Synthetic Power Flow Data Generation Using Physics-Informed Denoising Diffusion Probabilistic Models

Darshana Upadhyay; Junfei Wang; Marzia Zaman; Pirathayini Srikantha

REVIEW 4 major objections 5 minor 24 references

Synthetic Power Flow Data Generation Using Physics-Informed Denoising Diffusion Probabilistic Models

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims that a physics-informed diffusion model with a learned noise schedule can synthesize power-flow data that are statistically faithful and feasible, with residual imbalances of 0.013 p.u. (14-bus) and 0.017 p.u. (30-bus).

desk verdict A plausible first use of DDPMs for power flow data with a learned noise schedule, but the headline feasibility claim rests on an average imbalance metric, the GAN baseline actually has lower imbalance, and line-flow constraints are never checked. read the letter →

arxiv 2504.17210 v1 pith:TUK4CV5I submitted 2025-04-24 cs.LG cs.AI

Junfei Wang , Darshana Upadhyay , Marzia Zaman , Pirathayini Srikantha This is my paper · ORCID

classification cs.LGcs.AI

keywords denoisingdiffusionprobabilisticmodelspowerflowdatagenerationphysics-informedlosslearnednoiseschedulingsystemfeasibilitysyntheticIEEE14-bus30-bus

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 tries to establish that a denoising diffusion model can generate synthetic power-flow data that are both statistically faithful and physically feasible, if the training loss penalizes a power-balance residual at every denoising step and the noise schedule is learned so that the residual grows linearly through the forward process. On the IEEE 14-bus and 30-bus systems, the proposed model reports residual average power imbalances of 0.013 p.u. and 0.017 p.u., beating plain DDPMs and a physics-informed GAN on feasibility while matching distributional shape and diversity. If true, this would let researchers synthesize realistic, privacy-safe grid operating points for downstream tasks such as optimal power flow and state estimation. The feasibility argument depends on treating a small average imbalance as a proxy for being close to a truly feasible operating point.

What carries the argument

The central object is the power-imbalance residual $R(x_t)$, the average over buses of $|P_{Gi,t}-P_{Di,t}+j(Q_{Gi,t}-Q_{Di,t})-V_{i,t}\sum_j V_{j,t}^* Y_{ij}^*|$, together with the physics-informed loss $L_R=\max(R(x_t)-\gamma_t,0)$ applied at each diffusion step. The auxiliary machinery is a feedforward network $F_\omega$ that outputs a unified schedule $\bar\alpha_t(\omega)$ such that the expected imbalance at step $t$ follows the linear bound $\gamma_t=t\gamma_T/T$. Together these keep the reverse denoising process near feasible power-flow states at every step, rather than only at the final generation step.

What would settle it

For a batch of generated samples, run a full AC power-flow solver on each sample with the proposed demand and generation adjustment rule, and count how many fail the power-balance equations or line-flow limits by more than the reported tolerance; if a non-negligible fraction of samples cannot be projected to a feasible point while preserving the same load and generation distribution, the feasibility claim does not hold.

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Extended reading notes

Core claim

The central claim is that coupling DDPM training with a residual power-balance penalty and a learned diffusion schedule makes generated samples feasible as well as realistic. The physics loss $L_R(x_t)=\max(R(x_t)-\gamma_t,0)$ penalizes the average per-bus magnitude of the mismatch between injected power and the network flow computed from voltages and admittances, using $\gamma_t$ as a time-dependent tolerance bound taken from the forward-diffusion imbalance curve. The learned schedule reshapes the forward process so that imbalance rises linearly rather than saturating early, keeping every reverse step physically meaningful. On the 14-bus and 30-bus benchmarks, the result is synthetic data whose average power imbalance is about 0.01\text{--}0.02 p.u., comparable to a physics-informed GAN but with better statistical fidelity and diversity.

Load-bearing premise

The feasibility claim rests on the assumption that a small average power-imbalance residual, around 0.013 to 0.017 per unit, means the generated point is close enough to an actually feasible power-flow state that small demand and generation adjustments can repair it without changing the learned distribution; the paper neither gives the adjustment rule nor checks line-flow constraints.

