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Hierarchical Graph Neural Network for Probabilistic Power Flow with Grid Topology Contingencies

Jinbao Wang, Jun Liu, Shiliang Zhang, Xuehui Ma, Lin Yan, Haolin Liu, Sabita Maharjan

arXiv:2610.13495v1eess.SYcs.GR

TL;DR

PPF surrogates face a generalization gap when outages change the grid’s electrical coupling, while conventional PPF also requires repeated nonlinear solves. TDD–HBGNN reconstructs a hierarchy for each topology and combines a sensitivity-based prior with learned nonlinear corrections; across the tested unseen contingencies, it lowers phase-angle errors and favors joint voltage–angle error against the benchmarks.

  • Problem

    Existing approaches do not explicitly represent how topology changes reshape uncertainty propagation and inter-nodal coupling across local, regional, and system scales.

    Source: §I · pp. 1–2

  • Method

    TDD–HBGNN reconstructs FTUs, DHVCs, and a CBSG for each topology, combining a sensitivity-based physical prior with learned nonlinear residuals.

    Source: §I · p. 2, §V · p. 11

  • Results

    2.47–4.20 all-bus phase-angle ratios against the strongest benchmarks, with all 24 system–contingency–baseline summaries favoring TDD–HBGNN in joint voltage–angle error.

    Source: §V · p. 11

  • Takeaways & Limitations

    Across the tested IEEE and Polish systems and unseen N–1 and N–2 contingencies, every system–contingency–baseline summary favored TDD–HBGNN in joint voltage–angle error.

    Source: §V · p. 11

    Future work should extend the sensitivity prior to strongly nonlinear and unbalanced regimes and test genuine zero-shot transfer on related utility networks.

    Source: §V · p. 11

Abstract

from arXiv · show

Data-driven surrogates reduce the nonlinear solves required for probabilistic power flow (PPF), but models tied to a fixed bus--branch graph often generalize poorly when outages change electrical coupling. This paper proposes a topology-decomposition-driven hierarchical bidirectional graph neural network (TDD--HBGNN) for PPF under unseen topologies. For each realized topology, the network is decomposed into fundamental topological units (FTUs), dynamically homogeneous voltage clusters (DHVCs), and a core backbone sub-grid (CBSG), with equivalent-admittance representations preserving electrical ports between units. A sensitivity-based linear prior captures the first-order voltage-magnitude and phase-angle response, while the HBGNN learns topology-dependent nonlinear residuals. Impedance- and uncertainty-aware bottom-up aggregation condenses local perturbations, while mode-modulated top-down broadcasting restores system context at buses. We evaluate our approach on the IEEE 118-bus and Polish 2746-bus systems under unseen N--1 and N--2 contingencies against six baseline methods. Compared with these methods, TDD--HBGNN achieves 2.47--4.20-fold lower all-bus phase-angle errors and lower phase-angle errors in every held-out event. Its voltage-magnitude advantage is concentrated near the outaged branches, where paired error ratios reach 1.75--2.58 on the IEEE system and 1.87--1.97 in Polish N--2 events. All system--contingency--baseline comparisons favor TDD--HBGNN in joint voltage--angle error. Moreover, shared hierarchical operators keep the model size at 0.658 million parameters across both grids and yield 0.161--1.793~ms forward latency per probabilistic case. These results demonstrate system-wide angular accuracy, outage-local voltage accuracy, and parameter-efficient scaling across unseen topologies.

I. INTRODUCTION

Probabilistic power flow characterizes bus-voltage magnitudes and phase angles under uncertain injections, but repeated power-flow solves are computationally burdensome. This paper addresses limited generalization across changing topologies with a decomposed hierarchical model that combines a sensitivity-based baseline with learned nonlinear corrections.

