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PowerModelsGAT-AI: Physics-Informed Graph Attention for Multi-System Power Flow with Continual Learning
Chidozie Ezeakunne, Jose E. Tabarez, Reeju Pokharel, Anup Pandey
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
Last extracted: 7/20/2026, 1:32:42 PM
Summary
The paper introduces PowerModelsGAT-AI (PMGAT-AI), a physics-informed graph attention network designed to solve alternating current (AC) power flow equations in real-time. The model addresses limitations of existing Graph Neural Networks (GNNs) by enabling unified multi-system learning across 14 benchmark systems (4 to 6,470 buses) and employing bus-type-aware masking to handle different node types (PQ, PV, Slack). Key innovations include a physics-informed loss function with power-mismatch penalties, and a continual learning framework utilizing experience replay and elastic weight consolidation (EWC) to mitigate catastrophic forgetting when adapting to new systems. The model achieves high accuracy (0.89% normalized mean absolute error for voltage magnitudes) and demonstrates that learned attention weights correlate with physical branch parameters.
Entities (10)
Relation Signals (8)
PowerModelsGAT-AI → solves → AC Power Flow
confidence 95% · PowerModelsGAT-AI, a physics-informed graph attention network that predicts bus voltages and generator injections.
PowerModelsGAT-AI → uses → Graph Attention Network
confidence 92% · We present PowerModelsGAT-AI, a physics-informed graph attention network
PowerModelsGAT-AI → evaluatedon → Benchmark Systems
confidence 90% · We evaluate the model on 14 benchmark systems
PowerModelsGAT-AI → mitigates → Catastrophic Forgetting
confidence 90% · our experience replay and elastic weight consolidation strategy keeps error increases below 2%
Experience Replay → partof → PowerModelsGAT-AI
confidence 88% · our experience replay and elastic weight consolidation strategy
Elastic Weight Consolidation → partof → PowerModelsGAT-AI
confidence 88% · our experience replay and elastic weight consolidation strategy
PowerModelsGAT-AI → employs → Bus-Type-Aware Masking
confidence 87% · The model uses bus-type-aware masking to handle different bus types
PowerModelsGAT-AI → outperforms → Newton-Raphson
confidence 85% · classical Newton-Raphson solvers can be slow under stressed conditions
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Abstract
Abstract:Solving the alternating current power flow equations in real time is essential for secure grid operation, yet classical Newton-Raphson solvers can be slow under stressed conditions. Existing graph neural networks for power flow are typically trained on a single system and often degrade on different systems. We present PowerModelsGAT-AI, a physics-informed graph attention network that predicts bus voltages and generator injections. The model uses bus-type-aware masking to handle different bus types and balances multiple loss terms, including a power-mismatch penalty, using learned weights. We evaluate the model on 14 benchmark systems (4 to 6,470 buses) and train a unified model on 13 of these under N-2 (two-branch outage) conditions, achieving an average normalized mean absolute error of 0.89% for voltage magnitudes and R^2 > 0.99 for voltage angles. We also show continual learning: when adapting a base model to a new 1,354-bus system, standard fine-tuning causes severe forgetting with error increases exceeding 1000% on base systems, while our experience replay and elastic weight consolidation strategy keeps error increases below 2% and in some cases improves base-system performance. Interpretability analysis shows that learned attention weights correlate with physical branch parameters (susceptance: r = 0.38; thermal limits: r = 0.22), and feature importance analysis supports that the model captures established power flow relationships.
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- Source: https://arxiv.org/abs/2603.16879v1
- Canonical: https://arxiv.org/abs/2603.16879v1
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Research presented in this work was supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory under project number 20250854ECR. author: Chidozie Ezeakunne (e-mail: cezeakunne@lanl.gov). PowerModelsGAT-AI: Physics-Informed Graph Attention for Multi-System Power Flow with Continual Learning CHIDOZIE EZEAKUNNE1,2 JOSE E. TABAREZ1 REEJU POKHAREL1 and ANUP PANDEY1 Los Alamos National Laboratory, Los Alamos, NM, USA Department of Physics, University of Central Florida, Orlando, FL 32816 USA Abstract Solving the alternating current power flow equations in real time is essential for secure grid operation, yet classical Newton–Raphson solvers can be slow under stressed conditions. Existing graph neural networks for power flow are typically trained on a single system and often degrade on different systems. We present PowerModelsGAT-AI, a physics-informed graph attention network that predicts bus voltages and generator injections. The model uses bus-type-aware masking to handle different bus types and balances multiple loss terms, including a power-mismatch penalty, using learned weights. We evaluate the model on 14 benchmark systems (4 to 6,470 buses) and train a unified model on 13 of these under N−2N\!-\!2 (two-branch outage) conditions, achieving an average normalized mean absolute error of 0.89%0.89\% for voltage magnitudes and R2>0.99R^2>0.99 for voltage angles. We also show continual learning: when adapting a base model to a new 1,354-bus system, standard fine-tuning causes severe forgetting with error increases exceeding 1000%1000\% on base systems, while our experience replay and elastic weight consolidation strategy keeps error increases below 2%2\% and in some cases improves base-system performance. Interpretability analysis shows that learned attention weights correlate with physical branch parameters (susceptance: r=0.38r=0.38; thermal limits: r=0.22r=0.22), and feature importance analysis supports that the model captures established power flow relationships. Index Terms: alternating current power flow, continual learning, graph attention networks, physics-informed machine learning. I Introduction The alternating current (AC) power flow problem is a foundational computational tool for secure and economic power system operation and planning. The AC power flow equations describe the nonlinear relationship between bus voltages, power injections, and network admittances, and are fundamental to standalone power flow analysis, optimal power flow (OPF), and security-constrained formulations [CARPENTIER19793, frank2016introduction, bose2011optimal]. The goal is to determine unknown bus voltages and power injections from specified inputs, which vary by bus type (Table I). TABLE I: Bus-Type Unknown Targets and Supervision Masks. Bus Type Known in iy_i Unknown Targets Mask [Vm,δ,Pg,Qg][V_m,δ,P_g,Q_g] PQ (Load) Pg,QgP_g,Q_g Vm,δV_m,δ [1, 1, 0, 0] PV (Generator) Pg,VmP_g,V_m δ,Qgδ,Q_g [0, 1, 0, 1] Slack Vm,δV_m,δ Pg,QgP_g,Q_g [0, 0, 1, 1] For a system with N buses, let Vi=Vm,iejδiV_i=V_m,ie^j _i be the complex voltage at bus i, where Vm,iV_m,i is the voltage magnitude, δi _i is the voltage angle, and j=−1j= -1. Given the bus-admittance matrix ∈ℂN×NY ^N× N, the net complex power injection SiinjS_i^inj at each non-slack bus i satisfies Siinj=Pg,i−Pd,i+j(Qg,i−Qd,i)=Vi(∑k=1NYikVk)∗,S_i^inj=P_g,i-P_d,i+j(Q_g,i-Q_d,i)=V_i ( _k=1^NY_ikV_k )^\!*, (1) where Pg,iP_g,i and Qg,iQ_g,i are the active and reactive power generation at bus i, Pd,iP_d,i and Qd,iQ_d,i are the active and reactive power demand, and (⋅)∗(·)^* is the complex conjugate. Decomposing the admittance Yik=Gik+jBikY_ik=G_ik+jB_ik into its conductance GikG_ik and susceptance BikB_ik, (1) yields the familiar pair of nonlinear equations per non-slack bus: Pg,i−Pd,i P_g,i-P_d,i =Vm,i∑k=1NVm,k[Gikcos(δi−δk) =V_m,i _k=1^NV_m,k\! [G_ik ( _i- _k) . +Biksin(δi−δk)], .+B_ik ( _i- _k) ], (2) Qg,i−Qd,i Q_g,i-Q_d,i =Vm,i∑k=1NVm,k[Giksin(δi−δk) =V_m,i _k=1^NV_m,k\! [G_ik ( _i- _k) . −Bikcos(δi−δk)]. .-B_ik ( _i- _k) ]. (3) In standard AC power flow analysis, one solves for the unknown state variables given specified power injections and voltage setpoints. OPF augments these equations with an economic objective and additional constraints [wood2013power, jiang2024advancements], but OPF is not the focus of this work. We explicitly treat the complete per-bus state i=[Vm,i,δi,Pg,i,Qg,i],y_i=[V_m,i,\, _i,\,P_g,i,\,Q_g,i], (4) and design PMGAT-AI to infer the unknown components of iy_i for each bus type (PQ, PV, and Slack) as summarized in Table I. Thus, a single model predicts bus voltages (Vm,δV_m,δ at PQ buses) and generator injections (Pg,QgP_g,Q_g at slack buses) within a unified AC power flow formulation. Historically, operators executed AC power flow primarily for offline planning, day-ahead scheduling, and periodic security assessment, for which the computational cost of iterative Newton-type solvers was acceptable [frank2016introduction, stott1974fast]. This assumption is increasingly strained. High penetrations of inverter-based renewable energy sources and distributed energy resources introduce faster fluctuations, bidirectional flows, and operating points closer to security boundaries [mohagheghi2018survey, wang2023ac, feng2024safe]. As a result, AC power flow analysis is becoming a near real-time operational requirement, with operators requiring frequent solutions for contingency analysis and corrective actions [mohagheghi2018survey, feng2024safe]. Under these stressed or ill-conditioned conditions, convergence and robustness of Newton–Raphson solvers can degrade [iwamoto1981load, dehghanpour2018survey]. Due to the trade-off between computational speed and solution reliability, there has been significant research interest in data-driven, machine-learning-based alternatives for predicting AC power flow, with some extensions to OPF and state estimation [donon2020neural, lopez2023power, lin2024powerflownet, ugwumadu2025powermodel, hu2024adaptive, taghizadeh2024multi, yang2024probabilistic, bottcher2023solving, deihim2024initial, owerko2024unsupervised, lin2022elegnn]. Earlier ML architectures modeled the problem using engineered features and either multilayer perceptrons or convolutional neural networks, with the downside that they used little or no explicit graph-structure information from power systems [jiang2024advancements, pineda2025beyond]. This limitation often constrained those models to specific topologies, with limited generalization beyond training conditions and across systems [pineda2025beyond, khaloie2025review]. Graph neural networks (GNNs) have since emerged as an effective framework for power grids, with buses as nodes, and physical branches (lines and transformers) represented as edges [wu2020comprehensive, liao2021review]. Message passing in a GNN mirrors the physical dependence of each bus on its electrical neighbors, and existing works have applied GNNs to AC power flow prediction [lin2024powerflownet, pham2022neural, talebi2025graph, ugwumadu2025powermodel], OPF [owerko2024unsupervised, arowolo2025towards], contingency analysis [nakiganda2025graphneuralnetworksfast], and power system state estimation [lin2022elegnn]. Meanwhile, physics-informed GNNs for power systems incorporate AC power flow residuals or related constraints as differentiable penalties in the loss function, encouraging physically consistent predictions [karniadakis2021physics, raissi2019physics, lin2022elegnn, eeckhout2024improved, li2025physics]. However, four gaps remain: 1. System-specific models and limited transferability. Most existing GNNs for power flow are trained on a single, fixed system with varied operating conditions, requiring a separate specialized model for each system and contingency regime [donon2020neural, lopez2023power, lin2024powerflownet]. These models often fail to generalize under topology changes and are impractical for deployment where operators must maintain multiple models across systems and conditions [pineda2025beyond]. 