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Performance evaluation of two-dimensional compound elliptic lens concentrators using a yearly distributed insolation model
Solar Energy Materials & Solar Cells 57 (1999) 9—19
Performance evaluation of two-dimensional compound elliptic lens concentrators using a yearly distributed insolation model K. Yoshioka*, A. Suzuki, T. Saitoh Division of Electronics and Information Technology, Tokyo A&T University, 2-24-16 Nakamachi, Koganei, Tokyo 184-8588, Japan
Abstract Optical performance of a two-dimensional compound elliptic lens (2D-CEL) for a photovoltaic static concentrator module has been studied as a function of half-acceptance angle using an yearly distributed insolation model. The maximum yearly averaged optical concentration ratio of 1.75 was obtained for global radiation when the 2D-CEL was installed at a tilt angle equal to the latitude of Tokyo (N35°). Also, the 2D-CEL was found to be more advantageous for collection of direct and diffuse radiations than a conventional, compound parabolic concentrator. In the case where the 2D-CEL is installed at a tilt angle of 20—35°, the maximum yearly integrated irradiance collected on the receiver was found to be 1.45 at a suitable half-acceptance angle. 1999 Elsevier Science B.V. All rights reserved. Keywords: Two-dimensional compound elliptic tens (2D-CEL); Two-dimensional compound parabolic concentrator (2D-CPC); Optical concentration ratio (OCR)
1. Introduction In Japan, dissemination of residential PV systems is a key point for the growth of PV markets. By a subsidy policy started in 1994, the residential PV systems have steadily increased. Presently, PV modules for the residential PV systems are almost crystalline Si ones, but there are still two issues of production cost and Si feedstock
* Corresponding author. E-mail: [email protected] 0927-0248/99/$ — See front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 4 8 ( 9 8 ) 0 0 1 6 1 - 5
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K. Yoshioka et al./Solar Energy Materials & Solar Cells 57 (1999) 9–19
shortage. A static concentrator is one of the approaches to solve these issues because a Si PV module can be produced by less use of costly Si solar cells and does not need a complex tracking system [1,2]. In addtion, under the Japanese climate including much diffuse light, a static concentrator is considered to be effective for collection of diffuse light due to relatively wide acceptance angle. A two-dimensional compound parabolic concentrator (2D-CPC) by Winston et al. is well known as a typical static concentrator, but it has a problem that its height is too high to use in PV module application. The authors have developed a twodimensional compound elliptic lens (2D-CEL) for a photovoltaic static concentrator module [3]. The 2D-CEL has only a small reflective loss at a boundary surface between materials with different refractive indices because the 2D-CEL is formed of only a transparent resin. In addition, the ratio of a concentrator height to a reciever width is not so large as that of 2D-CPC. One of the important things in designing a static concentrator is to determine a half-acceptance angle. The half-acceptance angle should be determined in terms of an electric power generated pattern or maximized yearly electric power for a module installation angle. Thus, the halfacceptance angle should be determined by predicting yearly integrated irradiance collectable by a concentrator from insolation data. In this paper, an yearly distributed insolation model is used for evaluation. By using the insolation model, yearly integrated light intensity distribution on the celestial hemisphere can be easily obtained without an insolation database. Using the yearly distributed insolation model, optical performance of the 2D-CEL is calculated at half-acceptance angles from 10° to 80° separately for direct and diffuse radiation. The same calculation is also performed for the 2D-CPC to compare those performances. This characterizes the optical feature of the 2D-CEL for direct and diffuse radiation separately. In addition, relationship between the yearly averaged optical concentration ratio and half-acceptance angle for the 2D-CEL is investigated for several inclined surfaces with various tilt angles.
