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This document is an online supplement for Sparse, Dense, and Compact Linearizations of the AC OPF. Here we present complete derivations of the formulations examined, details of the lazy constraint algorithm, and full computational results supporting Sparse, Dense, and Compact Linearizations of the AC OPF.
We report a prominent challenge to the safe and optimal operation of the modern power grid arises due to growing uncertainties in loads and renewables. Stochastic optimal power flow (SOPF) formulations provide a mechanism to handle these uncertainties by computing dispatch decisions and control policies that maintain feasibility under uncertainty. Most SOPF formulations consider simple control policies such as affine policies that are mathematically simple and resemble many policies used in current practice. Motivated by the efficacy of machine learning (ML) algorithms and the potential benefits of general control policies for cost and constraint enforcement, we put forth a deep neural network (DNN)-based policy that predicts the generator dispatch decisions in real time in response to uncertainty. The weights of the DNN are learnt using stochastic primal–dual updates that solve the SOPF without the need for prior generation of training labels and can explicitly account for the feasibility constraints in the SOPF. The advantages of the DNN policy over simpler policies and their efficacy in enforcing safety limits and producing near optimal solutions are demonstrated in the context of a chance constrained formulation on a number of test cases.
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In this work we address the problem of accelerating complex power-grid simulation through machine learning ( ML). Specifically, we develop a framework, Smart-PGSim,which generates multitask-learning (MTL) neural network (NN)models to predict the initial values of variables critical to the problem convergence. MTL models allow information sharing when predicting multiple dependent variables while including customized layers to predict individual variables. We show that,to achieve the required accuracy, it is paramount to embed domain-specific constraints derived from the specific power-grid components in the MTL model. Smart-PGSim then employs the predicted initial values as a high-quality initial condition for the power-grid numerical solver (warm start), resulting in both higher performance compared to state-of-the-art solutions while maintaining the required accuracy. Smart-PGSim brings 2.60×speedup on average (up to 3.28×) computed over 10,000 problems, without losing solution optimality.
Increasing levels of renewable generation motivate a growing interest in data-driven approaches for AC optimal power flow (AC OPF) to manage uncertainty. However, a lack of disciplined dataset creation and benchmarking prohibits useful comparison between approaches in the literature. To instigate confidence, models must be able to reliably predict solutions across a wide range of operating conditions. This paper develops the OPF-Learn package for Julia and Python which uses a computationally efficient approach to create representative datasets that span a wide spectrum of the AC OPF feasible region. Load profiles are uniformly sampled from a convex set that contains the AC OPF feasible set. For each infeasible point found, the convex set is reduced using infeasibility certificates, found by utilizing properties of a relaxed formulation. The framework is shown to generate datasets which are more representative of the entire feasible space versus traditional techniques seen in the literature, improving machine learning model performance.
OPFLearn.jl is a Julia package for creating datasets for machine learning approaches to solving AC optimal power flow (AC OPF). It was developed to provide researchers with a standardized way to efficiently create AC OPF datasets that are representative of more of the AC OPF feasible load space compared to typical dataset creation methods. The OPFLearn dataset creation method uses a relaxed AC OPF formulation to reduce the volume of the unclassified input space throughout the dataset creation process. Over time this input space tightens around the relaxed AC OPF feasible region to increase the percentage of feasible load profiles found while uniformly sampling the input space. Load samples are processed using AC OPF formulations from PowerModels.jl. More information on the dataset creation method can be found in our publication, "OPF-Learn: An Open-Source Framework for Creating Representative AC Optimal Power Flow Datasets". To use OPFLearn.jl a PowerModels network data dictionary is required (can be loaded from Matpower ".m" files) to define the network the dataset is being created for.
OPFLearn.jl is a Julia package for creating datasets for machine learning approaches to solving AC optimal power flow (AC OPF). It was developed to provide researchers with a standardized way to efficiently create AC OPF datasets that are representative of more of the AC OPF feasible load space compared to typical dataset creation methods. The OPFLearn dataset creation method uses a relaxed AC OPF formulation to reduce the volume of the unclassified input space throughout the dataset creation process. Over time this input space tightens around the relaxed AC OPF feasible region to increase the percentage of feasible load profiles found while uniformly sampling the input space. Load samples are processed using AC OPF formulations from PowerModels.jl. More information on the dataset creation method can be found in our publication, "OPF-Learn: An Open-Source Framework for Creating Representative AC Optimal Power Flow Datasets". To use OPFLearn.jl a PowerModels network data dictionary is required (can be loaded from Matpower ".m" files) to define the network the dataset is being created for.
