Evaluation of models to predict insolation on tilted surfaces

An empirical study was performed to evaluate the validity of various insolation models which employ either an isotropic or an anisotropic distribution approximation for sky light when predicting insolation on tilted surfaces. Data sets of measured hourly insolation values were obtained over a 6-month period using pyranometers which received diffuse and total solar radiation on a horizontal plane and total radiation on surfaces tilted toward the equator at 37° and 60° angles above the horizon. Data on the horizontal surfaces were used in the insolation models to predict insolation on the tilted surface; comparisons of measured vs calculated insolation on the tilted surface were examined to test the validity of the sky light approximations. It was found that the Liu-Jordan isotropic distribution model provides a good fit to empirical data under overcast skies but underestimates the amount of solar radiation incident on tilted surfaces under clear and partly cloudy conditions. The anisotropic-clear-sky distribution model by Temps and Coulson provides a good prediction for clear skies but overstimates the solar radiation when used for cloudy days. An anisotropic-all-sky model was formulated in this effort which provided excellent agreement between measured and predicted insolation throughout the 6-month period.     

Incident solar radiation on Argentina from the geostationary satellite GOES: Comparison with ground measurements

Daily values of the solar radiant energy incident on earth's surface constitute a quantity of increasing importance, not only in nonconventional energy development but also for agricultural, weather, and climate monitoring and predictions. In Argentina, the solar global radiation has been measured since 1978 through a network of pyranometers distributed all over the country. Simultaneously, in the area limited by 21° and 41°S latitude and 53° and 67°W longitude, insolation determinations have been made using the geostationary satellite GOES from May 1982 to June 1984. Therefore, it was possible to compare the simultaneous hourly and daily values of solar global radiation corresponding to the northern part of the country during 1982 and 1983. In this article, the results of the comparison are presented. It is shown that the standard error of the satellite-derived hourly and daily insolation values when compared against pyranometers is about 25%, and in the range of 15 to 20% of the mean values, respectively. Therefore, operational estimation of surface insolation in the region using GOES data appears normally feasible.     

The two-solarimeter method for insolation on inclined surfaces

In calculating insolation values on inclined surfaces one must usually start from global insolation values on a horizontal surface. If in addition to these values the global insolation on an inclined surface is known, the inaccuracies of such calculations can be substantially reduced. In general this is the case if the global insolation on two differently inclined solarimeters is given. When these global values are given, a mathematical relation between them and the direct and diffuse components on a tilted surface can be established. This method, called the “two-solarimeter method”, is described in this paper. It is shown how the two-solarimeter method for a horizontal surface can be optimized in combination with a correlation procedure. For three vertical surfaces the calculated results are compared with measured values. A detailed error analysis suggests that the two-solarimeter method may become an attractive alternative to the more complicated traditional ways of determining the direct and diffuse components (i.e. measurement of the global and diffuse insolation on a horizontal surface).     

Correlation between hourly diffuse and global radiation for New Delhi

Estimation of hourly insolation on tilted surfaces is required in simulation of solar energy systems. This necessitates splitting of hourly global horizontal insolation into diffuse and direct components. Many models have been developed for this purpose, and the aim of this study is to develop a correlation between hourly diffuse ratio and clearness index for New Delhi. The correlation is developed on the basis of measured data for two years. The performance of this correlation was checked by calculating the difference between computed and measured diffuse radiation. The correlation proved to perform quite well. A comparison of the present correlation with those for other locations showed that such correlations are location dependent.     

Hourly vs daily method of computing insolation on inclined surfaces

Insolation on south-facing inclined planes has been computed using hourly values of total and diffuse radiation, obtained from experimental data. Such a computation procedure is then compared with the widely used method by Liu and Jordan for obtaining daily insolation on surfaces tilted toward the equator. Very small differences are noted between the results obtained by the two methods. These differences are mainly due to three factors; (a) Liu and Jordan's formulation uses a theoretical day-length while the hourly method uses day-length as incicated by the radiation data; (b) hourly method takes into account the asymmetries of total and diffuse radiation around solar noon while the daily method implicitly assumes symmetry of the same; (c) the daily method assumes uniform atmospheric transmissivity to beam radiation throughout the day. On the other hand, the hourly method assumes constant atmospheric transmissivity for one hour only.     

