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Optimizing Samp ling Rate of P&O M PPT TechniqueN.Femia, G.Petrone, G.Spagnuolo , r

D.I.I.I.E. - University of S alemo, Italy1-84084, Fisciano (SA) - TALYEmail: [email protected], [email protected],[email protected]

Abslrocl-This paper show s that the efficiency of the Perturband Observe (P&O) Maximum Power Point Tracking (MPPT)control technique can be improved by optimizing its samplinginterval T. according to the converters dynamics. Duringsunny days, when the maximum power point of th ephotovoltaic (PV) array moves very slowly, the samplinginterval T. must be set as short as possible without causinginstability. If the algorithm samples the array voltage andcurrent too quickly, it is subjected to possible mistakes causedby the transient behavior of the PV array+converter system,thus missing temporarily the MPP. A s a consequence, thealgorithm can be confused, the energy efficiency decays, andthe operating point can become unstable, entering disorderedbehaviors. The solution proposed in this paper lies in choosingT. according to the converters dynamics. The choice of thevalue of T. according to the proposed approach ensures athree-level steady-state duty-cycle swing around the MPP,whatever the duty-cycle step-size and the irradiance level are.As an example, a boost MPPT battery charger has beenstudied.

I. INTRODUCTIONA photovoltaic (PV) array under uniform irradiance exhibitsa current-voltage characteristic with a unique maximumpower point (MPP) where the array produces maximumoutput power, which changes as a consequence of thevariation of the irradiance level and of the panelstemperature [I] . The issue of maximum power pointhacking (MPPT) has been addressed in different ways in theliterature [2-IO]: fuzzy logic, neural networks, pilot cellsand DSP based implementations have been proposed. But,especially for low-cost implementations, the Perturb andObserve (P&O) and INcremental Conductance (INC) [2]techniques are widely used. In a typical P&O MPPTalgorithm, the operating voltage of the PV array is perturbedby changing the duty-cycle in a given direction (increase ordecrease) and the power drawn from the PV m a y is probed:if it increases, then the operating vo ltage is further perturbedin the same direction, whereas, if it decreases, then thedirection of operating voltage perturbation is reversed. Adrawback of P&O is that the operating point oscillatesaround the MPP, even during sunny days when theirradiance is slowly varyin g giving rise to the waste of someamount of available energy. Several improvements of theP&O algorithm have been proposed in order to reduce theamplitude of o scillations around the MPP in ,steady state, atthe price of slowing down the speed of response of thealgorithm to changing atmospheric conditions and loweringthe algorithm efficiency during cloudy days. The INC

M.VitelliD.I.I. -Second University of Naples, ItalyReal C asa dellAnnun ziata, Aversa (CE), ltalyEmail: [email protected]

algorithm seeks to overcome such limitations. However, asdiscussed in 1121, because of noise and measurement andquantization errors, also the INC operating voltage oscillatesaround the MPP. Both methods can he confused duringthose time intervals characterized by changing atmosphericconditions, since the operating point can move away fromthe MP P instead ofc lo se to it [2]. In [I21 it is shown that theP&O method, when properly optimized, leads to anefficiency which is equal to that obtainable by the INCmethod; however, no guidelines or general rules areprovided therein allowing the identification of the optimalvalues of P&O parameters which are instead chosen throughtrial and error tests. This paper shows that the efficiency ofP&O MPPT control technique can be improved byoptimizing its sampling rate according to the convertersdynamics. As an example, a boost MPPT converter (fig.1)has been studied.

Fig. I . A boost MPPT converter schematic.Let T, and Ad (>0) be respectively the sampling interval andthe magnitude of the duty-cycle perturbation of the P&OMPPT algorithm . The duty cycle perturbation at the (k+l)-t hsampling is given by:d((k+l)T,)= d(kT.)?Ad==d(kTJ+(d(kT&d((k- l)T&+g(p((k+l)TJ-p(kT.))

