

600

800

1000Radiation λ (W/m2)

time (s)

Fig. 16. PI control performances in presence of radiation changes.

Table 3MPPT error variation.

Noise

mean (%)T¼333.15; l¼1000 W/m2 T¼318.15; l¼1000 W/m2

Extracted

power P(W)

MPP

(W)

MPPT

error (%)

Extracted

power P(W)

MPP MPPT

error (%)

1 41.34 44.08 6.21 48.3 48.78 0.98

2 40.84 7.35 47.6 2.42

5 37.35 15.29 44.15 9.5

10 32.15 27.06 38.51 21.05

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

noise mean in % of nominal value

MPPT Error in % of MPP

T = 333.15K λ = 1000W/m2

T = 318.15K λ = 1000W/m2

Fig. 17. MPPT error in function of noise mean.

0 0.1 0.2 0.3 0.40

20

40

60

PV Power P(W)

X: 0.2195Y: 37.34

0 0.1 0.2 0.3 0.40

10

20

PV Voltage Vg (V)

0 0.1 0.2 0.3 0.4

0

0.2

0.4

0.6

0.8Duty ratio μ

time (s)0 0.1 0.2 0.3 0.4

300

310

320

330

Temperature T (K)

time (s)

Fig. 15. PI control performances in presence of temperature changes.

H. El Fadil, F. Giri / Control Engineering Practice 19 (2011) 513–521520

turn depends on sensors accuracy. As a matter of fact, the betterthe sensors accuracy is the higher their prices. The presentadaptive control strategy constitutes an efficient and low costalternative.

5. Conclusion

The problem of achieving MPPT in PV battery-charger systems hasbeen addressed. A cheap solution, not resorting to climatic variablessensors, is developed. It involves an adaptive controller designedusing the backstepping technique, based on system model (5a)–(5b)that accounts for the boost converter dynamics and the batterymodel. The adaptive feature is necessary to cope with the uncertaintyand change of temperature and radiation. The obtained voltageadaptive controller consists of the adaptive voltage reference gen-erator (8), the adaptive update laws (23a)–(23b) and the controllaw (25). This is formally shown to meet its control objectives, despitethe system parameter uncertainty (Theorem 1). Specifically, theclosed-loop system is globally asymptotically stable implying max-imum power point tracking. This theoretical performance is checkedby simulations which also illustrate the supremacy of the proposedsolution compared to simpler ones. The persistent excitation condi-tion (26) turned out to be crucial for the achievement of the MPPT.This is not the case in standard adaptive control problems.

Appendix A. Proof of Theorem 1

Proof of Part 1. Using (25) and (23a)–(23b), (19) can be rewrit-ten as follows:

_z2 ¼�c2z2�z1þw21~y1þw22

~y2 ðA1Þ

Finally, from (17), (A1) and (23a)–(23b) one obtains the followingoverall closed-loop system:

_z ¼ AzzþWT ~y ðA2Þ

_~y ¼�GWz ðA3Þ

where Az is a skew symmetric matrix defined as follows:

Az ¼�c1 1

�1 �c2

" #ðA4Þ

and

W ¼w11 w21

w12 w22

" #ðA5Þ

Page 9: Climatic sensorless maximum power point tracking in PV generation systems

H. El Fadil, F. Giri / Control Engineering Practice 19 (2011) 513–521 521

is the regressor matrix, and

z¼ ½z1, z2�T ðA6Þ

~y ¼ ½ ~y1, ~y2�T ðA7Þ

G¼ ½g1, g2�T ðA8Þ

Using (22), (23a)–(23b) and (25), one obtains the followingderivative of the Lyapunov function V:

_V ¼�c1z21�c2z2

2 ðA9Þ

which shows that the equilibrium ðz, ~yÞ ¼ 0is globally asymptoticallystable. Furthermore, it follows using Lasalle’s Invariance theorem, it

further follows that the state ðz, ~yÞ converges to the largest invariant

set M of (A2)–(A3) contained in E¼ f½zT ~yT�T A IR4=z¼ 0g, i.e. the set

where _V ¼ 0. This means, in particular, that z(t)-0 as t-N.

