Statistical approach to the proposition and validation of daily diffuse irradiation models

This paper explores the prospects of using sunshine duration and cloud cover in estimations of daily diffuse irradiation besides the conventional use of global irradiation, where all the parameters are gathered from typical ground-based measurements and proposes optimal region-based models. Data from eight locations across four countries are used for model proposition and subsequent evaluation. Daily sunshine fraction (SF) and daily cloudiness factor (CF) are used along with daily clearness index (K_t) by inter-combination to develop a series of diffuse ratio (K) empirical models for each site. Various statistical tools are employed to establish the criterion of best performing model. Each model’s performance is initially assessed, based on the data it is derived from, and then validated against an independent dataset. This validation is demonstrated by two means: first by testing the models developed for one site against another site in the same region and, secondly by testing the models derived from one section of data against a reserved section from the same site covering a different period of time. The accuracy of prediction is evaluated using three statistical measures (AS, SD, t-statistic). The final assessment also includes calculated versus measured diffuse irradiation plots and indicators such as percent MBD and percent RMSD for potential models. It was found that a model based on K_t and SF (and/or CF) performs better than a model based on K_t alone within the same data set. However, if these models are tested against the data belonging to a different period of time or a different site, the improvement is less significant. Such model-specificity can be attributed to the fact that the proposed models involve more than one measured parameter, hence greater uncertainty, as against the single-input K_t model. Given the climatological variants that differ from site-to-site and measurement uncertainties owing to the poor meteorological standards within the same dataset, validation of such models becomes a challenging task. Nevertheless, it is found that the K_t, SF model is an optimum choice when estimating diffuse radiation for independent data, as it yields improved results even over the local K–K_t model. Thus, this investigation establishes the improvement in estimation of daily diffuse irradiation that can possibly be achieved through incorporating effective variables along with global radiation for both local as well as independent sites.     

Modeling the hourly solar diffuse fraction in Taiwan

Using the data for global and diffuse radiation in Tainan, Taiwan, for the years of 2011 and 2012, respectively, four correlation models with five predictors: the hourly clearness index (k_t), solar altitude, apparent solar time, daily clearness index and a measure of persistence of global radiation level, are constructed to relate the hourly diffuse fraction on a horizontal surface (d) to the clearness index. Two models use a single logistic equation for all k_t values, Eqs. (6), (7), and the other two models use a set of piece-wise linear equations for four k_t intervals, Eqs. (8), (9). The proposed models are compared respectively with the fourteen models available in the literature, in terms of the four statistical indicators: the mean bias error, the root-mean-square error, the t-statistic and the Bayesian Information Criterion, using the out-of-sample dataset for Tainan, Taiwan. It is concluded from the analysis that the proposed piece-wise linear models perform well in predicting the diffuse fraction, while the performances of the proposed logistic models are more case-dependent. Among those fourteen models considered in this study, the models developed by Erbs et al., Chandrasekaran and Kumar, and Boland et al. have competitive performances as the proposed piece-wise linear models do, when applying to the prediction of diffuse fraction in Tainan, Taiwan.     

Models for obtaining solar radiation from other meteorological data

The application of solar energy requires a knowledge of long-term solar radiation and daylight data. Because of the limited availability of measured data, various formulae have been derived to compute the solar irradiance using other, more commonly available, weather data. In this article two such models are presented, MRM (meteorological radiation model) and CRM (cloud-cover radiation model). MRM requires hourly data for sunshine duration, dry- and wet-bulb temperature; while CRM requires only cloud amount. Both models can generate hour-by-hour data for horizontal global, diffuse, and beam irradiance. A brief comparison of the two models is presented. Results showed that MRM has the advantage over CRM, on account of its consistency with the measured data. Both models are now available via the Internet in the form of electronic spreadsheets.     

Global,direct and diffuse solar radiation on horizontal and tilted surfaces in Jeddah,Saudi Arabia

