How does electrical devices consume power?
The power line may be supplying a fixed voltage, like 220V, but how much current the device draws is up to the device.  Power is voltage times current, so you can see how the device determines the power by determining the current.

For exmaple, in order to take 1 kW from a 220V line, a device would have to draw 1kW / 220V = 4.5A.  If this were purely a resistor, it would be 220V / 4.5A = 48&Omega;.  A 24 &Omega; resistor would take twice that power (2 kW), and a 96 &Omega; resistor half that (500 W).  All these can be connected at the same time to the same 220 V line and each will still take the power calculated above.

At some upper limit, the 220 V line can't supply more current.  At that point something has to give, like the voltage sag.  In houses, a fuse will blow or a breaker pop before there is any significant sag.