Estimating Density Functions for Probabilistic Power Flow Using Invertible Neural Networks

arXiv:2604.00673 · eess.SY, cs.SY · Submitted 2026-04-01 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: We're looking at the paper: "Estimating Density Functions for Probabilistic Power Flow Using Invertible Neural Networks".

Dev: Here's the gist of the paper. The gist The proposed framework achieves state-of-the-art performance both as an accurate PF surrogate and as an efficient PPF density estimator by learning an explicit invertible mapping between stochastic power injections and system voltages using invertible neural networks.

Rosa: Problem Formulation The objective is to obtain an explicit approximation of the joint probability density of the resulting voltage magnitude and phase angle at each bus, Poi(

v i,θ i: ), i ∈ 1 to N (5).

Dev: This is achieved by considering a distribution of power injections Pw(

p,q: ) and buslevel densities Pwi(

p i,q i: ) known and modeled using GMMs

7: ,

24: (3).

Rosa: The PF formulation establishes a mapping between the active/reactive power

p,q: and the bus voltage magnitudes and phase angles

v,θ: (4).

Title and authors: Rosa: So we're looking at this paper titled "Estimating Density Functions for Probabilistic Power Flow Using Invertible Neural Networks". It sounds a bit heavy, but it’s about tackling probabilistic power flow, or PPF.

Dev: Yeah, PPF is essential because we need to figure out the uncertainty in modern power systems with all these renewable sources and flexible loads. This paper proposes a new way to get those voltage probability density functions without doing repeated power flow simulations during inference.

Taro: It’s interesting how they try to approximate complex, non-Gaussian voltage distributions without making any strict assumptions about the distribution beforehand. That sounds like it could handle a lot of real-world system behavior.

Rosa: Exactly. The core idea is to learn an explicit mapping between the power injections and the resulting voltages, and then use a change-of-variables theorem combined with scenario sampling to approximate those joint densities at each bus, which they call Poi(

v i,θ i: ), i equals one to N.

Dev: So it’s not just about getting one voltage distribution, it’s about getting the density function for every single bus voltage magnitude and phase angle simultaneously. That is a lot of data to model correctly.

Taro: I wonder how they handle the physical constraints of power systems when they are learning this mapping, especially since they restrict the domain to physically feasible solutions under normal steady-state operating conditions.

Rosa: Right, so they’re focusing on that restricted domain where the power flow equation has a unique solution and is almost bijective onto its image. The whole point is to avoid those expensive Monte Carlo methods or simplified approximations that often lose accuracy.

Dev: It sounds like they are building a framework that can directly approximate these complex voltage distributions efficiently, which would significantly reduce the data handling burden compared to conventional sampling-based PPF methods.

The paper's summary: Rosa: In this paper, "Estimating Density Functions for Probabilistic Power Flow Using Invertible Neural Networks", they introduce a novel PPF density approximation framework that avoids repeated power flow simulations during inference and can approximate complex and non-Gaussian voltage probability density functions without restrictive distributional assumptions.

Dev: The main contribution is the IMNF model, which they describe as an INN-based surrogate designed specifically for power flow modeling. Unlike some other neural network approaches that need separate models for the forward and inverse power flow, IMNF uses architectural invertibility to do both transformations with a single model.

Taro: So it’s a bidirectional capability that allows them to perform both mappings directly, which is what they call direct voltage density approximation in PPF. That seems like a big deal for building robust system models.

Rosa: They build this by combining an invertible PF surrogate with the change-of-variables theorem and scenario-based marginalization to approximate the marginal density Poi(

v i,θ i: ) using samples from known power injection distributions Pw/i(s/i).

Dev: They systematically present their mathematical foundation, how they model the density, their training strategy, and their scenario-sampling procedure. It sounds like a very thorough approach to making this work practically.

Taro: I’m curious about that scenario sampling part. They use a scheme called LSS which combines quasi-Monte Carlo point sets and Latin Hypercube Sampling to draw scenarios from the known power injection distributions Pw/i(·).

