Light Rail electrification
One design of light rail electrification is: 1. AC from a power company is rectified to DC.


Correct. DC is most popular on light rail such as trams underground or suburban trains. 750 V DC and 1500 V DC are common.


  
  In the train the DC is inverted back to AC for use by trains AC motors, etc. 
  


It may be. DC traction motors are very common and thyristor control allows pulse-width modulation to vary the power with low losses.



Figure 1. PWM varying from 80% on to 20% on, 80% and off.

AC motors require invertors to convert the DC to AC. This increases the complexity of the electronics but eliminates brushes and their maintenance from the motors.


  
  Then changed back to DC to return to the substation.
  


The trains are switched to regenerative mode during braking and the power fed back into the overhead line. It would be unusual for the substation to accept it as they are generally simple rectifiers and so won't allow current back onto the AC grid. Instead, the current is available to other trains on the line and is used to power those and reduce energy taken from the grid.

Braking sequence would typically be:


Switch to regeneration. On DC motors this would mean reversing the field winding polarity.
Use the PWM control to determine the braking effort.




Figure 1. Dublin Area Rapid Tranisit uses 1500 V DC overhead power. Source: Bahn Bilder.


The regenerative braking will cause line voltage to rise. This can be felt on the DART (Dublin Area Rapid Transit), for example, because the carriage heaters are fed directly from the overhead supply. If the thermostat is closed there is a noticeable blast of heat from under the seat as the voltage increases.
If line voltage exceeds a certain limit - say 2200 V for a 1500 V line - because there are no other units demanding power or they are too far away (and line resistance causes a high voltage drop) then switch to resistive braking.
Below a certain speed the regen braking will become too weak and the controls will switch to resistive braking (regardless of line voltage). The power is just dumped into large resistors.
At some point this will become too weak and the control system will blend in the air brakes.



  Can't PWM be done in exactly the same way with AC? – silverscania.


Not "exactly". Where one semi-conductor switch would do for a DC motor (in practice it would be multiple in parallel). A three-phase motor requires a three-phase bridge circuit with six switching elements. This is obviously more complex.

Another big advantage with DC traction is that of starting torque. This is maximum at zero speed and for many rapid transit systems top speed isn't as important as the acceleration and deceleration rates. Parts of the Paris Metro, for example, enchance the trains' acceleration and deceleration by having a downward gradient on both sides of the underground stations. Free deceleration is given by gravity on the way in and free acceleration on the way out.

Using the DART as an example again, the DC motor is given a high field current at start up to generate high starting torque. As the motor accelerates the back-EMF will rise so that acceleration starts to drop off. You can hear a change in the motor whine at about 30 kph when the motor is switched to "weak-field" and there is a controlled surge in power which will take the trains up to 100 kph - their maximum operating speed.

On regenerative braking the initial braking will be done on weak-field and when the speed drops enough that the output voltage and current is too low to give braking effort the regen will be switched to weak-field to get some more useful energy out of the kinetic energy of the train.

I don't know enough to comment on the controls required to have the same control with AC motors other than to say that it will be more complex.



Other considerations

Another point worth mentioning is that the traction system must not interfere with signalling track circuits used to prove lines clear. The use of DC traction precludes the use of DC track circuits as any current imbalance between the return rails may induce a voltage difference enough to energise a track relay and indicate "track clear" when it is, in fact, occupied. Similarly, since the DC is derived from 50 Hz supply that frequency can't be used either. Instead the track circuits are run on 83.3 Hz supply (least common multiple = 250 Hz) and detection is by vane relay. The chopper control for the motor PWM is run at a fixed frequency of several hundred hertz and not harmonically related to 50 or 83.3 Hz. The chopper frequency is monitored and the traction system is disabled if it goes out of tolerance.

As AC motors would require a variable-frequency drive and avoiding signalling system interference would be far more complex.