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Solar Energy 86 (2012) 2029–2040

Photovoltaic model and converter topology considerations for MPPTpurposes

Thomas Bennett ⇑, Ali Zilouchian, Roger Messenger

Department of Computer and Electrical Engineering and Computer Science, Florida Atlantic University, 777 Glades Rd., Boca Raton, FL, USA

Received 12 January 2012; accepted 6 April 2012Available online 1 May 2012

Communicated by: Associate Editor Elias Stefanakos

Abstract

In this paper a few variations of the single diode photovoltaic (PV) model are developed by removing the series and/or parallel resis-tors. The models are expected to yield similar results due to the large parallel resistance and small series resistance. Nonetheless, the IVand power curves are compared. The models are then compared in the context of different converter systems. If the behavior is similar,then one may be justified in analyzing Maximum Power Point Tracking (MPPT) systems by using a simpler PV model than that normallyused. In particular the single diode model with no resistors allows the current (or voltage) to be explicitly defined in terms of the voltage(or current), whereas the model using two resistors does not. The different converter topologies were also compared for each model to seeif the dynamic behavior of the PV system depended on the converter used. For the most part the results were fairly consistent regardlessof the PV model, PV module, or converter used. For example, low voltages on the IV curve yielded non-constant steady state behaviorfor all system configurations. However, there were some inconsistencies which could be worth further consideration, as they may affectthe design of the MPPT algorithm.� 2012 Elsevier Ltd. All rights reserved.

Keywords: Photovoltaic (PV) model; Maximum Power Point Tracking (MPPT); DC/DC converter

1. Introduction

The need for sustainable energy sources is well known,and has received much attention and funding in recentyears DOE (2011). There are many competing technolo-gies, such as solar, geothermal, wind, hydro, biomass, aswell as the use of fuel cells. Solar energy, and photovoltaicsin particular, have one of the highest potentials (Bennettand Zilouchian, 2011). Photovoltaic systems may includepower conditioning circuitry, as well as battery back up.

Photovoltaic systems typically employ either a DC/DCor DC/AC converter, with or without battery chargingcapabilities, that make use of MPPT. The MPPT

0038-092X/$ - see front matter � 2012 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.solener.2012.04.005

⇑ Corresponding author. Tel.: +1 5613024914.E-mail addresses: tbenne17@fau.edu (T. Bennett), zilouchi@fau.edu

(A. Zilouchian).

algorithms require voltage and current measurements asinputs, but due to transient behavior, these measured val-ues may not be representative of the system’s state (anddue to energy storage components, values measured atany one point are likely not even on the IV curve of thePV module). It seemed important to delve into this in moredetail.

To better understand the transient behavior, muchattention was given in trying to derive useful mathematicalequations, both for the PV model, as well as the PV-con-verter system. For this, it seemed reasonable to begin withthe simplest models first. This work raised another ques-tion. Could simpler models be used that still yield realisticresults? Ultimately the goal is to be able to implement arobust and highly efficient MPPT system. The aboveconsiderations and their relation to that ultimate goal isthe subject of this paper.

2030 T. Bennett et al. / Solar Energy 86 (2012) 2029–2040

The PV model that appears to be used most often is theone which is represented by a current source, a diode, andtwo resistors (Nelson, 2003). Others use the no resistor ver-sion, as well as a single resistor version. In Section 2, thesedifferent model versions will be developed. In this sectiononly the IV curves and the power curves of these differentmodels will be compared. This is to verify that they behavesimilarly. If the results are much different, this would showa great sensitivity to the model used, and would perhapsindicate that a more advanced model would be required.

Though the IV and power curves may be similar in allthe PV models considered in the preceeding section, doesthat mean that when these models are used with a con-verter that the behavior will also be similar? In Section 3,buck, boost, NonInverting BuckBoost, and Cuk converterswill be used to see if the dynamic behavior of the systemchanges significantly depending on the PV model used.The MPPT algorithms introduced in the literature are trea-ted as converter independent. In (Hussein et al., 1995) theincremental conductance method is introduced withoutany specified converter, though it is implemented using abuck converter. In (Femia et al., 2005) a boost converteris used, but the authors indicate it should work with otherconverters. Hence, in addition to comparing the modelsagainst each other using each fixed converter, the convert-ers will also be compared for each fixed model. This shouldrequire little extra work, but will allow one to verify thatMPPT algorithms are indeed converter independent.

