solar radiation modelling in the urban context

Solar Energy 77 (2004) 295–309

www.elsevier.com/locate/solener

Solar radiation modelling in the urban context

Darren Robinson *, Andrew Stone

BDSP Partnership Ltd, Summit House, 27 Sale Place, London W2 1YR, UK

Received 13 January 2004; received in revised form 6 May 2004; accepted 19 May 2004

Available online 17 June 2004

Communicated by: Associate Editor Pierre Ineichen

Abstract

This paper describes alternative methods for predicting surface irradiance in the urban context. In this the focus is on

means of accounting for the effects of nearby obstructions on reducing direct sky radiation and on contributing reflected

radiation. The first two methods involve abstracting the urban skyline into an effective canyon using isotropic and

anisotropic tilted surface irradiance models. The third predicts the irradiance contribution from two hemispheres which

are discretised into patches––given the radiance of the sky and dominant obstructions (if these exist) and associated

view factors––so that we have a new simplified radiosity algorithm (SRA). Results from the three methods (isotropic

canyon (IC), anisotropic canyon (AC) and simplified radiosity algorithm (SRA)) are compared with a ‘truth model’

under the following circumstances: (i) unobstructed sky, (ii) sky obstructed by black surfaces, (iii) sky obstructed by

grey diffusely reflecting surfaces. Results show conclusively that the SRA offers superior accuracy at comparable speed

to the canyon models. The SRA also compares well with a ray tracing program, it can handle urban scenes of arbitrary

geometric complexity and is readily amenable for inclusion into standard computer programs that require surface

irradiance as an input.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Urban; Solar radiation; Model; Theory; Validation

1. Introduction

To help to optimise the energy and environmental

performance of buildings, dynamic thermal simulation

programs are commonly deployed. These programs are

now relatively mature. Their core heat transfer mecha-

nisms have been extensively validated and they have a

growing user base in both academic and commercial

circles. Consequently, their capabilities have grown

considerably, in some cases to include simultaneous

simulation of plant systems, mass flow and electrical

power flow networks not to mention embedded com-

putational fluid dynamics (CFD) domains (Clarke,

* Corresponding author. Tel.: +44-207-298-8383; fax: +44-

207-298-8393.

E-mail address: [email protected] (D. Robinson).

0038-092X/$ - see front matter � 2004 Elsevier Ltd. All rights reserv

doi:10.1016/j.solener.2004.05.010

2001). However, these programs are presently only

weakly coupled to the urban context in which the vast

majority of the population in developed countries works

and resides. More specifically, (i) current dynamic

thermal simulation programs consider obstructions to

views of the sun but either ignore obstructions to the sky

vault or associated reflections from them, (ii) no clear

basis for translating synoptic meteorological measure-

ments to the local urban context has yet been developed

(e.g. local and background (heat island) influences on air

temperature), (iii) buildings are assumed to operate

independently, so that there is no basis for handling

centralised resource management (e.g. centralised energy

centres). There are also issues that fall beyond the scope

of conventional dynamic thermal simulation programs,

such as centralised water and waste management.

These are some of the challenges that are currently

being addressed by Project SUNtool––an EC funded

ed.

mail to: [email protected]

296 D. Robinson, A. Stone / Solar Energy 77 (2004) 295–309

research project to develop a new integrated resource

flow modelling and associated educational tool to sup-

port the design of sustainable urban neighbourhoods

(Robinson et al., 2003). This paper is concerned with the

first of the urban coupling issues identified above: the

prediction of solar radiation in a way that is sensitive to

obstructions both to the sun and to the sky vault, with

due regard for reflection.

The motivation for this work is to improve the res-

olution of solar radiation predictions in urban contexts.

Consider a room at the base of a tall street canyon of

constant height H (say H=W ¼ 2:5). For a point in the

centre of a window plane that is say 2 m above the

ground and for a street width W of 10 m, we have an

equivalent urban horizon angle (UHA) (Baker and

Steemers, 1994) of �66�. In other words over two thirds

of the sky vault is obscured. Now if our street is in

London, UK (in which case our window is never directly

insolated) and we assume isotropic skies then our win-

dow receives less than 1/10th the direct irradiation that a

standard thermal simulation program (with an isotropic

sky model) would predict (Fig. 1)! This has clear

implications not only for energy consumption but also

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90

Urban horizon angle, u (deg)

Idβ

/ id

h

β=0˚β=30˚

β=60˚

β=90˚

β=120˚β=150˚

Fig. 1. Ratio of incident and horizontal diffuse irradiance as a

function of urban horizon angle for surfaces at 30� increments

of slope.

for HVAC system (over)sizing and for identifying whe-

ther passive cooling options are feasible.

2. Alternative model formulations––theory

This section describes three methods to account for

sky obstructions. The first two are based on abstracting

real urban skylines into an effective canyon which re-

duces sky views but also contributes reflected radiation

to our receiving surface. This approach is used in con-

junction with both isotropic and anisotropic tilted sur-

face irradiance models. The third approach is more

physically rigorous. Using a continuous absolute radi-

ance distribution the irradiance incident on our receiving

plane is a function of its view to discrete regions of the

sky, so that the geometric and photometric integrity of

the problem is retained.