I. INTRODUCTION

  • Traditional PPF requires repeated power-flow calculations, whose computational burden grows with system scale, uncertainty scenarios, and stochastic-variable dimensionality.This burden limits online analysis and rapid decision-making for large grids.Source: §I · p. 1
  • Existing models do not explicitly represent how topological perturbations reshape uncertainty propagation and coupling across local, regional, and system scales.The paper frames the open questions as how topology changes reshape stochastic mappings and how to combine dominant physical response with topology-dependent corrections.Source: §I · p. 2
  • TDD–HBGNN reconstructs a grid hierarchy for each topology and combines a sensitivity-based linear response with HBGNN-learned nonlinear residuals.Its hierarchy comprises CBSGs, DHVCs, and FTUs; bottom-up aggregation and top-down broadcasting link these levels to bus-wise voltage-magnitude and phase-angle quantiles.Source: §I · p. 2

II. PROBLEM FORMULATION AND FRAMEWORK OF THE PROPOSED METHOD

  • The PPF formulation treats active and reactive nodal injections as random variables and represents their uncertainty with quantile sets.It predicts voltage-magnitude and phase-angle quantiles at each bus, using output quantile grids drawn from the input quantile levels.Source: §II · pp. 2–3
  • When topology changes, the mismatch between training and real-time data reduces traditional data-driven model accuracy.The figure contrasts a changed physical mapping with a data-driven branch that cannot adapt to it.Source: §II · p. 3, Figure 1 · p. 3

III. THE PROPOSED METHOD

The proposed method decomposes each grid into electrically meaningful hierarchical units, then uses their structure to represent both local dynamics and system-wide context. Its decomposition combines electrical and topological features, with equivalent-admittance representations preserving each unit’s connections to the rest of the grid.

A. Grid topological decomposition

  • The decomposition organizes the grid as a CBSG and multiple DHVCs, then subdivides these components into minimal FTUs.The hierarchy reflects electrical dynamics and topological structures such as series, parallel, and mesh connections.Source: A. Grid topological decomposition · p. 3, §III · p. 3
  • CBSG identification uses stable voltage profiles and electrical support, combining static node weights with Granger-causality-based dynamic relationships.The node weights include short-circuit capacity, voltage-reactive power sensitivity, and betweenness centrality; Net Causal Strength distinguishes disturbance sources from sinks.Source: A. Grid topological decomposition · p. 3
  • DHVCs are formed by clustering non-backbone buses using residual-voltage affinity while requiring each cluster to remain topologically connected.Residual voltages are conditioned on the backbone; the affinity incorporates correlation, electrical impedance distance, and topological hop-count distance.Source: A. Grid topological decomposition · pp. 3–4
  • FTU construction uses graph-theoretic decomposition and shields low-impedance edges to avoid electrically nonsensical structures.The process removes tap-changing and switch-labeled edges before identifying bridges, series chains, and biconnected blocks.Source: A. Grid topological decomposition · p. 4
  • Each unit is represented by an equivalent-admittance matrix over its boundary ports, with internal parameters supporting recovery of nodal quantities.The resulting hierarchy gives local disturbances a neighboring-node representation while backbone-transmitted power-flow laws carry global regulation.Source: A. Grid topological decomposition · p. 4

B. Hierarchical bidirectional graph neural network

HBGNN exchanges information bidirectionally across nodes, FTUs, DHVCs, and the CBSG to represent interactions at multiple spatial scales. A shared backbone predicts voltage and angle quantiles using a sensitivity-based linear baseline plus learned nonlinear corrections.

B. Hierarchical bidirectional graph neural network

  • HBGNN compresses the grid through four levels—nodes, FTUs, DHVCs, and CBSG—using bottom-up aggregation and top-down broadcasting.The shared backbone predicts voltage-magnitude and phase-angle quantiles through separate output heads.Source: B. Hierarchical bidirectional graph… · p. 4, Figure 2 · p. 5
  • The prediction combines base-case states and sensitivity-matrix responses to injection changes with HBGNN nonlinear corrections.The sensitivity matrices are fitted by least squares in normalized space; the architecture depicts a quantile-prediction module with a linearization baseline.Source: B. Hierarchical bidirectional graph… · p. 4, Figure 2 · p. 5

1) Bottom-up aggregation:

Transmission-edge messages are propagated at the node level, then pooled upward through topology, cluster, and backbone using physics-guided operators.

