Physics-Informed Learning of Proprietary Inverter Models for Grid Dynamic Studies

Kyung-Bin Kwon; Marcelo Elizondo; Ramij R. Hossain; Sayak Mukherjee

REVIEW 4 major objections 5 minor 8 references

Physics-Informed Learning of Proprietary Inverter Models for Grid Dynamic Studies

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A physics-informed latent neural ODE can emulate proprietary grid-forming inverter dynamics from measured trajectories, cutting voltage RMSE by over 30% and frequency RMSE by over 60% compared with an RNN baseline.

desk verdict Useful architecture idea undercut by an observability mismatch: the emulator is trained on internal controller states that a black-box unit would not reveal, and the baseline comparison is weak. read the letter →

arxiv 2507.15259 v1 pith:LPBPXBYG submitted 2025-07-21 eess.SY cs.LGcs.SY

Kyung-Bin Kwon , Sayak Mukherjee , Ramij R. Hossain , Marcelo Elizondo This is my paper · ORCID

classification eess.SYcs.LGcs.SY

keywords physics-informedmachinelearninglatentneuralODEgrid-forminginvertersinverter-basedresourcesproprietarymodelemulationpowersystemdynamicsimulationdroopcontrolblack-boxdynamics

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

The paper claims that a hybrid model—a latent neural ODE with a generic droop-control inverter model embedded as a physics prior—can emulate the dynamic behavior of a grid-forming inverter whose internal controls are kept secret by the manufacturer. Using only six measured trajectories from the black-box unit, the model learns the unmodeled residual dynamics and produces voltage and frequency predictions that track the true inverter better than a pure recurrent neural network. If this works in practice, utilities and grid operators can build accurate dynamic simulators for stability studies, gain tuning, and digital twins without waiting for manufacturers to disclose their controllers. The numerical demonstration on a grid-forming inverter reports a 33% reduction in voltage RMSE and a 62% reduction in frequency RMSE compared with the RNN baseline.

What carries the argument

The central object is the Physics-Informed Latent Neural ODE (PI-LNM), a latent ODE whose state is a concatenation of a neural state $z$ and a physics state $z_r$. The physics state evolves under the representative droop model $\dot{z}_r = \hat{f}_{gfm}(z_r, z, t)$, while the neural state evolves under a learned function $f_\theta$; an ODE-RNN encoder maps the observed sequence to the distribution of the initial latent state, and an MLP decoder maps latent states back to observations. Training maximizes the ELBO in Eq. (5). The physics block is what carries the argument: it anchors the latent dynamics to known droop behavior, so the neural part only has to learn the residual between the generic model and the true proprietary controller.

What would settle it

Train PI-LNM on load-step responses, then replay a three-phase fault or line trip with the same manufacturer model and compare trajectories with the emulator; if the voltage or frequency error grows to match or exceed the RNN baseline, the claim that physics guidance generalizes fails.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that the unknown manufacturer controller dynamics — written $\hat{f}_{gfm}$ — can be approximated by a physics-informed latent neural ODE model. The proposed PI-LNM splits the latent state into a learned part driven by a neural network $f_\theta$ and a physical part driven by $\hat{f}_{gfm}$, the generic droop-controlled grid-forming model REGFM_A1 with active-power/frequency and reactive-power/voltage droop. Training maximizes the evidence lower bound of a latent ODE with an ODE-RNN encoder, using 2000 load-step trajectories sampled every 0.01 seconds. With 24 latent variables and a tolerance of ±20% error in the droop and gain parameters used in the physics block, the learned model reproduces the true inverter trajectories; test RMSE drops from $6.0 \times 10^{-3}$ p.u. to $4.0 \times 10^{-3}$ p.u. for voltage and from $5.8 \times 10^{-3}$ Hz to $2.2 \times 10^{-3}$ Hz for frequency relative to an RNN without physics guidance.

Load-bearing premise

The method depends on the assumption that the true proprietary controller behaves like the generic droop model plus a residual that a 24-dimensional neural ODE can learn from six measured variables; if the real controller has a fundamentally different structure, such as current limiting or hidden states, the learned residual will not represent it.

Editorial extensions

If this is right

If the learned emulator generalizes beyond the training load steps, utilities can replace black-box manufacturer models in dynamic simulations with a compact surrogate that still respects known droop physics.
Because the physics block is explicit, the learned emulator can be interrogated for internal quantities such as angle, frequency, and voltage, which are useful for stability analysis and control gain tuning without manufacturer disclosure.
The reported gains over the RNN baseline support the claim that embedding physical priors improves both accuracy and learning efficiency for inverter dynamics.
The ±20% parameter mismatch tolerated by the physics prior suggests the approach works in the realistic operator situation where droop and gain parameters are only approximately known.
The latent-ODE structure permits continuous-time rollouts at arbitrary step sizes, which fits the variable-resolution needs of grid dynamic simulations.

