Abstract

We analyze the energy collection properties of holographic planar concentrator systems. The effects of solar variation on daily and annual energy collection are evaluated. Hologram diffraction efficiency, polarization, crosstalk in cascaded elements, and constraints imposed by the radiance theorem, as well as solar illumination characteristics, are considered. A planar holographic solar concentrator configuration is designed and modeled to maximize energy collection efficiency during the course of a year without the need for tracking. Results indicated that nearly 50% of the available energy illuminating hologram areas can be collected by photovoltaic cells without the need of tracking.

© 2010 Optical Society of America

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References

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  1. X. Deng, W. A. Schiff, “Amorphous silicon-based solar cells,” in Handbook of Photovoltaic Science and Engineering, A. Luque and S. Hegedus, eds. (Wiley, 2002), pp. 505–566.
  2. W. N. Shafarman, L. Stolt, “Cu(InGa)Se2 Solar Cells,” in Handbook of Photovoltaic Science and Engineering, A. Luque and S. Hegedus, eds. (Wiley, 2002), pp. 567–616.
  3. B. E. McCandless, J. R. Sites, “Cadmium telluride solar cells,” in Handbook of Photovoltaic Science and Engineering, A. Luque and S. Hegedus, eds. (Wiley, 2002), pp. 617–700.
  4. M. A. Green, Third Generation Photovoltaics (Springer, 2003), pp. 2–4.
  5. J. Ludman, “Holographic solar concentrator,” Appl. Opt. 21, 3057–3058 (1982).
    [CrossRef] [PubMed]
  6. C. G. Stojanoff, “Review of the technology for the manufacturing of large-format DCG holograms for technical applications,” Proc. SPIE 3011, 267–278 (1997).
    [CrossRef]
  7. W. H. Bloss, M. Griesinger, E. R. Reinhardt, “Dispersive concentrating systems based on transmission phase holograms for solar applications,” Appl. Opt. 21, 3739–3742 (1982).
    [CrossRef] [PubMed]
  8. J. Ludman, J. Riccobon, “Holographic solar concentrator for terrestrial photovoltaics,” in Conference Record of the 1994 IEEE First World Conference on Photovoltaic Energy Conversion (IEEE, 1994).
  9. A. G. Imenes, D. R. Mills, “Spectral beam splitting technology for increased conversion efficiency in solar concentrating systems: a review,” Sol. Energy Mat. Sol. Cells 84, 19–69 (2004).
    [CrossRef]
  10. E. U. Wagemann, K. Froehlich, J. Schulat, H. Schuette, C. G. Stojanoff, “Design and optimization of a holographic concentrator for two-color PV operation,” Proc. SPIE 2017, 252–263 (1993).
    [CrossRef]
  11. C. Bainier, C. Hernandez, D. Courjon, “Solar concentrating systems using holographic lenses,” Solar Wind Technol. 5, 395–404 (1988).
    [CrossRef]
  12. W. T. Welford, R. Winston, “Nonconventional optical sytems and the brightness theorem,” Appl. Opt. 21, 1531–1533 (1982).
    [CrossRef] [PubMed]
  13. R. Alferness, S. K. Case, “Coupling in doubly exposed, thick holographic gratings,” J. Opt. Soc. Am. 65, 730–739 (1975)
    [CrossRef]
  14. J. R. Riccobono, J. E. Ludman, “Solar holography,” in Holography for the New Millennium (Springer, 2002).
  15. Prism Solar Technologies, www.prismsolar.com. Private communication with Glenn Rosenberg CTO. Mass production cost of holographic collector ∼$4.00/m2 versus ∼$200−$400/m2 for silicon feedstock.
  16. E. Lorenzo, “Energy collected and delivered by PV modules,” in Handbook of Photovoltaic Science and Engineering, A. Luque and S. Hegedus, eds. (Wiley, 2002), pp. 906–930.
  17. D. R. Myers, K. Emery, C. Gueymard, “Terrestrial solar spectral modeling tools and applications for photovoltaic devices,” in Conference Record of the Twenty-Ninth IEEE Photovoltaic Specialists Conference, (IEEE, 2002).
    [CrossRef]
  18. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998).
  19. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  20. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1998).
  21. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  22. L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, 1981).
  23. Z. Li, D. Psaltis, W. Liu, W. R. Johnson, G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE 5694, 33–40 (2005).
    [CrossRef]
  24. A. Luque, Solar Cells and Optics for Photovoltaic Concentration (Institute of Physics, 1989), pp. 305, 355.
  25. R. Winston, J. C. Minaño, P. Benitez, Nonimaging Optics (Elsevier, 2005) pp. 43–68.
  26. P. Benites, J. C. Miñano, “Concentrator optics for next- generation photovoltaics,” in Next Generation Photovoltaics, A. Marti and A. Luque, eds. (Institute of Physics, 2004), pp. 285–322.
  27. J. Nilsson, M. Brogren, A. Helgesson, A. Roos, B. Karlsson, “Biaxial model for the incidence angle dependence of the optical efficiency of photovoltaic systems with asymmetric reflectors,” Solar Energy 80, 1199–1212 (2006).
    [CrossRef]