2.1. Isotropic canyon model

Some dynamic thermal simulation programs assume

that the sky is isotropic (the vault has the same radiance

in all directions). Under this assumption and given the

diffuse horizontal irradiance Idh, the diffuse irradiance

incident on an unobstructed plane of slope b (06 b6 p,where 0 is horizontal facing up, p=2 is vertical and p is

horizontal facing down) from the sky is:

Idb ¼ Idhð1þ cos bÞ=2 ð1aÞ

Furthermore, ignoring any obscuration of the sky vault

by our receiving plane and given the global horizontal

irradiance (Igh), the corresponding radiation received

from the ground (IqG;b) of reflectance qg is given by (1b):

IqG;b ¼ Ighqgð1� cosbÞ=2 ð1bÞ

Now, if as in the previous example (Section 1) we

combine our solar obstructions into an equivalent con-

tinuous skyline of angular height u (from the perpen-

dicular) to our point of reference, then (1a) can be

modified to give the corresponding obstruction-sensitive

diffuse irradiance due to the sky:

Idb ¼ Idh½1þ cosðb þ uÞ�=2 ð2Þ

This expression introduces a subtle complexity however,

since we need to compose a description of some paral-

lelogram which has the same characteristics, with due

regard for the angle of incidence of reflected (and indeed

obscured sky) radiation as the real, inevitably more

complex, skyline (Appendix A).

Notwithstanding the complexity of deriving u, a

convenient result of (2) is that expressed graphically

(Fig. 1) one can determine a priori the extent to which

diffuse surface irradiance (or indeed irradiation) due to

Fig. 2. Street canyon geometry, as viewed from a receiving

surface.

D. Robinson, A. Stone / Solar Energy 77 (2004) 295–309 297

the sky is reduced by urban obstructions––albeit under

the assumption of sky isotropy.

Another source of complexity is the treatment of

reflected radiation in a way in which the energy received

by each reflecting surface is sensitive to other obstruc-

tions to the sky vault (Fig. 2). For example, the ground

plane’s view of the sky is obstructed by the urban skyline

that opposes our receiving plane as well as by the urban

skyline of which our receiving plane is part (views to this

ground plane are also now reduced, in proportion to the

view factor to the below horizontal vertical surface).

Each urban skyline (or wall of our equivalent street

canyon) is similarly obstructed by the opposing wall.

Given some global irradiance Igb incident on an up-

per obstructing surface of diffuse reflectance q the cor-

responding energy received by our surface of interest

(IqU;b) is given by (3a): 1

IqU;b ¼ Igbq½cos b � cosðb þ uÞ�=2 ð3aÞ

and that received due to reflection from the opposing

wall below the horizontal plane (IqL;b) is:

IqL;b ¼ Igbq½cosðb � uÞ � cos b�=2 ð3bÞ

for which tan u is simply the quotient of the mean

height of our receiving plane and the canyon width

ðz=wÞ. All that remains then is to determine the radiation

reflected onto our receiving plane from the ground

(IqG;b).

1 Subscripts q and U refer to ‘reflected’ and ‘from upper

surface’ respectively. An L would refer to lower reflecting

surfaces.

IqG;b ¼ Igbq½1� cosðb � uÞ�=2 ð4Þ

Now, in deriving the energy incident on surfaces 1 and 2

identified in Fig. 3, it is first necessary to solve for that

received directly from the sky (using (2)) and the sun.

For this latter case, the standard approach used in dy-

namic thermal simulation programs (e.g. ESP-r (Clarke,

2001)) is to discretise the surface of interest into a reg-

ular grid of cells. At hourly intervals say for the central

day of each month, a test is then performed at each cell

centroid to determine whether the solar disc can be

‘seen’. The ratio of the number of successful hits to the

number of cells is an approximation of the proportion of

the surface that is directly insolated (r). Using a pre-

processed look up table to extract the appropriate r for

time t, the surface irradiance is simply: Ibb ¼ Ibnrt cos n,for direct normal irradiance Ibn and angle of incidence n.

The energy incident on the ground is a special case, in

that knowledge of both sides of the canyon is required,

and is given by (5) below.

Id;gnd ¼ Idhðcos d1 þ cos d2Þ=2þ q1Igb;1ð1� cos d1Þ=2þ q2Igb;2ð1� cos d2Þ=2 ð5Þ

With these contributions known, reflected radiation may

be modelled by iteratively solving for expressions (3)–

(5).

Finally, there are some geometric exceptions to (2)–

(5) which limit their range of applicability. In particular:

• The maximum surface tilt is limited by the urban

horizon angle: b þ u� p26

p2(corresponding to situa-

tions where the surface has tilted beyond the vertical

and is facing towards the ground such that the top of

the equivalent obstruction is not visible).

• Care must be taken with surfaces that are tilted by

less than p2as it is assumed that the surface can see

the sky over the entire altitude range from the calcu-

lated urban horizon to p2past the surface normal (Fig.

