Wbal model

Work Balance & Power Output Modeling

Interactive Wbal simulation

The Wbal model tracks the W' balance (Wbal): how much of W' an athlete has left at each moment of a session. Critical power (CP) is the horizontal asymptote of the power–duration curve, and W' is the finite amount of work that can be performed above CP (both are introduced in Module 3: see Critical power and W′M3). Wbal starts at W', falls while power is above CP, and recovers while power is below CP.

The Wbal model and its recovery time constant were introduced by Skiba et al., 2012 (Medicine & Science in Sports & Exercise). This page uses the differential form of the model (a differential equationM4, solved step by step with finite differencesM4 as in Module 4), published by Skiba et al., 2015 (European Journal of Applied Physiology).

Explore how Wbal evolves during high-intensity interval training (HIIT) sessions. Adjust the parameters below to see how session design and athlete characteristics affect Wbal depletion and recovery.

For a model that replaces this single balance with ten exponentially weighted means of power, one per time scale, see the Workout Reserve model.

Excel implementation

You can replicate the Wbal model in Excel as follows.

Set up your spreadsheet

1. Column A: Time (t) in seconds: 0, 1, 2, 3, ...
2. Column B: Power (P) values for your session
3. Column C: Wbal calculations
4. Column D (optional): Tau Wp, the recovery time constant of the original (integral) form of the model. The differential form used here does not need it, so you can leave this column out.

Mathematical implementation

The Wbal model uses a different equation depending on whether power is above or below CP. In the equations, the spreadsheet and the chart, W' is written Wp.

When Power ≤ CP (Recovery):

When Power > CP (Depletion):

Excel formula implementation

Use this formula in cell C3 for $W_{bal,k+1}$:

=IF(B2<=CP, C2+(CP-B2)*((Wp-C2)/Wp), C2-(B2-CP))

Here CP and Wp are named cells. Type your CP (in W) and W' (in J) into two empty cells, then give them the names CP and Wp (select the cell and type the name in the Name Box, left of the formula bar, or use Define Name on the Formulas tab). Put W' in cell C2 as the starting Wbal.

Copy this formula down column C for all time points.

Key points for Excel implementation

• Row indices: $k$ is the current row, $k+1$ the next row
• IF statement: checks whether $P_k \leq CP$ to choose between recovery and depletion
• Recovery rate: depends on the current Wbal: the lower the Wbal, the faster the recovery (the factor $(Wp - W_{bal,k})/Wp$ in the recovery equation)
• Depletion rate: a linear subtraction of the power above CP

Model parameters

• CP (critical power): the horizontal asymptote of the power–duration curve; above it Wbal is depleted, below it Wbal recovers
• Wp (W'): the finite amount of work that can be performed above CP, also called the curvature constant of the power–duration relationship
• Interval parameters: length, power and number of intervals
• Recovery parameters: length (30–300 s) and power (0–150 W) of the recovery between intervals

Understanding the model

The model has three regimes:

• Above CP: $W_{bal}$ depletes linearly at rate $P_k - CP$
• Below CP: $W_{bal}$ recovers exponentially
• At CP: $W_{bal}$ remains constant (sustainable intensity)