Convexity of Energy-Like Functions: Theoretical Results and Applications to Power System Operations

TL;DR

This paper investigates the convexity properties of energy functions in power systems, providing theoretical insights and practical tools for ensuring operational safety and solution uniqueness in modern, renewable-rich grids.

Contribution

It generalizes the energy function concept, characterizes its convexity domain, and demonstrates its application to power flow analysis in power system operations.

Findings

The energy function is convex within a large, practically relevant domain.

Convexity can certify solution uniqueness or non-existence of power flow solutions.

Applications demonstrated on IEEE bus system models.

Abstract

Power systems are undergoing unprecedented transformations with the incorporation of larger amounts of renewable energy sources, distributed generation and demand response. All these changes, while potentially making power grids more responsive, efficient and resilient, also pose significant implementation challenges. In particular, operating the new power grid will require new tools and algorithms capable of predicting if the current state of the system is operationally safe. In this paper we study and generalize the so-called energy function as a tool to design algorithms to test if a high-voltage power transmission system is within the allowed operational limits. In the past the energy function technique was utilized primarily to access the power system transient stability. In this manuscript, we take a new look at energy functions and focus on an aspect that has previously received little attention: Convexity. We characterize the domain of voltage magnitudes and phases within which the energy function is convex. We show that the domain of the energy function convexity is sufficiently large to include most operationally relevant and practically interesting cases. We show how the energy function convexity can be used to analyze power flow equations, e.g. to certify solution uniqueness or non-existence within the domain of convexity. This and other useful features of the generalized energy function are described and illustrated on IEEE 14 and 118 bus models.