Fault detection is one of the most important research topics till today in the field of VSC-HVDC grids protection. In this subject, fast and reliable identification of faults is the most demanding issue, which still requires further study. In this respect, this paper proposes an index for fault detection, which is based on correlation between the powers of DC-link capacitor and the protected line. It is proved by an analytical time-domain transient expression that there is a direct correlation between these two powers under fault conditions. To implement this scheme, the routing of the discharged power of DC-link capacitor into the protected line is the key technique. To do so, the Pearson correlation factor is used in this paper, which efficiently captures the level and direction of the correlation between these two powers. The proposed method benefits from different salient features of very fast fault detection, accurate discrimination of faulted line from healthy lines, covering high-resistance faults, noise immunity, robustness against severe power reversal conditions, robustness against low sampling frequencies, not requiring extra sensor and communication infrastructure, etc. To evaluate the performance of the proposed approach, a complete set of fault scenarios are simulated on a meshed multi-terminal MMC-based VSC-HVDC network.