Modelling of Compound Parabolic Concentrators for Photovoltaic Applications

In this paper ways of using compound parabolic concentrators as primary optical elements for concentrated photovoltaics are evaluated. The problems related to these classical non-imaging optical elements for photovoltaics applications have been evaluated by modelling different types of linear and point focus concentrators. Particular consideration is given to the issues of manufacturability and cost. The non-uniformity of the flux resulting at the concentrator exit aperture has been considered and some solutions are proposed in order to reduce adverse effects on performance, as well as to increase the angular tolerance of the system.


Introduction
Concentrator photovoltaics (CPV) systems [1,2] in use today can be divided , in first instance, into two main categories: Fresnel lens refractors and parabolic reflectors. Both can be either point focus (3D) or linear focus (2D) concentrators. The concept of the co mpound parabo lic concentrator (CPC) as a primary concentrator has received some attention in the field of build ing integrated PV, but only for lo w concentration (<5x) non-tracking applications [1][2][3] w it h few e xcep t io ns [4,5]. Fo r s o lar t rackin g applicat ions, CPCs o ffer the possib ility o f h igh so lar concentration ratios, in princip le approaching the theoretical limits [6,7]. Ho wever, one of the largest hurdles in the use of CPCs fo r p rimary opt ics in PV con cent rato rs is their unwieldy character and the necessary high material usage. This can in part be offset by reducing the length of the CPCs with the so called truncated CPCs, or T-CPCs, which use far less material with on ly a mino r reduct ion in concentration ratio and optical efficiency [6,8]. Despite this imp rovement , the su rface area o f the p rimary opt ical component remains high co mpared to a lens or a parabolic mirror. In this paper, so me new possibilit ies for cheap and easily manufactured CPCs will be d iscussed, as alternative of the more diffused concentrators based on lenses or parabolic troughs for the medium and mediu m-high levels of concentration positioned on trackers for large scale, field applications.
CPCs offer some technical advantages: compared to a classic parabolic reflector, a CPC can be used with a less precise tracking system, due to the flat optical efficiency response, opening up the possibility of using cheaper commercial trackers not normally suitable for CPV; moreover, co mpared to a Fresnel lens, the optical efficiency of a CPC is higher. The best designed lenses currently available show optical efficiencies <90% [9,10], wh ile the performances of CPCs can be limited with good approximation only by the reflectiv ity of the optical surface; indeed, the smoothness of the CPC's surface helps to strongly reduce the manufacturing defects limit ing mo re complex, structured designs. Therefore, optical efficiency can be higher than 90% with advanced reflective films or coatings, such as those discussed in this paper. Additionally, some of these materials permits to filtering unwanted portions of the solar spectrum, wh ich is advantageous in minimizing cooling requirements for the solar cell.
In common with most high concentration PV systems, the use of a flux ho mogenizer could be considered. As discussed in this paper, the flu x profile at the outlet aperture of a CPC is highly non-uniform, and therefore the impact on cell performance for a concentrator cell can be deleterious. The design of the homogenizer suited to a CPC is discussed.

