{"id":"1d12b6ce-341d-4826-b552-1e65d00c0c3d","arxiv_id":"2411.13213","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A Hammerstein-Wiener identification procedure produces high-accuracy surrogate models for three-phase grid-forming and grid-following inverters.","lead":"The paper shows that a standard black-box modeling technique, Hammerstein-Wiener, can approximate the behavior of grid-forming and grid-following inverters well enough for use in power system studies. This matters because grid operators often receive only black-box models from manufacturers, which until now could only be simulated in the time domain, not analyzed with stability tools.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Claimed eigenvalue/frequency-domain utility rests on unverified small-signal equivalence; time-domain NRMSE alone does not establish it.","rationale":"This paper is a credible application of standard Hammerstein-Wiener identification to inverter models, with a clear MIMO/MISO structure, a systematic search algorithm, and impressive time-domain validation fits. The microgrid experiment and residual analysis add useful evidence. However, the paper's stated purpose is to enable eigenvalue analysis and frequency-domain studies, and that claim is not supported by the validation. All reported accuracy metrics are time-domain fit measures (NRMSE, FPE, residual correlation); there is no comparison of the linearized identified model's eigenvalues or frequency response against the original black-box. For a nonlinear HW structure, a good time-domain fit does not guarantee that the linearized dynamics match the true system. Since the authors explicitly disclaim physical interpretability of the blocks, the missing small-signal validation is the exact condition needed for the central claim. The reader's identified weakest assumption (MISO decoupling) is also relevant and partially tested by the microgrid simulation, but the small-signal equivalence is entirely unaddressed. I therefore keep the CONDITIONAL verdict, adding the requirement that the authors provide a linearized-model comparison or frequency-domain validation, or explicitly temper the eigenvalue-analysis claim. This does not change the reader's overall verdict category, hence UNCHANGED.","tokens_in":8793,"tokens_out":6427,"duration_ms":73900,"concrete_test":"Using the identified GFM and GFL HW models from Sections IV-A/B and the original black-box models (Figs. 6 and 7), linearize each system at the nominal operating point (ud=1 pu, uq=0, id and iq at their steady-state values, f=50 Hz). For the HW model, linearize the static nonlinearities and combine with the linear block plus the 2π/s integrator and dq transforms from Fig. 5 to form a small-signal state-space model. For the black-box, linearize the full detailed model about the same operating point. Compare the eigenvalues and the 2×2 dq admittance (or impedance) matrix over 1–1000 Hz. If the dominant eigenvalues differ by more than 10% in damping ratio or the admittance response differs by more than 5 dB/10° at any frequency below 500 Hz, the identified model cannot support the claimed eigenvalue/frequency-domain studies without further refinement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central motivation is that black-box IBR models cannot be used for eigenvalue/frequency-domain analysis, and the identified HW model is offered as enabling such analysis. However, every validation presented — NRMSE (Eq. 2), residual analysis (Section IV-D), and the time-domain microgrid comparison (Section IV-C) — operates in the time domain. No result demonstrates that the linearization of the identified HW model reproduces the small-signal behavior of the original black-box. For a Hammerstein-Wiener structure (Fig. 2), small-signal properties at an operating point are governed by the slopes of the static nonlinearities and the linear dynamic block; these can be tuned to fit a transient waveform while giving an incorrect impedance magnitude/phase, resonant frequency, or damping. The conclusion concedes the identified blocks cannot be linked to physical subsystems, but it presents no frequency-domain or eigenvalue comparison to compensate. Thus the key use case of the identified model is asserted, not shown. This is load-bearing because if the linearized HW model differs from the black-box in the dq admittance or eigenvalues, the claimed expansion of analysis options in Section V fails, even though the reported NRMSE values are high.