Journal of Applied Science and Engineering, Vol. 18, No. 2, pp. 167-172 (2015)
DOI: 10.6180/jase.2015.18.2.09
Optimizing the Tilt Angle of Solar Collector under
Clear Sky by Particle Swarm Optimization Method
Feng-Jiao Liu1 and Tian-Pau Chang2*
1
Department of Electrical and Information Technology, Nan Kai University of Technology,
Nantou, Taiwan 542, R.O.C.
2
Department of Multimedia Animation and Application, Nan Kai University of Technology,
Nantou, Taiwan 542, R.O.C.
Abstract
Sufficient energy supply is an essential factor for a country’s economic development. The
utilization of solar energy plays very important role in this issue. Solar energy can be utilized through
various marketing devices such as solar collector, photovoltaic cell etc. The amount of solar energy
could be maximized if solar collector always faces the Sun throughout the day. In this paper, a famous
empirical model suitable for clear sky is applied to calculate the solar radiation for different locations
given geographic latitudes. A heuristic algorithm, particle swarm optimization, is adopted to determine
the best installation angle of solar collector in Taiwan under clear sky taking into consideration of
various time periods. The results show that the particle swarm optimization is a powerful method in
optimizing the design of solar collector; the optimal tilt angles are positive for most of the months of
year while negative for summer months from May to July. The annual best tilt angles for station Taipei,
Taichung, Tainan, Kaohsiung, Hualien and Taitung are 22.4°, 21.5°, 20.5°, 20.2°, 21.3° and 20.5°
respectively. Furthermore the annual received energy is greater than the one incident upon the ground
surface by 6.5%, 6.0%, 5.9%, 5.6%, 6.0% and 5.5% respectively.
Key Words: Solar Energy, Solar Radiation, Solar Collector, Optimal Angle, Particle Swarm
Optimization
1. Introduction
Power supply affects a lot a country’s economic activity. Currently fossil fuel is the most important part of
power supply in the world; however for the consideration of the issue of environment pollution, renewable energy has been paid more and more attentions including
solar energy, wind energy, biomass energy, etc. Among
them solar energy is the best choice for most of the areas
over the world [1-5]. Solar energy can be utilized through
solar collector or photovoltaic (PV) cell. To maximize
the collected energy, proper installing of the collector is
really needed. The collector’s azimuth and tilt angles affect the incidence of sunlight upon its surface [6-11]. In
*Corresponding author. E-mail: t118@nkut.edu.tw
the northern hemisphere, the optimal azimuth must be due
south or due north, but the best tilt angle depends on geographic latitude and local climate factors. In this paper,
solar radiation considered is the one predicted by an empirical model that might represent the worldwide weather
situation under clear sky [12-14]. Six locations in Taiwan, i.e. Taipei, Taichung, Tainan, Kaohsiung, Hualien
and Taitung are selected to be examples. As known, determining optimal tilt angle of collector for a particular
time period and location is always a tedious problem.
Herein the paper, the heuristic algorithm, particle swarm
optimization (PSO), is applied to search the best solutions
for the six locations. Solar energy is set to be the objective
function while doing searching in PSO. Though the PSO
heuristic algorithm has commonly been used in many
fields for its excellent ability; but it is rarely found in the
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Feng-Jiao Liu and Tian-Pau Chang
research about this topic. The energy received by the collector with the best tilt angle will be compared with that
by the horizontal ground surface for different months,
meanwhile corresponding gains are discussed as well.
The results of the present study would be helpful for the
practical applications.
where dn is the day number as counted from January 1st
throughout the year (1-365) (i.e. Julian day). r is the
ground reflection coefficient (albedo), which is about 0.7
for snow covered ground and 0.2 for snow free ground.
