Investigation of the relationship between ill-conditioned power flow and voltage collapse

Ill-conditioned power flow problems have been widely investigated and discussed in the literature. Typical approaches are to develop enhanced solution algorithms when a power flow case is found to diverge using the conventional Newton method. It is known that a genuine ill-conditioned problem is caused by the presence of a large condition number in the power flow Jacobian matrix. Since a large condition number is associated with small singular values and the voltage collapse is related to small eigenvalues, we therefore speculate that an ill-conditioned power flow problem is actually a voltage collapse problem. The objective of this letter is to demonstrate the relationship between power flow ill conditioning and voltage instability. Case studies of five classical ill-conditioned cases have confirmed that the ill conditioning of power flow indeed occurs-and only occurs-at the voltage collapse point.