SenseFlow: A Physics-Informed and Self-Ensembling Iterative Framework for Power Flow Estimation

## Abstract

Abstract Power flow estimation plays a vital role in ensuring the stability and reliability of electrical power systems, particularly in the context of growing network complexities and renewable energy integration.
However, existing studies often fail to adequately address the unique characteristics of power systems, such as the sparsity of network connections and the critical importance of the unique Slack node.
In this paper, we present SenseFlow, a novel physics-informed and self-ensembling iterative framework that integrates two main designs, the Physics-Informed Power Flow Network (FlowNet) and Self-Ensembling Iterative Estimation (SeIter), to carefully address the unique properties of the power system and thereby enhance the power flow estimation.
Specifically, SenseFlow enforces the FlowNet to gradually predict high-precision voltage magnitudes and phase angles through the iterative SeIter process.
On the one hand, FlowNet employs the Virtual Node Attention and Slack-Gated Feed-Forward modules to facilitate efficient global-local communication in the face of network sparsity and amplify the influence of the Slack node on angle predictions, respectively.
On the other hand, SeIter maintains an exponential moving average of FlowNet’s parameters to create a robust ensemble model that refines power state predictions throughout the iterative fitting process.
Experimental results demonstrate that SenseFlow outperforms existing methods, providing a promising solution for high-accuracy power flow estimation across diverse grid configurations 1 1 1 Code and logs will be available at https://github.com/zhenzhao/SenseFlow .

## 1 Introduction

Power flow estimation is a crucial task for maintaining the stable and reliable operations of electrical power systems [1, 2]. In practical power systems, any disturbance on a single bus can affect the overall balance of the system, necessitating a recalculation of the power flow to preserve stability. This makes power flow estimation not only essential but also highly frequent in operational contexts [3]. As shown in Figure [1(b)](https://arxiv.org/html/2505.12302v1#S1.F1.sf2), using the IEEE 39-bus system as an example, the network typically consists of three types of buses: multiple PQ (Load Bus) and PV (Generator Bus) nodes, and a single Slack node. The goal is to determine the voltage magnitudes and phase angles at each bus, adhering to the fundamental laws of power system dynamics.
While traditional methods like Newton-Raphson [4] and Gauss-Seidel [5] algorithms offer high accuracy, they face key challenges in modern power grids.
As power networks grow in scale and complexity, especially with the integration of renewables, these mathematical solvers become computationally inefficient, particularly under large contingency
(*e.g.*, N-K) analyses[6].
Additionally, their reliance on complete and well-conditioned parameters limits their applicability when critical information, such as nodes’ reactive power, is missing—an increasingly common issue in real-world scenarios [7].

Figure: (a) Statistics of different datasets.
Refer to caption: x1.png

In recent years, data-driven approaches, particularly deep learning techniques, have garnered significant attention for enhancing the accuracy and efficiency of power system analysis [8]. Among these approaches, Graph Convolutional Networks (GCNs) [9] have emerged as a prominent solution due to their effectiveness in handling graph-structured data, which aligns well with the inherent graph nature of power systems. However, despite their promise, many existing studies [10, 3, 7]
fall short of fully addressing the unique characteristics of power systems.
As depicted in Figure [1(a)](https://arxiv.org/html/2505.12302v1#S1.F1.sf1), one of the key overlooked features is the presence of only one single Slack bus in any size system, whose phase angle is used as a reference point for the entire system. The Slack bus is also the only node in the system that has both the known voltage magnitude and phase angle. Furthermore, power grids are fundamentally sparse networks: the number of edges typically scales linearly with the number of nodes (*i.e.*, $\mathcal{O}(N)$), which is considerably fewer than in fully connected graphs (*i.e.*, $\mathcal{O}(N^{2})$).
Such sparse connectivity limits information exchange between distant nodes, particularly concerning the Slack node, thereby posing a significant challenge for most GCN architectures that rely on graph connections for efficient node communication. To this end, we aim to employ physic-informed model designs that carefully integrate these distinct features to enhance power flow estimation.

On the other hand, most GCN-based methods follow an end-to-end fitting fashion [10, 11, 12], which significantly enhances analytical efficiency by directly mapping input graphs to desired flow estimations. While this streamlined process enables rapid power flow analysis, such methods often sacrifice accuracy, as these models may not adequately capture the intricate dependencies and dynamics. In contrast, traditional power flow analysis methods [4, 13, 14] typically employ iterative fitting techniques. These approaches gradually refine their predictions through successive approximations, improving the accuracy of voltage magnitudes and phase angles with each iteration. Aware of these limitations, we aim to incorporate an iterative process into the GCN-based framework for a refined balance between computational efficiency and high precision.

Inspired by these observed limitations, we propose a physics-informed and self-ensembling iterative framework for power flow estimation, dubbed as SenseFlow, which seamlessly integrates two novel designs, the Physics-Informed Power Flow Network (FlowNet) and the Self-Ensembling Iterative Estimation (SeIter).
FlowNet first adopts the Virtual Node Attention (VNA) module to aggregate the features of all nodes into a virtual node and apply cross-attention to distribute global information to individual PQ, PV, and Slack nodes. VNA facilitates efficient global-local communication without altering the original graph structure, ensuring that each node benefits from system-wide context.
We also design the Slack-Gated Feed-Forward (SGF) module in FlowNet to emphasize the Slack node’s significance by concatenating its features with PQ and PV nodes. A gated mechanism controls the Slack node’s influence, while a residual connection preserves local node characteristics and enhances the Slack node’s impact.
SeIter guides FlowNet to iteratively predict changes in voltage magnitude and phase angle, which gradually improves accuracy within each loop. During this process, an exponential moving average of FlowNet’s parameters maintains an ensemble model that generates more stable outputs, mitigating noise and fluctuations inherent in iterative training. Its outputs are then fed into the next training loop, forming a self-ensembling cycle that progressively refines predictions. FlowNet is trained using two losses: the ground-truth loss to align predictions with actual voltage and phase values, and the equation loss to enforce adherence to power balance equations. Our main contributions are summarized as follows,

- •
We propose a novel power flow estimation framework, SenseFlow, which integrates two novel designs FlowNet and SeIter to obtain high-accuracy power flow estimation iteratively.
- •
Our FlowNet carefully addresses the unique characteristics of power systems by designing the Virtual Node Attention and Slack-Gated Feed-Forward modules, which enhance global-local communication and highlight the Slack node’s influence effectively.
- •
Our SeIter strategy, equipped with a more stable and accurate self-ensembling model, progressively refines predictions into high-precision results.
- •
With physics-informed and graph-enhanced designs, SenseFlow achieves leading performance in power flow estimation across different-size grids, even with incomplete inputs.

