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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

Fault information in distributed-converter lines propagates along the conductor and is mixed at a limited number of measurement points, complicating detection, localization, and classification. The paper uses a geometric two-dimensional Laplace representation to encode localized faults in temporal and spatial signatures, with numerical experiments illustrating expected behavior within model assumptions.

  • Problem

    Detecting, locating, and classifying faults is difficult because localized fault information propagates along the line and arrives intermingled at a limited number of measurement points.

    Source: §1.1 · p. 3

  • Method

    The paper embeds a two-dimensional geometric Laplace transform in Cl(4,0), representing localized faults as residuals factored into temporal dynamics and a spatial rotor for integrated detection, localization, and classification.

    Source: §1.3 · p. 5, §6 · p. 29

  • Results

    Reproducible numerical experiments illustrate expected behavior within model assumptions, while the geometric representation yields interpretable fault signatures.

    Source: §8.1 · pp. 51–52

  • Takeaways & Limitations

    The representation is operationally convenient for describing space-time coupling, with distinctive value when two independent variables are effectively present and assumptions are reasonably fulfilled.

    Source: §8.1 · p. 52

    With scarce sensors relative to relevant spatial modes, spatial-phase reconstruction becomes underdetermined, and discrete-sensor reconstruction of modal amplitudes is not developed.

    Source: §8.2 · p. 52

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 introduction motivates fault diagnosis in distributed-converter DC lines as a spatio-temporal monitoring problem and presents a geometric two-variable Laplace framework that separates temporal and spatial phase. Its contribution is the operational use of that embedding for diagnosis, within an incremental model whose performance remains conditioned by observability and measurement conditions.

1.1 Applied context: DC lines with distributed converters

  • Distributed-converter DC lines are spatio-temporal systems, and fault information propagates along the conductor to only a limited number of measurement points.The model assumes small-signal incremental dynamics, a spatially continuous line, and converter injections represented as distributed actuation; it does not model large-signal transients or converter switching.Source: §1.1 · p. 3

1.2 Mathematical motivation: why a single complex variable does not suffice

  • Two geometric imaginary units, B_t and B_x, distinguish temporal and spatial phase in a commutative bicomplex subalgebra, making fault position readable as an argument in the B_x plane.A two-variable complex transform already distinguishes the coordinates; the proposed geometric embedding instead places the phases in distinguishable planes within one structure.Source: §1.2 · p. 4

1.3 Related work and delimitation of the novelty

  • The bicomplex Laplace transform itself is known; the novelty claimed is its geometric embedding and operational use for diagnosis in distributed-converter networks.This small-signal monitoring niche complements fault-location methods oriented toward pronounced short-circuit transients or dedicated injection infrastructure.Source: §1.3 · pp. 4–5

1.4 Contributions

  • The main contribution is an integrated framework that factorizes a localized fault residual as F_f(s_t)e^(-s_x x_f), enabling detection, classification, and location from its spatial geometric argument.The introduction also identifies a DC-line dispersion model, exact spatial-sampling theory for discrete converter sites, distributed control by admittance shaping, and numerical validation as contributions.Source: §1.4 · pp. 5–6, §1.3 · p. 5

1.5 Document structure

  • Diagnosis performance is conditional on spatial observability, sensor number and placement, measurement noise, line boundary conditions, and incremental-model quality.The introduction cautions that geometric phase separation does not provide a universal diagnosis guarantee.Source: §1.5 · p. 6

2 Minimal geometric algebra

Section 2 defines a four-real-dimensional commutative subalgebra of Cl(4, 0) generated by two commuting bivectors and establishes the algebraic operations used by the space–time transform. Its diagonalization, exponential, and argument constructions support componentwise calculations and independent temporal and spatial phases.

2.1 Definition of the subalgebra

  • A is generated by Bt = e1e2, Bx = e3e4, and Btx = BtBx, with elements uniquely represented as a + bBt + cBx + dBtx; it is an associative, unital, commutative four-dimensional subalgebra.The disjoint orthogonal vector pairs underlying Bt and Bx guarantee their commutativity.Source: §2.1 · pp. 6–7

2.2 Product and commutativity

  • The generators’ multiplication makes A commutative, and mapping Bt to i, Bx to j, and Btx to ij identifies it with Segre’s bicomplex algebra.This isomorphism allows bicomplex diagonalization and holomorphic-function theory to be used while retaining the bivectors’ geometric interpretation.Source: §2.2 · p. 7