2. Computational bottleneck in hybrid GNN-solver frameworks. Another line of work combines GNNs with iterative solvers, where the GNN provides a warm start or predictions validated by a physics-based criterion [deihim2024initial, shamseldein2026hybrid, owerko2024unsupervised, ugwumadu2025powermodel]. However, the speed advantage depends on the robustness of the trained model; frequent solver fallback on stressed or contingency cases diminishes the gain. A highly robust standalone model is therefore a prerequisite for effective hybrid deployment, and this work focuses on improving standalone predictive performance. 3. Catastrophic forgetting in continual learning. Real-world grids are nonstationary, and several works advocate on-the-fly fine-tuning where difficult cases are solved with a trusted AC solver and used to update the model [ugwumadu2025powermodel, jiang2024advancements]. However, naive fine-tuning can lead to catastrophic forgetting, where performance on previously learned tasks collapses [kirkpatrick2017overcoming, carta2022catastrophic]. This challenge is recognized as open for deep graph networks [carta2022catastrophic], yet explicit forgetting mitigation remains limited in published AC power flow GNN studies [ugwumadu2025powermodel]. 4. Limited physical interpretability. Although interpretability has received some attention in GNN-based power system modeling [lin2024powerflownet, varbella2024powergraph], systematic validation that learned representations reflect known electrical principles remains limited. Established explainability methods such as GNNExplainer [ying2019gnnexplainer] and integrated gradients [sundararajan2017axiomatic] are available for graph learning, yet without such validation it remains unclear whether predictions rely on physically meaningful features or spurious correlations. To address these gaps, we introduce PowerModelsGAT-AI (hereafter PMGAT-AI), a physics-informed model for solving AC power flow equations. I-A Contributions Our contributions are: 1. Unified multi-system learning for AC power flow. We evaluate PMGAT-AI on 14 standard benchmarks (4 to 6,470 buses), including system-specific baselines across all systems and a unified model trained across 13 systems. This establishes a single framework for unified learning across heterogeneous grid topologies and operating conditions. 2. Bus-type-aware prediction formulation. We formulate AC power flow prediction at the bus level as learning the unknown subset of i=[Vm,δ,Pg,Qg]y_i=[V_m,\,δ,\,P_g,\,Q_g] under a bus-type-aware supervision mask (Table I). This enables one model to jointly infer bus voltages and generator injections without leaking specified inputs into supervised losses. 3. Physics-informed graph-attention training objective. PMGAT-AI combines edge-aware graph attention with a differentiable power-mismatch constraint derived from (1). We optimize the four supervised targets and the physics term jointly using graph-normalized losses with homoscedastic uncertainty weighting, avoiding manual loss-weight tuning. 4. Continual learning with forgetting mitigation and physics-informed alignment. For adaptation to new systems or operating conditions, we integrate experience replay with elastic weight consolidation (EWC) to mitigate catastrophic forgetting. Separately, we analyze learned attention and feature attributions, showing alignment with known electrical relationships. We organize the rest of the paper as follows. Sec. I describes dataset generation and graph construction. Sec. I presents the model architecture, loss design, and optimization. Sec. IV describes the continual learning framework. Secs. V and VI report the evaluation setup and results. Sec. VII presents the interpretability analysis, and we conclude in Sec. VIII. The appendices provide feature definitions, contingency statistics, detailed per-system results, and continual learning analysis. I Dataset Generation and Preprocessing We generated a dataset from 14 standard power systems111All system names follow the pandapower naming conventions [pandapower2018, matpower2011]; variations in size and naming (e.g., case4gs, case14, case_illinois200) reflect the original system identifiers. spanning different scales using pandapower for AC power flow simulations [pandapower2018]. These systems enable cross-system training and evaluation. For each system, we extract bus-state vectors i=[Vm,δ,Pg,Qg]y_i=[V_m,δ,P_g,Q_g], per-bus input features ix_i, and branch-level edge features ije_ij (feature definitions in Appendix A); bus-type masks define which components are supervised. A full list of the systems considered and contingency statistics are provided in Appendix B [matpower2011, josz2016acpf, fliscounakis2013contingency]. I-A Systems and Contingencies We generate scenarios by sampling diverse operating conditions and topologies. To prevent overfitting to global load correlations, we use a hierarchical randomization scheme for active power loads. The load at bus i is scaled by ρi=γ⋅Rr(i)⋅Ji _i=γ· R_r(i)· J_i, where: • γ∼U[0.7,1.3]γ U[0.7,1.3] is a global scaling factor. • Rr(i)∼U[0.75,1.25]R_r(i) U[0.75,1.25] is a regional factor shared by buses with similar nominal voltage. • Ji∼U[0.95,1.05]J_i U[0.95,1.05] is a local jitter factor. Generator active power setpoints are scaled globally by factors sampled from U[0.8,1.2]U[0.8,1.2]. Generators are modeled with sufficient reactive capability to maintain scheduled voltage setpoints. To ensure the model learns generalized power flow physics rather than memorizing specific network impedances, we apply random jitter to the branch parameters. For every scenario, the electrical properties (resistance R, reactance X, and susceptance B) of all lines and transformers are independently scaled by a factor sampled from U[0.9,1.1]U[0.9,1.1] (i.e., ±10%± 10\% variation). Finally, to ensure robustness against topology changes, we generate N−kN\!-\!k contingencies by applying random branch outages. We consider up to two branch outages (N−2N\!-\!2) when generating contingency scenarios. I-B Graph Construction We construct a directed graph =(,ℰ)G=(V,E) where each bus is a node and each physical branch (line, transformer, series impedance, switch) is represented by reciprocal edges (i→j)(i\!→\!j) and (j→i)(j\!→\!i). This directed representation preserves direction-dependent branch parameters (Sec. I-C). Preprocessing yields node features ix_i and edge features ije_ij. We also add a self-loop (i→i)(i\!→\!i) for every node; its edge attributes carry diagonal network-admittance terms, while bus shunt admittance (Gsh,Bsh)(G_sh,B_sh) is retained as node-level input. I-C Node and Edge Features Node features For each bus i, we provide a feature vector i∈ℝ23x_i ^23 that combines operational and topological information, together with a one-hot bus-type indicator: • Nominal Values and Setpoints: Nominal voltage (VnV_n), voltage setpoint (VmsetV_m^set), and active power setpoint (PgsetP_g^set). • Known Injections: Active and reactive load (Pd,QdP_d,Q_d). • Physical Limits: Voltage and reactive power limits (Vmmin,Vmmax,Qgmin,QgmaxV_m ,V_m ,Q_g ,Q_g ). • Physics Parameters: System base power (SbaseS_base), frequency (f), and bus shunt admittance (Gsh,BshG_sh,B_sh). • Topological Features: 1st- and 2nd-hop degree, weighted distance to slack, electrical betweenness, and aggregate neighbor net injection. • Neighbor Indicators: Binary flags for 1st- and 2nd-hop neighbors being generators or slack buses. The aggregate neighbor net injection feature is computed from generator setpoints and load inputs only, not from solved output targets. Edge features Each directed branch edge (i→j)(i\!→\!j) has a 77-dimensional attribute vector obtained by concatenating series admittance (YijserY^ser_ij), branch type (ijt_ij), and thermal limit (IijmaxI^max_ij): ij=[ℜ(Yijser),ℑ(Yijser)⏟admittanceij⏟type (one-hot)∈ℝ4Iijmax⏟thermal limit]∈ℝ7.e_ij= [\ (Y^ser_ij),\ (Y^ser_ij)_admittance\; t_ij_type (one-hot) ^4\; I^max_ij_thermal limit\ ] ^7. Here, ℜ(⋅) (·) and ℑ(⋅) (·) are the real and imaginary parts. In implementation, YijserY^ser_ij is the directed off-diagonal branch-admittance edge term (including tap effects where applicable), so ℜ(Yijser) (Y^ser_ij) and ℑ(Yijser) (Y^ser_ij) are its conductance and susceptance components. IijmaxI^max_ij is a per-unit thermal-limit proxy computed as branch rating (MVA) normalized by system base power. For self-loops, the first two edge features use diagonal admittance terms, while type and rating features are set to zero. The nodal-balance terms YikY_ik in (1) are entries of the bus-admittance matrix Y assembled from network elements (including tap and shunt effects). Complete feature definitions are provided in Appendix A. I-D Bus-Type Masks and Data Validation We define a binary training mask, M∈0,1N×4M∈\0,1\^N× 4, to identify the learning targets for each bus i based on its type. A value of 11 indicates an unknown target to be predicted. Table I defines supervision over the model state vector i=[Vm,δ,Pg,Qg]y_i=[V_m,δ,P_g,Q_g]. Sampling balance We use a weighted random sampler with probability inversely proportional to the number of samples for each system to ensure balanced exposure across all systems (e.g., case14 and case_illinois200) during training. Feature and target scaling Node features are standardized (zero mean, unit variance) using statistics computed from the training set, and the same transformation is applied to the four target variables (Vm,δ,Pg,Qg)(V_m,δ,P_g,Q_g). Edge features are kept in physical units where appropriate. Predictions are inverse-transformed before evaluation and physics-loss computation, ensuring absolute-error metrics are reported in physical units, while normalized metrics are unitless. I Model Architecture and Training Figure 1: PowerModelsGAT-AI Architecture. The framework comprises (1) a bus-type-aware input stage with a supervision mask, (2) an encoder stack of 4 Pre-Norm Residual GATv2 blocks, (3) a shared decoding trunk, (4) multi-head outputs for Vm,δ,Pg,QgV_m,\,δ,\,P_g,\,Q_g, and (5) a physics-informed loss ℒphyL_phy. Red dashed lines indicate how predicted outputs and the supervision mask feed into the physics-informed loss ℒphyL_phy. We design PMGAT-AI (Fig. 1) as a deep GNN composed of stacked pre-norm residual blocks, followed by a