3). This limitation could be overcome by assigning

two UHAs to each surface.

2.2. Anisotropic canyon model

Several dynamic simulation programs now use

anisotropic sky radiation models, including those of (i)

Temps and Coulson (1977) and Klucher (1979), (ii) Hay

(1979), (iii) Muneer (1997), and (iv) Perez et al. (1990).

Of these the Perez model has been found to give the best

overall performance when applied to a wide variety of

locations (Jensen, 1994);

Idb ¼ Idh½ð1� F1Þð1þ cos bÞ=2þ F1a0=a1 þ F2 sin b� ð6Þ

in which the three terms within the square brackets cor-

respond to isotropic background scattering, circumsolar

Fig. 3. Geometric limitations to applicability of the isotropic model.


brightening and horizon brightening contributions

respectively.

One approach to producing predictions of diffuse

irradiance due to the sky which account for urban

obstructions is to adjust each term in brackets in (6), so

that (i) the first term is scaled in proportion to the angle

of elevation of the urban skyline, (ii) the second term is

multiplied by a Boolean operator B 2 to account for

whether the solar disc, expressed as a point source, can be

seen, and (iii) the final term is scaled by S according to the

proportion of the horizon band, of angular height b (say

6.5� as in the original Perez model), that is obstructed

(S ¼ max½0; ðb� uÞ=b�, where u ¼ 2p

R aw�p=2aw

u cos a da).Combining these influences, we have an obstruction-

sensitive implementation of the Perez anisotropic model

as given by (7).

Idb ¼ Idh½ð1� F1Þ ½1þ cosðb þ uÞ�=2þ B F1a0=a1 þ S F2 sinb� ð7Þ

There is a slight complication to this method due to the

fact that the horizon brightening coefficient F2 may be

negative. This is permitted because although it is not

physically realistic to consider a negative energy con-

tribution received from the horizon band, it is equivalent

to having a brighter region towards the zenith––as for

an overcast sky (Perez et al., 1987). When F2 is negativeit would not be correct to use the scaling factor defined

above as increasing u would lead to a decrease in the

magnitude of F2 and hence a reduction in the brightness

of the zenith region. Instead the CIE overcast sky can be

used to derive a different scaling based on the sky

brightness integrated over the visible range:

S ¼R p�bu ð1þ 2 sin cÞdc

p � b � u p � bR p�b

0ð1þ 2 sin cÞdc

¼ ðp � b � uÞ þ 2ðcos b þ cos uÞðp � bÞ þ 2ðcos b þ 1Þ p � b

p � b � uð8Þ

2 Or indeed some fraction, if we use a similar means for

determining views to the sun as for the direct calculation.

Note that this is only valid for overcast skies; although

the situation where F2 is negative and the sky is relatively

clear should never occur.

Contributions from reflecting surfaces may be solved

using the procedure described above, for the isotropic

canyon model ((3)–(5)). However, in using (5) it should

be noted that the contribution from horizon and cir-

cumsolar sources that are visible to the ground is ig-

nored; though this simplification is not likely to lead to

significant errors.

2.3. Simplified radiosity algorithm

Whilst the Perez anisotropic model is an undoubted

improvement over the isotropic case, it is nevertheless a

simplification of reality. Since the background contri-

bution is assumed to be isotropic and only small urban

obstructions would fully obscure the horizon band, the

sky’s anisotropy is concentrated into the circumsolar

point source, which can either be seen or not. Further-

more, whilst we may achieve a similar obstruction solid

angle in our mapping from real to continuous skylines,

the part of the sky vault that is being blocked out by a

particular obstruction may have quite a different radi-

ance to that in the isotropic case. The following is an

attempt to surmount these drawbacks.

Several models have been developed in the daylight-

ing field in recent years to represent the luminance of the

sky vault as a continuous function of the angular loca-

tion of a sky point relative to some reference, typically

the zenith or the sun. Also in the daylighting field, there

has been interest in predicting illuminances due to views

of discrete regions of the sky vault––as is typically the

case within rooms––and this has led to the development

of a now standard technique for sub-dividing the sky

vault into a series of patches. It is logical to utilise this

approach to predict the irradiance as opposed to illu-

minance at a plane. So, if we have some set of p sky

patches, each of which subtends a solid angle U (Sr) and

has radiance R (Wm�2 Sr�1) then, given the mean angle

of incidence n (rad) between the patch and our plane

together with the proportion of the patch that can be

Fig. 4. Radiance distribution for London, UK (using the Kew 1967 climate dataset in conjunction with the Perez all weather lumi-

nance distribution model).

3 Note that c, a should relate to the visible proportion of

partially obscured patches.


seen r (06 r6 1), we have the following general solution

for direct sky irradiance (Wm�2) (9):

Idb ¼Xp

i¼1

RiUiri cos ni ð9Þ

The patch view factor r in (9) may account for self-

obstructions as well as adjacent buildings, so that we

have an obstruction-sensitive method for deriving diffuse

irradiance due to the sky.

