We consider a sub-cooled liquid nitrogen (LN) circulation system for HTS power equipment. The planned circulation system consists of a sub-cool heat exchanger (subcooler) and a circulation pump. The sub-cooler will be connected to a neon turbo-Brayton cycle refrigerator with a cooling power of 2 kW at 65 K. Sub-cooled LN will be delivered into the sub-cooler by the pump and cooled within it. Sub-cooled LN is adequate fluid for cooling HTS power equipment, because its dielectric strength is high and it supports a large critical current. However, a possibility of LN solidification in the sub-cooler is a considerable issue. The refrigerator will produce cold neon gas of about 60 K, which is lower than the nitrogen freezing temperature of 63 K. Therefore, we designed two-stage heat exchangers which are based on a plate-fin type and a tube-intube type. Process simulations of those heat exchangers indicate that sub-cooled LN is not frozen in either sub-cooler. The plate-fin type sub-cooler is consequently adopted for its reliability and compactness. Furthermore, we found that a cooling system with a Brayton refrigerator has the same total cooling efficiency as a cooling system with a Stirling refrigerator.