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Abstract
The optical characteristics of a radially symmetrical core-shell spherical (CSSP) lens made of practical materials were investigated for microtracking concentrator photovoltaic (MTCPV) system applications. The full spectrum analysis results showed that a polymethyl methacrylate (PMMA)-water CSSP lens exhibits a higher optical efficiency at a longer focal length and a wider acceptance angle in MTCPV systems than PMMA homogenous spherical lenses and other reported concentrator systems. The lens-cell module efficiency with a single-junction GaAs solar cell was estimated to be ∼24% by optical analysis. The efficiency can be further improved by employing low-refractive-index core materials with a high transparency over the solar spectrum range along with multi-junction solar cells.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recently, III-V solar cells have achieved an efficiency of 30.5% with a single junction [1] and 46.0% with a multi-junction under concentrated sunlight [2]. These are the highest values reported for solar cells. High-concentration concentrator photovoltaic (CPV) systems using III-V solar cells have achieved a module efficiency of 36.7% when installed on-site [3]. However, one of the issues of high-concentration CPVs is that they require solar trackers. Pedestal solar trackers are often used for utility-scale high-concentration CPV systems but are unsuitable for building-integrated photovoltaics (BIPVs) because of their bulky structure. In addition, if the density of the tracker array is high, the self-shadow reduces the capacity factor of the system [4]. To this end, a microtracking CPV (MTCPV) system, wherein a tracking mechanism is integrated inside the module structure, has been proposed [5]. MTCPV systems can be mounted on fixed slopes or on walls and thus are suitable for BIPVs. MTCPV systems can be designed with various structures such as (a) two upper lenses on a solar-cell stage [6], (b) an upper lens and a waveguide on a solar-cell stage [7–14], (c) a solar-cell stage inserted between an upper lens and an under-mirror [15–17], (d) a resin-filled mirror under a solar-cell stage [18], (e) an upper wide angle aplanatic lens and a solar-cell stage [19], and (f) a gradient-index (GRIN) lens [20,21] and a solar-cell stage [22,23]. In most cases, the lens or the solar-cell stage is laterally or three-dimensionally actuated to ensure a constant focusing of the concentrated sunlight onto the solar cell. The recently reported prototype systems having the structures (c), (d), and (e) have experimentally demonstrated high module conversion efficiencies of up to ∼30% with acceptance half angles ranging from 20 to 60°. (f) is only a theoretical representation; the design requires a very low refractive index (RI) material, which cannot be easily found. The RI of MgF2, which is a typical low-RI solid material, is 1.39, which is even higher than that of water [24]. Porous materials can be used as low-RI materials [25–27]; however, the pore size of the material for visible light must be lower than several hundred nanometers [25]. Finding or fabricating a practical low-RI GRIN material is difficult.
High-concentration CPVs cannot harvest diffuse sunlight because of the theoretical limit of the solar concentration. This limits the usage of high-concentration CPVs only to areas that are rich in direct sunlight. To solve this issue, CPV/PV hybrid modules have been recently studied [28–31], in which diffuse sunlight is captured by an inexpensive solar cell, such as an Si solar cell, which is additionally placed around the III-V solar cell. To integrate this concept into an MTCPV system, the structures (e) and (f) are preferable owing to their transparency for diffuse sunlight. In the structures (c) and (d), even if an additional solar cell is used to capture the diffuse sunlight, the direct sunlight will be blocked by the additional solar cell and thus cannot be concentrated onto the III-V solar cell.
Herein, an alternative focusing system for the structures (e) and (f) is reported. A core-shell spherical (CSSP) lens made of practical materials, namely polymethyl methacrylate (PMMA) and water, is demonstrated, and its optical performance for GaAs solar cells in a MTCPV system is analyzed by conducting ray-tracing simulations. Typical spherical lenses generally exhibit spherical aberrations that aggravate ray convergence, i.e., optical efficiency, whereas a GRIN lens can overcome the spherical aberration problem. A CSSP lens is a type of simplified spherical GRIN lens composed of only two RI layers. The concept of CSSP lenses has been reported [32,33]. A preliminary optical simulation study on CSSP lenses for MTCPV systems suggested that CSSP lenses having a core with an RI lower than that of the shell enable longer focal lengths with smaller spherical aberrations [34]. Moreover, a spherical lens has no angle dependence in principle. These characteristics can improve the optical performance of MTCPVs, particularly at high angle of incidence (AOI). To clarify this, in this study, the solar concentration performance of the CSSP lens with the practical low-cost materials was investigated by conducting a full spectrum optical analysis. The results are compared with those of a homogenous PMMA spherical lens and other concentrator systems used in MTCPV systems.
