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Owing to the application of high-temperature superconductor (HTS) tapes, superconducting magnetic energy storage (SMES) magnets can be economical to run at temperatures around 20 K. The conduction-cooled SMES magnet has become a reality with the rapid development of cryocooler technology. In China, a 150 kJ/100 kW conduction-cooled SMES employed for power quality is now being developed by Huazhong University of Science and Technology (HUST) and Hubei Electric Power Company (HBEPC). In electromagnetic design, a hybrid structure that uses two kinds of HTS tapes is employed to increase the critical current, Ic, of the SMES magnet. This paper presents an electromagnetic and thermal design of the conduction-cooling system for the magnet. The relationship between Ic and temperature is analyzed, and the target parameters of the conduction cooling system are finalized. The design scheme of the cooling system is developed by analyzing thermal conductivity characteristics of HTS double-pancakes, thermal loads, thermal resistance, and cooling demands. Finite element analysis results show that the design scheme meets the requirements well in static conditions.