What is the science behind Performance Modelling applied to on-the-bike Power and Pace simulation, and can I trust it? These are important questions.

## Power & Speed

The relationship between power and speed is the foundation of Performance Modelling and we model it using long established certainties of physics, specifically Newton’s Second law relating to
conservation of energy. When a 70 kilo rider on a 7 kilo bike climbs a 5 per cent grade at a certain speed we know exactly what power he needs to overcome earth’s gravitational pull. Similar
principles apply to the calculation of power required to overcome aerodynamic drag, rolling resistance, and to accelerate. These principles have long been applied to cycling and with good
results.

Martin et al
showed how a mathematical model of road cycling power predicted required power to within 2% of power outputs recorded using SRM’s. Now consider that a well calibrated SRM claims accuracy of +/-2% and we’re
getting somewhere.

Assuming Newton’s physics are sound then the important factor in performance modelling becomes quantification of model parameters.

### Course Profile

The simplest way to use a power-speed model is to assume a very simple course of a certain distance with a certain average gradient between start and finish. Unfortunately this simple usage – still
adopted by many in ascent analysis – leads to accurate estimates of power or pace only in rare circumstances. Let us imagine an extreme example of why this doesn’t work – a rider climbing and descending
a hill. Applying the model simply the average gradient of this rider’s course is zero – pan flat – while this clearly isn’t so. Better results require a much more detailed representation of course profile.

We use GPS coordinates with corresponding geological survey elevation data to create a high resolution representation of the terrain making up a particular course. Typically this means a course is
decomposed into hundreds of sectors and we compute the power speed relationship for each one. A rider’s entry speed in sector S is the speed he exits sector S-1. Courses with significant changes
in altitude also require adjustments to air pressure which typically falls by 1mb every 10 metres of elevation gain.

*Example high resolution bike course profiles for the IronMan & 70.3 Worlds in Kona & Henderson...*
### Rider & Bike Weights

These are easy to measure using modern digital scales. The least error prone method to weigh a bike is to subtract the weight of a rider holding his bike from his weight alone.

### Aerodynamic Drag (CdA or “Drag Area”)

This is an important input.

- It can be estimated using anthropometric (body size) estimators established in scientific research.
Bassett el al.
determined a relationship between height, weight and frontal area (the A part of CdA) while
Heil
determined a relationship to drag coefficient (the Cd part of CdA).
- It can be estimated using aerodynamic field test protocols. We discuss and provide models to assist with the calculations in the main protocols - see
Aerodynamic Models.
- It can be measured precisely, consistent with the same physical principles underlying a cyclists power-speed relationship, in a wind tunnel.
- Component specific drag numbers are now widely available in the public domain - subject to some unknowns and a level of trust - from leading aerodynamic wheel and frame manufacturers.
- Drag interactions between rider, bike and wheel systems rely on common observations concluded from practical wind tunnel analysis.

### Atmospheric Variables

Air density is an important input in power-speed modelling as it influences the power a rider requires to overcome aerodynamic drag. Air pressure, temperature and humidity together define
density. These variables can all be sourced from simple weather reports. In the case of predictive event modelling we use a weather service to determine the most probable weather scenarios
at a particular location on a particular date.

### Wind Speed & Direction

Wind effects are the trickiest model inputs to get right. As with atmospheric variables we can rely on a weather service to envisage probable scenarios in the case of predictive event
modelling. In historic event analyses (see our

Pro Race Analysis) we apply what we call “Continuous Weather”.
This means we update wind variables in model calculations with the same resolution as historical time-of-day weather recordings and apply the appropriate winds to riders as they progress
through course sections using their event start time for synchronisation.

*A meteorologist's view of Race Day at Kona, 2012...*
### Coefficient of Rolling Resistance

There is now lots of data in the public domain relating to the Crr of specific tyre choices and how these scale from roller tests, through different inflation pressures, and onto an asphalt
road. We rely on this data to conduct power-speed modelling with good estimates of rolling resistance power demands.

These conclude the main considerations that promote accuracy in event modelling.