2. Insolation model and calculation procedure By using an yearly distributed insolation model [4], optical properties of the static concentrators can be calculated separately for direct and diffuse radiation. We will briefly introduce this model here for the sake of the readers’ convenience. Maps of yearly distributed direct and diffuse radiations were obtained by weighting all the microscopic areas dividing the celestial hemisphere into along azimuth and zenith angles at an equal interval. The map of the global solar radiation was obtained by adding each value on the already obtained, direct and diffuse radiation maps using a cloudiness factor determining the ratio between direct and diffuse radiation. The direct radiation component was calculated using two parameters, i.e. atmospheric transparency and declination change rate. Hence, creating the direct radiation map on the celestial hemisphere is equivalent to solving the weight function as a function of the azimuth and zenith angles of the sun. The direct radiation component has small weights around autumnal and vernal equinoxes due to the relatively
K. Yoshioka et al./Solar Energy Materials & Solar Cells 57 (1999) 9–19
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high change rate in the declination and large weights around winter and summer solstices due to the relatively low change rate. In addition, it had large weights at small zenith angles and small weights at large zenith angles due to difference in the air mass. As for the diffuse radiation, it was assumed that luminance is constant at all the points on the celestial hemisphere. A normalization constant, G , was introduced for an integrated value onto a horizontal-surface component of the direct radiation to be unity. Defining the weight function for the direct radiation, ¼ (u, h), in the micro scopic solid angle dX("sin h dh du), G was described by the following equation:
G " ¼ (u, h)cos h dX,
(1)
where h and u are the zenith and azimuth angles of the sun. The intensity on the celestial hemisphere per unit solid angle, ¸, was determined by integrating a value of diffuse radiation to be unity using the following equation:
1" ¸ cos h dX"p¸.
(2)
Thus, ¸"1/p is obtained. By introducing the cloudiness factor, c, defined as the ratio of diffuse to global solar radiation on the horizontal surface, the differential insolation dE in the microscopic solid angle dX around (u, h) was given by 1 ¼ (u, h) (1!c) dX#c dX. dE" p G By integrating the above equation over the celestial hemisphere,
(3)
¼ (u, h) c (1!c)# cos h dX"1. (4) G p In order to apply this model to the calculation of optical properties for the 2D-CEL, the optical concentration ratio (OCR) was introduced. The OCR was defined as the ratio of irradiance absorbed per unit cell area between the 2D-CEL and flat-plate models with the same receiver size in the case where sunlight with the same incidence angle and intensity enters each model shown in Fig. 1. It was assumed that both the models have Si solar cells with a refractive index of 3.5 at a wavelength of 900 nm and with an antireflection film with a refractive index of 2.3. The solar cells were attached to the receiver using silicone resin with a refractive index of 1.43. Geometrical and optical concentration ratios and the optical efficiency were different at every incidence angle for a static concentrator system. Two parameters, the zenith angle h and azimuth angle u were introduced to designate an incident direction. If the OCR for this direction was defined as a function of h and u, an yearly averaged OCR (OCR ) was given as Eq. (5) below under the assumption that the concentra 40 tor and flat-plate models are located on the horizontal surface. cos h dE"
OCR " OCR(u, h)cos h dE. 40
(5)
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K. Yoshioka et al./Solar Energy Materials & Solar Cells 57 (1999) 9–19
Fig. 1. Calculation models for flat-plate and concentrator models.
For the concentration model installed on an inclined surface, the calculation was carried out, by taking account of direct and diffuse radiations behind the inclined surface being intercepted.
3. Yearly distributed insolation and optical concentration maps for the 2D-CEL and 2D-CPC Using the insolation model, an yearly distributed insolation map was created for global solar radiation at Tokyo, Japan (N35°). The atmospheric transparency was set at 0.7 and the cloudiness factor at 0.45. Fig. 2 shows the yearly distributed insolation map for the global solar radiation. In this map, the radius direction, that is to say, the distance from the center of the circle corresponds to the zenith angle and the direction of the circumference corresponds to the azimuth. Brightness indicates light intensity on each point at every zenith and azimuth angles. Direct radiation is distributed in the scope of about $23° almost symmetrically with regard to the zenith angle of 35° in the due south. In particular, one can confirm that the sun path has bright intensity on summer and winter solstices and dark intensity on autumnal and vernal equinoxes. One can also confirm a brighter intensity at a large zenith angles and darker intensity at a small zenith angles. In order to evaluate yearly optical performance for the 2D-CEL, OCRs were calculated as a function of ray incidence angles using a ray-tracing method. In the case of treating of a three-dimensional ray, a incident direction is expressed by dividing into two angles. As shown in Fig. 3, a is defined as an angle between the zenith (Z-axis) and a projection vector of the incident ray on the Zenith-south (Z—X) plane and b is
K. Yoshioka et al./Solar Energy Materials & Solar Cells 57 (1999) 9–19
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Fig. 2. A map of yearly distributed insolation for global radiation at Tokyo, Japan (N35°).
