High performance Fresnel-based photovoltaic concentrator

Transcription

1 High performance Fresnel-based photovoltaic concentrator Pablo Benítez 1,2,*, Juan C. Miñano 1,2, Pablo Zamora 1, Rubén Mohedano 2, Aleksandra Cvetkovic 2, Marina Buljan 1, Julio Chaves 2, Maikel Hernández 2 1 Universidad Politécnica de Madrid (UPM), Cedint, Campus de Montegancedo 28223, Madrid, Spain 2 LPI, 24 Lincoln Ave., Altadena, CA 911 USA * pbenitez@cedint.upm.es 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. 21 Optical Society of America OCIS codes: (8.274) Geometric optical design; (3.6) Solar energy; ( ) Nonimaging optics; (22.177) 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, 24). 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), (29). 3. S. Kurtz, and M. J. O Neill, Estimating and controlling chromatic aberration losses for two-junction, twoterminal devices in refractive concentrator systems, 2th PVSC; pp , (1996). 4. W. Cassarly, Nonimaging Optics: Concentration and Illumination, in the Handbook of Optics, 2nd ed., pp , (McGraw-Hill, New York, 21). P. Benı tez, 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), (24). 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 , (2). 7. US and International patents pending by LPI, LLC, 24 Lincoln Avenue, Altadena, CA 911 USA 8. R. Leutz, and A. Suzuki, Nonimaging Fresnel Lenses, (Springer-Verlag, Berlin, 21). 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, 2) 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, (28). Note that in this reference Cg is defined using a circular active area instead of square, so the geometrical concentration 38x in it corresponds to 32x here. 12. L. W. James, Contractor Report SAND89 729, (1989). 13. D. Anderson, B. Bailor, D. Carroll, E. Schmidt, P. Tyjewski, M. Uroshevich, Alpha Solarco s Photovoltaic Development Concentrator Program, Contractor report SAND9 17, (199). 14. The same BK7 glass has been considered for all the SOE s under comparison. Though BK7 can be molded (see for instance, 8Hhttp:// is more common the use of, for instance, B27. The light absorption in B27 is slightly higher than in BK7, which causes that if the comparison is done using B27, the efficient of the RTP (whose optical path is longer) is penalized the most 9H. 1. 1Hhttp:// Hhttp:// 26 April 21 / Vol. 18, No. S1 / OPTICS EXPRESS A2

2 17. See, for instance: 12Hwww.sol3g.com, 13Hhttp:// and K. Araki et al., Development of a new X concentrator module with 3J cells-performance and. Reliability-, Proc. 31st IEEE PVSC, (2). 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, (29). 19. M. Victoria, C. Domínguez, I. Antón, G. Sala, Comparative analysis of different secondary optical elements for aspheric primary lenses, Opt. Express 17, (29). The design with highest CAP* in this reference has a rotationally-symmetric CPC-type TIR-based secondary. It achieved CAP* =.4 for a square aperture and square cell (for which the FK has a higher value of CAP* =.7.61). Moreover, it produces a poor irradiance uniformity. It also has the encapsulation problems discussed in Section. 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 (>) 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 sinα (1) Here C g 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 9% 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 ). It is remarkable that for a given concentrator architecture, the CAP is rather constant with C g. 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 C g, 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, () 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 2H), which encouraged us to further investigate Köhler integration, a well known method in illumination optics [4], usually, g 26 April 21 / Vol. 18, No. S1 / OPTICS EXPRESS A26

3 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 [] [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. b a c d c d b a Fresnel lens (POE) b a a d c b c d z Free form lens (SOE) z y x Solar cell x 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 26 April 21 / Vol. 18, No. S1 / OPTICS EXPRESS A27

4 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 2sinα s where α s is the solar radius, α s.26 ). 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 9 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 18 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. b c p c b b b 2sinα s 2sinα x x (a) (b) a d c a b d d p a b d c a x a a a p b b a x (c) (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 26 April 21 / Vol. 18, No. S1 / OPTICS EXPRESS A28

5 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<2 µm; (ii) SOE: made of BK7 glass (n 1.1) 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 C g = 62x and f/1, calculated at a single wavelength ( nm), the optical efficiency is 8%, 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 nm. Alternatively, an effective polychromatic optical efficiency can be defined as: conc conc conc { Isc, top Isc, middle Isc, bottom} {,,, } I min,, η = = (2) conc sc opt, polychrom 1 sun 1 sun 1 sun 1 sun Cg Isc Cg min Isc top, Isc middle, Isc bottom 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 C g, the AM1.d ASMT G173 spectrum, and the EQE of the sub-cells). The ratio of the currents I sc,top /I sc 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 nm and polychromatic) is negligible (<.1%) for both the uncoated 26 April 21 / Vol. 18, No. S1 / OPTICS EXPRESS A29

