TP-08: Solar Gains and Shading

Overview

Solar gains represent the heat energy entering a dwelling through its building envelope as a result of incident solar radiation. The calculation has two distinct paths: opaque elements absorb solar radiation at their external surfaces, while transparent elements transmit radiation through the glazing into the zone interior. Both paths account for the reduction of irradiance by shading objects.

The solar geometry and irradiance calculations follow BS EN ISO 52010-1:2017. Shading reduction factors follow the methodology in BS EN ISO 52016-1:2017, Annex F. The Perez anisotropic sky model is used to decompose diffuse radiation into sky, circumsolar, and horizon brightening components. Transmitted solar gains through glazing are calculated per BS EN ISO 52016-1:2017, section 6.5.7.

Inputs

Solar geometry and irradiance

Opaque element parameters

Transparent element parameters

Shading objects

Calculation

Solar Geometry

The solar position is calculated for each hour of the year using the site latitude, longitude, and time zone, following BS EN ISO 52010-1:2017.

Earth Orbit Deviation

$R_{dc}=\frac{360}{365}\times n_{day}$

Where $n_{day}$ is the day of the year (1 to 365 or 366).

Solar Declination

The solar declination ${\delta }_{sol}$ is derived from the earth orbit deviation:

${\delta }_{sol}=0.33281-22.984\cos (R_{dc})-0.3499\cos (2R_{dc})-0.1398\cos (3R_{dc})+3.7872\sin (R_{dc})+0.03205\sin (2R_{dc})+0.07187\sin (3R_{dc})$

Where $R_{dc}$ is in radians for the trigonometric functions. The result is in degrees.

Equation of Time

The equation of time $t_{eq}$ (in minutes) corrects for the eccentricity of the Earth's orbit and axial tilt. It is a piecewise function of the day of the year $n_{day}$:

Time Shift

$t_{shift}=t_z-\frac{\lambda }{15}$

Where $t_z$ is the time zone offset (hours) and $\lambda$ is the longitude (degrees).

Solar Time

$t_{sol}=n_{hour}-\frac{t_{eq}}{60}-t_{shift}$

Where $n_{hour}$ is the clock hour of the day (1-based).

Solar Hour Angle

$\omega =\frac{180}{12}\times (12.5-t_{sol})$

The result is clamped to the range $[-180,+180]$ degrees. The 12.5 term reflects that solar radiation is measured at the midpoint of each hour section.

Solar Altitude

${\alpha }_{sol}=\arcsin (\sin ({\delta }_{sol})\sin (\varphi )+\cos ({\delta }_{sol})\cos (\varphi )\cos (\omega ))$

Where $\varphi$ is the latitude. If ${\alpha }_{sol}<0.0001°$, it is set to zero.

Solar Azimuth

The solar azimuth ${\gamma }_{sol}$ (angle from south, eastwards positive, westwards negative) is calculated using auxiliary parameters:

$\sin ({\text{aux}}_1)=\frac{\cos ({\delta }_{sol})\sin (180-\omega )}{\cos ({\alpha }_{sol})}$

$\cos ({\text{aux}}_1)=\frac{\cos (\varphi )\sin ({\delta }_{sol})+\sin (\varphi )\cos ({\delta }_{sol})\cos (180-\omega )}{\cos ({\alpha }_{sol})}$

${\text{aux}}_2=\arcsin (\sin ({\text{aux}}_1))$

The azimuth is then determined by quadrant (BS EN ISO 52010-1:2017, Formula 16):

• If $\sin ({\text{aux}}_1)\ge 0$ and $\cos ({\text{aux}}_1)>0$: ${\gamma }_{sol}=180-{\text{aux}}_2$
• If $\cos ({\text{aux}}_1)<0$: ${\gamma }_{sol}={\text{aux}}_2$
• Otherwise: ${\gamma }_{sol}=-(180+{\text{aux}}_2)$

Air Mass

$m=\{\begin{array}{ll}\frac{1}{\sin ({\alpha }_{sol})}& \text{if }{\alpha }_{sol}\ge 10°\\ \frac{1}{\sin ({\alpha }_{sol})+0.15\times ({\alpha }_{sol}+3.885{)}^{-1.253}}& \text{otherwise}\end{array}$

Extraterrestrial Radiation

$G_{sol,c}=1367\times (1+0.033\cos (R_{dc}))\text{[W/m²]}$

Solar Angle of Incidence

The angle of incidence $\theta$ of the solar beam on a tilted surface of tilt $\beta$ and orientation $\gamma$ is:

$\cos (\theta )=\sin ({\delta }_{sol})\sin (\varphi )\cos (\beta )-\sin ({\delta }_{sol})\cos (\varphi )\sin (\beta )\cos (\gamma )+\cos ({\delta }_{sol})\cos (\varphi )\cos (\beta )\cos (\omega )+\cos ({\delta }_{sol})\sin (\varphi )\sin (\beta )\cos (\gamma )\cos (\omega )+\cos ({\delta }_{sol})\sin (\beta )\sin (\gamma )\sin (\omega )$

Irradiance on Tilted Surfaces

Direct Irradiance

$I_{dir}=\max (0,G_{sol,b}\cos (\theta ))\text{[W/m²]}$

Perez Model for Diffuse Irradiance

The Perez anisotropic sky model decomposes diffuse irradiance into three components using brightness coefficients from ISO 52010 Table 8.