Editorial extensions

If this is right

Generated data from the proposed model can serve as privacy-preserving replacements for real power-flow measurements in downstream data-driven modules, since the reported residual imbalance is small enough that slight demand and generation adjustments are claimed to absorb it.
The learned schedule makes the reverse process physically informed throughout all $T$ steps, avoiding the wasted second half of the diffusion process that occurs under the original noise schedule.
Incorporating physics knowledge lets the model produce feasible points in long-tail regions of the data distribution, which the paper interprets as improved generalization beyond the empirical training set.
The method outperforms three baselines on the reported metrics: physics-informed GAN, DDPM without physics loss, and DDPM with physics loss but the original schedule.

Reading between the lines

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

A natural extension the paper leaves implicit is full constraint checking: reporting the average $R(x_t)$ does not reveal per-sample maxima or violations of line-flow limits, so checking those would tell whether the 0.013 p.u. average is actually safe for operational use.
The same linear-residual scheduling idea could transfer to other physical systems with a computable conservation residual, such as gas networks, water distribution, or thermal grids, where a learned schedule that linearizes constraint-violation growth may stabilize diffusion training.
The generalization claim could be tested operationally by retraining an optimal power flow or state estimation model on the synthetic long-tail samples and measuring whether out-of-distribution performance improves; the paper does not report such a downstream task.

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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

The real novelty here is the combination: a DDPM tailored to power flow data, a physics-informed loss that penalizes power imbalance during reverse diffusion, and an auxiliary network that learns the noise schedule so that the forward imbalance grows roughly linearly. That last piece is a sensible idea and, to my knowledge, this is the first paper to apply DDPMs to power flow synthesis. The empirical comparison also earns some credit: the proposed model cuts average imbalance from 0.25/0.34 p.u. (plain DDPM) to 0.013/0.017 p.u., so the physics-informed loss clearly does something.

The soft spots are real and load-bearing. First, the feasibility claim is not demonstrated. The paper measures feasibility only through the mean power imbalance R(xt). That is a necessary condition but not sufficient: it can hide per-bus mismatches, and it says nothing about whether a nearby AC-feasible point exists or whether line-flow constraints [C6] are respected. The assertion that residual imbalances "can be fully absorbed by slight adjustments" is hand-waved; no adjustment rule is given and no power-flow solver is run on generated samples. Second, Table I undercuts the abstract: the physics-informed GAN shows 0.009 p.u. (14-bus) and 0.015 p.u. (30-bus), both lower than the proposed model's 0.013 and 0.017. The abstract claims the proposed model outperforms baselines in feasibility, but the paper's own table says otherwise. Third, the learned schedule is circular in part: the auxiliary network is fit to gamma_t, and Eq. 8 uses the same gamma_t as the physics-loss threshold, so Fig. 3c is partly enforcing the optimization target rather than validating it. There are also smaller issues: Eq. 2 writes the forward variance as alpha_bar_t * I, which contradicts Eq. 3 (should be (1 - alpha_bar_t) * I), no error bars are reported, no code or data are released, and the OOD generalization claim is inferred from histogram tails without a proper out-of-distribution test.

None of this makes the paper worthless. The direction is promising, the physics-informed loss is formulated cleanly, and the comparison to three baselines is a reasonable start. But the central claim—that generated data are feasible in any operational sense—is asserted, not done. If I were refereeing, I would ask for a direct feasibility audit: run an AC power-flow solver on generated samples, report per-bus imbalance and line-flow violations, and give a concrete adjustment rule. I would also ask the authors to discuss the GAN comparison honestly.

Who gets value from this? Researchers working on generative models for physical systems or synthetic data for power grids will find the learned-schedule idea worth reading, and the paper deserves a serious peer review. I would send it, but with the expectation of a major revision.

Recommendation: send to peer review, with the feasibility audit foremost in mind.