defined as an angle between the Z—X plane and the incident-ray vector. For comparison, they were also calculated for the 2D-CPC. Fig. 4a and Fig. 4b show distributed OCR maps for the 2D-CEL and 2D-CPC, designed at a half acceptance angle (HAA) of 30°, as a function of a amd b. The figures clearly show the difference in distribution tendency of optical concentration ratios between the 2D-CPC and the 2D-CEL. At a less than the HAA, the OCR of the 2D-CEL is a little lower than that of the 2D-CPC. At a greater than HAA, the OCRs of the 2D-CEL gradually decrease with the increase in the a while that of the 2D-CPC becomes 0 at any a more than the HAA. In addition, the 2D-CEL has a feature that an OCR at the fixed a over the HAA becomes larger with the increase in the b because the ratio of rays leaking when being at the small b are reachable at the receiver increases with the increase in the b. This tendency never appears for the 2D-CPC but is reasonable for the 2D-CEL as found earlier in our past work [5]. Because of the definition of the OCR of the 2D-CEL, OCRs around 80° in both a and b are quite large but in fact irradiance absorbed into each model is quite small due to reflection loss at the cell surface. From the above results, one can expect that the 2D-CEL is more effective for the collection of diffuse sunlight than the 2D-CPC.
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Fig. 3. Definition of angles, a and b to describe the direction of a ray entering the lens.
4. Relationship between optical performance and half-acceptance angle of the 2D-CELs Using the yearly distributed insolation map, yearly averaged optical properties of the 2D-CEL were calculated as a function of HAA, and compared with those of the 2D-CPC. Firstly, optical efficiencies were calculated by assuming that the model was installed at a tilt angle equal to a latitude of Tokyo in the due south direction. The solid lines in Fig. 5 indicated the ratio of optical efficiency of the 2D-CEL and 2D-CPC to the flat-plate model, and the dotted line indicated optical efficiency of the 2D-CEL for an incident sunlight. As seen from the figure, the 2D-CEL has higher values than the 2D-CPC up to about a HAA of 70°, and has a peak value of 1.0 at a HAA of 58°. On the other hand, the CPC approaches 1 asymptotically as the HAA increases. Fig. 6a and Fig. 6b show OCRs of direct, diffuse and global radiation for the 2D-CEL and 2D-CPC as a function of HAA. With regard to the 2D-CEL, the yearly averaged OCR for direct radiation attained a peak value of 2.0 at a HAA of 23°. OCR for diffuse radiation did not vary drastically at all for HAAs over 30°, and a peak value was attained at 1.5. Variation of OCR for global radiation was similar to that for the direct radiation, and a maximum value was 1.75 at a HAA of 23°. This value and the shpae of the curve for the global radiation were almost the same results as the ones calculated using a meteorological, HASP data for Tokyo, Japan [6]. Thus, the validity of this insolation model was reconfirmed. On the other hand, with regard to the 2D-CPC, the yearly averaged OCR for direct radiation had a maximum value of 1.85 at a HAA of 25°. OCR for diffuse radiation was almost 1.0 at any HAAs. Thus, variation in global solar radiation was the same as that for direct radiation. The maximum value was about 1.5 at a HAA of 25°.
K. Yoshioka et al./Solar Energy Materials & Solar Cells 57 (1999) 9–19
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Fig. 4. Distribution of optical concentration ratios for the two-dimensional compound elliptic lens and compound parabolic concentrator designed at a half-acceptance angle of 30° under the assumption that they are installed at a tilt angle equal to the latitude in due south. The incidence angles a and b are defined in Fig. 3.
From comparison between properties of the both concentrators, it was clarified that the 2D-CEL is not only more advantageous for the collection of diffuse sunlight but also can be obtained with yearly averaged OCR higher than the 2D-CPC.
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Fig. 5. Optical efficiency ratio of two-dimensional compound elliptic lens and two-dimensional compound parabolic concentrator to a flat-plate model as a function of half-acceptance angle.
Fig. 6. Optical concentration ratio of direct, diffuse and global solar radiations for the two-dimensional compound elliptic lens and two-dimensional compound parabolic concentrator as a function of halfacceptance angle.
