One consequence of an electric car boom, page-4

Sometimes back of the envelops calculations can be misleading. They do not take in to consideration future advances in battery technology. And the Tesla 85d (85kWh) is not the battery size of most EVs - the latest crop are 40kWh to 60kWh with ranges of between 300 and 400kms. The average daily drive for passenger cars is 38kms (ABS figures). That means these cars will be charged every ten days or so (same for the Tesla 85D with a range of 420kms. So every ten days we'll need 84GW or 8.4GW per day (based on an average battery size of 50kWh).

You may suggest this is all pie in the sky scenario but this is technology - it does not stand still and we know with tech that each iteration becomes more efficient and cheaper. Solid state batteries for example are on the horizon and Samsung has stated they will be commercially available by 2020 (https://asia.nikkei.com/Business/AC/Samsung-working-on-battery-breakthrough-for-powering-cars) Just about every battery manufacturer is working on solid state. The thing about solid state is batteries is they are at least a doubling of energy density (so double the range or half the battery size for the same range). With solid state batteries offering the same range for half the size that brings our daily additional requirement to 4.2 GW.

Places where they have looked specifically at the issue you have raised have quashed the alarm bells being put about (a lot coming from the fossil fuel Koch brothers) in regards to EVs. For example the UK has looked at various scenarios to see how much additional peak demand will be needed for their grid after the ban on ICE vehicles comes in to play in 2040 (http://fes.nationalgrid.com/media/1264/ev-myth-buster-v032.pdf). "The additional peak demand from EVs in that scenario is not 30 GW but more likely to be 5 GW."

Managed charging pilots are already underway in the US (https://www.fleetcarma.com/impact-growing-electric-vehicle-adoption-electric-utility-grids/) and have suggested the grid may benefit from EVs in vehicle to grid power smoothing.

The point you raised is valid and especially here in Australia we are very under prepared for the disruption that is coming but this has not been overlooked by more progressive (conservative) governments in other places.
Some additional reading to throw light on this issue can be found here (will EVs break the grid):
https://cleantechnica.com/2016/05/11/will-electric-cars-break-grid/

There is no doubt we will need additional electricity if the fleet changes to electric which now seems to be an inevitability globally. Smart charging, time of day charging by economic incentivization and managed charging will all play a part in ensuring the grid is capable of handling the additional loads. Not only that but with solar PV and residential storage getting cheaper every year many EV owners will opt to supply their own 'fuel'. The benefits is reduced CO2 and toxic emissions from our current fleet (with health costs in Australia around $4billion annually) will be a welcome change from our current strategic liability and economic burden from our dependence on overseas oil.