62 IEEE JOURNAL OF PHOTOVOLTAICS, VOL. 2, NO. 1, JANUARY 2012 Solar Profiles and Spectral Modeling for CPV Simulations Ian R. Cole, Thomas R. Betts, and Ralph Gottschalg Abstract—In this paper, a computer model for the simulation of solar flux distribution in the direct and circumsolar regions of the beam irradiation is presented. The model incorporates circumsolar ratios (CSRs) and spectral transmittance. It is used to demonstrate the importance of realistic solar flux distributions as source inputs in high-concentration concentrator photovoltaic simulations. Realistic solar flux distributions are achieved by treating the Sun as an inhomogeneous extended light source. Flux distributions incident on lenses of various entry apertures are generated and used to investigate the losses in incident flux resulting from tracking errors and CSR variation. It is found that, for a Seattle-based concentrating system with an entry aperture of 0.5◦ , a ∼5% loss in net annual incident flux occurs with a uniaxial tracking error of 0.3◦ , and a ∼20% loss occurs with a uniaxial tracking error of 0.4◦ . This analysis is at the primary concentrator stage; hence, it only considers the flux incident on the surface of the initial concentrating lens. The summation of flux at this primary stage does not account for further losses at the receiver because of optical misalignment. Index Terms—Circumsolar ratios (CSR), concentrator photovoltaic (CPV), solar profile, solar tracking, spectra. I. INTRODUCTION A. Solar Flux Intensity and Spectral Composition T entry aperture lenses in the subdegree range. The average radial angular extent of the central solar disk is ∼0.266◦ , varying with Sun–Earth distance in the range 0.262–0.272◦ [1]. Hence, the Sun occupies a significant proportion of a high-concentration system’s optical input range. This indicates that it is necessary to consider the Sun as an extended light source in simulating the performance of such systems. Such consideration permits a thorough investigation of the effects of optical misalignment. Furthermore, the effect of the variation of environmental factors on the optical illumination pattern that is formed at the receiver, which is not necessarily placed at the focal plane of the concentrating device, can be evaluated. High-concentration photovolatics (HCPV) systems take advantage of high-efficiency multijunction (MJ) cells. The relative expense of these high efficiency (>40%) converters is offset through the use of cheaper optical concentration systems. MJ cells achieve these high efficiencies through the partitioning of the solar spectrum. They comprise semiconductor junctions of varying bandgaps that are connected in series. As such, they are subject to the current mismatch problem. For a string of i power generating devices, the output power P is defined as HE output of any conversion system is a function of input. For solar energy conversion P = f (ϕ) (1) where P is the power, and ϕ is the solar flux. Concentrating photovoltaic (CPV) simulation programs today tend to make some assumptions regarding the distribution of solar flux. The solar resource is often described as a point source or as a pillbox, in which the solar flux is homogeneously distributed over a circular surface representing the effective relative size of the Sun. Such assumptions are unphysical and result in approximated system energy ratings and potentially in compromised system designs, overlooking the consequences of optical misalignment and changes in meteorological variables. In high-concentration CPV systems employing refractive optics, concentrations of ∼500–1000x are achieved with small Manuscript received July 12, 2011; revised September 7, 2011 and October 18, 2011; accepted November 7, 2011. Date of publication December 19, 2011; date of current version January 30, 2012. This work was supported by the Centre for Renewable Energy Systems Technology, Department of Electronic and Electrical Engineering, Loughborough University, Leicestershire, U.K. The authors are with the Centre for Renewable Energy Systems Technology, Loughborough University, Leicestershire, LE11 3TU, U.K. (e-mail: I.R.Cole@lboro.ac.uk; T.R.Betts@lboro.ac.uk; R.Gottschalg@lboro.ac.uk). