PowerModelsGAT-AI: Physics-Informed Graph Attention for Multi-System Power Flow with Continual Learning

Chidozie Ezeakunne; Jose E. Tabarez; Reeju Pokharel; Anup Pandey

eess.SYFeb 24, 2026

PowerModelsGAT-AI: Physics-Informed Graph Attention for Multi-System Power Flow with Continual Learning

Authors: Chidozie Ezeakunne, Jose E. Tabarez, Reeju Pokharel, Anup Pandey

Organizations: 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-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.

Figures & tables

Bus Type	Known in $\mathbf{y}_{i}$	Unknown Targets	Mask $[V_{m},\delta,P_{g},Q_{g}]$
PQ (Load)	$P_{g},Q_{g}$	$V_{m},\delta$	[1, 1, 0, 0]
PV (Generator)	$P_{g},V_{m}$	$\delta,Q_{g}$	[0, 1, 0, 1]
Slack	$V_{m},\delta$	$P_{g},Q_{g}$	[0, 0, 1, 1]

TABLE I: Bus-Type Unknown Targets and Supervision Masks.

Fig. 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

V_{m},\,\delta,\,P_{g},\,Q_{g}

, and (5) a physics-informed loss

\mathcal{L}_{\text{phy}}

. Red dashed lines indicate how predicted outputs and the supervision mask feed into the physics-informed loss

\mathcal{L}_{\text{phy}}

Panel of Figure 3 — Fig. 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.

	Base (Unadapted)		Naive		EWC+Replay
Target	NMAE%	R 2	NMAE%	R 2	NMAE%	R 2
$V_{m}$	12.98	$-2.36$	1.44	$0.945$	1.90	$0.907$
$\delta$	11.96	$-0.01$	0.61	$0.99$	0.68	$0.99$
$P_{g}$	20.79	$0.01$	0.42	$0.99$	0.45	$0.99$
$Q_{g}$	1.61	$0.02$	0.15	$0.994$	0.17	$0.994$

TABLE II: Target System ( case1354pegase ) Performance After Fine-Tuning.

State Variable	Naive	EWC+Replay
Voltage Magnitude ( $V_{m}$ )	12.00	0.26
Voltage Angle ( $\delta$ )	138.23	0.12
Active Power ( $P_{g}$ )	1891.82	1.25
Reactive Power ( $Q_{g}$ )	103.79	1.54

TABLE III: Average Knowledge Loss (

\bar{\mathcal{K}}_{\tau}^{\text{NMAE}}

) on Base Systems.

Panel of Figure 6 — Fig. 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.

Panel of Figure 7 — Fig. 4: Analysis of Branch Importance Scores. (a) Correlation between branch importance and physical branch parameters. (b) Distribution of branch importance scores across benchmark systems.

Panel of Figure 8 — Fig. 5: Global Stability of Bus Feature Sensitivity. Boxplots show the distribution of normalized bus feature sensitivity scores across 13 systems for (a) $V_{m}$ , (b) $\delta$ , (c) $P_{g}$ , and (d) $Q_{g}$ .

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Feature	Symbol	Description	Unit
Electrical Properties
Nominal Voltage	$V_{n}$	Rated voltage level of the bus	kV
Active Load	$P_{d}$	Active power demand at the bus	$\mathrm{\text{p.u.}}$
Reactive Load	$Q_{d}$	Reactive power demand at the bus	$\mathrm{\text{p.u.}}$
Active Generation Setpoint	$P_{g}^{\text{set}}$	Scheduled active power generation	$\mathrm{\text{p.u.}}$
Voltage Setpoint	$V_{m}^{\text{set}}$	Voltage magnitude setpoint (PV and Slack buses)	$\mathrm{\text{p.u.}}$

Appendix

TABLE IV: Node Feature Definitions (

\mathbf{x}_{i}\in\mathbb{R}^{23}

+ one-hot bus type). Complete specification of bus-level input features used by PowerModelsGAT-AI. See Sec. II-C for details.

Feature	Symbol	Description	Unit
Series Admittance (Edge Terms)
Conductance	$G$	Real part of the series-admittance edge term	$\mathrm{\text{p.u.}}$
Susceptance	$B$	Imaginary part of the series-admittance edge term	$\mathrm{\text{p.u.}}$
Branch Type (One-Hot)
Line	—	Transmission line	{0,1}
Transformer	—	Power transformer	{0,1}

Appendix

TABLE V: Edge Feature Definitions (

\mathbf{e}_{ij}\in\mathbb{R}^{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.

System	Total	PQ (Load)	PV (Gen)	Slack
case4gs	4	2	1	1
case5	5	1	3	1
case6ww	6	3	2	1
case9	9	6	2	1
case14	14	9	4	1
case30	30	24	5	1

Appendix

TABLE VI: Bus Type Distribution by System. Breakdown of bus counts per system, classified by bus type (PQ, PV, and Slack).

