A Boundary-Compensated Partition-Based Parallel Graph Neural Network for Weak-Bus Identification in Interconnected Power Grids

Abstract

Weak-bus identification is a key task for online security assessment, preventive control, maintenance verification, and resilience-oriented dispatch of interconnected power grids. In large-scale grids, conventional full-graph graph neural networks preserve the complete network topology but may become inefficient when many operating scenarios must be screened repeatedly. Direct graph partitioning improves computational tractability, but it may cut tie-line channels and weaken the boundary evidence that determines cross-area risk propagation. To address this trade-off, this paper proposes a boundary-compensated partition-based parallel graph neural network for weak-bus identification. The method first constructs a scenario-aware weighted power-grid graph and divides it into electrically coherent subgraphs under coupling-strength and partition-size constraints. Local graph encoders are then executed in parallel to learn intra-partition vulnerability representations. A boundary compensation module further restores cross-partition information by weighting tie-line neighbors according to electrical coupling, branch loading, and cross-area association. Standardized partition scores are finally fused into a whole-grid weak-bus ranking, and a composite learning objective jointly considers node-score regression, boundary consistency, and pairwise ranking stability. The method is evaluated on the IEEE 57-bus benchmark with mechanism-based node and branch vulnerability labels. Compared with the original full-graph GNN, the proposed method reduces the mean square error from 0.0359 to 0.0147, improves the Spearman rank coefficient from 0.248 to 0.446, and increases Hit@10 from 30% to 70%. Topological interpretation further shows that the identified weak buses are concentrated around high-risk branches such as 8-12, 12-14, 0-14, and 7-8, indicating that the proposed framework captures local aggregation, boundary transmission, and corridor-driven vulnerability propagation. The IEEE 57-bus benchmark is used as a focused validation case because it provides aligned node- and branch-level vulnerability evidence for evaluating weak-bus ranking behavior. Because the available aligned vulnerability evidence is concentrated in this medium-scale benchmark, the results should be interpreted as a focused validation of the proposed ranking mechanism rather than as a complete large-system scalability study.

1. Introduction

1.1. Motivation

Modern power grids are evolving toward wide-area interconnection, multi-voltage-level coordination, high renewable-energy penetration, flexible load participation, and frequent operating-mode transitions. Under this operating environment, the importance of a bus is no longer determined only by its local degree, injected power, or static position in the network. A bus may become vulnerable because it is adjacent to a heavily loaded branch, because it connects two electrically coupled regions, because it supports voltage at a regional boundary, or because its disturbance can trigger power-flow redistribution along a critical transfer corridor.

Weak-bus identification, therefore, has direct value for online security assessment and dispatch decision-making. In practical screening tasks, system operators often need to rank vulnerable buses before detailed contingency analysis, corrective control calculation, maintenance-schedule verification, or emergency resource allocation. The ranking result should be physically interpretable: it should not only indicate which bus is critical, but also explain whether the criticality comes from local stress, boundary transfer, branch overload, or cross-scenario exposure.

Graph neural networks provide a natural tool for learning from power-grid topology because buses and branches can be represented as nodes and edges. However, a direct full-graph GNN treats the whole grid as a single computational object. This is convenient for preserving global topology, but it may become inefficient when many operating scenarios, outage states, load-generation patterns, or maintenance combinations are evaluated in a rolling manner. Graph partitioning can reduce computational burden by decomposing the grid into smaller subgraphs, yet a naive partition may remove the very tie-line information that determines whether local stress can propagate to other regions.

The core motivation of this study is to reconcile computational scalability with physical completeness. The proposed framework partitions the power grid into electrically coherent subgraphs so that local encoders can be executed in parallel. At the same time, it explicitly models boundary nodes, cut edges, and cross-partition neighbors so that tie-line risk and corridor-driven propagation are retained in the final whole-grid ranking. This design is intended for security-screening tasks in which both fast inference and interpretable vulnerability localization are required.

1.2. Related Work

Blackout and cascading-risk studies first showed that power-grid vulnerability should be viewed as a propagation process rather than an isolated overload event. Carreras et al. [1] provided evidence for self-organized criticality in blackout time series, indicating that large disturbances may emerge from accumulated system stress. However, their work focused on statistical blackout behavior and did not produce an operational weak-bus ranking. Vaiman et al. [2] reviewed risk-assessment methodologies for cascading outages and clarified major challenges in modeling outage propagation, but the review did not provide a scalable graph-learning architecture for node-level prioritization. Alhelou et al. [3] surveyed blackout and cascading-event studies and summarized motivations for further research, whereas the discussion remained at the system-risk level and did not address boundary-compensated bus screening. Daeli and Mohagheghi [4] reviewed power-grid resilience against extreme events, but the resilience perspective was broader than the specific problem of ranking buses under topology-dependent electrical stress. Qin et al. [5] introduced underground energy storage into urban rail transit to improve energy efficiency and reliability, showing that storage resources can reshape operation under coupled transportation-energy constraints; nevertheless, that framework did not address transmission-grid weak-bus ranking or boundary propagation.

Network partitioning and parallel computation constitute another important research direction. Zhu and Bose [6] proposed a dynamic partitioning scheme for parallel transient-stability analysis, demonstrating that partitioning can accelerate large-scale power-system computation. Nevertheless, their partitioning objective was not connected to graph-neural vulnerability scoring. Yusof et al. [7] developed slow-coherency-based network partitioning, including load buses, which is useful for coherent-area analysis, but it does not directly generate weak-bus priorities. Qin et al. [8] studied aperiodic coordination scheduling of multiple pulsed power loads in shipboard integrated power systems, showing that strongly coupled energy networks require coordinated scheduling under intermittent high-power disturbances. However, that study did not formulate a partitioned graph-learning model for identifying electrically vulnerable buses. You et al. [9] used slow-coherency principles for islanding, whereas island construction and weak-bus identification have different operational objectives. Sarfi et al. [10] applied network partitioning theory to distribution-system reconfiguration for loss reduction, but loss minimization does not describe boundary-induced vulnerability propagation. Lemaitre and Thomas [11] demonstrated applications of parallel processing in power-system computation, yet the work predates modern representation learning. Dag and Alvarado [12] studied computation-free preconditioners for parallel solutions of power-system problems, focusing on numerical efficiency rather than learned node ranking.

Recent partitioning and uncertainty studies confirm that computational tractability and scenario representation remain relevant in modern power-system analysis. Wang et al. [13] formulated quantum annealing with integer slack variables for grid partitioning, but the output was a partition plan instead of a boundary-compensated vulnerability score. Hartmann et al. [14] developed quantum-annealing-based grid partitioning for parallel simulation, although graph-neural weak-bus ranking and node–line coupling interpretation were not considered. Li et al. [15] generated long-term renewable-energy scenarios using attention-based conditional generative adversarial networks, which is valuable for uncertainty modeling; however, a downstream topology-aware ranking model is still needed to convert scenario uncertainty into weak-bus priorities.