## 2 Related Work

Power flow analysis is a fundamental task in electrical power systems, which has been researched for decades [15]. Traditional methods, such as the Newton-Raphson Method [4], Gauss-Seidel Method [5], and Backward-Forward Sweep [13], provide promising estimation accuracy through iterative optimization procedures. However, these methods often struggle to scale effectively with larger and more complex power systems, particularly those incorporating renewable energy sources [3]. Consequently, research groups have increasingly shifted their focus towards data-driven approaches [8, 2, 16].
Studies along this line aim to fit the distribution of the collected historical data or simulated data for accurate and efficient power flow approximation. Considering the collinearity of the training data and the nonlinearity of the power flow model, [17] proposes a piecewise linear regression algorithm for model fitting. Similarly, [6] converts the nonlinear relationship of flow calculation into a higher dimension state space based on the Koopman operator theory. However, most of these works focused on the nonlinear fitting ability of the model and ignored the graph-structured topology nature of power systems, leading to unsatisfying estimation performance.

Graph Convolutional Networks (GCNs) [18, 19] are powerful models designed to handle graph-structured data and have demonstrated significant potential in addressing the graph topology in power systems[20, 12, 21].
The work by [22] highlights the promising capability of GCN to leverage the network structure of the data and approximates a specified optimal solution through an imitation learning framework.
Recent studies have incorporated the physical constraints of power systems into the loss design, enhancing estimation accuracy and robustness to the variations of
typologies [10, 23, 11, 7, 24].
For instance, [25] adopts a weakly supervised learning method based on power flow equations, which removes the requirement for labeled data but results in relatively lower accuracy than fully supervised approaches.
PowerFlowNet [10] introduces a joint modeling approach that simultaneously represents both buses and transmission lines, conceptualizing power flow estimation as a GNN node-regression problem.
However, none of these studies thoroughly examine the distinctive characteristics of power systems, such as network sparsity and the critical role of the slack node. Differently, we explore these features and deliberately incorporate them into our network designs.

Figure: (a) Iterative power flow estimation
Refer to caption: x3.png

## 3 SenseFlow

We first present an overview of SenseFlow and then describe the two main designs, the self-assembled iterative estimate and the physics-informed power flow network in Sec. [3.2](https://arxiv.org/html/2505.12302v1#S3.SS2) and Sec. [3.3](https://arxiv.org/html/2505.12302v1#S3.SS3), respectively.

### 3.1 Overview

Given a power system $\mathcal{G}$ with $N$ buses (nodes) and $E$ transmission lines (edges), the objective of power flow estimation is to determine the voltage magnitudes $V_{m,i}$ and phase angles $V_{a,i}$ at each bus $i\in\{1,2,\dots,N\}$, subject to the power balance equations that govern active and reactive power flows in the network. In terms of the training process, we have the active/reactive power for PQ nodes, *i.e.*, $P^{\text{PQ}}/Q^{\text{PQ}}$,
active power $P^{\text{PV}}$ and voltage magnitude $V_{m}^{\text{PV}}$ for PV nodes, and known voltage $V_{m}^{\text{Slack}}$ and phase angle $V_{a}^{\text{Slack}}$ for the Slack node, as well as the network topology encoded in the admittance matrix. Giving the ground-truth information on the PQ and PV nodes, including $V_{m}^{\text{PQ}}$, $V_{a}^{\text{PQ}}$, $V_{a}^{\text{PV}}$, our goal is to obtain corresponding accurate predictions.

Our proposed SenseFlow framework addresses the power flow estimation problem by seamlessly integrating physics-informed modeling with a self-ensembling iterative learning process. At its core, SenseFlow leverages both the unique structural features of power systems and the iterative refinement capabilities of ensembling models. Specifically, SenseFlow trains the proposed FlowNet via the SeIter strategy. FlowNet process input data $\mathcal{G}(N,E)$ with known features, $P^{\text{PQ}},Q^{\text{PQ}},P_{\text{PV}},V_{m}^{\text{PV}},V_{m}^{\text{Slack
}},V_{a}^{\text{Slack}}$ to predict the unknown values on the PV and PQ nodes, *i.e.*, the voltage magnitude $\hat{V}_{m}^{\text{PQ}}$ and phase angle $\hat{V}_{a}^{\text{PQ}}$ for the PQ nodes, and phase angle $\hat{V}_{a}^{\text{PV}}$ for the PV nodes. The training of FlowNet is guided by a ground-truth loss $\mathcal{L}_{g}t$, and the power balancing equation loss $\mathcal{L}_{equ}$,

$$ $\displaystyle\mathcal{L}=\mathcal{L}_{gt}+\lambda\mathcal{L}_{equ},$ (1) $$

where $\lambda$ is a scalar hyper-parameter to adjust the equation loss weight. The power balancing equation loss is applied to encourage minimal power changes,

$$ $\displaystyle\mathcal{L}_{equ}\!=\!\frac{1}{N_{\text{PQ}}}\sum_{i=1}^{N_{\text {PQ}}}\left(\left|\Delta P^{\text{PQ}}_{i}\right|\!+\!\left|\Delta Q^{\text{PQ }}_{i}\right|\right)\!+\!\frac{1}{N_{\text{PV}}}\sum_{j=1}^{N_{\text{PV}}} \left|\Delta P^{\text{PV}}_{j}\right|,$ (2) $$

where the calculations of $\Delta P$ and $\Delta Q$ are involved in the SeIter process. Through the SeIter strategy, SenseFlow refines its predictions by iteratively updating voltage magnitudes and phase angles. Similar to  [21, 7], we also use L1 loss for the ground-truth supervision,

$$ $\displaystyle\mathcal{L}_{gt}=$ $\displaystyle\frac{1}{N_{\text{PQ}}}\sum_{i=1}^{N_{\text{PQ}}}\left(\left|\hat {V}_{m,i}^{\text{PQ}}-V_{m,i}^{\text{PQ}}\right|+\left|\hat{V}_{a,i}^{\text{PQ }}-V_{a,i}^{\text{PQ}}\right|\right)+\frac{1}{N_{\text{PV}}}\sum_{j=1}^{N_{ \text{PV}}}\left|\hat{V}_{a,j}^{\text{PV}}-V_{a,j}^{\text{PV}}\right|.$ (3) $$

With in each loop in the SeIter process, the proposed FlowNet is trained by minimizing these two losses, progressively pushing the predictions toward higher accuracy. We will detail the SeIter process and physics-informed FlowNet in the following sections.

