IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 3, SEPTEMBER 2011
923
Evaluation of MPP Voltage and Power of mc-Si PV
Modules in Partial Shading Conditions
Evagelia V. Paraskevadaki and Stavros A. Papathanassiou, Senior Member, IEEE
Abstract—Photovoltaic (PV) system performance is influenced
by several factors, including irradiance, temperature, shading,
degradation, mismatch losses, soiling, etc. Shading of a PV array, in particular, either complete or partial, can have a significant
impact on its power output and energy yield, depending on array
configuration, shading pattern, and the bypass diodes incorporated
in the PV modules. In this paper, the effect of partial shading on
multicrystalline silicon (mc-Si) PV modules is investigated. A PV
module simulation model implemented in P-Spice is first employed
to quantify the effect of partial shading on the I–V curve and the
maximum power point (MPP) voltage and power. Then, generalized formulae are derived, which permit accurate enough evaluation of MPP voltage and power of mc-Si PV modules, without the
need to resort to detailed modeling and simulation. The equations
derived are validated via experimental results.
Isc,STC
IRsh
IRSTC
IRun -sh
Index Terms—Maximum power point (MPP), multicrystalline
silicon (mc-Si) photovoltaic (PV) modules, partial shading, photovoltaic power systems.
NG
NG sh
k
MPP
MPP1, 2
MPPT
n
n1 , n2
p1 , p2 , p3 , p4
NOMENCLATURE
a1 , a2
as 1 , as 2
AP , BP
AV , BV
Area 1
Area 2
b
cP 1 , cP 2
cV 1 , cV 2
IM PP,STC
Iph
Is 1 , Is 2
Irradiance and Temperature coefficient of
Iph .
Temperature coefficients for Is 1 and Is 2 .
Coefficients of first-order polynomial
representing cP 2 .
Coefficients of first-order polynomial
representing cV 2 .
Low voltage region of a dual peak P–V
curve.
High voltage region of a dual peak P–V
curve.
Correction factor.
Coefficients expressing dependence of MPP
power on irradiance and area of shade
(similar to cV 1 , cV 2 ).
Coefficients of VM PP dependence on
irradiance level and on area of shade.
PV module MPP current at STC.
PV cell photocurrent, proportional to
irradiance and temperature of the PV cell.
Saturation currents of diodes D1 , D2 .
Manuscript received November 27, 2010; accepted February 24, 2011. Date
of publication April 21, 2011; date of current version August 19, 2011. Paper
no. TEC-00465-2010.
The authors are with the School of Electrical and Computer Engineering, National Technical University of Athens, Athens 15773, Greece (e-mail:
evaparask@yahoo.gr; st@power.ece.ntua.gr).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TEC.2011.2126021
PM AX1,2
PM PP,STC
q
Rs
Rsh
ssh
STC
T
TR
Vbr
Vgap
VM PP1,2
VM PP,STC
w
Yc
ΔVD
ΔPD
PV module short circuit current at STC.
Irradiance incident on shaded cells (W/m2 ).
STC irradiance (IRSTC = 1000 W/m2 ).
Irradiance incident on unshaded cells
(W/m2 ).
Boltzmann’s constant (k = 1.381 ×
10−23 J/K).
Maximum power point.
Local MPP in Area 1 or 2.
Maximum power point tracking.
Breakdown coefficient.
Ideality factors of diodes D1 , D2 (n1 =
n2 = 1).
Number of cell groups (i.e., bypass diodes).
Number of cell groups experiencing
shading.
Coefficients of third-order polynomials
derived for AV , BV , AP and BP .
MPP power in Area 1 or 2.
MPP power of the PV module under STC.
Electron charge (q = 1.60217·10−19 C).
Series resistance of the PV cell equivalent
circuit.
Shunt resistance of the PV cell equivalent
circuit.
Shaded area of the PV module (per unit (p.u.)
of its total area).
Standard test conditions: irradiance
1000 W/m2 , cell junction temperature
25 ◦ C, and reference air mass 1.5 solar
spectral irradiance distribution.
Absolute temperature (K).
