⚡ electrical engineering

Branch-Level Energy Localization in Three-Phase Loads: Resolving Indeterminacy in Time-Domain

This paper establishes a branch-level energy-localization framework for three-phase loads that uniquely decomposes instantaneous power into dissipation and storage rates, resolves indeterminacy issues arising from multiple admissible topologies, and unifies classical electrical dualities under a generalized time-domain energetic principle.

Original authors: Francisco G. Montoya, Francisco de Leon, Francisco M. Arrabal-Campos, Alfredo Alcayde

Published 2026-06-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Francisco G. Montoya, Francisco de Leon, Francisco M. Arrabal-Campos, Alfredo Alcayde

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are standing outside a large, complex house (the electrical load) with a single window. You can see the electricity flowing in and out through the wires (voltage and current), and you can calculate the total power being used. However, you cannot see inside the house.

For decades, engineers have tried to guess what's happening inside that house just by looking at the window. They've developed many different theories to explain the energy flow. Some say, "Ah, that fluctuation is 'reactive power'!" Others say, "No, that's 'distortion'!" The problem is, these different theories often tell conflicting stories about where the energy is going, even though they are all looking at the same data.

This paper proposes a new way to look at the problem. Instead of guessing from the outside, it suggests we need to build a virtual model of the house's internal wiring first. Once we have a plausible model of the internal "rooms" (branches), we can finally see exactly where the energy is being used, stored, or lost.

Here is a breakdown of the paper's main ideas using simple analogies:

1. The "Black Box" Problem

Think of a three-phase electrical system like a three-lane highway merging into a single toll booth. You can count the total cars (current) and the speed (voltage) at the booth. But inside the highway, the cars might be merging, splitting, or looping around in complex ways.

Old Theories: Try to guess the internal traffic patterns just by looking at the toll booth. They often disagree. One theory might say "Car A is stuck in a loop," while another says "Car A is speeding up," even though the total number of cars is the same.
This Paper's Approach: It says, "Let's first build a map of the internal roads that could exist." Once we agree on a map (a "topology"), we can trace every single car and say exactly what it is doing at every second.

2. The Three Big Discoveries

The authors established three main rules (theorems) that change how we understand this:

Rule 1: The "One Map, One Truth" Rule
If you agree on a specific internal map (e.g., "The house has three rooms connected in a triangle"), then the math gives you one unique answer for where the energy is going. You can calculate exactly how much heat is generated in the resistor (Joule dissipation) and how much energy is sloshing back and forth in the inductor or capacitor (stored energy). There is no ambiguity if the map is fixed.
Rule 2: The "Many Maps" Paradox
Here is the twist: You can build two completely different maps that look identical from the outside.
- Map A might look like a triangle of wires (Delta).
- Map B might look like a star shape (Wye) with a fake center point.
  Both maps produce the exact same voltage and current at the toll booth. However, inside Map A, the energy might be stored in one set of components, while inside Map B, it's stored in a different set.
  The Lesson: The "truth" about where energy is stored depends on which map you choose. The paper argues that this isn't a failure of physics; it's just that the data from the outside isn't enough to force a single choice. You have to pick the map that matches the physical reality of the device you are testing.
Rule 3: The "Universal Translator"
The paper shows that even though Map A and Map B look different, they are actually two sides of the same coin. You can mathematically translate the energy story from one map to the other. It's like translating a story from English to French; the words change, but the plot (the total energy flow) stays the same. This unifies old electrical rules (like Norton and Thevenin) into one big, flexible rule that works even when the electricity is messy and changing fast.

3. Solving the "Ghost" Mysteries

The paper uses six test cases to show how this new method solves old puzzles:

The "Ghost" Current: In one case, a resistor was only connected between two wires, leaving the third wire empty. Old theories claimed there was "active current" flowing in the empty third wire (a ghost!). This paper says, "No, if you build the right internal map, you see the current is zero there. The ghost was just a math trick."
The "Fake" Storage: In another case, a switch turned a resistor on and off rapidly. Old theories saw the jagged waves and claimed there was "reactive power" or "energy storage" happening. This paper says, "Look closer. There are no batteries or magnets inside. It's just a resistor getting hot and cooling down. The 'storage' was an illusion created by looking at the wrong coordinates."
The Hidden Exchange: In a balanced system, the total energy storage might look like it's zero because the ups and downs cancel out. But this paper shows that inside the house, energy is actually rushing back and forth between the three phases. The "total" view hides the local activity.

4. The Bottom Line

The paper doesn't say the old theories are "wrong." It says they answer different questions.

Old Theories are like a weather report: "It's windy and rainy today." They are great for billing, sizing equipment, and general control.
This New Framework is like a forensic investigation: "The rain hit the roof here, ran down the gutter there, and soaked the basement." It answers the specific question: "Exactly where is the energy being used or stored at this exact moment?"

By forcing engineers to define the internal structure (the topology) first, this framework removes the confusion and "paradoxes" that have plagued electrical engineering for years. It turns a blurry, conflicting picture into a sharp, clear one—provided you know which internal map you are looking at.

Technical Summary: Branch-Level Energy Localization in Three-Phase Loads

Problem Statement
Power theory has long faced a tension between operational questions (which numerical quantities are useful for metering and control) and physical questions (which quantities correspond to actual energy flow, dissipation, or storage). While various theories (IEEE Std. 1459, CPC, instantaneous $p-q$ , FBD) provide operational indices, they often yield mutually incompatible interpretations of where energy is dissipated or stored, leading to "paradoxes." In three-phase systems, this difficulty is exacerbated by structural constraints: in three-wire connections, the neutral point is inaccessible, line currents are linear combinations of internal branch currents, and multiple admissible lumped topologies can reproduce identical terminal observables. Aggregation across phases can further conceal local energy exchanges, causing per-phase storage variations to vanish at the terminal boundary while remaining physically present.