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a Hammerstein-Wiener (HW) identification framework for black-box three-phase inverter-based resource (IBR) models, aiming to cover both grid-forming (GFM) and grid-following (GFL) modes. The black-box inverter is represented by a MIMO structure whose inputs are dq-frame currents and whose outputs are dq-frame voltages and frequency; this MIMO structure is decoupled into three independent MISO HW subsystems. The authors design excitation signals in the dq frame, define a grid-search algorithm over nonlinearity estimators, linear-block orders, delays, and search methods, and select models using NRMSE and final prediction error. The approach is evaluated on generic GFM and GFL simulation models, with reported validation NRMSE values of 95.23%, 96.89%, and 99.47% for GFM and 92.79%, 95.84%, and 93.74% for GFL. Additional evidence includes a residual analysis and a qualitative microgrid simulation comparison. The paper concludes that the identified HW models enable eigenvalue, frequency-domain, and small-signal stability studies that black-box models do not support.","tokens_in":9026,"tokens_out":3233,"duration_ms":35971,"significance":"If the central claims are correct, the paper would offer a practical route from time-domain black-box IBR simulations to structures more amenable to analytical studies. The work has several strengths: the evaluation uses separate estimation and validation data sets, which is standard and not circular; the residual analysis in Section IV-D provides a conventional model-quality check; the algorithm is described in sufficient detail to be reimplemented; and the authors are explicit about the arbitrary 92% threshold and about the non-interpretability of the identified blocks. However, the significance is currently limited by the narrow empirical base (two generic model implementations), the unsupported claim that the identified models enable eigenvalue/frequency-domain analysis, and the untested MISO decoupling assumption. These points are substantive because they bear directly on the paper's stated novelty and on the practical value of the proposed models.","major_comments":[{"comment":"The conclusion states that the identified HW model 'expands analysis options, enabling studies such as singular value analysis, advanced stability analysis, and frequency domain analysis,' but no validation supports this. All reported accuracy metrics (NRMSE in Eq. (2), the residual analysis in Section IV-D, and the microgrid comparison in Section IV-C) are time-domain. For a Hammerstein-Wiener structure, small-signal behavior at an operating point is governed by the local slopes of the static nonlinearities and the linear dynamic block, and a model can fit transient time-domain waveforms while misrepresenting dq admittance, resonant peaks, or eigenvalues. The authors should either add a frequency-domain or linearization-based comparison against the original black-box model (e.g., impedance/admittance magnitude and phase, or eigenvalue locations) or explicitly weaken the conclusion to only claim time-domain fidelity.","section":"Section V"},{"comment":"The decoupling of the MIMO inverter into three independent MISO Hammerstein-Wiener models is assumed rather than justified. The text cites design flexibility and reduced computation time, but it does not provide evidence that interactions among the outputs (for example, through the PLL or the current-control loop) are negligible. If such cross-coupling is strong, each MISO block may capture only a projection of the true dynamics, and the three-block model may not reproduce the full MIMO behavior. A concrete test would be to compare the three-MISO model against a coupled MIMO HW model on the same validation data, or to examine cross-correlations between the residual of one output and the other outputs' inputs.","section":"Section III-A, Fig. 4"},{"comment":"The reported high NRMSE values are obtained for only one generic GFM model and one generic GFL model, both from the same simulation framework and with the same assumed input/output structure. The conclusion in Section V that the proposed solution is 'independent of subsystem variations in inverter implementations' is therefore overreaching. At minimum, the paper should evaluate the procedure on inverter models with different control structures (e.g., different PLL designs, droop versus virtual-synchronous-machine GFM controls) or should restrict the claim to the tested class of models. Without such evidence, the generality claim is a statement of intent rather than a demonstrated property.","section":"Sections IV-A and IV-B"},{"comment":"The microgrid experiment is presented as confirming stability and accuracy, but the comparison in Fig. 13 is qualitative only. No quantitative error metric, such as NRMSE or maximum deviation, is reported for the online simulation, and the text does not state which aspects of the identified model are being challenged by this scenario. This limits the strength of the 'online-working model' claim and makes it difficult for a reader to judge whether the microgrid test genuinely validates the model in a closed-loop setting.","section":"Section IV-C"}],"minor_comments":[{"comment":"The values of the tolerances epsilon_1 and epsilon_2 are never specified. Since Algorithm 1 is intended to define a reproducible identification procedure, the paper should either state representative values or describe how they were chosen in the numerical study.","section":"Section III-C, Algorithm 1"},{"comment":"The fit threshold of 92% is acknowledged as arbitrary. This is acceptable for reporting the number of