Reading between the lines

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

A natural next test is to expose the trained emulator to event types absent from training—short circuits, line trips, islanding, or ramps—since the reported validation uses load steps at t = 0; generalization to faults is what would make the method useful for stability studies.
The same architecture should transfer to other proprietary resources by swapping the physics block, for example to grid-following inverters or wind and solar plant controllers, because the mechanism only assumes a nominal model plus a learnable residual.
If the true manufacturer controller contains hard nonlinearities such as current limiting or anti-windup, the learned residual may fail outside the training envelope; testing on large disturbances would reveal the boundaries of validity.
The non-autoregressive generative formulation could be extended to irregularly sampled or noisy field measurements, a regime where recurrent baselines typically degrade.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a reasonable hybrid architecture — latent ODE with a fixed droop-control wrapper — and the 33%/62% RMSE numbers are real for the test they ran. But the test doesn't match the motivating scenario: they train and test on internal states (voltage control error V_e and internal voltage magnitude V) that a proprietary unit would not give you at the point of interconnection. So the central practical claim, that you can emulate a black-box OEM inverter from externally measurable data, is not actually tested.

What's new: the specific PI-LNM block, embedding a nominal REGFM_A1 droop model as a physics prior alongside a learned latent ODE, is a reasonable combination not in the cited prior work. The paper is clear about the architecture and the training procedure, and using ±20% error on the approximate droop/gain parameters is a fair way to respect the unknown-parameter setting. Those are genuine positives.

Soft spots, in order of size. First, the observability mismatch. The observation vector in Section 3 includes V_e and V, which the text itself calls internal states. If an operator can only measure terminal quantities, the inverse problem is harder and this paper gives no evidence that the method works in that regime. That is a load-bearing gap, not a polish issue. Second, the RNN baseline is not converged (their own Fig. 3 shows that), so the comparison understates what a well-trained data-driven model could do; there are also no error bars or repeated runs. Third, the test scenario is structurally similar to training (load step at t=0, same bus), so the generalization claim isn't supported. The conclusion that physics 'enhances generalization' goes beyond the data. Fourth, no code or data release, which for a methods letter matters.

Where the paper is solid: the numbers reported are internally consistent, the method is reproducible in principle, and the idea of embedding a generic droop model as a fixed prior is sensible. The stress-test concern about internal states is correct and should be the main referee question.

Verdict: worth peer review, but only with major revision. The authors should either (a) redo the experiment using terminal observations only, or explicitly justify that V_e and V are available, e.g., through a cooperative OEM test setup, and (b) add a converged baseline, repeated runs, and out-of-sample tests. I'd send it to a reviewer who knows both power systems and neural ODEs; the risk is that the current version looks like a clean demonstration of an architecture tested on a generator of its own choosing.

Referee Report

4 major / 5 minor

Summary. The paper proposes a physics-informed latent neural ODE model (PI-LNM) to emulate the dynamics of proprietary grid-forming inverter (GFM) models. The architecture embeds a generic droop-controlled GFM model (REGFM_A1) as a physics prior in the latent ODE while using neural networks to learn residual dynamics. The method is trained on data generated from an emulator and compared against an RNN baseline in a load-step test. The reported results show a 33.3% RMSE improvement in voltage and a 62.1% improvement in frequency. The authors claim that embedding physics enhances learning efficiency and generalization, offering a practical pathway for black-box inverter emulation.

Significance. If the result holds, the proposed PI-LNM offers a promising direction for operators who need accurate dynamic models of inverters whose internal controls are proprietary. The combination of a latent ODE with a physically motivated prior is a sensible and potentially useful extension of existing latent-dynamics methods. However, the validation is narrow: the paper uses a single test scenario, does not compare against a demonstrably converged baseline, and—most importantly—uses internal states that the paper's own motivating scenario assumes are unavailable. The claimed practical benefit therefore rests on an untested observability assumption, and the generalization of the method beyond the specific emulator setup remains unestablished.

major comments (4)