Background
Descriptions of the CPC began appearing in literature in the mid-1960s [11,12]. As described by [6], the CPC was used for many different applications, ranging from high-energy physics to solar energy collection. In the field of solar energy, CPCs have mainly been used in solar thermal applications, most common ly as static linear collectors focusing light onto evacuated tubes at low concentration (~1.5x). There are applications where CPCs are used as the primary concentrator with photovoltaic cells, and other where they have been considered as secondary, non-imaging concentrator stage for some PV concentrator systems. Some projects have looked at the use of CPC troughs for combined PV-thermal (PV/T) applications [13]. In Sweden, Brogren firstly exp lored the use of CPCs for PV/T applications that require water for space heating [14], then further investigated [15,16].
One of the advantages that CPCs offer with respect to conventional imag ing systems (parabolic mirrors and some Fresnel lenses) is their higher tolerance to misalign ments with respect to the sun disk direct ion. Since CPCs approach the behaviour of ideal concentrators, their optical efficiency can be kept closed to unity up to the acceptance angle with a reduction factor for the entrance flu x of only the cosine of the misalign ment angle. As a consequence, for a given optical concentration ratio, they show the largest acceptance angle. The requirement on tracking accuracy is therefore lower, co mpared to other concentrators with the same concentration ratio.
Most Fresnel lens systems concentrate light onto single solar cells with a point focus approach, rather than onto dense arrays of series connected cells. The significant advantage of this approach is that the problems of cell current mis match are largely avoided (cells will still need to be series connected with other cells to build voltage, but, if the optical efficiency of each lens is the same, then cell currents should also be well matched). Single cells are able to tolerate a reasonably high degree of light non-uniformity, however, as discussed by [17,18], there can be a reduction in efficiency. In addition, when lenses for high concentration are used in conjunction with mu lti-junction cells, the effect of the non-uniformity can be a mo re serious problem because of the different light deflections for the different wavelengths converted by the cells in stack [19]. This problem is avoided for concentrators using reflective optics. Secondary flu x ho mogenisers can be emp loyed to give near uniform light distribution on the cells. They are frequently used for both lens systems [19,20] and parabolic d ishes [21][22][23]. The simp lest flu x ho mogenisers are rectangular boxes with reflect ive sidewalls (i.e. a kaleidoscope). Solid blocks made of plastic or glass, using the principles of total internal reflection, may realize the same design. However, care must be taken to avoid melting due to strongly focused spots of concentrated light.

CPC Design
The CPCs can be designed to concentrate light in either two or three d imensions. Obviously, the 2D-CPC has a lower concentration factor. According to [6], they can be designed following the Eq.s (1,2), as a function of the concentration ratio C(ND), the required acceptance angle θi and the refract ive index nout of the material at the exit aperture, for a CPC with ND d imensions: where L represents the length of the concentrator, ni is the refract ive index of the med iu m at the entry, (usually air, i.e. with n i = 1), ain and aout are the entrance and exit apertures radii respectively, as illustrated in the standard representation of Fig. 1. In o rder to utilize the advantage given by the refractive index at the outlet nout, it is important to have the solar cell in optical contact with a transparent, dielectric material with n> 1 as, for examp le, silicone; the interface should be matched to min imize the reflection losses at the receiver front surface. For PV, and in general for all the energy production applications, it is important to reduce the cost of the system to a min imu m; therefore, it is reasonable to consider CPCs filled with air rather than with materials capable to ensure higher concentration factors (with refract ive indexes greater than one). Even with this assumption, the highest theoretical limits for optical concentration in air is fairly high: ~216x for 2D concentrators, and ~46,000x for 3D concentrators [6].

