Improved Accuracy Models For Hourly Diffuse Solar Radiation

Improved Accuracy Models For Hourly Diffuse Solar Radiation T. Muneer e-mail: t.muneer@napier.ac.uk S. Munawwar e-mail: s.munawwar@napier.ac.uk Applied Energy Group, School of Engineering, Napier University, 10 Colinton Road, Edinburgh, United Kingdom Solar energy applications require readily available, site-oriented, and long-term solar data. However, the frequent unavailability of diffuse irradiation, in contrast to its need, has led to the evolution of various regression models to predict it from the more commonly available data. Estimating the diffuse component from global radiation is one such technique. The present work focuses on improvement in the accuracy of the models for predicting horizontal diffuse irradiation using hourly solar radiation database from nine sites across the globe. The influence of sunshine fraction, cloud cover, and air mass on estimation of diffuse radiation is investigated. Inclusion of these along with hourly clearness index, leads to the development of a series of models for each site. Estimated values of hourly diffuse radiation are compared with measured values in terms of error statistics and indicators like, R2, mean bias deviation, root mean square deviation, skewness, and kurtosis. A new method called “the accuracy score system” is devised to assess the effect on accuracy with subsequent addition of each parameter and increase in complexity of equation. After an extensive evaluation procedure, extricate but adequate models are recommended as optimum for each of the nine sites. These models were found to be site dependent but the model types were fairly consistent for neighboring stations or locations with similar climates. Also, this study reveals a significant improvement from the conventional k-kt regression models to the presently proposed models. 关DOI: 10.1115/1.2148972兴 Introduction Energy is the most crucial commodity of modern age. The everrising energy demand is nevertheless accompanied by a much faster increase in vulnerability and insecurity of its supply. It is high time that the actualization of renewable energies takes place to meet the everyday energy challenges from its current status of feeding only pilot-plant projects. Solar energy is one such promising renewable form that needs attention. Knowledge of solar radiation data is a prerequisite for the simulation and design of all solar energy systems. Architects, agriculturalists, air conditioning engineers, and energy-conscious designers of buildings also require such information. Most of the solar energy applications involve tilted surfaces. In order to effect the estimation of radiation on tilted surfaces, knowledge of both diffuse and direct components of global radiation falling on a horizontal surface is required. However, quite often, projects involving the utilization of solar energy are not supported by the required solar data at the place of interest, mainly due to the capital and maintenance costs that measuring instruments incur. Consequently, they need to be estimated from alternative information available at the site or a near by location. The infrequent diffuse and direct radiation measurements have led to the evolution of a number of regression models for their estimation based on global horizontal radiation. The latter component is routinely and more widely measured. Apart from this, even when data on both global and diffuse radiation are available, it is often necessary to estimate diffuse irradiation to fill the gaps in meteorological records. The present study focuses on the improvement in estimation techniques of hourly and subhourly horizontal diffuse radiation by developing regression models for hourly diffuse fraction 共k兲 as a function of not only hourly clearness index 共kt兲, but also two other synoptic parameters: sunshine fraction 共SF兲 and cloud cover 共CC兲 and a calculated parameter: air mass 共m兲. The influence of such parameters on the prediction of diffuse radiation is studied both Contributed by the Solar Energy Division of ASME for publication in the JOURSOLAR ENERGY ENGINEERING. Manuscript received June 17, 2004; final manuscript received January 31, 2005. Review conducted by Andy Walker. NAL OF 104 / Vol. 128, FEBRUARY 2006 qualitatively as well as quantitatively. All possible combinations are applied to generate different sets of models and thus select the best model for each of the nine sites under study, bearing in mind the optimum number of input variables and the gain in accuracy. We carried out all our analysis using the hourly or subhourly values of horizontal radiation, because detailed simulation of solar energy systems requires computation of hour by hour radiation rather than monthly average or daily values. Also, hourly values of radiation are much more precise in deducing the performance of a solar energy system. A Brief Review of Existing Procedures for the Estimation of Horizontal Diffuse Radiation Before we proceed, a quick review of the diffuse fraction estimation techniques proposed by a number of researchers in the past, would be helpful. To start with, Liu and Jordan proposed relationship between the daily diffuse fraction of the global radiation and daily clearness index 共ratio of global to extra terrestrial radiation兲. They developed a third degree polynomial regression between the above variables 关1兴. However, later on it was shown that the Liu and Jordan relationship is not universal and there was a need for individual models based on local climatic and geographical conditions. Page developed a linear relationship between k and kt based on the data from ten widely spread locations in the latitude range of 40° N to 40° S 关2兴. Following Liu and Jordan approach, Orgill and Holland used hourly data for Toronto for a period of four years 关3兴. Both, Bugler and Iqbal from their respective studies, deduced a significant effect of solar altitude in prediction of diffuse fraction 关4,5兴. Collares-Pereira and Rabl, using data from five US locations, proposed a fourth order polynomial regression between k and kt 关6兴. Erbs et al. developed a similar hourly regression using 65-month combined data for four locations in the U.S. and one location in Australia 关7兴. Spencer developed a regression for Australia using data from 12 different locations across the continent 关8兴. Muneer and Saluja, based on three-year data for five sites in the UK covering the whole latitude range, developed individual regressions 共for each site兲 and an overall regression for UK. They also investigated the effects of fractional possible sunshine and solar altitude on the regression Copyright © 2006 by ASME Transactions of the ASME Fig. 1 Geographical location of the sites under study they developed 关9兴. Skartveit and Osleth proposed regression equations of the hourly diffuse fraction as a function of the hourly solar elevation and clearness index 共kt兲 关10兴. Reindl et al. used data from five locations in the United States and Europe and deduced that the sine of solar altitude, ambient air temperature and relative humidity were important regressors, besides kt 关11兴. Vázquez et al. in their work showed the influence of scattering, absorption and air mass on the kd共ratio of diffuse to extraterrestrial radiation兲-kt relationships 关12兴. Another approach adopted by some researchers like Perez et al., to improve diffuse radiation modelling was to include readily available information, namely