[Section 3 (observation vector)] The paper's motivation is that OEMs do not disclose internal controls and parameters of inverters, leaving operators with only externally measurable quantities. However, the observation vector x := [θ_j, ω_j, V_e_j, V_j, P_n_j, Q_n_j] includes V_e_j and V_j, which the text explicitly identifies as the voltage control error and the internal voltage magnitude of the GFM model. These are internal controller states of the REGFM_A1-type model, not quantities available at the point of interconnection for a proprietary unit. The reported 33.3% and 62.1% RMSE improvements are therefore obtained under an observation model that the paper's own motivating problem rules out. The central claim should be conditioned on observability of these internal states, or the method should be demonstrated from terminal measurements alone (e.g., θ, ω, P_n, Q_n, and terminal voltage magnitude). As written, the paper does not provide evidence that the method works in the black-box scenario it is intended to address.
[Section 3 (baseline comparison)] The RNN baseline is reported as 'not converged' at iteration 2000 (Fig. 3), but no evidence is provided that the baseline was given a fair chance: there is no discussion of hyperparameter tuning, no training curves beyond 2000 iterations, and no error bars or multiple random seeds. The comparison in Table 1 may therefore reflect an undertrained baseline rather than a fundamental advantage of physics-informed learning. To support the claim that PI-LNM 'outperforms an RNN baseline,' the authors should show that the RNN has converged (e.g., via validation loss over a longer training horizon or early stopping) and should report mean and standard deviation of RMSE over several independent runs.
[Section 3 (data generation and emulator structure)] The paper states that training data is generated from an emulator with an 'assumed proprietary restricted GFM unit' but does not specify the structure of this actual GFM model. If the emulator is simply the same REGFM_A1 droop model with different parameter values, then the learned residual only corrects a parameter mismatch, which is a substantially easier problem than emulating a structurally different OEM controller (e.g., one with current limiting, different inner loops, or additional states). The representability assumption behind Eq. (3) is therefore untested. Please specify the emulator's actual model structure and, ideally, test the method against a structurally different proprietary model to support the general claim of black-box emulation.
[Section 3 (validation breadth)] The test consists of a single load-step event at one bus, observed for 5 seconds. The conclusion claims that PI-LNM 'can effectively mimic the dynamics of the actual GFM' and that the approach 'improves dynamic simulation accuracy,' but no evidence is given for other event types (faults, setpoint changes, variations in grid strength) or for statistical significance over multiple trials. The paper should include additional test scenarios, at minimum a few different load-step magnitudes and possibly a different event type, and report the distribution of errors, to support the broader claims in the abstract and conclusion.

minor comments (5)

[Section 2, Eq. (1)] The notation in Eq. (1) uses x_f and ẋ_f = N_θ(ˆf_gfm(x_f, V)), but the proposed framework is then described with latent states z and z_r in Eq. (3). Please clarify how Eq. (1) relates to the latent-state formulation, or unify the notation.
[Section 3, Fig. 2] The paper refers to the proposed model as 'LNM' in the caption of Fig. 2 and as 'PI-LNM' throughout the rest of the text; please use a single consistent name.
[Section 3] The sentence 'we assume that the exact values of the GFM parameters f_GFM are unknown' uses the symbol f_GFM, which is not defined; the intended meaning is presumably the parameter set of the actual GFM model. Please define all symbols in one place.
[Abstract] The phrase 'a novel physics-informed neural ordinary differential equations-based framework' is grammatically awkward; consider 'a physics-informed neural-ODE-based framework.'
[Section 3, Fig. 5] The test-time performance figure is not described in the text; please add a sentence explaining what the plotted curves are and how the comparison should be read.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the physics prior is an independently specified generic GFM model, and the neural parameters are fitted to separately generated emulator data; the result does not reduce to its inputs by construction.

full rationale

The paper's central derivation is self-contained. The proposed PI-LNM embeds a nominal droop-control GFM model, taken from the WECC generic REGFM_A1 specification, into a latent neural ODE. The neural parameters are learned by maximizing the ELBO on trajectories generated from a separate emulator that represents the assumed proprietary unit. The physics prior is not fitted to the target data; rather, the prior uses droop and gain parameters intentionally perturbed by ±20% from the true values, and the neural component learns the residual. This is a standard physics-informed learning setup rather than a circular one. The reported RMSE improvements are measured against an RNN baseline on rollouts from the trained model, not against quantities used to fit the physics block. No fitted parameter is renamed as a prediction, and no uniqueness theorem is invoked to force the choice of architecture. The only self-citation is Ref. [4], used to note that synchronous generator dynamics are well-established; this is peripheral and not load-bearing for the GFM emulation claim. A reviewer concern about the observability of internal states such as voltage control error and internal voltage magnitude is a legitimate scope limitation, but it is not circularity: the paper does not rely on an assumption that these states are unavailable in its test setup, and the claimed reductions do not follow by construction from the training data. Overall, the derivation is not equivalent to its inputs, and no circular step was found.