2006 (1)

J. Nilsson, M. Brogren, A. Helgesson, A. Roos, B. Karlsson, “Biaxial model for the incidence angle dependence of the optical efficiency of photovoltaic systems with asymmetric reflectors,” Solar Energy 80, 1199–1212 (2006).
[CrossRef]

2005 (1)

Z. Li, D. Psaltis, W. Liu, W. R. Johnson, G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE 5694, 33–40 (2005).
[CrossRef]

2004 (1)

A. G. Imenes, D. R. Mills, “Spectral beam splitting technology for increased conversion efficiency in solar concentrating systems: a review,” Sol. Energy Mat. Sol. Cells 84, 19–69 (2004).
[CrossRef]

1997 (1)

C. G. Stojanoff, “Review of the technology for the manufacturing of large-format DCG holograms for technical applications,” Proc. SPIE 3011, 267–278 (1997).
[CrossRef]

1993 (1)

E. U. Wagemann, K. Froehlich, J. Schulat, H. Schuette, C. G. Stojanoff, “Design and optimization of a holographic concentrator for two-color PV operation,” Proc. SPIE 2017, 252–263 (1993).
[CrossRef]

1988 (1)

C. Bainier, C. Hernandez, D. Courjon, “Solar concentrating systems using holographic lenses,” Solar Wind Technol. 5, 395–404 (1988).
[CrossRef]

1983 (1)

1982 (3)

1975 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Alferness, R.

Bainier, C.

C. Bainier, C. Hernandez, D. Courjon, “Solar concentrating systems using holographic lenses,” Solar Wind Technol. 5, 395–404 (1988).
[CrossRef]

Bearman, G.

Z. Li, D. Psaltis, W. Liu, W. R. Johnson, G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE 5694, 33–40 (2005).
[CrossRef]

Benites, P.

P. Benites, J. C. Miñano, “Concentrator optics for next- generation photovoltaics,” in Next Generation Photovoltaics, A. Marti and A. Luque, eds. (Institute of Physics, 2004), pp. 285–322.

Benitez, P.

R. Winston, J. C. Minaño, P. Benitez, Nonimaging Optics (Elsevier, 2005) pp. 43–68.

Bloss, W. H.

Brogren, M.

J. Nilsson, M. Brogren, A. Helgesson, A. Roos, B. Karlsson, “Biaxial model for the incidence angle dependence of the optical efficiency of photovoltaic systems with asymmetric reflectors,” Solar Energy 80, 1199–1212 (2006).
[CrossRef]

Case, S. K.

Cooke, D. J.

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, 1981).

Courjon, D.

C. Bainier, C. Hernandez, D. Courjon, “Solar concentrating systems using holographic lenses,” Solar Wind Technol. 5, 395–404 (1988).
[CrossRef]

Deng, X.

X. Deng, W. A. Schiff, “Amorphous silicon-based solar cells,” in Handbook of Photovoltaic Science and Engineering, A. Luque and S. Hegedus, eds. (Wiley, 2002), pp. 505–566.

Emery, K.

D. R. Myers, K. Emery, C. Gueymard, “Terrestrial solar spectral modeling tools and applications for photovoltaic devices,” in Conference Record of the Twenty-Ninth IEEE Photovoltaic Specialists Conference, (IEEE, 2002).
[CrossRef]

Froehlich, K.

E. U. Wagemann, K. Froehlich, J. Schulat, H. Schuette, C. G. Stojanoff, “Design and optimization of a holographic concentrator for two-color PV operation,” Proc. SPIE 2017, 252–263 (1993).
[CrossRef]

Gaylord, T. K.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1998).

