Residential Solar Panel Performance Improvement based on Optimal Intervals and Optimal Tilt Angle Morteza Sarailoo1, Student Member, IEEE, Shahrokh Akhlaghi1, Student Member, IEEE, Mandana Rezaeiahari2, Hossein Sangrody1, Student Member, IEEE, 1
Electrical and Computer Engineering Department, Binghamton University, State University of New York, Binghamton, NY 13902, USA {msarail1,sakhlag1, habdoll1}@binghamton.edu 2 Industrial Engineering Department, Binghamton University, State University of New York, Binghamton, NY 13902, USA,
[email protected] Abstract—In this paper, daily and yearly performance of a solar panel for residential usage is improved. To this end, an optimization problem is proposed and solved to find the optimal periods for a given number of intervals. It is shown that a limited number of tilt angle adjustments can significantly increase the maximum power production of a solar panel if the intervals are chosen properly. Therefore, this paper proposes a three steps algorithm to find the optimal period for each interval. First, the optimal tilt angle for a given interval is computed by using a Bee Algorithm (BA). Second, an optimization problem is formed and solved to get new periods for all intervals, which is guaranteed to have a better performance than the previous periods. And finally, a stopping criterion is checked to decide whether the previous step has to be repeated or the obtained periods are acceptable. The effects of the initial intervals on the results of the proposed algorithm are also investigated. The effectiveness of the proposed approach is studied at the Binghamton area, NY, US. The results show improvement in the received solar power by using the optimal intervals. Index Terms--Solar panel performance, optimal tilt angle, optimal interval.
I. INTRODUCTION Implementation of renewable energy resources, such as solar, geothermal, wind and etc., are growing rapidly due to much lower impact on environment. The renewable energy resources are offering clean and domestic source of energy. Solar energy as a primary source of renewable energy is becoming increasingly popular due to being a ubiquitous and immense source of energy which can provide abundant heat and electric power [1]. Solar panels and solar collectors are among the most promising renewable energy technologies. Solar radiation has a great influence on the performance of the solar panels and solar collectors. The amount of solar radiation incident on solar panels and solar collectors is significantly affected by their orientation and tilt angle [2]-[3]. The solar panels and solar collectors generally are oriented toward the equator, i.e. toward the south in northern hemisphere and toward the north in southern hemisphere.
The tilt angle of the solar panels plays an essential rule in capturing maximum solar radiation. The optimal tilt angle of a solar panel depends on the daily, monthly and yearly path of the sun. The accurate determination of the optimal tilt angle is essential for the maximum energy production for the location of interest. The optimal tilt angle is calculated by searching for a tilt angle for which the maximum total radiation during a specific period of time and a specific location is collected. The calculation of the optimal tilt angle depends on latitude, solar radiation characteristics, utilization period, and climatic condition at the particular site. A simple and secure solution to capture the maximum daily energy is to use a solar tracking system. Yet, the trackers are mechanical, their associated facilities are costly, and they consume energy. A more practical solution is to find the optimal tilt angles for a specific number of intervals and manually orient the solar collectors during each interval. Finding the optimal tilt angle and optimal duration for an interval can be formulated as optimization problems, which are challenging to solve. In recent years, many studies have been conducted on calculation of the optimal tilt angle of solar panels and solar collectors in different locations around the world. Saraf and Hamad [4] studied the relation between the tilt angle and harvested energy from solar collectors on a flat plate in Iraq by calculating the harvested energy for various tilt angles. They concluded that adjusting the tilt angle 8 times a year can capture the same amount of energy as when the tilt angle is adjusted daily. Skeiker [5] calculated the