B787 Captain reported loss of Center Hydraulic System fluid in flight. The flight crew continued to landing at destination airport. After shutting down engines on the runway; the aircraft was towed to the gate.

Narrative

Loss of hydraulic fluid in the center systemDuring cruise at FL 380 at XA00z we had EICAS message HYD PRESS C1 and did the associated checklist. Finishing that checklist noticed HYD QTY for center system was .69 (early in flight qty was around .85) and slowly decreasing. Over 15 minutes quantity went to 0.0 and we performed HYD PRESS SYS C checklist. Because of slow continued leak and initial pump failure of C1 we assumed leak was between the reservoir and C1 pump. With the loss in this area we assumed no nose wheel steering would be available during and after landing. We talked with Dispatch and Maintenance and decided that landing at our destination was our safest option. Flight Attendants were briefed and did not prep for evacuation. We told ATC about our system failure and [requested priority handling]. We informed crash/fire/rescue to be standing by for our landing and we would stop on the runway. (We descended at 250 kts. and at 10;000 ft. put the gear down with alternate extension. Alternate gear extension was a big concern of ours because the manual says a DC pump uses center system fluid to release the up locks and we did not know how much fluid was left in the system. Because of possible gear extension problems that was the first thing we did to configure for landing.) (Clean maneuvering speed for our weight was 227 kts. and flaps 1 load relief is 225 kts. in secondary mode; so we had to slow below 225 kts. to extend flaps. With nose gear doors open the cockpit was very loud and communication was more challenging from 250 kts. too 180 kts.)More system knowledge of the landing gear unlock would have been helpful because the loss of all hydraulic quantity in the center system gave us concern that the gear may not extended. Our manual only says a DC pump uses center system fluid to release the gear uplocks. Using my old system knowledge of the 767 I knew the uplocks do not use a lot of fluid but we were not sure the gear would extend on this aircraft. Because of this we extended the gear early to give us time to prepare for a gear unsafe condition if the gear did not free fall using the alternate gear extension.Winds were a direct headwind of 10 kts. down the runway. Flaps 20 approach and landing was performed with manual speed brakes and manual differential braking. Rudder was effective until 80 kts. and differential brakes were used to stay on center line. (Auto brakes were not used because I wanted instant differential braking available)Engines were shut down and aircraft was towed to parking where passengers de-planed normally. HYD fluid was on the belly of the aircraft from the main landing gear wheel well all the way to the tail.