… from stable to unstable manifold.

Elis Nycander, https://www.mail-archive.com/matpower-l@cornell.edu/msg06048.html, brought up a case where the operating point moves from stable to unstable manifold. Once on the unstable manifold, the continuation curve moves in the direction of lambda increase, which seems intuitively opposite to expected behavior (progress along the decreasing lambda direction). The reason for the lambda increase direction when on unstable manifold is because the tangent direction was always set towards lambda increase. This fix checks whether the operating point is on stable or unstable manifold (by checking the sign of minimum eigen value) and changes the direction accordingly. When on stable manifold, an increasing lambda direction is set, while a decreasing lambda direction on unstable manifold.

Relevant discussion can be found here MATPOWER#21

…y mistake).

…s for issues pointed out in PR MATPOWER#25.

i) Change the tangent direction after switching based on the sign of the product of current tangent direction and the
sign of	the smallest real eigen value for Jacobian. This resolves issues where the PV curve seems to turn
unnaturally (go in the opposite direction) after PV->PQ switching. 9-bus test case.
ii) Use tolerance 1e-3 for eigen value calculation. This resolves the crashing of the failing test for 300-bus case w/Qlims.
iii) Check for buses having negative V-Q sensitivity. This is done by checking for positive dQ_dV elements in the tangent vector z. If found, previous tangent direction is retained. Otherwise, a new direction is calculated as per (i). This resolves the failing 300-bus w/Q lims. (Thanks J. Marin!)

All test cases pass with the above fixes.

…on direction after potential change from stable to unstable manifold.

(or vice versa)