2.3 Idempotent decomposition and diagonalization

  • Two orthogonal idempotents decompose each z uniquely as z = ζ1e1 + ζ2e2, with ζ1 and ζ2 ordinary Bt-complex numbers, so products and analytic operations act componentwise.The decomposition identifies A with the product algebra CBt × CBt.Source: §2.3 · p. 8

2.4 Inverse and zero divisors

  • An element z is invertible exactly when both idempotent components ζ1 and ζ2 are nonzero; otherwise it lies on one of two isotropic cones, and A has zero divisors.The idempotents are nonzero zero divisors because e1e2 = 0.Source: §2.4 · pp. 8–9

2.5 Exponential and elementary functions

  • Because Bt and Bx commute, the transform kernel factors into temporal and spatial exponentials whose phases rotate in independent bivector planes.These exponentials provide the kernel used to define the transform in section 3.Source: §2.5 · p. 10

2.6 Geometric arguments and auxiliary norm

  • For a pure spatial rotor, ArgBx recovers its angle modulo 2π, allowing spatial position to be read as a phase; the auxiliary Euclidean norm is not multiplicative.The norm is used only to control convergence and magnitude, not as an algebraic absolute value.Source: §2.6 · p. 10

2.7 Relation to the implementation

  • The implementation’s Geo class stores elements as [a, b, c, d] and implements the product, inverse, exponential, arguments, and norm; inversion flags ζ1 = 0 or ζ2 = 0 as degenerate.Inversion uses diagonalization or solves the real-coordinate multiplication-matrix system.Source: §2.7 · pp. 10–11

3 Two-dimensional geometric Laplace transform and operational properties

The section defines a commutative bicomplex Laplace transform for time and space, then establishes convergence and operational properties for later modeling and fault diagnosis. Its geometric representation keeps temporal and spatial phases separately readable, while its single-variable case is the ordinary complex Laplace transform.

3.1 Definition and factorized kernel

  • The transform uses commuting geometric variables s_t = ρ_t + B_tω and s_x = ρ_x + B_xk, yielding a factorized kernel with separate temporal and spatial attenuation and oscillation.The function takes values in the commutative subalgebra generated by 1, B_t, B_x, and B_tx.Source: §3.1 · p. 11

3.2 Region of convergence

  • For functions of exponential order, the transform converges absolutely when ρ_t > σ_t and ρ_x > σ_x, with convergence reducible to two ordinary complex Laplace transforms in the idempotent representation.The Euclidean component norm is not submultiplicative, so the convergence argument uses the product norm bound in the idempotent basis.Source: §3.2 · p. 12

3.3 Linearity; 3.4 Transform of the partial derivatives

  • On common convergence regions, the transform is linear and maps time and space partial derivatives to algebraic multiplications with initial-condition and boundary-condition terms, which vanish under zero conditions.The derivative identities hold for continuously differentiable functions when the function and both partial derivatives are of exponential order.Source: §3.4 · p. 13, §3.3 · p. 13

3.5 Transform of a separable mode; 3.7 Correspondence with the implementation

  • A separable space–time mode transforms into a product of temporal and spatial pole factors, whose planes encode temporal dynamics and spatial propagation; the implementation returns this closed form.The stated convergence conditions are ρ_t > −α_t and ρ_x > −α_x.Source: §3.5 · p. 14, §3.7 · p. 15

3.6 Comparison with the ordinary complex Laplace transform

  • The transform packages temporal and spatial phases in distinct planes of one bicomplex element, while a single-variable geometric transform adds no mathematical content beyond the ordinary Laplace transform.The construction assumes commuting generators; an anticommuting version would require Baker–Campbell–Hausdorff corrections and is left for future work.Source: §3.6 · p. 15

4 Distributed converter-line model and dispersion relation

This section develops a two-variable geometric-transform model of a distributed-converter DC line, deriving its transfer function and dispersion relation and examining finite-line modes and discrete converter placement. It describes how placement limits independent actuation to at most the P lowest modes, with higher-mode content aliasing into that band and modes whose nodes contain every converter remaining uncontrollable.

