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Optimization of Slope Angles of Photovoltaic Arrays for Different Seasons
CHAPTER
OPTIMIZATION OF SLOPE ANGLES OF PHOTOVOLTAIC ARRAYS FOR DIFFERENT SEASONS
2.16 Ahmet Senpinar Firat University, Elazig, Turkey
1. INTRODUCTION Sun, wind, geothermal, biomass, and wave energy are some of the available alternative or renewable energy sources. These sources have a much lower negative effect on the environment than conventional energy sources [1]. Among these, solar energy is particularly vital for human health and the environment as it is abundant, renewable, and clean. Energy obtained from the sun on earth per unit time is known as the solar constant and is represented by GSC. The value of the solar constant as accepted by the World Radiation Center is 1367 W/m2 (1.96 cal/cm2 min) [2]. The sun is a gaseous body, with a mass of approximately 2 1030 kg and a diameter of 1.39 109 m. The distance from the sun to earth is approximately 1.49 1011 m [3]. Solar radiation has many advantages as a renewable energy source, particularly in terms of its abundance and freedom from pollution. It is utilized for generating both thermal energy and electricity directly using photovoltaic (PV) cells. Thus, solar insolation data are a significant parameter for the design and calibration of solar energy applications [4e6]. For example, calculations of solar insolation on horizontal surfaces are widely used for simulations, modelling, and sizing of solar processes. Data on solar radiation, which are measured hourly and daily, have significant importance for PV system designs, meteorology, solar maps, and engineering applications. Some researchers also use sky clearness to measure surface global solar irradiance [7,8]. PV systems are implemented in multiple ways, and the application of this technology is expanding throughout the world [9,10]. Current applications range from simple domestic PV energy systems to those for illumination, cooling, pumping water, plants for PV electricity generation, hybrid systems, space and telecommunication systems, etc. [11,12]. Evidently, the use of solar electricity has a significant place among energy resources in supplying future energy requirements. During the past few years, photovoltaic solar systems have become one of the most popular renewable energy sources in Europe [13,14]. Exergetic, Energetic and Environmental Dimensions. http://dx.doi.org/10.1016/B978-0-12-813734-5.00028-7 Copyright © 2018 Elsevier Inc. All rights reserved.
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CHAPTER 2.16 OPTIMIZATION OF SLOPE ANGLES
Standalone PV systems are typically used in rural and remote areas to provide necessary electricity. Building-integrated photovoltaics can be an appropriate alternative source to receive solar energy [15]. Determination of the optimal slope angle and orientation for a fixed PV array is important for maximizing its energy. Several studies have been conducted by various researchers to determine the optimal location for solar radiation collection using different empirical models [16e20]. The calculation of the optimal slope angle for a PV array at any location and time of the year is necessary before its installation [21]. A key aspect in the efficiency of a solar array is its slope angle relative to the horizontal. The slope angle varies with season and location. In general, PV systems in the northern hemisphere are mounted facing due south at a certain angle. Variations of values for the slope angle have been proposed. Many studies have used angles calculated by Ø þ 20 degrees [22], Ø þ (10 / 30) degrees [23], Ø þ 10 degrees [24], and Ø 10 degrees [25], whereas other researchers suggest two values for the slope angle, such as Ø 20 degrees [26], Ø 8 degrees [27], or Ø 5 degrees [28], where Ø is the latitude angle of the region, with “þ” for winter and “” for summer. For optimal performance on any given day, a fixed array should be mounted on the ground with a horizontal angle of (Ø d) degrees [29], where d is the declination angle known as the angle between the direction of the sun and the equator plane. This study recognizes the importance of seasonally adjusted slope angles to the optimal application of PV arrays in different locations across Turkey. The optimal slope angle of a fixed PV array is obtained for different periods and latitudes in the northern hemisphere. The optimal slope angle must be well determined to ensure system efficiency. If the slope angle of a PV system is chosen appropriately, the output power of the PV system increases. The optimal slope angle changes according to the time and location. The slope angles of some cities are presented, and calculations are performed to identify the optimal slope angle and orientation for PV arrays. Daily, monthly, and seasonal average slope angles can be calculated using the MATLAB software program. Graphical results associated with each city are presented.
2. MATHEMATICAL MODEL 2.1 PV ARRAY
Solar energy is utilized in two ways as thermal and electrical energy. One means of collecting solar energy as electricity is to use PV cells to generate electrical energy with solar radiation. The current generated by the PV cell is proportional to the effect of solar radiation on the cell. The power obtained from a PV cell is low because the current and voltage obtained from a single PV cell are also low. Therefore, to obtain adequate output power, PV cells are connected in series to form PV modules. In many applications, the power from one module is inadequate for the load. If higher voltages or currents than those available from a single module are required, modules must be connected into arrays [1]. When the modules are connected in parallel, their currents increase. When they are connected in series, their voltages increase. The current and voltage vary depending on the amount of sunlight shining on the PV cell. The IeV equation is then I ¼ Il I0 eðqVÞ=ðkTÞ 1 (1) where Il is the component of the PV cell current due to photons.