Estimation of hourly solar radiation for India

The ASHRAE constants predict high values of the hourly beam radiation and very low values of the hourly diffuse radiation when used to predict radiation at Indian locations. Hence a procedure has been developed for the estimation of direct, diffuse and global hourly solar radiation on a horizontal surface for any location in India. To calculate hourly solar radiation, an exponential curve, similar to the one used by ASHRAE, was fitted to the measured solar radiation data of six cities from different regions of India. The statistical analysis was carried out for the data computed using ASHRAE constants and the set of constants obtained for India using the measured data of four different Indian cities selected randomly. Three statistical indicators were used to compare the accuracy of the developed procedure. The results show that ASHRAE constants are not suitable to estimate hourly solar radiation in India. Hourly solar radiation estimated by constants obtained for India are fairly comparable with measured data. The mean percentage error with Indian constants for these four Indian cities was found as low as 2.27, −6.29 and −6.09% for hourly beam, diffuse and global radiation, respectively.     

Quality control and estimation hourly solar irradiation on inclined surfaces in South Korea

The measured data of global solar irradiation on a horizontal surface, the number of bright sunshine hours, and the amount of cloud cover for major cities of South Korea, during the period (1986–2005) are analyzed. Quality controls tests were carried out to eliminate spurious data and inaccurate measurements resulting from the cosine response error of the pyranometers. Accordingly, in six locations of South Korea where hourly global solar irradiation is currently measured, the hourly global solar irradiation was calculated using two Sunshine Fraction Radiation Model (SFRM) and a Cloud Cover Radiation Model (CRM), were compared and analyzed. Data for Ulsan, where hourly global solar irradiation is not currently measured, can be calculated from the data of the six other locations. The total solar irradiation on horizontal surfaces is separated into the beam and diffuses components. Empirical correlations for estimating the diffuse solar irradiation incident on horizontal surfaces have been proposed. The purpose of our work consists of predicting the solar irradiation on inclined surfaces based on horizontal measurements. To reach this goal, two studies were performed. They consisted of quantifying the accuracy of various well-known models. The first type of models calculated diffuse horizontal irradiations from global ones and the second type computed global irradiations on inclined planes from diffuse and global components on a horizontal surface. The proposed model can provide an alternative to building designers in estimating the solar irradiation on inclined surfaces where only the horizontal measurements are available.     

The dependence on solar elevation of the correlation between monthly average hourly diffuse and global radiation

In the present work, the dependence on of the correlation between and is studied, , , and respectively being the monthly average hourly values of the global, diffuse, and extraterrestrial radiation, all of them on a horizontal surface, and the solar elevation at midhour. The dependence is studied for Uccle for the following sky conditions. Condition A: clear skies (fraction of possible sunshine = 1) and the maximum values of direct radiation measured during the period considered (each of the hours before or after the solar noon for which radiation is received); Condition B: corresponding to all the values of radiation measured when the sunshine fraction is 1 during the period considered; Condition C: corresponding to all the data collected, independently of the state of the sky; Condition D: corresponding to overcast skies ( ).From the available values of and (monthly average hourly direct radiation on a horizontal surface), values of and for 5° ≤ ≤ 45° and Δ = 5° are calculated using Newton's divided difference interpolation formula. The interpolated values occupy three clearly different regions in the plot, one for each of the conditions A,B, and C. For Condition A and each value of best linear fits with high correlation coefficients are obtained for the correlation. The influence of the Linke turbidity factor on the correlation for Condition A is studied for 5° ≤ ≤ 35°, Δ = 5°.     

Direct and diffuse insolation using approximation methods applied to horizontal surface insolation

Approximation methods estimate the A, B and C required to calculate the direct and diffuse solar insolation by the ASHRAE equations. Theoretical, experimental and meteorological records of horizontal surface insolation under clear sky conditions are used to study the behavior of the approximation methods in solving the equation I_H exp (BM) = A(sin α + C). Instantaneous data read from the chart trace of horizontal surface insolation gave better results than the cumulative horizontal surface radiation measured from solar noon. With careful selection of data, a probable error of ±5 per cent for A and ±10 per cent for B is not unreasonable.     

