Optimization of the Secondary Optics for Photovoltaic Units with Fresnel Lenses

8/7/2019 Optimization of the Secondary Optics for Photovoltaic Units with Fresnel Lenses

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OPTIMIZATION OF THE SECONDARY OPTICS FOR PHOTOVOLTAIC UNITS WITH FRESNEL LENSES

V.M.Andreev1, V.A.Grilikhes1, A.A.Soluyanov2, E.V.Vlasova2, M.Z.Shvarts1

1 Ioffe Physico-Technical Institute, 26 Polytechnicheskaya str., St.-Petersburg, 194021, Russia

Tel.: +7(812) 292 7394; fax: +7(812) 297 1017; e-mail: [email protected] Technoexan LTD, 26 Polytechnicheskaya str., St.-Petersburg, 194021, Russia

ABSTRACT: The aim of the presented investigations was choosing the optimal parameters of secondary optics for

PV modules with flat Fresnel lens concentrators. The following types of secondary optics are under consideration: an open

truncated tetrahedral equilateral pyramid with specular walls and a kaleidoscope with a flat or convex top surface, which

ensure achieving high optical efficiency of a two-element optical system, lowering its sensitivity to the Sun tracking

inaccuracy and increasing uniformity of concentrated radiation distribution on a solar cell. To solve the problem raised,

simulation mathematical model for calculating the optical-power characteristics (OPC) of the “Fresnel lens-secondary

concentrator” system have been developed. As a result of the investigations carried out, the optimum parameters of the

secondary optics have been determined (tilt angle of walls and height of the pyramid or kaleidoscope, curvature radius of the

semispherical input surface), and the best optimal version has been selected for the developed high concentration flat Fresnel

lens.

Keywords: Fresnel Lens, Secondary Optics, Multijunction Solar Cell

1 INTRODUCTION

The economical estimations carried out in recent

years show that the installations with solar concentrators

and multijunction solar cells (SC) can ensure the

minimum cost of “solar” electrical power at high (up to

1000 X) average radiation concentration ratio [1, 2]. The

main obstacle to achieve the required radiation

concentration ratio in using Fresnel lenses (FL) as

concentrators is the chromatic aberration, which “smears

out” the concentrated radiation and decreases the average

level of the SC irradiance [3-8]. A corresponding choiceof the FL optimal parameters allows decreasing the

negative effect of the chromatic aberration, but to

eliminate it completely in a system comprising only the

primary concentrator is practically impossible [5-9].

Accounting for the nonuniformity of the concentrated

radiation distribution on the SC surface, one can obtain,

due to reducing the cell size, the higher average

concentration without essential losses in the total power

of radiation passed though the lens. However, in

choosing the SC dimensions, it is necessary accounting

for two contradicting tendencies: in increasing the

average concentration ratio the “FL - SC” system optical

efficiency (ηopt ) decreases and, on the contrary, the

average concentration ratio decreases with the rise of ηopt .The final choice is determined by the economical

reasoning with allowing for the design specific features

of a photovoltaic module and requirements to the

accuracy of mutual location of all its elements, and also

with allowing for requirements to the precise orientation

to the Sun of the photovoltaic system, which is especially

important for keeping the power production in real

operation conditions under action of wind and vibrations.

One of the ways for lowering the effect of the

chromatic aberration and inaccurate orientation on the

concentrator system power efficiency and for rising the

average level of the radiation concentration without

reducing the “FL - SC” system optical efficiency is the

application of the secondary concentrating optics locateddirectly before the SC surface.

As the secondary optics, specular or refracting

elements of different form and also their combinations

can be used. Among the most often met representative

secondary optical elements are a specular pyramidal

concentrator, a compound concentrator and a glass

kaleidoscope [10-14].

In this work, the results of optimizing parameters of

the secondary concentrator for the following design

versions are presented:

- open equilateral pyramid with specular walls (further

– specular pyramid);

- kaleidoscope (equilateral truncated glass pyramid

with a flat or convex top surface).

