A simple solar radiation model for computing direct and diffuse spectral fluxes

A solar spectral model that describes the solar radiation flux on a clear day at any given location was developed and tested. The model computes spectral fluxes of global, global photon, direct and diffuse solar radiation incident at the surface. Input parameters describe location and atmospheric characteristics. Location is described by latitude, altitude, slope orientation and surface albedo. Atmospheric characteristics described are turbidity, precipitable water vapor and total ozone content. The model was constructed using a one-layer homogeneous atmosphere with refinements, which are:

1. (1) Use of climatological data to predict the total ozone content, if it is not known.

2. (2) A more advanced treatment of infrared solar radiation (0.8–4.5 μm) absorption.

3. (3) A more complex scheme for predicting diffuse radiation.

4. (4) The capability of handling a spectral albedo.

5. (5) Inclusion of albedo dependence on zenith angle.

Input parameters are minimized and several simplifying features are incorporated for ease of handling variables not routinely measured. Turbidity and total ozone content are treated as climatological estimates if specific location measurements are not available. Precipitable water vapor can be predicted using surface vapor pressure, since the sounding network is not dense. These features allow researchers outside the field of solar radiation to use the model.Because complete measurements with needed location and atmospheric characteristics could not be found, the validity of the model was tested by comparing it with a more complex, multilayered atmosphere model by Dave et al. [3]. Calculated fluxes of total direct, diffuse, and global radiation from the model presented were 11.5, 20.1 and 13.2 per cent lower, respectively, when they were adjusted for differences in extra-terrestrial solar radiation fluxes. Direct spectral fluxes closely agreed in spectral composition, with slight exception in the 0.8–0.95 μm region. Diffuse spectral flluxes were slightly higher in the UV region lower in the rest of the spectrum than were those in the multilayered atmosphere model. A sensitivity analysis of the model was also conducted: the most influential inputs were found to be latitude, slope orientation and turbidity, and the least influential was total ozone content.The hourly integrated values for the model compared very well with measured values for clear days at Davis. Discrepancies between predicted and measured values were due to lack of local turbidity coefficients.     

An atmospheric model for computing direct and diffuse solar radiation

This paper proposes an atmospheric model, which extends the computation of the direct radiation given by Cole's model to the cloudy sky and shows a method to calculate the diffuse radiation.Therefore the monthly average values of the global radiation incident on a horizontal surface at Palermo are computed by this method. These values are compared with the experimental data provided by the Istituto di Idraulica Agraria, Università di Palermo and generally exhibit a mean deviation not more than 10 per cent. The deviations become lower than 8 per cent taking into account the effect of the underlying surface albedo.In order to better verify the validity of the proposed method, it should be suitable to extend its application to other locations provided with actinometric stations. This should allow to use this method with more realibility to predict the radiation incident on the locations lacking in actinometric data.     

A model for calculating direct and diffuse solar radiation

A new model is presented for computing both direct and diffuse solar radiation for a cloudy sky, based on the model of King and Buckius for the direct component for a clear sky. The model is employed to calculate the daily global insolation incident on a horizontal surface at Ibadan for the year 1977. The results are presented in the form of both weekly and monthly averages, and compared with the experimental data provided by the International Institute of Tropical Agriculture (I.I.T.A.), Ibadan. Two values of cloudiness coefficient k (= 1.0 and 0.75) are used in the calculations, with the case of k = 0.75 being superior and for which the deviations from the data do not exceed 15 per cent.     

A simple clear skies model for the luminous efficacy of diffuse solar radiation on inclined surfaces

In the present work we study the luminous efficacy of diffuse solar radiation incident on vertical surfaces for a clear sky and mean hourly values of diffuse irradiance and diffuse illuminance. We develop a model easy to use, similar to a model previously obtained for horizontal surfaces. To develop the present model for vertical surfaces we assume that the slope of the surface influences diffuse illuminance and diffuse irradiance in the same way. As a consequence of this hypothesis, the luminous efficacy of diffuse solar radiation is assumed to be the same for both horizontal and inclined surfaces.     

Applicability of a simple model for computing diffuse solar radiation to locations of the European, African, Asian and North American areas

It is tested if the correlation type D = K(s/S)^−0.25 (sin h_n)^1.55, 0.2 s/S 0.9 previously proposed by Coppolino for Italian locations using K = 7.0, allows a reliable estimation of the monthly mean daily diffuse solar radiation D (MJ m⁻² day⁻¹), from only the monthly mean daily relative sunshine s/S and the noon altitude of the sun on the 15th of the month h_n (degrees) at locations of the European, African, Asian and North American areas. The test is performed at 14 stations in the above areas displaced at various latitudes L, elevations above sea level E and geographical situations, using K = 7.0 for the stations where 0.48 (s/S)_m, 0.63, K = 8.5 for those where 0.40 (s/S)_m < 0.48 or (s/S)_m > 0.63, and K = 10.0 for those where (s/S)_m < 0.40; (s/S)_m is the yearly mean value of s/S. K = 7.0 is chosen for five stations provided only with sunshine measurements and for three of the other nine stations provided with diffuse radiation measured data as well. K = 8.5 is chosen for five of the above nine stations and K = 10.0 for the remaining one. Moreover the prediction validity of the tested correlation is compared with that of two equations commonly quoted in literature: one of D = f(K_t, G) type and the other of D = f (s/S, G) type, where K_t = G/G₀. G and G₀ are the global and extraterrestrial monthly mean daily solar radiation, respectively. The test results point to a good agreement between the measured data of D and values from the above (D = f(s/S, h_n)) equation for eight of the above nine stations (i.e. with the exception of one with E = 2210 m), and deviations are generally low among D-values from the three considered equations, respectively, for the other five stations. Consequently the tested D = f(s/S, h_n) equation, with K-value chosen by the above guidelines, can be considered applicable, with good reliability, to all locations of the above areas provided 30°S < L < 55°N and E < 2000 m.     

