Download High performance Fresnel-based photovoltaic

High performance Fresnel-based photovoltaic
concentrator
Pablo Benítez 1,2,*, Juan C. Miñano 1,2, Pablo Zamora1, Rubén Mohedano2,
Aleksandra Cvetkovic2, Marina Buljan1, Julio Chaves2, Maikel Hernández2
1
Universidad Politécnica de Madrid (UPM), Cedint, Campus de Montegancedo 28223, Madrid, Spain
2
LPI, 2400 Lincoln Ave., Altadena, CA 91001 USA
*
[email protected]
Abstract: In order to achieve competitive system costs in mass-production,
it is essential that CPV concentrators incorporate sufficient manufacturing
tolerances. This paper presents an advanced concentrator optic comprising a
Fresnel lens and a refractive secondary element, both with broken rotational
symmetry, an optic producing both the desired light concentration with high
tolerance (high acceptance angle) as well as an excellent light
homogenization by Köhler integration. This concentrator compares well
with conventional Fresnel-based CPV concentrators.
©2010 Optical Society of America
OCIS codes: (080.2740) Geometric optical design; (350.6050) Solar energy; (220.4298)
Nonimaging optics; (220.1770) Concentrators
References and links
1.
P. Benitez, and J. C. Miñano, “Concentrator Optics for the next generation photovoltaics”. Chap. 13 of A. Marti
& A. Luque. Next Generation Photovoltaics: High Efficiency through Full Spectrum Utilization, (Taylor &
Francis, CRC Press, London, 2004).
2. A. Braun, B. Hirsch, E. A. Katz, J. M. Gordon, W. Guter, and A. W. Bett, “Localized radiation effects on tunnel
diode transitions in multi-junction concentrator solar cells,” Sol. Energy Mater. Sol. Cells 93(9), 1692–1695
(2009).
3. S. Kurtz, and M. J. O’Neill, “Estimating and controlling chromatic aberration losses for two-junction, twoterminal devices in refractive concentrator systems”, 25th PVSC; pp.361–367, (1996).
4. W. Cassarly, “Nonimaging Optics: Concentration and Illumination”, in the Handbook of Optics, 2nd ed., pp.
2.23–2.42, (McGraw-Hill, New York, 2001)
5. P. Benıtez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff,
“Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
6. J. C. Miñano, M. Hernandez, P. Benítez, J. Blen, O. Dross, R. Mohedano, and A. Santamaría, “Free-form
integrator array optics”, in Nonimaging Optics and Efficient Illumination Systems II, SPIE Proc., R. Winston &
T.J. Koshel ed. Vol. 5942–12, (2005).
7. US and International patents pending by LPI, LLC, 2400 Lincoln Avenue, Altadena, CA 91001 USA
http://www.lpi-llc.com/.
8. R. Leutz, and A. Suzuki, Nonimaging Fresnel Lenses, (Springer-Verlag, Berlin, 2001).
9. R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. C. Bortz, “Nonimaging Optics”,
(Elsevier-Academic Press, New York, 2005).
10. http://www.concentrix-solar.de/fileadmin/user_upload/Download/Technical_Data_Sheets_Q3-2009.pdf.
11. G. Peharz, J. Jaus, P. Nitz, T. Schmidt, T. Schult, and A. W. Bett, “Development of refractive secondary optics
for flatcon® modules”, 23rd European Photovoltaic Solar Energy Conference, 1DV.3.34, (2008). Note that in this
reference Cg is defined using a circular active area instead of square, so the geometrical concentration 385x in it
corresponds to 302x here.
12. L. W. James, Contractor Report SAND89–7029, (1989).
13. D. Anderson, B. Bailor, D. Carroll, E. Schmidt, P. Tyjewski, M. Uroshevich, “Alpha Solarco’s Photovoltaic
Development Concentrator Program”, Contractor report SAND95–1557, (1995).
14. The same BK7 glass has been considered for all the SOE’s under comparison. Though BK7 can be molded (see
for instance, 8Hhttp://www.rpoptics.com/index.php?page=rpo-moldable-glass-data) is more common the use of,
for instance, B270. The light absorption in B270 is slightly higher than in BK7, which causes that if the
comparison is done using B270, the efficient of the RTP (whose optical path is longer) is penalized the most 9H.
15. 10Hhttp://www.amonix.com/technology/index.html.
16. 11Hhttp://www.guascorfoton.com/home_en.php.
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Received 19 Jan 2010; revised 26 Feb 2010; accepted 26 Feb 2010; published 26 Apr 2010
26 April 2010 / Vol. 18, No. S1 / OPTICS EXPRESS A25
17. See, for instance: 12Hwww.sol3g.com, 13Hhttp://www.solfocus.com/, and K. Araki et al., “Development of a new
550X concentrator module with 3J cells-Performance and. Reliability-”, Proc. 31st IEEE PVSC, (2005).
18. P. Zamora, A. Cvetkovic, M. Buljan, M. Hernández, P. Benítez, J.C. Miñano, O. Dross, R. Alvarez, A.
Santamaría, “Advanced PV Concentrators”, 34th IEEE PVSC, (2009).
19. M. Victoria, C. Domínguez, I. Antón, G. Sala, “Comparative analysis of different secondary optical elements for
aspheric primary lenses,” Opt. Express 17, 6487–6492 (2009). The design with highest CAP* in this reference
has a rotationally-symmetric CPC-type TIR-based secondary. It achieved CAP* = 0.54 for a square aperture and
square cell (for which the FK has a higher value of CAP* = 0.57–0.61). Moreover, it produces a poor irradiance
uniformity. It also has the encapsulation problems discussed in Section 5.
