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Energy Express

  • Editor: Bernard Kippelen
  • Vol. 18, Iss. S2 — Jun. 21, 2010
  • pp: A126–A132
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Evolutionary algorithm for optimization of nonimaging Fresnel lens geometry

N. Yamada and T. Nishikawa  »View Author Affiliations


Optics Express, Vol. 18, Issue S2, pp. A126-A132 (2010)
http://dx.doi.org/10.1364/OE.18.00A126


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Abstract

In this study, an evolutionary algorithm (EA), which consists of genetic and immune algorithms, is introduced to design the optical geometry of a nonimaging Fresnel lens; this lens generates the uniform flux concentration required for a photovoltaic cell. Herein, a design procedure that incorporates a ray-tracing technique in the EA is described, and the validity of the design is demonstrated. The results show that the EA automatically generated a unique geometry of the Fresnel lens; the use of this geometry resulted in better uniform flux concentration with high optical efficiency.

© 2010 OSA

1. Introduction

Currently, concentrating photovoltaics (CPV) is one of the key technologies being used to promote the use of photovoltaic electricity. The main purpose of CPV is to enable the use of a low-cost concentrator for concentrating solar radiation, which is then supplied to a highly efficient (high-cost) photovoltaic (PV) cell with a small cell area. Selecting the best concentrator and designing an optimal optical system is still an important issue, and hence, concentrators including mirrors and/or lenses have been studied [1

1. A. Antonini, M. A. Butturi, P. Di Benedetto, D. Uderzo, P. Zurru, E. Milan, M. Stefancich, M. Armani, A. Parretta, and N. Baggio, “Rondine® PV concentrators: field results and developments,” Prog. Photovolt. Res. Appl. 17(7), 451–459 (2009). [CrossRef]

5

5. A. W. Bett, C. Baur, F. Dimroth, G. Lange, M. Meusel, S. van Riesen, G. Siefer, V. M. Andreev, V. D. Rumyantsev, and N. A. Sadchikov, “FLATCONTM-modules: technology and characterization,” in Proceedings of IEEE Conference on WCPEC-3, (IEEE, 2003), pp. 634−637.

]. A nonimaging Fresnel lens is one of the most frequently used optical elements in CPV-based concentrators because of its high optical efficiency, small thickness, and low cost of fabrication using plastics. However, it is fairly difficult for designers to optimize the geometry of a nonimaging Fresnel lens. The optimization is generally achieved by adopting a time-consuming trial-and-error method.

With this background, the objective of this study was to introduce a versatile optimization method to overcome the abovementioned problem. The evolutionary algorithm (EA), by performing numerous calculations, automatically generates the best possible optical geometry that would yield the desired performance. The advantage of the present method is demonstrated by presenting an example of the optimization of a linear nonimaging Fresnel lens for achieving uniform flux concentration.

2. Evolutionary algorithm

3. Optimization of nonimaging Fresnel lens by EA

3.1 Model and method

One of the problems associated with the nonimaging Fresnel lens is insufficient uniformity of energy flux distribution on a PV cell because of chromatic aberration [11

11. R. Leutz, and A. Suzki, Nonimaging Fresnel Lenses Design and Performance of Solar Concentrators (Springer Verlag, 2001).

]. This problem often necessitates the addition of a secondary optical element such as a light-guiding glass plate, compound parabolic concentrator, or homogenizer to the system by coupling the element with the PV cell [12

12. R. Winston, J. Miano, and P. Bentez, Nonimaging Optics (Academic Press, 2005)

,13

13. M. Victoria, C. Domínguez, I. Antón, and G. Sala, “Comparative analysis of different secondary optical elements for aspheric primary lenses,” Opt. Express 17(8), 6487–6492 (2009). [CrossRef] [PubMed]

]. In this study, the authors attempted to optimize the geometry so as to achieve uniform concentration without using a secondary optical element. The proposed model of the nonimaging Fresnel lens is shown in Fig. 2
Fig. 2 Model of nonimaging Fresnel lens: Half of the lens geometry is omitted because of symmetry.
. The specifications of the model are listed in Table 1

Table 1. Specifications of Nonimaging Fresnel Lens

table-icon
View This Table
. Only a cross section of the lens is optimized to elucidate the potential of the present method. Furthermore, the calculation is simplified by assuming that no volumetric absorption loss occurs inside the lens. The size of the model was decided to be similar to that of the Fresnel lens found in the literature [5

5. A. W. Bett, C. Baur, F. Dimroth, G. Lange, M. Meusel, S. van Riesen, G. Siefer, V. M. Andreev, V. D. Rumyantsev, and N. A. Sadchikov, “FLATCONTM-modules: technology and characterization,” in Proceedings of IEEE Conference on WCPEC-3, (IEEE, 2003), pp. 634−637.

