arXiv subjects

Francisco G. Montoya

Publications and source records attributed to Francisco G. Montoya.

13 recordsLinked to original sources

arXiv2026

Multivariable Geometric Laplace Transform and Fault Detection in Distributed-Converter Lines

Monitoring a DC line with many distributed power converters is a genuinely spatio-temporal problem: the information about a localized fault travels along the whole conductor and reaches a few measurement points mixed with the dynamics of the line itself. This paper develops a two-dimensional geometric Laplace transform (t,x) -> (s_t,s_x) over a commutative subalgebra of the geometric algebra Cl(4,0), isomorphic to Segre's bicomplex numbers, in which two bivectors B_t and B_x act as independent imaginary units for the temporal and the spatial phase. Because the two phases live in algebraically distinguishable planes, a fault at position x_f leaves a transformed residual that factorizes as F_f(s_t) e^{-s_x x_f}: its temporal nature stays in the first factor and its location can be read as a geometric argument of the second. On this representation we build a transmission-line model of the converter line and its space-time dispersion relation, a distributed control by admittance shaping, including an exact treatment of discrete converter sites (spatial sampling, aliasing, and a per-converter droop realization that is exact on the sub-Nyquist band), and a fault diagnosis chain that detects, localizes and classifies injection-loss, shunt, sensor and local-controller faults, extends to multiple simultaneous faults with automatic order selection, and distinguishes the outage of a plant from a cable defect. As an integral object the transform is known in bicomplex analysis, and with a single independent variable it reduces to the complex Laplace transform; the contribution lies in its geometric embedding and in its operational use for fault diagnosis in distributed-converter networks. All results are reproduced by an accompanying open implementation.

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arXiv2026

Can a Dynamic Internal Field Govern a Transformer's Cognition? Certifiability, not Superiority, in Homeostatic Compute Control

An intelligent system does not merely reason: it governs its own reasoning - how much to compute, when to stop, which module to activate. Can that role be played by a dynamic internal field - a low-dimensional homeostatic state with explicit physics and certified stability - that modulates cognition without performing it? Ours is a field on the module graph governed by a family of PDEs on the graph Laplacian, advancing with an adaptive-depth reasoner. We certify the stability of the integrator of the whole family - an integrator certificate, not a closed-loop one. New, and proved here: a discrete Schur-Cohn criterion for Verlet with velocity coupling, necessary and sufficient per latent root, with no commutation hypothesis. The answer is threefold: substance no, structure only in part, certifiability yes. The type of the field's physics is irrelevant for accuracy: wave, diffusion, gated mixtures and a 2D Navier-Stokes substrate tie. A twenty-seed preregistered deconfounding campaign bounds the structural claim: at equalized caps the second-order effect is strong in one family (+0.087 [+0.042, +0.132], t=4.0) but is not detected in the other (+0.014 [-0.013, +0.040], n.s.), so part of the original contrast was capacity, not order; and a matched-interface GRU is indistinguishable in the first and nominally exceeds the field in the second (-0.035 [-0.067, -0.002]). What distinguishes the field is not capability but that its one-step operator admits an exact runtime stability check - a difference of kind, not of existence: learned recurrences carry certificates too, sufficient and conservative ones. A kill-gate with a positive control finds no evidence for the field as evidence accumulator (Delta AUC +0.0007 [-0.0065, +0.0079] vs a 0.03 threshold). A dynamic internal field is a viable, certifiable compute governor, but not an enhancer of cognition: it modulates, it does not think.

cs.AI↗

arXiv2026

Where Cognition Lives: Dissecting Emergent from Computed Function in a Minimal Complete Cognitive Architecture

A cognitive architecture is more than the module that reasons: it must also decide how long to think and what deserves the effort. We built a minimal but complete system - a recurrent reasoner with adaptive halting, a homeostatic control field, and a value module - and asked of each part: does this function emerge from gradient descent, or must it be computed? Competence emerges. Stopping appears to emerge too, and to be worth more than everything decidable in advance, but that appearance is instrumentation: payoff at matched mean compute climbs from 0.467 (uniform) through 0.546 (difficulty) to 0.698 (ex-ante value), and the further climb to 0.921 (posterior self-observation) does not survive audit. PonderNet-style halting returns a halting-weighted mixture of hidden states while forced-depth baselines return one, and the language head is trained on the mixture alone; equalizing the readout annihilates the apparent advantage of native execution (residual +0.000 [0.000, 0.000]). Value does not emerge: trained couplings capture zero of a payoff an explicit allocator captures completely (+0.151, routing correlation +0.79), so the second-order decisions that pay must be computed, at least where value is orthogonal to content, as here by construction. On a frozen LLM actuator the same instruments show self-consistency voting to be a measured bound (+0.0236 [+0.0150, +0.0326]) and inter-sample agreement nearly worthless as a stopping signal, its mass concentrating on wrong answers. Every null we assert carries a mechanism and a positive control, and the protocol is part of the contribution. Executing our own falsifiable prediction, value under commitment pays +0.1312 [+0.1124, +0.1502] in a cliff-cost family, some seven times the smooth-family estimate - not because the cliff shifts information ex ante, but because it multiplies the attainable range fivefold (5.1x [3.4, 8.2]).