The datasets are resulting from OPFLearn.jl, a Julia package for creating AC OPF datasets. The package was developed to provide researchers with a standardized way to efficiently create AC OPF datasets that are representative of more of the AC OPF feasible load space compared to typical dataset creation methods. The OPFLearn dataset creation method uses a relaxed AC OPF formulation to reduce the volume of the unclassified input space throughout the dataset creation process. The dataset contains load profiles and their respective optimal primal and dual solutions. Load samples are processed using AC OPF formulations from PowerModels.jl. More information on the dataset creation method can be found in our publication, "OPF-Learn: An Open-Source Framework for Creating Representative AC Optimal Power Flow Datasets" and in the package website: https://github.com/NREL/OPFLearn.jl.
We report that alternating current optimal power flow (AC-OPF) is a fundamental tool in electric utilities to determine optimal operation of the various resources. Typically, the AC-OPF problem uses power balance formulation containing voltages and power equations. Yet, there is no comprehensive comparison of the different AC-OPF formulations, especially for large-scale networks. This paper presents a detailed comparative evaluation of different formulations of the AC-OPF problem on networks ranging from 9-bus to 25,000 buses. Three different formulations: 1) power balance with polar voltages, 2) power balance with Cartesian voltages, and 3) current balance with Cartesian voltages are discussed in detail by comparing their characteristics, and numerical and computational performance.
This dataset supports research on graph foundation models for optimal power flow (OPF) on electric grids using HydraGNN. It contains heterogeneous graph representations of PGLib-OPF cases spanning systems from 14 to 13,659 buses, together with packed HDF5 datasets for pretraining, feasibility classification, and N-1 contingency analysis. The release includes OPF solution data, downstream fine-tuning datasets, pretrained HeteroSAGE and HeteroHEAT model checkpoints, hyperparameter-optimization summaries across multiple heterogeneous GNN architectures, and aggregated fine-tuning results for sample-efficiency studies. The dataset is designed to enable scalable training, evaluation, and transfer-learning studies for OPF surrogate modeling, including node-level AC-OPF solution prediction, graph-level prediction, feasibility classification, operating-condition generalization, and contingency-response tasks.
In this paper, we develop a optimal transmission switching (OTS) heuristic based on DC optimal power flow (OPF) and assess the efficacy of the approach when implemented within AC OPF. Traditional formulations of the OTS problem can result in hundreds or thousands of binary variables for large networks, making the OTS problem challenging to solve on fast timescales even for relatively small networks. Here, we identify which constraints and therefore which variables are constraining the DC OPF feasible region, and rank them based on their impact on the cost function. We develop a heuristic algorithm which iteratively removes these constraints and solves a series of standard DC OPF problems. The heuristic is tested on a variety of PGlib networks and the results show that the algorithm can provide substantial cost decreases without having to solve any mixed integer programs. Additionally, we provide insights about the OTS problem, including identifying scenarios outside congestion where OTS can prove useful. Lastly, the performance of the DC-based heuristic is shown when the line switching decisions are implemented within AC OPF.
Simulations have shown that while semi-definite relaxations of AC optimal power flow (AC-OPF) on three-phase radial networks with only wye connections tend to be exact, the presence of delta connections seem to render them inexact. This paper shows that such inexactness originates from the non-uniqueness of relaxation solutions and numerical errors amplified by the non-uniqueness. This finding motivates two algorithms to recover the exact solution of AC-OPF in unbalanced distribution networks featuring both wye and delta connections. In simulations using IEEE 13, 37 and 123-bus systems, the proposed algorithms provide exact optimal solutions up to numerical precision.
Here we present a decomposition approach for obtaining good feasible solutions for the security-constrained, alternating-current, optimal power flow (SC-AC-OPF) problem at an industrial scale and under real-world time and computational limits. The approach was designed while preparing and participating in ARPA-E’s Grid Optimization Competition (GOC) Challenge 1. The challenge focused on a near-real-time version of the SC-AC-OPF problem, where a base operating point is optimized, taking into account possible single-element contingencies, after which the system adapts its operating point following the response of automatic frequency droop controllers and voltage regulators. Our solution approach for this problem relies on state-of-the-art nonlinear programming algorithms, and it employs nonconvex relaxations for complementarity constraints, a specialized two-stage decomposition technique with sparse approximations of recourse terms and contingency ranking and prescreening. The paper describes and justifies our approach and outlines the features of its implementation, including functions and derivatives evaluation, warm-starting strategies, and asynchronous parallelism. We discuss the results of the independent benchmark of our approach by ARPA-E’s GOC team in Challenge 1, where it was found to consistently produce high-quality solutions across a wide range of network sizes and difficulty, and conclude by outlining future extensions of the approach.
