Abstract

The two-stage line-to-point focus solar concentrator with tracking secondary optics is introduced. Its design aims to reduce the cost per m2 of collecting aperture by maintaining a one-axis tracking trough as the primary concentrator, while allowing the thermodynamic limit of concentration in 2D of 215× to be significantly surpassed by the implementation of a tracking secondary stage. The limits of overall geometric concentration are found to exceed 4000× when hollow secondary concentrators are used, and 6000× when the receiver is immersed in a dielectric material of refractive index n=1.5. Three exemplary collectors, with geometric concentrations in the range of 5001500× are explored and their geometric performance is ascertained by Monte Carlo ray-tracing. The proposed solar concentrator design is well-suited for large-scale applications with discrete, flat receivers requiring concentration ratios in the range 5002000×.

© 2013 Optical Society of America

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  1. M. Brunotte, A. Goetzberger, and U. Blieske, “Two-stage concentrator permitting concentration factors up to 300× with one-axis tracking,” Sol. Energy 56, 285–300 (1996).
    [CrossRef]
  2. A. Mohr, T. Roth, and S. W. Glunz, “BICON: high concentration PV using one-axis tracking and silicon concentrator cells,” Prog. Photovoltaics 14, 663–674 (2006).
    [CrossRef]
  3. A. Rabl, “Comparison of solar concentrators,” Sol. Energy 18, 93–111 (1976).
  4. T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” Sol. Energy Mater. Sol. Cells 116, 238–251 (2013).
    [CrossRef]
  5. T. Cooper, G. Ambrosetti, A. Petretti, and A. Steinfeld, “A 500  kW 550X quasi-2-axis tracking CPV system based on an inflated parabolic trough with tracking secondary optics,” presented at the 21st International Photovoltaic Science and Engineering Conference (PVSEC 21), Fukuoka, Japan, 28November–2 December, 2011.
  6. P. Good, G. Zanganeh, G. Ambrosetti, M. C. Barbato, A. Pedretti, and A. Steinfeld, “Towards a commercial parabolic trough CSP system using air as heat transfer fluid,” SolarPACES 2013, Las Vegas, Nevada (2013).
  7. J. A. Duffie and W. A. Beckman, Solar Engineering of Thermal Processes (Wiley, 2006).
  8. D. Y. Goswami, F. Kreith, and J. F. Kreider, Principles of Solar Engineering (CRC Press, 2000).
  9. A. Rabl, Active Solar Collectors and Their Applications (Oxford University, 1985).
  10. R. Winston, J. C. Minano, P. G. Benitez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier Academic, 2005).
  11. P. Bendt, A. Rabl, H. W. Gaul, and K. A. Reed, “Optical analysis and optimization of line focus solar collectors,” (Solar Energy Research Institute, U.S. Department of Energy, 1979).
  12. A. N. Cox, Allen’s Astrophysical Quantities, 4th ed. (Springer, 2000).
  13. D. R. Mills and G. L. Morrison, “Compact linear fresnel reflector solar thermal powerplants,” Sol. Energy 68, 263–283 (2000).
    [CrossRef]
  14. R. Winston and W. Zhang, “Pushing concentration of stationary solar concentrators to the limit,” Opt. Express 18, A64–A72 (2010).
    [CrossRef]
  15. M. Collares-Pereira, J. M. Gordon, A. Rabl, and R. Winston, “High concentration two-stage optics for parabolic trough solar collectors with tubular absorber and large rim angle,” Sol. Energy 47, 457–466 (1991).
    [CrossRef]
  16. H. Ries and J. M. Gordon, “Double-tailored imaging concentrators,” Proc. SPIE 3781, 129–134 (1999).
    [CrossRef]
  17. J. M. Gordon and D. Feuermann, “Optical performance at the thermodynamic limit with tailored imaging designs,” Appl. Opt. 44, 2327–2331 (2005).
    [CrossRef]
  18. D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334, 787–796 (2002).
    [CrossRef]
  19. N. Ostroumov, J. M. Gordon, and D. Feuermann, “Panorama of dual-mirror aplanats for maximum concentration,” Appl. Opt. 48, 4926–4931 (2009).
    [CrossRef]
  20. P. Benítez, J. C. Miñano, and J. Blen, “Squeezing the étendue,” in Illumination Engineering: Design with Nonimaging Optics, R. J. Koshel, ed. (Wiley, 2013).
  21. G. Ambrosetti, J. Chambers, T. Cooper, and A. Pedretti, “Solar collector having a pivotable concentrator arrangement,” WO patent application2013/078567 A2 (filed 6June2013), assigned to Airlight Energy IP SA.
  22. J. C. Miñano and J. C. González, “New method of design of nonimaging concentrators,” Appl. Opt. 31, 3051–3060 (1992).
    [CrossRef]
  23. J. Chaves, Introduction to Nonimaging Optics (CRC Press, 2008).
  24. J. C. Miñano, P. Benítez, A. Cvetkovic, and R. Mohedano, “SMS 3D design method,” in Illumination Engineering: Design with Nonimaging Optics, R. J. Koshel, ed. (IEEE, 2013).
  25. R. Leutz and A. Suzuki, Nonimaging Fresnel Lenses: Design and Performance of Solar Concentrators (Springer, 2001).
  26. I. M. Bassett and G. W. Forbes, “A new class of ideal non-imaging transformers,” Opt. Acta 29, 1271–1282 (1982).
    [CrossRef]
  27. R. P. Friedman and J. M. Gordon, “Optical designs for ultrahigh-flux infrared and solar energy collection: monolithic dielectric tailored edge-ray concentrators,” Appl. Opt. 35, 6684–6691 (1996).
    [CrossRef]
  28. R. Winston, “Dielectric compound parabolic concentrators,” Appl. Opt. 15, 291–292 (1976).
    [CrossRef]
  29. X. H. Ning, R. Winston, and J. Ogallagher, “Dielectric totally internally reflecting concentrators,” Appl. Opt. 26, 300–305 (1987).
    [CrossRef]
  30. J. C. Miñano, P. Benítez, and J. C. González, “RX: a nonimaging concentrator,” Appl. Opt. 34, 2226–2235 (1995).
    [CrossRef]
  31. J. C. Miñano, J. C. González, and P. Benítez, “A high-gain, compact, nonimaging concentrator: RXI,” Appl. Opt. 34, 7850–7856 (1995).
    [CrossRef]
  32. J. Chaves and M. Collares-Pereira, “Ultra flat ideal concentrators of high concentration,” Sol. Energy 69, 269–281 (2000).
    [CrossRef]
  33. J. Petrasch, “A free and open source Monte Carlo ray-tracing program for concentrating solar energy research,” in Proceedings of the ASME 4th International Conference on Energy Sustainability (ES2010) (ASME, 2010).
  34. T. Cooper, F. Dähler, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Performance of compound parabolic concentrators with polygonal apertures,” Solar Energy 95, 308–318 (2013).
    [CrossRef]