Green, M. A.

M. A. Green, Third Generation Photovoltaics (Springer, 2003), pp. 2–4.

Griesinger, M.

Gueymard, C.

D. R. Myers, K. Emery, C. Gueymard, “Terrestrial solar spectral modeling tools and applications for photovoltaic devices,” in Conference Record of the Twenty-Ninth IEEE Photovoltaic Specialists Conference, (IEEE, 2002).
[CrossRef]

Helgesson, A.

J. Nilsson, M. Brogren, A. Helgesson, A. Roos, B. Karlsson, “Biaxial model for the incidence angle dependence of the optical efficiency of photovoltaic systems with asymmetric reflectors,” Solar Energy 80, 1199–1212 (2006).
[CrossRef]

Hernandez, C.

C. Bainier, C. Hernandez, D. Courjon, “Solar concentrating systems using holographic lenses,” Solar Wind Technol. 5, 395–404 (1988).
[CrossRef]

Imenes, A. G.

A. G. Imenes, D. R. Mills, “Spectral beam splitting technology for increased conversion efficiency in solar concentrating systems: a review,” Sol. Energy Mat. Sol. Cells 84, 19–69 (2004).
[CrossRef]

Johnson, W. R.

Z. Li, D. Psaltis, W. Liu, W. R. Johnson, G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE 5694, 33–40 (2005).
[CrossRef]

Karlsson, B.

J. Nilsson, M. Brogren, A. Helgesson, A. Roos, B. Karlsson, “Biaxial model for the incidence angle dependence of the optical efficiency of photovoltaic systems with asymmetric reflectors,” Solar Energy 80, 1199–1212 (2006).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Li, Z.

Z. Li, D. Psaltis, W. Liu, W. R. Johnson, G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE 5694, 33–40 (2005).
[CrossRef]

Liu, W.

Z. Li, D. Psaltis, W. Liu, W. R. Johnson, G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE 5694, 33–40 (2005).
[CrossRef]

Lorenzo, E.

E. Lorenzo, “Energy collected and delivered by PV modules,” in Handbook of Photovoltaic Science and Engineering, A. Luque and S. Hegedus, eds. (Wiley, 2002), pp. 906–930.

Ludman, J.

J. Ludman, “Holographic solar concentrator,” Appl. Opt. 21, 3057–3058 (1982).
[CrossRef] [PubMed]

J. Ludman, J. Riccobon, “Holographic solar concentrator for terrestrial photovoltaics,” in Conference Record of the 1994 IEEE First World Conference on Photovoltaic Energy Conversion (IEEE, 1994).

Ludman, J. E.

J. R. Riccobono, J. E. Ludman, “Solar holography,” in Holography for the New Millennium (Springer, 2002).

Luque, A.

A. Luque, Solar Cells and Optics for Photovoltaic Concentration (Institute of Physics, 1989), pp. 305, 355.

McCandless, B. E.

B. E. McCandless, J. R. Sites, “Cadmium telluride solar cells,” in Handbook of Photovoltaic Science and Engineering, A. Luque and S. Hegedus, eds. (Wiley, 2002), pp. 617–700.

Mills, D. R.

A. G. Imenes, D. R. Mills, “Spectral beam splitting technology for increased conversion efficiency in solar concentrating systems: a review,” Sol. Energy Mat. Sol. Cells 84, 19–69 (2004).
[CrossRef]

Minaño, J. C.

R. Winston, J. C. Minaño, P. Benitez, Nonimaging Optics (Elsevier, 2005) pp. 43–68.

Miñano, J. C.

P. Benites, J. C. Miñano, “Concentrator optics for next- generation photovoltaics,” in Next Generation Photovoltaics, A. Marti and A. Luque, eds. (Institute of Physics, 2004), pp. 285–322.

Moharam, M. G.

Myers, D. R.

D. R. Myers, K. Emery, C. Gueymard, “Terrestrial solar spectral modeling tools and applications for photovoltaic devices,” in Conference Record of the Twenty-Ninth IEEE Photovoltaic Specialists Conference, (IEEE, 2002).
[CrossRef]

Nilsson, J.

J. Nilsson, M. Brogren, A. Helgesson, A. Roos, B. Karlsson, “Biaxial model for the incidence angle dependence of the optical efficiency of photovoltaic systems with asymmetric reflectors,” Solar Energy 80, 1199–1212 (2006).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998).

