###### Abstract  
Fault location technology is crucial for enhancing the efficiency of fault maintenance and ensuring the safety of the power supply in small current grounding systems. To address the challenge that traditional single-end positioning methods experience when identifying the reflected wave head and that the adaptability of wave head calibration methods is typically limited, a single-end location method of modulus wave velocity differences based on marine predator algorithm optimized multivariate variational mode decomposition (MVMD) and morphological gradient is proposed. Firstly, the minimum comprehensive entropy kurtosis ratio is used as the fitness function, and the marine predator algorithm is used to realize the automatic optimization of the mode number and penalty factor of the multivariate variational mode decomposition. Therefore, with the goal of decomposing the traveling wave characteristic signals with the most significant traveling wave characteristic information and the lowest noise component, the line-mode traveling wave and the zero-mode traveling wave are accurately decomposed. Secondly, the intrinsic mode function component with the smallest entropy kurtosis ratio is selected as the line-mode traveling wave characteristic signal and the zero-mode traveling wave characteristic signal, respectively, and the arrival time of the wave head is accurately calibrated by combining the morphological gradient value. Finally, the fault distance is calculated by the modulus wave velocity difference location formula and compared with the variational mode decomposition-Teager energy operator (VMD-TEO) method and the empirical mode decomposition_first-order difference method. The results show that the proposed method has the highest accuracy of positioning results, and the algorithm time is significantly reduced compared with the VMD-TEO method, and it has strong adaptability to different line types of faults, different fault initial conditions, and noise interference. fault location; multivariate variational mode decomposition; small current grounding systems; marine predator algorithm; modulus wave velocity difference figure  
may lead to substantial equipment deterioration or pose significant safety risks. Hence, developing precise and dependable fault localization techniques is crucial for enhancing maintenance capabilities and ensuring power distribution reliability [2]. At present, fault localization techniques primarily encompass three methodologies: the traveling wave method, the impedance method [3; 4], and signal injection approaches [5; 6]. The traveling wave method has garnered particular attention due to its minimal susceptibility to variables such as fault transition resistance and system operation mode. This method branches into two categories mainly: single-end methods and double-end methods. While double-end methods offer reliable initial wave head detection, their accuracy heavily depends on precise clock synchronization between endpoints and requires additional terminal ranging equipment, resulting in elevated implementation costs. Although single-end methods [7] present cost advantages and eliminate synchronization requirements, they face challenges in accurately identifying the reflected wave head due to adjacent line interference and impedance variations. To address these limitations, reference [8] introduced an innovative ranging technique leveraging zero-mode and line-mode wave velocity differentials. This approach only requires initial wave head detection in both modes, offering enhanced reliability through simplified principles. Reference [9] developed constraint relationships between zero-line-mode temporal differences, velocity disparities, and transmission distances, utilizing particle swarm optimization (PSO) to determine optimal fault locations. Reference [10] further refined this methodology by implementing an enhanced truncation error correction algorithm.