2013

T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” Sol. Energy Mater. Sol. Cells 116, 238–251 (2013).
[CrossRef]

T. Cooper, F. Dähler, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Performance of compound parabolic concentrators with polygonal apertures,” Solar Energy 95, 308–318 (2013).
[CrossRef]

2010

2009

2006

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

2005

2002

D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334, 787–796 (2002).
[CrossRef]

2000

D. R. Mills and G. L. Morrison, “Compact linear fresnel reflector solar thermal powerplants,” Sol. Energy 68, 263–283 (2000).
[CrossRef]

J. Chaves and M. Collares-Pereira, “Ultra flat ideal concentrators of high concentration,” Sol. Energy 69, 269–281 (2000).
[CrossRef]

1999

H. Ries and J. M. Gordon, “Double-tailored imaging concentrators,” Proc. SPIE 3781, 129–134 (1999).
[CrossRef]

1996

M. Brunotte, A. Goetzberger, and U. Blieske, “Two-stage concentrator permitting concentration factors up to 300× with one-axis tracking,” Sol. Energy 56, 285–300 (1996).
[CrossRef]

R. P. Friedman and J. M. Gordon, “Optical designs for ultrahigh-flux infrared and solar energy collection: monolithic dielectric tailored edge-ray concentrators,” Appl. Opt. 35, 6684–6691 (1996).
[CrossRef]

1995

1992

1991

M. Collares-Pereira, J. M. Gordon, A. Rabl, and R. Winston, “High concentration two-stage optics for parabolic trough solar collectors with tubular absorber and large rim angle,” Sol. Energy 47, 457–466 (1991).
[CrossRef]

1987

1982

I. M. Bassett and G. W. Forbes, “A new class of ideal non-imaging transformers,” Opt. Acta 29, 1271–1282 (1982).
[CrossRef]

1976

A. Rabl, “Comparison of solar concentrators,” Sol. Energy 18, 93–111 (1976).

R. Winston, “Dielectric compound parabolic concentrators,” Appl. Opt. 15, 291–292 (1976).
[CrossRef]

Ambrosetti, G.

T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” Sol. Energy Mater. Sol. Cells 116, 238–251 (2013).
[CrossRef]

T. Cooper, F. Dähler, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Performance of compound parabolic concentrators with polygonal apertures,” Solar Energy 95, 308–318 (2013).
[CrossRef]

G. Ambrosetti, J. Chambers, T. Cooper, and A. Pedretti, “Solar collector having a pivotable concentrator arrangement,” WO patent application2013/078567 A2 (filed 6June2013), assigned to Airlight Energy IP SA.

T. Cooper, G. Ambrosetti, A. Petretti, and A. Steinfeld, “A 500  kW 550X quasi-2-axis tracking CPV system based on an inflated parabolic trough with tracking secondary optics,” presented at the 21st International Photovoltaic Science and Engineering Conference (PVSEC 21), Fukuoka, Japan, 28November–2 December, 2011.

P. Good, G. Zanganeh, G. Ambrosetti, M. C. Barbato, A. Pedretti, and A. Steinfeld, “Towards a commercial parabolic trough CSP system using air as heat transfer fluid,” SolarPACES 2013, Las Vegas, Nevada (2013).

Barbato, M. C.

P. Good, G. Zanganeh, G. Ambrosetti, M. C. Barbato, A. Pedretti, and A. Steinfeld, “Towards a commercial parabolic trough CSP system using air as heat transfer fluid,” SolarPACES 2013, Las Vegas, Nevada (2013).

Bassett, I. M.

I. M. Bassett and G. W. Forbes, “A new class of ideal non-imaging transformers,” Opt. Acta 29, 1271–1282 (1982).
[CrossRef]

Beckman, W. A.

J. A. Duffie and W. A. Beckman, Solar Engineering of Thermal Processes (Wiley, 2006).

Bendt, P.

P. Bendt, A. Rabl, H. W. Gaul, and K. A. Reed, “Optical analysis and optimization of line focus solar collectors,” (Solar Energy Research Institute, U.S. Department of Energy, 1979).

Benitez, P. G.

R. Winston, J. C. Minano, P. G. Benitez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier Academic, 2005).

Benítez, P.

J. C. Miñano, P. Benítez, and J. C. González, “RX: a nonimaging concentrator,” Appl. Opt. 34, 2226–2235 (1995).
[CrossRef]

J. C. Miñano, J. C. González, and P. Benítez, “A high-gain, compact, nonimaging concentrator: RXI,” Appl. Opt. 34, 7850–7856 (1995).
[CrossRef]

J. C. Miñano, P. Benítez, A. Cvetkovic, and R. Mohedano, “SMS 3D design method,” in Illumination Engineering: Design with Nonimaging Optics, R. J. Koshel, ed. (IEEE, 2013).

P. Benítez, J. C. Miñano, and J. Blen, “Squeezing the étendue,” in Illumination Engineering: Design with Nonimaging Optics, R. J. Koshel, ed. (Wiley, 2013).

Blen, J.

P. Benítez, J. C. Miñano, and J. Blen, “Squeezing the étendue,” in Illumination Engineering: Design with Nonimaging Optics, R. J. Koshel, ed. (Wiley, 2013).

Blieske, U.

M. Brunotte, A. Goetzberger, and U. Blieske, “Two-stage concentrator permitting concentration factors up to 300× with one-axis tracking,” Sol. Energy 56, 285–300 (1996).
[CrossRef]

Bortz, J. C.

R. Winston, J. C. Minano, P. G. Benitez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier Academic, 2005).

Brunotte, M.

M. Brunotte, A. Goetzberger, and U. Blieske, “Two-stage concentrator permitting concentration factors up to 300× with one-axis tracking,” Sol. Energy 56, 285–300 (1996).
[CrossRef]

Cadruvi, M.

T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” Sol. Energy Mater. Sol. Cells 116, 238–251 (2013).
[CrossRef]

Chambers, J.

G. Ambrosetti, J. Chambers, T. Cooper, and A. Pedretti, “Solar collector having a pivotable concentrator arrangement,” WO patent application2013/078567 A2 (filed 6June2013), assigned to Airlight Energy IP SA.

Chaves, J.

J. Chaves and M. Collares-Pereira, “Ultra flat ideal concentrators of high concentration,” Sol. Energy 69, 269–281 (2000).
[CrossRef]

J. Chaves, Introduction to Nonimaging Optics (CRC Press, 2008).

Collares-Pereira, M.