Methodology
The paper proposes a branch-level energy-localization framework for three-phase loads operating in the quasi-static lumped regime. The methodology relies on the following pillars:

Energy-Balance Axiom: The framework treats Tellegen's theorem as the working axiom, reducing the Poynting integral form to a circuit-level statement. Instantaneous terminal power ( $p_{term} = \mathbf{v}^\top \mathbf{i}$ ) is equated to the algebraic sum of branch contributions: Joule dissipation ( $p_R$ ) and the time rates of change of magnetic ( $W'_L$ ) and electric ( $W'_C$ ) stored energy.
Admissible Equivalents: The framework requires an identified lumped equivalent topology (e.g., delta-parallel or wye-series) with parameters determined via time-domain identification machinery (invoked from companion work).
Decomposition: Once a topology and parameters are identified, the instantaneous terminal power is decomposed uniquely into branch-level components:
$p_{term} = \sum_{r \in R} R_r i_r^2 + \sum_{\ell \in L} L_\ell i_\ell i'_\ell + \sum_{c \in C} C_c v_c v'_c$
This decomposition is distinct from defining new power components; it is an equality of scalar functions derived from constitutive relations.

Key Contributions
The paper establishes three formal results and validates them through six test cases:

Branch-Level Localization Theorem (Theorem 1): For a fixed admissible topology and identified parameters, the decomposition of terminal power into Joule dissipation and stored-energy rates is unique.
Topology-Indeterminacy Theorem (Theorem 2): Multiple admissible topologies (e.g., a delta-parallel equivalent vs. a wye-series equivalent with a virtual neutral) can reproduce identical terminal data ( $v, i$ ) while assigning distinct energy localizations to their internal branches. This establishes that branch-level localization is a property of the model, not solely of the data.
Generalized Energetic Duality Theorem (Theorem 3): Classical electrical dualities (Norton-Thevenin, series-parallel, $L \leftrightarrow C$ ) are shown to be restrictions of a single time-domain principle. For any branch terminal pair, every admissible topology family can be parameterized (using time-varying parameters) to reproduce the same branch energy decomposition. Classical dualities emerge only when parameters degenerate to constants under LTI sinusoidal regimes.

Results and Test Cases
The framework was exercised on six synthetic test cases, demonstrating its ability to resolve classical paradoxes:

Balanced RL Load: While aggregated terminal power and total magnetic energy are constant, the framework reveals that per-phase magnetic energy oscillates and storage rates exchange cyclically between phases. This exposes how aggregated invariance hides internal per-phase exchange.
Single-Branch Resistive Load (Open Phase): In a load with only one conducting branch, classical theories (CPC, $p-q$ ) generate "ghost" currents or non-zero reactive components in open phases. The branch-level framework correctly identifies zero storage and zero dissipation in the open branches, attributing all energy to the single active resistor.
Switched-Resistive Nonlinear Load: For a purely resistive load with TRIAC switching, classical theories report significant distortion, scattered, void, and reactive power components. The framework identifies these as coordinate-domain artifacts, showing that $W'_L$ and $W'_C$ are identically zero, and all energy is Joule dissipation via time-varying conductance.
Three-Wire Topology Indeterminacy: A numerical instance confirms that a delta-parallel model and a wye-virtual series model yield identical terminal power but distinct internal dissipation and storage profiles. Both are admissible, but the choice depends on the physical nature of the load.
Fluctuating-Phase Load: In a balanced system where fundamental reactive coordinates vanish, the framework detects manifest per-phase storage exchange that is invisible to fixed-coordinate projections.
Unbalanced Four-Wire Nonlinear Load: The framework successfully localizes hysteresis loss in a hysteretic inductor (via an effective time-varying conductance), linear RLC behavior, and switched resistive behavior within a single section, providing a unified decomposition for structurally heterogeneous branches.

Significance and Positioning
The paper positions this framework as complementary to existing standards and theories (IEEE Std. 1459, CPC, $p-q$ , FBD), not as a replacement.

Resolution of Paradoxes: The paper argues that classical paradoxes are not failures of physics but symptoms of describing systems using terminal-only coordinates. By identifying a physically admissible lumped equivalent, the framework returns the location, sign, and time profile of every storage and dissipation contribution, causing apparent paradoxes to vanish.
Operational vs. Physical: Classical theories are acknowledged as valid for operational tasks (metering, compensation design, converter control). However, they answer different questions than the branch-level framework, which specifically asks: "Where, inside the load, is energy dissipated or stored at any instant?"
Modeling Decision: The paper emphasizes that in three-wire systems without a measured neutral, the selection of a topology (delta vs. wye) is a modeling decision. The framework provides the tools to quantify the consequences of this decision, ensuring that claims of physical interpretation are accompanied by the specific topology used.

In summary, the paper provides a rigorous circuit-level method to map instantaneous terminal power to internal physical processes, clarifying that energy localization is contingent upon the chosen admissible topology and that classical non-active power components often represent coordinate artifacts rather than physical storage processes.

1. The "Black Box" Problem

2. The Three Big Discoveries

3. Solving the "Ghost" Mysteries

4. The Bottom Line

Technical Summary: Branch-Level Energy Localization in Three-Phase Loads

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