Electric Power System
modules are generally
deliver their maximum power at a particular solar radiation power density, typically 1000W/m²
(these units are preferable when dealing with conversion to electric
power). This is the commonly accepted typical average power density of solar radiation reaching the earth's surface when the sun
is directly overhead and the sky is clear. Outside the atmosphere the solar radiation power density in a plane perpendicular to the sun's rays, called the solar constant, has been measured at 1366W/m² with a variation of only a few
the course of a year. In reaching the earth's surface however the solar
radiation suffers from absorption mainly by oxygen and water vapour in
the air, as well as by transient atmospheric phenomena such as clouds,
pollutants. This can vary considerably from day to day. Despite this, for the purposes of
choosing the size of the solar modules we
can obtain a reasonable estimate of the solar radiation power that falls on a solar module placed
a given orientation and at a given latitude on a clear day.
The basis of the model is to assume a solar radiation power density at the earth's surface of 1060W/m² when the sun is directly overhead, and to estimate from the solar constant the amount of absorption that the radiation has experienced in traversing the atmosphere. Then using a bit of geometry we can recompute the amount of absorption that is experienced when the direction of the sun's rays is not vertical. From this we get the total energy that is available to a solar module over a day. We are particularly interested in the minimum that is available on a clear day during the year, as this will form the basis of our design. It is not possible to account realistically for cloudy days without access to extensive meteorological data. We should expect there to be times when the module output is very small and we will need to resort to other means to top up our batteries. Fortunately for us in South Australia the number of cloudy days is relatively low even in mid winter.
The solar radiation model that we use relies simply on the fact that the sun's rays experience greater absorption when they arrive at the Earth's surface at an oblique angle to the vertical. The mathematics behind the model is given in this short paper, and a simple C++ program is provided to generate csv output suitable for importing to a spreadsheet. The model is not apparently available on the Internet anywhere, so this may be useful for someone wishing to investigate further. Note that scattering from the atmosphere is not taken into account at this stage. It represents about 10% of the radiation reaching the earth's surface on a clear day but can be significant in terms of design considerations for cloudy days.
The model is useful for comparison purposes and for getting a "feel" for the behaviour of a solar module system. The first plot below shows how the power output of a solar module varies over an afternoon for a module fixed to the sun at noon (lower dotted curve), and a sun-following module (upper solid curve). The latitude is that of Adelaide (-34.929°) and the output shown is a percentage of the rated power output of the module.
The plot below shows the variation of the total daily output in kWH/m² as predicted by our model, of a perfect (100% efficiency) solar collector as a function of latitude when the sun is directly over the northern tropic. This represents the worst case for a clear day for the southern hemisphere and is the basis for our solar module system design. For the northern hemisphere simply reverse the sign of the latitude. Many of the conclusions are discussed in the excellent website by the Rainbow Power Company, who also provide a solar system calculator. Unfortunately it is difficult to work out what atmospheric conditions are included in their calculations. They appear to use an average measured insolation which would include cloudy conditions of various types. This may not be the best measure for a given application.
The top trace is proportional to the length of the day. It represents the output from a reference model in which the module receives 1 kW/m² throughout the daylight hours. The second trace is the output from a module that tracks the sun. The third trace (dotted) is that for a module that is oriented fixed so that it faces the sun at midday. The lowest trace is that for a horizontal module. The latter is used in measurements of the daily global solar exposure which is measured by the Bureaux of Meterology. This gives us an opportunity to check our model. The graph below shows the model predictions (solid line) and the measured (dotted line) monthly average of the daily global solar exposure at Armidale (latitude -30.515°). NSW. The agreement is surprisingly good given the assumptions that were used. The dip in the measurements in the latter part of the year is possibly due to a seasonally high incidence of cloudy days.
A useful conclusion we can draw from these plots is that at the latitude of Adelaide (-34.929°) a solar module will only deliver half its specified output over the shortest day of the year, if it follows the sun. However the difference in output for a fixed module is only very slightly less than that for a following module, being only 13% lower. Although we haven't computed the cost (in energy) of running a sun following module, it would appear to be unlikely to make any gains worthwhile unless we are particularly interested in harvesting the addition power output of a following module during the summer months.