Electropolishing (EP) has become a standard procedure for treating the inner surfaces of superconducting radio-frequency (SRF) cavities composed of pure niobium. In this study, a new EP facility was employed for the surface treatment of both 1.3 GHz and 3.9 GHz single-cell cavities at the Wuxi Platform. The stable "cold EP" mode was successfully implemented on this newly designed EP facility. By integrating the cold EP process with a two-step baking approach, a maximum accelerating gradient exceeding 40 MV/m was achieved in 1.3 GHz single-cell cavities. Additionally, an update to this EP facility involved the design of a special cathode system for small-aperture structures, facilitating the cold EP process for 3.9 GHz single-cell cavities. Ultimately, a maximum accelerating gradient exceeding 25 MV/m was attained in the 3.9 GHz single-cell cavities after undergoing the cold EP treatment. The design and commissioning of the EP device, as well as the electropolishing and vertical test results of the single-cell cavities, will be detailed herein. These methods and experiences are also transferable to multi-cell cavities and elliptical cavities of other frequencies.