A theoretical framework for Ångström equation. Its virtues and liabilities in solar energy estimation

• A self-consistent derivation of the Ångström equation is carried out.
• The theoretical assessment on its performance is well supported by the measured data.
• The variability in cloud transmittance is a major source of uncertainty for estimates.
• The degradation in time and space of the empirical equations calibration is assessed.

The relation between solar irradiation and sunshine duration was investigated from the very beginning of solar radiation measurements. Many studies were devoted to this topic aiming to include the complex influence of clouds on solar irradiation into equations. This study is focused on the linear relationship between the clear sky index and the relative sunshine proposed by the pioneering work of Ångström. A full semi-empirical derivation of the equation, highlighting its virtues and liabilities, is presented. Specific Ångström – type equations for beam and diffuse solar irradiation were derived separately. The sum of the two components recovers the traditional form of the Ångström equation. The physical meaning of the Ångström parameter, as the average of the clouds transmittance, emerges naturally. The theoretical results on the Ångström equation performance are well supported by the tests against measured data. Using long-term records of global solar irradiation and sunshine duration from thirteen European radiometric stations, the influence of the Ångström constraint (slope equals one minus intercept) on the accuracy of the estimates is analyzed. Another focus is on the assessment of the degradation of the equation calibration. The temporal variability in cloud transmittance (both long-term trend and fluctuations) is a major source of uncertainty for Ångström equation estimates.