2. MATHEMATICAL MODEL
509
2.2 THE LATITUDE AND LONGITUDE OF ANY POINT ON EARTH The sun shines at different angles at different times in different places on earth. To determine a point on earth, certain information is needed. This information is gathered by a process of dividing the earth into a grid of latitudes and longitudes. The 0 degree meridian longitude passes through the former site of the Royal Astronomical Observatory in Greenwich, England, and is called the prime meridian. Points east of the prime meridian have negative longitudes, and points west of it have positive ones. The angle Ø on the earth’s surface measured north or south of the equator to a point is its latitude. Latitude values increase toward the poles, with the north pole being þ90 degrees and the south pole 90 degrees. Observers at different latitudes will see the sun take different paths across the celestial sphere. Fig. 1 shows the sun paths for the year as seen on the equator. Fig. 2 shows the paths seen at the north pole. This is useful to determine the position of the sun in the sky at any time of the year and at any latitude. The solar PV systems consist of two groups: fixed and tracking systems. Fixed systems are systems in which the array of solar cells is placed with a specific fixed slope. The slope angle changes according to the season and region. In general, PV systems in the northern hemisphere are mounted facing due south at a certain angle [2]. Tracking arrays follow the sun to maximize the incident beam radiation on their surfaces. Tracking control is based on angles of incidence and surface azimuth angles. Solar tracking systems are more expensive and complex than fixed systems. In this chapter, a PV system will be analyzed. Here, an effort will be made to note the areas where a PV system is open to the discretion of the designer. The system’s reliability, performance, and
FIGURE 1 Visualization of the sun paths across the sky for different latitudes on equator [3].
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CHAPTER 2.16 OPTIMIZATION OF SLOPE ANGLES
FIGURE 2 The sun paths for different latitudes at the north pole [3].
advantage are among the most common concerns. The system is related to optimal slope angles of PV arrays in different seasons. This system is advantageous, economical, and sustainable.
2.3 CALCULATION OF SOLAR ANGLES PV arrays can be mounted to track the sun, but fixed systems must be maintained at a certain angle to the horizontal to fully exploit available sunlight at the location. If this slope angle is determined well, the amount of insolation and the generated energy increase. To maximize energy, solar panels, such as photovoltaic modules, are usually oriented toward the equator with an optimal slope angle from the horizon, which depends on climatic conditions and site latitude [16e18,30]. Slope angle and location are also important considerations because energy demand dictates the design and operation of a standalone PV cell system and the number of modules and batteries to be used. Thus, the performance of a system is subject to load, insolation level, and module characteristics. Slope angle can be determined using the meridians of longitude and latitude for any location. As latitude increases, the curvature of the earth has the effect of orienting the observer away from the sun. An array is sloped toward the equator to compensate for this effect. The earth revolves around the sun once a year in an elliptical orbit that is almost circular, with the earth to sun distance varying by approximately 3% from a mean distance of 150 million km. The earth is closest to the sun in the summer season and farthest away in the winter season because the rotational axis of the earth is inclined at 23.44 degrees to the axis of the orbital plane [2]. Thus, in winter, the earth is sloped with the northern hemisphere away from the sun, and in summer, the northern hemisphere is sloped toward the sun. This phenomenon is referred to as the slope or declination angle d of the axis relative to the suneearth line. Some angles are shown in Fig. 3A and B.
2. MATHEMATICAL MODEL
511
FIGURE 3 Some angles for a sloped surface (A and B).
Because these angles constantly change with the seasons relative to the position of a location, it is necessary to compute the optimal slope angle 12 times a year to provide monthly optimal slope angles that can also be used to calculate seasonal angles if necessary. Some of the angles to consider when calculating an optimum are as follows [1,2]: Latitude angle (Ø): Latitude is defined with respect to an equatorial reference plane, and it changes north positive, 90 degrees Ø 90 degrees. Values north of the equator are positive and those south are negative. Equinox: This is the time when the lengths of day and night are equal. March 20 is known as the vernal equinox and September 23 as the autumnal equinox. On these dates, the sun’s rays are parallel to the equator. Declination angle (d): The declination angle is the angular position of the sun at solar noon with respect to the plane of the equator, north positive: 23.45 degrees d 23.45 degrees. The variation of the declination angle through the year is shown in Fig. 4. The declination d can be found from the equation of Cooper: ð360$ð284 þ nÞÞ d ¼ 23:45 sin degrees (2) 365 where n represents the day of the year (n ¼ 1, for 1 January) [2]. The day of the year n can be obtained with the help of Table 1. Fig. 3A and B shows the zenith, slope, solar azimuth angle, surface azimuth angle, and solar altitude angle for a sloped surface. The declination angle is zero during equinox dates (March 20 and September 23) because the incidence angle is parallel to the equator. In addition, the value of the declination angle is 23.45 degrees on summer solstice and 23.45 degrees on winter solstice. Zenith angle (qz): The zenith angle (qz) is the angle between the vertical and the line to the sun and is calculated as follows [2]: cosqz ¼ cosd cos cosu þ sind sin
(3)
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CHAPTER 2.16 OPTIMIZATION OF SLOPE ANGLES
FIGURE 4 Changes in the declination angle as annual.
Table 1 Suggested Average Days for Months and Values of n by Months [2] Months
n for ith Day of Month
Date
n, Day of Year
d, Declination
January February March April May June July August September October November December
i 31 þ i 59 þ i 90 þ i 120 þ i 151 þ i 181 þ i 212 þ i 243 þ i 273 þ i 304 þ i 334 þ i
10 11 10 10 11 10 11 10 10 11 10 11
10 42 69 100 131 161 192 222 253 284 314 345
22.03 14.58 4.80 7.53 17.78 23.01 22.10 15.36 4.21 8.10 17.91 23.12
where u is the solar hour angle and is determined for 24 h time by the (Senpinar) formula as follows [1]: u ¼ ððhour 60 þ minuteÞ 720Þ=4 degrees
(4)
2. MATHEMATICAL MODEL
513
FIGURE 5 Changes in the solar hour angle as 24 h time.