The measured data of global and diffuse solar radiation on a horizontal surface, the number of bright sunshine hours, mean daily ambient temperature, maximum and minimum ambient temperatures, relative humidity and amount of cloud cover for Jeddah (lat. 21°42′37′′N, long. 39°11′12′′E), Saudi Arabia, during the period (1996–2007) are analyzed. The monthly averages of daily values for these meteorological variables have been calculated. The data are then divided into two sets. The sub-data set I (1996–2004) are employed to develop empirical correlations between the monthly average of daily global solar radiation fraction (H/H₀) and the various weather parameters. The sub-data set II (2005–2007) are then used to evaluate the derived correlations. Furthermore, the total solar radiation on horizontal surfaces is separated into the beam and diffuses components. Empirical correlations for estimating the diffuse solar radiation incident on horizontal surfaces have been proposed. The total solar radiation incident on a tilted surface facing south H_t with different tilt angles is then calculated using both Liu and Jordan isotropic model and Klucher’s anisotropic model. It is inferred that the isotropic model is able to estimate H_t more accurate than the anisotropic one. At the optimum tilt angle, the maximum value of H_t is obtained as ∼36 (MJ/m² day) during January. Comparisons with 22 years average data of NASA SSE Model showed that the proposed correlations are able to predict the total annual energy on horizontal and tilted surfaces in Jeddah with a reasonable accuracy. It is also found that at Jeddah, the solar energy devices have to be tilted to face south with a tilt angle equals the latitude of the place in order to achieve the best performance all year round.     

Comparative study of various correlations in estimating hourly diffuse fraction of global solar radiation

Proper design and performance predictions of solar energy systems require accurate information on the availability of solar radiation. The diffuse-to-global solar radiation correlation, originally developed by Liu and Jordan, has been extensively used as the technique providing accurate results, although it is latitude dependent. Thus, in the present study, empirical correlations of this type were developed to establish a relationship between the hourly diffuse fraction (k_d) and the hourly clearness index (k_t) using hourly global and diffuse irradiation measurements on a horizontal surface performed at Athalassa, Cyprus. The proposed correlations were compared against 10 models available in the literature in terms of the widely used statistical indicators, rmse, mbe and t test. From this analysis, it can be concluded that the proposed yearly correlation predicts diffuse values accurately, whereas all candidate models examined appear to be location-independent for diffuse irradiation predictions.     

Contribution to the study of solar radiation in Abu Dhabi

Measurements of global solar radiation for 10 yr in Abu Dhabi are used to examine the suitability of a method of prediction of global solar radiation. It shows a good degree of accuracy. A correlation between the clearness index, H/H₀, and the fraction of maximum possible number of sunshine hours is obtained and found to be H/H₀ = 0.307 + 0.312n/N. The diffuse solar radiation is predicted by two methods which show almost similar results. Correlation between the diffuse solar radiation as a fraction of the global radiation, H_d/H, and the fraction possible number of sunshine hours, n/N, is established using both methods, for comparison sake, since measurements of diffuse solar radiation are not available. The comparison shows a good agreement between the two correlations. The relation between the diffuse solar radiation as a fraction of the global radiation H_d/H, and the clearness index, H/H₀, is established and compared with those obtained elsewhere in order to justify their use, since direct measurements are not available. The relation shows a latitude dependency.     

Evaluation of clear-sky conditions in high altitude sites

The correct assessment of cloud conditions in any site is important for determining the characteristics of solar resource availability. The criterion proposed by Iqbal has been widely used for this purpose, through the classification of the daily clearness index K_t. However, evidence was found to suggest that this criterion may not be applicable to data measured in high altitude sites, because partially cloudy days are incorrectly classified as clear-sky days. This paper analyzes the cloudiness state of the sky by observing the values of the clearness index K_t and the clear-sky index K_c. These indices were obtained from reliable global solar irradiance data measured over a year in three places of Argentina located at different altitudes (25, 1200 and 3700 m.a.s.l). ESRA and ARG-P models were used to estimate daily clear-sky global solar irradiation values H_c. They also reveal that the most probable value of K_c which corresponds to an actual clear-sky day of a particular location and time depends on the model used to estimate the clear-sky global solar radiation values. Thus, it was confirmed that, unlike the index K_c, the Iqbal's criterion does not correctly classify the cloud cover status from data measured in high altitude sites (>1000 m.a.s.l.).     

Practical models to estimate horizontal irradiance in clear sky conditions: Preliminary results

The Argentinean Northwest (ANW) is a high altitude region located alongside Los Andes Mountains. The ANW is also one of the most insolated regions in the world due to its altitude and particular climate. However, the characterization of the solar resource in the region is incomplete as there are no stations to measure solar radiation continuously and methodically. With irradiance data recently having been measured at three sites in the Salta Province, a study was carried out that resulted in a practical model to quickly and efficiently estimate the horizontal irradiance in high altitude sites in clear sky conditions. This model uses the altitude above sea level (A) as a variable and generates a representative clearness index as a result (k_t-R) that is calculated for each site studied. This index k_t-R is then used with the relative optical air mass and the extraterrestrial irradiance to estimate the instantaneous clearness index (k_t). Subsequently, the index k_t-R is corrected by introducing the atmospheric pressure in the definition of relative optical air mass proposed by Kasten. The results are satisfactory as errors in the irradiance estimations with respect to measured values do not exceed 5% for pressure corrected air masses AM_c < 2. This model will be used in a feasibility study to locate sites for the installation of solar thermal power plants in the ANW. A prototype of a CLFR solar power plant is being built in the INENCO Campus, at the National University of Salta.     

