Abstract

We introduce a design methodology for nonimaging, single-reflection mirrors with polygonal inlet apertures that generate a uniform irradiance distribution on a polygonal outlet aperture, enabling a multitude of applications within the domain of concentrated photovoltaics. Notably, we present single-mirror concentrators of square and hexagonal perimeter that achieve very high irradiance uniformity on a square receiver at concentrations ranging from 100 to 1000 suns. These optical designs can be assembled in compound concentrators with maximized active area fraction by leveraging tessellation. More advanced multi-mirror concentrators, where each mirror individually illuminates the whole area of the receiver, allow for improved performance while permitting greater flexibility for the concentrator shape and robustness against partial shading of the inlet aperture.

© 2017 Optical Society of America

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References

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  1. W. A. Beckman, P. Schoffer, W. R. Hartman, and G. O. G. Löf, “Design considerations for a 50-watt photovoltaic power system using concentrated solar energy,” Sol. Energy 10, 132–136 (1966).
    [Crossref]
  2. A. Luque and V. Andreev, Concentrator Photovoltaics, Vol. 130 in Springer Series in Optical Sciences (Springer, 2007).
  3. G. Zubi, J. L. Bernal-Agustín, and G. V. Fracastoro, “High concentration photovoltaic systems applying III-V cells,” Renew. Sustain. Energy Rev. 13, 2645–2652 (2009).
    [Crossref]
  4. M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 48),” Prog. Photovoltaics 24, 905–913 (2016).
    [Crossref]
  5. M. Buljan, J. Mendes-Lopes, P. Benítez, and J. C. Miñano, “Recent trends in concentrated photovoltaics concentrators’ architecture,” J. Photon. Energy 4, 40995 (2014).
    [Crossref]
  6. H. Baig, K. C. Heasman, and T. K. Mallick, “Non-uniform illumination in concentrating solar cells,” Renew. Sustain. Energy Rev. 16, 5890–5909 (2012).
    [Crossref]
  7. T. Cooper, High-Concentration Solar Trough Collectors and Their Application to Concentrating Photovoltaics (ETH Zurich, 2014).
  8. A. Cuevas and S. López-Romero, “The combined effect of non-uniform illumination and series resistance on the open-circuit voltage of solar cells,” Sol. Cells 11, 163–173 (1984).
    [Crossref]
  9. E. T. Franklin and J. S. Coventry, “Effects of highly non-uniform illumination distribution on electrical performance of solar cells,” in ANZSES Solar Conference, New Castle, Australia (2002).
  10. 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]
  11. A. Minuto, G. Timò, P. Groppelli, and M. Sturm, “Concentrating photovoltaic multijunction (CPVM) module electrical layout optimisation by a new theoretical and experimental ‘Mismatch’ analysis including series resistance effects,” in 35th IEEE Photovoltaic Specialists Conference (PVSC) (2010), pp. 3081–3086.
  12. M. M. Chen, J. B. Berkowitz-Mattuck, and P. E. Glaser, “The use of a kaleidoscope to obtain uniform flux over a large area in a solar or arc imaging furnace,” Appl. Opt. 2, 265 (1963).
    [Crossref]
  13. P. E. Glaser, M. M. Chen, and J. Berkowitz-Mattuck, “The flux redistributor An optical element for achieving flux uniformity,” Sol. Energy 7, 12–17 (1963).
    [Crossref]
  14. H. Ries, J. M. Gordon, and M. Lasken, “High-flux photovoltaic solar concentrators with kaleidoscope-based optical designs,” Sol. Energy 60, 11–16 (1997).
    [Crossref]
  15. K. Kreske, “Optical design of a solar flux homogenizer for concentrator photovoltaics,” Appl. Opt. 41, 2053–2058 (2002).
    [Crossref]
  16. R. Winston, “Simple Köhler homogenizers for image-forming solar concentrators,” J. Photon. Energy 1, 15503 (2011).
    [Crossref]
  17. B. M. Coughenour, T. Stalcup, B. Wheelwright, A. Geary, K. Hammer, and R. Angel, “Dish-based high concentration PV system with Köhler optics,” Opt. Express 22, A211–A224 (2014).
    [Crossref]
  18. A. Köhler, “Ein neues Beleuchtungsverfahren für mikro-photographische Zwecke,” Zeitschrift für wissenschaftliche Mikroskopie 10, 433–440 (1893).
  19. P. G. Benítez, J. C. Miñano, P. Zamora, R. Mohedano, A. Cvetkovic, M. Buljan, J. Chaves, and M. Hernández, “High performance Fresnel-based photovoltaic concentrator,” Opt. Express 18, A25–A40 (2010).
    [Crossref]
  20. R. Leutz, A. Suzuki, A. Akisawa, and T. Kashiwagi, “Flux uniformity and spectral reproduction in solar concentrators using secondary optics,” in ISES Solar World Congress (SWC) (2001), pp. 775–784.
  21. J. C. Miñano and J.-C. González, “New method of design of nonimaging concentrators,” Appl. Opt. 31, 3051–3060 (1992).
    [Crossref]
  22. The geometric concentration limit for full collection with a one-reflection 3D mirror such as a parabolic dish is given by Cg,1,max,3D=sin (2ϕ)2/sin (2θsun)2−1, where ϕ is the mirror rim angle [23], with a maximum for ϕ=45° of 11561× with θsun=4.65  mrad. The geometric concentration limit of a two-stage system is Cg,tot,max,3D=Cg,1,max,3DCg,2,max,3D with Cg,2,max,3D=cos (ϕ)2/sin (θsun)2 [24]. It is maximized for ϕ=14.86° at a value of 43191×, which is close to the theoretical limit Cg,ideal,3D=1/sin (θsun)2−1=46247×. While practical designs, especially if designed for high irradiance uniformity, fall short of these theoretical limits by a considerable margin, the comparison of the limits provides a good concept of the fundamental difference in achievable concentration.
  23. R. Winston, J. C. Miñano, P. G. Benítez, N. Shatz, and J. C. Bortz, Nonimaging Optics (2005).
  24. M. Schmitz, T. Cooper, G. Ambrosetti, and A. Steinfeld, “Two-stage solar concentrators based on parabolic troughs: asymmetric versus symmetric designs,” Appl. Opt. 54, 9709–9721 (2015).
    [Crossref]
  25. A. Giannuzzi, E. Diolaiti, M. Lombini, A. De Rosa, B. Marano, G. Bregoli, G. Cosentino, I. Foppiani, and L. Schreiber, “Enhancing the efficiency of solar concentrators by controlled optical aberrations: method and photovoltaic application,” Appl. Energy 145, 211–222 (2015).
    [Crossref]
  26. H. Chayet, O. Kost, R. Moran, and I. Lozovsky, “Efficient, low cost dish concentrator for a CPV based cogeneration system,” AIP Conf. Proc. 1407, 249–252 (2011).
    [Crossref]
  27. M.-H. Tan, K.-K. Chong, and C.-W. Wong, “Optical characterization of nonimaging dish concentrator for the application of dense-array concentrator photovoltaic system,” Appl. Opt. 53, 475–486 (2014).
    [Crossref]
  28. P. Shirley and K. Chiu, “A low distortion map between disk and square,” J. Graph. Tools 2, 45–52 (1997).
    [Crossref]
  29. J. Petrasch, “A free and open source Monte Carlo ray tracing program for concentrating solar energy research,” in Proceedings ASME 4th International Conference on Energy Sustainability (2010), Vol. 2, pp. 125–132.
  30. S. Morita, Y. Nishidate, and T. Nagata, “Ray-tracing simulation method using piecewise quadratic interpolant for aspheric optical systems,” Appl. Opt. 49, 3442–3451 (2010).
    [Crossref]
  31. T. Nagata, “Simple local interpolation of surfaces using normal vectors,” Comput. Aided Geom. Design 22, 327–347 (2005).
    [Crossref]
  32. For a concentrator design with a circular mirror, the first mapping step (ΓP1→D) can simply be omitted. A nodal grid can be generated directly on the disk and ΓD→P2 directly produces the image on the receiver. The technique for regular grid generation on a disk used in this paper is outlined in Appendix A.
  33. R. Angel, T. Stalcup, B. Wheelwright, S. Warner, K. Hammer, and M. Frenkel, “Shaping solar concentrator mirrors by radiative heating,” Proc. SPIE 9175, 91750B (2014).
    [Crossref]
  34. R. Angel, B. Cuerden, and A. Whiteside, “Lightweight dual-axis tracker designs for dish-based HCPV,” AIP Conf. Proc. 220, 220–223 (2014).
    [Crossref]
  35. K. Stephens and J. R. P. Angel, “Comparison of collection and land use efficiency for various solar concentrating field geometries,” Proc. SPIE 8468, 846804 (2012).
    [Crossref]
  36. Obstruction by the receiver is neglected throughout this section. Otherwise, the fundamental limit for convex single-reflection concentrators with axial symmetry is Cg,1,max,3D=sin2(2ϕP1+)/sin2(2θsun)−1 [37,23].
  37. D. A. Harper, R. H. Hildebrand, R. Stiening, and R. Winston, “Heat trap: an optimized far infrared field optics system,” Appl. Opt. 15, 53–60 (1976).
    [Crossref]
  38. When accounting for obstruction by the receiver, the full-collection concentration ratio with a square receiver is Cg,1,max,square=Cg,design/(2wfringe(Cg,design)1/2+1)2−1.
  39. The shaded fraction of the inlet area is 1/Cg.
  40. A. Rabl, “Comparison of solar concentrators,” Sol. Energy 18, 93–111 (1976).
    [Crossref]
  41. K. G. T. Hollands, “A concentrator for thin-film solar cells,” Sol. Energy 13, 149–163 (1971).
    [Crossref]
  42. N. Fraidenraich, “Design procedure of V-trough cavities for photovoltaic systems,” Prog. Photovoltaics 6, 43–54 (1998).
    [Crossref]
  43. K. Shanks, S. Senthilarasu, and T. K. Mallick, “High-concentration optics for photovoltaic applications,” in High Concentrator Photovoltaics: Fundamentals, Engineering and Power Plants, P. Pérez-Higueras and E. F. Fernández, eds. (Springer, 2015), pp. 85–113.
  44. The secondary optic was modeled as ideal (wall reflectance ρ=100%) to provide results independent of mirror quality. The attenuation by the secondary mirror, however, is negligible. The average number of reflections of rays through the secondary optics was determined to be 9% with the method outlined in [45]. For example, if the mirror reflectance was 90%, then only 1−0.90.09=1% of the rays would be absorbed.
  45. T. Cooper, F. Dähler, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Performance of compound parabolic concentrators with polygonal apertures,” Sol. Energy 95, 308–318 (2013).
    [Crossref]
  46. In reality, with the presented method, it is not possible that all mirrors are perfectly flat simultaneously. Mirrors in the concentrator center will always be slightly less concave than mirrors on the concentrator edge if the same area in the focal plane is to be illuminated. This is due to the inherent coma of focusing concave concentrators, i.e., off-axis rays reflected from the concentrator edge intersect the focal plane further away from the optical axis than rays reflected at the concentrator center. When increasing the number of mirrors, there is a first design point where the innermost mirror is flat while all other mirrors are still concave. Conversely, for a slightly higher number of mirrors, if the outermost mirror becomes flat, all remaining mirrors are convex. However, for an intermediate design, all mirrors can reasonably well be approximated as flat.
  47. B. Delaunay, “Sur la sphère vide,” Bull. l’Académie des Sci. l’URSS 12, 793–800 (1934).