### 3.2 Self-ensembling Iterative Estimation

The self-ensembling iterative estimation (SeIter) diverges from conventional end-to-end learning approaches. Instead of directly fitting inputs to final voltage magnitudes and phase angles, SeIter gradually enforces the trainable module to approach the ground truth with the help of a self-ensembling prediction. As shown in Figure [2(a)](https://arxiv.org/html/2505.12302v1#S2.F2.sf1), the trainable model focuses on fitting the incremental changes in voltage and phase angle, allowing for refined adjustments with each cycle. This process enables the model to achieve high accuracy that end-to-end approaches cannot reach.

In the SeIter, each iteration, denoted as the $\eta$th loop, involves a dual approach that focuses on both training the FlowNet model and refining the estimates for voltage magnitude and phase angle for the future loop. On the one hand, as shown in Figure [2(a)](https://arxiv.org/html/2505.12302v1#S2.F2.sf1), the input data is first utilized to train the FlowNet, parameterized by $\theta_{s}$, by minimizing the ground truth loss $\mathcal{L}_{gt}$. Second, the input data is subjected to the power balance equations, which yield incremental changes in active power $\Delta P$ and reactive power $\Delta Q$. The objective here is to minimize the equation loss $\mathcal{L}_{equ}$, which is designed to ensure that the total power variations approach zero. Let $\psi(V_{m},V_{a},\mathcal{G})$ denote the Power balancing equations, the active and reactive power changes $\Delta P_{i}$ and $\Delta Q_{i}$ at the bus $i$ can be calculated by,

$$ $\displaystyle\Delta P_{i}$ $\displaystyle=P_{i}-\sum_{j=1}^{N}|V_{m,i}||V_{m,j}|(G_{ij}\cos(V_{a,i}-V_{a,j })+B_{ij}\sin(V_{a,i}-V_{a,j})),$ (4) $\displaystyle\Delta Q_{i}$ $\displaystyle=Q_{i}-\sum_{i=1}^{N}|V_{m,i}||V_{m,j}|(G_{ij}\sin(V_{a,i}-V_{a,j })-B_{ij}\cos(V_{a,i}-V_{a,j})),$ (5) $$

where $G_{ij}$ and $B_{ij}$ represent the conductance and susceptance of the line connecting
buses $i$ and $j$.

On the other hand, as shown in Figure [2(b)](https://arxiv.org/html/2505.12302v1#S2.F2.sf2), the input data is processed through the Self-Ensembling Inference module, which maintains an ensembling model, parameterized by $\theta_{t}$, updated by exponential moving averaging (EMA) of the FlowNet parameters, *i.e.*,

$$ $\displaystyle\theta_{t}\leftarrow\alpha\theta_{t}+(1-\alpha)\theta_{s},$ (6) $$

where $\alpha$ is a momentum parameter. The ensembling model acts as a stable reference point, providing an output that reflects the accumulated knowledge from the iterative training process. Its output is further used as the input for the subsequent iteration, *i.e.*, the $(\eta+1)$th loop. This self-ensembling iterative estimation allows the trainable model to benefit from the progressively refined output of the ensembling model, thus enhancing its learning capabilities and improving the overall convergence.

Figure: Figure 3: Illustration of our FlowNet, which mainly consists of two main modules, the Virtual Node Attention (VNA) and Slack-Gated Feed-Forward (SGF). THe whole hetero-graph is fed into the network. The VNA creates a virtual node by combining and pooling the features of all nodes, then uses cross-attention to selectively communicate global information to each node type. This enhances the interaction between global and local information, preserving the graph structure while improving the model’s ability to capture system-wide dependencies. The SGF combines the slack node’s features with each node’s features through a gated feed-forward network, enhancing the slack node’s influence on other nodes while preserving the original node characteristics via a residual connection.
Refer to caption: x5.png

### 3.3 Physics-informed Power Flow Network

As shown in Figure [3](https://arxiv.org/html/2505.12302v1#S3.F3), our proposed FlowNet is built upon two fundamental modules: the Virtual Node Attention (VNA) and Slack-Gated Feed-Forward (SGF). VNA enables each node to perceive global changes without disrupting the underlying graph structure, while SGF enhances the influence of the slack node on each PQ and PV node, fostering accurate phase angle predictions.

*Virtual Node Attention*. Our VNA is specifically designed to address the sparsity issue by providing each node with the ability to sense and respond to global system variations. This design ensures that each local node can dynamically adjust its state in response to changes in the overall system, thus accurately capturing the inter-dependencies that are essential for maintaining the stability and reliability of power systems. With the VNA, we enable a more comprehensive and adaptive modeling of global interactions, ensuring that the system-wide impact of local changes is appropriately reflected. Specifically, We obtain the virtual node representation by contacting all the node features $F_{\text{PQ}},F_{\text{PV}},F_{\text{Slack}}$ without breaking the graph structure,

$$ $\displaystyle F_{\text{afuse}}$ $\displaystyle\!=\!\text{Linear}(\text{Concat}(F_{\text{PQ}},F_{\text{PV}},F_{ \text{Slack}})),$ (7) $\displaystyle F_{\text{Vnode}}$ $\displaystyle\!=\!\text{Concat}(\text{AvgPool}(F_{\text{afuse}}),\text{MaxPool }(F_{\text{afuse}})).$ (8) $$

Meanwhile, we can obtain the updated node representation after the graph neural network,

$$ $\displaystyle F_{\star}$ $\displaystyle=\text{GCN}(F_{\star}),\star\in\{\text{PQ, PV, Slack}\}$ (9) $$

where GCN denotes multi-layer graph convolutional network (*e.g.*, GraphConv [26], GAT [27]). Subsequently, we attend the global information to each type of the power node via the cross attention,

$$ $\displaystyle F_{\star}\!$ $\displaystyle=\!\text{LayerNorm}(F_{\star}\!+\!\text{softmax}\left(\frac{F_{ \star}\cdot F_{\text{Vnode}}^{T}}{\sqrt{d_{k}}}\right)F_{\text{Vnode}}),$ $$

where $d_{k}$ is the dimension of $F_{\text{Vnode}}$. In this way, our VNA module preserves the original graph structure and bridges the connection between each node and the whole system without implicitly introducing auxiliary nodes and edges.