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

The method rests on the assumption that a nominal droop-control model plus a learned latent residual can capture proprietary GFM behavior, and on the sufficiency of the measured state for forecasting. The approximate +/-20% parameter error and hand-set hyperparameters are inputs to the pipeline, not fitted outputs. The neural weights are learned from emulator-generated data, so the empirical claim is not circular, but it is also not independently reproducible from the paper.

free parameters (3)

Neural network weights of encoder, decoder, and latent ODE f_theta = learned from 2000 training samples
Trained by maximizing the ELBO; these fitted parameters define the learned dynamics and therefore the central claim.
Approximate GFM droop and gain parameters m_p, m_q, k_pv, k_iv = nominal values perturbed by +/-20%
Used as fixed physics prior in Eq. (4); the choice affects the physics-informed component and is a modeling assumption.
Hyperparameters: latent dimension, batch size, initial learning rate, iterations = 24, 200, 0.02, 2000
Set by hand in Section 3 with no sensitivity study. The reported improvements may depend on these choices.

assumptions (4)

domain assumption The true proprietary GFM dynamics can be decomposed as the nominal REGFM_A1 droop dynamics plus a residual learnable in a 24-dimensional latent ODE.
Eq. (3) and Step 2 of Section 2.2; the paper does not test cases where the proprietary controller has a fundamentally different structure.
domain assumption The measured state vector x = [theta_j, omega_j, V_e_j, V_j, P_n_j, Q_n_j] is a sufficient observation for the proprietary dynamics and for the learned latent decoder.
Assumed in Section 3 data generation; if the proprietary model has unobservable internal states that affect the future, the decoder may not reconstruct them.
standard math Numerical ODE solutions during training and evaluation are accurate enough not to bias the learned model.
ODESolve in Eqs. (2)-(4) is treated as exact; no solver tolerances or integration errors are reported.
domain assumption Load-step disturbances at one bus are representative of grid dynamic studies and of the OEM's proprietary model behavior.
Validation in Section 3 uses random load steps only; no faults, topology changes, or multi-machine interactions are considered.

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Cite this review

Pith. "Pith review of Physics-Informed Learning of Proprietary Inverter Models for Grid Dynamic Studies." pith.science (2026). https://pith.science/paper/LPBPXBYG

@misc{pith2026250715259,
  author       = {Pith},
  title        = {Pith review of: Physics-Informed Learning of Proprietary Inverter Models for Grid Dynamic Studies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LPBPXBYG}},
  note         = {Machine review of arXiv:2507.15259}
}

read the original abstract

This letter develops a novel physics-informed neural ordinary differential equations-based framework to emulate the proprietary dynamics of the inverters -- essential for improved accuracy in grid dynamic simulations. In current industry practice, the original equipment manufacturers (OEMs) often do not disclose the exact internal controls and parameters of the inverters, posing significant challenges in performing accurate dynamic simulations and other relevant studies, such as gain tunings for stability analysis and controls. To address this, we propose a Physics-Informed Latent Neural ODE Model (PI-LNM) that integrates system physics with neural learning layers to capture the unmodeled behaviors of proprietary units. The proposed method is validated using a grid-forming inverter (GFM) case study, demonstrating improved dynamic simulation accuracy over approaches that rely solely on data-driven learning without physics-based guidance.

Figures

Figures reproduced from arXiv: 2507.15259 by the authors.

**Figure 2.** PI-LNM training performance [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Insufficient training performance of RNN-based (not physics-informed) modeling framework [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Trained PI-LNM GFM model integrated to simulation [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Test-time performance evaluation 4 Conclusion In this paper, we developed a PI-LNM framework to emulate proprietary GFM dynamics by embedding nominal droop-control priors into a neural-ODE architecture. The proposed PI-LNM combines an ODE-RNN encoder with a physics-au…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

8 extracted references · 6 canonical work pages

[1]

UNIFI specifications for grid-forming inverter-based resources (Version 2),

B. Kroposki et al., “UNIFI specifications for grid-forming inverter-based resources (Version 2),” National Renewable Energy Laboratory (NREL), Golden, CO, USA, Technical Report NREL/TP- 5D00-89269, 2024. 6

work page 2024
[2]

Generative adversarial nets,

I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y . Bengio, “Generative adversarial nets,” inAdvances in Neural Information Processing Systems , 2014, pp. 2672–2680

work page 2014
[3]

Physics-informed neural networks: A deep learn- ing framework for solving forward and inverse problems involving nonlinear partial differential equations,

M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks: A deep learn- ing framework for solving forward and inverse problems involving nonlinear partial differential equations,” Journal of Computational Physics, vol. 378, pp. 686–707, 2019

work page 2019
[4]

