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How to use
Pick a preset or set CP, W' and the session program (warm-up, interval power and duration, recovery, repeats). Press Play and watch the W' tank drain during efforts and refill during recovery. The planned effort is shaded; the amber line is the power actually ridden. Use the playback slider to run the session faster, Space to pause, R to restart. Compare the two recovery models, then try a 6% climb or a lighter rider to see how speed and drag change while the physiology stays the same.
The science
The critical power model (Monod and Scherrer, 1965) says that above a power CP an athlete spends a fixed finite work capacity W' (kJ), and below CP the effort is sustainable for a long time. Work to exhaustion at constant power P > CP in time t is
W = CP t + W' => t_lim = W' / (P - CP) (hyperbolic power-duration curve P = CP + W'/t)
Morton (1996) and Skiba (2012) turned this into a battery: while P > CP the balance drains, dW'bal/dt = -(P - CP), and while P < CP it refills. This page offers two refill models. Skiba 2015 (differential): dW'bal/dt = (CP - P)(W'0 - W'bal)/W'0, which has the exact solution W'bal(t) = W'0 - (W'0 - W'1) exp(-(CP-P) t / W'0). Skiba 2012: exponential refill with time constant tau = 546 exp(-0.01 DCP) + 316 s, DCP = CP - P_recovery in watts, an empirical fit to recovery experiments in a small group of subjects.
Road speed. The speed shown is solved from the power balance eta P = (m g (sin th + Crr cos th) + 0.5 rho CdA v^2) v with rho = 1.2 kg/m3, Crr = 0.004, drivetrain efficiency 0.976, by Newton iteration; it is a flat steady-state estimate and ignores wind, cornering and acceleration.
Checks. (1) Simulated time to exhaustion at 300 W equals W'/(P-CP) = 400 s. (2) After a 30 s effort the balance after 400 s of recovery at 150 W equals the exponential solution. (3) The balance removed in a hard effort equals the integral of (P - CP) dt. (4) Work to exhaustion at 400 W equals CP t + W'. (5) The speed solver reproduces the input power.
Honest limits. CP and W' are fitted parameters, not fixed biology: they move with fatigue, heat, altitude and the duration range used to fit them (the model is poor for efforts shorter than about 2 minutes or longer than about 30 minutes). The recovery models are empirical, and real recovery is slower after deep depletion. When the balance reaches zero this page simply limits power to CP, which is a simplification of the collapse a real athlete shows. This is an educational model, not training advice.