A system for supplying extra low voltage electrical energy for at least one electrical traction vehicle running on a track and comprising: - wheels linked to said vehicle and rolling on said track, and - at least one traction chain of said vehicle acting on the wheels and comprising in a manner known per se, an electric motor and its control, - at least one extra low voltage power supply means installed in the immediate vicinity of the track, - two power supply rails (41, 42) or similar power supply elements, parallel to each other, adjacent or distant, of which a first is linked to a terminal of said power supply means and the second is linked to another terminal of said power supply means, - at least one first electrical energy collection means on board the vehicle and placed in moving contact with said first rail, - at least one second electrical energy collection means also on board the vehicle and placed in moving contact with said second rail, - at least one electrical energy storage means on board the vehicle, - at least one on-board electric supply means connected to said electrical energy collecting means (51, 52), and connected on the one hand to said storage means and on the other hand to said traction, said system being characterized in that on-board power supply means comprises at least one on-board energy management computer device, and said storage means comprises at least super-capacitors whose charge / discharge is controlled by the computer and in that: 1 °) the computer manages the starting phase of the vehicle by supplying the necessary energy to the traction chain by unloading the super-capacitors to the motor through its control; 2 °) the computer manages the stroke phase on the road of the vehicle by driving the recharging of the super-capacitors from the external power supply while the vehicle is in motion; 3 °) the computer manages the braking phase of the vehicle by controlling the recharging of the super-capacitors from the energy generated by the traction chain during braking; 4 °) the computer manages the station stopping phase of the vehicle by controlling the charging of the super-capacitors from the external power supply while the vehicle is stopped, the calculator managing the end of charging when the super-capacitors have reached their maximum capacity. - wheels linked to said vehicle and rolling on said track, and - at least one traction chain of said vehicle acting on the wheels and comprising in a manner known per se, an electric motor and its control, - at least one extra low voltage power supply means installed in the immediate vicinity of the track, - two power supply rails (41, 42) or similar power supply elements, parallel to each other, adjacent or distant, of which a first is linked to a terminal of said power supply means and the second is linked to another terminal of said power supply means, - at least one first electrical energy collection means on board the vehicle and placed in moving contact with said first rail, - at least one second electrical energy collection means also on board the vehicle and placed in moving contact with said second rail, - at least one electrical energy storage means on board the vehicle, - at least one on-board electric supply means connected to said electrical energy collecting means (51, 52), and connected on the one hand to said storage means and on the other hand to said traction, 1 °) the computer manages the starting phase of the vehicle by supplying the necessary energy to the traction chain by unloading the super-capacitors to the motor through its control; 2 °) the computer manages the stroke phase on the road of the vehicle by driving the recharging of the super-capacitors from the external power supply while the vehicle is in motion; 3 °) the computer manages the braking phase of the vehicle by controlling the recharging of the super-capacitors from the energy generated by the traction chain during braking; 4 °) the computer manages the station stopping phase of the vehicle by controlling the charging of the super-capacitors from the external power supply while the vehicle is stopped, the calculator managing the end of charging when the super-capacitors have reached their maximum capacity.