m et al, 2011) 
to account. The 
ulated with the 
at any specific 
ith is measured 
ith will be close 
> to zero degree 
s depends upon 
ards the Sun 
  
phere measured 
the surface by 
nce of surface 
ed. The motion 
wer received by 
tion is the most 
arallel and it is 
ray has an angle 
must pass these 
(D 
applied to avoid 
llation. So, for 
ninimum length 
  
and width of a PV array must be fulfilled. For rectangular 
shaped surface: 
L » (2) 
Ww 2w, 
where |, = length of surface 
w, = width of surface 
1, 7 minimum length requirement for PV array 
wg 7 minimum width requirement for PV array 
Irregular shaped surface are placed in a bounding box 
(approximated by a smaller box) determined by south-west and 
north-east corner or upper-left and bottom-right corner. So, for 
irregular shaped surface: 
> 3) 
W, DUM 
where I; = length of bounding box 
w, = width of bounding box 
2.3 Surface subdivision 
In this step each surface of the city model is subdivided into 
triangles. Each triangle is given a subdivision index. The 
triangles extracted from potential surfaces are further 
subdivided into smaller triangles. This subdivision for potential 
surface is done in two steps: thin triangle subdivision and 
regular triangle subdivision. 
23.1 Thin triangle subdivision: If a triangle has an angle 
less than 30 degree then this subdivision is applied. Here the 
thin triangle is divided into two triangles by joining the middle 
point of the longest side and the opposite vertex. The resulting 
triangles are assigned with thin triangle subdivision indices. The 
resulting triangles are further passed through the triangle 
subdivision process. 
23.2 Regular triangle subdivision: If thin triangle 
subdivision process is passed then the triangles are passed 
through this regular triangle subdivision process. A maximum 
distance between two points is assumed as the resolution (i.e. 
10 cm). Each sides of the triangle are divided into two parts. 
The middle points are connected to form four smaller triangles. 
Then each resulting triangle is further passed through this 
process until the smallest side of the triangle is less than 
resolution. The final triangles are assigned with regular triangle 
subdivision indices. 
24 Sun's ray calculation 
Centroids of final subdivided triangles are assumed as the target 
points for which shadow will be calculated. Whether the 
triangle will be in shadow or not will be determined by this 
point. The quality of result will be determined by the 
distribution of these points, which depends on the resolution. 
For each target point, a distant point is measured in sun's 
direction (which varies with time) at a minimum distance and 
above the top most point in the city model. 
  
2.5 Potential shadow caster filtering 
For each point, the whole city models are divided into four 
quadrants, divided by the north-south and east west axis. The 
quadrant, which contains the sun, is marked as active quadrant 
and surfaces, which have at least one vertex in this area, are 
selected as potential shadow caster surface. Surfaces, which are 
below the target point, are further filtered from the selection by 
comparing the elevation or height of each vertex of the surface 
with the target point. 
2.6 Shadow calculation 
This is the main step, where shadow is calculated. A line-plane 
intersection check is performed here. A line can be expressed 
as: 
LU, la): (4) 
teR 
where ll; = target point (Xa,Ya,Za) 
I, = distant point (xb,yb,zb) 
A triabgle can be expressed as: 
P, (B - B)u* (P, - Bv; (5) 
u,veR 
where P,y= Vertex (xx, Yw Ze); k = 0,1,2 
At intersection point the point on line will be equal to the point 
on surface so by solving the equation in matrix form: 
x x X x, UxpX) 7X,—X$]4 
Ya Vale, M Vo Va Yo 
=, 20 Z TZ, S720. 22 Zo 
t,u,v e [0.1] (6) 
u+v<l 
where 1, = length of surface 
ws; = width of surface 
ln = minimum length requirement for PV array 
Wm = minimum width requirement for PV array 
If the line lies upon the surface or parallel then /, — [,, P; — Py, 
P, — P, will be linearly independent. In this case, if line lies 
upon the surface then the point must be also marked as 
shadowed. To determine if the line lays upon the surface a 
further line-line intersection check is performed for each side of 
the triangle with the sun’s ray by solving equation of two lines. 
If two lines are M, + (M, — Mjt; and N, + (N, — Nt, then t; 
and /; can be obtained by solving the equations for x and y. If 
the value for t;and f; also satisfies the equations for z then there 
is an intersection. Then it is checked if the intersection point 
lies within the lines. If all requirements are fulfilled for 
intersection then the point is marked as shadowed point.