Irradiance transmittance ratio of a shade
material.
Breakdown voltage.
Diode band gap voltage.
MPP voltage in Area 1 or 2.
MPP voltage of the PV module under STC.
IRsh in p.u. of STC value.
Generic variable for anyone of AV , BV , AP ,
BP .
Forward voltage drop of a bypass diode.
Conduction losses of a bypass diode.
I. INTRODUCTION
NE of the main considerations in commercial applications of photovoltaic (PV) systems is their performance
under nonideal operating conditions, including operation under
O
0885-8969/$26.00 © 2011 IEEE
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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 3, SEPTEMBER 2011
reduced irradiance incident on the PV modules due to shading.
Shading may occur uniformly over the whole surface of the PV
generator (be it a single module, an array or a multiarray installation), or it may affect only a part of it, albeit still with a
significant impact on its power and energy output.
Under partial shading conditions, PV cells belonging to the
same string experience different irradiance. The resulting P–V
characteristic curves then deviate from their standard form, depending on the electrical characteristics of the PV modules and
the shadow pattern and intensity. When PV modules with more
than one bypass diodes are utilized, partial shading leads to the
formation of multipeak power curves [1], which hinders the efficient maximum power point (MPP) tracking operation. The
resulting energy losses, when a local instead of the global maximum is tracked, can be significant, leading to the deterioration
of overall system performance.
The effect of partial shading on PV module and array characteristic curves has been analyzed in several publications [2]–[4].
The effect of bypass diodes in the PV modules or the array
circuit is studied in [5] and [6], where I–V and P–V curves
are extracted for several PV system configurations. Simulation models suitable for the investigation of partial shading
impact on PV array characteristic curves and on the overall
system performance are proposed in [7]–[12]. Performance deterioration under partial shading conditions due to MPP tracking inefficiency is discussed in [13] and [14]. Output power
changes of grid-connected PV systems due to moving clouds,
inducing rapid changes in irradiance incident on the PV array, is studied in [15]–[17]. Methods have been proposed for
improving PV system performance under partial shading conditions, mainly focusing on alterations of PV array configuration
in terms of bypass diode number [18] and string connection
patterns [19].
Already published work focuses mainly on understanding
partial shading phenomena and their effects, as well as on their
analysis using suitable PV cell electrical equivalents [20]–[23].
The issue of predicting the resulting MPP power and voltage,
given the shading pattern and intensity, remains still open, although it is fundamental for evaluating PV system performance
under nonideal operating conditions. Currently, to answer this
question one needs to resort to detailed electrical equivalent
models for the whole PV system, which is not realistic in most
practical situations.
Main objective of this paper, besides analyzing the effect of
partial shading on PV module performance, is to provide a simple and easily applicable method for calculating shadow effects
on the main electrical characteristics of a PV module. For this
purpose, simple, semiempirical formulae are introduced, which
provide the MPP power and voltage of multicrystalline silicon (mc-Si) PV modules. Application of the formulae requires
knowledge only of basic data-sheet information and provides
the power and voltage of each peak (local MPP point) of the
multipeak P–V characteristic of a PV module, for any shading
pattern and irradiance level, without resorting to the application of detailed electrical models and simulations. Knowledge
of the basic form of the P–V curve allows direct conclusions to
be drawn regarding MPP tracking (MPPT) effectiveness. The
Fig. 1.
Equivalent circuit of a PV cell based on the two-diode model [3].
equations derived in this paper for single PV modules also provide the basis for addressing PV strings and arrays, at a next step,
in order to account more accurately for the effect of shading on
the energy yield of PV installations.
To derive the formulae presented in this paper, four PV modules from the PV module database [24] [as it was initially available by the Sandia National Laboratories (SNL)] are used as a
study case and a detailed electrical equivalent of the PV module
is employed to conduct simulations of a large variety of shading
patterns and intensities. Then, the equations are validated by
experimental results, using three entirely different commercial
PV modules.
In Section II, the PV module equivalent circuit is described.