J. Chaves and M. Collares-Pereira, “Ultra flat ideal concentrators of high concentration,” Sol. Energy 69, 269–281 (2000).
[CrossRef]

M. Collares-Pereira, J. M. Gordon, A. Rabl, and R. Winston, “High concentration two-stage optics for parabolic trough solar collectors with tubular absorber and large rim angle,” Sol. Energy 47, 457–466 (1991).
[CrossRef]

Cooper, T.

T. Cooper, F. Dähler, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Performance of compound parabolic concentrators with polygonal apertures,” Solar Energy 95, 308–318 (2013).
[CrossRef]

T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” Sol. Energy Mater. Sol. Cells 116, 238–251 (2013).
[CrossRef]

T. Cooper, G. Ambrosetti, A. Petretti, and A. Steinfeld, “A 500  kW 550X quasi-2-axis tracking CPV system based on an inflated parabolic trough with tracking secondary optics,” presented at the 21st International Photovoltaic Science and Engineering Conference (PVSEC 21), Fukuoka, Japan, 28November–2 December, 2011.

G. Ambrosetti, J. Chambers, T. Cooper, and A. Pedretti, “Solar collector having a pivotable concentrator arrangement,” WO patent application2013/078567 A2 (filed 6June2013), assigned to Airlight Energy IP SA.

Cox, A. N.

A. N. Cox, Allen’s Astrophysical Quantities, 4th ed. (Springer, 2000).

Cvetkovic, A.

J. C. Miñano, P. Benítez, A. Cvetkovic, and R. Mohedano, “SMS 3D design method,” in Illumination Engineering: Design with Nonimaging Optics, R. J. Koshel, ed. (IEEE, 2013).

Dähler, F.

T. Cooper, F. Dähler, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Performance of compound parabolic concentrators with polygonal apertures,” Solar Energy 95, 308–318 (2013).
[CrossRef]

Duffie, J. A.

J. A. Duffie and W. A. Beckman, Solar Engineering of Thermal Processes (Wiley, 2006).

Feuermann, D.

Forbes, G. W.

I. M. Bassett and G. W. Forbes, “A new class of ideal non-imaging transformers,” Opt. Acta 29, 1271–1282 (1982).
[CrossRef]

Friedman, R. P.

Gaul, H. W.

P. Bendt, A. Rabl, H. W. Gaul, and K. A. Reed, “Optical analysis and optimization of line focus solar collectors,” (Solar Energy Research Institute, U.S. Department of Energy, 1979).

Glunz, S. W.

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

Goetzberger, A.

M. Brunotte, A. Goetzberger, and U. Blieske, “Two-stage concentrator permitting concentration factors up to 300× with one-axis tracking,” Sol. Energy 56, 285–300 (1996).
[CrossRef]

González, J. C.

Good, P.

P. Good, G. Zanganeh, G. Ambrosetti, M. C. Barbato, A. Pedretti, and A. Steinfeld, “Towards a commercial parabolic trough CSP system using air as heat transfer fluid,” SolarPACES 2013, Las Vegas, Nevada (2013).

Gordon, J. M.

Goswami, D. Y.

D. Y. Goswami, F. Kreith, and J. F. Kreider, Principles of Solar Engineering (CRC Press, 2000).

Kreider, J. F.

D. Y. Goswami, F. Kreith, and J. F. Kreider, Principles of Solar Engineering (CRC Press, 2000).

Kreith, F.

D. Y. Goswami, F. Kreith, and J. F. Kreider, Principles of Solar Engineering (CRC Press, 2000).

Leutz, R.

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

Lynden-Bell, D.

D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334, 787–796 (2002).
[CrossRef]

Mills, D. R.

D. R. Mills and G. L. Morrison, “Compact linear fresnel reflector solar thermal powerplants,” Sol. Energy 68, 263–283 (2000).
[CrossRef]

Minano, J. C.

R. Winston, J. C. Minano, P. G. Benitez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier Academic, 2005).

Miñano, J. C.

J. C. Miñano, J. C. González, and P. Benítez, “A high-gain, compact, nonimaging concentrator: RXI,” Appl. Opt. 34, 7850–7856 (1995).
[CrossRef]

J. C. Miñano, P. Benítez, and J. C. González, “RX: a nonimaging concentrator,” Appl. Opt. 34, 2226–2235 (1995).
[CrossRef]

J. C. Miñano and J. C. González, “New method of design of nonimaging concentrators,” Appl. Opt. 31, 3051–3060 (1992).
[CrossRef]

P. Benítez, J. C. Miñano, and J. Blen, “Squeezing the étendue,” in Illumination Engineering: Design with Nonimaging Optics, R. J. Koshel, ed. (Wiley, 2013).

J. C. Miñano, P. Benítez, A. Cvetkovic, and R. Mohedano, “SMS 3D design method,” in Illumination Engineering: Design with Nonimaging Optics, R. J. Koshel, ed. (IEEE, 2013).

Mohedano, R.

J. C. Miñano, P. Benítez, A. Cvetkovic, and R. Mohedano, “SMS 3D design method,” in Illumination Engineering: Design with Nonimaging Optics, R. J. Koshel, ed. (IEEE, 2013).

Mohr, A.

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

Morrison, G. L.

D. R. Mills and G. L. Morrison, “Compact linear fresnel reflector solar thermal powerplants,” Sol. Energy 68, 263–283 (2000).
[CrossRef]

Ning, X. H.

Ogallagher, J.

Ostroumov, N.

Pedretti, A.

T. Cooper, F. Dähler, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Performance of compound parabolic concentrators with polygonal apertures,” Solar Energy 95, 308–318 (2013).
[CrossRef]

G. Ambrosetti, J. Chambers, T. Cooper, and A. Pedretti, “Solar collector having a pivotable concentrator arrangement,” WO patent application2013/078567 A2 (filed 6June2013), assigned to Airlight Energy IP SA.

P. Good, G. Zanganeh, G. Ambrosetti, M. C. Barbato, A. Pedretti, and A. Steinfeld, “Towards a commercial parabolic trough CSP system using air as heat transfer fluid,” SolarPACES 2013, Las Vegas, Nevada (2013).

Petrasch, J.

J. Petrasch, “A free and open source Monte Carlo ray-tracing program for concentrating solar energy research,” in Proceedings of the ASME 4th International Conference on Energy Sustainability (ES2010) (ASME, 2010).

Petretti, A.

T. Cooper, G. Ambrosetti, A. Petretti, and A. Steinfeld, “A 500  kW 550X quasi-2-axis tracking CPV system based on an inflated parabolic trough with tracking secondary optics,” presented at the 21st International Photovoltaic Science and Engineering Conference (PVSEC 21), Fukuoka, Japan, 28November–2 December, 2011.