Now, means for predicting the distribution of energy

within the sky vault date back to the original CIE

overcast sky model (Moon and Spencer, 1942) and early

formulations of the CIE clear sky model (Kittler, 1967)

through more recent clear sky models (CIE, 1973, 1996)

to more general sky definitions (Perez et al., 1993; CIE,

2002). Of the latter models, which cover the range of sky

types from overcast through intermediate to clear, the

Perez model is more amenable to incorporation within a

computer program since there is a straightforward basis

for selecting different sky types and predicting the cor-

responding energy distribution. This is presently not the

case with the CIE general sky formulation.

In the case of the Perez model the luminance of a

given sky point is expressed relative to some reference

point thus: ‘v ¼ f ðZ; hÞ, where Z is the zenith angle of the

considered sky point and h the angle between this point

and the sun. Conveniently, by integrating this relative

luminance throughout the sky vault we have an equiva-

lent diffuse horizontal illuminance E0 ¼R 2p0

R p=20

f ðZ; hÞ�cos c sin c dadc (for azimuth a and altitude c). The

quotient of our diffuse horizontal irradiance Idh and this

equivalent illuminance E0 defines a hemispherical nor-

malisation factor v which, when multiplied by a sky

point’s relative luminance, gives the corresponding

absolute radiance, so that the radiance of some point

within the ith patch is

Ri ¼ f ðZ; hÞiv ð10Þ

Turning now to sky vault discretisation, the technique

proposed by Tregenza and Sharples (1993) has found

widespread favour. With this method the sky vault is

split into seven azimuthal strips in which the azimuthal

range Da of the composite patches tends to increase

towards the zenith (12�, 12�, 15�, 15�, 20�, 30�, 60�), atwhich there is a single patch so that there are 145 in

total. The intention is that the patches subtend similar

solid angles, these being determined by the azimuthal

range (rad) of the ith patch and the corresponding

maximum and minimum heights of elevation, using an

expression of the form:

Ui ¼ Daiðsin ci;max � sin ci;minÞ ð11Þ

With a method of discretising the sky vault into a series

of patches of known solid angle, we can calculate an

approximate hemispherical radiance normalisation fac-

tor numerically, based on the radiance at the centroid of

each patch: v ¼ Idh=P145

i¼1 ‘viUi sin ci (Fig. 4).With the view factor known (0, 1 or some fraction for

self-obstructed patches) all that remains is to calculate

the angle of incidence from the patch centroid to our

plane of interest n:

cos n ¼ cos ci cos a0i sinb þ sin ci cosb ð12Þ

in which ci is the altitude of the patch centroid, b is the

plane tilt and a0i is the azimuth angle from the patch

centroid to the plane surface normal (a0i ¼ ai � aw).

3

One of the key attractions of this technique for pre-

dicting diffuse sky radiation is that the radiance distri-

bution of the sky vault can be pre-processed and indeed

archived in an 8760 by 145 length file, in the case of an

hourly annual simulation. Similarly, sky patch view

factors for each of j surfaces may be pre-processed––so

that one can envisage a geometric pre-processing task

taking place within some graphical user interface once

the built vicinity has been described, and subsequent

Fig. 5. Below horizontal reflections from an inverted sky dome (left) and isotropic ground plane (right).

4 Some radiosity programs merge adjacent emitting cells into

a large cell of equivalent radiance to reduce the number of

computations in predicting the energy incident on each (and

every) receiving cell for each iteration. Our approach is similar,

we just combine angularly rather than physically adjacent

(sub)surfaces.


computation may be conducted by a separate processor

which reads the radiance distribution into a two-

dimensional array and refers to this when reading in the

view factors for each j surface at the kth hour.

For the purposes of testing our model, we have

developed a simplified algorithm that discretises each

patch using a regular 10 by 10 grid. A test is then per-

formed to determine the n number of cell centroids that

can be seen by our surface (i.e. whether centroids are

beyond the limits of obstructing polygons) and a view

factor-incidence angle product is derived:

ri cos ni ¼ 10�2Xn

j¼1

cos nj ð13Þ

Now to solve for radiation from surfaces that obscure

views to the vault IqU;b we have that:

IqU;b ¼X145i¼1

ðIgbq=pÞiUið1� riÞ cos ni ð14Þ

in which the final two terms may be solved using a

corollary of (13), based on the 102 � n centroids that

lie within the bounds of obstructing polygons

(ð1� riÞ cos ni ¼ 10�2P102�n

k¼1 cos nk). All that remains

then is to determine the irradiance Igb of the obstructing

surface(s) of known reflectance q in the ith patch. For

this we make an assumption that the radiance of the

largest surface obstructing the patch is similar to that of

the other surfaces obstructing the patch. Using our

simple patch discretisation algorithm, we identify this

surface as being that which produces the largest view

factor-incidence angle. This too can be determined as a

pre-process, so that for each patch with a non-unity view

factor, a surface identifier is parsed to the solver. This

then implies that the irradiance incident on each

obstructing surface is calculated, but since much of the

computation is dealt with as a pre-process the compu-

tational overheads of this approach are reasonably

modest. In the first instance then we calculate, for all

obstructing surfaces, the beam irradiance using the

method described in Section 2.1 and the diffuse irradi-

ance due to the sky using (9). This direct (sun+ sky)

irradiance is then used for the first pass of the reflection

calculation, so that (14) in the first instance represents a

single bounce of diffusely reflected radiation (assuming

all surfaces to be Lambertian). This process may be re-

peated, but now using a contribution from (14), n times

for all obstructing surfaces until we reach some con-

verged solution. If the radiance of each surface is held in

memory and we have many surfaces to solve for (e.g. in

the case of multiple buildings as in SUNtool), then the

apparent overhead of the iterative reflection algorithm is

generously shared so that the resultant run time is rea-

sonably modest.