2016 (1)

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 48),” Prog. Photovoltaics 24, 905–913 (2016).
[Crossref]

2015 (2)

M. Schmitz, T. Cooper, G. Ambrosetti, and A. Steinfeld, “Two-stage solar concentrators based on parabolic troughs: asymmetric versus symmetric designs,” Appl. Opt. 54, 9709–9721 (2015).
[Crossref]

A. Giannuzzi, E. Diolaiti, M. Lombini, A. De Rosa, B. Marano, G. Bregoli, G. Cosentino, I. Foppiani, and L. Schreiber, “Enhancing the efficiency of solar concentrators by controlled optical aberrations: method and photovoltaic application,” Appl. Energy 145, 211–222 (2015).
[Crossref]

2014 (5)

B. M. Coughenour, T. Stalcup, B. Wheelwright, A. Geary, K. Hammer, and R. Angel, “Dish-based high concentration PV system with Köhler optics,” Opt. Express 22, A211–A224 (2014).
[Crossref]

M. Buljan, J. Mendes-Lopes, P. Benítez, and J. C. Miñano, “Recent trends in concentrated photovoltaics concentrators’ architecture,” J. Photon. Energy 4, 40995 (2014).
[Crossref]

M.-H. Tan, K.-K. Chong, and C.-W. Wong, “Optical characterization of nonimaging dish concentrator for the application of dense-array concentrator photovoltaic system,” Appl. Opt. 53, 475–486 (2014).
[Crossref]

R. Angel, T. Stalcup, B. Wheelwright, S. Warner, K. Hammer, and M. Frenkel, “Shaping solar concentrator mirrors by radiative heating,” Proc. SPIE 9175, 91750B (2014).
[Crossref]

R. Angel, B. Cuerden, and A. Whiteside, “Lightweight dual-axis tracker designs for dish-based HCPV,” AIP Conf. Proc. 220, 220–223 (2014).
[Crossref]

2013 (2)

T. Cooper, F. Dähler, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Performance of compound parabolic concentrators with polygonal apertures,” Sol. 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]

2012 (2)

H. Baig, K. C. Heasman, and T. K. Mallick, “Non-uniform illumination in concentrating solar cells,” Renew. Sustain. Energy Rev. 16, 5890–5909 (2012).
[Crossref]

K. Stephens and J. R. P. Angel, “Comparison of collection and land use efficiency for various solar concentrating field geometries,” Proc. SPIE 8468, 846804 (2012).
[Crossref]

2011 (2)

R. Winston, “Simple Köhler homogenizers for image-forming solar concentrators,” J. Photon. Energy 1, 15503 (2011).
[Crossref]

H. Chayet, O. Kost, R. Moran, and I. Lozovsky, “Efficient, low cost dish concentrator for a CPV based cogeneration system,” AIP Conf. Proc. 1407, 249–252 (2011).
[Crossref]

2010 (2)

2009 (1)

G. Zubi, J. L. Bernal-Agustín, and G. V. Fracastoro, “High concentration photovoltaic systems applying III-V cells,” Renew. Sustain. Energy Rev. 13, 2645–2652 (2009).
[Crossref]