Dimensionless clearness parameter:

$\epsilon =\frac{(G_{sol,d}+G_{sol,b})/G_{sol,d}+K\cdot {\alpha }_{sol,rad}^3}{1+K\cdot {\alpha }_{sol,rad}^3}$

Where $K=1.014$ rad${}^{-3}$ (ISO 52010 Table 9) and ${\alpha }_{sol,rad}$ is the solar altitude in radians. When $G_{sol,d}=0$, $\epsilon$ is set to 999.

Dimensionless sky brightness parameter:

$\Delta =\frac{m\cdot G_{sol,d}}{G_{sol,c}}$

Circumsolar brightness coefficient:

$F_1=\max (0,f_{11}+f_{12}\cdot \Delta +f_{13}\cdot {\theta }_z)$

Where ${\theta }_z$ is the solar zenith angle in radians and $f_{11},f_{12},f_{13}$ are looked up from ISO 52010 Table 8 based on the value of $\epsilon$.

Horizontal brightness coefficient:

$F_2=f_{21}+f_{22}\cdot \Delta +f_{23}\cdot {\theta }_z$

Where $f_{21},f_{22},f_{23}$ are from the same table. $F_2$ is not clamped to zero because overcast skies produce a negative horizon brightening term.

Brightness coefficients (ISO 52010 Table 8):

Diffuse irradiance components on the tilted surface:

Sky diffuse:

$I_{dif,sky}=G_{sol,d}\cdot (1-F_1)\cdot \frac{1+\cos (\beta )}{2}$

Circumsolar:

$I_{dif,circ}=G_{sol,d}\cdot F_1\cdot \frac{a}{b}$

Where $a=\max (0,\cos (\theta ))$ and $b=\max (\cos (85°),\cos ({\theta }_z))$.

Horizon brightening:

$I_{dif,hor}=G_{sol,d}\cdot F_2\cdot \sin (\beta )$

Ground Reflection Irradiance

$I_{ref}=(G_{sol,d}+G_{sol,b}\sin ({\alpha }_{sol}))\cdot {\rho }_{ground}\cdot \frac{1-\cos (\beta )}{2}$

Calculated Direct and Diffuse Irradiance

For the purpose of shading, the circumsolar component is reassigned from diffuse to direct:

$I_{sol,dir}=I_{dir}+I_{dif,circ}$

$I_{sol,dif}=I_{dif,sky}+I_{dif,hor}+I_{ref}$

The total unshaded irradiance on the tilted surface is:

$I_{sol,total}=I_{sol,dir}+I_{sol,dif}$

Shading Reduction Factors

Shading reduces the irradiance reaching a surface. Two reduction factors are calculated separately for the direct and diffuse components, following BS EN ISO 52016-1:2017, Annex F.

Outside Solar Beam Test

Before computing shading from objects, the model checks whether the surface is within view of the solar beam. A surface is outside the solar beam if:

• The difference between the surface orientation and solar azimuth exceeds $\pm 90°$, or
• The difference between the surface tilt and solar altitude exceeds $\pm 90°$.

When the surface is outside the solar beam, the direct shading factor $F_{sh,dir}$ is set to 1.0 (i.e. no adjustment required, since the direct irradiance calculation already yields zero for surfaces facing away from the sun).

Direct Shading Reduction Factor

The direct shading factor $F_{sh,dir}$ represents the fraction of the surface area that remains sunlit. It accounts for shading from distant obstacles, distant overhangs, nearby overhangs, side fins, and nearby semi-transparent obstacles.

Obstacle shadow height. The height of shadow cast on the surface by an obstacle of height $H_{obst}$ at horizontal distance $L_{obst}$:

$H_{shade,obst}=\max (0,H_{obst}-H_{base}-L_{obst}\tan ({\alpha }_{sol}))$

Overhang shadow height. The height of shadow cast by an overhang of height $H_{ovh}$ at distance $L_{ovh}$:

$H_{shade,ovh}=\max (0,H_k+H_{base}-H_{ovh}+L_{ovh}\tan ({\alpha }_{sol}))$

Nearby overhang shadow (on windows). For an overhang of depth $D$ and gap distance $d$ directly above a window:

$H_{shade,ovh}=\frac{D\cdot \tan ({\alpha }_{sol})}{\cos ({\gamma }_{sol}-\gamma )}-d$

Side fin shadow width. For a side fin of depth $D$ and gap distance $d$:

$W_{shade,fin}=D\cdot \tan ({\gamma }_{sol}-\gamma )-d$

Right-hand fins shade only when ${\gamma }_{sol}-\gamma \le 0$; left-hand fins shade only when ${\gamma }_{sol}-\gamma \ge 0$.