Calculation of the secondary optics parameters was

carried out for a system based on a square Fresnel lens of urethane with side dimension of 40 mm, focal distance

f = 70 mm and profile step of 0.3 mm. Investigations

were carried out for the AM 1.5D LAOD spectrum in the

range of 340 – 920 mm. The lens profile was preliminary

optimized by the procedure presented in [15].

A square solar cell of 1.4 mm x 1.4 mm in size

located in the primary lens focal plane was considered as

a receiver of the radiation concentrated by the “FL-

secondary concentrator” system. At the constructional-

geometrical parameters of a FL enumerated above,

accurate system orientation to the Sun and absence of the

secondary optics, the average concentration ratio C av on

such a receiver is 700 X, the system optical efficiency is

85.8 %. The values mentioned are obtained withaccounting for the mean-statistical value of the FL tooth

tilt angle deviation (σ) equal to 6 min. of arc and served

as the references in comparing optical systems having the

secondary optics of different types both between each

other and with a system without the secondary optics.

2 MATHEMATICAL MODEL

For each type of the secondary optics, new

mathematical models and algorithms for calculating the

optical-power characteristics (OPC) of the “FL-

secondary concentrator” system have been developed.

The mathematical models and algorithms allow tracingthe path of beams, emerging from the primary lens,

which are passing through all elements of the secondary

optical system up to their getting a receiver. In this case,

dispersion in passing beams through the system

23rd European Photovoltaic Solar Energy Conference, 1-5 September 2008, Valencia, Spain

126



refracting elements is simulated, and all main types of

energy losses on their surface and in the bulk are taken

into account: beams going by the top (input) surface of

the secondary concentrator, Fresnel losses on the

kaleidoscope top and bottom surfaces both with allowing

for an antireflective coating and without it,counterreflection of beams (not towards a SC but towards

the secondary optics element top surface), absorption in

the material bulk of the refracting elements, losses at

specular single and multiple reflection from the walls of

the pyramid and kaleidoscope.

With the use of the developed models and algorithms

for each type of the secondary optics, calculations of

concentrated radiation distribution over the SC surface,

of dependencies of the average concentration ratio C av

for a SC and of the system optical efficiency η opt on the

acceptance angle ν have been carried out.

It should be noted that, at predetermined parameters

of the primary lens and SC dimensions, changes in the

C av and η opt values result from the effect on them of onlythe secondary optics parameters, and both these values

(i.e. C av and η opt ) may equally be used for comparative

estimation of the efficiency of systems with different

type secondary concentrators. In taking this into account,

as an indicator of the system efficiency in choosing

optimum values of the secondary optics parameters, the

system optical efficiency η opt was used, since this value

was the most convenient in plotting and understanding

the dependencies.

Since the secondary concentrator parameters’

optimum values obtained for the conditions of accurate

and inaccurate orientations do not coincide in most cases,

it was necessary to formulate the criterion of their choice

in such a way that this contradiction could be resolved.For this purpose, the concession principle was used. This

principle implies that the system optimization is carried

out independently over all criteria available, and the

choice of optimal parameters is done according to one of

them, which is accepted as main one. For the rest of the

criteria such concession values are taken that determine

the acceptable level in reducing the system efficiency by

these criteria.

According to this principle, optimization of the

systems with secondary concentrators of different types

was carried out twice: by the maximum η opt criterion at

accurate orientation and by that at a typical value of the

acceptance angle ν =1°. As the main criterion, the optical

efficiency maximum at ν =1° was chosen, and the

concession value was taken equal to 10%. Optimization

of the systems with different types of secondary

concentrators by both criteria was carried out by the pull-

down method.

The calculations carried out have shown that for all

versions of the secondary optics the presence of an air

gap between its bottom surface and a SC results in

deterioration of the concentrating system efficiency.

Taking into account this circumstance, the versions of the

secondary optics with in an air gap were excluded from

consideration at this stage of the work.