4.1 Modeling assumptions

  • The line is linearized about a nominal operating point, with small incremental signals, uniform parameters, zero initial and applicable boundary conditions, and a continuous-in-space injection approximation.The regularity and integrability conditions needed for the transform are assumed; discrete converters are treated separately.Source: §4.1 · p. 16

4.2 Line equations

  • The transmission-line model uses per-unit-length r, ℓ, c, and g, where g may be negative for converters with constant-power control.The model relates voltage drop to series impedance and current balance to capacitive current, conductance, and converter injection.Source: §4.2 · p. 16

4.3 Transformation and distributed transfer function

  • The two-variable transform yields a distributed voltage transfer function from converter injection wherever the characteristic determinant is invertible; its zeros define the line’s space–time eigenmodes.Commutativity of the bicomplex algebra permits the algebraic system to be solved with ordinary determinant methods.Source: §4.3 · p. 17

4.4 Dispersion relation

  • For an ideal lossless line, k^2 = ℓcω^2, so disturbances propagate at a frequency-independent phase velocity without dispersion or phase distortion.For generic losses, the pure spatial-plane equation is not an exact dispersion root; a real branch exists there only when rc + ℓg = 0.Source: §4.4 · p. 18
  • For generic losses, the lossy curve is only a classical projection, not an exact root on the pure spatial plane; that restriction is exact on the locus rc + ℓg = 0.Exact treatment off that locus requires spatial attenuation or the idempotent decomposition.Source: §4.4 · pp. 18–19

4.5 Finite line: modes and modal denominator

  • On a finite line with zero voltage at both ends, sine modes have modal transfer functions Z/Dn, whose denominator roots determine each mode’s poles, stability, and damping.Each spatial mode has its own modal denominator.Source: §4.5 · pp. 19–20, Figure 2 · p. 20

4.6 Discrete converters: point injection and spatial sampling

  • Discrete converter injections transform into finite sums of spatial rotors; on the uniform midpoint grid, modes 1 through P remain orthogonal, while modes 2Pj are invisible and higher modes alias.Mode P carries twice the weight of the other orthogonal modes; sampled shapes are 4P-periodic in mode order.Source: §4.6 · pp. 20–22
  • For a C2 target injection density sampled by midpoint-grid converters, each fixed modal coefficient converges with error O(P^-2); for constant density, the comparison is exact in closed form.For constant density, both coefficients vanish for even modes.Source: §4.6 · p. 22
  • With P converter sites, at most the P lowest modes can be actuated independently; injected content in modes n > P folds back onto the band n ≤ P.This is the spatial Nyquist limit of actuation.Source: §4.6 · p. 22
  • Mode n is uncontrollable from the converter set, whatever the control law, if every converter sits on one of its nodes.On the uniform grid xp = pL/(P + 1), mode n = P + 1 and its multiples are exactly uncontrollable.Source: §4.6 · p. 22
  • The reference implementation includes routines for converter positions, sampling, modal injection and coupling, controllability scores, aliasing, and discrete-droop poles.The implementation correspondence is intended to support numerical reproduction and verification of the analytical results.Source: §4.6 · p. 23

5 Distributed control via admittance shaping

This section develops control laws that shape the line’s effective shunt admittance and target the closed-loop determinant’s zeros, covering constant, space-time, and modal designs alongside their discrete realization and implementability. The formulas and exact discrete-droop results map to implementation modules for numerical validation.

Control law and closed-loop admittance

  • Choosing K makes the closed-loop determinant’s zeros the design variables, because the controller adds to the line’s shunt admittance.The closed-loop admittance is Yc(st, sx) = g + cst + K(st, sx).Source: Control law and closed-loop admittance · p. 23

5.1 Distributed virtual conductance

  • Positive virtual conductance kv raises the effective conductance from g to g + kv and increases modal damping, but strict passivity requires g + kv > 0.In the stated base case, g = −0.05 S/m and kv = 0.04 S/m leave the branch incrementally active, although the control stabilizes the modes considered.Source: §5.1 · p. 24
  • On a uniform midpoint grid, droop realizes the continuous virtual-conductance loop exactly for modes n ≤ P−1; mode P sees doubled gain, while higher modes alias and couple.For non-uniform placements, exactness is lost, though Voronoi-cell gains preserve convergence to the continuous loop as P grows.Source: §5.1 · pp. 24–25

5.2 Space-time dynamic controller

  • The space-time controller combines virtual conductance, temporal damping, spatial-gradient coupling, and a mixed term that selectively damps traveling waves without penalizing quasi-static modes.Spatial terms require neighbor-related measurements; with those terms disabled, the controller reduces to K = k0 + ktst.Source: §5.2 · p. 26

