Abstract: The key to enhancing the sensitivity of derivative-based traveling wave protection lies in shortening the sampling step. However, achieving a higher sampling frequency poses challenges to engineers and imposes limitations on the protection performance. For this reason, in this paper we start from the relationship between the electrical quantities at the boundary of the flexible straight line, and utilize the Lagrange median theorem to realize the complement of voltage traveling wave data, thereby increasing the sampling frequency. In addition, the amount of fault current change during line fault is negatively correlated with the transition resistance, so the adaptive protection threshold can be constructed based on the supplement of the voltage data. Combined with the above, in this paper we propose an adaptive traveling wave protection method for flexible DC lines based on the supplement of the voltage data. The hardware-in-the-loop test results indicate that the proposed protection can effectively identify faults inside and outside the zone with high sensitivity at a sampling frequency of 10 kHz, and the feasibility of the protection is verified.