Sky Radiance Model[LINK]
In EnergyPlus the calculation of diffuse solar radiation from the sky incident on an exterior surface takes into account the anisotropic radiance distribution of the sky. For this distribution, the diffuse sky irradiance on a surface is given by:
AnisoSkyMultipliersurface⋅DiffuseSolarIrradiance
Where
Diffuse Solar Irradiance is the diffuse solar irradiance from the sky on the ground.
surface is the surface being analyzed.
AnisoSkyMultiplier is determined by surface orientation and sky radiance distribution, and accounts for the effects of shading of sky diffuse radiation by shadowing surfaces such as overhangs. It does not account for reflection of sky diffuse radiation from shadowing surfaces.
The sky radiance distribution is based on an empirical model based on radiance measurements of real skies, as described in Perez et al., 1990. In this model the radiance of the sky is determined by three distributions that are superimposed (see Figure 1)
An isotropic distribution that covers the entire sky dome;
A circumsolar brightening centered at the position of the sun;
A horizon brightening.
The proportions of these distributions depend on the sky condition, which is characterized by two quantities, clearness factor and brightness factor, defined below, which are determined from sun position and solar quantities from the weather file.
The circumsolar brightening is assumed to be concentrated at a point source at the center of the sun although this region actually begins at the periphery of the solar disk and falls off in intensity with increasing angular distance from the periphery.
The horizon brightening is assumed to be a linear source at the horizon and to be independent of azimuth. In actuality, for clear skies, the horizon brightening is highest at the horizon and decreases in intensity away from the horizon. For overcast skies the horizon brightening has a negative value since for such skies the sky radiance increases rather than decreases away from the horizon.
Variables in Anisotropic Sky Model and Shadowing of Sky Diffuse Radiation
Isky 
Solar irradiance on surface from sky 
W/m2

 
Ihorizon 
Solar irradiance on surface from sky horizon 
W/m2

 
Idome 
Solar irradiance on surface from sky dome 
W/m2

 
Icircumsolar 
Solar irradiance on surface from circumsolar region 
W/m2

 
Ih 
Diffuse horizontal solar irradiance 
W/m2

 
S 
Surface tilt 
radians 
Surface(SurfNum)%Tilt*DegToRadians 
a, b 
intermediate variables 
 
 
F1, F2

Circumsolar and horizon brightening coefficients 
 
F1, F2 
α 
Incidence angle of sun on surface 
radians 
IncAng 
Z 
Solar zenith angle 
radians 
ZenithAng 
Δ 
Sky brightness factor 
 
Delta 
ε 
Sky clearness factor 
 
Epsilon 
m 
relative optical air mass 
 
AirMass 
Io 
Extraterrestrial solar irradiance 
W/m2

 
I 
Direct normal solar irradiance 
W/m2

Material%Thickness 
κ 
constant = 1.041 for Z in radians 
radians 
 
Fij 
Brightening coefficient factors 
 
F11R, F12R, etc. 
Rcircumsolar 
Shadowing factor for circumsolar radiation 
 
SunLitFrac 
Rdome 
Shadowing factor for sky dome radiation 
 
DifShdgRatioIsoSky 
Rhorizon 
Shadowing factor for horizon radiation 
 
DifShdgRatioHoriz 
E 
Sky radiance 
W/m2

 
θ 
Azimuth angle of point in sky 
radians 
Theta 
φ 
Altitude angle of point in sky 
radians 
Phi 
Ii 
Irradiance on surface from a horizon element 
W/m2

