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REVIEW 4 major objections 5 minor 22 references

Identification of Black-Box Inverter-Based Resource Control Using Hammerstein-Wiener Models

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2411.13213 v1 pith:DQBXKSHH submitted 2024-11-20 eess.SY cs.SY

Aldin Dželo , Amer Mešanović , Mirsad Cosovic This is my paper · ORCID
classification eess.SYcs.SY
keywords inverter-basedresourcesHammerstein-Wienermodelsblack-boxmodelidentificationgrid-forminginvertersgrid-followingeigenvalueanalysisdqreferenceframenonlinearsystem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

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.

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 (4)
  1. [Section V] 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.
  2. [Section III-A, Fig. 4] 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.
  3. [Sections IV-A and IV-B] 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.
  4. [Section IV-C] 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.
minor comments (5)
  1. [Section III-C, Algorithm 1] 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.
  2. [Section IV-B] 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.
  3. [Section IV-D, Fig. 14] 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.
  4. [General] 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.
  5. [Section V] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Hammerstein-Wiener identification is a standard fit-then-validate procedure, and the paper's only support gap (claimed frequency-domain utility) is non-circular.

full rationale

The derivation chain is: define a Hammerstein-Wiener structure (Fig. 2), decouple the MIMO inverter into three MISO subsystems (Section III-A, Fig. 4), generate estimation and validation data from black-box simulations (Section III-B), fit parameters by minimizing the loss function (Eq. 1), select structure by a space search over nonlinearity estimators and linear-block orders using NRMSE (Eq. 2) and FPE (Eq. 3) on estimation data (Algorithm 1), and then evaluate on separate validation data with residual analysis (Section IV-D). None of these steps define a target quantity in terms of itself or use a fitted parameter as its own prediction: the reported validation NRMSE values (95.23%, 96.89%, 99.47% for GFM; 92.79%, 95.84%, 93.74% for GFL) are comparisons against held-out waveforms, not algebraic consequences of the fitting objective. The MIMO-to-MISO decoupling and dq transformation are structural assumptions, not circular reductions. There are no load-bearing self-citations (references [15]-[19] are external prior work). The one legitimate weakness is in Section V, where the paper claims the identified model 'expands analysis options, enabling studies such as singular value analysis, advanced stability analysis, and frequency domain analysis' without showing that the linearization of the fitted HW model reproduces the black-box small-signal behavior; this is an omitted demonstration or correctness risk, not a circular derivation. Therefore the circularity score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper contributes an algorithmic framework; its dependencies are standard system identification assumptions plus modeling choices. The main free parameters are the arbitrary threshold and test signal ranges, which affect reported results but not the central methodology.

free parameters (3)
  • Fit threshold = 92% NRMSE
    Arbitrarily chosen in Section IV to count 'possible model structures' and define success; affects the reported number of solutions.
  • Test signal voltage and frequency ranges = 0.9-1.1 pu, +/-0.5 Hz
    Hand-chosen operating envelope in Section III-B that defines the excitation used for identification.
  • Epsilon values in Algorithm 1 = Not specified
    Govern the accuracy-complexity trade-off in model selection; their values are not given, making the selection criteria partly undefined.
assumptions (4)
  • domain assumption The HW structure (static input/output nonlinearities plus linear dynamics) can represent the nonlinear dynamics of the inverter.
    Section III assumes this structure without structural justification; the good fits suggest it holds for the tested models.
  • domain assumption The MIMO inverter can be decoupled into three independent MISO HW subsystems.
    Section III-A decouples the MIMO structure into MISO models, assuming output channels are sufficiently independent.
  • domain assumption The dq transformation using the measured frequency is sufficient for identification.
    Section III-B uses measured frequency to compute the rotation angle, effectively making the frequency an input to the model.
  • domain assumption The generic GFM and GFL models from references [15]-[17] are representative of IBR implementations.
    Section IV evaluates the approach on two specific model formulations and generalizes from them.

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Cite this review

Pith. "Pith review of Identification of Black-Box Inverter-Based Resource Control Using Hammerstein-Wiener Models." pith.science (2026). https://pith.science/paper/DQBXKSHH

@misc{pith2026241113213,
  author       = {Pith},
  title        = {Pith review of: Identification of Black-Box Inverter-Based Resource Control Using Hammerstein-Wiener Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DQBXKSHH}},
  note         = {Machine review of arXiv:2411.13213}
}
read the original abstract

The development of more complex inverter-based resources (IBRs) control is becoming essential as a result of the growing share of renewable energy sources in power systems. Given the diverse range of control schemes, grid operators are typically provided with black-box models of IBRs from various equipment manufacturers. As such, they are integrated into simulation models of the entire power system for analysis, and due to their nature, they can only be simulated in the time domain. Other system analysis approaches, like eigenvalue analysis, cannot be applied, making the comprehensive analysis of defined systems more challenging. This work introduces an approach for identification of three-phase IBR models for grid-forming and grid-following inverters using Hammerstein-Wiener models. To this end, we define a simulation framework for the identification process, and select suitable evaluation metrics for the results. Finally, we evaluate the approach on generic grid-forming and grid-following inverter models showing good identification results.

Figures

Figures reproduced from arXiv: 2411.13213 by the authors.

Figure 1
Figure 1. Generalized control structure of GFM/GFL converter. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Basic HW structure. iq id ud uq f MIMO GFM/GFL control [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Identified model MIMO structure. idq udq0 f HW IDENTIFIED MODEL 2π s dq0 θt abc PCC Grid dq0 θt abc [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figures from the paper (7 more)
Figure 6
Figure 6. Figure 6: Control structure example of GFM inverter model formulation. [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]
Figure 7
Figure 7. Figure 7: Control structure example of GFL inverter model formulation. [PITH_FULL_IMAGE:figures/full_fig_p004_7.png]
Figure 8
Figure 8. Figure 8: Inputs (GFM mode) estimation data for normal operating mode. [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]
Figure 10
Figure 10. Figure 10: Comparison between actual model and identified model outputs [PITH_FULL_IMAGE:figures/full_fig_p005_10.png]
Figure 11
Figure 11. Figure 11: Comparison between actual model and identified model outputs [PITH_FULL_IMAGE:figures/full_fig_p006_11.png]
Figure 12
Figure 12. Figure 12: Validation microgrid scheme for model in simulation. [PITH_FULL_IMAGE:figures/full_fig_p006_12.png]
Figure 14
Figure 14. Figure 14: Residual analysis results for the identified model. [PITH_FULL_IMAGE:figures/full_fig_p007_14.png]

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