Although you hear this myth perpetrated by people, politicians or organisations that want to halt or delay the advent of EVs, research and real world data refute the idea of a calamitous future for the grid. It is true of course that an EV fleet will require additional grid capacity but beware the rantings of people with vested interest that the sky will fall in. EVs are among the most flexible loads on the grid and with time of charging utilised in a smart manner (through using smart meters for example) the current grid is able to sustain up to an 80% switch to EVs according to modelling using real Australian data.

Real world data shows grids are coping well in places where EV uptake has been quite substantial and certainly at a much greater rate than in Australia. Various strategies such as smart metering and off-peak incentives will be employed to ensure a grid collapse will not happen. Many households in Australia also have solar panels and an increasing number have battery storage systems. There are over 2 million households in Australia (20.3%) with roof-top solar representing 7.9GW (gigawatts) of capacity – roughly 4.5 times the capacity of the coal fired AGL Energy’s Liddell coal fired power plant near Muswellbrook in NSW. Charging from household solar means energy directly from the sun to an EV.

Vehicle to Grid
Car batteries can be used to supplement grid power if demand is high with car owners paid by grid operators to make power available from their car batteries. This is already the case in Japan, Germany and the US where Vehicle to Grid trials are underway.

Rather than being a burden on the grid, electric vehicles have the potential to be a huge Virtual Power Plant (VPP) able to be called upon by the grid when needed. Below are some links regarding this.
https://www.nrdc.org/experts/noah-garcia/good-news-evs-are-not-crashing-grid
http://www.synapse-energy.com/sites/default/files/EVs-Not-Crashing-Grid-17-025_0.pdf
https://www.evse.com.au/blog/evse/
https://www.energynetworks.com.au/news/energy-insider/networks-ready-surge-electric-vehicles