variability in global irradiance between three consecutive hours 关13兴 共Fig. 1兲. The Need to Improve the Conventional Regression Between k and kt The very basic model, which shows a polynomial relationship between k and kt, does not provide an accurate estimation of diffuse fraction of global radiation. There definitely remain other potential parameters that exert a significant influence on the diffuse fraction and consequently on diffuse radiation. A typical k-kt plot shown in Fig. 2 displays a considerable scatter of data points for this chosen site 共Bracknell, UK兲. Any suitable regression fit 共linear, quadratic or cubic兲 does not and cannot account for the whole data. For example, a kt value of 0.5 has observed values of k = 0.35 and 0.87, respectively, in clear and overcast regimes 共marked by highlighted points in the figure兲. Whereas, cubic regression fit for the same data when kt is 0.5, gives a value of k = 0.61, which overestimates the clear sky observation by 74% and underestimates the corresponding overcast value by 30%. Likewise, quadratic regression fit yields a k = 0.63 for kt = 0.5, yielding errors of a similar order of magnitude. Linear fit, as can be inspected visually, is not in the least better than the other two. For a typical data point, with kt = 0.5, the diffuse horizontal radiation from quadratic regression model was calculated to be 141 W / m2 while the actual observed value was as low as 79 W / m2. This yields a large error of 78%, again emphasizing the fact, that diffuse radiation is very poorly predicted from the k-kt model. Furthermore, the value of R2 = 0.893 obtained for the above quadratic k-kt fit shows that 10.7% of the variation is unexplained 共the definition of R2 shall be explained later兲 and thus other factors have to be incorporated. Journal of Solar Energy Engineering As discussed above, various regressions have been developed in the past taking in account one or some of the following: atmospheric turbidity, atmospheric aerosols, relative humidity, ambient temperature, solar elevation, SF, m, CC, and albedo. For the present work, daily SF 共ratio of number of bright sunshine hours to day length兲 or hourly or sub-hourly sunshine fraction, as the case may be, CC and m were investigated as to explore their influence, if any, on the diffuse radiation estimation. While m can be calculated from the knowledge of solar altitude, measurement of SF and CC in contrast to the diffuse radiation is widely undertaken along with other meteorological data such as humidity and temperature. In the UK itself, some 600 sites measure CC, out of which 230 sites measure sunshine duration while diffuse radiation is measured at relatively fewer sites. When the sky is completely covered by clouds 共eight oktas of cloud cover兲, the sunshine fraction is zero, and the radiation received on the earth’s surface is completely diffuse in nature. Likewise, for completely clear skies, i.e., with nil cloud cover and sunshine fraction equal to unity, the global radiation chiefly comprises of beam compo- Fig. 2 k-kt regression analysis for Bracknell quality controlled database. Dashed curve represents linear fit „R2 = 0.855…, thick solid line is for quadratic polynomial fit „R2 = 0.893…, and thin solid line for cubic polynomial fit „R2 = 0.903…. FEBRUARY 2006, Vol. 128 / 105 Table 1 Geographical and data base information of the meteorological sites under study Country India Japan Spain UK Location Latitude 共 ⴰN兲 Longitude Chennai Mumbai New Delhi Pune Fukuoka Sapporo Gerona Madrid Bracknell 13 19.12 28.6 18.53 33.52 43.05 41.97 40.45 51.26 80.18° E 72.85° E 77.20° E 73.85° E 130.48° E 141.33° E 2.88° E 3.73° W 0.46° W Elevation 共masl兲 16 14 236 559 69 40 100 680 Period of measured data % age of data points omitted due to quality controla 1990–1994 1990–1994 1989–1998 1990–1994 1992–1994 1991–1993 1995–2001 1999–2001 1990–1995 3.4 5.4 4.7 5.7 5.4 7.7 1.8 4.4 2.1 Number of data points After quality controlb For regression analysisc 13,047 12,569 24,671 11,716 19,650 13,563 27,328 11,162 22,111 4297 4037 24,671 4008 19,560 13,563 4863 4041 22,111 a This represents the percentage of data rejected due to the presently used method of quality control on radiation data sets. This number excludes the data points for which sun shine duration was missing. With the exception of Fukuoka, Sapporo, and Bracknell, for all other sites daily sunshine fraction rather than hourly was taken for each hour of that day. c This number represents those data points for which cloud cover information was available. Although radiation data was hourly, cloud cover was recorded only at certain hours of the day, so we could retain only those hours. b nent. Hence, it is logical to say that both SF and CC can potentially play a significant role in the determination of the diffuse fraction of global radiation. Also, it was reported by Gopinathan and Soler 关14兴 that, when both clearness index and relative sunshine duration are used together in multiple regressions the estimated values of monthly mean daily diffuse radiation is better than when they are used separately. Regarding the third parameter presently under investigation i.e., m, it is known that the scattering effects of greater air mass render considerable portion of global radiation as diffuse component at lower solar altitudes 关15兴. This implies that for a given value of kt an increase in m can lead to a corresponding increase in k. We preferred air mass to solar altitude because the former is a more appropriate parameter for the characterization of atmosphere, e.g., turbidity and aerosol loading. The Present Data Base Data Assimilation. The present work is comprised of data sets from nine locations spread over four countries and two continents. Of those, six are from Asia: four 共Chennai, Mumbai, New Delhi, and Pune兲 from the Indian subcontinent and two Japanese 共Fukuoka and Sapporo兲 sites in the far east. The remaining three sites are from the western part of the world, i.e., Europe. These include two Spanish 共Gerona and Madrid兲 and a UK site 共Bracknell兲. Table 1 lists all the locations and their geographical information. Data for all the four Indian sites were provided by Indian Meteorological Department. The solar radiation data were hourly based, with daily number of bright sunshine hours. Also, cloud cover was provided at 8.30,11.30,14.30, and 17.30 h of local clock time for Chennai, Mumbai, and Pune, while for New Delhi no cloud data were available to the present research team. Data for Japanese sites were 10 min values 共courtesy: Kyushu University, Japan兲. Sunshine duration was measured every 10 min. Again, no cloud cover data were obtainable for these two locations. Madrid solar radiation data were hourly 共courtesy: Instituto Nacional De Meteorologia兲. Daily sunshine fraction and cloud cover data for both Madrid and Gerona were provided at 7, 13, and 18 h. Gerona data provided by University of Gerona, was originally 5 min based, which we later integrated into hourly values. Bracknell 共UK兲 data was obtained from the UK Metrological Office. All the data including radiation values, sunshine fraction, and cloud cover for Bracknell were provided at an hourly level. The selection of the above sites not only covers different longitudes and latitudes from the northern hemisphere, but also features varying climates and topographies, a mix of tropical, subtropical, temperate, and cold. 106 / Vol. 128, FEBRUARY 2006 Data Processing and