Training Stress Balance Formulas


This is the commonly used method for modelling your fitness and fatigue. It works on the basis that your current fitness is the total of all training done to date with a decay factor to reflect that the impact of a session on current fitness decays with time. Similarly your fatigue is impacted by all training done with a decay factor applied. Training Stress Balance (or form) is just the difference between them.

Terms we have are:
Chronic Training Load   = CTL – considered a measure of your fitness
Acute Training Load       = ATL – considered a measure of your fatigue
Training Stress Balance = TSB = CTL – ATL – considered a measure of your form.

The formula for Critical Training Load is

This is the generalised formula. Typically
and both are set to 42.

The formula for Acute Training Load is exactly the same with just different decay and impact constants:
and both are set to 7.
The simplified formula becomes:
All the same observations for CTL apply to this.

Using the typical numbers of 42 and 7 for CTL and ATL constants we get the following graph showing how CTL and ATL decay over time with no additional training. Training stress balance is merely the difference. ie:
100TSSDecayTSB = CTL – ATL

The graph shows the decay with TSB becoming positive after about 14 days. ProvidedATLDecayVCTLDecay then ATL will decay faster than CTL. These factors though appear meaningless. To make more sense of them we should look at their half life. ie how many days it will take for CTL / ATL to decay to half it’s value.

CTL / ATL Half-life
First note that in the case of no additional training we have:
To establish the half-life we want to find t such that
substituting in we get:
Noting that
and substituting we get:
By the same logic we get the ATL half life formula:
Using the typical numbers of 42 and 7 we get:
– Half life of CTL is ~ 29 days
– Half life of ATL is ~ 5 days
This is saying that in the absence of further training your fitness will half in 29 days and your fatigue will half in 5 days. If the software you use allows you to adjust these factors then this can give you some insight in to what appropriate numbers may be for each sport. Note that if the decay and impact factors are kept the same adjusting these will also change the effect (impact) of a given training dose.
This is saying that 100 TSS will add ~2.35 to CTL and ~ 13.31 to ATL

Replacement TSS
Consider what TSS is required to maintain CTL from yesterday. This is achieved by setting:
Not that in the typical case where
Substituting this we get
So to maintain your CTL from yesterday todays TSS has to be the same as yesterdays CTL. This really surprised me as in all my time using this formula I had not realised this simple fact. As ATL is just the same formula with a different factor then provided
we can also conclude that to maintain the same ATL you need to do TSS today thats equal to yesterdays ATL. This means if you want to continuously increase your CTL each day you need to do a TSS thats greater than yesterdays CTL. Clearly there’s a limit to how long you could keep this going.

Contant TSS
Now consider what happens if you apply the same TSS everyday for ever! Well, for at least a long time.
Now consider the case where
Substituting we get
Now. Lets pull out the first term of the first sum and the last term of the second sum so that both sums run from 1 to t. This gives us
Note that the middle two terms are the same and cancel out. Also the first term is just TSS as anything raised to the power of zero is 1. This gives us
Now remember from earlier that the decay factor is between 0 and 1. Which means as
Thus under constant TSS CTL will get ever closer to TSS. Similarly for ATL. Thus under constants TSS load both will approach TSS and thus TSB will approach Zero (from below. ie always negative. As per graph)


Replacing Lost CTL
So given a period of inactivity how much TSS will it take over a given number of days to achieve a new level of TSS. The answer to this question is something I’ve wanted to know regularly this year as I’ve had numerous trips where I’ve not been able to cycle.

NB: this part of this document is really me just documenting the derivation of the formula used in my code.

Lets define some terms
First lets calculate what the starting CTL will have decayed to by the end. So
Lets look at the CTL at the end of the constant TSS per day using the formula derived above under “Constant TSS”
Note that the sum has a “-1” term at the upper limit. This is because the first day of activity is day 0. So n days would be summed from 0 to (n-1).
The CTL at the end is the sum of these and to find target TSS we set that to the target CTL.
Lets just check this matches the result we had before. So set inactive days to zero and activity days to 1. This is our original case and thus the answer should be yesterdays CTL.
Noting that in the case we used early the impact and decay constants were the same which means
Note again this makes sense. With no days inactive this again reduces to TSS required is CTL from yesterday.
Here’s the approximate replacement TSS required in a single hit depending on number days inactivity:

0 days – TTS = CTL
1 day   – TSS ~ 2 x CTL
2 days – TSS ~ 2.9 x CTL
3 days – TSS ~ 3.9 x CTL
7 days – TSS ~ 6.5 x CTL

Clearly more than a couple of days inactivity and replacing lost CTL in one hit starts becoming inadvisable.
For comparison the equivalent TSS if we replace over 4 days are

0 days – TTS = CTL
1 day   – TSS ~1.2 x CTL
2 days – TSS ~ 1.5 x CTL
3 days – TSS ~ 1.7 x CTL
7 days – TSS ~ 2.5 x CTL

To give this context. If your CTL was 100 then to replace 1 days inactivity would take 4 days of 120 TSS per day, 2 days – 150 for 4 days, 3 days – 170 for 4 days and a weeks inactivity would require 4 days of 250 TSS per day. Given these figures again get quite high and require back to back training days I then considered what would be required if you train every other day.

The above formula gives us a TSS for everyday. I want to find the equivalent TSS every other day.
CTL after d days for every day training is given by the following formula where time t runs from 0 to (d-1):
ReplacementTSSTableSo what sort of results do we get?


Starting CTL of 100. We then have 7 days inactivity and then have up to 21 days training to bring the CTL back to that level. Here’s a table showing the TSS required per day or on alternate days.  Notice how on the evens days the alternate days TSS required increases. This makes sense as the last day has no training so some of the CTL gains have decayed. This clearly illustrates how it can be time consuming to regain lost CTL after a longer than a few days period of no training.ReplacementTSSGraph


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1 Response to Training Stress Balance Formulas

  1. Pingback: Banister Training Impulse Equation | Steven Lords Website

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