Despite metallic plasmonic excitations can enhance the performance of ultra-thin solar cells however these socalled plasmonic solar cells suffer from a large resistive (Ohmic) loss caused by metallic elements. In this work, we report on a new design that uses graphene nanoribbons (GNRs) in a two-dimensional (2D) grating form at the top of the semiconductor-on-insulator (SOI) solar cells aimed to reduce the resistive loss. The results showed that GNRs can remarkably reduce the resistive loss compared to the SOI cell with Ag nanograting, while keeping all other cell's parameters, comparable with those of Ag SOI cell. Optical absorption and short-circuit current density of the graphene cells showed, respectively, enhancements of 18 and 1.7 times when optimizations were done with respect to width and the grating period. Our calculations showed that the graphene solar cells dissipate at most 5% of incident sunlight power as narrow and tiny peaks around 508 nm, which is noticeably lower than those of Ag solar cells with high and broad band peaks with the maximum values of 29% at 480 nm and 24% at 637 nm.