Identifiability Limits of Forced Oscillation Sources in Power Systems

I. INTRODUCTION

• The central question is whether available measurements can uniquely distinguish declared candidate sources before a physical index is selected.The paper treats localization as an information-level identifiability problem, not as a new source-localization method.Source: §I · p. 1

B. Descriptor-System Representation for Power Systems

• The descriptor-system formulation covers interventions in dynamic and algebraic equations, signal and measurement channels, and first-order parameter modulation.Its channels include direct effects on measured signals, and sinusoidal parameter modulation with no first-order signature is locally invisible at the fundamental frequency.Source: §I · p. 1, B. Descriptor-System Representation… · p. 2, Table I · p. 2

A. Physical Intervention Sites

• Source location is tied to a declared physical intervention in a fixed realization, distinguishing region, unit, and internal mechanism.Candidates may correspond to signal paths, summing junctions, equation residuals, or physical parameters; terminal measurements can identify a unit while leaving internal channels equivalent.Source: A. Physical Intervention Sites · p. 2

C. Coordinate Dependence and Standing Assumptions

• A one-coordinate source injection can become a multi-coordinate injection after a nonsingular state transformation, while its named physical intervention remains unchanged.The analysis assumes one active candidate, nonzero constant forcing, known or consistently estimated frequency, a known model and measurement set, steady harmonic response, and a regular descriptor pencil.Source: C. Coordinate Dependence and Standing… · p. 3

A. Projective Harmonic Signatures

• Unknown complex forcing amplitude and phase make each candidate’s harmonic response identifiable only up to a nonzero complex scalar.The resulting projective signature represents the candidate independently of its forcing gain.Source: A. Projective Harmonic Signatures · p. 3
• The candidate dictionary is partitioned into observational-equivalence classes, and the finest data-supported result is the active candidate’s class.Candidates with the same ray cannot be distinguished from steady-state phasor measurements.Source: A. Projective Harmonic Signatures · p. 3

B. Necessary and Sufficient Condition

• Exact single-frequency localization holds if and only if candidate signatures are nonzero and pairwise non-collinear.Collinear signatures can produce the same measured phasor under different unknown forcing coefficients; a zero signature is unobservable.Source: B. Necessary and Sufficient Condition · p. 3
• The paper specializes input-distinguishability to physical interventions in power-system DAEs and uses projective identifiability as a reference for assessing feature maps.It relates the rank test to unknown-input observer and fault-isolation theory, –.Source: B. Necessary and Sufficient Condition · pp. 3–4
• With one scalar phasor, two or more nonzero candidates cannot be uniquely localized when their forcing amplitudes and phases are arbitrary.This limitation concerns steady single-frequency scalar measurements, not transient or multifrequency information.Source: B. Necessary and Sufficient Condition · p. 4

C. Hierarchical Identifiability

• Region localization can be exact even when units within that region are equivalent, and a unit can be identified while its internal candidates remain indistinguishable.The hierarchy separates identifying an area, a contributing resource, and a root control variable.Source: C. Hierarchical Identifiability · p. 4

A. Candidate Residuals

• The weighted residual over unknown forcing amplitude yields a point estimate and a thresholded set-valued estimate from the candidate dictionary.In the exact noise-free case, the residual is zero for candidates in the true equivalence class and positive outside it.Source: A. Candidate Residuals · p. 4
• The algorithm returns a candidate only when residual-margin and model-validity tests pass; otherwise it returns an ambiguity set.Its workflow constructs signatures and their separation matrix offline, then evaluates candidate residuals using the measured harmonic phasor.Source: A. Candidate Residuals · p. 4

• Strong pairwise separation does not establish identifiability of the full dictionary; reliability depends on the global minimum separation and the uncertainty margin.A candidate with zero signature is unobservable, and no universal numerical threshold for adequate separation is implied.Source: B. Separation and Deterministic… · p. 4
• A candidate is uniquely recovered when every competing noiseless ray is more than 2ε away, since bounded error can move the observation at most ε from the true ray.The guarantee applies to measurement error and aggregate response mismatch relative to a fixed nominal candidate dictionary.Source: B. Separation and Deterministic… · p. 4
• Under stochastic noise with unbounded support, no finite-data method guarantees deterministic 100% point accuracy; confidence levels, error probabilities, or asymptotic consistency are relevant instead.Source: B. Separation and Deterministic… · p. 4