candidate structures, but the repeated use of the phrase 'satisfying' results should be hedged, since the threshold does not have a statistical justification independent of the chosen validation data.","section":"Section IV-B"},{"comment":"The residual analysis discussion would benefit from explicitly noting that the nonzero autocorrelation at lag 0 is expected by definition and is not by itself an indication of model deficiency.","section":"Section IV-D, Fig. 14"},{"comment":"Some symbols are introduced informally: in Eq. (1), the weighting matrix W(theta) is described only as positive semidefinite, and the parameterization of the nonlinearities in Fig. 2 is not fully specified. A short remark on how W(theta) is set in the numerical study would aid reproducibility.","section":"General"},{"comment":"The sentence 'We believe this identification framework will deliver an upturn in this matter' is vague and could be replaced by a concrete statement of the intended next validation steps.","section":"Section V"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal and addresses a relevant practical problem. The main risk is that the central value proposition — that the identified HW models enable eigenvalue and frequency-domain analysis — is asserted but not demonstrated, and the generality claim rests on only two generic test cases. I believe these issues can be addressed within the manuscript's scope by adding frequency-domain/linearization validation, testing additional model implementations, and softening the generality claims. I do not see a fundamental flaw in the proposed identification pipeline itself."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a legitimate new application of Hammerstein-Wiener identification to three-phase grid-forming and grid-following inverters, and the reported time-domain fits are strong. The problem is that the authors motivate the whole thing with eigenvalue and frequency-domain analysis being impossible for black-box models, then never show that the identified HW model actually reproduces the black-box's small-signal behavior. That claim is asserted in the conclusion and supported only by time-domain NRMSE and a time-domain microgrid demo. The stress-test note is right: for an HW structure, the linearization is governed by the slopes of the static nonlinearities and the linear block, and a good transient fit does not by itself guarantee the right dq admittance or eigenvalues. If the authors want to sell the model as enabling eigenvalue analysis, they need to compare the linearized HW model against the original black-box on frequency response or small-signal eigenvalues.\n\nWhat is genuinely new: prior HW identification of inverters was single-phase and steady-state. Here they do a systematic MIMO-to-MISO decomposition in the dq frame, a structured space search over nonlinearity types, linear block orders, and delays, and they include residual analysis. The validation fits (95.23/96.89/99.47% for GFM, 92.79/95.84/93.74% for GFL) are credible, and the microgrid demonstration is a nice sanity check. I also give them credit for being candid: they state the 92% threshold was arbitrary and that the identified blocks cannot be linked to physical subsystems.\n\nSoft spots, in order: (1) the missing small-signal validation is the big one—it's the difference between a good time-domain surrogate and a model that enables the analysis they promise. (2) The generality claim rests on two generic models; that's not enough to say regardless of operating mode or inverter implementation. (3) No comparison against other black-box structures (NARX, ANN), so we don't know if the HW choice is actually better or just adequate. (4) No code or data, which makes reproduction harder. The MISO decoupling assumption is worth questioning too, since real inverters have cross-coupling through PLL and current loops, but the good fits suggest it is not a problem for these two cases.\n\nOverall, the identification procedure appears sound as far as it goes. The paper deserves a serious referee; the main thing a referee should push for is either a small-signal validation or a toned-down claim. If the authors add that, this becomes a useful reference for black-box inverter surrogate modeling. I wouldn't cite it in its current form because the key promised capability is unverified.