Certifying the Nonexistence of Feasible Path Between Power System Operating Points

TL;DR

This work addresses the problem of certifying the nonexistence of a feasible path between power-system operating points when the OPF feasible space is nonconvex and potentially disconnected. It introduces a novel algorithm that first identifies two feasible points connected by a line containing an infeasible point, then certifies that all points on a hyperplane normal to that line are infeasible using a QC-relaxed feasibility problem with optimization-based bound tightening; infeasibility of this hyperplane proves that the two points lie in different components. This results in the first rigorous certifications of disconnected OPF feasible spaces across several test cases, illustrating the method's effectiveness and its complementary role to feasible-path algorithms. By leveraging QC relaxation, McCormick envelopes, and bound tightening, the approach provides a practical tool for operators to detect unavoidable constraint violations during transitions and informs strategies for feasible-space decomposition and secure Point-of-Operation selection.

Abstract

By providing the optimal operating point that satisfies both the power flow equations and engineering limits, the optimal power flow (OPF) problem is central to power systems operations. While extensive research has focused on computing high-quality OPF solutions, assessing the feasibility of transitioning between operating points remains challenging since the feasible spaces of OPF problems may consist of multiple disconnected components. It is not possible to transition between operating points in different disconnected components without violating OPF constraints. To identify such situations, this paper introduces an algorithm for certifying the infeasibility of transitioning between two operating points within an OPF feasible space. As an indication of potential disconnectedness, the algorithm first seeks an infeasible point on the line connecting a pair of feasible points. The algorithm then certifies disconnectedness by using convex relaxation and bound tightening techniques to show that all points on the plane that is normal to this line are infeasible. Using this algorithm, we provide the first certifications of disconnected feasible spaces for a variety of OPF test cases.

Figures (3)

• Figure 1: Illustrations of convex and non-convex feasible regions. On the left, all points along the line segment between feasible points $A$ and $B$ are feasible. This is true for all possible pairs of feasible points. In contrast, on the right, the infeasible point $C$ is on the line between points $A$ and $B$. Points $A$, $B$, and $C$ confirm non-convexity of this region.
• Figure 2: Illustrative examples of applying the proposed algorithm to two test cases from Narimani_ACC. The OPF feasible spaces, which are computed using the method in molzahn_opf_spaces, are shown in gray. The pink regions correspond to the QC relaxation's feasible spaces (without bound tightening). The method from molzahn-nonconvexity_search identifies non-convexities in both feasible spaces via the feasible points $A$ and $B$ and the infeasible point $C$ on the line connecting $A$ and $B$. The yellow hyperplane is normal to the line connecting $A$ and $B$ and passes through point $C$. Applying bound tightening to \ref{['feasibility_problem']} certifies infeasibility of all points on the yellow hyperplane in Fig. \ref{['f:FS_cyclicthreebusFS']}, and thus this feasible space is disconnected. Conversely, the OPF problem's feasible space intersects the yellow hyperplane in Fig. \ref{['f:FS_AcyclicthreebusFS']} and thus \ref{['feasibility_problem']} is feasible. Accordingly, the proposed algorithm does not indicate disconnectedness of this OPF feasible space.
• Figure 3: Feasible space projection for the 9-bus test system from bukhsh_tps. Although the OPF problem's feasible space shown by the gray regions is disconnected, the proposed algorithm fails to certify disconnectedness since the yellow plane that is normal to the line segment between point $A$ and $B$ passes through the component of the feasible space in the lower-left corner of the figure. Conversely, applying bound tightening to the green hyperplane which is rotated by $45^\circ$ around the $P_2$ axis certifies disconnectedness.