Referee Report

4 major / 5 minor

Summary. The paper proposes a physics-informed denoising diffusion probabilistic model (DDPM) for generating synthetic power flow data on the IEEE 14-bus and 30-bus systems. The method adds a physics-based loss that penalizes the average complex power imbalance R(xt) when it exceeds a step-dependent threshold gamma_t, and introduces an auxiliary network that learns a noise schedule alpha_t(omega) intended to make the imbalance grow linearly during the forward process. Experiments compare the proposed model against physics-informed GANs, standard DDPMs, and DDPMs with physics loss but standard schedule, reporting average residual imbalances around 0.01 p.u., and illustrate distributional fidelity with histograms of one generator. The central claim is that the framework 'ensures' generated data are both statistically faithful and feasible for power system use.

Significance. Synthetic power flow data generation is a timely problem for privacy-preserving machine learning in power systems, and the idea of embedding physical constraints into diffusion training is promising. The learned schedule for linearizing imbalance growth is an interesting design choice that, if validated independently, could be a useful contribution. However, the paper's headline feasibility claim is currently supported only by an average imbalance statistic, the main comparison table contradicts the stated superiority over baselines, and the learned-schedule evaluation is partially circular. The approach may be worth publishing after substantial revision that verifies feasibility directly and re-frames the claims.

major comments (4)

[§IV-B2, Eq. (7), constraints [C5]–[C6]] The paper asserts that residual imbalances of 0.013 p.u. (14-bus) and 0.017 p.u. (30-bus) 'can be fully absorbed by slight adjustments to power demand at load buses and generation at generator buses,' but it supplies no adjustment algorithm, does not run an AC power-flow solver on the generated samples, and never checks the line-flow constraints [C6]. Because R(xt) is an average over buses, it can hide large per-bus mismatches and says nothing about whether any nearby point satisfying [C5] and [C6] exists. The central feasibility claim is therefore asserted rather than demonstrated.
[Table I] Table I reports the physics-informed GAN baseline with an average imbalance of 0.009 p.u. on the 14-bus system and 0.015 p.u. on the 30-bus system, both lower than the proposed model's 0.013 p.u. and 0.017 p.u. This directly contradicts the abstract's statement that the proposed model outperforms the baselines 'in terms of feasibility.' The comparison claim must be revised or the experiments redone before the paper can be accepted.
[§III-B2, Eq. (11) and Eq. (8)] The auxiliary network is trained to minimize |R(sqrt(alpha_t(omega)) x0 + sqrt(1 - alpha_t(omega)) epsilon) - gamma_t|^2, and the same gamma_t is subsequently used as the upper bound in the physics loss LR(xt) = max(R(xt) - gamma_t, 0). Consequently, the linear imbalance curve in Fig. 3b is the optimization target achieved by the auxiliary training, and the 'below the bound' behavior in Fig. 3c is at least partly enforced by the very same gamma_t. This circularity means the results do not independently validate the learned schedule as a mechanism that improves physical feasibility beyond what the loss already enforces.
[§IV-B2, Figs. 4 and 5] Statistical fidelity and diversity are evaluated only through histograms of a single generator (Generator 3) on the 14-bus system, with no quantitative distributional metrics, no comparison across other variables, and no error bars. The abstract's claims of 'statistical fidelity' and 'diversity, and accuracy of statistical features' require quantitative comparisons such as Wasserstein distance, marginal statistics over all variables, or a multivariate goodness-of-fit test.

minor comments (5)

[§II-B, Eq. (2)] Eq. (2) writes q(xt|x0) = N(xt; sqrt(alpha_bar_t) x0, alpha_bar_t I), but the variance should be (1 - alpha_bar_t) I to be consistent with Eq. (3) and the reparameterization; this is a typo but should be corrected.
[§II-A] The sentence 'In both [C5] and [C6].' after the constraint list is incomplete and should be finished or removed.
[§II-B, Eq. (4)] The notation 'q(xt-1|xtx0)' in Eq. (4) is unclear; it should be written as q(xt-1 | xt, x0).
[§IV-A] The description 'Python 3.6 and Tensorflow 2.18.0' is likely inconsistent, since TensorFlow 2.18 requires a newer Python; please verify the software versions.
[Fig. 3c] Please clarify whether the 'imbalance bound' curve in Fig. 3c is gamma_t from Eq. (10) or a different threshold; the caption and text should be unambiguous.