C. Sensor Placement

• Choose measurements to maximize worst-case projective separation while retaining adequate sensitivity; this is a design principle, not a PMU-placement optimization performed in the paper.Source: C. Sensor Placement · p. 5
• Large modal responses alone do not ensure localization when candidates share the same normalized spatial signature.Source: C. Sensor Placement · p. 5

V. A DIAGNOSTIC FRAMEWORK FOR SOURCE LOCALIZATION

• The four-step diagnostic framework tests raw-measurement identifiability, robust separation under uncertainty, preservation by feature extraction, and consistency between the interpretation and the oscillation mechanism.It is not intended to rank or replace DEF or other mechanism-based methods.Source: §V · p. 5

A. Measurement and Feature Hierarchy

• The localization pipeline maps a candidate's raw trajectory or harmonic measurement through a deterministic feature map Φ before applying decision rule D; at one frequency, unknown amplitude and phase scale the raw signature.Source: A. Measurement and Feature Hierarchy · p. 5
• DEF and CDEF form terminal branch and bus features from voltage-current relations, so they may miss source information expressed primarily through a volt-var relation.The implemented bus decision selects the largest signed score, or the least-negative score if all are negative.Source: A. Measurement and Feature Hierarchy · p. 5

B. Information Upper Bound for Feature-Based Localization

• A deterministic feature map cannot make overlapping raw-measurement hypotheses universally distinguishable, and a many-to-one map can collapse distinct raw inputs into the same feature.Source: B. Information Upper Bound for… · p. 5
• Distinct wide-area voltage-current patterns can yield the same scalar DEF value, terminal ranking, or projected CDEF score, indicating feature-projection loss rather than identical raw responses.A richer CDEF representation may preserve distinctions lost by scalar DEF, but it remains constrained by the same measurements.Source: B. Information Upper Bound for… · p. 6

1) Uncorrectable branch—structural internal ambiguity:

• When two internal interventions have identical projective signatures, unknown amplitude and phase absorb their complex gain difference, leaving the same steady downstream measurements.No alternative terminal feature can correct this structural ambiguity.Source: 1) Uncorrectable branch—structural… · p. 6
• The ambiguity can be removed only with additional information, such as an intermediate measurement, distinguishable onset-transient dynamics, or restrictions on the source waveform or internal realization.Source: 1) Uncorrectable branch—structural… · p. 6

2) Correctable branch—mechanism mismatch or feature-projection loss:

• A mechanism mismatch or overly aggressive feature projection can hide distinctions that an information-preserving feature could retain.A voltage–var measure may better expose relevant information for volt–var mechanisms than the voltage–current pairing underlying conventional DEF; it cannot overcome a raw-data identifiability limit.Source: 2) Correctable branch—mechanism… · p. 6
• A Q–V measure is not necessarily effective for mechanical-power, angle, or current-control mechanisms, and weak oscillatory visibility can leave it poorly conditioned.It also cannot resolve internal ambiguity; mechanism-based features are corrections within the raw-data identifiability limit, not substitutes for an identifiability test.Source: 2) Correctable branch—mechanism… · p. 6

D. Poor conditioning and model error

• Distinct candidate rays can remain identifiable in principle yet be practically indistinguishable when their separation is comparable to measurement, phasor-extraction, or model error.A small margin can let modest changes in filtering, terminal data, resistance compensation, or feature projection change the ranking.Source: D. Poor conditioning and model error · pp. 6–7
• Model perturbations can reduce realized separation or reverse residual ordering, while improved calibration, sensor placement, or uncertainty-aware set-valued decisions can restore a reliable margin.These changes need not alter the structural identifiability class.Source: D. Poor conditioning and model error · p. 6

E. Four-Step Diagnostic Procedure and Failure Classes

• Exact localization requires nonzero, pairwise projectively distinct candidate signatures; the four diagnostic steps then test raw identifiability, robust separation, feature preservation, and mechanism-consistent interpretation.Nearly collinear signatures can be identifiable in principle but exchange residual rankings under modest error; distinct raw responses can also collapse to the same feature.Source: E. Four-Step Diagnostic Procedure and… · p. 7

A. Common Numerical Protocol

• The numerical studies test measurement distinguishability, robustness to measurement and model uncertainty, and whether terminal features preserve raw distinctions under a common base model, operating point, forcing frequency, and measurement set.Reference simulations construct each candidate dictionary; separate test events and settled-window harmonic estimates evaluate localization.Source: A. Common Numerical Protocol · p. 7
• The model-based benchmark selects the candidate with the minimum normalized residual, while terminal comparisons fix DEF, CDEF, or Q–V implementations and their feature settings.Under uncertainty, the locator may retain multiple candidates, indicating practical ambiguity rather than exact equivalence unless signatures are analytically identical up to complex scale.Source: A. Common Numerical Protocol · p. 7

