This is already proven and you can find it plenty of places on the net. An electric engine is far more efficient than the internal combustion engine for this use. Powerplants are (or can be) more efficent, the logistics of moving electricity around vs gas, the heat created at powerplants (fueled by fossile fuels) can be sent to other customers rather than wasted. Up here most of the power is also created by hydro electric plants.
There is no maybe about it. The overall co2 and other harmful gasses would go WAY down if we switched to electric cars. Noise pollution is another factor that is less important yet it is a factor. That is not the reason for us getting one though. They are way cheaper to run, free parking, no toll both fees and you can drive in the public transport lanes. Add insurance cost and subsidies and there is no contest for OUR kind of driving.
Oh and our reports also sais that even if every norwegian switched to electric cars overnight it would only add some 20% strain on the powergrid, and it could handle it with only minor improvments in those areas with the oldest and most strained grids.