2. Modeling the PV module

Perhaps the most popular model used to represent the PVmodule is the current source in parallel with a diode, with aparallel and series resistor. This is illustrated in Fig. 1.Models where one resistor (Celik and Acikgoz, 2007; Duru,2006; Liu and Huang, 2011, or no resistors Messenger andVentre, 2004; Hussein et al., 1995; Yu et al., 2004) are used,can also be found. These simpler models are really just spe-cial cases of the two resistor model. However, they greatlyreduce the mathematics needed, and so should be consid-ered separately.

The equation of the circuit in Fig. 1 is:

i ¼ Ig � Is evþiRs

a � 1� �

� vþ iRs

Rpð1Þ

In this equation i is the PV current, v is the PV voltage,Rs is the series resistor, Rp is the parallel resistor, Ig is thelight-generated current, Is is the diode’s saturation current,

Fig. 1. A circuit representation of a PV module.

and a = AkT/q, where A is the diode ideality factor, k isBoltzmann’s constant, T is the temperature, and q is thecharge of an electron.

The derivative of this curve, with respect to voltage, isalso useful, and is easily found to be:

didv¼ �

Isa e

vþiRsa þ 1

Rp

1þ IsRsa e

vþiRsa þ Rs

Rp

ð2Þ

The two modules that will be used in this paper are the195 W BP-SX3195, and the 305 W Sunpower-305. Thespecs are given in Table 1.

These two modules have a substantial difference in thecurrent to voltage ratio, allowing them to be useful PV rep-resentatives (the size is less important since everything canbe scaled). They also have fairly different fill factors, as canbe easily verified.

The module specifications are given for standard testconditions (STCs), and so Ig, Is, a, and the resistors areall constant on the curve containing those points. Thereare three points from the spec sheet on the STC IV curve:(Voc, 0), (0, Isc), and (Vm, Im).

These points along with (1) yield three equations:

0 ¼ Ig � Is eV oc

a � 1� �

� V oc

Rpð3Þ

Isc ¼ Ig � Is eIscRs

a � 1� �

� IscRs

Rpð4Þ

Im ¼ Ig � I s eV m�ImRs

a � 1� �

� V m þ ImRs

Rpð5Þ

However, it is also know that dP ¼ dðIV Þ ¼ 0) dIdV ¼ �I

V at(Vm, Im). So, Eq. (2) gives another equation:

Im

V m� Is

ae

V mþImRsa 1� Rs

Im

V m

� �� 1

Rp1� Rs

Im

V m

� �¼ 0 ð6Þ

Since the no-resistor model is the simplest model to ana-lyze, it will be considered first. After this, two one resistormodels will be considered before returning to the two resis-tor model.

2.1. No resistor model

The no resistor model, NRM, is the model obtained byremoving the series and parallel resistors from the modelshown in Fig. 1 (i.e. set Rs = 0 and Rp =1). For theNRM, Eqs. (1) and (2) are simpler. Equation (1), for exam-ple, reduces to

i ¼ Ig � Is eva � 1

� �ð7Þ

For this model, the four equations based on the spec sheetare:

0 ¼ Ig � Is eV oc

a � 1� �

ð8Þ

Isc ¼ Ig ð9Þ

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9BP IV Curve

Voltage

Cur

rent

10 15 20 25 306

6.5

7

7.5

8

8.5

9BP IV Curve

Voltage

Cur

rent NRM

PRMSRMTRM

Fig. 2. The IV curves for the different models for the BP module.

Table 1Two PV modules used for testing.

Model Voc (V) Isc (A) Vm (V) Im (X)

BP-SX3195 30.7 8.6 24.4 7.96SP-305 64.2 5.96 54.7 5.58

T. Bennett et al. / Solar Energy 86 (2012) 2029–2040 2031

Im ¼ Ig � Is eV m

a � 1� �

ð10Þ

Is ¼aIm

V meV m=að11Þ

However, for this model, only three parameters areneeded (a, Ig, and Is), and so one of the equations needsto be dropped. The last equation, 10, was dropped for whatfollows. After combining these equations to solve for a, thefollowing is found:

Im ¼ Isc � IsceV m=a � 1

eV oc=a � 1ð12Þ

Note that an exact solution for a is not obtained. How-ever, using some of Matlab’s built in functions, such asfsolve or fzero, a can be solved for, and so too all the otherparameters. (A good approximation which allows an exactequation exists, a � V m�V oc

lnð1�Im=IscÞ, but the implicit equationabove was easy enough to solve for). See Table 2 for thederived parameter values. See Figs. 2–5 for the IV andpower curves for the BP and SP modules.

