ML-OPF-Bench: Benchmarking Machine Learning for Optimal Power Flow

Xinyi Liu; Xuan He; Danny H. K. Tsang; Yize Chen

eess.SYOct 3, 2026

ML-OPF-Bench: Benchmarking Machine Learning for Optimal Power Flow

Authors: Xinyi Liu, Xuan He, Danny H. K. Tsang, Yize Chen

Organizations: Dartmouth College · Hong Kong University of Science and Technology (Guangzhou) · University of Alberta

Abstract

Machine Learning (ML) methods promise a fast solution process for Optimal Power Flow (OPF). While inconsistent test cases, implementations, and evaluation metrics across existing studies make it challenging to determine which algorithmic advances are most critical for real-world deployment. To this end, we propose ML-OPF-Bench, a unified benchmark for AC- and DC-OPF that evaluates representative ML algorithms under a consistent pipeline, stress-tests them across system sizes, distribution shifts, and resource budgets, and ranks them with a multi-objective framework. We find that prediction accuracy alone is not a reliable indicator of operational feasibility. Under heavily loaded, congested conditions, even the strongest in-distribution performers lose their advantage, while feasibility is maintained largely by post-processing that enforces the target constraints rather than by the underlying pure ML predictor. Data scaling shows that prediction accuracy and constraint violations follow different trajectories, whereas compute scaling shows that returns diminish and that larger models do not consistently perform better. These results expose critical trade-offs among ML methods' speed, accuracy, and feasibility, and offer practical guidance for future ML-OPF design. We open-source the benchmark as an extensible Python package for integrating new learning-based OPF algorithms and evaluating them under the same standard as the existing baselines.

Figures & tables

Figure 1: Structural overview of the ML-OPF Benchmark.

Paradigm	Method	Training Objective	Constraint Handling
Pure Data-Driven	LR [ 10 ]	Least squares	No explicit constraint penalties
	DNN [ 13 ; 29 ; 28 ]	MSE	No explicit constraint penalties
	GNN [ 27 ]	MSE	Graph features; no constraint penalties
Physics-Informed	Constraint Penalty (CP) [ 29 ; 28 ]	MSE + violation penalties	Fixed-weight penalties in loss
	Multiplier Updates (MU) [ 13 ]	MSE + weighted violations	Adaptive multipliers on violations
	KKT Embedded (KKT) [ 25 ; 26 ]	Primal/dual + KKT residuals	Optimality residuals in loss

Table 1: Overview of benchmarked methods and their core characteristics.

Notation	Definition
$\mathcal{G},\mathcal{G}_{\text{ns}}$	Set of generators/non-slack generators
$\mathcal{B}$	Set of buses
$\mathcal{L}$	Set of transmission lines
$N_{g},N_{b},N_{l},N_{d}$	Number of generators/buses/lines/loads
$N$	Number of evaluation samples
$P_{g},Q_{g}$	Active/reactive power generation

Table 2: Notations for ML-OPF-Bench.


	$\displaystyle\mathcal{O}(P_{d},Q_{d})=\mathop{\min}_{P_{g},Q_{g},V,\theta}\sum_{i=1}^{N_{g}}\left(c_{2,i}P_{g,i}^{2}+c_{1,i}P_{g,i}+c_{0,i}\right)$		(1a)
subject to:

		$\displaystyle P_{g,i}-P_{d,i}=V_{i}\sum_{j\in\mathcal{B}}V_{j}(G_{ij}\cos\theta_{ij}+B_{ij}\sin\theta_{ij}),\forall i\in\mathcal{B}$	(1b)
		$\displaystyle Q_{g,i}-Q_{d,i}=V_{i}\sum_{j\in\mathcal{B}}V_{j}(G_{ij}\sin\theta_{ij}-B_{ij}\cos\theta_{ij}),\forall i\in\mathcal{B}$	(1c)
		$\displaystyle P_{g,i}^{\min}\leqslant P_{g,i}\leqslant P_{g,i}^{\max},\hfill\forall i\in\mathcal{G}$	(1d)

Table 4

Test Case	$N_{b}$	$N_{g}$	$N_{d}$	$N_{l}$	$N_{n}$ (AC)	$N_{n}$ (DC)	Batch
30-bus	30	6	21	41	64/32	32/16	32
118-bus	118	54	99	186	256/128	128/64	64
300-bus	300	69	191	411	512/256	256/128	128

Table 3: Test-system sizes and network configurations.

