Spectral Graph Analysis of the Geometry of Power Flows in Transmission Networks

Power flows in transmission networks are driven by the structure of the network and the spatial distribution of generators and loads. Understanding the interplay between grid, generators, and loads is crucial for efficient and robust planning and management of electrical networks. Using the spectral properties of the graph Laplacian, we show that we can express power flows on the basis of Laplacian eigenvectors and reveal the dominant modes for nodal voltages and branch powers. The bus voltages are dominated by low rank modes. The power in the lines depends on the nodal distribution of generators/loads, the line impedances, and the gradient of the eigenvectors across the branches. The most loaded lines and their associated dominant mode are then identified. A modification of the bus powers is finally proposed to better share the load between the lines and, hence, to decrease the system vulnerability. Results provided for several IEEE transmission test systems prove the relevance of the approach and suggest practical guidelines to improve the operation of power systems.