It's best to have a backup heating source during a power outage such as a vent free natural gas or propane heater, or a vented propane or natural gas heater that doesn't require power to operate.
Another option for heat during a power outage are kerosene heaters or propane portable heaters or even a wood burning stove.
If you have no other heat source then insulate the water lines inside and under your house or in your basement and then leave the faucets dripping at a slow steady stream which will prevent the pipes from freezing during the power outage and even when it's just cold outside.
Leave both the hot and cold water dripping in all faucets and you shouldn't have any problems with the water lines freezing during cold weather or during the power outage.
Your water bill will increase a little bit but if your water pipes burst then you'll have a more expensive plumbing repair bill and cause damage to your home should the water pipes burst and leak.