Eigenvalues and eigenvectors of the transfer matrix involved in the calculation of geomagnetically induced currents in an electric power transmission network

Geomagnetically induced currents (GIC) flowing in power grids can be calculated from matrix equations whose input data are the geoelectric field and the network parameters. The transfer matrix between the “perfect-earthing” (pe) currents and the earthing GIC are discussed in this paper by considering its eigenvalues and eigenvectors. The pe currents include the influence of the geoelectric field whereas the transfer matrix only depends on the network data. It is shown that an eigenvalue equals one or the corresponding eigenvector satisfies the condition that the sum of its pe currents is zero. Using physical arguments, we conclude that all eigenvalues of the transfer matrix are non-negative and between zero and one. This statement is proved mathematically for a three-node network and supported by numerical computations for the Finnish 400 kV GIC test model. Special attention is paid to the norm of the earthing GIC, which gives an idea of the risk of GIC to a power grid. This norm seems to have the lower and upper limits practically equal to the smallest and largest (≠1) eigenvalue of the transfer matrix multiplied by the norm of the pe currents.