BOOST-RPF: Boosted Sequential Trees for Radial Power Flow

1. Problem Statement

Accurate and scalable power flow analysis is critical for modern distribution systems, particularly with the proliferation of distributed energy resources (DERs) and the need for real-time control.

• Limitations of Classical Solvers: Traditional methods like Newton-Raphson (NR) offer high fidelity but suffer from high computational costs, convergence issues in heavily loaded networks, and a lack of scalability for real-time applications.
• Limitations of Current ML Approaches: Existing machine learning (ML) models typically treat voltage prediction as a global graph regression task, predicting all bus voltages simultaneously. While these models (e.g., Graph Neural Networks, MLPs) show good in-distribution accuracy, they struggle significantly with Out-of-Distribution (OOD) generalization. They often fail when applied to unseen network topologies or sizes because they memorize specific grid configurations rather than learning the underlying recursive physics.

2. Methodology: BOOST-RPF

The authors propose BOOST-RPF, a novel framework that reformulates power flow analysis from a global graph problem into a sequential path-based learning problem.

Core Concept

Instead of predicting the entire grid state at once, the method decomposes radial networks into root-to-leaf paths. It leverages the physical reality that voltage drops in radial feeders are governed by repeated applications of local physical relationships between parent and child buses.

Key Technical Components

1. Grid-to-Sequence Transformation:

   • The grid is modeled as a Directed Acyclic Graph (DAG) starting from the slack bus.
   • The state of a child node $j$ is conditionally dependent on its immediate parent $i$: $V_j = f(V_i, Z_{ij}, S_j, S_{agg,j})$.
   • This allows the model to learn local voltage-drop mappings rather than global correlations.

2. Feature Engineering:

   • Backward Power Accumulation: A leaf-to-root traversal calculates the total downstream demand ($P_{agg}, Q_{agg}$) for each node to proxy the current flowing through the branch.
   • Physics-Informed Baselines: The model utilizes LinDistFlow to generate a first-order linear baseline for voltage magnitude and angle.
   • Feature Vector: Inputs include parent voltage/angle, branch impedance, local injections, aggregated downstream demand, and the LinDistFlow baseline.

3. Learning Architectures (XGBoost Variants):
   The authors utilize Gradient-Boosted Decision Trees (XGBoost) trained on three different targets:

   • Variant A (Absolute): Predicts absolute voltage directly.
   • Variant B (Parent Residual): Predicts the direct voltage drop from the parent ($V_i - V_j$).
   • Variant C (Physics-Informed Residual): Predicts the deviation from the LinDistFlow linear approximation ($V_{LDF} - V_{true}$).

4. Inference Strategy:

   • Inference is performed via a Breadth-First Search (BFS) traversal.
   • The model predicts sequentially along paths, using the output of the previous node as the input for the next.
   • This ensures $O(N)$ computational complexity and allows for size-agnostic application.

3. Key Contributions

• Paradigm Shift: Transforms power flow from a global regression task to a sequential learning task, aligning the model architecture with the recursive physics of radial power flow.
• Superior Generalization: By training on local edge-level data (voltage drops) rather than global snapshots, the model achieves size- and topology-agnostic performance. It demonstrates strong zero-shot generalization to unseen feeders.
• Sample Efficiency: Decomposing a single grid into multiple root-to-leaf paths multiplies the available training data, significantly improving sample efficiency compared to global models.
• Physics-Informed Learning: The "Parent Residual" and "Physics-Informed Residual" variants allow the model to leverage known physics for first-order behavior while learning only the high-order nonlinearities and losses from data.

4. Experimental Results

The method was benchmarked against analytical solvers (DistFlow), Graph Neural Networks (ARMA-GNN), and global MLPs using the Kerber Dorfnetz and the ENGAGE dataset (heterogeneous LV networks).

• Accuracy (In-Distribution):

  • The XGB-Parent variant achieved state-of-the-art results, with a Voltage Magnitude RMSE of $8.42 \times 10^{-4}$ p.u. on the fixed Kerber grid, significantly outperforming neural baselines (which had errors an order of magnitude higher).
  • It consistently outperformed DistFlow, which struggled with nonlinearities in this specific setting.

• Generalization (Out-of-Distribution):

  • GlobalMLP Failure: The GlobalMLP failed catastrophically on unseen topologies (RMSE > 1.0 p.u.), rendering it unusable.
  • Sequential Robustness: BOOST-RPF (specifically XGB-Parent) maintained high precision on unseen networks, with an RMSE of $1.34 \times 10^{-2}$ p.u., demonstrating robustness where global models failed.
  • GNN Performance: ARMA-GNN showed some generalization but suffered significant performance degradation compared to its in-distribution results.

• Efficiency:

  • Training Speed: XGBoost models trained in ~12 seconds (CPU), whereas neural networks took 216s (MLP) and 1420s (GNN) (GPU).
  • Inference Scaling: The method exhibits linear $O(N)$ scaling. While global models have constant inference time, the sequential approach scales predictably and remains fast enough for real-time applications (e.g., ~14ms for 130 nodes).
  • Error Accumulation: Analysis showed that error accumulation along long feeder paths is negligible (marginal drift of $0.0008$ p.u. over 18 hops).

5. Significance and Impact

• Real-Time Applicability: BOOST-RPF offers a scalable, fast, and accurate alternative for Distribution System Operators (DSOs), suitable for real-time hosting capacity analysis and dynamic voltage control.
• Bridging the Gap: It successfully bridges the gap between the interpretability and physical consistency of classical analytical solvers and the high performance of data-driven ML.
• Scalability: The method is inherently scalable to networks of arbitrary size without retraining, addressing a major bottleneck in current ML-based power flow solvers.
• Future Potential: The framework opens avenues for extending sequential learning to meshed grids and implementing lower-level parallelization (e.g., C++) to further reduce latency to $O(\log N)$.

In conclusion, BOOST-RPF demonstrates that reformulating power flow as a sequential learning problem using gradient-boosted trees yields superior accuracy, generalization, and efficiency compared to both classical solvers and modern deep learning approaches.