# GRAPH THEORETIC VOLTAGE STABILITY

#### Ayusman Roy

### A power system will eventually reach its point of static voltage collapse under a sufficiently large transfer of active power from a source to a sink. As the transfer increases, voltage collapse can occur due to two different mechanisms: a) the power flow Jacobian matrix becomes singular (saddle-node bifurcation), or b) a control limit (reactive power generator limit, tap changer limit, etc) is encountered, causing a shift to a new system of algebraic equations, which does not have a power flow solution (limit-induced bifurcation).

### This project explores the relation between saddle-node bifurcation voltage collapse and complex flow limits of individual transmission elements. Necessary conditions for power system voltage collapse are derived. First, when a transfer of power takes place in a power system, at least one line must reach its static transfer stability limit (STSL) before the point of collapse is encountered. Second, for a point-to-point transfer, a path from the source to the sink, formed by lines all of which have reached their STSL limits, must be formed in the network before the point of collapse is reached.

### While collapse phenomena depend on the conditions of the entire network, topological methods are promising alternatives, providing mechanisms to monitor voltage collapse by looking at the behavior of the flow of active and reactive power in individual transmission elements. The central topic of this project is therefore the relation between system-wide voltage stability limits and the limits of individual transmission devices.

Last revised on Aug. 25, 2011