Vector Geometric Algebra in Power Systems: An Updated Formulation of Apparent Power under Non-Sinusoidal Conditions

Abstract

Traditional electrical power theories and one of their most important concepts—apparent power—are still a source of debate, because they present several flaws that misinterpret the power-transfer and energy-balance phenomena under distorted grid conditions. In recent years, advanced mathematical tools such as geometric algebra (GA) have been introduced to address these issues. However, the application of GA to electrical circuits requires more consensus, improvements and refinement. In this paper, electrical power theories for single-phase systems based on GA were revisited. Several drawbacks and inconsistencies of previous works were identified, and some amendments were introduced. An alternative expression is presented for the electric power in the geometric domain. Its norm is compatible with the traditional apparent power defined as the product of the RMS voltage and current. The use of this expression simplifies calculations such as those required for current decomposition. This proposal is valid even for distorted currents and voltages. Concepts are presented in a simple way so that a strong background on GA is not required. The paper included some examples and experimental results in which measurements from a utility supply were analysed.

New mathematical model based on geometric algebra for physical power flow in theoretical two-dimensional multi-phase power circuits

“…Electric and magnetic field in 2D using geometric algebra. Recently, Geometric Algebra has been successfully applied in physics 27,28 and electrical engineering 6,22,[29][30][31] . It enables a reformulation of electromagnetic Maxwell's equations in a very compact and unified form 28 .…”

Geometric Foundations of Power Theory for Multiphase AC Systems in the Frequency Domain

“…This work extends and elaborates on the results of [23] and [24] by presenting the theoretical framework in multi-phase electrical systems using symmetrical components. Specifically, three-phase circuits are thoroughly analyzed in detail.…”

“…ELECTRICAL SYSTEMS Any periodic electrical signal (voltage or current) can be expressed through the orthonormal basis in the vector space of Fourier functions ϕ = {1, √ 2 cos kωt, √ 2 sin kωt} m k=1 , where m is the number of harmonics under consideration. This basis is isomorphic [23] to the Euclidean basis σ = {σ k } 2m k=0 , with the following one-to-one mapping…”

The Frenet Frame as a Generalization of the Park Transform

“…These works define the active and reactive power as the dot and cross (or wedge in multi-phase circuits) products, respectively, of voltage and currents. A variety of recent works with same starting point have focused on the analysis of electric circuits using the formalism provided by geometric algebra [24]- [27].…”

“…where T, N and B are called tangent, normal and binormal vectors, respectively. The vectors in (24) are orthonormal, i.e. T = N × B and N = B × T, and satisfy the following set of differential equations [14]:…”