Rosa: That LSS approach is what they use to compute the final density in Eq. (14b), involving samples of s/it and the Jacobian determinant of g minus i,s/i times the partial derivative of oi over oi. It’s a clever way to get that marginal density approximation.

Dev: So it’s not just a black box neural network; it has this specific mathematical structure that allows it to use those samples effectively to approximate the target densities without having to solve the power flow equations every time you want an answer.

The paper's improvements: Rosa: The paper highlights several improvements. First, they propose the IMNF model, which is presented as the first INN-based surrogate specifically designed for PF modeling. They emphasize that this architecture exploits architectural invertibility to perform both mappings using a single model.

Dev: That’s significant because conventional neural network approaches often require you to maintain separate models for power flow and inverse power flow, so having one bidirectional model is a key improvement. It simplifies the overall structure considerably.

Taro: And they mention that this bidirectional capability enables direct voltage density approximation, which means you don't need a separate step to invert the mapping after getting a forward prediction. That streamlines the process for inference.

Rosa: They also improve things by using LSS-based sampling, showing that it consistently outperforms Monte Carlo based approaches in terms of sample efficiency when estimating density functions.

Dev: And they show this performance improvement across different system types, like the IEEE-thirty-nine bus system, where they achieved absolute decreases of zero point two three two eight in JSD and zero point zero nine one one in TVD relative to the Linear benchmark.

Taro: It’s interesting that even with just fifty scenarios using LSS-fifty they got performance better than the PLinear benchmark, with absolute decreases of zero point zero zero eight nine in JSD and zero point zero one four five in TVD. That shows how efficient the sampling can be when you have a good distribution to work with.

Rosa: They also note that this framework demonstrates superior performance when the underlying system has stronger nonlinear characteristics compared to near-linear regimes, which is where these models really shine compared to simpler architectures like Nice or RealNVP.

Conclusion: Dev: So, putting it all together for "Estimating Density Functions for Probabilistic Power Flow Using Invertible Neural Networks", this framework achieves state-of-the-art performance both as an accurate PF surrogate and as an efficient PPF density estimator.

Rosa: That means it’s not just a good substitute for solving power flow problems, but it's also a highly efficient way to get the voltage density information needed for planning.

Taro: The key is that they showed that using LSS-based sampling consistently outperforms Monte Carlo approaches in terms of sample efficiency, which makes this method much more practical for real-world application where you can’t run endless simulations.

Dev: Right, and it also proves that the IMNF model itself is a powerful tool because it handles the asymmetry of learning difficulty between the forward and inverse mappings through bidirectional training.

Rosa: So, overall, this paper proposes a novel framework for voltagedensity approximation in PPF that is state-of-the-art as both a standalone PF solver and as an efficient density estimator.

Dev: It suggests that future research should look at how to optimize these INN architectures because they aren't fully optimized for the specific needs of PPF tasks, and the asymmetry of learning difficulty between forward and inverse mappings is something we still need to investigate further.

Taro: I think looking at how this framework handles real-world failure modes when you push it outside the lab is important, Rosa.

Rosa: Definitely. We’ll keep an eye on this IMNF model as more people start using it in actual trials outside the lab and see how it performs when the world misbehaves.

Weijie Xia, James Ciyu Qin, Edgar Mauricio Salazar Duque, Hongjin Du, Peter Palensky, Giovanni Sansavini, Pedro P. Vergara

Delft University of Technology · ETH Zurich

eess.SY, cs.SY

Submitted: 2026-04-01

Updated: 2026-10-20

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 75/100

The gist: The gist The proposed framework achieves state-of-the-art performance both as an accurate PF surrogate and as an efficient PPF density estimator by learning an explicit invertible mapping between