cs.AI↗

arXiv2026

Geometric Time-Domain Identification of Three-Phase Load Equivalents from Terminal Measurements

This paper presents a geometric time-domain method for identifying three-phase load equivalents from instantaneous voltage and current measurements at the point of common coupling. Measured waveforms are interpreted as trajectories in Euclidean signal spaces, and load-equivalent parameters are recovered from the geometry of those trajectories. The method extends a previously published single-phase geometric identification formulation to three- and four-wire systems and places special emphasis on the three-wire case, where no neutral voltage is measured and the terminal data must satisfy coupled Kirchhoff constraints. The main advance over the earlier analytical formulation is a sampled-data implementation based on local time windows, normalized matrix equations, harmonic-projection derivative and primitive coordinates, explicit geometric identifiability tests, passivity constraints, and energy/Kirchhoff residuals. The method does not force a model when the measured trajectory lacks enough information; instead, it reports low-rank or ill-conditioned windows as low-confidence evidence. Numerical simulations with clean data, measurement noise, window-length sweeps, and sensor delay show that the method accurately identifies informative three-phase trajectories and exposes structurally degenerate cases such as pure single-frequency excitation for higher-order three-wire models. For a given admissible topology the identified circuit closes the instantaneous terminal energy balance of the measured load over the analysis window.

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arXiv2026

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

This paper develops a branch-level energy-localization framework for three-phase loads. The instantaneous terminal power of an admissible lumped equivalent is decomposed uniquely as Joule dissipation plus magnetic and electric stored-energy rates, branch by branch. Three formal results are established: a Branch-Level Localization Theorem (uniqueness given an admissible topology); a Topology-Indeterminacy Theorem (multiple admissible topologies reproduce identical terminal data with distinct localizations); and a Generalized Energetic Duality Theorem that organizes classical electrical dualities (Norton-Thevenin, series--parallel, L vs C, R vs G) as restrictions to Linear Time Invariant (LTI) sinusoidal regimes of a single time-domain principle in which constant-parameter equivalence is replaced by time-varying parameters. The framework is exercised on six test cases including the de Leon--Cohen open-phase paradox, switched-resistive loads, three-wire delta-versus-wye-virtual indeterminacy, fluctuating-phase loads, and a four-wire nonlinear load with hysteretic, linear, and switched branches. The framework is positioned as complementary to IEEE Std. 1459, CPC, instantaneous p-q, and Fryze-Buchholz-Depenbrock: each answers a different question, and the apparent paradoxes vanish once the question is posed precisely.

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arXiv2025

Beyond Phasors: Solving Non-Sinusoidal Electrical Circuits using Geometry

Classical phasor analysis is fundamentally limited to sinusoidal single-frequency conditions, which poses challenges when working in the presence of harmonics. Furthermore, the conventional solution, which consists of decomposing signals using Fourier series and applying superposition, is a fragmented process that does not provide a unified solution in the frequency domain. This paper overcomes this limitation by introducing a complete and direct approach for multi-harmonic AC circuits using Geometric Algebra (GA). In this way, all non-sinusoidal voltage and current waveforms are represented as simple vectors in a $2N$-dimensional Euclidean space. The relationship between these vectors is characterized by a single and unified geometric transformation termed the \textit{rotoflex}. This operator elevates the concept of impedance from a set of complex numbers per frequency to a single multivector that holistically captures the circuit response, while unifying the magnitude scale (flextance) and phase rotation (rotance) across all harmonics. Thus, this work establishes GA as a structurally unified and efficient alternative to phasor analysis, providing a more rigorous foundation for electrical circuit analysis. The methodology is validated through case studies that demonstrate perfect numerical consistency with traditional methods and superior performance.

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arXiv2025

Understanding the Geometry of Faulted Power Systems under High Penetration of Inverter-Based Resources via Ellipse Fitting and Geometric Algebra

Power systems with high penetration of inverter-based resources (IBR) present significant challenges for conventional protection schemes, with traditional distance protection methods failing to detect line-to-line faults during asymmetric conditions. This paper presents a methodology for electrical fault detection and classification using ellipse fitting and geometric algebra applied to voltage and current space curves. The approach characterizes electrical faults by fitting ellipses to voltage vector data, enabling fault detection with only a quarter-cycle. The method employs bivector components for line-to-ground fault classification, while ellipse parameters identify line-to-line and three-phase faults. The geometric representation preserves voltage or current curve shapes in three-dimensional space, overcoming Clarke transform limitations when zero-sequence components are present. Validation using simulations and laboratory experiments demonstrates accurate fault identification and magnitude estimation, providing enhanced power system protection capabilities.

