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

Francisco Manuel Arrabal-Campos, Francisco G. Montoya, Santiago Sánchez-Acevedo, Raymundo E. Torres-Olguin, Alfredo Alcayde

arXiv:2609.02431v1eess.SYeess.SP

TL;DR

Distributed-converter DC lines require fault monitoring across time and space, although measurements are available only at limited points. The paper develops a geometric two-variable Laplace framework with separate temporal and spatial bivector planes, then uses it for model-based detection, localization, and classification. The reported framework includes exact discrete-site treatment and reproducible validation, within stated modeling and observability limits.

  • Problem

    Fault information propagates through distributed-converter DC lines and becomes mixed at limited measurement points, complicating detection, localization, and classification.

  • Method

    The paper embeds a two-dimensional Laplace transform in a commutative Cl(4,0) subalgebra, factorizing localized residuals as F_f(s_t)e^{-s_x x_f} so temporal behavior and position remain separable.

  • Results

    The framework integrates fault detection, localization, and classification with exact discrete-converter sampling treatment and reproducible numerical experiments within the model assumptions.

  • Takeaways & Limitations

    The geometric representation provides an operationally unified description of space-time coupling and interpretable fault signatures for distributed-converter networks.

  • Takeaways & Limitations

    Diagnosis depends on spatial observability, sensor number and placement, measurement noise, boundary conditions, and the incremental model; internal converter dynamics are abstracted away.

Abstract

from arXiv · show

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.

1 Introduction and motivation

The paper frames distributed-converter DC-line monitoring as a spatio-temporal fault-diagnosis problem and proposes a geometric transform whose separate temporal and spatial phases support localization. Its novelty is the operational use of this embedding, rather than the underlying bicomplex integral itself.

  • Applied context: DC lines with distributed converters: Distributed converters interact through line dynamics, so voltage and current depend jointly on time and position.
  • Applied context: DC lines with distributed converters: Monitoring must detect, locate, and classify faults whose propagated information is mixed at limited measurement points.
  • Scope and assumptions: The model assumes small-signal incremental dynamics, a spatially continuous line, and converters represented as distributed injections.
  • Geometric transform: The transform embeds temporal and spatial phases in distinguishable bivector planes, labeling temporal information by B_t and spatial information by B_x.
  • Fault diagnosis: A localized fault factors as F_f(s_t)e^{-s_x x_f}, allowing its position to be read through the geometric spatial argument Arg_Bx.
  • Contributions: The framework combines fault detection, localization, and classification with a distributed-converter line model, exact discrete-site sampling theory, and reproducible numerical validation.

2 Minimal geometric algebra

The paper constructs a four-dimensional commutative subalgebra of Cl(4,0) from two commuting bivectors and identifies it with Segre’s bicomplex numbers. Idempotent decomposition reduces algebraic operations to componentwise ordinary complex operations, while zero divisors and a nonmultiplicative norm constrain analysis.

  • Algebra definition: The algebra A is the four-dimensional commutative subalgebra generated by 1, B_t, B_x, and B_tx within Cl(4,0).
  • Bicomplex structure: Mapping B_t to i and B_x to j gives an R-algebra isomorphism between A and Segre’s bicomplex numbers.
  • Idempotent diagonalization: Every z in A decomposes uniquely as z=ζ_1e_1+ζ_2e_2, with ζ_1 and ζ_2 ordinary complex numbers in the B_t plane.
  • Componentwise operations: Products, inverses, and exponentials act componentwise in the idempotent representation, provided both complex components are nonzero for inversion.
  • Algebraic limitations: A contains zero divisors, so noninvertibility occurs on the component conditions ζ_1=0 or ζ_2=0 and affects later pole and residue analysis.
  • Algebraic limitations: The auxiliary Euclidean component norm is not multiplicative, so it supports convergence and magnitude control but cannot serve as an algebraic absolute value.

3 Two-dimensional geometric Laplace transform and operational properties

The two-dimensional geometric Laplace transform is defined on the commutative bivector subalgebra with separate temporal and spatial spectral variables. Its idempotent representation reduces convergence and operations to classical complex components, while the commuting-plane structure enables the later space-time and fault-diagnosis framework.