Global transmissivity and diffuse fraction of solar radiation for clear and cloudy skies as measured and as predicted by bulk transmissivity models

Measurements of cloudiness and of global, direct, and diffuse radiation taken over a 13 mo period at Davis, CA, are analyzed in terms of global transmission (K_t) and diffuse fraction (K_d) for clear sky conditions and for various cloudiness conditions. A number of global transmission clear sky models are compared with observations for ranges of total water column and turbidity and some are found to give representative values for the global radiation at the ground.The dependence of the diffuse fraction on global transmission is found to be best represented by linear formulae—with different dependencies found for clear and cloudy conditions.Global transmission models are also compared with observations for cloudy conditions and found to give representative values of cloud transmissivities if climatological differences in the cloudiness at the measurement site and those sites used to calibrate the cloud models are considered.These results support the use of routine instantaneous surface meteorological data to calculate the most likely instantaneous global and diffuse radiation on a horizontal surface in the absence of any radiation measurements. These calculated irradiances are best used for solar energy system dynamic modeling in which system responses to typical sequences in meteorological conditions are being examined.     

Comparing diffuse radiation models with one predictor for partitioning incident PAR radiation into its diffuse component in the eastern Mediterranean basin

In the photosynthesis process, solar radiation energy is converted to chemical energy by using atmospheric CO₂. That is, almost all living species depend on energy produced through photosynthesis for their nourishing components thus making photosynthesis vital to the earth's life. Nevertheless, the knowledge of photosynthetic photon flux density Q_P (PAR, 400–700 nm) is important in several applications dealing with plants physiology, biomass production, natural illumination in greenhouses and agricultural research. This study aiming to explore the applicability of several diffuse radiation empirical models, hourly measurements of diffuse PAR and global PAR irradiation collected at Athens (37°N, 23°E, 250 m above MSL) from 1 January 2000 to 31 December 2002, are employed. These data were used to establish an empirical model relating the spectral diffuse fraction, k_dP (ratio of the diffuse-to-global PAR) with the fractional transmission of global PAR k_tP (ratio of the global PAR-to-extraterrestrial solar PAR). The performance of the proposed empirical model was further compared with those of twelve other diffuse–global correlation models available in the literature in terms of the widely used statistical indicators mbe, rmse and t-test. From the overall analysis, it can be concluded that the proposed model predicts diffuse PAR values accurately, whereas most of the candidate empirical models examined here appear to be location-independent for the diffuse PAR predictions.     

Validation of five global radiation models with measured daily data in China

Two sunshine based and three air temperature based global radiation models are calibrated using daily data in Jan. 1 1994–Dec. 31 1998 at 48 stations all over China. The Nash–Sutcliffe equation (NSE) is used as the model evaluation criterion. The sunshine based models are suitable for daily global radiation estimation. The averaged NSE value of the Angström model is 0.83, and the maximum value is 0.91. The maximum NSE value of the Bahel model is 0.92 with an averaged value of 0.84. The models that use air temperature as the input variable are not suitable for daily global radiation estimation in China. The averaged NSE values of the three air temperature based models (Bristow–Campbell model, Allen model and Hargreaves model) are not larger than 0.47. A logarithmic relationship between the daily global radiation/daily extra-terrestrial solar radiation (R_G/R_A) and the temperature difference between the maximum and minimum daily air temperature (T_M−T_m) is found in the present study. A new daily global radiation model that is a function of R_A, sunshine hours and T_M−T_m is designed, which gives an averaged NSE value of 0.85 and a maximum value of 0.92.     