1. Introduction
Minimizing energy cost (€/kWh) is necessary for the success of concentrated photovoltaic
energy (CPV). Key to minimizing this cost is an efficient and low cost optical design, goals
best met with the fewest elements and the maximum tolerances, but always maintaining the
high concentration (>500) that offsets the cost of expensive high-efficiency multi-junction
solar cells. A useful merit function for a CPV optic is the concentration-acceptance product
[1], which we define as:
CAP = C g sin α
(1)
Here Cg is the geometric concentration, defined as the ratio of concentrator aperture area to
solar cell area, α is the acceptance angle, defined as the incidence angle at which the
concentrator collects 90% of the on-axis power (the optical losses are considered in this
computation). The CAP value is limited by thermodynamics to an absolute upper bound [1],
given by the value of the refractive index of the material surrounding the solar cell (typically,
about 1.4-1.5).
It is remarkable that for a given concentrator architecture, the CAP is rather constant with
Cg. This means that if we know the acceptance angle at a given concentration, Eq. (16)H(1)
can be used to estimate the acceptance angle that we would obtain if we designed that
concentrator architecture for a different concentration.
For a given Cg, the acceptance angle α measures the total tolerance available to apportion
among the different elements of the system: (1) shape errors and roughness of the optical
surfaces, (2) concentrator module assembly, (3) array installation, (4) tracker structure finite
stiffness, (5) sun-tracking accuracy and (6) solar angular diameter. Each of these items can be
expressed as a fraction of the tolerance angle, so that all together comprise the tolerance
budget. Alternatively, for a given acceptance angle, a higher CAP allows a higher
concentration, consequently reducing cell size and cost.
Besides concentration and tolerance angle, an optical PV concentrator must achieve other
goals. One of them is irradiance uniformity on the cell. This is important because the cell
efficiency depends on it, but also to assure the long term cell and concentrator reliability. High
flux contrast across the cell surface causes series resistance losses, although this is less
dramatic in multi-junction solar cells than in silicon cells. Another limitation specific to multijunction cells, however, comes from tunnel diodes having to be kept operating in their
tunneling regime, which limits the irradiance at any point of the cell [2]. Additionally, if
different wavelengths have a different irradiance distribution (which has been referred to as
chromatic aberration [3] of the irradiance), the cell efficiency can be significantly affected,
due to local current mismatch between the top and middle junctions.
There are two candidates for attaining good irradiance uniformity on the solar cell: a
kaleidoscope homogenizer and the Köhler integrator. With a kaleidoscope homogenizer, the
solar cell is glued to one end and the light enters the other, most of the light reaching the cell
after bouncing off the kaleidoscope walls. The light distribution on the cell can be made
uniform given sufficient length. Kaleidoscope homogenizers for CPV, however, have
manufacturing drawbacks (see the discussion in section 20H5), which encouraged us to
further investigate Köhler integration, a well known method in illumination optics [4], usually,
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Received 19 Jan 2010; revised 26 Feb 2010; accepted 26 Feb 2010; published 26 Apr 2010
26 April 2010 / Vol. 18, No. S1 / OPTICS EXPRESS A26
comprising multiple elements with separate functions: some that collect and concentrate the
radiation (condensers) and others that produce the Köhler integration (fly-eye lens arrays).
In this work we present the application of the LPI’s design methods [5] [6] [7] to generate
a Köhler integrator array comprising free-form optical surfaces (i.e. neither rotationally nor
linearly symmetric). It combines the above two functions (concentration and homogenization)
using only two active optical surfaces, producing both high concentration and high acceptance
angle (i.e., high CAP).
2. LPI’s Fresnel Kohler (FK) concentrator design
We have developed a Köhler-based CPV optical device with a flat Fresnel lens as the Primary
Optical Element (POE), and a single refractive surface as the Secondary Optical Element
(SOE). The concentration and Köhler integration functions are achieved by the lower surface
of the Fresnel lens (which has the facets, as per usual in solar applications [8]) together with
the upper surface of the SOE. We will perform the integration with a four-unit array: POE and
SOE are divided in four symmetric parts, with each quarter of the POE corresponding to a
quarter of the SOE.
From a manufacturing point of view, Fresnel-Kohler optical surfaces are very similar to a
conventional flat Fresnel lens with a glass dome over the cell. This means that they can be
manufactured with the same techniques (continuous roll embossing, hot embossing,
compression molding, etc. for the POE; glass molding for the SOE). Thus their production
cost is essentially the same as these well known methods.
c’ d
b a
c
d’
d
c’
b’ a’
Fresnel
lens (POE)
a’
b
Free form lens (SOE)
z
a
b’
c
d’
z
x
x
y
Solar cell
Fig. 1. (Left) 3D view of the LPI’s four-fold Fresnel-Kohler (FK) concentrator: maroon rays
show how on-axis rays uniformly illuminate the cell while green rays illustrate how a point of
the primary is imaged on the cell, and. (Right) 2D schematic drawing of the edge-ray mapping
in an ideal FK concentrator.
Fig. 1(left) shows a 3D view of the four-unit Fresnel Köhler (FK) concentrator. The
Fresnel lens focuses on-axis rays (in maroon) close to the SOE surface. Then the SOE
distributes all these rays over the cell surface in a homogeneous way. In addition, rays exiting
a certain point of the primary (in green) are approximately imaged on the cell.
Our design procedure is based on the edge-ray theorem of Nonimaging Optics [9],
according to which an ideal design must map the edge rays of the input ray bundle onto those
of the output ray bundle. In order to illustrate such a mapping and show how the solar
homogenization is produced, a simplified 2D version is shown in Fig. 1 (right) and its
corresponding phase-space representations in Fig. 2(a) & (b) (not to scale, in the sense that if
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Received 19 Jan 2010; revised 26 Feb 2010; accepted 26 Feb 2010; published 26 Apr 2010
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drawn to scale, (a) would be a very narrow horizontal stripe and (b) a very narrow vertical
stripe). In the phase space at a reference line z = constant, a ray crossing that line at coordinate
point x is represented as a phase-space point (x,p), where p is the direction cosine with respect
to the x axis multiplied by the local refractive index [9]. The concentrator transforms the ray
bundles from the phase space x-p at the entry aperture (Fig. 2(a)) to the phase space x’-p’ at
the cell (Fig. 2(b)). The inactive vertical facets of the flat Fresnel lens would introduce gaps in
the phase space representation at the cell (a phenomenon usually called “dilution”), but the
Figure omits them for the sake of clarity.
