Abstract

Planar microtracking provides an alternate paradigm for solar concentration that offers the possibility of realizing high-efficiency embedded concentrating photovoltaic systems in the form factor of standard photovoltaic panels. Here, we investigate the thermodynamic limit of planar tracking optical concentrators and establish that they can, in principal, achieve the sine limit of their orientationally-tracked counterparts provided that the receiver translates a minimum distance set by the field of view half-angle. We develop a phase space methodology to optimize practical planar tracking concentrators and apply it to the design of a two surface, catadioptric system that operates with > 90% optical efficiency over a 140° field of view at geometric gains exceeding 1000×. These results provide a reference point for subsequent developments in the field and indicate that planar microtracking can achieve the high optical concentration ratio required in commercial concentrating photovoltaic systems.

© 2016 Optical Society of America

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References

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  1. R. Winston, J. C. Miñano, P. G. Benítez, and N. Shatz, Nonimaging Optics (Elsevier Academic Press, 2005).
  2. J. J. O’Gallagher, Nonimaging Optics in Solar Energy, Synthesis Lectures on Energy and the Environment (Morgan & Claypool Publishers, 2008).
  3. F. Duerr, Y. Meuret, and H. Thienpont, “Tracking integration in concentrating photovoltaics using laterally moving optics,” Opt. Express 19, A207 (2011).
    [Crossref] [PubMed]
  4. F. Duerr, Y. Meuret, and H. Thienpont, “Tailored free-form optics with movement to integrate tracking in concentrating photovoltaics,” Opt. Express 21, A401 (2013).
    [Crossref] [PubMed]
  5. J. S. Price, X. Sheng, B. M. Meulblok, J. A. Rogers, and N. C. Giebink, “Wide-angle planar microtracking for quasi-static microcell concentrating photovoltaics,” Nat. Commun. 6, 6223 (2015).
    [Crossref] [PubMed]
  6. J. H. Karp, E. J. Tremblay, and J. E. Ford, “Planar micro-optic solar concentrator,” Opt. Express 18, 1122 (2010).
    [Crossref] [PubMed]
  7. J. M. Hallas, K. A. Baker, J. H. Karp, E. J. Tremblay, and J. E. Ford, “Two-axis solar tracking accomplished through small lateral translations,” Appl. Opt. 51, 6117 (2012).
    [Crossref] [PubMed]
  8. J. H. Karp, E. J. Tremblay, J. M. Hallas, and J. E. Ford, “Orthogonal and secondary concentration in planar micro-optic solar collectors,” Opt. Express 19, A673 (2011).
    [Crossref] [PubMed]
  9. P. Kotsidas, V. Modi, and J. M. Gordon, “Nominally stationary high-concentration solar optics by gradient-index lenses,” Opt. Express 19, 2325 (2011).
    [Crossref] [PubMed]
  10. H. Apostoleris, M. Stefancich, and M. Chiesa, “Tracking-integrated systems for concentrating photovoltaics,” Nat. Energy 1, 16018 (2016).
    [Crossref]
  11. M. A. Green, “Commercial progress and challenges for photovoltaics,” Nat. Energy 1, 15015 (2016).
    [Crossref]
  12. J. Price, C. Giebink, X. Sheng, and J. Rogers, “Putting CPV on rooftops,” Compound Semiconductor 21, 44 (2015).
  13. J. C. Miñano, P. Benítez, and J. C. González, “RX: a nonimaging concentrator,” Appl. Opt. 34, 2226 (1995).
    [Crossref] [PubMed]
  14. R. K. Luneburg and M. Herzberger, Mathematical Theory of Optics (University of California, 1964).
  15. R. Winston, “Light collection within the framework of geometrical optics,” J. Opt. Soc. Am. 60, 245 (1970).
    [Crossref]
  16. J. Chaves, Introduction to Nonimaging Optics (CRC, 2008).
    [Crossref]
  17. J. M. Gordon, Solar Energy: The State of the Art (Routledge, 2001).
  18. J. Nelson, The Physics of Solar Cells (Imperial College, 2003).
    [Crossref]
  19. D. Schurig, “An aberration-free lens with zero F-number,” New J. Phys. 10, 115034 (2008).
    [Crossref]
  20. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129 (2010).
    [Crossref]
  21. R. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16, 1190 (1995).
    [Crossref]
  22. C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23, 550 (1997).
    [Crossref]
  23. E. Jones, T. Oliphant, P. Peterson, and others, “SciPy: Open source scientific tools for Python,” (2001), http://www.scipy.org .
  24. A. J. Grede and N. C. Giebink, “patsms,” Zenodo (2016). http://dx.doi.org/10.5281/zenodo.159214
  25. O. Korech, B. Hirsch, E. A. Katz, and J. M. Gordon, “High-flux characterization of ultrasmall multijunction concentrator solar cells,” Appl. Phys. Lett. 91, 064101 (2007).
    [Crossref]

