Direct-Beam Solar Irradiance Approximation with Trigonometric Series

Direct-Beam Solar Irradiance (DBSI) is a location-specific periodic function affected by three major factors: the Sun's energy output (or Sun's luminosity), the Earth's orbital motion, and the Earth's atmosphere. Quantitatively, Earth's atmosphere affects the incoming radiant energy the most and on the shortest time scale. Daily between 25% (for low Sun's zenith distances) to over 80% (for Sun close to the horizon) of incoming solar energy is stopped or altered by the atmosphere. The Earth's orbital motion causes approximately ±4% annual variation in the incoming radiant energy, while the Sun's luminosity varies only ±0.3% over an 11-year-long cycle. In this work an expansion of the DBSI into a fast-converging trigonometric series is presented. The frequencies of individual harmonic terms are expressed as linear combinations of two fundamental frequencies: 2π/24h frequency of the Sun’s local hour angle and ~2π/365day frequency of the Sun’s ecliptic longitude. The amplitudes and phases of individual harmonic terms depend on Earth’s obliquity and orbital parameters – the semi-major axis, eccentricity, and the longitude of perihelion. The general form of the DBSI approximation is given in Equation (1) where coefficients L_𝑙𝑜 and S_{𝑙𝑜,𝑡𝑜} are the trigonometric series that depend on the ecliptic Sun’s longitude l_o and local hour angle t_o. ℒ_𝑡,𝜆 is the Sun’s luminosity, and Θ_𝑡,𝜆,φ is the atmospheric transmissivity (which, for simplified applications, can be replaced by the Earth-atmosphere albedo). This expansion of DBSI is different from similar Fourier series expansions and has the advantage of being fully parameterized in terms of orbital elements, obliquity, and geographic coordinates. Application of this method eliminates the need for the pre-calculated, location-specific tables of coefficients required by the Fourier series. The trigonometric series have global coverage with arbitrary accuracy in capturing the seasonal effects on DBSI. The accuracy of truncated series, including the third order harmonic term, varies annually only within ±0.4 W/m² for any latitude (Figure 1). The accuracy of series that include the second order term varies in range ±1.4 W/m².

The most important properties of the introduced DBSI approximating series are time continuity, global coverage, explicit dependence of amplitudes on Earth’s obliquity and orbital elements, and arbitrarily high accuracy. These properties of the DBSI approximations make them suitable for climate studies such as DBSI sensitivity to changes in orbital parameters and obliquity, as well as various engineering application such as photovoltaic energy production and building energy efficiency.

We present a detailed derivation of the DBSI approximating series, discuss their domain of applicability and provide benchmarking examples against the number of currently used approaches [1- 4].

References

[1] Lumb, F. E., 1964., "The Influence of Clouds on Hourly Amount of Total Solar Irradiance at the Sea Surface Level", Quart., J. Roy. Meteor. Soc. 90, 43-56.

[2] Reed, R. K., 1977 “On Estimating Insolation Over the Ocean” J. Phys. Oceanography, 7, 482-485.

[3] Attwater, M. A., Ball, J. T. 1980., "A Surface Radiation Model for Cloudy Atmospheres", Monthly Weather Review, 109, 878-888.

[4] Bird, R., 1984., "A Simple Solar Spectral Model for Direct Normal and Diffuse Horizontal and Irradiance", Solar Energy, 32, 461-471.