For comparison purposes, the edge-ray mapping produced by a conventional imaging-type
concentrator is shown in Fig. 2(c) & (d)). At the entry aperture of both systems (Fig. 2(a) and
(c)), the sun is represented by a yellow bar (which extends for all the x values of the entry
aperture with a vertical width ∆p≈2sinαs where αs is the solar radius, αs≈0.265°). That yellow
bar in slightly decentered vertically, which means that the sun is slightly off the on-axis
direction. For both concentrators the spatial uniformity across the entry aperture is very good,
while the angular uniformity over said aperture is poor within the acceptance angle. The
mapping of the imaging concentrator produces a 90° phase-space rotation, so the angular nonuniformity at the entry aperture is reproduced as a spatial non-uniformity of the cell irradiance
(in Fig. 2(d) the yellow bar extends along the entire angular range but occupies a small range
of x’ values on the cell). In contrast, Köhler integration produces an approximately 180°
phase-space rotation, so that spatial uniformity at the entry aperture generates spatial
uniformity of cell illumination (in Fig. 2(b) the yellow bar extends along the entire x’ range
values on the cell). Note that the red rays of the left side of Fig. 1 (right) correspond to the red
phase-space sectors in Fig. 2(a) & (b), and the same for the blue rays and sectors of the same
two Figures.
p
c c’
b
2sinαs
b’
b
b’
x
2sinα
a
d d’
x
a
a’
a’
(c)
(a)
p’
d
c
a’
p’
a
b
d’ x’
x’
b’
c’
(b)
b’
b
a
a’
(d)
Fig. 2. Phase space representation of the Kohler integration in 2D geometry (the actual design
is in 3D) of (a) & (b) the Fresnel Kohler concentrator, (b) & (c) a classical imaging
concentrator. The sun is represented by the yellow bar.
Since we only have two surfaces available to design (the SOE surface and the lower
surface of the Fresnel lens), there are insufficient degrees of freedom to achieve the ideal
mapping of edge rays that is shown in Fig. 2(a) & (b), especially when considering the actual
problem is three-dimensional. As a consequence, the design relaxes the edge ray mapping
requirement for maximum CAP. Additionally, for manufacturability purposes the POE can be
composed of four symmetric sectors of a rotationally symmetric Fresnel lens with its
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Received 19 Jan 2010; revised 26 Feb 2010; accepted 26 Feb 2010; published 26 Apr 2010
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symmetry axes parallel to the z axis (see Fig. 1(a)). Fig. 3 shows a close-up rendering of the
four-way center of the Fresnel lens on the left and an actual glass-molded SOE on the right.
Fig. 3. (Left) Close-up rendering of the center of the LPI’s FK Fresnel lens, showing the four
sectors. (Right) Photograph of an LPI’s FK secondary made by glass molding with a ring to be
used as holder.
3. Simulation results for the FK concentrator
The various FK concentrator designs and their ray-tracing simulations have the following
features: (i) Fresnel lens: made of PMMA (n≈1.49), facet draft angle = 2°, vertex radius = 3
µm, facet height<250 µm; (ii) SOE: made of BK7 glass (n≈1.51) coupled to the cell with a
transparent silicone rubber of n≈1.41 (e.g., Sylgard 182 of Dow Corning); (iii) High efficiency
(≈38%) commercial triple-junction cell, with the values of the external quantum efficiency
provided by the manufacturer and considered to be independent of the incidence angle on the
cell. Absorption in dielectric materials and Fresnel reflections are considered, but surface
scattering is neglected. The f-number of the FK concentrator will be varied, and it is defined
to be the ratio of the distance between the cell and Fresnel lens to the Fresnel lens diagonal
(therefore, it is a purely geometrical value, without the usual optical meaning).
3.1 Optical efficiency
Concentrator optical efficiency can be defined as the ratio of power on the cell to the power on
the entry aperture when the sun is exactly on-axis. For an FK concentrator with Cg = 625x and
f/1, calculated at a single wavelength (555 nm), the optical efficiency is 85%, without an AR
coating on the SOE. In order to estimate the performance limit when there is an AR coating,
the optical efficiency for a perfect AR coating (i.e., zero Fresnel reflection by the SOE) is 89%
for f/1 at 555 nm.
Alternatively, an effective polychromatic optical efficiency can be defined as:
ηopt , polychrom =
conc
conc
min { I scconc
I scconc
,top , I sc , middle , I sc ,bottom }
=
1 sun
1 sun
Cg I sc1 sun Cg min { I sc1 sun
, top , I sc , middle , I sc , bottom }
(2)
Equation (41)H(2) is computed by integrating over the wavelength using the curves shown in
Fig. 4(a): the optical efficiency versus wavelength of that FK design, and the lossless spectral
photocurrent densities (calculated as the product of Cg, the AM1.5d ASMT G173 spectrum,
and the EQE of the sub-cells). The ratio of the currents Isc,top/Isc in that FK concentrator under
that standard spectrum is nearly the same (1% variation) as that ratio without the concentrator
losses. These calculations also show that the difference between the two optical efficiencies
(monochromatic @ 555 nm and polychromatic) is negligible (<0.1%) for both the uncoated
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Received 19 Jan 2010; revised 26 Feb 2010; accepted 26 Feb 2010; published 26 Apr 2010
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100
90
0.04
80
0.03
70
60
50
0.02
40
30
0.01
20
0
10
0
300
550
800 1050 1300 1550 1800
Wavelength (nm)
(a)
90
Optical efficiency (%)
Top cell
Middle cell
Bottom cell
Spectral optical efficiency
0.05
Spectral optical efficiency (%)
o)
Photocurrent density (A/nm))
and perfect AR coated case of that FK concentrator. Therefore, we conclude that for the FK
concentrator the monochromatic calculation satisfactorily estimates the optical efficiency.