In Table 3, the difference between where the spec sheetsays the MPP should be and where the maximum is in

Table 2Parameter values for different PV modules and models.

Model Ig (A) Is (A) a (V) Rp (X) Rs (X)

BP NRM 8.6 2.7307e�5 2.4249 1 0SP NRM 5.96 4.9709e�8 3.4512 1 0BP PRM 8.6 1.5772e�4 2.8034 �78.702 0SP PRM 5.96 2.2302e�8 3.3102 1214.1 0BP SRM 8.600 4.8592e�7 1.8395 1 .1911SP SRM 5.960 2.6016e�7 3.7883 1 �.1663BP TRM 8.631 9.1454e�12 1.1148 101.62 .36701SP TRM 5.963 1.5968e�10 2.6396 501.28 .25060

the NRM due to dropping Eq. 10 is shown. This modelappears to have done fairly well. This can also be seen inFigs. 4 and 5.

For both of these modules, it was confirmed that (Voc,0), (0, Isc), and (Vm, Im) (from the spec sheet) were pointson the curve.

2.2. One resistor models

Since four equations can be derived, and a one resistormodel has four parameters, these seem like worthy modelsto consider. Dropping the series resistor maintains an exactIV equation, as well as Ig = Isc. So, this is a good one tostart with. For simplicity, the model with the parallel resis-tor (and without the series) will be called the Parallel Resis-tor Model, PRM, and the one with the series (and withoutthe parallel) resistor will be called the Series Resistor

Model, SRM.

� First, the PRM is considered: The equations needed hereare similar to the ones derived previously. There are fourequations, and four parameters to solve for. Combiningall four equations into one equation in terms of a wasdone, and the equation was solved for using Matlab.The results are recorded in Table 2. For the BP modulea negative resistor value was obtained: R = �78.7021 X.It is also interesting to note, that with a negative R

value, the IV curve has a positive slope for low voltagevalues, as shown in Fig. 2.Another approach was also used to verify the results.Since the NRM worked well, the NRM parameters weretaken as a starting point, and then Rp was decreased frominfinity, and new parameter values were obtained. Theerror of Vm and Im were observed to see if these errorsgo to zero for a given Rp value (recall that we had anerror in the NRM as shown in Table 3). It is importantto keep in mind that for each change in the resistor value,all the other parameters are changed as well. SeeFig. 6.By increasing Vm for the BP module, one is ableto cause the slope of the IV curve to decrease again.Increasing Im will cause it to decrease again. It would

0 10 20 30 40 50 600

1

2

3

4

5

6

SP IV Curve

Voltage

Cur

rent

45 50 55 604.5

5

5.5

6SP IV Curve

Voltage

Cur

rent NRM

PRMSRMTRM

Fig. 3. The IV curves for the different models for the SP module.

Table 3Comparison of NRM MPP vs desired MPP.

Model Vm (V) Im (X) Max power (W)

BP-Actual 24.4 7.96 194.22BP-NRM 24.8 7.83 194.56SP-Actual 54.7 5.58 305.23SP-NRM 54.4 5.61 305.28

2032 T. Bennett et al. / Solar Energy 86 (2012) 2029–2040

likely be possible to find the ratio of Vm/Voc to Im/Isc

(perhaps also as a function of the fill factor) in whichthe slope becomes zero. However, this is not treated here.� Next, the SRM is considered: This time two equations in

two variables (a and R) was used, as solving for one var-iable was difficult. The results are recorded in Table 2.Note that this time the SP module yielded a negativeresistor value. Also, similar to last time, a second methodwas used, where the effects of Rs on the Vm and Im errorswere observed. This time Rs was increased (anddecreased) from zero. The results are shown in Fig. 7.

