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

TL;DR

SenseFlow tackles the core challenge of accurate power flow estimation in sparse grids with a single Slack bus by integrating physics-informed constraints with an iterative self-ensembling learning loop. The framework combines FlowNet, featuring Virtual Node Attention and Slack-Gated Feed-Forward modules, with Self-Ensembling Iterative Estimation (SeIter) that maintains an EMA teacher model to progressively refine predictions through multiple loops. Losses incorporate both ground-truth alignment and physics-based power-balance equations, enabling predictions of voltage magnitudes and phase angles that respect system dynamics. Empirical results on IEEE benchmarks (39/118/300 buses) show state-of-the-art accuracy and notable speedups for contingency analyses, with robust performance under incomplete input data. This approach promises improved reliability and scalability for power grid state estimation in increasingly complex and renewable-rich networks.

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, which poses significant challenges in achieving high-accuracy estimations. 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.

Figures (4)

• Figure 1: (a) Comparison of the number of nodes and edges across various IEEE standard systems (IEEE 39-Bus, 118-Bus, and 300-Bus), which reveals two key points: 1) there is only one slack node present in each system, and 2) the network exhibits relatively sparse connectivity. (b) Schematic diagram of the IEEE 39-Bus system with typical three different types of nodes and edges. The diagram also shows the parameters to be solved in the power flow calculation, including the phase angle of PV nodes, and the voltage and phase angle of PQ nodes, alongside the known values including the voltage and phase angle of the slack node and the voltage of the PV nodes.
• Figure 2: Illustration of our Self-ensembling Iterative Estimation (SeIter). (a) In the $\eta$-th loop, the trainable FlowNet ($\theta_s$) receives the input voltage magnitudes $V_m(\eta)$ and phase angles $V_a(\eta)$ from the previous loop and the changes in active and reactive power $\Delta P$ and $\Delta Q$ calculated by the power balancing equations $\psi$. The net is trained by two loss functions: the ground-truth loss, $\mathcal{L}_{gt}$, which aligns the predictions with the actual data, and the equation loss, $\mathcal{L}_{equ}$, which ensures the model adheres to the physical laws governing the system. (b) The Self-Ensembling Inference module prepares the updated data for the next loop. It leverages the self-ensembling teacher model ($\theta_t$) to generate predictions, which serve as the input for the trainable model in the subsequent $\eta + 1$ loop, where $\theta_t$ is updated by the exponential moving averaging of $\theta_s$.
• 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.
• Figure 4: (a) we examine the impact of the iterative loop on the IEEE 39-Bus system. The number of iterations is set to 8 by default, considering the increased inference effort with larger loops. (b) we investigate the missing Q values settings on IEEE 39-Bus and 118-Bus. These settings, without complete known information, cannot be addressed by conventional calculation methods. (c) we compare the total calculation time of N-2 contingency analysis on IEEE standards.