5. Relationship between optical performance and half-acceptance angle on several inclined surfaces Normally, a photovoltaic module is installed at a tilt angle equal to a latitude at the location where it is used. During installation on a roof, a photovoltaic module is not always installed at a tilt angle equal to the latitude. Thus, it is interesting to investigate the optical performance of the 2D-CELs inclined at different tilt angles for residential PV application. Prior to mentioning the performance of the 2D-CEL, the direct, diffuse and global solar radiation was calculated for inclined surfaces with different angles in the direction of due south in Tokyo, Japan, using the insolation model. In this model, a global solar insolation was normalized to be unity on a horizontal surface. As shown in Fig. 7, direct solar insolation became the maximum value at a tilt angle of 30°, smaller by 5° than the latitude. On the other hand, diffuse solar insolation
K. Yoshioka et al./Solar Energy Materials & Solar Cells 57 (1999) 9–19
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Fig. 7. Direct, diffuse and global solar insolations for inclined surfaces in due south direction as a function of tilt angle. Global solar radiation on a horizontal surface is assumed to be 1.0.
Fig. 8. Yearly integrated irradiance collected on the receiver of the two-dimensional compound elliptic lens as a function of half-acceptance angle for inclined surfaces. Global solar insolation is assumed 1.0 for a horizontal surface.
took the maximum value on a horizontal surface. As a result, a maximum global solar insolation of 1.16 was obtained on inclined surface with a tilt angle of 22°. Thus, one can understand that the tile angle for an inclined surface to install the 2D-CEL is not always suitable at a tilt angle equal to the latitude. Under the assumption that global solar insolation is unity on a horizontal plane, yearly integrated irradiance collected on the receiver of the 2D-CEL was calculated as a function of HAA for several inclined surfaces. As seen from Fig. 8, there is no big difference among the yearly integrated irradiance curves for the inclined surfaces tilled between 20 and 35°. But, an HAA to maximize yearly integrated irradiance was 15° for the inclined surface of 25°, 20° for that of 30°, and 20—25° for that of 35°. This result indicated that an optimum HAA, to maximize yearly integrated irradiance, shifts to a higher HAA with the tilt angle. The maximum value did not change so much with
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K. Yoshioka et al./Solar Energy Materials & Solar Cells 57 (1999) 9–19
Fig. 9. Cross-sectional schematic of the two-dimensional compound elliptic lenses with half-acceptance angles of 20—30°.
the variation of the tilt angles. In addition, from the shape of the curve, it was found that yearly integrated irradiance was not affected so much by the tilt angle variation to maximize the performance of the 2D-CEL. The figure also suggests that utilization of the concentrator is not effective on the inclined surfaces with tilt angles except for 20—35°. Finally, we discuss the effect of error in the dimension of the 2D-CEL on optical performance briefly. In manufacturing a molded lens or cut-out lens, there is the possibility to cause error in the dimension by manual-polish process of a cast or the lens. Fig. 9 shows the cross-sectional schematics of the 2D-CEL with a HAA 20—30°. According to calculation, the HAA is considered to shift one degree per about 3—5% in the mean error ratio to ½ coordinate at each X coordinate although it depends on largeness of an HAA. But even if the shift of one degree happens, the last paragraph tells us that the optical properties were not affected by the shift of a HAA within a few degrees while designing an HAA to maximize an OCR.
6. Conclusion In this work, optical performance of the 2D-CEL was evaluated using an yearly distributed insolation model. An insolation map for Tokyo was drawn for the evaluation. The maximum yearly averaged OCR of 1.75 was obtained for global radiation when the 2D-CEL was installed at a tilt angle equal to a latitude of Tokyo. As compared with the 2D-CPC, it was clarified that the 2D-CEL is not only more advantageous for the collection of diffuse sunlight, but also can obtain more yearly averaged OCR. Optical performance of the 2D-CEL was investigated also for several inclined surfaces with different tilt angles. Under the assumption that global solar insolation is
K. Yoshioka et al./Solar Energy Materials & Solar Cells 57 (1999) 9–19
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unity on a horizontal surface, the yearly integrated irradiance collected on the receiver of the 2D-CEL was calculated as a function of HAA. Consequently, there was not so much difference in the shapes and maximum yearly integrated irradiance curves for inclined surfaces with tilt angles of 20—35°. This result gave us the following useful knowledge: Yearly integrated irradiance is not so affected by varying the tilt angle. In this insolation model, variation in the cloudiness factor throughout the year and distributed diffuse radiation around circumsolar disk radiation was not taken into account. These issues will be studied in our future task.
Acknowledgements This work was supported by the New Energy and Industrial Technology Development Organization as a part of the New Sunshine Project of the Ministry of International Trade and Industry.