 
Iij 
Irradiance on surface from a sky dome element 
W/m2

 
SF 
Sunlit fraction 
 
FracIlluminated 
I′ 
Sky solar irradiance on surface with shadowing 
W/m2

 
Sky Diffuse Solar Radiation on a Tilted Surface[LINK]
The following calculations are done in subroutine AnisoSkyViewFactors in the SolarShading module.
In the absence of shadowing, the sky formulation described above gives the following expression for sky diffuse irradiance, Isky, on a tilted surface:
Isky=Ihorizon+Idome+Icircumsolar
where
Ihorizon=irradiance on surface from sky horizon=IhF2sinSIdome=irradiance on surface from sky dome=Ih(1−F1)(1+cosS)/2Icircumsolar=irradiance on surface from circumsolar region=IhF1a/b
AnisoSkyMult is then Isky/DifSolarRad.
In the above equations:
Ihorizon = horizontal solar irradiance (W/m2)
S = surface tilt (radians)
a = max(0,cosα)
b = max(0.087, cosZ)
F1 = circumsolar brightening coefficient
F2 = horizon brightening coefficient
where
α = incidence angle of sun on the surface (radians)
Z = solar zenith angle (radians).
The brightening coefficients are a function of sky conditions; they are given by:
F1=F11(ε)+F12(ε)Δ+F13(ε)ZF2=F21(ε)+F22(ε)Δ+F23(ε)Z
Here the sky brightness factor is:
Δ=Ihm/Io
where
m = relative optical air mass
Io = extraterrestrial irradiance (taken to have an average annual value of 1353 W/m2);
and the sky clearness factor is
ε=(Ih+I)/Ih+κZ31+κZ3
where
I = direct normal solar irradiance
κ = 1.041 for Z in radians
The factors Fij are shown in the following table. The Fij values in this table were provided by R. Perez, private communication, 5/21/99. These values have higher precision than those listed in Table 6 of Perez et al., 1990.
Fij Factors as a Function of Sky Clearness Range
F11 
0.0083117 
0.1299457 
0.3296958 
0.5682053 
0.8730280 
1.1326077 
1.0601591 
0.6777470 
F12 
0.5877285 
0.6825954 
0.4868735 
0.1874525 
0.3920403 
1.2367284 
1.5999137 
0.3272588 
F13 
0.0620636 
0.1513752 
0.2210958 
0.2951290 
0.3616149 
0.4118494 
0.3589221 
0.2504286 
F21 
0.0596012 
0.0189325 
0.0554140 
0.1088631 
0.2255647 
0.2877813 
0.2642124 
0.1561313 
F22 
0.0721249 
0.0659650 
0.0639588 
0.1519229 
0.4620442 
0.8230357 
1.1272340 
1.3765031 
F23 
0.0220216 
0.0288748 
0.0260542 
0.0139754 
0.0012448 
0.0558651 
0.1310694 
0.2506212 
Shadowing of Sky Diffuse Solar Radiation[LINK]
Sky diffuse solar shadowing on an exterior surface is calculated as follows in subroutine SkyDifSolarShading in the SolarShading module. The sky is assumed to be a superposition of the three Perez sky comp1onents described above.
For the horizon source the following ratio is calculated by dividing the horizon line into 24 intervals of equal length:
Rhoriz=Irradiance from horizon with obstructionsIrradiance from horizon without obstructions=24∑i=1IiSFi24∑i=1Ii
where Ii is the unobstructed irradiance on the surface from the ith interval, SFi is the sunlit fraction from radiation coming from the ith interval, and the sums are over intervals whose center lies in front of the surface. SFi is calculated using the beam solar shadowing method as though the sun were located at the ith horizon point. Here:
Ii=E(θi)dθcosαi
where
E (θi) = radiance of horizon band (independent of θ)
dθ = 2π/24 = azimuthal extent of horizon interval (radians)
θi = 0O, 15O, … , 345O
αi = incidence angle on surface of radiation from θi
The corresponding ratio for the isotropic sky dome is given by:
Rdome=Irradiance from dome with obstructionsIrradiance from dome without obstructions=24∑i=16∑j=1IijSFij24∑i=16∑j=1Iij
where (i,j) is a grid of 144 points (6 in altitude by 24 in azimuth) covering the sky dome, Iij is the unobstructed irradiance on the surface from the sky element at the ijth point, SFij is the sunlit fraction for radiation coming from the ijth element, and the sum is over points lying in front of the surface. Here:
Iij=E(θi,ϕj)cosϕjdθdϕcosαij
where
E (θi,ϕj) = sky radiance (independent of θ and ϕ for isotropic dome)
dθ = 2π/24 = azimuthal extent of sky element (radians)
dϕ = (π/2)/6 = altitude extent of sky element (radians)
θi = 0O, 15O, … , 345O
ϕj = 7.5O, 22.5O, … , 82.5O
αj = incidence angle on surface of radiation from (θi,ϕj)
Because the circumsolar region is assumed to be concentrated at the solar disk, the circumsolar ratio is:
Rcircumsolar=Irradiance from circumsolar region with obstructionsIrradiance from circumsolar without obstructions=SFsun