\n\nRecommendation: send to peer review with a request for revision, focusing on the frequency-domain and eigenvalue gap.","headline":"Solid time-domain Hammerstein-Wiener identification for two generic inverter models, but the paper's promised eigenvalue and frequency-domain payoff is never actually demonstrated.","tokens_in":9532,"tokens_out":3055,"would_cite":false,"duration_ms":34012,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proposes a single Hammerstein-Wiener identification procedure that fits black-box three-phase grid-forming and grid-following inverter models with validation NRMSE above 92%, enabling eigenvalue and frequency-domain analysis of…","keywords":["inverter-based resources","Hammerstein-Wiener models","black-box model identification","grid-forming inverters","grid-following inverters","eigenvalue analysis","dq reference frame","nonlinear system identification"],"falsifier":"Run the same identification pipeline on a three-phase inverter known to have strong cross-coupling between the d and q axes, for example through its phase-locked loop or current-control loops, and check whether validation NRMSE drops below the 92% threshold or residual cross-correlation leaves the confidence interval; if it does, the MISO decoupling is the fragile step.","tokens_in":8628,"feed_emoji":"⚡","tokens_out":7233,"duration_ms":66515,"temperature":0.7,"pith_summary":"Grid operators receive inverter models from manufacturers as black boxes that can only be simulated in the time domain, which blocks eigenvalue and frequency-domain stability analysis. The paper introduces a general Hammerstein-Wiener identification procedure that replaces such black-box three-phase grid-forming and grid-following inverter models with compact input-output models. The procedure treats the inverter as a multi-input multi-output block with current inputs $i_d,i_q$ and voltage/frequency outputs $u_d,u_q,f$, splits it into three single-output Hammerstein-Wiener subsystems, and estimates them in the dq frame. Validation fits are reported as 95.23%, 96.89%, and 99.47% NRMSE for grid-forming and 92.79%, 95.84%, and 93.74% for grid-following inverters, with residual tests within confidence bounds. If correct, the approach lets operators perform stability studies on mixed inverter and synchronous-machine networks without knowing vendor control details.","feed_headline":"Identified models give black-box inverters eigenvalue analysis","feed_subtitle":"Validation fits reach 99% for grid-forming and 96% for grid-following, unlocking stability studies.","key_machinery":"The central object is the Hammerstein-Wiener model: a linear dynamic block sandwiched between two memoryless nonlinearities. The paper's load-bearing move is to impose a fixed MIMO wrapper on every inverter, with $i_d,i_q$ as inputs and $u_d,u_q,f$ as outputs, and then decouple it into three independent MISO Hammerstein-Wiener models, one per output. Each MISO model is fit by a space search over nonlinearity estimators, polynomial degree, linear block numerator and denominator order and delay, and numerical search method, with NRMSE and final prediction error as selection criteria. The dq frame, the rotating reference frame aligned with the grid angle, and the fixed wrapper let the same procedure run for both grid-forming and grid-following modes, while the MISO decoupling keeps the search tractable.","core_discovery":"The central claim is that one mode-independent Hammerstein-Wiener structure can reproduce the external behavior of three-phase grid-forming and grid-following inverters accurately enough for analysis. The black-box is represented with terminal currents as inputs and terminal voltages plus frequency as outputs, and this MIMO mapping is decoupled into three MISO Hammerstein-Wiener models, each built from a linear transfer function between two memoryless nonlinearities. Identification is performed in the dq rotating frame on estimation data gathered under normal operating conditions, with voltage variations between 0.9 and 1.1 pu and frequency variations within ±0.5 Hz, and the best model is chosen by trading NRMSE fit against Akaike's final prediction error. On validation data the reported fits are 95.23%, 96.89%, and 99.47% for the grid-forming example and 92.79%, 95.84%, and 93.74% for the grid-following example, and residual autocorrelation and cross-correlation stay within the 99% confidence bounds. The authors state that the identified blocks are not physically interpretable as inverter subsystems; the value is that the overall model enables eigenvalue, singular-value, and frequency-domain studies that black-box time-domain models do not allow.","pith_inferences":["A natural test of the MISO decoupling is to run the same pipeline on an inverter whose phase-locked loop or current-control loops strongly couple the d and q axes; if validation NRMSE falls below the 92% threshold, the independence assumption is the fragile step.","The paper's 92% fit threshold is arbitrary; a more direct test of the stated motivation would compare the eigenvalues of the identified model against the linearization of the true black-box model at the same operating point.","Because frequency is an output in both modes, the identified model may transfer to other synchronization schemes such as droop or virtual-synchronous-machine control, though the paper does not demonstrate that transfer."],"forward_implications":["Grid operators can run eigenvalue analysis, singular-value analysis, and frequency-domain studies on identified models of black-box inverter-based resources, analyses that time-domain-only vendor models do not support.","The same identification procedure applies to both grid-forming and grid-following inverters without changing the MIMO wrapper or the search algorithm.","The identified model can