References [1] A. Luque, Solar Cells and Optics for Photovoltaic Concentration, Adam Hilger, Bristol and Philadelphia, 1989. [2] S. Bowden, S.R. Wenham, M.A. Green, Prog. Photovolt. 3 (1995) 413. [3] K. Yoshioka, K. Endoh, M. Kobayashi, A. Suzuki, T. Saitoh, Sol. Energy Mater. Sol. Cells 34 (1994) 125. [4] A. Suzuki, S. Kobayashi, Solar Energy 54 (1995) 327. [5] A. Suzuki, M. Kobayashi, K. Yoshioka, T. Saitoh, Design and optical features of a non-imaging lens and its application to solar energy systems, Proc. 1st Int. Conf. of New Energy Systems and Coversions, Universal Academy Press, 1993, p. 295. [6] K. Yoshioka, M. Kobayashi, A. Suzuki, K. Endoh, T. Saitoh, An optimum design and properties of a static concentrator with a non-imaging, Proc. of 1st World Conf. on Photovolt. Energy Conversion, 1994, pp. 1119—1122.
Performance evaluation of two-dimensional compound elliptic lens concentrators using a yearly distributed insolation model K. Yoshioka*, A. Suzuki, T. Saitoh Division of Electronics and Information Technology, Tokyo A&T University, 2-24-16 Nakamachi, Koganei, Tokyo 184-8588, Japan
Abstract Optical performance of a two-dimensional compound elliptic lens (2D-CEL) for a photovoltaic static concentrator module has been studied as a function of half-acceptance angle using an yearly distributed insolation model. The maximum yearly averaged optical concentration ratio of 1.75 was obtained for global radiation when the 2D-CEL was installed at a tilt angle equal to the latitude of Tokyo (N35°). Also, the 2D-CEL was found to be more advantageous for collection of direct and diffuse radiations than a conventional, compound parabolic concentrator. In the case where the 2D-CEL is installed at a tilt angle of 20—35°, the maximum yearly integrated irradiance collected on the receiver was found to be 1.45 at a suitable half-acceptance angle. 1999 Elsevier Science B.V. All rights reserved. Keywords: Two-dimensional compound elliptic tens (2D-CEL); Two-dimensional compound parabolic concentrator (2D-CPC); Optical concentration ratio (OCR)
1. Introduction In Japan, dissemination of residential PV systems is a key point for the growth of PV markets. By a subsidy policy started in 1994, the residential PV systems have steadily increased. Presently, PV modules for the residential PV systems are almost crystalline Si ones, but there are still two issues of production cost and Si feedstock
* Corresponding author. E-mail: [email protected] 0927-0248/99/$ — See front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 4 8 ( 9 8 ) 0 0 1 6 1 - 5
10
K. Yoshioka et al./Solar Energy Materials & Solar Cells 57 (1999) 9–19
shortage. A static concentrator is one of the approaches to solve these issues because a Si PV module can be produced by less use of costly Si solar cells and does not need a complex tracking system [1,2]. In addtion, under the Japanese climate including much diffuse light, a static concentrator is considered to be effective for collection of diffuse light due to relatively wide acceptance angle. A two-dimensional compound parabolic concentrator (2D-CPC) by Winston et al. is well known as a typical static concentrator, but it has a problem that its height is too high to use in PV module application. The authors have developed a twodimensional compound elliptic lens (2D-CEL) for a photovoltaic static concentrator module [3]. The 2D-CEL has only a small reflective loss at a boundary surface between materials with different refractive indices because the 2D-CEL is formed of only a transparent resin. In addition, the ratio of a concentrator height to a reciever width is not so large as that of 2D-CPC. One of the important things in designing a static concentrator is to determine a half-acceptance angle. The half-acceptance angle should be determined in terms of an electric power generated pattern or maximized yearly electric power for a module installation angle. Thus, the halfacceptance angle should be determined by predicting yearly integrated irradiance collectable by a concentrator from insolation data. In this paper, an yearly distributed insolation model is used for evaluation. By using the insolation model, yearly integrated light intensity distribution on the celestial hemisphere can be easily obtained without an insolation database. Using the yearly distributed insolation model, optical performance of the 2D-CEL is calculated at half-acceptance angles from 10° to 80° separately for direct and diffuse radiation. The same calculation is also performed for the 2D-CPC to compare those performances. This characterizes the optical feature of the 2D-CEL for direct and diffuse radiation separately. In addition, relationship between the yearly averaged optical concentration ratio and half-acceptance angle for the 2D-CEL is investigated for several inclined surfaces with various tilt angles.
