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 168 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. References [1] Nijegorodov, N., Devan, K. R. S., Jain, P. K. and Carlsson, S. “Atmospheric Transmittance Models and an Analytical Method to Predict the Optimum Slope of an Absorber Plate, Variously Orientated at any Latitude,” Renewable Energy, Vol. 4, No. 5, pp. 529-543 (1994). doi: 10.1016/0960-1481(94)90215-1 [2] Shu, N., Kameda, N., Kishida, Y. and Sonoda, H., “Experimental and Theoretical Study on the Optimal Tilt Angle of Photovoltaic Panels,” J. Asian Architecture and Building Engineering, Vol. 5, pp. 399-405 (2006). doi: 10.3130/jaabe.5.399 [3] Yakup, M. Ab. H. M. and Malik, A. Q., “Optimum Tilt Angle and Orientation for Solar Collector in Brunei Darussalam,” Renewable Energy, Vol. 24, pp. 223234 (2001). doi: 10.1016/S0960-1481(00)00168-3 [4] Cheng, C. L., Chan, C. Y. and Chen, C. L., “An Empirical Approach to Estimating Monthly Radiation on South-Facing Tilted Planes for Building Application,” Energy, Vol. 31, pp. 2940-2957 (2006). doi: 10.1016/ j.energy.2005.11.015 [5] Chen,Y. M., Lee, C. H. and Wu, H. C., “Calculation of the Optimum Installation Angle for Fixed Solar-Cell Panels Based on the Genetic Algorithm and the Simulated-Annealing Method,” IEEE Trans. on Energy Conversion, Vol. 20, No. 2, pp. 467-473 (2005). doi: 10. 1109/TEC.2004.832093 [6] Markvart, T., “Solar Electricity,” Second Edition, John Wiley & Sons Ltd, pp. 5-18 (2000). [7] Chow, T. T. and Chan, A. L. S., “Numerical Study of Desirable Solar-Collector Orientations for the Coastal Region of South China,” Applied Energy, Vol. 79, pp. 249-260 (2004). doi: 10.1016/j.apenergy.2004.01.001 [8] Saraf, G. R. and Hamad, F .A. W., “Optimum Tilt Angle for a Flat Plate Solar Collector,” Energy Conversion Management, Vol. 28, pp. 185-191 (1988). doi: 10.1016/0196-8904(88)90044-1 [9] Shariah, A., Al-Akhras, M. A. and Al-Omari, I. A., “Optimizing the Tilt Angle of Solar Collectors,” Renewable Energy, Vol. 26, pp. 587-598 (2002). doi: 10. 1016/S0960-1481(01)00106-9 [10] Chang, T. P., “Performance Evaluation for Solar Collectors in Taiwan,” Energy, Vol. 34, pp. 32-40 (2009). doi: 10.1016/j.energy.2008.09.016 [11] Gunerhan, H. and Hepbasli, A., “Determination of the Optimum Tilt Angle of Solar Collectors for Building Applications,” Building and Environment, Vol. 42, pp. 779-783 (2007). doi: 10.1016/j.buildenv.2005.09.012 [12] Huang, B. J. and Sun, F. S., “Feasibility Study of One Axis Three Positions Tracking Solar PV with Low Concentration Ratio Reflector,” Energy Conversion and Management, Vol. 48, pp. 1273-1280 (2007). doi: 10.1016/j.enconman.2006.09.020 [13] Chang, T. P., “Performance Analysis of Tracked Panel According to Predicted Global Radiation,” Energy Conversion and Management, Vol. 50, pp. 2029-2034 (2009). doi: 10.1016/j.enconman.2009.04.007 [14] Morcos, V. H., “Optimum Tilt Angle and Orientation for Solar Collectors in Assiut, Egypt,” Renewable Energy, Vol. 4, No. 3, pp. 291-298 (1994). doi: 10.1016/ 0960-1481(94)90032-9 [15] Kennedy, J. and Eberhart, R., “Particle Swarm Optimization,” in Proc. IEEE Int. Conf. Neural Networks, pp. 1942-1948 (1995). doi: 10.1109/ICNN.1995.488968 Manuscript Received: Nov. 22, 2013 Accepted: Apr. 20, 2015