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JPHOTOV.2011.2177445 P = Σi Vi Im in (2) where V represents voltage, and Im in is the lowest current in the stack. Currents in each junction of the device are induced by photons of wavelength bands associated with energies higher than the junction bandgap. Thus, MJ cells are especially sensitive to the spectral composition of solar flux. Given the spectral sensitivity of MJ cells, spectral variation is a highly important consideration in CPV modeling. Treating the Sun as an inhomogeneously distributed extended light source allows for the consideration and analysis of both spatial and spectral inhomogeneities in system inputs. If the spectrally resolved refractive indices of system materials are accounted for, the effects of chromatic aberration on the optical illumination pattern that is formed on the receiver are investigable. B. Circumsolar Ratios and the Sun as an Extended Light Source Throughout the 1970s and 1980s, the Lawrence Berkeley Laboratory (LBL), Berkeley, CA, conducted research into the properties of solar profiles. For this research, 11 sites across the U.S. were chosen, exhibiting different atmospheric characteristics such as altitude, proximity to sources of large particulates, and humidity. Over 200 000 solar profiles were collated in the 2156-3381/$26.00 © 2011 IEEE COLE et al.: SOLAR PROFILES AND SPECTRAL MODELING FOR CPV SIMULATIONS measurement process [2]. The data logged in these experiments are freely available online [3]. The statistical variation of angularly resolved flux distributions in the beam region leads to an important variable for consideration in solar flux analysis—the circumsolar ratio (CSR). The CSR is defined as the ratio of the flux contained in the circumsolar region (the solar aureole) to the flux contained within the entirety of the solar disk (the central solar disk plus the solar aureole). The radial angular extent of the circumsolar region is generally around 43.6 mrad (2.49◦ ), although the actual extent varies with atmospheric parameters. The central solar limit is the edge of the solar disk, the effective apparent size of the Sun because of real Sun–Earth size and distance relationships. The radial angular extent of the central solar region has a generally accepted average of 4.65 mrad (0.266◦ ) [4], although varies throughout the year from 4.58–4.74 mrad. These dimensions highlight that circumsolar radiation can be of a significant energetic contribution, even in subdegree entry aperture systems ϕaureole (3) C= ϕb eam where C is the CSR, ϕaureole is the flux in the circumsolar region, and ϕb eam is the net flux from the circumsolar region and the central solar disk. The CSR can be thought of simply as the percentage of beam irradiation that is contained in the circumsolar region. Buie et al. analyzed this data with a view to identify statistical laws linking the measured parameters [5], [6]. The formulae extracted from their analysis describe the relationship between CSRs and radiated flux intensity with angular deviation θ from the center of the solar disk    cos(0.326θ) , 0 ≤ θ ≤ 0.266◦ (4a) ϕ(θ) = cos(0.308θ)   κ γ ◦ ◦ e θ , 0.266 < θ ≤ 2.49 κ = 0.9 ln(13.5C)C −0.3 γ = 2.2 ln(0.52C)C 0.43 − 0.1 Fig. 1. 63 CSR variation and frequency plots for Edinburgh and Almeria. ognizes that CSR variation has a more complicated relationship with meteorological variables and is also dependent on variables such as temperature, humidity, and cloud cover [7]. Gueymard has published recent work on solar profiles, comparing the effects of atmospheric changes, such as cloud type and aerosol optical depth, on solar profile with respect to flux intensity and spectral contribution [8]. The potential contributory benefit of up to 5% incident power collection clearly indicates that circumsolar radiation is an important consideration in system design. It also puts into question the use of beam irradiance alone when analyzing system output as the performance ratios obtained will be unrealistically optimistic. (4b) C. Spatially Distributed Spectra (4c) There exists software today capable of generating separate spectral profile data for the central solar and circumsolar beam irradiance [9]. Used in conjunction with the aforementioned solar profile model, spectral distribution can be incorporated with intensity distribution in solar flux modeling. Generated solar profiles are split into central solar and circumsolar regions and simple model of the atmospheric radiative transfer of sunshine (SMARTS)-generated spectra distributed appropriately according to relative