Evaluating Photovoltaic Hosting Capacity (PVHC) is an essential step in the process of integrating solar energy into power grids, particularly when focusing on the distribution network (DN) as the primary integration target. PVHC needs to be investigated, especially in cases where the grids are unbalanced, and their operational conditions vary spatially and temporally. This motivation prompted us to propose a scalable model tailored to this application. In this paper, we applied linearization to the alternating current optimal power flow (AC-OPF) and solar inverters, transforming the original problem into a mixed-integer linear programming (MILP) problem. Additionally, we accounted for the battery energy storage system (BESS) as a time-coupling factor for calculating PVHC. We then compared the PVHC results between the IEEE-13 bus and SMART-DS San Francisco (SFO) cases and discussed the extent to which BESS can enhance the PVHC of a DN. Furthermore, we designed a web-based graphical visualization for the SFO case, enabling user interaction with raw data and simulation results on a map through a graphical user interface (GUI). In summary, our results and findings provide valuable insights for future three-phase unbalanced AC-OPF PVHC practices and their visualization.
With the increasing complexity and data availability in modern power systems, learning-based approaches to AC Optimal Power Flow (AC OPF) have garnered significant attention. In particular, the structure of smart grids lends itself naturally to graph-based representations, where Graph Neural Networks (GNNs) can capture spatial and relational dependencies. This paper investigates attention-based GNN architectures tailored to heterogeneous graph representations of electric grids. We evaluate two major paradigms: relational attention, which distinguishes between edge types during message passing, and meta-path attention, which captures high-level semantics through multi-hop, typed paths. Using a large corpus of public AC OPF scenarios, we benchmark representative models of each type of attention. Our results demonstrate the benefits of heterogeneous attention-based models in accurately capturing grid dynamics; heterogeneous attention models achieve superior performance in both standard and perturbed settings. The findings highlight the importance of semantic-aware architectures for improving prediction robustness and interpretability in power system applications.
As electric grids decarbonize and distributed energy resources (DERs) become increasingly prevalent, interconnection assessments must evolve to reflect operational variability and control flexibility. This paper highlights key modeling limitations observed in practice and reviews approaches for modeling uncertainty. It then introduces a Probabilistic Deliverability Assessment (PDA) framework designed to complement and extend existing procedures. The framework integrates scenario-based AC optimal power flow (AC OPF), corrective dispatch, and optional multi-temporal constraints. Together, these form a structured methodology for quantifying DER utilization, deliverability, and reliability under uncertainty in load, generation, and topology. Outputs include interpretable metrics with confidence intervals that inform siting decisions and evaluate compliance with reliability thresholds across sampled operating conditions. A case study on Puerto Rico’s publicly available bulk power system model demonstrates the framework’s application using minimal input data, consistent with current interconnection practice. Across staged fossil generation retirements, the PDA identifies high-value DER sites and regions requiring additional reactive power support. Results are presented through mean dispatch signals, reliability metrics, and geospatial visualizations, demonstrating how the framework provides transparent, data-driven siting recommendations. The framework’s modular design supports incremental adoption within existing workflows, encouraging broader use of AC OPF in interconnection and planning contexts.
This paper proposes an endogenous interface pricing model for day-ahead transmission–distribution co-optimization that co-determines the interface locational marginal price (LMP) and the transmission–distribution exchange, ensuring price–dispatch consistency while optimally scheduling discrete distribution controls. The formulation couples a DC optimal power flow (OPF) with a branch-flow AC OPF that schedules distributed energy resources (DERs), tap-changer settings, capacitor banks (CBs), and multi-period energy storage systems (ESSs) under feeder voltage and current limits, and is solved as a mixed-integer second-order cone program (MISOCP). In a T14–D33 system, coordinated device scheduling recovers about 90% of the distribution-to-transmission export achievable in a reference case that ignores distribution network (DN) limits, while satisfying a 1.05 p.u. voltage upper bound. In a T39–D34/D37/D123 system, a sequential decoupled benchmark produces interface LMP distortions up to 12.5% and a 7.28% mismatch in net export energy, whereas the proposed model removes these distortions and the associated settlement mismatches. Second-order cone (SOC) relaxation gaps remain below $10^{-3}$ in all cases.