Comparison of Solar Panel Models for Grid
Integrations Studies
E. M. da S. Brito, A. F. Cupertino, L. P. Carlette,
D. O. Filho, H. A. Pereira
Gerência de Especialistas em Sistemas Elétricos de Potência
Universidade Federal de Viçosa, Viçosa, Brasil
erick.brito@ufv.br, allan.cupertino@yahoo.com.br,
luan.carlette@ufv.br, delly@ufv.br, heverton.pereira@ufv.br
Abstract— Photovoltaic systems are highly dependent on climatic
conditions in which they are submitted. The incident solar
irradiance and temperature are the main factors impacting on
the power generated by a solar panel. This paper presents three
different models of a solar panel and compare, through
simulations, their accuracies and efficiencies, and also shows the
advantages and applications of each model. Simulations for each
model connected to the grid were also made through a controlled
inverter. This inverter keeps the voltage at the terminals of the
panel in a constant value equal to its maximum power point,
provided by the manufacturer. In the end, it is possible to see
that two of the presented models have almost the same behavior
while the third one has some discrepancy.
I.
P. F. Ribeiro
Technische Universiteit Eindhoven
Department of Electrical Engineering,
5600 MB Eindhoven, The Netherlands
pfribeiro@ieee.org
Photovoltaic panels have an intermediary behave between a
current and a voltage source. Moreover, variations in the
incident solar irradiance and temperature have a great impact
on the generated power as illustrated in Figure 1.
The modeling of photovoltaic panels can be considered a
multi physical modeling because it considers the temperature at
the panel, the radiation incident on it, and its electric power
generation. In this domain, it becomes necessary to make the
study of each subset, considering their effects in cascade that
influence on each other [3], [4].
INTRODUCTION
The technological progress has changed the lifestyle of the
modern human and the energy demand prompted motivates
new researches for renewable energy sources. Among
these are solar and wind power systems [1].
It has been happened a significant growth in the installed
power on solar photovoltaic systems. In 2011 the value was of
69.68 GWp, a growth of 76 % in relation to the previous year.
More than 95% of this power is generated by grid-connected
systems [2].
(a)
Although its strong growth, renewable sources does
not produce a significant amount of energy compared to
the world demand. This is due to lack of tax incentives,
making these sources less competitive in the market. Many
countries do not have specific legislation for photovoltaic
systems connected to the grid, for instance. Nevertheless,
expectations and forecasts for the future indicate another
reality.
Even though the sun represents an almost unlimited source
of clean energy, an obstacle to develop the photovoltaic (PV)
panels’ technology is low efficiency, compared to
hydroelectric and nuclear plants. The energy production is
further reduced on cloudy days or in shaded situations and it is
even worse in the night time when there is no generation.
The authors would like to thank to CNPQ, FAPEMIG and CAPES for
their assistance and financial support.
(b)
Figure 1. Effect of incident irradiance (a) and temperature (b) in I x V curves
of a solar panel.
Some parameters used in the modeling are informed by
manufactures and they are in TABLE I.
TABLE I.
MAIN PARAMETERS OF A SOLAR PANEL
Symbol
Maximum Power (W)
Maximum Power Voltage (V)
Maximum power current (A)
Short circuit current (A)
Temperature coefficient of (A/K)
Temperature coefficient of (V/K)
II.
METHODOLOGY
A. Model 1 – Equivalent Circuit
There is in literature a simplified model of PV used in some
works [5], shown in Figure 2. The resistances in series and in
parallel represent the voltage drop when the charge migrates
from the electrical contacts and the reverse leakage current of
the diode, respectively. is the open circuit voltage and is
a constant continuous current source. The parameters of the
model are calculated according to the electrical characteristics
of the photovoltaic panel. This model has some
divergences compared to the characteristic curves provided by
manufacturers. The temperature is not a parameter.
Consequently the PV curve does not change when variations
on temperature happen.
=
=
− − = Figure 2. Equivalent circuit of a photovoltaic panel.