]. The ray-tracing method was used to evaluate the optical performance of geometries, i.e., to calculate the evaluation index of geometries. The propagation of rays follows Snell’s law and Fresnel equations. In the proposed model, the incident angle range is set to θin = ± 0.25°, which is the same as that in a solar disk. It should be noted that the present optimization was carried out by taking into account the effects of three representative refractive indices: n = 1.4790 (λ = 1.20 μm), 1.4835 (0.80 μm), and 1.5095 (0.40 μm); these indices were selected on the basis of the optical data for PMMA, which is a dominant material used in the fabrication of the Fresnel lens.

3.2 Evaluation index

4. Results

EA optimization was carried out by using a standard personal computer. For comparison, the conventional nonimaging Fresnel lens designed using the methodology proposed by Leutz [6

6. D. R. Reid and G. S. Smith, “Design and optimization of Fresnel zone plates using a genetic algorithm and a full electromagnetic simulator,” Microw. Opt. Technol. Lett. 51(9), 2223–2227 (2009). [CrossRef]

] was included in the generation of the initial geometries. The relationship between E and the generation (iteration) step for F = 0.3 is shown in Fig. 4
Fig. 4 Evaluation index E vs. generation (iteration) step for F = 0.3. E is a nondimensional value. E = 0.60 for the conventional Fresnel lens. E = 0.75 for the EA-generated Fresnel lens at the 100th generation. The ideal value of E is 1.0.
. The calculation was terminated at the 100th generation. The value of E for the EA-generated Fresnel lens before the first generation is same as that for the conventional Fresnel lens (E = 0.60). As expected, the value of E increased with the generation step and became E = 0.75 at the 100th generation.

Figure 5
Fig. 5 Energy distribution on PV cell surface of (a) conventional Fresnel lens, (b) EA-generated Fresnel lens at the 100th generation for F = 0.3, and (c) EA-generated Fresnel lens at the 100th generation for F = 0.5. λ = 0.40 μm (blue), λ = 0.80 μm (green), and λ = 1.20 μm (red); the average of these three wavelengths is shown in black. The corresponding optical efficiency is (a) η = 84.0%, (b) η = 82.7%, and (c) η = 77.4%.
shows the energy distribution on the PV cell surface (between points A and B shown in Fig. 2). The concentration for three wavelengths, i.e., for the three refractive indices, is plotted; the average concentration for the three wavelengths is also determined. Figure 5(a) shows the energy distribution for the conventional Fresnel lens; this distribution represents a typical energy distribution influenced by chromatic aberration. In contrast, Fig. 5(b) and (c) shows the energy distribution for the EA-generated Fresnel lens at the 100th generation for F = 0.3 and F = 0.5, respectively. It can be clearly observed that the average energy distribution in the EA-generated Fresnel lens is more uniform than that in the conventional one. The uniformity of the energy distribution at each wavelength in the EA-generated lens is also better than that in the conventional one. For F = 0.5, a highly uniform concentration is observed. The optical efficiency of the conventional Fresnel lens is η = 84.0%, whereas that of the EA-generated Fresnel lens is η = 82.7% (F = 0.3) and 77.4% (F = 0.5). Optical efficiency-uniformity trade-off relation was examined by tuning F value and is clearly represented in Fig. 7
Fig. 7 Optical efficiency - uniformity trade-off by tuning F value 0.1 - 0.9. Result for a wider incident angle θin = ± 0.5° is additionally plotted.
. Additional result for a wider incident angle θin = ± 0.5° was also plotted in Fig. 7. It is obvious that the wider acceptance angle resulted in the lower optical efficiency. Although a trade-off between uniformity and optical efficiency is unavoidable, one can obtain the best possible geometry with a minimal decline of optical efficiency by choosing an appropriate value of F. It is unlikely that there exists a geometry that has an optical efficiency much higher than that of the present EA-generated geometry for the same uniformity; however, it is difficult to definitively confirm this fact. Because the actual amount of electricity generated depends on the characteristics of the installed PV cell, the value of F should be selected according to the cell characteristics.

