modem range
ASIC - Your real problem is signal to noise ratio. The AC power line is exceptionally noisy, and the noise is not easily predicted.
Atmospheric noise (lightning, static discharge from wind, etc.), switching noise (appliances turning on and off, commutating noise from rotating machinery, etc), external RF (welding machines, induced radio and TV signals, home electronics, etc.), signal bypass (home and industrial broadband noise filters, transformers, line capacitance, etc.) - all are unpredictable in amplitude, frequency, and signal characteristic. Each of the noise sources will diminish your range for a detectable signal.
No, it is improbable you can get 3km out of you modem using a single driving point - regardless of power. As a matter of fact, you'll be lucky to get 50m.
Commercial applications depending on power line data transmission use high power booster repeaters at regular intervals along the signal path. Unless you are prepared to do that, you are better off transmitting the data over a public radio frequency through the atmosphere. You can encrypt the signal if necessary, and you can use modulation schemes that are noise tolerant. It would take less power to use atmospheric transmission over 3km than to try and overcome the noise on a 3km power line.