[Paper Review] Geometric Algebra Power Theory in Time Domain

Francisco G. Montoya, Javier Roldán‐Pérez+3 more|arXiv (Cornell University)|Feb 13, 2020

Power Quality and Harmonics23 references4 citations

TL;DR

This paper proposes a time-domain power theory using geometric algebra (GA) and the Hilbert transform to overcome limitations in existing methods under voltage distortion, unbalanced loads, or non-linear conditions. The approach enables compact, physically meaningful current decomposition across single- and three-phase systems, with validation showing accurate separation of active, reactive, and unbalanced components in both sinusoidal and non-sinusoidal scenarios.

ABSTRACT

In this paper, the power flow in electrical systems is modelled in the time domain by using Geometric Algebra and the Hilbert Transform. The use of this mathematical framework overcomes some of the limitations shown by the existing methodologies under distorted supply or unbalanced load. In such cases, the derived instantaneous active current may not be the lowest RMS current in all circuit conditions and could contain higher levels of harmonic distortion than the supply voltage. Moreover, they cannot be applied to single phase systems. The proposed method can be used for sinusoidal and non-sinusoidal power supplies, non-linear loads, single- and multi-phase systems, and it provides meaningful engineering results with a compact formulation. Several examples have been included to prove the validity of the proposed theory.

Motivation & Objective

• To address the limitations of existing time-domain power theories in handling voltage distortion, unbalanced loads, and non-linearities.
• To develop a unified framework applicable to both single-phase and multi-phase systems with consistent engineering interpretation.
• To enable physically meaningful current decomposition for load compensation purposes under any supply or load condition.
• To provide a compact, mathematically rigorous formulation that simplifies power flow analysis in complex electrical networks.

Proposed method

• Utilizes geometric algebra (GA) to represent voltages and currents as multivectors in a vector space with orthonormal bases.
• Applies the Hilbert transform to generate analytic signals, enabling quadrature component extraction and phase synchronization in the time domain.
• Defines instantaneous power as a geometric product between voltage and current multivectors, decomposing it into scalar (active power) and bivector (unbalanced/reactive) components.
• Derives current decomposition via projection: active current as $\bm{i}_p = \frac{\bm{u}}{\|\bm{u}\|^2} M_p$, and unbalanced current as $\bm{i}_q = \bm{i} - \bm{i}_p$.
• Uses the geometric product $\bm{ab} = \bm{a} \cdot \bm{b} + \bm{a} \wedge \bm{b}$ to separate scalar (active) and bivector (reactive/unbalanced) power contributions.
• Applies the method to three-phase systems with balanced and unbalanced resistive loads, validating results through time-domain current decomposition and power component analysis.

Experimental results

Research questions

• RQ1 Can geometric algebra and the Hilbert transform enable a unified time-domain power theory applicable to both single-phase and multi-phase systems under arbitrary voltage and load conditions?
• RQ2 How can current components be meaningfully decomposed in the time domain to distinguish active, reactive, and unbalanced components when traditional methods fail?
• RQ3 Does the proposed formulation maintain consistency and physical relevance in the presence of harmonic distortion and unbalanced voltages?
• RQ4 Can the method accurately reproduce known power flow results in standard test cases, such as balanced three-phase resistive loads?
• RQ5 What is the mathematical structure of power decomposition in geometric algebra that supports engineering interpretation of current components?

Key findings

• The proposed method successfully decomposes current into active ($\bm{i}_p$) and unbalanced ($\bm{i}_q$) components, with $\bm{i}_p = \frac{G}{3}\bm{u}$ for a balanced resistive load.
• In a balanced three-phase resistive load, the Budeanu reactive power $\bar{M}_q = 0$, and the current decomposition matches the Fryze current, confirming consistency with established theories.
• The unbalanced current $\bm{i}_q$ is shown to consist of zero-sequence ($\bm{i}_0$) and negative-sequence ($\bm{i}_-$) components, with explicit expressions derived for each.
• For $U = 230$ V, $\omega = 1$ rad/s, and $G = 1$ Ω, the time-domain active current is $i_p(t) = \frac{\sqrt{2}GU}{3} \begin{bmatrix} \cos\omega t \\ \cos(\omega t - 120^\circ) \\ \cos(\omega t + 120^\circ) \end{bmatrix}$, matching expected symmetrical values.
• The unbalanced current $\bm{i}_q$ is decomposed into $\bm{i}_0 + \bm{i}_-$, with $\bm{i}_0$ representing zero-sequence and $\bm{i}_-$ representing negative-sequence components.
• The method provides a compact, consistent, and physically interpretable formulation across all tested conditions, including non-sinusoidal and unbalanced scenarios.