Modeling Solar Radiation at the Earth’s Surface

# Annually and Monthly Classification Analysis

The thresholds Dj and Djj have first been determined for the sites of Tahifet and Imehrou. For this purpose, the heuristic method and the statistical one has been used, Table 2.3 gathers the thresholds obtained with the two methods. We notice that the empirical and statistical thresholds are very close. Since the empirical approach is very expensive in time to build histograms and to carry out their meticulous exam­ination, we chose the statistical thresholds to classify the days of the studied sites. The obtained thresholds for all sites are illustrated by Table 2.4.

Table 2.5 gives the distribution of the probability of occurrence of daily solar irradiances for each class obtained from our classification. For Tahifet and Imehrou daily irradiances of class I have the largest probability of occurrence as compared to irradiances of the two other classes. These results confirm the pre-eminence of days with clear sky for the two sites; this is due to the climate of the south Algerian which is characterized by irradiances rarely fluctuated. However Class III (completely cov­ered sky) is preponderant for the Californian sites. Class I is also important, whereas class II has less frequency of occurrence. These results demonstrate that the two

Table 2.3 Fractal dimension thresholds obtained with the two methods: heuristic and statistic for Tahifet and Imehrou sites

 Site Dj (heuristic) Dj (statistic) Djj (heuristic) Djj (statistic) Tahifet 1.14 1.10 1.34 1.25 Imehrou 1.12 1.10 1.27 1.25

Table 2.4 Statistical Fractal dimension thresholds for all studied sites

 Site Di Dii Tahifet 1.14 1.34 Imehrou 1.12 1.27 Golden 1.35 1.49 Boulder 1.35 1.50 Palo Alto 1.19 1.37

Table 2.5 Probability of occurrence of daily solar irradiance shapes of each class

 Site Class I (%) Class II (%) Class III(%) Tahifet 58 16 26 Imehrou 62 16 21 Golden 24 22 53 Boulder 26 22 52 Palo Alto 49 17 34

studied sites are characterized by disturbed climate since the overcast sky days are preponderant at the two sites. At Palo Alto, classes I and III are pre-eminent which demonstrate that this site has a climate fairly disturbed.

On the accompanying CD, Tables of day’s class are included for each studied site.

To validate the classification results, the average of the fractal dimension <D>, of clearness index <KT > and their standard deviations o(D) and o(KT) have been computed for each class. They are summarized by Table 2.6.

These statistical properties show that our classification method leads to homo­geneous groupings of the studied days since the standard deviations of D and KT are weak compared to their averages. Indeed, in all the sites o(KT) is lower than 10% for all classes and except for Golden and Boulder we note the same thing for o(D) but only for classes I and II. The more important value of this standard devi­ation for class III (upper than 10%) is due to the fact that this class contains rainy days whose irradiance signals have a regular form thus a fractal dimension near to 1 like already explained. For example, the shape of solar daily irradiance of class III

Table 2.6 Mean value and standard deviation of D and KT for the different classes of days

 Site Golden Boulder Tahifet Imehrou Palo Alto Class I II III I II III I II III I II III I II III

Average <D> 1.15 1.43 1.47 1.17 1.43 1.48 1.03 1.24 1.42 1.02 1.19 1.40 1.06 1.27 1.46

<KT> 0.70 0.63 0.46 0.69 0.64 0.47 0.66 0.60 0.45 0.69 0.62 0.50 0.70 0.61 0.33

Standard a{D) 0.12 0.03 0.14 0.12 0.04 0.13 0.04 0.05 0.13 0.03 0.04 0.14 0.06 0.05 0.15

deviation a(KT) 0.07 0.07 0.18 0.08 0.08 0.12 0.04 0.04 0.12 0.04 0.04 0.13 0.07 0.05 0.17 (see Fig. 2.8) corresponds to a rainy day in Golden. Its fractal dimension is equal to 1.11 and its related KT is 0.47. Using D, this daily irradiance should be classified in class II. But, when using D and KT together it is categorized as class III. The fairly high values of o(D) for the class I in the two sites of Colorado is explained by the high value of Dj due to the irradiances character of these sites which is very fluctuating.

In order to better characterize the three classes obtained our statistical analysis was refined by carrying out it on a monthly scale. In Table 2.7 monthly results of the frequency of each class, averages and standard deviations of the two parameters: D and KT are presented. Table 2.7 shows that the distribution of the classes differs from a site to another.

