Modeling Solar Radiation at the Earth’s Surface

Conditioned Distributions kt

We have also analysed the PDFs and the CDFs of the kt conditioned by the kt hourly average value represented, respectively, by f(kt|kH) and F(kt|kH) expressions.

In order to obtain the 1-minute conditional probability distributions of kt, we have computed the hourly average values of solar global irradiance corresponding to the three-year period of data available at Armilla (Granada, Spain). Particularly, we have classified the data in intervals of 0.01, centered at k^ = (0.3,0.35,0.4,0.45,0.5,0.55, 0.6,0.65,0.7,0.75). The 1-minute data have been classified according to these cri­teria. These intervals are grouped into two teams. The first includes the intervals with hourly average values centered at 0.3, 0.4, 0.5, 0.6 and 0.7, including 91575 values. The second corresponds to distribution of values around 0.35, 0.45, 0.55, 0.65 and 0.75, and includes 98113 data points. The second group has been reserved for validation purposes.

Figure 3.20 shows the density probability distributions of kt for given k^ values (0.30-0.7) and their respective CDFs. These distributions show a marked unimodal­ity that contrasts with the bimodality that characterises the distributions conditioned by the optical air mass. This fact can be explained in terms of the reduced range of the kt values associated with a given k^. On the contrary, when the intervals are defined as a function of the optical air mass f(kt|ma) the distributions tend to be bimodal.

The distributions present a marked symmetry around a central value that is close to the corresponding k^ values. This feature is more marked for k^ in the range 0.45-0.65, while the distribution corresponding to k^ out of this range show a slight asymmetry. For values of k^ below 0.45, there is an asymmetry toward higher values, indicating that k^ in this range can be the result of a combination of very low and very high instantaneous values of kt. This can be related to transient conditions under partial cloud cover, with clouds close to the Sun position that, in a short time, can block the Sun or enhance the Sun direct beam due to reflections from the edges of the clouds.

Another relevant feature is the range ofkt instantaneous values associated with a given k^. Excluding the higher values of kH, the range of kt instantaneous values is rather wide. This indicates that, for these categories, we included partially-covered skies characterised by a great variability of instantaneous kt values, especially if the clouds are close to the Sun. For higher kH, the range of the kt instantaneous values is reduced, indicating that these higher hourly values are associated with cloudless-sky conditions. A rather narrow range of kt values characterises these distributions.

For the distributions corresponding to intermediate values of kt, associated with partially-cloudy skies, we observe the highest kt values. This is a result of multiple

reflections from the clouds located close to the position of Sun. Under these conditions, the reflections from the cloud edges lead to an increment of global irra - diance due to the enhancement of the diffuse component.

Considering the shape of the curves, we have approximated the probability dis­tributions using a function based on Boltzmann’s statistics. This function has been used previously for modelling 1-minute distributions conditioned by the air mass.

To account for the asymmetry of the analysed distributions, we also use the pa­rameter (P) in the above function. The modified equation reads as follows:

This function provides also reasonable fits, even for distributions that exhibit a high degree of asymmetry.

For this kind of distributions, the dependence of the coefficients kt0, X and P on k^ may be formulated by means of polynomial functions:

kt0 = -0,006 + 1,010 kH withR2 = 0.999, (3.52)

X = 11,284 + 1150,37 (kH)7205,withR2 = 0.935, (3.53)

P = 0,293 + 6,093 k"H - 15,643 (kH)2,withR2 = 0.984, (3.54)

The performance of these functions depends on kj1 and the ratio P/X, that provide information about the distribution asymmetry. Note that this ratio presents a sign change about kj1 = 0.45. The values of the parameter A must satisfy the normalisa­tion condition and the computation of the integrand:

kt f( kt| k"H)dkt

0

must return the corresponding value of kj1. The A parameter can be fitted by a polynomial function:

A = 2.81307 - 237859 kH + 143.72282 (kH)2 - 456.57093 (kH)3 +

+ 771.8157 (kH)4 - 656.1663 (kH)5 + 220.62361 (kH)6 (3.55)

for the range 0 to 1; although the experimental values that we found for kj1 are in the range 0.3 to 0.8. The associated R2 value is about 0.999 when considering the range 0.05 < kH < 0.95.