Efficient Resonance Mode Analysis (RMA) Algorithms for Resonance Studies in Large-Scale Power Transmission Grids

This repository provides a MATLAB implementation of efficient Resonance Mode Analysis (RMA) algorithms for resonance studies in large-scale transmission grids.

The code focuses exclusively on the numerical solution of the RMA problem, i.e., the efficient computation of dominant eigenvalues and their eigenvectors of the nodal admittance (or impedance) matrix over a frequency scan. The construction of the nodal admittance matrix for AC transmission grids is included only as a necessary preliminary step for applying the RMA algorithms; however, the algorithms are applicable to any nodal admittance matrix, including purely AC, purely DC, or hybrid AC/DC transmission grid models.

A benchmark test dataset, based on detailed information from IEEE n-bus and synthetic power system test cases, and fully compatible with the provided MATLAB code, is available in a separate Zenodo repository entitled “Dataset of IEEE and Synthetic Power System Test Cases”. This dataset can be used to evaluate and compare the performance of the implemented algorithms.

In addition to resonance identification, the provided RMA tools support stability assessment based on the Positive-Mode-Damping (PMD) stability criterion, enabling the detection of poorly damped or unstable modes in large-scale power transmission systems.

The following RMA-based approaches are implemented:

1. Faster RMA (f_RMA), based on a modified shifted-inverse power iteration method, enabling fast and robust convergence towards critical resonance modes for all frequencies.
2. Lanczos-based RMA (L_RMA), using a non-Hermitian Lanczos method, particularly suited for large-scale and sparse transmission grids and capable of computing both right and left eigenvectors.
3. Arnoldi-based RMA (A_RMA), provided as an alternative Krylov-subspace approach. While computationally efficient, this method does not directly provide left eigenvectors for large sparse matrices, which may limit its applicability in certain resonance studies. For frequencies of interest where left eigenvectors are required (e.g., for modal participation factors or sensitivity analyses), the above non-Hermitian Lanczos method is used.

The implemented algorithms are designed to significantly reduce the computational burden associated with conventional RMA (including r_RMA), while preserving accuracy in the identification of resonance frequencies, modal impedances, and damping characteristics. The code is intended for research and educational use in power quality and resonance studies of transmission and multi-terminal grids.

The methodology implemented in this repository is based on and extends the following publications:

• O. Cartiel, J. J. Mesas, L. Sainz, and A. Fabregas, “A Faster Resonance Mode Analysis Approach Based on a Modified Shifted-Inverse Power Iteration Method,” in IEEE Transactions on Power Delivery, vol. 38, no. 6, pp. 4145-4156, Dec. 2023, doi: 10.1109/TPWRD.2023.3318431.
• O. Cartiel, J. J. Mesas and L. Sainz, “Efficient Resonance Mode Analysis-Based Methodology for Resonance Studies in Multi-Terminal Transmission Grids,” in IEEE Transactions on Power Delivery, vol. 40, no. 1, pp. 287-300, Feb. 2025, doi: 10.1109/TPWRD.2024.3493381.
• O. Cartiel, J. J. Mesas, Ll. Monjo, and L. Sainz, “Computational time efficiency analysis for resonance studies in transmission grids and microgrid clusters,” in Mathematics and Computers in Simulation, vol. 243, May 2026, doi: 10.1016/j.matcom.2025.12.009.

The Arnoldi-based RMA, based on MATLAB function eigs(), implementation included in this repository is provided as an additional methodological extension that has not been explicitly presented in the above publications.

This repository includes a figure illustrating the comparative computational efficiency of the implemented RMA methods for large-scale systems. Detailed performance analyses and the corresponding benchmarking methodologies can be found in the related journal publications listed above.