As it can be observed from Table 2.7, Class III days have high frequency of oc­currence for the sites Golden and Boulder, reaching a maximum in May and June. Only for the month September for Golden and February for Boulder class I have higher frequency of occurrences which are 51.6% and 39.3%, respectively. How­ever, in Tahifet and Imehrou class I has higher frequency of occurrences for all the months, reaching maximum values in October and minimum in May and June.

In Palo Alto on the other hand we notice a seasonal distribution of the days. Indeed, class I presents high values in winter (January, February, November and December) where the maximum is detected in December and class III high values in summer (June-September).

These results are confirmed by the transition probabilities between two consec­utive days having the same or different classes. For the two sites of Algeria, while transition probabilities from class I to the same class were quite high (65% and 40%), all other transitions were low. However for Golden and Boulder all transition probabilities are quite close in the ranges of 5 to 20%.

 Fig. 2.8 An example of a rainy day with an enough regular shape, D = 1.11 and KT = 0.47

 Site Golden Boulder Tahifet Imehrou Palo Alto Class I II III I II III I II III I II III I II III January Freq (%) 19.4 25.8 54.8 19.4 22.6 58.0 64.5 12.9 22.6 58.1 12.9 29.0 12.9 22.6 64.5 1.20 1.44 1.39 1.16 1.45 1.43 1.02 1.25 1.47 1.02 1.17 1.42 1.18 1.25 1.49 a{D) 0.15 0.03 0.16 0.10 0.05 0.14 0.23 0.06 0.10 0.03 0.02 0.12 0.01 0.05 0.14 0.60 0.58 0.36 0.65 0.62 0.40 0.69 0.60 0.34 0.66 0.60 0.39 0.59 0.59 0.28 a(KT) 0.05 0.05 0.14 0.07 0.04 0.16 0.15 0.05 0.15 0.03 0.05 0.15 0.06 0.03 0.13 February Freq (%) 21.4 14.3 64.3 39.3 25.0 35.7 48.3 17.2 34.5 44.8 34.5 24.1 25.9 25.9 48.1 1.13 1.43 1.37 1.21 1.43 1.37 1.03 1.26 1.40 1.04 1.17 1.41 1.16 1.27 1.35 a{D) 0.16 0.04 0.13 0.13 0.04 0.16 0.23 0.05 0.07 0.04 0.03 0.19 0.01 0.05 0.10 0.66 0.65 0.41 0.68 0.66 0.42 0.69 0.62 0.49 0.66 0.64 0.52 0.63 0.60 0.25 o(Kt) 0.07 0.06 0.14 0.12 0.11 0.18 0.15 0.02 0.13 0.03 0.05 0.09 0.03 0.05 0.14 March Freq (%) 12.9 35.5 51.6 19.4 29.0 51.6 54.8 12.9 32.3 45.2 12.9 41.9 45.2 32.3 22.6 1.13 1.43 1.48 1.19 1.42 1.45 1.01 1.24 1.43 1.04 1.19 1.39 1.07 1.29 1.47 a{D) 0.10 0.03 0.16 0.11 0.04 0.19 0.01 0.07 0.18 0.04 0.06 0.15 0.05 0.06 0.15 0.69 0.66 0.43 0.72 0.70 0.50 0.65 0.61 0.39 0.67 0.57 0.41 0.68 0.62 0.38 o(Kt) 0.02 0.04 0.23 0.09 0.08 0.21 0.04 0.03 0.11 0.04 0.04 0.12 0.04 0.04 0.12 April Freq (%) 13.3 30.0 56.7 16.7 30.0 53.3 60.0 16.7 23.3 76.7 0.00 23.3 20.0 13.3 66.7 1.19 1.42 1.48 1.17 1.41 1.49 1.03 1.19 1.46 1.02 1.44 1.08 1.28 1.48 a{D) 0.18 0.02 0.10 0.15 0.04 0.11 0.04 0.04 0.08 0.03 0.11 0.06 0.02 0.14 0.76 0.64 0.49 0.74 0.66 0.51 0.67 0.63 0.51 0.70 0.53 0.68 0.65 0.46 a(KT) 0.07 0.08 0.19 0.10 0.09 0.18 0.03 0.01 0.10 0.04 0.17 0.07 0.09 0.13
 Table 2.7 Monthly characteristics of each class obtained for the various sites

 2 Fractal Classification of Typical Meteorological Days from Global Solar Irradiance

## Modeling Solar Radiation at the Earth’s Surface

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