Ray-Trace Evaluation of a Fisheye Lens as Static Concentrator

Abstract

The cost of electricity generated by photovoltaic panels is still far superior to that generated by conventional sources. This is mainly due to the high cost of solar cells and its low efficiency, which require the use large areas of solar panels for electricity generation. In order to reduce costs of infrastructure concentration photovoltaic systems (CPV) have been proposed, which require very precise solar tracking, which in turn limits the attainable cost reduction. To address this last point many different concentration systems that don’t require solar tracking are being developed in different parts of the globe.

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1876-6102 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Selection and/or peer
-review under responsibility of ISES.
doi: 10.1016/j.egypro.2015.06.055
Energy Procedia 57 ( 2014 ) 3091 – 3099
ScienceDirect
2013 ISES Solar World Congress
Ray-Trace Evaluation of a Fisheye Lens as Static
Concentrator
David Pérez-Márquez
a *
and G. Ramos
a
a
Instituto Politécnico Nacional, CICATA-IPN, Cerro Blanco 141, Col. Colinas del Cimatario, 76090 Querétaro, QRO, México
Abstract
The cost of electricity generated by photovoltaic panels is still far superior to that generated by conventional sources.
This is mainly due to the high cost of solar cells and its low efficiency, which require the use large areas of solar
panels for electricity generation. In order to reduce costs of infrastructure concentration photovoltaic systems (CPV)
have been proposed, which require very precise solar tracking, which in turn limits the attainable cost reduction. To
address this last point many different concentration systems that don't require solar tracking are being developed in
different parts of the globe.
In this paper we evaluate the possibility o
f using a fisheye lens as static CPV. This type of lens is widely used in
photography, having a wide acceptance angle. The evaluation is carried out using optical ray tracing CAD software,
comparing the energy collected by a flat surface with that captured by a similar surface using arrangement of lenses.
The evaluation is performed over an entire day, for different days throughout the year, including solstices and
equinoxes. The energy concentration factor, using a lens with an acceptance angle of 60 ° is 1.75X to 1.6X depending
on the time of year
© 2013 The Authors. Published by Elsevier Ltd.
Selection and/or peer-review under responsibility of ISES
Keywords: Solar concentrator; fisheye; Ray tracing; Concentration photovoltaic; Solar optical efficiency
* Corresponding author. Tel.: +52 4625993880
E-mail address: dperezm0805@ipn.mx
Available online at www.sciencedirect.com
© 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Selection and/or peer-review under responsibility of ISES.

3092 David Pérez-Márquez and G. Ramos / Energy Procedia 57 ( 2014 ) 3091 – 3099
The relatively high cost of solar photovoltaic systems h
as been an obstacle for the massive penetration
of the market. Solar concentration is seen as a solution, since it requires a smaller area of solar cell.
Concentration is usually used in combined with high performance cells using high concentration ratios
and require expensive solar tracking systems and costly maintenance. The on-roof installation of these
s
ystems is also nontrivial and requires special preparation to hold the heavy structure and withstand rain
and wind [1]. To get around these inconveniences, the
use of static solar concentrators is suggested. Many
of these systems started its development in the 70's but were abandoned afterwards.
Starting this century, the need of finding alternative ener
gy sources and solutions has led to come back
to these ideas. In addition, current computing power allows simulations closer to reality. This is the case
of ray tracing, which has reduced the cost of design of systems, increasing the number of developments in
the field of solar concentration. Current designs are usually limited to an acceptance angle of 60 º [2,3],
enabling these systems to collect only during 8 hours in the equinox and 3 hours in the solstice. Seeking
for a better performance, in this paper the ability of a "Fisheye" lens to concentrate the solar radiation is
evaluated. This type of lens can produce 180º images, a feature which is used here to have acceptance
angles greater than 60 °.
In our region the photovoltaic systems have little penetratio
n of the market, mainly because of the high
cost of photovoltaic cells. Systems with solar tracking have an even lower acceptance. Having a
photovoltaic system of the concentrating type that produces a high output throughout the day without the
need of solar tracking and costly maintenance could help the transition to alternative energies in our
region.
Nomenclature
C
eff
Instantaneous effective concentration.
C
eff av
Effective Concentration averaged for solar irradiance between 8 am and 4 pm.
G Solar constant (W/m
2
)
G
F
Solar irradiance on a surface with concentrator (W/m
2
)
G
woF
Solar irradiance on a surface without concentrator (W/m
2
)
ω Hour angle
1. Clear sky radiation simulation
Throughout this paper OptiCAD software is used f
or the calculations. This software can perform
raytracing and radiometric evaluations. In this paper the sun is modeled as a light source that varies its
position with time, according to the apparent movement of the sun. The apparent movement of the sun
has been calculated [4, 5] considering a location with 0º latitude and for the solstices and equinoxes. The
position of the sun has been calculated in one hour intervals, from 08 hrs. to 16 hrs., schematically shown
in fig. 1.