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arXiv2025

General Reference Frame Identification and Transformation in Unbalanced Power Systems

Coordinate transformations provide dimensional reduction benefits across power system analysis, electric machine modeling, and power electronic converter control. This paper introduces a novel transformation based on Geometric Algebra that directly identifies the plane containing unbalanced quantity loci through bivector analysis. The method provides a direct transformation valid for any degree of unbalance in $n$-phase, $(n+1)$-wire sinusoidal systems, requiring only two voltage or current measurements at different time instants. Through pure geometric reasoning, we demonstrate that our approach generalizes existing techniques while extending naturally to multi-dimensional systems. Experimental validation using real-time digital simulation and physical laboratory testing confirms the method's effectiveness under realistic conditions. Power electronics converter control implementation demonstrates significant practical advantages, eliminating zero component oscillations present in Clarke transformation under unbalanced conditions and enabling more effective control architectures. The combination of computational efficiency, robustness, and practical applicability represents a significant advancement for power system control applications.

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arXiv2025

Experimental Verification of a Time-Domain Load Identification Method for Single-Phase Circuits

This paper presents experimental validation of a time-domain load parameter determination method for single-phase circuits. The verification is performed in a state-of-the-art smart grid laboratory equipped with power hardware and real-time emulators. The proposed method enables the identification of circuit parameters using only instantaneous voltage and current measurements at the point of common coupling. The experimental setup includes a range of test cases covering linear and non-sinusoidal single-phase conditions. Voltage and current waveforms are acquired, preprocessed, and used to calculate the relevant circuit parameters. The experimental results demonstrate a high degree of accuracy and robustness, with minimal percentage errors across all test cases. The identified parameters show excellent agreement with the theoretical expectations, confirming the validity and applicability of the proposed method to identify the load of single-phase systems. This validation highlights the potential of the method for improved monitoring, control, and protection of smart grids, paving the way for future extensions to three-phase systems and real-time implementations.

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arXiv2022

A Geometric Procedure for Computing Differential Characteristics of Multi-phase Electrical Signals using Geometric Algebra

This paper presents exploratory investigations on the concept of generalized geometrical frequency in electrical systems with an arbitrary number of phases by using Geometric Algebra and Differential Geometry. By using the concept of Darboux bivector it is possible to find a bivector that encodes the invariant geometrical properties of a spatial curve named electrical curve. It is shown how the traditional concept of instantaneous frequency in power networks can be intimately linked to the Darboux bivector. Several examples are used to illustrate the findings of this work.

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arXiv2020

A new electromechanical analogy approach based on electrostatic coupling for vertical dynamic analysis of planar vehicle models

Analogies between mechanical and electrical systems have been developed and applied for almost a century, and they have proved their usefulness in the study of mechanical and electrical systems. The development of new elements such as the inerter or the memristor is a clear example. However, new applications and possibilities of using these analogues still remain to be explored. In this work, the electrical analogues of different vehicle models are presented. A new and not previously reported analogy between inertial coupling and electrostatic capacitive coupling is found and described. Several examples are provided to highlight the benefits of this analogy. Well-known mechanical systems like the half-car or three three-axle vehicle models are discussed and some numerical results are presented. To the best of the author's knowledge, such systems were never dealt with by using a full electromechanical analogy. The mechanical equations are also derived and compared with those of the electrical domain for harmonic steady state analysis.

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arXiv2020

A new approach to single-phase systems under sinusoidal and non-sinusoidal supply using geometric algebra

The aim of this work is to present major upgrades to existing power theories based on geometric algebra for single-phase circuits in the frequency domain. It also embodies an interesting new approach with respect to traditionally accepted power theories, revisiting power concepts in both sinusoidal and non-sinusoidal systems with linear and nonlinear loads for a proper identification of its components to achieve passive compensation of true non-active current. Moreover, it outlines traditional power theories based on the apparent power $S$ and confirms that these should definitively be reconsidered. It is evidenced that traditional proposals based on the concepts of Budeanu, Fryze and others fail to identify the interactions between voltage and current harmonics. Based on the initial work of Castro-Núñez and others, new aspects not previously included are detailed, modified and reformulated. As a result, it is now possible to analyze non sinusoidal electrical circuits, establishing power balances that comply with the principle of energy conservation, and achieving optimal compensation scenarios with both passive and active elements in linear and non-linear loads.

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arXiv2020

How to overcome the limitations of p-q Theory: Geometric Algebra Power Theory to the rescue

This paper investigates the recent advances in Geometric Algebra-based power theory (GAPoT) and how this tool provides new insights to solve the flaws of one of the most widespread theory in the time domain, the Instantaneous Reactive Power theory (IRP) and its further enhancements. GAPoT can be applied to single-phase and multi-phase systems to obtain an optimal current decomposition under any distorted voltage source supply and load condition. This could be the case in microgrids or smart grids. Moreover, it is possible to define different strategies based on instantaneous or averaged quantities depending on whether the voltage supply conditions are sinusoidal and symmetrical or not. Several examples illustrate how GAPoT is able to overcome the limitations of IRP theory.

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