$G$-deviations of polygons and their applications in Electric Power Engineering

TL;DR

This paper introduces the concept of G-deviations for polygons in metric spaces, provides formulas for affine transformations on the complex plane, and applies these to assess power quality in electrical engineering.

Contribution

It develops a new mathematical framework for measuring polygon deviations under group actions and applies it to evaluate three-phase power quality.

Findings

Formulas for G-deviation between polygons on the complex plane.

New measures of triangle asymmetry based on G-deviation.

Application of these measures to assess electric power quality.

Abstract

For any metric space $X$ endowed with the action of a group $G$, and two $n$-gons $\vec x=(x_1,\dots,x_n)\in X^n$ and $\vec y=(y_1,\dots,y_n)\in X^n$ in $X$, we introduce the $G$-deviation $d(G\vec x,\vec y\,)$ of $\vec x$ from $\vec y$ as the distance in $X^n$ from $\vec y$ to the $G$-orbit $G\vec x$ of $\vec x$ in the $n$-th power $X^n$ of $X$. For some groups $G$ of affine transformations of the complex plane, we deduce simple-to-apply formulas for calculating the $G$-derviation between $n$-gons on the complex plane. We apply these formulas for defining new measures of asymmetry of triangles. These new measures can be applied in Electric Power Engineering for evaluating the quality of 3-phase electric power. One of such measures, namely the affine deviation, is espressible via the unbalance degree, which is a standard characteristic of quality of three-phase electric power.