But if there are fewer trains there is also less opportunity for regeneration so more power use per train.
This is not the 50s, a substation can quite easily be equipped with a grid tie inverter pack.
Or in the relatively high voltage-headroom scenario (750V substation voltage I used for initial calculations gives you a full 150V of headroom according to the standard) we have available the overvoltage will be able to drive significant regeneration currents over multiple substation distances, allowing the line to capture most regeneration energy even if you only equipment one substation with regeneration equipment to save money.
If the substations are further apart there will be more resistance losses because of greater average distance from the substation. And if the substation distance and rail resistance remain the same, the loss of power is directly proportional to the current so the percentage loss which I quoted does not depend on the power demand.
The loss of power is proportional to the
square of the current.
P = I^2 x R remember.
While the total power that enters the conductor is dependant on the current linearly, so the proportion of the losses increases.
If I have a one amp, ~100V load that I have to drive through a one ohm resistance I will drop 1V and only 99V will drop over the load. 1% was lost.
But if I have a two amp, ~100V load that I have to drive through the same resistance I will drop 2V and only 98V will drop over the load. 2% was lost.
The percentage lost will increase linearly to the average current.
A mains supply that can run a few lights and PIDs isn't going to be suitable for traction power, and unless a higher voltage supply is close by the voltage drop in the 400V has to be added to that in the third rail.
The big big saving in using the 400V supply is that it drastically relaxes the earthing and safety requirements as you no longer need a giant earthing mat or similar. You use simple industrial switchgear rather than fancy HV stuff.
This has been done on various tram projects.
400V supplies are available that can support several hundred amps, and since it would feed an on site rectiformer the exact voltage of the input is irrelevant. Since the voltage will have to be retained inside the norms for an electricity supply and you will be charged on that basis.
Even a standard rural three phase supply will be capable of something like 375A.
What would that do for power loss, resilience, and fault detection?
Thanks to reduced currents the power loss will be less than in an urban setting - and the low currents will still permit resilience.
Fault detection is problematic in all systems like this - having fewer substations doesn't really hurt much since the limiting factor will be the breakers in the paralleling huts.
It's only needed every few tens of miles, and if necessary inverters can be used to balance the load. A supply covering a large area will be feeding several trains with less peaking (relatively speaking) than a DC substation which effectively only feeds a couple of trains once the resistance of the third rail is taken into account.
The supply will still have to be sized for the worst case of all trains at maximum throttle at once - the system is sufficiently small that I can think of several conditions that will stop all the trains at once. (Signal failure for example).
Additionally the DC substations will be spread over a wider series of electrical connections that will make the peaking less of a problem. The substations could be spread over 20 miles in a Borderlands route.
You will load the entire line with multiple ~1600kW trainset loads onto one point.
That will cause all sorts of problems.
And I have considered the whole three-phase rectifier attached to single phase inverter concept for rural 25kV substations - it unfortunately has all sorts of horrific problems that renders it impractical.
You end up with giant capacitor banks otherwise you will get problems with only really loading two of the phases properly, and lots and lots of harmonic distortions. Improvements in power electronics will help (SiC MOSFETs for example) - but that also helps DC systems by allowing simpler synchronous inverter-rectifier topologies and removing the need for big line frequency transformers at substations.
It also allows for such craziness as variable voltage DC substations - so that when a train is drawing power you keep the voltage at the top of the range, but when the line voltage rises as a result of regeneration the substation reduces its holding voltage to provide the headroom.
Significant savings will be available doing that - since the normal substation voltage can increase to ~900V.