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4 The model proposes a symmetric behavior with respect to Equator so the latitude dependence appears in absolute value. For validating the model, measurements of solar global and diffuse irradiance at a oneminute basis in Badajoz (Spain) were performed. Hourly averages of these measurements were calculated and the Spencer's model was applied to those data. Additionally, the regression coefficients were calculated and compared to those proposed by Spencer for our latitude. Finally, the new approach has been validated with an independent data set from the same radiometric station. Statistical parameters such as root mean square error (RMSE), mean percentage error (MPE) and mean absolute bias error (MABE) have been used in order to analyze the goodness of model results. These statistics are defined as: 100 1,,, 7 100,,, 8, 100,,, 9, where N is the total number of measurements,, is the ith estimate and, is the ithmeasurement. The statistic RMSE is a measure of the differences between the model and experimental measurements, MPE gives an estimate of the percentage by which a model underestimates or overestimates the experimental results, a positive value indicates that the model overestimates the experimental data while a negative value indicates underestimation. MABE is a measure of the goodness of the fit. Small values of these statistics represent a better fit of the model. 4. Results In this section we will discuss the results obtained applying Spencer s model to the data obtained in our station. Analysis is performed in two ways: (a) analysing the results obtained using the original coefficients propounded by Spencer for our latitude and (b) obtaining new values for the regression coefficients of Spencer model expressions and comparing with the previous ones. In the first step, the parameters a, b, c and d have been calculated for our latitude by the expressions (6) and then, values have been estimated by expression (5). In Fig. 1 it can be observed how the original model (orange line) reflects the general trend of the data but tends to be overestimated with respect to experimental data, especially for low values of. kd Fig. 1. Original model and new approach. This is confirmed by the positive value of the statistic MPE, 17.3%, while for MABE and RMSE were obtained 32.1% and 12.6% respectively. The high value of statistical MABE highlights the wide dispersion of experimental data on the model. This dispersion is mainly due to the great diversity of possible situations that may lead to the same value of. The clearness index is a measure of the attenuation of radiation by the atmosphere. However, such attenuation may be due to different causes such as cloud cover or the action of the aerosols and gases on the radiation. These factors have different influence on the radiation dispersion so that the proportion of diffuse can take very different values for the same radiation attenuation. In a second step values for a, b, c and d coefficients of Spencer model were obtained by kt modelo original nuevo ajuste datos experimentales Opt. Pura Apl. 45 (1) 5 9 (2012) 8 Sociedad Española de Óptica

5 regression analysis. These results are shown in Table I with those propounded in the original model calculated with Eq. (6). The relative differences between them are also shown. These coefficients correspond to the functional form of the model in the range of values below 0.7, where the data dispersion is higher. Other possible causes of the differences between the coefficients given by the model and those obtained by the new approach could be the wide range of latitudes in which data were taken for the original model development or the different instruments used in the measurement stations. The estimates with the new coefficients (red line) have been represented in Fig. 1. This graph shows the improvement of this model compared to the original. The differences between them are smaller when the value of clearness index,, increases. A quantitative comparison of both models can be done with the statistics RMSE, MPE and MABE, listed in Table II. In this table it is observed as statistical values for the new approach are slightly better than those obtained for the original model, but the new approach overestimates de experimental data. Table I Spencer model coefficients. Coefficient Original New Relative model approach difference a % b % c % d % This new approach has been validated with an independent data set as described in section 2. In this validation the values obtained for the statistical RMSE, MPE and MABE were 12.4%, 15.8% and 31.8% respectively. These values are similar to those obtained for the approach. 5. Conclusions The model proposed by Spencer for the estimation of hourly portion of diffuse solar irradiance from clearness index has been implemented and tested. In addition, a new approach of this model has been obtained, by fitting the functional form given by the model. This functional form reflects the general behavior of the data. Both, the original model and the new approach overestimate the experimental data although the latter does to a lesser extent. The coefficients a, b, c and d estimated from the original model for our station and the new ones calculated by regression analysis, have relative differences of 12%, 16%, 20% and 0.3% respectively. The statistics RMSE, MPE and MABE, obtained with the new approach are lower than those found using the original model. Acknowledgements This work was supported by the research project CGL C03 02/CLI (Ministerio de Ciencia e Innovación). Original model New approach Table II Statistics for original and approach models. RMSE (%) MPE (%) MABE (%) Opt. Pura Apl. 45 (1) 5 9 (2012) 9 Sociedad Española de Óptica

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