A three-state model for the probability distribution of instantaneous solar radiation, with applications

Abstract

Based on a new set of data covering a 13-month period and a wide range of the average clearness indices, the probability distribution for instantaneous solar radiation is modeled. The data’s probability density function is shown to be capably represented by a linear superposition of three (truncated) normal distributions. The variable of the distribution is the “normalized clearness index”, κ, which is the clearness index kt divided by its value under clear sky conditions—a quantity now available for most locations on the globe. It is shown that usingκ as the variable makes the distribution independent of the air mass. The only other parameter in the distribution is the mean value, κ¯, of κ. The latter quantity fixes the parameters and the respective weights of the normal distributions, through empirically-derived fits. A physical interpretation of these findings is presented, whereby each normal distribution is associated with one of the three possible states of the atmosphere: namely (a) clear sky conditions, (b) overcast conditions, and (c) all other conditions. This interpretation predicts that the mean of the clear-sky state normal curve will be unity, and this has been confirmed for the data in hand. Also lending support to the three-state interpretation, is its ability (demonstrated in this paper) to predict the fraction of time of bright sunshine. Based on this new three-state model, a preliminary first-principle model for the mean diffuse fraction is given, as well as expressions for mean radiation on inclined surfaces and the instantaneous utilizability. The instantaneous utilizability is found to be greater than that for hourly or daily time periods and to depend in a significant way on the clear sky value of kt.