Pravettoni, M.

T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” Sol. Energy Mater. Sol. Cells 116, 238–251 (2013).
[CrossRef]

Rabl, A.

M. Collares-Pereira, J. M. Gordon, A. Rabl, and R. Winston, “High concentration two-stage optics for parabolic trough solar collectors with tubular absorber and large rim angle,” Sol. Energy 47, 457–466 (1991).
[CrossRef]

A. Rabl, “Comparison of solar concentrators,” Sol. Energy 18, 93–111 (1976).

A. Rabl, Active Solar Collectors and Their Applications (Oxford University, 1985).

P. Bendt, A. Rabl, H. W. Gaul, and K. A. Reed, “Optical analysis and optimization of line focus solar collectors,” (Solar Energy Research Institute, U.S. Department of Energy, 1979).

Reed, K. A.

P. Bendt, A. Rabl, H. W. Gaul, and K. A. Reed, “Optical analysis and optimization of line focus solar collectors,” (Solar Energy Research Institute, U.S. Department of Energy, 1979).

Ries, H.

H. Ries and J. M. Gordon, “Double-tailored imaging concentrators,” Proc. SPIE 3781, 129–134 (1999).
[CrossRef]

Roth, T.

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

Shatz, N.

R. Winston, J. C. Minano, P. G. Benitez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier Academic, 2005).

Steinfeld, A.

T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” Sol. Energy Mater. Sol. Cells 116, 238–251 (2013).
[CrossRef]

T. Cooper, F. Dähler, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Performance of compound parabolic concentrators with polygonal apertures,” Solar Energy 95, 308–318 (2013).
[CrossRef]

T. Cooper, G. Ambrosetti, A. Petretti, and A. Steinfeld, “A 500  kW 550X quasi-2-axis tracking CPV system based on an inflated parabolic trough with tracking secondary optics,” presented at the 21st International Photovoltaic Science and Engineering Conference (PVSEC 21), Fukuoka, Japan, 28November–2 December, 2011.

P. Good, G. Zanganeh, G. Ambrosetti, M. C. Barbato, A. Pedretti, and A. Steinfeld, “Towards a commercial parabolic trough CSP system using air as heat transfer fluid,” SolarPACES 2013, Las Vegas, Nevada (2013).

Suzuki, A.

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

Winston, R.

R. Winston and W. Zhang, “Pushing concentration of stationary solar concentrators to the limit,” Opt. Express 18, A64–A72 (2010).
[CrossRef]

M. Collares-Pereira, J. M. Gordon, A. Rabl, and R. Winston, “High concentration two-stage optics for parabolic trough solar collectors with tubular absorber and large rim angle,” Sol. Energy 47, 457–466 (1991).
[CrossRef]

X. H. Ning, R. Winston, and J. Ogallagher, “Dielectric totally internally reflecting concentrators,” Appl. Opt. 26, 300–305 (1987).
[CrossRef]

R. Winston, “Dielectric compound parabolic concentrators,” Appl. Opt. 15, 291–292 (1976).
[CrossRef]

R. Winston, J. C. Minano, P. G. Benitez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier Academic, 2005).

Zanganeh, G.

P. Good, G. Zanganeh, G. Ambrosetti, M. C. Barbato, A. Pedretti, and A. Steinfeld, “Towards a commercial parabolic trough CSP system using air as heat transfer fluid,” SolarPACES 2013, Las Vegas, Nevada (2013).

Zhang, W.

Appl. Opt.

Mon. Not. R. Astron. Soc.

D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334, 787–796 (2002).
[CrossRef]

Opt. Acta

I. M. Bassett and G. W. Forbes, “A new class of ideal non-imaging transformers,” Opt. Acta 29, 1271–1282 (1982).
[CrossRef]

Opt. Express

Proc. SPIE

H. Ries and J. M. Gordon, “Double-tailored imaging concentrators,” Proc. SPIE 3781, 129–134 (1999).
[CrossRef]

Prog. Photovoltaics

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

Sol. Energy

A. Rabl, “Comparison of solar concentrators,” Sol. Energy 18, 93–111 (1976).

M. Brunotte, A. Goetzberger, and U. Blieske, “Two-stage concentrator permitting concentration factors up to 300× with one-axis tracking,” Sol. Energy 56, 285–300 (1996).
[CrossRef]

D. R. Mills and G. L. Morrison, “Compact linear fresnel reflector solar thermal powerplants,” Sol. Energy 68, 263–283 (2000).
[CrossRef]

M. Collares-Pereira, J. M. Gordon, A. Rabl, and R. Winston, “High concentration two-stage optics for parabolic trough solar collectors with tubular absorber and large rim angle,” Sol. Energy 47, 457–466 (1991).
[CrossRef]

J. Chaves and M. Collares-Pereira, “Ultra flat ideal concentrators of high concentration,” Sol. Energy 69, 269–281 (2000).
[CrossRef]

Sol. Energy Mater. Sol. Cells

T. Cooper, M. Pravettoni, M. Cadruvi, G. Ambrosetti, and A. Steinfeld, “The effect of irradiance mismatch on a semi-dense array of triple-junction concentrator cells,” Sol. Energy Mater. Sol. Cells 116, 238–251 (2013).
[CrossRef]

Solar Energy

T. Cooper, F. Dähler, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Performance of compound parabolic concentrators with polygonal apertures,” Solar Energy 95, 308–318 (2013).
[CrossRef]

Other

J. Petrasch, “A free and open source Monte Carlo ray-tracing program for concentrating solar energy research,” in Proceedings of the ASME 4th International Conference on Energy Sustainability (ES2010) (ASME, 2010).

P. Benítez, J. C. Miñano, and J. Blen, “Squeezing the étendue,” in Illumination Engineering: Design with Nonimaging Optics, R. J. Koshel, ed. (Wiley, 2013).

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T. Cooper, G. Ambrosetti, A. Petretti, and A. Steinfeld, “A 500  kW 550X quasi-2-axis tracking CPV system based on an inflated parabolic trough with tracking secondary optics,” presented at the 21st International Photovoltaic Science and Engineering Conference (PVSEC 21), Fukuoka, Japan, 28November–2 December, 2011.

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Figures (39)

Fig. 1.
Fig. 1.

LTP concentrator (b) differs from a traditional line-focus trough concentrator (a) in that the focal line is split into a number of point-like foci spanning along the length of the trough concentrator.

Fig. 2.
Fig. 2.

Skew angle ϑ , defined as the angle between the aperture normal n ^ (optical axis) of a trough concentrator and the vector pointing to the center of the solar disk s ^ , is a central concept in designing LTP concentrators.

Fig. 3.
Fig. 3.

