Solar Panel Rows in Profile

What is modelled

A small ground-mounted photovoltaic array on a Blender-built site: rows of six 2 m x 1 m modules (2.4 kW per row at 20 % efficiency). You choose the place and date, the tilt and direction (or a tracker), the row pitch and the number of rows. The page computes where the sun is, how much light reaches each panel, how much of the rear rows the front row shades, and the electrical power and the day's energy.

Equations

• Declination δ = 23.45° sin[360 (284 + n)/365], hour angle ω = 15°(t − 12); the sun vector in (east, north, up) is (−cosδ sinω, sinδ cosφ − cosδ sinφ cosω, sinφ sinδ + cosφ cosδ cosω). Noon elevation is 90° − |φ − δ|; day length from cosω0 = −tanφ tanδ.
• Clear-sky beam: DNI = E0 · 1361 · τAM0.678 with the Kasten-Young air mass; diffuse DHI = 0.1 DNI. On the panel: beam DNI cosθ · IAM (IAM = 1 − 0.05(1/cosθ − 1)), isotropic sky diffuse DHI (1 + cos b)/2 and ground-reflected albedo · GHI (1 − cos b)/2.
• Cell temperature Tc = Tair + (NOCT − 20) G/800 (NOCT 45 C), power P = A η (G/1000)[1 − 0.0035 (Tc − 25)] · 0.92 (0.92 stands for wiring, mismatch and inverter).
• Row shading: in the vertical plane perpendicular to the row axis, the front row's two edges are projected along the sun ray onto the rear panel; the overlap with the panel is the shaded slant fraction. Rows do not shade each other when the pitch exceeds p = L (cos b + sin b / tan a) (a = profile angle of the sun). Each module has three bypass-diode strings along its slant: a string that is even partly shaded loses all its beam power, diffuse light still arrives.
• Trackers: for a horizontal north-south axis the rotation that points the panel at the sun is tan R = sE/sU; backtracking turns the row back from that angle just far enough that no row shades another; the dual-axis panel simply faces the sun.

Checks run in the page

• Sun position scanned every 6 seconds: noon elevation and day length against the closed forms for five places and dates.
• Daily extraterrestrial radiation, integrated numerically, against the closed-form formula for five latitudes.
• Shaded fraction from edge projection against a 40,000-ray Monte Carlo for five random geometries, plus the zero-shading pitch formula.
• Single-axis tracker angle found by brute-force search against the closed form; backtracking leaves 0 % shading over a whole day; the dual-axis normal points exactly at the sun.

Honest simplifications

• Clear-sky model only: no clouds, haze changes or snow; DHI = 0.1 DNI is a rough ratio (real clear-sky diffuse varies with air mass and turbidity). The isotropic diffuse model under-estimates circumsolar light on tilted panels. The Meinel-type atmosphere uses one transmittance slider, not site data.
• Rows are infinitely long for the shading geometry (no end effects); the ground is flat. Dual-axis arrays are placed at least 7.5 m apart and their mutual shading is neglected. Soiling, spectral effects, degradation and inverter clipping are not modelled; the 0.92 factor is a typical lump, not a measurement.
• Temperature uses a simple sinusoidal day and the NOCT rule; real cell temperature also depends on wind and mounting. A fixed-tilt "best angle" chart ignores shading and is for one clear day, not a year; the annual optimum tilt for a place is lower.
• The numbers are illustrative of how design choices trade against each other, not a yield forecast for any real site.

• Pick Winter shading: steep tilt, tight rows: the front row shadows the rear panels; red bars show the shaded slant and the strings go dark.
• Switch to Single-axis with backtracking and let the day run: the rows lie back at low sun so nobody shades anybody.
• Open the last graph: dual-axis tracking gives the most energy per rated kW, but it needs the most land.
• Move the pitch slider: energy per land area has a maximum at an intermediate pitch.