2016 (2)

H. Apostoleris, M. Stefancich, and M. Chiesa, “Tracking-integrated systems for concentrating photovoltaics,” Nat. Energy 1, 16018 (2016).
[Crossref]

M. A. Green, “Commercial progress and challenges for photovoltaics,” Nat. Energy 1, 15015 (2016).
[Crossref]

2015 (2)

J. Price, C. Giebink, X. Sheng, and J. Rogers, “Putting CPV on rooftops,” Compound Semiconductor 21, 44 (2015).

J. S. Price, X. Sheng, B. M. Meulblok, J. A. Rogers, and N. C. Giebink, “Wide-angle planar microtracking for quasi-static microcell concentrating photovoltaics,” Nat. Commun. 6, 6223 (2015).
[Crossref] [PubMed]

2013 (1)

2012 (1)

2011 (3)

2010 (2)

J. H. Karp, E. J. Tremblay, and J. E. Ford, “Planar micro-optic solar concentrator,” Opt. Express 18, 1122 (2010).
[Crossref] [PubMed]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129 (2010).
[Crossref]

2008 (1)

D. Schurig, “An aberration-free lens with zero F-number,” New J. Phys. 10, 115034 (2008).
[Crossref]

2007 (1)

O. Korech, B. Hirsch, E. A. Katz, and J. M. Gordon, “High-flux characterization of ultrasmall multijunction concentrator solar cells,” Appl. Phys. Lett. 91, 064101 (2007).
[Crossref]

1997 (1)

C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23, 550 (1997).
[Crossref]

1995 (2)

R. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16, 1190 (1995).
[Crossref]

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

1970 (1)

Apostoleris, H.

H. Apostoleris, M. Stefancich, and M. Chiesa, “Tracking-integrated systems for concentrating photovoltaics,” Nat. Energy 1, 16018 (2016).
[Crossref]

Baker, K. A.

Benítez, P.

Benítez, P. G.

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

Byrd, R.

R. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16, 1190 (1995).
[Crossref]

Byrd, R. H.

C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23, 550 (1997).
[Crossref]

Chaves, J.

J. Chaves, Introduction to Nonimaging Optics (CRC, 2008).
[Crossref]

Chiesa, M.

H. Apostoleris, M. Stefancich, and M. Chiesa, “Tracking-integrated systems for concentrating photovoltaics,” Nat. Energy 1, 16018 (2016).
[Crossref]

Duerr, F.

Ford, J. E.

Giebink, C.

J. Price, C. Giebink, X. Sheng, and J. Rogers, “Putting CPV on rooftops,” Compound Semiconductor 21, 44 (2015).

Giebink, N. C.

J. S. Price, X. Sheng, B. M. Meulblok, J. A. Rogers, and N. C. Giebink, “Wide-angle planar microtracking for quasi-static microcell concentrating photovoltaics,” Nat. Commun. 6, 6223 (2015).
[Crossref] [PubMed]

González, J. C.

Gordon, J. M.

