Computer Applications in Electrical Engineering
Vol. 12
2014
Reflector shape design for small light
sources using merit function
Krzysztof Wandachowicz
Poznań University of Technology
60-965 Poznań, ul. Piotrowo 3a
e-mail: [email protected]
The article presents results of research on the calculation of the shape of a mirror
reflector which ensures the highest possible average distribution and uniformity ratio
of illuminance. Optimization calculations were carried out for different numbers
of interpolation nodes with the use of merit function.
KEYWORDS: luminaire calculation, optimization, merit function
1. Introduction
Optimization algorithms employ an iterative procedure of repeated evaluation
of objective function. A search takes place for the function extreme, which may
assume the values of luminous intensity or illuminance, or the difference
between the currently calculated value and the assumed value. The shape of
optical elements of a luminaire changes during the optimization process.
In the research completed so far I have presented the method and examples of
calculations in various configurations of shapes and sizes of reflectors, as well as
surfaces of illuminating sources of light [1, 2]. Among else, calculations were
completed for a reflector featuring a top opening and a bottom opening in the
shape of a square (Fig. 1). The shape of the four walls is generated by a profile
curve described with Hermite interpolation [3, 4]. The so-called LED module is
placed in the top opening. The module was created on the basis of technical data
of the Fortimo LED DLM 2000 module. The module's illuminating element is a
circle-shaped surface, approximately 6 cm in diameter, covered in luminophore.
The luminous distribution of this surface is almost Lambertian. The luminous
flux value is 2000 lm. The surface source of light, whose diameter is 6 cm, has
dimensions similar to the reflector's dimensions, whose top opening is 6 cm
wide, the bottom opening is approximately 10 cm wide and whose height is
about 8 cm.
The designed reflector is placed at the height of 3 meters above the middle of
a square area, whose width is 3 meters, and serves to ensure the highest possible
illuminance (at the assumed uniformity). The efficiency of illumination defined
as the ratio of the luminous flux incident on the illuminated surface, to the sum
551
K. Wandachowicz / Reflector shape design for small light sources using ...
of luminous fluxes of the lamp in the lighting installation, may serve to evaluate
the properties of the designed luminaire. Taking into account the calculated,
average value of illuminance (116,4 lx) and the area of the illuminated surface (9
sq m), the illumination efficiency is equal to 52%. This is not a very high value.
However, given the large area of the source of light, comparable to the size of
the reflector, it is impossible in this case to limit the emitting luminous flux only
to the area to be illuminated. The luminous flux will be also emitted outside this
area. This phenomenon is intensified by the mismatch of the shape of the source
of light (a circle) and the shape of the opening of the reflector (a square). To
ensure proper uniformity of the illuminance in the illuminated, square surface,
the output opening should be also a square. The light point figure generated on
the reflector's surface, if it is to fill up the entire surface of the reflector, will
require that some of the fluxes will have to be radiated outside the angle
determined by the direction of observation. Despite the difficulty of the task, the
operational efficiency of the designed luminaire is 85%.
Fig. 1. The figure presents a model of a reflector with a top and a bottom opening in the shape
of a square; obtained luminosity curve (continuous line - plane C0-C180;
dotted line - plane C45-C225)
The task discussed above proved difficult to complete as it was necessary to
obtain a complicated shape of the photometric solid and the mismatch of the
shape of the illuminating surface of the source of light and the bottom opening
of the reflector. The obtained illumination efficiency value may seem low.