Compilation and evaluation of solar and wind energy resources in Sudan

A number of years worth of data concerning the solar radiation on a horizontal surface, sunshine duration and wind speed in Sudan have been compiled, evaluated and presented in this article.Measurements of global solar radiation on a horizontal surface at 16 stations for several years are compared with predictions made by several independent methods. In the first method the Angstrom formula was used to correlate relative global solar irradiance to the corresponding relative duration of bright sunshine.Regression coefficients are obtained and used for prediction of global solar irradiance. The predicted values were consistent with measured values (± 8.01% variation).In the second method, by Barbaro et al. [Solar Energy, 1978, 20, 431] sunshine duration and minimum air mass were used to drive an empirical correlation for the global radiation. The predicted values compared well with measured values (± 12% variation).The diffuse solar irradiance is estimated. The results of two formulas have close agreement. A radiation map of Sudan was prepared from the estimated radiation values. The annual daily mean global radiation ranges from 3.05 to 7.62 kW h m⁻² per day.Routine wind data from 70 stations were analyzed. Monthly averaged wind speed and average powers were determined for each station. The derived annual average speeds range from 1.53 to 5.07 m s⁻¹. Maximum extractable average wind powers were found to vary between 1.35 and 49.5 W m⁻². A wind map of Sudan was also prepared.Sudan possessed a relatively high abundance of sunshine and moderate wind speed. It is concluded that Sudan is blessed with abundant solar and wind energy.     

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.     

Clear-sky irradiance predictions for solar resource mapping and large-scale applications: Improved validation methodology and detailed performance analysis of 18 broadband radiative models

The intrinsic performance of 18 broadband radiative models is assessed, using high-quality datasets from five sites in widely different climates. The selected models can predict direct, diffuse and global irradiances under clear skies from atmospheric data, and have all been (or still are) involved in large-scale applications, for instance to prepare solar resource maps and datasets, or to evaluate solar radiation in GIS software. The input data to the models include accurate aerosol and water vapor measurements by collocated sunphotometers, if needed. Cloud occurrences are meticulously scrutinized through the use of various tools to avoid cloud contamination of the test data. The intrinsic performance of the models is evaluated by comparison between their predictions and measurements at high frequency (1-minute time step at four sites, 3-minute at one site). The total expanded uncertainty of these measurements is estimated at 3% for direct irradiance, and 5% for diffuse and global irradiance.Various statistics are calculated to evaluate the systematic and random differences between the data series, as well as the agreement between the cumulative distribution functions. In the latter case, stringent statistics based on the Komolgorov–Smirnov (KS) test are used. Large differences in performance are apparent between models. Those that require more atmospheric inputs perform usually better than simpler models. Whereas many models can predict the global horizontal irradiance within uncertainty limits similar to those of the radiation measurements, the prediction of direct irradiance is less accurate. Moreover, the prediction of diffuse horizontal irradiance is particularly deficient in most models. The cumulative distribution functions also denote areas of concern.A ranking of all models is proposed, based on four statistical indicators: mean bias difference (MBD), root mean square difference (RMSD), total uncertainty with 95% confidence limits (U₉₅), and the newly introduced Combined Performance Index (CPI), which optimally combines two KS indices with RMSD. For direct irradiance, consistently high rankings are obtained with five models (REST2, Ineichen, Hoyt, Bird, and Iqbal-C, in decreasing order of performance) that require a relatively large number of atmospheric inputs. The inferior performance of models requiring little or no atmospheric inputs suggests that large-scale solar resource products derived from them may be inappropriate for serious solar applications. Additionally, prediction uncertainties under ideal clear-sky conditions can propagate and affect all-sky predictions as well—resulting in potential biases in existing solar resource maps at the continent scale, for instance.     

Estimation of global solar radiation in arid and semi-arid climates of East and West Iran

In Iran, most of the models used so far, have provided solar estimation for a few specific locations based on the short-term solar observations. Using different radiation models, (e.g. Sabbagh, Paltridge–Proctor, Daneshyar) and various input parameters (e.g. cloud cover, sunshine duration, relative humidity, temperature, and altitude) we developed a general height-depended formula for the prediction of the direct and diffuse monthly average daily solar radiation for 64 mountainous arid and semi-arid locations in West and East Iran. The models mentioned are modified and new coefficients are defined for the diffuse component based on the long-term observed diffuse data. Model results are validated against up to 13-year daily solar observations at 10 solar radiation sites. In comparison with the previous studies, the newly developed method performs more accurate estimation (less than 3% MPE error) in the arid and semi-arid regions. Comparison of the model results indicates that calibration of the coefficients made to the diffuse formula against the longer period experimental data can improve the estimations of global solar radiation.     