Graph learning and advanced energy-system scheduling have recently expanded the modeling toolbox for power and integrated energy systems. Lin et al. [16] used graph neural networks for spatial–temporal residential load forecasting, indicating that graph structure can improve power-system learning; however, forecasting load demand differs from identifying high-risk buses. Cao et al. [17] proposed an explainable graph neural network for reliability evaluation of electricity–hydrogen systems, but the target was multi-energy reliability rather than bus-level vulnerability screening. Ebtia et al. [18] introduced a dual-graph GNN for distribution-network topology detection, which demonstrates topology-aware learning but does not rank buses according to vulnerability. Wang et al. [19] developed a dynamic-carbon-market-driven multi-stage scheduling strategy for hydrogen integrated energy systems, yet the scheduling model did not address topological vulnerability or boundary-node compensation. Jin et al. [20] used graph neural networks to learn active constraints in power-system scheduling, whereas the learned object was a scheduling surrogate rather than a weak-bus score. Qin et al. [21] established a non-isothermal dynamic model and collaborative optimization strategy for multi-energy systems considering pipeline energy storage, while the representation was not designed for graph-partitioned electrical risk ranking. Huang et al. [22] designed a recurrent graph convolutional network for transient-stability assessment, but stability classification and boundary-aware weak-bus prioritization require different output structures.

Recent graph-learning studies have extended power-system risk assessment from static prediction to operationally coupled and topology-aware settings. Zhang et al. [23] used GNNs for system-, zone-, and branch-level operational risk assessment under evolving unit commitment. Gorka et al. [24] used statistically augmented GNNs to estimate cascading blackout severity from initial grid states. Yang et al. [25] embedded AC power-flow sensitivities in a physics-guided GNN for probabilistic power-flow analysis. Scenario-generation studies have also moved toward graph and diffusion models: Zhang et al. [26] generated extreme-weather source-load scenarios with a multi-scale conditional graph diffusion model, and Zhang et al. [27] used an extreme-value-enhanced diffusion model for photovoltaic scenario generation. These studies reinforce the value of graph representations for risk estimation, probabilistic analysis, and scenario modeling, but they do not directly recover boundary evidence for weak-bus ranking after electrical partitioning. Downstream congestion-management studies further indicate that weak-bus ranking can provide a prescreening signal for storage and demand-response actions, as discussed by Abdolahi et al. [28].

Based on the above literature, several unresolved issues remain. First, existing vulnerability and cascading-risk studies explain system-level propagation but rarely provide a scalable node-level learning model that can be executed repeatedly in rolling scenarios. Second, partitioning and parallel-computation methods reduce computational burden, but they do not automatically preserve the boundary evidence carried by tie lines, overloaded branches, and voltage-support buses. Third, graph-learning studies in power systems often focus on forecasting, topology detection, reliability evaluation, scheduling surrogates, or stability classification, while weak-bus identification requires a ranking-oriented output with node–line coupling interpretation. Fourth, scenario generation and integrated-energy scheduling studies provide important operating contexts, but a mechanism is still needed to map scenario-dependent electrical stress into interpretable bus priorities. These unresolved issues motivate a boundary-compensated partition-based GNN that combines parallel local learning, cross-partition information recovery, whole-grid score fusion, and topology-based explanation.

1.3. Manuscript Positioning and Main Contribution

This paper positions weak-bus identification as a partition-aware, boundary-compensated, and scenario-weighted ranking problem. The method is not a simple replacement of a full-graph GNN by several independent subgraph models. Instead, it decomposes the grid for parallel representation learning, then reconstructs boundary evidence before producing a whole-grid ranking. The main contributions are as follows.

The main contributions of this work are summarized as follows.

• A scenario-aware graph formulation is developed for weak-bus identification. The formulation incorporates electrical coupling, branch availability, boundary-node sets, cross-partition coupling, regional imbalance, and mechanism-driven vulnerability labels.
• A partition-based parallel GNN architecture is proposed. Electrically coherent subgraphs are encoded independently, enabling parallel local representation learning while preserving the physical meaning of intra-partition stress propagation.
• A boundary compensation mechanism is introduced. Cross-partition neighbors are weighted by electrical coupling, branch loading, and association significance so that tie-line information lost by graph cutting can be restored in boundary-node representations.
• A whole-grid fusion and ranking strategy is established. Standardized partition outputs are reliability-weighted and fused into a unified ranking, and the learning objective combines node regression, edge/boundary consistency, and pairwise ranking stability.
• An IEEE 57-bus case study is conducted with node and branch vulnerability evidence. The results quantify score fitting, rank correlation, short-list retrieval, component contribution, and node–line coupling in a transparent benchmark setting.

1.4. Paper Organization

The remainder of this paper is organized as follows. Section 2 formulates the weak-bus identification problem and defines scenario-aware partitioned graph construction. Section 3 presents the partition-based parallel GNN architecture and local vulnerability scoring mechanism. Section 4 describes boundary compensation, score fusion, multi-scenario aggregation, and the learning objective. Section 5 reports the IEEE 57-bus case study, including quantitative comparison, ranking analysis, topology visualization, node–line coupling evidence, and engineering implications. Section 6 concludes the paper and discusses future work.

2. Problem Formulation and Mechanism-Driven Partitioned Graph Construction

The main symbols repeatedly used in the mathematical formulation are summarized in

Table 1. Nomenclature and main variables used in the model.

Symbol	Definition
$G^{\xi }=(V,E,W^{\xi },X^{\xi },y^{\xi })$	Scenario-aware power-grid graph under operating scenario $\xi .$
$V, E$	Bus set and branch/transformer set.
$a_{ij}^{\xi }$	Binary branch-availability coefficient; 1 denotes an in-service branch and 0 denotes an unavailable branch.
$t_{ij}$	Binary tie-line indicator; 1 denotes a branch connecting two regions or partitions and 0 denotes an internal branch.
${\delta }_i^b$	Binary boundary-bus indicator; 1 denotes a bus adjacent to a cut edge or cross-partition neighbor and 0 denotes an internal bus.
$z_{ik}$	Binary partition-membership indicator before boundary duplication.
$n_0$	Target partition size used to determine the number and size of local subgraphs.
$\alpha , \beta , \chi ; \omega ; \lambda$	Preset or validation-selected weights for edge construction, vulnerability labels, and the composite loss.

.

2.1. Weak-Bus Identification as a Ranking Problem

For a power grid under operating scenario $\xi$, weak-bus identification aims to estimate a vulnerability score for each bus and produce an ordered list of critical buses. The output should be suitable for dispatch screening, which means that ranking stability is at least as important as pointwise score accuracy. The scenario-aware grid is represented as

$$G^{\xi }=\left(V,E,A^{\xi },X^{\xi },Y^{\xi }\right)$$

(1)

where $G^{\xi }$ denotes the power-grid graph under scenario $\xi$, $V$ denotes the bus set, $E$ denotes the branch and transformer set, $A^{\xi }$ denotes the scenario-dependent weighted adjacency matrix, $X^{\xi }$ denotes the feature matrix, and $Y^{\xi }$ denotes the vulnerability-label vector.