However, it cannot be compared to efficiencies of currently produced luminaires
that serve tasks similar to the tasks served by the designed luminaire, since
similar solutions were not identified in the market. Examples of luminaires with
reflectors, whose purpose is to illuminate surfaces with the designed uniformity,
552
K. Wandachowicz / Reflector shape design for small light sources using ...
are presented. Still, in these situations simple tasks are most often presented,
where the luminous intensity in the reflector's axis is equal to the luminous
intensity of the source of light, and the reflector's purpose is to ensure proper
luminous intensity value at larger angles. In the discussed example the luminous
intensity in the reflector's axis coming from the surface of the source of light
provides illuminance of just 70 lux from a distance of 3 meters. This is not
enough to complete the designed target, namely a high value of average
illuminance (over 100 lx) and uniformity of 0,7. The reflector must participate in
the creation of a luminous intensity, both in near zero angles (the reflector's axis)
and in critical angles defined by the edge of the illuminated surface.
2. Model of reflector with small size light source
In the example described in the previous item, a light source of large
dimensions in relation to the reflector's size was applied. Moreover, the shape of
the source of light (circle) was mismatched with the shape of the output opening
of the reflector (square). Consequently, a relatively low lighting efficiency was
achieved. The following example is to show whether the use of a smaller light
source leads to increased illumination efficiency.
The curve that determines the reflector's profile passes through P points that
create the so-called interpolation nodes (Fig. 2.). 3rd degree Hermite
polynomials are used to interpolate the reflector profile's shape between
interpolation nodes [3, 4]. The reflector model is created by rotating the profile
curve around the Z axis. The location of the initial point P1 has been set as fixed.
The remaining points change their locations both in the Z axis and in the X axis.
In this case, the operation of the optimization algorithm may result in the change
of height and width of the reflector. The identification of location of
interpolation nodes in the X axis is done with a single variable dx that is
responsible for changing the value of the coordinate X of the last point P5. The
coordinates of the remaining points are calculated taking into account identical
distance Δx in the X between the subsequent points. Three models of the
reflector were created. The first one uses four points that are interpolation nodes,
for the second one this number was increased to five, and for the third one the
number was increased to six. The application of a higher number of interpolation
zones will allow a more accurate control of the shape of the reflector's profile.
This is of special importance in the case of small-size sources, where minor
differences in the reflector's shape may lead to relatively serious changes in the
shape of the luminous intensity curve. The optimization algorithm operates on
five variables (z2, z3, z4, z5, dx) in the first case, on six variables in the second
case (z2, z3, z4, z5, z6, dx) and on seven variables in the third case (z2, z3, z4, z5, z6,
z7, dx).
553
K. Wandachowicz / Reflector shape design for small light sources using ...
In the top opening a model of a source of light was placed, whose diameter is
6 mm (ten times less than in the previously discussed example), and the
luminous flux has a value of 2000 lumen. It was assumed that the distribution of
light on the surface of the light source is Lambertian.
In the case of small-size sources, it is crucially important to determine the
initial size of the reflector that will be subject to correction by the optimization
algorithm in the calculation process. Wrongly made initial assumption may lead to
outcomes that are very far from the optimal one. The line creating the critical
angle g of flux radiation should cross the edge of the illuminated surface (Fig. 2).
Dx
P1
z2
z3
Dx
x2
Dx
x3
Dx
x4
x5
ub
P2
Z
P3
Pi
z4
light source
X
Pp
P4
lb
P5
z5
Pk
ag
dx
Fig. 2. Reflector's profile with interpolation nodes P1÷P5, the ub and lb lines determine
the permissible boundaries of location of interpolation nodes; critical angle g of flux radiation
Table 1. Initial coordinates of points determining the reflector's size
Name
Initial point Pp
End point Pk
Coordinates
x = 0.01 m
z = 0.00 m
x = 0.05 m ± 0.02 m
z = -0.10 m ÷ -0.80 m
Assuming the above conditions, initial coordinates of points determining the
reflector's size were identified (Table 1). The initial point Pp does not change its
position during the optimization algorithm, while the end point Pk changes its
position in both axis X and axis Z. Consequently, the operation of the algorithm
leads to the change of the size of the reflector.