1) Bottom-up aggregation:

  • At the node level, dual-pathway gated message passing sends admittance-weighted messages along transmission edges.Source: 1) Bottom-up aggregation · p. 5
  • Three operators pool features upward: impedance-weighted attention to topology, ε-aware attention to cluster, and mean aggregation to backbone.The hierarchy maps node-to-topology, topology-to-cluster, and cluster-to-backbone.Source: 1) Bottom-up aggregation · p. 5
  • Node-to-topology mapping uses an attention score and associated impedance, while uncertainty is aggregated at the topology level.Source: 1) Bottom-up aggregation · p. 5

2) Top-down broadcasting:

Global context returns to buses through hierarchical unpooling and mode-modulated broadcasting, whose outputs are fused to form refined node features for quantile prediction.

2) Top-down broadcasting:

  • Hierarchical unpooling progressively maps features from level l to level l−1, while mode-modulated broadcasting lets nodes selectively aggregate backbone signals by mode affinity.The mode projection vector belongs to a backbone node, and a projection maps backbone features into node feature space.Source: 2) Top-down broadcasting · p. 5
  • A gating mechanism fuses the unpooling and broadcasting pathways.Source: 2) Top-down broadcasting · p. 5
  • After K bidirectional rounds, each node has a refined feature vector for quantile prediction.Source: 2) Top-down broadcasting · p. 5

3) Quantile prediction and loss function:

The model forms voltage and angle quantiles by adding HBGNN residuals to linearization baselines, with monotone quantile outputs and physics-informed training constraints.

3) Quantile prediction and loss function:

A. Evaluation design and statistical protocol

The evaluation tests unseen branch outages on two grids using forecast-implied PV uncertainty, event-level partitions, and paired statistical comparisons across shared evaluation protocols.

A. Evaluation design and statistical protocol

  • The study evaluates IEEE 118-bus and Polish 2746-bus systems under N–1 and N–2 outages, with load uncertainty, generator redispatch, and cascading failures outside scope.PV forecasts provide 3,328 sets of 11 quantiles; each set generates 1,000 realizations, while loads and scheduled non-PV generation remain at base-case values.Source: A. Evaluation design and statistical… · p. 6
  • Mutually exclusive outage events are split 8:1:1 after excluding nonconvergent contingencies, and cases are aggregated within events before event-level inference.Retained event counts are 98/113 and 230/88 for N–1/N–2 on the 118- and 2746-bus systems, respectively.Source: A. Evaluation design and statistical… · p. 6
  • Learned models generally use hidden width 64 and AdamW, with validation-based learning-rate reduction and early stopping within 2,000 epochs.The stated setup uses batch size 64, initial learning rate 10^-3, and weight decay 10^-4.Source: A. Evaluation design and statistical… · p. 6
  • Six baselines span dense, conventional graph, high-order, recurrent, model-driven, and spectral methods; all share pinball loss and monotone heads, while TDD–HBGNN adds physics and linear-consistency terms.Spatial diagnostics use MD-GCN on bus 118 and DNN on bus 2746 as stored global references.Source: A. Evaluation design and statistical… · p. 6
  • D1 averages error across the stored distribution, endpoint errors measure the two extreme quantiles rather than coverage, and nonuniform grids use trapezoidal integration.Source: A. Evaluation design and statistical… · p. 6
  • Paired ratios favor TDD–HBGNN above one; the protocol reports their median, event win rate, and 95% interval from 20,000 paired event-bootstrap replicates.Ablations use paired Wilcoxon tests with Holm correction.Source: A. Evaluation design and statistical… · p. 6

B. Topology generalization and residual structure

Across unseen topologies, TDD–HBGNN improves joint voltage–angle error against every baseline, with its clearest advantages in phase angles system-wide and voltage near outages. Controlled comparisons show that error scaling depends on the method and does not simply follow contingency order or grid size.