Psaltis, D.

Z. Li, D. Psaltis, W. Liu, W. R. Johnson, G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE 5694, 33–40 (2005).
[CrossRef]

Reinhardt, E. R.

Riccobon, J.

J. Ludman, J. Riccobon, “Holographic solar concentrator for terrestrial photovoltaics,” in Conference Record of the 1994 IEEE First World Conference on Photovoltaic Energy Conversion (IEEE, 1994).

Riccobono, J. R.

J. R. Riccobono, J. E. Ludman, “Solar holography,” in Holography for the New Millennium (Springer, 2002).

Roos, A.

J. Nilsson, M. Brogren, A. Helgesson, A. Roos, B. Karlsson, “Biaxial model for the incidence angle dependence of the optical efficiency of photovoltaic systems with asymmetric reflectors,” Solar Energy 80, 1199–1212 (2006).
[CrossRef]

Schiff, W. A.

X. Deng, W. A. Schiff, “Amorphous silicon-based solar cells,” in Handbook of Photovoltaic Science and Engineering, A. Luque and S. Hegedus, eds. (Wiley, 2002), pp. 505–566.

Schuette, H.

E. U. Wagemann, K. Froehlich, J. Schulat, H. Schuette, C. G. Stojanoff, “Design and optimization of a holographic concentrator for two-color PV operation,” Proc. SPIE 2017, 252–263 (1993).
[CrossRef]

Schulat, J.

E. U. Wagemann, K. Froehlich, J. Schulat, H. Schuette, C. G. Stojanoff, “Design and optimization of a holographic concentrator for two-color PV operation,” Proc. SPIE 2017, 252–263 (1993).
[CrossRef]

Shafarman, W. N.

W. N. Shafarman, L. Stolt, “Cu(InGa)Se2 Solar Cells,” in Handbook of Photovoltaic Science and Engineering, A. Luque and S. Hegedus, eds. (Wiley, 2002), pp. 567–616.

Sites, J. R.

B. E. McCandless, J. R. Sites, “Cadmium telluride solar cells,” in Handbook of Photovoltaic Science and Engineering, A. Luque and S. Hegedus, eds. (Wiley, 2002), pp. 617–700.

Solymar, L.

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, 1981).

Stojanoff, C. G.

C. G. Stojanoff, “Review of the technology for the manufacturing of large-format DCG holograms for technical applications,” Proc. SPIE 3011, 267–278 (1997).
[CrossRef]

E. U. Wagemann, K. Froehlich, J. Schulat, H. Schuette, C. G. Stojanoff, “Design and optimization of a holographic concentrator for two-color PV operation,” Proc. SPIE 2017, 252–263 (1993).
[CrossRef]

Stolt, L.

W. N. Shafarman, L. Stolt, “Cu(InGa)Se2 Solar Cells,” in Handbook of Photovoltaic Science and Engineering, A. Luque and S. Hegedus, eds. (Wiley, 2002), pp. 567–616.

Wagemann, E. U.

E. U. Wagemann, K. Froehlich, J. Schulat, H. Schuette, C. G. Stojanoff, “Design and optimization of a holographic concentrator for two-color PV operation,” Proc. SPIE 2017, 252–263 (1993).
[CrossRef]

Welford, W. T.

Winston, R.

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

J. Opt. Soc. Am. (2)

Proc. SPIE (3)

C. G. Stojanoff, “Review of the technology for the manufacturing of large-format DCG holograms for technical applications,” Proc. SPIE 3011, 267–278 (1997).
[CrossRef]

E. U. Wagemann, K. Froehlich, J. Schulat, H. Schuette, C. G. Stojanoff, “Design and optimization of a holographic concentrator for two-color PV operation,” Proc. SPIE 2017, 252–263 (1993).
[CrossRef]

Z. Li, D. Psaltis, W. Liu, W. R. Johnson, G. Bearman, “Volume holographic spectral imaging,” Proc. SPIE 5694, 33–40 (2005).
[CrossRef]

Sol. Energy Mat. Sol. Cells (1)

A. G. Imenes, D. R. Mills, “Spectral beam splitting technology for increased conversion efficiency in solar concentrating systems: a review,” Sol. Energy Mat. Sol. Cells 84, 19–69 (2004).
[CrossRef]

Solar Energy (1)

J. Nilsson, M. Brogren, A. Helgesson, A. Roos, B. Karlsson, “Biaxial model for the incidence angle dependence of the optical efficiency of photovoltaic systems with asymmetric reflectors,” Solar Energy 80, 1199–1212 (2006).
[CrossRef]

Solar Wind Technol. (1)

C. Bainier, C. Hernandez, D. Courjon, “Solar concentrating systems using holographic lenses,” Solar Wind Technol. 5, 395–404 (1988).
[CrossRef]

Other (15)

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1998).