2.3. Two resistor model

The two resistor model, TRM, is the model with bothseries and parallel resistors, as shown in Fig. 1. As stated

0 5 10 15 20 25 300

50

100

150

200BP Power Curve

Voltage

Pow

er (w

atts

)

Fig. 4. The power curves for the diff

previously, this is a popular model and is used in most ofthe papers on MPPT. The papers (Kennerud, 1969; Phanget al., 1984; Chan and Phang, 1987) discuss different meansof obtaining parameter values for this equation. Theseauthors use five points on Eqs. (1) and (2):

� The open circuit voltage, Voc.� The short circuit current, Isc.� di

dv

��i¼Im¼ � Im

V m.

� dvdi

��v¼V oc

.� dv

di

��i¼Isc

.

The last two points require measurements that are notincluded on a typical PV spec sheet. In (Phang et al.,1984; Chan and Phang, 1987) some approximations aremade to make the variables easier to solve for (a numericalmethod is still required however). Another paper (Seraet al., 2007) attempts to solve for all the parameters usingonly values obtainable from a PV spec sheet. The first threepoints from above are used, but the last two are replacedby using (Im, Vm) as a point on the curve, and an approx-imation from (Chan and Phang, 1987), dI

dV

��I¼Isc� � 1

Rsh

(where Rsh = Rp). The point (Im, Vm) is certainly on a specsheet. However, the approximation was treated as anequality in their paper, and even in the original paper there

21 22 23 24 25 26 27175

180

185

190

195BP Power Curve

Voltage

Pow

er (w

atts

)

NRMPRMSRMTRM

erent models for the BP module.

0 10 20 30 40 50 600

50

100

150

200

250

300SP Power Curve

Voltage

Pow

er (w

atts

)

53 53.5 54 54.5 55 55.5 56302

302.5

303

303.5

304

304.5

305

305.5

306SP Power Curve

Voltage

Pow

er (w

atts

)

NRMPRMSRMTRM

Fig. 5. The power curves for the different models for the SP module.

0 500 1000 1500 2000−0.2

−0.18

−0.16

−0.14

−0.12

Im E

rror

(A)

Rp (ohms)

Errors for different Rp values in BP PRM

0.4

0.5

0.6

0.7

0.8

Vm

Err

or (V

)

Im ErrorVm Error

0 500 1000 1500 2000−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

Im E

rror

(A)

Rp (ohms)

Errors for different Rp values in SP PRM

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Vm

Err

or (V

)

Im ErrorVm Error

Fig. 6. The effects of Rp on two different modules.

−0.4 −0.2 0 0.2 0.4−0.5

0

0.5

Im E

rror

(A)

Rs (ohms)

Errors for different Rs values in BP SRM

−2

0

2

Vm

Err

or (V

)

Im ErrorVm Error

−0.4 −0.2 0 0.2 0.4−0.1

0

0.1

Im E

rror

(A)

Rs (ohms)

Errors for different Rs values in SP SRM

−1

0

1V

m E

rror

(V)

Im ErrorVm Error

Fig. 7. The effects of Rs on two different modules.

T. Bennett et al. / Solar Energy 86 (2012) 2029–2040 2033

is little justification. The approach outlined below will alsotry to use only PV specs, but will not use any simplifiedequations.

Combining Eqs. (3)–(5), (such as by solving for Ig in (3),Is in (4), and then plugging into (5)) yield the followingequation:

c1 1� eIscRs�V oc

a

� �þ c2 1� e

V mþImRs�V oca

� �¼ 0 ð13Þ

where

c1 ¼ V m � V oc þ ðRs þ RpÞIm

c2 ¼ V oc � I scðRs þ RpÞ

Eq. (13) has three unknowns: a, Rs and Rp. Based on(Kennerud, 1969; Sera et al., 2007; Phang et al., 1984),typically 0 < Rs < 1 and 100 < Rp < 10000. Based on thisand the values of the PV specs, one should expect that c1

2034 T. Bennett et al. / Solar Energy 86 (2012) 2029–2040

is positive and c2 is negative. One should also see that thefirst exponential term has an exponent that is negative sincetypically Voc > Isc and Rs < 1.