− 1# −
= % + ∆'(
Where , and are calculated by (1), (2) and (3):
!
+ The variable is calculated by (5):
Open circuit voltage (V)
= − Parameter
B. Model 2 – Mathematical Model
This model is based on equations of photovoltaic panels.
Through equations it was possible to simulate the solar panel.
The equation of the current in the solar panel is:
(1)
(2)
(3)
(4)
(5)
Where is the current in the nominal conditions,
calculated by equation (6); ∆' = ' − ') (T is the solar panel
temperature and T+ is the nominal solar panel temperature); e are the values of incident solar irradiance and the
reference irradiance (W/m²), respectively. The variable 0 is
the temperature coefficient of the short circuit current (A/K).
= + (6)
The reverse leakage current in the diode, I is:
= + ∆'
%3
678 ∆9(
:
2 45
3 ;
!
−1
(7)
is the nominal short-circuit current, is the nominal
open circuit voltage and is the coefficient of the open circuit
voltage (V/K). The variable <is the ideality constant of the
diode, contained in the range1 ≤ < ≤ 1.5. is strongly
dependent on temperature and equation (7) is an alternative
way to express this dependency showing a linear variation in
open circuit voltage due to . This equation also simplifies
the model canceling the errors around the open circuit voltage
points and consequently in other points of the IxV curve.
Finally, @ is calculated by:
@ =
A'
(8)
Where A is the Boltzmann’s constant, ' is the temperature
of the panel (K) and is the electron charge.
In [6] and [7] it is proposed an algorithm for adjusting and . The method is based on the fact that there is an only
pair { , } in which the maximum power calculated by the IV model E is equal to the maximum experimental power
from the datasheet . Using E = in equation
(4), it will be obtained [6]:
= % + (
− F
EG EG #
!
− 1HH − (9)
The interactive process is shown in Figure 3. The initial
values of and are [6]:
EJ = 0
− P
I
EJ = − − (10)
case, and calculated by (12
12) and (13), where
default value.
'
L QRS = N QRS = L,) #
')
L,) = ^_`a
exp b
+ ∆'
exp W
C. Model 3 – Solar Cell Multi Physical Model
The multi-physics
physics modeling represents the influence of
various phenomena in which a real system is subjected. There
is in Matlab 7.10.0/Simulink®, in Simscape library, a multi
physical model of a solar cell. This model is defined by 16
parameters including some temperature coefficients and the
reverse leakage current in both diodes, for example. This
model also contains two diodes which better represent the non
nonlinear characteristic of the cell. Moreover, it is possible do
observe the change in the series and
nd parallel resistances with
temperature. The equation of the current in the solar panel is:
= − L 2
−N 2
,5MM
!
,5MM
!
− 1;
;−
− 1; −
+ ,
OO
,
OO
(11)
,
OO and ,
OO are the series and parallel resistance of
each cell, respectively. is calculated by (5),
( L and N are
the reverse leakage current in each diode that are equal, in this
= 3, as a
cde 1 1
− #f
<A ') '
:< g − 1
@
' 9h\L
,
OO QRS = ,
OO #
')
Figure 3. Algorithm of the method used to adjust the I x V model[6].
9Z[\L
(12)
(13)
(14)
' 9hiL
,
OO QRS = ,
OO #
')
(15)
Y,
OO
,
OO = 2
;
Y,
OO
(16)
'X1 and '1 are the exponent that represent the
influence of temperature on resistance values
values. In this work, and are constant, independent of temperature, so 'X1 and
'1are zero. The resistance values of the panel were
estimated by an alternative
ative method based on [6]. The
resistances for each cell can be obtained by:
Y,
OO
,
OO = 2
;
Y,
OO
(17)
Where Y,
OO and Y,
OO
OO are the number of solar cells in
series and in parallel of the panel,
panel respectively.
Comparing equations (7)
( and (13) it is possible to see the
main difference between Model 2 and 3. In Model 2 there is
one more coefficient for temperature influence, .