5.3 Design by target denominator

  • For any target determinant, the unique controller is obtained by solving for K, provided Z(st) is invertible in the region of interest.Implementation must also account for temporal causality, spatial locality, finite-difference error, converter saturation, and communication delays.Source: §5.3 · pp. 26–27

5.4 Modal control on a finite line

  • On a finite line with zero end voltages, modal feedback independently controls each mode; its temporal gains shift effective capacitance by ktn and conductance by k0n.Source: §5.4 · p. 27

Assignment of temporal geometric poles

  • Matching each mode’s denominator to a prescribed geometric pole pair yields modal gains, physically implementable only when the effective capacitance ct and conductance gk are positive.The target pair is parameterized by αn for damping and βn for temporal geometric frequency.Source: Assignment of temporal geometric poles · p. 28
  • The control and exact discrete-droop formulas map one-to-one to implementation functions, enabling numerical validation.Mappings include virtual_conductance, design_K_from_delta, modal_gain_for_target_poles, modal_coupling_matrix, and discrete_droop_poles; validation is described in section 7.Source: Assignment of temporal geometric poles · p. 29

6 Localized faults: detection, localization, and classification

The section frames localized-fault diagnosis as model-based residual generation: a fault is a space-time singularity whose transform encodes position in a spatial rotor.

6 Localized faults: detection, localization, and classification

  • A localized fault appears as a space-time singularity whose transformed residual encodes fault position in a spatial rotor.The approach is presented as model-based fault diagnosis and residual generation.Source: §6 · p. 29
  • The balance residual represents the anomalous source, whereas the observed response is its propagated effect through the line.Source: §6 · p. 29
  • Voltage-only measurements require transfer-function removal to recover the residual signature; otherwise, a sensor anomaly can be mistaken for a physical fault.The required operation is described as prewhitening or modal deconvolution.Source: §6 · p. 29

6.1 Distributed residual and the signature of a localized fault

The section defines a distributed balance residual that acts as an equivalent fault source and shows how its transform separates a localized fault’s temporal signature from its position. It distinguishes additive injection faults from shunt disturbances and cautions that measured voltage responses include line propagation.

6.1 Distributed residual and the signature of a localized fault

  • The balance residual is R(t, x) = ∂_x i + c ∂_t v + g v + K[v] − r_u; it is zero ideally, with a statistical norm threshold used under noise or model error.The residual is defined using the distributed control operator K and known excitation r_u.Source: §6.1 · p. 30
  • The balance residual is the equivalent fault source and supports localization, but the measured voltage response also includes line propagation.Source: §6.1 · p. 30
  • A shunt is a local-admittance disturbance rather than an independent source; after localization, its admittance can be estimated from fault current and local voltage, though strong shunts alter that voltage.For strong shunts, classification is an estimate within the already-faulted system, not an independent external-source estimate.Source: §6.1 · p. 30
  • A localized additive fault is modeled as a spatial Dirac delta with a temporal current signature; its transformed residual factors as F_f(s_t)e^{-s_x x_f}.The temporal signature appears in the first factor, while position appears in the spatial rotor e^{-B_x k x_f}.Source: §6.1 · p. 30

6.2 From the residual to the measured signals

The section connects the physical residual to voltage measurements through the line’s propagation operator, showing that voltage-based fault localization requires prewhitening. It also addresses the noise sensitivity of exact inversion and the modal treatment needed for finite lines.

6.2 From the residual to the measured signals

  • Voltage measurements at finite converter or sensor locations do not directly provide the continuum residual, so the propagation from residual to voltage must be made explicit.The section relates the transformed residual voltage to the balance residual through the closed-loop model.Source: §6.2 · p. 31
  • For a point fault, voltage-based localization requires prewhitening with Z(st)Vres(st, sx), which ideally recovers Ff(st)e^-sxxf.The operation is sensitive to zeros of Z, zeros or near-zeros of ∆c, noise, and parameter uncertainty.Source: §6.2 · p. 31
  • For finite lines with boundary conditions, spatial modes or a boundary-compatible Green’s function replace the homogeneous-line response; localization must use modes unless reflections are negligible.A long-line approximation or local windows with negligible reflections retain the alternative free-phase localization approach.Source: §6.2 · p. 31

Regularized prewhitening

  • Exact deconvolution R = H^-1Vres amplifies noise near zeros of H or algebraic zero divisors, motivating a Tikhonov-type regularized inversion.The conjugation and modulus are handled using the real 4 × 4 multiplication-matrix representation to accommodate the algebra structure.Source: Regularized prewhitening · p. 31
  • With λ = 0 and invertible H, exact inversion is recovered, while λ > 0 keeps the solution bounded when H is ill-conditioned; finite lines also admit a modal regularization.The modal formulation uses Gn(st) = Z(st)/Dn(st) and Vn = GnAn.Source: Regularized prewhitening · p. 32

6.3 Localization by geometric spatial phase

This section localizes faults by reading the geometric spatial phase of residual signatures at multiple wavenumbers, and gives a robust least-squares alternative when noise, reflections, or phase ambiguity limit direct phase estimation.

