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The current distribution and stability of low temperature superconductor (LTS) and high temperature superconductor (HTS) hybrid superconductor, made of NbTi wire and YBCO coated-conductor, are described. The current distribution, quench propagation velocity (QPV) and the minimum quench energy (MQE) are numerically analysed by finite element method (FEM) based on the different power-law models since the value in LTS is much larger than that in HTS. It is shown that the transport current in LTS is larger than in HTS if both of them have the same critical currents, and the ratio of transport current in LTS to that in HTS decreases with increase of the total transport current near their critical currents. The stability of the hybrid NbTi/YBCO composite conductor is also simulated, which indicates that MQE in the hybrid conductor is much higher than in NbTi wire but smaller than in YBCO conductor. However, the quench propagation velocity is in the range of low-temperature superconductor through the high-temperature superconductor. With advantages of high engineering current density and improvement of stability, the hybrid superconductor is expected to have potential application in large scale magnet and conducting-cooled magnet particularly.