optimal tilt angle of the solar collectors by using a mathematical model in the Syrian area. It was concluded that adjusting the tilt angle 12 times a year can capture approximately the maximum solar radiation. Benghanem [6] determined the optimal tilt angle of the solar collectors in Saudi Arabia to get the maximum solar radiation. He concluded that, monthly adjustment can achieve 8% more surface radiation in compare with yearly adjustment. In US, Lave and Kleissl [7] calculated the optimal tilt angle and the azimuth angle of solar panels. The result showed that a solar panel with a fixed tilted angle gets 10-25% higher irradiations with increasing latitude. In Turkey, Bakirci [8] found that the variation of the optimal tilt angle throughout a year is between the 0o and 65o . In [9], authors determined the
optimal tilt angle of solar panels in Taiwan by using nonlinear time varying particle swarm optimization. Despotovic and Nedic [10] estimated the optimal tilt angle of a solar panel to get the maximum total radiation by searching for the values of slope and orientation. Akhlaghi et. al. [11] calculated the optimal number of intervals and their corresponding optimal tilt angles for residential solar panels at different locations in the US. As it has been reported by most researchers, the maximum solar power can be harvested only when the solar panel includes a tracking system. An alternative, especially for residential solar panels, is to adjust the tilt angle manually couple of times during a year. Researchers have suggested different number of intervals, from 4 to 12 [4]-[6], [11]; however, no one has proposed a systematic algorithm to optimally divide a year into the suggested number of intervals. In this paper a novel algorithm is proposed for increasing the overall and daily output of a solar panel by finding the optimal periods for a given number of intervals. It is shown that because a limited number of tilt angle adjustments is sufficient to achieve a significant improvement, it is not necessary to implement a continuous solar tracking system. The proposed novel algorithm is used to find the optimal periods for the desired number of intervals, iteratively, starting from the optimal tilt angles for the initial intervals. The optimal tilt angle for a given interval is computed by using BA. It is shown that the proposed algorithm is independent of the selection of initial intervals. The proposed approach is applied at Binghamton area, NY, US. The rest of the paper is organized as follows. In Section II, solar model, that captures relation between orientation of a solar panel and its received solar radiation, is presented in details. The optimal tilt angle using the BA is formulated and presented in Section III. In Section IV, the novel approach for determination of the optimal intervals is presented. The results of using the proposed algorithm at Binghamton area, NY, US are presented in Section V. Finally, conclusions are drawn in Section VI. II. SOLAR RADIATION MODEL This section introduces the basic of solar radiation and model to determine the solar radiation on the surface of solar panels. The model presented in this section is for northern hemisphere. The model for southern hemisphere can be obtained similarly. On average, the solar radiation incident outside the earth’s atmosphere which is called “extraterrestrial radiation” is around 1367 W/m2 [6]. A. Solar Geometry The sun’s position in the sky can be described by zenith angle ( q ) which is the angle of the sun relative to a line vertical to the earth’s surface. The zenith angle can be calculated as (Shown in Fig. 1): cos q = sin j sin d + cos j cos d cos w
(1) where j is the latitude of the location and w is the hour angle which is the angle between the longitude of the location on the earth’s surface and the longitude that is parallel to sun’s beam. Since the earth revolves every 24 h, therefore w varies
by 360o / 24 h = 15o per hour with morning negative and afternoon positive. The declination ( d ) of the sun is the angle between a plane perpendicular to a line between the sun and earth and the earth’s axis. It should be noted that the earth’s axis is tilted around 23.45o with respect to its orbit around the sun. The declination angle of the sun can be approximated as. d =
23.45p æ 2p (284 + n) ö sin ç ÷ 180 365 è ø
(2)
where n is the nth day of the year [12], for example the January 1st is 1.
a
g
Fig. 1.
