It's not a problem, once you look at the big picture. Let's pull together the facts and do a little math...
Over
here, the US used 136.78 billion gallons of gas in 2014.
If you look at
this, the total US grid electricity production for 2014 was 14.78 Quads of electricity.
These two looked at together show that US drivers drove about 2.98 trillion miles in 2014.
Incidentally, that suggests 21.8 mpg overall average (2.98 trillion miles/136.78 billion gallons.)
A quad of electricity is a quadrillion BTUs - at 3412.14 BTUs per kWh, it is about 293 GWh per quad.
For the sake of this exercise, I'll assume that all those 2.98 trillion miles are suddenly being driven by RWD 85 kWh Model Ss. Not only is this a more viable car for most people than the others, it's also one of the least efficient pure EVs in the current market (due to the weight and performance gearing) - so it should give a conservative answer.
Over
here, the EPA rates that car at 38 kWh per 100 miles including charging losses.
2.98 trillion miles at 38 kWh per 100 miles gives 1.132 PWh - 1.1 trillion kWh. That's a huge total, right?
But here's the perspective on that - it's only 3.86 quads. In the electricity chart I gave you above, we spent 4.79 quads (1.4 PWh) on residential electricity usage alone in 2014.
With that perspective and considering that the current grid experiences a 2:1 ratio of consumption from the middle of the afternoon to the middle of the night, it should be clear that the grid as a whole should be fine.
There could be local cases where residential neighborhoods that were already near the edge might need some improvement in the infrastructure, but it should be minimal and localized - and the variable nature of EV charging could help a lot...
If the cars involved are indeed Model Ss, with always on net connectivity and extensive software, there's no reason they can't be tied to the power company directly through the web. What if the power company gave you a 20% discount on all your charging power in exchange for the right to choose exactly when and how fast you charge, with a guarantee that they'd always give you a battery charged to your charging target before you leave in the morning?
That sort of deal would be a no-brainer for me, and I'd think most of us - but the power company would probably come out ahead on it as well. If they have a bunch of cars charging, they can ramp the rates up and down or suspend/initiate charging a lot faster than they can spin up most types of standby power to respond to load changes - and the distributed nature of the load would let them balance the response across transmission lines too.
The next step, vehicle to grid (V2G) where they could actually use a little power from you battery to meet the demand, requires a lot more infrastructure and control and may not happen, but just the flexibility to shape what would become ~21% of their overall usage would be a huge boon and might save them (and hopefully thereafter you) money on meeting spinning reserve requirements if the system for adjusting the charging is robust enough to convince regulators.
Of course, there aren't 216.8 million Model Ss to drive those 2.98 trillion miles, and there won't be for some time. We certainly have the ability to build enough car bodies, and I believe that with all the other things we use them for, we probably have enough capacity to build the power electronics and the drive motors. The problem is batteries...
Tesla sees this coming, and that's why they are building the Gigafactory. If you look
here, you'll see that Tesla is hoping to build 35 Gwh of batteries at the Gigafactory in 2020 and every year thereafter. That's more than the entire industry produced in 2013 - and still a drop in the bucket for total conversion.
Those 216.8 million Model Ss up above need 18.4 TWh of capacity - 526 years of production at 35 GWh per year. Even if they were 216.8 million Leafs instead, they'd still need nearly 5 TWh - two orders of magnitude more than the Gigafactory can produce.
After spending a while trying to digest
this chart, I think it's telling me that in 2014 we introduced about 16.8 million new cars into the fleet. (Which means the average car lasts 12.9 years in service somewhere.) If we were building only EVs, we'd need 1.4 TWh of production (or, eventually, recycling of batteries from old EVs) for Model Ss or 370 GWh for Leafs - between 11 and 40 times the planned Gigafactory capacity.
Lithium is fairly plentiful, and "mining" it is both relatively easy and not horrible for the environment (mostly achieved by concentrating salts from ground water, especially geothermal type deep underground water,) but we're going to need to do a whole lot of it, even though the Model S battery only has about 20 pounds of Lithium in the ~1200 pound pack.