Quality Control. A closer examination of the data sets provided by meteorological stations often reveals lack of quality data sometimes even for extended periods of time. This can be attributed to erroneous measurements due to a combination of factors like equipment error and uncertainty and operation related problems and errors. A more detailed account of such errors is provided in the work of Muneer and Fairooz on quality control of solar radiation data 关16兴. A code in FORTRAN was specifically developed to perform quality checks on the data and identify and eliminate the spurious data values, if any. The code takes as input: site elevation, latitude, longitude, and local time meridian. Also, logging related information is required, both solar time or local civil time is accepted. As a preliminary step, the code performs solar position calculations for each data entry. This includes the calculation of declination angle 共DEC兲, solar hour angle 共␻兲, apparent solar time 共AST兲, solar altitude 共SOLALT兲, and finally the calculation of solar azimuth 共SOLAZM兲. The first level test proceeds by eliminating entries that show a SOLALT less than 7°. For entries that have passed the first test, the day number and the horizontal extraterrestrial irradiation 共E兲 are calculated. Finally, hourly clearness index 共kt兲 and hourly diffuse fraction 共k兲 are calculated as kt = G/E and k = D/G 共1兲 Note that G and D are the measured hourly global and diffuse horizontal radiation values, respectively. Then follows the second test that logically retains the data for which clearness index and the diffuse ratio are positive and have values between zero and one 共inclusive兲 0⬍k⬍ =1 and 0 ⬍ kt ⬍ = 1 共2兲 At the third stage, global and diffuse irradiation are compared with their corresponding Page-model upper and lower boundaries. The Page model is based on the work undertaken for the production of the European Solar Radiation Atlas 关17兴 and the CIBSE guide 关18兴 on weather and solar data. According to his work, the overcast and clear-sky diffuse irradiance set, respectively, the upper and lower limits for diffuse radiation. For global radiation the upper limit is set by global clear-sky model, i.e., it ought to be less than or equal to the clear day global horizontal irradiation. A fourth test involves the construction of k-kt quality control envelope. This is a statistical procedure that requires estimation of kt banded mean, weighted mean 共k̄兲 and standard deviations of “k” values 共␴k兲. Typically the kt range of data may be divided in, say, ten bands of equal width. For each band the above-mentioned Transactions of the ASME Fig. 3 Scatter plot of crude data for Fukuoka with the defining curves „k̄ ± 2.0␴k… enveloping only the quality data „91.2% of the total data… statistics is obtained. From this information an envelope may be drawn that connects those points that, respectively, represent the top 共k̄ + 2␴k兲 and bottom 共k̄ − 2␴k兲 curves. The data lying within this envelope represent quality data—free of any measurement related errors 关16兴. Figure 3 shows one such sample plot for Fukuoka. The procedure described below completes the quality control method adopted for all the nine databases. Once the envelope constituted by the upper and lower boundaries is identified, it is possible to fit a polynomial for a mathematical description of the envelope of acceptance. A seconddegree polynomial was found to be adequate. Thus, the upper and lower boundaries are, respectively, represented as A共kt兲 = Max共1,a1kt2 + b1kt + c1兲 共3兲 B共kt兲 = Min共0,a2kt2 + b2kt + c2兲 共4兲 Note that any given polynomial may generate data that can go beyond the physical limits of k, which lie between 0 and 1. The formulation given in Eqs. 共3兲 and 共4兲 satisfy the above constraints. Furthermore, due care has to be taken to incorporate the “shoulder” effect caused by the intersection of the upper and lower polynomials with the respective k = 1 共upper兲 and k = 0 共lower兲 limits for the plot. In Fig. 3共b兲, C 共kt兲 and D 共kt兲 represent the lines of intersection for upper and lower bounds, respectively. By visual inspection of the plot it is possible to ascertain the intersection points. For all the locations we have worked with, those points have been below kt = 0.4 共upper bound兲, and between kt = 0.75, and kt = 1 共lower bound兲. The last item to be mentioned in this context is that in certain cases there may be a need for the control of the lower-bound polynomial with respect to its upper limit. Notice that within Fig. 3共b兲 an “unconstrained flow” of the B 共kt兲 curve would exclude a small proportion of otherwise good data belonging to heavy overcast regime. A cutoff shown as E 共kt兲 line is thus required, once again by visual inspection. Therefore, procedure of quality control can now be completed with the envelope of acceptance fully defined. It can, however, be noted that since the development of quality control envelope is based on visual study of the nature of plot, the cutoffs and even the type of polynomial curve can vary from site to site. For example, a third degree polynomial was adopted to define the quality control envelope for Mumbai 共India兲. Table 1 gives the percentage of data omitted for each site using the above described quality control procedure. While processing the database it was found that data sets from some of the sites showed a characteristic droop in the top leftJournal of Solar Energy Engineering Fig. 4 k-kt plot of quality-controlled Chennai database: „a… prior to diffuse radiation correction, „b… postdiffuse radiation correction hand corner of the k-kt data plot, like the one presented for Chennai in Fig. 4共a兲. Ideally, the data in that section of the plot would be expected to attain the limiting value of k = 1 as kt → 0. This flaw evidently indicates that the shade-ring correction factor has not been applied to diffuse irradiance measurements. Such uncorrected diffuse radiation was found not only for Chennai but all other Indian sites viz., Mumbai, New Delhi, and Pune and also in the data sets from Madrid and the two Japanese sites, i.e., Sapporo and Fukuoka. Appropriate measures were adopted to correct the diffuse radiation sets for the shadow band error in accordance with a method proposed by the authors of this article 关19兴. Figure 4共b兲 gives the Chennai k-kt plot after applying shade ring correction in diffuse radiation data. For the sites where only daily sunshine duration was given rather than an hourly fraction, the former was used within the hourly regression models. Air mass 共m兲 for each hour of the data was calculated using Kasten’s formula, which provides an accuracy of 99.6% for zenith angles up to 89° 关20兴 m = 关sin SOLALT + 0.50572共SOLALT + 6.07995兲−1.6364兴−1 共5兲 Eventually, the regression analyses were carried out for only those hours of each database for which both sunshine fraction and cloud cover were provided by the respective meteorological stations 共see Table 1兲. FEBRUARY 2006, Vol. 128 / 107 Modeling of Diffuse Radiation Effect of Clearness Index on Diffuse Ratio. It is a wellknown fact that clearness index 共kt兲 strongly influences the diffuse ratio 共k兲 and, hence, it is an integral parameter for the estimation of diffuse radiation. For that reason, we start from the basic k-kt model and then retain kt as the main regressor as we go along developing models with different combination sets of all the four parameters, namely, kt, SF, CC, and m. In the present analysis, we explored k-kt