A fast current-based MPPT technique based on sliding mode control

Enrico Bianconi(l), Javier Calvente(2), Roberto Giral(2), Giovanni Petrone(3) Carlos Andres Ramos-Paja(4), Giovanni Spagnuolo(3) and Massimo Vitelli(5)

(1) Bitron Industrie S.p.A., Grugliasco (TO) - Italy

(2) Departament d'Enginyeria Electronica, Electrica i Automatica - Universitat Rovira i Virgili - Spain

(3) Dip. Ingegneria dell'Informazione e Ingegneria Elettrica (D.I.I.I.E.), University of Salerno - Italy

(4) Facultad de Minas - Universidad Nacional de Colombia - Sede Medellin - Colombia

(5) Dip. Ingegneria dell'Informazione (D.!.I.), Second University of Naples - Italy

E-mail: enrico.bianconi@gru.bitron-ind.com. javier.calvente@urv.cat, roberto.giral@urv.cat

gpetrone@unisa.it, caramosp@unal.edu.co, gspagnuolo@unisa.it, vitelli@unina.it

Abstract-This paper introduces a novel maximum power point tracking technique aimed at maximizing the power pro­duced by photovoltaic systems. The largest part of the approaches presented in literature are based on the sensing of the pho­tovoltaic generator voltage. On the contrary, in this paper a current-based technique is proposed: the sensing of the current in the capacitor placed in parallel with the photovoltaic source is one of the innovative aspects of the proposal. A dual control loop based on the sliding mode control ensures a very fast tracking of irradiation variations. Features of the proposed algorithm are supported by a theoretical analysis and simulations. The technique described in this paper is patent pending.

I. INTRODUCTION

Maximum Power Point Tracking (MPPT) function is one of

the key factors in any PhotoVoltaic (PV) system. Perturb and

Observe (P&O) MPPT technique is often preferred in com­

mercial products because it makes the implementation in low

cost digital controllers quite simple. Significant efforts have

been producing in order to improve the electrical efficiency

of the PV power processing system, often forgetting that the

total efficiency is the product of the electrical efficiency by

the MPPT efficiency. The latter is affected by the steady

state and transient performances of the algorithm, thus in

presence of an almost constant and of an abrupt change in the

irradiation level. As demonstrated in [1], the P&O performance

optimization can be achieved by means of a suitable choice

of the values of the two parameters affecting the tracking

performances, that are the amplitude and frequency of the

perturbations applied to the PV voltage by means of a switch­

ing converter. In [1] it was demonstrated that in steady state

conditions the best performances are ensured by choosing the

smallest possible value of the perturbations amplitUde. This

choice minimizes the displacement from the maximum power

point when irradiation level is almost constant, but penalizes

the tracker promptness in presence of abrupt changes in the

irradiation level. In fact, the analytical result shown in [1]

clearly puts into evidence that the greater irradiation variation

to be tracked, the larger the perturbation amplitude must be

978-1-4244-9312-8/11/$26.00 ©2011 IEEE

chosen. In [1] it has also been shown that a further increase

in the value of the perturbation amplitude becomes mandatory

if the PV voltage is affected by some disturbance. This is

the case, for instance, of the low frequency oscillation, at a

frequency that is twice that one of the grid, backpropagating

to the PV voltage in systems connected to the AC mains.

In this paper a current-based P&O MPPT strategy is pro­

posed. It allows to have a very prompt tracking of the varia­

tions in the irradiance value and, additionally, it guarantees the

rejection the low frequency voltage oscillations affecting the

PV array and due to the inverter operation in grid-connected

systems.

The main peculiar aspect of the proposed approach is in the

fact that it is a current-based technique. The largest part of

the MPPT algorithms presented in literature is voltage-based,

because of the logarithmic dependency between the irradiation

level and the PV voltage. In fact, the linearity between the

irradiance level and the PV current would be very useful for

a fast MPPT, but irradiance drops would require a suitable

control.

In literature, few examples of current-based MPPT ap­

proaches can be found. For instance, in [2] a model-based

approach, depending on the assigned PV module characteris­

tic, is presented. In [3] the PV array current is not sensed, but it

is reconstructed through a sliding-mode observer of the dc/dc

converter's input inductor current and fed into the controller

to generate the maximum power point reference voltage.

