Maximum Power Point Tracking using the OptimalDuty Ratio for DCDC Converters and LoadMatching in Photovoltaic Applications
Eduardo I. OrtizRivera,
Member IEEE
University of Puerto RicoMayagüezDepartment of Electrical & Computer EngineeringPO Box 9042, Mayagüez, PR 006819042eduardo.ortiz@ece.uprm.edu
Abstract
The purpose of this paper is to present an alternativemaximum power point tracking, MPPT, algorithm for aphotovoltaic module, PVM, to produce the maximum power,
P
max
,using the optimal duty ratio,
D
, for different types of dcdcconverters and load matching. The proposed algorithm has theadvantages of maximizing the efficiency of the power utilization,can be integrated to other MPPT algorithms without affectingthe PVM performance, is excellent for RealTime applicationsand is a robust analytical method, different from the traditionalMPPT algorithms which are more based on trial and error, orcomparisons between present and past states. The procedure tocalculate the optimal duty ratio for a buck, boost and buckboostconverters, to transfer the maximum power from a PVM to aload, is presented in the paper. Additionally, the existence anduniqueness of optimal internal impedance, to transfer themaximum power from a photovoltaic module using loadmatching, is proved. Finally, results are presented in the paper.
I.
I
NTRODUCTION
Solar energy is one of the most important alternativesenergies with applications in urban areas, motor drives,satellites, [1][8] etc. A photovoltaic module, PVM, is the keycomponent to convert solar energy into electric energy [1]. Inaddition to the PVM, a typical dc photovoltaic systemconfiguration consists of storage capacitance, dcdc converter,and batteries [2]. In most of the applications, it is alwaysdesired to obtain the maximum power from a PVM, due thefact that the PVM operates at the highest efficiency [3][4].The maximum power point tracker, MPPT, is the typicalalgorithm to calculate the maximum power,
P
max
, provided bya PVM [3][6].In the past, many authors described different variations of the MPPT algorithm, [3][12] and the applications to controldcdc converters in energy conversion, [8][14]. Unfortunately,most of the existing MPPT methods to estimate the maximum power are based on trial and error algorithms where thevoltage is increased until the maximum power is achieved, better known as the hillclimbing method [5][7]. Other MPPTalgorithms compare the last sampled voltage and currentversus the presently sampled voltage and current to see whichstate will produce the maximum power [11]. Additionally, theliterature offers other types of MPPT algorithms such thatrippled based method [7], fuzzy logic [9] and lookup tablemethods [10].Disadvantages with these MPPT algorithms are that discretealgorithms require several iterations to calculate the optimalsteadystate duty ratio [13]. Some of them are not designed for quick changes in the weather conditions [13]. Also, for nonanalytical methods, the time for the iterations will depend onthe initial conditions and can create bifurcation problems [14][16]. In this paper, an analytical method for load matching is proposed using the optimal duty ratio for a dcdc converter totransfer the maximum power to the load. For load matching,the internal resistance
Ri
for a PVM is used. The paper isdivided into five sections which are the PVM model, optimalduty ratio for a dcdc converter, simulations and experimentalresults, and conclusions.II.
A
NALYTICAL
P
HOTOVOLTAIC
M
ODULE
M
ODEL
The PVM model based on the manufacturer data sheets [17]will be used to obtain the optimal duty ratio,
D.
The PVMmodel takes into consideration the temperature,
T
, andeffective irradiance,
E
i
, over the PVM and the Standard TestConditions, STC, i.e.
T
N
is 25
o
C and
E
iN
is 1000 W/m
2
. Themanufacturer data sheet will provide the temperature constantfor the voltage,
TCV
, the temperature constant for the current,
TCi
, the open circuit voltage under STC,
V
oc
, short circuitcurrent under STC,
I
sc
, and the PVM characteristic constant,
b
.Also, most of the manufacturers will provide the open circuitvoltage,
V
max
, when
E
i
is more than 1200 W/m
2
and
T
is 25
o
Cand the open circuit voltage,
V
min
, when
E
i
is less than 200W/m
2
and
T
is 25
o
C [17].
V
max
is approximately
1.03
.
V
oc
and
V
min
is approximately
0.85
.