*Slack-Gated Feed-Forward*. Our SGA effectively enhances the influence of the slack node in power system modeling by concatenating its feature representation with the feature representations of each PQ or PV node. The combined features are then processed through a gated feed-forward network, allowing the slack node’s influence to be dynamically adjusted based on the current state of the node. Moreover, a residual connection is added, incorporating the original node features to ensure that local characteristics are preserved while enhancing the model’s ability to accurately capture phase angle relationships throughout the system. Taking the PV node as an example, we have,

$$ $\displaystyle F_{\text{sfuse}}$ $\displaystyle=\text{Linear}(\text{Concat}(F_{\text{PQ}},F_{\text{Slack}})) \odot\sigma(\text{Linear}(\text{Concat}(F_{\text{PQ}},F_{\text{Slack}})))$ (10) $\displaystyle F_{\text{PQ}}$ $\displaystyle=\text{LayerNorm}(F_{\text{PQ}}+\text{Linear}(F_{\text{sfuse}})).$ (11) $$

To construct the complete model, as shown in Figure [3](https://arxiv.org/html/2505.12302v1#S3.F3), we stack $K$ layers of these blocks, allowing for deeper feature extraction and representation learning. Finally, outputs from all blocks are concatenated and then fed into a predictor to estimate the voltage states.

**Table 1: Performance comparison on the IEEE 39-Bus and IEEE 118-Bus system in terms of the root mean squared error (RMSE), where lower values indicate better performance. “$+$SeIter" indicates the application of our proposed self-ensembling iterative estimation process to the corresponding method. The best results are highlighted in bold.**
| Method | IEEE 39-Bus | IEEE 118-Bus |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
| PQ${}_{\text{Vm}}$ | PQ${}_{\text{Va}}$ | PV${}_{\text{Va}}$ | PQ${}_{\text{Vm}}$ | PQ${}_{\text{Va}}$ | PV${}_{\text{Va}}$ |  |
| GraphConv | 0.00724108 | 0.10125969 | 0.12637231 | 0.00192112 | 0.03769957 | 0.03597135 |
| $+$ SeIter | 0.00119714 | 0.01047964 | 0.01123176 | 0.00012959 | 0.00414722 | 0.00395151 |
| GINEConv | 0.00768264 | 0.10450818 | 0.12893821 | 0.00194470 | 0.03934504 | 0.03760612 |
| $+$ SeIter | 0.00148465 | 0.01158457 | 0.01231665 | 0.00013017 | 0.00463016 | 0.00444507 |
| SageConv | 0.00755344 | 0.10445687 | 0.12901593 | 0.00192444 | 0.04449241 | 0.04275129 |
| $+$ SeIter | 0.00160647 | 0.01383852 | 0.01506530 | 0.00020297 | 0.00740649 | 0.00720649 |
| ResGatedGraphConv | 0.00694495 | 0.10085707 | 0.12677170 | 0.00130103 | 0.03659180 | 0.03513782 |
| $+$ SeIter | 0.00086821 | 0.00854116 | 0.00910770 | 0.00007869 | 0.00301738 | 0.00288408 |
| GatConv | 0.00808900 | 0.10591513 | 0.13207403 | 0.00262339 | 0.04434326 | 0.04388360 |
| $+$ SeIter | 0.00531134 | 0.03361177 | 0.03675321 | 0.00070121 | 0.01037427 | 0.01344837 |
| TransformerConv | 0.00722702 | 0.10429660 | 0.13010464 | 0.00147067 | 0.04356860 | 0.04153621 |
| $+$ SeIter | 0.00086707 | 0.00917627 | 0.01005978 | 0.00012955 | 0.00477584 | 0.00457271 |
| FlowNet (ours) | 0.00453724 | 0.04653547 | 0.05373371 | 0.00115526 | 0.01273561 | 0.01269017 |
| $+$ SeIter (*i.e.*, SenseFlow) | 0.00078161 | 0.00608600 | 0.00609802 | 0.00009817 | 0.00102664 | 0.00103545 |

## 4 Experiment

### 4.1 Datasets

We construct our dataset based on standard IEEE test cases (39-Bus, 118-Bus, and 300-Bus) using Matpower [28], following approaches similar to [21] and [23]. To simulate diverse scenarios, we introduce variations in power injections, branch characteristics, and grid topology. Specifically, we apply uniform noise to the active and reactive power loads ($P$ and $Q$), adjusting them to range between 50% and 150% of their original values. Likewise, branch features are perturbed with uniform noise, ranging from 90% to 110% of their baseline values. To examine different grid topologies, we randomly disconnect one or two transmission lines in each sample. All load bus voltage magnitudes are initialized at 1 P.U., and phase angles are set relative to the slack bus reference angle. We generate 100,000 samples for the 39-Bus and 118-Bus systems, and 500,000 samples for the 300-Bus system. 20% of the records are reserved as test sets, with strictly distinct grid topologies from the training data. Voltage magnitudes and phase angles are calculated in units of P.U. and radians, respectively.

### 4.2 Implementation details

In our experiments, we utilized a batch size of 256 and employed the Adam optimizer with a learning rate set at 0.001, which follows a cosine decay schedule down to 1e-5 over a total of 100 epochs. Regarding feature embedding sizes, we set them to 128 for the IEEE 39-Bus and 118-Bus systems, while a size of 256 was used for the IEEE 300-Bus system. To effectively integrate information, we stacked a block that combines VNA and SGF modules a total of four times. Our models are trained and inferred using an iterative fitting approach with eight loops to enhance the accuracy. All code was implemented in PyTorch 2.1, and both training and testing were conducted on the 40GB A100 GPU.

