Almost all Police Stations and CSI labs around the country have either back up generators or Uninteruptable Power Supplies (UPS) or both. This is to ensure that a blackout will not hamper police response, or the processing of time-sensitive evidence.
In addition to the error about the police generators, when power goes out to the degree shown, cell phones will power on but the towers they utilize (that create the "cells") will not, so the phones will not work.
When Greg and Morgan are figuring out the blood types of different people and different pieces of evidence, they are using several different bottles to complete the task. In reality, there are only 3 bottles needed: blue for Anti-A antisera, yellow for Anti-B antisera, and a clear solution for determining Rh typing. Very basic and easy to do.
Almost instantaneously to the power going out, Capt. Brass announces storms have taken out three power stations and half the state is without power. This information would not have been available that quickly.
Using adapters cell phone batteries can be recharged using automotive power ports. So they can keep the phones charged up though it maybe inconvenient. Later a character claims that he is using his vehicle to charge up his phone.