To complete the model, we must consider reflections

from below the horizontal plane (IqL;b). For this we can

follow a similar principal to that established for above

horizontal reflections. We do this using an upside down

vault (Fig. 5) for which expression (14) simplifies to:

IqL;b ¼X145i¼1

ðIgbq=pÞiUi cos ni ð15Þ

and the reflection algorithm described in the previous

paragraph for 1; 2; . . . ; n bounces applies. In fact, for

computational efficiency the upper and lower reflections

may be carried out as part of the same routine. Alter-

natively, for large numbers of surfaces it may be more

efficient to solve for reflection in matrix form (Appendix

B). In either case, by combining obstructions into

aggregated patch occlusions whose radiance is given by

the dominant surface and solving for reflections between

these patch occlusions––(9)–(15) collectively describe a

simplified radiosity algorithm (SRA). 4

Fig. 6. Diffuse irradiation surface plots for isotropic (upper left) and anisotropic (upper right) canyon models and corresponding

difference image (bottom).


Alternatively, on the basis that model errors are

likely to be relatively weakly sensitive to errors in

accounting for the irradiance contribution from surfaces

below the horizontal plane, this contribution may be

represented by an equivalent isotropic ground plane

whose reflectance is a function of the below horizontal

obstructing surfaces. In order to calculate the irradiance

reflected by this ground plane, some means of deter-

mining the irradiance incident on it is required. This may

be achieved simply by passing a UHA for each surface,

which allows a sky view factor for the ground plane to

be determined (as with the isotropic model). The sim-

plification of an isotropic sky can then be used to

determine the ground plane irradiance. The error

introduced by this assumption will most likely be greater

than for the anisotropic canyon model, since the ground

reflected component will generally be greater in this case

(as the ground is consistently adjacent to the surface).

5 For all that follows, results have been produced for climate

data relating to Kew, UK for 1967.

3. Comparisons I––unobstructed sky

To verify the implementation of the sky models and

associated solar geometry results from ESP-r (Clarke,

2001) for the isotropic and anisotropic canyon models

and RADIANCE (Ward Larsen and Shakespeare,

1997)––with the additional module Gendaylit––for the

SRA were compared, using equivalent inputs. Having

achieved similar results, it is interesting to compare their

relative performance. 5 A useful approach to this is to

generate and compare irradiation surface plots (after

Robinson, 2003), based on solving for annual irradia-

tion (in this case diffuse only) incident on a plane for 5�bins of surface azimuth and altitude (entailing a total of

1387 annual calculation sets).

Taking first of all the isotropic model we see (Fig. 6)

the expected iso-contour banding, as the irradiation re-

sponds only to plane altitude. The anisotropic model on

the other hand depicts a concentration of energy about

the southern orientation. Perhaps most striking is the

irradiation difference plot which exemplifies the value in

considering sky anisotropy (in thermal simulation pro-

grams for example). Here we see that the isotropic

model considerably under-predicts the incident irradia-

tion on south facing tilted surfaces (by as much as 15%)

and over-predicts that received by north facing tilted

surfaces (by as much as 23%).

The difference between results from the (radiance

distribution based) SRA and IC models is further

exaggerated (Fig. 7), as there are now additional sources

Fig. 7. SRA irradiation surface plot (upper) and irradiation difference plots based on the isotropic (lower left) and anisotropic (lower

right) canyon models.

Table 1

Run time to produce an irradiation surface plot for an unob-

structed view of the sky vault together with associated RMS

error compared with RADIANCE

Model RMS error

(Wm�2)

Run time (s)

Isotropic canyon 32.4 7

Anisotropic canyon 11.8 34

SRA 0.6 2.11· 103SRA (archived radiance

distribution)

0.6 22

RADIANCE 0.0 4.99· 106


of anisotropy––within the treatment of the horizon band

and background scattering––so that south and north

facing surface irradiation are under-predicted and over-

predicted by as much as 26% and 18% respectively. The

contribution of this more rigorous treatment of sky

anisotropy can be appreciated by reference to the

(SRA � IC) difference plot shown lower right in Fig. 7,

as south facing vertical surfaces receiving correspond-

ingly more irradiation (ca. +5%) than those facing north

(ca. �4%), when compared to the anisotropic tilted

surface model.

