NOVELTY - Converting biogas to a gas rich in methane comprises: (a) mixing a carbon dioxide-comprising biogas with steam to form a mixture comprising carbon dioxide, methane and steam; (b) electrolyzing the mixture comprising carbon dioxide, methane and steam in a high temperature solid oxide electrolyzer cell unit, to obtain a gas comprising mainly hydrogen and carbon monoxide; and (c) catalytically converting hydrogen and carbon monoxide in the gas comprising hydrogen and carbon monoxide to methane in one or more methanation steps to obtain a gas rich in methane.    USE - The process and system are useful for converting biogas to a gas rich in methane (claimed).    ADVANTAGE - The process and system are environment friendly and efficiently upgrade biogas to substitute natural gas (SNG) by means of high temperature solid oxide electrolysis cell (SOEC) electrolysis and SNG technology.    DETAILED DESCRIPTION - An INDEPENDENT CLAIM is system for converting biogas to a gas rich in methane, comprising: optionally a digester for formation of biogas from biomass; a high temperature solid oxide electrolyzer cell unit in series with one or more methanation reactors located downstream the solid electrolyzer cell unit, the methanation reactor immediately downstream the high temperature solid oxide electrolyzer cell unit being at least one adiabatic reactor, and a non-adiabatic methanation reactor located downstream the adiabatic reactor; and means for regulating the temperature and pressure of the process gas.    DESCRIPTION OF DRAWING(S) - The drawing shows a schematic view of the process.    Biogas (1)    Compressor (2)    Exchanger (3, 6)    Water (4, 22)    Pump (5)    Stream of biogas and steam (7)    Desulfurizer (8)    Feed preheater (9)    SOEC (10)    Exit gas (12)    Heat exchanger (13, 18, 21)    Hydrogen and carbon monoxide (14)    Zinc oxide bed (15)    Adiabatic methanation reactor (16)    Methane (17)    Temperature controlled methanator (19)    Gas obtained (20)    Gas rich in methane (23)