regression equations with k as a polynomial function of kt up to third order. The quadratic fit was found to be the most adequate for all the nine sites. A plot of the data of Bracknell with three regression fits, as an example, is shown in Fig. 2. Although, as expected, the coefficient of determination increases with an increase in complexity of equation, the increase from quadratic to cubic regression model is not as significant as the improvement from a linear to quadratic fit. Moreover, the R2 static should always be used with caution in determining the best model. That is to say, that R2 does not always measure the appropriateness of the model, since it can be artificially inflated by adding higher order polynomial terms in the regression equation. Furthermore, even though R2 is relatively high, this does not necessarily imply that regression model will provide accurate predictions of future observation 关21兴. Also, note that if a cubic or quartic relationship was chosen it would have led to an unreasonable number of coefficients due to the interactions between the four parameters of kt, SF, CC, and m. Therefore, we select the second-degree polynomial k-kt relationship as the optimum and use it throughout our analysis for the sake of simplicity as well as uniformity 共6兲 k = a10 + a11kt + a12kt2 Effect of Sunshine Fraction, Cloud Cover, and Air Mass on Diffuse Ratio: Individual Basis with Clearness Index as the Common Parameter. The following regression equation is used to investigate the effect of hourly clearness index and sunshine fraction or cloud cover or air mass on the hourly diffuse ratio k = 共ai0 + ai1X + ai2X 兲 + 共bi0 + bi1X + bi2X 兲kt 2 2 + 共ci0 + ci1X + ci2X2兲kt2 共7兲 where i = 2 , 3, or 4 and X is SF, CC, or m, respectively. It is worthwhile to note here, that although the expression given above is the quadratic form of regression, two models based on both quadratic as well as linear function, were developed for each of the parameters. Effect of Clearness Index, Sunshine Fraction, Cloud Cover, and Air Mass on Diffuse Ratio: Combined Basis. Under this section, regression equations involving a combination of two variables and later on a tripartite variable combination, besides kt, were explored. All the possible combinations were applied, not neglecting of course, the linear and quadratic models for each parameter k = 共a50 + a51X + a52X2 + a53Y + a54Y 2兲 + 共b50 + b51X + b52X2 + b53Y + b54Y 2兲kt + 共c50 + c51X + c52X2 + c53Y + c54Y 2兲kt2 共8兲 where i = 5 , 6, or 7, X and Y are SF and CC, or SF and m, or CC, and m, respectively. The regression equation proposed to evaluate the effect of the independent variables kt, SF, CC, and m taken all together is represented in the form given by Eq. 共9兲 k = 共a80 + a81SF + a82SF2 + a83CC + a84CC2 + a85m + a86m2兲 + 共b80 + b81SF + b82SF2 + b83CC + b84CC2 + b85m + b86m2兲kt + 共c80 + c81SF + c82SF2 + c83CC + c84CC2 + c85m + c86m2兲kt2 共9兲 108 / Vol. 128, FEBRUARY 2006 Regression analysis for the models given by Eqs. 共6兲–共13兲 was carried out for each location. Notice that although regression models presented here are in quadratic form, a linear fit was also tried and the better of the two was selected for each parameter of each site. That is to say, if a linear model in SF was found to deliver comparable or better accuracy than a corresponding quadratic one, the former model was chosen. To avoid repetition the equations with same parameters but linear fit are not presented here. More so, quadratic form gives the reader an explicit idea of the scope of a given expression. Furthermore, for each data point under investigation the calculated value of diffuse radiation was obtained from the above k models. This calculated value was then subtracted from observed diffuse radiation. The errors thus generated, having the same units as diffuse radiation, i.e., W / m2, were plotted against their frequency of occurrence for each model. In view of economy of length of publication, such error histograms for eight short listed models of only two sites, Mumbai and Gerona, are presented here 共Figs. 5 and 6兲. It is evident from the latter figures that with an increase in number of parameters error histograms improve, i.e., the peakedness of the error distribution increase around the zero error bands. This trend was observed indisputably for all the sites. The simplistic k-kt model or “model 1” has the widest error distribution thus reflecting poor model performance. There is a significant increase in performance 共more than 40%兲 from model 1 to model 8, in context of error frequency in the range of −10 to +10 W / m2 for Mumbai and −5 to +5 W / m2 for Gerona as shown in Figs. 5 and 6, respectively. Although, the error histograms for other sites are not presented here but other indicators, discussed in the following section, convey the error statistics of their models quite adequately. Statistical Tests and Comparison Indices To investigate a given model’s performance the following statistical indicators were presently employed: • • • • • coefficient of determination 共R2兲 mean bias deviation 共MBD兲 root mean square deviation 共RMSD兲 skewness kurtosis R2 is the ratio of explained variation to the total variation. It lies between zero and one. A high value of R2, thus indicating a lower unexplained variation, is desirable R2 = 兺 共Y c − Y m兲 2 / 兺 共Y o − Y m兲 2 共10兲 R2 is often used to judge the adequacy of a regression model but it should not be the sole criterion 共as discussed earlier兲 for choosing a particular model. MBD provides a measure of the overall long-term trend of a given model. Positive values of MBD indicate under estimation while negative values imply over estimation by the proposed model. Nevertheless, within a data set overestimation of one observation can cancel underestimation of another. A MBD nearest to zero is desired. MBD is presently computed as MBD = 兺 共Y o − Y c兲/n 共11兲 Root mean square of deviation provides a means of comparison of the actual deviation between the predicted and the measured values. RMSD is always positive. A lower absolute value of RMSD indicates a better model. However, it is does not always present a true picture of the data as a whole. That is to say, only a few large errors in the data sets can increase the overall RMSD substantially RMSD = 关兺 共Y o − Y c兲2/n 兴 1/2 共12兲 Skewness is defined as a measure of the lack of symmetry in a Transactions of the ASME Fig. 5 Error histograms of calculated diffuse radiation of selectively chosen eight models for Mumbai distribution. A positively skewed distribution tails off to the high end of the scale while negative skew tails off the low end of the scale. If the distribution is normal or, in other words, has no skewness, then the skewness statistic will be zero. If the distribution has a positive skew, then the skewness statistic will be positive and likewise, negative for a distribution having a negative skew. Skewness is expressed as Skewness = 共mean-mode兲/standard deviation 共13兲 Kurtosis is defined as a measure of the degree of peakedness in Journal of Solar Energy Engineering the distribution, relative to its width. The kurtosis statistic will be zero 共mesokurtic兲 for a normal distribution, positive for peaked distributions 共leptokurtic兲 and negative for flat distributions 共platykurtic兲. Numerical representation of kurtosis based on quartiles and percentiles is given by Kurtosis = Q/共P90 − P10兲 共14兲 where Q = 共Q3 − Q1兲 / 2 is the semi-inter quartile range. FEBRUARY 