	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
case6ww	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

Appendix

TABLE VII: Contingency Distribution for System-Specific Models (

N\!-\!2

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

TABLE VIII: Contingency Distribution for Unified Models (

N\!-\!2

Training and Test Sets). Each system contributes 2,000 scenarios. Comparisons show the stability of the contingency distribution across training and test sets.

System	Target	MSE	RMSE	MAE	R 2	NMAE%
case4gs	$V_{m}$ [ $\mathrm{\text{p.u.}}$ ]	$1.134\text{\times}{10}^{-07}$	$3.368\text{\times}{10}^{-04}$	$2.672\text{\times}{10}^{-04}$	$0.9996$	$0.2652$
	$\delta$ [ $\mathrm{\SIUnitSymbolDegree}$ ]	$4.121\text{\times}{10}^{-04}$	$0.020\,30$	$0.014\,81$	$0.9999$	$0.1274$
	$P_{g}$ [ $\mathrm{\text{p.u.}}$ ]	$6.434\text{\times}{10}^{-04}$	$0.025\,36$	$0.018\,49$	$0.9996$	$0.2794$
	$Q_{g}$ [ $\mathrm{\text{p.u.}}$ ]	$5.654\text{\times}{10}^{-04}$	$0.023\,78$	$0.018\,32$	$0.9995$	$0.3571$
case5	$V_{m}$ [ $\mathrm{\text{p.u.}}$ ]	$1.290\text{\times}{10}^{-07}$	$3.592\text{\times}{10}^{-04}$	$3.017\text{\times}{10}^{-04}$	$0.9996$	$0.3619$
	$\delta$ [ $\mathrm{\SIUnitSymbolDegree}$ ]	$0.001\,159$	$0.034\,05$	$0.026\,28$	$0.9998$	$0.2556$

Appendix

TABLE IX: System-Specific Baseline Performance (

N\!-\!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	R 2	NMAE%
case4gs	$V_{m}$ [ $\mathrm{\text{p.u.}}$ ]	$2.597\text{\times}{10}^{-07}$	$5.096\text{\times}{10}^{-04}$	$3.502\text{\times}{10}^{-04}$	$0.9997$	$0.1353$
	$\delta$ [ $\mathrm{\SIUnitSymbolDegree}$ ]	$0.002\,268$	$0.047\,62$	$0.032\,80$	$0.9999$	$0.0763$
	$P_{g}$ [ $\mathrm{\text{p.u.}}$ ]	$3.522\text{\times}{10}^{-04}$	$0.018\,77$	$0.015\,17$	$0.9998$	$0.2552$
	$Q_{g}$ [ $\mathrm{\text{p.u.}}$ ]	$3.339\text{\times}{10}^{-04}$	$0.018\,27$	$0.014\,41$	$0.9997$	$0.2031$
case5	$V_{m}$ [ $\mathrm{\text{p.u.}}$ ]	$2.116\text{\times}{10}^{-07}$	$4.600\text{\times}{10}^{-04}$	$3.145\text{\times}{10}^{-04}$	$0.9996$	$0.2614$
	$\delta$ [ $\mathrm{\SIUnitSymbolDegree}$ ]	$0.031\,61$	$0.1778$	$0.098\,41$	$0.9987$	$0.2410$

Appendix

TABLE X: System-Specific Contingency Performance (

N\!-\!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\!-\!2

) contingencies.

System	Target	MSE	RMSE	MAE	R 2	NMAE%
case4gs	$V_{m}$ [ $\mathrm{\text{p.u.}}$ ]	$2.147\text{\times}{10}^{-06}$	$0.001\,465$	$0.001\,032$	$0.9978$	$0.5636$
	$\delta$ [ $\mathrm{\SIUnitSymbolDegree}$ ]	$0.1484$	$0.3852$	$0.2606$	$0.9938$	$0.6885$
	$P_{g}$ [ $\mathrm{\text{p.u.}}$ ]	$0.006\,864$	$0.082\,85$	$0.059\,96$	$0.9953$	$1.0250$
	$Q_{g}$ [ $\mathrm{\text{p.u.}}$ ]	$0.009\,688$	$0.098\,43$	$0.076\,50$	$0.9914$	$1.2720$
case5	$V_{m}$ [ $\mathrm{\text{p.u.}}$ ]	$2.283\text{\times}{10}^{-06}$	$0.001\,511$	$0.001\,147$	$0.9946$	$1.1010$
	$\delta$ [ $\mathrm{\SIUnitSymbolDegree}$ ]	$0.4478$	$0.6692$	$0.4484$	$0.9832$	$1.0560$

Appendix

TABLE XI: Unified PowerModelsGAT-AI Performance Across All 13 Systems (

N\!-\!2

Contingencies). Per-system breakdown for results discussed in Sec. VI-C .

