The vertical U-shaped tube natural circulation steam generator is used to exchange heat between the primary and secondary circuits in nuclear power plants (NPP). It is the key equipment, which occupies a relatively large volume inside the nuclear-powered ship. The increase of the output power of the steam generator (SG) generally leads to the increase of its volume generally, which brings certain difficulties to its arrangement in a narrow space. Reducing the volume of the SG while meeting its thermal load has important theoretical and practical significance for the design and operation of NPP. In this paper, a case study of multi-objective optimization is carried out for SG, in order to obtain the best parameter configuration combination of SG volume and the power of the NPP, and realize multi-objective optimization of SG volume and output power. The optimization results indicate that there is still a positive correlation between the volume of the optimized equipment and the power of the device; If the volume of the SG remains unchanged, the maximum power can be increased by about 33%; while the power is kept constant, the volume can be reduced by up to 24%, which can meet the needs of high-power devices with a smaller equipment volume.