## Physiology

When we extend Performance Modelling to account for energy expenditure, energy budgeting – effectively a carbohydrate and fat fuel demand analysis – and to develop variable power and
optimal pacing strategies consistent with physiological limitations we rely on models of physiology.

Gas analysis used to determine an athlete’s fuel utilisation at varying work intensities is one of the most common and well understood disciplines in sport and exercise science.

Physiological limitations relating to pacing rely on the Critical Power concept of Monod & Scherrer, the Fatigue Curve concept of Riegel, and other emerging models relating to anaerobic work
capacity recharge rates.

Much of the science behind Pre-Race Performance Modelling is also relevant to retrospective Race Reviews and Results Analysis. We use the same principles to set up a
course and event scenario, although the benefit of a retrospective view allows us to be more precise with many parameters. In the case of individual Race Review bringing
in the data from an actual race power file allows an extra level of analysis in two areas:

## Realised Parameter Solutions

We can normally solve any 2 of 4 “uncertain” parameters that have affected a riders time in a given event. Choices are:

### CdA & Crr Given Observed Wind Speed & Direction

Estimating the twin parameters of CdA & Crr is a common but advanced use of power data in cycling. Typical methodologies incude the Regression (Martin) Method
and the Virtual Elevation (Chung) Method. Calculations for both are available at

fastaerolab.com.
Ultimately the goal is to use the physical equations defining the relationship between cycling speed, power, and several other known inputs to
solve for the values of these most important parameters to cycling performance. Successfully estimating these parameters is highly valuable:
retrospectively in terms of evaluating the effectiveness of component and position choices (in the CdA case) and tyre and inflation choices
(in the Crr case) and prospectively as an aid to performance modelling.

Our solution schema applies the Virtual Elevation method although in the context of holistic ride modelling where wind bearings and velocities
are non-zero, observed weather station values. These will not be equivalent to the wind effects experienced by the rider but may offer a
better solution than the assumption of a windless environment. We iteratively solve the CdA and Crr parameters explaining a particular
ride performance using a multivariate solution algorithm which best fits a virtual elevation penalty function to actual road elevations
determined from geological survey data.

*Calculating the CdA and Crr or Wind Vectors that explain a riders time-on-course is a lot like calculating the fastest drive through every state in
the US - the right optimisation algorithm makes it easy.*
### Wind Speed & Direction given Supplied CdA & Crr

For riders who know or have a good prior estimate of their CdA and Crr we can alternatively solve the wind effects experienced during an event.
Sometimes this information can provide useful insights into the effect of changeable weather conditions on race outcomes or on differential
performance through distinct race sectors. On a more fundamental level this analysis can be appplied to answer the question “how closely did
reported weather conditions reflect the winds experienced by a rider”?

Again we use holistic event modelling, relying heavily on high resolution sectorisation of the course and a multivariate solution algorithm
to solve for effective winds.

### CdA & Wind Speed given supplied Crr & Wind Direction

Often in ride modelling we have a higher confidence around the coefficient of rolling resistance (Crr) attributed to the riders bike than to
the CdA applicable to the rider and bike combination on the day. And we have a higher confidence around the predominant wind bearing than
around momentary wind strengths. These conditions are ideal to consider the solution of realised CdA and Wind Speed given a known Crr and
Wind Direction.

## Power & Energy Distributions

Most cycling power meters tell us how much power a rider is using but not where that power is going. This is a shame, because it is a common
adage that one should “maximise the power you can produce but minimise the power you have to produce”. Knowing where the power you have to
produce is going is an important step toward this goal. We syncronise a riders race power and speed data with the course sectors it relates
to and in this way we can post-compute the distribution of power usage and energy expenditure among: aerodynamic drag, rolling resistance,
acceleration, and other minor components. An accompanying “unexplained” figure is exactly that and expresses the error in the solution. The
distributuion of power consumption is calculated as in our

Power Components Model.

*Rider energy expenditure by components on an undulating course. Note the gravitational energy gain due to height loss and dominance of aerodynamic drag...*