Sizing a rooftop PV array
solar panel sizinghow many kWh will my solar makepeak sun hoursPV yield estimate
Sizing a rooftop array: sun angle on the tilted plane, peak sun hours, the derate a hot roof costs, and the kilowatt-hours in a year.
Solar Declination Angle (Cooper's Equation)
The angle between the sun's rays and the plane of the equator on a given day of the year. Cooper's 1969 approximation, the one line of solar geometry every collector calculation begins from.
Angle of Incidence on a Tilted Surface
How far off square the noon sun strikes an equator-facing collector tilted at angle β, at latitude φ, on a day of declination δ. Set the tilt to φ − δ and the answer is zero: the rays arrive dead on.
Peak Sun Hours from Insolation
A day's solar energy on a surface, re-expressed as the number of hours of full 1000 W/m² sun that would have delivered the same total. The tidy fiction that turns a curve across the sky into a single number a designer can multiply by.
Solar Panel Power Output
The electrical power a photovoltaic module delivers: the irradiance falling on it, times its area, times its conversion efficiency. The definition that lies behind every nameplate wattage on the back of a panel.
PV Temperature Derate
What a photovoltaic module actually delivers once its cells are hotter than the 25 °C the nameplate was measured at. The correction that explains why a panel makes less on the hottest afternoon of the year than on a bright cold morning.
PV Array Energy Yield
The energy a photovoltaic array delivers over a period: its DC rating, times the peak sun hours the site receives, times a performance ratio covering every loss between the glass and the meter. NREL's PVWatts model in one line.
How they fit together
Two questions have to be answered in order: how much sun reaches this particular slope, and what fraction of it the modules will actually turn into kilowatt-hours. The first two formulas settle the geometry. Cooper's declination equation takes the day of the year and gives the sun's angle north or south of the equator, swinging ±23.45° across the year, and angle of incidence on a tilted surface combines it with your latitude and your roof's pitch. The rule of thumb that tilt should equal latitude falls out of these two, and so does the reason it is only a rule of thumb: it optimises the annual total, and a system sized against a winter heating load or a summer cooling load wants a steeper or shallower roof than that.
Peak sun hours is the bridge between the geometry and the output, and it is the term most often misread. It is not hours of daylight. It is the insolation on the plane expressed as an equivalent number of hours at exactly 1000 W/m², so a site with 14 hours of daylight and 5.2 peak sun hours is behaving normally. That reference irradiance is chosen to match the conditions modules are rated at, which is what makes the last step a simple multiplication.
Then the losses. Panel output from area and efficiency is the ideal figure, and temperature derate is the first and largest correction — roughly −0.35 to −0.45% per degree above 25 °C for silicon. The trap is which temperature to use: the coefficient applies to cell temperature, not ambient, and a module in full sun runs 25 to 30 °C above the air around it. A 35 °C afternoon is a 60 °C cell and 12 to 15% off the nameplate, which is why a system quoted at STC never once produces its rated power on a hot clear day. Array energy yield finishes the job with a performance ratio that sweeps up everything else: inverter losses, wiring, soiling, mismatch, shading. Between 0.75 and 0.85 for a healthy installation. Anyone quoting you 0.95 has forgotten something, and anyone measuring 0.65 has a dirty array or a shaded string.