•Optimal design is determined for a power-to-methane system.•Performance analysis is performed for the system under off-design conditions.•ANN model is established for off-design performance prediction.•ANN-based optimization is conducted to obtain optimal off-design performances. Power-to-methane (PtM) is a prospective solution to the mismatching between the supply and consumption of renewable energy resources (RES) by converting renewable power into methane. However, the continuous fluctuation of RES causes the PtM system to deviate from the design condition in the vast majority of cases, and thus it is significantly vital to study the operating characteristics of the PtM system under off-design conditions. This paper proposes a comprehensive investigation framework from design to off-design steps for the performance improvement of a PtM system combining solid oxide electrolysis cell with methanation reactor, and solar energy is selected as renewable energy input. Firstly, the system with the total exergy efficiency (ηEX,tot) of 11.83% and levelized cost of exergy (LCOE) of 150.76 $/MWh is selected as the optimal design condition based on the homogeneous assessment from both thermodynamic and economic aspects, by means of Non-dominated sorting genetic algorithm-II. Then, based on the optimal design point, the off-design performances are quantitatively investigated under varying solar radiation and key operating parameters, in terms of synthetic natural gas (SNG) yield and ηEX,tot. The results indicate that with the increment in solar radiation, the SNG yield rises, while the ηEX,tot increases first and then decreases. Finally, the multi-objective optimization based on the Artificial Neural Network models is implemented for the system under off-design conditions to acquire the best trade-off between hourly SNG yield and ηEX,tot. The off-design optimization solutions reveal that the hourly optimal SNG yield is located in the range of 275.06 kW to 946.53 kW, achieving a total annual SNG yield of 1697 MWh/y, and the hourly optimal ηEX,tot mainly varies in the range of 10.40-11.40%.