Three types of LTP solar concentrators classified by the degrees of freedom of the secondary concentrator stage: (a) fixed secondary optics; (b) discrete-switching secondary optics; and (c) continuous-tracking secondary optics.

Fig. 4.
Fig. 4.

One-axis tracker can be oriented in the cardinal frame ( X , Y , Z ) by defining its axis vector a ^ parallel to the tracking axis. The sun vector s ^ is defined by the solar altitude and azimuth angles α s and γ s , respectively. The axis vector may be analogously defined by the axis altitude angle α a and the axis azimuth angle γ a .

Fig. 5.
Fig. 5.

Parked ( x 0 , y 0 , z 0 ) and tracking ( x , y , z ) coordinate frames for a generic one-axis tracking collector. For a horizontal N–S tracker, x 0 points east, y 0 points north and z 0 points up. For a horizontal E–W collector, x 0 points south, y 0 points east and z 0 points up. For a polar tracker, y 0 is parallel to the Earth’s axis and x 0 points east. The tracking frame is rotated about the y axis by the tracking angle ψ .

Fig. 6.
Fig. 6.

Schematic representation of skew dilation. For ϑ 0 , the angular radius of the sun in the transverse x z plane of a trough concentrator appears larger than at normal incidence. Adapted from [11].

Fig. 7.
Fig. 7.

Schematic of the path traced by a set of parallel skew rays in the transverse x z and axial y z planes of a trough concentrator. The rays arrive as a parallel bundle and are concentrated in the transverse plane. Since they reflect off different parts of the primary mirror, they approach the focus with different directions in the axial plane.

Fig. 8.
Fig. 8.

Secondary concentrator must have acceptance angles of ± θ i , 2 , ax = α + θ sun in the axial plane y z and ± θ i , 2 , trans = Φ + θ sun in the transverse plane. Since Φ α , the acceptance angle is much smaller in the axial plane than in the transverse plane. The solution adopted here is to use a linear 2D design concentrating only in the axial y z plane, with x being the symmetry (extrusion) axis. In this case, we forego the possibility to concentrate in the transverse direction, but arrive at a simple secondary design that achieves θ i , 2 , trans = 90 ° .

Fig. 9.
Fig. 9.

Direction of a ray r ^ reaching the primary focus is determined by the skew angle ϑ and the polar angle φ .

Fig. 10.
Fig. 10.

Axial beamspread α as a function of the skew angle ϑ for different primary rim angles Φ , computed from Eq. (15). The curves are symmetric about the abscissa. The dashed curve shows the locus of critical skew ϑ crit and critical beamspread α crit from Eqs. (16) and (17).

Fig. 11.
Fig. 11.

Maximum overall concentration as a function of primary rim angle for tracking-secondary LTP collectors based on N–S one-axis aplanatic, parabolic, and cylindrical primaries at a latitude of 30°.

Fig. 12.
Fig. 12.

Maximum overall concentration as a function of latitude for tracking-secondary LTP collectors based on N–S one-axis aplanatic, parabolic and cylindrical primaries of optimal rim angle ( Φ opt 10 ° ).

Fig. 13.
Fig. 13.

Mirror profiles for (a) parabolic and (b) semi-parabolic primary concentrators for various rim angles Φ = ( 1 / 2 ) ( Φ 2 Φ 1 ) . The parabolic and semi-parabolic concentrators have the same secondary concentration limits for equal rim angles, but the semi-parabolic trough is more compact. For a two-wing design, the aspect ratios of the semi-parabolic troughs are half the values listed here.

Fig. 14.
Fig. 14.

Dimensioning of the primary and secondary concentrators of a tracking-secondary LTP concentrator. The secondary inlet aperture a i , 2 may be chosen freely, but generally must be much smaller than the primary inlet aperture a i , 1 , such that the secondaries are at a feasible scale. With a i , 2 specified, the receiver shape is defined by Eq. (29). For most cases, the resulting receiver shape will be rectangular. Schematic is not to scale.

Fig. 15.
Fig. 15.

Required secondary concentrator tracking range as a function of latitude for a LTP collector with a N–S one-axis primary with rim angle Φ = 10.5 ° . Calculated from Eq. (14).

Fig. 16.
Fig. 16.

In order to not interfere with adjacent concentrators while tracking, the secondary concentrator must fit within the envelope traced out by the dashed lines, which are circular arcs of radius a i , 2 centered at the edges of the inlet aperture. Since the relative motion of the adjacent concentrators is independent of the position of the tracking axes, the choice made here to place it at the center of the inlet aperture does not affect the generality of the resulting tracking envelope.

Fig. 17.
Fig. 17.

Interference envelope for a concentrator with a convex cylindrical inlet is constructed from three circular arcs described by Eqs. (30)–(32), and poses stricter limits on the concentrator shape than does the case of a flat inlet, shown in Fig. 16. The right pane shows the formation of the envelope for a concentrator with a curved inlet of arc angle Θ = 150 ° for σ max = 60 ° . As with the flat inlet, the position of the tracking axis does not affect the shape of the interference envelope.

Fig. 18.
Fig. 18.

Secondary stage based on asymmetrically truncated CPCs. This particular example has θ i = 6 ° and is designed to track from σ min = 30 ° to σ max = 60 ° . The CPCs are arranged along the focal line such that for σ = 0 , the right edge of each concentrator is aligned with the left edge of the adjacent concentrator. The geometric concentration of this example design is C g , 2 , ax = a i / a o = 7 x which is 25% lower than the thermodynamic limit 1 / sin θ i due to the truncation.

Fig. 19.
Fig. 19.

Geometric concentration of asymmetrically truncated CPCs designed to permit a rotation of σ max in the clockwise direction for different acceptance angles. The long branch is untruncated.

Fig. 20.
Fig. 20.

Secondary stage based on multifoliate LPCs. The LPC always fits within the interference envelope and can therefore track from σ = 90 ° to + 90 ° with no interference. It essentially achieves the concentration limits derived in Section 2.D.

Fig. 21.
Fig. 21.

Secondary stage based on DTERCs. The shape of the DTERC is well matched to the tracking envelope of Fig. 17. The design shown has an acceptance angle of θ i = 5 ° and can track to ± 50 ° without interference. The maximum slope angle for this design is Ψ = 62 ° cf. Fig. 22.

Fig. 22.
Fig. 22.

By increasing the slope of the aspheric lens at the inlet of the DTERC it becomes more compact and can thus track to a larger angle. This design curve shows the required slope angle Ψ to achieve a specified maximum tracking angle σ max for different acceptance angles.

Fig. 23.
Fig. 23.