intensity levels. The amalgamation of these two methods results in a solar profile generation model that is a powerful tool for CPV modeling. where C is the CSR. Normalized relative flux variation with angular distance from the solar center is here defined for two regions of space: the central solar and circumsolar regions. Flux variation in the central solar region is independent of the CSR, C, whereas flux variation in the circumsolar region is related to C through variables κ and γ. On a logarithmic plot of flux intensity I versus angular deviation θ, κ represents the I intercept and γ represents the gradient dI/dθ. More recent measurements were made by Neumann et al. [1] who conducted investigations into the frequency and variation of CSRs at three different terrestrial locations, namely Cologne, Germany; Almeria, Spain; and Odeillo, France. Such data are an important acquisition for CPV simulations as it allows for the algorithmic implementation of CSR variation. The inclusion of circumsolar radiation can, in some cases, lead to an increase in energy incident on a receiver by up to 5.26%. Currently, only variation with direct normal irradiance (DNI) bins is considered. This will act well as a first approximation, although Myers rec- II. RESULTS A. Circumsolar Ratio Variation and Frequency The application of CSR and DNI relationships from Neumann et al. [1] to hourly insolation data generated by the popular meteorological database, i.e., meteonorm [10], enables the approximation of location-specific CSR variation. Fig. 1 shows a comparison of CSR variation and frequency plots for the 64 IEEE JOURNAL OF PHOTOVOLTAICS, VOL. 2, NO. 1, JANUARY 2012 Fig. 2. Example of a solar profile with CSR = 0.4. Flux intensity is relatively normalized to a maximum of 1. Fig. 3. (red). Spectral compositions of solar profiles for CSRs of 0.4 (blue) and 0 locations Edinburgh, U.K. (55◦ 55’, 3◦ , 10’) (annual DNI ∼800 kWh), and Almeria, Spain (36◦ 50’, −2◦ 27’) (annual DNI ∼1900 kWh). Fig. 1 shows that a location typically described as unsuitable for HCPV, i.e., Edinburgh, has variation solar irradiation profiles weighted in the high CSR region, whereas an area much more suitable region for HCPV, i.e., Almeria, has solar profile variation weighted in the low CSR region. B. Solar Profiles Formula (4a)–(4c) has been used to generate 2-D solar profiles describing the spatial distribution of solar flux intensity. An example solar profile can be found in Figs. 2 and 3. Fig. 2 shows a spatially distributed solar profile that is generated with a CSR of 0.4. The relative flux intensity is displayed Fig. 4. Example of a sky-patch visible to 0.5◦ aperture lens normal to the sun. in color, normalized to a maximum intensity. Data of this form can be used as an input into a ray tracer for the simulation of optical illumination pattern that is formed at the receiver by concentration systems. The spectral distribution of the beam irradiation is, here, considered spatially binomial. Two spectral trends, defining the spectrally resolved radiation profiles of the central solar and circumsolar regions, are generated through the SMARTS program [9]. The standard AM1.5 spectrum is generated and then split into the two regions. These trends are appropriately applied to the flux intensity values of each data point, i.e., each simulated light ray. A CSR of 0.4 is something that can mostly be considered an extremity in geographical locations considered suitable for CPV. Fig. 3 compares the spectral content of a solar profile with a CSR of 0.4 with the other extremity—a solar profile with a CSR of 0. Fig. 3 shows extremities in solar spectra variation, the spectral shift is highlighted with the calculation of average photon energy (APE) [11]. The APE of a typical solar profile for CSR of 0 is 2.07 eV and for a CSR of 0.4 is 1.85 eV. Most spectra will fall somewhere between these two extremities. C. System Level Analysis The energy incident on a CPV system is effectively initially restricted by the entry aperture angle of the primary lens. The system field of view is then defined as that area of sky visible to the lens. Figs. 4 and 5 show a sky-patch visible to a 0.5◦ aperture lens with a CSR of 0.4. Fig. 4 shows the spatial intensity distribution of the sky-patch visible to a 0.5◦ aperture lens normal to the Sun. Each point in the image has a spectrally resolved intensity map, the net contribution of all points in this sky patch is shown in Fig. 5. Fig. 5 shows the net spectral distribution of the sky-patch visible to a 0.5◦ aperture lens normal to the Sun. Here, energy incident on the system amounts to 95% of the beam irradiation. COLE et al.: SOLAR PROFILES AND SPECTRAL MODELING FOR CPV SIMULATIONS Fig. 5. 