multi-target output head. I-A Pre-Norm Residual GNN Encoder The model maps input node features ix_i to a hidden dimension dhiddend_hidden using a linear projection. The resulting embeddings (0)h^(0) are processed by L pre-norm residual blocks (where normalization precedes each layer transformation). For the intermediate layers ℓ∈1,…,L−1 ∈\1,…,L-1\, the update rule includes a non-linearity and dropout to deepen the representation: (norm) ^(norm) =GraphNorm((ℓ−1),) =GraphNorm(h^( -1),b) (5) (ℓ) ^( ) =GNN−Layer((norm),ℰ,) =GNN-Layer(h^(norm),E,e) (6) (ℓ) ^( ) =(ℓ−1)+Dropout(ELU(head(ℓ)(ℓ))) =h^( -1)+Dropout\! (ELU\! (W_head^( )m^( ) ) ) (7) where GraphNorm(⋅,)GraphNorm(·,b) normalizes node features within each graph (identified by batch vector b) with a learnable shift [cai2021graphnorm], GNN−LayerGNN-Layer is the GATv2 convolution defined in Sec. I-B, and head(ℓ)W_head^( ) projects the concatenated attention heads back to dhiddend_hidden. The final layer (ℓ=L =L) prepares features for the readout head, aggregating attention heads by averaging rather than concatenation, and omitting the final activation: (L)=Projres((L−1))+GNN−Layer(GraphNorm((L−1)),ℰ,)h^(L)=Proj_res(h^(L-1))+GNN-Layer(GraphNorm(h^(L-1)),E,e) (8) where ProjresProj_res aligns dimensions for the residual connection if necessary. I-B GATv2 with Edge Features The GNN−LayerGNN-Layer uses GATv2 [brody2021attentive] with 7-dimensional edge features ije_ij integrated into the edge-aware attention mechanism. For each of H attention heads (indexed by m), the computation is: ij(m) _ij^(m) =t(m)i+s(m)j+e(m)ij, =W_t^(m)h_i+W_s^(m)h_j+W_e^(m)e_ij, e~ij(m) e_ij^(m) =(m)⊤LeakyReLU(ij(m)), =a^(m) LeakyReLU\! (z_ij^(m) ), αij(m) _ij^(m) =softmaxj(e~ij(m)), =softmax_j\! ( e_ij^(m) ), i(m) _i^(m) =∑j∈(i)αij(m)s(m)j. = _j (i) _ij^(m)\,W_s^(m)h_j. (9) Configuration We set the hidden dimension dhidden=128d_hidden=128 and employ L=4L=4 GATv2 layers with H=4H=4 heads. Intermediate layer heads are concatenated, while the final layer heads are averaged. Physics-informed self-loops are explicitly included in the graph topology during construction (Sec. I-C) rather than added implicitly by the layer. I-C Multi-Target Output Head A multi-head architecture maps the final node embedding i(L)h^(L)_i to the four-dimensional output vector ^i y_i. A shared multilayer perceptron (MLP) trunk first extracts a latent representation ξi _i: ξi=ELU(trunki(L)+trunk) _i=ELU(W_trunkh^(L)_i+b_trunk) (10) This representation is projected by four separate linear layers to produce the state variable predictions: V^m,i V_m,i =Vm⊤ξi+bVm =w_V_m _i+b_V_m δ^i δ_i =δ⊤ξi+bδ =w_δ _i+b_δ P^g,i P_g,i =Pg⊤ξi+bPg =w_P_g _i+b_P_g Q^g,i Q_g,i =Qg⊤ξi+bQg =w_Q_g _i+b_Q_g (11) This design uses a shared learned feature space while producing task-specific outputs. I-D Loss Functions and Physics Constraints We formulate the training objective as a multi-task learning problem. Let the state at bus i be i=[Vm,i,δi,Pg,i,Qg,i]y_i=[V_m,i,\, _i,\,P_g,i,\,Q_g,i]. We define five loss components ℒτL_τ for τ∈=Vm,δ,Pg,Qg,phyτ =\V_m,δ,P_g,Q_g,phy\, where phy is the physics-mismatch penalty, which are dynamically balanced (Sec. I-E). A naive mean-squared error (MSE) over all buses implicitly biases training toward more frequent bus types (typically PQ buses) and larger systems. To reduce this bias from bus-type frequency and graph size, we compute the loss per graph using masks that include only unknown targets for each bus type (Table I), then average across graphs. Let Mi,τ∈0,1M_i,τ∈\0,1\ be the binary mask indicating if variable τ is an unknown target for bus i. For a batch of N_G graphs, the loss for state variable τ is computed by first normalizing the error within each graph g, and then averaging across graphs: ℒτ=1N∑g=1N[∑i∈gMi,τ⋅fτ(y^i,τ,yi,τ)∑i∈gMi,τ+ε]L_τ= 1N_G _g=1^N_G [ _i _gM_i,τ· f_τ( y_i,τ,y_i,τ) _i _gM_i,τ+ ] (12) where gV_g is the set of nodes in graph g and ε is a stability term. This formulation ensures that a small 14-bus system contributes equally to the gradient update as a 200-bus system, preventing bias toward larger systems. Node-wise error functions fτf_τ For voltage magnitudes and power injections (τ∈Vm,Pg,Qgτ∈\V_m,P_g,Q_g\), fτf_τ is the standard squared error on standardized targets: fτ(y^,y)=(y^−y)2f_τ( y,y)=( y-y)^2 (13) For voltage angles, standard MSE is not suitable because angles are periodic. To address this, we optimize the wrapped residual in degrees; in our experiments, this improved convergence stability and final predictive performance relative to a radian-domain loss. Let Δi=δ^i(rad)−δi(rad) _i= δ_i^(rad)- _i^(rad) be the angular difference in radians. The wrapped loss is fδ(δ^i,δi)=(180πatan2(sinΔi,cosΔi))2f_δ( δ_i, _i)= ( 180π\,atan2\! ( _i, _i ) )^2 (14) where atan2(sinΔ,cosΔ)atan2( , ) returns the principal value in (−π,π](-π,π], so the error reflects the shortest arc. Physics-Informed Power Mismatch We enforce physical consistency by minimizing bus-level power-mismatch errors derived from the nodal power balance in (1). Specifically, we penalize the deviation between the hybrid net injection Siinj,hybS^inj,hyb_i and the calculated flow S^iflow S^flow_i. To prevent information leakage, we define Siinj,hybS^inj,hyb_i by mixing model predictions with reference data according to the bus mask: Siinj,hyb=(Pg,ihyb−Pd,i)+j(Qg,ihyb−Qd,i)S^inj,hyb_i=(P_g,i^hyb-P_d,i)+j(Q_g,i^hyb-Q_d,i) (15) where the hybrid generation terms combine the predicted (P^g,Q^g P_g, Q_g) and known (Pg,QgP_g,Q_g) values: Pg,ihyb P_g,i^hyb =Mi,PgP^g,i+(1−Mi,Pg)Pg,i, =M_i,P_g P_g,i+(1-M_i,P_g)P_g,i, (16) Qg,ihyb Q_g,i^hyb =Mi,QgQ^g,i+(1−Mi,Qg)Qg,i. =M_i,Q_g Q_g,i+(1-M_i,Q_g)Q_g,i. (17) Here, Pd,iP_d,i and Qd,iQ_d,i are ZIP-equivalent loads evaluated at V^m,i V_m,i. The calculated flow is the nodal power balance (1) evaluated at predicted voltages, where V^i=V^m,iejδ^i V_i= V_m,ie^j δ_i is the complex voltage constructed from the predicted magnitude and angle: S^iflow=V^i∑k∈(i)Yik∗V^k∗ S^flow_i= V_i _k (i)Y_ik^* V_k^* (18) The physics loss is the mean squared modulus of the mismatch, computed in 64 bit64\,bit precision for numerical stability and averaged per graph as in (12): ℒphy=1N∑g=1N[1|g|∑i∈g|Siinj,hyb−S^iflow|2]L_phy= 1N_G _g=1^N_G [ 1|V_g| _i _g |S^inj,hyb_i- S^flow_i |^2 ] (19) I-E Multi-Task Optimization The training objective involves minimizing five distinct loss components with different scales and optimization difficulty. Naive summation with fixed weights requires extensive hyperparameter tuning and can cause one loss to dominate the gradient updates over the others. To address this, we treat the problem as a multi-task learning challenge and employ homoscedastic uncertainty weighting [kendall2018multi]. This method interprets the relative weight of each task as a learnable uncertainty parameter στ _τ, derived from maximizing the Gaussian likelihood of the multi-task objective. Let =Vm,δ,Pg,Qg,phyC=\V_m,δ,P_g,Q_g,phy\ be the set of task indices defined in Sec. I-D, comprising the four supervised state variables and the physics mismatch. The unified training objective ℒtotal(,)L_total( θ, σ) is: ℒtotal(,)=∑τ∈12στ2ℒτ()+∑τ∈logστL_total( θ, σ)= _τ 12 _τ^2L_τ( θ)+ _τ _τ (20) where θ represents the model parameters and =στ∈ σ=\ _τ\_τ are learnable noise scalars optimized simultaneously with the model parameters. The term logστ _τ acts as a regularizer to prevent the trivial solution στ→∞ _τ→∞. This approach dynamically balances the supervised losses and the physics-informed constraint without manual tuning, with learned weights that adapt to reflect the relative difficulty of each prediction task. In implementation, uncertainty weighting is activated after an initial static-loss warmup period (50 epochs). I-F Optimization and Early Stopping We minimize (20) using the AdamW optimizer with an initial learning rate η0=10−3 _0=10^-3 and weight decay 10−510^-5. The learning rate follows a schedule comprising a linear warmup for the first EwE_w epochs, followed by cosine annealing decay [loshchilovstochastic] to ηmin=10−5 _ =10^-5 for the remainder of the training budget E. We trained on an NVIDIA RTX 4000 Ada GPU (20 GB20\,GB VRAM). We used a baseline batch size of 128, reducing this value for large-scale systems and during multi-system learning to accommodate memory constraints. The implementation is based on PyTorch and PyTorch Geometric, with NumPy, SciPy, NetworkX, Matplotlib, and scikit-learn for data processing, graph utilities, visualization, and evaluation [paszke2019pytorch, fey2025pyg, harris2020array, scipy2020nmeth, SciPyProceedings_11, Hunter:2007, scikit-learn]. We use a two-phase early stopping strategy. After the main validation metric plateaus, we keep a checkpoint only if it reduces the physics mismatch ℒphyL_phy and does not reduce overall performance by more than 5 %5\, 37 . This gives a final model that stays physically consistent while maintaining low supervised loss. IV Continual Learning for On-the-Fly Adaptation Operational power grids are dynamic; topological changes (switching, outages) or extreme loading conditions can effectively alter the data distribution. To adapt to new scenarios and new systems without full retraining, we combine experience replay and EWC within a continual-learning framework to mitigate catastrophic forgetting. Experience Replay To enable the model to learn from both old and new scenarios or systems, we employ experience replay. We maintain a replay buffer rep⊂oldD_rep _old of scenarios from the systems on which the model was originally trained, sampled uniformly at random. During adaptation, the new data newD_new may consist of stressed scenarios from existing systems or scenarios from a new system. Because shifts are larger when adapting across structurally different systems, we mix newD_new with replay data repD_rep to reduce forgetting while learning the new target systems. We construct a mixed training set mix=new∪repD_mix=D_new _rep, ensuring that the model retains performance on base systems while adapting to the new systems or operating conditions. Elastic Weight Consolidation (EWC) While experience replay helps mitigate data distribution shifts, it does not prevent important model parameters from changing. To mitigate this, we augment the loss with EWC [kirkpatrick2017overcoming], which constrains parameters that are critical for performance on base systems. Let ∗ θ^* be the parameters learned during initial training. We compute the diagonal Fisher Information Matrix F, where each element FjF_j approximates the importance of parameter θj _j for maintaining performance on base systems. The regularization penalty is: ℒEWC()=λewc∑jFj(θj−θj∗)2L_EWC( θ)= _ewc _jF_j( _j-θ^*_j)^2 (21) This regularization constrains parameters with high FjF_j (critical for power flow solutions on base systems), while allowing low-sensitivity parameters to adapt to new systems or operating conditions. The final on-the-fly objective unifies the learning signal from the mixed data batch ℬ∼mixB _mix with the EWC penalty: ℒonline=ℒtotal(,,ℬ)+ℒEWC()L_online=L_total( θ, σ,B)+L_EWC( θ) (22) where ℒtotalL_total is the multi-task objective from (20). V Evaluation Setup V-A Dataset and Evaluation Regimes PMGAT-AI is evaluated on 14 benchmark systems ranging from small-scale systems (e.g., case4gs, case14) to large-scale systems (e.g., case1354pegase, case6470rte). We define two distinct evaluation regimes: V-A1 Regime 1: System-Specific Baselines We establish baselines by training independent PMGAT-AI models for each of the 14 systems, using 10,000 scenarios per system. We evaluate two operating regimes: (1) Normal operation (N−0N\!