## Solar photovoltaics

The UK government has a scheme, the Feed in Tariff (FIT) scheme, in which one can earn money in return for generating electricity. In examining whether or not this is a worthwhile thing to do, it seemed a useful exercise to write an Applet that can give an indication of how the output of a solar photovoltaic device varies with time of day, month, year, latitude, orientation, etc. This turns out to be quite a complicated calculation because there are a number of paramaters that are not well defined. The applet below calculates the output of a 4 kW installation, which is the maximum size for a domestic UK user to be paid the maximum price for electricity generated. The applet shows how the output (in green) varies with time of day (Greenwich mean time (actually the very slightly different "Oxford mean time"), not British summer time) for any day of the year, based on the average bright sunshine hours for Oxford. It also shows the declination(height) (in red) and orientation of the sun, the hours of sunset and sunrise and six other pieces of information. The first four are the average annual, monthly, daily power and sunshine hours. Then there is daylight hours which is self-explanatory. The maximum power is the power that would be obtained if the sun shone all day long in a clear sky as a useful reference. The functions of the various sliders are self evident. The output, declination of the sun, and azimuth are drawn as circles that are moved by the "time" slider. The main uses of the applet are to help to judge effects of roof orientation and tilt on projected output and to optimize the daily use of the generated electricity, which affects the cost savings. The calculation is a large one and may be jerky if run on a computer that is not very powerful. The assumptions in the calculation are explained below.

The well defined part of the calculation of the level of insolation is the geometric part, which depends on the azimuth and declination (height in the applet) of the sun and the orientation and tilt of the roof. There are also standard models for loss of sunlight by absorption and scattering in the atmosphere. There are several good websites that explain these calculations, notably the PVCDROM site of Christiana Honsberg and Stuart Bowden. Based on only the above factors the calculation gives the power output on a clear sunny day and this can be checked by appropriate monitoring of an installation. The power output is dominated by the azimuth and declination, as can be seen by comparing the variation of the output with the height of the sun with time of day (the open circles show the value of each of these at the time of day chosen by the time slider). Maximum output is at noon and the maximum is sharp. In terms of optimizing one's use of the electrical output, this means that one should think about trying to maximize one's use of electrical appliances at midday (see below). Atmospheric absorption is important because the pathlength of the sunlight through the atmosphere is much less, for example, between noon in June and noon in March. The much less well defined parts of the calculation are to do with the weather itself. In general average hours of bright sunshine are available for most cities. These data are particularly extensive for Oxford. The difficulty can be well illustrated by these data which give 6.5 hours as the average bright sunshine in June. However, there are more than 16 hours between sunrise and sunset in June and the distribution of the 6.5 hours in the day, which is not known, then becomes important. The approach in the literature seems to be to divide the light into non-bright sunshine and the bright sunshine outputs and determine each with a single empirical parameter. These parameters are place and season dependent. This is a particularly unsuitable description for the UK where cloud conditions mostly give rise to a continuum of light intensities rather than divide into two. Nevertheless, this is what has been done in the applet. Final tuning of the parameters has been to make the average annual output match the estimates from firms that instal solar pv. The variation with latitude uses data from 6 cities at more or less equally spaced latitudes from Edinburgh to Cairo, all fitted by polynomials for each month. The data is, however, much less reliable than that for Oxford. The starting latitude is that of Oxford. There is significant scope for improving the accuracy of the calculation by comparing the observed output with the prevailing weather conditions. There is one factor that has not been included and that is the temperature response of the pv panels. For example, the electrical resistance of silicon increases with temperature and reduces the efficiency. While this factor is known, the temperature itself will be sensitive to weather conditions that cannot be quantified.

The economics of installing solar photovoltaics in the UK are as follows. When the pv is installed (by a government accredited installer) a generation meter, which measures how much electricity is generated by the installation, is also installed. Regardless of what happens to this electricity, the supplying electricity company pays an index linked tax free 43.3p per unit (kWh) (the FIT payment (feed-in-tariff)). At present, there is normally no way of measuring how much of the generated electricity feeds back into the grid. It is therefore assumed that 50% is used and 50% is exported. If 50% is actually used this will save that amount of electricity and reduce one's electricity bill accordingly. In addition 3.1p per unit is paid for the assumed 50% exported. Thus, looking at the figures calculated by the applet (these are based on installers' estimates and there are good reasons to believe that they are accurate overall estimates), the annual generation is about 3400 units. This earns an income of 3400x43.3 = £1472. There is a further payment of 3.1p for 1700 units = £52. If 50% is actually used there is also a saving on electricity used. Taking the current price as about 15p per unit, this is worth another 1700x15 = £255. Finally, the annual income for a 4 kW installation on an unshaded roof facing due South is just under about £1800 per annum. Of course, it is up to the user to optimize their usage to ensure that they achieve this usage, for example, by using dishwashers, washing machines, driers, etc. at times of day when the generation is optimum. Eventually, smart meters will be introduced, which will measure the actual amounts used, but this will not significantly affect the optimum way to use the generated electricity. The cost of installation is about £16000 and will probably come down. There are other issues such as maintenance, how long will the panels last, etc. These are best asked directly of firms that instal the pv panels.