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Given the recent shortages of liquid helium, cryogen-free operation of superconducting devices, such as programmable Josephson voltage standard (PJVS) systems, has become preferable worldwide, and a necessity in some locations. However, reliable operation on a cryocooler is heavily dependent on the ability to create a constant temperature that is low enough to allow the PJVS junctions to operate uniformly. In this work, we systematically investigated as a function of temperature the performance of NIST 10 V PJVS chips employing Nb/NbxSi1-x/Nb superconducting junctions. Additionally, we addressed the major factors limiting the performance of a cryocooled PJVS: adequate attenuation of the coldhead temperature oscillations and the minimization thermal gradients between the chip and the cryocooler. Through the development of a robust and reproducible method for soldering chips to a Cu carrier (package), we increased the thermal conductances within the packaging to their practical maximum values. This, in addition to the incorporation of a passive two-stage thermal filter, allows us to confidently predict that the required cooling power for the successful cryogen-free operation of the NIST 10 V PJVS is ~ 0.5 W at 4 K.