The study case PV modules used in the analysis are presented
in Section III. The effect of partial shading on typical mc-Si PV
modules is explored in Section IV. Equations for evaluating the
MPP voltage and power of a PV module under partial shading
are introduced in Section V for STC irradiance conditions and
then they are extended to other irradiance levels in Section VI.
The accuracy and validity of the proposed equations is experimentally validated in Section VII.
II. PV CELL AND MODULE EQUIVALENT
Various electrical equivalents have been developed to represent the solar cell, the single- and double-diode models being
the most widely used [25]. Since the single-diode model does
not represent precisely the IV characteristic of a PV cell, in
this paper the two-diode equivalent is employed, including an
extension term to describe operation in the negative voltage region [3], as shown in Fig. 1. This equivalent is described by the
following equations:
I = Iph − Is1 (eV +IRs/n 1 V T − 1) − Is2 (eV +IRs/n 2 V T − 1)
−n
V + IRs
V + IRs
−
− b(V + IRs ) 1 −
(1)
Rsh
Vbr
Iph = IRun -sh (a1 + a2 T )
3 −V g a p /V T
(2)
Is1 = as1 T e
(3)
Is2 = αs2 T 5/2 e−V g a p /2V T
(4)
VT =
kT
.
q
(5)
The last term in (1) is related to the second current source of the
PV cell equivalent circuit, accounting for the cell behavior in
PARASKEVADAKI AND PAPATHANASSIOU: EVALUATION OF MPP VOLTAGE AND POWER OF mc-Si PV MODULES
925
TABLE I
DATASHEET CHARACTERISTICS OF THE FOUR STUDY-CASE PV MODULES
reverse bias, which is of key importance for the study of partial
shading [26], [27].
To develop a PV module equivalent, the model of the PV
cell is implemented in P-Spice and individual cell circuits are
connected in series. Using such a modular cell-based model, a
detailed analysis is possible, setting parameter values for each
cell independently and allowing for the simulation of various
shading scenarios of the PV module. In this study, the cells
within a PV module are considered to be identical, both in
direct and reverse bias behavior.
In commercial PV modules, a number of bypass diodes are
connected across groups of PV cells, to prevent development
of high negative voltages on the cells and hot-spot phenomena,
due to partial shading and mismatch of PV cell characteristics
at reverse bias operation, [1], [6], [7]. The modular cell-based
PV module equivalent employed in this paper permits proper
representation of any number of bypass diodes connected across
cell groups in a straightforward manner.
Fig. 2. I–V characteristics of study-case Module 2 (see Table I) at different
irradiance conditions (Tc = 25 ◦ C). Solid lines: simulation results. Dot points:
approximation of measurements, based on the Sandia model, [24], [28].
III. STUDY-CASE PV MODULES
To study the behavior of PV modules under partial shading
conditions and derive equations describing their performance,
the four mc-Si PV modules of Table I have been selected as study
cases from the PV Module Database [24]. The selected modules present widely different characteristics in terms of nominal
power, cell number, bypass diodes, etc.
To simulate the operation of a PV module using the equivalent
circuit model of Section II, values need to be identified for all of
its parameters. Such values are often selected via a curve fitting
procedure to I–V and P–V curves measured under a single set
of irradiance and temperature conditions, while the accuracy of
the model over the wide range of irradiance values experienced
in an actual operation of the PV module is rarely taken into
consideration.
In this study, selection of PV cell model parameters takes into
account the behavior at reduced irradiance conditions, which is
crucial in the analysis of partial shading performance. For each
module, a single set of PV cell model parameters was determined, with the objective of achieving a good approximation of
the actual I–V curve, not only at STC irradiance (1000 W/m2 ),
but also at 500 and 100 W/m2 . As an approximation of the actual
I–V curves of the PV modules, the five I–V points derived from
the Sandia model [28] per irradiance level are utilized. In this
process, it was observed that a set of parameter values achieving best fit at STC might not be satisfactory at lower irradiance.
Fig. 3.
Shading patterns for examining partial shading effects.
The simulated I–V curves achieve a very good overall accuracy,
the normalized root-mean-squared error (NRMSE) of the current being lower than 5% and close to 1% for irradiance values
higher than 500 W/m2 . In Fig. 2, the I–V curve of Module 2 is
indicatively presented along with the points calculated by the
Sandia model.
