Solar radiation utilizability

The nonlinear behaviour of many solar energy systems requires consideration of the fluctuating nature of solar radiation. Utilizability curves constructed for this purpose consider the radiation fluctuation on the basis of hourly or daily insolation values. Solar collectors, however, do not see radiation fluctuations in terms of hourly or daily insolation values, but respond to instantaneous radiation fluctuations caused by clouds. The fluctuation of instantaneous and short-time integrated radiation has been shown to differ significantly from that of daily insolation. This paper presents utilizability curves constructed from the cumulative probability distribution of instantaneous terrestrial insolation values. It is shown that hourly and daily utilizability curves give a conservative estimate of long-term performance. Experimental measurements of the collector performance further suggest the use of instantaneous utilizability curves in place of daily curves. The effect of the collector time constant on utilizability is discussed.     

Statistical properties of hourly global radiation

A model is described which generates synthetic daily sequences of hourly radiation values, on the horizontal plane, for any location, with the daily clearness index K_t as input. The model assumes that for each K_t and solar hour the probability density of the hourly clearness index k_t is (simply) a truncated Gaussian function. A first-order autoregressive model is fitted for the k_t variable, normalised using parametrisations for the time-dependent average and standard deviation values. Values generated by this ARMA (1,0) model can then be transformed backwards to generate synthetic sequances of k_t values. Using a diffuse fraction correlation and a tilted radiation model, the horizontal global data can be transformed to any desired plane, thus providing solar system designers with the necessary hourly data for the accurate sizing of every type of solar system, including stand-alone, high solar fraction and passive ones.     

Analysis of short-term solar radiation data

Solar radiation data are available for many locations on an hourly basis. Simulation studies of solar energy systems have generally used these hourly values to estimate long-term annual performance, although solar radiation can exhibit wide variations during an hour. Variations in solar radiation during an hour, such as on a minute basis, could result in inaccurate performance estimates for systems that respond quickly and non-linearly to solar radiation. In addition, diffuse fraction regressions and cumulative frequency distribution curves have been developed using hourly data and the accuracy of these regressions when applied to short-term radiation has not been established. The purpose of this research is to investigate the inaccuracies caused by using hourly rather than short-term (i.e., minute and 3 min) radiation data on the estimated performance of solar energy systems. The inaccuracies are determined by examination of the frequency distribution and diffuse fraction relationships for short-term solar radiation data as compared to existing regressions and by comparing calculated radiation on tilted surfaces and utilizability based on hourly and short-term radiation data.     

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.     

The determination of hourly insolation on an inclined plane using a diffuse irradiance model based on hourly measured global horizontal insolation

Using only measured hourly values of global insolation on a horizontal surface, a method has been developed for computing the corresponding hourly values of insolation on a surface inclined at any angle and oriented in any direction. The method uses a solar radiation model in which the diffuse component is calculated from global horizontal radiation using three different relationships; the appropriate equation is selected according to the value of the ratio of measured hourly global insolation to hourly global insolation computed for clear sky conditions. The method has been checked using measured hourly values in Melbourne over a 5-yr period of insolation on both a horizontal surface and a plane inclined at 38° to the horizontal facing north. The differences between the computed hourly values and the measured hourly values are found to be approximately normally distributed about zero with a standard deviation of 0.16 MJ m⁻². This method is particularly useful for predicting the heat output of inclined solar flat plate collectors when only measured global horizontal insolation is available, which is often the case. Good agreement was found between the predicted output of a typical collector using measured 38° insolation and the computed hourly values using this method. Since the method has been checked only against Melbourne data it should be applied elsewhere with caution, but it is believed to have general application.     