2005 (1)

T. Nagata, “Simple local interpolation of surfaces using normal vectors,” Comput. Aided Geom. Design 22, 327–347 (2005).
[Crossref]

2002 (1)

1998 (1)

N. Fraidenraich, “Design procedure of V-trough cavities for photovoltaic systems,” Prog. Photovoltaics 6, 43–54 (1998).
[Crossref]

1997 (2)

P. Shirley and K. Chiu, “A low distortion map between disk and square,” J. Graph. Tools 2, 45–52 (1997).
[Crossref]

H. Ries, J. M. Gordon, and M. Lasken, “High-flux photovoltaic solar concentrators with kaleidoscope-based optical designs,” Sol. Energy 60, 11–16 (1997).
[Crossref]

1992 (1)

1984 (1)

A. Cuevas and S. López-Romero, “The combined effect of non-uniform illumination and series resistance on the open-circuit voltage of solar cells,” Sol. Cells 11, 163–173 (1984).
[Crossref]

1976 (2)

1971 (1)

K. G. T. Hollands, “A concentrator for thin-film solar cells,” Sol. Energy 13, 149–163 (1971).
[Crossref]

1966 (1)

W. A. Beckman, P. Schoffer, W. R. Hartman, and G. O. G. Löf, “Design considerations for a 50-watt photovoltaic power system using concentrated solar energy,” Sol. Energy 10, 132–136 (1966).
[Crossref]

1963 (2)

M. M. Chen, J. B. Berkowitz-Mattuck, and P. E. Glaser, “The use of a kaleidoscope to obtain uniform flux over a large area in a solar or arc imaging furnace,” Appl. Opt. 2, 265 (1963).
[Crossref]

P. E. Glaser, M. M. Chen, and J. Berkowitz-Mattuck, “The flux redistributor An optical element for achieving flux uniformity,” Sol. Energy 7, 12–17 (1963).
[Crossref]

1934 (1)

B. Delaunay, “Sur la sphère vide,” Bull. l’Académie des Sci. l’URSS 12, 793–800 (1934).

1893 (1)

A. Köhler, “Ein neues Beleuchtungsverfahren für mikro-photographische Zwecke,” Zeitschrift für wissenschaftliche Mikroskopie 10, 433–440 (1893).

Akisawa, A.

R. Leutz, A. Suzuki, A. Akisawa, and T. Kashiwagi, “Flux uniformity and spectral reproduction in solar concentrators using secondary optics,” in ISES Solar World Congress (SWC) (2001), pp. 775–784.

Ambrosetti, G.

M. Schmitz, T. Cooper, G. Ambrosetti, and A. Steinfeld, “Two-stage solar concentrators based on parabolic troughs: asymmetric versus symmetric designs,” Appl. Opt. 54, 9709–9721 (2015).
[Crossref]

T. Cooper, F. Dähler, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Performance of compound parabolic concentrators with polygonal apertures,” Sol. 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]

Andreev, V.

A. Luque and V. Andreev, Concentrator Photovoltaics, Vol. 130 in Springer Series in Optical Sciences (Springer, 2007).

Angel, J. R. P.

K. Stephens and J. R. P. Angel, “Comparison of collection and land use efficiency for various solar concentrating field geometries,” Proc. SPIE 8468, 846804 (2012).
[Crossref]

Angel, R.

R. Angel, B. Cuerden, and A. Whiteside, “Lightweight dual-axis tracker designs for dish-based HCPV,” AIP Conf. Proc. 220, 220–223 (2014).
[Crossref]

B. M. Coughenour, T. Stalcup, B. Wheelwright, A. Geary, K. Hammer, and R. Angel, “Dish-based high concentration PV system with Köhler optics,” Opt. Express 22, A211–A224 (2014).
[Crossref]

R. Angel, T. Stalcup, B. Wheelwright, S. Warner, K. Hammer, and M. Frenkel, “Shaping solar concentrator mirrors by radiative heating,” Proc. SPIE 9175, 91750B (2014).
[Crossref]

Baig, H.

H. Baig, K. C. Heasman, and T. K. Mallick, “Non-uniform illumination in concentrating solar cells,” Renew. Sustain. Energy Rev. 16, 5890–5909 (2012).
[Crossref]

Beckman, W. A.

W. A. Beckman, P. Schoffer, W. R. Hartman, and G. O. G. Löf, “Design considerations for a 50-watt photovoltaic power system using concentrated solar energy,” Sol. Energy 10, 132–136 (1966).
[Crossref]

Benítez, P.

M. Buljan, J. Mendes-Lopes, P. Benítez, and J. C. Miñano, “Recent trends in concentrated photovoltaics concentrators’ architecture,” J. Photon. Energy 4, 40995 (2014).
[Crossref]

Benítez, P. G.

Berkowitz-Mattuck, J.

P. E. Glaser, M. M. Chen, and J. Berkowitz-Mattuck, “The flux redistributor An optical element for achieving flux uniformity,” Sol. Energy 7, 12–17 (1963).
[Crossref]

Berkowitz-Mattuck, J. B.

Bernal-Agustín, J. L.

G. Zubi, J. L. Bernal-Agustín, and G. V. Fracastoro, “High concentration photovoltaic systems applying III-V cells,” Renew. Sustain. Energy Rev. 13, 2645–2652 (2009).
[Crossref]

Bortz, J. C.

R. Winston, J. C. Miñano, P. G. Benítez, N. Shatz, and J. C. Bortz, Nonimaging Optics (2005).

Bregoli, G.

A. Giannuzzi, E. Diolaiti, M. Lombini, A. De Rosa, B. Marano, G. Bregoli, G. Cosentino, I. Foppiani, and L. Schreiber, “Enhancing the efficiency of solar concentrators by controlled optical aberrations: method and photovoltaic application,” Appl. Energy 145, 211–222 (2015).
[Crossref]

Buljan, M.

M. Buljan, J. Mendes-Lopes, P. Benítez, and J. C. Miñano, “Recent trends in concentrated photovoltaics concentrators’ architecture,” J. Photon. Energy 4, 40995 (2014).
[Crossref]

P. G. Benítez, J. C. Miñano, P. Zamora, R. Mohedano, A. Cvetkovic, M. Buljan, J. Chaves, and M. Hernández, “High performance Fresnel-based photovoltaic concentrator,” Opt. Express 18, A25–A40 (2010).
[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]

Chaves, J.

Chayet, H.

H. Chayet, O. Kost, R. Moran, and I. Lozovsky, “Efficient, low cost dish concentrator for a CPV based cogeneration system,” AIP Conf. Proc. 1407, 249–252 (2011).
[Crossref]

Chen, M. M.

P. E. Glaser, M. M. Chen, and J. Berkowitz-Mattuck, “The flux redistributor An optical element for achieving flux uniformity,” Sol. Energy 7, 12–17 (1963).
[Crossref]

M. M. Chen, J. B. Berkowitz-Mattuck, and P. E. Glaser, “The use of a kaleidoscope to obtain uniform flux over a large area in a solar or arc imaging furnace,” Appl. Opt. 2, 265 (1963).
[Crossref]

Chiu, K.

P. Shirley and K. Chiu, “A low distortion map between disk and square,” J. Graph. Tools 2, 45–52 (1997).
[Crossref]

Chong, K.-K.

Cooper, T.

