An equation for the coupling resistance influence on (the many) radial link power reserves

The concept of power system stability reserve margin as a metric for the distance to the risk of blackout has evoked speculations during investigations of the global dynamics of events recently reported for stressed systems worldwide. Among the speculations made is the possible existence of `many´ such stability reserves. In this paper, it is shown that differences in the steady-state stability limit (SSSL) of a power system component are attributed to incrementally small variations of the resistance element in an impedance coupling configuration that are only detectable through the evolving voltages of that coupling. The results of the analytical solution to the boundary problem involves identification of the necessary conditions for maximum power that can be stated with reference to either sending or receiving-end voltages; with the sufficient conditions hence the complete solution being provided by conditions at the other end. The relationships for the two ends connecting internal voltages and sectional impedance constraints evolve in high-order algebraic polynomial expressions in any one of system variables. As a result, the receiving-end constraint relationship that may serve as an index for the on-line monitoring of maximum power reveals the existence of critical transmission states that couldn´t have been observed through equations without the resistance element. An impedance-voltage domain plot for the voltage-based index would therefore describe catastrophe boundaries ´in-waiting (for the sending voltage) to happen´; and hence, a ´metric´ for the distance to the risk of blackout. While case SSSL sensitivity calculations based on a stability index function for a coupled four-node system configuration are given , an algebraic equation for the ´approximate´ three-node circuit that will verify the associated constant-Q results within a viable range of sending voltage magnitudes is presented. The analysis results will further apply to the conventional tw- - o-node circuit model for which conditions of the evolving ´internal´ node will tend to those of the receiving end as the elements of the last section tend to zero.