Localization by geometric spatial phase

  • Apply phase localization to the residual or prewhitened signal, not directly to an uncompensated voltage response.The pure-rotor assumption is appropriate for an infinite line or half-axis, a local long-line approximation, or windows where reflections are negligible.Source: §6.3 · p. 32
  • With two distinct wavenumbers, the residual ratio cancels the temporal signature, leaving a spatial rotor whose phase gives (k2 - k1)xf modulo 2π.This requires an invertible temporal signature and a first residual that is not close to a zero divisor.Source: §6.3 · p. 32
  • For finite lines dominated by reflections and boundary conditions, the free spatial phase is not a pure rotor, so phase localization is only a local or long-line approximation.The global problem on 0 ≤ x ≤ L instead calls for the boundary-consistent modal estimator of subsection 6.4.Source: §6.3 · p. 32
  • For noisy or imperfectly prewhitened data, project onto the spatial plane and threshold its norm; with several wavenumbers, prefer the least-squares fit.The fit estimates the temporal signature and position by minimizing the residual mismatch over an admissible position interval.Source: §6.3 · pp. 32–33
  • Two-wavenumber phase estimates are ambiguous modulo 2π/(k2 - k1), so choose wavenumbers whose ambiguity period covers the line or fit over [0, L] with several wavenumbers.Source: §6.3 · p. 33

6.4 Modal localization in a finite line

In a finite line, boundary conditions select spatial modes, and modal residual amplitudes share a temporal signature while encoding fault position through their spatial weights. Two modes yield a closed-form localization under stated conditions, while multi-mode fitting and observability checks address noise and identifiability limits.

6.4 Modal localization in a finite line

  • Normalized modal amplitudes share the temporal signature and differ only in spatial weight, provided each modal transfer Gn(st) is nonzero and well-conditioned.Finite-line boundary conditions select spatial modes, for which the section introduces an orthonormal basis to avoid ambiguous normalization.Source: §6.4 · p. 33
  • With k1 = π/L and k2 = 2π/L, two modal amplitudes can localize xf in (0, L) when A1 is nonzero and their common signature can be canceled.The amplitudes are A1 = Ffψ1(xf) and A2 = Ffψ2(xf).Source: §6.4 · p. 34
  • The two-mode closed-form diagnostic is not the main choice under noise: it can fail near modal nodes, requires A1 ≠ 0, and may be ambiguous with few modes.A several-mode fit can produce a confidence metric; when modal-pattern energy is negligible, confidence should be set to 0 rather than reported spuriously high.Source: §6.4 · p. 34
  • A fault is unobservable if all observed modes vanish at its position, and localization is non-identifiable when distinct positions have proportional modal vectors.Localization requires the modal-pattern map to be injective up to scale; the observability energy vanishes at line ends and common nodes, while the normalized pattern reveals degeneracies.Source: §6.4 · p. 34

6.5 Extension: multiple faults

The section extends single-fault modeling to simultaneous faults through sparse recovery, implements complementary recovery methods, and describes super-resolution and automatic fault-count estimation with stated limits.

6.5 Extension: multiple faults

  • By linearity, J simultaneous point faults yield a sparse recovery problem for jointly identifying each fault’s amplitude and location, in continuum or finite-line modal form.On a fixed spatial grid, a shared modal dictionary represents candidate locations, while a sparsity-promoting penalty selects fault amplitudes.Source: §6.5 · pp. 34–35

Implemented recovery

  • Simultaneous OMP selects locations using residual correlation across four algebra components and refines them off-grid by VARPRO, while group-LASSO uses FISTA.OMP needs no λ tuning and is accurate for well-separated faults; faults too close relative to observed-mode resolution are difficult to separate because dictionary columns become nearly collinear.Source: Implemented recovery · p. 35

