On the Network Topology Dependent Solution Count of the Algebraic Load Flow Equations

Abstract

A large amount of research activity in power systems areas has focused on developing computational methods to solve load flow equations where a key question is the maximum number of isolated solutions. Though several concrete upper bounds exist, recent studies have hinted that much sharper upper bounds that depend the topology of underlying power networks may exist. This paper establishes such a topology dependent solution bound which is actually the best possible bound in the sense that it is always attainable. We also develop a geometric construction called adjacency polytope which accurately captures the topology of the underlying power network and is immensely useful in the computation of the solution bound. Finally we highlight the significant implications of the development of such solution bound in solving load flow equations.2. Algebraic formulation. In this paper, we focus on the mathematical abstraction of a power network which is captured by a graph G = (B, E) together with a complex matrix Y = (Y ij ). Here B is the finite set of nodes representing the "buses", E is the set of edges (also called lines or branches) representing the connections between buses, and the matrix Y is the nodal admittance matrix 1 which assigns a nonzero †

Directed acyclic decomposition of Kuramoto equations

“…This construction has been studied in the context of Kuramoto model and power flow equations. 16,19 Similar constructions have also appeared in other contexts. [20][21][22] The adjacency polytope bound is a simplification and relaxation of the Bernshtein-Kushnirenko-Khovanskii bound.…”

Counting Equilibria of the Kuramoto Model Using Birationally Invariant Intersection Index

“…, L G,n , its direct computation, in general, remains a difficult problem. Using a construction known as the "adjacency polytope bound" developed in [11,13], the primary contribution of this paper is the computation of explicit formulas for the birationally invariant intersection index [L G,1 , . .…”

“…A relaxation of the BKK bound was developed in the context of algebraic synchronization equations [11] as well as the closely related "power-flow equations." [13].…”

“…However, previous work has not addressed the normalized volume of these polytopes. A simple observation [11,13] is that the adjacency polytope bound (or simply, AP bound) is indeed an upper bound for answers of Problem 2 and 3: Proposition 6. Given a graph G containing vertices {0, 1, .…”

Holomorphic Embedding Based Continuation Method for Identifying Multiple Power Flow Solutions

“…13 No existing literature claims complete solution sets for larger IEEE test cases. 14 A more recent progress in [16] successfully reduced the computational time of case14 to 5 minutes, however, the proposed HEBC is still much faster.…”