Spectral solution of load flow equations

The load-flow equations are the main tool to operate and plan electrical networks. For transmission or distribution networks these equations can be simplified into a linear system involving the graph Laplacian and the power input vector. We show, using spectral graph theory, how to solve this system efficiently. This spectral approach gives a new geometric view of the network and power vector. This formulation yields a Parseval-like relation for the $L_2$ norm of the power in the lines. Using this relation as a guide, we show that a small number of eigenvector components of the power vector are enough to obtain an estimate of the solution. This would allow fast reconfiguration of networks and better planning.