M. Schmitz, T. Cooper, G. Ambrosetti, and A. Steinfeld, “Two-stage solar concentrators based on parabolic troughs: asymmetric versus symmetric designs,” Appl. Opt. 54, 9709–9721 (2015).
[Crossref]

T. Cooper, F. Dähler, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Performance of compound parabolic concentrators with polygonal apertures,” Sol. 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, High-Concentration Solar Trough Collectors and Their Application to Concentrating Photovoltaics (ETH Zurich, 2014).

Cosentino, G.

A. Giannuzzi, E. Diolaiti, M. Lombini, A. De Rosa, B. Marano, G. Bregoli, G. Cosentino, I. Foppiani, and L. Schreiber, “Enhancing the efficiency of solar concentrators by controlled optical aberrations: method and photovoltaic application,” Appl. Energy 145, 211–222 (2015).
[Crossref]

Coughenour, B. M.

Coventry, J. S.

E. T. Franklin and J. S. Coventry, “Effects of highly non-uniform illumination distribution on electrical performance of solar cells,” in ANZSES Solar Conference, New Castle, Australia (2002).

Cuerden, B.

R. Angel, B. Cuerden, and A. Whiteside, “Lightweight dual-axis tracker designs for dish-based HCPV,” AIP Conf. Proc. 220, 220–223 (2014).
[Crossref]

Cuevas, A.

A. Cuevas and S. López-Romero, “The combined effect of non-uniform illumination and series resistance on the open-circuit voltage of solar cells,” Sol. Cells 11, 163–173 (1984).
[Crossref]

Cvetkovic, A.

Dähler, F.

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

De Rosa, A.

A. Giannuzzi, E. Diolaiti, M. Lombini, A. De Rosa, B. Marano, G. Bregoli, G. Cosentino, I. Foppiani, and L. Schreiber, “Enhancing the efficiency of solar concentrators by controlled optical aberrations: method and photovoltaic application,” Appl. Energy 145, 211–222 (2015).
[Crossref]

Delaunay, B.

B. Delaunay, “Sur la sphère vide,” Bull. l’Académie des Sci. l’URSS 12, 793–800 (1934).

Diolaiti, E.

A. Giannuzzi, E. Diolaiti, M. Lombini, A. De Rosa, B. Marano, G. Bregoli, G. Cosentino, I. Foppiani, and L. Schreiber, “Enhancing the efficiency of solar concentrators by controlled optical aberrations: method and photovoltaic application,” Appl. Energy 145, 211–222 (2015).
[Crossref]

Dunlop, E. D.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 48),” Prog. Photovoltaics 24, 905–913 (2016).
[Crossref]

Emery, K.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 48),” Prog. Photovoltaics 24, 905–913 (2016).
[Crossref]

Foppiani, I.

A. Giannuzzi, E. Diolaiti, M. Lombini, A. De Rosa, B. Marano, G. Bregoli, G. Cosentino, I. Foppiani, and L. Schreiber, “Enhancing the efficiency of solar concentrators by controlled optical aberrations: method and photovoltaic application,” Appl. Energy 145, 211–222 (2015).
[Crossref]

Fracastoro, G. V.

G. Zubi, J. L. Bernal-Agustín, and G. V. Fracastoro, “High concentration photovoltaic systems applying III-V cells,” Renew. Sustain. Energy Rev. 13, 2645–2652 (2009).
[Crossref]

Fraidenraich, N.

N. Fraidenraich, “Design procedure of V-trough cavities for photovoltaic systems,” Prog. Photovoltaics 6, 43–54 (1998).
[Crossref]

Franklin, E. T.

E. T. Franklin and J. S. Coventry, “Effects of highly non-uniform illumination distribution on electrical performance of solar cells,” in ANZSES Solar Conference, New Castle, Australia (2002).

Frenkel, M.

R. Angel, T. Stalcup, B. Wheelwright, S. Warner, K. Hammer, and M. Frenkel, “Shaping solar concentrator mirrors by radiative heating,” Proc. SPIE 9175, 91750B (2014).
[Crossref]

Geary, A.

Giannuzzi, A.

A. Giannuzzi, E. Diolaiti, M. Lombini, A. De Rosa, B. Marano, G. Bregoli, G. Cosentino, I. Foppiani, and L. Schreiber, “Enhancing the efficiency of solar concentrators by controlled optical aberrations: method and photovoltaic application,” Appl. Energy 145, 211–222 (2015).
[Crossref]

Glaser, P. E.

P. E. Glaser, M. M. Chen, and J. Berkowitz-Mattuck, “The flux redistributor An optical element for achieving flux uniformity,” Sol. Energy 7, 12–17 (1963).
[Crossref]

M. M. Chen, J. B. Berkowitz-Mattuck, and P. E. Glaser, “The use of a kaleidoscope to obtain uniform flux over a large area in a solar or arc imaging furnace,” Appl. Opt. 2, 265 (1963).
[Crossref]

González, J.-C.

Gordon, J. M.

H. Ries, J. M. Gordon, and M. Lasken, “High-flux photovoltaic solar concentrators with kaleidoscope-based optical designs,” Sol. Energy 60, 11–16 (1997).
[Crossref]

Green, M. A.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 48),” Prog. Photovoltaics 24, 905–913 (2016).
[Crossref]

Groppelli, P.

A. Minuto, G. Timò, P. Groppelli, and M. Sturm, “Concentrating photovoltaic multijunction (CPVM) module electrical layout optimisation by a new theoretical and experimental ‘Mismatch’ analysis including series resistance effects,” in 35th IEEE Photovoltaic Specialists Conference (PVSC) (2010), pp. 3081–3086.

Hammer, K.

R. Angel, T. Stalcup, B. Wheelwright, S. Warner, K. Hammer, and M. Frenkel, “Shaping solar concentrator mirrors by radiative heating,” Proc. SPIE 9175, 91750B (2014).
[Crossref]

B. M. Coughenour, T. Stalcup, B. Wheelwright, A. Geary, K. Hammer, and R. Angel, “Dish-based high concentration PV system with Köhler optics,” Opt. Express 22, A211–A224 (2014).
[Crossref]

Harper, D. A.

Hartman, W. R.

W. A. Beckman, P. Schoffer, W. R. Hartman, and G. O. G. Löf, “Design considerations for a 50-watt photovoltaic power system using concentrated solar energy,” Sol. Energy 10, 132–136 (1966).
[Crossref]

Heasman, K. C.

H. Baig, K. C. Heasman, and T. K. Mallick, “Non-uniform illumination in concentrating solar cells,” Renew. Sustain. Energy Rev. 16, 5890–5909 (2012).
[Crossref]

Hernández, M.

Hildebrand, R. H.

Hishikawa, Y.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 48),” Prog. Photovoltaics 24, 905–913 (2016).
[Crossref]

Hollands, K. G. T.

K. G. T. Hollands, “A concentrator for thin-film solar cells,” Sol. Energy 13, 149–163 (1971).
[Crossref]

Kashiwagi, T.

R. Leutz, A. Suzuki, A. Akisawa, and T. Kashiwagi, “Flux uniformity and spectral reproduction in solar concentrators using secondary optics,” in ISES Solar World Congress (SWC) (2001), pp. 775–784.

Köhler, A.

A. Köhler, “Ein neues Beleuchtungsverfahren für mikro-photographische Zwecke,” Zeitschrift für wissenschaftliche Mikroskopie 10, 433–440 (1893).

