The azimuth angle and the elevation angle at solar noon are the two key angles which are used to orient photovoltaic modules. However, to calculate the sun's position throughout the day, both the elevation angle and the azimuth angle must be calculated throughout the day. These angles are calculated using "solar time". In conventional time keeping, regions of the Earth are divided into certain time zones. However, in these time zones, noon does not necessarily correspond to the time when the sun is highest in the sky. Similarly, sun rise is defined as the stage when the sun rises in one part of the time zone. However, due to the distance covered in a single time zone, the time at which the sun actually clears the horizon in one part of the time zone may be quite different to the "defined" sun rise (or what is officially recognized as the time of sun rise). Such conventions are necessary otherwise a house one block away from another would actually be different in time by several seconds. Solar time, on the other hand is unique to each particular longitude. Consequently, to calculate the sun's position, first the local solar time is found and then the elevation and azimuth angles are calculated.
Local Solar Time (LST) and Local Time (LT)
Twelve noon local solar time (LST) is defined as when the sun is highest in the sky. Local time (LT) usually varies from LST because of the eccentricity of the Earth's orbit, and because of human adjustments such as time zones and daylight saving.
Local Standard Time Meridian (LSTM)
The Local Standard Time Meridian (LSTM) is a reference meridian used for a particular time zone and is similar to the Prime Meridian, which is used for Greenwich Mean Time. The LSTM is illustrated below.
The (LSTM) is calculated according to the equation:
where ΔTGMT is the difference of the Local Time (LT) from Greenwich Mean Time (GMT) in hours.
Equation of Time (EoT)
The equation of time (EoT) (in minutes) is an empirical equation that corrects for the eccentricity of the Earth's orbit and the Earth's axial tilt.
in degrees and d is the number of days since the start of the year. The time correction EoT is plotted in the figure below.
Time Correction Factor (TC)
The net Time Correction Factor (in minutes) accounts for the variation of the Local Solar Time (LST) within a given time zone due to the longitude variations within the time zone and also incorporates the EoT above.
The factor of 4 minutes comes from the fact that the Earth rotates 1° every 4 minutes.
Local Solar Time (LST)
The Local Solar Time (LST) can be found by using the previous two corrections to adjust the local time (LT).
Hour Angle (HRA)
The Hour Angle converts the local solar time (LST) into the number of degrees which the sun moves across the sky. By definition, the Hour Angle is 0° at solar noon. Since the Earth rotates 15° per hour, each hour away from solar noon corresponds to an angular motion of the sun in the sky of 15°. In the morning the hour angle is negative, in the afternoon the hour angle is positive.
The declination of the sun is the angle between the equator and a line drawn from the centre of the Earth to the centre of the sun. The declination angle is given by:
Where d is the number of days since the start of the year.
A more accurate version is diven by:
The elevation angle (used interchangeably with altitude angle) is the angular height of the sun in the sky measured from the horizontal. Confusingly, both altitude and elevation are also used to describe the height in meters above sea level. The elevation is 0° at sunrise and 90° when the sun is directly overhead (which occurs for example at the equator on the spring and fall equinoxes).
Sunrise and Sunset
To calculate the sunrise and sunset time the elevation is set to zero and the elevation equation above is rearranged to give:
these equations can be simplified as:
The azimuth angle is the compass direction from which the sunlight is coming. At solar noon, the sun is always directly south in the northern hemisphere and directly north in the southern hemisphere. The azimuth angle varies throughout the day as shown in the animation below. At the equinoxes, the sun rises directly east and sets directly west regardless of the latitude, thus making the azimuth angles 90° at sunrise and 270° at sunset.