2006, Vol. 128 / 109 Fig. 6 Error histograms of calculated diffuse radiation of selectively chosen eight models for Gerona Though, practically speaking, the actual values of the skewness and kurtosis statistics rarely turn out to be exactly zero. For the present analysis, R2 is obtained for the regression analysis of k so it directly relates to the diffuse fraction. All the other parameters are evaluated for calculated diffuse radiation: MBD and RMSD enable insight in the performance evaluation of diffuse radiation prediction, whereas, skewness and kurtosis characterize the symmetry and shape of the distribution. Since, in our study it 110 / Vol. 128, FEBRUARY 2006 is the error distribution we are referring to, the best model will have lowest absolute skewness and highest kurtosis. For a minimum of eight models 共16 models for Bracknell, eight models for Delhi, nine models each for the Japanese sites, and 18 models for each of the remaining sites兲 broadly classified for each of the nine sites, it becomes increasingly difficult to find a model that clearly stands out as the best from the rest. While some models might show promise on a few of the statistical tests under Transactions of the ASME discussion, others may indicate a better performance in terms of other measures of evaluation. Also, we have no means to determine which statistical parameter of the given five influences the model performance the most. Hence, a methodology should be devised such that, provided all the indicators were considered to bear more or less similar effect, it can aid in the selection of an overall “best” model. The present work proposes one such technique of evaluation. For our analysis of the best model, it is essential that all the parameters follow a unidirectional trend. That is to say, if a higher R2 suggests better model performance then an increase in all the other four parameters should also be indicative of the same. However, this is not the case here. Whereas, a high R2 and kurtosis indicate better performance, a high, RMSD and skewness suggest the opposite. Moreover, both skewness and MBD can have either negative or positive values, so an absolute value that is closest to zero is desired. Therefore, we need to devise a tool to facilitate a discrete comparison and help in the assessment of all the presently developed models. For that purpose, we introduce a term called “accuracy score” which sums up all the credits accounted by each test of assessment. Before starting to test the model accuracy, the range of variabilities in the statistical tests needs to be addressed. MBD and RMSD have the same units as diffuse radiation 共W / m2兲, while R2, kurtosis, and skewness are dimensionless quantities. So, in order to sum up the statistics that vary substantially both in terms of values as well as units, we ought to normalize them. We need to make them dimensionless fractions ranging between zero and one to provide a commonality for the addition process. In order to this, we list down all the developed models and their respective statistics, illustrated by the left half of the tables 共Tables 2–5兲. The other half of each table gives the corresponding “algebraic equivalents.” The calculation of such equivalents is a step-by-step procedure. The very first step is to make MBD and skewness absolute so that all the parameters are now positive. As a second step, we divide every item of each parameter by the maximum value of that parameter 共within a given table兲 to convert it into a ratio. This gives the algebraic equivalents for their corresponding R2 and kurtosis. However, as we have already noted MBD, RMSD, and skewness follow an opposite trend to that of R2 and kurtosis, the third step thus involves the subtraction of resulting ratios of MBD, RMSD, and skewness from one, to yield their algebraic equivalents. This is done in order to achieve a unanimous trend for all the parameters. The fourth step involves the addition of all the five algebraic equivalents of statistical parameters for each model, thus obtaining its accuracy score 共AS兲. This is represented in the second-last column of Tables 2–5. Numerically, it is expressed as 2 AS = 关Ri2/Ri,max 兴 + 关1 − abs共MBDi兲/abs共MBDi兲max兴 + 关1 − RMSDi/RMSDi,max兴 + 关1 − abs共Skewi兲/abs共Skewi兲max兴 + 关Kurtosisi/Kurtosisi,max兴 共15兲 The highest accuracy score reflects the best model. Nevertheless, at the same time the increase in complexity of the model cannot be overlooked. Therefore, an optimum model has to be chosen: a model whose prediction accuracy considerably overweighs the involved intricacy. Accuracy score is plotted against the model number for each site 共Fig. 7兲. On the basis of assessment of accuracy scores, an optimum model is recommended for each site 共highlighted figures in Tables 2–5兲. Since the four Indian sites are widely located within the subcontinent, the most appropriate models were found to be different for different locations. Both for Chennai and New Delhi model 12 gave the best AS, and likewise another model 共model 16兲 was found to be consistent for Mumbai and Pune. Note that due to no information being available on CC data, the models for New Delhi are numbered intermittently in order to match with the corresponding models for other Indian sites 共Table 2兲. For Japan it was found that optimum model for Fukuoka 共model 7兲 and SapJournal of Solar Energy Engineering poro 共model 8兲 share a commonality with respect to the parameters involved. For Spain, while model 9 suffices for Madrid, Gerona data fits best with a more complex model 17. Model 11 was found to be more than adequate for Bracknell. However, analyzing the models on a comparative scale, it was found that a single model could more than adequately estimate the diffuse radiation for the locations within a given region, which do not differ widely in their local climate. For instance, not only does model 12 appropriately reflect Chennai data set, it may also effectively be used for estimating diffuse radiation for the other Indian sites. Likewise, model 8 reasonably represents the solar climate of Japanese sites, considering the fact that Fukuoka and Sapporo lie at the two extreme ends of the country. In the vein of the above argument we may deduce that model 17 can be employed to estimate diffuse radiation for Spanish sites, as is evident from the accuracy scores of Gerona and Madrid. However, further evaluation work has to be undertaken for a fuller validation. As for UK, only one site 共Bracknell兲 was under study so we cannot generalize our result for that country. Inspection of accuracy plots reveal that almost all recommended models present a hike of twice the accuracy score from their worst counterparts for most of the sites. This, in itself, signifies the consequent improvement in diffuse prediction with the inclusion of synoptic parameters and air mass along with kt. The minimum increase in accuracy score from the poorest model to the most adequate one is 35% 共for Sapporo, Fig. 7共b兲兲. On the other hand, AS of optimum model of sites like Pune increases three fold compared to the minimum, Fig. 7共a兲. Note that the lack of monotonic behavior of the accuracy curves suggests that an increase in degree of the polynomial function or addition of another independent parameter does not necessarily improve the model performance. There is always a unique set of arguments that gives the best regression fit or highest AS for the site concerned. So one model, which is optimum