The solar hour angle at noon is zero. Fig. 5 shows the variation of the solar hour angle for 24 h. Solar altitude angle (as): The solar altitude angle is the angle between the horizontal and the line to the sun; it is complementary to the zenith angle and calculated as follows: as þ qz ¼ 90 degrees and as ¼ ð90 qz Þ degrees
(5)
Incidence angle (q): The incidence angle (q) is the angle between the beam radiation on a surface and the normal angle to that surface and is calculated as follows [2]: cos q ¼ cos qz cos b þ sin qz sin b cosðgs gÞ
(6)
where g represents the surface azimuth angle. Solar azimuth angle (gs): The solar azimuth angle (gs) is the angular displacement from south of the projection of beam radiation on the horizontal plane. Displacements east of south are negative and west of south are positive; the solar azimuth angle changes in the range of 180 to 180 degrees. For north or south latitudes between 23.45 and 66.45 degrees, gs will be between þ90 and 90 degrees. To calculate gs, we must know the sun’s position [2]. A general formula for gs, from Braun and Mitchell 0 [31], is conveniently written in terms of gs , a pseudo surface azimuth angle in the first or fourth quadrant: 0
gs ¼ a1 a2 gs þ 180 a3 ðð1 a1 a2 Þ=2Þ
(7)
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CHAPTER 2.16 OPTIMIZATION OF SLOPE ANGLES
FIGURE 6 Solar azimuth angle, zenith angle, and change in solar altitude angle.
where 0
sin gs ¼ ððsin u cos dÞ=sinqz Þ
(8)
where a1, a2, and a3 are constants related to sunrise and sunset; g represents the surface azimuth angle, the deviation of the projection on a horizontal plane of the normal to the surface from the local meridian, 180 g 180 degrees [2]. Fig. 6 shows changing of the zenith angle and solar altitude angle according to the solar azimuth angle. The value of the solar azimuth angle is zero at noon. Slope angle (b): The slope angle (b) is the angle between the plane of the surface in question and the horizontal, and its value changes according to 0 b 180 degrees [2]. tan b ¼ tan qz jcos gs j
(9)
3. RESULTS AND DISCUSSION
3.1 GEOGRAPHICAL LOCATION AND INSOLATION LEVEL Turkey is geographically in the northern hemisphere between latitudes 36e42 degrees (N) and longitudes 26e45 degrees (E) [32]. There is a 19 degrees longitude difference between locations at the easternmost and westernmost ends of the country. Although Turkey has good insolation potential, levels vary among locations. The yearly average total radiation time in Turkey has been calculated as 2640 h per year (7.2 h/day), and the total average annual solar radiation is 1311 kWh/m2 per year (3.6 kWh/m2 day, Table 2) [33].
3. RESULTS AND DISCUSSION
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Table 2 Monthly Solar Energy Amount of Turkey Monthly Total Solar Energy Months
(Kcal/cm2-month)
(kWh/m2-month)
Sunshine Duration (h/month)
January February March April May June July August September October November December TOTAL (per year) AVERAGE
4.45 5.44 8.31 10.51 13.23 14.51 15.08 13.62 10.60 7.73 5.23 4.03 112.74 308.0 cal/cm2-day
51.75 63.27 96.65 122.23 153.86 168.75 175.38 158.40 123.28 89.90 60.82 46.87 1311 3.6 kWh/m2-day
103.0 115.0 165.0 197.0 273.0 325.0 365.0 343.0 280.0 214.0 157.0 103.0 2640 7.2 h/day
3.2 APPLICATIONS OF OPTIMAL SLOPE ANGLES FOR DIFFERENT SEASONS Solar radiation data are used in several forms. The most detailed information available is beam and diffuse solar radiation on a horizontal surface, per hour. Daily data are often available, and hourly radiation can be estimated from daily data. Data for the amount of monthly total solar radiation on a horizontal surface are used in some process design methods [2]. Some solar collectors “track” the sun by moving in prescribed ways to minimize the incidence angle of beam radiation on their surfaces and thus maximize the incident beam radiation [2]. Installation and operation of these collectors requires relevant data on the angles of incidence and the surface azimuth angles. Tracking PV systems are classified by their motions. Rotation can be about a single axis (horizontal eastewest, horizontal northesouth, or parallel to the Earth’s axis), or it can be about two axes. For a plane rotating about a horizontal eastewest axis with a single daily adjustment, the beam radiation is normal to the surface at noon each day [2], cos q ¼ sin2 d þ cos2 cos u
(10)
The slope of this surface can be calculated as follows: b ¼ j dj
(11)
For a plane rotated about a horizontal eastewest axis with continuous adjustment to minimize the angle of incidence [2], 1=2 cos q ¼ 1 cos2 d sin2 u (12)
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CHAPTER 2.16 OPTIMIZATION OF SLOPE ANGLES
the slope of this surface is tan b ¼ tan qz jcos gs j
(13)
where gs is the solar azimuth angle. If the slope of an array is set at an angle qz to the horizontal, the radiation on the array is normal to the surface at noon. Thus, an array would be exposed to the maximum level of solar radiation available at that location [34]. The slope of an array can also be seasonally adjusted. In summer or winter, the slope angle of an array is different from the other seasons; however, the slope of an array at any time of year can be set as an optimal value for the rest of the season. For the best average slope to achieve optimal summer and winter and thus annual performance, an array should be mounted at (Ø 15) and (Ø þ 15) degrees, respectively, with a slope angle of (0.9 Ø) degrees [29]. Daily and monthly average slope angles can be calculated using the MATLAB software program. For optimal seasonal performance, one simply chooses the average value of slope angle for the season. Using monthly average slope angles, the annual average optimal slope angle can be determined. Periodic adjustment of the slope angle can be economically advantageous at higher latitudes and help maximize the performance and generate efficiency of fixed systems. The amount of insolation received at different locations across Turkey varies according to geographical position and local climatic conditions. Thus, the researcher calculated an optimal slope angle for 13 cities across Turkey using data on insolation levels and meteorological records from 2014. The meteorological data for the 13 cities are shown in Tables 3 and 4 along with the average monthly and seasonal optimal slope angles. First, monthly average values for each city were calculated. The annual average slope angle value (0.9 Ø) degrees was then calculated using the monthly average values [29]. Fig. 7AeD presents the average values of optimal slope angles for some cities in Turkey. Fig. 8 presents seasonal average values.