  • Transform definition: The transform maps functions of time and space into A using s_t=ρ_t+B_tω and s_x=ρ_x+B_xk.
  • Kernel factorization: Commutativity makes the kernel factor into temporal and spatial exponentials whose phases rotate in independent bivector planes.
  • Idempotent representation: Idempotent decomposition turns the kernel into two ordinary complex exponentials, reducing convergence analysis to two classical complex Laplace transforms.
  • Norm and convergence: Because the component norm is not submultiplicative, convergence proofs use an exact idempotent-basis product bound rather than a naive norm-product estimate.
  • Region of convergence: For exponential-order inputs, absolute convergence holds when ρ_t>σ_t and ρ_x>σ_x, with identical attenuation in both idempotent components.
  • Operational properties: Transforming partial derivatives produces algebraic multiplications together with initial-condition and boundary-condition transforms, as in the classical Laplace case.
  • Interpretation and scope: The separated temporal and spatial poles encode temporal dynamics and spatial propagation independently, supporting the geometric reading of fault position.
  • Interpretation and scope: The implemented theory is restricted to commuting bivectors; a noncommutative extension would require Baker–Campbell–Hausdorff corrections and remains future work.

4 Distributed converter-line model and dispersion relation

The paper models a distributed-converter DC line as a spatio-temporal system and uses a two-dimensional geometric Laplace transform to derive its transfer function and dispersion relation. It also gives an exact spatial-sampling analysis for discrete converter sites, including aliasing and controllability limits.

  • Discrete converter sites: The continuous injection approximation has a quantitative convergence rate, while uniform droop feedback is exact on the sub-Nyquist band.These results specify when the distributed description approximates or exactly represents discrete converter behavior.
  • Model assumptions: The line is linearized around an operating point and represented with uniform per-unit-length resistance, inductance, capacitance, and incremental conductance.The incremental conductance may be negative for constant-power-controlled converters.
  • Geometric transform: The geometric Laplace transform maps temporal and spatial derivatives to s_t and s_x, producing an algebraic system and distributed transfer function wherever the determinant is invertible.The transform uses distinct bivectors B_t and B_x for the temporal and spatial variables and assumes zero initial and boundary conditions.
  • Dispersion relation: The dispersion relation is defined by Δ(s_t, s_x) = 0, linking temporal and spatial variables to describe disturbance propagation along the line.In the ideal lossless case, k = √(ℓc)ω is linear, yielding a nondispersive line with constant phase velocity and no phase distortion.
  • Dispersion relation: For generic losses with rc + ℓg ≠ 0, restricting the spatial variable to s_x = B_xk does not generally produce an exact real dispersion root; spatial attenuation or idempotent decomposition is required.A real branch exists for all frequencies exactly on the Heaviside-type locus rc + ℓg = 0, where k^2 = ℓcω^2 − rg.
  • Discrete converter sites: Discrete converter sites create spatial sampling effects: sampled modes are 4P-periodic, modes n = 2Pj are invisible, and higher modes alias to canonical modes in {1, …, P}.At most the P lowest modes can be actuated from P converter sites, with additional uncontrollable modes when converters lie on mode nodes.

5 Distributed control via admittance shaping

The control law shapes the line’s effective shunt admittance, allowing closed-loop dispersion to be designed through the controller. Uniform discrete droop realizes the continuous virtual-conductance model exactly below the spatial Nyquist limit, while higher spatial complexity introduces locality, bandwidth, and implementation constraints.

  • Admittance shaping: The controller acts additively on shunt admittance, replacing Y(st) with the closed-loop admittance Yc(st,sx)=Y(st)+K(st,sx).The closed-loop dynamics are encoded in the zeros of the resulting determinant, so choosing K amounts to placing those zeros.
  • Virtual conductance: Proportional feedback K=kv increases the effective conductance from g to g+kv and shifts modal behavior toward greater damping.The effective shunt branch is strictly dissipative only when g+kv>0.
  • Discrete realization: Uniform midpoint-grid droop is exactly equivalent to the continuous virtual conductance for modes n≤P−1, while mode P receives doubled gain 2kv.For N≤P−1, the coupling matrix is C=kv diag(1,...,1); for N=P, its final diagonal entry is 2kv.
  • Discrete realization: For N>P, spatial sampling aliases modes and couples each mode to its alias partner with full diagonal strength.For example, Cn,2P−n=kv when the alias mode is included.
  • Implementation boundaries: Nonuniform converter placement preserves convergence toward the continuous loop as P grows but loses exact diagonalization, while high-order spatial control reduces locality.Neighbor-based realization of spatial derivatives also limits effective spatial bandwidth, and saturation or communication delays can compromise stability if unaccounted for.

