Equilibrium Analysis of Power Systems

Synchronization in complex networks of phase oscillators: A survey

“…For arbitrary network topologies and weights the equilibrium and potential energy landscape of the oscillator network (1) has been studied by different communities, see (Tavora and Smith, 1972a;Korsak, 1972;Araposthatis et al, 1981;Baillieul and Byrnes, 1982;Mehta and Kastner, 2011). We particularly recommend the article (Araposthatis et al, 1981), where various surprising and counter-intuitive examples are reported.…”

“…Likewise, for the transient analysis, the ∞ -type contraction Lyapunov function (32) is a powerful analysis concepts for complete graphs and still needs to be extended to arbitrary connected graphs. Regarding the potential and equilibrium landscape, a few interesting and still unresolved conjectures can be found in Tavora and Smith (1972a); Araposthatis et al (1981); Baillieul and Byrnes (1982); Mehta and Kastner (2011);Korsak (1972) and pertain to the number of (stable) equilibria and topological properties of the equilibrium set. Finally, the complex networks, nonlinear dynamics, and statistical physics communities found various interesting scaling laws in their statistical and numerical analyses of random graph models, such as conditions depending on the spectral ratio λ 2 /λ n of the Laplacian eigenvalues, interesting results for correlations between the degree deg i and the natural frequency ω i , and degree-dependent synchronization conditions (Nishikawa et al, 2003;Moreno and Pacheco, 2004;Restrepo et al, 2005;Boccaletti et al, 2006;Gómez-Gardeñes et al, 2007;Arenas et al, 2008;Kalloniatis, 2010;Skardal et al, 2013).…”

Synchronous solutions of power systems: existence, uniqueness and stability

“…In particular, we may set T = S where S is defined in Eq. (10). Then p(a) is one-to-one on T, and w*, a* is the only equilibrium point of system (8) with a* e T.…”

“…(10)). It is known (see [10]) that some equilibrium points outside S may be stable, a fact which may be established by a continuity argument. However, it appears to be often assumed that stable equilibrium points occur in or near the region S.…”

“…The region S is not the largest region containing the origin in which dp/da is positive definite. If we assume a conjecture of Tavora and Smith [10] that each connected region in which dp/da is positive definite is convex, then system (8) has at most one equilibrium point in any such region. The author was unable to follow the uniqueness proof of Tavora and Smith.…”