Complex-Vector Power and Cross-Phase Unbalance in Three-Phase Systems

Francisco Casado-Machado; Juan Carlos Bravo-Rodr\'iguez; Juan Carlos del-Pino-L\'opez

arxiv: 2604.25473 · v1 · submitted 2026-04-28 · 📡 eess.SY · cs.SY

Complex-Vector Power and Cross-Phase Unbalance in Three-Phase Systems

Juan Carlos Bravo-Rodr\'iguez , Juan Carlos del-Pino-L\'opez , Francisco Casado-Machado This is my paper

Pith reviewed 2026-05-07 15:01 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords three-phase systemsunbalanced powercomplex vector powercross-phase unbalancephasor representationapparent powerFortescue transformationfour-wire systems

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The pith

Three-phase power gains a complex-vector form whose cross-product term explicitly captures cross-phase unbalance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a representation of power in unbalanced three-phase systems by supplementing the classical dot-product complex power with the cross product of voltage and current phasors. This keeps the usual active and reactive terms while introducing an explicit cross-phase unbalance vector that isolates antisymmetric interphase relations. Apparent power thereby splits into intraphase and cross-phase contributions whose overall norm remains unchanged under the power-invariant Fortescue transformation. The same structure extends to four-wire systems through equivalent coordinates that preserve the effective apparent-power norm. A reader would care because scalar apparent-power values hide these interphase effects, and the new vector makes them directly visible using only ordinary complex arithmetic.

Core claim

The central claim is that complex power in sinusoidal steady-state three-phase systems can be expressed as the sum of a dot-product term and a cross-product term between voltage and current phasors. The dot-product term recovers the conventional active and reactive power, while the cross-product term supplies a vector that encodes cross-phase unbalance arising from antisymmetric interphase relations. This decomposition separates apparent power into intraphase and cross-phase parts, preserves the apparent-power norm under power-invariant Fortescue transformation, and extends to three-phase four-wire systems by means of equivalent coordinates that maintain the effective apparent-power norm. No

What carries the argument

The cross product of voltage and current phasors, which produces the cross-phase unbalance vector that isolates antisymmetric interphase power relations.

If this is right

Apparent power separates into intraphase and cross-phase contributions.
The norm of apparent power is preserved under the power-invariant Fortescue transformation.
The formulation extends to three-phase four-wire systems while preserving the effective apparent-power norm for the chosen voltage reference.
Numerical cases exist in which a non-negligible fraction of apparent power is tied to cross-phase unbalance and cannot be inferred from active and reactive power alone.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The unbalance vector could serve as a direct input for control algorithms that target interphase compensation without altering the total apparent-power magnitude.
Because only standard complex numbers are required, the descriptor can be computed inside existing phasor-domain simulation packages without new data structures.

Load-bearing premise

The cross-product term between voltage and current phasors produces a physically interpretable unbalance descriptor that stays consistent with established scalar apparent-power definitions under all sinusoidal steady-state conditions.

What would settle it

A concrete sinusoidal steady-state operating point in which the magnitude of the proposed complex-vector power differs from the conventional apparent power, or in which the derived cross-phase unbalance vector does not correspond to observable phase asymmetries.

Figures

Figures reproduced from arXiv: 2604.25473 by Francisco Casado-Machado, Juan Carlos Bravo-Rodr\'iguez, Juan Carlos del-Pino-L\'opez.