Close faults: super-resolution

  • VARPRO [14] analytically eliminates amplitudes while optimizing continuous fault positions, so the grid initializes the search but does not constrain its final resolution.A grid-free alternative estimates harmonic poles from the modal sequence using matrix pencil [19] or ESPRIT, then maps their arguments to fault positions; it can resolve below the classical Rayleigh limit ∼L/N.Source: Close faults: super-resolution · pp. 35–36
  • The fault count can be estimated from the Hankel signal rank, which is 2J for J real sinusoids, using MDL by default or AIC.A numerical-rank guard prevents round-off singular values from inflating the estimate in exact or nearly exact data; reliability is not guaranteed and depends on modes, SNR, separation, noise, pencil size, and the maximum-order bound.Source: Close faults: super-resolution · p. 36
  • Recovery is reliable below L/N at fixed noise, with VARPRO more robust at the closest separations and ESPRIT offering a lower-cost, grid-free alternative.Both methods’ resolution depends on the number of observed modes and the signal-to-noise ratio; deeply sub-Rayleigh recovery and order selection remain challenging.Source: Close faults: super-resolution · p. 37

6.6 Fault classification

Fault classification follows localization and uses the temporal signature and, where relevant, the equivalent local admittance. A physical-consistency check helps distinguish line faults from sensor anomalies, while discrete-converter site matching can distinguish plant outages from cable defects.

Classification procedure

  • Check spatial-fit consistency before classifying a localized residual, since sensor anomalies can resemble localized sources but fail to match coherent line propagation.A robust relative-error test rejects the physical-fault hypothesis when its error exceeds a threshold or modal confidence is low, labeling the anomaly as sensor fault or model inconsistency.Source: §6.6 · pp. 37–38

(a) Open circuit or injection loss

  • An injection loss beginning at detection time has Ff(st) = −I0/st; test whether stFf(st)e^(st tf) is approximately constant.Source: §6.6 · p. 37

(b) Local shunt or short circuit

  • For a local shunt, estimate the equivalent admittance only when bVf is not small and localization is consistent, then compare relative dispersion of Yf, Yf/st, and Yf st.The least-dispersed quantity guides subclassification; the implementation distinguishes resistive, capacitive, and inductive shunts.Source: §6.6 · pp. 37, 39

(c) Sensor fault

  • A sensor fault alters measured signals without introducing a physical line source, so it can create an apparent residual without a propagation-coherent response.Source: §6.6 · p. 38

(d) Local controller fault

  • A controller gain change is formally indistinguishable from an anomalous local admittance, so separating it from a physical shunt requires internal-state or other local telemetry.Source: §6.6 · p. 38

Remark 6.15 (Discrete converters: residual and site identification)

  • For P discrete converters, residual subtraction preserves Ff(st)e^-s_xx_f; nearest-site matching identifies injection-loss anomalies as plant outages and off-site faults as cable defects.The distance threshold δ must exceed localization uncertainty but remain below half the site spacing; the rule is blind to cable faults within δ of a site and degrades with dense converter sets or noisy measurements. It assumes known dispatch and site locations, and additive injection-loss modeling does not capture an outage’s removed droop-feedback term.Source: §6.6 · pp. 38–39

7 Numerical results … 7.4 Localization of a fault

The numerical case study evaluates modal behavior and fault localization on a distributed-converter line, using directly executed simulations and analysis under controlled conditions, including noise. It shows open-loop instability removed by distributed control and evaluates two fault-position estimators.

7 Numerical results

  • The numerical results come from direct execution of the simulation and analysis code and illustrate modal behavior, fault localization, and classification under controlled conditions, including noise.The scripts that generate the results are indicated in theorem 7.3.Source: §7 · p. 39

7.1 Base case and distributed parameters

  • The 100 m base-case line has nominal distributed conductance g = −0.05 S/m, representing active energy-injecting converters and possible open-loop instability.The virtual conductance kv is provided by the distributed admittance-shaping control.Source: §7.1 · p. 39

7.2 Modal analysis: first mode

  • For the first mode, the characteristic polynomial’s negative constant term follows from active conductance g < 0 and anticipates a positive real, unstable pole.The quadratic coefficient coincides with ℓc.Source: §7.2 · p. 40