Model evaluation and optimum collector slope for a tropical country

Various models proposed in the literature, viz. the Beam method, proposed by Morse & Czarnecki; two methods proposed by Liu & Jordan, using hourly values and average daily values of solar radiation: and Klucher's method are compared in this paper using measured hourly values of global and diffuse radiation on a horizontal surface and the total radiation on a tilted surface. It was observed that the model proposed by Klucher estimates the radiation more consistently with the experimental observations for clear as well as cloudy sky conditions.

This method was used for predicting the year round radiation availability on (i) a fixed collector inclined at an optimum tilt, (ii) a partially tracking collector in which the slope is fixed but following the sun's azimuth angle, and (iii) a fully tracking collector. In this analysis, measured values of hourly global and diffuse solar radiation on a horizontal plane for four widely separated Indian stations, viz. Delhi (28°38′N), Poona (18°29′N), Calcutta (22°36′N), and Madras (13°8′N) as per India Meteorological Department classification for climatic analysis, have been used.

With the model proposed, the radiation availability on different slopes for all the twelve months of the year for a fixed and partially tracking collector have been evaluated and optimum tilts for various seasons of a year for different solar energy applications for all the four Indian stations have been determined. For a typical winter month at New Delhi in the semi-arid zone of India, a fixed collector, a partially tracking collector and a fully tracking collector receive 60, 80 and 95% more radiation compared to that received on a horizontal plane respectively. Another observation is that the variation of the insolation received on inclined surfaces is not at all sensitive to the angle of tilt in the neighbourhood of the optimum angle of operation.     

Estimation of half-hour solar radiation values from hourly values

A statistical Markovian insolation model for predicting the time-sequence of half-hour solar radiation values on a horizontal surface which uses the hourly insolation values is developed. The hourly transition density function, governing the diurnal evolution of the hourly solar radiation values, is used for obtaining the half-hour transition density function. A transition density function is a measure of the probability of the event at the next immediate hour of interest when the event at the present hour is given. The estimation of half-hour transition density function is done through the fundamental decomposition theorem for the density function. This assumes a set of well defined intermediate states. As a first approximation, the half-hour transition density function is assumed to be temporally stationary. Furthermore, it is assumed that the cumulative probability distribution functions of the normalised initial hour solar radiation value and the normalised initial half-hour solar radiation values are not significantly different; the hourly, or half-hourly solar flux values are normalised by the corresponding extraterrestrial solar flux values. The validity of these assumptions is established through the successful time-sequence predictions of the half-hour insolation values. The time-sequence aspect of solar radiation values is proved by comparing the predicted joint cumulative distribution functions for several successive normalised half-hour values, with the corresponding distribution function for the recorded values. In order to prove that the predicted and actual distribution functions are from the same set, the non-parametric statistical test proposed by Kolmogorov and Smirnov has been used.     

Evaluation of models for predicting insolation on slopes within the Colorado alpine tundra

This empirical study evaluates insolation predictions for the Colorado tundra from models based upon isotropic and anisotropic distribution approximations for diffuse sky and reflected solar radiation. The data set of hourly insolation values was obtained from 40 locations on an alpine ridge by simultaneous measurement of direct beam irradiance and total insolation to the horizontal ridge crest and two nearby sloping surfaces. Six models are used to predict insolation, two based on the isotropic distribution of diffuse solar radiation and four on anisotropic diffuse distribution fields. Three models (one isotropic and two anisotropic) employ measurements of normal direct beam irradiance while the other 3 models incorporate the correlation between the “clearness index” and the ratio of diffuse sky to total insolation. The precision of insolation estimates from models using the correlation method is only slightly less than from other models. Accounting for the increase of diffuse radiation in the circumsolar sky improves insolation predictions. However, inclusion of additional regions of diffuse radiation anisotropy decreases model accuracy. Errors of insolation estimates for the alpine tundra from all models vary in a systematic manner as a function of relative azimuth and ground slope angles.     