Key concepts

Probabilistic Power Flow (PPF): PPF is a method used in power systems to understand the uncertainty in system voltages by modeling the randomness of power injections. The goal is to determine the probability of observing certain voltage magnitudes and phase angles at different buses under varying power conditions.
Change of Variables Theorem: This mathematical theorem allows for transforming complex probability densities. It enables the framework to convert a known density function from one domain (power injections) into an unknown density function in another domain (bus voltages) using an invertible mapping, which is essential for approximating the marginal voltage distribution.
Invertible Mixed Neural Flow (IMNF): IMNF is a hybrid neural network structure consisting of three layers: Linear, Exp, and Spline. This architecture is designed to learn an explicit, invertible mapping between power injections and system voltages. Its invertibility allows it to accurately calculate the density of voltages without needing repeated power flow calculations during inference.
Latin Supercube Sampling (LSS): LSS is a sampling technique used to efficiently draw samples from a complex input distribution, like the power injection distribution. The framework uses LSS to generate scenarios, proving that this method is more sample-efficient than standard Monte Carlo methods for estimating the final voltage density.

Terminology

Accuracy — 2× in this paper · explained in 1 episode
Core — 1× in this paper · explained in 2 episodes
Monte Carlo — 1× in this paper

Summary

The gist The proposed framework achieves state-of-the-art performance both as an accurate PF surrogate and as an efficient PPF density estimator by learning an explicit invertible mapping between stochastic power injections and system voltages using invertible neural networks.

Problem Formulation

The objective is to obtain an explicit approximation of the joint probability density of the resulting voltage magnitude and phase angle at each bus, Poi([v i,θ i]), i ∈ 1 to N (5). This is achieved by considering a distribution of power injections Pw([p,q]) and buslevel densities Pwi([p i,q i]) known and modeled using GMMs [7], [24] (3). The PF formulation establishes a mapping between the active/reactive power [p,q] and the bus voltage magnitudes and phase angles [v,θ] (4). The domain is restricted to physically feasible solutions under normal steady-state operating conditions where the PF mapping admits a unique solution and can be regarded as almost bijective onto its image [25], [26] (3).

Methodology: Change of Variables Theorem

The framework utilizes the Change of Variable Theorem to transform probability densities through an invertible and differentiable mapping f: R d → R d (8a) or equivalently, Px(x) = Pz(z) / det ∂f−1(x)/∂x (8b). To address the mismatch between full system distribution and marginal voltage distribution, a conditional PF mapping gi,s/i([p i,q i]):= πi f([p i,q i],s/i) is defined for each fixed feasible s/i (9). The conditional Jacobian of this mapping is expressed as ∂gi,s/i([p i,q i]) / ∂ [p i,q j] = (A−BD−1C)−1 (11). By applying the theorem to the conditional density over s/i, the marginal density Poi([v i,θ i]) is approximated by Eq. (14b) involving samples of s/it and the Jacobian determinant of g−1 i,s/i(oi) ∂oi / ∂oi (14b).

Proposed Framework: Invertible Mixed Neural Flow (IMNF)

The core component is the IMNF model, which is a hybrid INN-based model built by stacking three elementary invertible layers: a Linear layer, an Exp layer, and a Spline layer [19] (3). For bus i, all three layers act on the vector x i = [p i0,q i0], conditioned on the scenario s/i collecting the power injections at every other bus (3).

Linear Layer: This layer applies a trainable, invertible 2×2 matrix A to the input vector without dependence on s/i (18), with its Jacobian determinant being detA (20).
Exp Layer: This affine coupling transformation updates one coordinate using an NN-predicted, exponentiated scale and shift that depend on the other coordinate and on s/i, then alternates the roles of the two coordinates (21), (22). The determinant of the full Exp layer Jacobian is exp(s1(p i0,s/i)) + s2(q i1,s/i) (26).
Spline Layer: This layer replaces the affine coupling of the Exp layer with an elementwise monotonic rational–quadratic spline (fsf(·)) (27).

The final IMNF model fimnf is constructed by stacking these layers: fimnf = f1sf ◦ ··· ◦ fLsfcp ◦ f1ll ◦ f1sfcp ◦ ··· ◦ fLsfcp (·) (27). The determinant of the full Jacobian Jf imnf is simply the product of the per-layer determinants, detJf imnf = Lsf∏l=1 detJf l sf × Lsfcp∏l=1 detJf l ll × detJf l sfcp (28).

