Will a large amount of constant-power loads make an electrical grid unstable?
Yes, potentially a lot of constant-power loads will destabilize a feedback
loop that keeps the grid voltage steady.   This is because constant power
is symbolized by 
$$power = V \times I = constant$$
 while resistive (like a heater) loads
are symbolized by $$power = V \times I = V^2 \times constant$$   

The grid must deliver extra power whenever it sees a voltage dip.
The heater load makes a V versus I function that has a strong
power decrease when V goes low, i.e. it helps the grid
feedback control (lower power load is like higher power supply).   The constant-power load, however, has a 
proportional current-load INCREASE when V goes low, which means
it requires a larger current increase than the voltage deficiency
alone would indicate.   It is thus fighting the power-grid voltage
regulation mechanism.

Fighting is normal, but you don't want the power-grid to lose that fight.