For comparison, we evaluated our approach against popular graph networks commonly used in power system analysis, including GraphConv [26], GINEConv [29], SageConv [30], ResGatedGraphConv [31], GatConv [27, 32], and TransformerConv [33]. The metrics for comparison focused on the root mean square error (RMSE) of voltage and phase angle predictions for PQ nodes, as well as phase angle predictions for PV nodes.

**Table 2: Performance comparison on IEEE 300-Bus system. All notations are the same as in Table [1](https://arxiv.org/html/2505.12302v1#S3.T1).**
| Method | w/o SeIter | w/ SeIter | Param. |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| PQ${}_{\text{Vm}}$ | PQ${}_{\text{Va}}$ | PV${}_{\text{Va}}$ | PQ${}_{\text{Vm}}$ | PQ${}_{\text{Va}}$ | PV${}_{\text{Va}}$ |  |  |
| GraphConv | 0.00088801 | 0.01430215 | 0.01519640 | 0.00018706 | 0.00177910 | 0.00158296 | 8.422M |
| GINEConv | 0.00086362 | 0.01507135 | 0.01591485 | 0.00022936 | 0.00213723 | 0.00190035 | 4.227M |
| SageConv | 0.00091070 | 0.01600684 | 0.01706942 | 0.00025046 | 0.00243048 | 0.00223354 | 8.422M |
| ResGatedGraphConv | 0.00051189 | 0.01318974 | 0.01419000 | 0.00013985 | 0.00147823 | 0.00124201 | 17.105M |
| GatConv | 0.00291205 | 0.01372243 | 0.02828574 | 0.00039533 | 0.00292789 | 0.00362555 | 34.112M |
| TransformerConv | 0.00053179 | 0.01439258 | 0.01582433 | 0.00016199 | 0.00206462 | 0.00216299 | 55.083M |
| SenseFlow (ours) | 0.00093000 | 0.00417808 | 0.00473750 | 0.00010600 | 0.00086501 | 0.00077378 | 21.844M |

**Table 3: Ablation studies on our SenseFlow. We examine the effectiveness of the self-ensembling iterative estimation process (SeIter) and the main components of our proposed FlowNet, including the block fusion, Virtual Node Attention (VNA) and Slack-Gated Feed-Forward (SGF). Results are reported on the IEEE 39-Bus. Improvements over the baseline are marked in blue.**
| SeIter | FlowNet | RMSE $\downarrow$ |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Base | Fusion | VNA | SGF | PQ${}_{\text{Vm}}$ | PQ${}_{\text{Va}}$ | PV${}_{\text{Va}}$ |  |
|  | ✓ |  |  |  | 0.00914456 | 0.12542769 | 0.14066372 ($0.0$) |
|  | ✓ | ✓ |  |  | 0.00774126 | 0.10663362 | 0.12872563 ($\downarrow 0.01193809$) |
|  | ✓ | ✓ | ✓ |  | 0.00561620 | 0.05007443 | 0.05717816 ($\downarrow 0.08348556$) |
|  | ✓ | ✓ |  | ✓ | 0.00658822 | 0.07125929 | 0.07577518 ($\downarrow 0.06488854$) |
|  | ✓ | ✓ | ✓ | ✓ | 0.00453724 | 0.04653547 | 0.05373371 ($\downarrow 0.08693001$) |
| ✓ | ✓ |  |  |  | 0.00102207 | 0.01159813 | 0.01238586 ($0.0$) |
| ✓ | ✓ | ✓ |  |  | 0.00112893 | 0.01129249 | 0.01206334 ($\downarrow 0.00032252$) |
| ✓ | ✓ | ✓ | ✓ |  | 0.00098343 | 0.00697184 | 0.00771528 ($\downarrow 0.00467058$) |
| ✓ | ✓ | ✓ |  | ✓ | 0.00100311 | 0.01011067 | 0.01089543 ($\downarrow 0.00149043$) |
| ✓ | ✓ | ✓ | ✓ | ✓ | 0.00078161 | 0.00608600 | 0.00609802 ($\downarrow 0.00628784$) |

### 4.3 Estimation Performance

Table [1](https://arxiv.org/html/2505.12302v1#S3.T1) presents a performance comparison between our proposed method, SenseFlow (comprising FlowNet and SeIter), and other advanced graph convolution approaches on the IEEE 39-Bus and IEEE 118-Bus systems. The results demonstrate that SenseFlow significantly outperforms the other methods across both systems. In the IEEE 39-Bus system, SenseFlow achieves the lowest root mean square error (RMSE) for voltage predictions at PQ and PV nodes, with magnitude error at 0.0007816 and phase angle errors at 0.00608600 and 0.00609802, showcasing its high-precision predictive capabilities. In the IEEE 118-Bus system, SenseFlow also exhibits exceptional performance. While it may not be the absolute best for magnitude predictions of PQ nodes, it remains very competitive and shows remarkable superiority in the more challenging phase angle predictions compared to other methods.
Notably, our self-ensembling iterative estimation (SeIter) strategy enhances prediction performance significantly across different methods. For instance, it improves phase angle prediction errors by an order of magnitude in traditional graph convolution methods like GraphConv, GINEConv, and SageConv, while also providing substantial gains in more advanced architectures such as ResGatedGraphConv, GatConv, and TransformerConv. Overall, the combination of the FlowNet architecture and the SeIter strategy positions SenseFlow as a highly effective approach for power system state estimation.

We investigate the estimation performance of our SenseFlow on the more complex and larger IEEE 300-Bus system in Table [2](https://arxiv.org/html/2505.12302v1#S4.T2). We can clearly observe that our SenseFlow with SeIter can obtain the state-of-the-art (SOTA) performance, evidenced by consistently lower RMSE values across different metrics. In the absence of our SeIter strategy, we achieved a significant reduction in phase angle prediction error from approximately 0.013 to around 0.004 compared to the second-best method, ResGatedGraphConv. When SeIter is incorporated, SenseFlow emerges as the only method capable of reducing phase angle errors below 1e-3, showcasing its superior performance in this context.