-\!0), where models are trained and tested on randomized load and generation profiles with fixed topology; and (2) Contingency (N−2N\!-\!2), where models are retrained on datasets with randomized load and generation profiles and up to two random branch outages per scenario to improve robustness (Appendix B, Table VII). V-A2 Regime 2: Multi-System and Continual Learning This regime evaluates multi-system learning under N−2N\!-\!2 contingency conditions. Because batching multiple large graphs is memory-intensive, case6470rte is excluded from unified training. We sample 2,000 scenarios per system, resulting in 26,000 scenarios for the unified model trained on 13 systems and 24,000 for the base model trained on 12 systems. We first train the unified model on all 13 systems (Table XI). This model is also used for interpretability analysis in Sec. VII. Then, we evaluate continual learning by pre-training a base model on 12 systems (excluding case1354pegase) and adapting it to case1354pegase using the EWC+Replay strategy in Sec. IV. Detailed contingency statistics are provided in Appendix B, Table VIII. For very small systems (notably case4gs and case9), feasible N−2N\!-\!2 samples are absent due to islanding or failure of AC power flow to converge; these systems contribute only N−0N\!-\!0 and N−1N\!-\!1 samples. V-B Evaluation Protocol We randomly split all the datasets into training (80%), validation (10%), and test (10%) sets, stratified by system, then evaluate all models on the held-out test sets. Metrics are computed only for unknown state variables as defined by each bus type (Table I). This prevents known fixed setpoints from artificially inflating performance metrics. We use the following metrics to evaluate different aspects of model performance: • Absolute and Squared Error: We report root mean squared error (RMSE) and mean absolute error (MAE). For voltage angles, errors are wrapped to [−180∘,180∘][-180 ,180 ] before computation. • Scale-Normalized Error: We report normalized mean absolute error (NMAE), defined as MAE divided by the range of target values for each target. • Coefficient of Determination (R2R^2): We report R2R^2. For voltage angles, R2R^2 is computed with wrapped residuals and a circular-mean reference so that it is consistent with the angle MAE and RMSE definitions. • Knowledge Loss: We quantify forgetting with Knowledge Loss (K). For state variable τ∈Vm,δ,Pg,Qgτ∈\V_m,δ,P_g,Q_g\, error metric μ (e.g., NMAE, MAE), and previously learned system s: τμ(s)=Errτ,μpost(s)−Errτ,μpre(s)K_τ^μ(s)=Err_τ,μ^post(s)-Err_τ,μ^pre(s) (23) where ErrpreErr^pre and ErrpostErr^post are errors before and after fine-tuning on the new task. Positive K indicates forgetting; negative K indicates improved post-adaptation performance. Across base systems S, we report: ¯τμ=1||∑s∈τμ(s) K_τ^μ= 1|S| _s K_τ^μ(s) (24) VI Results Results are organized by the two regimes defined in Sec. V. VI-A Regime 1: Baseline System Performance (N−0N\!-\!0) The detailed per-system metrics in Appendix C, Table IX establish baseline performance under normal N−0N\!-\!0 conditions. For voltage magnitudes (VmV_m), MAE on standard IEEE benchmark systems (e.g., case14, case30) is on the order of 10−4p.u.10^-4~$ p.u.$, corresponding to an NMAE of approximately 0.2%0.2\%–0.3%0.3\%. For the larger case6470rte system, NMAE is approximately 0.98%0.98\%, and R2R^2 remains above 0.940.94, showing that the model captures system-level voltage variation. We observe that voltage angle (δ) error increases with system size (MAE ≈0.05∘≈ 0.05 for case14 versus ≈5.0∘≈ 5.0 for case6470rte). A likely contributor is reference-angle drift in larger networks [caro2016uncertainty]. Since nodal power balance depends on angle differences between connected buses ((2), (3)), reference-angle drift can increase absolute angle MAE without degrading the inter-bus differences that govern power flow. Even so, angle R2R^2 remains high (≈0.99≈ 0.99 for case6470rte). Active (PgP_g) and reactive (QgQ_g) predictions at slack and PV buses also show low error, with PgP_g R2R^2 consistently above 0.99 across systems. Overall, these metrics show that the model preserves physically consistent power flow relationships across the evaluated systems and scenarios. VI-B Regime 1: Robustness to Contingencies (N−2N\!-\!2) To assess robustness to topology changes within a system, Regime 1 also evaluates each system under N−2N\!-\!2 contingencies, where random branch outages redistribute branch flows and bus-voltage profiles, creating many topology combinations and a harder prediction task. Complete per-system results are provided in Appendix C, Table X. Compared with the N−0N\!-\!0 baseline, we observe marginal error increases (e.g., in case14, voltage-magnitude MAE increases from 3.3×10−43.3× 10^-4 to 7.2×10−4p.u.7.2× 10^-4~$ p.u.$), but errors remain small. Voltage NMAE stays below 0.5%0.5\% for most systems, and angle R2R^2 stays near 0.990.99. Even in the challenging case6470rte system under N−2N\!-\!2 contingency, angle R2R^2 is ≈0.99≈ 0.99, close to its N−0N\!-\!0 value. This is consistent with the GATv2 dynamic attention mechanism reweighting branch importance as connectivity, loading, and branch attributes change. VI-C Regime 2: Multi-System and Continual Learning We evaluate PMGAT-AI across multiple trained systems and for adaptation to new systems. We first assess a unified model trained on 13 systems, then evaluate continual adaptation by fine-tuning a base model trained on 12 systems to case1354pegase. VI-C1 Unified 13-System Performance We analyze the unified model trained on 13 systems. Detailed results are provided in Appendix D, Tables XI and XII. Compared with system-specific models, the unified model can be used across multiple systems, with a small increase in the per-system error. For example, on case14, voltage NMAE increases from about 0.15% for the system-specific model (Regime 1) to about 0.76% for the unified model. (a) Training Performance (b) Test Performance Figure 2: PowerModelsGAT-AI Performance across 13 Power Systems. Parity plots compare model predictions (y-axis) against the Newton–Raphson reference solution (x-axis) for all unknown variables in the unified 13-system training and test sets; color intensity represents point density. Across the 13 trained systems, the average voltage NMAE is 0.89%0.89\%, and active-power R2R^2 exceeds 0.970.97 (Fig. 2), with Fig. 2a and Fig. 2b showing similar prediction versus reference trends across all predicted variables. Per-system plots are provided in the Supplementary Material (Figs. S1–S13). At the bus level, prediction error is typically lowest in small-scale systems such as case4gs and case30, where predictions closely match the reference values. In larger systems such as case1354pegase, target variables (especially δ, PgP_g, and QgQ_g) span wider ranges across buses and scenarios. As a result, absolute errors can be larger even when NMAE remains relatively low. Branch outages further widen these ranges and increase operating stress, which increases angle and power-injection deviations. Additional higher-error examples are provided in the Supplementary Material (Fig. S14). VI-C2 Continual Learning: Adaptation Without Forgetting While unified training works well, in practice new systems may need to be incorporated without full retraining. Prior work highlights persistent generalization and scalability challenges under changes in system size and topology [pineda2025beyond, khaloie2025review]. To quantify this, we evaluated a base model trained on 12 systems against the held-out case1354pegase system. As shown in Table I, the unadapted base model performs poorly on the target system, with voltage NMAE of 12.98%12.98\%, angle NMAE of 11.96%11.96\%, and negative R2R^2 values meaning predictions worse than the mean. These gaps make clear that cross-system generalization is not automatic and requires explicit adaptation strategies. We evaluated all models on identical test sets and used λewc=0.5 _ewc=0.5 with a replay ratio of 0.3 to balance adaptation and forgetting. We compared this setting against a naive baseline (λewc=0 _ewc=0, no replay). TABLE I: Target System (case1354pegase) Performance After Fine-Tuning. Base (Unadapted) Naive EWC+Replay Target NMAE% R2 NMAE% R2 NMAE% R2 VmV_m 12.98 −2.36-2.36 1.44 0.9450.945 1.90 0.9070.907 δ 11.96 −0.01-0.01 0.61 0.990.99 0.68 0.990.99 PgP_g 20.79 0.010.01 0.42 0.990.99 0.45 0.990.99 QgQ_g 1.61 0.020.02 0.15 0.9940.994 0.17 0.9940.994 Target-system performance is shown here; base-system forgetting is summarized in Table I. TABLE I: Average Knowledge Loss (¯τNMAE K_τ^NMAE) on Base Systems. State Variable Naive EWC+Replay Voltage Magnitude (VmV_m) 12.00 0.26 Voltage Angle (δ) 138.23 0.12 Active Power (PgP_g) 1891.82 1.25 Reactive Power (QgQ_g) 103.79 1.54 Values are in percentage points of NMAE change (¯τNMAE K_τ^NMAE), averaged across 12 base systems. Lower is better; ¯≈0 K≈ 0 indicates minimal forgetting on base systems. On case1354pegase, both strategies reach R2>0.99R^2>0.99 for voltage angles and power injections (Table I). Naive fine-tuning gives slightly better target VmV_m NMAE (1.44%1.44\% versus 1.90%1.90\%), but it causes severe forgetting on base systems (¯PgNMAE≈1892% K_P_g^NMAE≈ 1892\%, ¯δNMAE≈138% K_δ^NMAE≈ 138\%; Table I). Naive fine-tuning is not suitable for long-term cross-system adaptation. Our EWC+Replay strategy addresses this problem. Although target performance is slightly lower than naive fine-tuning, it maintains low knowledge loss across all state variables (¯VmNMAE=0.26% K_V_m^NMAE=0.26\%, ¯δNMAE=0.12% K_δ^NMAE=0.12\%, ¯PgNMAE=1.25% K_P_g^NMAE=1.25\%, ¯QgNMAE=1.54% K_Q_g^NMAE=1.54\%). The same trend appears with MAE (¯δMAE=0.16∘ K_δ^MAE=0.16 versus 74∘74 for naive; see Appendix E). Taken together, these results confirm that the model can adapt to new systems while minimizing forgetting on base systems. Visual summaries are provided in Supplementary Material (Figs. S15 and S16). VII Interpretability Analysis To assess PMGAT-AI’s ability to capture established physical relationships from learned representations rather than relying on spurious statistical correlations, we analyze the model from two perspectives: branch importance, quantified using learned attention weights, and bus feature sensitivity, assessed using integrated gradients. Attention weights are useful for interpretation, but they do not by themselves establish causal influence [jain-wallace-2019-attention, wiegreffe-pinter-2019-attention]. However, consistent correlations with known electrical parameters provide supporting evidence of alignment with electrical relationships. VII-A Identification of Critical Topological Structures To quantify topological influence, we use the learned