Figure 2: Active-load distributions at Bus 12 of the IEEE 118-bus system under typical conditions, larger load variance, and the API scenario.

		30-bus				118-bus				300-bus
Category	Method	MAE Pg	Viol total	Cost gap	Infer	MAE Pg	Viol total	Cost gap	Infer	MAE Pg	Viol total	Cost gap	Infer
Data-Driven	LR	0.000	0.000	0.000	0.119	1.013	0.076	-0.004	0.121	2.413	0.490	-0.003	0.128
	DNN	0.162	0.000	0.040	0.454	0.711	0.030	0.001	0.439	1.789	0.991	-0.152	0.456
	GNN	1.369	0.003	-0.034	2.950	1.073	0.021	-0.003	2.906	3.171	1.347	0.015	3.008
Physics-Informed	CP	1.118	0.003	-0.109	0.448	0.364	0.005	0.011	0.451	1.170	0.550	0.019	0.481
	MU	1.134	0.003	-0.079	0.463	0.377	0.004	0.014	0.483	1.925	0.185	0.928	0.473

Table 4: Performance of DC-OPF methods across test systems.

		30-bus					118-bus					300-bus
Category	Method	MAE Pg	MAE V	Viol total	Cost gap	Infer	MAE Pg	MAE V	Viol total	Cost gap	Infer	MAE Pg	MAE V	Viol total	Cost gap	Infer
Data-Driven	LR	0.040	0.005	0.001	0.000	9.720	0.518	0.066	0.093	-0.002	17.921	4.565	0.248	0.572	-0.028	26.831
	DNN	0.240	0.015	0.006	0.018	9.621	0.306	0.041	0.076	-0.012	11.060	3.995	0.216	0.451	0.701	17.967
	GNN	3.245	3.576	0.553	0.514	22.332	0.958	3.702	1.295	0.403	33.259	6.566	3.924	4.791	0.062	68.736
Physics-Informed	CP	1.970	0.253	0.014	-0.086	9.666	1.088	0.122	0.078	-0.426	11.267	10.718	0.797	0.686	1.011	17.868
	MU	0.569	0.032	0.005	-0.121	9.973	1.341	0.074	0.118	0.078	11.663	8.067	0.595	2.692	0.029	18.957

Table 5: Performance of AC-OPF methods across test systems.

Figure 3: Pairwise win-rate matrices $W_{jk}$ for AC-OPF (left) and DC-OPF (right). Each cell gives the win rate (%) of the row method $j$ against the column method $k$ over $U=12$ comparison units, where one unit is one (system, objective) pair formed by the three test systems (30/118/300-bus) and the four ranking objectives ( $\mathrm{MAE}_{P_{g}}$ , $\mathrm{Viol}_{\mathrm{total}}$ , $|\mathrm{Cost}_{\mathrm{gap}}|$ , inference time).

Rank	Method	30-bus	118-bus	300-bus	Mean	Marginal (%)
DC-OPF
1	LR	1	1	1	1.00	73.33
2	KKT	2	2	1	1.67	60.00
3	CP	3	1	1	1.67	57.50
4	NGT	3	1	1	1.67	56.25
5	E-NGT	2	2	1	1.67	55.83

Table 6: In-distribution ranking.

Panel of Figure 4 — Figure 4: Generalization degradation on the 118-bus system under two test-time load-distribution shifts. Each marker is the degradation factor $\Delta=E_{\mathrm{shift}}/E_{\mathrm{typ}}$ and the vertical axis is logarithmic. Marker colour identifies the method and marker shape encodes the ML paradigm.

Panel of Figure 5 — Figure 5: Data-scaling behaviour on the 118-bus system as the training-set size grows.

