Modeling Solar Radiation at the Earth’s Surface

Fractal Classification of Solar Irradiance Methodology

Our method classification uses the fractal dimension as a basic criterion to achieve the classification of the solar irradiance and to yield different types of days, i. e., clear sky day, covered sky day, a cloudy day, etc. Our research reveals that some daily solar irradiance signals have the same fractal dimension but corresponding to days with different weather conditions. Indeed, a uniformly cloudy day and a sunny one have regular irradiance shapes and practically the same value for D but have daily different clearness indexes. That is why the daily clearness index KT is calculated along with D as a second criterion in the categorization algorithm which allows sorting daily irradiances into three classes according to the following classification:

Class I: Clear sky day

1 < D < Dj and KT > (Kt )J

Class II: Partially cloudy sky

Dj < D < Djj and Kt > (Kt )j

Class III: Completely cloudy sky

D > Djj or D < Djj and KT < (Kt )j

Dj, Djj, are the thresholds for D and (Kt )j is the one for KT for the different classes.

The thresholds for D and KT are new parameters to be determined in order to achieve the classification of the irradiances. The value 0.5 is chosen for (Kt )j; this value permits to distinguish the covered sky day class from the one of clear sky day. Indeed, experimental results reveals that for some days of class III (covered sky day), the fractal dimension D is closer to 1, this is due to the fact that these days are so covered that the corresponding irradiance curve is regular but the clearness index is very low (lower than 0.5).

To determine the thresholds of the fractal dimension Dj and Djj we first used a heuristic approach then a statistical one. The heuristic approach consists in analyz­ing all daily solar irradiances shapes and their corresponding fractal dimension. For each day of the year the histograms of the irradiance signals are constructed. These histograms are built by class of 100W/m2. By observing their various forms, i. e. preponderance of low or high frequencies, we noted that there were three kinds of histograms (Maafi and Harrouni 2003):

• Histograms in the shape of J

• Histograms in the shape of U

• Histograms in the shape of L

By identifying the relations of classification established above with these three types of histograms, we can determine the D thresholds correspondent to the three classes. (An example of these histograms is given for Tahifet on the accompanying CD).

The statistical method is based on the cumulative distribution function (CDF) FX (x). This latter describes the probability distribution of a real-valued random vari­able X. For every real number x, the CDF of X is the probability that the random variable X takes on a value less than or equal to x. Thus, the two thresholds of D correspond respectively to the fractal dimension whose the cumulative distribution functionFX(x) are:

Fx(x) = maxFW>-minF<x» and (2.24)

F (x) = 2 (max(Fx(x)) - mm(FX (x)))

Data Bank

The experimental database contains global irradiance data measured at five sites of different climates. Two south Algerian sites: Tahifet (Tamanrasset) and Imehrou (Illizi), two sites of Colorado: Golden and Boulder and the last site is Palo Alto lo­cated in California. The geographical coordinates of these sites are given in Table 2.1.

Algerian sites data are recorded from the operation of two stand-alone photo­voltaic power installations during 1992-year on a 10°-tilted surface with a time step of 10 minutes. These systems have been installed by the National Company from

Electricity and Gaz (SONELGAZ). For Colorado sites the irradiance data have been collected during the year 2003 on a horizontal surface. These data are provided by MIDC (Measurement and Instrumentation Data Center) (MIDC, 2007). Data at Palo Alto have been recorded from the operation of one grid-connected system during 2003-year on a 30°-tilted surface with a time step of 15 minutes. This PV system was installed in May 2000 by CPAU (City of Palo Alto utilities) (CPAU, 2007).

By integrating the measured irradiances we determined the daily irradiation. Then, we calculated the daily clearness index KT using Eq. (2.6). The measured daily global and extraterrestrial irradiation together with daily clearness index for all studied sites are included on the accompanying CD.

H0 is calculated by using Eq. (2.7) for irradiations received on horizontal surface (Boulder and Golden) and Eq. (2.9) for irradiations on tilted plane (Tahifet, Imehrou and Palo Alto).