A simple, solar spectral model for direct-normal and diffuse horizontal irradiance

A spectral model for cloudless days that uses simple mathematical expressions and tabulated look-up tables to generate direct-normal and diffuse horizontal irradiance is presented. The model is based on modifications to previously published simple models and comparisons with rigorous radiative transfer codes. This model is expected to be more accurate than previous simple models and applicable to a broader range of atmospheric conditions. The primary significance of this model is its simplicity, which allows it to be used on small desk-top computers. The spectrum produced by this model is limited to 0.3–4.0 μm wavelength with an approximate resolution of 10 nm.     

A comparison of methods for estimating hourly diffuse solar radiation from global solar radiation

Four methods of estimating hourly diffuse irradiation from hourly global irradiation are compared, using global and diffuse irradiation data from the Australian Bureau of Meteorology for five widely separated Australian locations. A development of one of these methods with constants derived from the data for each place is found to perform best when judged on a criterion of absolute error, and this performance in maintained when constants averaged over the five locations are used. The derived constants are also given for all other Australian locations for which both global and diffuse data are available, and a method is suggested for deriving suitable values of the constants for places situated between 20° and 45° S for which only global radiation data are available. A previously reported dependence of the proportion of diffuse radiation on latitude is demonstrated.     

Computing diffuse fraction of global horizontal solar radiation: A model comparison

For simulation-based prediction of buildings’ energy use or expected gains from building-integrated solar energy systems, information on both direct and diffuse component of solar radiation is necessary. Available measured data are, however, typically restricted to global horizontal irradiance. There have been thus many efforts in the past to develop algorithms for the derivation of the diffuse fraction of solar irradiance. In this context, the present paper compares eight models for estimating diffuse fraction of irradiance based on a database of measured irradiance from Vienna, Austria. These models generally involve mathematical formulations with multiple coefficients whose values are typically valid for a specific location. Subsequent to a first comparison of these eight models, three better performing models were selected for a more detailed analysis. Thereby, the coefficients of the models were modified to account for Vienna data. The results suggest that some models can provide relatively reliable estimations of the diffuse fractions of the global irradiance. The calibration procedure could only slightly improve the models’ performance.     

A simple hourly clear-sky solar radiation model based on meteorological parameters

An hourly solar radiation model, based on observed meteorological data, was developed and tested. As a means of comparison, the Watt and Bird models and the Solmet regression models were also tested. Several conclusions were drawn about the parameterization of solar radiation-depleting parameters. It was determined that a reasonable estimate of lower layer aerosol extinction can be determined using humidity, visibility, and mixing height. The parameterization of water vapor absorption obtained by atmospheric rather than laboratory observations was found to give better model results.     

A simple hourly all-sky solar radiation model based on meteorological parameters

An hourly solar radiation model for cloudy skies, based on meteorological data, was developed andtested. As a means of comparison, the SOLMET regression and Watt models were also tested. The present model was examined for individual cloud types using measured solar radiation to judge the effectiveness of the model in the presence of particular clouds.     

Models of diffuse solar radiation

For some locations both global and diffuse solar radiation are measured. However, for many locations, only global is measured, or inferred from satellite data. For modelling solar energy applications, the amount of radiation on a tilted surface is needed. Since only the direct component on a tilted surface can be calculated from trigonometry, we need to have diffuse on the horizontal available. There are regression relationships for estimating the diffuse on a tilted surface from diffuse on the horizontal. Models for estimating the diffuse radiation on the horizontal from horizontal global that have been developed in Europe or North America have proved to be inadequate for Australia [Spencer JW. A comparison of methods for estimating hourly diffuse solar radiation from global solar radiation. Sol Energy 1982; 29(1): 19–32]. Boland et al. [Modelling the diffuse fraction of global solar radiation on a horizontal surface. Environmetrics 2001; 12: 103–16] developed a validated model for Australian conditions. We detail our recent advances in developing the theoretical framework for the approach reported therein, particularly the use of the logistic function instead of piecewise linear or simple nonlinear functions.Additionally, we have also constructed a method, using quadratic programming, for identifying values that are likely to be erroneous. This allows us to eliminate outliers in diffuse radiation values, the data most prone to errors in measurement.     

