This is the eighth live-blog of my spring 2026 DERs class.
The last two posts discussed modeling DERs using linear ordinary differential equations and applied that modeling approach to batteries, the canonical DER. This post focuses on the one big DER we’ll study this semester that we won’t model with differential equations: Solar energy.
As They Might Be Giants explained in 1993, the sun is a mass of incandescent gas.
Well, actually… No. Technically, a gas consists of neutrally-charged atoms or molecules, stuff with the same number of protons as electrons. The sun consists of charged stuff — mainly hydrogen and helium nuclei — making the sun a miasma of incandescent plasma, as They Might Be Giants correctly re-explained in 2009.
Ah, technically correct. The best kind of correct.
Anyway, the sun is a giant nuclear fusion reactor that mashes hydrogen nuclei into helium nuclei, releasing lots of energy. That energy propagates through space as electromagnetic waves — and/or as photons, if you’re into the whole wave/particle duality thing — which we perceive on earth as sunlight.
Sunlight is the underlying source of almost all flavors of energy that humans use on earth. This obviously includes electricity from solar photovoltaics and solar thermal power, but also hydropower (via the water cycle), wind power (driven by temperature differences, largely due to different amounts of sunlight hitting the equator and the poles), heat and power from biomass (via photosynthesis) and, indirectly, from fossil fuels (via biomass rotting and pressure-cooking underground for a few million years).
Nuclear power is one exception. All nuclear power plants on earth today use fission: Splitting heavy atoms into lighter atoms, releasing lots of energy. Although the sun itself is a giant nuclear fusion reactor, it is not the source of the heavy atoms that humans split on earth for nuclear fission power. Those can mostly be traced back to ancient stars that exploded when they died. Suns, but not our sun. Nuclear power plants generate about 19% of US electricity.
Geothermal power is another exception. It involves extracting energy from hot underground rocks, usually by piping cool water over them to get back hot water or steam. What makes those rocks hot? Mainly nuclear fission reactions deep underground. Geothermal power plays a small (0.4% of electricity generation), but potentially growing, role in our energy systems.
The third exception is tidal power, extracting kinetic energy from tides driven by gravitational interactions between the oceans and the moon. Tidal power today is mostly limited to small research and demonstration projects.
What does all this have to do with DERs? Not much, but I think it’s pretty cool.
Back to the sun: After beaming through space at the speed of light for eight minutes and twenty seconds, around 1.36 kW/m2 of sunlight hits earth’s upper atmosphere. That number is called the solar constant. It showed up in the simple climate model in the lecture slides on linear dynamical systems.
The amount of sunlight that reaches a square meter of earth’s surface (known as solar irradiance) is less than the solar constant. It’s highest on clear days and lowest on cloudy days. In most locations, solar irradiance peaks at about 1 kW/m2 on clear summer days.
Thanks to astronomers’ adventures in spherical geometry, we have extremely accurate formulas for the sun’s position in the sky at any moment of any day of any season, as viewed from any place on earth. Given a local weather forecast, these formulas let us predict the solar irradiance incident on any arbitrarily oriented surface. That lets us predict useful things such as solar photovoltaic power output and heat gains from sunshine through windows (the key to passive solar building design). Those formulas are the main topic of the lecture on solar energy.