II. BASICS OF THE PROPOSED CONTROL TECHNIQUE

Fig. 1 shows a possible control strategy operating on the

inductor current of the DCIDC converter in order to regulate

the PV current value. The role of the input capacitor Gin is in

absorbing the switching ripple affecting the inductance current,

so that the PV current is controlled through the average

inductor current value.

In the circuit node connecting the PV array and the boost

converter input inductance L and capacitance Gin (see Fig. 1),

59

PVarray

��-----�I"'I i J.:' 41 ' ,. Vpv: 1 -'-:.)l.::.J

: : l... ............ :-''''' {�y,�-EJ

, , , , , , L�!!.

Figure I. System scheme.

the Kirchhoff current law holds:

ipv = iCin + iL

The current controller gives the reference current irer

so that the steady-state is ensured when ivr = O.

(1)

(2)

According to the classical DC/DC converters current control

theory, the signal u(t) driving the MOSFET is a function of

the error signal ei(t) = ire!(t) - idt), so that the inductor

current iL is regulated according to the current reference signal

ire!' By using (1) and (2), the following simplification is thus

possible:

ire! = iCin + iL + ivr

ei = iCin + iL + ivr - iL

ei = iCin + ivr

(3)

(4)

(5)

where the control objective ire! = iL (see Fig. 1) leads to

ei = 0, so that it is equivalent to:

and the steady-state condition results in iCin = O.

PVarTay Inver1er

L-__ Jh:-r:�--�----�---Lt:-L�� : I \ :

Figure 2.

'N ! ,�!" v ·

8 j�� ' .. :-6��-EJ

System scheme based on input capacitor current control.

(6)

The simplified control objective (6) reveals that the control

structure of Fig. 1 can be simplified as in Fig. 2, with the

inner control loop which is now aimed at regulating the input

capacitance current icin. This simplification reveals important

because the practical implementation of the scheme shown in

Fig. 1 would require two high-bandwidth current sensors for

ipv and iL, both affecting the MOSFET control signal u(t) value. Instead, in the scheme of Fig. 2, u(t) depends on the

Isc 1 G

I.v

DCIOC converter

Figure 3. Small-signal model of the system.

iCin instantaneous value only. Consequently, only one high­

bandwidth current sensor is required because the other one

is a low-bandwidth one dedicated to ipv, whose variations

are dictated by the slow MPPT controller dynamics. Another

advantage of the system shown in Fig. 2 is in its easier analysis

with respect to that one depicted in Fig. 1.

III. SLIDING-MODE-BASED MPPT

The sliding surface SURF is given in (7):

SURF = -icin - ivr = 0 (7)

It is defined in order to fulfill the control objective (6). The

current loop reference ivr in the scheme of Fig. 2 is given

by Gv (s) . The definition of the sliding surface SURF as

in (7) suggests that, in sliding-mode operation, the capacitor

current iCin changes in order to reject the perturbations on the

bulk capacitor voltage Vb and to track the perturbations in the

irradiance level. Thus, the fact that SURF does not depend on

those variables ensures the MPPT operation and the rejection

of the low frequency disturbances, at a frequency that is the

double of the grid frequency, in single phase AC applications.

Two conditions must be fulfilled in order to ensure the

sliding-mode operation [4]:

SURF = 0 dSURF

= 0 dt

(8)

(9)

From the first condition (8), and by accounting for the

characteristic equation of the input capacitance:

. dvpv ZCin =Cin· �

the following condition is obtained:

dvpv dt

(10)

(11)

Equation (11) is valid in sliding-mode condition and gives a design criterion for the external voltage controller.