replace the black-box in a microgrid simulation and remains stable and accurate when the voltage reference is varied.","Because the procedure ranks candidate structures by NRMSE and final prediction error, it can return a simpler model with acceptable accuracy rather than the most complex one."],"supporting_citations":[{"why":"Supplies the Hammerstein-Wiener model structure and the identification methods used to fit each MISO block.","marker":"[18]"},{"why":"Provides the loss function, NRMSE and final prediction error definitions, and the numerical search implementations used in the identification toolbox.","marker":"[19]"},{"why":"Justifies the generalized MIMO representation of grid-forming and grid-following converters with voltage and frequency as outputs.","marker":"[15]"},{"why":"Supplies the generic grid-forming and grid-following inverter model formulations used as the black-box examples in the evaluation.","marker":"[17]"},{"why":"Supports the test-signal design and the decision to identify separate operating modes rather than one global model.","marker":"[20]"},{"why":"Supplies the residual autocorrelation and cross-correlation validation tests applied to the identified model.","marker":"[21]"},{"why":"Motivates the need for identified models by describing stability analysis challenges in grids with mixed inverter-based resources and synchronous machines.","marker":"[16]"}],"fun_headline_variants":["Hammerstein-Wiener tames black-box inverters for eigenvalue analysis","Black-box inverters get a key: Hammerstein-Wiener unlocks eigenvalue studies","Hammerstein-Wiener identification turns black-box IBRs into analyzable models","From black-box to eigenvalue-ready: Hammerstein-Wiener IBR identification"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that each output (d-axis voltage, q-axis voltage, and frequency) can be modelled independently from the two current inputs, so an inverter with strong coupling between those output channels would not be captured by the three separate Hammerstein-Wiener subsystems.","fun_headline_variants_meta":{"raw":{"variants":["Hammerstein-Wiener tames black-box inverters for eigenvalue analysis","Black-box inverters get a key: Hammerstein-Wiener unlocks eigenvalue studies","Hammerstein-Wiener identification turns black-box IBRs into analyzable models","From black-box to eigenvalue-ready: Hammerstein-Wiener IBR identification"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000324,"raw_usage":{"total_tokens":1830,"prompt_tokens":966,"completion_tokens":864,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":582,"completion_tokens_details":{"reasoning_tokens":777}},"tokens_in":582,"tokens_out":864,"duration_ms":7942,"temperature":1.0,"reasoning_tokens":777,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:40:42.012963+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same identification pipeline on a three-phase inverter known to have strong cross-coupling between the d and q axes, for example through its phase-locked loop or current-control loops, and check whether validation NRMSE drops below the 92% threshold or residual cross-correlation leaves the confidence interval; if it does, the MISO decoupling is the fragile step.","supporting_citations":[{"cited_title":"Identification of hammerstein–wiener models,","cited_arxiv_id":null,"evidence_quote":"Supplies the Hammerstein-Wiener model structure and the identification methods used to fit each MISO block."},{"cited_title":"Ljung, System identification toolbox: User’s guide","cited_arxiv_id":null,"evidence_quote":"Provides the loss function, NRMSE and final prediction error definitions, and the numerical search implementations used in the identification toolbox."},{"cited_title":"Grid- forming converters: Control approaches, grid-synchronization, and future trends-A review,","cited_arxiv_id":null,"evidence_quote":"Justifies the generalized MIMO representation of grid-forming and grid-following converters with voltage and frequency as outputs."},{"cited_title":"Modeling of grid-forming and grid- following inverters for dynamic simulation of large-scale distribution systems,","cited_arxiv_id":null,"evidence_quote":"Supplies the generic grid-forming and grid-following inverter model formulations used as the black-box examples in the evaluation."},{"cited_title":"Isermann and M","cited_arxiv_id":null,"evidence_quote":"Supports the test-signal design and the decision to identify separate operating modes rather than one global model."},{"cited_title":"System identification through the eyes of model validation,","cited_arxiv_id":null,"evidence_quote":"Supplies the residual autocorrelation and cross-correlation validation tests applied to the identified model."},{"cited_title":"Grid-forming inverter- based resource research landscape: Understanding the key assets for renewable-rich power systems,","cited_arxiv_id":null,"evidence_quote":"Motivates the need for identified models by describing stability analysis challenges in grids with mixed inverter-based resources and synchronous machines."}],"review_version":1}