###### Abstract
The accurate detection of traveling wave characteristics and wave head identification remains a significant technical challenge. Research [11] employed db4 mother wavelet's initial decomposition scale for traveling wave head extraction. While wavelet transform-based multi-resolution analysis effectively identifies traveling wave singularities, the effectiveness of fault feature extraction heavily depends on decomposition parameters and wavelet basis selection. Addressing these limitations, reference [12] developed an alternative approach combining empirical mode decomposition (EMD) with fast Fourier transform (FFT). Though EMD eliminates the need for wavelet basis and scale selection, it suffers from significant modal aliasing and boundary effects. To overcome these challenges, reference [13] introduced a methodology utilizing variational mode decomposition (VMD) alongside Teager energy operator (TEO) for wave head calibration. The multivariate variational mode decomposition (MVMD) extends VMD's capabilities from single to multiple dimensions, maintaining resistance to modal aliasing while efficiently processing multivariable data, thus enabling faster fault localization. Reference [14] introduced a novel wave head identification technique incorporating MVMD and kurtosis measurements, utilizing kurtosis values to calibrate high-frequency traveling wave signals post-MVMD. However, this method's accuracy is notably influenced by the modal decomposition parameter \(K\) and penalty coefficient \(\alpha\). Reference [15] addressed this by developing a comprehensive metric incorporating correlation coefficients, residual signal permutation entropy, and instantaneous energy loss ratios to optimize these parameters. Nevertheless, this optimization process remains complex and requires recalibration when line parameters or initial fault conditions shift, limiting the algorithm's adaptability. To address these persistent challenges in current traveling wave positioning methodologies, this paper presents an innovative single-end positioning approach that combines marine predator algorithm (MPA)-optimized MVMD with morphological gradient analysis. Firstly, the generation principle of traveling waves and phase-mode transformation is introduced. Secondly, a wave head calibration method based on parameter optimization MVMD and morphological gradient is proposed. Then, the fault location formulas under single line and mixed line are proposed. Finally, the influence of line type, initial fault condition, and noise on the proposed method is verified by simulation and compared with different algorithms. The research focuses on the following three key aspects:  
1. MPA considers that the minimum comprehensive entropy kurtosis ratio is introduced to optimize the decomposition mode number \(K\) and penalty factor \(\alpha\) of MVMD. The MVMD algorithm optimized by MPA can decompose the traveling wave characteristic signals with the most significant singularity and the lowest noise component as the target, so as to accurately decompose the line-mode traveling wave and the zero-mode traveling wave. 2. Compared with the independent processing of the two traveling waves, only one parameter optimization and one MVMD are needed, which effectively shortens the extraction time of traveling wave characteristic signals while retaining the advantages of VMD noise interference robustness and is not affected by modal aliasing. 3. Focusing on the problem of large positioning error caused by inconsistent wave velocity of hybrid line in small current grounding system, an equivalent method based on modulus wave velocity ratio is proposed. The hybrid line is equivalent to a single line for ranging, which effectively improves the accuracy of positioning results.