for one site, may not be of an equal importance for the other site. This may also be due to a disparity of accuracy of measurement for the given data set at different sites. Another reason could be the inconsistency of frequency of measurement of SF and CC data as compared to radiation values. That is to say, the total daily SF for majority of sites was regressed against hourly values of radiation and the incorporation of CC, measured at three to six hourly intervals, did not always coincide with the instance when radiation data was recorded. The results would have been much more improved if SF and CC were recorded concurrently with global and diffuse radiation. The regression coefficients of optimum models for each site are presented in Table 6. Note that some models involve only a linear function for some of the given independent parameters and hence the respective second order coefficients are left blank. Equations 共11兲–共13兲 can be referred in this respect. An overall model is indicated for the country in question within the table. Diffuse ratio was calculated and plotted against the actual k-kt envelope for all the nine sites. One such plot is shown here for Bracknell 共Fig. 8兲. In the figure, the light-colored scatter points form the “real envelope” which is the actual data points, the darkcolored envelope is the “computed envelope” which represents the estimated values using the optimum model and the black curve is the locus of quadratic k-kt regression. It is evident from the figure that the proposed model works adequately for the entire k-kt range with the computed envelope overlapping the real envelope. Plots of other sites also revealed similar overlapping trends, which is a strong indication of the accuracy of the models. Moreover, over/ under estimation of diffuse ratio 共k兲 is considerably reduced, because k estimates from the proposed model amply cover the broad scatter owing to the actual data points. This implies that k can be calculated with reasonable accuracy for different sky conditions within the same kt range, unlike the single return-value k-kt relationship. It can be attributed to the fact that improved models include other parameters as well 共apart from kt兲 which play a FEBRUARY 2006, Vol. 128 / 111 Table 2 Statistical indicators, comparative indices and accuracy scores for India: Chennai, Mumbai, New Delhi, and Pune Algebraic equivalents of R2 MBD 共W / m2兲 RMSD 共W / m2兲 K = f共qKt兲 K = f共qKt , linSF兲 K = f共qKt , qSF兲 K = f共qKt , linCC兲 K = f共qKt , qCC兲 K = f共qKt , lin m兲 K = f共qKt , q m兲 K = f共qKt , linSF, linCC兲 K = f共qKt , linSF, qCC兲 K = f共qKt , qSF, qCC兲 K = f共qKt , linSF, lin m兲 K = f共qKt, linSF, q m兲 K = f共qKt , qSF, q m兲 K = f共qKt , qCC, lin m兲 K = f共qKt , qCC, q m兲 K = f共qKt , linSF, qCC, lin m兲 K = f共qKt , linSF, qCC, q m兲 K = f共qKt , qSF, qCC, q m兲 0.42 0.67 0.67 0.43 0.67 0.45 0.52 0.66 0.72 0.72 0.68 0.70 0.71 0.69 0.71 0.74 0.75 0.76 −2.81 −8.58 −8.41 −2.82 −3.22 −0.52 −1.04 −3.48 −3.42 −3.50 −0.90 −0.59 −0.55 −0.89 −0.58 −0.97 −0.48 −0.45 98.84 77.87 77.69 98.44 76.32 95.62 90.12 76.58 69.92 69.95 73.63 69.13 68.65 73.89 70.86 67.17 63.54 63.31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 K = f共qKt兲 K = f共qKt , linSF兲 K = f共qKt , qSF兲 K = f共qKt , linCC兲 K = f共qKt , qCC兲 K = f共qKt , lin m兲 K = f共qKt , q m兲 K = f共qKt , linSF, linCC兲 K = f共qKt , linSF, qCC兲 K = f共qKt , qSF, qCC兲 K = f共qKt , linSF, lin m兲 K = f共qKt , linSF, q m兲 K = f共qKt , qSF, q m兲 K = f共qKt , qCC, lin m兲 K = f共qKt , qCC, q m兲 K = f共qKt, linSF, qCC, lin m兲 K = f共qKt , linSF, qCC, q m兲 K = f共qKt , qSF, qCC, q m兲 0.82 0.87 0.87 0.82 0.88 0.82 0.83 0.87 0.88 0.89 0.88 0.88 0.88 0.88 0.88 0.89 0.89 0.89 1.45 −1.67 −1.92 1.10 −1.94 1.37 0.37 −1.76 −2.48 −2.55 0.26 0.12 0.11 0.38 0.16 0.06 0.10 0.10 58.18 49.41 48.80 56.77 47.54 58.15 57.35 48.59 45.71 45.65 48.70 48.60 47.79 46.88 46.72 44.73 44.64 44.44 1 2 3 6 7 11 12 13 K = f共quadKt兲 K = f共qKt , linSF兲 K = f共qKt , qSF兲 K = f共qKt , lin m兲 K = f共qKt , q m兲 K = f共qKt , linSF, lin m兲 K = f共qKt, linSF, q m兲 K = f共qKt , qSF, q m兲 0.81 0.87 0.87 0.82 0.82 0.87 0.87 0.87 1.75 −1.12 −1.33 1.26 0.27 0.50 0.12 0.13 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 K = f共qKt兲 K = f共qKt , linSF兲 K = f共qKt , qSF兲 K = f共qKt , linCC兲 K = f共qKt , qCC兲 K = f共qKt , lin m兲 K = f共qKt , q m兲 K = f共qKt , linSF, linCC兲 K = f共qKt , linSF, qCC兲 K = f共qKt , qSF, qCC兲 K = f共qKt , linSF, lin m兲 K = f共qKt , linSF, q m兲 K = f共qKt , qSF, q m兲 K = f共qKt , qCC, lin m兲 K = f共qKt , qCC, q m兲 K = f共qKt, linSF, qCC, lin m兲 0.49 0.77 0.80 0.75 0.79 0.53 0.60 0.82 0.83 0.84 0.78 0.78 0.80 0.79 0.80 0.83 0.86 −0.98 −1.19 −0.26 −0.26 0.65 −0.66 −0.95 −0.78 −0.92 −0.29 −0.44 −0.35 0.06 −0.29 −0.20 No. Model type 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 112 / Vol. 128, FEBRUARY 2006 Kurtosis R2 MBD RMSD Skew Kurtosis AS 0.30 1.24 1.27 0.29 1.14 0.38 0.60 1.42 1.64 1.70 1.37 1.56 1.53 1.04 0.98 1.54 1.55 1.58 0.56 0.88 0.89 0.56 0.89 0.59 0.69 0.87 0.95 0.96 0.89 0.93 0.94 0.91 0.93 0.97 1.00 1.00 0.67 0.00 0.02 0.67 0.63 0.94 0.88 0.59 0.60 0.59 0.89 0.93 0.94 0.90 0.93 0.89 0.94 0.95 0.00 0.21 0.21 0.00 0.23 0.03 0.09 0.23 0.29 0.29 0.26 0.30 0.31 0.25 0.28 0.32 0.36 0.36 0.61 0.38 0.24 0.61 0.37 0.39 0.33 0.44 0.10 0.00 0.70 0.90 0.76 0.67 0.85 0.39 0.59 0.49 0.17 0.73 0.75 0.17 0.67 0.22 0.35 0.83 0.96 1.00 0.81 0.92 0.90 0.61 0.58 0.90 0.91 0.93 2.02 2.19 2.11 2.02 2.78 2.17 2.34 2.96 2.90 2.84 3.55 3.97 3.84 3.34 3.58 3.47 3.79 3.73 0.91 0.92 1.04 0.91 1.01 0.95 0.88 0.95 1.04 1.09 1.10 1.09 1.25 1.19 1.18 1.28 1.28 1.35 New Delhi 2.22 3.47 3.83 2.22 4.18 2.49 2.53 3.55 4.65 4.85 4.08 4.06 4.62 4.66 4.67 5.54 5.56 5.94 0.91 0.97 0.98 0.92 0.98 0.92 0.93 0.97 0.99 0.99 0.98 0.98 0.98 0.99 0.99 1.00 1.00 1.00 0.43 0.34 0.25 0.57 0.24 0.46 0.86 0.31 0.03 0.00 0.90 0.95 0.96 0.85 0.94 0.98 0.96 0.96 0.00 0.15 0.16 0.02 0.18 0.00 0.01 0.16 0.21 0.22 0.16 0.16 0.18 0.19 0.20 0.23 0.23 0.24 0.32 0.32 0.23 0.33 0.25 0.30 0.35 0.30 0.23 0.19 0.19 0.19 0.08 0.12 0.13 0.05 0.05 0.00 0.37 0.59 0.65 0.37 0.70 0.42 0.43 0.60 0.78 0.82 0.69 0.68 0.78 0.78 0.79 0.93 0.94 1.00 2.04 2.37 2.26 2.22 2.36 2.10 2.57 2.35 2.24 2.22 2.92 2.98 2.97 2.94 3.04 3.20 3.18 3.20 47.69 38.86 38.41 47.61 47.09 38.74 38.56 38.08 0.52 0.29 0.22 0.50 0.34 0.50 0.42 0.36 Pune 0.75 1.27 1.28 0.83 0.72 1.65 1.55 1.59 0.93 0.99 0.99 0.93 0.94 0.99 1.00 1.00 0.00 0.36 0.24 0.28 0.85 0.72 0.93 0.93 0.00 0.19 0.19 0.00 0.01 0.19 0.19 0.20 0.00 0.44 0.57 0.03 0.34 0.04 0.19 0.30 0.46 0.77 0.78 0.51 0.44 1.00 0.94 0.96 1.38 2.74 2.78 1.75 2.57 2.94 3.24 3.40 90.83 60.92 58.45 64.76 58.35 87.65 85.06 54.77 52.04 51.13 60.27 60.17 57.68 57.28 56.94 51.34 0.67 0.24 0.11 0.61 −0.15 0.72 0.68 0.06 −0.20 −0.19 0.30 0.29 0.22 −0.13 −0.12 −0.13 0.35 3.24 2.62 8.12 1.84 0.47 0.68 4.59 2.72 2.35 3.41 3.34 2.64 2.15 2.11 2.82 0.58 0.92 0.95 0.90 0.94 0.63 0.72 0.97 0.98 0.99 0.92 0.93 0.96 0.94 0.95 0.99 0.28 0.18 0.00 0.78 0.78 0.45 0.44 0.20 0.35 0.23 0.75 0.63 0.71 0.95 0.75 0.83 0.00 0.33 0.36 0.29 0.36 0.04 0.06 0.40 0.43 0.44 0.34 0.34 0.36 0.37 0.37 0.43 0.07 0.67 0.85 0.16 0.79 0.00 0.06 0.92 0.73 0.74 0.59 0.60 0.69 0.82 0.84 0.82 0.04 0.40 0.32 1.00 0.23 0.06 0.08 0.57 0.33 0.29 0.42 0.41 0.33 0.26 0.26 0.35 0.98 2.49 2.48 3.12 3.09 1.17 1.37 3.05 2.82 2.69 3.02 2.91 3.04 3.35 3.18 3.42 Skew Chennai 0.19 −0.31 −0.38 0.20 −0.32 0.31 0.34 −0.28 −0.45 −0.50 −0.15 −0.05 −0.12 −0.17 −0.07 −0.31 −0.21 −0.26 Mumbai Transactions of the ASME „Continued.