Fig. 4(b) shows the optical efficiency versus the f-number for different FK concentrator
designs, all for concentration 625x. Efficiency is slightly less for low f-numbers (at f/0.8 the
predicted effective optical efficiency is still 84% without an AR coating on the SOE, and 88%
with one), and the monochromatic versus polychromatic cases lead to very similar values, for
all f-numbers analyzed.
88
85
83
Monochromatic; no AR
Polychromatic; no AR
Monochromatic; perfect AR
Polychromatic; perfect AR
80
78
75
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
f/#
(b)
Fig. 4. (a) Lossless spectral photocurrent densities and spectral optical efficiency of an LPI’s
FK concentrator with Cg = 625x and f/1, (b) Monochromatic (at 555 nm) and polychromatic
optical efficiencies as a function of the f-number of the FK concentrator, considering no AR
coating on the SOE and perfect AR coating (i.e. no Fresnel reflection on the SOE).
3.2 Concentration-acceptance angle product (CAP)
In order to compute the CAP of a given concentrator design, parallel-ray inputs at various
incidence angles will determine the acceptance angle α defined in Section 1. An example is an
FK concentrator with Cg = 625x and f/1. Figure 5(a) shows the monochromatic (@ 555 nm)
power impinging the solar cell as a function of the incidence angle of the parallel rays on the
entry aperture, relative to on-axis incidence. Two cases are shown: no AR coating on the SOE,
and a perfect one on it. The acceptance angle is defined as the 90% values, which means that
this concentrator has α = ± 1.43° without an AR coating on the SOE, and α = ± 1.52° with a
perfect one. The increased acceptance angle is because the Fresnel losses on the surface of the
SOE of the FK concentrator are higher when the sun is off-axis than when it is on-axis. Note
that since the system does not have rotational symmetry, the curves in Fig. 5(a) depend on the
azimuth at which the rays are traced. In the Figure they are parallel to two sides of the POE,
which is the worst case for that FK concentrator (i.e., any other azimuth leads to a greater or
equal acceptance angle).
Figure 6 shows on the left the concentration angle product (CAP) of FK concentrator
designs, as a function of Fresnel lens f-number, calculated for monochromatic illumination (at
555 nm). All designs have been calculated for a geometrical concentration Cg = 625x. The
CAP is approximately constant in most of the f-number values (CAP≈0.63 with no AR coating
on the SOE, CAP≈0.66 with perfect AR coating), dropping slightly for f/0.80. It is remarkable
that, as mentioned before, the use of AR coating on the SOE can increase not only the optical
efficiency (Fig. 4.(b)) but also the CAP.
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monochromatic; perfect AR
monochromatic; no AR
Cell photocurrent (normalized))
Optical power on cell (normalized))
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Incidence angle (deg)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
polychromatic; perfect AR
polychromatic; no AR
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Tracking error angle (deg)
(a)
(b)
Fig. 5. For an LPI’s FK concentrator with Cg = 625x and f/1: (a) monochromatic (@ 555 nm)
power impinging the solar cell as a function of the incidence angle of the parallel rays on the
entry aperture, (b) Cell photocurrent as a function of the concentrator tracking error angle for
AM1.5d ASMT G173 spectrum (therefore, it accounts for the finite size of the sun and the EQE
of the three subcells to find which one is limiting). Both SOE’s, without AR coating and with
perfect AR coating, are considered. All curves are relative to normal incidence.
0.7
0.7
0.65
0.65
0.6
0.6
CAP *
CAP
Figure 6 shows also a further merit function (blue lines) that we call effective CAP (or
CAP*), which is defined by substituting in Eq. (49)H(1) for the acceptance angle α by a
tracking-error angle α* (which we can call effective acceptance angle) at which the cell
photocurrent reduces to 90% of on-axis (see Fig. 5(b)). Therefore, CAP* accounts for the
finite size of the sun and the comparison of the products of the solar spectrum and the EQE of
the three subcell photocurrents to find which one is limiting. The utility of the CAP* is that it
can be experimentally measured. In Fig. 6 we see that the CAP* of the FK concentrator shows
good independence from the f-number (CAP*≈0.57 with no AR coating; CAP*≈0.61 with
perfect AR coating on the SOE). The good behavior of the FK concentrator for small fnumbers is very interesting when compact CPV modules are desired.
0.55
0.5
0.5
CAP
CAP perfect
perfect AR
AR
CAP
CAP no AR
0.45
0.55
CAP*
CAP* perfect AR
CAP*
no AR
AR
CAP* no
0.45
0.4
0.4
0.8 0.9
1
1.1 1.2 1.3 1.4 1.5
f /#
0.8 0.9
1
1.1 1.2 1.3 1.4 1.5
f /#
Fig. 6. Concentration acceptance angle products CAP and CAP* for the LPI’s FK concentrator
with no AR coating and with a perfect AR coating on the SOE versus the POE f-number (f/#).
Although Fig. 6 has been calculated for Cg = 625x, it is remarkable that for a given
f/number, the CAP and CAP* of the FK concentrator design remain approximately constant
when the concentration is changed. For instance, for f/1 the FK concentrator with no AR
coating on the SOE has Cg = 625x and α = ± 1.43° (which corresponds to the values
CAP≈0.63 mentioned before). If the FK concentrator is designed for Cg = 1000x, the resulting
acceptance angle can be well estimated to be α = ± 1.43°(625/1000)1/2 = ± 1.13°. Table 1
summarizes other examples of such calculations. (This approximation is not accurate for α*<
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± 0.7° because the sun radius and POE chromatic aberration do not scale, but those values are
out of the range shown in Table 1).