554
K. Wandachowicz / Reflector shape design for small light sources using ...
3. Optimization of reflector’s shape
The model of the reflector is placed at the height of 3 meters above the
middle of a round area whose diameter is 3 meters. Fifteen calculation points are
distributed along the radius of the illuminated surface, with a pitch of 0,1 m
(from 0,05 m to 1,45 m). The calculation of the arithmetic mean from
calculation points in this layout is not equivalent to the calculation of the average
value of illuminance on the circle's area. For that purpose, the value of the
average illuminance Eav is calculated from the dependence (1). Assuming
distances between grid points to be 0.1 m, the fraction in the formula (1) which
is the sum of the relation of the squares of radii of the grid will have the form of
a sequence of values from 1 to 29 that change with a pitch of 2.
n
E śr   Ei
ri  ri1 2
r12
i 1
n

ri  ri1 2
i 1
r12
(1)
where: Ei - value of illuminance in point i, ri - radius determining grid
boundaries, i=1, 2, …, n, r0=0 (Fig. 3).
Y
i+3
i+2
i+1
i
ri
ri+1 ri+2 ri+3 X
Fig. 3. Presentation of layout of calculation points along the radius of the illuminated surface
The objective function with penalty function has been used in the research so
far. The application of the penalty function was to reject solutions not satisfying
the assumed criterion of uniformity of illumination [1]. In the described example
the objective function in the form of the weighted sum of squares of differences
between the calculated value and the expected value has been introduced (2),
hereinafter the merit function.
n
2
F ( X )   Wi H i  H ci 
(2)
i 1
where: Hi - the value of a specific parameter (e.g. the value of illuminance)
calculated from the operation of the optimization algorithm, Hci - the expected
value of the specific parameter (e.g. illuminance), Wi – the weight determining
the share of a given parameter in the value of the objective function.
555
K. Wandachowicz / Reflector shape design for small light sources using ...
In this particular case the new form of the objective function is built as
follows:
– for i=(1, 2, …, 15): Hi – the value of illuminance in point i; Hci =300;
Wi+1 = Wi +2, W1 =1,
– for i=(16, 17, …, 20): Hi – the value of illuminance in point i; Hci =0;
Wi =100,
– for i=21: Hi – the value of uniformity ratio of illumination calculated for
points i=(1, 2, …, 15); Hci =0,7; Wi =1000 for Hi < Hci, Wi =0 for Hi > Hci.
The first fifteen calculation points are distributed with a pitch of 0,1 m within
the illuminated area. The subsequent points (from 16 to 20) are situated outside
the illuminated area (from 1,55 m to 1,95 m). The optimization algorithm
attempts to achieve a shape of the reflector which would ensure:
– illuminance whose value is closest to 300 lx on the illuminated area,
– illuminance whose value is closest to 0 lx outside the illuminated area,
– solutions ensuring uniformity ratio of at least 0,7 are awarded bonus points.
A genetic algorithm with the following solutions was used for optimization
calculations (the search for the minimum of the objective function) [5, 6]:
– floating point representation - for approximating the algorithm to the
objective's space, two points located close to each other in the representation
space will also be close to each other in the objective's space,
– scaling of objective function by appointing ranks equalizes the score of less
adapted individuals, while maintaining a high diversity within the population,
– linear selection carries the two best solutions over to the subsequent
generation,
– heuristic crossing and mutation with Gaussian distribution (magnitude of
mutation decreases with each new generation),
– strategy of alteration of population diversity [7].
Table 2 presents a list of results obtained for a model with four (4p), five (5p)
and six (6p) interpolation nodes.