2. Insolation model and calculation procedure By using an yearly distributed insolation model [4], optical properties of the static concentrators can be calculated separately for direct and diffuse radiation. We will briefly introduce this model here for the sake of the readers’ convenience. Maps of yearly distributed direct and diffuse radiations were obtained by weighting all the microscopic areas dividing the celestial hemisphere into along azimuth and zenith angles at an equal interval. The map of the global solar radiation was obtained by adding each value on the already obtained, direct and diffuse radiation maps using a cloudiness factor determining the ratio between direct and diffuse radiation. The direct radiation component was calculated using two parameters, i.e. atmospheric transparency and declination change rate. Hence, creating the direct radiation map on the celestial hemisphere is equivalent to solving the weight function as a function of the azimuth and zenith angles of the sun. The direct radiation component has small weights around autumnal and vernal equinoxes due to the relatively
K. Yoshioka et al./Solar Energy Materials & Solar Cells 57 (1999) 9–19
11
high change rate in the declination and large weights around winter and summer solstices due to the relatively low change rate. In addition, it had large weights at small zenith angles and small weights at large zenith angles due to difference in the air mass. As for the diffuse radiation, it was assumed that luminance is constant at all the points on the celestial hemisphere. A normalization constant, G , was introduced for an integrated value onto a horizontal-surface component of the direct radiation to be unity. Defining the weight function for the direct radiation, ¼ (u, h), in the micro scopic solid angle dX("sin h dh du), G was described by the following equation:
G " ¼ (u, h)cos h dX,
(1)
where h and u are the zenith and azimuth angles of the sun. The intensity on the celestial hemisphere per unit solid angle, ¸, was determined by integrating a value of diffuse radiation to be unity using the following equation:
1" ¸ cos h dX"p¸.
(2)
Thus, ¸"1/p is obtained. By introducing the cloudiness factor, c, defined as the ratio of diffuse to global solar radiation on the horizontal surface, the differential insolation dE in the microscopic solid angle dX around (u, h) was given by 1 ¼ (u, h) (1!c) dX#c dX. dE" p G By integrating the above equation over the celestial hemisphere,
(3)
¼ (u, h) c (1!c)# cos h dX"1. (4) G p In order to apply this model to the calculation of optical properties for the 2D-CEL, the optical concentration ratio (OCR) was introduced. The OCR was defined as the ratio of irradiance absorbed per unit cell area between the 2D-CEL and flat-plate models with the same receiver size in the case where sunlight with the same incidence angle and intensity enters each model shown in Fig. 1. It was assumed that both the models have Si solar cells with a refractive index of 3.5 at a wavelength of 900 nm and with an antireflection film with a refractive index of 2.3. The solar cells were attached to the receiver using silicone resin with a refractive index of 1.43. Geometrical and optical concentration ratios and the optical efficiency were different at every incidence angle for a static concentrator system. Two parameters, the zenith angle h and azimuth angle u were introduced to designate an incident direction. If the OCR for this direction was defined as a function of h and u, an yearly averaged OCR (OCR ) was given as Eq. (5) below under the assumption that the concentra 40 tor and flat-plate models are located on the horizontal surface. cos h dE"
OCR " OCR(u, h)cos h dE. 40
(5)
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K. Yoshioka et al./Solar Energy Materials & Solar Cells 57 (1999) 9–19
Fig. 1. Calculation models for flat-plate and concentrator models.
For the concentration model installed on an inclined surface, the calculation was carried out, by taking account of direct and diffuse radiations behind the inclined surface being intercepted.
3. Yearly distributed insolation and optical concentration maps for the 2D-CEL and 2D-CPC Using the insolation model, an yearly distributed insolation map was created for global solar radiation at Tokyo, Japan (N35°). The atmospheric transparency was set at 0.7 and the cloudiness factor at 0.45. Fig. 2 shows the yearly distributed insolation map for the global solar radiation. In this map, the radius direction, that is to say, the distance from the center of the circle corresponds to the zenith angle and the direction of the circumference corresponds to the azimuth. Brightness indicates light intensity on each point at every zenith and azimuth angles. Direct radiation is distributed in the scope of about $23° almost symmetrically with regard to the zenith angle of 35° in the due south. In particular, one can confirm that the sun path has bright intensity on summer and winter solstices and dark intensity on autumnal and vernal equinoxes. One can also confirm a brighter intensity at a large zenith angles and darker intensity at a small zenith angles. In order to evaluate yearly optical performance for the 2D-CEL, OCRs were calculated as a function of ray incidence angles using a ray-tracing method. In the case of treating of a three-dimensional ray, a incident direction is expressed by dividing into two angles. As shown in Fig. 3, a is defined as an angle between the zenith (Z-axis) and a projection vector of the incident ray on the Zenith-south (Z—X) plane and b is
K. Yoshioka et al./Solar Energy Materials & Solar Cells 57 (1999) 9–19
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Fig. 2. A map of yearly distributed insolation for global radiation at Tokyo, Japan (N35°).