B. Topology generalization and residual structure

  • Near outages, bus 118 local V ratios span 1.75–2.58 and θ ratios 3.23–4.54, with TDD–HBGNN winning every θ event.For bus 2746 N–2, local V and θ ratios are 1.97 and 3.97; D1 and endpoint errors show the local advantage is not restricted to one quantile.Source: B. Topology generalization and… · p. 6
  • Across the full network, phase-angle ratios are 2.47–4.20 and favor TDD–HBGNN in every event, while all-bus voltage is near parity on bus 118 and favors DNN on bus 2746.The reported bus 2746 all-bus V D1/T.05,.95 ratios are 0.50/0.64 under N–1 and 0.42/0.48 under N–2.Source: B. Topology generalization and… · p. 6
  • In bus 118 spatial attribution, the leading five and ten buses contribute 27.93% and 42.47% of Joint error, whereas bus 2746 has more diffuse shares of 5.73% and 10.10%.The leading bus 118 contributors are buses 82, 83, and 96; the remote region carries 54.24% of its error, while bus 2746’s remote-region error and response enrichments are 1.013 and 0.970.Source: B. Topology generalization and… · p. 7, Figure 4 · p. 7
  • On bus 118, TDD–HBGNN reduces V/θ MAE from 2.46 × 10−4 p.u./0.704° to 5.42 × 10−5 p.u./0.0619°, although only 45.8% of buses improve in V.The 95th-percentile and maximum errors fall by factors of 5.87 and 3.72; on bus 2746, 58.9%/99.3% of buses improve in V/θ.Source: B. Topology generalization and… · p. 7, Figure 5 · p. 8

C. Controlled scaling and all-baseline advantage

  • On bus 118, matched N–2 displacement rises by factors of 4.32 for V and 2.68 for θ, while TDD–HBGNN error ratios are 0.83/1.05/0.94 for V/θ/Joint.The Joint interval [0.86, 1.01] includes parity; across methods, scaling behavior is method-dependent, and contingency order is not a universal measure of learning difficulty.Source: C. Controlled scaling and all-baseline… · p. 8, Figure 6 · p. 8
  • All 24 system–contingency–baseline summaries favor TDD–HBGNN in Joint error, with 95% intervals excluding parity.On bus 118, Joint ratios span 1.54–61.5 for N–1 and 1.68–60.2 for N–2; the smallest reported advantage is DNN on bus 2746 N–2 at 1.09.Source: C. Controlled scaling and all-baseline… · p. 8, Figure 7 · p. 8
  • Across the two evaluated systems, TDD–HBGNN’s bus 2746/bus 118 Joint ratios are 0.82 under N–1 and 0.80 under N–2, but results from only two systems cannot establish a universal grid-size relationship.Joint cross-system ratios fall below parity for DNN, GNN, and PowerFlowNet but exceed parity for GNS, MD-GCN, and PG-GNN.Source: C. Controlled scaling and all-baseline… · p. 8
  • For selected N–2 local voltage tails, median P95 errors are 13.25 × 10−5 p.u. versus 35.79–127.40 × 10−5 on bus 118, and 1.03 × 10−5 versus 3.14–22.81 × 10−5 on bus 2746.Paired ratios span 2.39–9.66 on bus 118 and 2.05–12.76 on bus 2746.Source: C. Controlled scaling and all-baseline… · p. 9, Figure 8 · p. 9

D. Finite-grid alignment and selective reliability

Finite-grid diagnostics show that endpoint and interior quantile errors can diverge, while the physics risk head improves reliability metrics in most settings. These diagnostics also bound the conclusions: lower physical-unit error does not establish calibration, and physics augmentation has an exception.

Finite-grid alignment

  • 16 of 24 bus 118 N–2 coordinate medians exceed one, but eight favor a baseline and no V endpoint is robustly favorable on both descriptors.GNN and GNS trade endpoint ratios of 7.85 and 3.84 for interior ratios of 0.87 and 0.82.Source: D. Finite-grid alignment and selective… · p. 9, Figure 9 · p. 9
  • On bus 118, DNN, PowerFlowNet, and MD-GCN are worse on both θ descriptors, while GNN and GNS trade endpoint gains for smaller interior residuals.On bus 2746, DNN reaches 0.40/3.42, while GNN, GNS, and PG-GNN have endpoint ratios of 21.58/21.51/16.51 and interior ratios of 0.55/0.41/0.16.Source: D. Finite-grid alignment and selective… · p. 9