### 4.4 Ablation Study

*Impact of different components of SenseFlow.* The ablation studies in the Table [3](https://arxiv.org/html/2505.12302v1#S4.T3) demonstrate the effectiveness of the key components in our SenseFlow, including the self-ensembling iterative estimation (SeIter), block fusion, Virtual Node Attention (VNA), and Slack-Gated Feed-Forward (SGF). Without the SeIter process, introducing the Fusion, VNA, and SGF results in RMSE reductions of 0.01193809, 0.08348556, and 0.06488854, respectively, for the phase angle predictions of PV nodes compared to the baseline. When these components are combined, forming the complete FlowNet, the RMSE is further reduced to 0.0537, an overall improvement of 0.0869. More notably, the addition of SeIter (with a default loop count of 8) significantly decreases all RMSE metrics by approximately 10-fold.
As a result, our complete SenseFlow achieves an RMSE of less than 1e-3 for the voltage magnitude estimation and less than 1e-2 for the phase angle estimation, demonstrating its substantial improvements and overall effectiveness.

*Scaling iterative loops.* Figure [4(a)](https://arxiv.org/html/2505.12302v1#S4.F4.sf1) investigate the effect of scaling iterative loops on the estimation performance. Specifically, transitioning from a single loop (loop = 1) to multiple loops (loop $>$ 1) significantly enhances the accuracy of voltage magnitude and phase angle predictions, with up to 12 loops reducing the phase angle error by nearly two orders of magnitude. As the number of loops increases, prediction errors continue to decrease, highlighting the benefits of iterative refinement. However, this improvement comes at the cost of increased training and inference costs. To balance accuracy and computational efficiency, we adopt 8 loops as the default, which ensures a phase angle prediction error below 1e-2 while minimizing computational overhead.

*Comparison with mathematical methods.*
Calculation time.
We compare the total time required to perform full N-2 contingency simulations on three IEEE test systems using PyPower (a traditional solver) and our Senseflow ( on single GPU-A100-40G). While performance is similar for small systems, as shown in Figure [4(c)](https://arxiv.org/html/2505.12302v1#S4.F4.sf3), Senseflow achieves 3–5 speedup on larger networks. This demonstrates the model’s superior scalability and suitability for high-volume, large-scale contingency analysis.
Estimation with incomplete inputs.
SenseFlow is capable of accurately estimating voltage states with missing parameters, a challenge that conventional methods cannot address due to incomplete information.
As shown in Figure [4(b)](https://arxiv.org/html/2505.12302v1#S4.F4.sf2), with missing inactive power in the grid, our method still achieves estimation performance at the $10^{-3}$ level, even when more than 10% of PV nodes lack Q-values on the 39-Bus system. Compared to TransformerConv and GatConv with complete information in Table [1](https://arxiv.org/html/2505.12302v1#S3.T1), SenseFlow still obtains lower estimation errors with Q-value missing on the 118-Bus system.

Figure: (a) Scaling Loops
Refer to caption: x6.png

## 5 Conclusion

In this paper, we emphasize the critical features of power systems for power flow analysis, *i.e.*, the unique phase angle-referencing Slack node and the sparse network structure. We propose SenseFlow, a physics-informed and self-ensembling iterative framework for power flow estimation. By integrating the proposed FlowNet and SeIter strategy, SenseFlow effectively addresses these characteristics and enhances the prediction accuracy of voltage states through iterative refinements.
Experimental results demonstrate that SenseFlow achieves state-of-the-art performance in power flow estimations. Ablation studies further validate the effectiveness of proposed strategies.
Impact: Our study provides a promising data-driven solution that supports smarter and more resilient power grid management amid increasingly complex energy environments.
Limitation: In real-world operation, power systems often encounter more complex challenges, such as noisy measurements, data redundancy, and asynchronous or missing inputs. While SenseFlow shows robustness with missing information, handling these practical issues remains an open problem.

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## Peer Reviews

confidence: 5
contribution: 2
initial_rating: 2
presentation: 2
questions: 1. It’s not clear whether you’re solving power flow or state estimation — sometimes the paper reads like one, sometimes the other (especially when talking about missing Q or uncertain data). Can you clarify exactly what the task is and what inputs/outputs the model sees?

2. The slack bus treatment is physically confusing. The slack only absorbs system mismatch; it’s not a “global messenger.” Why is a special Slack-Gated module needed, and how would this extend to the more realistic distributed slack formulation where all generators share mismatch?

3. The model uses hundreds of thousands of samples for training — that’s a lot. How data-hungry is SenseFlow? Would performance degrade sharply with smaller datasets? The  out-of-distribution tests information is limited to N-1 or N-2 line removals. How much these lines were loaded on avg? Were the least or most loaded lines? How much voltage distribution changed once you remove these lines? 

4. The SeIter refinement loop looks like a learned Gauss–Seidel update. Does it ever diverge or oscillate on difficult cases?

5. Power flow is a steady state problem. How are you relating it with stability?
soundness: 2
strengths: The paper makes an interesting attempt to tailor graph attention mechanisms specifically for power flow by emphasizing the role of the slack bus and physical dependencies. Also, the iterative, self-ensembling refinement idea shows awareness of the classical Gauss–Seidel/Newton–Raphson structure, which is conceptually appealing.
summary: The paper proposes SenseFlow, a GNN-based iterative estimator for AC power-flow state (voltage magnitudes and angles). The model contains (i) a Physics-informed Power Flow Network with Virtual Node Attention  and Slack-Gated Feed-Forward modules to increase global communication and emphasize the Slack node, and (ii) a Self-Ensembling Iterative Estimation training/inference loop that EMA-ensembles parameters across iterations. Experiments on multiple IEEE system datasets with perturbed injections/branches and random line outages are reported by authors. SenseFlow shows much smaller RMSE than competing GNNs and claims advantages (estimation under missing Q, faster N-2 contingency throughput vs PyPower, and scalability).
weaknesses: The authors seem to have missed the fundamental point of the power flow formulation — bus types (PQ, PV, slack) are not intrinsic physical categories but numerical conveniences to make the equations solvable. Also the slack bus itself merely absorbs system mismatch, not a source of “global influence” that must be learned. In reality, system works in distributed slack fashion (Dhople et. al. ) where all generators shares a mismatch of power, compared to schedule dispatch. Also, existing GNNs already capture network sparsity and coupling through the incidence or admittance matrix, so the claimed need for a special slack-aware attention is conceptually weak. Note that Gaussian Process–based models with physically informed kernels have already demonstrated how both local and global relationships in power systems can be modeled effectively and with far fewer samples (Bandong et.al., Pareek et.al.). Here, the network requires hundreds of thousands of samples  which raises concerns about scalability and true learning of physical structure.