To describe the set of the parameters being optimized

in Fig. 1, as an example, the kaleidoscope with a convex

top surface is presented. Parameters of this typeconcentrator being optimized are: pyramid height h,

inclination angle of its walls θ and radius of curvature of

the covering input surface Rs. In practice, more

convenient for analysis is the dimensionless parameter

R (relative radius), which is the ratio of Rs to the radius

of a circle circumscribed about the pyramid top base R:

R =Rs /R. At R =1, the covering lens is a semisphere, at

R >1 - a sphere segment, thickness of which decreases

with R and at R→∞ - the top surface becomes flat and

kaleidoscope is shaped in a form of truncated tetrahedral

equilateral pyramid.

h

Rs θ

R

a b

Figure 1: Kaleidoscope secondary concentrator: a – front

view; b – view from the top convex surface

Thus, the optimized parameters for specular pyramid

are the pyramid height h and the inclination angle of

walls θ , and those for the kaleidoscope -h, θ and R ,

correspondingly.

Other initial data used in simulating are:

- reflectance of the pyramid specular walls taken equal

to 0.93;

- material of the kaleidoscope – optical glass with

corresponding dependence of its refractive index on

wavelength;

- antireflection coating on the kaleidoscope input

surface.

Below presented are the results of parameters’optimization for the enumerated above secondary

concentrators.

3 RESULTS OF OPTIMIZATION OF SECONDARY

OPTICS PARAMETERS

3.1 Specular pyramid

Figure 2 presents dependencies of the optical

efficiency η opt on the angle of inclination of the pyramid

walls θ for different values of h at ν = 0° and at ν = 1°.

16 18 20 22 24 26

60

65

70

75

80

ν = 1o

Optical efficiency, %

Inclination angle θ , degrees

89 10 mm

ν = 0o

4 mm

6 mm

10 mm

h =18 mm

4 mm

6 mm

12 mm

12 mm

h =18 mm

Figure 2: Theoretical dependencies of the optical

efficiency (η opt ) on the inclination angle of walls (θ ) andheight (h) of a specular pyramid at the acceptance angles

ν = 0° (top graph) and ν =1° (bottom graph). h values

are indicated on a plot


127



It follows from the analysis of the dependencies that

in both cases the optimum value of θ lies in the range of

16 - 17°. Existence of the optimum by θ results from the

contradicting effect of this parameter on the value of the

radiation flow of the primary lens intercepted by the

secondary concentrator and on the value of losses atcounterreflection from the pyramid walls. Increasing the

pyramid height results in rising η opt , which is caused by

the increase in the coefficient of interception of the

primary lens radiation by the secondary concentrator

input surface. At the same time, with increasing the

pyramid height, the increase in η opt becomes less

noticeable, and at h > 18 mm, it practically stops, which

results from the rise of the amount of specular reflections

from the pyramid walls and energy losses associated with

this.

3.2 Kaleidoscope with flat top surface

Fig. 3 presents dependences of the optical efficiencyη opt on a SC on the angle of inclination of the

kaleidoscope walls θ for different h values at ν = 0° and

ν = 1°. It follows from the dependencies that more

substantial effect on this type secondary concentrator

efficiency comes from the kaleidoscope height h. This

influence is explained by two factors. First, the

kaleidoscope height affects directly the value of losses at

absorption of the sunlight in it. Second, with increasing

the kaleidoscope height, the number of reflections from

its walls rises, in most cases only the first reflection

being total (due to the total internal reflection effect), and

all following ones being accompanied by a drastic rise of

the energy losses.

