Estimation of solar radiation
for buildings with complex
architectural layouts
Arch. Stoyanka Ivanova
University of Architecture, Civil Engineering and Geodesy
Sofia, Bulgaria
1 Introduction
If we want the “solar friendly” architectural
thinking to be accepted widely, then we
have to think how it could be applied still in
the beginning of the architectural creative
process – when the idea of the building is
shaped and especially in the process of
creating of the architectural layout.
It’s possible to use a computer program
not only to draw a desired layout, but
also to create it. It’s hard to simulate the
architectural creative process, but with the
increase of the present computer might
this is not already a “mission impossible”.
1 Introduction
The program ArchiPlan is created to
assist the architect in the process of
searching of initial architectural idea –
architectural layout.
The program generates hundreds of 2D
orthogonal architectural layouts for just
few seconds.
1 Introduction
In the beginning the architect describes the
elements (rooms) of the desired layout:
 names of rooms
 square surface of each room
 functional relations between rooms
 requirements for each element for exposure
 desired grid, etc.
1 Introduction
Example (15 rooms)
Code
1
2
11
12
13
14
21
22
24
25
26
27
31
32
34
Room Name
Foyer
Great Room
Hall 1
Kitchen
Breakfast
Dining
Hall 2
Master Bedroom
Master Bath
Closet
Closet
Hall
Hall 3
Bedroom 2
Bathroom
Area sq.m
Exposure
9
30
3
14
9
9
5
18
9
3
3
3
1
16
9
North
South
North
West
North
1 Introduction
Matrix of functional relations between rooms
Code
1
2
11
12
13
14
21
22
24
25
26
27
31
32
34
Room Name
Foyer
Great Room
Hall 1
Kitchen
Breakfast
Dining
Hall 2
Master Bedroom
Master Bath
Closet
Closet
Hall
Hall 3
Bedroom 2
Bathroom
1
0
3
0
0
0
1
1
0
0
0
0
0
0
0
0
2
3
0
1
0
1
0
0
1
0
0
0
0
0
0
0
11 12 13 14 21 22 24 25 26 27 31 32 34
0 0 0 1 1 0 0 0 0 0 0 0 0
1 0 1 0 0 1 0 0 0 0 0 0 0
0 1 0 1 0 0 0 0 0 0 1 0 0
1 0 1 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 2 0 0 0
0 0 0 0 0 0 0 0 0 2 0 0 0
0 0 0 0 0 1 1 2 2 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 1 0 1
0 0 0 0 0 0 0 0 0 0 0 1 0
1 Introduction
With these data the program ArchiPlan
generates 602 architectural layouts
for less than 4 seconds…
1 Introduction
Exemplary architectural layouts,
generated by the program ArchiPlan:
1 Introduction
These layouts have to be evaluated, so the
program can offer the architect only the best of
them. The program uses the following criteria to
rank all generated layouts:
 compactness (as a planning quality)
 construction’s requirements
 energy effectiveness (energy gains
and losses)
The weight of these three criteria could vary,
regarding the architect’s final goal.
1 Introduction
The program ArchiPlan sorts the
generated architectural layouts on
compactness…
Let’s see worst 5 and top 10 layouts…
2 Methodology
The energy effectiveness of an architectural
layout has two sides – energy losses and
energy gains.
In this too early moment of the architectural
process the energy losses could be evaluated
with the compactness of the building - the
surface-area-to-volume ratio - a proportion
between the total exterior surface and internal
volume.
As less is this number, as more compact is the
building and lower the energy losses.
2 Methodology
 Since June 30th a new calculation of supposed
energy losses was added to the program. It uses
a difference between inside and outside air
temperature and supposed heat transmission
coefficient.
 The energy effectiveness is calculated as follows:
Ee  (Gains Losses) / Gains
2 Methodology
To evaluate energy gains, the program
ArchiPlan estimates the quantity of
incoming solar energy [Wh] to the
