An Introduction to Solar Geometry for Solar Cell Optimization
Solar geometry is a crucial tool to help find the best tilt and orientation of solar cells and to evaluate the impact of shadows. This article introduces some key basic concepts associated with solar geometry.
We may be fans of electronics but if we are to manage PV technology correctly, we also need to utilize a few concepts from astronomy, geography, and celestial mechanics.
Astronomy and Celestial Motion
Astronomy is the science that deals with the size, constitution, motion, and relative position of celestial bodies. All celestial bodies are constantly in motion. The movement of other entities as seen by an observer on our planet is referred to as apparent motion.
Celestial bodies have two principal motions:
- Rotation: The spinning motion around an axis within the body.
- Revolution: The motion of a body in its elliptical orbit around another body.
When the earth makes one complete turn around the sun, it is called apparent solar day.
In each planet's orbit, the point nearest the sun is called the perihelion while the point farthest from it is called the aphelion. The line joining perihelion and aphelion is called the line of apsides (the major axis of the orbit).
Utilizing Latitude and Longitude on Earth
Definitions of latitude and longitude are essential to locate positions on Earth.
Latitude is a measurement of location north or south of the Equator. It is the angular distance from the Equator, measured northward or southward along a meridian from 0° at the Equator to 90° at the poles.
Lines of latitude are known as parallels or parallels of latitude, and they connect all points of equal latitude. There are three types of latitude: astronomical, geodetic, and geocentric. Geocentric is the type most commonly used.
Longitude is the arc of a parallel or the angle at the pole between the prime meridian, passing through Greenwich, UK, and the meridian of a point on the earth, measured eastward or westward through 180°.
Latitude and longitude of the earth.
A great circle is the line of intersection of a sphere and a plane through the center of the sphere. The Equator and meridians are great circles.
A small circle is the line of intersection of a sphere and a plane not passing through the center of the sphere. All parallels, except the Equator, are small circles.
The meridian is a great circle through the geographical poles of the earth. All meridians meet at the poles, and their planes intersect each other in a line called the polar or earth axis.
The association of meridians of longitude and parallels of latitude creates a grid to find locations. For example, a place located at 30° N, 90° W, means 30° of arc north of the Equator and 90° of arc west of the Greenwich meridian (see New Orleans’ position in Figure 1 below).
Figure 1. Utilizing latitude and longitude allows for accurate, universal locating. Image courtesy of Encyclopedia Britannica.
Altitude and Azimuth
Altitude or elevation is the angular distance of a star above the horizon.
In navigation, Azimuth is the angular difference between north and any other horizontal direction (the bearing) when referred to as a celestial body. It is measured clockwise around the horizon from 0° at the north through 360°.
In PV technology, azimuth is the solar panel east-west orientation in degrees. The degree of azimuth indicates the array:
- At 0°, the array is facing the Equator in both the northern and southern hemispheres
- At 90°, the array is facing due west
- At -90°, the array is facing due east
A compass angle shows 180° for south, 90° for east, and 270° for west.
The Celestial Equator and Declination
Astronomers call the sky as seen from the earth the celestial sphere or globe. Imagine the sky as a large hollow ball with the earth at the center and the stars on the inside surface. As the earth rotates, the stars seem to move.
The Celestial Equator is the intersection of the celestial sphere and the extended plane of the Equator. Like the Earth’s Equator, it is a circle that divides the celestial globe into a northern and a southern celestial hemisphere.
Image courtesy of Encyclopedia Britannica.
Declination is the angular distance of a body north or south of the celestial Equator. The arc of a circle from the celestial Equator to a point on the celestial sphere measured northward or southward through 90°.
Tag declination N or (+) from the Equator to the north pole, and S or (-) to the south pole.
Declination is the celestial equivalent of latitude since it is the angular distance of a celestial body north or south of the Celestial Equator. The Celestial Equator’s declination is 0° (just as the earth's Equator is latitude 0°), and measures the declination of each star from there, in degrees.