(1 )Ad must b e properly chosen: low ering Ad red uces thesteady-state losses caused by the oscillation of the arrayoperating point around the MPP but makes the algorithmless efftcient in case of rapidly changing atmosphericconditions. The optim al choice of Ad in situation s where wehave to account for both the sources and convertersdynamics, is discussed in detail in the paper [13]. The caseof quickly varying MPP occurs in cloudy days only. There isa more general problem, which occurs even during sunny

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days where the MPP moves very slowly, connected to thechoice of the sampling interval T, of the P&O MPPTalgorithm. Indeed, T, must be set as short as possiblewithout causing instability. In fact, considering a slowly-varying MPP, if the algorithm samples the array voltage andcurrent too quickly, it is subjected to possible mistakescaused by the transient behavior of the PV array+convertersystem, thus missing temporarily the MPP. As aconsequence, the algorithm can be confused, the energyefficiency decays, and the operating point can becomeunstable, entering disordered behaviors [121. The solutionproposed in this paper lies in choosing T, according to theconverter's dynamics, so that after each duty-cycleperturbation the system is allowed reaching steady-stateoperation before the next duty-cycle step variation.

I I . THE MODELAt MPP the adapted load resistance R is equal to theabsolute value RMppof the differential resistance of the PVarray. If the operating point of the PV array is close to theMPP, the power drawn by the PV array can be expressed as:

The relation between the PV array terminal current andvoltage is:

From eq. (3) we obtain:r 1-1

so that:r 1-1

and, finally:1

where V M p p nd IMpIB are the PV array MPP voltage andcurrent respectively. In the neighborhood of the MPP,assuming vpv = V, + tpvnd i p v = I + ipvwe have:vsv+R..ipv p = P M p p B = V M p p j l M p pVMPpipv tpVI,, +tpv ipv

(7)(3 )PV + R , .iPVip v = I , - & . ( e T v ~ -1)- RI,

where R, and Rh are series and shunt resistancesrespectively, IH is the light induced current, q is the diodeideality factor, I, is the diode saturation current and VT is thethermal voltage [I]. IH depends on the irradiance level S andon the array temperature T, while I, and VT epend on Tonly [I]. Let the system be perturbed by a small duty-cyclestep. If the oscillations of the operating point are smallcompared to the MPP then w e get:

Symbols with hats in the linearized equation (4) representsmall-signal variations around the steady state values of thecorresponding quantities. At constant (or slowly-varying)irradiance level, it is i = 0. Moreover, at steady-state, dueto the relatively high thermal inertia of the PV array [ I ] , it is

zz 0 . Eq. (4 ) can therefore be rewritten as:

From eqs. (5)-(7), and considering that V = R,,,I,,, atMP P , we get:i, =VMppiPv CPV:[MPP Cpvipvso that:

- 2( 8 ), = CPVI IMPP--)+Cpv;pv'MPP = --PVR. MPP R MPP

Eq. (8) show s that tht: dynam ics of tzv nd that of 6 is thesame. The small signal equivalent circuit of the systemunder study can be solved to find the small-signal control-to-array voltage transfer function GM and the load-to-mayvoltage transfer function Gwload, such thatipvG v ~ d + G v ~ l , ~ a d,,, . The transfer function Gv$dgives the fluctuation!; of the array voltage caused by loadvariations; such fluctuations can confuse the MPPTalgorithm, which is not able to distinguish between anayvoltage oscillations #caused by the load or c aused by the

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modulation of the duty. As the P&O algorithm directly actson d, our attention focuses on G d.The case of a boost battery charger is considered, for whichGVpdassumeshe following expression:

P

The Bode diagrams of Gvpd re shown in fig. 3 4 with: L=600 pH, Ci=IOO pF, V,=350 V, RMpp=45 R (a t S=lOOOW/m2), RMPP=120 C2 (at S=350 W/m2), 5= 0.0986 (atS=lOOO W/m2), = 0.0816 (at S=350 W/m2), o.= 4082rads,. The 14 m2 PV array is mad e by a series of 14 panel.Given the transfer function ( IO) , the response of cpv o asmall duty-cycle s tep perturbation of amplitu de Ad is:

( 1 1 )From eqs. (8) and (1 I) , the response of i to the step dutycycle perturbatio n Ad can be approx imated as in eq.(12):

(12)Therefore, the time T, after which fi will be confinedin theregion [-( +E)p2Ad2/Rmp, -(1-&)p2Ad2/R~pp], enteredaround the steady-state value -p2Ad2/RMpp141, is given by:

111. SIMULATION RESULTSThe responses of fi and i,, to a small duty cycle stepperturbation (Ad=0.01) are shown respectively in fig. 3b and3c.The detailed view of the PV array voltage and of the dutycycle for the P& O controlled boost battery charger of fig. 4shows that, if T.>T,, a sufficiently low value of E ensuresthat the P&O MPPT algorithm is not contksed by thetransient behavior of the system.