From this, it must also be that the 2nd exponential termis negative, or there is no solution. This gives

V m þ ImRs � V oc < 0) 0 < Rs <V oc � V m

Im

This may be useful in obtaining an upper limit for Rs, help-ing to guarantee a solution. At this point, one can then usea range of Rs values starting at 0 (which is the PRM), andincreasing Rs towards its maximum (not quite to its max,otherwise the parameter values become extreme, since Rs

is approaching the point where there is no solution). A plotof Rs versus Rp is shown in Fig. 8.

At this point there are an infinite number of solutions.However, to better distinguish this model from theNRM, PRM, and SRM, a point was taken where Rs andRp are both significant. See Table 2 for the values chosenfor each module. Clearly more information is needed inorder to obtain a less arbitrary solution.

The change in voltage and current with respect to tem-perature could now be used in attempts to make some ofthe parameters temperature dependent. However, in thispaper everything is done at STC, so it is not necessary tomodify the PV parameters from their constant values.

3. Comparing the models with different converters

There were some slight differences in how the IV curveslooked in the different models, but for the most part theywere, as expected, similar. However, it seemed worthwhileto see how they compare when used in conjunction with aconverter, as would be used in an MPPT system. Topolo-gies using both the buck converter (Hussein et al., 1995)and the boost converter (Femia et al., 2005) or both(Hua and Shen, 1998) can be found in the literature. Forthis paper, simple buck and boost converters with a fixedvoltage output will be used, though often times a resistor

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−500

0

500

Rs (ohms)

Rp

(ohm

s)

Rp vs Rs for BP TRM

Fig. 8. The effects of Rs on

in parallel with a capacitor is used for the load. Resultsfrom the noninverting buckboost, and the Cuk will alsobe shown. Due to the large number of potential figures,only the NRM and TRM will be used.

3.1. Buck converter

To simplify the analysis, a fixed voltage output wasassumed. In the typical buck converter circuit, there is nocapacitor in parallel with the input voltage. During thetime when the switch is off the input provides no power.However, for a solar panel a capacitor in parallel withthe PV module is needed, so that even when the switch ofthe converter is off, the solar panel is providing power.Otherwise, when the switch is off, the solar panel wouldnot be providing energy from the sun. (In this case the sun-light would probably be absorbed and converted to heat,which would negatively affect the efficiency during theon-time as well.)

Only one solar module was used for these simulations.The results obtained here should also apply to a systemwhere the converter covers an array of modules (just scale).

The complete system, using the two resistor model, isgiven in Fig. 9. Note that the switch occurs in zero timein what follows, so that the switch shown is acceptable(otherwise a make before break switch could be used).

3.1.1. System equations

The following equations represent the circuit pictured inFig. 9, where d is 0 or 1 depending on the switch.

i� diL ¼ Cdvdt

ð14Þ

dv� vo ¼ LdiL

dtð15Þ

When d = 0, Eq. (14) yields the approximation

vðt þ hÞ ¼ vðtÞ þ hC

iðtÞ ð16Þ

0 0.5 1 1.50

200

400

600

800

1000

1200

1400

Rs (ohms)

Rp

(ohm

s)

Rp vs Rs for SP TRM

two different modules.

Fig. 9. The TRM PV model connected to a buck converter.

T. Bennett et al. / Solar Energy 86 (2012) 2029–2040 2035

and 15 yields the exact solution

iLðt þ hÞ ¼ iLðtÞ �vout

Lh ð17Þ

From the IV equation one is able to obtain i(t + h).However, for the TRM this means a lot of computation,since it would have to solve an implicit function every time.Another method would be to find the current by using thederivative of the IV equation allowing an explicit equationfor i(t + h). The potential problem with this is that after awhile the (i, v) points that are obtained might slowly devi-ate from the IV curve. To remedy this, at the end of everyperiod, the implicit IV equation was used to make sure thepoints were on the curve. For one 30 s simulation, it tookthis method roughly 30 s compared to the almost 1500 srequired for the implicit method. The results were com-pared, and were indistinguishable from each other. Hence,the latter, quicker method, was used. (The same could besaid about the SRM).

Due to the capacitor and inductor in our circuit,i(T�) = i(T+), iL(T�) = iL(T+), and v(T�) = v(T+), whereT represents a point in time when the switch was turnedon or off in the circuit.

