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Transmission lines (TL) are exposed to diverse climatic conditions, which make them prone to faults in different phases simultaneously or after a short delay at the same or different locations. These complicated faults can cause failure or mal-operation of the conventional distance protection schemes. This research work presents a solution to identify and classify faults in long TLs under challenging situations. In this algorithm, the fundamental components of voltage and current signals from both ends of the TLs have been considered to evaluate the admittance-based fault index to discriminate the internal fault from the external ones and also to identify the faulty phase. The proposed algorithm has been tested under different dynamic conditions viz. swing condition, close-in fault, presence of series compensator, load angle and source strength variation, etc., to ascertain the robustness and reliability of the algorithm. The significant contribution of this algorithm is that it can accurately detect and classify the complex high impedance faults including evolving and cross-country faults in the transmission systems. [ABSTRACT FROM AUTHOR]