q
gn
b
j
Zenith, azimuth and hour angles
The title angle ( b ) of solar panel is the angle between the surface plane and the horizontal which varies from 0o to 90o . The azimuth angle ( g ) is the sun’s position with respect to the north-south axis which varies from -180o to 180o . B. Solar time, local time, sunrise and sunset angles Solar time refers to the sun’s location relative to the observer, which is different for various longitudes where the solar time is calculated. Whereas the local time is same in the entire time zone. To adjust the solar time, the term (LongsmLongLocal)/15 must be added to the local time. Longsm is the longitude of the standard meridian of the time zone of observer and LongLocal is the observer longitude in degree. The sunrise and sunset occur when the sun is at the horizon when the cosine of zenith angle is zero. From equation (1) the sunrise ( wsr ) and sunset ( wss ) can be achieved as: wsr , ss = cos -1 (- tan j tan d )
(3)
C. Global, Beam, Diffuse and Reflected Radiation Global solar radiation can be categorized into direct or beam radiation (HB), diffuse radiation (HD) and reflect radiation (HR). Direct or beam solar radiation (HB) is the incident solar radiation which directly comes from the sun and reaches the earth’s surface without being considerably scattered. To calculate the direct beam radiation several models have been proposed by researchers [7]-[10]. The direct beam radiation received on inclined surface can be estimated by following equation. H B = ( H g - H d ) Rb
(4)
where Hg and Hd are the monthly mean daily global radiation [12] and diffuse radiation on a horizontal surface. Rb is the ratio of the direct beam radiation between tilted surface and
horizontal surface. Rb for the northern hemisphere (-) and southern hemisphere (+) can be estimated as follows: Rb =
cos(j m b ) cos d sin wss + wss sin(j m b ) sin d cos j cos d sin wss + wss sin j sin d
(5)
where the wss is the sunset hour angle for the tilted surface, given by equation (3). Diffuse radiation (HD) is the sunlight radiation which is scattered by the clouds or particles. Several methods to estimate the ratio of diffuse solar radiation between tilted surface and horizontal surface have been proposed in the literature which can be classified into isotropic and anisotropic [5], [12]-[13]. Diffuse radiation can be estimated as: H D = Rd H d
(6)
To choose between different models, the reader may refer to [1] for more information. In this paper, the diffuse radiation for clear atmosphere is calculated based on Liu and Jordan’s model [10] as: Rd =
1 + cos( b ) 2
(7)
be used for solving the optimization problem (10). In addition, because (10) is continuous and it is not possible to implement such cost function using heuristic algorithms following optimization problem is solved instead of (10). Maximize b
å
k
H T ( k Dt )
(11)
In this paper, BA is used for solving (11). BA has been used by many researchers for solving various optimization problems [17]. Using the BA, a function BA(.) is introduced for finding the optimal tilt angle of a given interval Si as βi,opt= BA(Si). The data for solar radiation on different locations are available at National Renewable Energy Laboratory (NREL) website. For prediction of the optimal tilt angle, here only an average of the last 20 years observation data has been used; however one can use more advance techniques such as quantile regression [18]. Fig. 2 Shows the result of using BA for computation of optimal tilt angle at Binghamton, NY, US for different intervals, namely daily, monthly, seasonal and yearly. It can be seen that optimal tilt angle varies significantly during a year, thus using the yearly or seasonal optimal tilt angles will result in smaller harvested power.
Reflected radiation (HR) is reflected from the ground significantly when the ground is covered by snow. This radiation only strikes by tilted panels and can be calculated by [13]: HR = Hg r
1 - cos( b ) 2
(8)
where r is the ground albedo. If no information is available, based on the literature it can be assumed to be 0.2 [10]-[14]. According to the aforementioned equations, total radiation (HT) on the surface of a solar panel normally estimated as: HT = H B + H D + H R 1 - cos( b ) = ( H g - H d ) Rb + Rd H d + H g r 2
(9)
Based on (9), in order to estimate the global solar radiation on tilted surface, the direct and diffuse component of global radiation are needed. III. OPTIMAL TILT ANGLE COMPUTATION USING BEE ALGORITHM Optimal tilt angle problem can be formulated as a search for the best tilt angle (β) that can provide the maximum solar radiation (HT) at the surface of a solar panel over a given period of time, as in (10). Maximize b
ò
t1
t0
H T (t )dt
(10)
It is possible to also add some inequality constraints to guarantee that during some given intervals total harvested solar radiation should be greater than a minimum value. Because of highly nonlinear relationship between tilt angle (β) and the total of the solar radiation over a time interval at the surface of a solar panel (HT), this optimization problem cannot be directly solved using mathematical programming such as convex optimization. Therefore a heuristic algorithm, such as particle swarm [15], genetic algorithm [16], or BA [17], has to
Fig. 2. Optimal tilt angle for different intervals, daily, monthly, seasonal and yearly at Binghamton, NY, US.