The equations for when d = 1 are similar.The equation relating the input and output voltage for a

buck converter is ideally given by v = vout/d (where d is theduty cycle). This will be seen to be true for duty cycles thatyield PV voltages near the maximum power voltage (as willbe seen below).

The problem with this equation is that a small enoughduty cycle will violate it, since the PV voltage is capped.In (Enrique et al., 2007), the authors compare the buck,boost, and buckboost converter topologies and state a sim-ilar result, but in terms of resistance values (they used aresistor load). In fact, with the circuit as it is, the batterywould start draining and current would actually flow intothe module. In the implementation of the buck, as well asother converters, a diode is used which would prevent backflow. Alternatively, one could use a blocking diode1 if theconverter implementation does not make use of a diode.Regardless of the approach, the PV current will not beallowed to drop below zero in this model or the others(or allow battery discharge).

The capacitor and inductor sizes should be small enoughto allow for quick transient, but large enough prevent largefluctuations of voltage and current when switching. Eqs. 14

1 Blocking diodes are not used as often in practice at present due in largepart to better quality control. Still, one could potentially be used here ifnecessary.

and 15 can be used to get a handle on these values. A veryhigh frequency would also reduce fluctuation, and thusallow for even smaller capacitor and inductor values. Forthis paper the period of the system is Ts = 0.001 s, to allowfor quicker simulations. In other words, the switch is off(d = 0) for (1 � d)Ts seconds and on (d = 1) for dTs

seconds.For the BP module, Vout = 12 V, and for the SP module

Vout = 24.

3.1.2. Results at STC

In Figs. 10–13 one can see the effects of the capacitorand duty cycle on transient and steady state behavior.The inductor had a similar effect as the capacitor, and sochanges to its value are not shown.

Note that for low voltages, the steady state has alarge ripple. Also, for voltages too close to Voc, thecurrent is very near zero, making it unlikely that the con-verter will stay in continuous mode. So, the PV voltagemust keep some distance from Voc. However, one canget rather close to Voc and stay in continuous mode. Thisis due to the sharp decrease in current near the open cir-cuit voltage, which means that there is significant currentup until the voltage is very close to Voc. This is evidentin the figures of BP, where for a duty cycle of 40%, thevoltage goes to 30 V, and stays below Voc, which is30.7 V.

Another interesting result is that the TRM convergesquicker for lower voltages, and the NRM for highervoltages.

Consider Fig. 12. The bottom curve oscillates betweenabout 24 and 36 V. Increasing either C and/or L by a factorof 100 only increased the period of the oscillation. Also,decreasing the period Ts by a factor of 100 seemed to haveno noticeable effect at all. Notice that the curve is stoppingat 24 V. This is the minimum allowable value based on Vout

(due to blocking diode). It is interesting to note that it doesnot appear to be clipped at 24 V though, but rather is a nicesinusoidal curve. Increasing the duty cycle from 80% to90% yields a curve that oscillates between 24 and 29 V. Thiscurve is also sinusoidal, rather than appearing to beclipped. Stopping at 24 V probably also explains why inFig. 13 the lowest voltage curve seems better behaved thanthe 2nd lowest voltage. As a result it would still seem thatthe lower the voltage, the worse the steady state, exceptwhen partially “corrected” by a blocking diode. The factthat large inductors or currents do not seem to help, seemto imply that as the voltage gets very low, the oscillationsbecome significantly worse.

Another example is with the BP NRM. Using a dutycycle of 80%, output voltage of 6 V, but large (C =L = 1) capacitor and inductor values, the simulation wasallowed to run for a long time. See Fig. 14 for 6000 s ofsimulation time. In fact, though not shown, even after24 h the curve is still converging. So larger capacitor andinductor values to try to smooth out the horrible transientsat low voltages will not solve the problem.

0 5 10 15 20 25 3010

15

20

25

30

BP NRM with Buck Converter

time (s)

PV

Vol

tage

0 5 10 15 20 25 3010

15

20

25

30

BP NRM with Buck Converter

time (s)

PV

Vol

tage

Fig. 10. A plot of the BP NRM buck circuit for duty cycles of 40%, 60% and 80%, two different C values, and Vout = 12.

0 5 10 15 20 25 3010

15

20

25

30

BP TRM with Buck Converter

time (s)

PV

Vol

tage

0 5 10 15 20 25 3010

15

20

25

30

BP TRM with Buck Converter

time (s)

PV

Vol

tage

Fig. 11. A plot of the BP TRM buck circuit for duty cycles of 40%, 60% and 80%, two different C values, and Vout = 12.

