The crew is committed to testing the life support systems of the ship, which is equipped with unique weaponry that has no analogues in the entire Russian Navy.
The next few months will see the crew of 200 testing and debugging the vessel's life support systems and weapons, a task that will include the firing of missiles, artillery and torpedo drills.
Displacing at 3,850 tons, the Admiral Grigorovich is due to set sail for a Black Sea Fleet base after the end of the exercises.
The 125-meter (409-feet) ship's primary functions include air defense, escorting, and submarine hunting.
The vessel is also equipped with advanced electronic warfare systems, including air search and surface search radar, as well as a fire control radar.
In early July, the Admiral Grigorovich took part in the 7th International Maritime Defense Show in St. Petersburg.