where SFsun is the beam sunlit fraction. The total sky diffuse irradiance on the surface with shadowing is then:
I′sky=RhorizonIhorizon+RdomeIdome+RcircumsolarIcircumsolar
If the Sky Diffuse Modeling Algorithm (ShadowCalculation object) is set to SimpleSkyDiffuseModeling, then Rhorizon and Rdome are calculated once for each surface since they are independent of sun position, or else if the Sky Diffuse Modeling Algorithm is set to DetailedSkyDiffuseModeling, then Rhorizon and Rdome are calculated every timestep for each surface.
With shadowing we then have:
AnisoSkyMult = I’sky/DifSolarRad.
Shadowing of Sky LongWave Radiation[LINK]
EnergyPlus calculates the sky longwave radiation incident on exterior surfaces assuming that the sky longwave radiance distribution is isotropic. If obstructions such as overhangs are present the sky longwave incident on a surface is multiplied by the isotropic shading factor, Rdome, described above. The longwave radiation from these obstructions is added to the longwave radiation from the ground; in this calculation both obstructions and ground are assumed to be at the outside air temperature and to have an emissivity of 0.9.
Sky Radiance Model[LINK]
In EnergyPlus the calculation of diffuse solar radiation from the sky incident on an exterior surface takes into account the anisotropic radiance distribution of the sky. For this distribution, the diffuse sky irradiance on a surface is given by:
AnisoSkyMultipliersurface⋅DiffuseSolarIrradiance
Where
Diffuse Solar Irradiance is the diffuse solar irradiance from the sky on the ground.
surface is the surface being analyzed.
AnisoSkyMultiplier is determined by surface orientation and sky radiance distribution, and accounts for the effects of shading of sky diffuse radiation by shadowing surfaces such as overhangs. It does not account for reflection of sky diffuse radiation from shadowing surfaces.
The sky radiance distribution is based on an empirical model based on radiance measurements of real skies, as described in Perez et al., 1990. In this model the radiance of the sky is determined by three distributions that are superimposed (see Figure 1)
An isotropic distribution that covers the entire sky dome;
A circumsolar brightening centered at the position of the sun;
A horizon brightening.
Schematic view of sky showing solar radiance distribution as a superposition of three components: dome with isotropic radiance, circumsolar brightening represented as a point source at the sun, and horizon brightening represented as a line source at the horizon. [fig:schematicviewofskyshowingsolarradiance]
The proportions of these distributions depend on the sky condition, which is characterized by two quantities, clearness factor and brightness factor, defined below, which are determined from sun position and solar quantities from the weather file.
The circumsolar brightening is assumed to be concentrated at a point source at the center of the sun although this region actually begins at the periphery of the solar disk and falls off in intensity with increasing angular distance from the periphery.
The horizon brightening is assumed to be a linear source at the horizon and to be independent of azimuth. In actuality, for clear skies, the horizon brightening is highest at the horizon and decreases in intensity away from the horizon. For overcast skies the horizon brightening has a negative value since for such skies the sky radiance increases rather than decreases away from the horizon.
Sky Diffuse Solar Radiation on a Tilted Surface[LINK]
The following calculations are done in subroutine AnisoSkyViewFactors in the SolarShading module.
In the absence of shadowing, the sky formulation described above gives the following expression for sky diffuse irradiance, Isky, on a tilted surface:
Isky=Ihorizon+Idome+Icircumsolar
where
Ihorizon=irradiance on surface from sky horizon=IhF2sinSIdome=irradiance on surface from sky dome=Ih(1−F1)(1+cosS)/2Icircumsolar=irradiance on surface from circumsolar region=IhF1a/b
AnisoSkyMult is then Isky/DifSolarRad.
In the above equations:
Ihorizon = horizontal solar irradiance (W/m2)
S = surface tilt (radians)
a = max(0,cosα)
b = max(0.087, cosZ)
F1 = circumsolar brightening coefficient
F2 = horizon brightening coefficient
where
α = incidence angle of sun on the surface (radians)