David Pérez-Márquez and G. Ramos / Energy Procedia 57 ( 2014 ) 3091 – 3099
3093
Fig. 1. Schematic diagram of positions of the sun and illumination used to model the Sun under OptiCAD.
The light source was limited to have an illumination an
gle of 0.5º, and was placed far enough to
illuminate all our fisheye lens setup, as schematically shown in fig. 1. The energy of each ray and the
number of rays were select so, that they reproduced the irradiance of 950 W/m
2
at noon. The simulation
of the clear sky irradiance was benchmarked against the result of analytical formulae that can be found
else
where [4,5]. The number of rays used in the simulations was held low enough to allow reasonable
computing times.
Using 10000 rays per square meter an accordance of 99,55% was found between simulated and
anal
ytically calculated results. Fig. 2 shows graphically the comparison of these results.
Only at noon a small difference can be appreciated, f
or all other times of day, the difference cannot be
appreciated in the graph. Table 1 shows the numerical results. No correction is introduced to account for
air mass.
Fig. 2. Comparison between analytical calculation and OptiCAD Irradiance simulation using 10000 rays.

3094 David Pérez-Márquez and G. Ramos / Energy Procedia 57 ( 2014 ) 3091 – 3099
Daytime (h)
8
9
10
11
12
13
14
15
16
Analytically calculated
irradiance (W/m
2
)
475
675
823
918
950
918
823
672
475
OptiCAD simulated
irradiance (W/m
2
)
474.1
669.9
822.1
913.5
943.6
913.5
822.1
669.9
474.1
Table 1. Numerical results of the comparison between analytically calculated irradiance and OptiCAD simulated irradiance.
2. Effective Concentration (C
eff
)
In reference [2] an effective concentration is defined for a system that has multiple reflections. Since we
w
ant to compare the solar energy collected on a flat surface with the energy collected on the same surface
but with fisheye system on-top, we will define an effective concentration. Being G
woF
the energy collected
on a given horizontal flat surface, for a specific irradiation direction, without any Fisheye and G
F
the
energy collected with Fisheye on-top, C
eff
is defined according to:
C
eff
=
G
woF
G
F
(1)
So, a C
eff
larger than 1 indicates an improvement in solar collection due to the fisheye system.
In order to evaluate the overall performance of each optical co
nfiguration along a day, an averaged value
C
eff
av
is calculated, integrating the instantaneous C
eff
values for solar illumination between 8 am and 4
pm.
3. Lens model
The name fisheye lens refers to any lens capable of imaging the entire hemisphere in object space onto a
f
inite circle on the focal plane [6]. It is formed by an arrangement of individual imaging lenses as in the
example shown in fig. 3. As in our case we don't require the imaging capability we can consider only the
first two first lenses, as these provide the larg
est concentration in the arrangement.
Each lens is formed by two spherical surfaces
with different radii. Considering the x-axis as the optical
axis, the 2D cross-section of the inner surface is given by:
(
x− x
0
)
2
+
(
y− y
0
)
2
=R
in
2
(2)
With
x
0
=0
and
y
0
=0
(circumference in the origin and with
x< 0
(only negative values). The outer
surface is also spherical, with larger radius, but with the center displaced with respect to the former circle.
The 2D cross-section is given by:

David Pérez-Márquez and G. Ramos / Energy Procedia 57 ( 2014 ) 3091 – 3099
3095
0º
45º
90º
Fig. 3 Schematic cross section of a typical Fisheye setup. For a comprehensive reference see for example [5]
(
x− x
1
)
2
+
(
y− y
0
)
2
=R
out
2
(3)
With
x
1
>x
0
and
y
0
=0
(circumference with center displaced to the right but in the optical axis), with
x< 0
(only negative values), and
R
out
>R
in
The lens with axial symmetry is formed by the rotation of the 2D shape, and the lens with cylindrical
s
ymmetry is formed by the lateral displacement of the 2D shape.
3.1. Lens Design
Two different fisheye configurations have been considered
, one having cylindrical symmetry and the
other having axial symmetry.
A slight modification of the original Fisheye configuration of fig. 3 has been used to ease fabrication and
m
ounting. Both lenses have been designed to rest at the same surface. Fig. 4 left shows a 3D render of the
cylindrical fisheye lens, formed with two individual lenses with cylindrical symmetry. The outer lens
covers a surface of 1 m
2
, while the second (inner) only a surface of 0.5 m
2
. In order to evaluate the
performance, the collecting surface has a rectangular shape 1 m long and 0.2 m wide.
The second system has axial symmetry, as shown in fig
. 4 right. The larger lens has a 1m diameter (0.78
m
2
area), while the inner lens a diameter of 0.5 m. In this case the collecting surface is a disc of 0.125m
2
of area.

3096 David Pérez-Márquez and G. Ramos / Energy Procedia 57 ( 2014 ) 3091 – 3099
Fig. 4. Isometric 3D render of the two lens systems: cylindrical fisheye (left) and axial fisheye (right)
4. Results
4.1 Raytracing
Fig.5. Raytracing on a two lens fisheye system for perpendicular incidence (0º or 12 pm)
Collecting surface
A
B
B
A
C
D
Collecting surface

David Pérez-Márquez and G. Ramos / Energy Procedia 57 ( 2014 ) 3091 – 3099
3097
Fig. 6. Raytracing on a two lens fisheye system for inclined incidence (45º or 09 am)
Fig. 7 Ratio of collection RC as function of the incidence angle.
Fig. 5 shows a cross sectional view of the raytracing of the cylindrical fisheye, together with the position
of the collecting surface for perpendicular incidence (noon). There are different situations we can
identify. The ray falling along the optical axis passes the lens system without deviation. Form this
position up to the position “A” in fig 5 all rays falling on the first lens are deviated and reach the second
lens, but many of them don't reach the collecting surface. Only the rays closer to the axis than position
“B” pass through the two lens system and reach the collecting surface. Fig. 6 shows a similar analysis
w
ith the rays having a falling angle of 45º. While rays between positions A and B are directed to the
second lens, only rays between positions C and D are actually collected. At higher incidence angles (not
shown here) only a few rays reach the collecting surface.
5.2 Acceptance angle
Figure 7 shows the C
eff
as function of the incidence angle, computed for a cylindrical fisheye lens with the
cylindrical axis oriented along the south-north direction, for a day during the equinox. At 0º incidence the
C
eff
amounts 1.6 approx., as the incidence angle increases, the C
eff
value increases slightly reaching a
maximum of 1.7 at an incidence angle of 45º. For larger angles, C
eff
decreases rapidly reaching 1 at an
angle of 52.5º and zero at 70º. According to this figure, the best collection occurs at 45º and the fisheye
lens offers an improvement in solar collection for angles smaller than 52.5º (daytimes between 08 am and
4 pm).
5.3 Performance
In order to evaluate the expected perform
ance of this system in the course of a year, two typical situations
have been simulated: the equinox and the solstices, as any other situation lies between them. Since we are
considering latitude of 0°, both solstices will have the same angular deviation. The calculated C
eff
is