2D Concentrator Systems
2D-PV concentrator systems have been extensively studied, both theoretically and experimentally, with both reflecting mirrors [24,25] as well as with lenses [26,27]. The 2D-CPCs are not common ly used in PV applications because the length of the two parabolic reflective walls appears to be excessive for large scale purposes. For example, fo r a concentration factor of 30x on a 4-cm wide cell (the same size used in the EUCLIDES pro ject [24]), the length of an ideal CPC co llector results 19m long. Even an halved-CPC is too long fo r any p ractical applications. The 2D-CPC is an ideal concentrator in terms of light concentration factor for a given acceptance angle, but, for PV applications, the ideal characteristics for the optical Parabola 2 L efficiency are not strictly required. The necessity for the optical systems, in fact, is to operate at an incident angle range for the imp inging rad iation at which its efficiency is the highest. In general, for the CPCs, the enhancing of the concentration factor leads to an increasing of the object length; besides, the higher the concentration ratio, the lower the angular acceptance of the system. The shortening of the CPC involves a s mall loss of concentration and a small gain of angular tolerance, if the truncation is produced in the region of the parabola where the sloping is lower, i.e. fro m the entrance aperture. So, it is possible to design a truncated CPC concentrator far shorter than the ideal one for a given concentration factor, reducing the length of an ideal CPC of higher concentration ratio and of lower angular acceptance, achieving a structure with higher angular acceptance respect to the ideal one and considerably shorter. Eq.
(3) defines a T-CPC, with the main parameters given in Fig 2, as exhaustively described in [2].
To avoid the problem of excessively large lenses and long focal d istance, PV lens concentrators typically consist of a number of small modules rather than a single large lens. CPCs designs could also be suitable to such a configuration. If very narrow solar cells were used for a 2D CPC, the length of the collector would be suitable for industrial fabrication technologies, and for tracking systems similar to those currently used for lens arrays. Suitable cells for th is purpose are, for examp le, the concentrator Sliver™ cells developed at the Australian Nat ional University. Sliver cells have a width of about 1mm, and could work efficiently under a concentration factor of about 30x [28,29].  2) shows that an increase in the refract ive index n out increases the optical performances of the concentrator. By partially filling the evacuated solid, the object can accept rays otherwise rejected, due to the refraction of light at the air-silicone interface. This effect is shown in the ray traces in Fig. 3. The figure shows the ray trace close to the exit aperture of a truncated 2D-CPC, 15-cm long, with an exit total aperture 2×a out = 1 mm and a concentration factor of 30×, for a ray beam misaligned at 0.6° and with the solar angular divergence of 0.26°. Fig. 3a shows the outlet of the concentrator without the dielectric, wh ile Fig. 3b shows the ray trace of the same rays when the concentrator is filled with a material with refractive index n = 1.49 (i.e. PMMA for λ = 600 n m) for a length of 25 mm starting fro m the exit. The latter configurat ion is able to tolerate misalign ment up to 0.6°. The structure behaves as a simp lified form of a two-stage CPC [3], while the surface curvature of the object is like that of a single CPC.  The partially filled, truncated CPC can be analysed as a two stage CPC, where the first stage is a T-CPC with a low exit angle θ out1 and with an exit material with n>1 (θ out1 ≅ 15° and n = 1.49 in the example o f Fig. 3b), and the second stage is another T-CPC with an exit angle θ out2 ; this last exit angle can be selected a little lo wer than 90°, in order to achieve the higher level of concentration for the selected angular acceptance. Because of the rays outgoing with the higher exit angle are the rays incoming with the higher angle of incidence respect to the optical axis of the system, the θ out1 corresponds at the inlet angular acceptance for the second stage. The acceptance angle is here defined as the highest entrance angle for wh ich all the light is transferred to the exit aperture.
As the truncation of the considered objects reduces their lengths, the incident, acceptance angle θ i has a smaller value for a given concentration factor C than the case of two ideal, longer CPCs, series connected. The incidence acceptance angle θ i,ideal for two comp lete CPCs series connected is derived fro m the relationship given in Eq. (4), with an assumed total concentration factor C. Consequently, the transmission-angle curve hasn't a cut off angle for incident beams in correspondence of the acceptance value as for ideal concentrators, but it has a slope for θ>θ i , as shown, for the considered case, in Fig. 4.
This kind of concentrator, because of its particular form, requires protective glass at the inlet aperture, to avoid the detrimental effect of dirty deposition on the large concave area. This element could be positioned on the co mplete structure, with an antireflect ion coating on it, usually acting as self-clean ing surface as well, to reduce the optical losses for the Fresnel reflection at its interfaces. Considering the different cases of presence of uncoated dielectric surfaces, the optical efficiencies obtained by simu lation with the software TracePro Where i i ,φ θ represent the angles of incidence for the incoming rad iation, in spherical coordinates, while s s ,φ θ are the angles indicating the scattering direction. L s is the scattered radiance, while E i is the incident irradiance. This optical property has been introduced to consider the slight effect of the light diffusion at the reflector surfaces. Because of the Fresnel reflection at the interfaces of materials of different refract ion indexes, portion of the incident light flu x is back reflected at the interfaces of the protective glass and of the encapsulant; this factor of losses, common with every concentrator system using lenses, can be strongly reduced depositing an antireflection layer on the surfaces, which are, in this case, all planar.
The very thin and long illu minated area of this proposed design has the additional advantage of a very high perimeter/surface ratio for the PV device, which permits to cool down the cells using passively, maximizing the thermal spreading effect at the receiver level.
The necessity of flu x unifo rmity on a single cell significantly depends on the particular kind of cell emp loyed; indeed, the cell size, the contact pattern and coverage, the doping levels and the external circu it configurat ions play all an important role. In the supposed case of Sliver cell used for the 2D-CPCs system, there's a fairly high tolerance for non-uniformity on the device, because the emitter contact is placed on the side of the device, and its small dimension is in the direction perpendicular to the inco ming radiat ion, which is of the order of the electrons diffusion length, for Si with lifetime higher than 200µs. For sy mmetrical reasons, the uniformity along the long dimension of the device is ensured, so uniform light could be expected along the string of series connected cells. A flu x profile along the short side of the cell is graphed in Fig. 5 Table 1. Optical efficiency of a reflective, 2D-CPC of 30× concentration factor, for different misalignment angles and for different characteristics of fabrication: (a) no protective glass and no encapsulant; (b) no protective glass, 25mm of encapsulant, without ARC; (c) 5mm of protective glass and 25 mm of encapsulant, without ARC; (d) 5mm of protective glass, 25mm of encapsulant and single MgF2 ARC layer on each interface. The specular reflectance of the surfaces adopted is 94.87%, while glass and encapsulant have been modelled with refraction index n = 1.49