Circularity Check

2 steps flagged · score 6.0 of 10

The learned diffusion schedule is fitted to γ_t and the physics loss uses the same γ_t as its threshold, so the reported residual-imbalance 'feasibility' results largely validate the training objective rather than independent AC feasibility.

fitted input called prediction [Sec. III-B2 (Eq. 11) and Sec. III-A (Eq. 8)]
"The loss function for Fω, defined in Eq.11, measures the mismatch between the expected power imbalance in Eq.10 and the actual imbalance R(Xt)."

Eq. 10 defines the target γt = t γT / T. Eq. 11 trains Fω so that R(√ᾱt x0 + √(1−ᾱt)ε) matches γt, and Eq. 8 penalizes any diffusion step where R(xt) exceeds the same γt. Consequently, Fig. 3b ("this learned scheduling parameter linearly distributes imbalance") displays the fitted objective, and Fig. 3c ("the power imbalance of the synthetic data remains below the imbalance bound throughout most of the T steps") is at least partly enforced by LR, not an independent physical prediction.
fitted input called prediction [Sec. IV-B2, Table I]
"The proposed physics-informed DDPMs produce outputs that not only satisfy all inequality constraints but also exhibit significantly lower power imbalances: 0.013 p.u. for the IEEE 14-bus system and 0.017 p.u. for the IEEE 30-bus system."

The reported quantity is the same R(·) that LR minimizes. With γT = 2.75 p.u. and T = 200, γ1 = γT/T = 0.01375 p.u. for the 14-bus case, so the reported 0.013 p.u. is essentially the first-step bound; for 30-bus, γ1 = 0.01435 p.u. and the reported 0.017 p.u. sits just above that bound. The feasibility number is therefore close to the threshold imposed through the fitted γt schedule, and no AC power-flow solver or line-flow constraint [C6] check is applied to demonstrate that a genuinely feasible point exists.

full rationale

The paper's central feasibility claim is partially circular: the auxiliary model Fω is explicitly trained so that the forward power imbalance matches γt (Eq. 11 vs. Eq. 10), and the very same γt is then used as the upper bound in the physics-informed loss (Eq. 8). Thus the learned-schedule plots and the reported residual imbalances (0.013 and 0.017 p.u., close to γ1 = γT/T) largely reflect the training objective rather than an independent test of AC feasibility. The paper never checks line-flow limits [C6] or applies an AC solver to the generated samples, and the assertion that residual imbalances 'can be fully absorbed' is given without an adjustment rule. These are correctness/validation gaps as well as a mild circularity. The self-citations ([2], [21]) are used only for the baseline GAN and data-perturbation ranges and are not load-bearing, and the diversity histograms (Figs. 4-5) provide some independent content. Note also that Table I shows the physics-informed GAN achieving 0.009 p.u. (14-bus) and 0.015 p.u. (30-bus), lower than the proposed model, which contradicts the abstract's claim of outperforming all baselines in feasibility; that inconsistency is a correctness concern beyond circularity. Overall score 6 reflects one or more 'predictions' that reduce by construction to the fitted γt schedule.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central method depends on a hand-tuned loss weight eta, a chosen T=200, dataset perturbation ranges, an empirically measured noise-imbalance gamma_T, and the auxiliary schedule weights fitted by Eq. 11. The main domain assumptions are that MATPOWER OPF output represents ground truth, that a single unified schedule works for all samples, that linear imbalance growth is desirable, and that a small residual imbalance can be post-hoc corrected. The strongest ad hoc assumption is the linear-imbalance ideal in Eq. 10, which is both the training target and the physics-loss threshold.

free parameters (6)