Source/acceptance matching of a DTERC for a LTP collector with and α crit = 0.5 ° and σ max = 43.9 ° . To track to σ max , the slope angle of the inlet must be Ψ = 55 ° (cf. Fig. 22). The curved refractive inlet causes the ideal acceptance map (dashed) to be distorted into the football-shaped map (see Appendix B.3.1). To envelope the source map, the acceptance angle of the DTERC must be considerably oversized, to θ i , 2 = 1.7 ° + θ sun . The acceptance maps shown in the plot do not account for the finite angular size of the sun.

Fig. 24.
Fig. 24.

Schematic of: (a) Design 1; (b) Design 2; and (c) Design 3 showing the path traced by rays incident at skew angle of ϑ = 0 ° (normal incidence). Only a small section of the primary has been illuminated for clarity.

Fig. 25.
Fig. 25.

Acceptance efficiency η acc of three exemplary tracking-secondary LTP collectors as a function of skew angle ϑ as simulated by Monte Carlo ray-tracing: (a) Design 1—asymmetric parabolic trough with asymmetrically truncated CPC secondaries, C g , tot = 526 x ; (b) Design 2—aplanatic trough with DTERC secondaries, C g , tot = 1482 x ; (c) Design 3—parabolic trough primary with LPC secondaries, C g , tot = 770 x . The dashed lines indicate the maximum and minimum skew angles at the latitude for which each system was designed.

Fig. 26.
Fig. 26.

One-axis tracker minimizes the incidence angle ϑ for any given sun position by tracking such that the sun vector s ^ , the axis vector a ^ , and the normal vector n ^ , are all coplanar. The incidence (skew) angle is found from cos ϑ = s ^ · n ^ , or more conveniently from sin ϑ = s ^ · a ^ .

Fig. 27.
Fig. 27.

(a) In 3-space, a ray is defined by a point Q ( x , y , z = 0 ) and a direction v ^ ( L , M , N = [ 1 M 2 N 2 ] 1 / 2 ) ; (b) In L M space, a ray is represented solely by L and M , the x and y components of its unit direction vector (direction cosines), respectively. The rays u and v , shown in Cartesian coordinates in (a), have the representation in L M space in (b). Any ray having the same direction would appear identically in L M space, regardless of the point Q where it strikes aperture.

Fig. 28.
Fig. 28.

(a) Source map of the solar disk at a skew angle of ϑ for a one-axis tracker. M points in the direction of the tracking axis; (b) the effective source map for a one-axis tracker. The effective source map is formed by superimposing the individual source maps (a) over all sun positions seen by the collector. In both schematics, the angular size of the solar disk has been intentionally oversized to emphasize the shape.

Fig. 29.
Fig. 29.

Schematic of a ray v ^ reaching the inlet aperture of trough concentrator and its projection into the transverse ( x z ) plane. For an ideal trough concentrator, the ray is accepted if the angle θ proj between optical axis and the projection of the ray is less than or equal to the acceptance angle, regardless of the value of M .

Fig. 30.
Fig. 30.

Matching of the acceptance map of a hollow (reflective) trough concentrator with acceptance angle of θ i to the effective source map of a generic one-axis tracker. The design point indicates the region for which the strictest requirements on the acceptance angle occur.

Fig. 31.
Fig. 31.

Effective source map at the secondary inlet aperture for a primary rim angle of Φ = 60 ° , shown for skew angles ranging from ϑ = 0 ° to + 90 ° . The map is symmetric about the L axis for negative skew angles, which are omitted for clarity. The acceptance map of a hollow ( n = 1 ) ideal 2D concentrator, sized to envelope the effective source map, is shown. The line at critical skew is indicated and shows how a hollow 2D concentrator can be sized, based on the critical beamspread alone. The acceptance map of an ideal 2D dielectric-filled concentrator ( n = 1.5 ) is also shown.

Fig. 32.
Fig. 32.

Ray v ^ is incident on the curved inlet of a linear concentrator. The ray has a fixed angle θ proj with respect to the optical ( z ) axis in the cross sectional ( y z ) plane. If the ray is a meridian ray ( L = 0 ), it refracts according to Snell’s law with n 1 and n 2 resulting in ray r ^ meridian having an angle χ 2 to the normal. If the ray is a nonmeridian ray ( L 0 ), it refracts according to Snell’s law with the apparent refractive indices n 1 * and n 2 * , resulting in the more strongly refracted ray r ^ nonmeridian having an angle χ 2 * to the normal. If the remainder of the optical interactions suffered by r ^ nonmeridian are reflections, then this ray will have the same projected path as an equivalent meridian ray v ^ app , having angle θ app with respect to the optical axis.

Fig. 33.
Fig. 33.

Acceptance map for a dielectric-filled concentrator with a curved refractive inlet for different maximum slope angles Ψ and an acceptance angle of θ i = 30 ° . All rays having directions inside the map are accepted by the concentrator.

Fig. 34.
Fig. 34.

Intersection of edge-ray pencils reflected from the edges of an asymmetric parabolic trough. The w Φ 2 beam originates from Φ 2 , and the w Φ 1 beam originates from Φ 1 < Φ 2 . For sufficiently small θ i , 1 , A B C D is a parallelogram with centroid F .

Fig. 35.
Fig. 35.

Primary, secondary, and overall concentration for a tracking-secondary LTP concentrator based on a semi-parabolic trough primary ( Φ 1 = 0 ° , Φ 2 = 60 ° ) as a function of the focal plane tilt angle. The system is designed for full collection at a latitude of ϕ = 30 ° . The secondary concentration is maximal for τ = ( 1 / 2 ) ( Φ 2 + Φ 1 ) = Φ av . The primary concentration is maximal for τ = τ * = 43.9 ° as calculated from Eq. (C6). Because the maximum secondary concentration is much more sensitive to the tilt angle, the total concentration is always maximal for the tilt angle τ = Φ av , which maximizes secondary concentration.

Fig. 36.
Fig. 36.

Generic schematic of a two-mirror aplanat showing major design parameters. For this particular design, the parameters are: s = 0.7 , K = 0.35 , Φ = 45 ° , and δ = 19.5 ° .

Fig. 37.
Fig. 37.

Two compact small rim angle two-mirror aplanats with: (a)  Φ = 10.5 ° with s = 0.084 , K = 0.084 ; and (b)  s = 0.101 and K = 0.101 . Both designs have a final focus at the vertex of the primary mirror and approach the compactness limit of AR = 0.25 . The truncation angle is chosen to eliminate blocking and shading ( δ = 1 ° for both designs).

Fig. 38.
Fig. 38.

Schematic of a multifoliate LPC with optimal leaf spacing showing major design parameters.

Fig. 39.
Fig. 39.

Total number of leaves required for a LPC as a function of the leaf constant c = 6 tan 2 θ d ( 1 f ) , where f is the fraction of rays within the acceptance angle that are accepted. The parameter g is the fraction of the inlet aperture for which the leaves do not meet the specified f .