0 5 10 15 20 25 3020

25

30

35

40

45

50

55

60

65SP NRM with Buck Converter

time (s)

PV

Vol

tage

0 5 10 15 20 25 3020

25

30

35

40

45

50

55

60

65SP NRM with Buck Converter

time (s)

PV

Vol

tage

Fig. 12. A plot of the SP NRM buck circuit for duty cycles of 40%, 60% and 80%, two different C values, and Vout = 24.

2036 T. Bennett et al. / Solar Energy 86 (2012) 2029–2040

For low voltages, the current is nearly fixed. In fact,replacing the PV module with a constant current sourceyields similar oscillating behavior that is achieved withthe PV model at low voltages. Hence, this behavior canlikely be explored in the easier context of a current sourceinput.

3.2. Boost

Just like for the buck system, a fixed DC output isused. In this case, the output needs to be a higher voltage.Here though, an inductor is not placed on the output, inseries with the fixed voltage, as two inductors switched

0 5 10 15 20 25 3020

25

30

35

40

45

50

55

60

65SP TRM with Buck Converter

time (s)

PV

Vol

tage

0 5 10 15 20 25 3020

25

30

35

40

45

50

55

60

65SP TRM with Buck Converter

time (s)

PV

Vol

tage

Fig. 13. A plot of the SP TRM buck circuit for duty cycles of 40%, 60% and 80%, two different C values, and Vout = 24.

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Fig. 14. The TRM PV model connected to a buck converter.

Fig. 15. The TRM PV model connected to a boost converter.

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Fig. 16. A plot of the BP NRM boost circuit for duty cyc

T. Bennett et al. / Solar Energy 86 (2012) 2029–2040 2037

into series would dictate potentially two different currents,and so would not work. See Fig. 15 for the TRM Boostconverter circuit.

3.2.1. System equations

The system equation for this circuit is simple:

v� ð1� dÞvo ¼ Ldidt

ð18Þ

For the BP module, Vout = 36 V, and for the SP moduleVout = 72 V. Also, the inductors used for the SP module arelarger, as indicated in the figures. This was due to the poorconvergence for the given duty cycles of 30%, 50%, and70%, used for both modules.

3.2.2. Results at STC

The results for the boost converter are shown in Figs.16–19. Like the buck system, the behavior is far worse atlower voltage values. Also like the buck system, the NRMis slightly better for voltages closer to the knee. Note thatthe inductor values were different between the BP and SPmodules. For the buck circuits the same values were usedfor both modules, but this time it was necessary to changethe values in order to obtain results that were not too

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les of 30%, 50%, and 70% for two different L values.

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Fig. 17. A plot of the BP TRM boost circuit for duty cycles of 30%, 50%, and 70% for two different L values.

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Fig. 18. A plot of the SP NRM boost circuit for duty cycles of 30%, 50%, and 70% for two different L values.

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Fig. 19. A plot of the SP TRM boost circuit for duty cycles of 30%, 50%, and 70% for two different L values.

2038 T. Bennett et al. / Solar Energy 86 (2012) 2029–2040

unstable. The boost converter equation V = (1 � D)Vout

holds for the convergent cases, as one would hope.Notice that for the values chosen for this converter sys-

tem, the inductor seemed to have an effect on convergenceas seen in Fig. 16. This held true even when the duty cyclewas changed from 70% to 80%. Even for the buck systemthe inductor and capacitor helped, though it seems moreobvious with the boost converter. Still, there seems to be

a point at which the operating voltage point gets so lowthat no reasonable capacitor or inductor value can smoothout the curve, as was shown in Fig. 14 previously.

3.3. Other converters

A non-inverting buckboost converter, as well as a Cukconverter were also tested (see (Erickson and Maksimovic,

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Converter Converter

Fig. 20. The NRM and TRM models of the BP module using the NonInverting BuckBoost converter.

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Fig. 21. The NRM and TRM models of the BP module using the Cuk converter.