7.3 Open-loop and closed-loop modal poles

  • Control shifts all five modes to real part −175.000, making the first mode −175.000 ± 219.251 Bt and eliminating its open-loop unstable pole at +8.240.The virtual conductance kv = 0.04 S/m shifts g to g + kv = −0.01 S/m, still negative, while stabilizing the modes considered.Source: §7.3 · p. 40, Table 2 · p. 40, Figure 3 · p. 41

7.4 Localization of a fault

  • The fault-localization case uses an injection loss at xf = 35 m with signature Ff = −2 evaluated at st = 1, comparing phase-ratio and modal least-squares estimators.The phase estimator uses the residual phase in the Bx direction; the modal estimator fits the modal decomposition by least squares.Source: §7.4 · p. 41
  • Both estimators locate the fault at 35.00 m with 0.00 m error without noise; with noise, both remain below 0.2 m error over the 100 m line.Across noise-seed sweeps, mean error is ∼0.02 m for the modal estimator and ∼0.05 m for the phase estimator; the modal estimator is more accurate in roughly two thirds of cases, not systematically.Source: §7.4 · p. 42

7.5 Fault classification · 7.6 Monte Carlo validation of the classification · 7.7 Recovery of multiple faults

The paper evaluates spectral-signature fault classification, validates it across noisy Monte Carlo trials, and tests sparse recovery of simultaneous faults, including close faults and automatic fault-count selection.

7.5 Fault classification

  • Classification uses noisy measurement-derived modal and phase estimates, relative dispersion across signature models, and a physical-consistency check.A non-constant local-voltage spectrum, Vf(s) = V0/(1 + τs), breaks the injection-loss/inductive-shunt degeneracy because both signatures are proportional to 1/s.Source: §7.5 · pp. 42–43
  • The classifier assigned injection_loss, shunt_resistive, and sensor_fault labels to the three evaluated cases, with physical consistency only for the first two.The injection_loss fit dispersion was 0.025 versus 0.293 for shunt_inductive; the shunt_resistive score was 0.098 versus 0.770 for injection_loss. The sensor case had relative error 1.006, modal confidence 0.184, and estimated position 82.2 m rather than 35 m.Source: §7.5 · p. 43, Table 3 · p. 44
  • The consistency test rejects the tested incoherent sensor anomaly, but structured sensor bias that imitates spatial propagation remains unevaluated.In the resistive-shunt case, the consistency margin was narrow (∼0.21 against tolerance 0.25), and legitimate faults were rejected in about 12% of noise realizations; the sensor test covers only noise that does not respect propagation.Source: §7.5 · pp. 43–44

7.7 Recovery of multiple faults

  • For two simultaneous faults, OMP recovered positions within 0.005 m and 0.033 m, while group-LASSO placed peaks at 24.98 m and 70.04 m.With N = 24 modes and σ = 0.02, OMP recovered amplitudes of 1.08 and 0.67; group-LASSO amplitudes were slightly shrunk by the ℓ1 penalty.Source: §7.7 · p. 45

7.6 Monte Carlo validation of the classification

  • 0.920 overall accuracy was achieved across 400 realizations; physical-fault precision was 1.000 for injection_loss, shunt_resistive, and shunt_capacitive, and sensor_fault recall was 1.000.The experiment used 100 realizations per class with σ = 0.02 and homogeneous signature magnitudes. The main errors were genuine resistive and capacitive shunts rejected as sensor_fault (15% and 17%, respectively), reducing sensor_fault precision to 0.758.Source: §7.6 · pp. 44–45, Table 4 · p. 46

7.7 Recovery of multiple faults

  • Both OMP+VARPRO and ESPRIT achieved 100% success at 2 m separation, below the approximate 4.2 m Rayleigh resolution; OMP+VARPRO was more robust at 1.5 m and 1 m.At 1.5 m and 1 m, success was 100% and 78% for OMP+VARPRO, versus 82% and 20% for ESPRIT; both methods fail well below modal resolution.Source: §7.7 · pp. 45–46
  • MDL order selection hit rates were 97–100% for J = 0, 82–98% for J = 1, 72–95% for J = 2, and 62–90% for J = 3.These rates cover well-separated faults (∆x ≥ 8 m) across two noise models and σ ∈ [0.01, 0.05]; reduced separation or deeply sub-Rayleigh conditions make order selection ambiguous.Source: §7.7 · p. 46

7.8 Discrete converters: exactness, aliasing, and localization

The discrete-converter study validates second-order convergence toward the continuous model and shows that exact droop matching depends on modal-band confinement. Its diagnosis pipeline distinguishes mid-span cable faults from converter outages by localization relative to converter sites, with performance qualified by dispatch error and a blind zone near sites.

