Cycling Power Formulas

Formula:

Worked Example:

JavaScript Implementation:

function calculateTSS(durationSeconds, normalizedPower, ftp) {
  const intensityFactor = normalizedPower / ftp;
  const tss = (durationSeconds * normalizedPower * intensityFactor) / (ftp * 3600) * 100;
  return Math.round(tss);
}

// Example usage:
const tss = calculateTSS(7200, 235, 250);
// Returns: 177

Algorithm (30-second rolling average):

Why the 4th Power?

The quartic (4th power) relationship reflects the non-linear physiological cost of variable efforts. A ride with surges and recoveries costs more energy than steady power at the same average.

JavaScript Implementation:

function calculateNormalizedPower(powerData) {
  // powerData is array of 1-second power values

  // Step 1: Calculate 30-second rolling averages
  const rollingAvgs = [];
  for (let i = 29; i < powerData.length; i++) {
    const window = powerData.slice(i - 29, i + 1);
    const avg = window.reduce((sum, p) => sum + p, 0) / 30;
    rollingAvgs.push(avg);
  }

  // Step 2: Raise to 4th power
  const powered = rollingAvgs.map(p => Math.pow(p, 4));

  // Step 3: Average of 4th powers
  const avgPowered = powered.reduce((sum, p) => sum + p, 0) / powered.length;

  // Step 4: Take 4th root
  const np = Math.pow(avgPowered, 0.25);

  return Math.round(np);
}

// Example usage:
const powerData = [/* 1-second power array */];
const np = calculateNormalizedPower(powerData);
// Returns: NP in watts

Formula:

Interpretation Ranges:

IF Range	Effort Level	Example Workout
< 0.75	Recovery / Easy	Active recovery ride, Zone 1-2
0.75 - 0.85	Endurance	Long steady ride, aerobic base
0.85 - 0.95	Tempo	Sweet spot training, tempo intervals
0.95 - 1.05	Threshold	FTP intervals, time trial effort
1.05 - 1.15	VO₂max	5-minute intervals, criterium race
> 1.15	Anaerobic	Short sprints, attacks, MTB bursts

Example Calculation:

function calculateIF(normalizedPower, ftp) {
  return (normalizedPower / ftp).toFixed(2);
}

// Example:
const if_value = calculateIF(235, 250);
// Returns: 0.94

Formula:

Interpretation by Discipline:

Discipline	Typical VI	Meaning
Road TT / Steady Effort	1.00 - 1.05	Very consistent power, optimal pacing
Road Racing	1.05 - 1.10	Some surges, generally steady
Criterium	1.10 - 1.20	Frequent accelerations and attacks
Mountain Bike XC	1.15 - 1.30+	Highly variable, constant surges

Example Calculation:

function calculateVI(normalizedPower, averagePower) {
  return (normalizedPower / averagePower).toFixed(2);
}

// Example:
const vi_road = calculateVI(240, 230);  // Returns: 1.04
const vi_mtb = calculateVI(285, 235);   // Returns: 1.21

Formulas (Exponentially Weighted Moving Averages):

Metric Definitions:

• CTL (Chronic Training Load): 42-day exponentially weighted average - represents fitness
• ATL (Acute Training Load): 7-day exponentially weighted average - represents fatigue
• TSB (Training Stress Balance): Form = Fitness - Fatigue

Worked Example (7-Day Training Block):

TSB Interpretation:

TSB Range	Status	Action
< -30	High Risk	Overtraining warning - reduce load
-30 to -10	Training Hard	Building fitness, monitor recovery
-10 to +5	Optimal	Normal training zone
+5 to +15	Race Ready	Peak form - race this weekend
> +25	Detraining	Loss of fitness - increase load

JavaScript Implementation:

function calculatePMC(workouts) {
  // workouts = [{date: "YYYY-MM-DD", tss: number}, ...]
  let ctl = 0, atl = 0;
  const results = [];

  workouts.forEach(workout => {
    // Update CTL (42-day time constant)
    ctl = ctl + (workout.tss - ctl) / 42;

    // Update ATL (7-day time constant)
    atl = atl + (workout.tss - atl) / 7;

    // Calculate TSB (yesterday's CTL - today's ATL for traditional calculation)
    // For simplicity here using current values
    const tsb = ctl - atl;

    results.push({
      date: workout.date,
      tss: workout.tss,
      ctl: Math.round(ctl * 10) / 10,
      atl: Math.round(atl * 10) / 10,
      tsb: Math.round(tsb * 10) / 10,
      status: getTSBStatus(tsb)
    });
  });

  return results;
}

function getTSBStatus(tsb) {
  if (tsb < -30) return "High Risk";
  if (tsb < -10) return "Training Hard";
  if (tsb < 5) return "Optimal";
  if (tsb < 15) return "Race Ready";
  return "Detraining";
}

// Example usage:
const workouts = [
  {date: "2025-01-01", tss: 100},
  {date: "2025-01-02", tss: 50},
  {date: "2025-01-03", tss: 120},
  // ... more workouts
];

const pmc = calculatePMC(workouts);
// Returns array with CTL, ATL, TSB for each day

Complete Formula:

Component Formulas:

Constants & Variables:

• CdA = Drag coefficient × frontal area (m²)
  • Typical road bike hoods: 0.35-0.40 m²
  • Drops: 0.32-0.37 m²
  • TT position: 0.20-0.25 m²
• ρ = Air density (1.225 kg/m³ at sea level, 15°C)
• V = Velocity (m/s)
• m = Total mass (rider + bike, kg)
• g = Gravity (9.81 m/s²)
• θ = Gradient angle (radians or degrees converted)
• Crr = Rolling resistance coefficient (~0.004 for good road tires)
• a = Acceleration (m/s²)

Worked Example (Flat Road TT):

JavaScript Implementation:

function calculatePowerRequired(params) {
  const {
    velocityKph,
    CdA = 0.32,              // m²
    rho = 1.225,             // kg/m³
    mass = 83,               // kg (rider + bike)
    gradientPercent = 0,     // %
    Crr = 0.004,             // rolling resistance
    accelerationMps2 = 0     // m/s²
  } = params;

  // Convert velocity to m/s
  const V = velocityKph / 3.6;

  // Convert gradient to angle
  const theta = Math.atan(gradientPercent / 100);

  // Calculate each component
  const P_aero = CdA * 0.5 * rho * Math.pow(V, 3);
  const P_gravity = mass * 9.81 * Math.sin(theta) * V;
  const P_rolling = Crr * mass * 9.81 * Math.cos(theta) * V;
  const P_kinetic = mass * accelerationMps2 * V;

  return {
    total: Math.round(P_aero + P_gravity + P_rolling + P_kinetic),
    aero: Math.round(P_aero),
    gravity: Math.round(P_gravity),
    rolling: Math.round(P_rolling),
    kinetic: Math.round(P_kinetic)
  };
}

// Example: TT at 40 km/h
const power_tt = calculatePowerRequired({
  velocityKph: 40,
  CdA: 0.22,
  mass: 83,
  gradientPercent: 0
});
// Returns: { total: 221, aero: 185, gravity: 0, rolling: 36, kinetic: 0 }

// Example: 8% climb at 15 km/h
const power_climb = calculatePowerRequired({
  velocityKph: 15,
  CdA: 0.38,
  mass: 75,
  gradientPercent: 8
});
// Returns: { total: 265, aero: 27, gravity: 244, rolling: 11, kinetic: 0 }