6 Localized faults: detection, localization, and classification

The fault-diagnosis framework models localized faults as anomalous residual sources and separates the residual from the propagated voltage response. This distinction supports localization while preventing sensor anomalies from being confused with physical faults.

  • Fault-diagnosis problem: A localized fault appears as a space-time singularity whose transformed residual contains a spatial rotor encoding its position.The framework is model-based and targets detection, localization, and classification along the distributed converter line.
  • Residual versus response: The balance residual represents the equivalent anomalous source, whereas the observed voltage is its propagated response through the line.These are distinct diagnostic objects rather than interchangeable measurements.
  • Residual versus response: Voltage-only diagnosis requires removing the line transfer function through prewhitening or modal deconvolution before using the residual signature.Without this separation, a sensor anomaly can be mistaken for a localized physical fault.

6.1 Distributed residual and the signature of a localized fault

The paper defines a distributed balance residual that vanishes in healthy operation and equals the anomalous fault injection otherwise. A point fault’s transform factorizes into temporal signature and spatial location components.

  • Distributed residual: The balance residual R(t,x)=∂xi+c∂tv+gv+K[v]−ru is zero under ideal healthy operation and is thresholded statistically under noise or model error.The residual is constructed from the distributed model, controller, and known excitation.
  • Distributed residual: A physical fault enters the line equation as an anomalous injection uf(t,x), making the balance residual the equivalent fault source.This identity is the basis for localizing faults from R.
  • Localized fault model: An additive localized fault is modeled at xf with temporal signature ff(t) and a spatial Dirac delta.The temporal signature has units of incremental current; local shunts instead represent multiplicative local-admittance disturbances.
  • Localized fault signature: The transformed fault signature separates temporal behavior Ff(st) from position through the spatial rotor e−Bxkxf on the oscillatory spatial axis.The temporal part appears in Ff(st), while the position is read from the spatial argument; this applies to the residual, not directly to measured voltage.

6.2 From the residual to the measured signals

Because sensors observe propagated voltages rather than the balance residual directly, diagnosis requires inverting or regularizing the closed-loop propagation operator. Finite boundaries, noise, zeros, and model uncertainty constrain this recovery.

  • Measured signals: Finite converter networks usually provide voltage or current measurements at discrete sensors, so the residual’s propagation to measured voltage must be modeled explicitly.The residual is generally not observed directly over the spatial continuum.
  • Measured signals: The residual voltage is related to the balance residual through the closed-loop line transfer factors, and a point-fault signature is recovered ideally after prewhitening.The required operation includes multiplication by the appropriate propagation inverse.
  • Finite-line effects: For finite lines, boundary conditions require spatial modes or a compatible Green’s function instead of unrestricted free-phase localization in k.Free-phase localization remains applicable only under long-line or local-window approximations where reflections are negligible.
  • Deconvolution limits: Exact deconvolution is sensitive to zeros or ill-conditioning of the propagation operator, noise, and parametric uncertainty.Direct inversion can amplify noise near zeros of H or algebraic zero divisors.
  • Deconvolution limits: Regularized Tikhonov-type deconvolution keeps the recovered solution bounded when the propagation operator is ill-conditioned.With λ=0 and invertible H, exact inversion is recovered; λ>0 provides bounded recovery.

6.3 Localization by geometric spatial phase

Geometric spatial-phase localization recovers a fault position from residuals whose temporal signature and spatial shift factorize. Robust estimation searches over admissible positions while accounting for noise and phase ambiguity.