P. Kotsidas, V. Modi, and J. M. Gordon, “Nominally stationary high-concentration solar optics by gradient-index lenses,” Opt. Express 19, 2325 (2011).
[Crossref] [PubMed]

O. Korech, B. Hirsch, E. A. Katz, and J. M. Gordon, “High-flux characterization of ultrasmall multijunction concentrator solar cells,” Appl. Phys. Lett. 91, 064101 (2007).
[Crossref]

J. M. Gordon, Solar Energy: The State of the Art (Routledge, 2001).

Green, M. A.

M. A. Green, “Commercial progress and challenges for photovoltaics,” Nat. Energy 1, 15015 (2016).
[Crossref]

Hallas, J. M.

Herzberger, M.

R. K. Luneburg and M. Herzberger, Mathematical Theory of Optics (University of California, 1964).

Hirsch, B.

O. Korech, B. Hirsch, E. A. Katz, and J. M. Gordon, “High-flux characterization of ultrasmall multijunction concentrator solar cells,” Appl. Phys. Lett. 91, 064101 (2007).
[Crossref]

Karp, J. H.

Katz, E. A.

O. Korech, B. Hirsch, E. A. Katz, and J. M. Gordon, “High-flux characterization of ultrasmall multijunction concentrator solar cells,” Appl. Phys. Lett. 91, 064101 (2007).
[Crossref]

Korech, O.

O. Korech, B. Hirsch, E. A. Katz, and J. M. Gordon, “High-flux characterization of ultrasmall multijunction concentrator solar cells,” Appl. Phys. Lett. 91, 064101 (2007).
[Crossref]

Kotsidas, P.

Kundtz, N.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129 (2010).
[Crossref]

Lu, P.

C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23, 550 (1997).
[Crossref]

R. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16, 1190 (1995).
[Crossref]

Luneburg, R. K.

R. K. Luneburg and M. Herzberger, Mathematical Theory of Optics (University of California, 1964).

Meulblok, B. M.

J. S. Price, X. Sheng, B. M. Meulblok, J. A. Rogers, and N. C. Giebink, “Wide-angle planar microtracking for quasi-static microcell concentrating photovoltaics,” Nat. Commun. 6, 6223 (2015).
[Crossref] [PubMed]

Meuret, Y.

Miñano, J. C.

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

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

Modi, V.

Nelson, J.

J. Nelson, The Physics of Solar Cells (Imperial College, 2003).
[Crossref]

Nocedal, J.

C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23, 550 (1997).
[Crossref]

R. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16, 1190 (1995).
[Crossref]

O’Gallagher, J. J.

J. J. O’Gallagher, Nonimaging Optics in Solar Energy, Synthesis Lectures on Energy and the Environment (Morgan & Claypool Publishers, 2008).

Price, J.

J. Price, C. Giebink, X. Sheng, and J. Rogers, “Putting CPV on rooftops,” Compound Semiconductor 21, 44 (2015).

Price, J. S.

J. S. Price, X. Sheng, B. M. Meulblok, J. A. Rogers, and N. C. Giebink, “Wide-angle planar microtracking for quasi-static microcell concentrating photovoltaics,” Nat. Commun. 6, 6223 (2015).
[Crossref] [PubMed]

Rogers, J.

J. Price, C. Giebink, X. Sheng, and J. Rogers, “Putting CPV on rooftops,” Compound Semiconductor 21, 44 (2015).

Rogers, J. A.

J. S. Price, X. Sheng, B. M. Meulblok, J. A. Rogers, and N. C. Giebink, “Wide-angle planar microtracking for quasi-static microcell concentrating photovoltaics,” Nat. Commun. 6, 6223 (2015).
[Crossref] [PubMed]

Schurig, D.

D. Schurig, “An aberration-free lens with zero F-number,” New J. Phys. 10, 115034 (2008).
[Crossref]

Shatz, N.

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

Sheng, X.

