Athlete summary
W′bal is how much of your anaerobic reserve (W′) you have left right now.
- Riding above CP spends W′ at
(power − CP)joules per second. - Riding below CP rebuilds W′ with time constant τW′ — faster when you are deeper into the hole. Analysis and session design label whether τ is fitted (your calibration) or generalised (population equation).
- Balance is always clamped between 0 and W′.
Deep digs (low min W′bal) explain why the last climb feels impossible even when average power looked “fine.” Session design keeps a v1 heuristic reserve (~10% of W′ in hand; — not a Skiba optimum) so there is room for a finish sprint or a missed shift in the bunch. Unsustainable VO2 prescriptions are rewritten (shorter work / fewer reps) or demoted rather than shipped with a decorative “W′bal-shaped” badge.
After a prior-work gate, analysis may switch the heavy↔severe boundary to CP_fatigued (clamp to W′_fatigued when known) — see Fatigued Critical Power and W′ after prior work. If fatigued CP is unknown, fresh CP stays.
τW′ — fitted vs generalised
Recovery in the ODE is dW′bal/dt = (W′ − W′bal) / τW′ below CP.
Welburn, Pugh, Bailey & Ferguson (Eur J Appl Physiol 2025, doi 10.1007/s00421-025-05912-0) showed that none of five published generalised τ equations predicted W′BAL ≈ 0 kJ at task failure on intermittent depletion. An individualised τ fitted from a 20:10 to-exhaustion (equivalent) protocol — assuming failure = 0 kJ — did.
| Mode | When | Notes |
|---|---|---|
| Fitted | Athlete has measurement tau_wprime (s) | Only athlete-specific path. Fit via fit_tau_wprime_from_exhaustion from a 20:10 TE (failure ≈ 0 kJ). Do not invent τ from CP/LT1 correlations. |
| Generalised Pugh Reg (sole default) | No fitted τ yet | τ = 5184 · DCP^(−0.700) — Welburn short-recovery ranking. Always labeled generalised. |
| Skiba2 (alternate constant) | Explicit tau_mode="skiba2" only | τ = W′ / DCP — Clarke Table 1 goldens / historical form. Not a second product default. |
| Skiba1 | — | Out of the product path (under-predicts / hits 0 too early). |
Surfaces (activity detail, design trajectory) expose wprime_tau_source / tau_label so a coach never confuses a generalised “empty” dig with a fitted reserve.
Where Koyaro uses it
| Context | Role |
|---|---|
| Ride analysis / streams | wprime_balance channel + dig summary (τ source) |
| Race debrief | Replay matches and finale W′ in hand |
| Route pacing | Prospective constraint so climbs do not empty W′ |
| VO2 session design | Adjust recovery / reps / work length until planned trajectory stays above the v1 reserve heuristic; refuse/demote if none |
| Limiter session design (M5) | FTP → long near-FTP blocks; 5-min / default VO2 → long severe intervals via W′bal when CP/W′ set (demote when unsustainable; HIT quality = T@90% VO2max); sprint → neuromuscular openers; attacking/short climbs → optional fixed-%MAP 30/15 (not crowned over long VO2) |
| Race effort budget | Forward surge script (attacks) before race day |
Prescription and debrief share the same ODE so the number you plan with is the number you review.
Authoring heuristics (not ODE parameters)
| Constant | Default | Status |
|---|---|---|
| Min W′ reserve in hand | 10% | v1 heuristic — Skiba justifies the ODE, not this floor |
| W′ spend per VO2 bout | 40% | v1 heuristic — recoverable repeats under the ODE; not a cited optimum |
Do not invent alternate formulas to “clear” these; retune only with an ADR / issue.
ODE vs integral (refinement note)
Koyaro implements the W′BAL-ODE differential form popularized for streaming and coaching (Clarke et al. 2021 Table 1 goldens; Skiba differential recovery).
Skiba et al. (2012) also describe an integral recovery formulation that is mathematically related but more elaborate to evaluate per-sample. For 1 Hz power streams, interval design, and race-surge scripts, the ODE is the practical standard: same CP/W′ inputs, transparent depletion/recovery, and literature-tested numeric behavior.
Optional adoption of the integral form for a specific use case is tracked in (🔬 deferred). It would belong as a refinement behind the same athlete CP/W′ anchors — not a second competing “W′bal” product metric. This issue is about τ, not the recovery form.
Literature
- Welburn, Pugh, Bailey, Ferguson — W′ reconstitution modelling during intermittent exercise performed to task failure (Eur J Appl Physiol, 2025), doi 10.1007/s00421-025-05912-0. Citation key:
welburn-2025-wprime-tau. - Skiba et al. — W′bal modeling (Med Sci Sports Exerc, 2012), including the integral recovery discussion.
- Clarke et al. — W′BAL-ODE for applied physiology / coaching (Int J Sports Physiol Perform, 2021), Table 1 reference values used as golden tests (Skiba2 form as documented alternate).
- Pugh et al. 2022 — Reg τ equation (default generalised path per Welburn 2025).