Figure 6
Fig. 6 Geometry of nonimaging Fresnel lens. (a) Conventional Fresnel lens, (b) EA-generated Fresnel lens at the 100th generation for F = 0.3, and (c) EA-generated Fresnel lens at the 100th generation for F = 0.5. The y-scale is reduced to 1/10th of the original scale.
shows the geometry of the conventional and EA-generated Fresnel lenses corresponding to the energy distribution shown in Fig. 5. The geometry illustrated in this figure is a top-ranked geometry among 200 EA-generated geometries rated by index E at the 100th generation. The top 20 geometries differ slightly from each other. It should be noted that the y-scale in Fig. 6 is reduced to 1/10th of the original scale. The actual slopes of these geometries are gentler than those shown in Fig. 6. The EA-generated Fresnel lens has an irregular geometry; however, the authors confirmed that a Fresnel lens can still be fabricated using this geometry because limits can be imposed on the prism angles of the EA-generated Fresnel lens such that the lens can be practically fabricated using techniques such as pressing. By using EA optimization, such fabrication limits can easily be set on the automatically generated geometry.

5. Conclusion

The EA optimization method developed in this study can be used to automatically generate a Fresnel lens geometry with uniform flux concentration, without adopting trial-and-error procedures and without relying on considerably difficult theoretical approaches. The authors conclude that the EA is one of the powerful and versatile methods that can be used to select the best possible geometry according to one’s requirements. For example, one can apply this method to determine the best possible geometry that yields not only uniform concentration but also a wider acceptance angle. Furthermore, more complicated systems with multiple optical elements, e.g., a Fresnel lens combined with a secondary or higher optical element, can also be optimized by using this method. The generation of a three-dimensional geometry would enable the design of a wide range of practical solar concentrators. The effectiveness of the method used for the generation of such a three-dimensional geometry can be increased with the use of supercomputing.

Acknowledgments

This work was partially supported by the Japan Science and Technology Agency under the FY2009 program Research for Promoting Technological Seeds (No.05-B09).

References and links

1.

A. Antonini, M. A. Butturi, P. Di Benedetto, D. Uderzo, P. Zurru, E. Milan, M. Stefancich, M. Armani, A. Parretta, and N. Baggio, “Rondine® PV concentrators: field results and developments,” Prog. Photovolt. Res. Appl. 17(7), 451–459 (2009). [CrossRef]

2.

A. Mohr, T. Roth, and S. W. Glunz, “BICON: high concentration PV using one-axis tracking and silicon concentrator cells,” Prog. Photovolt. Res. Appl. 14(7), 663–674 (2006). [CrossRef]

3.

G. Peharz and F. Dimroth, “Energy payback time of the high-concentration PV system FLATCON®,” Prog. Photovolt. Res. Appl. 13(7), 627–634 (2005). [CrossRef]

4.

M. Yamaguchi, T. Takamoto, and K. Araki, “Super high-efficiency multi-junction and concentrator solar cells,” Sol. Energy Mater. Sol. Cells 90(18–19), 3068–3077 (2006). [CrossRef]

5.

A. W. Bett, C. Baur, F. Dimroth, G. Lange, M. Meusel, S. van Riesen, G. Siefer, V. M. Andreev, V. D. Rumyantsev, and N. A. Sadchikov, “FLATCONTM-modules: technology and characterization,” in Proceedings of IEEE Conference on WCPEC-3, (IEEE, 2003), pp. 634−637.

6.

D. R. Reid and G. S. Smith, “Design and optimization of Fresnel zone plates using a genetic algorithm and a full electromagnetic simulator,” Microw. Opt. Technol. Lett. 51(9), 2223–2227 (2009). [CrossRef]

7.

W. G. Chen, C. M. Uang, and C. H. Jou, “Optimal design of an irregular Fresnel lens for multiple light sources using a three-layered Hierarchical Genetic Algorithm,” Opt. Express 15(16), 9918–9935 (2007). [CrossRef] [PubMed]

8.

W. Chen and C. Uang, “Better reading light system with light-emitting diodes using optimized Fresnel lens,” Opt. Eng. 45(6), 063001 (2006). [CrossRef]

9.

N. Yamada, and T. Ogawa, “Optimization method of concentrating optics by using evolutionary algorithm,” presented at the Second International Workshop on Concentrating Photovoltaic Optics and Power Darmstadt (2009), http://www.concentrating-pv.org/papers.html.