V
oc
. This model considers the usefuldata given by the manufacturer while no additional parametersare required, i.e. thermal voltage, diode reverse saturationcurrent, band gap for the material, etc. Additionally, the PVMmodel is continuous and differentiable with respect to thevoltage. The static PVM model is given in (1). The opencircuit voltage at any
T
or
E
i
,
Vx
is given by (2) and iscalculated when the current of operation is zero.
Ix
, the shortcircuit current at any
T
or
E
i
, is calculated when the voltage of operation is zero and is given by (2).
9781424418749/08/$25.00 ©2008 IEEE
987
0510152025024681012Voltage (V)
P V M o d u l e P o w e r ( W )
PV CurveVopVxPmax051015202500.10.20.30.40.50.60.7IV CurveVoltage (V)
P V M o d u l e C u r r e n t ( A )
VopVxVop. Iop = PmaxIop0510152025051015202530RiV CurveVoltage (V)
P V M o d u l e R e s i s t a n c e ( o h m s )
VopRop = Vop / Iop
(1)After substituting (2) and (3) into (1), a simplified PVMmodel is obtained and is given in (4). The PVM power isdescribed by (5). Finally, the PVM internal resistance,
Ri
, iscalculated by dividing the input voltage,
V
, by the current,
I(V)
,and is given by (6). The PVM internal resistance,
Ri
, will beused for the load matching. Typically, the batteries have aninternal resistance between 0.20.7
W
[18] and a short circuitcould be very dangerous for the battery. The PVM internalresistance is much larger and the value depends on the voltageand power drawn from the PVM. Also, a PVM is a currentlimited system hence can be shortcircuited without damage atdifference of the batteries. Finally, if the
Ri
is equal to the loadresistance then
P
max
can be transfer to the load but if bothresistances differ the power will be less than
P
max
[18].(2)(3)(4)(5)(6)Figures 1, 2 and 3 show the PV, RV and IV curves for aPVM SX10 [20] and their relationship between the internalresistance, optimal voltage and maximum power. Figure 1shows that there is a unique maximum point for the maximum power which will be produced when the PVM voltage is equalto the optimal voltage,
Vop
, which is unique. Then, using thefact that
P
max
is produced when the PVM voltage is equal to
Vop
, and because the derivative of the resistance with respectto the voltage is always positive, then it can be seen that thereis a unique optimal internal resistance,
Rop
. So if a PVM isoperating at
Vop
then the PVM will transfer
P
max
to the load.Finally, with this fact in mind, the next section will describehow to analytically calculate the optimal duty ratio for a dcdcconverter to transfer the maximum power to the load.
Fig. 1
PV
curve for the SX10 and their relationship between
P
max
, and
Vop
.Fig. 2
RiV
curve for the SX10 and their relationship between
Vop
and
Rop
.Fig. 3
IV
curve for the SX10 and their relationship between
Vop
and
Iop
.
( )( )
( )
( ) ( )
⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡⎟⎟⎟⎟⎟ ⎠ ⎞⎜⎜⎜⎜⎜⎝ ⎛ −⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ ⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ −−⋅⋅−−+−⋅⋅⋅
−⋅−−⋅−⋅+⋅=
bV V V V E E V V V T T TCV
E E bV bT T TCi I
E E V I
ociN i N iiN N sciN i
1lnexpexp11exp11)(
minmaxmaxminmaxmax
( )( )
⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ ⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ −−⋅⋅−−
+−⋅⋅=
minmaxmaxminmaxmax
lnexp
V V V V E E V V V T T TCV
E E Vx
ociN i N iiN
( )( )
N sciN i
T T TCi I
E E Ix
−⋅+⋅=
( )
⎥⎦⎤⎢⎣⎡⎟ ⎠ ⎞⎜⎝ ⎛ −⋅−⋅−−=
bVxbV b IxV I
1exp11exp1)(
( )
⎥⎦⎤⎢⎣⎡⎟ ⎠ ⎞⎜⎝ ⎛ −⋅−⋅−−⋅=
bVxbV b IxV V P
1exp11exp1)(
( )
⎥⎦⎤⎢⎣⎡⎟ ⎠ ⎞⎜⎝ ⎛ −⋅−⋅−⋅−=
bVxbV IxbV V V Ri
1exp11exp)(
988
( )( )( )( )
222222
11exp111exp111
D Ri DbVxbVi Ix DbVi D Ii DVi D Io DVi D IoVo Ro
−⋅=⎥⎦⎤⎢⎣⎡⎟ ⎠ ⎞⎜⎝ ⎛ −⋅−⋅⋅−
⎟ ⎠ ⎞⎜⎝ ⎛ ⎟ ⎠ ⎞⎜⎝ ⎛ −−⋅⋅=⋅−⋅=⋅−⋅−==
Rop Ro Ro D
+=
VopVoVo D
+=
( )
( )
f Ro Rop Ro f Rop Ro f D R L
⋅<+⋅⋅⋅=⋅−⋅=
