This is the first of three papers that propose and study a new concept of mooring turbine generators with the aim of resolving key difficulties in realizing ocean current power generation. The concept of Cross-stream Active Mooring (CSAM) features a hydro sail system that allows deployment of generator turbines, from anchoring points on shore or on shallow seafloors, across current stream to access current core flowing over deep seas or over seabed not suitable for anchoring construction. The CSAM can increase system power capacity by changing horizontal positions of generator turbines to track meandering current core, and can also change system depth to avoid storms. New anchoring designs of improved efficiency and implementation methods for resolving seafloor geological issues in the Kuroshio off southeast Taiwan are also included. This first paper presents the basic concept, basic analytical model and prediction of key performance parameters, conceptual designs of the hydro sail and multiple-unit linear array, and potential benefits of the proposed system. The second paper discusses construction of the tethers, mooring of power cables and 2D formation designs. The third paper discusses anchoring designs and constructions, large scale deployment, failure mode designs and system costs.