7.8 Discrete converters: exactness, aliasing, and localization

  • For midpoint-grid converters, coefficient error decreases at the expected O(P^-2) rate as the number of plants increases.For mode n = 1, the error falls from 3.82 × 10^-2 to 5.88 × 10^-4 over P = 4, 8, 16, 32, with observed dyadic rates 2.016, 2.004, and 2.001; modes n = 3 and 5 settle to rates 2.011 and 2.031.Source: §7.8 · pp. 46–47
  • With modal content confined to the retained band, discrete droop matches continuous closed-loop poles to 7.1 × 10^-13 s^-1 for sub-Nyquist modes n ≤ 7.When all N = 12 modes are retained, alias rows shift modes 4–7 by 2.0 to 14.4 s^-1; mode n = 8 instead has doubled gain. On the uniform node grid, mode 9 is numerically uncontrollable (γ9 = 1.3 × 10^-15), while random placements converge toward the continuous loop in the quadrature sense as P increases.Source: §7.8 · pp. 47–48
  • With δ = 1 m and σ = 0.02, localization correctly attributes a mid-span cable fault and a plant-3 outage, including across 200 noisy trials per scenario.The cable fault at 35 m is estimated at 35.003 m by the modal estimator and 34.971 m by the phase estimator; the noisy plant-3 outage estimate is 31.220 m, 0.030 m from its site. The rule is blind to cable faults within δ of a site, covering 16% of the line in this configuration.Source: §7.8 · p. 48
  • For the 35 m cable fault, median localization error rises from 0.018 m with exact dispatch to 0.049 m at ε = 1% and 0.250 m at ε = 5%.The corresponding 90th percentiles are 0.041, 0.121, and 0.593 m; larger or correlated dispatch errors can erode outage-versus-cable separation.Source: §7.8 · pp. 48–49

8 Conclusions, limitations and future work

The paper presents a geometric space–time framework whose main operational contribution is integrated fault diagnosis in distributed-converter lines. Its guarantees depend on modeling assumptions, and future work targets broader models and experimental validation.

8.1 Conclusions

  • The commutative subalgebra separates temporal and spatial phases; the two-variable transform supports the homogeneous line’s dispersion model and distributed admittance-shaping control.The transform establishes operational rules that turn the problem’s partial differential equations into algebraic relations in (s_t, s_x).Source: §8.1 · pp. 50–51
  • Localized faults factor into a temporal signature and a spatial rotor, enabling localization by geometric phase or modal weights and classification by fault signature.The classification covers open circuit or injection loss, shunt or short circuit, sensor faults and local-controller faults.Source: §8.1 · p. 51
  • The main contribution is integrated detection, localization and classification of faults, with distinctive value when time and space remain independent variables and the stated assumptions hold.The authors claim no universal superiority over established line-analysis or fault-detection techniques.Source: §8 · p. 49, §8.1 · pp. 51–52

8.2 Limitations

  • The guarantees assume homogeneous line parameters, idealized converter injections, known dispatch and sites, immediate communications, sufficient spatial observability and unsaturated linear converters.Discrete converter sites are treated exactly, and uniform droop feedback is exact on the sub-Nyquist band; internal plant loops, switching and filters remain outside the model.Source: §8.2 · pp. 52–53
  • Closely spaced faults remain difficult to identify; the commutative treatment excludes intrinsically coupled planes, and the single-variable transform reduces to classical Laplace.Evidence is analytic and numerical only; quantitative conclusions remain model predictions pending physical-plant validation.Source: §8.2 · p. 53

8.3 Future work

  • Future work would extend the framework to noncommuting planes, variable or piecewise line parameters, and converter models with internal loops, switching and filters.The latter would examine how converter bandwidths interact with spatial modes and exact discrete droop.Source: §8.3 · p. 53
  • Further diagnosis work targets multiple-fault order selection in difficult data regimes and improved robustness to measurement noise, sensor scarcity and communication delays.The current methods’ reliability depends on mode count, SNR, fault separation, pencil size and the maximum fault bound; deeply sub-Rayleigh cases remain open.Source: §8.3 · p. 54
  • Laboratory validation on a DC microgrid is proposed to compare the model’s predictions with hardware behavior.The work describes its current development as internally consistent and reproducible, but still requiring experimental data.Source: §8.3 · p. 54
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