Abstract

Static photovoltaic concentrators, which see the sky as an extended distribution of radiance, are analyzed in a general way. The rules for achieving the highest energy on the cell are derived and the appropriate figures of merit are defined. It is concluded that casting increasingly high values of energy on the cell, which would be bifacial, require collecting a lower portion of the total sky energy. The corresponding figures of merit for the concentrators of the CPC family are analyzed, concluding that a better type of concentrator should be developed for photovoltaic applications.

© 1983 Optical Society of America

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References

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  1. A. Luque, Sol. Cells 3, 355 (1981).
    [CrossRef]
  2. A. Luque, A. Cuevas, J. M. Ruiz, Sol. Cells 2, 151 (1980).
    [CrossRef]
  3. A. Luque, J. Eguren, Solid State Electron. 25, 797 (1982).
    [CrossRef]
  4. J. C. Miñano, A. Luque, Sol. Cells 8, 297 (1983).
    [CrossRef]
  5. W. T. Welford, R. Winston, Optics of Non-Imaging Concentrators (Academic, New York, 1978).

1983

J. C. Miñano, A. Luque, Sol. Cells 8, 297 (1983).
[CrossRef]

1982

A. Luque, J. Eguren, Solid State Electron. 25, 797 (1982).
[CrossRef]

1981

A. Luque, Sol. Cells 3, 355 (1981).
[CrossRef]

1980

A. Luque, A. Cuevas, J. M. Ruiz, Sol. Cells 2, 151 (1980).
[CrossRef]

Cuevas, A.

A. Luque, A. Cuevas, J. M. Ruiz, Sol. Cells 2, 151 (1980).
[CrossRef]

Eguren, J.

A. Luque, J. Eguren, Solid State Electron. 25, 797 (1982).
[CrossRef]

Luque, A.

J. C. Miñano, A. Luque, Sol. Cells 8, 297 (1983).
[CrossRef]

A. Luque, J. Eguren, Solid State Electron. 25, 797 (1982).
[CrossRef]

A. Luque, Sol. Cells 3, 355 (1981).
[CrossRef]

A. Luque, A. Cuevas, J. M. Ruiz, Sol. Cells 2, 151 (1980).
[CrossRef]

Miñano, J. C.

J. C. Miñano, A. Luque, Sol. Cells 8, 297 (1983).
[CrossRef]

Ruiz, J. M.

A. Luque, A. Cuevas, J. M. Ruiz, Sol. Cells 2, 151 (1980).
[CrossRef]

Welford, W. T.

W. T. Welford, R. Winston, Optics of Non-Imaging Concentrators (Academic, New York, 1978).

Winston, R.

W. T. Welford, R. Winston, Optics of Non-Imaging Concentrators (Academic, New York, 1978).

Sol. Cells

A. Luque, Sol. Cells 3, 355 (1981).
[CrossRef]

A. Luque, A. Cuevas, J. M. Ruiz, Sol. Cells 2, 151 (1980).
[CrossRef]

J. C. Miñano, A. Luque, Sol. Cells 8, 297 (1983).
[CrossRef]

Solid State Electron.

A. Luque, J. Eguren, Solid State Electron. 25, 797 (1982).
[CrossRef]

Other

W. T. Welford, R. Winston, Optics of Non-Imaging Concentrators (Academic, New York, 1978).

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Figures (9)

Fig. 1
Fig. 1

Yearly averaged sky radiance of Madrid, Spain. The darkened region represents the rays coming from the ground: (a) hemisphere containing the local noon sun positions; (b) opposite hemisphere. Levels in W/m2 sr.

Fig. 2
Fig. 2

Spherical coordinates of a ray reaching the concentrator's entry aperture. Also the geometrical meaning of the p,q coordinates of this ray is shown. In general n = 1 at the entry aperture. The bundle of rays reaching the point x1, y1 which is effectively collected is also represented. The area of the region representing this bundle in the p-q plane is called local acceptance area a(x1, y1).

Fig. 3
Fig. 3

Typical shape of the function ψ ad r ̅ ( â ) showing that it is an increasing function of negative curvature passing through the origin. The straight line tangent to ψ ad r ̅ ( â ) has the equation ψ = ψ ad r ̅ ( â ) + ( a â ) r r ̅ ( â ).

Fig. 4
Fig. 4

Power flux collected by an optimally oriented adapted concentrator vs the average acceptance area â for two types of source. A value of the intercept factor does not correspond to the same value of power flux for the two sources [see Eq. (35)] since ψe(π) is different.

Fig. 5
Fig. 5

Elevation angle of the normal to the entry aperture of the optimally oriented adapted concentrator vs the average acceptance area.

Fig. 6
Fig. 6

Radiance of the isoradiance line which bounds the region of the sky collected by an optimally oriented adapted concentrator vs the average acceptance area.