attention coefficients from the final GATv2 layer. For each branch (i,j)(i,j), we define a branch importance score as the mean attention weight α¯ij=1||∑c∈(1H∑m=1Hαij(m,c)), α_ij= 1|D| _c ( 1H _m=1^H _ij^(m,c) ), (25) averaged across all H attention heads and all test scenarios c∈c . We use these scores for topology visualization (Fig. 3). For correlation analysis (Fig. 4a), we compute feature–attention correlations from branch-level attention within each system and report the equal-weight average across systems. Fig. 3 illustrates the learned importance scores for two representative systems, case14 (Fig. 3a) and case57 (Fig. 3b). The model does not attend uniformly to all branches. Instead, it assigns higher importance to a subset of key branches. In case14, the model assigns higher importance to key branches (dark red). Transformer branches show mixed importance: several receive low scores, whereas others are among the most influential connections. The model responds to system-specific coupling rather than simply down-weighting transformers. (a) case14 (b) case57 Figure 3: Learned Branch Importance Maps. Darker edges indicate higher importance. (a) case14: the model assigns higher importance to key branches. (b) case57: importance scores vary across the system. VII-B Physical Drivers of Branch Importance (a) Feature Correlation (b) Distribution by System Figure 4: Analysis of Branch Importance Scores. (a) Correlation between branch importance and physical branch parameters. (b) Distribution of branch importance scores across benchmark systems. To interpret why some branches receive higher importance, we compute Pearson correlations (r) between branch-importance scores and raw electrical branch parameters (Fig. 4a). Correlations are computed within each system and then averaged with equal system weight. Three patterns hold across systems: 1. Susceptance (ℑ(Yser) (Y^ser)): Susceptance has the strongest positive correlation (r≈0.38r≈ 0.38). This reflects AC power flow relationships, where real-power flow is strongly tied to branch reactance and susceptance and shared among parallel branches according to their impedances [stott1974fast, grainger1994power]. 2. Thermal Limit (ImaxI^max): Thermal limit is also positively correlated (r≈0.22r≈ 0.22). Although thermal rating is not a direct admittance term, higher-rated branches are often electrically influential, so this association is physically consistent. The effect is system-dependent, and in some systems with near-constant branch ratings the thermal-limit correlation within the system is weak as expected. 3. Conductance (ℜ(Yser) (Y^ser)): Conductance shows a negative correlation (r≈−0.27r≈-0.27). In this dataset, branches with larger effective resistive components tend to receive lower branch-importance scores. To separate the individual contributions of each branch property, we also fit a multivariate regression with conductance, susceptance, thermal limit, and branch type as joint predictors of branch-importance score. Thermal limit has the largest standardized coefficient (βstd≈0.36 _std≈ 0.36), followed by susceptance (βstd≈0.15 _std≈ 0.15), while conductance has a near-zero independent effect. This pattern is clearer in lines, where resistance and reactance are less correlated than in transformers. VII-C Branch Importance Distribution by System Fig. 4b compares branch-importance distributions across systems. Larger systems show stronger concentration near low branch-importance scores. For case1354pegase, the median importance is 0.147, and 46.24% of branches have scores at or below 0.1. In case14 and case30, the medians are higher (0.297 and 0.298), and the fractions of branches with scores at or below 0.1 are smaller (11.50% and 15.34%). For high branch-importance scores, case1354pegase still has a non-negligible subset: 15.87% of branches are at or above 0.9, compared with 2.55% in case14 and 3.29% in case30. Intermediate systems such as case118 and case300 also show moderate low-score concentration, with 26.56% and 24.62% of branches, respectively, at or below 0.1. Overall, this points to sparser branch weighting in larger networks. case5 is an outlier among the smaller systems, with a split distribution across branches (30.34% at or below 0.1 and 22.76% at or above 0.9). In this system, two of six branches have series susceptance roughly three to five times the median, while the remaining four are tightly clustered near it. With only six branches, each one represents about 17% of the distribution, so this impedance contrast produces a pronounced split. In comparison, case4gs has similarly few branches but nearly uniform susceptance, and case14 has comparable impedance spread but 20 branches, so neither shows the same bimodality. VII-D Bus Feature Sensitivity and Physics Alignment (a) Voltage Magnitude (VmV_m) Sensitivity (b) Voltage Angle (δ) Sensitivity (c) Active Power (PgP_g) Sensitivity (d) Reactive Power (QgQ_g) Sensitivity Figure 5: Global Stability of Bus Feature Sensitivity. Boxplots show the distribution of normalized bus feature sensitivity scores across 13 systems for (a) VmV_m, (b) δ, (c) PgP_g, and (d) QgQ_g. Having analyzed edge-level importance, we now examine node-level feature sensitivity. We use integrated gradients (IG) [sundararajan2017axiomatic] to quantify how input bus features influence each predicted state variable. Using the dataset mean as the baseline, we define the feature-importance score for feature ϕφ as ℐϕ=1||∑s∈[1Ns∑n=1Ns|IGϕ(n(s))|],I_φ= 1|S| _s [ 1N_s _n=1^N_s |IG_φ\! (x_n^(s) ) | ], (26) where NsN_s is the total number of node samples from test scenarios in system s. We then normalize these scores as ℐ¯ϕ=ℐϕ/∑ϕ′ℐϕ′ I_φ=I_φ/ _φ I_φ , so the relative feature importances sum to unity. Fig. 5 shows these normalized sensitivities across all 13 systems. At the median level, the top contributors are VmsetV_m^set for VmV_m (≈0.15≈ 0.15), PgsetP_g^set for δ (≈0.17≈ 0.17), PgsetP_g^set for PgP_g (≈0.22≈ 0.22), and Bus Type for QgQ_g (≈0.10≈ 0.10). These results show three patterns expected from power system physics: VII-D1 P-δ Coupling For predicting Voltage Angle (δ) and Active Power Generation (PgP_g) (Figs. 5b and 5c), the primary drivers are Active Generation Setpoint (PgsetP_g^set) and Active Load (PdP_d). For δ, their median normalized sensitivities are ≈0.17≈ 0.17 and ≈0.14≈ 0.14. For PgP_g, they are ≈0.22≈ 0.22 and ≈0.16≈ 0.16. This ranking reflects the P-δ coupling in AC power flow, where real-power balance strongly influences voltage angles [wood2013power]. VII-D2 Q-V Coupling For Voltage Magnitude (VmV_m) prediction (Fig. 5a), the Voltage Setpoint (VmsetV_m^set) is the dominant driver (median sensitivity ≈0.15≈ 0.15), followed by Bus Type and Reactive Load (QdQ_d). The observed order is expected from the Q-V relationship: voltage magnitudes at PV buses are directly controlled by their setpoints, while at PQ buses they respond to local reactive injections [kundur1994stability]. The prominence of Bus Type suggests that the model distinguishes between bus categories when inferring voltage profiles. For Reactive Power Generation (QgQ_g) prediction (Fig. 5d), Bus Type (≈0.10≈ 0.10) and Voltage Setpoint (VmsetV_m^set, ≈0.08≈ 0.08) are the strongest median contributors, followed by Active Generation Setpoint (PgsetP_g^set, ≈0.07≈ 0.07). Since QgQ_g is predicted only at PV and Slack buses (Table I), this result matches bus-type-specific reactive-power constraints. VII-D3 Cross-System Feature Sensitivity The stability of these rankings across systems ranging from 14 to 1354 buses supports cross-system consistency. In particular, Bus Type remains stable across targets, with compact IQR values (≈0.05≈ 0.05 for VmV_m, ≈0.04≈ 0.04 for δ, and ≈0.04≈ 0.04 for QgQ_g), reflecting consistent use of PQ, PV, and Slack bus-type structure across systems. Together, the branch-importance maps and feature attributions provide a practical diagnostic view of which branches and bus features drive model outputs under different operating conditions. VIII Conclusion This work presented PowerModelsGAT-AI, a unified framework that solves AC power flow across the 13 systems in the trained set using physics-informed graph attention. The key contributions include: (i) system-specific baselines across 14 power systems (4 to 6,470 buses) and a unified multi-system model trained on 13 of these; (i) a mask-aware formulation that predicts bus voltages and generator injections for all bus types; (i) a physics-informed loss incorporating power mismatch constraints; and (iv) a continual-learning strategy (EWC+Replay) that mitigates catastrophic forgetting during adaptation to new systems. Experimental results show that PMGAT-AI achieves strong predictive performance, with an average voltage magnitude NMAE of 0.89%0.89\% across the unified 13-system benchmark set under the N−2N\!-\!2 evaluation regime. PMGAT-AI maintains robust angle prediction, with R2>0.99R^2>0.99 on the largest transmission systems tested. Analysis of learned attention weights reveals physically meaningful patterns, including stronger concentration of branch importance in larger systems (Sec. VII). Potential directions for future research include extending the contingency dataset and training to higher-order outages (up to N−4N\!-\!4), incorporating system security metrics such as the overall performance index (OPI) for contingency ranking and analysis, and adapting the framework to solve the OPF problem. Another direction is to scale unified training to larger systems using gradient checkpointing or distributed training. Appendix A Feature Definitions This appendix details the node and edge feature sets used in PMGAT-AI. Complete node and edge feature definitions are provided in Table IV and Table V, respectively. TABLE IV: Node Feature Definitions (i∈ℝ23x_i ^23 + one-hot bus type). Complete specification of bus-level input features used by PowerModelsGAT-AI. See Sec. I-C for details. Feature Symbol Description Unit Electrical Properties Nominal Voltage VnV_n Rated voltage level of the bus kV Active Load PdP_d Active power demand at the bus p.u. p.u. Reactive Load QdQ_d Reactive power demand at the bus p.u. p.u. Active Generation Setpoint PgsetP_g^set Scheduled active power generation p.u. p.u. Voltage Setpoint VmsetV_m^set Voltage magnitude setpoint (PV and Slack buses) p.u. p.u. Shunt Conductance GshG_sh Shunt conductance at the bus p.u. p.u. Shunt Susceptance BshB_sh Shunt susceptance at the bus p.u. p.u. Physical Limits Min Reactive Limit QgminQ_g Minimum reactive power generation capacity p.u. p.u. Max Reactive Limit QgmaxQ_g Maximum reactive power generation capacity p.u. p.u. Min Voltage Limit VmminV_m Minimum allowable voltage magnitude p.u. p.u. Max Voltage Limit VmmaxV_m Maximum allowable voltage magnitude p.u. p.u. System Parameters Base Power SbaseS_base System base power for per-unit conversion MVA Frequency f System nominal frequency Hz Topological Features Node Degree (1-hop) — Number of directly connected buses count Node Degree (2-hop) — Number of buses within two hops count Electrical Distance to Slack — Shortest impedance-weighted path length to a slack bus a.u. Electrical Betweenness — Betweenness centrality (impedance-weighted) normalized Aggregate Neighbor Net Injection — Sum of neighboring net injections p.u. p.u. Total Adjacent Admittance — Sum of inverse impedance-based edge weights over incident branches p.u. p.u. Neighbor Indicators (Binary) Generator Neighbor (1-hop) — Any directly connected bus has a generator 0,1 Generator Neighbor (2-hop) — Any 2-hop neighbor has a generator 0,1 Slack Neighbor (1-hop) — Any directly connected bus is a slack bus 0,1 Slack Neighbor (2-hop) — Any 2-hop neighbor is a slack bus 0,1 Categorical Bus Type — Power flow bus classification (one-hot encoded) PQ, PV, Slack TABLE V: Edge Feature Definitions (ij∈ℝ7e_ij ^7). Branch-level attributes for each directed branch edge, encoding series admittance (implemented as directed off-diagonal branch-admittance edge terms), one-hot branch type, and thermal limit. Feature Symbol Description Unit Series Admittance (Edge Terms) Conductance G Real part of the series-admittance edge term p.u. p.u. Susceptance B Imaginary part of the series-admittance edge term p.u. p.u. Branch Type (One-Hot) Line — Transmission line 0,1 Transformer — Power transformer 0,1 Impedance — Series impedance element 0,1 Switch — Switching device 0,1 Thermal Limit Thermal Limit ImaxI^max Per-unit thermal-limit proxy computed as branch rating (MVA) normalized by system base power p.u. p.u. Appendix B Dataset Contingency Statistics This appendix reports contingency statistics of the dataset. Table VI lists the static bus-type composition per system, and Tables VII and VIII summarize the training and test contingency distributions. The N−kN\!