building outside surface (exterior walls and
roof) for the whole heating season.
In this so early stage is convenient to use
2.5D model – with only horizontal and
vertical surfaces.
2 Methodology
Two methods of solar energy estimation
are created.
The final goal of the first method is to
rank the hundreds of generated architectural
layouts considering their energy
effectiveness.
In this case is important to receive quick
results, even with some little inexactness.
This is why we need to simplify computation
of the summary solar radiation, received by
all exterior surfaces.
2 Methodology
The second method is created to estimate
the solar radiation, received by each
particular vertical wall of a layout.
The goal is to allow the architect to study
how a specific architectural layout can
utilize the incoming solar energy in
best way. Such information could help
him to find the best positions for the wall
apertures - windows and glazed doors on
the walls of a complex architectural layout.
3.1 Quick analysis of summary beam radiation
The exterior surface of our 2.5D model is projected
on a plane, normal to solar beam.
3.1 Quick analysis of summary solar beam radiation
Area of roof’s projection:
A1  Aroof * sin ho
Area of walls’ projection:
A2  df * dh * cos ho
df – front of projection:
df  Max(Pr) Min(Pr)
Pri  X i * cos Az  Yi * sin Az
Beam radiation:
RB  B0c * ( A1  A2 )
3.2 Quick analysis of summary diffuse radiation
For the diffuse radiation RD [W] is valid the formula:
RD   Dic * Ai 
i
 Dic [W.m-2] - the diffuse irradiance on each external
surface (walls or roof) of the building
 Ai - area of this surface [m2]
Hofierka and Šúri in their work “The solar radiation model
for Open source GIS” described how to estimate the diffuse
irradiance on horizontal and inclined surfaces, when they are
sunlit, potentially sunlit or shadowed. But this model has to
be extended for the cases, when one or more near objects
with limited size hide parts of the sky, as it is for the buildings
with complex architectural layouts.
3.2 Quick analysis of summary diffuse radiation
According Hofierka for inclined surfaces in shadow:
Dic  Dhc * F ( N )
 Dhc is diffuse irradiance on a horizontal surface
 F(γN) is a function, which convert the diffuse irradiance
on a horizontal surface into diffuse irradiance on an inclined
surface under anisotropic clear sky:
F ( N )  ri ( N )  [sin N   N * cos N   * sin ( N / 2)]* N
2
 ri(γN)=(1+cos γN)/ 2 is a fraction of the sky dome, viewed
by an inclined surface.
For unobstructed surfaces in shadow of our 2.5D model,
N=0.25227 according Muneer and Hofierka:
a) when γN=0 (for horizontal surface): F(γN)=1
b) when γN=π/2 (for vertical surface): F(γN)=0.356
3.2 Quick analysis of summary diffuse radiation
Let’s consider this wall configuration:
3.2 Quick analysis of summary diffuse radiation
For the wall A0 a part of the sky is hidden by both other walls.
I propose such equation to the mentioned model:
Dic  Dhc * DFi
 DFi is a function, which transforms the value of the diffuse
irradiance on a horizontal surface to a value of diffuse
irradiance, incoming from partially shaded anisotropic sky,
onto an inclined surface (in our example it’s vertical):
DFi  F ( N ) * C1,2  VT1  VT2
3.2 Quick analysis of summary diffuse radiation
 α1 and α2 are azimuths of the
furthest ends of the neighbor
walls A1 and A2
 C1,2 is a correction
coefficient for the visible part
of the sky in horizontal
direction, between azimuths α1
and α2
 VT1 and VT2 are the
correction values due of the
spherical triangles T1 and T2
above the walls A1 and A2 on
the stereographic projection.
3.2 Quick analysis of summary diffuse radiation
For estimation of C1,2 this equation is proposed:
C1, 2
sin  2  sin 1