  • Phase localization: The residual factorizes as Ff(st)e^(-Bxkmxf), separating the temporal fault signature from its spatial location.Two wavenumbers convert the relative phase into a pure spatial rotor whose Bx argument encodes position modulo 2π.
  • Phase localization: Phase localization assumes an invertible fault signature and a residual close to a pure spatial rotor.The method is intended for infinite or locally long lines, reflection-free windows, or prewhitened balance residuals.
  • Robustness: With noise or model error, projecting onto the spatial plane is safer than applying ArgBx to the full bicomplex element.When several wavenumbers are available, the robust least-squares fit is preferred.
  • Robust estimation: The robust estimator solves linearly for the four real components of F at each candidate position, then performs a bounded one-dimensional search.The minimizer returns both the estimated position and temporal signature.
  • Limitations: Phase estimation is ambiguous modulo 2π/(k2−k1), so wavenumbers or multi-wavenumber fitting must cover the physical interval.For a line of length L, the ambiguity period should exceed the admissible position range, or the global fit should search [0,L].

6.4 Modal localization in a finite line

Modal localization represents a finite-line point fault through spatial-mode weights. Ratios and multi-mode fits can estimate position, but observability, modal nodes, noise, and normalization constrain reliability.

  • Finite-line modes: A finite line uses boundary-condition-selected spatial modes and an orthonormal basis for modal projections.The basis avoids ambiguous normalization factors in finite-line calculations.
  • Normalization: Normalization conventions cancel in localization ratios but must be handled explicitly for severity estimation and physical modal projections.Mixing projection conventions can introduce a spurious L/2 severity factor.
  • Modal representation: Each modal amplitude shares the temporal signature while differing by a spatial weight, enabling position estimation from multiple modes.The modal residual is normalized using the residual-to-voltage transfer Gn when that transfer is nonzero and well-conditioned.
  • Two-mode localization: Two-mode localization uses A1=Ffψ1(xf) and A2=Ffψ2(xf), with k1=π/L and k2=2π/L, assuming A1≠0.The common temporal signature is canceled to recover xf within (0,L).
  • Limitations: The closed-form modal diagnostic is unreliable near modal nodes, when A1=0, or when too few modes create ambiguity.A multi-mode fit can instead provide a confidence metric.
  • Observability: A fault is unobservable when all observed mode shapes vanish at its position, and positions with proportional modal vectors are not identifiable without extra information.The modal-observability energy vanishes at line ends and common nodes, so localization requires injectivity up to scale.

6.5 Extension: multiple faults

Multiple faults produce a linear superposition of modal signatures and can be recovered as a sparse identification problem. Grid, off-grid, and spectral methods address separation and resolution, while order selection remains conditional.

  • Multiple-fault model: For J simultaneous faults, joint identification of amplitudes and positions becomes sparse recovery over a modal dictionary.Each candidate position carries four real algebra components, with group sparsity enforcing common support.
  • Recovery methods: Simultaneous OMP selects shared-support dictionary columns, refits amplitudes by least squares, and refines selected positions off-grid with VARPRO.It needs no λ tuning and is accurate for well-separated faults.
  • Recovery methods: Group-LASSO provides a convex grid-based alternative using ℓ2,1 regularization and FISTA group soft-thresholding.The group norm imposes common support across the four algebra components.
  • Resolution limits: Recovery is reliable below L/N at fixed noise but degrades as faults approach, with VARPRO somewhat more robust in the hardest regime.The resolution limit depends on observed modes and signal-to-noise ratio.
  • Super-resolution: VARPRO refinement is not limited by the initialization grid because amplitudes are eliminated analytically while positions are optimized continuously.This refinement is the practical super-resolution mechanism of the OMP route.
  • Super-resolution: ESPRIT treats modal amplitudes as a sum of sinusoids, estimates poles from a Hankel signal subspace, and maps pole phases to grid-free positions.It offers a lower-cost, lower-memory alternative that can resolve faults closer than classical resolution limits.
  • Automatic order selection: MDL or AIC estimates the number of faults from Hankel singular-value structure, with a rank guard preventing round-off singular values from inflating the order.Reliability depends on modes, SNR, spacing, pencil size, and max_faults; deeply sub-Rayleigh cases may remain ambiguous.

6.6 Fault classification

Fault classification follows localization by interpreting the temporal signature and local admittance, while testing physical consistency against line propagation. Discrete site matching further separates plant outages from cable defects.