J. S. Price, X. Sheng, B. M. Meulblok, J. A. Rogers, and N. C. Giebink, “Wide-angle planar microtracking for quasi-static microcell concentrating photovoltaics,” Nat. Commun. 6, 6223 (2015).
[Crossref] [PubMed]

J. Price, C. Giebink, X. Sheng, and J. Rogers, “Putting CPV on rooftops,” Compound Semiconductor 21, 44 (2015).

Smith, D. R.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129 (2010).
[Crossref]

Stefancich, M.

H. Apostoleris, M. Stefancich, and M. Chiesa, “Tracking-integrated systems for concentrating photovoltaics,” Nat. Energy 1, 16018 (2016).
[Crossref]

Thienpont, H.

Tremblay, E. J.

Winston, R.

R. Winston, “Light collection within the framework of geometrical optics,” J. Opt. Soc. Am. 60, 245 (1970).
[Crossref]

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

Zhu, C.

C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23, 550 (1997).
[Crossref]

R. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16, 1190 (1995).
[Crossref]

ACM Trans. Math. Softw. (1)

C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23, 550 (1997).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

O. Korech, B. Hirsch, E. A. Katz, and J. M. Gordon, “High-flux characterization of ultrasmall multijunction concentrator solar cells,” Appl. Phys. Lett. 91, 064101 (2007).
[Crossref]

Compound Semiconductor (1)

J. Price, C. Giebink, X. Sheng, and J. Rogers, “Putting CPV on rooftops,” Compound Semiconductor 21, 44 (2015).

J. Opt. Soc. Am. (1)

Nat. Commun. (1)

J. S. Price, X. Sheng, B. M. Meulblok, J. A. Rogers, and N. C. Giebink, “Wide-angle planar microtracking for quasi-static microcell concentrating photovoltaics,” Nat. Commun. 6, 6223 (2015).
[Crossref] [PubMed]

Nat. Energy (2)

H. Apostoleris, M. Stefancich, and M. Chiesa, “Tracking-integrated systems for concentrating photovoltaics,” Nat. Energy 1, 16018 (2016).
[Crossref]

M. A. Green, “Commercial progress and challenges for photovoltaics,” Nat. Energy 1, 15015 (2016).
[Crossref]

Nat. Mater. (1)

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129 (2010).
[Crossref]

New J. Phys. (1)

D. Schurig, “An aberration-free lens with zero F-number,” New J. Phys. 10, 115034 (2008).
[Crossref]

Opt. Express (5)

SIAM J. Sci. Comput. (1)

R. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16, 1190 (1995).
[Crossref]

Other (8)

E. Jones, T. Oliphant, P. Peterson, and others, “SciPy: Open source scientific tools for Python,” (2001), http://www.scipy.org .

A. J. Grede and N. C. Giebink, “patsms,” Zenodo (2016). http://dx.doi.org/10.5281/zenodo.159214

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

J. J. O’Gallagher, Nonimaging Optics in Solar Energy, Synthesis Lectures on Energy and the Environment (Morgan & Claypool Publishers, 2008).