4. CONCLUSION A growing number of studies reflect the growing interest in renewable energy sources, also known as the energy sources of the future. These include solar energy systems, the efficiency of which is subject to multiple factors including the accurate prediction of the optimal slope angle for an array at different times of the year. The prediction of solar radiation is quite important for many solar applications and is affected by geographic location and climatic conditions. Tables 3 and 4 show how optimal slope angles vary according to geographical locations across Turkey. Monthly, seasonal, and annual values are shown. As winter and summer values vary considerably in Turkey, the efficiency of PV arrays can be maximized if they are mounted according to these seasonal variations. It is, thus, economically beneficial to adjust the slope angle of a PV system monthly in latitudes such as those found in Turkey, calculated according to seasonal values. The mathematical model can be used to calculate the appropriate monthly or seasonal optimal slope angle for any location on earth.
Table 3 Seasonal and Annual Average Values of Optimum Slope Angles for 13 Different Cities in Turkey Annual Values ( )
Seasonal Values ( ) Latitude ( )
Longitude ( )
Spring ( )
Summer ( )
Autumn( )
Winter ( )
Ankara Elazig _ Istanbul
39.56 38.68 41.01 38.21 36.52 39.55 40.09 37.01 41.17 39.45 38.43 37.05 37.12
32.52 39.14 28.58 38.19 36.12 44.02 26.24 35.18 36.20 37.02 35.30 37.22 38.22
30.92 30.04 32.37 29.57 27.88 30.91 31.45 28.37 32.53 30.81 29.79 28.41 28.48
20.40 19.52 21.85 19.05 17.36 20.39 20.93 17.85 22.01 20.29 19.27 17.89 17.96
48.52 47.64 49.97 47.17 45.48 48.51 49.05 45.97 50.13 48.41 47.39 46.01 46.08
58.64 57.76 60.09 57.29 55.60 58.63 59.17 56.09 60.25 58.53 57.51 56.13 56.20
Malatya Hatay(Dortyol) Igdir Canakkale Adana Samsun Sivas Kayseri Gaziantep Mugla
35.60 34.82 36.91 34.38 32.86 35.59 36.08 33.30 37.05 35.50 34.58 33.34 33.40
4. CONCLUSION
Cities
517
Table 4 Monthly Average Values of Optimum Slope Angles for 13 Different Cities in Turkey Cities Ankara Elazig _ Istanbul Malatya Hatay (Dortyol) Igdir Canakkale Adana Samsun Sivas Kayseri Gaziantep Mugla
January ( )
February ( )
March ( )
April ( )
May ( )
June ( )
July ( )
August ( )
September ( )
October ( )
November ( )
December ( )
60.40 59.52 61.85 59.05 57.36
52.88 52.00 54.33 51.53 49.84
41.94 41.06 43.39 40.59 38.90
30.06 29.18 31.51 28.71 27.02
20.75 19.87 22.20 19.40 17.71
16.48 15.60 17.93 15.13 13.44
18.45 17.57 19.90 17.10 15.41
26.26 25.38 27.71 24.91 23.22
37.56 36.68 39.01 36.21 34.52
49.40 48.52 50.85 48.05 46.36
58.61 57.73 60.06 57.26 55.57
62.65 61.77 64.10 61.30 59.61
60.39 60.93 57.85 62.01 60.29 59.27 57.89 57.96
52.87 53.41 50.33 54.49 52.77 51.75 50.37 50.44
41.93 42.47 39.39 43.55 41.83 40.81 39.43 39.50
30.05 30.59 27.51 31.67 29.95 28.93 27.55 27.62
20.74 21.28 18.20 22.36 20.64 19.62 18.24 18.31
16.47 17.01 13.93 18.09 16.37 15.35 13.97 14.04
18.44 18.98 15.90 20.06 18.34 17.32 15.94 16.01
26.25 26.79 23.71 27.87 26.15 25.13 23.75 23.82
37.55 38.09 35.01 39.17 37.45 36.43 35.05 35.12
49.39 49.93 46.85 51.01 49.29 48.27 46.89 46.96
58.60 59.14 56.06 60.22 58.50 57.48 56.10 56.17
62.64 63.18 60.10 64.26 62.54 61.52 60.14 60.21
(A)
(B)
Graphics of Adana and Istanbul
(C)
(D) ( )
Graphics of Samsun and Hatay Dortyol
Graphics of Gaziantep and Igdir
Graphics of Ankara and Mugla
FIGURE 7 Average optimum slope angles for some cities in Turkey (AeD). 70
25
60 20 50 40 30
15
10
20 5 10 0
FIGURE 8 Seasonal average optimum slope angles for 13 cities in Turkey.
