The Distribution of the Number of Real Solutions to the Power Flow Equations

A Fixed-Point Algorithm for the AC Power Flow Problem

“…Assumption 2 is a standing assumption for the rest of this section. It states that the load is inductive, and restricts the amount of loading in the system 4 . Now, define the compact and convex set…”

Polynomial Systems, Homotopy Continuation, and Applications

“…The visible regions which are blue, red, green, purple, and yellow correspond to parameter values with 0, 2, 4, 6, and 8 solutions, respectively. We refer to the article [18] for detailed explanations and many other interesting experimental results obtained using homotopy continuation methods.…”

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Systems of multivariate polynomial equations are ubiquitous throughout mathematics. They also appear prominently in scientific applications such as kinematics [20,22], computer vision [11,15], power flow engineering [18], and statistics [12]. Numerical homotopy continuation methods are a fundamental tool for both solving these systems and determining more refined information about their structure.In this article, we offer a brief glimpse of polynomial homotopy continuation methods: the general theory, a few applications, and some software packages that implement these methods.

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Implementing real polyhedral homotopy

“…These algorithms work by continuously deforming the solutions from an "easy" polynomial system into the desired one. While there exist many off-the-shelf homotopy continuation solvers that find all complex solutions, many applications, such as to power systems engineering [15], economics [12], and statistics [16], only require knowledge of the real solutions. In general, there are many more complex solutions than real ones, leading to wasted computation.…”