The EEH was developed simultaneously with the Emergency Command and Conn Hologram in 2380, after tests have shown that the Engineering department of a Starfleet facility or starship would benefit from an emergency holographic program as much as a Starfleet sickbay or infirmary did with the EMH.
Essentially, the EEH was a "backup chief engineer", intended to assume responsibility for the repair and maintenance of all ship's or station's systems in the event that the current engineering crew would be incapacitated for some reason. The EEH possesses over 4 million engineering subroutines from the engineering databases of all the leading Chief Engineers in Starfleet, and could incorporate them into its plans for the performance of its tasks.
There are five (5) EEH in the Emergency Holographic Complement of the fleet carrier classes and the heavy battleship classes of Starfleet and of Vanguard Command. In all other starships, there is only one (1) EEH in the Emergency Holographic Complement.