Table 1. Monochromatic acceptance angle α and effective acceptance angle α* of the
LPI’s FK concentrators at different geometrical concentration factors.
Cg
No AR coating on SOE
α
α*
± 1.43°
± 1.30°
± 1.13°
± 1.03°
± 0.92°
± 0.84°
625
1,000
1,500
Perfect AR coating on SOE
α
α*
± 1.52°
± 1.39°
± 1.20°
± 1.10°
± 0.98°
± 0.90°
3.3 Irradiance and intensity on the cell
600
600
500
500
400
400
a.u
a.u
As mentioned in the Introduction, the spatial non-uniformity of cell irradiance reduces the I-V
curve fill factor, due to higher series resistance losses, and possibly also due to the current
density being limited by a tunnel junction. This limitation may not occur for specific cells [1],
but it is not guaranteed by the manufacturer to never happen. Moreover, chromatic nonuniformity of cell irradiance can produce an additional fill factor loss [3] due to a local current
mismatch between the top and middle junctions. Again, quantification of the efficiency losses
due to these non-uniformities depends on the specific cell design and manufacturer.
The FK concentrator produces excellent irradiance uniformity on the cell, with negligible
chromatic aberration of that irradiance. Figure 7 shows two irradiance distributions of an FK
concentrator of Cg = 625x, f/1, no AR coating of SOE, when the sun is on-axis with Direct
Normal Irradiance (DNI) of 850W/m2. The distribution on the left is the irradiance when the
solar spectrum is restricted to the top subcell spectrum (360-690 nm), while the distribution on
the right corresponds to the middle subcell spectral range (690-900 nm).
300
300
200
200
5
100
5
100
0
0
-5
-5
-5
-5
5
5
(a)
(b)
Fig. 7. Irradiance distribution on the cell for the LPI’s FK concentrator with Cg = 625x, f/1, no
AR coating on SOE, when the sun is on axis and the solar spectrum is restricted to: (a) the topsubcell range (360-690 nm), and (b) the middle-subcell range (690-900 nm).
This result may seem surprising, since it is known that a conventional Fresnel lens has
considerable chromatic aberration (which certainly does affect the CAP*, as already
discussed). In the FK concentrator, however, the chromatic aberration of the Fresnel lens
barely affects the irradiance, thanks to its Kohler integration basis: White light that is split by
the Fresnel lens into a range of different focal points will be refocused at the same point of the
cell, after refraction by the SOE.
Another interesting feature is the angular intensity distribution of the light reaching the
cell. Figure 8 shows the cumulative distribution of light power on the cell with off-normal
angle. When the sun is perfectly tracked, the cell is entirely illuminated within a cone of ± 30°
or less from normal, and within a cone of ± 40° for a tracking error of 1°. Since solar cell
reflectivity is usually higher at large incidence angles (>50-60°) than at normal incidence, the
intensity profile of the FK concentrators guarantees optimum light coupling into the cell.
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Encircled power
100
90
80
70
60
50
40
30
20
10
0
Perfectly tracked sun
Sun miss-tracked 1 deg
0
10
20
30
40
50
60
70
80
90
Angle with the normal to the cell (deg)
Fig. 8. Cumulative distribution of light power received on the cell with off–normal angle for
the LPI’s FK concentrator with parameters Cg = 625x, f/1, when the sun is perfectly tracked
and when the tracking error is 1° (this design has α = ± 1.43 deg).
4. Comparison with other Fresnel-based designs
Comparing the FK concentrator with other five more conventional CPV concentrators designs
that use flat Fresnel lens as a primary, as shown schematically in Fig. 9. All the concentrators
will have the same POE entry aperture area (625 cm2), and concentration ratio will be varied
(and thus cell size will vary accordingly). Both the Fresnel lens and the solar cell are square.
Fig. 9. Diagonal cross-section of the Fresnel-based concentrators considered in the comparison.
SOE’s and cells are not to scale, but the concentrator depth to POE diagonal ratio is. From left
to right: Fresnel (no SOE), Spherical dome, SILO, XTP, RTP, FK concentrator.
From left to right in Fig. 9, first is a Fresnel lens concentrator with no secondary, which is
a type of system being used for instance by Concentrix Solar [10]. This concentrator, if
designed for maximum CAP at acceptance angles significantly higher that the angular radius
of the solar disc, would produce a highly non-uniform illumination (it performs approximately
as an imaging concentrator; see Fig. 2(c)&(d)). Therefore, this concentrator must sacrifice
CAP to reduce the peak concentration to moderate enough values.
Second is a hemispherical glass dome centered on the cell surface and in optical contact
with it. The purpose of the hemispherical centered dome is to increase the CAP, since it
produces an optical linear magnification about 1.5 (the refractive index of the SOE), i.e., the
cell is optically made 1.5 times bigger. Higher magnifications are possible by decentering the
spherical dome, but then the optical efficiency will be lower due to higher Fresnel reflections.
In any case, the compromise between CAP and peak concentration also apply for these domed
designs. This type of dome concentrator has been considered as a candidate for improving the
CAP of Concentrix design [11].
Third, we selected for the comparison the SILO secondary [12], which was proposed by
Sandia Labs in the late 80’s and later commercialized by Alpha Solarco Inc [13]. It comprises
a conventional rotational Fresnel lens as the primary and a single-surface rotationally
symmetric lens as a secondary which images the primary onto the cell. Thus it resembles the
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Kohler integration idea, but it is made of a single unit (instead of four). The fourth SOE of
Fig. 8 is a hollow reflective truncated pyramid (XTP), which is the type of SOE being used by
Amonix [15] and Guascor Foton [16]. And finally, the fifth is the dielectric-filled truncated
pyramid (RTP), which works by total internal reflection. This type of SOE is used in several
commercial products [17].