		N-0 (Normal)			N-1 Contingency			N-2 Contingency
System	Target	MSE	MAE	R 2	MSE	MAE	R 2	MSE	MAE	R 2
case4gs	$V_{m}$ [ $\mathrm{\text{p.u.}}$ ]	$8.439\text{\times}{10}^{-07}$	$0.000\,726$	$0.9972$	$3.296\text{\times}{10}^{-06}$	$0.001\,300$	$0.9975$	–	–	–
	$\delta$ [ $\mathrm{\SIUnitSymbolDegree}$ ]	$0.055\,31$	$0.1771$	$0.9873$	$0.2326$	$0.3360$	$0.9944$	–	–	–
	$P_{g}$ [ $\mathrm{\text{p.u.}}$ ]	$0.006\,174$	$0.056\,99$	$0.9955$	$0.007\,497$	$0.062\,61$	$0.9951$	–	–	–
	$Q_{g}$ [ $\mathrm{\text{p.u.}}$ ]	$0.007\,493$	$0.069\,66$	$0.9927$	$0.011\,66$	$0.082\,61$	$0.9902$	–	–	–
case5	$V_{m}$ [ $\mathrm{\text{p.u.}}$ ]	$1.123\text{\times}{10}^{-06}$	$0.000\,845$	$0.9964$	$2.462\text{\times}{10}^{-06}$	$0.001\,220$	$0.9940$	$3.641\text{\times}{10}^{-06}$	$0.001\,473$	$0.9930$

Appendix

TABLE XII: Unified PowerModelsGAT-AI Performance Across Systems under

N\!-\!0

N\!-\!1

, and

N\!-\!2

Contingencies. Comparison of Mean Squared Error (MSE), Mean Absolute Error (MAE), and Coefficient of Determination (

R^{2}

) for voltage magnitude, voltage angle, and power generation.

System	$\mathcal{K}_{V_{m}}^{\text{NMAE}}$		$\mathcal{K}_{\delta}^{\text{NMAE}}$		$\mathcal{K}_{P_{g}}^{\text{NMAE}}$		$\mathcal{K}_{Q_{g}}^{\text{NMAE}}$
	Naive	EWC+Replay	Naive	EWC+Replay	Naive	EWC+Replay	Naive	EWC+Replay
case4gs	$19.53$	$0.05$	$258.30$	$-0.05$	$1433.74$	$1.14$	$40.41$	$1.25$
case5	$9.91$	$0.69$	$219.00$	$0.04$	$721.58$	$1.21$	$12.90$	$0.80$
case6ww	$18.37$	$0.24$	$245.35$	$0.20$	$2887.41$	$1.62$	$121.43$	$1.88$
case9	$10.38$	$0.21$	$41.72$	$0.10$	$1493.63$	$1.58$	$110.11$	$3.20$
case14	$7.50$	$0.29$	$59.09$	$0.14$	$2912.61$	$2.26$	$303.94$	$2.32$

Appendix

TABLE XIII: Per-System Knowledge Loss (

\mathcal{K}_{\tau}^{\text{NMAE}}

) After Fine-Tuning on case1354pegase . Per-system values are reported for Naive and EWC+Replay across all base systems.

System	$\mathcal{K}_{V_{m}}^{\text{MAE}}$ [ $\mathrm{\text{p.u.}}$ ]		$\mathcal{K}_{\delta}^{\text{MAE}}$ [ $\mathrm{\SIUnitSymbolDegree}$ ]		$\mathcal{K}_{P_{g}}^{\text{MAE}}$ [ $\mathrm{\text{p.u.}}$ ]		$\mathcal{K}_{Q_{g}}^{\text{MAE}}$ [ $\mathrm{\text{p.u.}}$ ]
	Naive	EWC+Replay	Naive	EWC+Replay	Naive	EWC+Replay	Naive	EWC+Replay
case4gs	$0.04$	$0.00$	$97.77$	$-0.02$	$83.87$	$0.07$	$2.43$	$0.08$
case5	$0.01$	$0.00$	$92.95$	$0.02$	$72.30$	$0.12$	$3.65$	$0.23$
case6ww	$0.05$	$0.00$	$77.70$	$0.06$	$67.03$	$0.04$	$4.39$	$0.07$
case9	$0.03$	$0.00$	$38.91$	$0.10$	$56.05$	$0.06$	$1.96$	$0.06$
case14	$0.02$	$0.00$	$36.21$	$0.08$	$82.02$	$0.06$	$10.61$	$0.08$

Appendix

TABLE XIV: Per-System Knowledge Loss (

\mathcal{K}_{\tau}^{\text{MAE}}

) in Physical Units After Fine-Tuning on case1354pegase . Per-system values are reported for Naive and EWC+Replay across all base systems.

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