Tables (2)

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Table 1. Minimum and Maximum Skew Angles for Common One-Axis Trackersa

Tables Icon

Table 2. Design Table for Three Exemplary LTP Concentrators with Tracking Secondary Optics

Equations (107)

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sin ϑ = cos α s cos α a cos ( γ s γ a ) + sin α s sin α a .
C g , max , 2 D = 1 / sin θ i ,
θ sun = arcsin ( R sun / d sun ) ,
sin θ eff = R sun / ( d sun cos ϑ ) = sin θ sun sec ϑ .
sin θ i = sin θ sun sec ϑ max .
C g , 1 , parab. = sin Φ cos ( Φ + θ i , 1 ) sin θ i , 1 1 .
C g , 1 , circ. = 4 sin ( 1 2 Φ ) 3 sin ( 1 2 Φ ) sin ( 3 2 Φ 2 θ i , 1 ) 1 ,
C g , 1 , aplanat = sin Φ sin θ i , 1 .
v ^ = s ^ = [ 0 sin ϑ cos ϑ ] .
r ^ = [ L M N ] = [ cos ϑ sin φ , sin ϑ , cos ϑ cos φ ] ,
tan β = M N = tan ϑ sec φ .
β min = ϑ ,
β max = arctan ( tan ϑ sec Φ ) .
σ = 1 2 ( β max + β min ) = 1 2 [ arctan ( tan ϑ sec Φ ) + ϑ ] .
α = 1 2 ( β max β min ) = 1 2 [ arctan ( tan ϑ sec Φ ) ϑ ] = σ ϑ ,
tan ϑ crit = ( cos Φ ) 1 / 2 = cos 1 / 2 Φ .
α crit = 1 2 [ arctan ( 1 / cos 1 / 2 Φ ) arctan ( cos 1 / 2 Φ ) ] = 45 ° ϑ crit = 45 ° arctan ( cos 1 / 2 Φ ) .
θ i , 2 = α crit + θ sun ,
C g , 2 , ax = n sin θ i , 2 = n sin ( 45 ° ϑ crit + θ sun ) = n sin [ 45 ° arctan ( cos 1 / 2 Φ ) + θ sun ] .
C g , tot = C g , 1 C g , 2 , ax ,
x = 2 f tan ( 1 2 φ ) z = 2 f cos φ / ( 1 + cos φ ) ,
AR = Δ z Δ x ,
AR = f 4 f tan ( 1 2 Φ ) = 1 4 cot ( 1 2 Φ ) ,
Φ = 1 2 Δ Φ = 1 2 ( max φ min φ ) = 1 2 ( Φ 2 Φ 1 ) .
AR = f 2 f tan ( 1 2 Φ 2 ) = 1 2 cot Φ
C g , 1 , asym.parab. ( τ = Φ av ) = 1 2 csc θ i , 1 [ tan ( 1 2 Φ 2 ) tan ( 1 2 Φ 1 ) ] ( 1 + cos Φ 2 ) cos Φ ,
w 2 = a o , 1 = a i , 1 / C g , 1 .
a i , 2 / a o , 2 = C g , 2 ,
w 2 a o , 2 = a i , 1 / C g , 1 a i , 2 / C g , 2 = a i , 1 a i , 2 C g , 2 C g , 1 .
x 1 = a i , 2 cos ξ 1 2 a i , 2 z 1 = a i , 2 sin ξ } 0 ξ arccos [ sin ( Θ 2 ) ] ,
x 2 = ( a i , 2 R ) cos ξ z 2 = ( a i , 2 R ) sin ξ + d } arccos [ sin ( Θ 2 ) ] ξ σ max ,
x 3 = R cos ξ + a i , 2 cos σ max z 3 = R sin ξ + a i , 2 sin σ max + d } arccos [ sin ( Θ 2 ) ] ξ σ max ,
R = 1 2 a i , 2 csc ( 1 2 Θ ) ,
d = 1 2 a i , 2 cos ( 1 2 Θ ) .
l DTERC = ( a i + a o ) tan ( arccos [ sin ( θ i χ ) n ] Ψ ) ,
η acc ( ϑ ) = Φ e , receiver Φ e , aperture = E e , receiver A receiver E e , aperture A aperture = E e , receiver C g , t o t × DNI × cos ϑ ,
s ^ = [ cos α s sin γ s , cos α s cos γ s , sin α s ] ,
α s = arcsin ( cos ϕ cos δ cos ω + sin ϕ sin δ ) ,
γ s = sgn ω | arccos ( sin α s sin ϕ sin δ cos α s cos ϕ ) | ,
sin γ s = cos δ sin ω cos α s .
a ^ = [ cos α a sin γ a , cos α a cos γ a , sin α a ] .
s ^ · ( n ^ × a ^ ) = 0 ,
cos ϑ = s ^ · n ^ .
sin ϑ = s ^ · a ^ ,
sin ϑ = cos α s cos α a cos ( γ γ a ) + sin α s sin α a .
sin ϑ = sin δ ( sin ϕ sin α a cos ϕ cos α a cos γ a ) + cos δ [ cos ϕ cos ω sin α a + cos α a ( sin ϕ cos ω cos γ a + sin ω sin γ a ) ] .
sin ϑ = cos α s cos γ s cos ϕ sin α s sin ϕ .
ϑ = δ .
sin ϑ = cos α s sin γ s .
sin ϑ = cos α s cos δ sin ω cos α s = cos δ sin ω .
cos ω ss = tan ϕ tan δ .
sin ϑ = cos α s cos γ s .
sin ϑ = sin ϕ cos δ cos ω cos ϕ sin δ .
cos ω ss = tan ϕ tan δ = tan ϕ tan ε ,
ϑ min = arcsin ( sin ϕ cos ε tan ϕ tan ε cos ϕ sin ε ) = arcsin ( sin δ sec ε ) .
ϑ max = arcsin ( sin ϕ cos ε + cos ϕ sin ε ) = | ϕ | + ε .
v ^ ( ξ ) = [ sin θ sun sin ξ sin θ sun cos ξ cos θ sun ] [ sin θ sun sin ξ sin θ sun cos ξ 1 ] ,