2221
22min1
( )
RoRop Ro f C
RoC f DVoripple
+⋅⋅=⋅⋅=
11
1
Ro Rop
>
( ) ( )( )( )
( )
⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ ⎟ ⎠ ⎞⎜⎝ ⎛ −⋅+−⋅
⎥⎥⎦⎤⎢⎢⎣⎡−−⋅−⋅+⋅
⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ ⎟ ⎠ ⎞⎜⎝ ⎛ −−⋅⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ ⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ ⎟ ⎠ ⎞⎜⎝ ⎛ −⋅−⋅+⋅
⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ ⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ −−⋅⋅−−+−⋅⋅
==
bbbbT T TCi I
E E bbbbbV V V V E E V V V T T TCV
E E IopVop Rop
N sciN iociN i N iiN
1exp11exp111exp11expln1lnexp
minmaxmaxminmaxmax
III.
O
PTIMAL
D
UTY
R
ATIO FOR A DC

DC
C
ONVERTER TO
O
BTAIN THE
M
AXIMUM
P
OWER
Consider a PVM connected to a buckboost converter tosupply power to a resistive load. The objective is to calculatethe optimal duty ratio,
D
, so the PVM will supply
P
max
. Theanalysis will be done using the steadystate conditions for a buckboost converter, where all the components are ideal, theinductor current is continuous, the capacitor is large enough toassume a constant output voltage and the switch is closed for time
D/f
and open for
(1D)/f
, [21]. An advantage of the buck boost converter is that the magnitude of the output voltage can be either greater than or less than the source voltage,depending on the duty ratio of the switch [21], making itexcellent for photovoltaic applications where the weather conditions are changing very fast. The only minor disadvantage for the buck boost converter is the polarityreversal on the output.The first step for load matching will be done using therelationship between the voltage input and output for a buck boost converter relationship. The load resistance
Ro
can beseen as voltage output,
Vo
, divided by current output, Io.Using the last information, the relationship between the inputresistance,
Ri
, and the output resistance,
Ro
, is given by (7).If
V
is
Vop
, hence
Ri
is
Rop
, the optimal duty cycle,
D
, can besolved. The optimal duty ratio,
D
, is obtained and onlydepends on
Ro
and
Rop
. Switching at the optimal duty ratioguarantees that the power supplied to load is
P
max
.(7)Using (7), the optimal duty ratio,
D
, as a relationship of theoptimal resistance,
Rop
, and output resistance,
Ro
, can besolved and is given in (8). Additionally, if the power input andthe power output are equal to
P
max
,
D
can be expressed as arelationship between the optimal voltage,
Vop
, and the outputvoltage,
Vo
, as given by (9).(8)(9)The minimum inductance
L
1min
for the buckboost converter to preserve the continuous current mode is given in (10). Thevoltage output ripple is calculated in (11). The same type of procedure is done to calculate the duty cycle for the buck converter or boost converter.(10)(11)Table 1 shows the conditions and optimal duty ratio for a buck converter, boost converter and buckboost converter.Form Table 1, the only disadvantage of using a buck or boostconverter is the restriction in the values of
Rop
and
Ro
for both cases.
TABLE IO
PTIMAL
D
FOR
D
IFFERENT DC

DC
C
ONVERTERS FOR
L
OAD
M
ATCHING
Finally, this method for load matching can be integrated toother algorithms such that the linear reoriented coordinatesmethod, LRCM, which is described in details in [22]. TheLRCM is an analytical method used to calculate
Vop
and
Iop
,then
P
max
is calculated. Using the LRCM, the optimalresistance,
Rop
, is calculated under any changes in
T
or
Ei
andis given in (12). Also,
Rop
can be calculated using
Ix
and
Vx
,then the optimal duty ratio is calculated and used on the dcdcconverter to transfer
P
max
from the PVM to the load.(11)
DCDCconverter
D
for any
Po
D
when
Pi = Po = P
max
RequiredBuckBoost NoneBoostBuck
Rop Ro
>
Ri Ro D
=
Ri Ro Ro D
+=
VopVoVo D
+=
VopVo D
=
Ro Ri D
−=
1
VoVop D
−=
1
(12)
989
IV.