2
where α1=arctg(dx1/dy1) and α2=arctg(dx2/dy2).
If dy1=0 then α1=-π/2 and VT1=0.
If dy2=0 then α2=π/2 and VT2=0.
If both dy1 and dy2 are 0, then C1,2=1, VT1 and VT2
are 0.
In all other cases the values of VT1 or VT2 are
greater than 0.
3.2 Quick analysis of summary diffuse radiation
In the figure are illustrated the variable x1a and
angle β1, which are necessary to estimate the value
of VT1. AT1 is the area of segment T1 in the same
figure.
x1a 
dx1
dx1  dy1
2
2
 sin 1
1  arctg(dz1 / dx1)
AT 1 
 * (1  x1a )
4

1  x1a * arctg ( x1a * dz1 / dx1 )
2
3.2 Quick analysis of summary diffuse radiation
I propose the following approximate equation of VT1
to take into consideration at least partially the
anisotropic nature of the diffuse radiation:
AT 1
VT1 
* F ( N )
 /2
The transforming function DFi (corrected F(γN)) has
to be used to calculate diffuse radiation also on
sunlit or partially shaded surfaces. Within this quick
method I estimate Dic only once for the center of
each examined vertical wall.
3.3 Quick analysis of solar radiation under real sky
 Finally the computation of overcast radiation for
inclined surfaces is analogous to the procedure
described for clear-sky model (Hofierka and Šúri),
using the modified values of beam and diffuse
irradiance because of the concrete values of beam and
diffuse components of the clear-sky index (Kbc, Kdc).
 The layouts with best energy effectiveness are
compact with southerly exposure of the larger
dimension of the building and flat south vertical wall.
As higher is the percent of diffuse radiation under the
real sky, as more important is the compactness.
3.3 Quick analysis of solar radiation under real sky
The program ArchiPlan sorts the
generated architectural layouts on
energy effectiveness…
Let’s see worst 5 and top 10 layouts…
4.1 Detailed analysis of beam and diffuse radiation
When the rank list of all generated architectural
layouts is ready, the architect could examine
the best of them, with the help of a detailed
analysis of received solar radiation by each
vertical wall to find best place for windows
and glazed doors on each floor.
4.1.1 Detailed analysis of incoming beam radiation
According detailed methodology in “Solar radiation
and shadow modeling with adaptive triangular
meshes” by Montero et al:
Bic  B0c * sin exp * Lf
 B0c is the beam irradiance normal to the solar
beam
 δexp is the solar incidence angle between the sun
and an inclined surface
 Lf is calculated lighting factor for the concrete
wall, day and time (0 – completely shaded wall; 1 - sunlit wall;
between 0 and 1 – partially shadowed wall)
4.1.2 Detailed analysis of incoming diffuse radiation
The quantity of diffuse radiation for each
fragment of each wall is different.
So for a detailed analysis each wall is
divided in horizontal and vertical
directions and the program applies to
these small fragments the estimations,
described in the previous section.
4.1.2 Detailed analysis of incoming diffuse radiation
For this wall configuration:
DFi – the value of the diffuse transforming function for the wall
with length b and height h, is:


xj
xj
1 M N 
zk

DFi  F ( N ) *
arctan

*
arctan
* (dx * dz)



2
2
2
2
bh j 1 k 1 
zk

c

x
c

x
j
j


dx  b / M
dz  h / N
x j  ( j  0.5) * dx
zk  (k  0.5) * dz
 M is number of fragments in horizontal direction
 N is number of fragments in vertical direction
 dx is size of a fragment in horizontal direction
 dz is size of a fragment in vertical direction
 (xj, zk) – center of a fragment of this vertical wall
4.1.2 Detailed analysis of incoming diffuse radiation
Instead with this double sum, the exact value of diffuse
transforming function for the same vertical wall can be
estimated with this double definite integral:
h b
1  
x
x
z

DF  F ( N ) *
arctan 
* arctan




2
2
bh 0  0 
z
c x
c2  x2
 
dx dz

 
The value of the integral is:
1
D1  D2  D3 
DF  F ( N ) *
bh
b2  h2
b h2
b2
2
2
D1 
* ln(b  h )  bh * arctan  * ln h  * ln b
4
h 2
2
h2  b2  c 2
c 2  h2
h
2
2
2
D2 
* ln(h  b  c ) 
* ln(h2  c2 )  h * b2  c2 * arctan
4
4
b2  c 2
b2  c 2
c2
h
2
2
D3  
* ln(b  c )  * ln c  hc * arctan
4
2
c
4.1.2 Detailed analysis of incoming diffuse radiation
For different wall configurations the double
integral and its result looks also different.
There are 4 basic double integrals and
many combinations between them for
different wall configurations .
I still work on some combinations.
4.2 Balance principles of
incoming and received diffuse radiation
Balance principle 1: The quantity of diffuse radiation,
crossed an opening (with area Aopening ), is equal to sum of the
quantities of diffuse radiation, received by the surfaces (with
areas Ai ), which are behind this opening.
From this is easy to come to:
Aopening * DFopening   Ai * DFi
i
 DFopening is value of the diffuse transforming function for the planar
surface of the opening
 DFi is the value for each receiving surface behind the opening
DFi  ( Aj * DFj ) / Ai
j
 DFj and Aj are value of the transforming function and area of a small
fragment j of the surface i
4.2.1 Balance principle 1 - example
Opening
EFGH
Wall
ABFE
Wall
BCGF
Wall
CDHG
Wall
ADHE
Base
ABCD
4.2.1 Balance principle 1 - example
For isotropic sky and AB=2, BC=1, AE=3:
Aopening * DFopening   Ai * DFi
2=2 i
Surface
Wall ABFE
Wall BCGF
Wall CDHG
Wall ADHE
Base ABCD
Opening EFGH
DF
0.102713
0.107796
0.102713
0.107796
0.060331
1
Area
6
3
6
3
2
Area*DF
0.616281
0.323388
0.616281
0.323388
0.120663
Sum:
2
2
2
All values of DF are calculated with different combinations
of mentioned double integral.
4.2.2 Balance principle 2 of
incoming and received diffuse radiation
Balance principle 2: The quantity of diffuse radiation,
crossed two or more openings (with area Ak ), is equal to sum
of the quantities of diffuse radiation, received by the surfaces
(with areas Ai ), which are behind each of the openings.
From this is easy to come to:


k  Ak * DFk   i  Ai * DFi   i  Ai * k DFik 
 DFk is value of the diffuse transforming function for opening k
 DFik is the value for the surface i, generated by the incoming diffuse
radiation through opening k.
 DF has to be defined twice for the planes of vertical openings - for
inside and outside face of examined volume.
 The value of the transforming function of the surface i with fragments
j, for the incoming diffuse radiation from all openings, is:
 