It was assumed to be 0.2 in this study [3,4]. The clear
sky radiation on a horizontal surface, Ic, is the sum of Icb
and Icd [12-14]:
2. Solar Radiation
(7)
The radiation incident on a south facing collector tilted
at an angle (b) to the horizontal surface, as shown in Figure 1, is made up of three components: the direct beam,
diffusion, and reflection from the ground. The sum of these
three terms is called global radiation (It) [12-14]:
where Icb is the clear sky beam radiation on a horizontal
surface which can be calculated using the expression
derived by:
(8)
(1)
where b is positive for collectors facing due south but
negative for due north and zero for the horizon. The geometric factor (Rb) is the ratio of beam radiation on a tilted
surface to that on a horizontal surface and is given by:
where tb is the atmospheric transmittance of beam radiation and is equal to the ratio of the beam radiation on a
horizontal surface (Gb) to the air mass zero (AM0) extraterrestrial radiation (Go). Hence, tb = Gb/Go, and can
be established empirically:
(9)
(2)
where q is the instantaneous angle between the direct
beam and the normal vector of collector, and qz is zenith
angle:
The constants ao = ro a o* , a1 = r1 a 1* and k = rk k* for a
standard atmosphere with a visibility of 23 km can be
calculated from the following relationship by assuming
that the observation altitude is less than 2.5 km:
(3)
(4)
where f is the geographic latitude, w is the solar hour
angle which changes by 15 degrees per hour (and is zero
at solar noon, negative in the morning and positive in
the afternoon, e.g. -45° for 9 AM and +30° for 2 PM), y
is azimuth of the Sun, W is azimuth of the collector measured from due south, positive in the morning and negative in the afternoon. The solar declination (d) is the angle between the lines joining the centers of the Sun, the
Earth and the equatorial plane:
(5)
(6)
Figure 1. Geometry of solar collector.
Optimizing the Tilt Angle of Solar Collector under Clear Sky by Particle Swarm Optimization Method
(10)
where H is the altitude of the observer in kilometers.
The correction parameters ro, r1 and rk are related to climate conditions as summarized in Table 1 [12-14].
The accumulated extraterrestrial beam radiation at
AM0 for a period of time can be obtained from the integral:
(11)
where, Sc is the solar constant (1367 W/m2). The diffusion component of clear sky radiation on a horizontal
surface, Icd, can be estimated using the model:
(12)
(13)
where td is the atmospheric transmittance of diffuse radiation.
3. Particle Swarm Optimization
Particle swarm optimization is one of the famous meta-heuristic algorithms proposed firstly by Kennedy and
Eberhart [15], inspired mainly by the social behavior patterns of animals that live and interact within large groups.
PSO begins with random population of particles in search
space and incorporates swarming behaviors such as fish
schooling, bird flocking, etc. It stochastically assigns direction and velocity vectors to each particle, each particle
then moves or flies through the search space following
its velocity vector, which is adjusted by the directions and
velocities of other particles in its neighborhood. How
much influence a particular particle has on other particles
169
is estimated by its objective function. These localized interactions with neighboring particles propagates through
the entire swarm of potential solutions, each particle keeps
track of its own coordinates. This process is repeated until some condition is met. On the other hand, the PSO
finds the global best solution by simply adjusting the trajectory of each individual particle toward its own best location and toward the best particle of the entire swarm at
each iteration (generation). As shown in Figure 2, the position and the velocity vector of the ith particle in N dimensional search space can be expressed as Xi = [xi1,
xi2,…, xiN] and Vi = [vi1, vi2,…, viN] respectively. The
amount of solar energy captured by solar collector is set
to be the objective function (e):
(14)
The position of particle represents the candidate solution of the tilt angle (b) of solar collector. Note that the
tilt angle (b) is the only searching parameter while doing
PSO, other variables such as the study period, geographic
latitude, atmospheric parameter and so on are available
in this paper. According to the defined objective function,
the best position for the ith particle (local best) is Pi =
[pi1, pi2,…, piN], and the fittest particle found so far (global best) is Pg = [pg1, pg2,…, pgN]. The new velocities and
positions of the particles for the next fitness evaluation
are updated as follows:
(15)
(16)
Table 1. Atmospheric parameters for different climate
types
Climate type
Tropical
Mid-latitude summer
Sub-arctic summer
Mid-latitude winter
ro
r1
rk
0.95
0.97
0.99
1.03
0.98
0.99
0.99
1.01
1.02
1.02
1.01
1.00
Figure 2. Position and velocity vectors of particle in search
space.
