A growth in electric vehicles on European roads will need a corresponding  growth in electricity production to serve charging of these cars, both in peak production power and in total annual power. Let’s take italy for example which currently has a total year consumption of around 300 TWh (terna). Considering :

• Charging efficency (tesla) : 0.85 %
• KWh/km (tesla model x) :  0.1875
• Average Km/year : 10200

This accounts for 2250 KWh/year for each electric car. Considering 2 milion new cars per year and a 0.5 % of electric cars sold (10000) this means an additional 22 GWh/year to be produced to allow charging them.
So your electric car will consume like a small household (depending on your mileage) and personal/distributed production of electricity (domestic + accumulation) will be necessary.

If you want to really dig deeper : here.