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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.

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Inputs

NameSymbolUnitsDescription
UTC Date-Time—datetimeUniversal Coordinated Time
Day of YearnndaysFractional days since January 1 (0 at midnight on Jan 1, 365 or 366 at end of Dec 31)

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Outputs

NameSymbolUnitsDescription
Extraterrestrial Direct Normal IrradianceDNIextraDNI_{extra}W/m²Solar irradiance at top of atmosphere perpendicular to sun’s rays

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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).

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Calculation Method

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

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Step 1: Calculate Day Angle

First, compute the fractional day of year nn from the UTC date-time: n=UTCDate−January 1 of current yearn = \text{UTCDate} - \text{January 1 of current year} where n=0n = 0 for January 1 at 00:00 UTC and n=365n = 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=2πn365.25\zeta = \frac{2\pi n}{365.25} The constant 365.25 accounts for the average year length including leap years.

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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

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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).

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Typical Values Throughout the Year

DateApproximate DayDNIextraDNI_{extra} (W/m²)% Deviation
January 3 (perihelion)3~1412+3.3%
April 393~1362-0.4%
July 4 (aphelion)185~1322-3.3%
October 4277~1368+0.1%

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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

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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.