14 16 18 20 22 24 26 28 30 32

56

63

70

77

8384858687

ν =0o

ν =1o

2 mm

4 mm

6 mm

8 mm


Inclination angle θ , degrees

h =10 mm

8 mm

4 mm6 mm

2 mm

Figure 3: Theoretical dependencies of the optical

efficiency η opt on the inclination angle of walls (θ ) and

height (h) of a kaleidoscope with flat top surface at the

acceptance angles ν = 0° and ν = 1°. h values are

indicated on a plot

The increase of these losses is associated with that, at

the second and following reflections, the angle of the

beam incidence on the “glass-air” demarcation line

becomes, as a rule, smaller than the limiting one, and

more and more rising part of its energy passes into the

refracted beam, and, due to this, is lost.The effect of the angle of inclination of the

kaleidoscope walls θ on its efficiency, as in the case of

the specular pyramid, is associated with contradicting

effect of this parameter on the primary lens intercepted

radiation flow value and on the value of losses at

reflection from the “glass -air” demarcation line, which

was discussed above. At a small kaleidoscope height, a

large part of beams is reflected once (i.e., without losses)

practically independently on the angle θ in a wide range

of its variation. This explains a comparatively weak dependence of the kaleidoscope efficiency on this

parameter in the region of small h value.

The maximum η opt value at accurate orientation is

achieved at h = 3 mm and the quite wide range of

inclination angles of walls θ = 18-30°.

At the presence of misorientation, the contradicting

influence of different factors on the kaleidoscope

efficiency becomes to be more complicated. This results

in appearing the optical efficiency η opt maximum in the

region of h = 8-10 mm and θ = 17-18° (see Fig. 3).

3.3 Kaleidoscope with convex top surface

The process of optimizing the kaleidoscope withconvex top surface parameters by the pull-down method

has been organized in the form of three home loops: by

the specular pyramid height h (outer loop), by the angle

of inclination of its walls θ and by the curvature relative

radius of the input surface R (inner loop). Iterations by

variable parameters inside each cycle were performed up

to achievement of the local maximum of the chosen

system efficiency. The optimum values of the secondary

concentrator parameters correspond to the largest value

of the system efficiency among obtained local maxima.

Fig. 4 presents dependencies of η opt on the inclination

angle of the kaleidoscope walls θ for its heights h from 2

to 10 mm at four values of the curvature relative radiusof the input surface R and at the acceptance angle

ν = 0°, and Fig. 5 presents the same dependencies for

ν = 1º.

It follows from the analysis of the presented

dependencies (see Fig. 4 and Fig. 5) that, to achieve

maximum values of the optical efficiency and,

correspondingly, maximum values of the average

concentration ratio in the misorientation conditions, the

curvature relative radius of the kaleidoscope top surface

must be in the limits of R ≈1.5 - 2 at inclination angle of

walls of 22 - 28º and height of 4 - 8 mm.

4 FLUX DENSITY DISTRIBUTION ON A SC

Beside capability of the secondary optics to raise the

average concentration ratio and optical efficiency, its

important property is a possibility to change the flux

density distribution character and to create uniform

irradiance of the SC surface with the aim to compensate

the negative effect of the radiation redistribution on the

multijunction SC characteristics. For this reason, to

choose an optimal “FL – secondary concentrator” optical

system, it is necessary to compare the optical-power

characteristics (OPC) of these system with the aim to

choose a version ensuring the most uniform irradiance

distribution on a SC. In the given case, of great interest is

the analysis of OPCs obtained in a system with akaleidoscope at different orientation conditions, since

possibilities to change the irradiance character by a

specular pyramid appear to be insignificant compared

with distribution produced by a primary lens.


128



18 20 22 24 26 28 30 3270

72

74

76

78

80

82

84

86

88

8 mm

6 mm

h =2 mm

4 mm


18 20 22 24 26 28 30 3270

72

74

76

78

80

82

84

86

88

Optical efficie

ncy, %

h =2 mm

4 mm6 mm

8 mm

10 mm

18 20 22 24 26 28 30 3270

72

74

76

78

80

82

84

86

88

Optical efficie

ncy, %

h =2 mm

4 mm6 mm

8 mm10 mm

18 20 22 24 26 28 30 3270

72

74

76

78

80

82

84

86

88


Inclination angle θ, degrees

h =2 mm

4 mm6 mm

8 mm

10 mm

Figure 4: Theoretical dependencies of optical efficiency

(η opt ) on the height (h) and the inclination angle of walls

(θ ) of a kaleidoscope with convex top surface at the

acceptance angle ν =0°. The curvature relative radius R is equal to: a – 1.1 , b –1.5 , c – 2 , d – 4