Table 2. List of obtained results
Model description
Eav [lx]
Uniformity Emin/Eav
4p
246,7
0,69
5p
251,3
0,89
6p
250,5
0,79
The best result was obtained for a model of a reflector with five interpolation
nodes. The obtained value of average illuminance is 251,3 lx, and the surface of
the illuminated area is 7,1 m2 (the area of a circle whose diameter is 3 m). The
efficiency of lighting with the use a luminaire with a calculated reflector whose
556
K. Wandachowicz / Reflector shape design for small light sources using ...
reflection coefficient is ρ = 0,9 is equal to 89%. This value significantly exceeds
the value of efficiency of the reflector with an opening in the shape of a square
with a large light source. In this way the previous conjectures on the impact of a
large area of light source and the mismatch of the shape of the reflector's
opening with the shape of the source of light on the obtained value of lighting
efficiency were confirmed. The luminaire's efficiency is 91%, meaning that only
2% of the flux is radiated outside the illuminated area. Increasing the number of
interpolation nodes to six did not improve the result.
Figure 4 presents the calculated shape of a reflector profile. The optimization
algorithm led to achieving a triple curvature profile and the profile curve has
two, distinctively marked parts. The upper part responsible for illuminating the
central zone of the illuminated area and the bottom part responsible for
illuminating zones at the perimeter of the illuminated area.
Fig. 4. Drawing of the calculated reflector profile, discretization of the reflector area,
view from the side and view from the back
Figure 5 shows the distribution of illuminance along the radius of the
illuminated surface. Figure 6 presents the luminosity curve of a luminaire with a
reflector (the sum of luminosity of the LED and the reflector), whose shape was
calculated with the optimization algorithm. The lack of monotonicity in the
central part of the curve is caused by the complicated, triple curvature shape of
the reflector. There is an evident distinction in the curve that ensures
illumination of the central and extreme zone of the illuminated area.
557
K. Wandachowicz / Reflector shape design for small light sources using ...
300
250
E [lx]
200
150
100
50
0
0.0
0.5
1.0
r [m]
1.5
2.0
Fig. 5. Distribution of illuminance along the radius of the illuminated surface
Fig. 6. Luminous intensity distribution of the calculated luminaire
4. Conclusions
The conducted tests show that the new form of the objective function leads to
increased efficiency of the optimization algorithm. This is of key importance
when a minor change in the data leads to significant changes in the result. At the
same time it was proven that the low value of lighting efficiency achieved with a
large-size light source does not result from wrong operation of the optimization
algorithm. A high value of lighting efficiency was obtained for a small-size light
source whose shape was adjusted to the reflector shape.
Increasing the number of interpolation nodes from four to five improves the
obtained results. However, increasing it further (to six, in this case) has no
significant impact on the obtained result and may lead prolonged calculation
time.
558
K. Wandachowicz / Reflector shape design for small light sources using ...
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
Wandachowicz K.: Optymalizacja kształtu odbłyśnika zapewniającego uzyskanie
zakładanego poziomu równomierności oświetlenia. Przegląd Elektrotechniczny
5a/2012. Warszawa, Sigma-Not. 181-183.
Wandachowicz K., Kuczko W.: Weryfikacja metody obliczania odbłyśników
opraw oświetleniowych. Przegląd Elektrotechniczny, nr 1/2014, Warszawa,
Sigma-Not, PL ISSN 0033-2097, 281-284
Wandachowicz K.: Obliczanie profilu odbłyśnika z wykorzystaniem interpolacji
Hermite'a. Materiały konferencyjne: XV Conference Computer Applications in
Electrical Engineering, ZKwE'2010, Poznań, 19-21.04.2010, 231-232.
Fritsch F. N., Carlson R. E.: Monotone Piecewise Cubic Interpolation. SIAM
Journal on Numerical Analysis, 17 (1980), 238-246.
Michalewicz Z.: Algorytmy genetyczne + struktury danych = programy
ewolucyjne. WNT Warszawa 2003.
Global Optimization Toolbox User’s Guide. The MathWorks, Inc.
Wandachowicz K.: Optymalizacja profilu odbłyśnika z zastosowaniem strategii
zmiany różnorodności populacji, XXII Krajowa Konferencja Oświetleniowa
Technika Świetlna 2013, Warszawa 21-22 listopada 2013, materiały
konferencyjne.
559