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To generate continuously repetitive EUV and soft X-ray pulses with various wavelengths from laser-produced plasmas, a one-dimensionally translating substrate system with a closed He gas cryostat that can continuously supply various cryogenic targets for ∼10 Hz laser pulses has been developed. The system was successfully operated at a lowest temperature of 15 K and at a maximum up-down speed of 12 mm/s. Solid Ar, Kr, and Xe layers were formed, and their growth rates and the laser crater sizes on them were studied. By optimization of the operational parameters in accordance with our design rule, it was shown that stable output power was achieved continuously from the plasma emission at frequencies of 1–10 Hz. The average soft X-ray and EUV powers obtained were 19 mW at 3.2 nm, 33 mW at 10.0 nm, and 66 mW at 10.8 nm, with 10% bandwidths, from the Ar, Kr, and Xe solid targets, respectively, with a laser power of 1 W. We will be able to achieve higher frequencies using a high beam quality laser that produces smaller craters, and can expect higher powers. Although only Ar, Kr, and Xe gases were tested in this study, the target system achieved a temperature of 15 K and can thus solidify almost all target gases, apart from H and He, and can continuously supply the solid target. The use of various target materials will enable expansion of the EUV and soft X-ray emission wavelength range.