With the exception of SILO, we have selected concentrators that are being used in current
commercial products, or that have been considered as next-generation designs. We have
included the SILO in spite of it not being commercialized already, because it is conceptually
related to our FK concentrator.
The secondaries and cells shown in Fig. 9 are not to scale, but the concentrator ratio of
system height to POE diagonal (f-number) are, with Table 2 listing their numerical values.
The true-scale relative size comparison among the different SOE’s can be seen in the crosssectional drawings of Fig. 10. All these concentrators have the same POE entry aperture area
(625 cm2) and the same acceptance angle (α = ± 1°).
Table 2. Ratio of the distance between the cell and Fresnel lens to the Fresnel lens
diagonal (i.e., f-number) of the selected Fresnel-based concentrators under comparison
Concentrator
f-number
No SOE
1.5
Spherical dome
1.5
SILO
0.8-1.5
XTP
1.3
RTP
0.85
FK
0.8-1.5
Fresnel-RTP
55
50
45
40
35
30
Fresnel-XTP
25
20
SILO
15
FK concentrator
10
Spherical
Dome
5
-25
-20
-15
-10
-5
0
(mm)
5
10
15
20
25
Fresnel (no SOE)
Fig. 10. Cross section of the Secondary Optical Elements of the Fresnel-based concentrators of
Fig. 8. All these concentrators have the same POE entry aperture area (625 cm2) and the same
acceptance angle (α = ± 1 deg). The cross section of their corresponding cells, which should be
centered at the origin, are shown displaced downward to make them visible.
In the next sections, ray-trace simulations of these concentrators will be presented. We
have performed them for all but the SILO, for which the data has been calculated from the
information in [12]. All the concentrators have been given the following parameters: (i)
Fresnel lens: area = 625 cm2, made of PMMA (n≈1.49), facet draft angle = 2°, vertex radius =
3 µm, facet height<250 µm; (ii) Dielectric SOE: made of BK7 glass (n≈1.51) [14] with edge
radii 300 microns; (iii) Reflective SOE: made of first-surface silver coating on an aluminum
sheet; (iv) conformal coating on the solar cell of transparent silicone rubber of n≈1.41, which
protects the cell and adheres it to the dielectric SOE (when this applies); (v) High-efficiency
(≈38%) commercial triple-junction cell, with the external quantum efficiencies provided by
the manufacturer and considered to be independent of the incidence angle on the cell.
Absorption in dielectric materials and Fresnel reflections are considered, but not surface
scattering.
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4.1 Optical efficiency
The optical efficiency (both monochromatic and polychromatic) of the types of concentrator
under comparison is very similar, except when the f-number is very low. This occurs in the
RTP, which has approximately a 4% lower absolute efficiency than the others. This
calculation neglects the fact that when no dielectric SOE is used (i.e., when no SOE is used
and the XTP) the conformal coating may reduce the apparent reflectance of the cell grid-lines,
due to light trapping by total internal reflection on the conformal coating-air interface. The
amount of light trapping depends on the grid-lines’ surface properties and geometry, and its
quantification therefore depends on the specific grid line technology.
There are still some aspects not accounted for by the ray traces, which may affect optical
efficiencies in more refined models. First, total internal reflection (TIR) is sensitive to surface
scattering, dirt and surface corrosion of the glass of the RTP secondary (with otherwise 100%
efficiency). Second, perfect application of the silicone coupler is assumed, but imperfections
are difficult to avoid in the case of the RTP secondary in low-cost high-volume production (a
further discussion is given in Section 73H5).
It is also noticeable that the average ray path length inside the RTP secondary is about
twice as long as in the tolerance-equivalent FK concentrator SOE. This reduces the material
candidates to be used for the RTP to a few highly transparent glasses (as BK7), while a wider
variety of materials is available for the SOE of the FK concentrator.
4.2 Concentration-acceptance angle product
0.7
0.7
0.6
0.6
CAP * perfect AR
CAP * no AR
Figure 11 shows the effective concentration acceptance angle product CAP* for the selected
concentrators versus the f-number in the case that no AR coating is applied to the SOE’s. We
can see that the FK concentrator is superior to all of them, being the RTP the second
concentrator in the ranking.
0.5
0.4
0.3
0.2
FK
Spherical dome
Fresnel-XTP
0.1
SILO
Fresnel-RTP
No SOE
0.5
0.4
0.3
0.2
0.1
0
FK
Spherical dome
Fresnel-XTP
SILO
Fresnel-RTP
No SOE
0
0.8 0.9
1
1.1 1.2 1.3 1.4 1.5
f /#
0.8 0.9
1
1.1 1.2 1.3 1.4 1.5
f /#
Fig. 11. Effective concentration acceptance angle product (CAP*) for the concentrators under
comparison when the SOE has no AR coating SOE (left) and when a perfect AR coating is
assumed (right). The curves of the LPI’s FK concentrator are the same as those shown in Fig. 6.
Since we have stipulated that the entry aperture of the Fresnel lens is the same for all
concentrators (625 cm2), we will say that two concentrators are tolerance-equivalent if they
have the same effective acceptance angle α*. Table 3 shows that the FK’s superior CAP*
makes it able to concentrate significantly more than the any other configuration with the same
tolerance. This was previously illustrated when representing the cells size in Fig. 10, which
corresponds to the first row in Table 3.
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Table 3. Geometrical concentration at which the concentrators under comparison are
tolerance-equivalent (α* = ± 1°). For this table the f-number of the FK and SILO has
been fixed to 1.0 and 1.2, respectively.