A further basis for comparison that is interesting is

RMS error and how this relates to run time (Table 1), to

aid with selection of a model of greatest efficacy. The

first three rows show the expected reduction in RMS

error (as compared with RADIANCE 6) with improved

treatment of sky anisotropy, but at increasing compu-

tational cost. However, as noted earlier, it is sensible to

pre-process this radiance distribution (as well as patch

view factors) and to refer to these values, held in mem-

6 This table, perhaps unfairly, assumes that RADIANCE

should be our truth. For this unobstructed case this is not so, as

we have a direct analytical solution for the problem (ignoring

minor discretisation errors) and RADIANCE’s predictions

contain numerical errors––though these may be small, depend-

ing upon the settings used.

ory, at each time step. Consequently the run time for the

SRA improves to below that of the anisotropic tilted

surface model with an accuracy that is comparable to

RADIANCE.

4. Comparisons II––obstructed sky

The process of testing the incorporation of obstruc-

tion sensitivity into our diffuse radiation models has

been carried out in two phases. In the first instance we

have constructed representations of obstructions that

are black (non-reflective)––so that we test only the cor-

responding reduction to sky radiation. Following from

this we model grey obstructions, which are highly

Fig. 8. Obscured sky patches (left) from which a corresponding plane (centre) and an equivalent plane of constant height (right) was

produced.


reflective to exaggerate their influence on results. As in

Section 3 we have used RADIANCE as our truth model.

4.1. Black obstructions

For convenience our obstructing surface is simply a

cut out of the Tregenza sky (Fig. 8––left), so that whole

patches are obscured, for use within the SRA. These are

selected to test means for obscuring circumsolar and

horizon brightening. In the first instance the angular

coordinates of each patch vertex are translated to planar

coordinates relating to a surface at a distance of 20 m

from our viewpoint, for use within RADIANCE (see

Fig. 8––centre 7). From this we use the procedure de-

scribed in Appendix A to derive a UHA of an equivalent

continuous horizon, from which a plane of constant

height is found (Fig. 8––right), for use within the iso-

tropic and anisotropic canyon models.

In the first instance a test was performed to ensure

that the process for determining the UHA and from this

the constant height plane was correct. For this the

varying height plane was modelled using RADIANCE

and the result (Idb) used with a re-arrangement of (2) to

give u. The numerically derived value for u (17.04�) wasfound to be within 0.3� of that predicted by RADI-

ANCE (16.75�)––well within the error tolerance dis-

cussed in Appendix A––so that we can be reasonably

confident in the validity of the comparisons of models.

Two means of accounting for visibility of the cir-

cumsolar point source in (7) have been tested. In the

first instance this is based simply on whether the

solar altitude (cs) exceeds the altitude of the constant

height obstruction at the solar azimuth (as) (i.e.

cs > u cosðaw � asÞ). In the second case we use a look up

7 So that for a perpendicular distance d to our plane a vertex

(of angular height c and azimuthal offset Da) height h in y is

given by h ¼ d tan c= cosDa and the offset distance d in x by

d ¼ d tanDa.

table of pre-processed solar views based on the real

geometry. 8

The IC model tends to under-predict the surface

irradiance (Fig. 9) as the models were run for a south

facing surface; shown on the probability plot (Fig. 10) as

the relatively large area to the left of the origin. It should

be noted that relative error for the isotropic model peaks

at around zero––due to occasions when the circumsolar

region (in RADIANCE’s representation of the Perez all

weather luminance distribution model) is completely

obscured by the obstruction. In that situation the sky

radiance distribution will be close to isotropic (as most

of the horizon band is also blocked by the obstruction).

The corresponding RMS error is 23.45%.

The AC model with simple circumsolar visibility

tends to over-predict the surface irradiance as the south

facing surface ‘sees’ the sun whenever it is at an altitude

greater than u cosDa, whereas with the real obstruction

the circumsolar region is not visible around midday,

when the circumsolar region tends to have the largest

influence. The RMS error in this case is 15.6%. This is

reduced to 9.43% with the improved treatment of cir-

cumsolar views.

Excellent agreement is seen between the results from

the ray tracer and the SRA, with an RMS error of 0.3%.

4.2. Grey obstructions

The inclusion of reflected radiation has been tested

incrementally; in the first instance considering diffuse

radiation only and in the latter for global radiation, in

which case views to the sun from each surface are solved

for. The corresponding results are described below.

8 Note that the circumsolar visibility is determined using a

sun view factor for a whole surface rather than a point. This is

because the view factor for a point would be a Boolean value

(as the sun is assumed to be a point source), which leads to large

inaccuracies when the data is interpolated.

Fig. 9. Comparison of IC (upper left) and AC models without (upper right) and with (lower left) improved circumsolar views and SRA

(lower right) with RADIANCE predictions, using the Perez all weather luminance distribution model.

Fig. 10. Model error probability distributions (relative to RADIANCE) for a black obstructing surface.


4.2.1. Diffuse radiation

In order to test the validity of the way in which

reflecting surface radiances were derived in the canyon-

based models, some initial comparisons with RADI-

ANCE using an isotropic sky model were carried out,

isolating the ground and opposing surface contributions

respectively (by adjusting their reflectances). For these

tests, the surface was split into five horizontal slices (Fig.