170
Feng-Jiao Liu and Tian-Pau Chang
where C1 and C2 are the acceleration coefficients representing the cognitive and social parameter respectively,
rand1( ) and rand2( )are random numbers uniformly distributed within the range of [0,1].
4. Results and Discussion
Figure 3 shows the monthly optimal tilt angles under
clear sky for the six locations studied. Table 2 lists the
angle values for different time periods. It can be seen that
the angles during winter season reveal larger value than
Figure 3. Monthly optimal tilt angle of solar collector for six
locations studied.
other seasons, it is true because the Sun’s apparent position becomes lower in winter; those angles are about 50°.
The angles in summer season from May to July present
negative values implying that the best azimuth of the collector must be toward north; this is because the Sun’s
moving path is mostly staying in the northern sky in summer. The yearly best tilt angle is 22.4°, 21.5°, 20.5°, 20.2°,
21.3° and 20.5° for Taipei, Taichung, Tainan, Kaohsiung,
Hualien and Taitung respectively; they are a little smaller than the respective latitude. It is worth here to mention that the results obtained by the present paper are very
consistent with those by conventional trial-and-error
method shown in [10].
Table 3 shows the solar energy received for different
time periods. The energy during summer months is significantly larger than that during winter because of the longer sunshine time in summer, regardless of the locations
studied. Meanwhile the amount of solar energy incident
upon the tilted collector with its optimal tilt angles is always greater than that upon the horizontal surface; the
annual gains are 6.5%, 6.0%, 5.9%, 5.6%, 6.0% and
5.5%, respectively, for the six studied locations.
From the analysis aforementioned, it is worth to note
that the global radiation actually observed in Taiwan is
far less than the one predicted by the empirical model,
which might represent the average radiation worldwide
in clear sky. The average clearness index of the sky in
Taiwan is about 0.35 whereas the one calculated using
Table 2. Optimal tilt angle for different time periods (degree)
Taipei
(25.08°)*
Taichung
(24.15°)
Tainan
(23.00°)
Kaohsiung
(22.57°)
Hualien
(23.98°)
Taitung
(22.75°)
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
50.6
41.5
27.8
11.3
.-1.7
.-7.5
.-4.5
06.5
21.9
37.3
48.4
52.9
49.5
40.5
26.7
10.5
.-2.4
.-8.1
.-5.5
05.4
20.9
36.3
47.4
52.1
48.3
39.4
25.3
09.5
.-3.6
.-9.3
.-6.4
04.5
19.9
35.0
46.2
50.9
48.3
39.1
25.0
09.2
.-4.0
.-9.7
.-7.1
04.2
19.3
34.8
45.6
50.4
49.3
40.3
26.4
10.3
.-2.5
.-8.4
.-5.6
05.5
20.8
36.1
47.1
52.0
48.3
39.2
25.3
09.1
.-3.7
.-9.6
.-6.8
04.0
19.3
34.8
46.1
50.8
Yearly
22.4
21.5
20.5
20.2
21.3
20.5
Period
* Geographic latitude (Northern).
