The invention provides a system for operating a drilling rig on a drilling vessel, the system comprising a drilling tower (2) having at least two segments (4, 5), a first segment (4) is fixedly connected to the drilling vessel (1), a second segment (5) is connected to the first segment (4), said second segment (5) is adapted to move longitudinally relative the first segment (4) by raising and lowering means. A tool (8) is suspended from at least one wire (16), said at least one wire (16) is extending over at least one sheave (7a, 7b) on top of the second segment (5), said at least one wire (16) is coupled to with a passive compensator arrangement (20) and a winch (14), said passive compensator (20) is connected to the winch (14).