From the second sliding-mode condition (9), and by con­

sidering that iCin = ipv - iL, it results that:

dSURF =

diL _ dipv _ divr = 0

dt dt dt dt (12)

By neglecting the series and parallel resistances of the PV

array, namely Rs = 0 and Rsh = 00, the PV current can be

approximated by the expression:

60

mode control of the converter is preserved:

(13) disc vpv- L- > 0 (20)

where Vpv is the PV voltage, IR and a are parameters depend­

ing on the PV modules used, isc is the short-circuit current

for a given irradiance level, which has also an approximated

proportional relation with the irradiance isc = ks . S [5]. In

fact, it is:

isc = isc,STC· -S

S . (1

+ G.[ . (Tpv - TpV,STc)) (14)

STC

The analysis of the equation (12) can be done by using the

small-signal model shown in Fig. 3, wherein the switching

converter is represented by a current source and the PV gen­

erator is given as a Norton model including the photoinduced

current source and the differential conductance G which is

calculated in the generator's operating point. By looking at

Fig. 3 and (13), it results:

(15)

As a result, the sliding-mode condition given in (12) can be

thus rewritten as: d�; + G. d:v _ d;;c _ d;�T = 0 (16)

In addition, the classical boost converter model gives the

following relationship:

diL VPV

dt L

Vb· (1 - U) L

(17)

where the value u = 1 is used in the MOSFET ON state and

the value u = 0 in the OFF state.

In order to obtain the constraints that must be fulfilled

in order to ensure the sliding-mode operation, the equivalent

control technique [6] can be used. The constraints on the inputs

and states are defined by ensuring that the average control

signal ueq fulfills the inequalities 0 < ueq < 1. From equations

(11 )-( 17) the following equivalent control equation is obtained:

VPV _ Vb· (1 - Ueq ) + G. dvpv _ disc _ divT = 0 (18)

L L dt dt dt

From (7) and (12) it is deduced that the sliding-mode

equilibrium point is defined by {ipv = iL, iVT = O}. In (18),

at the equilibrium point it is dVd�v = 0, while the constraints

to be fulfilled in terms of maximum slope values of iVT and

irradiance, 4ftt and �� = ks· 4!tt, ensuring the sliding-mode

operation are calculated by using the superposition principle.

Thus, by putting d�tr = 0 in (18), the effect of the isc variations ensuring that 0 < ueq < 1 leads to the inequality:

vpv- L� 0< dt <1

Vb (19)

so that the two following inequalities ensure that the sliding

dt disc

vpv - L-- < Vb dt

(21)

Finally, the constraint to be fulfilled on the isc slope in order

to guarantee the proper sliding mode operation is obtained:

Vpv - Vb disc Vpv

L < dt < L (22)

It is worth noting that the lower bound for the isc slope

in (22) is negative because of the adoption of a boost con­

verter. Especially if the step-up dc/dc converter is designed

for guaranteeing a high boosting factor, e.g. in PV module

dedicated applications, the quantity vPvL-

vb is deeply negative.

This means that the larger the converter's voltage boosting

factor, the faster negative short circuit current variation can be

tracked without loosing the sliding mode behavior. Due to the

proportionality between isc and the irradiance S, condition

(22) can be translated into a constraint for the maximum

irradiance variation the MPPT technique is able to track:

Vpv - Vb dS Vpv --=-=:--:--..:.. < - < --

L· ks dt L· ks (23)

Inequality (23) reveals that the maximum irradiance vari­

ation that can be tracked without losing the sliding-mode

control is bounded by the inductor current derivatives in the

OFF and ON MOSFET states. This means that the inductance

value L can be properly designed in order to follow the

expected irradiance profile and variations, according to the

specific applications. For instance, stationary PV power plants

will be subjected to slow irradiance variations, while in PV

applications dedicated to sustainable mobility fast irradiance

variations would be tracked.

By fixing 4!tt = 0 in (18), the effect of variations on iVT

can be accounted for, so that the following constraint on 4ftt is obtained:

(24)

which again shows that the maximum iVT slope value that

can be tracked without missing the sliding-mode control

depends on the inductor current derivatives in the OFF and

ON MOSFET states. In this case, the voltage controller can

be designed to fulfill this dynamic constraint.

It is worth noting that, due to the symmetry of the expression

(18), constraints (23) and (24) show the same boundaries.

When both (22) and (24) conditions are fulfilled, the con­

verter is in sliding-mode control and therefore the dynamic

of the system is given by (11). The transfer function Gv/i(s) between the input capacitor voltage, that is the PV voltage,

vCin and the current reference iVT provided by the voltage

controller is:

(25)

61

A. PI controller design The Gv design is performed by means of a traditional PI

compensator. By taking into account the PI transfer function GPI(s) given in (26) and the voltage error Ev(s) definition

(27) that compensates the negative sign appearing in (25),

the closed loop transfer function T( s) of the system can be

expressed as in (28):

ki PI(s) = kp + ­

s Ev(s) = -(Vref(s) - Vcin(s))

T(s) = kps + ki

GinS2 + kps + ki

(26)

(27)

(28)

The transfer function T( s) in (28) is designed by accounting

for a classical relation between the rising time t R of the closed

loop voltage and the minimum switching period Tsw.