6 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 62x. Efficiency is slightly less for low f-numbers (at f/.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. Photocurrent density (A/nm) ) Top cell Middle cell Bottom cell Spectral optical efficiency Wavelength (nm) (a) Spectral optical efficiency (%) o ) Optical efficiency (%) Monochromatic; no AR Polychromatic; no AR Monochromatic; perfect AR Polychromatic; perfect AR f/# (b) Fig. 4. (a) Lossless spectral photocurrent densities and spectral optical efficiency of an LPI s FK concentrator with Cg = 62x and f/1, (b) Monochromatic (at 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 C g = 62x and f/1. Figure (a) shows the monochromatic (@ 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 9% values, which means that this concentrator has α = ± 1.43 without an AR coating on the SOE, and α = ± 1.2 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. (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 nm). All designs have been calculated for a geometrical concentration C g = 62x. The CAP is approximately constant in most of the f-number values (CAP.63 with no AR coating on the SOE, CAP.66 with perfect AR coating), dropping slightly for f/.8. 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. 26 April 21 / Vol. 18, No. S1 / OPTICS EXPRESS A3

7 Optical power on cell (normalized) ) ) monochromatic; perfect AR monochromatic; no AR Incidence angle (deg) Cell photocurrent (normalized) polychromatic; perfect AR polychromatic; no AR Tracking error angle (deg) (a) (b) Fig.. For an LPI s FK concentrator with C g = 62x and f/1: (a) monochromatic (@ 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.d 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. 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 9% of on-axis (see Fig. (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*.7 with no AR coating; CAP*.61 with perfect AR coating on the SOE). The good behavior of the FK concentrator for small f- numbers is very interesting when compact CPV modules are desired CAP. CAP *...4 CAP perfect AR CAP no AR..4 CAP* perfect AR CAP* no AR f /# 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 C g = 62x, 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 C g = 62x and α = ± 1.43 (which corresponds to the values CAP.63 mentioned before). If the FK concentrator is designed for C g = 1x, the resulting acceptance angle can be well estimated to be α = ± 1.43 (62/1) 1/2 = ± Table 1 summarizes other examples of such calculations. (This approximation is not accurate for α*< 26 April 21 / Vol. 18, No. S1 / OPTICS EXPRESS A31

8 ±.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. C g No AR coating on SOE Perfect AR coating on SOE α α* α α* 62 ± 1.43 ± 1.3 ± 1.2 ± , ± 1.13 ± 1.3 ± 1.2 ± 1.1 1, ±.92 ±.84 ±.98 ± Irradiance and intensity on the cell 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 = 62x, f/1, no AR coating of SOE, when the sun is on-axis with Direct Normal Irradiance (DNI) of 8W/m 2. The distribution on the left is the irradiance when the solar spectrum is restricted to the top subcell spectrum (36-69 nm), while the distribution on the right corresponds to the middle subcell spectral range (69-9 nm). a.u (a) - a.u (b) - Fig. 7. Irradiance distribution on the cell for the LPI s FK concentrator with C g = 62x, f/1, no AR coating on SOE, when the sun is on axis and the solar spectrum is restricted to: (a) the topsubcell range (36-69 nm), and (b) the middle-subcell range (69-9 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 ± 3 or less from normal, and within a cone of ± 4 for a tracking error of 1. Since solar cell reflectivity is usually higher at large incidence angles (>-6 ) than at normal incidence, the intensity profile of the FK concentrators guarantees optimum light coupling into the cell. 26 April 21 / Vol. 18, No. S1 / OPTICS EXPRESS A32

9 Encircled power Perfectly tracked sun Sun miss-tracked 1 deg 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 C g = 62x, 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 (62 cm 2 ), 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 [1]. 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. (the refractive index of the SOE), i.e., the cell is optically made 1. 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 8 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 26 April 21 / Vol. 18, No. S1 / OPTICS EXPRESS A33

10 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 [1] 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. 1. All these concentrators have the same POE entry aperture area (62 cm 2 ) 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 No SOE Spherical dome SILO XTP RTP FK f-number Fresnel-RTP Fresnel-XTP SILO Spherical Dome FK concentrator (mm) Fresnel (no SOE) Fig. 1. 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 (62 cm 2 ) 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 = 62 cm 2, made of PMMA (n 1.49), facet draft angle = 2, vertex radius = 3 µm, facet height<2 µm; (ii) Dielectric SOE: made of BK7 glass (n 1.1) [14] with edge radii 3 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. 26 April 21 / Vol. 18, No. S1 / OPTICS EXPRESS A34

