Editor's Note: Scientific American's George Musser will be chronicling his experiences installing solar panels in Solar at Home (formerly 60-Second Solar). Read his introduction here and see all posts here.

Solar homeowners' favorite topic of conversation is the performance of their arrays. As part of the sales pitch, the installer estimates how much power you'll generate, and most systems come with a meter (separate from the utility meter) to monitor the power output continuously. But how can you tell whether your array is really living up to expectations? That simple question set me off onto a mathematical hunt that other solar homeowners might enjoy -- and which would make a good term-paper project for a high-school science class.

There are lots of tools out there to estimate the performance of a solar array. NOAA provides an applet and Excel spreadsheet to determine the sun angle at any time of year based on the astronomical geometry, with corrections for atmospheric refraction and the offset between clock and solar time. The PV Watts website combines solar geometry with historical weather data to calculate the expected performance for a panel at a given location with a given orientation. PV Performance also factors in the properties of the solar cells and the inverter that converts their DC output to household AC power. For site-specific issues such as shading, you need to make more detailed measurements. The New Jersey state inspector who signed off on my system used a nifty 360-degrees light meter called the Solar Pathfinder.

So much for the predictive tools. To measure the actual output minute-by-minute, I have two independent energy monitors that are installed in my electrical service panel: the TED5000 and the Locus. They display the total array electrical output through a web interface.

Useful though they are, these devices don't separate out the myriad factors affecting power output: solar geometry, weather, diffuse vs. direct illumination, shading, reflection off the glass surface of the panels, temperature dependence of solar cell output, solar cell and inverter efficiency, and so forth. These effects work in concert. For instance, if the temperature gets so hot that it cuts the output of the solar cells, the total array output can drop low enough that the inverter chokes.

To weigh some of these effects quantitatively, I put together my own spreadsheet to calculate the expected power output for June 30th -- a nice, mild cloud-free day when the air temperature at our site stayed fairly constant, reducing the confounding effects of weather. Those who want to choose a similarly opportune day can look up their local weather records at Weather Underground.

Over the course of a day, the intensity of sunlight varies sinusoidally with time. The amplitude and phase of this sinusoid changes with the seasonal cycle, but in retrospect I didn't really need to worry about that. I could just have drawn a cosine curve and stretched and shifted it to match the midday peak power. Between 9 a.m. and 3 p.m., the array output followed a sinusoid almost perfectly (see the blue curve in the above plot).

In the early morning and mid- to late-afternoon, however, the output fell off more rapidly than the illumination conditions alone would explain. So I went down the list of other factors.

First, I took a bit more care with the geometry of my roof. Usually, panels in the Northern Hemisphere are assumed to point in the general direction of south, tilted upward. My roof, however, also slopes toward the west, so my panels are tilted along both their long and short axes. It took some linear algebra to account for this orientation. The effect was to give the sinusoid a slight skew.

Second, sunlight passes through a greater mass of air in the early morning and late afternoon, attenuating it. Bradley Hibberd, director of engineering at Borrego Solar Systems, pointed me toward a model adopted by the American Society of Heating, Refrigeration and Air Conditioning Engineers (ASHRAE). (For those who care about the technical details, the model uses the Beer–Lambert law, whereby the solar intensity is reduced by the exponential of the secant of the angle between the sun and the zenith.) This helped to explain why the power fell off so dramatically outside the peak hours.

Third, I considered losses due to sunlight glinting off the panel glass. A paper by Sandia Labs has a graph showing how reflection becomes a serious issue when the incidence angle of the sun's rays is greater than 55 degrees, which in my case meant before 9 a.m. and after 4 p.m. The drop-off goes roughly as the cube of the angle. Incorporating this into my spreadsheet further reduced the off-peak power.

Finally, I took into account the efficiency of my inverter. Below a certain power level, the inverter isn't able to convert the DC to AC very effectively, giving you the double-whammy of less sunlight and less power per unit sunlight. I approximated this fall-off using as a power law.

Between inverter, reflectance, and atmospheric attenuation, I was able to account for the diminished array output during the morning hours (see red curve). That left the abrupt decrease in the mid-afternoon to explain. I suspect it might be shading. A tree and chimney are located just west of the array, and although they're not very tall, even partial shading would cause a sharp loss of power. The panels are wired in electrical series, so that whatever affects one of them affects all of them. The Solar Pathfinder measurements showed somewhat more shading in the west than in the east, which should cut into the power output beginning about 3 p.m. One of these days when I find myself at home in the mid-afternnoon, I'll go up to the roof to see what's going on.

My analysis was really just the first step and I'm not sure that I've really isolated the important effects among all the possible ones. I'd love to hear how other people have sought to explain the performance of their systems!

Graph courtesy of George Musser