defined as an angle between the Z—X plane and the incident-ray vector. For comparison, they were also calculated for the 2D-CPC. Fig. 4a and Fig. 4b show distributed OCR maps for the 2D-CEL and 2D-CPC, designed at a half acceptance angle (HAA) of 30°, as a function of a amd b. The figures clearly show the difference in distribution tendency of optical concentration ratios between the 2D-CPC and the 2D-CEL. At a less than the HAA, the OCR of the 2D-CEL is a little lower than that of the 2D-CPC. At a greater than HAA, the OCRs of the 2D-CEL gradually decrease with the increase in the a while that of the 2D-CPC becomes 0 at any a more than the HAA. In addition, the 2D-CEL has a feature that an OCR at the fixed a over the HAA becomes larger with the increase in the b because the ratio of rays leaking when being at the small b are reachable at the receiver increases with the increase in the b. This tendency never appears for the 2D-CPC but is reasonable for the 2D-CEL as found earlier in our past work [5]. Because of the definition of the OCR of the 2D-CEL, OCRs around 80° in both a and b are quite large but in fact irradiance absorbed into each model is quite small due to reflection loss at the cell surface. From the above results, one can expect that the 2D-CEL is more effective for the collection of diffuse sunlight than the 2D-CPC.
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Fig. 3. Definition of angles, a and b to describe the direction of a ray entering the lens.
4. Relationship between optical performance and half-acceptance angle of the 2D-CELs Using the yearly distributed insolation map, yearly averaged optical properties of the 2D-CEL were calculated as a function of HAA, and compared with those of the 2D-CPC. Firstly, optical efficiencies were calculated by assuming that the model was installed at a tilt angle equal to a latitude of Tokyo in the due south direction. The solid lines in Fig. 5 indicated the ratio of optical efficiency of the 2D-CEL and 2D-CPC to the flat-plate model, and the dotted line indicated optical efficiency of the 2D-CEL for an incident sunlight. As seen from the figure, the 2D-CEL has higher values than the 2D-CPC up to about a HAA of 70°, and has a peak value of 1.0 at a HAA of 58°. On the other hand, the CPC approaches 1 asymptotically as the HAA increases. Fig. 6a and Fig. 6b show OCRs of direct, diffuse and global radiation for the 2D-CEL and 2D-CPC as a function of HAA. With regard to the 2D-CEL, the yearly averaged OCR for direct radiation attained a peak value of 2.0 at a HAA of 23°. OCR for diffuse radiation did not vary drastically at all for HAAs over 30°, and a peak value was attained at 1.5. Variation of OCR for global radiation was similar to that for the direct radiation, and a maximum value was 1.75 at a HAA of 23°. This value and the shpae of the curve for the global radiation were almost the same results as the ones calculated using a meteorological, HASP data for Tokyo, Japan [6]. Thus, the validity of this insolation model was reconfirmed. On the other hand, with regard to the 2D-CPC, the yearly averaged OCR for direct radiation had a maximum value of 1.85 at a HAA of 25°. OCR for diffuse radiation was almost 1.0 at any HAAs. Thus, variation in global solar radiation was the same as that for direct radiation. The maximum value was about 1.5 at a HAA of 25°.
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Fig. 4. Distribution of optical concentration ratios for the two-dimensional compound elliptic lens and compound parabolic concentrator designed at a half-acceptance angle of 30° under the assumption that they are installed at a tilt angle equal to the latitude in due south. The incidence angles a and b are defined in Fig. 3.
From comparison between properties of the both concentrators, it was clarified that the 2D-CEL is not only more advantageous for the collection of diffuse sunlight but also can be obtained with yearly averaged OCR higher than the 2D-CPC.
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Fig. 5. Optical efficiency ratio of two-dimensional compound elliptic lens and two-dimensional compound parabolic concentrator to a flat-plate model as a function of half-acceptance angle.
Fig. 6. Optical concentration ratio of direct, diffuse and global solar radiations for the two-dimensional compound elliptic lens and two-dimensional compound parabolic concentrator as a function of halfacceptance angle.