The T( s) structure gives the following relations:

kp = 2GinPWn

ki = GinWn2

(29)

(30)

so that, by using the equivalent time constant definition T = _1_

PWn In sliding-mode, the duty cycle D(t) and switching fre-

quency fsw(t) will oscillate around their nominal values Do

and fswo, respectively, in order to cancel out the bulk ca­

pacitor voltage oscillations !:,.Vb(t) according to the following

formulas:

D(t) = 1-VPV

VbO + !:,.Vb(t)

f () - VPV . D(t) sw t -

H. L

VPV Do=l- ­

VbO

f - Vpv· Do swO - H. L

B. MPPT refinements: input and output signals filtering

(31)

(32)

The MPPT controller adopted in this example is a traditional

Perturb and Observe (P&O) one. This means that the P&O

output generates a step change in the vref signal of the voltage

controller, thus violating the constraint on �. In order to

avoid this drawback, the P&O output is filtered thus generating

a vref dynamic behavior that is comparable with that one of

the closed loop system. In this way, the first order filter G fv (s) given in (33) is used,

G () Vref(s) 1

fv s = P&O(s) = Tfs + 1

where P&O(s) is the P&O control signal.

(33)

There is a need to filter signals VPV and ipv at the MPPT

input too. In fact, the possibility of reducing the amplitude of

the perturbations given by the P&O algorithm determines the

corresponding reduction of the vpv and ipv perturbations,

with a significant detrimental effect of the switching converter

operation. As for the MPPT output, the PV current and voltage

measurements used to calculate the PV power needed for the

P&O controller can be filtered by using the same G fv (s) to re­

move the switching frequency components without degrading

the dynamic response of the system, thus avoiding the P&O

controller to be confused by the switching ripple. The !:,.vref perturbation generated by the P&O controller can be adopted

equal to the voltage ripple, which is mitigated due to the PV

voltage filtering. The PV voltage ripple !:,.vPv is calculated as

[7]:

!:,. _ H . Tsw,max

vpv -8Gin

IV. SIMULATION RESULTS

(34)

Some simulation results have been obtained in PSIM en­

vironment. In Fig. 4 the complete PSIM schematic has been

shown.

Figure 4. Complete PSIM schematic.

The input capacitance value has been assumed equal to

Gin = 50 p,F, the input inductance L = 410 p,H, with

a desired inductor current ripple H = 4 A. The minimum

PV voltage value has been fixed at VPV,min = 100 V, and a bulk capacitor Gb = 22 p,F with an average voltage

VbO = 450 V has been considered. According to (32), the

minimum switching frequency is equal to 40 kHz. With such

values, from (34) it results that !:,.vref = !:,.vPv = 0 .2 V. In order to put into evidence one of the main features of

the proposed approach, i.e. the ability of rejecting the low

frequency voltage variations backpropagating from the bulk

voltage towards the PV voltage, !:,.Vb(t) has been assumed to

oscillate in the range [-160, 160] V, so that !:,. Vb = 160 V. This can be obtained by considering a small bulk capacitance

Gb = 22 p,F. Considerations done above about the PI controller design

lead to a maximum switching period Tsw,max = 25 p,s so that, by choosing a traditional tR/Tsw,max = 8 value,

the closed loop rising-time is tR = 200 p,s, which is the

first design consideration for the PI controller. The second

design consideration is related to the damping of the system,

which can be adjusted to the p = 0 .7 traditional value.

By approximating tR as 4T, where T is the equivalent time

constant of the closed loop system, it results that the PI

controller that ensures the defined t Rand p is the following

62

one:

( ) s + 2040 8 .16 PI s = 2· --.:.--­

s (35)

which also provides an infinite gain margin and a phase margin

equal to 65 .2°.

The PI design from the t R specification also allows to define

the MPPT controller period Ta equal to the Vpv stabilization

time, that is 1 .5· tR approximatively. This Ta value ensures

that the PV power has reached its steady state when the MPPT

controller measures it, thus avoiding the MPPT deception [1].