Training Design and Scenario Sampling

The IMNF model is trained using a mean squared error (MSE) as the primary loss function, combining three terms: lossw2o,i = MSE[fimnf([p i0,q i0],s/i),[v i,θ i]] (31a), losso2w = MSE[f−1imnf([v i,θ i],s/i),[p i0,q j0]] (31b), and lossjac, i = MSE[det Jf imnf, det J i PF] (32). The overall training objective is a weighted combination of these terms: loss = ω lossw2o+ (1−ω)losso2w+λ lossjac, where ω ∈ [0,1], λ ≥ 0 (33). A bidirectional training strategy enforces consistency in both the forward and inverse transformation to improve robustness and accelerate convergence.

To compute the final density in Eq. (14b), scenarios s/i t are drawn from Pw/i(·) using LSS, a scheme that combines quasi–Monte Carlo (QMC) point sets and Latin Hypercube Sampling (LHS) [29], [30] (38).

Experimental Results

The proposed framework shows the best performance experimentally, combining the advantages of both FCPFlow and SplineFlow (6). For PPF simulation, the JSD and TVD metrics consistently decrease as the number of sampled scenarios increases for both MC-based and LSS-based density estimation (8). For transmission systems, a clear performance advantage over benchmark approaches is demonstrated in the IEEE-39 bus system, achieving absolute decreases of 0.2328 in JSD and 0.0911 in TVD relative to the Linear benchmark (9). The required number of scenarios depends on both the complexity of the underlying distribution and the scale of the system (10). For example, using only 50 scenarios (LSS-50) can already yield a performance better than the PLinear benchmark, with absolute decreases of 0.0089 in JSD and 0.0145 in TVD (8). The framework demonstrates superior performance when the underlying system exhibits stronger nonlinear characteristics compared to near-linear regimes (11).

Conclusion

In this paper, we propose a novel framework for voltagedensity approximation in PPF (6). We first demonstrate that the proposed IMNF model achieves state-of-the-art performance as a standalone PF solver compared to existing benchmark methods (7). We then show that the complete framework attains state-of-the-art performance in voltage-density approximation for PPF (10). Furthermore, we investigate how the number of sampled scenarios affects the final approximation accuracy and demonstrate that LSS-based sampling consistently outperforms MCbased approaches in terms of sample efficiency (8). Despite these promising results, the application of INNs to PPF still warrants further investigation (10). In particular, we observe that commonly used INN architectures, primarily designed for generative modeling via maximum likelihood, are not fully optimized for PPF tasks (10). A key challenge lies in the asymmetry of learning difficulty between the forward and inverse mappings, which motivates the use of bidirectional training (33). We therefore anticipate that future research will further explore both the theoretical foundations and practical design of INN-based frameworks for PF and PPF applications.

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Improvements for AI systems

Bidirectional training strategy for IMNF improves robustness and accelerates convergence by enforcing consistency in both forward and inverse transformation, which is crucial because an accurate point mapping does not by itself guarantee an accurate density estimate.
The IMNF model can directly approximate voltage probability density functions without repeatedly solving the PF equations during inference, thereby reducing computational and data-handling burdens of conventional sampling-based PPF.
The framework can evaluate complex and non-Gaussian voltage probability density functions without repeated PF computations during inference, leveraging the change-of-variables theorem by approximating the marginal density using samples from known power injection distributions Pw/i(s/i).
By employing a hybrid architecture combining Linear, Exp, and Spline layers, the IMNF model can capture nonlinear PF transformations more effectively than simpler INN architectures like Nice or RealNVP, leading to superior performance as a PF surrogate.
The system can be used for high-dimensional integration of power injection distributions through the Latin Supercube Sampling (LSS) procedure, which is shown to consistently outperform MC-based approaches in terms of sample efficiency for density estimation.