D. Connecting the Models to the Grid
Figure 4 shows a schematic of the grid connected
photovoltaic system through a controlled
control
inverter. The control
of the inverter keeps the voltage at the panel’s terminal in a
Figure 4. Simulated grid-connected photovoltaic system.
fixed value equal to the maximum power point (provided by
the manufacturer) and it injects the generated power in the
grid with the power factor close to the unity.
The controlled variables were written in direct and
quadrature axis. The control loops of the inverter are shown in
Figure 5, there is the reactive power loop that controls the
power factor and a loop for regulate the DC bus voltage. The
current control loops use proportional controllers and the
external loops use proportional-integral controllers [8]. The
controller’s gains were adjusted by the poles allocation
method.
A Synchronous Reference Frame – PLL (SRF-PLL) circuit
is used as synchronism technique to connect the system to the
grid. This structure estimates the grid voltage angle for the
control of the inverter. To reduce the harmonics generated by
IGBT’s switching a LCL filter was used. The design of this
component is in [9].
TABLE II.
KYOCERA SM-48KSM PARAMETERS FOR 1000 W/M²
AND 25 ºC.
Parameter
TABLE III.
Value
48l
18.6
2.59p
22.1
2.89p
−0.070/
1.66rp/
*STC: AM1.5 spectrum
KYOCERA SM-48KSM PARAMETERS FOR 800 W/M²
AND 47 ºC.
Parameter
Value
34l
17.1
2.02p
20.5
2.26p
−0.070/
1.66rp/
*STC: AM1.5 spectrum
III.
Figure 5. Control loops of the inverter.
RESULTS
In order to validate the three mentioned models, it was
used the Kyocera panel, model SM-48KSM whose parameters
are shown in TABLE II. The curves for each model were
collected for different situations, varying the climate
parameters such as temperature and radiation in order to verify
the efficiency and accuracy of each one. Using the data shown
in TABLE II, TABLE III and the estimation method from [6],
the models were simulated for the nominal operating
conditions. The final values for the resistance for Model 2 are
shown in (18) and for Model 3 it is shown in (19). The
different values are justified because the reverse leakage
current of each model is calculated in a different way, as can
be seen in (7) and (12).
= 0.186200Ω
s = 108.602693Ω P
3
(18)
= 0.090600Ω
s = 92.939664Ω P
2.5
(19)
TABLE IV and TABLE V show the relative errors between
the parameters obtained by the graphics and the provided by
the manufacturer, for 1000 W/m² and 25°C. Some
manufactures also indicate the panels’ data for other values of
radiation and the most common is 800 W/m² and 47ºC. Thus,
simulations were made to these new values and the results are
shown in Figure 7.
TABLE IV. ERRORS (%) ON THE PARAMETERS OF EACH MODEL,
COMPARED WITH MANUFACTURES DATA - THE SITUATION OF
1000 W/M² AND 25 ºC.
1.5
1
Model 1
Model 2
Model 3
0.5
0
0
5
10
15
Voltage(V)
20
25
20
25
(a)
55
50
Model 1
Model 2
Model 3
45
40
35
Power(W)
Analyzing Figure 6, Model 1 follows a solar panel
behavior, but not its accuracy. It presented a different shape of
the curve, more linear, compared to the other ones, showing a
wide divergence from reality. Furthermore, it was observed
that for different values of irradiance the open circuit voltage in
Model 1 was constant. Despite being the less accurate, Model 1
has the fastest simulation and a small number of equations.
Model 3’s simulation is slower than Model 2 and it has a
satisfactory performance when compared to the Model 1, since
its waveform was more accurate and its open circuit voltage
decreased with a reduction of incident radiation. Besides,
Model 3 presents the characteristics of each solar cell, what is
an advantage.