(a)Step Response Dd=O.Ol

(b )Step Response Dd=O.Ol0

-1VP I v-2

-3

4

-54 I II Irradiance350Wlm

Irradiance1000Wlrn jl0.004 0.008 0.012 0.016t i m l s l

(C)Fig. 3. Duty cycle step responses.In this case, the duty-cycle assumes only three differentvalues: dMpp-Ad, d MPP , Mpp+A dfig.5.a) and the operatingpoint takes only three different positions on the PV arraycharacteristic of fig.4.b: point C (on the right of the MPP),point B (close to the h4PP) and point A (on the lefl of theMPP). It is worth noting that point B is not perfectlycoincident with the M PP because of the discretization of d;

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of course, the lower Ad the lower either the distancebetween B and the MPP or the speed of response of th eMPPT to changing atmospheric conditions.PV Voltaae IV 1

.0.4 0.45 0.5 0.55D ubI 10.4 0.45 0.5 0.55time [ se c ]Operating Points1871P Powel[ w I

Fig. 4. &=O.I , T.=O.Ols, Ad=0.005, S=lOOO W/m2The choice of the value of T, according to the proposedapproach ensures a three-level steady-state duty-cycle swingaround the MPP, whatever duty-cycle step-size Ad andirradiance level S are settled, as shown in the plots of theduty-cycle reported in fig. 5, obtained with T.=O.Ols,Ad=0.005 (fig.S.a), Ad=0.001 (fig.5.b) and with anirradiance step change from S=350 W/mz to S=IOOO W/m2.Fig. 5.c shows, that a lower value of T (Ta=0.0033s) leadsto a worse behavior o f the system characterized, at bothirradiance levels S=350 Wlmand S=1000 Wlm, by a wid erswing of the operating point around the MPP. This leads to alower efficiency with respect to the corresponding case(T.=O.Ol s) shown in fig. 5(b). Moreover, in fig. 5(c) atlower irradiance level, a non-repetitive duty-cycle behavioris also evident, in agreement with the experimental resultsreported in [12].Whenever a resistive load is considered, the expression of

r

the control-to-PV a m y voltage transfer function GVpd,ha tin the case of the baost battery charger assumes the form(IO), becomes as in (14). Duty

(a)Duty

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In equation (14), CO is the output capacitance, D is thequiescent value of the duty cycle at the MPP and VPv,,,,, isthe photovoltaic array voltage at the maximum pow er point.In fig.6 the ac model of the system of fig.1 is shown: it lkbeen obtained after the linearization of the photovoltaicarray characteristic and of the sw itching cell.

L -

Fig. 6. The ac model used.It is easy to show that if:C ~ R , , ~ ~ % L (15.1)CdCi (15.2)then (14) reduces to (IO), so that the eqs. (10)-(13) are stillvalid. When inequalities (15) are not verified or whenconsidering the effect of parasitics that of cou rse render theexpression of the transfer function G, much morecomplex, eqs. 10-13 are no more valid. However, whatremains valid, is the strategy proposed to optimize the P& Oalgorithm; such a strategy relies on the choice of T.according to the dynamic behaviour of the system compo sedby the dc-dc converter and PV array. Such an approach canbe applied to other converters and/or operating modes;whenever the analytical approach is unpractical orunaffordable, it is always possible to get numerically, bymeans of suitable simulations, the threshold value that Tamust exceed in order to optimize the P& O mppt technique.

P

IV. CONCLUSIONSIn this paper a theoretical analysis allowing the optimalchoice of the value of the sampling period T. to be adoptedwhen using the P&O mppt algorithm has been carried out.The idea underlying the proposed optimization approach liesin the customization of T. to the dynam ic behaviour of thewhole system composed by the specific converter and PVarray adopted. A s an example, a boost battery charger hasbeen studied in detail.

ACKNOWLEDGEMENTSThis work has been supported by MIUR and University ofSalemo grants.

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