The ranking task is expressed as

$${\mathcal{R}}^{\xi }={sort}_{i\in V}\left(S_i^{\xi }\right)$$

(2)

where ${\mathcal{R}}^{\xi }$ denotes the weak-bus ranking under scenario $\xi$, and $S_i^{\xi }$ denotes the predicted vulnerability score of bus $i$. The sorting operator ranks buses in descending order of vulnerability. In a dispatch application, the top-ranked buses can be used as candidates for detailed power-flow verification, preventive control, or monitoring reinforcement.

2.2. Scenario-Aware Node and Edge Features

The feature vector of bus $i$ combines structural, electrical, and scenario-dependent indicators:

$$x_i^{\xi }=\left[d_i,c_i^{\mathrm{b}\mathrm{e}\mathrm{t}},c_i^{\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}},P_i^{\xi },Q_i^{\xi },\Delta V_i^{\xi },r_i^{\mathrm{b}\mathrm{d}},z_i^{\xi }\right]$$

(3)

where $x_i^{\xi }$ denotes the feature vector of bus $i$, $d_i$ denotes the topological degree, $c_i^{\mathrm{b}\mathrm{e}\mathrm{t}}$ denotes weighted betweenness centrality, $c_i^{\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}}$ denotes the clustering coefficient, $P_i^{\xi }$ and $Q_i^{\xi }$ denote active and reactive injections, $\Delta V_i^{\xi }$ denotes voltage-deviation magnitude, $r_i^{\mathrm{b}\mathrm{d}}$ denotes the boundary-node indicator, and $z_i^{\xi }$ denotes a scenario descriptor such as outage state, load level, or renewable-output condition.

The scenario-dependent edge weight between buses $i$ and $j$ is defined as

$$A_{ij}^{\xi }=u_{ij}^{\xi }\left({\alpha }_1|y_{ij}|+{\alpha }_2{\displaystyle \frac{\left|F_{ij}^{\xi }\right|}{F_{ij}^{max}}}+{\alpha }_3{\chi }_{ij}^{\mathrm{t}\mathrm{i}\mathrm{e}}\right)$$

(4)

where $u_{ij}^{\xi }$ denotes the branch availability coefficient, $\left|y_{ij}\right|$ denotes the admittance magnitude, $F_{ij}^{\xi }$ denotes the active power flow, $F_{ij}^{max}$ denotes the branch capacity limit, ${\chi }_{ij}^{\mathrm{t}\mathrm{i}\mathrm{e}}$ denotes whether the branch is a tie line, and ${\alpha }_1$, ${\alpha }_2$, and ${\alpha }_3$ denote nonnegative weighting coefficients. Equation (4) makes the graph sensitive not only to topology, but also to branch loading and inter-area transfer significance.

After normalization of the three terms in (4), the coefficients were selected from a physically constrained validation grid. In the IEEE 57-bus experiment, $\alpha =0.45, \beta =0.35, \chi =0.20$; thus, admittance coupling remains dominant while branch loading and tie-line status still affect edge construction.

2.3. Electrical-Coupling-Based Partitioning

The grid is divided into $K$ electrically coherent partitions:

$$\{V_1,V_2,\dots ,V_K\},{\bigcup }_{k=1}^K{V}_k=V,V_p\cap V_q=\mathrm{\varnothing },p\ne q$$

(5)

where $V_k$ denotes the bus set of partition $k$. Before boundary compensation is applied, each bus belongs to one and only one partition. A size-constrained electrical partitioning objective is used:

$${min}_{\pi }{J}_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}={\sum }_{\left(i,j)\in E,\pi (i)\ne \pi (j\right)}{A}_{ij}^{\xi }+\lambda {\sum }_{k=1}^K{\left(\left|V_k\right|-n_0\right)}^2$$

(6)

where $\pi \left(i\right)$ denotes the partition index of bus $i$, $J_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}$ denotes the partitioning objective, $\left|V_k\right|$ denotes the size of partition $k$, $n_0$ denotes the target partition size, and $\lambda$ controls the balance between cut-edge cost and partition-size balance.

The first term in (6) discourages cutting branches with strong electrical coupling or heavy loading. The second term prevents one partition from becoming excessively large, which would reduce the benefit of parallel computation. This objective reflects the engineering requirement that partitions should be computationally manageable while still respecting physical coupling.

For a general network, $n_0$ is selected from the parallel memory budget and the required one-hop boundary context. A practical initialization sets $n_0$ between 40 and 120 buses and computes $K=ceil\left({\displaystyle \frac{N}{n_0}}\right)$; K is then adjusted when a partition becomes electrically disconnected or excessively small. In the IEEE 57-bus case, $n_0$ = 40 yields two balanced partitions.

2.4. Boundary Set and Cross-Partition Coupling

After partitioning, the cut edges and their terminal buses form the boundary set. For two partitions $p$ and $q$, the cross-partition branch set is

$$E_{pq}^{\xi }=\left\{\right(i,j)\in E\mid i\in V_p,j\in V_q,p\ne q\}$$

(7)

where $E_{pq}^{\xi }$ denotes branches connecting partitions $p$ and $q$. The corresponding coupling strength is

$$c_{pq}^{\xi }={\sum }_{(i,j)\in E_{pq}^{\xi }}{A}_{ij}^{\xi }$$

(8)

where $c_{pq}^{\xi }$ measures the total electrical interaction between two partitions. Partition pairs with negligible coupling do not require intensive boundary exchange, whereas strongly coupled pairs should preserve cross-partition information during learning.

The retained boundary-pair set is defined as

$${\Omega }_{\xi }=\left\{\right(p,q)\mid c_{pq}^{\xi }\ge {\tau }_c\}$$

(9)

where ${\Omega }_{\xi }$ denotes the set of partition pairs requiring boundary modeling, and ${\tau }_c$ denotes the coupling threshold. This threshold prevents the boundary module from being dominated by weak or inactive cross-partition links.

2.5. Boundary-Risk Indicator

For a retained partition pair, boundary risk should reflect branch loading, boundary voltage stress, and cross-partition coupling. The integrated boundary-risk indicator is defined as

$$R_{pq}^{\xi }={\displaystyle \frac{c_{pq}^{\xi }}{{max}_{(m,n)\in {\Omega }_{\xi }}{c}_{mn}^{\xi }+ϵ}}\left({\displaystyle \frac{1}{\left|E_{pq}^{\xi }\right|}}{\sum }_{(i,j)\in E_{pq}^{\xi }}{\rho }_{ij}^{\xi }+{\eta }_{pq}^{\xi }{\displaystyle \frac{1}{\left|B_{pq}^{\xi }\right|}}{\sum }_{i\in B_{pq}^{\xi }}{v}_i^{\xi }\right)$$

(10)

where $R_{pq}^{\xi }$ denotes the integrated boundary risk between partitions $p$ and $q$, ${\rho }_{ij}^{\xi }$ denotes the loading ratio or overload degree of branch $i-j$, $B_{pq}^{\xi }$ denotes the boundary-bus set, $v_i^{\xi }$ denotes the voltage-limit violation magnitude of boundary bus $i$, ${\eta }_{pq}^{\xi }$ denotes the voltage-risk conversion coefficient, and $ϵ$ denotes a small positive constant.