3098 David Pérez-Márquez and G. Ramos / Energy Procedia 57 ( 2014 ) 3091 – 3099
displayed in fig, 8. For the cylindrical Fisheye we co
nsidered two situations: with the cylindrical axis
oriented along the north-south direction or along the east-west direction. For the axial Fisheye any
orien
tation is equivalent.
In the equinox (fig.8. left) the cylindrical fisheye orien
ted along the north south direction shows the
behavior we already described in fig. 7. Integrating the C
eff
from 8 am to 4 pm, this configuration has a
C
eff
av
of 1.49. For the Orientation East-West, the lowest C
eff
of 1.5 is found at noon, and for higher
angles C
eff
grows steadily reaching a value of C
eff
=2.3 at approx. 8 am. Again, integrating the values a
C
eff
av
of 1.75 is obtained.
The axial fisheye shows also improvement in the solar collection. At noon it has a C
eff
of roughly 1.8,
keeping this value almost constant for daytimes between 11 am. and 1 pm. C
eff
values larger than 1
between 8:20 am. and typically 3:40 pm. The overall day performance is C
eff
av
= 1.5
Fig. 8 right shows a similar evaluation for the solstice. Here the axial Fisheye shows a similar behavior as
in the equinox, but the decay of C
eff
with daytime is faster. Stating with C
eff
=1.77 at noon, it rapidly
decays to C
eff
values smaller than one already at 10 am. Nevertheless it reaches C
eff
av
value of 1.28.
The cylindrical Fisheye in the solstice has a similar beha
vior as in the equinox but with an inferior
performance. In north-south orientation at noon a C
eff
= 1.57 is reached, the maximum being C
eff
=1.63.
The value of C
eff
=1 is reached approx. at 8:30 am. The overall performance of this device in this
orientation is C
eff
av
= 1.44. A similar analysis for the east-west orientation gives an overall performance
of C
eff
av
=1.66. These results are summarized in Table 2.
Fig. 8 C
eff
as function of the daytime for the equinox (left) and solstice (right) for the axial Fisheye and the cyl. Fisheye oriented
along the north-south and east-west directions, respectively.
Cylindrical Fisheye
Axial Fisheye
Orientation
north-south
east-west
C
eff av
at Equinox
1.49
1.75
1.5
C
eff av
at Solstice
1.44
1.66
1.28
Table 2. C
eff
av
values of the different Fisheye configurations for the Equinox and Solstice.

David Pérez-Márquez and G. Ramos / Energy Procedia 57 ( 2014 ) 3091 – 3099
3099
The use of a fisheye lens over a flat surface offers an
improvement in solar collection in all cases,
allowing good collection between 9 am and 3 pm. According to table 2, the cylindrical fisheye in east-
west orientation offers the best all-year performance. The fisheye concentrator has a wide collection angle
of
90 ° (-45° to +45°) with an effective collection of higher than 1.28. The curved surface offer the
ad
ditional advantage, that rain can help maintaining the cover cleaner than in a flat surface, reducing the
need of maintenance.
These results are encouraging to continue with the de
velopment of static concentrator systems, that have
potential of reducing investment costs and can help the establishment of photovoltaic systems in our
region.
Conclusions
Two Fisheye configurations have been proposed and ev
aluated as static solar concentrators. The systems
consist of two lens designs. The proposed systems have an angle of capture of 90° (-45° to+45°) which
allo
ws collection over 8 hours during the solstice.
The all-day performance of these systems range from 1.28 to 1.75 for different days along the year. The
in
stantaneous performance of these systems can be as high as 2.4 at specific daytimes at which the angle
of incidence is around 40°.
The cylindrical configuration performs better than the a
xial configurations. The cylindrical configuration
in east-west orientation shows the best overall performance (C
eff av
>1.66) over the entire year.
Acknowledgements
The authors will acknowledge the support of Concyteq-IPN for the realization of this work.
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