2. MTCPV concept
Figure 1 shows the tracking motion of a MTCPV system consisting of a spherical lens array and a solar-cell stage. The solar-cell stage moves to ensure a constant focusing of the concentrated sunlight onto the solar cell. Unlike aspheric lenses, the concentration performance of spherical lenses is independent of the AOI, as long as the concentrating rays do not interfere between the adjacent lenses. In this configuration, a long focal length is desirable to achieve a wide tracking angle and a high optical efficiency. If the focal length is short, the solar-cell stage will hit the lens array at a low AOI, and sunlight having a high AOI cannot be tracked. Figure 2 shows the schematic of the mechanical angle limit θlimit. Figure 2 shows the schematic of the mechanical angle limit θlimit, which is defined as the angle when the top end surface of the solar-cell stage contacts the bottom end surface of the spherical lens array. Clearly, a longer focal length f shown in Fig. 2(a) enables a wider θlimit than a shorter focal length f shown in Fig. 2(b). In addition, a shorter focal length would increase the reflection loss at the solar cell surface as the solar cell must receive the concentrated rays at a high incidence angle.
Fig. 2. Mechanical angle limit of MTCPV with a spherical lens for long focal length (a) and short focal length (b).
3. PMMA–water CSSP lens
Figure 3 shows the optical simulation model of the CSSP lens. This model comprises a high-RI-material shell and a low-RI-material core. A three-dimensional ray-tracing simulation was performed using LightTools 8.5.0, in which the Fresnel reflection loss and wavelength dependence of the RI and absorptivity were considered for the entire solar spectrum range. The receiver is a blackbody with a radius of 0.5 mm and a geometrical concentration ratio of 100. The shell radius was maintained at 5 mm, and the core radius rcore was varied between 0 and 5 mm. The light source emits rays with a divergence angle of ±0.256° (view angle of a solar desk) perpendicular to the solar cell. The optical efficiency is defined as the ratio of the concentrated rays on the receiver to the incident rays on the lens aperture. The ray-tracing is carried out for each wavelength, and the energy of the rays is determined by integrating over the considered spectrum range. For each geometry, the focal length f that gives the highest optical efficiency was determined by sweeping f.
In the CSSP lens, the combination of high RI for the shell and low RI for the core affects the optical performance. The RI of solid glass ranges from 1.45 to 1.85 at 589.3 nm [35], whereas that of various plastics are as follows: PMMA: 1.491; PC: 1.585; and PS: 1.592 at 587.6 nm [36]. In contrast, many liquids have a low RI: water: 1.333 at 550 nm [37]; methanol: 1.3288 at 532 nm; acetonitrile: 1.3422 at 532 nm [38]; and fluorocarbon liquid FC72: 1.248 at 630 nm and 30 °C [39]. PMMA and water were selected as the practical materials for the MTCPV system owing to their availability, ease of fabrication, low cost, less bubble formation by evaporation, and safety. Water evaporation can be suppressed by controlling the pressure of the encapsulated water. Water generally shows a high absorptivity for infrared light; thus, it is not suitable as a concentrator material when multi-junction solar cells are used. The optical absorption of water in the IR range is due to hydrogen bonding, whereas in the VIS-IR range, it is related to atomic bond vibration. To use multi-junction solar cells, the core material should not exhibit atomic bonding vibration in the VIS-IR range. Fluorocarbon liquids do not have atomic bonds for absorption and are transparent in the wavelength range of 230–7000 nm. Several organic solvents exhibit only narrow-band absorption because of C-H bonds. However, these materials have problems such as the incompatibility with resins, evaporation, and ozone depletion. Exploring better low-RI materials to improve the conversion efficiency of MTCPV systems with CSSP lenses and multi-junction solar cells remains an issue. Therefore, in this study, the feasibility of a PMMA–water CSSP lens for a single-junction GaAs solar cell was investigated as a first-step approach.
Figure 4 shows the spectral absorptivity of a 1-cm-thick layer of water [37], the external quantum efficiency (EQE) of a reported single-junction GaAs solar cell EQEλ having a cell conversion efficiency of 30.5% under 258 suns [1], and air mass (AM) 1.5D standard direct solar spectrum Iλ. This graph indicates that the water absorptivity when λ ≤ 900 nm is below 10%. However, when λ ≥ 925 nm, the water absorptivity rapidly increases. The typical band gap of a GaAs solar cell is 1.43 eV, corresponding to λ = 867 nm. The high EQE range of the GaAs solar cell and the high solar spectrum intensity range are consistent with the low absorptivity range of water.