Kost, O.

H. Chayet, O. Kost, R. Moran, and I. Lozovsky, “Efficient, low cost dish concentrator for a CPV based cogeneration system,” AIP Conf. Proc. 1407, 249–252 (2011).
[Crossref]

Kreske, K.

Lasken, M.

H. Ries, J. M. Gordon, and M. Lasken, “High-flux photovoltaic solar concentrators with kaleidoscope-based optical designs,” Sol. Energy 60, 11–16 (1997).
[Crossref]

Leutz, R.

R. Leutz, A. Suzuki, A. Akisawa, and T. Kashiwagi, “Flux uniformity and spectral reproduction in solar concentrators using secondary optics,” in ISES Solar World Congress (SWC) (2001), pp. 775–784.

Löf, G. O. G.

W. A. Beckman, P. Schoffer, W. R. Hartman, and G. O. G. Löf, “Design considerations for a 50-watt photovoltaic power system using concentrated solar energy,” Sol. Energy 10, 132–136 (1966).
[Crossref]

Lombini, M.

A. Giannuzzi, E. Diolaiti, M. Lombini, A. De Rosa, B. Marano, G. Bregoli, G. Cosentino, I. Foppiani, and L. Schreiber, “Enhancing the efficiency of solar concentrators by controlled optical aberrations: method and photovoltaic application,” Appl. Energy 145, 211–222 (2015).
[Crossref]

López-Romero, S.

A. Cuevas and S. López-Romero, “The combined effect of non-uniform illumination and series resistance on the open-circuit voltage of solar cells,” Sol. Cells 11, 163–173 (1984).
[Crossref]

Lozovsky, I.

H. Chayet, O. Kost, R. Moran, and I. Lozovsky, “Efficient, low cost dish concentrator for a CPV based cogeneration system,” AIP Conf. Proc. 1407, 249–252 (2011).
[Crossref]

Luque, A.

A. Luque and V. Andreev, Concentrator Photovoltaics, Vol. 130 in Springer Series in Optical Sciences (Springer, 2007).

Mallick, T. K.

H. Baig, K. C. Heasman, and T. K. Mallick, “Non-uniform illumination in concentrating solar cells,” Renew. Sustain. Energy Rev. 16, 5890–5909 (2012).
[Crossref]

K. Shanks, S. Senthilarasu, and T. K. Mallick, “High-concentration optics for photovoltaic applications,” in High Concentrator Photovoltaics: Fundamentals, Engineering and Power Plants, P. Pérez-Higueras and E. F. Fernández, eds. (Springer, 2015), pp. 85–113.

Marano, B.

A. Giannuzzi, E. Diolaiti, M. Lombini, A. De Rosa, B. Marano, G. Bregoli, G. Cosentino, I. Foppiani, and L. Schreiber, “Enhancing the efficiency of solar concentrators by controlled optical aberrations: method and photovoltaic application,” Appl. Energy 145, 211–222 (2015).
[Crossref]

Mendes-Lopes, J.

M. Buljan, J. Mendes-Lopes, P. Benítez, and J. C. Miñano, “Recent trends in concentrated photovoltaics concentrators’ architecture,” J. Photon. Energy 4, 40995 (2014).
[Crossref]

Miñano, J. C.

M. Buljan, J. Mendes-Lopes, P. Benítez, and J. C. Miñano, “Recent trends in concentrated photovoltaics concentrators’ architecture,” J. Photon. Energy 4, 40995 (2014).
[Crossref]

P. G. Benítez, J. C. Miñano, P. Zamora, R. Mohedano, A. Cvetkovic, M. Buljan, J. Chaves, and M. Hernández, “High performance Fresnel-based photovoltaic concentrator,” Opt. Express 18, A25–A40 (2010).
[Crossref]

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

R. Winston, J. C. Miñano, P. G. Benítez, N. Shatz, and J. C. Bortz, Nonimaging Optics (2005).

Minuto, A.

A. Minuto, G. Timò, P. Groppelli, and M. Sturm, “Concentrating photovoltaic multijunction (CPVM) module electrical layout optimisation by a new theoretical and experimental ‘Mismatch’ analysis including series resistance effects,” in 35th IEEE Photovoltaic Specialists Conference (PVSC) (2010), pp. 3081–3086.

Mohedano, R.

Moran, R.

H. Chayet, O. Kost, R. Moran, and I. Lozovsky, “Efficient, low cost dish concentrator for a CPV based cogeneration system,” AIP Conf. Proc. 1407, 249–252 (2011).
[Crossref]

Morita, S.

Nagata, T.

Nishidate, Y.

Pedretti, A.

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

Petrasch, J.

J. Petrasch, “A free and open source Monte Carlo ray tracing program for concentrating solar energy research,” in Proceedings ASME 4th International Conference on Energy Sustainability (2010), Vol. 2, pp. 125–132.

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.

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

Ries, H.

H. Ries, J. M. Gordon, and M. Lasken, “High-flux photovoltaic solar concentrators with kaleidoscope-based optical designs,” Sol. Energy 60, 11–16 (1997).
[Crossref]

Schmitz, M.

Schoffer, P.

W. A. Beckman, P. Schoffer, W. R. Hartman, and G. O. G. Löf, “Design considerations for a 50-watt photovoltaic power system using concentrated solar energy,” Sol. Energy 10, 132–136 (1966).
[Crossref]

Schreiber, L.

A. Giannuzzi, E. Diolaiti, M. Lombini, A. De Rosa, B. Marano, G. Bregoli, G. Cosentino, I. Foppiani, and L. Schreiber, “Enhancing the efficiency of solar concentrators by controlled optical aberrations: method and photovoltaic application,” Appl. Energy 145, 211–222 (2015).
[Crossref]

Senthilarasu, S.

K. Shanks, S. Senthilarasu, and T. K. Mallick, “High-concentration optics for photovoltaic applications,” in High Concentrator Photovoltaics: Fundamentals, Engineering and Power Plants, P. Pérez-Higueras and E. F. Fernández, eds. (Springer, 2015), pp. 85–113.

Shanks, K.

K. Shanks, S. Senthilarasu, and T. K. Mallick, “High-concentration optics for photovoltaic applications,” in High Concentrator Photovoltaics: Fundamentals, Engineering and Power Plants, P. Pérez-Higueras and E. F. Fernández, eds. (Springer, 2015), pp. 85–113.

Shatz, N.

R. Winston, J. C. Miñano, P. G. Benítez, N. Shatz, and J. C. Bortz, Nonimaging Optics (2005).

Shirley, P.

P. Shirley and K. Chiu, “A low distortion map between disk and square,” J. Graph. Tools 2, 45–52 (1997).
[Crossref]

Stalcup, T.

R. Angel, T. Stalcup, B. Wheelwright, S. Warner, K. Hammer, and M. Frenkel, “Shaping solar concentrator mirrors by radiative heating,” Proc. SPIE 9175, 91750B (2014).
[Crossref]

B. M. Coughenour, T. Stalcup, B. Wheelwright, A. Geary, K. Hammer, and R. Angel, “Dish-based high concentration PV system with Köhler optics,” Opt. Express 22, A211–A224 (2014).
[Crossref]

Steinfeld, A.