**Major Concerns:** 

1. **Slack bus understanding and discussion:** 
The SGF block assumes a single slack bus and distributes its features globally. In practice, this is not physically meaningful under uncertainty — real systems use distributed slack where all generators share imbalance. This has been studied extensively (e.g., Dhople et al.). The current approach ignores this and risks non-physical generalization. Further, I am not sure how can you make NRLF converge for 50-150% uncertainty in load with single slack bus. A slack bus in these systems with this much load variation (specially when you hit 150%) but be supplying way beyond its capacity, which is far away from real situations and thus biasing the data towards slack-bus attention artificially., 

2. *Related work (GP, distributed slack, classical solvers):*  Gaussian process–based power flow/state estimation works;Distributed slack formulations; Modern scalable solvers such as nonlinear algebraic multigrid and HELM are not discussed. See _Machine Learning for Solving Power Flow Equations_ at https://energy.hosting.acm.org/wiki/index.php/ML_OPF_wiki . This makes the contribution positioning unclear.

3. **Experimental issues:** Extremely large training sets (up to 500k samples) with no analysis of data efficiency. Think like these, why should I not just solve power flow on a particular instance if I need to solve these many to train. Also no comparison to modern numerical solvers; only older GNN baselines and PyPower runtime is given + No report of FLOPs, wall-clock, or inference cost relative to solver accuracy.

4. **Lack of theoretical grounding**
The iterative SeIter process lacks any convergence or stability analysis. It is unclear whether it always improves residuals or can diverge.


**Minor Concerns**

– Missing hyperparameters: number of layers, K, attention heads, etc.
– Equations (7–12) need clearer explanation or pseudocode.
– Several typos and unclear notation (e.g., PV vs PQ example).

Dhople et. al. : Reexamining the Distributed Slack Bus, IEEE Transactions on Power Systems ( Volume: 35, Issue: 6, November 2020)
Bandong et.al. : https://arxiv.org/pdf/2505.15950

GP Works+Local Kernel + Topologies: Pareek et.al. Data-efficient strategies for probabilistic voltage envelopes under network contingencies, Sustainable Energy, Grids and Networks. (https://arxiv.org/abs/2310.00763)

---

confidence: 5
contribution: 2
initial_rating: 4
presentation: 3
questions: 1. What is the main technical innovation of SenseFlow compared to prior ML-based power flow solvers, especially those using end-to-end learning with physical projection?
2. Does the method offer any formal guarantees regarding convergence, feasibility, or error bounds for the predicted solutions?
3. How does the model address the non-uniqueness of power flow solutions during training and inference, and is there any mechanism to ensure consistency or physical meaningfulness in the solutions it learns?
4. Could existing OPF learning methods be applied to the problem considered in this paper, and if so, what is the unique advantage of SenseFlow? Is the proposed approach extendable to OPF tasks, or are there limitations?
5. How well does the model generalize to unseen grid topologies or operating conditions?
6. What is the computational cost of the iterative self-ensembling process, and is the method scalable to large systems?
7. Can the model’s predictions be interpreted or trusted by system operators, and is there a way to quantify uncertainty in the outputs?
8. Have ablation studies or sensitivity analyses been carried out to assess the contribution of each architectural component and the impact of hyperparameter choices?
soundness: 3
strengths: 1. The problem addressed is timely and relevant to the increasing complexity of modern power systems.
2. The presentation is clear, and the technical pipeline is well-structured.
3. The proposed architecture and iterative refinement are technically valid and leverage recent advances in graph neural networks.
summary: This paper proposes SenseFlow, a physics-informed graph neural network framework for estimating power flow in electrical grids, combining system topology with physical constraints. The approach iteratively refines predictions using self-ensembling and dual loss functions to enforce physical feasibility.
weaknesses: 1. The paper’s core contribution is not clearly distinguished from existing machine learning approaches to power flow problems, particularly end-to-end methods that enforce physical correctness.
2. There is no theoretical analysis or guarantee provided for the prediction quality, convergence speed, or robustness of the proposed method.
3. The approach does not explicitly address the issue that power flow equations can have multiple feasible solutions, raising concerns about the uniqueness and consistency of the learned mapping.
4. The connection between the proposed method and the broader literature on learning optimal power flow (OPF) problems is insufficiently discussed, leaving unclear whether OPF methods could handle the considered problem as well or better.

---

confidence: 4
contribution: 2
initial_rating: 2
presentation: 2
questions: ## Questions

- In practice, there is often more than one slack bus in a power system. How does the proposed method handle this situation?
- How does the proposed method perform on samples where some bus voltages or line flows are at the security constraint boundary? For these samples, the PFA tool should accurately determine whether the operational variables exceed the boundary, rather than relying solely on the MSE evaluation.
- PFA is usually treated as a post-processing task of state estimation. Why do the authors consider the reactive power at buses in PFA to be unmeasurable, when it should be determinable through state estimation?
- During the iterative process, the calculation of deviations in bus injection powers depends on accurate line parameters. If the line parameters are inaccurate or missing, how significant would the impact be on the learning and inference of the entire model?
- The essence of the proposed method is to replace the computation of the Jacobian matrix. How significant is the impact of the accumulated prediction errors from the neural network on the results?
- Some arrows in Figure 3 are disconnected, please correct them.
- In the generation of the dataset, what do the line features refer to? Please clarify.
soundness: 2
strengths: ## Strengths

- This paper embeds the neural network into the iterative process of the traditional PFA method, which differs from most existing end-to-end learning methods and makes the model’s training more conservative.
- This paper distinguishes between different types of buses in PFA in order to learn representations that are more effective for PFA.
summary: This paper proposes a data-driven power flow analysis (PFA) method. The authors designed a learning mechanism embedded in the iterative process, along with a neural network architecture that accounts for the influence of the slack bus. Finally, several small standard IEEE cases are used to verify the proposed method’s advantage in terms of PFA accuracy.
weaknesses: ## Weakness

- The term “power flow estimation” can be easily misunderstood, and from the perspective of power systems, the motivation of this paper is unclear. PFA has already been extensively studied, and even in large-scale systems with tens of thousands of buses, fast batch PFA can be achieved through high-performance computing techniques. So, why is a data-driven PFA necessary?
- The description of the correlation between the two components in SenseFlow is not very clear; they appear to be two components that could operate completely independently.
- The test cases are too small in scale and do not include renewable energy sources, making it difficult to demonstrate the practicality of the proposed method.