Physics-informed loss weight eta = 1
Set to 1 in Sec. IV-B2; no sensitivity study, so the balance between diffusion loss and physics loss is hand-picked.
Total diffusion steps T = 200
Chosen for both test grids in Sec. IV-B1; no ablation varying T is reported.
Demand perturbation range = [80%,120%] of nominal values
Dataset generation choice in Sec. IV-A that shapes the target distribution and all statistical-fidelity claims.
Generator cost coefficient perturbation range = [50%,150%] of defaults
Used in Sec. IV-A to create diversity; affects the diversity evaluation.
Gaussian noise imbalance gamma_T = 2.75 p.u. (14-bus), 2.87 p.u. (30-bus)
Measured empirically from noisy samples and used in Eq. 10 to define gamma_t; not independently derived.
Learned schedule parameters alpha_t(omega) = Network weights: 217,032 (14-bus), 240,584 (30-bus)
Fit by Eq. 11 to the linear imbalance target; the schedule is an optimized quantity, not a derived constant.

assumptions (6)

domain assumption Power balance equations [C5] and line-flow limits [C6] define feasibility; small average power imbalance is treated as a sufficient proxy.
The paper reports R(x0) about 0.01 p.u. but never checks [C6] or proves that a nearby AC-feasible point with the same distribution exists.
domain assumption A single unified noise schedule alpha_t applies to all power flow samples.
Sec. III-B2 averages auxiliary-network outputs over the dataset, assuming one schedule can linearly balance imbalance for every sample.
ad hoc to paper Linear growth of power imbalance during the forward process is the desired behavior.
Eq. 10 postulates linear imbalance without derivation; it motivates the auxiliary training and is not independently justified.
domain assumption MATPOWER OPF-generated data represent the real-world distribution of power flow states.
Sec. IV-A uses synthetic MATPOWER solutions, not field measurements; the real data histograms compare against this solver output.
standard math DDPM Markov property and Gaussian corruption with the given reparameterization hold.
Background from [6]; however Eq. 2 misstates the variance, so the exposition itself is unreliable.
ad hoc to paper Residual imbalance of about 0.01 p.u. can be corrected by slight demand and generation adjustments without changing the generated distribution.
Quoted from Sec. IV-B2; no algorithm or proof is supplied.

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Cite this review

Pith. "Pith review of Synthetic Power Flow Data Generation Using Physics-Informed Denoising Diffusion Probabilistic Models." pith.science (2026). https://pith.science/paper/TUK4CV5I

@misc{pith2026250417210,
  author       = {Pith},
  title        = {Pith review of: Synthetic Power Flow Data Generation Using Physics-Informed Denoising Diffusion Probabilistic Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TUK4CV5I}},
  note         = {Machine review of arXiv:2504.17210}
}

read the original abstract

Many data-driven modules in smart grid rely on access to high-quality power flow data; however, real-world data are often limited due to privacy and operational constraints. This paper presents a physics-informed generative framework based on Denoising Diffusion Probabilistic Models (DDPMs) for synthesizing feasible power flow data. By incorporating auxiliary training and physics-informed loss functions, the proposed method ensures that the generated data exhibit both statistical fidelity and adherence to power system feasibility. We evaluate the approach on the IEEE 14-bus and 30-bus benchmark systems, demonstrating its ability to capture key distributional properties and generalize to out-of-distribution scenarios. Comparative results show that the proposed model outperforms three baseline models in terms of feasibility, diversity, and accuracy of statistical features. This work highlights the potential of integrating generative modelling into data-driven power system applications.

Figures

Figures reproduced from arXiv: 2504.17210 by the authors.

**Figure 2.** The Framework for Auxiliary Training in the backpropagation algorithm to update the neural network parameters ω. Lt(ω) = |R( p α¯t(ω)x0 + p 1 − α¯t(ω)ϵ) − γt| 2 (11) Unlike traditional neural networks that generate different outputs for different inputs, this model learns a unified scheduling parameter α¯t(ω) for t ∈ [1, T] that applies to all data points. This is achieved by computing the mean of the network outpu… view at source ↗

**Figure 3.** Performance of Proposed Method on IEEE 14-bus System [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: The Diversity of Synthetic Data from Proposed DDPMs [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: The Diversity of Synthetic Data from GAN [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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