0
Spring Autumn Winter Summer
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CHAPTER 2.16 OPTIMIZATION OF SLOPE ANGLES
NOMENCLATURE Io q V qz u as q g gs b d
Reverse saturation current, ampere (A) Electron electric charge (1.6 1019 C) Voltage (volt, V) The zenith angle (degrees) The solar hour angle (degrees) The solar altitude angle (degrees) The incidence angle (degrees) The surface azimuth angle (degrees) The solar azimuth angle (degrees) Slope angle (degrees) Declination angle (degrees)
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OPTIMIZATION OF SLOPE ANGLES OF PHOTOVOLTAIC ARRAYS FOR DIFFERENT SEASONS
2.16 Ahmet Senpinar Firat University, Elazig, Turkey
1. INTRODUCTION Sun, wind, geothermal, biomass, and wave energy are some of the available alternative or renewable energy sources. These sources have a much lower negative effect on the environment than conventional energy sources [1]. Among these, solar energy is particularly vital for human health and the environment as it is abundant, renewable, and clean. Energy obtained from the sun on earth per unit time is known as the solar constant and is represented by GSC. The value of the solar constant as accepted by the World Radiation Center is 1367 W/m2 (1.96 cal/cm2 min) [2]. The sun is a gaseous body, with a mass of approximately 2 1030 kg and a diameter of 1.39 109 m. The distance from the sun to earth is approximately 1.49 1011 m [3]. Solar radiation has many advantages as a renewable energy source, particularly in terms of its abundance and freedom from pollution. It is utilized for generating both thermal energy and electricity directly using photovoltaic (PV) cells. Thus, solar insolation data are a significant parameter for the design and calibration of solar energy applications [4e6]. For example, calculations of solar insolation on horizontal surfaces are widely used for simulations, modelling, and sizing of solar processes. Data on solar radiation, which are measured hourly and daily, have significant importance for PV system designs, meteorology, solar maps, and engineering applications. Some researchers also use sky clearness to measure surface global solar irradiance [7,8]. PV systems are implemented in multiple ways, and the application of this technology is expanding throughout the world [9,10]. Current applications range from simple domestic PV energy systems to those for illumination, cooling, pumping water, plants for PV electricity generation, hybrid systems, space and telecommunication systems, etc. [11,12]. Evidently, the use of solar electricity has a significant place among energy resources in supplying future energy requirements. During the past few years, photovoltaic solar systems have become one of the most popular renewable energy sources in Europe [13,14]. Exergetic, Energetic and Environmental Dimensions. http://dx.doi.org/10.1016/B978-0-12-813734-5.00028-7 Copyright © 2018 Elsevier Inc. All rights reserved.
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CHAPTER 2.16 OPTIMIZATION OF SLOPE ANGLES
Standalone PV systems are typically used in rural and remote areas to provide necessary electricity. Building-integrated photovoltaics can be an appropriate alternative source to receive solar energy [15]. Determination of the optimal slope angle and orientation for a fixed PV array is important for maximizing its energy. Several studies have been conducted by various researchers to determine the optimal location for solar radiation collection using different empirical models [16e20]. The calculation of the optimal slope angle for a PV array at any location and time of the year is necessary before its installation [21]. A key aspect in the efficiency of a solar array is its slope angle relative to the horizontal. The slope angle varies with season and location. In general, PV systems in the northern hemisphere are mounted facing due south at a certain angle. Variations of values for the slope angle have been proposed. Many studies have used angles calculated by Ø þ 20 degrees [22], Ø þ (10 / 30) degrees [23], Ø þ 10 degrees [24], and Ø 10 degrees [25], whereas other researchers suggest two values for the slope angle, such as Ø 20 degrees [26], Ø 8 degrees [27], or Ø 5 degrees [28], where Ø is the latitude angle of the region, with “þ” for winter and “” for summer. For optimal performance on any given day, a fixed array should be mounted on the ground with a horizontal angle of (Ø d) degrees [29], where d is the declination angle known as the angle between the direction of the sun and the equator plane. This study recognizes the importance of seasonally adjusted slope angles to the optimal application of PV arrays in different locations across Turkey. The optimal slope angle of a fixed PV array is obtained for different periods and latitudes in the northern hemisphere. The optimal slope angle must be well determined to ensure system efficiency. If the slope angle of a PV system is chosen appropriately, the output power of the PV system increases. The optimal slope angle changes according to the time and location. The slope angles of some cities are presented, and calculations are performed to identify the optimal slope angle and orientation for PV arrays. Daily, monthly, and seasonal average slope angles can be calculated using the MATLAB software program. Graphical results associated with each city are presented.
2. MATHEMATICAL MODEL 2.1 PV ARRAY
Solar energy is utilized in two ways as thermal and electrical energy. One means of collecting solar energy as electricity is to use PV cells to generate electrical energy with solar radiation. The current generated by the PV cell is proportional to the effect of solar radiation on the cell. The power obtained from a PV cell is low because the current and voltage obtained from a single PV cell are also low. Therefore, to obtain adequate output power, PV cells are connected in series to form PV modules. In many applications, the power from one module is inadequate for the load. If higher voltages or currents than those available from a single module are required, modules must be connected into arrays [1]. When the modules are connected in parallel, their currents increase. When they are connected in series, their voltages increase. The current and voltage vary depending on the amount of sunlight shining on the PV cell. The IeV equation is then I ¼ Il I0 eðqVÞ=ðkTÞ 1 (1) where Il is the component of the PV cell current due to photons.