---

confidence: 5
contribution: 2
initial_rating: 2
presentation: 1
questions: * Please comment on the distributed slack setting, where the active power mismatch is distributed across the entire network. This more closely matches the reality of transmission grids, where multiple units adjust their output to ensure active power balance.
* Please explain the difference between parameters $\theta_{s}$ and $\theta_{t}$: they seem to refer the PPFNet parameters (l 233) and the "ensembling model" (l.247), but the latter is updated using the former.
* What does PPFNet actually output? Figure 1a suggests that it outputs $V_{m}, V_{a}$ (from which one evaluates the loss function), but Figure 2b suggests that it outputs a correction $\Delta V_{m}, \Delta V_{a}$?
* Please provide some measure of performance variance (e.g. standard deviation of loss across 10 different random seeds) in numerical results
* Please provide a comparison of the self-ensembling mechanism against a gradient-descent based mechanism whereby the initial prediction is used to warm-start a gradient-based algorithm that adjust $V_{m}, V_{a}$ to minimize the power imbalance. I expect the corresponding per-iteration cost to be substantially cheaper than the self-ensembling model.
* Please provide more details regarding the pypower implementation, especially regarding the use of parallelization using multiple CPU cores. The computational comparison should take into account the fact that an A100 GPU is about 100x more expensive than a single CPU core (using cloud computing price estimates of ~1c/hr per CPU vs ~1\$/hr per A100 GPU)
* Since the ML models are evaluated on GPUs, a GPU-enabled batch power flow solver like ExaPF is a more relevant baseline (see https://exanauts.github.io/ExaPF.jl/stable/)
soundness: 2
strengths: * The paper introduces an attention-based mechanism to propagate information efficiently across the power network graph, which can potentially mitigate the fact that power network graphs have high diameter.
* The "self-ensembling" mechanism adds an iterative correction scheme to mitigate potential prediction errors.
summary: This paper seeks to solve power flow equations using a combination of an attention-based graph neural network architecture with an iterative self-corrective scheme. The proposed methodology is evaluated on small power grids with up to 300 buses, and is shown to yield accurate voltage predictions and modest speedups compared to a classical CPU-based power flow solver.

Overall, I find the paper hard to follow, due to a lack of mathematical clarity, multiple typos in the document, and several missing details regarding the experiments.
The numerical experiments should be drastically expanded (using power grids with several thousands of buses) to demonstrate the relevance of the proposed approach. In addition, performance evaluation should be more robust (e.g. by reporting performance variability and additional accuracy metrics)
weaknesses: * The paper does not state the mathematical problem it is trying to solve. Namely, Section 3.1 only includes an English description, it would be beneficial for non-power systems experts to include a complete mathematical statement (i.e., definitions + math equations) of the problem to be solved.
* The following works are relevant for the literature review:
  * [Neural networks for power flow: Graph neural solver](https://doi.org/10.1016/j.epsr.2020.106547): one of the first works on graph neural networks for solving power flow
  * [https://doi.org/10.1016/j.epsr.2024.110738](https://doi.org/10.1016/j.epsr.2024.110738): proposes topology augmentation to mitigate the limitations of message-passing in sparse power network graphs.
* The proposed approach relies on a supervised learning loss for training, which requires labeled data that can be expensive to compute (especially for large-scale grids)
* Numerical results
  * The numerical experiments consider very small, artificial networks with artificially-sampled data. Experiments should consider industry-size networks (i.e. several thousands of buses)
  * The performance evaluation focuses on errors on voltage magnitude and angles, which does not capture potential mismatches in reactive power injections at PV buses
  * Numerical results do not report computing times (i.e. labeling, training and inference times) for ML baselines
  * The ablation studies do not consider removing the "fusion" component in itself.
* There are several typos in the manuscript, e.g.: non-matched closing parenthesis `)` on line 202, subscript `gt` at line 204. Please avoid abbreviations like "the net"

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. In practice, why is the speed of power flow estimation important? Is it for timely prediction and reaction to avoid, e.g., cascading failure? What level of estimation latency is acceptable? Can SenseFlow achieve that level of latency? 
2. Can this work be generalized to other domains? 
3. As real-world power systems are affected by noise, could you discuss how the network handles noise?
4. The iterative approach reminds me of Neural ODE, which interprets ResNet as a discrete step of an ODE. The paper also uses iterative evaluations to fit the trajectory of an ODE. Considering a power grid is governed by differential equations in theory, it would be insightful if the authors can discuss theoretically if SenseFlow can be accommodated into the frame of Neural ODE -- is it a neural-network-based differential equation solver? 

Chen, Ricky TQ, et al. "Neural ordinary differential equations." Advances in neural information processing systems 31 (2018).
soundness: 3
strengths: 1. This work targets power flow estimation, achieving significantly higher estimation quality compared to neual-network-based baselines, and lower estimation time compared to traditional numerical solver (PyPower). 
2. This work covers the scenario of missing data in a power system, which is difficult to handle in numerical solvers. 
3. The domain specific technologies proposed in this work are reasonable and effective as experimental results demonstrate.
summary: The authors propose SenseFlow that combines two components designed in this work: PPFNet and SeIter. PPFNet (Physics-informed Power Flow Network) includes Virtual Node Attention (VNA) and Slack-Gated Feed-Forward (SGF) modules, where VNA condense all nodes into a virtual node to distill global information and distribute the information through cross-attention; SGF emphasizes the slack node (only one slack node in each dataset) by combining its features with each node's features. SeIter (Self-Ensembling Iterative Estimation) is an iterative strategy that refines the PPFNet estimations for multiple loops to approach ground truth.
weaknesses: 1. The motivation may be more clearly stated. 
2. The impact of this work may be generalized to other domains. 
3. Missing robustness discussion.