Proof of Part 2. Now, let us prove by contradiction that theinvariant set is limited to the origin, i.e. M¼{ [0 0 0 0]T}. To thisend, suppose that

½0 0 lT�AM for some lA IR2�f0g ðA10Þ

Let ½zT ð0Þ ~yTð0Þ� ¼ ½0 0 lT

�. As M is invariant, it follows that for allt40:

zðtÞ ¼ 0 ðand so _zðtÞ ¼ 0Þ ðA11Þ

which, together with (A2), yields

W ~y ¼ 0 for all t40 ðA12Þ

On the other hand, using (A3) one obtains_~y ¼ 0 and so

~yðtÞ ¼ la0 ðfor all tZ0Þ ðA13Þ

Now, multiplying (A12) from the left by WT and integrating bothsides, yields

0¼ lim inft-1

Z t

0WðtÞT WðtÞ ~yðtÞdt¼ lim inf

t-1

Z t

0WðtÞT WðtÞdt

� �l

ðA14Þ

which implies that l¼0, in view of (26). But thiscontradicts (A10). Hence, the invariant set M is reduced to theorigin and consequently the state vector ðz, ~yÞ does converge tozero. This ends the proof of the theorem. &

References

Applebaum, J. (1987). The quality of load matching in a direct-coupling photo-voltaic system. IEEE Transactions on Energy Conversion, 2(4), 534–541.

Braunstein, A., & Zinger, Z. (1981). On the dynamic optimal coupling of a solar cellarray to a load and storage batteries. IEEE Transactions on Power Apparatus andSystems, 100(3), 1183–1188.

Chiang, S. J., Chang, K. T., & Yen, C. Y. (1998). Residential photovoltaic energystorage system. IEEE Transactions on Industrial Electronics, 45(3), 385–394.

Chu, C.-C., & Chen, C.-L. (2009). Robust maximum power point tracking methodforphotovoltaic cells: A sliding mode control approach. Solar Energy, 83,1370–1378.

Eakburanawat, J., & Boonyaroonate, I. (2006). Development of a thermoelectricbattery-charger with microcontroller-based maximum power point trackingtechnique. Applied Energy, 83, 687–704.

El Fadil, H., Giri, F., El Magueri, O., & Chaoui, F. Z. (2009). Control of DC–DC powerconverters in the presence of coil magnetic saturation. Control EngineeringPractice, 17, 849–862.

Enrique, J. M., Duran, E., Sidrach-de-Cardona, M., & Andujar, J. M. (2007).Theoritical assessment of the maximum power point tracking efficiency ofphotovoltaic facilities with different converter topologies. Solar Energy, 81,31–38.

Fangrui, L., Shanxu, D., Fei, L., Bangyin, L., & Yong, K. (2008). A variable step sizeINC MPPT method for PV systems. IEEE Transactions on Industrial Electronics,55(7), 2622–2628.

Gow, J. A., & Manning, C. D. (1999). Development of a photovoltaic array model foruse in power electronics simulation studies. IEE Proceedings on Electric PowerApplications, 146(2), 193–200.

Henriques, J., Gil, P., Cardoso, A., Carvalho, P., & Dourado, A. (2010). Adaptiveneural output regulation control of a solar power plant. Control EngineeringPractice, 18(10), 1183–1196.

Hiyama, T., & Kitabayashi, K. (1997). Neural network based estimation of max-imum power generation from module using environmental information. IEEETransactions on Energy Conversion, 12(3), 241–247.

Houssamo, I., Locment, F., & Sechilariu, M. (2010). Maximum power tracking forphotovoltaic power system: Development and experimental comparison oftwo algorithms. Renewable Energy, 35, 2381–2387.

Hussein, K. H., Muta, I., Hoshino, T., & Osakada, M. (1995). Maximum photovoltaicpower tracking: An algorithm for rapidly changing atmospheric conditions. IEEProceedings on Generation, Transmission and Distribution, 142(1), 59–64.

Ioannou, P., & Fidan, B. (2006). Adaptive control tutorial. PA, USA: SIAM.Khouzam, K. Y. (1990). Optimum load matching in direct-coupled photovoltaic

power systems: Application to resistive loads. IEEE Transactions on EnergyConversion, 5(2), 265–271.

Krein, P. T., Bentsman, J., Bass, R. M., & Lesieutre, B. (1990). On the use of averagingfor analysis of power electronic system. IEEE Transactions on Power Electronics,5(2), 182–190.

Krstic, M., Kanellakopoulos, I., & Kokotovic, P. V. (1995). Nonlinear and adaptivecontrol design. New York, NY: John Wiley & Sons.

Leyva, R., Alonso, C., Queinnec, I., Cid-Pastor, A., Lagrange, D., & Martınez-Salamero, L. (2006). MPPT of photovoltaic systems using extremum-seekingcontrol. IEEE Transactions on Aerospace and Electronic Systems, 42(1),249–258.

Luque, A., & Hegedus, S. (2003). Handbook of photovoltaic science and engineering.John Wiley & Sons, Ltd..

Tan, Y. T., Kirschen, D. S., & Jenkins, N. (2004). A model of PV generation suitablefor stability analysis. IEEE Transactions on Energy Conversion, 19(4), 748–755.