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, 1981).

X. Deng, W. A. Schiff, “Amorphous silicon-based solar cells,” in Handbook of Photovoltaic Science and Engineering, A. Luque and S. Hegedus, eds. (Wiley, 2002), pp. 505–566.

W. N. Shafarman, L. Stolt, “Cu(InGa)Se2 Solar Cells,” in Handbook of Photovoltaic Science and Engineering, A. Luque and S. Hegedus, eds. (Wiley, 2002), pp. 567–616.

B. E. McCandless, J. R. Sites, “Cadmium telluride solar cells,” in Handbook of Photovoltaic Science and Engineering, A. Luque and S. Hegedus, eds. (Wiley, 2002), pp. 617–700.

M. A. Green, Third Generation Photovoltaics (Springer, 2003), pp. 2–4.

J. Ludman, J. Riccobon, “Holographic solar concentrator for terrestrial photovoltaics,” in Conference Record of the 1994 IEEE First World Conference on Photovoltaic Energy Conversion (IEEE, 1994).

J. R. Riccobono, J. E. Ludman, “Solar holography,” in Holography for the New Millennium (Springer, 2002).

Prism Solar Technologies, www.prismsolar.com. Private communication with Glenn Rosenberg CTO. Mass production cost of holographic collector ∼$4.00/m2 versus ∼$200−$400/m2 for silicon feedstock.

E. Lorenzo, “Energy collected and delivered by PV modules,” in Handbook of Photovoltaic Science and Engineering, A. Luque and S. Hegedus, eds. (Wiley, 2002), pp. 906–930.

D. R. Myers, K. Emery, C. Gueymard, “Terrestrial solar spectral modeling tools and applications for photovoltaic devices,” in Conference Record of the Twenty-Ninth IEEE Photovoltaic Specialists Conference, (IEEE, 2002).
[CrossRef]

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998).

A. Luque, Solar Cells and Optics for Photovoltaic Concentration (Institute of Physics, 1989), pp. 305, 355.

R. Winston, J. C. Minaño, P. Benitez, Nonimaging Optics (Elsevier, 2005) pp. 43–68.

P. Benites, J. C. Miñano, “Concentrator optics for next- generation photovoltaics,” in Next Generation Photovoltaics, A. Marti and A. Luque, eds. (Institute of Physics, 2004), pp. 285–322.

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Figures (21)

Fig. 1
Fig. 1

Daily and seasonal operation of a single hologram of the HSC. The seasonal trajectory consequence of the variation of the declination angle of the Sun. The daily trajectory is caused by Earth rotation.

Fig. 2
Fig. 2

Sun direct irradiance dependence on the Sun zenith angle.

Fig. 3
Fig. 3

SR using solar spectrum AM 1.5 and a monocrystalline Si PV.

Fig. 4
Fig. 4

Setup used to measure DE under seasonal and daily incident angle conditions.

Fig. 5
Fig. 5

(Top) Measured DE for incident zenith angle and wavelengths. (Bottom) Measured DE as a function of zenith angle for four wavelengths.

Fig. 6
Fig. 6

(Top) ACWA results for DE as a function incident zenith angle and wavelengths. (Bottom) ACWA DE as a function of zenith angles at four different wavelengths.

Fig. 7
Fig. 7

Measured DE as a function of the wavelength and daily rotation angles (ω) for six tilt angles (β).

Fig. 8
Fig. 8

RCWA results for DE as a function of the wavelength and daily rotation angles (ω) computed for six tilt angles (β).

Fig. 9
Fig. 9

ACWA results for DE as a function of the wavelength and daily rotation angles (ω) computed for six tilt angles (β).

Fig. 10
Fig. 10

Power concentration factor for a hologram with response shown in Fig. 2, tilted at latitude angle, during day 41.