Fig. 4. Spectral absorptivity of a 1-cm-thick layer of water αλ, EQE of the GaAs solar cell EQEλ, and AM 1.5D standard direct solar spectrum (ASTM G173) Iλ.
Figure 5 shows the results of the ray-tracing simulation performed under the conditions listed in Table 1. Simulation condition in two spectrum ranges, i.e., 280–2000 nm (full solar spectrum range) and 280–867 nm (in which the GaAs solar cell has a high EQE), by considering the optical constants of PMMA and water [37]. ηopt-2000 and f2000 and ηopt-867 and f867 represent the highest optical efficiency and the corresponding focal length for the ranges of 280–2000 nm and 280–867 nm, respectively. The results strongly depend on rcore. rcore = 0 indicates that the lens is homogenous and spherical and is made of only PMMA. Both ηopt-2000 and ηopt-867 present a valley when 2 < rcore < 3, and a second peak is observed when rcore = 3.7 mm and f = 8.7 mm for which ηopt-867 reaches 79.8%, which is greater than ηopt-2000. The highest value is obtained for ηopt-2000 when rcore = 0 (homogenous) and f = 5.9 mm; however, the efficiency is 73.0%, which is lower than 79.8%. The results indicate that the PMMA–water CSSP lens achieves a higher optical efficiency at a longer focal length than the PMMA homogenous spherical lens in the spectrum range of 200–867 nm owing to less volumetric absorption loss. For rcore < 2.4 mm, ηopt-867 and ηopt-2000 are approximately identical. This is because the incident light does not pass through the small core material (water) but through the highly transparent shell material (PMMA) over the solar spectral range.
Fig. 5. Simulated optical efficiency and the corresponding focal length for different core radii of the PMMA–water CSSP lens.
Table 1. Simulation conditions
4. Estimation of solar module efficiency
4.1 Short-circuit current density of a GaAs solar cell
The results, given in Section 3, indicate that the transparency of the PMMA–water CSSP lens is suitable for the GaAs solar cell. In this section, the short-circuit current density of the GaAs solar cell with the PMMA–water CSSP lens is estimated using the following equation.
Figure 6 shows the simulated spectral optical efficiencies ηopt,λ for three rcore cases in a λ range of 300–1300 nm and an f range of 5–10 nm. The optical efficiency is highest in the λ range of 450–600 nm when rcore = 3.7 mm. A lower rcore gives a higher optical efficiency at shorter f. The f exhibiting high optical efficiency at short wavelengths is slightly lower than that at long wavelengths because of the higher RI of water at short wavelengths. The optical efficiency when λ > 900 nm decreases because of the absorption loss by water.
Fig. 6. Spectral optical efficiencies for different focal lengths when rcore = 3.5 mm (a), 3.7 mm (b), and 4.0 mm (c).
Figure 7 shows the contour of Jsc calculated using Eq. (1) for different values of rcore and f. Jsc was calculated with the simulated ηopt,λ (Fig. 6) taking β = 1. As a result, the value of Jsc is highest, i.e., 22.9 mA/cm2, when rcore = 3.7 mm and f = 8.7 mm. The contour reveals that the region giving over 90% of the highest Jsc is in the ranges of − 0.1 ≤ f ≤ +0.7 mm and 0 ≤ rcore ≤ +0.2 mm as the deviation from the highest point.
Fig. 7. Estimated short-circuit current density of a GaAs solar cell Jsc with a PMMA–water CSSP lens for various core radii rcore and focal lengths f.
4.2 Open-circuit voltage, fill factor, and module efficiency
To predict the lens-cell module efficiency, in addition to Jsc, the open-circuit voltage Voc and the fill factor F.F. must be estimated. In [1], Voc and F.F were found to be 1.23 V and 0.861, respectively, at 258 suns for a GaAs solar cell. Another study reported the performance of a GaAs solar cell for different solar concentrations [40]; Table 2 lists the results at 1, 110, and 248 suns. By a linear approximation of the values from 110 to 248 suns, which is close to 100 suns (this study) and 258 suns ([1]), Voc and F.F. for the present lens was estimated to be 1.23 × (1.115 / 1.137) = 1.21 V and 0.861 × (86.64/86.74) = 0.860, respectively. Thus, the lens-cell module efficiency of the present lens with rcore = 3.7 mm and f = 8.7 mm was found to be 23.8% by multiplying Jsc, Voc, and F.F. This efficiency is regarded as a first approximation. For more accurate prediction, F.F. should be determined by an electrical simulation with a solar cell circuit model that can incorporate the effect of light intensity distribution at the solar cell surface [41–43].