M. Schmitz, T. Cooper, G. Ambrosetti, and A. Steinfeld, “Two-stage solar concentrators based on parabolic troughs: asymmetric versus symmetric designs,” Appl. Opt. 54, 9709–9721 (2015).
[Crossref]

T. Cooper, F. Dähler, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Performance of compound parabolic concentrators with polygonal apertures,” Sol. 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]

Stephens, K.

K. Stephens and J. R. P. Angel, “Comparison of collection and land use efficiency for various solar concentrating field geometries,” Proc. SPIE 8468, 846804 (2012).
[Crossref]

Stiening, R.

Sturm, M.

A. Minuto, G. Timò, P. Groppelli, and M. Sturm, “Concentrating photovoltaic multijunction (CPVM) module electrical layout optimisation by a new theoretical and experimental ‘Mismatch’ analysis including series resistance effects,” in 35th IEEE Photovoltaic Specialists Conference (PVSC) (2010), pp. 3081–3086.

Suzuki, A.

R. Leutz, A. Suzuki, A. Akisawa, and T. Kashiwagi, “Flux uniformity and spectral reproduction in solar concentrators using secondary optics,” in ISES Solar World Congress (SWC) (2001), pp. 775–784.

Tan, M.-H.

Timò, G.

A. Minuto, G. Timò, P. Groppelli, and M. Sturm, “Concentrating photovoltaic multijunction (CPVM) module electrical layout optimisation by a new theoretical and experimental ‘Mismatch’ analysis including series resistance effects,” in 35th IEEE Photovoltaic Specialists Conference (PVSC) (2010), pp. 3081–3086.

Warner, S.

R. Angel, T. Stalcup, B. Wheelwright, S. Warner, K. Hammer, and M. Frenkel, “Shaping solar concentrator mirrors by radiative heating,” Proc. SPIE 9175, 91750B (2014).
[Crossref]

Warta, W.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 48),” Prog. Photovoltaics 24, 905–913 (2016).
[Crossref]

Wheelwright, B.

R. Angel, T. Stalcup, B. Wheelwright, S. Warner, K. Hammer, and M. Frenkel, “Shaping solar concentrator mirrors by radiative heating,” Proc. SPIE 9175, 91750B (2014).
[Crossref]

B. M. Coughenour, T. Stalcup, B. Wheelwright, A. Geary, K. Hammer, and R. Angel, “Dish-based high concentration PV system with Köhler optics,” Opt. Express 22, A211–A224 (2014).
[Crossref]

Whiteside, A.

R. Angel, B. Cuerden, and A. Whiteside, “Lightweight dual-axis tracker designs for dish-based HCPV,” AIP Conf. Proc. 220, 220–223 (2014).
[Crossref]

Winston, R.

R. Winston, “Simple Köhler homogenizers for image-forming solar concentrators,” J. Photon. Energy 1, 15503 (2011).
[Crossref]

D. A. Harper, R. H. Hildebrand, R. Stiening, and R. Winston, “Heat trap: an optimized far infrared field optics system,” Appl. Opt. 15, 53–60 (1976).
[Crossref]

R. Winston, J. C. Miñano, P. G. Benítez, N. Shatz, and J. C. Bortz, Nonimaging Optics (2005).

Wong, C.-W.

Zamora, P.

Zubi, G.

G. Zubi, J. L. Bernal-Agustín, and G. V. Fracastoro, “High concentration photovoltaic systems applying III-V cells,” Renew. Sustain. Energy Rev. 13, 2645–2652 (2009).
[Crossref]

AIP Conf. Proc. (2)

H. Chayet, O. Kost, R. Moran, and I. Lozovsky, “Efficient, low cost dish concentrator for a CPV based cogeneration system,” AIP Conf. Proc. 1407, 249–252 (2011).
[Crossref]

R. Angel, B. Cuerden, and A. Whiteside, “Lightweight dual-axis tracker designs for dish-based HCPV,” AIP Conf. Proc. 220, 220–223 (2014).
[Crossref]

Appl. Energy (1)

A. Giannuzzi, E. Diolaiti, M. Lombini, A. De Rosa, B. Marano, G. Bregoli, G. Cosentino, I. Foppiani, and L. Schreiber, “Enhancing the efficiency of solar concentrators by controlled optical aberrations: method and photovoltaic application,” Appl. Energy 145, 211–222 (2015).
[Crossref]

Appl. Opt. (7)

Bull. l’Académie des Sci. l’URSS (1)

B. Delaunay, “Sur la sphère vide,” Bull. l’Académie des Sci. l’URSS 12, 793–800 (1934).

Comput. Aided Geom. Design (1)

T. Nagata, “Simple local interpolation of surfaces using normal vectors,” Comput. Aided Geom. Design 22, 327–347 (2005).
[Crossref]

J. Graph. Tools (1)

P. Shirley and K. Chiu, “A low distortion map between disk and square,” J. Graph. Tools 2, 45–52 (1997).
[Crossref]

J. Photon. Energy (2)

R. Winston, “Simple Köhler homogenizers for image-forming solar concentrators,” J. Photon. Energy 1, 15503 (2011).
[Crossref]

M. Buljan, J. Mendes-Lopes, P. Benítez, and J. C. Miñano, “Recent trends in concentrated photovoltaics concentrators’ architecture,” J. Photon. Energy 4, 40995 (2014).
[Crossref]

Opt. Express (2)

Proc. SPIE (2)

K. Stephens and J. R. P. Angel, “Comparison of collection and land use efficiency for various solar concentrating field geometries,” Proc. SPIE 8468, 846804 (2012).
[Crossref]

R. Angel, T. Stalcup, B. Wheelwright, S. Warner, K. Hammer, and M. Frenkel, “Shaping solar concentrator mirrors by radiative heating,” Proc. SPIE 9175, 91750B (2014).
[Crossref]

Prog. Photovoltaics (2)

N. Fraidenraich, “Design procedure of V-trough cavities for photovoltaic systems,” Prog. Photovoltaics 6, 43–54 (1998).
[Crossref]

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 48),” Prog. Photovoltaics 24, 905–913 (2016).
[Crossref]

Renew. Sustain. Energy Rev. (2)

G. Zubi, J. L. Bernal-Agustín, and G. V. Fracastoro, “High concentration photovoltaic systems applying III-V cells,” Renew. Sustain. Energy Rev. 13, 2645–2652 (2009).
[Crossref]

H. Baig, K. C. Heasman, and T. K. Mallick, “Non-uniform illumination in concentrating solar cells,” Renew. Sustain. Energy Rev. 16, 5890–5909 (2012).
[Crossref]

Sol. Cells (1)

A. Cuevas and S. López-Romero, “The combined effect of non-uniform illumination and series resistance on the open-circuit voltage of solar cells,” Sol. Cells 11, 163–173 (1984).
[Crossref]

Sol. Energy (6)

W. A. Beckman, P. Schoffer, W. R. Hartman, and G. O. G. Löf, “Design considerations for a 50-watt photovoltaic power system using concentrated solar energy,” Sol. Energy 10, 132–136 (1966).
[Crossref]

P. E. Glaser, M. M. Chen, and J. Berkowitz-Mattuck, “The flux redistributor An optical element for achieving flux uniformity,” Sol. Energy 7, 12–17 (1963).
[Crossref]

H. Ries, J. M. Gordon, and M. Lasken, “High-flux photovoltaic solar concentrators with kaleidoscope-based optical designs,” Sol. Energy 60, 11–16 (1997).
[Crossref]

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

K. G. T. Hollands, “A concentrator for thin-film solar cells,” Sol. Energy 13, 149–163 (1971).
[Crossref]

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

Sol. Energy Mater. Sol. Cells (1)

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]

Zeitschrift für wissenschaftliche Mikroskopie (1)

A. Köhler, “Ein neues Beleuchtungsverfahren für mikro-photographische Zwecke,” Zeitschrift für wissenschaftliche Mikroskopie 10, 433–440 (1893).