Statistical analysis of solar measurements in Algeria using beta distributions

A method of smoothing solar data by beta probability distributions is implemented in this paper. In the first step, this method has been used to process daily sunshine duration data recorded at thirty-three meteorological stations in Algeria for eleven year periods or more. In the second step, it has been applied to hourly global solar irradiation flux measured in Algiers during the 1987/89 period. For each location and each month of the year, beta probability density functions fitting the monthly frequency distributions of the daily sunshine duration measurements are obtained. Both the parameters characterising the resulting beta distributions are then mapped, enabling us to build the frequency distributions of sunshine duration for every site in Algeria. In the case of solar radiation for Algiers, the recorded data have been processed following two different ways. The first one consists in sorting the hourly global solar irradiation data into eight typical classes of the daily clearness index. The second one is based on the repartition of these data per month. The results of the first classification show that for each class of daily clearness index, the hourly data under consideration are modelled by only one beta distribution. When using the second classification, linear combinations of two beta distributions are found to fit the monthly frequency distributions of the hourly solar radiation data.     

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.     

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.     

The probability density and autocorrelation of short-term global and beam irradiance

For individual hours, a characteristic bimodal pattern of short-term global and beam irradiance is frequently observed, with modes at high and low irradiances and with low probabilities near the hourly averages. For such hours, averaging over the hour will imply smoothing of quite significant variations within the hour. Models for the probability density distributions of short-term (5 min or less) irradiances are presented in this paper. These distributions are not unique functions of the hourly averages, but depend heavily also on the irradiance variability within the hour. This intrahour variability is found to depend on the averaging time and also on the interhour variability among three hourly averages, namely, the hour in question, the preceding and the deceding hour. The distribution differences between 5 min averages and instantaneous values are, however, negligible for most practical purposes. The lag one autocorrelation is evaluated as a function of averaging time, and a first order autoregressive model is presented. With hourly averages as the only input the probability density and autoregressive model in combination produce time series of short-term intrahour averages having realistic distributions and autocorrelation structure.     

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.     

Mesoscale mapping of available solar energy at the earth's surface by use of satellites

Ground measurements of solar radiation are too sparse to determine important mesoscale differences that can be of major significance in solar power station location and for other purposes. A method is presented for use of cloud images in the visual spectrum from the SMS/GOES geostationary satellites to determine the hourly distribution of sunshine on the mesoscale. Cloud coverage and density as a function of time of day and season are evaluated through the use of digital data processing techniques. Low density cirrus clouds are less detrimental to solar energy collection than other types; and clouds in the morning and evening are less detrimental than those during midday hours of maximum insolation.Seasonal geographic distributions of cloud cover/sunshine are converted to joules of solar radiation received at the earth's surface through relationships developed from long-term measurements of these two parameters at six widely distributed stations. The technique can be used to generate maps showing the geographic distribution of total solar radiation on the mesoscale which is received at the earth's surface.     

Experimental investigation for total equivalent temperature difference (TETD) values of building walls and flat roofs

An experimental investigation for determining total equivalent temperature difference (TETD) values of building walls and flat roofs was performed. The TETD values are functions of the time lag, decrement factor and sol–air temperature. The time lag and decrement factor depend on the highest and lowest temperatures at the inner and outer surfaces of the walls or roofs, and the time periods involved in reaching these temperatures. Also, the sol–air temperature depends on essentially solar radiation and outside air temperature. For these reasons, two testing rooms each consisting of four walls and one flat roof, an air conditioner, thermocouples, data logger and a computer were constructed to measure all required temperatures. Inside and outside air temperatures, and surface temperatures of each wall and roof layers were measured in each minute and saved on the computer over a period of 24 h in the summer season of Gaziantep, Turkey. Data for the hourly solar radiation on the walls were computed using hourly measured solar radiation data on horizontal surface. The TETD values for eight different walls and two different roofs commonly used in Turkey were computed by using the measured temperatures and solar radiation flux. The TETD values for the walls and the roofs were also obtained for selected cities in Turkey by utilizing their outside air temperature and solar radiation inputs.     