In this example Ta = 300 f.Ls. As for the signal filtering, the time constant in (33)is set to

Tf = T = 50 f.Ls. Fig. 5 shows the results of the system simulation (Fig.

2) by using the sliding-mode and Gv controllers. First of

all, simulation results put into evidence that the large low

frequency oscillations affecting the bulk voltage have been

rejected at the PV terminals. Moreover, the sudden irradiance

variations, having an instantaneous effect on the PV short

circuit current, have been correctly tracked, with the track­

ing system permanently tracking the maximum power point

even at the very high rate of irradiance variation the control

technique has been subjected to. The two features listed here

above make the control strategy shown in Fig. 2 suitable for

all the PV applications in which the adoption of electrolytic

capacitors at the dc bus must be avoided andlor wherever an

excellent MPPT performance is required. Current literature

(e.g. in ( [8])) puts into evidence that electrolytic capacitors

are the bottleneck of any PV power processing system because

they affect its lifetime significantly. The extraordinary fast

MPPT capability opens to the PV generators controlled by

means of the proposed technique the doors of a large range of

applications for which the sudden irradiation changes are the

rule, e.g. in sustainable mobility and for the PV integration on

cars, trucks, buses, ships and so on.

Fig. 6 shows the magnification in two different time in­

tervals of the waveforms shown in Fig. 5. The first time

window is [23.5, 27.2] ms: plots show the system behavior

while approaching the steady state operation: Fig. 6(a) puts

into evidence the proper design of the P&O parameters leading

to a three points behavior of the PV voltage. The same figure

shows the waveform of the closed loop voltage reference vref and the accurate tracking performed by both the Gv(s) and the

sliding-mode controller. Fig. 6(b) shows that the PV current

waveform is free of the 100Hz oscillations due to the inverter

operation, thus demonstrating that the control technique has

been able to stop the back propagation of the large oscillations

of amplitude LlVb affecting the converter's output voltage. Fig.

6(c) shows the input capacitor current and the current reference

control signal waveforms. Finally, Fig. 6(d) shows that the

MPPT technique has been able to drive the system toward the

maximum PV power corresponding the short circuit current

of lO A settled for this simulation example.

Similarly, Figs. 6(e)-(h) show a magnification of the system

behavior in the time interval [333.6, 337.9] ms, when a high

irradiance transient generating a fast change in the short circuit

current of 80 % has been imposed. At the beginning, with

isc = 10 A, the converters works in continuous conduction

mode and the proper operation of the control algorithm, that is

the vref and the PV maximum power point tracking, is evident.

Afterwards, at t = 335 ms, the PV short circuit current has been

forced to drop suddenly at isc = 2 A, thus simulating a steep

reduction in the irradiance level. As a consequence of this, the

converter enters in discontinuous conduction mode at t > 335

ms (Fig. 6(f)): in this case the switching frequency of the

system is reduced, but the voltage controller Gv (s) still drives

the PV voltage to follow the MPPT controller reference. Fig.

6(e) shows the satisfactory PV voltage behavior in continuous

and discontinuous conduction modes, and Fig. 6(h) puts into

evidence the fast MPPT response under a constant and time

varying irradiance levels.

As a final consideration, the upper and lower limits of

the slopes � and � as appear in (22) and (24) have

been calculated for the numerical example considered in this

section. It results that:

disc divr I -805 Alms < - + -

d < 293 A ms

dt t (36)

This inequality shows that the most restrictive condition

appears when the irradiance level, and thus the short circuit

current, is subjected to a positive variation. By keeping into

account the transfer function (35) of the PI voltage controller

adopted in this numerical example and by considering the

sliding-mode eqUilibrium point (Vein = vref, V�t = 0), the derivative of ivr is given as follows:

(37)

The perturbation amplitude imposed by the adopted P&O

MPPT controller is Llvref = 0 .2 V, which results, by account­

ing for the filter dynamics (33), in a maximum slope of 2.1

V Ims. This can be expressed in terms of the boundaries of

� as follows:

min (Tt) = -4 .2 Alms ,Llvref > 0 } (38) max (Tt-) = 4 .2 Alms ,Llvref < 0

which is two orders of magnitude lower than the limits (36)

that ensure that the sliding-mode control is still operating

correctly. This means that this system is able to track PV short

circuit current perturbations with slopes between [-800, 290]

Alms, thus corresponding to very fast irradiance perturbations

that are approximately in the range [-80, 29] W l(m2f.Ls). Such

remark confirms the inherent bent of the proposed technique

for applications characterized by uncommon irradiance slopes.