**Figure 1.** Figure 1: Example 2: Unbalanced supply with unequal load and conductor view at source ↗

read the original abstract

Unbalanced three-phase systems still lack a compact phasor-domain representation of power that makes phase asymmetry explicit while remaining consistent with established apparent-power definitions. This paper addresses that point through a complex-vector power formulation for sinusoidal steady-state operation. The proposed representation supplements the classical dot-product expression of complex power with the cross product of voltage and current phasors, thereby retaining the usual active and reactive terms while making explicit a cross-phase unbalance vector that captures antisymmetric interphase relations. In this way, apparent power is separated into intraphase and cross-phase contributions, and its norm is preserved under the power-invariant Fortescue transformation. The formulation is extended to three-phase four-wire systems by introducing equivalent coordinates that preserve the effective apparent-power norm for the chosen voltage reference. Only standard complex numbers and matrices are required. Numerical examples show operating conditions in which a non-negligible part of the apparent-power structure is associated with cross-phase unbalance and cannot be inferred from active and reactive power alone. The proposed formulation thus provides a compact phasor-based descriptor of unbalance that complements established apparent-power theories by making explicit a component that is not accessible from scalar apparent-power representations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adds a cross-product term to complex power so the unbalance between phases shows up explicitly as a vector while the total norm stays the same under Fortescue, but the four-wire version ties that vector to the chosen voltage reference.

read the letter

The core move is to write apparent power as the dot product of voltage and current phasors plus their cross product. The dot part recovers the usual active and reactive power; the cross part gives a vector that points to the antisymmetric interphase relations. They show that the Euclidean norm of this combined object is invariant under the power-invariant Fortescue transform, which keeps the formulation compatible with existing scalar definitions. Numerical cases illustrate operating points where the cross term is sizable and invisible from the scalars alone. That separation is the actual novelty and it is cleanly executed with only ordinary complex arithmetic and matrices. The four-wire extension introduces equivalent coordinates that preserve the effective norm for a chosen voltage reference. The abstract does not claim the cross-phase vector itself is invariant under a change of reference, and the stress-test concern is therefore on target: if the vector rotates or changes magnitude when you pick a different neutral point, it stops being an intrinsic descriptor of the system and becomes an artifact of the coordinate choice. The paper would need to demonstrate that the decomposition remains stable across common reference selections, or else qualify the claim more narrowly. This work is for power-systems engineers who already use phasors and Fortescue and want a compact way to flag cross-phase unbalance without abandoning classical apparent-power numbers. A reader looking for a modest, self-contained extension that stays inside standard tools will find it useful. The derivations look standard and the examples are concrete, so the paper is coherent on its own terms. I would send it to peer review; the central construction is straightforward and the examples are helpful, but the four-wire invariance question needs explicit checking before the vector can be treated as reference-independent.

Referee Report

1 major / 2 minor

Summary. The paper proposes a complex-vector power formulation for sinusoidal steady-state three-phase systems. It supplements the classical dot-product expression for complex power with the cross product of voltage and current phasors, retaining active and reactive terms while introducing an explicit cross-phase unbalance vector that captures antisymmetric interphase relations. Apparent power is thereby separated into intraphase and cross-phase contributions, with its norm preserved under the power-invariant Fortescue transformation. The formulation is extended to three-phase four-wire systems via equivalent coordinates that preserve the effective apparent-power norm for the chosen voltage reference. Only standard complex numbers and matrices are used, and numerical examples illustrate operating conditions where a non-negligible portion of apparent-power structure arises from cross-phase unbalance.

Significance. If the cross-phase unbalance vector proves to be reference-independent and physically interpretable while remaining consistent with scalar apparent-power norms, the work would supply a compact phasor-domain descriptor that makes explicit an unbalance component inaccessible from established scalar representations. Strengths include the use of standard operations, norm preservation under Fortescue transformation, and the separation of intra- versus cross-phase contributions without introducing free parameters.

major comments (1)

[Four-wire extension / equivalent coordinates] The four-wire extension (abstract and § on equivalent coordinates) states that equivalent coordinates preserve the effective apparent-power norm 'for the chosen voltage reference.' However, the manuscript does not demonstrate that the resulting cross-phase unbalance vector remains invariant under different reference choices (e.g., neutral point versus one phase). If the vector changes while the norm is held fixed, it encodes an arbitrary coordinate choice rather than an intrinsic property of antisymmetric interphase relations, weakening the central claim that the vector is a physically interpretable descriptor independent of established scalar definitions.

minor comments (2)

[Formulation] Clarify the precise definition of the cross-phase unbalance vector (e.g., its scaling and units) in the main text so that readers can directly compare it to classical unbalance factors.
[Numerical examples] The numerical examples would benefit from an explicit table comparing the proposed vector magnitude against standard scalar unbalance indices for the same operating points.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the single major comment below and have revised the manuscript to clarify the reference dependence in the four-wire case while preserving the core contribution.