As it will be shown shortly in Section V (Fig. 3), there is a nonlinear relation between the number of intervals and the aggregated received power during a year, i.e., the increase rate of the aggregated power during a year significantly decreases as the number of intervals increases. In other words, investing in an automatic tracking system may not result in a great advantage over manually adjusting the optimal tilt angle a couple of times during a year. IV. PROPOSED APPROACH FOR FINDING THE OPTIMAL INTERVALS As mentioned in introduction, scholars have suggested to adjust the optimal tilt angle of a solar panel a fixed number of times over a year, i.e. 4-12 times, in order to improve the performance of solar panels [4]-[6], [11]. Still, the optimal intervals for tilt angle adjustment has NOT been studied yet. Therefore, this paper proposed a systematic approach to determine the optimal periods for the desired number of intervals (N). The proposed approach is summarized in three steps.
Step (0) - Initialization A year is equally divided into the desired number of intervals N and sets Si0 are formed i = 1,…, N. Set Si0 includes all days belonging to the ith interval. The optimal tilt angle of each set bi0,opt is calculated by solving (11) over the period of the interval, and its corresponding day identified.
d i0,opt
area. It should be noted; this result may change for different areas.
in that interval is
Step (I) - optimal Intervals Each day of the year is reassigned to one of the N intervals in a way to minimize a weighted multi-objective optimization problem (12). k N Mi
Minimize
åå æçè l b i =1 j =1
subject to
åM i
k i
k -1 i ,opt
-1 - b kj + (1 - l ) dik,opt - d kj ö÷ ø
Fig. 3. Aggregated solar power during a year as a function of the number of intervals, at Binghamton, NY, US.
(12)
= 365, I Sik = Æ,
"d qk Î S kp , d qk ¹ f ( S kp ), d qk ¹ g ( S kp ) ® d qk-1, d qk+1 Î S kp
Where sets Sik are optimization variables, k represents iteration of the algorithm, M ik denotes the total number of the days in the set Sik , functions f(.) and g(.) determine the first and last days belonging to the set Sik , and λ is a forgetting factor which 0 < λ kmax stops; otherwise for each set Sik , obtained from previous step, find the optimal tilt angle bik,opt and its corresponding day dik,opt in that interval. Go to step (I). i,opt
Remark 1. Here a procedure is proposed to solve the optimization problem (12). This producer takes advantage of the direct connection between the output of a solar panel and its tilt angle. When the tilt angle of a solar panel gets closer to the optimal tilt angle of a day, the outputs of the solar panel also gets closer to its maximum for that day. Thus, following procedure is proposed. Starting from dk1=dk-11,opt, find j=argmini{λ| βk-11,opt –BA(dk1)|+(1- λ)| dk-11,opt - dk1 |,…, λ| βk1 k k-1 k k-1 k i,opt –BA(d 1)|+(1- λ)| d i,opt - d 1 |,…, λ| β N,opt –BA(d 1)|+(1λ)| dk-1N,opt - dk1|}, and assign dk1 to the set Skj. Set dk2=dk-11,opt +1 and repeat the previous procedure until all days of a year has been assigned to one of the N sets Sk1,…, SkN. V. CASE STUDY In this section to evaluate the effectiveness of the proposed approach, the solar radiation data of Binghamton, NY, US is used as a case study. Based on simulation studies, the effect of different number of intervals on the total obtained solar power in Binghamton area during a year has been studied. Fig. 3 shows the effect of increasing the number of intervals on the total power using the optimal tilt angle for each interval. From this figure, it is obvious that, after four intervals there is not big achievement on the total received power in Binghamton
Fig. 4. Daily solar power during a year for different intervlas at Binghamton, NY, US. 70