A monthly probability distribution function of daily global irradiation values appropriate for both tropical and temperate locations

In this paper it is shown that the generalized cumulative distribution curves of Liu and Jordan are not suitable for tropical locations. However, the Bendt et al. probability density function, originally proposed to predict the cumulative distribution curves of North American locations, can still be used for tropical locations provided the value of the maximum clearness index is suitably adjusted. But this probability density function is unrealistic at high irradiation values, which may result in an overestimation of calculated solar system performance. This shortcoming can be overcome by using a higher order probability density function proposed by Saunier. This function is a generalised single parameter function, namely the ratio of the monthly mean clearness index to the maximum clearness index, and seems to be suitable without ad hoc modification for any type of climate. Finally, the problem of estimating the value of the maximum clearness index has also been addressed and a simple model is proposed to evaluate it.     

A Probabilistic Approach to Photovoltaic Generator Performance Prediction

This paper describes a methodology to predict the performance of photovoltic (PV) generator based on long term climatological data and expected cell performance. The methodology uses long term historical data on insolation to calculate the probability distribution function parameters for each hour of a typical day of any season, week or day. Once the probability distribution function parameters are calculated, they are used to evaluate the predicted hourly, daily, weekly and seasonal capacity factors of a particular design of a PV panel/array at a particular site. Long term insolation data from Sterling, Virginia have been utilized with Solarex SX-110 panel designs to predict PV array performance.     

One-minute global irradiance probability density distributions conditioned to the optical air mass

Knowledge of the temporal variability of the solar irradiance is important to study solar energy systems involving thermal and photovoltaic processes. The differences between hourly and instantaneous values of the clearness index considerably affect the utilizability of photovoltaic systems. In this work, we analyzed the probability density distributions of one-minute values of global irradiance, conditioned to the optical air mass, considering those as an approximation to the instantaneous distributions. The study reveals that the bimodality that characterizes these distributions increases with optical air mass. We propose the use of a functional form based on Boltzmann's statistics in order to describe these distributions. This function can be used for the generation of synthetic radiation data. Expressing the distribution as a sum of two functions provides an appropriate modeling of the bimodality feature that can be associated with the existence of two levels of irradiation corresponding to two extreme atmospheric situations, cloudless and cloudy conditions.     

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 frequency distribution of daily insolation values

Frequency distributions of insolation values are needed in order to derive simple correlations for predicting the performance of solar energy systems. In this paper the frequency distribution of daily total hemispherical solar irradiation values on the horizontal surface is derived from measured data for 90 locations in the U.S. The results can be approximated by generalized distribution curves which depend only on the clearness index, defined as ratio of terrestrial over extraterrestrial insolation. The results agree well with the curves originally derived by Liu and Jordan, except for a correction at high insolation values. The deviation of individual locations from the generalized curves is examined. A breakdown according to time of year reveals some seasonal variation. The theoretical distribution corresponding to random insolation sequences is also derived; it agrees very well with the generalized frequency distribution curves.     

On the complexity of radiation models for PV energy production calculation

Several authors have analysed the changes of the probability density function of the solar radiation with different time resolutions. Some others have approached to study the significance of these changes when produced energy calculations are attempted. We have undertaken different transformations to four Spanish databases in order to clarify the interrelationship between radiation models and produced energy estimations. Our contribution is straightforward: the complexity of a solar radiation model needed for yearly energy calculations, is very low. Twelve values of monthly mean of solar radiation are enough to estimate energy with errors below 3%. Time resolutions better than hourly samples do not improve significantly the result of energy estimations.     

A statistical approach for sub-hourly solar radiation reconstruction

This paper proposes a computational-statistics based approach for solar radiation reconstruction at sub-hourly intervals. A dimensionless form of stochastic variable, V, which is defined as the difference between the theoretical global solar radiation in clear-sky conditions and the actual solar radiation, normalized by the clear-sky global solar radiation, is introduced and adopted in this work. The probability density function of V is calculated from historical data using a Gaussian kernel density estimator. With the developed model, the only input information required for the reconstruction procedure is the cloud condition of the sky (i.e., fair, partly cloudy, overcast, and rain/snow etc.). A case study in simulating solar radiation in Singapore is conducted to validate the accuracy of the model. The calculated results agree well with the measured data. The normalized root mean square error (NRMSE) is on average 23.4% and 7.2% for the one-minute temporal resolution and hourly integral values, respectively.