## Calculation of the Suns Position

The position of the sun is essential for many further calculations for solar energy systems. The two angles sun height (solar altitude or elevation) 7s and solar or sun azimuth as define the position of the sun. However, definitions for these angles and the symbols used vary in the literature. The convention used

 Reference Symbol N NE E SE S SW W NW ISES, 2001; NREL, 2000 Y 0o 45o 90o 1350 180e 225o 270o 315e CEN, 1999 for ^>0a Y 180o 225o 270o 3150 0e 45o 90o 135e CEN, 1999 for qi<0b Y 0o 270o 270o 2250 180e 135e 90o 45o DIN, 1985, this book as 0o 45o 90o 1350 180e 225o 270o 315e

Note: q>, latitude; a north of the equator; b south of the equator

Note: q>, latitude; a north of the equator; b south of the equator in this book defines the sun height as the angle between the centre of the sun and the horizon seen by an observer. The azimuth angle of the sun describes the angle between geographical north and the vertical circle through the centre of the sun. EN ISO 9488 defines the solar azimuth as the angle between the apparent position of the sun and south measured clockwise at the northern hemisphere and between the apparent position of the sun and north measured anticlockwise in the southern hemisphere (CEN, 1999). This can cause confusion when comparing calculations with different angle definitions. Table 2.8 shows the different solar azimuth angle definitions of various sources.

The sun height and solar azimuth depend on the geographical location of the observer, the date, time and time zone. The position of the sun is strongly influenced by the angle between the equatorial plane of the Earth and the rotational plane of the Earth around the sun, called the solar declination. The solar declination 6 varies between +23°26.5' and -23°26.5' over a year. Since the orbit of the earth around the sun is not circular, the length of a solar day also changes throughout the year. Usually, the so-called equation of time eqt takes this into consideration. Many algorithms have been developed to calculate the position of the sun. A relatively simple algorithm is described overleaf (DIN, 1985).

Figure 2.10 Definitions of the Angles Describing the Position of the Sun Used in this Book

number of days of the year the solar declination becomes:

8(y') = {0.3948 - 23.2559 • cos(y'+9.1°) - 0.3915 • cos(2 x y'+5.4°) -0.1764 • cos(3 • y'+105.2°)}°

and the equation of time eqt(y') = [0.0066 + 7.3525 • cos(y'+85.9°) + 9.9359 •

cos(2 • y'+108.9°) + 0.3387 • cos(3 • y'+105.2°)]min

is calculated. With the Local time, the Time zone (e.g. Greenwich mean time GMT = 0, Central European Time CET = 1 h, Pacific Standard Time PST = -8 h) and the longitude A, the mean local time MLT becomes:

MLT = Local time - Time zone + 4 • A • min/° (2.16)

Adding the equation of time eqt to the mean local time MLT provides the Solar time:

Solar time = MLT + eqt (2.17) With the latitude y of the location and the hour angle m:

m = (12.00 h - Solar time) • 15°/h (2.18) the angle of solar altitude (sun height) ys and angle of solar azimuth aS become:

Ys = arcsin(cosm • cosy • cos8 + siny • sin8) (2.19)

sin • sin a>c - sin 8 180° - arccos ----- if Solar time s: 12.00 h cos yS • cos (p i ys • sin (ps - sii cos yS • cos q>

Table 2.9 Latitude p and Longitude X of Selected Locations

Bergen Berlin London Rome LA Cairo Bombay Upington Sydney p 60.40° 52.47° 51.52° 41.80° 33.93° 30.08° 19.12° -28.40° -33.95

X 5.32° 13.30° -0.11° 12.58°-118.40° 31.28° 72.85° 21.27° 151.18°

Table 2.9 shows angles of latitude and longitude for selected locations.

Other algorithms such as the SUNAE algorithm (Walraven, 1978; Wilkinson, 1981; Kambezidis and Papanikolaou, 1990) or the NREL SOLPOS algorithm (NREL, 2000) have improved accuracies, mainly at low solar altitudes. These algorithms include the refraction of the beam irradiance by the atmosphere. However, they are also much more complex than the algorithm described above. The CD-ROM of the book contains the code for these algorithms.

Solar position or sun-path diagrams are used to visualize the path of the sun in the course of a day. These diagrams show sun height and azimuth for every hour of the selected days with a curve drawn through the points. Figure 2.11 shows the solar position diagram for Berlin and Figure 2.12 that for Cairo. For clarity, these diagrams show only five months from the first half of the year; the corresponding months for the second half of the year have nearly symmetrical curves.

Solar position diagrams for southern latitudes look similar, except that the south position is in the centre of the diagram instead of north. Solar position diagrams for locations between the northern and southern tropics are different in that the sun is in the south at solar noon for some months and in the north for others.

1 60

ne se sw w

ne se sw

1 60

 21 June ^ 21 April . Time (GMT*1h) 21 March J "-- . . yi i\J4 21 Feb JÍ9 14 V 21 Dec Ü jfs XV-.. \l5 \j6 7 U^gia ^V» ifí >. 4 7 III_ r5 / t ¿ 6 S >íf if \_,7\ X 16~ >1

Solar azimuth angle in degrees Figure 2.11 Solar Position Diagram for Berlin, Germany (52.5°N)

m bo

sw w nw

sw w nw

m bo

 21 June _^ n GMT + 2h 21 April j^Tr Ä.__ 13 21 Mar -J ■~/l0 i 2 iJ3 \ yí \ " 4 21 Feb -^J y/io r 12 ^—aJJ \M V „_ \l 14 1 4 \ \5 l> 15 Y8 f f9 J y S J ? (flO 1 if¡ y_ ? 1 ¡ r Y 8 J ilt j / 21 Dec r a fS \'6\ i V17 \ f 7 / / B 1_p 5 f < LS-tf-f-- \ H V 5 90 ■ azimuth angle in degrees Figure 2.12 Solar Position Diagram for Cairo, Egypt (30.1°N)