all right.
lets start by assuming for the sake of convenience everyone's average battery capacity is ~60kwh (the longer range version of the nissan leaf)
to charge that overnight (forget fast chargers, we'll be nice and say it takes 12hrs to charge from empty to full) you'll be drawing 5kw from the charger.
so, lets just forget tower blocks, and houses without driveways and assume everyone has a 5kw charger available to them to do this.
doing a straight swap from IC to electric we'll assume we have ~38million cars on the roads (ie the number of cars doesn't increase from current levels), lets be super generous here and call half of those garage queens so 19million cars used daily (RAC has 15 million as the figure for daily commuters, so including non-workers and leisure use we're not too far off)
so, 19 million cars at 5kw each is 95Gw so that's our worst-case peak load. realistically not everyone's going to be charging at the same time but given we're talking a few hours to charge there's going to be a hell of a lot of overlap at the start of the night when people plug their car in and hit the hay.
given in 2014 the average power demand for electricity was 34Gw, we're gonna need 20 more stations on the scale of hinkley point c to meet that.
and that's just the macro scale, on the more local scale there are plenty of places where a hamlet of 10 cars dropping a 50kw load on the grid is going to kill it.
19 million cars need 60kWh per day?
Average UK mileage is 7900 miles per annum across 38 million cars (300bn miles per year total). So either those hypothetical cars have truly abysmal efficiency, or the average daily mileage for all 38m cars just jumped five-fold.
If we assume a "Well to Wheel" efficiency of 3 miles per kWh, and an average 12 hour overnight charging schedule, then we need roughly 22.85GW average spare overnight capacity to keep the UK's entire domestic car fleet operational (based on 21.64 miles/day average use). If charging were spread evenly across 24 hours, we'd only need 11.4GW of spare capacity. The true figure is likely somewhere between the two. In absolute terms, we would need to produce around 100TWh of extra electricity per annum.
That's a crude calculation. It assumes no variance in daily average mileage, and it assumes every vehicle does 100% of its charging during the same 12 hour overnight period. But it's in line with the National Grid's own, far more in depth analysis on the situation. By 2040, the Grid expects that the UK will need another 8GW of peak capacity above 2017/2018 numbers (so basically the same peak capacity we had 10 years ago). Nothing hugely dramatic.
The key to EVs is to find ways to level out demand so there is minimal variance. The National Grid want to be able to control charging speed, so that they can ensure no interruption in supply for businesses and households, regardless of the number of cars that are plugged in. They want V2G rolled out as part of a wider energy storage solution, to help smooth out supply. And they want "time of use" electricity tariffs for consumers, to encourage consumers to charge their vehicles when there is plenty of spare grid capacity. The average driver of a 60kWh EV needs a full charge once every 10 days. So outside of rapid charging or preparing for a long journey, the vast majority of drivers shouldn't need to charge during peak (domestic, commercial, industrial) demand periods.
If we weren't managing demand at all, then we'd need to cater for 38 million cars demanding 7kWh at the same time (plus extra to account for rapid charging). We'd need something like 350GW of generation capacity. So clearly we do need to manage demand. Doing so is a lot more efficient than building huge numbers of power stations which spend most of their life idle.