IV. EFFECT OF PARTIAL SHADING ON PV MODULE
CHARACTERISTIC CURVES
The effect of partial shading on PV module characteristic
curves is examined by shading an increasing number of cells,
as shown in Fig. 3. Cell temperature is assumed to be constant
at 25 ◦ C. Unshaded cells are considered fully illuminated at
1000 W/m2 . Irradiance on shaded cells is considered uniform
and varies from 0 to 1000 W/m2 with a step of 100 W/m2 .
In Fig. 4, the P–V curves of Module 2 are plotted for the
two shading scenarios of Fig. 3, along with the power curve at
unshaded STC conditions. Irradiance on shaded cells is IRsh =
400 W/m2 . Each shaded P–V curve can be divided into two areas. In Area 1, the bypass diode of cell group B (where the
shaded cells are located, Fig. 3) is conducting, effectively shorting out of the circuit the whole group, eliminating, thus, the
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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 3, SEPTEMBER 2011
shading is confined within one group of cells (<18 cells, for
Module 2 of Table I), a global maximum of 33.5 W at 7.5 V
is obtained for IRsh < 400 W/m2 , corresponding to VM PP1 of
Fig. 4. This maximum remains constant in terms of voltage and
power, as it represents the performance of the unshaded cell
groups when the bypass diode of the shaded group is conducting. For IRsh values greater than 400 W/m2 , the maximum at
VM PP2 becomes global and, hence, VM PP returns step wise to
the region of unshaded operation, reaching values up to 20%
greater than VM PP ,STC for this particular PV module.
In any case, as the process of module shading evolves, VM PP
can be subjected to large steps, obstructing the efficient operation of conventional MPP trackers, which would lock to the
local maximum of VM PP2 that is closer to the predisturbance
normal operating conditions and is much easier to be tracked,
compared to the much lower VM PP1 values [11].
Fig. 4. Location of MPP on the P–V characteristic of PV Module 2 (see
Table I) in the case of partial shading. Shading scenarios as in Fig. 3.
effect of shading. In this case, the PV module practically operates as an equivalent 18-cell module (only cell group A is
active), if losses on the bypass diode are ignored. In Area 2, on
the other hand, higher voltages appear, because the bypass diode
of group B is not conducting and the PV module characteristic
is dominated by the series-connected shaded cells.
At STC, the maximum power of the PV module is
PM PP,STC = 77 W at VM PP,STC = 16.7 V. When shading occurs, a significant reduction of the output power is noted and
two maxima appear on the P–V curve. The first (MPP1) at
VM PP1 = 7.5 V, with a power output of PM AX1 = 34 W, is located very far from the normal MPP region and it is practically
independent of the shaded area and irradiance, as the group of
shaded cells is short-circuited. The second maximum (MPP2a
or MPP2b) depends on the shading scenario, i.e., the number
of shaded cells and their irradiance. In Fig. 4, when one cell
is shaded, VM PP2a = 19.6 V and PM AX2a = 38.44 W, hence,
point MPP2a becomes the global maximum of the P–V curve.
When 18 cells are under shade, VM PP2b = 17.7 V and PM AXb =
32.83 W, and therefore, the global maximum lies now at point
MPP1. The difference between PM AX2a and PM AXb is small,
because the power reduction in Area 2 does not depend strongly
on shade surface but on the irradiance incident on shaded cells.
The voltage of the Area 2 MPP, VM PP2a or VM PP2b , can differ
considerably from the STC value, especially for small areas of
shade (VM PP2a is 17% higher than VM PP,STC , while VM PP2b
is about 6% higher).