  • Classification basis: Classification uses the estimated temporal signature Ff(st) and, when appropriate, the equivalent local admittance Yf(st).The spatial fit must be physically consistent before assigning a fault type.
  • Injection loss: Injection loss is identified by a stopped converter contribution, with time-origin signature Ff(st)=−I0/st for a constant-current loss.The test stFf(st)e^(sttf) should be approximately constant.
  • Shunt faults: A local shunt introduces current proportional to local voltage, and its admittance subtype is selected by dispersion across Yf, Yf/st, or Yfst.The estimate requires non-small local voltage and consistent spatial localization.
  • Sensor faults: Sensor anomalies can create apparent residuals without producing a response coherent with the line propagation operator.A physical-consistency error or low modal confidence rejects the localized physical-fault hypothesis.
  • Controller faults: A local controller-gain change is formally indistinguishable from an anomalous local admittance without internal state, setpoint, switch-current, or local telemetry data.These additional measurements are needed to separate controller faults from physical shunts.
  • Plant versus cable: In discrete-converter lines, matching an injection-loss estimate to a known site diagnoses a plant outage, whereas an off-site estimate indicates a cable defect.The threshold δ must lie between localization uncertainty and half the site spacing; the rule is blind to cable faults within δ of a site.
  • Scope boundary: The additive outage model captures lost feed-forward dispatch but not the additional loss of droop-feedback injection when the plant also participates in feedback.That omitted term has the form of a controller-fault admittance perturbation.

7 Numerical results

The numerical case study evaluates modal stability under virtual-conductance control and tests fault localization using phase and modal estimators. Control stabilizes the considered modes, while both estimators recover the fault position exactly without noise and degrade under measurement noise.

  • Base case: The 100 m base line uses negative distributed conductance g = −0.05 S/m, representing converter energy injection and enabling open-loop instability.The parameter kv denotes the virtual conductance supplied by the distributed admittance-shaping controller.
  • Modal analysis: first mode: The first-mode characteristic polynomial has a negative constant term caused by active conductance g < 0, anticipating a positive-real-part unstable pole.The quadratic coefficient equals ℓc = 1×10−8.
  • Open-loop and closed-loop modal poles: After virtual conductance, all first five modes have real part −175.000, and mode 1 becomes −175.000 ± 219.251 Bt instead of retaining a right-half-plane root.The control shifts the effective conductance from g to g + kv = −0.01 S/m.
  • Open-loop and closed-loop modal poles: The modal time response changes from divergence of the fundamental mode in open loop to a damped response in closed loop.The pole map depicts the corresponding crossing of the imaginary axis before control.
  • Localization of a fault: For an injection-loss fault at xf = 35 m, both phase and modal estimators recover 35.00 m with 0.00 m error and, for the modal estimator, confidence 1.000 without noise.With noise standard deviation σ = 0.05, a single realization favored the modal estimator, while a seed sweep reported mean errors of approximately 0.02 m and 0.05 m, respectively.

7.5 Fault classification

The classifier estimates fault signatures from noisy modal measurements and combines relative-fit dispersion with a physical-consistency check to distinguish physical faults from sensor anomalies.

  • Classification procedure: Classification uses measured modal amplitudes, estimated fault signatures, local voltage, relative dispersion across candidate models, and a physical-consistency test.The local voltage spectrum is chosen non-constant to separate injection loss from inductive shunt, whose signatures both scale as 1/s.
  • Classification results: The injection_loss case achieves the smallest relative dispersion, 0.025, versus 0.293 for shunt_inductive and 0.802 for shunt_resistive.The classifier assigns injection_loss with physical consistency; the nonzero residual reflects noisy estimates.
  • Classification results: The shunt_resistive case achieves relative dispersion 0.098, compared with 0.770 for injection_loss, 0.918 for shunt_capacitive, and 1.015 for shunt_inductive.The correct hypothesis is separated from alternatives, although the physical-consistency margin is narrow in this case.
  • Classification results: The sensor_fault case is rejected as a localized physical fault because relative error is 1.006, modal confidence is 0.184, and the estimated position is 82.2 m.The consistency gate prevents the fit from being interpreted as a physical fault at the actual location.
  • Monte Carlo validation: Across 400 Monte Carlo realizations, overall accuracy is 0.920, while physical-fault labels have precision 1.000 when they pass consistency.The main errors are conservative rejections: 15% of resistive and 17% of capacitive faults are labeled sensor_fault; sensor-anomaly recall is 1.000 and precision is 0.758.
  • Multiple-fault recovery: The two-fault sparse-recovery experiment recovers positions at 25 m and 70 m with errors of 0.005 m and 0.033 m, and amplitudes 1.08 and 0.67.Simultaneous OMP uses common support and off-grid VARPRO refinement; the study reports reliable recovery for well-separated faults.

