Okay, so ready for Part 2 of this "modelling HFR travel times on the existing Havelock alignment" exercise (Refer to post #
7,265 for Part 1)?
3. Modelling
3.1 Speed limits
As I've already covered in a
previous post, applicable speed limits are mostly determined by the tracks curvature (in combination with the superelevation applied to the curves) and is determined with the following formula:
Quoted from:
my Master Thesis (p. 60)
The resulting speed limits can be found in the following table:
Source: own calculations, first presented in post
#7,260
In terms of the curves, I reviewed my .kmz file (
here the newest version) and found another 10 curves (of which 2 have a radius of less than 550 meters), for a total of now 280 curves (or one every 1.4 km):
View attachment 272255
Source: own calculations with track geometry data estimated by using the circle drawing function built into Google Earth Pro
As you can see in above table, a superelevation of
5 inches only allows HFR trains to keep their top speed (assumed at 110 mph) in curves of the highest category (i.e. a radius of 3000 meters), which means that more than 90% of all curves require the train to slow down and that means in the case of just under 80% of all curves to slow down below 76.4 mph (i.e. the average speed which is required to achieve a travel time of exactly 3:15 hours between Ottawa and Toronto). Conversely, with a superelevation of
10 inches, HFR trains could also keep their top speed in curves with a radius of 2400 or 1700 meters and would only have to slow down in just under 80% of all curves and below the 76.4 mph in only just over 40% of all curves.
3.2 Uniform acceleration
Without access to specialist software (one of the main limitations of my Master Thesis!), the only realistic way to calculate travel times is to assume uniform acceleration, i.e. by distinguishing between three different types of movement: acceleration (where the train speed increases by exactly the acceleration value), constant movement (where the speed is unchanged) and breaking (where the train speed decreases by exactly the deceleration value). The main advantage is that the acceleration and breaking behavior is reduced to only two variables, which can be applied in the following two textbook formulae:
View attachment 272254
Quoted from:
my Master Thesis (p. 77)
Despite ignoring factors like gradients, this still get's quite complicated, but anyone really interested in this can read about it in Chapter 6 of you-know-what...
3.3 Fixed blocks vs. variable block
An important point in my Thesis was to compare the effects of fixed vs. variable blocks on achievable train frequency, as train capacity was an important consideration. If you recall what I just explained to Paul about PZB vs. LZB, variable blocks are basically a system with stationary signals (which in Germany would be protected with PZB magnets) and a system without signals, where moving authority and speed limits are communicated directly to the train (which in Germany would be the responsibility of LZB). For our purposes, however, we are more concerned about static speed limits (i.e. those permanently imposed to reflect track curvature than those temporarily imposed by signals) than about how long a train remains in a block, and therefore, it would be more accurate to talk about whether we assume speed limits which apply to individual curves or zones (which combine multiple curves, like Paul suggested).
Given that VIA's future fleet is based on the Siemens Charger locomotives, which run for Amtrak and (when it resumes service again) Brightline with mandated PTC, which requires a continuous and proactive enforcement of speed-limits (i.e. like LZB and unlike PZB), I believe it is reasonable to assume that the block lengths for speed limits would be fixed rather than variable, which means that I calculate speed limits with a granularity of 80 meters (i.e. 20 blocks per mile) rather than variable blocks with lengths of anything between 100 meters and multiple kilometers, which means that applicable speed limits may change a dozen time within a minute of a train's runtime, which would be far beyond what a human driver could process and safely reflect in his choice of acceleration and deceleration commands to the train's traction motors and breaks...
***
Having laid out the fundamentals of modelling train runtimes, I will show how to implement and solve the model to estimate the travel times for the Havelock alignment:
4. Model Solving
- Ignoring s-curves
- Respecting s-curves
In the meanwhile, please let me know if I lost you somewhere...
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