Fig. 11
Fig. 11

HSC consisting in four HOEs (A, B, A , B per unit cell (dashed box).

Fig. 12
Fig. 12

Configurations to reduce crosstalk.

Fig. 13
Fig. 13

DE for s- (solid curve) and p-(dashed curve) polarizations for the configurations shown in Fig. 12.

Fig. 14
Fig. 14

ECF as a function of thickness and index modulation at a 20 ° tilt angle.

Fig. 15
Fig. 15

Changes on Bragg wavelength along the year.

Fig. 16
Fig. 16

ECE for a single holographic collector as a function of the day of the year at a latitude of 32 ° .

Fig. 17
Fig. 17

ECF for the HSC composed of four HOEs: A, B, A , and B , as shown in Fig. 11.

Fig. 18
Fig. 18

Wavelength dispersion at the PV cell is used to calculate the required H, D H , and D P V .

Fig. 19
Fig. 19

Diffracted angles (inside the medium) for hologram B as a function of the incident angles (in air) for three wavelengths: 550 (solid gray curve), 700 (solid curve), and 850 nm (dashed curve). The circle triangle, and square indicate the Bragg condition for each wavelength.

Fig. 20
Fig. 20

Concentration profile C ( y ) over the PV cell for four different incident angles.

Fig. 21
Fig. 21

Derating parameter computed for three assumed values of series resistivity and different angles of incidence.

Equations (28)

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

δ ( d ) 23.45 sin ( 2 π 365 [ d + 284 ] ) .
cos ( θ [ t , d ] ) = sign ( ϕ ) sin ( δ [ d ] ) sin ( α ) + cos ( δ [ d ] ) cos ( α ) cos ( w [ t ] ) ,
cos ( 90 ° ψ [ t , d ] ) = cos ( θ [ t , d ] ) sin ( | ϕ | ) sin ( δ [ d ] ) sin ( θ [ t , d ] ) cos ( ϕ ) sign ( ϕ ) ,
SR ( λ , θ ) = S ( λ , θ ) R ( λ , θ ) .
u d = ( u x i K x ) x + ( u y i K y ) y + u z d z , u d = u x d x + u y d y + u z d z ,
u i = k 2 [ sin ( θ i ) cos ( ψ i ) , sin ( θ i ) sin ( ψ i ) , cos ( θ i ) ]
u d = k 2 [ sin ( θ d ) cos ( ψ d ) , sin ( θ d ) sin ( ψ d ) , cos ( θ d ) ]
u z d = k 3 2 | u x d | 2 | u y d | 2 ,
k 2 = 2 π n 2 λ , k 3 = 2 π n 3 λ .
K = 2 π n 2 λ c [ 0 , sin ( θ c 1 ) sin ( θ c 2 ) , cos ( θ c 1 ) cos ( θ c 2 ) ] ,
DE = sin 2 ( ν 2 + ξ 2 ) 1 + ξ ν 2 2 ,
ν = π Δ n t H λ c R c s ,
ϑ = 2 ( u i · K ) | K | 2 k 2 ,
ξ = ϑ d H 2 c s ,
C g = A c A P V = ( 2 D H + D P V ) / D P V ,
P P V ( t , d ) = F o e A P V cos [ θ ( t , d ) ] T [ θ ( t , d ) ] SR [ λ , θ ( t , d ) ] d λ .
T ( θ ) = 1 0.5 [ R p ( θ ) + R s ( θ ) ] ,
P H ( t , d ) = F o e A H cos [ θ ( t , d ) ] T [ θ ( t , d ) ] λ SR [ λ , θ ( t , d ) ] DE [ θ ( t , d ) , ψ ( t , d ) , λ ] d λ ,
PCF ( t , d ) = 100 P PV ( t , d ) + P H ( t , d ) P PV ( t , d ) % .
η D O ( θ , ψ ) = PCF ( θ , ψ ) C g .
C effective = η o C g = PCF ( θ max , ψ max ) .
ECF ( d ) = 100 [ P P V ( t , d ) + P H ( t , d ) ] d t P P V ( t , d ) d t % .
ECE ( d ) = 100 ECF ( d ) C g % .
D H / H tan ( | θ A s | ) ,
D P V / H tan ( | θ A l | ) ,
2 D H / H tan ( | θ B s | ) ,
( D P V + D H ) / H tan ( | θ B l | ) ,
D f = FF _ non _ uniform FF _ uniform .

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