Current(A)
2
30
25
20
15
10
Model 1
Model 2
Model 3
4.2
0
0
∆Pm(%)
3.6
0.3
0.5
∆Voc (%)
1.9
0
0.4
∆Isc (%)
4.2
0
0.8
∆Vmp (%)
0.4
0
0.8
TABLE V. ERRORS (%) ON THE PARAMETERS OF EACH MODEL,
COMPARED WITH MANUFACTURES DATA – THE SITUATION OF
800 W/M² AND 47 ºC.
Models
Model 1
Model 2
Model 3
18.6
0.6
0.5
∆Pm(%)
11.7
0.6
0.8
∆Voc (%)
0.4
3.5
3.1
∆Isc (%)
17.8
0.5
1.3
∆Vmp (%)
5
∆Imp (%)
1.0
1.0
1.5
0
0
5
10
15
Voltage(V)
(b)
Figure 6. Curves I x V (a) and P x V (b) of the three panels for an irradiance
of 1000 W/m² and temperature 25 ºC.
3
∆Imp (%)
2.5
2
Current(A)
Models
1.5
Model 1
Model 2
Model 3
1
0.5
0
0
5
10
15
Voltage(V)
(a)
20
25
3.5
50
Model 1
Model 2
Model 3
2.5
Current(A)
Power(W)
40
3
30
2
1.5
20
1
10
0
Model 2
Model 3
0.5
0
0
5
10
15
Voltage(V)
20
25
0
5
10
15
Voltage(V)
20
25
(b)
Figure 7. Curves I x V (a) and P x V (b) of the three panels for a irradiance of
800 W/m² and temperature 47 ºC
Besides irradiance another very influential climatic factor
in the efficiency of photovoltaic panels is the temperature. In
this context, the Model 1 is not able to simulate changes in
temperature, showing a major limitation in its applications. In
order to compare I x V and P x V curves of the Models 2 and 3
for different values of temperature, simulations were made for
25 ° C and 75 ° C.
Figure 8. I x V curves for 25 ºC (a) and for 75 ºC (b) of the Models 2 and 3,
with radiation of 1000 W/m².
50
45
Model 2
Model 3
40
35
Power(W)
(b)
It was noticed that both of the curves follow the same
behavior, it is hard to see the two curves due to their proximity.
Although the manufacturer does not provide data for different
temperature values, it can be observed in Figure 8 that the
temperature variation causes a slight increase in the shortcircuit current and it reduces the open circuit voltage.
30
25
20
15
10
5
0
0
5
10
15
Voltage(V)
20
25
20
25
(a)
3
45
Model 2
Model 3
40
2.5
35
30
Power(W)
Current(A)
2
1.5
1
25
20
15
Model 2
Model 3
0.5
10
5
0
0
5
10
15
Voltage(V)
(a)
20
25
0
0
5
10
15
Voltage(V)
(b)
Figure 9. P x V curves for 25 ºC (a) and for 75 ºC (b) of the Models 2 and 3,
with irradiance of 1000 W/m².
As a comparison criterion, it was simulated panels
connected to the grid. A controlled inverter maintains the
output voltage of the panels in a constant value. Figure 10(a)
and (b) shows the generated power and voltage in the panel.
The slight difference in the injected power is justified by
Figure 11, where the given by the manufacture is
highlighted and it is possible to see that the for Models 2
and 3 is almost the same in this point, but in Model 1 the is
a little higher.
Injected Power(W)
50
40
20
Model 1
Model 2
Model 3
10
0
0
0.02
0.04
0.06
time(s)
0.08
20
10
0
Voltage(V)
20
25
Figure 11. P x V curves of the three panels for a irradiance of 1000 W/m²
and temperature 25 ºC, highlighting the voltage in the maximum power point.
Another analyzed characteristic of the model is the
simulation time. Using a personal computer with an Intel CPU
of 2.1GHz processor speed and 3GB of RAM, the time to
simulate the grid-connected system, for each model, is shown
in TABLE VI. It is possible to see that despite Model 2 and 3
are very similar, Model 3 takes much more time to simulate
than Model 2.
COMPARISON BETWEEN SIMULATION TIME FOR THE
MODELS CONNECTED TO THE GRID
Models
Simulation time (s)
Model 1
Model 2
Model 3
19.0
22.55
1140.35
CONCLUSIONS
The advantages of high simplicity of the Model 1 and the
study of individual cells in Model 3 should not be disregarded.
