Analysis of the load flow behaviour near a Jacobian singularity

This paper presents new theoretical results about the behaviour 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 neighbourhood 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 new closed form relation derived here is theoretically important since it can predict and explain the main load flow phenomena observed through simulation analysis near a singularity. These are: (i) the non-existence of solutions for certain injection changes, (ii) the bifurcation of the voltages into two nearby solutions, (iii) the sudden collapse of voltages for small injection changes, (iv) the nature of the collapse, that is, which buses are more susceptible to the collapse. Numerical simulations support the validity of the theoretical result by comparing the closed form analytic relation near a singularity with exact load flow simulations.