Other (15)

A. Minuto, G. Timò, P. Groppelli, and M. Sturm, “Concentrating photovoltaic multijunction (CPVM) module electrical layout optimisation by a new theoretical and experimental ‘Mismatch’ analysis including series resistance effects,” in 35th IEEE Photovoltaic Specialists Conference (PVSC) (2010), pp. 3081–3086.

A. Luque and V. Andreev, Concentrator Photovoltaics, Vol. 130 in Springer Series in Optical Sciences (Springer, 2007).

E. T. Franklin and J. S. Coventry, “Effects of highly non-uniform illumination distribution on electrical performance of solar cells,” in ANZSES Solar Conference, New Castle, Australia (2002).

T. Cooper, High-Concentration Solar Trough Collectors and Their Application to Concentrating Photovoltaics (ETH Zurich, 2014).

Obstruction by the receiver is neglected throughout this section. Otherwise, the fundamental limit for convex single-reflection concentrators with axial symmetry is Cg,1,max,3D=sin2(2ϕP1+)/sin2(2θsun)−1 [37,23].

When accounting for obstruction by the receiver, the full-collection concentration ratio with a square receiver is Cg,1,max,square=Cg,design/(2wfringe(Cg,design)1/2+1)2−1.

The shaded fraction of the inlet area is 1/Cg.

For a concentrator design with a circular mirror, the first mapping step (ΓP1→D) can simply be omitted. A nodal grid can be generated directly on the disk and ΓD→P2 directly produces the image on the receiver. The technique for regular grid generation on a disk used in this paper is outlined in Appendix A.

R. Leutz, A. Suzuki, A. Akisawa, and T. Kashiwagi, “Flux uniformity and spectral reproduction in solar concentrators using secondary optics,” in ISES Solar World Congress (SWC) (2001), pp. 775–784.

The geometric concentration limit for full collection with a one-reflection 3D mirror such as a parabolic dish is given by Cg,1,max,3D=sin (2ϕ)2/sin (2θsun)2−1, where ϕ is the mirror rim angle [23], with a maximum for ϕ=45° of 11561× with θsun=4.65  mrad. The geometric concentration limit of a two-stage system is Cg,tot,max,3D=Cg,1,max,3DCg,2,max,3D with Cg,2,max,3D=cos (ϕ)2/sin (θsun)2 [24]. It is maximized for ϕ=14.86° at a value of 43191×, which is close to the theoretical limit Cg,ideal,3D=1/sin (θsun)2−1=46247×. While practical designs, especially if designed for high irradiance uniformity, fall short of these theoretical limits by a considerable margin, the comparison of the limits provides a good concept of the fundamental difference in achievable concentration.

R. Winston, J. C. Miñano, P. G. Benítez, N. Shatz, and J. C. Bortz, Nonimaging Optics (2005).

J. Petrasch, “A free and open source Monte Carlo ray tracing program for concentrating solar energy research,” in Proceedings ASME 4th International Conference on Energy Sustainability (2010), Vol. 2, pp. 125–132.

In reality, with the presented method, it is not possible that all mirrors are perfectly flat simultaneously. Mirrors in the concentrator center will always be slightly less concave than mirrors on the concentrator edge if the same area in the focal plane is to be illuminated. This is due to the inherent coma of focusing concave concentrators, i.e., off-axis rays reflected from the concentrator edge intersect the focal plane further away from the optical axis than rays reflected at the concentrator center. When increasing the number of mirrors, there is a first design point where the innermost mirror is flat while all other mirrors are still concave. Conversely, for a slightly higher number of mirrors, if the outermost mirror becomes flat, all remaining mirrors are convex. However, for an intermediate design, all mirrors can reasonably well be approximated as flat.

K. Shanks, S. Senthilarasu, and T. K. Mallick, “High-concentration optics for photovoltaic applications,” in High Concentrator Photovoltaics: Fundamentals, Engineering and Power Plants, P. Pérez-Higueras and E. F. Fernández, eds. (Springer, 2015), pp. 85–113.

The secondary optic was modeled as ideal (wall reflectance ρ=100%) to provide results independent of mirror quality. The attenuation by the secondary mirror, however, is negligible. The average number of reflections of rays through the secondary optics was determined to be 9% with the method outlined in [45]. For example, if the mirror reflectance was 90%, then only 1−0.90.09=1% of the rays would be absorbed.

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Figures (18)