Concentrator
Cg (no AR on SOE)
Cg (perfect AR on SOE)
No SOE
104
104
Spherical dome
257
257
SILO
248
278
XTP
425
425
RTP
677
677
FK
1,057
1,210
4.3 Cost
Table 3 lists the concentration for effective acceptance angle α* of ± 1° of the different
configurations of Fig. 9. Since the different configurations have the same aperture area and
acceptance angle, it is safe to assume that cost savings will come only from reduced cell area
and reduced SOE costs. The key question for the FK concentrator is to evaluate whether the
reduced cell cost allowed by its higher concentration capability is balanced by the cost of its
secondary. This question also arises for the other concentrators, so it is interesting to compare
all of them in terms of cost per POE unit area (which is representative if neglecting their
differences in optical efficiencies; see Section 4.1). For the FK concentrator and SILO, the fnumbers were fixed to 1.0 and 1.2, respectively, for this cost calculation.
For the cost model we have assumed the same Fresnel lens price for all the designs (their
area is the same, 625 cm2, and all can be made with the same technology). We have also
considered that the encapsulation cost is common to all designs, and that the rest of the
module (heat spreader, by-pass diode, electrical connection, heat sink, housing, etc.) is also
common to all concentrators. Since all of them have the same acceptance angle we have
considered that the cost of module assembly, array installation, structure and tracker design
are the same for all the cases. Therefore, the module cost will differ only in the cost of the
triple-junction cell plus the cost of the SOE. We have not used real quotations from suppliers
for such costs, but rather estimations of high-volume market prices in volume.
For the triple-junction cell, all the concentrators use the same cell technology and the cell
cost is proportional to its active area, with areal cost $7/cm2. This proportionality probably is
not true for small enough cells (area below 30-50 mm2), because the non-active area of the
cell becomes relatively larger and because the higher cost to test and bin the individual cells.
Since, however, we are considering in this exercise a POE of 625 cm2, cells smaller than 50
mm2 imply geometrical concentrations above 1250x, which is outside the range considered
here.
Among the different SOEs there is a clear distinction between reflective and refractive
elements. The cost of the XTP reflexive SOE has been considered proportional to its reflective
area, with an areal cost of $0.004/cm2. For the dielectric SOEs, the cost assumes no AR
coating, and the estimated values are shown in Fig. 12 as a function the SOE height h. Dometype SOEs refers to FK, SILO and spherical-dome configurations (for this cost estimation, the
outer part of the spherical dome, which is not active, is assumed to be truncated to make its
volume comparable to that of the FK and SILO).
Figure 13 shows the addition of the SOE and solar cell costs per unit area of the Fresnel
lens as a function of the effective acceptance angle α*. The comparison between different
concentrators should be made at the same effective acceptance angle α* (i.e., they are
tolerance-equivalent).
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Dielectric SOE cost ($)
4
3
2
RTP
1
Dome-type SOE's
0
20
30
40
50
60
70
80
Height (mm)
Fig. 12. Estimated SOE cost as a function of its height for the RTP SOE and for the Dome-type
SOE’s (FK, SILO and spherical dome). The dots correspond to the SOE’s of Fig. 14.
83HFor instance, if we specify α* = ± 1°, the FK concentrator has an SOE + cell cost
around $110/m2, while for the RTP, the XTP, and the SILO the costs are $150/m2, $200/m2,
and $340/m2, respectively. Note that this is because the cell in the FK is smaller (thanks to its
higher CAP*). It is remarkable that if no SOE is used the cell cost is $670/m2 (outside the
graph!) for α* = ± 1°, because to achieve such an α* the Cg must be as low as 104 (for this
reason systems with no SOE usually are designed for α* values smaller than ± 1° [10], which
increase the costs of assembly, installation, etc.).
SOE + cell cost per POE aperture area unit ($/m2)
XT
P
Sph
250
No
SOE
300
e ri
cal
do
me
SIL
O
350
P
RT
200
150
I’
LP
100
0.25
0.5
0.75
1
s
c
FK
on
r
ato
r
t
n
ce
1.25
1.5
Effective acceptance angle α* (deg)
Fig. 13. SOE and solar cell cost per Fresnel lens unit area as a function of the effective
acceptance angle α*. The SOE’s have no AR coating.
Figure 14 shows the SOE and active area of the cells in the two most cost-effective cases
with α* = ± 1°: the FK and RTP concentrators. Both concentrators have the same POE area
(625 cm2) and the cost advantage of the FK is mainly due to the smaller cell active area
required (shown as blue squares): 92 mm2 in the RTP versus 59 mm2 in the FK concentrator.
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Fresnel-RTP concentrator
SOE+cell cost ≈ $150/m2
LPI’s FK concentrator
SOE+cell cost ≈ $110/m2
Fig. 14. Cost comparison (per POE unit area) of the set formed by the SOE (green) and solar
cell (blue squares) between the RTP and FK with the same POE area (625 cm2) and the same
effective acceptance angle α* = ± 1°. The cost advantage of the FK resides on the smaller cell
active area needed: 92 mm2 in the RTP versus 59 mm2 in the FK concentrator. The costs of
these two SOE’s are highlighted with spots in Fig. 12.
The previous cost calculations considered that the refractive surfaces of the dielectric
SOEs are not AR coated. Since both optical efficiencies (Section 3.1 and 4.1) and CAP*
(Sections 3.2 and 4.2) are improved in the FK, it is expected that the AR coating application
will be cost effective, but this calculation will not be considered here. Note that since not only
the CAP* but efficiency is improved, its cost effectiveness cannot be computed just per unit
area of the POE, but the total system cost ($/W) needs to be known as well.
4.4 Irradiance
When comparing the irradiance distribution of the different selected concentrators, we can
group the FK concentrator, RTP, and SILO as devices producing good homogenization, and
the XTP, spherical dome, and the case without SOE as devices producing poor
homogenization, especially the last two when their design is done to maximize CAP*. This is
the case illustrated for the spherical dome in Fig. 15(a), which is the irradiance distribution
obtained for the Cg = 625x centered spherical dome designed for maximum CAP* (that gives
0.266) referred to in Fig. 11. The peak irradiance is ≈6,000 suns (@DNI = 850 W/m2), which
seems rather high but might be acceptable [2]. It will be probably be somewhat lower in
practice, however, (this ray-trace model does not include scattering and optical manufacturing
errors), but is instructive to compare it with the FK concentrator of the same Cg = 625x (Fig.