5. Relationship between optical performance and half-acceptance angle on several inclined surfaces Normally, a photovoltaic module is installed at a tilt angle equal to a latitude at the location where it is used. During installation on a roof, a photovoltaic module is not always installed at a tilt angle equal to the latitude. Thus, it is interesting to investigate the optical performance of the 2D-CELs inclined at different tilt angles for residential PV application. Prior to mentioning the performance of the 2D-CEL, the direct, diffuse and global solar radiation was calculated for inclined surfaces with different angles in the direction of due south in Tokyo, Japan, using the insolation model. In this model, a global solar insolation was normalized to be unity on a horizontal surface. As shown in Fig. 7, direct solar insolation became the maximum value at a tilt angle of 30°, smaller by 5° than the latitude. On the other hand, diffuse solar insolation
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Fig. 7. Direct, diffuse and global solar insolations for inclined surfaces in due south direction as a function of tilt angle. Global solar radiation on a horizontal surface is assumed to be 1.0.
Fig. 8. Yearly integrated irradiance collected on the receiver of the two-dimensional compound elliptic lens as a function of half-acceptance angle for inclined surfaces. Global solar insolation is assumed 1.0 for a horizontal surface.
took the maximum value on a horizontal surface. As a result, a maximum global solar insolation of 1.16 was obtained on inclined surface with a tilt angle of 22°. Thus, one can understand that the tile angle for an inclined surface to install the 2D-CEL is not always suitable at a tilt angle equal to the latitude. Under the assumption that global solar insolation is unity on a horizontal plane, yearly integrated irradiance collected on the receiver of the 2D-CEL was calculated as a function of HAA for several inclined surfaces. As seen from Fig. 8, there is no big difference among the yearly integrated irradiance curves for the inclined surfaces tilled between 20 and 35°. But, an HAA to maximize yearly integrated irradiance was 15° for the inclined surface of 25°, 20° for that of 30°, and 20—25° for that of 35°. This result indicated that an optimum HAA, to maximize yearly integrated irradiance, shifts to a higher HAA with the tilt angle. The maximum value did not change so much with
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Fig. 9. Cross-sectional schematic of the two-dimensional compound elliptic lenses with half-acceptance angles of 20—30°.
the variation of the tilt angles. In addition, from the shape of the curve, it was found that yearly integrated irradiance was not affected so much by the tilt angle variation to maximize the performance of the 2D-CEL. The figure also suggests that utilization of the concentrator is not effective on the inclined surfaces with tilt angles except for 20—35°. Finally, we discuss the effect of error in the dimension of the 2D-CEL on optical performance briefly. In manufacturing a molded lens or cut-out lens, there is the possibility to cause error in the dimension by manual-polish process of a cast or the lens. Fig. 9 shows the cross-sectional schematics of the 2D-CEL with a HAA 20—30°. According to calculation, the HAA is considered to shift one degree per about 3—5% in the mean error ratio to ½ coordinate at each X coordinate although it depends on largeness of an HAA. But even if the shift of one degree happens, the last paragraph tells us that the optical properties were not affected by the shift of a HAA within a few degrees while designing an HAA to maximize an OCR.
6. Conclusion In this work, optical performance of the 2D-CEL was evaluated using an yearly distributed insolation model. An insolation map for Tokyo was drawn for the evaluation. The maximum yearly averaged OCR of 1.75 was obtained for global radiation when the 2D-CEL was installed at a tilt angle equal to a latitude of Tokyo. As compared with the 2D-CPC, it was clarified that the 2D-CEL is not only more advantageous for the collection of diffuse sunlight, but also can obtain more yearly averaged OCR. Optical performance of the 2D-CEL was investigated also for several inclined surfaces with different tilt angles. Under the assumption that global solar insolation is
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unity on a horizontal surface, the yearly integrated irradiance collected on the receiver of the 2D-CEL was calculated as a function of HAA. Consequently, there was not so much difference in the shapes and maximum yearly integrated irradiance curves for inclined surfaces with tilt angles of 20—35°. This result gave us the following useful knowledge: Yearly integrated irradiance is not so affected by varying the tilt angle. In this insolation model, variation in the cloudiness factor throughout the year and distributed diffuse radiation around circumsolar disk radiation was not taken into account. These issues will be studied in our future task.
Acknowledgements This work was supported by the New Energy and Industrial Technology Development Organization as a part of the New Sunshine Project of the Ministry of International Trade and Industry.
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