read point-by-point responses

Referee: The four-wire extension (abstract and § on equivalent coordinates) states that equivalent coordinates preserve the effective apparent-power norm 'for the chosen voltage reference.' However, the manuscript does not demonstrate that the resulting cross-phase unbalance vector remains invariant under different reference choices (e.g., neutral point versus one phase). If the vector changes while the norm is held fixed, it encodes an arbitrary coordinate choice rather than an intrinsic property of antisymmetric interphase relations, weakening the central claim that the vector is a physically interpretable descriptor independent of established scalar definitions.

Authors: We acknowledge that the cross-phase unbalance vector depends on the chosen voltage reference, as the equivalent coordinates are explicitly constructed relative to it and the manuscript qualifies norm preservation as holding 'for the chosen voltage reference.' The manuscript does not claim or demonstrate full invariance under arbitrary reference changes such as neutral versus phase. To address this, we will revise the equivalent-coordinates section to include an explicit statement of reference dependence together with a short numerical illustration showing how vector components vary while the apparent-power norm remains fixed. This clarification aligns the four-wire treatment with the fact that scalar apparent-power definitions themselves can depend on reference choice in four-wire systems; the vector still supplies an explicit, coordinate-consistent descriptor of antisymmetric interphase relations for the selected reference. revision: yes

Circularity Check

0 steps flagged

No circularity: new formulation defined via standard vector operations without reduction to inputs or self-citations

full rationale

The paper introduces a complex-vector power by supplementing the classical dot-product expression with the cross product of voltage and current phasors, explicitly defining the cross-phase unbalance vector as the resulting antisymmetric term. This is a constructive proposal rather than a derivation that reduces to prior fitted values or self-referential definitions. It invokes the standard power-invariant Fortescue transformation (external to the paper) and extends to four-wire systems via equivalent coordinates chosen to preserve the apparent-power norm for a selected reference, without claiming reference-independence. No load-bearing steps rely on self-citations, ansatzes smuggled from prior work, or renaming of known results; numerical examples serve only to illustrate the defined quantities. The derivation chain is self-contained as an original representation consistent with but not tautological to scalar apparent-power norms.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The formulation rests on domain assumptions about sinusoidal steady-state operation and the properties of the Fortescue transformation; the cross-phase unbalance vector is derived from vector algebra rather than introduced as an independent postulate.

axioms (2)

domain assumption Operation is restricted to sinusoidal steady-state conditions
Explicitly stated in the abstract as the regime for the formulation.
standard math The Fortescue transformation is power-invariant
Invoked to claim that the norm of apparent power is preserved.

invented entities (1)

cross-phase unbalance vector no independent evidence
purpose: To capture antisymmetric interphase relations not visible in scalar active and reactive power
Derived directly from the cross product of voltage and current phasors; no independent falsifiable prediction outside the formulation is provided in the abstract.

pith-pipeline@v0.9.0 · 5524 in / 1528 out tokens · 49088 ms · 2026-05-07T15:01:01.345469+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

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C. L. Fortescue, “Method of symmetrical co-ordinates applied to the solution of polyphase networks,”Proc. Amer. Inst. Electr. Eng., vol. 37, no. 6, pp. 629–716, Dec. 1918

work page 1918
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work page 1920
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A. E. Emanuel, “On the definition of power factor and apparent power in unbalanced polyphase circuits with sinusoidal voltage and currents,”IEEE Trans. Power Deliv., vol. 8, no. 3, pp. 841–852, 1993, doi: 10.1109/61.252612

work page doi:10.1109/61.252612 1993
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Phasor total unbalance power: Formulation and some properties,

V . Le´on-Mart´ınez, J. Montan˜ana-Romeu, A. Cazorla-Navarro, J. Giner- Garcia, and J. Roger-Folch, “Phasor total unbalance power: Formulation and some properties,” inProc. IEEE Instrum. Meas. Technol. Conf. (IMTC), 2007, pp. 1–6

