Skip to main content

​
Summary

The Extraterrestrial Irradiance model calculates the solar irradiance incident at the top of earth’s atmosphere on a surface perpendicular to the sun’s rays. PlantPredict uses a second-order Fourier series expansion based on earth’s orbital mechanics to compute the extraterrestrial as a function of day of year. This value is essential for calculations, models, and corrections throughout the prediction engine.

​
Inputs

​
Outputs

​
Detailed Description

The extraterrestrial irradiance varies throughout the year due to earth’s elliptical orbit around the sun. The earth-sun distance changes by approximately ±1.7% from its mean value, causing the solar irradiance to vary by approximately ±3.4% (since irradiance varies as the inverse square of distance).

​
Calculation Method

PlantPredict implements a Fourier series approximation of the extraterrestrial irradiance based on Spencer’s equation.

​
Step 1: Calculate Day Angle

Compute the fractional day nfn_f (days) from the UTC date-time: nf=UTCDate−January 1 of current yearn_f = \text{UTCDate} - \text{January 1 of current year} where nf=0n_f = 0 for January 1 at 00:00 UTC and nf=365n_f = 365 (or 366366 for leap years) for December 31 at 24:00 UTC. The value includes the fractional time of day. The day angle ζ\zeta (in radians) represents the fractional progress through the year: ζ=2π nf365.25\zeta = \frac{2\pi \, n_f}{365.25} The constant 365.25 accounts for the average year length including leap years.

​
Step 2: Calculate Extraterrestrial Irradiance

The extraterrestrial direct normal irradiance is computed using a second-order Fourier series expansion: DNIextra=G0×(1.00011+0.034221cos⁡(ζ)+0.00128sin⁡(ζ)+0.000719cos⁡(2ζ)+0.000077sin⁡(2ζ))\begin{aligned} DNI_{extra} = G_0 \times \Big(&1.00011 + 0.034221 \cos(\zeta) + 0.00128 \sin(\zeta) \\ &+ 0.000719 \cos(2\zeta) + 0.000077 \sin(2\zeta) \Big) \end{aligned} where:
  • G0=1367G_0 = 1367 W/m² is the solar constant (mean extraterrestrial irradiance at 1 astronomical unit = 149,597,870.7 km), per the World Meteorological Organization (WMO) standard
  • ζ\zeta is the day angle in radians
  • The Fourier coefficients are derived from the Spencer (1971) equation as presented in Duffie & Beckman, Solar Engineering of Thermal Processes

​
Physical Interpretation

The terms in the Fourier series represent:
  1. Constant term (1.00011): Slight adjustment to solar constant
  2. Annual variation (cos⁡(ζ),sin⁡(ζ))(\cos(\zeta), \sin(\zeta)): Primary effect of earth’s elliptical orbit
  3. Semi-annual variation (cos⁡(2ζ),sin⁡(2ζ))(\cos(2\zeta), \sin(2\zeta)): Higher-order orbital effects
The dominant variation is the annual term with amplitude ±3.4%, corresponding to earth’s perihelion (closest approach, ~January 3) and aphelion (farthest point, ~July 4).

​
Typical Values Throughout the Year

​
Downstream Models

The extraterrestrial irradiance DNIextraDNI_{extra} is used by:
  • Clearness Index — atmospheric transmittance calculation
  • Transposition Models (Perez, Hay-Davies) — anisotropy index
  • Models (DISC, DIRINT) — DNI/DHI estimation from GHI
  • Spectral Models — atmospheric optical depth

​
References

  • Spencer, J. W. (1971). Fourier series representation of the position of the sun. Search, 2(5), 172–176.
  • Iqbal, M. (1983). An Introduction to Solar Radiation. Academic Press. ISBN: 0-12-373750-8.
  • Duffie, J. A., & Beckman, W. A. (2013). Solar Engineering of Thermal Processes (4th ed.). John Wiley & Sons. ISBN: 978-0-470-87366-3. DOI: 10.1002/9781118671603