65 Spectral content of the normal sky patch. Modeling the Sun as a point source or pillbox distribution, here, would show the net incident flux amounting to 100% of beam irradiation. The extended light source analysis shows a 5% loss in incident energy that would otherwise be ignored. Normality with the Sun is achieved with a perfect solar tracking system. Although perfect tracking is often simulated in CPV, it is realistically an implausible achievement. One of the major causes of poor system CPV performance is tracking error and the associated ramifications [12]. These effects can be investigated with the generation of misaligned solar images. Fig. 6. lens. Example of a sky-patch abnormal to the sun visible to 0.5◦ aperture Fig. 7. Spectral content of the abnormal sky patch. D. Tracking Error and Incident Irradiance Losses The generation of solar profiles permits the investigation of performance degradation because of tracking errors with the inclusion of misalignment parameters. Abnormal field-of-view sky patches are generated. Appropriate tolerance limits that are required by tracking systems can be defined. An example of a sky-patch visible to a 0.5◦ aperture lens abnormal to the Sun with a uniaxial tracking error of 0.141◦ (equivalent to a 0.1◦ error in each tracking axis, azimuthal and elevatory) is shown in Figs. 6 and 7. Fig. 6 shows an optical image that is the effective input to an optical concentration system with a 0.5◦ aperture lens abnormal to the Sun. Each point in the image has a spectrally resolved intensity map, the net contribution of all points in this sky patch is shown in Fig. 7. Fig. 7 shows the net spectral distribution of the sky-patch visible to a 0.5◦ aperture lens abnormal to the Sun. Here, the energy incident on the system amounts to 94% of the solar beam irradiation. Modeling the Sun as a point source or pillbox distribution, here, would show the net incident flux amounting to 100% of the solar beam irradiation. Thus, the extended light source analysis shows a 6% loss in incident energy that would otherwise be ignored. There is also a slight spectral shift. As the solar abnormality increases, so does the effective incident solar energy. When the tracking error reaches ∼0.25◦ , the central solar region begins to escape the input range of the concentration system. Beyond this point, the effective incident energy drops dramatically with further abnormality, and greater spectral shifts are seen. The effect of tracking error on incident energy loss varies with CSR; this variation is shown for a 0.5◦ aperture system in Table I. E. Effect on Annual Incident Energy Harvest The effect of using extended source descriptions on the annual incident energy harvest for Seattle, WA (annual DNI ∼1100 kWh), is shown. Meteonorm [10] was used to generate a set of hourly insolation information. Appropriate CSRs for each hourly step were then chosen using the analysis of Neumann [1]. The variation and frequency of CSRs for Seattle are shown in Fig. 8 Fig. 8 shows Seattle as having a CSR variation and frequency with an almost equal weighting of extremely high and low CSRs. Seattle is considered as comparatively (see Fig. 1) better for CPV than Edinburgh but worse than Almeria. 66 IEEE JOURNAL OF PHOTOVOLTAICS, VOL. 2, NO. 1, JANUARY 2012 TABLE I NORMALIZED IRRADIATION COLLECTION FOR CSR VERSUS TRACKING ERROR FOR A 0.5◦ APERTURE LENS Fig. 9. Effect of tracking error and primary lens aperture on annual incident energy harvest. Fig. 8. CSR variation and frequency plot for Seattle. Each hourly dataset was used to construct an effective system input image for a given primary lens aperture. The tracking error was set to a constant value and varied for each year-long simulation. The net incident flux at each time step was calculated. This was integrated over the entire year and compared with a complete beam collection (normalized to 1). Fig. 9 shows the effect of a continual tracking error on the incident energy harvests for Seattle-based systems with 0.5◦ , 0.75◦ , and 1◦ primary lens apertures. Fig. 9 shows that a 0.5◦ aperture CPV system in