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P. A. Blasco, R. Montoya-Mira, J. M. Diez, and R. Montoya, “An alternate representation of the vector of apparent power and unbalanced power in three-phase electrical systems,”Appl. Sci., vol. 10, no. 11, Art. no. 3756, Jun. 2020, doi: 10.3390/app10113756

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F. G. Montoya, J. Ventura, A. Alcayde, and F. M. Arrabal-Campos, “Geometric power theory in the frequency domain for non-sinusoidal and unbalanced multiphase AC systems,”Int. J. Electr. Power Energy Syst., vol. 165, Art. no. 110454, 2025, doi: 10.1016/j.ijepes.2025.110454

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J. L. Willems and J. A. Ghijselen, “The relation between the generalized apparent power and the voltage reference,” inProc. 6th Int. Workshop on Power Definitions and Measurements under Nonsinusoidal Conditions, Milan, Italy, Oct. 13–15, 2003, pp. 37–45

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Effects of the selected point of voltage reference on the apparent power measurement in three-phase Star Systems,

V . Le´on-Mart´ınez, J. Monta˜nana-Romeu, E. Pe ˜nalvo-L´opez, and C. M. ´Alvarez-Bel, “Effects of the selected point of voltage reference on the apparent power measurement in three-phase Star Systems,”Appl. Sci., vol. 10, no. 3, Art. no. 1036, 2020, doi: 10.3390/app10031036

work page doi:10.3390/app10031036 2020

[1] [1]

Method of symmetrical co-ordinates applied to the solution of polyphase networks,

C. L. Fortescue, “Method of symmetrical co-ordinates applied to the solution of polyphase networks,”Proc. Amer. Inst. Electr. Eng., vol. 37, no. 6, pp. 629–716, Dec. 1918

work page 1918

[2] [2]

Polyphase power representation by means of symmetri- cal coordinates,

C. L. Fortescue, “Polyphase power representation by means of symmetri- cal coordinates,”Trans. Amer. Inst. Electr. Eng., vol. 39, pp. 1481–1484, 1920

work page 1920

[3] [3]

The FBD-method, a generally applicable tool for analyzing power relations,

M. Depenbrock, “The FBD-method, a generally applicable tool for analyzing power relations,”IEEE Trans. Power Syst., vol. 8, no. 2, pp. 381–387, 1993

work page 1993

[4] [4]

Currents’ physical components (CPC) in circuits with nonsinusoidal voltages and currents. Part 2: Three-phase three-wire linear circuits,

L. S. Czarnecki, “Currents’ physical components (CPC) in circuits with nonsinusoidal voltages and currents. Part 2: Three-phase three-wire linear circuits,”Electr. Power Qual. Util. J., vol. 12, no. 1, pp. 3–13, 2006

work page 2006

[5] [5]

On the definition of power factor and apparent power in unbalanced polyphase circuits with sinusoidal voltage and currents,

A. E. Emanuel, “On the definition of power factor and apparent power in unbalanced polyphase circuits with sinusoidal voltage and currents,”IEEE Trans. Power Deliv., vol. 8, no. 3, pp. 841–852, 1993, doi: 10.1109/61.252612

work page doi:10.1109/61.252612 1993

[6] [6]

The Buchholz-Goodhue apparent power definition: The practical approach for nonsinusoidal and unbalanced systems,

A. E. Emanuel, “The Buchholz-Goodhue apparent power definition: The practical approach for nonsinusoidal and unbalanced systems,”IEEE Trans. Power Deliv., vol. 13, no. 2, pp. 344–348, 1998

work page 1998

[7] [7]

Apparent power definitions for three-phase systems,

A. E. Emanuel, “Apparent power definitions for three-phase systems,” IEEE Trans. Power Deliv., vol. 14, no. 3, pp. 767–772, 1999

work page 1999

[8] [8]

1459–2010, Mar

IEEE Standard Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions, IEEE Std. 1459–2010, Mar. 2010

work page 2010

[9] [9]