7.8 Discrete converters: exactness, aliasing, and localization

The discrete-converter validation tests continuum convergence, exact droop realization on the retained modal band, and residual-based localization that distinguishes plant outages from cable faults.

  • Continuum limit: For a smooth injection density, midpoint discretization converges with the expected O(P −2) rate as the number of converter sites increases.For mode n = 1, coefficient error decreases from 3.82 × 10−2 to 5.88 × 10−4, with observed dyadic rates near 2.
  • Exact discrete droop: With per-converter gains κp = kvL/P and modal content confined to the retained band, maximum pole deviation for sub-Nyquist modes n ≤7 is 7.1 × 10−13 s−1.This validates the exact discrete droop realization under the theorem’s sub-Nyquist validity condition.
  • Plant-versus-cable diagnosis: For a cable fault at 35 m, modal and phase estimators locate it at 35.003 m and 34.971 m, so a 1 m site threshold assigns it to the cable.The estimated position is 3.72 m from the nearest converter site, beyond the threshold.
  • Plant-versus-cable diagnosis: For a plant-3 outage at 31.25 m, the noisy estimate is 31.220 m, only 0.030 m from the site, and the site rule flags plant 3.The noiseless localization error is 8.5 × 10−9 m.
  • Plant-versus-cable diagnosis: Over 200 noisy trials per scenario, the site rule correctly identifies plant 3 outages and mid-span cable faults in every trial.The rule is blind to cable faults within δ = 1 m of a converter site, covering 16% of the line in this configuration.
  • Sensitivity to dispatch knowledge: Dispatch uncertainty increases cable-fault localization error: median error rises from 0.018 m with exact dispatch to 0.250 m at ε = 5%.The corresponding 90th percentile reaches 0.593 m at ε = 5%, and larger or correlated dispatch errors can erode outage/cable separation.

8 Conclusions, limitations and future work

The paper’s operational contribution is an integrated diagnosis framework built on geometric separation of temporal and spatial phases, while its guarantees remain bounded by modeling, observability, communication, and validation constraints.

  • Conclusions: The framework embeds temporal and spatial phases in a commutative bicomplex subalgebra, enabling geometric and modal localization of distributed-converter faults.The spatial phase is read through Arg_Bx or modal weights, while the temporal and spatial planes remain distinguishable.
  • Conclusions: Admittance shaping is formulated in the transformed domain as a distributed control design for modifying the line’s effective admittance.The design is expressed through conditions on the geometric variables (s_t, s_x).
  • Conclusions: The diagnosis chain detects, localizes, and classifies injection-loss, shunt, sensor, and local-controller faults using signatures in the geometric domain.The paper presents this integrated detection, localization, and classification chain as its main operational contribution.
  • Limitations: The analysis does not claim universal superiority over established fault-detection techniques, and quantitative conclusions remain model predictions pending physical-plant validation.Communication delays and converter saturation can also reduce stability margins or invalidate the linear superposition used by the control and residual interpretation.
  • Limitations: The guarantees assume homogeneous line parameters, incremental distributed-injection modeling, and converter reductions that omit internal loops, switching, filters, and closed-loop transients.Discrete converter sites are treated exactly for smooth-density homogenization and uniform droop on the sub-Nyquist band, but internal converter dynamics remain outside the model.
  • Limitations: Localization is constrained by sensor number and placement, measurement noise, boundary conditions, model quality, and the spatial separation of simultaneous faults.Very close faults create nearly collinear dictionary columns, while scarce sensors can make spatial-phase reconstruction underdetermined.
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