Analysis of the load flow behaviour near a Jacobian singularity

The authors present theoretical results about the behavior of the load flow solution near a Jacobian singularity. The principal result is the derivation of an analytic closed-form relation between the specified injections and the resulting voltages in the neighborhood of a singularity. This result is a companion to the conventional load flow sensitivity analysis which is valid only if the operating point is not at a Jacobian singularity. The closed-form relation derived is theoretically important since it can predict and explain the main load flow phenomena observed through simulation analysis near a singularity. These are: the nonexistence of solutions of certain injection changes, the bifurcation of the voltages into two nearby solutions, the sudden collapse of voltages for small injection changes, and the nature of the collapse, that is, which buses are more susceptible to the collapse. Numerical simulations supported the validity of the theoretical result by comparing the closed-form analytic relation near a singularity with exact load flow simulations