On Adapting Test Function Methods for Fast Detection of Fold Bifurcations in Power Systems

Abstract

Test function methods are well known techniques to monitor a continuation method to determine if a fold (saddle-point) bifurcation point has been reached. Two such test functions due to [Abbott, 1978] and [Seydel, 1979] are considered here in a unified framework and their connection to the tangent vector index method from power system community is shown. The challenge in power systems applications is to adapt a test function method to produce a performance index that approximates the fold parameter λ* without the need of following the solution path to the fold point (x*, λ*). We demonstrate that the tangent vector index method, due to its connection with the mathematically justifiable test function methods, can lead to a useful performance index.

Probabilistic load margins of power systems embedded with wind farms

“…On the possible way to reduce the size of the original problem is to characterize the singularity of the Jacobian matrix by one equation only. Such alternative characterizations have been studied [6], [4], [5]. We will use the bordered matrices technique to find a substitute function that avoids the above difficulties [2].…”

“…In our new direct method, an extended system of side + 1 will be formulated. In addition to power flow equations, an implicit test function [4], [5] is constructed in the extended system. In numerical aspect, we propose a new method with only one LU factorization is needed in each iteration [2].…”

Avoiding fold bifurcations with the help of new proximity indices

“…These formulas are used in the following manner:Ṙ 1,1 is computed by (13). Next, elements W 1, j are computed by (14) for j = 2, .…”

“…, n, etc (notice that W is skewsymmetric). In general, having computed W i−1, j , the value ofṘ i,i is computed using (13), and afterwards the values of W i, j for j = i + 1, . .…”

“…Scalar indices and test functions have been proposed in the literature to monitor this proximity ( [4,13,14]). The problem of calculating the closest fold bifurcation of (1) has been extensively researched by the power systems community ( [1,7,9]).…”