Besides electrical and thermal characteristics, the simulation
time is a critical factor since Model 3 might become
impracticable using a lot of solar cells. There will be situations
where each of these models will be very important in the
development of works in PV studies and it is up to the
researcher to decide which one fits better.
19
0.02
10
15
Voltage(V)
The results show that the mathematical model (Model 2) is
more accurate if compared to the equivalent circuit model
(Model 1) and to the panel made by individual cells association
(Model 3). It allows obtaining the results closer to the reality.
Model 1
Model 2
Model 3
0
5
IV.
20
19.5
0
0.1
(a)
18.5
30
TABLE VI.
30
Model 1
Model 2
Model 3
40
Power(W)
Figure 10(b) shows the smooth operation of the inverter to
keep the output voltage of the panel at a constant value. This
fact was evidenced for the three analyzed models. Note in
Figure 10(a) that in steady state the power in all the Models is
close to the maximum power provided by the manufacturer.
The only difference in Model 1 is the overshoot due to the
different dynamics behavior that comes from the linearization
of the model.
50
0.04
0.06
time(s)
0.08
0.1
(b)
Figure 10. Power (a) and Voltage (b) in the panel connected to the grid.
Obtaining a model of PV panels with high accuracy is vital
for future works in the photovoltaic solar engineering area.
These Models make it possible to develop works and
techniques to improve the performance and applicability of PV
systems in the context of global renewable energy production.
Luan Peterle Carlette was born in Cachoeiro de
Itapemirim, Brazil. He is student of Electrical
Engineering at Federal University of Viçosa, Viçosa,
Brazil. He works with Power Systems, especially
with photovoltaic energy and control applied to
converters.
ACKNOWLEDGMENT
The authors would like to thank to CNPQ, FAPEMIG and
CAPES for their assistance and financial support.
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University in Montreal, Canada. He has been working
as an Associate Professor of the Energy Area of the
Agricultural Engineering Department of Federal
University of Viçosa, Minas Gerais, Brazil, since
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BIOGRAPHIES
Erick Matheus da Silva Brito was Born in Feira de
Santana, Brazil. He is student of Electrical
Engineering at Federal University of Viçosa, Viçosa,
Brazil. He works with Power Systems, especially
with photovoltaic energy.
Allan Fagner Cupertino was born in Visconde do
Rio Branco, Brazil. He is student of Electrical
Engineering at Federal University of Viçosa, Viçosa,
Brazil. Currently is integrant of GESEP, where
develop works about power electronics applied in
renewable energy systems. His research interests
include solar photovoltaic, wind energy, control
applied on power electronics and grid integration of
dispersed generation.
1988.
Paulo F. Ribeiro received B.S. degree in electrical
engineering from the Federal Univesity of
Pernambumco, Brazil, in 1975, and the Ph.D. degree
in electrical engineering from the University of
Manchester, Manchester, U.K., in 1985. Presently, he
is a Professor at Calvin College, Grand Rapids,
Michigan. In 2010 he was a visiting professor at the
Eindhoven
University of
technology, The
Netherlands. Dr. Rbeiro is active in the IEEE,.
International Council on Large Electric Systems (CIGRE), and International
Electrotechnical Comission (IEC) technical working groups
Heverton Augusto Pereira was born in São Miguel
do Anta, Brazil, in 1984. He received the B.S. degree
in electrical engineering from the Federal University
of Viçosa (UFV),Viçosa, Brazil, in 2007, the M.S.
degree in electrical engineering from the
Universidade Estadual de Campinas (UNICAMP),
Campinas, Brazil, in 2009, and currently is Ph.D.
student from the Federal University of Minas Gerais
(UFMG), Belo Horizonte,
Brazil. He is currently an Assistant Professor with the Department of Electric
Engineering, Federal University of Viçosa, Brazil. His research interests are
wind power, solar energy and power quality.