ONE MINUTE kb AND kd PROBABILITY DENSITY DISTRIBUTIONS CONDITIONED TO THE OPTICAL AIR MASS

In this work we have analysed and modelled one-minute probability distribution function of solar direct and diffuse irradiance conditioned to the optical air mass. For this purpose, we have used one-minute data acquired in a radiometric station located in southeastern Spain (37.13° N, 3.63° W, 687 m a.m.s.l.). The study has been made over the dimensionless indices k_b and k_d. We have found marked bimodality in the k_b distribution function and asymmetry associated with the unimodal function of one-minute k_d values. In spite of these differences, we have modelled both distribution functions using a functional form based on Botlzmann's statistics. These functions have been used in a previous study devoted to modelling the clearness index for global radiation, k_t. In order to model the asymmetry that is evident in the k_d and k_b distribution functions, the functional forms have been modified by the inclusion of an additional parameter. The adjustable parameters included in the model equations present a dependence on the optical air mass.     

Dependence of one-minute global irradiance probability density distributions on hourly irradiation

This paper analyzes the behavior of one minute global irradiance distributions as a function of hourly average solar global irradiance. For this purpose, we have used the clearness index k_t which describes the atmospheric transmittance. Our interest is in characterizing the intrahourly variability of solar global irradiance and the behavior of the instantaneous values as a function of hourly values of solar global irradiance. The distributions are unimodal and show a marked symmetry around a central value that is close to the corresponding hourly average value. The probability density functions have been modeled using functions based on the Boltzmann statistic used in recent studies of the one minute distributions of k_t conditioned to the optical air mass. These functions provide good fit of the distributions and are analytically integrable and can be inverted analytically. The one minute global irradiance data used in this study have been recorded during a three year period in a radiometric station located in south eastern Spain.     

Monthly ratios of PAR to global solar radiation measured at northern Tibetan Plateau, China

Using narrowband and broadband solar radiation measurements collected at Wudaoliang (WDL) site in northern Tibetan Plateau (NTP) from September 1993 to December 1998, the relationship between monthly photosynthetically active radiation (Q_P) and global solar radiation (R_S) values is analyzed. Temporal variability of the ratio (Q_P/R_S) and its further dependence on several meteorological variables are presented. The narrowband ratio exhibited diurnal and seasonal variability with high values during morning and afternoon hours and low values around noon in winter time, whereas during summer period the ratio was decreased from morning to afternoon. The ratio (Q_P/R_S) was correlated positively with several atmospheric parameters such as water vapor pressure and low-level cloud amount; in contrast, the ratio was negatively correlated with clearness index, relative sunshine duration and atmospheric turbidity. It was also found that both the relative sunshine duration and water vapor pressure are the most influential parameters on the estimations of the spectral PAR ratio. Finally, an empirically derived model is proposed for estimating monthly average PAR values over the northern Tibetan Plateau. Verification results further ensured that the proposed model predicts monthly global PAR values accurately.     

Estimation of total cloud cover from solar radiation observations at Lake Rotorua, New Zealand

The DYRESM-CAEDYM model is a valuable tool for simulating water temperature for biochemical studies in aquatic ecosystem. The model requires inputs of surface short-wave radiation and long-wave radiation or total cloud cover fraction (TC). Long-wave radiation is often not measured directly so a method to determine TC from commonly measured short-wave solar irradiance (E₀) and theoretical short-wave solar irradiance under a clear sky (E_c) has broad application. A more than 17-year (15 November 1991 to 20 February 2009) hourly solar irradiance data set was used to estimate the peak solar irradiance for each ordinal date over one year, which was assumed to be representative of solar irradiance in the absence of cloud. Comparison between these daily observed values and the modelled clear-sky solar radiation over one year was in close agreement (Pearson correlation coefficient, r = 0.995 and root mean squared error, RMSE = 12.54 W m⁻²). The downloaded hourly cloudiness measurements from 15 November 1991 to 20 February 2009 was used to calculate the daily values for this period and then the calculated daily values over the 17 years were used to calculate the average values for each ordinal date over one year. A regression equation between (1 − E₀/E_c) and TC produced a correlation coefficient value of 0.99 (p > 0.01, n = 71). The validation of this cloud cover estimation model was conducted with observed short-wave solar radiation and TC at two sites. Values of TC derived from the model at the Lake Rotorua site gave a reasonable prediction of the observed values (RMSE = 0.10, r = 0.86, p > 0.01, n = 61). The model was also tested at Queenstown (South Island of New Zealand) and it provided satisfactory results compared to the measurements (RMSE = 0.16, r = 0.67, p > 0.01, n = 61). Therefore the model’s good performance and broad applicability will contribute to the DYRESM-CAEDYM accuracy of water temperature simulation when long-wave radiation is not available.