… Table 2 Algebraic equivalents of No. Model type 17 18 K = f共qKt , linSF, qCC, q m兲 K = f共qKt , qSF, qCC, q m兲 R2 MBD 共W / m2兲 RMSD 共W / m2兲 Skew Kurtosis R2 MBD RMSD Skew Kurtosis AS 0.83 0.84 −0.29 −0.25 51.31 50.36 −0.13 −0.09 2.78 2.40 0.99 1.00 0.76 0.79 0.44 0.45 0.83 0.87 0.34 0.30 3.35 3.41 critical in the estimation of diffuse ratio. However, the model poses limitations in the high kt-low k range. For instance, if the same model is employed in that range, k can attain a zero value at a kt of 0.9 共Fig. 8兲. Practically speaking, extremely clear-sky conditions are very rare. This is also supported by the fact that none of the data points from the nine databases under study fell in that range. Nonetheless, there has to be a more physically based modeling condition that does not give negative k values when kt exceeds a certain limit. The method adopted to modify this limitation is the use of a constant value of k, for a kt exceeding the maximum limit. By observation, two points from the actual k-kt envelope at the bottom right end that contain the minimum k and maximum kt are selected. Interpolation of the corresponding k values of the two points determines the average k value. For those rare cases of points that may exceed kt, max, the latter, constant value of k is prescribed. Thus, the kt constraint is identified and a constant k value is calculated for four selected sites 共see Table 7兲 which were chosen as representing optimum regression models for the respective regions/countries. Calculated diffuse horizontal radiation against its observed value was also plotted for each site. Again, due to brevity of space only the plots of four sites representative of each country are presented here in Fig. 9. The improvement from the conventional k-kt model is evident with the mere addition of SF. However, the improvement from the intermediate to the optimum model is not that prominent for Fukuoka and Bracknell. This may be attributed to the fact that the optimum model only includes one more additional parameter m. The latter was selected as it significantly improved the statistics, in particular, MBD. It is important to note that the above described procedure for comparative analysis of developed models and proposal for optimum models henceforth, is only based on the available data. The models can be modified and adjusted according to the siteavailable data. The major assumptions in our approach were: 共1兲 Unlike the radiation data, the quality of other intrinsic parameters 共SF and CC兲 measurements was not queried. 共2兲 Approximation of daily SF and subdaily CC data to fit the frame of present regression analysis for hourly radiation. 共3兲 Assumption in the accuracy score system, used for the evaluation of best models, that all of the statistics contribute equally towards the model accuracy. Future work may consider the allocation of weights to the above statistical tests for a more refined accuracy score. Conclusion This work demonstrated the influence of synoptic elements, sunshine fraction, cloud cover, and air mass on the regressions used for estimation of hourly diffuse radiation. It was concluded that they have a significant bearing on the model accuracy and can be effectively used along with global radiation to improve the diffuse radiation modeling. Optimum models for each site were proposed that provided reasonable accuracy and minimal intricacy. The models were found to be region specific. An overall optimum model was also recommended for each of the given countries. Generally, models, which included all the parameters, gave highest accuracy scores. It was shown presently that incorporation of the three parameters 共SF, CC, and m兲 in addition to the kt variable yields a significant improvement over the conventional k-kt models. Table 3 Statistical indicators, comparative indices, and accuracy scores for Japan: Fukuoka, Sapporo Algebraical equivalents of R2 MBD 共W / m2兲 RMSD 共W / m2兲 K = f共quadKt兲 K = f共qKt , linSF兲 K = f共qKt , qSF兲 K = f共qKt , lin m兲 K = f共qKt , q m兲 K = f共qKt , linSF, lin m兲 K = f共qKt, linSF, q m兲 K = f共qKt , qSF, lin m兲 K = f共qKt , qSF, q m兲 0.88 0.92 0.92 0.89 0.89 0.92 0.92 0.92 0.92 2.84 1.06 0.97 0.45 0.02 0.21 0.03 0.23 0.03 53.11 44.64 44.14 52.20 52.28 44.53 44.51 44.01 43.97 K = f共quadKt兲 K = f共qKt , linSF兲 K = f共qKt , qSF兲 K = f共qKt , lin m兲 K = f共qKt , q m兲 K = f共qKt , linSF, lin m兲 K = f共qKt , linSF, q m兲 K = f共qKt, qSF, lin m兲 K = f共qKt , qSF, q m兲 0.85 0.91 0.92 0.85 0.85 0.92 0.92 0.92 0.92 −0.07 −2.59 −2.51 0.02 0.07 −0.06 0.05 −0.07 0.04 55.61 44.42 43.27 55.54 55.47 42.34 42.29 41.15 41.12 Model type 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 No. Journal of Solar Energy Engineering Kurtosis R2 MBD RMSD Skew Kurtosis AS Fukuoka 1.55 0.83 0.78 1.25 1.18 0.71 0.66 0.68 0.61 Sapporo 8.32 4.64 4.71 8.00 8.21 4.72 4.73 4.77 4.80 0.96 1.00 1.00 0.96 0.97 1.00 1.00 1.00 1.00 0.00 0.63 0.66 0.84 0.99 0.93 0.99 0.92 0.99 0.00 0.16 0.17 0.02 0.02 0.16 0.16 0.17 0.17 0.00 0.46 0.50 0.19 0.24 0.54 0.58 0.56 0.61 1.00 0.56 0.57 0.96 0.99 0.57 0.57 0.57 0.58 1.96 2.80 2.89 2.98 3.20 3.20 3.29 3.23 3.35 1.18 0.42 0.36 1.40 1.48 1.30 1.37 1.24 1.29 6.30 4.45 4.99 7.56 8.20 6.77 7.05 7.43 7.64 0.92 0.99 0.99 0.92 0.92 0.99 0.99 1.00 1.00 0.97 0.00 0.03 0.99 0.97 0.98 0.98 0.97 0.98 0.00 0.20 0.22 0.00 0.00 0.24 0.24 0.26 0.26 0.20 0.71 0.75 0.05 0.00 0.12 0.08 0.17 0.13 0.77 0.54 0.61 0.92 1.00 0.83 0.86 0.91 0.93 2.87 2.44 2.61 2.89 2.90 3.15 3.15 3.30 3.31 Skew FEBRUARY 2006, Vol. 128 / 113 Table 4 Statistical indicators, comparative indices, and accuracy scores for Spain: Gerona, Madrid Algebraic equivalents of R2 MBD 共W / m2兲 RMSD 共W / m2兲 K = f共qKt兲 K = f共qKt , linSF兲 K = f共qKt , qSF兲 K = f共qKt , linCC兲 K = f共qKt , qCC兲 K = f共qKt , lin m兲 K = f共qKt , q m兲 K = f共qKt , linSF, linCC兲 K = f共qKt , qSF, qCC兲 K = f共qKt , linSF, lin m兲 K = f共qKt , linSF, q m兲 K = f共qKt , qSF, q m兲 K = f共qKt , linCC, lin m兲 K = f共qKt , linCC, q m兲 K = f共qKt , qCC, q m兲 K = f共qKt , linSF, linCC, lin m兲 K = f共qKt, linSF, linCC, q m兲 K = f共qKt , qSF, qCC, q m兲 0.88 0.91 0.91 0.92 0.93 0.90 0.90 0.93 0.93 0.92 0.92 0.92 0.93 0.93 0.93 0.93 0.93 0.94 5.50 3.45 3.26 2.04 1.99 0.80 0.17 1.89 1.83 0.75 0.12 0.12 0.45 0.05 0.04 0.50 0.05 0.04 36.41 31.54 31.27 28.55 28.51 31.00 30.58 27.74 27.57 27.61 26.98 26.78 26.49 26.15 26.18 25.59 25.17 25.06 K = f共qKt兲 K = f共qKt , linSF兲 K = f共qKt , qSF兲 K = f共qKt , linCC兲 K = f共qKt , qCC兲 K = f共qKt , lin m兲 K = f共qKt , q m兲 K = f共qKt , linSF, linCC兲 K = f共qKt, qSF, qCC兲 K = f共qKt , linSF, lin m兲 K = f共qKt , linSF, q m兲 K = f共qKt , qSF, q m兲 K = f共qKt , linCC, lin m兲 K = f共qKt , linCC, q m兲 K = f共qKt , qCC, q m兲 K = f共qKt , linSF, linCC, lin m兲 K = f共qKt , linSF, linCC, q m兲 K = f共qKt , qSF, qCC, q m兲 0.87 0.92 0.92 0.94 0.94 0.89 0.89 0.94 0.94 0.92 0.92 0.93 0.94 0.94 0.94 0.94 0.94 0.94 6.76 2.10 1.69 0.47 0.48 1.99 0.09 0.24 0.17 1.13 0.12 0.13 0.64 0.08 0.08 0.60 0.09 0.09 40.20 32.42 31.60 28.34 28.19 36.47 35.36 27.53 27.20 31.69 30.67 30.28 28.54 28.05 27.83 27.77 27.27 26.95 No. Model type 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Kurtosis R2 MBD RMSD Skew Kurtosis AS Gerona 1.37 1.32 1.32 1.15 1.15 0.95 0.76 1.08 1.05 0.87 0.62 0.63 0.79 0.59 0.59 0.70 0.47 0.45 Madrid 3.28 4.09 4.15 4.66 4.77 2.81 2.65 4.74 4.70 3.66 3.56 3.33 4.11 3.91 4.08 4.33 4.25 4.16 0.94 0.97 0.97 0.99 0.99 0.96 0.97 0.99 0.99 0.98 0.99 0.99 0.99 1.00 1.00 1.00 1.00 1.00 0.00 0.37 0.41 0.63 0.64 0.85 0.97 0.66 0.67 0.86 0.98 0.98 0.92 0.99 0.99 