Coherency-Aware Learning Control of Inverter-Dominated Grids: A Distributed Risk-Constrained Approach,

K.-B. Kwon, R. Hossain, S. Mukherjee, K. Chatterjee, S. Kundu, S. Nekkalapu, and M. Elizondo, “Coherency-Aware Learning Control of Inverter-Dominated Grids: A Distributed Risk-Constrained Approach,” IEEE Control Systems Letters, vol. 8, pp. 2120–2125, 2024

work page 2024
[5]

Model specification of droop-controlled, grid-forming inverters (REGFM_A1),

W. Du et al, “Model specification of droop-controlled, grid-forming inverters (REGFM_A1),” Pa- cific Northwest National Lab, USA, Tech. Rep., 2023

work page 2023
[6]

On Approximating the Dynamic Response of Synchronous Generators via Operator Learning: A Step Towards Building Deep Operator-based Power Grid Simulators

C. Moya, G. Lin, T. Zhao, and M. Yue, “On approximating the dynamic response of synchronous generators via operator learning: A step towards building deep operator-based power grid simula- tors,” arXiv preprint arXiv:2301.12538, 2023

work page Pith review arXiv 2023
[7]

Latent ODEs for irregularly-sampled time series,

Y . Rubanova, R. T. Q. Chen, and D. Duvenaud, “Latent ODEs for irregularly-sampled time series,” in Advances in Neural Information Processing Systems, vol. 32, 2019. [Online]. Available: https://arxiv.org/abs/1907.03907

arXiv 2019
[8]

Neural Ordinary Differential Equations,

R. T. Q. Chen, Y . Rubanova, J. Bettencourt, and D. Duvenaud, “Neural Ordinary Differential Equations,” in Advances in Neural Information Processing Systems , vol. 31, 2018. [Online]. Available: https://arxiv.org/abs/1806.07366 7

arXiv 2018

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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UNIFI specifications for grid-forming inverter-based resources (Version 2),

B. Kroposki et al., “UNIFI specifications for grid-forming inverter-based resources (Version 2),” National Renewable Energy Laboratory (NREL), Golden, CO, USA, Technical Report NREL/TP- 5D00-89269, 2024. 6

work page 2024

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Generative adversarial nets,

I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y . Bengio, “Generative adversarial nets,” inAdvances in Neural Information Processing Systems , 2014, pp. 2672–2680

work page 2014

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Physics-informed neural networks: A deep learn- ing framework for solving forward and inverse problems involving nonlinear partial differential equations,

M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks: A deep learn- ing framework for solving forward and inverse problems involving nonlinear partial differential equations,” Journal of Computational Physics, vol. 378, pp. 686–707, 2019

work page 2019

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Coherency-Aware Learning Control of Inverter-Dominated Grids: A Distributed Risk-Constrained Approach,

K.-B. Kwon, R. Hossain, S. Mukherjee, K. Chatterjee, S. Kundu, S. Nekkalapu, and M. Elizondo, “Coherency-Aware Learning Control of Inverter-Dominated Grids: A Distributed Risk-Constrained Approach,” IEEE Control Systems Letters, vol. 8, pp. 2120–2125, 2024

work page 2024

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Model specification of droop-controlled, grid-forming inverters (REGFM_A1),

W. Du et al, “Model specification of droop-controlled, grid-forming inverters (REGFM_A1),” Pa- cific Northwest National Lab, USA, Tech. Rep., 2023

work page 2023

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On Approximating the Dynamic Response of Synchronous Generators via Operator Learning: A Step Towards Building Deep Operator-based Power Grid Simulators

C. Moya, G. Lin, T. Zhao, and M. Yue, “On approximating the dynamic response of synchronous generators via operator learning: A step towards building deep operator-based power grid simula- tors,” arXiv preprint arXiv:2301.12538, 2023

work page Pith review arXiv 2023

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Latent ODEs for irregularly-sampled time series,

Y . Rubanova, R. T. Q. Chen, and D. Duvenaud, “Latent ODEs for irregularly-sampled time series,” in Advances in Neural Information Processing Systems, vol. 32, 2019. [Online]. Available: https://arxiv.org/abs/1907.03907

arXiv 2019

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Neural Ordinary Differential Equations,

R. T. Q. Chen, Y . Rubanova, J. Bettencourt, and D. Duvenaud, “Neural Ordinary Differential Equations,” in Advances in Neural Information Processing Systems , vol. 31, 2018. [Online]. Available: https://arxiv.org/abs/1806.07366 7

arXiv 2018