2. MATHEMATICAL MODEL
509
2.2 THE LATITUDE AND LONGITUDE OF ANY POINT ON EARTH The sun shines at different angles at different times in different places on earth. To determine a point on earth, certain information is needed. This information is gathered by a process of dividing the earth into a grid of latitudes and longitudes. The 0 degree meridian longitude passes through the former site of the Royal Astronomical Observatory in Greenwich, England, and is called the prime meridian. Points east of the prime meridian have negative longitudes, and points west of it have positive ones. The angle Ø on the earth’s surface measured north or south of the equator to a point is its latitude. Latitude values increase toward the poles, with the north pole being þ90 degrees and the south pole 90 degrees. Observers at different latitudes will see the sun take different paths across the celestial sphere. Fig. 1 shows the sun paths for the year as seen on the equator. Fig. 2 shows the paths seen at the north pole. This is useful to determine the position of the sun in the sky at any time of the year and at any latitude. The solar PV systems consist of two groups: fixed and tracking systems. Fixed systems are systems in which the array of solar cells is placed with a specific fixed slope. The slope angle changes according to the season and region. In general, PV systems in the northern hemisphere are mounted facing due south at a certain angle [2]. Tracking arrays follow the sun to maximize the incident beam radiation on their surfaces. Tracking control is based on angles of incidence and surface azimuth angles. Solar tracking systems are more expensive and complex than fixed systems. In this chapter, a PV system will be analyzed. Here, an effort will be made to note the areas where a PV system is open to the discretion of the designer. The system’s reliability, performance, and
FIGURE 1 Visualization of the sun paths across the sky for different latitudes on equator [3].
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CHAPTER 2.16 OPTIMIZATION OF SLOPE ANGLES
FIGURE 2 The sun paths for different latitudes at the north pole [3].
advantage are among the most common concerns. The system is related to optimal slope angles of PV arrays in different seasons. This system is advantageous, economical, and sustainable.
2.3 CALCULATION OF SOLAR ANGLES PV arrays can be mounted to track the sun, but fixed systems must be maintained at a certain angle to the horizontal to fully exploit available sunlight at the location. If this slope angle is determined well, the amount of insolation and the generated energy increase. To maximize energy, solar panels, such as photovoltaic modules, are usually oriented toward the equator with an optimal slope angle from the horizon, which depends on climatic conditions and site latitude [16e18,30]. Slope angle and location are also important considerations because energy demand dictates the design and operation of a standalone PV cell system and the number of modules and batteries to be used. Thus, the performance of a system is subject to load, insolation level, and module characteristics. Slope angle can be determined using the meridians of longitude and latitude for any location. As latitude increases, the curvature of the earth has the effect of orienting the observer away from the sun. An array is sloped toward the equator to compensate for this effect. The earth revolves around the sun once a year in an elliptical orbit that is almost circular, with the earth to sun distance varying by approximately 3% from a mean distance of 150 million km. The earth is closest to the sun in the summer season and farthest away in the winter season because the rotational axis of the earth is inclined at 23.44 degrees to the axis of the orbital plane [2]. Thus, in winter, the earth is sloped with the northern hemisphere away from the sun, and in summer, the northern hemisphere is sloped toward the sun. This phenomenon is referred to as the slope or declination angle d of the axis relative to the suneearth line. Some angles are shown in Fig. 3A and B.
2. MATHEMATICAL MODEL
511
FIGURE 3 Some angles for a sloped surface (A and B).
Because these angles constantly change with the seasons relative to the position of a location, it is necessary to compute the optimal slope angle 12 times a year to provide monthly optimal slope angles that can also be used to calculate seasonal angles if necessary. Some of the angles to consider when calculating an optimum are as follows [1,2]: Latitude angle (Ø): Latitude is defined with respect to an equatorial reference plane, and it changes north positive, 90 degrees Ø 90 degrees. Values north of the equator are positive and those south are negative. Equinox: This is the time when the lengths of day and night are equal. March 20 is known as the vernal equinox and September 23 as the autumnal equinox. On these dates, the sun’s rays are parallel to the equator. Declination angle (d): The declination angle is the angular position of the sun at solar noon with respect to the plane of the equator, north positive: 23.45 degrees d 23.45 degrees. The variation of the declination angle through the year is shown in Fig. 4. The declination d can be found from the equation of Cooper: ð360$ð284 þ nÞÞ d ¼ 23:45 sin degrees (2) 365 where n represents the day of the year (n ¼ 1, for 1 January) [2]. The day of the year n can be obtained with the help of Table 1. Fig. 3A and B shows the zenith, slope, solar azimuth angle, surface azimuth angle, and solar altitude angle for a sloped surface. The declination angle is zero during equinox dates (March 20 and September 23) because the incidence angle is parallel to the equator. In addition, the value of the declination angle is 23.45 degrees on summer solstice and 23.45 degrees on winter solstice. Zenith angle (qz): The zenith angle (qz) is the angle between the vertical and the line to the sun and is calculated as follows [2]: cosqz ¼ cosd cos cosu þ sind sin
(3)
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CHAPTER 2.16 OPTIMIZATION OF SLOPE ANGLES
FIGURE 4 Changes in the declination angle as annual.
Table 1 Suggested Average Days for Months and Values of n by Months [2] Months
n for ith Day of Month
Date
n, Day of Year
d, Declination
January February March April May June July August September October November December
i 31 þ i 59 þ i 90 þ i 120 þ i 151 þ i 181 þ i 212 þ i 243 þ i 273 þ i 304 þ i 334 þ i
10 11 10 10 11 10 11 10 10 11 10 11
10 42 69 100 131 161 192 222 253 284 314 345
22.03 14.58 4.80 7.53 17.78 23.01 22.10 15.36 4.21 8.10 17.91 23.12
where u is the solar hour angle and is determined for 24 h time by the (Senpinar) formula as follows [1]: u ¼ ððhour 60 þ minuteÞ 720Þ=4 degrees
(4)
2. MATHEMATICAL MODEL
513
FIGURE 5 Changes in the solar hour angle as 24 h time.