The global MPP of the PV module can be either of the two
local MPPs, depending on the shading scenario. Hence, the variation of the global VM PP can be significant for a PV module
with bypass diodes, exhibiting large step variations as the number of shaded cells and the shaded irradiance IRsh change. This
is demonstrated in Fig. 5, where the global VM PP and PM PP of
Module 2 are plotted as a function of the irradiance IRsh incident on the shaded area, assuming different numbers of shaded
cells, from one up to the entire module area (36 cells). When
V. EVALUATION OF MPP VOLTAGE AND POWER AT STC
A. Empirical Expressions for MPP Voltage and Power
In order to quantify the location of the MPP on the P–V
curve of the module for a given shading pattern, mathematical
formulae are derived in this section, which allow an accurate
enough calculation of the resulting MPP voltage and power,
without the need to resort to the full modeling and simulation
of the PV module.
Following the discussion of the previous section, when the
bypass diode of the shaded group of cells is conducting, the PV
module characteristic will resemble that of the unshaded groups
of cells. Hence, the respective MPP voltage (VM PP1 in Area 1,
Fig. 4) will be close to the reference voltage of the unshaded
cells, minus the voltage drop across the bypass diode. When the
shaded cells are not bypassed, the respective local maximum
(VM PP2 in Area 2, Fig. 4) will deviate from the reference (STC)
value due to the effect of the series-connected shaded cells. To
quantify this effect, the irradiance incident on the shaded cells
and the respective area need to be taken into account.
The following equations are introduced for VM PP1 and
VM PP2 in the general case of a dual-peak P–V curve, as in
Fig. 4, assuming an irradiance of 1000 W/m2 on the unshaded
part of the PV module
VM PP,STC
+ ΔVD
(6)
VM PP1 = VM PP,STC − NG sh
NG
VM PP2 = cV 1 VM PP,STC (1 + cV 2 ).
(7)
In (7), the term cV 1 VM PP ,STC gives the module MPP voltage
if all cells were illuminated at the irradiance IRsh of the shaded
part. Multiplication factor 1 + cV 2 adapts the MPP voltage to
the fact that only a part of the module area operates under shade,
i.e., at irradiance IRsh . Coefficients cV 1 and cV 2 obtain values
1 and 0, respectively, when no cell is shaded.
Similarly, output powers PM AX1 and PM AX2 , corresponding
to VM PP1 and VM PP2 , can be estimated by
PM PP,STC
+ ΔPD (8)
PM AX1 = PM PP,STC − NG sh
NG
PARASKEVADAKI AND PAPATHANASSIOU: EVALUATION OF MPP VOLTAGE AND POWER OF mc-Si PV MODULES
Fig. 5.
927
Variation of (a) global VM P P and (b) global PM A X of Module 2, Table I, for different shading scenarios.
TABLE II
COEFFICIENTS cv 1 AND cP 1 FOR THE STUDY CASE PV MODULES
PM AX2 = cP 1 PM PP,STC (1 + cP 2 ).
(9)
The terms related to the bypass diodes in (6) and (8) are now
quantified. Concerning the forward voltage drop ΔVD a typical
value of 0.7–0.8 V is sufficiently accurate. Power losses ΔPD
are given by the product of diode voltage and current, which in
turn depends on IRsh , as well as on the irradiance on unshaded
cells. Since (6)–(9) are valid for STC conditions in the unshaded
part of the module, the diode power loss can be approximated
by
IRsh
Isc,STC .
ΔPD = ΔVD IM PP,STC −
(10)
IRSTC
Equation (10) reflects the fact that the bypass diode conducts the
difference between the currents of the fully illuminated and the
shaded cells. The latter, when the bypass diode is ON, operate
at near short-circuit conditions (actually at a slightly negative
voltage, due to the forward voltage drop of the diode) and, hence,
their current can be approximated as (IRsh /IRSTC )·Isc,STC .
B. Evaluation of Coefficients
In this section, specific values and empirical expressions are
derived for coefficients cV 1 , cV 2 , cP 1 , and cP 2 of (7) and (9),
using the study case PV modules of Table I. The coefficient
values, thus, derived are also suitable for application to other
PV modules, as shown by experimental results in Section VII.
Coefficient cV 1 is a function of the irradiance on the shaded
part of the module. Specific values calculated for the study case
modules are given in Table II, along with the mean value for
all four modules. Coefficient cP 1 , also shown in Table II, varies
almost linearly with irradiance IRsh on the shaded cells. Its value
is very close to the p.u. IRsh (IRsh /IRSTC ), differences observed
mainly at low irradiance due to reduced cell efficiency at such
conditions.