3D Concentrator Systems
In the case of 3D-CPCs it's possible to consider a system assembly similar to that used with a 3D-lenses concentrator. An illustration of an array of these 3D-CPCs objects is shown in Fig. 6. One important characteristics for PV applications of these 3D concentrators is the very high non-homogeneity in the spatial flu x distribution produced at the exit aperture, as shown, for examp le, in Fig. 7, for a truncated CPC with a concentration ratio of 115x and a length of 30 cm, with an incident radiation directed along the optical axis of the concentrator, with the solar angular distribution of 0.26°.
A method to correct this effect is to emp loy a light mixer to redistribute the light on the exit area. To achieve this result it is necessary to break the symmetry of the system as described in [19,32]. The strong non-linearity introduced by these changes of geometry produces a chaotic behaviour in the determin istic path of the rays. A well known method is the use of a kaleidoscope with squared section and reflective walls at the CPC outlet. Depending on the mixer unit length it is possible to achieve different levels of uniformity for the illu mination flu x on the target area. For practical purposes it is important to find a t rade-off between the length of the kaleidoscope and the level of flu x unifo rmity; indeed, using a non-ideal reflector, the optical losses introduced by each reflection on the mixer walls significantly reduce the concentrator optical efficiency. Moreover, if the kaleidoscope and a portion of the CPC is filled with a dielectric, as previously described for 2D-CPCs in order to increase the angular acceptance of the concentrator, a material with a very low absorption coefficient has to be selected. Considering a reflector with a 94.87% of specular reflectance, 5% absorbance and 0.13% of integrated BRDF as before, and a dielectric with the PMMA optical properties which co mpletely fills the 3-cm long kaleidoscope and fills the CPC outlet for 1.4 cm, the simulated perfo rmances are reported in Tab. 2, for different incident angles of a beam with the solar d ivergence. Fro m the results in the Tab. 2, the energy loss due to mult iple reflect ions at the kaleidoscope walls is evident. Indeed, the fraction of inco ming rays achieving the exit aperture is close to 1 (column 4), but a significant part of the radiation energy is absorbed, even for a fairly good reflector with the characteristics specified before. The variation in the flu x uniformity as function of the mixer length for normal incidence of the solar radiat ion for the truncated CPC unit is considered without dielectric filling, and is reported in Fig. 8.   Table 2. Optical efficiency and fraction of collected rays for the 115×, 3D-CPC with a 3-cm long kaleidoscope at the exit aperture, for different misalignment angles; the results are considered without and with the partial filling of the output of the structure with a transparent dielectric with n = 1,49. The specular reflectance of the surfaces adopted in the model is 94.87%