Computation of diffuse solar radiation

A technique for computing the spectral and angular (both the zenith and azimuthal) distribution of the solar energy reaching the surface of earth and any other plane in the atmosphere has been developed. Here the computer code LOWTRAN is used for getting the atmospheric transmittances in conjunction with two approximate procedures: one based on the Eddington method and the other on van de Hulst's adding method, for solving the equation of radiative transfer to obtain the diffuse radiation in the cloud-free situation. The aerosol scattering phase functions are approximated by the Hyeney-Greenstein functions. When the equation of radiative transfer is solved using the adding method, the azimuthal and zenith angle dependence of the scattered radiation is evaluated, whereas when the Eddington technique is utilized only the total downward flux of scattered solar radiation is obtained. Results of the diffuse and beam components of solar radiation received on surface of earth compare very well with those computed by other methods such as the more exact calculations using spherical harmonics and when atmospheric conditions corresponding to that prevailing locally in a tropical location (as in India) are used as inputs the computed values agree closely with the measured values.     

A simple model for estimating the transmittance of direct solar radiation through clear atmospheres

The clear-day all-wavelength transmittance τ of solar radiation directly through the 1962 standard atmosphere to a surface at altitude A is found to fit a simple mixed-gray-gas model (1 black, 1 gray, 1 clear) with a maximum error of 0.4 per cent. The relation is τ= a_o+a₁e^{k/cos z} where z is the zenith angle of the sun. The constant a₀, a₁ and k are functions only of altitude and of haze model (visibility range). Allowance is made for the effect of four climate types (tropical, midlatitude summer and winter, and subarctic summer) by modifying the 3 constants by ratios which depend on climate type only (not on altitude or haze model) and which lie between 0.92 and 1.04. This simple model should be capable of being combined with sunshine or cloud records, plus generalizations covering diffuse radiation, to yield predictions of insolation for use in design optimizations.     

A simple model for sky radiation measurements

In the proposed model the sky hemisphere as a radiating source has been replaced by a single point radiating source and a diffuse background. The point source produces insolation H on a horizontal surface, its beam coming from direction (α,?), α being the azimuth and ? the zenith angle. The diffuse radiation comes from all directions equally and produces insolation D on a horizontal surface. For estimating all four parameters H, α, ? and D of the model, a four-segment photo-cell was developed and tested. The method is suitable for calculating the optimal orientation (azimuth and tilt) of a solar collector, depending on local conditions.     

A brief review of models for computing solar radiation on inclined surfaces

Modelling of solar energy systems requires estimation of the hourly radiation incident on surfaces of different tilts and orientations. Most meteorological stations report radiation values on a daily rather than hourly basis. Again, the value of the diffuse component may or may not be reported. Hence, different strategies may be encountered, for each location, with the common goal of computing hourly radiation on inclined surfaces.

In this study, five different schemes are presented to achieve this goal. In each scheme, certain correlations are required which are discussed in detail. The study is divided into five sections, each dealing with a specific type of correlation. These sections are: (i) correlations between daily diffuse and global radiation; (ii) correlations between hourly/daily global radiation; (iii) correlations between hourly/daily diffuse radiation; (iv) correlations between hourly diffuse and global radiation; and (v) models for computing diffuse sky radiation on inclined surfaces. The last section deals extensively with the anisotropic nature of sky diffuse radiation.

The important aspects of all correlation studies are highlighted, and the relative merits and demerits of their results are brought to light.

Mathematical expressions, where available, for models/correlations are provided so that the reader will have access to a comprehensive study. This information should be useful for modelling purposes in which computation of radiation on surfaces of different orientations and tilts is required.     

A hybrid model for estimating global solar radiation

In view of the site-dependence of Ångström correlation, this study developed a hybrid model to estimate global radiation H. Unlike Ångström correlation H=(α+βS/S₀)H₀, this model suggested that H=(a+b S/S₀)H_b+(c+d S/S₀)H_d, H_b and H_d are effective beam radiation and effective diffuse radiation, which imply latitude, elevation and seasonal effect on radiation. H_b and H_d are calculated by an arithmetic model derived from spectral model. The hybrid model was designed for estimating monthly mean daily global radiation with hourly-recorded bright sunshine time, and its applicability was verified at observatories in Japan.     

A note on diffuse solar radiation on a tilted surface

We are presenting a note on comparative study of diffuse component of solar radiation on tilted surface with different angle of inclination. The monthly mean daily diffuse radiation have been calculated on inclined surfaces from the data of horizontal surface using the Circumsolar and Isotropic model and the anisotropic model of Klucher and Hay at Lucknow (Latitude 26.75°, Longitude 80.85°), Uttar Pradesh, India. These calculated results are compared with our measured value. The comparison shows that Klucher model who considered the effect of cloudy sky conditions gives comparatively good estimations particularly at low inclination angles.     

A model for the diffuse fraction of hourly global radiation

Hourly values of diffuse and beam irradiance are often required in cases where, at best, only global irradiance is available. For use in the evaluation of the climatological potential for solar energy utilization in Norway, an analytical model is proposed expressing the hourly diffuse fraction of global irradiance in terms of hourly solar elevation and clearness index. This model, developed for average snow-free close to sea level conditions in Norway, fairly well fits the average picture of an extensive independent data base.