Calculation of coefficients cV 2 and cP 2 is realized via application of the PV module simulation model for the four studycase modules, under various partial shading scenarios (shaded
areas and irradiances). Estimates of cV 2 and cP 2 are then extracted using (7) and (9)
cV 2 =
VM PP2 − cV 1 VM PP,STC
cV 1 VM PP,STC
(11)
cP 2 =
PM AX2 − cP 1 PM PP,STC
.
cP 1 PM PP,STC
(12)
Based on this procedure the plots of Fig. 6(a) are derived,
depicting the variation of cV 2 in terms of irradiance and shaded
area for all four PV modules. Coefficient cV 2 presents a similar
variation for all irradiance values, being rather independent of
the module characteristics, especially when IRsh is greater than
100 W/m2 and the shaded area is larger than 10%. The cV 2
values are generally positive, signifying that VM PP2 is greater
than cV 1 ·VM PP ,STC , i.e., the MPP voltage of the PV module
when its total surface is uniformly illuminated at IRsh . The
increase of the MPP voltage is more marked at heavy shade
conditions. Yet, when a very small number of cells are under
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Fig. 6.
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 3, SEPTEMBER 2011
(a) cV
2
and (b) cP 2 variation with shaded area and irradiance incident on the shaded cells for the study-case PV modules of Table I.
TABLE III
COEFFICIENTS OF THE THIRD DEGREE POLYNOMIALS OF (15) APPROXIMATING
AV , BV , AP , AND BP
severe shading (IRsh < 100 W/m2 ), then cV 2 declines rapidly,
obtaining negative values. Similar remarks apply for cP 2 , shown
in Fig. 6(b).
The almost linear dependence of cV 2 and cP 2 on shaded area,
for a given irradiance level, permits a simple approximation,
valid for all irradiance levels higher than approx. 100 W/m2 and
shade areas greater than 10%
cV 2 (ssh ) = AV ssh + BV
(13)
cP 2 (ssh ) = AP ssh + BP .
(14)
Values of coefficients AV , BV , AP , and BP depend on the irradiance IRsh on the shaded cells. This dependence can in turn be
approximated by a polynomial expression, such as
YC = p1 w3 + p2 w2 + p3 w + p4
(15)
where Yc is any one of AV , BV , AP , BP , and w is the p.u. IRsh .
Coefficients p1 , p2 , p3 , and p4 are given in Table III, valid for
any of the four study case PV modules.
Using the formulae derived in this section, location of the
two MPPs on the module P–V curve is now possible, based only
on the shading pattern, i.e., the shaded area and the respective
irradiance IRsh , for practically any realistic shading scenario.
VI. EXTENSION TO OTHER IRRADIANCE LEVELS
The formulae derived so far and their coefficients refer to the
operation of the PV module at STC irradiance (1000 W/m2 ).
Although this serves well as a reference case and is convenient
for analyzing a wide range of conditions, in practice partial
TABLE IV
DATASHEET CHARACTERISTICS OF COMMERCIAL PV MODULES USED IN THE
EXPERIMENTAL VALIDATION
shading will mostly occur at other than reference conditions.
In such a case, the equations derived in Section V are still
applicable, after the modifications introduced in the following.
To extend the analysis to other irradiance levels, an extended
set of simulations has been performed for different shading
patterns and intensities, assuming an irradiance of 600 and
200 W/m2 on the unshaded cells. MPP voltages and powers
calculated with the P-Spice model of Section II were then compared to those obtained from (6)–(9).
Concerning the MPP voltage, (6) and (7) can still be utilized,
their error being lower than 3% when IRsh > 0.5·IRun -sh . Under
heavy shade conditions (IRsh < 0.5·IRun -sh ), the MPP voltage
error is still acceptable (around 5%), as long as the shaded
cell irradiance remains higher than 100 W/m2 (the identified
coefficient values are not accurate below this threshold). To
evaluate maximum power, a simple modification of (8) and (9)
is necessary, to account for the almost linear dependence of the
unshaded cell output power on irradiance.
