Quantifying the uncertainty of machine learning models is crucial for numerous applications, particularly in large-scale real-world scenarios where prediction sets, rather than point predictions, enable more flexible and informed decision making. Uncertainty quantification (UQ) methods are essential for characterizing the unpredictibility arising in various real-world problems across science and engineering. Initially proposed by Vovk et al. [2005], CP is a popular distribution-free method for UQ, largely due to its ability to provide finite-sample coverage guarantees and its computational efficiency. Most studies in CP focus on constructing prediction intervals based on a fitted mean model. This work introduces a novel setting where the value of interest is estimated using only a pair of valid upper and lower bounds, instead of a mean model. While valid bounds themselves provide perfect coverage by definition, they can sometimes be overly conservative. By slightly reducing the coverage level, these bounds can be tightened, resulting in significantly smaller intervals with theoretical guarantees and greater utility for decision making.

The nonlinear, non-convex AC Optimal Power Flow (AC-OPF) problem is fundamental for power systems operations. The intrinsic complexity of AC-OPF has fueled a growing interest in the development of optimization proxies for the problem, i.e., machine learning models that predict high-quality, close-to-optimal solutions. More recently, dual conic proxy architectures have been proposed, which combine machine learning and convex relaxations of AC-OPF, to provide valid certificates of optimality using learning-based methods. Building on this methodology, this paper proposes, for the first time, a dual conic proxy architecture for the semidefinite (SDP) relaxation of AC-OPF problems. Although the SDP relaxation is stronger than the second-order cone relaxation considered in previous work, its practical use has been hindered by its computational cost. The proposed method combines a neural network with a differentiable dual completion strategy that leverages the structure of the dual SDP problem. This approach guarantees dual feasibility, and therefore valid dual bounds, while providing orders of magnitude of speedups compared to interior-point algorithms. The paper also leverages self-supervised learning, which alleviates the need for time-consuming data generation and allows to train the proposed models efficiently. Numerical experiments are presented on several power grid benchmarks with up to 500 buses. The results demonstrate that the proposed SDP-based proxies can outperform weaker conic relaxations, while providing several orders of magnitude speedups compared to a state-of-the-art interior-point SDP solver.

Recent years have witnessed increasing interest in optimization proxies, i.e., machine learning models that approximate the input-output mapping of parametric optimization problems and return near-optimal feasible solutions. Following recent work by (Nellikkath & Chatzivasileiadis, 2021), this paper reconsiders the optimality verification problem for optimization proxies, i.e., the determination of the worst-case optimality gap over the instance distribution. The paper proposes a compact formulation for optimality verification and a gradient-based primal heuristic that brings substantial computational benefits to the original formulation. The compact formulation is also more general and applies to non-convex optimization problems. The benefits of the compact formulation are demonstrated on large-scale DC Optimal Power Flow and knapsack problems.

Optimal Power Flow (OPF) is a valuable tool for power system operators, but it is a difficult problem to solve for large systems. Machine Learning (ML) algorithms, especially Neural Networks-based (NN) optimization proxies, have emerged as a promising new tool for solving OPF, by estimating the OPF solution much faster than traditional methods. However, these ML algorithms act as black boxes, and it is hard to assess their worst-case performance across the entire range of possible inputs than an OPF can have. Previous work has proposed a mixed-integer programming-based methodology to quantify the worst-case violations caused by a NN trained to estimate the OPF solution, throughout the entire input domain. This approach, however, does not scale well to large power systems and more complex NN models. This paper addresses these issues by proposing a scalable algorithm to compute worst-case violations of NN proxies used for approximating large power systems within a reasonable time limit. This will help build trust in ML models to be deployed in large industry-scale power grids.

This paper presents Dual Lagrangian Learning (DLL), a principled learning methodology that combines conic duality theory with the representation power of ML models. DLL leverages conic duality to provide dual-feasible solutions, and therefore valid Lagrangian dual bounds, for parametric linear and nonlinear conic optimization problems. The paper introduces differentiable conic projection layers, a systematic dual completion procedure, and a self-supervised learning framework. The effectiveness of DLL is demonstrated on linear and nonlinear parametric optimization problems for which DLL provides valid dual bounds within 0.5% of optimality.