70
60
60
50
50
40 30
Degree
Sik if |β
k-1
Degree
k
Interval 1 Interval 2 Interval 3 Interval 4
20
30
Interval 1 Interval 2 Interval 3 Inetrval 4
20
10 0 0
40
10
20
40
60 Iteration
80
100
0 0
20
40
60
80
100
Iteration
Fig. 5. Optimal tilt angles of intervals as a function of the algorithm iteration, left) starting from seasonal intervals, right) starting from equally divided year into 4 intervals, at Binghamton, NY, US
In order to show the effects of the initial intervals on the results of the proposed approach, the proposed algorithm has been applied to two different sets of initial intervals for N = 4; dividing a year into 4 equal intervals and seasonal intervals. Fig. 5 shows the optimal tilt angle of each interval as a function of the algorithm iteration for seasonal intervals and equally divided initial intervals. In addition, Fig. 6 and Fig. 7 present the daily solar power for both sets of initial intervals and optimal intervals. It is clear that regardless of the initial intervals, the daily solar power of the optimal intervals are
almost the same, although the algorithm converges to two different sets of optimal intervals. Total Radiation based on optimal tilt angle for a period
1
4
8500
2
3
8000
W/m
2
7500
7000
finding the optimal periods was applied to the data obtained for Binghamton area, NY, US with 4 desired intervals. It was shown that regardless of the initial intervals, the proposed algorithm always results in almost same daily solar power for the optimal intervals. In addition, it was observed that using the proposed appraoch with a reasonable number of intervals, i.e. 3-5, results in almost same solar power as using daily adjustment of solar panels. VII. REFERENCES
6500
[1] 6000
4
2
1
3
5500 Jan
Feb
Mar
Apr
May
Jun
Jul Day
Aug
Sep
Oct
Nov
Dec
Fig. 6. Daily solar power for seasonal intervals, blue circles, and optimal intervals, red crosses. The blue dashed vertical lines denote the boundery of the seasonal intervals and black dash-dotted vertical lines denote the boundary of the optimal intervals, at Binghamton, NY, US.
[2]
[3]
[4]
8500
2
1
3
4
8000
[5]
W/m2
7500
[6] 7000
[7]
6500 6000
[8] 5500
4
1
2
Jan Feb Mar Apr May Jun Jul Day
3
4
Aug Sep Oct Nov Dec
Fig. 7. Daily solar power for equally divided year into 4 intervals, blue circles, and optimal intervals, red crosses. The blue dashed vertical lines denote the boundery of the equally divided year into 4 intervals and black dashed-dot vertical lines denote the boundary of the optimal intervals, at Binghamton, NY, US.
The results show an improvement in the total recived solar power in a year after using the optimal intervals instead of the initial intervals, approximately +20 kW/m2. In addition in Fig. 6 and Fig. 7, it can be seen that optimal intervals provides a smooth change in the recieved solar power over a year in comparsion with the results of the initial intervals that contain niches and sudden drops. Comparing the results in Fig. 6 and Fig. 7 with the results in Fig. 4 also shows that using 4 optimal intervals results in almost same behavior and outcome as using daily intervals.
[9]
[10]
[11]
[12] [13] [14]
[15] [16]
VI. CONCLUSION In this paper a novel approach was proposed to improve the performance of the residential solar panels by finding the optimal periods for a desired number of intervals. It was shown that as the number of intervals increases the rate of the increase of the overall received solar power significantly decreases. Thus, for residential solar panels the tilt angle of the solar panel can be manually adjusted instead of using a tracking system. The proposed approach in this paper for
[17]
[18]
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