Strange (wrong?) TSS(s) evaluation

FredJ

Hi all,

I hope this post doesn’t make any duplicate. I read a lot of other ones related to TSS calculation too, but no one seems to describe the same “pathology”.

A few weeks ago I ran the same track twice in the week, beeing proud the 2nd time because I ran significantly faster. But I noticed that the TSS of this fast run was lower than the one of the slow run :

               date     km   D+  duration  km/h    tss
2025-12-15 04:49:02  22.08  166  02:15:27   9.8  121.0
2025-12-23 04:59:50  22.04  157  02:03:45  10.7  115.5

Note: except from the speed, all data come from my Suunto Ambit 3 Peak, used without heart rate tracking. Thus I “tss” should mean tss[r] (aka rTSS).

According to me the 2nd run should give a higher TSS. I see that the measured distances and denivelation are both slightly lower for the 2nd run, but I don’t think this should counter the big difference (almost 10%) in the duration and thus the speed.

To complete deeper this analysis, I gathered all the fit files of the activities made on the same track and with the same watch. Then I plot the calculated TSS as a function of the duration, which gives me this figure :

I don’t mind the few strange values which are out of the “main line” (they represent around 10% of the 32 selected points), but once again I don’t understand its shape.

The definition of TSS should produce a descending line (see my demonstration at the end of the message), not an ascending one.
So what does happen here ?

Bonus : I suspect that this behaviour can be generalized over other workouts even if the analysis is more difficult when comparing different tracks.
I found two other interesting recordings from my last summer holidays :

               date    km      D+  duration  km/h    tss
2025-07-13 06:30:25  22.6  1487.0  03:46:42   6.0  185.2
2025-07-15 08:28:33  12.0   806.0  04:57:16   2.4  201.0

The first run is a competition ran fast though, not at my maximum.
The second one is a very gentle walk with my wife and my kids. Both recorded with the same watch using the same sport mode.
So do I need to stop running fast if I want to progress more ?

PS : my “proof” of why the plotted curve should decrease (let’s do some maths as we were still at school) :

Neglecting the constants used for normalisation, raw definition of TSS[r] can be expressed as formula (1) source: trainingpeaks

(1) TSS = duration . NGP . IF / FTP

but because IF = NGP/FTP we can simplify it as :

(2) TSS = duration . NGP² / FTP²

Now let’s define ND as the “Normalized Distance” to match with NGP as NGP = ND / duration. Then we get :

(3) TSS = ND² / FTP² / duration

Because all the selected workouts are recorded on the very same track, ND is a constant through the plotted data.
And I assume that FTP doesn’t change over the time because I use neither pace nor heart levels (I can’t remember when I modify them for the last time, considering that I really did it once). And according to me this is confirmed by the chosen colorscale on the plotted data.
So for this example we end with :

(4) TSS = Const / duration

which tells us that : lower the duration, higher the TSS.
All but suprising !