R
ESULTS
Figure 4 present the algorithm to validate and test the proposed load matching method. Figure 5 shows an integratedPV system using a pyranometer to measure the irradiancelevel and thermocouples to measure the temperature over thePVM surface. The integrated PV system has a Sharp ND208U1 PVM [23] with
P
max
is 208 W,
Rop
is 2.65
Ω
,
Vop
is23.48 V,
Ix
is 0.75 A,
Vx
is 30 V and
b
is 0.1, connected to adc bus with capacitance 400
μ
F. The dcdc converter is a 50kHz buckboost converter with inductance 100
μ
H andcapacitance 400
μ
F, and the resistive load is 0.75
Ω
. Theobjective of the presented PV system is to supply 208W (i.e.the maximum power) produced by the PVM to the resistiveload. Figure 6 shows the transient results simulations for theintegrated PV system and the simulations were done usingSimulink. These results show how a PVM can be controlledand deliver
P
max
using the proposed load matching strategy.V.
C
ONCLUSIONS
In this paper new contributions to the field of solar energyconversion were presented. The first contribution is the use of the optimal duty ratio and load matching to transfer themaximum power of the PVM to the load. The proposedmethod can be integrated with other algorithms such asLRCM to calculate the PVM internal resistance. Also, sincethe derivative of
Ri(V)
with respect to the voltage is positive,existence and uniqueness was proven, and the optimal internalresistance,
Rop
, which will transfer the maximum power to theload was calculated. Also, the optimal duty ratios for differenttypes of dcdc converters for a PVM to supply
P
max
werederived. Finally, the method is extremely efficient, easy to program in a DSP and applicable to other nonlinear power sources such as fuel cells.R
EFERENCES
[1] Streetman, B.G.; Baerjee, S. “Solid State Electronic Devices” 5th Edition.[2] Markvart, T.; Arnold, R.J.; “Integration of photovoltaic convertors into the public electricity supply network” IEE Colloquium on Power Electronics for Renewable Energy, June 16 1997 Page(s):4/1  4/4[3] Jiang, JA.; Huang TL.; Hsiao, YT.; Chen CH.; “Maximum Power Tracking for Photovoltaic Power System” Tamkang Journal of Scienceand Engineering, Vol.8, No 2, 2005, Page(s): 147153[4] Kuo Y.Ch.; Liang T.J.; Chen J.F.; “Novel maximum powerpointtracking controller for photovoltaic energy conversion system” IEEETransactions on Industrial Electronics, Volume: 48, Issue: 3, June 2001Pages: 594 – 601.[5] Lim Y. H.; Hamill, D.C.; “Simple maximum power point tracker for photovoltaic arrays” Electronics Letters , Volume: 36 Issue: 11 , 25 May2000, Page(s): 997 999[6] Swiegers, W.; Enslin, J.H.R.; “An integrated maximum power pointtracker for photovoltaic panels” Proceedings of the IEEE InternationalSymposium on Industrial Electronics, 1998. ISIE '98. Volume 1, 710July 1998 Page(s): 40 – 44[7] Jain, S.; Agarwal, V.; “A new algorithm for rapid tracking of approximatemaximum power point in photovoltaic systems” IEEE Power Electronics Letters, Vol. 2, Issue 1, March 2004 Page(s):16  19[8] Hua, Ch.; Shen, Ch.; “Study of maximum power tracking techniques andcontrol of DC/DC converters for photovoltaic power system” Power Electronics Specialists Conference, 1998. PESC 98 Record. 29th AnnualIEEE Volume 1, May 1722, 1998 Pages: 86 – 93.Fig. 4 Algorithm to calculate the optimal duty ratio given
E
i
and
T
.Fig. 5 Integrated PV system using load matching and the optimal duty ratiogiven
Ei
and
T
.
Pi
P
max
= 208WVi
Vop
= 23.48V
Po
P
max
= 208WVo
Vo
= 12.49V
Fig. 6 Power supplied by the PVM, voltage supplied by the PVM, BuckBoost converter power output, and BuckBoost converter voltage output.
990