The conversion coefficient is given by

$${\eta }_{pq}^{\xi }={\displaystyle \frac{{\overline{\rho }}_{pq}^{\xi }}{{\overline{v}}_{pq}^{\xi }+ϵ}}$$

(11)

where ${\overline{\rho }}_{pq}^{\xi }$ denotes the average boundary branch-loading risk, and ${\overline{v}}_{pq}^{\xi }$ denotes the average boundary voltage-violation risk. Equation (11) places branch-overload risk and voltage-support risk on comparable numerical scales. This avoids a situation in which one risk component dominates the boundary score only because of unit magnitude.

Thus, the boundary-risk indicator in (10) does not introduce an additional empirical scaling constant; it uses the adaptive conversion in (11) to place branch-loading and voltage-risk terms on comparable scales.

2.6. Mechanism-Driven Vulnerability Labels

The vulnerability label of bus $i$ is decomposed into local, boundary, contingency, and propagation components:

$$N_i={\omega }_1{C}_i^{\mathrm{l}\mathrm{o}\mathrm{c}}+{\omega }_2{C}_i^{\mathrm{b}\mathrm{d}}+{\omega }_3{R}_i^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}+{\omega }_4{R}_i^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}}$$

(12)

where $N_i$ denotes the mechanism-driven vulnerability label, $C_i^{\mathrm{l}\mathrm{o}\mathrm{c}}$ denotes local structural and electrical criticality, $C_i^{\mathrm{b}\mathrm{d}}$ denotes boundary-transfer criticality, $R_i^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}$ denotes post-contingency consequence risk, $R_i^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}}$ denotes propagation risk, and ${\omega }_1$, ${\omega }_2$, ${\omega }_3$, and ${\omega }_4$ denote contribution weights.

This decomposition is important because weak buses are not necessarily the buses with the largest local load. A bus located at a tie-line terminal can be critical because it couples regional stress, even if its own injection is moderate. Similarly, a bus adjacent to a high-risk branch may become important because branch overload can amplify its voltage or transfer sensitivity.

The four components in (12) are min–max normalized before weighting. The IEEE 57-bus experiment uses $({\omega }_1, {\omega }_2, {\omega }_3, {\omega }_4)=(0.30,0.25,0.25,0.20)$, so local and boundary evidence receive slightly higher emphasis while contingency and propagation effects remain represented.

The mechanism-driven label is computed before GNN training through a deterministic workflow. First, scenario power-flow data, branch availability states, branch-loading ratios, voltage-deviation magnitudes, partition-boundary indicators, and contingency consequence records are collected for the studied operating scenario. Second, four raw component scores are calculated for each bus: the local component summarizes normalized structural and electrical stress around the bus; the boundary component transfers the risk of cut edges and tie-line neighbors to their terminal buses; the contingency component reflects the severity of post-contingency violations associated with the bus or its adjacent branches; and the propagation component accumulates one-hop neighboring branch and bus risk to represent local risk diffusion. Third, each component is scaled by min–max normalization over all buses in the same scenario. Finally, the normalized components are combined with the weights in (12) to obtain the supervised target label used by the node-regression and ranking losses.

Thus, the label is not produced by the proposed GNN itself. It is an interpretable reference score derived from physical quantities and risk records, and the GNN is trained to approximate this mechanism-driven reference while preserving ranking consistency. This separation prevents information leakage from the learned model into the target labels and makes the weak-bus ranking reproducible for the IEEE 57-bus case.

3. Partition-Based Parallel Graph Neural Network

3.1. Overall Framework

Figure 1 illustrates the proposed boundary-compensated partition-based parallel GNN. The workflow contains graph construction, electrical partitioning, local subgraph encoding, boundary compensation, whole-grid fusion, and ranking interpretation.

The key principle is to keep local computation parallel without treating partitions as isolated systems. The partitioning stage reduces the graph size seen by each local encoder. The compensation stage then restores cross-partition evidence for boundary buses. The final fusion stage converts multiple local predictions into one comparable whole-grid ranking.

3.2. Local Subgraph Encoder

For partition $k$, the local scenario subgraph is written as

$$G_k^{\xi }=\left(V_k^{\xi },E_k^{\xi },A_k^{\xi },X_k^{\xi }\right)$$

(13)

where $G_k^{\xi }$ denotes the subgraph of partition $k$, $V_k^{\xi }$ denotes the active bus set, $E_k^{\xi }$ denotes the active internal branch set, $A_k^{\xi }$ denotes the internal weighted adjacency matrix, and $X_k^{\xi }$ denotes the local feature matrix.

The normalized adjacency matrix is constructed as

$${\tilde{A}}_k^{\xi }={\left(D_k^{\xi }\right)}^{-1/2}\left(A_k^{\xi }+I\right){\left(D_k^{\xi }\right)}^{-1/2}$$

(14)

where ${\tilde{A}}_k^{\xi }$ denotes the normalized adjacency matrix, $D_k^{\xi }$ denotes the degree matrix of $A_k^{\xi }+I$, and $I$ denotes the identity matrix. The self-loop term $I$ preserves each bus’s own operating features during message passing.

The first graph-convolution layer maps the input feature matrix into a hidden representation:

$$H_k^{1,\xi }=\sigma \left({\tilde{A}}_k^{\xi }{X}_k^{\xi }{W}_0\right)$$

(15)

where $H_k^{1,\xi }$ denotes the first-layer embedding matrix, $W_0$ denotes the learnable weight matrix, and $\sigma (·)$ denotes the activation function. For deeper layers, the propagation rule becomes

$$H_k^{l+1,\xi }=\sigma \left({\tilde{A}}_k^{\xi }{H}_k^{l,\xi }{W}_l\right)$$

(16)

where $H_k^{l,\xi }$ and $H_k^{l+1,\xi }$ denote the hidden embeddings before and after the $\left(l+1\right)$-th layer, and $W_l$ denotes the learnable weight matrix of layer $l$.

Equations (14)–(16) show how local voltage stress, branch loading, and neighboring-bus features propagate inside each partition. Since different partitions are encoded independently, the computational workload can be distributed across parallel inference tasks. However, the local encoder alone cannot fully represent buses whose risk is determined by cut edges or adjacent partitions.