10.

L. Jiao and L. Wang, “A novel genetic algorithm based on immunity,” IEEE Trans. Syst. Man Cybern. 30(5), 552–561 (2000). [CrossRef]

11.

R. Leutz, and A. Suzki, Nonimaging Fresnel Lenses Design and Performance of Solar Concentrators (Springer Verlag, 2001).

12.

R. Winston, J. Miano, and P. Bentez, Nonimaging Optics (Academic Press, 2005)

13.

M. Victoria, C. Domínguez, I. Antón, and G. Sala, “Comparative analysis of different secondary optical elements for aspheric primary lenses,” Opt. Express 17(8), 6487–6492 (2009). [CrossRef] [PubMed]

OCIS Codes
(220.1770) Optical design and fabrication : Concentrators
(350.6050) Other areas of optics : Solar energy
(220.4298) Optical design and fabrication : Nonimaging optics

ToC Category:
Solar Concentrators

History
Original Manuscript: March 25, 2010
Revised Manuscript: May 14, 2010
Manuscript Accepted: May 16, 2010
Published: May 19, 2010

Citation
N. Yamada and T. Nishikawa, "Evolutionary algorithm for optimization of nonimaging Fresnel lens geometry," Opt. Express 18, A126-A132 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-S2-A126


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References

  1. A. Antonini, M. A. Butturi, P. Di Benedetto, D. Uderzo, P. Zurru, E. Milan, M. Stefancich, M. Armani, A. Parretta, and N. Baggio, “Rondine® PV concentrators: field results and developments,” Prog. Photovolt. Res. Appl. 17(7), 451–459 (2009). [CrossRef]
  2. A. Mohr, T. Roth, and S. W. Glunz, “BICON: high concentration PV using one-axis tracking and silicon concentrator cells,” Prog. Photovolt. Res. Appl. 14(7), 663–674 (2006). [CrossRef]
  3. G. Peharz and F. Dimroth, “Energy payback time of the high-concentration PV system FLATCON®,” Prog. Photovolt. Res. Appl. 13(7), 627–634 (2005). [CrossRef]
  4. M. Yamaguchi, T. Takamoto, and K. Araki, “Super high-efficiency multi-junction and concentrator solar cells,” Sol. Energy Mater. Sol. Cells 90(18–19), 3068–3077 (2006). [CrossRef]
  5. A. W. Bett, C. Baur, F. Dimroth, G. Lange, M. Meusel, S. van Riesen, G. Siefer, V. M. Andreev, V. D. Rumyantsev, and N. A. Sadchikov, “FLATCONTM-modules: technology and characterization,” in Proceedings of IEEE Conference on WCPEC-3, (IEEE, 2003), pp. 634−637.
  6. D. R. Reid and G. S. Smith, “Design and optimization of Fresnel zone plates using a genetic algorithm and a full electromagnetic simulator,” Microw. Opt. Technol. Lett. 51(9), 2223–2227 (2009). [CrossRef]
  7. W. G. Chen, C. M. Uang, and C. H. Jou, “Optimal design of an irregular Fresnel lens for multiple light sources using a three-layered Hierarchical Genetic Algorithm,” Opt. Express 15(16), 9918–9935 (2007). [CrossRef] [PubMed]
  8. W. Chen and C. Uang, “Better reading light system with light-emitting diodes using optimized Fresnel lens,” Opt. Eng. 45(6), 063001 (2006). [CrossRef]
  9. N. Yamada, and T. Ogawa, “Optimization method of concentrating optics by using evolutionary algorithm,” presented at the Second International Workshop on Concentrating Photovoltaic Optics and Power Darmstadt (2009), http://www.concentrating-pv.org/papers.html .
  10. L. Jiao and L. Wang, “A novel genetic algorithm based on immunity,” IEEE Trans. Syst. Man Cybern. 30(5), 552–561 (2000). [CrossRef]
  11. R. Leutz, and A. Suzki, Nonimaging Fresnel Lenses Design and Performance of Solar Concentrators (Springer Verlag, 2001).
  12. R. Winston, J. Miano, and P. Bentez, Nonimaging Optics (Academic Press, 2005)
  13. M. Victoria, C. Domínguez, I. Antón, and G. Sala, “Comparative analysis of different secondary optical elements for aspheric primary lenses,” Opt. Express 17(8), 6487–6492 (2009). [CrossRef] [PubMed]

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