Eventually, the generalized form of the (6)–(9), valid for any
irradiance level incident on the PV module, is the following,
after substituting coefficients cV 2 and cP 2 from (13) and (14)
VM PP1 = VM PP,STC − NG sh
VM PP,STC
+ ΔVD
NG
VM PP2 = cV 1 VM PP,STC (1 + AV ssh + BV )
(16a)
(16b)
PARASKEVADAKI AND PAPATHANASSIOU: EVALUATION OF MPP VOLTAGE AND POWER OF mc-Si PV MODULES
Fig. 7.
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Experimental and simulated I–V curve of (a) Module A and (b) Module C (see Table IV) under unshaded and shaded conditions.
PM AX1 =
PM PP,STC − NG sh
+ΔVD
IM PP,STC −
PM PP,STC
NG
IRsh
Isc,STC
IRSTC
TABLE V
IRRADIANCE AND TEMPERATURE CONDITIONS FOR THE MEASUREMENTS
PRESENTED IN TABLE VI
IRun -sh
IRSTC
(16c)
PM AX2 = [cP 1 PM PP,STC (1 + AP ssh + BP )]
IRun -sh
. (16d)
IRSTC
Values for cV 1 and cP 1 are given in Table II, while coefficients
AV , BV , AP , and BP are related to IRsh via (15), using p1 , p2 ,
p3 , and p4 values from Table III.
Equation (16) permit direct estimation of the MPP voltage
and power of a mc-Si PV module operating under partial shading conditions, at any module irradiance, shading intensity and
shade area, using only basic datasheet information and dispensing with the need to resort to accurate simulations and the associated copious procedure of model parameter identification.
Limitations of the proposed equations are related to the assumption of uniform irradiance over the shaded cells and the fact
that they refer to P–V curves with two peaks. Nevertheless, the
PV module simulation model and the rationale and extraction
methodology presented could be applied to curves with three
or more local peaks (arising when more levels of shade exist or
when different groups of cells are unequally shaded), after the
necessary modifications are performed to the empirical formulae. The restriction for irradiance levels higher than 100 W/m2
on the shaded cells is of no real consequence, because such
low levels would result only from shading obstacles almost in
contact with the PV module.
VII. EXPERIMENTAL VALIDATION
To experimentally validate the PV module model of
Section II and the empirical relations of Sections V and VI,
three commercial mc-Si PV modules available in the laboratory
have been utilized. Their datasheet characteristics are presented
in Table IV. To generate shading patterns, shades were placed
directly on each PV module, covering part of its surface. Two
shading materials were used with transmittance rates TR of
28% and 64%, reducing irradiance in the shaded area to 28% or
64% of the unshaded value. Transmittance rates were calculated
from the module short-circuit current when totally covered by
the shading material.
For each shading scenario, the I–V curve of the module was
measured using a continuously variable resistor connected to its
terminals. Measuring equipment and PV modules were placed
outdoors and all experiments were realized under clear sky conditions. PV module irradiance and temperature were measured
by means of a reference PV cell, operating in unshaded conditions. Irradiance on shaded cells is given by the measured
irradiance multiplied by the TR of the shading material. Temperature of shaded cells was not directly measured. However,
shading of the module surface was sustained for short intervals,
justifying the assumption that the temperature of shaded cells
does not deviate significantly from the unshaded ones.
A. PV Cell/Module Equivalent Circuit
To validate the suitability of the PV cell/module electrical
equivalent of Section II, I–V curves of the three modules were
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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 3, SEPTEMBER 2011
TABLE VI
COMPARISON OF MEASUREMENTS AND SIMULATION RESULTS FOR THE PV MODULES OF TABLE IV
measured under several operating conditions and compared to
the simulations. The PV cell/module equivalent proved accurate in all cases, permitting reproduction of the module characteristics with very low error. Selected results are presented
in Fig. 7, both for unshaded and partially shaded operation.