Misalignment angle (°)
Optical efficiency (Without partial filling of dielectrics)  The variat ion of the optical efficiency of the concentrator with the kaleidoscope length, for the reflector of the described properties, is reported in Fig. 10a for the cases of partial filled and of empty objects; diversely, the correspondent transmission-angle curves are in Fig. 10b.
To reduce the length of the mixer, a structured surface with V-shaped grooves can be employed, as described by Leutz [33]. Such a design increases the chaotic behaviour of the light rays path, working as an efficient mixing tricks to permit a length reduction. Another imp rovement of the optical efficiency can be achieved for structures with a lower concentration factor; indeed, in these cases, the average exit angle for the rays is lower and consequently also the number of reflection on the kaleidoscope walls. Ho wever, in o rder to achieve a high optical efficiency for real 3D objects the solution adopting a metal coated reflective kaleidoscope does not seem effective. An alternative solution adopts a kaleidoscope made of a transparent dielectric material working for total internal reflect ions (TIR). In such a way this part of the structure doesn't give a performance reduction strongly related to its length as in the previous cases with metalized, reflective surfaces. In Tab. 3 the optical efficiency of the T-CPC 30-cm long with a 4-cm long kaleidoscope made of a material with the optical properties of highly transparent glass, coated with a single layer of MgF2 as antireflection, is reported fro m simu lations with the TracePro ® software.

Materials
Because of the particular geo metries required fo r the surface profiles, the fabrication of the structure can be done by plastic moulding. Co mputer controlled machining tools can work surface profiles with the CPCs curvature, with a precision level of 0.01mm; the smooth curvature required for these objects takes out the fabrication problems, own of Fresnel lenses, of achieving very sharp corners. The reflectance of the surface can be ensured by metallization with A l, Ag or applying reflective films; in any case the reflective coating must be properly covered with poly meric layers acting as protective barriers against moisture.
The large interest in high reflective, lo w cost materials for solar concentrator, both for PV as well as for thermal application has lead to a large body of literature on this issue. Reflective materials have very good optical properties, even for large scale and low cost production [34][35][36]. For the here modelled structures, both reflective adhesive films as well as evaporated metal coatings directly deposited on the concentrator surfaces can be evaluated. For the part icular geometries of the CPCs, the specular reflectance of the surfaces has to be evaluated at high angles of incidence for the light beam. Metallic reflectors have high insensitivity to the light impinging angle, as shown in the measured results in Fig. 11 for a glass coated with silver, tested for two different light wavelengths. The peak reflectance at higher angles in Fig. 11b is due to the Fresnel reflection. Nevertheless, mu lti-layer poly meric films also demonstrate very high reflectance for all the incidence angles [38].  Figure 11. Experimental results of specular reflectance for a silvered mirror at different angles of light incidence, for two different wavelengths, 543 nm (a) and 1063 nm (b). The measurements have been carried out at the glassed side of the mirror The transparent, dielectric material here used for the simu lations has refractive index n = 1.49. By varying the material it is possible to change the refractive properties in order to manage the angular acceptance.

Conclusions
The use of some CPC designs as primary concentrators for CPV has been described. Both 2D and 3D CPC structures have been evaluated and some particular solutions have been selected for possible photovoltaic applications. Historically, the large reflective area required for CPCs has limited their use to being secondary collectors or concentrators for low level of concentration, but, considering the very low price of currently available, high efficiency film reflectors, or the possibility of industrially coating small size structures with high reflective metals, this family o f optical objects can be considered as a competitive choice for CPV applications.
The industrial develop ment of very narrow linear concentrator cells has opened up the possibility of linear micro -concentrators. The part icular shape of this kind of cells is suitable for linear concentrators, where each cell represents an element of a string of cells along a trough. The small width of the cells allo ws the use of CPCs, a class of concentrators not normally emp loyed for large scale photovoltaic applications because of their intrinsically large dimensions, despite the fact that they have almost ideal non-imaging optical properties. By mov ing toward very small devices, it is possible to achieve concentrators of reasonable size with the inherent advantages of this class of optical object, i.e. their good tolerance at misalign ment errors and the possibility o f emp loying lo w cost but with very high reflective materials leading to high optical efficiency. Moreover, the very thin width of the cell permits efficient cooling at med iu m level concentration ranges, increasing the overall system efficiency.
3D-CPCs can be employed in the range of 100×, permitting very h igh optical efficiency (closed to 90%) for real devices produced with available industrial technology. The detrimental effect of the high non-uniformity in the light distribution at the target can be corrected with low optical losses, using a kaleidoscopic transparent dielectric material, acting for total internal reflections, working as light guide, and for mixing the radiation concentrated by the truncated CPC.