modelIrradianceVariableLocation

Information

Analytical model of the solar irradiance on an inclined plane without considering weather conditions. By default, only the direct irradiance is considered. The sun position (sun height and sun azimuth) is calculated according to the DIN 5034 algorithm, as described in [Quaschning2015], section 2.5, eqs. (2.22) to (2.28). The angle of incidence on the inclined plane is calculated by eq. (2.32) and the direct irradiance on the inclined plane by eq. (2.35).

Atmospheric attenuation

If useAirMass = false (default), the direct normal irradiance is equal to irradianceRef, independent of the sun height, i.e., the atmospheric attenuation is not considered. All transmission factors of eq. (2.17) of [Quaschning2015] are equal to one in this case. As a consequence, the irradiance of an inclined plane facing the rising or setting sun changes abruptly at sunrise and sunset.

If useAirMass = true, the direct normal irradiance depends on the relative optical air mass airMass. The air mass is calculated by the approximation of [Kasten1989]:

  airMass = 1 / (sin(sunHeight) + 0.50572 * (sunHeight/deg + 6.07995)^(-1.6364))

The extraterrestrial irradiance varies over the year due to the varying distance between sun and earth, see eq. (2.10) of [Quaschning2015]:

  irradianceExtraterrestrialActual = irradianceExtraterrestrial * (1 + 0.0334 * cos(0.9855° * dayOfYear - 2.7198°))

The parameter irradianceExtraterrestrial is the solar constant, 1360.8 W/m² according to eq. (2.9) of [Quaschning2015]. The attenuation of the direct normal irradiance is determined by the empirical relationship of [Meinel1976]:

  directIrradianceNormal = irradianceExtraterrestrialActual * 0.7^(airMass^0.678)

In this case, the parameter irradianceRef is not used. The direct irradiance smoothly approaches (almost) zero at sunrise and sunset. For the sun at the horizon, the air mass is approximately 38 and the direct normal irradiance is approximately 1.5 % of the extraterrestrial irradiance. For the sun below the horizon, the direct normal irradiance is zero.

Diffuse and ground reflected irradiance

If useDiffuse = false (default), diffuse and ground reflected irradiance are not considered. If useDiffuse = true, the diffuse irradiance on the horizontal is determined for a clear sky according to [Liu1960]:

  diffuseIrradianceHorizontal = irradianceExtraterrestrialActual * sin(sunHeight) * (0.271 - 0.294 * tau_b)
  tau_b = directIrradianceNormal / irradianceExtraterrestrialActual

The diffuse irradiance on the inclined plane is calculated by the isotropic sky model and the ground reflected irradiance by the ground reflectance rho, see [Duffie2013]:

  diffuseIrradianceInclined = diffuseIrradianceHorizontal * (1 + cos(gamma)) / 2
  reflectedIrradianceInclined = rho * (directIrradianceHorizontal + diffuseIrradianceHorizontal) * (1 - cos(gamma)) / 2

The output irradiance is the sum of the direct, diffuse and ground reflected irradiance on the inclined plane. The clear sky model is intended to be used with useAirMass = true. Clouds are not considered, i.e., the results represent a sunny day without clouds.

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