V. CONCLUSIONS

In this paper a novel, patent pending technique for the

maximum power point tracking of photovoltaic systems has

been introduced. The approach is based on the sliding mode

control technique and is based on the sensing of the current

drained by the capacitor which is usually put in parallel

with the photovoltaic generator. The technique implementation

63

i IL-:�:�:�:;;;;;;:;;;;;;:;;;;;:"-------:�EiJ�:� 0.1 0.2 0.3 0.

4 0.5 0.6 0.7 0.8 0.9

Time(s) (e)

r�t ' '" : ' : · o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

(d) 600 �1II!I!!IUI!IIl!!Il!l!ll!IlIlU! ...... � ......... � ......... � ...... JI!!1!11!I1I1I1!I1!I!!lIII!I!�

:D !�� �imiiiiliiliimilimiliil\lir"'''�'''''''''�'''''''''�'''''''''�miiiili�liimm�mliiml 3 00'-

--0

,.... '--..

,.

0."-

2 --

0...

,.3--

--'

0.4----,O.

L.5

--.,..

0.6,.--...,.O

L

.7

--0

,....

8--..

,

.0

.

�

g-

---'

Time(s)

Figure 5. Simulation of the system of Fig. 2 using the sliding-mode and Gv controllers.

('1

L:�I�.� II Ii er! 23.5 24 24.5 25 25.5 26 26.5 27

Time[ms] (dl

(0)

:�8f:S 334 334.5 335 335.5 336 336.5 337 Time (ms)

(II

1':1 � 334 334.5 335 335.5 336 336.5 337

Time [msj

334 334.5

(gl

f�1 \ c=g: 334 334.5 335 335.5 336 336.5 337

Time [ms] Time [msJ

Figure 6. Zoom of of Fig. 5.

requires few components and allows to track uncommonly fast

irradiance variations and is able to reject the low frequency

disturbances affecting the bulk voltage in grid connected

applications and backpropagating towards the photovoltaic

generator. Simulation results confirm the attractiveness of the

proposed method. The authors are confident that in the final

paper also some experimental results will be available.

REFERENCES

[1] N. Femia, G. Petrone, G. Spagnuolo, and M. Vitelli, "Optimization of perturb and observe maximum power point tracking method," Power Electronics, IEEE Transactions on, vol. 20, no. 4, pp. 963 - 973, 2005.

[2] H. T. Duro, "A maximum power tracking algorithm based on impp =

f(pmax) function for matching passive and active loads to a photovoltaic generator," Solar Energy, vol. 80, no. 7, pp. 812 - 822,2006.

[3] E. limenez-Brea, A. Salazar-L1inas, E. Ortiz-Rivera, and 1. Gonzalez­Llorente, "A maximum power point tracker implementation for photo­voltaic cells using dynamic optimal voltage tracking," feb. 2010, pp. 2161 -2165.

[4] S.-c. Tan, Y. Lai, and C. Tse, "General design issues of sliding-mode controllers in dc dc converters," Industrial Electronics, IEEE Transactions on, vol. 55, no. 3, pp. 1160 -1174, 2008.

[5] U.Eicker, Solar Technologies/or Buildings. Wiley, 2003. [6] H. Sira-Ramirez, "Sliding motions in bilinear switched networks," Cir­

cuits and Systems, IEEE Transactions on, vol. 34, no. 8, pp. 919 - 933, Aug. 1987.

[7] R. W. Erickson and D. Maksimovic, Fundamentals 0/ Power Electronics, 2nd ed. Springer, 2001.

[8] G. Petrone, G. Spagnuolo, R. Teodorescu, M. Veerachary, and M. Vitelli, "Reliability issues in photovoltaic power processing systems," Industrial Electronics, IEEE Transactions on, vol. 55, no. 7, pp. 2569 -2580, july 2008.

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