Fig. 1.
Fig. 1. (top) Area-conserving, low-distortion mapping ΓP1D of a point pP1 on a polygon with NP1 sides and area AP1=1 to a point pD on a disk with area AD=1, illustrated for the case NP1=6 (hexagon). (bottom) Area-conserving, low-distortion mapping ΓDP2 of a point pD on a disk with area AD=1 to a point pP2 on a polygon with NP2 sides and area AP2=1, illustrated for the case NP2=4 (square). By combining the two steps, an area-conserving, low-distortion mapping ΓP1P2 between two polygons with different numbers of sides can be achieved.
Fig. 2.
Fig. 2. Schematic illustrating the scaling and translation of the mirror and the receiver.
Fig. 3.
Fig. 3. Schematic outlining the surface optimization procedure for an on-axis mirror. (a) Situation before the optimization. All nodes x1,i on the mirror surface lie at z1,i=0. The tangent vectors t^i of the mirror surface are horizontal and, with the exception of the central node, not perpendicular to the normal vectors n^i, required to reflect an on-axis incident ray r^i by r^o,i to the corresponding node on the image x2,i; (b) Situation at the end of the optimization. Eq. (7) is fulfilled (t^j and n^j are perpendicular) at each node x1,j (with minimized rms error).
Fig. 4.
Fig. 4. Schematics of exemplary single-mirror on-axis concentrators. (a) Square mirror (NP1=4); (b) hexagonal mirror (NP1=6); (c) circular mirror (NP1). All concentrators have inlet area A=1, focal length f=1 and a square receiver (NP2=4).
Fig. 5.
Fig. 5. Schematics of potential applications of tessellated polygonal concentrators for CPV. (a) Rectangular compound design, where several square on-axis concentrators, each with its own receiver, are mounted together on the same tracker; (b) similar design with hexagonal concentrators and a circular outer perimeter; (c), (d) concentrator panel, where miniature hexagonal concentrators are tessellated and protected by a transparent cover, which is also used to hold in place the CPV cells.
Fig. 6.
Fig. 6. Exemplary multi-mirror designs based on square (NP1=4) mirrors, where each mirror is designed such that it uniformly distributes incident rays over the complete receiver (image area). (a) Square concentrator with a similar perimeter to Fig. 4(a) but composed of four individual mirrors; (b) same design with nine mirrors; (c) rectangular concentrator composed of eight mirrors and having an aspect ratio of 2, which can be beneficial for the shading efficiency in a field of dishes; (d) circular dish composed of 37 square mirrors, e.g., allowing very large inlet apertures and having the structural advantages of circular perimeters.
Fig. 7.
Fig. 7. Concentrator where multiple off-axis mirrors redirect radiation to multiple adjacent receivers in the focal plane.
Fig. 8.
Fig. 8. Mapping of a uniform grid of equally-spaced nodes from a hexagon (NP1=6) (a) to a square (NP2=4) (c) via the intermediate of a disk (b), as used for the shape optimization of the hexagonal concentrator from Fig. 4(c). All geometries have an equal surface area AP1=AD=AP2=1. The area of each element made up of three neighboring nodes is conserved throughout the mapping procedure. The colors of the nodes indicate their initial region within the hexagon.
Fig. 9.
Fig. 9. Mirror profile difference between optimized hexagonal concentrators with Cg,design=100×, 500×, and 1000× and a parabolic dish, normalized with the mirror apothem a1. (a) Contour plot of the profile difference; (b) radial difference along the direction of minimal offset (x>0, y=0; solid lines) and the direction of maximal offset (x>0, y=tan(π/6)x; dashed lines). With increasing Cg,design, the mirror shape approaches that of the parabola as the nodes on the image merge into a single focal point.
Fig. 10.
Fig. 10. Irradiance distribution on a square receiver (NP2=4) produced with a square (NP1=4), hexagonal (NP1=6) and disk (NP1) primary mirror with Cg,design=100× (top row); 500× (middle row); and 1000× (bottom row). The receiver coordinates are normalized by the apothem of the scaled image, a2. While the square primary produces a very uniform distribution for all Cg,design, artifacts of the mapping are visible for the circular and hexagonal primary at low concentrations.
Fig. 11.
Fig. 11. Cell-to-cell irradiance uniformity on a 6×6 cell array in the receiver plane plotted versus the intercept factor achieved within this array, for hexagonal mirrors with Cg,design=100×, 500×, and 1000×. The circles (•) indicate the intercept factor and uniformity for a receiver having the size of the image area. The downward pointing triangles (▾) show the influence on the intercept factor and uniformity when taking shading of incident radiation by the receiver into account, and upward pointing triangles (▴) indicate the improvement that is possible by using a simple secondary concentrator design. The dotted lines indicate irrational receiver designs to the left of the uniformity peak. For every such receiver, there exists a larger receiver on the opposite side of the peak that achieves the same uniformity with a higher intercept factor.
Fig. 12.
Fig. 12. Irradiance distributions without (left) and with (right) secondary concentrator and including shading effects, for a hexagonal concentrator with Cg,design=500×. The receiver (secondary outlet aperture) has the exact dimensions of the image area; the irradiance outside this area is disregarded. (top) Local irradiance distributions. Most of the fringe radiation that would otherwise impinge outside the receiver can be redirected to the receiver edge. As a result, both the average irradiance and the uniformity are substantially increased. While in the case without secondary stage the shading by the receiver can be neglected, the shading produced in the receiver center by obstruction of the receiver and secondary optics is significant; (bottom) cell-averaged irradiance on a 6×6 cell array. As the low central irradiance caused by shading is distributed over four cells, the average irradiance is only marginally affected.
Fig. 13.
Fig. 13. Contour plots of the normalized mirror profile difference to a parabolic dish of (a) a single-, (b) four-, and (c) a nine-mirror concentrator with square aperture and Cg,design=500×.
Fig. 14.
Fig. 14. Comparison of the irradiance distributions produced by the four- and nine-mirror concentrators to the distribution produced by the single-mirror concentrator. (top) Local irradiance; (bottom) irradiance difference with respect to the single-mirror irradiance, normalized by the design concentration. The multi-mirror designs redirect more rays towards the inside of the image area.
Fig. 15.
Fig. 15. Comparison of the irradiance distribution on the receiver produced by square (a) single-, (b) four- and (c) nine-mirror concentrators where a corner corresponding to one-ninth (top row) and one-fourth (bottom row) of the inlet aperture is obstructed. The fractions indicate the portion of the unshaded irradiance received by the different areas on the receiver.
Fig. 16.
Fig. 16. (a) Schematic showing the generation of a uniform grid on a regular polygon with apothem aP1, illustrated for the example of a hexagon (NP1=6) and k=6 nodes along each basis vector (total number of nodes N=91). The red nodes are created by the basis {uv} in the first region. The remaining nodes are obtained by rotation of {uv} around the origin; (b) schematic showing the generation of an approximately uniform grid on a disk with radius RD and k=6 nodes along x>0 (marked in red). The total number of nodes is N=95. Using Delaunay triangulation of the nodes, triangular equal-area elements can be generated.
Fig. 17.
Fig. 17. Schematic showing the approximation of the surface using the mid-nodes of an edge connecting two nodes. The surface element ijk is obtained by Delaunay triangulation of the surface nodes. The tangent vectors to the surface needed for the optimization can more easily and universally be computed on the mid-nodes of the element edges.
Fig. 18.
Fig. 18. Influence of number of nodes n on the irradiance intercepted within the image area, for a hexagonal primary mirror with Cg,design=500×, A1=1 and f=1. N=500 provides reasonably good results for performance metrics, while N2500 is required for smooth, high-resolution irradiance maps.

Tables (5)

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Table 1. Average Irradiance E within the Image Area for a Square (NP1=4), Hexagonal (NP1=6) and Disk (NP1) Primary Mirror with Cg,design=100×, 500×, and 1000×, and a Square Receiver (NP2=4)

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Table 2. Intercept Factor, Cell-To-Cell Uniformity and Standard Deviation with a Receiver Having the Size of the Image Area, for Hexagonal Concentrators with Cg,design=100×, 500×, and 1000×

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Table 3. Offset of Actual Rim Angles on a Hexagonal Concentrator Compared to the Parabolic Dish

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Table 4. Major and Minor Rim Angles for Different Polygonal Concentratorsa

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Table 5. Offset of Actual Rim Angles on a Hexagonal Concentrator Compared to the Parabolic Dish

Equations (14)

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pD=ΓP1D(pP1)=[xDyD]=rD[sin(φD)cos(φD)],with{rD=(RD/aP1)x1φD=(aP12/RD2)(y1/x1).
pP2=ΓDP2(pD)=[x2y2]=[(aP2/RD)rD(RD/aP2)rDφD].
pP2=ΓP1P2(pP1)=ΓDP2(ΓP1D(pP1))=[(aP2/aP1)x1(aP1/aP2)y1].
x1,n=[x1,n+x1,cy1,n+y1,cz1,n+z1,c],
x2,n=[Cg,design1/2x2,nCg,design1/2y2,nf].
n^n=r^i+r^o,nr^i+r^o,n.
t^nn^n=0.
γi=Q˙iQ˙tot=AiE(x,y)dxdyAtotE(x,y)dxdy.
U=min(Eav,i)Eav,
N=1+i=1k1ni=1+i=1k1round(2πi).
tijn^ij=0(zn^,i+zn^,j)(zjzi)=(xn^,i+xn^,j)(xixj)+(yn^,i+yn^,j)(yiyj),
A·z=b,with{(A)m,n=(zn^,i+zn^,j)(δnjδni)(z)n=zn(b)m=(xn^,i+xn^,j)(xixj)+(yn^,i+yn^,j)(yiyj),
tan(ϕP1)=aP1f1/(4f)aP12=4aP1f4f2aP12.
tan(ϕP1+)=4RP1f4f2RP12.

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