3.3. Initial Local Vulnerability Decoding

After $L$ graph-convolution layers, the local embedding of bus $i$ in partition $k$ is decoded into an initial vulnerability score:

$$s_{i,k}^{0,\xi }=f_{\theta }\left(h_{i,k}^{L,\xi }\right)$$

(17)

where $s_{i,k}^{0,\xi }$ denotes the initial score, $f_{\theta }(·)$ denotes the score decoder implemented by a multi-layer perceptron, and $h_{i,k}^{L,\xi }$ denotes the final local embedding of bus $i$.

The initial score describes how vulnerable a bus appears when only intra-partition evidence is observed. This score is useful for local risk screening, but it may be biased for boundary buses. For example, a bus connected to a high-risk tie line may be underestimated if the adjacent partition is removed from the local receptive field.

3.4. Parallel Execution and Computational Interpretation

If the original graph contains $\left|V\right|$ buses and is divided into $K$ partitions, the approximate local graph-convolution workload can be expressed as

$${\mathcal{C}}_{\mathrm{p}\mathrm{a}\mathrm{r}}\approx {max}_{1\le k\le K}\left(L|E_k|d_h\right)+{\mathcal{C}}_{\mathrm{b}\mathrm{d}}$$

(18)

where ${\mathcal{C}}_{\mathrm{p}\mathrm{a}\mathrm{r}}$ denotes the parallel computational cost, $L$ denotes the number of graph-convolution layers, $\left|E_k\right|$ denotes the number of internal edges in partition $k$, $d_h$ denotes the hidden dimension, and ${\mathcal{C}}_{\mathrm{b}\mathrm{d}}$ denotes the boundary-compensation cost.

For a full-graph GNN, the corresponding workload is, approximately,

$${\mathcal{C}}_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}\approx L\left|E\right|d_h$$

(19)

where $\left|E\right|$ denotes the number of branches in the whole grid. Equations (18) and (19) show why partitioning is beneficial when several partitions can be processed concurrently. The remaining challenge is to ensure that ${\mathcal{C}}_{\mathrm{b}\mathrm{d}}$ is small enough to preserve efficiency while still large enough to recover physically important boundary information.

The implementation-level resource profile in

Table 2. Runtime and memory profile for the IEEE 57-bus experiment.

Model	Training Time (s)	Inference Time (ms/Graph)	Peak CPU Memory (MB)	GPU Memory	Relative Training Speedup
Original full-graph GNN	18.6	3.2	234	N/A; CPU execution	1.00×
Partitioned GNN without boundary compensation	11.4	1.9	151	N/A; CPU execution	1.63×
Proposed boundary-compensated partition GNN	13.2	2.3	165	N/A; CPU execution	1.41×

reports the runtime and memory use of the IEEE 57-bus experiment.

Because the IEEE 57-bus benchmark is small, the absolute times are reported primarily for implementation transparency. The computational benefit is expected to become more visible when more subgraphs can be evaluated concurrently in larger systems.

4. Boundary Compensation, Score Fusion, and Learning Strategy

4.1. Boundary Relevance Score

For a boundary bus $i$ and its cross-partition neighbor $j$, the relevance score is defined as

$${\varphi }_{ij}^{\xi }=b_1{A}_{ij}^{\xi }+b_2{\displaystyle \frac{\left|F_{ij}^{\xi }\right|}{F_{ij}^{max}}}+b_3{c}_j^{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}+b_4{v}_j^{\xi }$$

(20)

where ${\varphi }_{ij}^{\xi }$ denotes the relevance of neighbor $j$ to boundary bus $i$, $A_{ij}^{\xi }$ denotes the electrical coupling weight, $\left|F_{ij}^{\xi }\right|/F_{ij}^{max}$ denotes the branch loading ratio, $c_j^{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}$ denotes cross-partition association significance, $v_j^{\xi }$ denotes voltage-risk magnitude at neighbor bus $j$, and $b_1$, $b_2$, $b_3$, and $b_4$ denote weighting coefficients.

This definition gives higher compensation priority to neighbors located on electrically strong, heavily loaded, or voltage-stressed cross-partition channels. It also keeps the compensation mechanism interpretable because every term corresponds to a physical risk factor.

The four weights in (20) are 0.35, 0.30, 0.20, and 0.15 for electrical coupling, loading ratio, cross-partition association, and neighbor voltage risk, respectively.

4.2. Boundary Compensation Weight and Representation Update

The relevance scores are normalized by a Softmax operation:

$$w_{ij}^{\mathrm{b}\mathrm{d},\xi }={\displaystyle \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left({\varphi }_{ij}^{\xi }\right)}{{\sum }_{r\in {\mathcal{N}}_{\mathrm{b}\mathrm{d}}\left(i\right)}\mathrm{e}\mathrm{x}\mathrm{p}\left({\varphi }_{ir}^{\xi }\right)}}$$

(21)

where $w_{ij}^{\mathrm{b}\mathrm{d},\xi }$ denotes the normalized boundary compensation weight, and ${\mathcal{N}}_{\mathrm{b}\mathrm{d}}\left(i\right)$ denotes the set of cross-partition boundary neighbors of bus $i$. The weight is larger when neighbor $j$ contributes stronger boundary evidence.

The compensated representation is then computed as

$$h_i^{\mathrm{*},\xi }=h_i^{\xi }+{\gamma }_i^{\xi }{\sum }_{j\in {\mathcal{N}}_{\mathrm{b}\mathrm{d}}\left(i\right)}{w}_{ij}^{\mathrm{b}\mathrm{d},\xi }{W}_{\mathrm{b}\mathrm{d}}{h}_j^{\xi }$$

(22)

where $h_i^{\mathrm{*},\xi }$ denotes the boundary-compensated representation of bus $i$, $h_i^{\xi }$ denotes the original local representation, $W_{\mathrm{b}\mathrm{d}}$ denotes the boundary transformation matrix, $h_j^{\xi }$ denotes the representation of neighbor bus $j$, and ${\gamma }_i^{\xi }$ denotes the boundary-gating coefficient.

The gating coefficient is defined as

$${\gamma }_i^{\xi }=sigmoid\left(a_1{r}_i^{\mathrm{b}\mathrm{d}}+a_2{\overline{\rho }}_i^{\xi }+a_3{\overline{v}}_i^{\xi }\right)$$

(23)

where $r_i^{\mathrm{b}\mathrm{d}}$ denotes the boundary-bus indicator, ${\overline{\rho }}_i^{\xi }$ denotes the average loading risk of boundary branches adjacent to bus $i$, ${\overline{v}}_i^{\xi }$ denotes the voltage-risk indicator, and $a_1$, $a_2$, and $a_3$ denote learnable or preset coefficients. This gate prevents unnecessary compensation for purely internal buses and strengthens compensation for boundary buses under high stress.

The gate in (24) uses 0.50, 0.30, and 0.20 for boundary membership, adjacent boundary-branch loading risk, and voltage risk, respectively.

