With this knowledge, a set of power balance equations in the rectangular form is proposed in this paper. Space allows for a faster solution for the set of equations. Represents square of bus voltage magnitudes, has lower order terms, behaves better and is less nonlinear. A solution space forĮquations that includes power, such as a set of equations of bus power balance, when constructed by variables that Power of a constant impedance load is proportional to the square of bus voltage magnitude. Numerical simulations support the validity of the theoretical result by comparing the closed-form analytic relation near a singularity with exact load flow simulations These are: the nonexistence of solutions for 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.
The new 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. 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 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. New theoretical results above the behavior of the load flow solution near a Jacobian singularity are presented.
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