The science
Every case-study file computes P for one source and integrates E = Pt. The lab keeps those exact power laws and adds what makes them useful: availability over a day, a battery with a finite capacity and charge-path efficiency, and a load that must be met.
Equations as delivered in the case-study files
P(t)=ηAG sin(πt/T) [W]; E(t)=ηAG T(1−cos(πt/T))/π [J].
P=G A η [W]; dE/dt=P [J/s]; E(t)=E₀+Pt [J].
P=η(ρ A u³/2) [W]; dE/dt=P [J/s]; E(t)=Pt [J].
Pnet = ηPₕ − Pₗ [W]; dE/dt=Pnet [J/s]; E(t)=E₀+Pnet t [J].
P=ρ g Q H η [W] with ρ=1000 kg/m³; dE/dt=P [J/s]; E(t)=E₀+Pt [J].
What this lab adds
dE_batt/dt = ηc·P(t) − P_load, clamped to 0 ≤ E ≤ E_cap [W]
Solar day profile: P(t) = P_peak sin(π (t − 6 h)/12 h) for 06:00–18:00, so ∫P dt = P_peak·24/π [Wh]
Betz limit for a wind turbine: η ≤ 16/27 ≈ 0.593
Assumptions in the delivered files
- Ideal time-compressed half-sine illumination during scenario duration T, fixed area and efficiency.
- Constant irradiance, uniform illumination and constant conversion efficiency; shadows, heating and inverters omitted.
- Steady uniform wind at the rotor, fixed effective extraction fraction below Betz limit; no gust turbulence, cut-in/out or generator dynamics.
Honest simplifications of this lab
- Weather is not random: a steady, daylight half-sine or 09:00–17:00 availability profile is assumed, not measured data.
- The battery has no self-discharge, depth-of-discharge limit or ageing; the charge path has one efficiency number.
- The solar efficiency does not vary with temperature; the hydro head and wind speed are constant.
- Battery size and load are expressed relative to the mean source power so one scale works for every preset; they are lab choices, not values from the files.
Independent check The page recomputes the 24 h energy with closed forms (P_peak·24/π, the battery balance, the unmet-load formula); the test file repeats this with an independent minute-by-minute integrator and exact power laws.
Educational model. Case-study presets reuse the parameter sets, titles and captions of ChatGPT-generated simulations delivered for MySimulator; the physics core here is a richer shared model.