Selective reliability

  • The physics risk head raises ρ to 0.438–0.877, lowers nAUSE to 0.118–0.573, and gives 3.92–7.05-fold worst-decile lift; both ρ and nAUSE improve in seven of eight settings.Figure 10 compares raw interval width, a generic risk head, and the physics risk head; a validation-fixed threshold flags 7.0–11.0% of unseen cases and recalls 34.0–72.8% of high-tail-error cases.Source: D. Finite-grid alignment and selective… · p. 10, Figure 10 · p. 10
  • For bus 2746 N–2 voltage, physics augmentation changes ρ from 0.519 to 0.438 and nAUSE from 0.390 to 0.494, while lift changes from 3.70 to 3.65.A large nonlinear correction can reflect topology redistribution, so treating correction magnitude only as a risk indicator can create false positives.Source: D. Finite-grid alignment and selective… · p. 10

E. Component evidence and deployment boundary

Ablations show that both the prior–correction structure and repeated hierarchical exchange contribute to accuracy, while shared operators keep model size fixed across grids. TDD–HBGNN also supports amortized screening, with high-risk cases routed to power-flow analysis.

Component evidence

  • Removing either the learned correction or linear prior yields median D1 ratios of 2.18–5.99 or 1.66–12.04, while structural variants yield 1.46–7.80 and remain significant after Holm correction.MLP’s global V-D1 ratios of 1.10 and 1.02 are not significant, though its N–2 local ratio is 1.56, angle ratios exceed 2.18, and voltage-tail ratios exceed 1.39.Source: E. Component evidence and deployment… · p. 10, Figure 11 · p. 10
  • Flattening, bottom-up-only propagation, and one exchange round remove distinct parts of the hierarchical information path, including compression, global broadcasting, or re-aggregation of corrected states.The ablations support both multiscale compression and repeated bidirectional exchange.Source: E. Component evidence and deployment… · p. 10

Deployment boundary

  • TDD–HBGNN retains 0.658 million parameters across both grids, while DNN grows from 2.150 to 45.723 million but remains faster in batch inference.The fixed count comes from shared fixed-width hierarchical operators; it does not imply constant training cost, since larger grids require more unit evaluations and activation memory.Source: E. Component evidence and deployment… · p. 10, Table III · p. 11
  • TDD–HBGNN training takes 2.220/0.665 hours on bus 118 and 2.46/2.96 hours on bus 2746; its MCS times are 1,068/1,080 and 1,890/1,914 ms per case.MCS cases use 1,000 AC-PF samples; one-solve AC-PF values represent only the deterministic kernel, not equivalent probabilistic output.Source: E. Component evidence and deployment… · pp. 10–11
  • The results support amortized screening, with high-risk cases routed to AC power flow or full probabilistic analysis.Source: E. Component evidence and deployment… · p. 11

V. CONCLUSION

TDD–HBGNN combines topology-specific hierarchical decomposition with a sensitivity prior and learned nonlinear residuals. Across unseen contingencies, it achieves strong system-wide angular and joint voltage–angle results, while future work remains in broader regimes and calibration.

Conclusion

  • For each topology, TDD–HBGNN reconstructs FTUs, DHVCs, and a CBSG, combines a sensitivity prior with learned nonlinear residuals, and exchanges information bottom-up and top-down.Shared fixed-width operators process the reconstructed hierarchy across grid sizes and within-system unseen contingencies.Source: §V · p. 11
  • 2.47–4.20-fold lower all-bus phase-angle errors and favorable joint voltage–angle error in all 24 system–contingency–baseline summaries were observed across the two test systems.The IEEE hotspot concentrates 42.47% of Joint error in the leading ten buses and 30.63% in leading branches; the corresponding Polish shares are 10.10% and 9.72%.Source: §V · p. 11
  • Future work should extend the sensitivity prior to strongly nonlinear and unbalanced regimes and include uncertainty from loads and non-PV dispatch.The authors also identify topology-aware calibration, out-of-distribution detection, adaptive thresholds, solver fallback, and zero-shot transfer as future work.Source: §V · p. 11
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