0.91 0.99 0.99 0.00 0.13 0.14 0.22 0.22 0.15 0.16 0.24 0.24 0.24 0.26 0.26 0.27 0.28 0.28 0.30 0.31 0.31 0.00 0.04 0.04 0.16 0.17 0.31 0.45 0.21 0.23 0.37 0.55 0.54 0.43 0.57 0.57 0.49 0.66 0.67 0.69 0.86 0.87 0.98 1.00 0.59 0.56 0.99 0.99 0.77 0.75 0.70 0.86 0.82 0.86 0.91 0.89 0.87 1.63 2.38 2.43 2.97 3.01 2.87 3.10 3.09 3.12 3.22 3.52 3.47 3.47 3.66 3.70 3.60 3.85 3.85 1.26 1.23 1.06 0.36 0.40 0.67 0.23 0.38 0.35 1.00 0.61 0.54 0.39 0.17 0.20 0.48 0.26 0.27 2.30 4.80 4.51 4.06 4.37 1.87 2.01 4.22 4.50 4.47 3.91 3.90 4.15 4.15 4.46 4.30 4.24 4.58 0.92 0.98 0.98 0.99 1.00 0.94 0.95 1.00 1.00 0.98 0.98 0.98 0.99 1.00 1.00 1.00 1.00 1.00 0.00 0.69 0.75 0.93 0.93 0.71 0.99 0.96 0.97 0.83 0.98 0.98 0.91 0.99 0.99 0.91 0.99 0.99 0.00 0.19 0.21 0.29 0.30 0.09 0.12 0.31 0.32 0.21 0.24 0.25 0.29 0.30 0.31 0.31 0.32 0.33 0.00 0.02 0.16 0.71 0.68 0.47 0.82 0.70 0.72 0.20 0.52 0.57 0.69 0.87 0.84 0.62 0.79 0.79 0.48 1.00 0.94 0.85 0.91 0.39 0.42 0.88 0.94 0.93 0.81 0.81 0.87 0.87 0.93 0.90 0.89 0.95 1.40 2.88 3.04 3.78 3.82 2.60 3.30 3.86 3.96 3.16 3.53 3.60 3.75 4.02 4.06 3.74 3.98 4.06 Skew Table 5 Statistical indicators, comparative indices, and accuracy scores for Bracknell „UK… Algebraic equivalents of No. Model type 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 K = f共qKt兲 K = f共qKt , linSF兲 K = f共qKt , qSF兲 K = f共qKt , linCC兲 K = f共qKt , qCC兲 K = f共qKt , lin m兲 K = f共qKt , q m兲 K = f共qKt , linSF, linCC兲 K = f共qKt , qSF, qCC兲 K = f共qKt , linSF, lin m兲 K = f共qKt, qSF, q m兲 K = f共qKt , qCC, lin m兲 K = f共qKt , qCC, q m兲 K = f共qKt , linSF, qCC, lin m兲 K = f共qKt , linSF, qCC, q m兲 K = f共qKt , qSF, qCC, q m兲 R2 MBD 共W / m2兲 RMSD 共W / m2兲 Skew Kurtosis R2 MBD RMSD Skew Kurtosis AS 0.89 0.94 0.94 0.92 0.92 0.90 0.90 0.95 0.95 0.94 0.94 0.92 0.93 0.95 0.95 0.95 2.86 2.15 2.03 0.86 0.95 0.55 −0.02 1.24 1.25 0.28 0.01 0.37 −0.02 0.25 0.01 0.00 33.85 28.61 28.40 29.49 29.02 32.25 31.93 26.25 26.13 27.71 27.40 29.21 28.44 25.88 25.73 25.56 1.06 1.06 0.96 0.70 0.74 0.51 0.23 0.77 0.74 0.62 0.37 0.52 0.28 0.49 0.29 0.28 5.43 6.13 6.15 5.26 5.28 4.80 4.54 5.91 6.07 6.37 6.25 4.97 4.79 5.90 5.76 5.95 0.94 0.99 0.99 0.97 0.97 0.95 0.95 1.00 1.00 0.99 0.99 0.97 0.98 1.00 1.00 1.00 0.00 0.25 0.29 0.70 0.67 0.81 0.99 0.57 0.56 0.90 1.00 0.87 0.99 0.91 1.00 1.00 0.00 0.15 0.16 0.13 0.14 0.05 0.06 0.22 0.23 0.18 0.19 0.14 0.16 0.24 0.24 0.24 0.00 0.01 0.10 0.34 0.30 0.52 0.79 0.28 0.30 0.41 0.65 0.51 0.74 0.54 0.73 0.74 0.85 0.96 0.97 0.83 0.83 0.75 0.71 0.93 0.95 1.00 0.98 0.78 0.75 0.93 0.90 0.93 1.79 2.36 2.50 2.96 2.92 3.08 3.50 3.00 3.04 3.49 3.82 3.27 3.62 3.61 3.87 3.92 114 / Vol. 128, FEBRUARY 2006 Transactions of the ASME Fig. 7 Evaluation of given models by means of presently developed accuracy scoring system. „a… India, „b… Japan, „c… Spain, „d… UK. Table 6 The regression coefficients for the recommended models, refer to Eqs. „10…–„13… India Chennaia a60 a61 a62 a63 a64 b60 b61 b62 b63 b64 c60 c61 c62 c63 c64 a 0.9508 0.1172 – −0.0609 0.0087 2.2681 −2.2769 – −1.3345 0.2065 −4.0137 1.8349 – 2.5424 −0.3669 Mumbai a80 a81 a82 a83 a84 a85 a86 b80 b81 b82 b83 b84 b85 b86 c80 c81 c82 c83 c84 c85 c86 0.9665 0.2862 – −0.0118 −0.0001 0.0016 – −0.0856 −1.8640 – 0.1277 0.0013 −0.0831 – −1.3538 1.8634 – −0.1266 −0.0014 0.2164 – Pune a80 a81 a82 a83 a84 a85 a86 b80 b81 b82 b83 b84 b85 b86 c80 c81 c82 c83 c84 c85 c86 0.8747 0.0040 – 0.0426 −0.0047 −0.0256 – −0.1501 −1.4875 – −0.1360 0.0346 −0.0261 – −0.5558 1.3729 – 0.1020 −0.0309 0.1566 – Japan New Delhi a60 a61 a62 a63 a64 b60 b61 b62 b63 b64 c60 c61 c62 c63 c64 0.845 0.357 – −0.010 −0.002 1.025 −2.736 – −0.033 0.013 −2.121 2.560 – 0.069 −0.006 Fukuoka a60 a61 a62 a63 a64 b60 b61 b62 b63 b64 c60 c61 c62 c63 c64 0.9701 0.3739 – 0.0088 −0.0010 0.4060 −2.0874 – −0.0277 −0.0007 −1.3740 1.7343 – −0.0879 0.0168 Spain Sapporoa a60 a61 a62 a63 a64 b60 b61 b62 b63 b64 c60 c61 c62 c63 c64 0.9311 −0.3966 0.7583 0.0091 – 0.7206 −2.2515 −0.3544 −0.1259 – −1.8335 5.1822 −2.8571 0.2712 – Geronaa a80 a81 a82 a83 a84 a85 a86 b80 b81 b82 b83 b84 b85 b86 c80 c81 c82 c83 c84 c85 c86 1.0585 0.0079 – −0.0329 – 0.0517 −0.0084 −1.2333 −0.4660 – 0.2375 – −0.1092 0.0278 0.3954 0.4238 – −0.2421 – −0.1544 −0.0026 UK Madrid a50 a51 a52 a53 a54 b50 b51 b52 b53 b54 c50 c51 c52 c53 c54 0.9956 0.0922 −0.0096 0.0157 −0.0031 −0.8392 −0.9612 −0.1230 0.0260 0.0210 −0.5441 1.6523 −0.3052 −0.0452 −0.0213 Bracknell a60 a61 a62 a63 a64 b60 b61 b62 b63 b64 c60 c61 c62 c63 c64 0.8991 −0.6826 0.6482 0.0277 −0.0017 0.8799 −0.6656 −0.3144 −0.1580 0.0031 −1.7511 2.7865 −1.9243 0.0438 0.0124 Optimum regression model for diffuse radiation for the given country. Nomenclature AS AST CC DEC ⫽ ⫽ ⫽ ⫽ accuracy score apparent solar time cloud cover earth’s declination Journal of Solar Energy Engineering D ⫽ diffuse horizontal hourly irradiation, W / m2 E ⫽ extraterrestrial horizontal hourly irradiation, W / m2 G ⫽ global horizontal hourly irradiation, W / m2 k ⫽ hourly diffuse ratio FEBRUARY 2006, Vol. 128 / 115 Table 7 Recommended constant k values for kt exceeding kt,max. k* is the corresponding k-value for kt,max. k, constant is the average of k* and k,min. Region kt,max k* k,min k,constant Chennai, India Sapporo, Japan Gerona, Spain Bracknell, UK 0.99 0.93 0.82 0.83 0.38 0.33 0.08 0.28 0.09 0.11 0.06 0.09 0.24 0.22 0.07 0.19 Fig. 8 Calculated vs measured k-kt plot for Bracknell. Lightand dark-colored data points represent actual and calculated values, respectively. Solid black curve is the locus of quadratic k-kt regression. Fig. 9 The above plots show a significant improvement in the estimation of diffuse radiation from the basic k-kt model through an intermediate „k-kt, SF… to the eventual models selected as optimum. „a… Pune, „b… Fukuoka, „c… Gerona, „d… Bracknell. 116 / Vol. 128, FEBRUARY 2006 Transactions of the ASME kt k̄ m MBD n P90 P10 Q1 Q3 R2 RMSD SF SHA Skew SOLALT SOLAZM Yc Ym Yo Greek ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ hourly clearness index weighted mean of k values air mass mean bias deviation number of data points 90th percentile 10th percentile first quartile third quartile coefficient of determination root mean square deviation sunshine fraction solar hour angle skewness solar altitude solar azimuth calculated value of dependent variable mean of the dependent variable observed value of dependent variable ␴k ⫽ standard deviations of k values ␻ ⫽ solar hour angle Prefix lin ⫽ linear q, quad ⫽ quadratic Subscripts m c i i,max ⫽ ⫽ ⫽ ⫽ measured calculated ith variable maximum value of all i variables References 关1兴 Liu, B. 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A., 1990, “Diffuse Fraction Correlations,” Sol. Energy, 45共1兲, pp. 1–7. 关12兴 Vázquez, M., Ruiz, V., and Perez, R., 1991, “The Roles of Scattering, Absorption, and Air Mass on the Diffuse-to-Global Correlations,” Sol. Energy, 47共3兲, pp. 181–188. 关13兴 Perez, R., Ineichen, P., Maxwell, E., Seals, R., and Zelenka, A., 1992, “Dynamic Global to Direct Irradiance Conversion Models,” ASHRAE Trans., 98共1兲, pp. 354–369. 关14兴 Gopinathan, K. K., and Soler, A., 1996, “The Determination of Monthly Mean Hourly Diffuse Radiation on Horizontal Surfaces Using Equations Based on Hourly Clearness Index, Sunshine Fraction and Solar Elevation,” Int. J. Sol. Energy, 18, pp. 115–124. 关15兴 De Miguel, A., Bilbao, J., Aguiar, R., Kambezidis, H., and Negro, E., 2001, “Diffuse Solar Radiation Model Evaluation in the North Mediterranean Belt Area,” Sol. 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