15(b)), with an irradiance plateau of ≈460 suns (@DNI = 850 W/m2). Note that it is also quite
remarkable that their effective acceptance angles are very different: α* = ± 0.61° for the
spherical dome and α* = ± 1.30° for the FK concentrator.
The irradiance homogeneity in XTP, spherical dome, and the case without SOE can be
improved by an adequate Fresnel lens design, but that improvement is done at the expense of
reducing the CAP*.
Regarding spectral inhomogeneity of the irradiance, the same group is weak: XTP,
spherical dome, and the case without SOE (especially for low f-numbers). Even though they
can be designed to minimize that aberration at the nominal POE to cell distance, the balance is
rapidly lost when that distance is modified (as occurs if the Fresnel lens parquet shows
warping or bowing). On the other hand, the FK concentrator, RTP, and SILO are not that
sensitive to the variation of that distance.
To complete this comparison, another irradiance related issue occurs inside the SOE. For
the cases of the RTP and SILO, the Fresnel lens produces a region of high solar radiation
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density near the top of the SOE (see schematic ray-trace of Fig. 9). In the early 90’s Alpha
Solarco reported problems of degradation of the SILO glass due to UV absorption by
undesired particle traces (solarization) [13]. By now glass technology has improved, and this
risk may be already solved, but even so the 4-fold FK concentrator divides that risk by four
(since the high radiation is split in the four units of the SOE).
6000
6000
5000
5000
4000
4000
3000
3000
2000
2000
5
1000
5
1000
0
0
-5
-5
-5
-5
5
5
(a)
(b)
Fig. 15. Irradiance distributions, in suns (1 sun = 1kW/m2), on the cell for the (a) spherical
dome with f/1.5 and the (b) f/1 FK concentrator (this is the average of the graphs in Fig. 7).
Both have geometrical Cg = 625x; and α* = ± 0.61° for the spherical dome and α* = ± 1.30°
for the LPI’s FK concentrator.
5. Practical considerations for mass-production
There are several features of the SOE of the FK concentrator that make it advantageous with
respect to other dielectric SOE’s that use flow-line type [9] total internal reflectors, as for
instance the RTP.
First, all the surfaces of the RTP are optically active (see Fig. 16). This makes it complex
to hold it without introducing optical losses and compromising the mechanical stability (some
CPV manufacturers hold the RTP from the joint to the cell, which compromises the
requirements for the RTP-cell coupling material). In contrast, a large portion of the SOE
surface has no optical function, and consequently can be used to make that mechanical fixing
and to introduce alignment features.
Another problematic issue is that the optical coupling of the cell and the SOE is very
critical for the RTP, because lateral spillage of the silicone rubber causes significant optical
losses from leakage through it. If to avoid spillage the joint is under-filled, the resulting air
gap produces optical losses too. These losses cannot be quantified until full production is
achieved. This particularly discourages the use of RTP for small cells (<5mm side). In the FK
concentrator, however, the overflow does not affect the optical performance, which greatly
simplifies the joint coupling. Particularly, the optical coupling and mechanical fixturing
functions can be totally separated in the FK concentrator.
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refraction
Every surface is
optically active
Total
Internal
Reflection
(TIR)
refraction
Only these
surfaces are
optically active
Lost ray
Silicone optical coupler
Fig. 16. Practical aspects of the R-TP secondary versus the LPI’s FK secondary.
6. Conclusions
The FK concentrator is a reliable and robust high-performance device that allows mass
production thanks to its high tolerance and practical manufacturability aspects. The FK optical
surfaces, from the manufacturing point of view, are very similar to a conventional flat Fresnel
lens and a conventional dome. This means that they can be manufactured with the same
techniques (continuous roll embossing, hot embossing, compression molding, etc. for the
POE; glass molding for the SOE) and that their production cost is essentially the same, but
their optical performance (CAP*) is much better.
The FK concentration acceptance angle products, CAP and CAP*, are the highest among
the concentrators based on flat Fresnel lenses considered here, and are also superior to other
designs reported in [19]. However, two designs using mirrors as POE have been reported with
a higher CAP than the FK [18]. The irradiance uniformity obtained by the FK concentrator is
excellent, without the chromatic aberration so typical of other Fresnel concentrators. This
minimizes the losses associated with such inhomogeneities, while diminishing the reliability
risks associated with elevated local concentrations. It is also remarkable that the FK
concentrator maintains its high CAP* for rather small f-numbers, which is very interesting if
compactness of the CPV module is desired.
The FK concentrator is in our opinion an excellent candidate to make low-cost highconcentration CPV modules that combine very high electrical efficiency at the array level (and
not only at the single cell level) with cost-effective assembly and installation.
The free-form Köhler array solutions are not limited to the present FK concentrator. Even
for the flat Fresnel lens as a primary, multiple options for the secondary are envisaged for
future improvements. The true innovation in these new designs is that they are free-form
Köhler integrating arrays. This degree of freedom enables the design of optical surfaces that
can perform different functions at the same time (improving the device performance without
affecting its cost). This allows good irradiance uniformity and high tolerance angle at high
concentration values.
Acknowledgments
The LPI’s FK device presented in this paper is protected under US and International patents
pending by LPI, LLC, 2400 Lincoln Avenue, Altadena, CA 91001 USA http://www.lpillc.com/. The authors thank the Spanish Ministry MCI (project SIGMASOLES PSE-4400002009-8) and the Madrid Regional Agency IMADE (project PIE/469/2009) for their partial
support. The authors also thank Bill Parkyn for his help in editing the paper.
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