Panel of Figure 6 — Figure 6: Prediction accuracy under compute budgets on the 118-bus system. Each coloured curve reports the test error of an architecture’s best validation checkpoint available within each budget. Colour identifies hidden depth, and marker shape identifies width. The black curve reports the architecture selected by validation loss at each budget; annotations indicate depth $\times$ width.

Form	Method	Metric	$A$	$\alpha$	$R^{2}_{\log}$	$R^{2}_{\mathrm{raw}}$	$\alpha_{\mathrm{NLS}}$	$R^{2}_{\mathrm{raw,NLS}}$
AC	DNN	MAE $P_{g}$ (%)	0.349	-0.555	0.974	0.995	-0.568	0.996
AC	KKT	MAE $P_{g}$ (%)	0.719	-0.584	0.966	0.939	-0.508	0.965
AC	LR	MAE $P_{g}$ (%)	0.534	-0.048	0.826	0.837	-0.051	0.839
AC	MU	MAE $P_{g}$ (%)	0.960	-0.487	0.786	0.480	-0.328	0.648
AC	DNN	Viol $P_{g}$	0.009	-0.356	0.742	0.692	-0.280	0.754
AC	KKT	Viol $P_{g}$	0.011	-0.613	0.920	0.914	-0.508	0.965

Table 14

Figure 7: Additional AC scaling metrics: generator-voltage MAE, reactive-power violations, and voltage violations. Markers are observations; lines follow the common log-space fitting rule.

Figure 8: Validation histories for all neural scaling runs. Markers show printed epoch/loss pairs.

Form	Depth	Width	Parameters	Stop epoch	Best epoch	TFLOPs	Train (s)
AC	2	64	23851	214	194	0.365	58.41
AC	2	128	55787	250	230	1.023	65.71
AC	2	256	144235	171	151	1.870	47.60
AC	2	512	419435	228	208	7.509	62.11
AC	3	64	28011	333	313	0.684	107.31
AC	3	128	72299	250	230	1.370	78.25

Table 17

Figure 9: Iso-error maps for 118-bus DNNs, separated by formulation and hidden depth. Dots are distinct measured checkpoints; Red stars mark checkpoints selected by validation loss across all architectures at the prescribed compute budgets. Colours show test

P_{g}

MAE, interpolated between measured checkpoints in log–log coordinates for compute

C

and parameter count

P

. Interpolation is linear in

\log_{10}

MAE.

Form	Budget (TFLOPs)	Actual (TFLOPs)	Depth $\times$ width	Checkpoint epoch	$P_{g}$ MAE (%)
AC	0.077	0.077	$2\times 64$	45	0.896
AC	0.254	0.252	$2\times 64$	148	0.493
AC	0.835	0.831	$4\times 128$	121	0.360
AC	2.748	1.868	$4\times 128$	252	0.206
AC	9.037	1.868	$4\times 128$	252	0.206
DC	0.072	0.072	$3\times 64$	45	0.843

Table 19

Case	Scenario	Method	MAE $P_{g}$	Viol line	Gap (%)	Infer (ms)	Train (s)	Epoch/steps
30	TYP	AS	0.000	0.000	0.000	4.442	12.73	33
30	TYP	CP	1.118	0.003	-0.109	0.448	33.50	72
30	TYP	DNN	0.162	0.000	0.040	0.454	14.53	51
30	TYP	E-NGT	0.225	0.001	-0.069	0.379	2732.66	2500
30	TYP	GNN	1.369	0.003	-0.034	2.950	164.73	118
30	TYP	KKT	0.058	0.000	-0.015	0.438	85.84	63

Table 20

Case	Scenario	Method	MAE $P_{g}$	MAE $V$	Viol $P_{g}$	Viol $Q_{g}$	Viol $V$	Viol line	Gap (%)	PF (%)	Infer (ms)	Train (s)	Epoch/steps
30	TYP	AS	20.187	0.068	0.000	0.042	0.000	0.000	6.654	100.00	11.005	13.58	28
30	TYP	CP	1.970	0.253	0.001	0.000	0.009	0.005	-0.086	100.00	9.666	10240.01	272
30	TYP	DNN	0.240	0.015	0.001	0.004	0.000	0.000	0.018	100.00	9.621	18.35	62
30	TYP	E-NGT	32.507	0.105	0.104	0.004	0.000	0.007	-5.565	100.00	10.903	6952.06	2500
30	TYP	FR	0.000	0.001	0.000	0.000	0.000	0.000	0.000	100.00	232.519	10240.01	272
30	TYP	GNN	3.245	3.576	0.002	0.490	0.000	0.061	0.514	100.00	22.332	85.57	66