Optimizing Solar Radiation by Calculating the Sun’s Position
Solar radiation intensity fluctuates a lot depending on the sun's location and the inclination of the surface that is receiving the radiation. The radiation received on a surface fluctuates daily. Setting a surface (such as a solar panel) perpendicular to the sun rays allows for maximum radiation. So it’s no surprise that knowing the sun’s position along the day and the year is of paramount importance.
The sun's apparent motion is affected by three main factors:
- Time of year
- Time of day
These factors are in turn affected by Earth’s equinoxes and solstices. In the next section, let’s look at how these work in Earth’s hemispheres. Note that the angles and dates shown are approximate but close enough for the purpose of creating an understanding.
The Sun’s Yearly Path: Equinoxes and Solstices
The ecliptic is the path the sun appears to take among the stars due to the annual revolution of the earth in its orbit.
Since the inclination of the earth's axis is 23.5°, the sun has an apparent motion of 47° during the year. Along with the sun's path, declination changes from 23.5° by June 21st to -23.5° around December 22nd for the northern hemisphere. In the southern hemisphere, the situation is the inverse: in June, the declination is – 23.5° and in December is 23.5°. The declination on any given day in the southern hemisphere has the opposite sign to the northern hemisphere.
The tilt of the earth's axis causes the seasons and changes the day length. Without the tilt, day and night would be equal all year long with no seasons.
There are four specific positions on earth’s orbit that define sun motion. The celestial Equator and the ecliptic intersect in two points, at opposite locations, leaving one half of the ecliptic above the Equator (north of it) and the other half below (south of it).
The sun is on the Equator on the two days that it reaches these locations. As expected, day and night are of the same length, referred to as equinoxes (from the Latin aequus, equal, and nox, night). These days are on or about September 23rd and March 21st which mark the beginning of the autumnal equinox (fall) and vernal equinox (spring), respectively.
Image courtesy of physics.weber.edu.
In the northern hemisphere, the apparent sun motion starts from the vernal equinox, on or about March 21st. As the sun continues movement, its distance from the celestial equator increases (its northern declination). Hence, it is longer above the horizon than below it.
On or around June 21st, after one-quarter of its yearly course, the sun reaches its most considerable distance from the Equator (northern declination of 23.5°), where the day is longest, and the night is shortest. This is called summer solstice (from Latin sol, sun, and stare, stand still). From this moment, declination and day length decrease, and, at the autumnal equinox, on or about September 23rd, day and night are equal again.
From the autumnal equinox, the sun's declination becomes negative, i.e., southern, and the sun stays more time below the horizon than above, causing the shorter days and longer nights. On or about December 22nd, the sun is farthest south of the Equator with declination - 23.5°, with the shortest day and the longest night. This position is called the winter solstice. From this time, the days will be longer and the nights shorter. At the vernal equinox, the yearly cycle commences again.
Again, this cycle works in the opposite order in the southern hemisphere.
The Sun’s Daily Path
The sun follows a daily path with a maximum height at solar noon. Solar noon refers to the instant the sun passes an observer's meridian and reaches its highest position in the sky. Despite popular belief, this usually does not take place at noon.
There is a disparity between civil time and solar time. For applications of flat solar panels, this difference is meaningless. For solar thermal collectors, however, the exact sun position should be known, and the time difference becomes of paramount importance.
Placing Solar Panels According to Location
The differences in temperature during the seasons is affected in the sun's altitude and the time it remains above the horizon. The next figure shows a comparison of the surface covered by the same amount of sunlight during summer and winter.
Image courtesy of physics.weber.edu.
Sunlight is most direct at the Equator and slanted for higher latitudes north or south as seen in the image below.
Image courtesy of the Lunar and Planetary Institute.
A site’s latitude rules the sun's location. The higher the latitude (i.e., the farther from the Equator), the lower the sun's path over the horizon.
In the northern hemisphere at latitudes above 23.5°, the sun never reaches the Zenith and is always at the south. For this reason, we place the solar panels facing south. In the southern hemisphere at latitudes below -23.5°, the sun is always at the north, so we position solar panels facing north.
This article went over the basics of sun motion as well as geographical and astronomical concepts needed for solar PV applications.