DFi     Aj *  DFjk   / Ai
k

 j 
4.2.2 Balance principle 2 - example
Opening
EFGH
Opening
ABFE
Surface
ABFE
Wall
BCGF
Wall
CDHG
Wall
ADHE
Base
ABCD
4.2.2 Balance principle 2 - example
For isotropic sky and AB=2, BC=1, AE=3:


k  Ak * DFk   i  Ai * DFi   i  Ai * k DFik 
Surface
Surface ABFE
Wall BCGF
Wall CDHG
Wall ADHE
Base ABCD
DF1 (EFGH) DF2 (ABFE)
0.102713
0
0.107796 0.159498
0.102713 0.237788
0.107796 0.159498
0.060331 0.308140
Opening 1 (EFGH)
Opening 2 (ABFE)
5=5
DF
0.102713
0.267294
0.340502
0.267294
0.368472
Area
6
3
6
3
2
Area*DF
0.616281
0.801883
2.043010
0.801883
0.736943
5
1
0.5
Sum:
2
6
Sum:
5
2
3
4.2.2 Balance principle 2 - example
Here is another viewpoint to the same example:
Wall
BCGF
Surface
Surface ABFE
Wall BCGF
Wall CDHG
Wall ADHE
Base ABCD
DFv
0
0.190983
0.309017
0.190983
0.323893
Wall
CDHG
Wall
ADHE
DFt
DF=DFv+DFt Area
0.102713 0.102713
6
0.076311 0.267294
3
0.031485 0.340502
6
0.076311 0.267294
3
0.044578 0.368472
2
Opening 1 (EFGH)
1
Sum:
2
Opening 2 (ABFE)
0.5
6
Sum:
Opening
ABFE
Opening
EFGH
DFv*Area
0
DFt*Area
0.616281
0.572949
0.228934
1.854102
0.188908
0.572949
0.228934
0.647787
0.089157
0.616281
0.801883
2.043010
0.801883
0.736943
3
0.646776
5
DF*Area
2
3
5
Here we have 2 interesting equalities:
3=3
5=5
The summary product of DF and Area for these 3 vertical walls is 3.646776
4.2.2 Balance principle 2 - example
RD   Dhc * Ai * DFi  Dhc *   Ai * DFi 
i
i
This is the main practical advantage from both balance principles.
It helps to calculate very easy the summary diffuse radiation on a
complex wall configuration. The main part of it (pink values) comes
across an endless high vertical opening, so for these values we
can simplify the outline of the layout in this way:
5 Discussion
 Both equations from the balance principles can be proved for
isotropic sky analytically with already mentioned double
integrals. It’s really beautiful mathematics, just “music of the
projected spheres”.
 I wasn’t able to prove both balance rules for anisotropic sky
numerically. The proof is not obvious and will be more difficult.
It could be possible with better knowledge of sky’s anisotropy.
 In spite of this the consequences of the second balance rule
can be implemented into calculations for obstructed
anisotropic sky and to receive values with good approximation.
Under real sky this inexactness will be even more unimportant.
 Both balance principles and corresponding equations could
also be applied to beam and total radiation, to sunlit or
partially shadowed surfaces, and for different sky types.
6 Conclusion and future work
 It’s proposed a numerical model for estimation of the
solar radiation on the external vertical walls of a building
with complex architectural layout. This could help the
architect to think more about the solar gains and to
choose the best solar friendly architectural layout.
 Further research is needed to precise the calculation of
the diffuse radiation from anisotropic sky for spherical
triangles above the vertical walls.
 The balance principles and corresponding equations are
a good start base for further improvement of this model
for both isotropic and anisotropic sky.
 Calculations of reflected radiation and analysis of solar
radiation in summer will be added in the future.
7 References
1.
2.
3.
Muneer, T., Solar radiation model for Europe. Building services
engineering research and technology, 1990, vol. (11)
Hofierka, J., Šúri M., The solar radiation model for Open source GIS:
implementation and applications, Open source GIS - GRASS users
conference, Trento, Italy, 2002
Montero, G. et al, Solar radiation and shadow modeling with
adaptive triangular meshes, Sol. Energy, 2009,
doi:10.1016/j.solenar.2009.01.004
Thank you!