Phasor total unbalance power: Formulation and some properties,

V . Le´on-Mart´ınez, J. Montan˜ana-Romeu, A. Cazorla-Navarro, J. Giner- Garcia, and J. Roger-Folch, “Phasor total unbalance power: Formulation and some properties,” inProc. IEEE Instrum. Meas. Technol. Conf. (IMTC), 2007, pp. 1–6

work page 2007

[10] [10]

Formulation of phasor unbalance power: Application to sinusoidal power systems,

J. M. Diez, P. A. Blasco, and R. Montoya, “Formulation of phasor unbalance power: Application to sinusoidal power systems,”IET Gener. Transm. Distrib., vol. 10, no. 16, pp. 4178–4186, 2016

work page 2016

[11] [11]

Formulations for the apparent and unbalanced power vectors in three-phase sinusoidal systems,

V . Le´on-Mart´ınez and J. Montan ˜ana-Romeu, “Formulations for the apparent and unbalanced power vectors in three-phase sinusoidal systems,” Electr. Power Syst. Res., vol. 160, pp. 37–43, 2018

work page 2018

[12] [12]

An alternate representation of the vector of apparent power and unbalanced power in three-phase electrical systems,

P. A. Blasco, R. Montoya-Mira, J. M. Diez, and R. Montoya, “An alternate representation of the vector of apparent power and unbalanced power in three-phase electrical systems,”Appl. Sci., vol. 10, no. 11, Art. no. 3756, Jun. 2020, doi: 10.3390/app10113756

work page doi:10.3390/app10113756 2020

[13] [13]

Instantaneous power quantities in polyphase systems—A geometric algebra approach,

H. Lev-Ari and A. M. Stankovi ´c, “Instantaneous power quantities in polyphase systems—A geometric algebra approach,” inProc. IEEE Energy Convers. Congr. Expo. (ECCE), 2009, pp. 592–596

work page 2009

[14] [14]

Geometric power theory in the frequency domain for non-sinusoidal and unbalanced multiphase AC systems,

F. G. Montoya, J. Ventura, A. Alcayde, and F. M. Arrabal-Campos, “Geometric power theory in the frequency domain for non-sinusoidal and unbalanced multiphase AC systems,”Int. J. Electr. Power Energy Syst., vol. 165, Art. no. 110454, 2025, doi: 10.1016/j.ijepes.2025.110454

work page doi:10.1016/j.ijepes.2025.110454 2025

[15] [15]

Generalized instantaneous reactive power theory for three-phase power systems,

F. Z. Peng and J.-S. Lai, “Generalized instantaneous reactive power theory for three-phase power systems,”IEEE Trans. Instrum. Meas., vol. 45, no. 1, pp. 293–297, 1996

work page 1996

[16] [16]

Symbolic representation of general alternating waves and of double frequency vector products,

C. P. Steinmetz, “Symbolic representation of general alternating waves and of double frequency vector products,”AIEE Trans., vol. XVI, pp. 269– 296, 1899

work page

[17] [17]

A. E. Emanuel,Power Definitions and the Physical Mechanism of Power Flow. Hoboken, NJ, USA: Wiley–IEEE Press, 2011

work page 2011

[18] [18]

A. I. Petroianu,Bridging Circuits and Fields: Foundational Questions in Power Theory. Boca Raton, FL, USA: CRC Press, 2021

work page 2021

[19] [19]

The relation between the generalized apparent power and the voltage reference,

J. L. Willems and J. A. Ghijselen, “The relation between the generalized apparent power and the voltage reference,” inProc. 6th Int. Workshop on Power Definitions and Measurements under Nonsinusoidal Conditions, Milan, Italy, Oct. 13–15, 2003, pp. 37–45

work page 2003

[20] [20]

Effects of the selected point of voltage reference on the apparent power measurement in three-phase Star Systems,

V . Le´on-Mart´ınez, J. Monta˜nana-Romeu, E. Pe ˜nalvo-L´opez, and C. M. ´Alvarez-Bel, “Effects of the selected point of voltage reference on the apparent power measurement in three-phase Star Systems,”Appl. Sci., vol. 10, no. 3, Art. no. 1036, 2020, doi: 10.3390/app10031036

work page doi:10.3390/app10031036 2020