This paper introduces Dual Interior Point Learning (DIPL) and Dual Supergradient Learning (DSL) to learn dual feasible solutions to parametric linear programs with bounded variables, which are pervasive across many industries. DIPL mimics a novel dual interior point algorithm while DSL mimics classical dual supergradient ascent. DIPL and DSL ensure dual feasibility by predicting dual variables associated with the constraints then exploiting the flexibility of the duals of the bound constraints. DIPL and DSL complement existing primal learning methods by providing a certificate of quality. They are shown to produce high-fidelity dual-feasible solutions to large-scale optimal power flow problems providing valid dual bounds under 0.5% optimality gap.

The Optimal Power Flow (OPF) problem is integral to the functioning of power systems, aiming to optimize generation dispatch while adhering to technical and operational constraints. These constraints are far from straightforward; they involve intricate, non-convex considerations related to Alternating Current (AC) power flow, which are essential for the safety and practicality of electrical grids. However, solving the OPF problem for varying conditions within stringent time frames poses practical challenges. To address this, operators resort to model simplifications of varying accuracy. Unfortunately, better approximations (tight convex relaxations) are often computationally intractable. This research explores machine learning (ML) to learn convex approximate solutions for faster analysis in the online setting while still allowing for coupling into other convex dependent decision problems. By trading off a small amount of accuracy for substantial gains in speed, they enable the efficient exploration of vast solution spaces in these complex problems.

In recent years, there has been significant interest in the development of machine learning-based optimization proxies for AC Optimal Power Flow (AC-OPF). Although significant progress has been achieved in predicting high-quality primal solutions, no existing learning-based approach can provide valid dual bounds for AC-OPF. This paper addresses this gap by training optimization proxies for a convex relaxation of AC-OPF. Namely, the paper considers a second-order cone (SOC) relaxation of ACOPF, and proposes a novel dual architecture that embeds a fast, differentiable (dual) feasibility recovery, thus providing valid dual bounds. The paper combines this new architecture with a self-supervised learning scheme, which alleviates the need for costly training data generation. Extensive numerical experiments on medium- and large-scale power grids demonstrate the efficiency and scalability of the proposed methodology.

This paper considers optimization proxies for Optimal Power Flow (OPF), i.e., machine-learning models that approximate the input/output relationship of OPF. Recent work has focused on showing that such proxies can be of high fidelity. However, their training requires significant data, each instance necessitating the (offline) solving of an OPF for a sample of the input distribution. To meet the requirements of market-clearing applications, this paper proposes Bucketized Active Sampling (BAS), a novel active learning framework that aims at training the best possible OPF proxy within a time limit. BAS partitions the input distribution into buckets and uses an acquisition function to determine where to sample next. By applying the same partitioning to the validation set, BAS leverages labeled validation samples in the selection of unlabeled samples. BAS also relies on an adaptive learning rate that increases and decreases over time. Experimental results demonstrate the benefits of BAS.

The growing penetration of intermittent, renewable generation in US power grids, especially wind and solar generation, results in increased operational uncertainty. In that context, accurate forecasts are critical, especially for wind generation, which exhibits large variability and is historically harder to predict. To overcome this challenge, this work proposes a novel Bundle-Predict-Reconcile (BPR) framework that integrates asset bundling, machine learning, and forecast reconciliation techniques. The BPR framework first learns an intermediate hierarchy level (the bundles), then predicts wind power at the asset, bundle, and fleet level, and finally reconciles all forecasts to ensure consistency. This approach effectively introduces an auxiliary learning task (predicting the bundle-level time series) to help the main learning tasks. The paper also introduces new asset-bundling criteria that capture the spatio-temporal dynamics of wind power time series. Extensive numerical experiments are conducted on an industry-size dataset of 283 wind farms in the MISO footprint. The experiments consider short-term and day-ahead forecasts, and evaluates a large variety of forecasting models that include weather predictions as covariates. The results demonstrate the benefits of BPR, which consistently and significantly improves forecast accuracy over baselines, especially at the fleet level.