Ok, Good news, it didn't take fire in my house and let's say that in fact, you will need more than one wire, there is no 12V regulator on board, so you have to add one (you can find some very cheap on eaby, at ~3 euros.).
After some measurment, the VOUT pin on the LCD display is about 17.80V when the battery is fully charged and the robot is at the base station.
Here is how I repair it :
First of all, the very nice blank screen

- The wonderfull blank screen
To repair it without changing the LCD, you will need :
- a Red wire
- a Black wire
- a green wire
- a Power Regulator like this one :
https://www.ebay.fr/itm/LM2596S-Module-DC-DC-Step-Down-Ajustable-Module-Abaisseur-Regulateur-de-Tension/263330885857?_trkparms=aid%3D111001%26algo%3DREC.SEED%26ao%3D1%26asc%3D20160908105057%26meid%3D1df35fca2b144204a915a9f38cb26a2a%26pid%3D100675%26rk%3D5%26rkt%3D15%26sd%3D201125671881%26itm%3D263330885857&_trksid=p2481888.c100675.m4236&_trkparms=pageci%3Ad988c03e-c3cd-11e8-9edb-74dbd18038e1%7Cparentrq%3A24c17bf91660ac88c2ccd32cfff6f962%7Ciid%3A1- a double tape rubber glue
- an antistatic plastic (simply use the one you will receive with the DC-DC power board)
- Some talent
1 - Remove the batterys of your robot, and open it.
2 - unscrew and unplug the LCD board.
3 - you will have to solder the green wire on the capacitor C5, the side closed to the LCD ribbon cable like this :

- C5 Green wire soldered
Yes I now my capacitor are very sadly soldered, before to find out the issue with VOUT, I tired to replace some capacitor, this was without any effect.
4 - Solder the red and black wire on the main board, the red wire goes to the first pin of U23, the black wire on the capacitor C143, side closed to the plastic like on this picture :

- PowerWire
5 - Solder the other side of the red and black wire to your power PCB (black is -, red is + ) like this :

- Power DC Board
6 - STOP NOW AND READ CAREFULLY : Now, you will have to adjust the voltage regulator, DO NOT connect the green wire to the Power Regulator now. Place your battery back and place your Robot in charge and wait until the 100% charge of the battery, you will measure about 24V on the Power PCB input (between the red and black wire). Use a screw driver to adjust the ouptut voltage to 14.0V, you must stay at the charging station while doing that to ensure that you receive the 24V coming from the charger.
Why 14.0V ? I read the LCD datasheet and see that this value is inside the accepted power input for the VOUT pin.
We can set 17.80V like at the origin but it will perfectly work like that with 14.0V and we limit the current.
7 - Now, remove again the battery, take the robot and you can solder the green wire to the + output of the power regulator (like on the picture above).
8 - We have now to corretcly place the power regulator inside the robot. Use a piece of double tape rubber that you will glu on the two chip U11 and the one in front of it like on this picture :

- RubberTape
9 - Glu the antistatic plastic on it and add another piece of rubber on it :

- AntiStatic plastic and rubber
10 - Place your DC Regulator on it :

- Power DC Board in place
11- Place the wire in manner that they will not be scratched by anything while closing the robot, place your battery, Enjoy !

- Unblanked Screen
You will need to adjust the contrast because the power source is not 17.80 anymore but lower (14.0V).
I don't know how long it will work but at the moment, it works.
Hope this will help you guys.
By the way, don't forget to replace the C10 capacitor as it can cause trubbles later (dimmed light green led).