Z = solar zenith angle (radians).
The brightening coefficients are a function of sky conditions; they are given by:
F1=F11(ε)+F12(ε)Δ+F13(ε)ZF2=F21(ε)+F22(ε)Δ+F23(ε)Z
Here the sky brightness factor is:
Δ=Ihm/Io
where
m = relative optical air mass
Io = extraterrestrial irradiance (taken to have an average annual value of 1353 W/m2);
and the sky clearness factor is
ε=(Ih+I)/Ih+κZ31+κZ3
where
I = direct normal solar irradiance
κ = 1.041 for Z in radians
The factors Fij are shown in the following table. The Fij values in this table were provided by R. Perez, private communication, 5/21/99. These values have higher precision than those listed in Table 6 of Perez et al., 1990.
Shadowing of Sky Diffuse Solar Radiation[LINK]
Sky diffuse solar shadowing on an exterior surface is calculated as follows in subroutine SkyDifSolarShading in the SolarShading module. The sky is assumed to be a superposition of the three Perez sky comp1onents described above.
For the horizon source the following ratio is calculated by dividing the horizon line into 24 intervals of equal length:
Rhoriz=Irradiance from horizon with obstructionsIrradiance from horizon without obstructions=24∑i=1IiSFi24∑i=1Ii
where Ii is the unobstructed irradiance on the surface from the ith interval, SFi is the sunlit fraction from radiation coming from the ith interval, and the sums are over intervals whose center lies in front of the surface. SFi is calculated using the beam solar shadowing method as though the sun were located at the ith horizon point. Here:
Ii=E(θi)dθcosαi
where
E (θi) = radiance of horizon band (independent of θ)
dθ = 2π/24 = azimuthal extent of horizon interval (radians)
θi = 0O, 15O, … , 345O
αi = incidence angle on surface of radiation from θi
The corresponding ratio for the isotropic sky dome is given by:
Rdome=Irradiance from dome with obstructionsIrradiance from dome without obstructions=24∑i=16∑j=1IijSFij24∑i=16∑j=1Iij
where (i,j) is a grid of 144 points (6 in altitude by 24 in azimuth) covering the sky dome, Iij is the unobstructed irradiance on the surface from the sky element at the ijth point, SFij is the sunlit fraction for radiation coming from the ijth element, and the sum is over points lying in front of the surface. Here:
Iij=E(θi,ϕj)cosϕjdθdϕcosαij
where
E (θi,ϕj) = sky radiance (independent of θ and ϕ for isotropic dome)
dθ = 2π/24 = azimuthal extent of sky element (radians)
dϕ = (π/2)/6 = altitude extent of sky element (radians)
θi = 0O, 15O, … , 345O
ϕj = 7.5O, 22.5O, … , 82.5O
αj = incidence angle on surface of radiation from (θi,ϕj)
Because the circumsolar region is assumed to be concentrated at the solar disk, the circumsolar ratio is:
Rcircumsolar=Irradiance from circumsolar region with obstructionsIrradiance from circumsolar without obstructions=SFsun
where SFsun is the beam sunlit fraction. The total sky diffuse irradiance on the surface with shadowing is then:
I′sky=RhorizonIhorizon+RdomeIdome+RcircumsolarIcircumsolar
If the Sky Diffuse Modeling Algorithm (ShadowCalculation object) is set to SimpleSkyDiffuseModeling, then Rhorizon and Rdome are calculated once for each surface since they are independent of sun position, or else if the Sky Diffuse Modeling Algorithm is set to DetailedSkyDiffuseModeling, then Rhorizon and Rdome are calculated every timestep for each surface.
With shadowing we then have:
AnisoSkyMult = I’sky/DifSolarRad.
Shadowing of Sky LongWave Radiation[LINK]
EnergyPlus calculates the sky longwave radiation incident on exterior surfaces assuming that the sky longwave radiance distribution is isotropic. If obstructions such as overhangs are present the sky longwave incident on a surface is multiplied by the isotropic shading factor, Rdome, described above. The longwave radiation from these obstructions is added to the longwave radiation from the ground; in this calculation both obstructions and ground are assumed to be at the outside air temperature and to have an emissivity of 0.9.
Documentation content copyright © 19962021 The Board of Trustees of the University of Illinois and the Regents of the University of California through the Ernest Orlando Lawrence Berkeley National Laboratory. All rights reserved. EnergyPlus is a trademark of the US Department of Energy.
This documentation is made available under the EnergyPlus Open Source License v1.0.