-\!k label gives the number of removed branches per scenario: N−0N\!-\!0 (normal), N−1N\!-\!1 (single-branch outage), and N−2N\!-\!2 (double-branch outage). For case4gs and case9, N−2N\!-\!2 samples are absent (see Sec. V). TABLE VI: Bus Type Distribution by System. Breakdown of bus counts per system, classified by bus type (PQ, PV, and Slack). System Total PQ (Load) PV (Gen) Slack case4gs 4 2 1 1 case5 5 1 3 1 case6w 6 3 2 1 case9 9 6 2 1 case14 14 9 4 1 case30 30 24 5 1 case_ieee30 30 24 5 1 case39 39 29 9 1 case57 57 50 6 1 case118 118 64 53 1 case_illinois200 200 162 37 1 case300 300 231 68 1 case1354pegase 1354 1094 259 1 case6470rte 6470 6017 452 1 TABLE VII: Contingency Distribution for System-Specific Models (N−2N\!-\!2 Training and Test Sets). N-0 (%) N-1 (%) N-2 (%) System Train Test Train Test Train Test case4gs 49.5 50.1 50.5 49.9 — — case5 36.1 33.7 35.7 34.4 28.2 31.9 case6w 35.0 32.1 33.1 33.2 31.9 34.7 case9 60.5 60.0 39.5 40.0 — — case14 36.3 35.4 33.7 34.4 30.0 30.2 case30 36.3 38.5 34.1 33.2 29.6 28.3 case_ieee30 37.6 37.4 32.9 34.0 29.5 28.6 case39 44.8 42.7 33.3 32.5 21.9 24.8 case57 36.3 35.3 33.0 33.9 30.7 30.8 case118 34.2 34.9 34.1 32.7 31.7 32.4 case_illinois200 46.2 45.5 31.7 31.6 22.1 22.9 case300 43.6 39.9 32.0 36.0 24.4 24.1 case1354pegase 44.0 45.7 32.3 30.6 23.7 23.7 case6470rte 44.6 43.3 33.0 31.0 22.4 25.7 TABLE VIII: Contingency Distribution for Unified Models (N−2N\!-\!2 Training and Test Sets). Each system contributes 2,000 scenarios. Comparisons show the stability of the contingency distribution across training and test sets. N-0 (%) N-1 (%) N-2 (%) Model Train Test Train Test Train Test Unified (13 Systems) 40.3 41.8 36.2 35.1 23.5 23.1 Base (12 Systems) 39.5 41.5 37.3 35.6 23.2 22.8 Appendix C System-Specific Baseline Results This appendix provides detailed baseline performance metrics for each system under normal (N−0N\!-\!0) and contingency (N−2N\!-\!2) conditions. TABLE IX: System-Specific Baseline Performance (N−0N\!-\!0). Per-system metrics for system-specific models described in Sec. VI-A. Each model is trained and evaluated on its respective system under normal operating conditions (no element outages). System Target MSE RMSE MAE R2 NMAE% case4gs VmV_m [p.u. p.u.] 1.134×10−071.134×10^-07 3.368×10−043.368×10^-04 2.672×10−042.672×10^-04 0.99960.9996 0.26520.2652 δ [° ] 4.121×10−044.121×10^-04 0.020 300.020\,30 0.014 810.014\,81 0.99990.9999 0.12740.1274 PgP_g [p.u. p.u.] 6.434×10−046.434×10^-04 0.025 360.025\,36 0.018 490.018\,49 0.99960.9996 0.27940.2794 QgQ_g [p.u. p.u.] 5.654×10−045.654×10^-04 0.023 780.023\,78 0.018 320.018\,32 0.99950.9995 0.35710.3571 case5 VmV_m [p.u. p.u.] 1.290×10−071.290×10^-07 3.592×10−043.592×10^-04 3.017×10−043.017×10^-04 0.99960.9996 0.36190.3619 δ [° ] 0.001 1590.001\,159 0.034 050.034\,05 0.026 280.026\,28 0.99980.9998 0.25560.2556 PgP_g [p.u. p.u.] 0.002 3450.002\,345 0.048 420.048\,42 0.038 400.038\,40 0.99940.9994 0.37470.3747 QgQ_g [p.u. p.u.] 0.015 510.015\,51 0.12450.1245 0.091 500.091\,50 0.99930.9993 0.28290.2829 case6w VmV_m [p.u. p.u.] 2.935×10−072.935×10^-07 5.418×10−045.418×10^-04 4.285×10−044.285×10^-04 0.99950.9995 0.27220.2722 δ [° ] 9.686×10−049.686×10^-04 0.031 120.031\,12 0.023 830.023\,83 0.99990.9999 0.12080.1208 PgP_g [p.u. p.u.] 3.147×10−053.147×10^-05 0.005 6100.005\,610 0.004 1790.004\,179 0.99990.9999 0.14750.1475 QgQ_g [p.u. p.u.] 1.152×10−041.152×10^-04 0.010 730.010\,73 0.008 2370.008\,237 0.99950.9995 0.22660.2266 case9 VmV_m [p.u. p.u.] 1.395×10−061.395×10^-06 0.001 1810.001\,181 9.378×10−049.378×10^-04 0.99830.9983 0.53610.5361 δ [° ] 0.002 5770.002\,577 0.050 760.050\,76 0.039 020.039\,02 0.99990.9999 0.08870.0887 PgP_g [p.u. p.u.] 1.261×10−041.261×10^-04 0.011 230.011\,23 0.008 1490.008\,149 0.99980.9998 0.21530.2153 QgQ_g [p.u. p.u.] 2.511×10−042.511×10^-04 0.015 850.015\,85 0.012 830.012\,83 0.99430.9943 0.96790.9679 case14 VmV_m [p.u. p.u.] 1.748×10−071.748×10^-07 4.181×10−044.181×10^-04 3.279×10−043.279×10^-04 0.99960.9996 0.23980.2398 δ [° ] 0.002 9050.002\,905 0.053 900.053\,90 0.041 360.041\,36 0.99990.9999 0.16230.1623 PgP_g [p.u. p.u.] 5.692×10−055.692×10^-05 0.007 5450.007\,545 0.005 6250.005\,625 0.99980.9998 0.18230.1823 QgQ_g [p.u. p.u.] 1.912×10−041.912×10^-04 0.013 830.013\,83 0.009 7160.009\,716 0.99930.9993 0.27510.2751 case30 VmV_m [p.u. p.u.] 2.735×10−072.735×10^-07 5.230×10−045.230×10^-04 4.047×10−044.047×10^-04 0.99920.9992 0.31050.3105 δ [° ] 0.003 2750.003\,275 0.057 230.057\,23 0.041 850.041\,85 0.99970.9997 0.17790.1779 PgP_g [p.u. p.u.] 4.123×10−054.123×10^-05 0.006 4210.006\,421 0.004 7620.004\,762 0.99980.9998 0.22300.2230 QgQ_g [p.u. p.u.] 2.017×10−042.017×10^-04 0.014 200.014\,20 0.009 1140.009\,114 0.99950.9995 0.20450.2045 case_ieee30 VmV_m [p.u. p.u.] 4.821×10−074.821×10^-07 6.944×10−046.944×10^-04 5.270×10−045.270×10^-04 0.99900.9990 0.31620.3162 δ [° ] 0.045 520.045\,52 0.21340.2134 0.16680.1668 0.99790.9979 0.56880.5688 PgP_g [p.u. p.u.] 1.861×10−041.861×10^-04 0.013 640.013\,64 0.010 830.010\,83 0.99960.9996 0.33000.3300 QgQ_g [p.u. p.u.] 4.971×10−044.971×10^-04 0.022 300.022\,30 0.015 270.015\,27 0.99840.9984 0.41120.4112 case39 VmV_m [p.u. p.u.] 2.339×10−062.339×10^-06 0.001 5290.001\,529 0.001 0740.001\,074 0.99800.9980 0.31350.3135 δ [° ] 0.18480.1848 0.42990.4299 0.25900.2590 0.99980.9998 0.14560.1456 PgP_g [p.u. p.u.] 0.013 860.013\,86 0.11770.1177 0.090 670.090\,67 0.99990.9999 0.19300.1930 QgQ_g [p.u. p.u.] 0.010 800.010\,80 0.10390.1039 0.068 300.068\,30 0.99750.9975 0.27560.2756 case57 VmV_m [p.u. p.u.] 2.629×10−062.629×10^-06 0.001 6220.001\,622 0.001 1380.001\,138 0.99960.9996 0.20480.2048 δ [° ] 0.030 250.030\,25 0.17390.1739 0.12730.1273 0.99960.9996 0.18350.1835 PgP_g [p.u. p.u.] 0.001 3780.001\,378 0.037 120.037\,12 0.029 480.029\,48 0.99990.9999 0.19480.1948 QgQ_g [p.u. p.u.] 0.002 1220.002\,122 0.046 070.046\,07 0.036 210.036\,21 0.99820.9982 0.53550.5355 case118 VmV_m [p.u. p.u.] 5.940×10−075.940×10^-07 7.707×10−047.707×10^-04 5.771×10−045.771×10^-04 0.99840.9984 0.39720.3972 δ [° ] 0.17970.1797 0.42390.4239 0.31860.3186 0.99950.9995 0.21630.2163 PgP_g [p.u. p.u.] 0.046 690.046\,69 0.21610.2161 0.19120.1912 0.99950.9995 0.43640.4364 QgQ_g [p.u. p.u.] 0.008 4080.008\,408 0.091 690.091\,69 0.066 360.066\,36 0.99760.9976 0.23700.2370 case_illinois200 VmV_m [p.u. p.u.] 7.409×10−077.409×10^-07 8.607×10−048.607×10^-04 6.508×10−046.508×10^-04 0.99840.9984 0.41590.4159 δ [° ] 0.092 010.092\,01 0.30330.3033 0.22620.2262 0.99900.9990 0.32160.3216 PgP_g [p.u. p.u.] 0.003 7980.003\,798 0.061 630.061\,63 0.053 830.053\,83 0.99990.9999 0.21850.2185 QgQ_g [p.u. p.u.] 2.261×10−042.261×10^-04 0.015 040.015\,04 0.009 1670.009\,167 0.99850.9985 0.12110.1211 case300 VmV_m [p.u. p.u.] 6.682×10−056.682×10^-05 0.008 1750.008\,175 0.003 9780.003\,978 0.97470.9747 0.62700.6270 δ [° ] 4.0294.029 2.0072.007 1.4781.478 0.99780.9978 0.41530.4153 PgP_g [p.u. p.u.] 0.043 940.043\,94 0.20960.2096 0.15790.1579 0.99910.9991 0.59450.5945 QgQ_g [p.u. p.u.] 0.015 920.015\,92 0.12620.1262 0.085 290.085\,29 0.99790.9979 0.18180.1818 case1354pegase VmV_m [p.u. p.u.] 1.804×10−051.804×10^-05 0.004 2470.004\,247 0.002 8020.002\,802 0.97250.9725 0.85640.8564 δ [° ] 6.8616.861 2.6192.619 1.9551.955 0.99720.9972 0.61580.6158 PgP_g [p.u. p.u.] 8.6278.627 2.9372.937 2.5702.570 0.99940.9994 0.54510.5451 QgQ_g [p.u. p.u.] 3.9843.984 1.9961.996 1.0981.098 0.99660.9966 0.10770.1077 case6470rte VmV_m [p.u. p.u.] 7.382×10−057.382×10^-05 0.008 5920.008\,592 0.006 5040.006\,504 0.94410.9441 0.98460.9846 δ [° ] 53.2653.26 7.2987.298 4.9684.968 0.98920.9892 1.38001.3800 PgP_g [p.u. p.u.] 6.0716.071 2.4642.464 2.0022.002 0.99810.9981 0.95680.9568 QgQ_g [p.u. p.u.] 0.12590.1259 0.35480.3548 0.17210.1721 0.99090.9909 0.14830.1483 TABLE X: System-Specific Contingency Performance (N−2N\!-\!2). Per-system metrics for system-specific models described in Sec. VI-B. Each model is trained and evaluated on scenarios with up to two simultaneous branch outages, testing robustness under severe (N−2N\!