Electric Car Range Lab: Aerodynamic Drag, Rolling Resistance, Regeneration and Drive Cycles

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speed - wheel power - charge - range here -

Stories (presets)

The science

A generic electric car, a teaching model. No real vehicle or brand is represented; coefficients are textbook-order values for a mid-size electric hatchback. Pack, motor and tyre numbers are illustrative, not data for any product.

Road-load force at the wheels: F = ½ ρ C_dA (v+w)|v+w| + C_rr(v,T) m g cosθ + m g sinθ + m_eff a, wheel power P = F v. Rotating inertia is lumped as m_eff = 1.03 m. Air density from the ideal gas law ρ = p/(R T) with a barometric altitude term. C_rr rises ~0.5 %/°C below 20 °C and ~15 % from 0 to 100 km/h (empirical, after Wong, Gillespie).
Why range ~ 1/v² only at high speed: energy per km is E/d = F/η + P_aux/v. The aerodynamic part of F grows as v², but rolling resistance and the auxiliary (climate) term do not, so range follows 1/v² only where drag dominates. At low speed the fixed auxiliary power makes range fall again: there is a best-range speed.
Drivetrain (motor + inverter + gear) as a smooth loss model: P_loss = P₀ + k_i|T| + k_cuT² + k_feω^1.5 + k_wω² + k_g|P_mech|, so η(T,ω) = P_mech/(P_mech+P_loss) peaks near 96 % at mid load and drops at light load (fixed losses) and at high torque (copper loss). Fixed gear ratio 9.0, wheel radius 0.33 m. The numbers imitate published efficiency-map shapes; they are not measured.
Battery: usable energy = nominal × capacity factor(T) × (SOC_start − 5 % reserve). Cold lowers usable capacity (~0.6 %/°C below 20 °C, approximation) and raises internal resistance. Pack power losses R I² with I = (V − √(V²−4RP))/2R, open-circuit voltage 340–420 V by SOC. Thermal management and heating of the cells are not modelled separately; use the climate-power slider.
Regeneration: when the wheel force must be negative, the recovered share is limited by the smallest of: the regen-strength setting, motor torque / inverter power (and a fade below ~9 km/h), tyre grip on the driven axle μ N_driven with dynamic load transfer N = m g (0.5 − (h/L)(a/g)) for rear drive, and the battery charge acceptance (taper above 80 % SOC and strongly reduced when cold). The rest is done by friction brakes (heat). Round trip: the energy you get back is less than what you spent accelerating, because both directions have drivetrain and battery losses.
Drive cycles are synthetic and generic (urban stop-start, highway, mixed, mountain). They are not any regulatory test cycle. The mountain route climbs about 700 m and descends again. Grade uses the local slope of a smooth elevation profile; cornering and vehicle pitch dynamics are ignored.
Experiments: A compares the time-stepped model with the closed form R = E_use/(F v/η + P_aux)·v (it neglects pack resistance and the SOC dependence of voltage, hence the small difference). B runs each cycle with regen on and off, and checks an idealised limit (lossless, unlimited regen) against the exact result ΔE = ∫(F_aero+F_roll)v dt + m g Δh + ΔKE.
Illustrations: airflow streaks are a hand-made deflection around an ellipsoid, not CFD; the force arrows are to scale in a compressed (square-root) length.

Typical references: Wong, Theory of Ground Vehicles; Gillespie, Fundamentals of Vehicle Dynamics; Ehsani et al., Modern Electric, Hybrid Electric and Fuel Cell Vehicles; MacKay, Sustainable Energy — without the hot air (range and drag).