Fig. 7
Fig. 7

Maximum optical concentration achievable with a given intercept factor for two values of n. The dots represent the Co and I of several cylindrical CPCs with bifacial collector. Some of them are filled with a refractive medium of n = 1.5 and the others are empty. The dot corresponding to an ideal monofacial flat panel is also shown. The straight lines correspond to concentrators with the same geometrical concentration. (a) Corresponds to the multilevel model and (b) to the three-level model. The number associated with each dot is the semi-acceptance angle Φ.

Fig. 8
Fig. 8

Degree of isotropy of CPCs filled with a medium with n = 1.5 and of empty ones vs the semiacceptance angle Φ.

Fig. 9
Fig. 9

Adaptation factor of the studied CPCs under (a) the multilevel model and (b) the three-level model.

Equations (46)

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W e = e dxdy 0 2 π d ϕ 0 π / 2 d θ sin θ R ( θ , ϕ ) cos θ ,
p = n cos ϕ sin θ ,
q = n sin ϕ sin θ ,
W e = e H R ( p , q ) dx dy dp dq = R sph A e π ,
W c = e H R sph t ( x , y , p , q ) dx dy dp dq .
a ( x , y ) = H t ( x , y , p , q ) dp dq .
W c = R sph e a ( x , y ) dx dy = R sph A e â ,
â = 1 A e a ( x , y ) dx dy
W c = 2 π n 2 g A c R sph ,
g = â 2 π n 2 A e A c = â 2 π n 2 C g ,
α ( p , q ) = 1 A e e H t ( x , y , p , q ) dx dy
â = H α ( p , q ) dp dq .
C o = 2 n 2 g ,
I = â π .
a ( x , y ) = H t ( x , y , p , q ) dp dq = â ,
Ψ ad r ̅ ( â ) = W c A e = 1 A e e H R ( p , q ) t ( x , y , p , q ) dx dy dp dq = ( â , r ̅ ) R ( p , q ) dp dq
Ψ ad r ̅ ( â ) = 0 â da ( d a , r ̅ ) R ( p , q ) dp dq = 0 â R r ̅ ( a ) da .
d Ψ ad r ̅ ( a ) da = R r ̅ ( a ) ,
Ψ ad r ̅ ( a ) Ψ ad r ̅ ( â ) + ( a â ) R r ̅ ( â )
â R r ̅ ( â ) Ψ ad r ̅ ( â ) .
Ψ c Ψ m r ̅ ( â ) .
Ψ m r ̅ ( â ) = 0 π f ( a ) Ψ ad r ̅ ( a ) da ,
â = 0 π af ( a ) da .
0 π f ( a ) da = 1 ,
Ψ m r ̅ ( â ) < Ψ ad r ̅ ( â ) ,
Ψ c Ψ ad r ̅ ( â ) .
C o = A e ψ c a c ψ e ( π ) = ψ c ψ e ( π ) C g .
C o = ψ c ψ e ( π ) 2 π n 2 g â ,
C o 2 π n 2 g ψ e ( π ) ψ ad r ̅ ( â ) â 2 π n 2 g ψ e ( π ) ψ e ( â ) â = C om ( â ) g .
d C om ( a ) d a = 2 π n 2 ψ e ( π ) a [ d ψ e ( a ) / d a ] ψ e ( a ) a 2 .
ψ e ( a ) = ψ ad r ̅ ( a ) and d [ ψ e ( a ) ] / d a = d [ ψ ad r ̅ ( a ) ] / d a
d [ C om ( a ) ] d a = 2 π n 2 ψ e ( π ) a { d [ ψ ad r ̅ ( a ) ] / d a } ψ ad r ̅ ( a ) a 2 .
I = ψ c ψ e ( π ) .
C o = I C g .
I ψ a d r ̅ ( â ) ψ e ( π ) ψ e ( â ) ψ e ( π ) .
d ψ e 1 ( x ) d x = 1 d ψ e ( a ) / d a .
â ψ e 1 [ I ψ e ( π ) ] .
C o C om { ψ e 1 [ I ψ e ( π ) ] } g = C OM ( I ) g = 2 n 2 I ψ e 1 [ I ψ e ( π ) ] g .
C o < C OM ( I ) n 2 n 2 g ,
C OM ( I ) n 2 = 2 π I ψ e 1 [ I ψ e ( π ) ] .
C o M = 2 π n 2 1 ψ e 1 [ ψ e ( π ) ] ,
ψ e 1 [ I ψ e ( π ) ] = ψ e 1 [ ψ e ( π ) ] I ,
C g = 2 n sin Φ .
g = 2 π { sin 1 [ cos Φ ( n 2 sin 2 Φ ) 1 / 2 ] + sin 1 [ ( n 2 1 n 2 sin 2 Φ ) 1 / 2 sin Φ ] n sin Φ } .
γ = C o g C o M ( I ) ; C o = γ g C o M ( I ) ;
γ = ψ e 1 ( ψ c ) â .

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