R. K. Luneburg and M. Herzberger, Mathematical Theory of Optics (University of California, 1964).

J. Chaves, Introduction to Nonimaging Optics (CRC, 2008).
[Crossref]

J. M. Gordon, Solar Energy: The State of the Art (Routledge, 2001).

J. Nelson, The Physics of Solar Cells (Imperial College, 2003).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Comparison between traditional orientation-based tracking (a), where the optical system rotates to face the Sun, and planar tracking (b), where the optics remain fixed and the receiver translates laterally to follow the moving focal point.
Fig. 2
Fig. 2 Diagram showing the geometry of a generic orientational concentrator in (a) and a µPT concentrator in (c). In each case, the source has a finite angular spread equal to 2α and is incident on the input aperture (−uixui) immersed in a medium with a refractive index n. The output aperture has a width 2uo and a refractive index n′; its center is laterally offset at xc in the µPT case. (b) Phase space transformation for an ideal orientational concentrator, which maps the horizontal input rectangle to the vertical output rectangle while preserving its area. (d) The phase space transformation for an ideal µPT concentrator is similar, except that in this case, the input associated with each incidence angle of the source (horizontal orange rectangle) must be mapped to a different lateral output position (vertical orange rectangle) for all incidence angles within the concentrator acceptance range (represented by the horizontal gray rectangle).
Fig. 3
Fig. 3 (a) Output phase space for the cylindrical singlet lens (R-Cyl.) shown in (b) and the nonimaging concentrator (RX-SMS) shown in (c), with input and output refractive indices, n = 1 and n′ = 1.5, respectively. The dashed rectangles in (a) denote the total available output phase space and the color scale denotes the incidence angle associated with each point. The ratio of the colored area to that of the bounding rectangle is the phase fill of the concentrator. (d) Phase filling fraction for each optic as a function of increasing angular acceptance together with the maximum 2D concentration ratio that is possible for µPT operation. The phase fill of the RX-SMS design reaches unity, corresponding to operation at the sine limit, when the angular acceptance narrows to the angular width of the source (i.e. the solar disk, with 0.27° half-angle) that it is designed for.
Fig. 4
Fig. 4 Optimized two-dimensional RX µPT concentrator with a refractive index, n′ = 1.5. This design collects all light from the solar disk over a ±60° incidence angle range with a receiver size, uo = 0.078ui, which is 25× larger than that given by the sine limit. (b) Output phase space associated with this design, showing the combination of large phase fill and vertical color bands characteristic of a good µPT concentrator.
Fig. 5
Fig. 5 (a) Concentration ratio for two families of optimized 3D RX concentrators with wide (±70°) and narrow (±30°) incident angle acceptance as a function of their refractive index. The minimum optical efficiency of these designs is ηopt > 0.9, neglecting absorption and reflection losses. As emphasized by the upward arrows, each point is the result of an optimization and therefore marks a lower bound. (b) Phase fill fractions of the 3D RX concentrators. The comparatively flat trend for the ±70° family suggests a global geometric optimum over this index range whereas the large difference between the n′ = 1.7 and n′ = 1.8 outliers in the ±30° acceptance case indicates that there is still significant room for improvement within this family.
Fig. 6
Fig. 6 Series of spot diagrams for the wide-angle n′ = 1.9 concentrator from Fig 5(a) for different incidence angles at (z = 0), above (z = 4 × 10−2ui), and below (z = 4 × 10−2ui) its focal plane. Astigmatic aberration is clearly evident at oblique angles, effectively limiting the receiver size to the circle of least confusion.
Fig. 7
Fig. 7 (a) Focal spot diagram showing the impact of refractive index dispersion on an RX concentrator made of acrylic plastic over the wavelength range λ = 400 to 1800 nm. This design has a geometric gain G = 308 and operates with a solar spectrum-weighted optical efficiency, ηopt > 0.9, neglecting absorption by the acrylic. (b) Including acrylic absorption linearly decreases the optical efficiency of this concentrator as its overall size increases.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

u i u i Δ p x ( β ) d x = x c u o x c + u o Δ p x ( x ) d x .
Δ p x ( β ) = n [ sin ( β + α ) sin ( β α ) ] = 2 n sin α cos β ,
Δ p x ( x ) = 2 n .
u o = u i n n sin α cos β .
C 2 D = u i u o = n n sin α ,
u i u i p ( β + δ β / 2 ) p ( β δ β / 2 ) d x = x c δ x / 2 x c + δ x / 2 Δ p ( x ) d x u i u i 2 n sin ( δ β / 2 ) cos β d x = x c δ x / 2 x c + δ x / 2 2 n d x .
d x d β = u i n n cos β ,
x c = u i n n sin β .
η = η max [ 1 + k b T q V OC , max ln ( cos β ) ] ,
P F = x c ( β α ) x c ( β + α ) Δ p d x ( p max p min ) ( x max x min ) ,
X = σ x ( p ) d p p max p min ,

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