Fig. 11. Geometry of the test case illustrating surface discretisation (left) and ground plane flanges (right).


11). In the first instance, the IC model was found to

slightly under-predict (by 3.4%), as brightening of the

floor towards the ends of the canyon was ignored (recall

that for the IC model the surface is abstracted to a

continuous obstruction). In the latter case, the IC model

somewhat over-predicted (by 1.8%) as the dominant

reflecting surface for below horizontal obstructions

tends to be that which is perpendicular (owing to the

angle of incidence of lower surfaces), so that the dark-

Fig. 12. Grey obstruction comparison of IC (upper left), AC with

ground plane (lower left) and SRA with inverted sky dome (lower ri

weather luminance distribution model.

ening of surfaces close to the floor tends to be ignored.

Note that the above horizontal surfaces tend to have

relatively un-obscured views of the sky. These errors,

which tend towards cancelling, were assumed to be

reasonable, so that comparisons progressed to determine

the validity of the models themselves. As before, model

predictions are compared with RADIANCE using the

Perez all weather luminance distribution model (Fig.

12).

improved circumsolar views (upper right), SRA with isotropic

ght) models with RADIANCE predictions, using the Perez all


As with the black obstruction case, the IC model

tends to under-predict surface irradiance for this south

facing surface. Only results for the improved treatment

Fig. 13. Model error probability distribut

Fig. 14. Global irradiance comparisons of IC (upper left), AC with im

sky (lower) models with RADIANCE using the Perez all weather lum

of circumsolar brightening in the AC model are pre-

sented, as the trend towards improved prediction reso-

lution is similar to that observed previously. For this

ions for a grey obstructing surface.

proved circumsolar views (upper right) and SRA with inverted

inance distribution model for south facing grey obstructions.

9 Note that interpolation errors should be inversely propor-

tional to surface size.

Fig. 15. Global irradiance comparisons for North facing grey obstructions.


model, there is also a tendency to under-predict (as the

background source is isotropic), but the scatter is re-

duced due to improvements in modelling the circumso-

lar region––although its concentration into a point

source leads to an over-prediction strand.

It is clear from the lower charts in Fig. 12 that the

method of calculating below horizontal reflected irradi-

ance using an isotropic ground plane leads to a sys-

tematic under-prediction. This is principally due to

errors involved with deriving the mean radiance of the

ground plane and in simplifying the treatment of the

angle of incidence of reflected energy on our receiving

surface. The SRA implementation (with an inverted sky

dome) has no such limitations, so that the agreement is

correspondingly good (Fig. 13).

4.2.2. Global radiation

To complete the comparisons between models, the

effects of direct insolation on both receiving and

reflecting surfaces has been accounted for. As noted in

Section 2.1, this is based on scaling incident beam irra-

diance according to the proportion of the surface that

can ‘see’ the sun. It is desirable to reduce the number of

hours at which the sun views are calculated to limit pre-

processing computation and on-line storage demands.

Typically, hourly view factors are calculated for one day

per month between and including the solstices. To re-

duce errors between these points in time to within

reasonable limits, two-dimensional interpolation is re-

quired 9 (to account for solar to clock time conversion

as well as days between months). Two interpolation

methods have been tested––linear and three-point

Lagrangian. For these two cases the RMS error for our

south facing surface is 10 and 6.8 Wm�2 respectively,

based on the SRA, so that the modest increase in com-

plexity is warranted.

Fig. 14 compares results for the three models,

assuming Lagrangian interpolation of solar views. Due

to the magnitude of direct irradiance, the degree of

scatter due to the IC and AC models is much reduced.

Nevertheless, the relative accuracy of the SRA––for

which the visible scatter is due to remaining solar view

interpolation errors––is clearly apparent. With a

diminished solar influence Fig. 15, produced for a north

facing surface, illustrates a substantial variation in

accuracy between the models. Differences between the

models for other orientations can be expected to lie

between these extremes.


5. Conclusions

Three models have been described which account for

the influence of obstructions in reducing sky radiation

and contributing reflected radiation. The first two ab-

stract adjacent urban obstructions into an effective

canyon, with an opposing face that is parallel to the

receiving plane and infinitely long. This has been

implemented using both isotropic and anisotropic tilted

surface irradiance models. The third is more physically

meaningful, since the geometry of the urban skyline is

preserved. For this, two hemispheres (above and below

the horizontal) are discretised into patches for which the

radiance of the sky and dominant obstructions (if these

exist) and associated view factors are used to determine

the corresponding contribution to surface irradiance––

so that we have a simplified radiosity algorithm (SRA).

The ray tracing program RADIANCE has been adopted

as a truth with which to compare predictions from these

models. From this study we conclude that

• The current general practice of ignoring obstructions

to the sky vault leads to considerable error in predict-

ing surface irradiance.

• Differences in computation time between the three

implementations are relatively small.

• For the UK climate, the isotropic canyon model

tends to considerably under-predict for south aspects

and over-predict for northern aspects, particularly in

unobstructed situations. The anisotropic canyon

model follows a similar but dampened trend.