Prediction of hourly diffuse solar radiation from measured hourly global radiation on a horizontal surface

A statistical procedure has been employed to develop correlations between the hourly global horizontal radiation and its diffuse component. Several years', hourly radiation data from three Canadian stations and two French stations have been employed for this purpose. The relationships have been developed in dimensionless form which predict for particular solar altitudes when is given.Under heavily cloudy conditions or when the sky is completely covered ( ), diffuse radiation increases linearly with the global radiation. In this region, solar altitude has no bearing on the fraction of diffuse radiation.As goes beyond 0.35, the effect of solar altitude begins to appear and the region immediately following this may be considered as partly-cloudy-skies conditions. In the beginning of this region, the diffuse component increases briefly with the increase in global radiation and then begins to decrease as the partly cloudy skies become clearer. At particular solar altitudes, a minimum value of the diffuse radiation is reached. The value of where reaches its minimum value varies with solar altitude.The region beyond which a minimum value of is reached may be considered as mainly-clear-sky conditions. In this region, increases again with , lower solar altitudes giving a higher percentage of diffuse radiation.Under partly cloudy skies and under clear skies, solar altitudes lower than 30° had a marked effect on the fraction of diffuse radiation. However, solar altitudes greater than 30° had minimal influence on the fraction of diffuse radiation.     

Estimation of global solar irradiation on horizontal and inclined surfaces based on the horizontal measurements

Solar radiation data are essential in the design of solar energy conversion devices. In this regard, empirical models were selected to estimate the global solar radiation on horizontal and inclined surfaces. The hourly solar radiation data measured at the study area during the period of 2004-2007, were used to calculate solar radiations using selected models. The selected models were compared on the basis of statistical methods. Based on the results, a new model, H/H_o = 0.19490 + 0.4771(n/N) + 0.02994 exp(n/N) has been developed, based on Kadir Bakirci linear exponential model. This is highly recommended to estimate monthly mean daily global solar irradiation, on a horizontal surface. Further, a model to convert horizontal solar global radiation to that of radiation on a tilted surface is also presented. It is based upon a relatively simple model proposed by Olmo et al. which requires only measurements of horizontal solar radiation. The developed model appears to give excellent results and has the advantage of being relatively simple for applications. The present work will help to improve the state of knowledge of global solar radiation to the point where it has applications in the estimation of global solar radiation, both on horizontal and inclined surfaces.     

Two estimation methods for monthly mean hourly total irradiation on tilted surfaces from monthly mean daily horizontal irradiation from solar radiation data of Ajaccio, Corsica

The sizing of a photovoltaic or a thermal solar system is generally based on monthly mean values of daily solar radiation on tilted surfaces. Many authors have demonstrated that it will be better to use monthly mean values of hourly radiation, particularly taking into account the Sun's position and to predict long-term performances of solar systems. (Liu and Jordan, 1963; Clark et al., 1984). Moreover, for most of the sites around the world, only monthly mean values of daily horizontal total irradiation are available for use in such calculations. We propose, by using well-known correlations in the literature, to estimate these monthly mean values of hourly total irradiation on tilted planes from monthly mean values of daily total horizontal irradiation, using three steps:

• — determination of monthly mean value of hourly total horizontal irradiation;

• — determination of monthly mean value of hourly diffuse horizontal irradiation;

• — determination of monthly mean value of hourly total irradiation on tilted planes.

In the first step, using the Collares Pereira and Rabl correlation, the root mean square error (RMSE) between correlated and experimental calculated data is 8%. In the second step, we used two methods: the first one utilizes the Erbs correlation and the second one is based on a local correlation which has been developed in our centre. Both of them gave identical results with an RMSE lower than 9%. We calculated monthly mean values of hourly total irradiation on three tilted planes (30°, 45° and 60°) and we compared these results with the experimental ones, obtaining a RMSE respectively of less than 10%. The method is then validated by these results.     

Comparison of diffuse/global ratios calculated from one-minute, hourly and daily solar radiation data

One-minute values of direct, diffuse and global radiation have been continuously collected at Davis, California (38.5°N, 121.1°W) since 1 January, 1979. These datasets are quality controlled to insure the most accurate and reliable data possible. Analysis of one-minute data has provided an opportunity to evaluate some of the bias that may be inherent in statistical representation of solar radiation data. A simple mean and standard deviation do not adequately describe the variation in the data and we show that a more representative treatment includes the box and whisker analysis. In this the mean, median, first and third quartiles, and the maximum and minimum ranges are presented. It is possible to compute the variability between days more completely with this technique while the means may be very close. This has application to evaluation of solar collectors as a better method of evaluating theire efficiency. This is applied to diffuse/global ratios which show a seasonal dependence although some clear winter days have ratios close to clear summer values; however, the first and third quartile and median distinctly separate these days. Analysis of solar radiation data should be conducted with caution as shown by these results.A simple model is proposed to compute hourly global values from the integrated daily total. Comparisons of calculated with measured hourly values indicated less than a 10 per cent error between 0700 to 1600 with the maximum value being slightly underestimated. This procedure allows one to evaluate solar collectors with only daily values and presents a method for thoroughly evaluating our solar resources.     

