Modeling Solar Radiation at the Earth’s Surface
Fractal Treatment of Solar Irradiances
Figure 2.5 presents two examples of the log-log lines permitting the estimation of the fractal dimension of irradiance curves. This figure shows that the log-log points are grouped around the fitting line which demonstrates the self affinity of the studied solar irradiances.
The fractal dimensions obtained from the slopes of the log-log lines for all sites are given in the accompanying CD. Figure 2.6 gives representative examples for the daily irradiation values corresponding to different fractal dimensions from three classes. As can be observed there is good correspondences between the shapes of the signals and the corresponding fractal dimensions.
Figure 2.7 gives the annual evolution of the monthly average of D for the studied sites. This figure shows clearly that D fluctuates.
In order to quantify this fluctuation we calculated the annual average <D> of the fractal dimension and the corresponding standard deviation a which are tabulated in Table 2.2. These values suggest that the solar irradiances of Tahifet and Boulder exhibit the similar fluctuations (<D> = 1.16 for Tahifet and 1.13 for Imehrou).
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Fig. 2.5 Two examples (Golden site) of log-log plots fitted by the least-squares estimation with their slopes which represent the estimated fractal dimension |
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Fig. 2.6 Typical daily irradiance values for the three classifications and fractal dimensions for the sites under consideration a) Golden, b) Boulder, c) Tahifet, d) Imehrou, e) Palo Alto
This is also observed for Golden and Boulder (<D> = 1.38 for Golden and 1.39 for Boulder). To compare the degree of fluctuation of the solar irradiances of the different sites we can refer to the values of <D>. Hence, the two sites of Colorado are fluctuating, those of the Algerian sites fluctuate slightly, they are practically regular, and in Palo Alto irradiances are fairly fluctuating. However, the analysis of D month by month permits the detection of the months where the fluctuations of the irradiances are most intense - June and December for Tahifet, March and June for
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Fig. 2.7 Annual variation of the monthly means of the estimated fractal dimension D, the straight solid line represents the annual mean of the fractal dimension <D> |
Table 2.2 Annual averages <D> and standard deviations о of the estimated fractal dimensions
|
Site |
<D> |
о(%) |
|
Tahifet |
1.16 |
19 |
|
Imehrou |
1.13 |
17 |
|
Golden |
1.38 |
18 |
|
Boulder |
1.39 |
18 |
|
Palo alto |
1.23 |
20 |
Imehrou, May and June for Golden and Boulder, January and December for Palo Alto - and those where these irradiances are very regular - October for the two sites Tahifet and Imehrou, September for Golden and Boulder, June for Palo Alto. These informations are very useful to refine the sizing of photovoltaic systems. Indeed, the anomalies in the operating of the photovoltaic systems installed in these sites appear during these months. There is for example for Tahifet excess of energy in October and storage is requested in June and December much more than in other months.