The solar hour angle at noon is zero. Fig. 5 shows the variation of the solar hour angle for 24 h. Solar altitude angle (as): The solar altitude angle is the angle between the horizontal and the line to the sun; it is complementary to the zenith angle and calculated as follows: as þ qz ¼ 90 degrees and as ¼ ð90 qz Þ degrees
(5)
Incidence angle (q): The incidence angle (q) is the angle between the beam radiation on a surface and the normal angle to that surface and is calculated as follows [2]: cos q ¼ cos qz cos b þ sin qz sin b cosðgs gÞ
(6)
where g represents the surface azimuth angle. Solar azimuth angle (gs): The solar azimuth angle (gs) is the angular displacement from south of the projection of beam radiation on the horizontal plane. Displacements east of south are negative and west of south are positive; the solar azimuth angle changes in the range of 180 to 180 degrees. For north or south latitudes between 23.45 and 66.45 degrees, gs will be between þ90 and 90 degrees. To calculate gs, we must know the sun’s position [2]. A general formula for gs, from Braun and Mitchell 0 [31], is conveniently written in terms of gs , a pseudo surface azimuth angle in the first or fourth quadrant: 0
gs ¼ a1 a2 gs þ 180 a3 ðð1 a1 a2 Þ=2Þ
(7)
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CHAPTER 2.16 OPTIMIZATION OF SLOPE ANGLES
FIGURE 6 Solar azimuth angle, zenith angle, and change in solar altitude angle.
where 0
sin gs ¼ ððsin u cos dÞ=sinqz Þ
(8)
where a1, a2, and a3 are constants related to sunrise and sunset; g represents the surface azimuth angle, the deviation of the projection on a horizontal plane of the normal to the surface from the local meridian, 180 g 180 degrees [2]. Fig. 6 shows changing of the zenith angle and solar altitude angle according to the solar azimuth angle. The value of the solar azimuth angle is zero at noon. Slope angle (b): The slope angle (b) is the angle between the plane of the surface in question and the horizontal, and its value changes according to 0 b 180 degrees [2]. tan b ¼ tan qz jcos gs j
(9)
3. RESULTS AND DISCUSSION
3.1 GEOGRAPHICAL LOCATION AND INSOLATION LEVEL Turkey is geographically in the northern hemisphere between latitudes 36e42 degrees (N) and longitudes 26e45 degrees (E) [32]. There is a 19 degrees longitude difference between locations at the easternmost and westernmost ends of the country. Although Turkey has good insolation potential, levels vary among locations. The yearly average total radiation time in Turkey has been calculated as 2640 h per year (7.2 h/day), and the total average annual solar radiation is 1311 kWh/m2 per year (3.6 kWh/m2 day, Table 2) [33].
3. RESULTS AND DISCUSSION
515
Table 2 Monthly Solar Energy Amount of Turkey Monthly Total Solar Energy Months
(Kcal/cm2-month)
(kWh/m2-month)
Sunshine Duration (h/month)
January February March April May June July August September October November December TOTAL (per year) AVERAGE
4.45 5.44 8.31 10.51 13.23 14.51 15.08 13.62 10.60 7.73 5.23 4.03 112.74 308.0 cal/cm2-day
51.75 63.27 96.65 122.23 153.86 168.75 175.38 158.40 123.28 89.90 60.82 46.87 1311 3.6 kWh/m2-day
103.0 115.0 165.0 197.0 273.0 325.0 365.0 343.0 280.0 214.0 157.0 103.0 2640 7.2 h/day
3.2 APPLICATIONS OF OPTIMAL SLOPE ANGLES FOR DIFFERENT SEASONS Solar radiation data are used in several forms. The most detailed information available is beam and diffuse solar radiation on a horizontal surface, per hour. Daily data are often available, and hourly radiation can be estimated from daily data. Data for the amount of monthly total solar radiation on a horizontal surface are used in some process design methods [2]. Some solar collectors “track” the sun by moving in prescribed ways to minimize the incidence angle of beam radiation on their surfaces and thus maximize the incident beam radiation [2]. Installation and operation of these collectors requires relevant data on the angles of incidence and the surface azimuth angles. Tracking PV systems are classified by their motions. Rotation can be about a single axis (horizontal eastewest, horizontal northesouth, or parallel to the Earth’s axis), or it can be about two axes. For a plane rotating about a horizontal eastewest axis with a single daily adjustment, the beam radiation is normal to the surface at noon each day [2], cos q ¼ sin2 d þ cos2 cos u
(10)
The slope of this surface can be calculated as follows: b ¼ j dj
(11)
For a plane rotated about a horizontal eastewest axis with continuous adjustment to minimize the angle of incidence [2], 1=2 cos q ¼ 1 cos2 d sin2 u (12)
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CHAPTER 2.16 OPTIMIZATION OF SLOPE ANGLES
the slope of this surface is tan b ¼ tan qz jcos gs j
(13)
where gs is the solar azimuth angle. If the slope of an array is set at an angle qz to the horizontal, the radiation on the array is normal to the surface at noon. Thus, an array would be exposed to the maximum level of solar radiation available at that location [34]. The slope of an array can also be seasonally adjusted. In summer or winter, the slope angle of an array is different from the other seasons; however, the slope of an array at any time of year can be set as an optimal value for the rest of the season. For the best average slope to achieve optimal summer and winter and thus annual performance, an array should be mounted at (Ø 15) and (Ø þ 15) degrees, respectively, with a slope angle of (0.9 Ø) degrees [29]. Daily and monthly average slope angles can be calculated using the MATLAB software program. For optimal seasonal performance, one simply chooses the average value of slope angle for the season. Using monthly average slope angles, the annual average optimal slope angle can be determined. Periodic adjustment of the slope angle can be economically advantageous at higher latitudes and help maximize the performance and generate efficiency of fixed systems. The amount of insolation received at different locations across Turkey varies according to geographical position and local climatic conditions. Thus, the researcher calculated an optimal slope angle for 13 cities across Turkey using data on insolation levels and meteorological records from 2014. The meteorological data for the 13 cities are shown in Tables 3 and 4 along with the average monthly and seasonal optimal slope angles. First, monthly average values for each city were calculated. The annual average slope angle value (0.9 Ø) degrees was then calculated using the monthly average values [29]. Fig. 7AeD presents the average values of optimal slope angles for some cities in Turkey. Fig. 8 presents seasonal average values.