Techniques for the precise estimation of hourly values of global, diffuse and direct solar radiation

The method usually used to compute solar radiation, when no measured data are available, is the well-known regression technique relating mean daily totals of global and diffuse solar radiation with the mean duration of sunshine. Using this method and taking into account the first order multiple reflections between the ground and the atmosphere, regression parameters were obtained from the monthly mean values of daily totals of global solar radiation and sunshine at a network of 16 stations in India. Daily values of global and diffuse solar radiation were then computed for 121 stations, where sunshine data are available for periods of 6–28 yr, using interpolated values of the regression parameters. Where no sunshine data were available, global and diffuse solar radiation were computed from cloud observations, using the inverse relationship between sunshine and cloudiness. Further, using the empirical relationship between daily totals and the corresponding hourly values of global and diffuse solar radiation, two sets of curves were prepared valid for the whole country, using which mean hourly values of global and diffuse radiation could be deduced from the corresponding daily totals, with a high degree of accuracy. The paper discusses the validity of the techniques used for computing daily and hourly values of global and diffuse solar radiation from sunshine and cloud amounts at an extended network of 145 stations in India and stresses the fact that such techniques are successful, only if accurate data on both radiation and sunshine are available at a widely distributed network of stations for a minimum period from at least 5 to 6 yr, using carefully calibrated and well-maintained instruments of the required quality. Theoretical models have also been used to compute clear sky noon values of global, diffuse and direct solar radiation from the solar constant, allowing for attenuation by atmospheric constituents such as ozone, water vapour, dust and aerosols. Using a simple model, calculations of global and diffuse solar radiation on clear days were made for 145 stations from values of the solar constant and measured values of ozone, water vapour and atmospheric turbidity. A method of extending the technique to overcast skies and partly clouded skies is discussed. The values of the mean annual transmission factor for global solar radiation under cloud-free conditions using the two methods show excellent agreement and establishes the soundness of the regression technique on one hand and the reliability of the theoretical model used for computing clear sky radiation, on the other.     

A model for the calculation of solar global insolation

A theoretical model is described that is designed to give the total global insolation falling on the earth's surface and the transmission of the atmosphere. It is compared to a model by Braslau and Dave[1] and found to agree to within a few percent in all cases. Climatogical values of total pricipitable water, turbidity, and surface albedo are required as the model inputs, and the sources of these data are described. The model has been applied to 26 stations in the National Weather Service (NWS) pyranometer network, where measured true solar noon atmospheric transmission values are available, as part of the NOAA program to rehabilitate the old pyranometer observations. For three of these stations where reliable true solar noon irradiance and transmission values are available, the model calculations and observations are compared. At 18 locations the calculated and measured daily mean insolation values are compared for clear days. At one location (Boulder, Colorado) calculated and measured radiation climatologies for all weather conditions are compared. In all comparisons the model and observations differ by no more than 2.7 per cent, which is within the experimental accuracy (±5 per cent) of the pyranometers. Possible sources of errors are discussed.     

Stochastic simulation model of hourly total solar radiation

A stochastic simulation model of hourly global solar radiation is presented in this paper. It is developed by introducing the concept of “time dependent frequency distribution” (TDFD) of hourly insolation values. In this model the two most critical aspects of time series simulation, i.e., the reproduced time series values which have the appropriate time dependent frequency distribution for the parameter being simulated and the correlation between successive values, are taken into account. The elimination of the TDFD of the data and the transformation of the data distribution to a Gaussian distribution (required for the stationary time series analysis) were carried out using a mapping technique. The autocorrelation function of the transformed data showed that the produced time series is stationary. Then, an antimapping coefficient matrix is developed, which provides a simple yet an effective simulation device. The described model has been applied in Athens (Greece) where hourly insolation data covering a period of two years are used. The theoretical results obtained using this simulation model, regarding both the TDFD and the correlation, are in agreement with the measured data.