Optimizing the Tilt Angle of Solar Collector under Clear Sky by Particle Swarm Optimization Method
171
Table 3. Solar energy upon the collector for different time periods (kWh/m2)
Period
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Yearly
Collector
Taipei
Taichung
Tainan
Kaohsiung
Hualien
Taitung
Tilted
Horizon
Tilted
Horizon
Tilted
Horizon
Tilted
Horizon
Tilted
Horizon
Tilted
Horizon
Tilted
Horizon
Tilted
Horizon
Tilted
Horizon
Tilted
Horizon
Tilted
Horizon
Tilted
Horizon
Tilted
Horizon
Tilted/Horizon
187.7
128.8
179.4
141.6
209.6
190.1
213.1
209.6
230.8
230.7
228.4
226.9
232.5
231.7
221.5
220.6
204.1
192.5
200.9
166.3
183.4
131.2
182.2
119.8
2331.30
2189.10
1.065
189.3
132.2
181.1
144.3
210.5
191.8
213.3
210.3
230.4
230.5
228.1
226.3
231.9
231.4
221.7
220.8
204.6
193.6
202.1
169.2
185.2
134.3
184.6
123.1
2339.90
2207.20
1.060
191.5
133.0
183.6
145.2
212.3
192.4
213.9
211.3
230.9
230.6
228.5
227.5
232.5
232.2
221.9
221.4
205.2
193.8
203.5
170.5
185.9
136.0
185.2
124.2
2350.30
2218.50
1.059
192.8
137.7
183.4
148.4
211.5
195.2
213.7
211.5
230.0
229.5
227.2
224.7
231.5
229.1
221.4
221.2
205.5
195.6
203.7
173.2
188.5
139.9
188.4
129.1
2355.70
2231.20
1.056
189.7
132.8
181.3
144.8
210.7
192.4
213.3
210.6
230.3
230.3
228.0
226.0
231.8
230.7
221.7
221.2
204.7
193.9
202.3
169.9
185.7
134.9
185.1
123.9
2342.40
2210.80
1.060
192.6
135.4
183.2
145.9
211.6
193.6
214.2
211.4
230.2
229.2
228.2
223.9
231.6
228.7
221.9
221.0
205.8
195.5
203.1
173.4
188.4
139.7
188.5
130.5
2353.60
2230.70
1.055
the empirical model reaches about 0.65. As stated in [2,
10], the existence of clouds or aerosols in the sky would
make the solar collector need a flatter tilt angle to capture
more diffuse component of solar radiation.
The PSO heuristic algorithm can be an effective alternative of conventional trial-and-error method in searching
optimal tilt angle for solar collector, even though it is seldom found in the literature of this field. From a series of
tests of the present paper, an important notice must be
proposed here, i.e. the superiority of PSO will be more
significant while the time period considered in determining the collector’s optimum angle becomes longer.
5. Conclusions
In this paper, the particle swarm optimization method
was applied to search the optimal tilt angles for solar col-
lectors in Taiwan. Herein, solar radiation considered is
calculated by using empirical model suitable for clear sky
situation. The results would be useful for practical applications. Conclusions are summarized as below:
(1) The optimal tilt angles are positive (i.e. facing toward
south) for most of the months of year whereas negative (facing north) for summer months from May to
July.
(2) The annual optimal tilt angles for locations Taipei,
Taichung, ,Tainan, Kaohsiung, Hualien and Taitung
are 22.4°, 21.5°, 20.5°, 20.2°, 21.3° and 20.5°, respectively.
(3) The annual received energy of the collector with each
optimal angle is greater than the one incident upon
the ground surface; corresponding gains are 6.5%,
6.0%, 5.9%, 5.6%, 6.0% and 5.5%, respectively, for
the six studied locations.
172
Feng-Jiao Liu and Tian-Pau Chang
(4) The superiority of PSO becomes more significant
while the time period of searching is longer. The PSO
is a powerful method in optimizing the design of solar collector.
Acknowledgements
This study was supported partly by the National Science Council under contract NSC99-2221-E-252-011. The
authors would also deeply appreciate the Central Weather
Bureau for providing observation data and deeply thank
Dr. Wu CF and Dr. Huang MW, researchers of the Institute of Earth Sciences, Academia Sinica, Taiwan, for their
precious comments.
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Manuscript Received: Nov. 22, 2013
Accepted: Apr. 20, 2015