The shaded heights and widths are clamped to the surface dimensions. The remaining sunlit area is:

$H_{k,sun}=\max (0,H_k-H_{k,obst}-H_{k,ovh})$

$W_{k,sun}=\max (0,W_k-W_{k,finR}-W_{k,finL})$

$F_{sh,dir}=\frac{H_{k,sun}\times W_{k,sun}}{H_k\times W_k}$

For nearby semi-transparent obstacles, the factor is further reduced. The transparency of the obstacle allows a fraction of the shaded area to transmit direct radiation:

$H_{k,sun,obs}=\max (0,H_k-H_{k,obst,new}-H_{k,ovh})+\min (H_{k,obst,new},H_k-H_{k,ovh})\times T_{obs}$

Where $T_{obs}$ is the transparency of the obstacle. The minimum of all computed $F_{sh,dir}$ values across all nearby obstacles is taken.

Diffuse Shading Reduction Factor

Diffuse shading accounts for the reduction of sky view caused by shading objects. The method distinguishes between remote (environment) shading and nearby (window-attached) shading objects.

Remote obstacle shading. The ground plane is divided into azimuthal shading segments. For each segment, the model calculates the overlap between the segment arc and the 180-degree forward-facing arc of the surface. The sky view factor contribution of each segment is reduced based on the angular height of obstacles and overhangs within that segment.

The reduced sky view factor $F_{sky,new}$ is accumulated across all segments. The remote diffuse shading factor is:

$F_{sh,dif,rem}=1-\frac{F_{sky}-F_{sky,new}}{F_{sky}}$

Nearby object shading. For window-attached objects (overhangs, side fins, reveals, and nearby obstacles), geometric view factors are calculated per ISO 52016 Annex F, section F.6.3. The model computes view factor ratios using the depth and distance of each object relative to the window dimensions:

$P_1=\frac{D}{H}\text{(overhang)}{P}_1=\frac{D}{W}\text{(side fin)}$

$P_2=\frac{d}{H}\text{(overhang)}{P}_2=\frac{d}{W}\text{(side fin)}$

View factors for side fins and overhangs follow the formulations in ISO 52016 equations F.15 to F.18. The minimum shading factor across all object types is taken as the final nearby diffuse shading factor $F_{sh,dif,ws}$.

Combined diffuse factor. The overall diffuse shading reduction factor is the minimum of the remote and nearby factors:

$F_{sh,dif}=\min (F_{sh,dif,rem},F_{sh,dif,ws})$

Reveal Expansion

Window reveal shading is modelled by expanding a reveal object into three separate objects: one overhang and two side fins (left and right), each with the same depth and distance as the reveal.

Surface Irradiance

The total irradiance reaching a surface after shading is:

$I_{surf}=I_{sol,dir}\cdot F_{sh,dir}+I_{sol,dif}\cdot F_{sh,dif}\text{[W/m²]}$

When both $I_{sol,dir}$ and $I_{sol,dif}$ are zero, the shading factors are set to zero and the surface irradiance is zero.

Solar Gains on Opaque Elements

For opaque elements (walls, roofs), solar radiation is absorbed at the external surface. The absorbed solar flux enters the external surface node (node 0) of the element's node model (see TP-07: Thermal Mass):

${\dot{q}}_{sol,opaque}={\alpha }_{sol}\cdot (I_{sol,dif}\cdot F_{sh,dif}+I_{sol,dir}\cdot F_{sh,dir})\text{[W/m²]}$

Where ${\alpha }_{sol}$ is the solar absorption coefficient of the external surface.

This term appears in the heat balance equation for the external surface node (BS EN ISO 52016-1:2017, equation 41), alongside the convective and radiative exchange with the external environment and the thermal radiation to the sky.

The longwave thermal radiation to the sky is:

${\dot{q}}_{sky}=F_{sky}\cdot h_{re}\cdot \Delta T_{sky}\text{[W/m²]}$

Where:

• $F_{sky}=\frac{1+\cos (\beta )}{2}$ is the sky view factor
• $h_{re}=4.14$ W/(m²K) is the external radiative heat transfer coefficient
• $\Delta T_{sky}=11$ K is the difference between external air temperature and sky temperature (default for intermediate climatic region, BS EN ISO 52016-1:2017, Table B.19)

Solar Gains on Transparent Elements

For transparent elements (windows, glazed doors), solar radiation is transmitted through the glazing into the zone interior.

g-Value Correction

The total solar energy transmittance is corrected for the angle of incidence using a constant correction factor (BS EN ISO 52016-1:2017, Appendix B, Table B.22):

$g_{corrected}=F_w\cdot g$

Where $F_w=0.90$ is the default correction factor for non-scattering glazing.