In Fig. 7(a), shading of Module A is performed using Material 1 (TR = 28%), with a shaded area ssh of 33.3%. Irradiance on the unshaded cells was 742 W/m2 and on shaded
cells 475 W/m2 , while the I–V curve for unshaded operation
corresponds to 723 W/m2 . The simulation model presents a
NRMSE lower than 2%. In Fig. 7(b), Module C and Material 2 (TR = 64%) are used. Irradiance on unshaded cells was
1020 W/m2 and on shaded cells 285 W/m2 . The NRMSE of
the model is lower than 2.5% in all cases. During the measuring procedure, module temperatures varied in the range
of 40–47 ◦ C.
PARASKEVADAKI AND PAPATHANASSIOU: EVALUATION OF MPP VOLTAGE AND POWER OF mc-Si PV MODULES
Fig. 8.
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Measured P–V curves of Module A (see Table IV) for different shading scenarios.
B. Equations for MPP Voltage and Power Estimation
To validate the applicability of (16) to other mc-Si PV modules, partial shading operation of the experimental modules was
realized under the irradiance and temperature conditions presented in Table V. Several shading patterns were applied, using
the two shading materials to create shaded areas from 17% to
83% of the module surface, as presented in Table VI. In each
case, the P–V curves of the modules were measured to obtain
the voltage and power at the two MPP points of the curve.
Indicative P–V curves obtained for Module A are illustrated
in Fig. 8.
Then, the MPP voltages and powers were calculated for each
case, applying (16). For this purpose, module characteristics
from the respective datasheets were used and the values obtained
from (16) were referred to the module operating temperatures
of Table V, using the temperature coefficients provided in the
datasheets. All experimental and calculation results, together
with the respective errors, are summarized in Table VI. Blank
cells exist in Table VI because the first peak (MPP1) voltage
and power were not recorded for certain shading scenarios of
Modules B and C, as the minimum value of the variable resistor
used in the experiments was not low enough.
From Table VI, it is clear that the proposed formulae provide a fairly accurate approximation, the error being lower than
5% in most cases. Higher errors do exist and may be attributed
to several factors, besides the approximate nature of (16), including shaded-cell temperature differences, shading surface
abnormalities, as well as to the fact that the PV modules of the
experiments were in operation for some years and, therefore,
their actual characteristics may deviate from datasheet values
due to degradation.
VIII. CONCLUSION
In this paper, the effect of partial shading on PV module performance is analyzed in order to provide a simple and easily
applicable methodology for the calculation of shadow effects
on the main electrical characteristics of a PV module. A double diode PV cell model, with an extension term for reverse
bias operation, is utilized for the simulation of four study case
mc-Si PV modules. Semiempirical formulae are then derived
for the MPP voltage and power under partial shading, taking
into account the module characteristics and the shading pattern.
The proposed methodology is validated by experimental results,
using three different mc-Si PV modules, measured in outdoor
conditions, and proved to provide a good approximation in most
practical situations.
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Evagelia V. Paraskevadaki received the Diploma degree in electrical engineering from the National Technical University of Athens, Athens, Greece, in 2007,
where she is currently working toward the Ph.D. degree.
Her current research interests include renewable energy technologies, especially photovoltaics, PV system design, and simulation and distributed generation integration into the electrical grid.
Ms. Paraskevadaki is a member of the Technical Chamber of Greece.
Stavros A. Papathanassiou (S’93–M’98–SM’10) received the Diploma degree
in electrical engineering and the Ph.D degree from the National Technical University of Athens (NTUA), Athens, Greece, in 1991 and 1997, respectively.
He was with the Distribution Division of the Public Power Corporation of
Greece, in power quality and distributed generation studies, being responsible
for the elaboration of DG interconnection guidelines. In 2002, he became a
Member of the Faculty in the Electric Power Division, NTUA, where he is
currently an Assistant Professor. His research interests include wind turbine and
PV technology and the integration of DG to the grid.
Dr. Papathanassiou is a member of the CIGRE. He is also a Registered Professional Engineer and a member of the Technical Chamber of Greece. Since
2009, he has been a Member of the Board of the Hellenic Transmission System
Operator.