11 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 1% 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 73H). 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 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. CAP * no AR FK.1 Spherical dome Fresnel-XTP SILO Fresnel-RTP No SOE f /# CAP * perfect AR FK SILO Spherical dome Fresnel-RTP Fresnel-XTP No SOE 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 (62 cm 2 ), 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. 1, which corresponds to the first row in Table April 21 / Vol. 18, No. S1 / OPTICS EXPRESS A3

12 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. and 1.2, respectively. Concentrator No SOE Spherical dome SILO XTP RTP FK C g (no AR on SOE) ,7 C g (perfect AR on SOE) , 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 f- numbers were fixed to 1. 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, 62 cm 2, 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/cm 2. This proportionality probably is not true for small enough cells (area below 3- mm 2 ), 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 62 cm 2, cells smaller than mm 2 imply geometrical concentrations above 12x, 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 $.4/cm 2. 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). 26 April 21 / Vol. 18, No. S1 / OPTICS EXPRESS A36

13 Dielectric SOE cost ($) RTP 1 Dome-type SOE's 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 HFor instance, if we specify α* = ± 1, the FK concentrator has an SOE + cell cost around $11/m 2, while for the RTP, the XTP, and the SILO the costs are $1/m 2, $2/m 2, and $34/m 2, 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 $67/m 2 (outside the graph!) for α* = ± 1, because to achieve such an α* the C g must be as low as 14 (for this reason systems with no SOE usually are designed for α* values smaller than ± 1 [1], which increase the costs of assembly, installation, etc.) SOE + cell cost per POE aperture area unit ($/m 2 ) No SOE SILO Spherical dome XTP RTP LPI s FK concentrator 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 (62 cm 2 ) and the cost advantage of the FK is mainly due to the smaller cell active area required (shown as blue squares): 92 mm 2 in the RTP versus 9 mm 2 in the FK concentrator. 26 April 21 / Vol. 18, No. S1 / OPTICS EXPRESS A37

14 Fresnel-RTP concentrator LPI s FK concentrator SOE+cell cost $1/m 2 SOE+cell cost $11/m 2 Fig. 14. Cost comparison (per POE unit area) of the set formed by the SOE (green) and solar cell (blue squares) between the RTP and FK with the same POE area (62 cm 2 ) and the same effective acceptance angle α* = ± 1. The cost advantage of the FK resides on the smaller cell active area needed: 92 mm 2 in the RTP versus 9 mm 2 in the FK concentrator. The costs of these two SOE s are highlighted with spots in Fig. 12. The previous cost calculations considered that the refractive surfaces of the dielectric SOEs are not AR coated. Since both optical efficiencies (Section 3.1 and 4.1) and CAP* (Sections 3.2 and 4.2) are improved in the FK, it is expected that the AR coating application will be cost effective, but this calculation will not be considered here. Note that since not only the CAP* but efficiency is improved, its cost effectiveness cannot be computed just per unit area of the POE, but the total system cost ($/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. 1(a), which is the irradiance distribution obtained for the C g = 62x centered spherical dome designed for maximum CAP* (that gives.266) referred to in Fig. 11. The peak irradiance is 6, suns (@DNI = 8 W/m 2 ), 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 C g = 62x (Fig. 1(b)), with an irradiance plateau of 46 suns (@DNI = 8 W/m 2 ). Note that it is also quite remarkable that their effective acceptance angles are very different: α* = ±.61 for the spherical dome and α* = ± 1.3 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 26 April 21 / Vol. 18, No. S1 / OPTICS EXPRESS A38

15 density near the top of the SOE (see schematic ray-trace of Fig. 9). In the early 9 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) (a) (b) - Fig. 1. Irradiance distributions, in suns (1 sun = 1kW/m 2 ), on the cell for the (a) spherical dome with f/1. and the (b) f/1 FK concentrator (this is the average of the graphs in Fig. 7). Both have geometrical C g = 62x; and α* = ±.61 for the spherical dome and α* = ± 1.3 for the LPI s FK concentrator.. 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 (<mm 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. 26 April 21 / Vol. 18, No. S1 / OPTICS EXPRESS A39

16 Total Internal Reflection (TIR) refraction Every surface is optically active Lost ray refraction Only these surfaces are optically active Silicone optical coupler 6. Conclusions Fig. 16. Practical aspects of the R-TP secondary versus the LPI s FK secondary. 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, 24 Lincoln Avenue, Altadena, CA 911 USA The authors thank the Spanish Ministry MCI (project SIGMASOLES PSE ) and the Madrid Regional Agency IMADE (project PIE/469/29) for their partial support. The authors also thank Bill Parkyn for his help in editing the paper. 26 April 21 / Vol. 18, No. S1 / OPTICS EXPRESS A4