Stability tests were performed on two test coils wound with two kinds of large-scale composite superconductor, respectively. One is a original LHD conductor, which consists of a NbTi/Cu Rutherford cable, a pure aluminum stabilizer, and a copper sheath around the composite. Another is an Al-less test conductor, which is a LHD conductor without the aluminum stabilizer and a half of the copper sheath. This paper describes mainly on the latter Al-less test conductor comparing with the LHD conductor. The conditions of the tests were with the magnetic flux densities from 3 T to 7 T and at the bulk liquid helium temperatures 4.2, 2.2 K (sub-cooled He I) and 2.0, 1.8 K (He II) at atmospheric pressure. Asymmetrical normal zone propagation was observed even in the Al-less test conductor. The one-side propagation of the normal zone was also observed as was in the LHD conductor. The asymmetrical propagation is due not only to aluminum stabilizer but to the copper sheath and asymmetrical configuration. However, "traveling normal zone" observed in the LHD conductor was not seen in the Al-less test conductor. The current range for the one-side propagation is narrower than that for the LHD conductor. It is confirmed that the Al-stabilizer deeply affects the asymmetrical propagation.