-\!2) contingencies. System Target MSE RMSE MAE R2 NMAE% case4gs VmV_m [p.u. p.u.] 2.597×10−072.597×10^-07 5.096×10−045.096×10^-04 3.502×10−043.502×10^-04 0.99970.9997 0.13530.1353 δ [° ] 0.002 2680.002\,268 0.047 620.047\,62 0.032 800.032\,80 0.99990.9999 0.07630.0763 PgP_g [p.u. p.u.] 3.522×10−043.522×10^-04 0.018 770.018\,77 0.015 170.015\,17 0.99980.9998 0.25520.2552 QgQ_g [p.u. p.u.] 3.339×10−043.339×10^-04 0.018 270.018\,27 0.014 410.014\,41 0.99970.9997 0.20310.2031 case5 VmV_m [p.u. p.u.] 2.116×10−072.116×10^-07 4.600×10−044.600×10^-04 3.145×10−043.145×10^-04 0.99960.9996 0.26140.2614 δ [° ] 0.031 610.031\,61 0.17780.1778 0.098 410.098\,41 0.99870.9987 0.24100.2410 PgP_g [p.u. p.u.] 0.002 9530.002\,953 0.054 340.054\,34 0.044 740.044\,74 0.99930.9993 0.43430.4343 QgQ_g [p.u. p.u.] 0.012 280.012\,28 0.11080.1108 0.081 390.081\,39 0.99930.9993 0.27110.2711 case6w VmV_m [p.u. p.u.] 4.495×10−064.495×10^-06 0.002 1200.002\,120 7.581×10−047.581×10^-04 0.99770.9977 0.16980.1698 δ [° ] 0.10000.1000 0.31620.3162 0.080 970.080\,97 0.99650.9965 0.13250.1325 PgP_g [p.u. p.u.] 2.264×10−042.264×10^-04 0.015 050.015\,05 0.009 4370.009\,437 0.99930.9993 0.29030.2903 QgQ_g [p.u. p.u.] 3.512×10−043.512×10^-04 0.018 740.018\,74 0.011 030.011\,03 0.99900.9990 0.21350.2135 case9 VmV_m [p.u. p.u.] 2.578×10−062.578×10^-06 0.001 6060.001\,606 0.001 0750.001\,075 0.99820.9982 0.27950.2795 δ [° ] 0.017 840.017\,84 0.13360.1336 0.077 210.077\,21 0.99980.9998 0.05920.0592 PgP_g [p.u. p.u.] 1.150×10−041.150×10^-04 0.010 730.010\,73 0.008 8470.008\,847 0.99980.9998 0.22450.2245 QgQ_g [p.u. p.u.] 3.321×10−043.321×10^-04 0.018 220.018\,22 0.013 870.013\,87 0.99550.9955 0.66930.6693 case14 VmV_m [p.u. p.u.] 2.792×10−062.792×10^-06 0.001 6710.001\,671 7.185×10−047.185×10^-04 0.99550.9955 0.14750.1475 δ [° ] 0.11730.1173 0.34240.3424 0.17390.1739 0.99770.9977 0.25190.2519 PgP_g [p.u. p.u.] 3.380×10−043.380×10^-04 0.018 390.018\,39 0.013 070.013\,07 0.99910.9991 0.40770.4077 QgQ_g [p.u. p.u.] 2.854×10−042.854×10^-04 0.016 890.016\,89 0.011 440.011\,44 0.99890.9989 0.28540.2854 case30 VmV_m [p.u. p.u.] 7.611×10−077.611×10^-07 8.724×10−048.724×10^-04 6.365×10−046.365×10^-04 0.99820.9982 0.23400.2340 δ [° ] 0.018 020.018\,02 0.13420.1342 0.078 630.078\,63 0.99880.9988 0.21080.2108 PgP_g [p.u. p.u.] 6.472×10−056.472×10^-05 0.008 0450.008\,045 0.005 6440.005\,644 0.99970.9997 0.24470.2447 QgQ_g [p.u. p.u.] 3.211×10−043.211×10^-04 0.017 920.017\,92 0.011 710.011\,71 0.99920.9992 0.25460.2546 case_ieee30 VmV_m [p.u. p.u.] 2.870×10−062.870×10^-06 0.001 6940.001\,694 0.001 0850.001\,085 0.99600.9960 0.30400.3040 δ [° ] 0.27360.2736 0.52310.5231 0.27670.2767 0.99330.9933 0.39630.3963 PgP_g [p.u. p.u.] 3.698×10−043.698×10^-04 0.019 230.019\,23 0.012 280.012\,28 0.99920.9992 0.36180.3618 QgQ_g [p.u. p.u.] 6.803×10−046.803×10^-04 0.026 080.026\,08 0.016 730.016\,73 0.99780.9978 0.44540.4454 case39 VmV_m [p.u. p.u.] 7.331×10−067.331×10^-06 0.002 7080.002\,708 0.001 5750.001\,575 0.99430.9943 0.43750.4375 δ [° ] 3.5773.577 1.8911.891 0.51710.5171 0.99580.9958 0.22160.2216 PgP_g [p.u. p.u.] 0.007 9320.007\,932 0.089 060.089\,06 0.064 920.064\,92 0.99990.9999 0.13740.1374 QgQ_g [p.u. p.u.] 0.025 040.025\,04 0.15820.1582 0.083 140.083\,14 0.99420.9942 0.36050.3605 case57 VmV_m [p.u. p.u.] 1.238×10−051.238×10^-05 0.003 5180.003\,518 0.001 9350.001\,935 0.99820.9982 0.30960.3096 δ [° ] 0.10760.1076 0.32810.3281 0.20520.2052 0.99880.9988 0.19840.1984 PgP_g [p.u. p.u.] 0.006 1190.006\,119 0.078 220.078\,22 0.058 630.058\,63 0.99930.9993 0.41100.4110 QgQ_g [p.u. p.u.] 0.004 1320.004\,132 0.064 280.064\,28 0.050 930.050\,93 0.99650.9965 0.71650.7165 case118 VmV_m [p.u. p.u.] 9.187×10−079.187×10^-07 9.585×10−049.585×10^-04 6.478×10−046.478×10^-04 0.99760.9976 0.34250.3425 δ [° ] 0.48690.4869 0.69780.6978 0.46660.4666 0.99890.9989 0.28830.2883 PgP_g [p.u. p.u.] 0.048 020.048\,02 0.21910.2191 0.20200.2020 0.99950.9995 0.44490.4449 QgQ_g [p.u. p.u.] 0.008 3820.008\,382 0.091 550.091\,55 0.067 750.067\,75 0.99760.9976 0.24230.2423 case_illinois200 VmV_m [p.u. p.u.] 2.812×10−062.812×10^-06 0.001 6770.001\,677 0.001 1110.001\,111 0.99450.9945 0.41200.4120 δ [° ] 0.21530.2153 0.46400.4640 0.29570.2957 0.99790.9979 0.30410.3041 PgP_g [p.u. p.u.] 0.002 0420.002\,042 0.045 190.045\,19 0.032 990.032\,99 0.99990.9999 0.13070.1307 QgQ_g [p.u. p.u.] 3.278×10−043.278×10^-04 0.018 110.018\,11 0.010 150.010\,15 0.99790.9979 0.11570.1157 case300 VmV_m [p.u. p.u.] 5.132×10−055.132×10^-05 0.007 1640.007\,164 0.003 8000.003\,800 0.98110.9811 0.50100.5010 δ [° ] 8.3688.368 2.8932.893 1.9521.952 0.99560.9956 0.57130.5713 PgP_g [p.u. p.u.] 0.060 410.060\,41 0.24580.2458 0.17580.1758 0.99880.9988 0.67120.6712 QgQ_g [p.u. p.u.] 0.026 480.026\,48 0.16270.1627 0.10130.1013 0.99650.9965 0.23370.2337 case1354pegase VmV_m [p.u. p.u.] 1.345×10−051.345×10^-05 0.003 6670.003\,667 0.002 4520.002\,452 0.97920.9792 0.53580.5358 δ [° ] 4.6154.615 2.1482.148 1.6001.600 0.99800.9980 0.50980.5098 PgP_g [p.u. p.u.] 2.4702.470 1.5721.572 1.3581.358 0.99980.9998 0.29310.2931 QgQ_g [p.u. p.u.] 5.0155.015 2.2392.239 1.1401.140 0.99570.9957 0.11660.1166 case6470rte VmV_m [p.u. p.u.] 9.468×10−059.468×10^-05 0.009 7300.009\,730 0.007 3330.007\,333 0.92680.9268 1.11101.1110 δ [° ] 36.1636.16 6.0136.013 4.3584.358 0.99180.9918 1.21101.2110 PgP_g [p.u. p.u.] 2.8962.896 1.7021.702 1.3451.345 0.99900.9990 0.63570.6357 QgQ_g [p.u. p.u.] 0.14500.1450 0.38070.3807 0.20120.2012 0.98930.9893 0.16410.1641 Appendix D Unified Model Per-System Results This appendix reports the per-system performance of the unified model, which uses a single set of shared weights to solve the AC power flow across all 13 systems. TABLE XI: Unified PowerModelsGAT-AI Performance Across All 13 Systems (N−2N\!-\!2 Contingencies). Per-system breakdown for results discussed in Sec. VI-C. System Target MSE RMSE MAE R2 NMAE% case4gs VmV_m [p.u. p.u.] 2.147×10−062.147×10^-06 0.001 4650.001\,465 0.001 0320.001\,032 0.99780.9978 0.56360.5636 δ [° ] 0.14840.1484 0.38520.3852 0.26060.2606 0.99380.9938 0.68850.6885 PgP_g [p.u. p.u.] 0.006 8640.006\,864 0.082 850.082\,85 0.059 960.059\,96 0.99530.9953 1.02501.0250 QgQ_g [p.u. p.u.] 0.009 6880.009\,688 0.098 430.098\,43 0.076 500.076\,50 0.99140.9914 1.27201.2720 case5 VmV_m [p.u. p.u.] 2.283×10−062.283×10^-06 0.001 5110.001\,511 0.001 1470.001\,147 0.99460.9946 1.10101.1010 δ [° ] 0.44780.4478 0.66920.6692 0.44840.4484 0.98320.9832 1.05601.0560 PgP_g [p.u. p.u.] 0.038 830.038\,83 0.19710.1971 0.12360.1236 0.99190.9919 1.23301.2330 QgQ_g [p.u. p.u.] 0.14500.1450 0.38070.3807 0.27810.2781 0.99110.9911 0.98240.9824 case6w VmV_m [p.u. p.u.] 5.255×10−065.255×10^-06 0.002 2920.002\,292 0.001 6410.001\,641 0.99640.9964 0.56150.5615 δ [° ] 0.30850.3085 0.55540.5554 0.37980.3798 0.98260.9826 1.19901.1990 PgP_g [p.u. p.u.] 0.003 7060.003\,706 0.060 880.060\,88 0.046 150.046\,15 0.98680.9868 1.98801.9880 QgQ_g [p.u. p.u.] 0.008 9960.008\,996 0.094 850.094\,85 0.068 980.068\,98 0.96870.9687 1.90901.9090 case9 VmV_m [p.u. p.u.] 6.174×10−066.174×10^-06 0.002 4850.002\,485 0.001 7280.001\,728 0.99560.9956 0.55190.5519 δ [° ] 0.60660.6066 0.77890.7789 0.53230.5323 0.99430.9943 0.57080.5708 PgP_g [p.u. p.u.] 0.004 1120.004\,112 0.064 130.064\,13 0.051 780.051\,78 0.99250.9925 1.38001.3800 QgQ_g [p.u. p.u.] 0.004 3660.004\,366 0.066 080.066\,08 0.050 670.050\,67 0.93640.9364 2.84702.8470 case14 VmV_m [p.u. p.u.] 1.181×10−051.181×10^-05 0.003 4370.003\,437 0.001 8820.001\,882 0.97940.9794 0.76430.7643 δ [° ] 0.69640.6964 0.83450.8345 0.52490.5249 0.98780.9878 0.85660.8566 PgP_g [p.u. p.u.] 0.007 3300.007\,330 0.085 620.085\,62 0.063 350.063\,35 0.98160.9816 2.24902.2490 QgQ_g [p.u. p.u.] 0.010 580.010\,58 0.10290.1029 0.073 910.073\,91 0.96040.9604 2.11702.1170 case30 VmV_m [p.u. p.u.] 4.063×10−064.063×10^-06 0.002 0160.002\,016 0.001 4340.001\,434 0.99010.9901 0.58990.5899 δ [° ] 0.41220.4122 0.64200.6420 0.47020.4702 0.96650.9665 1.68201.6820 PgP_g [p.u. p.u.] 0.003 1260.003\,126 0.055 910.055\,91 0.043 470.043\,47 0.98480.9848 2.07902.0790 QgQ_g [p.u. p.u.] 0.004 6860.004\,686 0.068 450.068\,45 0.050 120.050\,12 0.98940.9894 1.14601.1460 case_ieee30 VmV_m [p.u. p.u.] 1.701×10−051.701×10^-05 0.004 1250.004\,125 0.002 3490.002\,349 0.97820.9782 0.54110.5411 δ [° ] 1.2331.233 1.1101.110 0.52590.5259 0.97810.9781 0.72430.7243 PgP_g [p.u. p.u.] 0.013 330.013\,33 0.11540.1154 0.053 120.053\,12 0.97280.9728 1.53101.5310 QgQ_g [p.u. p.u.] 0.011 160.011\,16 0.10560.1056 0.070 140.070\,14 0.96570.9657 1.82301.8230 case39 VmV_m [p.u. p.u.] 4.844×10−054.844×10^-05 0.006 9600.006\,960 0.003 9520.003\,952 0.96660.9666 1.13001.1300 δ [° ] 7.0987.098 2.6642.664 1.6601.660 0.99240.9924 0.74710.7471 PgP_g [p.u. p.u.] 0.048 400.048\,40 0.22000.2200 0.15710.1571 0.99960.9996 0.34080.3408 QgQ_g [p.u. p.u.] 0.13140.1314 0.36250.3625 0.22750.2275 0.97290.9729 0.98470.9847 case57 VmV_m [p.u. p.u.] 2.840×10−052.840×10^-05 0.005 3290.005\,329 0.003 2400.003\,240 0.99570.9957 0.58850.5885 δ [° ] 0.71600.7160 0.84620.8462 0.57900.5790 0.99130.9913 0.86740.8674 PgP_g [p.u. p.u.] 0.029 380.029\,38 0.17140.1714 0.14810.1481 0.99630.9963 1.18201.1820 QgQ_g [p.u. p.u.] 0.022 780.022\,78 0.15090.1509 0.11590.1159 0.98150.9815 1.83601.8360 case118 VmV_m [p.u. p.u.] 4.980×10−064.980×10^-06 0.002 2320.002\,232 0.001 6330.001\,633 0.98720.9872 1.14501.1450 δ [° ] 2.4902.490 1.5781.578 1.1341.134 0.99430.9943 0.84870.8487 PgP_g [p.u. p.u.] 0.048 000.048\,00 0.21910.2191 0.16470.1647 0.99950.9995 0.37960.3796 QgQ_g [p.u. p.u.] 0.073 070.073\,07 0.27030.2703 0.19140.1914 0.97910.9791 0.78900.7890 case_illinois200 VmV_m [p.u. p.u.] 9.892×10−069.892×10^-06 0.003 1450.003\,145 0.002 2090.002\,209 0.97910.9791 1.14701.1470 δ [° ] 0.83690.8369 0.91480.9148 0.62550.6255 0.99060.9906 0.95060.9506 PgP_g [p.u. p.u.] 0.019 690.019\,69 0.14030.1403 0.097 860.097\,86 0.99920.9992 0.41830.4183 QgQ_g [p.u. p.u.] 0.007 6980.007\,698 0.087 740.087\,74 0.061 420.061\,42 0.94440.9444 0.88820.8882 case300 VmV_m [p.u. p.u.] 1.061×10−041.061×10^-04 0.010 300.010\,30 0.005 6100.005\,610 0.95800.9580 0.99660.9966 δ [° ] 15.2915.29 3.9103.910 2.7852.785 0.99050.9905 0.93220.9322 PgP_g [p.u. p.u.] 0.14640.1464 0.38260.3826 0.27880.2788 0.99680.9968 1.07301.0730 QgQ_g [p.u. p.u.] 0.17180.1718 0.41450.4145 0.27470.2747 0.97650.9765 0.68680.6868 case1354pegase VmV_m [p.u. p.u.] 8.194×10−058.194×10^-05 0.009 0520.009\,052 0.006 6730.006\,673 0.87240.8724 2.27702.2770 δ [° ] 12.8512.85 3.5853.585 2.7372.737 0.99440.9944 0.86320.8632 PgP_g [p.u. p.u.] 1.8181.818 1.3481.348 1.0881.088 0.99990.9999 0.23590.2359 QgQ_g [p.u. p.u.] 6.9636.963 2.6392.639 1.4071.407 0.99430.9943 0.16200.1620 All Systems (Pooled) VmV_m [p.u. p.u.] 7.038×10−057.038×10^-05 0.008 3890.008\,389 0.005 5690.005\,569 0.95860.9586 0.88520.8852 δ [° ] 10.5910.59 3.2543.254 2.2922.292 0.99430.9943 0.71080.7108 PgP_g [p.u. p.u.] 0.16820.1682 0.41020.4102 0.18280.1828 0.99980.9998 0.03960.0396 QgQ_g [p.u. p.u.] 3.9163.916 1.9791.979 0.86330.8633 0.99420.9942 0.09940.0994 TABLE XII: Unified PowerModelsGAT-AI Performance Across Systems under N−0N\!-\!0, N−1N\!-\!1, and N−2N\!