Table 21

Size	Scenario	Method	MAE $P_{g}$	Viol $P_{g}$	Viol line	Gap (%)	Infer (ms)	Train (s)	Epoch/steps
1000	TYP	DNN	1.790	0.001	0.043	-0.002	0.437	0.86	48
1000	TYP	KKT	1.378	0.022	0.039	-0.035	0.456	15.36	218
1000	TYP	LR	1.008	0.074	0.004	-0.003	0.118	0.01	–
1000	TYP	MU	1.740	0.000	0.033	0.039	0.458	2.62	80
1000	TYP	QP	2.187	0.000	0.000	0.566	5.042	1.79	61
5000	TYP	DNN	1.014	0.035	0.014	0.007	0.442	8.68	123

Table 22

Size	Scenario	Method	MAE $P_{g}$	MAE $V$	Viol $P_{g}$	Viol $Q_{g}$	Viol $V$	Viol line	Gap (%)	PF (%)	Infer (ms)	Train (s)	Epoch/steps
1000	TYP	DNN	1.286	0.120	0.018	0.098	0.002	0.013	0.018	100.00	11.178	5.67	363
1000	TYP	KKT	2.443	0.205	0.039	0.194	0.002	0.027	0.011	100.00	11.399	112.63	309
1000	TYP	LR	0.615	0.081	0.031	0.096	0.001	0.003	-0.002	100.00	17.832	0.82	–
1000	TYP	MU	2.082	0.147	0.013	0.086	0.002	0.034	0.006	100.00	11.843	22.69	80
1000	TYP	QC	2.988	0.233	0.000	0.029	0.000	0.049	0.141	100.00	21.419	2.62	104
5000	TYP	DNN	0.535	0.056	0.017	0.074	0.001	0.006	-0.002	100.00	11.228	13.08	175

Table 23

Formulation	Case	Solver time (ms)
DC	30	14.88
DC	118	419.91
DC	300	3325.96
AC	30	67.32
AC	118	432.57
AC	300	14093.14

Table 24

System	Method	Samples	MAE ${}_{P_{g}}$ (%)	MAE ${}_{V_{m}}$ (%)	Direct MAE ${}_{Q_{g}}$ (%)	PF MAE ${}_{Q_{g}}$ (%)	Direct MAE θ ( ∘ )	PF MAE θ ( ∘ )
30-bus	MU	1000	0.569	0.032	0.407	2.550	0.029	0.016
	KKT	1000	0.624	0.027	0.587	2.536	0.035	0.007
118-bus	MU	1000	1.341	0.074	3.620	5.058	0.241	0.269
	KKT	1000	0.669	0.107	3.100	8.359	0.185	0.254
300-bus	MU	985	8.067	0.595	7.665	13.664	12.282	15.973
	KKT	1000	4.150	0.522	4.383	10.563	2.849	4.925

Table 25

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AC optimal power flow (ACOPF) is foundational yet computationally expensive in power grid operations, driving learning-based surrogates for large-scale grid analysis. These surrogates, however, often fail to generalize across network topologies, a critical gap for deployment on grids not seen during training and for routine operational what-if studies. We introduce LUMINA-Bench, a comprehensive benchmark suite for ACOPF surrogate learning covering multi-topology pretraining, transfer, and adaptation. The benchmark evaluates homogeneous and heterogeneous architectures under single- and multi-topology learning settings using unified metrics that capture both predictive accuracy and physics-informed constraint violations. We additionally compare constraint-aware training objectives, including MSE, augmented Lagrangian, and violation-based Lagrangian losses, to characterize accuracy-robustness trade-offs across settings. Data processing, training, and evaluation frameworks are open-sourced as the LUMINA suite to support reproducibility and accelerate future research on feasibility-aware OPF surrogates.

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