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The authors have developed a small-sized superconducting magnetic energy storage (SMES) magnet with the nominal storage capacity of 1 MJ, which provides electric power with high quality to sensitive electric loads. In large magnets such as the SMES magnets the stability, which is determined by several factors, e.g., conductor's cooling condition and operating current, magnet's winding structure, is a crucial problem. For the stable and compact SMES magnet development, two kA class conductors with different copper ratios were selected as candidate conductors. The effect of the cooling condition, the copper ratio, and the conductor's size upon the recovery currents was investigated experimentally. The test results indicate that the recovery current characteristics of the strands vary considerably according to their insulation method. In the fully insulated strands with a low copper ratio, the recovery current densities range from 10 to 20% of their engineering critical current densities. The recovery current density of the 30-conductor with a cooling channel is about a factor of 1.8 higher than that without a cooling channel.