T. Bennett et al. / Solar Energy 86 (2012) 2029–2040 2039

2001) for a description). Due to space constraints, only theBP module will be shown, as the SP results were similar.These can be seen in Figs. 20 and 21. Once again lowvoltages are worse, though the TRM does better. TheNonInverting BuckBoost seemed to do better. However,the Cuk converter has two inductors, and so it is not aperfect comparison. Nonetheless, some changes in param-eters did not seem to yield a better result.

4. Conclusion

The figures clearly show that there are some slight differ-ences in the IV curve characteristics of the different PVmodels. In Fig. 2, one curve shows a positive slope beforethe knee. However, any effects on the power curve were lessobvious, and that is the curve that an MPPT algorithmtries to control. Nonetheless, even with these small differ-ences one could see some significant differences in the

dynamic behavior of the converter systems. If one wereto consider a more advanced two diode model, as well assome possible internal capacitance, more differences maybe discovered. On the other hand, the converters used withthe models should realistically have some nonidealitieswhich could potentially smooth out some differences. Thishas not yet been explored. Still, it seems worthwhile toexplore this more to make sure there are no importantimplications in designing an MPPT converter.

For an MPPT algorithm, the point on the IV curveneeds to be constantly measured. If it turns out that itmay take a long time to measure this, due to a long tran-sient, this could influence some properties of the algorithm.For example, one may wish to use smaller step sizes with aperturbation to prevent operating at too low of a voltage.Or perhaps the better solution would be to verify moreoften that one is not operating at a low voltage, by perhapschecking Voc (more often).

2040 T. Bennett et al. / Solar Energy 86 (2012) 2029–2040

Having a transient that can last an indefinite time meansthat one cannot simply have the MPPT algorithm wait Tseconds before assuming steady state is reached. Further-more, as seen, it is possible to be at steady state, but stillhave a very large ripple (as seen for voltage values belowthe knee). As a result, changing voltages may not meantransient at all. It could mean terrible steady state, stillbeing at transient, or environmental changes. This behav-ior suggests that it might be a good idea to begin the MPPTalgorithm by first measuring Voc and then starting atroughly 80% of that value (to be near Vm).

Though this does not seem that likely, considering thedifferent frequencies at which the voltage could oscillatewhile at steady state, one may need to take care with thesampling frequency to avoid aliasing. In particular, onecould measure a constant voltage, though the actual volt-age is oscillating. Even if the voltage was changed usingwrong information, the change may prevent the aliasingfrom reoccurring, meaning it may not be a big issue. Still,it is something that one may need to consider (if one doesnot prevent the algorithm from running at low voltages).

Though there were some slight differences, for the mostpart the converter did not determine dynamic behavior, butrather the point on the IV curve where the converter isoperating. One difference may be that the threshold as towhat makes a voltage “too low” is different. However, ifone keeps close to the knee of the IV curve, this shouldnot be too large an issue. This seems to support the notionthat MPPT algorithms are mostly converter independent.

Also, though BP and SP had noticeably different out-comes when deriving the PV models (they had negativeresistor values in different models), they behaved similarlyin the different converter systems. This supports the ideathat the control algorithms used in MPPT should notdepend heavily on the solar panels being used. This wouldthen seem to indicate that converter systems need not bebuilt dependent on the modules being used in the system.

In the perturb and observe control algorithm used byone of the authors (algorithm not included in this paper),perturbations start to the left. The reasoning was that theslope of the power curve is less on that side, and so mayallow for operating at a higher power during the search.However, as is evident, this may imply a longer transient,and so a longer time before the next perturbation. Conse-quently, it may be worth looking into modifying the algo-rithm to perturb to the right.

Some preliminary results using a resistor/capacitor out-put, rather than a fixed voltage, seemed to show betterbehavior at low voltages. Consequently, the load may playan important role in the MPPT algorithm as well. Morework will need to be done to see the extent of the loadeffects. Some other preliminary results have come fromadding converter losses (switching losses – due to on resis-tance of transistor and diode losses, as well as inductor

resistance). These system losses appear to remove muchof the oscillatory behavior even at low voltages. Thismay suggest that this is more of an issue when modelingwith ideal components, rather than an issue in a practicalcircuit.

In short, for the most part, especially when operatingaround the maximum power point, the MPPT algorithmdoes not appear to depend on the module, model, or con-verter used. However, there were enough slight differencesto make this not fully inconclusive.

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