Seattle operating with a 0.3◦ tracking error will only collect 95% of the incident beam irradiation. It is important to note that the numbers presented here represent an initial loss, effectively the energetic loss at the primary lens. This does not include any further losses because of component misalignment. All losses presented here will be amplified at the receiver. III. CONCLUSION The simulation of high-concentration CPV systems employing optical concentration systems with small angle entry apertures can benefit greatly from realistic source descriptions, i.e., spatially distributed solar profiles. It can be argued that this actually calls into question the lone use of simple beam irra- diance for CPV system modeling, as such approximations give unrealistically high values of performance ratios. The spectral sensitivity of MJ cells highlights the importance of incorporating spatially distributed spectra in CPV modeling. This importance is elevated when dealing with systems employing small aperture concentration systems. It is shown that the CSR will change the incident spectrum. Optimizing the cell structure to include this effect can gain further energy at no additional cost. An investigation into the combined effects of tracking abnormality and CSRs has been undertaken. The results demonstrate the significance of the inclusion of such variables in a CPV simulation. Realistic tracking errors (0.3–0.4◦ ) can result in a 5–20% of incident energy for a 0.5◦ aperture system. Such information ought to be a primary consideration in system design and the definition of acceptable tracking tolerance levels. The inclusion of CSR variation, multidimensional solar profiling, and spatially distributed spectra in CPV modeling will improve the accuracy of system energy ratings and predictions and help better system design. REFERENCES [1] A. Neumann, A. Witzke, S. Jones, and G. Schmitt, “Representative terrestrial solar brightness profiles,” J. Solar Energy Eng., vol. 124, pp. 198–204, 2002. [2] J. Noring, D. Grether, and A. Hunt, “Circumsolar radiation data: The Lawrence Berkeley laboratory,” Nat. Renewable Energy Lab., Golden, CO, Rep. NREL/TP-262-4429, 1991. [3] Circumsolar radiation data: The Lawrence Berkeley laboratory reduced data base. (Oct. 2010). Renewable Resource Data Centre, Nat. Renewable Energy Lab., Golden, CO [Online]. Available: http://rredc.nrel.gov/solar/old_data/circumsolar [4] S. Puliaev, J. Penna, E. Jilinski, and A. Andrei, “Solar diameter observations at Observatorio Nacional in 1998–1999,” Astron. Astrophys. Suppl. Series, vol. 143, pp. 265–267, 2000. [5] D. Buie, C. Dey, and S. Bosi, “The effective size of the solar cone for solar concentrating systems,” Solar Energy, vol. 74, pp. 417–427, 2003. [6] D. Buie, A. Monger, and C. Dey, “Sunshape distributions for terrestrial solar simulations,” Solar Energy, vol. 74, pp. 113–122, 2003. [7] D. Myers, “Solar radiation modeling and measurements for renewable energy applications: Data and model quality,” Nat. Renewable Energy Lab., Golden, CO, Rep. NREL/CP-560-33620, 2003. COLE et al.: SOLAR PROFILES AND SPECTRAL MODELING FOR CPV SIMULATIONS [8] C. Gueymard, “Spectral circumsolar radiation contribution to CPV,” in Proc. 6th Int. Conf. Concentrating Photovoltaic Syst. Conf., Freiberg, Germany, 2010, pp. 316–319. [9] C. Gueymard, “SMARTS, A simple model of the atmospheric radiative transfer of sunshine: Algorithms and performance assessment,” Florida Solar Energy Center, Cocoa, FL, Tech. Rep. FSEC-PF-270-95, 1995. [10] Meteonorm. (Jul. 2011). Global solar radiation database. [Online]. Available: http://www.meteonorm.com/pages/en/meteonorm.php [11] T. R. Betts, C. N. Jardine, R. Gottschalg, D. G. Infield, and K. Lane, “Impact of spectral effects on the electrical parameters of multijunction amorphous silicon cells,” in Proc. 3rd World Conf. Photovoltaic Energy Convers., Osaka, Japan, 2003, pp. 1756–1759. [12] B. Stafford, M. Davis, J. Chambers, M. Martinez, and D. Sanchez, “Tracking accuracy: Field experience, analysis, and correlation with meteorological conditions,” in Proc. 34th IEEE Photovoltaic Spec. Conf., Philadelphia, PA, 2010, pp. 002256–002259. 67 Ian R. Cole, photograph and biography not available at the time of publication. Thomas R. Betts, photograph and biography not available at the time of publication. Ralph Gottschalg, photograph and biography not available at the time of publication.