4.3. Boundary-Corrected Vulnerability Score

The boundary-corrected score is obtained by decoding the compensated representation:

$$s_{i,k}^{1,\xi }=f_{\theta }\left(h_i^{\mathrm{*},\xi }\right)$$

(24)

where $s_{i,k}^{1,\xi }$ denotes the corrected vulnerability score of bus $i$ in partition $k$. Compared with the initial score in (17), this score contains both local subgraph evidence and cross-partition boundary information.

This module is the main difference between the proposed framework and a naive partitioned GNN. Without boundary compensation, graph cutting may remove the neighboring buses and branches that explain why a bus is vulnerable. With (20)–(24), tie-line loading, voltage stress, and cross-partition association are explicitly reintroduced before whole-grid ranking.

4.4. Partition Score Standardization and Reliability Weighting

Scores generated by different local encoders may have different numerical scales. Partition-level standardization is therefore applied:

$${\hat{s}}_{i,k}^{\xi }={\displaystyle \frac{s_{i,k}^{1,\xi }-{\mu }_k^{\xi }}{{\sigma }_k^{\xi }+ϵ}}$$

(25)

where ${\hat{s}}_{i,k}^{\xi }$ denotes the standardized score, ${\mu }_k^{\xi }$ denotes the mean score of partition $k$, ${\sigma }_k^{\xi }$ denotes the score standard deviation, and $ϵ$ prevents division by zero.

The reliability of partition $k$ is evaluated as

$$q_k^{\xi }=c_1{Acc}_k^{\xi }+c_2{Con}_k^{\xi }-c_3{Var}_k^{\xi }$$

(26)

where $q_k^{\xi }$ denotes the partition reliability score, ${Acc}_k^{\xi }$ denotes local prediction accuracy, ${Con}_k^{\xi }$ denotes boundary consistency, ${Var}_k^{\xi }$ denotes excessive output variance, and $c_1$, $c_2$, and $c_3$ denote weighting coefficients.

The whole-grid fusion weight is obtained by

$${\lambda }_k^{\xi }={\displaystyle \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(q_k^{\xi }\right)}{{\sum }_{r=1}^K\mathrm{e}\mathrm{x}\mathrm{p}\left(q_r^{\xi }\right)}}$$

(27)

where ${\lambda }_k^{\xi }$ denotes the fusion weight of partition $k$. A partition receives a higher weight when its prediction is accurate, boundary-consistent, and stable.

In (26), the reliability weights are 0.50, 0.30, and 0.20 for local accuracy, boundary consistency, and output-variance penalty, respectively.

4.5. Whole-Grid Score and Multi-Scenario Aggregation

The whole-grid vulnerability score under scenario $\xi$ is defined as

$$S_i^{\xi }={\sum }_{k=1}^K{\lambda }_k^{\xi }{\hat{s}}_{i,k}^{\xi }$$

(28)

where $S_i^{\xi }$ denotes the fused vulnerability score of bus $i$. If a bus belongs to only one partition, the summation is interpreted as the contribution of its own partition. If boundary sharing or virtual boundary duplication is used, multiple partition contributions can be fused.

For a set of operating scenarios, the expected vulnerability score is

$$S_i^{\mathrm{a}\mathrm{l}\mathrm{l}}={\sum }_{\xi \in \Xi }{p}_{\xi }{S}_i^{\xi }$$

(29)

where $S_i^{\mathrm{a}\mathrm{l}\mathrm{l}}$ denotes the multi-scenario expected score, $\Xi$ denotes the scenario set, and $p_{\xi }$ denotes the probability or importance weight of scenario $\xi$. Equation (29) allows high-probability routine scenarios and low-probability severe scenarios to be integrated into one dispatch-oriented ranking.

4.6. Composite Learning Objective

The learning objective combines node regression, boundary consistency, and ranking stability:

$$L=L_{\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}}+\tau L_{\mathrm{b}\mathrm{d}}+\zeta L_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}}$$

(30)

where $L$ denotes the total loss, $L_{\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}}$ denotes node-score regression loss, $L_{\mathrm{b}\mathrm{d}}$ denotes boundary consistency loss, $L_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}}$ denotes pairwise ranking loss, and $\tau$ and $\zeta$ denote loss-weighting coefficients.

The node regression loss is

$$L_{\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}}={\displaystyle \frac{1}{\left|V\right|}}{\sum }_{i\in V}{\left(S_i−N_i\right)}^2$$

(31)

where $S_i$ denotes the predicted score and $N_i$ denotes the mechanism-driven label. The boundary consistency loss is

$$L_{\mathrm{b}\mathrm{d}}={\displaystyle \frac{1}{\left|{\Omega }_{\xi }\right|}}{\sum }_{(p,q)\in {\Omega }_{\xi }}{\left({\overline{S}}_{pq}^{\mathrm{b}\mathrm{d}}−R_{pq}^{\xi }\right)}^2$$

(32)

where ${\overline{S}}_{pq}^{\mathrm{b}\mathrm{d}}$ denotes the average predicted score of boundary buses associated with the partition pair $\left(p,q\right)$, and $R_{pq}^{\xi }$ denotes the boundary-risk indicator in (10). This term encourages boundary-bus predictions to agree with branch- and voltage-based boundary evidence.

The pairwise ranking loss is

$$L_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}}={\sum }_{(i,j)\in {\Omega }_N}max\left(0,1-\left(S_i−S_j\right)sign\left(N_i−N_j\right)\right)$$

(33)

where ${\Omega }_N$ denotes the sampled bus-pair set, and $sign(·)$ indicates whether one bus has a higher target vulnerability than another. The ranking loss is important because dispatch screening depends on the ordering of the top buses, not only on the exact numerical scores.

In (30), the node-regression term has a unit weight, whereas the boundary-consistency and pairwise ranking terms use weights of 0.30 and 0.20, respectively.

5. Case Studies and Experimental Analysis

5.1. Benchmark System and Data Source

The IEEE 57-bus benchmark is adopted as the representative case for detailed evaluation. The system is a medium-scale transmission benchmark with a compact but nontrivial topology, making it suitable for analyzing weak-bus clusters, high-risk branches, and boundary-aware graph learning. The processed dataset contains mechanism-based node vulnerability labels and branch vulnerability labels. The complete standard IEEE 57-bus topology is used for visualization so that predicted weak buses can be interpreted together with high-risk branches.

In this case study, the mechanism-driven labels are fixed before model training. The branch-risk records and normalized node-level component scores described in Section 2.6 are used to build the target labels, and the GNN predictions are evaluated against these fixed labels rather than against labels fitted after training.

Table 3. Benchmark configuration of the IEEE 57-bus case.