B. Validation on a Two-Area System

• All eight independent Kundur test events uniquely identified the correct source in the well-separated 0.4-Hz governor-reference positive control.The minimum pairwise projective separation was 0.952, and the minimum best-to-second-best normalized squared-residual margin was 0.710.Source: B. Validation on a Two-Area System · p. 8
• For a 0.4-Hz voltage-reference event, raw signatures identified G1, while DEF identified it in two of four windows and the in-phase Q–V score did so in all four.The result does not establish a unique cause for each incorrect DEF window.Source: B. Validation on a Two-Area System · p. 8
• The fixed largest-signed-slope rule assigned all four well-separated G2 and G4 governor-reference trajectories to G2 despite distinct four-bus DEF vectors.For the G4 cases, all slopes were negative, so the rule selected the least-negative terminal; the study illustrates decision-stage information loss.Source: B. Validation on a Two-Area System · p. 8, Figure 3 · p. 8
• With only the line 7–8 active-power fundamental, G3 and G1 signatures were exactly indistinguishable, while adding the line 10–11 current angle raised their separation to 0.988.This illustrates measurement-induced ambiguity and its removal through sensor augmentation.Source: B. Validation on a Two-Area System · p. 8
• Area aggregation reduced G2/G3 DEF feature separation to 0.562, yet pairwise matching still correctly identified both held-out trajectories.The study cautions that raw-data and feature-space distances have different scaling and uncertainty levels and should not share a robustness threshold.Source: B. Validation on a Two-Area System · p. 8
• At 0.7 Hz, the G3/G4 separation was 3.08×10^-5 and the nominal residual margin was 4.47×10^-10; two of six ±1% perturbations misclassified the test event.With a residual-set tolerance of 10^-5, the nominal event and all six perturbed cases retained G1–G4 as admissible candidates, reflecting sensitivity rather than exact nonidentifiability.Source: B. Validation on a Two-Area System · pp. 8–9

C. Studies on IEEE–NASPI Contest Models

• All 21 IEEE–NASPI Case 1 test events were localized to the correct bus across three settled windows per event.The dictionary included 103 in-service generator interventions spanning 48 buses; the result establishes bus-level localization, not unique identification among channels within a unit.Source: 2) Case 1—end-to-end recovery · p. 9

3) Case 4—candidate-set/source-label coverage:

• With four terminal measurement pairs, both source buses are correctly identified in all three windows across 12 trajectories per source, but restricting candidates to eight measured-branch buses excludes bus 2438.The raw signatures have projective separation 0.905 versus maximum within-source variation 0.0162; the DEF responses remain distinct, so the failure is at the decision stage.Source: 3) Case 4—candidate-set/source-label… · p. 9

4) Case 7—mechanism dependence:

• Raw harmonic signatures identify bus 2634 in all 36 settled windows for both forcing mechanisms, while a mismatched feature–mechanism pairing yields as few as 0/36 correct decisions.This demonstrates mechanism dependence of the tested features, not a ranking among methods.Source: 4) Case 7—mechanism dependence · p. 9, Table III · p. 10
• Under exciter forcing, DEF and CDEF both select bus 2604, although CDEF improves bus 2634’s median rank from 66.5 to 6.CDEF changes the retained representation, but larger aggregated scores at bus 2604 still determine the same top-1 decision.Source: 4) Case 7—mechanism dependence · p. 9

5) Case 3—measurement-induced ambiguity:

• One current-magnitude harmonic coefficient makes buses 1032 and 1131 projectively equivalent, producing ambiguity in all 48 individual-window decisions.Extending measurements to 58 voltage/current channels separates the signatures and correctly identifies the source in all 48 decisions; the full terminal layout increases separation, while generator measurements add little.Source: 5) Case 3—measurement-induced ambiguity · p. 10, Figure 4 · p. 10

VII. DISCUSSION AND CONCLUSION

• Exact localization is conditional on the declared dictionary, measurements, model, frequency, and single-source assumption; closely spaced projective rays remain sensitive to measurement and model errors.The studies assume steady-state harmonic measurements for one constant-amplitude sinusoidal source; transient, time-varying, nonlinear, and multiple-source cases require separate formulations.Source: §VII · p. 10