Transmitted Solar Gains

$Q_{sol,trans}=g_{corrected}\cdot I_{surf}\cdot A\cdot (1-F_f)\text{[W]}$

Where:

• $I_{surf}$ is the total surface irradiance after shading W/m²
• $A$ is the total window area m²
• $F_f$ is the frame area fraction

Window Treatment Reduction

When window treatments (curtains, blinds, shutters) are present and in the closed position, the transmitted solar gains are reduced:

$Q_{sol,treated}=Q_{sol,trans}-Q_{sol,trans}\cdot r_{trans}$

Where $r_{trans}$ is the transmittance reduction factor of the treatment (per BS EN 13125:2001). Multiple treatments are applied sequentially.

The open/closed state of each treatment is determined by a control logic per BS EN ISO 52016-1:2017, Annex G and Tables B.23/B.24. The controls evaluate time-based schedules and surface irradiance thresholds:

1. If a time-based control specifies the treatment as open or closed, that takes precedence.
2. If no time-based control is active, irradiance-based logic applies: the treatment closes when surface irradiance exceeds a closing threshold and opens when it falls below an opening threshold.
3. For automatically controlled treatments, a delay period applies between closing and re-opening.

Distribution of Solar Gains in the Zone Heat Balance

Transmitted solar gains through glazing enter the zone heat balance with a prescribed convective/radiative split (BS EN ISO 52016-1:2017):

• Convective fraction $f_{sol,c}=0.1$, applied at the zone air node
• Radiative fraction $(1-f_{sol,c})=0.9$, distributed equally across all internal surface nodes

The convective portion appears in the zone air node heat balance equation (equation 38 of BS EN ISO 52016-1:2017):

$\dots +f_{sol,c}\cdot Q_{sol}+\dots$

The radiative portion appears in the internal surface node heat balance equation (equation 39) for each building element:

$\dots +\frac{(1-f_{sol,c})\cdot Q_{sol}}{A_{el,total}}+\dots$

Where $A_{el,total}$ is the total area of all building elements in the zone.

Absorbed solar radiation on opaque surfaces does not use this split. It enters the external surface node (node 0) directly, as described above.

Outputs

Assumptions

• The Perez anisotropic sky model is used for all sky conditions. The model is parameterised by the brightness coefficients in ISO 52010 Table 8, which are optimised for mid-latitude climates.
• The g-value correction factor $F_w$ is fixed at 0.90 for all glazing types. The angle-dependent method for scattering glazing (ISO 52016 Appendix E) is not applied.
• Ground reflectivity is a per-timestep input, defaulting to 0.2 (ISO 52010 Annex B). Snow cover or varying ground surfaces are not modelled by default but can be supplied through the weather data.
• The sky temperature offset $\Delta T_{sky}$ is fixed at 11 K (intermediate climatic region). Climate-specific values are not applied.
• The sky view factor $F_{sky}$ for a surface of tilt $\beta$ is $(1+\cos \beta )/2$. Reduction of the sky view by surrounding obstacles is not included in this factor; it is handled separately through the diffuse shading calculation.
• The convective fraction of transmitted solar gains is fixed at $f_{sol,c}=0.1$. The remaining 0.9 is distributed equally by area across all internal surface nodes in the zone.
• The projected height of a surface is calculated as $H\sin (\beta )$, clamped to a minimum of 0.01 m for horizontal surfaces (BS EN ISO 52010-1 Table 7, footnote d).
• Adjacent conditioned spaces, adjacent unconditioned spaces, and ground floor elements have zero solar absorption coefficient and no solar interaction.
• When the sun is below the horizon (solar altitude effectively zero), direct beam irradiance and circumsolar irradiance are zero. The air mass formula uses the extended Kasten and Young approximation for low solar altitudes.
• Window reveal shading is modelled as equivalent to one overhang and two side fins with identical depth and distance.

Cross-references

• TP-01: Overview and Climate Data, provides the weather file containing direct beam radiation, diffuse horizontal radiation, and ground reflectivity
• TP-03: External Conditions, supplies the external air temperature used in the external surface node heat balance alongside the solar term
• TP-04: Space Heating Demand, receives transmitted solar gains and absorbed solar flux for the zone heat balance matrix
• TP-05: Fabric Heat Loss, defines the node model structure, surface resistances, and the thermal conductances through which absorbed solar energy propagates
• TP-07: Thermal Mass, determines the heat capacity at each node, which governs the dynamic response to solar gains
• TP-18: PV and Battery, uses the same solar irradiance calculations on tilted surfaces for photovoltaic generation