v ( ϑ ; ξ ) = R x ( ϑ ) v ( ξ ) = [ sin θ sun sin ξ cos θ sun sin ϑ + sin θ sun cos ϑ cos ξ cos θ sun cos ϑ sin θ sun sin ϑ cos ξ ] [ sin θ sun sin ξ sin ϑ + sin θ sun cos ϑ cos ξ cos ϑ sin θ sun sin ϑ cos ξ ] .
sin θ proj = L ( M 2 + N 2 ) 1 / 2 sin θ i .
L 2 sin 2 θ i + M 2 1 .
sin 2 θ sun sin 2 θ i , 1 + sin 2 ϑ max = 1 .
sin θ i , 1 = sin θ sun sec ϑ max ,
r ^ = [ L M N ] = R x ( σ ) r ^ = [ 1 0 0 0 cos σ sin σ 0 sin σ cos σ ] [ cos ϑ sin φ sin ϑ cos ϑ cos φ ] = [ cos ϑ sin φ sin ϑ cos σ cos ϑ cos φ sin σ sin ϑ sin σ + cos ϑ cos φ cos σ ] ,
L = cos ϑ sin φ M = sin ϑ cos σ + cos ϑ cos φ sin σ ,
L 2 n 2 + M 2 sin 2 α = 1 .
α = σ ϑ
cos 2 ϑ sin 2 Φ n 2 + ( sin ϑ cos σ cos ϑ sin σ cos Φ ) 2 sin 2 α = 1
sin θ proj = M ( 1 L 2 ) 1 / 2 .
n * = ( n 2 L 2 ) 1 / 2 .
sin χ 2 * = ( n 1 2 L 2 ) 1 / 2 ( n 2 2 L 2 ) 1 / 2 sin χ 1 = sin ( Ψ θ proj ) ( 1 L 2 n 2 L 2 ) 1 / 2 .
sin χ 1 * = sin ( Ψ θ app ) = n sin χ 2 * ,
sin ( θ app Ψ ) = n sin ( θ proj Ψ ) ( 1 L 2 n 2 L 2 ) 1 / 2 .
M = ( 1 L 2 ) 1 / 2 sin ( arcsin { sin ( θ i Ψ ) n [ ( 1 L 2 ) / ( n 2 L 2 ) ] 1 / 2 } + Ψ ) ,
| L | min { [ sin θ i sin ( 2 Ψ θ i ) sin 2 Ψ sin 2 ( Ψ θ i ) / n 2 ] 1 / 2 { n 2 [ 1 + cos ( 2 Ψ 2 θ i ) ] n 2 + cos ( 2 Ψ 2 θ i ) 1 } 1 / 2 .
w φ = r ( φ ) sin θ i = 2 f sin θ i 1 + cos φ = f sin θ i sec 2 ( 1 2 φ ) .
H D = w Φ 2 sin ( Δ Φ ) = f sin θ i sec 2 ( 1 2 Φ 2 ) csc ( Δ Φ ) .
F D 2 = H D 2 + H F 2 2 H D H F cos ( Δ Φ ) ,
F D = f sin θ i csc ( Δ Φ ) [ sec 4 ( 1 2 Φ 1 ) + sec 4 ( 1 2 Φ 2 ) 2 sec 2 ( 1 2 Φ 1 ) sec 2 ( 1 2 Φ 2 ) cos ( Δ Φ ) ] 1 / 2 .
sin ( H F D ) = H D sin ( Δ Φ ) / F D .
cos ( Φ 2 τ * ) = sec 2 ( 1 2 Φ 2 ) sin ( Δ Φ ) [ sec 4 ( 1 2 Φ 1 ) + sec 4 ( 1 2 Φ 2 ) 2 sec 2 ( 1 2 Φ 1 ) sec 2 ( 1 2 Φ 2 ) cos ( Δ Φ ) ] 1 / 2 .
F D = F B = csc ( 90 ° Φ 2 + τ ) w Φ 2 = 2 f sin θ i ( 1 + cos Φ 2 ) cos ( τ Φ 2 ) .
F D + = F B + = 2 f sin θ i ( 1 + cos Φ 1 ) cos ( τ Φ 1 ) .
a i , 1 = 2 f ( tan 1 2 Φ 2 tan 1 2 Φ 1 ) .
C g , 1 , asym.parab. = a i , 1 2 F D = 1 2 csc θ i [ tan ( 1 2 Φ 2 ) tan ( 1 2 Φ 1 ) ] ( 1 + cos φ ) cos ( τ φ ) where : φ = { Φ 2 for τ < τ * Φ 1 for τ > τ * Φ 2 or Φ 1 for τ = τ * .
C g , APT ( τ = Φ av ) = 1 2 csc θ i [ tan ( 1 2 Φ 2 ) tan ( 1 2 Φ 1 ) ] ( 1 + cos Φ 2 ) cos Φ .
r ^ = R y ( τ ) r ^ = [ cos τ 0 sin τ 0 1 0 sin τ 0 cos τ ] [ cos ϑ sin φ sin ϑ cos ϑ cos φ ] = [ cos ϑ sin ( φ τ ) sin ϑ cos ϑ cos ( φ τ ) ] ,
Φ av = 1 2 ( Φ 1 + Φ 2 ) .
tan β = M N = tan ϑ sec ( φ τ ) .
β min = arctan [ tan ϑ sec ( Φ av Φ av ) ] = ϑ ,
β max = arctan [ tan ϑ sec ( Φ av Φ 1 ) ] = arctan ( tan ϑ sec Φ ) .
x p = sin φ z p = s cos 2 ( 1 2 φ ) + g ( φ ) cos 4 ( 1 2 φ ) [ 1 K f ( φ ) ] / s .
x s = 2 s K f ( φ ) tan ( 1 2 φ ) K f ( φ ) tan 2 ( 1 2 φ ) + g ( φ ) z s = x s cot φ ,
g ( φ ) = s ( 1 s ) tan 2 ( 1 2 φ ) ,
f ( φ ) = | g ( φ ) / s | s / ( s 1 ) .
δ φ Φ upward-facing absorber 180 ° Φ φ 180 ° δ downward-facing absorber ,
a o , 1 = sin θ i , 1 .
a i , 1 = x p ( Φ ) x p ( δ ) = sin Φ sin δ .
C g , 1 = a i , 1 a o , 1 = sin Φ sin δ sin θ i , 1 .
f 1 t 6 r cot 2 θ d .
t j = 6 tan 2 θ d ( 1 f ) r j = c r j ,
c = 6 tan 2 θ d ( 1 f ) .
r j + 1 = r j t j + 1 = r j c r j + 1 r j + 1 = 1 1 + c r j = k r j r j = r 0 k j = 1 2 a i k j .
r j + 1 = r j t j + 1 = 1 c t j t j + 1 = 1 c t j + 1 t j + 1 = 1 1 + c t j = k t j t j = t 0 k j .
1 2 a i ( 1 g ) = 1 2 a i 1 2 a i k n n = log g log k .
t min = t 0 k n = 1 2 a i c k n .
m = a i k n t min = floor ( 1 c ) ,
N = m + n = floor ( 1 c ) + ceiling ( log g log [ 1 / ( 1 + c ) ] ) .

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