Item	Value	Description
Benchmark system	IEEE case 57	Medium-scale transmission benchmark
Nodes	57	Buses in the standard network topology
Network branches	78	Lines and transformers reconstructed from the standard topology
Branch-risk labels	63	Available branch vulnerability records
Main task	Weak-bus ranking	Node-level vulnerability prioritization
Baseline	Original full-graph GNN	Whole-grid graph learning without partition compensation
Proposed method	Partition-based GNN	Parallel subgraph learning with boundary compensation

summarizes the case-study configuration.

The IEEE 57-bus case is not used merely as a numerical benchmark. Its moderate size allows node–line relationships to be visualized clearly. This is important because a weak-bus identification method should be judged not only by prediction metrics, but also by whether the predicted buses are physically connected to vulnerable branches and transfer paths.

5.2. Evaluation Metrics and Baseline

The baseline is the original full-graph GNN, which processes the entire network as one graph and outputs node vulnerability scores. The proposed method first partitions the grid, encodes each subgraph in parallel, compensates boundary buses, and then fuses local scores into a whole-grid ranking.

The evaluation metrics include mean square error, Spearman’s rank correlation, and top-k hit ratios. Mean square error measures numerical score fitting. Spearman’s correlation measures global ranking consistency. Hit@3, Hit@5, and Hit@10 measure whether the model retrieves the most critical reference buses within a short candidate list.

The detailed protocol is reported together with the case configuration in Section 5.3.

5.3. Experimental Protocol and IEEE 57-Bus Case Configuration

The IEEE 57-bus experiment is configured as a deterministic transductive ranking benchmark.

Table 4. Experimental protocol, hyperparameters, and preset coefficients.

Item	Setting Used in the IEEE 57-Bus Case Study	Related Equations
Dataset and scenarios	One processed IEEE 57-bus graph with 57 buses, 78 branches, node vulnerability labels, and branch vulnerability labels.	-
Evaluation protocol	Deterministic transductive ranking on the labeled graph; MSE, Spearman, Hit@3, Hit@5, and Hit@10 are computed against mechanism-driven labels.	-
Random seed	42 for partitioning and model initialization.	-
Optimizer and training	Adam optimizer; learning rate 0.002; 150 maximum epochs; early-stopping patience of 30 epochs.	(30)
Encoder size	Two graph-convolution layers; hidden dimension 32; ReLU activation; sigmoid score decoder.	(14)–(17)
Partitioning	Target partition size $n_0$ = 40; two subgraphs for IEEE 57; one-hop boundary context.	(6)
Edge-weight coefficients	$\alpha =0.45, \beta =0.35, \chi =0.20$ after feature normalization.	(4)
Boundary-risk processing	Branch-loading and voltage-risk terms are normalized by the adaptive conversion coefficient in (11).	(10)
Label-component weights	$({\omega }_1, {\omega }_{2 },{\omega }_3, {\omega }_4)=(0.30,0.25,0.25,0.20)$.	(12)
Boundary relevance weights	0.35, 0.30, 0.20, and 0.15 for coupling, loading, association, and voltage-risk terms.	(20)
Fusion and loss weights	Reliability weights 0.50/0.30/0.20; boundary loss weight 0.30; ranking-loss weight 0.20.	(26), (30)

summarizes the data, optimization settings, architecture, and preset coefficients used in the case study.

Table 5. Performance comparison between the original full-graph GNN and the proposed partition-based GNN.

Metric	Original Full-Graph GNN	Partition-Based GNN	Change
Nodes	57	57	0
Network branches	78	78	0
MSE	0.0359	0.0147	−0.0212
Spearman	0.248	0.446	+0.198
Hit@3	67%	67%	stable
Hit@5	40%	60%	+20 percentage points
Hit@10	30%	70%	+40 percentage points

compares the baseline and the proposed method on the IEEE 57-bus benchmark.

To isolate the contribution of each component,

Table 6. Component ablation on the IEEE 57-bus weak-bus ranking task.

Configuration	MSE	Spearman	Hit@3	Hit@5	Hit@10
Original full-graph GNN	0.0359	0.248	67%	40%	30%
Partitioned GNN without boundary compensation	0.0238	0.319	67%	40%	50%
Partitioned GNN with boundary compensation but without ranking loss	0.0171	0.389	67%	60%	60%
Proposed method with boundary compensation and ranking loss	0.0147	0.446	67%	60%	70%

reports an ablation study over partitioning, boundary compensation, and the pairwise ranking term.

The ablation indicates that partitioning improves score fitting, boundary compensation improves top-list retrieval, and the pairwise ranking term improves the overall rank correlation and Hit@10.

The proposed method substantially improves score fitting and ranking quality. The MSE decreases from 0.0359 to 0.0147, indicating that partition-aware local learning reduces the deviation between predicted scores and mechanism-based labels. The Spearman coefficient increases from 0.248 to 0.446, showing that the overall ranking becomes more consistent with the reference ordering. The most important improvement appears in Hit@10, which increases from 30% to 70%. This means that the proposed model retrieves most of the important weak buses within the top-10 screening set.

The Spearman coefficient of 0.446 nevertheless indicates moderate, not complete, whole-network ranking consistency, and Hit@3 remains unchanged at 67%. The result should therefore be interpreted as an improvement in short-list retrieval and local risk-cluster identification rather than as full ordering accuracy over all buses.

Figure 2 compares the three evaluation dimensions without mixing their physical scales. The MSE panel reports score-fitting error, the Spearman panel reports whole-network ranking consistency, and the Hit@k panel reports short-list retrieval quality. The proposed method reduces the prediction error and improves Hit@5 and Hit@10, whereas Hit@3 remains unchanged.

The improvement is not only a numerical phenomenon. In a dispatch workflow, Hit@10 represents whether the operator’s short candidate list contains the buses that deserve further contingency verification. A 40-percentage-point increase in Hit@10, therefore, has practical value because it reduces the probability that critical buses are omitted from subsequent detailed analysis.

5.4. Weak-Bus Ranking Analysis

Table 7. Top-ranked weak buses in the IEEE 57-bus system.

Rank	Reference Bus	Reference Score	Original GNN Bus	Original Score	Partition-GNN Bus	Partition Score
1	12	0.753	8	0.523	8	0.483
2	14	0.608	12	0.518	12	0.477
3	8	0.437	35	0.470	11	0.433
4	0	0.408	11	0.459	14	0.403
5	3	0.356	34	0.432	10	0.403
6	11	0.349	37	0.414	13	0.372
7	2	0.316	40	0.389	9	0.372
8	5	0.232	55	0.380	7	0.370
9	18	0.212	39	0.371	5	0.370
10	7	0.210	33	0.370	0	0.353

compares the top-ranked buses from the reference labels, the original full-graph GNN, and the proposed method.

The proposed method retrieves seven of the reference top-10 weak buses, including buses 12, 14, 8, 0, 11, 5, and 7. The predicted top buses are concentrated around the same local transfer region as the reference weak buses. In contrast, the original full-graph GNN identifies several buses outside the dominant weak-bus cluster, which explains its lower Hit@10 and lower Spearman coefficient.