18 20 22 24 26 28 30 3270

72

74

76

78

80

82

84

86

88

8 mm

6 mm

2 mm

h =4 mm


18 20 22 24 26 28 30 3270

72

74

76

78

80

82

84

86

88

10 mm

8 mm6 mm

h =2 mm

4 mm

Optical efficie

ncy, %

18 20 22 24 26 28 30 3270

72

74

76

78

80

82

84

86

88

10 mm8 mm

6 mm

h =2 mm

4 mm

Optical efficie

ncy, %

18 20 22 24 26 28 30 3270

72

74

76

78

80

82

84

86

88

h =10 mm

8 mm

6 mm

4 mm

Optical efficie

ncy, %

Inclination angle θ, degrees

Figure 5: Theoretical dependencies of optical efficiency

(η opt ) on the height (h) and the inclination angle of walls

(θ ) of a kaleidoscope with convex top surface at the

acceptance angle ν =1°. The curvature relative radius R is equal to: a – 1.1 , b –1.5 , c – 2 , d – 4

a

b

c

d

a

b

c

d


129



Combined data on the optimum values of the

secondary concentrator parameters obtained by

simulation are presented in Table 1. It should be taken

into account that the optimum secondary concentrator

parameters correspond to the criterion of the maximum

η opt at the acceptance angle ν = 1º for a system withpreassigned parameters of the primary lens and SC

dimensions.

Table I: Data for the optimum types of the secondary

optics: 1-7 - kaleidoscope with convex (1-6) and flat (7)

top surface, 8 - specular pyramid

R h,

mmθ ,

degree

C loc, X

ν = 0º

C loc, X

ν = 1º

η opt , %

ν = 0º

η opt , %

ν = 1º

1 1.1 4 27 5620 8230 85.3 82.5

2 1.1 6 22 870 1220 84.0 79.2

3 1.5 6 25 5560 7880 84.8 83.1

4 2 6 25 3760 3880 84.9 83.3

5 2 4 28 3450 2690 85.8 81.7

6 4 10 20 1500 1660 83.3 81.3

7 ∞ 10 18 720 970 83.5 77.0

8 - 18 17 2340 1690 89.3 76.6

Comparison of OPCs show that the uniform

irradiance distribution can be achieved at the following

kaleidoscope parameters: h = 6 mm, θ = 22º, R = 1.1

(Fig. 6). However, in this case, the optical losses at

misorientation become significant, which result in

efficiency drop down to ~79% (Table I). The optical

losses can be to some extent decreased, if the version

with h = 10 mm, θ = 20º, R =4 (Table I and Fig. 6) isaccepted, the illumination distribution nonuniformity

rising.

Maximum optical efficiencies are achieved in the

system with R ≈1.5 - 2 (see Table I), but in these cases

it is impossible to compensate the pronounced

nonuniformity of the irradiance distribution at any

combination of h and θ parameters. The most acceptable

version with a maximum optical efficiency at the level of

85.8% at precise orientation, that comparable with

optical efficiency for a system without the secondary

optics, is achieved at the kaleidoscope parameters h = 4

mm, θ = 28º, R = 2 (the local concentration in the center

is 3450X). Passing to a position with ν = 1º for the

mentioned kaleidoscope configuration leads to forming

the more uniform light distribution (see Fig.6) in

conserving the efficiency at the level of 81.7 %.

5 COMPARISON OF THE SYSTEM EFFICIENCY

WITH THE OPTIMUM SECONDARY OPTICS.

Fig. 7 presents dependencies of the optical efficiency

of the concentrating system of the considered above type

and configuration (see Table I) on the orientation

accuracy.

Comparison shows that in a system with a specular

pyramid, due to low optical losses at refraction from the

specular walls, an advantage compared withkaleidoscope of any configuration is ensured only at the

acceptance angles less than 0.5º. At greater angles the

optical efficiency of such a system drops down to 76.6%.