Table 2. Reported Power Generation Performance of GaAs Solar Cells Under Varied Solar Concentration [40]
5. Angular optical efficiency in an MTCPV system
Although the spherical lens has no angular dependency in principle, the optical efficiency varies with the AOI because the angle of the concentrated sunlight incident on the solar cell varies in the MTCPV system, as shown in Fig. 1. In this section, the AOI dependency of ηopt-867 is analyzed using the same ray-tracing model described in Section 3. Figure 8 shows the angular optical efficiency of the PMMA–water CSSP lens (rcore = 3.7 mm) in comparison with that of a PMMA homogenous spherical lens (rcore = 0 mm). The upper contours show the AOI dependency of ηopt-867 for various f. The bottom graphs show the AOI dependency of ηopt-867 for specific focal lengths (f = 5.8, 8.7, and 9.2 mm). The CSSP lens with f = 9.2 mm exhibits the widest acceptance angle of ±57.7°, and the homogenous spherical lens (f = 5.8 mm) exhibits an acceptance angle of ±30.4°. This reveals that the CSSP lens achieves a wider acceptance angle at longer f. This characteristic contributes to an increase in θlimit of the MTCPV system, as shown in Fig. 2.
Fig. 8. Angular optical efficiency of a PMMA–water CSSP lens ((a) and (c)) in comparison with that of a PMMA homogenous spherical lens ((b) and (d)). Upper graphs: contours for different AOIs and focal lengths. Bottom graphs: characteristics for specific focal lengths.
To compare the present result with those reported on MTCPV systems, we selected reports that satisfied the following conditions: (i) The AOI dependency of the optical efficiency or short-circuit current of a solar cell is measured and/or simulated, and (ii) the concentration ratio exceeds 50 suns. Table 3 lists such reports. (b1), (b2), (c1), (c2), (d), (e) correspond to the type of optical system described in Section 1(b)–(e), respectively. The mechanical angle limits of the reported studies indicate the highest AOI in the measurement result.
Table 3. List of the Reported MTCPVs in which AOI Dependency is Measured and/or Simulated
Figure 9 shows the AOI dependency of the reported MTCPV systems in comparison with the present study. The solid lines indicate the simulated values, and the markers indicate the measured values. The vertical axis represents the optical efficiency (or short-circuit current) normalized by the optical efficiency (or short-circuit current) at an AOI of 0° (ηopt_0). Although this comparison is not completely fair because the concentration ratio of each system is different, it may outline the characteristics of each system. The structure (d) shows the highest optical efficiency until ∼35°, though the concentration ratio is relatively low. The present CSSP lens and the structure (e) exhibit the second highest optical efficiency in the widest AOI range, outperforming the structure (c2). The structure (c1) exhibits the highest optical efficiency in the range of 35 − 55° although slightly underperforming in 10 − 30°. The structure (b1) and (b2) exhibits the lowest performance, probably because of the angular characteristic of the total internal reflection in the waveguide. Because the MTCPV system with the CSSP lens and (e) have a more simpler structure than (c1), (c2), and (d), they are better candidates for further research and development.
6. Conclusions
An optical analysis of a CSSP lens made of practical materials was conducted for MTCPV applications. The PMMA–water CSSP lens exhibited a higher optical efficiency, a longer focal length, and a wider acceptance angle than the conventional PMMA homogenous spherical lenses and some of the reported concentrator systems. Therefore, the CSSP lens has a good potential for MTCPV applications. However, the estimated lens-cell module efficiency with a GaAs single-junction solar cell was ∼24%, which is not significant compared with the reported module efficiency of MTCPV systems with multijunction solar cells (∼30%). The use of a multijunction solar cell helps obtain higher module efficiencies. This requires low-RI core materials with a high transparency over the entire solar spectrum range. Improving the optical efficiency is another solution toward achieving a higher module efficiency. The optical efficiency can be improved by shrinking the size of the spherical lens and/or by installing a secondary optical element (SOE) on the solar cell surface to capture more rays and direct them toward the solar cell. SOEs have been frequently used in conventional CPVs [44] but have not been designed for the present MTCPV concept. Furthermore, the PMMA–water CSSP lens and tracking concept can be applied to other emerging optical applications such as optical wireless power transmission (OWPT). An experimental demonstration of OWPT with a GaAs solar cell resulted in a cell conversion efficiency of 52.7% at a laser wavelength of 824 nm and a laser beam irradiance of 4 suns [45]. The present study reported a high optical efficiency (∼80%) at the same wavelength.
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