• In obstructed situations, the re-use of solar visibility

profiles to determine the visibility of the circumsolar

point source improves the accuracy of the anisotropic

canyon model reasonably significantly. On a related

note, it is recommended that Lagrangian interpola-

tion should be used to determine solar visibility from

look up tables based on hourly solar trajectory dis-

cretisation at monthly intervals.

• The SRA is considerably more accurate than the

alternatives tested. This approach also offers longev-

ity, since the technique may be used in conjunction

with future relative sky luminance/radiance distribu-

tion models.

• Finally, the core of the SRA may be adapted to pre-

dict illuminance, both internal and external. One such

implementation will be reported in a future paper.

The proposed SRA will be incorporated within a new

sustainable urban neighbourhood modelling tool

(SUNtool) as part of an EC-funded research project. In

principal this model is amenable for inclusion within any

software application for which obstruction-sensitive

irradiance predictions are important, such as dynamic

thermal simulation and renewable (solar) energy mod-

elling programs.

Acknowledgements

The funding for this work by the European Com-

mission’s Directorate General for Transport and Energy

is gratefully acknowledged.

Appendix A. Determination of UHA

The equivalent UHA u of some arbitrary arrange-

ment of adjacent obstructing surfaces may be found

from the following expression:

1

2cosðb þ uÞ ¼ 1

p

Z ZS

cos n dx ðA:1Þ

where n is the angle of incidence of some small

obstruction element and dx its solid angle. The accuracy

with which u can be solved for with this expression,

using some numerical procedure may be determined as

follows. Suppose it is required that j DIdb

Idhj6 0:01, where

DIdb is the error in the calculated incident diffuse irra-

diance. This error can be approximated by

DIdb ¼ dIdb

dudu ¼ � sinðb þ uÞ

2Idh du

where du is the error in the calculation of u.The worst case will therefore be when sinðb þ uÞ ¼

�1. Therefore 12du6 0:01, i.e. u must be calculated to an

accuracy of 0.02 rad, or 1.16�.In order to calculate the reflected component of

irradiance, a value for the reflectance of the equivalent

obstruction is required. This may be determined either

by averaging over all of the obstructing surfaces or by

simply taking the reflectance of the main obstructing

surface (i.e. the surface that makes the greatest contri-

bution toR RS

cos n dx).

Appendix B. Solution by matrix inversion

The equations presented in Section 2 are sufficient to

calculate the irradiance on a surface using various sky

models and given certain geometric information. The

solution of the equations however is not straightfor-

ward. The inclusion of reflections means that the irra-

diance of a particular surface potentially depends on the

irradiance of many other surfaces, particularly in the

case of the SRA, where up to 290 different surfaces could

be involved in the reflection calculation (one for each

patch).

An obvious way to solve this system of equations is

by iteration, using the component of irradiance received

directly from the sky and the sun as a starting condition.

This approach is reasonably physically realistic, simu-

lating the effect of successive reflections (albeit in a crude


way). An alternative approach would be to formulate

the problem as a matrix equation and solve by inversion.

For the SRA this would give Id ¼ AIg þ BR; where

Ig ¼ Id þ Ib, is a vector listing the global irradiance on

each surface, and R a vector giving the radiance of each

sky patch.

Rearranging:

Id ¼ ðI � AÞ�1ðAIb þ BRÞ ðB:1Þ

The matrix A is square and describes how the direct

component of irradiance falling on each surface is

eventually distributed around the n surfaces in the world

(entry ði; jÞ in the array describes the proportion of di-

rect insolation on surface j that is reflected to surface i):

A ¼

q1k1;1p

q2k1;2p qnk1;n

p

q1k2;1p

. .. ..

.

..

. . .. ..

.

q1kn;1p

q2kn;2p qnkn;n

p

2666664

3777775

ðB:2Þ

where qi is the reflectance of surface i, and ki;j describes ascaling factor for the effect of the energy reflected from

surface j to surface i. If surface j obstructs m sky patches

when viewed from surface i, denoted by x1; x2; . . . ; xm,then

ki;j ¼Xmk¼1

Ui;xk ð1� ri;xk � rself ;xk Þ cos ni;xk

(after (13)) (ri;xk is the view factor from surface i to sky

patch xk and Ui;xk is the solid angle of sky patch xk fromsurface i).

Matrix B describes the contribution from each sky

patch (of unit radiance) to the irradiance received by

each surface within the world:

B¼

U1;1r1;1 cosn1;1 U1;2r1;2 cosn1;2 U1;pr1;p cosn1;p

U2;1r2;1 cosn2;1. .. ..

.

..

. . .. ..

.

U1;nr1;n cosn1;n U2;nr2;n cosn2;n Un;prn;p cosnn;p

266664

377775

ðB:3Þ

The matrices ðI � AÞ�1A and ðI � AÞ�1B need only be

computed once for any given geometry. Therefore at each

time step once the direct component of irradiance has

been solved for, only two matrix multiplications and an

addition are required to solve for the diffuse component.

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