The top reference bus, Bus 12, is ranked second rather than first because the boundary-enhanced representation assigns a slightly higher score to Bus 8. Bus 8 is directly attached to the highest-risk branch 8-12 and to the high-risk branch 7-8, so the learned ranking emphasizes local branch-risk aggregation at Bus 8. This explains the remaining mismatch in the first two positions despite the improved Hit@10.

The ranking result indicates that the proposed model does not simply assign high scores to isolated nodes. Instead, it identifies a coherent weak-bus group around branches 8-12, 12-14, 0-14, and 7-8. This is consistent with the boundary-compensation design: a bus is considered vulnerable when its local representation and adjacent branch evidence jointly indicate elevated risk.

5.5. Topological Interpretation and Node–Line Coupling

Figure 3 presents a topology-centered visualization of the IEEE 57-bus case. Node color indicates a reference vulnerability. Dark red rings indicate reference top-10 weak buses. Blue rings indicate predicted top-10 buses of the partition-based GNN. Orange lines indicate the top-10 high-risk branches.

The visualization separates normal branches, high-risk branches, reference weak buses, and predicted weak buses through distinct visual channels. The result shows that vulnerable buses are concentrated around a compact risk cluster rather than being scattered randomly across the network. This spatial pattern supports the interpretation that weak-bus vulnerability is formed by node–line coupling and local transfer-corridor stress.

Table 8. Top-ranked high-risk branches in the IEEE 57-bus system.

Rank	From Bus	To Bus	Branch Vulnerability
1	8	12	1.000
2	12	14	0.780
3	0	14	0.741
4	7	8	0.715
5	2	14	0.692
6	11	12	0.507
7	2	3	0.433
8	0	1	0.367
9	1	2	0.303
10	11	15	0.275

lists the top-ranked high-risk branches.

The top branch-risk list explains why buses 8, 12, 14, 0, 11, and 7 appear in the weak-bus ranking. Bus 12 is located between the highest-risk branch 8-12 and the second-ranked branch 12-14. Bus 14 connects branches 12-14, 0-14, and 2-14. Bus 8 is attached to both 8-12 and 7-8. Therefore, the predicted weak buses are not arbitrarily high-score points; they are physically associated with high-risk branches.

Table 9. Coupling relationship between representative weak buses and high-risk branches.

Weak Bus	Associated High-Risk Branches	Topological Interpretation	Support for the Proposed Method
12	8-12, 12-14, 11-12	Center of the highest-risk branch cluster	Retrieved as a top weak bus and consistent with branch-risk evidence
14	12-14, 0-14, 2-14	Connected to several transfer paths	Confirms branch-risk aggregation around bus 14
8	8-12, 7-8, 8-9	Directly attached to the highest-risk branch	Explains why the model ranks this bus first
0	0-14, 0-1	Adjacent to a major high-risk tie branch	Shows boundary-like transfer influence
11	11-12, 11-15	Close to the central weak-bus group	Supports local aggregation and compensation effects
7	7-8, 5-7	Connected to a high-risk branch terminal	Indicates neighboring risk propagation

summarizes representative node–line coupling relationships.

The coupling evidence supports the main modeling assumption. If a graph is partitioned without compensation, the model may weaken the influence of branches such as 8-12 or 12-14 when their terminals fall near partition boundaries. Boundary compensation reintroduces this evidence into node embeddings, making the ranking more consistent with the physical risk cluster.

5.6. Engineering Implications

The IEEE 57-bus case provides three engineering implications. First, partition-based graph learning can improve weak-bus ranking when boundary information is explicitly restored after graph partitioning. Second, improved Hit@10 is relevant to operational practice because dispatchers often examine a short candidate list before conducting more expensive contingency or power-flow analysis. Third, topology-based visualization helps explain why predicted weak buses are critical: they are coupled with high-risk branches and local transfer paths.

In online security assessment, the proposed method can be used as a fast prescreening layer. Local subgraph encoders identify regional stress patterns. Boundary compensation prevents tie-line terminals from being underestimated. Whole-grid fusion then produces a single ranking that can guide preventive control, monitoring reinforcement, or maintenance verification.

5.7. Limitations and Future Research

The current case study is based on the available IEEE 57-bus vulnerability labels and processed branch-risk evidence. Although the proposed method improves MSE, the Spearman correlation, Hit@5, and Hit@10, the whole-network ranking quality remains moderate, and scalability still requires validation on IEEE 118-bus, IEEE 300-bus, and multi-scenario datasets. In addition, the current boundary relevance score uses physically interpretable factors, including electrical coupling, branch loading, cross-partition association, and voltage risk. Future work will extend the validation to larger systems, incorporate renewable-energy scenario generation and uncertainty quantification, report confidence intervals for weak-bus rankings, and explore attention-based adaptive compensation under explicit physical constraints.

This evidence establishes the behavior of the labeling, compensation, and fusion mechanisms on a medium-scale labeled network. Larger regional grids may introduce additional sparsity, heterogeneous operating areas, and denser boundary interactions, so their influence on ranking robustness and computational gain should be assessed separately.

6. Conclusions

This paper proposed a boundary-compensated partition-based parallel graph neural network for weak-bus identification in interconnected power grids. The method constructs a scenario-aware weighted graph, partitions it into electrically coherent subgraphs, learns local vulnerability representations in parallel, restores cross-partition information through boundary compensation, and fuses standardized partition outputs into a whole-grid ranking.

The mathematical formulation links each stage of the method to a physical meaning. The partitioning objective reduces cut-edge cost while maintaining computational balance. The boundary-risk indicator combines cross-partition coupling, branch loading, and voltage stress. The compensation mechanism weights cross-partition neighbors according to their relevance and updates boundary-bus embeddings. The learning objective integrates node regression, boundary consistency, and pairwise ranking stability.

The IEEE 57-bus case study showed that the proposed method reduces MSE from 0.0359 to 0.0147, improves the Spearman rank coefficient from 0.248 to 0.446, and increases Hit@10 from 30% to 70% compared with the original full-graph GNN. Topological analysis further showed that predicted weak buses are concentrated around high-risk branches such as 8-12, 12-14, 0-14, and 7-8. This node–line coupling evidence confirms that the method captures local aggregation, boundary transmission, and corridor-driven vulnerability propagation.

Overall, the proposed framework provides a scalable and interpretable modeling direction for rolling security assessment, while the present experimental evidence is limited to the IEEE 57-bus labeled benchmark. After further validation on larger systems and richer operating scenarios, the method can support preventive control, online screening, maintenance verification, and resilience-oriented dispatch. Future research will extend the validation to larger IEEE systems, incorporate probabilistic scenarios, and develop adaptive boundary-compensation mechanisms with explicit physical constraints.

A natural next step is to test the framework on IEEE 118-bus, IEEE 300-bus, and multi-area scenarios, where partition size, boundary density, and scenario diversity may affect both computational gain and ranking robustness.