0

500

1000

1500

2000

ν =0o

ν =1o

0

500

1000

1500

2000

2500

3000

3500

Concentration ratio, X

ν =1o

ν =0o

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.80

500

1000

1500

2000

Radius, mm

ν =1o

ν =0o

Figure 6: Distribution of the concentrated radiation over

the solar cell radius for the kaleidoscope systems of

following configurations (see Table I): a – 2, b – 5, c - 6

For the kaleidoscope with the convex top surface(h = 6 mm, θ = 22º, R = 1.1 ), in which, at a precise

orientation, a uniform irradiance distribution is ensured

(see Fig. 6), the optical efficiency lowers from 83.4% at

ν = 0º down to 80% at ν = 0.95º with some increase in

illumination difference over the SC surface (see Fig. 7).

A similar situation takes place also in using kaleidoscope

with a flat top surface at h = 10 mm and θ = 18º.

However, in the present case, the acceptable values of the

optical efficiency (more than 80%) are conserved at the

acceptance angle less than 0.85º.

0.0 0.2 0.4 0.6 0.8 1.076

78

80

82

84

86

88

90


Acceptance angle ν , degrees

2

4

56

7

8

Figure 7: Theoretical dependencies of the opticalefficiency on the acceptance angle ν for the systems with

the secondary optics of different configurations indicated

in Table I

a

b

c

Type


130



In going to the systems with higher values of optical

efficiency at ν = 1º, it should be searched for a

compromise between 2 and 4% reduction of optical

losses and a possible negative effect of significant

differences in illumination over the cell surface on the

operation of a multijunction SC (local concentration inthe spot are from 1500X to 8000X).

6 CONCLUSION

Mathematical models and algorithms for calculating

optical-power characteristics of the “Fl-secondary

concentrator” system have been developed. The results of

a theoretical investigation of the effect of design

parameters of secondary concentrators on the

concentrated on the SC surface radiation distribution

character and on the average radiation concentration ratio

and the concentration system optical efficiency for

different orientation angles are presented. Optimumparameters for the secondary concentrators of three types

have been determined: an open lateral pyramid with

specular walls and a kaleidoscope with the flat or convex

top surface. It has been shown that it is impossible in the

considered “Fl-secondary concentrator” systems to

ensure a uniform distribution of irradiance with

conserving a high optical efficiency and weak sensitivity

of the concentration system characteristics to the

inaccuracy of the system orientation to the Sun.

It has been found that, for the considered primary

Fresnel lens with preassigned design–geometrical

parameters, there exist versions of manufacturing the

secondary concentrator of the kaleidoscope type (see

Table I, version 2 and 7), at which a uniform irradiancedistribution on the SC surface takes place at precise

orientation to the Sun (see Fig. 6), and the optical

efficiency exceeds 83.4 %. In the misorientation

conditions an insignificant difference in illumination over

the SC surface arises at conserving the optical efficiency

at the level higher 80% at angles 0.7º and 0.5º for the

kaleidoscope version 2 and 7, correspondingly.

In going to the systems with higher optical

efficiency at ν = 1º, it should be searches for a

compromise between the negative effect of arising

differences in illumination of a cell (local concentrations

in the spot from 1500X to 8000X) and the 2-4% decrease

in optical losses.

It is obvious that the “Fresnel lens -kaleidoscope” optical system version of 2 and 6 types

(see Table I and Fig. 6) creating a more uniform

irradiance distribution are, by the sum total of effects,

preferential at their matching with multijunction SCs,

even in spite of insignificant (at the level of 2%) drop of

the optical efficiency compared with maximally

achievable rated values.

7 ACKNOWLEDGMENT

The authors wish to thank to N.Kh.Timoshina for the

technical assistance.

This work was partly supported by Russian

Foundation on Basic Research (Grants № 05-08-33603and № 07-08-13616) and by FULLSPECTRUM Project

(Contract SES6-CT-2003-502620).

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Optimization of the Secondary Optics for Photovoltaic Units with Fresnel Lenses