4. CONCLUSION A growing number of studies reflect the growing interest in renewable energy sources, also known as the energy sources of the future. These include solar energy systems, the efficiency of which is subject to multiple factors including the accurate prediction of the optimal slope angle for an array at different times of the year. The prediction of solar radiation is quite important for many solar applications and is affected by geographic location and climatic conditions. Tables 3 and 4 show how optimal slope angles vary according to geographical locations across Turkey. Monthly, seasonal, and annual values are shown. As winter and summer values vary considerably in Turkey, the efficiency of PV arrays can be maximized if they are mounted according to these seasonal variations. It is, thus, economically beneficial to adjust the slope angle of a PV system monthly in latitudes such as those found in Turkey, calculated according to seasonal values. The mathematical model can be used to calculate the appropriate monthly or seasonal optimal slope angle for any location on earth.
Table 3 Seasonal and Annual Average Values of Optimum Slope Angles for 13 Different Cities in Turkey Annual Values ( )
Seasonal Values ( ) Latitude ( )
Longitude ( )
Spring ( )
Summer ( )
Autumn( )
Winter ( )
Ankara Elazig _ Istanbul
39.56 38.68 41.01 38.21 36.52 39.55 40.09 37.01 41.17 39.45 38.43 37.05 37.12
32.52 39.14 28.58 38.19 36.12 44.02 26.24 35.18 36.20 37.02 35.30 37.22 38.22
30.92 30.04 32.37 29.57 27.88 30.91 31.45 28.37 32.53 30.81 29.79 28.41 28.48
20.40 19.52 21.85 19.05 17.36 20.39 20.93 17.85 22.01 20.29 19.27 17.89 17.96
48.52 47.64 49.97 47.17 45.48 48.51 49.05 45.97 50.13 48.41 47.39 46.01 46.08
58.64 57.76 60.09 57.29 55.60 58.63 59.17 56.09 60.25 58.53 57.51 56.13 56.20
Malatya Hatay(Dortyol) Igdir Canakkale Adana Samsun Sivas Kayseri Gaziantep Mugla
35.60 34.82 36.91 34.38 32.86 35.59 36.08 33.30 37.05 35.50 34.58 33.34 33.40
4. CONCLUSION
Cities
517
Table 4 Monthly Average Values of Optimum Slope Angles for 13 Different Cities in Turkey Cities Ankara Elazig _ Istanbul Malatya Hatay (Dortyol) Igdir Canakkale Adana Samsun Sivas Kayseri Gaziantep Mugla
January ( )
February ( )
March ( )
April ( )
May ( )
June ( )
July ( )
August ( )
September ( )
October ( )
November ( )
December ( )
60.40 59.52 61.85 59.05 57.36
52.88 52.00 54.33 51.53 49.84
41.94 41.06 43.39 40.59 38.90
30.06 29.18 31.51 28.71 27.02
20.75 19.87 22.20 19.40 17.71
16.48 15.60 17.93 15.13 13.44
18.45 17.57 19.90 17.10 15.41
26.26 25.38 27.71 24.91 23.22
37.56 36.68 39.01 36.21 34.52
49.40 48.52 50.85 48.05 46.36
58.61 57.73 60.06 57.26 55.57
62.65 61.77 64.10 61.30 59.61
60.39 60.93 57.85 62.01 60.29 59.27 57.89 57.96
52.87 53.41 50.33 54.49 52.77 51.75 50.37 50.44
41.93 42.47 39.39 43.55 41.83 40.81 39.43 39.50
30.05 30.59 27.51 31.67 29.95 28.93 27.55 27.62
20.74 21.28 18.20 22.36 20.64 19.62 18.24 18.31
16.47 17.01 13.93 18.09 16.37 15.35 13.97 14.04
18.44 18.98 15.90 20.06 18.34 17.32 15.94 16.01
26.25 26.79 23.71 27.87 26.15 25.13 23.75 23.82
37.55 38.09 35.01 39.17 37.45 36.43 35.05 35.12
49.39 49.93 46.85 51.01 49.29 48.27 46.89 46.96
58.60 59.14 56.06 60.22 58.50 57.48 56.10 56.17
62.64 63.18 60.10 64.26 62.54 61.52 60.14 60.21
(A)
(B)
Graphics of Adana and Istanbul
(C)
(D) ( )
Graphics of Samsun and Hatay Dortyol
Graphics of Gaziantep and Igdir
Graphics of Ankara and Mugla
FIGURE 7 Average optimum slope angles for some cities in Turkey (AeD). 70
25
60 20 50 40 30
15
10
20 5 10 0
FIGURE 8 Seasonal average optimum slope angles for 13 cities in Turkey.
0
Spring Autumn Winter Summer
520
CHAPTER 2.16 OPTIMIZATION OF SLOPE ANGLES
NOMENCLATURE Io q V qz u as q g gs b d
Reverse saturation current, ampere (A) Electron electric charge (1.6 1019 C) Voltage (volt, V) The zenith angle (degrees) The solar hour angle (degrees) The solar altitude angle (degrees) The incidence angle (degrees) The surface azimuth angle (degrees) The solar azimuth angle (degrees) Slope angle (degrees) Declination angle (degrees)
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