How is power demand variation of supercomputers handled at power plants
Do large supercomputers communicate with power stations to notify them
  of demand changes?


No. Within most countries power stations are linked together to share loading. In addition there are stations and generators on standby ready to go online as demands require. Adjustable phase shift networks can trim the line voltage to stay within expected values.

Supercomputers consume a tiny fraction of the power that the aluminum and steel industries do, which often have dedicated power feeds from the closest generators.

For decades now most (but not all) countries have dynamic power grids that compensate for loads, sharing of power, voltage trim,.. etc. An exception would be the Large Hadron Collider (LHC) on the France-Swiss border. This monster of a machine is run only in the summer to avoid overloading the power grid during winter, when the local towns need power for heating.