-\!2 Contingencies. Comparison of Mean Squared Error (MSE), Mean Absolute Error (MAE), and Coefficient of Determination (R2R^2) for voltage magnitude, voltage angle, and power generation. N-0 (Normal) N-1 Contingency N-2 Contingency System Target MSE MAE R2 MSE MAE R2 MSE MAE R2 case4gs VmV_m [p.u. p.u.] 8.439×10−078.439×10^-07 0.000 7260.000\,726 0.99720.9972 3.296×10−063.296×10^-06 0.001 3000.001\,300 0.99750.9975 – – – δ [° ] 0.055 310.055\,31 0.17710.1771 0.98730.9873 0.23260.2326 0.33600.3360 0.99440.9944 – – – PgP_g [p.u. p.u.] 0.006 1740.006\,174 0.056 990.056\,99 0.99550.9955 0.007 4970.007\,497 0.062 610.062\,61 0.99510.9951 – – – QgQ_g [p.u. p.u.] 0.007 4930.007\,493 0.069 660.069\,66 0.99270.9927 0.011 660.011\,66 0.082 610.082\,61 0.99020.9902 – – – case5 VmV_m [p.u. p.u.] 1.123×10−061.123×10^-06 0.000 8450.000\,845 0.99640.9964 2.462×10−062.462×10^-06 0.001 2200.001\,220 0.99400.9940 3.641×10−063.641×10^-06 0.001 4730.001\,473 0.99300.9930 δ [° ] 0.11650.1165 0.25470.2547 0.98230.9823 0.39630.3963 0.45700.4570 0.97980.9798 0.95070.9507 0.69750.6975 0.98380.9838 PgP_g [p.u. p.u.] 0.040 150.040\,15 0.10950.1095 0.99290.9929 0.032 580.032\,58 0.11120.1112 0.99170.9917 0.044 500.044\,50 0.15700.1570 0.98910.9891 QgQ_g [p.u. p.u.] 0.14620.1462 0.25530.2553 0.99330.9933 0.15440.1544 0.30500.3050 0.98980.9898 0.13240.1324 0.27830.2783 0.98730.9873 case6w VmV_m [p.u. p.u.] 1.773×10−061.773×10^-06 0.001 0350.001\,035 0.99750.9975 4.544×10−064.544×10^-06 0.001 6340.001\,634 0.99680.9968 1.085×10−051.085×10^-05 0.002 4880.002\,488 0.99520.9952 δ [° ] 0.095 650.095\,65 0.24360.2436 0.99000.9900 0.19990.1999 0.33280.3328 0.98830.9883 0.71400.7140 0.61430.6143 0.97330.9733 PgP_g [p.u. p.u.] 0.001 9250.001\,925 0.034 080.034\,08 0.99430.9943 0.003 5680.003\,568 0.045 660.045\,66 0.98860.9886 0.006 3070.006\,307 0.063 200.063\,20 0.96200.9620 QgQ_g [p.u. p.u.] 0.003 9780.003\,978 0.046 810.046\,81 0.98280.9828 0.007 9370.007\,937 0.067 570.067\,57 0.97460.9746 0.017 090.017\,09 0.10120.1012 0.94920.9492 case9 VmV_m [p.u. p.u.] 2.590×10−062.590×10^-06 0.001 2560.001\,256 0.99670.9967 1.103×10−051.103×10^-05 0.002 3690.002\,369 0.99450.9945 – – – δ [° ] 0.26420.2642 0.34350.3435 0.99430.9943 1.0701.070 0.78660.7866 0.99430.9943 – – – PgP_g [p.u. p.u.] 0.003 3550.003\,355 0.047 080.047\,08 0.99380.9938 0.005 0180.005\,018 0.057 140.057\,14 0.99100.9910 – – – QgQ_g [p.u. p.u.] 0.002 8150.002\,815 0.041 960.041\,96 0.92880.9288 0.006 4460.006\,446 0.062 320.062\,32 0.92450.9245 – – – case14 VmV_m [p.u. p.u.] 1.966×10−061.966×10^-06 0.001 0900.001\,090 0.99440.9944 5.298×10−065.298×10^-06 0.001 6950.001\,695 0.99010.9901 3.043×10−053.043×10^-05 0.002 9860.002\,986 0.96370.9637 δ [° ] 0.15010.1501 0.30260.3026 0.99330.9933 0.49760.4976 0.49130.4913 0.99080.9908 1.5401.540 0.81110.8111 0.98250.9825 PgP_g [p.u. p.u.] 0.002 6900.002\,690 0.040 520.040\,52 0.99350.9935 0.005 7990.005\,799 0.058 300.058\,30 0.98660.9866 0.014 330.014\,33 0.094 930.094\,93 0.95610.9561 QgQ_g [p.u. p.u.] 0.006 6670.006\,667 0.061 840.061\,84 0.97680.9768 0.007 8080.007\,808 0.067 070.067\,07 0.96690.9669 0.018 170.018\,17 0.095 320.095\,32 0.93540.9354 case30 VmV_m [p.u. p.u.] 2.014×10−062.014×10^-06 0.001 1000.001\,100 0.99410.9941 3.901×10−063.901×10^-06 0.001 4040.001\,404 0.98990.9899 6.273×10−066.273×10^-06 0.001 7950.001\,795 0.98720.9872 δ [° ] 0.21460.2146 0.36850.3685 0.98160.9816 0.40010.4001 0.47550.4755 0.96680.9668 0.62260.6226 0.56500.5650 0.95280.9528 PgP_g [p.u. p.u.] 0.002 0010.002\,001 0.036 090.036\,09 0.99130.9913 0.003 5960.003\,596 0.045 850.045\,85 0.98270.9827 0.003 9390.003\,939 0.050 540.050\,54 0.97740.9774 QgQ_g [p.u. p.u.] 0.004 4930.004\,493 0.048 150.048\,15 0.99180.9918 0.004 3250.004\,325 0.048 250.048\,25 0.98970.9897 0.005 2910.005\,291 0.054 220.054\,22 0.98570.9857 case_ieee30 VmV_m [p.u. p.u.] 3.447×10−063.447×10^-06 0.001 4610.001\,461 0.99270.9927 2.036×10−052.036×10^-05 0.002 6110.002\,611 0.97160.9716 3.503×10−053.503×10^-05 0.003 4720.003\,472 0.97320.9732 δ [° ] 0.15550.1555 0.31250.3125 0.99250.9925 2.6482.648 0.65110.6511 0.97070.9707 1.4111.411 0.72500.7250 0.97950.9795 PgP_g [p.u. p.u.] 0.001 6080.001\,608 0.030 640.030\,64 0.99640.9964 0.037 210.037\,21 0.078 990.078\,99 0.94000.9400 0.006 4600.006\,460 0.063 100.063\,10 0.98290.9829 QgQ_g [p.u. p.u.] 0.005 9660.005\,966 0.059 060.059\,06 0.98180.9818 0.018 270.018\,27 0.081 770.081\,77 0.94730.9473 0.011 720.011\,72 0.075 430.075\,43 0.96040.9604 case39 VmV_m [p.u. p.u.] 2.408×10−052.408×10^-05 0.002 9830.002\,983 0.98090.9809 3.573×10−053.573×10^-05 0.003 5950.003\,595 0.97000.9700 9.783×10−059.783×10^-05 0.005 7470.005\,747 0.95020.9502 δ [° ] 4.2504.250 1.3441.344 0.99490.9949 6.5286.528 1.5791.579 0.99240.9924 11.8211.82 2.2082.208 0.98920.9892 PgP_g [p.u. p.u.] 0.034 200.034\,20 0.13010.1301 0.99970.9997 0.038 270.038\,27 0.14660.1466 0.99970.9997 0.078 830.078\,83 0.20300.2030 0.99940.9994 QgQ_g [p.u. p.u.] 0.074 990.074\,99 0.18500.1850 0.98580.9858 0.13530.1353 0.22550.2255 0.96610.9661 0.20790.2079 0.29010.2901 0.96020.9602 case57 VmV_m [p.u. p.u.] 1.058×10−051.058×10^-05 0.002 3990.002\,399 0.99820.9982 4.346×10−054.346×10^-05 0.003 7070.003\,707 0.99390.9939 4.065×10−054.065×10^-05 0.004 1590.004\,159 0.99440.9944 δ [° ] 0.34310.3431 0.45420.4542 0.99530.9953 0.78140.7814 0.63620.6362 0.99070.9907 1.3361.336 0.74450.7445 0.98630.9863 PgP_g [p.u. p.u.] 0.027 390.027\,39 0.15140.1514 0.99640.9964 0.029 200.029\,20 0.14910.1491 0.99640.9964 0.039 820.039\,82 0.16140.1614 0.99510.9951 QgQ_g [p.u. p.u.] 0.018 400.018\,40 0.10380.1038 0.98520.9852 0.024 220.024\,22 0.12000.1200 0.97970.9797 0.029 040.029\,04 0.13290.1329 0.97720.9772 case118 VmV_m [p.u. p.u.] 3.825×10−063.825×10^-06 0.001 5050.001\,505 0.98990.9899 5.186×10−065.186×10^-06 0.001 6560.001\,656 0.98700.9870 6.001×10−066.001×10^-06 0.001 7450.001\,745 0.98460.9846 δ [° ] 1.4271.427 0.91670.9167 0.99640.9964 2.8892.889 1.2131.213 0.99350.9935 3.1943.194 1.2771.277 0.99280.9928 PgP_g [p.u. p.u.] 0.024 360.024\,36 0.12130.1213 0.99970.9997 0.060 790.060\,79 0.18570.1857 0.99940.9994 0.062 020.062\,02 0.19500.1950 0.99940.9994 QgQ_g [p.u. p.u.] 0.066 880.066\,88 0.18580.1858 0.98060.9806 0.073 270.073\,27 0.19210.1921 0.98020.9802 0.079 190.079\,19 0.19630.1963 0.97610.9761 case_illinois200 VmV_m [p.u. p.u.] 6.539×10−066.539×10^-06 0.001 9790.001\,979 0.98780.9878 1.333×10−051.333×10^-05 0.002 3680.002\,368 0.96730.9673 1.193×10−051.193×10^-05 0.002 4900.002\,490 0.97260.9726 δ [° ] 0.44420.4442 0.52020.5202 0.99560.9956 0.80680.8068 0.65560.6556 0.98970.9897 1.9271.927 0.86510.8651 0.97630.9763 PgP_g [p.u. p.u.] 0.020 260.020\,26 0.093 250.093\,25 0.99930.9993 0.016 910.016\,91 0.10240.1024 0.99920.9992 0.024 860.024\,86 0.10930.1093 0.99880.9988 QgQ_g [p.u. p.u.] 0.008 0280.008\,028 0.062 000.062\,00 0.94980.9498 0.007 5870.007\,587 0.061 240.061\,24 0.93940.9394 0.006 8990.006\,899 0.059 850.059\,85 0.93570.9357 case300 VmV_m [p.u. p.u.] 1.027×10−041.027×10^-04 0.005 3890.005\,389 0.95630.9563 1.106×10−041.106×10^-04 0.005 8370.005\,837 0.96070.9607 1.060×10−041.060×10^-04 0.005 6720.005\,672 0.95590.9559 δ [° ] 10.7910.79 2.4872.487 0.99240.9924 21.6921.69 3.3043.304 0.98830.9883 14.0714.07 2.5872.587 0.99050.9905 PgP_g [p.u. p.u.] 0.11390.1139 0.25270.2527 0.99720.9972 0.21340.2134 0.32930.3293 0.99570.9957 0.10610.1061 0.25210.2521 0.99770.9977 QgQ_g [p.u. p.u.] 0.16030.1603 0.26860.2686 0.97830.9783 0.18920.1892 0.28060.2806 0.97460.9746 0.16760.1676 0.27680.2768 0.97600.9760 case1354pegase VmV_m [p.u. p.u.] 8.222×10−058.222×10^-05 0.006 7020.006\,702 0.87040.8704 7.873×10−057.873×10^-05 0.006 5570.006\,557 0.87890.8789 8.487×10−058.487×10^-05 0.006 7460.006\,746 0.86870.8687 δ [° ] 12.7112.71 2.7042.704 0.99470.9947 13.4413.44 2.8472.847 0.99440.9944 12.4212.42 2.6752.675 0.99350.9935 PgP_g [p.u. p.u.] 1.7111.711 1.0581.058 0.99990.9999 2.5222.522 1.3021.302 0.99980.9998 1.2391.239 0.90630.9063 0.99990.9999 QgQ_g [p.u. p.u.] 6.7916.791 1.4061.406 0.99430.9943 7.1377.137 1.4141.414 0.99410.9941 6.9876.987 1.3971.397 0.99430.9943 Appendix E Detailed Continual Learning Results This appendix reports the per-system knowledge-loss values after fine-tuning (Table XIII and Table XIV). Bar-chart visualizations of the same results are provided in the Supplementary Material (Figs. S15 and S16). TABLE XIII: Per-System Knowledge Loss (τNMAEK_τ^NMAE) After Fine-Tuning on case1354pegase. Per-system values are reported for Naive and EWC+Replay across all base systems. System VmNMAEK_V_m^NMAE δNMAEK_δ^NMAE PgNMAEK_P_g^NMAE QgNMAEK_Q_g^NMAE Naive EWC+Replay Naive EWC+Replay Naive EWC+Replay Naive EWC+Replay case4gs 19.5319.53 0.050.05 258.30258.30 −0.05-0.05 1433.741433.74 1.141.14 40.4140.41 1.251.25 case5 9.919.91 0.690.69 219.00219.00 0.040.04 721.58721.58 1.211.21 12.9012.90 0.800.80 case6w 18.3718.37 0.240.24 245.35245.35 0.200.20 2887.412887.41 1.621.62 121.43121.43 1.881.88 case9 10.3810.38 0.210.21 41.7241.72 0.100.10 1493.631493.63 1.581.58 110.11110.11 3.203.20 case14 7.507.50 0.290.29 59.0959.09 0.140.14 2912.612912.61 2.262.26 303.94303.94 2.322.32 case30 12.5712.57 0.220.22 406.13406.13 −0.03-0.03 7737.947737.94 1.611.61 47.5947.59 1.641.64 case_ieee30 6.436.43 0.200.20 99.5399.53 0.150.15 3280.433280.43 2.232.23 287.24287.24 2.272.27 case39 8.738.73 0.310.31 33.4233.42 0.120.12 322.10322.10 0.150.15 12.6812.68 0.670.67 case57 22.1422.14 0.240.24 111.06111.06 0.240.24 824.90824.90 1.791.79 239.30239.30 2.222.22 case118 9.299.29 0.290.29 42.3442.34 0.120.12 208.89208.89 0.370.37 23.7123.71 0.790.79 case_illinois200 13.6213.62 0.250.25 116.10116.10 0.080.08 292.57292.57 0.290.29 31.7331.73 0.910.91 case300 5.585.58 0.140.14 26.7226.72 0.320.32 586.10586.10 0.760.76 14.4614.46 0.570.57 TABLE XIV: Per-System Knowledge Loss (τMAEK_τ^MAE) in Physical Units After Fine-Tuning on case1354pegase. Per-system values are reported for Naive and EWC+Replay across all base systems. System VmMAEK_V_m^MAE [p.u. p.u.] δMAEK_δ^MAE [° ] PgMAEK_P_g^MAE [p.u. p.u.] QgMAEK_Q_g^MAE [p.u. p.u.] Naive EWC+Replay Naive EWC+Replay Naive EWC+Replay Naive EWC+Replay case4gs 0.040.04 0.000.00 97.7797.77 −0.02-0.02 83.8783.87 0.070.07 2.432.43 0.080.08 case5 0.010.01 0.000.00 92.9592.95 0.020.02 72.3072.30 0.120.12 3.653.65 0.230.23 case6w 0.050.05 0.000.00 77.7077.70 0.060.06 67.0367.03 0.040.04 4.394.39 0.070.07 case9 0.030.03 0.000.00 38.9138.91 0.100.10 56.0556.05 0.060.06 1.961.96 0.060.06 case14 0.020.02 0.000.00 36.2136.21 0.080.08 82.0282.02 0.060.06 10.6110.61 0.080.08 case30 0.030.03 0.000.00 113.55113.55 −0.01-0.01 161.79161.79 0.030.03 2.082.08 0.070.07 case_ieee30 0.030.03 0.000.00 72.2672.26 0.110.11 113.83113.83 0.080.08 11.0511.05 0.090.09 case39 0.030.03 0.000.00 74.2774.27 0.270.27 148.48148.48 0.070.07 2.932.93 0.150.15 case57 0.120.12 0.000.00 74.1374.13 0.160.16 103.31103.31 0.220.22 15.1015.10 0.140.14 case118 0.010.01 0.000.00 56.5756.57 0.170.17 90.6790.67 0.160.16 5.755.75 0.190.19 case_illinois200 0.030.03 0.000.00 76.4076.40 0.050.05 68.4568.45 0.070.07 2.192.19 0.060.06 case300 0.030.03 0.000.00 79.8279.82 0.950.95 152.26152.26 0.200.20 5.785.78 0.230.23 Conflict of Interest None of the authors have a conflict of interest to disclose. References Chidozie Ezeakunne received the M.Sc. degree in Physics from the University of Central Florida in 2025, where he is currently pursuing the Ph.D. degree in Physics. He is a Graduate Intern with Los Alamos National Laboratory. His research interests include scientific machine learning, graph-based modeling, computational physics, and data-driven methods for complex physical and engineering systems. JOSE E. TABAREZ received his Ph.D. from New Mexico State University in Electrical Engineering in 2019. At New Mexico State University he was an EMUP and Sandia National Laboratories fellow. He is currently a research and development engineer at Los Alamos National Laboratory. His focus is on power system optimization and analysis, and modeling effects on power systems from geomagnetic disturbances. Reeju Pokharel is a staff scientist in the Materials Science and Technology Division at Los Alamos National Laboratory. She received her Ph.D. in materials science from Carnegie Mellon University in 2013. Her research focuses on data science for 3D imaging and integrating physics constraints into generative machine learning models to provide real-time feedback during dynamic imaging experiments at light sources. ANUP PANDEY received his M.S. and Ph.D. degrees in Physics from Ohio University, Athens, OH, in 2014 and 2017, respectively. He served as a postdoctoral research associate at Oak Ridge National Laboratory from 2017 to 2019. His research interests focus on the application of artificial intelligence and physics-informed machine learning to critical infrastructure systems.